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abstract: 'In this work, we prove that any asymptotically stable Markov-Feller operator possesses the e-property everywhere outside at most a meagre set. We also provide an example showing that this result is tight. Moreover, an equivalent criterion for the e-property is proposed.'
author:
- 'Ryszard Kukulski[^1]'
- 'Hanna Wojew[ó]{}dka-[Ś]{}ciko'
bibliography:
- 'markov\_ec\_rk\_hws.bib'
title: 'The e-property of asymptotically stable Markov-Feller operators'
---
=1
[**Keywords:**]{} Markov operator, asymptotic stability, e-property, equicontinuity, Feller property\
[**2010 AMS Subject Classification:**]{} 37A30, 60J05\
Introduction {#introduction .unnumbered}
============
Asymptotic behaviour of random Markov dynamical systems, including mainly the existence of stationary distributions, along with asymptotic stability of Markov operators acting on measures, associated with these systems, has been widely studied over the years. One of the first results concerning asymptotics of Markov-Feller operators evolving on Polish metric spaces have been obtained by T. Szarek (cf. [@szarek_03] or [@szarek_06]). Later on, even more interesting articles on this topic have been published (see e.g. [@lasota_szarek_06; @szarek_worm_11; @czapla_horbacz_14; @wedrychowicz_18], just to name a few). In most of them, the so-called lower-bound technique for equicontinuous families of Markov-Feller operators has been applied to prove asymptotic stability of these operators. We say that a regular Markov operator $P$, with dual operator $U$, has the e-property in the set of functions $\mathcal{R}$ if the family of iterates $(U^nf)_{n\in\mathbb{N}_0}$ is equicontinuous for all $f\in\mathcal{R}$. Most often, $\mathcal{R}$ is assumed to be the set of all bounded Lipschitz functions, as e.g. in [@worm_10; @szarek_worm_11; @hille_szarek_ziem_17], although it can also viewed as the set of all bounded continuous functions, as in this paper (for the convenience of the reader, both these cases are discussed and compared in Remark \[rem:1\]). Markov operators with the e-property are widely applied e.g. in the theory of partial differential equations (cf. [@lasota_szarek_06; @komorowski_peszat_szarek_10; @szarek_14] and the papers relating to the equation of a passive tracer [@szarek_sleczka_urbanski_10] or a non-linear heat equation driven by an impulsive noise [@kapica_szarek_sleczka_11]). On the other hand, similar techniques as those described above have been also applied to establish the existence of a unique stationary distribution in a stochastic model for an autoregulated gene [@hille_horbacz_szarek_16]. Incidentally, let us draw the attention of the reader to the fact that asymptotic stability, or even exponential ergodicity, for a general class of Markov operators may be also proved using a quite different concept, based on the application of an asymptotic coupling (cf. [@czapla_18; @hairer_02; @hairer_cloez_15; @kapica_sleczka_20; @czapla_horbacz_w-s_19]).
Knowing the criteria on asymptotic stability of Markov-Feller operators possessing the e-property, one may ask about a reverse relation, studied e.g. in [@hille_szarek_ziem_17], where the authors prove that any asymptotically stable Markov-Feller operator with an invariant measure such that the interior of its support is non-empty satisfies the e-property. In this paper, we generalize this result (formulated as ). To be more precise, we prove that any asymptotically stable Markov-Feller operator possesses the e-property everywhere except at most a meagre set (Theorem \[th-2\]). Moreover, in Theorem \[th-1\], we propose an equivalent condition for the e-property for asymptotically stable Markov-Feller operators. Namely, we prove that any asymptotically stable Markov-Feller operator has the e-property if and only if it has the e-property in at least one point of the support of its invariant measure. Our mains results, that is, Theorems \[th-2\] and \[th-1\], then naturally imply [@hille_szarek_ziem_17 Theorem 2.3]. Indeed, it is clear, according to Theorem \[th-2\], that, whenever the interior of the support of an invariant measure of a Markov-Feller operator $P$ is non-empty, then there exists at least one point belonging to this support, at which $P$ has the e-property. This, in turn, implies, due to Theorem \[th-1\], that $P$ possesses the e-property at any point.
In the final part of the paper, we present two examples. In , we define an asymptotically stable Markov-Feller operator such that the set of points, at which it does not possess the e-property, is a dense set. Such an example yields that the main result of this paper, formulated as Theorem \[th-2\], is tight. In , on the other hand, we construct an asymptotically stable Markov-Feller operator, for which the set of points not possessing the e-property is uncountable.
The outline of the paper is as follows. Section \[sec:1\] contains notation and basic definitions relating mainly to the theory of Markov operators. In Section \[sec:main\_results\], we state the main results of this article, as well as conduct their proofs. Section \[sec:examples\] is devoted to the above-mentioned examples, which complete the discussion on the e-property of asymptotically stable Markov-Feller operators.
Preliminaries {#sec:1}
=============
Let $(S,\rho)$ be a Polish metric space. By $B(x,\epsilon)$ we denote an open ball in $S$ centered at $x\in S$ and of radius $\epsilon>0$. Closure and interior of any set $A \subset S$ shall be denoted by $\mathtt{Cl}(A)$ and $\mathtt{Int} (A)$, respectively.
Let $(S,\rho)$ be endowed with a Borel $\sigma-$algebra ${\ensuremath{\mathtt{Bor}}}(S)$. Now, let ${\ensuremath{\mathcal{B}}}_b(S)$ be a family of all real-valued, bounded and Borel measurable functions on $S$, equipped with the supremum norm $\|f\|:=\sup_{x\in S}|f(x)|$, and let ${\ensuremath{\mathcal{C}}}_b(S)$ and ${\ensuremath{\mathcal{L}}}_b(S)$ be the subfamilies of ${\ensuremath{\mathcal{B}}}_b(S)$ consisting of continuous and Lipschitz continuous functions, respectively. By ${\ensuremath{\mathcal{L}}}_{FM}(S)$ we mean the special subfamily of ${\ensuremath{\mathcal{L}}}_b(S)$ whose components satisfy $f(S) \subset [0,1]$ and $\mathtt{Lip}(f)\le 1$, where $\mathtt{Lip}(f)$ denotes the Lipschitz constant of $f$.
Let us further consider the set ${\ensuremath{\mathcal{M}}}(S)$ of finite Borel measures, defined on the measurable space $(S,{\ensuremath{\mathtt{Bor}}}(S))$, and its subset ${\ensuremath{\mathcal{M}}}_1(S)$ consisting of probability measures. Moreover, we will also consider the linear space ${\ensuremath{\mathcal{M}}}_s(S)$ of finite signed Borel measures, i.e. $${\ensuremath{\mathcal{M}}}_s(S)=\{\mu=\mu_+ - \mu_-: \; \mu_+,\mu_- \in {\ensuremath{\mathcal{M}}}(S) \}.$$ Let us equip this space with the Fortet-Mourier norm $\| \cdot \|_{FM}$, defined by $$\|\mu\|_{FM}=\sup_{f \in {\ensuremath{\mathcal{L}}}_{FM}(S)} |{\ensuremath{\left\langle f,\mu \right\rangle}}|, \quad \mu \in {\ensuremath{\mathcal{M}}}_s(S),$$ where ${\ensuremath{\left\langle f,\mu \right\rangle}} := \int_S f(x) \mu(dx)$ for any $f
\in {\ensuremath{\mathcal{B}}}_b(S)$, $\mu \in {\ensuremath{\mathcal{M}}}_s(S)$. The support of any measure $\mu \in {\ensuremath{\mathcal{M}}}(S)$ shall be defined as usual, that is $$\text{supp} \; \mu=\{x \in S: \mu(B(x,\epsilon))>0 \mbox{ for any } \epsilon>0\}.$$ The operator ${\ensuremath{\mathcal{P}}}: {\ensuremath{\mathcal{M}}}(S) \rightarrow {\ensuremath{\mathcal{M}}}(S)$ is called Markov if
- ${\ensuremath{\mathcal{P}}}( \lambda \mu_1 + \mu_2)= \lambda {\ensuremath{\mathcal{P}}}(\mu_1)+ {\ensuremath{\mathcal{P}}}(\mu_2)$ for any $\lambda \ge 0$ and any $\mu_1, \mu_2 \in {\ensuremath{\mathcal{M}}}(S)$,
- ${\ensuremath{\mathcal{P}}}\mu (S) = \mu(S)$ for any $\mu \in {\ensuremath{\mathcal{M}}}(S)$.
We say that a Markov operator $P$ is regular, provided that there exists a linear map (the so-called dual operator) $U: {\ensuremath{\mathcal{B}}}_b(S) \rightarrow {\ensuremath{\mathcal{B}}}_b(S)$ such that $${\ensuremath{\left\langle f,{\ensuremath{\mathcal{P}}}\mu \right\rangle}}={\ensuremath{\left\langle Uf,\mu \right\rangle}} \quad \text{for any } f\in
{\ensuremath{\mathcal{B}}}_b(S),\mu\in{\ensuremath{\mathcal{M}}}(S).$$ In this work, we will focus on Markov-Feller operators, that is, regular Markov operators that fulfill the property $U({\ensuremath{\mathcal{C}}}_b(S)) \subset {\ensuremath{\mathcal{C}}}_b(S)$.
Let us also indicate that, given a transition probability function , that is, a map for which $\pi(x,\cdot):{\ensuremath{\mathtt{Bor}}}(S)\to[0,1]$ is a probability measure for any fixed $x\in
S$ and $\pi(\cdot,A):S\to[0,1]$ is a Borel measurable function for any fixed $A\in{\ensuremath{\mathtt{Bor}}}(S)$, one may define a regular Markov operator $P$, along with its dual operator $U$, as follows: $$\begin{aligned}
\label{pi_P_U}
\begin{aligned}
&P\mu(A)=\left\langle\pi(\cdot,A),\mu\right\rangle
\quad\text{for any }A\in{\ensuremath{\mathtt{Bor}}}(S),\;\mu\in{\ensuremath{\mathcal{M}}}(S)\\
&Uf(x)=\left\langle f,\pi(x,\cdot)\right\rangle
\quad\text{for any }x\in S,\;f\in{\ensuremath{\mathcal{B}}}_b(S).
\end{aligned}
\end{aligned}$$
As we have already mentioned before, we will study the relation between two properties of Markov-Feller operators: asymptotic stability and the . We say that a sequence $(\mu_n)_{n \in {\ensuremath{\mathbb{N}}}}$ of finite Borel measures on $S$ converges weakly to a measure $\mu \in {\ensuremath{\mathcal{M}}}(S)$ (which we denote by $\mu_n \xrightarrow{\omega} \mu$), as $n\to\infty$, if for any $f\in {\ensuremath{\mathcal{C}}}_b(S)$ we have $$\lim_{n \to \infty}{\ensuremath{\left\langle f,\mu_n \right\rangle}}= {\ensuremath{\left\langle f,\mu \right\rangle}}.$$ A Markov operator ${\ensuremath{\mathcal{P}}}$ is said to be asymptotically stable if there exists a unique measure such that ${\ensuremath{\mathcal{P}}}\mu_* = \mu_*$ ($\mu_*$ is then called an invariant measure of $P$) and ${\ensuremath{\mathcal{P}}}^n\mu \xrightarrow{\omega} \mu_*$, as $n \to \infty$, for each measure $\mu \in {\ensuremath{\mathcal{M}}}_1(S)$. A Markov operator ${\ensuremath{\mathcal{P}}}$ has the e-property in a set of functions $\mathcal{R}$ at a point $z \in S$, if for any $f \in \mathcal{R}$ the following holds: $$\label{e-prop}
\lim_{x \rightarrow z}\sup_{n \in {\ensuremath{\mathbb{N}}}} |U^nf(x)-U^nf(z)|=0.$$ If the above equality holds for each $z \in S$, then we say that ${\ensuremath{\mathcal{P}}}$ has the in $\mathcal{R}$. Usually, $\mathcal{R}$ is assumed to be one of the following family of functions: ${\ensuremath{\mathcal{C}}}_b(S)$, ${\ensuremath{\mathcal{L}}}_b(S)$, ${\ensuremath{\mathcal{L}}}_{FM}(S)$.
\[rem:1\] In this paper, the notion of the e-property in ${\ensuremath{\mathcal{L}}}_{FM}(S)$ shall be needed in the proof of Theorem \[th-2\]. Let us, however, observe that whenever is satisfied for every $f\in {\ensuremath{\mathcal{L}}}_{FM}(S)$, it is also satisfied for every , due to the linearity of $U$. This implies that the notions of the in ${\ensuremath{\mathcal{L}}}_{FM}(S)$ and ${\ensuremath{\mathcal{L}}}_b(S)$ coincide. Nevertheless, if no further assumptions are imposed, a regular Markov operator does not need to have the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$ at any point, even if it possesses the e-property in ${\ensuremath{\mathcal{L}}}_{FM}(S)$ at every point. Indeed, in the case where $S={\ensuremath{\mathbb{R}}}$ and a Markov operator ${\ensuremath{\mathcal{P}}}$ is given by ${\ensuremath{\mathcal{P}}}\mu = \mu \circ T^{-1}$ for $\mu \in {\ensuremath{\mathcal{M}}}(S)$, with $T: S \to S$ defined by $T(x)=x+1$ for $x \in S$, one can see that ${\ensuremath{\mathcal{P}}}$ has the e-property in ${\ensuremath{\mathcal{L}}}_{FM}(S)$. In spite of that, for each $z \in S$ and a function $f_z \in {\ensuremath{\mathcal{C}}}_b(S)$, given as $$f_z(x)=(n+2)^2 (x-(z+n)) (z+n+2/(n+2) - x)$$ for $x \in \left[z+n,z+n+2/(n+2) \right)$, $n \in {\ensuremath{\mathbb{N}}}$, and $f_z(x)=0$ everywhere else in $S$, we obtain $$\begin{split}
&\limsup_{m \to \infty} \sup_{n \in
{\ensuremath{\mathbb{N}}}}|U^nf_z(z+1/(m+2))-U^nf_z(z)| \\
&\qquad\geq \limsup_{m \to \infty} |f_z(z+m+1/(m+2))-f_z(z+m)|=1,
\end{split}$$ which means that ${\ensuremath{\mathcal{P}}}$ does not have the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$ at any $z \in S$.
On the other hand, let us indicate that the notion of the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$ coincides with the corresponding ones in ${\ensuremath{\mathcal{L}}}_{FM}(S)$ and ${\ensuremath{\mathcal{L}}}_b(S)$, provided that a given Markov operator is asymptotically stable, which fact shall be proved later on in Lemma \[lem-2\].
Main results {#sec:main_results}
============
In this section, we will formulate and prove the main results of this paper.
\[th-2\] Let ${\ensuremath{\mathcal{P}}}$ be an asymptotically stable Markov-Feller operator. The set of points, where ${\ensuremath{\mathcal{P}}}$ does not have the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$, is a meagre set, while the set of points, at which $P$ possesses the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$, is dense.
Before we prove Theorem \[th-2\], let us first establish a few lemmas, to which we will refer in the main proof.
\[prop-1\] Let ${\ensuremath{\mathcal{P}}}$ be a regular Markov operator, which has the e-property in ${\ensuremath{\mathcal{L}}}_{FM}(S)$ at $z\in S$. Then any map $S\ni x\mapsto{\ensuremath{\mathcal{P}}}^n
\delta_x\in\mathcal{M}_1(S)$, $n \in {\ensuremath{\mathbb{N}}}$, is a continuous function in the space ${\ensuremath{\mathcal{M}}}_s(S)$, equipped with the weak topology, at $z \in S$.
Fix $n \in {\ensuremath{\mathbb{N}}}$, and consider a sequence $(x_m)_{m \in {\ensuremath{\mathbb{N}}}}$ of points converging to $z \in S$. Due to the e-property of ${\ensuremath{\mathcal{P}}}$ in ${\ensuremath{\mathcal{L}}}_{FM}(S)$ at $z\in S$, for each $g \in {\ensuremath{\mathcal{L}}}_{FM}(S)$, we have $$\lim_{m \to
\infty} \sup_{n \in {\ensuremath{\mathbb{N}}}} |U^ng(x_m)-U^ng(z)|=0,$$ which implies that ${\ensuremath{\left\langle g,{\ensuremath{\mathcal{P}}}^n \delta_{x_m} \right\rangle}} \to {\ensuremath{\left\langle g,{\ensuremath{\mathcal{P}}}^n \delta_{z} \right\rangle}}$, as $m \to \infty$. Finally, due to the Portmanteau Theorem ([@klenke_13 Theorem 13.16]), we receive ${\ensuremath{\mathcal{P}}}^n \delta_{x_m} \xrightarrow{\omega} {\ensuremath{\mathcal{P}}}^n \delta_z$, as $m \to \infty$.
\[lem-1\] If $f \in {\ensuremath{\mathcal{C}}}_b(S)$ and $K \subset S$ is an arbitrary compact set, then, for each $\epsilon>0$, there exists $L \in {\ensuremath{\mathcal{L}}}_b(S)$ such that $\| f|_K - L|_K\| \leq \epsilon$ and $\|L\| \leq \|f\|$.
Choose an arbitrary $f\in {\ensuremath{\mathcal{C}}}_b(S)$, and fix $\epsilon>0$. Function $f|_K$ is uniformly continuous. Hence, for $\delta<\epsilon$, there exists $r>0$ such that for any $x,y \in K$, satisfying $\rho(x,y)\leq r$. Due to the compactness of $K$, it is possible to find a finite cover of $K$, i.e. $$K=\bigcup_{i=1}^N \left(\mathtt{Cl}(B(x_i,r/2)) \cap K \right) \quad
\text{for some } N \in {\ensuremath{\mathbb{N}}},$$ where $x_1,\ldots,x_N \in K$.
Let us define a family of real functions $\{L^{c,l}:c,l>0 \}$ given by the formula $$L^{c,l}(x)=\sum_{i=1}^N p_i^{c,l}(x) f(x_i)\quad\text{for }x\in S,$$ where $$\begin{split}
p_i^{c,l}(x)&=\frac{d_i^l(x) + c/N}{\sum_{j=1}^N d_j^l(x) + c},\\
d_i^l(x)&=\frac{1}{\max\{l (\rho(x,x_i)-r/2),0\}+1}.
\end{split}$$ Note that, for any $c,l>0$, $(p_i^{c,l}(x))_{i=1}^N$ is a probability vector, whence $\|L^{c,l}\|\leq \|f\|$. The function $x \mapsto \max\{l(\rho(x,x_i)-r/2),0\}+1 \geq 1$ is Lipschitz continuous, and so is a function $x \mapsto d_i^l(x) \in [0,1]$. Therefore, we see that $x \mapsto 1/(\sum_{j=1}^N d_j^l(x)+c)$ is Lipschitz continuous. Moreover, it is bounded. Finally, we observe that $p_i^{c,l}$ is Lipschitz continuous, too, and so is a function $L^{c,l}$.
Take an arbitrary $x \in K$. Without loss of generality, we can assume that $\rho(x,x_1) \leq r/2$, and there exists $J \in \{1,\ldots,N\}$ such that $\rho(x,x_i) \leq r$ for any $i=1,\ldots,J$, and $\rho(x,x_i)>r$ for any $i>J$. We therefore obtain $$\begin{split}
|L^{c,l}(x)-f(x)| &\leq \sum_{i=1}^N p_i^{c,l}(x) |f(x_i)-f(x)|
\leq
\sum_{i=1}^J
p_i^{c,l}(x) \delta + \sum_{i=J+1}^N p_i^{c,l}(x) 2 \|f\| \\
&\leq \delta + 2 \|f\|
\sum_{i=J+1}^N \frac{d_i^l(x) + c/N}{\sum_{j=1}^N d_j^l(x) + c}.
\end{split}$$ Note that $d_1^l(x)=1$, and also, for $i > J$, we have $$d_i^l(x) \leq \frac{1}{lr/2+1}.$$ Finally, we get $$\begin{split}
|L^{c,l}(x)-f(x)| &\leq \delta + 2 \|f\|
\sum_{i=J+1}^N \frac{d_i^l(x) + c/N}{\sum_{j=1}^N d_j^l(x) + c}
\\ & \leq \delta + 2
\|f\| (N-J) \left(\frac{1}{lr/2+1} + \frac{c}{N}\right)\\
&\leq \delta + 2
\|f\| (N-1) \left(\frac{1}{lr/2+1} + \frac{c}{N}\right),
\end{split}$$ and $$\lim_{c \to 0, l \to \infty} \delta + 2
\|f\| (N-1) \left(\frac{1}{lr/2+1} + \frac{c}{N}\right) = \delta <
\epsilon,$$ which means that it is possible to choose $c_0,l_0>0$ so that $$\delta + 2
\|f\| (N-1) \left(\frac{1}{l_0r/2+1} + \frac{c_0}{N}\right)<\epsilon.$$ Consequently, the function $L \coloneqq L^{c_0,l_0}$ satisfies $\|L|_K-f|_K\|
\leq
\epsilon,$ and the proof is completed.
\[lem-2\] A regular Markov operator ${\ensuremath{\mathcal{P}}}$, which is asymptotically stable and has the e-property in ${\ensuremath{\mathcal{L}}}_{FM}(S)$ at $z \in S$, possesses also the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$ at $z \in S$.
By assumption, for $z\in S$ and any $g\in {\ensuremath{\mathcal{L}}}_{FM}(S)$, we have $$\lim_{x \to z} \sup_{n \in {\ensuremath{\mathbb{N}}}} |U^ng(x)-U^ng(z)|=0.$$ The above equality also holds for any $g\in {\ensuremath{\mathcal{L}}}_b(S)$, as it was already explained in Remark \[rem:1\].
Choose an arbitrary function $f \in {\ensuremath{\mathcal{C}}}_b(S)$, and let $(x_m)_{m\in {\ensuremath{\mathbb{N}}}}$ be a sequence of points from $S$, converging to $z $, as $m \to \infty$. Let us further consider the family of probability measures $$\mathcal{M}= \{\mu_*\} \cup \{ {\ensuremath{\mathcal{P}}}^n \delta_z: n \in {\ensuremath{\mathbb{N}}}\} \cup
\{{\ensuremath{\mathcal{P}}}^n
\delta_{x_m}: n,m \in {\ensuremath{\mathbb{N}}}\},$$ where $\mu_*$ stands for the unique invariant measure of ${\ensuremath{\mathcal{P}}}$. We want to prove that ${\ensuremath{\mathcal{M}}}$ is compact in the space ${\ensuremath{\mathcal{M}}}_s(S)$, equipped with the weak topology. Choose an arbitrary sequence of measures $(\mu_k)_{k \in {\ensuremath{\mathbb{N}}}} \subset {\ensuremath{\mathcal{M}}}$. There are three non-trivial cases to consider:
- The sequence $(\mu_k)_{k \in {\ensuremath{\mathbb{N}}}}$ contains infinitely many elements either of the sequence $({\ensuremath{\mathcal{P}}}^n
\delta_z)_{n \in {\ensuremath{\mathbb{N}}}}$ or $({\ensuremath{\mathcal{P}}}^n \delta_{x_m})_{n \in {\ensuremath{\mathbb{N}}}}$ for some fixed $m \in {\ensuremath{\mathbb{N}}}$. Due to the asymptotic stability of ${\ensuremath{\mathcal{P}}}$, both these sequences converge weakly to $\mu_*$, as $n \to \infty$.
- The sequence $(\mu_k)_{k \in {\ensuremath{\mathbb{N}}}}$ contains infinitely many elements of the sequence $({\ensuremath{\mathcal{P}}}^n\delta_{x_m})_{m \in {\ensuremath{\mathbb{N}}}}$ for some fixed $n \in {\ensuremath{\mathbb{N}}}$. By Proposition \[prop-1\], we have ${\ensuremath{\mathcal{P}}}^n \delta_{x_m} \xrightarrow{\omega} {\ensuremath{\mathcal{P}}}^n \delta_{z},$ as $m \to \infty$.
- The sequence $(\mu_k)_{k \in {\ensuremath{\mathbb{N}}}}$ contains infinitely many elements of the sequence $({\ensuremath{\mathcal{P}}}^{n_m} \delta_{x_m})_{m \in {\ensuremath{\mathbb{N}}}}$, where $n_m \nearrow \infty$, as $m \to \infty$. Then, for any $g \in{\ensuremath{\mathcal{L}}}_{FM}(S)$, we have $$\begin{split}
\lim_{m \to \infty} |{\ensuremath{\left\langle g,{\ensuremath{\mathcal{P}}}^{n_m} \delta_{x_m} \right\rangle}}-{\ensuremath{\left\langle g,\mu_* \right\rangle}}|
&\leq
\lim_{m \to \infty} |{\ensuremath{\left\langle g,{\ensuremath{\mathcal{P}}}^{n_m}
\delta_{x_m} \right\rangle}}-{\ensuremath{\left\langle g,{\ensuremath{\mathcal{P}}}^{n_m}
\delta_z \right\rangle}}| \\
&+\lim_{m \to \infty} |{\ensuremath{\left\langle g,{\ensuremath{\mathcal{P}}}^{n_m}
\delta_z \right\rangle}}-{\ensuremath{\left\langle g,\mu_* \right\rangle}}| =0,
\end{split}$$ where the last equality follows from the asymptotic stability of ${\ensuremath{\mathcal{P}}}$. Due to the Portmanteau Theorem, we further obtain ${\ensuremath{\mathcal{P}}}^{n_m} \delta_{x_m}
\xrightarrow{\omega} \mu_*,$ as $m \to \infty$.
Using the Prokhorov theorem ([@billingsley_99 Theorem 5.1 & 5.2]), we obtain that the family ${\ensuremath{\mathcal{M}}}$ is tight. Hence, for an arbitrairly fixed $\epsilon>0$, there exists a compact set $K \subset S$ such that for any $\mu \in {\ensuremath{\mathcal{M}}}$ we have $\mu(K') \leq \epsilon$.
According to Lemma \[lem-1\], for a given function $f\in{\ensuremath{\mathcal{C}}}_b(S)$, there exists a function $L \in {\ensuremath{\mathcal{L}}}_b(S)$, which satisfies $\|f|_K - L|_K \|\leq \epsilon$ and $\|L\| \leq \|f\|$, whence we get $$L|_K+f|_{K'}-\epsilon \leq f\leq L|_K+f|_{K'}+\epsilon.$$ This, in turn, implies $$\begin{split}
&|{\ensuremath{\left\langle f,{\ensuremath{\mathcal{P}}}^n \delta_{x_m}-{\ensuremath{\mathcal{P}}}^n
\delta_z \right\rangle}}| \\
&\qquad\leq |{\ensuremath{\left\langle L|_K,{\ensuremath{\mathcal{P}}}^n
\delta_{x_m}-{\ensuremath{\mathcal{P}}}^n \delta_z \right\rangle}}|+ |{\ensuremath{\left\langle f|_{K'},{\ensuremath{\mathcal{P}}}^n
\delta_{x_m} \right\rangle}}|
+|{\ensuremath{\left\langle f|_{K'},{\ensuremath{\mathcal{P}}}^n \delta_z \right\rangle}}| +2 \epsilon \\
&\qquad\leq |{\ensuremath{\left\langle L,{\ensuremath{\mathcal{P}}}^n
\delta_{x_m}-{\ensuremath{\mathcal{P}}}^n \delta_z \right\rangle}}|+ |{\ensuremath{\left\langle L|_{K'},{\ensuremath{\mathcal{P}}}^n
\delta_{x_m}-{\ensuremath{\mathcal{P}}}^n \delta_z \right\rangle}}|+ 2\epsilon(1+\|f\|)\\
&\qquad\leq |{\ensuremath{\left\langle L,{\ensuremath{\mathcal{P}}}^n
\delta_{x_m} \right\rangle}}-{\ensuremath{\left\langle L,{\ensuremath{\mathcal{P}}}^n \delta_z \right\rangle}}|+ 2\epsilon(1+2\|f\|).
\end{split}$$ Finally, we obtain $$\begin{split}
&\limsup_{ m \to \infty } \sup_{n \in {\ensuremath{\mathbb{N}}}}
|U^nf(x_m)-U^nf(z)| \\
&\qquad\leq
\limsup_{ m
\to
\infty } \sup_{n \in {\ensuremath{\mathbb{N}}}} |U^nL(x_m)-U^nL(z)|+
2\epsilon(1+2\|f\|)\\
&\qquad= 2\epsilon(1+2\|f\|),
\end{split}$$ which, in view of the fact that $\epsilon$ was chosen arbitrarily, proves that ${\ensuremath{\mathcal{P}}}$ has the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$ at $z\in S$.
Now, we are ready to prove the assertion of Theorem \[th-2\].
Let $\mu_* \in {\ensuremath{\mathcal{M}}}_1(S)$ be a unique invariant measure of ${\ensuremath{\mathcal{P}}}$. By $\widehat{O}_x$ we will denote an open neighborhood of $x \in S$, containing $x$. For each $k \in {\ensuremath{\mathbb{N}}}$ let us define an open set $$O_k=\left\{x \in S: \, \exists_{\widehat{O}_x} \, \exists_{n_x
\in {\ensuremath{\mathbb{N}}}} \, \forall_{n \geq n_x} \, \forall_{y \in \widehat{O}_x} \quad
\|{\ensuremath{\mathcal{P}}}^n
\delta_y - \mu_*\|_{FM} \leq 1/k \right\}.$$ We want to show, that, for any $k\in {\ensuremath{\mathbb{N}}}$, the set $O_k$ is dense in $S$. Let us take $x_0\in S$ and $\epsilon>0$. Define $$\begin{split}
&Y\coloneqq \overline{B(x_0,\epsilon)},\\
&Y_{k,n} \coloneqq \left\{x \in Y: \forall_{m \geq n} \; \|{\ensuremath{\mathcal{P}}}^m \delta_x - \mu_* \|_{FM}
\leq 1/k \right\}, \quad k,n \in {\ensuremath{\mathbb{N}}}.
\end{split}$$ One can note that $Y$, as a closed subset of a Polish space, is a Polish space, and the sets $Y_{k,n}$, $k,n \in {\ensuremath{\mathbb{N}}}$, are closed by the Feller property of $P$. By assumption, ${\ensuremath{\mathcal{P}}}$ is asymptotically stable, so ${\ensuremath{\mathcal{P}}}^n
\delta_x \xrightarrow{\omega} \mu_*$, as $n \to \infty$, which is equivalent to $\|{\ensuremath{\mathcal{P}}}^n \delta_x - \mu_*\|_{FM} \to 0$ (cf. [@bogachev_07 Theorem 8.3.2.]). That means that $Y= \bigcup_{n \in {\ensuremath{\mathbb{N}}}} Y_{k,n}$ for any $k\in{\ensuremath{\mathbb{N}}}$. By the Baire category theorem, Polish spaces are necessarily Baire spaces, so, for each $k\in{\ensuremath{\mathbb{N}}}$, there exists $N \in {\ensuremath{\mathbb{N}}}$ such that $\mathtt{Int} (Y_{k,N}) \not= \emptyset$. Obviously, $\mathtt{Int} (Y_{k,N}) \subset Y$ and $\mathtt{Int} (Y_{k,N}) \subset O_k$, so the intersection of $O_k$ with an arbitrarily small open ball $Y$ is non-empty. This, in turn, implies that, for any $k\in{\ensuremath{\mathbb{N}}}$, the set $O_k$ is dense in $S$.
Let us now define $$C \coloneqq \bigcap_{k \in {\ensuremath{\mathbb{N}}}} O_k.$$ Fix $z\in C$. Obviously, $z\in O_k$ for any $k\in{\ensuremath{\mathbb{N}}}$. Further, let $\epsilon>0$ and $k \in {\ensuremath{\mathbb{N}}}$ be such that $1/k \leq \epsilon/2$, and note that, by the definition of $O_k$, there exist an open neighbourhood $\widehat{O}_z$ of $z$, as well as $n_z \in{\ensuremath{\mathbb{N}}}$ such that for any $n\geq n_z$ and any $x\in\widehat{O}_z$ one has $$\|{\ensuremath{\mathcal{P}}}^n \delta_x - \mu_*\|_{FM} \leq 1/k \leq \epsilon/2.$$ Due to the asymptotic stability of ${\ensuremath{\mathcal{P}}}$, there exists $n_0 \in {\ensuremath{\mathbb{N}}}$ such that for any $n \geq n_0$. Using the Feller property of ${\ensuremath{\mathcal{P}}}$, we can conclude that there exists another neighborhood $\widetilde{O}_z$ of point $z$ such that for every $n < \max(n_0,n_z)$ and every $x \in \widetilde{O}_z$ we have $\|{\ensuremath{\mathcal{P}}}^n \delta_x
-
{\ensuremath{\mathcal{P}}}^n \delta_z \|_{FM} \leq \epsilon.$ Hence, for $x \in \widehat{O}_z \cap
\widetilde{O}_z $ we obtain $$\begin{split}
& \sup_{n \in {\ensuremath{\mathbb{N}}}} \|{\ensuremath{\mathcal{P}}}^n \delta_x - {\ensuremath{\mathcal{P}}}^n \delta_z
\|_{FM} \\
&\qquad\leq
\max\left(\epsilon, \sup_{n \geq \max(n_0,n_z)} \|{\ensuremath{\mathcal{P}}}^n \delta_x - \mu_*
\|_{FM}+\sup_{n \geq \max(n_0,n_z)} \|{\ensuremath{\mathcal{P}}}^n \delta_z - \mu_* \|_{FM}\right)
\\ &\qquad\leq \epsilon.
\end{split}$$ Keeping in mind that $\epsilon>0$ and $z\in C$ were chosen arbitrarily, we end up with the following equality: $$\lim_{x \to z} \sup_{n \in {\ensuremath{\mathbb{N}}}} \| {\ensuremath{\mathcal{P}}}^n \delta_x - {\ensuremath{\mathcal{P}}}^n \delta_z
\|_{FM}=0\quad\text{for }z\in C,$$ which means that ${\ensuremath{\mathcal{P}}}$ has the e-property in ${\ensuremath{\mathcal{L}}}_{FM}(S)$ at any $z \in
C$. Further, Lemma \[lem-2\] yields that $P$ also enjoys the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$ at each $z\in C$. As a consequence, the set of points, where ${\ensuremath{\mathcal{P}}}$ does not have the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$, constituting a subset of $C^{\prime}=\bigcup_{k\in{\ensuremath{\mathbb{N}}}}O_k^{\prime}$, where $O_k^{\prime}$, $k\in{\ensuremath{\mathbb{N}}}$, are nowhere dense, is a meagre set. Moreover, using again the Baire category theorem, we obtain that the set $C$ is dense.
\[th-1\] Let ${\ensuremath{\mathcal{P}}}$ be an asymptotically stable Markov-Feller operator with an invariant measure $\mu_* \in {\ensuremath{\mathcal{M}}}_1(S)$. Then ${\ensuremath{\mathcal{P}}}$ has the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$ if there exists at least one point $z \in \text{supp}\, \mu_*$, at which ${\ensuremath{\mathcal{P}}}$ has the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$.
Before we present the sketch of the proof of Theorem \[th-1\], let us quote the main result of [@hille_szarek_ziem_17].
[@hille_szarek_ziem_17 Theorem 2.3]\[th-szarek\] Let ${\ensuremath{\mathcal{P}}}$ be an asymptotically stable Markov-Feller operator and let $\mu_*$ be its invariant measure. If , then ${\ensuremath{\mathcal{P}}}$ satisfies the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$.
The proof of Theorem \[th-szarek\] is based on the application of the following lemma:
[@hille_szarek_ziem_17 Lemma 2.4]\[lem-szarek\] Let ${\ensuremath{\mathcal{P}}}$ be an asymptotically stable Markov-Feller operator whose unique invariant measure is denoted by $\mu_* \in {\ensuremath{\mathcal{M}}}_1(S)$. If $\mathtt{Int} (\mathtt{supp} \, \mu_*) \not= \emptyset$, then for an arbitrary function $f \in {\ensuremath{\mathcal{C}}}_b(S)$ and any $\epsilon>0$, there exist a ball $B \subset \mathtt{supp} \, \mu_*$ and a constant $N \in {\ensuremath{\mathbb{N}}}$ such that for any $n \geq N$ and any $x \in B$, we have $|U^nf(x)-{\ensuremath{\left\langle f,\mu_* \right\rangle}}| \leq \epsilon$.
Following the proof of Lemma \[lem-szarek\], one may observe that its assertion holds (excluding condition $B \subset \mathtt{supp} \, \mu_*$) without assuming $\mathtt{Int} (\mathtt{supp} \, \mu_*) \not= \emptyset$. On the other hand, after analyzing the proof of Theorem \[th-szarek\], we come to the conclusion that the e-property of $P$ in ${\ensuremath{\mathcal{C}}}_b(S)$ can be relatively easily obtained under the following condition: $$\label{eq-th-1-2}
\forall_{f \in {\ensuremath{\mathcal{C}}}_b(S)} \, \forall_{\epsilon>0} \, \exists_{B
\subset S: \mu_*(B)>0}
\, \exists_{N \in {\ensuremath{\mathbb{N}}}} \, \forall_{n \ge N} \, \forall_{x \in B} \quad
|U^nf(x)-{\ensuremath{\left\langle f,\mu_* \right\rangle}}| \le \epsilon.$$ Hence, proving that $\mu_*(B)>0$ is still crucial, and it shall to be done here without assuming that $\mathtt{Int} (\mathtt{supp} \, \mu_*) \not= \emptyset$.
In view of the above, it suffices now to show that any asymptotically stable Markov-Feller operator ${\ensuremath{\mathcal{P}}}$, possessing the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$ at some point , satisfies .
Let $f \in
{\ensuremath{\mathcal{C}}}_b(S)$ and $\epsilon>0$. Then, by assumption, there exist $z \in
\text{supp } \, \mu_*$ such that $$\lim_{x \rightarrow z} \sup_{n \in {\ensuremath{\mathbb{N}}}} |U^nf(x)-U^nf(z)|=0.$$ As a consequence, setting $B \coloneqq \widehat{O}_z$, where $\widehat{O}_z$ is an open neighbourhood of $z$ such that for any $x \in \widehat{O}_z$ we have $$\sup_{n \in {\ensuremath{\mathbb{N}}}} |U^nf(x)-U^nf(z)| \leq \frac{\epsilon}{2},$$ we get $\mu_*(B)>0$. By the asymptotic stability of ${\ensuremath{\mathcal{P}}}$, we can take $N \in {\ensuremath{\mathbb{N}}}$ such that for $n \geq N$ the following holds: $$|U^nf(z)-{\ensuremath{\left\langle f,\mu_* \right\rangle}}|\le \epsilon/2.$$ Finally, for any $x \in B$ and any $n \geq N$, we obtain $$|U^nf(x)-{\ensuremath{\left\langle f,\mu_* \right\rangle}}| \le |U^nf(x)-U^nf(z)| + |U^nf(z)-{\ensuremath{\left\langle f,\mu_* \right\rangle}}|
\le
\epsilon,$$ which gives , and hence completes the sketch of the proof.
Examples {#sec:examples}
========
In this section we shall present two important examples.
In we construct an asymptotically stable Markov-Feller operator $P$ such that the set of points, at which $P$ does not possess the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$, is dense, and hence it is a non-trivial meagre set. The main aim of presenting such an example is to justify that the main result of this paper, formulated as Theorem \[th-2\], is tight.
In we construct an asymptotically stable Markov-Feller operator ${\ensuremath{\mathcal{P}}}$ such that the set of points, where ${\ensuremath{\mathcal{P}}}$ does not possess the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$, has positive Lebesgue measure, and so, it is, in particular, uncountable.
Example 1 {#ex1 .unnumbered}
---------
Let $S$ be a unit sphere in ${\ensuremath{\mathbb{R}}}^2$ endowed with the Euclidean metric, i.e. $$S=\{\phi(x)=(cos(2 \pi x), \sin(2 \pi x)): x \in [0,1)\}.$$ Then $S$ is obviously a Polish space. Note that every number $x \in
[0,1)$ can be uniquely represented, using binary numeral system, as $$\sum_{i=1}^{\infty} \frac{1}{2^i} e_i (x) \eqqcolon
[e_1(x),e_2(x),\ldots]_2, \quad e_i(x) \in \{0,1\},$$ according to the following convention: $$\forall_{x \in [0,1)} \quad \exists_{(i_k)_{k \in {\ensuremath{\mathbb{N}}}}
\subset {\ensuremath{\mathbb{N}}}:\; i_k \nearrow \infty} \quad \forall_{k \in {\ensuremath{\mathbb{N}}}} \quad
e_{i_k}(x)=0.$$ Since $\phi:[0,1)\to S$ is a bijection, from now on, we will identify any $\phi(x)\in S$ with $x\in[0,1)$.
Let us further define $\pi: [0,1) \times {\ensuremath{\mathtt{Bor}}}([0,1)) \to {\ensuremath{\mathbb{R}}}$ by the following formula: $$\pi(y,\cdot)=\begin{cases} \delta_{2y}(\cdot) & \quad \text{for }
0\leq y<1/2\\
\frac{1}{2} \delta_0(\cdot) + \frac{1}{2}
\delta_{2y-1}(\cdot) & \quad \text{for } 1/2\leq y < 1.
\end{cases}$$ Equivalently, (in the binary notation) we can write $$\begin{split}
\pi\left([0,e_1(x),e_2(x),\ldots]_2,\cdot\right)&=\delta_{[e_1(x),e_2(x),\ldots]_2}(\cdot)\\
\pi\left([1,e_1(x),e_2(x),\ldots]_2,\cdot\right)&=\frac{1}{2}
\delta_0(\cdot) + \frac{1}{2}
\delta_{[e_1(x),e_2(x),\ldots]_2}(\cdot) \quad \text{for } x \in S.
\end{split}$$ Note that $\pi$ is a transition probability function, so we can define ${\ensuremath{\mathcal{P}}}$ and $U$ to be a Markov operator and its dual operator, respectively, both generated by $\pi$, according to the rule given in .
Let $f \in {\ensuremath{\mathcal{C}}}_b(S)$. For any $y \in [0,1/2)$ a map $y\mapsto
Uf(y)=f(2y)$ is continuous on the set $(0,1/2)$ and right continuous in $0$. Similarly, for any $y \in
[1/2,1)$ a function $y\mapsto Uf(y)=(f(0)+f(2y-1))/2$ is continuous on $(1/2,1)$ and right continuous in $1/2$. Moreover, if $y \nearrow 1$, then , and if $y \nearrow 1/2$, then $Uf(y) \to f(0)=Uf(1/2)$. That means that ${\ensuremath{\mathcal{P}}}$ is a Markov-Feller operator. One can also observe that $\delta_0$ is an invariant measure of ${\ensuremath{\mathcal{P}}}$.
Now, choose arbitrarily $\mu \in {\ensuremath{\mathcal{M}}}_1(S)$, $A \in {\ensuremath{\mathtt{Bor}}}(S)$ and $\epsilon>0$. Let $N \in {\ensuremath{\mathbb{N}}}$ be such that $2^{-N}<\epsilon,$ and, for any $n \in
{\ensuremath{\mathbb{N}}}$, define $$B_n=\left\{ x \in [0,1):\; \sum_{i=1}^n e_i(x) \geq N \;\vee\;
\sum_{i=n+1}^\infty
e_i(x)=0 \right\}.$$ Note that the sets $B_n$, $n \in {\ensuremath{\mathbb{N}}}$, form a non-decreasing sequence, and also $[0,1)=\bigcup_{n \in {\ensuremath{\mathbb{N}}}} B_n.$ Take such a set $B_K$ for which $\mu(B_K') \leq \epsilon$. Moreover, introduce $$m_{k,x}=\sum_{i=1}^k e_i(x)\quad\text{and}\quad
y_{k,x}=2^kx-\lfloor 2^kx \rfloor\quad
\text{for any }k\in{\ensuremath{\mathbb{N}}},\;x\in S.$$ Further, let $B_K=B_K^+ \cup
B_K^-$, where $B_K^+$ and $B_K^-$ are disjoint sets, determined by $$x \in B_K^+ \iff \sum_{i=K+1}^\infty e_i(x)=0,$$ $$x \in B_K^- \iff \sum_{i=1}^K e_i(x) \geq N \;\wedge\; x \not\in B_K^+ .$$ Note that, if $x \in B_K^+ $, then ${\ensuremath{\mathcal{P}}}^K \delta_x = \delta_0$, while in the case where $x \in B_K^-
$ we get . Hence, for $n
\geq K$, we obtain $$\begin{split}
{\ensuremath{\mathcal{P}}}^n \mu (A)
=&\int\limits_{B_K^+} {\ensuremath{\mathcal{P}}}^n
\delta_x(A) \mu(dx)+\int\limits_{B_K^-} {\ensuremath{\mathcal{P}}}^n \delta_x(A)
\mu(dx)+\int\limits_{B_K'}
{\ensuremath{\mathcal{P}}}^n \delta_x(A) \mu(dx)\\
=&\delta_0(A)\mu(B_K^+)+\int\limits_{B_K^-} \left( 1-2^{-m_{K,x}}
\right)
\delta_0(A) \\
&+ 2^{-m_{K,x}} {\ensuremath{\mathcal{P}}}^{n-K}\delta_{y_{K,x}}(A) \mu(dx)+\int\limits_{B_K'}
{\ensuremath{\mathcal{P}}}^n \delta_x(A) \mu(dx),
\end{split}$$ which implies $$\begin{split}
{\ensuremath{\mathcal{P}}}^n\mu(A) &\leq \delta_0(A)\mu(B_K^+)+\delta_0(A)
\mu(B_K^-)+2^{-N}\mu(B_K^-)+\mu(B_K') \\ &\leq \delta_0(A)
\mu(B_K)+2\epsilon\leq \delta_0(A)+2\epsilon,
\end{split}$$ and $$\begin{split}
{\ensuremath{\mathcal{P}}}^n\mu(A)&\geq \delta_0(A)\mu(B_K^+)+\int\limits_{B_K^-} \left(
1-2^{-m_{K,x}}
\right)
\delta_0(A) \mu(dx) \\ &\geq \delta_0(A) \mu(B_K^+)
+\left(1-2^{-N}\right)
\delta_0(A) \mu(B_K^-)\\
&\geq (1-\epsilon) \delta_0(A) \mu(B_K)\geq (1-\epsilon)^2
\delta_0(A) \geq
\delta_0(A) - 2 \epsilon.
\end{split}$$ We have shown that ${\ensuremath{\mathcal{P}}}^n \mu (A) \to \delta_0(A)$, as $n \to \infty$, so, by the Portmanteau Theorem, we obtain that ${\ensuremath{\mathcal{P}}}$ is an asymptotically stable operator.
Let us now investigate the e-property of ${\ensuremath{\mathcal{P}}}$ in ${\ensuremath{\mathcal{C}}}_b(S)$. Take $z
\in S$ such that $\sum_{i \in {\ensuremath{\mathbb{N}}}} e_i(z)= \infty$, and let , $\epsilon>0$. Further, choose $N \in {\ensuremath{\mathbb{N}}}$, satisfying $2\|f\| 2^{-N} \leq
\epsilon$. Note that there exists $K \in {\ensuremath{\mathbb{N}}}$ such that $e_K(z)=0$ and $m_{K,z} \geq N$. We can define the neighborhood $\widehat{O}_z$ of $z$ consisting of points $x\in S$ such that $e_i(z)=e_i(x)$ for $i \leq K$. Then, for any $x \in \widehat{O}_z$, we get $$\begin{split}
\sup_{n \geq K}|U^nf(x)-U^nf(z)|
=&\sup_{n \geq
0} \left|{\ensuremath{\left\langle f,(1-2^{-m_{K,x}})\delta_0+2^{-m_{K,x}} {\ensuremath{\mathcal{P}}}^n
\delta_{y_{K,x}} \right\rangle}} \right. \\
&- \left.
{\ensuremath{\left\langle f,(1-2^{-m_{K,z}})\delta_0+2^{-m_{K,z}}
{\ensuremath{\mathcal{P}}}^n
\delta_{y_{K,z}} \right\rangle}}\right|\\
=&\sup_{n \geq 0} \left|{\ensuremath{\left\langle f,2^{-m_{K,z}}
{\ensuremath{\mathcal{P}}}^n\delta_{y_{K,x}}-2^{-m_{K,z}} {\ensuremath{\mathcal{P}}}^n
\delta_{y_{K,z}} \right\rangle}}\right|\\
\leq &2
\|f\|2^{-N}\leq \epsilon.
\end{split}$$ Using the Feller property of ${\ensuremath{\mathcal{P}}}$, and recalling that $\epsilon>0$ was chosen arbitrarily, we obtain that ${\ensuremath{\mathcal{P}}}$ has the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$ at $z \in S$.
Now, take the point $z \in S$, satisfying $\sum_{i \in {\ensuremath{\mathbb{N}}}}
e_i(z)< \infty$, and let be such that $f(0)=0$ and $f(1/2)=1$. Choose $K \in {\ensuremath{\mathbb{N}}}$, for which . Further, fix the sequence of points $x_n=z+2^{-K-n}$, $n
\in {\ensuremath{\mathbb{N}}}$, that converges to $z$, as $n \to \infty$. Then we obtain the following conclusion: $$\begin{split}
\limsup_{n \to \infty} \sup_{k \in {\ensuremath{\mathbb{N}}}}
&|U^kf(x_n)-U^kf(z)|
\geq\limsup_{n \to
\infty} |U^{K+n-1}f(x_n)-U^{K+n-1}f(z)|\\
&=\limsup_{n \to
\infty}
\left|{\ensuremath{\left\langle f,(1-2^{-m_{K,z}})\delta_0+2^{-m_{K,z}}{\ensuremath{\mathcal{P}}}^{n-1}
\delta_{2^{-n}} \right\rangle}}\right|
=2^{-m_{K,z}},
\end{split}$$ and, as a consequence, we see that $P$ does not have the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$ at any dyadic rational point.
Example 2 {#ex2 .unnumbered}
---------
Let $S=[-2,-1] \cup [0,1]$ and consider the Smith-Volterra-Cantor (SVC) set constructed on the interval $[-2,-1]$. For the sake of constructing an appropriate Markov operator, let us first discuss the construction of the SVC set.
We start by deleting an open interval $w_{1,1}$, having length $1/4$, from the middle of $ [-2,-1]$. The union of the remaining intervals forms a closed set, say $C_1$. Suppose now, that in the $n-$th step the sets $C_1,\ldots,C_{n}$ are well-defined and satisfy $C_n \subset C_{n-1}
\subset \ldots
\subset C_1$. The set $C_n$ is a union of $2^n$ disjoint and closed intervals $c_{n,k}$, $k \in \{1,\ldots, 2^n\}$. Moreover, we have $$|c_{n,k}|=\frac{2^n+1}{2 \times 4^n}.$$ To obtain $C_{n+1}$, we delete $w_{n+1,k}$, that is, an open interval of the length $4^{-(n+1)}$, from the middle of any $c_{n,k}$. Then, $C_{n+1}$ is a closed set consisting of $2^{n+1}$ disjoint intervals, having length $$|c_{n+1,k}|=\frac{|c_{n,k}|-4^{-(n+1)}}{2}=\frac{2^{n+1}+1}{2 \times
4^{n+1}}.$$ The SVC set is defined as $C=\bigcap_{n \in {\ensuremath{\mathbb{N}}}} C_n.$ One can show that $C$ is a closed, nowhere dense set, having Lebesgue measure equal to $1/2$.
Let us now define a function $T: [-2,-1] \to [0,1]$. Provided that for some $a,b \in {\ensuremath{\mathbb{R}}}$, $|b-a|=4^{-n}$, we may introduce maps , $n \in {\ensuremath{\mathbb{N}}}$, $1 \leq k \leq 2^{n-1}$, given by $$T_{n,k}(x)=\frac{4^{2n+1}}{n}(b-x)(x-a) \quad \text{for } x \in (a,b).$$ Next, for any $n \in {\ensuremath{\mathbb{N}}}$, let us define $T_n:[-2,-1] \to [0,1]$ as $$T_n(x)=\sum_{k=1}^{2^{n-1}} T_{n,k}(x) {\ensuremath{\mathds{1}}}_{w_{n,k}}(x) \quad \text{for
} x \in [0,1].$$ Functions $T_n$, $n \in {\ensuremath{\mathbb{N}}}$, are continuous. Moreover, they have disjoint supports and $T_n([-2,-1]) \subset [0,1/n]$ for any $n \in {\ensuremath{\mathbb{N}}}$. That means that the series $\sum_{n \in {\ensuremath{\mathbb{N}}}} T_n$ fulfills the Cauchy criterion, so it converges to some continuous function, say $T$.
Define $\pi: S \times {\ensuremath{\mathtt{Bor}}}(S) \to {\ensuremath{\mathbb{R}}}$ by the formula $$\begin{split}
\pi(x,A)&=\delta_{T(x)}(A), \quad x \in [-2,-1]\\
\pi(x,A)&=\delta_{2x} (A), \quad x \in [0,1/2) \\
\pi(x,A)&=(2x-1)\delta_0 (A) + (2-2x)\delta_1 (A), \quad x \in
[1/2,1]
\end{split}$$ where $A \in {\ensuremath{\mathtt{Bor}}}(S)$. As $\pi$ is a transition probability function, we can define ${\ensuremath{\mathcal{P}}}$ to be a Markov operator generated by $\pi$, according to the rule given in . One can show, that ${\ensuremath{\mathcal{P}}}$ enjoys the Feller property, and notice that $\delta_0$ is an invariant measure of ${\ensuremath{\mathcal{P}}}$. Moreover, for an arbitrary $\mu \in {\ensuremath{\mathcal{M}}}_1(S)$, we have $\text{supp} \, {\ensuremath{\mathcal{P}}}\mu \subset
[0,1]$. Let $\mu \in {\ensuremath{\mathcal{M}}}_1(S)$, $A \in {\ensuremath{\mathtt{Bor}}}(S)$ and $\epsilon>0$. Note that there exists $N\in{\ensuremath{\mathbb{N}}}$ such that the set $$B=\{0\} \cup [2^{-N},1]$$ satisfies ${\ensuremath{\mathcal{P}}}\mu (B') \leq \epsilon$. Further, observe that if $x
\geq 1/2$, then ${\ensuremath{\mathcal{P}}}^2 \delta_x=\delta_0$. Analogously, for $x \in
[1/4,1/2)$ we have ${\ensuremath{\mathcal{P}}}^3 \delta_x=\delta_0$, and so on. Then, for $n \geq
N+2$ we obtain $$\begin{split}
{\ensuremath{\mathcal{P}}}^n \mu (A) &=\int\limits_B {\ensuremath{\mathcal{P}}}^{n-1} \delta_x(A) {\ensuremath{\mathcal{P}}}\mu
(dx)+\int\limits_{B'}
{\ensuremath{\mathcal{P}}}^{n-1}
\delta_x(A) {\ensuremath{\mathcal{P}}}\mu (dx)\\ &=\delta_0(A){\ensuremath{\mathcal{P}}}\mu(B)+\int\limits_{B'}
{\ensuremath{\mathcal{P}}}^{n-1}
\delta_x(A)
{\ensuremath{\mathcal{P}}}\mu (dx),
\end{split}$$ which implies $$\delta_0(A)-\epsilon \leq \delta_0(A){\ensuremath{\mathcal{P}}}\mu(B)\leq {\ensuremath{\mathcal{P}}}^n \mu(A) \leq
\delta_0(A)+{\ensuremath{\mathcal{P}}}\mu(B') \leq
\delta_0(A)+\epsilon$$ for any $n\geq N+2$. Using the Portmanteau Theorem, we obtain that ${\ensuremath{\mathcal{P}}}$ is asymptotically stable.
Now, let us investigate the e-property of ${\ensuremath{\mathcal{P}}}$ in ${\ensuremath{\mathcal{C}}}_b(S)$. Fix $z > 0$ and $f\in
{\ensuremath{\mathcal{C}}}_b(S)$. There exists $M \in {\ensuremath{\mathbb{N}}}$ such that $z > 1/2^M$. Whenever $y \geq 1/2^M$, we have $$\sup_{n \geq M+1} |U^nf(y)-U^nf(z)|=|f(0)-f(0)|=0.$$ The Feller property of ${\ensuremath{\mathcal{P}}}$ then yields that ${\ensuremath{\mathcal{P}}}$ has the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$ at any point $z>0$. If $z \in [-2,-1] \cap C'$, then, by the construction of $T$, there exists a neighborhood $\widehat{O}_z$ of $z$ such that, for some $m \in {\ensuremath{\mathbb{N}}}$ and any $y \in \widehat{O}_z$, we have $T(y)>1/2^m$, but this, according to the reasoning presented above, leads us to the conclusion that at every point of $[-2,-1] \cap C'$ operator ${\ensuremath{\mathcal{P}}}$ has the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$.
We will show that at any point of $ C \cup \{0\}$ (which has measure $1/2$) the operator ${\ensuremath{\mathcal{P}}}$ does not have the e-property in ${\ensuremath{\mathcal{C}}}_b(S)$. Let’s start with $z=0$. Take $f(x)=x$. Then, $$\lim_{k \to \infty} \sup_{n \in {\ensuremath{\mathbb{N}}}} \left|U^nf(1/2^k)-U^nf(0)\right| \geq \lim_{k
\rightarrow \infty} U^k f(1/2^k) = f(1)=1.$$ Taking $z \in C$, we can choose such a point $x\in C'$, which is arbitrarily close to $z$. Then, for some $n_0 \in {\ensuremath{\mathbb{N}}}$, we have $T(x) \in [1/2^{n_0+1},1/2^{n_0}]$, and therefore we finally obtain $$\left|U^{n_0+1}f(x)-U^{n_0+1}f(z)\right|=\left|f\left(2^{n_0} T(x)\right)-f(0)\right| \geq 1/2.$$
Acknowledgments {#acknowledgments .unnumbered}
===============
Ryszard Kukulski acknowledges the support from the National Science Centre, Poland, under project number 2016/22/E/ST6/00062. The work of Hanna Wojewódka-Ściżko is supported by the Foundation for Polish Science (FNP) under grant number POIR.04.04.00-00-17C1/18-00.
[^1]: ryszard.kukulski@gmail.com
|
---
abstract: |
Define ${\|n\|}$ to be the *complexity* of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that ${\|n\|}\ge 3\log_3 n$ for all $n$, leading this author and Zelinsky to define the *defect* of $n$, ${\delta}(n)$, to be the difference ${\|n\|}-3\log_3 n$. Meanwhile, in the study of addition chains, it is common to consider $s(n)$, the number of small steps of $n$, defined as $\ell(n)-\lfloor\log_2 n\rfloor$, an integer quantity. So here we analogously define $D(n)$, the *integer defect* of $n$, an integer version of ${\delta}(n)$ analogous to $s(n)$. Note that $D(n)$ is not the same as $\lceil
{\delta}(n) \rceil$.
We show that $D(n)$ has additional meaning in terms of the defect well-ordering considered in [@paperwo], in that $D(n)$ indicates which powers of $\omega$ the quantity ${\delta}(n)$ lies between when one restricts to $n$ with ${\|n\|}$ lying in a specified congruence class modulo $3$. We also determine all numbers $n$ with $D(n)\le 1$, and use this to generalize a result of Rawsthorne [@Raws].
author:
- Harry Altman
date: 'August 1, 2018'
title: 'Integer Complexity: The Integer Defect'
---
Introduction {#intro}
============
The *complexity* of a natural number $n$, denoted ${\|n\|}$, is the least number of $1$’s needed to write it using any combination of addition and multiplication, with the order of the operations specified using parentheses grouped in any legal nesting. For instance, $n=11$ has a complexity of $8$, since it can be written using $8$ ones as $$11=(1+1+1)(1+1+1)+1+1,$$ but not with any fewer than $8$. This notion was implicitly introduced in 1953 by Kurt Mahler and Jan Popken [@MP], and was later popularized by Richard Guy [@Guy; @UPINT].
Integer complexity is approximately logarithmic; it satisfies the bounds $$\label{eq1}
3 \log_3 n= \frac{3}{\log 3} \log n\le {\|n\|} \le \frac{3}{\log 2} \log n ,\qquad n>1.$$ The lower bound can be deduced from the results of Mahler and Popken, and was explicitly proved by John Selfridge [@Guy]. It is attained with equality for $n=3^k$ for all $k \ge1$. The upper bound can be obtained by writing $n$ in binary and finding a representation using Horner’s algorithm. It is not sharp, and the constant $\frac{3}{\log2} $ can be improved for large $n$ [@upbds].
Based on the lower bound, this author and Zelinsky [@paper1] introduced the notion of the *defect* of $n$, denoted ${\delta}(n)$, which is the difference ${\|n\|}-3\log_3 n$. Subsequent work [@paperwo] showed that the set of defects is in fact a well-ordered subset of the real line, with order type $\omega^\omega$.
However, it is worth considering the result of Selfridge in more detail:
\[selfridge\] For any $k\ge1$, let $E(k)$ be the largest number that can be made with $k$ ones, i.e., the largest $n$ with ${\|n\|}\le k$. Then:
1. If $k=1$, then $E(k)=1$.
2. If $k\equiv 0\pmod{3}$, then $E(k)=3^{k/3}$.
3. If $k\equiv 1\pmod{3}$ and $k>1$, then $E(k)=4\cdot3^{(k-4)/3}$.
4. If $k\equiv 2\pmod{3}$, then $E(k)=2\cdot3^{(k-2)/3}$.
(This result is also a special case of the results of Mahler and Popken [@MP].) From this one can of course derive the lower bound ${\|n\|}\ge
3\log_3 n$, but what if one wanted an integer version? We make the following definition:
Given a natural number $n$, we define $L(n)$ to be the largest $k$ such that $E(k)\le n$.
With this, we define:
For a natural number $n$, we define the *integer defect* of $n$, denoted $D(n)$, to be the difference ${\|n\|}-L(n)$.
Because of Theorem \[selfridge\], $L(n)$ is quite easy to compute (see Proposition \[computL\]), and hence if one knows ${\|n\|}$ then $D(n)$ is also easy to compute. Note that while we consider $D(n)$ to be an integer analogue of ${\delta}(n)$, it is not in general equal to $\lceil {\delta}(n) \rceil$; see Theorem \[dtoD\] for the precise relation. However it’s not immediately obvious that $D(n)$ has any actual significance. In fact, however, the integer defect of a number tells you about its position in the well-ordering of defects.
$L(k)$ is not the best lower bound we can get from Theoerem \[selfridge\]; that would instead be the smallest $k$ such that $E(k)\ge n$, which we might denote $L'(n)$. ($L'(n)$ could also be defined as the minimum of ${\|m\|}$ over all $m\ge n$.) For reasons that will become clear later, though, we will prefer to discuss $L$ rather than $L'$. In any case, $L'(n)=L(n)+1$ unless $n=E(k)$ for some $k$, in which case $L'(n)=L(n)=k$, so one can easily convert any results expressed in the one formulation to the other. One could consider a similar $D'(n)$ as well, but we will not do that either.
The sets $\mathscr{D}^0$, $\mathscr{D}^1$, and $\mathscr{D}^2$ and the main result {#thm1sec}
----------------------------------------------------------------------------------
As has been noted above, if we define $\mathscr{D}$ to be the set of all defects, then as a subset of the real line this set is well-ordered and has order type $\omega^\omega$. However, more specific theorems are proved in [@paperwo]. We will need the following definition:
\[dadef\] If $a$ is a congruence class modulo $3$, we define $$\mathscr{D}^a = \{ {\delta}(n) : {\|n\|}\equiv a\pmod{3},~~n\ne 1\}.$$
The number $n=1$ is excluded from $\mathscr{D}^1$ because it is dissimilar to other numbers whose complexity is congruent to $1$ modulo $3$. Unlike other numbers which are $1$ modulo $3$, the number $1$ cannot be written as $3j+4$ for some $j\ge0$, and so the largest number that can be made with a single $1$ is simply $1$, rather than $4\cdot 3^j$.
In fact the sets $\mathscr{D}^a$ for $a=0,1,2$ are disjoint, and so together with $\{1\}$ form a partition of $\mathscr{D}$.
Moreover in [@paperwo] it was proved:
\[wothm\] For $a=0,1,2$, the sets $\mathscr{D}^a$ are all well-ordered, each with order type $\omega^\omega$.
It is these sets, the $\mathscr{D}^a$, that $D(n)$ will tell us about the position of ${\delta}(n)$ in. We show:
\[mainthm\] Let $n>1$ be a natural number Let $\zeta$ be the order type of $\mathscr{D}^{{\|n\|}}\cap[0,{\delta}(n))$. Then $D(n)$ is equal to the smallest $k$ such that $\zeta<\omega^k$.
As mentioned above, $D(n)$ is easy to compute, so this theorem gives an way to easily compute where around ${\delta}(n)$ falls in the ordering on $\mathscr{D}^a$.
We will also prove a version of this theorem for the stable integer defect; see Sections \[secstab\] and \[secdefn\].
It’s worth comparing this theorem to what was already known. It was proved in [@paperwo] that the limit of the initial $\omega^k$ elements of $\mathscr{D}$ is equal to $k$. This raises the question – just what is the limit of the initial $\omega^k$ elements of $\mathscr{D}^a$? It was further shown in [@paperwo] that when $k\equiv a\pmod{3}$ this limit is equal to $k$, but what about otherwise?
In this paper we will answer this question:
\[chgoverpt\] The limit of the initial $\omega^k$ elements of $\mathscr{D}^a$ is equal to $k$ if $k-a\equiv 0\pmod{3}$; it is equal to $k+{\delta}(2)$ if $k-a\equiv 1\pmod{3}$; and it is equal to $k+2{\delta}(2)$ if $k-a\equiv 2\pmod{3}$.
In fact, Theorem \[chgoverpt\] will be used to prove Theorem \[mainthm\]. See Section \[mainsec\] for more general statements. Further generalizations will appear in a future paper [@stab].
Generalizing Rawsthorne’s theorem {#rawsintro}
---------------------------------
We know how to compute $E(k)$, the highest number of complexity at most $k$ (or exactly $k$), but what about the next highest? This question was answered by Daniel Rawsthorne [@Raws] in 1989:
\[rawsthm\] For any $k\ge 8$, the highest number of complexity at most $k$ other than $E(k)$ itself is $\frac{8}{9}E(k)$, and this number has complexity exactly $k$.
In this paper we generalize this result. First, a definition:
Given $r\ge 0$ and $k\ge 1$, we define $E_r(k)$ to be the $r$’th largest number of complexity at most $k$. We will $0$-index here, so that by definition $E_0(k)=E(k)$, and Theorem \[rawsthm\] gives a formula for $E_1(k)$.
Then, with this, we show:
\[tablethm\] Given $r\ge 0$, and $a$ a congruence class modulo $3$, there exists $K_{r,a}>1$ and $h_{r,a}\in\mathbb{Q}$ such that for $k\ge K_{r,a}$ with $k\equiv a
\pmod{3}$, we have $E_r(k)=h_{r,a} E(k)$, and these $h_{r,a}$ and $K_{r,a}$ are as given by Tables \[table0\], \[table2\], and \[table1\]. Moreover, for such $r$ and $k$, we have $E_r(k)>E(k-1)$ and therefore ${\|E_r(k)\|}=k$ (and thus for such $r$ and $k$, $E_r(k)$ is not just the $r$’th largest number with complexity at most $k$, but the $r$’th largest number with complexity exactly $k$).
$r$ $h_{r,0}$ $K_{r,0}$
------------------------------------- ------------- -----------
$0$ $1$ $3$
$1$ $8/9$ $6$
$2$ $64/81$ $12$
$3$ $7/9$ $12$
$4$ $20/27$ $12$
$5$ $19/27$ $12$
$6$ $512/729$ $18$
$7$ $56/81$ $18$
$8$ $55/81$ $18$
$9$ $164/243$ $18$
$10$ $163/243$ $18$
$(\mathrm{for}\: n\ge6) \quad 2n-1$ $2/3+2/3^n$ $3n$
$(\mathrm{for}\: n\ge6) \quad 2n$ $2/3+1/3^n$ $3n$
: Table of $h_r$ and $K_r$ for $k\equiv 0\pmod{3}$.[]{data-label="table0"}
$r$ $h_{r,2}$ $K_{r,2}$
------------------------------------ ------------------------- -----------
$0$ $1$ $2$
$1$ $8/9$ $8$
$2$ $5/6$ $8$
$3$ $64/81$ $14$
$4$ $7/9$ $14$
$5$ $20/27$ $14$
$6$ $13/18$ $14$
$7$ $19/27$ $14$
$8$ $512/729$ $20$
$9$ $56/81$ $20$
$10$ $37/54$ $20$
$11$ $55/81$ $20$
$12$ $164/243$ $20$
$13$ $109/162$ $20$
$14$ $163/243$ $20$
$(\mathrm{for}\: n\ge 6)\quad3n-3$ $2/3+2/3^n$ $3n+2$
$(\mathrm{for}\: n\ge 6)\quad3n-2$ $2/3+1/(2\cdot3^{n-1})$ $3n+2$
$(\mathrm{for}\: n\ge 6)\quad3n-1$ $2/3+1/3^n$ $3n+2$
: Table of $h_r$ and $K_r$ for $k\equiv 2\pmod{3}$.[]{data-label="table2"}
$r$ $h_{r,1}$ $K_{r,1}$
------------------------------------ --------------------- -----------
$0$ $1$ $4$
$1$ $8/9$ $10$
$2$ $5/6$ $10$
$3$ $64/81$ $16$
$4$ $7/9$ $16$
$5$ $41/54$ $16$
$(\mathrm{for}\: n\ge 4)\quad n+2$ $3/4+1/(4\cdot3^n)$ $3n+4$
: Table of $h_r$ and $K_r$ for $k\equiv 1\pmod{3}$ with $k>1$.[]{data-label="table1"}
Note that Tables \[table0\], \[table2\], and \[table1\] don’t list the regular pattern in the $h_{r,a}$ until such point as $K_{r,a}$ also becomes regular; for tables based solely on $h_{r,a}$, see Tables \[table0r\], \[table2r\], and \[table1r\].
What does Theorem \[tablethm\] have to do with integer defect? Well, the numbers $h_{r,a}E(k)$ appearing in this theorem are almost exactly the numbers $n$ with $D(n)\le1$; see Proposition \[coincicor\] for a precise statement.
After all, by Theorem \[mainthm\], the numbers $n$ with $D(n)\le 1$ are precisely those $n$ whose ${\delta}(n)$ lie in the initial $\omega$ of $\mathscr{D}^{{\|n\|}}$. So if one fixes a particular $k$, then going down the set of $n$ with ${\|n\|}=k$ corresponds to going up the set of defects ${\delta}(n)$ of $n$ with ${\|n\|}=k$; and assuming $k$ is large enough relative to how far up or down you want to go, this is just looking at $\mathscr{D}^k$. And if we count up one at a time, then – again, assuming $k$ is sufficiently large relative to how far out we count – we will stay within the initial $\omega$ of $\mathscr{D}^k$. So with a classification of numbers $n$ such that $D(n)\le 1$, one can determine the $E_r(k)$. (Indeed, one can also do the reverse.)
Note that Theorem \[rawsthm\] also works for $k=6$, so if one wants to break it down by the residue of $k$ modulo $3$, one could say it works for $k\ge 6$ with $k\equiv 0\pmod{3}$, for $k\ge 8$ with $k\equiv 2\pmod{3}$, and for $k\ge
10$ with $k\equiv 1\pmod{3}$. (Indeed, this is what we have done in Tables \[table0\], \[table2\], and \[table1\].) Note how all three of these correspond to $k$ exactly large enough for $E(k)$ to be divisible by $9$, as per the last part of Theorem \[tablethm\].
One thing worth noting here is that the formulae for $E_0(k)$ and $E_1(k)$, as originally proven by Selfridge and Rawsthorne respectively, were both originally proven directly by induction on $k$. Whereas here we have proven Theorem \[tablethm\] by a different method, namely, analysis of defects. (Although this analysis of defects in turn depends on Rawsthorne’s formula for $E_1(k)$ to serve as a base case; see [@paper1].) This raises the question of whether a similar inductive proof for general $E_r(k)$ could be done now that the formulae for them are known. (In fact this author originally proved these formulae by a different method entirely, that of analyzing certain transformations of expression, so other methods certainly are possible.)
Low-defect polynomials and numbers of low defect {#intropoly}
------------------------------------------------
In order to prove Theorem \[mainthm\], we make use of the idea of low-defect polynomials from [@paperwo; @theory]. A low-defect polynomial is a particular type of multilinear polynomial; see Section \[polysec\] for details. In [@paperwo] it is proved that, given any positive real number $s$, one can write down a finite set of low-defect polynomials ${{\mathcal S}}$ such that every number $n$ with ${\delta}(n)<s$ can be written in the form $f(3^{n_1},\ldots,3^{n_d})3^{n_{d+1}}$ for some $f\in{{\mathcal S}}$; and that, moreover, such an $n$ can always be represented “efficiently” in such a fashion. Moreover, one can choose ${{\mathcal S}}$ such that for any $f\in{{\mathcal S}}$, one has $\deg f\le s$. (Note that the degree of a low-defect polynomial is always equal to the number of variables it is in, since low-defect polynomials are multilinear and always include a term containing all the variables.)
Using this fact about low-defect polynomials, this author proved in [@paperwo] that the set $\mathscr{D}$ is well-ordered with order type $\omega^\omega$, as well as the more specific Theorem \[wothm\] mentioned above, and other results mentioned above such as that the limit of the initial $\omega^k$ defects is equal to $k$. However, this is not enough to prove the more specific theorems shown in this paper, such as Theorem \[chgoverpt\]. But in [@theory] an improvement was shown, that we can in fact take ${{\mathcal S}}$ such that for all $f\in\mathscr{T}$, one has ${\delta}(f)\le s$; here ${\delta}(f)$ is a number that bounds above ${\delta}(n)$ for any $n$ represented by $f$ in the fashion described above; again, see \[polysec\] for more on this.
On top of that, it was shown in [@theory] that ${\delta}(f)\ge \deg f +
{\delta}(m)$, where $m$ is the leading coefficient of $f$. Putting this together, one gets the inequality $$\deg f + {\delta}(m) \le s.$$ It’s this stronger inequality that allows us to prove Theorem \[mainthm\], where the inequality $\deg f\le s$ would not be enough. To see why this inequality is so helpful, say we’re given $s$ and we pick ${{\mathcal S}}$ as described above. Then if $f\in{{\mathcal S}}$, one of two things must be true: Either $\deg f<\lfloor
s\rfloor$, in which case $f$ does not make much of a contribution to $\mathscr{D}\cap [0,s)$ compared to polynomials of higher degree; or $\deg
f=\lfloor s\rfloor$, in which case ${\delta}(m)$ is at most the fractional part of $r$, a number which is less than $1$. Since there are only finitely many defects below any given number less than $1$, this puts substantial constraints on $m$ and therefore on $f$, in ways that the weaker inequality $\deg f\le s$ does not. This allows us to prove Theorem \[chgoverpt\].
Note that the method we use to turn the results of [@theory] into Theorem \[mainthm\] actually has much more power than we use in this paper; but an exploration of the full power of this method would take us too far away from the subject of $D(n)$, and so will be detailed in a future paper [@stab].
A quick note on stabilization {#secstab}
-----------------------------
An important property satisfied by integer integer complexity is the phenomenon of stabilization. Because one has ${\|3^k\|}=3k$ for $k>1$, as well as that ${\|2\cdot3^k\|}=2+3k$ and ${\|4\cdot3^k\|}=4+3k$, one might hope that in general the equation ${\|3n\|}={\|n\|}+3$ holds for all $n>1$. Unfortunately that is not the case; for instance, for $n=107$, one has ${\|107\|}=16$, but ${\|321\|}=18$. Another counterexample is $n=683$, for which one has ${\|683\|}=22$, but ${\|2049\|}=23$. There are even cases where ${\|3n\|}<{\|n\|}$, such as $n=4721323$, which has ${\|3n\|}={\|n\|}-1$.
And yet the initial hope is not entirely in vain. In [@paper1], it was proved:
\[basicstab\] For any natural number $n$, there exists $K\ge 0$ such that, for any $k\ge K$, $${\|3^k n\|}=3(k-K)+{\|3^K n\|}.$$
Based on this, we define:
A number $m$ is called *stable* if ${\|3^k m\|}=3k+{\|m\|}$ holds for every $k \ge 0$. Otherwise it is called *unstable*.
So, we can restate Theorem \[basicstab\] by saying, for any $n$, there is some $K$ such that $3^K n$ is stable.
This allows us to define stable or stabilized analogues of many of the concepts and discussed above, and prove stabilized analogues of the theorems discussed in Section \[thm1sec\]. See Sections \[subsecdft\] and \[secdefn\] for the relevant definitions, and Section \[mainsec\] for the versions of the main theorems generalized to cover the stabilized case as well.
Discussion: Comparison to addition chains
-----------------------------------------
In order to make sense of Theorem \[mainthm\], it is helpful to introduce an analogy to addition chains, a different notion of complexity which is similar in spirit but different in detail. An *addition chain* for $n$ is defined to be a sequence $(a_0,a_1,\ldots,a_r)$ such that $a_0=1$, $a_r=n$, and, for any $1\le k\le r$, there exist $0\le i, j<k$ such that $a_k = a_i + a_j$; the number $r$ is called the length of the addition chain. The shortest length among addition chains for $n$, called the *addition chain length* of $n$, is denoted ${\ell}(n)$. Addition chains were introduced in 1894 by H. Dellac [@Dellac] and reintroduced in 1937 by A. Scholz [@aufgaben]; extensive surveys on the topic can be found in Knuth [@TAOCP2 Section 4.6.3] and Subbarao [@subreview].
The notion of addition chain length has obvious similarities to that of integer complexity; each is a measure of the resources required to build up the number $n$ starting from $1$. Both allow the use of addition, but integer complexity supplements this by allowing the use of multiplication, while addition chain length supplements this by allowing the reuse of any number at no additional cost once it has been constructed. Furthermore, both measures are approximately logarithmic; the function ${\ell}(n)$ satisfies $$\log_2 n \le {\ell}(n) \le 2\log_2 n.$$
A difference worth noting is that ${\ell}(n)$ is actually known to be asymptotic to $\log_2 n$, as was proved by Brauer [@Brauer], but the function ${\|n\|}$ is not known to be asymptotic to $3\log_3 n$; the value of the quantity $\limsup_{n\to\infty} \frac{{\|n\|}}{\log n}$ remains unknown.
Nevertheless, there are important similarities between integer complexity and addition chains. As mentioned above, the set of all integer complexity defects is a well-ordered subset of the real numbers, with order type $\omega^\omega$. We might also define the notion of *addition chain defect*, defined by $${\delta}^{\ell}(n):={\ell}(n)-\log_2 n;$$ for as shown in [@adcwo], the well-ordering theorem for integer complexity has an analogue for addition chains:
Let $\mathscr{D}^{\ell}$ denote the set $\{ {\delta}^{\ell}(n) : n \in \mathbb{N}
\}$. Then considered as a subset of the real numbers, $\mathscr{D}^{\ell}$ is well-ordered and has order type $\omega^\omega$.
More commonly, however, it is not ${\delta}^{\ell}(n)$ that has been studied, but rather $s(n)$, the number of *small steps* of $n$, which is defined to be ${\ell}(n)-\lfloor\log_2\rfloor$, or equivalently $\lceil{\delta}^{\ell}(n)\rceil$. The quantity $D(n)$ that we introduce seems to play a role in integer complexity similar to $s(n)$ in the study of addition chains. Now, unlike with $s(n)$ and ${\delta}^{\ell}(n)$, $D(n)$ is not simply $\lceil{\delta}(n)\rceil$; for instance, $D(56)=1$ even though ${\delta}(56)>1$. (Although Theorem \[dtoD\] will show how $D(n)$ is in a certain sense almost $\lceil{\delta}(n)\rceil$.) But, there are further analogies.
Analogous to Theorem \[basicstab\], we have (from [@adcwo]) the following:
\[adcstab\] For any natural number $n$, there exists $K\ge 0$ such that, for any $k\ge K$, $${\ell}(2^k n)=(k-K)+{\ell}(2^K n).$$
So we define a number $n$ to be *${\ell}$-stable* if for any $k$, one has ${\ell}(2^k n)=k+{\ell}(n)$; then Theorem \[adcstab\] says that for any $n$, there is some $K$ such that $2^K n$ is ${\ell}$-stable. This allows us to formulate a stabilized version of the previous analogy – and of the ones to follow.
In [@adcwo], this author conjectured:
For each whole number $k$, $\mathscr{D}^{\ell}\cap[0,k]$ has order type $\omega^k$.
In other words, this conjecture states that the limit of the initial $\omega^k$ addition chain defects is equal to $k$. If true, this would mean that $s(n)$ plays the same role for $\mathscr{D}^{\ell}$ as $D(n)$ does for the $\mathscr{D}^a$, that $s(n)$ is the smallest $k$ such that the order type of $\mathscr{D}^{\ell}\cap[0,{\delta}^{\ell}(n))$ is less than $\omega^k$.
One similarly based on conjectures in [@adcwo] gets analogies between $D_{{st}}(n)$ and $s_{{st}}(n)$ and how they determine position in $\mathscr{D}^a_{{st}}$ and $\mathscr{D}^{\ell}_{{st}}$, respectively; see Section \[secdefn\] for definitions of these.
It’s worth noting here one important difference between these two cases: in the integer complexity case, we need to split things into congruence classes modulo $3$ based on ${\|n\|}$. This has no analogue in the addition chain case. The difference comes from a difference in certain fundamental inequalities that these quantities obey. Integer complexity obeys ${\|3n\|}\le{\|n\|}+3$, with equality if and only if ${\delta}(3n)={\delta}(n)$. The addition chain analogue of this is that one has ${\ell}(2n)\le{\ell}(n)+1$, with equality if and only if ${\delta}^{\ell}(2n)={\delta}^{\ell}(n)$. The result [@adcwo; @paper1] is that if we have two numbers $m$ and $n$ with ${\delta}^{\ell}(n)={\delta}^{\ell}(m)$, then one must have $m=2^k n$ for some $k\in\mathbb{Z}$; and if we have two numbers $m$ and $n$ with ${\delta}(n)={\delta}(m)$, then one must have $m=3^k n$ for some $k\in\mathbb{Z}$. However in the latter case we must also have ${\|m\|}\equiv{\|n\|}\pmod{3}$; this is why the sets $\mathscr{D}^a$ are disjoint. In the addition chain case there is no such congruence requirement; ${\ell}(n)$ and ${\ell}(m)$ need only be congruent modulo $1$, which is no requirement at all, so splitting up $\mathscr{D}^{\ell}$ in a similar manner does not make sense. The set $\mathscr{D}^{\ell}$ already covers the one and only congruence class that exists in the addition chain case.
But it is not only our primary theorem but also our secondary theorem here that has an analogues for addition chains, and in this case the analogy does not rely on any conjectures. While the hypothesis that the order type of $\mathscr{D}^{\ell}\cap[0,k]$ is equal to $\omega^k$ remains a conjecture, that this holds for $k\le 2$ – and in particular that it holds for $k=1$ – was proven in [@adcwo]. This means that just as we can look at the first $\omega$ elements of each $\mathscr{D}^a$ in order to determine the $r$’th-highest number of complexity $k$, we can look at the first $\omega$ elements of $\mathscr{D}^{\ell}$ to determine the $r$’th-highest number of addition chain length $k$ (or at most $k$, which in these cases is the same thing). (Again, here $k$ must be sufficiently large relative to $r$. Also, again here we are using the convention that $r$ starts at $0$ rather than $1$.)
Specifically, it’s an easy corollary of the classification of numbers with $s(n)\le 1$ (due to Gioia et al. [@sub1962]) that:
For $k\ge r+1$ (or for $k\ge0$ when $r=0$), the $r$’th-largest number of addition chain length $k$ is $(\frac{1}{2}+\frac{1}{2^{r+1}})2^k$.
Obviously here the fraction $\frac{1}{2}+\frac{1}{2^{r+1}}$ plays the role of the $h_r$ and $r+1$ plays the role of $K_r$; unlike with integer complexity, there are no irregularities here, just a single straightforward infinite family. (And note how the analogue of the $K_r$ increases in what is mostly steps of $1$, rather than mostly steps of $3$ like the actual $K_r$, because once again with addition chains there’s only one congruence class.) For more on the analogy between integer complexity and addition chains, particularly with regard to their sets of defects, one may see [@theory].
Integer complexity, well-ordering, and low-defect polynomials {#polysec}
=============================================================
In this section we summarize the results of [@paperwo; @theory; @paper1] that we will need later regarding the defect ${\delta}(n)$; the stable complexity ${\|n\|}_{{st}}$ and stable defect ${\delta}_{{st}}(n)$ described below; and low-defect polynomials.
The defect and stability {#subsecdft}
------------------------
First, some basic facts about the defect:
\[oldprops\] We have:
1. For all $n$, ${\delta}(n)\ge 0$.
2. For $k\ge 0$, ${\delta}(3^k n)\le {\delta}(n)$, with equality if and only if ${\|3^k n\|}=3k+{\|n\|}$. The difference ${\delta}(n)-{\delta}(3^k n)$ is a nonnegative integer.
3. A number $n$ is stable if and only if for any $k\ge 0$, ${\delta}(3^k
n)={\delta}(n)$.
4. If the difference ${\delta}(n)-{\delta}(m)$ is rational, then $n=m3^k$ for some integer $k$ (and so ${\delta}(n)-{\delta}(m)\in\mathbb{Z}$).
5. Given any $n$, there exists $k$ such that $3^k n$ is stable.
6. For a given defect $\alpha$, the set $\{m: {\delta}(m)=\alpha \}$ has either the form $\{n3^k : 0\le k\le L\}$ for some $n$ and $L$, or the form $\{n3^k :
0\le k\}$ for some $n$. The latter occurs if and only if $\alpha$ is the smallest defect among ${\delta}(3^k n)$ for $k\in \mathbb{Z}$.
7. If ${\delta}(n)={\delta}(m)$, then ${\|n\|}={\|m\|} \pmod{3}$.
8. ${\delta}(1)=1$, and for $k\ge 1$, ${\delta}(3^k)=0$. No other integers occur as ${\delta}(n)$ for any $n$.
9. If ${\delta}(n)={\delta}(m)$ and $n$ is stable, then so is $m$.
Parts (1) through (8), excepting part (3), are just Theorem 2.1 from [@paperwo]. Part (3) is Proposition 12 from [@paper1], and part (9) is Proposition 3.1 from [@paperwo].
We will want to consider the set of all defects:
We define the *defect set* $\mathscr{D}$ to be $\{{\delta}(n):n\in{{\mathbb N}}\}$, the set of all defects.
We also defined $\mathscr{D}^a$, for $a$ a congruence class modulo $3$, in Definition \[dadef\] earlier.
The paper [@paperwo] also defined the notion of a *stable defect*:
We define a *stable defect* to be the defect of a stable number, and define $\mathscr{D}_{{st}}$ to be the set of all stable defects. Also, for $a$ a congruence class modulo $3$, we define $\mathscr{D}^a_{{st}}=\mathscr{D}^a \cap
\mathscr{D}_{{st}}$.
Because of part (9) of Theorem \[oldprops\], this definition makes sense; a stable defect $\alpha$ is not just one that is the defect of some stable number, but one for which any $n$ with ${\delta}(n)=\alpha$ is stable. Stable defects can also be characterized by the following proposition from [@paperwo]:
\[modz1\] A defect $\alpha$ is stable if and only if it is the smallest $\beta\in\mathscr{D}$ such that $\beta\equiv\alpha\pmod{1}$.
We can also define the *stable defect* of a given number, which we denote ${\delta}_{{st}}(n)$.
For a positive integer $n$, define the *stable defect* of $n$, denoted ${\delta}_{{st}}(n)$, to be ${\delta}(3^k n)$ for any $k$ such that $3^k n$ is stable. (This is well-defined as if $3^k n$ and $3^\ell n$ are stable, then $k\ge \ell$ implies ${\delta}(3^k n)={\delta}(3^\ell n)$, and $\ell\ge k$ implies this as well.)
Note that the statement “$\alpha$ is a stable defect”, which earlier we were thinking of as “$\alpha={\delta}(n)$ for some stable $n$”, can also be read as the equivalent statement “$\alpha={\delta}_{{st}}(n)$ for some $n$”.
Similarly we have the stable complexity:
For a positive integer $n$, define the *stable complexity* of $n$, denoted ${\|n\|}_{{st}}$, to be ${\|3^k n\|}-3k$ for any $k$ such that $3^k n$ is stable.
We then have the following facts relating the notions of ${\|n\|}$, ${\delta}(n)$, ${\|n\|}_{{st}}$, and ${\delta}_{{st}}(n)$:
\[stoldprops\] We have:
1. ${\delta}_{{st}}(n)= \min_{k\ge 0} {\delta}(3^k n)$
2. ${\delta}_{{st}}(n)$ is the smallest $\alpha\in\mathscr{D}$ such that $\alpha\equiv {\delta}(n) \pmod{1}$.
3. ${\|n\|}_{{st}}= \min_{k\ge 0} ({\|3^k n\|}-3k)$
4. ${\delta}_{{st}}(n)={\|n\|}_{{st}}-3\log_3 n$
5. ${\delta}_{{st}}(n) \le {\delta}(n)$, with equality if and only if $n$ is stable.
6. ${\|n\|}_{{st}}\le {\|n\|}$, with equality if and only if $n$ is stable.
7. ${\|3n\|}_{{st}}= {\|n\|}_{{st}}+3$
8. If ${\delta}_{{st}}(n)={\delta}_{{st}}(m)$, then ${\|n\|}_{{st}}\equiv{\|m\|}_{{st}}\pmod{3}$.
Statements (1)-(6) are just Propositions 3.5, 3.7, and 3.8 from [@paperwo]. Statement (7) follows from the definition of stable complexity; if $3^k n$ is stable, then ${\|3n\|}_{{st}}={\|3^k n\|}-3(k-1)={\|3^k n\|}-3k+3={\|n\|}_{{st}}+3$. To prove statement (8), note that if ${\delta}_{{st}}(n)={\delta}_{{st}}(m)$, then by statement (2) one has ${\delta}(n)\equiv{\delta}(m)\pmod{1}$, and so by Propostion \[oldprops\], one has that $n=m3^k$ for some $k\in\mathbb{Z}$, and so ${\|n\|}_{{st}}={\|m\|}_{{st}}+3k$.
Note, by the way, that just as $\mathscr{D}_{{st}}$ can be characterized either as defects ${\delta}(n)$ with $n$ stable or as defects ${\delta}_{{st}}(n)$ for any $n$, $\mathscr{D}^a_{{st}}$ can be characterized either as defects ${\delta}(n)$ with $n$ stable and ${\|n\|}\equiv a\pmod{3}$, or as defects ${\delta}_{{st}}(n)$ for any $n$ with ${\|n\|}_{{st}}\equiv a\pmod{3}$.
Three defects that will be particularly important in this paper are the smallest three defects:
\[small3\] $$\mathscr{D}\cap [0,2{\delta}(2)] = \{0,{\delta}(2),2{\delta}(2)\}.$$
Proposition 37 from [@paper1] tells us that the only leaders with defect less than $3{\delta}(2)$ are $3$, $2$, and $4$, which respectively have defects $0$, ${\delta}(2)$, and $2{\delta}(2)$.
Low-defect polynomials {#polysubsec}
----------------------
As has been mentioned in Section \[intropoly\], we are going to represent the set of numbers with defect at most $r$ by substituting in powers of $3$ into certain multilinear polynomials we call *low-defect polynomials*. We will associate with each one a “base complexity” to form a *low-defect pair*. In this section we will review the basic properties of these polynomials. First, their definition:
\[polydef\] We define the set $\mathscr{P}$ of *low-defect pairs* as the smallest subset of ${{\mathbb Z}}[x_1,x_2,\ldots]\times {{\mathbb N}}$ such that:
1. For any constant polynomial $k\in {{\mathbb N}}\subseteq{{\mathbb Z}}[x_1, x_2, \ldots]$ and any $C\ge {\|k\|}$, we have $(k,C)\in \mathscr{P}$.
2. Given $(f_1,C_1)$ and $(f_2,C_2)$ in $\mathscr{P}$, we have $(f_1\otimes
f_2,C_1+C_2)\in\mathscr{P}$, where, if $f_1$ is in $d_1$ variables and $f_2$ is in $d_2$ variables, $$(f_1\otimes f_2)(x_1,\ldots,x_{d_1+d_2}) :=
f_1(x_1,\ldots,x_{d_1})f_2(x_{d_1+1},\ldots,x_{d_1+d_2}).$$
3. Given $(f,C)\in\mathscr{P}$, $c\in {{\mathbb N}}$, and $D\ge {\|c\|}$, we have $(f\otimes x_1 + c,C+D)\in\mathscr{P}$ where $\otimes$ is as above.
The polynomials obtained this way will be referred to as *low-defect polynomials*. If $(f,C)$ is a low-defect pair, $C$ will be called its *base complexity*. If $f$ is a low-defect polynomial, we will define its *absolute base complexity*, denoted ${\|f\|}$, to be the smallest $C$ such that $(f,C)$ is a low-defect pair. We will also associate to a low-defect polynomial $f$ the *augmented low-defect polynomial* $${\hat{f}} = f\otimes x_1;$$ if $f$ is in $d$ variables, this is $fx_{d+1}$.
In this paper we will only concern ourselves with low-defect pairs $(f,C)$ where $C={\|f\|}$, so in the remainder of what follows, we will mostly dispense with the formalism of low-defect pairs and just discuss low-defect polynomials.
Note that the degree of a low-defect polynomial is also equal to the number of variables it uses; see Proposition \[polystruct\]. Also note that augmented low-defect polynomials are never themselves low-defect polynomials; as we will see in a moment (Proposition \[polystruct\]), low-defect polynomials always have nonzero constant term, whereas augmented low-defect polynomials always have zero constant term. We can also observe that low-defect polynomials are in fact read-once polynomials as discussed in for instance [@ROF].
Note that we do not really care about what variables a low-defect polynomial is in – if we permute the variables of a low-defect polynomial or replace them with others, we will still regard the result as a low-defect polynomial. From this perspective, the meaning of $f\otimes g$ could be simply regarded as “relabel the variables of $f$ and $g$ so that they do not share any, then multiply $f$ and $g$”. Helpfully, the $\otimes$ operator is associative not only with this more abstract way of thinking about it, but also in the concrete way it was defined above.
In [@paperwo] were proved the following propositions about low-defect polynomials:
\[polystruct\] Suppose $f$ is a low-defect polynomial of degree $d$. Then $f$ is a polynomial in the variables $x_1,\ldots,x_d$, and it is a multilinear polynomial, i.e., it has degree $1$ in each of its variables. The coefficients are non-negative integers. The constant term is nonzero, and so is the coefficient of $x_1\cdots x_d$, which we will call the *leading coefficient* of $f$.
This is Proposition 4.2 from [@paperwo].
\[basicub\] If $f$ is a low-defect polynomial of degree $d$, then $${\|f(3^{n_1},\ldots,3^{n_d})\|}\le {\|f\|}+3(n_1+\ldots+n_d).$$ and $${\|{\hat{f}}(3^{n_1},\ldots,3^{n_{d+1}})\|}\le {\|f\|}+3(n_1+\ldots+n_{d+1}).$$
This is a combination of Proposition 4.5 and Corollary 4.12 from [@paperwo].
The above proposition motivates the following definition:
Given a low-defect polynomial $f$ (say of degree $d$) and a number $N$, we will say that $f$ *efficiently $3$-represents* $N$ if there exist nonnegative integers $n_1,\ldots,n_d$ such that $$N=f(3^{n_1},\ldots,3^{n_d})\ \textrm{and}\
{\|N\|}={\|f\|}+3(n_1+\ldots+n_d).$$ We will say ${\hat{f}}$ efficiently $3$-represents $N$ if there exist $n_1,\ldots,n_{d+1}$ such that $$N={\hat{f}}(3^{n_1},\ldots,3^{n_{d+1}})\ \textrm{and}\
{\|N\|}={\|f\|}+3(n_1+\ldots+n_{d+1}).$$ More generally, we will also say $f$ $3$-represents $N$ if there exist nonnegative integers $n_1,\ldots,n_d$ such that $N=f(3^{n_1},\ldots,3^{n_d})$. and similarly with ${\hat{f}}$.
Note that previous papers [@paperalg; @paperwo; @theory] instead spoke of a low-defect pair $(f,C)$ efficiently $3$-representing a number $N$; however, as mentioned in those papers, it is only possible for some $(f,C)$ to efficiently $3$-represent a number $N$ if in fact $C={\|f\|}$, so there is no loss here.
In keeping with the name, numbers $3$-represented by low-defect polynomials, or their augmented versions, have bounded defect. Let us make some definitions first:
Given a low-defect polynomial $f$ we define ${\delta}(f)$, the defect of $f$, to be ${\|f\|}-3\log_3 m$, where $m$ is the leading coefficient of $f$.
Given a low-defect polynomial $f$ of degree $d$, we define $${\delta}_f(n_1,\ldots,n_d) =
{\|f\|}+3(n_1+\ldots+n_d)-3\log_3 f(3^{n_1},\ldots,3^{n_d}).$$
Then we have:
\[dftbd\] Let $f$ be a low-defect polynomial of degree $d$, and let the numbers $n_1,\ldots,n_{d+1}$ be nonnegative integers.
1. We have $${\delta}({\hat{f}}(3^{n_1},\ldots,3^{n_{d+1}}))\le {\delta}_f(n_1,\ldots,n_d),$$ and the difference is an integer.
2. We have $${\delta}_f(n_1,\ldots,n_d)\le{\delta}(f),$$ and if $d\ge 1$, this inequality is strict.
3. The function ${\delta}_f$ is strictly increasing in each variable, and $${\delta}(f) = \sup_{n_1,\ldots,n_d} {\delta}_f(n_1,\ldots,n_d).$$
This is a combination of Proposition 4.9 and Corollary 4.14 from [@paperwo] and Proposition 2.15 from [@theory].
Importantly, the set of defects coming from a low-defect polynomial of degree $r$ has order type approximately $\omega^r$; if rather than the actual defects we use ${\delta}_f$, then this is exact. More formally:
\[oldwo\] Let $f$ be a low-defect polynomial of degree $d$. Then:
1. The image of ${\delta}_f$ is a well-ordered subset of $\mathbb{R}$, with order type $\omega^d$.
2. The set of ${\delta}(N)$ for all $N$ $3$-represented by the augmented low-defect polynomial ${\hat{f}}$ is a well-ordered subset of $\mathbb{R}$, with order type at least $\omega^d$ and at most $\omega^d(\lfloor \delta(f)
\rfloor+1)<\omega^{d+1}$. The same is true if $f$ is used instead of the augmented version ${\hat{f}}$.
This is a combination of Propositions 6.2 and 6.3 from [@paperwo].
The second part of the above proposition follows from the first by means of theorems about cutting and pasting of well-ordered sets, ultimately due to Carruth [@carruth]. In particular:
\[cutandpaste\] We have:
1. If $S$ is a well-ordered set and $S=S_1\cup\ldots\cup S_n$, and $S_1$ through $S_n$ all have order type less than $\omega^k$, then so does $S$.
2. If $S$ is a well-ordered set of order type $\omega^k$ and $S=S_1\cup\ldots\cup S_n$, then at least one of $S_1$ through $S_n$ also has order type $\omega^k$.
One may see [@carruth] or [@wpo] for proofs of these.
We will need in particular the following variant:
\[interleave\] Suppose $\alpha$ is an ordinal and $S$ is a well-ordered set which can be written as a finite union $S_1 \cup \ldots \cup S_k$ such that:
1. The $S_i$ all have order types at most $\omega^\alpha$.
2. If a set $S_i$ has order type $\omega^\alpha$, it is cofinal in $S$.
Then the order type of $S$ is at most $\omega^\alpha$. In particular, if at least one of the $S_i$ has order type $\omega^\alpha$, then $S$ has order type $\omega^\alpha$.
A proof of this can be found in [@adcwo] where it is Proposition 5.4.
As was noted above, we have ${\delta}(f(3^{n_1},\ldots,3^{n_d})\le
{\delta}_f(n_1,\ldots,n_d)$. Importantly, though, for certain low-defect polynomials $f$, namely, those with ${\delta}(f)<\deg f+1$, we can show that equality holds for “most” choices of $(n_1,\ldots,n_d)$ in a certain sense.
Specifically:
\[dump\] Let $f$ be a low-defect polynomial of degree $d$ with ${\delta}(f)<d+1$. Define its “exceptional set” to be $$S:=\{(n_1,\ldots,n_d): {\|f(3^{n_1},\ldots,3^{n_d})\|}_{{st}}<{\|f\|}+3(n_1+\ldots+n_d)\}$$ Then the set $\{{\delta}(f(3^{n_1},\ldots,3^{n_d})):(n_1,\ldots,n_d)\in S\}$ has order type less than $\omega^d$, and therefore so does the set $\{{\delta}({\hat{f}}(3^{n_1},\ldots,3^{n_{d+1}})):(n_1,\ldots,n_d)\in S\}$. In particular, for $a\not\equiv {\|f\|}\pmod{3}$, the set $$\{{\delta}({\hat{f}}(3^{n_1},\ldots,3^{n_{d+1}})):(n_1,\ldots,n_{d+1})\in
\mathbb{Z}^{d+1}_{\ge 0}\} \cap \mathscr{D}^a$$ has order type less than $\omega^d$. Meanwhile, the set $$\{{\delta}(f(3^{n_1},\ldots,3^{n_d})):(n_1,\ldots,n_d)\notin
S\}$$ has order type at least $\omega^d$, and thus so does the set $$\{{\delta}(f(3^{n_1},\ldots,3^{n_d})):(n_1,\ldots,n_d)\in \mathbb{Z}^d_{\ge
0}\} \cap \mathscr{D}_{{st}}^{{\|f\|}};$$ moreover, the supremum of this latter set is equal to ${\delta}(f)$.
Most of this is direct from Proposition 7.2 from [@paperwo]; the only parts not covered in the statement there there are the statement about $\{{\delta}({\hat{f}}(3^{n_1},\ldots,3^{n_{d+1}})):(n_1,\ldots,n_d)\in S\}$, the statement regarding $a\not\equiv {\|f\|}\pmod{3}$, and the final statement.
The first of these follows directly from the first part, because $${\delta}({\hat{f}}(3^{n_1},\ldots,3^{n_{d+1}}))\le{\delta}(f(3^{n_1},\ldots,3^{n_d}))$$ with the difference being an integer, and that integer can certainly be no more than ${\delta}(f(3^{n_1},\ldots,3^{n_d}))\le{\delta}(f)$. Thus the set $$\{{\delta}({\hat{f}}(3^{n_1},\ldots,3^{n_{d+1}})):(n_1,\ldots,n_{d+1})\in
\mathbb{Z}^{d+1}_{\ge 0}\} \cap \mathscr{D}^a$$ can be covered by finitely many translates of $\{{\delta}(f(3^{n_1},\ldots,3^{n_d})):(n_1,\ldots,n_d)\in S\}$ and so by Proposition \[cutandpaste\] has order type less than $\omega^d$.
For the statement about $$\{{\delta}({\hat{f}}(3^{n_1},\ldots,3^{n_{d+1}})):(n_1,\ldots,n_{d+1})\in
\mathbb{Z}^{d+1}_{\ge 0}\} \cap \mathscr{D}^a$$ with $a\not\equiv C\pmod{3}$, if ${\|{\hat{f}}(3^{n_1},\ldots,3^{n_d})\|}\equiv a\not\equiv {\|f\|} \pmod{3}$, then in particular this means that $${\|{\hat{f}}(3^{n_1},\ldots,3^{n_{d+1}})\|} \ne {\|f\|}+3(n_1+\ldots+n_{d+1})$$ which means that $${\|{\hat{f}}(3^{n_1},\ldots,3^{n_{d+1}})\|} < {\|f\|}+3(n_1+\ldots+n_{d+1})$$ and therefore that $${\|f(3^{n_1},\ldots,3^{n_d})\|}_{{st}}< {\|f\|}+3(n_1+\ldots+n_d),$$ i.e., that $(n_1,\ldots,n_d)\in S$. Applying what was proved in the previous paragraph now proves the statement.
As for the final statement, the set $\{{\delta}(f(3^{n_1},\ldots,3^{n_d})):(n_1,\ldots,n_d)\in \mathbb{Z}^d_{\ge
0}\} \cap \mathscr{D}_{{st}}^{{\|f\|}}$ contains ${\delta}_f(\mathbb{N}^d\setminus S)$ (one may see the proof in [@paperwo]) which in turn contains ${\delta}_f(\mathbb{N}^d)\setminus{\delta}_f(S)$. Since the image of ${\delta}_f$ has order type $\omega^d$ while ${\delta}_f(S)$ has order type less than $\omega^d$ – similarly to above, this follows by the initial statement and Proposition \[cutandpaste\] – it follows that ${\delta}_f(\mathbb{N}^d)\setminus{\delta}_f(S)$ has order type $\omega^d$ and thus is cofinal in the image of ${\delta}_f$, and thus has supremum ${\delta}(f)$; and the same is true of the larger set $\{{\delta}(f(3^{n_1},\ldots,3^{n_d})):(n_1,\ldots,n_d)\in \mathbb{Z}^d_{\ge 0}\}
\cap \mathscr{D}_{{st}}^{{\|f\|}}$ which is also bounded above by ${\delta}(f)$.
Finally, one more property of low-defect polynomials we will need is the following:
\[polycpxbd\] Let $f$ be a low-defect polynomial, and suppose that $a$ is the leading coefficient of $f$. Then ${\|f\|}\ge {\|a\|} + \deg f$. In particular, ${\delta}(f) \ge {\delta}(a) + \deg f$.
This is Proposition 3.24 from [@theory].
With this, we have the basic properties of low-defect polynomials.
Note that one reason nothing is lost here by discarding the formalism of low-defect pairs is that the low-defect pairs $(f,C)$ we will (implicitly) concern ourselves with in this paper are ones that satisfy $C-3\log_3 m<\deg
f+1$, where $m$ is the leading coefficient of $f$. However, by Proposition \[polycpxbd\], $$\deg f \le {\delta}(f) \le C-3\log_3 m < \deg f + 1,$$ thus $C-{\|f\|}=(C-3\log_3 m)-{\delta}(f)<1$ and so $C={\|f\|}$. So if we were to use low-defect pairs, we would only be using pairs where $C={\|f\|}$, so we lose nothing by making this assumption.
Good coverings {#buildsec}
--------------
We need one more set of definitions before we can state the theorem that will be used as the basis of the proof of the main theorem. We define:
A natural number $n$ is called a *leader* if it is the smallest number with a given defect. By part (6) of Theorem \[oldprops\], this is equivalent to saying that either $3\nmid n$, or, if $3\mid n$, then ${\delta}(n)<{\delta}(\frac{n}{3})$, i.e., ${\|n\|}<3+{\|\frac{n}{3}\|}$.
Let us also define:
For any real $s\ge0$, define the set of [*$s$-defect numbers*]{} $A_s$ to be $$A_s := \{n\in\mathbb{N}:{\delta}(n)<s\}.$$ Define the set of [*$s$-defect leaders*]{} $B_s$ to be $$B_r:= \{n \in A_s :~~n~~\mbox{is a leader}\}.$$
These sets are related by the following proposition from [@paperwo]:
\[arbr\] For every $n\in A_s$, there exists a unique $m\in B_s$ and $k\ge 0$ such that $n=3^k m$ and ${\delta}(n)={\delta}(m)$; then ${\|n\|}={\|m\|}+3k$.
Because of this, if we want to describe the set $A_r$, it suffices to describe the set $B_r$. Now we can define:
\[goodcover\] For a real number $s\ge0$, a finite set ${{\mathcal S}}$ of low-defect polynomials will be called a *good covering* for $B_s$ if every $n\in B_r$ can be efficiently $3$-represented by some polynomial in ${{\mathcal S}}$ (and hence every $n\in Asr$ can be efficiently represented by some ${\hat{f}}$ with $f\in {{\mathcal S}}$) and if for every $f\in{{\mathcal S}}$, ${\delta}(f)\le s$, with this being strict if $\deg f=0$.
This allows us to state the main theorem from [@theory]:
\[theory\] For any real number $s\ge 0$, there exists a good covering of $B_s$.
This is Theorem 4.9 from [@theory] rewritten in terms of Definition \[goodcover\], and using low-defect polynomials instead of pairs. (Any low-defect pairs $(f,C)$ with $C>{\|f\|}$ can be filtered out of a good covering, since such a pair can never efficiently $3$-represent anything.)
Note that by Proposition \[polycpxbd\], if $f$ is in a good covering of $B_s$ with leading coefficient $m$, we must have ${\delta}(m)+\deg f\le s$.
The integer defect {#secdefn}
==================
In this section we state some basic facts about $D(n)$, what it means, and how it may be computed.
Let us start by giving another interpretation of what $D(n)$ means:
\[Dinterp\] For a natural number $n$, $$D(n) = |\{k: n<E(k)\le E({\|n\|})\}|.$$
That is to say, $D(n)$ measures how far down $n$ is among numbers with complexity ${\|n\|}$, measured by how many values of $E$ one passes as one counts downwards towards $n$ from the largest number also having complexity ${\|n\|}$.
By definition, $L(n)$ is the largest $k$ such that $E(k)\le n$. Since $E(k)$ is strictly increasing, the number of $k$ such that $n<E(k)\le E({\|n\|})$ is equal to the difference ${\|n\|}-L(n)$, i.e., $D(n)$.
So for instance, one has that $D(n)=0$ if and only if $n$ is of the form $E(k)$ for some $k$, i.e., $n$ is the largest number of its complexity; while $D(n)\le
1$ if and only if $n>E({\|n\|}-1)$, i.e., $n$ is greater than all numbers of lower complexity. Numbers $n$ with $D(n)\le 1$ will be discussed more in Section \[Erk\].
As for properties of the integer defect, it behaves largely analogously to the real defect:
\[oldpropsD\] We have:
1. For all $n$, $D(n)\ge 0$.
2. For all $n>1$, $L(3n)=L(n)+3$.
3. For $n>1$ and $k\ge 0$, one has $D(3^k n)\le D(n)$, with equality if and only if ${\|3^k n\|}=3k+{\|n\|}$.
4. A number $n>1$ is stable if and only if for any $k\ge 0$, $D(3^k n)=D(n)$.
Statement (1) is just the statement that $L(n)\le{\|n\|}$; this follows from the definition of $L(n)$ as $E({\|n\|})\ge n$ and so (as $E(k)$ is increasing) one must have $L(n)\le {\|n\|}$. And once statement (2) is established, statements (3) and (4) then follow from that and may be proved in exactly the same way their analogous statements in Theorem \[oldprops\] are proved. This leaves just statement (2) to be proved. Note that, for any $k>1$, $E(k+3)=3E(k)$. Therefore, for any $k>1$, $E(k+3)\le
3n$ if and only if $E(k)\le n$, and so $L(3n)=L(n)+3$; the only possible exception to this would be if one had $L(n)=1$, which happens only when $n=1$.
Note that while the theorem that for any $n$ there is some $k$ such that $3^k n$ is stable was originally proven using the defect ${\delta}(n)$, it could also just as well be proven using the integer defect $D(n)$.
We can also of course define a stable variant of $D(n)$:
For a positive integer $n$, we define the stable integer defect of $n$, denoted $D_{{st}}(n)$, to be $D(3^k n)$ for any $k$ such that $3^k n$ is stable.
Note that Proposition \[oldpropsD\] shows that this is well-defined. We then have:
\[stoldpropsD\] We have:
1. $D_{{st}}(n)= \min_{k\ge 0} D(3^k n)$
2. For $n>1$, $D_{{st}}(n)={\|n\|}_{{st}}-L(n)$
3. $D_{{st}}(n) \le D(n)$, with equality if and only if $n$ is stable or $n=1$
4. For $n>1$, $D(n)-D_{{st}}(n)={\delta}(n)-{\delta}_{{st}}(n)=
{\|n\|}-{\|n\|}_{{st}}$
With the exeption of (4), of which no analogue has previously been mentioned, these all follow from Proposition \[oldpropsD\] and their proofs are exactly analogous to those of the statements in Proposition \[stoldprops\]; meanwhile (4) follows immediately from (2) and the definition of $D(n)$.
We then also have the analogue of Proposition \[Dinterp\]:
For a natural number $n>1$, $$D_{{st}}(n) = |\{k: n<E(k)\le E({\|n\|}_{{st}})\}|.$$
Once again, by definition, $L(n)$ is the largest $k$ such that $E(k)\le n$. And since $E(k)$ is strictly increasing, the number of $k$ such that $n<E(k)\le
E({\|n\|}_{{st}})$ is equal to the difference ${\|n\|}_{{st}}-L(n)$, which by Proposition \[stoldpropsD\] is $D_{{st}}(n)$.
It may seem strange that $1$ needs to be excluded, given that its special status goes away when stabilized. However, ${\|1\|}_{{st}}=0$, and $E(0)$ is not defined, so $n=1$ must still be excluded from the theorem statement.
Note, by the way:
\[D=0prop\] For any natural number $n$, $D(n)=0$ if and only if $D_{{st}}(n)=0$.
It’s immediate that a number $n$ with $D(n)=0$ is stable and so has $D_{{st}}(n)=0$ (unless $n=1$, in which case one still has $D_{{st}}(n)=0$). For the reverse, a number $n$ has $D_{{st}}(n)=0$ if and only if there is some $k$ such that $D(3^k
n)=0$. However, as the numbers $n$ with $D(n)=0$ are precisely those numbers of the form $3^k$, $2\cdot3^k$, and $4\cdot 3^k$, we see that if $n$ has $D_{{st}}(n)=0$, it must itself be of one of these forms, and thus have $D(n)=0$.
See Corollaries \[D=1cor\] and \[2stab\] for related statements.
Having discussed what $D(n)$ is and how it acts, let’s finally discuss how it may be computed. The quantity $D(n)$ is just the difference ${\|n\|}-L(n)$. We know how to compute ${\|n\|}$, although not necessarily quickly; see [@anv] for the currently best-known algorithm for computing complexity, and [@miller] for the best-known bounds on its runtime. But the other half, computing $L(n)$, is very simple and can be done much quicker, because it’s given by the following formula:
\[computL\] For a natural number $n$, $$L(n) = \max\{3\lfloor\log_3n\rfloor,\
3\left\lfloor\log_3\frac{n}{2}\right\rfloor+2,\
3\left\lfloor\log_3\frac{n}{4}\right\rfloor+4,1\}.$$
The quantity $L(n)$ is by definition the largest $k$ such that $E(k)\le n$. The largest such $k$ congruent to $0$ modulo $3$ is $3\lfloor\log_3n\rfloor$ (so long as this quantity is positive; otherwise there is none), the largest such $k$ congruent to $2$ modulo $3$ is $3\lfloor\log_3\frac{n}{2}\rfloor+2$ (with the same caveat), the largest such $k>1$ congruent to $1$ modulo $3$ is $3\lfloor\log_3\frac{n}{4}\rfloor+4$ (again with the same caveat), and of course the largest such $k$ equal to $1$ is $1$. So the largest of these is $L(n)$ (and any of them that are not valid positive and thus not a valid $k$ will not affect the maximum).
Let us make here a definition that will be useful later:
For a natural number $n$, define $R(n)=\frac{n}{E({\|n\|})}$. We also define $R_{{st}}(n)$ to be $R(3^k n)$ for any $k$ such that $3^k n$ is stable, or equivalently (for $n>1$) as $\frac{n}{E({\|n\|}_{{st}})}$.
This is easily related to the defect, as was done in an earlier paper [@paperwo]:
\[dRformulae\] We have, for $n>1$, $$\delta(n)=\left\{ \begin{array}{ll}
-3\log_3 R(n) & \mathrm{if}\quad {\|n\|}\equiv 0\pmod{3}, \\
-3\log_3 R(n) +2\,\delta(2)
& \mathrm{if}\quad {\|n\|}\equiv 1\pmod{3}, \\
-3\log_3 R(n) +\delta(2)
& \mathrm{if}\quad {\|n\|}\equiv 2\pmod{3},
\end{array} \right.$$ and the same relation (without the $n>1$ restriction) holds between $R_{{st}}(n)$, ${\|n\|}_{{st}}$, and ${\delta}_{{st}}(n)$.
The relation between $R(n)$ and ${\delta}(n)$ is just Proposition A.3 from [@paperwo], and the proof for the stable case is exactly analogous.
Now we see that in addition to being easy to compute $L(n)$, it’s also simple to determine $D(n)$ from ${\delta}(n)$, at least if we know the value of ${\|n\|}$ modulo $3$, which technically is implicit in ${\delta}(n)$. First, a definition:
Let $a$ be a congruence class modulo $3$ and $k$ be a whole number. Define $${t_{a}}(k) = \left\{
\begin{array}{ll}
k &\textrm{if}\ k\equiv a\pmod{3}\\
k+{\delta}(2) &\textrm{if}\ k\equiv a+1\pmod{3}\\
k+2{\delta}(2) &\textrm{if}\ k\equiv a+2\pmod{3}\\
\end{array}
\right.$$
Now:
\[dtoD\] Let $n>1$ be a natural number. Then $D(n)$ is equal to the smallest $k$ such that ${\delta}(n)\le {t_{{\|n\|}}}(k)$. Moreover, if $n$ is any natural number, $D_{{st}}(n)$ is equal to the smallest $k$ such that ${\delta}_{{st}}(n)\le
{t_{{\|n\|}_{{st}}}}(k)$.
Since two numbers with the same defect also have the same complexity modulo $3$ (and ${\delta}(n)=1$ if and only if $n=1$), and the analogous statement is also true of stable complexity and defect, in particular we have that if ${\delta}(n)={\delta}(m)$ then $D(n)=D(m)$, and if ${\delta}_{{st}}(n)={\delta}_{{st}}(m)$ then $D_{{st}}(n)=D_{{st}}(m)$.
Note in addition that since ${\delta}(n)={\delta}(m)$ implies ${\delta}_{{st}}(n)={\delta}_{{st}}(m)$ (see statement (2) in Proposition \[stoldprops\]) one has that if ${\delta}(n)={\delta}(m)$ then $D_{{st}}(n)=D_{{st}}(m)$.
Theorem \[dtoD\] makes precise how $D(n)$ is “almost $\lceil{\delta}(n)\rceil$”. It is, as was noted in the introduction, not the same, but it is the smallest $k$ such that ${\delta}(n)\le {t_{{\|n\|}}}(k)$, where ${t_{{\|n\|}}}(k)$ may not be exactly $k$ but never differs from it by more than $2{\delta}(2)<0.215.$
We prove only the non-stabilized case as the stabilized case is exactly analogous. We assume $n>1$.
From Proposition \[Dinterp\], we can see that $D(n)$ is determined by $R(n)$ and the value of ${\|n\|}$ modulo $3$. Specifically, $$D(n) = \left| \left\{ k : R(n) < \frac{E(k)}{E({\|n\|})} \le 1 \right\}
\right|,$$ so $D(n)$ is the number of values of $\frac{E(k)}{E({\|n\|})}$ in $(R(n),1]$. What are the values of this? They can be obtained as products of values $\frac{E(k)}{E(k+1)}$; this is equal to $2/3$ when $k\equiv1\ \textrm{or}\ 2
\pmod{3}$ (for $k>1$) and to $3/4$ when $k\equiv0\pmod{3}$.
Thus, if ${\|n\|}\equiv0\pmod{3}$, $D(n)$ will increase whenever $R(n)$ passes a value of the sequence $1, \frac{2}{3}, \frac{4}{9}, \frac{1}{3}, \frac{2}{9},
\frac{4}{27}, \frac{1}{9}, \ldots$; if ${\|n\|}\equiv1\pmod{3}$, whenever it passes a value of the sequence $1,
\frac{3}{4}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{6}, \frac{1}{9}, \ldots$; and if ${\|n\|}\equiv2 \pmod{3}$, whenever it passes a value of the sequence $1, \frac{2}{3}, \frac{1}{2},
\frac{1}{3}, \frac{2}{9}, \frac{1}{6}, \frac{1}{9}, \ldots$. (These sequences are just the sequences obtained by taking products of one of the three shifts of the periodic sequence $\frac{2}{3},
\frac{2}{3}, \frac{3}{4}, \frac{2}{3}, \frac{2}{3}, \frac{3}{4}\ldots$; note that regardless of which shift is used, the repeating part of the sequence always has a product of $\frac{1}{3}$, and so the product sequences will always consist of three interwoven geometric sequences each with ratio $\frac{1}{3}$.)
It just remains, then, to convert these values of $R(n)$ to their equivalents in defects, which can be done with Proposition \[dRformulae\]. Once this is done one finds that the values of ${\delta}(n)$ where $D(n)$ increases are precisely those listed in the definition of ${t_{{\|n\|}}}$, which completes the proof.
Theorem \[dtoD\] will form half the proof of Theorem \[mainthm\], and its stable analogue, Theorem \[mainthmstab\]; it tells us that the values of $D(n)$ “switch over” when ${\delta}(n)$ is of the form $k$, $k+{\delta}(2)$, or $k+2{\delta}(2)$ depending on the congruence class of $k-{\|n\|}$ modulo $3$. The other half the proof is, of course, Theorem \[chgoverpt\] (and its stable analogue, Theorem \[stabchgoverpt\]), which will tell us that these changeover points are exactly the limits of the initial $\omega^k$ defects in $\mathscr{D}^a$ (or $\mathscr{D}^a_{{st}}$).
The order interpretation of $D(n)$ {#mainsec}
==================================
In this section we aim to prove Theorem \[chgoverpt\] using the methods described in Section \[intropoly\]; combined with Theorem \[dtoD\] from the previous section, this will prove Theorem \[mainthm\]. Really, we want to prove generalizations:
\[stabchgoverpt\] For any $k\ge0$ and $a$ a congruence class modulo $3$, the order type of $\mathscr{D}^a\cap[0,{t_{a}}(k)]$ and the order type of $\mathscr{D}_{{st}}^a\cap[0,{t_{a}}(k)]$ are both equal to $\omega^k$.
\[mainthmstab\] Let $n>1$ be a natural number. Let $\zeta$ be the order type of $\mathscr{D}^{{\|n\|}}\cap[0,{\delta}(n))$. Then $D(n)$ is equal to the smallest $k$ such that $\zeta<\omega^k$. The same is true if we replace ${\delta}(n)$ by ${\delta}_{{st}}(n)$, $\mathscr{D}^{{\|n\|}}$ by $\mathscr{D}^{{\|n\|}_{{st}}}_{{st}}$, and $D(n)$ by $D(n)_{{st}}$.
Note that the proofs in this section will rely heavily on the results in Sections \[polysubsec\] and \[buildsec\]. Before we prove these, though, we will need a slight elaboration on Proposition \[dump\]:
\[dumptype\] Let $f$ be a low-defect polynomial of degree $d$ with ${\delta}(f)<d+1$. Then the order type of the set of all ${\delta}(N)$ for $n$ $3$-represented by ${\hat{f}}$ is exactly $\omega^d$.
By Proposition \[dump\], $\{{\delta}({\hat{f}}(3^{n_1},\ldots,3^{n_d})):(n_1,\ldots,n_d)\in S\}$ has order type less than $\omega^d$. Meanwhile, also by Proposition \[dump\], the set $$\{{\delta}(f(3^{n_1},\ldots,3^{n_d})):(n_1,\ldots,n_d)\notin S\}$$ has order type at least $\omega^d$, and is cofinal in $[0,{\delta}(f))$ (or $[0,{\delta}(f)]$ if $\deg f=0$) and therefore in the set of all ${\delta}(N)$ for $n$ $3$-represented by ${\hat{f}}$. But in fact, for $(n_1,\ldots,n_d)\notin S$, one has ${\delta}({\hat{f}}(3^{n_1},\ldots,3^{n_{d+1}})={\delta}_f(n_1,\ldots,n_d)$, and so this set (even when $f(3^{n_1},\ldots,3^{n_d})$ is replaced by ${\hat{f}}(3^{n_1},\ldots,3^{n_{d+1}})$) is a subset of the image of ${\delta}_f$, which by Proposition \[oldwo\] has order type $\omega^d$. So the conditions of Proposition \[interleave\] apply, and the union of these two sets, the set of all ${\delta}(n)$ for $N$ $3$-represented by ${\hat{f}}$, has order type at most $\omega^d$. We already know by Proposition \[oldwo\] it has order type at least $\omega^d$, so this proves the claim.
We now prove the main theorems of this section.
We need to show that the order type of $\mathscr{D}^a\cap[0,{t_{a}}(k)]$, as well as the order type of $\mathscr{D}_{{st}}^a\cap[0,{t_{a}}(k)]$, are both equal to $\omega^k$. This proof breaks down into two parts, an upper bound and a lower bound. Since $\mathscr{D}_{{st}}^a\subseteq\mathscr{D}^a$, it suffices to prove the upper bound for $\mathscr{D}^a\cap[0,{t_{a}}(k)]$, and the lower bound for $\mathscr{D}_{{st}}^a\cap[0,{t_{a}}(k)]$.
We begin with the upper bound. First, we observe that ${t_{a}}(k)$ is not itself an element of $\mathscr{D}^a$ for any $k>0$. We can see this as neither $k+{\delta}(2)$ nor $k+2{\delta}(2)$ is a defect for any $k>0$ (such a defect would have to come from some number $n$ satisfying $3^\ell n=2$ or $3^\ell n=4$ for $\ell>0$, which is impossible), and similarly no nonzero integer is a defect except $k=1$, which though an element of $\mathscr{D}$ is by definition excluded from all three $\mathscr{D}^a$. Thus $\mathscr{D}^a\cap[0,{t_{a}}(k)]=\mathscr{D}^a\cap[0,{t_{a}}(k))$ and we may concern ourselves with the order type of the latter.
Now we take a good covering ${{\mathcal S}}$ of $B_{{t_{a}}(k)}$ as per Theorem \[theory\]. For any $f\in{{\mathcal S}}$ with leading coefficient $m$, we have the inequality ${\delta}(m)+\deg f\le {\delta}(f)\le {t_{a}}(k)$. In particular, for any $f\in {{\mathcal S}}$, we have $\deg f\le \lfloor {t_{a}}(k)\rfloor = k$.
Suppose now that $\deg f=k$; then there is more we can say. For in this case, we have ${\delta}(m)\le {t_{a}}(k)-k\le2{\delta}(2)$. Thus ${\delta}(m)\in\{0,{\delta}(2),2{\delta}(2)\}$ by Proposition \[small3\]. Note that by their respective definitions, ${\delta}(f)\equiv{\delta}(m)\pmod{1}$; and, as noted above, ${\delta}(f)\ge\deg f=k$, and so ${\delta}(f)=k+{\delta}(m)\in\{k,k+{\delta}(2),k+2{\delta}(2)\}$. Note that ${\delta}(f)=k+{\delta}(m)$ means that $$k+{\|m\|}-3\log_3 m = {\|f\|} - 3\log_3 m$$ and therefore ${\|f\|}=k-{\|m\|}$. Moreover, if ${\delta}(m)=0$, then $m$ is of the form $3^\ell$ (for some $\ell>0)$ and ${\|m\|}=3\ell$, if ${\delta}(m)={\delta}(2)$ then $m$ is of the form $2\cdot3^\ell$ with ${\|m\|}=2+3\ell$, and if ${\delta}(m)=2{\delta}(2)$ then $m$ is of the form $4\cdot3^\ell$ with ${\|m\|}=4+3\ell$; from this we can conclude that, modulo $3$, $${\|f\|}\equiv\left\{\begin{array}{ll}
k & \textrm{if}\ {\delta}(f)=k \\
k-2 & \textrm{if}\ {\delta}(f)=k+{\delta}(2) \\
k-1 & \textrm{if}\ {\delta}(f)=k+2{\delta}(2) \\
\end{array}\right.$$
Now, let $T_f=\{{\delta}({\hat{f}}(3^{n_1},\ldots,3^{n_{d+1}})):n_1,\ldots,n_{d+1}\ge0\} \cap
\mathscr{D}^a$, where $d=\deg f$. Then by the assumption that ${{\mathcal S}}$ is a good covering of $B_{{t_{a}}(k)}$, we have that $$\mathscr{D}^a\cap[0,{t_{a}}(k)) = \bigcup_{f\in{{\mathcal S}}} T_f.$$ We want to show that the conditions of Proposition \[interleave\] hold for the sets $T_f$, so that we can conclude that $\mathscr{D}^a\cap[0,{t_{a}}(k))$ has order type at most $\omega^k$. If $\deg f<k$, then, by Proposition \[oldwo\], $T_f$ has order type less than $\omega^k$, and thus so does $T_f\cap\mathscr{D}^a$. Meanwhile, if $\deg f=k$, then since ${\delta}(f)\le
{t_{a}}(k)<k+1$, we can apply Proposition \[dumptype\] to conclude that the set of ${\delta}(N)$ for $N$ $3$-represented by ${\hat{f}}$ has order type $\omega^k$. However, if ${\delta}(f)\ne {t_{a}}(k)$, then by the previous paragraph and Proposition \[dump\], we see that while this has order type $\omega^k$, $T_f$, which is its intersection with $\mathscr{D}^a$, has order type less than $\omega^k$.
It remains to check, then, that when $\deg f=k$ and ${\delta}(f)={t_{a}}(k)$, that the set $T_f$ is cofinal in $\bigcup_{f\in{{\mathcal S}}}
T_f=\mathscr{D}^a\cap[0,{t_{a}}(k))$, or in other words, just that it’s cofinal in $[0,{t_{a}}(k))$. But this follows from Proposition \[dump\], which in fact goes further and states that $T_f\cap\mathscr{D}^a_{{st}}$ is cofinal in $[0,{\delta}(f))=[0,{t_{a}}(k))$.
Thus, applying Proposition \[interleave\], we conclude that $\mathscr{D}^a\cap[0,{t_{a}}(k))$ has order type at most $\omega^k$. This proves the upper bound.
To prove the lower bound, let’s consider the low-defect polynomial $$f = (\cdots((mx_1+1)x_2+1)\cdots)x_k + 1$$ (for a particular $m$ to be chosen shortly) which has ${\|f\|}={\|m\|}+k$. (The upper bound on ${\|f\|}$ is immediate and the lower bound follows from Proposition \[polycpxbd\].) For the value of $m$, we take $$m=\left\{\begin{array}{ll}
3 &\textrm{if}\ k-a\equiv 0\pmod{3}\\
4 &\textrm{if}\ k-a\equiv 2\pmod{3}\\
2 &\textrm{if}\ k-a\equiv 1\pmod{3},
\end{array}\right.$$ so that ${\|m\|}\equiv a-k\pmod{3}$ and ${\|f\|}\equiv a\pmod{3}$, meaning $\mathscr{D}_{{st}}^{{\|f\|}}=\mathscr{D}_{{st}}^a$.
Then ${\delta}(f)={t_{a}}(k)$ and so in particular ${\delta}(f)<k+1$, meaning once again we can apply Proposition \[dump\] to conclude that the set $$\{{\delta}(f(3^{n_1},\ldots,3^{n_k})):(n_1,\ldots,n_k)\in \mathbb{Z}^k_{\ge 0}\}
\cap \mathscr{D}_{{st}}^{{\|f\|}}$$ has order type at least $\omega^k$. Since this set is bounded above by ${\delta}(f)={t_{a}}(k)$ and $\mathscr{D}_{{st}}^{{\|f\|}}=\mathscr{D}_{{st}}^a$, we conclude that the order type of $\mathscr{D}_{{st}}^a\cap[0,{t_{a}}(k))$ is at least $\omega^k$. This completes the proof.
In particular this encompasses Theorem \[chgoverpt\].
This is just a rephrasing of Theorem \[stabchgoverpt\] with the application to $\mathscr{D}^a_{{st}}$ omitted.
Having proven Theorem \[stabchgoverpt\], we can now combine it with Theorem \[dtoD\] to obtain Theorem \[mainthmstab\] and Theorem \[mainthm\]:
By Theorem \[dtoD\], $D(n)$ is equal to the smallest $k$ such that ${\delta}(n)\le
{t_{{\|n\|}}}(k)$. However, since the order type of $\mathscr{D}^{{\|n\|}}\cap[0,{t_{{\|n\|}}}(k))$ is equal to $\omega^k$, one has that $\zeta<\omega^k$ if and only if ${\delta}(n)<{t_{{\|n\|}}}(k)$. Thus $D(n)$ is equal to the smallest $k$ such that $\zeta<\omega^k$. The proof for the stabilized version is similar.
This is just the special case of Theorem \[mainthmstab\] where we only consider ${\delta}(n)$ and not ${\delta}_{{st}}(n)$.
Numbers $n$ with $D(n)\le 1$ {#Erk}
============================
In the previous section we showed that the numbers with integral defect at most $k$ correspond to the initial $\omega^k$ defects in each of $\mathscr{D}^0$, $\mathscr{D}^1$, and $\mathscr{D}^2$. In this section we take a closer look at the initial $\omega$, the numbers with integral defect at most $1$, and use this to generalize Theorem \[rawsthm\].
Let’s start by listing all the numbers with integral defect at most $1$:
\[defect1\] A natural number $n$ satisfies $D(n)\le1$ if and only if it can be written in one of the following forms:
1. $1$, of complexity $1$
2. $2^a 3^k$ for $a\le10$, of complexity $2a+3k$ (for $a$, $k$ not both zero)
3. $2^a(2^b3^\ell+1)3^k$ for $a+b\le 2$, of complexity $2(a+b)+3(\ell+k)+1$ (for $b$, $\ell$ not both zero).
By Theorem \[dtoD\], any $n$ with $D(n)\le 1$ must have ${\delta}(n)\le1+2{\delta}(2)$. Theorem 31 from [@paper1] gives a classification of all numbers $n$ with ${\delta}(n)<12{\delta}(2)$, together with their complexities; since $12{\delta}(2)>1+2{\delta}(2)$, any $n$ with $D(n)\le 1$ may be found among these. (One may also use the algorithms from [@paperalg] to find such a classification.) It is then a straightforward matter to determine which of the $n$ listed there have $D(n)\le 1$.
This has an important corollary:
\[D=1cor\] For any natural number $n$, $D(n)=1$ if and only if $D_{{st}}(n)=1$.
From Theorem \[defect1\], we see that if $D(n)\le 1$ then we also have $D(3^k
n)\le 1$, and if $D(3^k n)\le 1$ then we have $D(n)\le 1$; this shows that $D(n)\le 1$ if and only if $D_{{st}}(n)\le 1$. Combining this with Proposition \[D=0prop\] proves the claim.
From this we can conclude:
\[2stab\] For any natural number $n>1$, if $D(n)\le 2$ then $n$ is stable (and so $D_{{st}}(n)\le 2$).
If $D(n)=0$ or $D(n)=1$, this is Proposition \[D=0prop\] or Corollary \[D=1cor\], respectively. If $D(n)=2$, then for any $k\ge 0$, if we had $D(3^k n)<2$, then, by Proposition \[D=0prop\] and Corollary \[D=1cor\], we would have $D(n)<2$, contrary to assumption; thus $D(3^k n)=2$ for all $k\ge
0$, i.e., $n$ is stable (by Proposition \[stoldpropsD\]).
Note that the converse, that if $D_{{st}}(n)\le 2$ then $D(n)\le 2$, does not hold; for instance, we can consider $107$, which has $D_{{st}}(107)=2$ but $D(107)=3$, or $683$, which has $D_{{st}}(683)=2$ but $D(683)=4$. (It is easy to verify that these numbers have stable integer defect at most $2$ because $D(321)=D(2049)=2$; that these numbers do then have stable integer defect equal to $2$ and not any lower can then be inferred from Corollary \[2stab\]. Alternately, the stable complexity, and thus stable integer defect, may be computed with the algorithms from [@paperalg].)
However, for our purposes, the most important consequence of Corollary \[D=1cor\] is the following rephrasing of it:
\[v3lem\] Let $k>1$ be a natural number and suppose $h$ is a value of $R$ corresponding to a defect in the initial $\omega$ of $\mathscr{D}^k$. Then if $hE(k)$ is a natural number $n$, one has ${\|n\|}=k$, and, moreover, $n>E(k-1)$.
Suppose $hE(k)$ is a natural number $n$. We must have $n>1$ because having $h=1/E(k)$ for $k>1$ would by Proposition \[dRformulae\] correspond to a defect which is a nonzero integer, and these (by Proposition \[stoldprops\]) do not exist.
Then there is, by defintion of $h$, some number $m>1$ with ${\|m\|}\equiv
k\pmod{3}$ and $R(m)=h$, i.e., $m=hE({\|m\|})$. Since ${\|m\|}\equiv k\pmod{3}$ we see that $m=n3^\ell$ for some $\ell\in\mathbb{Z}$, where $\ell=\frac{{\|m\|}-k}{3}$. But also we have $D(m)\le 1$. Therefore, whether $\ell\ge0$ or $\ell\le0$, we must have $D_{{st}}(n)\le 1$, and so, by Proposition \[D=0prop\] and Corollary \[D=1cor\], we have $D(n)\le 1$. Then by Proposition \[oldpropsD\], we have ${\|m\|}={\|n\|}+3\ell$. From the definition of $\ell$ we also have ${\|m\|}=k+3\ell$ and thus we conclude that ${\|n\|}=k$. And since $D(n)=1$ this means (by Proposition \[Dinterp\]) that $n>E(k-1)$.
We can now prove Theorem \[tablethm\]:
Suppose we want to determine the $r$’th largest number of complexity $k$. This is equivalent to determining the $r$’th largest value of $R(n)=\frac{n}{E(k)}$ that occurs among numbers $n$ of complexity $k$, which is equivalent to determining the $r$’th smallest defect ${\delta}(n)$ that occurs among numbers $n$ of complexity $k$.
Now, we can easily determine the initial values $\alpha_0,\ldots,\alpha_r$ of $\mathscr{D}^k$; let $h_0,\ldots,h_r$ be the corresponding values of the function $R$, as given by Proposition \[dRformulae\]. (For instance, for a way of getting $h_0,\ldots,h_r$ directly rather than going by means of defects, one may take the numbers $n$ given in Theorem \[defect1\], group them by the residues of ${\|n\|}$ modulo $3$, then sort them in decreasing order by $R(n)$; note that the values of $R(n)$ obtained this way for any one congruence class of ${\|n\|}$ modulo $3$ will have reverse order type $\omega$.) One may see Tables \[table0r\], \[table2r\], and \[table1r\] for tables of the resulting values of $h$. Then certainly, the $r$’th largest number of complexity $k$ is at most $h_r E(k)$, because the set of values of $R(n)$ occuring for $n$ with ${\|n\|}=k$ is a subset of the values of $R(n)$ occuring for $n>1$ with ${\|n\|}\equiv k\pmod{3}$. However, it will only be exactly the $r$’th largest number of complexity $k$ if all of $h_1$ through $h_r$ do indeed occur for some $n$ with ${\|n\|}=k$.
But, by Proposition \[v3lem\], this is equivalent to just requiring that all of the numbers $h_0E(k),\ldots,h_rE(k)$ are indeed whole numbers (and moreover when this does occur one will have $h_i E(k)>E(k-1)$). In other words, this is the same as requiring $$k \ge \left\{
\begin{array}{ll}
-3\min_{s\le r} v_3(h_s) & \textrm{if}\ k\equiv0\pmod{3} \\
-3\min_{s\le r} v_3(h_s) + 4 & \textrm{if}\ k\equiv1\pmod{3} \\
-3\min_{s\le r} v_3(h_s) + 2 & \textrm{if}\ k\equiv2\pmod{3}.
\end{array}
\right.$$
So we have our $h_{r,a}$, and we can take $K_{r,a}$ to be given by this formula. (Although since for $K_{0,0}$ it may not may make much sense to take $K_{0,0}=0$, one may wish to take $K_{0,0}=3$ instead, as we have done in Table \[table0\].)
Combining this with Tables \[table0r\], \[table2r\], and \[table1r\] yields Tables \[table0\], \[table2\], and \[table1\], and proves the theorem.
While in the proof of Theorem \[tablethm\] we have referred to facts proved in Section \[mainsec\], none of the techniques deployed in that section are necessary for the proof. For instance, one can easily verify the values of the $\overline{\mathscr{D}^a}(\omega)$ by directly determining the initial $\omega$ elements without needing to determine it for all $\omega^k$; indeed Tables \[table0r\], \[table2r\], and \[table1r\] essentially do this directly from Theorem \[defect1\].
$r$ $h$ Corresponding leader
------------------------------------- ------------- -----------------------
$0$ $1$ $3=3^1 2^0=2^1 3^0+1$
$1$ $8/9$ $8=2^3=2^1(3^1+1)$
$2$ $64/81$ $64=2^6$
$3$ $7/9$ $7=2^13^1+1$
$4$ $20/27$ $20=2^1(3^2+1)$
$5$ $19/27$ $19=2^13^2+1$
$6$ $512/729$ $512=2^9$
$(\mathrm{for}\: n\ge4) \quad 2n-1$ $2/3+2/3^n$ $2^1(3^{n-1}+1)$
$(\mathrm{for}\: n\ge4) \quad 2n$ $2/3+1/3^n$ $2^1 3^{n-1}+1$
: Table of $h_r$ for $k\equiv 0\pmod{3}$.[]{data-label="table0r"}
$r$ $h$ Corresponding leader
------------------------------------ ------------------------- ----------------------
$0$ $1$ $2=2^1$
$1$ $8/9$ $16=2^4=2^2(3^1+1)$
$2$ $5/6$ $5=2^23^0+1$
$3$ $64/81$ $128=2^7$
$4$ $7/9$ $14=2^1(2^13^1+1)$
$5$ $20/27$ $40=2^2(3^2+1)$
$6$ $13/18$ $13=2^23^1+1$
$7$ $19/27$ $38=2^1(2^13^2+1)$
$8$ $512/729$ $1024=2^{10}$
$(\mathrm{for}\: n\ge 4)\quad3n-3$ $2/3+2/3^n$ $2^2(3^{n-1}+1)$
$(\mathrm{for}\: n\ge 4)\quad3n-2$ $2/3+1/(2\cdot3^{n-1})$ $2^23^{n-1}+1$
$(\mathrm{for}\: n\ge 4)\quad3n-1$ $2/3+1/3^n$ $2^1(2^1 3^{n-1}+1)$
: Table of $h_r$ for $k\equiv 2\pmod{3}$.[]{data-label="table2r"}
$r$ $h$ Corresponding leader
------------------------------------ --------------------- ----------------------
$0$ $1$ $4=2^2=3^1+1$
$1$ $8/9$ $32=2^5$
$2$ $5/6$ $10=3^2+1$
$3$ $64/81$ $256=2^8$
$(\mathrm{for}\: n\ge 2)\quad n+2$ $3/4+1/(4\cdot3^n)$ $3^{n+1}+1$
: Table of $h_r$ for $k\equiv 1\pmod{3}$ with $k>1$.[]{data-label="table1r"}
As a final note, it is worth making formal a statement mentioned in Section \[rawsintro\], that the numbers $hE(k)$ coming from Theorem \[tablethm\] are almost exactly the $n$ with $D(n)\le 1$:
\[coincicor\] A number $n$ has $D(n)\le 1$ if and only if there are some $\ell\ge0$, $k\ge
1$, and $r\ge 0$ such that $k\ge K_{r,k}$ and $3^\ell n = h_{r,k} E(k)$.
We already know that if $k\ge K_{r,k}$ then, if we let $m=h_{r,k}E(k)$, that $m>E(k-1)=E({\|m\|}-1)$, i.e., $D(m)\le 1$, and so if $m=3^\ell n$, then $D(n)\le 1$ by Corollary \[D=1cor\].
Conversely, if $D(n)\le 1$, let $h=R(n)$; then by the construction of the $h_{r,a}$ in the proof of Theorem \[tablethm\], and the fact that the values of $R(n)$ for numbers $n$ with ${\|n\|}$ in a fixed congruence class modulo $3$ have reverse order type $\omega$, there is some $r$ such that $h=h_{r,{\|n\|}}$. We may then take any $k\ge K_{r,{\|n\|}}$ with $k\equiv {\|n\|}\pmod{3}$; then $3^\ell n = h_{r,{\|n\|}}E(k) = h_{r,k}E(k)$ for $\ell=\frac{k-{\|n\|}}{3}$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
Work of the author was supported by NSF grants DMS-0943832 and DMS-1101373.
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|
---
author:
- 'A. Papitto'
- 'A. D’Aì'
- 'S. Motta'
- 'A. Riggio$^{,}$'
- 'L. Burderi'
- 'T. Di Salvo'
- 'T. Belloni'
- 'R. Iaria'
bibliography:
- 'biblio.bib'
title: The spin and orbit of the newly discovered pulsar
---
Introduction
============
The dense environment of a globular cluster and the resulting frequent stellar encounters [@Mey97] make the production of binary systems hosting a compact object very efficient. Terzan 5 is probably one of the densest and metal-richest cluster in our Galaxy [@Chn02; @Ort07], as clearly indicated by the large number of rotation-powered millisecond pulsars discovered there ($\simgt30$, @Rns05 [@Hss06]). The cluster also contains at least 28 discrete X-ray sources, a substantial number of which can be identified as quiescent low-mass X-ray binaries [LMXB, @Hnk06]. According to the recycling scenario [see, e.g. @BhtvdH91], the population of rotation-powered millisecond pulsars and low-mass X-ray binaries (LMXB) share an evolutionary link, because the former are thought to be spun up by the accretion of mass and angular momentum in an LMXB. Accreting pulsars in LMXB are usually found with periods clustering in two distinct groups. So far, 13 sources have been discovered with spin periods lower than 10 ms and were therefore termed accretion-powered millisecond pulsars [see, e.g., @WijvdK98]. However, a smaller number of sources are found with longer periods and correspondingly higher estimates of the neutron star (NS) magnetic field.
So far, the only bright transient LMXB known in the cluster Terzan 5 was the burster EXO 1745-248 [@Mks81]. The first detection of a new outburst of a source in this cluster was made on 2010 October 10.365 with [*INTEGRAL*]{} [@Brd10] and was tentatively attributed to EXO 1745-248. Follow-up [*Swift*]{} observations refined the source position, and a comparison with the position of sources previously known thanks to [*Chandra*]{} observations of the cluster suggested instead a different association [@Hnk10; @Knn10; @Poo10]. The X-ray transient is then considered a newly discovered source and named as ( in the following). A coherent signal at a period of 90.6 ms was detected thanks to observations performed with the [ in the following, @Str10]. A signal at the same period is present also during the several bursts that the source shows [@Alt10], while burst oscillations have never been observed from NS rotating at periods exceeding a few ms. A sudden decrease of the flux was tentatively attributed to an eclipse of the source by the companion [@Str10]. However, eclipses were not observed during subsequent observations, and the earlier flux decrease was identified with a lunar occultation [@Str10bis S10 in the following].
Below we present the first analysis of the properties of the coherent signal emitted by this source, using and [*Swift*]{} observations performed between 2010 October 10 and November 6, and give a refined orbital and timing solution of the pulsar with respect to those first proposed (@Ppt10, P10 in the following, S10).
Observations and data analysis {#sec:obs}
==============================
After the source discovery on 2010 October 10.365 [@Brd10], started monitoring the source on MJD 55482.010 (October 13.010; all the epochs reported in this paper are given with regard to the Barycentric Dynamical Time, TDB, system). We present an analysis of the observations performed until MJD 55506.359 (ObsId 95437-01-12-01), for a total exposure of 206 ks. In this time interval, a large number ($>$ 300) of X-ray bursts are observed with a recurrence time decreasing from $\simgt26$ to $\sim3$ min as the X-ray flux increases. The analysis of the bursts shown by the source, as well as of its aperiodic timing properties, will appear in a companion paper. The 2.5-25 keV light curve recorded by PCU2 of the Proportional Counter Array (PCA) on-board RXTE, with the burst intervals removed and background subtracted, is plotted in Fig.\[fig1\]. The count rate increases during the first days of the outburst, reaching a peak value of $\sim1700$ s$^{-1}$ at MJD 55487.5, and then decreases to a value of $\sim 800$ s$^{-1}$ with an exponential decay time scale of $\sim5$ d. Quite complex dipping-like structures also appear especially between MJD 55485 and 55490 as the flux shows sudden variations up to 75$\%$ on timescales of $\sim10$ min.
The combined X-ray spectrum observed by the top layer of the PCU2 of the PCA (2.5–25 keV), and by Cluster A of the High Energy X-ray Timing Experiment (HEXTE, 22–50 keV), can be well modelled by a sum of a blackbody and a Comptonized component, which we model with `compps` [@PouSve96]. Throughout the observations considered here, the spectrum softens significantly as the source evolves towards higher luminosities. The unabsorbed total flux, extrapolated in the 0.1-100 keV band, rises from $0.47(3)\times
10^{-8}$ to a maximum level of $1.89(4)\times10^{-8}$ erg cm$^{-2}$ s$^{-1}$, observed during the observation of MJD 55487.5. All the uncertainties on the fluxes given here are quoted at a 90% confidence level.
To analyse the spin and orbital properties of the source we consider data taken by the PCA in Event ($122\mu$s temporal resolution) and Good Xenon ($1\mu$s temporal resolution) packing modes. We discard 10s prior, and 100s after the onset of each type-I X-ray burst. The time series were also preliminarly barycentred with respect to the solar system barycentre using the available orbit files and assuming the best [*Chandra*]{} estimate of the source position, RA=17$^{h}$ 48$^{m}$ 4.831$^{s}$, DEC=-24$^{\circ}$ 46’ 48.87”, with an error circle of 0.06” (1$\sigma$ confidence level, @Hnk06 [@Poo10]). A coherent signal at a frequency of 11.045(1) Hz \[equivalent to a period of 90.539(8) ms\] is clearly detected in the power spectrum at a Leahy-normalised power of $\simeq 1.5\times10^4$. A first orbital solution is obtained modelling the observed modulation caused by the orbital motion, $$P(t)= P_0\left\{1+\frac{2\pi x}{P_{orb}} [\cos{m}+e\cos(2m-\omega)]\right\}.$$ Here $P_0$ is the barycentric spin period of the source, $x=a\sin{i}/c$ the semi-major axis of the NS orbit, $P_{orb}$ the orbital period, $m=2\pi(t-T*)/P_{orb}$ the mean anomaly, $T^*$ the epoch at which $m=0$, $e$ the eccentricity and $\omega$ the longitude of the periastron measured from the ascending node. The periods $P(t)$ are estimated by performing an epoch-folding search on 1.5 ks long data segments (for a total of 127) around the periodicity indicated by the power spectrum. The resulting variance profiles are fitted following @Leh87 and the uncertainties affecting the period estimates are evaluated accordingly. The best-fitting orbital solution we obtain with this technique is shown in the leftmost column of Table \[tab\].
The time series were then corrected for the orbital motion with these parameters and folded around the best estimate of the spin period over 300s data segments (for a total of 717). The pulse profiles could generally be modelled using up to three harmonics. The pulsed fraction is observed to greatly vary in between the first (MJD 55482.010 to 55482.043) and the other observations. During the former the pulse fractional amplitude of the first harmonic is very high \[$A_1\simeq0.252(2)$\], while the second harmonic is detected at a much lower amplitude \[$A_2=0.016(2)$\]. In subsequent observations the amplitude of the first harmonic drastically decreases to values between 0.02 and 0.04, whereas the second harmonic amplitude remains stable and a third harmonic is sometimes requested by the profile modelling. To show this, we plot in Fig. \[fig3\] the pulse profiles calculated over the observations performed during MJD 55482 (solid line) and MJD 55483 (dashed line), with the latter profile shown at a magnified scale. The pulsed fraction decrease is evident, as are the variation of the shape of the peak at rotational phase $\sim0.75$.
To increase the accuracy of our timing solution we model the temporal evolution of the phases evaluated on the first and second harmonic of the pulse profiles with the relation: $$\label{eq:phases}
\phi(t)=\phi_0+(\nu_0-\nu_{F})\;(t-T_{ref})+\frac{1}{2}{<\dot{\nu}>}(t-T_{ref})^2+R_{orb}(t).$$ Here $T_{ref}$ is the reference epoch for the timing solution, $\nu_0$ is the pulsar frequency at the reference epoch, $\nu_F=1/P_F$ is the folding frequency and $<\dot{\nu}>$ is the average spin frequency derivative. The term $R_{orb}(t)$ describes the phase residuals induced by a difference between the actual orbital parameters, namely $x$, $P_{orb}$, $T^*$, $e\sin{\omega}$ and $e\cos{\omega}$, and those used to correct the time series [see e.g. @DeeBoyPrv81]. Once a new set of orbital parameters is found, it is used to correct the time series, and the resulting phases are modelled again with Eq.(\[eq:phases\]). This procedure is iterated until the phase residuals are normally distributed around zero.
Periods 1$^{st}$ harm. ph. 2$^{nd}$ harm. ph.
--------------------------------- ------------ -------------------- --------------------
$\nu_0\:$-11.044885 ($\mu$Hz) $<5$ +0.64(1) +0.17(1)
$<\dot{\nu}>$ ($10^{-12}$ Hz/s) $<16$ $1.22(1)$ $1.68(1)$
a$\sin{i}$/c (lt-s) 2.498(5) 2.4967(3) 2.4973(2)
P$_{orb}$ (hr) 21.2744(8) 21.2745(1) 21.27454(8)
T\* - 55481.0 (MJD) 0.7805(4) 0.78033(6) 0.78048(4)
e $<0.02$ $<7\times10^{-4}$ $<6\times10^{-4}$
f($M_1$,$M_2$,i) (M$_{\odot}$) 0.0213(2) 0.0212587(8) 0.021275(5)
$\chi^2$/dof 156/121 7879/662 3287/566
: Spin and orbital parameters of .\[tab\]
Although no residual modulation at the orbital period is observed, the phases of the first harmonic are strongly affected by timing noise. The reduced $\chi^2$ we obtain modelling their evolution with Eq.(\[eq:phases\]) is extremely large ($\simeq 11.9$ over 662 d.o.f.). Such a behaviour is most probably caused by to pulse shape changes like the one shown in Fig \[fig3\]. The second harmonic phases appear to be less affected by timing noise, resulting in a reduced $\chi^2_r=5.8$ (566 d.o.f.). We argue that the second harmonic phases are better fitted with respect to the first harmonic because of the greater stability of this component [as already observed in some accreting millisecond pulsars, see, e.g., @Brd06; @Rgg08]. The best-fitting parameters calculated over the phase evolution of the first and second harmonic are quoted in the central and rightmost column of Table \[tab\]. In Fig. \[fig2\] we show the phases of both harmonics, when the observations corrected for the orbital motion of the source are folded around $P_F=90.539645$ ms. The phase evolution is clearly driven by at least a quadratic component. A consequence of timing noise is that the spin frequency and its derivative, estimated over the two harmonic components, are significantly different. We quote conservatively a value of $\nu_0=11.0448854(2)$ Hz that overlaps both frequency estimates, and use a spin frequency derivative between 1.2–1.7$\times10^{-12}$ Hz s$^{-1}$ in the discussion below. However, it is worthwhile to note that the orbital parameters are entirely consistent between the two harmonic solutions, which supports the reliability of these estimates. The solution we obtained is entirely compatible with, but more precise than, those proposed by P10[^1] and S10. Given the accuracy of the source position considered here (0.06”), the systematic uncertainties introduced by the position indetermination on the measured values of spin frequency and of its derivative [e.g. @Brd07] are $\sigma_{pos,\nu}\simlt3\times10^{-10}$ Hz and $\sigma_{pos,\dot{\nu}}\simlt6\times10^{-17}$ Hz s$^{-1}$, respectively. Finally, as the cluster moves towards the solar system at a velocity of $85\pm10$ km s$^{-1}$ [@Fer09], the measured value of the spin frequency is affected by a systematic offset of $\sim +3\times10^{-3}$ Hz, though this is unimportant when making conclusions about the source properties.
In order to extend the range of fluxes at which the source was observed and pulsations were detected, we also analysed three [ *Swift*]{} observations (Obs. 00031841002, 00031841003 and 00031841004) in which the XRT observed in Windowed Timing (WT) mode, with a temporal resolution of 1.7ms. The [*Swift*]{} XRT started monitoring the source on MJD 55479.737, more than two days before , with the first observation in WT mode starting at MJD 55479.802 for 2 ks. After applying barycentric corrections for the satellite orbit, then correcting for the source orbital motion and selecting photons from a 50 pixel wide box around the source position, a pulsation is clearly detected at a period of $P_S=90.5395(2)$ ms by means of an epoch-folding search. The XRT signal is particularly strong and consistent with that seen by during its first observation, with a pulse profile modelled by a sinusoid of amplitude 0.23(2). Pulsations were also searched for in the subsequent $\sim1$ ks long XRT WT observations starting on MJD 55484.767 and 55485.357. Only a weak signal was detected in the latter at a fractional amplitude of 0.018(4), still compatible with that seen by at those later times.
Discussion and conclusions
==========================
We reported on the spin and orbital properties of the newly discovered accreting pulsar, . Its 90.6 ms period makes it the first confirmed accreting pulsar in the range 10–100 ms. Pulsations were detected in all observations performed by , as well as in two out of the three [*Swift*]{} observations performed in WT mode presented here.The pulsed fraction is observed to drastically change on a timescale of $\simlt 1$ d, after the observation performed on MJD 55482. While previously both [*Swift*]{} and observations revealed a strong signal dominated by a first harmonic component of fractional amplitude as large as $0.25$, later observations at higher fluxes performed by both satellites never detected an amplitude $\simgt 0.03$. Simultaneously the pulse shape changes and becomes more complex. This behaviour is suggestive of a change of the geometrical properties of the flow in the accretion columns above the NS hot spots.
Because pulsations are detected throughout the observations shown here, an estimate of the NS magnetic field strength can be made. For accretion to proceed and pulsations to appear in the X-ray light curve, the inner disc radius, $R_{in}$, has to lie in between the NS radius, R$_{NS}$, and the corotation radius, $R_{C}$, defined as the distance from the NS at which the velocity of the magnetosphere equals the Keplerian velocity of the matter in the disc. For a larger accretion radius, accretion would be inhibited or severely reduced by the onset of a centrifugal barrier. For a pulsar spinning at 90.6 ms, $R_C=(GMP^2/4\pi^2)^{1/3}=338 m_{1.4}^{1/3}$ km, where $m_{1.4}$ is the NS mass in units of 1.4 M$_{\odot}$. Defining the inner disc radius in terms of the pressure equilibrium between the disc and the magnetosphere, one obtains $R_{in}\simeq 160 \;
m_{1.4}^{1/7}\;R_6^{-2/7}\;L_{37}^{-2/7}\mu_{28}^{4/7}$ km [@Brd01], where $R_6$ is the NS radius in units of 10 km, $L_{37}$ the accretion luminosity in units of $10^{37}$ erg s$^{-1}$, and $\mu_{28}$ the magnetic dipole moment of the NS in units of $10^{28}$ G cm$^{3}$. Extrapolating the fluxes observed by to the 0.1–100 keV band, and assuming a distance of $d=5.5\pm0.9$ kpc to Terzan 5 [@Ort07], we estimate a maximum and minimum bolometric luminosities of $1.7(1)\times10^{37}$ d$_{5.5}^2$ erg s$^{-1}$ and $6.8(1)\times10^{37}$ d$_{5.5}^2$ erg s$^{-1}$, during the time covered by the observations considered here. Assuming that the X-ray luminosity is a good tracer of the accretion power and imposing $R_{NS}<R_{in}\simlt R_{C}$, we obtain $$0.02\: \: m_{1.4}^{-1/4}\: R_{6}^{9/4}\: d_{5.5} \simlt \mu_{28} \simlt 4.8 \:\: m_{1.4}^{1/3}\: R_{6}^{1/2}\: d_{5.5}.$$ The upper limit on the magnetic dipole can be reduced considering that pulsations are detected also in a Swift observation taking place $\sim2$ d earlier than the first observation. @Bzz10 estimated the source flux in that observation as $4.5(2)\times10^{-10}$ erg cm$^{-2}$ s$^{-1}$ (1–10 keV). This value is a factor $\sim4$ lower than the value obtained extrapolating the spectrum of the first observation in the same energy band. Assuming that this ratio holds also for the bolometric luminosity of the source, we get to an upper limit on the magnetic dipole moment of $\simeq 2.4\times10^{28}$ G cm$^{-3}$. The limits thus derived translate to a magnetic surface flux density between $\sim 2\times10^8$ and $\sim 2.4\times10^{10}$ G. The upper bound of this interval can be overestimated because the exact flux at which the pulsations appeared is unknown at present. Monitoring the presence of coherent pulsations as a function of the flux when the source fades will probably allow us to derive a tighter constraint. @Alt10bis have also reported the presence of a kHz QPO at $\sim815$ Hz (10–50 keV) during the observations performed on MJD 55487. Under the hypothesis that this feature originates in the innermost part of the accretion disc, it indicates an inner disc radius $R_{in}\simlt \:20 \:m_{1.4}^{1/3}$ km. As the luminosity we estimated during that day is $6.8(1)\times10^{37}$ d$_{5.5}^2$ erg s$^{-1}$, this would imply a magnetic field $\simlt7\times 10^8$ d$_{5.5}$ G if the disc is truncated at the magnetospheric radius.
Despite the presence of timing noise, the analysis of the phase evolution over the $\sim24$ d time interval presented here clearly indicates the need for a quadratic component to model these phases. Interpreting this component as a tracer of the NS spin evolution, we thus conclude that the source spins up while accreting. Values of the spin-up rate between 1.2 and 1.7$\times10^{-12}$ Hz s$^{-1}$ are found, depending on the harmonic considered. This discrepancy is probably due to the effect of timing noise. These values are compatible with those expected for a NS accreting the Keplerian disc matter angular momentum given the observed luminosity, $\dot{\nu}\simeq1.5\times10^{-12}$ $(L_{37}/5)$ $(R_{in}/70km)^{1/2}$ $I_{45}^{-1}$ $R_6$ $m_{1.4}$ Hz s$^{-1}$. Here $I_{45}$ is the NS moment of inertia in units of $10^{45}$ g cm$^2$. The observed spin period and the magnetic field we estimated place this source between the population of “classical” ($B\simgt 10^{11}$G, $P\simgt 0.1$ s) and millisecond ($B\simeq 10^8$–$10^9$ G, $P\simeq 1.5$–$10$ms) rotation-powered pulsars. The observation of a significant spin-up at rates compatible with those predicted by the recycling scenario further supports the identification of this source as a slow, mildly recycled pulsar. We note that the only other two accreting pulsars with similar, though significantly different parameters, are GRO J1744-28 ($P_S=467ms$, $B\simeq2.4\times10^{11}$ G, @Cui97), and 2A 1822-371 ($P_S=590$ ms, $B\simlt 10^{11}$ G, @Jnk01).
The orbital parameters we measured for the NS in allow us to derive constraints on the nature of its companion star. With a mass function of $f(M_2;M_1,i)\simeq 0.02$ M$_{\odot}$, a minimum mass for the companion can be estimated to be as low as 0.41 M$_{\odot}$ for an inclination of 90$^{\circ}$, and an NS mass of 1.4 M$_{\odot}$. Since the source shows no eclipses, the inclination is most probably $\simlt80^{\circ}$, [ and]{} the lower limit increases to $m_2=0.16+0.26 m_{1.4}$, where $m_2$ is the mass of the companion star in solar units. An upper limit can be obtained if the companion star [ is assumed not to]{} overfill its Roche lobe. Using the relation given by @Egg83 and the third Kepler law to relate the Roche Lobe radius to the orbital period and to the companion mass, $R_{L2}\simeq
0.55(GM_{\odot})^{1/3}(P_{orb}/2\pi)^{2/3}m_{1.4}\:q^{2/3}(1+q)^{1/3}/[0.6q^{2/3}+\log(1+q^{1/3})]$, where $q=M_2/M_1$, and assuming the companion star follows a main sequence mass-radius relation, $R_2/R_{\odot}\approx(M_2/M_{\odot})$, yields a maximum mass of 2.75 M$_{\odot}$ for the companion when $m_{1.4}=1$. This upper limit is indeed higher than the maximum mass expected for a main sequence star belonging to one of the two stellar populations found by @Fer09 in Terzan 5. One has in fact $m_2\simlt0.95$ if the companion of belongs to the older population [t=10 Gyr, @Dan10], while $m_2\simlt 1.2$ and $m_2\simlt 1.5$ if it belongs to a younger population of 6 and 4 Gyr, respectively (D’Antona, priv. comm.). We conclude that a reasonable upper limit for the companion-star mass is 1.5 M$_{\odot}$, possibly a main sequence or a slightly evolved star.
This work is supported by the Italian Space Agency, ASI-INAF I/088/06/0 contract for High Energy Astrophysics, as well as by the operating program of Regione Sardegna (European Social Fund 2007-2013), L.R.7/2007, “Promotion of scientific research and technological innovation in Sardinia”. We thank F. D’Antona for providing the mass estimates of main sequence stars in Terzan 5, and the referee for the prompt reply and useful comments.
[^1]: There is an offset between the values of frequency and epoch of mean longitude quoted by P10 and those presented here, as theirs were not referred to the TDB reference system.
|
IFT-UAM/CSIC-02-23\
[**hep-th/0206159**]{}\
June $17$th, $2002$
[**Tomás Ortín**]{}\
1truecm
0.2cm [*Instituto de Física Teórica, C-XVI, Universidad Autónoma de Madrid\
E-28049-Madrid, Spain*]{}
[*and*]{}
[*I.M.A.F.F., C.S.I.C., Calle de Serrano 113 bis\
E-28006-Madrid, Spain*]{}
[**Abstract**]{}
> The definition of “Lie derivative” of spinors with respect to Killing vectors is extended to all kinds of Lorentz tensors. This Lie-Lorentz derivative appears naturally in the commutator of two supersymmetry transformations generated by Killing spinors and vanishes for Vielbeins. It can be identified as the generator of the action of isometries on supergravity fields and its use for the calculation of supersymmetry algebras is revised and extended.
Introduction {#introduction .unnumbered}
============
Spinors are defined by their transformation properties under $SO(n_{+},n_{-})$ (“Lorentz”) transformations and, thus, they can only be introduced in the tangent space of curved spaces using the formalism introduced by Weyl in Ref. [@kn:Weyl4]. This formalism makes use of Vielbeins $\{e_{a}{}^{\mu}\}$ which form an orthonormal basis in tangent space
$$e_{a}{}^{\mu}e_{b}{}^{\nu} g_{\mu\nu}=\eta_{ab}\, ,
\hspace{1cm}
(\eta_{ab})={\rm diag}\,(+\cdots+,-\cdots -)\, ,$$
and it is covariant w.r.t. local transformations that preserve this orthonormality: local $SO(n_{+},n_{-})$ transformations. These are the only transformations that act non-trivially on any Lorentz tensor[^1] $T$ in the representation $r$
$$T^{\prime}(x^{\prime}) = \Gamma_{r}[g(x)] T(x)\, ,$$
where $\Gamma_{r}[g(x)]$ is the representation of the position-dependent group element $g(x)$ that can be constructed by exponentiation
$$\Gamma_{r}[g(x)]=
{\rm exp}[{\textstyle\frac{1}{2}} \sigma^{ab}(x)
\Gamma_{r}\left(M_{ab}\right)]$$
where $\Gamma_{r}\left(M_{ab}\right)$ are the $SO(n_{+},n_{-})$ generators in the representation $r$. For the contravariant vector and spinor representations
$$\Gamma_{v}\left(M_{ab}\right){}^{c}{}_{d}
=2\eta_{[a}{}^{c}\eta_{b]d}\, ,
\hspace{1cm}
\Gamma_{s}\left(M_{ab}\right)
={\textstyle\frac{1}{2}}\Gamma_{[a}\Gamma_{b]}\, ,
\hspace{1cm}
\{\Gamma_{a},\Gamma_{b}\}=2\eta_{ab}\, .$$
Local Lorentz covariance is required in order to be able to gauge away the additional degrees of freedom that the Vielbein ($d^{2}$) has, as compared with the metric ($d(d+1)/2$), and it is achieved by introducing a covariant derivative $\mathcal{D}_{\mu}$ which acts on $T$ according to
$$\mathcal{D}_{\mu}T\equiv \left(\partial_{\mu}
-\omega_{r\, \mu}\right) T\, ,
\hspace{1cm}
\omega_{r\, \mu} \equiv
{\textstyle\frac{1}{2}}\omega_{\mu}{}^{ab}
\Gamma_{r}\left(M_{ab}\right) \, .$$
$\omega_{\mu}{}^{ab}$ is the $SO(n_{+},n_{-})$ (spin) connection, i.e. it is antisymmetric $\omega_{\mu}{}^{ab}=
-\omega_{\mu}{}^{ba}$, which implies
$$\label{eq:constantmetric}
\mathcal{D}_{\mu}\eta_{ab}=0\, .$$
and transforms according to
$$\omega_{r\, \mu}^{\prime}=\Gamma_{r}[g(x)]
\omega_{r\, \mu} \Gamma^{-1}_{r}[g(x)]
+(\partial_{\mu}\Gamma_{r}[g(x)] )\Gamma^{-1}_{r}[g(x)]\, .$$
In order to have only one connection the spin connection is related to the affine connection $\Gamma_{\mu\nu}{}^{\rho}$ by the Vielbein postulate
$$\nabla_{\mu} e_{a}{}^{\nu} = \partial_{\mu} e_{a}{}^{\nu}
+\Gamma_{\mu\rho}{}^{\nu}e_{a}{}^{\rho}
-e_{b}{}^{\nu}\omega_{\mu a}{}^{b}=0\, ,$$
($\nabla_{\mu}$ denotes the total (general and Lorentz) covariant derivative) which implies
$$\label{eq:relationbetweentheconnections}
\omega_{\mu\, a}{}^{b} = \Gamma_{\mu a}{}^{b}
-e_{a}{}^{\nu}\partial_{\mu}e_{\nu}{}^{b}\, .$$
The Vielbein postulate together with Eq. (\[eq:constantmetric\]) imply that the affine connection is metric compatible and the spin connection is completely determined by the Vielbein, their inverses, and the contorsion tensor $K_{abc}$:
$$\omega_{abc}=-\Omega_{abc} +\Omega_{bca} -\Omega_{cab}+K_{abc}\, ,
\hspace{1cm}
\Omega_{abc} =e_{a}{}^{\mu} e_{b}{}^{\nu}\partial_{[\mu|}e_{c |\nu]}\, .$$
In Weyl’s formalism Lorentz tensors (in particular spinors) behave, then, as scalars under general coordinate transformations (g.c.t.’s). However, we can use Weyl’s formalism just to work in curvilinear coordinates in Minkowski spacetime and we would find that a standard Lorentz transformation becomes a g.c.t. that does not act on the spinorial indices. This looks strange, but it is unavoidable in Weyl’s formalism.
On the other hand, Lorentz transformations and g.c.t.’s do not commute and the result of a g.c.t. on a Lorentz tensor is strongly frame-dependent. This shows up in the standard Lie derivative (an infinitesimal g.c.t.) which does not transform Lorentz tensors into Lorentz tensors. Thus, while the Lie derivative of the metric $g_{\mu\nu}$ with respect to a Killing vector field vanishes (by definition) the Lie derivative of the inverse Vielbein $e^{a}{}_{\mu}$ associated to the same metric with respect to the same Killing vector in general does not.
For spinors $\psi$ this situation was solved in Ref. [@kn:Kos][^2] where a [*spinorial Lie derivative*]{} $\mathbb{L}_{k}$ with respect to Killing vector fields $k$ was defined by[^3]
$$\label{eq:LLonspinors}
\mathbb{L}_{k}\psi \equiv k^{\mu}\mathcal{D}_{\mu}\psi
+{\textstyle\frac{1}{4}} \mathcal{D}_{[a}k_{b]}\Gamma^{ab}\psi\, .$$
This derivative is a derivation that transforms spinors into spinors (it is Lorentz-covariant) and satisfies the property
$$[\mathbb{L}_{k_{1}},\mathbb{L}_{k_{2}}]\, \psi
=\mathbb{L}_{[k_{1},k_{2}]}\, \psi\, ,$$
but, most importantly, the second term defines an action of certain g.c.t.’s (the isometries) on the spinorial indices: an infinitesimal Lorentz transformation with parameter $\frac{1}{2}\mathcal{D}_{[a}k_{b]}$. This is precisely what one expects on physical grounds. Let us consider an example: Minkowski spacetime $g_{\mu\nu}=\eta_{\mu\nu}$ with the obvious Vielbein $e_{a}{}^{\mu}=\delta_{a}{}^{\mu}$ and the infinitesimal g.c.t. generated by the Killing vector
$$k^{\mu} = \sigma^{\mu}{}_{\nu}x^{\nu}\, ,$$
where $\sigma^{\mu\nu}=-\sigma^{\nu\mu}$ and constant. This is a standard infinitesimal Lorentz transformation and, indeed, we find
$$\mathbb{L}_{k}=k^{\mu}\partial_{\mu}\psi
+{\textstyle\frac{1}{4}}\sigma_{ab}\Gamma^{ab}\psi\, ,$$
which is the result that one would obtain in the standard spinor formalism.
The spinorial Lie derivative can be seen as a Lorentz covariantization of the standard Lie derivative (first term) supplemented by an infinitesimal local Lorentz transformation that trivializes the holonomy. It is clear that these ideas can be extended to other Lorentz tensors and we can define a Lie-lorentz derivative which is first a Lorentz covariantization of the standard Lie derivative supplemented by an infinitesimal Lorentz transformation that trivializes the holonomy. The parameter of this transformation has to be exactly the same as in the spinorial case and, thus, we arrive to the following definition.
Definition and Properties of the Lie-Lorentz Derivative
=======================================================
On pure Lorentz tensors $T$ we define the Lie-Lorentz derivative with respect to the Killing vector $k$ by
$$\mathbb{L}_{k}T \equiv k^{\rho}\nabla_{\rho}T +{\textstyle\frac{1}{2}}
\nabla_{[a}k_{b]}\, \Gamma_{r}(M^{ab})T\, .$$
On tensors that also have world indices $T_{\mu_{1}\cdots\mu_{m}}{}^{\nu_{1}\cdots\nu_{n}}$
$$\begin{array}{rcl}
\mathbb{L}_{k}T_{\mu_{1}\cdots\mu_{m}}{}^{\nu_{1}\cdots\nu_{n}}
& \equiv &
k^{\rho}\nabla_{\rho}T_{\mu_{1}\cdots\mu_{m}}{}^{\nu_{1}\cdots\nu_{n}}
-\nabla_{\rho}k^{\nu_{1}}
T_{\mu_{1}\cdots\mu_{m}}{}^{\rho\nu_{2}\cdots\nu_{n}}
-\cdots \\
& & \\
& &
+\nabla_{\mu_{1}}k^{\rho}
T_{\rho\mu_{2}\cdots\mu_{m}}{}^{\nu_{1}\cdots\nu_{n}}
+\cdots
+{\textstyle\frac{1}{2}}
\nabla_{[a}k_{b]}\, \Gamma_{r}(M^{ab})
T_{\mu_{1}\cdots\mu_{m}}{}^{\nu_{1}\cdots\nu_{n}}\, . \\
\end{array}$$
In all the cases that we are going to consider $\nabla_{\mu}$ is the full (affine plus Lorentz) torsionless covariant derivative satisfying the Vielbein postulate.
In the following $T_{1},T_{2}$ will be two mixed tensors of any kind, $k_{1},k_{2}$ any two conformal Killing vector fields and $a^{1},a^{2}$ two arbitrary constants. The Lie-Lorentz derivative has the following basic properties:
1. Leibnitz rule:
$$\label{eq:Leibnitz}
\mathbb{L}_{k}(T_{1}T_{2})= \mathbb{L}_{k}(T_{1})T_{2}
+T_{1} \mathbb{L}_{k}T_{2}\, .$$
Thus, it is a derivation.
2. The commutator of two Lie-Lorentz derivatives
$$\label{eq:bracket}
[\mathbb{L}_{k_{1}},\mathbb{L}_{k_{2}}]\, T
=\mathbb{L}_{[k_{1},k_{2}]}\, T\, ,$$
where $[k_{1},k_{2}]$ is their Lie bracket.
3. The Lie-Lorentz derivative is linear in the vector fields
$$\mathbb{L}_{a^{1}k_{1}+a^{2}k_{2}}\, T = a^{1}\, \mathbb{L}_{k_{1}} T
+a^{2}\, \mathbb{L}_{k_{2}}T\, ,$$
and, thus, the Lie-Lorentz derivative with respect to the conformal killing vector fields forms a representation of the Lie algebra of conformal isometries of the manifold.
Some immediate consequences of the definition and basic properties are:
1. The Lie-Lorentz derivative of the Vielbein is
$$\label{eq:LLvielbein}
\mathbb{L}_{k}e^{a}{}_{\mu}
={\textstyle\frac{1}{d}}\nabla_{\rho}k^{\rho}e^{a}{}_{\mu}\, ,$$
and vanishes when $k$ is a Killing vector. In this case, we have the desirable property
$$\mathbb{L}_{k}\xi^{a} = e^{a}{}_{\mu} \mathcal{L}_{k}\xi^{\mu}\, .$$
2. The Lie-Lorentz derivative of gamma matrices is zero.
$$\label{eq:LLgamma}
\mathbb{L}_{k}\gamma^{a}=0\, .$$
3. \[item:1\] As a consequence of Eqs. (\[eq:Leibnitz\]), (\[eq:LLvielbein\]) and (\[eq:LLgamma\]), the Lie-Lorentz derivative with respect to Killing vectors preserves the Clifford action of vectors $v$ on spinors $\psi$ $v\cdot \psi \equiv
v_{a}\Gamma^{a} \psi=\not\!v \psi$ :
$$\label{eq:preserve1}
[\mathbb{L}_{k},\not\!v]\, \psi = [k,v]\cdot \psi\, .$$
4. \[item:2\] Also for Killing vectors $k$ it preserves the covariant derivative
$$\label{eq:preserve2}
[\mathbb{L}_{k},\nabla_{v}]\, T = \nabla_{[k,v]}\, T\, .$$
5. \[item:3\] All these properties imply that the Lie-Lorentz derivative with respect to Killing vectors preserves the supercovariant derivative of supergravity theories, at least in the simplest cases that we are going to examine next.
We stress that the properties \[item:1\], \[item:2\], \[item:3\] are only valid for Killing (not just conformal Killing) vectors.
The Lie-Lorentz Derivative and Supersymmetry
============================================
The Lie-Lorentz derivative occurs naturally in supergravity theories.
To start with, let us consider the local on-shell supersymmetry algebra, in particular the commutator of two infinitesimal, local supersymmetry transformations $\delta_{Q}(\epsilon)$ in $N=1,d=4$ supergravity
$$\delta_{Q}(\epsilon)\, e^{a}{}_{\mu} =
-i\bar{\epsilon}\gamma\psi_{\mu}\, ,
\hspace{1cm}
\delta_{Q}(\epsilon)\, \psi_{\mu} =
{\cal D}_{\mu}\epsilon\, .$$
which is usually written in the form
$$\label{eq:commutator1}
[\delta_{Q}(\epsilon_{1}), \delta_{Q}(\epsilon_{2})] =\delta_{\rm gct}(\xi)
+\delta_{\rm LL}(\sigma) +\delta_{Q} (\epsilon)\, ,$$
where $\delta_{\rm gct}(\xi)$ is an infinitesimal general coordinate transformation with parameter
$$\xi^{\mu} = -i\bar{\epsilon}_{1}\gamma^{\mu}\epsilon_{2}\, .$$
and is given by $\delta_{\rm gct}(\xi)=-\pounds_{\xi}$, $\delta_{\rm LL}(\sigma)$ is an infinitesimal local Lorentz transformation with parameter
$$\sigma^{ab} = \xi^{\nu}\omega_{\nu}{}^{ab}\, ,$$
and where
$$\epsilon = \xi^{\mu}\psi_{\mu}\, .$$
We are interested in obtaining the global superalgebra from the commutators of all the symmetry transformations of the theory. There is no unique superalgebra associated to a given supergravity theory, rather there are superalgebras associated to given bosonic solutions of the supergravity theory and their (super) symmetries. Solutions with a high degree of (super) symmetry are usually considered vacua of the supergravity theory and their associated superalgebras are of special interest.
Let us, then, consider a given vacuum solution of the $N=1,d=4$ supergravity equations of motion admitting Killing spinors $\varepsilon$ and Killing vectors $k$
$$\mathcal{D}_{\mu} \varepsilon =0\, ,
\hspace{1cm}
\nabla_{(\mu}k_{\nu)}=0\, .$$
On this vacuum, the commutator Eq. (\[eq:commutator1\]) should reduce to the commutator of two supercharges $Q_{1,2}=\bar{\varepsilon}_{1,2}Q$ which should be proportional to translations $\sim k^{a}P_{a}$ where $k^{a}=-i
\bar{\varepsilon}_{1}\gamma^{a}\varepsilon_{2}$ is a Killing vector if, as we have assumed, $\varepsilon_{1,2}$ are Killing spinors. However, the commutator Eq. (\[eq:commutator1\]) does not give that result if we naively interpret $\delta_{\rm gct}(k)$ as an infinitesimal translation since there is another term $\delta_{\rm
LL}$ whose meaning is unclear.
Observing that, actually, all the Killing vectors $k^{a}=-i
\bar{\varepsilon}_{1}\gamma^{a}\varepsilon_{2}$ are covariantly constant, we can write instead
$$[\delta_{Q}(\varepsilon_{1}), \delta_{Q}(\varepsilon_{2})]=
\delta_{P}(k)\, ,
\hspace{1cm}
k^{a}=-i \bar{\varepsilon}_{1}\gamma^{a}\varepsilon_{2}$$
where now we have identified the generator of the vacuum isometries acting on the supergravity fields
$$\delta_{P}(k) \equiv -\mathbb{L}_{k}\, .$$
This commutator has an immediate interpretation in terms of the global superalgebra. On the other hand, in this form, on account of Eq. (\[eq:LLvielbein\]), it is evident the the commutator of two supersymmetry transformations generated by two Killing spinors leaves invariant the Vierbein, as it should be.
To check that these definitions and interpretations actually make sense, let us consider a more complicated case: gauged $N=2,d=4$ supergravity, whose supersymmetry transformation rules are
$$\delta_{Q}(\epsilon)\, e^{a}{}_{\mu} =
-i\bar{\epsilon}\gamma^{a}\psi_{\mu}\, ,
\hspace{1cm}
\delta_{Q}(\epsilon)\, A_{\mu} =
-i\bar{\epsilon}\sigma^{2}\psi_{\mu}\, ,
\hspace{1cm}
\delta_{Q}(\epsilon)\, \psi_{\mu} =
\tilde{\mathcal{D}}_{\mu}\epsilon\, ,$$
where
$$\label{eq:supercovariantderivativeungauged}
\tilde{\nabla}_{\mu}= \hat{\mathcal{D}}_{\mu} +ig A_{\mu}\sigma^{2}
+{\textstyle\frac{1}{4}}\not\!\tilde{F}\gamma_{\mu}\sigma^{2}\, ,$$
is the [*supercovariant derivative*]{} and
$$\hat{\mathcal{D}}_{\mu} =
\mathcal{D}_{\mu} -{\textstyle\frac{ig}{2}}\gamma_{\mu}\, ,$$
is the $AdS_{4}\sim SO(2,3)$ covariant derivative. The commutator of two $N=2,d=4$ supersymmetry transformations is usually written in this form
$$\label{eq:commutator2}
[\delta_{Q}(\epsilon_{1}), \delta_{Q}(\epsilon_{2})] =\delta_{\rm gct}(\xi)
+\delta_{\rm LL}(\sigma) +\delta_{e}(\Lambda) +\delta_{Q} (\epsilon)\, ,$$
where the parameter of the infinitesimal local Lorentz transformations is now
$$\sigma^{ab} =
\xi^{\mu}\omega_{\mu}{}^{ab} -g\bar{\epsilon}_{2}\gamma^{ab}\epsilon_{1}
-i \bar{\epsilon}_{2}\left(\tilde{F}^{ab}
-i\gamma_{5}{}^{\star}\tilde{F}^{ab}\right)
\sigma^{2}\epsilon_{1}\, ,$$
and where $\delta_{e}(\Lambda)$ are $U(1)$ gauge transformations of the vector field and charged gravitino with parameter
$$\Lambda = -i\bar{\epsilon}_{2}\sigma^{2}\epsilon_{1}
+\xi^{\nu}A_{\nu}\, .$$
It is easy to see that on a vacuum solution admitting Killing spinors $\varepsilon$ and Killing vectors $k$
$$\tilde{\nabla}_{\mu} \varepsilon =0\, ,
\hspace{1cm}
\nabla_{(\mu}k_{\nu)}=0\, .$$
the commutator Eq. (\[eq:commutator2\]) can be written in the form
$$[\delta_{Q}(\varepsilon_{1}), \delta_{Q}(\varepsilon_{2})]=
\delta_{P}(k) +\delta_{e}(\chi)\, ,
\hspace{1cm}
k^{a}=-i \bar{\varepsilon}_{1}\gamma^{a}\varepsilon_{2}\, ,
\hspace{.5cm}
\chi = -i\bar{\varepsilon}_{2}\sigma^{2}\varepsilon_{1}
+k^{\nu}A_{\nu}\, .$$
The interpretation is again immediate. This commutator should vanish on all the fields of the theory. It clearly does on the Vierbein. On the vector and gravitino fields, it tells us that the Lie derivative vanishes up to a gauge transformation.
Let us consider now the remaining commutators.
Since we are identifying the bosonic generators of the the bosonic subalgebra of the supersymmetry algebra of the vacuum with the Lie-Lorentz derivative with respect to the Killing vector fields of the solution, we are going to get as bosonic subalgebra the Lie algebra of isometries of the vacuum, on account of Eq. (\[eq:bracket\]):
$$[\delta_{P}(k_{1}),\delta_{P}(k_{2})]
=\delta_{P}([k_{1},k_{2}])\, .$$
The commutator of local supersymmetry transformations and g.c.t.’s. these are difficult to compute, as a matter of principle, since the standard Lie derivative of Lorentz tensors is not a Lorentz tensor and then it does not make sense to perform on the transformed tensor a further supersymmetry transformation. Further, while we have a prescription to calculate the effect of an infinitesimal g.c.t. on any geometrical object, we do not know how to calculate the effect of an infinitesimal supersymmetry transformation of geometrical objects which are not fields of our theory (for instance, the vector that generates the infinitesimal g.c.t.).
Thus, it is necessary to give a prescription to calculate these commutators. In Ref. [@Figueroa-O'Farrill:1999va] the following prescription was proposed:
$$[\delta_{Q}(\varepsilon),\delta_{P}(k)]
=\delta_{Q}(\mathbb{L}_{k}\varepsilon)\, .$$
This prescription is based on the property (checked for certain geometrical Killing spinors in that reference) that the Lie-Lorentz derivative preserves the supercovariant derivative and, therefore, transforms Killing spinors into Killing spinors.
This property also holds in the simple theories we are considering here: on account of Eqs. (\[eq:preserve1\]),(\[eq:preserve2\]), we get
$$[\mathbb{L}_{k},\hat{\mathcal{D}}_{v}] \epsilon
=\hat{\mathcal{D}}_{[k,v]} \epsilon\, ,$$
(so the $AdS_{4}$ covariant derivative is preserved and the Lie-Lorentz derivative transforms $N=1,AdS_{4}$ Killing spinors into Killing spinors [@Figueroa-O'Farrill:1999va]) and, if
$$\label{eq:thecondition}
\mathbb{L}_{k}A_{a}=0\, ,$$
we find that the $N=2,AdS_{4}$ supercovariant derivative is also preserved
$$[\mathbb{L}_{k},\tilde{\mathcal{D}}_{v}] \epsilon
=\tilde{\mathcal{D}}_{[k,v]} \epsilon\, ,$$
and the Lie-Lorentz derivative transforms again Killing spinors into Killing spinors.
The condition Eq. (\[eq:thecondition\]) is satisfied in most cases in which $k$ is a Killing vector (up to a gauge transformation). If it was not satisfied in any gauge, the Killing vector $k$ would be an isometry but not a symmetry of the complete supergravity background and would not be a generator of the vacuum supersymmetry algebra. Thus, it must always be satisfied.
It is clear that these results can be generalized to higher dimensions and supersymmetries.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank E. Bergshoeff and specially P. Meessen for interesting conversations, the Institute for Theoretical Physics of the University of Groningen for its hospitality and financial support and M.M. Fernández for her continuous support. This work has been partially supported by the Spanish grant FPA2000-1584.
[30]{}
H. Weyl, [*Z. Phys.*]{} 330 56 (1929). Translated in Ref. [@kn:OR]
L. O’Raifeartaigh, Princeton University Press, Princeton, New Jersey (1997).
Y. Kosmann, [*Annali di Mat. Pura Appl.*]{} **(IV) 91** (1972) 317-395.
D. J. Hurley and M. A. Vandyck, J. Phys. A [**27**]{} (1994) 4569.
M. A. Vandyck, Gen. Rel. Grav. [**20**]{} (1988) 261. M. A. Vandyck, Gen. Rel. Grav. [**20**]{} (1988) 905.
J. M. Figueroa-O’Farrill, Class. Quant. Grav. [**16**]{} (1999) 2043 \[arXiv:hep-th/9902066\].
[^1]: We will call Lorentz tensor any object transforming in some finite-dimensional representation of the Lorentz group like vectors and spinors.
[^2]: For a different approach an further references, see, e.g. [@Hurley:cf]. The Lie derivative on spinors has been studied and used in the derivation of supersymmetry algebras in Refs. [@Vandyck:ei; @Vandyck:gc] and more recently in Ref. [@Figueroa-O'Farrill:1999va] whose results to revise and extend.
[^3]: We use the symbol $\mathbb{L}_{k}$ to avoid confusion with the standard Lie derivative that we keep denoting by $\mathcal{L}_{k}$. On spinors, then, $\mathcal{L}_{k}\psi =k^{\mu}\partial_{\mu}\psi$.
|
---
abstract: 'We study the effect of spin injection into a Luttinger liquid. The spin-injection-detection setup of Johnson and Silsbee is considered; here spins injected into the Luttinger liquid induce, across an interface with a ferromagnetic metal, either a spin-dependent current ($I_s$) or a spin-dependent boundary voltage ($V_s$). We find that the spin-charge separation nature of the Luttinger liquid affects $I_s$ and $V_s$ in a very different fashion. In particular, in the Ohmic regime, $V_s$ depends on the spin transport properties of the Luttinger liquid in essentially the same way as it would in the case of a Fermi liquid. The implications of our results for the spin-injection-detection experiments in the high $T_c$ cuprates are discussed.'
address: 'Department of Physics, Rice University, Houston, TX 77251-1892'
author:
- Qimiao Si
title: '**Spin Injection into a Luttinger Liquid**'
---
\[ @twocolumnfalse
\]
Spin-charge separation has long been proposed to describe the normal state of the high $T_c$ cuprate superconductors[@Anderson1]. Existing experimental results cited as evidences for spin-charge separation are mostly on transport properties[@Anderson2]. Since only charge transport properties have so far been measured, the inference about the coupling, or lack thereof, between the underlying spin and charge excitations is indirect. It would appear natural that spin transport, when combined with charge transport, should be useful in this context. Indeed we have recently proposed to probe spin-charge separation using a comparison between the temperature dependence of the yet-to-be-measured spin resistivity and that of the known electrical resistivity[@Si]. Several factors point to the feasibility of experimentally measuring spin transport in the cuprates using the spin-injection-detection technique[@Johnson1; @Johnson2; @Johnson3]. First of all, progresses in the preparation of the manganite-cuprate heterostructures appear to have led to the first demonstration of spin injection into the cuprates (albeit in the superconducting state)[@Goldman; @Venkatesan]. Secondly, the spin-diffusion length in the cuprates has been estimated to fall in the range required by this technique[@Si].
In light of the new experiments on the high $T_c$ cuprates, it is important to understand how spin injection into a non-Fermi liquid differs from spin injection into a Fermi liquid. The effects of spins injected into a non-interacting electron system has been studied in the past by Johnson and Silsbee and others[@Johnson1; @Johnson2; @Johnson3; @Fert; @Hershfield], following the initial proposals for spin injection and detection[@Aronov; @Silsbee].
Here we address the influence of spin-charge separation in the bulk metal on the boundary voltage/current measured in a spin-injection experiment. For definiteness, the one-dimensional Luttinger liquid[@Voit] is used as a prototype for a spin-charge-separated metal. Crucial for our analysis is the fact that, the interface transport involves the binding of spinons and holons which then tunnel as a whole from the Luttinger liquid into the ferromagnetic metal. We find that the spin-charge-separation nature of the Luttinger liquid affects the boundary voltage and boundary current in a very different fashion.
Figs. \[fig:setup\]a) and \[fig:setup\]b) illustrate two specific geometries involving a single channel Luttinger liquid (LL). The Luttinger liquid is in contacts with two itinerant ferromagnets, FM1 and FM2. The magnetization of FM1 is chosen as the $\uparrow$ direction. The magnetization of FM2 is either parallel ($\sigma=\uparrow$) or antiparallel ($\sigma=\downarrow$) to that of FM1. One passes an electrical current ($I$) across the FM1-LL interface. This current serves to inject non-equilibrium magnetization into the Luttinger liquid. For a given $\sigma$, $I_{\sigma}$ represents the induced current across the LL-FM2 interface in a closed circuit and $V_{\sigma}$ is the induced boundary voltage ($V_{\sigma}$) in an open circuit. The spin-dependent current, $I_s$, is defined as the difference between the induced current when the magnetizations of the two ferromagnets are in parallel and that when they are antiparallel. Likewise, the spin-dependent voltage, $V_s$, is the difference between the induced boundary voltages in the corresponding two cases. The setups illustrated in Figs. \[fig:setup\]a) and \[fig:setup\]b) differ in two regards. First of all, while the Luttinger liquid is in point contacts with both of the ferromagnets in Fig. \[fig:setup\]a), in Fig. \[fig:setup\]b) it is in contact with FM2 over an extended spatial region. Secondly, unlike in Fig. \[fig:setup\]a) the injection and detection loops in Fig. \[fig:setup\]b) are closed through contacts with LL far away both FM1 and FM2. The setup of Fig. 1b) is perhaps easier to implement experimentally. On the other hand, that of Fig. 1a) is easier to analyze theoretically. To illustrate the basic principle, in the rest of this paper we will focus on Fig. \[fig:setup\]a).
The Hamiltonian of the Luttinger liquid can be written as $$\begin{aligned}
H_{lut} = && H_{\rho} + H_{s} + H'\nonumber\\
H_{\rho} = && { 1 \over 2\pi} v_{\rho} \int dx ~~[ K_{\rho} (\pi
\Pi_{\rho})^2 + {1 \over K_{\rho}} (\partial_x \phi _{\rho})^2] \nonumber\\
H_{s} = && { 1 \over 2\pi} v_{s} \int dx ~~[ K_{s} (\pi \Pi_{s})^2
+ {1 \over K_{s}} (\partial_x \phi _{s})^2]
\label{hamlut0}\end{aligned}$$ where $H_{\rho}$ and $H_{s}$ are respectively the Hamiltonian for the charge ($\rho$) and spin ($s$) bosons, $\phi_{\rho}$ and $\phi_{s}$; $\Pi_{\rho}$ and $\Pi_{s}$ are the corresponding conjugate momenta. The charge and spin velocities, $v_{\rho}$ and $v_{s}$, and Luttinger liquid parameters, $K_{\rho}$ and $K_{s}$, are determined by the forward scattering interactions. We consider the case when the spin-rotational invariance is preserved so that $K_s=1$.
We focus on the regime where spin transport inside the Luttinger liquid is diffusive, with a spin diffusion constant $D_s$. The diffusive transport is the result of the dissipative terms in the Hamiltonian, $H'$, which also lead to a finite spin relaxation time $T_1$. The precise form of $H'$ is however unimportant for our purpose in this paper and is left unspecified here. (It will, of course, determine the specific temperature dependences of $D_s$ and $T_1$.) To simplify the discussion, we assume that the FMs are half-metallic ferromagnets. We will also neglect the electron interactions inside the ferromagnets[@magnon]. The free electron Hamiltonians for FM1 and FM2 are respectively $$\begin{aligned}
H_1 = \sum_k \epsilon_k^1 c_{k\uparrow}^{\dagger} c_{k\uparrow}
\label{ham.left}\end{aligned}$$ and $$\begin{aligned}
H_2 = \sum_k \epsilon_k^2 c_{k,\sigma}^{\dagger} c_{k,\sigma}
\label{ham.right}\end{aligned}$$ where $\epsilon_k^1$ and $\epsilon_k^2$ are the corresponding energy dispersions.
Since we will consider only the cases when the magnetizations of the two ferromagnets are either parallel or anti-parallel with each other, we need only the kinetic equation for the longitudinal component of the magnetization. We introduce $m(x)$ to denote the deviation of the steady state magnetization density from the corresponding equilibrium value. The FM1-LL and LL-FM2 interfaces are located at $x=0$ and $x=d$, respectively. The non-equilibrium magnetization density $m(x)$ satisfies the following[@transverse] $$\begin{aligned}
-D_s {\partial ^ 2 m \over \partial x^2} = -{ {m(x)} \over T_1}
\label{kinetic}\end{aligned}$$ where $T_1$ is the longitudinal spin relaxation time. The boundary conditions will depend on the details of the interface. Leaving more general cases for elsewhere[@Si2], we assume that no spin-flip scattering exists at the interface. In this case, the spin current $j_s$ is conserved across the interface. Given that $j_s (x=0^-) = \mu_B I/e $, where $\mu_B$ is the Bohr magneton and $e$ the electron charge, the boundary condition at the FM1-LL interface is, $$\begin{aligned}
-D_s \partial m / \partial x |_{x=0} = \mu_B I/e
\label{bc.spincurrent}\end{aligned}$$ Likewise, at the LL-FM2 interface, $$\begin{aligned}
-D_s \partial m / \partial x |_{x=d} = \mu_B I_{\sigma}/e
\label{bc.spincurrent2}\end{aligned}$$
The induced current $I_{\sigma}$ depends on the drop of the non-equilibrium magnetization across the LL-FM2 interface, which is equal to $\Delta m = m(d)$. We separately discuss the closed circuit and open circuit cases.
Consider, first, the case of a closed circuit. The drop of non-equilibrium magnetization, $\Delta m$, leads to a drop in the effective magnetic field, $$\begin{aligned}
\Delta H = \Delta m / \chi
\label{Delta.H}\end{aligned}$$ across the LL-FM2 interface. Here, $\chi$ is the uniform spin susceptibility of the Luttinger liquid. $\Delta H$ provides a driving force for spinons to move across the interface. Since only electrons can tunnel across the barrier, spinons bind with holons and move across the interface as a whole. A finite electrical current, $I_{\sigma}^{closed}$, is then induced by $\Delta H$. To calculate $I_{\sigma}^{closed}$, we follow the general procedure of Kane and Fisher[@Kane-Fisher] and integrate out all the degrees of freedom of the Luttinger liquid except at the site of contact, $x=d$. This leads to an effective action entirely determined by the boson field $\phi_{\rho} \equiv
\phi_{\rho}(d)$ and $\phi_{s} \equiv \phi_{s}(d)$: $$\begin{aligned}
S_{site} = K_{\rho} {1 \over \beta} \sum_{\omega_n} |\omega_n|
|\phi_{\rho} (\omega_n) | ^ 2
+
{1 \over \beta} \sum_{\omega_n} |\omega_n|
|\phi_{s} (\omega_n) | ^ 2
\label{action.site}\end{aligned}$$ where $\beta$ is the inverse of temperature and $\omega_n$ the bosonic Matsubara frequencies. In deriving this on-site action, we have neglected the effect of the non-equilibrium magnetization, $m(x)$. This is appropriate for the Ohmic regime $k_BT > \mu_B m/\chi $. We are now faced with a problem of one[@multi-mode] retarded impurity – whose dynamics is controlled by Eq. (\[action.site\]) – coupled to a three dimensional ferromagnetic metal. This effective impurity problem is illustrated in Fig. 2. The impurity problem can also be written in a Hamiltonian form, by introducing a fictitious bosonic bath.
From the one-particle tunneling Hamiltonian, one can construct respectively the charge current operator $J$ and spin current operator $J_M$. They are as follows, $$\begin{aligned}
J/e = &&J_M / {\mu_B} \nonumber \\
= &&{t \over \sqrt{2\pi a}}
[F_{\sigma -}^{\dagger}
{\rm e}^{-i{ {\phi_{\rho} + \theta_{\rho}} \over \sqrt{2}}
-i\sigma {{\phi_{s} + \theta_{s}} \over \sqrt{2}}} \nonumber\\
&&+
F_{\sigma +}^{\dagger}
{\rm e}^{-i{{-\phi_{\rho} + \theta_{\rho}} \over \sqrt{2}}
-i\sigma {{-\phi_{s} + \theta_{s}} \over \sqrt{2}}}]
c_{\sigma} -H.c.
\label{current}\end{aligned}$$ where $F_{\sigma \pm}^{\dagger}$ are the Klein operators for the left and right moving branches and $a$ is a lattice cutoff[@Voit; @Klein]. Here, $c_{\sigma}$ is the annihilation operator for the Wannier orbital of the conduction electrons at the contact, and $t$ is the tunneling matrix. That the interface charge and spin current operators are simply related to each other[@half-metallic] in spite of the separated spin and charge excitations in the bulk Luttinger liquid reflects the simple physics that, only a bare electron tunnels across the interface. We can now calculate $I_{\sigma}^{closed}$ in the Ohmic regime using the Kubo formalism, $$\begin{aligned}
I_{\sigma}^{closed} = (-\Delta H)
\lim_{\omega \rightarrow 0} {{-{\rm Im}{\pi_{JJ_M} (\omega+i0^+) }}
\over \omega }
\label{Iind.1}\end{aligned}$$ where $\pi_{JJ_M}$ is the charge current-spin current correlation function. Similar to the Kane-Fisher problem, for repulsive interactions $K_{\rho} <1$, the tunneling term is an irrelevant coupling in the renormalization group sense. We can then calculate $\pi_{JJ_M}$ perturbatively in $t$. The result is as follows, $$\begin{aligned}
I_{\sigma}^{closed} = C (k_BT)^{1/2K_{\rho} - 1/2}
e \mu_B \Delta m /\chi
\label{Iind1}\end{aligned}$$ where $C=c N_F t^2 / W^{1/2K_{\rho}+1/2}$. Here $N_F$ is the density of states of FM2 at the Fermi energy, $W$ is a typical bare energy scale associated with the electrons in the Luttinger liquid, and $c$ is a constant of order unity. The induced spin-dependent current across the LL-FM2 interface, $I_s = I_{\uparrow}^{closed} - I_{\downarrow}^{closed}$ can now be determined, from Eqs. (\[kinetic\],\[bc.spincurrent\],\[bc.spincurrent2\],\[Iind1\]). The result is as follows, $$\begin{aligned}
I_{s} / I = C (k_B T)^{1/2K_{\rho} - 1/2}
{ \mu_B^2 \over \chi} {T_1 \over \delta_s \sinh (d / \delta_s)}
\label{Iind.cc}\end{aligned}$$ where $\delta_s = \sqrt{D_s T_1}$ is the spin-diffusion length of the Luttinger liquid.
We now turn to the open circuit case. Here, in order to balance the current induced by the magnetization drop, a boundary voltage, $V_{\sigma}$, develops across the LL-FM2 interface. The induced current in this case is $$\begin{aligned}
I_{\sigma}^{open} = &&
(-\Delta H )
\lim_{\omega \rightarrow 0} {{-{\rm Im}{\pi_{JJ_M} (\omega+i0^+)}
\over \omega}} \nonumber\\
&&+ V_{\sigma} \lim_{\omega \rightarrow 0}
{{-{\rm Im}{\pi_{JJ} (\omega+i0^+)}}
\over \omega}
\label{Iind.2}\end{aligned}$$ which leads to $$\begin{aligned}
I_{\sigma}^{open} = C (k_BT)^{1/2K_{\rho} - 1/2} e
(\mu_B \Delta m/\chi + e V_{\sigma})
\label{Iind2}\end{aligned}$$ Setting $I_{\sigma}^{open}=0$ in Eq. (\[Iind.2\]), and combining with Eqs. (\[kinetic\],\[bc.spincurrent\],\[bc.spincurrent2\]), we arrive at the following expression for the spin-dependent boundary voltage, $V_s = V_{\uparrow}- V_{\downarrow}$: $$\begin{aligned}
V_s / I = {\mu_B^2 \over e^2 \chi } {T_1 \over {\delta_s \sinh (d /
\delta_s )}}
\label{Vs}\end{aligned}$$
Eqs. (\[Iind.cc\],\[Vs\]) are the main results of this work. Several comments are in order.
First of all, it is instructive to see how our results reduce to those for free electrons[@Johnson1; @Johnson2; @Johnson3; @Fert; @Hershfield] when electron interactions are reduced to zero. For non-interacting electrons, spin diffusion $D_s$ is reduced to the usual electron diffusion constant $D = v_F^2 \tau$, where $v_F$ is the Fermi velocity and $\tau$ the transport scattering time. In addition, $\chi = \mu_B^2 N_F^P$, where $N_F^P$ is the density of states at the Fermi energy. Straightforward manipulation leads to $V_s/I = \rho \delta_s^0 / \sinh (d/\delta_s^0)$, where $\rho$ is the electrical resistivity of the bulk metal and $\delta_s^0=\sqrt{D T_1}$. In addition, for non-interacting electrons, $K_{\rho}=1$, and our expression for $I_s$ is reduced to $I_s/I = (e^2N_F N_F^P t^2) \rho \delta_s / \sinh (d/\delta_s)$.
Secondly, we note that for the Luttinger liquid, the temperature dependence of $I_s$ is not solely determined by that of the bulk spin diffusion and relaxation properties of the Luttinger liquid. There is an additional temperature-dependent factor, with a power which explicitly depends on the Luttinger liquid parameter $K_{\rho}$. This additional temperature-dependent factor, however, cancels out in $V_s$. This last result can ultimately be traced to the fact that the interface charge current and spin current are directly related to each other, in spite of the spin-charge separation nature of the bulk Luttinger liquid. On this ground, we expect the expression for $V_s$ to be valid very generally, so long as one particle processes dominate the interface transport and no strong-coupling (in $t$) phenomena[@Fradkin] take place. The boundary voltage $V_s$ is hence more useful than $I_s$ for the purpose of extracting bulk spin transport properties of strongly correlated metals, including the high $T_c$ cuprates.
The general expression for the spin-dependent boundary voltage suggests the following procedure to measure the temperature dependence of the spin transport scattering rate, $1/\tau_{tr,spin}$, of correlated metals. The latter is the quantity of interest in probing spin-charge separation[@Si]. In strongly interacting metals, it is likely that the dissipations for both the spin current and total spin come primarily from electron-electron interactions. The spin relaxation time and spin transport relaxation rate are then proportional to each other: ${1 \over T_1}
\approx (\lambda_{so})^2 {1 \over \tau_{tr,spin}}$, where $\lambda_{so}$ is the dimensionless spin-orbit coupling constant. When the sample thickness $d$ is small compared to the spin diffusion length $\delta_s$, the temperature dependence of $\chi V_s/I$ is then directly proportional to the temperature dependence of $\tau_{tr,spin}$. On the other hand, for thickness much larger than the spin diffusion length, the temperature dependence of $\ln (\chi V_s / I )$ is directly proportional to that of $1/\tau_{tr,spin}$. This procedure does not require measurement in a series of samples of different thicknesses – as was necessary in the case of simple metals[@Johnson2] – and may be experimentally easier to implement.
To summarize, we have studied the effects of spin injection into a Luttinger liquid. Our conclusion that the temperature dependence of the boundary voltage depends on the bulk spin-transport properties only is expected to be generally applicable and of direct relevance to the spin-injection-detection experiments in the cuprates.
I would like to thank N. Andrei, M. Johnson, Q. Li and A. J. Rimberg for useful discussions, and the Aspen Center for Physics and the CM/T group at NHMFL/FSU for hospitality and support during different stages of this work. The work has also been supported by NSF Grant No. DMR-9712626, a Robert A. Welch Foundation grant and an A. P. Sloan Fellowship.
P. W. Anderson, Science [**235**]{}, 1196 (1987); Phys. Rev. Lett. [**64**]{}, 1839 (1990).
P. W. Anderson, [*The Theory of Superconductivity in the High-$T_c$ Cuprates*]{} (Princeton University Press, Princeton, NJ 1997).
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M. Johnson and R. H. Silsbee, Phys. Rev. Lett. [**55**]{}, 1790 (1985).
M. Johnson, Phys. Rev. Lett. [**70**]{}, 2142 (1993) and references therein.
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Z. W. Dong, R. Ramesh, T. Venkatesan, M. Johnson, Z. Y. Chen, S. P. Pai, V. Talyansky, R. P. Sharma, R. Shreekala, C. J. Lobb and R. L. Greene, Appl. Phys. Lett. [**71**]{}, 1718 (1997).
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For a review see J. Voit, Rep. Prog. Phys. [**58**]{}, 977 (1995).
Having super-ohmic spectral functions, the spin waves in the ferromagnets are unimportant for our analysis.
The kinetic equation for the transverse magnetization would be more complex, due to the Leggett-Rice effect. A. J. Leggett and M. J. Rice, Phys. Rev. Lett. [**20**]{}, 586 (1968); A. J. Leggett, J. Phys. C[**3**]{}, 448 (1970).
Q. Si, unpublished.
C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. [**68**]{}, 1220 (1992); Phys. Rev. B[**46**]{}, 15233 (1992).
With more than one transverse channels, the effective problem describes multiple, spatially separated, impurities coupled to a three dimensional metal. N. Andrei and Q. Si, unpublished.
G. Kotliar and Q. Si, Phys. Rev. [**B**]{}53, 12373 (1996).
When FM2 is not half-metallic, the relationship between $J$ and $J_M$ will be more complex. The conclusions are, however, essentially the same as give here[@Si2].
See, e.g., N. P. Sandler, C. de C. Chamon, and E. Fradkin, Phys. Rev. [**B**]{}57, 12324 (1998) and references therein.
|
---
abstract: 'The initial mass function (IMF), binary fraction and distributions of binary parameters (mass ratios, separations and eccentricities) are indispensable input for simulations of stellar populations. It is often claimed that these are poorly constrained significantly affecting evolutionary predictions. Recently, dedicated observing campaigns provided new constraints on the initial conditions for massive stars. Findings include a larger close binary fraction and a stronger preference for very tight systems. We investigate the impact on the predicted merger rates of neutron stars and black holes. Despite the changes with previous assumptions, we only find an increase of less than a factor 2 (insignificant compared with evolutionary uncertainties of typically a factor $10-100$). We further show that the uncertainties in the new initial binary properties do not significantly affect (within a factor of $2$) our predictions of double compact object merger rates. An exception is the uncertainty in IMF (variations by a factor of $6$ up and down). No significant changes in the distributions of final component masses, mass ratios, chirp masses and delay times are found. We conclude that the predictions are, for practical purposes, robust against uncertainties in the initial conditions concerning binary parameters with exception of the IMF. This eliminates an important layer of the many uncertain assumptions affecting the predictions of merger detection rates with the gravitational wave detectors aLIGO/aVirgo.'
author:
- 'S. E. de Mink and K. Belczynski'
bibliography:
- 'my\_bib.bib'
title: 'Merger rates of double neutron stars and stellar origin black holes: The Impact of Initial Conditions on Binary Evolution Predictions'
---
Introduction
============
These are very exciting times for gravitational wave astrophysics. The direct detection of the gravitational wave signal of the merger of two compact objects, neutron stars (NS) or black holes (BH) is anticipated before the end of this decade.
Gravitational waves are a natural consequence of the theory of General Relativity [@Einstein1918]. They are perturbations of the spacetime metric propagating at the speed of light, which are generated, for example, during the inspiral of two compact objects. Indirect evidence for the existence of gravitational waves was provided by the orbital decay of the Hulse-Taylor pulsar [@Hulse+1975; @Taylor+1989] and later with stronger constraints by the double pulsar [@Burgay+2003; @Lyne+2004]. Direct detection of the gravitational wave signal of the inspiral of NS-NS, BH-NS or BH-BH binaries and the subsequent merger and ring down is expected to happen in the next few years, now the advanced ground-based gravitational wave detectors aLIGO and Virgo are coming online [@Abbott+2009; @Caron+1997].
The initial LIGO/Virgo observations were concluded in 2010 without detection, but they provided upper limits on the merger rates [@Abadie+2012]. The advanced version of the detectors will be approximately 10 times more sensitive than the initial versions, expanding the detection volume and thus the chance of detection by a factor of about a thousand [@Aasi+2013].
The first science run of advanced LIGO is scheduled for late 2015 [@Aasi+2013a]. A detection during the first science run is not considered to be likely, but one or more detections are anticipated in the next few years as the sensitivity increases to a range of 200 Mpc for double neutron stars. This translates to $\sim 0.2$–$200$ expected detections double neutron star mergers per year [@Aasi+2013a]. Even without detections, the new upper limits will become astrophysical interesting as they start to rule out the models that predict the highest merger rates [@Mandel+2010; @Belczynski+2012; @Stevenson+2015].
Obtaining reliable predictions of the merger rates of relativistic compact objects has been a very large challenge, as reviewed by [@Abadie+2010]. The rates quoted above are derived semi-empirically using the observed binary pulsars in our Galaxy [@Phinney1991; @Kalogera+2004]. The large uncertainties result from the small size of the sample of observed binary pulsars and the pulsar luminosity distribution. The implicit assumption is made that the observed sample is representative for the Galactic population. Further constraints come from short gamma-ray bursts, but the rate estimates depend on the uncertain luminosity function and opening angle of the jets [e.g. @Fong+2012]. Type Ibc supernovae have served so far as ultimate upper limits [@Kim+2010]. The merger rates for BH-NS and BH-BH systems are even more problematic as we lack direct observational evidence of their existence. However, some immediate progenitors for BH-BH [e.g. @Bulik+2011] and BH-NS systems [e.g. @Grudzinska+2015] have been proposed.
Even though theoretical predictions suffer from even larger uncertainties, they have been crucial to estimate the merger rates involving black holes. They further provide the expected distribution of properties for all merger types including for example the delay times, component and chirp masses. More importantly, the theoretical predictions are providing the tool for future comparison with the detections, crucial to identify what gravitational wave signals teach us about the astrophysics of the progenitor systems [e.g. @Stevenson+2015]. Several groups have presented similar estimates and studies in the past decade [e.g. @Lipunov+1997; @Bethe+1998; @De-Donder+1998; @Bloom+1999; @Grishchuk+2001; @Nelemans+2001; @Voss+2003; @Dewi+2003; @Nutzman+2004; @De-Donder+2004; @Pfahl+2005; @Postnov+2006; @Yungelson+2006; @OShaughnessy+2008; @OShaughnessy+2010; @Dominik+2012; @Dominik+2013; @Mennekens+2014; @Dominik+2015].
For all predictions we can distinguish several sources of uncertainty: (I) the adopted *initial conditions* in the simulations, (II) the uncertainties in the *physics of the stellar evolution and binary interaction*, (III) the uncertainties associated with the *normalization* of the merger rates, which include the mapping to the appropriate star formation history of the detection volumes and (IV) modeling of the *detectability* of gravitational waves from the predicted merger events.
Recent observations have provided new constraints on the initial conditions and primordial binary properties [@Kobulnicky+2007; @Kiminki+2012; @Chini+2012; @Sana+2012; @Sana+2013; @Sana+2014; @Kobulnicky+2014; @Moe+2015; @Moe+2015a; @Dunstall+2015], for a review see [@Duchene+2013]. The studies show that the initial conditions for young massive stars differ substantially from the initial conditions that have typically been adopted to simulate compact object mergers coming from binary evolutionary channels.
Among the most striking findings are (i) the large fraction massive stars that have a companion close enough to interact by exchanging mass before they die and (ii) the preference for very tight binaries with orbital periods of a few days, implying that a large fraction of massive stars will interact even before leaving the main sequence. Further findings include (iii) the confirmation of a flat distribution of mass ratios, ruling out a distribution that is strongly peaked to equal masses and (iv) a distribution of eccentricities that favors circular systems in stark contrast with the typically adopted thermal eccentricity distribution which favors eccentric systems.
The first two conclusions depend on adopted model of stellar evolution and in particular they require at least modest radial expansion for the majority of massive stars. The amount by which stars expand is considerably uncertain for high mass stars. Note that some very rapidly rotating massive stars may actually decrease in size as they evolve [@Yoon+2005], possibly even preventing interaction with companion by mass transfer [@de-Mink+2009a] in very tight systems. For example, let us consider the fraction of massive binary systems in which mass transfer starts already during the main sequence evolution of the primary star. This depends on the maximum radius that a star reaches on the main sequence which depends on a number of rather uncertain processes such as convection and overshooting, [e.g. calibrations by @Pols+1997; @Ribas+2000; @Brott+2011], further mixing processes such as those induced by rotation [e.g. @Yoon+2005; @Brott+2011a; @Ekstrom+2012; @Szecsi+2015], mass loss by stellar winds and eruptions [review by @Smith2014] and the possible density inversions in the outer layers of massive stars which can result in inflated envelopes[e.g. @Yusof+2013; @Kohler+2015 and Jiang et al. 2015, subm.]. Varying the maximum radius of main sequence stars by $30\%$ up and down and allowing for uncertainties in the initial distributions, we find a large variation for this fraction, $20\%-50\%$[^1].
A further caveat to keep in mind is that the observations are limited to regions nearby such as our own Galaxy and the Magellanic Clouds which only cover metallicities down to about one fifth of the solar value. Also the regime of the highest mass stars is not well probed. @Sana+2012 observations cover stars in mass range $15 - 60 \msun$, but most of these are towards the lower end of this mass range. For low metallicities and higher star masses we have no direct constraints and the uncertainties in the evolutionary models start to play a larger role. Despite several attempts, no trends with metallicity or environment are found so far [e.g. @Bastian+2010; @Moe+2013]. We extend @Sana+2012 distributions in our study to the entire range that can produce double compact objects ($M_{\rm zams}=5 -150 \msun$) and to a broad range of metallicities ($Z=0.002 - 0.02$) as discussed in Section 2.
In this study we focus on the binary formation channels [in contrast to the dynamical formation channels that may occur in dense star clusters, e.g. @Ivanova+2008; @Banerjee+2010; @Aarseth2012; @Clausen+2013; @Samsing+2014; @Ramirez-Ruiz+2015]. We investigate two questions: (i) What are the implications of the new initial conditions? and (ii) How robust are the predictions against the allowed variations in the new initial conditions resulting from the observational uncertainties.
For this purpose we perform a comparative population synthesis study where we use the recent work by @Dominik+2012 as a reference. We simulate the evolution of massive binary systems following the evolutionary channels for the formation of double compact objects. In Sect. \[sec:initdist\] we describe the new and old initial conditions. In Sect. \[sec:method\] we give a brief description of our computational method, the physical assumptions and computation of the merger rates. In Sect. \[sec:dist\] we compare the old and new initial distributions for the entire simulated populations and for the progenitors of double compact object mergers. In Sect. \[sec:rates\] we discuss the impact on the resulting merger rates. Finally, in Sect. \[sec:concl\] we present our discussion and conclusions.
Initial Distributions {#sec:initdist}
======================
New standard model (N) and its variations\[sec:initnew\]
--------------------------------------------------------
Recent dedicated observing campaigns have provided new constraints on the binary properties of young massive stars. Here, we investigate the impact of the distributions obtained by the work of [@Sana+2012]. This study is based on an intense spectroscopic monitoring spanning a decade surveying all O-type stars in six nearby ($\lesssim 3{\ensuremath{\,\rm{kpc}}}$) very young (about 2Myr old) open star clusters and associations. Even though the sample may seem of modest size it exceeds previous studies in completeness in terms of the fraction of systems for which orbital solutions have been obtained. It provided an average of 20 radial velocity measurements for all 71 systems in the open clusters/associations that contain at least one O-star. This dataset includes orbital solutions for several long period systems (between 100 and 1000 days) which are very challenging as they require a long term observing campaign. This sample allowed a robust derivation of the underlying distribution of binary parameters after correction for incompleteness and biases.
The very young ages, relatively low densities and velocity dispersion of the stars, imply that the effects of stellar evolution and dynamical interactions are minimal. This makes this sample the most suitable to provide constraints on the primordial binary properties and thus the initial conditions for our simulations.
The primary stars in this sample have spectral types ranging from O9.7 to O3, which correspond approximately to a mass range from 15 to 60 [$\,M_\odot$]{}. This is appropriate for the progenitor systems of double compact object mergers that involve at least one black hole. Lower mass binaries may dominate the formation of double neutron stars.
The most suitable study for this mass range is provided by @Dunstall+2015 for the early B-type stars in the 30 Dor region and the Cygnus OB2 sample analyzed by @Kobulnicky+2014, which contains stars down to spectral types of B2.5V, which approximately corresponds to 8[$\,M_\odot$]{}. [@Wright+2015] infer an age spread for this region of 1-7 Myr. Statistically the findings by @Kobulnicky+2014 and @Dunstall+2015 are consistent with the distributions by @Sana+2012, although the results by @Kobulnicky+2014 and @Dunstall+2015 favor a flatter period distribution and a lower close binary fraction, similar to the old initial distributions that we use as reference (see Sec. 2.2). Whether this is a sign of a trend with decreasing primary mass, or whether the distribution of ages play a role is not clear.
Sourcing from our conclusions, we find that the changes in the period distribution have a rather small impact on double compact merger rates. Thus we apply the @Sana+2012 distributions as our new initial conditions for both O and early B stars for consistency.
#### Orbital periods
For the distribution of orbital periods, $p$, we adopt the distribution @Sana+2012 which significantly favors short period systems. Such a preference had been observed in previous surveys uncorrected for biases [e.g. @Mason+2009], but was generally interpreted as being the result of selection effects. @Sana+2012 demonstrated that this preference remains even after carefully correcting for observational biases. We adopt $$f_p (\log p) \propto ( \log p )^\pi, \quad \quad \text{for $ \log p\in
[0.15, 5.5]$}$$ where $p$ is given in days. For our standard simulation we adopt $\pi=-0.5$. We change the slope of orbital period distribution from $\pi=-0.75$ in model N-p1 to $\pi=-0.35$ in model N-p2.
Note that the spectroscopic observations can only reliable probe systems with $\log p \lesssim 3.5$. However, wider systems can still produce double compact object mergers as we will show in the following section. For wider systems we adopt the simplest assumption we can take and extrapolate the distribution, since we have no reasons to believe that the binary fraction suddenly drops beyond $\log p = 3.5$.
This assumption is consistent with the recent findings by interferometric studies of nearby Galactic massive stars. @Sana+2014 provided a large systematic survey probing companions of O stars at angular separations between 1 and 100 mili-arcseconds. For unevolved massive stars (O stars with luminosity class V) the detected companion fraction reaches 100% at 30 mili-arcseconds. The physical separations this corresponds to will remain uncertain until more accurate distance measurements become available. Roughly it corresponds to physical separations of 60-6000 AU. In our preferred units for the orbital separation $a$ this corresponds to $\log a ({\ensuremath{\,R_\odot}}) =\,$4.1-6.1. Considering systems with typical masses we find that our extrapolation of the orbital period distribution and binary fraction are consistent with these observations.
#### Mass ratios
For the distribution of mass ratios, which we define as the mass of the initially less massive star over the mass of the more massive star, i.e. $q \equiv M_2/M_1$, we use $$f_q (q) \propto q^\kappa, \quad \quad \text{for $q \in [0.1, 1]$}
\label{eq:iqfNew}$$ where $\kappa = -0.1 \pm 0.6$ according @Sana+2012. We adopt $\kappa = 0$ such that the distribution becomes uniform distribution, which has also been found in several recent studies such as @Kobulnicky+2007 and @Kobulnicky+2014. To consider the uncertainties we consider lower and upper limits of $\kappa=0.5$ in model N-q1 to $\kappa=-0.7$ in model N-q2.
We note that the most recent observations rule out the presence of a so-called twin population of equal mass systems [@Pinsonneault+2006]. The idea of the possible existence of such a population gained interest as it favors the formation of double compact object mergers, in particular through the so-called double core formation channel proposed by @Dewi+2006. However, the claimed evidence of such a population has been demonstrated to be the result of observational biases towards equal mass systems that were not accounted for [@Lucy2006; @Sana+2013; @Cantrell+2014].
#### Eccentricities
For the eccentricity distribution in the very young open clusters @Sana+2012 finds $$f_e (e) \propto e^\eta, \quad \quad \text{for $e \in [0.0, 0.9]$}$$ with $\eta=-0.42\pm0.17$. The very short period systems ($p \lesssim 4$ days) show a larger degree of circularization as expected from the short time scale for tidal circularization [@Zahn1975]. Unfortunately the data sample is not large enough to provide a reliable separation dependent eccentricity distribution. Instead we adopt this distribution as initial distribution for our simulations, independently of the period. We explicitly follow the effects of tides in our simulations, which quickly circularizes the shortest period systems, consistent with the observations. As we will show later this assumption is justifiable as the variations in the eccentricity result in only minimal changes in the rates. In our standard simulation N we adopt ${\eta =
-0.42}$. We consider uncertainties by changing the the slope to $\eta=-0.59$ in model N-e1 to $\eta=-0.25$ in model N-e2.
#### Binary fraction
The spectroscopic survey by @Sana+2012 yields a binary fraction of $f_{\rm SB} = 0.7\pm0.1$ after correcting for biases. The binary fraction here is defined as $$f_{\rm bin} \equiv \frac{N_{\rm bin}}{N_ {\rm bin} + N_{\rm single} }$$ where $N_{\rm bin}$ and $N_{\rm single}$ are the number of binary systems and the number of single stars respectively.
This fraction refers only to systems that have orbital parameters within the considered boundaries , i.e. $q \in [0.1, 1]$, $\log p {\rm {(days)}}
\in [0.15, 3.5]$ and $e \in [0.0, 0.9]$. The remainder of the sample ($\sim 30\%$) consists of stars of unknown nature, including wider binaries, binaries with more extreme mass ratios and possibly genuine single stars. Motivated by the interferometric survey [@Sana+2014] we designate the remainder as wide binaries. We extrapolate the original [@Sana+2012] period distribution that extends to $\log p = 3.5$ to $\log p = 5.5$. Such extrapolation results in a binary fraction, including wide systems, of $f_{\rm bin} = 1$. We also consider a reduced binary fraction of $f_{\rm bin}=0.85$ in model N-f1 and $f_{\rm bin}=0.7$ in model N-f2.
#### Masses
The observed distribution of primary masses in the sample of @Sana+2012 is consistent with a standard @Kroupa2002 mass function. Although we only simulate the evolution of massive binaries, for normalization we adopt the three component power law for the primary mass or single star $m_1$, given in solar units [$\,M_\odot$]{}$$f_{m_1}(m_1) \propto \begin{cases}
m_1^{-1.3}, &\text{for $ m_1 \in [0.08, 0.5]$}\\
m_1^{-2.2}, &\text{for $ m_1 \in [0.5, 1.0]$}\\
m_1^{-\alpha}, &\text{for $m_1 \in [1,150]$}
\end{cases}
\label{eq:imf}$$ We adopt $\alpha = 2.7$ in our standard model, consistent with the field star population [@Kroupa+1993; @Kroupa+2003]. To allow for uncertainties we consider $\alpha=3.2$ in model N-m1 and $\alpha=2.2$ in model N-m2.
Old standard model (O) and its variations \[sec:initold\]
---------------------------------------------------------
We adopt the @Dominik+2012 reference model as the old standard model O. In their work systems are drawn from an initial distribution of separations instead of orbital periods. This distribution is assumed to be flat in the log [@Abt1983; @Opik1924] $$f_a (\log a) \propto {\rm constant}, \quad \quad \text{for $ \log
a\in [a_{\rm min}, 5]$}.$$ The lower boundary $a_{\rm min}$ is assumed to be a function of the stellar radii of the primary and secondary star at the zero age main-sequence (ZAMS), $$a_{\rm min}\equiv \frac{2 R_{\rm ZAMS,1} + 2 R_{\rm ZAMS,2}}{1-e}$$ This was adopted to ensure that at zero age both stars are well within their Roche lobes at closest approach, i.e. the periastron distance $d_{\rm per}=a (1-e)$. The distribution of mass ratios is a pseudo-flat distribution. The only difference with the new distribution concerns the boundaries. @Dominik+2012 consider $q
\in [q_{\rm min},1]$ with $q_{\rm min}=0.08/m_1$ to avoid selecting secondaries below H-burning limit. For the distribution of eccentricities a thermal-equilibrium distribution is adopted [@Heggie1975], which favors eccentric systems: $$f_e (e) \propto e, \quad \quad \text { for $e \in [0,1]$}.$$
The binary fraction adopted by @Dominik+2012 is $f_{\rm bin}=0.5$, corresponding to the parameter boundaries specified above. For comparison we also provide estimates based on the old distribution but assuming a high binary fraction of $f_{\rm bin} = 1$ in model O-f1.
The distribution of primaries masses (and single star masses) is identical to the one we adopt in the new reference model, given in Eq \[eq:imf\].
Computational method {#sec:method}
====================
For the purpose of this study we use the binary population synthesis code [StarTrack]{} [@Belczynski+2002; @Belczynski+2008], which has been used extensively to simulate the evolutionary channels and formation rates of compact objects [www.syntheticuniverse.org]{}. The suite of our simulations is available on this website.
This code belongs to a family of very fast and stable binary evolutionary codes, that provide powerful tools to explore the vast multi-dimensional space of the initial parameters that determine the fate of a binary system. These codes also enable the exploration of the effect of model uncertainties. These codes relay on precomputed grids of detailed stellar models and approximate treatments of the physical processes as described below. Despite the approximations, studies with this and similar codes based on the same philosophy have enabled insights into the many exotic phenomena resulting from low and high mass interacting binaries
A full description of the [StarTrack]{} code used in this study and assumptions concerning the treatment of stellar evolution and binary interaction can be found in the papers above @Belczynski+2002 [@Belczynski+2008], @Dominik+2012 and references therein. Below we provide a summary of the main assumptions relevant for this study.
Physical assumptions
--------------------
This study is a comparative study of the impact of the initial conditions. @Dominik+2012 is used as our reference model. We use @Dominik+2012, also when the assumptions are uncertain.
#### Stellar evolution
The code is based on detailed, but non rotating, single stellar evolutionary models by @Pols+1998. For this purpose the code utilizes the fit formula to the detailed models by @Hurley+2000 with several adaptations described in @Belczynski+2002. The effects of mass loss and accretion are simulated using algorithms originally developed by [@Tout+1997; @Hurley+2002] to account for effects such as rejuvenation of the accreting star when appropriate.
#### Stellar wind mass loss
We employ updated wind mass loss rates, which include winds mass loss from early-type stars [@Vink+2001], Wolf-Rayet stars [@Hamann+1998; @Vink+2005] and enhanced mass loss rates for Luminous Blue Variables calibrated to account for the observed distribution of masses of known black holes [@Belczynski+2010; @Orosz+2011]. As a result the simulations allow for the formation of black holes with masses up to 15[$\,M_\odot$]{} in a solar metallicity environment. For sub-solar metallicity environments ($Z =
0.006$), black holes up to 30[$\,M_\odot$]{} are produced, consistent with the mass of the most massive known stellar-origin black hole in the IC10 X-1 system [@Prestwich+2007; @Silverman+2008]. Under these assumptions the simulations predict the formation of black holes with masses up to 80[$\,M_\odot$]{} in low metallicity ($Z=0.0002$) environments for stars below initial mass of $150\msun$.
#### Roche-lobe overflow and common envelope evolution
To determine whether mass transfer through Roche-lobe overflow is stable we consider the stellar type of the donor and mass ratio as outlined in [@Belczynski+2008]. In the case of stable mass transfer, we assume that half of the mass lost by the donor is accreted by the companion. The remainder is leaving the system carrying a specific angular momentum of $2\pi a^2 /P$, where $a$ and $P$ are the orbital separation and period following @Podsiadlowski+1992. Common envelope evolution is accounted for using the classical energy balance formalism [@Webbink1984] adopting a value of $\alpha_{\rm CE} =1$ for the envelope efficiency parameter. The parameter $\lambda$ describing the binding energy of the envelope is taken from fits by @Xu+2010 [@Xu+2010a], implemented as described in @Dominik+2012.
#### Remnant masses for compact objects
To obtain the final mass of the compact objects, we consider the formation of a proto-neutron star and subsequent fall back of material. These are inferred to be a function of the carbon oxygen core mass as outlined in Section 4.2 of @Fryer+2012. The employed scheme utilizes the rapid supernova model that is able to reproduce the apparent mass gap between NSs and BHs [@Belczynski+2012b]. We also account for the formation of neutron stars by electron capture supernovae [e.g. @Nomoto1984; @Podsiadlowski+2004] as outlined in Section 4.4 of @Fryer+2012. For single stars this effectively results in the formation of neutron stars for initial stellar masses above 7.6[$\,M_\odot$]{}. The transition from neutron star to black hole is set at a mass of 2.5[$\,M_\odot$]{} for the resulting compact object. For high metallicity ($Z=0.02$) this makes single stars above initial mass of $21\msun$ to form BHs.
#### Supernova kicks
We account for the classical Boersma-Blaauw kick [@Boersma1961; @Blaauw1961] resulting from mass loss during the explosion. In addition, newly born compact objects receive natal kicks which are given by a Maxwellian distribution with a 1D rms of $\sigma = 265 {\ensuremath{\,\rm{km}\,\rm{s}^{-1}}}$, based on the observed distributions of radio pulsars [@Hobbs+2005]. The kicks are lowered proportional to the amount of fall back as described in Section 4.5 of @Fryer+2012. An exception is made for neutron stars formed through electron capture supernovae for which no natal kicks are assumed in this simulation. As a result of this the mergers of double neutron stars are biased toward neutron stars formed through electron capture supernovae.
#### Submodel A and B
To consider the large uncertainties concerning binaries that begin Roche-lobe overflow while the donor star crosses the Hertzsprung gap, we consider two extreme cases. In submodel A we consider all common envelope formation channels including donor stars in any post main sequence evolutionary stage. In submodel B we explicitly exclude all evolutionary channels where the common envelope is initiated by a Hertzsprung gap donor star. Submodel B can be considered as a more conservative estimate of the merger rates for NS-NS, BH-NS, and BH-BH binaries [@Belczynski+2007a]. A detailed study and revision of criteria for the onset of common envelope will be presented shortly (Belczynski, Pavlovskii & Ivanova, in prep.).
#### Metallicity
With the reach of advanced LIGO/Virgo mergers will originate from all sorts of environments with high and low metallicity. We therefore consider two representative metallicities $Z=0.02$ (typical of solar neighborhood) and $Z=0.002$ (typical of small starburst galaxies). The physically consistent way to derive merger rates would be to start from an appropriate star formation history in the Universe, adopt a model of metallicity evolution with redshift and then calculate broad range of population synthesis models with varying metallicity in order to obtain current rate of mergers [@OShaughnessy+2008; @OShaughnessy+2010; @Dominik+2013; @Dominik+2015]. Given the scope of this study we adopt a simpler approach and take an even mix of these two compositions as a crude representation of metallicity distribution in local Universe [e.g. @Panter+2008]. Although simplified, this is sufficient to judge the overall impact of the change in initial conditions and their associated uncertainties.
#### Difference with respect to @Dominik+2012
Since the publication of the population synthesis calculations by @Dominik+2012, it was found that a technical bug was introduced during one of the annual code updates, related to the treatment of tidal locking. For detached binaries, in which both stars reside well within their Roche lobe, the effect of tides on the stellar and orbital spin is negligible. In such systems, the spins of the stars are typically not synchronized with the orbital motion. As the stars evolve they change their spin period, for example as a result of evolutionary expansion. At some point the spin and orbital period may become comparable. Such a system appears to be synchronized, at least momentarily, even though tides are not effective. The simulations by @Dominik+2012 incorrectly treated these systems as tidally locked, resulting in an incorrect further orbital evolution for specific cases. In general, this resulted in shrinking of the binary system with the overall effect of over predicting the double compact object merger rates.
For majority of the cases the differences are small: within factor of $\sim 2$, i.e. comparable to differences arising from the change of initial conditions. This can be seen when comparing the results before the correction given in Table 2 and 3 of @Dominik+2012, the standard model marked “S” in their paper and our results for the same assumptions, but after correction for the bug, as we provide in our Table 1, marked as the old standard model, “O”.
In two cases, both for solar metallicity submodels B, involving binary merges with black holes the differences are larger. The BH-NS Galactic merger rate was revised from 0.2 [$\,\rm{Myr}^{-1}$]{}[@Dominik+2012] to 0.06 [$\,\rm{Myr}^{-1}$]{}(current study). The BH-BH Galactic merger rate was revised from 1.9 [$\,\rm{Myr}^{-1}$]{}[@Dominik+2012] to 0.22 [$\,\rm{Myr}^{-1}$]{}(current study).
Although the changes for these high metallicity channels are substantial, we repeat that the overall rate is completely dominated by the low metallicity channels. Therefore the impact of this correction on the overall compact merger rates, masses and delay times presented by [@Dominik+2012; @Dominik+2013; @Dominik+2015] is negligible.
{width="\textwidth"}
Computation of the merger rates {#sec:ratesmethod}
-------------------------------
We randomly draw binary systems from the initial distribution functions, once for high metallicity ($Z=0.02$) and once for low metallicity ($Z=0.002$). For computational efficiency we only select binary systems that are massive enough, within some safety margin, to potentially produce a double compact object involving a neutron star or a black hole. For primaries we take $M_1\ge5$[$\,M_\odot$]{} and for secondaries $M_2\ge3$[$\,M_\odot$]{}.
Using the [StarTrack]{} population synthesis code we evolve $N_{\rm sim}
= 2\times 10^6$ of such binary systems. We estimate the mass formed in our simulations. To obtain the total mass in stars down to the lower stellar mass limits we integrate over the full extent of the initial mass function ($0.08-150\msun$) and mass ratio distribution. We include the appropriate mass for single stars when needed. For this we assume that for each binary system with a primary mass $M_1$, there are $(1-f_{\rm bin})/f_{\rm bin}$ single stars with a mass $M = M_1$
We present our rates in terms of the rate for a fiducial Galaxy, ${\cal R}_{\rm gal}$. We chose to adopt the exact same method as @Dominik+2012 to allow for comparison. We therefore assume a $10$Gyr continuous star formation rate $\eta_{\rm SFR} = 3.5{\ensuremath{\,M_\odot}}\,\myr^{-1}$. The resulting total mass formed in stars is within a factor of two of the estimates for the present-day stellar mass in the Milky Way[^2]. The star formation history of our galaxy is not well known [e.g. @Wyse2009] but there is evidence for several discrete epochs of enhanced of star formation [e.g. @Cignoni+2006]. The assumption of continuous star formation may be a rather crude approximation for the Milky Way itself. It may however serve as the rate for a fiducial Milky Way-like galaxy obtained after averaging over the LIGO detection volume.
In Table \[tab:Zdep\] we provide the rates for the different simulations. These rates can be converted into approximate volumetric rates, as used by the LIGO/Virgo collaboration, with the following expression. $${\cal R}_{\rm vol} = 10 \ {\rm yr}^{-1} \ {\rm \Gpc}^{-3}\
\left[
\frac{\rho}{0.01 \Mpc^{-3}}
\right]
\left[
\frac{ {\cal R}_{\rm gal}}{ \myr^{-1} }
\right]$$ where $\rho$ is the local density of Milky Way like galaxies and $R_{\rm gal}$ the Galactic merger rate.
The detailed choices on how to normalize the rates are, to some extent, arbitrary. This is a result of the uncertainties in the low mass binary fraction and the poorly constrained distribution of their mass ratios. We chose to adopt very simple assumptions and we assume for the sake of this study that both binary fraction and mass ratio distribution adopted in a given model for massive stars hold true for the entire considered mass range ($0.08-150\msun$).
Our Monte Carlo simulations are subject to statistical fluctuations due to they finite size. We estimate this by counting the total number of double compact objects that merge with 10 Gyr in each simulation, $N_{\rm x}$, considering NS-NS, BH-NS, BH-BH separately. We use $1/\sqrt {N_x}$ as an approximation for the statistical uncertainty.
Results: distributions {#sec:dist}
=======================
Comparison of the initial distributions \[sec:results:initdist\]
------------------------------------------------------------------
The effective initial distributions for all systems with a primary mass of at least 7[$\,M_\odot$]{} are shown in Fig. \[initdist\_all\] for the old and new assumptions. Systems with lower primary masses are not capable of producing double neutron stars in our simulations. The histograms are shown for a metallicity $Z=0.02$ and normalized to unity. The main differences between the old and new assumption concern (a) the period/separation distribution and (b) the eccentricity distribution.
#### Periods and separations
The main difference between old and new assumptions concerns the tightest systems. The new distributions show a clear preference for short periods / small separations as can be seen in central top and bottom panels of Fig. \[initdist\_all\]. The old distribution is flat for large separations but tight systems ($a \lesssim 100 {\ensuremath{\,R_\odot}}$ ) are strongly suppressed. This is a result of the adopted lower limit $a_{\min}$, which depends on the eccentricity and the stellar radii ( see Sect. \[sec:initdist\]). Such a turnover is not observed for massive binaries. Even though we do not know at present how such close systems form, they exist and must be accounted for in the simulations as we do in the new initial conditions.
#### Eccentricities
The second main difference concerns the eccentricities (depicted in the top right panel of Fig. \[initdist\_all\]). The distribution adopted in the old simulations is the thermal distribution which strongly favors eccentric systems. The new distribution favors (near) circular systems. The high eccentricities in the old simulations allow much wider systems to still interact as discussed below.
#### Periastron separations
The distribution of periastron separations $d_{\rm per} = a(1-e)$, i.e. the distance of closest approach at zero age, is shown in the bottom right panel in Fig. \[initdist\_all\]. To first order this is the most relevant parameter governing when systems start to interact by tides and later mass transfer. The old simulations favor wider and more eccentric systems, while the new simulations favor tighter and more circular systems. By coincidence, these changes partially compensate each other.
The old assumptions resulted in a nearly flat distribution of periastron separations extending from $\log d_{\rm per} (\rsun) = 1$-$5$, being nearly flat between $\log d_{\rm per} (\rsun) = 1.5$-$4$ and turning over on both ends. Instead the new distributions peak at short periastron separations of about $10$-$50\rsun$ and gradually decay for higher separations.
However, in the regime of relevance for the formation of double compact objects ($\log d_{\rm per} (\rsun) = 1.5$-$4$, as we will see in the next section) both, old and new, periastron distribution are rather similar. We note only a slight domination of old distribution in this regime. The distributions in this figure are normalized to unity. The larger overall binary fraction in the new simulations will compensate for this.
#### Primary masses and mass ratios
In both simulations the mass ratios are drawn from a flat seed distribution (the bottom left panel of Fig. \[initdist\_all\]). They only differ in the adopted boundaries. In the old simulation companion masses are drawn down to the hydrogen burning limit. In the new simulations the minimum mass ratio adopted is ${q_{\rm min} = 0.1}$. In our simulations with old initial distributions we found no double compact objects resulting from systems with such extreme mass ratio. This difference therefore only has a very minor effect on the final normalization.
The distributions of initial primary masses $M_1$ (shown in the top left panel of Fig. \[initdist\_all\]) are identical in both simulations. The only differences are stochastical in nature.
Comparison of the birth properties of the progenitors of double compact object mergers \[sec:progdist\]
-------------------------------------------------------------------------------------------------------
{width="\textwidth"} {width="\textwidth"}
{width="65.00000%"}
{width="50.00000%"}{width="50.00000%"}
Only a very small subset of the simulated binaries produces two compact objects that are bound and that will merge within 10Gyr. The birth properties of this subset are shown in Fig. \[initdist\_progenitors\], where we compare the impact of the old and new assumptions for the initial conditions.
Note that the histograms are normalized to unity to allow comparison of the shape of the distribution, the resulting rates are discussed below. We separately show results for low metallicity (Z = 0.002, top panels) and high metallicity (Z = 0.02, bottom panels).
Since the trends observed when comparing the old and new initial distributions are very similar for submodel A and B, we chose to only show the results submodel B here. This is the more conservative submodel which excludes progenitor systems in which a Hertzsprung gap star acts as donor star during CE phase, see Section 3.1. In practice this means that on average the progenitors in model B result from wider systems and show a slightly stronger preference for equal mass progenitor systems. Data for submodel A (as well as data subdivided by the double compact object type) can be obtained from the online data repository for readers that are interested. More insight in the differences between submodel A and B can be found in [@Dominik+2012].
#### Primary masses
The distribution of initial primary masses of the progenitors consists of separate components. The narrow low mass peak around $10\msun$ is generated by the progenitors of double neutron stars. The high mass component is only prominent in the low metallicity simulations. This component consists of the progenitors of mergers that involve a black hole.
The minimum initial mass to form a black hole is $40\msun$ for the primary star of a double compact object progenitor at $Z=0.02$. This is much higher than the minimum mass for a single star (or star in a very wide non interacting binary) to form a black hole in our simulations, which is $21\msun$ at $Z=0.02$. The difference is the result of the additional mass loss experienced in binary systems due to Roche lobe overflow and common envelope ejection from BH progenitor. At lower metallicity these minimum masses decrease. At $Z=0.002$ we find a minimum initial mass of about $20\msun$ for the primary star of a double compact object progenitor involving a black hole.
The difference between the old and new simulations is very small. At low metallicity there is a slight tendency towards lower mass progenitors (favoring NS-NS progenitors) in the new simulations.
#### Mass ratios
For both the old and new simulations we find that the progenitors exclusively result from systems with mass ratios larger than about $M_2/M_1 \gtrsim 0.4$ at low metallicity and about 0.6 at high metallicity. For both metallicities there is a preference for progenitors with very equal masses, i.e. 0.9 and higher.
The difference between the old and new simulations is marginal, unsurprisingly given the similarity of the input distribution. A very slight tendency for the old simulations to favor more equal mass systems is observed at high metallicity, while the opposite behavior is observed at low metallicity. However, the differences are insignificant in comparison with the model uncertainties.
#### Periods, separations and eccentricities
The initial period and separation distributions of the progenitor systems show a clear difference between the old and new initial conditions. For both metallicities the old simulations are biased towards wider systems than the new simulations. At high metallicity the distributions peak near $150\rsun$ in the new simulations and near $300\rsun$ in the old simulations. At low metallicity the distribution is clearly bimodal peaking around $800$ and $8000\rsun$ in the new simulations.
The distribution of initial eccentricities of merger progenitors closely reflects the shape of the input distribution of the population, although in both cases systems with larger eccentricities are over represented.
#### Periastron separations
Although the distributions of initial separations $a$ and eccentricities $e$ are very different between the old and new simulations, the distribution of initial periastron separations $d_{\rm per}$ of merger progenitors is nearly indistinguishable.
For low metallicity the distribution is again bimodal. Detailed inspection of the evolutionary paths of the systems that populate both peaks show that the tighter systems first evolve through a stable mass transfer phase. The wider systems first evolve through a common envelope phase. These evolutionary channels are identified and described in earlier work ([@Dominik+2012]; see their Table 5 and associated text).
Both channels exist in the old and new simulations, since the input physics assumptions are not changed. The similarity in the periastron separation distributions implies that the relative contribution of both channels did not significantly change when switching from the old to new assumptions.
Distributions of final properties
-----------------------------------
The distribution of the final properties predicted by our simulations that are of interest for gravitational wave searches include the delay times, component masses, mass ratios and chirp masses. Detailed models of the anticipated waveforms will enable the inference on several parameters, such as component masses [@Aasi+2013b; @Veitch+2015] and if multiple events are detected constraints on the properties of the population can be inferred [@Mandel2010; @Mandel+2015].
We find that the distributions of final properties are remarkably insensitive to the assumed initial conditions. The predicted distribution of the component masses, mass ratios, chirp masses and delay times show no significant changes when the initial conditions are varied within current uncertainties. They are almost identical to those presented by [@Dominik+2012], to which we refer for detailed discussion.
The insensitivity of rate predictions results from the (coincidental) similarity of the initial periastron separation distribution and the mass ratio distribution, which remained practically unchanged. These two parameters are key in determining the fate of the binary system (and thus merger rates). However, the physical properties of merging binaries are not sensitive to changes in initial conditions.
There are relatively few formation channels for double compact object mergers. For a given model of evolution, these formation channels originate from a rather narrow volume of initial parameter space (“the formation volume”). Change of initial conditions alters the number of the progenitor binaries in the formation volume, but has very little effect on the binary merger properties. This insensitivity is not just connected to the specific initial conditions considered here, but is of more fundamental nature. It is a robust result that has also been found in connection to other compact object binaries. For example by @Kalogera+1998 in the context of low-mass X-ray binaries, and see also @Belczynski+2002.
The robustness of the final distributions against initial condition uncertainties considered here, but also wider variations, is reassuring in the light of anticipated gravitational wave detections. It removes one layer of uncertainties and allows for more optimism concerning the potential of such detections for constraining more interesting aspects such as the physics uncertainties.
The final properties of our simulations are available through our online data files available at ([www.syntheticuniverse.org]{}).
Results: Merger Rates {#sec:rates}
=====================
In Table \[tab:Zdep\] we list the double compact object merger rates for our new standard model N, the old standard model O, and variations on the models that explore the effect of uncertainties in initial distributions. All rates are expressed as the rate for a fiducial Milky Way Galaxy as detailed in Sect \[sec:ratesmethod\]. We give the results both for submodel A and the more conservative submodel B. Merger rates are also presented separately for low ($Z=0.002$) and high metallicity ($Z=0.02$) stellar and binary evolution.
In Table \[tab:Mix\] we present merger rates that correspond to a fiducial Galaxy with an even mix ($50\%$–$50\%$) of low-metallicity ($Z=0.002$) and high-metallicity ($Z=0.02$) stars. Absolute mixed merger rates are given only for our new standard model (an average of low- and high-metallicity fiducial Galactic rates from Table \[tab:Zdep\]) and can be readily obtained for any other model. For all other models only relative change of rates with respect to our new standard is given. A graphical representation of the changes relative to the new standard model is given in Figure \[fig:uncertainties\].
The overall conclusions are: [*(i)*]{} merger rates increase slightly with the new assumptions, and they are within a factor of $2$ of old standard rates; [*(ii)*]{} merger rates change by less than factor of $2$ within associated uncertainties embedded in new initial distributions concerning binary parameters (the only exception being the IMF). These conclusions hold for all types of double compact object mergers (NS-NS, BH-NS and BH-BH) and the changes are typically much smaller than the maxima listed above. In the following section we discuss the various trends.
Impact of the new initial conditions
------------------------------------
The adoption of the new assumptions for the initial conditions leads to higher estimates for all merger rates. The largest change is found when adopting model A. The old rates are 39% lower than the new rates for NS-NS mergers, 29% lower for BH-NS mergers and 27% lower for BH-BH mergers. An overview of the relative changes is given in Tab. \[tab:Mix\] using the new standard rate as reference.
When adopting submodel B the observed changes are even smaller in the case of BH-NS and BH-BH mergers. The submodel B excludes progenitor channels which involve common envelope evolution with a Hertzsprung gap donor [@Belczynski+2007a]. These typically come from progenitor binaries with smaller initial separations where the old and new distributions deviate most strongly. Excluding these progenitors reduces the difference between old and new simulations. For the BH-NS in submodel B the changes are so small that they are of the same order as the expected statistical fluctuations resulting from the finite number of systems in our simulations. These average statistical fluctuations are shown in Figure \[fig:uncertainties\] as an errorbars.
The main reason for the slight increase in the merger rate is the large binary fraction adopted in the new initial conditions ($f_b=1$). The old distributions with the high binary fraction of $f_b =1$ (model O-f1) give merger rates that are higher (up to $\sim 50\%$ in the case of BH-NS mergers in submodel B) than for the new standard model (see Table 2).
Variations due to uncertainties in the new initial conditions
-------------------------------------------------------------
All relative merger rate changes are smaller than $34\%$ for [*all*]{} types of double compact objects when varying the initial conditions concerning binary properties (models N-p1, N-p2, N-q1, N-q2, N-e1, N-e2, N-f1, N-f2; see Tab. \[tab:Mix\]). This is not true for variations in the IMF, which we will discuss at the end of this paragraph.
The initial distribution of mass ratios that rather steeply falls off with mass ratio (model N-q2) is the cause of merger rate decrease of the order of $\sim 30\%$ for NS-NS and BH-BH mergers. A distribution favoring equal mass systems at birth mildly favors the formation of NS-NS systems and BH-BH systems. The BH-NS mergers show the same trend, but the changes are comparable to the Poison fluctuations. Varying the initial period distribution has a smaller effect, below $20\%$ and approaching our statistical uncertainties. A flatter period distributions, i.e. one that favors wider systems (model N-p2), results in a more effective formation of compact object mergers. A period distribution favoring short period systems more strongly (model N-p1) mainly adds systems that interact or merge prematurely, preventing the formation of a double compact object system. Varying the eccentricity distribution within the ranges given by @Sana+2012 has no significant effect on the merger rates. All variations found are less than $5\%$ for all submodels and all types of mergers considered. These variations are dominated by the statistical fluctuations in our Monte Carlo simulations. A reduction of the total binary fraction to $f_{\rm bin} = 0.85$ [which roughly corresponds to the one sigma lower limit of intrinsic spectroscopic binary fraction, $f_{\rm sp} = 0.7 \pm 0.1$, derived by @Sana+2012] leads to a reduction of all compact object merger rates by $10\%$ (model N-f1). A further reduction of the total binary fraction to $f_{\rm bin} = 0.7$ (model N-f2), which roughly corresponds two the two sigma lower limit on the observational constraints, reduces the merger rates by $15\%$.
Significant change of the power law exponent of the IMF for stars more massive than $1\msun$ generates the largest change of double compact merger rates in our suite of models. We allow for a change of the exponent by $0.5$ up and down from our standard value $\alpha=2.7$. Note that this affects the mass distribution of the primary stars in binary systems and single stars for those models where $f_{\rm bin} < 1$ (model O-f1, N-f1, N-f2). Merger rates decrease within a factor of $2.7$ for NS-NS mergers, a factor of $4.8$ for BH-NS mergers and a factor of $5.8$ for BH-BH mergers for our most steep IMF (model N-m1) in respect to our new standard model (N). Merger rates increase within a factor of $2.2$ for NS-NS mergers, a factor of $4.1$ for BH-NS mergers and a factor of $4.7$ for BH-BH mergers for our most flat IMF (model N-m2). These changes are due to two effects. First, the steeper the IMF the fewer double compact objects form in a population of binary stars. The heaviest BH-BH mergers are affected the most. Second, the slope of high-mass IMF has a significant impact on the total mass of the fiducial Galaxy. For the same number of massive binary stars that we draw from initial distributions ($N_{\rm sim}=2 \times 10^6$) the corresponding total mass of simulation (over the entire stellar mass range) is $\sim 2.5$ higher and $\sim 2.5$ smaller for our model with the steep and flat IMF, respectively as compared with our adopted standard IMF.
[l cc | cc | cc c cc | cc | cc r]{} &&&&& &&&&&&&\
N & 12.8 & 3.68 & 4.45 & 2.32 & 98.3 & 15.6 & & 36.9 & 12.5 & 1.77 & 0.06 & 9.73 & 0.22 & new standard\
&&&&& &&&&&&&\
O & 8.17 & 2.43 & 3.17 & 2.24 & 72.2 & 13.3 && 22.3 & 7.82 & 1.24 & 0.01 & 6.98 & 0.22 & old standard\
O-f1 & 13.2 & 3.92 & 5.11 & 3.60 & 116 & 21.5 && 35.9 & 12.6 & 1.99 & 0.02 & 11.3 & 0.36 & old, $f_{\rm bin}=1.0$\
&&&&& &&&&&&&\
N-p1 & 12.5 & 3.34 & 4.72 & 2.23 & 90.8 & 13.3 && 35.9 & 11.6 & 1.46 & 0.05 & 9.46 & 0.29 & new, $(\log p)^{-0.75}$\
N-p2 & 13.4 & 3.67 & 4.64 & 2.75 & 102 & 17.5 && 37.3 & 12.6 & 2.00 & 0.05 & 9.87 & 0.20 & new, $(\log p)^{-0.35}$\
&&&&& &&&&&&&\
N-q1 & 16.3 & 4.28 & 5.15 & 2.53 & 113 & 18.8 && 44.7 & 15.6 & 1.48 & 0.07 & 11.7 & 0.24 & new, $q^{0.5}$\
N-q2 & 8.65 & 2.52 & 4.00 & 2.23 & 69.8 & 11.1 && 25.7 & 8.17 & 1.32 & 0.06 & 6.84 & 0.13 & new, $q^{-0.7}$\
&&&&& &&&&&&&\
N-e1 & 13.2 & 3.71 & 4.49 & 2.40 & 99.0 & 16.0 && 38.8 & 12.9 & 1.70 & 0.06 & 9.93 & 0.22 & new, $e^{-0.59}$\
N-e2 & 12.7 & 3.41 & 4.66 & 2.37 & 95.3 & 15.7 && 35.5 & 12.0 & 1.57 & 0.04 & 9.90 & 0.12 & new, $e^{-0.25}$\
&&&&& &&&&&&&\
N-f1 & 11.6 & 3.34 & 4.03 & 2.09 & 89.1 & 14.1 && 33.4 & 11.4 & 1.60 & 0.06 & 8.84 & 0.20 & new, $f_{\rm bin}=0.85$\
N-f2 & 10.9 & 3.13 & 3.78 & 1.97 & 83.4 & 13.2 && 31.3 & 10.6 & 1.51 & 0.05 & 8.26 & 0.19 & new, $f_{\rm bin}=0.7$\
&&&&& &&&&&&&\
N-m1 & 4.70 & 1.43 & 0.99 & 0.49 & 17.2 & 2.77 && 13.4 & 4.65 & 0.31 & 0.01 & 1.54 & 0.04 & new, ${m_1}^{-3.2}$\
N-m2 & 26.5 & 7.80 & 17.9 & 9.51 & 457 & 66.9 && 81.3 & 26.9 & 7.86 & 0.32 & 51.4 & 1.19 & new, ${m_1}^{-2.2}$\
\[tab:Zdep\]
[l cc | cc | cc | c r]{} N & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & new standard\
& [**(24.85)**]{} & [**(8.090)**]{} & [**(3.110)**]{} & [**(1.190)**]{} & [**(54.02)**]{} & [**(7.910)**]{} & [ absolute rates \[Myr$^{-1}$\]]{}\
&&&&&&\
O & 0.613 & 0.633 & 0.709 & 0.945 & 0.733 & 0.855 & old standard\
O-f1 & 0.988 & 1.021 & 1.141 & 1.521 & 1.178 & 1.382 & old, $f_{\rm bin}=1.0$\
&&&&&&\
N-p1 & 0.974 & 0.923 & 0.994 & 0.958 & 0.928 & 0.859 & new, $(\log p)^{-0.75}$\
N-p2 & 1.020 & 1.006 & 1.068 & 1.176 & 1.036 & 1.119 & new, $(\log p)^{-0.35}$\
&&&&&&\
N-q1 & 1.227 & 1.229 & 1.066 & 1.092 & 1.154 & 1.204 & new, $q^{0.5}$\
N-q2 & 0.691 & 0.661 & 0.855 & 0.962 & 0.709 & 0.710 & new, $q^{-0.7}$\
&&&&&&\
N-e1 & 1.046 & 1.027 & 0.995 & 1.034 & 1.008 & 1.025 & new, $e^{-0.59}$\
N-e2 & 0.970 & 0.952 & 1.002 & 1.013 & 0.974 & 1.000 & new, $e^{-0.25}$\
&&&&&&\
N-f1 & 0.905 & 0.911 & 0.905 & 0.903 & 0.907 & 0.904 & new, $f_{\rm bin}=0.85$\
N-f2 & 0.849 & 0.849 & 0.850 & 0.849 & 0.848 & 0.846 & new, $f_{\rm bin}=0.7$\
&&&&&&\
N-m1 & 0.364 & 0.376 & 0.209 & 0.210 & 0.173 & 0.178 & new, ${m_1}^{-3.2}$\
N-m2 & 2.169 & 2.145 & 4.141 & 4.130 & 4.706 & 4.304 & new, ${m_1}^{-2.2}$\
\[tab:Mix\]
Discussion and Conclusion {#sec:concl}
=========================
Recently, [@Sana+2012] have presented measurements of initial binary parameters for massive stars. These measurements, in some aspects, are in stark contrast with so far used assumptions in evolutionary studies of massive stars [e.g. @Belczynski+2002; @Voss+2003; @Dominik+2012; @Mennekens+2014]. The old standard prescriptions (a flat distribution in log separation, a thermal eccentricity distribution strongly favoring eccentric systems and typically an adopted binary fraction of 50% counting systems with separations up to $10^5$[$\,R_\odot$]{}) may still apply for low- and intermediate-mass stars, but the new measurements show very different initial binary parameter distributions for massive stars (strong overabundance of very tight systems, a binary fraction that reaches $100\%$ when including systems up to about $10^{4.5}\rsun$ and mostly circular systems).
We find a small increase in the NS-NS, BH-NS and BH-BH merger rate as a result of the new initial conditions (due to the increased binary fraction), with the old rates being up to $40$% lower than the new rates. Secondly, we study the impact of the allowed variations of the new binary parameter distributions within the observational uncertainties. We find that they have negligible impact (typical variations of a few percent, always less than a factor of $2$). All changes are negligible in comparison with the evolutionary model uncertainties [which are typically $1$-$2$ orders of magnitude; e.g., @Dominik+2012].
The only exception to our above conclusion are the variations of double compact merger rates due to the uncertainty in the slope of the IMF for stars more massive than $1\msun$. These variations can be as high as factor of $\sim 3, 4, 6$ up and down for NS-NS, BH-NS and BH-BH mergers, respectively from our new reference model that employs the best estimates on the initial conditions for massive binaries. This is due to the change of the overall mass of the stellar mass content which is used to normalize the rates and to the diminishing number of double compact object progenitors with the increasing steepness of the IMF.
The small impact of the new distributions of binary properties on the merger rates is somewhat counter-intuitive. This result stems from the fact that major changes in the initial binary parameter distributions cancel out in the regime important for close double compact object formation. Although, new distributions show a much stronger bias toward close binary orbits in comparison to the old distributions, the periastron distance distribution is similar to the one generated by old distributions in the regime $1.5<\log(d_{\rm per}/\rsun)<4$ (see Fig. \[initdist\_all\]). This is because of the strong preference for very eccentric systems adopted in the older distributions, for which there is no supported in the new observations (mostly circular orbits). This regime of periastron separations plays a crucial role in the formation of stellar-mass double compact object mergers (see Fig. \[initdist\_progenitors\]). The periastron distance is key in determining when tides become important and when the primary star fills its Roche lobe for the first time. Since the distribution of component masses is very similar, it is the periastron separation that largely determines the future evolution of the binary system. It happens (by pure coincidence) that the old prescriptions and the new measurements result in rather similar orbital separations at the time of the first mass transfer for progenitors of double compact objects. Additionally, the observational uncertainties of the new estimates of initial binary parameters for massive stars are not large enough to cause significant NS-NS, BH-NS and BH-BH merger rate changes. Similarly, we find that the distributions of final properties (delay-times and final component masses) are insensitive to the variations of initial conditions. The summary of our merger rate analysis is given in Fig. \[fig:uncertainties\].
Our study allows to eliminate one layer of uncertainties involved in population synthesis predictions for double compact object mergers. Such predictions are infamous for their number of poorly constrained model parameters. Recent years of intensive studies of double compact object formation have proven that it is extremely hard (yet, not impossible) to identify a well posed non-degenerate problem with population synthesis. This fact does not reflect the intrinsic weakness of the population synthesis method, but rather the lack of understanding of some basic processes involved in the evolution of massive single (e.g., convection, rotation, supernova/core collapse) and binary stars (e.g., tidal interactions, common envelope evolution).
The reduction of the modeling uncertainties, like the one presented here, is crucial for advancement of our understanding of the remaining poorly constrained physical processes involved in double compact object formation. The further reduction of the uncertainties may potentially allow gravitational-wave observatories to assess and constrain physical processes that are thus far beyond the reach of electromagnetic observations and theoretical studies.
We would like to thank Hugues Sana, Ilya Mandel, Cole Miller, Matteo Cantiello and Colin Norman for many stimulating discussion that lead to this paper. We would also like to thank the anonymous referee, Ylva G[ö]{}tberg, Abel Schootemeijer and Manos Zapartas for their feedback on the manuscript.
The authors acknowledge Carnegie Observatories for providing the computational resources, the Caltech LIGO center for support of an extended visit by KB and the hospitality of University of Washington during 2014 INT workshop. The authors acknowledge support by the European Council (SdM) through a Marie Sklodowska-Curie Reintegration Fellowship (SdM), H2020-MSCA-IF-2014, project id 661502, by NASA through an Einstein Fellowship grant (SdM), PF3-140105, the Polish Science Foundation Master 2013 Subsidy (KB), Polish NCN grant Sonata Bis 2, DEC-2012/07/E/ST9/01360 (KB), the National Science Foundation Grant No. PHYS-1066293 (KB), the hospitality of the Aspen Center for Physics (KB) and a grant from the Simons Foundation (KB).
[^1]: This estimate was done with the [StarTrack]{} population synthesis code described in Section 3 considering systems with primary masses in the range $15-150\msun$. The initial distributions of period and eccentricity were altered within limits listed in Section 2.1 assuming a binary fraction of $100\%$.
[^2]: Estimates for the total stellar mass by @Flynn+2006 yields $4.85 -5.5 \times 10^{10}{\ensuremath{\,M_\odot}}$ and $6.43 \pm 0.63 \times 10^{10}{\ensuremath{\,M_\odot}}$ by @McMillan2011.
|
---
abstract: 'In many real-world settings, we are interested in learning invariant and equivariant functions over nested or multiresolution structures, such as a set of sequences, a graph of graphs, or a multiresolution image. While equivariant linear maps and by extension multilayer perceptrons (MLPs) for many of the individual basic structures are known, a formalism for dealing with a hierarchy of symmetry transformations is lacking. Observing that the transformation group for a nested structure corresponds to the “wreath product” of the symmetry groups of the building blocks, we show how to obtain the equivariant map for hierarchical data-structures using an intuitive combination of the equivariant maps for the individual blocks. To demonstrate the effectiveness of this type of model, we use a hierarchy of translation and permutation symmetries for learning on point cloud data, and report state-of-the-art on [[<span style="font-variant:small-caps;"></span>]{}]{} and [[<span style="font-variant:small-caps;"></span>]{}]{}, two of the largest real-world benchmarks for 3D semantic segmentation.'
author:
- |
Renhao Wang, Marjan Albooyeh[^1]\
University of British Columbia,\
`{renhaow,albooyeh}@cs.ubc.ca`\
Siamak Ravanbakhsh\
McGill University & Mila\
`siamak@cs.mcgill.ca`\
bibliography:
- 'refs.bib'
title: Equivariant Maps for Hierarchical Structures
---
Introduction {#intro}
============
In designing deep models for structured data, equivariance (invariance) of the model to transformation groups has proven to be a powerful inductive bias, which enables sample efficient learning. A widely used family of equivariant deep models constrain the feed-forward layer so that specific transformations of the input lead to the corresponding transformations of the output. A canonical example is the convolution layer, in which the constrained MLP is equivariant to translation operations. Many recent works have extended this idea to design equivariant networks for more exotic structures such as sets, exchangeable tensors and graphs, as well as relational and geometric structures.
This paper considers a nested hierarchy of such structures, or more generally, any hierarchical composition of transformation symmetries. These hierarchies naturally appear in many settings: for example, the interaction between nodes in a social graph may be a sequence or a set of events. Or in diffusion tensor imaging of the brain, each subject may be modeled as a set of sequences, where each sequence is a fibre bundle in the brain. The application we consider in this paper models point clouds as 3D images, where each voxel is a set of points with coordinates relative to the center of that voxel.
![[]{data-label="fig:demo"}](figures/wreath_hierarchy2.pdf){width=".9\textwidth"}
To get an intuition for a hierarchy of symmetry transformations, consider the example of a sequence of sequences – [*e.g.*, ]{}a text document can be viewed as a sequence of sentences, where each sentence is itself a sequence of words. Here, each inner sequence as well as the outer sequence is assumed to possess an “[independent]{}” translation symmetry. Contrast this with symmetries of an image (2D translation), where all inner sequences (say row pixels) translate together, so we have a total of two translations. This is the key difference between the *wreath product* of two translation groups (former) and their *direct product* (latter). It is the wreath product that appears in nested structures. As is evident from this example, the wreath product results in a significantly larger set of transformations, and therefore provides a stronger inductive bias.
We are interested in application of equivariant/invariant deep learning to this type of nested structure. The building blocks of equivariant and invariant MLPs are *equivariant linear maps* of the feedforward layer; see \[fig:demo\]. In particular, we show how to obtain the closed form of the equivariant linear map for the hierarchical structure from equivariant maps for the individual symmetry group. We observe that the number of independent linear operators grows with the “sum” of independent operators for individual building blocks. In contrast, this number grows with the “product” of independent operators on blocks when using direct product, confirming the stronger bias in the wreath product setting.
In the following, after discussing related works in \[sec:related\], we give a short background on equivariant MLPs in \[sec:preliminaries\]. \[sec:method\] starts by giving the closed form of equivariant maps for direct product of groups before moving to the more difficult case of wreath product in \[sec:wreath\]. Finally, \[sec:application\] applies this idea to impose a hierarchical structure on 3D point clouds. We show that the equivariant map for this hierarchical structure achieves state-of-the-art performance on the largest benchmark datasets for 3D semantic segmentation.
Related Works {#sec:related}
=============
Group theory has a long history in signal processing [@holmes1987mathematical], where in particular Fourier transformation and group convolution for Abelian groups have found tremendous success over the past decades. However, among non-commutative groups, wreath product constructions have been the subject of few works. @rockmore1995fast give efficient procedures for fast Fourier transforms for wreath products. In a series of related works @foote2000wreath [@mirchandani2000wreath] investigate the wreath product for multi-resolution signal processing. The focus of their work is on the wreath product of cyclic groups for compression and filtering of image data.
Group theory has also found many applications in machine learning [@kondor2008group], and in particular deep learning. Design of invariant MLPs for general groups goes back to @shawe1989building. More recently, several works investigate the design of equivariant networks for general finite [@ravanbakhsh_symmetry] and infinite groups [@cohen2016group; @kondor2018generalization; @cohen2019general]. In particular, use of the wreath product for design of networks equivariant to a hierarchy of symmetries is briefly discussed in [@ravanbakhsh_symmetry]. Equivariant networks have found many applications in learning on various structures, from image and text [@lecun1995convolutional], to sets [@zaheer2017deep; @qi2017pointnet], exchangeable matrices [@hartford2018deep], graphs [@kondor2018covariant; @maron2018invariant; @albooyeh2019incidence], and relational data [@graham2019deep], to signals on spheres [@cohen2018spherical; @kondor2018clebsch]. A large body of works investigate equivariance to Euclidean isometries; [*e.g.*, ]{} [@dieleman2016exploiting; @thomas2018tensor; @weiler2019general].
When it comes to equivariant models for compositional structures, contributions have been sparse, with most theoretical work focusing on semidirect product (or more generally using induced representations from a subgroup) to model tensor-fields on homogeneous spaces [@cohen2016steerable; @cohen2019general]. Direct product of symmetric groups have been used to model interactions across sets of entities [@hartford2018deep] and in generalizations to relational data [@graham2019deep]. In a recent work, [@Maron2020OnLS] study equivariant networks for sets of symmetric elements; however, they use direct product rather than wreath product discussed here. In the following we specifically contrast the use of direct product with wreath product for compositional structures.
Preliminaries {#sec:preliminaries}
=============
Group Action
------------
A group ${{\ensuremath{\mathcal{G}}}\xspace}= \{{{\ensuremath{\mathcal{g}}}}\}$ is a set equipped with a binary operation, such that the set is closed under this operation, ${{\ensuremath{\mathcal{g}}}}{{\ensuremath{\mathcal{h}}}}\in {{\ensuremath{\mathcal{G}}}\xspace}$, the operation is associative, ${{\ensuremath{\mathcal{g}}}}({{\ensuremath{\mathcal{h}}}}{{\ensuremath{\mathcal{k}}}}) = ({{\ensuremath{\mathcal{g}}}}{{\ensuremath{\mathcal{h}}}}) {{\ensuremath{\mathcal{k}}}}$, there exists identity element ${\ensuremath{\mathcal{e}}} \in {{\ensuremath{\mathcal{G}}}\xspace}$, and a unique inverse for each ${{\ensuremath{\mathcal{g}}}}\in {{\ensuremath{\mathcal{G}}}\xspace}$ satisfying ${{\ensuremath{\mathcal{g}}}}{{\ensuremath{\mathcal{g}}}}^{-1} = {\ensuremath{\mathcal{e}}}$. The action of ${{\ensuremath{\mathcal{G}}}\xspace}$ on a finite set ${{\ensuremath{\mathbb{N}}}}$ is a function $\alpha: {{\ensuremath{\mathcal{G}}}\xspace}\times {{\ensuremath{\mathbb{N}}}}\to {{\ensuremath{\mathbb{N}}}}$ that transforms the elements of ${{\ensuremath{\mathbb{N}}}}$, for each choice of ${{\ensuremath{\mathcal{g}}}}\in {{\ensuremath{\mathcal{G}}}\xspace}$; for short we write ${{\ensuremath{\mathcal{g}}}}\cdot n$ instead of $\alpha({{\ensuremath{\mathcal{g}}}}, n)$. Group actions preserve the group structure, meaning that the transformation associated with the identity element is identity ${\ensuremath{\mathcal{e}}} \cdot n = n$, and composition of two actions is equal to the action of the composition of group elements $({{\ensuremath{\mathcal{g}}}}{{\ensuremath{\mathcal{h}}}}) \cdot n = {{\ensuremath{\mathcal{g}}}}\cdot ({{\ensuremath{\mathcal{h}}}}\cdot n)$. Such a set, with a ${{\ensuremath{\mathcal{G}}}\xspace}$-action defined on it is called a ${{\ensuremath{\mathcal{G}}}\xspace}$-set. The group action on ${{\ensuremath{\mathbb{N}}}}$ naturally extends to ${{\ensuremath{\mathbf{\MakeLowercase{{x}}}}}}\in {\mathds{R}}^{|{{\ensuremath{\mathbb{N}}}}|}$, where it defines a permutation of indices ${{\ensuremath{\mathcal{g}}}}\cdot (x_1,\ldots,x_N) {\ensuremath{\doteq}}(x_{{{\ensuremath{\mathcal{g}}}}\cdot 1}, \ldots, x_{{{\ensuremath{\mathcal{g}}}}\cdot N})$. We often use a permutation matrix ${{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}} \in \{0,1\}^{N \times N}$ to represent this action – that is ${{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}} {{\ensuremath{\mathbf{\MakeLowercase{{x}}}}}}= {{\ensuremath{\mathcal{g}}}}\cdot {{\ensuremath{\mathbf{\MakeLowercase{{x}}}}}}$.
Equivariant Multilayer Perceptrons
----------------------------------
A function $\phi: {\mathds{R}}^{{\ensuremath{\mathbb{N}}}}\to {\mathds{R}}^{{\ensuremath{\mathbb{M}}}}$ is equivariant to a given actions of group ${{\ensuremath{\mathcal{G}}}\xspace}$ iff $\phi({{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}} {{\ensuremath{\mathbf{\MakeLowercase{{x}}}}}}) = \tilde{{{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}} \phi({{\ensuremath{\mathbf{\MakeLowercase{{x}}}}}})$ for any ${{\ensuremath{\mathbf{\MakeLowercase{{x}}}}}}\in {\mathds{R}}^N$ and ${{\ensuremath{\mathcal{g}}}}\in {{\ensuremath{\mathcal{G}}}\xspace}$. That is, a symmetry transformation of the input results in the corresponding symmetry transformation of the output. Note that the action on the input and output may in general be different. In particular, when $\tilde{{{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}} = {{\ensuremath{{\mathbf{\MakeUppercase{{I}}}}}}}_{{\ensuremath{\mathbb{M}}}}$ for all ${{\ensuremath{\mathcal{g}}}}$ – that is, the action on the output is trivial – equivariance reduces to invariance. Here, ${{\ensuremath{{\mathbf{\MakeUppercase{{I}}}}}}}_{{\ensuremath{\mathbb{M}}}}$ is the $M \times M$ identity matrix. For simplicity and motivated by practical design choices, in the following we assume the same action on the input and output.
For a feedforward layer $\phi: {{\ensuremath{\mathbf{\MakeLowercase{{x}}}}}}\mapsto \sigma({{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}{{\ensuremath{\mathbf{\MakeLowercase{{x}}}}}})$, where $\sigma$ is a point-wise non-linearity and ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}\in {\mathds{R}}^{N \times N}$, the equivariance condition above simplifies to commutativity condition ${{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}} {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}= {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}{{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}} \forall {{\ensuremath{\mathcal{g}}}}\in {{\ensuremath{\mathcal{G}}}\xspace}$. This imposes a symmetry on ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}$ in the form of parameter-sharing [@wood1996representation; @ravanbakhsh_symmetry]. While we can use computational means to solve this equation for any finite group, an efficient implementation requires a closed form solution. Several recent works derive the closed form solutions for interesting groups and structures. Note that in general the feedforward layer may have multiple input and output channels, with identical ${{\ensuremath{\mathcal{G}}}\xspace}$-action on each channel. This only require replicating the parameter-sharing pattern in ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}$, for each combination of input and output channel. An Equivariant MLP is a stack of equivariant feed-forward layers, where the composition of equivariant layers is also equivariant. Therefore, our task in building MLPs equivariant to finite group actions is reduced to finding equivariant linear maps in the form of parameter-sharing matrices satisfying ${{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}} {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}= {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}{{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}} \forall {{\ensuremath{\mathcal{g}}}}\in {{\ensuremath{\mathcal{G}}}\xspace}$; see \[fig:param\_sharing\](a.1,a.2) for ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}$ that are equivariant to circular translation (a.1) and symmetric group ${{\ensuremath{\mathcal{S}}}}_4$; the group of all permutations of 4 objects (a.2).
Equivariant Map for Product Groups {#sec:method}
==================================
In this section we formalize the *imprimitive action* of the wreath product, which is used in describing the symmetries of hierarchical structures. We then introduce the closed form of linear maps equivariant to the wreath product of two groups. With hierarchies of more than two levels, one only needs to iterate this construction. To put this approach in perspective and to make the distinction clear, first we present the simpler case of direct product.
[r]{}[0.3]{} {width="30.00000%"}
Equivariant Linear Maps for Direct Product of Groups {#sec:direct}
----------------------------------------------------
The easiest way to combine two groups is through their [direct product]{} ${{\ensuremath{\mathcal{G}}}\xspace}= {{\ensuremath{\mathcal{H}}}}\times {{\ensuremath{\mathcal{K}}}}$. Here, the underlying set is the Cartesian product of the input sets and group operation is $({{\ensuremath{\mathcal{h}}}}, {{\ensuremath{\mathcal{k}}}}) ({{\ensuremath{\mathcal{h}}}}', {{\ensuremath{\mathcal{k}}}}') {\ensuremath{\doteq}}({{\ensuremath{\mathcal{h}}}}{{\ensuremath{\mathcal{h}}}}', {{\ensuremath{\mathcal{k}}}}{{\ensuremath{\mathcal{k}}}}')$. If ${{\ensuremath{\mathbb{P}}}}$ is an ${{\ensuremath{\mathcal{H}}}}$-set and ${{\ensuremath{\mathbb{Q}}}}$ a ${{\ensuremath{\mathcal{K}}}}$-set then the group ${{\ensuremath{\mathcal{G}}}\xspace}= {{\ensuremath{\mathcal{H}}}}\times {{\ensuremath{\mathcal{K}}}}$ naturally acts on ${{\ensuremath{\mathbb{N}}}}= {{\ensuremath{\mathbb{P}}}}\times {{\ensuremath{\mathbb{Q}}}}$ using $({{\ensuremath{\mathcal{h}}}}, {{\ensuremath{\mathcal{k}}}}) \cdot (p,q) {\ensuremath{\doteq}}({{\ensuremath{\mathcal{h}}}}\cdot p, {{\ensuremath{\mathcal{k}}}}\cdot q)$; see \[fig:direct\_product\]. This type of product is useful in modeling the Cartesian product of structures. The following claim characterises the equivariant map for direct product of two groups using the equivariant map for building blocks.
\[claim:1\] Let ${{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{h}}}})}}}$ represent ${{\ensuremath{\mathcal{H}}}}$-action, and let ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{H}}}}\in {\mathds{R}}^{P \times P}$ be an equivariant linear map for this action. Similarly, let ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{K}}}}: {\mathds{R}}^{Q \times Q}$ be equivariant to ${{\ensuremath{\mathcal{K}}}}$-action given by ${{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}})}}}$ for ${{\ensuremath{\mathcal{k}}}}\in {{\ensuremath{\mathcal{K}}}}$. Then, the product group ${{\ensuremath{\mathcal{G}}}\xspace}= {{\ensuremath{\mathcal{H}}}}\times {{\ensuremath{\mathcal{K}}}}$ naturally acts on ${\mathds{R}}^N = {\mathds{R}}^{P Q}$ using ${{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}} = {{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{h}}}})}}} \otimes {{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}})}}}$, and the Kronecker product ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{G}}}\xspace}} = {{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}}_{{\ensuremath{\mathcal{H}}}}\otimes {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{K}}}}$, is a ${{\ensuremath{\mathcal{G}}}\xspace}$-equivariant linear map.
Note that the claim is not restricted to permutation action.[^2] The proof follows from the *mixed-product property* of the Kronecker product[^3], and the equivariance of ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{H}}}}$ and ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{K}}}}$: $$\begin{aligned}
{{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}} {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{G}}}\xspace}} &= ({{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{h}}}})}}} \otimes {{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}})}}}) ({{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}}_{{\ensuremath{\mathcal{H}}}}\otimes {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{K}}}})
= ({{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{h}}}})}}} {{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}}_{{\ensuremath{\mathcal{H}}}}) \otimes ({{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}})}}} {{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}}_{{\ensuremath{\mathcal{K}}}}) \\
&= ( {{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}}_{{\ensuremath{\mathcal{H}}}}{{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{h}}}})}}}) \otimes ({{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}}_{{\ensuremath{\mathcal{K}}}}{{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}})}}}) = ({{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}}_{{\ensuremath{\mathcal{H}}}}\otimes {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{K}}}}) ({{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{h}}}})}}} \otimes {{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}})}}}) = {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{G}}}\xspace}} {{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}} \quad \forall {{\ensuremath{\mathcal{g}}}}\in {{\ensuremath{\mathcal{G}}}\xspace}.\end{aligned}$$
An implication of the tensor product form of ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{H}}}}\times {{\ensuremath{\mathcal{K}}}}}$ is that the number of independent linear operators of the product map (free parameters in the parameter-sharing) is the product of the independent operators of the building blocks.
D-dimensional convolution is a Kronecker (tensor) product of one-dimensional convolutions. The number of parameters grows with the product of kernel width across all dimensions; \[fig:param\_sharing\](a.1) shows the parameter-sharing for circular 1D convolution ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{C}}}}_4}$, and (b.1) shows the parameter-sharing for the direct product ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{C}}}}_3 \times {{\ensuremath{\mathcal{C}}}}_4}$.
@hartford2018deep introduce a layer for modeling interactions across multiple sets of entities, [*e.g.*, ]{}a user-movie rating matrix. Their model can be derived as the Kronecker product of 2-parameter equivariant layer for sets [@zaheer2017deep]. The number of parameters is therefore $2^D$ for a rank $D$ exchangeable tensor. \[fig:param\_sharing\](b.1) shows the 2-parameter model ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{\S_4}$ for sets, and (c.1) shows the parameter-sharing for the direct product ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{S}}}}_3 \times {{\ensuremath{\mathcal{S}}}}_4}$.
Wreath Product Action and Equivariance to a Hierarchy of Symmetries {#sec:wreath}
-------------------------------------------------------------------
[r]{}[0.3]{} {width="30.00000%"}
Let us start with an informal definition. Suppose as before ${{\ensuremath{\mathbb{P}}}}$ and ${{\ensuremath{\mathbb{Q}}}}$ are respectively an ${{\ensuremath{\mathcal{H}}}}$-set and a ${{\ensuremath{\mathcal{K}}}}$-set. We can attach one copy of ${{\ensuremath{\mathbb{Q}}}}$ to each element of ${{\ensuremath{\mathbb{P}}}}$. Each of these inner sets or fibers have their own copy of ${{\ensuremath{\mathcal{K}}}}$ acting on them. Action of ${{\ensuremath{\mathcal{H}}}}$ on ${{\ensuremath{\mathbb{P}}}}$ simply permutes these fibers. Therefore the combination of all ${{\ensuremath{\mathcal{K}}}}$ actions on all inner sets combined with ${{\ensuremath{\mathcal{H}}}}$-action on the outer set defines the action of the wreath product on ${{\ensuremath{\mathbb{P}}}}\times {{\ensuremath{\mathbb{Q}}}}$. \[fig:wreath\] demonstrates how one point $(p,q)$ moves under this action. Next few paragraphs formalize this.
#### Semidirect Product.
Formally, wreath product is defined using *semidirect* product which is a generalization of direct product. In part due to its use in building networks equivariant to Euclidean isometries, application of semidirect product in building equivariant networks is explored in several recent works; see [@cohen2019general] and citations therein. In semidirect product, the underlying set (of group members) is again the product set. However the group operation is more involved. The (external) semi-direct product ${{\ensuremath{\mathcal{G}}}\xspace}= {{\ensuremath{\mathcal{K}}}}\rtimes_\gamma {{\ensuremath{\mathcal{H}}}}$, requires a *homomorphism* $\gamma: {{\ensuremath{\mathcal{H}}}}\to \mathrm{Aut}({{\ensuremath{\mathcal{K}}}})$ that for each choice of ${{\ensuremath{\mathcal{h}}}}\in {{\ensuremath{\mathcal{H}}}}$, re-labels the elements of ${{\ensuremath{\mathcal{K}}}}$ while preserving its group structure. Using $\gamma$, the binary operation for the product group ${{\ensuremath{\mathcal{G}}}\xspace}= {{\ensuremath{\mathcal{K}}}}\rtimes_\gamma {{\ensuremath{\mathcal{H}}}}$ is defined as $({{\ensuremath{\mathcal{h}}}}, {{\ensuremath{\mathcal{k}}}}) ({{\ensuremath{\mathcal{h}}}}', {{\ensuremath{\mathcal{k}}}}') = ({{\ensuremath{\mathcal{h}}}}{{\ensuremath{\mathcal{h}}}}', {{\ensuremath{\mathcal{k}}}}\gamma_{{\ensuremath{\mathcal{h}}}}({{\ensuremath{\mathcal{k}}}}'))$. A canonical example is the semidirect product of translations (${{\ensuremath{\mathcal{K}}}}$) and rotations (${{\ensuremath{\mathcal{H}}}}$), which identifies the group of all rigid motions in the Euclidean space. Here, each rotation defines an automorphism of translations ([*e.g.*, ]{}moving north becomes moving east after $90^\circ$ clockwise rotation). Now we are ready to define the wreath product of two groups. As before, let ${{\ensuremath{\mathcal{H}}}}$ and ${{\ensuremath{\mathcal{K}}}}$ denote two finite groups, and let ${{\ensuremath{\mathbb{P}}}}$ be an ${{\ensuremath{\mathcal{H}}}}$-set. Define ${{\ensuremath{\mathcal{B}}}}$ as the direct product of $P = |{{\ensuremath{\mathbb{P}}}}|$ copies of ${{\ensuremath{\mathcal{K}}}}$, and index these copies by $p \in {{\ensuremath{\mathbb{P}}}}$: ${{\ensuremath{\mathcal{B}}}}= {{\ensuremath{\mathcal{K}}}}_1 \times \ldots \times {{\ensuremath{\mathcal{K}}}}_p \times \ldots \times {{\ensuremath{\mathcal{K}}}}_{P}$. Each member of this group is a tuple ${{\ensuremath{\mathcal{b}}}}= ({{\ensuremath{\mathcal{k}}}}_1,\ldots,{{\ensuremath{\mathcal{k}}}}_p, \ldots {{\ensuremath{\mathcal{k}}}}_{P})$. Since ${{\ensuremath{\mathbb{P}}}}$ is an ${{\ensuremath{\mathcal{H}}}}$-set, ${{\ensuremath{\mathcal{H}}}}$ also naturally acts on ${{\ensuremath{\mathcal{B}}}}$ by permuting the *fibers* ${{\ensuremath{\mathcal{K}}}}_p$. The semidirect product ${{\ensuremath{\mathcal{B}}}}\rtimes {{\ensuremath{\mathcal{H}}}}$ defined using this automorphism of ${{\ensuremath{\mathcal{B}}}}$ is called the wreath product, and written as ${{\ensuremath{\mathcal{K}}}}\wr {{\ensuremath{\mathcal{H}}}}$. Each member of the product group can be identified by the pair $({{\ensuremath{\mathcal{h}}}}, {{\ensuremath{\mathcal{b}}}})$, where ${{\ensuremath{\mathcal{b}}}}$ as a member of the *base* group itself is a $P$-tuple. This shows that the order of the wreath product group is $|{{\ensuremath{\mathcal{K}}}}|^{P} |{{\ensuremath{\mathcal{H}}}}|$ which can be much larger than the direct product group ${{\ensuremath{\mathcal{K}}}}\times {{\ensuremath{\mathcal{H}}}}$, whose order is $|{{\ensuremath{\mathcal{K}}}}| |{{\ensuremath{\mathcal{H}}}}|$.
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\[3pt\] ![[]{data-label="fig:param_sharing"}](figures/C4.png "fig:"){width=".15\linewidth"} ![[]{data-label="fig:param_sharing"}](figures/C3xC4.png "fig:"){width=".15\linewidth"} ![[]{data-label="fig:param_sharing"}](figures/S3xS4.png "fig:"){width=".15\linewidth"} ![[]{data-label="fig:param_sharing"}](figures/C3xS4.png "fig:"){width=".15\linewidth"} ![[]{data-label="fig:param_sharing"}](figures/S3xC4.png "fig:"){width=".15\linewidth"}
[(**a.1**)]{} ${{\ensuremath{\mathcal{C}}}}_4$ (**b.1**) ${{\ensuremath{\mathcal{C}}}}_3 \times {{\ensuremath{\mathcal{C}}}}_4$ (**c.1**) ${{\ensuremath{\mathcal{S}}}}_3 \times {{\ensuremath{\mathcal{S}}}}_4$ (**d.1**) ${{\ensuremath{\mathcal{C}}}}_3 \times {{\ensuremath{\mathcal{S}}}}_4$ (**e.1**) ${{\ensuremath{\mathcal{S}}}}_3 \times {{\ensuremath{\mathcal{C}}}}_4$
\[3pt\] ![[]{data-label="fig:param_sharing"}](figures/S4.png "fig:"){width=".15\linewidth"} ![[]{data-label="fig:param_sharing"}](figures/C3wrC4.png "fig:"){width=".15\linewidth"} ![[]{data-label="fig:param_sharing"}](figures/S3wrS4.png "fig:"){width=".15\linewidth"} ![[]{data-label="fig:param_sharing"}](figures/C3wrS4.png "fig:"){width=".15\linewidth"} ![[]{data-label="fig:param_sharing"}](figures/S3wrC4.png "fig:"){width=".15\linewidth"}
(**a.2**) ${{\ensuremath{\mathcal{S}}}}_4$ (**b.2**) ${{\ensuremath{\mathcal{C}}}}_3 \wr {{\ensuremath{\mathcal{C}}}}_4$ (**c.2**) ${{\ensuremath{\mathcal{S}}}}_3 \wr {{\ensuremath{\mathcal{S}}}}_4$ (**d.2**) ${{\ensuremath{\mathcal{C}}}}_3 \wr {{\ensuremath{\mathcal{S}}}}_4$ (**e.2**) ${{\ensuremath{\mathcal{S}}}}_3 \wr {{\ensuremath{\mathcal{C}}}}_4$
\[3pt\]
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### Imprimitive Action of Wreath Product
If in addition to ${{\ensuremath{\mathbb{P}}}}$ being an ${{\ensuremath{\mathcal{H}}}}$-set, ${{\ensuremath{\mathcal{K}}}}$ action on ${{\ensuremath{\mathbb{Q}}}}$ is also defined, the wreath product group acts on ${{\ensuremath{\mathbb{P}}}}\times {{\ensuremath{\mathbb{Q}}}}$ (making it comparable to the direct product). Specifically, $({{\ensuremath{\mathcal{h}}}}, {{\ensuremath{\mathcal{k}}}}_1,\ldots, {{\ensuremath{\mathcal{k}}}}_{P}) \in {{\ensuremath{\mathcal{K}}}}\wr {{\ensuremath{\mathcal{H}}}}$ acts on $(p,q) \in {{\ensuremath{\mathbb{P}}}}\times {{\ensuremath{\mathbb{Q}}}}$ as follows: $$({{\ensuremath{\mathcal{h}}}}, {{\ensuremath{\mathcal{k}}}}_1,\ldots, {{\ensuremath{\mathcal{k}}}}_{P}) \cdot (p,q) {\ensuremath{\doteq}}({{\ensuremath{\mathcal{h}}}}\cdot p, {{\ensuremath{\mathcal{k}}}}_{{{\ensuremath{\mathcal{h}}}}\cdot p} \cdot q).$$ Intuitively, ${{\ensuremath{\mathcal{H}}}}$ permutes the copies of ${{\ensuremath{\mathcal{K}}}}$ acting on each ${{\ensuremath{\mathbb{Q}}}}$, and itself acts on ${{\ensuremath{\mathbb{P}}}}$. We can think of ${{\ensuremath{\mathbb{P}}}}$ as the outer structure and ${{\ensuremath{\mathbb{Q}}}}$ as the inner structure; see \[fig:wr\_prod\].
Consider our early example where both ${{\ensuremath{\mathcal{H}}}}= {\ensuremath{\mathcal{C}}}_P$ and ${{\ensuremath{\mathcal{K}}}}= {\ensuremath{\mathcal{C}}}_Q$ are cyclic groups with regular action on ${{\ensuremath{\mathbb{P}}}}\cong {{\ensuremath{\mathcal{H}}}}$, ${{\ensuremath{\mathbb{Q}}}}\cong {{\ensuremath{\mathcal{K}}}}$. Each member of the wreath product ${\ensuremath{\mathcal{C}}}_Q \wr {\ensuremath{\mathcal{C}}}_P$, acts by translating each inner sequence using some ${\ensuremath{\mathcal{c}}} \in {\ensuremath{\mathcal{C}}}_Q$, while ${\ensuremath{\mathcal{c}}}' \in {\ensuremath{\mathcal{C}}}_P$ translates the outer sequence by ${\ensuremath{\mathcal{c}}}'$.
Let the $P \times P$ permutation matrix ${{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{h}}}})}}}$ represent the action of ${{\ensuremath{\mathcal{h}}}}\in {{\ensuremath{\mathcal{H}}}}$, and the $Q \times Q$ matrices ${{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}}_1)}}}, \ldots, {{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}}_P)}}}$ represent the action of ${{\ensuremath{\mathcal{k}}}}_p$ on $ {{\ensuremath{\mathbb{Q}}}}$. Then the action of ${{\ensuremath{\mathcal{g}}}}\in {{\ensuremath{\mathcal{K}}}}\wr {{\ensuremath{\mathcal{H}}}}$ on (the vectorized) ${{\ensuremath{\mathbb{P}}}}\times {{\ensuremath{\mathbb{Q}}}}$ is the following $P Q \times P Q$ permutation matrix: $$\begin{aligned}
\label{eq:wreath_perm_matrix}
{{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}} =
\begin{bmatrix}
{{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{h}}}})}}}_{1,1} {{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}}_1)}}}, &\ldots, & {{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{h}}}})}}}_{1,P} {{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}}_1)}}}\\
\vdots & \ddots & \vdots \\
{{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{h}}}})}}}_{P,1} {{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}}_P)}}}, &\ldots, & {{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{h}}}})}}}_{P,P} {{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}}_P)}}}\\
\end{bmatrix}
= \sum_{p=1}^P {\ensuremath{{\mathbf{\MakeUppercase{{1}}}}}}_{p,{{\ensuremath{\mathcal{h}}}}\cdot p} \otimes {{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}}_p)}}}\end{aligned}$$ where ${\ensuremath{{\mathbf{\MakeUppercase{{1}}}}}}_{p,{{\ensuremath{\mathcal{h}}}}\cdot p}$ is a $P \times P$ matrix whose only nonzero element is at row $p$ and column ${{\ensuremath{\mathcal{h}}}}\cdot p$ with the value of 1; and ${{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}_{p,p'}{\ensuremath{^{({{\ensuremath{\mathcal{h}}}})}}}$ is the element at row $p$ and column $p'$ of the permutation matrix ${{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{h}}}})}}}$. In the summation formulation, the Kronecker product ${\ensuremath{{\mathbf{\MakeUppercase{{1}}}}}}_{p,{{\ensuremath{\mathcal{h}}}}\cdot p} \otimes
{{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}}_p)}}}$ puts a copy of permutation matrix ${{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{k}}}}_p)}}}$ as a block of ${{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}}$. Note that the resulting permutation matrix is different from the Kronecker product; in this case we have $P+1$ (permutation) matrices participating in creating ${{\ensuremath{{\mathbf{\MakeUppercase{{G}}}}}}}{\ensuremath{^{({{\ensuremath{\mathcal{g}}}})}}}$, compared with two matrices in the vanilla Kronecker product.
### Equivariant Map
Consider a hierarchical structure, potentially with more than two levels of hierarchy, such as a set of sequences of images. Moreover, suppose that we have an equivariant map for each individual structure. The question answered by the following theorem is: how to use the equivariant map for each level to construct the equivariant map for the entire hierarchy? For now we only consider two levels of hierarchy; extension to more levels follows naturally, and is discussed later.
\[th:main\] Let ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{H}}}}\in {\mathds{R}}^{P \times P}$ be the matrix of an ${{\ensuremath{\mathcal{H}}}}$-equivariant map, and let ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{K}}}}\in {\mathds{R}}^{Q \times Q}$ be equivariant to ${{\ensuremath{\mathcal{K}}}}$-action. Then $$\begin{aligned}
\label{eq:wreath_prod_w}
{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{K}}}}\wr {{\ensuremath{\mathcal{H}}}}} {\ensuremath{\doteq}}{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{H}}}}\otimes ({\ensuremath{\mathbf{\MakeLowercase{{1}}}}}_Q {\ensuremath{\mathbf{\MakeLowercase{{1}}}}}^\top_Q) + {\ensuremath{{\mathbf{\MakeUppercase{{I}}}}}}_P \otimes {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{K}}}}, \quad \text{where} \quad
{\ensuremath{\mathbf{\MakeLowercase{{1}}}}}_Q = \overbrace{[1,\ldots,1]^\top}^{Q\; \text{times}}\end{aligned}$$ is a $P Q \times P Q$ matrix of a linear map equivariant to the imprimitive action of ${{\ensuremath{\mathcal{K}}}}\wr {{\ensuremath{\mathcal{H}}}}$ on ${{\ensuremath{\mathbb{P}}}}\times {{\ensuremath{\mathbb{Q}}}}$. Here, ${\ensuremath{{\mathbf{\MakeUppercase{{I}}}}}}_P$ is the $P\times P$ identity matrix.
Proof is in . Assuming sets of independent equivariant linear operators $\{{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{K}}}}\}$ and $\{{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{H}}}}\}$, from \[eq:wreath\_prod\_w\] it is evident that the *number of independent linear operators* equivariant to ${{\ensuremath{\mathcal{K}}}}\wr {{\ensuremath{\mathcal{H}}}}$ is the sum of those of the building blocks. These operators may be combined using any parameter to create a parameterized linear map, which in turn can be expressed using parameter-sharing in a vanilla feedforward layer. This in contrast with direct product in which the number of free parameters has a product form.
Consider two of the most widely used equivariant maps, translation equivariant convolution ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{C}}}}_P}$, and ${{\ensuremath{\mathcal{S}}}}_{{\ensuremath{\mathbb{Q}}}}$-equivariant map which has the form ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{S}}}}_Q} = w_1 {\ensuremath{{\mathbf{\MakeUppercase{{I}}}}}}_Q + w_2$ [@zaheer2017deep]. There are four combinations of these structures in a two level hierarchical structure: 1) set of sets ${{\ensuremath{\mathcal{S}}}}_Q \wr {{\ensuremath{\mathcal{S}}}}_P$; 2) sequence of sequences ${{\ensuremath{\mathcal{C}}}}_Q \wr {{\ensuremath{\mathcal{C}}}}_P$; 3) set of sequences ${{\ensuremath{\mathcal{C}}}}_Q \wr {{\ensuremath{\mathcal{S}}}}_P$; 4) sequence of sets ${{\ensuremath{\mathcal{S}}}}_Q \wr {{\ensuremath{\mathcal{C}}}}_P$. \[fig:param\_sharing\](b.2-e.2) show the parameter-sharing matrix for these hierarchical structures, assuming a full kernel in 1D circular convolution.
The reader is invited to contrast the three parameter layer for sets of sets, with the four parameter layer for interactions across sets in \[fig:param\_sharing\](c.1,c.2). Similarly, a model for sequence of sequences has far fewer parameters than a model for an image, as seen in \[fig:param\_sharing\](b.1, b.2).
### Deeper Hierarchies and Combinations with Direct Product
With more than two levels, the symmetry group involves more than one wreath product, which means that the equivariant map for the hierarchy is given by a recursive application of \[th:main\]. For example, the equivariant map for $({{\ensuremath{\mathcal{K}}}}\wr {{\ensuremath{\mathcal{H}}}}) \wr {{\ensuremath{\mathcal{U}}}}$, in which ${{\ensuremath{\mathcal{U}}}}$ acts on ${\ensuremath{\mathbb{R}}}$, and $ {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{U}}}}$ is the corresponding equivariant map, is given by ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{({{\ensuremath{\mathcal{K}}}}\wr {{\ensuremath{\mathcal{H}}}}) \wr {{\ensuremath{\mathcal{U}}}}} = {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{U}}}}\otimes ({\ensuremath{\mathbf{\MakeLowercase{{1}}}}}_{PQ} {\ensuremath{\mathbf{\MakeLowercase{{1}}}}}^\top_{PQ}) + {\ensuremath{{\mathbf{\MakeUppercase{{I}}}}}}_R \otimes {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{K}}}}\wr {{\ensuremath{\mathcal{H}}}}}$, where ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{K}}}}\wr {{\ensuremath{\mathcal{H}}}}}$ is in turn given by \[eq:wreath\_prod\_w\]. Note that wreath product is associative, and so the iterative construction above leads the same equivariant map as ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{K}}}}\wr ({{\ensuremath{\mathcal{H}}}}\wr {{\ensuremath{\mathcal{U}}}})}$.
We can also mix and match this construction with that of direct product in \[claim:1\]; for example, to produce the map for exchangeable tensors (product of sets), where each interaction is in the form of an image (hierarchy) – [*i.e.*, ]{}the group is $({{\ensuremath{\mathcal{C}}}}_R \times {{\ensuremath{\mathcal{C}}}}_V) \wr ({{\ensuremath{\mathcal{S}}}}_P \times {{\ensuremath{\mathcal{S}}}}_Q)$.
Efficient Implementation
------------------------
With equivariant maps for direct product of groups, efficient implementation of ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{K}}}}\times {{\ensuremath{\mathcal{H}}}}}$ using efficient implementation for individual blocks ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{K}}}}}$, and ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{H}}}}}$ is non-trivial – [*e.g.*, ]{}consider 2D convolution. In contrast, it is possible to use a black-box implementation of the parameter-sharing layers for ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{K}}}}}$, and ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{H}}}}}$ to construct ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{K}}}}\wr{{\ensuremath{\mathcal{H}}}}}$. To this end, let ${{\ensuremath{\mathbf{\MakeLowercase{{x}}}}}}\in {\mathds{R}}^{PQ}$ be the input signal, and $\operatorname{mat}({{\ensuremath{\mathbf{\MakeLowercase{{x}}}}}}) \in {\mathds{R}}^{P \times Q}$ denote its matrix form; for example in a set of sets, each column is an inner set in this matrix form. Throughout, we are assuming a single input-output channel, relying on the idea that having multiple channels simply corresponds to replicating the linear map for each input-output channel combination. Then we can rewrite \[eq:wreath\_prod\_w\] as $$\begin{aligned}
\label{eq:efficient}
{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{{\ensuremath{\mathcal{K}}}}\wr {{\ensuremath{\mathcal{H}}}}} {{\ensuremath{\mathbf{\MakeLowercase{{x}}}}}}= \operatorname{vec}
\left (
\left ( {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{H}}}}\left ( \operatorname{mat}({{\ensuremath{\mathbf{\MakeLowercase{{x}}}}}}){\ensuremath{\mathbf{\MakeLowercase{{1}}}}}_Q \right ) \right ) {\ensuremath{\mathbf{\MakeLowercase{{1}}}}}^\top_Q + \operatorname{mat}({{\ensuremath{\mathbf{\MakeLowercase{{x}}}}}}) {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{K}}}}\right ),\end{aligned}$$ where the first multiplication $\operatorname{mat}({{\ensuremath{\mathbf{\MakeLowercase{{x}}}}}}){\ensuremath{\mathbf{\MakeLowercase{{1}}}}}_Q$ pools over columns (inner structures), and after application of the ${{\ensuremath{\mathcal{H}}}}$-equivariant map ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{H}}}}$ to the pooled value, the result is broadcasted back using right-multiplication by ${\ensuremath{\mathbf{\MakeLowercase{{1}}}}}^\top_Q$. The second term simply transforms each inner structure using ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{{\ensuremath{\mathcal{K}}}}$. The overall operation turns out to be simple and intuitive: [the inner equivariant map is applied to individual inner structures, and the outer equivariant map is applied to pooled values and broadcasted back.]{}
Consider a coarse pixelization of an image into small patches. \[eq:efficient\] gives the following recipe for a layer equivariant to independent translations within each patch as well as global translation of the coarse image: 1) apply convolution to each patch independently; 2) pool over each patch, apply convolution to the coarse image, and broadcast back to individual pixels in each patch. 3) add the contribution of these two operations. Notice how pooling over regions, a widely used operation in image processing, appears naturally in this approach. One could also easily extend this to more levels of hierarchy for larger images.
Application: Point-Cloud Segmentation {#sec:application}
=====================================
In this section we consider a simple application of the theory above to large-scale point-cloud segmentation. The layer is the 3D version of equivariant linear map for sequence of sets, which is visualized in \[fig:param\_sharing\](d.2). Using a hierarchical structure is beneficial compared to both the set model and 3D convolution. In particular, the set model lacks any prior of the Euclidean nature of the data, while the 3D convolution, in order to preserve resolution requires a fine-grained voxelization where each point appears in one voxel. The past few years have seen a growing body of work on learning with point cloud data; see [@guo2019deep] for a survey. Many methods use hierarchical aggregation and pooling; this includes the use of furthest point clustering for pooling in [[<span style="font-variant:small-caps;"></span>]{}]{}[@qi2017pointnet++], use of concentric spheres for pooling in [[<span style="font-variant:small-caps;"></span>]{}]{}, or KD-tree guided pooling in [@klokov2017escape]. Several works extend convolution operation to maintain both translation and permutation equivariance in one way or another [@boulch2019generalizing; @wang2018deep; @atzmon2018point]; see also [@graham2017submanifold; @choy20194d]. Here, objective is not to introduce a radically new procedure, but to show the effectiveness of approach discussed in previous sections in deep model design from first principles. Indeed, we are able to achieve state-of-the-art in several benchmarks for large point-clouds.
Equivariance to a Hierarchy of Translations and Permutations
------------------------------------------------------------
Let ${{\ensuremath{{\mathbf{\MakeUppercase{{X}}}}}}}\in {\mathds{R}}^{N \times C}$ be the input point cloud, where $N$ is the number of points and $C$ is the number of input channels (for concreteness, in this section are including the channel dimension in our formulae.) In addition to 3D coordinates, these channels may include RGB values, normal vectors or any other auxiliary information. Consider a voxelization of the point cloud with a resolution $D$ voxels per dimension, and consider the hierarchy of translation symmetry across voxels and permutation symmetry within each voxel. We may also replace 3D coordinates with relative coordinates within each voxel. Let $\Pi \in \{0,1\}^{D^3 \times N}$ with one non-zero per column identify the voxel membership. Then the combination of equivariant set layer ${{\ensuremath{{\mathbf{\MakeUppercase{{X}}}}}}}{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_{1} + {\ensuremath{\mathbf{\MakeLowercase{{1}}}}} {\ensuremath{\mathbf{\MakeLowercase{{1}}}}}^\top {{\ensuremath{{\mathbf{\MakeUppercase{{X}}}}}}}{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_2$ [@zaheer2017deep] with 3D convolution using the pool-broadcast interpretation given in \[eq:efficient\], results in the following wreath-product equivariant linear layer $$\begin{aligned}
\label{eq:conv}
\phi({{\ensuremath{{\mathbf{\MakeUppercase{{X}}}}}}}) &= {{\ensuremath{{\mathbf{\MakeUppercase{{X}}}}}}}{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_1 + \Pi^\top \big ( {{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_3 \ast (\Pi {{\ensuremath{{\mathbf{\MakeUppercase{{X}}}}}}}) \big), \end{aligned}$$ where ${{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_1 \in {\mathds{R}}^{C \times C'}$, $\ast$ denotes the convolution operation, ${{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}}_3 \in {\mathds{R}}^{K^3 \times C \times C' } $ is the convolution kernel, with kernel width $K$, and $C'$ output channels. Here, multiplication with $\Pi$ and $\Pi^\top$ performs the pooling and broadcasting from/to points within voxels, respectively. Note that we have dropped the “set” operation $\Pi^\top ((\Pi {{\ensuremath{{\mathbf{\MakeUppercase{{X}}}}}}}) {{{\ensuremath{{\mathbf{\MakeUppercase{{W}}}}}}}_2})$ that pools over each voxel, multiplies by the weight and broadcasts back to the points. This is because it can be absorbed in the convolution operation, and therefore it is redundant. In the equation above pooling operation can replace summation (implicit in matrix multiplication) with any other commutative operation; see [@murphy2018janossy; @zhang2018end; @xu2018powerful], we use mean-pooling for the experiments. While the layer of \[eq:conv\] already achieves state-of-the-art in the experiments, we also consider adding an (equivariant) attention mechanism for further improvement; see for details.
Empirical Results {#sec:experiments}
-----------------
We evaluate our model on two of the largest real world point cloud segmentation benchmarks, [[<span style="font-variant:small-caps;"></span>]{}]{} [@hackel2017semantic3d] and the Stanford Large-Scale 3D Indoor Spaces ( [[<span style="font-variant:small-caps;"></span>]{}]{}) [@armeni20163d], as well as a dataset of virtual point cloud scenes, the [[<span style="font-variant:small-caps;"></span>]{}]{} benchmark [@gaidon2016virtual]. In all cases we report new state-of-the-art. The architecture of [[<span style="font-variant:small-caps;"></span>]{}]{} is a stack of equivariant layer \[eq:conv\] plus ReLU nonlinearity and residual connections. [[<span style="font-variant:small-caps;"></span>]{}]{} also adds the attention mechanism. [^4] Details on architecture and training, as well as further analysis of results appear in .
\[tab:wreath-results\]
#### Outdoor Scene Segmentation - [[<span style="font-variant:small-caps;"></span>]{}]{}
[@hackel2017semantic3d] is the largest LiDAR benchmark dataset, consisting of 15 training point clouds and 15 tests point clouds with withheld labels, amassing altogether over 4 billion labeled points from a variety of urban and rural scenes. In particular, rather than working with the smaller [[<span style="font-variant:small-caps;"></span>]{}]{} variation, we run our experiments on the full dataset ([[<span style="font-variant:small-caps;"></span>]{}]{})[^5]. \[tab:wreath-results\] reports mIoU unweighted mean intersection over union metric (mIoU), as well as the overall accuracy (OA) for various methods.
#### Indoor Scene Segmentation - Stanford Large-Scale 3D Indoor Spaces ([[<span style="font-variant:small-caps;"></span>]{}]{})
[@armeni20163d] consists of various 3D RGB point cloud scans from an assortment of room types on six different floor in three buildings on the Stanford campus, totaling almost 600 million points. \[tab:wreath-results\] show that we achieve the best overall accuracy as well as mean intersection over union. This is in spite of the fact that our competition use extensive data-augmentation also for this dataset. In addition to random jittering and subsampling employed by both [[<span style="font-variant:small-caps;"></span>]{}]{} and [[<span style="font-variant:small-caps;"></span>]{}]{}, [[<span style="font-variant:small-caps;"></span>]{}]{} also uses random dropout of RGB data.
#### Virtual Scene Segmentation - Virtual KITTI ([[<span style="font-variant:small-caps;"></span>]{}]{})
[@gaidon2016virtual] contains 35 monocular photo-realistic synthetic videos with fully annotated pixel-level labels for each frame and 13 semantic classes in total. Following [@engelmann2017exploring], we project the 2D depth information within these synthetic frames into 3D space, thereby obtaining semantically annotated 3D point clouds. Note that [[<span style="font-variant:small-caps;"></span>]{}]{} is significantly smaller than either [[<span style="font-variant:small-caps;"></span>]{}]{} or [[<span style="font-variant:small-caps;"></span>]{}]{}, containing only 15 millions points in total.
Conclusion {#conclusion .unnumbered}
==========
This paper presents a procedure to design neural networks equivariant to hierarchical symmetries and nested structures. We describe how wreath product can formulate the symmetries in these settings, contrast its use case with direct product, and identify linear maps equivariant to the wreath product of groups as a function of equivariant maps for the building blocks, using additional pool-and-broadcast operations. We consider one of the many use cases of this approach to design a deep model for large-scale semantic segmentation of point cloud data, where we are able to achieve state-of-the-art using a simple architecture.
Broader Impact {#broader-impact .unnumbered}
==============
As deep learning finds its way in various real-world applications, the practitioners are finding more constrains in representing their data in formats and structures amenable to existing deep architectures. The list of basic structures such as images, sets, and graphs that we can approach using deep models has been growing over the past few years. The theoretical contribution of this paper substantially expands this list by enabling deep learning on a hierarchy of structures. This could potentially unlock new applications in data-poor and structure-rich settings. The task we consider in our experiments is deep learning with large point-cloud data, which is finding growing applications, from autonomous vehicles to geographical surveys. While this is not a new task, our empirical results demonstrate the effectiveness of the proposed methodology in dealing with hierarchy in data structure.
[^1]: Currently at Borealis AI. Work done while at UBC.
[^2]: The tensor product of irreducible representation of two distinct finite groups is an irreducible representation for the product group. Therefore this construction of equivariant maps can be used with a decomposition into irreducible representations for the general linear case.
[^3]: $({\ensuremath{{\mathbf{\MakeUppercase{{A}}}}}} \otimes {\ensuremath{{\mathbf{\MakeUppercase{{B}}}}}}) ({\ensuremath{{\mathbf{\MakeUppercase{{C}}}}}} \otimes {\ensuremath{{\mathbf{\MakeUppercase{{D}}}}}}) = ({\ensuremath{{\mathbf{\MakeUppercase{{A}}}}}} {\ensuremath{{\mathbf{\MakeUppercase{{C}}}}}}) \otimes ({\ensuremath{{\mathbf{\MakeUppercase{{B}}}}}} {\ensuremath{{\mathbf{\MakeUppercase{{D}}}}}})$
[^4]: We have released a [[<span style="font-variant:small-caps;"></span>]{}]{} implementation of our models. at [[<span style="font-variant:small-caps;"></span>]{}]{}
[^5]: The leader-board can be viewed at <http://www.semantic3d.net/view_results.php?chl=1>
|
---
abstract: 'We study cosmological perturbations in the context of an interacting dark energy model, in which the cosmological term decays linearly with the Hubble parameter, with concomitant matter production. A previous joint analysis of the redshift-distance relation for type Ia supernovas, barionic acoustic oscillations, and the position of the first peak in the anisotropy spectrum of the cosmic microwave background has led to acceptable values for the cosmological parameters. Here we present our analysis of small perturbations, under the assumption that the cosmological term, and therefore the matter production, are strictly homogeneous. Such a homogeneous production tends to dilute the matter contrast, leading to a late-time suppression in the power spectrum. Nevertheless, an excellent agreement with the observational data can be achieved by using a higher matter density as compared to the concordance value previously obtained. This may indicate that our hypothesis of homogeneous matter production must be relaxed by allowing perturbations in the interacting cosmological term.'
author:
- 'H. A. Borges$^1$, S. Carneiro$^{1,2}$, J. C. Fabris$^3$ and C. Pigozzo$^1$'
title: Evolution of density perturbations in decaying vacuum cosmology
---
Introduction
============
The cosmological constant problem has acquired a renewed importance since several independent observations have been pointing to the presence of a negative pressure component in the cosmic fluid [@Pad]. From the point of view of quantum field theories, the natural candidate for such a dark energy is the quantum vacuum. Since, at the macroscopic level, it has the symmetry of the background, its energy-momentum tensor has the form $T_{\mu}^{\nu}=
\Lambda g_{\mu}^{\nu}$, where $\Lambda$ is a scalar function of coordinates. This leads, in the case of an isotropic and homogeneous space-time and co-moving observers, to the equation of state $p_{\Lambda} = - \rho_{\Lambda} = - \Lambda$, where $\Lambda$ may be, in general, a function of time. In the case of a constant $\Lambda$, the vacuum contribution plays the role of a cosmological constant in Einstein’s equations.
However, the estimation of the vacuum energy density by quantum field theories in the flat space-time leads, after some regularization procedure, to a very huge result when compared to the observed value. A possible way out of this difficult is to argue that such a result is valid only in a flat background, in which the very Einstein equations predict a null total energy-momentum tensor. Therefore, the huge vacuum density should be canceled by a bare cosmological constant, like in a renormalization process. Now, if we could obtain the vacuum density in the FLRW space-time, after the subtraction of the Minkowskian result it would remain an effective time-dependent $\Lambda$ term, which decreases with the expansion.
The idea of a time-dependent cosmological term has found different phenomenological implementations [@Ozer], being a subject of renewed interest in recent years [@Schutzhold; @Horvat; @Fabris]. A general feature of all those approaches is the production of matter, concomitant with the vacuum decay in order to assure the covariant conservation of the total energy [@Barrow]. Indeed, in the FLRW space-time, the Bianchi identities lead to the conservation equation $$\dot{\rho}_T+3H(\rho_T+p_T)=0, \label{continuidade}$$ where $\rho_T$ and $p_T$ stand for the total energy density and pressure, respectively, and $H = \dot{a}/a$ is the Hubble parameter. By writing $\rho_T = \rho_m + \Lambda$ and $p_T = p_m - \Lambda$ (where $\rho_m$ and $p_m$ are the energy density and pressure of matter), the above equation reduces to $$\dot{\rho}_m + 3H(\rho_m + p_m)=-\dot{\Lambda},
\label{continuidade3}$$ which shows that, in the case of a varying $\Lambda$, matter is not independently conserved[^1].
An important point to be clarified in this kind of model is the homogeneity of matter production. Of course, in a strictly homogeneous space-time the production is homogeneous, since $\rho_m$ and $\Lambda$ depends only on time. But, in the presence of density perturbations, is the new matter produced homogeneously, or just where matter already exists [@Dirac]? In the case of a homogeneous production, the new matter tends to dilute the density perturbations, leading to a suppression of the density contrast. In some models, this suppression is strong enough to impose very restrictive observational limits to them [@Opher].
In this paper we will analyze the evolution of density perturbations in a particular, spatially flat, cosmological model with vacuum decay [@Borges; @Jailson]. It can be based on a phenomenological prescription for the variation of $\Lambda$ with time [@SC], given by $\Lambda \approx (H + m)^4 - m^4$, where $m$ is a characteristic energy that can be identified with the scale of the QCD vacuum condensation, the latest cosmological vacuum transition. Although it can be corroborated by holographic arguments [@SC; @SC2], based on the thermodynamics of de Sitter space-times, here we will take it just as a phenomenological ansatz. In the limit of very early times, we have $\Lambda \approx H^4$, which provides a non-singular inflationary solution [@SC].
In the opposite limit of large times we have $\Lambda = \sigma H$, with $\sigma \approx m^3$. This scaling law for the vacuum density was also suggested in [@Schutzhold], on the basis of different arguments. It leads to a cosmological scenario in qualitative agreement with the standard one [@Borges], with an initial radiation era followed by a long phase dominated by dust. This dust phase tends asymptotically to a de Sitter universe, with the deceleration/acceleration transition occurring some time before the present epoch. On the other hand, a quantitative analysis has shown a good accordance with supernova observations, leading to age and matter density parameters inside the limits imposed by other independent observations [@Jailson].
Since the radiation phase we obtain is indistinguishable from the standard one, our analysis will be initially focused on the evolution of density perturbations of non-relativistic matter in the dust-dominated phase, considering wavelengths inside the horizon[^2]. In this way, it will be possible to make use of a generalization of the Newtonian linear treatment of the problem, which includes the effects of matter production [@Waga]. We will show that, even in the case of a homogeneous vacuum decay, the contrast suppression is important only for late times, not affecting the process of galaxy formation. On the other hand, it dominates for future times, and we will discuss how this behavior can possibly alleviate another problem related to the cosmological term: the cosmic coincidence problem.
Subsequently, a relativistic analysis will be performed, in order to construct the matter power spectrum. Again, the hypothesis of homogeneous matter production will be used, leading as well to a consequent power suppression. A second interesting difference as compared to the $\Lambda$CDM model is a shift of the spectrum turnover to the left, that is, to smaller wavenumbers. The late-time suppression is not very sensitive to the value used for the matter density, a feature that can already be noted from the Newtonian analysis. On the other hand, the correction of the turnover position, by taking a higher matter density, displaces all the spectrum to the right, compensating the late-time power suppression. In this way we can obtain an excellent fit of data, but with a higher matter density in comparison with the standard case.
The article is organized as follows. In next section we review the main features of our interacting model. In Section III we perform the Newtonian analysis of evolution of density perturbations in the matter era. In Section IV the matter power spectrum is constructed, on the basis of a simplified relativistic calculation. In Section V the reader can find our concluding remarks.
The model
=========
The Friedmann equations in the spatially flat case are given by (\[continuidade\]) and $\rho_T = 3 H^2$. Let us take $\rho_T =
\rho_m + \Lambda$, $p_T = p_m - \Lambda$ and $p_m = (\gamma-1)
\rho_m$, with constant $\gamma$. Let us also take the ansatz $\Lambda = \sigma H$, with $\sigma$ constant and positive. We obtain the evolution equation $$\label{evolucao}
2\dot{H} + 3\gamma H^2 - \sigma \gamma H = 0.$$
The solution, for $\rho_m, H > 0$, is given by [@Borges] $$\label{a}
a = C \left[\exp\left(\sigma \gamma t/2\right) -
1\right]^{\frac{2}{3\gamma}},$$ where $a$ is the scale factor, $C$ is an integration constant, and a second one was taken equal to zero in order to have $a = 0$ for $t =
0$.
In the radiation phase, taking $\gamma = 4/3$ and the limit of early times ($\sigma t << 1$), we have $$\label{asmall}
a \approx \sqrt{2C^2\sigma t/3}.$$ This is the same scaling law we obtain in the standard case, leading to $H \approx 1/2t$. In the same limit we then have $\rho_m = \rho_T
- \Lambda = 3H^2 - \sigma H \approx 3H^2 = \rho_T$. By using (\[asmall\]) we then obtain $$\label{rhosmall}
\rho_T \approx \rho_m \approx \frac{\sigma^2 C^4}{3a^4} \approx
\frac{3}{4t^2},$$ i.e., the same variation law for radiation one obtains in the standard model, which shows that, during the radiation era, both the cosmological term and the matter production can be dismissed.
On the other hand, in the matter era we obtain, by doing $\gamma =
1$, $$\label{adust}
a = C \left[\exp\left(\sigma t/2\right) - 1\right]^{\frac{2}{3}}.$$ Taking again the limit of early times, we have $$\label{adustsmall}
a \approx C(\sigma t/2)^{2/3},$$ as in the Einstein-de Sitter solution. It is also easy to see that, in the opposite limit $t \rightarrow \infty$, (\[adust\]) tends to the de Sitter solution.
With the help of (\[adust\]), and by using $\Lambda = \sigma H$ and $\rho_m = 3H^2 - \sigma H$, it is straightforward to derive the matter and vacuum densities as functions of the scale factor. One has $$\label{rhodust}
\rho_m = \frac{\sigma^2 C^3}{3a^3} + \frac{\sigma^2
C^{3/2}}{3a^{3/2}},$$ $$\label{Lambdadust}
\Lambda = \frac{\sigma^2}{3} + \frac{\sigma^2 C^{3/2}}{3a^{3/2}}.$$ In these expressions, the first terms give the standard scaling of matter (baryons included) and vacuum densities, being dominant in the limits of early and very late times, respectively. The second ones are owing to the process of matter production, being important at an intermediate time scale.
With (\[rhodust\]) and (\[Lambdadust\]) we obtain, for the total energy density and pressure[^3], $$\rho_T = \frac{\sigma^2}{3} \left[ \left( \frac{C}{a} \right) ^{3/2}
+ 1 \right] ^2,$$ $$p_T = - \sqrt{\frac{\sigma^2}{3}} \rho_T^{1/2}.$$
From (\[adust\]) we can also derive the Hubble parameter as a function of time in the matter era. It is given by $$\label{H}
H = \frac{\sigma/3}{1-\exp(-\sigma t/2)}.$$ With this expression, and by using (\[adust\]) and (\[rhodust\]), it is not difficult to obtain the present age of the universe, given, in terms of the age parameter, by $$\label{age}
H_0 t_0 = \frac{2\ln\Omega_{m0}}{3(\Omega_{m0}-1)},$$ where $\Omega_{m0} = \rho_{m0}/3H_0^2$ is the relative matter density at present.
Finally, with the help of (\[adust\]) and (\[H\]), we can express $H$ as a function of the redshift $z = a_0/a - 1$, which leads to $$\label{Hz}
H(z) = H_0 \left[1-\Omega_{m0}+\Omega_{m0}(z+1)^{3/2}\right].$$
With this function we have analysed the redshift-distance relation for type Ia supernovas [@Jailson], obtaining data fits as good as with the flat $\Lambda$CDM model. With the Supernova Legacy Survey (SNLS) [@SNLS] - the most confident survey we have so far - the best fit is given by $h=0.70 \pm 0.02$ and $\Omega_{m0} = 0.32
\pm 0.05$ (with $2\sigma$), with a reduced $\chi$-square $\chi^2_r =
1.01$ (here, $h \equiv H_0/$($100$km/s.Mpc)). On the other hand, a joint analysis of the Legacy Survey, baryonic acoustic oscillations and the position of the first peak of CMB anisotropies has led to the concordance values $h=0.69 \pm 0.01$ and $\Omega_{m0} = 0.36 \pm
0.01$ (with $2\sigma$), with $\chi^2_r = 1.01$ [@Jailson2]. With these results one can obtain, from (\[age\]), a universe age $t_0
\approx 15.0$ Gyr, inside the interval allowed by age estimations of globular clusters [@age].
Newtonian evolution of density perturbations
============================================
The Newtonian equation for the evolution of density perturbations in a pressureless fluid can be generalized in order to account for matter production [@Waga]. In this generalized form, it is given by $$\label{Waga}
\frac{\partial^2\delta}{\partial t^2} + \left(2H +
\frac{\Psi}{\rho_m}\right) \frac{\partial\delta}{\partial t} -
\left[ \frac{\rho_m}{2} - 2H\frac{\Psi}{\rho_m} -
\frac{\partial}{\partial t}\left(\frac{\Psi}{\rho_m}\right)\right]
\delta = 0.$$ Here, $\delta = \delta \rho_m/\rho_m$ is the density contrast of the pressureless matter, and $\Psi$ is the source of matter production, defined as $$\dot{\rho}_m+3H(\rho_m+p_m)=\Psi. \label{continuidade2}$$ In the case of a constant $\Lambda$, $\Psi = 0$, and (\[Waga\]) reduces to the usual non-relativistic equation for the linear evolution of the contrast. In our case, on the other hand, $\Psi = -
\dot{\Lambda} = - \sigma \dot{H}$, as can be seen from (\[continuidade3\]).
Equation (\[Waga\]) is derived on the basis of two main assumptions [@Waga]. The first one is that the produced particles have negligible velocities as measured by observers co-moving with the cosmic fluid. This is a reasonable hypothesis, since we are dealing with a non-relativistic phase of universe expansion, when $H$ (and so $\Lambda$) varies slowly enough. The second assumption is that the vacuum component $\Lambda$ is strictly homogeneous, which means that matter production is homogeneous as well. This stronger hypothesis is totally ad hoc at the present stage of the model development and, as we will see, leads to a suppression of the contrast at large times.
In order to solve (\[Waga\]) for our case, it is convenient to introduce the new variable $$\label{x}
x=\exp(-\sigma t/2).$$ After calculating $\rho_m$, $H$ and $\Psi$ as functions of $x$ with the help of (\[adust\]), (\[rhodust\]) and (\[H\]), equation (\[Waga\]) takes the form $$\label{contraste}
3x^2(x-1)^2\delta''+4x(x-1)\delta'-2(3x-2)\delta=0,$$ where the prime means derivative with respect to $x$. It is possible to show that, in the limit of early times, it reduces to the evolution equation for the contrast in the Einstein-de Sitter model, as should be.
The general solution of (\[contraste\]) can be written as $$\label{geral}
\delta = \frac{x}{x-1} \left\{ C_1 + C_2 \left[ \frac{2}{3}
\beta(x,1/3,2/3) + x^{1/3} (x-1)^{2/3} \right] \right\},$$ where $C_1$ and $C_2$ are integration constants to be determined by initial conditions, and $\beta(x,a,b)$ is the incomplete beta function, defined as $$\label{beta}
\beta(x,a,b)=\int_0^x\; y^{a-1}\; (1-y)^{b-1}\; dy.$$
This $\beta$ function can be expanded in a Laurent series around $x=1$, leading to $$\label{Lorrent}
\beta(x,1/3,2/3) \approx \beta(1,1/3,2/3) - \frac{3}{2}(x-1)^{2/3}.$$ In this way, with the help of (\[Lorrent\]) and (\[x\]) we can expand (\[geral\]) around $t = 0$, obtaining $$\label{expansao}
\delta \approx \frac{D_1}{t} + D_2 t^{2/3},$$ which is precisely the general solution obtained in the Einstein-de Sitter model, as expected. The new arbitrary constants are given by $$\begin{aligned}
\label{D}
D_1 &=& -\frac{2}{\sigma} \left[ C_1 + \frac{2}{3}\beta(1,1/3,2/3)
C_2 \right],\\ D_2 &=& \frac{(4\sigma)^{2/3}}{15} C_2.\end{aligned}$$
If, in the early time approximation (\[expansao\]), we want to retain just the growing mode, proportional to $t^{2/3}$, we must choose $D_1 = 0$. Then, our general solution (\[geral\]) reduces to $$\label{solution}
\frac{\delta}{C_2} =
\frac{2\,x\;[\beta(1,1/3,2/3)-\beta(x,1/3,2/3)]}{3(1-x)} -
\frac{x^{4/3}}{(1-x)^{1/3}}.$$ The above solution can be expressed as functions of $t$ or $a$, with the help of (\[x\]) and (\[adust\]). It can also be expressed as a function of the redshift, by using the relation $$x = \frac{\Omega_{m0} (1+z)^{3/2}}{1-\Omega_{m0}+\Omega_{m0}
(1+z)^{3/2}},$$ which can be derived with the help of (\[adust\]), (\[rhodust\]) and (\[H\]).
Figures 1 and 2 show the density contrast (\[solution\]) as a function of $a$ and $z$, respectively. We have taken $a_0 = 1$, and used for the matter density parameter the best-fit value we have obtained from the SNLS analysis [@Jailson], $\Omega_{m0} =
0.32$. The integration constant $C_2$ was chosen so that for the time of last scattering ($z \approx 1100$) one has $\delta \approx
10^{-5}$, as imposed by anisotropy observations of the cosmic microwave background [@WMAP]. For the sake of comparison, we have also plotted the evolution of the density contrast in the Einstein-de Sitter solution and in the spatially flat $\Lambda$CDM model with $\Omega_{m0} = 0.27$.
In our case the density contrast grows monotonically with time until $z \approx 0.6$, after which it decreases monotonically, tending to zero in the limit $t \rightarrow \infty$. The consequences of such a suppression at large times will be discussed in our Conclusions, where a possible relation with the cosmic coincidence problem will be outlined. The important point here is that the evolution of $\delta$ in our case is indistinguishable from its behavior in the $\Lambda$CDM case until $z \approx 5$, that is, along the entire era of galaxy formation. On the other hand, the late-time suppression leads to a present contrast approximately $1/3$ of the standard one, a difference that will be manifest in the power spectrum, as we will see now.
The power spectrum
==================
The shape of the spectrum depends on several parameters. But one of the most important is given by the moment of equilibrium between radiation and matter, $\Omega_R = \Omega_m$, where $\Omega_R$ and $\Omega_m$ are the respective density parameters (relative to the critical density). In the $\Lambda$CDM model, we have $$\begin{aligned}
\Omega_R = \frac{\Omega_{R0}}{a^4} = \Omega_{R0}(1 + z)^4,\\
\Omega_m = \frac{\Omega_{m0}}{a^3} = \Omega_{m0}(1 + z)^3,\end{aligned}$$ where $\Omega_{R0}$ and $\Omega_{m0}$ are the density parameters for radiation and matter today. The redshift at equilibrium is then given by $$1 + z_{eq} = \frac{\Omega_{m0}}{\Omega_{R0}}.$$ Following [@lahav], we fix, for the $\Lambda$CDM model, $$\Omega_{m0}h^2 = 0.127, \quad \Omega_{R0}h^2 = 4.1\times10^{-5}.$$ This implies $$1 + z_{eq} = 3097.$$ Remark that this value, as a matter of fact, is independent of $h$.
Now, we can analyse the moment the perturbations enter in the horizon. This is obtained by inspecting the perturbed equations. In general, it can be written as $$\ddot\delta + 2\frac{\dot a}{a}\dot\delta +
\biggr\{v_s^2\frac{k^2}{a^2} - \frac{3}{2}\biggr(\frac{\dot
a}{a}\biggl)^2\biggl\}\delta = 0.$$ In this equation, $\delta$ is the density contrast, and $v_s^2 =
\frac{\partial p}{\partial\rho}$ represents the sound velocity in unities of $c$ (the velocity of light). The presence of a first derivative term is related to the [*friction*]{} due to the expansion of the universe, while the two last terms describe the interplay between the pressure, that avoids the collapse, and the gravitational attraction, that drives the collapse. When the first of these terms dominates, the perturbation does not grow; when the second one dominates, the perturbation increases. Ignoring numerical factors of order of unities, related to the sound velocity, equation of state etc, the condition that separates both regimes is $$k = \frac{a}{d_H}, \quad d_H = \frac{c}{H} = \frac{c\,a}{\dot a}.$$ In this expression, $d_H$ is the Hubble radius. Of course, this is just an estimation.
For the $\Lambda$CDM model, we have $$d_H = \frac{c}{H_0}\biggr\{\Omega_{m0}(1 + z)^3 + \Omega_{R0}(1 +
z)^4 + \Omega_{\Lambda0}\biggl\}^{-1/2},$$ where $\Omega_{\Lambda0}$ is the density parameter for the cosmological term today. Hence, we have $$[(1 + z) k\,l_{H0}]^2 = \Omega_{m0}(1 + z)^3 + \Omega_{R0}(1 + z)^4
+ \Omega_{\Lambda0},$$ where $l_{H0}$ is the Hubble’s radius today, $l_{H0} =
3000\,h^{-1}$Mpc. In general, for large values of $z$ the term $\Omega_{\Lambda0}$ can be ignored. In doing so, and using the expression above for $z = z_{eq}$, we find the formula (7.39) of reference [@dodelson], $$\label{Dodelson}
k_{eq} = \sqrt{\frac{2}{\Omega_{R0}}} \frac{\Omega_{m0}}{l_{H0}}.$$
Using, besides the values of $\Omega_{m0}$ and $\Omega_{R0}$ already quoted, also $h = 0.7$, we obtain $$k_{eq} = 0.013.$$ We notice that, using the BBKS transfer function for the $\Lambda$CDM model [@jerome], the turning point is also located at $k = 0.013$.
Now, the observations cover scales from $k_{min}h^{-1} = 0.010$ until $k_{max}h^{-1} = 0.185$. Using the parameters above, we find that these modes entered in the Hubble horizon at $$k_{min} \rightarrow z_1 = 2077; \quad k_{max} \rightarrow z_2 =
59143.$$ That is, essentially, all modes entered in the radiation dominate era.
Turning to the present interacting model, the main modifications are the following:
1. The expression governing the moment the modes enter in the Hubble horizon is given by $$\label{Hzalto}
\biggl [k\,l_{H0}\,(1 + z)\biggr]^2 = \frac{1}{\Omega_{m0} +
\Omega_{\Lambda0}}\biggl[\Omega_{\Lambda0} + \Omega_{m0}(1 +
z)^{3/2}\biggr]^2 + \Omega_{R0}(1 + z)^4,$$ with $\Omega_{m0} + \Omega_{\Lambda0} \approx 1$. This is an approximate expression obtained from (\[Hz\]) by adding a conserved radiation density to the Friedmann equation $3H^2 =
\rho_T$.[^4]
2. An inspection of (\[Hz\]) for high $z$, when $\Lambda$ and the matter production are dismissable, shows that $\Omega_m(z) = \Omega_{m0}^2 (1+z)^3$. In other words, we have the same scaling of conserved matter as in the standard model, but with an extra factor $\Omega_{m0}$. This is owing to the matter production between $t(z)$ and $t_0$: in order to have the same matter density today, we need a smaller density at high redshifts. As a consequence, the redshift of equilibrium between matter and radiation is now given by $z_{eq} =
\Omega_{m0}^2/\Omega_{R0}$, while for the correspondent wave number we obtain, instead of (\[Dodelson\]), $$\label{Dodelson2}
k_{eq} = \sqrt{\frac{2}{\Omega_{R0}}} \frac{\Omega_{m0}^2}{l_{H0}}.$$ Note the extra factor $\Omega_{m0}$ as compared to the corresponding $\Lambda$CDM expression. As this factor is smaller than unity, this means that the turnover of the spectrum is moved to the left, that is, to smaller $k$’s as compared to the standard model.
3. The matter density parameter and the Hubble parameter are not the same as before. In the subsequent analysis we will use $\Omega_{m0} = 0.32$ and $h = 0.7$ (the type Ia supernovas best fitting [@Jailson]).
Now, the results are the following:
1. The equilibrium occurs at $z_{eq} = 2263$, which implies $k_{eq} = 0.007$;
2. The mode $k_{min}$ enters in the Hubble horizon at $z_1 = 3469$, while the mode $k_{max}$ at $z_2 = 81404$.
As already noticed, the results indicate that the spectrum is displaced to the left, implying that there is a power suppression with respect to the $\Lambda$CDM model. Moreover, there is, as we have seen in the previous section, an additional power suppression during the matter dominated phase. Hence, essentially, we must expect that the power spectrum displays, in what concerns matter agglomeration, an expressive power suppression in comparison with the $\Lambda$CDM model.
However, we can displace the spectrum to the right, instead of displace it to the left, if the values of $\Omega_{m0}$ and/or $h$ are increased. For example, for $\Omega_{m0} = 0.48$ and $h = 0.73$, the $k_{eq}$ occurs at $0.016$, with $z_{eq} = 5094$. Moreover, $k_{min}$ enters in the Hubble horizon at $z_1 = 2589$ and $k_{max}$ at $z_2 = 80020$. The substantial displacement to the right of $k_{eq}$ compensates the smaller growing of perturbations during the matter dominated phase. So, the general features of the power spectrum are reproduced for larger values of $\Omega_{m0}$ as compared to the $\Lambda$CDM model.
A precise derivation of the spectrum is a very though calculation, since the Einstein-Boltzmann coupled system must be considered. A complete analysis for the $\Lambda$CDM model leads to the so-called BBKS transfer function [@jerome], which gives the spectrum today as function of a given primordial spectrum. For the scale invariant spectrum, favored by the primordial inflationary scenario, the BBKS transfer function is given by $$P_m(k) = |\delta_m(k)|^2 =
AT(k)\frac{g^2(\Omega_{m0})}{g^2(\Omega_T)}k,$$ where $A$ is a normalization of the spectrum (which can be fixed by the spectrum of anisotropy of the cosmic microwave background radiation), $T(k)$ is given by $$\begin{aligned}
T(k) = \frac{\ln(1 + 2.34q)}{2.34q}\biggr[1 + 3.89q + (16.1q)^2 + (5.64q)^3
+ (6.71q)^4\biggl]^{-\frac{1}{4}}, \\
q = \frac{k}{h\Gamma}{\text Mpc}^{-1}, \quad \Gamma =
\Omega_{dm0}he^{- \Omega_{b0} - \frac{\Omega_{b0}}{\Omega_{dm0}}},\end{aligned}$$ and where $\Omega_{m0}$, $\Omega_{dm0}$, $\Omega_{b0}$ and $\Omega_T$ are, respectively, the present density parameters of pressureless (baryonic + dark) matter, dark matter, baryons and the total energy. The function $g(\Omega)$ is defined by $$g(\Omega) = \frac{5}{2}\Omega\biggr[\Omega^{\frac{4}{7}} -
\Omega_{\Lambda0} + \biggr(1 + \frac{\Omega}{2}\biggl)\biggr(1 +
\frac{\Omega_{\Lambda0}}{70}\biggl)\biggl]^{- 1}.$$ The transfer function defined above represents the fitting of the complete numerical evaluation.
A simplified version of the transfer function, which keeps all its essential features, can be obtained by integrating the perturbed equations for the coupled system containing radiation and pressureless matter, from a very high redshift until today [@weinberg; @winfried]. The starting point is given by the Einstein equations and the conservation law for the energy-momentum tensor: $$\begin{aligned}
R_{\mu\nu} &=& 8\pi G\sum_i\biggr\{T^i_{\mu\nu}
- \frac{1}{2}g_{\mu\nu}T^i\biggl\},\\
{T^{\mu\nu}_i}_{;\mu} &=& 0,\end{aligned}$$ where the indice denotes the $i\,th$ fluid component. One of them will be radiation. The other one will be the pressureless matter in the $\Lambda$CDM case, or the vacuum-matter interacting fluids in our case (remember that, in our case, the pressureless matter is not independently conserved, since it interacts with vacuum). Introducing the perturbations, $g_{\mu\nu} = g^0_{\mu\nu} +
h_{\mu\nu}$, $\rho_i = \rho_i^0 + \delta\rho$, $p_i = p_i^0 + \delta
p_i$, with ($g^0_{\mu\nu}$, $\rho_i^0$, $p_i^0$) being the background solutions, and imposing the synchronous coordinate condition $h_{\mu 0} = 0$, we end up with the following set of coupled equations: $$\begin{aligned}
\label{p1}
\ddot h + 2\frac{\dot a}{a}\dot h &=& \rho_m \delta_m + 2\rho_R \delta_R,\\
\label{p2}
\dot\delta_m - \frac{\dot\Lambda}{\rho_m}\delta_m &=& \frac{\dot h}{2},\\
\label{p3}
\dot\delta_R + \frac{4}{3}\biggr\{\frac{v}{a} - \frac{\dot h}{2}\biggl\} &=& 0,\\
\dot v &=& \frac{k^2}{4a}\delta_R,\end{aligned}$$ where $h = h_{kk}/a^2$, $\delta_m$ and $\delta_R$ are the density contrast for matter and radiation respectively, $v$ is connected with the peculiar velocities of the perturbed radiative fluid, and in the $\Lambda$CDM case $\dot{\Lambda}$ is, evidently, zero.
We now eliminate the variable $\dot h$ using (\[p2\]), divide all the expressions by $H_0^2$ and rewrite the resulting equations in terms of the redshift $z$, which becomes the new dynamical variable. In the $\Lambda$CDM case the system of equations is reduced to $$\begin{aligned}
\delta_m'' - \frac{g[z]}{f[z]}\frac{\delta_m'}{1 + z} =
\frac{3}{2f[z]}\biggr\{\Omega_{m0}(1 + z)\delta_m
+ 2\Omega_{R0}(1 + z)^2\delta_R\biggl\},\\
\delta_R' - \frac{4}{3}\biggr\{\frac{v}{\sqrt{f[z]}} +
\delta_m'\biggl\} = 0,\\
v' = -
\biggr(\frac{k\,l_{H0}}{2}\biggl)^2\frac{\delta_R}{\sqrt{f[z]}},\end{aligned}$$ where the primes indicate derivative with respect to the redshift $z$. The background functions $f[z]$ and $g[z]$ are given by $$\begin{aligned}
f[z] &=& \frac{\dot a^2}{a^2} = \Omega_{m0}(1 + z)^3 +
\Omega_{R0}(1 + z)^4 + \Omega_{\Lambda0}, \\
g[z] &=& \frac{\ddot a}{a} = - \frac{1}{2} \Omega_{m0}(1 + z)^3 -
\Omega_{R0}(1 + z)^4 + \Omega_{\Lambda0}.\end{aligned}$$ Integrating, for example, from $z = 10^8$ (when the initial spectrum is supposed to be scale invariant, i.e., $\delta_m, \delta_R \propto
\sqrt{k}$) until today, $z = 0$, we can reproduce the BBKS transfer function with about $10\%$ of precision.
We can perform the same calculation for the present model, finding the following set of perturbed equations: $$\begin{aligned}
\delta_m'' - \biggr\{\frac{\Omega_\Lambda'}{\Omega_m} +
\frac{g_1[z]}{f_1[z]}\frac{1}{(1 + z)^2}\biggl\}\delta_m' +
\biggr\{\frac{g_1[z]}{f_1[z]}\frac{\Omega_\Lambda'}{\Omega_m}\frac{1}{(1
+ z)^2} - \frac{\Omega_\Lambda''}{\Omega_m} +
\frac{\Omega_m'\Omega_\Lambda'}{\Omega_m^2}\biggl\}\delta_m
= \nonumber\\
\frac{3}{2}\frac{1}{f_1[z](1 + z)^4}\biggr\{\Omega_m\delta_m +
2\Omega_R\delta_R\biggl\},\\
\delta_R' - \frac{4}{3}\biggr\{\frac{v}{(1 + z)\sqrt{f_1[z]}} +
\delta_m' - \frac{\Omega_\Lambda'}{\Omega_m} \delta_m \biggl\} = 0,\\
v' = - \biggl(\frac{k\,l_{H0}}{2}\biggl)^2\frac{\delta_R}{(1 +
z)\sqrt{f_1[z]}}.\end{aligned}$$ In these equations, we use the following definitions: $$\begin{aligned}
f_1[z] &=& \dot a^2 = \frac{1}{(\Omega_{\Lambda0} + \Omega_{m0})(1 +
z)^2}\biggr\{\Omega_{\Lambda0}
+ \Omega_{m0}(1 + z)^{3/2}\biggl\}^2\nonumber\\ &+& (1 + z)^2\Omega_{R0},\\
g_1[z] &=& \ddot a = - \frac{(1 + z)^2}{2}f_1'[z], \\
\Omega_{m}(z) &=&
\frac{\Omega_{\Lambda0}\Omega_{m0}}{\Omega_{\Lambda0} +
\Omega_{m0}}\biggr\{\frac{\Omega_{m0}}{\Omega_{\Lambda0}}(1 + z)^3 +
(1 + z)^\frac{3}{2}\biggl\}, \\
\Omega_\Lambda(z) &=& \frac{\Omega_{\Lambda0}^2}{\Omega_{\Lambda0} +
\Omega_{m0}}\biggr\{1 + \frac{\Omega_{m0}}{\Omega_{\Lambda0}}(1 +
z)^\frac{3}{2}\biggl\}.\end{aligned}$$
![The matter power spectra as given by the BBKS transfer function (blue), the approximative numerical analysis used here for $\Lambda$CDM (red) and for the interacting model (violet). The data come from the 2dFGRS galaxy survey program [@2dFGRS]. It has been used $\Omega_{m0} = 0.36$ for the interacting model.](NewFigure3.ps)
![The matter power spectra as given by the BBKS transfer function (blue), the approximative numerical analysis used here for $\Lambda$CDM (red) and for the interacting model (violet). The data come from the 2dFGRS galaxy survey program [@2dFGRS]. It has been used $\Omega_{m0} = 0.48$ for the interacting model.](NewFigure4.ps)
In figures $3$ and $4$ we display the results for the exact transfer function for the $\Lambda$CDM model (blue), the corresponding numerical approximation (red) and the approximative transfer function for the present model (violet). The observational data come from the 2dFGRS galaxy survey program [@2dFGRS]. In the case of the interaction model we used, in Figure $3$, $\Omega_{m0} = 0.36$, the concordance value obtained from the joint analysis of type Ia supernovas, BAO and CMB [@Jailson2]. In Figure $4$, on the other hand, we have used $\Omega_{m0} = 0.48$. We see that in the first case there is a substantial suppression of power, while in the second case, where the dark matter parameter has been increased, the agreement is excellent.
Hence, concerning the matter power spectra, the interacting model with homogeneous matter production requires an almost double quantity of dark matter with respect to the $\Lambda$CDM model.
Conclusions
===========
In spite of the physical plausibility of a time dependent cosmological term, a complete theoretical development of this idea, including the microscopic details of the vacuum-matter interaction, is still lacking. On the other hand, macroscopic approaches depend on some phenomenological hypothesis, leading some times to diverse prescriptions for the vacuum decay[^5]. For this reason, a careful comparison with current observations is very important, playing the role of corroborating or ruling out the different models.
We have already analysed the supernova observations [@Jailson], obtaining good fits and cosmological parameters in accordance with other independent tests, as the age of globular clusters and dynamical limits to the matter density [@dynamical]. Other precise tests, as the position of the first acoustic peak of the cosmic microwave background and the baryonic acoustic oscillations have also been performed [@Jailson2], showing a good concordance when jointed to the supernova analysis.
In the present paper we have studied the evolution of matter density perturbations, in particular the contrast suppression associated to the process of matter production. We have shown that, even in the case of a homogeneous production, the evolution of the contrast is the same as in the standard recipe along the entire era of galaxy formation, diverging from the later only for $z < 5$.
On the other hand, the suppression would be dominant for future times, and this may have an interesting relation with another problem related to the cosmological term, namely the approximate coincidence between the present densities of matter and dark energy. Indeed, we can see from figures 1 and 2 that the matter contrast has its maximum just before today, when matter and vacuum give similar contributions to the total density. The largest structures formed until now tend to disaggregate in the future, and their existence then coincides with the time of approximate equality between the matter and vacuum densities.
This could alleviate the cosmic coincidence problem, if galaxies also follow such a process. However, we should remember that galaxies have left the linear regime of growth a long time ago, and that now their evolution is non-linear, driven essentially by their self-gravitation. Therefore, an explanation of the cosmic coincidence in the terms above will depend on a non-linear study of density perturbations in the context of the present model. Only such an investigation would tell us whether the contrast suppression described here can affect smaller structures like galaxies.
Its also important to have in mind that the homogeneity of the matter production, implicit in the derivation of solution (\[solution\]) and in our simplified relativistic treatment, is just an [*ad hoc*]{} hypothesis, to be verified from both the theoretical and observational viewpoints. For a constant, non-interacting vacuum term it is certainly true, but not necessarily in the present case. Any inhomogeneity of the vacuum density around matter distributions may lead to an inhomogeneous production, reducing in this way the contrast suppression. This would allow us to fit the observed power spectrum with a smaller matter density, closer to the concordance value obtained in [@Jailson2]. Whether the matter contrast will still have a maximum around the present time, with the discussed implications for the coincidence problem, is a matter of investigation. A relativistic study of this case, that is, with $\delta \Lambda \neq
0$, is already in progress.
Acknowledgements {#acknowledgements .unnumbered}
================
H. A. Borges and C. Pigozzo were supported by Capes. S. Carneiro and J. C. Fabris are partially supported by the Brazilian Council for Scientific Research (CNPq).
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[^1]: Properly speaking, we should also consider the pressure and energy associated to the very process of matter production, that is, the energy-momentum tensor of the interaction between matter and vacuum. In this sense, decaying vacuum models do not differ essentially from interacting dark energy models [@Zimdahl], with the scalar function $\Lambda$ replaced by a scalar field interacting with matter. Nevertheless, if the vacuum decays into non-relativistic particles, as we will consider here, the interaction term can be neglected, and the above decomposition may be considered a good approximation.
[^2]: As already commented, we will assume that the vacuum is decaying into non-relativistic particles, in order to avoid any conflict with CMB observations and with the observed coldness of dark matter. We will also suppose that only dark matter is produced, since the baryon content is well constrained by nucleosynthesis. Evidently, these assumptions cannot be verified without a microscopic theory of the vacuum-matter interaction.
[^3]: These are the same expressions we obtain for a generalized Chaplygin gas (characterized by the equation of state $p_{ch} = - A/
\rho_{ch}^{\alpha}$ [@Julio]), if we choose $\alpha = - 1/2$ and $A = \sqrt{\sigma^2/3}$ (see [@Sandro] for a detailed discussion about this and other curious equivalences between dark energy models). Note, however, that the oscillations in the evolution of density perturbations characteristic of a Chaplygin gas [@Waga2] are not present in our case, as we will see below.
[^4]: Note that the inclusion of conserved radiation changes the dynamics, and, consequently, the production of matter, $\Lambda(z)$ and $\rho_m(z)$ also change. Therefore, the exact generalization of (\[Hz\]) requires a reanalysis of the dynamics. Nevertheless, as $\Omega_{R0} \approx 10^{-4} << 1$, when the vacuum and the matter production begin to have importance, the radiation is negligible, and vice-versa. In this way, (\[Hzalto\]) can be considered a very good approximation. Indeed, a numerical analysis in the range $0 < z < 10^4$ has shown that the difference between (\[Hzalto\]) and the exact $H(z)$ is as small as $0.01\%$.
[^5]: For example, the linear dependence of $\Lambda$ on the Hubble parameter we use here contrasts with the quadratic dependence used in reference [@Fabris]. In that work, the quadratic dependence is due to the computation of quantum effects of matter field in a cosmological background, which leads to a running cosmological term. The authors also used the matter power spectrum data to constrain the fundamental parameters of the quantum model.
|
---
abstract: '[We have studied the spectral and timing behaviour of the atoll source 4U 1608–52. We find that the timing behaviour of 4U 1608–52 is almost identical to that of the atoll sources 4U 0614+09 and 4U 1728–34. Recently Muno, Remillard & Chakrabarty (2002) and Gierlinski & Done (2002) suggested that the atoll sources trace out similar three–branch patterns as the Z sources. The timing behaviour is not consistent with the idea that 4U 1608–52 traces out a three–branched Z shape in the color–color diagram along which the timing properties vary gradually.]{}'
author:
- 'S. van Straaten$^1$, M. van der Klis$^1$ and M. Méndez$^2$'
title: 'The atoll source 4U 1608–52 is not a Z source!'
---
2[cm$^2$ ]{} 1[s$^{-1}$ ]{}
Introduction and Data Analysis
==============================
In this work we use all available data from RXTE’s PCA to simultaneously study the spectral and timing properties in the transient low mass X-ray binary 4U 1608–52. We calculate a color–color diagram. As the energy spectrum of a source changes, it moves through the diagram. To study the timing we calculate Fourier power density spectra and fit them with the multi–Lorentzian fit function; a sum of Lorentzian components plus an occasional power law to fit the very low frequency noise [@bpk02; @vstr02]. It has been recently proposed [@muno02; @gier02] that the atoll sources trace out similar three–branch patterns as the Z sources; one of our goals in this work is to test this hypothesis.\
Stepping through the color–color diagram
========================================
As a first step we go through the data in chronological order and look at the timing properties (per observation) and the position of the source in the color–color diagram. The obtained lightcurve and color–color diagram for 4U 1608–52 can be grouped into 3 parts. The first part ranges from 1996 March 3 to December 28 (the decay of the 1996 outburst, see [@berg96]), the second part from 1998 February 3 to September 29 (the 1998 outburst, see [@mendez98]) and the third part from 2000 March 6 to May 10. In practice most data was available for the second part of the lightcurve (the 1998 outburst) so we will present the results for the second part of the lightcurve first. The results can then serve as a template for the rest of the data.
For the 1998 outburst we find 7 different color diagram position/power spectral classes and we can confirm the result of [@mendez98] that the color–color diagram shows the classical atoll C shape (see [@hk89]). For the decay of the 1996 outburst we find one additional class that was different from those observed during the 1998 outburst. The color–color diagram deviates from the C shape and if we sort the classes by characteristic frequency, the color–color diagram seems to follow an $\epsilon$ shape instead of the classical atoll shape (see also below). In the third part of the data the source countrates were low and in most cases it was impossible to identify any power spectral features, therefore the classification for this part of the data was solely done on position in the color–color diagram.
Combining the power spectra
===========================
To improve the statistics of the power spectra we add up all the continuous time intervals in each of the 8 classes. To avoid doubling of the lower kilohertz QPO peak we split the class marked with the filled circles up into three parts depending on lower kilohertz QPO frequency. In Figure \[fig:cc\_int\] we show the resulting 10 intervals marked from A to J in the color–color diagram.\
We fit the power density spectrum of each interval of Figure \[fig:cc\_int\] with the multi–Lorentzian fit function. In Figure \[fig:freq\_freq\] we show the characteristic frequencies of the Lorentzians used to fit the power spectra of 4U 1608–52 plotted versus the characteristic frequency of the Lorentzian identified as the upper kilohertz QPO, together with the results of [@vstr02] for 4U 1728–34 and 4U 0614+09. The results of the multi–Lorentzian fit to 4U 1608–52 are remarkably similar to those of 4U 1728–34 and 4U 0614+09.
Interval C in Figure \[fig:cc\_int\] represents a deviation from the classical atoll shape (see also [@muno02] and [@gier02]). According to the interpretation of [@muno02] and [@gier02] interval C would represent the analogon to the horizontal branch of the Z sources. As in Z sources the characteristic frequencies of the timing features increase along the Z starting at the horizontal branch, the measured frequencies in C should then be lower than those in A and B. Instead we find that all characteristic frequencies clearly increase from A to J so that the frequencies in C are intermediate between those in B and D. This is not consistent with the idea that 4U 1608-52 traces out a three-branched Z shape in the color-color diagram,
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by NWO SPINOZA grant 08–0 to E.P.J. van den Heuvel, by the Netherlands Organization for Scientific Research (NWO), and by the Netherlands Research School for Astronomy (NOVA). This research has made use of data obtained through the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA/Goddard Space Flight Center.
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|
---
abstract: 'We consider UL and LU stochastic factorizations of the transition probability matrix of a random walk on the integers, which is a doubly infinite tridiagonal stochastic Jacobi matrix. We give conditions on the free parameter of both factorizations in terms of certain continued fractions such that this stochastic factorization is always possible. By inverting the order of the factors (also known as a Darboux transformation) we get new families of random walks on the integers. We identify the spectral matrices associated with these Darboux transformations (in both cases) which are basically conjugations by a matrix polynomial of degree one of a Geronimus transformation of the original spectral matrix. Finally, we apply our results to the random walk with constant transition probabilities with or without an attractive or repulsive force.'
address:
- |
Manuel D. de la Iglesia\
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510, Ciudad de México, México.
- |
Claudia Juarez\
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510, Ciudad de México, México.
author:
- 'Manuel D. de la Iglesia'
- Claudia Juarez
date:
-
-
title: The spectral matrices associated with the stochastic Darboux transformations of random walks on the integers
---
Introduction
============
The main goal of this paper is to describe the spectral matrices associated with the discrete Darboux transformations of the one-step transition probability matrix $P$ of a random walk on the integers ${{\mathbb Z}}$. This transition probability matrix is a doubly infinite tridiagonal stochastic matrix, also known as a Jacobi matrix or Jacobi operator (see below) acting on the space $\ell_{\pi}^2({{\mathbb Z}})$ for certain sequence $\pi=(\pi_n)_{n\in{{\mathbb Z}}}$. The case of random walks on the nonnegative integers ${{\mathbb Z}}_{\geq0}$ has been recently considered by F.A. Grünbaum and one of the authors of this paper in [@GdI3]. The main motivation for this UL and LU stochastic factorization is to divide the probabilistic model associated with the random walk into two different and simpler experiments, and combine them together to obtain a simpler description of the original probabilistic model (see applications to urn models in [@GdI3]).
We start by analyzing the conditions under we can perform a stochastic UL factorization of the form $P=P_UP_L$ or a stochastic LU factorization of the form $P=\widetilde P_L\widetilde P_U$, where all factors (bidiagonal matrices) are also stochastic matrices. In the case of the UL factorization, as the situation of random walks on ${{\mathbb Z}}_{\geq0}$, we still have one free parameter. But one important difference now is that for the LU factorization we also have one *free parameter*, something that did not happen for random walks on ${{\mathbb Z}}_{\geq0}$, where the factorization was unique. In both cases, the factorization, if it can be achieved in terms of stochastic factors, will represent a family of factorizations of the original transition probability matrix $P$. In [@GdI3] it is shown that this free parameter has to be bounded from above by certain continued fraction if we want to guarantee that the factors are still stochastic matrices. In our case, since we are dealing with doubly infinite stochastic matrices, this free parameter (in both cases) has to be not only bounded from above but also bounded from below by another continued fraction which is built from the negative states of the original random walk. This will be the content of Section \[sec2\]. UL and LU factorizations of stochastic matrices have been considered earlier in the literature (see for instance [@Gr1; @Gr2; @Hey; @Vig]) but these factorizations are different from the one we try to consider here where all matrices involved are stochastic (see [@GdI3] for an extended discussion about this matter).
Once we have a stochastic UL or LU factorization of a tridiagonal stochastic matrix we can make use of the so-called *discrete Darboux transformation*, consisting of inverting the order of the factors. The new matrices $\widetilde P=P_LP_U$ and $\widehat P=\widetilde P_U\widetilde P_L$ will also be doubly infinite tridiagonal and stochastic matrices. Since both factorizations come with one free parameter, we will have a family of new random walks different in general from the original one. These discrete Darboux transformations have been studied before in the context of the theory of orthogonal polynomials, in particular in the description of some families of Krall polynomials (see [@GH; @GHH; @SZ; @Yo; @Ze]). It has played an important role in the study of integrable systems (see [@MS]). An important issue that has not been considered before, as far as the authors know, is how to relate the spectral matrix $\Psi(x)$ associated with $P$ with the spectral matrices $\widetilde\Psi(x)$ or $\widehat\Psi(x)$ associated with $\widetilde P$ or $\widehat P$, respectively. By spectral matrix we mean that the spectral analysis of $P$ comes now with *three measures* $\psi_{\alpha,\beta}, \alpha,\beta=1,2,$ (two positive and one signed measure, and $\psi_{12}=\psi_{21}$ due to the symmetry) and they can be written in a $2\times 2$ spectral matrix of the form $$\begin{aligned}
\Psi(x)&=\begin{pmatrix} \psi_{11}(x) & \psi_{12}(x)\\ \psi_{12}(x) & \psi_{22}(x) \end{pmatrix},\end{aligned}$$ which turns out to be a proper weight matrix in the context of the theory of matrix-valued orthogonal polynomials. In fact, a random walk in ${{\mathbb Z}}$ can be viewed as a special type of discrete-time *quasi-birth-and-death process* with state space ${{\mathbb Z}}_{\geq0}\times\{1,2\}$. These processes can be defined in general in state spaces of the form ${{\mathbb Z}}_{\geq0}\times\{1,\ldots, N\}$ for $N\geq1$ a positive integer (see [@LaR; @Neu] for general references). The spectral analysis of these processes has been considered for instance in [@DRSZ; @G2; @G1; @GdI2; @dIR] to mention a few.
For random walks on $\mathbb{Z}_{\geq0}$ it is very well-known that the spectral measure associated with the Darboux transformation of a UL factorization is given by a so-called *Geronimus transformation* of the original spectral measure, while for the LU factorization is given by a *Christoffel transformation*. For random walks on $\mathbb{Z}$ we will show that the spectral matrices associated with these Darboux transformations (in both cases) are conjugations of the form $$\widetilde \Psi (x)=\bm S_0(x) \Psi_S (x) \bm S_0^*(x),\quad \widehat \Psi (x)=\bm T_0(x) \Psi_T (x) \bm T_0^*(x),$$ where $\Psi_S (x), \Psi_T (x)$ are Geronimus transformations of the original spectral matrix $\Psi$ and $\bm S_0(x), \bm T_0(x)$ are certain matrix polynomials of degree one (see Theorems \[thmorto\] and \[thmorto2\] in Section \[sec3\]). In [@GdI4] a first attempt has been done to study stochastic Darboux transformations of block tridiagonal stochastic matrices, which are the transition probability matrices of discrete-time quasi-birth-and-death processes. In that paper the authors only consider one (Jacobi type) example previously introduced in [@GPT1] (see also [@GPT2; @GPT3]). The UL factorization depends now on one free matrix-valued parameter and it is not clear how to transform the corresponding spectral weight matrices associated with the discrete Darboux transformation in general. In the case of random walks on ${{\mathbb Z}}$ we will only have one free (real) parameter. Our results can give some insights about how to compute the spectral matrix of the Darboux transformations for tridiagonal stochastic block matrices in general. For a different application of the Darboux transformation in the context of the noncommutative bispectral problem see [@GHY; @G3; @Zu] and references therein.
Once we have the spectral matrix it is easy to analyze the corresponding random walk in terms of the two independent families of polynomials which arise as a solution of the eigenvalue equation. For the case of random walks on $\mathbb{Z}_{\geq0}$ this was first done in a series of papers by S. Karlin and J. McGregor (inspired by work by W. Feller and H.P. McKean) in the 1950s (see [@KMc2; @KMc3; @KMc6]) where they studied first continuous-time birth-and-death processes and then the case of discrete-time random walks. Apart from an explicit expression of the $n$-step transition probabilities and the invariant measure, it is possible to study some other probabilistic properties using spectral methods such as recurrence, absorbing times, first return times or limit theorems. In the last section of [@KMc6] one can find the first attempt to perform the spectral analysis of a random walk on ${{\mathbb Z}}$ using orthogonal polynomials. We will recall this approach at the beginning of Section \[sec3\]. After that, apart from [@Ber], there are not so many references concerning the spectral analysis of doubly infinite Jacobi operators acting on $\ell_{\pi}^2({{\mathbb Z}})$. In [@Pru], W.E. Pruitt studied the case of birth-and-death processes on ${{\mathbb Z}}$ (also known as bilateral birth-and-death processes). An example of this approach can be found in the last section of [@ILMV]. A more theoretical work about the spectral theory of Jacobi operators acting on $\ell_{\pi}^2({{\mathbb Z}})$ was given by D.E. Masson and J. Repka in [@MR] and revisited recently in [@DIW].
Finally, we will apply our results to two examples in Section \[sec4\]. The first one is the random walk on ${{\mathbb Z}}$ with constant transition probabilities, while the second one is the random walk on ${{\mathbb Z}}$ with constant transition probabilities but allowing an attractive or repulsive force to or from the origin. In both cases we study the conditions under we get a stochastic UL and LU factorization, give the corresponding spectral matrices and the spectral matrices associated with both discrete Darboux transformations. As a final remark we will show that it is possible to choose certain values of the free parameters such that the Darboux transformation (both from the UL or the LU factorization) is *invariant*, i.e. we get the same random walk after we perform the Darboux transformation. This phenomenon it is not possible for Darboux transformations of random walks on ${{\mathbb Z}}_{\geq0}$.
Stochastic UL and LU factorization on the integers {#sec2}
==================================================
Let $\{X_t : t=0,1,\ldots\}$ be an irreducible random walk on the integers $\mathbb{Z}$ with transition probability matrix $P$ given by $$\label{QZ}
P=
\left(
\begin{array}{ccc|cccc}
\ddots&\ddots&\ddots&&&\\
&c_{-1}&b_{-1}&a_{-1}&&&\\
\hline
&&c_0&b_0&a_0&&\\
&&&c_1&b_1&a_1&\\
&&&&\ddots&\ddots&\ddots
\end{array}
\right).$$ The matrix $P$ is stochastic, i.e. all entries are nonnegative and $$c_n+b_n+a_n=1,\quad n\in{{\mathbb Z}}.$$ Since the random walk is irreducible then we have that $0<a_n,c_n<1, n\in{{\mathbb Z}}$. A diagram of the transitions between the states is given by
Let us perform a UL factorization of $P$ in the following way $$\label{QZUL}
P=\left(\begin{array}{cc|cccc}
\ddots&\ddots&&\\
0&y_{-1}&x_{-1}&&\\
\hline
&0&y_0&x_0&\\
&&0&y_1&x_1\\
&&&&\ddots&\ddots
\end{array}
\right)\left(\begin{array}{ccc|ccc}
\ddots&\ddots&&&\\
&r_{-1}&s_{-1}&0&\\
\hline
&&r_0&s_0&0\\
&&&r_1&s_1&0\\
&&&&\ddots&\ddots
\end{array}
\right)=P_UP_L,$$ where $P_U$ and $P_L$ are also stochastic matrices. This means that all entries of $P_U$ and $P_L$ are nonnegative and $$\label{stpul}
x_n+y_n=1,\quad s_n+r_n=1,\quad n\in{{\mathbb Z}}.$$ A direct computation shows that $$\begin{aligned}
\nonumber a_n&=x_ns_{n+1},\\
\label{ULd}b_n&=x_nr_{n+1}+y_ns_n,\quad n\in{{\mathbb Z}},\\
\nonumber c_n&=y_nr_n.\end{aligned}$$ By the irreducibility conditions we immediately have that $0<x_n,y_n,s_n,r_n<1, n\in{{\mathbb Z}}$. The Markov chain associated with $P_U$ is a pure birth random walk on ${{\mathbb Z}}$ with diagram
while $P_L$ is a pure death random walk on ${{\mathbb Z}}$ with diagram
As in the case of random walks on ${{\mathbb Z}}_{\geq0}$ (see [@GdI3]) we can compute all entries of $P_U$ and $P_L$ in terms of only *one free parameter*, namely $y_0$. Indeed, for nonnegative values of the indices and $y_0$ fixed we can compute $x_0, s_1, r_1, y_1, x_1, s_2, r_2, y_2,\ldots$ recursively using and . Similarly, for negative values of the indices and $y_0$ fixed we can compute $r_0, s_0, x_{-1}, y_{-1}, r_{-1}, s_{-1}, x_{-2}, y_{-2},\ldots$ recursively using again and .
Following the same steps as in Lemma 2.1 of [@GdI3] we have that for a decomposition like in , i.e. $P=P_UP_L$, the matrix $P_U$ is stochastic if and only if the matrix $P_L$ is stochastic. We also have, following , the following relations $$\label{relys}
y_n=\frac{c_n}{1-s_n},\quad s_{n+1}=\frac{a_{n}}{1-y_n},\quad n\in{{\mathbb Z}}.$$ Although we can compute all coefficients $x_n, y_n, s_n, r_n$ in terms of one free parameter $y_0$, we can not infer anything about the positivity of these coefficients. This will be the goal of the next theorem. As it was done in Theorem 2.1 of [@GdI3] for random walks on ${{\mathbb Z}}_{\geq0}$ the matrices $P_U$ and $P_L$ are stochastic if and only if $0\leq y_0\leq H$, where $H$ is the continued fraction given below by . The difference now for random walks on ${{\mathbb Z}}$ is that the free parameter $y_0$ will be also bounded below by another number $H'$ defined by a continued fraction generated by the probabilities of the negative states of the random walk. We recommend the reference [@Wa] for the reader unfamiliar with continued fractions (as well as [@GdI3] for the case of random walks on $\mathbb{Z}_{\geq0}$).
Let $H$ and $H'$ be the continued fractions generated by alternatively choosing $a_n$ and $c_n$ in different directions, i.e. $$\label{cfUL}
H=1-\cfrac{a_0}{1-\cfrac{c_1}{1-\cfrac{a_1}{1-\cfrac{c_2}{1-\cdots}}}},\quad H'=\cfrac{c_0}{1-\cfrac{a_{-1}}{1-\cfrac{c_{-1}}{1-\cfrac{a_{-2}}{1-\cdots}}}}.$$ In a different notation $$H=1-\cFrac{a_0}{1}-\cFrac{c_1}{1}-\cFrac{a_1}{1}-\cFrac{c_2}{\cdots},\quad H'=\cFrac{c_0}{1}-\cFrac{a_{-1}}{1}-\cFrac{c_{-1}}{1}-\cFrac{a_{-2}}{\cdots}.$$ For each continued fraction, consider the corresponding sequence of *convergents* $(h_{n})_{n\geq0}$ and $(h_{-n}')_{n\geq0}$, given by $$\label{hhh}
h_{n}=\frac{A_n}{B_n},\quad h_{-n}'=\frac{A_{-n}'}{B_{-n}'}.$$ Recall that the convergents of a continued fraction $H (H')$ are the sequence of truncated continued fractions of $H (H')$ and these are always rational numbers. In [@GdI3] (see also [@Wa]) it is proved that the numbers $A_n,B_n$ can be recursively obtained using the following formulas $$\begin{aligned}
A_{2n}&=A_{2n-1}-c_nA_{2n-2},\quad n\geq1,\quad A_{2n+1}=A_{2n}-a_nA_{2n-1},\quad n\geq0,\quad A_{-1}=1,\quad A_0=1,\\
B_{2n}&=B_{2n-1}-c_nB_{2n-2},\quad n\geq1,\quad B_{2n+1}=B_{2n}-a_nB_{2n-1},\quad n\geq0,\quad B_{-1}=0,\quad B_0=1.\end{aligned}$$ In the same way the numbers $A_{-n}',B_{-n}'$ can be recursively obtained using $$\begin{aligned}
A_{-2n}'&=A_{-2n+1}'-a_{-n}A_{-2n+2}',\quad n\geq1,\quad A_{-2n-1}'=A_{-2n}'-c_{-n}A_{-2n+1}',\quad n\geq0,\quad A_{1}'=-1,\quad A_0'=0,\\
B_{-2n}'&=B_{-2n+1}'-a_{-n}B_{-2n+2}',\quad n\geq1,\quad B_{-2n-1}'=B_{-2n}'-c_{-n}B_{-2n+1}',\quad n\geq0,\quad B_{1}'=0,\quad B_0'=1.\end{aligned}$$ Using these relations it is not hard to prove that $$\label{rels2}
\begin{split}
A_{-2n}'B_{-2n-1}'-A_{-2n-1}'B_{-2n}'&=-c_0a_{-1}c_{-1}\cdots a_{-n}c_{-n},\quad n\geq0,\\
A_{-2n-1}'B_{-2n-2}'-A_{-2n-2}'B_{-2n-1}'&=-c_0a_{-1}c_{-1}\cdots c_{-n}a_{-n-1},\quad n\geq0.
\end{split}$$ Then we have the following:
\[thmfracUL\] Let $H$ and $H'$ be the continued fractions given by and the corresponding convergents $h_n$ and $h_{-n}$ defined by . Assume that $$\label{AnBn}
0<A_n<B_n,\quad\mbox{and}\quad0<A_{-n}'<B_{-n}',\quad n\geq1.$$ Then both $H$ and $H'$ are convergent. Moreover, let $P=P_UP_L$ as in . Assume that $H'\leq H$. Then, both $P_U$ and $P_L$ are stochastic matrices if and only if we choose $y_0$ in the following range $$\label{yy0r}
H'\leq y_0\leq H.$$
The convergence of $H$ and the upper bound for $y_0$ is proved in Theorem 2.1 of [@GdI3]. From and using the assumptions we have that $$\begin{aligned}
h_{-2n-2}'-h_{-2n-1}'&=\frac{A_{-2n-2}'}{B_{-2n-2}'}-\frac{A_{-2n-1}'}{B_{-2n-1}'}=\frac{c_0a_{-1}c_{-1}\cdots c_{-n}a_{-n-1}}{B_{-2n-1}'B_{-2n-2}'}>0,\quad n\geq0,\\
h_{-2n-1}'-h_{-2n}'&=\frac{A_{-2n-1}'}{B_{-2n-1}'}-\frac{A_{-2n}'}{B_{-2n}'}=\frac{c_0a_{-1}c_{-1}\cdots a_{-n}c_{-n}}{B_{-2n}'B_{-2n-1}'}>0,\quad n\geq0.\end{aligned}$$ Therefore we have $$0=h_0'<h_{-1}'<h_{-2}'<\cdots<h_{-2n}'<h_{-2n-1}'<h_{-2n-2}'<\cdots<1,$$ that is, $(h_{-n}')_{n\geq0}$ is a bounded strictly increasing sequence, so it is convergent to $H'$.
For the lower bound for $y_0$, assume first that both $P_U$ and $P_L$ are stochastic matrices, so that $0<x_n,y_n,s_n,r_n<1,n\in{{\mathbb Z}}$. Then we have, using , that $$\begin{aligned}
s_0=&1-\frac{c_0}{y_0}>0 \Leftrightarrow \frac{c_0}{y_0}<1 \Leftrightarrow c_0<y_0 \Leftrightarrow h_{-1}'<y_0,\end{aligned}$$ and $$\begin{aligned}
y_{-1}=&1-\frac{a_{-1}}{s_0}>0 \Leftrightarrow \frac{a_{-1}}{s_0}<1 \Leftrightarrow a_{-1} <s_0
\Leftrightarrow a_{-1}< 1-\frac{c_0}{y_0}\\& \quad \Leftrightarrow 1-a_{-1}>\frac{c_0}{y_0} \Leftrightarrow \frac{1}{1-a_{-1}}<\frac{y_0}{c_0} \Leftrightarrow \frac{c_0}{1-a_{-1}}<y_0 \Leftrightarrow h_{-2}'<y_0.\end{aligned}$$ Following the same argument we have $$\begin{split}
y_{-n}=1-\frac{a_{-n}}{s_{-n+1}}>0& \Leftrightarrow a_{-n}<s_{-n+1} \Leftrightarrow 1-a_{-n}>\frac{c_{-n+1}}{y_{-n+1}} \Leftrightarrow \frac{c_{-n+1}}{1-a_{-n}}<y_{-n+1} \\
&\Leftrightarrow \frac{c_{-n+1}}{1-a_{-n}}<1-\frac{a_{-n+1}}{s_{-n+2}} \Leftrightarrow \frac{a_{-n+1}}{1-{\displaystyle}\frac{c_{-n+1}}{1-a_{-n}}}<s_{-n+2} \\
&\cdots \Leftrightarrow \cFrac{a_{-n+n-1}}{1}-\cFrac{c_{-1}}{1}-\cFrac{a_{-2}}{1}\cdots -\cFrac{a_{-n}}{1} <s_{-n+n}\\
&\Leftrightarrow \cFrac{a_{-1}}{1}-\cFrac{c_{-1}}{1}-\cFrac{a_{-2}}{1}\cdots -\cFrac{a_{-n}}{1} <1-\frac{c_0}{y_0}\\
&\Leftrightarrow \cfrac{c_0}{1}- \cFrac{a_{-1}}{1}-\cFrac{c_{-1}}{1}-\cFrac{a_{-2}}{1}\cdots -\cFrac{a_{-n}}{1} <y_0 \\
&\Leftrightarrow h_{-2n}'<y_0.
\end{split}$$ Therefore we always have $$0=h_0'<h_{-n}'<H'\le y_0,\quad n\geq0,$$ and we get the lower bound for $y_0$. On the contrary, if holds, in particular we have that $h_{-n}<H'\leq y_0\leq H<h_n$ for every $n\geq0$. Following the same steps as before and Theorem 2.1 of [@GdI3], using an argument of strong induction, will lead us to the fact that both $P_U$ and $P_L$ are stochastic matrices with the conditions that $0<x_n,y_n,s_n,r_n<1,n\in{{\mathbb Z}}$.
Consider now the LU factorization of the stochastic matrix $P$ given by in the following way $$\label{QZLU}
P=\left(\begin{array}{ccc|ccc}
\ddots&\ddots&&&\\
&\tilde r_{-1}&\tilde s_{-1}&0&\\
\hline
&&\tilde r_0&\tilde s_0&0\\
&&&\tilde r_1&\tilde s_1&0\\
&&&&\ddots&\ddots
\end{array}
\right)\left(\begin{array}{cc|cccc}
\ddots&\ddots&&\\
0&\tilde y_{-1}&\tilde x_{-1}&&\\
\hline
&0&\tilde y_0&\tilde x_0&\\
&&0&\tilde y_1&\tilde x_1\\
&&&&\ddots&\ddots
\end{array}
\right)=\widetilde P_L\widetilde P_U,$$ where again $\widetilde P_L$ and $\widetilde P_U$ are also stochastic matrices, i.e. all entries of $\tilde P_L$ and $\tilde P_U$ are nonnegative and $$\label{stplu}
\tilde r_n+ \tilde s_n=1, \quad \tilde y_n +\tilde x_n=1,\quad n \in {{\mathbb Z}}.$$ Now, a direct computation shows that $$\begin{aligned}
\nonumber a_n&=\tilde s_n \tilde x_n,\\
\label{LUd}b_n&=\tilde r_n\tilde x_{n-1}+\tilde s_n\tilde y_n,\quad n\in{{\mathbb Z}},\\
\nonumber c_n&=\tilde r_n\tilde y_{n-1}.\end{aligned}$$ By the irreducibility condition we have that $0<\tilde r_n$, $\tilde s_n$, $\tilde y_n$, $\tilde x_n<1$, $n\in {{\mathbb Z}}$. Now there is an *important difference* if we compare with the case of random walks on ${{\mathbb Z}}_{\geq0}$, where the LU factorization is unique. In this case there will be also one *free parameter*, namely $\tilde r_0$, from where we can compute all the coefficients of the matrices $\widetilde P_L$ and $\widetilde P_U$. If we fix $\tilde r_0$, for the nonnegative values of the indices we can compute $\tilde s_0$, $\tilde x_0$, $\tilde y_0$, $\tilde r_1$, $\tilde s_1$, $\tilde x_1$, $\tilde y_1$, $\tilde r_2$, $\dots$ recursively from and , while for the negative values of the indices we can compute $\tilde y_{-1}$, $\tilde x_{-1}$, $\tilde s_{-1}$, $\tilde r_{-1}$, $\tilde y_{-2}$, $\tilde x_{-2}$, $\tilde s_{-2}$, $\tilde r_{-2}$, $\dots$ recursively again from and .
Following the same steps as in Lemma 2.1 of [@GdI3] we have that for a decomposition like in , i.e. $P=\widetilde P_L \widetilde P_U$, the matrix $\widetilde P_L$ is stochastic if and only if the matrix $\widetilde P_U$ is stochastic and by , the following relations holds $$\label{relyslu}
\tilde r_n=\frac{c_n}{1-\tilde x_{n-1}},\quad \tilde x_{n}=\frac{a_{n}}{1-\tilde r_n},\quad n\in{{\mathbb Z}}.$$
Although we can compute all coefficients $\tilde r_n$, $\tilde s_n$, $\tilde y_n$ and $\tilde x_n$ in terms of one free parameter $\tilde r_0$, we can not infer anything about the positivity of these coefficients. However we have an analogue of Theorem \[thmfracUL\] where now the free parameter $\tilde r_0$ will be bounded by the same continued fractions.
\[thmfracLU\] Let $H$ and $H'$ be the continued fractions given by and the corresponding convergents $h_n$ and $h_{-n}$ defined by . Assume conditions . Then both $H$ and $H'$ are convergent. Moreover, let $P=\widetilde P_L\widetilde P_U$ as in . Assume that $H'\leq H$. Then, both $\widetilde P_L$ and $\widetilde P_U$ are stochastic matrices if and only if we choose $\tilde r_0$ in the following range $$\label{rr0r}
H'\leq \tilde r_0\leq H.$$
The proof is similar to the proof of Theorem \[thmfracUL\] but using instead of .
Stochastic Darboux transformations and the associated spectral matrices {#sec3}
=======================================================================
In the previous section we have shown under what conditions a doubly infinite stochastic matrix $P$ like in can be decomposed as a UL (or LU) factorization where both factors are still stochastic matrices. In both factorizations we have one free parameter. We knew about this fact for the UL factorization, but now for the LU factorization the phenomenon is new. Once we have a UL (or LU) factorization we can perform what is called a *discrete Darboux transformation*. The Darboux transformation has a long history but probably the first reference of a discrete Darboux transformation like we study here appeared in [@MS] in connection with the Toda lattice. We explain now what a Darboux transformation is in our context.
If $P=P_UP_L$ as in , then by inverting the order of the factors we obtain another tridiagonal matrix of the form $$\label{QZULdar}
\widetilde P=P_LP_U=\left(\begin{array}{ccc|ccc}
\ddots&\ddots&&&\\
&r_{-1}&s_{-1}&0&\\
\hline
&&r_0&s_0&0\\
&&&r_1&s_1&0\\
&&&&\ddots&\ddots
\end{array}
\right)\left(\begin{array}{cc|cccc}
\ddots&\ddots&&\\
0&y_{-1}&x_{-1}&&\\
\hline
&0&y_0&x_0&\\
&&0&y_1&x_1\\
&&&&\ddots&\ddots
\end{array}
\right).$$ The new coefficients of the matrix $\widetilde P$ are given by $$\begin{aligned}
\nonumber\tilde{a}_n&=s_nx_{n},\\
\label{coeffDLU}\tilde{b}_n&=r_nx_{n-1}+s_ny_n,\quad n\in{{\mathbb Z}},\\
\nonumber\tilde{c}_n&=r_ny_{n-1}.\end{aligned}$$ The matrix $\widetilde{P}$ is actually stochastic, since the multiplication of two stochastic matrices is again a stochastic matrix. Therefore it gives a *family* of new random walks $\{\widetilde X_t : t=0,1,\ldots\}$ on the integers ${{\mathbb Z}}$ with coefficients $(\tilde{a}_n)_{n\in{{\mathbb Z}}}$, $(\tilde{b}_n)_{n\in{{\mathbb Z}}}$ and $(\tilde{c}_n)_{n\in{{\mathbb Z}}}$ depending on a free parameter $y_0$.
The same can be done for the LU factorization of the form $P=\widetilde P_L\widetilde P_U$. Indeed, the new random walk is given by $$\label{QZLUdar}
\widehat P=\widetilde P_U\widetilde P_L=\left(\begin{array}{cc|cccc}
\ddots&\ddots&&\\
0&\tilde y_{-1}&\tilde x_{-1}&&\\
\hline
&0&\tilde y_0&\tilde x_0&\\
&&0&\tilde y_1&\tilde x_1\\
&&&&\ddots&\ddots
\end{array}
\right)\left(\begin{array}{ccc|ccc}
\ddots&\ddots&&&\\
&\tilde r_{-1}&\tilde s_{-1}&0&\\
\hline
&&\tilde r_0&\tilde s_0&0\\
&&&\tilde r_1&\tilde s_1&0\\
&&&&\ddots&\ddots
\end{array}
\right),$$ where the new coefficients are $$\label{coeffDUL}
\begin{split}
\hat{a}_n&=\tilde x_{n}\tilde s_{n+1},\\
\hat{b}_n&=\tilde x_{n}\tilde r_{n+1}+\tilde y_n\tilde s_n,\quad n\in{{\mathbb Z}},\\
\hat{c}_n&=\tilde y_{n}\tilde r_n.
\end{split}$$ Again, the matrix $\widehat{P}$ is stochastic, so we have a *family* of new random walks $\{\widehat X_t : t=0,1,\ldots\}$ on the integers ${{\mathbb Z}}$ with coefficients $(\hat{a}_n)_{n\in{{\mathbb Z}}}$, $(\hat{b}_n)_{n\in{{\mathbb Z}}}$ and $(\hat{c}_n)_{n\in{{\mathbb Z}}}$ depending on one free parameter $\tilde r_0$.
In terms of a model driven by urn experiments both factorizations may be thought as two urn experiments, Experiment 1 and Experiment 2, respectively. We first perform the Experiment 1 and with the result we immediately perform the Experiment 2. The model for the Darboux transformation will be reversing the order of both experiments. For more details about these urn models see [@GdI3; @GdI4].
Now we will focus in the following question: given the spectrum of the doubly infinite matrix $P$, how can we compute the spectrum of the Darboux transformations $\widetilde{P}$ and $\widehat{P}$? It turns out that both transformations will be related with what is called a *Geronimus transformation* (see below). Before that let us introduce some notation to study the spectral measures associated with the original random walk $P$.
We will follow the last section of [@KMc6]. For $P$ like in consider the eigenvalue equation $ x q^\alpha(x) = P q^\alpha(x) $ where $q^\alpha(x)=(\cdots, Q_{-1}^\alpha(x),Q_0^\alpha(x),Q_1^\alpha(x),\cdots)^T, \alpha=1,2$. For each $ x $ real or complex there exist two polynomial families of linearly independent solutions $ Q_n ^ {\alpha} (x), \alpha = 1,2, n \in {{\mathbb Z}}, $ depending on the initial values at $n = 0 $ and $ n = -1 $. These polynomials are given by $$\begin{aligned}
\nonumber Q_0^1(x)&=1,\quad Q_{0}^2(x)=0,\\
\label{TTRRZ}Q_{-1}^1(x)&=0,\quad Q_{-1}^2(x)=1,\\
\nonumber xQ_n^{\alpha}(x)&=a_nQ_{n+1}^{\alpha}(x)+b_nQ_n^{\alpha}(x)+c_nQ_{n-1}^{\alpha}(x),\quad n\in{{\mathbb Z}},\quad\alpha=1,2.\end{aligned}$$ Observe that $$\label{degqs}
\begin{split}
\deg(Q_n^1)&=n,\quad n\geq0,\hspace{.95cm}\deg(Q_n^2)=n-1,\quad n\geq1,\\
\deg(Q_{-n-1}^1)&=n-1,\quad n\geq1,\quad\deg(Q_{-n-1}^2)=n,\quad n\geq0.
\end{split}$$ From the three-term recurrence relation it is possible to compute the leading coefficients of the polynomials $Q_n^{\alpha}(x), \alpha =1,2, n\in {{\mathbb Z}}$. Indeed, for $n\geq0,$ we have $$\begin{aligned}
\label{lead1}Q_n^1(x)&=R_n^1x^n+\mathcal{O}(x^{n-1}),\quad R_0^1=1,\quad R_n^1=(a_0\cdots a_{n-1})^{-1},\quad n\geq1,\\
\nonumber Q_{-n-1}^1(x)&=L_{n-1}^1x^{n-1}+\mathcal{O}(x^{n-2}),\quad L_{n-1}^1=-a_{-1}(c_{-1}\cdots c_{-n})^{-1},\quad n\geq1,\end{aligned}$$ and $$\begin{aligned}
\nonumber Q_n^2(x)&=R_{n-1}^2x^{n-1}+\mathcal{O}(x^{n-2}),\quad R_{n-1}^2=-c_0(a_0\cdots a_{n-1})^{-1},\quad n\geq1,\\
\label{lead2}Q_{-n-1}^2(x)&=L_{n}^2x^n+\mathcal{O}(x^{n-1}),\quad L_0^2=1,\quad L_{n}^2=(c_{-1}\cdots c_{-n})^{-1},\quad n\geq1.\end{aligned}$$ Let us define the *potential coefficients* as $$\label{potcoeff}
\pi_0=1,\quad \pi_n=\frac{a_0a_1\cdots a_{n-1}}{c_1c_2\cdots c_n},\quad \pi_{-n}=\frac{c_0c_{-1}\cdots c_{-n+1}}{a_{-1}a_{-2}\cdots a_{-n}},\quad n\geq1.$$ In particular we have that $\pi P=\pi$, i.e. $\pi=(\pi_n)_{n\in{{\mathbb Z}}}$ is an invariant vector of $P$. In the Hilbert space $\ell^2_{\pi}({{\mathbb Z}})$ the matrix $P$ gives rise to a self-adjoint operator of norm $\leq1$, which we will denote by $P$, abusing the notation. This result is a consequence of Corollary 2.2 of [@MR] since the coefficients $(a_n)_{n\in{{\mathbb Z}}}$ and $(c_n)_{n\in{{\mathbb Z}}}$ are probabilities and therefore $0<a_n,c_n<1$. Applying the spectral theorem *three times*, there exist three unique measures $\psi_{11}(x), \psi_{22}(x)$ and $\psi_{12}(x)$ (since $\psi_{12}(x)=\psi_{21}(x)$ as a consequence of $P$ being self-adjoint and the symmetry of the inner product) supported on the interval $[-1,1]$ such that $$\label{ortoZ}
\sum_{\alpha,\beta=1}^{2}\int_{-1}^{1}Q_i^{\alpha}(x)Q_j^{\beta}(x)d\psi_{\alpha\beta}(x)=\frac{\delta_{i,j}}{\pi_j},\quad i,j\in{{\mathbb Z}}.$$
The measures $\psi_{11}$ and $\psi_{22}$ are positive (in fact $\psi_{11}$ is a probability measure but $\psi_{22}$ is not, since $\int_{-1}^1d\psi_{22}(x)=1/\pi_{-1}$). The measure $\psi_{12}$ is a signed measure satisfying $0=\int_{-1}^1d\psi_{12}(x)$. For simplicity let us assume that the three measures are continuously differentiable with respect to the Lebesgue measure, i.e. $d\psi_{\alpha\beta}(x)=\psi_{\alpha\beta}(x)dx, \alpha,\beta=1,2$, abusing the notation. These 3 measures can be written in matrix form as the $2\times2$ matrix $$\label{2spmt}
\Psi(x)=\begin{pmatrix} \psi_{11}(x) & \psi_{12}(x)\\ \psi_{12}(x) & \psi_{22}(x) \end{pmatrix},$$ so that the orthogonality relations can be written in matrix form as $$\label{ortoZ2}
\int_{-1}^{1}\left(Q_i^1(x),Q_i^2(x)\right)\Psi(x)\begin{pmatrix}Q_j^1(x)\\Q_j^2(x)\end{pmatrix}dx=\frac{\delta_{i,j}}{\pi_j},\quad i,j\in{{\mathbb Z}}.$$ The matrix $\Psi(x)$ in is called the *spectral matrix* associated with $P$. The orthogonality conditions are valid for any indexes $ i, j \in {{\mathbb Z}}$. With this information we can compute the $n$-step transition probabilities of the random walk $\{X_t : t=0,1,\ldots\}$, given by the so-called Karlin-McGregor integral representation formula (see [@KMc6]) $$\label{KmcG1}
P_{ij}^{(n)}\doteq\mathbb{P}(X_n=j \;| X_0=i)=\pi_j\int_{-1}^{1}x^n\left(Q_i^1(x),Q_i^2(x)\right)\Psi(x)\begin{pmatrix}Q_j^1(x)\\Q_j^2(x)\end{pmatrix}dx,\quad i,j\in{{\mathbb Z}}.$$ It is possible to relabel the states in such a way that all the information of $ P $ can be collected in a semi-infinite block tridiagonal matrix $ \bm P $ with blocks of size $ 2 \times2 $. Indeed, after the new labeling $$\{0,1,2,\ldots\}\to\{0,2,4,\ldots\},\quad\mbox{and}\quad\{-1,-2,-3,\ldots\}\to\{1,3,5,\ldots\},$$ we have that $P$ (doubly infinite tridiagonal) is equivalent to a matrix $\bm P$ (semi-infinite block tridiagonal) of the form $$\bm P=\begin{pmatrix}
B_0&A_0&&&\\
C_1&B_1&A_1&&\\
&C_2&B_2&A_2&\\
&&\ddots&\ddots&\ddots
\end{pmatrix},$$ where $$\begin{aligned}
B_0&=\begin{pmatrix} b_0 & c_0\\a_{-1} & b_{-1}\end{pmatrix},\quad B_n=\begin{pmatrix} b_n & 0\\ 0 & b_{-n-1}\end{pmatrix},\quad n\geq1,\\
A_n&=\begin{pmatrix} a_n & 0\\ 0 & c_{-n-1}\end{pmatrix},\quad n\geq0,\quad C_n=\begin{pmatrix} c_n & 0\\ 0 & a_{-n-1}\end{pmatrix},\quad n\geq1.\end{aligned}$$ The random walk generated by $\bm P $ can be interpreted as a walk that takes values in the two-dimensional state space ${{\mathbb Z}}_{\geq0}\times \{1,2 \}. $ These type of processes are called discrete-time *quasi-birth-and-death processes*. In general these processes allow transitions between all two-dimensional adjacent states (see [@LaR; @Neu] for a general reference).
If we define the matrix-valued polynomials $$\label{2QMM}
\bm Q_n(x)=\begin{pmatrix} Q_n^1(x) & Q_n^2(x) \\ Q_{-n-1}^1(x) & Q_{-n-1}^2(x)\end{pmatrix},\quad n\geq0,$$ then we have $$\begin{aligned}
x\bm Q_0(x)&=A_0\bm Q_{1}(x)+B_0\bm Q_0(x),\quad \bm Q_0(x)=I_{2\times2},\\
x\bm Q_n(x)&=A_n\bm Q_{n+1}(x)+B_n\bm Q_n(x)+C_n\bm Q_{n-1}(x),\quad n\geq1,\end{aligned}$$ where $I_{2\times2}$ denotes the $2\times2$ identity matrix. Observe that $\deg(\bm Q_n)=n$ and the leading coefficient is a nonsingular matrix (by and ). The matrix orthogonality is defined in terms of the (matrix-valued) inner product $$\label{ortoZ3}
\int_{-1}^{1}\bm Q_n(x)\Psi(x)\bm Q_m^*(x)dx=\begin{pmatrix} 1/\pi_n & 0 \\ 0 & 1/\pi_{-n-1}\end{pmatrix}\delta_{nm},$$ where $A^*$ is the Hermitian transpose of a matrix $A$ and $\pi=(\pi_n)_{n\in{{\mathbb Z}}}$ is given by . In this case we have (see [@DRSZ; @G2]) the Karlin-McGregor integral representation formula where the $2\times2$ block entry $(i,j)$ is given by $$\bm P_{ij}^{(n)}=\left(\int_{-1}^{1}x^n\bm Q_i(x)\Psi(x)\bm Q_j^*(x)dx\right)\begin{pmatrix} \pi_j & 0 \\ 0 & \pi_{-j-1}\end{pmatrix},\quad i,j\in{{\mathbb Z}}_{\geq0}.$$
\[remrec\] In Corollaries 4.1 and 4.2 of [@DRSZ] one can find some results concerning recurrence for discrete-time quasi-birth-and-death processes. Applying these to the case of random walks on ${{\mathbb Z}}$ we have that the random walk is *recurrent* if and only if $\int_{-1}^1\psi_{\alpha,\beta}(x)/(1-x)dx=\infty$ for some $\alpha,\beta=1,2,$ and it is *positive recurrent* if and only if one of the measures $\psi_{\alpha,\beta}(x),\alpha,\beta=1,2,$ has a jump at the point 1.
In the following lemma we will give a characterization of the orthogonality of the vector-valued polynomials $\left(Q_n^1(x),Q_n^2(x)\right)$ in terms of monomials.
\[lemorto\] Let $(Q_n^\alpha)_{n\in{{\mathbb Z}}}$ be the polynomials defined by . Then the vector-valued polynomials $\left(Q_n^1(x),Q_n^2(x)\right)$, $n\in{{\mathbb Z}}$ are orthogonal in the sense of if and only if for $n\geq0$ we have $$\label{cond1}
\int_{-1}^1\left(Q_n^1(x),Q_n^2(x)\right)\Psi(x)x^jdx=
\begin{cases}
(0,0),&\mbox{for}\quad j=0,1,\ldots,n-1,\\
(\alpha_n,0),\alpha_n\neq0,&\mbox{for}\quad j=n,
\end{cases}$$ and $$\label{cond2}
\int_{-1}^1\left(Q_{-n-1}^1(x),Q_{-n-1}^2(x)\right)\Psi(x)x^jdx=
\begin{cases}
(0,0),&\mbox{for}\quad j=0,1,\ldots,n-1,\\
(0,\beta_n),\beta_n\neq0,&\mbox{for}\quad j=n.
\end{cases}$$ Moreover $\alpha_0=1, \alpha_n=c_1\cdots c_n, n\geq1$ and $\beta_n=c_0^{-1}a_{-1}\cdots a_{-n-1},n\geq0$.
The orthogonality conditions are equivalent to the matrix orthogonality . Since $\bm Q_n(x)$ in is a matrix polynomial of degree $n$ with nonsingular leading coefficient the orthogonality is equivalent to $\int_{-1}^1\bm Q_n(x)\Psi(x) x^jdx=0_{2\times2}$ for $j=0,1,\ldots,n-1,$ where $0_{2\times2}$ denotes the $2\times2$ null matrix, and $\int_{-1}^1\bm Q_n(x)\Psi(x) x^ndx$ is a nonsingular (diagonal) matrix. Looking at the rows of these expressions we get and . The values of $\alpha_n$ and $\beta_n$ can be computed using , and .
Now that we have studied the spectral properties of the doubly infinite matrix $P$ in let us study the spectral matrices associated with the Darboux transformations $\widetilde P$ in and $\widehat P$ in . We will analyze both cases separately.
Darboux transformation for the UL case
--------------------------------------
Consider the discrete Darboux transformation $\widetilde P$ in with probability coefficients $(\tilde{a}_n)_{n\in{{\mathbb Z}}}$, $(\tilde{b}_n)_{n\in{{\mathbb Z}}}$ and $(\tilde{c}_n)_{n\in{{\mathbb Z}}}$ given by . Before defining the corresponding polynomials associated with $\widetilde P$ let us introduce an auxiliary family of polynomials $S_n^\alpha(x)$ given by the relation $s^\alpha(x)=P_Lq^\alpha(x)$, where $q^\alpha(x)=(\cdots, Q_{-1}^\alpha(x),Q_0^\alpha(x),Q_1^\alpha(x),\cdots)^T,$ and $s^\alpha(x)=(\cdots, S_{-1}^\alpha(x), S_0^\alpha(x),S_1^\alpha(x),\cdots)^T, \alpha=1,2$, i.e. $$\label{qenqhat}
S_n^\alpha(x)=s_n Q_n^\alpha (x)+ r_n Q_{n-1}^\alpha(x),\quad n \in \mathbb{Z}, \quad \alpha=1,2.$$ From the UL factorization we also have $P_U s^\alpha(x)= xq^\alpha(x)$, that is $$\label{qen1}
xQ_n^\alpha (x)=x_n S_{n+1}^\alpha(x)+y_n S_n^\alpha(x), \quad n\in \mathbb{Z}, \quad\alpha=1,2.$$ Evaluating at $x=0$ we get recursively $$\begin{split}\label{q0hat}
S_{n}^\alpha (0)&=(-1)^{n}\frac{y_0\dots y_{n-1}}{x_0 \dots x_{n-1}}S_0^\alpha (0), \quad n\ge 1,\\
S_{-n-1}^\alpha (0)&=(-1)^{n+1} \frac{x_{-1}\dots x_{-n-1}}{y_{-1}\dots y_{-n-1}} S_0^\alpha(0), \quad n\ge 0,
\end{split}$$ where $$\label{relacons}
S_0^\alpha(0)=\begin{cases} s_0, \quad \text{if}\quad \alpha=1, \\ r_0, \quad \text{if} \quad \alpha=2. \end{cases}$$ The equations establish a direct relation between the polynomials $(S_n^\alpha)_{n\in{{\mathbb Z}}}, \alpha=1,2,$ given by $$\label{relq0}
s_0 S_n^2(0)=r_0 S_n^1(0), \quad n\in \mathbb{Z}.$$ Another useful relation follows using and which gives the polynomials $(S_n^\alpha)_{n\in \mathbb{Z}}$ in terms of $(Q_n^\alpha)_{n\in \mathbb{Z}}$. Indeed, for $n\geq0,$ $$\begin{split}
S_{n+1}^\alpha(x)&=\frac{x}{x_n} Q_n^\alpha(x)-\frac{y_n}{x_n}S_n^\alpha (x)= \frac{x}{x_{n}}Q_n^\alpha(x) +\frac{S_{n+1}^\alpha(0)}{ S_{n}^\alpha(0)}\left[ \frac{x}{x_{n-1}} Q_{n-1}^\alpha(x)+\frac{S_{n}^\alpha(0)}{ S_{n-1}^\alpha(0)}S_{n-1}^\alpha (x) \right]\\
&= x\left[ \frac{Q_n^\alpha(x)}{x_n}+ \frac{S_{n+1}^\alpha(0)}{ S_{n}^\alpha(0)} \frac{Q_{n-1}^\alpha(x)}{x_{n-1}} \right] + \frac{S_{n+1}^\alpha(0)}{ S_{n-1}^\alpha(0)} S_{n-1}^\alpha(x)= \cdots =x\sum_{j=0}^{n} \frac{S_{n+1}^\alpha(0)}{ S_{j+1}^\alpha(0)}\frac{Q_j^\alpha(x)}{x_j} +S_{n+1}^\alpha(0),
\end{split}$$ since $S_0^\alpha(x)$ is constant (see ). Therefore $$\label{qhatrec1}
\begin{split}
S_n^\alpha(x)&=S_{n}^\alpha(0) \left[ 1+x\sum_{j=0}^{n-1} \frac{Q_j^\alpha(x) }{ S_{j+1}^\alpha(0) x_j} \right], \quad n\ge 1.
\end{split}$$ Similarly $$\label{qhatrec2}
\begin{split}
S_{-n-1}^\alpha(x)&=S_{-n-1}^\alpha(0) \left[ 1+x\sum_{j=0}^{n} \frac{Q_{-j-1}^\alpha(x) }{ S_{-j-1}^\alpha(0) y_{-j-1}} \right], \quad n\ge 0.
\end{split}$$ Observe that this auxiliary family $(S_n^\alpha)_{n\in \mathbb{Z}}$ does not satisfy the same initial conditions as the family $(Q_n^\alpha)_{n\in \mathbb{Z}}$ since by we have $$\begin{split}
S_0^1(x)=&s_0, \qquad\qquad S_0^2(x)=r_0,\\
S_{-1}^1(x)=&-\frac{x_{-1}s_0}{y_{-1}}, \quad S_{-1}^2(x)=\frac{x-x_{-1}r_0}{y_{-1}}.
\end{split}$$ The degrees of the polynomials $(S_n^\alpha)_{n\in \mathbb{Z}}$ are also not the same as the degrees of the polynomials $(Q_n^\alpha)_{n\in \mathbb{Z}}$, since $$\label{degqss}
\begin{split}
\deg(S_n^1)&=n, \quad n\ge 0, \quad \deg(S_n^2)=n,\quad n\ge 0,\\
\deg(S_{-n-1}^1)&=n, \quad n\ge 0, \quad \deg(S_{-n-1}^2)=n+1,\quad n\ge 0.
\end{split}$$ Therefore the family of matrix polynomials $$\label{S00}
\bm S_n(x)=\begin{pmatrix} S_n^1(x)&S_n^2(x) \\ S_{-n-1}^1(x) & S_{-n-1}^2(x)\end{pmatrix}, \quad n\geq0,$$ has degree $n+1$ and singular leading coefficient. Now we will define a new family of polynomials which will turn out to be the associated family of the Darboux transformation $\widetilde P$. For $n\geq0$ define $$\label{qtilde}
\widetilde{\bm Q}_n(x)=\bm S_n(x) \bm S_0^{-1}(x),\quad n\geq0,$$ where $$\label{S000}
\bm S_0(x)=\begin{pmatrix} s_0& r_0 \\ -{\displaystyle}\frac{x_{-1}s_0}{y_{-1}}& {\displaystyle}\frac{x-x_{-1}r_0}{y_{-1}}
\end{pmatrix}.$$ Following the same representation as in we can define the functions $(\widetilde Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2$, which turn out to be polynomials, as the following proposition shows.
\[proppol\] Let $\widetilde{\bm Q}_n(x), n\geq0,$ be the matrix function defined by . Then, for $n\geq0$, $\widetilde{\bm Q}_n(x)$ is a matrix polynomial of degree exactly $n$ with nonsingular leading coefficient and $\widetilde{\bm Q}_0(x)=I_{2\times2}$.
Computing the inverse of $\bm S_0(x)$ we have $$\bm S_0^{-1}(x)=\frac{1}{x}\begin{pmatrix} {\displaystyle}\frac{x-x_{-1}r_0}{s_{0}}& -{\displaystyle}\frac{y_{-1}r_0}{s_{0}} \\ x_{-1}& y_{-1}\end{pmatrix}.$$ Observe that $|\bm S_0(x)|=\frac{xs_0}{y_{-1}}$, so the inverse is not well-defined at $x=0$. We will show that we can avoid this problem using the properties of the polynomials $(S_n^\alpha)_{n\in \mathbb{Z}}$. Indeed, from we have $$\label{qtilss}
\begin{split}
\widetilde Q_n^1(x)&=\frac{S_n^1(x)}{s_0}+\frac{x_{-1}}{xs_0}\left(s_0S_n^2(x)-r_0S_n^1(x)\right),\quad n\in{{\mathbb Z}},\\
\widetilde Q_n^2(x)&=\frac{y_{-1}}{xs_0}\left(s_0S_n^2(x)-r_0S_n^1(x)\right),\quad n\in{{\mathbb Z}}.
\end{split}$$ A straightforward computation using , and gives $$\begin{aligned}
s_0S_0^2(x)-r_0S_0^1(x)&=0,\\
s_0S_n^2(x)-r_0S_n^1(x)&=x\sum_{j=0}^{n-1}\frac{1}{x_j}\left(s_0\frac{S_n^2(0)Q_j^2(x)}{S_{j+1}^2(0)}-r_0\frac{S_n^1(0)Q_j^1(x)}{S_{j+1}^1(0)}\right),\quad n\geq1,\\
s_0S_{-n-1}^2(x)-r_0S_{-n-1}^1(x)&=x\sum_{j=0}^{n}\frac{1}{y_{-j-1}}\left(s_0\frac{S_{-n-1}^2(0)Q_{-j-1}^2(x)}{S_{-j-1}^2(0)}-r_0\frac{S_{-n-1}^1(0)Q_{-j-1}^1(x)}{S_{-j-1}^1(0)}\right),\quad n\geq0.\end{aligned}$$ Therefore from these relations we can see that $(\widetilde Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2$ are indeed polynomials. A close look to the degrees of $(Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2,$ in gives that $\deg\left((s_0S_n^2(x)-r_0S_n^1(x))/x\right)=n-1,n\geq1$. Therefore from we get $\deg(\widetilde Q_n^1)=n, n\geq0$ and $\deg(\widetilde Q_n^2)=n-1,n\geq1$. On the other hand $\deg\left((s_0S_{-n-1}^2(x)-r_0S_{-n-1}^1(x))/x\right)=n,n\geq0$. Therefore $\deg(\widetilde Q_{-n-1}^2)=n, n\geq0$. Finally, $\widetilde Q_{-n-1}^1$ is in principle a polynomial of degree $n$, but we will see that in fact is a polynomial of degree $n-1$. Indeed, call $\Lambda_n$ the coefficient of $x^n$ in $\widetilde Q_{-n-1}^1$. Then, using , , and , we have $$\Lambda_n=\frac{1}{s_0}\left(-\frac{a_{-1}r_{-n-1}}{c_{-1}\cdots c_{-n-1}}+\frac{x_{-1}s_0}{y_{-n-1}c_{-1}\cdots c_{-n}}\right)=\frac{1}{s_0c_{-1}\cdots c_{-n}}\left(-\frac{a_{-1}r_{-n-1}}{c_{-n-1}}+\frac{x_{-1}s_0}{y_{-n-1}}\right)=0.$$ Therefore $\deg(\widetilde Q_{-n-1}^1)=n-1, n\geq1$. The fact that $\widetilde{\bm Q}_0(x)=I_{2\times2}$ comes from the definition .
The previous proposition shows that the *Darboux polynomials* $(\widetilde Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2,$ satisfy the same initial conditions and degree conditions than the original polynomials $(Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2$. They also satisfy the three-term recurrence relation $$\label{TTRRZUL}
\begin{split}
\widetilde Q_0^1(x)&=1,\quad \widetilde Q_{0}^2(x)=0,\\
\widetilde Q_{-1}^1(x)&=0,\quad \widetilde Q_{-1}^2(x)=1,\\
x\widetilde Q_n^{\alpha}(x)&=\tilde a_n\widetilde Q_{n+1}^{\alpha}(x)+\tilde b_n\widetilde Q_n^{\alpha}(x)+\tilde c_n\widetilde Q_{n-1}^{\alpha}(x),\quad n\in{{\mathbb Z}},\quad\alpha=1,2,
\end{split}$$ where the Darboux coefficients $(\tilde{a}_n)_{n\in{{\mathbb Z}}}$, $(\tilde{b}_n)_{n\in{{\mathbb Z}}}$ and $(\tilde{c}_n)_{n\in{{\mathbb Z}}}$ are defined by . This is a consequence of writing the polynomials $(\widetilde Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2,$ in terms of $(S_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2,$ (see ) and the fact that the polynomials $(S_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2,$ satisfy the same three-term recurrence relation (with different initial conditions) by construction (see ).
We will now show one the the main results of this paper, namely how to compute the spectral matrix associated with the Darboux random walk $\widetilde P$ in and prove that $(\widetilde Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2,$ are the corresponding orthogonal polynomials. We define first the potential coefficients associated with $\widetilde P$ given by $$\label{cofpotUL}
\tilde \pi_0=1, \quad \tilde \pi_n=\frac{\tilde a_0 \cdots \tilde a_{n-1}}{\tilde c_1 \cdots \tilde c_{n}}, \quad \tilde \pi_{-n}=\frac{\tilde c_0 \cdots \tilde c_{-n+1}}{ \tilde a_{-1} \cdots \tilde a_{-n}}, \quad n\ge 1.$$
\[thmorto\] Let $\{X_t: t=0, 1, \dots\}$ be the random walk on ${{\mathbb Z}}$ with transition probability matrix $P$ given by and $\{\widetilde X_t : t=0, 1, \dots\}$ the Darboux random walk on ${{\mathbb Z}}$ with transition probability matrix $\widetilde P$ given by . Assume that $M_{-1}={\displaystyle}\int_{-1}^{1} {\displaystyle}\frac{\Psi(x)}{x}dx$ is well-defined (entry by entry), where $\Psi(x)$ is the original spectral matrix (see ). Then the polynomials $(\widetilde Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2,$ defined by (see also ) are orthogonal with respect to the following spectral matrix $$\label{esptil}
\widetilde \Psi (x)=\bm S_0(x) \Psi_S (x) \bm S_0^*(x),$$ where $\bm S_0(x)$ is defined by and $$\label{esphat}
\Psi_S(x)=\frac{y_0}{s_0}\frac{\Psi(x)}{x}+ \left[ \begin{pmatrix}1/s_0&0 \\ 0 &1/r_0 \end{pmatrix}- \frac{y_0}{s_0}M_{-1}\right] \delta_0(x).$$ Moreover, we have $$\label{qusnorms}
\int_{-1}^{1} \widetilde{\bm Q}_n (x) \widetilde \Psi (x) \widetilde{\bm Q}_m^*(x) dx=\begin{pmatrix} 1/\tilde \pi_n &0\\ 0& 1/\tilde \pi_{-n-1}\end{pmatrix} \delta_{n,m},$$ where $(\tilde \pi_n)_{n\in \mathbb{Z}}$ are the potential coefficients defined by .
Let $(Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2,$ be the polynomials defined by , which are orthogonal with respect to the original spectral matrix $\Psi$. By Lemma \[lemorto\] we have the orthogonality conditions and . Since $(\widetilde Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2,$ satisfies the same initial and degree conditions than $(Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2,$ we will use Lemma \[lemorto\] to prove that $(\widetilde Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2,$ are orthogonal with respect to $\widetilde \Psi (x)$ in .
Assume first that $n\geq1$. Then we have, using , and , that $$\begin{split}
\int_{-1}^{1} \left(\widetilde Q_n^1(x), \widetilde Q_n^2(x)\right) \widetilde \Psi (x)x^jdx &= \int_{-1}^{1} \left(\widetilde Q_n^1(x), \widetilde Q_n^2(x)\right) x\widetilde \Psi(x) x^{j-1}dx\\
&= \int_{-1}^{1} \left(\widetilde Q_n^1(x), \widetilde Q_n^2(x)\right) \bm S_0(x) x\Psi_S(x) \bm S_0^*(x) x^{j-1}dx\\
&=\frac{y_0}{s_0} \int_{-1}^{1} \left(S_n^1(x), S_n^2(x)\right) \bm S_0^{-1}(x) \bm S_0(x) \Psi(x) \bm S_0^*(x) x^{j-1}dx \\
&=\frac{y_0}{s_0} \int_{-1}^{1} \left(S_n^1(x), S_n^2(x)\right) \Psi(x) \bm S_0^*(x) x^{j-1}dx. \\
\end{split}$$ Now, using , the above expression can be written as $$\begin{split}
\int_{-1}^{1} \left(\widetilde Q_n^1(x), \widetilde Q_n^2(x)\right) \widetilde \Psi (x)x^jdx&= \frac{s_n y_0}{s_0} \int_{-1}^{1} \left(Q_n^1(x), Q_n^2(x)\right) \Psi(x) \bm S_0^*(x) x^{j-1}dx \\
&\quad+ \frac{r_ n y_0}{s_0} \int_{-1}^{1} \left(Q_{n-1}^1(x), Q_{n-1}^2(x)\right) \Psi(x) \bm S_0^*(x) x^{j-1}dx.
\end{split}$$ Writing $\bm S_0(x)=A+ xB$, where $A$ and $B$ are given by $$\label{q0ext}
A=\begin{pmatrix} s_0& r_0\\ -{\displaystyle}\frac{x_{-1}s_0}{y_{-1}} & -{\displaystyle}\frac{x_{-1}r_0}{y_{-1}}\end{pmatrix},\quad B=\begin{pmatrix} 0& 0\\ 0 & {\displaystyle}\frac{1}{y_{-1}}\end{pmatrix},$$ the above expression can be written as $$\begin{split}
\int_{-1}^{1} \left(\widetilde Q_n^1(x), \widetilde Q_n^2(x)\right)& \widetilde \Psi (x)x^jdx= \frac{s_n y_0}{s_0}\left[\int_{-1}^{1} \left(Q_n^1(x), Q_n^2(x)\right) \Psi(x) A^* x^{j-1}dx+\int_{-1}^{1} \left(Q_n^1(x), Q_n^2(x)\right) \Psi(x) B^* x^{j}dx\right]\\
&\quad+ \frac{r_ n y_0}{s_0}\left[\int_{-1}^{1} \left(Q_{n-1}^1(x), Q_{n-1}^2(x)\right) \Psi(x) A^* x^{j-1}dx+\int_{-1}^{1} \left(Q_{n-1}^1(x), Q_{n-1}^2(x)\right) \Psi(x) B^* x^{j}dx\right].
\end{split}$$ Using we have that the first term of the sum vanishes for $j=1,\ldots,n$, the second term vanishes for $j=0,\ldots,n-1$, the third term vanishes for $j=1,\ldots,n-1$ and the fourth term vanishes for $j=0,\ldots,n-2$. Therefore the above expression vanishes for $j=1,\ldots,n-2$. For $j=n-1$ the only term that does not vanish is the fourth one. But in this case we have, using and , that $$\label{Bbss}
\begin{split}
\int_{-1}^{1} \left(\widetilde Q_n^1(x), \widetilde Q_n^2(x)\right) \widetilde \Psi (x)x^{n-1}dx&= \frac{r_ n y_0}{s_0}\int_{-1}^{1} \left(Q_{n-1}^1(x), Q_{n-1}^2(x)\right) \Psi(x) B^* x^{n-1}dx\\
&=\frac{r_ n y_0}{s_0} (\alpha_{n-1}, 0) \begin{pmatrix} 0& 0\\ 0 & {\displaystyle}\frac{1}{y_{-1}}\end{pmatrix}=(0,0).
\end{split}$$ For $j=0$ we have, using and , that $$\begin{split}
\int_{-1}^{1} \left(\widetilde Q_n^1(x), \widetilde Q_n^2(x)\right) \widetilde \Psi (x)dx&= \int_{-1}^{1} \left(S_n^1(x), S_n^2(x)\right) \bm S_0^{-1}(x) \bm S_0(x) \Psi_S(x) \bm S_0^*(x)dx \\
&= \left[ \int_{-1}^{1} \left(S_n^1(x), S_n^2(x)\right) \Psi_S(x) dx \right] A^* + \left[ \int_{-1}^{1} \left(S_n^1(x), S_n^2(x)\right) x\Psi_S(x) dx \right] B^*.
\end{split}$$ The second term of the sum of the above expression vanishes as a consequence of , and . Indeed, for $n\geq2,$ we have $$\begin{split}
&\left[ \int_{-1}^{1} \left(S_n^1(x), S_n^2(x)\right)x\Psi_S(x) dx \right] B^*=
\left[\frac{y_0}{s_0} \int_{-1}^{1} \left(S_n^1(x), S_n^2(x)\right) \Psi(x) dx \right] B^*\\
&\hspace{3cm}= \left[ \frac{s_ n y_0}{s_0} \int_{-1}^{1} \left(Q_n^1(x), Q_n^2(x)\right) \Psi(x)dx + \frac{r_n y_0}{s_0} \int_{-1}^{1} \left(Q_{n-1}^1(x), Q_{n-1}^2(x)\right) \Psi(x)dx\right] B^*=(0,0).
\end{split}$$ For $n=1$ we can use the same argument as in and get again $(0,0)$. Now, using , we can write $$\left(S_n^1(x), S_n^2(x)\right)=x \sum_{j=0}^{n-1} \frac{1}{x_j} \left( \frac{S_n^1(0)}{S_{j+1}^1(0)} Q_j^1(x) , \frac{S_n^2(0)}{S_{j+1}^2(0)} Q_j^2(x) \right)+ \left(S_n^1(0),S_n^2(0)\right).$$ Substituting this in the remaining integral we get $$\begin{aligned}
\int_{-1}^{1} \left(\widetilde Q_n^1(x), \widetilde Q_n^2(x)\right)& \widetilde \Psi (x)dx= \left[ \int_{-1}^{1} \left(S_n^1(x), S_n^2(x)\right) \Psi_S(x) dx \right] A^*\\
&= \left(S_n^1(0),S_n^2(0)\right)\left[ \begin{pmatrix} 1/s_0 &0 \\ 0 & 1/r_0\end{pmatrix} -\frac{y_0}{s_0}M_{-1}\right] A^* \\
& \hspace{.5cm} +\frac{y_0}{s_0} \sum_{j=0}^{n-1} \frac{1}{x_j} \int_{-1}^{1} \left( \frac{S_n^1(0)}{S_{j+1}^1(0)} Q_j^1(x) , \frac{S_n^2(0)}{S_{j+1}^2(0)} Q_j^2(x) \right) \Psi (x)A^*dx+\frac{y_0}{s_0}\left(S_n^1(0),S_n^2(0)\right) M_{-1} A^*\\
&=\left(S_n^1(0),S_n^2(0)\right) \begin{pmatrix} 1/s_0 &0 \\ 0 & 1/r_0\end{pmatrix} A^*+\frac{y_0}{s_0 x_0}\frac{S_n^1(0)}{S_{1}^1(0)} \int_{-1}^1\left(Q_0^1(x),Q_0^2(x)\right)\Psi(x)A^*dx.\end{aligned}$$ The third step is a consequence of ${\displaystyle}\frac{S_n^1(0)}{S_{j+1}^1(0)}=\frac{S_n^2(0)}{S_{j+1}^2(0)}$ using and the orthogonality properties. Since ${\displaystyle}\int_{-1}^1\left(Q_0^1(x),Q_0^2(x)\right)\Psi(x)dx=(1,0)$, $S_{1}^1(0)=-s_0y_0/x_0$, and $A$ is given by then we have $$\begin{split}
\int_{-1}^{1} \left(\widetilde Q_n^1(x), \widetilde Q_n^2(x)\right)& \widetilde \Psi (x)dx=\left(S_n^1(0),S_n^2(0)\right) \begin{pmatrix} 1/s_0 &0 \\ 0 & 1/r_0\end{pmatrix} A^*-\frac{1}{s_0^2}\left(S_n^1(0),0\right)A^*\\
&=\left( (1-1/s_0)\frac{S_n^1(0)}{s_0} , \frac{S_n^2(0)}{r_0} \right) \begin{pmatrix} s_0& -{\displaystyle}\frac{x_{-1}s_0}{y_{-1}}\\ r_0 & -{\displaystyle}\frac{x_{-1}r_0}{y_{-1}}\end{pmatrix}\\
&=\left( (-r_0/s_0)S_n^1(0)+S_n^2(0), -\frac{x_{-1}}{y_{-1}} \left( (-r_0/s_0)S_n^1(0) +S_n^2(0) \right) \right)=(0,0),
\end{split}$$ as a consequence of .
For $n\leq-1$ the proof is similar but now using and . Therefore we have proved for $n\neq m$. Observe that this implies in particular that the family of vector-valued polynomials $\left(S_n^1(x),S_n^2(x)\right), n\in{{\mathbb Z}},$ is also orthogonal (for $n\neq m$) with respect to the weight matrix $\Psi_S(x)$ in . For $n=m$, using this fact and , , , and , we have that $$\begin{split}
\int_{-1}^{1} &\left(\widetilde Q_n^1(x), \widetilde Q_n^2(x)\right) \widetilde \Psi(x) \left(\widetilde Q_n^1(x) , \widetilde Q_n^2(x)\right)^*dx= \int_{-1}^{1} \left(S_n^1(x),S_n^2(x)\right) \Psi_S(x) \left(S_n^1(x), S_n^2(x)\right) ^* dx\\
&= \int_{-1}^{1} \left[ \frac{x}{y_n}\left( Q_n^1(x), Q_n^2(x)\right)-\frac{ x_n}{ y_n} \left(S_{n+1}^1(x), S_{n+1}^2(x)\right) \right] \Psi_S(x) \left(S_n^1(x), S_n^2(x)\right)^*dx\\
&= \frac{1}{y_n}\int_{-1}^{1} x \left( Q_n^1(x), Q_n^2(x)\right) \Psi_S(x) \left(S_n^1(x),S_n^2(x)\right)^*dx - \frac{x_n}{y_n} \int_{-1}^{1} \left(S_{n+1}^1(x) , S_{n+1}^2(x)\right)\Psi_S(x) \left(S_n^1(x) , S_n^2(x)\right)^*dx \\
&= \frac{y_0}{y_n s_0}\int_{-1}^{1} \left( Q_n^1(x) , Q_n^2(x)\right)\Psi(x)\left[ s_n\left( Q_n^1(x) , Q_n^2(x)\right)^* + r_n\left( Q_{n-1}^1(x) , Q_{n-1}^2(x)\right)^* \right] \\
&= \frac{ s_n y_0}{y_n s_0 }\int_{-1}^{1} \left( Q_n^1(x) , Q_n^2(x)\right) \Psi(x) \left( Q_n^1(x) , Q_n^2(x)\right)^*dx + \frac{ r_n y_0}{y_n s_0}\int_{-1}^{1} \left( Q_n^1(x) , Q_n^2(x)\right) \Psi(x) \left( Q_{n-1}^1(x) , Q_{n-1}^2(x)\right)^* dx \\
&= \frac{ s_n y_0}{y_n s_0 }\int_{-1}^{1} \left( Q_n^1(x) , Q_n^2(x)\right)\Psi(x) \left( Q_n^1(x) , Q_n^2(x)\right)^*dx =\frac{ s_n y_0}{y_n s_0 } \frac {1}{\pi _n}= \frac{1}{\tilde \pi_n}.
\end{split}$$ The last step follows using , and the definition of $\pi_n$ and $\tilde \pi_n$ in and , respectively.
The spectral matrix $\widetilde\Psi$ associated with the Darboux transformation given in the previous theorem is a *conjugation* by a matrix polynomial of degree 1 (namely $\bm S_0(x)$) of a *Geronimus transformation* of the original spectral matrix $\Psi$. This phenomenon was already present, except for the conjugation, in the case of the Darboux transformation of transition probability matrices on ${{\mathbb Z}}_{\geq0}$ for the UL factorization (see [@GdI3]).
In this paper we have only considered doubly infinite stochastic Jacobi matrices of the form , i.e. all entries are nonnegative and $a_n+b_n+c_n=1,n\in{{\mathbb Z}}$. Certainly the same approach can be used to more general doubly infinite Jacobi matrices as long as we are in the conditions of Corollary 2.2 of [@MR], i.e. to guarantee that the Jacobi matrix is self-adjoint. The spectral matrix associated with the Darboux transformation will be similar, up to constants, to the one given in Theorem \[thmorto\].
Darboux transformation for the LU case
--------------------------------------
Consider now the discrete Darboux transformation $\widehat P$ in with probability coefficients $(\hat{a}_n)_{n\in{{\mathbb Z}}}$, $(\hat{b}_n)_{n\in{{\mathbb Z}}}$ and $(\hat{c}_n)_{n\in{{\mathbb Z}}}$ given by . We will see that this case is similar to the one in the previous subsection, so we will only give the important formulas necessary to prove the main results. As in the UL case we need to introduce the auxiliary family of polynomials $T_n^\alpha(x)$ given by the relation $t^\alpha(x)=\widetilde P_Uq^\alpha(x)$, where $q^\alpha(x)=(\cdots,Q_{-1}^\alpha(x),Q_0^\alpha(x),Q_1^\alpha(x),\cdots)^T,$ and $t^\alpha(x)=(\cdots, T_{-1}^\alpha(x), T_0^\alpha(x),T_1^\alpha(x),\cdots)^T, \alpha=1,2$, i.e. $$\label{qenqhat2}
T_n^\alpha(x)=\tilde y_n Q_n^\alpha (x)+ \tilde x_n Q_{n+1}^\alpha(x),\quad n \in \mathbb{Z}, \quad \alpha=1,2.$$ From the LU factorization we also have $\widetilde P_L t^\alpha(x)= xq^\alpha(x)$, that is $$\label{qen2}
xQ_n^\alpha (x)=\tilde r_n T_{n-1}^\alpha(x)+\tilde s_n T_n^\alpha(x), \quad n\in \mathbb{Z}, \quad\alpha=1,2.$$ Evaluating at $x=0$ we get recursively $$\begin{split}\label{q0hat2}
T_{n}^\alpha (0)&=(-1)^{n+1}\frac{\tilde r_0\dots\tilde r_{n}}{\tilde s_0 \dots \tilde s_{n}}T_{-1}^\alpha (0), \quad n\ge 1,\\
T_{-n-1}^\alpha (0)&=(-1)^{n} \frac{\tilde s_{-1}\dots \tilde s_{-n}}{\tilde r_{-1}\dots \tilde r_{-n}} T_{-1}^\alpha(0), \quad n\ge 0,
\end{split}$$ where $$\label{relacons2}
T_{-1}^\alpha(0)=\begin{cases} \tilde x_{-1}, \quad \text{if}\quad \alpha=1, \\ \tilde y_{-1}, \quad \text{if} \quad \alpha=2. \end{cases}$$ The equations establish a direct relation between the polynomials $(T_n^\alpha)_{n\in{{\mathbb Z}}}, \alpha=1,2,$ given by $$\label{relq02}
\tilde x_{-1} T_n^2(0)=\tilde y_{-1} T_n^1(0), \quad n\in \mathbb{Z}.$$ Again, the polynomials $(T_n^\alpha)_{n\in \mathbb{Z}}$ can be written in terms of the polynomials $(Q_n^\alpha)_{n\in \mathbb{Z}}$ as follows $$\label{qhatrec3}
\begin{split}
T_n^\alpha(x)&=T_{n}^\alpha(0) \left[ 1+x\sum_{j=0}^{n} \frac{Q_j^\alpha(x) }{T_{j}^\alpha(0) \tilde s_j} \right], \quad n\ge 1,\\
T_{-n-1}^\alpha(x)&=T_{-n-1}^\alpha(0) \left[ 1+x\sum_{j=0}^{n-1} \frac{Q_{-j-1}^\alpha(x) }{T_{-j-2}^\alpha(0) \tilde r_{-j-1}} \right], \quad n\ge 0.
\end{split}$$ This auxiliary family $(T_n^\alpha)_{n\in \mathbb{Z}}$ does not satisfy the same initial conditions as the family $(Q_n^\alpha)_{n\in \mathbb{Z}}$. In fact we have $$\begin{split}
T_0^1(x)=\frac{x-\tilde r_0\tilde x_{-1}}{\tilde s_{0}},& \hspace{0.5cm} T_0^2(x)=-\frac{\tilde r_0\tilde y_{-1}}{\tilde s_{0}},\\
T_{-1}^1(x)=\tilde x_{-1},&\hspace{0.5cm} T_{-1}^2(x)=\tilde y_{-1}.
\end{split}$$ The degrees of the polynomials $(T_n^\alpha)_{n\in \mathbb{Z}}$ are $$\label{degqss2}
\begin{split}
\deg(T_n^1)&=n+1, \quad n\ge 0, \quad \deg(T_n^2)=n,\quad n\ge 0,\\
\deg(T_{-n-1}^1)&=n-1, \quad n\ge 0, \quad \deg(T_{-n-1}^2)=n,\quad n\ge 0.
\end{split}$$ Therefore the family of matrix polynomials $$\label{S002}
\bm T_n(x)=\begin{pmatrix} T_n^1(x)&T_n^2(x) \\ T_{-n-1}^1(x) & T_{-n-1}^2(x)\end{pmatrix}, \quad n\geq0,$$ has degree $n+1$ and singular leading coefficient. Now we will define a new family of polynomials which will turn out to be the associated family of the Darboux transformation $\widehat P$. For $n\geq0$ define $$\label{qtilde2}
\widehat{\bm Q}_n(x)=\bm T_n(x) \bm T_0^{-1}(x),\quad n\geq0,$$ where $$\label{S0002}
\bm T_0(x)=\begin{pmatrix} {\displaystyle}\frac{x-\tilde r_0\tilde x_{-1}}{\tilde s_{0}}& {\displaystyle}-\frac{\tilde r_0\tilde y_{-1}}{\tilde s_{0}} \\ \tilde x_{-1}&\tilde y_{-1}
\end{pmatrix}.$$ Following the same representation as in we can define the functions $(\widehat Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2$, which turn out to be polynomials, as the following proposition shows.
Let $\widehat{\bm Q}_n(x), n\geq0,$ be the matrix function defined by . Then, for $n\geq0$, $\widehat{\bm Q}_n(x)$ is a matrix polynomial of degree exactly $n$ with nonsingular leading coefficient and $\widehat{\bm Q}_0(x)=I_{2\times2}$.
Now, since $$\bm T_{0}^{-1}(x)=\frac{1}{x}\begin{pmatrix} \tilde s_0 & \tilde r_0 \\ -{\displaystyle}\frac{\tilde s_{0}\tilde x_{-1}}{\tilde y_{-1}}& {\displaystyle}\frac{x-\tilde r_0\tilde x_{-1}}{\tilde y_{-1}}\end{pmatrix},$$ we have from $$\label{qtilss2}
\begin{split}
\widehat Q_n^1(x)&=\frac{\tilde s_{0}}{x\tilde y_{-1}}\left(\tilde y_{-1}T_n^1(x)-\tilde x_{-1}T_n^2(x)\right),\quad n\in{{\mathbb Z}},\\
\widehat Q_n^2(x)&=\frac{T_n^2(x)}{\tilde y_{-1}}+\frac{\tilde r_{0}}{x\tilde y_{-1}}\left(\tilde y_{-1}T_n^1(x)-\tilde x_{-1}T_n^2(x)\right),\quad n\in{{\mathbb Z}}.
\end{split}$$ From here the proof follows the same lines as in the proof of Proposition \[proppol\] but now using and .
Again these new *Darboux polynomials* $(\widehat Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2,$ satisfy the same initial conditions and degree conditions than the original polynomials $(Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2,$ as well as a three-term recurrence relation of the form but with Darboux coefficients $(\hat{a}_n)_{n\in{{\mathbb Z}}}$, $(\hat{b}_n)_{n\in{{\mathbb Z}}}$ and $(\hat{c}_n)_{n\in{{\mathbb Z}}}$ given by .
Let us now prove the analogue of Theorem \[thmorto\] for the spectral matrix for the polynomials $(\widehat Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2$. We define first the potential coefficients associated with $\widehat P$ given by $$\label{cofpotUL2}
\hat \pi_0=1, \quad \hat \pi_n=\frac{\hat a_0 \cdots \hat a_{n-1}}{\hat c_1 \cdots \hat c_{n}}, \quad \hat \pi_{-n}=\frac{\hat c_0 \cdots \hat c_{-n+1}}{ \hat a_{-1} \cdots \hat a_{-n}}, \quad n\ge 1.$$
\[thmorto2\] Let $\{X_t : t=0, 1, \dots\}$ be the random walk on ${{\mathbb Z}}$ with transition probability matrix $P$ given by and $\{\widehat X_t : t=0, 1, \dots\}$ the Darboux random walk on ${{\mathbb Z}}$ with transition probability matrix $\widehat P$ given by . Assume that $M_{-1}={\displaystyle}\int_{-1}^{1} {\displaystyle}\frac{\Psi(x)}{x}dx$ is well-defined (entry by entry), where $\Psi(x)$ is the original spectral matrix (see ). Then the polynomials $(\widehat Q_n^\alpha)_{n\in \mathbb{Z}}, \alpha=1,2,$ defined by are orthogonal with respect to the following spectral matrix $$\label{esptil2}
\widehat \Psi (x)=\bm T_0(x) \Psi_T (x) \bm T_0^*(x),$$ where $\bm T_0(x)$ is defined by and $$\label{esphat2}
\Psi_T(x)=\frac{\tilde s_0}{\tilde y_0}\frac{\Psi(x)}{x}+ \left[\frac{\hat{a}_{-1}}{\hat{c}_0}\begin{pmatrix}1/\tilde x_{-1}&0 \\ 0 &1/\tilde y_{-1} \end{pmatrix}- \frac{\tilde s_0}{\tilde y_0}M_{-1}\right] \delta_0(x).$$ Moreover, we have $$\label{qusnorms2}
\int_{-1}^{1} \widehat{\bm Q}_n (x) \widehat \Psi (x) \widehat{\bm Q}_m^*(x) dx=\begin{pmatrix} 1/\hat \pi_n &0\\ 0& 1/\hat \pi_{-n-1}\end{pmatrix} \delta_{n,m},$$ where $(\hat \pi_n)_{n\in \mathbb{Z}}$ are the potential coefficients defined by .
The proof follows the same lines as the proof of Theorem \[thmorto\]. For $n\geq1$ and $j=1,\ldots,n-1$ we have, using , , and , that $$\begin{split}
\int_{-1}^{1} \left(\widehat Q_n^1(x), \widehat Q_n^2(x)\right)& \widehat \Psi (x)x^jdx= \frac{\tilde y_n \tilde s_0}{\tilde y_0}\left[\int_{-1}^{1} \left(Q_n^1(x), Q_n^2(x)\right) \Psi(x) A^* x^{j-1}dx+\int_{-1}^{1} \left(Q_n^1(x), Q_n^2(x)\right) \Psi(x) B^* x^{j}dx\right]\\
&\quad+ \frac{\tilde x_n \tilde s_0}{\tilde y_0}\left[\int_{-1}^{1} \left(Q_{n+1}^1(x), Q_{n+1}^2(x)\right) \Psi(x) A^* x^{j-1}dx+\int_{-1}^{1} \left(Q_{n+1}^1(x), Q_{n+1}^2(x)\right) \Psi(x) B^* x^{j}dx\right].
\end{split}$$ where $A$ and $B$ (from $\bm T_0(x)=A+ xB$), are given now by $$\label{q0ext2}
A=\begin{pmatrix} -{\displaystyle}\frac{\tilde r_0\tilde x_{-1}}{\tilde s_{0}}& -{\displaystyle}\frac{\tilde r_0\tilde y_{-1}}{\tilde s_{0}}\\ \tilde x_{-1} &\tilde y_{-1} \end{pmatrix},\quad B=\begin{pmatrix} {\displaystyle}\frac{1}{\tilde s_{0}}& 0\\ 0 & 0\end{pmatrix},$$ Using we have that the first term of the sum vanishes for $j=1,\ldots,n$, the second term vanishes for $j=0,\ldots,n-1$, the third term vanishes for $j=1,\ldots,n+1$ and the fourth term vanishes for $j=0,\ldots,n$. Therefore the above expression vanishes for $j=1,\ldots,n-1$.
For $j=0$ we have, using and , that $$\begin{split}
\int_{-1}^{1} \left(\widehat Q_n^1(x), \widehat Q_n^2(x)\right) \widehat \Psi (x)dx&= \int_{-1}^{1} \left(T_n^1(x), T_n^2(x)\right) \bm T_0^{-1}(x) \bm T_0(x) \Psi_T(x) \bm T_0^*(x)dx \\
&= \left[ \int_{-1}^{1} \left(T_n^1(x), T_n^2(x)\right) \Psi_T(x) dx \right] A^* + \left[ \int_{-1}^{1} \left(T_n^1(x), T_n^2(x)\right) x\Psi_T(x) dx \right] B^*.
\end{split}$$ As before, the second term of the sum of the above expression vanishes as a consequence of , and . Now, using , we can write $$\left(T_n^1(x), T_n^2(x)\right)=x \sum_{j=0}^{n} \frac{1}{\tilde s_j} \left( \frac{T_n^1(0)}{T_{j}^1(0)} Q_j^1(x) , \frac{T_n^2(0)}{T_{j}^2(0)} Q_j^2(x) \right)+ \left(T_n^1(0),T_n^2(0)\right).$$ Substituting this in the remaining integral we get $$\begin{aligned}
\int_{-1}^{1} \left(\widehat Q_n^1(x), \widehat Q_n^2(x)\right)& \widehat \Psi (x)dx= \left[ \int_{-1}^{1} \left(T_n^1(x), T_n^2(x)\right) \Psi_T(x) dx \right] A^*\\
&= \left(T_n^1(0),T_n^2(0)\right)\left[\frac{\hat{a}_{-1}}{\hat{c}_0}\begin{pmatrix} 1/\tilde x_{-1} &0 \\ 0 & 1/\tilde y_{-1}\end{pmatrix} -\frac{\tilde s_0}{\tilde y_0}M_{-1}\right] A^* \\
& \hspace{.5cm} +\frac{\tilde s_0}{\tilde y_0} \sum_{j=0}^{n} \frac{1}{\tilde s_j} \int_{-1}^{1} \left( \frac{T_n^1(0)}{T_{j}^1(0)} Q_j^1(x) , \frac{T_n^2(0)}{T_{j}^2(0)} Q_j^2(x) \right) \Psi (x)A^*dx+\frac{\tilde s_0}{\tilde y_0} \left(T_n^1(0),T_n^2(0)\right) M_{-1} A^*\\
&=\frac{\hat{a}_{-1}}{\hat{c}_0}\left(T_n^1(0),T_n^2(0)\right) \begin{pmatrix} 1/\tilde x_{-1} &0 \\ 0 & 1/\tilde y_{-1}\end{pmatrix} A^*+\frac{1}{\tilde y_0} \frac{T_n^1(0)}{T_{0}^1(0)}\int_{-1}^1\left(Q_0^1(x),Q_0^2(x)\right)\Psi(x)A^*dx.\end{aligned}$$ The third step is a consequence of ${\displaystyle}\frac{T_n^1(0)}{T_{j}^1(0)}=\frac{T_n^2(0)}{T_{j}^2(0)}$ using and the orthogonality properties. Since ${\displaystyle}\int_{-1}^1\left(Q_0^1(x),Q_0^2(x)\right)\Psi(x)dx=(1,0)$, $T_{0}^1(0)=-\tilde r_0\tilde x_{-1}/\tilde s_0$, $\hat{a}_{-1}/\hat{c}_0=\tilde x_{-1}\tilde s_{0}/\tilde y_0\tilde r_0$ and $A$ is given by then we have $$\begin{split}
\int_{-1}^{1} \left(\widehat Q_n^1(x), \widehat Q_n^2(x)\right)& \widehat \Psi (x)dx=\frac{\tilde x_{-1}\tilde s_{0}}{\tilde y_0\tilde r_0}\left(T_n^1(0),T_n^2(0)\right) \begin{pmatrix} 1/\tilde x_{-1} &0 \\ 0 & 1/\tilde y_{-1}\end{pmatrix} A^*-\frac{\tilde s_0}{\tilde y_0\tilde r_0\tilde x_{-1}}\left(T_n^1(0),0\right)A^*\\
&= \left(\frac{\tilde s_0}{\tilde y_0\tilde r_0}(1-1/\tilde x_{-1})T_n^1(0), \frac{\tilde x_{-1}\tilde s_0}{\tilde r_0\tilde y_0\tilde y_{-1}} T_n^2(0) \right) \begin{pmatrix} -{\displaystyle}\frac{\tilde r_0\tilde x_{-1}}{\tilde s_{0}}&\tilde x_{-1} \\ -{\displaystyle}\frac{\tilde r_0\tilde y_{-1}}{\tilde s_{0}}&\tilde y_{-1} \end{pmatrix}\\
&=\left( \frac{1}{\tilde y_0}(\tilde y_{-1}T_n^1(0)-\tilde x_{-1}T_n^2(0)), -\frac{\tilde s_0}{\tilde y_0\tilde r_0}(\tilde y_{-1}T_n^1(0)-\tilde x_{-1}T_n^2(0)) \right)=(0,0),
\end{split}$$ as a consequence of . For $n\leq-1$ the proof is similar but now using and . Finally, using , , , and , we have that $$\begin{split}
\int_{-1}^{1} &\left(\widehat Q_n^1(x), \widehat Q_n^2(x)\right) \widehat \Psi(x) \left(\widehat Q_n^1(x) , \widehat Q_n^2(x)\right)^*dx= \int_{-1}^{1} \left(T_n^1(x),T_n^2(x)\right) \Psi_T(x) \left(T_n^1(x), T_n^2(x)\right) ^* dx\\
&= \int_{-1}^{1} \left[ \frac{x}{\tilde s_n}\left( Q_n^1(x), Q_n^2(x)\right)-\frac{\tilde r_n}{ \tilde s_n} \left(T_{n-1}^1(x), T_{n-1}^2(x)\right) \right] \Psi_T(x) \left(T_n^1(x), T_n^2(x)\right)^*dx\\
&= \frac{1}{\tilde s_n}\int_{-1}^{1} x \left( Q_n^1(x), Q_n^2(x)\right) \Psi_T(x) \left(T_n^1(x),T_n^2(x)\right)^*dx\\
&= \frac{\tilde s_0}{\tilde s_n \tilde y_0}\int_{-1}^{1} \left( Q_n^1(x) , Q_n^2(x)\right)\Psi(x)\left[ \tilde y_n\left( Q_n^1(x) , Q_n^2(x)\right)^* + \tilde x_n\left( Q_{n+1}^1(x) , Q_{n+1}^2(x)\right)^* \right] \\
&= \frac{\tilde y_n\tilde s_0}{\tilde s_n \tilde y_0}\int_{-1}^{1} \left( Q_n^1(x) , Q_n^2(x)\right) \Psi(x) \left( Q_n^1(x) , Q_n^2(x)\right)^*dx=\frac{\tilde y_n\tilde s_0}{\tilde s_n \tilde y_0}\frac {1}{\pi _n}= \frac{1}{\hat \pi_n}.
\end{split}$$ The last step follows using , and the definition of $\pi_n$ and $\hat \pi_n$ in and , respectively.
The spectral matrix $\widehat\Psi$ associated with the Darboux transformation given in the previous theorem is *again* a conjugation by a matrix polynomial of degree 1 (namely $\bm T_0(x)$) of a *Geronimus transformation* of the original spectral matrix $\Psi$. This phenomenon is different from a Darboux transformation of a transition probability matrix on ${{\mathbb Z}}_{\geq0}$ for the LU factorization (see [@GdI3]), where the associated spectral measure is given by a *Christoffel transformation*, i.e. multiplying the original measure by the polynomial $x$.
Examples {#sec4}
========
Random walk on ${{\mathbb Z}}$ with constant transition probabilities
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Consider $P$ as in with coefficients $$a_n=a,\quad b_n=b,\quad c_n=c,\quad n\in\mathbb{Z},\quad a+b+c=1,\quad a,c>0, b\geq0.$$ For the UL factorization, the continued fractions can be computed explicitly. Indeed, using Proposition 4.1 of [@GdI3] we have that $$\label{FFF1}
H=\frac{1}{2}\left(1+c-a+\sqrt{(1+c-a)^2-4c}\right),$$ as long as $a\leq(1-\sqrt{c})^2$ (to ensure convergence). On the other hand, we have $H'=c/H$. Therefore we have $$\label{FFF2}
H'=\frac{1}{2}\left(1+c-a-\sqrt{(1+c-a)^2-4c}\right).$$ It is also possible to see, under the conditions on the parameters, that $0\leq H'\leq H\leq1$. Therefore, according to Theorem \[thmfracUL\], as long as $H'\leq y_0\leq H$ we always have a stochastic UL factorization where both factors are stochastic matrices. All formulas simplify considerably when $y_0=H$ (or $y_0=H'$). Indeed, in this case we have $$\label{coeff}
\begin{split}
y_n&=H,\quad x_n=1-H,\\
s_n&=1-\frac{c}{H},\quad r_n=\frac{c}{H},
\end{split}$$ while if $y_0=H'$ we have the same formulas but replacing $H'$ by $H$. It is also remarkable that in these cases the coefficients of the Darboux transformation *remain invariant*, i.e. the random walk $\widetilde P$ is exactly the same as the original random walk $P$. This phenomenon is not possible for Darboux transformations for random walks on ${{\mathbb Z}}_{\geq0}$.
As for the LU factorization, following Theorem \[thmfracLU\], we have a stochastic LU factorization if and only if we choose the free parameter $\tilde r_0$ in the range $H'\leq \tilde r_0\leq H$. In this case we also have that if we choose $\tilde r_0$ either $H$ or $H'$ then the coefficients of the Darboux transformation *remain invariant*, i.e. the random walk $\widehat P$ is exactly the same as the original random walk $P$.
The spectral matrix associated with this example appeared for the first time in the last section of [@KMc6] (for the case of $b=0$, i.e. the symmetric random walk) along with a method to compute the spectral matrix using Stieltjes transforms and the spectral measures associated with the positive and negative states of the original random walk. A combination of this method and Proposition 4.2 of [@GdI3] shows that the spectral matrix of the original random walk is given by only an absolutely continuous part, i.e. $$\label{WW1}
\Psi(x)=\frac{1}{\pi\sqrt{(x-\sigma_-)(\sigma_+-x)}}\begin{pmatrix} 1 & {\displaystyle}\frac{x-b}{2c}\\{\displaystyle}\frac{x-b}{2c}& a/c\end{pmatrix},\quad x\in[\sigma_-,\sigma_+],\quad \sigma_{\pm}=1-\left(\sqrt{a}\mp\sqrt{c}\right)^2.$$ Therefore we get the Karlin-McGregor formula for the $n$-step transition probabilities of the random walk $P$. Also, using Remark \[remrec\], we have that the random walk is always transient except for the case $a=c$. The random walk is never positive recurrent since the spectral matrix does not have a jump at the point 1, so for the case $a=c$ the random walk is null recurrent.
A straightforward computation shows that the moment $M_{-1}$ of $\Psi$ is given by $$\label{Mm11}
M_{-1}=\begin{pmatrix} {\displaystyle}\frac{1}{\sqrt{\sigma_-\sigma_+}} &{\displaystyle}\frac{1}{2c}\left(1-{\displaystyle}\frac{b}{\sqrt{\sigma_-\sigma_+}}\right)\\{\displaystyle}\frac{1}{2c}\left(1-{\displaystyle}\frac{b}{\sqrt{\sigma_-\sigma_+}}\right)& {\displaystyle}\frac{a}{c\sqrt{\sigma_-\sigma_+}}\end{pmatrix}.$$ In order for $M_{-1}$ to be well-defined we need to assume that $\sigma_->0$, i.e. $\sqrt{a}+\sqrt{c}<1$, or, in other words $a<(1-\sqrt{c})^2$, which is the condition for convergence of the continued fractions $H$ and $H'$. With this information we can compute the spectral matrices associated with the Darboux transformation $\widetilde P$ in (for the UL factorization) and $\widehat P$ in (for the LU factorization), both depending on one free parameter.
For the UL case we have, using Theorem \[thmorto\], that $$\widetilde \Psi(x)=\bm S_0(x) \Psi_S (x) \bm S_0^*(x),$$ where $$\bm S_0(x)=\begin{pmatrix} s_0& r_0 \\ -{\displaystyle}\frac{x_{-1}s_0}{y_{-1}}& {\displaystyle}\frac{x-x_{-1}r_0}{y_{-1}}
\end{pmatrix}=\begin{pmatrix} {\displaystyle}\frac{y_0-c}{y_0}& {\displaystyle}\frac{c}{y_0} \\ -{\displaystyle}\frac{a(y_0-c)}{y_0(1-a)-c}& {\displaystyle}\frac{x(y_0-c)-ac}{y_0(1-a)-c}
\end{pmatrix},$$ and $$\Psi_S(x)=\frac{y_0}{y_0-c}\left(y_0\frac{\Psi(x)}{x}+ \left[ \begin{pmatrix}1&0 \\ 0 &{\displaystyle}\frac{y_0-c}{c} \end{pmatrix}-y_0M_{-1}\right] \delta_0(x)\right),$$ where $\Psi$ and $M_{-1}$ are defined by and , respectively. Observe that the only free parameter is $y_0$. A straightforward computation gives that $$\widetilde\Psi(x)=\frac{1}{\pi x\sqrt{(x-\sigma_-)(\sigma_+-x)}}\left[\widetilde A+\widetilde Bx+\widetilde Cx^2\right]+\widetilde{\bm M}\delta_0(x),$$ where $$\begin{aligned}
\widetilde A&=\frac{(H'-y_0)(H-y_0)}{s_0y_0}\begin{pmatrix}1&-x_{-1}/y_{-1} \\ -x_{-1}/y_{-1}&(x_{-1}/y_{-1})^2 \end{pmatrix},\\
\widetilde B&=\begin{pmatrix}1&-{\displaystyle}\frac{by_0}{2cy_{-1}} \\ -{\displaystyle}\frac{by_0}{2cy_{-1}} &{\displaystyle}\frac{(y_0b-c(1-c))x_{-1}^2}{acy_{-1}^2} \end{pmatrix},\quad \widetilde C=\frac{y_0}{2cy_{-1}}\begin{pmatrix}0&1\\1&0\end{pmatrix},\\
\widetilde{\bm M}&=\frac{(y_0-H')(H-y_0)}{s_0y_0\sqrt{\sigma_-\sigma_+}}\begin{pmatrix}1&-x_{-1}/y_{-1} \\ -x_{-1}/y_{-1}&(x_{-1}/y_{-1})^2 \end{pmatrix}.\end{aligned}$$ From here we clearly see that if we choose $y_0$ in the range $H'\leq y_0\leq H$, then $\widetilde{\bm M}$ is a positive semidefinite matrix, so $\widetilde\Psi$ is a proper *weight matrix*. Another interesting case, as we mentioned earlier, is when we choose either $y_0=H$ or $y_0=H'$. In these cases we have $\widetilde A=0_{2\times2}, \widetilde{\bm M}=0_{2\times2}$ and following we get $$\widetilde B=\begin{pmatrix}1&-{\displaystyle}\frac{b}{2c} \\ -{\displaystyle}\frac{b}{2c}&a/c\end{pmatrix},\quad \widetilde C=\frac{1}{2c}\begin{pmatrix}0&1\\1&0\end{pmatrix}.$$ Therefore we recover the original weight matrix , as we predicted before. From the spectral matrix $\widetilde\Psi$ we get the Karlin-McGregor formula for the $n$-step transition probabilities of the random walk $\widetilde P$. The recurrence of the Darboux random walk is not affected by the transformation.
For the LU case we have, using Theorem \[thmorto2\] and after some computations, that $$\widehat\Psi(x)=\frac{1}{\pi x\sqrt{(x-\sigma_-)(\sigma_+-x)}}\left[\widehat A+\widehat Bx+\widehat Cx^2\right]+\widehat{\bm M}\delta_0(x),$$ where $$\begin{aligned}
\widehat A&=\frac{\tilde r_0(H'-\tilde r_0)(H-\tilde r_0)}{\tilde x_{-1}\tilde s_0^2}\begin{pmatrix}1&-\tilde s_{0}/\tilde r_{0} \\ -\tilde s_{0}/\tilde r_{0}&(\tilde s_{0}/\tilde r_{0})^2 \end{pmatrix},\\
\widehat B&=\frac{\tilde r_0\tilde y_0}{\tilde x_{-1}\tilde s_0}\begin{pmatrix}1&-{\displaystyle}\frac{b}{2\tilde y_{0}\tilde r_0} \\ -{\displaystyle}\frac{b}{2\tilde y_{0}\tilde r_0} &{\displaystyle}\frac{\tilde s_0 \tilde x_{-1}}{\tilde y_0\tilde r_0} \end{pmatrix},\quad \widehat C=\frac{1}{2\tilde s_0\tilde x_{-1}}\begin{pmatrix}0&1\\1&0\end{pmatrix},\\
\widehat{\bm M}&=\frac{\tilde r_0(\tilde r_0-H')(H-\tilde r_0)}{\tilde s_0^2\tilde x_{-1}\sqrt{\sigma_-\sigma_+}}\begin{pmatrix}1&-\tilde s_{0}/\tilde r_{0} \\ -\tilde s_{0}/\tilde r_{0}&(\tilde s_{0}/\tilde r_{0})^2 \end{pmatrix}.\end{aligned}$$ Again we clearly see that if we choose $\tilde r_0$ in the range $H'\leq \tilde r_0\leq H$, then $\widehat{\bm M}$ is a positive semidefinite matrix, so $\widehat\Psi$ is a proper *weight matrix*. If we choose either $\tilde r_0=H$ or $\tilde r_0=H'$, then we recover the original weight matrix , as we predicted before. Finally, from the spectral matrix $\widehat\Psi$ we get the Karlin-McGregor formula for the $n$-step transition probabilities of the random walk $\widehat P$ and the recurrence of the Darboux random walk is not affected by the transformation.
Random walk on ${{\mathbb Z}}$ with constant transition probabilities and an attractive or repulsive force
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Consider $P$ as in with $$a_n=a,\quad c_{n}=c,\quad n\geq0,\quad a_{-n}=c,\quad c_{-n}=a,\quad n\geq1,\quad b_n=b,\quad n\in\mathbb{Z},$$ and, as before, $a+b+c=1, a,c>0, b\geq0$. Observe that the probabilities $a$ and $c$ are interchanged for nonnegative and negative states of the random walk. Therefore if $a<c$ we have a random walk where the origin is an attractive state. On the contrary, if $a>c$, then the origin is a repulsive state.
Again, the continued fractions can be computed explicitly. $H$ is the same as before, i.e. $$\label{FFF1}
H=\frac{1}{2}\left(1+c-a+\sqrt{(1+c-a)^2-4c}\right),$$ as long as $a\leq(1-\sqrt{c})^2$ (to ensure convergence). On the other hand, we have $$H'=\frac{c}{1-{\displaystyle}\frac{c}{H}}.$$ Rationalizing we get $$\label{FFF2}
H'=\frac{c}{2a}\left(1+a-c-\sqrt{(1+c-a)^2-4c}\right).$$ It is easy to see that $H'>0$ if and only if $a>0$. On the other hand, now it is not true that $H'\leq H$ for all values of the parameters $a$ and $c$ such that $a\leq(1-\sqrt{c})^2$. In fact, as $a$ gets closer to 0, we have that there are some values of $c$ such that $H'>H$. A closer look to the inequality $H'\leq H$ shows that $a$ must be in the following range $$\label{rangg}
\begin{cases}
0<a\leq(1-\sqrt{c})^2,&\mbox{if}\quad 0<c\leq1/4,\\
0<a\leq{\displaystyle}\frac{1-2c}{2},&\mbox{if}\quad 1/4\leq c<1.
\end{cases}$$ Now, if we choose $y_0=H$, we observe that the positive states of the original random walk remain invariant under the Darboux transformation , while if $y_0=H'$, then the negative states of the original random walk remain invariant, but the rest of coefficients are difficult to compute.
As for the LU factorization, following Theorem \[thmfracLU\], we have a stochastic LU factorization if and only if we choose the free parameter $\tilde r_0$ in the range $H'\leq \tilde r_0\leq H$. As before, in order for $H'\leq H$, $a$ must be in the range . Similar behavior holds if we assume either $\tilde r_0=H$ or $\tilde r_0=H'$ for the random walk $\widehat P$.
The spectral matrix associated with this example appeared for the first time in Section 6 of [@G1] (for the case of $b=0$, i.e. the symmetric random walk). As before, we can compute the spectral matrix for the original random walk $P$. Now in this case the spectral matrix is given by an absolutely continuous and a discrete part, which we write as $\Psi(x)=\Psi_c(x)+\Psi_d(x)$. The absolutely continuous part is given by $$\Psi_c(x)=\frac{(a+c)\sqrt{(x-\sigma_-)(\sigma_+-x)}}{2\pi c(1-x)(x-2b+1)}\begin{pmatrix} 1 & {\displaystyle}\frac{x-b}{a+c}\\{\displaystyle}\frac{x-b}{a+c}& 1\end{pmatrix},\quad x\in[\sigma_-,\sigma_+],$$ where $\sigma_{\pm}$ are defined in . The discrete part is given by $$\Psi_d(x)=\frac{c-a}{2c}\left[\begin{pmatrix}1&-1\\-1&1\end{pmatrix}\delta_{2b-1}(x)+\begin{pmatrix}1&1\\1&1\end{pmatrix}\delta_{1}(x)\right]\chi_{\{c>a\}},$$ where $\chi_{A}$ is the indicator function. Therefore we get the Karlin-McGregor formula for the $n$-step transition probabilities of the random walk $P$. Also, using Remark \[remrec\], the random walk is always recurrent and positive recurrent if $c>a$ since for that case the spectral matrix has a jump at the point 1.
Now the computation of the moment $M_{-1}$ of $\Psi$ is more complicated since we have an absolutely continuous and a discrete part. Also one of the Dirac deltas of the discrete part can be located at $x=0$ if $b=1/2$, so that $M_{-1}$ may have a different expression in that case. Nevertheless, after some computations, we obtain $$\label{Mm12}
M_{-1}=\begin{pmatrix} \mu_{-1}&{\displaystyle}\frac{\gamma-b\mu_{-1}}{a+c}\\{\displaystyle}\frac{\gamma-b\mu_{-1}}{a+c}& \mu_{-1}\end{pmatrix}+\frac{c-a}{c(2b-1)}\begin{pmatrix}b&-(a+c)\\-(a+c)&b\end{pmatrix}\chi_{\{c>a\}},$$ where $$\mu_{-1}=\frac{1}{2c(2b-1)}\left((a+c)\sqrt{\sigma_-\sigma_+}-b|a-c|\right),\quad \gamma=\begin{cases} 1,&\mbox{if}\quad c\leq a,\\a/c,&\mbox{if}\quad c> a.\end{cases}$$ In order for $M_{-1}$ to be well-defined we need to assume that $\sigma_->0$, i.e. $\sqrt{a}+\sqrt{c}<1$, or, in other words $a<(1-\sqrt{c})^2$, which is the condition for convergence of the continued fractions $H$ and $H'$. For the case of $b=1/2$ ($c=1/2-a$) we obtain $$\label{Mm122}
M_{-1}=\begin{pmatrix} {\displaystyle}\frac{4a}{|4a-1|}& 2\left(\gamma-{\displaystyle}\frac{2a}{|4a-1|}\right)\\2\left(\gamma-{\displaystyle}\frac{2a}{|4a-1|}\right)&{\displaystyle}\frac{4a}{|4a-1|}\end{pmatrix}+\frac{1-4a}{1-2a}\begin{pmatrix}1&1\\1&1\end{pmatrix}\chi_{\{a<1/4\}}.$$
For the UL case we have, using Theorem \[thmorto\] and after some computations, that $$\widetilde\Psi(x)=\frac{\sqrt{(x-\sigma_-)(\sigma_+-x)}}{2\pi cx(1-x)(x-2b+1)}\left[\widetilde A+\widetilde Bx+\widetilde Cx^2\right]+\widetilde{\bm M}_0\delta_0(x)+\widetilde{\bm M}_{2b-1}\delta_{2b-1}(x)+\widetilde{\bm M}_1\delta_1(x),$$ where $$\begin{aligned}
\widetilde A&=\frac{(a+c)(\alpha_+-y_0)(\alpha_--y_0)}{s_0y_0}\begin{pmatrix}1&-x_{-1}/y_{-1} \\ -x_{-1}/y_{-1}&(x_{-1}/y_{-1})^2 \end{pmatrix},\quad \alpha_{\pm}=\frac{c}{a+c}\left(1\pm\sqrt{1-2a-2c}\right),\\
\widetilde B&=\begin{pmatrix}2c&{\displaystyle}\frac{(c(1-2c)-by_0)x_{-1}}{cy_{-1}} \\ {\displaystyle}\frac{(c(1-2c)-by_0)x_{-1}}{cy_{-1}} &{\displaystyle}\frac{2(by_0-c(1-c))x_{-1}^2}{cy_{-1}^2}\end{pmatrix},\quad \widetilde C=\frac{y_0s_0x_{-1}}{cy_{-1}}\begin{pmatrix}0&1\\1&{\displaystyle}\frac{(a-c)x_{-1}}{cy_{-1}}
\end{pmatrix},\\
\widetilde{\bm M}_0&=\frac{a(\bar{H'}-\bar{H})(y_0-H')(H-y_0)}{c(2b-1)s_0y_0}\begin{pmatrix}1&-x_{-1}/y_{-1} \\ -x_{-1}/y_{-1}&(x_{-1}/y_{-1})^2 \end{pmatrix},\\
\widetilde{\bm M}_{2b-1}&=\frac{c-a}{2c(2b-1)}\begin{pmatrix}(s_0-r_0)^2&-(s_0-r_0)(s_{-1}-r_{-1})\\ -(s_0-r_0)(s_{-1}-r_{-1})&(s_{-1}-r_{-1})^2 \end{pmatrix}\chi_{\{c>a\}},\quad \widetilde{\bm M}_1=\frac{c-a}{2c}\begin{pmatrix}1&1\\1&1\end{pmatrix}\chi_{\{c>a\}}.\end{aligned}$$ Here $\bar{H}$ and $\bar{H'}$ in $\widetilde{\bm M}_0$ are the radical conjugates of $H$ and $H'$, respectively, and we are implicitly assuming that $b\neq 1/2$. If $b=1/2$ then the Geronimus transformation is not well-defined for the Dirac delta at $x=0$. However, it is possible to define the spectral matrix in terms of the *derivative* of the Dirac delta at $x=0$. Indeed, the spectral matrix for the Darboux transformation is given in this case by $$\label{spmat12}
\widetilde\Psi(x)=\frac{\sqrt{(x-\sigma_-)(\sigma_+-x)}}{2\pi cx^2(1-x)}\left[\widetilde A+\widetilde Bx+\widetilde Cx^2\right]+\widetilde{\bm M}_0\delta_0(x)-\widetilde{\bm M}_{0}'\delta_{0}'(x)+\widetilde{\bm M}_1\delta_1(x),$$ where $\widetilde A,\widetilde B,\widetilde C$ and $\widetilde{\bm M}_1$ are the same as before writing $b=1/2$ and $c=1/2-a$ and $$\begin{aligned}
\widetilde{\bm M}_0'&=\lim_{b\to1/2}(2b-1)\widetilde{\bm M}_{2b-1},\\
\widetilde{\bm M}_0&=\eta\frac{(y_0-H')(H-y_0)}{s_0y_0}\begin{pmatrix}1&-x_{-1}/y_{-1} \\ -x_{-1}/y_{-1}&(x_{-1}/y_{-1})^2 \end{pmatrix},\quad\eta=\begin{cases} {\displaystyle}\frac{1}{2(1-2a)(1-4a)},&\mbox{if}\quad a<1/4,\\{\displaystyle}\frac{a}{4a-1},&\mbox{if}\quad a>1/4.\end{cases}\end{aligned}$$ If $a=1/4$ then the moment is not well-defined. Observe that in this case we are in the situation of the previous example. From the spectral matrix $\widetilde\Psi$ we get the Karlin-McGregor formula for the $n$-step transition probabilities of the random walk $\widetilde P$. The recurrence of the Darboux random walk is not affected by the transformation.
For the LU case we have, using Theorem \[thmorto2\] and after some computations, that
$$\widehat \Psi(x)=\frac{ \sqrt{(x-\sigma_-)(\sigma_+-x)}}{2\pi cx(1-x)(x-2b+1)}\left[\widehat A+\widehat Bx+\widehat Cx^2\right]+\widehat {\bm M}_0\delta_0(x)+\widehat {\bm M}_{2b-1}\delta_{2b-1}(x)+\widehat {\bm M}_1\delta_1(x),$$
where $$\begin{aligned}
\widehat A&=\frac{(a+c)(\beta_+-\tilde s_0)(\beta_--\tilde s_0)}{\tilde s_0\tilde y_0}\begin{pmatrix}1&-\tilde s_0/\tilde r_0 \\ -\tilde s_0/ \tilde r_0 &(\tilde s_0/\tilde r_0)^2 \end{pmatrix},\quad \beta_{\pm}=\frac{a+c\sqrt{2b-1}}{a+c},\\
\widehat B&=\frac{1}{\tilde y_0\tilde r_0}\begin{pmatrix} {\displaystyle}\frac{2\tilde r_0}{\tilde s_0}(c(1-c)-p\tilde r_0) & (a-c)\tilde r_0-c(1-2c) \\ (a-c)\tilde r_0-c(1-2c) & 2c\tilde s_0\tilde x_{-1}\end{pmatrix},\quad \widehat C=\frac{\tilde y_{-1}}{\tilde y_0}\begin{pmatrix} {\displaystyle}\frac{a-c}{\tilde y_{-1}} &1\\1&0
\end{pmatrix},\\
\widetilde{\bm M}_0&=\frac{a(\bar{H'}-\bar{H})(\tilde r_0-H')(H-\tilde r_0)}{c(2b-1)\tilde s_0\tilde y_0}\begin{pmatrix}1&-\tilde s_{0}/\tilde r_0 \\ -\tilde s_{0}/\tilde r_0&(\tilde s_{0}/\tilde r_0)^2 \end{pmatrix},\quad \widehat{\bm M}_1=\frac{c-a}{2c}\begin{pmatrix}1&1\\1&1\end{pmatrix}\chi_{\{c>a\}},\\
\widehat {\bm M}_{2b-1}&=\frac{c-a}{2c(2b-1)}\begin{pmatrix}(\tilde x_0-\tilde y_0)^2&-(\tilde x_0-\tilde y_0)(\tilde x_{-1}-\tilde y_{-1})\\ -(\tilde x_0-\tilde y_0)(\tilde x_{-1}-\tilde y_{-1})&(\tilde x_{-1}-\tilde y_{-1})^2 \end{pmatrix}\chi_{\{c>a\}}.\end{aligned}$$ Similar results hold for the case $b=1/2$. From the spectral matrix $\widehat\Psi$ we get the Karlin-McGregor formula for the $n$-step transition probabilities of the random walk $\widehat P$ and the recurrence of the Darboux random walk is not affected by the transformation.
Observe that if we assume $a=c$ then we recover the previous example and the Darboux transformation is invariant if we choose the free parameter $y_0=H=\frac{1}{2}(1+\sqrt{1-4a})$ or $y_0=H'=\frac{1}{2}(1-\sqrt{1-4a})$ (same for the LU factorization). We have not found any other choice of the free parameter $y_0$ (or $\tilde r_0$) such that the Darboux transformation is invariant. Nevertheless there are some values of the parameters where we can guarantee that the Darboux transformation is *almost invariant*. Indeed, for $0<a<1/2$ consider $c=1/2-a$. With this choice we always have $b=1/2$. The values of the continued fractions and depend on the value of $a$. We have two situations:
- If $0<a\leq1/4$, then $H=H'=1-2a$. Therefore the only choice of the parameter $y_0$ in order to have a stochastic factorization is $y_0=1-2a$. For this value we always have $$\label{coeff2}
\begin{split}
y_n&=1-2a, \quad n\geq0,\quad y_{-n}=2a, \quad n\geq1,\\
s_n&=r_n=1/2,\quad x_n=1-y_n,\quad n\in{{\mathbb Z}}.
\end{split}$$ Therefore the transition probabilities of the Darboux transformation are exactly the same as the original case except for the state 0, where we have $$\label{newc}
\tilde c_0=a,\quad \tilde a_0=a,\quad \tilde b_0=1-2a.$$ The random walk generated by $\widetilde P$ is almost the same as the original one except for the state 0. The spectral matrix is given in this case by $$\label{specs}
\widetilde\Psi(x)=\frac{\sqrt{(x-\sigma_-)(\sigma_+-x)}}{2\pi cx(1-x)}\left[\widetilde B+\widetilde Cx\right]+\widetilde{\bm M}_1\delta_1(x),\quad \sigma_{\pm}=1/2\pm\sqrt{2a(1-2a)},$$ where $$\begin{aligned}
\widetilde B&=(1-2a)\begin{pmatrix}1&-{\displaystyle}\frac{1-2a}{2a} \\ -{\displaystyle}\frac{1-2a}{2a}&{\displaystyle}\frac{(1-2a)^2}{4a^2}\end{pmatrix},\quad \widetilde C=\frac{1-2a}{2a}\begin{pmatrix}0&1\\1&-{\displaystyle}\frac{1-4a}{2a}
\end{pmatrix},\quad \widetilde{\bm M}_1=\frac{1-4a}{2(1-2a)}\begin{pmatrix}1&1\\1&1\end{pmatrix}.\end{aligned}$$ Similar results hold for the LU case.
- If $1/4<a<1/2$, then $H=1/2$ and $H'=(1-2a)/4a$. Therefore the parameter $y_0$ can be chosen in the range $$(1-2a)/4a\leq y_0\leq 1/2.$$ If we take $y_0=1-2a$ (which satisfies the previous bounds) then we are in the same situation of the previous case, i.e. we have for the sequences $x_n,y_n,s_n,r_n$ and the transition probabilities of the Darboux transformation are exactly the same as the original case except for the state 0, where we have again . The spectral matrix is then given by . If $y_0=1/2$ then we get invariance on the positive states of the Darboux random walk but not on the negative states nor state 0. On the contrary, if $y_0=(1-2a)/4a$, then we get invariance on the negative states of the Darboux random walk but not on the nonnegative states. The spectral matrix can be computed from . Similar results hold for the LU case.
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\[fourier-GUTenberg package\]
\#1[other]{}
undefined
backslashchar [symbols]{}[110]{}[largesymbols]{}[178]{} = [symbols]{}[110]{}[largesymbols]{}[178]{}
=2mu =2.5mu plus 1mu minus 2.5mu =3.5mu plus 2.5mu 850
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---
abstract: 'Inspired by the combinatorial constructions in earlier work of the authors that generalized the classical Alexander polynomial to a large class of spatial graphs with a balanced weight on edges, we show that the value of the Alexander polynomial evaluated at $t=1$ gives the weighted number of the spanning trees of the graph.'
address:
- ' Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo 153-8914, Japan '
- ' Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong '
author:
- Yuanyuan Bao
- Zhongtao Wu
title: Alexander polynomial and spanning trees
---
Introduction
============
In [@BW], we studied an Alexander polynomial $\Delta_{(G,c)}(t)$ for a certain class of spatial graphs in the 3-sphere $S^3$. Having a standard definition in terms of abelian covers of graph complement [@bao Section 5], the invariant is foremost a topological invariant that naturally generalizes the classical Alexander polynomial for knots and links. On the other hand, the equivalent definitions in terms of Kauffman states and MOY calculus discovered by the authors reveal several interesting combinatorial flavour of the invariant. In particular, it is shown that the value of the Alexander polynomial evaluated at $t=1$ is unchanged under crossing changes of the graph diagrams. Consequently, for a spatial graph $(G, c)$, $\Delta_{(g,c)} :=\Delta_{(G,c)}(1)$ is an intrinsic invariant of the underlying abstract graph $(g, c)$ of $(G, c)$.
In this paper, we go one step further and relate the invariant $\Delta_{(g,c)}$ with a certain count of spanning trees of the graph. In order to state the main result precisely, we introduce a few notations and terms first.
Given a vertex $r$ in a connected directed graph $\Gamma$, an [*oriented spanning tree of $\Gamma$ rooted at $r$*]{} is a spanning subgraph $T$ that satisfies the following $3$ conditions:
1. Every vertex $v\neq r$ has in-degree $1$.
2. The root $r$ has in-degree $0$.
3. $T$ has no oriented cycle.
Denote ${\mathcal}{T}_r(\Gamma)$ the set of all oriented spanning trees of $\Gamma$ rooted at $r$. One can then count the number of such spanning trees. If there is in addition a weight function $w: E\rightarrow \mathbb{Z}$ on the edge set, we can count instead the weighted number of spanning trees.
\[Def:weightednumber\] Define the weight of each spanning tree $T$ by $$\label{treeweight}
w(T) := \prod_{e\in E(T)} w(e),$$ where $E(T)$ is the edge set of $T$. Then, the [*weighted number of spanning trees rooted at $r$*]{} is: $$\label{weightednumber}
N(\Gamma, w, r):=\sum_{T\in {\mathcal}{T}_r(\Gamma)} w(T).$$
In this paper, we will be mostly interested in weight functions satisfying a certain balanced property. We review the related definitions below and refer the reader to [@BW Definition 2.1] for more details.
\[Def:moygraph\]
1. An *abstract MOY graph* is a directed graph that equipped with *a positive balanced weight/coloring* $c: E\to \mathbb{N}$ such that for each vertex $v$, $$\label{balancedcoloring}
\sum_{\text{$e$: pointing into $v$}} c(e)=\sum_{\text{$e$: pointing out of $v$}} c(e).$$
2. An [*MOY graph diagram*]{} in $\mathbb{R}^2$ is an immersion of an abstract MOY graph into $\mathbb{R}^2$, with crossing information and a *transverse orientation*: through each vertex $v$, there is a straight line $L_v$ that separates the edges entering $v$ and the edges leaving $v$.
(-1, -1.75) \[->-\] to (0, -0.75); (-0.5, -1.75) \[->-\] to (0, -0.75); (0, -0.75) \[->\] to (1, 0.25); (1, -1.75) \[->-\] to (0, -0.75); (0, -0.75) \[->\] to (-1, 0.25); (0, -0.75) \[->\] to (0.5, 0.25); (0, -0.75) node\[circle,fill,inner sep=1.5pt\]; (-0.7, -0.75)–(0.7, -0.75); (1.25, -0.75) node[$L_v$]{}; (0.1, -1.5) node[$......$]{}; (-0.1, 0) node[$......$]{};
\[fig:e3\]
An MOY graph $(G,c)$ is an equivalence class of MOY graph diagrams of $(g, c)$ under a certain topological equivalence relation (a.k.a. the [*Reidemeister moves*]{}). The Alexander polynomial $\Delta_{(G,c)}(t)$ is defined using an MOY graph diagram and proved to be a topological invariant for the equivalence class $(G,c)$. [**Convention.**]{} Throughout this paper, we only study connected graphs. As notational convention, we use $\Gamma, w$, and $w(T)$ to denote a general directed graph, a weight on $\Gamma$, and the weight of a spanning tree $T$ of $\Gamma$, respectively. In contrast, we reserve the letters $g, c$ and $c(T)$ for an abstract MOY graph, its balanced weight/coloring, and the weight of a spanning tree $T$ of $g$, respectively.\
With the balanced property on the weight function $c$, one can show that the weighted number of spanning trees of a given abstract MOY graph $(g, c)$ is in fact independent of the choice of the root $r$ (Proposition \[BalancedWeightCount\]). Thus we denote this number by $N(g, c)$, and our main theorem identifies it with the value $\Delta_{(g,c)}$.
\[MainTheorem\] For an abstract MOY graph $(g,c)$, we have $$\Delta_{(g,c)}=N(g,c).$$
As a corollary, we can establish the non-vanishing property for the Alexander polynomial $\Delta_{(G,c)}(t)$ as a consequence of the existence of spanning trees, thus generalizing an earlier result of the authors [@BW Theorem 5.6], which treated the case that $G$ is plane.
\[Nonvanishing\] Suppose $G$ is a connected MOY graph with a positive balanced weight $c$. Then $\Delta_{(G,c)}(1)>0$. In particular, this implies $\Delta_{(G,c)}(t)\neq 0$.
[**Acknowledgements.**]{} We would like to thank Xian’an Jin for helpful discussions. The first named author is partially supported by JSPS KAKENHI Grant Number JP20K14304. The second named author is partially supported by grant from the Research Grants Council of Hong Kong Special Administrative Region, China (Project No. 14309017 and 14301819).
Matrix tree theorem
===================
Kirchhoff’s matrix tree theorem is a classical result that allows one to determine the number of spanning trees by simply computing the determinant of an appropriate matrix associated to the graph. In this section, we review the theorem in the weighted directed graph setting. As an application, we prove the independence of the weighted number of spanning trees on the choice of root for balanced weight.
\[Def:Laplacian\] Suppose $(\Gamma, w)$ is a weighted directed graph with vertex set $V=\{v_1, v_2, \cdots, v_n\}$. The $n\times n$ [*Laplacian matrix*]{} $L$ is given by $$L_{ij}=\begin{cases}
-a_{ij} & \text{if } i\neq j \\
\sum\limits_{k=1}^n a_{kj} & \text{if } i=j
\end{cases}$$ where $$a_{ij}=\begin{cases}
\sum\limits_{\{e \,|\, e \text{ is an edge from } v_i \text{ to } v_j\}}w(e) & \text{if } i\neq j \\
0 & \text{if } i=j
\end{cases}.$$
Fix a vertex $v_r$ in $\Gamma$, and let $L_r$ be the Laplacian matrix of $\Gamma$ with the $r^{th}$ row and column removed. The matrix tree theorem asserts:
\[MatrixTreeTheorem\]
Let $(\Gamma, w)$ be a weighted directed graph. Then $$\det(L_r)=N(\Gamma, w, v_r),$$ where the right hand side is the weighted number of the oriented spanning trees rooted at $v_r$.
A proof of the above theorem can be found, for example, in [@Ch][@Tutte]. In general, the weighted number of oriented spanning trees with different roots are not necessarily the same. Nonetheless, for the most relevant case to our paper, namely, a balanced weight/coloring (Definition \[Def:moygraph\]), $N(\Gamma, w, v_r)$ is independent of the choice of root.
\[BalancedWeightCount\] Suppose $(\Gamma, w)$ is a directed graph with a balanced weight. We have $$N(\Gamma, w, v_i)=N(\Gamma, w, v_j)$$ for all $v_i, v_j \in V$. In other words, the weighted number of oriented spanning trees is independent of the choice of the root $v_r$.
Recall that a balanced weight means $\sum\limits_{\text{$e$: pointing into $v$}} w(e)=\sum\limits_{\text{$e$: pointing out of $v$}} w(e)$ for each vertex $v$. In terms of the Laplacian matrix $L$ in Definition \[Def:Laplacian\], this is equivalent to the identity $\sum\limits_{k=1}^n a_{kj}=\sum\limits_{k=1}^n a_{jk}$ for all $j$; so $L$ has the property that every row and every column sums up to $0$. From $\sum\limits_j L_{ij}=0$, we can readily show that the cofactors of the elements of any particular row of $L$ are all equal. From $\sum\limits_i L_{ij}=0$, we can likewise deduce that the cofactors of the elements of any particular column of $L$ are all equal. Hence, all cofactors of $L$ are equal, and the statement follows from Theorem \[MatrixTreeTheorem\].
\[ExampleSpanningTree\] We consider the directed graph $\Gamma$ with a balanced weight $w$ indicated by numbers drawing near the edges, which is Fig. \[Fig:gamma\].
(0, 2) node\[circle,fill,inner sep=1.5pt\]; (1, 0) node\[circle,fill,inner sep=1.5pt\]; (-1, 0) node\[circle,fill,inner sep=1.5pt\];
(1, 0) \[->-\] to (0, 2); (-1, 0) \[->-\] to (0, 2); (-1, 0) \[->-\] to (1, 0);
(0,2) \[->-\] to \[out=30,in=45\] (1,0); (0,2) \[->-\] to \[out=150,in=135\] (-1,0); (0, 2.2) node [$v_1$]{}; (-1.2, -0.2) node [$v_2$]{}; (1.3, -0.2) node [$v_3$]{};
(0, -0.3) node [$k$]{}; (0.1, 1) node [$i+k$]{}; (-0.7, 1) node [$j$]{}; (1.3, 1.2) node [$i$]{}; (-1.5, 1.2) node [$j+k$]{};
(0, 2) node\[circle,fill,inner sep=1.5pt\]; (1, 0) node\[circle,fill,inner sep=1.5pt\]; (-1, 0) node\[circle,fill,inner sep=1.5pt\];
(0,2) \[->-\] to \[out=30,in=45\] (1,0); (0,2) \[->-\] to \[out=150,in=135\] (-1,0); (0, 2.3) node [$v_1$]{}; (-1.2, -0.2) node [$v_2$]{}; (1.3, -0.2) node [$v_3$]{};
(1.3, 1.2) node [$i$]{}; (-1.7, 1.2) node [$j+k$]{};
(0, 2) node\[circle,fill,inner sep=1.5pt\]; (1, 0) node\[circle,fill,inner sep=1.5pt\]; (-1, 0) node\[circle,fill,inner sep=1.5pt\];
(0,2) \[->-\] to \[out=150,in=135\] (-1,0);
(-1, 0) \[->-\] to (1, 0); (0, 2.3) node [$v_1$]{}; (-1.2, -0.2) node [$v_2$]{}; (1.3, -0.2) node [$v_3$]{};
(0, -0.3) node [$k$]{}; (-1.7, 1.2) node [$j+k$]{};
(0, 2) node\[circle,fill,inner sep=1.5pt\]; (1, 0) node\[circle,fill,inner sep=1.5pt\]; (-1, 0) node\[circle,fill,inner sep=1.5pt\];
(-1, 0) \[->-\] to (0, 2);
(0,2) \[->-\] to \[out=30,in=45\] (1,0);
(0, 2.3) node [$v_1$]{}; (-1.2, -0.2) node [$v_2$]{}; (1.3, -0.2) node [$v_3$]{};
(1.3, 1.2) node [$i$]{}; (-0.8, 1) node [$j$]{};
(0, 2) node\[circle,fill,inner sep=1.5pt\]; (1, 0) node\[circle,fill,inner sep=1.5pt\]; (-1, 0) node\[circle,fill,inner sep=1.5pt\];
(-1, 0) \[->-\] to (0, 2);
(-1, 0) \[->-\] to (1, 0); (0, 2.3) node [$v_1$]{}; (-1.2, -0.2) node [$v_2$]{}; (1.3, -0.2) node [$v_3$]{};
(-0.8, 1) node [$j$]{}; (0, -0.3) node [$k$]{};
(0, 2) node\[circle,fill,inner sep=1.5pt\]; (1, 0) node\[circle,fill,inner sep=1.5pt\]; (-1, 0) node\[circle,fill,inner sep=1.5pt\];
(1, 0) \[->-\] to (0, 2); (-1, 0) \[->-\] to (1, 0); (0, 2.3) node [$v_1$]{}; (-1.2, -0.2) node [$v_2$]{}; (1.3, -0.2) node [$v_3$]{};
(0, -0.3) node [$k$]{}; (-0.1, 1) node [$i+k$]{};
(0, 2) node\[circle,fill,inner sep=1.5pt\]; (1, 0) node\[circle,fill,inner sep=1.5pt\]; (-1, 0) node\[circle,fill,inner sep=1.5pt\];
(1, 0) \[->-\] to (0, 2);
(0,2) \[->-\] to \[out=150,in=135\] (-1,0); (0, 2.2) node [$v_1$]{}; (-1.2, -0.2) node [$v_2$]{}; (1.3, -0.2) node [$v_3$]{};
(-0.1, 1) node [$i+k$]{};
(-1.7, 1.2) node [$j+k$]{};
We can check that the weighted numbers of spanning trees rooted at either vertex are all equal to $(i+k)(j+k)$, as illustrated in Fig. \[Fig: root1\], \[Fig: root2\] and \[Fig: root3\]. On the other hand, the Laplacian matrix is:
$$L=\begin{pmatrix}
i+j+k & -(j+k) & -i \\
-j & j+k & -k \\
-(i+k) & 0 & i+k
\end{pmatrix}.$$
It is a straightforward calculation to see that all the matrix cofactors are also equal to $(i+k)(j+k)$.
Spanning trees and Kauffman states
==================================
In this section, we prove Theorem \[MainTheorem\] for a planar MOY graph $(g,c)$, that is, there exists an MOY graph diagram $G$ of $g$ in the plane without intersections between the interior of edges. Our strategy is to express $\Delta_{(g,c)}=\Delta_{(G,c)}(1)$ using the Kauffman state sum formulation and then make an explicit bijection of Kauffman states to the oriented spanning trees in (\[weightednumber\]) for the weighted sum.
From now on, $G$ denotes a plane MOY graph diagram in $\mathbb{R}^2$ of the graph $g$. In [@BW Section 2], the authors defined the Kauffman state sum for general MOY graph diagrams; we do not need the full generality here, and instead, will only focus on the simpler plane diagram case, following [@BW Section 5.3].
Starting from the plane diagram $G$, we can obtain a [*decorated diagram*]{} $(G, \delta)$ by putting a base point $\delta$ on one of the edges in $G$ and drawing a circle around each vertex of $G$. Then
1. $\operatorname{Cr}(G)$: denotes the set of crossings which are the intersection points around each vertex between the incoming edges with the circle. Such a crossing is said to be generated by the edge. (In Example \[Ex1\], there are two crossings around $v_1$ generated by the edges with weights $j$ and $i+k$, respectively.)\
2. $\operatorname{Re}(G)$: denotes the set of regions, including the [*regular regions*]{} of $\mathbb{R}^{2}$ separated by $G$ and the [*circle regions*]{} around the vertices. Note that there is exactly one circle region around each vertex. [*Marked regions*]{} are the regions adjacent to the base point $\delta$, and the others are called [*unmarked regions*]{}. (In Example \[Ex1\], there are $2$ marked regions and $5$ unmarked regions.)\
3. Corners: There are $3$ corners around a crossing , and we call the one inside the circle region the [*north*]{} corner, the one on the left of the crossing the [*west*]{} corner and the one on the right the [*east*]{} corner, as illustrated below. Note also that every corner belongs to a unique region in $\operatorname{Re}(G)$.\
(0, 0.5) ellipse (1.5cm and 0.8cm); (0,-1) \[->-\] to (0,-0.3); (-0.4, -0.6) node [W]{}; (0.4, -0.6) node [E]{}; (0, 0) node [N]{};
Calculating the Euler characteristic of $\mathbb{R}^2$ using $G$ shows $$\vert \operatorname{Re}(G) \vert = \vert \operatorname{Cr}(G) \vert+2.$$ Also, a generic base point $\delta$ is adjacent to two regions, which will be denoted by $R_u$ and $R_{v}$. Note that since we only consider a graph equipped with a positive balanced weight, $R_u$ and $R_v$ must be distinct.
\[states\] A [*Kauffman state*]{} for a decorated diagram $(G, \delta)$ is a bijective map $$s: \, \operatorname{Cr}(G)\rightarrow \operatorname{Re}(G)\backslash \{R_u, R_{v}\},$$ which sends a crossing in $\operatorname{Cr}(G)$ to one of its corners. Let $S(G, \delta)$ denote the set of all Kauffman states.
\[Def:KauffmanStateSum\] Suppose $(G, \delta)$ is a decorated plane diagram with $n$ crossings $C_1, C_2, \cdots, C_n$ in $\operatorname{Cr}(G)$ and $n+2$ regions $R_1, R_2, \cdots, R_{n+2}$ in $\operatorname{Re}(G)$. We assume that the base point $\delta$ is on an edge $e_1$ with weight $i_1$.
1. We define a local contribution $P_{C_p}^{\triangle}(t)$ as in Fig. \[fig:f2\], which is a polynomial in $t$.
(0, 0.5) ellipse (1.5cm and 0.8cm); (0,-2) \[->-\] to (0,-0.3); (-0.5, -0.6) node [$t^{-i/2}$]{}; (0.5, -0.6) node [$t^{i/2}$]{}; (0, 0.2) node [$[i]$]{}; (0.3, -1.5) node [$i$]{};
(0, 0.5) ellipse (1.5cm and 0.8cm); (0,-2) \[->-\] to (0,-0.3); (0, 0.2) node [$t^{i_1/2}$]{}; (0.4, -1.3) node [$i_1$]{}; (0, -1.7) node [$*$]{}; (-0.3, -1.7) node [$\delta$]{};
Here, $\triangle$ represents a corner around $C_p$, and $$[i]:= \frac{t^{i/2}-t^{-i/2}}{t^{1/2}-t^{-1/2}}=t^{\frac{i-1}{2}}+\cdots +t^{\frac{1-i}{2}}.$$
2. For each Kauffman state $s$, let $$P_s(t):= \prod_{p=1}^{n} P_{C_p}^{s(C_p)}(t).$$
3. The [*Kauffman state sum*]{} is defined as $$\label{equation: planar}
\Delta_{(G, c)}(t):=\sum_{s\in S(G, \delta)} P_s(t).$$
The function $\Delta_{(G, c)}(t)$ is a topological invariant of $(G,c)$ well-defined up to $t^k$ and is independent of the choice of $\delta$.
Now, we are ready to prove Theorem \[MainTheorem\] for the planar graph case. The key observation is the remarkable similarity in the formula of the weighted number of spanning trees in Definition \[Def:weightednumber\] and the formula of the Kauffman state sum in Definition \[Def:KauffmanStateSum\]. Note that when one substitutes $t=1$ in Equation (\[equation: planar\]), the value $\Delta_{(G, c)}(1)$ is expressed as a sum of the value $P_s(1)$ over all Kauffman states $s$, where each $P_s(1)$ is a product of local contributions $P_{C_p}^{\triangle}(1)$ as in Fig. \[fig:f3\]. Our goal is to describe an explicit bijection between the set of rooted spanning trees ${\mathcal}{T}_r(G)$ with the set of Kauffman states $S(G, \delta)$, and then identify the weights $c(e)$ and $c(T)$ with the local contributions $P_{C_p}^{\triangle}(1)$ and $P_s(1)$, respectively.
(0, 0.5) ellipse (1.5cm and 0.8cm); (0,-2) \[->-\] to (0,-0.3); (-0.5, -0.6) node [$1$]{}; (0.5, -0.6) node [$1$]{}; (0, 0.2) node [$i$]{}; (0.3, -1.5) node [$i$]{};
\[Correspondence\] Suppose $(G,c)$ is a plane MOY graph diagram where the base point $\delta$ is on an edge that enters the vertex $r$. Then, there is a canonical bijective map $$\phi: {\mathcal}{T}_r(G) \longrightarrow S(G, \delta)$$ so that each oriented spanning tree $T$ rooted at $r$ has a one-to-one correspondence with a certain Kauffman state $s\in S(G, \delta)$. Moreover, the weight of each spanning tree $c(T)$ is equal to the corresponding term $P_s(1)$. Consequently, $\Delta_{(G,c)}(1)=N(G,c).$
Recall that for each spanning tree $T$ of the plane graph $G$, there is a canonical dual spanning tree $T^*$ in the dual graph $G^*$ consisting of all edges which are duals of the edges not in $T$. We then construct the Kauffman state $s$ in the following way:
1. For the edge $e_1$ where the base point $\delta$ is on, assign the crossing generated by $e_1$ to its north corner inside the circle region around the vertex $r$.
2. For each edge $e$ in the oriented spanning tree $T$ that enters the vertex $v$, assign the crossing generated by $e$ to its north corner inside the circle region around the vertex $v$.
3. For all other crossings, there is a unique way of assigning one of the east and west corners: Starting from the vertices in $G^*$ dual to the two regions $R_u$ and $R_v$, one can travel to all other vertices in $G^*$ along edges of $T^*$. In each step that we traverse $e$ on the dual edge $e^*$ from $v_1^*$ and $v_2^*$, assign the crossing generated by $e$ to the corner that belongs to the regular region dual to $v_2^*$.
The above construction may be easier to understand if one looks instead at the more concrete pictures in Example \[Ex1\] below. Since every vertex $v\neq r$ of an oriented spanning tree $T$ has in-degree 1, and vertices in $T^*$ have a one-to-one correspondence with regular regions of $\mathbb{R}^2$ separated by $G$, one can see that $s$ thus defined gives a bijective map between $\operatorname{Cr}(G)$ and $\operatorname{Re}(G)\backslash \{R_u, R_{v}\}$; so it is a Kauffman state by Definition \[states\]. Therefore, the map $\phi: {\mathcal}{T}_r(G) \longrightarrow S(G, \delta) $ is well-defined.
To show that $\phi$ is bijective, we construct an inverse map $\psi: S(G, \delta) \longrightarrow {\mathcal}{T}_r(G)$. Given a Kauffman state $s$, let $F\subset E$ be the set of edges so that $s$ assigns the crossing generated by those edges to their north corners. By definition $e_1 \in F$, recalling that $e_1$ is the edge with base point $\delta$. Then $E-F$ is the set of edges so that $s$ assigns the crossing generated by those edges to their east or west corners. Let $T$ be the subgraph of $G$ generated by $F-\{e_1\}$, and let $T^*$ be the subgraph of $G^*$ generated by $(E-F)\cup \{e_1\}$. We want to show that $T$ is an oriented spanning tree rooted at $r$. To this end, note that the size of $T$ is by definition $|F|-1=|V|-1$ since $s$ is a Kauffman state. It is also clear that every vertex $v\neq r$ has in-degree $1$ and the root $r$ has in-degree $0$. Thus, it suffices to show that $T$ does not have a cycle.
We prove by contradiction. Suppose $C$ is a cycle in $T$. Then $C$ bounds a disk $D$ in $\mathbb{R}^2$. Without loss of generality, we assume that $D\cap \mathrm{Int}(e_1) =\emptyset$, and therefore the marked regions $R_u, R_v$ are not contained in $D$. Consider the subgraph $G'=G\cap D$. Let $a$ be the number of vertices of $G'$, and let $b$ be the number of edges of $G'$. By Euler’s formula, the number of regular regions of $G$ inside $D$ is $b-a+1$. Together with the additional $a$ circle regions intersecting $D$, the total number is $$\#( \text{regions})= (b-a+1)+a=b+1.$$ Meanwhile, the total number of crossings in $D$ is $$\#( \text{crossings} \diaCircle )= b.$$ Since $\partial D=C\subset T$, the Kauffman state $s$ assigns the crossing generated by edges of the cycle $C$ to their north corners (circle regions intersecting the boundary of $D$). It follows that $s$ must map $b$ crossings in $D$ onto $b+1$ regions in $D$, which is impossible.
Thus, we proved $T$ is a spanning tree, and we define $\psi(s)=T$. Clearly, $\psi$ is the inverse of $\phi$. It is straightforward to see that the weight of each spanning tree $c(T)$ is equal to the corresponding term $P_s(1)$. This proves the theorem.
\[Ex1\]
The graph in Example \[ExampleSpanningTree\] is in fact an MOY graph diagram, so we can compute its Alexander polynomial. With the base point $\delta$ on the edge of weight $k$, we obtain a decorated diagram and find exactly one Kauffman state $s$, as indicated by $\bullet$ in Fig. \[fig:example\] (left). The associated spanning tree $T$ rooted at $v_3$ and its dual spanning tree $T^*$ specified by Theorem \[Correspondence\] are marked in thick red in Fig. \[fig:example\] (right).
(0, 1.5) node\[circle,fill,inner sep=1pt\]; (1, -0.5) node\[circle,fill,inner sep=1pt\]; (-1, -0.5) node\[circle,fill,inner sep=1pt\];
(1, -0.5) \[->-\] to (0, 1.5); (-1, -0.5) \[->-\] to (0, 1.5); (-1, -0.5) \[->-\] to (1, -0.5);
(0,1.5) \[->-\] to \[out=30,in=45\] (1,-0.5); (0,1.5) \[->-\] to \[out=150,in=135\] (-1,-0.5); (0, 1.7) node [$v_1$]{}; (-1.2, -0.6) node [$v_2$]{}; (1.2, -0.6) node [$v_3$]{};
(0, -0.8) node [$k$]{}; (0.1, 0.5) node [$i+k$]{}; (-0.7, 0.5) node [$j$]{}; (1.3, 0.7) node [$i$]{}; (-1.5, 0.7) node [$j+k$]{};
(0, 1.5) circle (0.4); (1, -0.5) circle (0.4); (-1, -0.5) circle (0.4);
(0.2, -0.5) node[$*$]{}; (0.3, -0.3) node[$\delta$]{};
(0.8, -0.5) node [$\bullet$]{}; (0.1, 1.3) node [$\bullet$]{}; (-1.15, -0.3) node [$\bullet$]{}; (-0.3, 1.1) node [$\bullet$]{}; (1.1, 0) node [$\bullet$]{};
(0, 2) node\[circle,fill,inner sep=1.5pt\]; (1, 0) node\[circle,fill,inner sep=1.5pt\]; (-1, 0) node\[circle,fill,inner sep=1.5pt\];
(1, 0) \[->-, color=red, line width=0.7mm\] to (0, 2); (-1, 0) \[->-\] to (0, 2); (-1, 0) \[->-\] to (1, 0);
(0,2) \[->-\] to \[out=30,in=45\] (1,0); (0,2) \[->-, color=red, line width=0.7mm\] to \[out=150,in=135\] (-1,0); (0, 2.2) node [$v_1$]{}; (-1.2, -0.1) node [$v_2$]{}; (1.2, -0.1) node [$v_3$]{};
(-0.8, 2) \[color=red, line width=0.7mm\] node [$T$]{};
(0, 1) node[$\circ$]{}; (0, -1) node[$\circ$]{}; (0.7, 1.35) node[$\circ$]{}; (-0.7, 1.35) node[$\circ$]{};
(0.7,1.35) to \[out=0,in=90\] (1.8,0) to \[out=270, in=0\] (0,-1);
(-0.7,1.35) to \[out=180,in=90\] (-1.8,0) to \[out=270, in=180\] (0,-1);
(0, 1) – (0, -1); (0, 1) – (-0.7, 1.35); (0, 1) – (0.7, 1.35);
(0.3, -0.5) \[color=red\] node [$T^*$]{};
According to Definition \[Def:KauffmanStateSum\], $$\Delta_{(G,c)}(t)=P_s(t)= t^{k/2}\cdot t^{-j/2}\cdot t^{i/2}\cdot [i+k]\cdot [j+k],$$ as $s$ is the unique Kauffman state. In particular, we see that $$P_s(1)=(i+k)(j+k)=c(T).$$
Spanning trees and skein relations
==================================
To establish Theorem \[MainTheorem\] for arbitrary graphs, our strategy is to prove a set of skein relations and reduce the general case to the plane graph case, which was just confirmed in the previous section.
Let $(g, c)$ be an abstract MOY graph, and let $G$ be an MOY graph diagram of $g$ on $\mathbb{R}^2$. In general, $G$ may have double points corresponding to crossings of type (positive crossing) and (negative crossing). Since neither of the invariant $\Delta_{(g, c)}$ or $N(g, c)$ depends on the types of crossings, hereafter, we simply use to represent a double point (for either positive or negative crossing) in $G$. We begin with a lemma.
\[insertvertex\] Let $(g, c)$ be an abstract MOY graph. We obtain a new graph $(g', c')$ by inserting a vertex $v'$ of degree 2 into an edge $e$ of $g$. Then we have $$N(g', c')=c(e)N(g, c),$$ where $c'$ denotes the induced balanced weight on $g'$ from $c$.
Suppose $e$ in $g$ is separated into two edges $e_1$ and $e_2$ in $g'$, and $e_1$ is the edge pointing to $v'$. $$\begin{aligned}
\begin{tikzpicture}
\draw (-1, 0) [->-] to (1, 0);
\draw (1,0.5) node {$v'$};
\draw (1, 0) [->-] to (3, 0);
\draw (1,0) node[circle,fill,inner sep=1pt]{};
\draw (0,-0.5) node {$e_1$};
\draw (2,-0.5) node {$e_2$};
\end{tikzpicture}\end{aligned}$$ For any root vertex $r$ in $g$, consider $\mathcal {T}_r(g)$ and $\mathcal {T}_{r}(g')$, the set of all oriented spanning trees of $g$ and $g'$ rooted at $v$, respectively. There is a canonical one-to-one correspondence between $\mathcal {T}_r(g)$ and $\mathcal {T}_{r}(g')$: if $T\in \mathcal {T}_r(g)$ contains $e$, let $T'=(T-\{e\})\cup \{e_1, e_2\}$; if $T \in \mathcal {T}_r(g)$ does not contain $e$, let $T'=T\cup \{e_1\}$. In either case, we have $c'(T')=c(e)c(T)$. Taking the sum over all trees gives the lemma.
\[skeinrelation\] We have the following skein relations for the weighted number of spanning trees, where $N(G)$ represents $N(g, c)$ if $G$ is a graph diagram with underlying graph $(g, c)$. In each equality, the graph diagrams are identical outside the local diagrams shown there. When $i=j$, ignore the edge with weight $j-i$. $$\begin{aligned}
&\text{When $i\leq j$:}\\
&N\left(\begin{tikzpicture}[baseline=-0.65ex, thick, scale=0.6]
\draw (-1, -1) [->] -- (1, 1) node[above]{$i$};
\draw (1,-1) [->] to (-1,1) node[above]{$j$};
\end{tikzpicture}\right)
=\frac{-1}{i\cdot j}\cdot
N\left(\begin{tikzpicture}[baseline=-0.65ex, thick, scale=1.2]
\draw (0,-1) [->-] to (0, 0.33);
\draw (0, 0.33) [->] to (0,1);
\draw (1,-1) [->-] to (1, -0.33);
\draw (1, -0.33) [->] to (1,1);
\draw (0,0.33) [-<-] to [out=270,in=180] (0.5,0) to [out=0,in=90] (1,-0.33);
\draw (0,1.25) node {$j$};
\draw (0,-1.25) node {$i$};
\draw (1,1.25) node {$i$};
\draw (0.9,-1.25) node {$j$};
\draw (0.5,0.3) node {$j-i$};
\draw (1, -0.33) node[circle,fill,inner sep=1pt]{};
\draw (0, 0.33) node[circle,fill,inner sep=1pt]{};
\end{tikzpicture}\right)
+ \, \frac{1}{i\cdot (i+j)}\cdot
N\left(
\begin{tikzpicture}[baseline=-0.65ex, thick, scale=1.2]
\draw (0,-1) [->-] to [out=90,in=270] (0.5,-0.33);
\draw (0.5, -0.33) [->-] to [out=90,in=270] (0.5,0.33);
\draw (0.5, 0.33) [->] to [out=90,in=270] (0,1);
\draw (1,-1) [->-] to [out=90,in=270] (0.5,-0.33);
\draw (1,1) [<-] to [out=270,in=90] (0.5,0.33);
\draw (0, -1.25) node {$i$};
\draw (0.9,-1.25) node {$j$};
\draw (0,1.25) node {$j$};
\draw (1,1.25) node {$i$};
\draw (1,0) node {$i+j$};
\draw (0.5, -0.33) node[circle,fill,inner sep=1pt]{};
\draw (0.5, 0.33) node[circle,fill,inner sep=1pt]{};
\end{tikzpicture}\right).\\
&\text{When $j< i$:}\\
&\left(\begin{tikzpicture}[baseline=-0.65ex, thick, scale=0.6]
\draw (-1, -1) [->] -- (1, 1) node[above]{$i$};
\draw (1,-1) [->] to (-1,1) node[above]{$j$};
\end{tikzpicture}\right)
=\frac{-1}{i\cdot j}\cdot
\left(\begin{tikzpicture}[baseline=-0.65ex, thick, scale=1.2]
\draw (0,-1) [->-] to (0, -0.33);
\draw (0, -0.33) [->] to (0,1);
\draw (1,-1) [->-] to (1, 0.33);
\draw (1, 0.33) [->] to (1,1);
\draw (0,-0.33) [->-] to [out=90,in=180] (0.5,0) to [out=0,in=270] (1,0.33);
\draw (0,1.25) node {$j$};
\draw (0,-1.25) node {$i$};
\draw (1,1.25) node {$i$};
\draw (0.9,-1.25) node {$j$};
\draw (0.5,0.3) node {$i-j$};
\draw (1, 0.33) node[circle,fill,inner sep=1pt]{};
\draw (0, -0.33) node[circle,fill,inner sep=1pt]{};
\end{tikzpicture}\right)
+ \, \frac{1}{j\cdot (i+j)}\cdot
\left(
\begin{tikzpicture}[baseline=-0.65ex, thick, scale=1.2]
\draw (0,-1) [->-] to [out=90,in=270] (0.5,-0.33);
\draw (0.5, -0.33) [->-] to [out=90,in=270] (0.5,0.33);
\draw (0.5, 0.33) [->] to [out=90,in=270] (0,1);
\draw (1,-1) [->-] to [out=90,in=270] (0.5,-0.33);
\draw (1,1) [<-] to [out=270,in=90] (0.5,0.33);
\draw (0, -1.25) node {$i$};
\draw (0.9,-1.25) node {$j$};
\draw (0,1.25) node {$j$};
\draw (1,1.25) node {$i$};
\draw (1,0) node {$i+j$};
\draw (0.5, -0.33) node[circle,fill,inner sep=1pt]{};
\draw (0.5, 0.33) node[circle,fill,inner sep=1pt]{};
\end{tikzpicture}\right).\\\end{aligned}$$
We prove the first relation only and the second one can be proved analogously. Denote $G, G_1, G_2$ the diagram on the left hand side and the two diagram on the right hand side of the equality, respectively.
We assume without loss of generality that there are four vertices $a, b, c, d$ of degree $2$ in each of the diagram as shown below. If not, we will just insert the missing ones: Lemma \[insertvertex\] ensures that the invariants $N(G)$, $N(G_1)$ and $N(G_2)$ will change by a same factor.
$$\begin{aligned}
\begin{tikzpicture}[baseline=-0.65ex, thick, scale=1]
\draw (-1, -1) [->] -- (1, 1) node[above]{$i$};
\draw (1,-1) [->] to (-1,1) node[above]{$j$};
\draw (0.6,0.6) node[circle,fill,inner sep=1pt]{};
\draw (-0.6,0.6) node[circle,fill,inner sep=1pt]{};
\draw (0.6,-0.6) node[circle,fill,inner sep=1pt]{};
\draw (-0.6,-0.6) node[circle,fill,inner sep=1pt]{};
\draw (0.9,0.6) node{$b$};
\draw (-0.9,0.6) node{$a$};
\draw (-0.9,-0.6) node{$c$};
\draw (0.9,-0.6) node{$d$};
\draw (0,-2.2) node{$G$};
\end{tikzpicture}\quad\quad\quad
\begin{tikzpicture}[baseline=-0.65ex, thick, scale=1.4]
\draw (0,-0.7) [->-] to (0, 0.33);
\draw (0,-1) -- (0, -0.7);
\draw (0, 0.33) [->] to (0,1);
\draw (1,-0.7) [->-] to (1, -0.33);
\draw (1,-1) -- (1, -0.7);
\draw (1, -0.33) [->] to (1,1);
\draw (0,0.33) [-<-] to [out=270,in=180] (0.5,0) to [out=0,in=90] (1,-0.33);
\draw (0,1.25) node {$j$};
\draw (0,-1.25) node {$i$};
\draw (1,1.25) node {$i$};
\draw (0.9,-1.25) node {$j$};
\draw (0.5,0.3) node {$j-i$};
\draw (1, -0.33) node[circle,fill,inner sep=1pt]{};
\draw (0, 0.33) node[circle,fill,inner sep=1pt]{};
\draw (0,-0.7) node[circle,fill,inner sep=1pt]{};
\draw (-0.2,-0.7) node {$c$};
\draw (1,-0.7) node[circle,fill,inner sep=1pt]{};
\draw (1.2,-0.7) node {$d$};
\draw (0,0.7) node[circle,fill,inner sep=1pt]{};
\draw (-0.2,0.7) node {$a$};
\draw (1,0.7) node[circle,fill,inner sep=1pt]{};
\draw (1.2,0.7) node {$b$};
\draw (0.5,-1.8) node {$G_1$};
\draw (1.25,-0.33) node {$v_1$};
\draw (-0.25,0.33) node {$v_2$};
\end{tikzpicture}
\quad\quad\quad
\begin{tikzpicture}[baseline=-0.65ex, thick, scale=1.4]
\draw (0,-1) [->-] to [out=90,in=270] (0.5,-0.33);
\draw (0.5, -0.33) [->-] to [out=90,in=270] (0.5,0.33);
\draw (0.5, 0.33) [->] to [out=90,in=270] (0,1);
\draw (1,-1) [->-] to [out=90,in=270] (0.5,-0.33);
\draw (1,1) [<-] to [out=270,in=90] (0.5,0.33);
\draw (0, -1.25) node {$i$};
\draw (0.9,-1.25) node {$j$};
\draw (0,1.25) node {$j$};
\draw (1,1.25) node {$i$};
\draw (1,0) node {$i+j$};
\draw (0.5, -0.33) node[circle,fill,inner sep=1pt]{};
\draw (0.5, 0.33) node[circle,fill,inner sep=1pt]{};
\draw (0.5,-1.8) node {$G_2$};
\draw (0.93,-0.8) node[circle,fill,inner sep=1pt]{};
\draw (0.13,0.75) node[circle,fill,inner sep=1pt]{};
\draw (0.87,0.75) node[circle,fill,inner sep=1pt]{};
\draw (0.07,-0.8) node[circle,fill,inner sep=1pt]{};
\draw (1.1,0.75) node{$b$};
\draw (-0.1,0.75) node{$a$};
\draw (-0.1,-0.75) node{$c$};
\draw (1.1,-0.75) node{$d$};
\draw (0.7,-0.33) node{$v_1$};
\draw (0.7,0.33) node{$v_2$};
\end{tikzpicture}\end{aligned}$$
Proposition \[BalancedWeightCount\], which claims that the weighted number of spanning trees is independent of the choice of the root for a balanced weight, enables us to further simplify our argument. In each of $G, G_1, G_2$, we choose $b$ to be the root and analyze the shape of the corresponding spanning trees.
An oriented spanning tree of $G$ rooted at $b$ must contain the edge pointing to $c$, the edge pointing to $d$, the edge pointing out of $b$ and the edge $da$, which are highlighted in thick red as below. $$\begin{aligned}
\begin{tikzpicture}[baseline=-0.65ex, thick, scale=1][h!]
\draw [line width=0.7mm, red] (0.6,0.6) [->] -- (1, 1);
\draw (1,1) node[above]{$i$};
\draw [dotted] (-0.6,-0.6) -- (0.6,0.6);
\draw [line width=0.7mm, red] (-1, -1) -- (-0.6,-0.6);
\draw (-0.6,0.6) [->] to (-1,1) node[above]{$j$};
\draw [line width=0.7mm, red] (1, -1) -- (-0.6,0.6);
\draw (0.6,0.6) node[circle,fill,inner sep=1pt]{};
\draw (-0.6,0.6) node[circle,fill,inner sep=1pt]{};
\draw (0.6,-0.6) node[circle,fill,inner sep=1pt]{};
\draw (-0.6,-0.6) node[circle,fill,inner sep=1pt]{};
\draw (0.9,0.6) node{$b$};
\draw (-0.9,0.6) node{$a$};
\draw (-0.9,-0.6) node{$c$};
\draw (0.9,-0.6) node{$d$};
\end{tikzpicture}\end{aligned}$$
An oriented spanning tree of $G_1$ rooted at $b$ must contain the edge pointing to $c$, the edge pointing to $d$, the edge $dv_1$, the edge $v_2a$, the edge pointing out of $b$ and either one of the edges in the following 2 cases:
- the edge $v_1v_2$
- the edge $cv_2$,
which are highlighted in thick red as below.
$$\begin{aligned}
\begin{tikzpicture}[baseline=-0.65ex, thick, scale=1.5][h!]
\draw [dotted] (0,-0.7) [->-] to (0, 0.33);
\draw [line width=0.7mm, red] (0,-1) -- (0, -0.7);
\draw [line width=0.7mm, red] (0, 0.33) -- (0,0.7);
\draw (0, 0.7) [->] to (0,1);
\draw [line width=0.7mm, red] (1,-0.7) [->-] to (1, -0.33);
\draw [line width=0.7mm, red] (1,-1) -- (1, -0.7);
\draw [dotted](1, -0.33) -- (1,0.7);
\draw [line width=0.7mm, red] (1, 0.7) [->] to (1,1);
\draw [line width=0.7mm, red] (0,0.33) [-<-] to [out=270,in=180] (0.5,0) to [out=0,in=90] (1,-0.33);
\draw (0,1.25) node {$j$};
\draw (0,-1.25) node {$i$};
\draw (1,1.25) node {$i$};
\draw (0.9,-1.25) node {$j$};
\draw (0.5,0.3) node {$j-i$};
\draw (1, -0.33) node[circle,fill,inner sep=1pt]{};
\draw (0, 0.33) node[circle,fill,inner sep=1pt]{};
\draw (0,-0.7) node[circle,fill,inner sep=1pt]{};
\draw (-0.2,-0.7) node {$c$};
\draw (1,-0.7) node[circle,fill,inner sep=1pt]{};
\draw (1.2,-0.7) node {$d$};
\draw (0,0.7) node[circle,fill,inner sep=1pt]{};
\draw (-0.2,0.7) node {$a$};
\draw (1,0.7) node[circle,fill,inner sep=1pt]{};
\draw (1.2,0.7) node {$b$};
\draw (0.5,-1.8) node {(A)};
\draw (1.25,-0.33) node {$v_1$};
\draw (-0.25,0.33) node {$v_2$};
\end{tikzpicture}\quad\quad\quad
\begin{tikzpicture}[baseline=-0.65ex, thick, scale=1.5]
\draw [line width=0.7mm, red] (0,-0.7) [->-] to (0, 0.33);
\draw [line width=0.7mm, red] (0,-1) -- (0, -0.7);
\draw [line width=0.7mm, red] (0, 0.33) -- (0,0.7);
\draw (0, 0.7) [->] to (0,1);
\draw [line width=0.7mm, red] (1,-0.7) [->-] to (1, -0.33);
\draw [line width=0.7mm, red] (1,-1) -- (1, -0.7);
\draw [dotted](1, -0.33) -- (1,0.7);
\draw [line width=0.7mm, red] (1, 0.7) [->] to (1,1);
\draw [dotted] (0,0.33) [-<-] to [out=270,in=180] (0.5,0) to [out=0,in=90] (1,-0.33);
\draw (0,1.25) node {$j$};
\draw (0,-1.25) node {$i$};
\draw (1,1.25) node {$i$};
\draw (0.9,-1.25) node {$j$};
\draw (0.5,0.3) node {$j-i$};
\draw (1, -0.33) node[circle,fill,inner sep=1pt]{};
\draw (0, 0.33) node[circle,fill,inner sep=1pt]{};
\draw (0,-0.7) node[circle,fill,inner sep=1pt]{};
\draw (-0.2,-0.7) node {$c$};
\draw (1,-0.7) node[circle,fill,inner sep=1pt]{};
\draw (1.2,-0.7) node {$d$};
\draw (0,0.7) node[circle,fill,inner sep=1pt]{};
\draw (-0.2,0.7) node {$a$};
\draw (1,0.7) node[circle,fill,inner sep=1pt]{};
\draw (1.2,0.7) node {$b$};
\draw (0.5,-1.8) node {(B)};
\draw (1.25,-0.33) node {$v_1$};
\draw (-0.25,0.33) node {$v_2$};
\end{tikzpicture}\end{aligned}$$
An oriented spanning tree of $G_2$ rooted at $b$ must contain the edge pointing to $c$, the edge pointing to $d$, the edge $v_1v_2$, the edge $v_2a$, the edge pointing out of $b$ and either one of the edges in the following 2 cases:
- the edge $dv_1$,
- the edge $cv_1$,
which are highlighted in thick red as below.
$$\begin{aligned}
\begin{tikzpicture}[baseline=-0.65ex, thick, scale=1.4][h!]
\draw [line width=0.7mm, red] (0,-1) to [out=90,in=225] (0.13,-0.75);
\draw [dotted] (0.13,-0.75) [->-] to [out=45,in=270] (0.5,-0.33);
\draw [line width=0.7mm, red] (0.5, -0.33) [->-] to [out=90,in=270] (0.5,0.33);
\draw [line width=0.7mm, red] (0.5, 0.33) to [out=90,in=315] (0.13,0.75);
\draw (0.13,0.75) [->] to [out=135,in=270] (0,1);
\draw [line width=0.7mm, red] (1,-1) [->-] to [out=90,in=270] (0.5,-0.33);
\draw [line width=0.7mm, red] (0.87,0.75) [->] to [out=45,in=270] (1, 1);
\draw [dotted] (0.5, 0.33) to [out=90,in=225] (0.87,0.75);
\draw (0, -1.25) node {$i$};
\draw (0.9,-1.25) node {$j$};
\draw (0,1.25) node {$j$};
\draw (1,1.25) node {$i$};
\draw (1,0) node {$i+j$};
\draw (0.5, -0.33) node[circle,fill,inner sep=1pt]{};
\draw (0.5, 0.33) node[circle,fill,inner sep=1pt]{};
\draw (0.5,-1.8) node {$(\alpha)$};
\draw (0.93,-0.8) node[circle,fill,inner sep=1pt]{};
\draw (0.13,0.75) node[circle,fill,inner sep=1pt]{};
\draw (0.87,0.75) node[circle,fill,inner sep=1pt]{};
\draw (0.13,-0.75) node[circle,fill,inner sep=1pt]{};
\draw (1.1,0.75) node{$b$};
\draw (-0.1,0.75) node{$a$};
\draw (0,-0.65) node{$c$};
\draw (1.1,-0.75) node{$d$};
\draw (0.7,-0.33) node{$v_1$};
\draw (0.7,0.33) node{$v_2$};
\end{tikzpicture}\quad\quad\quad
\begin{tikzpicture}[baseline=-0.65ex, thick, scale=1.4]
\draw [line width=0.7mm, red] (0,-1) to [out=90,in=225] (0.13,-0.75);
\draw [line width=0.7mm, red] (0.13,-0.75) [->-] to [out=45,in=270] (0.5,-0.33);
\draw [line width=0.7mm, red] (0.5, -0.33) [->-] to [out=90,in=270] (0.5,0.33);
\draw [line width=0.7mm, red] (0.5, 0.33) to [out=90,in=315] (0.13,0.75);
\draw (0.13,0.75) [->] to [out=135,in=270] (0,1);
\draw [line width=0.7mm, red] (1,-1) to [out=90,in=315] (0.87,-0.75);
\draw [dotted] (0.87,-0.75) to [out=135,in=270] (0.5, -0.33);
\draw [line width=0.7mm, red] (0.87,0.75) [->] to [out=45,in=270] (1, 1);
\draw [dotted] (0.5, 0.33) to [out=90,in=225] (0.87,0.75);
\draw (0, -1.25) node {$i$};
\draw (0.9,-1.25) node {$j$};
\draw (0,1.25) node {$j$};
\draw (1,1.25) node {$i$};
\draw (1,0) node {$i+j$};
\draw (0.5, -0.33) node[circle,fill,inner sep=1pt]{};
\draw (0.5, 0.33) node[circle,fill,inner sep=1pt]{};
\draw (0.5,-1.8) node {$(\beta)$};
\draw (0.87,-0.75) node[circle,fill,inner sep=1pt]{};
\draw (0.13,0.75) node[circle,fill,inner sep=1pt]{};
\draw (0.87,0.75) node[circle,fill,inner sep=1pt]{};
\draw (0.13,-0.75) node[circle,fill,inner sep=1pt]{};
\draw (1.1,0.75) node{$b$};
\draw (-0.1,0.75) node{$a$};
\draw (0,-0.65) node{$c$};
\draw (1,-0.65) node{$d$};
\draw (0.7,-0.33) node{$v_1$};
\draw (0.7,0.33) node{$v_2$};
\end{tikzpicture}\end{aligned}$$
Note that an oriented spanning tree $T$ of $G$ rooted at $b$ corresponds to a unique tree $T_1$ of type (A) in $G_1$ and a unique tree $T_2$ of type ($\alpha$) in $G_2$, and vice versa. Hence, $$\mathcal{T}_b(G) \xleftrightarrow{1:1} \{ \text{type (A) in } \mathcal{T}_b(G_1) \} \xleftrightarrow{1:1} \{ \text{type ($\alpha$) in } \mathcal{T}_b(G_2) \} .$$ Under this correspondence, we can check that $$c(T)=\frac{-1}{i\cdot j} c(T_1)+\frac{1}{i \cdot (i+j)}c(T_2).$$
Similarly, an oriented spanning tree $T'_1$ of type (B) in $G_1$ corresponds to a unique tree $T'_2$ of type ($\beta$) in $G_2$, and vice versa. Hence, $$\{ \text{type (B) in } \mathcal{T}_b(G_1) \} \xleftrightarrow{1:1} \{ \text{type ($\beta$) in } \mathcal{T}_b(G_2) \}.$$ Under this correspondence, we have $$\frac{-1}{i\cdot j} c(T'_1)+\frac{1}{i \cdot (i+j)}c(T'_2)=0.$$
Finally, we sum up over all trees and apply the above two identities on their weights to obtain the desired equality. This completes the proof.
Now we are ready to prove our main theorem.
The key observation is that the skein relations in Proposition \[skeinrelation\] for $N(g,c)$ is the same as the ones for $\Delta_{(g, c)}=\Delta_{(G, c)}(1)$, obtained by substituting $t=1$ in [@BW Theorem 4.1 (iv)]. Note that Theorem \[MainTheorem\] for plane MOY graphs has been proved in Theorem \[Correspondence\], and a general MOY graph diagram can be related to plane graphs by a finite number of skein relations. It follows by induction that Theorem \[MainTheorem\] holds for arbitrary MOY graphs.
We conclude this section by proving Corollary \[Nonvanishing\]. This follows directly from the following two lemmas, since existence of spanning trees implies positivity of weighted number of spanning trees when the weight function is positive.
A connected directed graph with a balanced positive weight is strongly connected, i.e., every vertex is reachable from every other vertex by a directed path.
Suppose $\Gamma$ is a connected graph with a positive balanced weight $c$. Given a vertex $v\in V$, let $S \subset V$ be the set of all vertices that can be reached from $v$. If $S$ is a proper subset of $V$, then $V-S$ is not empty. As $\Gamma$ is a connected graph, there must be edges that connect vertices in $S$ with vertices in $V-S$. Let $F$ be the set of such edges. Applying the positivity and the balanced condition (\[balancedcoloring\]) of $c$ on all vertices in $S$, we can further see that there must be some edges in $F$ that are oriented from some vertices in $S$ to some vertices in $V-S$. Then there is a vertex in $V-S$ which is also reachable from $v$ by a directed path. This contradicts the definition of $S$, so we must have $S=V$.
Every strongly connected graph has an oriented spanning tree with any given root.
This is a standard result in graph theory. For any given root $r$, simply take a maximal oriented tree rooted at $r$. Such a tree must be spanning by the strongly connected assumption.
[10]{}
, [*Floer homology and embedded bipartite graphs*]{}, arXiv:1401.6608v4, (2018).
, [*An Alexander polynomial for MOY graphs*]{}, Selecta Math. (N.S) 26 (2020), Article number: 32
, [*A combinatorial proof of the all minors matrix tree theorem*]{}, SIAM J. Alg. Disc. Meth. 3 (1982), pp. 319–329.
, [*The Dissection of Equilateral Triangles into Equilateral Triangels*]{}, Math. Proc. Cambridge Philos. Soc. 44 (1948), pp. 463–482
|
---
abstract: 'Microwave-radiation induced giant magnetoresistance oscillations recently discovered in high-mobility two-dimensional electron systems are analyzed theoretically. Multiphoton-assisted impurity scatterings are shown to be the primary origin of the oscillation. Based on a theory which considers the interaction of electrons with electromagnetic fields and the effect of the cyclotron resonance in Faraday geometry, we are able not only to reproduce the correct period, phase and the negative resistivity of the main oscillation, but also to predict the secondary peaks and additional maxima and minima observed in the experiments. These peak-valley structures are identified to relate respectively to single-, double- and triple-photon processes.'
author:
- 'X.L. Lei and S.Y. Liu'
title: |
Radiation-induced magnetoresistance oscillation in a two-dimensional electron gas\
in Faraday geometry
---
The discovery of a new type of giant magnetorersistance oscillations in a high mobility two-dimensional (2D) electron gas (EG) subject to crossed microwave (MW) radiation field and a magnetic field,[@Zud01; @Ye; @Mani; @Zud03; @Mani05507] especially the observation of “zero-resistance” states developed from the oscillation minima,[@Mani; @Zud03; @Yang; @Dor] has revived tremendous interest in magneto-transport in 2D electron systems.[@Durst; @Andreev; @Anderson; @Xie; @Phil; @Koul] These radiation-induced oscillations of longitudinal resistivity $R_{xx}$ are accurately periodical in inverse magnetic field $1/B$ with period determined by the MW frequency $\omega$ rather than the electron density $N_{\rm e}$. The observed $R_{xx}$ oscillations exhibit a smooth magnetic-field variation with resistivity maxima at $\omega/\omega_c=j-\delta_{-}$ and minima at $\omega/\omega_c=j+\delta_{+}$ ($\omega_c$ is the cyclotron frequency, $j=1,2,3...$) having positive $\delta_{\pm}$ around $\frac{1}{4}$.[@Mani] The resistivity minimum goes downward with increasing sample mobility and/or increasing radiation intensity until a “zero-resistance” state shows up, while the Hall resistivity keeps the classical form $R_{xy}=B/N_{\rm e}e$ with no sign of quantum Hall plateau over the whole magnetic field range exhibiting $R_{xx}$ oscillation.
To explore the origin of the peculiar “zero-resistance” states, different mechanisms have been suggested.[@Durst; @Andreev; @Anderson; @Xie; @Phil; @Koul] It is understood that the appearance of negative longitudinal resistivity or conductivity in a uniform model suffices to explain the observed vanishing resistance.[@Andreev] The possibility of absolute negative photoconductance in a 2DEG subject to a perpendicular magnetic field was first explored 30 years ago by Ryzhii.[@Ryz; @Ryz86] Recent works[@Durst; @Anderson; @Xie] indicated that the periodical structure of the density of states (DOS) of the 2DEG in a magnetic field and the photon-excited electron scatterings are the origin of the magnetoresistance oscillations. Durst [*et al.*]{}[@Durst] presented a microscopic analysis for the conductivity assuming a $\delta$-correlated disorder and a simple form of the 2D electron self-energy oscillating with the magnetic field, obtaining the correct period, phase and the possible negative resistivity. Shi and Xie[@Xie] reported a similar result using the Tien and Gorden current formula[@Tien] for photon-assisted coherent tunneling. In these studies, however, the magnetic field is to provide an oscillatory DOS only and the high frequency (HF) field enters as if there is no magnetic field or with a magnetic field in Voigt configuration. The experimental setup requires to deal with the magnetic field ${\bf B}$ perpendicular to the HF electric field. In this Faraday configuration, the electron moving due to HF field, experiences a Lorentz force which gives rise to additional electron motion in the perpendicular direction. In the range of $\omega\sim\omega_c$, the electron velocities in both directions are of the same order of magnitude and are resonantly enhanced. This cyclotron resonance (CR) of the HF current response will certainly change the way how the photons assist the electron scattering.
In this Letter, we construct a microscopic model for the interaction of electrons with electromagnetic fields in Faraday geometry. The basic idea is that, under the influence of a spatially uniform HF electric field, the center-of-mass (CM) of the whole 2DEG in a magnetic field performs a cyclotron motion modulated by the HF field of frequency $\omega$. In an electron gas having impurity and/or phonon scatterings, there exist couplings between this CM motion and the relative motion of the 2D electrons. It is through these couplings that a spatially uniform HF electric field affects the relative motion of electrons by opening additional channels for electron transition between different states. Based on the theory for photon-assisted magnetotransport developed from this physical idea, we show that the main experimental results of the radiation-induced magnetoresistance oscillations can be well reproduced. We also obtain the secondary peaks and additional maxima and minima observed in the experiments.[@Zud03; @Dor]
For a general treatment, we consider $N_{\rm e}$ electrons in a unit area of a quasi-2D system in the $x$-$y$ plane with a confining potential $V(z)$ in the $z$-direction. These electrons, besides interacting with each other, are scattered by random impurities/disorders and by phonons in the lattice. To include possible elliptically polarized MW illumination we assume that a uniform dc electric field ${\bf E}_0$ and ac field ${\bf E}_t\equiv{\bf E}_s \sin(\omega t)+{\bf E}_c\cos(\omega t)$ of frequency $\omega$ are applied in the $x$-$y$ plane, together with a magnetic field ${\bf B}=(0,0,B)$ along the $z$ direction. In terms of the 2D CM momentum and coordinate of the electron system,[@Lei85; @Lei851; @Ting] which are defined as ${\bf P}\equiv\sum_j {\bf p}_{j\|}$ and ${\bf R}\equiv N_{\rm e}^{-1}\sum_j {\bf r}_{j\|}$ with ${\bf p}_{j\|}\equiv(p_{jx},p_{jy})$ and ${\bf r}_{j\|}\equiv (x_j,y_j)$ being the momentum and coordinate of the $j$th electron in the 2D plane, and the relative electron momentum and coordinate ${\bf p}_{j\|}'\equiv{\bf p}_{j\|}-{\bf P}/N_{\rm e}$ and ${\bf r}_{j\|}'\equiv{\bf r}_{j\|}-{\bf R}$, the Hamiltonian of the system can be written as the sum of a CM part $H_{\rm cm}$ and a relative-motion part $H_{\rm er}$ (${\bf A}({\bf r})$ is the vector potential of the ${\bf B}$ field), $$\begin{aligned}
H_{\rm cm}=\frac 1{2N_{\rm e}m}({\bf P}-N_{\rm e}e{\bf A}({\bf
R}))^2-N_{\rm e}e({\bf E}_{0}+{\bf E}_t)\cdot {\bf R},&&\\
H_{\rm er}=\sum_{j}\Big[\frac{1}{2m}\left({\bf p}_{j\|}'-e{\bf A}
({\bf r}_{j\|}')\right)^{2}
+\frac{p_{jz}^2}{2m_z}+V(z_j)\Big]\,\,&&\nonumber\\
+\sum_{i<j}V_c({\bf r}_{i\|}'-{\bf r}_{j\|}',z_i,z_j),\,\,\,\,\,\,&&\end{aligned}$$ together with couplings of electrons to impurities and phonons, $H_{\rm ei}$ and $H_{\rm ep}$. Here $m$ and $m_z$ are respectively the electron effective mass parallel and perpendicular to the plane, and $V_c$ stands for the electron-electron Coulomb interaction. Note that although in $H_{\rm cm}$ and $H_{\rm er}$ the CM and the relative electron motion are completely separated, the CM coordinate ${\bf R}$ enters $H_{\rm ei}$ and $H_{\rm ep}$.[@Lei85; @Lei851] Starting from the Heisenberg operator equations for the rate of change of the CM velocity $\dot{\bf V}=-i[{\bf V},H]+\partial{\bf V}/\partial t$ with ${\bf V}=-i[{\bf R},H]$, and that of the relative electron energy $\dot{H}_{\rm er}=-{\rm i}[H_{\rm er},H]$, we proceed with the determination of their statistical averages.
The CM coordinate ${\bf R}$ and velocity ${\bf V}$ in these equations can be treated classically, i.e. as the time-dependent expectation values of the CM coordinate and velocity,[@Lei85] ${\bf R}(t)$ and ${\bf V}(t)$, such that ${\bf R}(t)-{\bf R}(t^{\prime})
=\int_{t^{\prime}}^t{\bf V}(s)ds$. We are concerned with the steady transport under an irradiation of single frequency and focus on the photon-induced dc resistivity and the energy absorption of the HF field. These quantities are directly related to the time-averaged and/or base-frequency oscillating components of the CM velocity. At the same time, in an ordinary semiconductor the effect of higher harmonic current is safely negligible for the HF field intensity in the experiments. Hence, it suffices to assume that the CM velocity, i.e. the electron drift velocity, consists of a dc part ${\bf v}_0$ and a stationary time-dependent part that $${\bf V}(t)={\bf v}_0+{\bf v}_1 \cos(\omega t)+{\bf v}_2 \sin(\omega t).$$ This time-dependent CM velocity enters all the operator equations having couplings to impurities and/or phonons in the form of the following exponential factor, which can be expanded in terms of Bessel functions ${\rm J}_n(x)$: $$\begin{aligned}
\hspace*{-1.3cm}&&{\rm e}^{-{\rm i}{\bf q}\cdot \int_{t^{\prime }}^{t}{\bf V}(s)ds}
=\sum_{n=-\infty }^{\infty }{\rm J}_{n}^{2}(\xi ){\rm e}^{{\rm i}({\bf q}\cdot {\bf %%@
v}_0-n\omega) (t-t^{\prime
})}+\nonumber\\
&&\sum_{m\neq 0}{\rm e}^{{\rm i}m(\omega t-\varphi )}\sum_{n=-\infty }^{\infty
}{\rm J}_{n}(\xi ){\rm J}_{n-m}(\xi ){\rm e}^{{\rm i}({\bf q}\cdot {\bf v}_0-n\omega) %%@
(t-t^{\prime })}.\nonumber\end{aligned}$$ Here $\xi\equiv \sqrt{({\bf q}_\|\cdot {\bf v}_1)^2+({\bf q}_\|\cdot {\bf %%@
v}_2)^2}/\omega$ and $\tan \varphi=({\bf q}\cdot {\bf v}_2)/({\bf q}\cdot {\bf v}_1)$. On the other hand, for 2D systems having electron sheet density of order of 10$^{15}$ m$^{-2}$, the intra-band and inter-band Coulomb interactions are sufficiently strong that it is adequate to describe the relative-electron transport state using a single electron temperature $T_{\rm e}$. Except this, the electron-electron interaction is treated only in a mean-field level under random phase approximation (RPA).[@Lei85; @Lei851] For the determination of unknown parameters ${\bf v}_0$, ${\bf v}_1$, ${\bf v}_2$, and ${T_{\rm e}}$, it suffices to know the damping force up to the base frequency oscillating term ${\bf F}(t)= {\bf F}_0+{\bf F}_s\sin(\omega t)+{\bf F}_c\cos(\omega t)$, and the energy-related quantities up to the time-average term. We finally obtain the following force and energy balance equations: $$\begin{aligned}
0&=&N_{\rm e}e{\bf E}_{0}+N_{\rm e} e ({\bf v}_0 \times {\bf B})+
{\bf F}_0,\label{eqv0}\\
{\bf v}_{1}&=&\frac{e{\bf E}_s}{m\omega}+\frac{{\bf F}_s}{N_{\rm e}m\omega }
-\frac{e}{m\omega }({\bf v}_{2}\times
{\bf B}),\label{eqv1}\\
-{\bf v}_{2}&=&\frac{e{\bf E}_c}{m\omega}+\frac{{\bf F}_c}{N_{\rm e}m\omega }
-\frac{e}{m\omega }({\bf v}_{1}
\times {\bf B}),\label{eqv2}\end{aligned}$$ $$N_{\rm e}e{\bf E}_0\cdot {\bf v}_0+S_{\rm p}- W=0.
\label{eqsw}$$ Here $$\begin{aligned}
{\bf F}_{0}=\sum_{{\bf q}_\|}\left| U({\bf q}_\|%
)\right| ^{2}%
\sum_{n=-\infty }^{\infty }{\bf q}_\|{\rm J}_{n}^{2}(\xi )\Pi _{2}({\bf %
q}_\|,\omega_0-n\omega )\,\,\,\,\,&&\nonumber\\
+\sum_{{\bf q}}\left| M({\bf q})\right|
^{2}\sum_{n=-\infty
}^{\infty }{\bf q}_\|{\rm J}_{n}^{2}(\xi )\Lambda _{2}({\bf q},\omega_0+\Omega _{{\bf %%@
q}}-n\omega )&&
\label{eqf0}\end{aligned}$$ is the time-averaged damping force, $S_{\rm p}$ is the time-averaged rate of the electron energy-gain from the HF field, $\frac{1}{2}N_{\rm e}e({\bf E}_s\cdot{\bf v}_2+{\bf E}_c\cdot{\bf v}_1)$, which can be written in a form obtained from the right hand side of Eq.(\[eqf0\]) by replacing the ${\bf q}_\|$ factor with $n \omega$, and $W$ is the time-averaged rate of the electron energy-loss due to coupling with phonons, whose expression can be obtained from the second term on the right hand side of Eq.(\[eqf0\]) by replacing the ${\bf q}_\|$ factor with $\Omega_{\bf q}$, the energy of a wavevector-${\bf q}$ phonon. The oscillating frictional force amplitudes ${\bf F}_s\equiv {\bf F}_{22}-{\bf F}_{11}$ and ${\bf F}_c\equiv {\bf F}_{21}+{\bf F}_{12}$ are given by ($\mu=1,2$) $$\begin{aligned}
{\bf F}_{1\mu}=-\sum _{{\bf q}_\|}{\bf q}_\|\eta_{\mu}| U({\bf q}_\|%
)| ^{2}\sum_{n=-%
\infty }^{\infty }\left[ {\rm J}_{n}^{2}(\xi )\right] ^{\prime }\Pi _{1}(%
{\bf q}_\|,\omega_0-n\omega )&&\nonumber\\
-
\sum_{\bf q}{\bf q}_\|\eta_{\mu}| M({\bf q})|
^{2}\sum_{n=-\infty
}^{\infty }\left[ {\rm J}_{n}^{2}(\xi )\right] ^{\prime }\Lambda _{1}({\bf q%
}, \omega_0+\Omega _{{\bf q}}-n\omega ),&&\nonumber\\
{\bf F}_{2\mu}=\sum_{{\bf q}_\|}{\bf q}_\|\frac{\eta_{\mu}}
{\xi}| U({\bf q}_\|)| ^{2}%
\sum_{n=-\infty }^{\infty }2n{\rm J}_{n}^{2}(\xi )\Pi _{2}({\bf %
q}_\|,\omega_0-n\omega )\,\,&&\nonumber\\
+
\sum_{{\bf q}}{\bf q}_\|\frac{\eta_{\mu}}{\xi}| M({\bf q})|^{2}\sum_{n=-\infty
}^{\infty }2n{\rm J}_{n}^{2}(\xi )\Lambda _{2}({\bf q},\omega_0+\Omega _{\bf q}-n\omega %%@
).
&&\nonumber
\end{aligned}$$ In these expressions, $\eta_{\mu}\equiv {\bf q}_\|\cdot {\bf v}_{\mu}/\omega \xi$; $\omega_0\equiv {\bf q}_\|\cdot {\bf v}_0$; $U({\bf q}_\|)$ and $M({\bf q})$ stand for effective impurity and phonon scattering potentials, $\Pi_2({\bf q}_\|,\Omega)$ and $
\Lambda_2({\bf q},\Omega)=2\Pi_2({\bf q}_\|,\Omega)
[n(\Omega_{\bf q}/T)-n(\Omega/T_{\rm e})]
$(with $n(x)\equiv 1/({\rm e}^x-1)$) are the imaginary parts of the electron density correlation function and electron-phonon correlation function in the presence of the magnetic field. $\Pi_1({\bf q}_\|,\Omega)$ and $\Lambda_1({\bf q},\Omega)$ are the real parts of these two correlation functions. The effect of interparticle Coulomb interactions are included in them to the degree of level broadening and RPA screening.
The HF field enters through the argument $\xi$ of the Bessel functions in ${\bf F}_0$, ${\bf F}_{\mu\nu}$, $W$ and $S_{\rm p}$. Compared with that without the HF field ($n=0$ term only),[@Lei98] we see that in an electron gas having impurity and/or phonon scattering (otherwise homogeneous), a HF field of frequency $\omega$ opens additional channels for electron transition: an electron in a state can absorb or emit one or several photons and scattered to a different state with the help of impurities and/or phonons. The sum over $|n|\geq 1$ represents contributions of single and multiple photon processes of frequency-$\omega$ photons. These photon-assisted scatterings help to transfer energy from the HF field to the electron system ($S_{\rm p}$) and give rise to additional damping force on the moving electrons. Note that ${\bf v_1}$ and ${\bf v}_2$ always exhibit CR in the range $\omega\sim\omega_c$, as can be seen from Eqs.(\[eqv1\]) and (\[eqv2\]) rewritten in the form $$\begin{aligned}
&&{\bf v}_{1}=(1-{\omega_c^2}/{\omega^2})^{-1}\left\{
\frac{e}{m\omega}\left[{\bf E}_s+\frac{e}{m\omega }(
{\bf E}_c\times{\bf B})\right]\right.\nonumber\\
&&\hspace{1.0cm}+\left.\frac{1}{N_{\rm e}m\omega }
\left[{\bf F}_s+
\frac{e}{m\omega }(
{\bf F}_c\times
{\bf B})\right]\right\},\label{vv1}\\
&&{\bf v}_{2}=({\omega_c^2}/{\omega^2}-1)^{-1}\left \{
\frac{e}{m\omega}\left[{\bf E}_c-\frac{e}{m\omega }(
{\bf E}_s\times{\bf B})\right]\right.\nonumber\\
&&\hspace{1.0cm}+\left.\frac{1}{N_{\rm e}m\omega }
\left[{\bf F}_c-
\frac{e}{m\omega }
({\bf F}_s\times
{\bf B})\right]\right\}.\label{vv2}\end{aligned}$$ Therefore, $\xi$ may be significantly different from the argument of the corresponding Bessel functions in the case without a magnetic field or with a magnetic field in Voigt configuration.[@Lei98]
Eqs.(\[eqv0\])-(\[eqsw\]) can be used to describe the transport and optical properties of magnetically-biased quasi-2D semiconductors subject to a dc field and a HF field. Taking ${\bf v}_0=(v_{0x},0,0)$ in the $x$ direction, Eq.(\[eqv0\]) yields transverse resistivity $R_{xy}\equiv E_{0y}/N_{\rm e}ev_{0x}=B/N_{\rm e}e$, and longitudinal resistivity $
R_{xx}\equiv E_{0x}/N_{\rm e}ev_{0x}=-F_0/N_{\rm e}^2e^2v_{0x}$. The linear magnetoresistivity is then $$\begin{aligned}
R_{xx}&=&-\sum_{{\bf q}_\|}q_x^2\frac{|
U({\bf q}_\|)| ^2}{N_{\rm e}^2 e^2}\sum_{n=-\infty }^\infty {\rm J}_n^2(\xi)\left.
\frac {\partial \Pi_2}{\partial\, \Omega }\right|_{\Omega =n\omega }\nonumber\\
&&- \sum_{ {\bf q}} q_x^2\frac{\left| M ( {\bf
q})\right| ^2}{N_{\rm e}^2 e^2}\sum_{n=-\infty }^\infty {\rm J}_n^2(\xi)\left.
\frac {\partial \Lambda_2}{\partial\, \Omega }\right|_{\Omega =\Omega_{{\bf %%@
q}}+n\omega}.
\,\,\,\,\,\,\,\label{rxx}\end{aligned}$$ The parameters ${\bf v}_1$, ${\bf v}_2$ and $T_{\rm e}$ in (\[rxx\]) should be determined by solving equations (\[eqv1\]), (\[eqv2\]) and (\[eqsw\]) with vanishing ${\bf v}_0$. We see that although the transverse resistivity $R_{xy}$ remains the classical form, the longitudinal resistivity $R_{xx}$ can be strongly affected by the irradiation.
We calculate the unscreened $\Pi_2({\bf q}_{\|}, \Omega)$ function of the 2D system in a magnetic field by means of Landau representation:[@Ting] $$\begin{aligned}
&&\hspace{-0.7cm}\Pi _2({\bf q}_{\|},\Omega ) = \frac 1{2\pi
l_{\rm B}^2}\sum_{n,n'}C_{n,n'}(l_{\rm B}^2q_{\|}^2/2)
\Pi _2(n,n',\Omega),
\label{pi_2}\\
&&\hspace{-0.7cm}\Pi _2(n,n',\Omega)=-\frac2\pi \int d\varepsilon
\left [ f(\varepsilon )- f(\varepsilon +\Omega)\right ]\nonumber\\
&&\,\hspace{2cm}\times\,\,{\rm Im}G_n(\varepsilon +\Omega){\rm Im}G_{n'}(\varepsilon ),\end{aligned}$$ where $l_{\rm B}=\sqrt{1/|eB|}$ is the magnetic length, $
C_{n,n+l}(Y)\equiv n![(n+l)!]^{-1}Y^le^{-Y}[L_n^l(Y)]^2
$ with $L_n^l(Y)$ the associate Laguerre polynomial, $f(\varepsilon
)=\{\exp [(\varepsilon -\mu)/T_{\rm e}]+1\}^{-1}$ the Fermi distribution function, and ${\rm Im}G_n(\varepsilon )$ is the imaginary part of the Green’s function, or the DOS, of the Landau level $n$. The real part functions $\Pi_1({\bf q}_{\|},\Omega)$ and $\Lambda_1({\bf q}_{\|},\Omega)$ can be obtained from their imaginary parts via the Kramers-Kronig relation.
In principle, to obtain the Green’s function $G_n(\varepsilon )$, a self-consistent calculation has to be carried out with all the scattering mechanisms included.[@Leadley] In this Letter we do not attempt a self-consistent calculation of $G_n(\varepsilon)$ but choose a Gaussian-type function for the purpose of demonstrating the $R_{xx}$ oscillations ($\varepsilon_n$ is the energy of the $n$-th Landau level):[@Ando82] $${\rm Im}G_n(\varepsilon)=-(\pi/2)^{\frac{1}{2}}\Gamma^{-1}
\exp[-(\varepsilon-\varepsilon_n)^2/(2\Gamma^2)]$$ with a broadening parameter $\Gamma=(2e\omega_c\alpha/\pi m \mu_0)^{1/2}$, where $\mu_0$ is the linear mobility at temperature $T$ in the absence of the magnetic field and $\alpha > 1$ is a semiempirical parameter to take account the difference of the transport scattering time determining the mobility $\mu_0$, from the single particle lifetime.[@Mani; @Durst; @Anderson]
![The longitudinal magnetoresistivity $R_{xx}$ of a GaAs-based 2DEG subject to a lineraly polarized HF field $E_s\sin(\omega t)$. The parameters are: temperature $T=1$K, electron density $N_{\rm e}=3.0\times 10^{11}$cm$^{-2}$, zero-magnetic-field linear mobility $\mu_0=2.4\times 10^7$cm$^2$ V$^{-1}$ s$^{-1}$, and the broadening coefficient $\alpha=12$.[]{data-label="fig1"}](fig1.eps){width="45.00000%"}
The moderate microwave intensity for the $R_{xx}$ oscillation in these high-mobility samples yield only slight electron heating, which is unimportant as far as the main phenomenon is concerned and is neglected for simplicity. We consider scatterings from remote impurities as well as from acoustic phonons. After solving ${\bf v}_1$ and ${\bf v}_2$ from Eqs.(\[vv1\]) and(\[vv2\]) the magnetoresisivity $R_{xx}$ can be obtained directly from Eq.(\[rxx\]). At lattice temperature $T=1$K, the contribution from photon-assisted phonon scattering is minor. The role of acoustic phonons, however, becomes essential at elevated lattice temperatures. Calculations were carried out for linearly polarized MW fields with multiphoton processes included.
Fig.1 shows the calculated longitudinal resistivity $R_{xx}$ versus $\omega/\omega_c\equiv \gamma_c$ subject to a linearly polarized MW radiation of frequency $\omega/2\pi=100$GHz at four values of amplitude: $E_s=20, 45, 65$ and 80V/cm. Shubnikov-de Haas (SdH) oscillations show up strongly at high $\omega_c$ side, and gradually decay away as $1/\omega_c$ increases. All resistivity curves exhibit pronounced oscillation having main oscillation period $\gamma_c=1$ (they are crossing at integer points $\gamma_c=2,3,4,5$). The resistivity maxima locate around $\gamma_c=j-\delta_{-}$ and minima around $\gamma_c=j+\delta_{+}$ with $\delta_{\pm}\sim 0.23-0.25$ for $j=3,4,5$, $\delta_{\pm}\sim 0.17-0.21$ for $j=2$, and even smaller $\delta_{\pm}$ for $j=1$. The magnitude of the oscillation increases with increasing HF field intensity for $\gamma_c >1.5$. Resistivity gets into negative value for $E_s=80$V/cm around the minima at $j=1,2$ and 3, for $E_s=65$V/cm at $j=1$ and 2, and for $E_s=20$ and 45V/cm at $j=1$. All these features, which are in fairly good agreement with experimental findings,[@Zud01; @Mani05507; @Mani; @Zud03] are relevant mainly to single-photon ($|n|=1$) processes. Anomaly appears in the vicinity of $\gamma_c=1$, where the CR greatly enhances the effective amplitude of the HF field in photon-assisted scatterings and multiphoton processes show up. The amplitudes of the $j=1$ maximum and minimum no longer monotonically change with field intensity. Furthermore, there appears a shoulder around $\gamma_c=1.5$ on the curves of $E_s=45$ and 65V/cm, and it develops into a secondary peak in the $E_s=80$V/cm case. This has already been seen in the experiment (Fig.2 in Ref.). The valley between $\gamma_c=1.4$ and 1.8 peaks can descend down to negative as $E_s$ increasing further (Fig.1b). The appearance of the secondary peak is due to two-photon ($|n|= 2$) processes.
Radiation-induced resistivity behavior at $\gamma_c<1$ is shown more clearly in the $\omega/2\pi=60$GHz case. As seen in Fig.1c, a shoulder around $\gamma_c=$0.4-0.6 with a minmum at $\gamma_c=0.6$ can be indentified from the SdH oscillation background for all three curves, which is related to two-photon process. With increasing MW strength there appears a clear peak around $\gamma_c=0.68$ and a valley around $\gamma_c=0.76$. This peak-valley is mainly due to three-photon ($|n|=3$) process. In the case of $40$GHz, similar peak and valley also show up (Fig.1d).
Qualitatively, the main oscillation features come from the symmetrical property of the DOS function in a magnetic field. Since $G_n(\varepsilon-j\omega_c)=G_{n-j}(\varepsilon)$ for any integer $j$, the impurity contribution to $R_{xx}$ from the $n$-photon process, which is related to the weighted summation of the derivative $\Pi_2$ function over all the Landau levels at frequency $n\omega$ \[Eq.(\[rxx\])\], has an intrinsic periodicity characterized by $n\omega=j\omega_c$. The main oscillation of $R_{xx}$ shown in Fig.1a relates to single-photon process and characterized by $\omega=j\omega_c$. We have also performed calculation using a Lorentz-type DOS function and find that, although the oscillating amplitude and the exact peak and valley positions are somewhat different, the basic periodic feature of the radiation-induced magnetoresistivity oscillation remains.
This work was supported by the National Science Foundation of China, the Special Funds for Major State Basic Research Project, and the Shanghai Municipal Commission of Science and Technology.
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|
---
abstract: |
The Planck satellite in orbit mission ended in October 2013. Between the end of Low Frequency Instrument (LFI) routine mission operations and the satellite decommissioning, a dedicated test was also performed to measure the Planck telescope emissivity.\
The scope of the test was twofold: i) to provide, for the first time in flight, a direct measure of the telescope emissivity; and ii) to evaluate the possible degradation of the emissivity by comparing data taken in flight at the end of mission with those taken during the ground telescope characterization.\
The emissivity was determined by heating the Planck telescope and disentangling the system temperature excess measured by the LFI radiometers.\
Results show End of Life (EOL) performance in good agreement with the results from the ground optical tests and from *in-flight* indirect estimations measured during the Commissioning and Performance Verification (CPV) phase.\
Methods and results are presented and discussed.
author:
- 'F. Cuttaia \[1\]'
- 'L. Terenzi \[1\]'
- 'G. Morgante \[1\]'
- 'M. Sandri \[1\]'
- 'F. Villa \[1\]'
- 'A. De Rosa \[1\]'
- 'E. Franceschi \[1\]'
- 'M. Frailis \[2\]'
- 'S. Galeotta \[2\]'
- 'A. Gregorio \[2\]'
- 'P. Delannoy \[3\]'
- 'S. Foley \[3\]'
- 'B. Gandolfo \[3\]'
- 'A. Neto \[3\]'
- 'C. Watson \[3\]'
- 'F. Pajot \[4\]'
- 'M. Bersanelli \[5\]'
- 'R. C. Butler \[1\]'
- 'N. Mandolesi \[1,6\]'
- 'A. Mennella \[5\]'
- 'J. Tauber \[7\]'
- 'A. Zacchei \[2\]'
title: 'In-flight measurement of Planck telescope emissivity'
---
\[2001/04/25 1.1 (PWD)\]
Introduction {#sec:introduction}
============
The Planck satellite [@Tauber2010],[@Planck_coll_1_2011] was launched together with the Herschel spacecraft on an Ariane 5 from Europe’s spaceport in Kourou, French Guyana, on 14 May 2009. The two satellites were injected into an orbit around the Sun-Earth Lagrange point $L_2$. The duration of the nominal Planck mission was 15,5 months. Nevertheless Planck operated continuously for 1623 days, until 23 October 2013, with the Low Frequency Instrument (LFI). The High Frequency Instrument (HFI) [@Lamarre2010] operated until 13 January 2012 when the supply of $^3$He needed to cool the HFI bolometers to 0.1 K ran out. However, the HFI’s He Joule-Thomson cooler [@Planck_coll_2_2011] continued to operate normally to support the LFI pseudocorrelation radiometers [@Bersanelli2010] with the required thermal reference at $\sim$4K [@Valenziano2011] until mission end.\
The Planck de-orbiting started on 14 August 2013, when the first manoeuvre for the spacecraft departure from L2 was performed: this phase lasted until October 9th (final de-orbiting manoeuvre).\
In the period between October 4th and October 21st, the LFI functionality at End of Life (EOL) was verified: some tests, already performed during the CPV phase [@Gregorio2014] or during the ground calibration tests [@Mennella2011], [@Terenzi2010], [@Cuttaia2010], were repeated. New additional tests were also performed to verify or better characterize other features revealed during the mission.\
In particular, the procedure named Telescope Loss Test (TLT) was run: it was aimed at indirectly measuring the Planck telescope emissivity [@Tauber_2_2010] at the LFI frequencies at EOL, by operating the de-contamination heaters located on the primary and the secondary mirrors. This test was not foreseen at the beginning of the Planck mission and was decided upon only during the planning of Planck EOL phase, taking advantage of the LFI radiometers sensitivity [@Mennella_2_2011] and of our improved knowledge of the LFI properties and of systematic effects over the mission [@Planck_coll_2014], [@Planck_coll_2015].\
The test consisted in heating the primary and secondary mirrors by few Kelvin ($\sim 4K$) and then measuring the power excess measured by the LFI pseudo-correlation radiometers. The underlying basic assumption was that the measured excess would be mostly proportional to the telescope reflection loss (emissivity), provided that the other possible effects affecting the radiometric response were known and kept under control.\
The TLT procedure was succesfully run on 7 October 2013. Such a test was never performed before on a microwave space telescope: the high LFI instrumental sensitivity, a very good knowledge of systematic effects and the Planck Mission Operation Center (MOC) ability in controlling the telescope thermal response, were the key ingredients of its success.
The Planck Telescope {#sec:Planck_Telescope}
====================
he Planck telescope was designed to comply with the following high level opto-mechanical requirements:
- wide frequency coverage: about two decades, from 25 GHz to 1 THz;
- 100 squared degrees of field of view, wide focal region (400 X 600 mm);
- cryogenic operational environment between 40 K and 65 K.
The telescope optical layout was based on a dual reflector off-axis Gregorian design (Fig. \[fig:telescope\]). Both the primary and secondary mirrors were elliptical in shape. The size of the primary mirror rim was 1.9 X 1.5 meters; the rim of the secondary mirror was nearly circular with a diameter of about 1 meter.\
{width="100.00000%"}
. \[fig:telescope\]
The overall focal ratio was 1.1, and the projected aperture was circular with a diameter of 1.5 meters. The telescope field of view was $\pm5 ^{\circ}$ centred on the line of sight (LOS), which was tilted at about $3.7^{\circ}$ relative to the main reflector axis, and formed an angle of $85^{\circ}$ with the satellite spin axis, which was typically oriented in the anti-Sun direction during the survey.\
The Gregorian off-axis configuration ensured a small overall focal ratio (and thus small feeds), an unobstructed field of view, and low diffraction effects from the secondary reflector and struts.\
The core of primary and secondary mirrors was fabricated using Carbon Fiber Reinforced Plastic (CFRP) honeycomb sandwich technology (Fig. \[fig:PR\_honeycomb\]). The facesheets underwent reflective coating (Fig. \[fig:coating\_SR\]), following a procedure developed by EADS Astrium, consisting of three layers: 15 nm NiCr as adhesion layer, 550 nm Aluminium as reflective layer, $\sim$30 nm PLASIL as protection layer [@stute2005].\
This design was chosen to satisfy the requirements of low mass ( < 120 Kg including struts and supports), high stiffness, high dimensional accuracy, and low thermal expansion coefficient. Further details on the Planck optical system can be found in [@Planck_coll_2_2011] and in [@stute2004].
![Secondary Reflector: reflective coating[]{data-label="fig:coating_SR"}](PR_honeycomb.eps){width="5.75cm"}
![Secondary Reflector: reflective coating[]{data-label="fig:coating_SR"}](coating_SR.eps){width="5.3cm"}
The Telescope Loss Test {#sec:Telescope_Loss}
=======================
The TLT started on 7 October 2013 at 19:25:00 UTC, when *anti-contamination* heating was activated through heaters placed on the primary (PR) and secondary (SR) reflectors. The heaters were operated adapting to the test, in a cyclic fashion, the algorithm that was originally designed for de-contaminating the reflectors during the early launch phases.\
Temperatures for decontamination were monitored in real time by three dedicated sensors for each of the Planck reflectors (the three adjacent sensors in line in figs. \[fig:sensor\_PR\] and \[fig:sensor\_SR\]), while temperatures used for analysis are measured with a better resolution (about 0.25 K istead of 0.5 K) by two couples of nominal and redundant sensors (the four symmetrically distributed sensors in figs. \[fig:sensor\_PR\] and \[fig:sensor\_SR\]), for each reflector.\
On the basis of the emissivity measured during the ground tests (to be lower than 0.0006 at the LFI frequency) a minimum temperature change of 2K was required to unambiguously characterize the in-flight emissivity: nevertheless, the overall decontamination procedure was able to increase the temperature of the PR and SR by roughly 4K (averaged over the corresponding monitoring sensors), while the temperatures remained quite stable for the last 90 minutes of the test.\
Finally, both reflectors started to cooldown at a rate of less than 1K in 12 hours.\
Temperature profiles of PR and SR, caused by anti-contamination heaters activation and de-activation, are respectively shown in Fig. \[primary\_solo\] and in Fig. \[secondary\].
![Secondary Reflector: heaters harness and temperature sensors location scheme.[]{data-label="fig:sensor_SR"}](PR_sensors_Label.eps){width="5.75cm"}
![Secondary Reflector: heaters harness and temperature sensors location scheme.[]{data-label="fig:sensor_SR"}](SR_sensors_Label.eps){width="5.3cm"}
Emissivity Characterization {#sec:Emissivity}
===========================
The emissivity plays a crucial role in microwave telescopes, even more in spinning telescopes like Planck. Actually, the black-body thermal emission from the telescope is the cause of a higher system temperature; moreover, thermal fluctuations of the telescope can mimic the effect of changes in sky emission, critical expecially at fluctuation frequencies near the satellite spin frequency. For this reason the telescope emissivity was required, at beginning of life (BOL), to be lower than 0.6% .\
The telescope emissivity was expected to change during the mission due to UV irradiation and micrometeoroid impact, especially at the HFI frequencies.\
The emissivity was estimated on ground, by measuring the reflection loss of several samples from the Herschel telescope [@Fisher2005]. However, due to non-neglible differences between the Herschel and Planck telescopes, more accurate tests, based on a high-quality open Fabry-Perot resonator, were performed in 2008,directly on same Planck telescope samples, between 100 GHz and 380 GHz [@parshin2008], at the Institute of Applied Physics of the Russian Academy of Sciences (IAP RAS). Measures performed at low temperature (between 80 K and 110 K), showed that the reflectivity of mirror surfaces basically depends on: i) the quality of thin reflecting metal layers, ii) the coating, iii) the temperature . Results show an emissivity lower than the requirement, in the frequency range 100 GHz- 380 GHz.\
The emissivity was also measured indirectly in flight by the HFI, from thermal arguments: the background power in the bolometer bands, coming from the primary and secondary mirrors, was measured for each detector. Results, reported in [@Planck_coll_A2_2011] show an emissivity of about 0.07%, an order of magnitude lower than the requirement, obtained from the least squares fit of the computed in-band power from the two mirrors: emissivity is assumed to be frequency independent. As reported in [@Planck_coll_A2_2011], these results are affected by a large uncertainty (up to 100%) , especially at the two highest frequencies (545 GHz and 857 GHz), possibly due to calibration error in the bolometer plate temperature thermometer or thermal gradients between thermometer and bolometers location (Fig. \[fig:planck\_telescope\_emissivity\_w\_error\]).\
![Emissivity fit calculated by HFI from thermal arguments. The plot displays the Residual Bolometer Loading (pW) versus Frequency (GHz). All frequency channels are evenly constraining the emissivity of the mirrors (a common error on the bolometer plate temperature thermometer is considered): this leads to an estimate of 0.07 +/- 0.06% for each of the two mirrors, using a Rayleigh-Jeans law. Three cases are shown: best fit (0.07%, blue line), fit with positive error (0.07 + 0.06%, red line), fit with negative error (0.07 - 0.06%, orange line). Error bars correspond to experimental errors at each frequency for each detector. The best fit, and the two uncertainty curves, result from considering all points simultaneously.[]{data-label="fig:planck_telescope_emissivity_w_error"}](HFI_plot.eps){width="80.00000%"}
The TLT allowed to measure emissivity also in the Planck complementary frequency range covered by LFI radiometers. To first order, the mean differential power output for each of the four receiver diodes of the LFI radiometers can be written as (Eq.1, [@Mennella_2_2011]):
$$P_{\rm out}^{\rm diode} = a\, G_{\rm tot}\,k\,\beta \left[ \tilde{T}_{\rm sky} + T_{\rm noise} - r\left(
T_{\rm ref} + T_{\rm noise}\right) \right],
\label{eq_p0}$$
where $G_{\rm tot}$ is the total gain, $k$ is the Boltzmann constant, $\beta$ the receiver bandwidth and $a$ is the detector constant. $\tilde{T}_{\rm sky}$ and $T_{\rm ref}$ are, respectively, the apparent average sky antenna temperature and the reference load antenna temperature at the inputs of the first hybrid; $T_{\rm noise}$ is the receiver noise temperature. $\tilde{T}_{\rm sky}$ is the apparent sky signal entering the first hybrid after the two reflections on the primary and on the secondary mirros. The two reflections combine, attenuating the true sky signal and adding a spurious thermal signal proportional to the emissivity of the mirrors. The gain modulation factor (Eq.2, [@Mennella_2_2011]), $r$, is defined by: $$r = \frac{\tilde{T}_{\rm sky} + T_{\rm noise}}{T_{\rm ref} + T_{\rm noise}},
\label{eq_r}$$
In order to accurately characterize the telescope emissivity a good knowledge of the following quantities is mandatory:
- <span style="font-variant:small-caps;">PR and SR temperatures</span>: they affect $T_{\rm sky}$. To this aim, we must consider that the thermal sensors have limited resolution (about 0.2K).\
- <span style="font-variant:small-caps;">4K Reference Load (4KRL) stage thermal stability</span>: it affects $T_{\rm ref}$. Instabilities at 4KRL level impact on the differenced output of LFI radiometers, mimicking a change in the measured sky signal.\
- <span style="font-variant:small-caps;">LFI detectors calibration constants</span>: they affect $G_{\rm tot}$. Any error in the calibration constants propagates as a multiplicative error in the emissivity.\
- <span style="font-variant:small-caps;">Front End Unit (FEU) thermal stability</span>: it affects $G_{\rm tot}$. Instabilities at FEU level impact on the gain of the front end low noise amplifiers.\
- <span style="font-variant:small-caps;">Back End Unit (BEU) thermal stability</span>: it affects $G_{\rm tot}$ and $a$. Instabilities at BEU level can either impact on the gain and bias offset of the back end low noise amplifiers or on the radiometers power suppliers (controlling the FEU LNAs gain) or on both.\
A detailed analysis of these systematic effects is given in [@Planck_coll_2014], [@Planck_coll_2015].\
All these systematic effects were accounted for in the test preparation and execution and in the data analysis.\
The total signal transmitted from PR ans SR can be written as:
$$\begin{aligned}
T_{\rm PR}^{\rm out} \approx (1-\epsilon_{\rm 1})T_{\rm sky} + \epsilon_{\rm 1}T_{\rm PR}\\
T_{\rm SR}^{\rm out} \approx (1-\epsilon_{\rm 2})[(1-\epsilon_{\rm 1})T_{\rm sky}+\epsilon_{\rm 1}T_{\rm PR}]+\epsilon_{\rm 2}T_{\rm SR}\\
T_{\rm SR}^{\rm out}\equiv \tilde{T}_{\rm sky}
\label{eq:eq_T_reflectors} \end{aligned}$$
PR and SR were manufacured following a common procedure and using the same materials. For this reason, we can assume that:
$$\epsilon_{\rm 1} \approx \epsilon_{\rm 2} \sim \epsilon$$
Reducing the above equations, we get a simple expression relating the antenna temperature variation to thermal excess due to reflectors heating:
$$\Delta T^{\rm ant} \approx \epsilon (\Delta T_{\rm PR}+\Delta T_{\rm SR})$$
Data Analysis {#sec:Data_Analysis}
=============
The differenced output from each diode of the LFI detectors was correlated to the nominal temperature changes of the primary and secondary reflectors. The temperature associated to the reflectors was the average among the sensors respectively monitoring the PR and the SR.\
Data were calibrated averaging nominal gains calculated during one day in the late routine phase (day 1480 after launch) before the TLT. Calibration constant used are reported in the Appendix (Tab. \[cal\_const\]).\
The effect of signal fluctuations induced by the dipole modulation, caused by the Planck Telescope spinning, was also taken into account. Results, after dipole contribution removal, differ only negligibly from those before correction.\
The results were also corrected for radiometer susceptibility to temperature changes of: the front end unit (<span style="font-variant:small-caps;">FEU</span>), the back end enit (<span style="font-variant:small-caps;">BEU</span>) and the 4K stage (<span style="font-variant:small-caps;">4KRL</span>). Also with respect to these systematic effects, differences were negligible, because of the high thermal stability of the LFI during the TLT test.\
The LFI thermal behavior is shown in the following figures. Peak to peak variations are:
- lower than 0.05 K in the <span style="font-variant:small-caps;">FEU</span> (Fig. \[FEU\_temp\] );
- at the level of sensors resolution in the BEU (Fig. \[BEM\_tray\] and Fig. \[FEM\_tray\] show quantized signals );
- lower than 4 mK in the 4K Reference Load Unit (Fig. \[4K\]);
All the above effects do not show any correlations with temperature changes in PR and SR. Reference values for the thermal susceptibilities are those from [@Planck_coll_2014].\
![Fron end unit sensors positioned near feedhorn LFI28, LFI25, LFI26. They correspond to the three LFI Q band channels ([@Bersanelli2010])[]{data-label="FEU_temp"}](FEU_temp.eps "fig:"){width="8"}\
![Back end unit sensors positioned on BEM tray ([@Bersanelli2010])[]{data-label="BEM_tray"}](BEM-tray.eps "fig:"){width="8"}\
![Back End Unit sensors positioned on FEM tray ([@Bersanelli2010])[]{data-label="FEM_tray"}](FEM-tray.eps "fig:"){width="8"}\
![4K stage temperature[]{data-label="4K"}](4K.eps "fig:"){width="8"}\
The temperature variation of primary and secondary reflectors, averaged over the sensors monitoring each reflector, is displayed in Fig. \[primary\_solo\] and in Fig. \[secondary\]. The relevant quantity is not the absolute temperature, but instead the thermal change due to reflector heating.\
![Primary reflector temperature averaged between three PR sensors[]{data-label="primary_solo"}](Primary_solo.eps "fig:"){width="8"}\
![Secondary reflector temperature averaged between three SR sensors[]{data-label="secondary"}](Secondary.eps "fig:"){width="8"}\
The effect of PR and SR heating on radiometers is hereafter shown for three channels
LFI18, LFI25, LFI28
, representative of the full LFI frequency range (70 GHz, 44 GHz, 30 GHz respectively), in Fig. \[RCA1810\_reb\], Fig. \[RCA2501\_reb\], Fig. \[RCA2800\_reb\]: in order to simplify the visualization, data have been rebinned. The differential nature of the LFI radiometers and their high sensitivity ([@Mennella2011]) makes it possible to identify clearly the sky temperature excess due to reflectors heating.\
![differenced output change in the 70 GHz channel LFI-1810 caused by telescope heating during TLT[]{data-label="RCA1810_reb"}](RCA1810_reb.eps "fig:"){width="8"}\
![differenced output change in the 44 GHz channel LFI-2501 caused by telescope heating during TLT[]{data-label="RCA2501_reb"}](RCA2501_reb "fig:"){width="8"}\
![differenced output change in the 30 GHz channel LFI-2800 caused by telescope heating during TLT[]{data-label="RCA2800_reb"}](RCA2800_reb "fig:"){width="8"}\
Results {#sec:Results}
=======
Results are presented for each frequency channel in Tab. \[emissivity\_frequency\].\
Results are presented per radiometer (
Main,Side
)([@Bersanelli2010]) in Tab. \[emissivity\_rad\], showing for each channel the emissivity corresponding to the measured apparent sky temperature excess caused by telescope heating. Differenced outputs from coupled diodes are linearly combined as described in [@Planck_coll_2_2014].\
A different way to combine results is shown in Tab. \[emissivity\_channel\], where the measured excess is presented averaging over the LFI optically paired channels (observing the same region of the secondary reflector): this approach is aimed at accounting for possible inhomogeneities in the temperature of the reflectors. Channels have been paired basing on the scheme reported in Tab. \[emissivity\_channel\] (channel
LFI24
is not considered, as it is not paired to any other channels).
. \[emissivity\_frequency\]
---------- ---------- ----------
`70 GHz` 5.55E-04 1.19E-04
`44 GHz` 4.74E-04 8.74E-05
`30 GHz` 3.85E-04 5.54E-05
---------- ---------- ----------
: Telescope Emissivity. Results are displayed per frequency channel together with the associated uncertainties (calculated as standard deviatiation of emissivities of all radiometers sharing the same frequency
. \[emissivity\_rad\]
--------- ---------- ----------
`LFI18` 5.25E-04 6.19E-04
`LFI19` 6.18E-04 6.48E-04
`LFI20` 5.76E-04 6.87E-04
`LFI21` 5.51E-04 6.15E-04
`LFI22` 5.47E-04 5.51E-04
`LFI23` 2.18E-04 5.11E-04
`LFI24` 3.57E-04 5.32E-04
`LFI25` 4.79E-04 4.91E-04
`LFI26` 5.93E-04 3.93E-04
`LFI27` 3.03E-04 3.99E-04
`LFI28` 4.20E-04 4.18E-04
--------- ---------- ----------
: Telescope Emissivity. Results are displayed per radiometer. M and S correspond to `MAIN` and `SIDE` radiometers [@Bersanelli2010]
--------------- ----------
`LFI18-LFI23` 4.68E-04
`LFI19-LFI22` 5.91E-04
`LFI20-LFI21` 6.07E-04
`LFI25-LFI26` 4.89E-04
`LFI27-LFI28` 3.85E-04
--------------- ----------
: Telescope Emissivity. Results are displayed per paired channels, corresponding to feedhorns looking the same region of the telescope. `MAIN` and `SIDE` radiometers data have been averaged [@Bersanelli2010].[]{data-label="emissivity_channel"}
The telescope emissivity, per frequency channels (values from Tab. \[emissivity\_frequency\]), was compared to values reported in Appendix B of [@Tauber2010], where the measured dependence of the Reflection Loss (1-R) of a sample of Planck reflector material is shown at 110 K, as a function of frequency, in the range 100 GHz - 380 GHz.\
Differences in the Reflection Loss are expected between *in-flight* tests (TLT) and *on-ground* tests ([@Tauber2010]), due to the different temperature of the telescope: the largest differences are expected at high frequency [@parshin2008]. This frequency dependency allowed to superpose results from TLT test, obtained at 40K in the range 27 GHz - 77 GHz, to results from Fig.B1 -right in [@Tauber2010], obtained at 110 K and at 296 K in the range 100 GHz - 380 GHz. Comparison is shown in Fig. \[plot\_300\_110\]\
![In-flight Reflection Loss at 40K compared to data from[@Tauber2010]. They are respectively shown: experimental data from ground test @296 K (asterisc); experimental data from ground test @110 K; Experimental data from LFI in-Flight measurement (this work) at 40K (cross)[]{data-label="plot_300_110"}](plot_300-110.eps "fig:"){width="8"}\
<span style="font-variant:small-caps;">High Frequency Data Extrapolation to 40K</span>
A more accurate comparison among the data sets, at different temperatures, is obtained by extrapolating to 40K the high frequency data (100 GHz - 380 GHz) measured at 110 K.\
The temperature dependence of the Reflection Loss was modeled by taking into account the experimental evidences described in [@Serov2016], where the case of mirrors of highly pure aluminum (99.99% Al) was investigated at fixed frequency f = 150 GHz. When Al is cooled down to cryogenic temperature, experimental results highlight discrepancies with respect to the purely theoretical model, even when anomalous skin depth is considered. In the case of pure Al, below 150K, the Reflection Loss drops much smoother than predicted, showing an almost linear decrement down to 40K, and constant behaviour at lower temperature (plot 10 in [@Serov2016]); the measured Reflection Loss exceded the theoretical value by about 65%. Data can be fit with high accuracy (R=0.997) by a 4th order polynomial fit. Nevertheless, also a linear fit in the range 110 K : 300 K is able to predict with good accuracy the behaviour at least down to 40 K. (Fig. \[plot\_extrapolation\]).
![Reflection Loss at from data in [@Serov2016], at 150 GHz. Stars: experimental data. Triangles: forth order polynomial fit; solid line: linear fit in the range \[110K : 296K\]. Cross: extrapolation down to 40K.[]{data-label="plot_extrapolation"}](extrapolation.eps "fig:"){width="8"}\
At each sampled frequency, the slope was calculated by linearly fitting data in the range 296K-110K; Reflection Loss measured at 110K was hence extrapolated down to 40K. Comparison is presented in Fig. \[plot\_comparison\_40K\]; error bars for data at HFI frequencies, at 296K and 110K, were not available.\
The emissivity at LFI frequencies - at 40K - is, as expected, lower than the emissivity measured at 110K, at the HFI frequencies, and slightly higher than extrapolated data.\
Reflection Loss at cryogenic temperature depends on the purity level of the material; in addition, in the specific case of a Space Telescope, cleanliness of the mirrors at the end of mission and aging can play a crucial role. Despite everything, deviations of measured from extrapolated data can be considered negligible, as they are well within the error bars of the in-flight measurement.\
Results confirm the goodness of this approach and the quality of the Planck telescope, in space - where it was not measured until the end of the mission.
![Telescope Reflection Loss. Reflection Loss data are multiplied by a factor of 1000. Results from TLT test, at LFI frequencies are compared to the following data sets: (i) data @110K from Fig.B1 -right in [@Tauber2010] (<span style="font-variant:small-caps;">MEAS 110K</span>), red rombs; (ii) data extrapolated in temperature down to 40K and in frequency down to LFI frequencies (<span style="font-variant:small-caps;">EXT 40K</span>), solid line with triangles; (iii) indirect *in-flight* measure by HFI from thermal arguments (reported in [@Planck_coll_2_2011]: <span style="font-variant:small-caps;">HFI-Flight</span>), blue dashed-dot line: the behaviour is flat in frequency; error bars are reported.[]{data-label="plot_comparison_40K"}](comparison_40K_F2.eps "fig:"){width="12"}\
Conclusions {#sec:Conclusions}
===========
The End of Life (EOL) phase, before Planck satellite de-orbiting, represented a very useful step in completing the characterization of several instrumental properties measured before launch or during the early phases of the mission (CPV).\
The telescope total emissivity was measured only indirectly (Reflection Loss tests), on test samples of the Herschel telescope first, on samples of the Planck telescope finally. Until EOL, emissivity was measured only in a reduced frequency range covered by HFI (100 GHz -380 GHz), keeping the samples at a temperature (110 K) higher than the in-flight nominal temperature of the telescope (around 40 K).\
The presence of de-contamination heaters and temperature sensors on the primary and secondary reflectors permitted a dedicated measurement of the telescope emissivity at mission completion. The high sensitivity of the LFI radiometers, together with the optimal knowledge of LFI systematic effects, allowed the derivation of the telescope emissivity from the thermal excess measured by the LFI radiometers.\
The emissivity measured is consistent with the *on ground* Reflection Loss measured in the range 100 GHz - 380 GHz, extrapolated to the LFI frequencies. Slight deviations from the extrapolated curve are consistent with the improvements expected from the lower telescope temperature in flight. Extrapolation to 40K of ’*on ground*’ reflection Loss measured at higher temperatures showed that the telescope performance was not degraded at EOL w.r.t. BOL, and that the emissivity was about one order of magnitude better than the mission requirement.\
The measure of success of future CMB experiments is how we cope with the knowledge of the systematic effects. Telescope emissivity can represent a large source of systematic uncertainties, since bigger and bigger mirrors will be required to feed thousands of receivers, needed to meet the ambitious requirements of next CMB experiments. This method could be hence usefully implemented for future space experiment at mm sub-mm wavelengths to finely characterize the telescope emissivity during the mission, in nominal conditions, provided that a dedicated thermal control system, based on control loop heaters and on a network of high resolution thermometers, is present.
Appendix {#sec:Appendix}
========
. \[cal\_const\]
--------- ----------- -----------
`LFI18` 14.2561 21.68022
`LFI19` 26.69584 41.77317
`LFI20` 25.07979 30.41089
`LFI21` 44.25306 41.59089
`LFI22` 62.7823 60.77838
`LFI23` 34.74583 51.49897
`LFI24` 287.79424 178.85588
`LFI25` 126.36797 126.56825
`LFI26` 170.71762 144.40973
`LFI27` 12.79919 15.27546
`LFI28` 15.81183 19.22634
--------- ----------- -----------
: Calibration constants (K/V). Results are displayed per radiometer. M and S correspond to `MAIN` and `SIDE` radiometers
The Planck Collaboration acknowledges the support of: ESA; CNES, and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN, and JA (Spain); Tekes, AoF, and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); ERC and PRACE (EU). A description of the Planck Collaboration and a list of its members, indicating which technical or scientific activities they have been involved in, can be found at URL:\
http://www.cosmos.esa.int/web/planck/planck-collaboration.\
The Planck LFI project (including instrument development and operation, data processing and scientific analysis) is developed by an international consortium led by Italy and involving Canada, Finland, Germany, Norway, Spain, Switzerland, UK, and USA. The Italian contribution is funded by the Italian Space Agency (ASI) and INAF.\
We want to give special thanks to the Planck Mission Operations Center (MOC) for the professionality and helpfulness shown during the whole Planck mission and, with respect to this work, during the LFI EOL Test Campaign.\
J.A. Tauber et al., Planck pre-launch status: the planck mission, A&A 520 (2010) A1.
Planck Collaboration I. 2011, A&A, 536, A1
J.M. Lamarre et al., Planck pre-launch status: The HFI instrument, from specification to actual performance, A&A 520 (2010) A9.
M. Bersanelli et al., Planck pre-launch status: design and description of the low frequency instrument, A&A 520 (2010) A4.
Planck Collaboration,Planck early results. II. The thermal performance of Planck, 2011 A&A 536, A2
Planck-LFI: design and performance of the 4 Kelvin reference load unit,JINST 4 (2009) T12006.
Gregorio et al., In-flight calibration and verification of the Planck-LFI instrument,JINST 8 (2013) T07001.
Mennella et al., Planck early results. III. First assessment of the Low Frequency Instrument in-flight performance , A&A , 536, (2011) A3.
Mennella et al., A&A , 520, (2010) A5.
Terenzi et al., Thermal susceptibility of the Planck-LFI receivers, JINST 12 (2009), T12012.
Cuttaia et al., Planck-LFI radiometers tuning , JINST 12 (2009), T12013.
Tauber et al., Planck pre-launch status: the optical system, 2010 A&A 520, A2
The Planck Collaboration, Planck early results. III. The thermal performance of Planck, 2011 A&A 536, A2
The Planck Collaboration, Planck 2013 results. III. LFI systematic uncertainties, A&A 571, A3
Mennella et al., Planck early results. III. First assessment of the Low Frequency Instrument in-flight performance, 2011 A&A 536, A3
The Planck Collaboration, Planck 2015 results. III. LFI systematic uncertainties, 2015, A&A submitted, arXiv:1507.08853
Fischer, J., Klaassen, T., Hovenier, N., et al. 2005, Appl. Opt., 43, 3765
The Planck Collaboration, Planck 2013 results. II. Low Frequency Instrument data processing, 2014, A&A 571, AA2.
Parshin, V., & van der Klooster, C. 2008, 30th ESA Antenna Workshop
Serov, E., A., Parshin, V., and Bubnov, G., M., Reflectivity of Metals in the Millimeter Wavelength Range at Cryogenic Temperatures, IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 11, NOVEMBER 2016
Thomas Stute, The Telescope Reflectors For the ESA Mission Planck, Presentation at 28th ESA Antenna Workshop ESA/ESTEC Noordwijk, 02 June 2005
Thomas Stute; The Planck Telescope reflectors. Proc. SPIE 5495, Astronomical Structures and Mechanisms Technology, 1 (September 29, 2004); doi:10.1117/12.549268.
|
---
abstract: 'Reconstructing 3D scenes from multiple views has made impressive strides in recent years, chiefly by correlating isolated feature points, intensity patterns, or curvilinear structures. In the general setting – without controlled acquisition, abundant texture, curves and surfaces following specific models or limiting scene complexity – most methods produce unorganized point clouds, meshes, or voxel representations, with some exceptions producing unorganized clouds of 3D curve fragments. Ideally, many applications require structured representations of curves, surfaces and their spatial relationships. This paper presents a step in this direction by formulating an approach that combines 2D image curves into a collection of 3D curves, with topological connectivity between them represented as a 3D graph. This results in a **3D drawing**, which is complementary to surface representations in the same sense as a 3D scaffold complements a tent taut over it. We evaluate our results against truth on synthetic and real datasets.'
author:
- |
Anil Usumezbas\
SRI International\
- |
\
Ricardo Fabbri\
Polytechnic Institute\
[rfabbri@iprj.uerj.br]{}
- |
\
Benjamin B. Kimia\
Shool of Engineering\
[benjamin\_kimia@brown.edu]{}
nocite: '[@Kowdle:etal:ECCV10; @Chen:Klette:IVT2014; @Zhu:Luong:etal:CVPR2014; @Rao:etal:IROS2012; @Abuhashim:Sukkarieh:IROS2012; @Yang:Ahuja:TIP2012; @Chen:Klette:IVT2014; @Simoes:etal:SVR2014; @Kanazawa:Kanatani:etal:IPSJTCVA2014; @Heinly:Frahm:etal:CVPR2015; @Teney:Piater:3DIMPVT12; @Zhang:line:PHDThesis2013; @Lebeda:etal:ACCV2014; @Bertasius:Shi:Torresani:CVPR2015; @Fleming:etal:PNAS2011; @Kunsberg:Zucker:LNM2014; @Zucker:PIEEE2014; @Koenderink:Wagemans:etal:iPerception2013; @Ruizhe:Medioni:CVPR2014; @Shinozuka:Saito:VRIC14; @Fathi:etal:AEI2015; @Restrepo:etal:JPRS2014; @Calakli:etal:3DIMPVT2012; @Strecha:etal:CVPR2008; @Jensen:etal:CVPR14]'
title: 'From Multiview Image Curves to 3D Drawings$\,^*$'
---
16SubNumber[707]{}
Introduction
============
The automated 3D reconstruction of *general* scenes from multiple views obtained using conventional cameras, under uncontrolled acquisition, is a paramount goal of computer vision, ambitious even by modern standards. While a fully complete working system addressing all the underlying challenges is beyond current technology, significant progress has been made in the past few years using approaches that fall into three broad classes, depending on whether one focuses on correlating isolated points, surface patches, or curvilinear structures across views, as described below.
![ Our approach transforms calibrated views of a scene into a “3D drawing” – a graph of 3D curves meeting at junctions. Each curve is shown in a different color. *(Please zoom in to examine closely. The 3D model is available as supplementary data.)*[]{data-label="fig:recon:results"}](figs/00.pdf "fig:"){width="0.3\linewidth"} ![ Our approach transforms calibrated views of a scene into a “3D drawing” – a graph of 3D curves meeting at junctions. Each curve is shown in a different color. *(Please zoom in to examine closely. The 3D model is available as supplementary data.)*[]{data-label="fig:recon:results"}](figs/08.pdf "fig:"){width="0.3\linewidth"} ![ Our approach transforms calibrated views of a scene into a “3D drawing” – a graph of 3D curves meeting at junctions. Each curve is shown in a different color. *(Please zoom in to examine closely. The 3D model is available as supplementary data.)*[]{data-label="fig:recon:results"}](figs/14.pdf "fig:"){width="0.3\linewidth"}\
![ Our approach transforms calibrated views of a scene into a “3D drawing” – a graph of 3D curves meeting at junctions. Each curve is shown in a different color. *(Please zoom in to examine closely. The 3D model is available as supplementary data.)*[]{data-label="fig:recon:results"}](figs/20.pdf "fig:"){width="0.3\linewidth"} ![ Our approach transforms calibrated views of a scene into a “3D drawing” – a graph of 3D curves meeting at junctions. Each curve is shown in a different color. *(Please zoom in to examine closely. The 3D model is available as supplementary data.)*[]{data-label="fig:recon:results"}](figs/28.pdf "fig:"){width="0.3\linewidth"} ![ Our approach transforms calibrated views of a scene into a “3D drawing” – a graph of 3D curves meeting at junctions. Each curve is shown in a different color. *(Please zoom in to examine closely. The 3D model is available as supplementary data.)*[]{data-label="fig:recon:results"}](figs/46.pdf "fig:"){width="0.3\linewidth"}
![ Our approach transforms calibrated views of a scene into a “3D drawing” – a graph of 3D curves meeting at junctions. Each curve is shown in a different color. *(Please zoom in to examine closely. The 3D model is available as supplementary data.)*[]{data-label="fig:recon:results"}](figs/recon.png){width="1.1\linewidth"}
A vast majority of multiview reconstruction methods rely on correlating isolated interest points across views to produce an unorganized 3D cloud of points. The **interest-point-based approach** has been highly successful in reconstructing large-scale scenes with [*texture-rich images*]{}, in systems such as in Phototourism and recent large-scale 3D reconstuction work [@Heinly:Frahm:etal:CVPR2015; @Pollefeys:VanGool:etal:handheld:IJCV2004; @Argarwal:Snavely:etal:ICCV09; @Diskin:Vijayan:JEI2015]. Despite their manifest usefulness, these methods generally cannot represent smooth, textureless regions (due to the sparsity of interest points in image regions with homogeneous appearance), or regions that change appearance drastically across views. This limits their applicability, especially in man-made environments [@Simoes:etal:SVR2014] and objects such as cars [@Shinozuka:Saito:VRIC14], non-Lambertian surfaces such as that of the sea, appearance variation due to changing weather [@Baatz:Pollefeys:etal:ECCV12], and wide baseline [@Moreels:Perona:IJCV07].
Another approach matches intensity patterns across views using multiview stereo, producing denser point clouds or mesh reconstructions. **Dense multi-view stereo** produces detailed 3D reconstructions of objects imaged under controlled conditions by a large number of precisely calibrated cameras [@Furukawa:Ponce:CVPR2007; @Habbecke:Kobbelt:CVPR2007; @Hernandez:Schmitt:CVIU04; @Goesele:etal:ICCV07; @Seitz:etal:CVPR06; @Calakli:etal:3DIMPVT2012; @Restrepo:etal:JPRS2014]. For general, complex scenes with various kinds of objects and surface properties, this approach has shown most promise towards obtaining an accurate and dense 3D model of a given scene. Homogeneous areas, such as walls of a corridor, repeated texture, and areas with view-dependent intensities create challenges for these methods.
A smaller number of techniques correlate and reconstruct image **curvilinear structure** across views, resulting in 3D curvilinear structure. Pipelines based on straight lines (see [@Lebeda:etal:ACCV2014; @Zhang:line:PHDThesis2013; @Fathi:etal:AEI2015] for recent reviews), algebraic and general curve features [@Teney:Piater:3DIMPVT12; @Litvinov:etal:IC3D2012; @Fabbri:Kimia:CVPR10; @Fabbri:Giblin:Kimia:ECCV12; @Wendel:etal:CVWW2011; @Berthilsson:etal:IJCV2001; @Fabbri:Kimia:EMMCVPR2005] have been proposed, but some lack generality, [*e.g.*]{}, requiring specific curve models [@Carrasco:etal:LNCS2012]. The 3D Curve Sketch system [@Fabbri:PhD:2010; @Fabbri:Giblin:Kimia:ECCV12; @Fabbri:Kimia:CVPR10] operates on multiple views by pairing curves from two arbitrary “hypothesis views” at a time via epipolar-geometric consistency. A curve pair reconstructs to a 3D curve fragment hypothesis, whose reprojection onto several other “confirmation views” gathers support from subpixel 2D edges. The curve pair hypotheses with enough support result in an unorganized set of 3D curve fragments, the “3D Curve Sketch”. While the resulting 3D curve segments are visually appealing, they are fragmented, redundant, and lack explicit inter-curve organization.
![ **3D drawings for urban planning and industrial design.** A process from professional practice for communicating solution concepts with a blend of computer and handcrafted renderings [@Leggitt:drawing:book; @Yee:architectural:book]. New designs are often based off real object references, mockups or massing models for selecting viewpoints and rough shapes. These can be modeled *manually* in, [*e.g.*]{}, Google Sketchup (top-left), in some cases from reference imagery. The desired 2D views are rendered and *manually* traced into a reference curve sketch (center-left, bottom-left) easily modifiable to the designer’s vision. The stylized drawings to be presented to a client are often produced by *manually* tracing and painting over the reference sketch (right). Our system can be used to generate reference *3D curve drawings* from video footage of the real site for urban planning, saving manual interaction, providing initial information such as rough dimensions, and aiding the selection of pose, editing and tracing. The condensed 3D curve drawings make room for the artist to overlay his concept and harness imagery as a clean reference, clear from details to be redesigned. []{data-label="fig:architectural:drawing"}](figs/aerial-architectural-drawing.png){width="1\linewidth"}
The plethora of multiview representations, as documented above, arise because 3D structures are geometrically and semantically rich [@Zia:Stark:Schindler:IJCV2015; @Feng:Medioni:etal:SIGGRAPH14]. A building, for example, has walls, windows, doorways, roof, chimneys, etc. The structure can be represented by sample points ([*i.e.*]{}, unorganized cloud of points) or a surface mesh where connectivity among points is captured. This representation, especially when rendered with surface albedo or texture, is visually appealing. However, the representation also leaves out a great deal of semantic information: which points or mesh areas represent a window or a wall? Which two walls are adjacent? The representation of such components, or parts, requires an explicit representation of part boundaries such as ridges, as well as where these boundaries come together, such as junctions.
The same point can equally arise if objects in the scene were solely defined by their curve structures. A representation of a building by its ridges may usually give an appealing impression of its structure, but it fails to identify the walls, [*i.e.*]{}, which collection of 3D curves bound a wall and what its geometry is. Both surfaces and curves are important and needed across the board, [*e.g.*]{}, in applications such as robotics [@Carlson:etal:ICRA2014], urban planning and industrial design [@Yee:architectural:book; @Leggitt:drawing:book], Fig. \[fig:architectural:drawing\]. In general, image curve fragments are attractive because they have good localization, they have greater invariance than interest points to changes in illumination, are stable over a greater range of baselines, and are typically denser than interest points. Furthermore, the reflectance or ridge curves provide boundary condition for surface reconstruction, while occluding contour variations across views lead to surfaces [@Giblin:Motion:Book; @Liu:Cooper:etal:PAMI07; @Crispell:etal:LNCS2009]. Recent studies strongly support the notion that image curves contain much of the image information [@Koenderink:Wagemans:etal:iPerception2013; @Zucker:PIEEE2014; @Kunsberg:Zucker:LNM2014; @Cole:etal:SIGGRAPH09]. Moreover, curves are structurally rich as reflected by their differential geometry, a fact which is exploited both in recent computer systems [@Zucker:PIEEE2014; @Abuhashim:Sukkarieh:IROS2012; @Fabbri:Giblin:Kimia:ECCV12; @Fabbri:Kimia:CVPR10] and peception studies [@Fleming:etal:PNAS2011; @Zucker:PIEEE2014].
This paper develops the technology to process a series of (intrinsic and extrinsically) calibrated multiview images to generate a *3D curve drawing* as a graph of 3D curve segments meeting at junctions. The ultimate goal of this approach is to integrate the 3D curve drawing with the traditional recovery of surfaces so that 3D curves bound the 3D curve segments, towards a more semantic representation of 3D structures. The 3D curve drawing can also be of independent value in applications such as fast recognition of general 3D scenery [@Wendel:etal:CVWW2011], efficient transmission of general 3D scenes, scene understanding and modeling by reasoning at junctions [@Mattingly:etal:JVLC2015], consistent non-photorealistic rendering from video [@Chen:Klette:IVT2014], modeling of branching structures, among others [@Rao:etal:IROS2012; @Kowdle:etal:ECCV10; @Ruizhe:Medioni:CVPR2014].
The paper is organized as follows. In Section \[sec:improved:sketch\] we review the 3D curve sketch, identify three shortcomings and suggest solutions to each, resulting in the *Enhanced Curve Sketch*. Since the original 3D curve sketch was built around a few views at a time, it did not address fundamental issues surrounding integration of information from numerous views. Section \[sec:3d:drawing\] presents as our main contribution the multiview integration of information both at edge- and curve-level, which naturally leads to junctions. Section \[sec:results\] validates the approach using real and synthetic datasets.
Enhanced 3D Curve Sketch {#sec:improved:sketch}
========================
Image curve fragments formed from grouped edges are central to our framework. Each image $V^v$ at view $v = 1,\dots,N$ contains a number of curves ${\boldsymbol{\gamma}}_i^v$, $i=1,\dots,M^v$. Reconstructed 3D curve fragments are referred as ${\boldsymbol{\Gamma}}_k$, $k=1,\dots,K$, whose reprojection onto view $v$ is ${\boldsymbol{\gamma}}^{k,v}$. Indices may be omitted where clear from context.
The initial stage of our framework is built as an extension of the hypothesize-and-verify 3D Curve Sketch approach [@Fabbri:Kimia:CVPR10]. We use the same hypothesis generation mechanism with a novel verification step performing a finer-level analysis of image evidence and significantly reducing the fragmentation and redundancy in the 3D models.
Two image curves ${\boldsymbol{\gamma}}^{v_1}_{l_1}$ and ${\boldsymbol{\gamma}}^{v_2}_{l_2}$ are paired from two distinct views $v_1$ and $v_2$ at a time, the *hypothesis views*, provided they have sufficient epipolar overlap [@Fabbri:Kimia:CVPR10]. The verification of these $K$ curve pair hypotheses, represented as $\omega_k$, $k=1,\dots,K$ with the corresponding 3D reconstruction denoted as ${\boldsymbol{\Gamma}}_k$, gauges the extent of edge support for the reprojection ${\boldsymbol{\gamma}}^{k,v}$ of ${\boldsymbol{\Gamma}}_k$ onto another set of *confirmation views*, $v = v_{i_3},\dots,v_{i_n}$. An image edge in view $v$ suports ${\boldsymbol{\gamma}}^{k,v}$ if it is sufficiently close in distance *and* orientation. The total support a hypothesis $\omega$ receives from view $v$ is $$\label{eq:support}
S^v_{\omega_k} \doteq \int_{0}^{L^{k,v}} \phi({\boldsymbol{\gamma}}^{k,v}(s))ds,$$ where $L^{k,v}$ is the length of ${\boldsymbol{\gamma}}^{k,v}$, and $\phi({\boldsymbol{\gamma}}(s))$ is the extent of edge support at ${\boldsymbol{\gamma}}(s)$. A view is considered a *supporting view* for $\omega_k$ if $S^v_{\omega_k} > \tau_v$. Evidence from confirmation views is aggregated in the form $$\mathcal S_{\omega_k} \doteq \sum_{v = i_3}^{i_n} \left[ S^v_{\omega_k} > \tau_v \right]
S^v_{\omega_k}.$$ The set of hypotheses $\omega_k$ whose support $S_{\omega_k}$ exceeds a threshold are kept and the resulting ${\boldsymbol{\Gamma}}_k$ form the unorganized 3D curves.
Despite these advances, three major shortcomings remain: $(i)$ some 3D curve fragments are correct for certain portions of the underlying curve and erroneous in other parts, due to multiview grouping inconsistencies; $(ii)$ gaps in the 3D model, typically due to unreliable reconstructions near epipolar tangencies, where epipolar lines are nearly tangent to the curves; and $(iii)$ multiple, redundant 3D structures. We now document each issue and describe our solutions.
**Erroneous grouping:** inconsistent multiview grouping of edges can lead to reconstructed curves which are veridical only along some portion, which are nevertheless wholly admitted, Fig. \[fig:sample:localization\](a). Also, fully-incorrect hypotheses can accrue support coincidentally, as with repeated patterns or linear structures, Fig. \[fig:sample:localization\](b). Both issues can be addressed by allowing for selective local reconstructions: only those portions of the curve receiving adequate edge support from sufficient views are reconstructed. This ensures that inconsistent 2D groupings do not produce spurious 3D reconstructions. The shift from cumulative global to multi-view local support results in greater selectivity and deals with coincidental alignment of edges with the reconstruction hypotheses.
![(a) Due to a lack of consistency in grouping of edges at the image level, a correct 3D curve reconstruction, shown here in blue, can be erroneously grouped with an erroneous reconstruction, shown here in red, leading to partially correct reconstructions. When such a 3D curve is projected in its entirety to a number of image views, we only expect the correct portion to gather sustained image evidence, which argues for a hypothesis verification method that can distinguish between supported segments and outlier segments; (b) An incorrect hypothesis can at times coincidentally gather an extremely high degree of support from a limited set of views. The red 3D line shown here might be an erroneous hypothesis, but because parallel linear structures are common in man-made environments, such an incorrect hypothesis often gathers coincidental strong support from a particular view or two. Our hypothesis verification approach is able to handle such cases by requiring explicit support from a minimum number of viewpoints simultaneously. []{data-label="fig:sample:localization"}](figs/support-cases-sm.png){width="\linewidth"}
**Gaps:** The geometric inaccuracy of curve segment reconstructions nearly parallel to epipolar lines led [@Fabbri:Kimia:CVPR10] to break off curves at epipolar tangencies, creating 2D gaps leading to gaps in 3D. We observe, however, that while reconstructions near epipolar tangency are *geometrically unreliable*, they are *topologically correct* in that they connect the reliable portions correctly but with highly inaccurate geometry. What is needed is to flag curve segments near epipolar tangency reconstructions as geometrically unreliable. We do this by the integration of support in Equation \[eq:support\], giving significantly lower weight to these unreliable portions instead of fully discarding them, which greatly reduces the presence of gaps in the resulting reconstruction.
![ (a) Redundant 3D curve reconstructions (orange, green and blue) can arise from a single 2D image curve in the primary hypothesis view. If the redundant curves are put in one-to-one correspondence and averaged, the resulting curve is shown in (b) in purple. Our robust averaging approach, on the other hand, is able to get rid of that bump by eliminating outlier segments, producing the purple curve shown in (c). []{data-label="fig:robust:averaging"}](figs/robust-averaging.png){width="0.6\linewidth"}
**Redundancy:** A 2D curve can pair up with dozens of curves from other views, all pointing to the same reconstruction, leading to redundant pairwise reconstructions as partially overlapping 3D curve segments, each localized slightly differently. Our solution is to detect and reconcile redundant reconstructions. Since redundancy changes as one traverses a 3D curve, we reconcile redundancy at the local level: each 3D edge is in one-to-one correspondence with a 2D edge of its primary hypothesis view ([*i.e.*]{}, the first view from which it was reconstructed), hence 3D edges can be grouped in a one-to-one manner, all corresponding to a common 3D source. These are robustly averaged by data-driven outlier removal, where a Gaussian distribution is fit on all pairwise distances between corresponding samples, discarding samples farther than $2\sigma$ from the average, Fig. \[fig:robust:averaging\]. Robust averaging improves localization accuracy, removes redundancy, and elongates shorter curve subsegments into longer 3D curves.
![ A visual comparison of: (left) the curve sketch results [@Fabbri:Kimia:CVPR10], with (right) the results of our enhanced curve sketch algorithm presented in Section \[sec:improved:sketch\]. Notice the significant reduction in both outliers and duplicate reconstructions, without sacrificing coverage. []{data-label="fig:improved:qual"}](figs/curve-sketch-amsterdam.png "fig:"){height="4cm"} ![ A visual comparison of: (left) the curve sketch results [@Fabbri:Kimia:CVPR10], with (right) the results of our enhanced curve sketch algorithm presented in Section \[sec:improved:sketch\]. Notice the significant reduction in both outliers and duplicate reconstructions, without sacrificing coverage. []{data-label="fig:improved:qual"}](figs/enhanced-sketch-amsterdam.png "fig:"){height="4cm"}
From 3D Curve Sketch to 3D Drawing {#sec:3d:drawing}
==================================
Despite the visible improvements of the Enhanced 3D Curve Sketch of Section \[sec:improved:sketch\], Fig. \[fig:improved:qual\], curves are broken in many places, and there remains redundant overlap. The sketch representation as unorganized clouds of 3D curves are not able to capture the fine-level geometry or spatial organization of 3D curves, [*e.g.*]{}by using junction points to characterize proximity and neighborhood relations. The underlying cause of these issues is lack of integration across multiple views. The robust averaging approach of Section \[sec:improved:sketch\] is one step, anchored on one primary hypothesis view, but integrates evidence within that view only; a scene curve can be visible from multiple hypothesis view pairs, and some redundancy remains.
This lack of multiview integration is responsible for three problems observed in the enhanced curve sketch, Fig. \[fig:issues:remaining\]: $(i)$ localization inaccuracies, Fig. \[fig:issues:remaining\]b, due to use of partial information; $(ii)$ reconstruction redundancy, which lends to multiple curves with partial overlap, all arising from the same 3D structure, but remaining distinct, see Fig. \[fig:issues:remaining\]c; $(iii)$ excessive breaking because each curve segment arises from one curve in one initial view independently.
**Multiview Local Consistency Network:** The key idea underlying integration of reconstructions across views is the detection of a common image structure supporting two reconstruction hypotheses. Two 3D local curve segments depict the same single underlying 3D object feature if they are supported by the same 2D image edge structures. Since the identification of common image structure can vary along the curve, it must necessarily be a local process, operating at the level of a 3D local edge and not a 3D curve. Two 3D edge elements (edgels) depict the same 3D structure if they receive support from the same 2D edgels in a sufficient number of views, so 3D-2D links between a 2D edgel to the 3D edgel it supports must be kept. Typically, they share supporting image edges in many views; and the number of shared supporting edgels is the measure of strength for a 3D-3D link between them.
Formally, we define the Multiview Local geometric consistency Network (MLN) as pointwise alignments $\phi_{ij}$ between two 3D curves ${\boldsymbol{\Gamma}}_i$ and ${\boldsymbol{\Gamma}}_j$: let ${\boldsymbol{\Gamma}}_i(s_i)$ and ${\boldsymbol{\Gamma}}_j(s_j)$ be two points in two 3D curves, and define $$S_{ij} \doteq \{v : {\boldsymbol{\gamma}}^{i,v}(s_i) \text{ and } {\boldsymbol{\gamma}}^{j,v}(s_j) \text{ share local support}\}.$$ Then the a kernel function $\phi$ defines a consistency link between these two points, weighted by the extent of multiview image support $\phi_{ij}(s_i,s_j) \doteq
|S_{ij}|$. When the curves are sampled, $\phi$ becomes an adjacency matrix of a graph representing links between individual curve samples. The implementation goes through each image edgel which votes for a 3D curve point that has received support from it (see the supplementary material for details).
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**Multiview Curve-level Consistency Network:** The identification of 3D edges sharing 2D edges leads to high recall operating point with many false links due to accidental alignment of edge support. False positives can be reduced without affecting high recall by employing a notion of curve context for each 3D edgel: a link between two 3D edgels based on a supporting 2D edgel is more effective if the respective neighbors of the underlying 3D edge on the underlying 3D curve are also linked.
The curve context idea requires establishing new pairwise links between 3D curves using MLN, when there are a sufficient number of links with $\phi_{ij}>\tau_{\epsilon}$ between their constituent 3D edges (in our implementation, $\tau_{\epsilon}=3$ and we require 5 such edges or more). The linking of 3D curves is represented by the Multiview Curve-level Consistency network (MCCN), a graph whose nodes are the 3D curves ${\boldsymbol{\Gamma}}_j$ and the edges represent the presence of high-weight 3D edge links between these 3D curves. The [<span style="font-variant:small-caps;">mccn</span>]{}graph allows for a clustering of 3D curves by finding connected components; and once a link is established between two curves, there is a high likelihood of their edges corresponding in a regularized fashion, thus fewer common supporting 2D edges are required to establish a link between all their constituent 3D edges. This fact is used to perform gap filling, since even no edge support is acceptable to fill in small gaps and create a continuous and regularized correspondence if both neighbors of the gap are connected (see pseudocode in Supplementary Materials for details). The two stages in tandem, [*i.e.*]{}, high recall linking of 3D edges and use of curve context to reduce false positives leads to high recall and high precision, *i.e.*, all the 3D edges which need to be related are related and very few outlier connections remain.
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\[2\][\*]{}\[13mm\][![ The correspondence between 3D edge samples is skewed along a curve, a direct indication that these links cannot be used as-is when averaging and fusing redundant curve reconstructions. Instead, each point is assumed to be in correspondence with the point closest to it on another overlapping curve, during the iterative averaging step. Observe that corrections can be partial along related curves. []{data-label="fig:overlap:masks"}](figs/need-for-clusters.png "fig:"){width="0.55\linewidth"}]{} ![ The correspondence between 3D edge samples is skewed along a curve, a direct indication that these links cannot be used as-is when averaging and fusing redundant curve reconstructions. Instead, each point is assumed to be in correspondence with the point closest to it on another overlapping curve, during the iterative averaging step. Observe that corrections can be partial along related curves. []{data-label="fig:overlap:masks"}](figs/cluster7.png "fig:"){width="0.45\linewidth"}
![ The correspondence between 3D edge samples is skewed along a curve, a direct indication that these links cannot be used as-is when averaging and fusing redundant curve reconstructions. Instead, each point is assumed to be in correspondence with the point closest to it on another overlapping curve, during the iterative averaging step. Observe that corrections can be partial along related curves. []{data-label="fig:overlap:masks"}](figs/cluster7-corr.png "fig:"){width="0.45\linewidth"}
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**Integrating information across related edges:** The identification of a bundle of curves as arising from the same 3D source implies that we can improve the geometric accuracy of this bundle by allowing them to converge to a common solution. While this might appear straightforward, 3D edges are not consistenly distributed along related curves, yielding a skew in the correspondence of related samples, Fig. \[fig:overlap:masks\], sometimes not a one-to-one correspondence, Fig. \[fig:graph:organization\]a. This argues for averaging 3D curves and not 3D edge samples, which in turn requires finding a more regularized alignment between the 3D curves, without gaps; we find each curve samples’s closest point on the other curve.
![(a) A schematic of sample correspondence along two related 3D curves, showing skewed correspondences that may not be one-to-one. (b) A sketch of how two curves are integrated. Bottom row: a real case. []{data-label="fig:graph:organization"}](figs/graph-organization-caption.pdf "fig:"){width="0.86\linewidth"} ![(a) A schematic of sample correspondence along two related 3D curves, showing skewed correspondences that may not be one-to-one. (b) A sketch of how two curves are integrated. Bottom row: a real case. []{data-label="fig:graph:organization"}](figs/cluster32-1.png "fig:"){height="2.8cm"} ![(a) A schematic of sample correspondence along two related 3D curves, showing skewed correspondences that may not be one-to-one. (b) A sketch of how two curves are integrated. Bottom row: a real case. []{data-label="fig:graph:organization"}](figs/cluster32-after1.png "fig:"){height="2.8cm"} ![(a) A schematic of sample correspondence along two related 3D curves, showing skewed correspondences that may not be one-to-one. (b) A sketch of how two curves are integrated. Bottom row: a real case. []{data-label="fig:graph:organization"}](figs/cluster32-merged1.png "fig:"){height="2.8cm"}
When post averaging a sample with its closest points on related curves, the order of resulting averaged samples is not clear. The order should be inferred from the underlying curves, but this information can be conflicting, unless the distance between two curves is substantially smaller than the sampling distance along the curves. This requires first updating each curve’s geometry separately and iteratively, without merging curves until after convergence, Fig. \[fig:graph:organization\]d. This also improves the correspondence of samples at each iteration, as the closest points are continuously updated.
At each stage, the iterative averaging process simply replaces each 3D edge sample with the average of all closest points on curves related to it, Fig \[fig:graph:organization\]b–d. This can be formulated as evolving all 3D curves by averaging along the [<span style="font-variant:small-caps;">mccn</span>]{}using closest points. Formally, each ${\boldsymbol{\Gamma}}_i$ is evolved according to $$\frac{\partial{\boldsymbol{\Gamma}}_i}{\partial t}(s) = \alpha
{\operatornamewithlimits{avg}}_{\substack{(i,j)\in L\\ ({\boldsymbol{\Gamma}}_i, {\boldsymbol{\Gamma}}_j) \in \text{ \textsc{mccn}}}}
\{{\boldsymbol{\Gamma}}_j(r) : {\boldsymbol{\Gamma}}_j(r) = \text{cp}_j({\boldsymbol{\Gamma}}_i(s)) \},$$where $\text{cp}_i(\mathbf p)$ is the closest point in ${\boldsymbol{\Gamma}}_i$ to $\mathbf p$ and $L$ is the link set defined as follows: Let the set $S_{ij}$ of so-called strong local links between curves ${\boldsymbol{\Gamma}}_i$ and ${\boldsymbol{\Gamma}}_j$ be $$S_{ij} \doteq \{(s,t) : \phi_{ij}(s,t) \geq \tau_{\epsilon}, \phi_{ij} \in
\text{MLN}({\boldsymbol{\Gamma}}_1,\dots,{\boldsymbol{\Gamma}}_K) \}.$$ Then the set $L$ of the [<span style="font-variant:small-caps;">mccn</span>]{}is defined as $$L \doteq \{(i,j) : |S_{ij}| \geq \tau_{sl}\}.$$ In practice, the averaging is robust and $\alpha$ is chosen such that in one step we move to the average.
**3D Curve Drawing Graph:** Once all related curves have converged, they can be merged into single curves, separated by junctions where 3 or more curves meet. The order along the resulting curve is also dictated by closest points: The immediate neighbors of any averaged 3D edge are the two closest 3D edges to it among all converged 3D edges in a given [<span style="font-variant:small-caps;">mccn</span>]{}cluster.
This where junctions naturally arise: as two distinct curves may merge along one portion they may diverge at one point, leaving two remaining, non-related subsegments behind, Fig \[fig:graph:organization\]e. This is a *junction node* relating three or more curve segments, and its detection is done using the *merging primitives*, whose complete set are shown in Fig. \[fig:junction:topology\]. The intuition is this: a complex merging problem along the full length of two 3D curves actually consists of smaller, simpler and independent merging operations between different segments of each curve. A full merging problem between two complete curves can be expressed as a permutation of any number of simpler merging primitives. These primitives were worked out systematically to serve as the basic building blocks capable of constructing *all possible configurations* of our merging problem.
![The complete set of merging primitives, which were systematically worked out to cover all possible merging topologies between a pair of curves whose overlap regions are computedbeforehand. We claim that any configuration of overlap between two curves can be broken down into a series of these primitives along the length of one of the curves. The 5th primitive is representative of a “bridge” situation, where the connection at either end of the yellow curve can be any one of the first four cases shown, and 6th primitive is representative of a situation where only one end of the yellow curve connects to multiple existing curves, but not necessarily just two.[]{data-label="fig:junction:topology"}](figs/topology.pdf){width="0.6\linewidth"}
After iterative averaging, all resulting curves in any given cluster are processed in a pairwise fashion using these primitives: initialize the 3D graph with the longest curve in the cluster, and merge every curve in the cluster one by one into this graph. At each step, any number of these merging primitives arise and are handled appropriately. This process outputs the Multiview Curve Drawing Graph (MDG), which consists of multiple disconnected 3D graphs, one for each 3D curve cluster in the MCCN. The nodes of each graph are the junctions (with curve endpoints) and the links are curve fragment geometries. This structure is the final 3D curve drawing.
![(a) The four main issues with the enhanced curve sketch: (b) localization errors along the camera principal axis, which cause loss in accuracy if not corrected, (c) redundant reconstructions due to a lack of integration across different views, (d) the reconstruction of a single long curve as multiple, disconnected (but perhaps overlapping) short curve segments, and (e) the lack of connectivity among distinct 3D curves which naturally form junctions. (f) shows the 3D drawing reconstructed from this enhanced curve sketch, as described in Section \[sec:3d:drawing\]. Observe how each of the four bottlenecks have been resolved. Additional results are evaluated visually and quantitatively, and are reported in Section \[sec:results\] as well as Supplementary Materials. []{data-label="fig:issues:remaining"}](figs/rois.pdf "fig:"){width="0.49\linewidth"} ![(a) The four main issues with the enhanced curve sketch: (b) localization errors along the camera principal axis, which cause loss in accuracy if not corrected, (c) redundant reconstructions due to a lack of integration across different views, (d) the reconstruction of a single long curve as multiple, disconnected (but perhaps overlapping) short curve segments, and (e) the lack of connectivity among distinct 3D curves which naturally form junctions. (f) shows the 3D drawing reconstructed from this enhanced curve sketch, as described in Section \[sec:3d:drawing\]. Observe how each of the four bottlenecks have been resolved. Additional results are evaluated visually and quantitatively, and are reported in Section \[sec:results\] as well as Supplementary Materials. []{data-label="fig:issues:remaining"}](figs/recon-all.png "fig:"){width="0.49\linewidth"}
Experiments and Evaluation {#sec:results}
==========================
We have devised a number of large real and synthetic multiview datasets, available at [multiview-3d-drawing.sourceforge.net](multiview-3d-drawing.sourceforge.net).
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\[2\][\*]{}\[10mm\][![ Our publicly-available synthetic (left and top-right) and **real** structured lighting (bottom-right) 3D ground truths modeled and rendered using Blender for the present work. []{data-label="fig:synth:gt"}](figs/pavilion-GT.png "fig:"){width="0.57\linewidth"}]{}
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**The Barcelona Pavilion Dataset:** a realistic synthetic dataset we created for validating the present approach with control over illumination, geometry and cameras. It consists of: 3D models composing a large, mostly man-made, scene professionally composed by eMirage studios using the 3D modeling software Blender; ground-truth cameras fly-by’s around chairs with varied reflectance models and cluttered background; $(iii)$ ground-truth videos realistically rendered with high quality ray tracing under 3 extreme illumination conditions (morning, afternoon, and night); $(iv)$ ground-truth 3D *curve* geometry obtained by manually tracing over the meshes. This is the first synthetic 3D ground truth for evaluating multiview reconstruction algorithms that is realistically complex – most existing ground truth is obtained using either laser or structured light methods, both of which suffer from reconstruction inaccuracies and calibration errors. Starting from an existing 3D model ensures that our ground truth is not polluted by any such errors, since both 3D model and the calibration parameters are obtained from the 3D modeling software, Fig. \[fig:synth:gt\]. The result is the first publicly available, high-precision 3D curve ground truth dataset to be used in the evaluation of curve-based multiview stereo algorithms. For the experiments reported in the main manuscript we use 25 views out of 100 from this dataset, evenly distributed around the primary objects of interest, namely the two chairs, see Fig. \[fig:synth:gt\].
**The Vase Dataset:** constructed for this research from the <span style="font-variant:small-caps;">dtu</span> Point Feature Dataset with calibration and 3D ground truth from structured light [@Aan:Pedersen:etal:IJCV2012; @Jensen:etal:CVPR14]. The images were taken using an automated robot arm from pre-calibrated positions and our test sequence was constructed using views from different illumination conditions to simulate varying illumination. To the best of our knowledge, these are the most exhaustive public multiview ground truth datasets. To generate ground-truth for curves, we have constructed a GUI based on Blender to manually remove all points of the ground-truth 3D point-cloud that correspond to homogeneous scene structures as observed when projected on all views, Fig. \[fig:synth:gt\](bottom). What remains is a dense 3D point cloud ground truth where the points are restricted to be near abrupt intensity changes on the object, [*i.e.*]{} edges and curves. Our results on this real dataset showcase our algorithm’s robustness under varying illumination.
**The Amsterdam House Dataset:** 50 calibrated multiview images, also developed for this research, comprising a wide variety of object properties, including but not limited to smooth surfaces, shiny surfaces, specific close-curve geometries, text, texture, clutter and cast shadows, Fig. \[fig:recon:results\]. The camera reprojection error obtained by Bundler [@Argarwal:Snavely:etal:ICCV09] is on average subpixel. There is no ground truth 3D geometry for this dataset; the intent here is: to qualitatively test on a scene that is challenging to approaches that rely on, [*e.g.*]{}, point features; and to be able to closely inspect expected geometries and junction arising from simple, known shapes of scene objects.
**The Capitol High Building:** 256 HD frames from a high $270^\circ$ helicopter fly-by of the Rhode Island State Capitol [@Fabbri:Kimia:CVPR10]. Camera parameters are from the Matlab Calibration toolbox and tracking 30 points.
![The 3D drawing results on the Barcelona Pavilion, DTU Vase and Capitol Datasets. See Supplementary Materials for more extensive results and comparisons []{data-label="fig:pavilion:barcelona:capitol"}](figs/pavilion-midday-drawing-1.png "fig:"){height="3.2cm"} ![The 3D drawing results on the Barcelona Pavilion, DTU Vase and Capitol Datasets. See Supplementary Materials for more extensive results and comparisons []{data-label="fig:pavilion:barcelona:capitol"}](figs/vase-drawing-2.png "fig:"){height="3.0cm"} ![The 3D drawing results on the Barcelona Pavilion, DTU Vase and Capitol Datasets. See Supplementary Materials for more extensive results and comparisons []{data-label="fig:pavilion:barcelona:capitol"}](figs/capitol-drawing-1.png "fig:"){height="3.0cm"}
**Qualitative Evaluation:** The enhancements of Section \[sec:improved:sketch\] lead to significant improvements to the 3D curve sketch of [@Fabbri:Kimia:CVPR10] in increasing recall while maintaining precision. See Fig. \[fig:improved:qual\] for a qualitative comparison. When the clean clouds of curves are organized into a set of connected 3D graphs, the results are more accurate, more visually pleasing and not redundant, Fig. \[fig:issues:remaining\](f) and Fig. \[fig:pavilion:barcelona:capitol\]. Each of the issues in Fig. \[fig:issues:remaining\](a-e) have been resolved and spatial organization of 3D curves have been captured as junctions, represented by small white spheres.
![ Precision-recall curves for quantitative evaluation of 3D curve drawing algorithm: (a) Curve sketch, enhanced curve sketch and curve drawing results are compared on Barcelona Pavilion dataset with afternoon rendering, showing significant improvements in reconstruction quality; (b) A comparison of 3D curve drawing results on fixed and varying illumination version of Barcelona Pavilion dataset proves that 3D drawing quality does not get adversely affected by varying illumination; (c) 3D drawing improves reconstruction quality by a large margin in Vase dataset, which consists of images of a real object under slight illumination variation.[]{data-label="fig:drawing:quan"}](figs/PR-pav.png "fig:"){width="0.325\linewidth"} ![ Precision-recall curves for quantitative evaluation of 3D curve drawing algorithm: (a) Curve sketch, enhanced curve sketch and curve drawing results are compared on Barcelona Pavilion dataset with afternoon rendering, showing significant improvements in reconstruction quality; (b) A comparison of 3D curve drawing results on fixed and varying illumination version of Barcelona Pavilion dataset proves that 3D drawing quality does not get adversely affected by varying illumination; (c) 3D drawing improves reconstruction quality by a large margin in Vase dataset, which consists of images of a real object under slight illumination variation.[]{data-label="fig:drawing:quan"}](figs/PRillum.png "fig:"){width="0.325\linewidth"} ![ Precision-recall curves for quantitative evaluation of 3D curve drawing algorithm: (a) Curve sketch, enhanced curve sketch and curve drawing results are compared on Barcelona Pavilion dataset with afternoon rendering, showing significant improvements in reconstruction quality; (b) A comparison of 3D curve drawing results on fixed and varying illumination version of Barcelona Pavilion dataset proves that 3D drawing quality does not get adversely affected by varying illumination; (c) 3D drawing improves reconstruction quality by a large margin in Vase dataset, which consists of images of a real object under slight illumination variation.[]{data-label="fig:drawing:quan"}](figs/PR-vase.png "fig:"){width="0.325\linewidth"}
**Quantitative Evaluation:** Accuracy and coverage of 3D curve reconstructions is evaluated against ground truth. We compare 3 different results to quantify our improvements: $(i)$ Original Curve Sketch [@Fabbri:Kimia:CVPR10] run exhaustively on all views, $(ii)$ Enhanced Curve Sketch, Section \[sec:improved:sketch\], and $(iii)$ Curve Drawing, Section \[sec:3d:drawing\]. Edge maps are obtained using Third-Order Color Edge Detector [@Tamrakar:Kimia:ICCV07], and are linked using Symbolic Linker [@Yuliang:etal:CVPR14] to extract curve fragments for each view. Edge support thresholds are varied during reconstruction for each method, to obtain precision-recall curves. Here, [**precision**]{} is the percentage of accurately reconstructed curve samples: a ground truth curve sample is a true positive if its closer than a proximity threshold to the reconstructed 3D model. A reconstructed 3D sample is deemed a false positive if its not closer than $\tau_{prox}$ to any ground truth curve. This method ensures that redundant reconstructions aren’t rewarded multiple times. All remaining curve samples in the reconstruction are false positives. [**Recall**]{} is the fraction of ground truth curve samples covered by the reconstruction. A ground truth sample is marked as a false negative if its farther than $\tau_{prox}$ to the test reconstruction. The precision-recall curves shown in Fig. \[fig:drawing:quan\] *quantitatively* measure the improvements of our algorithm and showcase its robustness under varying illumination.
Conclusion
==========
We have presented a method to extract a 3D drawing as a graph of 3D curve fragments to represent a scene from a large number of multiview imagery. The 3D drawing is able to pick up contours of objects with homogeneous surfaces where feature and intensity based correlation methods fail. The 3D drawing can act as a scaffold to complement and assist existing feature and intensity based methods. Since image curves are generally invariant to image transformations such as illumination changes, the 3D drawing is stable under such changes. The approach does not require controlled acquisition, does not restrict the number of objects or object properties.
\
\
Full supplementary material, code and datasets available at [multiview-3d-drawing.sourceforge.net](multiview-3d-drawing.sourceforge.net)
This appendix presents additional descriptions, details and results that could not go into the main paper due to space constraints. This is a subset of the full supplementary materials available at [multiview-3d-drawing.sourceforge.net](multiview-3d-drawing.sourceforge.net). Section \[sec:more:dataset\] discusses the 3D ground truth benchmarks that were used in the quantitative evaluation of our results, and details the process with which 3D curvilinear ground truth models were obtained with the aid of Blender, for both synthetic and real data. In Section \[sec:moreresults\] we present additional figures for our results.
Obtaining Ground Truth for Quantitative Evaluation {#sec:more:dataset}
==================================================
Quantitative evaluation of 3D models reconstructed from a sequence of images is a non-trivial task due to the difficulties involved in obtaining clean and accurate ground truth 3D models for physical objects in the world, as well as precise calibration for each of the images in the sequence. The well-known Middlebury benchmark [@Seitz:etal:CVPR06] evaluates full surface reconstructions, and the ground truth 3D models are not made public; therefore it is not possible to appropriate them for quantitative evaluation of curve reconstructions. The EPFL benchmark [@Strecha:etal:CVPR08] makes the ground truth 3D models publicly available, but these datasets are limited in the number of views in the image sequence, as well object types and illumination conditions captured in the scene. In our case, the difficulty is compounded by the fact that our reconstruction is a wireframe representation, whereas almost all existing ground truth for multiview stereo is for evaluating dense surface reconstruction algorithms.
![ Our synthetic truth modeled and rendered using Blender for the present work. The bottom images are sample frames of three different videos for different illumination conditions. A fourth sequence is also used in the experiments, mixing up frames from the three conditions. []{data-label="fig:synth:gt:pavilion:frames"}](figs/day-sm.jpg "fig:"){width="0.993\linewidth"}\
![ Our synthetic truth modeled and rendered using Blender for the present work. The bottom images are sample frames of three different videos for different illumination conditions. A fourth sequence is also used in the experiments, mixing up frames from the three conditions. []{data-label="fig:synth:gt:pavilion:frames"}](figs/sunset-sm.jpg "fig:"){width="0.495\linewidth"} ![ Our synthetic truth modeled and rendered using Blender for the present work. The bottom images are sample frames of three different videos for different illumination conditions. A fourth sequence is also used in the experiments, mixing up frames from the three conditions. []{data-label="fig:synth:gt:pavilion:frames"}](figs/night-sm.jpg "fig:"){width="0.495\linewidth"}\
![ Our synthetic truth modeled and rendered using Blender for the present work. The bottom images are sample frames of three different videos for different illumination conditions. A fourth sequence is also used in the experiments, mixing up frames from the three conditions. []{data-label="fig:synth:gt:pavilion:frames"}](figs/chairframes-sm.pdf "fig:"){width="1.015\linewidth"}
Our first approach for reliable and fair evaluation of our 3D drawing algorithm is to utilize a synthetic 3D model and a rendering software to factor out calibration and reconstruction errors common among ground truth models obtained from real world objects. Here, the realistically-rendered images for this scene, Figure \[fig:synth:gt:pavilion:frames\], as well as the precisely calibrated views, are obtained using Blender. Three different illumination conditions were rendered, and these can be mixed up to test any given algorithm’s robustness under varying illumination, such as a slow sunset. This synthetic data was modeled after a real scene in Barcelona.
![ The full Barcelona Pavilion synthetic ground truth (top) and the bounding box (bottom) corresponding to Figure 10 in the paper. []{data-label="fig:synth:gt:pavilion:curves:fake"}](figs/pavilion-extras/keep/full-gt.png "fig:"){width="1\linewidth"} ![ The full Barcelona Pavilion synthetic ground truth (top) and the bounding box (bottom) corresponding to Figure 10 in the paper. []{data-label="fig:synth:gt:pavilion:curves:fake"}](figs/pavilion-extras/keep/bbx-gt.png "fig:"){width="1\linewidth"}
To the best of our knowledge, there is no popular, publicly-available multiview stereo ground truth that is based on a precise and complex 3D model and its rendered images. We have made two versions of our Barcelona Pavilion dataset available for the evaluation of 3D reconstruction algorithms: i) The full mesh version for evaluating dense surface reconstruction algorithms, ii) 3D curve version for evaluating curvilinear models, such as the 3D drawing presented in this work, Figure \[fig:synth:gt:pavilion:curves:fake\]. The latter version was obtained by a Blender-aided process of manually deleting surface meshes until only the outline of the objects remained, see Figure \[fig:synth:gt:pavilion:chair:overlay\] and Figure \[fig:synth:gt:pavilion:lotus\].
![ Process of deleting mesh edges to produce the desired ground truth edges. []{data-label="fig:synth:gt:pavilion:chair:overlay"}](figs/pavilion-extras/keep/chair-3d-overlay.png "fig:"){width="\linewidth"} ![ Process of deleting mesh edges to produce the desired ground truth edges. []{data-label="fig:synth:gt:pavilion:chair:overlay"}](figs/pavilion-extras/keep/chair-gt-blueprint.png "fig:"){width="\linewidth"}
![ Detail of our ground truth generation. Even minute objects were modeled by discarding internal mesh edges (blue). []{data-label="fig:synth:gt:pavilion:lotus"}](figs/pavilion-extras/keep/lotus-1.png "fig:"){width="0.49\linewidth"} ![ Detail of our ground truth generation. Even minute objects were modeled by discarding internal mesh edges (blue). []{data-label="fig:synth:gt:pavilion:lotus"}](figs/pavilion-extras/keep/lotus-3.png "fig:"){width="0.49\linewidth"} ![ Detail of our ground truth generation. Even minute objects were modeled by discarding internal mesh edges (blue). []{data-label="fig:synth:gt:pavilion:lotus"}](figs/pavilion-extras/keep/lotus-2.png "fig:"){width="0.49\linewidth"}
Although the Barcelona Pavilion dataset allows for a very precise and reliable way of evaluating 3D models, a point can be made about the necessity of testing any reconstruction algorithm in the context of real world objects and real camera imagery to get a real sense of its performance. Our second approach, therefore, is to appropriate one of the many scenes present in DTU Robot Dataset [@Aan:Pedersen:etal:IJCV2012] to the task of evaluating 3D curvilinear reconstructions. This is a significantly harder task than eliminating the surface meshes in the synthetic case, since the ground truth representation is a 3D point cloud, and no explicit distinction is made between curve outlines and surface geometry. We therefore use Blender to project the 3D point cloud ground truth for our selected scene onto several different images, correct for calibration errors to the best of our capacity, then remove all the internal surface points to end up with a subset of 3D points which are in the proximity of curved structures in the scene, **see the full supplementary materials**.
Additional Results {#sec:moreresults}
==================
In this Section we present more detailed figures for our results presented in the main paper, Figure \[fig:cap:draw\]; as well as visual comparisons to the results of PMVS [@Furukawa:Ponce:CVPR2007], Figure \[fig:cap:pmvs\]. **See the full supplementary materials for additional results.**
![ Curve drawing results for the Capitol High dataset. []{data-label="fig:cap:draw"}](figs/capitol-strip-sm.jpg "fig:"){width="\linewidth"} ![ Curve drawing results for the Capitol High dataset. []{data-label="fig:cap:draw"}](figs/capitol-drawing-1.png "fig:"){width="1.1\linewidth"} ![ Curve drawing results for the Capitol High dataset. []{data-label="fig:cap:draw"}](figs/capitol-drawing-2.png "fig:"){width="1.1\linewidth"}
![ Reference PMVS results for the Capitol High dataset. []{data-label="fig:cap:pmvs"}](figs/capitol-pmvs-1-sm.jpg "fig:"){width="\linewidth"} ![ Reference PMVS results for the Capitol High dataset. []{data-label="fig:cap:pmvs"}](figs/capitol-pmvs-2-sm.jpg "fig:"){width="\linewidth"}
Additional Details {#sec:details}
==================
Language
--------
We used C++ to implement the base system up to the enhanced curve sketch, using widely-available open source libraries, such as Boost ([www.boost.org](www.boost.org)), and VXL ([vxl.sourceforge.net](vxl.sourceforge.net)). The curve drawing stage is implemented in Matlab. The experiments ran on Linux but the code is very portable.
Additional Supplementary Material
---------------------------------
Other than this pdf document, **there is a full supplementary materials document**, as well as a supplementary materials package which contains, among others: i) Two mp4 vieos comparing reconstructions of Curve Sketch, Enhanced Curve Sketch, 3D Drawing and PMVS on Amsterdam House Dataset, and ii) A .PLY file which contains the 3D Drawing results on the Amsterdam House Dataset. You can view this model in MeshLab or any other software that supports .PLY file format.
Availability
------------
The C++ and Matlab source code are available at [multiview-3d-drawing.sourceforge.net](multiview-3d-drawing.sourceforge.net), as well as the ground truth datasets and additional supplementary material.
|
---
abstract: |
We characterise asymptotic behaviour of families of symmetric orthonormal polynomials whose recursion coefficients satisfy certain conditions, satisfied for example by the (normalised) Hermite polynomials. More generally, these conditions are satisfied by the recursion coefficients of the form $c(n+1)^p$ for $0<p<1$ and $c>0$, as well as by recursion coefficients which correspond to polynomials orthonormal with respect to the exponential weight $W(x)=\exp(-|x|^\beta)$ for $\beta>1$. We use these results to show that, in a Hilbert space defined in a natural way by such a family of orthonormal polynomials, every two complex exponentials $e_{\omega}(t)={\e}^{\ii \omega t}$ and $e_{\sigma}(t)={\e}^{\ii \sigma t}$ of distinct frequencies $\omega,\sigma$ are mutually orthogonal. We finally formulate a surprising conjecture for the corresponding families of non-symmetric orthonormal polynomials; extensive numerical tests indicate that such a conjecture appears to be true.\
**keywords:** orthogonal polynomials, unbounded recurrence coefficients, Christoffel functions, almost periodic functions, signal processing\
**AMS classification numbers:** 42C05, 41A60, 42A75
address: 'School of Computer Science and Engineering, University of New South Wales, Sydney, Australia; email: ignjat@cse.unsw.edu.au'
author:
- Aleksandar Ignjatović
title: Asymptotic behaviour of some families of orthonormal polynomials and an associated Hilbert space
---
[^1]
[^1]: This paper is dedicated to my wife Sharon Younghi Choi; without her love and patience this work would have never seen daylight. I also want to thank Jeff Geronimo, Doron Lubinsky, Paul Nevai, Vilmos Totik and especially the anonymous referees for their most valuable comments which have greatly improved this article.
|
---
abstract: 'Based on the author’s work in [@BerndPoly] we deduce that the invariant introduced by Bierstone and Milman in order to give a proof for constructive resolution of singularities in characteristic zero can be achieved purely by considering certain polyhedra and their projections.'
address: |
Bernd Schober\
Lehrstuhl Prof. Jannsen\
Fakultät für Mathematik\
Universität Regensburg\
Universitätsstr. 31\
93053 Regensburg\
Germany
author:
- Bernd Schober
title: A polyhedral approach to the invariant of Bierstone and Milman
---
[^1]
Introduction {#Intro .unnumbered}
============
In his celebrated paper [@Hiro64] Hironaka proved the existence of resolution of singularities for arbitrary dimensional algebraic varieties over fields of characteristic zero. The original proof is quite complicated and consists of more than 200 very technical pages. Moreover, the result is not constructive. Nowadays there are quite accessible and constructive proofs available, which are all based on Hironaka’s work. The first results in this direction were published about 25 years ago by Bierstone and Milman [@BMSimpleConstr], [@BMheavy] and Villamayor [@OrlandoConstr], [@OrlandoPatch]. More recent approaches are for example by Bravo, Encinas and Villamayor [@AnaSantiOrlando05], Cutkosky [@CutBook], Encinas and Hauser [@SantiHerwig], Hauser [@HerwigCharZero], Koll[á]{}r [@Kollar] and W[ł]{}odarczyk [@Wlo].
In positive characteristic Abhyankar was the first to show resolution of singularities for surfaces and he also proved the case of dimension 3 if the characteristic of the base field $ k $ is not $ 2, 3 $ or $ 5 $ ($ k $ algebraically closed!). Both results have been simplified by Cutkosky in [@Cut3fold] and [@CutSurf]. (For the precise references to Abhyankar’s original papers see also Cutkosky’s articles).
In the appendix of [@CGO] Hironaka gave an alternative approach to the resolution of hypersurfaces of dimension $ 2 $. There he made intensive use of the characteristic polyhedron of a singularity, which he introduced in [@HiroCharPoly]. Following his strategy Cossart, Jannsen and Saito [@CJS] extended the proof to arbitrary excellent schemes of dimension at most $ 2 $. (This includes in particular the arithmetic case over $ {\mathbb{Z}}$!). Again the characteristic polyhedron played a crucial role. Already several years before Lipman proved resolution of singularities for 2-dimensional excellent schemes in [@LipDim2]. But in contrast to [@CJS] his approach is not only using blow-ups but also normalizations. Therefore it is not clear how Lipman’s proof extends to the case where the scheme is embedded into a regular ambient scheme.
In [@CPmixed] Cossart and Piltant extend their previous work ([@CP1], [@CP2]) and prove the existence of a birational and global resolution in dimension $ 3 $ in the arithmetic case. Since their result is not given by a resolution algorithm, it is not clear that the resolution is achieved purely by blow-ups in regular centers and the problem of embedded resoultion of singularities in dimension $ 3 $ is still open.
There are programs which try to tackle the proof for arbitrary characteristic. But up to now none of them succeeded to show resolution in arbitrary dimension. By using so called alterations de Jong [@deJong] was able to prove a weaker form of resolution in positive characteristic for all dimensions (where the term “birational" has to be replaced by “generically finite").
In this article we focus on Bierstone and Milman’s approach [@BMheavy] to resolution of singularities in characteristic zero. (See also [@BMlight] for the hypersurface case).
Let $ X $ be a scheme of finite type over a field $ k $ of characteristic zero, which is embedded into a regular scheme $ Z $ (also of finite type over $ k $). The strategy for the proof of resolution of singularities in characteristic zero is to define an invariant $ {\mathrm{inv}}_X ( x ) $ for each $ x \in X $, which satisfies the following properties:
- $ {\mathrm{inv}}_X ( x ) $ has values in a totally ordered abelian group and is upper semi-continuous.
- If $ T $ denotes the locus where the maximal value of $ {\mathrm{inv}}_X ( x ) $ on $ Z $ is attained, then there is a canonical way to deduce from $ T $ the center of the next blow-up.
- This center is regular and has at most simple normal crossings with the exceptional divisors obtained by the preceding resolution process.
- After each such blow-up the invariant decreases strictly and after finitely many steps the singularities are resolved.
In order to define $ {\mathrm{inv}}_X ( x ) $ some important tools are needed. The most powerful of those used in the proof is the notion of maximal contact. Roughly speaking, a regular subscheme $ W $ of $ Z $ has maximal contact with $ X $ if its transform $ W ' $ after a sequence of blow-ups (with certain good centers) contains all the points of the transform of $ X $, where the singularities did not improve.
In characteristic zero maximal contact locally always exists. Whereas in positive characteristic there exist examples where maximal contact does not hold, see [@Nara] or Theorem 14.3 in [@CJS].
The notion of maximal contact leads to another important tool, the so called coefficient ideal [with respect to ]{}some regular subscheme $ W $. This is a local construction which yields a restriction to a smaller dimensional ambient scheme so that we can then apply induction on its dimension.
The invariant used in [@BMheavy] has the form $${\mathrm{inv}}_X ( x ) = ( \nu_1, s_1;\, \nu_2, s_2; \, \ldots;\, \nu_t, s_t;\, \nu_{t + 1} ),$$ $ \nu_1 = H_{X,x} $ is the Hilbert-Samuel function of $ X $, $ \nu_i \in {\mathbb{Q}}_ 0 \cup \{ \infty \} $, $ i \geq 2 $, are certain higher order multiplicities (sometimes also called residual orders) and $ s_i \in {\mathbb{Z}}_0 $ counts certain exceptional divisors. The starting point for this investigations has been the following problem, which we formulate here as
\[MainThm:nuPurelyPoly\] There is a purely polyhedral approach for obtaining the numbers $ \nu_i \in {\mathbb{Q}}_ 0 \cup \{ \infty \} $. This means we can get $ \nu_i $ by only considering certain polyhedra.
For our task we need an appropriate language. In this article we use Hironaka’s theory of pairs $ {\mathbb{E}}$ and idealistic exponents $ {\mathbb{E}}_\sim $. Here $ {\mathbb{E}}= ( J, b ) $ denotes a pair consisting of a quasi-coherent ideal sheaf $ J \subset {\mathcal{O}}_Z $ and a positive rational number $ b \in {\mathbb{Q}}_+ $. Two of them can be compared via an equivalence relation $ \sim $ which considers their behavior under blow-ups in permissible center. The latter are regular centers contained in the locus of order at least $ b $. An idealistic exponent $ {\mathbb{E}}_\sim $ denotes then an equivalence class of pairs [with respect to ]{}$ \sim $.
In his thesis [@BerndThesis] the author introduced the notion of characteristic polyhedra of pairs $ {\Delta( \, {\mathbb{E}}\,;\, u \, ) } $ and idealistic exponents [with respect to ]{}a certain system of regular elements $ ( u ) = ( u_1, \ldots, u_e ) $, see also [@BerndPoly] for a shorter account. By using this we obtain the connection to polyhedra. Since the polyhedra are not stable under $ \sim $ (Example \[Ex:PolyNotUnique\]) we investigate the information which we can get out of the polyhedra which is still intrinsic for the idealistic exponent $ {\mathbb{E}}_\sim $. More precisely, we define the number $ \delta_x ( {\Delta( \, {\mathbb{E}}\,;\, u \, ) } ) $ for a singular point $ x $ which recovers the order of the coefficient ideal in this setting (Proposition \[Prop:basicPoly\]).
In order to obtain a canonical resolution of singularities one needs to consider the exceptional divisors which were created under the preceding process; for example, the singular locus of $ X = V ( t^2 + x y z ) $ consists of the curves $ V ( t, x,y ) $, $ V ( t, x, z ) $ and $ V ( t,y, z) $. Since none of them is a better center than the other, we have to blow up the origin in order to obtain a canonical resolution. After blowing up $ V ( t, x,y,z) $ the situation in the $ X $-, the $ Y $- and the $ Z $-chart is the same as before.
To see the whole structure of the singularity it is reasonable to not only consider the upcoming resolution process in the equivalence relation $ \sim $, but also the preceding process (or the history). This leads to the definition of *[pairs with history]{}* $ ( {\mathbb{E}}, {\mathcal{E}}) $ and [[idealistic exponents with history]{}]{} $ ( {\mathbb{E}}, {\mathcal{E}})_{{\sim_{{\mathcal{E}}}}} $ where $ {\mathcal{E}}$ is a map remembering the exceptional divisors and reflecting the factorization of the exceptional components in $ {\mathbb{E}}$. The latter part is done via certain non-negative rational numbers $ ( d_1, \ldots, d_l ) $, which may vary under $ \sim $. In order to obtain intrinsic information, we extend $ \sim $ by additionally fixing these numbers and denote the resulting equivalence relation $ {\sim_{{\mathcal{E}}}}$. *Idealistic exponents with history* are then equivalence classes [with respect to ]{}$ {\sim_{{\mathcal{E}}}}$. After developing this notion in his thesis [@BerndThesis] the author recognized that it is in fact a slight variant of *NC-divisorial exponents* which Hironaka introduced in [@HiroThreeKey].
For a singular point $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $ we define $$\nu_x ( {\mathbb{E}}, {\mathcal{E}}; u ) := \delta_x ( {\mathbb{E}}; u ) - \sum_{i=1}^l d_i,$$ which turns out to be the key ingredient for the invariants $ \nu_i $ appearing in $ {\mathrm{inv}}_X ( x ) $. We have
\[MainThm:nu\] Let $ ({\mathbb{E}},{\mathcal{E}}) = ( \, (J,b),\, {\mathcal{E}}\, ) $ be a [[pair with history]{}]{} on some regular scheme $ Z $, $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $, $ (u, y) = (u_1, \ldots, u_d; y_1, \ldots, y_s) $ a [[r.s.p.]{}]{} such that $ ( y ) $ is part of a generating system of the directrix of $ {\mathbb{E}}$ at $ x $. Let $ {\mathcal{E}}:= {\mathcal{E}}(x) := {\mathcal{E}}_x ({\mathbb{E}}, u, y) $ be some fixed exceptional data of $ {\mathbb{E}}$ on $ V(y) $ at $ x $.
1. Then $ \nu_x ( {\mathbb{E}}; u ;y ) $ is independent of $ ( y ) $ and invariant under $ {\sim_{{\mathcal{E}}(x)}}$. Therefore we may also write $$\nu_x ( \, {\mathbb{E}},\, {\mathcal{E}};\, u \,) \,:=\, \nu_x (\, {\mathbb{E}};\, u ;\,y \,) .$$ and this is an invariant of the [[idealistic exponent with history]{}]{} $ ({\mathbb{E}},{\mathcal{E}})_{{\sim_{{\mathcal{E}}(x)}}} $.
2. Further they determine the entries $ \nu_i $, $ i \geq 2 $, of the invariant $ {\mathrm{inv}}_X ( x ) $ of Bierstone and Milman.
Since the definition of $ {\mathrm{inv}}_X ( x ) $ is quite complicated we mention a method to abbreviate its construction in the final section and further we explain how the generators behave in these steps.
Let us remark that we are considering only the situation in the local ring. The focus of this article lies on the construction of $ {\mathrm{inv}}_X ( x) $. Hence we neither regard extensions of all these constructions to open neighborhoods of $ x $ nor their gluing.
*Acknowledgement:* The results presented here are part of my thesis [@BerndThesis]. I thank my advisors Uwe Jannsen and Vincent Cossart for countless discussions on the topic and all their support. I’m grateful to the Laboratoire de Mathémathiques Versailles for their hospitality during my visit in September 2012 and my stay since October 2014.
Paris and Idealistic exponents
==============================
In this section we briefly introduce the language of pairs and idealistic exponents which we use throughout this paper. Their notion goes back to Hironaka [@HiroIdExp] (see also [@HiroThreeKey]) and has later been refined in different ways (e.g. to basic objects [@AnaSantiOrlando05], presentations [@BMheavy], marked ideals [@Wlo], singular mobile [@HerwigCharZero], or idealistic filtrations [@IFP1]).
Let $ Z $ be a regular irreducible scheme of finite type over $ {\mathbb{Z}}$.
\[Def:idexp\] A [*pair*]{} $ {\mathbb{E}}= (J,b) $ on $ Z $ is a pair consisting of a quasi-coherent ideal sheaf $ J \subset {\mathcal{O}}_Z $ and a positive rational number $ b \in {\mathbb{Q}}_+ $.
We define its order at a point $ x \in Z $ (not necessarily closed) as $$ \_x () = {
------------------------------------- -------------------------------------------
$ \frac{{\mathrm{ord}}_x (J) }{b} $ , if $ {\mathrm{ord}}_x (J) \geq b $ and
\[8pt\] $ 0 $ , else,
------------------------------------- -------------------------------------------
. $$ where $ {\mathrm{ord}}_x (J) = \sup \{ d \in {\mathbb{Z}}_{ \geq 0 } \cup \{\infty\} \mid J_x \subseteq {M}_x^d \} $ (and $ {M}_x $ denotes the maximal ideal in the local ring at $ x $). Further we define the singular locus (or support) of $ {\mathbb{E}}$ as $${\mathrm{Sing\,}}({\mathbb{E}})= \{ x \in Z \mid {\mathrm{ord}}_x ( J ) \geq b \}.$$
If $ Z = {\mathrm{Spec\,}}(R) $ is affine, then we also say $ {\mathbb{E}}$ is a pair on $ R $.
\[Def:perm\] Let $ {\mathbb{E}}= (J, b) $ be a pair on $ Z $. A blow-up $ \pi: Z' \to Z $ with center $ D $ is called [*permissible*]{} for $ {\mathbb{E}}$, if $ D $ is regular and $ D \subseteq {\mathrm{Sing\,}}({\mathbb{E}}) $. The transform of $ {\mathbb{E}}$ is then given by $ {\mathbb{E}}' = (J', b) $, where $ J' $ is defined via $ J {\mathcal{O}}_{Z'} = J' H^b $, where $ H $ denotes the ideal sheaf of the exceptional divisor.
If the blow-up with center $ D $ is permissible for $ {\mathbb{E}}$, then we also say that $ D $ is a permissible center for $ {\mathbb{E}}$.
For the sake of finding a resolution of singularities for a given pair $ {\mathbb{E}}$ it is reasonable to try to compare it with other pairs. Thus the idea of two pairs being equivalent is that they should be treated the same way if they undergo the same resolution process. For the precise definition we need the following
We define a [*local sequence of regular blow-ups (short [LSB]{}) over $ Z $*]{} as a sequence of the form $$\label{eq:LSB}
\begin{array}{ccccccccc}
Z=Z_0 \supset U_0 & \stackrel{\pi_1}{\longleftarrow} & Z_1 \supset U_1 & \stackrel{\pi_2}{\longleftarrow} & \ldots & \stackrel{\pi_{l-1}}{\longleftarrow} & Z_{l-1} \supset U_{l-1} & \stackrel{\pi_l}{\longleftarrow} & Z_l \\[5pt]
\hspace{55pt} \cup && \hspace{30pt} \cup && && \hspace{45pt} \cup \\[5pt]
\hspace{55pt} D_0 && \hspace{30pt} D_1 && \ldots && \hspace{45pt} D_{l-1}
\end{array}$$ where $ l \in {\mathbb{Z}}_+ \cup \{ \infty\} $, each $ U_i \subset Z_i $ is an open subscheme, $ D_i \subset U_i $ is a regular closed subscheme and $ \pi_{i+1}: Z_{i+1} \to U_i $ denotes the blow-up with center $ D_i $, $ 0 \leq i \leq l - 1 $.
Combining this with Definition \[Def:perm\] we say that the [[LSB]{}]{} (\[eq:LSB\]) is permissible for $ {\mathbb{E}}$ if each blow-up $ \pi_{i + 1} $ is permissible for $ {\mathbb{E}}_{ i } $, $ 0 \leq i \leq l - 1 $.
Let $ ( t ) = (t_1, \ldots, t_a) $ be a finite system of indeterminates. Then we use the notation $$Z[t] := Z \times_{\mathbb{Z}}{\mathbb{A}}^a_{\mathbb{Z}}= Z \times_{\mbox{\footnotesize Spec} ({\mathbb{Z}})} {\mathrm{Spec\,}}({\mathbb{Z}}[t]).$$ We consider the pair $ {\mathbb{E}}[t] = (J[t], b) $, where $ J[t] = J {\mathcal{O}}_{Z[t]} $.
\[Def:equiv\] Let $ {\mathbb{E}}_1 = (J_1, b_1) $ and $ {\mathbb{E}}_2 = (J_2, b_2) $ be two pairs on $ Z $. Then we define $${\mathbb{E}}_1 \subset {\mathbb{E}}_2$$ if the following condition holds: $$\label{eq:equiv}
\parbox{12.5cm}{
Let $ ( t ) = ( t_1, \ldots, t_a ) $ be an arbitrary finite system of indeterminates and let $ {\mathbb{E}}_i[t] = (J_i[t], b_i) $, $ i \in \{ 1,2 \} $.
If any {{LSB}} over $ Z[t] $ is permissible for $ {\mathbb{E}}_1[t] $, then it is also permissible for $ {\mathbb{E}}_2[t] $.
}$$ Further we say $ {\mathbb{E}}_1 $ and $ {\mathbb{E}}_2 $ are *equivalent*, $${\mathbb{E}}_1 \sim {\mathbb{E}}_2,$$ if both $ {\mathbb{E}}_1 \subset {\mathbb{E}}_2 $ and $ {\mathbb{E}}_1 \supset {\mathbb{E}}_2 $. By $ {\mathbb{E}}_1 \cap {\mathbb{E}}_2 \sim {\mathbb{E}}_3 $ we mean that a [[LSB]{}]{} over $ Z[t] $ is permissible for $ {\mathbb{E}}_3[t] $ if and only if it is permissible for $ {\mathbb{E}}_1[t] $ and $ {\mathbb{E}}_2 [t] $.
An [*idealistic exponent*]{} $ {\mathbb{E}}_\sim $ is the equivalence class of a pair $ {\mathbb{E}}$.
In other literature pairs are sometimes also called idealistic exponents, e.g. [@HiroThreeKey]. In order to avoid confusion when coming to result and the dependence on the choice of a representant of the equivalence class, we use the original terminology of [@HiroIdExp].
Let us recall some of the properties of pairs.
\[Prop:basicandmore\] Let $ {\mathbb{E}}= (J,b) $ and $ {\mathbb{E}}_i = (J_i,b_i) $, $ i \in \{1, 2, 3, 4 \}$, be pairs on $ Z $.
1. For every $ a \in {\mathbb{Z}}_+ $ we have $ (J^a, a b) \sim (J,b) $.
2. Let $ m \in {\mathbb{Z}}_+ $ with $ b_1 \mid m $ and $ b_2 \mid m $. Then $$(J_1,b_1) \cap (J_2,b_2) \sim \left(J_1^{\frac{m}{b_1}} + J_2^{\frac{m}{b_2}}, m\right).$$
3. We always have $ (J_1 J_2, b_1+ b_2) \supset (J_1,b_1) \cap (J_2,b_2) $. If further $ {\mathrm{Sing\,}}(J_i, b_i + 1) = \emptyset $ for $ i \in \{ 1, 2\} $, then the previous inclusion becomes an equivalence.
4. If $ {\mathbb{E}}_1 \subset {\mathbb{E}}_2 $ and $ {\mathbb{E}}_3 \subset {\mathbb{E}}_4 $, then $ {\mathbb{E}}_1 \cap {\mathbb{E}}_3 \subset {\mathbb{E}}_2 \cap {\mathbb{E}}_4 $. In particular, $ {\mathbb{E}}_1 \sim {\mathbb{E}}_2 $ implies by symmetry $ {\mathbb{E}}_1 \cap {\mathbb{E}}_3 \sim {\mathbb{E}}_2 \cap {\mathbb{E}}_3 $.
5. Let $ \pi : Z' \to Z $ be a permissible blow-up for $ {\mathbb{E}}_1 $ and $ {\mathbb{E}}_2 $. Then $ ({\mathbb{E}}_1 \cap {\mathbb{E}}_ 2)' \sim {\mathbb{E}}'_1 \cap {\mathbb{E}}'_2 $.
6. *(Numerical Exponent Theorem; [@HiroThreeKey], Theorem 5.1)* If $ {\mathbb{E}}_1 \subset {\mathbb{E}}_2 $, then $${\mathrm{ord}}_x ({\mathbb{E}}_1) \leq {\mathrm{ord}}_x ({\mathbb{E}}_2) \hspace{10pt}\mbox{ for all } x \in Z .$$ By symmetry $ {\mathbb{E}}_1 \sim {\mathbb{E}}_2 $ implies $ {\mathrm{ord}}_x ({\mathbb{E}}_1) = {\mathrm{ord}}_x ({\mathbb{E}}_2) $ for all $ x \in Z $. In particular, we get $ {\mathrm{Sing\,}}({\mathbb{E}}_1) = {\mathrm{Sing\,}}({\mathbb{E}}_2) $ if $ {\mathbb{E}}_1 \sim {\mathbb{E}}_2 $.
7. *(Diff Theorem; [@HiroThreeKey], Theorem 3.4)* Let $ {\mathcal{D}}$ be a left $ {\mathcal{O}}_Z $-submodule of $ {\mathrm{Diff}}^{\leq m}_{{\mathbb{Z}}}(Z) $ the (absolute) differential operators of $ {\mathcal{O}}_Z $ (resp. $ R $) on itself of order $ m \in {\mathbb{N}}_0 $. Then $$(J, b) \subset ({\mathcal{D}}J, b - m)$$ or equivalently $ (J, b) \sim ({\mathcal{D}}J, b - m) \cap (J,b) $.
This follows by [@BerndPoly], Lemma 1.6, Proposition 1.8, and Proposition 1.9.
Let $ {\mathbb{E}}= (J, b) $ be a pair on $ Z $ and $ x \in {\mathrm{Sing\,}}({\mathbb{E}}) $. Denote by $ ( R = {\mathcal{O}}_{Z,x}, {M}, {K}= R/{M}) $ the local ring at $ x $ and by $ T_x ( Z ) := {\mathrm{Spec\,}}(S) $, $ S := gr_x (Z) = \bigoplus_{a \geq 0} {M}^a/ {M}^{a+1} $, the tangent space of $ Z $ at $ x $. By abuse of notation we neglect in $ {\mathbb{E}}_x = (J_x, b) $ the index $ x $ and write in the situation on $ R $ also $ {\mathbb{E}}= (J,b) $.
The [*$ b $-initial form*]{} of $ f \in J $ ([with respect to ]{}$ {M}$) is defined as $$ in (f,b) := {
--------------------------- ---------------------------------
$ f \mod {M}^{ b + 1 } $, if $ b \in {\mathbb{Z}}_+ $,
\[5pt\] $ 0 $, if $ b \notin {\mathbb{Z}}_+ $.
--------------------------- ---------------------------------
. $$ Further we define the tangent cone $ {T}_x ({\mathbb{E}}) \subset T_x ( Z ) $ of $ {\mathbb{E}}$ at $ x $ as the subspace generated by the homogeneous ideal $ In_x (J,b) \subset gr_x (Z) $, where $$ In\_x (J,b) := In\_x() := {
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$ \langle J \mod {M}^{ b + 1 } \rangle = \langle in (f, b) \mid f \in J \rangle $, if $ b \in {\mathbb{Z}}_+ $,
\[5pt\] $ \langle 0 \rangle $, if $ b \notin {\mathbb{Z}}_+ $.
------------------------------------------------------------------------------------- ---------------------------------
. $$ For two pairs $ {\mathbb{E}}_1 = (J_1,b_1), {\mathbb{E}}_2 = (J_2,b_2) $ on $ Z $ we set $
In_x({\mathbb{E}}_1 \cap {\mathbb{E}}_2) = In_x({\mathbb{E}}_1) + In_x({\mathbb{E}}_2)
$.
In this setting we can define the directrix and the ridge of the homogeneous ideal $ In_x({\mathbb{E}}) $ which go back to Hironaka and Giraud.
Let $ {\mathbb{E}}= (J,b) $ be a pair on $ Z $. Then we define
1. the directrix of $ {\mathbb{E}}$ at $ x $ by $
{\mathrm{Dir}}_x ({\mathbb{E}}) := {\mathrm{Dir}}_x ({T}_x ({\mathbb{E}}))
$ as the smallest $ K $-subvectorspace $ T = \bigoplus_{ j = 1 }^r K Y_j \subset S_1 = \bigoplus_{ i = 1 }^n K U_i $ generated by elements $ Y_1,\ldots Y_r \in S_1 $ (homogeneous of degree one) such that $$(\, In_x({\mathbb{E}}) \,\cap\, K[Y_1,\ldots,Y_r ] \,)\,S = In_x({\mathbb{E}}) .$$ We call $ I{\mathrm{Dir}}( C ) := \langle Y_1, \ldots Y_r \rangle $ the ideal of the directrix.
2. The ridge (or faîte in French) $
{\mathrm{Rid}}_x ({\mathbb{E}}) := {\mathrm{Rid}}_x ({T}_x ({\mathbb{E}}))
$ of $ {\mathbb{E}}$ at $ x $ is the smallest additive subspace $ K[ \varphi_1, \ldots, \varphi_l ] \subset S $ generated by additive homogeneous polynomials $ \varphi_1, \ldots, \varphi_l \in S $ such that $$(\, In_x({\mathbb{E}}) \,\cap\, K[ \varphi_1, \ldots, \varphi_l ]\,)\,S = In_x({\mathbb{E}}) .$$ (Recall that a polynomial $ \varphi \in K [U] = S $ is called additive if for any $ x, y \in K^n $ we have $ \varphi ( x + y ) = \varphi ( x ) + \varphi ( y ) $). We call $ I{\mathrm{Rid}}( C ) := \langle \varphi_1, \ldots, \varphi_l \rangle $ the ideal of the ridge.
Hence $ Dir_x ({\mathbb{E}}) $ is the minimal $ K $-subspace such that $ In_x({\mathbb{E}}) $ is generated by elements in $ K[ Y_1, \ldots, Y_r ] $. We also say $ ( Y ) = ( Y_1, \ldots, Y_r ) $ defines the directrix and we implicitly assume that $ r $ to be minimal. By abuse of notation we denote the vector space in $ {\mathbb{A}}^n_k = {\mathrm{Spec\,}}( S ) $ corresponding to $ Dir_x ({\mathbb{E}}) $ also by $ Dir_x ({\mathbb{E}}) $.
Similarly we say $ ( \varphi_1, \ldots, \varphi_l) $ defines the ridge and identify $ {\mathrm{Rid}}(C) $ with the group subscheme which it defines in $ {\mathbb{A}}^N_K $
The directrix and the ridge are closely related. For example the previous definitions coincide if we are in the situation over a field of characteristic zero. In general, there exists a purely inseparable finite extension $ K'/K $ such that over $ K' $ the radical of the ideal of the ridge coincides with the ideal of the directrix. For more details on the ridge (and in particular an intrinsic definition) see [@GiraudEtude] and [@ComputeRidge].
In [@BerndPoly] the author introduced the following idealistic interpretation of these objects.
Let $ {\mathbb{E}}= (J,b) $ be a pair on $ Z $ and $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $. Let further $ I{\mathrm{Dir}}_x ( {\mathbb{E}}) = \langle Y_1, \ldots, Y_r \rangle $ and $ I{\mathrm{Rid}}_x ( {\mathbb{E}}) = \langle \varphi_1, \ldots, \varphi_l \rangle $ for elements $ Y_j $ homogeneous of degree one, $ 1 \leq j \leq r $, and additive homogeneous polynomials $ \varphi_i $ of order $ p^{d_i} $, $ 1 \leq i \leq l $. Then we define the following pairs on $ T_x (Z) = {\mathrm{Spec\,}}( gr_x (Z) ) $:
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$ {\mathbb{{T}}}_x ( {\mathbb{E}}) = (\, In_x ( {\mathbb{E}}),\, b \,) $ *(idealistic tangent cone of $ {\mathbb{E}}$ at $ x $),\
$ {\mathbb{D}\mathrm{ir}}_x ( {\mathbb{E}}) = (\, I{\mathrm{Dir}}_x ( {\mathbb{E}}),\, 1 \,) $ & *(idealistic directrix of $ {\mathbb{E}}$ at $ x $),\
$ {\mathbb{R}\mathrm{id}}_x ( {\mathbb{E}}) = \bigcap_{i = 1 }^l (\, \varphi_i ,\, p^{d_i} \,) $ & *(idealistic ridge of $ {\mathbb{E}}$ at $ x $) .***
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If we have two pairs on $ Z $, say $ {\mathbb{E}}_1 = (J_1,b_1) $ and $ {\mathbb{E}}_2 = (J_2,b_2) $, and $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}_1 \cap {\mathbb{E}}_2 ) $, then we set
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$ {\mathbb{{T}}}_x ( {\mathbb{E}}_1 \cap {\mathbb{E}}_2 ) = {\mathbb{{T}}}_x ( {\mathbb{E}}_1 ) \cap {\mathbb{{T}}}_x ({\mathbb{E}}_2 ) = (\, In_x ( {\mathbb{E}}_1 ),\, b_1 \,) \cap (\, In_x ( {\mathbb{E}}_2 ),\, b_2 \,) $,
\[5pt\] $ {\mathbb{D}\mathrm{ir}}_x ( {\mathbb{E}}_1 \cap {\mathbb{E}}_2 ) = {\mathbb{D}\mathrm{ir}}_x ( {\mathbb{E}}_1 ) \cap {\mathbb{D}\mathrm{ir}}_x ( {\mathbb{E}}_2 ) = (\, I{\mathrm{Dir}}_x ( {\mathbb{E}}_1 )+ I{\mathrm{Dir}}_x ( {\mathbb{E}}_2 ),\, 1 \,) $,
\[5pt\] $ {\mathbb{R}\mathrm{id}}_x ( {\mathbb{E}}_1 \cap {\mathbb{E}}_2 ) = {\mathbb{R}\mathrm{id}}_x ( {\mathbb{E}}_1 ) \cap {\mathbb{R}\mathrm{id}}_x ( {\mathbb{E}}_2 ) $.
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Let $ {\mathbb{E}}= (J,b) $, $ {\mathbb{E}}_i = (J_i,b_i) $ be pairs on $ Z $ and $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $. Then we have
1. $ {\mathbb{D}\mathrm{ir}}_x ( {\mathbb{E}}) \subset {\mathbb{R}\mathrm{id}}_x( {\mathbb{E}}) \subset {\mathbb{{T}}}_x({\mathbb{E}}) $.
2. $ {\mathrm{Dir}}_x ({\mathbb{E}}) = {\mathrm{Sing\,}}({\mathbb{D}\mathrm{ir}}_x({\mathbb{E}})) \subseteq {\mathrm{Sing\,}}({\mathbb{R}\mathrm{id}}_x({\mathbb{E}})) \subseteq {\mathrm{Sing\,}}({\mathbb{{T}}}_x({\mathbb{E}})) \subseteq T_x (Z) $.
3. Assume $ char ({K}) = 0 $ or $ b < char ({K}) $, where $ {K}$ denotes the residue field of $ Z $ at $ x $. Then $${\mathbb{D}\mathrm{ir}}_x ( {\mathbb{E}}) \sim {\mathbb{R}\mathrm{id}}_x( {\mathbb{E}}) \sim {\mathbb{{T}}}_x({\mathbb{E}}).$$ In particular, $ {\mathrm{Dir}}_x ({\mathbb{E}}) = {\mathrm{Sing\,}}({\mathbb{D}\mathrm{ir}}_x({\mathbb{E}})) = {\mathrm{Sing\,}}({\mathbb{R}\mathrm{id}}_x({\mathbb{E}})) = {\mathrm{Sing\,}}({\mathbb{{T}}}_x({\mathbb{E}})) $.
4. $ {\mathbb{E}}_1 \subset {\mathbb{E}}_2 $ implies $ {\mathbb{{T}}}_x ({\mathbb{E}}_1) \subset {\mathbb{{T}}}_x ({\mathbb{E}}_2) $, $ {\mathrm{Dir}}_x ({\mathbb{E}}_1) \subseteq {\mathrm{Dir}}_x ({\mathbb{E}}_2) $, and $ {\mathbb{R}\mathrm{id}}_x ({\mathbb{E}}_1) \subset {\mathbb{R}\mathrm{id}}_x ({\mathbb{E}}_2) $. By symmetry we get equivalence $ \sim $ and equality if $ {\mathbb{E}}_1 \sim {\mathbb{E}}_2 $.
These results are proven in [@BerndPoly], Lemma 2.11, Corollary 2.12, and Proposition 2.14.
Another important tool for the study of singularities at a point $ x \in Z $ is the coefficient ideal [with respect to ]{}a closed subscheme of maximal contact. We now recall its variant in the idealistic setting. But we do not restrict our attention to characteristic zero and moreover, we define the coefficient pair [with respect to ]{}any regular subvariety $ W = V(z) = V(z_1, \ldots, z_n ) $ containing $ x $ such that $ (z) $ is part of a [[r.s.p.]{}]{} for the local ring $ R $ of $ Z $ at $ x $.
\[Def:IdCoeffExp\] Let $ {\mathbb{E}}= (J,b) $ be a pair on $ Z $ and $ x \in Z $. Let $ (R = {\mathcal{O}}_{Z,x}, {M}, {K}) $ be the regular local ring of $ Z $ at $ x $. We consider a fixed system of elements $ (u) = (u_1, \ldots, u_d) $ which can be extended to a [[r.s.p.]{}]{} for $ R $. Let $ (z) = ( z_1, \ldots, z_s ) $ be elements of $ R $ such that $ (u,z) $ is a [[r.s.p.]{}]{} for $ R $. We define the [*coefficient pair $ {\mathbb{D}}_x ({\mathbb{E}}, u, z ) $ of $ {\mathbb{E}}$ at $ x $ [with respect to ]{}$ (z) $*]{} as the pair on $ W = {\mathrm{Spec\,}}( {K}[[ u ]] ) $ which is given by the following construction: Any $ f \in J_x $ may be written (in the $ {M}$-adic completion $ \widehat{ R } $) as $$f = f (u,z) = \sum_{B \in {\mathbb{Z}}^s_{ \geq 0 }} f_B (u)\, z^B$$ with $ f_B(u) \in {K}[[u]] $. Then we set $ {\mathbb{D}}(f,u,z) := \bigcap\limits_{\substack{B \in{\mathbb{Z}}^s_{ \geq 0 } \\[3pt] |B| < b}} ( f_B (u) , \, b -|B| ) $ and define further $${\mathbb{D}}_x ({\mathbb{E}}, u, z) := \bigcap_{ f \in J_x } {\mathbb{D}}(f, u, z) = \bigcap_{ l = 0 }^{ b - 1 } \;\; (\; I(l, u, z) ,\; b - l \;),$$ where $ I (l, u, z) = \langle \, f_B \mid f \in J_x, B \in {\mathbb{Z}}_{ \geq 0 }^s : |B| = l \,\rangle $.
Before coming to results on coefficient pairs let us recall the concept of maximal contact. Classical references for this are [@GiraudMaxZero] and [@AHV].
Let $ {\mathbb{E}}= (J,b) $ be a pair on $ Z $ and $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $. Let $ (z) = (z_1, \ldots, z_s) $ be a system of elements in the local ring $ R = {\mathcal{O}}_{Z,x} $ which can be extended to a [[r.s.p.]{}]{} for $ R $. We say $ W := V(z) $ has [*maximal contact with $ {\mathbb{E}}$ at $ x $*]{} if the following equivalence holds $${\mathbb{E}}_x = ( J_x, b ) \sim (z, 1) \cap (J_x,b).$$
\[Prop:moreCoef\] Let $ {\mathbb{E}}= {\mathbb{E}}_1 \subset {\mathbb{E}}_2 $ be two pairs on $ Z $, $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $, and $ ( u ,z ) = (u_1, \ldots, u_d; z_1, \ldots, z_s ) $ be a [[r.s.p.]{}]{} for $ (R = {\mathcal{O}}_{Z,x}, {M}, {K}) $. Then we have
1. $${\mathbb{D}}_x ({\mathbb{E}}_1 , u, z) \subset {\mathbb{D}}_x ({\mathbb{E}}_2 , u, z) .$$ By symmetry, $ {\mathbb{E}}_1 \sim {\mathbb{E}}_2 $ implies $ {\mathbb{D}}_x ({\mathbb{E}}_1 , u, z) \sim {\mathbb{D}}_x ({\mathbb{E}}_2 , u, z) $.
2. $$(z,1) \cap {\mathbb{E}}_x \sim (z,1) \cap {\mathbb{D}}_x ({\mathbb{E}}, u, z)$$
3. Let $ (y) = ( y_1, \ldots, y_s ) $ be another system extending $ ( u ) $ to a [[r.s.p.]{}]{} for $ R $ . Assume $ (z,1) \cap {\mathbb{E}}_x \subset (y,1) \cap {\mathbb{E}}_x $. Then $${\mathbb{D}}_x ({\mathbb{E}}, u, z) \subset {\mathbb{D}}_x ({\mathbb{E}}, u, y) .$$ By symmetry, $ (z,1) \cap {\mathbb{E}}\sim (y,1) \cap {\mathbb{E}}$ implies $ {\mathbb{D}}_x ({\mathbb{E}}, u, z) \sim {\mathbb{D}}_x ({\mathbb{E}}, u, y) $.
4. \[IE:maxContact\] Suppose the images of $ (z) $ in $ {M}/ {M}^2 $ define the $ {\mathrm{Dir}}_x ({\mathbb{E}}) $. Further assume $ char ({K}) = 0 $ or $ b < char ({K}) $.
Then there exists a system $ (y) = (y_1, \ldots, y_s) $ of elements in $ {\widehat{R}}$ such that we have for every $ j \in \{ 1, \ldots, s \} $:
1. The images of $ z_j $ and $ y_j $ in $ {M}/ {M}^2 $ coincide.
2. If we set $ ( \widetilde{u}^ {(j)} ) := (u, y_1, \ldots, y_{j-1}, y_{j+1}, \ldots, y_s ) $, then $${\mathbb{E}}_x \sim (y_j,1) \cap {\mathbb{D}}_x ({\mathbb{E}}, \widetilde{u}^{(j)}, y_j ) .$$ In particular, $ {\mathbb{E}}_x \sim (y,1) \cap {\mathbb{D}}_x ({\mathbb{E}}, u, y ) $, i.e. each $ V ( y_j ) $ (and thus $ V (y_1, \ldots, y_s ) $) has maximal contact with $ {\mathbb{E}}$ at $ x $.
3. There exist $ {\mathcal{D}}_{ j } \in {\mathrm{Diff}}_{K}^{ \leq b - 1 } ( {K}[Y] ) $ and $ F(j) \in In_x ({\mathbb{E}}) $ such that $ {\mathcal{D}}_{ j } (F (j)) = { \epsilon_j } \, Y_j $ for some units $ { \epsilon_j } \in R $. Further there are $ f(j) \in J \widehat{ R } $ which map in $ gr_x (Z) $ to $ F (j) $ and $ ( {\mathcal{D}}'_{ j } ( f (j) ), 1 ) \sim ( y_j, 1 ) $, where $ {\mathcal{D}}'_j $ denotes the differential operator on $ \widehat{ R } $ induced by $ {\mathcal{D}}_j $.
See [@BerndPoly], Theorem 3.2, Corollary 3.3, Proposition 3.4, and Lemma 3.6.
Characteristic polyhedra of pairs and idealistic exponents
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The aim of this article is to deduce the invariant of Bierstone and Milman only by considering certain polyhedra. In this section we recall the authors notion of characteristic polyhedra of pairs resp. idealistic exponents and their properties [@BerndPoly].
Let $ {\mathbb{E}}= ( J, b ) $ be a pair on $ Z $ and $ x \in {\mathrm{Sing\,}}({\mathbb{E}}) $. Denote by $ ( R = {\mathcal{O}}_{Z,x}, {M}, {K}) $ the local ring of $ Z $ at $ x $ and we write $ {\mathbb{E}}= (J,b) $ instead of $ {\mathbb{E}}_x = ( J_x, b ) $.
Fix a system $ (u) = ( u_1, \ldots, u_e)$ of elements in $ {M}$ which can be extended to a [[r.s.p.]{}]{} for $ R $. We consider various choices of a system $ (y) = (y_1, \ldots, y_r) $ such that $ (u, y) $ is a [[r.s.p.]{}]{} for $ R $.
Let $ ( f ) = ( f_1, \ldots, f_m ) $ be a set of generators of $ J $ and consider finite expansions of these $$\label{expansion}
f_i = \sum_{ (A,B) \in {\mathbb{Z}}^n_{ \geq 0 } } C_{A,B,i} \, u^A \, y^B$$ with coefficients $ C_{ A, B,i } \in R^\times \cup \{ 0 \} $.
\[Def:PolyNonIntrinsic\] For the given data we define the [*polyhedron $ \Delta ({\mathbb{E}}, u, y) = \Delta_x ({\mathbb{E}}, u, y) $ of $ {\mathbb{E}}= (J,b) $ at $ x $ [with respect to ]{}$ ( u, y ) $*]{} as the smallest closed convex subset of $ {\mathbb{R}}^e_{ \geq 0 } $ containing all elements of the set $$\left\{ \frac{ A }{ b - |B| } + {\mathbb{R}}^e_{ \geq 0 } \;\bigg|\; 1 \leq i \leq m \,\wedge\, C_{ A, B, i } \neq 0 \,\wedge\, | B | < b \right\}.$$
Let $ {\mathbb{E}}' $ be another pair on $ Z $ with $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}' ) $. Then $ \Delta ({\mathbb{E}}\cap {\mathbb{E}}', u, y) \subset {\mathbb{R}}^e_{ \geq 0 } $ denotes the smallest closed convex subset containing $ \Delta ({\mathbb{E}}, u, y) $ and $ \Delta ({\mathbb{E}}', u, y) $.
As it is shown in [@BerndPoly], Example 4.9, these polyhedra are not necessarily invariant under the equivalence relation $ \sim $. But still we see later how we can get intrinsic information on the idealistic exponent $ {\mathbb{E}}_\sim $ by using them. Since the mentioned example is crucial for our further investigations, we briefly recall it:
\[Ex:PolyNotUnique\] The origin of this example is [@BMexc], Example 5.14, p.788 and it has been slightly modified and worked out for our setting together with Vincent Cossart.
Let $ K = {\mathbb{C}}$, $ d \in {\mathbb{Z}}_+ , d \geq 2 $. We look at the origin of $ {\mathbb{A}}^4_{\mathbb{C}}$. Consider the two pairs $$\begin{array}{l}
{\mathbb{E}}_1 = (z^d - x^{d-1} y^{d-1},\, d) \;\cap\; (t,\, 1) \\[5pt]
{\mathbb{E}}_2 = (z^d - x^{d-1} y^{d-1},\, d) \;\cap\; (t^{d-1}-x^{d-2} y^{d-1},\, d-1 )
\end{array}$$ Then $ {\mathbb{E}}_1 $ and $ {\mathbb{E}}_2 $ are two equivalent pairs whose associated polyhedra differ!
The generating set of the polyhedron associated to $ {\mathbb{E}}_1 $ is $
V_1 = \left\{\, \left(\frac{d-1}{d},\, \frac{d-1}{d}\right) \right\}
$ and the one for $ {\mathbb{E}}_2 $ is $
V_2 = \left\{\, \left(\frac{d-1}{d},\, \frac{d-1}{d}\right);\, \left(\frac{d-2}{d-1},\, 1\right) \right\} .
$ Clearly the polyhedra do not coincide. For the details on the equivalence $ {\mathbb{E}}_1 \sim {\mathbb{E}}_2 $ see [@BerndPoly], Example 4.9.
An important invariant of the singularity of $ {\mathbb{E}}$ at $ x $ is the order of the coefficient pair [with respect to ]{}a system $ ( y ) $ which determines $ {\mathrm{Dir}}_x ( {\mathbb{E}}) $. Using the following definition this can be recovered from the polyhedron $ {\Delta( \, {\mathbb{E}}\, ; \, u \, ;\, y \, ) } $.
\[def:delta(Delta)\] Let $ \Delta \subset {\mathbb{R}}^n_{ \geq 0 } $ be any subset. We define $$\delta ( \Delta ) := \inf \{ \, |v| = v_1 + \ldots + v_n \mid v = (v_1, \ldots, v_n) \in \Delta \, \}.$$ If $ \Delta = \Delta ({\mathbb{E}}, u, y) $, then we set $ \delta_x (\Delta ({\mathbb{E}}, u, y)) := \delta ( \Delta ({\mathbb{E}}, u, y) )$.
Let $ {\mathbb{E}}$ be a pair on $ Z $, $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $, and $ ( u ,y ) $ a [[r.s.p.]{}]{} for $ R = {\mathcal{O}}_{Z,x} $. Then we have
1. The polyhedron $ \Delta ({\mathbb{E}}, u, y) $ associated to $ {\mathbb{E}}$ is a certain projection of the corresponding Newton polyhedron $ \Delta^N ( {\mathbb{E}}, u, y) $, where the latter is generated by the points $ ( A,B ) \in {\mathbb{Z}}^{e+r}_{\geq 0} $.
2. The polyhedron $ \Delta ({\mathbb{E}}, u, y) $ is independent of the chosen set of generators $ ( f ) = ( f_1, \ldots, f_m ) $.
3. The non-negative rational number $
\delta_x ( \Delta ({\mathbb{E}}, u, y) )
$ coincides with the order of the coefficient pair $ {\mathbb{D}}_x ({\mathbb{E}}, u, y) $.
4. Let $ {\mathbb{E}}_2 $ be another pair on $ Z $ which is equivalent to $ {\mathbb{E}}_1 := {\mathbb{E}}$.
1. Then $$\delta_x ( \Delta ({\mathbb{E}}_1, u, y) ) = \delta_x ( \Delta ({\mathbb{E}}_2, u, y) ) .$$
2. Let $ ( u, z ) $ be another choice for the [[r.s.p.]{}]{} and suppose $ (z,1) \cap {\mathbb{E}}\subset (y,1) \cap {\mathbb{E}}$. Then $$\delta_x ( \Delta ({\mathbb{E}}, u, y) ) = \delta_x ( \Delta ({\mathbb{E}}, u, z) ) .$$ This implies in particular that this number is independent of the choice the maximal contact coordinates.
See [@BerndPoly], Proposition 4.3, Corollary 4.4, Lemma 4.6, and Proposition 4.7.
Now we can give the definition of the characteristic polyhedron
Let $ {\mathbb{E}}= (J, b) $ be a pair on $ Z $, $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $, and let $ ( u ) = (u_1, \ldots, u_e ) $ be a system of regular elements that can be extended to a [[r.s.p.]{}]{} of $ R = {\mathcal{O}}_{Z,x} $. We define $${\Delta( \, {\mathbb{E}}\,;\, u \, ) } := \Delta_x (\, {\mathbb{E}}; \,u\,) := \bigcap_{(y)} {\Delta( \, {\mathbb{E}}\, ; \, u \, ;\, y \, ) },$$ where the intersection ranges over all systems $ ( y ) $ extending $ ( u ) $ to a [[r.s.p.]{}]{} of $ R $. We call $ {\Delta( \, {\mathbb{E}}\,;\, u \, ) } $ the [*characteristic polyhedron of the pair $ {\mathbb{E}}$ at $ x $ [with respect to ]{}$ ( u ) $*]{}.
\[Thm:BerndHiro\] Let $ {\mathbb{E}}= (J, b) $ be a pair on $ Z $ and $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $. Denote by $ ( u, y ) = ( u_1, \ldots, u_e; y_1, \ldots, y_r ) $ a [[r.s.p.]{}]{} for $ R = {\mathcal{O}}_{Z,x} $ such that the initial forms of $ ( y ) $ yield the whole directrix $ {\mathrm{Dir}}_x ( {\mathbb{E}}) $.
Then there exist elements $ ( y^* ) = ( y^*_1, \ldots, y^*_r ) $ in $ {\widehat{R}}$ such that $ ( u, y^* ) $ is a [[r.s.p.]{}]{} for $ {\widehat{R}}$, $ ( y^* ) $ yields $ {\mathrm{Dir}}_x ( {\mathbb{E}}) $, and $${\Delta( \, {\mathbb{E}}\, ; \, u \, ;\, y^* \, ) } = {\Delta( \, {\mathbb{E}}\,;\, u \, ) }$$
In [@CSCcompl] Cossart and the author are considering the question in which cases it is possible to attain the characteristic polyhedron without going to the completion.
The assumption in Theorem \[Thm:BerndHiro\] that the initial forms of $ ( y ) = ( y_1, \ldots, y_r ) $ yield the *whole* directrix $ {\mathrm{Dir}}_x ( {\mathbb{E}}) $ is crucial, see [@BerndPoly], Example 5.6. Hence it is in general not possible to make $ {\Delta( \, {\mathbb{E}}\, ; \, u \, ;\, y \, ) } $ (with our definitions) independent of the choice of the system $ ( y ) $ if the assumption on the directrix does not hold. But still we can say something on $ \delta ( {\Delta( \, {\mathbb{E}}\, ; \, u \, ;\, y \, ) } ) $, namely we always have $ \delta ( {\Delta( \, {\mathbb{E}}\, ; \, u \, ;\, y \, ) } ) = 1 $. For the precise statement see the proposition below.
\[Rk:CharPolyId\] As we have seen in Example \[Ex:PolyNotUnique\] $ {\Delta( \, {\mathbb{E}}\, ; \, u \, ;\, y \, ) } $ (and also $ {\Delta( \, {\mathbb{E}}\,;\, u \, ) } $) do not behave well under the equivalence relation $ \sim $. Therefore it is not clear what the characteristic polyhedron of an idealistic exponent $ {\mathbb{E}}_\sim $ should be. In the proposition below we see that we always get the same intrinsic information on $ {\mathbb{E}}_\sim $ by considering $ {\Delta( \, {\mathbb{E}}\,;\, u \, ) } $ for any representant $ {\mathbb{E}}$ of $ {\mathbb{E}}_\sim $. Thus we consider the collection of $ {\Delta( \, {\mathbb{E}}\,;\, u \, ) } $, where representants$ {\mathbb{E}}$ of $ {\mathbb{E}}_\sim $ vary, as the characteristic polyhedra of the idealistic exponent $ {\mathbb{E}}_\sim $. For some more discussion on this see [@BerndPoly], Remark 5.8.
Let us recall some result on the characteristic polyhedra and the information they provide.
\[Prop:basicPoly\] \[Prop:nuandsoon\] Let $ {\mathbb{E}}= (J,b) $ and $ {\mathbb{E}}_i = (J_i,b_i) $, $ i \in \{1, 2 \}$, be pairs on $ Z $ and $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $. Let $ (u,y) = ( u_1,, \ldots, u_{ d };y_1, \ldots, y_s ) $ be a [[r.s.p.]{}]{} for the regular local ring $ (R, {M}, {K}) $ of $ Z $ at $ x $.
1. Suppose $ V ( y ) $ has maximal contact with $ {\mathbb{E}}$ at $ x $. Then the polyhedron $
{\Delta( \, {\mathbb{E}}\, ; \, u \, ;\, y \, ) }
$ is independent of the choice of $ ( y ) $ with this property. This means if $ (z) \subset R $ is another extension of $ ( u ) $ and $ V ( z ) $ has maximal contact, then $${\Delta( \, {\mathbb{E}}\, ; \, u \, ;\, y \, ) } = {\Delta( \, {\mathbb{E}}\, ; \, u \, ;\, z \, ) } .$$
2. We abbreviate the notation by $ \Delta (J,b) := {\Delta( \, (J,b) \, ; \, u \, ;\, y \, ) } $.
1. If $ a \in {\mathbb{Z}}_+ $, then $ \Delta (J,b) = \Delta (J^a, ab) $.
2. Suppose $ b_1, b_2 \in {\mathbb{Z}}_+ $ and let $ m \in {\mathbb{Z}}_+ $ with $ b_1 \mid m $ and $ b_2 \mid m $. Then $$\Delta ( (J_1,b_1) \cap (J_2,b_2) ) = \Delta\left ( J_1^{\frac{m}{b_1}} + J_2^{\frac{m}{b_2}}, m \right).$$
3. Let $ M \in {\mathbb{Z}}^n_{ \geq 0 } $ and $ m := |M| $. Denote by $ {\mathcal{D}}_M \in {\mathrm{Diff}}^{\leq m}_{{K}} \left( \widehat{R} \right) $ the differential operator defined by $ {\mathcal{D}}_M \left( C_D \,(uy)^D \right) = \binom{D}{M} \,C_D\, (uy)^{D-M} $. We set $ {\mathcal{D}}_M^{log} := (uy)^M {\mathcal{D}}_M \in {\mathrm{Diff}}^{\leq m}_{{K}} \left( \widehat{R} \right) $. Then $$\Delta \big( \, (J, b) \cap ( {\mathcal{D}}_M^{log} J, b - m)\,\big) = \Delta ( J, b).$$
3. We define $$\hspace{30pt} \delta_x ( {\mathbb{E}}; u ) := \delta_x ( {\Delta( \, {\mathbb{E}}\,;\, u \, ) } ) = \min \{ |v| = v_1 + \ldots + v_e \mid v \in {\Delta( \, {\mathbb{E}}\,;\, u \, ) } \} \in \frac{1}{b!} \,{\mathbb{Z}}^e_{> 1}$$ Then $ \delta_x ( {\mathbb{E}}; u ) $ does not depend on $ ( y ) $ and is invariant under the equivalence relation $ \sim $. Therefore $ \delta_x ( {\mathbb{E}}; u )_\sim $ is an invariant of the idealistic exponent $ {\mathbb{E}}_\sim $ and $ ( u ) $.
4. Suppose that $ (y, u_{ e + 1 }, \ldots, u_d ) $, $ e < d $, yields the directrix $ {\mathrm{Dir}}_x ( {\mathbb{E}}) $. Then we have $$\delta_x ( \, {\Delta( \, {\mathbb{E}}\, ; \, u_1,\ldots, u_d \, ;\, y_1, \ldots, y_s \, ) } \, ) = 1$$
See [@BerndPoly], Proposition 6.1, Lemma 6.2, Theorem 6.3, and Lemma 6.4.
Now we define the key ingredient for the invariant of Bierstone and Milman.
\[Def:nu\] Let $ {\mathbb{E}}$ be a pair on $ Z $ and $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $. Fix a system of elements $ ( u_1, \ldots, u_d ) $ in $ R = {\mathcal{O}}_{Z,x} $ which can be extended to a [[r.s.p.]{}]{} for $ R $ and let $ ( y ) = ( y_1, \ldots, y_s ) $ be such an extension of $ (u) $.
1. For $ i \in \{ 1, \ldots, d \} $ we define $$d_i ( {\mathbb{E}}; u ;y ) := \min \{ v_i \mid v = ( v_1, \ldots, v_d ) \in {\Delta( \, {\mathbb{E}}\, ; \, u \, ;\, y \, ) } \} \in \frac{1}{b!} \, {\mathbb{Z}}^d_{\geq 0} .$$
2. Consider a subset $ I \subseteq \{ 1, \ldots, d \} $. Then we define $$\nu_I ( {\mathbb{E}}; u; y ) := \delta ( {\mathbb{E}}; u; y ) - \sum_{i \in I } d_i ( {\mathbb{E}}; u; y ) \in \frac{1}{b!}\, {\mathbb{Z}}^d_{\geq 0} .$$
As we see later the connection to the invariant of Bierstone and Milman is given if the subset $ I $ is related to the exceptional divisors of the preceding resolution process.
Example \[Ex:PolyNotUnique\] shows that the non-negative rational number $ \nu_I ( {\mathbb{E}}, u, y ) $ may change under the equivalence relation $ \sim $. Therefore it is not an invariant of the idealistic exponent! In order to take care of this we introduce in the next section idealistic exponents with history.
Pairs and idealistic exponents with history
===========================================
In this we give a refinement of the equivalence relation for pairs in order to make $ \nu_I ( {\mathbb{E}}, u, y ) $ an invariant. As the author discovered later this is a slight variant of so called NC-divisorial exponents which were introduced by Hironaka in [@HiroThreeKey].
To begin with, let us make the following
\[Obs:OneBlowUpExcept\] Let $ {\mathbb{E}}= ( J, b) $ be a pair on $ Z $, $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $ and denote by $ ( u, y ) = (u_1, \ldots, u_d; \, y_1, \ldots, y_s ) $ a [[r.s.p.]{}]{} for $ ( R = {\mathcal{O}}_{Z,x}, {M}, {K}) $ such that $ (y, u_{ e + 1 }, \ldots, u_d ) $, $ e \leq d $, defines the directrix $ {\mathrm{Dir}}_x ( {\mathbb{E}}) $.
1. Suppose that $ V ( y ) $ has maximal contact with $ {\mathbb{E}}$ at $ x $. Let $ D := V(y,u_{i(1)}, \ldots, u_{i(c)} ) $, $ 1 \leq i(1) < i(2) < \ldots < i(c) \leq d $ ($ c \leq d $), be a permissible center for $ {\mathbb{E}}$. Set $$\delta_D ({\mathbb{E}}; u; y) :=
\min \{ v_{i(1)} + \ldots + v_{i(c)} \mid v = ( v_1, \ldots, v_d ) \in \Delta ({\mathbb{E}}, u, y) \}$$ Since $ D $ is permissible for $ {\mathbb{E}}$, we have $ \delta_D ({\mathbb{E}}; u; y) \geq 1 $. By using that $ V ( y ) $ has maximal contact an easy computation shows that after the blow-up with center $ D $ the exceptional component does locally factor to the power $ \delta_D ({\mathbb{E}}; u; y) - 1 \geq 0 $ (for details see [@BerndThesis], Observation 2.5.10).
Suppose the exceptional divisor is locally given by $ V ( u_j ) $, then $$d_{ j } ({\mathbb{E}}'; u'; y') = \delta_D ({\mathbb{E}}; u; y) - 1,$$ where $ {\mathbb{E}}' $ denotes the transform of $ {\mathbb{E}}$ under the blow-up with center $ D $ and $ ( u', y' ) $ denotes a [[r.s.p.]{}]{} at a point after the blow-up with $ u_j' = u_j $.
2. During the resolution process we have to deal with the exceptional divisors $ E = \{ H_1, \ldots, H_l \} $. We require on the resolution algorithm that the divisors associated to $ E $ has at most simple normal crossing singularities, i.e. each each irreducible component is regular and they intersect transversally. By this condition we may suppose that, for every $ i \in \{ 1, \ldots, l \} $, $ H_i $ with $ x \in H_i $ is locally given by $ V ( x_i ) $, where $ ( x ) = ( x_1, \ldots, x_l ) \subset {M}$ is part of a [[r.s.p.]{}]{} for $ R $.
The observation implies the following: If $ I \subseteq \{ 1, \ldots, d \} $ in Definition \[Def:nu\] is the subset determined the exceptional divisors containing the point $ x $, then the numbers $ d_i ( {\mathbb{E}}; u; y ) $ are characterized by the preceding resolution process. Hence $$\nu_I ( {\mathbb{E}}; u; y ) = \delta ( {\mathbb{E}}; u; y ) - \sum_{i \in I } d_i ( {\mathbb{E}}; u; y ) \in \frac{1}{b!} \, {\mathbb{Z}}^d_{\geq 0}$$ is an invariant taking care not only on the upcoming resolution process, but also the preceding one.
This leads to
\[Def:IEH\] Let $ {\mathbb{E}}= (J,b) $ be a pair on $ Z $ and let $ E = \{ H_1, \ldots, H_l \} $, $ l \in {\mathbb{Z}}_+ $, be a set of irreducible divisors on $ Z $ such that the associated divisor has at most simple normal crossing singularities. As usual we denote for $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $ by $ (u, y) = (u_1, \ldots, u_d; y_1, \ldots, y_s) $ a [[r.s.p.]{}]{} for the regular local ring $ ( R = {\mathcal{O}}_{Z,x}, {M}, {K}) $ such that $ (y, u_{ e + 1 }, \ldots, u_d ) $, $ e \leq d $, defines the directrix $ {\mathrm{Dir}}_x ( {\mathbb{E}}) $.
1. We define the [*exceptional data map of $ {\mathbb{E}}$ associated to $ E $*]{} $${\mathcal{E}}:= {\mathcal{E}}_E : {\mathrm{Sing\,}}( {\mathbb{E}}) \to ( E \times {\mathbb{R}}_0 )^l$$ by sending $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $ to the exceptional data of $ {\mathbb{E}}$ on $ V ( y ) $ at $ x $ which is induced by $ E $, $${\mathcal{E}}(x) := {\mathcal{E}}_x ( \,{\mathbb{E}},\, u,\, y \,) := \{ \, (H_1, d_1), \, \ldots, \, (H_l, d_l)\, \},$$ where $ d_i $ is the number associated to $ H_i $ as in Observation \[Obs:OneBlowUpExcept\] if $ x \in H_i $ and $ d_i = 0 $ if $ x \notin H_i $ or $ H_i \supset V ( y ) $.\
We call $
({\mathbb{E}},{\mathcal{E}}) = ((J,b),{\mathcal{E}})
$ a *pair with history* on $ Z $.
2. Let $ {\mathbb{E}}_1 \sim {\mathbb{E}}_2 $ be two [equivalent]{} pairs on $ Z $ and $ E $ as above. Then the induced pairs with history $ ({\mathbb{E}}_1,{\mathcal{E}}_1) $ and $ ({\mathbb{E}}_2,{\mathcal{E}}_2) $ are defined to be [*equivalent at $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}_1 ) = {\mathrm{Sing\,}}( {\mathbb{E}}_2 ) $ [with respect to ]{}$ ( u, y ) $*]{} if $${\mathcal{E}}_1 (x) = {\mathcal{E}}_2 (x), \hspace{10pt} \mbox{ in particular the assigned numbers $ d_j $ coincide. }$$ In this case we write $ {\mathbb{E}}_1 {\sim_{{\mathcal{E}}(x)}}^{(u,y)} {\mathbb{E}}_2 $ or if there is no confusion possible we use only $
{\mathbb{E}}_1 {\sim_{{\mathcal{E}}(x)}}{\mathbb{E}}_2 .
$ Further we say $ ({\mathbb{E}}_1,{\mathcal{E}}_1) $ and $ ({\mathbb{E}}_2,{\mathcal{E}}_2) $ are *equivalent*, if they are equivalent at any $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}_1 ) = {\mathrm{Sing\,}}( {\mathbb{E}}_2 ) $ and we write $ {\mathbb{E}}_1 {\sim_{{\mathcal{E}}}}{\mathbb{E}}_2 $.\
An *[idealistic exponent with history]{}* $ ({\mathbb{E}},{\mathcal{E}})_{{\sim_{{\mathcal{E}}}}} $ denotes the equivalence class of a pair with history $ ({\mathbb{E}},{\mathcal{E}}) $ [with respect to ]{}the equivalence relation $ {\sim_{{\mathcal{E}}}}$.
For applications it is sometimes important to consider at the beginning of the resolution process a pair $ {\mathbb{E}}_0 $ together with a set $ E_0 $ of irreducible divisors which have at most simple normal crossings. Then we obtain already at this point a [[pair with history]{}]{} $ ( {\mathbb{E}}_0 , {\mathcal{E}}_{E_0} ) $ with non-trivial exceptional data.
Since we focus on the construction of the Bierstone-Milman invariant locally at a point $ x $, it suffices for our purposes to consider the equivalence $ {\sim_{{\mathcal{E}}(x)}}$ at $ x $.
Let $ ({\mathbb{E}},\,{\mathcal{E}}={\mathcal{E}}_E) $ (with $ E = \{ H_1, \ldots, H_l \} $) be an [[pair with history]{}]{} on $ Z $ as in the previous definition. A blow-up $ \pi : Z' \to Z $ with center $ D \subset Z $ is called [*permissible for $ ({\mathbb{E}},{\mathcal{E}}) $*]{}, if the following conditions hold:
1. $ \pi $ is permissible for $ {\mathbb{E}}$ in the sense of Definition \[Def:perm\] ($ D $ is regular and $ D \subseteq {\mathrm{Sing\,}}( {\mathbb{E}}) $).
2. $ D \cup H_1 \cup \cdots \cup H_l \subset Z $ has at most simple normal crossing singularities.
The transform of $ ({\mathbb{E}}, {\mathcal{E}}) $ under a permissible blow-up $ \pi $ is given by $ ( {\mathbb{E}}', {\mathcal{E}}' ) $, where $ {\mathbb{E}}' $ denotes the transform of $ {\mathbb{E}}$ under $ \pi $ and the exceptional data map $ {\mathcal{E}}' $ is defined by $ E' := \{ H'_1, \ldots, H'_l, H_{l + 1 } \} $ — here $ H'_i $ is the transform of $ H_i $ under the blow-up $ \pi $ and $ H_{ l + 1 } $ is the exceptional divisor corresponding to $ \pi $.
Analogous to before this leads to the definition of a [[LSB]{}]{} which is permissible for a given [[pair with history]{}]{}.
Let us have another look at Example \[Ex:PolyNotUnique\]. Suppose $ V ( x ) $ is exceptional. Then we get that the assigned number is $\, \dfrac{d - 1}{d} \,$ for $ {\mathbb{E}}_1 $ and it is $\, \dfrac{ d - 2}{d - 1}\, $ for $ {\mathbb{E}}_2 $. Hence they are not equivalent as [[pairs with history]{}]{}, because they have different exceptional data.
Therefore the Diff Theorem, as it is stated in Proposition \[Prop:basicandmore\], is in general not true for the equivalence relation $ {\sim_{{\mathcal{E}}(x)}}$. A weaker version, which is still valid for [[idealistic exponents with history]{}]{}, is given in Lemma \[Lem:BasicHist\].
\[Prop:nuinvariant\] \[Theorem \[MainThm:nu\] part (1)\] Let $ ({\mathbb{E}},{\mathcal{E}}) = ((J,b),{\mathcal{E}}) $ be an [[pair with history]{}]{} on $ Z $, $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $, $ (u, y) = (u_1, \ldots, u_d; y_1, \ldots, y_s) $ a [[r.s.p.]{}]{} for $ ( R = {\mathcal{O}}_{Z,x}, {M}, {K}) $, and let $ {\mathcal{E}}:= {\mathcal{E}}(x) := {\mathcal{E}}_x ({\mathbb{E}}, u, y) $ be some fixed exceptional data of $ {\mathbb{E}}$ on $ V(y) $ at $ x $.
Then $ \nu_x ( {\mathbb{E}}; u ;y ) $ is independent of $ ( y ) $ and invariant under $ {\sim_{{\mathcal{E}}(x)}}$. Therefore we may also write $$\nu_x ( \, {\mathbb{E}},\, {\mathcal{E}};\, u \,) \,:=\, \nu_x (\, {\mathbb{E}};\, u ;\,y \,) .$$ and this is an invariant of the [[idealistic exponent with history]{}]{} $ ({\mathbb{E}},{\mathcal{E}})_{{\sim_{{\mathcal{E}}(x)}}} $.
By Proposition \[Prop:nuandsoon\](3) $ \delta_x ( {\mathbb{E}}; u ) $ is independent of $ (y) $ and invariant under $ \sim $. Hence it is also invariant under $ {\sim_{{\mathcal{E}}(x)}}$. Further the exceptional data $ {\mathcal{E}}= \{ \, (H_1, d_1), \, \ldots, \, (H_l, d_l)\, \} $ is fixed under $ {\sim_{{\mathcal{E}}(x)}}$. The definition $ \nu_x ( {\mathbb{E}}; u ;y ) = \delta_x ( {\mathbb{E}}; u ) - \sum_{ i = 1 }^l d_i $. implies the assertion.
Let us see which of the properties stated in section 1 for $ \sim $ survive under the refined equivalence $ {\sim_{{\mathcal{E}}(x)}}$.
\[Lem:BasicHist\] Let $ {\mathbb{E}}= (J,b) $ be a pair on $ Z $, $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}) $, $ (u, y) $ a [[r.s.p.]{}]{} for $ ( R = {\mathcal{O}}_{Z,x}, {M}, {K}) $ as before, and let $${\mathcal{E}}( x )= {\mathcal{E}}_x ( \,{\mathbb{E}},\, u,\, y \,) = \{ \, (H_1, d_1), \, \ldots, \, (H_l, d_l)\, \}$$ be some fixed exceptional data of $ {\mathbb{E}}$ on $ V(y) $ at $ x $.
1. For every $ a \in {\mathbb{Z}}_+$ we have $ (J^a, a b) {\sim_{{\mathcal{E}}(x)}}(J,b) $.
2. Suppose there is another choice for $ ( y ) $, say $ ( z ) = (z_1, \ldots, z_s ) $, such that $ (z,1) \cap {\mathbb{E}}{\sim_{{\mathcal{E}}(x)}}(y,1) \cap {\mathbb{E}}$. Then $${\mathbb{D}}_x ({\mathbb{E}}, u, z) {\sim_{{\mathcal{E}}(x)}}{\mathbb{D}}_x ({\mathbb{E}}, u, y)$$ (both times with the induced exceptional data on $ V(y) $ and $ V(z) $).
3. If $ char ({K}) = 0 $ or if $ b < char ({K}) $, then there exists a choice for the system $ (y)= ( y_1, \ldots, y_s ) $ such that $${\mathbb{E}}_x {\sim_{{\mathcal{E}}(x)}}(y,1) \cap {\mathbb{D}}_x ({\mathbb{E}}, u, y )$$ (with the induced exceptional data on $ V(y) $).
4. Let $ {\mathcal{D}}_{ M, u }^{ log } = u^M {\mathcal{D}}_{ M, u } \in {\mathrm{Diff}}^{\leq m}_{{K}} \left( {K}[[u, y]] \right) $, $ M = (M_1, \ldots, M_d ) \in {\mathbb{Z}}^d_0 $ with $ |M| = m $, be the logarithmic differential operators given by $${\mathcal{D}}_{ M, u }^{log} \left( C_{A,B} \,u^A y^B \right) = \binom{A}{M} C_{A,B} \,u^{A} y^B.$$ Then $$(J, b) \cap ( {\mathcal{D}}_{ M, u }^{log} J, b - m) {\sim_{{\mathcal{E}}(x)}}(J, b)$$ (with the induced exceptional data on $ V(y) $). Moreover, if $ M_i = 0 $ for all $ i \in \{ 1, \ldots, d \} $ with $ d_i \neq 0 $ in $ {\mathcal{E}}( x ) $, then the analogous statement is true for $ {\mathcal{D}}_{ M, u } $.
Let $ {\mathbb{E}}_1 = (J_1, b_1) $, $ {\mathbb{E}}_2 = ( J_2, b_2 ) $ be two pairs on $ Z $ such that both are equipped with the same exceptional data $ {\mathcal{E}}_1 ( x ) = {\mathcal{E}}_2 ( x ) = {\mathcal{E}}( x) $ on $ V( y ) $ at $ x $ . Suppose $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}_1 ) \cap {\mathrm{Sing\,}}( {\mathbb{E}}_2 ) $.
1. Assume $ b_1, b_2 \in {\mathbb{Z}}_+ $ and let $ m \in {\mathbb{Z}}_+ $ be a positive integer such that $ b_1 \mid m $ and $ b_2 \mid m $. Then $ (J_1,b_1) \cap (J_2,b_2) {\sim_{{\mathcal{E}}(x)}}\left(J_1^{\frac{m}{b_1}} + J_2^{\frac{m}{b_2}}, m\right) $.
2. $ {\mathbb{E}}_1 {\sim_{{\mathcal{E}}(x)}}{\mathbb{E}}_2 $ implies
1. $ {\mathbb{E}}_1 \cap {\mathbb{E}}{\sim_{{\mathcal{E}}(x)}}{\mathbb{E}}_2 \cap {\mathbb{E}}$.
2. $ {\mathrm{ord}}_z ({\mathbb{E}}_1) = {\mathrm{ord}}_z ({\mathbb{E}}_2) $ for all $ z \in Z $. In particular, $ {\mathrm{ord}}_x ({\mathbb{E}}_1) = {\mathrm{ord}}_x ({\mathbb{E}}_2) $.
3. $ {\mathrm{Sing\,}}({\mathbb{E}}_1) = {\mathrm{Sing\,}}({\mathbb{E}}_2) $.
4. $ {\mathbb{{T}}}_x ({\mathbb{E}}_1) {\sim_{{\mathcal{E}}(x)}}{\mathbb{{T}}}_x ({\mathbb{E}}_2) $, $ {\mathbb{D}\mathrm{ir}}_x ({\mathbb{E}}_1) {\sim_{{\mathcal{E}}(x)}}{\mathbb{D}\mathrm{ir}}_x ({\mathbb{E}}_2) $ and $ {\mathbb{R}\mathrm{id}}_x ({\mathbb{E}}_1) {\sim_{{\mathcal{E}}(x)}}{\mathbb{R}\mathrm{id}}_x ({\mathbb{E}}_2) $.
5. $ {\mathbb{D}}_x ({\mathbb{E}}_1 , u, y) {\sim_{{\mathcal{E}}(x)}}{\mathbb{D}}_x ({\mathbb{E}}_2 , u, y) $ and $ (y,1) \cap {\mathbb{E}}{\sim_{{\mathcal{E}}(x)}}(y,1) \cap {\mathbb{D}}_x ({\mathbb{E}}, u, y) $ (both with the induced exceptional data on $ V(y) $ at $ x $).
These are easy consequences of Proposition \[Prop:basicandmore\], Proposition \[Prop:moreCoef\], Proposition \[Prop:basicPoly\] and study of the behavior of the exceptional data. For more details see [@BerndThesis], Lemma 2.6.7.
Further the behavior under permissible blow-ups is interesting for us.
Let $ {\mathbb{E}}_1 = (J_1, b_1) $, $ {\mathbb{E}}_2 = ( J_2, b_2 ) $ be two pairs on $ Z $, $ x \in {\mathrm{Sing\,}}( {\mathbb{E}}_1 ) \cap {\mathrm{Sing\,}}( {\mathbb{E}}_2 ) $ and $ (u, y) $ a [[r.s.p.]{}]{} for $ ( R = {\mathcal{O}}_{Z,x}, {M}, {K}) $. Suppose both have the same exceptional data $${\mathcal{E}}( x ) = \{ \, (H_1, d_1), \, \ldots, \, (H_l, d_l)\, \}$$ on $ V(y) $ at $ x $. Let $ \pi: Z' \to Z $ be a blow-up which is permissible for both pairs and $ x' \in {\mathrm{Sing\,}}( {\mathbb{E}}'_1 ) \cap {\mathrm{Sing\,}}( {\mathbb{E}}'_2 ) $ with $ \pi ( x') = x $. Then we have
1. $ ({\mathbb{E}}_1 \cap {\mathbb{E}}_ 2)' {\sim_{{\mathcal{E}}(x')}}{\mathbb{E}}'_1 \cap {\mathbb{E}}'_2 $.
2. $ {\mathbb{E}}_1 {\sim_{{\mathcal{E}}(x)}}{\mathbb{E}}_2 $ implies $ {\mathbb{E}}_1' {\sim_{{\mathcal{E}}(x')}}{\mathbb{E}}_2' $.
3. $ {\mathbb{E}}_1 {\sim_{{\mathcal{E}}(x)}}{\mathbb{E}}_2 $ is stable under extensions by $ {\mathbb{A}}^a_k $ ($ a \in {\mathbb{Z}}_+ $), i.e. stable under extensions of the [[r.s.p.]{}]{} by systems $ ( t ) = (t_1, \ldots, t_ a) $ corresponding to $ {\mathbb{A}}^a_k $
If $ x $ is not contained in the center of the blow-up $ \pi $, then the situation at $ x' $ did not change. In this case the lemma is trivially true. Thus let us assume that the center contains $ x $. Since the exceptional data are equal for $ {\mathbb{E}}_1 $ and $ {\mathbb{E}}_2 $, the same is true for $ {\mathbb{E}}_1[t] $ and $ {\mathbb{E}}_2 [t] $. Further it follows that the exceptional data for $ {\mathbb{E}}_1' $, $ {\mathbb{E}}'_2 $, $ ({\mathbb{E}}_1 \cap {\mathbb{E}}_ 2)' $ and $ {\mathbb{E}}'_1 \cap {\mathbb{E}}'_2 $ are always given by the transform of $ {\mathcal{E}}( x ) $. Hence the first part follows by Proposition \[Prop:basicandmore\]. The definition of $ \sim $ implies remaining two parts.
Setup
=====
Before we come to the construction of the invariant of Bierstone and Milman, we want to recall the setup which is stated at the beginning of [@BMheavy].
\[SetupB\] Let
- $ X $ be a scheme contained in a regular scheme $ Z $ of finite type over field $ k $ of characteristic zero, $ char ( k ) = 0 $, and $ x \in X $.
- Denote by $ ( R = {\mathcal{O}}_{Z,x}, {M}, {K}) $ the regular local ring at $ x $ and by $ J \subset R $ the ideal defining $ X $ locally at $ x $.
- Let $ ( u, y ) = (u_1, \ldots, u_e; y_1, \ldots, y_r ) $ be a [[r.s.p.]{}]{} for $ R $ such that the images of $ ( y ) $ in $ {M}/ {M}^2 $, $ ( Y ) = ( Y_1, \ldots, Y_r) $, define $ {\mathrm{Dir}}_x ( X ) $. ($ {\mathrm{Dir}}_x ( X ) $ denotes the directrix associated to the tangent cone $ T_x ( X ) $ whose defining ideal is $ \langle in_{M}( g ) \mid g \in J \rangle $).
- Let $ ( f ) = (f_1, \ldots, f_m ) $ be a normalized $ ( u ) $-standard base of $ J $ (for the definition see for example, [@BerndThesis], Definition 2.2.16(4)) and $ b_i = {\mathrm{ord}}_x ( f_i ) $, $ 1 \leq i \leq m $.
- We associate to this the pair $ {\mathbb{E}}$ on $ R $, $${\mathbb{E}}:= ( f_1, b_1 ) \cap \ldots ( f_m, b_m ).$$
- Further we choose $ ( y ) $ such that $ V(y) $ has maximal contact with $ {\mathbb{E}}$ at $ x $ and we may assume that $ R $ is complete — if not, then we pass to the $ {M}$-adic completion $ \widehat{ R } $ of $ R $.
——————————
For our purpose it is not crucial to give the precise definition of a normalized $ ( u ) $-standard basis. Therefore we skip this quite technical definition and only remark that these are generators $ ( f ) = ( f_1, \ldots, f_m ) $ of $ J $ such that $ f_i \notin \langle u \rangle $, ordered by the order of $ f \mod \langle u \rangle $, and moreover $ m $ is as small as possible.
In [@BerndThesis], Lemma 3.1.5, the author proved that the conditions of Setup \[SetupB\] imply the original assumptions of Bierstone and Milman [@BMheavy]. Since this part is not essential for the construction of the invariant we do not recall all the technical notation and refer the reader to [@BerndThesis], section 3.1.
The case without exceptional divisors
=====================================
Now we come to the description of the procedure which Bierstone and Milman use to determine their invariant $ {\mathrm{inv}}_X ( x ) $. For the hypersurface case see [@BMlight] and for the general case see [@BMheavy].
First, we do the easier case without considering the exceptional components and after that we investigate the general case.
Let $ k $, $ X \subset Z $, $ x \in X $, $ J \subset R $, $ ( u, y ) = (u_1, \ldots, u_e; y_1, \ldots, y_r ) $, $ ( f ) = (f_1, \ldots, f_m ) $ and $ {\mathbb{E}}= ( f_1, b_1 ) \cap \ldots ( f_m, b_m ) $ be as in Setup \[SetupB\].
\[Constr:NoExc\] Let the situation be as in Setup \[SetupB\]. For the moment let us forget about $ (u,y) $ and consider an arbitrary [[r.s.p.]{}]{} $ ( w ) = ( w_1, \ldots, w_n) $ for $ R $. Locally at $ x $ the scheme $ X $ is given by the pair $ {\mathbb{E}}$. We define $${\mathcal{G}}_1 ( x ) := {\mathbb{E}}= ( f_1, b_1 ) \cap \ldots ( f_m, b_m ) .$$ Choose $ i_0 \in \{ 1,\ldots, m\} $. Then $ {\mathrm{ord}}_{x} (f_{i_0}) = b_{i_0} $ and for simplicity we write $ (f, b) $ instead of $ (f_{i_0}, b_{i_0}) $. After a linear coordinate change we may assume that $\dfrac{ \partial^{b} f }{ \partial w_{n}^{b} } \neq 0$. Set $ N_1( x ) := V( z_1 ) $, where $$z_1 : = \frac{ \partial^{ b -1 } f }{ \partial w_{ n }^{ b-1 } }.$$ Then by the Diff Theorem, Proposition \[Prop:basicandmore\], $${\mathcal{G}}_1 ( x ) \sim {\mathcal{G}}_1 ( x ) \cap ( z_1, 1 )$$ and $ N_1 ( x ) $ has maximal contact with $ {\mathcal{G}}_1 ( x ) $ at $ x $. After another coordinate change we may suppose that $ w_n = z_1 $.
In the next step we consider the situation on $ N_1 ( x ) $, where we define the pair $ {\mathcal{H}}_1 ( x ) $ (on $ N_1 ( x ) $) by $${\mathcal{H}}_1 ( x ) := \bigcap_{i = 1}^m \; \bigcap_{ l := l(i) = 0 }^{ b_i - 1 } \, \left( \, \frac{ \partial^l f_i }{ \partial w_n^l } \Big|_{ V( z_1 ) }, \, b_i - l \, \right) .$$ This is the coefficient pair of $ {\mathcal{G}}_1 ( x ) $ [with respect to ]{}$ ( z_1 ) $ (Definition \[Def:IdCoeffExp\]). Then set $$\mu_2 := \mu_2 ( x ) :=
{\mathrm{ord}}_{x} ( \, {\mathcal{H}}_1 ( x ) \, ).$$ and in the case without looking at exceptional components $ \nu_2 := \nu_2 ( x ) := \mu_2 ( x ) $. Further we define $${\mathcal{G}}_2 ( x ) \,:=
\, \bigcap_{ j = 1 }^{m^{(2)}} \, (g_j,b_j) \,:=\, \bigcap_{ (h,b_h) \supset {\mathcal{H}}_1 (x) } \, \left( h, b_h \nu_2 \right) .$$ This completes the first step of the process (without exceptional divisors). Then we start again with $ {\mathcal{G}}_2 ( x) $ instead of $ {\mathcal{G}}_1 ( x ) $. By construction there exists $ j_0 \in \{1, \ldots, m^{(2)} \} $ such that $ {\mathrm{ord}}_{ x } ( g_{j_0} ) = b_{j_0} $.
If we start with $ ( u, y ) $ and not an arbitrary [[r.s.p.]{}]{} $ ( w ) $, then we can make our choices such that after $ r $ steps in the process $ z_j = y_j $ for all $ 1 \leq j \leq r $. By construction $ {\mathcal{H}}_r ( x ) = {\mathbb{D}}_{ x } ( {\mathbb{E}}, u, y ) $ and thus $ \nu_{ r + 1} = \mu_{ r + 1} = \delta_{ x } ( {\mathbb{E}}, u ) $.
Since $ \mu_{s + 1} ( x ) = {\mathrm{ord}}_{x} ( \, {\mathcal{H}}_s ( x ) \, ) = \delta_{x} ({\mathcal{G}}_1 ( x ); w_1, \ldots, w_{ n -s } ; z_1, \ldots, z_s ) $, we also get $ \mu_{s + 1 } ( x ) = 1 = \delta_{ x } ( {\mathbb{E}}; w_1, \ldots, w_{ n- s } ) $ if $ s < r $ or equivalently if $ d := n- s > e $ (Proposition \[Prop:nuandsoon\]).
After $ r $ steps we start over with $ {\mathcal{G}}_{r + 1 } ( x ) $ instead of $ {\mathcal{G}}_1 ( x ) $. We determine the directrix $ {\mathrm{Dir}}_x ( {\mathcal{G}}_{ r + 1 } ( x ) ) $, distinguish $ ( u ) = (u_1, \ldots, u_e ) $ into $$( u ) = ( \,u_1,\, \ldots,\, u_{ e^{ (2) } };\, y_{ r + 1 },\, \ldots,\, y_{ r^{ ( 2 ) } }\, ),$$ and so on.
This leads to
\[Obs:InvZero\] Let the situation be as in Setup \[SetupB\]. As in Construction \[Constr:NoExc\] set $ {\mathcal{G}}_1 ( x ) = ( f_1, b_1 ) \cap \ldots ( f_m, b_m ) $. In the case without exceptional components the invariant of Bierstone and Milman has the following form $$\begin{array}{cl}
{\mathrm{inv}}_X ( x )
&
= ( \nu_1, s_1;\, \nu_2, s_2; \, \ldots) =
\\[5pt]
&
= ( \nu_1, 0;\, 1, 0;\, \ldots;\, 1, 0;\, \nu_{ r^{(1)} + 1 }, 0;\, 1, 0;\, \ldots;\, 1, 0;\, \nu_{ r^{ ( 2 ) } + 1 }, 0;\, 1, 0;\, \ldots),
\end{array}$$ where $ r^{ ( 1 ) } := r $ and $ r^{ ( q ) } $, $ q \geq 3 $, is defined in the analogous way as $ r^{ ( 2 ) } $ in the previous remark. Set $ r^{ ( 0 ) } := 0 $. Then we have $ 0 = r^{ ( 0 ) } < r^{ ( 1 ) } < r^{ ( 2 ) } < \ldots \leq n $ and, for all $ q \geq 1 $, $ (Y_{ r^{ ( q - 1 ) } + 1 }, \ldots, Y_{ r^{ ( q ) } } ) $ yields $ {\mathrm{Dir}}_x (\, {\mathcal{G}}_{ r^{ ( q - 1 ) } + 1 } ( x ) \,) $ and with $ e^{ ( q ) } := n - r^{ ( q ) } $ we get $$\nu_{ r^{ ( q ) } + 1 } \,=\, \delta_{ x } (\, {\mathcal{G}}_{ r^{ ( q - 1 ) } + 1 } ( x ) ; \, u_1, \ldots, u_{ e^{ (q ) } } \, ) \, > \, 1.$$
Recall that we have already shown that $
\delta_{ x } (\, {\mathcal{G}}_{ r^{ ( q - 1 )} + 1 } ( x ) ; \, u_1, \ldots, u_{ e^{ (q ) } } \, )
$ is coming from some polyhedra and does neither depend on the choice of the representative for $ {\mathcal{G}}_{ r^{ ( q - 1 ) } + 1 } ( x ) $ as idealistic exponent nor on the choice of $ ( y ) $ (Proposition \[Prop:nuandsoon\]).
Putting everything together we get the following proposition, which implies Main Theorem \[MainThm:nuPurelyPoly\] in the special case without exceptional divisors.
\[Prop:MainThmNoExc\] Let the data be as in Setup \[SetupB\] and use the notation of Observation \[Obs:InvZero\]. Let $ J_{ r^{ ( q - 1 )} + 1 } \subset {K}[[ u_1, \ldots, u_{ e^{ ( q - 1 ) } } ]] $ be the ideal corresponding to $ {\mathcal{G}}_{ r^{ ( q - 1 )} + 1 } ( x ) $, $ q \geq 1 $. Let $ ( g ) = ( g_1, \ldots, g_l ) $ ($ l \in {\mathbb{Z}}_+ $) denote the generators of $ J_{ r^{ ( q - 1 )} + 1 } $ which we get from $ ( f ) = (f_1, \ldots, f_m ) $ via Construction \[Constr:NoExc\]. Set $ d_i := {\mathrm{ord}}_x ( g_i ) $ for $ 1 \leq i \leq l $.
For every $ i \in \{ 1, \ldots, l \} $, $ g_i $ has an expansion of the form $$\label{eq:ExpanFi}
g_i = g_i ( u^{ (q) }, y^{(q)} ) = G_i( y^{(q)} )+ \sum_{|B| < d_i} G_{B,i}( u^{ (q) } ) \cdot ( y^{(q)} )^B + g_i^\ast (u^{ (q) }, y^{ (q) } ),$$ where $ ( u^{ ( q ) }, y^{ ( q ) } ) = ( u_1, \ldots, u_{ e^{ ( q ) } } ; y_{ r^{ ( q - 1 ) } + 1 }, \ldots, y_{ r^{ ( q ) } } ) $, $ B \in {\mathbb{Z}}^{ r^{ ( q ) } - r^{ ( q - 1 ) } }_0 $ and with certain elements $ g_i^\ast (u,y) \in \langle y^{ ( q ) } \rangle^{ d_{ i } + 1 } $,
1. $ G_i ( y^{ ( q ) } ) \in {K}[ y^{ ( q ) } ] $ is a polynomial homogeneous of degree $ d_i $ and
2. $ G_{B,i}( u^{ ( q ) } ) \in {K}[[ u^{ ( q ) } ]] $ has order $ {\mathrm{ord}}_x ( \,G_{ B, i } \, ) > d_i -|B|$ at $ x $.
Further we have the properties (always $ 1 \leq i \leq l $ and $ B := B(i)\in{\mathbb{Z}}^{ r^{ ( q ) } - r^{ ( q - 1 ) } }_0 $)
1. $ {\mathcal{H}}_{r^{ ( q ) } } ( x ) = \left\{\, \left(\, G_{B,i}( u^{ ( q ) } ),\; d_i -|B| \,\right) \;\mid\; i, B : |B| < d_i \,\right\} $,\
$ {\mathcal{G}}_{ r^{ ( q ) } + 1 } ( x ) = \left\{\, \left(\, G_{B,i}( u^{ ( q ) } ),\; ( d_i -|B|)\cdot \delta^{ ( q ) } \,\right) \;\mid\; i, B : |B| < d_i \,\right\} $,\
$ \nu_{ r^{ ( q ) } + 1 } = \min \left\{\, \dfrac{{\mathrm{ord}}_x (\,G_{B,i} \,)}{ d_i -|B|} \,\;\Big|\,\; i, B : |B| < d_i \,\right\} = \delta^{ ( q ) }> 1 $,\
where $ \delta^{ ( q ) } := \delta_x ( \, {\mathcal{G}}_{ r^{ ( q - 1 ) } + 1 } ( x ),\, u^{ ( q ) } \, ) = \delta (\; \Delta_{ x }(\, {\mathcal{G}}_{ r^{ ( q - 1 ) } + 1 } ( x ) ;\, u^{ ( q ) } \, )\; ) $.
2. The polyhedron $ \Delta_{ x } (\, {\mathcal{G}}_{ r^{ ( q ) } + 1 } ( x ) ; u^{ ( q + 1 ) } \, ) = \Delta_{ x } (\, {\mathcal{G}}_{ r^{ ( q ) } + 1 } ( x ) ; \, u_1, \ldots, u_{ e^{ (q + 1 ) } } \, ) $ is a projection of $ \Delta_{ x } (\, {\mathcal{G}}_{ r^{ ( q - 1 ) } + 1 } ( x ) ; u^{ ( q ) } \, ) = \Delta_{ x } (\, {\mathcal{G}}_{ r^{ ( q - 1 ) } + 1 } ( x ) ; \, u_1, \ldots, u_{ e^{ (q ) } } \, ) $.
Let $ s \in {\mathbb{Z}}_ + $ with $ r^{ ( q - 1 ) } < s < r^{ ( q ) } $. We set $ ( u^{ ( q , s ) } ) := ( u^{ ( q ) }, y_{ s + 1 }, \ldots, y_{ r^{ ( q ) } } ) $ and $ ( y^{ ( q , s ) } ) := ( y_{ r^{ ( q - 1 ) } + 1 }, \ldots, y_s ) $. Then the statements analogous to (\[eq:ExpanFi\]) and (1)–(4) are true for $ ( u^{ ( q , s ) }, y^{ ( q , s ) } ) $ instead of $ ( u^{ ( q ) }, y^{ ( q ) } ) $. The only modification, which we have to do, is in (2): $ {\mathrm{ord}}_x ( \,G_{ B, i } ( u^{ ( q , s ) } ) \, ) \geq d_i -|B| $ and there exist at least one $ 1 \leq i \leq l $ and $ B := B(i)\in{\mathbb{Z}}^{ s - r^{ ( q - 1 ) } }_0 $ such that equality holds. By Proposition \[Prop:nuandsoon\] we have $ \delta^{ ( q , s ) } := \delta ( \, \Delta_x ( \, {\mathcal{G}}_{ r^{ ( q - 1 ) } + 1 } ( x );\, u^{ ( q,s )};\, y^{ ( q , s ) } \, )\, ) = 1 $.
For $ q = 1 $ we set $ J_{ r^{ ( q - 1 )} + 1 } = J_1 := J \subset {K}[[ u_1, \ldots, u_{e^{ ( 0) }} ]] = R $. (Recall that $ {e^{ ( 0) }} = n $ and we put $ ( u_1, \ldots, u_n ) := ( w_1, \ldots, w_n ) $).
Assertion *(1)*, *(2)* and $ \delta^{ ( q ) } > 1 $ follow since $ ( Y^{ ( q ) } ) = (Y_{ r^{ ( q - 1 ) } + 1 }, \ldots, Y_{ r^{ ( q ) } } ) $ yields $ {\mathrm{Dir}}_x (\, {\mathcal{G}}_{ r^{ ( q - 1 ) } + 1 } ( x ) \,) $.
Part *(3)* is a consequence of the definition of the coefficient pair (Definition \[Def:IdCoeffExp\]) and the construction of $ {\mathcal{H}}_{r^{ ( q ) } }, {\mathcal{G}}_{ r^{ ( q ) } + 1 } ( x ) $ and $ \nu_{ r^{ ( q ) } + 1 } $ (Construction \[Constr:NoExc\]).
Since $ V( y^{ ( q ) } ) $ has maximal contact, the polyhedron $$\Delta_{ x } (\, {\mathcal{G}}_{ r^{ ( q ) } + 1 } ( x ) ;\, u^{ ( q + 1 ) }; \, y_{ r^{ ( q - 1 )} + 1 },\, \ldots,\, y_{ r^{ ( q )} } \, ) = \Delta_{ x } (\, {\mathcal{G}}_{ r^{ ( q ) } + 1 } ( x ) ;\, u^{ ( q + 1 ) } \, )$$ is minimal (Proposition \[Prop:basicPoly\]) and this implies *(4)*.
The proof of the last part with $ s \in {\mathbb{Z}}_+ $, $ r^{ ( q - 1 ) } < s < r^{ ( q ) } $, is clear.
A similar description of $ {\mathcal{H}}_r ( x ) $ as above (with $ \nu_1 ( x) = H_{X,x} $ the Hilbert-Samuel function of $ X $) has already been proven in [@BMheavy], see loc. cit. Construction 4.18, and Theorem 9.4. Further they show how to get their invariant in the case without exceptional components by using “weighted initial exponents” and the “weighted diagram of initial exponents”, see loc. cit. Remark 3.25. But note that they do not give a polyhedral approach in the general case, where exceptional divisors also have to be considered.
So far we have to determine the generators $ ( g ) $ of the ideal $ J_{ r^{ ( q - 1 )} + 1 } $ step-by-step and apply the previous proposition. By introducing weights on the [[r.s.p.]{}]{} $ ( u, y) $ we are (at least in this special case) able to extend this result such that we get similar statements only with the use of the generators $ ( f ) $ of $ J $. Polyhedra of pairs in the quasi-homogeneous setting are discussed in [@BerndPoly], Remark 5.9.
\[Rk:quasihomogeneous\] Let the situation be as in Setup \[SetupB\] and as in Construction \[Constr:NoExc\] let $ {\mathcal{G}}_1 ( x ) = ( f_1, b_1 ) \cap \ldots ( f_m, b_m ) $. Recall that, if we do not consider exceptional components, then we have $${\mathrm{inv}}_X ( x )
= ( \nu_1, 0;\, 1, 0;\, \ldots;\, 1, 0;\, \nu_{ r^{(1)} + 1 }, 0;\, 1, 0;\, \ldots;\, 1, 0;\, \nu_{ r^{ ( 2 ) } + 1 }, 0;\, 1, 0;\, \ldots)$$ and with the notation of Proposition \[Prop:MainThmNoExc\] $ \nu_{ r^{(1)} + 1 } = \delta^{ ( 1 ) } > 1 $. At the beginning we separated the [[r.s.p.]{}]{} of the regular local ring $ R $ into $ ( u, y ) $, where the initial forms of $ ( y ) = ( y_1, \ldots, y_r ) $ build a minimal generating set for the ideal of $ {\mathrm{Dir}}_x ( {\mathcal{G}}_1 ( x ) ) $. The latter is the directrix associated to the homogeneous ideal $${{ I }^{ ( 0 ) } } := \langle in ( f_i, b_i ) \mid 1 \leq i \leq m \rangle.$$ Let $ L^{ ( 0 ) } := L_0 \in {\mathbb{L}}_+ $ be the positive linear form on $ {\mathbb{R}}^e $ which is given by $ L^{ ( 0 ) } ( v ) = |v | = v_1 + \ldots + v_e $ for $ v = ( v_1, \ldots, v_e ) \in {\mathbb{R}}^e $. We associate to such a linear form the valuation $ v_{ L^{ ( 0 ) } } $ on $ R $, where $$v_{ L^{ ( 0 ) } } (g) := \sup \{\, L^{ ( 0 ) } (A) + |B| \mid g \in u^A y^B R \,\}$$ for $ g \in R $.
Up to now the images of $ ( u, y ) $ in the graded ring $ gr_{M}( R ) $ were equipped with the standard grading. So the values are determined by the valuation $ v^{ ( 0 ) } $ on $ R $ with $${{ v }^{ ( 0 ) } } ( y_j ) = {{ v }^{ ( 0 ) } } (u_i) = 1
\hspace{10pt} \mbox{ and } \hspace{10pt}
{{ v }^{ ( 0 ) } } ( \lambda ) = 0$$ for $ j \in \{ 1, \ldots, r \} $, $ i \in \{ 1, \ldots, e \} $ and $ \lambda \in R^\times $. Recall that $ {{ r }^{ ( 1 ) } } = r $ and $ {{ e }^{ ( 1 ) } } = e $. We define the valuation $ {{ v }^{ ( 1 ) } } $ on $ R $ which assigns weights to the $ ( u, y ) $ as follows $${{ v }^{ ( 1 ) } } ( y_j ) = 1,
\hspace{10pt}
{{ v }^{ ( 1 ) } } (u_i) = \dfrac{ 1 }{ {{ \delta }^{ ( 1 ) } } }
\hspace{10pt} \mbox{ and } \hspace{10pt}
{{ v }^{ ( 1 ) } } ( \lambda ) = 0$$ for $ j \in \{ 1, \ldots, {{ r }^{ ( 1 ) } } \} $, $ i \in \{ 1, \ldots, {{ e }^{ ( 1 ) } } \} $ and $ \lambda \in R^\times $.
Let $ {{ L }^{ ( 1 ) } } \in {\mathbb{L}}_+ $ be the positive linear form on $ {\mathbb{R}}^{ {{ e }^{ ( 1 ) } } } $ given by $ {{ L }^{ ( 1 ) } } ( v_1, \ldots, v_e ) = \dfrac{ | v | }{ {{ \delta }^{ ( 1 ) } } } $ for $ v \in {\mathbb{R}}^{ {{ e }^{ ( 1 ) } } } $. Then $ {{ v }^{ ( 1 ) } } ( u^A y^B ) = c $, for some $ c \in {\mathbb{Z}}_+ $, if and only if $${{ L }^{ ( 1 ) } } ( A ) + | B | = \dfrac{ | A | }{ {{ \delta }^{ ( 1 ) } } } + | B | = c .
\eqno{(\ast)}$$ The last condition is equivalent to $ \dfrac{ | A | }{ c - | B | } = {{ \delta }^{ ( 1 ) } } $ if $ c - | B | \neq 0 $ . Therefore we get together with $ {{ v }^{ ( 1 ) } } ( f_i ) = b_i $ that $$in_{ {{ \delta }^{ ( 1 ) } } } (f_i, b_i )_{ u, y } \,=\, in (f_i, b_i )_{ u,y} + \sum_{ (A, B ) } \,\overline{ C_{A,B} } \,U^A\, Y^B
\,=:\, in (f_i, {{ L }^{ ( 1 ) } } )_{u,y} ,$$ where the sum ranges over those $ (A,B) \in {\mathbb{Z}}^{ e + r } $ which fulfill $ (\ast) $. Consider the quasi-homogeneous ideal $ {{ I }^{ ( 1 ) } } $ in the graded ring associated to $ {{ v }^{ ( 1 ) } } $, $${{ I }^{ ( 1 ) } } := \langle\, in (f_i, {{ L }^{ ( 1 ) } } )_{u,y} \mid 1 \leq i \leq m \,\rangle.$$ The directrix $ {\mathrm{Dir}}_x ( {{ I }^{ ( 1 ) } } ) $ corresponding to $ {{ I }^{ ( 1 ) } } $ is defined in the same way as for a homogeneous ideal; we only have to be careful with the grading. Modify $ ( y_1, \ldots, y_{ {{r}^{ ( 2 ) } } } ) $ such that their initial forms [with respect to ]{}$ {{ L }^{ ( 1 ) } } $ define $ {\mathrm{Dir}}_x ( {{ I }^{ ( 1 ) } } ) $. In the same way as we determine at the beginning $ ( {{y}^{ ( 1 ) } } ) = (y_1, \ldots, y_{ {{r}^{ ( 1 ) } } } ) $ ($ {{r}^{ ( 1 ) } } = r $), we can compute now $ ( {{y}^{ ( 2 ) } } ) = ( y_{ {{r}^{ ( 1 ) } } + 1 } , \ldots, y_{ {{r}^{ ( 2 ) } } } ) $, $ {{r}^{ ( 2 ) } } > {{r}^{ ( 1 ) } } $. Note that $ v_{ {{ L }^{ ( 1 ) } } } ( y_j ) = \dfrac{ 1 }{ {{ \delta }^{ ( 1 ) } } } $ for all elements in $ ( {{y}^{ ( 2 ) } } ) $.
Let $ {{ M }^{ ( 1 ) } } \in {\mathbb{L}}_+ $ be the positive linear form on $ {\mathbb{R}}^{ {{ r }^{ ( 2 ) } } } $ defined by $${{ M }^{ ( 1 ) } } ( v, w ) = {{ M }^{ ( 1 ) } } ( v_1, \ldots, v_{ {{ r }^{ ( 1 ) } } }, w_1, \ldots, v_{ {{ r }^{ ( 2 ) } } - {{ r }^{ ( 1 ) } } } ) = | v | + \frac{| w | }{ {{ \delta }^{ ( 1 ) } } }$$ for $ (v,w) \in {\mathbb{R}}^{ {{ r }^{ ( 2 ) } } } $. We expand $ f_i $ [with respect to ]{}$ ( {{u}^{ ( 2 ) } }; {{y}^{ ( 1 ) } }, {{y}^{ ( 2 ) } } ) $, $ i \in \{ 1, \ldots, m \} $, as in (\[eq:ExpanFi\]) $$f_i = f_i ( {{u}^{ ( 2 ) } }; {{y}^{ ( 12 ) } } )
= F_i( {{y}^{ ( 12 ) } } )
+ \sum_{ {{ M }^{ ( 1 ) } } ( B ) < b_i} F_{B,i} ( {{ u }^{ ( 2 ) } } ) \cdot ( {{y}^{ ( 12 ) } } )^B + f_i^\ast ( {{u}^{ ( 2 ) } }; {{y}^{ ( 12 ) } } ),$$ where we write $ ( {{y}^{ ( 12 ) } } ) $ for $ ( {{y}^{ ( 1 ) } }, {{y}^{ ( 2 ) } } ) $ and with some $ f_i^\ast ( {{u}^{ ( 2 ) } }; {{y}^{ ( 12 ) } } ) \in \langle {{y}^{ ( 12 ) } } \rangle^{ b_{ i } + 1 } $. Further the following properties hold
1. $ F_i ( {{y}^{ ( 12 ) } } ) \in {K}[ {{y}^{ ( 12 ) } } ] $ is a polynomial, quasi-homogeneous of degree $ b_i $ ([with respect to ]{}$ v_{ {{ L }^{ ( 1 ) } } } $).
2. $ F_{B,i}( {{u}^{ ( 2 ) } } ) \in {K}[[ {{u}^{ ( 2 ) } } ]] $ and $ v_{ {{ L }^{ ( 1 ) } } } ( \,F_{ B, i } \, ) > b_i -{{ M }^{ ( 1 ) } } ( B ) $.
3. $ {\mathcal{H}}_{r^{ ( 2 ) } } ( x ) = \left\{\, \left(\, F_{B,i}( {{u}^{ ( 2 ) } } ) ),\; b_i - {{ M }^{ ( 1 ) } } ( B ) \,\right) \;\mid\; i, B : {{ M }^{ ( 1 ) } } ( B ) < b_i \,\right\} $,\
$ {\mathcal{G}}_{ r^{ ( 2 ) } + 1 } ( x ) = \left\{\, \left(\, F_{B,i}( {{u}^{ ( 2 ) } } ) ),\; ( b_i -{{ M }^{ ( 1 ) } } ( B ))\cdot \delta^{ ( 2 ) } \,\right) \;\mid\; i, B : {{ M }^{ ( 1 ) } } ( B ) < b_i \,\right\} $,\
$ \nu_{ r^{ ( 2 ) } + 1 } = \min \left\{\, \dfrac{v_{ {{ L }^{ ( 1 ) } } } ( \,F_{ B, i } \, )}{ b_i -{{ M }^{ ( 1 ) } } ( B )} \,\;\Big|\,\; i, B : {{ M }^{ ( 1 ) } } ( B ) < b_i \,\right\} = \delta^{ ( 2 ) }> 1 $,\
where $ 1 \leq i \leq m $ and $ B := B(i)\in{\mathbb{Z}}^{ r^{ ( 2 ) } }_0 $ and $$\delta^{ ( 2 ) } := \delta_x ( \, {\mathcal{G}}_{ 1 } ( x ),\, u^{ ( 2 ) } ,\, {{y}^{ ( 12 ) } } \, ) = \delta (\; \Delta_{ x }^{(1)} (\, {\mathcal{G}}_{ 1 } ( x ) ,\, u^{ ( 2 ) } ,\, {{y}^{ ( 12 ) } } \, )\; ).$$ Here $ \Delta_{ x }^{(1)} (\, {\mathcal{G}}_{ 1 } ( x ) ,\, u^{ ( 2 ) } ,\, {{y}^{ ( 12 ) } } \, ) $ denotes polyhedron in the quasi-homogeneous setting induced by $ v_{ {{ L }^{ ( 1 ) } } } $.
One can show that the polyhedron $
\Delta_{ x }^{(1)} (\, {\mathcal{G}}_{ 1 } ( x ) ,\, u^{ ( 2 ) } ,\, {{y}^{ ( 12 ) } } \, )
$ is a certain projection of the characteristic polyhedron $ \Delta ( {\mathcal{G}}_{ 1 } ( x ) ,\, u^{ ( 1 ) } , {{y}^{ ( 1 ) } } \, ) = \Delta_x ( {\mathcal{G}}_{ 1 } ( x ) ,\, u^{ ( 1 ) } ) $ (and thus also one of the Newton polyhedron $ \Delta^N ( {\mathcal{G}}_{ 1 } ( x ) ,\, ( u^{ ( 2 ) } , {{y}^{ ( 12 ) } }) \, ) $).
Further, one can prove $$\Delta_{ x }^{(1)} (\, {\mathcal{G}}_{ 1 } ( x ) ,\, u^{ ( 2 ) } ,\, {{y}^{ ( 12 ) } } \, ) = \Delta_{ x } (\, {\mathcal{G}}_{ {{r}^{ ( 1 ) } } + 1 } ( x ) ,\, u^{ ( 2 ) } ,\, {{y}^{ ( 2 ) } } \, ) = \Delta_{ x } (\, {\mathcal{G}}_{ {{r}^{ ( 1 ) } } + 1 } ( x ) ,\, u^{ ( 2 ) } \, ).$$ In particular this implies that $
\Delta_{ x }^{(1)} (\, {\mathcal{G}}_{ 1 } ( x ) ,\, u^{ ( 2 ) } ,\, {{y}^{ ( 12 ) } } \, )
$ is minimal [with respect to ]{}the choices for $ ( {{y}^{ ( 12 ) } } ) $.
Note that $ {\mathcal{H}}_{r^{ ( 2 ) } } ( x ) $ is not the coefficient pair of $ {\mathcal{G}}_{ 1 } ( x ) $ [with respect to ]{}$ ( {{y}^{ ( 12 ) } } ) $ (Definition \[Def:IdCoeffExp\]), because in the definition of the latter we do not take care of the non-standard valuation $ {{ v }^{ ( 1 ) } } $. Of course, it is easy to extend the definition to this more general case. (But then the notation is getting more complicated $\ldots $).
Then we go on and define the new valuation $ {{v}^{ ( 2 ) } } $ on $ R $ by $$\begin{array}{ll}
{{ v }^{ ( 2 ) } } ( y_j ) = 1 ,
&
\mbox{ if } j \in \{ 1, \ldots, {{r}^{ ( 1 ) } } \},
\\[5pt]
{{ v }^{ ( 2 ) } } ( y_j ) = \dfrac{ 1 }{ {{ \delta }^{ ( 1 ) } } } ,
&
\mbox{ if } j \in \{ {{r}^{ ( 1 ) } } + 1 , \ldots, {{r}^{ ( 2 ) } } \} ,
\\[5pt]
{{ v }^{ ( 2 ) } } (u_i) = \dfrac{ 1 }{ {{ \delta }^{ ( 1 ) } } \, {{ \delta }^{ ( 2 ) } } } ,
&
\mbox{ for } i \in \{ 1, \ldots, {{e}^{ ( 2 ) } } \} ,
\\[5pt]
{{ v }^{ ( 2 ) } } ( \lambda ) = 0 ,
&
\mbox{ for } \lambda \in {K}.
\end{array}$$ Let $ {{ I }^{ ( 2 ) } } $ be the quasi-homogeneous ideal (in the graded ring associated to $ {{v}^{ ( 2 ) } } $) given by the initial forms of $ ( f_1, \ldots, f_m ) $ [with respect to ]{}$ {{v}^{ ( 2 ) } } $. Via its directrix we distinguish $ ( {{u}^{ ( 2 ) } } ) = ( {{u}^{ ( 3 ) } }, {{y}^{ ( 3 ) } } ) $, $ {{r}^{ ( 3 ) } } > {{r}^{ ( 2 ) } } $. The further procedure and the resulting statements are now clear.
Thus we obtain a new version of Proposition \[Prop:MainThmNoExc\], where we only use the generators $ ( f ) $ of $ J $. We achieve the result for $ s \in {\mathbb{Z}}_ + $ with $ r^{ ( q - 1 ) } < s < r^{ ( q ) } $ in the same way as in the first version.
The general case
================
In this section we consider the construction of the invariant introduced by Bierstone and Milman in the general case, where exceptional components are involved.
Let $ X $ be a scheme embedded in some regular scheme $ Z $ of finite type over a field $ k $, $ char ( k ) = 0 $. In the arbitrary case we have to consider the exceptional components, which arose during the preceding resolution of $ X $. Suppose we are in the year $ j $. Then we have a sequence $$\label{eq:bigseq}
\begin{array}{ccccccccccccc}
\emptyset=E_0 & & E_1 & & \ldots & & E_{i} & & \ldots & & E_{j-1} & & E_j \\[8pt]
Z=Z_0 & \stackrel{\pi_1}{\longleftarrow} & Z_1 & \stackrel{\pi_2}{\longleftarrow} & \ldots & \stackrel{\pi_{i}}{\longleftarrow} & Z_{i} & \stackrel{\pi_{i + 1 }}{\longleftarrow} & \ldots & \stackrel{\pi_{j-1}}{\longleftarrow} & Z_{j-1} & \stackrel{\pi_j}{\longleftarrow} & Z_j \\[5pt]
\hspace{1cm}\bigcup & & \bigcup & & & & \bigcup & & & & \bigcup & & \bigcup\\[5pt]
X=X_0 & \longleftarrow & X_1 & \longleftarrow & \ldots & \longleftarrow & X_{i} & \longleftarrow & \ldots & \longleftarrow & X_{j-1} & \longleftarrow & X_j \\
& & & & & & {\rotatebox{90}{\ensuremath{\in}}}& & & & {\rotatebox{90}{\ensuremath{\in}}}& & {\rotatebox{90}{\ensuremath{\in}}}\\
& & & & & & x_i & {\mathrel{\reflectbox{\ensuremath{\longmapsto}}}}& \ldots & {\mathrel{\reflectbox{\ensuremath{\longmapsto}}}}& x_{j-1} & {\mathrel{\reflectbox{\ensuremath{\longmapsto}}}}& x_j
\end{array}$$ where each $ \pi_{i+1} :Z_{i+1} \rightarrow Z_{i }$ is a blow-up in a regular center which is contained in the singular locus of $ X_i $ and has at most simple normal crossings with $ E_i $, $ E_i $ denotes the set of exceptional divisors on $ Z_i $ corresponding to the former blow-ups and $ X_i $ is the transform of $ X $ in $ Z_i $. (The last line is needed later).
Let $ x \in X_j $. We want to determine $${\mathrm{inv}}_X (x) = (\nu_1, s_1;\, \nu_2, s_2; \, \ldots) .$$ We denote by $ {\mathrm{inv}}_r (x ) $ resp. $ {\mathrm{inv}}_{ r+ \frac{1}{2} } (x) $ the invariant which is truncated after $ s_r $ resp. $ \nu_{r+1} $. Hence $${\mathrm{inv}}_r (x) = (\nu_1, s_1; \, \ldots; \, \nu_r, s_r)
\hspace{0.5cm} \mbox{ and } \hspace{0.5cm}
{\mathrm{inv}}_{ r + \frac{1}{2} } (x) = (\nu_1, s_1; \, \ldots; \, \nu_r, s_r;\, \nu_{r+1}).$$ Further we denote by $ E_j (x) $ the set of exceptional components passing through $ x $. The first invariant $ \nu_1 (x) = H_{X,x} $ is the Hilbert-Samuel function of $ X $ at $ x $. We first introduce how we get the terms $ s_i (x) $ and explain afterwards the precise construction of the $ \nu_i (x) $.
In order to define $ s_1 ( x ) $ we need the following notation: For $ i \leq j $ we denote by $ \pi_{ij} : Z_j \rightarrow Z_i $ the composition map, $ \pi_{ij} = \pi_{i + 1} \circ \pi_{ i +2 } \circ \cdots \circ \pi_{j-1} \circ \pi_j $ ($ \pi_{jj} := id_{Z_j} $), and $ x_i:=\pi_{ij} (x) $ is the image of $ x = x_j \in Z_j$ in $ Z_i $. Let $$i_1 := \min \left\{ \; k \in \{0,\ldots, j\} \,\mid\, {\mathrm{inv}}_{ \frac{1}{2} } (x) =\nu_1 (x) = \nu_1 (x_k) \; \right\}.$$ We define $ E^1 (x) \subseteq E_j (x) $ to be the set of those exceptional components which are the strict transform of an exceptional component in $ E_{i_1} ( x_{i_1} ) $, i.e. $$E^1 ( x) = \left\{ \; H \in E_j( x) \,\mid\, \exists \; H_0 \in E_{i_1} (x_{i_1})\,:\, H \mbox{ is the strict transform of } H_0 \;\right\}.$$ Then we set $$s_1( x ) := \# E^1 (x),$$ $${\mathcal{E}}_1 (x) := E_j (x) \setminus E^1 (x).$$ Suppose we know $ {\mathrm{inv}}_{ r+\frac{1}{2} }(x) = (\nu_1, s_1; \, \ldots; \, \nu_r, s_r;\, \nu_{ r + 1 } ) $ for some $ r \geq 1 $. Let $$i_{ r + 1 } := \min \left\{ \; k \in \{0,\ldots, j\} \,\mid\, {\mathrm{inv}}_{ r + \frac{1}{2} } (x) = {\mathrm{inv}}_{ r + \frac{1}{2} }(x_k) \; \right\} \geq i_r.$$ Then we define $ E^{ r + 1 }(x) \subseteq {\mathcal{E}}_r (x) \left( \,= {\mathcal{E}}_{r-1}(x) \setminus E^r(x)=E_j(x) \setminus \bigcup_{k=1}^r E^k(x) \,\right)$ to be the set of those exceptional components coming from the year $ i_{ r + 1 } $ and which we did not yet count in $ E^1(x), \ldots, E^r (x)$, $$E^{r+1}(x) = \left\{ \; H\in {\mathcal{E}}_r(x) \,\mid\, \exists \; H_0 \in E_{i_{r+1}} (x_{i_{r+1}})\,:\, H \mbox{ strict transform of } H_0 \;\right\}.$$ As above $$s_{r+1} (x) := \#E^{ r + 1 }(x),$$ $${\mathcal{E}}_{r+1} (x) = {\mathcal{E}}_{r}(x) \setminus E^{r+1}(x)=E_j(x) \setminus \bigcup_{k=1}^{r+1} E^k(x).$$
——————————
The exceptional components in $ E^k(x) $ are old, because they all arose before or in the year $ i_k $. The set $ {\mathcal{E}}_k(x) $ consists of new or young exceptional components which occurred after the year $ i_k $, where the value of the truncated invariant appeared for the first time. The sets $ E^k (x) $ and $ {\mathcal{E}}_k( x ) $ play an important role in the construction for $\nu_i$.
\[Constr:nu\] As already mentioned, the first term of the invariant $ \nu_1(x) = H_{X,x} $ is the Hilbert-Samuel function of $ X $ at $ x $.
Let $ ( f ) = ( f_1, \ldots, f_m ) $ and $ ( u, y ) $ be as in Setup \[SetupB\]. As in Construction \[Constr:NoExc\] the scheme $ X_j $ is locally at $ x $ given by the pair on $ R = {\mathcal{O}}_{ Z_j, x } $ $${\mathcal{G}}( x ) = ( f_1, b_1 ) \cap \ldots ( f_m, b_m ) .$$ (In order to avoid too many indices, we do not refer to the year $ j $). For the definition of $ \nu_i = \nu_i ( x ) $, $ i \in {\mathbb{Z}}_{\geq 2} $, it is important to know exactly what the ambient scheme and corresponding exceptional components are. In [@BMheavy] this is done by considering triples $
(\,N_{i-1}( x ),\, {\mathcal{G}}_i ( x ),\, {\mathcal{E}}_{i-1}( x ) \,)
$, where $ N_{i-1}( x ) $ is a regular ambient scheme contained in $ Z_j $, $ {\mathcal{G}}_i (x ) $ is a local description of $ X_j $ on $ N_{ i - 1 } ( x ) $ and $ {\mathcal{E}}_{i-1} ( x ) $ is an ordered set of exceptional divisors on $ Z_j $ which have simultaneously only normal crossing with $ N_{i-1}( x )$. In our language this means $ ( \, {\mathcal{G}}_i ( x ),\, {\mathcal{E}}_{i-1}( x ) \, ) $ is a [[pair with history]{}]{} on $ N_{i-1}( x ) $ (Definition \[Def:IEH\]), where we identify $ {\mathcal{E}}_{i-1}( x ) $ with the exceptional data which it defines together with $ {\mathcal{G}}_i ( x ) $ on $ N_{i-1}( x ) $.
At the beginning $ N_0 ( x) = {\mathrm{Spec\,}}( R ) $ is the germ of $ Z_j $ at $ x $ ($ R = {\mathcal{O}}_{ Z_j, x } $), $ {\mathcal{G}}_1( x ) = {\mathcal{G}}( x ) $ and $ {\mathcal{E}}_0 ( x ) = E_j( x ) $. (*Attention:* In [@BMheavy] $ {\mathcal{E}}_0( x ) = \emptyset $, but it seems to be more convenient to put $ {\mathcal{E}}_0( x ) = E_j ( x ) $, because $ {\mathcal{E}}_1( x ) \supseteq {\mathcal{E}}_2( x ) \supseteq \ldots $).
Start with the [[pair with history]{}]{} $$( \,{\mathcal{G}}_1(x),\,{\mathcal{E}}_0(x) \,) = (\, {\mathcal{G}}( x ) , E_j ( x ) \,)
\hspace{10pt}
\mbox{ on }
N_0( x )
\hspace{3pt}
\left(\mbox{resp. on }
R \right).$$ We determine $ E^1( x )$ and $ {\mathcal{E}}_1( x ) $ as described before and set $${\mathcal{F}}_1 ( x ):= {\mathcal{G}}_1( x ) \cap \left(E^1 (x),1 \right)$$ where $ \left( E^1 ( x ), 1 \right) = \bigcap_{ H \in E^1 ( x ) } \, (x_H, 1) $ and $ x_H $ denotes a local generator of $ H $. Thus we get the [[pair with history]{}]{} $$( \,{\mathcal{F}}_1(x),\,{\mathcal{E}}_1 (x) \,)
\hspace{10pt}
\mbox{ on }
R.$$ Note that also the exceptional data has changed. Not only that there are maybe less components, but also the assigned numbers may differ from those of the previous exceptional data. (For example, if $ E^1 ( x ) \neq \emptyset $, then all the assigned numbers in $ {\mathcal{E}}_1 ( x ) $ are zero, because $ E ( x ) $ defines a simple normal crossing divisor).
Using the method of Construction \[Constr:NoExc\], we choose the maximal contact hypersurface $ V ( y_1 ) $ ([without loss of generality ]{}let $ y_1 $ be as in Setup \[SetupB\]). Let $${\mathcal{H}}_1 ( x ) = {\mathbb{D}}_x ( \, {\mathcal{F}}_1 ( x );\, u_1, \ldots, u_e, y_2, \ldots, y_{ r } ;\, y_1 \, )$$ be the coefficient pair of $ {\mathcal{F}}_1 ( x ) $ [with respect to ]{}$ ( y_1 ) $.
If $ {\mathcal{F}}_1 ( x ) = ( f_1, b_1 ) \cap \ldots \cap ( f_q, b_q ) $ ($ q \in {\mathbb{Z}}_ + $, $ q \geq m $ and $ ( f_1, \ldots f_m ) $ as in Setup \[SetupB\]), then $${\mathcal{H}}_1 ( x ) =
\bigcap_{ i = 1 }^q
\bigcap_{ l := l(i) = 0 }^{ b_i - 1}
\,
\left(\, \frac{ \partial^l f_i }{ \partial y_1^l } \Big|_{ V(y_1) }, \, b_i - l \,\right)
.$$ We set $ N_1 ( x ) := V ( y_1 ) $ and get the [[pair with history]{}]{} $$( \,{\mathcal{H}}_1(x),\,{\mathcal{E}}_1 (x) \,)
\hspace{10pt}
\mbox{ on }
V ( y_1 ).$$ Again the exceptional data has changed, because we have to consider here $ {\mathcal{E}}_1 (x) $ as data on $ N_1 ( x ) = V ( y_1 ) $.
We put $ h_{i,l} := \dfrac{ \partial^l f_i }{ \partial y_1^l } \Big|_{ V(y_1) } $ for $ 1 \leq i \leq q $ and $ 0 \leq l \leq b_i - 1 $. Then we define (always $ i \in \{ 1, \ldots, q \} $ and $ l := l(i) \in \{ 0, \ldots, b_i - 1 \} $) $$\left\{\hspace{15pt}
\begin{array}{l}
\mu_2 ( x ) :=\min \left\{ \;\dfrac{ {\mathrm{ord}}_x ( h_{i,l} ) }{ b_i - 1 } \,\; \Big| \;\, i,\, l\, \right\} ,
\\[15pt]
\mu_{2,H} ( x ) := \min \left\{ \;\dfrac{ {\mathrm{ord}}_{H,x} ( h_{i,l} ) }{ b_i - l } \,\; \Big| \; \, i,\, l\, \right\} , \hspace{10pt} \mbox{ for } H \in {\mathcal{E}}_1( x ) ,
\\[15pt]
\nu_2 ( x ) := \,\mu_2 ( x ) \; - \sum\limits_{ H \in {\mathcal{E}}_1 ( x ) } \mu_{2,H} ( x ),
\end{array}
\hspace{15pt}\right.$$ where $ {\mathrm{ord}}_{H,x} ( h_{i,l} ) $ denotes the multiplicity of $ h_{i,l} $ along $ H $, i.e. if $ g_H $ is a local generator of $ H \in {\mathcal{E}}_1 ( x ) $, then $${\mathrm{ord}}_{H,x} ( h_{i,l} ) = \max \left\{ \, k \in {\mathbb{Z}}_0\cup \{\infty\} \;\big|\; g_H^k \mbox{ divides } h_{i,l} \,\right\}.$$ Clearly, $ \mu_{2,H} ( x ) $ coincides with the assigned number of $ H $ in the exceptional data of the [[pair with history]{}]{} $ ({\mathcal{H}}_1 ( x ), {\mathcal{E}}_1 (x ) ) $. Further we have $$\Delta_x^N ({\mathcal{H}}_1 ( x ), u_1, \ldots, u_e, y_2, \ldots, y_{ e } ) = \Delta_x ( \, {\mathcal{F}}_1 ( x );\, u_1, \ldots, u_e, y_2, \ldots, y_{ e } ;\, y_1 \, )$$ and $ \mu_2 ( x ) = \delta ( \Delta_x ( \, {\mathcal{F}}_1 ( x );\, u_1, \ldots, u_e, y_2, \ldots, y_{ e } ;\, y_1 \, ) ) $.
If $ \nu_2( x ) \in \{ 0, \infty\} $, then the process ends and the invariant is defined as $${\mathrm{inv}}_X (x) := {\mathrm{inv}}_{ 1\frac{1}{2} } ( x ) = (\nu_1, s_1;\, \nu_2).$$
Suppose $ 0 < \nu_2( x ) < 1 $. We consider $$D_2( x ) := \prod_{ H \in {\mathcal{E}}_1(x) } g_H^{\mu_{2,H} ( x ) },$$ where $ g_H $ denotes a local generator of $ H \in {\mathcal{E}}_1 ( x ) $. (We allow here fractional exponents; see also the remark below). Then by definition of the terms $ \mu_{2,H} (x) $, each $ h_{i,l} $, $ 1 \leq i \leq q $ and $ 0 \leq l \leq b_i - 1 $, can be written as $$h_{i,l} = D_2^{b_i - l } \cdot g_{i,l}$$ for some element $ g_{i,l} $. (Recall that $ b_i - l = b_{ h_{i,l}} $ is the number assigned to $ h_{i,l} $ in $ {\mathcal{H}}_1 ( x ) $). We define the new pair $$\label{Def:cG2(x)}
{\mathcal{G}}_2 ( x ) := \Bigg( \, \bigcap_{ i = 1 }^q \;\, \bigcap_{ l = l(i) = 0 }^{ b_i - 1 } (\, g_{i,l},\, ( b_i - l ) \cdot \nu_2\,) \,\Bigg) \cap (D_2 ( x ), 1 - \nu_2)
\hspace{10pt}
\mbox{ on }
V ( y_1 ).$$ This is our variant of the so called companion ideal, thus we call it the [*companion pair*]{}. Clearly, the exceptional data has changed again.
If $1 \leq \nu_2 (x) < \infty $, then the assigned number of the $ D_2 ( x ) $-component is not positive and hence can be omitted, ${\mathcal{G}}_2( x ) := \bigcap_{ i = 1 }^q \;\, \bigcap_{ l = 0 }^{ b_i - 1 } ( \,g_{i,l},\, ( b_i - l ) \cdot \nu_2\,) $.
Together we get for $ 0 < \nu_2 ( x ) < \infty $ the [[pair with history]{}]{} $$(\, {\mathcal{G}}_2(x),\; {\mathcal{E}}_1(x) \,)
\hspace{10pt}
\mbox{ on }
N_1 ( x ) = V ( y_1 ).$$ This completes the first step in the general procedure. Then we start again but this time with the [[pair with history]{}]{} $ (\, {\mathcal{G}}_2(x),\; {\mathcal{E}}_1(x) \,) $ instead of $ (\, {\mathcal{G}}_1(x),\; {\mathcal{E}}_0(x) \,) $.
——————————
\[Lem:ObjectsEquiv\] Let $ {\mathcal{G}}_1 ( x ) $ and $ {\mathcal{G}}'_1 ( x ) $ be two equivalent [[pair with history]{}]{}. Then $${\mathcal{F}}_1 ( x ) {\sim_{{\mathcal{E}}(x)}}{\mathcal{F}}'_1 ( x ),
\hspace{15pt}
{\mathcal{H}}_1 ( x ) {\sim_{{\mathcal{E}}(x)}}{\mathcal{H}}'_1 ( x )
\hspace{15pt}
\mbox{ and }
\hspace{15pt}
{\mathcal{G}}_2 ( x ) {\sim_{{\mathcal{E}}(x)}}{\mathcal{G}}'_2 ( x ),$$ where we have to consider the induced exceptional data. Thus these objects are invariants of the [[idealistic exponent with history]{}]{} corresponding to $ {\mathcal{G}}_1 ( x ) $.
The first and the second equivalence follow by Lemma \[Lem:BasicHist\].
The equivalence $ {\mathcal{G}}_2 ( x ) {\sim_{{\mathcal{E}}(x)}}{\mathcal{G}}'_2 ( x ) $ is clear for the cases
- $ {\mathcal{H}}_1 ( x ) = (J, b ) $ and $ {\mathcal{H}}'_1 ( x ) = ( J^a, a b ) $ for some $ a \in {\mathbb{Z}}_+ $.
- $ {\mathcal{H}}_1 ( x ) = (J_1, b ) \cap ( J_2, b) $ and $ {\mathcal{H}}'_1 (x) = ( J_1 + J_2, b ) $.
Thus we may assume $ {\mathcal{H}}_1 ( x ) = ( J, b ) $ and $ {\mathcal{H}}'_1 ( x ) = ( J', b ) $ (with the same assigned number $ b \in {\mathbb{Z}}_+ $). For an element $ h \in J $ we have defined $ g = g ( h ) $ via $ h = D_2^b \cdot g $. Set $ I := \langle \, g ( h ) \mid h \in J \, \rangle $, then $ J = D_2^b \cdot I $. (Here we identify $ D_2^b $ with the ideal which it generates in $ R $). Clearly, $ {\mathcal{G}}_2 ( x ) = ( I, \nu_2 b ) \cap ( D_2, 1 - \nu_2 ) $. We can do the same for $ {\mathcal{H}}_1' ( x ) $ and obtain the ideal $ I' $ with the analogous property.
If we can show $ ( I, \nu_2 b ) {\sim_{{\mathcal{E}}(x)}}( I', \nu_2 b ) $ (as [[pairs with history]{}]{} on $ R $), then the assertion follows. Since we have factored $ D_2 $, the assigned numbers in the induced exceptional data are all zero. Thus we only have to prove $$( I, \nu_2 b ) \sim ( I', \nu_2 b ) .$$ An extension of the [[r.s.p.]{}]{} by further independent elements does not change the situation. Hence we may assume that the extension is trivial. Further we have for any point $ x_0 \in {\mathrm{Spec\,}}( R ) $ $${\mathrm{ord}}_{ x_0 } ( I ) = {\mathrm{ord}}_{ x_0 } ( J ) - {\mathrm{ord}}_{x_0} ( D^b )
= {\mathrm{ord}}_{x_0} ( J' ) - {\mathrm{ord}}_{x_0} ( D^b ) = {\mathrm{ord}}_{ x_0 } ( I' ).$$ For the first (resp. third) equality we use $ J = D_2^b \cdot I $ (resp. $ J' = D_2^b \cdot I' $) and the second follows by $ ( J, b ) \sim ( J', b ) $. Therefore $ {\mathrm{Sing\,}}( I, b ) = {\mathrm{Sing\,}}( I', b ) $. After a permissible blow-up $ \pi: \widetilde{ Z } \to {\mathrm{Spec\,}}( R ) $ the transform $ ( \widetilde{I}, \nu_2 b ) $ of $ ( I, \nu_2 b ) $ is determined by $ I {\mathcal{O}}_{\widetilde{ Z }} = H^{\nu_2 b } \widetilde{I} $, where $ H $ denotes the ideal sheaf of the exceptional divisor. For the transform of $ J $ we have $ J {\mathcal{O}}_{\widetilde{ Z }} = H^b \widetilde{J} = H^{ (1 - \nu_2) b } \widetilde{D_2^b} \cdot H^{\nu_2 b } \widetilde{I} $. ($ \widetilde{D_2^b} $ denotes the transform of $ D_2^b $). Thus the situation is the same as before the blow-up, $ \widetilde{J} = \widetilde{D_2^b} \widetilde{I} $ and this is also true for $ J' $ and $ I' $. Together we get the desired equivalence $ ( I, \nu_2 b ) \sim ( I', \nu_2 b ) $.
Theorem \[MainThm:nuPurelyPoly\] and the second part of Theorem \[MainThm:nu\] boil down to
\[Prop:ThmA2ndB\] Let $ r \in {\mathbb{Z}}_+ $ , $ r \geq 1 $. Let $ {\mathcal{E}}_r ( x ) = \{ \, (H_1, d_1), \, \ldots, \, (H_j, d_j)\, \} $ be the exceptional data of the [[pair with history]{}]{} $ ( {\mathcal{H}}_r (x), {\mathcal{E}}_r (x ) ) $ on $ V ( y_1, \ldots, y_r ) $. Let $ ( u ) = ( u_1, \ldots, u_e ) $ be the remaining part of the [[r.s.p.]{}]{} for the local ring $ R $ of $ Z $ at $ x $. Then $$\mu_{ r + 1 } ( x ) = \delta (\, \Delta_x(\, {\mathcal{F}}_r ( x ) ; u; y_r \,) \, ) =: \delta_{ r + 1 }$$ and $ \nu_{ r + 1 } ( x) = \delta_{ r + 1 } - \sum_{ i = 1 }^j d_i $. Hence $ \nu_{ r + 1 } ( x ) $ coincides with $ \nu_x ( \,{\mathcal{F}}_r ( x ),\, {\mathcal{E}}_r ( x )_{\mathcal{H}};\, u \,) $, where the index $ {\mathcal{H}}$ should indicate that the exceptional data is the one of $ {\mathcal{H}}_r (x) $.
This follows by the definition of $ \mu_{ r + 1 } ( x ), \mu_{ r + 1, H } ( x ) $ and $ \nu_{ r + 1 } $ (for $ H \in {\mathcal{E}}_r ( x ) $) (see Construction \[Constr:nu\]).
Thus the invariant $ \nu_{ r + 1 } ( x ) $ can be achieved by purely considering polyhedra. By Proposition \[Prop:nuinvariant\] $ \nu_x ( \,{\mathcal{F}}_r ( x ),\, {\mathcal{E}}_r ( x )_{\mathcal{H}};\, u \,) $ and thus $ \nu_{ r + 1 } ( x ) $ is independent of the choice of a representative as [[idealistic exponent with history]{}]{} and also of the choice of $ ( y ) $ (for fixed $ ( u) $). Moreover, equivalent [[pairs with history]{}]{} determine the same invariant $ {\mathrm{inv}}_X ( x ) $ by Lemma \[Lem:ObjectsEquiv\]. This means $ {\mathrm{inv}}_X ( x) $ is really an invariant of the [idealistic exponent with history]{}.
All this seems now to be obvious. But keep in mind that before coming to this point we had to work hard in order to develop the theory of [[idealistic exponents with history]{}]{} and to prove important results for them.
1. If $ E_j( x)=\emptyset $, then we have $ s_i( x ) = 0 $ for all $ i $ and the above procedure coincides with the year zero case. In particular, if $ {\mathcal{E}}_k( x ) = \emptyset $ for some $ k $, then the remaining process coincides with the case described in the previous section.
2. In [@BMheavy], Remark 9.15, they slightly modify the construction of the invariant $ {\mathrm{inv}}_X ( x ) $ if $ ( {\mathcal{F}}_1 ( x ), {\mathcal{E}}_1 ( x ) ) $ can be embedded in a lower dimension ambient scheme, say for example into $ V (z_1, \ldots, z_a ) $ instead of $ N_0 ( x ) = {\mathrm{Spec\,}}( R ) $. In this case $ {\mathrm{inv}}_{ r + 1 } ( x ) := ( \nu_1, 0;\, 1, 0;\, \ldots;\, 1, 0 ) $. After this shift, they consider $ ( {\mathcal{F}}_1 ( x ), {\mathcal{E}}_1 ( x ) ) $ as an [[pair with history]{}]{} on $ V (z_1, \ldots, z_a ) $ (with the induced exceptional data) and continue as usual. This does not affect our considerations seriously.
3. In the construction we are not forced to start with $ E_0 = \emptyset $ (see (\[eq:bigseq\])). We could also require that there is additionally to $ X $ a simple normal crossing divisor $ E_0 $ on $ Z_0 $ given. This could be important for possible applications.
Recall that by construction we have $${\mathcal{H}}_1 ( x ) \,= \, \bigcap_{ i = 1 }^q \;\, \bigcap_{ l = l(i) = 0 }^{ b_i - 1 } (\, h_{i,l},\, b_i - l \,)$$ and $ \mu_2 ( x ) \geq 1$. If $ 0 < \nu_2(x) < 1$, then the transformation law (under permissible blow-ups) of the pair $\, \bigcap_{ i = 1 }^q \;\, \bigcap_{ l = 0 }^{ b_i - 1 } (\, g_{i,l},\, ( b_i - l ) \cdot \nu_2\,) $ is not consistent with that of $ \,\bigcap_{ i = 1 }^q \;\, \bigcap_{ l = 0 }^{ b_i - 1 } (\, h_{i,l},\, ( b_i - l ) \cdot \nu_2\,) $. Therefore we have to add $ (D_2 ( x ), 1 - \nu_2) $. (Recall that $ D_2 := D_2( x ) = \prod_{ H \in {\mathcal{E}}_1(x) } g_H^{\mu_{2,H} ( x ) } $ is the greatest common divisor of the $ ( h_{i,l}, b_i -l )_{i,l} $, which is a monomial in the new exceptional components $ {\mathcal{E}}_1( x ) $).\
More precisely, $ h_{i,l} = D_2^{b_i - l } \cdot g_{i,l} $ for every $ i, l $ and we defined $${\mathcal{G}}_2 ( x ) = \Bigg( \, \bigcap_{ i = 1 }^q \;\, \bigcap_{ l = l(i) = 0 }^{ b_i - 1 } (\, g_{i,l},\, ( b_i - l ) \cdot \nu_2\,) \,\Bigg) \cap (D_2 ( x ), 1 - \nu_2).$$ In the last part we have $ ( D_2, 1 - \nu_2 ) {\sim_{{\mathcal{E}}(x)}}( D_2^d, (1-\nu_2)\,d ) $ for all $ d \in {\mathbb{R}}_+ $. So $${\mathcal{G}}_2 ( x ) {\sim_{{\mathcal{E}}(x)}}\bigcap_{ i = 1 }^q \;\, \bigcap_{ l = l(i) = 0 }^{ b_i - 1 }
\Bigg(\, (\, g_{i,l},\, ( b_i - l ) \cdot \nu_2\,) \cap (\, D_2^{ b_i - l }, (1 - \nu_2)(b_i - l ) \,) \,\Bigg).$$ Suppose $ 0 < \nu < 1 $ ($ \nu := \nu_2 $). If a blow-up is permissible for the pair $ ( g, d \cdot\nu ) := (\, g_{i,l},\, ( b_i - l ) \cdot \nu_2\,) $, then we know $ {\mathrm{ord}}_x ( g ) \geq d \nu $. But $ 0 < \nu < 1 $ implies $ d \nu < d $ and hence $ {\mathrm{ord}}_x ( g ) < d $ might be possible! This means a center which is permissible for $ ( g, d \cdot\nu ) $ is not necessarily permissible for $ ( h, d ):= ( h_{i,l}, {b_i - l } ) $ ($ h = D_2^d \cdot g $).
Further the transform of $ (h,d) $ after a permissible blow-up is locally given by $ ( z_{exc}^{-d} \cdot \widetilde{h}, d )$, where $ z_{exc} $ denotes a local generator of the exceptional component and $ \widetilde{h} $ denotes the total transform. By using $ h = D_2^d \cdot g $ we get $$\bigg( z_{exc}^{-d} \cdot \widetilde{h}
= \left( z_{exc}^{-(1-\nu) d} \cdot \widetilde{D_2}^d \right)\;\cdot\; \left( z_{exc}^{-\nu d} \cdot \widetilde{g}\right),\;\; d \bigg),$$ where $ \widetilde{D_2} $ denotes the total transform of $ D_2 $ under the blow-up. If we consider only $ (g, d\nu) $, then the transformations are not consistent.\
In the case $ \nu \geq 1 $ we get $ d \nu \geq d $ and the transform of $ h $ is determined by the terms $ z_{exc}^{-d} \cdot g = z_{exc}^{(\nu-1) d} \cdot z_{exc}^{-d \nu} \cdot g $, where $ ( \nu - 1 ) d \geq 0 $; thus we have not to add $ D_2 $.
If $ \nu_t \in \{0, \infty\} $ for some $ t \in {\mathbb{Z}}_+ $, then $${\mathrm{inv}}_X( x ) = {\mathrm{inv}}_{ t -\frac{1}{2} } (x) = (\nu_1, s_1; \, \ldots; \, \nu_{t-1}, s_{t-1};\, \nu_t).$$ In the case $ \nu_t (x ) = \infty $ the center of the upcoming blow-up is $$N_{t-1} (x) =V (y_1,\ldots,y_{ t-1} ).$$ In every chart the invariant decreases, because all elements of $ ( y_1, \ldots, y_{t-1} ) $ are coming from certain initial forms.
If $ \nu_t (x ) = 0 $, then we set $ {\mathcal{G}}_t ( x ) = (D_t ( x ),\, 1) $. (This fits into the definition of these terms; the assigned numbers of the first part in (\[Def:cG2(x)\]) are $ 0 $, because $ \nu_t (x ) = 0 $, hence we can ignore it and get $ {\mathcal{G}}_t ( x ) = (D_t ( x ),\, 1) $). This is the monomial case. In [@BMlight], Remark 3.6, it is explained how to choose the center for the upcoming blow-up in this case.
Let us see what in each step of the general process by Bierstone and Milman happens to our polyhedra. Let $ r \in {\mathbb{Z}}_+ $ with $ 0 < r \leq n $ and let $ e = n - r $.
***From $\boldsymbol{{\mathcal{G}}_r (x)}$ to $\boldsymbol{{\mathcal{F}}_r (x) = {\mathcal{G}}_r (x) \cap ( E^r (x),1)} $:*** In this step we add $$\left( E^1 ( x ), 1 \right) = \bigcap_{ H \in E^1 ( x ) } \, (x_H, 1),$$ where $ x_H $ denotes a local generator of $ H $. Recall that $ s_r = s_r(x) = \#E^r (x) $. By construction $ E^r ( x ) \subseteq {\mathcal{E}}_{r-1} (x) $ has only normal crossings with $ N_{r-1} (x) $. Thus we can choose the [[r.s.p.]{}]{} $ ( u ) = ( u_1,\ldots, u_{e+1} ) $ for $ N_{r-1} (x) $ such that for all $ H \in E^r(x) $ the local generator is $ g_H = u_k $ for $ k \in I_r:= \{k_1, \ldots, k_{s_r} \} \subseteq \{1, \ldots, e+1 \} $. (In fact, we can choose the [[r.s.p.]{}]{} such that the analogous condition holds for every $ H \in {\mathcal{E}}_{r-1} (x) $).
Adding the old exceptional components $ ( E^r (x),1 ) $ corresponds to adding points to the generators of the polyhedron $ \Delta_x^N ( {\mathcal{G}}_r (x), u ) $. More precisely, the new points are $$\big\{
( \delta_{ \alpha k })_{ \alpha \in \{1,\ldots, e + 1 \} } =
(0,\,\ldots,\,0,\underset{ \displaystyle k }{ \underset{\displaystyle \uparrow}{1} }, \,0,\,\ldots,\,0) \;\mid\; k \in I_r =\{k_1, \ldots, k_{s_r} \}
\big\},$$ where $\delta_{\alpha k}$ denotes the usual Kronecker delta. Obviously, the number of new points for the polyhedron is $s_r$.
By the equivalence of $ \bigcap_{ H \in E^r(x) } ( x_H, 1) $ and $
\left( \prod_{ H \in E^r(x) } x_H,\, \sum_{ H \in E^r(x) } 1 = s_r \right)
$ this can also be reinterpreted as adding of only one point to the generators of the polyhedron, namely the one given by $$\left( \sum_{k\in I_r} \delta_{\alpha k} \cdot \frac{1}{s_r} \right)_{\alpha\in\{1,\ldots, e + 1 \}}
= \bigg( 0,\,\ldots,\,
0,\underset{\displaystyle k_1}{\underset{\displaystyle \uparrow}{\frac{1}{s_r}}},0,\,\ldots,\,
0,\underset{\displaystyle k_2}{\underset{\displaystyle \uparrow}{\frac{1}{s_r}}},0,\,\ldots,\,
0,\underset{\displaystyle k_{s_r}}{\underset{\displaystyle \uparrow}{\frac{1}{s_r}}},0,\,\ldots,\,
0 \bigg)$$ ([without loss of generality ]{}we may assume $ 1 \leq k_1 < k_2 < \ldots < k_{s_r} \leq e + 1 $).
In both cases the same variables are involved. Hence the ideal of the tangent cone (the directrix and the ridge) behaves in both cases the same way. This change is described by the initial forms of $ ( E^r( x ), 1 ) $.
Further observe: $ V ( u_{ k_1}, \ldots, u_{k_{s_r}} ) $ has maximal contact with $ {\mathcal{F}}_r ( x ) $ at $ x $. Clearly, the coefficient pairs [with respect to ]{}$ ( u_{ k_1}, \ldots, u_{k_{s_r}} ) $ coincide in both cases. Thus the projections of the polyhedra [with respect to ]{}$ ( u_{ k_1}, \ldots, u_{k_{s_r}} ) $ coincide, too.
——————————
***From $\boldsymbol{{\mathcal{F}}_r (x)}$ to $\boldsymbol{{\mathcal{H}}_r (x)}$:*** Suppose $ {\mathcal{F}}_r(x) = ( f_1, b_1) \cap \ldots \cap ( f_q, b_q) $, then there is at least one $ i \in \{ 1, \ldots, q \} $ such that $ b_i = {\mathrm{ord}}_x ( f_i ) $. We assume [without loss of generality ]{}that $ y_r := u_{ e + 1 } $ has maximal contact with $ {\mathcal{F}}_r(a) $ at $ x $. Hence in this step we project the polyhedron $ \Delta_x^N ( {\mathcal{F}}_r (x); u_1, \ldots, u_e, u_{ e +1 } ) \subset {\mathbb{R}}_0^{ e + 1} $ from the point $ ( 0, \ldots,0,1) \in {\mathbb{Z}}^{e +1 }_0 $ to $ {\mathbb{R}}_0^e $. The resulting polyhedron is $$\Delta_x ( {\mathcal{F}}_r (x); u_1, \ldots, u_e; y_r ) = \Delta_x^N ( {\mathcal{H}}_r (x); u_1, \ldots, u_e ) \subset {\mathbb{R}}_0^{ e }.$$
——————————
***From $\boldsymbol{{\mathcal{H}}_r ( x )}$ to $\boldsymbol{{\mathcal{G}}_{r+1} ( x )}$:*** Suppose $ {\mathcal{H}}_r(x) = ( h_1, b_1) \cap \ldots \cap ( h_p, b_p) $. The last step is rather consisting of three smaller steps. We determine $ D_{r+1} (x ) $ and write each $ (h_i,b_i) $ as $ h_i = D_{r+1}^{b_i} \cdot g_i $. We set $$\widetilde{{\mathcal{H}}_r} (x) := \bigcap_{ i = 1 }^p \; (\,g_i,\, b_i\,)
\hspace{15pt}
\mbox{ and }
\hspace{15pt}
\widetilde{{\mathcal{G}}_{r+1}} (x) := \bigcap_{ i = 1 }^p \;(\,g_i,\, b_i \, \nu_{r+1}\,) .$$ Further recall that $ {\mathcal{G}}_{r+1}(x) = \widetilde{{\mathcal{G}}_{r+1}} (x) \cap ( \, D_{r+1}(a), 1 - \nu_{r+1}\,) $. Then the smaller steps are the following
- ***From $\boldsymbol{{\mathcal{H}}_r (x)}$ to $\boldsymbol{\widetilde{{\mathcal{H}}_r}(x)}$:*** Since $ N_r (x) $ and $ {\mathcal{E}}_r(x ) $ have simultaneously only normal crossings, we can choose the coordinates $ (u_1, \ldots, u_e) $ of $ N_r (x) $ such that for all $ H \in {\mathcal{E}}_r( x ) $ the local generator is $ g_H = u_l $ for some $ l \in \{1, \ldots, e \} $. In this situation we set $ \mu_{r+1,l} := \mu_{r+1,H}(x) $. Put $$I_r :=\left\{\, l_1,\, \ldots,\, l_{m_r} \,\right\} := \left\{ \, l \in \{1, \ldots, e \} \;\mid \; \mu_{r+1,l} \neq 0 \, \right\} \subseteq \{1, \ldots, e \} .$$ Further we denote by $ T_r : {\mathbb{R}}^e \to {\mathbb{R}}^e $ the translation in the negative direction by the vector $${{w}^{ ( r ) } } :=
\bigg( 0,\,\ldots,\,
0,\,\underset{\displaystyle l_1}{\underset{\displaystyle \uparrow}{\mu_{r+1,1}}},\,0,\,\ldots,\,
0,\,\underset{\displaystyle l_2}{\underset{\displaystyle \uparrow}{\mu_{r+1,2}}},\,0,\,\ldots,\,
0,\,\underset{\displaystyle l_{m_r}}{\underset{\displaystyle \uparrow}{\mu_{r+1,m_r}}},\,0,\,\ldots,\,
0 \bigg).$$ This means a point $ v \in {\mathbb{R}}^e $ is sent to $ T_r ( v ) = v - {{w}^{ ( r ) } } $. Then we have for the Newton polyhedra $$T_r \left(\, \Delta_x^N ( \, {\mathcal{H}}_r ( x ) , u \,) \,\right) = \Delta_x^N (\, \widetilde{{\mathcal{H}}_r} ( x ) , u \,) \subseteq {\mathbb{R}}^e_0.$$
- ***From $\boldsymbol{\widetilde{{\mathcal{H}}_r} (x)}$ to $\boldsymbol{\widetilde{{\mathcal{G}}_{r+1}}(x)}$:*** In this step we multiply each point of the polyhedron $ \Delta_x^N (\, \widetilde{{\mathcal{H}}_r} ( x ) , u \,) $ by the factor $ \dfrac{ 1 }{ \nu_{r+1} } $ and get $ \Delta_x^N (\, \widetilde{{\mathcal{G}}_{r + 1 }} ( x ) , u \,) $.
This corresponds to the change of the valuation on the [[r.s.p.]{}]{} $ ( u ) $ for the regular local ring $ {K}[[ u ]] $ corresponding to $ V ( y ) $ (resp. of the [[r.s.p.]{}]{} $ ( u, y ) $ for $ R $), see Remark \[Rk:quasihomogeneous\].
- ***From $\boldsymbol{\widetilde{{\mathcal{G}}_{r+1}}(x)}$ to $\boldsymbol{{\mathcal{G}}_{r+1}(x)}$:*** The last step is similar to “*From ${\mathcal{G}}_r (x)$ to ${\mathcal{F}}_r (x)$*”. By definition $ {\mathcal{G}}_{r+1}(x) = \widetilde{{\mathcal{G}}_{r+1}}(x) \cap (D_{r+1}(x), 1 - \nu_{r+1}) $. Thus we add to the generators of $ \Delta_x^N (\, \widetilde{{\mathcal{G}}_{r + 1 }} ( x ) , u \,) $ the points associated to $$\left(D_{r+1}(x) = \prod_{ H \in {\mathcal{E}}_{r}(x) } g_H^{ \mu_{r+1,H}(x) },\, 1 - \nu_{r+1} \right),$$ where $ g_H $ is a local generator of $ H \in {\mathcal{E}}_{r}(x) $. (Recall that we defined $ \nu_{r+1} ( x ) = \mu_{r+1} ( x ) - \sum_{ H \in {\mathcal{E}}_{r}(x)} \mu_{r+1,H}(x) $). As in *(1)* we choose $ ( u ) = (u_1,\ldots, u_e) $ such that for all $ H \in {\mathcal{E}}_r(x) $ the local generator is $ g_H = u_l $ for some $ l \in \{1, \ldots, e \} $. Again we set $ \mu_{r+1,l} := \mu_{r+1,H}(x) $ and $$I_r :=\left\{\, l_1,\, \ldots,\, l_{m_r} \,\right\} := \left\{ \, l \in \{1, \ldots, e \} \;\mid \; \mu_{r+1,l} \neq 0 \, \right\} \subseteq \{1, \ldots, e \} .$$
Then $ (D_{r+1}(x), 1 - \nu_{r+1}) $ yields in $ \Delta_x^N (\, {\mathcal{G}}_{r + 1 } ( x ) , u \,) $ the point $$\bigg( 0,\,\ldots,\,
0,\underset{\displaystyle l_1}{\underset{\displaystyle \uparrow}{\frac{\mu_{r+1,1}}{1 - \nu_{r+1}} }},0,\,\ldots,\,
0,\underset{\displaystyle l_2}{\underset{\displaystyle \uparrow}{\frac{\mu_{r+1,2}}{1 - \nu_{r+1}} }},0,\,\ldots,\,
0,\underset{\displaystyle l_{m_r}}{\underset{\displaystyle \uparrow}{\frac{\mu_{r+1,m_r}}{1 - \nu_{r+1}} }},0,\,\ldots,\,
0 \bigg).$$
By definition $ \delta (\, \Delta_x^N (\, \widetilde{{\mathcal{H}}_r} ( x ) , u \,) \, ) = \mu_{ r + 1 } ( x ) $. By going from $ {\mathcal{H}}_r (x) $ to $ \widetilde{{\mathcal{H}}_r}(x) $ we send the assigned numbers in the exceptional data to zero. Therefore $ \delta (\, \Delta_x^N (\, \widetilde{{\mathcal{H}}_r} ( x ) , u \,) \, )= \delta (\, \Delta_x(\, {\mathcal{F}}_r ( x ) ; u; y_r \,) \,) = \nu_{ r + 1 } ( x ) $.
Simplification of the strategy
==============================
The construction of the invariant of Bierstone and Milman is quite complicated. Therefore it is hard to formulate a step-by-step result on the behavior of the generators of the ideal $ J $ as we did in Proposition \[Prop:MainThmNoExc\] and Remark \[Rk:quasihomogeneous\]. But we show now that in certain good situations the procedure becomes easier. In particular, we can sometimes make bigger steps.
For this we introduce the following
Fix $ r \in {\mathbb{Z}}_+ $. Let $ {\mathcal{I}}_r \in \{ {\mathcal{G}}_r (x) , {\mathcal{F}}_r ( x ) , {\mathcal{H}}_{r + 1} ( x ) \} $ and $ s \in {\mathbb{Z}}_+ $, $ s < r $.
1. We define the $ {\mathcal{G}}_s $-part of $ {\mathcal{I}}_r $ to be the part of $ {\mathcal{I}}_r $ which is by the construction coming from $ {\mathcal{G}}_s ( x ) $.
2. By the $ {{E}^{ ( s ) } } $-part (resp. $ {{D}^{ ( s ) } } $-part) of $ {\mathcal{I}}_r $ we denote the part which occurred by adding $ E^s ( x ), \ldots, E^r ( x ) $ (resp. $ D_{s + 1} ( x ), \ldots, D_r (x) $).
If $ s = 1 $, then we speak also of the $ {\mathcal{G}}$-part (resp. $ E $-part, resp. $ D $-part) of $ {\mathcal{I}}_r $ instead of the $ {\mathcal{G}}_1 $-part (resp. $ {{E}^{ ( 1 ) } } $-part, resp. $ {{D}^{ ( 1 ) } } $-part) of $ {\mathcal{I}}_r $.
(For $ {\mathcal{I}}_r = {\mathcal{G}}_r ( x ) $ we neglect $ E^r ( x ) $ in the definition of the $ {{E}^{ ( s ) } } $-part, because it has not been added yet.)
\[Obs:BigEstep\] In the definition of $ {\mathcal{F}}_1 ( x ) $ we add the old exceptional components $ (E^1 (x), 1 ) $ to $ {\mathcal{G}}_1 ( x ) $. This enables us to make sometimes more than one step in Construction \[Constr:nu\]: First, this may change the separation of the [[r.s.p.]{}]{} into $ ( u, y ) $ as in Setup \[SetupB\]. Thus let us consider an arbitrary [[r.s.p.]{}]{} $ ( t ) = (t_1, \ldots, t_n ) $ for $ R $. Further $ E^1 ( x )$ is a simple normal crossing divisor on $ N_0 ( x ) = Z_j $. Hence we can choose the [[r.s.p.]{}]{} $ ( t ) = (t_1, \ldots, t_n ) $ for $ R $ such that every $ H \in E^1(x) $ is locally given by some $ t_l = 0 $ for $ l \in \{1,\ldots,n\} $, say $ E^1( x ) $ is given by $ ( t_{l_1},\, \ldots,\, t_{l_{s_1}} ) $. Suppose $ s_1 = \# E^1( x )\geq 1 $. Set $ ( z ) = ( z_1, \,\ldots,\, z_{s_1} ) = ( t_{l_1},\, \ldots,\, t_{l_{s_1}} ) $. Then $ V ( z ) $ has maximal contact with $ {\mathcal{F}}_1 ( x ) $ at $ x $. (Recall that locally $ x $ is given by the maximal ideal of $ R $). So this is a possible choice for the first $ s_1 $ steps in definition of $ \nu_i ( x ) $. (After that we consider $ {\mathcal{H}}_{s_1}( x ) $ which determines $ \nu_{s_1 + 1} $). Since $ {\mathcal{E}}_1(x) $ and $ E^1(x) $ have simultaneously only normal crossings, we can require the additional property on $ ( t ) $ that $ {\mathcal{E}}_1 ( x ) $ is given by $ ( t_{ m_1 }, \ldots, t_ { m_p} ) $, where $ t_\iota \neq t_\rho $ for $ \iota \in \{ m_1, \ldots, m_p \} $ and $ \rho \in \{ l_1, \ldots, l_{s_1 } \} $. Thus we get for every $ i \in \{ 2,\ldots, s_1 \} $ (If $ s_1 = 1 $ then the previous set is empty):
- $ \mu_{i,H}( x ) = 0 $ for every $ H \in {\mathcal{E}}_i( x ) $, thus $ D_i( x ) = 1 $ and
- $ \nu_i( x ) = \mu_i( x ) = 1 $.
The first assertion holds, because we can not factor $ t_{ \iota } $ from $ t_\rho $ ($ \iota $ and $ \rho $ as above). The second part follows from the condition that $ V ( z ) $ has maximal contact with $ {\mathcal{F}}_1 ( x ) $ at $ x $.
Therefore we already know $ \nu_i $ up to the step $ i = s_1 $. Set $ d := s_1 $. In the procedure we also added $ E^2( x ), \ldots, E^{d}( x ) \subset {\mathcal{E}}_1 ( x ) $. Further $ s_q = \# E^q (x ) $ for $ q \in \{ 1, \ldots, d \} $ and $ {\mathcal{E}}_d( x ) = E_j( x ) \setminus \bigcup_{ l = 1 }^{ d } E^q ( x ) $. If the condition $$s_1 + \ldots + s_d - d \geq 1 \; \Leftrightarrow \; s_2 + \ldots + s_d \geq 1$$ holds, then $ D_{ d + 1 } ( x ) = 1 $, $ \nu_{ d + 1 }(x) = \mu_{ i + 1 }(x) = 1 $ and we can choose the next maximal contact $ V ( z_{ d + 1 } ) $ in the $ E $-part of $ {\mathcal{H}}_d ( x ) $.
*Convention: We choose the maximal contact variables in the $ E $-part until we get to the stage $ r > d $, where $ s_1 + \ldots + s_r - r = 0 $.*
This means the $ E $-part of $ {\mathcal{H}}_r( x ) $ is empty. (Recall that $ {\mathcal{H}}_r(x) $ determines $ \nu_{ r + 1 } ( x ) $). As above it follows for every $ i \in \{ 2,\ldots, r \}$:
- $ \mu_{i,H}( x ) = 0$ for every $ H \in {\mathcal{E}}_i( x ) $, thus $ D_i( x ) = 1 $ and
- $ \nu_i( x ) = \mu_i( x ) = 1 $.
In particular, $ {\mathcal{H}}_r( x ) $ is only given by the $ {\mathcal{G}}_1 $-part. This means, $ {\mathcal{H}}_r( x ) $ is the coefficient pair of $ {\mathcal{G}}_1 ( x ) $ [with respect to ]{}$ (z_1, \ldots, z_r ) $.
In general, we cannot assume $ s_1 > 0 $. So we set $$d := \min \left\{\, q \in {\mathbb{Z}}_+ \;|\; s_q \neq 0 \,\right\}.$$ Then $ E^d ( x ) \neq \emptyset $ and $ {\mathcal{F}}_d ( x ) = {\mathcal{G}}_d( x ) \cap ( E^d ( x ) , 1 ) $. We choose the maximal contact $ V (z_{d}) $ such that there is some $ H \in E^d( x ) $ which is locally given by $ V (z_{d}) $.
If $ s_d \geq 2 $, then the $ {{E}^{ ( d ) } } $-part of $ {\mathcal{H}}_{d} (x ) $ is non-empty. This implies $ \nu_{ d + 1 } ( x ) = \mu_{ d + 1} ( x ) = 1 $. In the next step of the procedure we multiply the assigned numbers by $ \nu_{ d + 1 } = 1 $, thus $ {\mathcal{G}}_{ d + 1 } ( x ) = {\mathcal{H}}_d( x ) $ and then we add $ E^{ d + 1 }( x ) $ in order to obtain $ {\mathcal{F}}_{ d + 1 }( x )$. We choose the maximal contact in the $ {{E}^{ ( d ) } } $-part and so on. This continues until we are at the step $$r:= \min \left\{\, l \in {\mathbb{Z}}_+ \mid l \geq d \;\wedge\; s_d + \ldots + s_l - (l-d+1) = 0 \,\right\}.$$
Putting everything together yields
\[Prop:EpartResult\] Let $ d, r \in {\mathbb{Z}}_+ $ be as above. For every $ i \in \{ d + 1, \ldots , r \} $ we get
- $ \mu_{i,H}( x ) = 0 $ for every $ H \in {\mathcal{E}}_i( x )$, thus $ D_i ( x ) = 1 $ and
- $ \nu_i( x ) = \mu_i( x ) = 1 $,
- the $ {{E}^{ ( d ) } } $-part of $ {\mathcal{H}}_r ( x ) $ (and $ {\mathcal{G}}_{r + 1} ( x ) $) is empty,
- hence $ {\mathcal{H}}_r ( x ) $ is the coefficient pair of $ {\mathcal{G}}_d ( x ) $ [with respect to ]{}$ (z_{ d }, \ldots, z_r ) $ and $ \mu_{ r + 1 } ( x ) = \delta ( \, \Delta_x ( {\mathcal{G}}_d ( x ), u , (z_{ d }, \ldots, z_r ) \, ) $, where $ ( u ) $ denotes the remaining elements of the [[r.s.p.]{}]{} $ ( t ) = ( u, z) $.
Further $ \nu_{ r + 1 } ( x ) $ is determined by $ \mu_{ r + 1 } ( x ) $ and the assigned numbers in the exceptional data of $ {\mathcal{H}}_r ( x ) $.
If $ s_d = 1 $ then $ r = d $ and the above statement is empty except for part [(iv)]{}.
Recall that we have constructed $ {\mathcal{G}}_2 ( x ) $ from $ {\mathcal{H}}_1 ( x ) $ by factoring out $ D_2 ( x ) $, $ h = D_2^{ b_h } g $ (where $ {\mathcal{H}}_1 ( x ) \subset ( h, b_h ) $). If $ D_2 = D_2 ( x ) = 1 $ is trivial, i.e. if the assigned numbers in the exceptional data are all zero, then $ {\mathcal{G}}_2 ( x ) = {\mathcal{H}}_1 ( x ) $. Together with the previous this leads to
\[Obs:BigDstep\] Set $$d := \min \left\{\, q \in {\mathbb{Z}}_+ \;|\; D_q = 1 \,\right\}
\hspace{10pt}
\mbox{ and }
\hspace{10pt}
r:= \min \left\{\, l \in {\mathbb{Z}}_+ \mid l > d \;\wedge\; D_q \neq 1 \,\right\}.$$ (For the steps before $ d $ we have to apply the usual procedure). Consider $ {\mathcal{G}}_d ( x ) $. Since $ D_d ( x ) = 1 $, we have $ {\mathcal{G}}_d ( x ) = {\mathcal{H}}_{ d - 1 } ( x ) $. If $ s_d = \# E^d ( x ) = 0 $, then the next step is as without exceptional divisors. On the other hand, if $ s_d \geq 1 $, then we can apply Observation \[Obs:BigEstep\] until the $ E $-part is empty. Note that we have during this process $ D_q = 1 $. Thus we have good control on these steps.
This works until we come to $ {\mathcal{H}}_{ r - 1 } ( x ) $. There $ D_r ( x ) \neq 1 $. By the convention of choosing first the exceptional components in the $ E $-part, the $ {{E}^{ ( d ) } } $-part of $ {\mathcal{H}}_{ r - 1 } ( x ) $ has to be empty. This implies that $ {\mathcal{H}}_{ r - 1 } ( x ) $ is only given by the $ {\mathcal{G}}_d $-part. (But it is not necessarily the coefficient pair of $ {\mathcal{G}}_d ( x ) $ [with respect to ]{}$ ( z_d, \ldots, z_{ r - 1 } ) $, because maybe not all $ \nu_i ( x ) $ are equal $ 1 $ for $ d < i < r $; nevertheless the situation is similar to Remark \[Rk:quasihomogeneous\] — see also Remark \[Rk:FormulaGenerators\] below).
We modify $ {\mathcal{H}}_{ r - 1 } ( x ) $ as described in Construction \[Constr:nu\] (factor out $ D_r ( x ) $ and then add $ ( D_r ( x ) , 1 - \nu_{ r } ) $) and obtain $ {\mathcal{G}}_r ( x ) $.
If $ \mu_{ r } ( x ) = 1 $, then $ (D_r (x), 1 - \nu_r) {\sim_{{\mathcal{E}}(x)}}\bigcap_H\, (g_H, 1) $, where the intersection is over those $ H \in {\mathcal{E}}_r(x) $ with $ \mu_{r,H}(x) \neq 0 $ and $ g_H $ denotes a local generator of $ H $. Then the same procedure as in the previous observation can be applied: First we choose the maximal contact only in the part coming from $ D_r ( x ) $ and after that we consider the $ {{E}^{ ( r ) } } $-part.
We can apply this until we get to the point, where $ D_{r'} ( x ) \neq 1 $ and $ \mu_{r'} ( x ) > 1 $. Then we have to apply the full procedure to construct $ \nu_{ r' } $ and we go back to the beginning of this observation.
Also recall that if the exceptional data $ {\mathcal{E}}_{ r' } ( x ) = \emptyset $ is empty, then the general procedure is the same as in the easy case without exceptional divisors.
Let us recapitulate the result.
\[Prop:DpartResult\] Let $ d, r, r' \in {\mathbb{Z}}_+ $ be as in the previous observation. (Not to be confused with the $ d, r $ in Proposition \[Prop:EpartResult\]; these are different integers). Then the case without exceptional divisors and Proposition \[Prop:EpartResult\] determine completely the procedure of Construction \[Constr:nu\] for the steps $ i \in \{ d, \ldots, r' - 1 \} $.
Note that Proposition \[Prop:EpartResult\] and Proposition \[Prop:DpartResult\] depend on the convention that we choose the maximal contact first in the $ E $-part of the given [[pair with history]{}]{}.
\[Rk:FormulaGenerators\] For the proof of our main theorem we did not need concrete formulas for $ {\mathcal{G}}_r ( x) $, $ {\mathcal{F}}_r ( x ) $ resp. $ {\mathcal{H}}_{ r + 1 } ( x ) $. Let us now briefly mention some results in this direction.
In order to simplify the situation we assume that $ D_2 = \ldots = D_r = 1 $ and $ D_{ r + 1 } \neq 1 $ for some $ r \in {\mathbb{Z}}_+ $. After $ r $ steps in the procedure we have distinguished the [[r.s.p.]{}]{} for $ R = {\mathcal{O}}_{Z_j, x } $ as $ ( u, z ) = (u_1, \ldots, u_e; z_1,\ldots, z_r) $ and further we know the terms $ \nu_2 = \mu_2 \geq 1, \ldots, \nu_r =\mu_r \geq 1 $ and $ \nu_{ r + 1 } $. ($ D_{ r + 1 } \neq 1 $ implies $ \nu_{ r + 1 } < \mu_{ r + 1 } $).
Define $ \beta_1 := 1 $ and $ \beta_j := ( \nu_2 \cdots \nu_j )^{-1} $ for $ j > 1 $. Recall that we choose (by the convention) the next maximal contact components in the $ E $-part until it is empty. Since the $ E $-part and $ {\mathcal{E}}_r $ have simultaneously only normal crossings, it follows that the $ E $-part of $ {\mathcal{H}}_r ( x ) $ is empty. ($ D_{ r +1 } ( x ) $ is determined by $ {\mathcal{H}}_r ( x ) $). Together with the definition of $ r $ this yields that $ {\mathcal{H}}_r ( x ) $ is completely determined by the $ {\mathcal{G}}_1 $-part. Suppose $ {\mathcal{G}}_1 ( x ) = \bigcap_{ i = 1 }^m ( f_i, b_i ) $ for some $ f_i \in R $. Then we can write for every $ i $ $$f_i ( u, z ) = F_{b_i, i} (z) + \sum_{L_{\nu(r)} (B) < b } F_{B,i}(u) \cdot z^B + f_i^\ast ( u,z ),$$ for some $ f_i^\ast ( u,z ) \in \langle z \rangle^{ b + 1 } $ and $ F_{b_i, i} (z) \in {K}[z] $ is ([with respect to ]{}$ L_{\nu(r)}(B) : = \sum_{j=1}^r \beta_j \, B_j $) quasi-homogeneous of degree $ b_i $. Further let $ i \in \{ 1, \ldots, m \} $ and $ B = B( i ) \in {\mathbb{Z}}^r_0 $ be such that $ L_\nu(B) < b_i $. Then $${\mathcal{H}}_s (x) =
\bigcap_{\mbox{\small $ i, B $ as above }}
\left(
F_{B, i }(u),\,
(b - L_\nu(B))\cdot \frac{\displaystyle 1}{\displaystyle \beta_s}
\right)
.$$ By definition, $ D_{r+1} \neq 1 $ and thus $ \nu_{r+1} < \mu_{r+1} $.
Further any element $ ( h_{B,i}, b_h) := \left( F_{B, i }(u),\, (b - L_\nu(B))\cdot (\beta_s)^{-1} \right) $ can be written in the form $ h_{B,i} = D_{ r + 1 }^{b_h} \cdot g_{B,i} $ and $${\mathcal{G}}_{r+1} (x) =
\left\{
\begin{array}{ll}
\bigcap_{i,B} ( g_{B,i}, b_h \cdot \nu_{ r + 1} ), & \mbox{ if } 1 \leq \nu_{r+1} < \infty,
\\[5pt]
\left(
\bigcap_{i,B} ( g_{B,i}, b_h \cdot \nu_{ r + 1} )
\right) \cap (D_{r+1}, 1- \nu_{r+1}) , & \mbox{ if } 0 < \nu_{r+1} < 1,
\\[5pt]
(D_{r+1}, 1- \nu_{r+1}), & \mbox{ if } \nu_{r+1} = 0.
\end{array}
\right.$$ (In the case $ \nu_{r+1}=\infty $ the center of the next blow-up is $ N_{r} ( x ) = V (z_1, \ldots, z_r)$ ).
Then we start again with $ {\mathcal{G}}_{ r + 1 }( x )$ instead of $ {\mathcal{G}}_1( x ) $. We define $$s' := \min \{\, l \in {\mathbb{Z}}_+ \mid l \geq r + 1 \,\wedge\, D_{ l + 1 } \neq 1 \,\}$$ and we use for the formulas $( g_{B,i}, b_h \cdot \nu_{ r + 1} ) $ (and $ (D_{r+1}, 1- \nu_{r+1}) $) instead of $ ( f_i, b_ i )$.
In general, let $ ( f ) = ( f_1, \ldots, f_m ) $ be generators of $ J $. Then $ {\mathcal{G}}_1 ( x ) = \bigcap_{ i = 1 }^m ( f_i, b_i ) $. Set $ ( f, b ) = ( f_i, b_i ) $ for some $ i $. After $ r $ steps in Construction \[Constr:nu\], we have determined $ ( z ) = (z_1,\dots,z_r) $, $ \nu_1,\nu_2,\ldots, \nu_{r+1} $ and $ D_2(x), \ldots D_{r+1}(x) $. By the definitions, $ ( b - b_1) \nu_2 - b_2 = \nu_2 \big( b - L_{ \nu ( 2 ) } ( b_1, b_2 ) \big) $. One can check that $ f $ can be written as $ f ( u,z ) = $ $$= F_b(z,D) + \sum_{ L_{\nu(r)} (B) < b} z^B \cdot D_2^{b-b_1}\cdot D_3^{(b-b_1)\nu_2 - b_2} \cdots D_{r+1}^{(\nu_2\cdots\nu_r)(b- L_{\nu(r)}(B))} \cdot F_B(u) + f^\ast,$$ for some $ f^\ast = f^\ast ( u, z) \in \langle z \rangle^{ b + 1 } $ and $ F_b(z,D) $ is ([with respect to ]{}$ L_{\nu(r)}(B) = \sum_{i=1}^r \beta_i \, b_i $) quasi-homogeneous of degree $ b $ in the variables $ z $ .
But the exceptional components $ D_2, \ldots, D_{r+1} $ are also involved in $ F_b(z,D) $. With the above formula we can give a description of the $ {\mathcal{G}}_1 $-part of $ {\mathcal{G}}_r(x) $, $ {\mathcal{F}}_r ( x ) $, $ {\mathcal{H}}_r(x )$ resp. $ {\mathcal{G}}_{r+1} (x) $ similar to the one in Lemma \[Prop:MainThmNoExc\] resp. Remark \[Rk:quasihomogeneous\]. But still there may be also an $ E $- and a $ D $-part.
[BHM10]{}
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[^1]: Supported by the Emmy Noether Programme “Arithmetic over finitely generated fields" (Deutsche Forschungsgemeinschaft) and by a Research Fellowship of the Deutsche Forschungsgemeinschaft.
|
---
abstract: 'Wearable devices have the potential to enhance sports performance, yet they are not fulfilling this promise. Our previous studies with 6 professional tennis coaches and 20 players indicate that this could be due the lack of psychological or mental state feedback, which the coaches claim to provide. Towards this end, we propose to detect the flow state, mental state of optimal performance, using wearables data to be later used in training. We performed a study with a professional tennis coach and two players. The coach provided labels about the players’ flow state while each player had a wearable device on their racket holding wrist. We trained multiple models using the wearables data and the coach labels. Our deep neural network models achieved around 98% testing accuracy for a variety of conditions. This suggests that the flow state or what coaches recognize as flow, can be detected using wearables data in tennis which is a novel result. The implication for the HCI community is that having access to such information would allow for design of novel hardware and interaction paradigms that would be helpful in professional athlete training.'
author:
- |
Cem Eteke[^1] Hayati Havlucu[^2] Nisa İrem Kırbaç Mehmet Cengiz Onbaşlı[^3] Aykut Coşkun Terry Eskenazi[^4] Oğuzhan Özcan Barış Akgün\
Koç University\
Istanbul, Turkey\
`{ceteke13, hhavlucu16, nkirbac, monbasli,`\
`aykut-coskun, teskenazi, oozcan, baakgun}@ku.edu.tr`\
bibliography:
- 'references.bib'
title: 'Flow From Motion: A Deep Learning Approach'
---
Introduction
============
**Wearable technologies** are rapidly advancing thanks to the developments in consumer electronics, with activity trackers leading the way. However, these devices have yet to fulfill their promise of revolutionizing the way we live. The abandonment rate is relatively high as well. There are many hypotheses out there for why this could be, from perceived “ugliness” of the device design to lack of features [@Lazar:2015:WWU:2750858.2804288]. Our research agenda is to tackle these both aspects and in this paper, we focus on the latter. We are specifically interested in detecting **players’ mental state** by using a wearable device. Our interviews with 6 professional coaches and 20 professional players show that: (1) players do not need/want wearable devices to track their fitness level since they are already self-aware in this respect and (2) their biggest concern is about the tracking and learning to regulate their **mental states** [@havlucu2017understanding].
In this study, we investigate whether wearable devices, specifically a commercially available activity tracker, can be used to detect more than an activity, such as a psychological state. To reach this goal, we set out to detect the **flow state**, the mental state of optimal performance of players as they play the game using wearables as a first step towards this end. Csikszentmihalyi [@csikszentmihalyi1990flow] defines the flow state as “*putting oneself in a state of optimal experience, the state in which people are so involved in an activity that nothing else seems to matter*”. The coaches we interviewed claim to be able to observe whether their players are in the flow state or not. This suggests that flow, or what coaches call being *in-the-zone* and *fall*, can be detected, at least for tennis.
Motivated by this, we performed a study involving an experienced coach working with professional tennis players and two of his students. Each player wore a wearable device which recorded data while the coach indicated when the players were in flow or not. Using the coach’s labels as targets and the recorded data as inputs, we trained multiple **machine learning** models. We reached around 98% testing accuracy using **deep neural networks** for a variety of conditions involving multiple data combinations. Our results show that the **flow state** can be detected using wearables data from an **Inertial Measuring Unit (IMU)**. To the best of our knowledge, this has never been demonstrated before.
Related Work
------------
Existing work on wearables data in sports mostly concentrate on activity recognition. Um et al. uses *deep learning* to classify exercise motion from large-scale wearable sensor data achieving 92.14% accuracy with a 3-layered Convolutional Neural Network (CNN) [@um2016exercise]. In [@chernbumroong2011activity], 5 activities including sitting, standing, lying, walking, and running are classified using Decision Trees and Artificial Neural Networks using a wrist-worn accelerometer. The authors use 4 separate feature sets from time and frequency domains achieving 94.13% accuracy with their best models. In [@connaghan2011multi], the authors try to classify tennis strokes - forehand, backhand, and serves - of the players using an IMU which is equipped with accelerometer, gyroscope and magnetometer sensors. They have achieved 90% accuracy using the fusion of accelerometer, gyroscope and magnetometer sensors. There are other studies that take advantge of IMU sensors in wearable devices. In [@spriggs2009temporal], the authors use a camera and IMU for temporally segmenting human motion into primitive actions. Our work focuses on detecting the flow state as opposed to a specific activity.
Existing literature about flow state detection includes different sensors and are concerned with different tasks. In [@bian2016framework], the heart rate, interbeat interval, heart rate variability (HRV), high-frequency HRV (HF-HRV), and respiratory rate are argued to be effective indicators of flow. In [@de2010psychophysiology], the authors try to find a relationship between subjective flow and psychophysiological measures while playing piano. They measure arterial pulse pressure, respiration, head movements (via a 3-axis accelerometer) and certain facial muscle activity. They did not find any significant relationship between flow and the head movements. In [@nacke2008flow], the authors use electroencephalography, electrocardiography, electromyography, galvanic skin response, and eye tracking equipment to detect the flow state of participants playing a video game. Our approach of detecting flow in a sports application using IMUs has not been done before.
------------------- ------------- -------------
(r)[2-3]{} [**Min**]{} [**Max**]{}
[GravityX]{} -1.0 1.0
[GravityY]{} -1.0 1.0
[GravityZ]{} -1.0 1.0
[AccelerationX]{} -17.1031 7.0596
[AccelerationY]{} -16.2396 16.6477
[AccelerationZ]{} -16.3296 16.8778
[RotationRateX]{} -26.3145 40.8265
[RotationRateY]{} -39.8416 32.0937
[RotationRateZ]{} -35.2566 25.3197
[AttitudeYAW]{} $-\pi$ $\pi$
[AttitudeROLL]{} $-\pi$ $\pi$
[AttitudePITCH]{} $-\pi/2$ $\pi/2$
------------------- ------------- -------------
: The collected motion data and their respective ranges.[]{data-label="tab:rawrange"}
Method
======
Data Collection
---------------
The data was collected using two Apple Watches (Series 2) linked to two Apple iPhones, worn by two tennis players on their racket holding wrists during a match. The flow labels were recorded separately by the players’ coach as a binary variable. The duration of the match was 74 minutes. The devices start recording before the match begins. In order to capture the data, we use a self-developed application that collects the raw data. The players locate themselves in a corner of the field and raise their hand for 3 seconds. The players then move down the line to the other corner and raise their hand for another 3 seconds. This procedure is done to synchronize the recordings. The motion data was collected for each player for every decisecond (i.e. with 10 Hz) throughout the match. Table \[tab:rawrange\] summarizes the data and their respective ranges in the recorded data.
The flow labels are recorded on a separate iPhone via another application. The coach observes the match and uses volume up and volume down keys to capture the flow state while speaking out the labels. The flow labels are also written down by one of the researchers next to the coach. This is done to cross-validate the recorded labels from the app in case the coach forgets or mis-presses the buttons on the device.
As our wearables data, we use motion data captured by an IMU sensor including gravity relative coordinate axes (3D), acceleration along these axes (3D), rotation rate (angular velocity) about these axes (3D), and attitude relative to the magnetic north reference frame (YAW, ROLL, PITCH). The players’ heart rates and GPS locations were also recorded to help with flow detection but large chunks of missing data and the poor accuracy of GPS hindered these useless.
Data Cleaning and Preprocessing
-------------------------------
{width="1\columnwidth"}
\[fig:flows\]
The motion data was pre-processed before being used. The duplicated entries were removed. Next, the data was averaged using a sliding window of size 5. Then, the entries were scaled between -1 and 1. Fig.The working data for the models contain 44,516 entries for each player, amounting to around 74 minutes. Player 1 and 2 are in flow 51.11% and 49.95% of the time respectively. Figure \[fig:flows\] shows the flow states of the players during the match using the pre-processed data. 1 denotes *flow* and -1 denotes *fall*.
Learning
========
------------ --------------- --------------
(r)[2-3]{} [**Train**]{} [**Test**]{}
[B-B]{} 0.9(P1+P2) 0.1(P1+P2)
[B-P1]{} 0.9P1+1.0P2 0.1P1
[B-P2]{} 1.0P1+0.9P2 0.1P2
[P1-P1]{} 0.9P1 0.1P1
[P1-P2]{} 1.0P1 1.0P2
[P2-P1]{} 1.0P2 1.0P1
[P2-P2]{} 0.9P2 0.1P2
------------ --------------- --------------
: Train/test split combinations used in models. (B: Both Players - P1: Player 1 - P2: Player 2)[]{data-label="tab:datacombination"}
The pre-processed data along with coach labels were used to train multiple binary-classifiers. We use 7 different data combinations and corresponding train-test splits to evaluate our models. These combinations are presented in Table \[tab:datacombination\].
Conventional methods, other than k-Nearest Neighbors (kNN) and random forests performed poorly, barely beating random choice (50% accuracy) as shown in Figure \[fig:modelsum\].
Due to the poor performance of the conventional approaches, we used convolutional neural networks (CNNs) and recurrent neural networks (RNN). To further capture the sequential nature of data, our input to these models are formed by combining 10 sequential data points in a sliding windows fashion, resulting in an input dimensionality of $10\times 12$.
The CNN model has three 2D convolutional layers. The first layer has an output size of 128 with a kernel size of $1\times 12$. The second and third layers have size 256 and 512 respectively with kernel size of 1, which results in weight sharing between each time step.
Activation function of each hidden unit is ReLU and batch normalization is applied after each convolution layer. The last convolutional layer is followed by a fully connected layer of size 128 with ReLU activation function and an output layer of size 2. Softmax function is applied to the output to get flow-state probabilities.
The RNN model has a single layer of Long Short-Term Memory (LSTM) with hidden unit of size 512 attached to a fully connected layer of size 128 with ReLU activation function and an output layer of size 2. Softmax function is applied to the output to get flow-state probabilities.
To train both models, we used the Adam optimizer with 0.001 learning rate with no decay and a mini-batch size of 64 to minimize cross-entropy loss. We included a 0.5 dropout rate before the output layer in both models. We further include a 0.25 dropout rate after the second convolution later in the CNN based model.
The kNN model uses 1 neighbor and the SVM model uses RBF kernel with $\gamma = 1/12$ and soft-margin cost of $C=1000$ (the latter selected via cross-validation). Figure \[fig:modelsum\] illustrates the testing accuracy of the models for the data combinations depicted in Table \[tab:datacombination\].
{width="1\columnwidth"}
\[fig:modelsum\]
Results
=======
*Coach’s perception of flow can be detected from IMU data in tennis with high accuracy.* The CNN and LSTM models reach around 98% testing accuracy. Even with a simple approach like kNN, we get around 75% accuracy.
*Flow cannot be generalized from a single player.* All the methods are around 50% when we look at the P1-P2 and P2-P1 parts of the Figure \[fig:modelsum\]. This shows that, data from just one player cannot be used to detect flow in others.
*Flow maybe generalized with more player data.* When we look at the B-B, B-P1 and B-P2 parts of the Figure \[fig:modelsum\], we can see that results are as good or better than the P1-P1, and the P2-P2 case. This shows that using more player data may improve flow detection accuracy. This suggests that with more data, we maybe able to detect flow in players we have not trained with but this needs further study.
*Deep neural network based models outperform conventional methods.* The models based on CNN and LSTM have the best results but CNN is slightly better than LSTM for combinations other than P1-P2 and P2-P1. SVM has a very poor performance, barely beating the random chance of 50% accuracy. The kNN approach with one neighbor is more successful but it is still not competitive[^5]. One reason is that these methods do not account for the sequential nature of the data and utilizing methods such as Hidden Markov Models (HMM) and Conditional Random Fields (CRF) may help. However, our preliminary trials with augmented states (concatenated multiple time steps) and HMMs lacked behind deep models.
We think that the flow signal is in the IMU data but we need sophisticated models with lots of data to detect it.
Implications and Future Work
============================
Our end-goal is to be able to detect flow state in professional tennis players. The novel results presented in the previous section strongly support this aim. There are two main directions to take this study; verify flow state detection and advance it and further use the successful detection results to develop devices and interaction paradigms to be used in training to regulate flow state.
Even though the results are highly encouraging, there are still certain challenges to be addressed. These are; (1) do we need to have training data for a player to be able to detect his/her flow state or can we collect enough data to be able to generalize cross professional tennis players? In other words, can we detect flow in a player we have no training data for? (2) do the movements of professional tennis players change over time and with training such that it affects flow detection in the future? In other words, would the data we collect now be used to detect flow in a player in the future as well? (3) is what perceived as flow by the coach is really flow and whether this matters or not?
To address challenges (1) and (2) we need to conduct further studies and collect more data. To address (2) specifically, we need to collect data from the same players over time. Collecting more data is our immediate future work. To address the first half of challenge (3), we need to be able to measure flow directly and to see whether the coach labels are correlated with the measurements. There is no easy way to do this with the current body of flow state knowledge. To address the second half of this challenge, we are planning to follow the first research direction and develop a wearable device for training and see if it works.
A problem that tennis players face is that tennis is a lonely sport and it is hard for them to recover after they lose concentration [@havlucu2017understanding]. It is important for these players to train mentally to be able to cope with such difficulties. Not all players get to train with capable coaches or get enough individual training time with them. A wearable device that can help with such mental training would be invaluable. Detecting whether the player is in flow state or not is the first step towards this end. For example, if the device detects that the player goes out-of-flow, it can interact with the user or provide feedback - which is necessary to maintain flow - to help the player get back in flow. We are going to conduct user studies to validate the device and our approach in general.
Conclusion
==========
In this study, we concentrate on the flow state, mental state of optimal performance, in tennis. We collect flow labels from a professional coach during a tennis match between two of his players and IMU data from the players themselves. We then train several models using this data.
Our findings show that flow, or what the coach perceived as flow, can be detected from IMU data. Most successful methods, two deep learning models, reach around 98% testing accuracy in a variety of data combinations. The results are the same or better if we have both players’ data in the training set. However, one player’s data cannot be used to detect flow in the other player. These findings about flow state detection is first in the field.
There are two immediate directions for this study. First, to address data collection and generalization challenges in flow state detection; and second, to develop devices and interaction paradigms to help professional tennis players to train to regulate their flow. We are interested in pursuing both of these directions simultaneously.
[^1]: Department of Computer Engineering
[^2]: Arçelik Research Center for Creative Industries
[^3]: Department of Electrical and Electronics Engineering
[^4]: Department of Psychology
[^5]: The story is similar with other conventional methods.
|
---
abstract: 'Near- and mid-IR absorption spectra of endohedral H$_2$@C$_{60}^+$ have been measured using He-tagging. The samples have been prepared using a ’molecular surgery’ synthetic approach and were ionized and spectroscopically characterized in the gas phase. In contrast to neutral C$_{60}$ and H$_2$@C$_{60}$, the corresponding He-tagged cationic species show distinct spectral differences. Shifts and line splittings in the near- and mid-IR regions indicate the influence of the caged hydrogen molecule on both the electronic ground and excited states. Possible relevance to astronomy is discussed.'
author:
- 'Dmitry V. Strelnikov'
- 'Juraj Jaš[í]{}k'
- Dieter Gerlich
- Michihisa Murata
- Yasujiro Murata
- Koichi Komatsu
- 'Jana Roithov[á]{}'
title: 'Near- and Mid-IR Gas-Phase Absorption Spectra of H$_2$@C$_{60}^+$-He'
---
Introduction
============
Endohedral H$_2$@C$_{60}$ can be synthesized in macroscopic quantities using a ‘molecular surgery’ approach [@Komatsu2005]. This has already led to numerous investigations of H$_2$@C$_{60}$, using methods such as NMR [@Carravetta2006; @Sartori2006] or IR absorption spectroscopy [@Mamone2009; @Room2013]. The H$_2$ inside C$_{60}$ has enough space to behave like an almost free gas-phase molecule; however, confinement leads to observable coupling of its’ rotational, vibrational and translational degrees of freedom [@Mamone2009]. Recently, five of the many diffuse interstellar bands (DIBs) [@Snow2006] have been assigned to C$_{60}^+$ using He-tagging spectroscopy in the Basel cryogenic ion trap [@Campbell2016]. This observation together with the fact that hydrogen is the most abundant element in the Universe, raises the questions of whether endohedral H$_2$@C$_{60}$ might be formed in Space as well and under which conditions [@Omont2016]. In contrast to the exohedral van der Waals complexes, endohedral H$_2$@C$_{60}^+$ can survive under ionizing conditions and higher temperatures, possibly present in some regions of Space. This stability of H$_2$@C$_{60}^+$ is demonstrated by the fact that this ion can be observed in electron ionization (EI) mass spectra of sublimed H$_2$@C$_{60}$. In order to find out whether H$_2$@C$_{60}^+$ is present in Space and provide guidance for astronomy, we investigated optical spectroscopic signatures of H$_2$@C$_{60}^+$ in the near IR range.
Experimental Methods
====================
The experiments have been performed using the cryogenic wire quadrupole ion trap of the Prague instrument ISORI (Infrared Spectroscopy of Reaction Intermediates), described in detail in @Jasik2013 [@Jasik2015a; @Jasik2015b]. Many details concerning the production and cooling of fullerene ions and tagging them with helium atoms can be found elsewhere[@Gerlich2018a; @Gerlich2018b]. Briefly, ions have been produced in a Finnigan Solids Probe EI-source with 65 eV electrons. Experiments have been performed with three samples, (i) 10 mg of 100% H$_2$@C$_{60}$, (ii) 10 mg of 80% H$_2$@C$_{60}$/20% C$_{60}$ and (iii) C$_{60}$ (99,5% purity). The endohedral probes have been synthesized using a ‘molecular surgery’ approach [@Komatsu2005]. The C$_{60}$ sample was obtained from SES Research. Ions, emerging from the source, are mass-selected by a quadrupole mass filter and transferred via a quadrupole bender and an octopole to the cryogenic quadrupole ion trap. The temperature of the cold head has been 3.7 K. The mass-selected ions were trapped and relaxed by collisions with helium buffer gas. A part of the ions formed complexes with helium atoms. The trapped ions were probed by IR photons from an OPO system (LaserVision) pumped by a Nd:YAG laser (Surelite EX from Continuum, 10 Hz repetition rate, 6ns pulse width). Using an injection seeder the line width of the OPO is smaller than 1.5 cm$^{-1}$. The wavelength is measured using a high-precision wavelength meter (HighFiness WS6-200). For the near-IR electronic transitions, a Sunlite EX OPO tunable System (Continuum) is used, pumped with a seeded PL 9010 (line width $<$ 0.1 cm$^{-1}$). Absorption of photons was monitored by counting the number of helium complexes via a second quadrupole filter followed by an ion detector. If a complex absorbs a photon the helium is eliminated. In order to correct for any background the light beam is blocked every other second with a mechanical shutter, resulting in numbers of unperturbed ions, $N_0$, while $N(\nu)$ is the number of complexes remaining after irradiation at frequency $\nu$. For the IR measurements $N(\nu)$ was obtained from 10 Laser pulses during one second, followed by the one second $N_0$ accumulation. For the near-IR measurements the number of the laser pulses was reduced to one. The laser pulse energy have been changing in the 0.017-0.08 mJ range for the IR measurements and in the 0.16-0.83 mJ range for the near-IR measurements. The laser beam diameters were 2 mm and 1 mm for the IR and near-IR measurements correspondingly. Laser scan velocities were 0.2 cm$^{-1}$/s for the IR and 0.03 nm/s for the near-IR.
We present the spectra either as attenuation spectra, plotting $1-N(\nu)/N_0$, or we correct in first order for non-linear attenuation by calculating relative cross section using the equation: $\sigma=-\Phi^{-1}\ln(N(\nu)/N_0)$, where $\Phi$ is number of photons per cm$^2$ the ions have been exposed to.
Raman measurements have been performed in Karlsruhe on powder samples of neutral H$_2$@C$_{60}$ and C$_{60}$ using a Raman spectrometer Kaiser Optics RXN1 (785 nm excitation, 5 cm$^{-1}$ spectral resolution).
Results and Discussion
======================
Vibrational spectroscopy.
-------------------------
Fig. \[raman\] compares Raman spectra of 80% H$_2$@C$_{60}$ (red) with those from C$_{60}$ bulk samples. A careful inspection reveals that the hydrogen inside the C$_{60}$ does not influence the Raman spectrum. The line positions agree within 0.4 cm$^{-1}$. Since Raman scattering probes the vibrational levels in the electronic ground state, the state of the neutral endohedral C$_{60}$ is the same as that of the empty C$_{60}$. Some weak interference in the H$_2$@C$_{60}$ spectrum is caused by the luminescence of traces of impurities. (The interference is an instrumental artifact. It could be clearly observed when measuring continuous spectra. Probably, it is generated in the optical fiber of the Raman probe.)
{width="\textwidth"}
Fig. \[IR\] compares our gas-phase spectra for H$_2$@C$_{60}^+$-He with C$_{60}^+$-He from @Gerlich2018a and with C$_{60}^+$ recorded in a cryogenic Ne-matrix [@Kern2013]. The two gas-phase spectra differ in the bands’ shifts and splittings. These differences reflect the influence of H$_2$ on the ground state of C$_{60}^+$. Contrary to the neutral systems (see above), the cations apparently interact more strongly with the H$_2$ molecule inside. The C$_{60}^+$ cation has a $D_{5d}$ ground state symmetry [@Kern2013], therefore, not all directions for translational motion of H$_2$ inside the cage are equivalent. H$_2$ can move further from the fullerene center of gravity along the $C_5$ symmetry axis. In addition, a delocalized positive charge on a fullerene cage enhances the van der Waals interaction with H$_2$.
{width="\textwidth"}
The structure of the IR absorptions in the helium tagged fullerene ions may originate from the interaction with the attached He atom or be caused by vibrational coupling of C-C modes, the details are discussed elsewhere [@Gerlich2018a].
The splittings of IR bands differ for H$_2$@C$_{60}^+$-He and C$_{60}^+$-He. The H$_2$@C$_{60}^+$-He has a reduced splitting in comparison to the empty C$_{60}^+$-He. In addition, the band at 1170 cm$^{-1}$ is either very weak or absent for H$_2$@C$_{60}^+$-He. These effects are apparently induced by the distortion from H$_2$, but the theoretical calculations do not offer a reasonable qualitative explanation of this effect. Suprisingly, the IR spectrum of H$_2$@C$_{60}^+$-He resembles more the IR spectrum of C$_{60}^+$ in neon matrix than that of C$_{60}^+$-He in the gas phase. A qualitative outcome of this measurement is the following: the more perturbation C$_{60}^+$ experiences, the larger blue shift and the smaller band splitting are observed. At present, we are not able to rationalize this by the DFT calculations (see Supporting information). The Lorentzian fit parameters of the experimental data: the band positions, relative heights and FWHMs are compared in Table \[IRtable\].
-------- -------- ------ -------- -------- ------ --------- -------- ------
$\nu$ rel. FWHM $\nu$ rel. FWHM $\nu$ rel. FWHM
height height height FWHM
1172.5 0.6 4.8
1177.8 0.7 3.9 1182.4 0.3 11.3
1190.7 0.2 6.1 1194.7 0.8 4.1
1210.7 0.5 3.6 1212.5 0.3 4.0
1216.3 1.4 8.7 1218.1 1.0 4.6
1222.4 0.5 2.8 1221.4 1.5 23.6 1223.7 0.8 4.0
1239.6 0.2 4.4
1272.5 0.1 4.2
1306.8 0.5 18.4 1317.4 0.5 35.5
1326.6 1.8 4.2 1333.8 2.1 13.3 1331.1 1.2 3.2
1336.4 0.7 4.5 1333.6 0.5 1.2
1344.6 0.3 3.2 1335.15 0.5 1.9
1354.9 0.2 7.2
1397.3 1.7 7.2 1403.5 6.4 11.0 1407.12 5.5 7.4
1401.6 1.6 6.8 1409.4 2.3 4.7
1406.3 1.1 3.0
1428.9 0.1 66.3
1534.1 0.4 7.9 1535.4 0.6 5.2
1544.1 1.4 6.4 1551.2 3.8 15.1 1557.6 2.2 9.2
1559.8 0.8 7.0 1563.6 0.9 5.0
1574.8 0.9 23.1
-------- -------- ------ -------- -------- ------ --------- -------- ------
\
^*a*^ Data from @Gerlich2018a; ^*b*^ Data from @Kern2013.
Electronic spectroscopy.
------------------------
The geometry of a linear quadrupole trap together with high laser fluence in our experimental setup allows relatively fast acquisition of overview spectra: the 860 – 970 nm region have been scanned in about one hour. Comparison of our data with @Campbell2016 [@Campbell2018] reveals minor differences in band positions (about 0.3 Å) but pronounced deviations in band widths and intensities (see Figs. S1,S2 in Supporting Information). These deviations originate from the non-linearities, caused by a high laser power in our measurements. The high laser power leads to the saturation of strong absorption lines, but allows to see weak absorptions. Although the attenuation for many saturated bands is about 0.8 (see below), these bands show definitively a saturation effect. The experimental geometry of the ISORI setup is such, that one has a complete overlap of the laser beam and the trapped ions. Currently, the saturation effects are not completely understood and require further dedicated experiments. Despite the deviations, the fast overview scan allows to obtain reliable absorption wavelengths. An estimate of the absorption intensities is more difficult. This can be seen from the fact that we observed more spectral bands of C$_{60}^+$-He than previously reported [@Campbell2016; @Campbell2018]. Some of the narrow lines have been not resolved, because of the 0.25 nm step wavelength sampling. For a better resolution one would need to scan slowly with a denser sampling.
{width="\textwidth"}
The electronic spectra of C$_{60}^+$-He and H$_2$@C$_{60}^+$-He show pronounced differences in band positions as well as in the band structures at higher energies (Fig. \[NIRsigma\]). Hence, importantly the near-IR absorption spectra of C$_{60}^+$ and H$_2$@C$_{60}^+$ can be clearly distinguished. We fitted the spectra with Lorentzian functions and present the obtained fit parameters in Table \[NIRtable\]. Because of the limited time for the measurements, we did not obtain absolute cross sections of the observed H$_2$@C$_{60}^+$-He absorption bands, which require power dependence measurements. However, we used identical experimental settings for both measurements: C$_{60}^+$-He and H$_2$@C$_{60}^+$-He. Thus the measured absorption intensities for both systems are comparable (Fig. \[NIRsigma\]), and therefore the absolute absorption cross sections should be also similar. For a direct comparison with astronomical data, one should consider a possible influence of an attached He atom on the absorption wavelengths. As in the case of C$_{60}^+$, we expect for an untagged H$_2$@C$_{60}^+$ a $\sim$0.7 Å blue shift[@Campbell2016He1-3] of the near-IR absorption wavelengths. In addition, the presented here vacuum wavelengths should be converted to the air values. Inspection of the present DIBs catalogues [@Hobbs2008; @Hamano2016] reveals no absorption bands, corresponding to H$_2$@C$_{60}^+$. The upper limit estimation for the presence of H$_2$@C$_{60}^+$ in the interstellar medium depends on the signal to noise ratio of the astronomical data. There is a complication arising from the atmospheric water absorption lines in the near-IR region. Using the astronomical data[@MaierDIBs2015_2; @Cox2017] we estimated the upper limit for the H$_2$@C$_{60}^+$ to C$_{60}^+$ ratio to be $\sim$0.1.
----------- -------- ------ ----------- -------- ------
$\lambda$ rel. FWHM $\lambda$ rel. FWHM
height height
962.33 0.0165 1.64 963.57 0.0170 0.68
956.42 0.0171 1.27 958.04 0.0145 1.01
942.89 0.0013 0.93
941.61 0.0069 0.76 943.14 0.0055 0.68
938.37 0.0016 0.80 940.63 0.0010 0.39
937.33 0.0061 0.41 939.01 0.0014 0.54
935.55 0.0073 0.81 936.80 0.0049 0.85
934.80 0.0048 0.33
933.38 0.0035 0.39 935.14 0.0032 0.63
930.60 0.0016 0.38 932.25 0.0016 0.27
924.85 0.0019 0.56 926.24 0.0018 0.23
921.07 0.0025 0.79 922.43 0.0015 0.45
919.63 0.0023 0.29 921.48 0.0013 0.31
919.08 0.0031 0.41 920.29 0.0019 0.40
918.49 0.0022 0.34 919.79 0.0007 0.49
915.83 0.0017 0.43 916.98 0.0014 0.52
913.77 0.0041 0.34 915.31 0.0020 0.31
912.83 0.0022 0.34 914.29 0.0016 0.40
910.53 0.0031 0.27 912.16 0.0015 0.28
907.31 0.0028 0.33 908.50 0.0020 0.59
----------- -------- ------ ----------- -------- ------
{width="\textwidth"}
Fig.\[NIRatt\] shows a section from Fig. \[NIRsigma\] as attenuation. For better comparison, the H$_2$@C$_{60}^+$-He spectrum has been blue-shifted by about 1.5 nm with respect to the C$_{60}^+$-He spectrum. Detailed inspection reveals that some of the lines in the H$_2$@C$_{60}^+$-He spectrum exhibit some splitting. Apparently, H$_2$ perturbs the excited state of C$_{60}^+$. This is, however, not surprising, since the ground state is also influenced by the presence of H$_2$ inside the cage. So far there is no theoretical work, properly describing and explaining the vibronic transitions of C$_{60}^+$ in the $^2$E$_{1g}$ state. With a hydrogen molecule inside the cage it is an even more challenging system for a theoretical treatment than C$_{60}^+$. TDDFT was done for several different orientations of H$_2$. The theory predicts the blue shifts up to 2.5 nm and also the splitting for some of the H$_2$ positions (see Supporting Information). Line shifts in the electronic spectrum, introduced by H$_2$ inside a fullerene are much larger, than those, induced by adsorption of Ne, Ar, N$_2$, H$_2$, D$_2$ outside the C$_{60}^+$ [@Holz2017]. The external molecule/atom adsorption on C$_{60}^+$ molecules mostly leads to red shifts in the near-IR spectra [@Holz2017]. Conversely, we observe a considerable blue shift in the spectrum of the endohedral H$_2$@C$_{60}^+$ ion.
Conclusion
==========
Gas-phase mid-IR and near-IR measurements show distinct spectral differences between H$_2$@C$_{60}^+$ and C$_{60}^+$. The correct quantitative description of the observed differences in the absorptions requires dedicated theoretical modeling. There are no known DIBs at the absorption positions of H$_2$@C$_{60}^+$. The estimated upper limit of the interstellar H$_2$@C$_{60}^+$/C$_{60}^+$ ratio is $\sim$0.1.
The project was funded by the European Research Council (ERC CoG No. 682275) and the Deutsche Forschungsgemeinschaft (KA 972/10-1). We also acknowledge support by KIT and Land Baden-Württemberg.
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Supporting Information {#supporting-information .unnumbered}
======================
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abstract: 'Machine learning has shown much promise in helping improve the quality of medical, legal, and economic decision-making. In these applications, machine learning models must satisfy two important criteria: (i) they must be causal, since the goal is typically to predict individual treatment effects, and (ii) they must be interpretable, so that human decision makers can validate and trust the model predictions. There has recently been much progress along each direction independently, yet the state-of-the-art approaches are fundamentally incompatible. We propose a framework for learning causal interpretable models—from observational data—that can be used to predict individual treatment effects. Our framework can be used with any algorithm for learning interpretable models. Furthermore, we prove an error bound on the treatment effects predicted by our model. Finally, in an experiment on real-world data, we show that the models trained using our framework significantly outperform a number of baselines.'
bibliography:
- 'paper.bib'
---
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abstract: 'Part-and-parcel of the study of “multiplicative number theory” is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we find out why such a result has been so elusive, and discover what can be proved along these lines and develop some limitations. For a fixed residue class $a$ we extend such averages out to moduli $\leq x^{\frac {20}{39}-\delta}$.'
address:
- |
AG: Département de mathématiques et de statistique\
Université de Montréal\
CP 6128 succ. Centre-Ville\
Montréal, QC H3C 3J7\
Canada; and Department of Mathematics\
University College London\
Gower Street\
London WC1E 6BT\
England.
- |
Mathematical Institute\
Radcliffe Observatory Quarter\
Woodstock Road\
Oxford OX2 6GG\
United Kingdom
author:
- Andrew Granville
- Xuancheng Shao
title: 'Bombieri-Vinogradov for multiplicative functions, and beyond the $x^{1/2}$-barrier'
---
[^1]
[^2]
[^3]
Formulating the Bombieri-Vingradov Theorem for multiplicative functions
=======================================================================
The Bombieri-Vingradov Theorem, a mainstay of analytic number theory, shows that the prime numbers up to $x$ are “well-distributed” in all arithmetic progressions mod $q$, for almost all integers $q\leq x^{1/2-\varepsilon}$. To be more precise, for any given sequence $f(1), f(2), \ldots$, we define $$\Delta(f,x;q,a):=\sum_{\substack{n\leq x \\ n\equiv a \pmod q}} f(n) - \frac 1{\varphi(q)} \sum_{\substack{n\leq x \\ (n,q)=1}} f(n).$$ The Bombieri-Vinogradov Theorem states that if $f$ is the characteristic function for the primes, then for any given $A>0$ there exists $B=B(A)>0$ such that if $Q\leq x^{1/2}/(\log x)^B$ then $$\label{eq:BVI}
\sum_{q\sim Q} \max_{a:\ (a,q)=1} | \Delta(f,x;q,a) | \ll \frac x {(\log x)^A},$$ where, here and henceforth, “$q\sim Q$” denotes the set of integers $q$ in the range $Q<q\leq 2Q$. The analogous result is known to hold when $f=\mu$, the Mobius function, and when $f$ is the characteristic function for the $y$-smooth numbers [@FT; @Har], and the literature is swarming with many other interesting examples besides. There are many proofs of the original Bombieri-Vinogradov Theorem: more modern proofs rely on bilinearity and Vaughan’s identity, for example see chapter 28 of [@Da], or the very elegant proof in Theorems 9.16, 9.17, and 9.18 of [@FI]. The extraordinary generality of the latter proof leads one to guess that something like the Bombieri-Vinogradov Theorem should be true for most sensible arithmetic functions $f$.
In this paper we focus on proving a Bombieri-Vinogradov type Theorem for multiplicative functions $f$ which take values within the unit circle, something that has been proved for several interesting examples, and one might guess is true in some generality. However one needs to be careful: if $f(n)=(n/3)$, the quadratic character mod 3, then $f$ is certainly not well-distributed in arithmetic progressions mod 3, nor in arithmetic progressions mod $q$, whenever 3 divides $q$. More generally if $f=\chi$ is a primitive character mod $r$, or even if $f$ is “close” to $\chi$, then $f$ is not well-distributed in arithmetic progressions mod $q$, whenever $r$ divides $q$. So this is a significant departure from the classical Bombieri-Vinogradov type Theorem, in that it seems likely that[^4] $$\sum_{q\sim Q} \max_{a:\ (a,q)=1} | \Delta(f,x;q,a) | \gg \max_{\substack{\chi \pmod r \\ \chi \ \text{primitive} \\ r>1}} \frac 1{\varphi(r)} |S_f(x,\chi)|,$$ where $$S_f(x,\chi):= \sum_{n\leq x} f(n) \overline{\chi}(n) .$$ Therefore if $f$ strongly correlates with some $\chi$ of “small" conductor (that is, $S_f(x,\chi)\gg x/(\log x)^C$ for some non-principal character $\chi$ of conductor $\ll (\log x)^D$, for for some fixed $C,D>0$) then cannot hold for all $A$. So to prove something like we need to assume that no such character exists. The usual way to formulate this involves equidistribution for moduli of small conductor (see e.g. (9.68) in [@FI]): For any given $A>0$ we have
**The $A$-Siegel-Walfisz criterion**: *If $(a ,q)=1$ then* $$\sum_{\substack{n\leq x \\ n\equiv a \pmod q \ }} f(n) - \frac 1{\varphi(q)} \sum_{\substack{n\leq x \\ (n,q)=1}} f(n) \ll_A \frac x {(\log x)^A}$$ for all $x \geq 2$. We say that $f$ *satisfies the Siegel-Walfisz criterion* if this holds for any fixed $A>0$.
Let $$F(s) = \sum_{n=1}^{\infty} \frac{f(n)}{n^{s}}\ \ \text{and} \ \
- \frac{F'(s)}{F(s)} = \sum_{n=2}^{\infty} \frac{\Lambda_f(n)}{n^{s}},$$ for Re$(s)>1$. Following [@GHS], we restrict attention to the class ${\mathcal C} $ of multiplicative functions $f$ for which $$|\Lambda_f(n)|\leq \Lambda(n) \ \ \textrm{for all} \ \ n\geq 1.$$ This includes most multiplicative functions of interest, including all $1$-bounded completely multiplicative functions. Two key observations are that if $f\in \mathcal C$ then each $|f(n)|\leq 1$, and if $F(s)G(s)=1$ then $g\in \mathcal C$.
\[Thm: SW\] Let $f$ be a multiplicative function with $f\in \mathcal C$, and assume that $f$ satisfies the $1$-Siegel-Walfisz criterion. Fix $\delta, \varepsilon>0$. If $Q \leq x^{1/2-\delta}$ then $$\sum_{q\sim Q} \max_{a:\ (a,q)=1} | \Delta(f,x;q,a) | \ll \frac{x} {(\log x)^{1-\varepsilon}} .$$
This bound is considerably weaker than the hoped-for bound in that we improve upon the “trivial bound”, $\ll x$, by only a factor $(\log x)^{1-\varepsilon}$ rather than by an arbitrary power of $\log x$. However we will show in Proposition \[LargePrimes2b\] that Corollary \[Thm: SW\] is, up to the $\varepsilon$-factor, best possible.
Taking exceptional characters into account
------------------------------------------
Even if the $1$-Siegel-Walfisz crieterion does not hold we can prove a version of the Bombieri-Vinogradov Theorem which takes account of the primitive characters $\psi$ for which $S_f(x,\psi)$ is large.
For any character $\psi \pmod r$, define $$\sigma_f(x,\psi) := \sup_{x^{1/2}< X \leq x} |S_f(X,\psi)/X|.$$ We order the primitive characters $\psi_1 \pmod {r_1},\ \psi_2 \pmod {r_2},\ldots $ with each $r_i\leq \log x$, so that $$\sigma_f(x,\psi_1) \geq \sigma_f(x,\psi_2) \geq \ldots$$ which more-or-less corresponds to the ordering of $|S_f(x,\psi)|$, at least if these get “large”.
Notice that $ \sum_{ {n\leq x,\ (n,q)=1}} f(n) = S_f(x,\chi_0)$, and so $$\label{expansion}
\Delta(f,x;q,a)=\ \frac 1{\varphi(q)} \sum_{\substack{ \chi \pmod q \\ \chi \ne \chi_0 }} \chi(a) S_f(x,\chi).$$ It therefore makes sense to reformulate the Bombieri-Vinogradov Theorem, so as to remove the largest value(s) of $|S_f(x,\chi)|$ from the sum in .[^5] If $|S_f(x,\chi)|$ is large then $\chi \pmod q$ is induced from some $\psi_j$ with $1\leq j\leq k$; and there is a character $\chi_j \pmod q$ induced by $\psi_j$, if and only if $r_j|q$. Therefore we define $$\Delta_k(f,x;q,a):=\sum_{\substack{n\leq x \\ n\equiv a \pmod q}} f(n) - \frac 1{\varphi(q)} \sum_{\substack{ 1\leq j\leq k \\ r_j|q}} \chi_j(a) S_f(x,\chi_j).$$ We believe that $$\sum_{q\sim Q} \max_{a:\ (a,q)=1} | \Delta_k(f,x;q,a) | \gg \max_{j\geq k+1}\ \frac 1{\varphi(r_{j})} |S_f(x,\psi_{j})|$$ (see the discussion in section \[ClaimJusitify\]). Moreover $f$ can be chosen so that this is $\gg x/\log x$ for any fixed $k$ (see section \[ManyChar\]), unconditionally. Nonetheless this reformulation, taking into account the characters that correlate well with $f$, can lead to upper bounds which are almost of this strength, as we now state.
\[Cor:Result2\] Fix $\delta, \varepsilon>0$ and let $k$ be the largest integer $\leq 1/\varepsilon^2$. For any $f\in {\mathcal C} $, and for any $Q \leq x^{1/2-\delta}$, we have $$\sum_{q\sim Q} \max_{a:\ (a,q)=1} | \Delta_k(f,x;q,a) | \ll \frac{x} {(\log x)^{1-\varepsilon}} .$$
Corollary \[Thm: SW\] will follow from Theorem \[Cor:Result2\].
Theorem \[Cor:Result2\] is close to “best possible" in that (as we show in section \[ManyChar\]), that for given integer $k$ there is an $ \varepsilon' \asymp \frac 1{\sqrt{k}}$, for which there exists $f\in {\mathcal C} $ such that $$\sum_{q\sim Q} \max_{a:\ (a,q)=1} | \Delta_{k}(f,x;q,a) | \gg \frac{x} {(\log x)^{1-\varepsilon'}} .$$
In [@GHS] it is proved that if $\log q\leq (\log x)^\delta$ where $\delta=\delta(\varepsilon)>0$, then $$\label{eq:GHS}
| \Delta_k(f,x;q,a) | \ll \frac 1{\varphi(q)} \frac{x} {(\log x)^{1-\varepsilon}}$$ whenever $(a,q)=1$, where $k$ is the largest integer $\leq 1/\varepsilon^2$.[^6] That is, $f$ is well-distributed in all arithmetic progressions mod $q$ for *all* $q\leq Q$ provided $x$ is very large compared to $Q$, and in Theorem \[Cor:Result2\] we have shown good distribution in all arithmetic progressions mod $q$ for *almost all* $q\leq Q$ provided $x>Q^{2+\delta}$, a much larger range for $q$ but at the cost of some possible exceptions. This is reminiscent of what we know about the distribution of prime numbers in arithmetic progressions, though here we have significantly weaker bounds on the error terms.
Limitations on the possible upper bounds
----------------------------------------
One might guess that if there are no characters of small conductor that strongly correlate with $f$ then perhaps one can significantly improve the upper bounds in Theorem \[Cor:Result2\] and Corollary \[Thm: SW\] (if one assumes an $A$-Siegel-Walfisz criterion for all $A>0$). Unfortunately there is another obstruction, in which the values of $f(p)$ for the large primes $p$ up to $x$, do their utmost to block equi-distribution:
\[LargePrimes2\] Let $g$ be a multiplicative function with each $|g(n)|\leq 1$, and suppose we are given $Q\leq x$ with $Q,x/Q\to\infty$ as $x\to \infty$. There exists a subset $\mathcal P$ of the primes in the interval $(x/2,x]$, that contains almost all of those primes, and a constant $\sigma\in \{ -1,1\}$, such that if $f(n)=g(n)$ for all $n\leq x$ other than $n\in \mathcal P$, and $f(p)=\sigma$ for all $p\in \mathcal P$, then $| \Delta(f,x;q,1) |\gg \pi(x)/\varphi(q)$ for at least half of the moduli $q\sim Q$. This implies that $$\label{LowerBd}
\sum_{q\sim Q} | \Delta(f,x;q,1) | \gg \frac{x} {\log x } .$$
It is widely believed that for any fixed ${\varepsilon}, A>0$ we have $$\label{ExtendPNT}
\pi(x;q,a) = \frac{\pi(x)}{\varphi(q)} \left( 1 +O\left( \frac 1{(\log x)^A} \right) \right)$$ whenever $(a,q)=1$ and $q\leq x^{1 -\varepsilon}$.
We now show that even assuming a strong Siegel-Walfisz criterion, we expect that one cannot significantly improve the upper bound in Corollary \[Thm: SW\].
\[LargePrimes2b\] Assume . Let $x^{2/5} < Q < x/2$. There exists a completely multiplicative function $f$, taking only values $-1$ and $1$, which satisfies the $A$-Siegel-Walfisz criterion for all $A>0$, but for which holds.
We have exhibited two fundamental obstructions to a Bombieri-Vinogradov Theorem for multiplicative functions:
(i) If $f$ correlates closely with a character of small conductor; or
(ii) If the values of $f(p)$ with $x/2<p\leq x$ conspire against equi-distribution.
Consequently, although we have been able to beat the “trivial bound” by a factor of $(\log x)^{1-\varepsilon}$ by taking (i) into account in Theorem \[Cor:Result2\], (ii) ensures that we cannot do much better in general.
These two obstructions have arisen before in the multiplicative functions literature, in Montgomery and Vaughan’s seminal work [@MV] on bounding exponential sums twisted by multiplicative coefficients. For this question, the contributions from obstruction (i) are identified precisely in [@BGS], but the sharpness of the bounds are inevitably restricted by obstruction (ii). However, in Proposition 1 of [@dlB], de la Bréteche showed that one can obtain much better bounds if one restricts attention to $f$ that are supported only on smooth numbers,[^7] since then obstruction (ii) is rendered irrelevant. We can do much the same here:
Multiplicative functions supported on smooth numbers
----------------------------------------------------
The key result generalizes (which was proved in [@GHS]) to error terms in which one “saves” an arbitrary power of $\log x$, for multiplicative functions supported on smooth numbers.
Given a finite set of primitive characters, $\Xi$, let $\Xi_q$ be the set of characters mod $q$ that are induced by the characters in $\Xi$, and then define $$\Delta_\Xi(f,x;q,a):= \sum_{\substack{n\leq x \\ n\equiv a \pmod q}} f(n) - \frac 1{\varphi(q)} \sum_{ \substack{ \chi \in \Xi_q }} \chi(a) S_f(x,\chi) .$$
\[MainCor\] Fix $\varepsilon>0$ and $0<\gamma\leq \frac 12-\varepsilon$, let $y= x^{\gamma}$, and suppose that $f\in \mathcal C$, and is only supported on $y$-smooth numbers. Fix $B\geq 0$. There exists a set, $\Xi$, of primitive characters $\psi \pmod r$ with each $r\leq R := x^{ \varepsilon/(3\log\log x)}$, containing $\ll (\log x)^{6B+7+ o(1)}$ elements, such that if $q\leq R$ and $(a,q)=1$ then $$| \Delta_{\Xi}(f,x;q,a) | \ll \frac 1{\varphi(q)} \frac{x} {(\log x)^{B}} .$$
By we know that if $B<1$ then Proposition \[MainCor\] holds for any $f\in \mathcal C$ (not just those supported on $y$-smooth numbers) with the size of $|\Xi|$ bounded only in terms of $B$.
We state two Bombieri-Vinogradov type Theorems that follow from this.
\[Keep Xi\] Fix $0< \varepsilon\leq \frac 1{10}, A, B$ with $B>0$ and $2A>6B+7$. Let $y=x^\varepsilon$. Let $f\in \mathcal C$ be a multiplicative function which is only supported on $y$-smooth integers. There exists a set, $\Xi$, of primitive characters, containing $\ll (\log x)^{6B+7+ o(1)}$ elements, such that for any $Q\leq x^{1/2}/y^{1/2}(\log x)^A$, we have $$\label{eq:boum5}
\sum_{q \sim Q} \max_{(a,q)=1} \left| \Delta_{\Xi}(f,x;q,a) \right| \ll \frac x{(\log x)^B}.$$
\[MathResult3\] Fix $0 < \varepsilon \leq \frac{1}{10}$. Let $y = x^{\varepsilon}$. Let $f\in \mathcal C$ be a multiplicative function which is only supported on $y$-smooth integers. Assume that the Siegel-Walfisz criterion holds for $f$. For any given $B>0$ there exists $A$ such that for any $Q\leq x^{1/2}/y^{1/2}(\log x)^A$, we have $$\sum_{q \sim Q} \max_{(a,q)=1} \left| \Delta(f,x;q,a) \right| \ll_B \frac x{(\log x)^B}.$$
Combining Corollary \[MathResult3\] with the machinery developed in [@Sha], one may prove for such multiplicative functions that their higher Gowers $U^k$-norms are $o(1)$ in progressions on average. This result will be stated and discussed in Section \[sec:higher-uk\].
Breaking the $x^{1/2}$-barrier
------------------------------
The main method used in our proofs is a modification of that developed by Green in [@Gre]; see also [@Sha] for using a similar argument to deal with higher Gowers norms. Green proved (a more general result which implies) that $$\sum_{\substack{q\sim Q \\ q \ \text{prime}}} | \Delta(f,x;q,1) | \ll \frac{x \log\log x} {(\log x)^2 } ,$$ for any $Q<x^{\frac{20}{39}-\varepsilon}$, remarkably breaking the $x^{1/2}$-barrier. In this case the issue of correlations with non-trivial characters of small conductor does not arise since no such character induces a character modulo a large prime (and Green is only summing over prime moduli). Nonetheless obstruction (ii) still applies and so Proposition \[LargePrimes2\], as well as the construction in section \[NoBV\], shows that Green’s result is more-or-less best possible (up to the $\log\log x$ factor). One can modify Green’s proof to include composite moduli by taking account of the characters $\psi_j$, as we have done here. This leads to the following extensions (for fixed $a$) of Corollary \[Thm: SW\] and Theorems \[Cor:Result2\], as well as Theorem \[Keep Xi\] and Corollary \[MathResult3\].
\[Cor:Result2+\] Let $f$ be a multiplicative function with $f\in \mathcal C$. Fix $\delta, \varepsilon>0$ and let $k$ be the largest integer $\leq 1/\varepsilon^2$. For any $1\leq |a|\ll Q\leq x^{\frac{20}{39}-\delta}$, we have $$\sum_{\substack{q\sim Q \\ (a,q)=1}} | \Delta_k(f,x;q,a) | \ll \frac{x} {(\log x)^{1-\varepsilon}} .$$ If $f$ satisfies the $1$-Siegel-Walfisz criterion then $$\sum_{\substack{q\sim Q \\ (a,q) = 1}} | \Delta(f,x;q,a) | \ll \frac{x} {(\log x)^{1-\varepsilon}} .$$
\[MathResult3+\] Fix $\delta, B>0$. Let $y = x^{\varepsilon}$ for some $\varepsilon > 0$ sufficiently small in terms of $\delta$. Let $f\in \mathcal C$ be a multiplicative function which is only supported on $y$-smooth integers. Then there exists a set, $\Xi$, of primitive characters, containing $\ll (\log x)^{6B+7+ o(1)}$ elements, such that for any $1 \leq |a| \ll Q \leq x^{\frac{20}{39}-\delta} $, we have $$\sum_{\substack{q \sim Q \\ (a,q) = 1}} \left| \Delta_{\Xi}(f,x;q,a) \right| \ll \frac x{(\log x)^B}.$$ If $f$ satisfies the Siegel-Walfisz criterion then $$\sum_{\substack{q \sim Q \\ (a,q)=1}} \left| \Delta(f,x;q,a) \right| \ll \frac x{(\log x)^B}.$$
The proofs of these last two results, which break the $x^{1/2}$-barrier, rely on a deep estimate of Bettin and Chandee [@BC] on bilinear Kloosterman sums, which is an impressive development going beyond the famous estimates of Duke, Friedlander and Iwaniec [@DFI].
Our focus in this last part of the paper is to go beyond the $x^{1/2}$-barrier by incorporating the necessary expedient of $x^\varepsilon$-smooth functions into our arguments. Can one go much further beyond the $x^{1/2}$-barrier using current technology, especially if $f$ is $y$-smooth for $y$ a lot smaller than $x^\varepsilon$? In a sequel to this paper, joint with Sary Drappeau, we will extend the last two results to the range $Q\leq x^{3/5-\varepsilon}$ and with a wide range for the smoothness parameter $y$, by incorporating somewhat different techniques into our arguments.
Notation {#notation .unnumbered}
--------
Let $w = w(x)$ be a parameter, which will typically be a fixed power of $\log x$. For any positive integer $q$, we have a unique decomposition $q = q_sq_r$ of $q$ into a $w$-smooth part $q_s$ and a $w$-rough part $q_r$, where $q_s = (q, \prod_{p\leq w}p^{\infty})$ is the largest $w$-smooth integer dividing $q$, and $q_r = q/q_s$ has no prime factors $\leq w$. Although the values of $q_r,q_s$ depend on the parameter $w$, we will not explicitly indicate this dependence as the choice of $w$ should always be clear from the context.
Smooth number estimates {#sec:smooth-prelim}
=======================
We call $n$ a $y$-*smooth integer* if all of its prime factors are $\leq y$. We let $P(n)$ denote the largest prime factor of $n$ so that $n$ is $y$-smooth if and only if $P(n)\leq y$.
We need several well-known estimates involving the distribution of smooth numbers (unless otherwise referenced, see [@Gra]). Let $\Psi(x,y)$ be the number of $y$-smooth integers up to $x$. If $y\leq (\log x)^{1+o(1)}$ then $\Psi(x,y)=x^{o(1)}$. Otherwise if $x\geq y\geq (\log x)^{1+\varepsilon}$ we write $x=y^u$ and then $$\Psi(x,y) = x /u^{u+o(u)} .$$ In particular if $y=(\log x)^A$ then $\Psi(x,y) =x^{1-\frac 1A + o(1)}$. Key consequences include if $x\geq y$ then $$\sum_{\substack{ x<n\leq 2x \\ P(n)\leq y }} \frac 1n,\quad \frac 1{\log y} \sum_{\substack{ n>x \\ P(n)\leq y }} \frac 1n = u^{-u+o(u)} + x^{-1+o(1)}.$$ One consequence of this is that if $Y\geq w^{(C+\varepsilon) \log x / \log\log x}$ then $$\label{tail sum}
\sum_{\substack{ n>Y \\ P(n)\leq w }} \frac xn \ll \frac x{(\log x)^C}.$$
Rather more precisely, we define $\rho(u)=1$ for $0\leq u\leq 1$, and then determine $\rho(u)$ from the differential-delay equation $\rho'(u)=-\rho(u-1)/u$ for all $u>1$. Then $$\label{HildEst}
\Psi(x,y) = x\rho(u) \left( 1+O\left( \frac {\log (u+1)}{\log y}\right)\right)$$ for $x\geq y\geq \exp( (\log\log x)^2)$.
Define $\alpha(x,y)$ to be the real number for which $$\sum_{p\leq y} \frac{\log p}{p^\alpha-1} = \log x;$$ one has $y^{1-\alpha}\asymp u\log u$. If $x\geq y\geq (\log x)^{1+\varepsilon}$ then $\alpha\gg \varepsilon$. We need the comparison bounds $$\label{alphaCompare1}
\Psi(x/d,y)= \left( 1 + O \left( \frac 1u \right) \right) \frac{\Psi(x,y)} {d^{\alpha}}$$ for $d=y^{O(1)}$ and, in general, $$\label{alphaCompare}
\Psi(x/d,y) \ll \frac{\Psi(x,y)} d \cdot d^{1-\alpha}$$ which follow from Theorem 2.4 of [@BT]. This last bound implies that if $y \geq (\log x)^{1+\varepsilon}$ then $\int_2^x \Psi(t,y)/t \ dt \ll_{\varepsilon} \Psi(x,y)$. The bound in is not useful for us when $u,d\ll 1$. To deal with this we use the estimate $\Psi(x,y) = x \rho(u) ( 1 +O(1-\alpha))$ derived from . Let $d=y^\theta$ with $\theta\ll 1$. Then we need to determine $$\log(\rho(u-\theta)/\rho(u)) = -\int_{u-\theta}^u (\rho'(t)/\rho(t)) dt = \int_{u-\theta}^u \xi(t) dt +O(\theta/u)$$ by (3.32) of [@BT], where $\xi(t)$ satisfies $e^\xi =t\xi+1$. Now $\xi(t) =\xi(u)+O(1/u)$ in our range so the integral equals $-\theta \xi(u)+O(\theta/u)$. By (3.3) of [@BT] we see that this equals $(1-\alpha)\log d+O(\theta/u + 1/(\log y)^2)$. We deduce that $$\label{alphaCompare2}
\Psi(x/d,y) =\frac { \Psi(x,y) }{ d^{\alpha+O(1/\log x)}} \left( 1 +O\left(\frac{\log u}{\log y} \right) \right).$$
Theorem 6 of [@FT] gives a version of the Bombieri-Vinogradov Theorem for $y$-smooth numbers: For any $A>0$ there exists a constant $B=B(A)$ such that $$\label{BVsmooth}
\sum_{q\leq \sqrt{x}/(\log x)^B} \max_{(a,q)=1} \left| \Psi(x,y;q,a) - \frac {\Psi_q(x,y)}{\varphi(q)} \right| \ll \frac x{(\log x)^A}$$ where $ \Psi_q(x,y)$ denotes the number of $y$-smooth integers up to $x$ that are coprime to $q$, and $ \Psi(x,y;q,a)$ denotes those that are $\equiv a \pmod q$. Then Theorem 1 of [@FT] gives the upper bound $$\label{SmoothUB}
\Psi_q(x,y) \ll \frac {\varphi(q)} q \Psi (x,y)$$ provided $x\geq y\geq \exp((\log\log x)^2)$ and $q\leq x$.
Corollary 2 of [@Hil] implies a good upper bound from smooth numbers in short intervals: For any fixed $\kappa>0$, $$\label{SmoothShorts}
\Psi(x+\frac xT,y)-\Psi(x,y) \ll_\kappa \frac {\Psi(x,y)}T \text{ for } 1\leq T\leq \min \{ y^\kappa, x\}.$$
The contribution of characters {#sec:contrib-char}
==============================
Comparing large character sums for a primitive character and the characters it induces
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Recall the definition of $\sigma_f(x,\psi)$ from the introduction. We now define $$\sigma_f(x,z,\psi) := \sup_{x/z< X \leq x} |S_f(X,\psi)/X|,$$ so that $\sigma_f(x,\psi) =\sigma_f(x,x^{1/2},\psi) $.
\[Lem:Chars\] Suppose that $f \in \mathcal{C}$. Let $z\geq \exp((\log\log x)^2)$ and $Q \leq x$. If $\chi \pmod q$ is induced by $\psi\pmod r$, where $r \leq q \leq Q$, then $$S_f(x,\chi) \ll _C x \sigma_f(x,z,\psi) \log\log x +\frac x{(\log x)^C} ,$$ and $$S_f (x,\psi) \ll _C x \sigma_f(x,z,\chi) \log\log x +\frac x{(\log x)^C} ,$$ for any given constant $C>0$. Furthermore, for $\psi\pmod{r}$ with $r \leq Q$ we have $$\sum_{\substack{r|q\sim Q \\ \chi \mod q \ \text{induced by} \ \psi}} \frac{|S_f(x,\chi)|}{\varphi(q)} \ll_C
\left(\sigma_f(x,z,\psi)\ +\frac 1{(\log x)^C}\right) \frac{x}{\varphi(r)} ;$$ and if $P(r)\leq w$ then $$\frac 1{\log w}\sum_{\substack{r|q\leq Q ,\ P(q)\leq w \\ \chi \mod q \ \text{induced by} \ \psi}} \frac{|S_f(x,\chi)|}{\varphi(q)} \ll_C
\left(\sigma_f(x,z,\psi)\ +\frac 1{(\log x)^C}\right) \frac{x}{\varphi(r)} .$$
Let $h(.)$ be the multiplicative function which is supported only on the prime powers $p^k$, for which $p$ divides $q$ but not $r$, with $(h*f\overline\psi)(p^k)=0$ if $k\geq 1$. Thus $h*f\overline\psi = f\overline\chi$, and note that $h\in \mathcal C$ as $f\in \mathcal C$, so that each $|h(m)|\leq 1$. Now $$S_f(x,\chi) = \sum_{m\leq x} h(m) S_f(x/m,\psi)$$ and therefore we obtain, as $|S_f(x/m,\psi)|\leq \sigma_f(x,z,\psi) x/m$ if $m\leq z$, $$\begin{split}
|S_f(x,\chi)| & \ll \sigma_f(x,z,\psi)\ x \sum_{\substack{m\leq z \\ p|m\implies p|q, \ p\nmid r } } \frac{1}m + x \sum_{\substack{z<m\leq x\\ p|m\implies p|q } } \frac{1}m\\
& \ll_C \sigma_f(x,z,\psi)\ x \cdot \prod_{p|q,\ p\nmid r} \frac p{p-1} + \frac x{(\log x)^C} ,
\end{split}$$ since the second sum is maximized when $q$ is the product of the primes $\ll \log Q\leq \log x$, and then the estimate follows from . The first term is also maximized when $q$ is the product of the primes $\ll \log Q$ in which case the Euler product is $\ll \log\log Q$.
In the other direction we have $$S_f(x,\psi) = \sum_{\substack{m\leq x \\ p|m\implies p|q, \ p\nmid r } } f(m) \overline{\psi(m)} S_f(x/m,\chi)$$ and the same argument leads to the second claimed result.
We now prove the fourth part of the Lemma (the third part is proved by a simple modification of this proof), by using the upper bound proved for $|S_f(x,\chi)|$ in the first part. The second term in the upper bound is, writing $q=rn$, $$\ll \frac{1}{\log w} \sum_{ \substack{ r|q\leq Q\\ P(q)\leq w}} \frac{1}{\varphi(q)} \frac x{(\log x)^C}
\ll \frac{1}{\log w} \sum_{ \substack{ n\leq Q/r\\ P(n)\leq w}} \frac{1}{\varphi(n)} \cdot \frac x{\varphi(r)(\log x)^C} \ll \frac x{\varphi(r)(\log x)^C}.$$ The first term in the upper bound is $$\begin{split}
& \ll \frac{1}{\log w} \sum_{ \substack{ r|q\leq Q\\ P(q)\leq w}} \frac{1}{\varphi(q)} \left( \sigma_f(x,z,\psi)\ x\sum_{\substack{m\leq z \\ p|m\implies p|q, \ p\nmid r } } \frac{1}m \right)\\
& \leq \sigma_f(x,z,\psi)\ \frac{1}{\log w} \sum_{ \substack{ r|q\leq Q\\ P(q)\leq w}} \frac{x}{\varphi(q)}\sum_{\substack{m\leq z \\ m|q, \ (m,r)=1} } \frac{\mu^2(m)}{\varphi(m)} \\
& \leq \sigma_f(x,z,\psi)\ x \sum_{\substack{m\leq z \\ (m,r)=1 } } \frac{\mu^2(m)}{\varphi(m)}
\frac{1}{\log w} \sum_{ \substack{ mr|q\leq Q\\ P(q)\leq w}} \frac{1}{\varphi(q)} \\
& \ll \sigma_f(x,z,\psi)\ \frac{x}{\varphi(r)} \sum_{\substack{m\leq z \\ (m,r)=1 } } \frac{\mu^2(m)}{\varphi(m)^2} \frac{1}{\log w} \sum_{ \substack{ n\leq Q/mr\\ P(n)\leq w}} \frac{1}{\varphi(n)} \ll \sigma_f(x,z,\psi)\ \frac{x}{\varphi(r)} , \\
\end{split}$$ writing $q=mrn$, and the claim follows.
Focusing on large character sums {#sec:large-char-sum-2}
--------------------------------
For fixed $B>0$, let $\Xi(B,Q)$ denote the set of primitive characters $\psi \pmod r$ with $r\leq Q$ for which $$\sigma_f(x,\psi) \geq \frac 1{(\log x)^B}.$$
\[No Exceptions\] Let $f\in \mathcal C$ and $B > 0$. (a) Suppose that $Q \leq x$. If $\chi \pmod q$ is a character with $q\leq Q$ and is not induced by any of the characters in $\Xi(B,Q)$, then $$S_f(x,\chi) \ll \frac {\log\log x}{(\log x)^B}.$$ (b) Now suppose that $\log Q=(\log x)^{o(1)}$ and $J\geq 2$ is a given integer with $B<1-1/\sqrt{J}$. Then $|\Xi(B,Q)|<J$ and $$\label{GHSprecise}
|\Delta_{\Xi(B,Q)}(f,x;q,a)| \ll \frac 1{\varphi(q)} \frac x {(\log x)^{B+o(1)}}$$ for any $q \leq Q$ and $(a,q) = 1$.
(a) If $\chi$ is induced from $\psi$ then $ \sigma_f(x,\psi) \leq 1/{(\log x)^B}$ by the hypothesis, and the result then follows from the first part of Lemma \[Lem:Chars\].
(b) Suppose that there are at least $J$ characters $\psi_j \pmod {r_j}$ in $\Xi(B,Q)$. Let $r=[r_1,\ldots,r_J]$ so that $\log r = (\log x)^{o(1)}$, and let $\chi_j$ be the character mod $r$ induced by $\psi_j$, so that, for each $j$, there exists $x^{1/2}<X_j\leq x$ for which $\sigma_f(X_j,\chi_j)\gg 1/ (\log x)^{B+o(1)}$ by the second part of Lemma \[Lem:Chars\]. However, by Theorem 6.1 of [@GHS], one of these is $\ll 1/(\log x)^{1-1/\sqrt{J}+o(1)}$, a contradiction.
Now Theorem 6.1 of [@GHS], applied to the set $S$ of $J-1$ characters $\chi \pmod q$ which give the $J-1$ largest values of $|S_f(x,\chi)|$, implies that $$|\Delta_{S}(f,x;q,a)| \ll \frac 1{\varphi(q)} \frac x {(\log x)^{1-1/\sqrt{J}+o(1)}} .$$ Write $\Xi = \Xi(B, Q)$. Now $ |S_f(x,\chi)| \ll x/{(\log x)^{B+o(1)}}$ for every $\chi\in S \setminus \Xi_q$ by (a), and also for every $\chi\in \Xi_q\setminus S$ by the definition of $S$, Theorem 6.1 of [@GHS], and the hypothesis $B < 1-1/\sqrt{J}$. This implies that $$|\Delta_{\Xi}(f,x;q,a) - \Delta_{S}(f,x;q,a)| \leq \frac 1{\varphi(q)} \sum_{\chi\in D} |S_f(x,\chi)| \ll \frac 1{\varphi(q)} \frac x {(\log x)^{B+o(1)}},$$ where $D$ is the symmetric difference of the sets $S$ and $\Xi_q$, and the result follows from adding the last two displayed equations.
\[MathResult2Cor\*\*\] Fix an integer $J\geq 2$, then $0<B<1-1/\sqrt{J}$ and let $w=(\log x)^{2B}$. For any $f\in \mathcal C$ we have $$\frac 1{\log w} \sum_{\substack{q\leq x \\ P(q)\leq w}} \max_{(a,q)=1}
|\Delta_{\Xi(B,\log x)}(f,x;q,a)| \ll \frac x {(\log x)^{B+o(1)}}.$$ Moreover $$\frac 1{\log w} \sum_{\substack{q\leq x \\ P(q)\leq w}} \max_{(a,q)=1}
|\Delta_{J-1}(f,x;q,a)| \ll \frac x {(\log x)^{B + o(1)}}.$$
Let $\Xi= \Xi(B,\log x)$, which has no more than $J$ elements by Corollary \[No Exceptions\](b). We begin by bounding the contributions of the values of $q>R:=\exp((\log\log x)^2)$: $$\begin{split}
\sum_{\substack{R<q\leq x \\ P(q)\leq w}} \max_{(a,q)=1} |\Delta_{\Xi}(f,x;q,a)| & \ll
\sum_{\substack{R<q\leq x \\ P(q)\leq w}} \frac xq +
\sum_{\chi \in \Xi} \sum_{\substack{R<q\leq x \\ P(q)\leq w\\ r_\chi|q}} \frac xq \\
&\ll x \sum_{\substack{q > R\\ P(q)\leq w }} \frac 1n + x \sum_{\chi \in \Xi} \frac{1}{r_{\chi}} \sum_{\substack{R/r_{\chi} < n \leq x/r_{\chi} \\ P(n) \leq w}} \frac{1}{n},
\end{split}$$ writing $q=nr_\chi$ in the second sum. The first term, and the contribution to the second term from those $\chi$ with $r_{\chi} \leq \sqrt{R}$ are both acceptable, by the estimates for smooth numbers. The contribution to the second term from those $\chi$ with $r_{\chi} > \sqrt{R}$ is $$\ll \frac{x }{\sqrt{R}} \sum_{\substack{n \leq x\\ P(n)\leq w}} \frac{1}{n} \ll \frac{x \log w}{\sqrt{R}} \ll \frac{x}{(\log x)^B},$$ which is also acceptable.
Finally, by Corollary \[No Exceptions\](b) we have that $$\frac 1{\log w} \sum_{\substack{q\leq R \\ P(q)\leq w}} \max_{(a,q)=1}
|\Delta_{\Xi}(f,x;q,a)| \ll \frac 1{\log w} \sum_{q: P(q) \leq w} \frac 1{\varphi(q)} \frac x {(\log x)^{B+o(1)}},$$ and this is $\ll x/(\log x)^{B+o(1)}$ as $ \sum_{q: P(q)\leq w} 1/{\varphi(q)}\ll \log w$. This completes the proof of the first part of the Corollary.
Now $\Xi= \{ \psi_1,\ldots, \psi_{k}\}$ for some $k<J$, by definition. Therefore $$|\Delta_{J-1}(f,x;q,a)| \leq |\Delta_{\Xi }(f,x;q,a)| + \frac 1{\varphi(q)}\sum_{\substack{k<j<J \\ r_j|q}} |S_f(x,\chi_j)| ,$$ and so the result follows from summing this over the $w$-smooth moduli $q$, using the last part of Lemma \[Lem:Chars\] for each $j$, along with the definition of $\Xi$.
Making use of the Siegel-Walfisz criterion
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\[Using SW\] Let $f\in \mathcal C$ and $Q \leq x$. For each $q \sim Q$ let $a_q\pmod q$ be a residue class with $(a_q,q) = 1$. Suppose that $\Xi$ is a set of primitive characters, containing $\ll (\log x)^{C}$ elements, such that $$\sum_{q \sim Q} \left| \Delta_{\Xi}(f,x;q,a_q) \right| \ll \frac x{(\log x)^B} .$$ If the $D$-Siegel-Walfisz criterion holds for $f$, where $D\geq B+C$, then $$\sum_{q \sim Q} \left| \Delta(f,x;q,a_q) \right| \ll \frac x{(\log x)^B}.$$
By definition we have $$| \Delta(f,x;q,a_q) | \leq | \Delta_{\Xi}(f,x;q,a_q) | + \frac 1{\varphi(q)} \sum_{ \substack{\chi \pmod {q} \\ \chi \in \Xi_{q},\ \chi\ne \chi_0 }} \left| S_f(x,\chi) \right| .$$ Summing this up over $q\sim Q$, and using the hypothesis, we deduce that $$\sum_{\substack{ q\sim Q }} | \Delta(f,x;q,a_q) |
\leq
\sum_{\substack{ \psi\in \Xi \\ \psi\ne 1}} \sum_{\substack{r_\psi|q\sim Q \\ \chi \pmod q \ \text{induced by } \psi } } \frac{|S_f(x,\chi)|}{\varphi(q)} +O\left( \frac{x} {(\log x)^{B}} \right).$$ The third part of Lemma \[Lem:Chars\] then implies that this is $$\leq
x \sum_{\substack{ \psi\in \Xi \\ \psi\ne 1}} \frac{1}{\varphi(r)} \left( \sigma_f(x, \psi) + (\log x)^{-D} \right) +O\left( \frac{x} {(\log x)^{B}} \right).$$ The $D$-Siegel-Walfisz criterion implies that for any non-principal $\psi\pmod{r}$, we have $$\frac 1{\varphi(r)} S_f(X,\psi) = \frac 1{\varphi(r)} \sum_{a \pmod r} \overline{\psi}(a) \Delta(f,X;r,a) \ll \frac X{(\log x)^{D}} ,$$ for $x^{1/2}<X\leq 2x$, and so $\sigma_f(x,\psi)/\varphi(r)\ll 1/(\log x)^{D}$. Therefore the above is $$\ll \# \Xi \cdot \frac x{(\log x)^{D}} + \frac x{(\log x)^B} \ll \frac x{(\log x)^B}$$ provided $D\geq B+C$.
Lower bounds {#ClaimJusitify}
------------
If $\chi\not\in \Xi_q$ then $$S_f(x,\chi) = \sum_{a \pmod q} \overline{\chi}(a) \Delta_\Xi(f,x;q,a) ,$$ and so $$\frac{ |S_f(x,\chi) |}{\varphi(q)} \leq \max_{(a,q)=1} |\Delta_\Xi(f,x;q,a) |.$$ Therefore $$\max_{(a,q)=1} |\Delta_\Xi(f,x;q,a) | \geq \max_{\chi\not\in \Xi_q} \frac{ |S_f(x,\chi) |}{\varphi(q)} ;$$ in particular we deduce that for any primitive $\psi\not\in \Xi$ we have $$\sum_{q\sim Q} \max_{(a,q)=1} |\Delta_\Xi(f,x;q,a) | \geq \sum_{\substack{q\sim Q \\ \chi \text{ induced by } \psi}} \frac{ |S_f(x,\chi) |}{\varphi(q)} .$$ From the identity $$S_f(x,\chi) = \sum_{m\leq x} h(m) S_f(x/m,\psi)$$ (see the proof of Lemma \[Lem:Chars\]) we might expect that if $|S_f(x ,\psi)|$ is large then each $|S_f(x,\chi)|$ should be too, though this is difficult to prove for every induced $\chi$. However we can do so when the smallest prime factor of $q$ that does not divide $r$ is $>L \log x$, where $L:=x/|S_f(x,\psi)|$:. Taking absolute values, and remembering the support of $h(.)$ we have $$\begin{split}
|S_f(x,\chi) | &\geq |S_f(x ,\psi) | - \sum_{\substack{m> 1 \\ p|m\implies p|q \\ (m,r)=1}} |S_f(x/m,\psi)| \geq
\frac xL - \sum_{\substack{m> 1 \\ p|m\implies p|q \\ (m,r)=1}} \frac xm\\
&\geq
\frac xL - x \left( \prod_{p|q,\ p\nmid r} \left(1 - \frac 1p\right)^{-1} -1 \right) \sim \frac xL= |S_f(x,\psi)|,\\
\end{split}$$ since $q$ has $o(\log x)$ prime factors, For such $q$ we also have $\varphi(q)\sim \varphi(r)q/r$. Therefore $$\sum_{\substack{q\sim Q \\ \chi \text{ induced by } \psi}} \frac{ |S_f(x,\chi) |}{\varphi(q)} \gg
\frac{ |S_f(x,\psi) |}{\varphi(r)} \sum_{\substack{q\sim Q \\ q=rn \\ p|n\implies p>L \log x}} \frac{ 1}{n} \gg
\frac{ |S_f(x,\psi) |}{\varphi(r)} \cdot \frac 1{\log (L\log x)} .$$ Therefore if $|S_f(x,\psi)|\gg x/(\log x)^A$ for some primitive $\psi$ then $$\sum_{q\sim Q} \max_{(a,q)=1} |\Delta_\Xi(f,x;q,a) | \gg \frac{ |S_f(x,\psi) |}{\varphi(r)} \cdot \frac 1{\log\log x} .$$ At worst, when all of the $|S_f(x,\psi)|$ with $\psi\not\in \Xi$ are small, one deduces that $$\sum_{q\sim Q} \max_{(a,q)=1} |\Delta_\Xi(f,x;q,a) | \gg \frac 1 {\log x} \max_{\substack{\psi \pmod r \\ \psi \ \text{primitive} \\ \psi\not\in \Xi}} \frac{ |S_f(x,\psi) |}{\varphi(r)} .$$
Formulating the key technical result
====================================
If $\chi \pmod r$ is in $\Xi$, we write $r=r_\chi$, and note that it induces a character mod $q$ if and only if $r$ divides $q$, and then the induced character is $\chi\xi_q$ where $\xi_q$ is the principal character mod $q$. We have the upper bound $|S_f(x,\chi)|\leq \sum_{n\leq x,\ (n,q)=1} 1\ll (\varphi(q)/q) x$ for $x\geq q$. Therefore $|\Delta_\Xi(f,x;q,a)|\ll \frac{1+|\Xi_q|}q x$ for $x\geq q$. Moreover $$\Delta_\Xi(f,x;q,a) = \frac 1{\varphi(q)} \sum_{ \substack{\chi \pmod q \\ \chi \not\in \Xi_q }} \chi(a) S_f(x,\chi).$$
\[MathResult2Cor\] Fix $\varepsilon>0$, let $Q \leq x^{1/2-\varepsilon}$, and let $w \geq 2$. Let $\Xi$ be a set of primitive characters, each with $w$-smooth conductors $\leq Q/\exp((\log w)^2)$, such that $$\label{CharHyp}
\sum_{\substack{\chi \in \Xi}} \frac 1{r_\chi} \ll w^{1/2}.$$ For any $1$-bounded multiplicative function $f$, we have $$\begin{split}
& \sum_{q \sim Q} \max_{(a,q )=1} \left| \Delta_{\Xi}(f,x;q,a) \right| \\
& \leq
\frac 1{\log w} \sum_{\substack{q_s\leq 2Q \\ P(q_s)\leq w}} \max_{(a,q_s)=1}
|\Delta_{\Xi}(f,x;q_s,a)|
+O\left( \frac x{w^{1/2}} + \frac{x \log\log x}{\log x} \right) .
\end{split}$$
The proof can be modified to allow any $Q \leq x^{1/2}/2$, though we would need to replace the $(\log\log x)/\log x$ by $(\log\log x)/\log (x/Q^2)$ on the right-hand side.
As we will justify below, Corollary \[MathResult2Cor\] is a consequence of:
\[MathResult2\] Fix $\varepsilon>0$, let $Q \leq x^{1/2-\varepsilon}$, and let $w \geq 2$. For any $1$-bounded multiplicative function $f$, we have $$\sum_{q \sim Q} \max_{(a,q)=1} \left|\sum_{\substack{n \leq x\\ n \equiv a\pmod{q}}} f(n) - \frac 1{q_r}\sum_{\substack{n \leq x\\ n \equiv a\pmod{q_s}}} f(n)\right| \ll \frac x{w^{1/2}} + \frac{x \log\log x}{\log x},$$ where $q_s,q_r$ are the $w$-smooth and the $w$-rough parts of $q$, respectively.
This will be proved in Section \[BilinBd\].
\[Deduction of Corollary \[MathResult2Cor\] from Theorem \[MathResult2\]\] Note that if $\psi\pmod r \in \Xi$ induces a character in $\Xi_q$, then it also induces a character in $\Xi_{q_s}$, since $r$ is $w$-smooth. We use the identity $$\begin{split}
\Delta_{\Xi}(f,x;q,a) &= \frac 1{q_r}\Delta_{\Xi}(f,x;q_s,a) + \sum_{\substack{n \leq x\\ n \equiv a\pmod{q}}} f(n) - \frac 1{q_r}\sum_{\substack{n \leq x\\ n \equiv a\pmod{q_s}}} f(n) \\
+ \sum_{\psi\in \Xi_{q_s}} &\psi(a) \left( \frac{S_f(x,\psi)-S_f(x,\psi\xi_q)}{ \varphi(q)} - \left( \frac{1}{ \varphi(q)} - \frac{1}{q_r\varphi(q_s)} \right)S_f(x,\psi) \right).
\end{split}$$ We sum the absolute value of this up over each $q\sim Q$ with $a=a_q$ which maximizes $|\Delta_{\Xi}(f,x;q,a)|$. The first term on the right-hand side is then, summing up over $q=q_rq_s\sim Q$, $$\begin{split}
& \leq \sum_{\substack{q_s\leq 2Q \\ P(q_s)\leq w}} \max_{(a,q_s)=1}
|\Delta_{\Xi}(f,x;q_s,a)| \sum_{\substack{q_r\sim Q/q_s\\ p|q_r\implies p>w}} \frac 1{q_r}\\
&\ll \frac 1{\log w} \sum_{\substack{q_s\leq 2Q \\ P(q_s)\leq w}} \max_{(a,q_s)=1}
|\Delta_{\Xi}(f,x;q_s,a)| + \sum_{\substack{q_s\sim Q \\ P(q_s)\leq w}} \max_{(a,q_s)=1}
|\Delta_{\Xi}(f,x;q_s,a)| .
\end{split}$$ The last term comes from those $q$ with $q_r=1$, in which case each $|\Delta_{\Xi}(f,x;q_s,a)|\ll(1+|\Xi_q|)\cdot x/ q_s$, and so $$\begin{split}
\sum_{\substack{q_s\sim Q \\ P(q_s)\leq w}} & \max_{(a,q_s)=1} |\Delta_{\Xi}(f,x;q_s,a)|
\ll
\sum_{\substack{q_s\sim Q \\ P(q_s)\leq w}} \frac{x}{q_s}+ \sum_{\chi \pmod r \in \Xi} \sum_{\substack{q_s\sim Q \\ P(q_s)\leq w\\ r|q_s}} \frac x{q_s} \\
& \ll x v^{-v+o(v)} + \sum_{\substack{\chi \pmod r \in \Xi\\ P(r)\leq w}} \frac xr \sum_{\substack{n\sim Q/r \\ P(n)\leq w }} \frac 1{n} \ll \left( 1 + \sum_{\substack{\chi \in \Xi}} \frac 1{r_\chi} \right) xv^{-v+o(v)}
\end{split}$$ writing $q_s=rn$ in the second sum, as $Q/r\geq \exp((\log w)^2)=w^v$, say, and this term is $ \ll x/{w^{1/2}} $ by .
By Theorem \[MathResult2\] we have $$\sum_{q \sim Q} \max_{(a,q)=1} \left|\sum_{\substack{n \leq x\\ n \equiv a\pmod{q}}} f(n) - \frac 1{q_r}\sum_{\substack{n \leq x\\ n \equiv a\pmod{q_s}}} f(n)\right| \ll \frac x{w^{1/2}} + \frac{x \log\log x}{\log x}.$$
Now since $|S_f(x,\psi)|\leq \sum_{n\leq x,\ (n,q_s)=1} 1 \ll (\varphi(q_s)/q_s)x$, we have $$\left( \frac{1}{ \varphi(q)} - \frac{1}{q_r\varphi(q_s)} \right)S_f(x,\psi) \ll \left( \frac {q_r}{\varphi(q_r) } - 1\right) \frac 1{q_r\varphi(q_s)} \cdot \frac {\varphi(q_s)}{q_s} x= \left( \frac {q_r}{\varphi(q_r)} - 1 \right) \frac x{q}.$$ Moreover, writing $n=ab$ where $p|a\implies p|q$ and $(b,q)=1$, we have $$\begin{split}
\frac{S_f(x,\psi)-S_f(x,\psi\xi_q)}{ \varphi(q)} &= \frac{1}{ \varphi(q)}\sum_{\substack{ab\leq x \\ a>1 }} (f\overline\psi)(a)(f\overline\psi)(b)\leq \frac{1}{ \varphi(q)}\sum_{\substack{1<a \leq x \\ p|a \implies p|q_r}} \sum_{ \substack{b\leq x/a \\ (b,q)=1}} 1 \\
&\ll \frac{1}{ \varphi(q)}\sum_{\substack{1<a \leq x/q \\ p|a \implies p|q_r}} \frac{\varphi(q)}{q} \frac xa+ \frac{1}{ \varphi(q)}\sum_{\substack{ x/q<a\leq x \\ p|a \implies p|q_r}} \frac xa \\
&\ll \left( \frac {q_r}{\varphi(q_r)} - 1 \right) \frac x{q} + x^{o(1)} \\
\end{split}$$ since there are $x^{o(1)}$ integers $a$ in an interval $[X,2X]$ with $X\geq x/q$, all of whose prime factors divide $q_r$ (to see this note that the worst case is when $q_r$ is the product of all the primes $\ll \log x$, and then this easily follows from our estimates for smooth numbers).
Therefore $$\begin{split}
& \sum_{q\sim Q} \left| \sum_{\psi\in \Xi_{q_s}} \psi(a) \left( \frac{S_f(x,\psi)-S_f(x,\psi\xi_q)}{ \varphi(q)} - \left( \frac{1}{ \varphi(q)} - \frac{1}{q_r\varphi(q_s)} \right)S_f(x,\psi) \right) \right| \\
& \ll
\sum_{\chi\in \Xi} \sum_{\substack{q\sim Q \\ r_\chi|q}} \left( \frac {q_r}{\varphi(q_r)} - 1 \right) \frac x{q} + \sum_{\chi \in \Xi} \sum_{\substack{q \sim Q \\ r_{\chi}|q}} x^{o(1)} \\
& = \sum_{\chi\in \Xi} \sum_{\substack{ q_s\leq 2Q \\ r_\chi|q_s}} \frac 1{q_s}
\sum_{\substack{q_r\sim Q/q_s }} \left( \frac {q_r}{\varphi(q_r)} - 1 \right) \frac x{q_r} + \sum_{\chi\in \Xi} \frac{Qx^{o(1)} }{r_\chi}\\
& \ll \sum_{\chi\in \Xi} \frac{\log w}{ r_\chi} \cdot
\frac x{w\log w} + \sum_{\chi\in \Xi} \frac{Qx^{o(1)} }{r_\chi} \ll \frac x{w} \cdot \sum_{\chi\in \Xi} \frac{1}{ r_\chi} \ll \frac x{w^{1/2}} ,
\end{split}$$ since $\sum_{\substack{ n\leq Q/r \\ P(n)\leq w}} 1/n\ll \log w $ where $q_s=r_\chi n$, and for $Q_s=Q/q_s$ we have $$\begin{split}
\sum_{q_r \sim Q_s} & \left( \frac {q_r}{\varphi(q_r)} - 1 \right) \frac x{q_r} = \sum_{q_r \sim Q_s} \frac x{q_r} \sum_{\substack{d>1 \\ d|q_r}} \frac{\mu^2(d)}{\varphi(d)}
= \sum_{\substack{ 1<d\leq 2Q_s\\ p|d\implies p>w}} \frac{\mu^2(d)}{\varphi(d)} \sum_{\substack{q_r \sim Q _s\\ d|q_r}} \frac x{q_r} \\ &\ll
x \sum_{\substack{ 1<d\leq 2Q_s\\ p|d\implies p>w}} \frac{\mu^2(d)}{d\varphi(d)}
\leq x \left( \prod_{p>w} \left( 1+\frac 1{p(p-1)} \right) -1 \right) \ll \frac x{w\log w}.
\end{split}$$ Collecting up the estimates above the result follows.
Using Ramaré’s weights
----------------------
Let $a_q \pmod q$ be the arithmetic progression with $(a,q)=1$ for which $$\left|\sum_{\substack{n \leq x \\ n \equiv a\pmod{q}}} f(n) - \frac{1}{q_r} \sum_{\substack{n \leq x \\ n\equiv a\pmod{q_s}}} f(n)\right|$$ is maximized, and then select $\xi_q$ with $|\xi_q|=1$ so that this equals $$\xi_q \left( \sum_{\substack{n \leq x \\ n \equiv a\pmod{q}}} f(n) - \frac{1}{q_r} \sum_{\substack{n \leq x \\ n\equiv a\pmod{q_s}}} f(n) \right) .$$ Therefore the left-hand side of the equation in Theorem \[MathResult2\] can be re-written as $$\sum_{n \leq x} f(n) F(n)$$ where, here and throughout, the function $F$ is defined by $$\label{F-value}
F(n) := \sum_{q\sim Q} \xi_q \left({\mathbf{1}}_{n \equiv a_q\pmod{q}} - \frac{1}{q_r} {\mathbf{1}}_{n\equiv a_q\pmod{q_s}}\right).$$
Sums like $$\sum_{n \leq x} f(n) F(n)$$ have been tackled in the literature using the Cauchy-Schwarz inequality and then studying bilinear sums, such as what happens after (17) in [@MV]. Here we develop a formal inequality, which is a variant of Proposition 2.2 of [@Gre].
\[prop:ramare\] Let $F: {\mathbb{Z}}\to {\mathbb{C}}$ be an arbitrary function. Let $2 \leq Y < Z < x^{1/9}$ be parameters, and write $u = (\log Z)/\log Y$. Then for any $1$-bounded multiplicative function $f$, we have $$\sum_{Z^9 \leq n \leq x} f(n) F(n) \ll \frac{T}{Y \log Y} + E_{\text{sieve}} + E_{\text{bilinear}},$$ where $$T = \max_{d \leq Z^2} d \sum_{\substack{n \leq x \\ d\mid n}} |F(n)|,$$ $$E_{\text{sieve}} = \sum_{n \leq x} |f(n)F(n)| {\mathbf{1}}_{(n,\prod_{Y \leq p < Z}p) = 1},$$ and $$E_{\text{bilinear}} = \max_{P \in [Y,Z)} \left(Px \cdot {\mathbb{E}}_{p,p' \sim P} \left| \sum_{m \leq \min(x/p, x/p')} F(pm) \overline{F(p'm)} \right|\right)^{1/2}.$$
We use Ramar[é]{}’s weight function: $$w(n) = \frac{1}{\#\{Y \leq p < Z: p \mid n\} + 1},$$ to obtain the identity $$\sum_{\substack{Y \leq p < Z \\ p^k\| n,\ k\geq 1}} w(n/p^k) = \begin{cases} 1 & \text{if }p \mid n\text{ for some }Y \leq p < Z \\ 0 & \text{otherwise.} \end{cases}$$ Therefore, letting $F(n)=0$ for $n<Z^9$, $$\begin{split}
\sum_{n \leq x} f(n) F(n) &\leq E_{\text{sieve}} + \sum_{\substack{n \leq x }} f(n) F(n) \sum_{\substack{Y \leq p < Z \\ p^k\|n}} w(n/p^k)\\
&\leq E_{\text{sieve}} + \sum_{\substack{n=mp^k \leq x\\ Y \leq p < Z \\ (p,m)=1}} w(m)f(m) \cdot f(p^k) F(mp^k),
\end{split}$$ writing $n=mp^k$. We replace each term where $p^2|n$, currently written as $n=mp^k$, by $n$ written as $m'p$. Given that $|w(.)|,|f(.)|\leq 1$ the difference is $$\leq 2\sum_{Y \leq p < Z} \sum_{\substack{n \leq x \\ p^2 \mid n}} |F(n)| \leq 2T \sum_{Y \leq p < Z} \frac{1}{p^2} \ll \frac{T}{Y\log Y}.$$ Therefore, $$\label{eq:Sigma}
\sum_{n \leq x} f(n) F(n) \leq \Sigma' + E_{\text{sieve}} + O\left( \frac{T}{Y\log Y} \right)$$ where $$\Sigma' := \sum_{m \leq x/Y} w(m) f(m) \sum_{\substack{Y \leq p < Z \\ p \leq x/m}} f(p) F(pm).$$ We divide the range $[Y,Z)$ for $p$, dyadically, into $$\Sigma'(P) := \sum_{m \leq x/P} w(m) f(m) \sum_{\substack{P \leq p < 2P \\ p \leq x/m}} f(p) F(pm)$$ for $P \in [Y, Z)$. Since $F(pm)$ is supported on $pm \geq Z^9$, we may add the restriction $m \geq Z^8$ to the sum. By Lemma 2.1 of [@Gre] we have $$\sum_{Z^8<m \leq x/P} w(m)^2 \ll \frac{x}{P (\log u)^2}.$$ Therefore, by the Cauchy-Schwarz inequality, we obtain $$|\Sigma'(P)|^2 \ll \frac{x}{P(\log u)^2} \sum_{m \leq x/P} \left|\sum_{\substack{P \leq p < 2P \\ p \leq x/m}} f(p) F(pm) \right|^2.$$ After expanding the square and changing the order of summation, we obtain $$\begin{aligned}
|\Sigma'(P)|^2 & \ll \frac{x}{P(\log u)^2} \sum_{P \leq p,p' < 2P} f(p) \overline{f(p')} \sum_{m \leq \min(x/p, x/p')} F(pm) \overline{F(p'm)} \\
& \ll \frac{x}{P(\log u)^2} \sum_{P \leq p,p' < 2P} \left| \sum_{m \leq \min(x/p, x/p')} F(pm) \overline{F(p'm)} \right| \\
& \ll \frac{x}{P (\log u)^2} \cdot \frac{\pi(P)^2}{Px} E_{\text{bilinear}}^2 = \left(\frac{E_{\text{bilinear}}}{(\log P)(\log u)}\right)^2.\end{aligned}$$ Taking square root and then summing dyadically over $P$, we obtain $\Sigma' \ll E_{\text{bilinear}}$. This completes the proof.
Bounding the sieve term and the bilinear term {#BilinBd}
---------------------------------------------
We now apply Proposition \[prop:ramare\] to the function $F$ in to prove Theorem \[MathResult2\]. Let $2 \leq Y < Z < x^{1/9}$ be two parameters. Recall the notation that $u = (\log Z)/\log Y$.
\[sievebound\] We have $$\sum_{n \leq x} |F(n)| {\mathbf{1}}_{(n, \prod_{Y\leq p < Z}p)=1} \ll \frac{x}{u}.$$
Using the trivial bound $$|F(n)| \leq \sum_{q\sim Q} \left({\mathbf{1}}_{n\equiv a_q\pmod{q}} + \frac{1}{q_r} {\mathbf{1}}_{n\equiv a_q\pmod{q_s}}\right),$$ we may bound the desired expression by $$\sum_{q\sim Q} \sum_{\substack{n \leq x \\ n \equiv a_q\pmod{q}}} {\mathbf{1}}_{(n,\prod_{Y \leq p < Z}p)=1} + \sum_{q\sim Q} \frac{1}{q_r} \sum_{\substack{n \leq x \\ n \equiv a_q\pmod{q_s}}} {\mathbf{1}}_{(n,\prod_{Y \leq p < Z}p)=1}.$$ By an upper bound sieve, the inner sum over $n$ in the first term is $$\ll \frac{x}{q} \prod_{\substack{Y \leq p < Z \\ p\nmid q}} \left(1 - \frac{1}{p}\right) \ll \frac{x}{u} \cdot \frac{1}{\varphi(q)}$$ for any $q \sim Q$. Thus the first term is $O(x/u)$ since $$\sum_{q \sim Q} \frac{1}{\varphi(q)} \ll 1.$$ The second term is dealt with similarly: bound the inner sum over $n$ by $O(x/u\varphi(q_s))$ and then the second term is $$\ll \frac{x}{u} \sum_{q \sim Q} \frac{1}{q_r\varphi(q_s)} \leq \frac{x}{u} \sum_{q\sim Q} \frac{1}{\varphi(q)} \ll \frac{x}{u}.$$ This completes the proof of the lemma.
\[BilinBound\] For any $P,Q \geq 2$ we have $${\mathbb{E}}_{p,p'\sim P} \left| \sum_{m \leq \min(x/p, x/p')} F(pm) \overline{F(p'm)} \right| \ll \frac{x}{P}\left(\frac{1}{w\log w} + \frac{P^{0.1} + \log x}{\pi(P)}\right) +Q^2 ,$$ where $p,p'$ denote primes.
By the definition of $F$, we change the order of summation to write the inner sum over $m$ as $$\Sigma(p,p') := \sum_{q,q'\sim Q} \xi_q \xi_{q'} \sum_{m \leq \min(x/p, x/p')} K(p,p',q,q';m),$$ where $K(p,p',q,q';m)$ is the expression $$\left({\mathbf{1}}_{pm\equiv a_q\pmod{q}} - \frac{1}{q_r} {\mathbf{1}}_{pm\equiv a_q\pmod{q_s}}\right) \left({\mathbf{1}}_{p'm\equiv a_{q'}\pmod{q'}} - \frac{1}{q_r'} {\mathbf{1}}_{p'm\equiv a_{q'}\pmod{q_s'}}\right).$$ The inner sum $\sum_m K(p,p',q,q';m)$ can be written as a sum of four sums, the first of which is $$\sum_{m \leq \min(x/p, x/p')} {\mathbf{1}}_{pm\equiv a_q\pmod{q}} \cdot {\mathbf{1}}_{p'm\equiv a_q'\pmod{q'}}.$$ This sum should have a main term of $$S(p,p',q,q') := \begin{cases} \frac{\min(x/p, x/p')}{[q,q']} & \text{if }(p,q) = (p',q') = 1\text{ and }p'a_q \equiv pa_q' \pmod{(q,q')}, \\ 0 & \text{otherwise,} \end{cases}$$ with an error of $O(1)$. Similarly for the other three sums. It follows that the sum $$\sum_{m \leq \min(x/p, x/p')} K(p,p',q,q';m) = g_0(p,p',q,q') + O(1),$$ where the main term $g_0(p,p',q,q')$ is defined by $$g_0(p,p',q,q') = S(p,p',q,q') - \frac{1}{q_r}S(p,p',q_s,q') - \frac{1}{q_r'} S(p,p',q,q_s') + \frac{1}{q_rq_r'} S(p,p',q_s,q_s').$$ The total contribution from the $O(1)$ error is $O(Q^2)$, which is acceptable. Thus it suffices to show that $${\mathbb{E}}_{p,p' \sim P} \sum_{q,q' \sim Q} |g_0(p,p',q,q')| \ll \frac{x}{P}\left(\frac{1}{w\log w} + \frac{P^{0.1}+\log x}{\pi(P)}\right).$$ Note that we have the upper bound $$\label{eq:g0bound1}
|g_0(p,p',q,q')| \ll \frac{x}{P} \cdot \frac{(q,q')}{qq'}.$$ Moreover, when $S(p,p',q,q')=0$ we have the (possibly) improved upper bound $$\label{eq:g0bound2}
|g_0(p,p',q,q')| \ll \frac{x}{P} \cdot \frac{(q_s,q_s')}{qq'}.$$
Case 0 {#case-0 .unnumbered}
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First consider the case when $p \mid q$ or $p' \mid q'$. Then $S(p,p',q,q') = 0$ and, so by , $${\mathbb{E}}_{p,p' \sim P} \sum_{\substack{q,q' \sim Q \\ p \mid q \text{ or } p' \mid q}} |g_0(p,p',q,q')| \ll
\frac{x}{P\pi(P)^2} \sum_{p,p' \sim P} \sum_{\substack{q,q' \sim Q \\ p \mid q \text{ or } p' \mid q}} \frac{(q_s,q_s')}{qq'},$$ which by symmetry is $$\ll
\frac{x}{P\pi(P)} \sum_{p\sim P} \sum_{\substack{q,q' \sim Q \\ p \mid q }} \sum_{\substack{ d\geq 1 \\ P(d)\leq w \\ d|q, d|q'}} \frac{d}{qq'} = \frac{x}{P\pi(P)} \sum_{p\sim P} \frac 1p \sum_{\substack{ d\geq 1 \\ P(d)\leq w }} \frac 1d
\sum_{\substack{r \sim Q/pd \\ r'\sim Q/d }}\frac{1}{rr'}$$ writing $q=pdr, q'=dr'$, which is $$\ll \frac{x}{P\pi(P)} \cdot \frac{1}{\log P} \cdot \log w \cdot 1^2 \ll \frac{x\log w}{P^2},$$ which is easily acceptable.
Case 1 {#case-1 .unnumbered}
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Now consider the case when $(p,q) = (p',q') = 1$ and $p'a_q \equiv pa_q' \pmod{(q,q')}$. Then $$g_0(p,p',q,q') = \min(x/p, x/p') \left(\frac{(q,q')}{qq'} - \frac{(q_s,q_s')}{qq'}\right).$$ It vanishes unless $(q_r,q_r') > 1$, and thus by it suffices to show that $$\Sigma_1:= \frac{1}{\pi(P)^2} \sum_{\substack{ q,q'\sim Q \\ (q_r,q_r')>1}} \sum_{\substack{ p,p'\sim P \\ (p,q) = (p',q') = 1 \\ p'a_q \equiv pa_q'\pmod{(q,q')} }}
\frac{(q,q')}{qq'} \ll \frac{1}{w\log w} + \frac{P^{0.1} + \log x}{\pi(P)}.$$ For fixed $q,q',p$, the constraint $p'a_q \equiv pa_{q'} \pmod{(q,q')}$ imposes a congruence condition on $p' \pmod{(q,q')}$, and the number of $p'$ satisfying it is $$\ll \begin{cases} \pi(P)/\varphi((q,q')) & \text{if }(q,q') \leq P^{0.9}, \\ P/(q,q') + 1 & \text{if }(q,q') > P^{0.9}. \end{cases}$$ Here the bound in the first case follows from Brun-Titchmarsh, and in the second case by dropping the primality condition on $p'$. Therefore $$\label{Bound1}
\Sigma_1\ll \frac{1}{Q^2} \sum_{\substack{ q,q'\sim Q\\ (q_r,q_r')>1}} \frac{(q,q')}{\varphi((q,q')) } +
\frac{1}{\pi(P) Q^2} \sum_{\substack{ q,q'\sim Q\\ (q,q') > P^{0.9} }} (P+ (q,q') ) .$$ In the second term let $d=(q,q')>P^{0.9}$ and then the sum is $$\leq \sum_{P^{0.9} < d \leq 2Q} (P+d) \sum_{\substack{q,q' \sim Q \\ d\mid (q,q')}} 1 \ll \sum_{P^{0.9} < d \leq 2Q} (P+d) \frac{Q^2}{d^2} \ll Q^2(P^{0.1} + \log x),$$ which is acceptable. The first term has the restriction that $(q_r,q_r') > 1$, which implies that there is some prime $p>w$ with $p \mid (q,q')$. Writing $q=pm$ and $q'=pm'$, we have $$\sum_{\substack{ q,q'\sim Q\\ (q_r,q_r')>1}} \frac{(q,q')}{\varphi((q,q')) } \leq \sum_{w < p \leq 2Q} \frac{p}{p-1} \sum_{m,m' \sim Q/p} \frac{(m,m')}{\varphi((m,m'))}.$$ Using the identity $$\frac{(m,m')}{\varphi((m,m'))} = \sum_{d \mid (m,m')} \frac{\mu^2(d)}{\varphi(d)},$$ we can bound the sum over $q,q'$ by $$\begin{split} \sum_{w < p \leq 2Q} \frac{p}{p-1} \sum_{d \leq 2Q/p} \frac{\mu^2(d)}{\varphi(d)} \sum_{\substack{m,m' \sim Q/p \\ d\mid (m,m')}} 1 & \ll \sum_{w < p \leq 2Q} \sum_{d} \frac{\mu^2(d)}{\varphi(d)} \left(\frac{Q}{pd}\right)^2 \\
& \ll Q^2 \sum_{p>w} \frac{1}{p^2} \ll \frac{Q^2}{w\log w}. \end{split}$$ This completes the task of bounding $\Sigma_1$.
Case 2 {#case-2 .unnumbered}
------
Finally consider the case when $(p,q) = (p',q') = 1$ and $p'a_q \not\equiv pa_q' \pmod{(q,q')}$. If we further have $p'a_q \not\equiv pa_q' \pmod{(q_s,q_s')}$, then $$S(p,p',q,q') = S(p,p',q_s,q') = S(p,p',q,q_s') = S(p,p',q_s,q_s') = 0,$$ and thus $g_0(p,p',q,q') = 0$. Hence we may impose the condition $p'a_q \equiv pa_q' \pmod{(q_s,q_s')}$. By it suffices to show that $$\Sigma_2:= \frac{1}{\pi(P)^2} \sum_{\substack{ q,q'\sim Q }} \sum_{\substack{ p,p'\sim P \\ (p,q) = (p',q') = 1 \\ p'a_q \not\equiv pa_q'\pmod{(q,q')}
\\ p'a_q\equiv pa_q'\pmod{(q_s,q_s')}}}
\frac{(q_s,q'_s)}{qq'} \ll \frac{1}{w\log w} + \frac{P^{0.1} + \log x}{\pi(P)}.$$ Note that the sum is nonempty only if $(q_r,q_r') > 1$. Arguing as in Case 1, the number of $p'$ satisfying the congruence condition on $p'\pmod{(q_s,q_s')}$ is $$\ll \begin{cases} \pi(P)/\varphi((q_s,q_s')) & \text{if }(q_s,q_s') \leq P^{0.9} \\ P/(q_s,q_s') + 1 & \text{if }(q_s,q_s') > P^{0.9}. \end{cases}$$ This leads to the upper bound $$\Sigma_2\ll \frac{1}{ Q^2} \sum_{\substack{ q,q'\sim Q\\ (q_r,q_r')>1}} \frac{(q_s,q_s')}{\varphi((q_s,q_s')) } +
\frac{1}{\pi(P) Q^2} \sum_{\substack{ q,q'\sim Q\\ (q_s,q_s') > P^{0.9} }} (P+ (q_s,q_s') ) .$$ Now as $(q_s,q_s')\leq (q,q')$ and $\frac{(q_s,q_s')}{\varphi((q_s,q_s')) } \leq \frac{(q,q')}{\varphi((q,q')) } $, we bound $\Sigma_2$ by the same quantity with which we bounded $\Sigma_1$ in , and the result follows.
Putting the pieces together
---------------------------
We now have the ingredients to deduce Theorem \[MathResult2\] from Proposition \[prop:ramare\].
\[Proof of Theorem \[MathResult2\]\] Recall that the left-hand side of the equation in Theorem \[MathResult2\] can be re-written as $$\sum_{n \leq x} f(n) F(n)$$ where the function $F$ is defined as in . We bound this by applying Proposition \[prop:ramare\]. Set $Y = (\log x)^4$ and $Z = x^{{\varepsilon}/2}$, so that $u = \log Z/(\log Y) \asymp \log x/(\log\log x)$. By Lemma \[sievebound\] we have $$E_{\text{sieve}} \ll \frac{x}{u} \ll \frac{x\log\log x}{\log x}.$$ By Lemma \[BilinBound\], the assumption $Q \leq x^{1/2-{\varepsilon}}$, and our choice of $Y$ and $Z$, we have $$E_{\text{bilinear}} \ll \frac x{\log x} + \frac{x}{(w\log w)^{1/2}} .$$ To bound $T$, provided $d\leq Z^2\leq x/Q$ we have $$\sum_{\substack{n \leq x \\ d\mid n}} |F(n)| \leq \sum_{q\sim Q} \sum_{\substack{n \leq x \\ n \equiv a_q\pmod{q}\\ d\mid n}} 1
+ \frac 1{q_r} \sum_{q\sim Q} \sum_{\substack{n \leq x \\ n \equiv a_q\pmod{q_s}\\ d\mid n}} 1 \ll \sum_{q\sim Q} \frac x{qd} \ll \frac xd,$$ so that $T\ll x$. The proof is completed by combining all these estimates together.
Good error terms for smooth-supported $f$ in arithmetic progressions
====================================================================
In the following result we will prove a good estimate for all $f$ supported on $y$-smooth integers. In this article we will only use this with $y$ a fixed power of $x$, but the full range will be useful in the sequel [@DGS].
\[Errorfaps1\] Fix $B\geq 0$ and $0<\eta<\frac 12$. Given $ (\log x)^{4B+5}\leq y=x^{1/u}\leq x^{1/2-\eta}$ let $$R=R(x,y):=\min\{ y^{ \frac{ \log\log\log x}{3\log u}}, x^{\frac \eta{3\log\log x}}\} \ (\leq y^{1/3 + o(1)}).$$ Suppose that $f\in \mathcal C$, and is only supported on $y$-smooth numbers. There exists a set, $\Xi$, of primitive characters $\psi \pmod r$ with $r\leq R$, containing $\ll u^{u+o(u)}(\log x)^{6B+7+ o(1)}$ elements, such that if $q\leq R$ and $(a,q)=1$ then $$| \Delta_{\Xi}(f,x;q,a) | \leq \frac 1{\varphi(q)} \sum_{ \substack{\chi \pmod q \\ \chi \not\in \Xi_q }} \left| S_f(x,\chi) \right| \ll \frac 1{\varphi(q)} \frac{\Psi(x,y)} {(\log x)^{B}} .$$
This immediately implies Proposition \[MainCor\].
Fix $\varepsilon>0$. In Proposition \[Errorfaps1\] we let $\Xi=\Xi(2B+2+\varepsilon)$, where we define $\Xi(C)$ to be the set of primitive characters $\psi \pmod r$ with $r\leq R$ such that there exists $x^{\eta}< X\leq x$ for which $$\label{Hyp3}
S_f(X,\psi) \geq \frac {\Psi(X,y)}{(u\log u)^4(\log x)^{C}} .$$ We prove that $\Xi(C)$ has $\ll u^{u+o(u)}(\log x)^{3C+1}$ elements in section \[LargeS\].
A further support restriction for $f$
-------------------------------------
We will deduce Proposition \[Errorfaps1\] from a similar (but stronger) result restricted to multiplicative functions $f$, which are only supported on those prime powers $p^k$ for which $p\in (q^2,y]$.
Fix a real number $A$. Let $\mathcal T_q^*(A)=\mathcal T_{q,f}(A ,y^2(q^2(\log x)^{2A})^{1/\alpha})$ where $\mathcal T_{q,f}(A,z)$ is the set of characters $\chi \pmod q$ for which there exists $x/z<X\leq x$ such that $$\label{Hyp2}
S_f(X,\chi) > \frac { \Psi(X,y)}{(u\log u)^4(\log x)^{2A+2+\varepsilon}} .$$
\[Errorfaps1q\] Fix $A\geq 0$ and $0<\eta<\frac 12$. If $(\log x)^{1+\varepsilon}\leq y=x^{1/u}\ll x^{1/2-\eta}$ and $q\leq \min\{ y^{1/2} , x^{ \eta/2} \}/(\log x)^{ A+2}$ with $(a,q)=1$, and $f\in \mathcal C$ is only supported on prime powers $p^k$ with $q^2<p\leq y$, then $$\label{smallsupport}
| \Delta_{\mathcal T_q^*(A)}(f,x;q,a) | \leq \frac 1{\varphi(q)} \sum_{ \substack{\chi \pmod q \\ \chi \not\in \mathcal T_q^*(A) }} \left| S_f(x,\chi) \right| \ll \frac 1{\varphi(q)} \frac{\Psi(x,y)} {(\log x)^{A}} .$$
By minor modifications of the proof of Proposition \[Errorfaps1q\] we will deduce the following:
\[ErrorfapsA=0\] Fix $0 < \eta < \tfrac 12$. If $(\log x)^{1+\varepsilon}\leq y\leq x^{1/2-\eta}$ and $q\leq \min\{ y^{1/2} ,\ x^{ \eta/2} \}$ with $(a,q)=1$, and $f\in \mathcal C$ is only supported on prime powers $p^k$ with $q^2<p\leq y$, then $$\sum_{ \substack{\chi \pmod q }} \left| S_f(x,\chi) \right| \ll \Psi(x,y) (\log x)^{3+\varepsilon} .$$
By Cauchying, the upper bound is $\ll q^{1/2} \Psi(x,y) $, so Corollary \[ErrorfapsA=0\] is better in a wide range.
In the next seven subsections we will prove Proposition \[Errorfaps1q\]. Throughout this section we will write $(u\log u)^2=:V$ for convenience, so that if $1\leq d\leq y^2$ then, as a consequence of , $$\label{SmoothCompare}
\Psi(x/d,y)\ll V\Psi(x,y)/d.$$
Harper’s identity and Perron’s formula
--------------------------------------
\[Proof of Proposition \[Errorfaps1q\]\] We use Harper’s identity: $$f(n)\log n = \Lambda_f(n) + \int_{\beta=0}^\infty n^{-\beta} \sum_{abm=n} \Lambda_f(a) a^\beta \Lambda_f(b) f(m) d\beta .$$ Multiplying through by $\overline\chi(n)$ and summing over $n$ we obtain $$\begin{split}
\sum_{x<n\leq 2x} f(n)\overline\chi(n) \log n &= \int_{\beta=0}^\infty \sum_{\substack{x<abm\leq 2x \\ a,b\leq y}} (abm)^{-\beta} \Lambda_f(a)\overline\chi(a) a^\beta \Lambda_f(b)\overline\chi(b) f(m)\overline\chi(m) d\beta \\
& + \text{Error},
\end{split}$$ where the error term is $O(y\log x/\log y)$ from the contribution of the $\Lambda_f(n)$, plus $$\ll \sum_{\substack{x<abm\leq 2x \\ a \text{ or } b> y\\ P(abm)\leq y}} \frac{\Lambda(a) \Lambda(b)}{\log (x/a)} \ll
\sum_{\substack{ ab \leq 2x \\ a \text{ or } b> y\\ P(ab)\leq y}} \frac{\Lambda(a) \Lambda(b)}{ \log (x/a)} \Psi(2x/ab,y).$$ Now if $a=p^i, b=q^j$ then the term with $a$ replaced by $ap$ or the term with $b$ replaced by $bq$ is no more than this term times a constant $<1$ that depends only on $\alpha$ (by and ). Therefore we can restrict our attention to where $a$ is a prime power $<y$ and $b$ is a prime power in $(y,y^2]$. Therefore the above is $$\begin{split}
&\ll \sum_{a\leq y < b\leq y^2} \frac{\Lambda(a) \Lambda(b)}{ \log x} \Psi(x/ab,y)\ll
\frac{V^{3/2} \Psi(x,y)} {\log x} \sum_{a\leq y < b\leq y^2} \frac{\Lambda(a) \Lambda(b)}{ ab} \\
& \ll \frac{\Psi(x,y)}{\sqrt{y}} \cdot \frac{V^{3/2}\log y}{\log x}
\ll \frac { \Psi(x,y) }{\varphi(q) (\log x)^{A-1}} ,
\end{split}$$ by , as $x^{1/2}>y\geq V^3 \varphi(q)^2 (\log x)^{2A-2}$.
We use Perron’s formula to try to work with the main term, applying it at $x$ and $2x$. Wlog we assume that $m$ is only supported on $(x/y^2,2x]$. Therefore our integrand equals, for $c=1+1/\log x$, $$\frac 1{2i\pi} \int_{Re(s)=c} \sum_{\substack{x/y^2<m\leq 2x \\ a,b\leq y}} (bm)^{-\beta} \Lambda_f(a)\overline\chi(a) \Lambda_f(b)\overline\chi(b) f(m)\overline\chi(m) \left( \frac{x}{abm} \right)^s \frac{2^s-1} s ds,$$ which we will truncate at a height $T$ where $$T=\varphi(q)U \ \text{with} \ U:= V(\log x)^{A+1+\varepsilon} .$$ The main term of the integrand then equals $$\frac 1{2i\pi} \int_{\substack{s=c+it \\ |t|\leq T}}
\sum_{a \leq y} \frac{ \Lambda_f(a)\overline\chi(a) } { a^{s}} \sum_{ b\leq y} \frac{ \Lambda_f(b)\overline\chi(b) } { b^{\beta+s}} \sum_{x/y^2<m\leq 2x} \frac{ f(m)\overline\chi(m) } { m^{\beta+s}}
\frac{(2^s-1)x^s} s ds ,$$ and, taking absolute values, we therefore have that the main part of our integral is $$\ll x \int_{\substack{s=c+it \\ |t|\leq T}} \int_{\beta=0}^\infty \left| \sum_{a \leq y} \frac{ \Lambda_f(a)\overline\chi(a) } { a^{s}} \right| \left| \sum_{ b\leq y} \frac{ \Lambda_f(b)\overline\chi(b) } { b^{\beta+s}} \right| \left| \sum_{x/y^2<m\leq 2x} \frac{ f(m)\overline\chi(m) } { m^{\beta+s}} \right| d\beta \frac{dt }{1+|t|}.$$
Truncation bounds; the contribution with $|t|>T$
------------------------------------------------
The usual formula for truncation means that for the integral for $n$ up to $x$ we have an error term of $$\ll \sum_{\substack{ a,b\leq y \\ x/y^2<m\leq 2x \\ P(m)\leq y}} (bm)^{-\beta} \Lambda (a)\ \Lambda (b) \left( \frac{x}{abm} \right)^{c} \min\left \{ 1 , \frac 1{T|\log (x/abm)|} \right\} .$$ Integrating over $\beta$, the $(bm)^{-\beta}$ contributes $1/\log x$. We have $1/2y^2\leq x/abm\leq y^2$ and so $(x/abm)^{c} \asymp x/abm$, so the error term above is $$\ll \frac x{\log x} \sum_{\substack{ a,b\leq y \\ x/y^2<m\leq 2x\\ P(m)\leq y}} \frac{\Lambda (a)}a \frac{\Lambda (b) }b \frac 1m \min\left \{ 1 , \frac 1{T|\log (x/abm)|} \right\} .$$ The contribution of the triples $abm$ with $abm\leq x/2$ or $>2x$ is $$\ll \frac x{T \log x} \sum_{\substack{ a,b\leq y \\ x/y^2<m\leq 2x\\ P(m)\leq y}} \frac{\Lambda (a)}a \frac{\Lambda (b) }b \frac 1m
\ll \frac {V\Psi(x,y)(\log y)^2}{T \log x },$$ by . Next we look at those $abm\in (x(1-\frac {k+1}T), x(1-\frac {k}T)]$ for some $k,\ 1\leq k\leq T/2$, which contribute $$\ll \frac x{k \log x} \sum_{\substack{ a,b\leq y }} \frac{\Lambda (a)}a \frac{\Lambda (b) }b
\sum_{\substack{\frac x{ab} (1-\frac {k+1}T) <m\leq \frac x{ab} (1-\frac {k}T)\\ P(m)\leq y}} \frac 1m \ll \frac {V\Psi(x,y)(\log y)^2}{kT \log x},$$ since by we have$$\sum_{\substack{\frac x{ab} (1-\frac {k+1}T) <m\leq \frac x{ab} (1-\frac {k}T)\\ P(m)\leq y}} \frac 1m \ll
\frac {\Psi(x/ab,y)}{T\cdot x/2ab} \ll \frac VT \frac{\Psi(x,y)}x .$$ The same argument works when $abm\in [x(1+\frac {k}T), x(1+\frac {k+1}T))$ for some $k,\ 1\leq k\leq T/2$. Finally if $abm\in (x(1-\frac {1}T), x(1+\frac {1}T))$, we get the same bound as with $k=1$ above, and so our total truncation error is $$\ll \frac {V\Psi(x,y)(\log y)^2\log T}{T \log x} \ll \frac {\Psi(x,y) }{\varphi(q) (\log x)^{A-1}} ,$$ by our choice of $T$. The same argument works for the integral up to $2x$.
Back to the main term
---------------------
Let $T_q=T_q(A, x)$ be the set of characters $\chi \pmod q$ for which there exists $x/y^2<X\leq x$, for which $$\label{Hyp5}
S_f(X,\chi) > { \Psi(X,y)}/{V^2(\log x)^{2A+2+\varepsilon}} .$$ Note that this is a weaker hypothesis than in that the range for $X$ is slightly shortened. Summing what remains of the main term over all $\chi \not\in T_q$, and dividing through by $\varphi(q)$ we obtain the bound $$\ll x \int_{\substack{s=c+it \\ |t|\leq T }} \int_{\beta=0}^\infty \frac 1{\varphi(q)} \sum_{ \chi \not\in T_q}
\left| \sum_{a \leq y} \frac{ \Lambda_f(a)\overline\chi(a) } { a^{s}} \right| \left| \sum_{ b\leq y} \frac{ \Lambda_f(b)\overline\chi(b) } { b^{\beta+s}} \right| \left| \sum_{\substack{x/y^2<m\leq 2x\\ P(m)\leq y}} \frac{ f(m)\overline\chi(m) } { m^{\beta+s}} \right| d\beta \frac{dt }{1+|t|}.$$
A more subtle truncation: $T\geq |t|>U$ {#subtle}
---------------------------------------
In this range we extend the character sum to all $\chi \pmod q$, and use the trivial upper bound, with $\beta'=\beta+\frac 1{\log x}$, to obtain $$\label{Trivial M}
\left| \sum_{\substack{x/y^2<m\leq 2x\\ P(m)\leq y}} \frac{ f(m)\overline\chi(m) } { m^{\beta+s}} \right| \leq
\sum_{\substack{x/y^2<m\leq 2x\\ P(m)\leq y}} \frac{ 1 } { m^{\beta+c}} \ll \frac{V \Psi(x,y) } {x \beta' (x/y^2)^{\beta'}} \ll
\frac{ V\Psi(x,y) \log x } { x (x/y^2)^{\beta }} .$$ Therefore our integral is $V\Psi(x,y) \log x$ times $$\ll \int_{\substack{s=c+it \\ T\geq |t|>U}} \int_{\beta=0}^\infty \frac 1{\varphi(q)} \sum_{ \chi \pmod q}
\left| \sum_{a \leq y} \frac{ \Lambda_f(a)\overline\chi(a) } { a^{s}} \right| \left| \sum_{ b\leq y} \frac{ \Lambda_f(b)\overline\chi(b) } { b^{\beta+s}} \right| \frac{d\beta} { (x/y^2)^{\beta }} \frac{dt }{1+|t|}.$$ We now use the Cauchy-Schwarz inequality, so that the square of this integral is $$\leq \int_{\substack{s=c+it \\ T\geq |t|>U}} \int_{\beta=0}^\infty \frac 1{\varphi(q)} \sum_{ \chi \pmod q}
\left| \sum_{a \leq y} \frac{ \Lambda_f(a)\overline\chi(a) } { a^{s}} \right|^2 \frac{d\beta} { (x/y^2)^{\beta }} \frac{dt }{1+|t|}$$ times $$\int_{\substack{s=c+it \\ T\geq |t|>U}} \int_{\beta=0}^\infty \frac 1{\varphi(q)} \sum_{ \chi \pmod q}
\left| \sum_{ b\leq y} \frac{ \Lambda_f(b)\overline\chi(b) } { b^{\beta+s}} \right|^2 \frac{d\beta} { (x/y^2)^{\beta }} \frac{dt }{1+|t|} .$$ We begin by noting that for $s=c+it$, $$\begin{split}
\frac 1{\varphi(q)} \sum_{ \chi \pmod q}
\left| \sum_{a\leq y}\frac{ \Lambda_f(a)\overline\chi(a) }{a^{s}} \right|^2 &=
\sum_{a,b\leq y}\frac{ \Lambda_f(a)\overline{\Lambda_f(b)}}{(ab)^{c}(a/b)^{it}} \frac 1{\varphi(q)} \sum_{ \chi \pmod q} \overline\chi(a) \chi(b) \\
&=
\sum_{\substack{a,b\leq y \\ a\equiv b \pmod q}}\frac{ \Lambda_f(a)\overline{\Lambda_f(b)}}{(ab)^{c}} (a/b)^{-it} .
\end{split}$$ Therefore, as $|\Lambda_f(a)\overline{\Lambda_f(b)}/(ab)^{c}|\leq \Lambda(a) \Lambda(b)/ab$, and as $f$ is only supported on primes $>q^2$, $$\int_{|t|=U}^T \frac 1{\varphi(q)} \sum_{ \chi \pmod q}
\left| \sum_{a\leq y}\frac{ \Lambda_f(a)\overline\chi(a) }{a^{s}} \right|^2 \frac{dt }{1+|t|} \leq
\sum_{\substack{q^2<a,b\leq y \\ a\equiv b \pmod q}}\frac{ \Lambda(a) \Lambda(b)}{ab}
\left| \int_{|t|=U}^T (a/b)^{-it} \frac{dt }{1+|t|} \right| .$$ Now we obtain two different bounds on this integral: Trivially $|(a/b)^{it}|=1$ so the integral is $\ll \log (T/U)\leq \log T$. Alternatively, working with $U<t\leq T$ (the integral for $-t$ being entirely analogous), for $a\ne b$ $$\int_{U}^T (a/b)^{-it} \frac{dt }{1+t} = \frac{(a/b)^{-it}}{i\log(b/a) (1+t)} \bigg|_{U}^T + \int_{U}^T \frac{(a/b)^{-it}}{i\log(b/a) } \frac{dt }{(1+t)^2} \ll \frac{1}{|\log(b/a)| U} .$$ Therefore, our original $a$-integral is $$\ll \sum_{\substack{q^2<a\leq b\leq y \\ a\equiv b \pmod q}}\frac{ \Lambda(a) \Lambda(b)}{ ab }
\min \left\{ \log T, \frac{1}{|\log(b/a)| U} \right\} \cdot
\int_{\beta=0}^\infty \frac{d\beta} { (x/y^2)^{\beta }} .$$ The $\beta$-integral is $\ll 1/\log x$. The sum over $a$ and $b$ needs to be partitioned into intervals: First where $a\leq b\leq a(1+1/U\log T)$ in which case we multiply by $\log T$; then where $a(1+2^j/U\log T)\leq b\leq a(1+2^{j+1}/U\log T)$ for $0\leq j\leq J$, with $2^J\asymp U\log T$, in which case the multiplier is $2^{-j}\log T$; and finally where $2^ka<b\leq 2^{k+1}a$ for $0\leq k\leq K$ where $2^K\asymp y/a$, whence the multiplier is $1/kU$. In the first two sets of intervals we use the trivial upper bound $\Lambda(b)\leq \log b\leq \log 2a$, and subsequently we use the Brun-Titchmarsh Theorem. Therefore for a given $a$ the sum is $$\ll \frac 1{U\varphi(q)} \sum_{q^2<a\leq y} \frac{ \Lambda(a)}a (J\log a + \log K)�\ll \frac {(\log x)^{2+\varepsilon}}{U\varphi(q)} \ll
\frac 1{V\varphi(q)(\log x)^{A-1}} .$$ as $T,y\leq x$, by our choice of $U$. Therefore the $a$-integral is $\ll 1/(V\varphi(q)(\log x)^{A})$. The same argument works for the $b$-integral (where we use that $c+\beta>1$ rather than $c>1$, and so the total contribution of this part of the $t$-integral is $$\ll \frac{ \Psi(x,y) }{ \varphi(q)(\log x)^{A-1} } ,$$ which is acceptable.
Consequences of
----------------
If $\chi\not\in T_q$ then, by , we have $$\begin{split}
\sum_{ x/y^2<m\leq X} f(m)m^{-it} \overline\chi(m) &= \int_{x/y^2}^X \frac{dS_f(h,\chi) }{h^{it}} =
\frac{ S_f(h,\chi) }{h^{it}}\bigg|_{x/y^2}^X + it \int_{x/y^2}^X \frac{ S_f(h,\chi) }{h^{1+it}}dh \\
& \ll
\frac { \Psi(X,y)}{V^2(\log x)^{2A+2+\varepsilon}} +\frac { |t|}{V^2(\log x)^{2A+2+\varepsilon}} \int_{x/y^2}^X \frac{\Psi(h,y)}h dh \\
& \ll \frac { (1+|t|) \Psi(X,y)}{V^2(\log x)^{2A+2+\varepsilon}} ,
\end{split}$$ as $\Psi(h,y)/h\ll (\Psi(X,y)/X)(X/h)^{1-\alpha}$, and $\int^X_{x/y^2} (X/h)^{1-\alpha} dh \ll X$.
Calling this new sum $S(X)$ then, for $\delta=c-1+\beta\geq 1/\log x$ where $ \beta\geq 0$, we have, using , $$\label{Hyp}
\begin{split}
\sum_{x/y^2<m\leq 2x} \frac{ f(m)\overline\chi(m) } { m^{c+\beta+it}} &=
\int_{x/y^2}^{2x} \frac{dS(X)}{X^{1+\delta}}
\ll \frac {(1+|t|)V \Psi(x,y)/x }{V^2(\log x)^{2A+2+\varepsilon}} \frac{1}{\delta (x/y^2)^{\delta}}\\
& \ll \frac {(1+|t|) \Psi(x,y)/x}{V(\log x)^{2A+1+\varepsilon} x^{\eta\beta}},
\end{split}$$ for $1\leq y\leq x^{(1-\eta)/2}$.
Bounds under for $ |t|\leq U$
------------------------------
If $\chi\not\in T_q$ then, by , our remaining main term is $$\ll \frac {\Psi(x,y)}{V(\log x)^{2A+1+\varepsilon} } \int_{\substack{s=c+it \\ |t|\leq U}} \left( \int_{\beta=0}^\infty x^{-\eta\beta} \left| \sum_{a \leq y} \frac{ \Lambda_f(a)\overline\chi(a) } { a^{s}} \right| \left| \sum_{ b\leq y} \frac{ \Lambda_f(b)\overline\chi(b) } { b^{\beta+s}} \right| d\beta\right) dt.$$ We now sum over all such $\chi\not\in T_q$ and divide by $\varphi(q)$, so that the inner integral becomes $$\int_{\beta=0}^\infty x^{-\eta\beta} \frac 1{\varphi(q)} \sum_{ \chi \pmod q} \left| \sum_{a \leq y} \frac{ \Lambda_f(a)\overline\chi(a) } { a^{s}} \right| \left| \sum_{ b\leq y} \frac{ \Lambda_f(b)\overline\chi(b) } { b^{\beta+s}} \right| d\beta.$$ We follow the plan of the previous subsection, Cauchying and then computing the separated integrals, but here we do not bring the integral over $t$ inside, simply using the bound $|(a/b)^{it}|=1$. Therefore the square of this inner integral is, by Cauchy-Schwarz, $$\leq \int_{\beta=0}^\infty x^{-\eta\beta} \frac 1{\varphi(q)} \sum_{ \chi \pmod q}
\left| \sum_{a\leq y}\frac{ \Lambda_f(a)\overline\chi(a) }{a^{s}} \right|^2 d\beta$$ times the analogous integral for $b$. Expanding the square in the first integral we have $$\frac 1{\varphi(q)} \sum_{ \chi \pmod q}
\left| \sum_{a\leq y}\frac{ \Lambda_f(a)\overline\chi(a) }{a^{s}} \right|^2
\leq \sum_{\substack{q^2<a,b\leq y \\ a\equiv b \pmod q}}\frac{ \Lambda (a) {\Lambda (b)}}{ab} ,$$ and then the integral over $\beta$ is $\ll 1/\log x$. We get to the same point with the $b$-integral. By the Brun-Titchmarsh theorem we have $$\sum_{\substack{q^2<a,b\leq y \\ a\equiv b \pmod q}}\frac{ \Lambda (a) {\Lambda (b)}}{ab} =
\sum_{\substack{q^2<a \leq y }}\frac{ \Lambda (a) }{a}
\sum_{\substack{q^2<b\leq y \\ b\equiv a \pmod q}}\frac{ {\Lambda (b)}}{b} \ll \frac{(\log y)^2}{\varphi(q) }
\ll \frac{(\log x)^2}{\varphi(q) }.$$ Our remaining main term, summed over all $\chi \not\in T_q$, divided through by $\varphi(q)$, is therefore $$\ll \frac { \Psi(x,y)}{V(\log x)^{{2A+1+\varepsilon}} } \int_{\substack{s=c+it \\ |t|\leq U}} \frac{\log x}{\varphi(q) } dt \ll \frac { \Psi(x,y)}{\varphi(q)(\log x)^{{A-1}} }.$$
Combining results and partial summation
---------------------------------------
Combining the above we have therefore proved that $$\frac 1{\varphi(q)} \sum_{ \substack{\chi \pmod q \\ \chi \not\in T_q(A,x) }} \left|\sum_{x<n\leq 2x} f(n)\overline\chi(n) \log n\right|
\ll \frac{ { \Psi(x,y)}}{\varphi(q) (\log x)^{A-1}} ,$$ provided $ x^{(1-\eta)/2}\geq y\geq V^3q^2(\log x)^{2A-2}$. Select integer $k$ so that $2^{k\alpha}\asymp q(\log x)^{A}$. Since $$\bigcup_{j=1}^k \ T_q(A,x/2^j) \subset \mathcal T_{q,f}(A,2^ky^2),$$ we deduce that $$\frac 1{\varphi(q)} \sum_{ \substack{\chi \pmod q \\ \chi \not\in \mathcal T_q(A,2^{k}y^2) }} \left|\sum_{X/2<n\leq X} f(n)\overline\chi(n) \log n\right|
\ll \frac{ { \Psi(X,y)}}{\varphi(q) (\log x)^{A-1}} ,$$ for $X=x,x/2,x/4,\ldots, x/2^{k-1}$. Summing these up, as $$\sum_{j=0}^{k-1} \Psi(x/2^j,y) \ll \Psi(x ,y) \sum_{j=0}^{k-1} \frac 1{2^{j\alpha}} \ll \Psi(x ,y),$$ and as $\Psi(x/2^k,y)\ll \Psi(x,y)/2^{k\alpha} \ll \Psi(x,y)/q(\log x)^{A} $, we deduce that $$\frac 1{\varphi(q)} \sum_{ \substack{\chi \pmod q \\ \chi \not\in \mathcal T_q(A,2^{k}y^2) }} \left|\sum_{ n\leq x} f(n)\overline\chi(n) \log n\right|
\ll \frac{ \Psi(x,y) }{\varphi(q) (\log x)^{A-1}} ,$$ provided $y\geq V^3q^2(\log x)^{2A-2}$ and $y^2(q(\log x)^A)^{1/\alpha}\leq x^{1-\eta}$.
We deduce that $$\frac 1{\varphi(q)} \sum_{ \substack{\chi \pmod q \\ \chi \not\in \mathcal T_q(A,2^{2k}y^2) }} \left|\sum_{ n\leq X} f(n)\overline\chi(n) \log n\right|
\ll \frac{\Psi(X,y) }{\varphi(q) (\log x)^{A-1}} ,$$ for all $X$ in the range $x/(q(\log x)^A)^{1/\alpha}\leq X\leq x$, provided $$y\geq V^3q^2(\log x)^{2A-2} \ \text{ and } y(q(\log x)^A)^{1/\alpha}\leq x^{(1-\eta)/2}.$$ Then holds by partial summation, as $\int_2^x (\Psi(t,y) /t)dt \ll \Psi(x,y)$.
Now as $y\geq (\log x)^{1+\varepsilon}$ we have $V\leq (\log x)^2$, so the first range hypothesis holds when $q\leq y^{1/2} /(\log x)^{ A+2}$. If $y\leq x^{1/2-\eta}$ then the second hypothesis holds when $q \leq x^{ \eta\alpha/2}/(\log x)^A$. Now this follows from the first range hypothesis if $y\leq x^\eta$; if $y$ is larger then this simplifies to $q \ll x^{ \eta/2}/(\log x)^A$.
Proof of Corollary \[ErrorfapsA=0\]
-----------------------------------
We take $T=\varphi(q)$ in the same argument, and extend the range in section \[subtle\] to $ |t|\leq T$, getting rid of the need for the final range. In the calculations in that section we replace $U$ by $U+1$ in our bounds, and end up with a contribution $\ll V\Psi(x,y)(\log x)^{2+\varepsilon}/\varphi(q) $. The new choice of $T$ means that the contribution with $|t|>T$ also is bounded by the same quantity, so, going through the same process, we end up with the upper bound $\ll V\Psi(x,y)(\log x)^{1+\varepsilon}/\varphi(q)\ll \Psi(x,y)(\log x)^{3+\varepsilon}/\varphi(q) $ on our sum, as desired.
Extension to $f$ supported on any primes $\leq y$
-------------------------------------------------
Given $f$ supported on primes $p \leq y$, define the multiplicative function $g$ with $g(p^k)=0$ if $p\leq q^2$, and $g(p^k)=f(p^k)$ if $p> q^2$. Therefore if $\psi \pmod r$ is a primitive character that induces $\chi \pmod q$ then $g(m) \overline{\chi}(m) = f(m) \overline{\psi}(m) 1_{(m,P)=1}$ where $P=\prod_{p\leq q^2} p$. Let $h(.)$ be the multiplicative function which is supported only on the prime powers $p^k$, for which $p\leq q^2$ but does not divide $r$, and defined so that $(h*f\overline\psi)(p^k)=0$ if $k\geq 1$. Thus $h*f\overline\psi = g\overline\chi$ and so $$\label{Ident}
S_{g}(x,\chi) = \sum_{m\leq x} h(m) S_{f}(x/m,\psi) .$$ We note that $h\in \mathcal C$ as $f\in \mathcal C$, so that each $|h(m)|\leq 1$.
Assume that $\psi$ is a primitive character mod $r$ with $r\leq R$, such that $$|S_{f}(X ,\psi)| \leq \Psi(X,y)/V^2(\log x)^C$$ for all $X$ in the range $x/z_0< X\leq x$ where $z_0= z_1 q^{\log\log x}$. Now let $x/z_1\leq X\leq x$. Then $|S_{f}(X/m,\psi)| \leq \Psi(X/m,y)/V^2(\log x)^C$ for $m\leq q^{\log\log x}$ and so, by and , we obtain $$\begin{split}
|S_{g}(X,\chi)| & \ll \ \frac {\Psi(X,y)}{V^2(\log x)^C} \sum_{\substack{m\leq q^{\log\log x} \\ P(m)\leq q^2}} \frac 1{m^\alpha}+ \Psi(X,y)\sum_{\substack{q^{\log\log x}<m\leq X\\ P(m)\leq q^2}} \frac 1{m^\alpha} \\
& \ll \ \frac {\Psi(X,y)}{V^2(\log x)^C} \exp \left( \sum_{p\leq q^2} \frac 1{p^\alpha}\right) + \Psi(X,y)\sum_{\substack{q^{\log\log x}<m\leq X\\ P(m)\leq q^2}} \frac 1{m^\alpha} .
\end{split}$$ Now if $$\label{Restrictq}
\log q \leq \frac{\log y \log\log\log x}{3\log u},$$ then $q^{2(1-\alpha)} \leq (\log\log x)^{2/3+o(1)}$ as $y^{1-\alpha}=u^{1+o(1)}$, and so $$\sum_{p\leq q^2} \frac 1{p^\alpha} \ll \log 1/\alpha + \frac{q^{2(1-\alpha)} }{(1-\alpha)\log q} \leq (\log\log x)^{2/3+o(1)}.$$ Also if $M=q^{2v}$ with $v\geq \frac 12\log\log x$ then $$\sum_{\substack{m\sim M \\ P(m)\leq q^2}} \frac 1{m^\alpha} \ll M^{1-\alpha} v^{-v+o(v)} \ll \left( q^{2(1-\alpha)}/v^{1+o(1)}\right)^v \ll (\log\log x)^{-\frac 13 (v+o(v))} .$$ Combining this information yields that $$|S_{g}(X,\chi)| \ll \frac {\Psi(X,y)}{V^2(\log x)^{C+o(1)}\ }$$ Therefore fails taking $C= 2A+2+2\varepsilon$, that is $\chi\not\in \mathcal T_{q,g}(A,z_1)$. Proposition \[Errorfaps1q\] then implies that, for $z_1=y^2(q^2(\log x)^{2A})^{1/\alpha}$, $$\label{smallsupport2}
\sum_{ \substack{\chi \pmod q \\ \chi \not\in \mathcal T_q^*(A)}} \left| S_g(x,\chi) \right| \ll \frac{\Psi(x,y)} {(\log x)^{A}} .$$
Write each $n\leq x$ as $n=rN$ with $P(r)\leq q^2$, and $p|N \implies p> q^2$, so that $f(n)=f(r)g(N)$, and therefore $$S_f(x,\chi) = \sum_{\substack{r\geq 1\\ P(r)\leq q^2}} f(r) S_g(x/r,\chi) .$$
Now assume that $\psi\not\in \Xi(C)$, so that $|S_{f}(X ,\psi)| \leq \Psi(X,y)/V^2(\log x)^C$ for all $x/z_2< X\leq x$ where $z_2= z_0 q^{\log\log x}$, so that holds with $x$ replaced by $X$, for every $X$ in the range $x/q^{\log\log x}\leq X\leq x$. Then we have, by , Corollary \[ErrorfapsA=0\] and , $$\begin{split}
\sum_{ \substack{\chi \pmod q \\ \chi \not\in \Xi_q(C)}} \left| S_f(x,\chi) \right| &\leq
\sum_{\substack{r\leq q^{\log\log x}\\ P(r)\leq q^2}} \sum_{ \substack{\chi \pmod q \\ \chi \not\in \Xi_q(C) }} |S_g(x/r,\chi) |
+ \sum_{\substack{r> q^{\log\log x}\\ P(r)\leq q^2}} \sum_{ \substack{\chi \pmod q }} |S_g(x/r,\chi) | \\
&\ll \frac{\Psi(x,y)} {(\log x)^{A}} \sum_{\substack{r\leq q^{\log\log x}\\ P(r)\leq q^2}} \frac 1{r^\alpha} + \Psi(x,y)(\log x)^{3+\varepsilon} \sum_{\substack{r> q^{\log\log x}\\ P(r)\leq q^2}} \frac 1{r^\alpha} \\
&\ll \frac{\Psi(x,y)} {(\log x)^{A+o(1)}} ,
\end{split}$$ proceeding for these two sums just as we did above.
The claimed estimate then follows by taking $A=B+\varepsilon$ and so $C= 2B+2+4\varepsilon$. To guarantee that $x/z_2\geq x^{\eta}$ we need that $$y(q(\log x)^A)^{1/\alpha} q^{\log\log x} \leq x^{(1-\eta)/2},$$ and we already have the restrictions $y\geq V^3q^2(\log x)^{2A-2}$ and . These all follow from the bounds on $q$ and $y$ given in the hypothesis.
Bounding the size of $\Xi(C)$ {#LargeS}
-----------------------------
The large sieve gives that $$\sum_{r\leq \sqrt{X}} \sum_{\psi \text{ primitive mod } r} \left| \sum_{ m\leq X} f(m) \overline\psi(m) \right|^2 \ll X \Psi(X,y) .$$ Therefore, for a given $X$ in the range $ x^{\eta}\leq X\leq x$, the number of exceptional $\psi$ with $r\leq R$ is $\ll (X/\Psi(X,y)) (\log x)^{2C} u^{O(1)}\ll u^{u+o(u)} (\log x)^{2C}$.
If holds for $X$, and $|X'-X|\ll X/(\log x)^{C}$ then holds for $X'$. So to obtain the full range for $X$ we sample at $X$-values spaced by gaps of length $\gg X/(\log x)^{C}$. Therefore the number of exceptional $\psi$ with $r\leq R$ in our range is $\ll u^{u+o(u)} (\log x)^{3C+1}$, as claimed.
Deduction of several Theorems {#sec:deduce-many-thms}
=============================
Proof of two Theorems on $f$ in arithmetic progressions
-------------------------------------------------------
\[Proof of Theorem \[Cor:Result2\] \] Let $J=k+1$ so that $\frac 12 \leq B=1-\varepsilon<1-1/\sqrt{J}$. Let $w=(\log x)^{2B}$ and $ \exp(C(\log\log x)^2)<Q\leq x^{1/2-\delta}$, with $C>4B^2$. Let $\Xi=\{ \psi_1,\ldots, \psi_{J-1}\}$ so that $\Delta_k=\Delta_{\Xi}$ and holds. Then, by Corollary \[MathResult2Cor\] and Corollary \[MathResult2Cor\*\*\] we deduce that $$\sum_{q \sim Q} \max_{(a,q )=1} \left| \Delta_{\Xi}(f,x;q,a) \right|
\ll \frac x {(\log x)^{B}} .$$
\[Deduction of Corollary \[Thm: SW\]\] We apply Proposition \[Using SW\] whose hypotheses are satisfied using Theorem \[Cor:Result2\] for any $C>0$, suitably adjusting the value of $\varepsilon$.
On average, supported only on the smooths
------------------------------------------
\[Proof of Theorem \[Keep Xi\]\] We proceed as in the proof of Theorem \[MathResult2\], but now with the parameters $Y=\exp((\log\log x)^{2}), Z=y$ and $w=(\log x)^{2A}$. Since $Q\leq x^{1/2}/y^{1/2}(\log x)^A$, Lemma \[BilinBound\] implies that $$E_{\text{bilinear}} \ll x \left(\frac{1}{(w\log w)^{1/2}} + \frac{Y^{0.1} + (\log x)^{1/2}}{\pi(Y)^{1/2}}\right) +QZ^{1/2}x^{1/2} \ll \frac x{(\log x)^A}.$$ Moreover, we have $$\frac{T}{Y \log Y} \ll \frac x{(\log x)^A}.$$ The key difference here is in the $E_{\text{sieve}}$ term. Since $f$ is only supported on the $y$-smooth integers, we now have $$E_{\text{sieve}} \leq \sum_{n \leq x} | F(n)| {\mathbf{1}}_{(n,\prod_{p>Y}p) = 1} \leq \sum_{q\sim Q} \left(\Psi(x,Y,a_q,q) + \frac 1{q_r} \Psi(x,Y,a_q,q_s)\right).$$ By , followed by , this quantity is $$\ll \sum_{q\sim Q} \frac 1{q} \Psi(x,Y) + \frac x{(\log x)^A} \ll \frac x{(\log x)^A}.$$ Thus from Proposition \[prop:ramare\] we deduce the following variant on Theorem \[MathResult2\]: $$\sum_{q \sim Q} \max_{(a,q)=1} \left| \sum_{\substack{n \leq x\\ n \equiv a\pmod{q}}} f(n) -\frac{1}{ q_r}
\sum_{\substack{n \leq x\\ n \equiv a\pmod{q_s}}} f(n) \right| \ll \frac x{(\log x)^A}.$$
Now we apply Proposition \[MainCor\] with $B$ chosen so that $6B+7<2A$, which implies that holds, provided $x^{1/2-{\varepsilon}}\geq Q\geq x^{ \varepsilon/(2\log\log x)}$. By Corollary \[MathResult2Cor\] suitably amended with this input, we then deduce that $$\label{eq:boum0}
\sum_{q \sim Q} \max_{(a,q)=1} \left| \Delta_{\Xi}(f,x;q,a) \right| \leq
\frac { 1}{\log w}
\sum_{\substack{ q_s\leq 2Q \\ P(q_s)\leq w}} \max_{(a,q_s)=1} | \Delta_{\Xi}(f,x;q_s,a) | + O\left( \frac x{(\log x)^A}\right) .$$
By Proposition \[MainCor\] we have, for $R= x^{ \varepsilon/(3\log\log x)}$ $$\label{eq:boum7}
\frac { 1}{\log w}
\sum_{\substack{ q_s\leq R \\ P(q_s)\leq w}} \max_{(a,q_s)=1} | \Delta_{\Xi}(f,x;q_s,a) | \ll
\frac { 1}{\log w}
\sum_{\substack{ q_s\leq R \\ P(q_s)\leq w}} \frac 1{\varphi(q_s)} \frac{x} {(\log x)^{B}} \ll \frac{x} {(\log x)^{B}} .$$ For the remaining $q_s$ we use the trivial upper bound $ | \Delta_{\Xi}(f,x;q ,a) |\ll (1+|\Xi_{q}|)x/q$, to obtain $$\begin{split}
& \frac { 1}{\log w}
\sum_{\substack{ R\leq q_s\leq 2Q \\ P(q_s)\leq w}} \max_{(a,q_s)=1} | \Delta_{\Xi}(f,x;q_s,a) | \ll
\frac { x}{\log w}
\sum_{\substack{ R\leq q\leq 2Q \\ P(q)\leq w}} \frac {1+|\Xi_{q}|}{q} \\
& \leq
\frac { x}{\log w} \left( \left( 1+ \sum_{\substack{\chi\in \Xi \\ r_\chi\leq Y} } \frac 1{r_\chi} \right) \sum_{\substack{ q\geq R/Y \\ P(q)\leq w}} \frac 1{q} + \sum_{\substack{\chi\in \Xi \\ r_\chi>Y} } \frac 1{r_\chi} \
\sum_{\substack{ q\geq 1 \\ P(q)\leq w}} \frac {1}{q} \right) \ll \frac{x} {(\log x)^{A}},
\end{split}$$ using , and that $ \sum_{\chi\in \Xi,\ r_\chi>Y } 1/{r_\chi} \leq |\Xi|/Y$, and our estimates on smooth numbers (as $R/Y>w^{\log\log x}$). We therefore deduce .
\[Deduction of Corollary \[MathResult3\]\] We apply Proposition \[Using SW\] whose hypotheses are satisfied using Theorem \[Keep Xi\] for any $C>6B+7$.
Breaking the $x^{1/2}$-barrier
==============================
To break the $x^{1/2}$-barrier, we need to reduce the $Q^2$ in the upper bound in Lemma \[BilinBound\]. This term arises in the estimates that the number of terms in various arithmetic progressions is the length of that progression plus $O(1)$. The idea now is to be more precise about all those “$O(1)$”s by using Fourier analysis to obtain some cancellation. We will be able to do this when the residue classes $a_q$ do not vary with $q$:
Let $1\leq |a|\ll Q\leq x^{\frac{20}{39}-\delta}$ for some small fixed $\delta>0$, and consider the residue classes $a\pmod q$ when $q \sim Q$ and $(a,q) = 1$. Let $F$ be the function defined by $$F(n) = \sum_{\substack{q \sim Q \\ (a,q)=1}} \xi_q \left({\mathbf{1}}_{n \equiv a\pmod{q}} - \frac{1}{q_r} {\mathbf{1}}_{n \equiv a\pmod{q_s}}\right)$$ when $n \neq a$, and set $F(a) = 0$. Writing $\tau$ for the divisor function, we have $|F(n)| \ll \tau(n-a) \leq x^{o(1)}$ for each $n \neq a$ and $n \leq x$, and so $ \|F\|_{\infty}\leq x^{o(1)}$. Moreover we have the $L^2$-bound: $$\sum_{n \leq x} |F(n)|^2 \ll \sum_{n \leq 2x} \tau(n)^2 \ll x(\log x)^3.$$
\[prop:bilinear-beyond1/2\] Let $F$ be defined as above, and fix $\delta>0$. If $1\leq |a|\ll Q\leq x^{\frac{20}{39}-\delta}$ and $P\leq x^\varepsilon$ for some very small $\varepsilon>0$ (depending on $\delta$) then $${\mathbb{E}}_{p,p'\sim P} \left| \sum_{m \leq \min(x/p, x/p')} F(pm) \overline{F(p'm)} \right| \ll \frac{x}{P}\left(\frac{1}{w\log w} + \frac{P^{0.1}+(\log x)^3}{\pi(P)}\right) .$$
We begin by noting that the contribution from those terms with $p=p'$ is bounded by $$\frac{1}{\pi(P)^2} \sum_{p\sim P} \sum_{m \leq x/p}|F(pm)|^2 \ll \frac{x}{P} \cdot \frac{(\log x)^3}{\pi(P)},$$ which is acceptable. Now fix $p \neq p'$ and analyze the inner sum over $m$. We insert a smooth weight into the sum over $m$: Let $\eta > 0$ be a small parameter to be chosen later. Let $\psi \colon {\mathbb{R}}\to [0,1]$ be a smooth approximation to the indicator function ${\mathbf{1}}_{[0,1]}$ with the following properties:
1. $\int\psi = 1$;
2. $\psi$ is supported on $[-\eta, 1+\eta]$, and $\psi=1$ on $[\eta,1-\eta]$;
3. $\|\psi^{(A)}\|_1 \ll_A \eta^{-A+1}$ for all $A \geq 1$.
By partial summation it follows that the Fourier transform $\widehat{\psi}$ decays rapidly: $$|\widehat{\psi}(h)| \ll_A (1+|h|)^{-A} \eta^{-A+1}$$ for all $A \geq 1$. We may replace the sum over $m$ by the smoothed version $$\Sigma(p,p') := \sum_{m \in {\mathbb{Z}}} \psi(m/M) F(pm) \overline{F(p'm)},$$ where $M = \min(x/p, x/p')$, at the cost of an error $$\ll \eta M \|F\|_{\infty}^2 \ll \eta x^{o(1)} \cdot \frac{x}{P}.$$ Let $0<\sigma\leq \min\{ \delta/2,1/75\}$ and assume $\varepsilon=\sigma/3$. If we choose $\eta = x^{-\sigma}$, then the error is acceptable since $P \leq x^{\varepsilon}$. By the definition of $F$, we have $$\Sigma(p,p') = \sum_{\substack{q,q'\sim Q \\ (a,q) = (a,q') = 1}} \xi_q \xi_{q'} \sum_{m \in {\mathbb{Z}}} \psi(m/M) K(p,p',q,q';m),$$ where $K(p,p',q,q';m)$ is the expression $$\left({\mathbf{1}}_{pm\equiv a\pmod{q}} - \frac{1}{q_r} {\mathbf{1}}_{pm\equiv a\pmod{q_s}}\right) \left({\mathbf{1}}_{p'm\equiv a\pmod{q'}} - \frac{1}{q_r'} {\mathbf{1}}_{p'm\equiv a\pmod{q_s'}}\right).$$ The inner sum over $m$ can be written as a sum of four terms, the first of which is $$\sum_{m \in {\mathbb{Z}}} \psi(m/M) \cdot {\mathbf{1}}_{pm\equiv a\pmod{q}} \cdot {\mathbf{1}}_{p'm\equiv a\pmod{q'}}.$$ The sum is empty unless $$p'a \equiv pa \pmod{(q,q')} \Leftrightarrow p' \equiv p\pmod{(q,q')}.$$ When this holds, the sum should have a main term of $M/[q,q']$ and an error term of $$g(q,q') = \sum_{m \in {\mathbb{Z}}} \psi(m/M) \cdot {\mathbf{1}}_{pm\equiv a\pmod{q}} \cdot {\mathbf{1}}_{p'm\equiv a\pmod{q'}} -
\frac{M}{[q,q']}.$$ After summing over $p,p'$, the contributions from the four main terms lead to exactly $\Sigma_1 + \Sigma_2$, which were treated in the proof of Lemma \[BilinBound\]: $$\Sigma_1 + \Sigma_2 \ll \frac{x}{P}\left(\frac{1}{w\log w} + \frac{P^{0.1}+\log x}{\pi(P)}\right).$$ It suffices to control the error terms by showing that $$E_1 := \sum_{\substack{q,q' \sim Q \\ (a,q) = (a,q') = 1}} \xi_q\xi_{q'} g(q,q') \ll x/P^2,$$ $$E_2 := \sum_{\substack{q,q' \sim Q \\ (a,q) = (a,q') = 1}} \xi_q\xi_{q'} q_r^{-1} g(q_s,q') \ll x/P^2,$$ $$E_2' := \sum_{\substack{q,q' \sim Q \\ (a,q) = (a,q') = 1}} \xi_q\xi_{q'} q_r'^{-1} g(q,q_s') \ll x/P^2,$$ $$E_3 := \sum_{\substack{q,q' \sim Q \\ (a,q) = (a,q') = 1}} \xi_q\xi_{q'} (q_rq_r')^{-1} g(q_s,q_s') \ll x/P^2,$$ for any fixed $p \neq p'$. By symmetry, it suffices to prove the bounds for $E_1, E_2, E_3$. We start by analyzing $g(q,q')$ for fixed $q,q'$.
\[lem:g(q,q’)\] Suppose that $1\leq |a|\ll Q\leq x^{\frac{20}{39}}$ and $P\leq x^\varepsilon$. Fix $q,q' \leq 2Q$ and $p,p'\sim P$ with $d = (q,q')$ satisfying $p \equiv p'\pmod{d}$. Write $q=d\ell$ and $q'=d\ell'$ so that $(\ell,\ell')=1$. Then $$g(q,q') = \frac{M}{q'} \sum_{0 < |h| \leq H} {{\rm e}}_{p'\ell}(kh \cdot \overline{p\ell'}) \int_{|u| \leq 2d/Q} \psi(\ell u) \ {{\rm e}}_{d\ell'}(-Mhu) {{\rm d}}u + O(x^{-9\sigma }),$$ where $H = x^{2\sigma}Q^2/(dM)$ and $k = (p'-p)a/d$, assuming $0<\sigma \leq 1/75$ and $\varepsilon = \sigma/3$.
Here, and in the sequel, the notation $\overline{p\ell'}$ denotes the multiplicative inverse of $p\ell'$ modulo $p'\ell$, when it appears inside ${{\rm e}}_{p'\ell}(\cdot)$. The definition of $g(q,q')$ involves $[q,q']=d\ell\ell'$. A key advantage of Lemma \[lem:g(q,q’)\] is that the two variables $\ell$ and $\ell'$ are separated apart from a term of the form ${{\rm e}}_\ell(*\overline{\ell'})$.
Let $r$ be the unique solution modulo $[q,q']$ to the simultaneous congruence conditions $$p r \equiv a\pmod{q}, \ \ p' r \equiv a\pmod{q'}.$$ Then $$g(q,q') = \sum_{m \in {\mathbb{Z}}} \psi\left(\frac{m}{M}\right) {\mathbf{1}}_{m \equiv r\pmod{[q,q']}} - \frac{M}{[q,q']}.$$ By the definition of the Fourier transform $\widehat{\psi}$ and a change of variables we have $$\widehat{\psi}\left(\frac{2\pi Mh}{[q,q']}\right) = \widehat{\psi}\left(\frac{2 \pi M h}{d\ell\ell'}\right) = \ell \int_{|u| \leq 2/\ell} \psi(\ell u) {{\rm e}}\left(-\frac{Mhu}{d\ell'}\right) {{\rm d}}u,$$ and by Poisson summation, we obtain $$g(q,q') = \frac{M}{[q,q']} \sum_{h \in {\mathbb{Z}}\setminus\{0\}} \widehat{\psi}\left(\frac{2\pi Mh}{[q,q']}\right) {{\rm e}}\left(\frac{rh}{[q,q']}\right).$$ Using the rapid decay of $\widehat{\psi}$, the contribution to $g(q,q')$ from those terms with $|h| \geq H$ is $$\ll_A 1\big/\left(\frac{ \eta MH}{[q,q']}\right)^{A-1} \ll x^{-9\sigma }$$ by the choices of $\eta$ and $H$ and letting $A=10$. We will prove that $$\label{AlmostId}
{{\rm e}}\left(\frac{rh}{[q,q']}\right) = {{\rm e}}_{d\ell\ell'}(rh) = {{\rm e}}_{p'\ell}(kh \cdot \overline{p\ell'}) + O\left( \frac {|h|}{Q}\right).$$ The total contribution of these error terms, over all $h$ with $1\leq |h|\leq H$, is therefore, $$\ll \frac {1}{Q} \bigg/\left(\frac{ \eta M}{[q,q']}\right)^{9} \ll
\frac {1}{Q} \left(\frac{Q^2P}{ \eta x}\right)^{9} \ll x^{-\frac{20}{39}} (x^{\frac{1}{39}+\sigma+\varepsilon})^{9}\ll x^{-9\sigma },$$ as $\sigma \leq 1/75$ and $\varepsilon = \sigma / 3$, which implies the result when we insert the formulas and estimates above into the sum for $g(q,q')$.
To establish , let $c\pmod{d}$ be the residue class with $pc \equiv p'c \equiv a\pmod{d}$, so that $r \equiv c\pmod{d}$. Make a change of variables $r = ds+c$, so that $${{\rm e}}_{d\ell\ell'}(rh) = {{\rm e}}_{\ell\ell'}(sh) {{\rm e}}_{d\ell\ell'}(ch)= {{\rm e}}_{\ell\ell'}(sh) +O(ch/d\ell\ell').$$ The value of $s$ is determined by the congruence conditions $$p(ds+c) \equiv a\pmod{q}, \ \ p'(ds+c) \equiv a\pmod{q'},$$ which can be rewritten as $$ps \equiv b \pmod{\ell}, \ \ p's \equiv b' \pmod{\ell'},$$ where $b = (a-pc)/d$ and $b' = (a-p'c)/d$. Since $\ell$ and $\ell'$ are coprime, the Chinese remainder theorem leads to $$s \equiv b(p\ell')^{-1}\pmod{\ell} \cdot \ell' + b'(p'\ell)^{-1}\pmod{\ell'}\cdot \ell \pmod{\ell\ell'}.$$ Hence $${{\rm e}}_{\ell\ell'}(sh) = {{\rm e}}_\ell(bh \cdot \overline{p\ell'}) {{\rm e}}_{\ell'}(b'h \cdot \overline{p'\ell}).$$ Now apply the reciprocity relation $$\frac{v^{-1}\pmod{u}}{u} + \frac{u^{-1}\pmod{v}}{v} \equiv \frac{1}{uv}\pmod{1}$$ with $u=p\ell'$ and $v = p'\ell$ to obtain $${{\rm e}}_{p\ell'}(w\overline{p'\ell}) {{\rm e}}_{p'\ell}(w\overline{p\ell'}) = {{\rm e}}_{pp'\ell\ell'}(w)$$ for any $w$. This implies that $${{\rm e}}_{\ell'}(b'h \cdot \overline{p'\ell}) = {{\rm e}}_{p\ell'}(pb'h \cdot \overline{p'\ell})
= {{\rm e}}_{p'\ell}(-b'h \cdot \overline{\ell'}) {{\rm e}}_{p'\ell\ell'}(b'h)
= {{\rm e}}_{p'\ell}(-b'h \cdot \overline{\ell'}) +O(b'h/p'\ell\ell') .$$ The main term is therefore $${{\rm e}}_\ell(bh \cdot \overline{p\ell'}) {{\rm e}}_{p'\ell}(-b'h \cdot \overline{\ell'})= {{\rm e}}_{p'\ell}(kh \cdot \overline{p\ell'}) ,$$ as claimed. For the error terms we note that $|c|\leq d$ and $|b'|\leq |a/d|+p'$; and therefore the error is $$\ll \left(1+ \frac{|a|}{dP}\right) \frac {|h|}{\ell\ell'} \ll \left(P^2+ |a|\right) \frac {|h|}{Q^2} \ll \frac {|h|}{Q}$$ as $|a|, x^{2\sigma}\leq Q$ and $|d|\leq P\leq x^\sigma$, as claimed. This completes the proof.
Now we perform the summation over $q,q'$:
\[lem:sumqq’\] Suppose that $1\leq |a|\ll Q\leq x^{\frac{20}{39}-\delta}$ and $P\leq x^{\varepsilon}$. For any sequences $\{\gamma(q)\}, \{\gamma'(q)\}$ with $|\gamma(q)| , |\gamma'(q)| \leq 1$ we have $$\sum_{\substack{q,q' \leq 2Q \\ (q,q') \mid p-p'}} \gamma(q) \gamma'(q') g(q,q') \ll x^{2\sigma+\varepsilon} Q^{2-\frac {1}{20}} +Q^2 x^{-9\sigma}
\ll \frac x{P^2},$$ assuming $\sigma = \min\{ \delta/2,1/75\}$ and $\varepsilon=\sigma/3$.
The total contribution of the error term from Lemma \[lem:g(q,q’)\] is $O(Q^2 x^{-9\sigma})$. For each integer $h,\ 1 \leq |h| \leq H:=x^{2\sigma}Q^2/dM$, each divisor $d$ of $p'-p$, and each fixed $|u|\leq 2d/Q$, we restrict our attention to those pairs with $(q,q') = d$. Write $q=d\ell$ and $q'=d\ell'$, and then define $$\alpha(n) = \begin{cases} \gamma(q)\psi(\ell u) & \text{if }n = p'\ell \text{ for some } \ell \leq 2Q/d\\ 0 & \text{otherwise},\end{cases}$$ so that $|\alpha(n)|\leq |\psi(\ell u) |\ll 1$, and $$\beta(n') = \begin{cases} (Q/q') \gamma'(q'){{\rm e}}_{d\ell'}(-Mhu) &
\text{if }n' = p\ell' \text{ for some } \ell\ne \ell' \leq Q/d\\ 0 & \text{otherwise}.
\end{cases}$$
Theorem 1 of [@BC] (with $A=1,\ \theta=kh$ and the roles of $\alpha$ and $\beta$ swapped) implies that $$\sum_{\substack{n\sim N,\ n'\sim N' \\ (n,n')=1}} \alpha(n) \beta(n') {{\rm e}}_n(kh \cdot \overline{n'}) \ll
\frac Qd \left( \frac{PQ}d \right)^{\frac {19}{20}+\varepsilon}$$ where $N=p'Q/d$ and $N'=pQ/d$, and $k = (p'-p)a/d$, as $|kh|/ (PQ/d)^2 \ll |a|/ x^{1-2\sigma}\ll 1$. Summing over $u$ and $h$, our sum is therefore $$\ll \frac{M}{Q} \sum_{d|p-p'} \frac{2d}{Q} \cdot \frac{x^{2\sigma}Q^2}{dM} \cdot \frac Qd \left( \frac{PQ}d \right)^{\frac {19}{20}+\frac{\varepsilon}{20}}
\ll Q x^{2\sigma} \left( PQ \right)^{\frac {19}{20}+\frac{\varepsilon}{20}}\ll x^{2\sigma+\varepsilon} Q^{2-\frac {1}{20}}.$$
\[Completion of the proof of Proposition \[prop:bilinear-beyond1/2\]\] Each of $E_1,E_2,E_3$ can be effectively bounded using Lemma \[lem:sumqq’\]. Indeed, for $E_1$ we apply Lemma \[lem:sumqq’\] with $$\gamma(q) = \begin{cases} \xi_q & \text{if }q \sim Q \\ 0 & \text{otherwise,} \end{cases} \ \ \gamma'(q') = \begin{cases} \xi_{q'} & \text{if }q' \sim Q \\ 0 & \text{otherwise.} \end{cases}$$ To bound $E_2$ we apply Lemma \[lem:sumqq’\] with $$\gamma(q_s) = \sum_{\substack{q_r \sim Q/q_s }} \xi_{q_rq_s} q_r^{-1}, \ \ \gamma'(q') = \begin{cases} \xi_{q'} & \text{if }q' \sim Q \\ 0 & \text{otherwise.} \end{cases}$$ Similarly for $E_3$.
Proof of Theorems \[Cor:Result2+\] and \[MathResult3+\]
-------------------------------------------------------
For $a$ fixed with $1\leq |a|\ll Q\leq x^{\frac{20}{39}-\delta}$ , we replace Lemma \[BilinBound\] by Proposition \[prop:bilinear-beyond1/2\] to obtain analogues of Theorem \[MathResult2\] and Corollary \[MathResult2Cor\]. The proof of Theorems \[Cor:Result2+\] and \[MathResult3+\] then follows in exactly the same way as the arguments in section \[sec:deduce-many-thms\].
Various examples
================
The large prime obstruction; the proof of Proposition \[LargePrimes2\]
----------------------------------------------------------------------
Let $\mathcal P$ be the set of primes $p \in (x/2,x]$ for which there does not exist an integer $q\sim Q$ with $q|p-1$. Let $f_+,f_-$ be the multiplicative functions for which $f_+(n)=f_-(n)=g(n)$ for all $n\leq x$ with $n\not\in \mathcal P$, and $f_+(p)=1$ and $f_-(p)=-1$ for all $p\in \mathcal P$. If $f \in \{f_+,f_-\}$ then $$\sum_{\substack{ n\leq x \\ n\equiv a \pmod q}} f(n) = \sum_{\substack{ n\leq x \\ n\equiv a \pmod q}} g(n) + \sum_{\substack{ x/2<p\leq x \\ p\equiv a \pmod q}} ( f(p)-g(p)) ,$$ and so $$\Delta(f,x;q,1)-\Delta(g,x;q,1) = \sum_{\substack{ x/2<p\leq x \\ p\equiv 1 \pmod q}} ( f(p)-g(p)) -
\frac 1{\varphi(q)} \sum_{\substack{ x/2<p\leq x }} ( f(p)-g(p)).$$ Since $f(p)=g(p)$ if $p\not \in \mathcal P$, the first term on the right-hand side of the last displayed equation vanishes, by the definition of $\mathcal P$. Therefore $$\Delta(f_+,x;q,1)-\Delta(f_-,x;q,1) = \frac {2\# \mathcal P} {\varphi(q)} \sim \frac {1} {\varphi(q)} \frac x{\log x}.$$ The asymptotic in the last step follows from Theorem 6 and Corollary 2 of Ford’s masterpiece [@For] which imply that almost all primes $p$ in the interval $x/2<p\leq x$ belong to $\mathcal P$. Thus for each $q$ we have either $|\Delta(f_+,x;q,1)|$ or $|\Delta(f_-,x;q,1)|$ is $\gg \pi(x)/\varphi(q)$. The conclusion follows immediately.
$f$ satisfying the Siegel-Walfisz criterion but not Bombieri-Vinogradov with prime moduli {#NoBV}
-----------------------------------------------------------------------------------------
We will assume in what follows. Let $x^{1/3}<Q<x^{1/2-\varepsilon}$, and let $\mathcal P$ now be the set of primes $p \in (x/2, x]$ for which there exists a prime $q\sim Q$ for which $q|p-1$. We now define the multiplicative function $f$ so that $f(p)=-1$ if $p\in \mathcal P$, and $f(n)=1$ if $n \leq x$ and $n\notin\mathcal P$. Therefore if $m<x/2$ and $(a,m)=1$ then $$\Delta(f,x;m,a) = \Delta(1,x;m,a) - 2 \Delta(1_{\mathcal P},x;m,a) = - 2 \Delta(1_{\mathcal P},x;m,a) +O(1).$$ Note that if $p\in \mathcal P$ then $p-1$ can have at most two prime factors from the set of primes $q\sim Q$. Therefore, by inclusion-exclusion we have $$\# \mathcal P = \sum_{\substack{q\sim Q \\ q \ \text{prime}}} \pi^*(x;q,1) -
\sum_{\substack{q_1<q_2\sim Q \\ q_1,q_2 \ \text{prime}}} \pi^*(x;q_1q_2,1)$$ where $\pi^*(x;q,a):=\pi(x;q,a)-\pi(x/2;q,a)$. Similarly if $m<Q$ (so that $(m,q)=1$, and therefore $\varphi(mq)=\varphi(m)\varphi(q)$), then $$\#\{ p\in \mathcal P: \ p\equiv a \pmod m\} = \sum_{\substack{q\sim Q \\ q \ \text{prime}}} \pi^*(x;qm,a_{qm}) - \sum_{\substack{q_1<q_2\sim Q \\ q_1,q_2 \ \text{prime}}} \pi^*(x;q_1q_2m,a_{q_1q_2m}) ,$$ where $a_{rm}$ is that residue class mod $rm$ which is $\equiv 1 \pmod r$ and $\equiv a \pmod m$ (for $r=q$ or $q_1q_2$). Comparing these two equations, we have $$\begin{split}
\Delta(1_{\mathcal P},x;m,a) &= \sum_{\substack{q\sim Q \\ q \ \text{prime}}} \left( \pi^*(x;qm,a_{qm})-\frac{\pi^*(x;q,1)}{\varphi(m)} \right) \\
& \hskip1in - \sum_{\substack{q_1<q_2\sim Q \\ q_1,q_2 \ \text{prime}}} \left( \pi^*(x;q_1q_2m,a_{q_1q_2m}) -\frac{\pi^*(x;q_1q_2,1)}{\varphi(m)} \right) \\
& \ll \frac{\pi(x)}{(\log x)^A} \sum_{\substack{q\sim Q \\ q \ \text{prime}}} \frac{1}{\varphi(mq)} + \frac{\pi(x)}{(\log x)^A} \sum_{\substack{q_1<q_2\sim Q \\ q_1,q_2 \ \text{prime}}} \frac{1}{\varphi(mq_1q_2)} \\ & \ll \frac 1{\varphi(m)} \frac{x}{(\log x)^{A+2}},
\end{split}$$ using under the assumption $4mQ^2\leq x^{1 -\varepsilon}$. This shows that the $A$-Siegel-Walfisz criterion for $f$ holds for every $A>0$, for the sums over $n\leq x$. The same argument works for sums over $n\leq X$, for any $X\leq x$, changing the definition of $\pi^*(X;q,a)$ to be $0$ if $X\leq x/2$, and $\pi^*(X;q,a):=\pi(X;q,a)-\pi(x/2;q,a)$ if $x/2<X\leq x$.
On the other hand, (or the unconditional Brun-Titchmarsh Theorem), implies that $$\# \mathcal P \ll \sum_{\substack{q\sim Q \\ q \ \text{prime}}} \frac{\pi^*(x)}{\varphi(q)} \ll \frac x{(\log x)^2} .$$ Therefore if $q\sim Q$ and $q$ is prime then $$\Delta(1_{\mathcal P},x;q,1) = \pi^*(x;q,1)- \frac{\# \mathcal P} {\varphi(q)} \sim \frac{\pi^*(x)}{\varphi(q)}.$$ Therefore if $q\sim Q$ and $q$ is prime then $$\Delta(f,x;q,1) \sim - \frac{1}{q} \cdot \frac x{\log x},$$ and so $$\sum_{q\sim Q} |\Delta(f,x;q,1)| \geq \sum_{\substack{q\sim Q \\ q \ \text{prime}}} |\Delta(f,x;q,1)| \gg \frac x{(\log x)^2} .$$ This doesn’t quite prove Corollary \[LargePrimes2b\], but it will be fixed in the next subsection.
$f$ satisfying the Siegel-Walfisz criterion but not Bombieri-Vinogradov; the proof of Corollary \[LargePrimes2b\] {#NoBV2}
------------------------------------------------------------------------------------------------------------------
Now we use a minor modification of the argument in the previous subsection to extend the estimate to all moduli. Let $x^{2/5}<Q<x^{1/2-\varepsilon}$, and $I=(x^{1/3},x^{2/5}]$. Let $\mathcal P$ now be the set of primes $p \in (x/2, x]$ for which there exists a prime $\ell \in I$ with $\ell |p-1$. Define the multiplicative function $f$ so that $f(p)=-1$ if $p\in \mathcal P$, and $f(n) = 1$ if $n \leq x$ and $n\notin\mathcal P$. So we again have that if $m<x/2$ and $(a,m)=1$ then $$\Delta(f,x;m,a) = - 2 \Delta(1_{\mathcal P},x;m,a) +O(1).$$ The argument of Section \[NoBV\] yields the Siegel-Walfisz criterion for $f$, assuming . We also have, assuming , that $$\# \mathcal P \leq \sum_{\substack{ \ell\in I \\ \ell \ \text{prime}}} \pi^*(x;\ell,1) = \{ 1+o(1)\}\sum_{\substack{ \ell\in I \\ \ell \ \text{prime}}} \frac{\pi^*(x)}{ \ell-1} = \{ c+o(1)\} \pi^*(x) ,$$ where $c=\log(6/5)<1/5$.
Now suppose that $q\sim Q$ (not necessarily prime) and that there exists a prime $\ell$ in the interval $I$ which divides $q$. If $x/2<p\leq x$ and $p\equiv 1 \pmod q$ then $\ell|q|p-1$ and so $p\in \mathcal P$. So we have $$\Delta(1_{\mathcal P},x;q,1) = \pi^*(x;q,1)- \frac{\# \mathcal P} {\varphi(q)} \geq (1-c+o(1)) \frac{\pi^*(x)}{\varphi(q)} \geq \frac 45 \cdot \frac{\pi^*(x)}{\varphi(q)},$$ and therefore $$- \Delta(f,x;q,1) \gg \frac{1}{\varphi(q)} \cdot \frac x{\log x}.$$ Now each such $q$ can have at most one such prime divisor $\ell$. Therefore $$\sum_{q\in Q} |\Delta(f,x;q,1)|
\gg \frac{x}{\log x} \sum_{\substack{\ell \in I \\ \ell\ \text{prime}}} \sum_{\substack{q \sim Q \\ \ell | q}} \frac{1}{\varphi(q)} \gg
\frac x{\log x} \sum_{\substack{ \ell\in I \\ \ell \ \text{prime}}} \frac 1 \ell \gg \frac x{\log x}.$$ This completes the proof of Corollary \[LargePrimes2b\].
$f$ that correlate to many characters {#ManyChar}
-------------------------------------
For a given integer $q$ suppose we are given constants $g(\chi)$ for each character $\chi \pmod q$, and define the multiplicative function $f$ by $$\Lambda_f(n) = c_q(n) \Lambda(n) , \ \text{where} \ c_q(n):= \sum_{\chi \pmod q} g(\chi) \chi(n)$$ depends only on $n\pmod q$. This implies that $$F(s) = \prod_{\chi \pmod q} L(s,\chi)^{g(\chi)}.$$ Moreover if we twist $f$ by character $\psi$, then $$F_\psi(s):= \sum_{n\geq 1} \frac{(f\overline\psi)(n) }{n^s} = \prod_{\chi \pmod q} L(s,\chi)^{g(\chi\psi)}.$$ By the Selberg-Delange Theorem (see, e.g., Theorem 5 of Section 5.5 in [@Ten]) one then has that $$|S_f(x,\psi)| \gg \frac x{ (\log x)^{1-\text{Re}(g(\psi))}} \ .$$
Now $f\in \mathcal C$ if and only if $|c_q(n)| \leq 1$ for all $n$. We will make the particular choice that $$g(\chi) = \frac 1{\varphi(q)} \sum_{a \pmod q} \overline{\chi}(a) e(a/q)$$ are Gauss sums (where $e(t):=e^{2i\pi t}$), so that $$\begin{split}
c_q(n)&= \frac 1{\varphi(q)}\sum_{\chi \pmod q} \chi(n) \sum_{a \pmod q} \overline{\chi}(a) e(a/q) \\
& = \sum_{a \pmod q} e(a/q) \frac 1{\varphi(q)} \sum_{\chi \pmod q} \chi(n)\overline{\chi}(a) = e(n/q) .
\end{split}$$ Therefore $f\in \mathcal C$ and $| g(\chi) |=\sqrt{q}/(q-1)$ whenever $\chi$ is non-principal and $q$ is prime. Moreover Katz showed (though see [@KZ] for a much easier proof) that the arguments of the $g(\chi)$ are equi-distributed around the unit circle. In particular the number of non-principal $\psi \pmod q$ for which $1-\text{Re}(g(\psi))>1-\frac 1{2\sqrt{q}}$ is $\sim q/3$.
One can deduce that for given integer $k$ there is an $ \varepsilon' \sim \frac 1{2\sqrt{3k}}$, for which there exists $f\in {\mathcal C} $ such that $$\sum_{q\sim Q} \max_{a:\ (a,q)=1} | \Delta_{k}(f,x;q,a) | \gg \frac{x} {(\log x)^{1-\varepsilon'}} .$$ This shows that Theorem \[Cor:Result2\] is close to “best possible".
Average of higher $U^k$-norms {#sec:higher-uk}
=============================
In this section, we investigate a higher order generalization of the Bombieri-Vinogradov inequality, which measures more refined distributional properties. This higher order version involves Gowers norms, a central tool in additive combinatorics. We refer the readers to [@TV06 Chapter 11] for the basic definitions and applications. In particular, $\|f\|_{U^k(Y)}$ stands for the $U^k$-norm of the function $f$ on the discrete interval $[0,Y]\cap {\mathbb{Z}}$.
For any arithmetic function $f: {\mathbb{Z}}\to {\mathbb{C}}$ and any residue class $a \pmod q$, denote by $f(q\cdot + a)$ the function $m \mapsto f(qm+a)$.
\[cor:uk-average\] Fix a positive integer $k$. Fix $B \geq 2$ and ${\varepsilon}> 0$. Let $2 \leq y \leq x^{1/10}$ be large. Let $f$ be a completely multiplicative function with each $|f(n)| \leq 1$, which is only supported on $y$-smooth integers. Assume that $$\label{eq:savelogpower}
\sum_{\substack{n \leq x \\ n \equiv a\pmod{q}}} f(n) \ll_C \frac{x}{(\log x)^C}$$ for any $(a,q)=1$ and any $C \geq 1$. Let $Q = x^{1/2}/y^{1/2}(\log x)^A$, where $A$ is sufficiently large in terms of $k,B,{\varepsilon}$. Then for all but at most $Q(\log x)^{-B}$ moduli $q \leq Q$, we have $$\max_{\substack{0 \leq a < q \\ (a,q)=1}} \|f(q \cdot +a)\|_{U^k(x/q)} \leq {\varepsilon}.$$
In view of Corollary \[MathResult3\], the hypothesis on $f$ implies that $$\label{eq:BV-f}
\sum_{q \leq Q} \max_{(a,q)=1} \left|\sum_{\substack{n \leq x \\ n \equiv a\pmod{q}}} f(n) \right| \ll \frac{x}{(\log x)^{B'}},$$ for any constant $B' = B'(B)$, provided that $A$ is chosen large enough. The conclusion of Corollary \[cor:uk-average\] follows by using this estimate in place of the Bombieri-Vinogradov inequality for the Liouville function $\lambda$ in the argument in [@Sha Section 2.2].
More precisely, if $q \leq Q$ is an exceptional moduli in the sense that $$\|f(q \cdot +a_q)\|_{U^k(x/q)} \geq {\varepsilon}$$ for some residue class $a_q\pmod{q}$, where $0 \leq a_q < q$ and $(a_q,q)=1$, then by the inverse theorem for Gowers norms [@GTZ] $f$ must correlate with a nilsequence of complexity $O_{{\varepsilon}}(1)$ on the progression $\{n \leq x : n \equiv a_q\pmod{q}\}$. Now apply [@Sha Lemma 2.4] and follow the deduction of [@Sha Theorem 1.2], with $f$ in place of $\lambda$. The number of exceptional moduli satisfying [@Sha Equation (2.3)] can be satisfactorily bounded using , and the rest of the argument already applies to a general multiplicative function $f$.
The hypothesis of Corollary \[cor:uk-average\] follows from assuming the Siegel-Walfisz criterion, along with $$\sum_{\substack{n \leq x }} f(n) \ll_C \frac{x}{(\log x)^C}$$ uniformly in $x$.
Corollary \[cor:uk-average\] only applies to smoothly supported $f$ since we need to save an arbitrary power of $\log x$ in the Bombieri-Vinogradov inequality for $f$ for the results in [@Sha] to be applicable. However we would guess that our estimate $$\max_{\substack{0 \leq a < q \\ (a,q)=1}} \|f(q \cdot + a)\|_{U^k(x/q)} = o(1)$$ holds on average over $q \leq x^{1/2-o(1)}$, for any $1$-bounded multiplicative function $f$ satisfying the $1$-Siegel-Walfisz criterion.
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[^1]: A.G. has received funding from the European Research Council grant agreement n$^{\text{o}}$ 670239, and from NSERC Canada under the CRC program.
[^2]: X.S. was supported by a Glasstone Research Fellowship.
[^3]: We would like to thank Dimitris Koukoulopoulos for a valuable discussion
[^4]: This is discussed in detail in section \[ClaimJusitify\], and proved there up to a factor of $\log x$.
[^5]: From this perspective, the usual formulation implicitly assumes that $\psi_1=1$, which is true whenever $f(n)\geq 0$ for all $n$.
[^6]: The definition of $\Delta_k(f,x;q,a)$ in [@GHS] involves removing the $k$ largest character sums for characters mod $q$, whereas here we use the characters mod $q$ induced from the largest character sums for characters mod $r$, for any $r|q$ with $r\leq \log x$. This is a minor technical difference, which is sorted out in Corollary \[No Exceptions\].
[^7]: *Smooth numbers* are integers with no large prime factors. That is, we restrict attention to multiplicative function $f(.)$ for which $f(p^k)=0$ for any prime $p>y$ and integer $k\geq 1$.
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abstract: 'We show how the viscous evolution of Keplerian accretion discs can be understood in terms of simple kinetic theory. Although standard physics texts give a simple derivation of momentum transfer in a linear shear flow using kinetic theory, many authors, as detailed by Hayashi & Matsuda 2001, have had difficulties applying the same considerations to a circular shear flow. We show here how this may be done, and note that the essential ingredients are to take proper account of, first, isotropy locally in the frame of the fluid and, second, the geometry of the mean flow.'
author:
- |
C.J. Clarke$^1$, J.E. Pringle$^{1,2}$\
$^1$Institute of Astronomy, Madingley Rd, Cambridge, CB3 0HA, UK\
$^2$Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
date: 'Submitted: July 2103'
title: Kinetic theory viscosity
---
9[GRS 1915+105]{} \[firstpage\]
accretion discs -circumstellar matter - stars:accretion
Introduction
============
Accretion discs play a central role in a wide range of astronomical environments, mediating the gas flows in the vicinity of object as diverse as AGN, binary stars and protostars (Pringle, 1981). In an accretion disc, the predominant flow is a circular shear flow, with angular velocity $\Omega(R)$ a function of radius $R$ from the central object. Accretion takes place because of the action of some form of dissipation which releases the free energy of the shear flow as heat, and so allows the disc material to fall deeper into the potential well of the central object. Simple physical energy arguments ([*e.g.*]{} Lynden-Bell & Pringle, 1974) indicate that the dissipative process must take the form of a stress which transports angular momentum outwards. Because the free energy of a circular shear flow is zero only if $d\Omega/dR = 0$, it follows that the relevant element of the stress tensor must be of the form $$T_{\phi R} \propto - d\Omega/dR,$$ where $\phi$ is the azimuthal coordinate.
This can be deduced from the standard derivation of Navier-Stokes stress to be found in the fluid dynamics textbooks. However, in a recent paper, Hayashi & Matsuda (2001) have drawn attention to the fact that attempts to provide a physical explanation of the above result in terms of simple kinetic theory have resulted in failure.
The simple kinetic explanation given in basic physics text books for the effect of viscosity on a simple linear shear flow, of the form ${\bf u} = (0,U(x))$ in Cartesian coordinates, relies on the fact that the kinetic particles conserve linear momentum between collisions. Thus particles crossing some fiducial plane $x = x_o$ tend to mix up and smooth out the momentum distribution of the fluid and so give rise to a stress of the form $T_{yx} \propto -dU/dx$.
However, as Hayashi & Matsuda (2001) point out, it is in the application of these simple concepts to a circular shear flow that the problems seem to arise. The simple generalisation that, in a circular shear flow, the kinetic particles conserve angular momentum $j = R^2
\Omega$ between collisions would imply, taken at face value, that the the movement of particles across some fiducial circle $R=R_o$ would tend to try to mix up and smooth out the distribution of angular momentum of the fluid, and thus that the stress would be proportional to $-dj/dR$ (see e.g. Madej & Paczynski 1977). From the arguments given above, this is clearly wrong. Not only would this predict a stress in the case when the shear $d\Omega/dR$ is zero, but for a standard Keplerian accretion disc for which $j \propto R^{1/2}$ it would transport angular momentum inwards rather than outwards. As detailed by Hayashi & Matsuda (2001) attempts to get round this and to produce the ’correct’ result have only succeeded by making mathematical errors in the derivation.
From all these problems, Hayashi & Matsuda (2001) conclude that although what they call the derivation ’with mathematical rigour’ ([*i.e.*]{} the usual Navier-Stokes argument) gives the correct answer, in order to obtain the correct answer using kinetic theory one must take account of such complications as Coriolis force. In this paper, we shall show that, although it is obviously necessary to include Coriolis force if one works in a frame co-rotating with the flow, one can obtain the correct result from straightforward kinetic theory in the inertial frame.
Before we do so, it is instructive to return to the ’mathematical’ relationship between stress and strain derived in the standard fluid textbooks (see, for example, Batchelor, 1967, Section 3.3). The basic point we wish to make is that the standard Navier-Stokes expression for momentum transfer ([*i.e.*]{} stress) is based on a simple [*physical*]{} argument. The argument may involve the use of tensors, which physicists tend to meet in courses on mathematical methods, but the argument itself is not a mathematical one. The point is that stress (momentum transfer) in a fluid (or in a solid) can be expressed as a second order tensor. This is a physical (and not a mathematical) statement, in exactly the same sense that the statement that a velocity is a first order tensor (i.e. a vector) is a physical, and not a mathematical, statement.[^1] For a fluid, the physical [*Ansatz*]{} is simply that the stress tensor must be physically related to the rate of strain tensor (which is a second order tensor which derives from the first derivatives of the velocity field, and thus incorporates information about the shear). The simplest assumption is that relationship between these two tensors is through a (fourth order) tensor which is isotropic. It is this assumption of the isotropy of the relationship, which is based on the physical assumption that the fluid itself is isotropic, which gives rise to the standard Navier-Stokes expression for the viscosity.[^2]
It is important to realise that these tensors ([*i.e.*]{} scalars, vectors, second order tensors) exist as physical quantities, independent of any coordinate system. The statement that there is a relationship between two of them is a physical statement. It is only when one has to calculate a particular element of the stress tensor, for example corresponding to linear momentum transfer in a linear shear flow, or to the angular momentum transfer in a circular shear flow, that one has to evaluate coordinate dependent expressions, which can get mathematically complicated. But when one does this, one finds that the flux of linear momentum in a linear shear flow just depends on $-dU/dx$, and that the angular momentum flux in a circular shear flow just depends on $-d\Omega/dR$. However, this result also enables us to draw another conclusion. The reason for the difference between the terms $dU/dx$ and $d\Omega/dR$ is due solely to the difference between the coordinate systems. That is, it comes from geometry alone. This implies that when looking for differences in derivations for simple kinetic theory formulae between the linear and circular shear flows, we need only concern ourselves with geometry, and not with dynamical complications such as Coriolis force. In addition, we also need to take note of the fact that the Navier-Stokes expression does depend critically on assumptions about isotropy of the fluid. Thus we should expect to have to make a similar assumption about the properties of our kinetic particles.
Kinetic Theory
==============
In this Section we compute the relationship between stress and (rate of) strain for two simple shear flows using kinetic theory. To keep the concepts and the algebra simple, we make a number of simplifying assumptions. We work in two dimensions only. That is, we assume that the kinetic particles (assumed to be identical, with mass $m$) move only in a two-dimensional plane. We assume that the net effect of the scattering processes within the fluid is that these particles are emitted at a constant rate at each point of the fluid, and that each particle is emitted with an identical velocity, $c$, relative to the local fluid.[^3]. We represent by $\dot N(\lambda) $ the number of particles emitted per unit area per unit time that travel a distance greater than $\lambda$ before colliding with another particle and we assume that $\dot N(\lambda) $ is independent of position. The requirement of isotropy, as discussed above, can now be imposed by making the assumption that this emission takes place at each place in the fluid isotropically [*in the frame of the fluid*]{} at that place.
Plane shear flow
----------------
In Cartesian $(x,y)$ coordinates we let the background fluid flow be of the form
$${\bf u} = ( 0 , U(x)).$$
We consider the stress acting on a line element, length $dl$, centred at point S, at position $ ( x_0, 0 )$, and lying in the $y-$direction, that is, with unit normal in the $x-$direction. For a linear shear flow, we consider the case
$$U(x) = U_0 +(x-x_0)U^\prime,$$
where $U^\prime \equiv dU/dx$ is a constant. Note that although we consider here a linear shear flow with constant shear, any linear shear flow can be treated as a constant shear for $x \approx x_0$ and with $U^\prime = dU/dx$, evaluated at $x = x_0$.
For convenience, we work in the frame comoving with the mean fluid flow at S. This frame is an inertial frame, and in this frame the mean flow is given by
$${\bf u} = ( 0 , (x-x_0) U^\prime).$$
We now focus on particles that are emitted from a point E that is located at a distance $\lambda$ from S and where the line SE makes an angle $\alpha$ with the negative $x-$axis (see Figure 1a). The critical point is that [*in the frame of the fluid at* ]{}E the particles that travel along ES are emitted at an angle $\theta$ to the $x-$axis where the relationship between $\theta$ and $\alpha$ is simply deduced by considering the ratio of the $x-$ and $y-$components of the particle velocity in the rest frame of S, [*i.e.*]{}
$$\tan \alpha = {{c \sin \theta - U^\prime \lambda \cos \alpha}\over{c \cos \theta}}.$$
We consider the limit that the shear velocity across a typical mean free path is much less than the random particle velocity ([*i.e.*]{} $U^\prime \lambda \ll c$), so that (to O$(U^\prime \lambda /c)$) we may write
$$\label{tantheta}
\tan \theta \sim \tan \alpha + U^\prime \lambda/c,$$
$$\cos \theta \sim \cos \alpha - \sin \alpha \cos^2 \alpha \, U^\prime \lambda/c,$$
and $$\label{sintheta}
\sin \theta \sim \sin \alpha + \cos^3 \alpha \,U^\prime \lambda/c.$$
In order to compute the rate of arrival of $y$-momentum at S due to particles originating near E, we consider a patch around E that is a portion of an annulus centred on S (radius $\lambda$, thickness $d\lambda$) where the patch subtends an angle $d \alpha$ at S (see Figure 1b). The rate of emission of particles from this patch that travel far enough before colliding to be able to reach S is simply $\dot N(\lambda) \lambda d\lambda d \alpha$. Such particles emanate isotropically from E [*in the rest frame of the fluid at* ]{}E. Thus the fraction, $f$, of such particles that impinge on the line element $dl$ at S is
$$f = {{dl \cos \alpha}\over{2 \pi L}},$$
where $L$ is the distance traveled by the particles between E and S [*in the rest frame of the fluid at* ]{}E. The relationship between $L$ and $\lambda$ is simply ascertained by noting that the relative velocity between E and S is zero in the $x-$direction and hence that the distances travelled in the $x-$direction between E and S in the two frames is thus equal (i.e. $L \cos \theta = \lambda \cos \alpha$) from which we deduce, to first order in $\lambda$,
$$L \sim \lambda (1 - \sin \alpha \cos \alpha U^\prime \lambda/c)^{-1},$$
and thus that
$$\label{f}
f \sim {{dl \cos \alpha(1 - \sin \alpha \cos \alpha U^\prime \lambda/c)}\over{2 \pi \lambda}}.$$
Finally, the $y-$velocity of each particle emitted from around E is
$$v_y = -U^\prime \lambda \cos \alpha + c \sin \theta,$$
which (using equation \[sintheta\]) may be approximated as
$$\label{vy}
v_y \sim c \sin \alpha - U^\prime \lambda \cos \alpha \sin^2 \alpha.$$
Thus the total rate of arrival of $y$-momentum at S from the patch at E is
$$\dot p_y = m v_y f \dot N(\lambda) \lambda d\lambda d \alpha.$$
Substituting for $f$ and $v_y$ from equations \[f\] and \[vy\] we may obtain, to the order of our approximation, the total rate of arrival of $y-$momentum at S by integrating over $\alpha$ and $\lambda$, in the form
$$\dot p_y \sim \int_{0}^{\infty} {{m \dot N(\lambda) dl }\over{2 \pi}} \int_{-\pi/2}^{\pi/2} \cos \alpha (1 - \sin \alpha \cos \alpha U^\prime \lambda/c)(c \sin \alpha - U^\prime \lambda \cos \alpha \sin^2 \alpha) d\alpha \, d\lambda.$$
Since we are only considering the momentum flux due to particles arriving from one side of the line element (that is, particles with $x < x_o$), then $\alpha $ runs from $-\pi/2$ to $\pi/2$.
$$\label{pydot}
\dot p_y \sim -{{m U^\prime dl}\over{8}} \int_{0}^{\infty} \dot N(\lambda) \lambda d\lambda.$$
\[We note that the simple arguments found in many physics text books for kinetic theory viscosity in a linear shear flow, do [*not*]{} correctly take account of the fact that the particle velocity distribution is isotropic in the frame of the fluid at the point of [*emission*]{} (i.e. last collision), rather than at the point where the momentum flux is measured (see for example, Jeans 1940). In a linear shear flow, this error only changes the coefficient in front of the viscosity coefficient and retains the correct form of the equation for the resulting viscous stress. In the circular case, however, we shall find that it is necessary to treat this subtlety correctly, even in order to obtain the correct functional form of the viscous stress\].
Circular shear flow
-------------------
We now carry out essentially the same analysis, but this time for a circular shear flow. We still work in an inertial frame, (that is, a frame in which the particle trajectories are straight lines) but in this case the underlying fluid flow is, in cylindrical polar coordinates $(R,\phi)$, of the form $${\bf u} = (0, R\Omega(R)).$$
We consider the stress acting on a (small) line element, length $dl$, centred at point S which in Cartesian coordinates, centred at the origin O ([*i.e.*]{} at $R=0$), lies at $(R_0, 0)$. The line element lies in the azimuthal direction, and so has unit normal in the radial $R-$direction. As before, we work in the [*inertial*]{} frame that co-moves with the mean fluid flow at S and again consider the limit that the shear across a mean free path $\lambda$ is much less than the random velocity $c$ ([*i.e.*]{} $ R \Omega^\prime \lambda \ll
c$).[^4] We note that, since $ R \Omega^\prime
\sim \Omega$, it is also the case that $\lambda \ll R$. We defer until the end of this Section a discussion of the constraints placed on $\lambda$ by our neglect of the curvature of particle orbits between collisions.
.
As before, we consider particles that are emitted in the vicinity of a point E, located at distance $\lambda$ from S where ES makes an angle $\alpha $ with the inward directed radius vector at S. The only difference from the foregoing analysis is that since the streamlines are now circular, the velocity of the mean flow at E with respect to that at S has non-zero components in both the $x-$ and $y-$directions (see Figure 2). Specifically, the point E is located at radius $R_{\rm
E} = R_0- \lambda \cos \alpha$ (to O$(\lambda/R)$) where the velocity of the mean flow (with respect to the origin) is in the azimuthal direction and of magnitude $ R_0 \Omega_0 - \lambda \cos \alpha
(\Omega_0 + R_0 \Omega^\prime)$ ($\Omega_0$ and $\Omega^\prime$ being respectively the angular velocity and its gradient evaluated at S). The radial vector at E makes an angle
$$\phi \sim {{\lambda}\over{R}} {{\sin \alpha}}$$
with the $x-$axis. Consequently the $x-$ and $y-$components of the mean flow velocity at E with respect to a frame co-moving with the mean flow at S are (to O$(\lambda/R)$)
$$\label{vxrel}
v_{x}^{\rm rel} \sim {{\lambda \Omega_0 \sin \alpha}},$$
and
$$v_{y}^{\rm rel} \sim - \lambda \cos \alpha (\Omega_0 +
R_0 \Omega^\prime).$$
As before, we can establish the relationship between the angles $\alpha$ and $\theta$ using
$$\tan \alpha = {{v_{y}^{\rm rel}+ c \sin \theta}\over{v_{x}^{\rm rel} + c \cos \theta}},$$
so that, in this case,
$$\tan \theta \sim \tan \alpha + (\Omega_0 +
R_0 \Omega^\prime) \lambda /c +
\lambda \Omega_0 \tan^2 \alpha /c,$$
$$\cos \theta \sim \cos \alpha - \sin \alpha \cos^2 \alpha
(\Omega_0 +
R_0 \Omega^\prime)\lambda /c - \lambda \Omega_0 \sin^3 \alpha /c,$$
and $$\sin \theta \sim \sin \alpha + \cos^3 \alpha (\Omega_0 +
R_0 \Omega^\prime) \lambda /c + \lambda \Omega_0 \sin^2 \alpha \cos \alpha /c.$$
We note the similarity of these three equations to equations \[tantheta\] – \[sintheta\], since in the circular flow $U^\prime$ corresponds simply to $(\Omega_0 + R_0 \Omega^\prime)$. In each equation there is, however, an additional term (proportional to $\Omega_0$) which takes care of the fact that the mean flow is not plane-parallel in this case, and that hence each emission point has a non-zero velocity with respect to S along the radial vector at S (see equation \[vxrel\]). This modification in the relationship between $\alpha$ and $\theta$ is crucial in explaining the different form of the viscous stress in the circular case.
We also need to modify the relationship between $L$ and $\lambda$ compared with what we had previously, which we can do most simply by equating the time of flight ES ($L/c$) with the time ($\lambda \cos
\alpha/(c \cos \theta + v_{x}^{\rm rel})$) required to traverse the distance $\lambda \cos \alpha$ that separates E and S in the $x-$direction. After some algebra we obtain the expression
$$L \sim \lambda (1 - \sin \alpha \cos \alpha
R_0 \Omega^\prime \lambda/c )^{-1},$$
and hence find in this case that
$$\label{fcirc}
f \sim {{dl \cos \alpha(1 - \sin \alpha \cos \alpha
R_0 \Omega^\prime \lambda/c)}\over{2
\pi \lambda}}.$$
Finally, the $y-$velocity of each particle emitted from E is
$$v_y =c \sin \theta + v_{y}^{\rm rel},$$
and hence, to the order to which we are working,
$$\label{vycirc}
v_y \sim c \sin \alpha - \sin^2 \alpha \cos \alpha \, R_0 \Omega^\prime \lambda$$
Comparison of equations \[f\] and \[vy\] with equations \[fcirc\] and \[vycirc\] shows that the expressions are identical except in as much as $U^\prime$ in the linear shear flow is replaced by $R_0
\Omega^\prime$ in the circular case. Thus (following the same procedure for finding $\dot p_y$ as previously – that is, by integrating contributions to the $y$-momentum flux from all points with $R< R_0$) we obtain (by analogy with equation \[pydot\]):
$$\label{pydotcirc}
\dot p_y \sim -{{m R \Omega^\prime dl}\over{8}} \int_{0}^{\infty} \dot N(\lambda)
\lambda d\lambda$$
We note that this analysis recovers the ‘correct’ answer (as given by the usual Navier-Stokes argument) that whereas the viscous stress in a linear shear flow depends on the velocity gradient, in the case of a circular flow the viscous stress instead depends on the gradient of [*angular*]{} velocity.
Finally, we turn to the issue of our neglect of the curvature of particle orbits between collisions. This neglect is always justified, for example, in the case of laboratory Couette flow, where the fluid is not subject to external long range forces. In the case that the acceleration of the mean flow [*is*]{} provided by a long range central force ([*i.e.*]{} by a force that is experienced by the particles between collisions, as in the case of a Keplerian accretion disc) then the change in ($x$-) velocity of a particle between collisions is $\delta v \sim
R \Omega^2 \lambda/c$. If $\delta v \ll c$ we can repeat the above analysis by adding $\delta v$ to $v_x^{rel}$ (equation 19) and hence modifying the subsequent equations relating $\theta$ to $\alpha$. We find that (to O($\delta v/c)$) the relation between $L$ and $\lambda$ (equation (25)) and hence the expression for $f$ (equation (26)) is unchanged by this addition but that $v_y$ (equation (28)) now contains an additional term ($\delta v \sin^3 \alpha/c \cos \alpha$). This term however makes a zero contribution to the $y$-momentum flux, when integrated over $\alpha $, since the contributions from $\pm \alpha$ cancel. We thus find that our analysis is independent of the nature of the central force providing the acceleration of the mean flow , provided that $\delta v \ll c$. [^5]. This requirement translates into a condition on $\lambda$ for a thin Keplerian disc of the form $\lambda \ll H^2/R$, which is more stringent than the condition ($\lambda \ll H$) derived above in order to justify our approach of expanding quantities to first order in $\lambda \Omega_0/c$.
Discussion
==========
We have shown that it is possible to obtain the correct expression for the momentum transfer in a both a linear and a circular shear flow using simple kinetic theory. This correct expression implies that an isotropic viscosity transfers momentum [*down*]{} an angular velocity gradient. We have noted that it is essential to take proper account of both the geometry and the fluid’s isotropy.
Our analysis allows us to understand the flaw in the simple heuristic argument applied to a circular shear flow (described in Section 1), which leads to the conclusion that since angular momentum is conserved along particle trajectories, it is the gradient in angular momentum which particle mixing tends to smooth out. Particles arrive at the reference patch, at point S, with momentum along the streamline ($y-$momentum) that derives from two sources – (i) the $y-$component of the random emission velocity (assumed isotropic in the local rest frame of the emission point) and (ii) the $y-$component of the streaming velocity of the emission point relative to the reference patch.
The usual argument given to explain the momentum transfer in a [*linear*]{} shear flow is simply that (ii) has the same sign for all the emission points on one side of the reference patch and so that momentum is added to, or subtracted from, the reference patch just depending on the sign of the velocity gradient in the mean flow. In fact, there is another contribution, equal in magnitude and sign to that described above, which may be understood by considering the particles that are incident at the reference patch from $\pm
\alpha$. From Figure 3 it may be seen that the $x-$velocity (and hence flux) of particles on the $+ \alpha$ side exceeds that on the $-
\alpha$ side ([*i.e.*]{} there is a greater flux of particles at the reference patch whose emission is prograde than retrograde). Since the $y-$momentum from (i) far exceeds that from (ii), this slight asymmetry in the fluxes of prograde and retrograde particles produces a net momentum flux that is equal to the more obvious source of momentum flux described above.
We may now apply the same considerations to the [*circular*]{} shear flow, where once again there are the two contributions to the $y-$momentum – (i) and (ii) described above. The important difference now is that the mean streaming velocity is no longer in the $y-$direction. The geometry of the circular arc ensures that this contributes a positive (negative) $x-$velocity for $\alpha > 0 \:
(\alpha < 0)$, respectively. As may be seen from Figure 4, particles that are incident from $+ \alpha$ (prograde particles) have a larger amplitude of both $v_x$ and $v_y$ than those from $- \alpha$. This ensures that the mean $y-$velocity of particles arriving at the reference patch from a particular emission streamline is not equal to the mean $y-$velocity of particles on that emission streamline. This disparity is because of the relative boost in the arrival flux of particles whose random velocity is prograde in the frame of the emission streamline, compared with those that are retrograde.
This now allows us to understand the behaviour of a Keplerian accretion disc at a qualitative level. The mean angular momentum of particles orbiting at radii less than that of the reference patch is less than that of the reference patch. If the distribution of angular momenta of the particles arriving at the reference patch merely reflected the distribution of angular momenta of particles on their parent emission streamlines, this would imply that the arrival of particles at the reference patch from smaller radii should exert a spindown torque. Instead, the relative boost in the [*arrival rate*]{} of particles that are emitted in the prograde direction, ensures that the average angular momentum of the particles arriving at the reference patch exceeds the average at the parent streamline. In the Keplerian case, this relative boost in the arrival rate of the prograde particles is enough to reverse the sign of angular momentum transfer. Thus, particles arriving at the reference patch from smaller radii exert a spin up torque, as required.
Acknowledgments {#acknowledgments .unnumbered}
===============
CJC gratefully acknowledges support from the Leverhulme Trust. JEP gratefully acknowledges continuing support from the STScI Visitors’ Program. We acknowledge useful discussions on this problem with Peter Scheuer and David Syer. We are indebted to the referee, Takuya Matsuda, for interesting comments and for pointing out several typographical errors in the manuscript.
Batchelor, G.K, 1967, [*An Introduction to Fluid Dynamics*]{}, Cambridge University Press.
Brahic, A., 1977, A&A, 54, 895
Bretherton, F.P., Turner, J.S., 1968, J. Fluid. Mech, 32, 449
Hayashi, E., Matsuda, T., 2001, Prog. Theor. Phys., 105, 531
Jeans, J., 1940. An Introduction to the Kinetic Theory of Gases, Cambridge University Press, p. 157
Kumar, P., Narayan, R., Loeb, A., 1995, ApJ.,453, 480
Lynden-Bell, D., Pringle, J.E., 1974, MNRAS, 168, 603
Madej, J. Paczynski, B., 1977 in ‘The Interactions of Variable Stars with their Environment’, Proc. IAU Coll. 42, eds. R. Kippenhahn, J. Rahe & W. Strohmeier, Bamberg: Remeis-Sternwarte, p. 313
Paczynski, B., 1978, Acta Astr., 28, 253
Pringle, J.E., 1981, ARAA, 19, 137
Torkelsson, U., Ogilvie, G.I., Brandenburg, A., Pringle, J.E., Nordland, Å., Stein, R. F., 2000, MNRAS, 318, 47
[^1]: The mathematical complications come in when one has to specify a particular tensor (or vector) by specifying the coordinate representation of that tensor in a particular coordinate frame, and get worse when one has to specify the representation of that same tensor in some other coordinate frame.
[^2]: This simplest assumption is not necessarily the correct one. For example it is possible that the viscosity might depend on the absolute orientation of the shear in some inertial frame ([*e.g.*]{} the stress due to convection in a rotating medium – see Kumar, Narayan & Loeb 1995, and references therein; the stress due to non-isotropic mixing – Bretherton & Turner, 1968; or the stress induced by the magneto-rotational instability – Torkelsson [*et. al.*]{} 2000).
[^3]: It is of course straightforward to generalise the analysis to take account of the particles being emitted with a distribution of velocities.
[^4]: For a Keplerian accretion disc the disc thickness is $H
\sim c/\Omega$, and thus this approximation implies that $\lambda \ll
H$. Thus our analysis does not cover astrophysical situations in which the mean free path is comparable to, or larger than the disc thickness (Paczynski, 1978; Brahic, 1977)
[^5]: Note that it is [*not*]{} necessary that $\delta v$ be much less than the shear velocity across a particle mean free path $\lambda \Omega_0$, and that indeed this latter condition [*cannot*]{} be met in a thin Keplerian accretion disc. We however find that the symmetry of the terms in the momentum flux involving $\delta v$ ensures that their contribution can be neglected, compared with those involving $\lambda \Omega_0$, even when $\delta v \gg \lambda \Omega_0$
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abstract: 'We describe a limitation in the expressiveness of the predictive uncertainty estimate given by mean-field variational inference (MFVI), a popular approximate inference method for Bayesian neural networks. In particular, MFVI fails to give calibrated uncertainty estimates in between separated regions of observations. This can lead to catastrophically overconfident predictions when testing on out-of-distribution data. Avoiding such overconfidence is critical for active learning, Bayesian optimisation and out-of-distribution robustness. We instead find that a classical technique, the linearised Laplace approximation, can handle ‘in-between’ uncertainty much better for small network architectures.'
bibliography:
- 'example\_paper.bib'
---
Introduction
============
Neural networks have been shown to be extremely successful for supervised learning. However, they are known to underestimate their uncertainty when trained by maximum likelihood or maximum a posteriori (MAP) methods. A neural network that returns reliable uncertainty estimates whilst maintaining the computational and statistical efficiency of standard networks would have numerous applications in active learning, reinforcement learning and critical decision-making tasks [@gal2017deep; @chua2018deep; @gal2016uncertainty].
A variety of techniques have been proposed to obtain uncertainty estimates for neural networks in computationally efficient ways [@mackay1992practical; @hinton1993keeping; @barber1998ensemble; @hernandez2015probabilistic; @gal2016dropout; @lakshminarayanan2017simple]. Among these, mean-field variational inference (MFVI) is a widely used approximate inference method that gives state-of-the-art performance in non-linear regression. On the commonly used UCI regression benchmark, MFVI with the reparameterisation trick [@blundell2015weight; @kingma2015variational] often outperforms Stochastic Gradient Langevin Dynamics, Probabilisitic Back-Propagation and ensemble methods [@bui2016deep; @tomczakneural], and is competitive with Monte Carlo Dropout.[^1]
Performance on the UCI datasets is usually measured by held out log-likelihood. This represents both accuracy and uncertainty quantification, since it penalises methods that are overconfident in addition to being inaccurate. It is therefore perhaps surprising that MFVI performs poorly on sequential decision making tasks that require good uncertainty quantification, such as contextual bandits [@riquelme2018deep]. To perform well, a method must ‘know what it knows, and what it doesn’t know’: it should have high confidence near clusters of observations, and low confidence elsewhere. More specifically, a well-calibrated network should predict with high uncertainty far from data, as well as *in regions between separated clusters of observations*. However, the current UCI benchmark is not suitable for evaluating ‘in-between’ uncertainty, as the test set is obtained by uniformly subsampling the full dataset. We therefore design another UCI benchmark to test for in-between uncertainty, by taking the ‘middle region’ of the full dataset as the test set. We find that although MFVI performs well on the standard UCI benchmark, it can fail catastrophically on the in-between version, showing the detrimental effect the mean-field approximation has on in-between uncertainty. In contrast, a classical technique, the linearised Laplace approximation [@mackay1992practical], performs well on both.
it should have high confidence near regions with a lot of observed data, and low confidence far from the data region and between separated clusters of data. Testing uncertainty quality in these different scenarios requires different kinds of datasets. We therefore test on two versions of the UCI datasets: one with the test set uniformly subsampled from the dataset, and one with the test set taken from the ‘middle’ of the dataset, to test for ‘in-between’ uncertainty, by which we mean the ability to express increased uncertainty in between well-separated regions of observations. We find that a classical technique, the linearised Laplace approximation [@mackay1992practical], performs well on both. However, we find that although MFVI performs well on the former, it can fail catastrophically on the latter, showing the detrimental effect the mean-field approximation has on in-between uncertainty.
A single number from a single kind of test set split cannot be indicative of performance in all of these scenarios. Here we show that the standard UCI dataset splits are not good tests of ‘in-between’ uncertainty, by which we mean the ability to express increased uncertainty in between well-separated regions of low uncertainty. This is a desirable property for any well-calibrated uncertainty estimate. Furthermore, we show that whereas MFVI fails to provide in-between uncertainty, a classical technique, the linearised Laplace approximation [@mackay1992practical], can. We provide comparisons on a 1D toy dataset and also on two versions of the UCI datasets: the standard train/test splits, and splits designed to test for in-between uncertainty.
Methods
=======
This paper focuses on two approximate inference techniques for Bayesian Neural Networks (BNNs): Variational Inference (VI) and the Laplace approximation. We consider networks whose output $f_{\theta}(\mathbf{x})$ given input $\mathbf{x}$ and parameters $\theta$ is interpreted as the mean of a Gaussian distribution with homoscedastic output noise variance $\sigma_o^2$. We place a diagonal Gaussian prior over $\theta$, here written as a column vector of all weights and biases in the network.
**Variational Inference**. Let the posterior distribution over $\theta$ given a dataset $\mathcal{D} = \{ (\mathbf{x}_n, y_n) \}_{n=1}^N$ be $p(\theta| \mathcal{D})$. VI approximates this posterior with a simpler distribution $q_{\phi}(\theta)$. The parameters $\phi$ are learned by optimising a simple Monte Carlo estimate of the Evidence Lower Bound (ELBO): $$\begin{aligned}
\mathcal{L}(\phi) &= \sum_{n=1}^N \mathbb{E}_{q_{\phi}}[ \log p(y_n|\theta, \mathbf{x}_n) ] - \mathrm{KL}(q_{\phi}(\theta)||p(\theta))\\
&\approx \frac{1}{M} \sum_{m=1}^M \sum_{n=1}^N \log p(y_n|\theta_m, \mathbf{x}_n) - \mathrm{KL}(q_{\phi}(\theta)||p(\theta)),\end{aligned}$$ where $\theta_m$ are sampled from $q_{\phi}(\theta)$. Optimising the ELBO minimises the KL-divergence between $q_{\phi}(\theta)$ and the true posterior. Once $\phi$ is learned, we can make predictions by Monte Carlo sampling from $q_{\phi}(\theta)$: $$\begin{aligned}
p(y^*|\mathbf{x}^*, \mathcal{D}) &= \mathbb{E}_{p(\theta|\mathcal{D})}[p(y^*|\mathbf{x}^*, \theta)] \nonumber\\
&\approx \mathbb{E}_{q_{\phi}(\theta)}[p(y^*|\mathbf{x}^*, \theta)]\nonumber\\
&\approx \frac{1}{M} \sum_{m=1}^M p(y^*|\mathbf{x}^*, \theta_m). \label{MC_predictive}\end{aligned}$$ A common and scalable choice for the form of $q_{\phi}(\theta)$ is the mean-field Gaussian approximation (MFVI), which is a fully factorised Gaussian distribution. Another choice is to let $q_{\phi}(\theta)$ be a full covariance Gaussian (FCVI). This is more flexible, but the number of variational parameters $\phi$ is now quadratic in the number of parameters in the network.
**Laplace Approximation**. The Laplace approximation [@denker1991transforming; @mackay1992practical] finds a mode $\theta_{\mathrm{MAP}}$ of the posterior, and sets the approximate posterior to $q(\theta) = \mathcal{N}(\theta; \mu, \Sigma)$ with $\mu = \theta_{\mathrm{MAP}}$. $\Sigma$ is set such that the curvature of $\log p(\theta|\mathcal{D})$ matches the curvature of the logarithm of the Gaussian approximation at $\theta_{\mathrm{MAP}}$, that is: $$\begin{aligned}
\Sigma = -\bigg[ \nabla_{\theta} \nabla_{\theta} \log p(\theta|\mathcal{D}) \big\rvert_{\theta = \theta_{\mathrm{MAP}}} \bigg]^{-1}.\end{aligned}$$ In words, $\Sigma$ is the negative inverse Hessian evaluated at the MAP solution. In practice we use the Gauss-Newton matrix, which is guaranteed to be positive semi-definite, and can be evaluated using only first derivatives: $$\begin{aligned}
\Sigma = -\bigg[ \frac{1}{\sigma_o^2} \sum_{n=1}^N g(\mathbf{x}_n) g(\mathbf{x}_n)^\mathsf{T} + \mathrm{diag}(p) \bigg]^{-1}.\end{aligned}$$ Here $g(\mathbf{x}_n) = \nabla_{\theta} f_{\theta}(\mathbf{x}_n) \big\rvert_{\theta = \theta_{\mathrm{MAP}}}$ and $p$ is a vector whose $i$th element is $1/\sigma^2_i$, where $\sigma^2_i$ is the prior variance of $\theta_i$.
Once $\theta_{\mathrm{MAP}}$ and $\Sigma$ are obtained, there are two different ways to make predictions. The first is to Monte Carlo sample from the approximate posterior as in equation (\[MC\_predictive\]). We refer to this method as Sampled Laplace (SL). Unfortunately, the Laplace approximation is known to cause severe underfitting [@lawrence2001variational]. An alternative procedure which empirically alleviates this is to linearise the output of the network about $\theta_{\mathrm{MAP}}$. This leads to a linear Gaussian model that can be solved exactly for the predictive distribution: $$\begin{aligned}
p(y^*|\mathbf{x}^*, \mathcal{D}) \approx \mathcal{N}(y^*; f_{\theta_{\mathrm{MAP}}}(\mathbf{x}^*), \sigma_o^2 + g(\mathbf{x}^*)^{\mathsf{T}}\Sigma \, g(\mathbf{x}^*));\end{aligned}$$ see Appendix \[laplace\_linearised\] for details. We refer to this method as Linearised Laplace (LL).
Finding $\theta_{\mathrm{MAP}}$ is identical to standard neural network training. Once at a mode, calculating the Gauss-Newton matrix requires one backward pass for each element of the dataset, which has a cost that scales linearly in the number of observations. Lastly the Laplace approximation requires inverting this matrix which has cubic cost in the number of parameters. This is still tractable for the smaller networks typically considered for regression on UCI datasets. Recent work has applied Kronecker-Factored Approximate Curvature (K-FAC) to obtain a scalable method; however they used sampling instead of linearisation and had to take steps to mitigate the underfitting problem inherent in the Laplace approximation [@ritter2018scalable].
Experiments {#experiment_section}
===========
To test for in-between uncertainty, we compare these methods on two tasks. The first is a synthetic 1D regression dataset formed by adding Gaussian noise to a sine wave and observing two separated clusters of input points. The second are the UCI regression datasets.
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**Synthetic 1D Dataset**. We plot results for four inference methods on the 1D dataset: MFVI, FCVI, LL and Hamiltonian Monte Carlo (HMC), in Figure \[tanh\_comparison\].[^2] We use a single hidden layer network with tanh non-linearities[^3] and 50 hidden units. Diagonal Gaussian priors are used for all networks, and the observation noise $\sigma_o$ is fixed to $0.1$, the true value. Details and additional results are in Appendix \[appendix\_extra\]. We see that MFVI fails to represent in-between uncertainty: its error bars are of similar magnitude in the data region and the in-between region. FCVI has larger uncertainty in the middle, but is slightly underconfident in the data region. LL and HMC show high confidence in the data region and increased uncertainty in between, showing that MFVI’s failure is rooted in approximate inference, not the model class.
There are several reasons for MFVI’s overconfidence. First, we show in Appendix \[convex\_variance\] that a single hidden layer BNN with ReLU activations and with deterministic input weights and mean-field (possibly non-Gaussian) output weights must have an output variance that is *convex* as a function of its input. Such a BNN is incapable of expressing increased uncertainty between regions of low uncertainty. This would not be the case if the output weights had an unrestricted distribution. Although this insight does not immediately apply to BNNs with tanh activations and mean-field input weights, it shows that the mean field assumption can in some cases severely restrict the complexity of uncertainty estimates a BNN can express in function space.
Second, MFVI fails to express increased in-between uncertainty because fitting data in the outer region whilst having increased uncertainty in-between requires strong *dependencies* in the approximate posterior. This is because in a mean-field distribution, any parameter uncertainty used to express increased in-between uncertainty leads to uncontrolled variations in the fit in the data region. The only way to have a good fit *and* increased in-between uncertainty is to have variations in one parameter compensated for by variations in others, such that the resulting function still passes through the data points. This is explained in detail in Appendix \[toy\_explanation\] via a synthetic example.
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**UCI Regression Datasets**. We now investigate the uncertainty quality of BNNs in real-world datasets. The UCI datasets are usually split into training and test sets by subsampling the dataset uniformly. In the first experiment, we use the standard splits also used in [@hernandez2015probabilistic; @bui2016deep; @mukhoti2018importance]. In our second experiment, we create custom splits to test for in-between uncertainty. For each of the $D$ input dimensions of $\mathbf{x}_n \in \mathbb{R}^D$, we sort the datapoints in increasing order in that dimension. We then remove the middle $1/3$ of these datapoints for use as a test set. The outer $2/3$ are the training set. $10\%$ of the training set is used as a validation set. Thus a dataset with $D$ inputs has $D$ splits. We refer to these as the ‘gap splits’. A satisfactory method would achieve good results on the standard splits (showing an ability to fit the data well) and avoid catastrophically poor results on the gap splits (showing increased in-between uncertainty). The results are shown in Figures \[error\_bars\_standard\] and \[error\_bars\_gap\].[^4]
For the standard splits MFVI and LL perform best. FCVI does poorly, likely due to optimisation difficulties. LL does much better than SL, which is surprising given linearisation adds another approximation. However, it appears to redeem the poor Gaussian approximation [@lawrence2001variational]. For the gap splits, MAP is competitive with Bayesian methods on power, protein and wine. However, on energy and naval MAP fails catastrophically, doing dozens or hundreds of nats worse than LL. The test sets of energy and naval thus show very different behaviour from their training sets, and good in-between uncertainty is required to prevent overconfident extrapolations. In this situation we would expect Bayesian methods to outperform MAP. However MAP and MFVI perform similarly poorly on energy and naval, showing that MFVI is overconfident. The only method that performs well on the standard splits and avoids any catastrophic results on the gap splits is LL.
Conclusions
===========
We have shown that MFVI fails to provide calibrated in-between uncertainty, and that the standard UCI splits fail to adequately test for it. However, the decades-old LL approximation performs far better in this regard. Although recent advances in variational inference have allowed BNNs to scale to larger architectures than ever before, in terms of uncertainty *quality* the mean-field approximation loses crucial expressiveness compared to the less scalable LL approximation. It is therefore key for the field of approximate inference to consider how the approximation of posteriors in parameter space affects the expressiveness of uncertainties in *function space*. Future work will investigate the conditions an approximate posterior must satisfy to reliably capture in-between uncertainty. It would also be natural to see if combining K-FAC Laplace [@ritter2018scalable] with linearisation leads to improved results.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank David R. Burt, Sebastian W. Ober and Ross Clarke for helpful discussions. AYKF gratefully acknowledges the Trinity Hall Research Studentship and the George and Lilian Schiff Foundation for funding his studies.
Linearised Laplace Approximation {#laplace_linearised}
================================
To obtain the linearised Laplace approximation, we linearise the output of the network about $\theta_{\mathrm{MAP}}$: $$\begin{aligned}
f_{\theta}(\mathbf{x}) \approx f_{\theta_{\mathrm{MAP}}}(\mathbf{x}) + g(\mathbf{x})^{\mathsf{T}}(\theta - \theta_{\mathrm{MAP}}). \label{linearisation}\end{aligned}$$ We now have the following approximating distributions: $$\begin{aligned}
p(\theta|\mathcal{D}) &\approx \mathcal{N}(\theta; \theta_{\mathrm{MAP}}, \Sigma),\\
p(y^*|\theta, \mathbf{x}^*) &\approx \mathcal{N}(y^*; f_{\theta_{\mathrm{MAP}}}(\mathbf{x}) + g(\mathbf{x})^{\mathsf{T}}(\theta - \theta_{\mathrm{MAP}}), \sigma_o^2).\end{aligned}$$ Since this is now a linear-Gaussian model, we can use standard formulas to obtain: $$\begin{aligned}
p(y^*|\mathbf{x}^*, \mathcal{D}) \approx \mathcal{N}(y^*; f_{\theta_{\mathrm{MAP}}}(\mathbf{x}^*), \sigma_o^2 + g(\mathbf{x}^*)^{\mathsf{T}}\Sigma \, g(\mathbf{x}^*)).\end{aligned}$$
Convex Variance Result {#convex_variance}
======================
Consider a single hidden layer BNN with input $\mathbf{x} \in \mathbb{R}^D$ and output $\mathbf{y} \in \mathbb{R}^K$ with a mean field distribution over the output weights and biases $(\mathbf{W}, \mathbf{b})$ but a point estimate for the input weights and biases $(\mathbf{U}, \mathbf{v})$. In detail: $$\begin{aligned}
y_k (\mathbf{x}) &= \sum_i W_{ki}\phi(a_i) + b_k,\\
a_i &= \sum_j U_{ij}x_j + v_i.\end{aligned}$$ We assume a fully factorised approximating distribution for the output weights such that: $$\begin{aligned}
q(\mathbf{W}, \mathbf{b}) = q(\mathbf{b})\prod_{k,i}q_{ki}(W_{ki}).\end{aligned}$$ We further assume that $\mathbf{U}$ and $\mathbf{v}$ are deterministic constants. Consider the variance of the output under this distribution: $$\begin{aligned}
\mathrm{Var}[y_k (\mathbf{x})] &= \sum_i \mathrm{Var}[W_{ki}] \phi(a_i)^2 + \mathrm{Var}[b_k]. \label{sum_variance}
\end{aligned}$$ Equation (\[sum\_variance\]) is justified since each weight is independent under $q$. This variance is a measure of the uncertainty in the output at $\mathbf{x}$ represented by the approximate posterior $q$. Consider the Hessian of this variance $\mathbf{H}$, where $H_{nm} = \partial_{x_n}\partial_{x_m} \mathrm{Var}[y_k(\mathbf{x})]$. Taking derivatives, we have: $$\begin{aligned}
H_{nm} &= \sum_i 2 \mathrm{Var}[W_{ki}] \big(\phi(a_i)\phi''(a_i) + \phi'(a_i)^2 \big) U_{in}U_{im}\\
\mathbf{H} &= \sum_i 2\mathrm{Var}[W_{ki}] \big(\phi(a_i)\phi''(a_i) + \phi'(a_i)^2 \big) \mathbf{u}_i \mathbf{u}^{\mathsf{T}}_i,
\end{aligned}$$ where $\mathbf{u}_i$ is the column vector whose elements are the $i$th row of $\mathbf{U}$. Since $\mathbf{H}$ is a sum of outer products, it will be positive semi-definite (PSD) if $\phi(a_i)\phi''(a_i) + \phi'(a_i)^2 \geq 0$ for all $i$. This is the case for ReLU nonlinearities. The first and second derivatives of the ReLU do not exist at $a_i = 0$. However if we consider $\phi''$ to be a bump function of arbitrarily small width and area 1, then all these derivatives exist and $\phi(a_i)\phi''(a_i)$ is non-negative. Since the Hessian of $\mathrm{Var}[y_k(\mathbf{x})]$ is PSD, it follows that $\mathrm{Var}[y_k(\mathbf{x})]$ is a convex function of $\mathbf{x}$.[^5] Therefore it is impossible for this kind of posterior to exhibit greater uncertainty in between two regions of low uncertainty.
To investigate the relevance of this result to the standard case where the input parameters $(\mathbf{U}, \mathbf{v})$ are not deterministic but are also mean-field, we train three ReLU BNNs on the 1D dataset in Figure \[tanh\_comparison\]: (i) mean-field VI on all parameters (MVFI), (ii) maximum-likelihood on the input parameters and mean-field on the output parameters (MFVI-output) and (iii) maximum-likelihood on the input parameters followed by Bayesian linear regression on the output parameters (BLR). Results are shown in Figure \[variance\_comparison\]. MFVI-output has convex variance, as predicted. MFVI also has convex variance, even though its input parameters are mean-field. BLR with its full-covariance Gaussian posterior shows increased in-between uncertainty even though its input weights are deterministic, showing that it is the mean-field assumption on the output parameters that is responsible for severely restricting the expressiveness of the predictive uncertainty. Further work is required to characterise the expressiveness of mean field distributions with deeper networks.
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![Predictive variances (without observation noise) on the 1D dataset. Black lines show $x$-locations of the data. []{data-label="variance_comparison"}](variance_comparison.pdf){width="0.9\columnwidth"}
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In practice when MFVI is used, the input weights to the hidden layer are not deterministic, but are also mean field. However this result sheds light on the difficulty of obtaining expressive uncertainty estimates with mean field posteriors. It shows that in addition to non-convex variances being difficult to obtain experimentally, they are theoretically impossible to obtain with a mean field distribution over just output weights. Note that for an unconstrained $q$ distribution over the output weights, $\mathrm{Var}[y_k(\mathbf{x})]$ is generally non-convex even with deterministic input weights. Therefore the convexity restriction is a direct consequence of the mean-field restriction on $q$. Further work is required to characterise the expressiveness of mean field distributions when input weights are also mean-field, and with deeper networks.
We plot the output variance of MFVI (on *all* parameters) on the 1D dataset from Section \[experiment\_section\] in Figure \[MFVI\_variance\]. For tanh the variance is close to zero everywhere, and for ReLU it is convex even though the input weights are also mean-field. We observed similar behaviour with deeper networks.
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![Predictive variance for MFVI with tanh (left) and ReLU (right) non-linearities.[]{data-label="MFVI_variance"}](MFVI_variance.pdf){width="0.9\columnwidth"}
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Analysis of Uncertainty in Toy Case {#toy_explanation}
===================================
To gain intuition for why MFVI fails to provide in-between uncertainty, we consider a toy example involving a single hidden layer network with two ReLU hidden units mapping $x \in \mathbb{R} \rightarrow y \in \mathbb{R}$: $$\begin{aligned}
y(x) = W_1 \phi (U_1 x + v_1) + W_2 \phi (U_2 x + v_2) + b.\end{aligned}$$ Here $W_1, W_2$ and $b$ are the output weights and bias, and $U_1, U_2$ and $v_1, v_2$ are the input weights and biases. Consider the case where $W_1, W_2, U_1, U_2$ are all deterministic and positive so that $y(x)$ is non-decreasing. Then: $$\begin{aligned}
y(x)=
\begin{cases}
b & \mathrm{(I)}\\
W_1U_1x + W_1v_1 + b & \mathrm{(II)}\\
(W_1U_1 + W_2U_2)x + W_1v_1 + W_2v_2 + b & \mathrm{(III)}
\end{cases}\end{aligned}$$ where $x< -\frac{v_1}{U_1}$ in region (I), $-\frac{v_1}{U_1} \leq x < -\frac{v_2}{U_2}$ in region (II) and $x\geq -\frac{v_2}{U_2}$ in region (III). Consider a simple observed dataset with many points at $(x_1, y_1)$ and $(x_2, y_2)$ where $x_2 > x_1$ and $y_2 > y_1$. A reasonable Bayesian posterior predictive would have low uncertainty around $x_1$ and $x_2$, but large uncertainty in between. To first fit this dataset with deterministic weights, we could place $x_1$ in region (I) and $x_2$ in region (III). Then to fit the $y$-values we must set $$\begin{aligned}
b &\approx y_1, \\
(W_1U_1 + W_2U_2)x_2 &+ W_1v_1 + W_2v_2 + b \approx y_2. \label{fit_equation}\end{aligned}$$
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![Samples from a 2-hidden unit neural network obtained by HMC. Notice how the position of the kinks varies between samples, leading to larger uncertainty in between the 2 datapoints $(x_1,y_1)$ and $(x_2,y_2)$, marked by black crosses. (For some of these samples, only one kink is between $x_1$ and $x_2$; the other is to the left of $x_1$.)[]{data-label="2_points"}](2_points_plain.pdf){width="0.8\columnwidth"}
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There are many settings of $W_1, U_1, W_2, U_2, v_1$ and $v_2$ that satisfy Equation \[fit\_equation\]. Consider choosing one such setting as a point estimate. To obtain a Bayesian method, we would now like to increase our uncertainty in the parameters. In particular, we should have relatively large uncertainty in the position of the ‘kinks’ $-\frac{v_1}{U_1}$ and $-\frac{v_2}{U_2}$ since they can take any values between $x_1$ and $x_2$ and fit the data equally well.[^6] This corresponds to having large uncertainty between two regions of low uncertainty ($x_1$ and $x_2$), as in Figure \[2\_points\]. To express this, we could relax the distribution over, say, $v_2$ from a delta function to a Gaussian with positive variance. However, injecting randomness in $v_2$ jeopardises the fit in region (III) since $v_2$ is involved in Equation \[fit\_equation\]. The only way to express predictive uncertainty between $x_1$ and $x_2$ and *still* fit the data is to have the values of $W_1, U_1, W_2, U_2$ and $v_1$ *compensate* for any change in $v_2$ such that Equation \[fit\_equation\] still holds. In other words, we need strong dependencies between the parameters to simultaneously fit the data regions and express predictive uncertainty in the in-between region.[^7]
The mean-field approximation assumes that there are no dependencies. Therefore any parameter randomness used to express increased in-between uncertainty leads to uncontrolled variations of $y(x)$ in the data region. Hence Equation \[fit\_equation\] is not satisfied. There are two possibilities: either the data fit will be poor or the variances will be minimised (leading to a large penalty in the KL term in the ELBO). In practice MFVI finds a solution that prunes out hidden units, allowing it to fit the data with the minimum number of variances set to zero [@trippe2018overpruning].
Extra Results and Experimental Details {#appendix_extra}
======================================
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**Synthetic 1D Dataset**. We use single hidden layer BNNs with 50 hidden units. We include results for ReLU activations in Figure \[relu\_comparison\]. To verify that MFVI’s lack of in-between uncertainty is due to approximate inference and not the model class, we include a Gaussian Process (GP) using the kernel for a BNN with infinitely many ReLU hidden units [@lee2017deep; @cho2009kernel]. Note LL shows strange discontinuous behaviour in its uncertainty. This is because the non-smooth ReLU function makes the gradient $g(\mathbf{x})$ in equation (\[linearisation\]) discontinuous. Similar behaviour is seen in [@snoek2015scalable].
We use independent $\mathcal{N}(0,1)$ priors on the biases and $\mathcal{N}(0,\frac{\omega^2}{H})$ priors on the weights; $H$ is the number of inputs to the weight matrix. This scaling is chosen so that the GP limit exists [@neal2012bayesian]. $\omega$ is set to $4$. To optimise Laplace, MFVI and FCVI we use ADAM [@kingma2014adam] with learning rate 0.001 and 20,000 epochs. We use the entire dataset for each batch. For MFVI, weight means were initialised from $\mathcal{N}(0,1/\sqrt{4n_{outputs}})$ and all variances were initialised to $10^{-5}$. Bias means were initialised to zero. The local reparameterisation trick [@kingma2015variational] was used. For FCVI, the Cholesky decomposition of the covariance matrix was parameterised as a lower triangular matrix, with the diagonal entries made positive by exponentiating them. The diagonal entries were initialised to $\log(0.05)$ and the off-diagonals were initialised to 0. The mean vector was initialised from $\mathcal{N}(0, 0.1)$. For both MFVI and FCVI we approximate the ELBO during training with 32 samples. For HMC, the number of leapfrog steps was chosen uniformly between 5 and 10, and the step size uniformly sampled from $[0.001, 0.0015]$. The chain was burned in for 10,000 iterations and samples were collected during the next 20,000 iterations. For MFVI, FCVI and HMC, the error bars in Figures \[tanh\_comparison\] and \[relu\_comparison\] were estimated with 100 samples.
**UCI Datasets**. All BNNs had 50 neurons per hidden layer. Inputs and outputs were normalised to zero mean and unit variance. Hyperparameters were optimised by grid search on a validation set that consisted of $10\%$ of the training set. The best hyperparameters were used to train again on the training set with validation set combined. This was repeated for each split. Minibatches were randomly selected from the training set with replacement. For MAP and Laplace, all parameters had independent $\mathcal{N}(0,\omega^2)$ priors. For Laplace, minibatch size was $100$. The hyperparameters optimised were: $\omega$: $[1, 2]$, learning rate: $[0.01, 0.001]$, number of epochs: $[40, 100, 200, 400]$. For MAP, the same ranges were searched, except the number of epochs was $[20, 40, 100]$, since we expected MAP to favour early stopping. For both methods, $\log(\sigma_o^2)$ was initialised to $-1$ and learned by maximum likelihood. For FCVI and MFVI, we used independent $\mathcal{N}(0,1)$ priors on all parameters. The hyperparameters searched for MFVI were: minibatch size: $[32, 100]$, learning rate: $[0.01, 0.001]$, number of epochs: $[500, 1000, 2000]$ for smaller datasets (boston, concrete, energy, wine, yacht) and $[50, 100, 200]$ for larger ones (kin8nm, naval, power, protein). The same ranges were used for FCVI except the learning rate was fixed to $0.001$. The ELBO was approximated with $32$ samples. For MFVI, weight means were initialised from $\mathcal{N}(0,1/\sqrt{4n_{out}})$ and all variances initialised to $10^{-5}$. Bias means were initialised to zero. For FCVI, the mean vector and covariance matrix of all the parameters in the network were optimised to maximise the ELBO. The Cholesky decomposition of the covariance matrix was parameterised directly as a lower triangular matrix, with the diagonal entries constrained to be positive by exponentiating them. The diagonal entries were initialised to $\log(10^{-5})$ and the off-diagonal entries were initialised to $0$. The mean vector was initialised randomly from $\mathcal{N}(0, 0.1)$. For MFVI, FCVI and sampled Laplace, test log-likelihoods were computed by sampling $100$ times from the approximate posterior. $\log(\sigma_o^2)$ was initialised to $-1$ and learned by optimising the ELBO.
Full results are given in Tables \[standard\_split\_table\] and \[gap\_split\_table\]. We also provide pairwise comparisons of LL versus MFVI-ReLU on the standard splits and the gap splits in Figures \[MFVIvslap\_standard\], \[MFVIvslap\_gap\] & \[MFVIvslap\_gap\_removed\]. Each point corresponds to one test log-likelihood. Each colour represents a different dataset. The histogram shows the log likelihood of the method on the $x$-axis minus that of the method on the $y$-axis. The dotted blue line is the line $y=x$.
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![Comparison of MFVI-ReLU and linearised Laplace tanh on the standard splits. Positive difference means Laplace performs better than MFVI.[]{data-label="MFVIvslap_standard"}](laplace_vs_MFVI_relu.pdf){width="0.9\columnwidth"}
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![Comparison of MFVI-ReLU and linearised Laplace tanh on the gap splits. Positive difference means Laplace performs better than MFVI. MFVI fails catastrophically on energy and naval.[]{data-label="MFVIvslap_gap"}](laplace_vs_MFVI_relu_gap.pdf){width="0.9\columnwidth"}
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![Same as Figure \[MFVIvslap\_gap\] but with energy and naval removed. Positive difference means Laplace performs better than MFVI. The two methods now perform comparably.[]{data-label="MFVIvslap_gap_removed"}](laplace_vs_MFVI_relu_gap_nonavalenergy.pdf){width="0.9\columnwidth"}
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[^1]: See [@mukhoti2018importance] for a recent strong baseline for Monte Carlo Dropout on the UCI regression datasets.
[^2]: SL is not shown as the fit is so poor that the error bars completely fill the figure. See [@lawrence2001variational] pp. 88 - 91.
[^3]: ReLUs caused problems with LL, see Appendix \[appendix\_extra\].
[^4]: FCVI was only run on one hidden layer due to its long training times, and only tanh was used for Laplace as ReLUs caused problems with linearisation - see Figure \[relu\_comparison\]. Full UCI results and experimental details in Appendix \[appendix\_extra\].
[^5]: This argument can be made rigorous by constructing a sequence of networks with non-linearities $\phi_n$ such that $\phi_n''$ is a triangular function at zero with area 1 and width $\epsilon/n$. Each network will have a convex output variance, and the variance of these networks converges pointwise to the variance of a ReLU network. Since a pointwise limit of convex functions is convex, the result holds for ReLU networks.
[^6]: Here we assume a reasonably broad prior such that the prior probabilities of the kink locations are roughly uniform over the range $[x_1, x_2]$.
[^7]: The in-between uncertainty seen in Figure 3 in [@duvenaud2015black] is seemingly an exception. However in that case radial basis function non-linearities were used. Since these have only local effects, the argument here does not apply.
|
---
abstract: 'In this report, structural, electronic, magnetic and transport properties of quaternary Heusler alloys CoRuMnGe and CoRuVZ (Z = Al, Ga) are investigated. All the three alloys are found to crystallize in cubic structure. CoRuMnGe exhibits L2$_1$ structure whereas, the other two alloys have B2-type disorder. For CoRuMnGe and CoRuVGa, the experimental magnetic moments are in close agreement with the theory as well as those predicted by the Slater-Pauling rule, while for CoRuVAl, a relatively large deviation is seen. The reduction in the moment in case of CoRuVAl possibly arises due to the anti-site disorder between Co and Ru sites as well as V and Al sites. Among these alloys, CoRuMnGe has the highest T$\mathrm{_C}$ of 560 K. Resistivity variation with temperature reflects the half-metallic nature in CoRuMnGe alloy. CoRuVAl shows metallic character in both paramagnetic and ferromagnetic states, whereas the temperature dependence of resistivity for CoRuVGa is quite unusual. In the last system, $\rho$ vs. T curve shows an anomaly in the form of a maximum and a region of negative temperature coefficient of resistivity (TCR) in the magnetically ordered state. The ab initio calculations predict nearly half-metallic ferromagnetic state with high spin polarization of 91, 89 and 93 % for CoRuMnGe, CoRuVAl and CoRuVGa respectively. In the case of CoRuMnGe, the XRD analysis reveals that the Co and Ru sites are equally probable. Hence, to investigate the electronic properties of the experimentally observed structure, the Co-Ru swap disordered structures of CoRuMnGe alloy are also simulated and it is found that the disordered structures retain half-metallic nature, high spin polarization with almost same magnetic moment as in the ideal structure. Nearly half-metallic character, high T$\mathrm{_C}$ and high spin polarization make CoRuMnGe alloy promising for room temperature spintronic applications.'
address: |
$^a$Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, Maharashtra, India\
$^b$ WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
author:
- 'Deepika Rani$^{a}$, Lakhan Bainsla$^{a,b}$, K. G. Suresh$^{a}$ and Aftab Alam$^a$'
bibliography:
- 'bib.bib'
title: 'Experimental and Theoretical Investigation on the Possible Half-metallic Behaviour of Equiatomic Quaternary Heusler Alloys: CoRuMnGe and CoRuVZ (Z = Al, Ga)'
---
Introduction
============
In the last few years, quaternary Heusler alloys received enormous interest due to their wide applications in the field of spintronics. Many of them are reported to show half-metallic behavior and thus have high spin polarization.[@doi:10.1063/1.4959093] In half-metallic materials, one of the spin bands exhibits metallic character whereas, the other spin band exhibits a gap at the Fermi level. Magnetic materials with high spin polarization and high Curie temperature are desirable to improve the performance of spintronic devices such as magnetic tunnel junctions,[@doi:10.1063/1.3330942; @doi:10.1063/1.2354026] spin injectors, spin transistors,[@doi:10.1002/pssc.200672894; @:/content/journals/10.1049/ip-cds_20045196] and spin valves.[@doi:10.1063/1.4821243; @1882-0786-7-3-033002] Heusler alloys are potential materials in this field because of their stable structure, high T$\mathrm{_C}$ , tunable electronic properties and high spin polarization. Equiatomic quaternary Heusler alloys X$\mathrm{X^\prime}$YZ (where X, $\mathrm{X^\prime}$ , Y are transition metals and Z is a main group element) crystallize in the space group no. 216 with F$\bar{4}$3m symmetry (Y-type structure with prototype LiMgPdSn).[@Y] Among the various reported quaternary Heusler alloys only a few crystallize in ordered Y-type structure. Controlling disorder and defects in this class of materials is still a big challenge for the applications, since disorder greatly affects the spin polarization.[@0953-8984-19-31-315215; @PhysRevB.74.104405; @doi:10.1063/1.4929252] There are a large number of reports on 3d- transition elements based quaternary Heusler alloys, but only a few 4d - based (Ru and Rh based) equiatomic quaternary Heusler alloys have been studied experimentally.[@BAINSLA2015631; @PhysRevB.96.184404; @0953-8984-24-4-046001] It would be interesting to see the effect of replacing one of the 3d- element by a 4d- element in quaternary Heusler alloys. For example, CoFeMnGe alloy was found to have considerable amount of DO$_3$ disorder[@doi:10.1063/1.4902831] and replacing Fe by a 4d element (Rh and Ru) is expected to improve the structure, as both CoRuMnGe and CoRhMnGe[@PhysRevB.96.184404] are found to crystallize in L2$_1$ structure.\
We synthesized CoRuMnGe (CRMG), CoRuVAl (CRVA) and CoRuVGa (CRVG) equiatomic quaternary Heusler alloys and investigated their structural, magnetic, electronic and transport properties. X-ray diffraction study reveals that in case of CRMG, 50 % disorder exists between the Co and Ru sites (i.e. 4c and 4d sites are equally probable for Co and Ru atoms), which reduces its symmetry to L2$_1$ structure. CRVA and CRVG alloys show B2 disorder. The experimental saturation magnetization values are in good agreement with the Slater-Pauling rule for CRMG and CRVG, but CRVA shows a relatively large deviation, which may be attributed to the disorder. This suggest that high spin polarization is not possible in CRVA alloy as Slater-Pauling rule is considered to be a prerequisite for half metallic nature.[@PhysRevB.66.174429] The temperature dependence of electrical resistivity is studied in detail for all the three alloys. In case of CRVG, an unconventional behavior is seen in the form of a maximum and a region with semiconducting behavior in the ferromagnetic state. To investigate the half-metallic behavior in these alloys, electronic structure calculations by ab initio method were performed using Perdew, Burke, and Ernzerhof (PBE) potential. We have also studied the Co-Ru swap disordered structures of CoRuMnGe alloy to get a deeper insight into the effect of disorder on its magnetic and electronic properties. Among these three alloys, CRMG is found to be a potential material for spintronic applications due to its stable structure, high spin polarization and high T$\mathrm{_C}$ value.
Experimental Details
====================
The polycrystalline alloys CoRuMnGe and CoRuVZ ( Z = Al, Ga) were prepared by arc melting the stoichiometric amounts of high purity (at least 99.9% purity) constituent elements in argon atmosphere. A Ti ingot was used as an oxygen getter to further reduce the contamination. 2% extra Mn was taken to compensate the weight loss due to Mn evaporation during melting.[@PhysRevB.96.184404] For better homogeneity the ingots formed were flipped and melted several times . After melting, the samples were sealed in a quartz tube and annealed for 7 days at 1073 K followed by furnace cooling.[@PhysRevApplied.10.054022] Room temperature X-ray diffraction patterns were taken using Cu-K$_\alpha$ radiation with the help of Panalytical X-pert diffractometer. Crystal structure analysis was done using FullProf Suite software. Magnetization isotherms at 5 K were obtained using a vibrating sample magnetometer (VSM) attached to the physical property measurement system (PPMS) (Quantum design) for fields up to 40 kOe. Thermo-magnetic curves in the higher temperature range were obtained using a VSM attached with high temperature oven, in a field of 100 Oe. Electrical resistivity measurements were done using the four probe method in PPMS, applying a 5 mA current.
Computational Details
=====================
Ab initio simulations were performed using a spin resolved density functional theory (DFT) implemented within Vienna ab initio simulation package (VASP) [@VASP] with a projected augmented-wave basis.[@PAW] The electronic exchange-correlation potential due to Perdew, Burke, and Ernzerhof (PBE) was used within the generalized gradient approximation (GGA) scheme. A $24^3$ k-mesh was used to perform the Brillouin zone integration within the tetrahedron method. A plane wave energy cutoff of 500 eV was used for all the calculations. All the structures are fully relaxed (cell volume, shape, and atomic positions of constituent atoms), with total energies (forces) converged to values less than $10^{-6}$ eV (0.01 eV/Å). In order to find the most stable crystallographic configuration, four atom primitive cell was used. In general, the possible non degenerate crystallographic configurations for any quaternary Heusler alloy (QHA) XX’YZ, by keeping Z at 4a site, are:
1. X at 4d, X$'$ at 4c and Y at 4b sites (type I),
2. X at 4b , X$'$ at 4d and Y at 4c sites (type II),
3. X at 4d , X$'$ at 4b and Y at 4c (type III)
Figure \[primitive\] shows the primitive cells of three distinct configurations of CoRuYZ (Y =Mn, V and Z = Ge, Al, Ga) quaternary Heusler alloys.\
To study the Co-Ru defects in CoRuMnGe, a $2\times2\times2$ supercell, formed from a four-atom primitive fcc cell of the most stable configuration has been considered. This supercell contains a total of 32 atoms with 8 atoms of each kind. Brillouin zone integrations were performed using $6^3$ k mesh for 32-atoms cell.
{width="0.9\linewidth"}
![Rietveld refined room temperature XRD patterns for (a) CoRuMnGe, (b) CoRuVAl and (c) CoRuVGa alloys. The insets I and II show the observed and calculated intensities of the superlattice reflections (111) and (200) with perfectly ordered LiMgPdSn-type and disordered ($\mathrm{L2_1}$ in case of CRMG and B2 in case of CRVG and CRVA) structure respectively.[]{data-label="XRD"}](fig2){width="0.7\linewidth"}
Results and Discussion
======================
Structural analysis
-------------------
![Isothermal magnetization curves at 5 K for CoRuMnGe, CoRuVAl and CoRuVGa alloys.[]{data-label="MH"}](fig3){width="0.5\linewidth"}
[l c c c c c c]{} & & & & & &\
Type& $\ $ $a_0$ (Å) $ \ $ & $m^{\mathrm{Co}}$ & $\ $ $m^{\mathrm{Ru}}$ $\ $ & $m^{\mathrm{Mn}}$ & $\ $ $m^{\mathrm{Total}}$ $ \ $ & $\Delta E_{rel}$(eV/atom)\
& & & & & &\
\
I & 5.89 & 0.92 & 0.09 & 3.03 & 4.03 & 0.00\
& & & & & &\
II & 5.81 & 0.75 & -0.08 & 1.05 & 1.70 & 0.25\
& & & & & &\
III & 5.85 & 1.23 & 0.87 & 0.89 & 2.98 & 0.33\
& & & & & &\
\[tab1\]
[l c c c c c c]{} & & & & & &\
Type& $\ $ $a_0$ (Å) $ \ $ & $m^{\mathrm{Co}}$ & $\ $ $m^{\mathrm{Ru}}$ $\ $ & $m^{\mathrm{V}}$ & $\ $ $m^{\mathrm{Total}}$ $ \ $ & $\Delta E_{rel}$(eV/atom)\
& & & & & &\
\
I & 5.88 & 0.72 & 0.02 & 0.28 & 0.96 & 0.00\
& & & & & &\
II & 5.95 & 1.57 & 0.40 & -0.32 & 1.67 & 0.38\
& & & & & &\
III & 5.95 & 1.58 & 0.67 & 0.36 & 2.63 & 0.37\
& & & & & &\
\[tab2\]
[l c c c c c c]{} & & & & & &\
Type& $\ $ $a_0$ (Å) $ \ $ & $m^{\mathrm{Co}}$ & $\ $ $m^{\mathrm{Ru}}$ $\ $ & $m^{\mathrm{V}}$ & $\ $ $m^{\mathrm{Total}}$ $ \ $ & $\Delta E_{rel}$(eV/atom)\
& & & & & &\
\
I & 5.91 & 0.72 & 0.02 & 0.29 & 0.99 & 0.00\
& & & & & &\
II & 5.95 & 1.57 & 0.45 & -0.14 & 1.92 & 0.28\
& & & & & &\
III & 5.96 & 1.53 & 0.73 & 0.35 & 2.64 & 0.32\
& & & & & &\
\[tab3\]
{width="\linewidth"}
![Temperature dependence of electrical resistivity($\rho$) for CoRuMnGe at zero field.[]{data-label="res"}](fig5){width="0.7\linewidth"}
![Temperature dependence of electrical resistivity($\rho$) for CoRuVAl at zero field.[]{data-label="res1"}](fig6){width="0.7\linewidth"}
![Temperature dependence of electrical resistivity($\rho$) for CoRuVGa at zero field.[]{data-label="res2"}](fig7){width="0.7\linewidth"}
Room temperature powder XRD patterns of CoRuMnSi, CoRuVAl and CoRuVGa alloys are shown in Fig. \[XRD\]. From the patterns, it is clear that all the three alloys exhibit cubic crystal structure. Rietveld refinement of the XRD data was done using FullProf Suite software.[@RR] The lattice parameters as deduced from refinement were found to be 5.88, 5.90 and 5.91 $\mathrm{\AA}$ for CRMG, CRVA and CRVG respectively. The quaternary Heusler alloys exhibit LiMgPdSn-type structure whose primitive cell contains four atoms at the Wyckoff positions 4a, 4b, 4c and 4d. For the quaternary Heusler alloy considering X at 4b, $\mathrm{X^\prime}$ at 4c, Y at 4d and Z at 4a Wyckoff positions, the structure factor for the superlattice reflections (111) and (200) can be written as [@PhysRevB.99.104429]
$${F_{111}} = 4[{(f_Y-f_Z)-i(f_X-f_{X'})}]$$
$${F_{200}} = 4[{(f_Y+f_Z)-(f_X-f_{X'})}]$$
where, ${f_X, f_{X'}, f_Y}$ and ${f_Z}$ are the atomic scattering factors for X, X’, Y and Z respectively.
{width="\linewidth"}
![$2\times2\times2$ supercell of type I configuration of CoRuMnGe alloy. []{data-label="CRMG"}](fig9){width="0.7\linewidth"}
![Density of states (DoS) for CoRuMnGe with (a) 12.5%, (b) 25 %, (c) 37.5 % and (d) 50 % Co-Ru swap disorder.[]{data-label="disorder"}](fig10){width="0.7\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
As per the equation (1), in the case of B2 disorder (Y & Z and X & X$^\prime$ atoms are randomly distributed), the intensity of the (111) peak should be low or absent. On the other hand, for a completely disordered structure i.e., A2-type (all the four atoms occupy random positions), both the superlattice peaks (111) and (200) should be absent. Thus, to determine the disorder, the intensities of superlattice peaks (111) and (200) play a crucial role. The insets I of Fig. \[XRD\] (a), Fig. \[XRD\] (b) and Fig. \[XRD\] (c) show the superlattice reflections with perfectly ordered LiMgPdSn-type structure for CRMG, CRVA and CRVG alloy respectively. It is clear that refinement considering ordered Y-type structure did not fit well. In case of CRMG, the XRD pattern fits well when disorder is considered between Co and Ru sites. Figure \[XRD\] (a) shows the rietveld refined XRD pattern for CRMG with 50 % disorder between Co and Ru sites i.e. the 4c and 4d sites are equally probable for Co and Ru atoms. Thus, due to 50% swap disorder between tetrahedral site atoms (as revealed from refinement), the crystal symmetry reduces to L2$_1$. In case of CRVG and CRVA, the absence of (111) peak indicates complete B2 disorder. Figure \[XRD\] (b) and \[XRD\] (c) show the rietveld refined XRD pattern for CRVA and CRVG respectively with 50 % disorder between Co & Ru sites and V & Al sites which results in complete B2 disorder. In this case, 4c & 4d sites are equally probable for Co & Ru atoms and 4a & 4b sites are equally probable for V & Al atoms. The insets II of Fig \[XRD\](b) and Fig \[XRD\](c) show the observed and calculated intensities of the superlattice reflections considering B2 disorder for CRVA and CRVG respectively. Thus, CRMG is found to have L2$_1$ structure, whereas CRVA and CRVG alloys tend to show B2-type disorder.
[c| c c c |c |c |c c]{} & & & & & &\
System & Co & Ru & Mn & $\mathrm{X_d}$ & $\mathrm{m_{total}}$($\mu_B$) &$\mathrm{m_{total}}$($\mu_B$/f.u.)\
\
Ordered & 0.91 & 0.08 & 3.03 & & 32.16 & 4.02\
Co-Ru swap & 0.96 & 0.03 & 3.07 & $\mathrm{Co_{Ru}}$: 1.02,& 32.10 & 4.01\
(12.5%)& 0.95 & 0.01 & 2.97 & $\mathrm{Ru_{Co}}$: -0.19 &\
& 0.97 & 0.04 & 3.09 & & &\
& 1.01 & 0.15 & 2.95 &\
& & -0.03 & 3.08 & &\
Co-Ru swap & 1.00 & -0.01 & 3.08 & $\mathrm{Co_{Ru}}$: 1.05,1.05& 32.08 & 4.01\
(25.0%)& 0.99 & -0.03 & 2.95 & $\mathrm{Ru_{Co}}$: -0.14,-0.15 &\
&0.98 & 0.08 & 3.04 & &\
& &-0.14 & 3.03 & &\
Co-Ru swap & 0.97 & -0.07 & 3.04 & $\mathrm{Co_{Ru}}$: 1.04,1.04,1.05& 32.05& 4.01\
(37.5%)& 1.01 & 0.00 & 3.01 & $\mathrm{Ru_{Co}}$: -0.10, -0.10, -0.11 &\
& 1.04 & 0.01 & 3.05 & &\
& & -0.04 & 2.96 & &\
& & & 3.03 & &\
Co-Ru swap & 1.05 & -0.06 & 3.04 & $\mathrm{Co_{Ru}}$: 1.03,1.04,1.05,1.03& 32.07 & 4.01\
(50%)& 1.04& -0.08 & 3.04 & $\mathrm{Ru_{Co}}$: -0.06,-0.06,-0.08,-0.06 &\
& 1.03 & -0.06 & 3.03 & &\
& & & 3.01 & &\
\[MM\]
[l c c c c c c]{} & & & & & &\
Alloy& $a_{exp}$ & ${M_S}$(S-P) & ${M_S}$(Theor.) & ${M_S}$(Exp) & P (Theor.) &\
& $(\mathrm{\AA)}$ & $({\mu_B}/f.u.$) & $({\mu_B}/f.u.$) & (${\mu_B}/f.u.$) & (%) &\
& & & & & &\
\
CRMG & 5.88 & 4.0 & 4.03 & 4.10 & 91\
& & & & & &\
CRVA & 5.90 & 1.0 & 0.96 & 0.53 & 89\
& & & & & &\
CRVG & 5.91 & 1.0 & 0.99 & 0.84 & 93\
& & & & & &\
\[tab4\]
Magnetic Properties
-------------------
Figure \[MH\] shows the variation of magnetization with field (M-H) at 5 K for CRMG, CRVA and CRVG. Quaternary Heusler alloys are known to follow the Slater-Pauling (S-P) rule according to which the total magnetic moment is directly related to the number of valence electrons in the unit cell as per the relation, [@PhysRevB.66.174429] $$M = ( N_v - 24 )\hspace{0.1cm}\mu_B/f.u.$$ CRMG has 28 valence electrons, whereas CRVA and CRVG have 25 valence electrons. Thus, as per the S-P rule CRMG, CRVA and CRVG are expected to have a magnetic moment of 4, 1 and 1 $\mu_B/f.u.$ respectively. From the M-H curves as shown in Fig. \[MH\], the calculated saturation magnetization values are found to be 4.10, 0.53 and 0.84 $\mu_B/f.u.$ for CRMG, CRVA and CRVG respectively. The small deviation in the magnetic moment from the S-P value in case of CRMG and CRVG alloys is due to the presence of small density of states in the minority spin channel at the Fermi level (See section [4.4]{} for details). This indicates a possibility of nearly half-metallic character in CRMG and CRVG alloys. A large deviation in the case of CRVA, may be attributed to the disorder. All the three alloys are found to be soft ferromagnetic with negligible hysteresis.\
Figure \[MT\] shows the thermo-magnetic (M-T) curves recorded under an applied field of 100 Oe. The Curie temperature, T$\mathrm{_C}$ is estimated by taking the minima of the $\frac{dM}{dT}$ vs. T curves. T$\mathrm{_C}$ for CRMG, CRVA and CRVG is found to be 560, 168 and 150 K respectively. A small increase in the magnetic moment in the lower temperature range is possibly due to anisotropic variations with temperature, since the M vs. T curves are measured in low fields of 100 Oe. Among the three alloys, CRMG is found to have the highest T$\mathrm{_C}$ and thus is suitable for room temperature spintronic applications.
Transport properties
--------------------
**1. CoRuMnGe**\
Figure \[res\] shows the temperature dependence of electrical resistivity ($\rho_{xx}$) for CRMG in the temperature range 5 - 350 K at zero field. The resistivity increases with temperature indicating the metallic nature. To further investigate the $\rho_{xx}$ vs. T behaviour, the resistivity curve was fitted using the power law given by: $$\rho(T) = \rho_{0}+ \rho(T) = \rho_{0} + BT^n$$ In previous reports on half-metallic Heusler compounds, different values of n are reported depending on the temperature range considered.[@PhysRevB.96.184404; @0022-3727-40-6-S01; @0022-3727-37-15-001; @doi:10.1063/1.126606; @PhysRevApplied.10.054022] In the temperature region $\mathrm{35 K < T < 100 K}$ (Region I), the value of n was found to be 1.35 whereas, for T $>$ 100 K (Region II), the resistivity varies almost linearly with n = 1.06. The linear dependence in the region II can be attributed to the electron-phonon scattering. In a half-metal, there are no minority spin charge carriers at E$_{F}$ due to complete spin polarization and thus, spin-flip scattering is usually not possible.[@1989JPCM1.2351O; @Kubo-1972] Due to this, the $\mathrm{T^2}$ term related to single magnon scattering is expected to be absent in resistivity. From the value of n, we can conclude that the dependence is not quadratic which indirectly confirms the half-metallic nature in CRMG alloy. Also, the residual resistivity and the residual resistivity ratio (RRR = ${\rho_{300 K} / \rho_{5 K}}$) values are found to be 1.75 $\mathrm{\mu \Omega}$ m and 1.63 respectively. The obtained RRR value is highest among the three alloys under study, indicating the possibility of least disorder.\
**2. CoRuVAl**\
Figure \[res1\] shows the temperature dependence of electrical resistivity ($\rho_{xx}$) for CRVA in the temperature range of 5 - 350 K at zero field (the arrow indicates the Curie temperature). The alloy shows metallic nature in both ferromagnetic and paramagnetic regions. It should be noted that below T$\mathrm{_C}$, the change in resistivity is mainly due to magnetic scattering, whereas in the paramagnetic state the magnetic component of resistivity saturates. Thus, for $\mathrm{T > T_C}$ (169 K), the change in resistivity with temperature is determined by electron-phonon scattering only, which is usually observed as a linear contribution in $\rho$(T) above the Debye temperature. Inset of Fig. \[res1\] shows the linear behaviour of $\rho$ vs. T for $\mathrm{T > T_C}$. The residual resistivity is found to be 1.25 $\mathrm{\mu \Omega}$ m and the RRR value is found to be 1.27.\
**3. CoRuVGa**\
Figure \[res2\] shows the temperature dependence of resistivity ($\rho_{xx}$) for CRVG in the temperature range of 5 - 350 K at zero field. In this case, the resistivity behavior is significantly different and shows unconventional features like: (a) large value of residual resistivity ($\rho_0$ = 575 $\mathrm{\mu \Omega}$ cm), (b) anomaly in the form of a maxima below T$\mathrm{_C}$, and (c) presence of region with semiconducting behavior i.e., with a negative coefficient of resistivity in the ferromagnetic state. The two main factors which determine the resistivity behavior in Heusler alloys are (1) conduction electron scattering mechanism and (2) effect of magnetic ordering on the electronic band structure near the Fermi level. In the low temperature region ($\mathrm{T < 80 }$ K), the resistivity decreases with decreasing temperature and hence shows the metallic nature. This decrease in resistivity can be attributed to the reduction in the magnetic scattering with decreasing temperature. Anomaly in the form of a maximum seen in resistivity in the magnetically ordered state is not something new. It has been observed in a few systems that, the effect of change in electronic structure strongly reflects in the temperature dependence of resistivity. This causes an anomaly in the form of a maximum because of the superposition of the electron-phonon and the magnetic contributions, when an abrupt decrease in magnetic contribution induced by vanishing spontaneous magnetization is superimposed by a linear increase in phonon contribution.[@DC] The electron-transport behavior in half-metallic ferromagnetic Heusler alloy Co$_2$CrGa was found to show a similar behavior as in our system CRVG. [@Kourov2013; @KOUROV2015839] Below the Curie temperature, the change in resistivity is mainly caused by the magnetic contributions. In case of half-metallic ferromagnets, the magnetic contribution to the conductivity is determined by considering two parallel conduction channels for electrons, one with spin-up and the other with spin down and the total magnetic contribution of conductivity can be written as, [@Kourov20131] $$\sigma_m = \sigma_{\downarrow} + \sigma_{\uparrow} \hspace{0.5cm}\mathrm{or} \hspace{0.5cm} \rho_m = \frac{\rho_{\downarrow} \rho_{\uparrow}} {\rho_{\downarrow} + \rho_{\uparrow}}$$ The conductivity of spin-up electrons is determined mainly by the scattering of the charge carriers. For the spin down electrons, the conductivity is dependent on the energy gap parameters in the electronic spectrum. Also, the energy gap parameters mainly depend on the spontaneous magnetization ($\mathrm{M_S}$).[@DC; @Kourov20131] At $\mathrm{T \ll T_C}$, $\mathrm{M_S}$ does not vary much with temperature and thus, the energy gap parameters and conductivity of spin down channel remain almost constant. Due to this, at low temperature, the variation in resistivity is mainly due to conduction of spin up electrons. At T $\rightarrow$ $\mathrm{T_C}$, the saturation magnetization vanishes which results in the disappearance of energy gap in the spin down channel and hence increase in its conductivity ($\mathrm{\sigma_{\downarrow}}$). As a result, $\mathrm{\rho_m}$ decreases. And thus, we observe a region of negative TCR near the Curie temperature. A negative TCR value has been reported in materials like V and Ti doped $\mathrm{Fe_3Ga}$, [@PhysRevB.44.12406] Ti, Mn, Cr doped $\mathrm{Fe_3Si}$ [@PhysRevB.48.13607] and was speculated to arise due to the existence of small electronic density at $\mathrm{E_F}$ and large spin disorder scattering. Similar behavior is also reported in various half-metallic ferromagnetic Heusler alloys where a region of negative TCR is observed and was explained on the basis of presence of a gap in the electronic spectrum near E$_F$.[@Kourov2013; @PhysRevB.72.012417; @Kourov2016; @MARCHENKOV2018211] It has also been observed that a negative TCR is generally seen in alloys when the electrical resistivity $\rho > 1.5 \mu \Omega$ m in the paramagnetic state.[@PhysRevB.44.12406; @PhysRevB.48.13607] For CoRuVGa, the resistivity in the paramagnetic state lies in the range of 5.79 to 6.20 $\mathrm{\mu \Omega}$ m and thus it fulfills this condition to show anomaly in the resistivity near the Curie temperature. In the paramagnetic state \[$\mathrm{T > T_C}$ (150 K), the resistivity increases almost linearly with temperature which can be attributed to the electron-phonon scattering since in this region $\mathrm{\rho_m}$ is constant. The residual resistivity value is found to be 5.75 $\mathrm{\mu \Omega}$ m, which is the highest among the three alloys under study. The RRR is found to be low (1.06) as compared to other two alloys.
Theoretical Results
-------------------
The magnetic configuration of the alloys was examined by simulating different initial magnetic (para-, ferro- and ferrimagnetic) arrangements. The results of the structural optimization for CRMG, CRVA and CRVG are displayed in Table \[tab1\], \[tab2\] and \[tab3\] respectively. It showed that all the three alloys are stable in configuration I with ferromagnetic ordering as it exhibits the lowest total energy. The most stable, (Type I) configuration has Co at 4d (0.75,0.75,0.75), Ru at 4c (0.25,0.25,0.25), Y (Mn,V) at 4b (0.5,0.5,0.5) and Z (Ge,Ga,Al) at 4a (0,0,0) sites. It is seen that, in case of CRMG, most of the magnetic moment is contributed by Mn (3 $\mu_B$), whereas Co contributes a small moment (1 $\mu_B$) and Ru has negligible moment. In case of CRVA and CRVG, moments are mainly contributed by Co and V, whereas Ru has negligible moment. To further study the electronic structure and magnetic properties of these alloys, the energetically most favourable Type I structure was used. Figure \[dos\] shows the calculated spin resolved band structure and density of states (DoS) for CRMG alloy, calculated at the experimental lattice parameter ($a_{elp}$). The alloy manifests electronic structure of a highly spin-polarized, nearly half-metal. The ideal half metal acquires 100 % spin polarization. In this case, due to the presence of small number of states (0.11 states/eV/f.u.) at E$\mathrm{_F}$ for the minority spin channel, the alloy has slightly low spin polarization(P = 91 %). The calculated moment is 4.03 $\mu_B/f.u.$, which is close to that of Slater-Pauling value (4 $\mu_B/f.u.$).
Going back to our XRD results, it was found that 4c and 4d fcc sites in CoRuMnGe are equally possible for Co and Ru atoms (see section 4.1). In order to get a deeper insight into the properties of this actual experimental structure, one should simulate a mixture of Co and Ru atoms at these two sites. One way to do this is to swap the position of Co and Ru atoms at 4c and 4d sites. To simulate such intrinsic disorder, a $2\times2\times2$ supercell of the primitive cell of the type I configuration of CRMG is constructed (See Fig. \[CRMG\]). In a $2\times2\times2$ supercell with 32 atoms (eight formula units), 12.5% swap disorder was simulated by exchanging one of the eight Co atom positions and one of the eight Ru atom positions. Similarly, 50% swap disorders were simulated by exchanging four of the eight Co atoms and four of the eight Ru atom positions. All possible configurations for replacement of Co by Ru and vice versa was checked, and energetically the most stable configuration is chosen to present the result here. The Co-Ru swaps almost give the same total magnetic moment as the ideal (no swap) case. The calculated DoS for disordered structures are shown in Fig. \[disorder\]. As seen from Fig. \[disorder\], half-metallicity in CoRuMnGe is quite robust against swapping disorder between Co ad Ru sites. In the ordered Y-type structure, Co and Ru atoms are surrounded by four Al and 4 Mn atoms as their nearest neighbors. On considering 50 % swap disorder between Co and Ru sites, which results in the disordered $L2_1$ type structure, it is seen that the local environment of Co and Ru atoms remains the same i.e. they are still surrounded by four Al and 4 Mn atoms, however there is a slight change in the bond length. It turns out that, with swap disorder, the local environment of the defected sites does not affect the exchange interaction much but causes a little change in the bond length (due to relaxation effect). This, in turn, changes the local moments (and hence the total cell moment) by a little amount only. Thus, in case of CoRuMnGe, the swap disorder between Co and Ru sites does not change the total magnetic moment as well as the density of states. To have a better understanding, we have calculated the local moments at the individual atomic sites, the values of which are shown in Table \[MM\]. Here $\mathrm{X_d}$ refers to the sites where Co and Ru atoms are swapped. In each case, the magnetic moment of Ru swapped with Co atom is changing from positive to negative (making it antiferromagnetically aligned) as compared to the ideal case,however Ru has negligible moment in both case. Also, the magnetic moment of Co swapped with Ru atoms slightly increases. Co-Ru swap defected structures almost maintain the same total magnetic moment as the ideal structure i.e. $\sim$ 4.0 $\mu_B$/f.u.\
The DoS plots for CRVA and CRVG alloys are shown in Fig. \[CRVA\_band\] and Fig. \[CRVG\_band\] respectively, which reveal a nearly half-metallic character. The spin polarization is found to be 89 and 93 % for CRVA and CRVG respectively. In the minority spin channel, a small number of states ($\mathrm{N_{\downarrow}}$ = 0.20 states/eV/f.u. for CRVA and 0.13 sates/eV/f.u. for CRVG) are present at $\mathrm{E_F}$, which is responsible for a slightly lower spin polarization as compared to the ideal half-metal. The magnetic moment is close to the expected value as per Slater-Pauling rule. To have a one-to-one comparison, the experimental lattice parameters, theoretical, simulated and experimental magnetic moments and simulated spin polarizations are tabulated in Table \[tab4\]. It is seen that the experimentally observed moment for CRVA is quite different from the calculated value. To understand the reason behind such difference, we have performed swap disorder and anti-site disorder calculations for CRVA, details of which are provided in the supplementary material.[@RR3] The presence of such disorder is justified in CRVA because of the B2-type disorder as predicted from XRD data. As in the case of CoRuMnGe, the swap disorder does not alter the net magnetic moment for CRVA as well and the total moment still remains close to 1.0 $\mu_B /f.u.$. The total magnetic moment, however, are extremely sensitive to the anti-site disorder (antisite between Co & Ru and V & Al). Interestingly, due to the change in the local environment in this case, the local atomic moments change drastically; sometimes even get quenched and/or antiferromagnetically aligned as compared to the completely ordered case. The latter is attributed to the itinerant character of magnetism in Co, Ru, and V-containing Heusler alloys, and to the frustration of antiferromagnetic exchange interactions, possibly accompanied by a tetragonal distortion. In fact, the exchange interaction in this case is reasonably long ranged, affecting the moment of atoms sitting far from the defected sites. Table S2 of the supplement file shows the simulated results for a $2\times2\times2$ supercell which reduces the net cell moment up to 0.57 $\mu_B /f.u.$, when antisite disorder between both (Co & Ru) and (V & Al) pairs are considered. Similar behavior has been reported in other Heusler alloys, where swap/anti-site disorder causes a decrease in moment.[@doi:10.1063/1.4972797; @JOUR; @doi:10.1063/1.4998308] When comparing CoRuVGa and CoRuVAl, the electronegativity value of Al (1.61) is similar to that of V(1.63). Due to this, there is a high probability of Al atom to occupy one of the octahedral sites (1/2, 1/2, 1/2) i.e. the disorder between V and Al is much more probable to occur due to their similar electronegativity values. This, however, is not the case for Ga (1.81) and V(1.63) in CoRuVGa. This is probably the reason for a stronger swap/antisite disorder in CRVA as compared to CRVG and hence a larger reduction in the moment in the former as compared to later.
Conclusion
==========
In conclusion, we have performed a detailed experimental and theoretical study on the structural, magnetic, transport and electronic properties of CoRu- based quaternary Heusler alloys CoRuMnGe, CoRuVAl and CoRuVGa. All the three alloy were found to crystallize in cubic structure. In the case of CRMG, the XRD analysis reveals that the tetrahedral sites (Co and Ru) are equally probable, due to which its crystal symmetry is reduced to L2$_1$. On the other hand, for the other two alloys CRVA and CRVG, the absence of (111) superlattice reflection gives the indication of B2-type disorder. For CRMG and CRVG, the experimental magnetic moment values are in good agreement with the theory as well as Slater-Pauling rule and for CRVA, a relatively large deviation is found. The reduction in the magnetic moment in case of CoRuVAl possibly arises due to the anti-site disorder between Co & Ru sites and V & Al sites. The resistivity measurements indirectly support the half-metallic behavior in CRMG alloy. The highest T$\mathrm{_C}$ of 560 K was found for CRMG alloy. CRVA was found to be metallic in the ferromagnetic as well as paramagnetic state. A strong dependence of magnetic and electronic structure is seen in the temperature dependence of resistivity for CRVG which shows a maximum and a region of negative TCR value in the ferromagnetic region. The ab intio calculations predict the nearly half-metallic ferromagnetic state with high spin polarization in all these alloys. Thus, nearly half-metallic character, high T$\mathrm{_C}$ and high spin polarization makes CoRuMnGe alloy promising for room temperature spintronic applications.
Acknowledgment {#acknowledgment .unnumbered}
==============
DR thank Council of Scientific and Industrial Research (CSIR), India for providing Senior Research Fellowship. AA acknowledge National Center for Photovoltaic Research and Education (NCPRE), IIT Bombay, India for possible funding to support this research.
References {#references .unnumbered}
==========
|
---
abstract: 'We study the effect on CPP GMR of changing the order of the layers in a multilayer. Using a tight-binding simple cubic two band model ($s$-$d$), magneto-transport properties are calculated in the zero-temperature, zero-bias limit, within the Landauer-Büttiker formalism. We demonstrate that for layers of different thicknesses formed from a single magnetic metal and multilayers formed from two magnetic metals, the GMR ratio and its dependence on disorder is sensitive to the order of the layers. This effect disappears in the limit of large disorder, where the results of the widely-used Boltzmann approach to transport are restored.'
address:
- |
School of Physics and Chemistry, Lancaster University, Lancaster, LA1 4YB, UK and\
DERA, Electronics Sector, Malvern, Worcs. WR14 3PS UK
- 'School of Physics and Chemistry, Lancaster University, Lancaster, LA1 4YB, UK'
- 'DERA, Electronics Sector, Malvern, Worcs. WR14 3PS, UK'
author:
- 'S. Sanvito[^1],'
- 'C.J. Lambert[^2],'
- 'J.H. Jefferson'
title: 'Breakdown of the resistor model of CPP-GMR in magnetic multilayered nanostructures'
---
[2]{}
[**PACS**]{}: 73.23-b, 75.70-i, 75.70Pa
Giant magnetoresistance (GMR) in transition metal magnetic multilayers [@gmr1; @gmr2] is a spin filtering effect which arises when the magnetizations of adjacent layers switch from an anti-parallel (AP) to a parallel (P) alignment. The resistance in the anti-aligned state is typically higher than the resistance with parallel alignment, the difference being as large as 100%. This sensitive coupling between magnetism and transport allows the development of magnetic field sensors with sensitivity far beyond that of conventional anisotropic magnetoresistance (AMR) devices. In the most common experimental setup, the current flows in the plane of the layers (CIP), and the resistance is measured with conventional multi-probe techniques. Measurements in which the current flows perpendicular to the planes (CPP) are more delicate because of the small resistances involved. Despite these difficulties the use of superconducting contacts [@msu], sophisticated lithographic techniques [@phil], and electrodeposition [@el1; @el2; @el3], makes such measurements possible (for recent reviews see references [@bau; @ans]).
A widely adopted theoretical approach to GMR is based on the semi-classical Boltzmann equation within the relaxation time approximation. This model has been developed by Valert and Fert [@fer1; @fert2], and has the great advantage that the same formalism describes both CIP and CPP experiments. In the limit that the spin diffusion length $l_{\mathrm sf}$ is much larger than the layer thicknesses (ie in the infinite spin diffusion length limit), this model reduces to a classical two current resistor network, with additional possibly spin-dependent scattering at the interfaces [@msu2]. Despite the undoubted success of this description recent experiments [@chris2; @msuord] have drawn attention to the possibility of new features which lie outside the theory. Two important and central predictions of this model are that the CPP GMR ratio is independent of the number of bilayers in the case that the total multilayer length is not constrained to be constant, and furthermore is independent of the order of the magnetic layers in the case of different magnetic species. An apparent violation of the first prediction has been observed in CIP and CPP measurements [@chris; @chris2], and of the second prediction in CPP measurements [@chris2; @msuord]. However a convincing theoretical explanation is lacking.
The aim of this letter is to provide a quantitative description of the breakdown of the resistor model in diffusive CPP multilayers in the limit of infinite spin-relaxation length. To illustrate this breakdown, consider a multilayer consisting of two independent building blocks, namely a (N/M) and a (N/M$^\prime$) bilayer, where M and M$^\prime$ represent magnetic layers of different materials or of the same material but with different thicknesses and N represents normal metal ‘spacer’ layers. From an experimental point of view M and M$^\prime$ must possess different coercive fields, in order to allow AP alignment. In the case of [@msuord] this is achieved by considering respectively Co and Ni$_{84}$Fe$_{16}$ layers with Ag as non-magnetic spacer, while in [@chris2] both the layers are Co (with Cu as spacer) but with different thicknesses (respectively 1nm and 6nm). Two kinds of multilayer can be deposited. The first, that we call type I, consists of a (N/M/N/M$^\prime$)$\times \mu$ sequence where the species M and M$^\prime$ are separated by an N layer and the group of four layers is repeated $\mu$ times. The second, that we call type II, consists of a (N/M)$\times \mu$(N/M$^\prime$)$\times \mu$ sequence, where the multilayers (N/M)$\times \mu$ and (N/M$^\prime$)$\times \mu$ are arranged in series. If the coercive fields of M ($H_{M}$) and M$^\prime$ ($H_{M^\prime}$) are different (eg $H_{M}<H_{M^\prime}$) and if N is long enough to decouple adjacent magnetic layers, the AP configuration can be achieved in both type I and type II multilayers by applying a magnetic field $H$ whose intensity is $H_{M}<H<H_{M^\prime}$. The AP configuration is topologically different in the two cases, because in type I multilayers it consists of AP alignment of adjacent magnetic layers (conventional AP alignment), while in type II multilayers it consists of the AP alignment between the (N/M)$\times \mu$ and (N/M$^\prime$)$\times \mu$ portions of the multilayer, within which the alignment is parallel (see figure \[ord-sc\]a and figure \[ord-sc\]b). From the point of view of a resistor network description of transport, the two configurations are equivalent, because they possess the same number of magnetic and non-magnetic layers, and the same number of N/M and N/M$^\prime$ interfaces. Hence the GMR ratio must be the same. In contrast the GMR ratio of type I multilayers is found experimentally to be larger than that of type II multilayers [@chris2; @msuord], and the difference between the two GMR ratios increases with the number of bilayers. In the case of [@chris2] the GMR ratio of both type I and type II multilayers increases with the number of bilayers, which again lies outside the resistor network model.
In this Letter we demonstrate for the first time that a description which incorporates phase-coherent transport over long length scales can account for such experiments. To illustrate this we have simulated type I and type II multilayers using a Co/Cu system with different thicknesses for the Co layers, namely $t_{\mathrm Cu}=10$AP, $t_{\mathrm Co}=10$AP, $t_{\mathrm Co}^\prime=40$AP. The technique for computing transport properties is based on a three dimensional simple cubic tight-binding model with nearest neighbor couplings and two degrees of freedom per atomic site. The general spin-dependent Hamiltonian is
$$H^\sigma=\sum_{i,\alpha}\epsilon^{\alpha\sigma}_{i}
c_{\alpha i}^{\sigma\dagger} c^\sigma_{\alpha i}
+\sum_{i,j,\alpha\beta}\gamma^{\alpha\beta\sigma}_{ij}
c_{\beta j}^{\sigma\dagger} c^\sigma_{\alpha i}
\;{,}
\label{spham}$$
where $\alpha$ and $\beta$ label the two orbitals (which for convenience we call $s$ and $d$), $i,j$ denote the atomic sites and $\sigma$ the spin. $\epsilon^{\alpha\sigma}_{i}$ is the on-site energy which can be written as $\epsilon^{\alpha}_{i}=\epsilon_0^{\alpha}+
\sigma h \delta_{\alpha {d}}$ with $h$ the exchange energy and $\sigma=-1$ ($\sigma=+1$) for majority (minority) spins. In equation (\[spham\]), $\gamma^{\alpha\beta\sigma}_{ij}=\gamma^{\alpha\beta}_{ij}$ is the hopping between the orbitals $\alpha$ and $\beta$ at sites $i$ and $j$, and $c^\sigma_{\alpha i}$ ($c_{\alpha i}^{\sigma\dagger}$) is the annihilation (creation) operator for an electron at the atomic site $i$ in an orbital $\alpha$ with a spin $\sigma$. $h$ vanishes in the non-magnetic metal, and $\gamma^{\alpha\beta}_{ij}$ is zero if $i$ and $j$ do not correspond to nearest neighbor sites. Hybridization between the $s$ and $d$ orbitals is taken into account by the non-vanishing term $\gamma^{sd}$. We have chosen to consider two orbitals per site in order to give an appropriate description of the density of states of transition metals and to take into account inter-band scattering occurring at interfaces between different materials. The DOS of a transition metal consists of a narrow band (mainly $d$-like) embedded in a broader band (mainly $sp$-like). This feature can be reproduced in the above two band model, as shown in reference [@noidd], where the appropriate choices for $\gamma^{\alpha\beta}_{ij}$ and $\epsilon^{\alpha}_{i}$ in Cu and Co are discussed.
We analyze the simplest generic model of disorder, introduced by Anderson within the framework of the localization theory [@and], which consists of adding a random potential $V_i$ to each on-site energy, with a uniform distribution of width $W$ ($-W/2\leq V \leq W/2$), centered on $V=0$
$$\tilde{\epsilon}^{\alpha\sigma}_{i}=\epsilon^{\alpha\sigma}_{i}+V
\;{.}
\label{and}$$
The conductances and GMR ratios are calculated within the Landauer-Büttiker theory of transport [@but] using a technique already presented elsewhere [@noi]. In figure \[leed\] we present the mean GMR ratio for type I (type II) multilayers GMR$_{\mathrm I}$ (GMR$_{\mathrm II}$) and the difference between the GMR ratios of type I and type II multilayers $\Delta$GMR=GMR$_{\mathrm I}$-GMR$_{\mathrm II}$, as a function of $\mu$ for different values of the on-site random potential. The average has been taken over 10 different random configurations except for very strong disorder where we have considered 60 random configurations. In the figure we display the standard deviation of the mean only for $\Delta$GMR because for GMR$_{\mathrm I}$ and GMR$_{\mathrm II}$ it is negligible on the scale of the symbols. It is clear that type I multilayers possess a larger GMR ratio than type II multilayers, and that both the GMR ratios and their difference increase for large $\mu$. These features are in agreement with experiments [@chris2; @msuord] and cannot be explained within the standard Boltzmann description of transport. The increase of the GMR ratio as a function of the number of bilayers is a consequence of enhancement of the spin asymmetry of the current due to disorder. In fact, despite the Anderson potential being spin-independent it will be more effective on the $d$ band than on the $s$ band, because the former possesses a smaller bandwidth. Since the minority spin sub-band is dominated by the $d$-electrons and the majority by the $s$-electrons, the disorder will suppress the conductance more strongly in the minority band than in the majority. Moreover, since the transport is phase-coherent, the asymmetry builds up with the length, resulting in a length-dependent increase of the GMR ratio. The different GMR ratios of type I and type II multilayers can be understood by considering the inter-band scattering. Both multilayers possess the same conductance in the P alignment, while the conductance of type I multilayer in the AP alignment is smaller than that of type II. The inter-band scattering is very strong when an electron crosses phase-coherently a region where the magnetizations have opposite orientations, and this occurs in each (N/M/N/M$^\prime$) cell for type I multilayers, while only in the central cell for type II multilayer (see figure \[ord-sc\]a and \[ord-sc\]b). Hence the contribution to the resistance in the AP alignment due to inter-band scattering is larger in type I than in type II multilayers. Finally when the elastic mean free path is comparable with a single Co/Cu cell one expects the resistor model to become valid. To illustrate this feature, figure \[leed\] shows that in the case of very large disorder ($W=1.5$eV), $\Delta$GMR vanishes within a standard deviation as predicted by the Valert and Fert theory.
As a second example in which the dependence of the GMR ratio on disorder changes when the multilayer geometry is varied, consider the system whose AP alignment is sketched in figure \[ord-sc\]c and \[ord-sc\]d. In this case M and M$^\prime$ are different materials chosen in such a way that the minority (majority) band of M possesses a good alignment with the majority (minority) band of M$^\prime$. Moreover the thickness of the N layers has been chosen in order to allow an AP alignment of the magnetizations of adjacent magnetic layers in both type I and type II multilayers. In this case both type I and type II multilayers exhibit conventional P and AP alignments, but their potential profile is quite different. In figure \[prof\] we present a schematic view of the potential profiles for type I and type II multilayers for both the spins in the P and AP configuration. A high barrier corresponds to large scattering and a small barrier corresponds to weak scattering. The dashed line represents the effective potential for material M and and the continuous line for material M$^\prime$. Figure \[prof\] illustrates that type I multilayers possess a high transmission spin-channel in the AP alignment, and hence the resulting GMR ratio will be negative. In contrast type II multilayers do not possess a high transmission channel (there are large barriers for all spins in both the P and AP configuration) and the sign of the GMR ratio will depend on details of the band structure of M and M$^\prime$. Consider the effects of disorder on these two kinds of multilayers. Using the same heuristic arguments as above we expect that the GMR ratio of type I multilayers will increase (become more negative) as disorder increases, in the case of disorder that changes the spin asymmetry of the current. This is a consequence of the fact that, in common with the conventional single-magnetic element, one of the spin sub-bands in the AP alignment is dominated by weak $s$-electrons (small barrier), which are only weakly affected by disorder. It is clear that this system is entirely equivalent to conventional single-magnetic element multilayers discussed above. In contrast for type II multilayers there are no spin sub-bands entirely dominated by the weak scattering (small barriers) $s$-electrons, and all spins in either the P and AP configuration will undergo scattering by the same number of high barriers. In this case the effect of disorder will be to increase all the resistances and this will result in a suppression of GMR. Moreover it is important to note that in the completely diffusive regime, where the resistances of the different materials may be added in series, the GMR ratio will vanish if $R_{\mathrm M}^{\uparrow(\downarrow)}\sim
R_{\mathrm M^\prime}^{\downarrow(\uparrow)}$, where $R_{\mathrm A}^{\uparrow(\downarrow)}$ is the spin-dependent resistance of the material A. To verify this prediction we have simulated both type I and type II multilayers using the parameters corresponding to Co and Fe$_{72}$V$_{28}$ of reference [@noidd], respectively for M and M$^\prime$, and corresponding to Cu for N. This choice was motivated by the fact that a reverse CPP-GMR has been obtained for (Fe$_{72}$V$_{28}$/Cu/Co/Cu)$\times\mu$ multilayers [@msufer]. The GMR ratio for type I and type II multilayers is shown in figure \[inverse\], which illustrates the remarkable result that the GMR ratio of type I multilayers increases with disorder, while for type II structures it decreases. As explained above this is due to an enhanced asymmetry between the conductances in the P and AP alignment for type I multilayers, and to a global increase of all the resistances for type II multilayers. As far as we know there are no experimental studies of the consequences of the geometry-dependent effect described above, and further investigation will be of interest, in order to clarify the rôle of the disorder in magnetic multilayers.
Despite the fact that GMR was discovered more than ten years ago, it continues to present fascinating insights into transport in magnetic heterostructures. In this Letter we have addressed a new issue which lies outside the widely-adopted Boltzmann description of GMR, namely that changing the order of magnetic multilayers can significantly alter the magnetoresistance [@chris2; @msuord]. We have shown that this effect is a consequence of phase coherence on a length scale greater than the layer thicknesses.
[**Acknowledgments**]{}: The authors want to thank D.Bozec, C. Marrows, B.Hickey and M.Howson from the University of Leeds for their suggestions and for the permission to discuss results not yet published. This work is supported by the EPSRC, the EU TMR Programme and the DERA.
M.N.Baibich, J.M.Broto, A.Fert, F.Nguyen Van Dau, F.Petroff, P.Etienne, G.Creuzet, A.Friederich, and J.Chazelas, Phys. Rev. Lett. [**61**]{}, 2472 (1988); G.Binasch, P.Grünberg, F.Sauerbach and W.Zinn, Phys. Rev. [**B 39**]{}, 4828 (1989) W.P.Pratt Jr., S.-F.Lee, J.M.Slaughter, R.Loloee, P.A.Schroeder, J.Bass, Phys. Rev. Lett. [**66**]{}, 3060 (1991) M.A.M.Gijs, S.K.J.Lenczowski, J.B.Giesbers, Phys. Rev. Lett. [**70**]{}, 3343 (1993) W.Schwarzacher, D.S.Lashmore, IEEE Trans. Magn. [**32**]{}, 3133 (1996) A.Blondel, J.P.Meier, B.Doudin, J.Ph.Ansermet, Appl. Phys. Lett. [**65**]{}, 3019 (1994) L.Piraux, S.Dubois, A.Fert, L.Belliard, Eur. Phys. J. [**B 4**]{}, 413 (1998) M.A.M.Gijs, G.E.W.Bauer, Adv. Phys., [**46**]{}, 285 (1997) J-Ph. Ansermet, J.Phys.: Cond. Matter [**C 10**]{}, 6027 (1998) T.Valet, A. Fert, Phys. Rev. [**B 48**]{}, 7099 (1993) A. Fert, J.-L. Duvail, T.Valet, Phys.Rev. [**B 52**]{}, 6513 (1995) S.-F.Lee, W.P.Pratt Jr., R.Loloee, P.A.Schroeder, J.Bass, Phys. Rev. [**B 46**]{}, 548 (1992) D. Bozec, C. Marrows, B. Hickey, M. Howson, private communication W.-C. Chiang, Q. Yang, W. P. Pratt Jr., R. Loloee, J. Bass, J. Appl. Phys. [**81**]{}, 4570 (1997) C. Marrows, PhD dissertation, University of Leeds, (1997) S.Sanvito, C.J.Lambert, J.H.Jefferson, preprint P.W.Anderson, D.J.Thouless, E. Abrahams, D.S.Fisher, Phys.Rev. [**B 22**]{}, 3519 (1980) M.Büttiker, Y.Imry, R.Landauer, and S.Pinhas, Phys. Rev. [**B 31**]{}, 6207 (1985) S.Sanvito, C.J.Lambert, J.H.Jefferson, and A.M.Bratkovsky, accepted for publication in Phys.Rev. [**B**]{}, also cond-mat/9808282 S.Y. Hsu, A. Barthélémy, P. Holody, R. Loloee, P.A. Schroeder, A. Fert, Phys. Rev. Lett. [**78**]{}, 2652 (1997)
[^1]: e-mail: sanvito@dera.gov.uk
[^2]: e-mail:c.lambert@lancaster.ac.uk
|
---
abstract: 'We revisit the problem of decay of a metastable vacuum induced by the presence of a particle. For the bosons of the ‘master field’ the problem is solved in any number of dimensions in terms of the spontaneous decay rate of the false vacuum, while for a fermion we find a closed expression for the decay rate in (1+1) dimensions. It is shown that in the (1+1) dimensional case an infrared problem of one-loop correction to the decay rate of a boson is resolved due to a cancellation between soft modes of the field. We also find the boson decay rate in the ‘sine-Gordon staircase’ model in the limits of strong and weak coupling.'
---
[**William I. Fine Theoretical Physics Institute\
University of Minnesota\
**]{}
FTPI-MINN-05/46-T\
UMN-TH-2418-05\
ITEP-TH-63/05\
November 2005\
[**Particle decay in false vacuum\
**]{} and [**M.B. Voloshin\
**]{} William I. Fine Theoretical Physics Institute, University of Minnesota,\
Minneapolis, MN 55455\
and\
Institute of Theoretical and Experimental Physics, Moscow, 117259\
Introduction
============
The decay of a metastable vacuum state is a quite universal problem in quantum field theory. The decay proceeds through nucleation and subsequent classical expansion of the bubbles of the true vacuum. The classical bubbles can exist only starting from a certain critical radius at which the energy loss due to the surface terms is compensated by the gain in the volume energy. The formation of the critical bubbles is thus a quantum tunneling process[@vko] which tunneling can be described by an Euclidean-space configuration of the field, called a ‘bounce’[@cc1]. The space-time nucleation rate of the critical bubbles, $w_0$, i.e. the probability of such nucleation per unit time and per unit volume, is proportional to the exponent of the classical action on the bounce configuration: $w_0 \propto \exp(-S_{cl})$, while the pre-exponential factor requires a calculation of the functional determinant at the bounce. The exponential factor is readily found[@vko; @cc1] in the so called thin wall limit, namely when the radius of the critical bubble is much bigger than the effective thickness of its wall. This limit is always realized at a small difference $\epsilon$ between the vacuum energy density of the false and the true vacua, with the other parameter determining $S_{cl}$ being the surface tension $\mu$ of the bubble wall, i.e. of the boundary between the metastable and the stable phases. The pre-exponential factor in the bubble nucleation rate is known in a closed form only in (1+1) dimensional models[@ks; @mv], where in the thin wall limit it is determined only by the parameter $\epsilon$, with only partial results found in (2+1) dimensions[@garriga; @mr; @mv2], and virtually no result known in the (3+1) dimensional case.
Similarly to the behavior in the decay of a metastable phase in a thermal setting, the presence of matter in the false vacuum generally provides ‘centers of nucleation’ for the bubbles of the true vacuum. Thus one can consider the false vacuum decay induced by the presence of a particle[@adl; @sv; @mv3], by particle collisions[@sv; @mv3], by matter with finite density[@gk], as well as by the matter being in a thermal equilibrium where the problem goes back to the more conventional thermodynamic setting[@langer]. In this paper we revisit the calculation of the bubble nucleation rate associated with the presence of a particle in the false vacuum. The particle-induced nucleation can also be viewed as the decay of the particle (albeit in the process the initial ‘vacuum’ state also gets destroyed), whose rate $\Gamma$ generically can be written in the form $\Gamma= K \, w_0$, where the constant $K$, which can be naturally called the ‘catalysis factor’, is the main subject of our consideration. The catalysis is most efficient for the particles which have zero modes localized on the boundary between the false and the true vacua. The reason for this behavior is that in this case the energy corresponding to the mass of the particle $m$ in the initial state is fully transferred to the bubble degrees of freedom, since in the final state the particle ends up as a zero mode localized on the bubble wall. This effectively corresponds to the upward shift by $m$ of the energy at which the tunneling takes place[@sv], and results in $K$ being proportional to the exponential factor $\exp( 2 \, m \, \tau)$, where $\tau$ is the (Euclidean) time on the tunneling trajectory. The exponential behavior due to the shift of the energy for the tunneling trajectory can be found explicitly both in (1+1) dimensions[@sv] and in higher-dimensional models[@mv3]. However the pre-exponential factor has been calculated only for the bosons of the master field in a (3+1) dimensional case[@adl], for which bosons the existence of the zero mode is always true. Here we calculate the pre-exponential behavior of the catalysis factor for the same bosons in lower dimensions, and also find a closed formula for this factor in (1+1) dimensions for a fermion, whose field has a zero mode on the inter-vacua boundary. The existence of such fermionic mode is a generic phenomenon and is guaranteed in the case where the mass term for the fermions changes sign across the bubble wall[@jr].
Our consideration, similarly to Ref.[@adl], is generally limited to models with weak coupling, which implies that the masses $m$ of the both types of considered particles are small in comparison with the scale of the surface tension $\mu$. In this case the deformation of the tunneling trajectory due to the energy shift by $m$[@sv; @mv3] can be neglected, so that, in particular, the tunneling time $\tau$ coincides with the radius $R \propto \mu/\epsilon$ of the critical bubble, $\tau=R$, as it does in the spontaneous vacuum decay[@cc1]. Furthermore, we also assume the applicability of the thin wall limit, which implies the condition $m \, R \gg 1$, and which is always valid in the limit of small $\epsilon$.
The catalysis factor $K$, as defined, has the dimension of the spatial volume. Thus it would be natural to compare the pre-exponential factor in $K$ with the spatial volume of the critical bubble of the radius $R$. Under our assumptions we find that for a fermion in (1+1) dimensions this factor is indeed of order $R$, while the catalysis factor for the bosons is enhanced in comparison with the volume of the bubble by inverse powers of the (small) coupling constant.
It should be noted that technically the bosonic catalysis factor in lower dimensions can be found by a straightforward application of the treatment of Ref.[@adl]. Such application is fully justified in a (2+1) dimensional case. However in (1+1) dimensions there is a potential complication in estimating the effect of the quantum fluctuations arising from an infrared behavior of the modes of the bosonic field over the bounce background. We demonstrate for this case that the large infrared terms in fact cancel due to the specific properties of the soft modes.
The material in the rest of the paper is organized as follows. In Sec.2 we briefly review the calculation of the spontaneous decay rate of false vacuum and present a calculation of the decay rate induced by a boson of the scalar field, which defines the vacuum states. In Sec.3 the problem of the infrared behavior of the one-loop correction to the calculated decay rate in (1+1) dimensions is considered and it is shown that this problem is resolved due to a cancellation of the contributions to this correction between the negative mode and the sum over the positive soft modes of the field of the bounce. In Sec.4 the catalysis factor is calculated for a fermion in a (1+1) model. We then discuss the decay of metastable states in the sine-Gordon model with added linear term, the so called ‘sine-Gordon staircase’. Using the equivalence[@stone] of this bosonic model and the massive fermionic Thirring model in an external electric field, we find the induced decay rate for both the weak coupling limit (Sec.5) and for the strong coupling limit (Sec.6), the latter corresponding to a weak coupling in the Thirring model. Finally, in Sec.7 we discuss possible implications of our calculation for other models.
Spontaneous and induced decay of false vacuum
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In what follows we assume a situation where the energy density of a scalar field $\phi$, the ‘master field’, has a local minimum at $\phi=\phi_+$, which is higher than in a neighboring minimum at $\phi=\phi_-$. The vacuum state defined by the former minimum is referred to as the false vacuum, while the latter is the true vacuum. A typical example of such situation is provided by the well known model of a scalar field with the potential V()=[\^2 8]{} ( \^2 - v\^2 )\^2 + a , \[pot\] where $\lambda$, $v$, and $a$ are constants. At $a=0$ the potential has two degenerate minima at $\phi_\pm = \pm v$, while at small positive $a$ the degeneracy is lifted in such a way that the minimum at $\phi_+$ has energy density bigger than that of $\phi_-$ by the amount $\epsilon \approx 2\, a\, v$. The vacuum state at $\phi_+$, being stable at $a \le 0$ becomes metastable at positive $a$ and decays by nucleation and subsequent expansion of the bubbles filled with the phase $\phi_-$. At small $a$ the surface density of the bubble wall can be approximated[@vko; @cc1] by the surface density of the soliton with the field profile (x) = v \[wall\] interpolating between the two degenerate vacua in the limit $a \to 0$: =\_[-v]{}\^[+v]{} d= [2 3]{} v\^3 . \[mu\] The mass of the scalar particles of the field $\phi$ propagating in either of the vacua is given (also in the limit $a \to 0$) as $m=\lambda \, v$. In a model with the total space-time dimensions equal to $d$ the ratio $m^{d-1}/\mu$ coincides with the dimensionless coupling constant for the perturbation theory in this model. We assume throughout this paper that this ratio is a small parameter, which thus corresponds to weak coupling.
In the Euclidean-space formulation of the problem of the false vacuum decay[@cc1; @cc2] the calculation of the spontaneous decay rate amounts to a semiclassical evaluation of the imaginary part of the energy of the false vacuum from the path integral Z= [N]{} e\^[-S\[,…\]]{} [D]{}…\[funint\] where the dots stand for other possible fields present in a specific model, ${\cal N}$ is the normalization factor, and the integration is performed with the condition that the field $\phi$ approaches its false vacuum value $\phi_+$ at the boundaries of the space-time normalization box. The decay rate is then given by $w_0 = 2 \, {\rm Im} (\ln Z)/VT$, where $VT$ is the space-time volume of the normalization box.
The action functional $S$ has a semiclassical saddle point at the configuration described by the bounce[@cc1]. In the thin wall limit the bounce is an $O(d)$ symmetric bubble with the field $\phi_-$ inside and $\phi_+$ outside, and the bubble wall, separating the two phases has the surface tension $\mu$. The action for the bounce in this approximation is given by S= A\_B - V\_B , \[act\] where $V_B$ is the $d$ dimensional volume of the bounce and $A_B$ is its $(d-1)$ dimensional surface area. The action (\[act\]) reaches its extremum on a spherical bounce with the radius $R=(d-1)\, \mu/\epsilon$, which is also the radius of the critical bubbles capable of classical expansion in the Minkowski space-time.
The spectrum of small deformations of the bounce around the extremum contains exactly one negative mode, corresponding to an overall variation of the radius. This mode in fact gives rise to the imaginary part[@cc1] of the path integral in eq.(\[funint\]). Furthermore this spectrum also contains $d$ translational zero modes, the integration over which introduces the factor of the space-time volume $VT$ in the contribution of the bounce to the energy of the vacuum state.
The decay rate of a particle of the field $\phi$ in the false vacuum can be calculated[@adl; @sv] by considering the imaginary part of the contribution of a bounce to the Euclidean-space propagator of the excitations $\sigma(x)=\phi(x)-\phi_+$ of the field $\phi$: D(x,y)=[1 Z]{} (x) (y) e\^[-S\[,…\]]{} [D]{}…\[prop\] in the limit of large separation $L=|x-y|$. Indeed, the contribution of the bounce to the correlator (\[prop\]), as shown in Fig.1a, has the generic form D(x,y) = [i 2]{} w\_0 d\^dz F(x-z,y-z) D\_0(x-z) D\_0 (y-z) , \[dprop\] where $D_0(x)$ is the free-particle propagator in the vacuum $\phi_+$, satisfying the equation ( - \^2 +m\^2 ) D(x) = \^[(d)]{}(x) \[prop0\] and the factor $(i/ 2)\, w_0 d^dz$ is the proper measure of integration over the coordinate $z$ of the center of the bounce, as follows from the consideration of the bounce contribution to (the imaginary part of) the vacuum energy.
Let us consider the contribution to the integral (\[dprop\]) arising from the configurations, where the bounce is far (in units of its radius) from either of the points $x$ and $y$, i.e. where $|x-z| \gg R$ and $|y-z| \gg R$. The propagators $D_0(x-z)$ and $D_0 (y-z)$ in the integral in eq.(\[dprop\]) describe the exponential attenuation of the correlation ($D(x) \sim \exp(-m
|x|)$ at large separations, while the form factor $F(x-z,y-z)$ does not have this exponential behavior. For this reason at $|x-y|=L \gg R$ the integrand in eq.(\[dprop\]) is maximized for $z$ lying on the straight line running between $x$ and $y$: $z_\nu=s \, (x_\nu-y_\nu)/L$, and the integration can be split into the longitudinal, over the parameter $s$ along this line, and the transversal, over $z_\perp$. The integration over $z_\perp$ can be done by the saddle point method, so that the form factor $F(x-z,y-z)$ can be replaced by its value at $z_\perp = 0$, and the essential configuration to be considered is the one shown in Fig.1b. As will be discussed few lines below, when the bounce is far from the endpoints of integration over $s$, i.e. $s \gg R$ and $L-s \gg R$, the value of the form factor in fact does not depend on $s$ and is a constant $F_0$. Since the contribution of the excluded regions around the endpoints is only of relative order $R/L$, the integral in eq.(\[dprop\]) can be replaced at large $L$ by D(x,y) = [i 2]{} w\_0 F\_0 d\^dz D\_0(x-z) D\_0 (y-z) , \[dprop1\] where $F_0$ should be calculated from the configuration shown in Fig.1b.
The expression (\[dprop1\]) for the modification $\delta D$ of the propagator by the bounce can be compared with the first-order correction to the propagator due to a small shift of mass by $\delta m^2$, $m^2 \to m^2 + \delta m^2$, in eq.(\[prop0\]). In the standard way one finds \_m D(x,y) = -m\^2 d\^dz D\_0(x-z) D\_0 (y-z) . \[dpropm\] Thus the contribution (\[dprop1\]) of the bounce to the propagator of the boson in the false vacuum is equivalent to an imaginary shift of the boson mass: $\delta m^2= - (i / 2)\, w_0 \, F_0$, which corresponds to the particle decay rate given by $\Gamma= F_0 w_0/(2m)$, so that the catalysis factor $K$ is found as K = [F\_0 2 m]{} . \[gamb\]
The factor $F_0$ can be readily found[@adl] for the discussed here case of the bosons of the classical field of the bounce. Indeed, consider the classical field just outside the bounce, i.e. at the distance $r > R$ from the center, such that $r-R \gg m^{-1}$, but still $r-R \ll R$. The former condition ensures that the field is described by its asymptotic approach to the vacuum value $\phi_+$, while the latter implies that in this region the curvature of the bounce wall can be neglected in a calculation of this asymptotic behavior. Thus one can consider instead the asymptotic behavior of the field in the limit $\epsilon \to 0$, i.e. of the field of the stable soliton separating two degenerate vacua. This asymptotic behavior has the form $\phi(x)-\phi_+ = - 2 v
\exp[-m \, (r-R)]$, where in the model described by the potential (\[pot\]) $v$ coincides with the corresponding parameter in the potential, while in a generic model $v \sim (\phi_+ - \phi_-)/2$. On the other hand in the $O(d)$-symmetric problem the asymptotic approach of the scalar field to its vacuum value is described by the solution of the linearized spherically-symmetric equation, equivalent to the homogeneous part of eq.(\[prop0\]), and reads as, (r)-\_+ = C D\_0(r) \[defc\] where the free boson propagator in $d$ dimensions has the well known expression in terms of the modified Bessel function $K_\nu(mr)$: D\_0(r)= [m\^[d/2-1]{} (2 )\^[d/2]{} r\^[d/2-1]{}]{} K\_[d/2-1]{}(mr) . \[d0\] The constant $C$ in the asymptotic expression (\[defc\]) is found by comparing the two expression for $\phi(x)-\phi_+$ in the discussed region just outside the bounce and using the standard asymptotic formula for the function $K_\nu(mr)$. In this way one finds C= - 4 (2 )\^[d/2-1]{} m\^[(3-d)/2]{} R\^[(d-1)/2]{} v e\^[m R]{} . \[cexp\] Using then the expression (\[defc\]) for the field with thus determined constant $C$, one finds the product of the classical fields $\sigma(x)
\sigma(y)$ in the integral in eq.(\[prop\]) in the configuration shown in Fig.1b, corresponding to the constant $F_0$ in eq.(\[dprop1\]) given by F\_0 = C\^2 = 16 (2 )\^[d-2]{} m\^[3-d]{} R\^[(d-1)]{} v\^2 e\^[2 m R]{} , \[fres\] which indeed does not depend on the position of the bubble along the straight line connecting the points $x$ and $y$ as long as both these points are sufficiently outside the bounce. The catalysis factor thus can be found from the relations (\[gamb\]) and (\[fres\]) in the form K=2\^[d+1]{} \^[(d-3)/2]{} ([d+1 2]{} ) m\^[2-d]{} v\^2 V\_[d-1]{} e\^[2 m R]{} , \[catb\] where $V_{d-1}= \pi^{(d-1)/2} \, R^{d-1}/ \Gamma[(d+1)/2]$ is the spatial ($d-1$ dimensional) volume of the critical bubble. As discussed in the Introduction, it is natural to compare the catalysis factor with this volume. The result in eq.(\[catb\]) shows that besides the classical exponential factor the catalysis is additionally enhanced by the factor $ m^{2-d} \, v^2$ in the pre-exponent, which is the inverse of the small dimensionless coupling in the theory.
Boson-induced decay in (1+1) dimensions
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The formula for the catalysis factor in eq.(\[catb\]) reduces in (3+1) dimensions to the result of Ref.[@adl], and in other dimensions it presents a rather straightforward generalization. There is however one point, of a special importance to a (1+1) dimensional case, related to the effect of the quantum fluctuations on the essentially classical result in eq.(\[catb\]). Generally, the effect of the quantum fluctuations (the loop correction) is expected to be suppressed by a power of the coupling constant as compared to the classical contribution. In the discussed problem this expectation is true at $d
>2$, however in a (1+1) dimensional problem this expectation is potentially jeopardized by an infrared behavior. Indeed the eigenvalues of the second variation of the action for the fluctuations of the shape of the bounce, described by the effective action (\[act\]), are proportional to $R^{-2}$. The modes with these eigenvalues are localized on the bounce boundary and describe the soft part of the spectrum of the modes of the field around the stationary bounce configuration, as opposed to the modes, whose eigenvalues start at $O(m)$, and those ‘hard’ modes describe the excitations propagating in the bulk as well as possible deformations of the profile of the field across the bounce wall. Let us estimate the contribution of an individual soft mode with the eigenvalue $c_n/R^2$ to the correlator (\[prop\]), with $c_n$ being a number. All such modes originate from local shifts of the wall of the bounce, so that the field profile of an individual mode is proportional to the radial derivative of the field of the bounce, $\phi'(r)$. The field $\sigma_n$ of a normalized to one mode in (1+1) dimensions is then parametrically estimated at the distance $r
> R$, such that $r-R \gg m^{-1}$, but still $r-R \ll R$, as \_n(r) \~[m v ]{} e\^[-m (r-R)]{} \~ e\^[-m (r-R)]{} , \[sest\] where it is taken into account that $\int(\phi')^2 \, dr \approx \mu \sim m \,
v^2$, and any numerical factors are dropped for a parametrical estimate. The contribution of such mode to the correlator (\[prop\]) is then proportional to \_n(r\_1) \_n(r\_2) \~c\_n\^[-1]{} (m R) e\^[2 m R]{} e\^[-m (r\_1+r\_2)]{} . \[s2est\] In this estimate the large factor $(m \, R)$ stands in place of the factor $v^2$ in the similar product for the classical field. Thus the magnitude of an individual mode contribution to the correlator relative to the classical part is described by the parameter $(m \, R)\over v^2$, which in-spite of the expected suppression by the dimensionless coupling $v^{-2}$ is infrared unstable at large $R$.
We will show however that the sum over the soft modes gives zero for this infrared contribution due to a cancellation between one negative and all positive modes. In order to demonstrate this we consider the parametrization of the shape of the bounce in the polar coordinates $(r, \theta)$ on a plane, so that the effective action (\[act\]) for the soft modes takes the form S=\_0\^[2]{} ( - [1 2]{} r\^2 ) d= [ \^2 ]{} + \_0\^[2]{} [2]{} (\^2 - \^2) d + O(\^4), \[act2\] where the latter expression shows the two first terms of expansion in the small deviation $\rho=r-R$ of the radial variable $r$ from its stationary value $R=\mu/\epsilon$, and the dot stands for the derivative over $\theta$. The quadratic part in this expression has one negative eigenmode $\rho = 1/\sqrt{2
\pi}$ and the spectrum of zero and positive double degenerate eigenmodes: \_n\^[(1)]{} = [1 ]{} n , [and]{} \_n\^[(2)]{} = [1 ]{} n ; (n=1,2,…) . \[modes\] The spectrum of the eigenvalues is proportional[^1] to $(n^2-1)$ with the negative mode corresponding to $n=0$.
Let us consider now the configuration shown in Fig.1b with the bounce located on the line connecting the points $x$ and $y$. Let the angle $\theta$ be defined as measured counterclockwise from the downward vertical connecting the center of the bounce with the point $x$, so that $\theta = \pi$ corresponds to the upward vertical connecting the same center with the point $y$. Clearly, the contribution of the fluctuations of $\rho$ to the propagator (\[prop\]) is proportional to \^2 \_n [\[\_n(0)+\_n()\]\^2 n\^2 -1]{} . \[sum0\] Note however that the sum $\rho(0)+\rho(\pi)$ is not vanishing only for the negative mode and for the positive modes of the first type, $\rho_n^{(1)}$, with even $n$, i.e. $n=2 k$. Thus the sum in eq.(\[sum0\]) is proportional to the numeric sum -[1 2]{} + \_[k=1]{}\^ =0 , \[sum2\] where the first term is due to the negative mode and the sum runs over the positive modes. The arithmetic identity (\[sum2\]) explicitly demonstrates that the infrared contribution in a (1+1) dimensional model cancels between the negative mode and the sum over the positive ones.
Let us also remark on the decay of a moving particle in the false vacuum. If particle moves with a constant velocity then the probability depends on the velocity through the standard Lorentz factor. However if it moves with the constant acceleration situation is more subtle since in the particle frame vacuum behaves as the thermal bath due to Unruh effect. The effective temperature is defined through the acceleration as T\_[eff]{}=[a 2]{} hence probability of the particle decay is modulated by the thermal effects. The most essential effect corresponds to the possible deformation of the classical bounce. Since temperature corresponds to the periodicity in the Euclidean time then the deformation of the bounce happens when period corresponding to the temperature becomes comparable to $2R$. That is deformation of the bubble emerges if the acceleration is larger then a\_[crit]{}=. and our approximation fails.
Fermion-induced decay in (1+1) dimensions
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The very existence of fermions in a model is known to modify (without any fermions being present in the initial state) the pre-exponential factor in the rate of the false vacuum decay in the situation where the complex fermion field $\psi$ has a zero mode on the boundary between the vacua (in the limit $\epsilon
\to 0$). Such situation takes place when the mass term for the fermion changes sign between the two vacua[@jr]. In particular the rate $w_0$ for the spontaneous decay of the false vacuum in (1+1) dimensions receives a factor of 2 in comparison with purely bosonic theory[@ks2; @mv4]. This doubling corresponds to the existence of two final states in the false vacuum decay in (1+1) dimensions viewed as a spontaneous creation of a kink-antikink pair: one state where both the kink and the antikink are created with the fermion zero mode empty, and the other state is where is a zero-energy fermion on the kink and a zero-energy antifermion on the antikink.
In what follows we assume that the interaction of the fermion field with the scalar field of the bounce is such that there exists a zero fermion mode on the kink separating the two vacua. In order to find the effect of a fermion on the probability of nucleation of a critical bubble we consider the bounce contribution to the fermion propagator $G(x,y)=\langle \psi(x) {\overline \psi} (y)\rangle $ in the configuration shown in Fig.1b. Clearly, the exponentially enhanced factor $\exp(2 \, m_f \, R)$, where $m_f$ is the mass of the fermion, arises from the contribution of the zero mode, whose propagation from the lower point of the bounce to the upper one does not contain any exponential attenuation. Assuming for definiteness that the fermion mass is positive (and equals $m_f$) in the false vacuum, and choosing the $\gamma$ matrices as $\gamma_1=\sigma_1$ and $\gamma_2=\sigma_2$, one finds the solution for the Dirac equation for the field of the zero mode of $\psi$ in the background scalar field $\phi(r)$ of the bounce, \_0 = 0 , \[dirac\] in the form \_0(r, )=C\_f { -\_R\^r m\[(r’)\] dr’ } () (
[c]{} e\^[-i /2 ]{}\
e\^[i /2 ]{}
) , \[psi0\] where $\chi(\ell)$ is a one-dimensional fermion field living on the bounce boundary and (nominally) depending on the length parameter $\ell = R \theta$ along the boundary. Notice that the classical equation for $\chi$ reads $\dot
\chi =0$. Finally, the constant $C_f$ in eq.(\[psi0\]) is the normalization factor relating the normalization of $\psi$ and $\chi$ and satisfying the condition 2 C\_f\^2 dr = 1 . \[cfact\] In what follows we rather use a related factor ${\tilde C}_f$ defined as the coefficient in the expression C\_f { -\_R\^r m\[(r’)\] dr’ } \_f , \[tcfact\] which is valid sufficiently far outside the bounce where $m_{min} \, (r-R) \gg
1$ with $m_{min}$ being the minimal mass scale in the model. Generally the factor ${\tilde C}_f$ can be estimated as \_f\^2 = [m\_f 2]{} f( [m\_f m]{} ) , \[mrat\] where $f$ is a dimensionless function of the ratio of $m_f$ to masses of other particles in the false vacuum. In the limit where $m_f$ is much smaller than other masses one has $f(0)=1$. In the model, where the scalar ‘master field’ is described by the potential (\[pot\]) and the mass of the fermion is proportional to $\phi$, the function $f$ can be found explicitly: f(u)=[2\^[2 u]{} ]{} [(u+1/2) (u+1)]{} . \[funcf\]
The contribution of the fermion zero mode on the bounce the configuration shown in Fig.1b can be written, using the asymptotic behavior of the zero mode (\[psi0\]), in terms of the one-dimensional propagator of the field $\chi$ on the boundary, $g(\ell_1, \ell_2)=\langle \chi(\ell_1) \chi^\dagger(\ell_2)
\rangle$, as G(x,y)=-[i 2]{} [[w\_0]{} 2]{} [d\^2z]{} [C]{}\_f\^2 e\^[2 m\_f R]{} R [e\^[-|x-y|]{} ]{} (1+\_1) g(0,R) . \[delg\] Notice that for a complex fermion there is only one path for propagation along of $\chi$ along the boundary from the bottom of the bounce to its top (assumed here for definiteness to be counterclockwise in terms of Fig.1b), corresponding to the final state, where the fermion is a bound state localized on the kink. The other path (clockwise) would be relevant for the vacuum decay induced by an anti-fermion, which in the final state is localized on the antikink. The expression in eq.(\[delg\]) contains an extra factor 1/2, due to the fact that, as mentioned before, the spontaneous nucleation rate $w_0$ in the theory with fermions contains extra factor of 2, due to the existence of two final states in the decay, which is to be compensated in the proper measure of integration over the coordinate of the center of the bounce $d^2z$. The propagator $g$ has a very simple explicit form in terms of the sign function: $g(\ell_1,\ell_2)= (1/2) \, {\rm sign(\ell_1- \ell_2)}$, so that $g(0,\pi
R)=-1/2$.
The expression in eq.(\[delg\]) can now be compared with the corresponding change of the free propagator $G_0$ under a shift $\delta m_f$ of the fermion mass: \_m G(x,y)=-m\_f d\^2 z G\_0(x-z) G\_0(z-y) -m\_f d\^2 z [m 4 ]{} (1+ \_1) [e\^[-|x-y|]{} ]{} , \[delmg\] where the asymptotic expression takes into account the explicit form of the free propagator: G\_0(x,y)=[1 2 ]{} (-\_i \_i +m ) K\_0(m\_f |x-y|) . \[g0\] Using this comparison and eq.(\[mrat\]) one finds the imaginary part of the fermion mass shift corresponding to the decay rate of the fermion \_f= [2]{} f( [m\_f m]{} ) R w\_0 (2 m\_f R) = [2]{} f( [m\_f m]{} ) ( -[ \^2 ]{} + 2 m\_f R ) . \[ferg\] Here in the latter transition are used the explicit expressions: $w_0 =
(\epsilon/\pi) \, \exp(-\pi \, \mu^2/\epsilon)$ and $R = \mu/\epsilon$. One can readily see that in the fermion case, as expected, the pre-exponent in the catalysis factor, $K_f= (\pi/ 2) \, f(m_f/m) \, R \, \exp (2 \, m_f \, R)$, is indeed of the order of the spatial size of the critical bubble.
Meson decay in weakly coupled sine-Gordon model
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In order illustrate the universality of the derived results let us discuss one more example where the particle decay in the false vacuum happens in two dimensions. We shall derive the probability of the decay of the electrically neutral meson bound state in the Thirring model with the pre-exponential accuracy.
Consider the sine-Gordon theory with the Lagrangian L\_[SG]{}=[1 2]{} ()\^2 + () and add term $(\epsilon \, \beta/2\pi)\, \phi$ which yields the situation with the metastable states. This theory upon the two-dimensional fermionization is equivalent to the massive Thirring model with the Lagrangian L\_[Th]{}= i|\_\^-[12]{} gj\^j\_ + |+ A\_0 j\_0 where ${{\beta}^2\over 4\pi}=(1+{g\over \pi})^{-1} $, and $j_\nu=
\bar{\psi}\gamma_{\nu}\psi$. One can identify $\mu$ with the soliton mass in the sine-Gordon model, and $\partial_{x} A_0=\epsilon$. In what follows we shall assume that $\beta^2< 4\pi$ which is the condition for the bound state of fermions to exist in the Thirring model. The solitons in the sine-Gordon model get mapped into the fermions in the Thirring model while the field $\phi$ gets mapped into the fermion-antifermion meson bound state.
The linear perturbation term in the sine-Gordon model corresponds to the constant electric field in the Thirring model realization, so that the problem of false vacuum decay can be discussed in both formulations. In the Thirring model it corresponds to the Schwinger pair production. The probability of the spontaneous vacuum decay in the sine-Gordon model and equivalent Schwinger process in Thirring model has been found in [@stone]. The one-loop result coincides with the general formula $w_{Th} = (\epsilon/2\pi) \, \exp(-\pi \, \mu^2/\epsilon)$, while in the special case of $\beta^2=4\pi$, corresponding to $g=0$, the exact result is found as w\_[Th]{} =-[2 ]{} (1 -e\^[- \^2 / ]{}) Note that in this case the bose-fermi equivalence allows to perform the summation over the multiple bounces in the sin-Gordon theory.
Now we can discuss the decay of the false vacuum in the presence of a particle corresponding to the field $\phi$ in the sine-Gordon model. In the weak coupling regime for the bosons, i.e. at small $\beta$, the soliton is much heavier then the boson particle, which corresponds to the situation where the external particle does not deform the classical bounce configuration. Hence the decay rate can be immediately read off eq.(\[catb\]). In this case the process corresponds in the Thirring model to the nonperturbative decay of the light electrically neutral meson in the electric field and the catalysis factor of this process is K\_[Th]{} = [32 \^2]{} e\^[[2 m\_[b]{} / ]{}]{} where $m_b$ is the meson mass. Note that this process is the two-dimensional counterpart of the induced Schwinger processes discussed in four dimensions in [@gss; @monin] in the exponential approximation.
Meson decay in strongly coupled sine-Gordon model
=================================================
The boson-fermion correspondence in this model actually allows to find the meson decay rate in the limit, opposite to what has been considered so far in this paper, namely for strongly coupled bosons, when the boson mass is close to the kink-antikink threshold. This limit corresponds to a small positive $g$, and the boson mass (at $\epsilon \to 0$) is $m_b=2\,\mu - \mu \, g^2$. The near-threshold dynamics of the soliton-antisoliton pair can be considered nonrelativistically as a motion of a pair with the reduced mass $\mu/2$ in the local potential $U(x)=-2 g \, \delta(x)$, which correctly reproduces the energy of the bound state (the boson). For a nonzero $\epsilon$ the nonrelativistic Hamiltonian for this system takes the form: H=[p\^2 ]{} - x -2 g (x) . \[hamnr\] The problem of the boson decay in the false vacuum is reduced in terms of this equivalent nonrelativistic system to that of ionization of the bound state in the external electric field $\epsilon$.
In order to solve the ionization problem we start with considering the Euclidean time propagator (“the heat kernel") defined as ${\cal K}(x, y; \tau)= \langle x | \exp(-H \, \tau)|y \rangle$, and the corresponding energy dependent Green’s function at the negative (i.e. below the threshold) energy $E=-\kappa^2/\mu$: G(x,y; -[\^2 ]{} ) = \_0\^(x, y; ) ( -[\^2 ]{} ) d\[nrgf\] at $x=0$ and $y=0$. We remind that if only the kinetic term is retained in the Hamiltonian (\[hamnr\]), i.e. at $\epsilon =0$ and $g=0$, these functions are ${\cal K}_0(0, 0; \tau) = (4 \pi \, \tau /\mu)^{-1/2}$ and $G_0(0,0;
-\kappa^2/\mu)=\mu/(2 \kappa)$. At $\epsilon = 0$ and a nonzero $g$ the Green’s function for the Hamiltonian (\[hamnr\]) is found as G\_[=0]{} (0,0; -\^2/) = [ G\_0(0,0; -\^2/) 1- 2 g G\_0(0,0; -\^2/)]{} = [ 2 ]{} [1 1- g /]{} . \[gg\] The latter expression contains explicitly the pole at $\kappa = \mu \, g$ corresponding to the bound state.
When both the $\epsilon$ and $g$ are nonzero the equation (\[gg\]) is modified by replacing the Green’s function $G_{\epsilon=0}$ by that for a nonvanishing $\epsilon$: $G_\epsilon$. The latter Green’s function can be expressed in terms of the corresponding propagator ${\cal K}_\epsilon(0, 0; \tau)$ which can be found in the textbook [@fh]: \_(x, y; ) = , \[ke\] G\_(0,0; -[\^2 ]{} ) = \_0\^ ( [\^2 12 ]{} \^3 -[\^2 ]{} ) d , \[ge\] and the pole position is thus determined from the equation 2 g G\_(0,0; -[\^2 ]{} ) =1 . \[pole\]
The peculiarity of the latter equation is that the integral in eq.(\[ge\]) is formally divergent, which is the usual situation in a calculation of the energy of an unstable state. In order to make physical sense, both that energy and the integral in eq.(\[ge\]) should be understood as a result of an analytical continuation in the parameters of the model from the region where the considered state is stable. In terms of eq.(\[ge\]) this corresponds to a continuation from the region of (formally) negative $\epsilon^2$ where the integral is convergent. The result of such analytical continuation to physical positive $\epsilon^2$ can be formulated as follows[@cc1]: The integration runs along the real axis of $\tau$ from $\tau =0$ to the value of $\tau$ where the integrand has minimum, i.e. to $\tau_0=2 \kappa/\epsilon$. From that point the contour of integration should be turned parallel to the imaginary axis of $\tau$, corresponding to the direction of the steepest descent (see Fig.2). This contour rotation gives rise to an imaginary part of the integral, and hence to an imaginary part of the energy of the resonant state, corresponding to the decay width of the resonance. Following this procedure one can readily find the real and the imaginary parts of the integral in eq.(\[ge\]) and reduce the equation (\[pole\]) for the position of the pole to the form g = 1 , which corresponds to the decay rate of the bound state = 2 g\^2 ( - [4 3]{} g\^3 [\^2 ]{} ) . \[gf\] It can be noted that the exponential factor in this formula is the standard WKB tunneling exponent in a linear potential, while the pre-exponential factor is a new result. The described derivation of the formula (\[gf\]) assumes that the integral in eq.(\[ge\]) can be evaluated in the saddle point approximation, which implies that the parameter in the exponent in eq.(\[gf\]) is large, i.e. that $g^3 \, \mu^2 \gg \epsilon$.
Discussion
==========
In this paper we have refined the calculations of the decay rate of the boson in the false vacuum and have found the decay rate of fermion in the false vacuum in (1+1) dimension with the pre-exponential factor. All calculations, except for the one in Sec.6, have been performed in the approximation when the back reaction of the external particle on the Euclidean bounce solution can be neglected.
The account of the back reaction amount to several new effects which are different for d=2 and $d>2$. In the (1+1) dimensional case the back reaction deforms the classical solution which deformation has been described classically in Ref.[@sv] however the calculation of the pre-exponential factor is beyond our approximation. Such calculation could be potentially interesting from the stringy perspective. Indeed, the worldsheet theory on nonabelian string in several models can be identified with the $CP^N$ model (see [@shifman] for a recent review) which has one true vacuum and a set of metastable ones. At large N this theory can be treated perturbatively and the issue of the decay of metastable vacua or in other terns exited strings can be discussed, Similarly one can discuss the fate the different excitations on the exited metastable string which is just the problem we have considered. In some situations the pre-exponential factor is of the prime importance since in some range of parameters the N dependence disappear from the exponent [@shifman]. However it is unclear if the regime with the negligible back reaction could exist in the worldsheet theory. It seems that the analysis similar to one in Section 6 could be applicable in this case.
In higher dimensions the situation is more complicated. The point is that the initial particle evolves along a trajectory in the complexified Minkowski space[@mv3]. At the first stage of the process the initial particle “produces” the oscillating bubble in the Minkowski space which later develops the path in the Euclidean space. The overlap of the initial particle and the bubble happens out of the real axis. Let us also note that there is a possibility of the resonant decay of the particle in the false vacuum when the particle mass coincides with the energy levels of the quantized bubbles in the Minkowski space.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of A.G. was supported in part by grants CRDF RUP2-261-MO-04 and RFBR-04-011-00646 . A.G. thanks FITP Institute at University of Minnesota where the part of the work has been done for the kind hospitality and support. He also thanks KITP at UCSB for the hospitality during the program “Mathematical Structures in String Theory” supported by grant NSF PHY99-07949. The work of MBV is supported in part by the DOE grant DE-FG02-94ER40823.
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J. S. Langer, Annals Phys. [**54**]{}, 258 (1969). R. Jackiw and C. Rebbi, Phys. Rev. D [**13**]{}, 3398 (1976). M. Stone, Phys. Rev. D [**14**]{}, 3568 (1976). V. Kiselev and K. Selivanov, Sov. J. Nucl. Phys. [**43**]{}, 153 (1986) \[Yad. Fiz. [**43**]{}, 239 (1986)\]. M. B. Voloshin, Yad. Fiz. [**43**]{}, 769 (1986). A. S. Gorsky, K. A. Saraikin and K. G. Selivanov, Nucl. Phys. B [**628**]{}, 270 (2002). A. K. Monin, JHEP [**0510**]{} (2005) 109 arXiv:hep-th/0509047. R. P. Feynman and A. R. Hibbs, [*Quantum Mechanics and Path Integrals*]{}, McGraw-Hill, NY 1965; Chapt.3. M. Shifman, Minnesota FTPI report FTPI-MINN-05-44, October 2005; arXiv:hep-ph/0510098.
[^1]: The proportionality coefficient is not important for this discussion. It can be noted however that in terms of the normalized modes for the field $\phi$ the eigenvalues are $(n^2-1)/R^2$.
|
---
abstract: 'We consider a real Gaussian process $X$ having a global unknown smoothness $({r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}}$), ${r_{{\scriptscriptstyle}0}}\in \n_0$ and ${\beta_{{\scriptscriptstyle}0}}\in]0,1[$, with $X^{({r_{{\scriptscriptstyle}0}})}$ (the mean-square derivative of $X$ if ${r_{{\scriptscriptstyle}0}}\ge 1$) supposed to be locally stationary with index ${\beta_{{\scriptscriptstyle}0}}$. From the behavior of quadratic variations built on divided differences of $X$, we derive an estimator of $({r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$ based on - not necessarily equally spaced - observations of $X$. Various numerical studies of these estimators exhibit their properties for finite sample size and different types of processes, and are also completed by two examples of application to real data.'
address:
- 'Avignon Université, LMA EA2151, 33 rue Louis Pasteur, F-84000 Avignon, France.'
- 'Université de Lyon, CNRS UMR 5208, Polytech Lyon-Université de Lyon 1, Institut Camille Jordan, 43 blvd du 11 novembre 1918, 69622 Villeurbanne Cedex, France.'
author:
- Delphine Blanke
- Céline Vial
bibliography:
- 'bv2014.bib'
title: Global smoothness estimation of a Gaussian process from regular sequence designs
---
Introduction
============
In many areas assessing the regularity of a Gaussian process represents still and always an important issue. In a straightforward way, it allows to give accurate estimates for approximation or integration of sampled process. An important example is the kriging, which becomes more and more popular with growing number of applications. This method consists in interpolating a Gaussian random field observed only in few points. Estimating the covariance function is often the first step before plug this estimates in the Kriging equations, see @St00. Usually the covariance function is assumed to belong to a parametric family, where these unknown parameters are linked to the sampled path regularity: for example the power model, which corresponds to a Fractional Brownian motion. Actually, many applications make use of irregular sampling and @St00 (chap. 6.9) gives an hint of how adding three points very near from the origin among the already twenty equally spaced observations improve drastically the estimation of the regularity parameter. In this paper, we defined an estimator of global regularity of a Gaussian process when the sampling design is regular, that is observation points correspond to quantile of some distribution, see section \[Framework\] for details. Taking into account a non uniform design is innovating regarding other existing estimators and makes sense as to the remark above.
A wide range of methods have been proposed to reconstruct a sample path from discrete observations. For processes satisfying to the so-called Sacks and Ylvisaker (SY) conditions, recent works include: @MG96 [orthogonal projection, optimal designs], @MGR97 [linear interpolation, optimal designs], @MGR98 [linear interpolation, adaptive designs]. Under Hölder type conditions, one may cite e.g. works of @Se96 [linear interpolation], @Se00 [Hermite interpolation splines, optimal designs], @SB98 [best approximation order]. Note that a more detailed survey may be found in the book by @Rit00. Another important topic, involving the knowledge of regularity and arising in above cited works, is the search of an optimal design. In time series context, @Ca85 analyzes three important problems (estimation of regression coefficients, estimation of random integrals and detection of signal in noise) for which he is looking for optimal design. The latter two problems involve approximations of integrals, where knowledge of process regularity is particularly important [see, e.g. @Rit00], we provide a detailed discussion on this topic in section \[AppInt\] together with additional references. Applications of estimation of regularity can be also find in @Ad90 where bounds of suprema distributions depend on the sample roughness, in @Is92 where the regularity is involved in the choice of the best wavelet base in image analysis or more generally in prediction area, see @Cu77 [@Li79; @Bu85].
Furthermore for a real stationary and non differentiable Gaussian process with covariance $\mathds{K}(s,t)=\mathds{K}({\lv t-s{\right\vert}},0)$ such that ${\displaystyle}\mathds{K}(t,0) = \mathds{K}(0,0) - A
{\lv t{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}} + o({\lv t{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}})$ as ${\lv t{\right\vert}} \to 0$, the parameter ${\beta_{{\scriptscriptstyle}0}}$, $0 < {\beta_{{\scriptscriptstyle}0}}< 1$, is closely related to fractal dimension of the sample paths. This relationship is developed in particular in the works by @Ad81 and @TT91 and it gave rise to an important literature around estimation of ${\beta_{{\scriptscriptstyle}0}}$. Note that this relation can be extended, e.g. for non Gaussian process in @HR94. The recent paper of @Gneit2012 gives a review on estimator of the fractal dimension for times series and spatial data. They also provide a wide range of application in environmental science, e.g. hydrology, topography of sea floor. Note that this paper is restricted to the case of $(n+1)$ equally spaced observations. In relation with our work, we refer especially to @CH94 for estimators based on quadratic variations and their extensions developed by @KW97. Still for this stationary framework, @CHP95 introduce a periodogram-type estimator whereas @FHW94 use the number of level crossings.
In this paper, our aim is to estimate the global smoothness $({r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$ of a Gaussian process $X$, supposed to be ${r_{{\scriptscriptstyle}0}}$-times differentiable (for some nonnegative integer ${r_{{\scriptscriptstyle}0}}$) where $X^{({r_{{\scriptscriptstyle}0}})}$ (the ${r_{{\scriptscriptstyle}0}}$-th mean-square derivative of $X$ for non-zero ${r_{{\scriptscriptstyle}0}}$) is supposed to be locally stationary with regularity ${\beta_{{\scriptscriptstyle}0}}$. The parameters $({r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$ being both unknown, we improve the previous works in several ways:
- not necessarily equally spaced observations of $X$ over some finite interval $[0,T]$ are considered,
- X is not supposed to be stationary not even with stationary increments,
- X has an unknown degree of differentiability, ${r_{{\scriptscriptstyle}0}}$, to be estimated,
- for ${r_{{\scriptscriptstyle}0}}\ge 1$, the coefficient of smoothness ${\beta_{{\scriptscriptstyle}0}}$ is related to the unobserved derivative $X^{({r_{{\scriptscriptstyle}0}})}$.
Our methodology is based on an estimator of ${r_{{\scriptscriptstyle}0}}$, say $\widehat{{r_{{\scriptscriptstyle}0}}}$, derived from quadratic variations of divided differences of $X$ and consequently, generalize the estimator studied by @BV11 for the equidistant case. In a second step, we proceed to the estimation of ${\beta_{{\scriptscriptstyle}0}}$, with an estimator $\widehat{\beta}_0$ which can be viewed as a simplification of that studied, in the case ${r_{{\scriptscriptstyle}0}}=0$, by @KW97. Also for processes with stationary increments and using a linear regression approach, @IL97 have proposed and studied an estimator of $H=2({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}})$ for equally spaced observations. As far as we can judge, our two steps procedure seems to be simpler and more competitive. We obtain an upper bound for ${\mathds{P}}(\widehat{{r_{{\scriptscriptstyle}0}}} \not = {r_{{\scriptscriptstyle}0}})$ as well as the mean square error of $\widehat{{r_{{\scriptscriptstyle}0}}}$ and almost sure rates of convergence of $\widehat{{\beta_{{\scriptscriptstyle}0}}}$. Surprisingly, these almost sure rates are comparable to those obtained in the case of ${r_{{\scriptscriptstyle}0}}$ equal to 0: by this way the preliminary estimation of ${r_{{\scriptscriptstyle}0}}$ does not affect that of ${\beta_{{\scriptscriptstyle}0}}$, even if $X^{({r_{{\scriptscriptstyle}0}})}$ is not observed. Next, in section \[AppInt\], we derive theoretical and numerical results concerning the important application of approximation and integration. We complete this work with an extensive computational study: we compare different estimators of ${r_{{\scriptscriptstyle}0}}$ and ${r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}}$ for processes with various kinds of smoothness, derive properties of our estimators for finite sample size, an example of consequence of the misspecification of ${r_{{\scriptscriptstyle}0}}$ is given, and an example of process with trend is also study. To end this part, we apply our global estimation to two well-known real data sets: Roller data @La94 and Biscuit data @BFV01.
The framework {#Framework}
=============
The process and design
----------------------
We consider a Gaussian process $ X=\{X(t), \, t\in [0,T] \}$ observed at $(n+1)$ instants on $[0,T]$, $T>0$, with covariance function $\mathds{K}(s,t)={{\mrm{Cov\,}}}(X(s),X(t))$. We shall assume the following conditions on regularity of $X$.
\[h21\] $X$ satisfies the following conditions.
- There exists some nonnegative integer ${r_{{\scriptscriptstyle}0}}$, such that $X$ is ${r_{{\scriptscriptstyle}0}}$-times differentiable in quadratic mean, denote $X^{({r_{{\scriptscriptstyle}0}})}$.
- The process $X^{({r_{{\scriptscriptstyle}0}})}$ is supposed to be locally stationary: $$\label{e21}
\lim_{h\to 0} \sup_{s,t\in[0,T],\\{\lv s-t{\right\vert}} \le h,\\s\not=t} {\lv \frac{{\mathds{E}}\big(X^{({r_{{\scriptscriptstyle}0}})}(s)
-X^{({r_{{\scriptscriptstyle}0}})}(t)\big)^2}{{\lv s-t{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}}} - d_0(t){\right\vert}} = 0$$ where ${\beta_{{\scriptscriptstyle}0}}\in ]0,1[$ and $d_0$ is a positive continuous function on $[0,T]$.
- For either $p=1$ or $p=2$, $\mathds{K}^{({r_{{\scriptscriptstyle}0}}+p,{r_{{\scriptscriptstyle}0}}+p)}(s,t)$ exists on $[0,T]^2\big\backslash\{s=t\}$ and satisfies for some $D_p>0$: $${\lv \mathds{K}^{({r_{{\scriptscriptstyle}0}}+p,{r_{{\scriptscriptstyle}0}}+p)}(s,t){\right\vert}} \le D_p{\lv s-t{\right\vert}}^{-(2p- 2{\beta_{{\scriptscriptstyle}0}})}.$$ Moreover, we suppose that $\mu\in C^{{r_{{\scriptscriptstyle}0}}+1}([0,T])$.
Note that the local stationarity makes reference to @Ber74’s sense. The condition A\[h21\]-(i) can be translated in terms of the covariance function. In particular the function $\mathds{K}$ is continuously differentiable with derivatives $\mathds{K}^{(r,r)}(s,t)= {{\mrm{Cov\,}}}(X^{(r)}(s), X^{(r)}(t)) $, for $r=1,\dotsc,{r_{{\scriptscriptstyle}0}}$. Also, the mean of the process $\mu(t):={\mathds{E}}X(t)$ is a ${r_{{\scriptscriptstyle}0}}$-times continuously differentiable function with ${\mathds{E}}X^{(r)}(t)= \mu^{(r)}(t)$, $r=0,\dotsc,{r_{{\scriptscriptstyle}0}}$. Conditions A\[h21\]-(iii-$p$) are more technical but classical ones when estimating regularity parameter, see @CH94 [@KW97].
These assumptions are satisfied by a wide range of examples, e.g. the ${r_{{\scriptscriptstyle}0}}$-fold integrated fractional Brownian motion or the Gaussian process with Matérn covariance function, i.e. $\mathds{K}(t,0)=\frac{\pi^{1/2}\phi}{2^{(\nu)-1}\Gamma(\nu+1/2)}(\alpha|t|)^{\nu}K_{\nu}(\alpha|t|)$, where $K_{\nu}$, is a modified Bessel function of the second kind of order $\nu$. The latter process gets a global smoothness equal to $(\lfloor\nu\rfloor,\nu-\lfloor\nu\rfloor)$, see @St00 p.31. Detailed examples, including different classes of stationary processes, can be found in @BV08 [@BV11].
Note that, for processes with stationary increments, the function $d_0$ is reduced to a constant. Of course, cases with non constant $d_0(\cdot)$ are allowed as well as processes with a smooth enough trend. In particular, for some sufficiently smooth functions $a$ and $m$ on $[0,T]$, the process $Y(t)=a(t)X(t)+m(t)$ will also fulfills Assumption A\[h21\], see lemma \[l61\] for details.
Let us turn now to the description of observation points. We consider that the process $X$ is observed at $(n+1)$ instants, denoted by $$0=t_{0,n} < t_{1,n} < \dotsb < t_{n,n}\le T$$ where the $t_k := t_{k,n}$ form a regular sequence design. That is, they are defined as quantiles of a fixed positive and continuous density $\psi$ on $[0,T]$: $$\int_{0}^{t_k} \psi(s) \mathrm{d}s = \frac{k{\delta_n}}{T}, \;\;\; k=0,\dotsc,n,$$ for ${\delta_n}$ a positive sequence such that ${\delta_n}\to 0$ and $n{\delta_n}\to T(-)$. Clearly, if $\psi$ is the uniform density on $[0,T]$, one gets the equidistant case. Some further assumptions on $\psi$ are needed to get some control over the $t_k$’s.
\[h22\] The density $\psi$ satisfies:
- ${\displaystyle}\inf_{t\in[0,T]} \psi(t)>0$,
- ${\forall\,}(s,t)\in [0,T]^2$, ${\lv \psi(s) - \psi(t){\right\vert}} \le L {\lv s-t{\right\vert}}^{\alpha}$, for some $\alpha \in ]0,1]$.
These hypothesis ensure a controlled spacing between two distinct points of observation, see Lemma \[l62\]. From a practical point of view, this flexibility may allow to recognize inhomogeneities in the process (e.g. presence of pics in environmental pollution monitoring, see @Gi87 and references therein) or else to describe situations where data are collected at equidistant times but become irregularly spaced after some screening (see for example the wolfcamp-aquifer data in @Cr93 p. 212).
The methodology
---------------
In this part, we want to give background ideas about the construction of our estimate of the global regularity $({r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$, when the process is observed on a non equidistant grid. The main idea is to introduce divided differences, quantities generalizing the finite differences, studied by @BV11 [@BV12]. Let us first recall that the unique polynomial of degree $r$ that interpolates a function $g$ at $r+1$ points $t_k,\dotsc,t_{k+ur}$ (for some positive integer $u$) can be written as: $$\begin{gathered}
g[t_k] + g[t_k,t_{k+u}](t-t_k) + g[t_k,t_{k+u},t_{k+2u}](t-t_k)(t-t_{k+u}) \\+ \dotsb + g[t_k,\dotsc,t_{k+ru}](t-t_k)\dotsm (t-t_{k+(r-1)u})
\label{e22}\end{gathered}$$ where the divided differences $g[\dots]$ are defined by $g[t_k] = g(t_k)$ and for $j=1,\dotsc,r$ (using the Lagrange’s representation): $$g[t_k,\dotsc,t_{k+ju}] = \sum_{i=0}^j \frac{g(t_{k+iu})}{\prod_{m=0,m\not=i}^j (t_{k+iu} - t_{k+mu})}.$$ In particular, we write $g[t_k,\dotsc,t_{k+ru}] = \sum_{i=0}^r b_{ikr}^{(u)} g(t_{k+iu})$ with $$\label{e23}
b_{ikr}^{(u)} := \frac{1}{\prod_{m=0,m\not=i}^r (t_{k+iu} - t_{k+mu})}.$$
These coefficients are of particular interest. In fact their first non-zero moments are of order $r$. We can also derive an explicit bound and an asymptotic expansion for $b_{irk}^{(u)}$, see lemma \[l63\] for details.
Then, for positive integers $r$ and $u$, we consider the $u$-dilated divided differences of order $r$ for $X$: $$\label{e24}
D_{r,k}^{(u)}\,X=\sum_{i=0}^{r} b_{ikr}^{(u)}X(t_{k+iu}), \;\;\; k=0,\dotsc,n- u r$$ with $b_{ikr}^{(u)}$ defined by . Note that, if $\psi(t) = T^{-1}{\mathds{1}}_{[0,T]}(t)$, the sequence of designs is equidistant, that is $t_{k,n}=k\delta_n$, and divided differences turn to be finite differences. More precisely, for the sequence $a_{i,r}= \binom{r}{i} (-1)^{r-i}$, let define the finite differences $\Delta_{r,k}^{(u)} = \sum_{i=0}^r a_{i,r} X((k+iu){\delta_n})$. Noticing that in the case of equally spaced observations, $$b_{ikr}^{(u)} = \frac{(u{\delta_n})^{-r}}{\prod_{m=0,m\not= i}^{r} (i-m)} = \frac{a_{i,r}}{r!}(u{\delta_n})^{-r},$$ we deduce the relation $D_{r,k}^{(u)}\,X= \frac{(u{\delta_n})^{-r}}{r!} \Delta_{r,k}^{(u)}\,X$. From now on, we set the $u$-dilated quadratic variations of $X$ of order $r$: ${\displaystyle}\overline{\big( D_r^{(u)}X\big)^2}= \frac{\sum_{k=0}^{n_r} \big( D_{r,k}^{(u)}\,X\big)^2}{n_r+1}$ with $n_r := n-ur$. Construction of our estimators is based on following asymptotic properties concerning the mean behavior of $\overline{\big( D_r^{(u)}X\big)^2}$: $$(P)\left\{\begin{tabular}{l}\textrm{the quantity ${\mathds{E}}\, \overline{\big(D_{r} ^{(u)}X\big)^2}$ is of order $(u/n)^{-2(p-{\beta_{{\scriptscriptstyle}0}})}$ for $r={r_{{\scriptscriptstyle}0}}+p$} \\
\textrm{for $p=1,2$, and gets a finite non zero limit when $r\leq {r_{{\scriptscriptstyle}0}}$.}
\end{tabular}\right.$$
See proposition \[p61\] for precise results. These results imply that a good choice of $r$ (namely $r={r_{{\scriptscriptstyle}0}}+1$ or ${r_{{\scriptscriptstyle}0}}+2$) could provide an estimate of ${\beta_{{\scriptscriptstyle}0}}$, at least with an adequate combination of $u$-dilated quadratic variations of $X$. To this end, we propose a two steps procedure:
- *Estimation of ${r_{{\scriptscriptstyle}0}}$.*\
Based on $D_{r,k}^{(1)}X$, we estimate ${r_{{\scriptscriptstyle}0}}$ with $$\label{e25}
\widehat{{r_{{\scriptscriptstyle}0}}} =\min\Big\{r\in\{2,\dotsc,m_n\} : \overline{\big( D_r^{(1)}X\big)^2}\geq
n^2b_n\Big\}-2.$$ If the above set is empty, we fix $\widehat{{r_{{\scriptscriptstyle}0}}}= l_0$ for an arbitrary value $l_0\not\in {\n}_{{\scriptscriptstyle}0}$. Here, $m_n\to\infty$ but if an upper bound $B$ is known for ${r_{{\scriptscriptstyle}0}}$, one has to choose $m_n=B+2$. The threshold $b_n$ is a positive sequence chosen such that : $n^{-2(1-{\beta_{{\scriptscriptstyle}0}})}b_n \rightarrow 0$ and $n^{2{\beta_{{\scriptscriptstyle}0}}}b_n\to \infty$ for all ${\beta_{{\scriptscriptstyle}0}}\in ]0,1[$. For example, omnibus choices are given by $b_n=(\ln n)^{\alpha}$, $\alpha\in{{\mathds{R}}}$.
- *Estimation of ${\beta_{{\scriptscriptstyle}0}}$.*\
Next, we derive two families of estimators for ${\beta_{{\scriptscriptstyle}0}}$, namely $\widehat{\beta}_n^{(p)}$, with either $p=1$ or $p=2$ and $u,v$ $(u<v)$ given integers: $$\widehat{\beta}_n^{(p)}:= \widehat{\beta}_n^{(p)}(u,v)= p+ \frac{1}{2}\frac{ \ln\Big(\overline{\big(
D_{\widehat{{r_{{\scriptscriptstyle}0}}}+p}^{(u)}X\big)^2}\Big) -\ln\Big( \overline{ \big( D_{\widehat{{r_{{\scriptscriptstyle}0}}}+p} ^{(v)}X\big)^2}\Big)}{\ln(u/v)}.$$
[\[RqKW\] @KW97 proposed estimators of $2{\beta_{{\scriptscriptstyle}0}}=\alpha$ based on ordinary and generalized least squares on the logarithm of the quadratic variations versus logarithm of a vector of values $u$, more precisely $$\hat{\alpha}^{(p)}=\frac{({{\mathbf 1}}\T W {{\mathbf 1}})({{\mathbf u}}\T W {{\mathbf Q}}^{(p)})-({{\mathbf 1}}\T W {{\mathbf u}})({{\mathbf 1}}\T W {{\mathbf Q}}^{(p)})}{({{\mathbf 1}}\T W {{\mathbf 1}})({{\mathbf u}}\T W {{\mathbf u}})-({{\mathbf 1}}\T W {{\mathbf u}})^2}$$ where ${{\mathbf 1}}$ is the $m$-vector of $1$s, ${{\mathbf u}}=(\ln(u),u=1,\ldots, m)\T$, ${{\mathbf Q}}^{(p)}=(\ln(\overline{\big(\Delta_{p}^{(u)}X\big)^2})$, $u=1,\ldots, m)\T$ and $W$ is either the identity matrix $I_m$ of order $m\times m$ or a matrix depending on $(n,{\beta_{{\scriptscriptstyle}0}})$ which converges to the asymptotic covariance of $n^{1/2}\big(\overline{(\Delta_{p}^{(u)}X)^2}-{\mathds{E}}\overline{(\Delta_{p}^{(u)}X)^2}\big)$. The ordinary least square estimator – corresponding to $W=I_m$– is denoted by $\widehat{\alpha}_{OLS}^{(p)}$, where $p$ is adapted to the regularity of the process (supposed to be known in their work). The choice $p=1$, with the sequence $(-1,1)$, leads to the estimator studied by @CH94. Remark that, for $(u,v)=(1,2)$, one gets $\widehat{\beta}_n^{(1)}= \widehat{\alpha}^{(0)}_\text{OLS}$ and $\widehat{\beta}_n^{(2)}= \widehat{\alpha}^{(1)}_\text{OLS}$ but, even in this equidistant case, new estimators may be derived with other choices of $(u,v)$ such as $(u,v)=(1,4)$ (which seems to perform well, see Section \[sub52\]).]{}
Asymptotic results {#Asymptres}
==================
In @BV11, an exponential bound is obtained for ${\mathds{P}}( \widehat{{r_{{\scriptscriptstyle}0}}} \not= {r_{{\scriptscriptstyle}0}})$ in the equidistant case, implying that, almost surely for $n$ large enough, $\widehat{{r_{{\scriptscriptstyle}0}}}$ is equal to ${r_{{\scriptscriptstyle}0}}$. Here, we generalize this result to regular sequence designs but also, we complete it with the average behavior of $\widehat{{r_{{\scriptscriptstyle}0}}}$.
\[t31\] Under Assumption A\[h21\] (fulfilled with $p=1$ or $p=2$) and A\[h22\], we have ${\displaystyle}{\mathds{P}}(\widehat{{r_{{\scriptscriptstyle}0}}} \not= {r_{{\scriptscriptstyle}0}}) = {{\mathcal}O} \Big( \exp\big( - {\varphi}_n(p)\big)\Big)$ and ${\displaystyle}{\mathds{E}}(\widehat{{r_{{\scriptscriptstyle}0}}} - {r_{{\scriptscriptstyle}0}})^2 = {\mathcal O} \Big( m_n^3 \exp\big(- {\varphi}_n(p) \big)\Big)$, where, for some positive constant $C_1({r_{{\scriptscriptstyle}0}})$, ${\varphi}_n(p)$ is defined by $${\varphi}_n(p) = C_1({r_{{\scriptscriptstyle}0}})\times \begin{cases} {\displaystyle}n{\mathds{1}}_{]0,\frac{1}{2}[}({\beta_{{\scriptscriptstyle}0}})+ n(\ln n)^{-1}
{\mathds{1}}_{\{\frac{1}{2}\}}({\beta_{{\scriptscriptstyle}0}})+n^{2-2{\beta_{{\scriptscriptstyle}0}}}{\mathds{1}}_{]\frac{1}{2},1[}({\beta_{{\scriptscriptstyle}0}}) \text{ if } p=1\\ {\displaystyle}n \text{ if } p=2.\end{cases}$$
Remark that one may choose $m_n$ tending to infinity arbitrary slowly. Indeed, the unique restriction is that ${r_{{\scriptscriptstyle}0}}$ belongs to the grid $ \{1,\dotsc,m_n\}$ for $n$ large enough. From a practical point of view, one may choose a preliminary fixed bound $B$, and, in the case where the estimator return the non-integer value $l_0$, replace $B$ by $B'$ greater than $B$.
The bias of ${\widehat{\beta}_n^{(p)}}$ will be controlled by a second-order condition of local stationarity, more specifically we have to strengthen the relation in: $$\label{e31}
\lim_{h\to 0} \sup_{\substack{s,t\in[0,T],\\{\lv s-t{\right\vert}} \le h,\\s\not=t}} {\lv \,
{\lv s-t{\right\vert}}^{-{\beta_{{\scriptscriptstyle}1}}}\Big(\frac{{\mathds{E}}\big(X^{({r_{{\scriptscriptstyle}0}})}(s)
-X^{({r_{{\scriptscriptstyle}0}})}(t)\big)^2}{{\lv s-t{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}}} - d_0(t)\Big)-d_1(t){\right\vert}} = 0$$ for a positive ${\beta_{{\scriptscriptstyle}1}}$ and continuous function $d_1$.
\[t32\] If relation , Assumption A\[h21\] with $p=1$ or $p=2$, and A\[h22\] are fulfilled, we obtain $$\limsup_{n\to\infty} V_n^{(p)} {\lv {\widehat{\beta}_n^{(p)}}- {\beta_{{\scriptscriptstyle}0}}{\right\vert}} \le C_1(p)\;\;\;\text{a.s.}$$ where $C_1(p)$ is some positive constant and $$\begin{aligned}
V_n^{ (1) }&= \min\Big(n^{{\beta_{{\scriptscriptstyle}1}}},\sqrt{\frac{n}{\ln n}} {\mathds{1}}_{]0,\frac{3}{4}[}({\beta_{{\scriptscriptstyle}0}})
+\frac{\sqrt{n}}{\ln n}{\mathds{1}}_{\{\frac{3}{4}\}}({\beta_{{\scriptscriptstyle}0}})+ \frac{n^{2(1-{\beta_{{\scriptscriptstyle}0}})}}{\ln
n}{\mathds{1}}_{]\frac{3}{4},1[}({\beta_{{\scriptscriptstyle}0}})\Big),\\
V_n^{ (2) }&= \min\Big(n^{{\beta_{{\scriptscriptstyle}1}}},\sqrt{\frac{n}{\ln n}} \Big).\end{aligned}$$
[For stationary gaussian processes, @KW97 give the mean square error and convergence in distribution of their estimator described in remark \[RqKW\]. They obtained the same rate up to a logarithmic order, due here to almost sure convergence, for both families $p=1$ and $p=2$. The asymptotic distribution is either of Gaussian or of Rosenblatt type depending on ${\beta_{{\scriptscriptstyle}0}}$ less or greater than $3/4$. @IL97 introduced an estimator of $H=2({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}})$ (for stationary increment processes) with a global linear regression approach, based on an asymptotic equivalent of the quadratic variation and using some adequate family of sequences. For ${r_{{\scriptscriptstyle}0}}=0$, their approach matches with the previous one, with $p=1$, in using dilated sequence of type $a_{jr}= \binom{r}{j} (-1)^{r-j}$. Assuming a known upper bound on ${r_{{\scriptscriptstyle}0}}$, they derived convergence in distribution to a centered Gaussian variable with rate depending on ${\beta_{{\scriptscriptstyle}0}}$–root-$n$ for ${\beta_{{\scriptscriptstyle}0}}\leq3/4$ and $n^{1/2-\alpha(2{\beta_{{\scriptscriptstyle}0}}-3/2)}$, where $\delta_n=n^{-\alpha}$ and $\alpha<1$, for ${\beta_{{\scriptscriptstyle}0}}>3/4$. In this last case, to obtain a gaussian limit they have to assume that ${r_{{\scriptscriptstyle}0}}$ is known, the observation interval is no more bounded, and the rate of convergence is lower than root-$n$.]{}
Approximation and integration {#AppInt}
==============================
Results for plug-in estimation
------------------------------
A classical and interesting topic is approximation and/or integration of a sampled path. An extensive literature may be found on these topics with a detailed overview in the recent monograph of @Rit00. The general framework is as follows : let $ X=\{X_t, \, t\in [0,1] \}$, be observed at sampled times $t_{0,n},\dotsc,t_{n,n}$ over \[a,b\], more simply denoted by $t_{0},\dotsc,t_{n}$. Approximation of $X(\cdot)$ consists in interpolation of the path on $[a,b]$, while weighted integration is the calculus of $\mathcal{I}_{\rho}=\int_a^b X(t) \rho(t) {\,{\mrm{d}}t}$ for some positive and continuous weight function $\rho$. These problems are closely linked, see e.g. @Rit00 p. 19-21. Closely to our framework of local stationary derivatives, we may refer more specifically to works of @PRW04 for approximation and @Be98 for integration. For sake of clarity, we give a brief summary of their obtained results. In the following, we denote by ${{\mathcal}H}({r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$ the family of Gaussian processes having ${r_{{\scriptscriptstyle}0}}$ derivatives in quadratic mean and ${r_{{\scriptscriptstyle}0}}$-th derivative with Hölderian regularity of order ${\beta_{{\scriptscriptstyle}0}}\in]0,1[$. For measurable $g_i(\cdot)$, we consider the approximation ${{\mathcal}A}_{n, g} (t) = \sum_{i=0}^n
X(t_i) g_i(t)$ and the corresponding weighted and integrated $L^2$-error $e_{\rho}({{\mathcal}A}_{n,g})$ with $e_{\rho}^2({{\mathcal}A}_{n,g}) = \int_a^b {\mathds{E}}{\lv X(t) - {{\mathcal}A}_{n, g}(t){\right\vert}}^2 \rho(t){\,{\mrm{d}}t}$. For $X\in{{\mathcal}H}({r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$ and known $({r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$, @PRW04 have shown that $$\begin{gathered}
0< c({r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}}) \le \varliminf_{n\to\infty} n^{{r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}}}\inf_{g} e_{\rho}({{\mathcal}A}_{n,g})
\\ \le \varlimsup_{n\to\infty} n^{{r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}}} \inf_{g} e_{\rho}({{\mathcal}A}_{n,g}) \le C({r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})<+\infty\end{gathered}$$ for equidistant sampled times $t_1,\dotsc,t_n$ and Gaussian processes defined and observed on the half-line $[0,+\infty[$. Of course, optimal choices of functions $g_i$, giving a minimal error, depend on the unknown covariance function of $X$.
For weighted integration, the quadrature is denoted by ${{\mathcal}Q}_{n, d}= \sum_{i=0}^n
X(t_i) d_i$ with well-chosen constants $d_i$ (typically, one may take $d_i = \int_a^b g_i(t){\,{\mrm{d}}t}$). For known $({r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$, a short list of references could be:
- @SY68 [@SY70] with ${r_{{\scriptscriptstyle}0}}=0$ or 1, ${\beta_{{\scriptscriptstyle}0}}=\frac{1}{2}$ and known covariance,
- @BC92 for arbitrary ${r_{{\scriptscriptstyle}0}}$ and ${\beta_{{\scriptscriptstyle}0}}=\frac{1}{2}$,
- @St95 for stationary processes and ${r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}}<\frac{1}{2}$,
- @Ri96 for minimal error, under Sacks and Ylvisaker’s conditions, and with arbitrary ${r_{{\scriptscriptstyle}0}}$.
Let us set $e^{2}_{\rho}({{\mathcal}Q}_{n,d})={\mathds{E}}{\lv I_{\rho} - {{\mathcal}Q}_{n, d}{\right\vert}}^2 $, the mean square error of integration. In the stationary case and for known ${r_{{\scriptscriptstyle}0}}$, @Be98 established the following exact behavior: If $\rho \in C^{{r_{{\scriptscriptstyle}0}}+3}([a,b])$ then for some given quadrature ${{\mathcal}Q}_{n,d^{\ast}({r_{{\scriptscriptstyle}0}})}$ on $[a,b]$, $$n^{{r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}}+\frac{1}{2}}\, e_{\rho}({{\mathcal}Q}_{n,d^{\ast}({r_{{\scriptscriptstyle}0}})})\xrightarrow[n\to\infty]{}c_{{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}}} (\int_a^b \rho^2(t) \psi^{-(2({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}})+1)}(t){\,{\mrm{d}}t})^{\frac{1}{2}}$$ where $\psi$ is the density relative to the regular sampling $\{t_{1},\dotsc,t_n\}$. Moreover, following @Ri96, it appears that this last result is optimal under Sacks and Ylvisaker’s conditions. Finally, @IL97b have proposed a quadrature, requiring only an upper bound on ${r_{{\scriptscriptstyle}0}}$, also with an error of order $ {{\mathcal}O} \big( n^{-({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}}+\frac{1}{2})} \big)$.
All these results shown the importance of well estimating ${r_{{\scriptscriptstyle}0}}$ and motivate ourself to focus on plugged-in interpolators, namely those using Lagrange polynomial of order estimated by $\widehat{{r_{{\scriptscriptstyle}0}}}$. More precisely, Lagrange interpolation of order $r\ge 1$ is defined by $$\label{e41}
{\widetilde}{X}_r(t)=\sum_{i=0}^{r} L_{i,k,r}(t) X \big( t_{kr+i} \big),\text{ with } L_{i,k,r}(t) =
\prod_{\substack{j=0\\j\not=i}}^{r}\frac{(t- t_{kr+j})}{t_{kr+i} - t_{kr+j}},$$ for ${\displaystyle}t\in {{\mathcal}I}_k := \Big[ t_{kr}, t_{kr+r}\Big]$, ${\displaystyle}k=0,\dotsc,\lfloor \frac{n}{r}\rfloor-2$ and ${\displaystyle}{{\mathcal}I}_{\lfloor \frac{n}{r}\rfloor-1} = \Big[ \lfloor t_{\lfloor(\frac{n}{r}\rfloor-1)r},T\Big]$.
Our plugged method will consist in the approximation given by ${{\mathcal}A}_{n,L}(t) = {\widetilde}{X}_{\max(\widehat{{r_{{\scriptscriptstyle}0}}},1)}(t)$, $t\in [0,T]$, and quadrature by ${{\mathcal}Q}_{n,L} = \int_0^T{\widetilde}{X}_{\widehat{{r_{{\scriptscriptstyle}0}}}+1}(t)\,\rho(t){\,{\mrm{d}}t}$. Indeed, Lagrange polynomials are of easy implementation and by the result of @PRW04 with known ${r_{{\scriptscriptstyle}0}}$, they reach the optimal rate of approximation, $n^{-({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}})}$ without requiring knowledge of covariance. Our following result shows also that the associate quadrature has the expected rate $n^{-({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}}+\frac{1}{2})}$. Indeed, in the weighted case and for $T>0$, we obtain the following asymptotic bounds in the case of a regular design.
\[t41\] Suppose that conditions A\[h21\](i)-(ii) and A\[h22\] hold, choose a logarithmic order for $m_n$ in and consider a positive and continuous weight function $\rho$.
- Under condition A\[h21\](iii-1), we have $$e_{\rho} ({\rm{app}}\big(\widehat{{r_{{\scriptscriptstyle}0}}})\big) := \Big(\int_0^T {\mathds{E}}{\lv X(t) - \tX_{\max(\widehat{{r_{{\scriptscriptstyle}0}}},1)}(t){\right\vert}}^2 \rho(t) {\,{\mrm{d}}t}\Big)^{1/2}
={{\mathcal}O}\big(n^{-({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}})}\big),$$
- if condition A\[h21\](iii-2) holds: $$e_{\rho} ({\rm{int}}\big(\widehat{{r_{{\scriptscriptstyle}0}}})\big):= \Big({\mathds{E}}{\lv \int_0^T (X(t) - \tX_{\widehat{{r_{{\scriptscriptstyle}0}}}+1}(t) ) \rho(t) {\,{\mrm{d}}t}{\right\vert}}^2\Big)^{1/2} ={{\mathcal}O} \big(n^{-({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}}+\frac{1}{2})}\big).$$
In conclusion, expected rates for approximation and integration are reached by plugged Lagrange piecewise polynomials. Of course if ${r_{{\scriptscriptstyle}0}}$ is known, this last result holds true with $\widehat{{r_{{\scriptscriptstyle}0}}}$ replaced by ${r_{{\scriptscriptstyle}0}}$.
Simulation results
------------------
The figure \[figinterpol\] is obtained using $1000$ simulated sample paths observed in equally spaced points on $[0,1]$. This figure illustrates results of approximation for different processes. The logarithm of empirical integrated mean square error (in short IMSE), i.e. $e_{1}^2(\rm{app}\big(\widehat{{r_{{\scriptscriptstyle}0}}})\big)$, is drawn as a function of $\ln(n)$ with a range of sample size from $ 25$ to $1000$. We may notice that we obtain straight lines with slope very near to $-H=-2({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}})$. Since the Ornstein-Uhlenbeck process is a scaled time-transformed Wiener process, intercepts are different contrary to stationary versus non-stationary continuous ARMA processes.
[![Logarithm of $e_{1}^2(\rm{app}\big(\widehat{{r_{{\scriptscriptstyle}0}}})\big)$, i.e. the IMSE in function of $\ln(n)$, for different processes. The dashed line corresponds to Brownian motion, long dashed line to O.U., dashed line to non stationary CARMA(2,1), dotted-dashed line to stationary CARMA(2,1) and solid line to non stationary CARMA(3,1). The small triangles near lines are here to indicate the theoretic slope.[]{data-label="figinterpol"}](resinterpol.eps "fig:"){width="9cm"}]{}
Numerical results
=================
In this section, to numerically compare our estimators with existing ones, we restrict ourselves to the equidistant case with the choice $\psi(t) = \frac{1}{T} {\mathds{1}}_{[0,T]}(t)$. As noticed before, we get for $a_{i,r}= \binom{r}{i} (-1)^{r-i}$ and $\Delta_{r,k}^{(u)} = \sum_{i=0}^r a_{i,r} X((k+iu){\delta_n})$, the relation $D_{r,k}^{(u)}\,X= \frac{(u{\delta_n})^{-r}}{r!} \Delta_{r,k}^{(u)}\,X$ implying in turn that $$\label{e51}
\widehat{H}_n^{(p)} = \frac{ \ln\Big(\overline{\big(
\Delta_{\widehat{{r_{{\scriptscriptstyle}0}}}+p}^{(u)}X\big)^2}\Big) -\ln\Big( \overline{ \big( \Delta_{\widehat{{r_{{\scriptscriptstyle}0}}}+p}
^{(v)}X\big)^2}\Big)}{\ln(u/v)}$$ is a consistent estimator of $H=2({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}})$. All the simulation results are obtained by simulation of trajectories using two different methods : for stationary processes or with stationary increments we use the procedure described in @WC94 and for CARMA (continuous ARMA) processes, we use @TC00. Each of them consists in $n$ equally spaced observation points on $[0,1]$ and 1000 simulated sample paths. All computations have been performed with the R software [@R12].
Results for estimators of ${r_{{\scriptscriptstyle}0}}$
-------------------------------------------------------
This section is dedicated to the numerical properties of two estimators of ${r_{{\scriptscriptstyle}0}}$. We consider the estimator introduced by @BV11, derived from in the equidistant case. An alternative, says $\tilde r_n$, based on Lagrange interpolator polynomials was proposed by @BV08. More precisely, for ${\delta_n}=n^{-1}$ et $T=1$, $\tilde r_n$ is defined by $$\begin{gathered}
\tilde{r}_n =\min\Big\{r\in\{1,\ldots,m_n\} :
\frac{1}{r{\widetilde}{n}_r}\sum_{k=0}^{r{\widetilde}{n}_r -1} \big(X\Big( {
\frac{2k+1}{n}}\Big)-\tXr\Big({\frac{2k+1}{n}}\Big)\big)^2 \\ \geq n^{-2r} b_n\Big\}-1\end{gathered}$$ where ${\widetilde}{n}_r= \lfloor \frac{n}{2r} \rfloor$ and $\tXr(s)$ is defined for all $s \in [0,1]$ and each $r\in \{1,\dotsc,m_n\}$ in the following way : there exist $k=0,\dotsc, {\widetilde}{n}_r-1$ such that for $t\in
{\mathcal I}_{2k} := [\frac{2kr}{n}, \frac{2(k+1)r}{n}]$, the piecewise Lagrange interpolation of $X(t)$, ${\widetilde}{X}_r(t)$, is given by ${\widetilde}{X}_r(t)=\sum\limits_{i=0}^{r} L_{i,k,r}(t) X \big( (kr+i)n^{-1} \big)$, with $L_{i,k,r}(t)=
\prod_{\substack{j=0\\j\not=i}}^{r}\frac{(t- (kr+j)n^{-1})}{(i-j)n^{-1}}$.
Both estimators use the critical value $b_n$ which is involved in detection of the jump. Here, due to convergence properties, we make the choice $b_n = (\ln n)^{-1}$. Table \[tabNS\] illustrates the strong convergence of both estimators and shows that this convergence is valid even for small number of observation points $n$, up to 10 for the estimator $\widehat{{r_{{\scriptscriptstyle}0}}}$. We may noticed that, in the case of bad estimation, our estimators overestimate the number of derivatives. Remark also that, for identical sample paths, $\widehat{{r_{{\scriptscriptstyle}0}}}$ seems to be more robust than $\tilde r_n$. This behavior was expected as the latter uses only half of the observations for the detection of the jump in quadratic mean. In these first results, processes have fractal index ${\beta_{{\scriptscriptstyle}0}}$ equals to $1/2$, but alternative choices of ${\beta_{{\scriptscriptstyle}0}}$ are of interest, so we consider the fractional Brownian motion (in short fBm) and the integrated fractional Brownian motion (in short ifBm), with respectively ${r_{{\scriptscriptstyle}0}}=0$ and ${r_{{\scriptscriptstyle}0}}=1$ and various values of ${\beta_{{\scriptscriptstyle}0}}$.
----------------------------------------------------------------------- ------- ------- ------- ------- ------- -------
event 10 25 10 25 10 25
$\tilde r_n={r_{{\scriptscriptstyle}0}}$ 0.995 1.000 0.913 1.000 0.585 0.999
$\tilde r_n={r_{{\scriptscriptstyle}0}}+1$ 0.005 0.000 0.087 0.000 0.415 0.001
$\widehat{{r_{{\scriptscriptstyle}0}}}={r_{{\scriptscriptstyle}0}}$ 1.000 1.000 1.000 1.000 0.999 1.000
$\widehat{{r_{{\scriptscriptstyle}0}}}={r_{{\scriptscriptstyle}0}}+1$ 0.000 0.000 0.000 0.000 0.001 0.000
----------------------------------------------------------------------- ------- ------- ------- ------- ------- -------
: [Value of the empirical probability that $\widehat{{r_{{\scriptscriptstyle}0}}}$ or $\tilde r_n$ equals ${r_{{\scriptscriptstyle}0}}$ or ${r_{{\scriptscriptstyle}0}}+1$ with $n=10$ or $25$. \[tabNS\]]{}
Table \[tabfBmifBm\] shows that $\widehat{{r_{{\scriptscriptstyle}0}}}$ succeeds in estimating the true regularity for ${\beta_{{\scriptscriptstyle}0}}$ up to 0.9. Of course the number of observations must be large enough and, even more important for large values of ${r_{{\scriptscriptstyle}0}}$ when ${\beta_{{\scriptscriptstyle}0}}\ge 0.95$. This latter result is clearly apparent when one compares the errors obtained for an ifBm with ${\beta_{{\scriptscriptstyle}0}}=0.95$ and a fBm with ${\beta_{{\scriptscriptstyle}0}}=0.95$. Finally, we can see once more that $\tilde r_n$ is less robust against increasing ${\beta_{{\scriptscriptstyle}0}}$, whereas our simulations have shown that, for $n=2000$ and each simulated path, the estimator $\widehat{{r_{{\scriptscriptstyle}0}}}$ is able to distinguish processes with regularity $(0,0.98)$ and $(1,0.02)$, an almost imperceptible difference!
Estimation of $H$ and ${\beta_{{\scriptscriptstyle}0}}$ {#sub52}
-------------------------------------------------------
This part is dedicated to the numerical properties of estimators $\widehat H_n^{(p)}$, for $p=1$ or $2$ using the values $u=1$ and $v=4$ (giving more homogeneous results than $u=1$ and $v=2$). It ends with real data examples.
### Quality of estimation
For the numerical part, we focus on the study of fBm, ifBm and, CARMA(3,1) with ${r_{{\scriptscriptstyle}0}}=2$, ${\beta_{{\scriptscriptstyle}0}}=0.5$. Table \[tabHp\] illustrates the performance of our estimators when ${\beta_{{\scriptscriptstyle}0}}$, ${r_{{\scriptscriptstyle}0}}$ are increasing: we compute the empirical mean-square error from our 1000 simulated sample paths and $n=1000$ equally spaced observations are considered. It appears that, contrary to $\widehat H_n^{(2)}$, the estimator $\widehat H_n^{(1)}$ slightly deteriorates for values of ${\beta_{{\scriptscriptstyle}0}}$ greater than 0.8. This result is in agreement with the rate of convergence of Theorem \[t32\], that depends on ${\beta_{{\scriptscriptstyle}0}}$ for this estimator. The bias is negative and seems to be unsensitive to the value of ${r_{{\scriptscriptstyle}0}}$ but the mean-square error is slightly deteriorated from ${r_{{\scriptscriptstyle}0}}=0$ to ${r_{{\scriptscriptstyle}0}}=1$ in both cases. Finally, for ${\beta_{{\scriptscriptstyle}0}}<0.8$, $H_n^{(1)}$ seems preferable to $\widehat H_n^{(2)}$, possibly due to a lower variance of this estimator. Nevertheless, both estimators perform globally well on these numerical experiments.
### Asymptotic properties
Results of Theorem \[t32\] are also illustrated in Table \[tabvitesse\] where we have computed the regression of $\ln({\mathds{E}}|\widehat{H}_n^{(p)} - H|)$ on $\ln n$ for various values of $n$ and ${\mathds{E}}|\widehat{H}_n^{(p)} - H|$ estimated from our 1000 simulated sample paths. As expected, the slope (corresponding to our arithmetical rate of convergence) is constant and approximatively equal to 0.5 for $\widehat{H}_n^{(2)}$ while, for $\widehat{H}_n^{(1)}$, the decrease is apparent for high values of ${\beta_{{\scriptscriptstyle}0}}$. Finally, Figure \[figboxplot\] illustrates the behavior of the estimators $\widehat H_n^{(p)}$ with $p=1$ or $2$, for different values of the regularity parameter ${\beta_{{\scriptscriptstyle}0}}$. As we can see, boxplots deteriorates only slightly for $n=100$ and $250$ when ${\beta_{{\scriptscriptstyle}0}}$ increases from $0.5$ to $0.8$ but the dispersion for $\widehat H_n^{(2)}$ is quite larger. For ${\beta_{{\scriptscriptstyle}0}}=0.95$, $\widehat H_n^{(2)}$ clearly outperforms $\widehat H_n^{(1)}$ with $n=500$ observations. Estimation appears more difficult for smaller values of $n$, but it is a quite typical behavior in our considered framework.
![Each boxplot corresponds to 1000 estimations of $H$ by $\widehat{H}^{(1)}_n$ on the left and $\widehat{H}^{(2)}_n$ on the right of the graph. Each realization consists in $n$ equally spaced observations on $[0,1]$ of a fBm with ${\beta_{{\scriptscriptstyle}0}}=0.5$ (top), ${\beta_{{\scriptscriptstyle}0}}=0.8$ (middle), ${\beta_{{\scriptscriptstyle}0}}=0.95$ (bottom), where $n=100,250,500,750,1000,1250,1500,2000$. The solid line corresponds to the real value of $H$.[]{data-label="figboxplot"}](boxplot-fBm05-Hbv1.eps){height="4.2cm" width="5.5cm"}
[![Each boxplot corresponds to 1000 estimations of $H$ by $\widehat{H}^{(1)}_n$ on the left and $\widehat{H}^{(2)}_n$ on the right of the graph. Each realization consists in $n$ equally spaced observations on $[0,1]$ of a fBm with ${\beta_{{\scriptscriptstyle}0}}=0.5$ (top), ${\beta_{{\scriptscriptstyle}0}}=0.8$ (middle), ${\beta_{{\scriptscriptstyle}0}}=0.95$ (bottom), where $n=100,250,500,750,1000,1250,1500,2000$. The solid line corresponds to the real value of $H$.[]{data-label="figboxplot"}](boxplot-fBm05-Hbv2.eps "fig:"){height="4.2cm" width="5.5cm"}]{}
\
[![Each boxplot corresponds to 1000 estimations of $H$ by $\widehat{H}^{(1)}_n$ on the left and $\widehat{H}^{(2)}_n$ on the right of the graph. Each realization consists in $n$ equally spaced observations on $[0,1]$ of a fBm with ${\beta_{{\scriptscriptstyle}0}}=0.5$ (top), ${\beta_{{\scriptscriptstyle}0}}=0.8$ (middle), ${\beta_{{\scriptscriptstyle}0}}=0.95$ (bottom), where $n=100,250,500,750,1000,1250,1500,2000$. The solid line corresponds to the real value of $H$.[]{data-label="figboxplot"}](boxplot-fBm08-Hbv1.eps "fig:"){height="4.2cm" width="5.5cm"}]{}
[![Each boxplot corresponds to 1000 estimations of $H$ by $\widehat{H}^{(1)}_n$ on the left and $\widehat{H}^{(2)}_n$ on the right of the graph. Each realization consists in $n$ equally spaced observations on $[0,1]$ of a fBm with ${\beta_{{\scriptscriptstyle}0}}=0.5$ (top), ${\beta_{{\scriptscriptstyle}0}}=0.8$ (middle), ${\beta_{{\scriptscriptstyle}0}}=0.95$ (bottom), where $n=100,250,500,750,1000,1250,1500,2000$. The solid line corresponds to the real value of $H$.[]{data-label="figboxplot"}](boxplot-fBm08-Hbv2.eps "fig:"){height="4.2cm" width="5.5cm"}]{}
\
![Each boxplot corresponds to 1000 estimations of $H$ by $\widehat{H}^{(1)}_n$ on the left and $\widehat{H}^{(2)}_n$ on the right of the graph. Each realization consists in $n$ equally spaced observations on $[0,1]$ of a fBm with ${\beta_{{\scriptscriptstyle}0}}=0.5$ (top), ${\beta_{{\scriptscriptstyle}0}}=0.8$ (middle), ${\beta_{{\scriptscriptstyle}0}}=0.95$ (bottom), where $n=100,250,500,750,1000,1250,1500,2000$. The solid line corresponds to the real value of $H$.[]{data-label="figboxplot"}](boxplot-fBm095-Hbv1.eps){height="4.2cm" width="5.5cm"}
[![Each boxplot corresponds to 1000 estimations of $H$ by $\widehat{H}^{(1)}_n$ on the left and $\widehat{H}^{(2)}_n$ on the right of the graph. Each realization consists in $n$ equally spaced observations on $[0,1]$ of a fBm with ${\beta_{{\scriptscriptstyle}0}}=0.5$ (top), ${\beta_{{\scriptscriptstyle}0}}=0.8$ (middle), ${\beta_{{\scriptscriptstyle}0}}=0.95$ (bottom), where $n=100,250,500,750,1000,1250,1500,2000$. The solid line corresponds to the real value of $H$.[]{data-label="figboxplot"}](boxplot-fBm095-Hbv2.eps "fig:"){height="4.2cm" width="5.5cm"}]{}
$\widehat{H}^{(1)}_n$$\widehat{H}^{(2)}_n$
### Impact of misspecification of regularity
Next, Table \[tabHro\] illustrates the impact of estimating $H$ when the order $r$ in quadratic variation is misspecified. In fact estimating ${\beta_{{\scriptscriptstyle}0}}$ requires the knowledge of ${r_{{\scriptscriptstyle}0}}$ or an upper bound of it. On the other hand, working with a too high value of ${r_{{\scriptscriptstyle}0}}$ may induce artificial variability in estimation, so a precise estimation of ${r_{{\scriptscriptstyle}0}}$ is important. Here, our numerical results show that, if the order $r$ of quadratic variation used for estimating ${\beta_{{\scriptscriptstyle}0}}$ is less than ${r_{{\scriptscriptstyle}0}}+1$, then the quantity estimated is $2r$ and not $2(r+{\beta_{{\scriptscriptstyle}0}})$.
### Processes with varying trend or non constant function $d_0$
![Wiener process (solid) and its locally stationary transformation (dashed) used in Table \[tabSelenjev\].\[fig32\]](Selenjev.eps){width="5cm"}
All previous examples are locally stationary with a constant function $d_0$. Processes meeting our conditions but with no stationary increments may be constructed using Lemma \[l61\]. As an example, from $Y$ a standard Wiener process (${r_{{\scriptscriptstyle}0}}=0$, ${\beta_{{\scriptscriptstyle}0}}= 0.5$) or an integrated one (${r_{{\scriptscriptstyle}0}}=1$, ${\beta_{{\scriptscriptstyle}0}}= 0.5$), we simulate $X(t)= (t^{{r_{{\scriptscriptstyle}0}}+0.7}+1)\,Y(t)$ having the regularity $({r_{{\scriptscriptstyle}0}},0.5)$ and $d_0(t)$ equaling to $(t^{{r_{{\scriptscriptstyle}0}}+0.7}+1)^2$. Figure \[fig32\] illustrates a Wiener sample path and its transformation. Results are summarized in Table \[tabSelenjev\]: comparing with Table \[tabHp\] (${\beta_{{\scriptscriptstyle}0}}=0.5$), it appears that the estimation is only slightly damaged for ${r_{{\scriptscriptstyle}0}}=1$ but of the same order when ${r_{{\scriptscriptstyle}0}}=0$. Other non stationary processes may also be obtained by adding some smooth trend. To this aim, we used same sample paths as for Table \[tabHp\] with the additional trend $m(t)=(1+t)^2$, see Figure \[Figtrend\]. We may noticed in Table \[tabtrend\] that we obtain exactly the same results for the estimator $\widehat H_n^{(2)}$ and that only a slight loss is observed for $\widehat H_n^{(1)}$.
![Sample path of a fBm with ${\beta_{{\scriptscriptstyle}0}}=0.8$ (dashed line) and the same with a trend $m(t)=(1+t)^2$ (solid line).\[Figtrend\]](Trendbis.eps){width="5cm"}
---------------------- -------------------------------------------------------------------------------- --------------------------------------------------------------------------------
\[-5pt\] Wiener Integrated Wiener
$\widehat H_n^{(1)}$ [$ \begin{matrix} 0.0021 \\[-5pt] {{\scriptstyle (-0.0032)}} \end{matrix} $]{} [$ \begin{matrix} 0.0032 \\[-5pt] {{\scriptstyle (0.0061)}} \end{matrix} $]{}
$\widehat H_n^{(2)}$ [$ \begin{matrix} 0.0043 \\[-5pt] {{\scriptstyle (-0.0042)}} \end{matrix} $]{} [$ \begin{matrix} 0.0058 \\[-5pt] {{\scriptstyle (-0.0091)}} \end{matrix} $]{}
---------------------- -------------------------------------------------------------------------------- --------------------------------------------------------------------------------
: [Value of MSE and bias (between brackets) for non constant $d_0(\cdot)$. \[tabSelenjev\]]{}
Real data
---------
Let us turn to examples based on real data sets. In this part, we compare our estimators of $H$ with those proposed by @CH94 [@KW97]. We compute estimated values by setting $(u,v)=(1,m)$ in with $m$ in $\{2,4,6,8,10\}$ while for $\widehat{\alpha}_{OLS}^{(p)}$, $p=1,2$, defined in Remark \[RqKW\] regression is carried out over ${{\mathbf u}}=(\ln(u),u=1,\ldots, m)\T$.
### Roller data
We first focus on roller height data introduced by @La94, which consists in $n=1150$ heights measure at 1 micron intervals along a drum of a roller. This example was already studied in @KW97: they noticed that local self similarity may hold at sufficiently fine scales, so the regularity ${r_{{\scriptscriptstyle}0}}$ was supposed to be zero. Indeed, our estimator $\widehat {r_{{\scriptscriptstyle}0}}$, directly used on the data with $b_n=1/\ln(n)$, gives $\widehat{r_{{\scriptscriptstyle}0}}=0$ (with a value of $n^{4-2}\overline{\big( \Delta_2 ^{(1)}X\big)^2}$ equal to $1172345$). Next, we compute the values obtained for the estimation of $H$ in Table \[tabroller\], where values of estimates proposed by @CH94 [@KW97] are also reported for comparison. It should be observed that our simplified estimators present a similar sensitivity to the choice of $m$.
$m$ $\widehat{H}_n^{(1)}$ $\widehat{\alpha}^{(0)}_{\text{OLS}}$ $\widehat{H}_n^{(2)}$ $\widehat{\alpha}^{(1)}_{\text{OLS}}$
----- ----------------------- --------------------------------------- ----------------------- ---------------------------------------
2 0.63 0.63 0.77 0.77
4 0.50 0.51 0.63 0.65
6 0.38 0.39 0.49 0.51
8 0.35 0.33 0.44 0.42
10 0.30 0.28 0.39 0.35
: Estimates in the roller height example \[tabroller\]
### Biscuit data
![(a) Curve drawing reflectance in function of wavelength, varying between $1100$ and $2498$. (b) Box-plots for both estimators on the left $\widehat H_n^{(1)}$, on the right $\widehat H_n^{(2)}$ for the $39$ curves and $(u,v)=(1,4)$.[]{data-label="figbiscuit"}](data-biscuit.eps){height="3.6cm"}
![(a) Curve drawing reflectance in function of wavelength, varying between $1100$ and $2498$. (b) Box-plots for both estimators on the left $\widehat H_n^{(1)}$, on the right $\widehat H_n^{(2)}$ for the $39$ curves and $(u,v)=(1,4)$.[]{data-label="figbiscuit"}](Boxplot-HH.eps){height="3.8cm"}
\(a) (b)
$m=2$ $m=4$ $m=6$ $m=8$ $m=10$
----------------------------- ------------- ------------- ------------- ------------- -------------
$\widehat{H}_n^{(1)}$ 3.60 (0.12) 3.67 (0.07) 3.65 (0.05) 3.62 (0.04) 3.59 (0.04)
$\alpha^{(1)}_{\text{OLS}}$ 3.60 (0.12) 3.67 (0.07) 3.66 (0.05) 3.63 (0.04) 3.60 (0.03)
$\widehat{H}_n^{(2)}$ 2.84 (0.45) 3.69 (0.30) 3.83 (0.24) 3.84 (0.19) 3.83 (0.16)
$\alpha^{(2)}_{\text{OLS}}$ 2.84 (0.45) 3.67 (0.31) 3.91 (0.23) 3.98 (0.18) 3.99 (0.14)
: Means of estimates in the biscuit example \[tab59\]
Now, in order to compare the (empirical) variances of these estimators, we consider a second example introduced by @BFV01. The experiment involved varying the composition of biscuit dough pieces and data consist in near infrared reflectance (NIR) spectra for the same dough. The 40 curves are graphed on the figure \[figbiscuit\]. Each represents the near-infrared spectrum reflectance measure at each $2$ nanometers from $1100$ to $2498$ nm, then $700$ observation points for each biscuit. According to @BFV01, the observation 23 appears as an outlier. We estimate ${r_{{\scriptscriptstyle}0}}$ for each of the left 39 curves, using the threshold $b_n=1$, which gives $\widehat{r_{{\scriptscriptstyle}0}}=1$ for each curve. Furthermore, the averaged mean quadratic variation $n^{2r-2}\overline{\big( D_r^{(1)}X\big)^2}$ equals to $0.33$ when $r=2$ and $122133$ when $r=3$, this explosion confirming the choice $\widehat{{r_{{\scriptscriptstyle}0}}}=3-2=1$. We turn to estimation of $H$, having in mind the comparison of our estimators together with $\alpha_\text{OLS}^{(p)}$ (where $p=1$ corresponds to the choice $(1,-2,1)$ for $a_{jr}$ and $p=2$ to the choice $(-1,3,-3,1)$). The results are summarized in Table \[tab59\] where it appears that, for order $\widehat{{r_{{\scriptscriptstyle}0}}}+2=3$, our estimator $\widehat{H}_n^{(2)}$ seems to be less sensitive toward high values of $m$. Also our simplified estimators present a similar variance to $\widehat{\alpha}_{OLS}^{(p)}$, $p=1,2$. To conclude this part, it should be noticed that for the 23rd curve, the choice $m=4$ gives $\widehat{H}_n^{(1)} = 3.64$ and $\widehat{H}_n^{(2)} = 3.55$. It appears that, in both cases, these values belong to the interquartile range obtained from the 39 curves, so at least concerning the regularity, the curve 23 should not be considered as an outlier.
Annexes
=======
Proofs of section \[Framework\]
-------------------------------
\[l61\] Let $Y$ be a zero mean process with given regularity $({r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$ and asymptotic function $d_0(t)\equiv
C_{{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}}}$ that satisfies A\[h21\](iii-$p$) ($p=1$ or 2). For a positive function $a\in C^{{r_{{\scriptscriptstyle}0}}+p}([0,T])$ and $m\in C^{{r_{{\scriptscriptstyle}0}}+p}([0,T])$, if $X(t)=a(t)Y(t)+m(t)$, then $X$ has regularity $({r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$ with asymptotical function $D_{{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}}}(t)
=a^2(t) C_{{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}}}$ and satisfies A\[h21\](iii-$p$).
See @Se00 and straightforward computation.
\[l62\] Under Assumptions A\[h22\], we get for $k=0,\dotsc,n-1$, and $i=1,\dotsc,n-k$: $$\begin{aligned}
t_{k+i} - t_k &\ge C_1 i {\delta_n}, \;\; C_1 = (T\sup_{t\in[0,T]} \psi(t))^{-1},
\label{e61}\\
\label{e62} t_{k+i} - t_k &\le C_2 i{\delta_n}, \;\; C_2 = (T\inf_{t\in[0,T]} \psi(t))^{-1},\\
\intertext{and if $i=1,\dotsc, i_{\max}$ with $i_{\max}$ not depending on $n$:}
t_{k+i} - t_k &= \frac{i{\delta_n}}{T\psi(t_k)} (1+ {{\mathcal}O} ({\delta_n}^{\alpha}))
\label{e63}\end{aligned}$$ where ${{\mathcal}O}(\dots)$ is uniform over $i$ and $k$.
Relations - are obtained with the mean-value theorem that induces, for $k=0,\dotsc,n-1$ and $i=1,\dotsc,n-k$: $$t_{k+i} -t_k = \frac{i{\delta_n}}{T\psi(t_k + \theta (t_{k+i} - t_k))}, \;\; 0<\theta <1.$$ To obtain the equivalence , one may write $t_{k+i} - t_k = \frac{i{\delta_n}}{T\psi(t_k)} (1+ R_n)$ with $R_n$ defined by $$R_n = \frac{\psi(t_k) - \psi(t_{k} + \theta (t_{k+i} - t_k))}{\psi(t_k + \theta (t_{k+i} - t_k))} \le \frac{L {\lv t_{k+i} - t_k{\right\vert}}^{\alpha}}{\inf_{t\in [0,T]} \psi(t)} = {{\mathcal}O}({\delta_n}^{\alpha})$$ by Assumption A\[h22\] and uniformly over $i$, $n$ and $k$ for $i=1,\dotsc,i_{\max}$.
\[l63\] We have under Assumption A\[h22\] and for $r\ge 1$, $i=0,\dotsc,r$,
1. for $p=0,\dotsc,r-1$ and convention $0^0=1$ $$\label{e64} \sum_{i=0}^r (t_{k+iu} - t_k)^p b_{ikr}^{(u)} = 0,$$
2. $$\label{e65} \sum_{i=0}^r (t_{k+iu} - t_k)^r b_{ikr}^{(u)} = 1,$$
3. $$\label{e66} {\lv b_{ikr}^{(u)}{\right\vert}} \le \frac{C_1^{-r}u^{-r}{\delta_n}^{-r},}{\prod_{m=0,m\not=i}^r {\lv i-m{\right\vert}}},$$
with $C_1$ given by ,
4. $$b_{ikr}^{(u)} = \frac{u^{-r} \psi^{r}(t_k)T^r{\delta_n}^{-r} }{\prod_{m=0,m\not=i}^r (i-m)}(1+ {{\mathcal}O}({\delta_n}^{\alpha}))\label{e67}$$
with ${{\mathcal}O}(\dots)$ uniform over $i$ and $k$.
The term $g[t_k,\dotsc,t_{k+ru}]= \sum_{i=0}^r b_{ikr}^{(u)} g(t_{k+iu})$ is the leading coefficient in the polynomial approximation of degree $r$ of $g$, given in the decomposition . Considering the polynomial $g(t) = (t-t_{k})^p$, we may immediately deduce the properties -, from uniqueness of relation . Next, - are direct consequences of Lemma \[l62\] and definition of $b_{ikr}^{(u)}$.
\[p61\] Under Assumption A\[h21\] and A\[h22\], one obtains:\
(i) for $r={r_{{\scriptscriptstyle}0}}+p$ with $p=1,2$: $$n^{-2(p-{\beta_{{\scriptscriptstyle}0}})}\,{\mathds{E}}\Big(\,\overline{\big( D_{{r_{{\scriptscriptstyle}0}}+p}^{(u)}X\big)^2}\,\Big) {\xrightarrow}[n\to\infty ]{ }u^{-2(p-{\beta_{{\scriptscriptstyle}0}})}\ell(p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$$ where ${\displaystyle}\ell (p,0,{\beta_{{\scriptscriptstyle}0}}) = -\frac{1}{2} \int_0^T d_0(t) \frac{\psi^{2p+1}(t)}{\psi^{2{\beta_{{\scriptscriptstyle}0}}}(t)} {\,{\mrm{d}}t}\sum_{i,j=0}^{p} \frac{{\lv i-j{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}}}{\prod_{\substack{m=0\\m\not=i}}^{p} (i-m) \prod_{\substack{q=0\\q\not=j}}^{p} (j-q)} $ while if ${r_{{\scriptscriptstyle}0}}\ge 1$, $$\label{e68} \ell (p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}}) =\frac{(-1)^{{r_{{\scriptscriptstyle}0}}+1} \int_0^T d_0(t) \frac{\psi^{2p+1}(t)}{\psi^{2{\beta_{{\scriptscriptstyle}0}}}(t)}{\,{\mrm{d}}t}}{2(2{\beta_{{\scriptscriptstyle}0}}+2{r_{{\scriptscriptstyle}0}})\dotsm(2{\beta_{{\scriptscriptstyle}0}}+1)} \sum_{i,j=0}^{{r_{{\scriptscriptstyle}0}}+p} \frac{{\lv i-j{\right\vert}}^{2({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}})}}{\prod_{\substack{m=0\\m\not=i}}^{{r_{{\scriptscriptstyle}0}}+p} (i-m) \prod_{\substack{q=0\\q\not=j}}^{{r_{{\scriptscriptstyle}0}}+p} (j-q) }$$
\(ii) for ${r_{{\scriptscriptstyle}0}}\ge 1$ and $r=1,\dotsc,{r_{{\scriptscriptstyle}0}}$: $${\mathds{E}}\Big(\,\overline{\big( D_r^{(u)}X\big)^2}\,\Big) {\xrightarrow}[n\to\infty ]{ } \frac{1}{(r!)^2} \int_0^T {\mathds{E}}\big(X^{(r)}(t)\big)^2 \psi(t){\,{\mrm{d}}t}.$$
[**A.**]{} Let us begin with general expressions of ${\displaystyle}{\mathds{E}}\Big(\,D_{r,k}^{(u)}\,X D_{r,\ell}^{(u)}\,X\Big)$ useful for the sequel. First for $\mathds{L}^{(p,p)}(s,t) = {\mathds{E}}\big(X^{(p)}(s)X^{(p)}(t)\big)$ ($p\ge 0$), the relation is equivalent to $$\label{e69}
\lim_{h\to 0} \sup_{\substack{s,t\in[0,T]\\ {\lv s-t{\right\vert}} \le h, s\not=t}} {\lv \frac{{\mathds{L}}^{({r_{{\scriptscriptstyle}0}},{r_{{\scriptscriptstyle}0}})}(s,s) +
{\mathds{L}}^{({r_{{\scriptscriptstyle}0}},{r_{{\scriptscriptstyle}0}})}
(t,t) - 2 {\mathds{L}}^{({r_{{\scriptscriptstyle}0}},{r_{{\scriptscriptstyle}0}})}(s,t)}{{\lv s-t{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}}} - d_0(t){\right\vert}} = 0.$$ For $(v,w)\in[0,1]^2$, we set $\dot{v}_{ik} = t_k +(t_{k+iu} - t_k)v$ and $\dot{w}_{j\ell} = t_{\ell} + (t_{\ell+ju} - t_{\ell})w$. Next, from the definition of $D_{r,k}^{(u)}\,X$ given in , we get $${\mathds{E}}(D_{r,k}^{(u)}\,X D_{r,\ell}^{(u)}\,X) = \sum\limits_{i,j=0}^{r} b_{ikr}^{(u)}b_{j\ell r}^{(u)}
{\mathds{L}}^{(0,0)}(t_{k+iu},t_{\ell+ju}).$$ For ${r_{{\scriptscriptstyle}0}}=0$ and since $\sum_{i=0}^{r} b_{ikr}^{(u)}=0$, we have: $$\begin{gathered}
\label{e610} {\mathds{E}}(D_{r,k}^{(u)}\,X D_{r,\ell}^{(u)}\,X) = \sum_{i,j=0}^{r}
\!b_{ikr}^{(u)}b_{j\ell r}^{(u)}
\Big\{ {\mathds{L}}^{(0,0)}(t_{k+iu},t_{\ell+ju})\\ -\frac12 {\mathds{L}}^{(0,0)}(t_{k+iu},t_{k+iu}) -\frac12
{\mathds{L}}^{(0,0)}(t_{\ell+ju},t_{\ell+ju}) \Big\}.\end{gathered}$$
If ${r_{{\scriptscriptstyle}0}}\ge 1$, we apply multiple Taylor series expansions with integral remainder. Next, the properties $\sum_{i=0}^{r}
b_{ikr}^{(u)} (t_{k+iu} - t_k)^p=0$ for $p=0,\dotsc, r-1$ (and convention $0^0=1$) induce : $$\begin{gathered}
\label{e611} {\mathds{E}}(D_{r,k}^{(u)}\,X D_{r,\ell}^{(u)}\,X) = \sum_{i,j=0}^{r} \!b_{ikr}^{(u)}b_{j\ell r}^{(u)}
(t_{k+iu} - t_k)^{\ra}(t_{\ell+ju} - t_\ell)^{\ra}\\ \times {\int\!\!\!\int}_{[0,1]^2} \!\frac{\!(1-v)^{\ra-1}\!(1-w)^{\ra-1}}{((\ra -
1)!)^2} {\mathds{L}}^{(\ra,\ra)}(\dot{v}_{ik},\dot{w}_{j\ell}) {\,{\mrm{d}}v}\!{\,{\mrm{d}}w}\end{gathered}$$ where we have set $\ra=\min({r_{{\scriptscriptstyle}0}},r)\ge 1$.
[**B.**]{} From expressions -, we are in position to derive the asymptotic behavior of $ {\mathds{E}}\big(\,\overline{(D_r^{(u)}\, X)^2}\big)$.
#### *Case ${r_{{\scriptscriptstyle}0}}\ge 1$, $r={r_{{\scriptscriptstyle}0}}+p$, $p=1$ or $p=2$.*
In this case, $\ra={r_{{\scriptscriptstyle}0}}\le r-1$. From and the property $\sum_{i=0}^{r} b_{ikr}^{(u)} (t_{k+iu} - t_k)^{{r_{{\scriptscriptstyle}0}}}=0$, we may write $$\begin{gathered}
\label{e612} {\mathds{E}}(D_{r,k}^{(u)}\,X)^2 = \sum_{i,j=0}^{r} b_{ikr}^{(u)}b_{jkr}^{(u)}
(t_{k+iu} - t_k)^{{r_{{\scriptscriptstyle}0}}} (t_{k+ju} - t_k)^{{r_{{\scriptscriptstyle}0}}}\!\!\int_0^1\!\!\!\int_0^1 \!\! \frac{((1-v)(1-w))^{{r_{{\scriptscriptstyle}0}}-1}}{(({r_{{\scriptscriptstyle}0}}- 1)!)^2}
\\ \times\Big\{
{\mathds{L}}^{({r_{{\scriptscriptstyle}0}},{r_{{\scriptscriptstyle}0}})}(\dot{v}_{ik},\dot{w}_{jk}) -\frac12 {\mathds{L}}^{({r_{{\scriptscriptstyle}0}},{r_{{\scriptscriptstyle}0}})}(\dot{v}_{ik},\dot{v}_{ik}) -\frac12
{\mathds{L}}^{({r_{{\scriptscriptstyle}0}},{r_{{\scriptscriptstyle}0}})}(\dot{w}_{jk},\dot{w}_{jk}) \Big\} {\,{\mrm{d}}v}\!{\,{\mrm{d}}w}\end{gathered}$$ Using the locally stationary condition , uniform continuity of $d_0(\cdot)$ on $[0,T]$ and the bound: ${\lv \dot{v}_{ik} - \dot{w}_{jk}{\right\vert}}\le t_{k+ru} - t_k \le C_1 ru {\delta_n}$ for $i=0,\dotsc,r$ and $j=0,\dotsc,r$, we may show that the predominant term for $ {\mathds{E}}\big(\,\overline{(D_r^{(u)}\, X)^2}\big)$ is given by: $$\begin{gathered}
\label{e613} \frac{-1}{2(n_r+1)} \sum_{k=0}^{n_r}\sum_{i,j=0}^{r} b_{ikr}^{(u)}b_{jkr}^{(u)}
(t_{k+iu} - t_k)^{{r_{{\scriptscriptstyle}0}}} (t_{k+ju} - t_k)^{{r_{{\scriptscriptstyle}0}}} \\ \times {\int\!\!\!\int}_{[0,1]^2} \frac{((1-v)(1-w))^{{r_{{\scriptscriptstyle}0}}-1}}{(({r_{{\scriptscriptstyle}0}}- 1)!)^2}
{\lv \dot{v}_{ik} - \dot{w}_{jk}{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}} d_0(t_k) {\,{\mrm{d}}v}{\,{\mrm{d}}w}.\end{gathered}$$ From the equivalents and , we can write the leading term of as a Riemann sum on $t_k$ to obtain $$\begin{gathered}
{\delta_n}^{2p - 2{\beta_{{\scriptscriptstyle}0}}} {\mathds{E}}\big(\,\overline{(D_r^{(u)}\, X)^2}\big) {\xrightarrow}[n\to\infty]{} - \frac{1}{2} (\frac{u}{T})^{-2p+2{\beta_{{\scriptscriptstyle}0}}} \int_0^T d_0(t) \psi^{2p+1-2{\beta_{{\scriptscriptstyle}0}}}(t) {\,{\mrm{d}}t}\\\times \sum_{i,j=0}^r \frac{(ij)^{{r_{{\scriptscriptstyle}0}}}}{\prod\limits_{\substack{m=0\\m\not=i}}^r (i-m) \prod\limits_{\substack{q=0\\q\not=j}}^r (j-q)} {\int\!\!\!\int}_{[0,1]^2} \!\! \frac{((1-v)(1-w))^{{r_{{\scriptscriptstyle}0}}-1}}{(({r_{{\scriptscriptstyle}0}}- 1)!)^2} {\lv iv - jw{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}}{\,{\mrm{d}}v}{\,{\mrm{d}}w}.\end{gathered}$$ Next by performing elementary but tedious multiple integrations by parts, we arrive at the following simpler form of $\ell (r,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$ given in , for $n{\delta_n}\to T$.
#### *Case ${r_{{\scriptscriptstyle}0}}=0$, $r={r_{{\scriptscriptstyle}0}}+1,_,{r_{{\scriptscriptstyle}0}}+2$.*
The proof is the same but starting from and $\ell=k$.
#### *Case ${r_{{\scriptscriptstyle}0}}\ge 1$ and $r=1,\dotsc,{r_{{\scriptscriptstyle}0}}$.*
In this case, $\ra = r$ and from the relation , one gets $$\begin{gathered}
{\mathds{E}}(D_{r,k}^{(u)}\,X)^2 = \sum\limits_{i,j=0}^{r} \!b_{ikr}^{(u)}b_{jkr}^{(u)}
(t_{k+iu}-t_k)^{r}(t_{k+ju}-t_k)^{r}\\ \times {\int\!\!\!\int}_{[0,1]^2}
\frac{((1-v)(1-w))^{r-1}}{((r - 1)!)^2} {\mathds{L}}^{(r,r)}(\dot{v}_{ik},\dot{w}_{jk}) {\,{\mrm{d}}v}\!{\,{\mrm{d}}w}.\end{gathered}$$ The result follows after Riemann summation with the help of uniform continuity of ${\mathds{L}}^{(r,r)}(\cdot,\cdot)$, $r=1,\dotsc,{r_{{\scriptscriptstyle}0}}$ and properties , .
Auxiliary results
-----------------
The following lemma gives some useful results on the asymptotic behavior of $\mathds{C}_r (k,\ell)$ and $\mathds{C}_r^2 (k,\ell)$ with $\mathds{C}_r (k,\ell) = {{\mrm{Cov\,}}}\big(D_{r,k}^{(u)}\,X,D_{r,\ell}^{(u)}\,X\big)$ with $n_r = n-ur$ and $u$ a positive integer.
\[l64\] Suppose that Assumption A\[h21\] and A\[h22\] are fulfilled.
\(i) Under the condition A\[h21\]-(iii-1) and for $r={r_{{\scriptscriptstyle}0}}+p$, $p=1$ or $p=2$, one obtains $$\max_{k=0,\dotsc,n_r} \sum_{\ell=0}^{n_r}
{\lv \mathds{C}_r (k,\ell){\right\vert}} = \begin{cases} {\mathcal O} ( n^{2p-2{\beta_{{\scriptscriptstyle}0}}}) &\text{ if } 0 < {\beta_{{\scriptscriptstyle}0}}<\frac{1}{2},\\
{\mathcal O} ( n^{2p-1}\ln n) &\text{ if } {\beta_{{\scriptscriptstyle}0}}=\frac{1}{2}, \\
{\mathcal O} ( n^{2p-1}) &\text{ if } \frac{1}{2}<{\beta_{{\scriptscriptstyle}0}}<1~;\end{cases}$$ and $$\sum_{k=0}^{n_r} \sum_{\ell=0}^{n_r}
\mathds{C}_r^2 (k,\ell) = \begin{cases} {\mathcal O} ( n^{4p-4{\beta_{{\scriptscriptstyle}0}}+1}) &\text{ if } 0 < {\beta_{{\scriptscriptstyle}0}}<\frac{3}{4},\\
{\mathcal O} ( n^{4p -2}\ln n) &\text{ if } {\beta_{{\scriptscriptstyle}0}}=\frac{3}{4}, \\
{\mathcal O} ( n^{4p-2}) &\text{ if } \frac{3}{4}<{\beta_{{\scriptscriptstyle}0}}<1.\end{cases}$$
\(ii) Under the condition A\[h21\]-(iii-2) and for $r={r_{{\scriptscriptstyle}0}}+2$, one obtains $$\max\limits_{k=0,\dotsc,n_r} \sum_{\ell=0}^{n_r}
{\lv \mathds{C}_r (k,\ell){\right\vert}} = {\mathcal O}(n^{4-2{\beta_{{\scriptscriptstyle}0}}}) \text{ and }
\sum_{k=0}^{n_r} \sum_{\ell=0}^{n_r}
\mathds{C}_r^2 (k,\ell) ={\mathcal O}(n^{9-4{\beta_{{\scriptscriptstyle}0}}}).$$
\(iii) If $r=1,\dotsc,{r_{{\scriptscriptstyle}0}}$ (with ${r_{{\scriptscriptstyle}0}}\ge 1$), then ${\displaystyle}\max_{k=0,\dotsc,n_r} \sum_{\ell=0}^{n_r}
{\lv \mathds{C}_r(k,\ell){\right\vert}} = {\mathcal O} (n) $ and ${\displaystyle}\sum_{k=0}^{n_r} \sum_{\ell=0}^{n_r}
\mathds{C}_r^2(k,\ell) = {\mathcal O} ( n^{2}). $
\(i) Setting $\mu(t)={\mathds{E}}\big(X(t)\big)$, $\mu$ is ${r_{{\scriptscriptstyle}0}}$-times differentiable and similarly to -, we get the expansion $$\begin{gathered}
\mathds{C}_r(k,\ell) = \sum_{i,j=0}^{r} b_{ikr}^{(u)}b_{j\ell r}^{(u)}
(t_{k+iu} - t_k)^{{r_{{\scriptscriptstyle}0}}} (t_{\ell+ju} - t_{\ell})^{{r_{{\scriptscriptstyle}0}}}\\ \times {\int\!\!\!\int}_{[0,1]^2}\frac{((1-v)(1-w))^{{r_{{\scriptscriptstyle}0}}-1}}{(({r_{{\scriptscriptstyle}0}}-
1)!)^2}\mathds{K}^{({r_{{\scriptscriptstyle}0}},{r_{{\scriptscriptstyle}0}})}(\dot{v}_{ik},\dot{w}_{j\ell}) {\,{\mrm{d}}v}{\,{\mrm{d}}w}\end{gathered}$$ for ${r_{{\scriptscriptstyle}0}}\ge 1$ while if ${r_{{\scriptscriptstyle}0}}=0$, $\mathds{C}_r(k,\ell) = \sum_{i=0}^{r}\sum_{j=0}^{r} b_{ikr}^{(u)}b_{j\ell r}^{(u)}
\mathds{K}(t_{k+iu},t_{\ell+ju})$.
#### *Case $r={r_{{\scriptscriptstyle}0}}+1$ or ${r_{{\scriptscriptstyle}0}}+2$*.
For ${r_{{\scriptscriptstyle}0}}\ge 1$, we have the bound: $$\max_{k=0,\dotsc,n_r} \sum_{\ell=0}^{n_r} {\lv \mathds{C}_r(k,\ell){\right\vert}} \le U_{1n}+ U_{2n} +
U_{3n}$$ with $U_{1n}= \max\limits_{k=ur+1,\dotsc,n_r} \sum\limits_{\ell=0}^{k-ur-1}\!\!{\lv \mathds{C}_r(k,\ell){\right\vert}}$, $U_{2n}= \max\limits_{k=0,\dotsc,n-2ur-1} \sum\limits_{\ell=k+ur+1}^{n_r}\!\!{\lv \mathds{C}_r(k,\ell){\right\vert}}$ and $U_{3n}= \max\limits_{k=0,\dotsc,n_r}\sum\limits_{\ell=\max(0,k-ur)}^{\min(n_r,k+ur)}{\lv \mathds{C}_r(k,\ell){\right\vert}}$. First, consider the sum $U_{1n}+U_{2n}$ where ${\lv k-\ell{\right\vert}}\ge ur+1$. Since $\sum_{i=0}^{r} b_{ikr}^{(u)} (t_{k+iu} - t_k)^{{r_{{\scriptscriptstyle}0}}}=0$ for $r={r_{{\scriptscriptstyle}0}}+1$ or $r={r_{{\scriptscriptstyle}0}}+2$, and $[t_k,\dot{v}_{ik}]$ is distinct from $[t_{\ell},\dot{w}_{j\ell}]$, we get $$\begin{gathered}
\label{e614}
\mathds{C}_r(k,\ell) =\sum_{i,j=0}^{r} b_{ikr}^{(u)}b_{jkr}^{(u)} (t_{k+iu} - t_k)^{{r_{{\scriptscriptstyle}0}}}(t_{\ell+ju} - t_{\ell})^{{r_{{\scriptscriptstyle}0}}} \\ \times
{\int\!\!\!\int}_{[0,1]^2} \frac{((1-v)(1-w))^{{r_{{\scriptscriptstyle}0}}-1}}{(({r_{{\scriptscriptstyle}0}}- 1)!)^2}
\int_{t_k}^{\dot{v}_{ik}}\int_{t_{\ell}}^{\dot{w}_{j\ell}}
\mathds{K}^{({r_{{\scriptscriptstyle}0}}+1,{r_{{\scriptscriptstyle}0}}+1)}(s,t) \,\mathrm{d}s\!{\,{\mrm{d}}t}\!{\,{\mrm{d}}v}\!{\,{\mrm{d}}w}.\end{gathered}$$ Condition A\[h21\]-(iii-1), together with the bounds and , gives a bound of ${\mathcal O} (n^{2p-2{\beta_{{\scriptscriptstyle}0}}} \sum_{i=1}^n i^{-2(1-{\beta_{{\scriptscriptstyle}0}})})$ for ${\lv U_{1n} +U_{2n}{\right\vert}}$, which is of order $n^{2(p-{\beta_{{\scriptscriptstyle}0}})}$ if $0 < {\beta_{{\scriptscriptstyle}0}}< \frac{1}{2}$, $n^{2(p-{\beta_{{\scriptscriptstyle}0}})} \ln n$ if ${\beta_{{\scriptscriptstyle}0}}= \frac{1}{2}$ and $n^{2p -1}$ if ${\beta_{{\scriptscriptstyle}0}}> \frac{1}{2}$. Next, for $ U_{3n}$ where ${\lv k-\ell{\right\vert}}\le ur$, we obtain that $U_{3n} = {{\mathcal}O} (n^{2(p-{\beta_{{\scriptscriptstyle}0}})})$ in a similar way as in the proof of Proposition \[p61\], and with the help of Cauchy-Schwarz inequality to control the terms depending on $\mu^{({r_{{\scriptscriptstyle}0}})}(t)$.
We proceed similarly for the case ${r_{{\scriptscriptstyle}0}}=0$, starting from the definition of $\mathds{C}_r(k,\ell)$ as well as for the study of $\sum_{k=0}^{n_r}\sum_{\ell=0}^{n_r} \mathds{C}_r^2(k,\ell)$ for which dominant terms are of order ${{\mathcal}O}(n^{1+4p - 4{\beta_{{\scriptscriptstyle}0}}}\sum_{i=1}^n i^{-4(1 - {\beta_{{\scriptscriptstyle}0}})})$.
\(ii) The condition A\[h21\]-(iii-2) and $r={r_{{\scriptscriptstyle}0}}+2$ allows to transform into $$\begin{gathered}
\label{e615}
\mathds{C}_r(k,\ell) =\sum_{i,j=0}^{r} b_{ikr}^{(u)}b_{jkr}^{(u)} (t_{k+iu} - t_k)^{{r_{{\scriptscriptstyle}0}}}(t_{\ell+ju} - t_{\ell})^{{r_{{\scriptscriptstyle}0}}} {\int\!\!\!\int}_{[0,1]^2} \frac{((1-v)(1-w))^{{r_{{\scriptscriptstyle}0}}-1}}{(({r_{{\scriptscriptstyle}0}}- 1)!)^2}\\
\times\int_{t_k}^{\dot{v}_{ik}}\int_{t_{\ell}}^{\dot{w}_{j\ell}} \int_{t_k}^t \int_{t_{\ell}}^s K^{({r_{{\scriptscriptstyle}0}}+2,{r_{{\scriptscriptstyle}0}}+2)}(y,z) {\,{\mrm{d}}y}{\,{\mrm{d}}z}\mathrm{d}s{\,{\mrm{d}}t}{\,{\mrm{d}}v}{\,{\mrm{d}}w}\end{gathered}$$ which gives that $$\max\limits_{k=0,\dotsc,n_r}\sum_{\ell=0}^{n_r} {\lv \mathds{C}_r(k,\ell){\right\vert}} = {{\mathcal}O}(n^{2(2-{\beta_{{\scriptscriptstyle}0}})} \sum_{i=1}^n i^{-4+2{\beta_{{\scriptscriptstyle}0}}}) = {{\mathcal}O}(n^{2(2-{\beta_{{\scriptscriptstyle}0}})})$$ for all ${\beta_{{\scriptscriptstyle}0}}\in]0,1[$ and ${r_{{\scriptscriptstyle}0}}\ge 1$. From , we also get that $\sum_{k=0}^{n_r}\sum_{\ell=0}^{n_r} \mathds{C}_r^2(k,\ell) = {{\mathcal}O}(n^{9-4{\beta_{{\scriptscriptstyle}0}}} \sum_{i=1}^n i^{-8+4{\beta_{{\scriptscriptstyle}0}}}) = {{\mathcal}O}(n^{9-4{\beta_{{\scriptscriptstyle}0}}})$ for all ${\beta_{{\scriptscriptstyle}0}}\in ]0,1[$.
\(iii) Results of these part, where ${r_{{\scriptscriptstyle}0}}\ge 1$, are consequences of $$\begin{gathered}
\mathds{C}_r(k,\ell) = \sum_{i,j=0}^r b_{ikr}^{(u)} b_{j\ell r}^{(u)} (t_{k+iu} - t_k)^r (t_{\ell+ju} - t_{\ell})^r \\ \times{\int\!\!\!\int}_{[0,1]^2} \frac{( (1-v)(1-w))^{r-1}}{(r-1)!^2} K^{(r,r)} (\dot{v}_{ik}, \dot{w}_{jl}) {\,{\mrm{d}}v}{\,{\mrm{d}}w}= {{\mathcal}O} (1)\end{gathered}$$ with uniform continuity of $K^{(r,r)}(\cdot,\cdot)$ for $r=1,\dotsc,{r_{{\scriptscriptstyle}0}}$ together with bounds and .
Next proposition gives a general exponential bound, involved in all our results.
\[p62\] Suppose that Assumption A\[h21\] and A\[h22\] are fulfilled. Let $\eta_{n}(r)$ be some given positive sequence and $u\in\bN\as$, then $$\mathds{P}\Big( {\lv \overline{ (D_r^{(u)} X)^2} - {\mathds{E}}\Big(\,\overline{ (D_r^{(u)} X)^2 }\,\Big){\right\vert}} \ge
\eta_n(r)\Big)$$ is of order: $$\begin{gathered}
{\mathcal O}\bigg( \exp\Bigl(-C(r) n\eta_{n}(r)\times \min\Bigl( \big(\max\limits_{0\leq k \leq
n_r}\sum\limits_{\ell=0}^{n_r}{\lv \, \mathds{C}_r(k,\ell){\right\vert}}\big)^{-1}, \frac{n\eta_n(r)}{\sum\limits_{k,\ell=0}^{n_r} \mathds{C}_r^2(k,\ell)}\Bigr)\Bigr)\bigg)\\+ {\mathcal O}\bigg( \frac{v_n^{1/2}(r)}{n\eta_n(r)} \exp\Big( -
C(r)\frac{n^2\eta_n^2(r)}{v_n(r)}\Big)\bigg)\end{gathered}$$ for some positive constant $C(r)$, not depending on $\eta_n(r)$ and $$\label{e616} v_n(r) := n \max\limits_{k=0,\dotsc,n_r} \big( {\mathds{E}}(D_{r,k}^{(u)}\,X)\big)^2 \max\limits_{k=0,\dotsc,n_r} \sum\limits_{\ell=0}^{n_r}{\lv \mathds{C}_r(k,\ell){\right\vert}}.$$
For all $r\ge 1$, we may bound $\mathds{P}\Big( {\lv \overline{ (D_r^{(u)} X)^2} - {\mathds{E}}\Big(\,\overline{ (D_r^{(u)} X)^2 }\,\Big){\right\vert}} \ge
\eta_n(r)\Big)$ by $S_1+S_2$ with $$S_1= {\mathds{P}}\Big( {\lv \sum_{k=0}^{n_r}(D_{r,k}^{(u)}\,X - {\mathds{E}}(
D_{r,k}^{(u)}\,X))^2 - {{\mrm{Var\,}}}(D_{r,k}^{(u)}\,X){\right\vert}} > \frac{(n_r+1)\eta_n(r)}{2}\Big)$$ and ${\displaystyle}S_2=
{\mathds{P}}\Big( {\lv \sum_{k=0}^{n_r}({\mathds{E}}(D_{r,k}^{(u)}\,X))\big( D_{r,k}^{(u)}\,X - {\mathds{E}}(D_{r,k}^{(u)}\, X)\big) {\right\vert}} >\frac{(n_r+1)\eta_n(r)}{4}\Big)$. First, let $\{Y_i\}_{i=1,\dotsc,d_n}$ be an orthonormal basis for the linear span of $\{ D_{r,k}^{(u)}\,X\}_{k=0,\dotsc,n_r}$ (so that $Y_i$ are i.i.d. with density ${\mathcal N}(0,1)$). We can write ${\displaystyle}D_{r,k}^{(u)}\,X-{\mathds{E}}\big(D_{r,k}^{(u)}\,X\big)=\sum_{i=1}^{d_n}d_{k,i}Y_i$ with $d_{k,i} = {{\mrm{Cov\,}}}\bigl(D_{r,k}^{(u)}\,X,Y_i\bigr)$. Next, if $Y=(Y_1,\ldots,Y_{d_n})\T$, we obtain $$\sum_{k=0}^{n_r} (D_{r,k}^{(u)}\,X - {\mathds{E}}D_{r,k}^{(u)}\,X)^2 = \sum_{i,j=1}^{d_n} c_{i,j} Y_iY_{j} =
Y\T CY \text{ and }
\sum_{k=0}^{n_r} {{\mrm{Var\,}}}(D_{r,k}^{(u)}\,X) = \sum_{i=1}^{d_n} c_{i,i}$$ with ${\displaystyle}c_{i,j}=\sum_{k=0}^{n_r}d_{ki}d_{kj}$. Next, for $ C=\bigl( c_{i,j}\bigr)_{\substack{i=1,\dotsc,d_n\\j=1,\dotsc,d_n}}$ and $D=\bigl(d_{k,j}\bigr)_{\substack{k=0,\ldots,n_r\\j=1,\dotsc,d_n}}$, one gets $C=D\T\,D$ where $C$ is a real, symmetric and positive semidefinite matrix. There exists an orthogonal matrix $P$ such that ${\rm
diag}(\lambda_1,\ldots,\lambda_{d_n})=P\T CP$, for $\lambda_i$ eigenvalues of $C$. Then we can transform the quadratic form as: $$\sum_{k=0}^{n_r} (D_{r,k}^{(u)}\,X - {\mathds{E}}(D_{r,k}^{(u)}\,X))^2 =\sum_{i=1}^{d_n}\lambda_i(P\T Y)_i^2$$ where $(P\T Y)_i$ denotes the $i$-th component of the ($d_n\times 1$) vector $P\T Y$. Since $\sum_{i=1}^{d_n} c_{i,i} = \sum_{i=1}^{d_n} \lambda_i$, we arrive at $$S_1= \mathds{P}\Bigl( \,\Bigl\vert \,\sum_{i=1}^{d_n} \lambda_i \bigl( (P\T Y)_i^2 -1\bigr)
\Bigr\vert\ge \frac{(n_r+1)\eta_{n}(r)}{2}
\Bigr).$$ Now, with the exponential bound of @HW71, we obtain for some generic constant $c$: $$S_1 \leq
2\exp\biggl(-c(n_r+1)\eta_{n}(r)\times \min\Bigl(\frac{1}{\max (\lambda_i)},
\frac{(n_r+1)\eta_n(r)}{\sum\lambda_i^2}\Bigr)\biggr).$$ Next, since $D\T D$ and $DD\T$ have the same non zero eigenvalues, $$\max_{i=1,\dotsc,d_n}\lambda_i
\le \max_{0\leq k \leq n_r}\sum_{\ell=0}^{n_r}{\lv \, \mathds{C}_r(k,\ell){\right\vert}}$$ and $
\sum\limits_{i=1}^{d_n} \lambda_i^2 = \sum\limits_{i=1}^{d_n} \sum\limits_{j=1}^{d_n} c_{ij}c_{ji} = \sum\limits_{k=0}^{n_r}\sum\limits_{\ell=0}^{n_r} (\sum\limits_{i=1}^{d_n} d_{ki}d_{li})^2
= \sum\limits_{k=0}^{n_r}\sum\limits_{\ell=0}^{n_r} \mathds{C}_r^2(k,\ell)$. Finally $S_1$ is bounded by $$2\exp\biggl(-c (n_r+1)\eta_{n}(r)\times \min\Bigl( \big(\max\limits_{0\leq k \leq
n_r}\sum\limits_{\ell=0}^{n_r}{\lv \, \mathds{C}_r(k,\ell){\right\vert}}\big)^{-1}, \frac{(n_r +1)\eta_n(r)}{\sum\limits_{k=0}^{n_r}\sum\limits_{\ell=0}^{n_r} \mathds{C}_r^2(k,\ell)}\Bigr)\biggr).$$ For $S_2$, we use the classical exponential bound on a Gaussian variable: $Y \sim {{\mathcal}N}(0,\sigma^2)$ implies that ${\mathds{P}}({\lv Y{\right\vert}}\ge {\varepsilon}) \le \min(1, \sqrt{\frac{2\sigma^2}{\pi{\varepsilon}^2}}) \exp ( - \frac{{\varepsilon}^2}{2\sigma^2}), \; {\varepsilon}>0.$ Here $Y = \sum\limits_{k=0}^{n_r} ({\mathds{E}}D_{r,k}^{(u)} X) ( D_{r,k}^{(u)} X - {\mathds{E}}D_{r,k}^{(u)} X)$ and we get easily that ${{\mrm{Var\,}}}(Y) \le v_n(r)$.
Proofs of section \[Asymptres\]
-------------------------------
**Proof of Theorem \[t31\]**\
Recall that $\widehat{{r_{{\scriptscriptstyle}0}}}$ is given by: $
\widehat{{r_{{\scriptscriptstyle}0}}} =\min\Big\{r\in\{2,\dotsc,m_n\}\;\;\;:\;\;\; B_n(r) \text{ holds} \Big\}-2
$ where the event $B_n(r)$ is defined by ${\displaystyle}B_n(r) = \big\{ \overline{
\big( D^{(1)}_r X\big)^2} \ge n^2 b_n\big\}$, and $\widehat{{r_{{\scriptscriptstyle}0}}} = \ell_0$ if $\cap_{r=2}^{m_n} B_n(r)$. The condition $m_n \to\infty$ guarantees that for $n$ large enough, ${r_{{\scriptscriptstyle}0}}+2\in\{2,\dotsc,m_n\}$. From this definition, we write $${\mathds{E}}\big(\widehat{{r_{{\scriptscriptstyle}0}}} - {r_{{\scriptscriptstyle}0}})^2 = \sum_{r=0}^{m_n -2} (r -{r_{{\scriptscriptstyle}0}})^2{\mathds{P}}\big(\widehat{{r_{{\scriptscriptstyle}0}}} = r\big) + (l_0 - {r_{{\scriptscriptstyle}0}})^2 {\mathds{P}}\big( \widehat{{r_{{\scriptscriptstyle}0}}}=l_0\big)$$ where ${\mathds{P}}\big(\widehat{{r_{{\scriptscriptstyle}0}}} = 0\big) = {\mathds{P}}\big(B_n(2) \big)$, $
{\mathds{P}}\big(\widehat{{r_{{\scriptscriptstyle}0}}} = r\big) = {\mathds{P}}\big(B_n^c(2)\cap \dotsb \cap B_n^c(r+1) \cap B_n(r+2) \big)$ if $r=1,\dotsc,m_n-2$, and ${\mathds{P}}\big(\widehat{{r_{{\scriptscriptstyle}0}}} = l_0\big) \le {\mathds{P}}\big( B_n^c({r_{{\scriptscriptstyle}0}}+2)\big)$. Then, for all ${r_{{\scriptscriptstyle}0}}\in\n_0$: $
{\mathds{E}}\big(\widehat{{r_{{\scriptscriptstyle}0}}} -{r_{{\scriptscriptstyle}0}}\big)^2 = {\mathcal O} \big( T_{1n}({r_{{\scriptscriptstyle}0}})\big) + {\mathcal O} \big(m_n^3 T_{2n}({r_{{\scriptscriptstyle}0}})\big)
$ where we have set $T_{1n}(0) = 0$, $T_{1n}({r_{{\scriptscriptstyle}0}}) = \sum\limits_{r=2}^{{r_{{\scriptscriptstyle}0}}+1} {\mathds{P}}\big(B_n(r)\big) \text{ (for } {r_{{\scriptscriptstyle}0}}\ge 1)$ and $ T_{2n}({r_{{\scriptscriptstyle}0}}) = {\mathds{P}}\big( B_n^c({r_{{\scriptscriptstyle}0}}+2) \big)$. Now, the study of $T_{1n}$ and $T_{2n}$ is derived from results of Lemma \[l62\], Lemma \[l63\], Proposition \[p61\] and Lemma \[l64\]. In particular, since $\mu\in C^{{r_{{\scriptscriptstyle}0}}+1}([0,T])$ we get: $${\mathds{E}}(D_{r,k}^{(u)} X) = \sum_{i=0}^r b_{ikr}{(u)} (t_{k+iu}-t_k)^{\ra} \int_0^1 \frac{(1-v)^{\ra-1}}{(\ra - 1)!} \mu^{(\ra)} (t_k + (t_{k+iu} - t_k)v) {\,{\mrm{d}}v}$$ which is ${{\mathcal}O} (n^{r -\ra})$ for $\ra=\min(r,{r_{{\scriptscriptstyle}0}}+1)$ implying that ${\mathds{E}}(D_{r,k}^{(u)} X)= {{\mathcal}O}(1)$ for $r=1,\dotsc,{r_{{\scriptscriptstyle}0}}+1$, ${\mathds{E}}(D_{r,k}^{(u)} X)= {{\mathcal}O}(n)$ for $r={r_{{\scriptscriptstyle}0}}+2$. Then one may bound $v_n(r)$ given in equation by ${{\mathcal}O}(n^2)$ if $r=1,\dotsc,{r_{{\scriptscriptstyle}0}}$, ${{\mathcal}O} \big(n^{3-2{\beta_{{\scriptscriptstyle}0}}}{\mathds{1}}_{]0,\frac{1}{2}[}({\beta_{{\scriptscriptstyle}0}}) + n^2\ln n{\mathds{1}}_{\{\frac{1}{2}\}}({\beta_{{\scriptscriptstyle}0}}) + n^2{\mathds{1}}_{]\frac{1}{2},1[}({\beta_{{\scriptscriptstyle}0}})\big)$ if $r={r_{{\scriptscriptstyle}0}}+1$ with A\[h21\]-(iii-1), ${{\mathcal}O} \big(n^{7-2{\beta_{{\scriptscriptstyle}0}}}{\mathds{1}}_{]0,\frac{1}{2}[}({\beta_{{\scriptscriptstyle}0}}) + n^6\ln n{\mathds{1}}_{\{\frac{1}{2}\}}({\beta_{{\scriptscriptstyle}0}}) + n^6{\mathds{1}}_{]\frac{1}{2},1[}({\beta_{{\scriptscriptstyle}0}})\big)$ if $r={r_{{\scriptscriptstyle}0}}+2$ with A\[h21\]-(iii-1), and ${{\mathcal}O} (n^{7-2{\beta_{{\scriptscriptstyle}0}}})$ if $r={r_{{\scriptscriptstyle}0}}+2$ and A\[h21\]-(iii-2) holds. Next after some calculations based on properties $n^{2{\beta_{{\scriptscriptstyle}0}}}b_n\to \infty$ and $n^{-2(1-{\beta_{{\scriptscriptstyle}0}})}b_n\to 0$, one may derive from Proposition \[p62\] that: $$T_{1n}({r_{{\scriptscriptstyle}0}}) = {\mathcal O} \Big(\exp\Big(- D({r_{{\scriptscriptstyle}0}}) b_n \big(n^{2{\beta_{{\scriptscriptstyle}0}}+1}{\mathds{1}}_{]0,\frac{1}{2}[}({\beta_{{\scriptscriptstyle}0}}) + \big(\frac{n^2}{\ln n}\big) {\mathds{1}}_{\{\frac{1}{2}\}}({\beta_{{\scriptscriptstyle}0}}) + n^2{\mathds{1}}_{]\frac{1}{2},1[}({\beta_{{\scriptscriptstyle}0}})\big)\Big) \Big).$$ Next, if A\[h21\]-(iii-1) holds $$T_{2n}({r_{{\scriptscriptstyle}0}}) = O \bigg( \exp\Big( - D({r_{{\scriptscriptstyle}0}}) \big(n{\mathds{1}}_{]0,\frac{1}{2}[}({\beta_{{\scriptscriptstyle}0}}) + \big(\frac{n}{\ln n}\big){\mathds{1}}_{\{\frac{1}{2}\}}({\beta_{{\scriptscriptstyle}0}}) + n^{2(1-{\beta_{{\scriptscriptstyle}0}})}{\mathds{1}}_{]\frac{1}{2},1[}({\beta_{{\scriptscriptstyle}0}}) \big)\Big) \bigg)$$ while, under A\[h21\]-(iii-2) and for all ${\beta_{{\scriptscriptstyle}0}}\in]0,1[$, $T_{2n}({r_{{\scriptscriptstyle}0}}) =O \big( \exp( - D({r_{{\scriptscriptstyle}0}}) n) \big)$. For $p=1,2$, we get that $T_{1n}({r_{{\scriptscriptstyle}0}}) = o\big( T_{2n} ({r_{{\scriptscriptstyle}0}})\big)$ and the mean square error follows. Finally, to obtain a bound for ${\mathds{P}}(\widehat{{r_{{\scriptscriptstyle}0}}} \not={r_{{\scriptscriptstyle}0}})$, it suffices to notice that $\{\widehat{{r_{{\scriptscriptstyle}0}}} = 0\} = B_n(2)$ for ${r_{{\scriptscriptstyle}0}}=0$ and $\{\widehat{{r_{{\scriptscriptstyle}0}}} = {r_{{\scriptscriptstyle}0}}\} = B_n^c(2) \cap \dotsb \cap B_n^c({r_{{\scriptscriptstyle}0}}+1) \cap B_n({r_{{\scriptscriptstyle}0}}+2)$ for ${r_{{\scriptscriptstyle}0}}\ge 1$, by this way ${\mathds{P}}(\widehat{{r_{{\scriptscriptstyle}0}}} \not= {r_{{\scriptscriptstyle}0}}) = T_{1n}({r_{{\scriptscriptstyle}0}}) + T_{2n}({r_{{\scriptscriptstyle}0}})= T_{2n}({r_{{\scriptscriptstyle}0}})(1+o(1))$.
**Proof of Theorem \[t32\]**\
We start the proof, with either $p=1$ or $p=2$, and thus denote by $\widehat{r_{\!p}}$ (resp. $r_{\!p}$) the quantity $\widehat{{r_{{\scriptscriptstyle}0}}}+p$ (resp. ${r_{{\scriptscriptstyle}0}}+p$). We set $$\begin{gathered}
\label{e617}
l_n(p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})= -\frac{1}{2n} \sum_{k=0}^{n}d_0(t_k)\psi^{2(p-{\beta_{{\scriptscriptstyle}0}})}(t_k) \sum_{i,j=0}^{r_{\!p}} \frac{(ij)^{{r_{{\scriptscriptstyle}0}}}}{\prod_{\substack{m=0\\m\not=i}}^{r_{\!p}} (i-m)\prod_{\substack{q=0\\q\not=j}}^{r_{\!p}} (j-q)}
\\ \times {\int\!\!\!\int}_{[0,1]^2} \frac{((1-v)(1-w))^{{r_{{\scriptscriptstyle}0}}-1}}{({r_{{\scriptscriptstyle}0}}- 1)!^2}{\lv iv-jw{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}} {\,{\mrm{d}}v}\!{\,{\mrm{d}}w},\end{gathered}$$ for all ${r_{{\scriptscriptstyle}0}}\ge 1$ while if ${r_{{\scriptscriptstyle}0}}=0$, $$\label{e618}
l_n(p,0,{\beta_{{\scriptscriptstyle}0}}) = -\frac{1}{2n} \sum_{k=0}^{n} d_0(t_k)\psi^{2(p-{\beta_{{\scriptscriptstyle}0}})}(t_k)\sum_{i,j=0}^{r_{\!p}}
\frac{ {\lv i-j{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}}}{\prod_{\substack{m=0\\m\not=i}}^{p} (i-m)\prod_{\substack{q=0\\q\not=j}}^{p} (j-q)}.$$ We study the convergence of $\widehat{\alpha}_p = 2(\widehat{\beta}_n^{(p)} -p)$ toward $\alpha_p = 2({\beta_{{\scriptscriptstyle}0}}-p)$, so that $$\widehat{\alpha}_p = \frac{\ln \big( \overline{(D_{\widehat{r_{\!p}}}^{(u)} X)^2}\big) - \ln\big(\overline{(D_{\widehat{r_{\!p}}}^{(v)} X)^2}\big)}{\ln(u/v)}.$$ We consider the following decomposition of $\ln(u/v)\widehat{\alpha}_p$: $$\begin{gathered}
\ln\big(\frac{n^{\alpha_p}}{n-u\widehat{r}_p+1} \sum_{k=0}^{n-u\widehat{r}_p}
\big( D_{\widehat{r_{\!p}},k}^{(u)}\,X\big)^2-u^{\alpha_p}l_n(p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})+u^{\alpha_p}l_n(p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})\big)\\
-\ln\big( \frac{n^{\alpha_p}}{n-v\widehat{r_{\!p}}+1} \sum_{k=0}^{n-v\widehat{r_{\!p}}} \big(
D_{\widehat{r_{\!p}},k}^{(v)}\,X\big)^2-v^{\alpha_p}l_n(p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})+v^{\alpha_p}l_n(p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})\big)
$$
Hence $\ln(u/v)(\widehat{\alpha}_p-\alpha_p)=F_n(u)-F_n(v)+o(F_n(u)+F_n(v))$ where $o(\cdot){\xrightarrow}[n\to\infty]{a.s.} 0$ as soon as $F_n(\cdot){\xrightarrow}[n\to\infty]{a.s.} 0$ with $$F_n(u)=\frac{n^{\alpha_p} \overline{\big(
D_{\widehat{r_{\!p}}}^{(u)}X\big)^2}-u^{\alpha_p}l_n(p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})}{u^{\alpha_p}l_n(p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})}=
\frac{F_{1,n,p}(u)+F_{2,n,p}(u)+F_{3,n,p}(u)}{u^{\alpha_p}l_n(p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})}$$ for ${\displaystyle}F_{1,n,p}(u) = n^{\alpha_p} \Big(\overline{\big( D_{\widehat{r_{\!p}}}^{(u)}X\big)^2}-\overline{ \big( D_{r_{\!p}}^{(u )}X\big)^2}\Big)$, ${\displaystyle}F_{2,n,p}(u) = n^{\alpha_p}\big( \overline{ \big( D_{r_{\!p}}^{(u)} X\big)^2}- {\mathds{E}}\overline{\big( D_{r_{\!p}}^{(u)}X\big)^2} \big)$ and ${\displaystyle}F_{3,n,p}(u) = n^{\alpha_p} {\mathds{E}}\Big(\,\overline{\big( D_{r_{\!p}}^{(u)}X\big)^2}\,\Big) - u^{\alpha_p} \, l_n(p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$.
#### [**(i) *Study of***]{} ${\displaystyle}F_{1,n,p}(u)$.
From Theorem \[t31\], we get that $\sum\limits_n{\mathds{P}}({\widehat{r_{{\scriptscriptstyle}0}}}\not= {r_{{\scriptscriptstyle}0}}) < \infty$, so, a.s. for $n$ large enough, $\widehat{{r_{{\scriptscriptstyle}0}}} = {r_{{\scriptscriptstyle}0}}$ and $F_{1,n,p}(u)\equiv 0$, $p=1$ or $p=2$.
#### [**(ii) *Study of $F_{2,n,p}(u)$.***]{}
We study $${\mathds{P}}\big( {\lv \overline{ \big( D_{r_{\!p}}^{(u)} X\big)^2}- {\mathds{E}}\overline{\big( D_{r_{\!p}}^{(u)}X\big)^2} {\right\vert}} > c_p n^{2(p-{\beta_{{\scriptscriptstyle}0}})} \psi_{np}^{-1}({\beta_{{\scriptscriptstyle}0}})\big)$$ for $c_p$ a positive constant, $\psi_{n2}({\beta_{{\scriptscriptstyle}0}}) \equiv \big(\frac{n}{\ln n}\big)^{\frac{1}{2}}$ and $$\psi_{n1}({\beta_{{\scriptscriptstyle}0}}) = \big(\frac{
n}{\ln n}\big)^{\frac{1}{2}} {\mathds{1}}_{]0,\frac{3}{4}[}({\beta_{{\scriptscriptstyle}0}}) +\big(\frac{n^{1/2}}{\ln n}\big){\mathds{1}}_{\{\frac{3}{4}\}}({\beta_{{\scriptscriptstyle}0}})+
\big(\frac{n^{2(1-{\beta_{{\scriptscriptstyle}0}})}}{\ln n}\big){\mathds{1}}_{]\frac{3}{4},1[}({\beta_{{\scriptscriptstyle}0}}).$$ We apply Lemma \[l64\] and Proposition \[p62\] with $p=1$ or $p=2$. After some calculations and the application of Borel Cantelli’s lemma with $c_p$ chosen large enough, we obtain that for $p=1$, almost surely, $
\varlimsup\limits_{n\to\infty} \psi_{np}({\beta_{{\scriptscriptstyle}0}}) {\lv F_{2,n,p}(u){\right\vert}} < +\infty
$ under the condition A\[h21\]-(iii-$p$), where $p=1$ or $2$.
#### **(iii) *Study of* $F_{3,n,p}(u)$.**
From and proceeding similarly as in , we get for ${r_{{\scriptscriptstyle}0}}\ge 1$, that $n^{\beta_1} \big(n^{\alpha_p}{\mathds{E}}\overline{\big(D^{(u)}_{r_{\!p}} X \big)^2} - u^{\alpha_p} \, l_n(p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}}) \big)$ could be decomposed into $B_{n1}+ B_{n2} + B_{n3}$ with $$\begin{gathered}
B_{n1}= -\frac{n^{\alpha_p + {\beta_{{\scriptscriptstyle}1}}}}{2(n-ur_p+1)} \sum_{k=0}^{n-ur_p} \sum_{i,j=0}^{r_{\!p}} b_{ikr}^{(u)} b_{jkr}^{(u)} (t_{k+iu} - t_k)^{{r_{{\scriptscriptstyle}0}}} (t_{k+ju} - t_k)^{{r_{{\scriptscriptstyle}0}}} \\ \times {\int\!\!\!\int}_{[0,1]^2} \frac{((1-v)(1-w))^{{r_{{\scriptscriptstyle}0}}-1}}{({r_{{\scriptscriptstyle}0}}- 1)!^2} {\lv \dot{v}_{ik} - \dot{w}_{jk}{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}1}}} \\ \times
\bigg\{ \frac{\frac{{\mathds{L}}^{({r_{{\scriptscriptstyle}0}},{r_{{\scriptscriptstyle}0}})}(\dot{v}_{ik},\dot{w}_{jk}) -\frac12 {\mathds{L}}^{({r_{{\scriptscriptstyle}0}},{r_{{\scriptscriptstyle}0}})}(\dot{v}_{ik},\dot{v}_{ik}) -\frac12
{\mathds{L}}^{({r_{{\scriptscriptstyle}0}},{r_{{\scriptscriptstyle}0}})}(\dot{w}_{jk},\dot{w}_{jk})}{{\lv \dot{v}_{ik} - \dot{w}_{jk}{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}}}-d_0(\dot{w}_{jk})}{{\lv \dot{v}_{ik} - \dot{w}_{jk}{\right\vert}}^{{\beta_{{\scriptscriptstyle}1}}}} - d_1(\dot{w}_{jk}) \bigg\} {\,{\mrm{d}}v}{\,{\mrm{d}}w}\end{gathered}$$ $$\begin{gathered}
B_{n2} = -\frac{n^{\alpha_p + {\beta_{{\scriptscriptstyle}1}}}}{2(n-ur_p+1)} \sum_{k=0}^{n-ur_p} \sum_{i,j=0}^{r_{\!p}} b_{ikr}^{(u)} b_{jkr}^{(u)} (t_{k+iu} - t_k)^{{r_{{\scriptscriptstyle}0}}} (t_{k+ju} - t_k)^{{r_{{\scriptscriptstyle}0}}} \\ \times {\int\!\!\!\int}_{[0,1]^2} \frac{((1-v)(1-w))^{{r_{{\scriptscriptstyle}0}}-1}}{({r_{{\scriptscriptstyle}0}}- 1)!^2} {\lv \dot{v}_{ik} - \dot{w}_{jk}{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}1}}} d_1(\dot{w}_{jk}),\end{gathered}$$ $$\begin{gathered}
B_{n3}= n^{{\beta_{{\scriptscriptstyle}1}}}\Big( \frac{-n^{\alpha_p}}{2(n-ur_p+1)} \sum_{k=0}^{n-ur_p} \sum_{i,j=0}^{r_{\!p}} b_{ikr}^{(u)} b_{jkr}^{(u)} (t_{k+iu} - t_k)^{{r_{{\scriptscriptstyle}0}}} (t_{k+ju} - t_k)^{{r_{{\scriptscriptstyle}0}}} \\ \times {\int\!\!\!\int}_{[0,1]^2} \frac{((1-v)(1-w))^{{r_{{\scriptscriptstyle}0}}-1}}{({r_{{\scriptscriptstyle}0}}- 1)!^2} {\lv \dot{v}_{ik} - \dot{w}_{jk}{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}} d_0(\dot{w}_{jk}) {\,{\mrm{d}}v}{\,{\mrm{d}}w}- u^{\alpha_p} l_n(p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})\Big)
$$ with $l_n(p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$ given by . Next, using Lemma \[l62\] and \[l63\] and the condition with uniform continuity of $d_1(\cdot)$, we get that $B_{n1} = o(1)$ and $B_{n2}$ has the limit: $$\begin{gathered}
-\frac{u^{\alpha_p+{\beta_{{\scriptscriptstyle}1}}}}{2} \sum_{i,j=0}^{r_{\!p}} \frac{(ij)^{{r_{{\scriptscriptstyle}0}}} \int_0^T d_1(t) \psi^{1-\alpha_p -{\beta_{{\scriptscriptstyle}1}}}(t) {\,{\mrm{d}}t}}{\prod_{\substack{m=0\\m\not=i}}^{r_{\!p}}(i-m)\prod_{\substack{q=0\\q\not=j}}^{r_{\!p}}(j-q)}
\\\times {\int\!\!\!\int}_{[0,1]^2} \frac{\big((1-v)(1-w)\big)^{{r_{{\scriptscriptstyle}0}}-1}}{(({r_{{\scriptscriptstyle}0}}- 1)!)^2}
{\lv iv -jw{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}1}}} {\,{\mrm{d}}v}\!{\,{\mrm{d}}w}.\end{gathered}$$ For the last term $B_{n3}$, one may show that it is of order ${{\mathcal}O} (n^{{\beta_{{\scriptscriptstyle}1}}-1})$. Finally, the case ${r_{{\scriptscriptstyle}0}}=0$ is treated similarly from .
#### **Conclusion.**
One may note that the determinist term, $l_n(p,{r_{{\scriptscriptstyle}0}},{\beta_{{\scriptscriptstyle}0}})$, defined in -, converges to the nonzero term: $$-\frac{1}{2} \sum_{i,j=0}^{r_{\!p}}\frac{ (ij)^{{r_{{\scriptscriptstyle}0}}}\int_0^T d_0(t) \psi^{-\alpha_p+1}(t){\,{\mrm{d}}t}}{\prod\limits_{\substack{m=0\\m\not=i}}^{r_{\!p}}(i-m)\prod\limits_{\substack{q=0\\q\not=j}}^{r_{\!p}}(j-q)}
{\int\!\!\!\int}_{[0,1]^2}\!\!\!\!\!\!\frac{((1-v)(1-w))^{{r_{{\scriptscriptstyle}0}}-1}}{({r_{{\scriptscriptstyle}0}}- 1)!^2}{\lv iv-jw{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}}\!\!{\,{\mrm{d}}v}\!{\,{\mrm{d}}w}$$ for ${r_{{\scriptscriptstyle}0}}\ge 1$ while if ${r_{{\scriptscriptstyle}0}}=0$, the limit is $ -\frac{1}{2} \sum_{i,j=0}^{p}
\frac{{\lv i-j{\right\vert}}^{2{\beta_{{\scriptscriptstyle}0}}}\int_0^T d_0(t) \psi^{-\alpha_p+1}(t){\,{\mrm{d}}t}}{\prod_{\substack{m=0\\m\not=i}}^{p}(i-m)\prod_{\substack{q=0\\q\not=j}}^{p}(j-q)}$.
Proofs of section \[AppInt\]
----------------------------
**Proof of Theorem \[t41\]**\
We set $\widetilde{{r_{{\scriptscriptstyle}0}}} = \max(\widehat{{r_{{\scriptscriptstyle}0}}},1)$ and, for $\widehat{{r_{{\scriptscriptstyle}0}}}$ and ${\widetilde}{X}_r(\cdot)$ respectively defined in and , we use the convention: $ {\widetilde}{X}_{\widetilde{{r_{{\scriptscriptstyle}0}}}}(\cdot)= {\widetilde}{X}_{m_n-1}(\cdot)$ and ${\widetilde}{X}_{\widehat{{r_{{\scriptscriptstyle}0}}}+1}(\cdot)= {\widetilde}{X}_{m_n}(\cdot)$ when $\widehat{{r_{{\scriptscriptstyle}0}}} = l_0$.
\(a) If ${\overline r}= \max(r,1)$ and $\overline{{r_{{\scriptscriptstyle}0}}}=\max({r_{{\scriptscriptstyle}0}},1)$, we get, for $n$ large enough such that ${r_{{\scriptscriptstyle}0}}\le m_n -2$, $$\begin{aligned}
\big(X(t) - {\widetilde}{X}_{\widetilde{{r_{{\scriptscriptstyle}0}}}}(t) \big)^2
&=\sum_{r=0}^{m_n -2} \big(X(t) - {\widetilde}{X}_{{\overline r}}(t) \big)^2{\mathds{1}}_{\{\widehat{{r_{{\scriptscriptstyle}0}}}=r\}}+ \big(X(t) - {\widetilde}{X}_{m_n-1}(t) \big)^2{\mathds{1}}_{\{\widehat{{r_{{\scriptscriptstyle}0}}}=l_0\}}
\\
&\le \big(X(t) - {\widetilde}{X}_{\overline{{r_{{\scriptscriptstyle}0}}}}(t) \big)^2+ {\mathds{1}}_{\{\widehat{{r_{{\scriptscriptstyle}0}}}\not={r_{{\scriptscriptstyle}0}}\}}\sum_{r=0,r\not={r_{{\scriptscriptstyle}0}}}^{m_n -1} \big(X(t) - {\widetilde}{X}_{{\overline r}}(t) \big)^2.\end{aligned}$$ By this way, $e_{\rho}^2 ({\mathrm{app}}\big(\widehat{{r_{{\scriptscriptstyle}0}}})\big)$ should be bounded by $$\int_0^T \!\!\!{\mathds{E}}\big(X(t) - {\widetilde}{X}_{\overline{{r_{{\scriptscriptstyle}0}}}}(t) \big)^2 \rho(t){\,{\mrm{d}}t}+ \big({\mathds{P}}(\widehat{{r_{{\scriptscriptstyle}0}}} \not={r_{{\scriptscriptstyle}0}})\big)^{\frac{1}{2}} \!\!\!\sum_{r=0,r\not={r_{{\scriptscriptstyle}0}}}^{m_n -1}\!\!\int_0^T \!\! \Big({\mathds{E}}\big(X(t) - {\widetilde}{X}_{{\overline r}}(t) \big)^4\Big)^{\frac{1}{2}} \rho(t){\,{\mrm{d}}t}$$ We make use of the exponential bound established for ${\mathds{P}}(\widehat{{r_{{\scriptscriptstyle}0}}} \not={r_{{\scriptscriptstyle}0}})$ in Theorem \[t31\] as well as the property ${\mathds{E}}(Y^4) \le 3 \big({\mathds{E}}(Y^2)\big)^2$ for a Gaussian r.v. $Y$. Moreover, ${\displaystyle}\sup_{t\in[0,T]}\Big({\mathds{E}}\big(X(t) - {\widetilde}{X}_{r}(t)\big)^2\Big) = \max\limits_{k=0,\dotsc,\lfloor \frac{n}{r}\rfloor - 1} \sup\limits_{t\in {{\mathcal}I}_k} \Big({\mathds{E}}\big(X(t) - {\widetilde}{X}_{r}(t)\big)^2\Big)$. If ${r_{{\scriptscriptstyle}0}}\ge 1$, we use the decomposition established in @BV08 [lemma 4.1] to obtain, for $t\in{{\mathcal}I}_k$ and $r^{\ast} = \min(r,{r_{{\scriptscriptstyle}0}})$: $$\begin{gathered}
{\mathds{E}}\big(X(t) - {\widetilde}{X}_{r}(t)\big)^2 = \sum_{i,j=0}^{r} L_{i,k,r}(t) L_{j,k,r}(t) \frac{(t_{kr+i}-t_{kr})^{\ra}(t_{kr+j}-t_{kr})^{\ra}}{((\ra-1)!)^2} \\\times {\int\!\!\!\int}_{[0,1]^2} \big((1-v)(1-w)\big)^{\ra -1} \Big\{ {\mathds{L}}^{(\ra,\ra)}(t_{kr}+ (t-t_{kr})v,t_{kr} + (t-t_{kr})w) \\- {\mathds{L}}^{(\ra,\ra)}(t_{kr}+ (t-t_{kr})v,t_{kr} + (t_{kr+j}-t_{kr}) w) \\- {\mathds{L}}^{(\ra,\ra)}(t_{kr}+ (t_{kr+i} - t_{kr}) v, t_{kr} + (t-t_{kr})w)\\ + {\mathds{L}}^{(\ra,\ra)}(t_{kr}+ (t_{kr+i} -t_{kr}) v,t_{kr} + (t_{kr+j} - t_{kr})w) \Big\} {\,{\mrm{d}}v}\!{\,{\mrm{d}}w}.\end{gathered}$$ If $r=1,\dotsc,{r_{{\scriptscriptstyle}0}}-1$, $({r_{{\scriptscriptstyle}0}}\ge 2)$, we obtain the uniform bound ${{\mathcal}O} \big( {\delta_n}^{2r+2} \big)$ by uniform continuity of ${\mathds{L}}^{(r+1,r+1)} (\cdot,\cdot)$ and results of Lemma \[l62\]. For $r={r_{{\scriptscriptstyle}0}},\dotsc,m_n$, we have $r^{\ast} = {r_{{\scriptscriptstyle}0}}$ so we apply the Hölderian regularity condition . Since $L_{i,k,r}(t) \le r^r$, we arrive at ${\displaystyle}\sup_{t\in[0,T]}{\mathds{E}}\big(X(t) - {\widetilde}{X}_{\overline{{r_{{\scriptscriptstyle}0}}}}(t)\big)^2 = {{\mathcal}O}\big({\delta_n}^{2({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}})}\big)$ for $r={r_{{\scriptscriptstyle}0}}$ while if $r={r_{{\scriptscriptstyle}0}}+1,\dotsc,m_n$, ${\displaystyle}\sup_{t\in[0,T]}{\mathds{E}}\big(X(t) - {\widetilde}{X}_{{\overline r}}(t)\big)^2 = {{\mathcal}O}\big(m_n^{2(m_n+{r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}})} {\delta_n}^{2({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}})}\big)$. The logarithmic order of $m_n$ yields the final result. In the case where ${r_{{\scriptscriptstyle}0}}= 0$, above results hold true starting from $$\begin{gathered}
{\mathds{E}}\big(X(t) - {\widetilde}{X}_{{\overline r}}(t)\big)^2 = \sum_{i,j=0}^{{\overline r}} L_{i,k,{\overline r}}(t) L_{j,k,{\overline r}}(t)\Big\{ {\mathds{L}}(t,t) - {\mathds{L}}(t,t_{k{\overline r}+j}) \\- {\mathds{L}}(t_{k{\overline r}+i},t) + {\mathds{L}}(t_{k{\overline r}+i},t_{k{\overline r}+j}) \Big\}.\end{gathered}$$
\(b) For $e_{\rho}^2 ({\rm{int}}\big(\widehat{{r_{{\scriptscriptstyle}0}}})\big)$, $\int_0^T \big( X(t) -\widetilde{X}_{r+1}\big)\rho(t){\,{\mrm{d}}t}$ is again a Gaussian variable, so in a similar way as for approximation, we get the following bound for this term: $$\begin{gathered}
\sqrt{3}\big({\mathds{P}}(\widehat{{r_{{\scriptscriptstyle}0}}} \not={r_{{\scriptscriptstyle}0}})\big)^{\frac{1}{2}} \sum_{r=0}^{m_n}
\Big(\sup_{t\in[0,T]}\Big({\mathds{E}}\big(X(t) - {\widetilde}{X}_{r+1}(t)\big)^2\Big)^{\frac{1}{2}}\Big)^2 \big(\int_0^T \rho(t){\,{\mrm{d}}t}\big)^2
\\+
\sum_{k=0}^{\lfloor \frac{n}{{r_{{\scriptscriptstyle}0}}+1}\rfloor-1}\sum_{\ell=0}^{\lfloor \frac{n}{{r_{{\scriptscriptstyle}0}}+1}\rfloor-1} \int_{{{\mathcal}I}_k}\int_{{{\mathcal}I}_{\ell}}{\mathds{E}}\big(X(t) - {\widetilde}{X}_{{r_{{\scriptscriptstyle}0}}+1}(t) \big) \big(X(s) - {\widetilde}{X}_{{r_{{\scriptscriptstyle}0}}+1}(s) \big)\rho(t)\rho(s)\,\mathrm{d}s\!{\,{\mrm{d}}t}.\end{gathered}$$
#### Study of the term ${\displaystyle}{\mathds{E}}\big(X(t) - {\widetilde}{X}_{{r_{{\scriptscriptstyle}0}}+1}(t) \big) \big(X(s) - {\widetilde}{X}_{{r_{{\scriptscriptstyle}0}}+1}(s) \big) $, $(s,t)\in {{\mathcal}I}_{\ell}\times {{\mathcal}I}_k$.
Denoting $\overline{r}={r_{{\scriptscriptstyle}0}}+1$ we get again from lemma 4.1 of @BV08 that ${\mathds{E}}\big(X(t) - {\widetilde}{X}_{\overline{r}}(t) \big) \big(X(s) - {\widetilde}{X}_{\overline{r}}(s) \big)$ is equal to: [$$\begin{gathered}
\sum_{i,j=0}^{\overline{r}} L_{i,k,\overline{r}}(t) L_{j,\ell,\overline{r}}(s) \frac{((t_{k \overline{r} +i}-t_{k \overline{r} })(t_{\ell \overline{r} +j}-t_{\ell \overline{r} }))^{{r_{{\scriptscriptstyle}0}}}}{(({r_{{\scriptscriptstyle}0}}-1)!)^2} {\int\!\!\!\int}_{[0,1]^2}\!\!\! \!\!\!{\,{\mrm{d}}v}\!{\,{\mrm{d}}w}\, ((1-v)(1-w))^{{r_{{\scriptscriptstyle}0}}-1}\\ \times
\Big\{
{\mathds{L}}^{({r_{{\scriptscriptstyle}0}},{r_{{\scriptscriptstyle}0}})} (t_{k \overline{r} }+(t-t_{k \overline{r} })v, t_{\ell \overline{r} }+(t-t_{\ell \overline{r} })w)
-{\mathds{L}}^{({r_{{\scriptscriptstyle}0}},{r_{{\scriptscriptstyle}0}})} (t_{k \overline{r} }+(t-t_{k \overline{r} })v, t_{\ell \overline{r} }+ (t_{\ell \overline{r} +j} - t_{\ell \overline{r} } )w)
\\- {\mathds{L}}^{({r_{{\scriptscriptstyle}0}},{r_{{\scriptscriptstyle}0}})} (t_{k \overline{r} }+ (t_{k \overline{r} +i} - t_{k \overline{r} } )v, t_{\ell \overline{r} }+(t-t_{\ell \overline{r} })w)
+ {\mathds{L}}^{({r_{{\scriptscriptstyle}0}},{r_{{\scriptscriptstyle}0}})} (t_{k \overline{r} }+ (t_{k \overline{r} +i} - t_{k \overline{r} } )v, t_{\ell \overline{r} }+(t-t_{\ell \overline{r} })w)\Big\}.\end{gathered}$$]{} For non-overlapping intervals ${{\mathcal}I}_k$ and ${{\mathcal}I}_{\ell}$, that is ${\lv k-l{\right\vert}}\ge 2$, we make use of Condition A\[h22\](2) four times, by adding and subtracting the necessary terms, noting that $$\begin{gathered}
\sum_{i,j=0}^{ \overline{r} } L_{i,k, \overline{r} }(t)L_{j,\ell, \overline{r} }(s) (t_{k \overline{r} +i} - t_{k \overline{r} })^{r_1}(t_{\ell \overline{r} +j} - t_{\ell \overline{r} })^{r_2}= (t-t_{k \overline{r} })^{ r_1}(s- t_{\ell \overline{r} })^{ r_2}.\end{gathered}$$ with either $r_i= \overline{r} -1$ or $r_i = \overline{r} $ for $i=1,2$. By this way, we get $$\begin{gathered}
\sum_{\stackrel{k,\ell=0}{{\lv k-\ell{\right\vert}}\ge 2}}^{\lfloor \frac{n}{\overline{r}}\rfloor-1} \int_{{{\mathcal}I}_k}\int_{{{\mathcal}I}_{\ell}}{\mathds{E}}\big(X(t) - {\widetilde}{X}_{\overline{r}}(t) \big) \big(X(s) - {\widetilde}{X}_{\overline{r}}(s) \big)\rho(t)\rho(s)\,\mathrm{d}s\!{\,{\mrm{d}}t}\\ = {{\mathcal}O}\Big({\delta_n}^{2({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}}+1)}\sum_{\stackrel{k,\ell=0}{{\lv k-\ell{\right\vert}}\ge 2}}^{\lfloor \frac{n}{\overline{r}}\rfloor-1} \big\lvert{\lv k-\ell{\right\vert}}-1\big\rvert^{-2(2-{\beta_{{\scriptscriptstyle}0}})} \Big)
$$ which is a ${{\mathcal}O}\Big({\delta_n}^{2({r_{{\scriptscriptstyle}0}}+{\beta_{{\scriptscriptstyle}0}})+1}\Big)$. For overlapping intervals ${{\mathcal}I}_k$ and ${{\mathcal}I}_{\ell}$, that is in the case where ${\lv k-l{\right\vert}} \le 1$, we make use of Cauchy-Schwarz inequality to obtain the same bound as above. Since the second part of $e_{\rho}^2 (\rm{int}\big(\widehat{{r_{{\scriptscriptstyle}0}}})\big) $ is negligible, we obtain the result.
|
---
abstract: |
The attractive Hubbard model in $d=2$ is studied through Monte Carlo simulations at intermediate coupling. There is a crossover temperature $%
T_{X} $ where a pseudogap appears with concomitant precursors of Bogoliubov quasiparticles that are [*not*]{} local pairs. The pseudogap in $A\left(
{\bf k}_{F},\omega \right) $ occurs in the renormalized classical regime when the correlation length is larger than the direction-dependent thermal de Broglie wave length, $\xi _{th}=\hbar v_{F}\left( {\bf k}\right) /k_{B}T.$ The ratio $T_{X}/T_{c}$ for the pseudogap may be made arbitrarily large when the system is close to a point where the order parameter has $SO\left(
n\right) $ symmetry with $n>2$. This is relevant in the context of $SO\left(
5\right) $ theories of high $T_{c}$ but has more general applicability.
address:
- |
$^{1}$Département de Physique and Centre de Recherche en Physique du\
Solide and $^{2}$Institut canadien de recherches avancées
- |
Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1.\
$^{3}$2100 Valencia Dr. apt. 406, Northbrook, IL 60062
author:
- 'S. Allen$^{1}$, H. Touchette$^{1}$, S. Moukouri$^{1}$, Y.M. Vilk$^{3}$, and A.-M. S. Tremblay$^{1,2}$[@email]'
date: 'April, revised August 1999'
title: Role of symmetry and dimension for pseudogap phenomena
---
=\#1[[\#1]{}]{} \#1[\#1]{} \#1[\#1 ]{} \#1[\#1 ]{} \#1[\#1 ]{} \#1\#2[[\#1 \#2 ]{}]{} \#1[[\#1 ]{}]{} \#1[[**\#1**]{}]{}
Normal state pseudogaps observed in angle-resolved photoemission experiments (ARPES)[@ARPES] and tunneling[@Tunnel] have become the focus of much of the research on high-temperature superconductors. Theoretically, strong-coupling models of this phenomenon include repulsive doped insulator models[@Lee], and attractive models with preformed local pairs[@Preformed; @Randeria92; @Stintzing]. At intermediate coupling, resonant pair scattering has been invoked[@Levin]. In weak coupling, antiferromagnetic or superconducting fluctuations also lead to pseudogap formation before long range order is established[@VT; @LongPaper; @VA; @Schmalian]. Finally, arguments about the relevance of phase fluctuations[@PhaseFluct] do not specify whether the small superfluid density comes from strong coupling or from thermal fluctuation effects.
Although the effect of superconducting fluctuations on the density of states was studied long ago[@1970], the relevance of these critical fluctuations on pseudogap phenomena in high temperature superconductors has been questioned mainly because, even in low dimension, the critical region should be quite small compared with the size of the pseudogap region observed in high temperature superconductors. Indeed, even for the Kosterlitz-Thouless (KT) transition in two dimensions, the range of temperatures where superconducting phase fluctuations occur is of order $%
k_{B}T_{c}^{2}/E_{F}$ unless disorder depresses $T_{c}$[@Sharapov]. Another question for the original critical-fluctuation calculations is that, contrary to density of states pseudogaps, a pseudogap can appear in the single-particle spectral weight $A\left( {\bf k}_{F}{\bf ,}\omega \right) $ only when the fluctuation-induced scattering rate at the Fermi surface $%
%TCIMACRO{\func{Im}}%
%BeginExpansion
\mathop{\rm Im}%
%EndExpansion
\Sigma \left( {\bf k}_{F},\omega =0\right) $ increases with decreasing temperature, a behavior that is opposite to that predicted by the traditional phase-space arguments of Fermi liquid theory.
The last objection has already been answered[@VT] by showing that in $%
d=2 $ the Fermi liquid arguments fail when one enters the so-called renormalized classical (RC) regime of fluctuations ($d=3$ is the upper critical dimension [@VT]). This RC regime is the one where critical slowing down leads to increasing dominance of classical fluctuations $\left(
\hbar \omega _{c}<k_{B}T\right) $ as temperature is lowered. However, the question of the large temperature range where pseudogaps appear has to be addressed theoretically both at the qualitative and at the quantitative levels. On the quantitative side, the dependence of the critical region on interaction strength and quasi two-dimensionality is not obvious, but it has been argued [@Preosti] that specific models can give large critical regions even at intermediate coupling. On the qualitative side, a factor that may considerably enlarge the size of the pseudogap regime is the proximity to a point where the order-parameter symmetry is $SO\left(
n\right) $ with $n>2$. Indeed, in $d=2$ the critical region can become much larger when $n>2$ since then the Mermin-Wagner theorem implies that the transition temperature is pushed down to $T=0$. We argue that both conditions, namely $d=2$ and higher symmetry, are generic for high-temperature superconducting materials. Indeed, in the underdoped region, where the pseudogap is largest, these materials are highly anisotropic (quasi two-dimensional) and it has been proposed that the order parameter may have both antiferromagnetic and superconducting character corresponding to approximate $SO\left( 5\right) $ symmetry[@Zhang].
The attractive Hubbard model may be used to illustrate the properties of pseudogaps that appear in such situations of approximate high-symmetry in $%
d=2$. For our purposes, the important characteristic of this model is that the long-wavelength critical behavior is as follows for all values of interaction $U$. At half-filling, there is a zero-temperature phase transition that breaks the finite-temperature $SO\left( 3\right) $ symmetry [@NoteSO(4)] while away from half-filling, there is a KT transition at finite temperature. The corresponding ground state breaks $SO\left( 2\right)
$ symmetry$.$ While the details of this model are clearly inappropriate for high-temperature superconductors, it is useful to illustrate a number of general points that should be applicable to models with transition temperatures that are pushed down from their mean-field value by a combination of low dimension and high order-parameter symmetry $SO\left(
n>2\right) $[@Zhang].
In this paper, we present Monte Carlo simulations for the $d=2$ Hubbard model at $U=-4t,$ a value that is slightly on the BCS side of the BCS to Bose-Einstein crossover curve $\left( U<U\left( T_{c}^{\max }\right) \right)
$[@Singer98]. We use units where nearest-neighbor hopping is $t=1,$ lattice spacing is unity, $\hbar =1$ and $k_{B}=1$. Previous numerical work charted the phase diagram[@Scalettar]. They have also investigated the pseudogap phenomenon mostly in strong coupling where, we stress, the Physics is different from the case discussed here[@Singer98; @Singer96; @Singer99]. On the weak-coupling side of the BCS to Bose-Einstein crossover, there have been numerical studies of KT superconductivity[@Moreo92] as well as several discussions of pseudogap phenomena in the spin properties and in the total density of states at the Fermi level[@Randeria92; @Trivedi95; @Singer96]. The only study of $A\left( {\bf k},\omega
\right) $ was restricted to regions far from the $SO\left( 3\right) $ symmetric point[@Singer99].
Here we establish a dynamical connection between the appearance of the RC regime in the pairing collective modes and pseudogap formation in single-particle quantities. In particular, we show that a) Close to a high symmetry point, pseudogaps can appear at a crossover temperature $T_{X}$ that scales with the mean-field transition temperature while the real transition may occur at much lower temperature, $T_{c},$ leading to a wide temperature range for the pseudogap. b) At the crossover temperature, one enters the RC regime where the characteristic frequency for fluctuations becomes smaller than the temperature. c) Pseudogap in weak-to-intermediate coupling do not require resonance in the two-particle correlations[@NotePairPseud]. d) To have a pseudogap in $A\left( {\bf k}_{F}{\bf ,}\omega
\right) $ for a given wave vector, it is not enough to have the collective mode (two-particle) correlation length satisfy $\xi $ $>1.$ It is necessary that $\xi $ becomes larger than the single-particle thermal de Broglie wavelength $\xi _{th}=v_{F}\left( {\bf k}\right) /T$. This implies in particular that even for an isotropic interaction, as temperature decreases a pseudogap opens first near the zone edge, where $v_{F}$ is small, and it opens last along the zone diagonal where $v_{F}$ is largest. This anisotropy would be amplified for an anisotropic interaction of $d-$wave type[@Preosti]. For $U=-4$, the condition $\xi $ $>\xi _{th}$ is realized near the zone edge for $\xi $ not so large. Analytical arguments for the above results have appeared elsewhere[@LongPaper; @VA; @Vilk].
Let us first recall a few results at half-filling, $\left\langle
n\right\rangle =1,$ where the chemical potential $\mu $ vanishes. There the canonical transformation $c_{i\downarrow }\rightarrow \left( -1\right)
^{i_{x}+i_{y}}c_{i\downarrow }^{\dagger }$ maps the attractive model onto the repulsive one at the same filling. The ${\bf q}=0,$ $s-$wave superconducting fluctuations and the ${\bf Q}=\left( \pi ,\pi \right) $ charge fluctuations are mapped onto the three components of antiferromagnetic spin fluctuations of the repulsive model and hence they are degenerate. Because of this degeneracy, the order parameter at half-filling has $SO\left( 3\right) $ symmetry[@NoteSO(4)], hence, by the Mermin-Wagner theorem, in two dimensions the phase transition is at $%
T_{c}=0$. Results for the attractive model are easily extracted from simulations of the canonically equivalent repulsive model. The pair structure factor $S_{\Delta }=\left\langle \Delta ^{\dagger }\Delta +\Delta
\Delta ^{\dagger }\right\rangle $ with $\Delta =\frac{1}{\sqrt{N}}%
\sum_{i=1}^{N}c_{i\uparrow }c_{i\downarrow }$ and the ${\bf Q=}\left( \pi
,\pi \right) $ charge structure factor $S_{c}=\left\langle \rho _{{\bf Q}%
}\rho _{-{\bf Q}}\right\rangle $ are identical, showing an increase as temperature decreases and then size-dependent saturation. The sudden rise of $S_{\Delta }$ as temperature decreases indicates a crossover to a RC regime with a concomitant opening of the pseudogap[@VT; @LongPaper; @Moukouri99] in $A\left( {\bf k}_{F,}\omega \right) $. The crossover temperature is a sizeable fraction of the mean-field transition temperature.
8.5cm
Slightly away from half-filling, the $SO\left( 3\right) $ symmetry is formally broken by the chemical potential[@NoteSO(4)] since it couples only to the charge part of the triplet. However, in the regime where the temperature is larger than the symmetry-breaking field, the symmetry is approximately satisfied.[@VA] For filling $\left\langle n\right\rangle
=0.95$, we have $T>\left| \mu \right| $ for the whole range of temperature shown in Fig.1(a) ($\mu \sim -0.07$ at $T=0.125$)$.$ On this figure, superconducting ($S_{\Delta },$ filled symbols) and charge ($S_{c},$ open symbols) structure factors are of comparable size when one enters the RC regime, showing that we have approximate $SO\left( 3\right) $ symmetry for various sizes $L\times L.$ The beginning of the RC regime, that occurs at a temperature nearly identical to the crossover temperature [@VT] $T_{X}$ identified at half-filling, is signaled by the increase in the magnitude of correlations. Eventually, the concomitant increase of $\xi $ leads to the size-dependence apparent at lower temperature in Fig.1(a). The plot in Fig.1(a) resembles the result at half-filling.[@White89; @LongPaper] The near equality of superconducting and charge fluctuations at the crossover to the RC regime should be contrasted with the case $\left\langle
n\right\rangle =0.8$ in Fig.1(b) where the charge fluctuations show basically no critical behavior when the superconducting correlations begin to do so. Hence, at this filling, there is little $SO\left( 3\right) $ symmetry left at $T_{X}.$ This is expected since the symmetry-breaking field $\left| \mu \left( T_{X}\right) \right| =\left| \mu \left( 0.25\right)
\right| =0.26$ is comparable to $T_{X}$. One basically enters directly into the RC regime of a $SO\left( 2\right) $ KT transition[@Scalettar]. In this regime $dT_{c}\left( n\right) /dn>0.$
Let us go back to the filling $\left\langle n\right\rangle =0.95.$ At this filling, it has been estimated[@Scalettar] that $T_{c}<0.1$ and $%
dT_{c}\left( n\right) /dn<0$. We already showed that at this filling, one has approximate $SO\left( 3\right) $ symmetry. One can confirm that the increase of $S_{\Delta }$ and $S_{c}$ at $T_{X}$ comes from RC fluctuations by calculating the spectral weight for superconducting fluctuations $\chi
_{\Delta }^{\prime \prime }$ (imaginary part of the $T-matrix$), $\chi
_{\Delta }^{\prime \prime }\left( \omega \right) =\int dt\frac{1}{2}%
\left\langle \left[ \Delta \left( t\right) ,\Delta ^{\dagger }\right]
\right\rangle e^{i\omega t}.$ The even part[@NotePaire] of $\chi
_{\Delta }^{\prime \prime }\left( \omega \right) /\omega $ was obtained from a Monte Carlo calculation of the imaginary-time quantity $\left\langle
\Delta \left( \tau \right) \Delta ^{\dagger }+\Delta ^{\dagger }\left( \tau
\right) \Delta \right\rangle $ followed by Maximum Entropy inversion[@Meshkov]. The even part of $\chi _{\Delta }^{\prime \prime }\left( \omega
\right) /\omega $ is plotted on Fig.2 for an $8\times 8$ system and various temperatures.
The symbols are for a $6\times 6$ system at $T=1/5.$ The maximum is at zero frequency, as for an overdamped mode $\chi _{\Delta }^{-1}\left( {\bf q}%
,\omega \right) \propto \xi ^{-2}$ $+q^{2}-i\omega /\omega _{0}$. The characteristic frequency, given by the half-width at half maximum, is $%
\omega _{c}=\omega _{0}/\xi ^{2}$ with $\omega _{0}$ a microscopic relaxation rate. There is a marked narrowing of the width in frequency as temperature decreases. One enters the RC regime when $\omega _{c}\simeq
1/4\simeq T_{X}$, a temperature larger than $T_{c}\left( <0.1\right) $. At $%
T=1/5$, the correlation length is becoming comparable with system size since the $6\times 6$ system gives a result that differs from $8\times 8$.
The effect of RC collective fluctuations on single-particle quantities is illustrated in Fig.3 (a) to (c) that show density plots of the single-particle spectral weight $A\left( {\bf k,}\omega \right) $ for an $%
8\times 8$ system at, respectively, $T=1/3,1/4,$ and $1/5.$
$\quad $When temperature reaches $T=1/4$ one notices that a minimum ([*pseudogap*]{}) around ${\bf k}=\left( 0,\pi \right) $, $\omega =0$ develops along with two maxima away from $\omega =0$. The latter maxima are [*precursors*]{} of the Bogoliubov quasiparticles of the ordered state[@LongPaper][@Kyung]. The pseudogap becomes deeper and deeper as temperature decreases, the distance between maxima remaining about constant, as observed in high temperature superconductors[@Tunnel]. The condition for the appearance of a pseudogap in $A\left( {\bf k,}\omega \right) $ is not only that we should be in the RC regime and in low dimension but also that $\xi $ $/\xi _{th}$ should be large[@VT; @LongPaper]. This illustrated by the fact that the pseudogap at ${\bf k=}\left( \pi /2,\pi
/2\right) ,$ where the Fermi velocity is larger, is not opened yet at $T=1/4$ despite the fact that at $T=1/5$ the pseudogap is of comparable size both around ${\bf k}=\left( 0,\pi \right) $ and at ${\bf k=}\left( \pi /2,\pi
/2\right) $, in concordance with the equality of the gaps at these two points in the zero-temperature spin-density wave state. From the slopes in Figs.3 (a) to (c), $v_{F}$ is clearly larger at $\left( \pi /2,\pi /2\right)
$ meaning that the condition $\xi $ $>\xi _{th}=v_{F}/T$ is harder to satisfy at this wave vector. Numerical estimates show that $\xi $ is nearly isotropic by contrast with $v_{F}.$ These estimates are consistent with the appearance of the pseudogap in $A\left( {\bf k,}\omega \right) $ when $\xi $ $\sim \xi _{th}.$
As in any numerical simulation, finite-size effects should be considered carefully. There are two important intrinsic lengths in this problem, namely $\xi _{th}$ and $\xi .$ When $L\ll \xi _{th},$ the system acts as if it was basically in the quantum zero-temperature limit of a finite system. We have checked that at $T=1/8,$ $A\left( {\bf k},\omega \right) $ shows real gaps, instead of pseudogaps, that appear at progressively higher temperature in systems of smaller size. For $T=1/3,1/4$ on the other hand, estimates of $%
\xi _{th}$ and of $\xi $ as well as calculations for $6\times 6$ systems and a few $10\times 10$ systems suggest that our numerical results for $A\left(
{\bf k,}\omega \right) $ on $8\times 8$ systems are free of appreciable size effects, i.e. $L$ $>\xi ,\xi _{th}$. This can also be checked by the size dependence of the results in Fig.1(a) and Fig.2. While size effects become important in $\chi _{\Delta }^{\prime \prime }\left( \omega \right) /\omega $ when $\xi $ exceeds $L$, as long as $\xi _{th}<L$ then it is possible to see thermally induced pseudogap effects in $A\left( {\bf k},\omega \right) $ even if $\xi >L$[@VT].
In summary, the qualitative phase diagram for the attractive Hubbard model in $d=2$ sketched in Fig.4 shows that near a point with high order parameter symmetry, the transition temperature decreases while the pseudogap temperature increases along with the mean-field transition temperature and the zero-temperature gap.
5.5cm
In the region where $dT_{c}/dn<0,$ the crossover to KT critical behavior occurs in the RC pseudogap regime when $T$ is less than the symmetry breaking field. In our simulations we would need larger system size to reach the KT critical regime. Contrary to the scenario of Ref.[@Levin], in our case dimension and symmetry contribute to create a wide pseudogap region, there is no critical coupling strength, and furthermore one enters the RC regime without sharp resonance in $\chi _{\Delta }^{\prime \prime
}\left( \omega \right) /\omega .$ Also, the precursors of Bogoliubov quasiparticles in $A\left( {\bf k},\omega \right) $ occur under conditions very different from those for strong-coupling Cooper pairs that are local and do not need low dimension or large $\xi /\xi _{th}$. Comparisons with non-perturbative many-body calculations should appear elsewhere[@Kyung].
In high $T_{c}$ superconductors the competition is between antiferromagnetism and superconductivity. Recent time-domain transmission spectroscopy experiments[@Orenstein98] suggest that the RC regime for the KT transition (hatched region in Fig.4) has been observed. Close enough to the transition there is dimensional crossover to $d=3$. For antiferromagnetic fluctuations, there are suggestions from NMR of a RC regime [@Pines], but there is no definite proof.
We acknowledge stimulating discussions with B. Kyung, R.J. Gooding, R.B. Laughlin, J. Orenstein, D. Sénéchal, S.C. Zhang and participants of the 1998 Aspen Summer Workshop on High-Temperature Superconductivity. We thank H.-G. Mattutis for discussions on Monte Carlo methods, and D. Poulin and L. Chen for numerous contributions to codes. Monte Carlo simulations were performed in part on an IBM-SP2 at the Centre d’Applications du Calcul Parallèle de l’Université de Sherbrooke. This work was partially supported by NSERC (Canada), and by FCAR (Québec).
electronic address: tremblay@physique.usherb.ca
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abstract: 'We study structural properties of truncated Weyl modules. A truncated Weyl module $W_N(\lambda)$ is a local Weyl module for $\lie g[t]_N = \lie g \otimes \frac{\mathbb C[t]}{t^N\mathbb C[t]}$, where $\lie g$ is a finite-dimensional simple Lie algebra. It has been conjectured that, if $N$ is sufficiently small with respect to $\lambda$, the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when $\lambda$ is a multiple of certain fundamental weights, including all minuscule ones for simply laced $\lie g$. We also take a further step towards proving the conjecture for all multiples of fundamental weights by proving that the corresponding truncated Weyl module is isomorphic to a natural quotient of a fusion product of Kirillov-Reshetikhin modules. One important part of the proof of the main result shows that any truncated Weyl module is isomorphic to a Chari-Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that $\lie g=\lie{sl}_2$ related to Demazure flags and chains of inclusions of truncated Weyl modules.'
address:
- 'Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Germany'
- 'Departamento de Matemática Pura e Aplicada, CCENS, Universidade Federal do Espírito Santo, Alegre - ES - Brazil, 25900-000'
- 'Departamento de Matemática, Universidade Estadual de Campinas, Campinas - SP - Brazil, 13083-859.'
author:
- 'Ghislain Fourier, Victor Martins, Adriano Moura'
title: On Truncated Weyl Modules
---
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[^2]
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Introduction
============
The concept of Weyl modules in the realm of finite dimensional representation theory of classical and quantum affine algebras was introduced by Chari and Pressley in [@cp:weyl]. The definition, given via generators and relations, was inspired by the similar notion in modular representation theory of algebraic groups. In the years that followed, the notion was extended to other algebras sharing some similarity with the affine Kac-Moody algebras such as algebras of the form $\lie g\otimes A$ where $\lie g$ is a symmetrizable Kac-Moody algebra or a Lie super algebra and $A$ is a commutative associative algebra with unit (see [@cls; @cfk:cat; @fl:multw; @frs; @jm:weyl; @nesa:sur] and references therein). After [@fl:multw], there is a distinction between two kinds of Weyl modules: the local and the global ones (both appeared in [@cp:weyl], but only the former under the terminology Weyl module). Since this work is concerned only with local Weyl modules, we shall simply say Weyl modules.
The context of the current algebra $\lie g[t]=\lie g\otimes\mathbb C[t]$ with $\lie g$ a finite-dimensional simple Lie algebra is the most studied for several reasons. On one hand, some questions about the structure of the Weyl modules for quantum and classical affine Kac-Moody algebras can be reformulated as questions about certain quotients of the graded Weyl modules for $\lie g[t]$. In particular, this connects the theory of Weyl modules to those of Demazure modules ([@cl:wdf; @foli:wdkr]) and fusion products (in the sense of [@fl:kosver]). On the other hand, the study of the category of graded finite-dimensional representations of the current algebra is motivated by applications in mathematical physics, algebraic geometry and geometric Lie theory, as well as combinatorics.
The definition of Weyl modules for generalized current algebras can be explained as follows. Given a triangular decomposition of $\lie g$, say $\lie g=\lie n^-\oplus\lie h\oplus \lie n^+$ where $\lie h$ is a Cartan subalgebra, and a $\mathbb Z_{\ge 0}$-graded, commutative, associative algebra with unit $A$, consider the induced decomposition: $$\lie g\otimes A=\lie n^- \otimes A\oplus \lie h \otimes A\oplus \lie n^+\otimes A.$$ Any linear functional $\lambda$ on $\lie h$ can be extended to one on $(\lie h \oplus \lie n^+)\otimes A$ by setting it to be zero on $\lie h\otimes A_+ \oplus \lie n^+\otimes A$, where $A_+$ is the ideal spanned by the positive degree elements. One can then consider the Verma type module $M(\lambda)$ associated to the above induced triangular decomposition, which is naturally $\mathbb Z_{\ge 0}$-graded. If $\lambda$ is a dominant integral weight, the irreducible quotient of $M(\lambda)$ is finite-dimensional: it is the corresponding finite-dimensional simple $\lie g$-module $V(\lambda)$ on which $\lie g \otimes A_+$ acts trivially. It turns out that $M(\lambda)$ has other finite-dimensional quotients and the largest are called Weyl modules, i.e. any finite-dimensional quotient of $M(\lambda)$ is a quotient of some of the Weyl modules.
If the zero graded piece of $A$ is $\mathbb C$, e.g. when $A=\mathbb C[t]$ or $\mathbb C[t]/t^N\mathbb C[t]$ (where $N$ is a natural number), there is a unique graded Weyl module for each $\lambda$, up to isomorphism. Therefore, the graded Weyl module is the universal highest-weight module of highest weight $\lambda$ in the corresponding category of finite-dimensional graded modules. We let $W(\lambda)$ and $W_N(\lambda)$ denote the graded Weyl module for $\lie g[t] = \lie g \otimes \mathbb C[t]$ and $\lie g[t]_N = \lie g \otimes \frac{\mathbb C[t]}{t^N\mathbb C[t]}$, respectively, and refer to $W_N(\lambda)$ as a truncated Weyl module. One motivation for studying the truncated Weyl modules comes from a conjecture stated in [@cfs:schur] related to Schur positivity which, as seen in [@fou:nhi; @kl:tor; @rav:dcvf], can be rephrased as a conjecture on the realization of truncated Weyl modules as fusion products of certain irreducible modules. In particular, a positive answer for this conjecture leads to a way of computing the characters of truncated Weyl modules, which is still an open problem in general.
To explain the conjecture, consider the set $P^+(\lambda , N)$ whose elements are $N$-tuples $$\bs \lambda = (\lambda _1, \ldots , \lambda _N)$$ of dominant integral weights adding up to $\lambda$. A partial order on $P^+(\lambda , N)$ was defined in [@cfs:schur] and an algorithm for computing its maximal elements was described in [@fou:order]. It turns out that all maximal elements are in the same orbit under the obvious action of the symmetric group and, hence, the character of the fusion products $$V(\lambda_1)*\cdots* V(\lambda_N)$$ is independent of the choice of a maximal element $\bs{\lambda}$. Recall also that, since $\lie g[t]_N$ is a graded quotient of $\lie g[t]$, every $\lie g[t]_N$-module can be regarded as a module for $\lie g[t]$. In particular, we regard $W_N(\lambda)$ as a $\lie g[t]$-module and, in that case, it is a quotient of $W(\lambda)$. Let $|\lambda|$ be the sum of the evaluations of $\lambda$ in the simple coroots of $\lie g$. The following was stated in [@fou:nhi]:
**Conjecture:** Suppose $\bs{\lambda} = (\lambda_1,\dots,\lambda_N)$ is a maximal element of $P^+(\lambda,N)$. If $N\le|\lambda|$, there exists an isomorphism of $\mathbb Z$-graded $\lie g[t]$-modules: $W_N(\lambda)\cong V(\lambda_1)*\cdots* V(\lambda_N)$.
The conjecture has been proved for particular values of $\lambda$ or $N$. For instance, it follows from the results of [@fou:nhi] for $N=2$, $\lie g=\lie{sl}_n$, and $\lambda$ any multiple of a fundamental weight. Other cases for general $N$, with restrictions on $\lambda$ or $\lie g$ were proved in [@kl:tor; @rav:dcvf]. Our main result (Theorem \[t:min\]) extends the list of known cases for which the conjecture holds. Namely, we prove it for every $N>1$ in the case that $\lambda$ is a multiple of a “small” fundamental weight. More precisely, a fundamental weight $\omega_i$ is “small’ if the coordinate of the highest root of $\lie g$ in the direction of the simple root $\alpha_i$ is $1$. Since all minuscule fundamental weights are small in this sense when $\lie g$ is simply laced, this gives an alternative proof for the case $\lie g=\lie{sl}_n$ and $N=2$ previously established in [@fou:nhi]. In fact, the proof presented here is completely different from that of [@fou:nhi] as it relies on the theory of Chari-Venkatesh (CV) and Kirillov-Reshetikhin (KR) modules.
The CV modules, introduced in [@cv:cvm], are graded quotients of Weyl modules associated to certain families of partitions indexed by the set $R^+$ of positive roots of $\lie g$. A fact used crucially in the proof of Theorem \[t:min\] is that every truncated Weyl module is isomorphic to a CV module (see Theorem \[t:cvtruncated\] which had been proved for $\lie g=\lie{sl}_2$ in [@kl:tor]). The associated family of partitions $\xi^\lambda_N$ is explicitly described as follows. Given $\alpha\in R^+$, let $h_\alpha$ be the associated coroot and let $q$ and $r$ be the quotient and remainder of the division of $\lambda(h_\alpha)$ by $N$. Then, the partition associated to $\alpha$ is $$((q+1)^{(r)}, q^{(N-r)}),$$ where the exponent indicates the number of times that part is repeated. Together with results from [@bcsv:dfcpmt; @cv:cvm; @cssw:demflag], Theorem \[t:cvtruncated\] leads to further results in the case that $\lie g = \lie {sl}_2$ related to Demazure flags and chains of inclusions of truncated Weyl modules. For instance, from the description of $\xi^\lambda_N$ and results from [@cv:cvm], one can immediately identify the truncated Weyl modules which are isomorphic to Demazure modules. Otherwise, the results of [@bcsv:dfcpmt; @cssw:demflag] allows us to study Demazure flags for truncated Weyl modules since every CV module (for $\lie g=\lie{sl}_2$) admits a Demazure flag.
The KR modules were originally considered in the quantum setting motivated by mathematical physics [@kr] and remain objects of intense studies (see [@her:krc; @naoi:tpkr] and references therein). They can also be seen as minimal affinizations, in the sense of [@cha:minr2], having a multiple of a fundamental weight as highest weight. The graded KR modules are the so called graded limits of the original KR modules, in the sense of [@mou:res]. The proof of Theorem \[t:min\] also relies on a result from [@naoi:tpkr] which gives a presentation in terms of generators and relations for fusion products of graded KR modules. This result also allows us to take a further step towards proving the conjecture with $\lambda$ being a multiple of any fundamental weight. Namely, in Proposition \[p:W=T\], we prove that a truncated Weyl module whose highest weight is a multiple of the fundamental weight $\omega_i$ is isomorphic to a “natural” quotient of the fusion product of the KR modules associated to the partition indexed by the corresponding simple root in the sense of Theorem \[t:cvtruncated\].
The text is organized as follows. In Section \[s:main\], after a brief review on simple Lie algebras, current algebras and their representations, fusion products, CV modules, and KR modules, we state our mains results. The proofs are given in Section \[s:pf\], after reviewing further results and properties of fusion products and CV modules which are needed in the arguments. The final section contains the aforementioned results related to Demazure flags and inclusions of truncated Weyl modules for $\lie g = \lie {sl}_2$.
The Main Results {#s:main}
================
We shall use the symbol $\diamond$ to mark the end of remarks, examples, and statements of results whose proofs are postponed. The symbol will mark the end of proofs as well as of statements whose proofs are omitted. The sets of integers and complex numbers will be denoted by $\mathbb Z$ and $\mathbb C$, respectively, and the notation $\mathbb Z_{\ge m}, m\in \mathbb Z$, is defined in the obvious way.
Current Algebras and Their Irreducible Modules
----------------------------------------------
Let $\lie g$ be a finite-dimensional simple Lie algebra over $\mathbb C$, fix a Cartan subalgebra $\lie h \subset \lie g$ as well as a Borel subalgebra $\lie b \supseteq \lie h$. Let $R, R^+ \subseteq \lie h^\ast$ be the sets of roots and positive roots, respectively, corresponding to these choices and denote by $\Delta = \{ \alpha _1, \ldots , \alpha _n\}$ the corresponding set of simple roots. Let also $\omega _1, \ldots , \omega _n$ denote the corresponding fundamental weights. For convenience, set $I = \{1, \ldots , n\}$. The root and weight lattices and their positive cones will be denoted by $Q, Q^+, P, P^+$, respectively. The highest root will be denoted by $\theta$ and the highest short root by $\vartheta$. We use the convention that, when $\lie g$ is simply laced, all roots are short and long simultaneously. Given $\eta=\sum_i a_i\alpha_i\in Q$ and $i\in I$, set $$\het_i(\eta) = a_i \qquad\text{and}\qquad \het(\eta) = \sum_{i\in I}\het_i(\eta).$$
Fix a Chevalley basis $\{x^\pm_\alpha, h_i:\alpha\in R^+,i\in I \}$ and set $h_\alpha = [x_\alpha^+,x_\alpha^-], \alpha\in R^+,$ and $x_i^\pm = x_{\alpha_i}^\pm, i\in I$. In particular, $h_i=h_{\alpha_i}$. Let $c_{i,j} = \alpha_j(h_i)$ be the Cartan matrix of $\lie g$ and $d_1,\dots,d_n$ be positive relatively prime integers such that $DC$ is symmetric, where $D={\rm diag}(d_1,\dots,d_n)$. Set also $$P_{sym}^+ = \{\lambda\in P^+: d_i|\lambda(h_i)\}=\bigoplus_{i\in I}\mathbb Z_{\ge 0}\ d_i\omega_i.$$ Recall that, for all $\alpha\in R^+$, $\lie g_{\pm \alpha}=\mathbb Cx_\alpha^\pm$ is the associated root space and, setting $\lie n^\pm = \sum_{\alpha\in R^+} \lie g_{\pm\alpha}$, we have $$\lie g = \lie n^-\oplus\lie h\oplus\lie n^+ \qquad\text{and}\qquad \lie b = \lie h\oplus\lie n^+.$$ Also, the vector subspace $\lie{sl}_\alpha$ spanned by $x_\alpha^\pm, h_\alpha$ is a subalgebra isomorphic to $\lie{sl}_2$.
For any Lie algebra $\lie l$ and commutative associative algebra $A$, the vector space $\lie l\otimes A$ can be equipped with a Lie algebra structure by setting $$[x\otimes a , y\otimes b]= [x,y] \otimes ab \qquad \mbox{for all}\qquad x, y \in \lie l, \quad a,b \in A.$$ If $A$ has an identity element, then the subspace $\lie l\otimes 1$ is a Lie subalgebra of $\lie l\otimes A$ isomorphic to $\lie l$ and we identify $\lie l$ with this subalgebra. If $A$ is graded, then $\lie g\otimes A$ inherits the gradation in the obvious way. In the case that $A=\mathbb C[t]$ is the polynomial ring in one variable, this algebra is called the current algebra over $\lie l$ and will be denote by $\lie l[t]$. If $A=\mathbb C[t]/(t^N)$ for some $N\in\mathbb Z_{\ge 0}$, the algebra $\lie l\otimes A$ is called the truncated current Lie algebra of nilpotence index $N$ and will be denoted by $\lie l[t]_N$. For convenience, we set $\lie l[t]_\infty=\lie l[t]$. Note that $$\label{e:triangdec}
\lie g\otimes A= \lie n^-\otimes A\ \oplus\ \lie h\otimes A\ \oplus\ \lie n^+\otimes A.$$
Given $a\in \mathbb C$, let $\ev_a: \lie g[t] \rightarrow \lie g$ be the evaluation map $x\otimes f(t) \mapsto f(a)x$, which is a Lie algebra homomorphism. Thus, if $V$ is a $\lie g$-module, we can consider the $\lie g[t]$-module $\ev_a V$ obtained by pulling-back the action of $\lie g$ to one of $\lie g[t]$ via $\ev_a$. Modules of this form are called evaluation modules. Note that $\ev_a V$ is simple if and only if $V$ is simple. By abuse of notation, we shall identify $V$ with $\ev_0 V$. Given a $\lie g$-module $V$ and $\mu\in P$, we denote the associated weight space by $V_\mu$. Set also $$|\mu| = \sum_{i\in I} \mu(h_i) \qquad\text{for}\qquad \mu\in P.$$
Given $\lambda\in P^+$, we will denote by $V(\lambda)$ an irreducible $\lie g$-module of highest weight $\lambda$ and by $V_a(\lambda)$ the corresponding evaluation module. Recall that $V(\lambda)$ is generated by a vector $v$ satisfying the following defining relations: $$\lie n^+v =0, \quad hv=\lambda(h)v, \quad (x_i^-)^{\lambda(h_i)+1}v = 0 \quad\text{for all}\quad h\in\lie h, i\in I.$$ In particular, when $v$ is regarded as an element of $V_a(\lambda)$, we have $$\label{e:defrelweylev}
\lie n^+[t]v=0 \qquad\text{and}\qquad (h\otimes t^r)v = a^r\lambda(h)v \quad\text{for all}\quad h\in\lie h, r\in\mathbb Z_{\ge 0},$$ which shows that $V_a(\lambda)$ is a highest-weight module with respect to the decomposition . Notice also that $V_a(\lambda)$ is a $\mathbb Z$-graded $\lie g[t]$-module if and only if $a=0$. Moreover, if $N>1$, $V_a(\lambda)$ factors to a $\lie g[t]_N$-module if and only if $a=0$. Given $k\ge 0, \lambda_1,\dots,\lambda_k\in P^+\setminus\{0\}$, and $a_1,\dots,a_k\in\mathbb C$, it is well-known that $$\label{e:irrastp}
V_{a_1}(\lambda_1)\otimes\cdots\otimes V_{a_k}(\lambda_k) \quad\text{is irreducible}\quad \Leftrightarrow\quad a_i\ne a_j \quad\text{for}\quad i\ne j.$$ Moreover, every irreducible finite-dimensional $\lie g[t]$-module is isomorphic to a unique tensor product of this form. Note that, if $v_j\in V(\lambda_j)_{\lambda_j}, 1\le j\le k$, and $v=v_1\otimes\cdots\otimes v_k$, it follows from that $$\label{e:defrelweylgen}
(h\otimes t^r)v = \left(\sum_{j=1}^k a_j^r\lambda_j(h)\right)v \qquad\text{for all}\qquad h\in\lie h,\ r\in\mathbb Z_{\ge 0}.$$
Denote by $\mathcal G$ and $\mathcal G_N$ the categories of graded finite-dimensional $\lie g[t]$-modules and $\lie g[t]_N$-modules, respectively, where the morphisms are those preserving grades. By pulling-back by the natural epimorphism of Lie algebras $\lie g[t]\to\lie g[t]_N$, every object from $\mathcal G_N$ can be regarded as an object in $\mathcal G$. Evidently, every object from $\mathcal G$ arises in this way for some $N\in\mathbb Z_{> 0}$. We shall denote the $k$-th graded piece of a $\mathbb Z$-graded vector space $V$ by $V[k]$. It will be convenient to introduce the notation $$V_+ = \bigoplus_{k>0} V[k].$$ For $m\in\mathbb Z$, we consider the functor $\tau_m$ on the category of $\mathbb Z$-graded vector spaces which shifts the grades by the rule $$\tau_m(V)[k] = V[k-m].$$ This functor induces endofunctors of $\mathcal G_N, N\in\mathbb Z_{>0}\cup\{\infty\}$, by shifting the grades and keeping the action of $\lie g[t]_N$ unaltered. For $\lambda\in P^+$ and $m\in\mathbb Z$, set $$V(\lambda,m) = \tau_m(V_0(\lambda)).$$ It follows from the comment after that a simple object in $\mathcal G_N$ is isomorphic to a unique module of the form $V(\lambda,m)$. For more on the simple finite-dimensional representations of algebras of the form $\lie g\otimes A$, see the survey [@nesa:sur] and references therein.
Given $a\in\mathbb C$, consider the Lie algebra automorphism $\zeta_a$ of $\lie g[t]$ induced by $t\mapsto t+a$. Then, given a $\lie g[t]$-module $V$, denote by $V_a$ the pullback of $V$ by $\zeta_a$. Note that, if $V\in\mathcal G$ and $a\ne 0$, then $V_a$ is not a graded module. One easily checks that, if $V=\ev_0W$ for some $\lie g$-module $W$, there exists an isomorphism of $\lie g[t]$-modules $V_a \cong \ev_a W$.
Truncated Local Weyl Modules
----------------------------
The local Weyl modules are certain modules for algebras of the form $\lie g\otimes A$ which are highest-weight with respect to the decomposition . Given a unital associative algebra $A$ and $\bs\omega\in(\lie h\otimes A)^*$, one can consider the Verma module $M(\bs\omega)$ which is generated by a vector $v$ satisfying the highest-weight relations $$(\lie n^+\otimes A) v = 0 \qquad\text{and}\qquad xv = \bs{\omega}(x)v \quad\text{for all}\quad x\in\lie h\otimes A$$ as defining relations. Under certain conditions on $\bs{\omega}$ (see [@cfk:cat] for details), $M(\bs{\omega})$ admits nonzero finite-dimensional quotients. In the case $A=\mathbb C[t]$, it follows from the classification of the finite-dimensional irreducible modules and that these conditions are equivalent to the existence of $k\ge 0, \lambda_1,\dots,\lambda_k\in P^+\setminus\{0\}$, and distinct $a_1,\dots,a_k\in\mathbb C$ such that $$\bs{\omega}(h\otimes t^r) = \sum_{j=1}^k a_j^r\lambda_j(h) \qquad\text{for all}\qquad h\in\lie h,\ r\in\mathbb Z_{\ge 0}.$$ In that case, $\bs{\omega}|_{\lie h}\in P^+$ and the local Weyl module $W(\bs{\omega})$ can be defined as the quotient of $M(\bs{\omega})$ by the submodule generated by $$(x_i^-)^{\bs{\omega}(h_i)+1} v \qquad\text{for all}\qquad i\in I.$$ It turns out that $W(\bs{\omega})$ is finite-dimensional and every other finite-dimensional quotient of $M(\bs{\omega})$ is also a quotient of $W(\bs{\omega})$. Note that $M(\bs{\omega})$ is graded if and only if $\bs\omega(\lie h[t]_+) = 0$ and it admits finite-dimensional quotients if and only if $\bs{\omega}|_\lie h\in P^+$. Moreover, in that case, the corresponding local Weyl module is also graded and we denote it simply by $W(\lambda)$ where $\lambda = \bs{\omega}|_\lie h$. In other words, the graded local Weyl module associated to $\lambda\in P^+$ is the $\lie g[t]$-module generated by a vector $v$ satisfying the following defining relations $$\label{e:defgrweyl}
\begin{aligned}
\lie n^+[t]v=0, & \qquad (h\otimes t^r)v = \delta_{r,0}\lambda(h)v \quad\text{for all}\quad h\in\lie h, r\in\mathbb Z_{\ge 0},\\ & \text{and}\qquad (x_i^-)^{\lambda(h_i)+1} v=0 \qquad\text{for all}\qquad i\in I.
\end{aligned}$$ Given $N\in\mathbb Z_{>0}$, we define the truncated local Weyl module $W_N(\lambda)$ as the $\lie g[t]_N$-module generated by a vector $v$ satisfying as defining relations. It follows from [@cfk:cat] that $W_N(\lambda)$ is finite-dimensional and every finite-dimensional graded highest-weight (with respect to ) $\lie g[t]_N$-module is a quotient of $W_N(\lambda)$ for a unique $\lambda\in P^+$. For convenience, we set $W_\infty(\lambda)=W(\lambda)$.
Since every $\lie g[t]_N$-module can be naturally regarded as a $\lie g[t]$-module, the universal property of $W(\lambda)$ immediately implies that we have an epimorphism of $\lie g[t]$-modules: $$W(\lambda)\twoheadrightarrow W_N(\lambda).$$ If $v\in W(\lambda)_\lambda\setminus\{0\}$, one can easily check that the kernel of this map is the submodule of $W(\lambda)$ generated by $$\label{e:relfor WN}
(x\otimes t^N)v \qquad\text{for}\qquad x\in\lie n^-.$$ The details can be found in [@mar:PhD]. Since $\lie h[t]_+v = 0$, it follows that $(x\otimes t^k)v$ is in the kernel for all $k\ge N$. In particular, we have epimorphisms of $\lie g[t]$-modules $$\label{e:truncproj}
W_{N}(\lambda)\twoheadrightarrow W_{N'}(\lambda) \quad\text{if}\quad N\ge N'.$$ It is well known that, for all $\alpha\in R^+$, $$\label{e:endofweyl}
(x_\alpha^-\otimes t^r)v = 0 \quad\text{if}\quad r\ge \lambda(h_\alpha).$$ Hence, $$W_N(\lambda)\cong W(\lambda) \quad\text{if}\quad N\ge\max\{\lambda(h_\alpha):\alpha\in R^+\}$$ or, equivalently, $$\label{e:Nlargedef}
W_N(\lambda)\cong W(\lambda) \quad\text{if}\quad N\ge\lambda(h_\vartheta).$$
Fusion Products
---------------
We now review the notion of fusion products defined in [@fl:kosver] (for more details see [@cl:wdf; @rav:dcvf]). If $V$ is a cyclic $\lie g[t]$-module and $v$ generates $V$, define a filtration on $V$ by $$F^rV = \sum _{0 \leq s \leq r} U(\lie g[t])[s] v$$ where $U(\lie g[t])[s]$ denotes the $s$-th graded piece of $U(\lie g[t])$ for the $\mathbb Z$-grading induced from that of $\mathbb C[t]$. For convenience of notation, we set $F^{-1}V$ to be the zero space. Then, $\displaystyle \gr\ V := \bigoplus _{r \geq 0} \dfrac{F^rV}{F^{r-1}V}$ becomes a cyclic graded $\lie g[t]$-module with action given by $$(x \otimes t^s)(w+ F^{r-1}V)= (x \otimes t^s)w+ F^{r+s-1}V$$ for all $x \in \lie g,$ $w \in F^rV$, $r, s \in \mathbb Z _{\geq 0}$.
Given $k\in\mathbb Z_{>0}$, a family of distinct complex numbers $a_1,\dots,a_k$, and generators $v_1, \dots, v_k$ of cyclic objects $V^1,\dots, V^k$ from $\mathcal G$, it was proved in [@fl:kosver] that $$V_{a_1}^1\otimes \cdots\otimes V_{a_k}^k$$ is generated by $v = v_1\otimes\cdots\otimes v_k$. The fusion product of $V^1,\dots, V^k$ with respect to the parameters $a_1,\dots,a_k$ (it also depends on the choices of cyclic generators), which will be denoted by $$V^1 _{a_1} * \cdots * V^k _{a_k},$$ is the associated graded $\lie g[t]$-module corresponding to the filtration on $V_{a_1}^1\otimes \cdots\otimes V_{a_k}^k$ as defined above. Note that we have an isomorphism of $\lie g$-modules $$V^1\otimes\cdots\otimes V^k \cong_{\lie g} V^1 _{a_1} * \cdots * V^k _{a_k}.$$ It was conjectured in [@fl:kosver] that, under certain conditions, the fusion product does not actually depend on the choice of the parameters $a_1,\dots,a_k$. Motivated by this conjecture, it is usual to simplify notation and write $V^1 * \cdots * V^k$ instead of $V^1 _{a_1} * \cdots * V^k _{a_k}$. This conjecture has been proved in some special cases (see [@cl:wdf; @ff:qchtp; @fl:kosver; @foli:wdkr; @kl:tor; @naoi:tpkr] and references therein). In all these special cases, each $V^j$ is a quotient of a graded local Weyl module and the cyclic generator $v_j$ is a highest-weight generator. All cases relevant to us are of this form and, hence, we make no further mention about the choice of cyclic generators. In particular, it is was proved in [@cl:wdf] for $ \lie g = \lie{sl}_n$, in [@foli:wdkr] for simply laced $\lie g$, and in [@naoi:tpkr] in general, that we have an isomorphism of graded $\lie g[t]$-modules: $$\label{e:weylfusion}
W(\lambda_1)*\cdots*W(\lambda_k) \cong W(\lambda) \quad\text{if}\quad \lambda = \lambda_1+\cdots+\lambda_k.$$ Note that this is equivalent to saying that, for all $\lambda\in P^+$, we have $$\label{e:weylfusfund}
W(\lambda)\cong W(\omega_1)^{*\lambda(h_1)}*\cdots*W(\omega_n)^{*\lambda(h_n)}.$$ We now recall a conjectural generalization of for truncated Weyl modules stated in [@cfs:schur; @fou:nhi; @kl:tor] which is the subject of the first of our main results.
Given $\lambda\in P^+$ and $N\in\mathbb Z_{>0}$, let $P^+(\lambda,N)$ be the subset of $(P^+)^N$ consisting of elements $\bs{\lambda} = (\lambda_1,\dots,\lambda_N)$ such that $$\lambda_1+\cdots+\lambda_N = \lambda.$$ Given $\alpha\in R^+$ and $1\le k\le N$, define $$r_{\alpha,k}(\bs{\lambda}) = \min\{(\lambda_{i_1}+\cdots+\lambda_{i_k})(h_\alpha): 1\le i_1<\cdots<i_k\le N\}.$$ Then, equip $P^+(\lambda,N)$ with the partial order defined by $$\bs{\mu}\le\bs{\lambda} \quad\Leftrightarrow\quad r_{\alpha,k}(\bs{\mu})\le r_{\alpha,k}(\bs{\lambda}) \quad\text{for all}\quad \alpha\in R^+,\ 1\le k\le N.$$ It turns out that the maximal elements of $P^+(\lambda,N)$ form a unique orbit under the obvious action of the symmetric group $S_N$ on $P^+(\lambda,N)$. Note that, if $N\ge|\lambda|$, an element of $P^+(\lambda,N)$ is maximal if and only if all its nonzero entries are fundamental weights. Motivated by this uniqueness and the results on Schur positivity from [@cfs:schur], the following conjecture was formulated by the first author, Chari and Sagaki, stated first in [@fou:nhi]:
\[cj:KusLit\] Let $N\in\mathbb Z_{>0}, \lambda\in P^+$, and suppose $\bs{\lambda} = (\lambda_1,\dots,\lambda_N)$ is a maximal element of $P^+(\lambda,N)$. If $N\le|\lambda|$, $W_N(\lambda)\cong V(\lambda_1)*\cdots* V(\lambda_N)$ as graded $\lie g[t]$-modules.
Recall that $\lambda\in P^+$ is said to be minuscule if $\{\mu\in P^+:\mu<\lambda\}=\emptyset$. All nonzero minuscule weights are fundamental weights and, if $\lie g$ is of type $A$, then all fundamental weights are minuscule. Conjecture \[cj:KusLit\] has been proved in the following special cases:
(1) for simply laced $\lie g$, $\lambda=m\theta$ for some $m\ge 0$, and $N=|\lambda|$ in [@rav:dcvf];
(2) for $\lie g$ of type $A$, $N=2$, and $\lambda=m\omega_i$ for some $i\in I$ in [@fou:nhi];
(3) for $\lambda=N\mu+\nu$ with $\mu\in P^+_{sym}$ and $\nu$ minuscule in [@kl:tor].
Our main result generalizes and provides an alternate proof for item (2) above:
\[t:min\] Suppose $i\in I$ satisfies $\het_i(\theta)=1$. Then, Conjecture \[cj:KusLit\] holds for $\lambda=m\omega_i$ for all $m\ge 0$.
The proof of Theorem \[t:min\] will rely on results about Kirillov-Reshetikhin and Chari-Venkatesh modules which we review in the next subsections.
Note that Conjecture \[cj:KusLit\] is not a complete generalization of since we may have $|\lambda|<\lambda(h_\vartheta)$ (cf. ). Regarding the region $|\lambda|\le N\le \lambda(h_\vartheta)$, it was proved in [@rav:dcvf] for simply laced $\lie g$ that $$W_N(m\theta)\cong W(\theta)^{*(N-m)}* V(\theta)^{*(2m-N)}.$$ We shall address this region in the future.
Truncated Weyl Modules as Chari-Venkatesh Modules
-------------------------------------------------
Given a sequence $\bsr m = (m_j)_{j\in\mathbb Z_{>0}}$ of nonnegative integers, we let $\supp(\bsr m)=\{j: m_j\ne 0\}$. We denote by $\mathscr P$ the set of non-increasing monotonic sequences with finite-support and refer to the elements of $\mathscr P$ as partitions. For any sequence $\bsr m$ with finite support, we denote by $\bsru m$ the partition obtained from $\bsr m$ by re-ordering its elements. Given $\bsr m\in\mathscr P$, set $$\ell(\bsr m) = \max\{j:m_j\ne 0\} \qquad\text{and}\qquad |\bsr m| = \sum_{j\ge 1} m_j.$$ If $|\bsr m|=m$, then $\bsru m$ is said to be a partition of $m$. We denote by $\mathscr P_m$ the set of partition of $m$. The element $m_j$ of a partition $\bsr m$ will be often referred to as the $j$-th part of $\bsr m$. Hence, $\ell(\bsr m)$, which is often referred to as the length of $\bsr m$, is the number of nonzero parts of $\bsr m$. Given distinct nonnegative integers $k_1>k_2\cdots>k_l$ and $a_1,\dots,a_l$, we denote by $$(k_1^{(a_1)},\dots, k_l^{(a_l)})$$ the partition where each $k_j$ is repeated $a_j$ times.
Given $\lambda\in P^+$, a family of partitions $\xi =(\xi(\alpha ))_{\alpha\in R^+}$ indexed by $R^+$ is said to be $\lambda$-compatible if $$\xi(\alpha )\in\mathscr P_{\lambda(h_\alpha)} \qquad\text{for all}\qquad \alpha\in R^+.$$ We will denote by $\mathscr P_\lambda$ the set of $\lambda$-compatible family of partitions. Given $\xi\in\mathscr P_\lambda$, the CV module $CV(\xi)$ is the quotient of $W(\lambda)$ by the submodule generated by $$\label{e:cvdef}
(x_\alpha^+ \otimes t)^s (x_\alpha^-)^{s+r} w, \quad \alpha \in R^+,\ s,r\ne 0,\ s+r > rk + \sum _{j > k} \xi(\alpha )_j$$ for some $k>0$. These modules were introduced in [@cv:cvm].
Fix $\lambda\in P^+$ and $N\in\mathbb Z_{>0}\cup\{\infty\}$ and, for each $\alpha\in R^+$, let $q_\alpha$ and $p_\alpha$ be the unique integers such that $$\label{e:eucdiv}
\lambda(h_\alpha) = Nq_\alpha + p_\alpha \quad\text{and}\quad 0\le p_\alpha<N,$$ where we understand $q_\alpha=0$ and $p_\alpha=\lambda(h_\alpha)$ if $N=\infty$. Then, consider the element $\xi_N^\lambda\in\mathscr P_\lambda$ given by $$\label{e:trxi}
\xi_N^\lambda(\alpha) = ((q_\alpha+1)^{(p_\alpha)},q_\alpha^{(N-p_\alpha)}).$$ The following theorem will be crucial in the proof of Theorem \[t:min\].
\[t:cvtruncated\] The modules $CV(\xi_N^\lambda)$ and $W_N(\lambda)$ are isomorphic graded $\lie g[t]$-modules.
This theorem was proved in [@kl:tor Theorem 4.3] for $\lie g=\lie {sl}_2$ and the general case follows easily from that case as we shall see in Section \[ss:cvtruncated\] (see also [@she Theorem 5]). Note that, if $N\ge\lambda(h_\vartheta)$, then $\xi_N^\lambda(\alpha)=(1^{(\lambda(h_\alpha))})$ for all $\alpha\in R^+$ and Theorem \[t:cvtruncated\] follows from the results of [@cv:cvm] (see also [@mar:PhD Proposition 2.2.5]). In Section \[s:sl2\], we will obtain further results about truncated Weyl modules in the case $\lie g=\lie{sl}_2$ as consequences of Theorem \[t:cvtruncated\] together with results from [@bcsv:dfcpmt; @cssw:demflag].
Kirillov-Reshetikhin Modules
----------------------------
The proof of Theorem \[t:min\] will rely on Theorem \[t:cvtruncated\] as well as on a result about fusion products of graded Kirillov-Reshetikhin (KR) modules. Following [@cha:fer], the graded KR module $KR(m\omega_i), m\in\mathbb Z_{\ge 0}, i\in I$, is defined as the quotient of $W(m\omega_i)$ by the submodule generated by $$(x_i^-\otimes t)v \qquad\text{with}\qquad v\in W(m\omega_i)_{m\omega_i}\setminus\{0\}.$$ These modules are the graded limits of the KR modules for quantum affine algebras (see [@cm:kr; @mou:res; @naoi:tpkr] and references therein). It follows from [@cha:fer] (see also [@cm:kr; @foli:wdkr]) that $$\label{e:skr}
\het_i(\theta) = 1 \qquad\Rightarrow\qquad KR(m\omega_i)\cong V(m\omega_i) \quad\text{for all}\quad m\ge 0.$$
Given $N>0, \bsr i = (i_1,\dots, i_N)\in I^N$ and $\bsr m = (m_1,\dots, m_N)\in\mathbb Z_{> 0}^N$, set $$\begin{gathered}
\notag
KR_{\bsr i}(\bsr m) = KR(m_1\omega_{i_1})*\cdots* KR(m_N\omega_{i_N})\\ \label{e:fusKR} \quad\text{and}\quad\\ \notag
V_{\bsr i}(\bsr m) = V(m_1\omega_{i_1})*\cdots* V(m_N\omega_{i_N}),\end{gathered}$$ for some choice of parameters in the definition of fusion products. In the case that $i_j=i$ for all $1\le j\le N$ and some $i\in I$, we write $KR_{i}(\bsr m)$ in place of $KR_{\bsr i}(\bsr m)$ and we similarly define $V_i(\bsr m)$. A presentation for $KR_{\bsr i}(\bsr m)$ in terms of generators and relations was obtained in [@naoi:tpkr]. In particular, the independence of $KR_{\bsr i}(\bsr m)$ on the choice of parameters for the fusion product follows. Together with Theorem \[t:cvtruncated\], this presentation will imply the following Corollary which, together with , leads to a proof of Theorem \[t:min\].
\[c:KR>>W\] Let $m,N>0, i\in I$, and $\bsr m = \xi^{m\omega_i}_N(\alpha_i)$. Then, there exists an epimorphism of graded $\lie g[t]$-modules $KR_i(\bsr m)\to W_N(m\omega_i)$.
We also prove a further step towards a proof of Theorem \[t:min\] without any hypothesis on $\lie g$ and $i$. To explain this result, introduce the following notation. For $i\in I, k\ge 0$, set $$R^+_{i,k} = \{\alpha\in R^+:\het_i(\alpha)=k\}.$$ Evidently, $R^+_{i,k}=\emptyset$ for $k\ge 2$ if $\het_i(\theta)=1$. Otherwise, by inspecting the root systems, one checks that $R^+_{i,k}$ has a unique minimal element for $k\ge 2$ (with respect to the standard partial order on $P$). Let $v$ be a highest-weight generator for $KR_i(\bsr m)$ and denote by $T_i(\bsr m)$ the quotient by the submodule generated by $$\label{e:Tidef}
(x_{\alpha}^-\otimes t^N)v \qquad\text{with}\qquad \alpha=\min R^+_{i,k},\ k\ge 2.$$ We will see that, if $\bsr m = \xi^{m\omega_i}_N(\alpha_i)$ and $N\le m$, there are epimorphisms of graded $\lie g[t]$-modules $$\label{e:theprojs}
T_i(\bsr m)\twoheadrightarrow W_N(m\omega_i)\twoheadrightarrow V_i(\bsr m).$$ The first of these epimorphisms is a consequence of Corollary \[c:KR>>W\] together with and is valid for all $m,N$ while the second is a corollary of Proposition \[fusiontrunc\] below. The third of our main results is:
\[p:W=T\] If $\bsr m = \xi^{m\omega_i}_N(\alpha_i)$ for some $i\in I,m\ge 0,N>0$, $W_N(m\omega_i)\cong T_i(\bsr m)$.
Proposition \[p:W=T\] says that the first arrow in is an isomorphism, while Conjecture \[cj:KusLit\] expects that the second is also an isomorphism for all $i\in I$ provided $N\le m$. This motivates:
\[cj:new\] Let $i\in I, m\ge 0$. Then, for every $\bsr m\in\mathscr P_m$, $V_i(\bsr m)$ is isomorphic to $T_i(\bsr m)$.
Note that item (1) after Conjecture \[cj:KusLit\], together with Proposition \[p:W=T\], proves this Conjecture for types $DE$, $\bsr m =\xi^{N\omega_i}_N(\alpha_i)$, and $i=i_\theta$ where $i_\theta\in I$ satisfies $\theta=m\omega_{i_\theta}$ for some $m>0$. Evidently, Conjecture \[cj:new\], together with , implies Conjecture \[cj:KusLit\].
Proofs {#s:pf}
======
More About CV Modules {#ss:CV}
---------------------
In this subsection we review some facts and technical tools concerning CV modules.
Given $f(u)\in U(\lie g[t])[[u]]$ and $s\in\mathbb Z_{\ge 0}$, let $f(u)_s$ be the coefficient of $u^s$ in $f$. Let also $$f(u)^{(r)}= \frac{1}{r!} f(u)^{r} \quad\text{for all}\quad r\ge 0.$$ For $\alpha\in R^+$, set $$\bsr x^\pm_\alpha(u) = \sum_{k\ge 0} (x^\pm_\alpha\otimes t^k)u^{k+1}$$ and $$x_\alpha^\pm(r,s) = (\bsr x_\alpha(u)^{(r)})_{r+s} \quad\text{for all}\quad r,s>0.$$ In other words, $$x_\alpha^\pm(r,s) = \sum _ {(b_p) \in S(r,s)} (x_\alpha^\pm \otimes 1)^{(b_0)} (x^\pm_\alpha\otimes t ) ^{(b_1)} \ldots (x^\pm_\alpha\otimes t^s)^{(b_s)},$$ where $$S(r, s) = \{(b_p)_{0 \leq p \leq s} :\, \, b_p \in \mathbb Z_{\geq 0}, \, \, \sum _{p = 0} ^s b_p = r, \, \, \sum _{p= 0 } ^s p b_p = s \}.$$ Given $k\geq 0$, define also $$\begin{gathered}
\notag
_kS(r,s) = \{ (b_p)_{0\leq p\leq s} \in S(r,s): \, \, \, b_p = 0 \, \, \, \mbox{if} \, \, \, p < k \} \\
\label{e:defcv2ndrel}\quad\text{and}\quad\\ \notag
_kx_\alpha^\pm(r,s) = \sum _{(b_p) \in _kS(r,s)} (x_\alpha^\pm\otimes t^k)^{(b_k)}\ldots (x^\pm_\alpha \otimes t^{s})^{(b_{s})}.\end{gathered}$$ Note that $$_kx^\pm_\alpha(r,kr)=(x^\pm_\alpha\otimes t^k)^{(r)}.$$ If $\alpha=\alpha_i$ for some $i\in I$, we may simplify notation and write $\bsr x_i^\pm (u)$, etc. The following is [@rav:dcvf Lemma 6].
\[l:rav\] Let $\lambda\in P^+, w\in W(\lambda)_\lambda\setminus\{0\}$, and $\xi\in\mathscr P_\lambda$. Then, $x_\alpha^-(r,s)w=0$ for all $\alpha\in R^+, r\ge \xi(\alpha)_1, s>0$ such that $s+r>rk + \sum_{j>k} \xi(\alpha)_j$ for some $k>0$.
Let $w$ and $\xi$ be as in the above lemma and denote by $CV'(\xi)$ the quotient of $W(\lambda)$ by the submodule generated by $$\label{e:CV2ndrel}
x_\alpha^-(r,s)w \quad\text{for all}\quad \alpha\in R^+,\ r,s>0 \quad\text{s.t.}\quad s+r > rk + \sum_{j>k} \xi(\alpha)_j$$ for some $k>0$. Consider also the quotient $CV''(\xi)$ of $W(\lambda)$ by the submodule generated by $$\label{e:CV3rdrel}
_kx^-_\alpha(r,s) w \quad\text{for all}\quad \alpha \in R^+,\ s, r, k>0 \quad\text{s.t.}\quad s+r > rk + \sum _{j>k} \xi(\alpha )_j.$$ The following was proved in [@cv:cvm].
\[p:cv2ndrel\] The modules $CV'(\xi)$ and $CV''(\xi)$ are isomorphic to $CV(\xi)$.
We will often denote by $v_\xi$ a nonzero element of $CV(\xi)_\lambda$ for $\xi\in\mathscr P_\lambda$. It follows from the previous proposition that $$(x^-_\alpha \otimes t^k)^{(r)}v_\xi = 0 \quad\text{for all}\quad\alpha \in R^+, k,r> \text{ s.t. } r> \sum _{j>k} \xi(\alpha)_j.$$ In particular, since $\sum _{j > k} \xi(\alpha)_j = 0$ for all $k\ge \ell(\xi(\alpha))$, $$\label{e:cv2ndrelparts}
(x^-_\alpha \otimes t^k)v = 0 \quad\text{for all}\quad \alpha \in R^+,\ k\ge \ell(\xi(\alpha)).$$
More About Fusion Products
--------------------------
Given a filtered $\lie g[t]$-module $V$, recall that $F^rV$ denotes the corresponding filtered piece. Quite clearly $$\label{e:tsas}
\left(x\otimes (t^s - f(t))\right)w \in F^{r+s-1}V \quad\text{for all}\quad x\in\lie g,\ r,s\in\mathbb Z,\ w\in F^rV,$$ and monic polynomial $f$ of degree $s$.
\[l:fusiontrunc\] Let $l>0$ and, for each $1\le j\le l$, let $V^j$ be a finite-dimensional cyclic graded $\lie g[t]$-module generated by a vector $v_j$. Let $x\in\lie g$, suppose $N_j\in\mathbb Z_{\ge 0}, 1\le j\le l$, satisfy $(x\otimes t^{N_j})v_j=0$, and set $\lambda=\lambda_1+\cdots+\lambda_l, N=N_1+\cdots+N_l$. Then, for any choice of distinct $a_1,\dots,a_l\in\mathbb C$, the vector $v_1*\cdots*v_l$ of $V_{a_1}^1 * \cdots * V_{a_l}^l$ satisfies $(x\otimes t^N)v_1*\cdots*v_l=0$.
Let $f(t) = \displaystyle \prod_{j=1}^{l} (t-a_j)^{N_j}$ and $v=v_1*\cdots*v_l$. By , we have $$(x\otimes t^N)v = (x\otimes f(t))v.$$ On the other hand, in $V^1_{a_1}\otimes\cdots\otimes V^l_{a_l}$, we have $$(x\otimes f(t))(v_1\otimes \cdots\otimes v_l) = \sum_{j=1}^l v_1 \otimes\cdots\otimes \left(x\otimes f(t+a_j)\right)v_j \otimes\cdots\otimes v_l = 0.$$
\[fusiontrunc\] Let $l>0$ and, for each $1\le j\le l$, let $V^j$ be a quotient of $W(\lambda_j)$ for some $\lambda_j\in P^+$ and $v_j\in V^j_{\lambda_j}\setminus\{0\}$. Suppose $N_j\in\mathbb Z_{\ge 0}, 1\le j\le l$, satisfy $$\label{e:fusiontrunc}
(\lie n^-\otimes t^{N_j}\mathbb C[t])v_j=0$$ and set $\lambda=\lambda_1+\cdots+\lambda_l$ and $N=N_1+\cdots+N_l$. Then, for any choice of distinct $a_1,\dots,a_l\in\mathbb C$, there exists an epimorphism of graded $\lie g[t]$-modules $W_N(\lambda)\to V_{a_1}^1 * \cdots * V_{a_l}^l$.
Denote by $v$ the image of $v_1\otimes \cdots\otimes v_l$ in $V_{a_1} (\lambda _1) * \ldots * V_{a_l}(\lambda_l)$. Then, quite clearly, $n^+[t]v=\lie h[t]_+v=0, h(v)=\lambda(h) v$ for all $h\in\lie h$, and $(x_i^-)^{\lambda(h_i)+1}v=0$ for all $i\in I$. Thus, by , it suffices to show that $(x_\alpha^-\otimes t^N)v = 0$ for all $\alpha\in R^+$. But this follows from and Lemma \[l:fusiontrunc\].
Recall the definition and, given $i\in I, N>0$, and $\bsr i\in I^N$, set $S_i(\bsr i) = \{j: i_j=i\}$.
\[t:naoifkr\] For every $N>0$, $\bsr i\in I^N$, and $\bsr m\in \mathbb Z_{\ge 0}^N$, the module $KR_{\bsr i}(\bsr m)$ is isomorphic to the quotient of $W(\lambda)$ by the submodule generated by $$ x_{\alpha_i}^-(r,s)\, v \quad\text{for all}\quad i\in I,\ r>0,\ s+r>\sum_{j\in S_i(\bsr i)}\min\{r,m_j\}.$$
Proof of Theorem \[t:cvtruncated\] {#ss:cvtruncated}
----------------------------------
Let $\xi = \xi^\lambda_N$ and $v_\xi\in CV(\xi_N^\lambda)_\lambda\setminus\{0\}$. It follows from that $$(x^-_\alpha \otimes t^N)v_\xi = 0 \quad\text{for all}\quad \alpha \in R^+,$$ and, hence, there exists a surjective homomorphism of $\lie g[t]$-modules $$\begin{aligned}
\label{3}
W_N(\lambda ) \twoheadrightarrow CV(\xi_N^\lambda ).\end{aligned}$$ To prove the converse, observe that Proposition \[p:cv2ndrel\] implies that it suffices to show that $$\begin{aligned}
_kx^-_\alpha(r,s) w=0 \quad\text{for all}\quad \alpha \in R_+, s, r, k \in \mathbb Z_{>0} \quad\text{s.t.}\quad s+r > rk + \sum _{j>k} \xi_N^\lambda(\alpha)_j, \end{aligned}$$ with $w\in W_N(\lambda)_\lambda$. Thus, by considering the subalgebra $\lie{sl}_\alpha[t]$, it suffices to prove Theorem \[t:cvtruncated\] for $\lie g=\lie{sl}_2$, which was done in [@kl:tor], as mentioned earlier (see also [@mar:PhD] for a slightly modified proof).
Proof of Theorem \[t:min\]
--------------------------
Given $\bsr i$ and $\bsr m$ as in Theorem \[t:naoifkr\], recall the definition of $S_i(\bsr i), i\in I$, just before Theorem \[t:naoifkr\], set $$\bsr m_i = (m_j)_{j\in S_i(\bsr i)} \quad\text{for all}\quad i\in I$$ and let $\xi_{\bsr i}^{\bsr m}\in\mathscr P_{\lambda}$ be given by $$\xi_{\bsr i}^{\bsr m}(\alpha) = \begin{cases} \bsru m_i, & \text{if }\alpha = \alpha_i \text{ for some } i\in I,\\ (1^{\lambda(h_\alpha)}), &\text{otherwise.} \end{cases}$$ The following corollary was observed in [@naoi:tpkr Remark 3.4(b)] and a proof can be found in [@mar:PhD Corollary 2.4.1].
For every $N>0$, $\bsr i\in I^N$, and $\bsr m\in \mathbb Z_{\ge 0}^N$, there is an isomorphism $CV(\xi_{\bsr i}^{\bsr m})\cong KR_{\bsr i}(\bsr m)$.
Let $\bsr m = \xi^{m\omega_i}_N(\alpha_i), m,N>0, i\in I$. Since $W_N(m\omega_i)\cong CV(\xi^{m\omega_i}_N)$ by Theorem \[t:cvtruncated\] and $KR_{i}(\bsr m)\cong CV(\xi_{\bsr i}^{\bsr m})$ with $\bsr i=(i^{(N)})$ by the previous corollary, we are left to show that there exists an epimorphism $$CV(\xi_{\bsr i}^{\bsr m}) \to CV(\xi^{m\omega_i}_N).$$ Letting $v\in CV(\xi^{m\omega_i}_N)_\lambda$, this is in turn equivalent to showing that $$x_\alpha^-(r,s)\, v = 0 \quad\text{for all}\quad \alpha\in R^+,\ r,s>0 \quad\text{s.t.}\quad s+r \geq 1+ rk + \sum_{j>k} \xi_{\bsr i}^{\bsr m}(\alpha)_j$$ for some $k>0$. Since $\xi_{\bsr i}^{\bsr m}(\alpha_i) = \xi^{m\omega_i}_N(\alpha_i)$ by definition, we can assume $\alpha$ is not simple, in which case $r \geq 1=\xi (\alpha)_1$ and we are done by Lemma \[l:rav\].
It follows from that all maps in are isomorphisms if $\het_i(\theta)=1$. Write $\xi^{m\omega_i}_N(\alpha_i) = (m_1,\dots,m_l)$ and let $$\bs{\lambda} = (m_1\omega_i,\dots,m_l\omega_l).$$ Note that $l=N$ if $N\le m$ while $l=m$ and $m_j=1$ for all $1\le j\le m$, otherwise. In order to prove Theorem \[t:min\], it remains to observe that $\bs{\lambda}$ is a maximal element of $P^+(m\omega_i,N)$. By [@cfs:schur Lemma 3.3], an element $\bs{\mu} = (\mu_1,\dots,\mu_k)\in P^+(m\omega_i,N)$ is maximal if and only if $$\max_{1\le j\le k} \mu_j(h_i) - \min_{1\le j\le k} \mu_j(h_i) \le 1.$$ Since $(m_1 - m_l) \leq 1$ by definition of $\xi^{m\omega_i}_N(\alpha_i)$, it follows that $\bs{\lambda}$ is indeed maximal.
Proof of Proposition \[p:W=T\]
------------------------------
Let $i\in I, m,N>0$, $\bsr m = \xi^{m\omega_i}_N(\alpha_i)$, $v\in KR_i(\bsr m)_{m\omega_i}$, and $u$ be the image of $v$ in $T_i(\bsr m)$. By the existence of the first epimorphism in , we are left to show that there exists an epimorphism $$W_N(m\omega_i)\to T_i(\bsr m).$$ In light of and the universal property of truncated Weyl modules mentioned after , to prove this, it suffices to show that $$\label{e:W>T}
(x_\alpha^-\otimes t^N)u=0 \qquad\text{for all}\qquad \alpha\in R^+.$$ Evidently, $$\het_i(\alpha)=0 \qquad\Rightarrow\qquad x_\alpha^-v=0$$ which implies for such roots. Next, let us show that $$\label{e:W>T1}
(x_\alpha^-\otimes t^N)v=0 \qquad\text{for all}\qquad \alpha\in R^+_{i,1},$$ which implies for such roots.
Let $\ell(\bsr m)=l$ and $m_1\ge \cdots\cdots m_l$ be its nonzero parts. Recall once more that $l=N$ if $N\le m$ and $l=m$ and $m_j=1$ for all $1\le j\le m$ otherwise. In any case, $l\le N$. Without loss of generality assume $v=v_1*\cdots*v_l$ with $v_j\in KR(m_j\omega_i)_{m_j\omega_i}\setminus\{0\}$. In particular, by definition, $$\label{e:W=T}
(x_i^-\otimes t)v_j =0 \qquad\text{for all}\qquad 1\le j\le l.$$ We proceed by induction on $\het(\alpha)$. If $\het(\alpha)=1$, then $\alpha=\alpha_i$ and , together with Lemma \[l:fusiontrunc\], implies that $(x_i^-\otimes t^l)v=0$ and, hence, $(x_i^-\otimes t^N)v=0$ since $l\le N$. If $\het(\alpha)>1$, we can write $\alpha=\beta+\gamma$ with $\beta,\gamma\in R^+$ and we may assume, without loss of generality, that $\het_i(\beta)=0$. It follows that $x_\beta^-v=0$ and, by induction hypothesis, $(x_\gamma^-\otimes t^l)v=0$. Hence, $[x_\beta^-,x_\gamma^-\otimes t^l]v=0$ and follows. Finally, assume $\alpha\in R^+_{i,k}, k\ge 2$. By definition , holds with $\alpha = \beta_k:=\min R^+_{i,k}, k\ge 2$. If $\alpha\in R^+_{i,k}\setminus\{\beta_k\}$, there exist $m\ge 1$ and $\gamma_j\in R^+_{i,0}, 1\le j\le m$ such that $$\beta_k + \sum_{j=1}^{s}\gamma_j\in R^+ \quad\text{for all}\quad 1\le s\le m \qquad\text{and}\qquad \alpha= \beta_k + \sum_{j=1}^{m}\gamma_j$$ Since $x_{\gamma_j}^-v=0$ for all $1\le j\le m$, $$[x_{\gamma_m}^-,\cdots[x_{\gamma_1}^-,x_{\beta_k}^-\otimes t^N]\cdots] u =0,$$ which completes the proof of Proposition \[p:W=T\].
We give the easiest example in support of Conjecture \[cj:new\]. Namely, let $\lie g$ be of type $B_2$ and assume $I=\{1,2\}$ is such that $\alpha_2$ is the short simple root. Consider the case $\bsr m = (m,1)$ for some $m\ge 1$. In this case, Conjecture \[cj:new\] states that $$\label{e:b2cj}
V(m\omega_2)* V(\omega_2) \cong (KR(m\omega_2)* KR(\omega_2)) / M$$ where $M$ is the submodule of $KR(m\omega_2)* KR(\omega_2)$ generated by $(x_{\theta}^-\otimes t^2)v$ with $v$ a highest-weight generator of $KR_2(\bsr m)$. It follows from [@cha:fer; @cm:kr] that, if $v_k\in KR(k\omega_2)_{k\omega_2}\setminus\{0\}$, then $$\label{e:krroots}
KR(k\omega_2) = \sum_{r=0}^{\lfloor k/2\rfloor} U(\lie n^-)(x_{\theta}^-\otimes t)^r v_k, \qquad (x_{\theta}^-\otimes t^2) v_k=0,$$ and $(x_{\alpha}^-\otimes t) v_k=0$ if $\alpha\ne\theta$. In particular, $$\label{e:krb}
KR(k\omega_2)[r]\cong V((k-2r)\omega_2) \qquad\text{for}\qquad 1\le r\le k/2$$ and is immediate when $m=1$.
One can check that $$\label{e:tpb2}
V(k\omega_2)\otimes V(\omega_2) \cong V((k+1)\omega_2)\oplus V(\omega_1+(k-1)\omega_2)\oplus V((k-1)\omega_2)$$ for all $k\ge 1$. Since $V_2(\bsr m)$ is a quotient of $W_2((m+1)\omega_2)$ isomorphic to $V(m\omega_2)\otimes V(\omega_2)$ as a $\lie g$-module, proceeding as in the proof of , it follows that $$V_2(\bsr m) = U(\lie n^-)(x_{\alpha_2}^-\otimes t)v' \oplus U(\lie n^-)(x_{\theta}^-\otimes t)v'$$ where $v'\in V_2(\bsr m)_{(m+1)\omega_2}\setminus\{0\}$. Moreover, one can check that this implies that $V_2(\bsr m)$ is the quotient of $W_2((m+1)\omega_2)$ by the submodule generated by $$(x_{\alpha_2}^-\otimes t)^2w, \quad (x_{\theta}^-\otimes t)^2w, \quad\text{and}\quad (x_{\alpha_2}^-\otimes t)(x_{\theta}^-\otimes t)w,$$ where $w\in W_2((m+1)\omega_2)_{(m+1)\omega_2}\setminus\{0\}$. Letting $u$ be the image of $v$ in $T_2(\bsr m)$, follows if one shows that $$\label{e:b2cj1}
(x_{\alpha_2}^-\otimes t)^2u =(x_{\theta}^-\otimes t)^2u= (x_{\alpha_2}^-\otimes t)(x_{\theta}^-\otimes t)u=0.$$ The relation $ (x_{\alpha_2}^-\otimes t)^2u =0$ follows from Theorem \[t:naoifkr\] for all $m$. Let us check the others for $m=2$. It follows from and that $$KR_2(\bsr m) \cong_\lie g V(3\omega_2)\oplus V(\omega_1+\omega_2)\oplus V(\omega_2)^{\oplus 2}.$$ One can then proceed as in the proof of to show that $$KR_2(\bsr m)= U(\lie n^-)v \oplus U(\lie n^-)(x^-_{\alpha_2}\otimes t)v \oplus U(\lie n^-)(x^-_{\theta}\otimes t)v \oplus U(\lie n^-)(x^-_{\theta}\otimes t^2)v$$ from where follows immediately.
Further Results for Rank One {#s:sl2}
============================
Demazure Flags {#ss:flags}
--------------
Given $\ell\in\mathbb Z_{\ge 0}$ and $\lambda\in P^+$, the $\lie g$-stable level-$\ell$ Demazure module $D(\ell,\lambda)$ is the quotient of $W(\lambda)$ by the submodule generated by $$\label{e:D(ell,lambda)}
\{(x_\alpha^-\otimes t^{s_\alpha}) v: \alpha\in R^+\}\cup \{(x_\alpha^-\otimes t^{s_\alpha-1})^{m_\alpha+1} v: \alpha\in R^+ \text{ s.t. } m_\alpha<\ell r^\vee_\alpha\},$$ where $v$ is a highest-weight generator of $W(\lambda)$ and $r^\vee_\alpha, s_\alpha$, and $m_\alpha$ are the integers defined by $$r^\vee_\alpha = \begin{cases}
1,& \text{ if } \alpha \text{ is long,}\\
r^\vee,& \text{ if } \alpha \text{ is short},
\end{cases}
\quad\text{and}\quad \lambda(h_\alpha) = (s_\alpha-1)\ell r^\vee_\alpha + m_\alpha, \quad 0<m_\alpha\le \ell r^\vee_\alpha,$$ where $r^\vee$ is the lacing number of $\lie g$. In particular, if $\lie g$ is simply laced, it follows from that $$\label{e:weyl=dem}
W(\lambda)\cong D(1,\lambda).$$ It is well known that there are epimorphisms of graded $\lie g[t]$-modules $$\label{e:projdems}
D(\ell,\lambda)\twoheadrightarrow D(\ell',\lambda) \qquad\text{for all}\qquad \lambda\in P^+,\ \ell\le\ell'.$$ In particular, $D(\ell,\lambda)\cong V_0(\lambda)$ if $\ell$ is sufficiently large.
Let $\xi_{\ell,\lambda}\in\mathscr P_\lambda$ be given by $$\xi_{\ell,\lambda}(\alpha) = ((\ell r^\vee_\alpha)^{(s_\alpha-1)},m_\alpha) \quad\text{for}\quad \alpha\in R^+.$$ It was shown in [@cv:cvm] that we have an isomorphism of graded $\lie g[t]$-modules: $$\label{e:Dem=CV}
D(\ell,\lambda)\cong CV(\xi_{\ell,\lambda}).$$ Set $$D(\ell,\lambda,m) = \tau_m D(\ell,\lambda).$$ A $\lie g[t]$-module $V$ admits a Demazure flag of level-$\ell$ if there exist $k>0, \lambda_j\in P^+, m_j\in\mathbb Z, j=1,\dots,k$, and a sequence of inclusions $$\label{e:demflag}
0 = V_0 \subset V_1 \subset \cdots \subset V_{k-1} \subset V_k = V \quad\text{with}\quad V_j/V_{j-1}\cong D(\ell, \lambda_j,m_j) \ \forall\ 1\le j\le l.$$ Let $\mathbb V$ be a Demazure flag of $V$ as in and, for a Demazure module $D$, define the multiplicity of $D$ in $\mathbb V$ by $$[\mathbb V:D] = \#\{1\le j\le l : V_j/V_{j-1}\cong D\}.$$ As observed in [@cssw:demflag Lemma 2.1], the multiplicity does not depend on the choice of the flag for fixed $\ell$ and, hence, by abuse of language, we shift the notation from $[\mathbb V:D]$ to $[V:D]$. Also following [@cssw:demflag], we consider the generating polynomial $$[V:D](t) = \sum_{m\in\mathbb Z}\ [V:\tau_mD]\ t^m \ \in\ \mathbb Z[t,t^{-1}].$$ In the category of non-graded $\lie g[t]$-modules we have $\tau_mD\cong D$ and, hence, one can also be interested in computing the ungraded multiplicity of $D$ in $V$ which is given by $$[V:D](1) = \sum_{m\in\mathbb Z}\ [V:\tau_mD].$$
Demazure Flags for Truncated Weyl Modules
-----------------------------------------
For the remainder of the paper, let $\lie g=\lie{sl}_2$, $q=q_{\alpha_1}$ and $p=p_{\alpha_1}$ as defined in and identify $\lambda\in P^+$ with $\lambda(h_1)$. Also, given $\xi\in\mathscr P_\lambda$, we write $\xi_j$ instead of $\xi(\alpha_1)_j$ for simplicity.
Theorem \[t:cvtruncated\] together with implies that $$W_N(\lambda)\cong \begin{cases}
D(q,\lambda),& \text{if } N|\lambda,\\
D(q+1,\lambda),& \text{if } p\in \{N-1,\lambda\}.
\end{cases}$$ Note that $p=\lambda$ if and only if $N>\lambda$. We will see below that $W_N(\lambda)$ is not a Demazure module for all other values of $p$. Observe that, if $N=2$, then $p\in\{0,1\}=\{0,N-1\}$ and, hence, $W_N(\lambda)$ is always a Demazure module. The following was proved in [@cssw:demflag].
\[t:CVDemflag\] let $\xi\in\mathscr P_\lambda$ for some $\lambda\in P^+$. Then, $CV(\xi)$ admits a level-$\ell$ Demazure flag if and only if $\ell\ge \xi_1$. In particular, $D(\ell,\lambda)$ admits a level-$\ell'$ Demazure flag iff $\ell'\ge\ell$.
Together with , this theorem implies that $W(\lambda)$ admits a level-$\ell$ Demazure flag for all $\ell\ge 1$. We have the following corollary of Theorems \[t:cvtruncated\] and \[t:CVDemflag\].
The module $W_N(\lambda)$ admits a level-$\ell$ Demazure flag if and only if $$\ell\ge \begin{cases}
q, &\text{if } N|\lambda,\\
q+1, &\text{otherwise.}
\end{cases}$$
In light of , it follows that, in order to show that $W_N(\lambda)$ is not a Demazure module for $p\ne 0,N-1$, it suffices to show that its level-$(q+1)$ Demazure flag has length bigger than 1. To see this, we will use the main tool of the proof of Theorem \[t:CVDemflag\]. Namely, it was shown in [@cv:cvm] that, if $\xi\in\mathscr P_\lambda$ has at least two nonzero parts, there exists a short exact sequence of $\lie g[t]$-modules $$\label{e:CVshortseq}
0 \rightarrow \tau _{(l-1)\xi_l} CV(\xi^-) \rightarrow CV(\xi) \rightarrow CV(\xi^+) \rightarrow 0,$$ where $l$ is the number of nonzero parts of $\xi$, $\xi^- = (\xi^- _1 \geq \ldots \geq \xi^-_{l-2} \geq \xi^-_{l-1} \geq 0)$ is given by $$\xi^-_j = \xi_j \quad\text{if}\quad j< l-1, \qquad \xi_{l-1}^- = \xi_{l-1} - \xi_l,$$ and $\xi ^+ = (\xi^+_1 \geq \ldots \geq \xi^+_{l-1} \geq \xi^+ _{l} \geq 0)$ is the unique partition associated to the $l$-tuple $$(\xi_1, \ldots , \xi_{l-2},\xi_{l-1}+1, \xi_l -1).$$ Note that $$\xi ^+ \in \mathscr P_\lambda \qquad\text{and}\qquad \xi^- \in \mathscr P_{\lambda -2\xi_l}.$$ One also easily checks that $$\xi=\xi_N^\lambda \quad\text{and}\quad p\ne 0,N-1 \quad\Rightarrow\quad \xi_1^\pm=q+1.$$ Hence, the length of a level-$(q+1)$ Demazure flag of $CV(\xi)$ is the sum of the lengths of level-$(q+1)$ Demazure flags of $CV(\xi^\pm)$, showing that $W_N(\lambda)$ is not a Demazure module. In the examples, $\soc(M)$ denotes the socle of a module $M$.
\[ex:p=N-2\] If $p=N-2\ne\lambda$ we have a length-$2$ flag: $$\begin{gathered}
0 \rightarrow D(q+1,\lambda-2q,(N-1)q) \rightarrow W_N(\lambda) \rightarrow D(q+1,\lambda) \rightarrow 0.\end{gathered}$$ Consider the case that $\lambda=4$ and $N=3$, so $p=q=1$ and the above sequence becomes $$\begin{gathered}
0 \rightarrow V(2,2) \rightarrow W_3(4) \rightarrow D(2,4) \rightarrow 0.\end{gathered}$$ One can easily check using that we have exact sequences $$\begin{gathered}
0 \rightarrow V(0,2) \rightarrow D(2,4)\rightarrow D(3,4) \rightarrow 0 \qquad\text{and}\qquad
0 \rightarrow V(2,1) \rightarrow D(3,4) \rightarrow V(4,0) \rightarrow 0.\end{gathered}$$ This implies that $\soc(D(2,4))\cong V(0,2)$ and $$\begin{gathered}
\soc(W_3(4))= W_3(4)[2]\cong V(2,2) \oplus V(0,2).\end{gathered}$$ This shows that, differently from the non truncated case, truncated Weyl modules may have non simple socle.
If $p=N-3\ne\lambda$, the flag has length $2$ or $3$. To see this, observe that $$\xi^+ = ((q+1)^{N-2},q,q-1) \qquad\text{and}\qquad \xi^- = ((q+1)^{N-3},q).$$ In particular, $CV(\xi^-)\cong D(q+1,\lambda-2q)$. If $q=1$ (i.e., $\lambda=2N-3$), we have a length-$2$ flag: $$\begin{gathered}
0 \rightarrow D(2,\lambda-2,N-1) \rightarrow W_N(\lambda) \rightarrow D(2,\lambda) \rightarrow 0.\end{gathered}$$ Otherwise, one easily checks using that $CV(\xi^+)$ has a lenght-$2$ flag: $$\begin{gathered}
0 \rightarrow D(q+1,\lambda-2(q-1),(N-1)(q-1)) \rightarrow CV(\xi^+) \rightarrow D(q+1,\lambda) \rightarrow 0.\end{gathered}$$
Consider the case $\lambda=5$ and $N=4$. Then, $p=q=1=N-3$ and we have the following exact sequence $$\begin{gathered}
0 \rightarrow D(2,3,3) \rightarrow W_4(5) \rightarrow D(2,5) \rightarrow 0.\end{gathered}$$ The grading series is described by
$D(2,5)$ $D(2,3,3)$
--- ------------------- ------------
0 $V(5)$
1 $V(3)$
2 $V(3)\oplus V(1)$
3 $V(1)$ $V(3)$
4 $V(1)$
It follows that $\soc(W_4(4))$ is isomorphic either to $V(1,4)$ or to $V(1,4)\oplus V(1,3)$. Let us show that it is former, i.e., $W_4(5)$ has simple socle. Indeed, the socle is not simple if and only if there exists nonzero $v\in W_4(5)[3]_1$ satisfying $$(x^+\otimes t^r)v = (h\otimes t^r)v = 0 \qquad\text{for all}\qquad r\ge 0.$$ To simplify notation, set $$x^\pm_r = x^\pm\otimes t^r \qquad\text{and}\qquad h_r = h\otimes t^r.$$ Let $w\in W_4(5)_5$ be nonzero. The weight subspace $W_4(5)[3]_1$ is spanned by $x_0^-x_3^-w$ and $x_2^-x_1^-w$. One can then easily check that a linear combination of such vectors is killed by $x^+_r$ for all $r\ge 0$ if and only if it is a scalar multiple of $$(x_2^-x_1-x_0^-x_3^-)w.$$ Since $$h_1(x_2^-x_1-x_0^-x_3^-)w = -2(x_2^-)^2w,$$ it suffices to check that $(x_2^-)^2w\ne 0$. The vectors $(x_2^-)^2w$ and $x^-_1x^-_3w$ clearly span $W_4(5)[4]_1$ which is not zero by the above table. On the other hand, Proposition \[p:cv2ndrel\] implies $x_{\alpha_1}^-(2,4)w$$=0$, i.e., $(x_2^-)^2w=-2x^-_1x^-_3w$ and, hence, the assertion follows.
The following characterization of the truncated Weyl modules having a Demazure flag of length $2$ is easily deduced from the computations of the above two examples.
Suppose $p\ne 0,N-1$. The level-$(q+1)$ Demazure flag of $W_N(\lambda)$ has lenght $2$ if and only if either $p=N-2\ne\lambda$ or $p=N-3$ and $q=1$.
By [@kona Lemma 2.3], since $W_N(\lambda)$ obviously has a simple head, its radical series coincide with its grading series. Example \[ex:p=N-2\] shows that truncated Weyl modules may not have simple socle and, hence, [@kona Lemma 2.3] does not guarantee that the socle series coincides with the grading series. However, that is the case in Example, \[ex:p=N-2\].
Given $a,b,\ell\in\mathbb Z_{\ge 0}$, let $$\xi_{a,b}^\ell = ((\ell+1)^{(a)},\ell^{(b)}), \qquad \lambda_{a,b}^\ell = \ell(a+b)+a,$$ and note that $$\label{e:abpq}
\xi_N^\lambda = \xi_{p,N-p}^q$$ or, equivalently, $$CV(\xi_{a,b}^\ell)\cong W_{a+b}(\lambda_{a,b}^\ell).$$ In particular, $$CV(\xi^{\ell-1}_{N,0})= CV(\xi^{\ell}_{0,N})\cong D(\ell,N\ell)\cong W_N(N\ell).$$ Note also that, for $\lambda=\lambda_{a,b}^1$ and $N=a+b, b\ne 0$, can be rewritten as $$\label{e:inc1}
0 \rightarrow \tau_{N-1} W_{N-1-\delta_{p,N-1}}(\lambda-2) \rightarrow W_N(\lambda) \rightarrow W_{N-1}(\lambda) \rightarrow 0.$$
Given $\mu\in P^+$, consider the function $$\gamma_{a,b}^\ell(\mu,t) = [ CV(\xi_{a,b}^\ell):D(\ell+1,\mu)](t).$$ Such functions were studied in [@bcsv:dfcpmt; @cssw:demflag], but a full understanding is still not achieved. For instance, it follows from [@cssw:demflag] that $$\label{e:csswmf2}
\gamma^1_{0,\lambda}(\lambda-2k,t) = [W(\lambda):D(2,\lambda-2k)](t) = t^{k\lceil \lambda/2\rceil} \tqbinom{\lfloor \lambda/2\rfloor}{k}_t$$ for all $0\le k\le\lfloor\lambda/2\rfloor$ where $$\label{e:qbin}
\qbinom{m}{k}_t = \prod_{j=0}^{k-1} \frac{1-t^{m-j}}{1-t^{k-j}} \quad\text{for}\quad 0\le k\le m.$$ Henceforth, assume $a>0$. Note that, $$\xi^\ell_{a,b} = \xi_{\ell+1,\lambda^\ell_{a,b}} \qquad\text{if}\qquad b=0,1$$ and, therefore, it follows from that $CV(\xi^\ell_{a,b})\cong D(\ell+1,\lambda^\ell_{a,b})$. In particular, $$\gamma_{a,b}^\ell(\mu,t)=\delta_{\lambda^\ell_{a,b},\mu} \qquad\text{if}\qquad b=0,1.$$ Hence, we can assume $b\ge 2$. In that case, it follows from that $$\gamma^1_{a,b}(\mu,t) = \gamma^1_{a-1,b+2}(\mu,t) - \gamma^1_{a-1,b}(\mu,t)t^{a+b},$$ which, combined with , gives a recursive procedure to compute $\gamma^1_{a,b}(\mu,t)$. However, one can use another approach leading to a formula without minus signs, as we shall see next.
Given a partition $\xi$, let $\xi^*$ be the partition obtained from $\xi$ by removing its largest part. In particular, if $\xi \in \mathscr P_\lambda$, then $$\xi^* \in \mathscr P_{\lambda-\xi_1}.$$ Note that, for $a>0$, we have $$\label{e:xiab*}
(\xi_{a,b}^\ell)^* = \xi_{a-1,b}^\ell$$ The following is [@cssw:demflag Lemma 3.8] $$\label{e:cssw38}
[CV(\xi):D(\xi_1, \mu)](t) = t^{\frac{\lambda - \mu}{2}}\ [CV(\xi^*):D(\xi_1, \mu - \xi_1)](t).$$ In particular, $$[CV(\xi):D(\xi_1, \mu)](t) = 0 \quad\text{if}\quad \xi_1>\mu.$$ Using and iterating we get $$\label{e:deltaiter}
\gamma_{a,b}^\ell(\mu,t)= t^{\frac{a}{2}(\lambda_{a,b}^\ell - \mu)}\ \gamma_{0,b}^\ell(\mu-a(\ell+1),t).$$ In particular, implies $$\label{e:truncDq}
[W_N(\lambda):D(q+1, \mu)](t) = t^{\frac{p}{2}(\lambda - \mu)}\ [D(q,q(N-p)):D(q+1,\mu-p(q+1))](t).$$
\[c:mult\] Let $\lambda\in P^+$ and $N>1$ be such that $N\le\lambda< 2N$. For all $0\le k\le\lambda/2$, we have $$[W_N(\lambda):D(2,\lambda-2k)](t)= \begin{cases}
t^{k\lceil \lambda/2\rceil} \tqbinom{N-\lceil \lambda/2\rceil}{k}_t, &\text{if } \, \, k \le N-\lceil \lambda/2\rceil,\\
0, &\text{otherwise.}
\end{cases}$$
Writing $\lambda=N+p$ with $0\le p<N$ and $\mu=\lambda-2k$, we have $$[W_N(\lambda):D(2,\mu)](t)=\gamma ^1_{p, N-p}(\mu,t).$$ Hence, by , $$\gamma^1_{p,N-p}(\mu,t) = t^{pk}\ \gamma^1_{0,N-p}(N-p - 2k,t) = t^{pk}\ \gamma^1_{0,\lambda-2p}(\lambda-2p - 2k,t).$$ In particular, this is zero if $2k>\lambda-2p = 2N-\lambda$. Otherwise, we are done using .
This corollary can be used to compute the length of the level-$2$ Demazure flag. Namely, recall from [@cssw:demflag Section 3.8] that, for every partition $\xi$, $$[CV(\xi):D(\ell,\mu)](t)\ne 0 \qquad\Rightarrow\qquad |\xi|-2\mu \in2\mathbb Z_{\ge 0}.$$ Then, if we consider the generating function $$L_\xi^\ell(x,t) = \sum_{k=0}^{\lfloor|\xi|/2\rfloor} [CV(\xi):D(\ell,|\xi|-2k)](t)\ x^k,$$ the length of the level-$\ell$ Demazure flags of $CV(\xi)$ is $$L_\xi^\ell := L_\xi^\ell(1,1).$$ In the case that $\xi=\xi^\lambda_N$ with $\lambda$ and $N$ as in Corollary \[c:mult\], we get $$\label{e:Lq=1}
L_\xi^2 = \sum_{k=0}^{N-\lceil \lambda/2\rceil} \binom{N-\lceil \lambda/2\rceil}{k} = 2^{N-\lceil \lambda/2\rceil}.$$
Similar results can be obtained for $\gamma^2_{a,b}$, i.e., for $2N\le\lambda<3N$, by using [@bcsv:dfcpmt Proposition 1.4] in place of . Since higher level analogues of are still unavailable, no analogue of Corollary \[c:mult\] and for $\ell N\le \lambda<(\ell+1)N$ with $\ell>2$ can be stated at this point.
Final Remarks {#ss:inc}
-------------
Let $\lambda\in P^+ = \mathbb{Z}_{\geq 0}$ and $w\in W(\lambda)_\lambda\setminus\{0\}$. It is known from the identification with affine Demazure modules (see [@cl:wdf; @foli:wdkr]) that the submodule generated by $W(\lambda)[\lambda-1]_{\lambda-2}$ or, equivalently, by the vector $(x^-\otimes t^{\lambda-1})w$, is isomorphic to $\tau_{\lambda-1}W(\lambda-2)$. In other words, we have an inclusion $$\tau_{\lambda-1} W(\lambda-2)\hookrightarrow W(\lambda).$$ Combining this with , it follows that we have an exact sequence $$\label{e:incnt}
0\to \tau_{\lambda-1} W(\lambda-2)\to W(\lambda)\to W_{\lambda-1}(\lambda)\to 0.$$ In other words, if $N= \lambda$ and $N'=N-1$, the kernel of the projection is, up to grade shift, isomorphic to $W(\lambda-2)\cong W_{N-2}(\lambda-2)$ and can be rewritten as $$\label{e:incnt'}
0\to \tau_{\lambda-1} W_{\lambda-2}(\lambda-2)\to W_\lambda(\lambda)\to W_{\lambda-1}(\lambda)\to 0.$$
Next, we make a few observations about two natural questions. On one hand, it would be interesting to understand the kernel of with $N<\lambda$, specially for $N'=N-1$. Note that gives the answer of this special case when $N<\lambda<2N$ and it coincides with the case $N=\lambda$ as seen in . Unfortunately, as we shall see in Example \[ex:kernotw\], the kernel is not always a truncated Weyl module and, in fact, may not even be a CV-module. On the other hand, in analogy to the non-truncated case, one would like to characterize all possible chains of inclusions of truncated Weyl modules. For instance, if $\lambda = qN+p$ with $0\le p<N$ as before and, either $p<N-1$ or $q=1$, then gives rise to the inclusion $$\tau _{(N-1)q} W_{N-2}(\lambda-2q) \hookrightarrow W_N(\lambda).$$ The corresponding quotient is a truncated Weyl module if and only if $q=1$ which is again. If $q>1$ and $p=N-1$, does not give rise to an inclusion of truncated Weyl modules, but a second application gives rise to the inclusion $$\tau_{N-2}\tau _{(N-1)q} W_{N-2}(\lambda-2(q+1)) \hookrightarrow W_N(\lambda).$$
Denote by $\pi_N^{N'}$ the projection and, for $N'=N-1$, simplify the notation and write $\pi_N$. Note $$\pi_N^{N'} = \pi_{N'+1}\circ\cdots\circ\pi_{N-1}\circ\pi_N \qquad\text{for all}\qquad N'<N.$$ Moreover, implies that $$\ker(\pi_N) = U(\lie n^-[t])(x^-\otimes t^{N-1})w_N$$ where $w_N\in W_N(\lambda)_\lambda\setminus\{0\}$. One easily checks that we have a surjective map $$\varpi_N:\tau_{N-1}W_N(\lambda-2) \twoheadrightarrow \ker(\pi_N).$$ Let $$\delta_N(\lambda) = \dim(W_N(\lambda)).$$ The following is [@cv:cvm Theorem 5(ii)].
\[t:cvfusion\] Let $\lambda\in P^+$, $\xi\in\mathscr P_\lambda$, and $l=\ell(\xi)$. For any choice of distinct $a_1,\dots,a_l\in\mathbb C$, there exists an isomorphism $$CV(\xi)\cong V_{a_1}(\xi_1) \ast \cdots \ast V_{a_l}(\xi_l)$$ of graded $\lie g[t]$-modules.
It follows from Theorems \[t:cvtruncated\] and \[t:cvfusion\] that $$\delta_N(\lambda) = (q+2)^p(q+1)^{N-p}.$$ Therefore, $\varpi_N$ is an isomorphism if and only if $$\label{e:kerNistw}
\delta_N(\lambda) - \delta_N(\lambda-2) = \delta_{N-1}(\lambda).$$ Noting that $$\xi_N^{\lambda-2} =
\begin{cases}
((q+1)^{(p-2)}, q^{(N-p+2)}), &\text{if } p\ge 2,\\
(q^{(N-2+p)},(q-1)^{(2-p)} ), &\text{if } p=0,1,
\end{cases}$$ while $$\begin{gathered}
\xi_{N-1}^{\lambda} = ((q+q'+1)^{(p')}, (q+q')^{(N-1-p')})\\ \quad\text{with}\quad\\ p+q = q'(N-1)+p',\ 0\le p'<N-1,\end{gathered}$$ one can rewrite in terms of the parameters $q$ and $p$. In particular, one can easily check that is always satisfied for $N=2$ and, hence, we have exact sequences $$0\to \tau_1 W_2(\lambda-2)\to W_2(\lambda)\to W_1(\lambda)\cong V(\lambda)\to 0.$$ Henceforth, assume $N>2$. For $q=1$, since $$N-1-\delta_{p,N-1}\ge \lambda-2 \qquad\Leftrightarrow\qquad p=0,1,$$ it follows that $\varpi_N$ is injective if and only if $\lambda =N,N+1$. Hence, we may assume $q>1$.
\[ex:kernotw\] The smallest example of non injective $\varpi_N$ happens with $N=3$ and $\lambda=6$. One can easily check that is not satisfied. Alternatively, note that $W_3(6)\cong D(2,6)$ has simple socle. If $\varpi_3$ were injective, then it would contain a submodule isomorphic to $\tau_2W_3(4)$ which does not have simple socle by Example \[ex:p=N-2\]. In fact, in this case, we see that there exists a short exact sequence $$0 \to V(0,2) \to W_3(4)\stackrel{\varpi_3}{\longrightarrow}\ker(\pi_3)\to 0$$ since $$\delta_3(6) - \delta_2(6) = 11 \qquad\text{and}\qquad \delta_3(4)=12.$$ It is now easy to see that $\ker(\pi_3)$ is not a CV-module in this case. Indeed, if it were, then $\ker(\pi_3)$ would be isomorphic to $CV(\xi)$ with $\xi$ being a partition of $4$. However, using Theorem \[t:cvfusion\], one easily sees that $\dim(CV(\xi))\ne 11$ for all such partitions.
The only inclusion of a truncated Weyl module in $W_3(6)$ comes from which reads $$0\to\tau_2 W_1(2)\to W_3(6)\to CV(\xi)\to 0 \qquad\text{with}\qquad \xi = (3,2,1).$$ In particular, $\soc(W_3(6))\cong V(2,2)\cong \tau_2W_1(2)$.
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[^1]:
[^2]: Part of this work was developed while the second author was a visiting Ph.D. student at the University of Cologne. He thanks Peter Littelmann and Deniz Kus for their guidance and support during that period and the University of Cologne for hospitality. He also thanks CAPES and CNPq (SWE 203324/2014-5) for the financial support.
[^3]: A. M. was partially supported by CNPq grant 304477/2014-1 and Fapesp grant 2014/09310-5.
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abstract: 'The presence of symmetries, be they discrete or continuous, in a physical system typically leads to a reduction in the problem to be solved. Here we report that neither translational invariance nor rotational invariance reduce the computational complexity of simulating Hamiltonian dynamics; the problem is still BQP complete, and is believed to be hard on a classical computer. This is achieved by designing a system to implement a Universal Quantum Interface, a device which enables control of an entire computation through the control of a fixed number of spins, and using it as a building-block to entirely remove the need for control, except in the system initialisation. Finally, it is shown that cooling such Hamiltonians to their ground states in the presence of random magnetic fields solves a QMA-complete problem.'
author:
- Alastair
title: The Computational Power of Symmetric Hamiltonians
---
Introduction
============
Within the study of quantum computation, it is desirable to find ‘natural’ problems for a quantum computer to solve i.e. those that exhibit intrinsically quantum properties which elucidate the power of the device. For example, Feynman first proposed the quantum computer as a device that naturally simulates Hamiltonian dynamics. More recently, it has become apparent that finding the ground state energies of Hamiltonians is QMA-complete [@KSV02a; @oliveira-2005; @Kempe; @gottesman; @Kay:07] i.e. this is a natural problem for the class where solutions can be efficiently verified on a quantum computer. The question that naturally arises with regards to both of these problems is that of the minimal properties that any such Hamiltonian must posses. The same question, viewed from another perspective demands when we should expect efficient classical approximations to the properties and dynamics of Hamiltonians.
It is often found that the presence of symmetries, such as translational or rotational invariance, vastly simplify a problem. It may be hoped that accounting for these properties could make classical simulations of ground state, thermal or dynamic properties tractable and yet pertinent to physical phenomena. A straightforward example of this reduction is the description of a two-qubit mixed state $\rho$. To do this ordinarily requires 15 real parameters. However, if the state is rotationally invariant, i.e. $(U\otimes U)\rho (U^\dagger\otimes U^\dagger)=\rho$ for all single-qubit unitaries $U$, then the family of possible states, the Werner states [@werner_state], is parametrised by a single number. Similarly, under the discrete permutation symmetry, $\text{SWAP}\cdot\rho\cdot\text{SWAP}=\rho$, the family of states only contains 10 real parameters. The hope that the introduction of translational invariance reduces the complexity of finding ground states underpins studies such as Matrix Product States [@Whi92a; @Whi93a; @DMRG_period].
In this paper, we are primarily concerned with the effects that these symmetries have on the ability to efficiently simulate Hamiltonian dynamics, and report that, in fact, the symmetries have no bearing; the problem is BQP-complete i.e. as hard as any quantum computation is to simulate on a classical computer. We will show this by constructing Hamiltonians that implement arbitrary quantum computations.
The first steps towards incorporating translational invariance, for both Hamiltonian evolution and ground state properties, were taken in [@Kay:07], which traded the spatial variation for another property such as the local Hilbert space dimension, which grew as $\text{poly}(N)$ for $N$ spins. Necessary to this construction was the inclusion of both a time label (clock) and position label at each site, so that all the information was available locally. Processing was then achieved by implementing a read head that moved backwards and forwards over the system data. Here, we use global control (GC) schemes [@Lloyd:1993a; @Kay:thesis] to remove the necessity of both time and position labels. In Sec. \[sec:2\], we will describe how to apply global commands through Hamiltonian evolution leading, in Sec. \[sec:3\], to the realisation of a translational invariant, nearest-neighbor Hamiltonian of fixed local Hilbert space dimension that implements arbitrary quantum computations, thus implying the classical intractability of simulation of the dynamics, of which there is already some evidence [@norbert]. The construction is easy to motivate because a GC scheme typically works by repeated application of a finite set of pulses “$A$", “$B$" etc., which are local gates applied uniformly to all qubits in the system. As such, we immediately lose the need for spatial resolution in our Hamiltonian. To remove the time resolution, the program sequence is written on the initial state of some of the spins instead of encoding it in the Hamiltonian. Proceeding to a proof of QMA-completeness for ground state properties still requires the introduction of spatially varying magnetic fields (Sec. \[sec:4\]), but this in turn has severe implications for the cooling of physical systems in the presence of random external fields.
A Universal Quantum Interface {#sec:2}
=============================
Lloyd and co-workers proposed the concept of a Universal Quantum Interface (UQI) [@UQI]. They proved that through manipulation of a single spin which is coupled to a larger system, the entire quantum system can be controlled, such that a quantum computation can be implemented on it. The proof was non-constructive, and there has recently been some interest in how such a device might be implemented [@Burgarth:07]. An example of one of the potential benefits of such a scheme is that it would in principle allow one to isolate the bulk system from the environment so that it’s much less susceptible to noise. In comparison to schemes for Hamiltonian evolution, the UQI protocol also incorporates the ability to prepare the initial state of the system, although we will not explore that aspect here. Control theory proves the existence of solutions and offers numerical techniques to determine high-fidelity control [@Burgarth:07]. In contrast, we realise an exact UQI constructively. This will allow us to introduce most of the concepts required throughout the paper. Indeed, we will use the UQI as a building block in the design of our computing Hamiltonians.
Consider a linear chain of $N$ 4-dimensional spins. These 4-level spins can be assigned a structure as the tensor product of two qubits, labelled $a$ and $q$. Systems $q$ will contain computational qubits, and systems $a$ will be used as a ‘read head’. If the spins are coupled by an interaction $$H_{\text{UQI}}={\mbox{$\textstyle \frac{1}{2}$}}\sum_{i=1}^{N-1}(X_i^aX_{i+1}^a+Y_i^aY_{i+1}^a)\otimes S_{i,i+1}^q, \label{eqn:uqi}$$ where $S_{i,i+1}^q$ is the swap operation between computational qubits $i$ and $i+1$, and $X$ and $Y$ are the standard Pauli operators, then control of just spins 1, 2 and $N$ is sufficient to realise universal quantum computation. We can understand this by transforming the Hamiltonian into a state transfer system. We are interested in the subspace when all the $a$ qubits are in the state ${\left | 0 \right\rangle}$, except for qubit $i$ which is in the state ${\left | 1 \right\rangle}$. We choose to denote that as ${\left | i \right\rangle}^a{\left | \psi_i \right\rangle}^q$, where ${\left | \psi_i \right\rangle}^q$ is the state of the computational qubits in that step, and satisfies ${\left | \psi_i \right\rangle}=S_{i-1,i}{\left | \psi_{i-1} \right\rangle}$. With this definition, one can see that $$H_{\text{UQI}}{\left | i \right\rangle}^a{\left | \psi_i \right\rangle}^q=
{\left | i-1 \right\rangle}^a{\left | \psi_{i-1} \right\rangle}^q+{\left | i+1 \right\rangle}^a{\left | \psi_{i+1} \right\rangle}^q$$ (${\left | 0 \right\rangle}^a$ and ${\left | N+1 \right\rangle}^a$ are taken to be $0$) which is just the same as the model of state transfer studied by Bose [@Bos03], except that one needs to apply local magnetic fields on spins 1 and $N$. In particular, Bose studied the evolution ${\left | 1 \right\rangle}^a{\left | \psi_{1} \right\rangle}^q\rightarrow{\left | N \right\rangle}^a{\left | \psi_{N} \right\rangle}^q$ [@Bos03; @Kay:2004c], where one can achieve an arrival probability of $O(N^{-2/3})$ within a time $O(N)$ [@Bos03]. Moreover, this transfer is heralded – by measuring system $a$ on qubit $N$, we know whether or not the transfer has occurred without disturbing the computational qubits. If the transfer is not finished, we simply wait and try again. With this strategy, one can achieve an evolution which fails with probability less than $\varepsilon$ in a time $O(M^{5/3}\log(1/\varepsilon))$ [@Bos03]. The setup is depicted in Fig. \[fig:uqi\].
![The basic scenario of a universal quantum interface. At each site, there are 2 qubit systems, $a$ and $q$. All the $a$s are initialised as ${\left | 0 \right\rangle}$. A ${\left | 1 \right\rangle}^a$ is input at site 1, and its arrival at site $N$ is monitored. The hopping Hamiltonian is such that as the excitation moves between site $i$ and $i+1$, a unitary $U_{i,i+1}$ is enacted on the computational qubits $q$. Thus, upon arrival at site $N$, the unitary $U_{N-1,N}\ldots U_{1,2}$ has been implemented. Furthermore, this transfer is efficient.[]{data-label="fig:uqi"}](uqi){width="45.00000%"}
The computational scheme now proceeds as follows. We assume the state to be initialised with all the $a$ qubits in ${\left | 0 \right\rangle}$, and the computational qubits are in state ${\left | \psi_1 \right\rangle}$. The read head is initialised by placing the $a$ qubit of spin 1 in the ${\left | 1 \right\rangle}$ state, and we perform a heralded transfer to spin $N$, which means that when it is observed that the excitation has arrived at ${\left | N \right\rangle}^a$, we reset the spin to ${\left | 0 \right\rangle}$, and the computational qubits have changed to $S_{N-1,N}\ldots S_{2,3}S_{1,2}{\left | \psi_1 \right\rangle}^q$, which is simply a cyclic permutation of the computational qubits. This can be repeated, allowing any computational qubit to be placed on spin 1. Similarly, by starting the excitation on spin 2, and removing it from spin $N$, the computational qubits $2$ to $N$ undergo a cyclic permutation, allowing us to place any single qubit on spin 2 without disturbing spin 1. One we have any arbitrary pair of qubits on spins 1 and 2, which requires no more than ${\mbox{$\textstyle \frac{1}{2}$}}N$ cycles, our control of these two qubits allows the implementation of an arbitrary one- or two-qubit gate, which is sufficient for universal quantum computation.
A Toolbox of UQI Gadgets
------------------------
Having demonstrated a relatively simple way to implement a UQI, we will now show how to modify the scheme to improve its capabilities. Our first observation is that we can replace the swap gate, $S^q_{i,i+1}$ in Eqn. (\[eqn:uqi\]) with any unitary $U_{i,i+1}^q$, and the transfer ${\left | 1 \right\rangle}^a_1\rightarrow{\left | 1 \right\rangle}^q_N$ implements the operation $U_{N-1,N}\ldots U_{2,3}U_{1,2}{\left | \psi_1 \right\rangle}^q$. The only difference is that we must ensure that the Hamiltonian remains Hermitian, which we do by writing it as $$H_{\text{UQI}}^{(1)}=\sum_i(\sigma^{-a}_i\sigma^{+a}_{i+1})\otimes U_{i,i+1}^q+(\sigma^{+a}_i\sigma^{-a}_{i+1})\otimes U_{i,i+1}^{\dagger q}.$$ The Hamiltonian remains translationally invariant provided $U_{i,i+1}$ is the same as $U_{1,2}$, just acting on different qubits.
The next step is to see how to implement $U_{i,i+1}$ only on every second qubit. We achieve this by increasing the dimension of system $a$ to 3. The idea is that instead of having a Hamiltonian where the read-head hops to the right, implementing $U$ as it goes, it will alternate its value between 1 and 2 as it hops, and will only implement $U$ if it’s hopping from 1 to 2, and not 2 to 1. Thus, the new Hamiltonian reads $$H_{\text{UQI}}^{(2)}=\sum_i{\left | 02 \right\rangle}{\left \langle 10 \right |}^a_{i,i+1}\otimes U_{i,i+1}^q+{\left | 01 \right\rangle}{\left \langle 20 \right |}^a_{i,i+1}\otimes \identity_{i,i+1}^q+h.c.$$ When we initialise the read-head on spin 1, whether it’s in the state ${\left | 1 \right\rangle}$ or ${\left | 2 \right\rangle}$ determines which pairs of computational qubits the $U$ is applied to, pairs $(2i-1,2i)$ or $(2i,2i+1)$, and also what state we should be checking for on spin $N$. If $N$ is odd, the read-head undergoes an even number of steps and arrives in the same state it started on spin 1.
Finally, we might like the choice of applying more than one different $U$, say $U$ and $V$. Again, we can achieve this with the system $a$ having 3 levels. $$H_{\text{UQI}}^{(3)}=\sum_i{\left | 01 \right\rangle}{\left \langle 10 \right |}^a_{i,i+1}\otimes U_{i,i+1}^q+{\left | 02 \right\rangle}{\left \langle 20 \right |}^a_{i,i+1}\otimes V_{i,i+1}^q+h.c.$$ By initialising the read-head on spin 1 as ${\left | 1 \right\rangle}^a$, the ${\left | 1 \right\rangle}^a$ hops from site to site, and as it hops it implements $U$. Similarly, initialising spin $1$ in the state ${\left | 2 \right\rangle}^a$ causes $V$ to be implemented.
These constructions are readily combined so that if we want to implement, say, two different global commands, each of which only acts on every second qubit, then we can construct a translationally invariant Hamiltonian that does it, using a read-head of dimension 5, and thus an overall spin dimension of 10. It is notationally convenient at this stage to decompose the 5-dimensional read-head at each site $i$ into $$(a_i\otimes r_i)\oplus n_i.$$ The state ${\left | n \right\rangle}$ (the equivalent of ${\left | 0 \right\rangle}^a$ in the previous notation), is the state which all the read-head systems are initialised in, except for one, and is used to indicate the absence of the read head. The systems $a_i$ and $r_i$ are both qubits. $a_i$ indicates if the read-head is ‘active’, and $r_i$ contains the ‘program information’, i.e. which of the two gates to implement. When we propagate the read-head to the right, the interaction that is implemented on the data qubits is conditioned on the value of the read-head, and whether the read-head is active. As the read-head hops, we flip the active setting, so that it only applies an operation on every second qubit.
Computation by Hamiltonian Evolution {#sec:3}
====================================
In [@shepherd], a global control scheme was developed based on just two nearest-neighbor gates, SWAP ($S_{i,i+1}$) and $$G_{i,i+1}=\identity\oplus(Z-iY)/\sqrt{2}.$$ These gates need to be applied to distinct qubit pairings $(2i-1,2i)$ and $(2i,2i+1)$ across the entire lattice. Evidently, this ties in very usefully with our UQI construction. If we implement these two gates within the Hamiltonian, then the 10 level system is capable of universal quantum computation, where we only need single-spin control of spins 1 and $N$ to perform the transfer of the read-head, and to specify the program sequence (i.e. the order in which global commands are implemented).
Our aim is now to construct a Hamiltonian that implements the entire evolution of the computation without even this basic level of control, but instead retaining the ability to prepare the system in some initial state. The first step is to remove the asymmetry between operations at either end of the chain; we introduce periodic boundary conditions and a special marker state. The read-head will be emitted from one side of the marker state, and will arrive at the other side. It will then be down to our preparation of the initial state to select where this marker state is. The second step is to incorporate the program sequence. Again, we will do this by writing the additional information in the initial state. This will require an additional two-level system at each site to denote whether that site constitutes a computational qubit or a spin that holds the program data. The previously mentioned marker will then be used to denote which of the program spins is the one that’s being actively implemented. Again, the read-head will be emitted from one side of this marker, but when it arrives on the other side, it will move the marker onto the next program spin (see Fig. \[fig:schematic2\]). The natural start and end points to the computation are when the active program marker is at either end of the program bus, and are detected by a change in the $l_i$ label between program and data spins.
![Schematic of the Hamiltonian’s mechanism. At each site, there is a label to specify if that site is a computational qubit or program qutrit. One location in the program bus is marked as the active region, and that value is stored in the read-head qubit $r_i$. The read-head applies a unitary to each computational qubit consecutively, controlled off its value. When it reaches the active program region, it exchanges its data with the next step in the program.[]{data-label="fig:schematic2"}](schematic2){width="45.00000%"}
To be specific, consider a 1D chain of spins of local dimension 31, which can be decomposed into several subsystems $$s_i=(q_i\otimes l_i\otimes ((a_i\otimes r_i)\oplus n_i))\oplus m_i. \label{eqn:hilbert_space2}$$ The system $q_i$ is a qutrit, serving two different purposes depending on the label of the qubit system $l_i$. If the label is ${\left | 0 \right\rangle}$, then ${\left | 0 \right\rangle}^q$, ${\left | 1 \right\rangle}^q$ encode a computational qubit (the system $q$ in the UQI construction), otherwise the qutrit $q_i$ contains program information – “skip”, $G$ or $S$. There is a single state which is not used yet, ${\left | 2 \right\rangle}^q{\left | 0 \right\rangle}^l$, and is reserved as the marker to denote which program trit is active. The single level $m_i$ is used to help moving the marker over a ‘skip’ label.
Whether a particular global gate works on pairings of qubits $(2i-1,2i)$ or $(2i,2i+1)$ is solely determined by the alignment of the active program trit with respect to the start of the block of data qubits, which is why we require skip; such that the relative alignment changes. Assuming that there are an odd number of spins in the system, if a read-head leaves a program trit in the active state, it returns in the inactive state. Consequently, an inactive read-head in the location of the active program trit can be used to indicate that the read-head should move the active trit to the next one.
The main term in the Hamiltonian $H_{prop}$ involves read-head propagation, and comes directly from the UQI construction. As such, we will not repeat it here. We must additionally incorporate a term to stop the read-head propagating if the state ${\left | 2 \right\rangle}^q{\left | 0 \right\rangle}^l$ is present i.e. $$\begin{aligned}
&\tilde H_{prop}^i=H_{prop}^i(\identity-{{\left | 20 \right\rangle}{\left \langle 20 \right |}}^{ql}_i)+& \nonumber\\
&\sum_x{\left | n20 \right\rangle}{\left \langle x120 \right |}^{raql}_i\otimes({\left | x0 \right\rangle}^{ra}{\left \langle n \right |}\otimes\identity^{ql})_{i+1}+h.c.& \nonumber\end{aligned}$$ Next, program manipulation (when the read-head is in the neighborhood of the currently active program state): $$\begin{aligned}
H_{prog}^i&=&\sum_{x,y}({\left | n \right\rangle}{\left \langle x0 \right |}^{ra}\otimes{\left | x+1 \right\rangle}{\left \langle 2 \right |}^q\otimes{\left | 1 \right\rangle}{\left \langle 0 \right |}^l)_i \nonumber\\
&&\!\!\!\!\otimes({\left | y1 \right\rangle}^{ra}{\left \langle n \right |}\otimes{\left | 2 \right\rangle}{\left \langle y+1 \right |}^q\otimes{\left | 0 \right\rangle}{\left \langle 1 \right |}^l)_{i+1}+h.c. \nonumber\end{aligned}$$ except that this doesn’t (yet) handle the skip. First, if we’re in a region where we’ve just arrived back from doing a loop, and should be moving onto a skip label $$\begin{aligned}
H_{s1}^i&=&\sum_x({\left | n \right\rangle}{\left \langle x0 \right |}^{ra}\otimes{\left | x+1 \right\rangle}{\left \langle 2 \right |}^q\otimes{\left | 1 \right\rangle}{\left \langle 0 \right |}^l)_i \nonumber\\
&&\otimes({\left | m \right\rangle}{\left \langle n \right |}{\left \langle 01 \right |}^{ql})_{i+1}+h.c. \nonumber\end{aligned}$$ and, secondly, the step over the skip label $$\begin{aligned}
H_{s2}^i&=&\sum_x({\left | n \right\rangle}{\left | 01 \right\rangle}^{ql}{\left \langle m \right |})_i\otimes \nonumber\\
&&({\left | x1 \right\rangle}^{ra}{\left \langle n \right |}\otimes{\left | 2 \right\rangle}{\left \langle x+1 \right |}^q\otimes{\left | 0 \right\rangle}{\left \langle 1 \right |}^l)_{i+1}+h.c. \nonumber\end{aligned}$$ Finally, the total Hamiltonian is $$H_T=\sum_i\tilde H_{prop}^i+H_{prog}^i+H_{s1}^i+H_{s2}^i, \label{eqn:ham_final}$$ which is entirely a sum of two-body terms, so it can be represented as $\sum_{i}h_{i,i+1}$. Moreover, for the permutation operator $$P=\sum_{i_1\ldots i_N=0}^{30}{\left | i_1i_2\ldots i_N \right\rangle}{\left \langle i_2i_3\ldots i_Ni_1 \right |},$$ we have that $$PH_TP^\dagger=H_T$$ i.e. the Hamiltonian is translationally invariant.
If there are $N$ computational qubits in the system, then any efficient quantum algorithm is described by $\text{poly}(N)$ bits, and the total number of qubits in the system is $M$, the sum of these two. Again, we can invoke the fact [@Bos03] that after an evolution time $O(M)$, the probability of having successfully completed the computation of $O(M^{-2/3})$, which is an efficient implementation. So, this translationally invariant, nearest neighbor Hamiltonian evolution can implement any quantum computation, starting from a separable state. The fact that it starts from a separable (although not translationally invariant) state is important since it ensures that we are not encapsulating the difficulty of the problem within the preparation of the initial state. We conclude that the problem is BQP-hard. However, it is also known how to simulate Hamiltonian evolution on a quantum computer [@mick], so the problem is BQP-complete. Thus, as strongly as we believe that quantum computation is more powerful than classical computation is how strongly we believe that simulation of Hamiltonian dynamics, even under the translational invariant restriction, is hard to simulate on a classical computer.
Rotational Invariance
---------------------
So far, we have seen how a 1D Hamiltonian with fixed local Hilbert space dimension and fixed range interactions, which is translationally invariant, can implement an arbitrary quantum computation. However, this discrete symmetry is not nearly as restrictive as the continuous symmetry of rotational invariance, which requires the Hamiltonian to satisfy $$U^{\otimes M}HU^{\dagger \otimes M}=H$$ for all single qubit unitaries $U$. We will now show how to build this into the Hamiltonian, retaining translational invariance and the ability to perform arbitrary quantum computations. The first step is to take the 31 dimensional construction $H_T$, Eqn. (\[eqn:ham\_final\]), and replace each spin with 10 qubits. Between these 10 qubits, there are several decoherence-free subsystems [@rob:revmod]. For an $N$ qubit system ($N$ even), there are decoherence-free subsystems which have $\binom{N}{N/2}(2j+1)/(N/2+j+1)$ levels for $j=0\ldots N/2$. These subsystems enable the storage of quantum information in a way that is not affected by collective decoherence $U^{\otimes N}$. Thus, encoding within one of these subsystems stores the information in a rotationally invariant way. We select any one of the 4 subsystems ($N=10$) that is large enough to encode the 31 levels. Transcribing $H_T$ into this new form automatically makes it rotationally invariant, although instead of being translationally invariant, it is periodic, with a repetition length of 10 qubits. We can thus write it as $$H_R=\sum_{i=1}^{M-1}h_{(10(i-1)\ldots 10i-1),(10i\ldots 10(i+1)-1)},$$ denoting the blocks of logical spins. A summary of this is depicted schematically in Fig. \[fig:ham\_conversion\]. Note that the initial state that the Hamiltonian evolution acts on must also be encoded. However, it is encoded into fixed sized blocks which are separable from each other. Thus, the initial state can still be efficiently represented on a classical computer, so we have not transformed the problem of simulation into the preparation of the initial state – it’s still contained within the Hamiltonian evolution.
![(a) Schematic depiction of a 1D array of 31 dimensional spins (s). The black regions denote the nearest-neighbor interactions of the Hamiltonian. (b) The same system made rotationally invariant by encoding the states of the spin in logical states on a block of qubits. The logical spins, and the periodic Hamiltonian interactions are depicted. To make the system translationally invariant requires the inclusion of terms (gray) where the interactions do not align with the logical qubits of the state. (c) To regain translational invariance in the Hamiltonian, we introduce a flag state (f) before each block of logical qubits, denoting the start of that block.[]{data-label="fig:ham_conversion"}](ham_conversion){width="45.00000%"}
In order to reintroduce translational invariance, we want to incorporate a local patterning of states that enables us to detect the alignment of the blocks of qubits (Fig. \[fig:ham\_conversion\](c)). The technique that we use is much clearer if we concentrate, initially, on restricting the arbitrary rotations to rotations about a single axis, $U=e^{i\theta Z}$. Our problem is that a translationally invariant Hamiltonian will be made up of sums of terms, each of which acts on a block of qubits, comprised of two logical spins. We need to make sure that if a Hamiltonian term is not perfectly aligned with the block-wise patterning of the initial state, then it does not contribute to the evolution. To do this, we introduce a patterning of the qubits which is still rotationally invariant, and yet flags the start of each block of spins. For $Z$-rotation invariant states, this can be done by taking each set of qubits that constitutes a logical spin, and introducing a qubit in the ${\left | 0 \right\rangle}$ state between each of them. This means that in the initial state, there is never a pair of neighboring qubits in the ${\left | 11 \right\rangle}$ state. Thus, we use this to flag that a block of logical qubits is starting. So, each logical spin now constitutes 23 qubits, of the form ${\left | 110q_10q_20q_30q_40\ldots q_{10}0 \right\rangle}$ where ${\left | q_1q_2\ldots q_{10} \right\rangle}$ was the previous logical spin encoded into a decoherence-free subsystem, and the Hamiltonian is of the form ${{\left | 11 \right\rangle}{\left \langle 11 \right |}}_{1,2}h_{(4,6,8\ldots 22),(27,29\ldots 45)}$.
![The rotationally invariant state that flags, and the Hamiltonian that detects the flag, at the start of the block of 10 qubits encoding a logical spin within a decoherence free subsystem. Section A makes sure that when the Hamiltonian and herald state are offset by a single qubit, the overlap is 0. Section B, where the states $P_3$ are repeated 8 times, handles a relative shift of an even number of qubits. Section C does the same for an odd number of qubits, and the states $P_3$ are repeated 9 times. The difference in the number of repetitions of $P_3$ handles an edge effect that arises otherwise.[]{data-label="fig:heralding"}](herald){width="45.00000%"}
For $Z$-rotations, we have the advantage that the states ${\left | 0 \right\rangle}$ and ${\left | 1 \right\rangle}$ are rotationally invariant (but not superpositions of them). For arbitrary rotations $U$, the construction of suitable flag states is much more involved, and is based on the fact that ${\left | \psi^- \right\rangle}=({\left | 01 \right\rangle}-{\left | 10 \right\rangle})/\sqrt{2}$ is a rotationally invariant two-qubit state. Hamiltonian terms can be constructed which detect the presence, $P^1={{\left | \psi^- \right\rangle}{\left \langle \psi^- \right |}}$, or absence, $P^3=\identity-{{\left | \psi^- \right\rangle}{\left \langle \psi^- \right |}}$, of such a state. There will be a correspondence between the Hamiltonian term that detects the flag state, and the flag state itself, the only difference is that if the Hamiltonian includes a term $P^3$, the flag state can be made out of any two-qubit pure state orthogonal to ${\left | \psi^- \right\rangle}$, such as ${\left | 00 \right\rangle}$. We proceed by observing that it is relatively easy to suppress misalignments between the flag state and the Hamiltonian term when the misalignment is by an even number of qubits, one simply ensures that a ${\left | \psi^- \right\rangle}$ in the flag state and a $P^3$ in the Hamiltonian align. This trick can be repeated for a misalignment by an odd number of qubits greater than 1. There are two concerns remaining. Firstly, whether there any edge effects arising and, secondly, how to deal with an offset of just one qubit. The first concern is overcome simply by using a large enough flag state, which is larger than the 10 qubits in the logical spin. An offset of one qubit is handled by incorporating four additional qubits in the flag state in the form $P^1_{1,3}\otimes P^3_{2,4}$. The interleaving of the two projectors ensures overlap after just a single shift. The entire state is depicted in Fig. \[fig:heralding\] and requires 43 qubits in total. The projector onto the state is written as $$\begin{aligned}
P^{\text{flag}}_{1\ldots 43}&=&P^1_{1,3}\otimes P^3_{2,4}\otimes P^1_{5,6}\otimes\bigotimes_{i=0}^7P^3_{2i+1,2i+8}\otimes \nonumber\\
&&\otimes P^1_{23}\otimes\identity_{24}\otimes \bigotimes_{i=0}^8P^3_{2i+25,2i+26}. \nonumber\end{aligned}$$ Thus, the overall translationally and rotationally invariant Hamiltonian acts on blocks of 106 qubits which are local on a 1D lattice. It is of the form $$\begin{aligned}
H_{RT}&=&\sum_{i=1}^{M-1}P^{\text{flag}}_{53(i-1)\ldots 53(i-1)+42}\otimes P^{\text{flag}}_{53i\ldots 53i+42}\otimes \nonumber\\
&&\otimes h_{(53(i-1)+43\ldots 53i-1),(53i+43\ldots 53(i+1)-1)}, \nonumber\end{aligned}$$ where the $h_{(),()}$ are the same as in $H_R$. The majority of the cost in terms of the range of the Hamiltonian terms is due to the flag state, which we have made little attempt to optimise; the important element is that the range of the terms is independent of $N$.
Ground State Properties in the Presence of Magnetic Fields {#sec:4}
==========================================================
In previous studies, Hamiltonian evolution has been used as a basis for classifying the problem of finding ground state energies of Hamiltonians with similar properties as QMA-complete. In the present case, this is not expected to be possible as there seems to be no way to encode the verifier’s computation while retaining translational invariance. However, by breaking the translational invariance of $H_T$, one arrives at similar results to [@gottesman], but only using local magnetic fields. To achieve this, we need to add several energy penalties; to detect the solution to the verifier circuit, to initialise ancillas in ${\left | 0 \right\rangle}$ and to prepare the initial state of the program tape for a specific computation corresponding to the verifier of the QMA problem. To implement these penalties, we need to be able to locally detect that we are either at the beginning or end of a computation, requiring an increase in the local Hilbert space dimension, such that the system $n$ has 2 levels. ${\left | 0 \right\rangle}^n$ will be used as before, to indicate that the read-head is inactive. ${\left | 1 \right\rangle}^n$ is a program command that will only get used once, as the first command. It is not necessary to program it, because two-body terms can readily detect the transition between the data and program spins. A further change is that the system $a$ must be increased to dimension 3, leaving the overall Hilbert space dimension as 49. The extra level in $a$ serves a dual purpose. Firstly, it can be used in the same way as ${\left | 1 \right\rangle}^n$, but to indicate the end of the computation, such that we can penalise the output qubit. Secondly, it is used to help ensure that the correct computational sequence occurs. We will adapt the Hamiltonian propagation such that if a read-head in either ${\left | 0 \right\rangle}^a$ or ${\left | 1 \right\rangle}^a$ arrives at the region of transition from data to program spins, it is converted into ${\left | 2 \right\rangle}^a$, which will continue to propagate through the program region until it gets to the active program spin, where it releases its information and gets reinitialised in ${\left | 1 \right\rangle}^a$. If the read-head reaches the end of the program region in the ${\left | 2 \right\rangle}^a$ state, it is deactivated, and the computation ends. In particular, this means that if the system were to be initialised without an active program label, the computation is much shorter than it would otherwise have been. The Hamiltonian is readily revised to take these alterations into account. An energy penalty for when the read-head passes a particular qubit in either ${\left | 1 \right\rangle}^n$ or ${\left | 2 \right\rangle}^a$ behaves exactly like the initial and final penalties that we require, so we simply use penalties $$H_{in}=\sum_i{{\left | 1 \right\rangle}{\left \langle 1 \right |}}^n_i\otimes{{\left | \bar x_i \right\rangle}{\left \langle \bar x_i \right |}}_i^q$$ where the tape value should be $x_i$ ($x_i=0$ for ancillas), and hence ${\bar x_i}$ implies a sum over all other possible program (data) states, including the active label ${\left | 0 \right\rangle}^l{\left | 2 \right\rangle}^q$. The final result term (to test the output of the verifier on qubit $o$) is similar: $$H_{out}={{\left | 2 \right\rangle}{\left \langle 2 \right |}}^a_o\otimes{{\left | 0 \right\rangle}{\left \langle 0 \right |}}^q_o.$$ We have to be sure that the computation is initialised correctly, with all the spins correctly arranged. This is achieved by adding a constant term $H_b$. On program spins, this term is ${{\left | 0 \right\rangle}{\left \langle 0 \right |}}^l\otimes({{\left | 0 \right\rangle}{\left \langle 0 \right |}}+{{\left | 1 \right\rangle}{\left \langle 1 \right |}})^q$, ensuring that they are never data qubits. On data spins, this term is the opposite, ${{\left | 1 \right\rangle}{\left \langle 1 \right |}}^l\otimes\identity^q+{{\left | 0 \right\rangle}{\left \langle 0 \right |}}^l\otimes{{\left | 2 \right\rangle}{\left \langle 2 \right |}}^q$. Taking all of this into account, one can directly apply the projection lemma of [@Kempe],
Let $H=H_1+H_2$ be the sum of two Hamiltonians operating on some Hilbert space $\cal H=\cal S+\cal S^\perp$. The Hamiltonian $H_2$ is such that $\cal S$ is the ground state eigenspace (with eigenvalue 0) and the eigenvectors in $\cal S^\perp$ have eigenvalue at least $J> 2\|H_1\|$. Then, $$\lambda(H_1|_{\cal S}) - \frac{\|H_1\|^2}{J-2\|H_1\|} \le \lambda(H).$$ $\lambda(H)$ denotes the smallest eigenvalue of $H$.
For example, with $J=8\|H_1\|^2+2\|H_1\|$, one obtains $\lambda(H_1|_{\cal S}) - \frac{1}{8} \le \lambda(H).$ We start with the total Hamiltonian $$H_{total}=-J_0H_T+J_bH_b+J_{in}H_{in}+J_{out}H_{out}+\kappa\identity,$$ where $\kappa=-J_0\lambda(-H_T)$. $H_T$ takes a slightly different form to standard proofs since instead of mapping to a Heisenberg chain, it maps to an XX model, and hence the eigenvalues are $2\cos(\pi m/(M+2))$ for $m=1\ldots M+1$. There is an energy gap for computations that are fewer than $M$ steps, as well as a gap to the first excited state, $$\begin{aligned}
-2\cos\left(\frac{\pi}{M+1}\right)+2\cos\left(\frac{\pi}{M+2}\right)&\geq&\frac{c}{M^2}=\Delta E \nonumber\\
-2\cos\left(\frac{2\pi}{M+2}\right)+2\cos\left(\frac{\pi}{M+2}\right)&\geq&\Delta E \nonumber\end{aligned}$$ for some constant $c>0$. In the case where ‘yes’ solutions exist, $\lambda(H_{total})=J_0\lambda(-H_T)+\kappa=0$. In the case where there are only ‘no’ solutions, we assign $H_2=-J_0H_T+J_bH_b+\kappa\identity$ to find that $
J\geq\min\left(J_b,J_0\Delta E\right),
$ and, furthermore, $
\lambda(H_{total})\geq \lambda(H_1|_{{\cal S}_0})-1/8,
$ provided $J\geq 8(J_{out}+J_{in})^2+2(J_{out}+J_{in})$, imposing a polynomial relation between $J_0,J_b$ and $J_{in},J_{out}$. Repeating the process on $\lambda(H_1|_{{\cal S}_0})$ with $H_1'=J_{out}H_{out}|_{{\cal S}_0}$, shows that provided $J_{in}\geq 8J_{out}^2+2J_{out}$, $$\lambda(H_1|_{{\cal S}_0})\geq \frac{J_{out}(1-\varepsilon)}{M+1}\sin^2\left(\frac{(M+1)\pi}{M+2}\right)-\frac{1}{8}$$ where the verifier circuit of our QMA problem accepts the result with probability less than $\varepsilon$. Thus, by selecting $
J_{out}=(M+1)\sin^{-2}\left(\frac{\pi}{M+2}\right)\leq c'M^3,
$ all the terms $J_{out},J_{in},J_b$ and $J_0$ are polynomial in $M$, and $$\lambda(H_{total})\geq \frac{3}{4}-\varepsilon.$$ Distinguishing the ground state energy of this Hamiltonian to within $1/\text{poly}(M)$ determines the existence of ‘yes’ solutions, and thus finding the ground state energy is QMA-complete. In comparison to [@gottesman], all of the spatially varying terms are local magnetic fields. An identical proof holds using $H_{RT}$, our qubit Hamiltonian which is both translationally and rotationally invariant, although the penalties are no longer local magnetic fields and are, instead, 53-body.
Conclusions
===========
Making use the GC scheme introduced in [@shepherd], we have developed three main results. Firstly, the Universal Quantum Interface was described, and used as a building block for the second part, which showed that the evolution of a fixed Hamiltonian which is translationally invariant on a nearest-neighbor chain and has fixed spin dimension can simulate any arbitrary quantum computation, thereby suggesting that the evolution is hard to simulate classically because it is a BQP-complete problem. Even simulations over short time scales, $O(\Delta E^{-1/4})$, reveal the solution since we can project onto the heralded outcome. We have also extended this result to include qubit Hamiltonians which are translationally and rotationally invariant, and still act on $O(1)$-nearest neighbors which are local on a 1D lattice.
Finally, finding the ground state of a translationally invariant Hamiltonian in the presence of a specific sequence of local magnetic fields is QMA-complete. This has a more useful interpretation as a specific example of a random magnetic field i.e. finding the ground state of a translationally invariant Hamiltonian on $M$ spins in the presence of a random local magnetic field of size $O(1/M^2)$ is QMA-complete. This has some important consequences for physical scenarios involving cooling. For example, if one were to couple a refrigerator to a quantum system in an attempt to cool it, if the coupling is too strong, it would take a prohibitively long time to reach the ground state of the system, and that ground state is not the same ground state when it’s not coupled to the refrigerator [^1]. Conversely, the weaker one couples the system to the refrigerator, the longer it takes to cool. This helps to provide a motivation for the use of topological and self-correcting systems, which are carefully designed such that local magnetic fields cannot affect the system degeneracy [@DKLP02a].
Since completing this work, we have been made aware of related work on evolution by translationally invariant Hamiltonians [@karl] and subsequent work [@followup] which reduces the local Hilbert space dimension in that case. This could presumably be used to reduce the number of levels required for the other results presented in this paper.
In the future, we intend to examine whether the present work enables any useful insights into the problem of determining ground states for translationally invariant systems. It certainly leads to some natural conjectures which we are working to prove. Also, the technique for converting translationally invariant Hamiltonians into translationally and rotationally invariant Hamiltonians may be usefully applied to show that the two ground state problems have the same computational complexity – it is certainly true that the eigenstates of $H_T$ map through to $H_{RT}$. It remains to prove that the ground state of $H_T$ maps to the ground state of $H_{RT}$.
AK would like to thank F. Verstraete for useful conversions and Clare College, Cambridge for financial support.
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[^1]: Although in the particular example that we have constructed, the ground state of the coupled system has significant overlap with the degenerate ground state space of the original Hamiltonian.
|
---
abstract: |
We introduce a filter-construction method for pulse processing that differs in two respects from that in standard optimal filtering, in which the average pulse shape and noise-power spectral density are combined to create a convolution filter for estimating pulse heights. First, the proposed filters are computed in the time domain, to avoid periodicity artifacts of the discrete Fourier transform, and second, orthogonality constraints are imposed on the filters, to reduce the filtering procedure’s sensitivity to unknown baseline height and pulse tails. We analyze the proposed filters, predicting energy resolution under several scenarios, and apply the filters to high-rate pulse data from gamma-rays measured by a transition-edge-sensor microcalorimeter.
Keywords
: baseline insensitivity, energy resolution, optimal filtering, pulse pile-up, pulse tail insensitivity
PACS numbers
: 07.20.Mc, 07.05.Kf, 84.30.Sk.
U.S. government publication
: Not subject to copyright.
author:
- 'B.K. Alpert'
- 'R.D. Horansky'
- 'D.A. Bennett'
- 'W.B. Doriese'
- 'J.W. Fowler'
- 'A.S. Hoover'
- 'M.W. Rabin'
- 'J.N. Ullom'
bibliography:
- 'filters.bib'
title: Filters for High Rate Pulse Processing
---
The extraction of physical quantities from noisy data streams is ubiquitous in the physical sciences. Examples include the determination of photon and particle energies or incidence times in nuclear and particle physics. Raw data records are invariably filtered to extract the quantity of interest with the highest signal-to-noise, and extensive effort has gone into filter development. One important example is so-called “optimal filtering,” for isolated pulses with amplitude proportional to photon energy. Filters constructed from the average pulse shape and the noise power spectral density are convolved with pulse records to estimate pulse amplitudes [@Szymkowiak93]. This filter is widely used in X-ray astrophysics [@Eckart12] and direct searches for weakly interacting dark matter [@Ahmed11].
Here, we propose and demonstrate a novel method for optimal-filter construction for pulse processing. The previous optimal filter is shown to be an example of a much larger class of filters that have new and useful properties. For example, optimal filters can be constructed that are orthogonal to exponential tails of prior pulses, prompted by the need to cope with high-rate pulse data. Many applications of high-resolution photon spectroscopy require very large photon counts for accurate characterization of an absorption or emission spectrum across a broad energy band. For example, isotopic analysis of nuclear materials for treaty verification requires approximately $10^9$ photons in a spectrum between 60 keV and 260 keV to achieve uncertainty of $10^{-3}$ [@Jethava09]. Low-temperature detectors can reach this goal, within limited collection periods, only through large arrays of elements operating at high photon count rates per element. Consequently, operation at high count rates is an active topic of research [@HuiTan08; @HuiTan09; @HuiTan11; @Alpert12].
The new framework departs from prior algorithms in two respects: (1) noise autocovariance is used in place of its mathematical dual, the noise power spectral density, to avoid the discrete Fourier transform (DFT) and enable the construction to be entirely in the time domain; and (2) the filter optimization is subject to explicit constraints beyond maximization of signal-to-noise ratio for isolated pulses, including for the filter length, orthogonality to constants, and orthogonality to exponentials of one or more decay rates. (The method is related to constrained optimization in some other contexts. For example, a similar approach has recently been developed for designing matched filters for wavefront sensing [@Gilles08].) Orthogonality to exponentials can reduce or eliminate sensitivity to tails of prior pulses. With these additional constraints imposed, the filters suffer some loss of sensitivity for isolated pulses compared to filters optimized for that case, but they compensate by retaining resolution with piled-up pulses and by avoiding DFT artifacts, including artificial periodicity.
#### Processing procedure. {#processing-procedure. .unnumbered}
Estimation of pulse amplitudes under standard filtering [@Szymkowiak93; @Moseley88] is optimized for isolated pulses. Each pulse is convolved with a filter and the maximum of the convolution (or a smoothed maximum as provided by a quadratic polynomial fit to several values near the maximum) provides an estimate of the pulse amplitude. The filter, in principle, is constructed to minimize the variance of this estimate, given a known pulse shape and known noise power spectrum.
#### Continuous time model. {#continuous-time-model. .unnumbered}
We assume a signal $f$ consists of a pair of pulses sitting on a baseline $$f(t)=a_0 s(t-t_0)+a_1 s(t-t_1)+b,\label{signal}$$ where $s$ is the pulse shape, $a_0$ and $a_1$ are the pulse amplitudes, $t_0$ and $t_1$ are the pulse arrival times, with $t_0<t_1$, and $b$ is the baseline. A noisy signal consists of signal plus noise, $m(t)=f(t)+\eta(t),$ where the noise $\eta$ is assumed to be a realization of a stationary stochastic process with a mean of zero and autocovariance $$R_\eta(\tau)=\int_{-\infty}^\infty \eta(t)\eta(t+\tau)dt.$$
#### Discrete time model. {#discrete-time-model. .unnumbered}
The measurement apparatus obtains an approximation $m_i$ of $m(i\Delta)$ for $i$ an integer, where $\Delta$ is the sample time spacing, as a convolution of $m$ with a response function $$m_i=\int_{-\infty}^\infty m(i\Delta-t)w(t)dt,$$ where $w$ is an approximate $\delta$-function centered at the origin with unit integral. We define $f_i,$ $s_i,$ and $\eta_i$ analogously. Our measurement model is then $$\begin{aligned}
m_i&=f_i+\eta_i\nonumber\\
&=a_0 s_{i-i_0}+a_1 s_{i-i_1}+b+\eta_i.\label{discrete}\end{aligned}$$ In this approximate model, arrival times $t_0=i_0\Delta,\;t_1=i_1\Delta$ are assumed aligned with the samples, and known, to avoid interpolation issues. The pulse shape $s=(s_0,\ldots,s_n,\ldots)^t$ is approximated by averaging many pulses to obtain the estimate $\hat{s}=(\hat{s}_0,\ldots,\hat{s}_n,\ldots)^t$, normalized so $\max\hat{s}=1$, and the noise autocovariance $r=(r_0,\ldots,r_n,\ldots)^t$, given by the expectation $$r_k=\mathbb{E}\left[\eta_i\eta_{i+k}\right]-\mathbb{E}\left[\eta_i\right]^2
=\mathbb{E}\left[\eta_i\eta_{i+k}\right]\label{r},$$ is approximated by averaging products of pulse-free samples of the sensor output to obtain the estimate $\hat{r}=(\hat{r}_0,\ldots,\hat{r}_n,\ldots)^t$.
#### Amplitude estimation. {#amplitude-estimation. .unnumbered}
The standard procedure assumes $a_0=0$, computes the discrete convolution $$(q\star m)_i=\sum_{j=0}^{n-1}q_{j}m_{i-j}$$ of a given filter $q=(q_0,\ldots,q_{n-1})^t$ with $\ldots,m_{-1},m_0,m_1,\ldots,$ the discrete convolution of $q$ with $\ldots,\hat{s}_{-1},\hat{s}_0,\hat{s}_1,\ldots,$ where $\hat{s}_i=0$ for $i<0,$ and estimates $a_1$ as the ratio of their maximums $$\hat{a}_1=\frac{\max_i(q\star m)_i}{\max_i(q\star \hat{s})_i}.$$
#### Estimate mean and variance. {#estimate-mean-and-variance. .unnumbered}
We seek the mean and variance of the amplitude estimate $\hat{a}_1.$ We have $$\mathbb{E}\left[(q\star m)_i\right]
=a_0\cdot (q\star s)_{i-i_0}+a_1\cdot (q\star s)_{i-i_1}+b\sum_{j=0}^{n-1}q_j.$$ We define $\bar{\imath}$ so that $(q\star s)_{\bar{\imath}}=\max_i(q\star s)_i.$ Under assumptions of orthogonality to the prior tail and to constants, $$(q\star s)_{\bar{\imath}+i_1-i_0}=0=\sum_{j=0}^{n-1}q_j,\label{constraints}$$ we have $$\begin{aligned}
\mathbb{E}\left[\hat{a}_1\right]&=\frac{\mathbb{E}\left[\max_i(q\star m)_i
\right]}{\max_i(q\star \hat{s})_i}\nonumber\\
&\approx\frac{\max_i\mathbb{E}\left[(q\star m)_i\right]}{
\max_i(q\star \hat{s})_i}
\approx\frac{a_1\cdot (q\star s)_{\bar{\imath}}}{(q\star\hat{s})_{\bar{\imath}}}
\approx a_1,\end{aligned}$$ where the approximations are equalities under somewhat restrictive conditions.
Toward a variance estimate, $$\begin{gathered}
\mathbb{E}\left[m_{i-j}m_{i-k}\right]=
\left(a_0 s_{i-i_0-j}+a_1 s_{i-i_1-j}+b\right)\\
\times\left(a_0 s_{i-i_0-k}+a_1 s_{i-i_1-k}+b\right)+r_{j-k},\end{gathered}$$ where $r_{j-k}$ is the noise autocovariance of (\[r\]). Now $$\begin{aligned}
{\rm Var}\left[\hat{a}_1\right]&=\mathbb{E}\big[{\hat{a}_1}^{\;2}\big]
-\mathbb{E}\left[\hat{a}_1\right]^2\nonumber\\
&=\frac{\mathbb{E}\big[\max_i(q\star m)_i^{\;2}\big]
-\mathbb{E}\left[\max_i(q\star m)_i\right]^2}{\max_i(q\star \hat{s})_i^{\;2}}
\nonumber\end{aligned}$$ $$\begin{aligned}
\hspace{0.49in} &\approx\frac{\max_i\mathbb{E}\big[(q\star m)_i^{\;2}\big]
-\max_i\mathbb{E}\left[(q\star m)_i\right]^2}{\max_i(q\star \hat{s})_i^{\;2}}
\nonumber\\
&=\frac{q^tRq}{[q^t\overline{s}]^2}\approx\frac{q^t\hat{R}q}{[q^t\overline{s}]^2}
\stackrel{\mathrm{def}}{=}\hat{\rm V}{\rm ar}\left[\hat{a}_1\right],\end{aligned}$$ where the variance estimate $\hat{\rm V}{\rm ar}\big[\hat{a}_1\big]$ is defined to be the last expression, $\hat{R}$ is the $n\times n$ estimated covariance matrix with $\hat{R}_{jk}=\hat{r}_{j-k}=\hat{r}_{|j-k|},$ and $\overline{s}=(\hat{s}_i,\hat{s}_{i-1},\ldots,\hat{s}_{i-n+1})^t$ is the length $n$ segment from $\hat{s}$ with $q^t\overline{s}=\max_i(q\star\hat{s})_i$.
 Two scenarios, one with pile-up, are shown (top). From the pulse shape and noise autocovariance, a filter orthogonal to an exponential of tail decay, $\tau=3.2$ ms, is computed (inset, separate vertical scales). Convolution of the filter with the signal yields peaks of essentially constant height (bottom) and nearly eliminates pile-up dependence.](filter_action){width="1.095\linewidth"}
[|l|r|r|r|r|]{} Dataset & & & &\
Duration (s) & 4249.50 & 3096.93 & 3182.13 & 4583.17\
Pulses triggered & 5496 & 6581 & 17872 & 60267\
Rate (Hz) & 1.29 & 2.13 & 5.62 & 13.15\
\
Discards: & & & &\
pulse starts ($>$1)& 143 & 201 & 1088 & 7610\
SQUID unlock & 42 & 92 & 585 & 3648\
early peak & 22 & 17 & 62 & 167\
pre-trigger rise & 8 & 21 & 93 & 368\
post-peak rise & 5 & 13 & 65 & 631\
97 keV (raw height) & 1095 & 1286 & 3213 & 9607\
\
Discards: & & & &\
pulse starts ($>$1)& 243 & 387 & 2463 & 17266\
SQUID unlock & 40 & 90 & 537 & 2985\
early peak & 19 & 17 & 58 & 138\
pre-trigger rise & 8 & 23 & 105 & 606\
post-peak rise & 19 & 36 & 223 & 1325\
97 keV (raw height) & 1067 & 1228 & 2938 & 7565\
#### Filter optimization. {#filter-optimization. .unnumbered}
This expression for the variance of the amplitude estimate enables us to design filters that minimize the estimated variance. $\hat{\rm V}{\rm ar}\big[\hat{a}_1\big]$ is minimized at a stationary point of the Lagrange function, $$\Lambda(q,\lambda)=q^t\hat{R}q-\lambda\left[q^t\overline{s}-1,\right]$$ where $\lambda$ is a Lagrange multiplier to ensure that the scale of $q$ satisfies $q^t\overline{s}=1=\max\overline{s}.$ Setting the partial derivatives of $\Lambda$ to zero and solving gives $$q=\frac{\hat{R}^{-1}\overline{s}}{\overline{s}^t\hat{R}^{-1}\overline{s}},
%\cdot\tilde{s},
\qquad\hat{\rm V}{\rm ar}\big[\hat{a}_1\big]=
%\tilde{s}^2\cdot
\left[\overline{s}^t \hat{R}^{-1}\overline{s}\right]^{-1}.
%\frac{\tilde{s}^2}{\overline{s}^t R^{-1}\overline{s}}.
\label{vanilla}$$ This solution depends on the choice, made above tacitly, of the length $n$ of the convolution filter $q$.
Extending beyond the prescription above, we optimize subject to stipulated constraints. Orthogonality to constants or exponentials of particular decay rates can be imposed by revising the Lagrange function. For orthogonality to $k$ vectors $V=\left[v_1\cdots v_k\right],$ $$\Lambda(q,\lambda,\gamma)=q^t\hat{R}q-\lambda\left[q^t\overline{s}-
1\right]-q^tV\gamma,$$ where $\gamma=(\gamma_1,\ldots,\gamma_k)^t$ are $k$ additional Lagrange multipliers. The solution is $$q=\hat{R}^{-1}\overline{V}\left(\overline{V}^t\hat{R}^{-1}\overline{V}\right)^{-1}e_1,
%\cdot\tilde{s},
\qquad\hat{\rm V}{\rm ar}\big[\hat{a}_1\big]=q^t\hat{R}q,$$ where $\overline{V}=[\overline{s}\; v_1 \cdots v_k]$ and $e_1=(1,0,\ldots,0)^t$ is of length $k+1.$
 Pulse spectrum is the absolute value of the discrete Fourier transform (DFT) of the average of pulses, near 97.431 keV line, from the highest-rate dataset. The noise spectrum is the average of the square of absolute value of DFT of pulse-free records of TES output. Records are 25.6 ms.](pulse_noise){width="1.0\linewidth"}
 Predicted resolution on an isolated pulse of four filters is shown. The filters, determined from average pulse shape and noise autocovariance (Fig. \[pulse\_noise\]), include the standard DFT-computed filter with lowest frequency bin set to zero [@Doriese09] and proposed filters orthogonal to constants and zero, one, or two exponentials ($\tau_1=6.0$ ms, $\tau_2=1.5$ ms).](filter_resolution){width="1.09\linewidth"}
 Energy histograms near the 97.431 keV line, from the 2.13 Hz dataset with 10.24 ms record length, are shown for the standard filter, the proposed filter orthogonal to constants, and the proposed filter orthogonal to constants and exponentials ($\tau=6$ ms). Color denotes the pulse arrival time lag since the previous pulse, averaged over the histogram bin, and illustrates that filtering errors, concentrated in heavily piled-up pulses, are nearly eliminated by the filter orthogonal to exponentials.](97keV2000_4000_2hz){width="1.07\linewidth"}
Orthogonality to exponentials of a particular decay time constant enables filters to be less sensitive to tails of prior pulses. Fig. \[convol\] illustrates the principle of these filters. Avoidance of the DFT, with an increase in filter computation cost that is very mild for filter lengths up to $n\approx 10^4,$ avoids false assumptions of signal and noise periodicity and yields nonperiodic filters.
#### Experiment. {#experiment. .unnumbered}
Measurements were taken at NIST of photons from a $^{153}$Gd source with a single transition-edge-sensor (TES) microcalorimeter [@Bennett12], at varied count rates (1.29, 2.13, 5.62, and 13.15 Hz), by placing the source at four different distances from the detector. Essentially all pulses were filtered; no attempt was made to selectively discard pulses to improve the energy resolution. In extraction of pulse records from the data streams, pulses were lost principally due to onset within the prior pulse record and to occasional SQUID mode unlock. Statistics for these measurements are summarized in Table \[summary\]. The following analysis focuses on pulses near the 97.431 keV gamma-ray emission line of $^{153}$Gd.
 Energy histograms as in Fig. \[hist1\], except from 13.15 Hz dataset. At this higher rate, the errors of the first two filters are much more significant, as is the improvement offered by the third.](97keV2000_4000_8hz){width="1.07\linewidth"}
The noise spectrum and, for comparison, the spectrum of the average pulse are plotted in Fig. \[pulse\_noise\]. The noise spectrum and the DFT of the average pulse are used to compute the standard filter. The autocovariance and the average pulse (shown above in Fig. \[convol\]) are used to compute the proposed filters and the predicted energy resolution of each. Fig. \[filtlen\] shows predicted resolution versus filter length for the proposed filters and the standard DFT-computed optimal filter, with the lowest frequency bin set to zero to reduce sensitivity to baseline drift. The standard filter and the proposed filter orthogonal to constants would agree, absent discretization and periodicity artifacts due to the DFT. This calculation is for isolated pulses; for piled-up pulses these two filters suffer bias problems that are significantly reduced by the filters orthogonal to exponentials. The filter orthogonal to two exponentials, however, due to the additional constraint, suffers significant loss of sensitivity at short to moderate filter lengths, and is not considered further here.
The performance of the other three filters is compared on measured pulses, and histograms near the 97.431 keV line are plotted for two different pulse rates in Fig. \[hist1\] and Fig. \[hist2\]. Each histogram was fit with a Gaussian plus a constant to determine the energy resolution. The histogram bins are colored based on the pulse arrival-time lag from the previous pulse, averaged over the bin, demonstrating that errors in processing are due mainly to closely piled-up pulses and are significantly ameliorated by the proposed filters orthogonal to exponentials. This effect is pronounced at the higher pulse rate, yielding much-enhanced peak height and reduced leakage for the filter orthogonal to exponentials.
In Fig. \[icr\_ocr\] the output pulse rate, for the energy range $97.431\pm 0.100$ keV, and energy resolution are compared for all four input count rates and the three types of filter, for both short and long pulse records. At the highest rate and for short pulse records, the filter orthogonal to both constants and exponentials ($\tau=6$ ms) offers 45 % higher output rate than the standard DFT-computed filter and 40 % higher than the filter orthogonal to constants alone, at better energy resolution than either one.
 The output pulse rate and energy resolution are compared across input count rates and three filter types ($\tau=6$ ms), for both short (10.24 ms) and long (25.60 ms) pulse records. We note that the maximum output pulse rate is considerably lower than the corresponding raw pulse rate, because many raw pulses are due to spectral features other than the 97.431 keV line.](icr_ocr2){width="1.08\linewidth"}
One important issue regarding the filters orthogonal to exponentials concerns their performance sensitivity to the choice of decay time constant $\tau$. The average pulse, for the TES microcalorimeter tested, was well-approximated over a 25.6 ms record by a linear combination of four exponentials, with decay time constants $\tau=$ 0.018 ms, 0.144 ms, 0.963 ms, and 2.514 ms. If just the tail is fit, however, the constants increase considerably. It is evident, therefore, that no single decay rate is optimal for all arrival-time lags. Nevertheless, for the full set of Poisson-distributed arrival time lags, the performance of the filters is only mildly sensitive to the choice of time constant. For the highest-rate data with short records, over the range $\tau=3,\ldots,10$ ms, the $97.431\pm0.100$ keV output pulse rate varied as $1.568\pm 0.044$ Hz (mean and one standard deviation) and the energy resolution varied as $128.3\pm 3.2$ eV, as compared with the $\tau=6$ ms values of 1.640 Hz and 125.7 eV.
#### Summary. {#summary. .unnumbered}
The proposed filter construction method, differing from the standard procedure by being computed in the time domain and enabling filter optimization subject to explicit length and orthogonality constraints, assumes linear superposition of pulses and simple exponential decay of pulse tails. Although these assumptions are satisfied rather imperfectly for the TES microcalorimeter tested, the method yields notable improvement over standard filtering. Our tests also point to additional, more specialized options, such as filters optimized for a particular interval of arrival time lags since the previous pulse. Such filters fit easily within this framework and underline that the new approach has implications for pulse processing in a broad range of applications.
We gratefully acknowledge support from the NIST Innovations in Measurement Science program, the DOE Office of Nuclear Nonproliferation Research and Development, and the DOE Office of Nuclear Energy.
|
---
abstract: |
[**Background:**]{} For many applications one wishes to decide whether a certain set of numbers originates from an equiprobability distribution or whether they are unequally distributed. Distributions of relative frequencies may deviate significantly from the corresponding probability distributions due to finite sample effects. Hence, it is not trivial to discriminate between an equiprobability distribution and non-equally distributed probabilities when knowing only frequencies.\
[**Results:**]{} Based on analytical results we provide a software tool which allows to decide whether data correspond to an equiprobability distribution. The tool is available at http:/$\!$/bioinf.charite.de/equifreq/.\
[**Conclusions:**]{} Its application is demonstrated for the distribution of point mutations in coding genes.
address: |
Charité, Institut für Biochemie,\
Monbijoustra[ß]{}e 2, D-10117, Berlin, Germany\
thorsten.poeschel@charite.de, cornelius.froemmel@charite.de, christoph.gille@charite.de\
$^*$corresponding author\
author:
- 'Thorsten Pöschel$^*$, Cornelius Frömmel, and Christoph Gille'
title: |
\
Online tool for the discrimination of equi-distributions
---
Background {#background .unnumbered}
==========
Assume a set of certain events occur with frequencies $M_i$, $i=1\dots
N$, with $\sum\limits_{i=1}^N M_i=M$, e.g., $M_i=\{4,5,2,3,2,9,3,3,5,12,4,6,4,\dots\}$. We ask the question whether the events obey an equiprobability distribution $p_i\equiv
1/N$. According to the general definition of probabilities $$p_i=\lim\limits_{M\rightarrow\infty}\frac{M_i}{M}\,,$$ for an equiprobability distribution and for large sample size $M$ it is expected to find each of the events approximately $M_i\equiv M/N$ times. For finite sample size, however, the frequencies $M_i$ may deviate considerably from this value (Fig. \[fig:freqs\]).
![Histogram of frequencies of $M=1000$ events which have been drawn according to an equiprobability distribution $p_i=1/N=1/100$. The dashed line displays the expectation value.[]{data-label="fig:freqs"}](unsort.eps){width="8cm"}
The deviation from the equidistribution becomes particularly obvious if we order the events according to their rank, i.e., the most frequently occurring event appears left at the abscissa, then the next frequent, etc. (Fig. \[fig:freqsordered\]).
![Same data as in Figure \[fig:freqs\] but in rank order. From the figure it might be erroneously concluded that the events do not obey an equidistribution. The distribution is deformed, however, exclusively due to finite sample effects.[]{data-label="fig:freqsordered"}](sort.eps){width="8cm"}
If we conclude naîvely from the observed frequencies to the probabilities, i.e., if we assume $p_i/p_j=M_i/M_j$, in the extreme case $M_{100}=3$ we end up with a relative error of 70%. In other words, from the frequencies measured in an experiment as shown in Figs. \[fig:freqs\] and \[fig:freqsordered\], it might be erroneously concluded that the events are strongly non-equally distributed.
Using the methods of statistics we can generate (predict) the rank ordered frequency distribution for given $N$ and $M$ under the precondition that the events are equidistributed [@TPJanf]. The predicted frequency distribution can then be compared with the distribution as measured in an experiment with the same values of $M$ and $N$. From the comparison it can be judged whether the events in the experiment obey an equidistribution.
Following this procedure we describe a tool which helps to decide whether a given set of frequencies complies with an equidistribution. For demonstration the tool is applied to the distribution of point mutations in human genes.
Implementation {#implementation .unnumbered}
==============
The numerical tool is available via the web address\
http:/$\!$/bioinf.charite.de/equifreq/. The underlying kernel program which computes the most probable frequency distribution is implemented in $C++$ and the user interface is written in PHP. The program source is available at this address.
Results and Discussion {#results-and-discussion .unnumbered}
======================
Mathematical method {#mathematical-method .unnumbered}
-------------------
We want to sketch briefly the derivation of the basic formula: Assume we distribute $M$ balls over $N$ urns according to an equidistribution. The probability $p(k_i,i)$ to find $k_i$ urns filled each with [exactly]{} $i$ balls is given by $$p(k_i,i) = \frac{M!}{N^M}\sum\limits_{j=k_i}^{\lfloor M/i\rfloor}
(-1)^{(j-k_i)}
\binom{j}{k_i}\frac{(N-j)^{(M-ji)}}{(i!)^{j} (M-ji)!}\,,
\label{eq:InclExcl}$$ where $\lfloor x\rfloor$ denotes the integer of $x$.
Note that the probability to find a number of $k_i$ urns which contain [*exactly*]{} $i$ balls is different form the probability to find the number of urns which contain [*at least*]{} $i$ balls which is a simple textbook problem, whereas the derivation of Eq. requires quite involved algebra. The relation between both probabilities is provided by the exclusion-inclusion principle [@vonMises; @JK]. For our purpose we need the number $\left<K_i\right>$ of urns filled with $i$ balls which are found on average, i.e., we need the first moments of the probabilities Eq. . These values can be found in closed form applying the method of generating functions for the descending factorial moments. The averages $\left<K_i\right>$ have been derived in a different context earlier, the details of the derivation can be found in [@JanfTP; @TPJanf]: $$\langle K_i\rangle=
N\binom{M}{i} \frac{1}{N^i} \left(1-\frac{1}{N}\right)^{(M-i)}\,.
\label{eq:Mom}$$ As an interesting detail of the solution, the average number of filled urns is given by the total number of urns minus the number of empty ones, $N^*=N-\left<K_0\right>$, i.e. [@TPJanf], $$\label{eq:Nstar}
\frac{N^*}{N}
=1-\left(1-\frac{1}{N}\right)^M
\approx 1-\exp\left(-\frac{M}{N}\right)\,.$$ Obviously, for small $M$ (numbers of balls) there is a significant number of urns which, on average, stay empty. Translating back to the language of biology we come to a surprising result: given a population of $N=1000$ species. If we investigate a number of $M=5000$ individuals, from Eq. we obtain $N^*\approx 993.3$, i.e., about 7 species are never found, although from naîve reasoning one expects each species occurring about 5 times.
The moments $\left<K_i\right>$ given in Eq. allow to reconstruct the rank ordered frequency distribution since they describe how many, on average, events do not occur (zero times), how many occur once, twice, etc. Hence, the desired rank ordered frequency distribution reads finally $$M_i^{\rm theo}=
\begin{cases}
0~~ \text{for} & N\ge i > N-\left<K_0\right>\\
1~~ \text{for} & N-\left<K_0\right> \ge i > N-\left<K_0\right> -\left<K_1\right> \\
&\dots\\
j~~ \text{for} & N-\sum\limits_{k=0}^{j-1} \left<K_k\right> \ge i
> N-\sum\limits_{k=0}^{j}\left<K_k\right>\,.
\end{cases}
\label{eq:Hauf}$$
We apply Eq. to predict the frequency distribution which arises from an equidistribution for different sample sizes $M$ and compare with direct numerical simulations, s. Figs. \[fig:prove\], \[fig:prove1\]. The predictions due to Eq. agrees well with the numerical experiment.
![Rank ordered frequency distribution for $N=100$ equally distributed events for different sample size $M$. The solid lines show the distribution as predicted by Eq. , the dashed lines show the distribution of independently drawn equidistributed random numbers from the interval $[1,N]$.[]{data-label="fig:prove"}](provesmall.eps){width="8cm"}
![Same as Fig. \[fig:prove\] but for larger sample size $M$.[]{data-label="fig:prove1"}](provelarge.eps){width="8cm"}
Exploration of experimental data {#exploration-of-experimental-data .unnumbered}
--------------------------------
The theoretical distribution of frequencies due to Eq. can be compared with experimentally obtained frequencies. From the distance between both (rank ordered) frequency distributions we can conclude whether the experimental data obey an equidistribution. To this end we have elaborated a web based tool (http:/$\!$/bioinf.charite.de/equifreq/). The user interface offers four alternative input masks which differ in the way the input file is generated:
1. The measured frequencies of each species $M_i$ are given directly.
2. The number of species $N$ and the total number of individuals $M$ are specified. Each individual is assigned a species by chance.
3. As for (2) the rank ordered frequencies are computed but with the generalization that each species is assigned an individual probability. The theoretical basis for this computation is not given here but will be published elsewhere [@biofreq].
4. The last input mask is intended for the investigation of the spatial distribution of point mutation in genes which is presently the most specialized application of the described program.
The program computes the expected frequency distribution due to Eq. with the assumption that the species obey an equiprobability distribution. Three output files are generated: [*freq*]{}, [*ktheo*]{} and [*kexp*]{}. The file [*freq*]{} contains the rank ordered frequencies as generated from the input data set (cases (1) and (4)) or randomly due to an equiprobability distribution (case (2)) or a general distribution (case (3)). [*ktheo*]{} contains the moments $\left<K_i\right>$ for each rank $i$, i.e. the expected number of individuals occurring $i$ times, due to Eq. for given numbers $N$ of species and $M$ of individuals. For cases (1) and (4) the values of $M$ and $N$ are extracted from the input data, for (2) and (3) they are provided by the user. (Note that these expectation values are real numbers in general.) The third column of line $i$ contains the value $M-\sum_{j=0}^i\left<K_i\right>$. The last file, [*kexp*]{} contains the same data as [*ktheo*]{}, but based on the input data (cases (1) and (4)) or on the randomly generated data (cases (2) and (3)), respectively. Besides the pure output files the program generates a number of visualizations (see section [*Example: Distribution of point mutation in genes*]{}). In order to compare the experimental data with the mathematical prediction both, the experimental data and the theoretical data, are plotted in the same chart. Congruence of both curves indicates that the experimental data obey an equidistribution (case (2)) or the specified distribution (case (3)), respectively.
It may occur that the curve of the rank ordered experimental data decays significantly slower than the corresponding theoretical curve due to Eq. . Since there is no distribution more homogeneous than the equidistribution this situation may occur either as a rare fluctuation (recall that the theoretical curve was generated according to the [*averaged*]{} occupation numbers, Eq. ). In such cases there is no probability distribution $\{p_i\}$ which reproduces the experiment [*on average*]{}. This case can be artificially evoked when the species in the input file occur with almost identical frequencies.
The difference between the experimental rank ordered frequency distribution and the corresponding theoretical distribution (Eq. ) evaluates the degree of coincidence of the input data with an equidistribution (case (1)) or with a specified distribution (case (3)). We define the score by $$\label{eq:score}
S=\sum_{i=0}^M \left|M_i-M_i^{\rm theo}\right|\,.$$ The significance of a particular difference score can be assessed by relating it to the distribution of difference scores. This distribution depends on $M$ and $N$.
Example: Distribution of point mutations in genes {#example-distribution-of-point-mutations-in-genes .unnumbered}
-------------------------------------------------
The increasing number of known point mutations and polymorphisms in many genes coding for pathogenetically important proteins offers the opportunity to apply statistical tests to correlate their type and location to evolutionary, biological and clinical features.
In each replication generation there occur mutations of the genome but frequently they remain unnoticed since they do not cause diseases. These so-called polymorphisms or variants may occur either in regions of the genome which are coding for amino acid sequences or in non-coding segments. Those changes of the DNA sequence that alter the amino acid sequence are frequently associated with diseases because the respective proteins cannot operate properly. Screenings for mutations using DNA of patients have been performed for many human diseases and the identified mutations are accessible in mutation databases [@database].
The detection of so-called mutation hot spots, i.e. sequence regions with many mutation positions, is important for the identification of the functional and genetical properties of the genetic code [@function]. These hot spots must be distinguished from statistical fluctuations that occur even when the probabilities for mutations are identical for each residue position. Moreover, the spatial distribution of point mutations in genes is of importance for the localization of coding and non-coding parts in the genome.
We wish to apply the described method to the investigation of the amino acid sequence of the cystic fibrosis transmembrane conductance regulator. The unperturbed gene (wild type) is given as a sequence of 1480 letters: [ MQRSPLEKASVVSKLFFSWTRPILRKGYRQRLELSDIYQIPSVDSADNLSEKLER…]{}, each standing for one amino acid [@cysticsequence]. In experiments there has been observed a large number of mutations, i.e., deviations from this sequence. Such mutations are available from data bases, e.g. [@database].
$\begin{array}[b]{c} M1V \\ M1K \\ \underbrace{M1I }{} \\ {\bf M} \end{array} $ $\begin{array}[b]{c} \underbrace{Q2X }{} \\ {\bf Q} \end{array} $ ${\bf R}$ $\begin{array}[b]{c} \underbrace{S4X }{} \\ {\bf S} \end{array} $ $\begin{array}[b]{c} \underbrace{P5L }{} \\ {\bf P} \end{array} $ ${\bf L}$ ${\bf E}$ ${\bf K}$ ${\bf A}$ $\begin{array}[b]{c} \underbrace{S10R }{} \\ {\bf S} \end{array} $ ${\bf V}$ ${\bf V}$ $\begin{array}[b]{c} \underbrace{S13F }{} \\ {\bf S} \end{array} $ $\begin{array}[b]{c} \underbrace{K14X }{} \\ {\bf K} \end{array} $ ${\bf L}$ ${\bf F}$ ${\bf F}$ ${\bf S}$ $\begin{array}[b]{c} W19C \\ \underbrace{W19X }{} \\ {\bf W} \end{array} $ ${\bf T}$ ${\bf R}$ ${\bf P}$ ${\bf I}$ ${\bf L}$ ${\bf R}$ ${\bf K}$ $\begin{array}[b]{c} G27X \\ \underbrace{G27E }{} \\ {\bf G} \end{array} $ $Y$ $R$ $\begin{array}[b]{c} \underbrace{Q30X }{} \\ {\bf Q} \end{array} $ $\begin{array}[b]{c} R31C \\ \underbrace{R31L }{} \\ {\bf R} \end{array} $ ${\bf L}$ ${\bf E}$ ${\bf L}$ ${\bf S}$ ${\bf D}$ ${\bf I}$ ${\bf Y}$ $\begin{array}[b]{c} \underbrace{Q39X }{} \\ {\bf Q} \end{array} $ ${\bf I}$ ${\bf P}$ $\begin{array}[b]{c} \underbrace{S42F }{} \\ {\bf S} \end{array} $ ${\bf V}$ $\begin{array}[b]{c} \underbrace{D44G }{} \\ {\bf D} \end{array} $ ${\bf S}$ $\begin{array}[b]{c} \underbrace{A46D }{} \\ {\bf A} \end{array} $ ${\bf D}$ ${\bf N}$ ${\bf L}$ $\begin{array}[b]{c} S50P \\ \underbrace{S50Y }{} \\ {\bf S} \end{array} $ ${\bf E}$ ${\bf K}$ ${\bf L}$ ${\bf E}$ ${\bf R}$…
The codes on top of the underbraces stand for the found mutations, e.g., $P5L$ means that at position 5 it has been found that the amino acid proline (P) was replaced by leucine (L).
We subdivided the sequence into 74 parts of equal length 20 and counted the number of point mutations in each part. This way we obtain the [*measured frequencies*]{} $M_i=\{2,2,4,5,4,5,2,2,2,3,2,1,0,0,1,4,\dots\}$ which serve as input data. (The subdivision into parts may be repeated with a different starting point which yields similar results.) Certainly, measured frequencies as small as given above do not allow for the application of the $\chi^2$-test. The measured frequencies are shown in Fig. \[fig:CFroh\].
![Observed numbers of point mutations. The sequence has been subdivided into 74 parts of length 20.[]{data-label="fig:CFroh"}](CFroh.eps){width="8cm"}
Obviously, based on this data it is not possible to decide à priori whether the frequencies are equidistributed.
After processing the data as described above we obtain the rank ordered measured distribution (bars in Fig. \[fig:CFresult\]).
![Results of the computation. The bars show the rank ordered frequencies, the line displays the expected frequency distribution which would be obtained if the point mutations were (equally) randomly distributed. The curves differ significantly, therefore, we conclude that the point mutations are not random.[]{data-label="fig:CFresult"}](CFresult.eps){width="8cm"}
The full line shows the expected (theoretical) frequency distribution due to Eq. which has been generated with the hypothesis that the positions of the point mutations are equidistributed. Both curves deviate significantly from each other, therefore, we conclude that the mutations are not equidistributed. This conclusion agrees with the hypothesis in ref. [@cystichypothesis].
Since the investigation of point mutation is an interesting field of application of the program we developed a separate input mask for this purpose (case (4) of the list in the previous section). The input syntax for this mode is described in detail in the online help file of the program.
Recently, it has been shown for point mutations in the human androgen receptor (AR) that the severity of the disease correlates with the local sequence conservation [@androgen]. Germline mutations in the gene of the androgen receptor lead to the androgen insensitivity syndrome (AIS). In addition it was found that somatic point mutations associated with prostate cancer are more frequently found at locations with higher sequence variation compared to germline mutations leading to complete AIS. The related prediction method SIFT [@sift] has been proposed recently. Both methods, SIFT and the method used in [@androgen] are based on the alignment of a large number of related proteins. Inspired by their observation we asked the question whether mutations in the androgen receptor are distributed randomly over the sequence depending on the association with AIS or prostate cancer. The disease-associated mutations in the AR were obtained from the AR gene mutation database [@AndrogenDatabase]. Multiple mutations at identical positions were counted only once. Those mutations resulting in single amino acid substitutions were included in the analysis. The test was performed for 61 mutations associated with prostate cancer and 86 mutations found in patients with complete AIS. To perform the analysis we divided the sequence of 919 amino acids into 46 intervals of length 20 and counted the number of mutations in each interval. As expected, the results for the two datasets were different: Cancer associated mutations are more disseminated than congenital mutations found in patients with AIS. For mutations associated with prostate cancer the bar chart of the rank ordered frequencies nearly follows the theoretical curve for equal probabilities (Figs. \[fig:androgen\], \[fig:androgen1\]) whereas for AIS associated mutations the bar chart deviates markedly from the theoretical curve.
![Distribution of missense mutations in the androgen receptor for germline mutations leading to AIS. The height of the bars reflects the rank ordered frequencies of mutations in sequence intervals of length 20. The thick line displays the expected frequencies which would be obtained if point mutations were randomly distributed.[]{data-label="fig:androgen"}](freq_CAIS.eps){width="8cm"}
![Distribution of missense mutations in the androgen receptor for somatic mutations associated with prostate cancer. Explanation see caption of Fig. \[fig:androgen\].[]{data-label="fig:androgen1"}](freq_CaP.eps){width="8cm"}
Based on this finding we hypothesize that mutagenesis in the germline is followed by a selection process so that only a portion of the mutations are found in patients while others lead to early embryonal or fetal death. Conversely, mutations associated with prostate cancer may persist and are recorded.
Conclusions {#conclusions .unnumbered}
===========
For small sample sizes the relative frequencies $M_i/M$ of occurrence of individuals of a certain species $i$ deviate significantly from the probabilities of occurrence $p_i$. With the assumption that the $N$ species occur with equal probability $p_i=1/N$ the expectation values $\left<K_j\right>$ of the numbers of events which are contained $j$ times ($j=0,\dots,M$) in a sample of $M$ individuals can be determined based on combinatorial algebra. These expectation values allow for a prediction of the rank ordered frequency distribution.
For many practical problems the amount of available data is insufficient to employ standard tests, such as $\chi^2$, to discriminate whether or not a certain set of events complies with an equiprobability distribution. For such situations which occur frequently in the biological sciences we have developed an online tool which is available at http:/$\!$/bioinf.charite.de/equifreq/. As demonstrated for the case of point mutations in the sequence of amino acids of the cystic fibrosis transmembrane conductance regulator and the androgen receptor, even for sample set sizes which are certainly not sufficient to decide this question directly from the observed frequencies (see Figs. \[fig:prove\], \[fig:prove1\]) this tool helps to make a reliable statement.
The proposed method may be generalized to arbitrary probability distributions provided there exists a hypothesis on the functional form of the distribution [@PER]. For mathematical reasons, however, (see [@biofreq]) it is more difficult to derive an equivalent to Eq. formula for non-equiprobability distributions, which is subject of current research.
Avalability and requirements {#avalability-and-requirements .unnumbered}
============================
- [**Project name:**]{} equifreq
- [**Project home page:**]{} http:/$\!$/bioinf.charite.de/equifreq/
- [**Operating systems:**]{} platform independent
- [**Programming language:**]{} C++
- [**Other requirements:**]{} none
- [**License:**]{} GNU GPL
- [**Any restrictions to use by non-academics:**]{} none
Authors’ contributions {#authors-contributions .unnumbered}
======================
TP worked out the statistical and combinatorial background, wrote the kernel C++-program and drafted the manuscript. CF and CG provided the biological expertise, collected relevant biological data and organized the biological relevant applications. CG wrote the PHP user interface. All authors contributed in writing the manuscript.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are grateful to W. Ebeling, J. Freund and R. Mrowka for helpful discussion. We thank the reviewers for their helpful remarks and recommendations. Particularly fruitful was the analysis of mutations in the androgen receptor.
[10]{} url \#1[`#1`]{}urlprefix
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Freund JA, Pöschel T: [**A statistical approach to vehicular traffic.**]{} [*Physica A*]{} 1995, 219:95-114.
Pöschel T, Ebeling W, Frömmel C, Ramírez R: [**Correction algorithm for finite sample statistics.**]{} [*European Physical Journal E*]{}, in press.
Cotton RG, Horaitis O: [**The [HUGO]{} mutation database initiative. [H]{}uman genome organization.**]{} [*Pharmacogenomics*]{} 2002, [**2**]{}:16-19.
Walker DR, Bond JP, Tarone RE, Harris CC, Makalowski W, Boguski MS, Greenblatt MS: [**Evolutionary conservation and somatic mutation hotspot maps of p53: correlation with p53 protein structural and functional features.**]{} [*Oncogene*]{} 1999, [**7**]{}:211-218.
Zielenski J, Rozmahel R, Bozon D, Kerem B, Grzelczak Z, Riordan JR, Rommens J, Tsui LC: [**Genomic [DNA]{} sequence of the cystic fibrosis transmembrane conductance regulator ([CFTR]{}) gene.**]{} [*Genomics*]{} 1991, [**10**]{}:214-228; (see entry [CFTR\_HUMAN]{} in the [SWISSPROT]{} database).
Rommens JM, Iannuzzi MC, Kerem B, Drumm ML, Melmer G, Dean M, Rozmahel R, Cole JL, Kennedy D, Hidaka N, Zsiga M, Buchwald M, Riordan JR, Tsui LC, Collins FS: [**Identification of the cystic fibrosis gene: chromosome walking and jumping.**]{} [*Science*]{} 1989, [**245**]{}:1059-1065.
Ng PC, Henikoff S: [**SIFT: Predicting amino acid changes that affect protein function.**]{} [*Nucleic Acids Research*]{} 2003, [**13**]{}:3812-3814. Mooney SD, Klein TE, Altman RB, Trifiro MA, Gottlieb B: [**A functional analysis of disease-associated mutations in the androgen receptor gene.**]{} [*Nucleic Acids Research*]{} 2003, [**31**]{}:e42. Gottlieb B, Lehvaslaiho H, Beitel LK, Lumbroso R, Pinsky L, Trifiro M: [**The Androgen Receptor Gene Mutations Database.**]{} [*Nucleic Acids Research*]{} 1998,[**26**]{}:234-238.
Pöschel T, Ebeling W, Rosé H: [**Guessing probability distributions from small samples.**]{} [*Journal of Statistical Physics*]{} 1995 [**80**]{}:1443-1452.
|
---
abstract: 'Numerous energy harvesting wireless devices that will serve as building blocks for the Internet of Things (IoT) are currently under development. However, there is still only limited understanding of the properties of various energy sources and their impact on energy harvesting adaptive algorithms. Hence, we focus on *characterizing the kinetic (motion) energy that can be harvested by a wireless node with an IoT form factor* and on *developing energy allocation algorithms* for such nodes. In this paper, we describe methods for estimating harvested energy from acceleration traces. To characterize the energy availability associated with *specific human activities* specific human activities (e.g., relaxing, walking, cycling), we analyze a motion dataset with over 40 participants. Based on acceleration measurements that we collected for over 200 hours, we study energy generation processes associated with *day-long human routines*. day-long human routines. We also briefly summarize our experiments with *moving objects*. moving objects. We develop energy allocation algorithms that *take into account practical IoT node design considerations*, and evaluate the algorithms using the collected measurements. Our observations provide insights into the design of motion energy harvesters, IoT nodes, and energy harvesting adaptive algorithms.'
author:
- |
Maria Gorlatova, John Sarik, Guy Grebla, Mina Cong, Ioannis Kymissis,\
Gil Zussman [^1] [^2]
- 'Maria Gorlatova, John Sarik, Guy Grebla, Mina Cong, Ioannis Kymissis, Gil Zussman [^3] [^4]'
- |
Maria Gorlatova, John Sarik, Guy Grebla, Mina Cong, Ioannis Kymissis, Gil Zussman\
\
\
\
{mag2206@, jcs2160@, guy@ee., mc3415@, johnkym@ee., gil@ee.}columbia.edu
title:
- 'Movers and Shakers: Kinetic Energy Harvesting for the Internet of Things'
- |
Movers and Shakers:\
Kinetic Energy Harvesting for the Internet of Things
- |
Movers and Shakers:\
Kinetic Energy Harvesting for the Internet of Things
---
[**Keywords:**]{} Energy harvesting; motion energy; measurements; low-power networking; Algorithms; Internet of Things.
Energy harvesting, motion energy, measurements, ultra-low-power networking, Internet of Things.
Introduction
============
Related Work {#sect:SourcesOverview}
============
Models & Measurement Setup {#sect:KineticEnergy}
==========================
Human Motion {#sect:perActivityEnergy}
============
Long-term Human Mobility {#sect:DailyEnergy}
========================
Object Motion Energy {#sect:Intuition}
====================
Energy-aware Algorithms {#sect:EnergyAlgorithms}
=======================
Conclusions {#sect:Conclusions}
===========
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by Vodafone Americas Foundation Wireless Innovation Project and NSF grants CCF-09-64497 and CNS-10-54856. We thank Sonal Shetkar for her contributions to the development of the measurement setup and study methodology, and Craig Gutterman for his contributions to preliminary data analysis. We additionally thank Chang Sun and Kanghwan Kim for their contributions. We also thank our shepherd Ranveer Chandra for valuable feedback.
**Code reference: what plots what**
- - Figure \[fig:FilterViewHarvesters\]: with [D:\\Dropbox\\KineticAndLightEnergyMeasurements\\MGcodeUpdatedOct2012\\PlotKBFilterResponce.m](D:\Dropbox\KineticAndLightEnergyMeasurements\MGcodeUpdatedOct2012\PlotKBFilterResponce.m)
- Figure \[fig:ParameterSearchSpace\]: with [MATLAB\_scripts/KBvaluesSearching\_3D](MATLAB_scripts/KBvaluesSearching_3D), followed by [KBvaluesSearching\_3D\_plotting.m](KBvaluesSearching_3D_plotting.m)
- Figure \[fig:policyPerformanceNode\]: with [MATLAB\_scripts/Policies\_differentTraces](MATLAB_scripts/Policies_differentTraces), which calls a plotting script that can also run independently, [MATLAB\_scripts/Policies\_differentTraces\_PLOT](MATLAB_scripts/Policies_differentTraces_PLOT)
- Figure \[fig:policyPerformanceSimplRepres\]: with [MATLAB\_scripts/Policies\_differentInputTypes](MATLAB_scripts/Policies_differentInputTypes), which calls a plotting script that can also run independently, [MATLAB\_scripts/Policies\_differentInputTypes\_PLOT](MATLAB_scripts/Policies_differentInputTypes_PLOT)
- Figure \[fig:MotionAsAfunctionOfTime\](a,b): TBD
- Figure \[fig:MotionAsAfunctionOfTime\](c): with [MATLAB\_scripts/WorkingWithEnergyPerSecond](MATLAB_scripts/WorkingWithEnergyPerSecond), setting file index to 15.
[1]{}
APPENDIX I {#appendix-i .unnumbered}
==========
[^1]: M. Gorlatova, J. Sarik, G. Grebla, M. Cong, I. Kymissis, and G. Zussman are with the Department of Electrical Engineering, Columbia University, New York, NY 10027. E-mail: {mag2206, jcs2160, gg2519, mc3415}@columbia.edu, {johnkym@ee, gil@ee}.columbia.edu
[^2]: Preliminary version of this paper will appear in Proc. ACM SIGMETRICS’14 [@MoversShakers2014].
[^3]: M. Gorlatova, J. Sarik, G. Grebla, M. Cong, I. Kymissis, and G. Zussman are with the Department of Electrical Engineering, Columbia University, New York, NY 10027. E-mail: {mag2206, jcs2160, gg2519, mc3415}@columbia.edu, {johnkym@ee, gil@ee}.columbia.edu
[^4]: Preliminary version of this paper will appear in Proc. ACM SIGMETRICS’14 [@MoversShakers2014].
|
---
abstract: 'Here we describe the least distributive lattice congruence $\eta$ on an idempotent semiring in general and characterize the varieties ${\mathbf{D}^{\bullet}}, {\mathbf{L}^{\bullet}}$ and ${\mathbf{R}^{\bullet}}$ of all idempotent semirings such that $\eta={\mathcal{D}^{\bullet}}, {\mathcal{L}^{\bullet}}$ and ${\mathcal{R}^{\bullet}}$, respectively. If $S \in {\mathbf{D}^{\bullet}}[{\mathbf{L}^{\bullet}}, {\mathbf{R}^{\bullet}}]$, then the multiplicative reduct $(S, \cdot)$ is a \[left, right\] normal band. Every semiring $S \in {\mathbf{D}^{\bullet}}$ is a spined product of a semiring in ${\mathbf{L}^{\bullet}}$ and a semiring in ${\mathbf{R}^{\bullet}}$ with respect to a distributive lattice.'
author:
- 'M. K. Sen$^{a}$, A. K. Bhuniya$^{b}$[^1] and R. Debnath$^c$'
title: 'On the idempotent semirings such that ${\mathcal{D}^{\bullet}}$ is the least distributive lattice congruence'
---
a\) Department of Pure Mathematics, University of Calcutta, Kolkata, India\
b) Department of Mathematics, Visva-Bharati, Santiniketan-731235, India.\
c) Department of Mathematics, Kurseong college, Kurseong-734203, India.
[*E-mail addresses*]{}: senmk6@yahoo.com$^a$, anjankbhuniya@gmail.com$^b$ and rajib.d6@gmail.com$^c$.
[**Keywords :**]{} idempotent semiring; least distributive lattice congruence; normal band, spined product, Malcev’s product.\
[**AMS Mathematics Subject Classification:**]{} 16Y60.
Introduction
============
A *semiring* $(S,+, \cdot)$ is an algebra with two binary operations + and $\cdot$ such that both the additive reduct $(S, +)$ and the multiplicative reduct $(S, \cdot)$ are semigroups and such that the following distributive laws hold: $$\begin{aligned}
x(y+z) = xy+xz \; \textrm{and} \; (x+y)z = xz+yz.\end{aligned}$$ If moreover, both the reducts $(S, +)$ and $(S, \cdot)$ are bands, then $S$ is called an *idempotent semiring*. Thus the class of all idempotent semirings is an equational class satisfying two additional identities: $$\begin{aligned}
x+x \approx x \; \textrm{and} \; x \cdot x \approx x.\end{aligned}$$ The variety of all idempotent semirings will be denoted by ${\mathbf{I}}$.
We now fix the notation for some varieties of idempotent semirings and give their determining identity within ${\mathbf{I}}$:
Notation Defining identity within ${\mathbf{I}}$ Notation Defining identity within ${\mathbf{I}}$
-------------------------- ----------------------------------------- -------------------------- -----------------------------------------
$\mathbf{R}^{+}$ $x+y+x \approx x$, $\mathbf{R}^{\bullet}$ $xyx \approx x$,
$\mathbf{LZ}^{+}$ $x+y \approx x$, $\mathbf{RZ}^{+}$ $x+y \approx y$,
$\mathbf{LZ}^{\bullet}$ $xy \approx x$, $\mathbf{RZ}^{\bullet}$ $xy \approx y$,
$\mathbf{LNB}^{\bullet}$ $xyz \approx xzy$, $\mathbf{RNB}^{\bullet}$ $xyz \approx yxz$,
${\mathbf{LQBI}}$ $x+xy+x \approx x$, ${\mathbf{RQBI}}$ $x+yx+x \approx x$,
${\mathbf{LN}}$ $x+xyx \approx x$, ${\mathbf{RN}}$ $xyx+x \approx x$,
${\mathbf{N}}$ $x+xyx+x \approx x$, $\mathcal{S}l^+$ $x+y \approx y+x$.
For an idempotent semiring $(S, +, \cdot)$ the Green’s relations ${\mathcal{L}}, {\mathcal{R}}$ and ${\mathcal{D}}$ on the additive \[multiplicative\] reduct $(S, +)[(S, \cdot)]$ will be denoted by ${\mathcal{L}^+}, {\mathcal{R}^+}$ and ${\mathcal{D}^+}[{\mathcal{L}^{\bullet}}, {\mathcal{R}^{\bullet}}, {\mathcal{D}^{\bullet}}]$. Since the multiplicative reduct $(S, \cdot)$ of an idempotent semiring $S$ is a band, ${\mathcal{D}^{\bullet}}, {\mathcal{L}^{\bullet}}$ and ${\mathcal{R}^{\bullet}}$ are given by: for $a, b \in S$, $$\begin{aligned}
& a {\mathcal{D}^{\bullet}}b \, \Leftrightarrow \, aba=a, \ bab=b, \\
& a {\mathcal{L}^{\bullet}}b \, \Leftrightarrow \, ba=b, \ ab=a \\
\textrm{and} \; & a {\mathcal{R}^{\bullet}}b \, \Leftrightarrow \, ab=b, \ ba=a.\end{aligned}$$ The relations ${\mathcal{D}^+}, {\mathcal{L}^+}$ and ${\mathcal{R}^+}$ are given dually.
The variety ${\mathbf{I}}$ of all idempotent semirings contains the variety ${\mathbf{D}}$ of all distributive lattices; and hence there is the least distributive lattice congruence on each idempotent semiring $S$. Throughout this article, we denote the least distributive lattice congruence on $S$ by $\eta_S$ or precisely by $\eta$. Then $(S/\eta, +)$ is a semilattice. Since $(S, +)$ is a band, ${\mathcal{D}^+}$ is the least congruence on the idempotent semiring $S$ such that $(S/{\mathcal{D}^+}, +)$ is a semilattice. Hence ${\mathcal{D}^+}\subseteq \eta$. In [@SGS], Sen, Guo and Shum characterized in depth the idempotent semirings $S$ such that $\eta = {\mathcal{D}^+}$.
[@SGS] Let $S$ be an idempotent semiring. Then the following conditions are equivalent.
1. ${\mathcal{D}^+}$ is the least distributive lattice congruence on $S$;
2. $S$ satisfies the identities: $$\begin{aligned}
x+xy+x & \approx x \label{bi1} \\
\textrm{and} \; x+yx+x & \approx x; \label{bi2}\end{aligned}$$
3. $S \in \mathbf{R}^{+} \circ {\mathbf{D}}$.
An idempotent semiring $S$ is said to be a *band semiring* if ${\mathcal{D}^+}$ is the least distributive lattice congruence on $S$, equivalently if it satisfies the identities (\[bi1\]) and (\[bi2\]). We will denote the variety of all band semirings by ${\mathbf{BI}}$. We define an idempotent semiring to be a *left \[right\] quasi-band semiring* if it satisfies the identity (\[bi1\]) \[(\[bi2\])\]. Thus ${\mathbf{BI}}={\mathbf{LQBI}}\cap {\mathbf{RQBI}}$. Additive reduct $(S, +)$ of a band semiring $S$ is a regular band [@WZG].
The idempotent semirings for which $\eta = {\mathcal{D}^{\bullet}}$ have been characterized by Pastijn and Zhao [@PZ2000]. There are several articles characterizing the variety ${\mathbf{I}}$ of idempotent semirings by Green’s relations [@PG], [@PZ2000], [@SGS] - [@ZSG].
We extend the problem to the idempotent semirings $S$ such that the least distributive lattice congruence $\eta$ on $S$ is either of the Green’s relations ${\mathcal{L}^+}$, ${\mathcal{R}^+}$, ${\mathcal{L}^{\bullet}}$ and ${\mathcal{R}^{\bullet}}$. For this, we first describe explicitly the least distributive lattice congruence $\eta$ on an idempotent semiring in general, which makes it possible to characterize the idempotent semirings for which $\eta={\mathcal{D}^+}$ $[{\mathcal{D}^{\bullet}}, {\mathcal{L}^+}, {\mathcal{L}^{\bullet}}, {\mathcal{R}^+}, {\mathcal{R}^{\bullet}}]$ in an unified and simple approach.
We present our results in two parts. The idempotent semirings $S$ such that $\eta={\mathcal{D}^+}$ $[{\mathcal{L}^+}, {\mathcal{R}^+}]$ have been characterized in [@BD]. The following result has some use in the last section of this articel.
[@BD] \[lp is the least lat cong\] Let $S$ be an idempotent semiring. Then the following conditions are equivalent.
1. ${\mathcal{L}^+}$ is the least distributive lattice congruence on $S$;
2. $S \in {\mathbf{LN}}$ and ${\mathcal{D}^{\bullet}}\subseteq {\mathcal{L}^+}$;
3. $S$ satisfies the identities: $$x+yxy \approx x; \label{identity for lp}$$
4. $S \in \mathbf{LZ}^{+} \circ {\mathbf{D}}$.
In Section 2 of this article, we characterize the least distributive lattice congruence on an idempotent semiring. The idempotent semirings $S$ such that $\eta={\mathcal{D}^{\bullet}}$ $[{\mathcal{L}^{\bullet}}, {\mathcal{R}^{\bullet}}]$ have been studied in Section 3. In Section 4, we show that the multiplicative reduct $(S, \cdot)$ of an idempotent semiring $S$ such that $\eta = {\mathcal{D}^{\bullet}}$ is a normal band, and such a semiring $S$ is a spined product of an idempotent semiring $S_1$ such that $\eta = {\mathcal{L}^{\bullet}}$ and an idempotent semiring $S_2$ such that $\eta = {\mathcal{R}^{\bullet}}$ with respect to a distributive lattice.
We refer to [@howie] for the information concerning semigroup theory, [@Golan] for background on semirings and [@McKenzie] for notions concerning universal algebra and lattice theory.
The least distributive lattice congruence on an idempotent semiring
===================================================================
Let $S$ be an idempotent semiring. Define a binary relation $\sigma$ on $S$ by: for $a, b \in S$, $$\begin{aligned}
a \sigma b & \ \ \textrm{if and only if} \\
& \hspace{2cm} aba=aba+a+aba \ \textrm{and} \ bab=bab+b+bab.\end{aligned}$$
\[congruence generated by sigma\] Let $S$ be an idempotent semiring. Then the congruence relation generated by $\sigma$ is the least distributive lattice congruence on $S$.
Let $a, b \in S$. Then we have $(a+b)(b+a)(a+b)+a+b+(a+b)(b+a)(a+b) = (a+b)(b+a+a+b+b+a)(a+b) = (a+b)(b+a)(a+b)$, and similarly $(b+a)(a+b)(b+a)=(b+a)(a+b)(b+a)+a+b+(b+a)(a+b)(b+a)$, which implies that $(a+b) \sigma (b+a)$. Also $(ab) \sigma (ba)$ for all $a, b \in S$.
Now, $a(a+ab)a+a+a(a+ab)a=(a+aba)+a+(a+aba)=a+aba=a(a+ab)a$ and $(a+ab)a(a+ab)+a+ab+(a+ab)a(a+ab)=(a+ab)a(a+ab)$ implies that $(a+ab) \sigma a$ for all $a, b \in S$. Thus any congruence on $S$ which contains $\sigma$ is a distributive lattice congruence on $S$.
Consider a distributive lattice congruence $\rho$ on $S$ and $a, b \in S$ such that $a \sigma b$. Then $aba=aba+a+aba$ implies that $b \rho (b+aba)=(b+aba+a+aba) \rho (b+a) \rho (a+b)$. Similarly from $bab=bab+b=bab$ we have $a \rho (a+b)$ and it follows that $a \rho b$. Thus $\sigma$ is contained in every distributive lattice congruence on $S$. Hence the result follows.
Now we show that $\sigma$ is not transitive on a semiring in general. For this consider the following example [@GPZ] of an idempotent semiring,
$\begin{array}{c|ccc}
+ & a & b & c \\
\hline
a & a & b & c \\
b & b & b & b \\
c & c & b & c
\end{array}$ $\begin{array}{c|ccc}
\cdot & a & b & c \\
\hline
a & a & a & a \\
b & b & b & b \\
c & a & b & c
\end{array}$
Then $a \sigma b$ and $b \sigma c$ but $a \overline{\sigma} c$ shows that $\sigma$ is not transitive.
The transitive closure $\sigma^{\ast}$ of $\sigma$ is given by: $$\begin{aligned}
a \sigma^{\ast} b & \ \ \textrm{if and only if there exists} \ x \in S \ \textrm{such that} \\
& \hspace{2cm} axbxa=axbxa+a+axbxa \ \textrm{and} \ bxaxb=bxaxb+b+bxaxb.\end{aligned}$$
A semiring $(S, +, \cdot)$ is called *almost idempotent* if $(S, +)$ is a semilattice and $a+a^2=a^2$ for every $a \in S$. In [@SB5], we proved that $\sigma^{\ast}$ is the least distributive lattice congruence on an almost idempotent semiring. Since every idempotent semiring $S$ with commutative addition is an almost idempotent semiring, $\sigma^{\ast}$ is the least distributive lattice congruence on $S$. The proof that $\sigma^{\ast}$ remains the least distributive lattice congruence on an idempotent semiring $S$ even if the addition in $S$ is not commutative, is almost similar and so we omit the proof here.
\[eta\] Let $S$ be an idempotent semiring. Then $\sigma^{\ast}$ is the least distributive distributive congruence on $S$.
Almost all varieties of idempotent semirings considered here are subvarieties of ${\mathbf{N}}$. For this, the following result will have an important roll throughout the rest of this article.
\[equivalent N\] An idempotent semiring $S \in {\mathbf{N}}$ if and only if it satisfies the identity: $$xz+xyz+xz \approx xz.$$
First assume that $S \in {\mathbf{N}}$ and $a, b, c \in S$. Then $ac = (a+abca+a)c=ac+abcac+ac=ac+abca(c+cabc+c)+ac=ac+abcac+abc+abcac+ac$ implies that $ac+abc+abcac+ac=ac$ \[add $abc+abcac+ac$ to both sides from right\]. Similarly this implies that $ac = ac+abc+ac$.
Converse follows directly.
\[sigma is the ldlc\] If an idempotent semiring $S \in {\mathbf{N}}$, then $\sigma$ is the least distributive lattice congruence on $S$.
Let $a, b \in S$ be such that $a \eta b$. Then there exists $x \in S$ such that $axbxa=axbxa+a+axbxa$ and $bxaxb=bxaxb+b+bxaxb$. Since $S \in {\mathbf{N}}$, $ab+axb+ab=ab$ and $ba+bxa+ba=ba$, by Lemma \[equivalent N\]. Then we have $$\begin{aligned}
aba & = a(ba+bxa+ba) && \\
& = aba+bxa+aba && \\
& = aba+(ab+axb+ab)xa+aba && \\
& = aba+abxa+axbxa+abxa+aba && \\
& = aba+abxa+axbxa+a+axbxa+abxa+aba && \\
& = aba+abxa+axbxa+a+(aba+abxa+ && \\
& \hspace{3cm} axbxa+a+axbxa+abxa+aba) && \text{since $(S, +)$ is a band} \\
& = aba+abxa+axbxa+a+aba && \\
& = aba+a+aba && \text{since $(S, +)$ is a band}.\end{aligned}$$ Similarly, $bab=bab+b+bab$. Thus $a \sigma b$ and so $\eta \subseteq \sigma$. Also $\sigma$ is contained in every distributive lattice congruence. Hence $\sigma = \eta$ is the least distributive lattice congruence on $S$.
The idempotent semirings such that $\eta={\mathcal{D}^{\bullet}}, {\mathcal{L}^{\bullet}}, {\mathcal{R}^{\bullet}}$
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In this section we show that the class of all idempotent semirings for which $\eta = {\mathcal{D}^{\bullet}}[{\mathcal{L}^{\bullet}}, {\mathcal{R}^{\bullet}}]$ is an equational class and so a variety. Here we find several systems of identities defining these varieties.
On an idempotent semiring $(S, +, \cdot)$ one may introduce the relations $\leq^{l}_{+}, \, \leq^{l}_{\cdot}, \leq^{r}_{+}, \, \leq^{r}_{\cdot}$ and $\leq_{+}, \, \leq_{\cdot}$ by the following: for $a, b \in S$, $$\begin{aligned}
& a \leq^{l}_{+} b \hspace{.5cm} \Leftrightarrow \hspace{.5cm} b=a+b \hspace{1cm}
a \leq^{l}_{\cdot} b \hspace{.5cm} \Leftrightarrow \hspace{.5cm} a=ba \\
& a \leq^{r}_{+} b \hspace{.5cm} \Leftrightarrow \hspace{.5cm} b=b+a \hspace{1cm}
a \leq^{r}_{\cdot} b \hspace{.5cm} \Leftrightarrow \hspace{.5cm} a=ab \\
& \hspace{1.4cm} \leq_{+} = \leq^{l}_{+} \cap \leq^{r}_{+} \hspace{.5cm} \textrm{and} \hspace{.5cm}
\leq_{\cdot} = \leq^{l}_{\cdot} \cap \leq^{r}_{\cdot}\end{aligned}$$ The relations $\leq^{l}_{+}, \, \leq^{l}_{\cdot}, \leq^{r}_{+}, \, \leq^{r}_{\cdot}$ are quasi-orders and the relations $\leq_{+}$ and $\leq_{\cdot}$ are partial orders [@nambooripad].
In [@PZ2000], Pastijn and Zhao characterized the idempotent semirings $S$ such that ${\mathcal{D}^{\bullet}}$ is the least distributive lattice congruence on $S$. Following result is already proved in [@PZ2000]. Use of Theorem \[eta\] shorten the proof which we would like to include here.
\[ddot least lat cong\] Let $S$ be an idempotent semiring. Then the following conditions are equivalent.
1. ${\mathcal{D}^{\bullet}}$ is the least distributive lattice congruence on $S$;
2. $S \in {\mathbf{N}}$ and ${\mathcal{D}^+}\subseteq {\mathcal{D}^{\bullet}}$;
3. $S$ satisfies the identity $$\label{identity for ddot}
x \approx xyx+x+xyx.$$
$\mathbf{(1) \ \Rightarrow \ (2):}$ Let $a, b \in S$. Since ${\mathcal{D}^{\bullet}}$ is a distributive lattice congruence, $(a+ab) {\mathcal{D}^{\bullet}}a$. Then we have $$\begin{aligned}
a=a(a+ab)a \, \Rightarrow \, & a=a+aba \\
\Rightarrow \, & a=a+aba+a\end{aligned}$$ and so $S \in {\mathbf{N}}$.
Now let $a, b \in S$ such that $a {\mathcal{D}^+}b$. Then $a=a+b+a$ and $b=b+a+b$ implies that $bab=bab+b+bab$ and $aba=aba+a+aba$, and so $a \sigma b$. Thus ${\mathcal{D}^+}\subseteq \sigma \subseteq \eta={\mathcal{D}^{\bullet}}$.\
$\mathbf{(2) \ \Rightarrow \ (3):}$ Similar to the proof of the Theorem 2.17 [@PZ2000].\
$\mathbf{(3) \ \Rightarrow \ (1):}$ Let $a, b \in S$ such that $a \eta b$. Since $S$ satisfied the identity (\[identity for ddot\], it follows that $S \in {\mathbf{N}}$ and so $\eta=\sigma$, by Theorem \[sigma is the ldlc\]. Hence we have $$aba=aba+a+aba \, \textrm{and} \, bab=bab+b+bab.$$ Since $S$ satisfies the identity (\[identity for ddot\]), it follows that $a=aba$ and $b=bab$. Thus $a {\mathcal{D}^{\bullet}}b$ and so $\eta \subseteq {\mathcal{D}^{\bullet}}$. Also ${\mathcal{D}^{\bullet}}\subseteq \sigma = \eta$. Therefore ${\mathcal{D}^{\bullet}}=\eta$ and so ${\mathcal{D}^{\bullet}}$ is the least distributive lattice congruence on $S$.
Now wecharacterize the idempotent semirings for which ${\mathcal{L}^{\bullet}}$ is the least distributive lattice congruence. First we prove the following lemma.
\[bi1 iff n and\] Let $S$ be an idempotent semiring. Then the following conditions are equivalent.
1. $S$ satisfies the identity (\[bi1\]);
2. $S \in {\mathbf{N}}$ and ${\mathcal{R}^{\bullet}}\subseteq {\mathcal{D}^+}$.
$\mathbf{(1) \, \Rightarrow \, (2):}$ Let $a, b \in S$ such that $a {\mathcal{R}^{\bullet}}b$. Then $a=ba$ and $b=ab$. Now $a=a+ab+a=a+b+a$ and $b=b+ba+b=b+a+b$ implies that $a {\mathcal{D}^+}b$. Thus ${\mathcal{R}^{\bullet}}\subseteq {\mathcal{D}^+}$. Also it follows trivially that $S \in {\mathbf{N}}$.\
$\mathbf{(2) \, \Rightarrow \, (1):}$ Let $a, b \in S$. Then $ab {\mathcal{R}^{\bullet}}aba$ implies that $ab {\mathcal{D}^+}aba$. This implies that $$\begin{aligned}
& aba+ab+aba=aba \\
\Rightarrow \, & a+aba+ab+aba+a=a+aba+a \\
\Rightarrow \, & a+aba+ab+aba+a=a, \hspace{1cm} (\textrm{since} \, S \in {\mathbf{N}}) \\
\Rightarrow \, & a+aba+ab+aba+a=a+aba+ab+a \\
\Rightarrow \, & a=a+aba+ab+a \\
\Rightarrow \, & a+ab+a=a+aba+ab+a \\
\Rightarrow \, & a+ab+a=a\end{aligned}$$ Thus $S$ satisfies the identity (\[bi1\]).
\[ldot least lat cong\] Let $S$ be an idempotent semiring. Then the following conditions are equivalent.
1. ${\mathcal{L}^{\bullet}}$ is the least distributive lattice congruence on $S$;
2. ${\mathcal{D}^+}\subseteq {\mathcal{L}^{\bullet}}$ and $S$ satisfies the identity (\[bi1\]);
3. $S \in {\mathbf{N}}$ and ${\mathcal{R}^{\bullet}}\subseteq {\mathcal{D}^+}\subseteq {\mathcal{L}^{\bullet}}$;
4. $\leq^{l}_{\cdot} \subseteq \leq_{+}$ for $S$;
5. $S$ satisfies the identity $$x \approx xy+x+xy; \label{identity for ldot}$$
6. $S$ satisfies the identity $$x \approx x(y+x+y).$$
Equivalence of (2) and (3) follows from the Lemma \[bi1 iff n and\]. Equivalence of (5) and (6) follows trivially.\
$\mathbf{(1) \, \Rightarrow \, (2):}$ Let $a, b \in S$. Then $(a+ab) {\mathcal{L}^{\bullet}}a$ implies that $a=a(a+ab)=a+ab$ and so $a=a+ab+a$ for all $a, b \in S$. Thus $S$ satisfies the identity (\[bi1\]). Since ${\mathcal{D}^+}$ is the least semilattice congruence on the additive reduct $(S, +)$, ${\mathcal{D}^+}\subseteq {\mathcal{L}^{\bullet}}$.\
$\mathbf{(2) \, \Rightarrow \, (5):}$ Let $a, b \in S$. Then $(a+b) {\mathcal{D}^+}(b+a)$ and so $(a+b) {\mathcal{L}^{\bullet}}(b+a)$. Therefore $$\begin{aligned}
& a+b=(a+b)(b+a)=ab+a+b+ba \\
\textrm{and} \; \; & b+a=(b+a)(a+b)=ba+b+a+ab\end{aligned}$$ Also $S$ satisfies the identity (\[bi1\]). Thus we have $$\begin{aligned}
& a=a(a+b)+a(b+a) \\
\Rightarrow \, & a=a(ab+a+b+ba)+a(ba+b+a+ab) \\
\Rightarrow \, & a=ab+a+ab+aba+ab+a+ab \\
\Rightarrow \, & ab+a+ab=ab+a+ab+aba+ab+a+ab \\
\Rightarrow \, & a=ab+a+ab\end{aligned}$$ for all $a, b \in S$.\
$\mathbf{(5) \, \Rightarrow \, (1):}$ For every $a, b \in S$, we have $a=ab+a+ab$. This implies that $a=aba+a+aba$. Therefore ${\mathcal{D}^{\bullet}}$ is the least distributive lattice congruence on $S$, by Theorem \[ddot least lat cong\]. Also for every $a, b \in S$, $$\begin{aligned}
& a=ab+a+ab \\
\Rightarrow \, & a=a+ab+a+ab+a \\
\Rightarrow \, & a=a+ab+a\end{aligned}$$ Now let $a, b \in S$ such that $a {\mathcal{D}^{\bullet}}b$. Then $a=aba$ and $b=bab$. Now $$\begin{aligned}
& a=aba \\
\Rightarrow \, & a+ab+a=aba+ab+aba \\
\Rightarrow \, & a=ab\end{aligned}$$ Similarly $b=ba$. Hence $a {\mathcal{L}^{\bullet}}b$ and so ${\mathcal{D}^{\bullet}}={\mathcal{L}^{\bullet}}$. Thus ${\mathcal{L}^{\bullet}}$ is the least distributive lattice congruence on $S$.\
$\mathbf{(4) \, \Rightarrow \, (5):}$ Let $a, b \in S$. Then $ab \leq^{l}_{\cdot} a$. This implies that $ab \leq_{+} a$. Then $ab+a=a=a+ab$ and so $a=ab+a+ab$. Thus $S$ satisfies the identity (\[identity for ldot\].\
$\mathbf{(5) \, \Rightarrow \, (4):}$ Let $a, b \in S$ such that $a \leq^{l}_{\cdot} b$. Then $a=ba$. Also $a, b \in S$ implies that $b=ba+b+ba$. This implies that $$\begin{aligned}
& b=a+b+a \\
\Rightarrow \, & b+a=b=a+b\end{aligned}$$ and so $a \leq_{+} b$. Therefore $\leq^{l}_{\cdot} \subseteq \leq_{+} $.
The left-right dual of this result is stated as follows. Since the proof is similar to the above theorem, we omit.
\[rdot least lat cong\] Let $S$ be an idempotent semiring. Then the following conditions are equivalent.
1. ${\mathcal{R}^{\bullet}}$ is the least distributive lattice congruence on $S$;
2. ${\mathcal{D}^+}\subseteq {\mathcal{R}^{\bullet}}$ and $S$ satisfies the identity (\[bi2\]);
3. $S \in {\mathbf{N}}$ and ${\mathcal{L}^{\bullet}}\subseteq {\mathcal{D}^+}\subseteq {\mathcal{R}^{\bullet}}$;
4. $\leq^{r}_{\cdot} \subseteq \leq_{+}$ for $S$;
5. $S$ satisfies the identity $$x \approx yx+x+yx; \label{identity for rdot}$$
6. $S$ satisfies the identity $$x \approx (y+x+y)x.$$
Joins and Malcev’s Products:
============================
The variety of all idempotent semirings $S$ such that ${\mathcal{D}^{\bullet}}[{\mathcal{L}^{\bullet}}, {\mathcal{R}^{\bullet}}]$ is the least distributive lattice congruence on $S$ will be denoted by $\mathbf{D}^{\bullet} [\mathbf{L}^{\bullet}, \mathbf{R}^{\bullet}]$. Thus it follows, by Theorem \[ddot least lat cong\], Theorem \[ldot least lat cong\] and Theorem \[rdot least lat cong\], that the varieties $\mathbf{D}^{\bullet}, \mathbf{L}^{\bullet}$ and $\mathbf{R}^{\bullet}$ are determined by the additional identities (\[identity for ddot\]), (\[identity for ldot\]) and (\[identity for rdot\]), respectively.
For subvarieties $\mathbf{V}$ and $\mathbf{W}$ of ${\mathbf{I}}$, the [*Mal’cev product*]{} $\mathbf{V} \circ \mathbf{W}$ of $\mathbf{V}$ and $\mathbf{W}$ (within ${\mathbf{I}}$) is the class of all idempotent semirings $S$ on which there exists a congruence $\rho$ such that $S/\rho \in \mathbf{W}$ and such that the $\rho$-classes belong to $\mathbf{V}$.
In this section, different characterizations of the varieties $\mathbf{D}^{\bullet}, \mathbf{L}^{\bullet}$ and $\mathbf{R}^{\bullet}$ by Malcev’s product are given.
\[vld=lzd v vd\]
1. $\mathbf{L}^{\bullet} = \mathbf{LZ}^{\bullet} \circ {\mathbf{D}}$.
2. $\mathbf{R}^{\bullet} = \mathbf{RZ}^{\bullet} \circ {\mathbf{D}}$.
\(i) First we assume that $S \in \mathbf{LZ}^{\bullet} \circ \mathbf{D}$. Then there exists a congruence relation $\delta$ on $S$ such that $S/\delta \in {\mathbf{D}}$ and such that each $\delta$-class belongs to $\mathbf{LZ}^{\bullet}$. Let $a, b \in S$. Since $S/\delta \in {\mathbf{D}}$, we have $(a+ab) \delta a$ and $(ab+a) \delta a$. Since the $\delta$-class of $a$ is a left zero band for the multiplication, we have $$a=a(ab+a)=ab+a \hspace{.5cm} \textrm{and} \hspace{.5cm} a=a(a+ab)=a+ab,$$ and so $a=(ab+a)+(a+ab)=ab+a+ab$. Thus $S$ satisfies the identity \[identity for ldot\]. Hence $S \in \mathbf{L}^{\bullet}$, by Theorem \[ldot least lat cong\]. Thus $\mathbf{LZ}^{\bullet} \circ {\mathbf{D}}\subseteq \mathbf{L}^{\bullet}$. The reverse inclusion follows trivially. Hence $\mathbf{L}^{\bullet} = \mathbf{LZ}^{\bullet} \circ {\mathbf{D}}$.\
(ii) The proof is left-right dual to (i).
Thus for $S \in \mathbf{L}^{\bullet}$ the multiplicative reduct $(S, \cdot)$ is a left regular band and for $S \in \mathbf{R}^{\bullet}$ the multiplicative reduct $(S, \cdot)$ is a right regular band.
For an idempotent semiring $S$ the following conditions are equivalent:
1. $S \in {\mathbf{LN}}$.
2. ${\mathcal{D}^{\bullet}}$ is a congruence on $S$ and $S/{\mathcal{D}^{\bullet}}\in {\mathbf{LZ}^{+}}\circ {\mathbf{D}}$.
$\mathbf{(1) \, \Rightarrow \, (2):}$ Let $S \in {\mathbf{LN}}$. Then $S \in {\mathbf{N}}$ and so ${\mathcal{D}^{\bullet}}$ is a congruence on $S$, by Theorem 2.11 [@PZ2000]. Hence $S/{\mathcal{D}^{\bullet}}\in {\mathbf{LN}}$ and ${\mathcal{D}^{\bullet}}_{S/{\mathcal{D}^{\bullet}}} \subseteq {\mathcal{L}^+}_{S/{\mathcal{D}^{\bullet}}}$. Thus $S/{\mathcal{D}^{\bullet}}\in {\mathbf{LZ}^{+}}\circ {\mathbf{D}}$.\
$\mathbf{(2) \, \Rightarrow \, (1):}$ Assume that $S/{\mathcal{D}^{\bullet}}\in {\mathbf{LZ}^{+}}\circ {\mathbf{D}}$. Then ${\mathcal{L}^+}_{S/{\mathcal{D}^{\bullet}}}$ is the least distributive lattice congruence on $S/{\mathcal{D}^{\bullet}}$, by Lemma \[lp is the least lat cong\]. Let $\overline{a}$ be the ${\mathcal{D}^{\bullet}}$-class of $a$ in $S$. Then for all $a, b \in S$, we have $$\begin{aligned}
(\overline{a}+\overline{a}\overline{b}) {\mathcal{L}^+}\overline{a} \, & \Rightarrow \, \overline{a}=\overline{a+ab} \\
& \Rightarrow \, a=a+aba\end{aligned}$$ and so $S \in {\mathbf{LN}}$.
1. ${\mathbf{LN}}= \mathbf{R}^{\bullet} \circ ({\mathbf{LZ}^{+}}\circ {\mathbf{D}})$.
2. ${\mathbf{RN}}= \mathbf{R}^{\bullet} \circ ({\mathbf{RZ}^{+}}\circ {\mathbf{D}})$.
(Revise) Let $S \in {\mathbf{LN}}$. Then by the above lemma, $S/{\mathcal{D}^{\bullet}}\in {\mathbf{LZ}^{+}}\circ {\mathbf{D}}$. Hence ${\mathbf{LN}}\subseteq \mathbf{R}^{\bullet} \circ ({\mathbf{LZ}^{+}}\circ {\mathbf{D}})$. If $S \in \mathbf{R}^{\bullet} \circ ({\mathbf{LZ}^{+}}\circ {\mathbf{D}})$ then there exists a congruence $\rho$ on $S$ such that $S/\rho \in {\mathbf{LZ}^{+}}\circ {\mathbf{D}}$ and each $\rho$-class is in $\mathbf{R}^{\bullet}$. Hence $S/\rho \in {\mathbf{LN}}$, by Lemma \[lp is the least lat cong\], and so $(a+aba) \rho a$ for all $a, b \in S$. Since each $\rho$-class is in $\mathbf{R}^{\bullet}$, it follows that $a=a+aba$. Thus ${\mathbf{LN}}= \mathbf{R}^{\bullet} \circ ({\mathbf{LZ}^{+}}\circ {\mathbf{D}})$.
Let $S_1$ and $S_2$ be two semirings and $D$ a distributive lattice. If there are two homomorphisms $\phi_1 : S_1 \longrightarrow D$ and $\phi_2 : S_2 \longrightarrow D$ onto $D$ then the semiring $S = \{(s_1, s_2) \in S_1 \times S_2 \mid \phi_1(s_1)=\phi_2(s_2)\}$ is called a spined product of the two semirings $S_1$ and $S_2$ with respect to the spine $D$.
We show that every semiring $S \in \mathbf{D}^{\bullet}$ is a spined product of a semiring $S_1 \in \mathbf{L}^{\bullet}$ and a semiring $S_2 \in \mathbf{R}^{\bullet}$. For this first we prove that the multiplicative reduct of each idempotent semiring $S \in \mathbf{D}^{\bullet}$ is a normal band. We do the groundwork by providing a sequence of useful lemmas.
Every idempotent semiring satisfies the following two identities: $$\begin{aligned}
& xyzx \approx xyzx+xyxzx+xyzx \label{regband1}
\\ \textrm{and} \hspace{.5cm} & xyxzx \approx xyxzx+xyzx+xyxzx. \label{regband2}\end{aligned}$$
Let $S$ be an idempotent semiring. Then ${\mathcal{D}^+}$ is a congruence on $S$ and $S/ {\mathcal{D}^+}\in \mathcal{S}l^+$. Hence the multiplicative reduct $(S/ {\mathcal{D}^+}, \cdot)$ is a regular band [@PZ2005]. Then for all $a, b, c \in S$, $abca {\mathcal{D}^+}abaca$ and so $$\begin{aligned}
& abca=abca+abaca+abca
\\ \textrm{and} \hspace{.5cm} & abaca=abaca+abca+abaca\end{aligned}$$
For an idempotent semiring $S$ the following conditions are equivalent:
1. $S \in \mathbf{D}^{\bullet}$;
2. $S$ satisfies both the identities: $$\begin{aligned}
xz \approx xz+xyz \hspace{.5cm} \textrm{and} \hspace{.5cm} xz \approx xyz+xz;\end{aligned}$$
3. $S$ satisfies the identity: $$\label{xz=xyz+xz+xyz}
xz \approx xyz+xz+xyz.$$
It is clear that ${\textbf (2) \Rightarrow (3)}$ and ${\textbf (3) \Rightarrow (1)}$. Assume that (1) holds. Then $xz\approx (xyzx+x+xyzx)(zxyz+z+zxyz)\approx xyzxzxyz+xyzxz+xyzxzxyz+xzxyz+xz+xzxyz+xyzxzxyz+xyzxz+xyzxzxyz\approx xyz+xyzxz+xyzxzxyz+xzxyz+xz+xzxyz+xyzxzxyz+xyzxz+xyz$. This implies that $xz \approx xz+xyz$ and $xz \approx xyz+xz$.
If $S\in{\mathbf{D}^{\bullet}}$ then $S$ satisfies the identity: $$xyzx\approx xzyx+xyzx+xzyx. \label{nbd primary}$$
Let $S \in {\mathbf{D}^{\bullet}}$. Then for $x,y,z \in S$, $$\begin{aligned}
xyzx & \approx (xzy+xy+xzy)zx \\
& \approx xzyzx+xyzx+xzyzx \\
& \approx xzy(zyx+zx+zyx)+xyzx+xzy(zyx+zx+zyx) \\
& \approx xzyx+xzyzx+xzyx+xyzx+xzyx+xzyzx+xzyx,\end{aligned}$$ which implies that $xyzx+xzyx \approx xyzx$ and hence $xzyx+xyzx+xzyx \approx xyzx$.
For $S \in {\mathbf{D}^{\bullet}}$ the multiplicative reduct $(S,\cdot)$ is a normal band.
Let $S \in {\mathbf{D}^{\bullet}}$ and $x,y,z,\in S$. Then $xyzx \approx xzyx+xyzx+xzyx$. Also we have $$\begin{aligned}
& xzyx \approx xzyx+xzyx+xzyx \\
\Rightarrow \;& xzyx \approx xzyx+xyzx+xzyx+xyzx+xzyx \\
\Rightarrow \;& xzyx+xyzx+xzyx \approx xzyx \\
\Rightarrow \;& xyzx \approx xzyx.\end{aligned}$$
For every idempotent $S$ the following conditions are equivalent:
- $S \in {\mathbf{LNB}^{\bullet}}\cap {\mathbf{D}^{\bullet}}$;
- $S$ satisfies the identity: $$xz\approx xzy+xz+xzy;$$
- $S \in \mathbf{L}^{\bullet}$.
Equivalence of $(2)$ and $(3)$ is trivial.\
**$(1) \Rightarrow (2)$ :** Assume that $S \in {\mathbf{LNB}^{\bullet}}\cap {\mathbf{D}^{\bullet}}$. Then we have $$xz \approx xyz+xz+xyz.$$ Since $S \in {\mathbf{LNB}^{\bullet}}$, $$xz \approx xzy+xz+xzy.$$ **$(2) \Rightarrow (1)$ :** Let $x,y,z \in S$. Then $xz \approx xzy+xz+xzy$ and $xz \approx xyz+xz+xyz$. Also we have $$\begin{aligned}
& xyz \approx xyzy+xyz+xyzy \\
\Rightarrow \; & xyz \approx (xzy+xy+xzy)zy+xyz+(xzy+xy+xzy)zy \\
\Rightarrow \; & xyz \approx xzy+xyzy+xzy+xyz+xzy+xyzy+xzy \\
\Rightarrow \; & xzy+xyz+xzy \approx xyz.\end{aligned}$$ Now $xzy \approx xzy+xzy+xzy$ implies that $$\begin{aligned}
& xzy \approx xzy+xyzy+xzy+xyzy+xzy \\
\Rightarrow \; & xzy \approx xzy+xyzy+xzy \\
\Rightarrow \; & xzy \approx xzy+xyzyz+xyzy+xyzyz+xzy \\
\Rightarrow \; & xzy \approx xzy+xyz+xzy.\end{aligned}$$ Thus we get $xyz \approx xzy$ and so $S \in {\mathbf{LNB}^{\bullet}}$. Also it follows that $S \in {\mathbf{D}^{\bullet}}$.
Thus the multiplicative reduct $(S, \cdot)$ of every idempotent semiring $S \in \mathbf{L}^{\bullet}$ is a left normal band. Similarly, the multiplicative reduct $(S, \cdot)$ of every idempotent semiring $S \in \mathbf{R}^{\bullet}$ is a right normal band.
In $\mathbf{D}^{\bullet}$ we have the following derivation: $$\begin{aligned}
& x \approx xyx+x+xyx \\
\Rightarrow \, & x \approx x+xyx+x+xyx+x \\
\Rightarrow \, & x \approx x+xyx+x\end{aligned}$$
An idempotent semiring $S\in \mathbf{D}^{\bullet}$ if and only if $S$ is a spined product of an idempotent semiring $S_1\in \mathbf{L}^{\bullet}$ and an idempotent semiring $S_2\in \mathbf{R}^{\bullet}$ with respect to a distributive lattice $D$.
Let $S \in \mathbf{D}^{\bullet}$. Then the multiplicative reduct $(S, \cdot)$ is a regular band. So both ${\mathcal{L}^{\bullet}}$ and ${\mathcal{R}^{\bullet}}$ are congruences on the multiplicative reduct $(S, \cdot)$. Let $a, b, c \in S$ and $a {\mathcal{L}^{\bullet}}b$. Then $a=ab, b=ba$ and $ca {\mathcal{L}^{\bullet}}cb, ac {\mathcal{L}^{\bullet}}bc$. Hence we have, $$\begin{aligned}
(a+c)(b+c) &= ab+ac+cb+c \\
&= a+ac+cbca+c \hspace{2cm} \textrm{(since $ca {\mathcal{L}^{\bullet}}cb$)} \\
&= a+ac+(c+cac+c)bca+c \\
&= a+ac+cbca+cacbca+cbca+c \\
&= a+ac+cbca+ca+cbca+c \hspace{2cm} \textrm{(since $ca {\mathcal{L}^{\bullet}}cb$ and ${\mathcal{L}^{\bullet}}\subseteq {\mathcal{D}^{\bullet}}$)}\\
&= a+ac+(cbc+c+cbc)a+c \\
&= a+ac+ca+c \\
&= (a+c)^2 \\
&= a+c\end{aligned}$$ and similarly $(b+c)(a+c)=b+c$. Thus $(a+c) {\mathcal{L}^{\bullet}}(b+c)$. Similarly $(c+a) {\mathcal{L}^{\bullet}}(c+b)$. Hence ${\mathcal{L}^{\bullet}}$ is a congruence on $S$. Now for any $a, b \in S$, $$\begin{aligned}
& a=aba+a+aba \\
\Rightarrow \, & a=a(ba+a+ba)\end{aligned}$$ implies that $a {\mathcal{L}^{\bullet}}(ba+a+ba)$. Hence $S/{\mathcal{L}^{\bullet}}\in \mathbf{R}^{\bullet}$. Similarly ${\mathcal{R}^{\bullet}}$ is a congruence on $S$ and $S/{\mathcal{R}^{\bullet}}\in \mathbf{L}^{\bullet}$. Denote $S_1=S/{\mathcal{L}^{\bullet}}$ and $S_2=S/{\mathcal{R}^{\bullet}}$. Also $D=S/\mathcal{D}^{\bullet}$ is a distributive lattice. Since ${\mathcal{L}^{\bullet}}$, ${\mathcal{R}^{\bullet}}\subseteq \mathcal{D}^{\bullet}$, it follows that $\phi_1:S_1\longrightarrow D$ and $\phi_2:S_2\longrightarrow D$ defined by $\phi_1(L_a)=D_a$ and $\phi_2(R_a)=D_a$ are well defined surjective homomorphisms, where $L_a \,[ R_a, D_a]$ is the ${\mathcal{L}^{\bullet}}$ \[ ${\mathcal{R}^{\bullet}}$, $\mathcal{D}^{\bullet} ]$ -class containing $a$. Thus $R = \{(L_a, R_a) \in S_1 \times S_2 \mid a \mathrel\mathcal{D}^{\bullet} b \}$ is a spined product of $S_1$ and $S_2$ with respect to $D$.
Then the mapping $\theta : S \longrightarrow R$ defined by $\theta(a)=(L_a, R_a)$ is a monomorphism. Again if $( L_a, R_a) \in R$, then we have $a \mathrel\mathcal{D}^{\bullet} b$. Also since $\mathcal{D}^{\bullet}={\mathcal{L}^{\bullet}}o {\mathcal{R}^{\bullet}}$, there exists $c\in S$ such that $a\mathrel{\mathcal{L}^{\bullet}}c$ and $c \mathrel{\mathcal{R}^{\bullet}}b$. This implies that $\theta(c)=(L_a, R_a)$. Thus $\theta$ is onto.
Thus $ \mathbf{D}^{\bullet} \subseteq \mathbf{L}^{\bullet} \vee \mathbf{R}^{\bullet} $. Also $\mathbf{L}^{\bullet} \subseteq \mathbf{D}^{\bullet}$ and $\mathbf{R}^{\bullet} \subseteq \mathbf{D}^{\bullet}$ implies that $ \mathbf{L}^{\bullet} \vee \mathbf{R}^{\bullet} \subseteq \mathbf{D}^{\bullet}$. Then we have the following corollary.
$ \mathbf{D}^{\bullet} = \mathbf{L}^{\bullet} \vee \mathbf{R}^{\bullet} $.
[**PROBLEMS**]{}
Characterize the class of all idempotent semirings $S$ such that $\sigma$ is the least distributive lattice congruence on $S$.
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[^1]: Corresponding author
|
---
abstract: 'This work presents the subtraction procedure and the Regge cut in the logarithmic Regge pole approach. The subtraction mechanism leads to the same asymptotic behavior as previously obtained in the non-subtraction case. The Regge cut, on the other hand, introduces a clear role to the non-leading contributions for the asymptotic behavior of the total cross section. From these results, one introduces some simple parameterization to fit the experimental data for the proton-proton and antiproton-proton total cross section above some minimum value up to the cosmic-ray. The fit parameters obtained are used to present predictions to the $\rho(s)$-parameter as well as to the elastic slope $B(s)$ at high energies.'
address: 'Universidade Federal de São Carlos, CCTS/DFQM, Sorocaba, São Paulo CEP 18052780, Brazil'
author:
- 'S. D. Campos'
title: Logarithmic Regge Pole
---
Introduction {#sec:intro}
============
The introduction of the complex angular momentum $l$ in the potential scattering gives rise to poles representing bound states or resonances for $l>0$ [@t.regge.nuovo.cim.14.951.1959; @t.regge.nuovo.cim.18.947.1960]. As well-known, the application of Regge’s original ideas to the high energy scattering of elementary particles results in a very successful theory. For example, the broad class of reactions that can be explained in a unified view led Donnachie and Landshoff to conjecture that the Regge theory would be part of the truth of the particle physics [@A.Donnachie.P.V.Landshoff.Phys.Lett.B.296.227.1992]. In other words: if such formalism is wrong, then it is wrong, probably, by some little misunderstanding.
The main entity in the Regge theory is the $\alpha(t)$ trajectory, which is a phenomenological input within the formalism. The location of a pole corresponds to a bound state, and the behavior of the scattering amplitude is controlled by this leading Regge pole. Then, a resonance with spin $j$ turns necessary to include the excitation $j+2$, $j+4$,..., keeping fixed the other quantum numbers. These particles lies on the Regge trajectory, obtained from the so-called Chew-Frautschi plot [@G.F.Chew.S.C.Frautschi.Phys.Rev.Lett.7.394.1961; @G.F.Chew.S.C.Frautschi.Phys.Rev.Lett.8.41.1962]. The Regge trajectory is assumed to be linear for light mesons and baryons [@A.J.G.Hey.R.L.Kelly.Phys.Rep.96.71.1983], whose parameters can be extracted from the scattering data as well as from the particle spectra. However, an important question without an answer is if these trajectories are linear everywhere or only in the asymptotic regime. Moreover, the analyticity and unitarity properties require that $\alpha(t)$ be a non-linear complex-valued function [@A.Degasperis.E.Predazzi.Nuovo.Cimento.A.65.764.1970; @A.I.Bugrij.G.Cohen.Tannoudji.L.L.Jenkovszky.N.A.Kobylinsky.Fortschritte.der.Physik.21.427.1973].
The Pomeranchuk theorem asserts the total cross section of particle-particle and particle-antiparticle at high energies should tend to the same limit [@I.Ia.Pomeranchuk.Sov.Phys.JETP.7.499.1958]. This theorem creates the need for the exchange of a state, with the quantum numbers of the vacuum, that not differs particles from antiparticles in the asymptotic energy regime. This is the role played by the leading Regge pole, also known as pomeron, in the particle scattering at high energies, and the reason supporting its existence is entirely phenomenological. The problem here is that $\alpha(t)$ for the pomeron can only be obtained from the experimental data since it has no particle spectrum. However, if one extrapolates the pomeron trajectory from negative to positive values, then one can predict glueball states with integer spin. The first mention to the pomeron as a pair of gluons in a color singlet is due to Low and Nussinov [@F.E.Low.Phys.Rev.D.12.163.1975; @S.Nussinov.Phys.Rev.Lett.34.1286.1975]. Nonetheless, there is no experimental evidence for the pomeron in present-day energies.
The odderon, the counterpart of the pomeron, can distinguish particles from antiparticles in the convenient energy range. The original idea of the odderon is due to Bouquet *et al.* [@A.Bouquet.B.Diu.E.Leader.B.Nicolescu.Nuovo.Cimento.29A.30.1975] and Joynson *et al.* [@D.Joynson.E.Leader.C.Lopez.B.Nicolescu.Nuovo.Cimento.30A.345.1975] and the proposition of the odderon as the exchange of three reggeized gluons is due to Bartels [@J.Bartels.Nucl.Phys.B175.365.1980], followed close by Refs. [@J.Kwiecinski.M.Praszalowicz.Phys.Lett.B94.413.1980; @T; @Jaroszewicz.J.Kwiecinski.Z.Phys.C12.167.1982], being a formal QCD prediction. The general belief is that the odderon produces the pronounced dip in the proton-proton scattering. On the other hand, it fills the dip in the case of the proton-antiproton. Differently from the pomeron, the odderon has a possible experimental evidence [@G.Antchev.etal.TOTEM.Coll.Eur.Phys.J.C79.785.2019] subject to discussions in the literature [@E.Martynov.B.Nicolescu.Phys.Lett.B778.414.2018; @V.A.Khoze.A.D.Martin.M.G.Ryskin.Phys.Lett.B780.352.2018; @A.Szczurek.P.Lebiedowicz.PoS.DIS2019.071.2019].
Since a long time ago, one knows that the Regge theory is valid in the perturbation theory. Indeed, the Regge trajectory also can be obtained from the Bethe-Salpeter equation [@B.W.Lee.R.F.Sawyer.Phys.Rev.127.2266.1962]. The BFKL equation also result in the pomeron [@V.S.Fadin.E.A.Kuraev.L.N.Lipatov.Sov.Phys.JETP44.443.1976; @Y.Y.Balitsky.L.N.Lipatov.Sov.J.Nucl.Phys.28.822.1978]. This leading Regge pole emerging in the perturbative QCD approach is called hard pomeron, and it is used to describe the behavior at small $x$ (the Bjorken scale) in deep inelastic scattering as well as in diffractive processes. The non-perturbative leading Regge pole, in contra-position, is called soft pomeron. Efforts towards a unified view of both pomeron pictures are being conduced [@J.Bartels.C.Contreras.G.P.Vacca.JHEP.01.004.2019].
However, not everything is a flower in Regge’s garden. The Froissart-Martin (FM) bound, for example, disagree with the rise of the total cross section given by the leading Regge pole [@m.froissart.phys.rev.123.1053.1961; @a.martin.nuovo.cim.42.930.1966]. The Regge theory predicts the total cross section asymptotically behaving as $s^{\alpha(0)-1}$, as $s\rightarrow\infty$. The FM bound predicts, however, a rising bounded by $\ln s^2$. The only way to ensure the validity of the Regge formalism in front of the FM bound is with a trajectory of less than 1. Nonetheless, the fitting procedures for the total cross section always produces $\alpha(0)>1$ [@A.Donnachie.P.V.Landshoff.Phys.Lett.B.296.227.1992; @R.C.Badatya.P.K.Patnaik.Pramana.15.463-474.1980]. The FM bound is a crucial formal result of the high energy physics and cannot be disregarded in any theoretical approach. The analyticity principle is also not be satisfied unless the trajectory of all particles lies on the Regge trajectory.
Recently, obeying the FM bound, a novel approach to the leading Regge pole was obtained by introducing a logarithmic representation for the leading Regge pole [@S.D.Campos.Phys.Scrip.xx.2020]. In the present work, one continues the logarithmic Regge approach introducing the subtraction and the cut problem in the logarithmic Regge framework.
The subtraction procedure, in the present formalism, cannot be used in its *pure version* since the fast decreasing caused by the subtraction $s^{-1}$ turns the approach useless. However, as shall be seen, there is a subtle approximation allowing the use of a less restrictive version of the subtraction mechanism. This approach will lead to a modified subtraction mechanism, whose consequence is that subtraction and non-subtraction cases produce the same functional form to the asymptotic scattering amplitude. The cut, on the other hand, seems to be a result of the sub-leading contributions. Then, this mechanism may be particularly important to explain the mixed energy region where the total cross section, for example, is controlled by the pomeron and odderon exchange.
Using naive parameterizations for the proton-proton and proton-antiproton total cross section, it is possible to understand the role of the pomeron at high energies within the logarithmic Regge approach. As obtained in [@S.D.Campos.Phys.Scrip.xx.2020], the double pomeron picture is favored for energies above 1.0 TeV. On the other hand, as shall be seen, starting the fitting procedures taking into account energies above $25.0$ GeV, then the pomeron assumes values greater than 2 suggesting the saturation of the Froissart-Martin bound. This problem can be solved by using the recent TOTEM measurement of $\rho(s)$ [@G.Antchev.etal.TOTEM.Coll.Eur.Phys.J.C79.785.2019].
Applying the derivative dispersion relation, one obtains the real part of the elastic scattering amplitude. The fitting parameters are taken from the fitting procedures, and the predictions for the $\rho(s)$-parameter are presented. Assuming a null subtraction constant (it interferes only in the low energy experimental data), the curves suggest a double-pomeron exchange to reproduce the $\rho(s)$ obtained in the TOTEM Collaboration. Then, using this constraint, one keeps the double-pomeron trajectory, obtaining a general description of the total cross section obeying the FM bound, and the correct prediction for the $\rho(s)$-parameter at high energies. Hereafter, $\rho(s)=\rho$.
The paper is organized as follows. Section \[sec:fr\] presents the experimental data set and the fitting procedure. In the section \[sec:rgt\], one presents the main results of [@S.D.Campos.Phys.Scrip.xx.2020] and develops the subtracted case as well as the Regge cut case. Section \[sec:dr\] presents a brief discussion about the $\rho$-parameter as well as predictions for the slope of the differential cross section. The critical remarks are the subject of the last section \[sec:critical\].
Experimental Data and Fitting Procedures {#sec:fr}
========================================
The main quantity in the forward elastic scattering is the total cross section, connected through the optical theorem with the imaginary part of the forward elastic scattering amplitude. Bearing this in mind, one considers here the experimental data for the proton-proton ($pp$) and proton-antiproton ($p\bar{p}$) total cross sections, $\sigma_{tot}^{pp}(s)$ and $\sigma_{tot}^{p\bar{p}}(s)$. As usual, $t$ is the squared momentum transfer and $s$ is the squared energy, both in the center-of-mass system.
These $pp$ and $p\bar{p}$ experimental data are used to form a joint data set since the Pomeranchuk theorem asserts they tend to the same limit if $s\rightarrow\infty$. This behavior, predicted to occur only at the asymptotic regime, seems yet to be started from energies above $\sqrt{s}\geq 25.0$ GeV. Figure \[fig:fig\_0\] shows the experimental data used in the fitting procedures for $\sigma_{tot}^{pp}(s)$ and for $\sigma_{tot}^{p\bar{p}}(s)$.
The goal of treating both data set as only one resides on the possibility that the absence of $pp$ experimental data, on some energy range, can be compensated by the existence of $p\bar{p}$ data in this range, and vice-versa. Moreover, there is no data selection in any set considered. The following experimental data set are used throughout this paper.
- The SET 1 is formed by the experimental data for $\sigma_{tot}^{pp}(s)$ and $\sigma_{tot}^{p\bar{p}}(s)$ above $\sqrt{s}=1.0$ TeV up to the cosmic-ray data.
- The SET 2 uses the experimental data for $\sigma_{tot}^{pp}(s)$ and $\sigma_{tot}^{p\bar{p}}(s)$ above $\sqrt{s}=1.0$ TeV, excluding the cosmic-ray experimental data.
- The SET 3 contains experimental data for $\sigma_{tot}^{pp}(s)$ and $\sigma_{tot}^{p\bar{p}}(s)$ above $\sqrt{s_c}$ GeV up to the cosmic-ray data.
- The SET 4 excludes the cosmic-ray data from the SET 3.
The energy cut $\sqrt{s_c}$, corresponds to the energy for which the total cross section achieves its minimum value [@S.D.Campos.C.V.Moraes.V.A.Okorokov.Phys.Scrip.2019]. For the SET 1 and 2, one uses $\sqrt{s_c}=25.0$ GeV. However, as shall be seen, the emergence of a term $\ln\ln(s/s_0)$ in the logarithmic Regge cut implies a change in $\sqrt{s_c}$ to avoid the emergence of negative or complex values. Then, for the SET 3 and 4, one uses $\sqrt{s_c}=15.0$ GeV, which is a consequence of the constraint $\sqrt{s}\geq\sqrt{e s_c}$, where $e$ is the Napier’s constant.
Using the derivative dispersion relations, one obtains the real part of the forward elastic scattering amplitude. Then, it is possible to present the predictions for the $\rho$-parameter based on the fitting results. One uses only the experimental data for $\rho$ above $25.0$ GeV. It is also shown predictions for the slope of the differential cross section.
All the experimental data were collected from Particle Data Group [@PDG-PhysRev-D98-030001-2018]. Moreover, $\sigma_{tot}^{pp}(s)$ at $\sqrt{s}=2.76$ TeV is from [@G_Antchev_TOTEM_; @Coll_Eur_Phys_J_C79_103_2019]. Hereafter, one uses only $\sigma_{tot}(s)$ to refer to both $\sigma_{tot}^{pp}(s)$ and $\sigma_{tot}^{p\bar{p}}(s)$ (the same for $\rho$).
The logarithmic leading Regge Pole {#sec:rgt}
==================================
In the Regge theory, the scattering amplitude is written as an analytic function of the angular momentum $J$. This representation is formulated in the high energy limit $s\rightarrow\infty$, and associates the asymptotic behavior of the scattering amplitude in the $s$-channel to the exchange of one-particle or more, represented in the $t$-channel by the leading Regge poles. One writes the scattering amplitude as $$\begin{aligned}
A(s,t)=\mathrm{Re}A(s,t)+i\mathrm{Im}A(s,t),\end{aligned}$$
and one also assumes that behavior of such a function, at very high energies, is given only by the absorptive part $A(s,t)\approx\mathrm{Im}A(s,t)$. As well-known, in the usual Regge pole formalism, one writes the asymptotic scattering amplitude as $$\begin{aligned}
\label{eq:regge_pole}
A(s,t)\rightarrow (\eta+e^{-i\pi\alpha(t)})\beta(t)(s/s_c)^{\alpha(t)}, ~~~s\rightarrow\infty,\end{aligned}$$
where $\eta=\pm 1$ is the signature related with the crossing symmetry $s\leftrightarrow u$, $\sqrt{s_c}$ some critical energy, and $\beta(t)$ is the residue function of the pole depending only on $t$. Using (\[eq:regge\_pole\]), one can attain the asymptotic form of the differential cross section $$\begin{aligned}
\label{eq:dif_1}
\frac{d\sigma}{dt}\approx (s/s_c)^{2\alpha(t)},\end{aligned}$$
and adopting the normalization $s\sigma_{tot}(s)=\mathrm{Im}A(s,t)\approx A(s,t)$, one has $$\begin{aligned}
\label{eq:totalregge}
\sigma_{tot}(s)\approx (s/s_c)^{\alpha(0)-1}.\end{aligned}$$
The mathematical disagreement between (\[eq:totalregge\]) and the FM bound given below $$\begin{aligned}
\label{eq:froissart}
\sigma_{tot}(s)\leq c\ln(s/s_c)^2,\end{aligned}$$
where $c$ is some real constant, is inevitable for $1<\alpha(0)$.
However, the Regge theory and the FM bound can agree with each other if it is imposed a constraint on the scattering angle as well as a mathematical approximation on the cosine series [@S.D.Campos.Phys.Scrip.xx.2020]. The scattering angle is restricted to the range $0\leq \cos \theta \leq 1$ for $|t|<\!\!< s$. In this case, one writes the following approximation for the cosine of the scattering angle [@S.D.Campos.Phys.Scrip.xx.2020] $$\begin{aligned}
\label{eq:cos_2}
\cos (\theta) = 1+\frac{2t}{s} \approx \ln \left(1+\sqrt{e}\left(1+\frac{2t}{s}\right)\right).\end{aligned}$$
Using the asymptotic properties of the Legendre polynomial and taking into account the approximation (\[eq:cos\_2\]), one writes $$\begin{aligned}
\label{eq:asymp_1}
P_l(s)\rightarrow \ln (s/s_c)^{l}.\end{aligned}$$
Indeed, the above result can be used to write the asymptotic scattering amplitude as a logarithmic leading Regge pole $$\begin{aligned}
\label{eq:asymp_2}
A(s,t)\rightarrow \ln(s/s_c)^{\alpha(t)},\end{aligned}$$
respecting the FM bound.
The very successful model of Donnachie and Landshoff [@A.Donnachie.P.V.Landshoff.Phys.Lett.B.296.227.1992], early proposed by Badatya and Patnaik [@R.C.Badatya.P.K.Patnaik.Pramana.15.463-474.1980], describes the hadronic exchanges remarkably well assuming a simple Regge pole $$\begin{aligned}
\label{eq:dl_1}
A(s,t)= C(t)s^{1.08+0.25t},\end{aligned}$$
where $C(t)$ is a constant depending only on $t$. The corresponding $\sigma_{tot}(s)$ of such parameterization is written using the optical theorem as $$\begin{aligned}
\label{eq:sigma_tot_1}
\sigma_{tot}(s)=C(0)s^{0.08},\end{aligned}$$
saturating the FM bound. This Regge pole corresponds to one-pole exchange, while the double-pole exchange leads to $\sigma_{tot}(s)\sim \ln(s/s_0)$. The intercept of the Regge pole in this model is defined as a linear function $$\begin{aligned}
\label{eq:intercept_1}
\alpha(t)=\alpha(0)+\alpha' t,\end{aligned}$$
where the intercept takes the value $\alpha(0)=\alpha_{\mathbb{P}}=1.08$ and the slope $\alpha'=0.25$ GeV$^{-2}$.
In the language of physics, this intercept corresponds to a soft pomeron - low momentum transfer - $\alpha_{\mathbb{P}}\approx 1.05\sim1.08$. In contrast, the hard pomeron, predicted to mediate diffractive processes - large momentum transfer - has a value $\alpha_{\mathbb{P}}\approx1.4\sim1.5$. This hard pomeron emerges due to the use of the Regge formalism as an analogy to explain the structure-function $F_2(x,Q^2)$ in terms of the Bjorken scale $x$ and the photon virtuality $Q^2$.
Note that both, the one-pomeron exchange in the Donnachie and Landshoff model or the double-pomeron exchange in the logarithmic Regge pole, leads to intercepts $\gtrsim 1$ [@A.Donnachie.P.V.Landshoff.Phys.Lett.B.296.227.1992; @S.D.Campos.Phys.Scrip.xx.2020]. However, the one-pomeron exchange with an intercept slightly above 1 violates the FM bound while for the double-pomeron exchange, this violation only occurs for an intercept above 2. If the intercept is equal 2, then the triple-pomeron exchange is favorable, $\sigma_{tot}\approx \ln(s/s_c)^2$. Neither the soft nor the hard pomeron has been discovered yet.
Non-subtraction case
--------------------
In [@S.D.Campos.Phys.Scrip.xx.2020], one considers only the non-subtraction case. Then, using the optical theorem one has a simple relation for the asymptotic total cross section $$\begin{aligned}
\label{eq:asymp_3}
\sigma_{tot}(s)\rightarrow \ln(s/s_c)^{\alpha_{\mathbb{P}}}.\end{aligned}$$
In the specified range for $\cos(\theta)$, it respects the FM bound if $\alpha_{\mathbb{P}}\leq 2$, providing a physical relation between the pomeron intercept, $\alpha_{\mathbb{P}}$, and the saturation of this bound. The soft pomeron, if it exists, is the particle allowing the maximum growth to the total cross section, obeying the FM bound. As shall be seen, the phenomenology associated with the $\rho$-parameter is crucial to get the correct pomeron intercept.
Using a simple parameterization for the total cross section $$\begin{aligned}
\label{eq:sig_tot_1}
\sigma_{tot}(s)=\beta\ln(s/s_c)^{\alpha_{\mathbb{P}}},\end{aligned}$$
where $\beta$ and $\alpha_{\mathbb{P}}$ are free fitting parameters, one can attain the pomeron intercept. Using the SET 1, one obtains the values for the fitting parameters shown in the first line of the Table \[tab:table\_1\]. Figure \[fig:fig\_1\]a shows the curve obtained from the fitting procedure using (\[eq:sig\_tot\_1\]). The intercept agrees with a double-pole pomeron exchange. The fitting results using the SET 2 are shown in the second line of the Table \[tab:table\_1\]. From the statistical point of view, the absence of the cosmic-ray data in the SET 2 practically does not alter the results.
SET $\alpha_{\mathbb{P}}$ $\beta$ (mb) $\chi^2/ndf$
------- --------------------------- ------------------- ---------------
1 1.05$\pm$0.05 7.54$\pm$0.92 1.26
2 1.04$\pm$0.05 7.72$\pm$0.98 1.31
: \[tab:table\_1\]Parameters obtained by using (\[eq:sig\_tot\_1\]) in the fitting procedures for the SET 1 and 2 and taking $\sqrt{s_c}=25.0$ GeV.
Subtraction case
----------------
The total cross section with one subtraction can be written using the following normalization of the optical theorem $$\begin{aligned}
\label{eq:sub_1}
s\sigma_{tot}(s)=\mathrm{Im}A(s).\end{aligned}$$
This normalization implies in the logarithmic leading Regge pole $$\begin{aligned}
\label{eq:sub_2}
\sigma_{tot}(s)\approx \frac{\ln(s/s_c)^{\alpha_{\mathbb{P}}}}{s}.\end{aligned}$$
First of all, one notices that to tame the fast decreasing of the total cross section entailed by the subtraction, it is necessary a $\alpha_{\mathbb{P}}$ far above from the expected saturation of the FM bound, $\alpha_{\mathbb{P}}\rightarrow 2$. Indeed, to give some physical meaning for $\alpha_{\mathbb{P}}$ is now a hard task. The effect of using $s$ in the above result can be observed by adopting the parameterization given below $$\begin{aligned}
\label{eq:sig_tot_2}
\sigma_{tot}(s)=\beta\frac{\ln(s/s_c)^{\alpha_{\mathbb{P}}}}{s}.\end{aligned}$$
The fitting results obtained by using the parameterization (\[eq:sig\_tot\_2\]) to the SET 1 and 2 are shown in the Table \[tab:table\_2\]. Of course, the subtracted case cannot be used as a realistic parameterization to describe the $\sigma_{tot}(s)$. Figure \[fig:fig\_1\]b shows the curve obtained from the fitting procedure using (\[eq:sig\_tot\_2\]).
SET $\alpha_{\mathbb{P}}$ $\beta$ (mb) $\chi^2/ndf$
------- --------------------------- ----------------------------------------------- ---------------
1 11.02$\pm$0.09 1.99$\times 10^{-5}\pm 4.9\times 10^{-6}$ 5.14
2 10.99$\pm$0.10 2.15$\times 10^{-5}\pm 5.7\times 10^{-6}$ 5.89
: \[tab:table\_2\]Fitting parameters obtained by using (\[eq:sig\_tot\_2\]) in the fitting procedure to the SET 1 and 2 and taking $\sqrt{s_c}=25.0$ GeV.
However, if we release the subtraction mechanism by introducing the $\delta$- index as a measurement of the deviation of the non-subtraction to the subtraction case, then one writes the parameterization $$\begin{aligned}
\label{eq:sub_3}
\sigma_{tot}(s)= \beta\frac{\ln(s/s_c)^{\alpha_{\mathbb{P}}}}{s^\delta}.\end{aligned}$$
Using the SET 1 and 2, one obtains very small values for the $\delta$-index (near zero). In Table \[tab:table\_3\] are displayed the fitting parameters. The $\delta$-index introduces an error in the fitting parameters higher than the non-subtraction case. However, the central value of each parameter is practically the same. The fitting result is shown in Figure \[fig:fig\_2\]a.
SET $\alpha_{\mathbb{P}}$ $\beta$ (mb) $\delta$ $\chi^2/ndf$
------- --------------------------- -------------------- ------------------- ----------------
1 1.05$\pm$0.82 7.55$\pm$8.07 0.00$\pm$0.08 1.29
2 1.04$\pm$1.07 7.72$\pm$10.70 0.00$\pm$0.11 1.77
: \[tab:table\_3\]Fitting parameters obtained by using (\[eq:sub\_3\]) in the fitting procedure to the SET 1 and 2 and taking $\sqrt{s_c}=25.0$ GeV.
To circumvent the need for the use of the $\delta$-index, one introduces the approximation [@S.D.Campos.Phys.Scrip.xx.2020] $$\begin{aligned}
\label{eq:app_1}
\frac{1}{(s/s_c)^{\delta}}\approx \frac{1}{a(\ln(s/s_c))^{\epsilon}}, \end{aligned}$$
which is a consequence of the fact that the inequality given below holds for $0<s_c\leq s$ and $0\leq\epsilon\leq\delta\in \mathbb{R}$ $$\begin{aligned}
\label{eq:app_2}
\ln(s/s_c)^\epsilon\leq (s/s_c)^\delta\Rightarrow\frac{1}{(s/s_c)^\delta}\leq\frac{1}{\ln(s/s_c)^\epsilon}.\end{aligned}$$
In particular, for some $0<a\in\mathbb{R}$, the approximation (\[eq:app\_1\]) can be used, resulting in the total cross section $$\begin{aligned}
\label{eq:sub_4}
\sigma_{tot}(s)\approx \beta'\ln(s/s_c)^{\alpha_{\mathbb{P}}'},\end{aligned}$$
where $\alpha_{\mathbb{P}}'=\alpha_{\mathbb{P}}-\epsilon$ and $\beta'=c/a$. The choice of $a$ is not unique, and in general, its value depends on the energy range where the experimental data are being analyzed. However, in the fitting procedure, it is absorbed by $\beta'$. Therefore, the expression (\[eq:sub\_4\]) corresponds to the subtraction case and it is exactly equal to the non-subtraction one given by (\[eq:sig\_tot\_1\]).
The cut case
------------
As well-known, the singularities play a fundamental role in the determination of the divergences of the partial-wave amplitude. Neglecting the signature of the partial-wave amplitude, one may write the singularities as $$\begin{aligned}
\label{eq:sing_1}
A_l(t)\propto\int_z^{\infty}\mathrm{disc}A(s,t)Q_l(z')dz'\end{aligned}$$
where $Q_l(z')$ is the Legendre function of the second kind and $\mathrm{disc}A(s,t)$ is the discontinuity across the $l$-plane cut. The easy way to obtain singularities from (\[eq:sing\_1\]) is to associate the $\mathrm{disc}A(s,t)$ with the Legendre function of first kind, $P_{\alpha_c(t)}(z)$, where $\alpha_c(t)$ is the cut. Then, in the simplest case, $$\begin{aligned}
\mathrm{disc}A(s,t)\propto P_{\alpha_c(t)}(z).\end{aligned}$$
In this picture, one uses the properties of the Legendre functions [@erdelyi_book] $$\begin{aligned}
\label{eq:leg_1}
\int_1^{\infty}P_{\alpha_c(t)}(z)Q_l(z)dz=\frac{1}{(l+1+\alpha_c(t))(l-\alpha_c(t))},\end{aligned}$$
revealing a pole for $l=\alpha_c(t)$ ($l\in\mathbb{R}$).
The Watson-Sommerfeld representation for the partial-wave expansion in the complex angular plane can be written as $$\begin{aligned}
\label{eq:ws_1}
A(s,t)_{s\rightarrow\infty}\propto R_p(s,t)+R_c(s,t)+\mathrm{vanishing}~\mathrm{terms},\end{aligned}$$
where $R_p(s,t)$ and $R_c(s,t)$ stands for the Regge pole and the Regge cut contributions for the scattering amplitude. The logarithmic Regge pole introduced in [@S.D.Campos.Phys.Scrip.xx.2020] gives for the first term in the r.h.s. of (\[eq:ws\_1\]) $$\begin{aligned}
\label{eq:reggepole}
R_p(s,t)\approx P_{\alpha(t)}(\cos(\theta))_{s\rightarrow\infty}^{s>\!>|t|}\longrightarrow \frac{\mathrm{\Gamma}(2l+1)}{\mathrm{\Gamma}^2(l+1)}\ln(s/s_c)^{\alpha(t)}\approx \ln(s/s_c)^{\alpha(t)}.\end{aligned}$$
If a branch point with a cut is encountered during the deformation of the contour in the complex momentum angular plane, then it contributes to the asymptotic behavior of the scattering amplitude. Then, one should take into account the cut contribution. One writes $$\begin{aligned}
\label{eq:cut_1}
R_c(s,t)\propto-\frac{1}{i}\int^{\alpha_{c}(t)}dl(2l+1)\mathrm{disc}A(l,t)\frac{P_l(-z)}{\sin\pi l},\end{aligned}$$
As stated above, the functional form of the discontinuity is not known *a priori*, and there is no phenomenological approach for it. Therefore, the only way to go through this point is by using an ansatz as performed in Ref. [@P.D.B.Collins.book.1977].
Using the property $\mathrm{Im}P_l(z)=-P_l(-z)\sin\pi l$, $z<-1$, in the logarithmic Regge approach, one adopts $\mathrm{disc}A(l,t)=(\alpha_c(t)-l)^{1+\beta(t)}$, obtaining the approximation $$\begin{aligned}
\label{eq:cut_2}
R_c(s,t)\approx-\ln(s/s_c)^{\alpha_c(t)}(\ln\ln(s/s_c))^{-(2+\beta(t))}\end{aligned}$$
where $-2<\beta(t)$ is a function of $t$ only. Taking into account the results (\[eq:reggepole\]) and (\[eq:cut\_2\]), then the asymptotic scattering amplitude is written as $$\begin{aligned}
\label{eq:final}
A(s,t)\approx \ln(s/s_c)^{\alpha(t)} - \ln(s/s_c)^{\alpha_c(t)}\ln\ln(s/s_0)^{-(2+\beta(t))}.\end{aligned}$$
Of course, the above result is strongly dependent on the form of $\mathrm{disc}A(l,t)$ and there is no information about $\beta(t)$. The Regge cut $\alpha_c(t)$ is also an unknown function of $t$. One suppose here, that $\alpha_c(t)$ and $\beta(t)$ are a real-valued functions as well as they assume finite values at $t=0$.
It is important to stress that $\sigma_{tot}(s)$, independently of the functional form of $\mathrm{disc}A(l,t)$, should obey the FM bound. Then, the rise of (\[eq:final\]) is controlled by $\ln(s/s_0)^2$. It is not difficult to see that (hereafter, $\alpha_c(0)=\alpha_c$) $$\begin{aligned}
0\leq 1-\ln(s/s_c)^{\alpha_c-\alpha_{\mathbb{P}}}\ln\ln(s/s_c)^{-(2+\beta(0))}\end{aligned}$$
implies that in the asymptotic regime $s\rightarrow\infty$ $$\begin{aligned}
\frac{\alpha_c-\alpha_{\mathbb{P}}}{2+\beta(0)}\leq 0 \Rightarrow \alpha_c\leq \alpha_{\mathbb{P}}\end{aligned}$$
for $\beta(0)>-2$. The inequality $\alpha_c\leq \alpha_{\mathbb{P}}$ implies the Regge cut is bounded by the Regge pole. Therefore, the Regge cut (\[eq:final\]) may be used to explain the experimental data behavior from the minimum value of the total cross section up to 1.0$\sim$2.0 TeV.
To take into account the Regge pole and the Regge cut contributions for the total cross section, one uses the simple parameterization $$\begin{aligned}
\label{eq:tcs_2}
\sigma_{tot}(s)=\beta_1 \ln(s/s_c)^{\alpha_{\mathbb{P}}}-\beta_2\ln(s/s_c)^{\alpha_c}\ln\ln(s/s_c)^{-(2+\beta(0))},\end{aligned}$$
to fit the SET 3 and 4. The fitting results are displayed in Table \[tab:table\_2\]. The Figure \[fig:fig\_2\]b shows the curve obtained for the parameterization (\[eq:tcs\_2\]). The role of the cut is now clear: it represents the mixed contributions below the logarithmic Regge pole dominance, above 1.0 TeV.
It is important to stress that there is no constraint on the fitting parameters. Then, one allows the pomeron intercept to assume any value necessary to fit the experimental data. The result, in a first glance, shows a pomeron intercept far above from the saturation of the FM bound - a supercritical value. Then, a control mechanism should be imposed to tame the growth of the total cross section as $s\rightarrow\infty$. This control mechanism, as shall be seen, is obtained by looking to the $\rho$-parameter experimental data.
The $\rho$-parameter and slope B(s,t) {#sec:dr}
=====================================
As well-known, the validity of the Cauchy theorem is crucial for the convergence of the integral dispersion relation. This kind of relationship allows the obtaining of the real part of the forward scattering amplitude from the imaginary part. First of all, one writes the scattering amplitude as the sum of the even (+) and odd (-) amplitudes as $$\begin{aligned}
\label{eq:scatt}
A(s,t)=A_+(s,t)\pm A_-(s,t)\end{aligned}$$
where $A_\pm(s,t)=\mathrm{Re}A_\pm(s,t)+i\mathrm{Im}A_\pm(s,t)$ are the crossing even (+) and odd (-) amplitudes. The integral form of the dispersion relations is a consequence of analyticity, unitarity, and crossing properties. In the non-subtraction case, it is simply written as ($t=0$) $$\begin{aligned}
\label{eq:dr_1}
\mathrm{Re}A_+(s)=\frac{2s}{i}P\int_{s_0}^\infty ds' \frac{\mathrm{Im}A_+(s')}{s-s'}\end{aligned}$$
where $P$ is the principal value of Cauchy integral. The convergence of the above integral can be ensured by using the subtraction procedure, i.e. by rewritten the scattering amplitude as $$\begin{aligned}
\label{eq:scatt_2}
\tilde{A}(s)=\left|\frac{A(s)}{s}\right|\end{aligned}$$
which result in the subtraction term $$\begin{aligned}
\mathrm{Re}\tilde{A}_+(s)=K+\frac{2s^2}{i}P\int_{s_0}^\infty ds' \frac{\mathrm{Im}\tilde{A}_+(s')}{s(s^2-s'^2)},\end{aligned}$$
$$\begin{aligned}
\mathrm{Re}\tilde{A}_-(s)=\frac{2s^2}{i}P\int_{s_0}^\infty ds' \frac{\mathrm{Im}\tilde{A}_-(s')}{(s^2-s'^2)},\end{aligned}$$
where $K$ is the subtraction constant. Of course, this method is valid only for a finite number of subtractions [@Y.S.Jin.A.Martin.Phys.Rev.135.B1375.1964]. However, the integral dispersion relations are very restrictive, since to know the value of the real part to a specific value one should know the value of the imaginary part in the whole plane. Then, despite its rigorous formulation, the use of integral dispersion relations is of little interest in the Regge theory.
On the other hand, the derivative dispersion relation can be used in the present case [@J.B.Bronzan.G.L.Kane.U.P.Sukhatme.Phys.Lett.B49.272.1974; @K.Kang.B.Nicolescu.Phys.Rev.D11.2461.1975]. These derivative relations can be written in the first-order approximation for the odd and even amplitudes as [@S.D.Campos.EPJC.47.171.2006] $$\begin{aligned}
\frac{\mathrm{Re}A_+(s,t)}{s}=\frac{K}{s}+\frac{\pi}{2}\frac{d}{d\ln s}\frac{\mathrm{Im}A_+(s)}{s},\end{aligned}$$
$$\begin{aligned}
\frac{\mathrm{Re}A_-(s,t)}{s}=\frac{\pi}{2}\left(1+\frac{d}{d\ln s}\right)\frac{\mathrm{Im}A_-(s)}{s}\end{aligned}$$
Considering (\[eq:sig\_tot\_1\]) and (\[eq:tcs\_2\]), it is possible to obtain real part of the scattering amplitude in the subtraction as well as in the Regge cut. Then, the real part of the scattering amplitude can be used to define the $\rho$-parameter $$\begin{aligned}
\rho(s)=\frac{\mathrm{Re}A(s)}{\mathrm{Im}A(s)}.\end{aligned}$$
Without loss of generality, one set $K=0$, since the influence of this parameter is restricted to the low energy region, and the main interest here is to understand the asymptotic regime. Then, all parameters were previously obtained from the fits to the total cross section.
In Figure \[fig:fig\_3\] one has the predictions to the $\rho$-parameter. The results from the logarithmic Regge pole (\[eq:sig\_tot\_1\]) are represented by the dotted line and by the dot-dashed line, SET 1 and 2, respectively. These curves represent the pure contribution coming from the double pomeron exchange. Therefore, these results are not able to reproduce the low energy behavior of the $\rho$-parameter. Using the parameterization (\[eq:tcs\_2\]), the solid and dashed lines, respectively, display the Regge cut contribution obtained from the fitting procedures for the SET 3 and 4. These predictions show behavior in the high energy regime not present in the experimental data. There is a fast-rising introduced by the experimental data below $\sqrt{s}=1.0$ TeV, which seems not present in the experimental data above this energy since the fittings for the SET 1 and 2 shown a pomeron intercept $\alpha_{\mathbb{P}}\approx 1.05$.
An attempt to solve this problem can be done by using the recent experimental data for the $\rho$-parameter at $\sqrt{s}=13.0$ TeV. These experimental data suggest a double pomeron intercept taming the rise of the total cross section. Then, taking into account the above discussion, one introduces the constraint $\alpha_{\mathbb{P}}\leq 1$ to reduce the fast-rising of $\sigma_{tot}(s)$ above 1.0 TeV, introduced by (\[eq:tcs\_2\]). The Table \[tab:table\_5\] shows the fitting parameters only for the SET 3. Despite the high value to $\chi^2/ndf$, the fitting parameters seems to be able to reproduce the growth of $\sigma_{tot}(s)$ as $s\rightarrow\infty$.
The Figure \[fig:fig\_4\]a shows the fitting results for $\sigma_{tot}(s)$ using the parameterization (\[eq:tcs\_2\]) under the double-pomeron constraint. In Figure \[fig:fig\_4\]b, one shows the prediction for the $\rho$-parameter. Despite this naive parameterization, one can observe that the only way to reproduce the high energy behavior of $\rho$ at $\sqrt{s}=13.0$ TeV is assuming a double pomeron exchange in the logarithmic Regge pole representation.
A possible understanding here is: the fast rise of $\sigma_{tot}(s)$ in the logarithmic Regge approach cannot be analyzed independently of the $\rho$-parameter. In particular, the $\rho$ value at $\sqrt{s}=13.0$ TeV is crucial to determine an upper bound to the pomeron intercept. It should be stressed that it is a feature of the Regge theory: the phenomenology.
Another physical quantity that can be predicted from the fitting procedures performed here is the (forward) slope of the elastic differential cross section $d\sigma/dt$, defined as $$\begin{aligned}
B(s,t\rightarrow 0)=\frac{d}{dt}\left(\ln{\frac{d\sigma}{dt}}\right)_{t=0}.\end{aligned}$$
In the present approach, using the simple asymptotic parameterization (\[eq:sig\_tot\_1\]), for example, one obtains for the forward case $$\begin{aligned}
\label{eq:slope}
B(s)=B_0+2\alpha'\ln(\ln s/s_0)\end{aligned}$$
Using the TOTEM result $B_0=19.9\pm0.3$ GeV$^{-2}$ at $\sqrt{s_0}=7.0$ TeV [@G.Antchev.etal.TOTEM.Coll.Europhys.Lett.101.21002.2013] and $\alpha'=0.25$ GeV$^{-2}$, one has the prediction for the elastic slope $B(s)=20.0\pm0.3$ GeV$^{-2}$ at $\sqrt{s}=13.0$ TeV. As well-known, the LHC result is $B(s)=20.36\pm0.19$ GeV$^{-2}$ [@G_Antchev_TOTEM_; @Coll_Eur_Phys_J_C79_103_2019], in accordance with the value predicted here. On the other hand, starting from the TOTEM result, one cannot reproduce, for example, the unexpected value for the slope encountered by E710 Collaboration, $B(s)=16.98\pm0.25$ GeV$^{-2}$ [@N.A.Amos.etal.E710.Coll.Phys.Lett.B247.124.1990].
The prediction capability of (\[eq:slope\]) is, of course, limited by the constraint $\sqrt{s}\geq\sqrt{e s_0}$. This implies, for example, that the next energy above the LHC at $\sqrt{s}=13.0$ TeV that one can predict the slope is at $\sqrt{s}\approx 22$ TeV (the same for the $\rho$-parameter). In this case, using the LHC result at $\sqrt{s}=13.0$ TeV, one obtains $B(s)=20.38$ GeV$^{-2}$, indicating a very slow increase for the slope. At the same energy, the slope from the usual leading Regge pole is $B(s)=20.88$ GeV$^{-2}$.
On the other hand, if instead (\[eq:sig\_tot\_1\]) one uses the parameterization considering the logarithmic Regge cut (\[eq:tcs\_2\]), then one has for the slope in the forward case $$\begin{aligned}
B(s)=B_0+3[(\alpha'+\alpha_c'(0))\ln(\ln s/s_0)-\beta'(0)\ln(\ln(\ln s/s_0))].\end{aligned}$$
Of course, the factor $\ln(\ln(\ln s/s_0))$ depends strongly on the values of $s$ and $s_0$. For example, for the E710 to TOTEM, $\ln(\ln(\ln s/s_0))=-7.49\times 10^{-4}$, and for the TOTEM to LHC $\ln(\ln(\ln s/s_0))=-1.54$. Therefore, one sets, for the sake of simplicity, $\beta'(0)=0$. Then, one writes for the slope $$\begin{aligned}
B(s)\approx B_0+3(\alpha'+\alpha_c'(0))\ln(\ln s/s_0).\end{aligned}$$
Using the TOTEM result $B_0=19.9\pm0.3$ GeV$^{-2}$ at $\sqrt{s_0}=7.0$ TeV and taking $\alpha_c'(0)=0.25$ GeV$^{-2}$, then one has for the LHC result at $\sqrt{s}=13.0$ TeV, $B(s)=20.22\pm0.3$ GeV$^{-2}$. Using the E710 result and $\alpha_c'(0)=0.25$ GeV$^{-2}$, one has $B(s)=18.47\pm 0.25$ GeV$^{-2}$ for the TOTEM at $\sqrt{s}=7.0$ TeV.
However, setting $\alpha_c'(0)=0.50$ GeV$^{-2}$, then for the LHC at $\sqrt{s}=13.0$ TeV, one gets $B(s)=20.38\pm0.3$ GeV$^{-2}$. Using the E710 at $\sqrt{s}=1.8$ TeV and the TOTEM at $\sqrt{s}=7.0$ TeV, then the slope is $B(7.0 TeV)=19.22\pm0.25$ GeV$^{-2}$. The last result is better than using $\alpha_c'(0)=0.25$ GeV$^{-2}$, but still below the TOTEM result.
On the other hand, assuming $B_0$ and $\alpha_c'(0)$ acts as free parameters, and setting $\alpha'=0.25$ GeV$^{-2}$, then, using only experimental data above 1.0 TeV ($pp$ and $p\bar{p}$), one obtains $\alpha_c'(0)=2.91\pm 0.25$ GeV$^{-2}$ and $B_0=-3.51\pm 1.73$ GeV$^{-2}$.
It is interesting to note that it is a very hard task to obtain the slope $B(s)$ since it is not extracted at $t=0$. In general, the slope is obtained considering very small ranges of $t\approx 0$, which implies Coulomb and Nuclear contributions to the scattering amplitude. For example, for the E710 Collaboration [@N.A.Amos.etal.E710.Coll.Phys.Lett.B247.124.1990], the interval for $t$ is $[0.04, 0.29]$ GeV$^2$ and for the TOTEM Collaboration one has $t\in[0.012, 0.2]$ GeV$^2$ [@G_Antchev_TOTEM_; @Coll_Eur_Phys_J_C79_103_2019]. Then, these experimental values may carry some dependence on the momentum transfer.
Conclusions {#sec:critical}
===========
One obtains here the leading Regge pole with one subtraction as well as the Regge cut, both in the logarithmic representation introduced previously in [@S.D.Campos.Phys.Scrip.xx.2020]. The fitting procedures for the subtraction case led to unrealistic results to $\sigma_{tot}(s)$, i.e. to a decreasing faster than the rise of the experimental data, turning the subtraction a problem in the logarithmic Regge pole.
Trying to understand this problem, one introduces a small parameter, the $\delta$-index, to measure how strong should be the subtraction in the approach performed here. The fitting procedures indicate $\delta$-index value near to zero (a very weak dependence on the subtraction). This result allowed the use of a logarithmic representation for the subtraction. Then, the subtraction and the non-subtraction cases can be described by the same logarithmic Regge pole.
The fitting procedures considering only energies above 1.0 TeV result in a pomeron intercept compatible with the double-pomeron value, $\alpha_{\mathbb{P}}\approx 1.04\sim1.05$. This value corroborates the Double-Logarithmic contributions, where the higher accuracy of calculations, the lower is the intercept, resulting in the best value is given by the intercept close to 1 [@B.I.Ermolaev.S.I.Troyan.Eur.Phys.J.C80.98.2020; @B.I.Ermolaev.S.I.Troyan.Acta.Phys.Pol.B12.979.2019].
The Regge cut in the original Regge formalism does not possess a clear role. However, in the present approach, the logarithmic Regge cut may represent the contributions coming from below the logarithmic Regge pole, $\alpha_{\mathbb{P}}$, when one adopts $\mathrm{disc}A(l,t)=(\alpha_c(t)-l)^{1+\beta(t)}$. Then, it can be used to explain the mixed region ($25.0 ~\mathrm{GeV} \leq \sqrt{s}\leq 1.0 ~ \mathrm{TeV}$), where the odderon and the pomeron compete as the leading contribution to the total cross section.
Then, one expects here that logarithmic Regge cut can describe the total cross section for $pp$ and $p\bar{p}$, from the minimum of the total cross section up to 1.0 TeV. However, assuming the parameters coming from the Regge cut can act as free fitting parameters, then they cause a supercritical value to the pomeron intercept, leading to the saturation of the FM bound. Notwithstanding, the dominance of the pomeron as the leading contribution at LHC energy seems to be an experimental fact [@L.Jenkovszky.R.Schicker.I.Szanyi.Int.J.Mod.Phys.E27.1830005.2018]. It is important to stress that the fitting procedures for the SET 1 and 2, using the parameterization (\[eq:sig\_tot\_1\]), furnish an important constraint on the pomeron intercept $\alpha_{\mathbb{P}}$. Moreover, the experimental data at the cosmic-ray energies seem to have a little influence on the pomeron intercept.
To solve this problem, it is necessary to use the experimental data for the $\rho$-parameter. In particular, one should use the experimental result at $\sqrt{s}=13.0$ TeV. This value seems to impose a double pomeron exchange, resulting in an intercept $\alpha_{\mathbb{P}}\leq 1$. When this experimental fact is used, then $\sigma_{tot}(s)$ rises below the saturation of FM bound. Of course, as stated in [@G.Antchev.etal.TOTEM.Coll.Eur.Phys.J.C79.785.2019], the values for the $\rho$-parameter at $\sqrt{s}=13.0$ TeV excluded all the models classified and published by COMPETE. Therefore, the slowing down of the $\sigma_{tot}(s)$ seems to be given by the $\rho$-parameter at $\sqrt{s}=13.0$ TeV.
The slope of the differential cross section can also be predicted by the logarithmic Regge pole. Using the slope obtained by the TOTEM Collaboration, one predicts $B(\sqrt{s}=13.0~ \mathrm{TeV})=20.0\pm0.3$ GeV$^{-2}$, which is in accordance with the result obtained by the LHC, $B(\sqrt{s}=13.0~ \mathrm{TeV})=20.36\pm0.19$ GeV$^{-2}$.
In the logarithmic Regge pole approach presented here, the Regge cut seems to has a clear role: it is responsible by the mixed region ($25.0~ \mathrm{GeV}\lesssim \sqrt{s}\lesssim 1.0~\mathrm{TeV}$) where the total cross section can be described by the odderon and the pomeron contributions. However, this result is strongly dependent on the discontinuities of the scattering amplitude. Unfortunately, there is no theoretical nor phenomenological information about discontinuity of $A(l,t)$.
Acknowledgments {#acknowledgments .unnumbered}
===============
SDC thanks to UFSCar by the financial support.
References {#references .unnumbered}
==========
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|
---
author:
- |
V.K. Kharchenko, Jaime Keller and S. Rodríguez-Romo $^{\ast}$\
Centro de Investigaciones Teóricas,\
Universidad Nacional Autónoma de México, Campus Cuautitlán\
Apdo. Postal 95, Cuautitlán Izcalli, Edo. de México, 54768 México
title: 'Actions of $GL_q(2,C)$ on $C(1,3)$ and its four dimensional representations'
---
[^1]
0.6cm [ A complete classification is given of all inner actions on the Clifford algebra $C(1,3)$ defined by representations of the quantum group $GL_q(2,C),$ $q^m\neq 1$ with nonzero perturbations. As a consequence of this classification it is shown that the space of invariants of every $GL_q(2,C)$-action of this type, which is not an action of $SL_q(2,C),$ is generated by 1 and the value of the quantum determinant for the given representation.]{}
Introduction
============
The development of the quantum group theory and its applications in mathematical physics (more exactly an interpretation of some results using physical intuitions) give us the hope that the notions of quantum actions, quantum invariants and quantum symmetries could play an important role in the “quantum" mathematics as the classical notions of invariants and symmetry do in classical mathematics and theoretical physics. In this paper we study in details actions and invariants of the quantum group $GL_q(2,C)$ on the space-time Clifford algebra $C(1,3),$ which is generated by vectors $\gamma _{\mu }, \mu =0,1,2,3$ with relations defined by the form $g_{\mu \nu}=(1,-1,-1,-1).$ This algebra has 16 matrix units $$\vbox{\halign{# \hfil &\ # \hfil \cr
$e_{11}=(1+\gamma_0)(1+i\gamma_{12})/4,$&
$e_{21}=(1+\gamma_0)(i\gamma_2-\gamma_1)\gamma_3/4,$\cr
$e_{31}=(1-\gamma_0)(1+i\gamma_{12})\gamma_3/4,$ &
$e_{41}=(1-\gamma_0)(\gamma_1-i\gamma_2)/4,\nonumber $\cr
$e_{12}=(1+\gamma_0)(\gamma_1+i\gamma_2)\gamma_3/4,$ &
$e_{22}=(1+\gamma_0)(1-i\gamma_{12})/4,\nonumber $\cr
$e_{32}=(1-\gamma_0)(\gamma_1+i\gamma_2)/4,$ &
$e_{42}=(1-\gamma_0)(i\gamma_{12}-1)\gamma_3/4,\nonumber $\cr
$e_{13}=-(1+\gamma_0)(1+i\gamma_{12})\gamma_3/4,$ &
$e_{23}=-(1+\gamma_0)(\gamma_1-i\gamma_2)/4,\nonumber $\cr
$e_{33}=(1-\gamma_0)(1+i\gamma_{12})/4,$ &
$e_{43}=(1-\gamma_0)(-\gamma_1+i\gamma_2)\gamma_3/4,\nonumber $\cr
$e_{14}=-(1+\gamma_0)(\gamma_1+i\gamma_2)/4,$ &
$e_{24}=(1+\gamma_0)(1-i\gamma_{12})\gamma_3/4,\nonumber $\cr
$e_{34}=(1-\gamma_0)(\gamma_1+i\gamma_2)\gamma_3/4,$ &
$e_{44}=(1-\gamma_0)(1-i\gamma_{12})/4.\nonumber $\cr }}
\label{matrix}$$ and therefore it is abstractly isomorphic to the algebra of 4 by 4 complex matrices.
The quantum group ${\it GL}_q(2,C)$ is generated by four, so-called, $q$-spinors; therefore the classification of representations of the $q$-spinor in $C(1,3)$ given in Theorem \[t2\] produces a basic tool for the investigation of the inner actions defined by representations of the algebraic structure of ${\it GL}_q(2,C)$. In the fourth section we find a necessary and sufficient condition for different representations to define equal or equivalent actions. Using this result in Sections 6, 7 we have obtained a complete classification of all inner actions defined by representations with nonzero “perturbations" (i.e. when two main $q$-spinors do not commute). Results are summarized in the Table given in Section 8. As a consequence of this classification we have shown that the space of quantum invariants of every $GL_q(2,C)$-action of this type, which is not an action of $SL_q(2,C),$ is generated by 1 and the value of the quantum determinant. Roughly speaking, it means that in this case the quantum determinants are the only quantum invariants.
We believe that the results of this classification, besides possible interpretations with physical intuition, could be usefull as a starting material for investigations of actions of more complicated quantum groups on another natural classical objects.
Preliminary notions
===================
The quantum group $GL_q(2,C)$ is a Hopf algebra whose algebraic structure is generated by five elements $a_{11}, a_{12}, a_{21}, a_{22}, d^{-1}$ which can be presented by the following diagram: $$\begin{picture}(80,80)
\put(15.29,2.39){\makebox(0,0)[cc]{$a_{21}$}}
\put(60.0,2.59){\makebox(0,0)[cc]{$a_{22}$}}
\put(60.0,45.46){\makebox(0,0)[cc]{$a_{12}$}}
\put(15.29,45.46){\makebox(0,0)[cc]{$a_{11}$}}
\put(22,45.46){\vector(1,0){28}}
\put(22,2.39){\vector(1,0){28}}
\put(15.29,38){\vector(0,-1){28}}
\put(58.32,38){\vector(0,-1){28}}
\put(20.0,8.0){\line(1,1){30}}
\put(25.0,33.0){$\cdot$}
\put(30.0,28.0){$\cdot$}
\put(35.0,23.0){$\cdot$}
\put(40.0,18.0){$\cdot$}
\put(45.0,13.0){$\cdot$}
\put(50.0,8.0){$\cdot$}
\end{picture}
\label{p1}$$ where by the arrows $x\rightarrow y$ are denoted the so-called $q$-spinors $xy=qyx$, by the straight line commuting elements and by dots, elements with nontrivial commutator $[a_{11}a_{22}]=(q-q^{-1})a_{12}a_{21}$ (for another quantum deformations of $GL_n$ see [@Art],[@Dem] [@Dip], [@Wor]). The comultiplication and the counit are defined as follows: $
\Delta(a_{ij})=\sum_{k=1}^{k=2}a_{ik}\otimes a_{kj},$ $\ \varepsilon (a_{ij})=\delta ^j_i.$ The antipode is given by the following formula $$S\left(\matrix{a_{11}&a_{12}\cr a_{21}&a_{22}\cr}\right)=
d^{-1}\left(\matrix{a_{22}&-q^{-1}a_{12}\cr -qa_{21}&a_{11}\cr}\right).
\label{ant}$$
The quantum group $SL_q(2,C)$ is defined as the factor-Hopf algebra of $GL_q(2,C)$ by the additional relation $d=1.$
An action of Hopf algebra $H$ on an algebra $A$ is characterized by the following two main formulae $$(hg)\cdot v=h\cdot (g\cdot v),
\label{mod}$$ $$h\cdot vw=\sum (h_{(1)}\cdot v)(h_{(2)}\cdot w),
\label{act}$$ where $h, g\in H, \ v, w\in A$ and $\Delta (h)=\sum h_{(1)}\otimes h_{(2)}$ (see details in [@Coh], [@Shn]).
The first formula shows that the action will be defined if it is defined for generators $a_{ij}$ of $GL_q(2,C)$, while the second is showing that for a definition of an action it is enough to set the action of $a_{ij}$ on the generators $\gamma_0$, $\gamma_1$, $\gamma_2$, $\gamma_3$ of the Clifford algebra ${\it C}(1,3)$. Thus, an action is set by formulas of the following type $$a_{ij}\cdot \gamma_k=f_{ijk}(\gamma_0, \gamma_1, \gamma_2, \gamma_3)
\label{fact}$$ where $f_{ijk}$ are some noncommutative polynomials in four variables.
Two actions $*$ and $\cdot $ of a Hopf algebra $H$ on an algebra $A$ are called [*equivalent*]{} if $
h*v=
\left( h\cdot v^{\zeta^{-1}}\right)^{\zeta},
$ where $\zeta$ is an automorphism of the algebra $A.$ By Skolem—Noether theorem every automorphism of $C(1,3)$ is given by a conjugation $w^{\zeta}=uwu^{-1}.$ Therefore equivalence of actions can be presented by the following formula $$a_{ij}*(uwu^{-1})=u(a_{ij}\cdot w)u^{-1}.
\label{fequiv}$$ This formula shows that for an action $*$ to be equivalent to the action $\cdot $ there exists a presentation $$a_{ij}*\gamma'_k=f_{ijk}(\gamma'_0, \gamma'_1, \gamma'_2, \gamma'_3)$$ with the same polynomials $f_{ijk}$ and a system of generators $\gamma'_0$, $\gamma'_1$, $\gamma'_2$, $\gamma'_3$ with the same relations.
Using Skolem—Noether theorem (see also [@Ko], or [@Mo] chapter 6.2.) it is easy to show that for every action there exists an invertible 2 by 2 matrix $M$ over $C(1,3)$ such that $$\left(\matrix{a_{11}\cdot v &a_{12}\cdot v \cr
a_{21}\cdot v &a_{22}\cdot v \cr}\right) =
M\left(\matrix{v&0\cr 0&v\cr }\right)M^{-1}.
\label{sko}$$
Inversely, if $M=\left(\matrix{m_{11}&m_{12}\cr m_{21}&m_{22}\cr }\right)$ is a 2 by 2 matrix with the additional condition that $a_{ij}\rightarrow A_{ij} =
m_{ij}$ represents a homomorphism of the algebraic structure of $GL_q(2,C)$ to $C(1,3),$ then $M$ is invertible and (\[sko\]) defines an action of $GL_q(2,C)$ on $C(1,3)$ (called an [*inner*]{} action).
For a given representation $a_{ij}\rightarrow A_{ij}$ we denote by $\Re $ an [*operator algebra*]{} i.e. a subalgebra of $C(1,3)$ generated by $A_{ij}.$ Recall that the algebra of invariants of an action is defined in the following way $$Inv=\{ v\in A | \forall h\in H \ \ \ h\cdot v=\varepsilon (h)v\} .
\label{inv}$$
. If an action of $GL_q(2,C)$ is given by the formula $({}{sko})$ then the algebra $Inv$ equals the centralizer of all components of $M.$ In particular the algebra $Inv$ for an inner action on $C(1,3)$ defined by a representation $a_{ij}\rightarrow A_{ij}$ equals the centralizer of $\Re $ in $C(1,3).$
[*Proof.*]{} An element $v$ is an invariant if an only if $a_{11}\cdot v=\varepsilon (a_{11})v=v,$ $a_{12}\cdot v=\varepsilon (a_{12})v=0,$$a_{21}\cdot v=\varepsilon (a_{21})v=0,$$a_{22}\cdot v=\varepsilon (a_{22})v=v.$ In matrix form this is equivalent to $diag(v,v)=M\, diag(v,v)M^{-1}.$ Therefore $diag(v,v)M=M\, diag(v,v),$ i.e. $vm_{ij}=m_{ij}v$ for all components $m_{ij}$ of $M.$ $\Box$
$q$-spinor representations
==========================
Let $(x,y)\in A^{2/0}_q$ be a $q$-spinor $xy=qyx$. If $x\rightarrow A$, $y\rightarrow B$ is its representation by $4\times 4$ matrices over complex numbers, then for every invertible $4\times 4$ matrix $u$ and nonzero number $\alpha,$ the map $x\rightarrow uAu^{-1}\alpha$, $y\rightarrow uBu^{-1}\alpha$ also defines a representation of the $q$-spinor. We consider this two representations as [*equivalent*]{} ones. Thus, under investigation of representations of a $q$-spinor, we can suppose that the matrix $A$ has a Jordan Normal form and one of it’s eigenvalues is equal to 1 (if $A\neq 0)$. For a given matrix $A,$ a set $B(A)$ of all matrices $B,$ such that $x\rightarrow A$, $y\rightarrow B$ is a representation of the $q$-spinor, forms a linear space. Therefore, it is natural to represent the space $B(A)$ by one of it’s basis: $\{B_1, B_2, ...\}.$
. Every representation of the $q$-spinor $(q^3, q^4\neq 1)$ by $4\times 4$ complex matrices, $x\rightarrow A$, $y\rightarrow B$, such that $A$ is invertible and $B(A)^2\neq 0,$ is equivalent to one of the following representations \[t2\]
$$1.\ A=diag(q^2, q, 1, 1), \;\;B_1=e_{12},\;B_2=e_{23},\;B_3=e_{24}\hfill
\label{f31}$$
$$2.\ A=diag(q^2, q, q, 1),\;\; B_1=e_{12},\;B_2=e_{13},\;B_3=e_{24},
\;B_4=e_{34}\hfill
\label{f32}$$
$$3. \ A=diag(q^2, q^2, q, 1),\;\; B_1=e_{13},
\;B_2=e_{23},\;B_3=e_{34}\hfill
\label{f33}$$
$$4. \ A=diag(q^3, q^2, q, 1),\;\;B_1=e_{12},\;B_2=e_{23},\;B_3=e_{34}\hfill
\label{f34}$$
$$5. \ A=diag(\alpha, q^2, q, 1),\;\;B_1=e_{23},\;B_2=e_{34},\
\alpha \neq 0, q^{-1}, 1, q, q^2, q^3\hfill
\label{f35}$$
$$6. \ A=diag(q^2, q^2, q, 1)+e_{12},\;\;B_1=e_{13},\;B_2=e_{34}\hfill
\label{f36}$$
$$7. \ A=diag(q^2, q, 1, 1)+e_{34},\;\;B_1=e_{24},\;B_2=e_{12}\hfill
\label{f37}$$
. Let a matrix $A$ have the form $A$=$\alpha E+U$, where $\alpha\neq 0$, $U$ is a nilpotent $n\times n$-matrix, $E$ is the identity matrix, $E$=diag$(1,1,...,1)$. If $AB$=$qBA$, $q\neq 1$, then $B=0$. \[2\]
[*Proof*]{}. If we denote $[x,y]_q$=$xy-qyx$ then $[A,B]_q=0$; i.e. $$[\alpha E+U,B]_q=\alpha(1-q)B+[U,B]_q.
\label{f40}$$ This formula implies $
[U,B]_q=\alpha(q-1)B.
$ Iterating this equality $m$ times we have $$[U,[U,[...[U,B]_q...]_q]_q=\alpha^m(q-1)^mB. \label{f42}$$ If $U^k=0$ and $m=2k$ then the left hand part of (\[f42\]) is equal to zero. Thus $\alpha^m(q-1)^mB$=$0$ and $B=0$. $\Box$
. If $x\rightarrow A$, $y\rightarrow B$ is a representation of the $q$-spinor and $C_1$, $C_2$ are matrices commuting with $A$, then $x\rightarrow A$, $y\rightarrow C_1BC_2$ is also a representation of the $q$-spinor. \[3\]
[*Proof*]{}. $A\cdot C_1BC_2$=$C_1A\cdot BC_2$=$C_1qBA\cdot C_2$= $qC_1BC_2\cdot A$.$\Box$
. If $A$ is a diagonal matrix then $B(A)$ has a basis consisting of matrix units. \[4\]
[*Proof*]{}. If $B=||\beta_{ij}||\in B(A)$, then $\beta_{ij}e_{ij}$= $e_{ii}Be_{jj}\in B(A)$ (since $e_{ii}A$=$Ae_{ii}$ for each diagonal matrix $A$ and we can use Lemma \[3\]). Thus, for every nonzero $\beta_{ij}$ the matrix unit $e_{ij}$ belongs to $B(A)$.$\Box$
[*Proof of Theorem*]{} \[t2\]. We can assume that the matrix $A$ has a Jordan Normal form and one of it’s eigenvalues is equal to 1. Assume firstly that $A$ is a diagonal matrix; i.e. it has four simplest Jordan Normal blocks: $A$=diag$(\alpha_1, \alpha_2, \alpha_3, \alpha_4)$, $\alpha_4=1$. In this case, by Lemma \[4\], it is enough to describe all matrix units from $B(A)$. We have $$Ae_{ij}=\alpha_ie_{ij}=\alpha_i\alpha^{-1}_j(\alpha_je_{ij})=
\alpha_i\alpha^{-1}_j(e_{ij}A).\label{f43}$$ This formula shows that $e_{ij}\in B(A)$ if and only if $\alpha_i\alpha^{-1}_j=q$; i.e. $$\alpha_i=q\alpha_j.\label{f44}$$ If $B(A)^2\neq 0$, then there exist indices $i, j, k$ such that $e_{ij}$, $e_{jk}\in B(A)$. A conjugation by matrix $T=E-e_{ii}-e_{jj}+e_{ij}+e_{ji}$ changes i’th and j’th entries in any diagonal matrix. Therefore we can suppose that $i=2,j=3,k=4$; i.e. (\[f44\]) implies $A$=diag$(\alpha, q^2, q, 1)$. If $\alpha\neq q^{-1}, 1,q, q^2, q^3$ then $B(A)$=$Ce_{23}+Ce_{34}$ and we have a representation (35). If $\alpha= q^{-1}, 1,q, q^2, q^3$ then we will obtain four possibilities which are equivalent to (\[f34\]), (\[f31\]), (\[f32\]), (\[f33\]) and (\[f34\]), respectively. Now, let us consider the cases such that $A$ has less than four blocks. By Lemma \[2\], $A$ cannot be a simplest Jordan Normal matrix; i.e. it has more than one block. This means that $A$ has a form $
A=diag(a,b),
$ where $a,b$ are either invertible $2\times 2$ matrices in Jordan Normal form or $a$ is an invertible simplest Normal Jordan $3\times 3$ matrix and $b$ is a nonzero complex number (and therefore we can suppose that $b=1$). Let us write $$B=
\left(
\begin{array}{cc}
\alpha & \beta \\
\gamma & \delta
\end{array}
\right)\neq 0,
\label{f46}$$ where $\alpha$, $\beta$, $\gamma$, $\delta$ are, respectively, either $2\times 2$ matrices or $\alpha$ is a $3\times 3$ matrix, $\delta$ is a complex number and $\beta$, $\gamma$, a column and a row, respectively. In both cases we have $$[A,B]_q=
\left(
\begin {array}{cc}
a\alpha-q\alpha a & a\beta-q\beta b \\
b\gamma-q\gamma a & b\delta-q\delta b
\end{array}
\right)=0.
\label{f47}$$ This implies, in particular, that $x\rightarrow a$, $y\rightarrow \alpha$ and $x\rightarrow b$, $y\rightarrow \delta$ are representations of $q$-spinors. Let us consider firstly the case when $a$ is a $3\times 3$ matrix. In this case, by Lemma \[2\], we have $\alpha=0$, $\delta=0$ and also $a\beta=q\beta$, $\gamma=q\gamma a$ (recall that $b=1$). Therefore for a $3\times 3$ matrix $\beta\gamma$ (this is the matrix product of a column by a row), we have $
a(\beta\gamma)=q\beta\gamma=q^2(\beta\gamma)a.
$ Again by Lemma \[2\] we obtain $\beta\gamma=0,$ therefore either $\beta=0$ or $\gamma=0$. Let $\beta=0$. Then $\gamma\neq 0$ and the equality $\gamma=q\gamma a$ can be written in the form $\gamma a$=$q^{-1}\gamma$. It means that $q^{-1}$ is an eigenvalue of $a$ and $\gamma$ is an eigenvector of $a$; i.e.
In this case $B(A)$=$Ce_{43}$ and $B(A)^2$=$0$. $a=q^{-1}E+e_{12}+e_{23},$ $ \gamma =\epsilon e_{43}.$ Let $\gamma=0,$ then $\beta\neq 0$ and the equality $a\beta=q\beta$ shows that $q$ is the eigenvalue of $a$ and $\beta$ is an eigenvector i.e. $a=qE+e_{12}+e_{23},$ $\beta =\epsilon e_{14}$ and $B(A)$=$Ce_{14},$ so $B(A)^2=0$.
Consider now the case when $a, b, \alpha, \beta, \gamma, \delta$ are $2\times 2$ matrices. Let us start with the case when both matrices $a$ and $b$ have a simplest Jordan Normal Form i.e. $a=\epsilon E+e_{12},$ $b=E+e_{34}$ By formula (\[f47\]) and Lemma \[2\] we have $\alpha=\delta=0$. If $$B'=
\left(
\begin{array}{cc}
\alpha' & \beta' \\
\gamma' & \delta'
\end{array}
\right)
\label{f54}$$ is another matrix from $B(A)$; then we also have $\alpha'=\delta'=0$ and by (\[f47\]) we have the relations $
a\beta'=q\beta' b,\;\;\; b\gamma'=q\gamma'a.
$ This relations and (\[f47\]) imply $
a(\beta\gamma')=q\beta b\gamma'=q^2(\beta\gamma')a
$ and by Lemma \[2\], $\beta\gamma'=0$. In the same way $
b(\gamma \beta')=q\gamma a\beta'=q^2(\gamma\beta')b
$ and by Lemma 2, $\gamma\beta'=0$. Now $$BB'=
\left(
\begin{array}{cc}
0 & \beta \\
\gamma & 0
\end{array}
\right)
\left(
\begin{array}{cc}
0 & \beta'\\
\gamma' & 0
\end{array}
\right)=
\left(
\begin{array}{cc}
\beta\gamma' & 0 \\
0 & \gamma\beta'
\end{array}
\right)=0,
\label{f58}$$ therefore $B(A)^2$=$0$.
Suppose now that one of the matrix $a,b$ is a simplest Jordan matrix while another is a diagonal matrix. A conjugation by $T=
\left(
\begin{array}{cc}
0 & E\\
E & 0
\end{array}
\right),$ where $E$ is the identity $2\times 2$ matrix, $E$=diag$(1,1)$, changes $A$=diag$(a,b)$ to diag$(b,a)$, so we can suppose that $
a=\epsilon E+e_{12},\;\;\;b=diag(\mu,1),
$ where $\epsilon, \mu\neq 0$.
By (\[f47\]) $x\rightarrow a$, $y\rightarrow \alpha$ is a representation of the $q$-spinor and Lemma \[2\] implies $\alpha=0$. In the same way $x\rightarrow b$, $y\rightarrow \delta$ is a representation of the $q$-spinor. Therefore if $\mu\neq q$, $q^{-1}$ then $\delta=0$ and we can repeat word by word the proof beginning from formula (\[f54\]) up to (\[f58\]) in order to show that $B(A)^2=0$. If $\mu=q^{-1}$ we can multiply matrices $A$ and $B$ by $q$ and conjugate them by the matrix $
diag(1,1,
\left(
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\right)).
$ We will obtain an equivalent representation with $\mu=q$. Thus, it is enough to consider the case $$a=\epsilon E+e_{12},\;\;\; b=diag(q,1)
\label{f61}$$ and by Lemma \[2\] $$\alpha=0,\;\;\; \delta=ce_{12}, \;\;\;c\in \hbox{\bf C}.
\label{f62}$$
For $\beta=
\left(
\begin{array}{cc}
\beta_{11} & \beta_{12}\\
\beta_{21} & \beta_{22}
\end{array}
\right),$ by (\[f47\]), we have $\alpha \beta =q\beta b\;$, which implies $$(\epsilon-q^2)\beta_{11}=-\beta_{21}, \ \
(\epsilon-q)\beta_{12}=-\beta_{22}
\label{f64}$$ $$(\epsilon -q^2)\beta_{21}=0, \ \ (\epsilon-q)\beta_{22}=0.
\label{f65}$$ If $\epsilon$=$q^2$ then the first equality of (\[f64\]) gives $\beta_{21}=0$, and if $\epsilon\neq q^2$ then the first equality of (\[f65\]) gives $\beta_{21}=0$. Therefore $\beta_{21}=0$ in any case. In the same way $\beta_{22}=0$ and (\[f64\]), (\[f65\]) are equivalent to $$(\epsilon-q^2)\beta_{11}=0 \ \ \ (\epsilon-q)\beta_{12}=0
\label{f66}$$ $$\beta_{21}=0, \ \ \ \beta_{22}=0.
\label{f67}$$ Analogously for the matrix $\gamma=
\left(
\begin{array}{cc}
\gamma_{11} & \gamma_{12} \\
\gamma_{21} & \gamma_{22}
\end{array}
\right)$ we have $b\gamma$=$q\gamma a$; which is equivalent to $$\gamma_{11}=0, \ \ \ (1-\epsilon)\gamma_{12}=0,
\label{f71}$$ $$\gamma_{21}=0, \ \ \ (1-q\epsilon)\gamma_{22}=0.
\label{f72}$$ Now if $\epsilon\neq q^{-1},1,q,q^2$ then by (\[f66\]),(\[f67\]) and (\[f71\]), (\[f72\]) $\beta=\gamma=0$ and the representation has the form $$A=diag\left(
\left(
\begin{array}{cc}
\epsilon & 1 \\
0 & \epsilon\\
\end{array}
\right), q, 1\right),\;\;\; B_1=e_{34}.
\label{f73}$$ This means $B(A)=Ce_{34}$ and $B(A)^2=0$. Finally, let us consider four last possibilities. 1. $\epsilon=q^{-1}$. By (\[f66\]) and (\[f67\]) we have $\beta=0$ and by (\[f71\]) and (\[f72\]), $\gamma=ce_{22}$. Together with (\[f61\]) and (\[f62\]) it follows $$A=diag\left(
\left(
\begin{array}{cc}
q^{-1} & 1 \\
0 & q^{-1}\\
\end{array}
\right),q,1\right), \;\;\;B_1=e_{42} , \;\;\;B_2=e_{34}.
\label{f74}$$ If we multiply $A$ by $q$ and conjugate it by $T$=diag$(1, q^{-1},1,1)$ we will obtain an equivalent representation $A$=$diag(1,1,q^2,q)+e_{12}$, $B_1$=$e_{42}$, $B_{2}$=$e_{34}$. Using conjugations by matrices $E-e_{ii}-e_{jj}+e_{ij}+e_{ji}$ we can change indices with the help of permutation $1\rightarrow 3$, $2\rightarrow 4$, $3\rightarrow 1$, $4\rightarrow 2$. In this case $e_{42}\rightarrow e_{24}$, $e_{34}\rightarrow
e_{12}$ and we have the representation (\[f37\]). 2. $\epsilon=1$. By (\[f66\]) and (\[f67\]) we again have $\beta=0$ and by (\[f71\]) and (\[f72\]), $\gamma=Ce_{12}$. Now relations (\[f61\]) and (\[f62\]) show that the representation has the form $$A=diag\left(
\left(
\begin{array}{cc}
1 & 1 \\
0 & 1\\
\end{array}
\right),q,1\right), \;\;\;B_1=e_{32} \;\;\;B_2=e_{34}.
\label{f75}$$ and therefore $B(A)^2=0$.
3\. $\epsilon=q$. By (\[f71\]) and (\[f72\]) we have $\gamma=0$ and by (\[f66\]) and (\[f67\]) $\beta=ce_{12}$. With (\[f61\]) and (\[f62\]) this implies the representation has the form $$A=diag\left(
\left(
\begin{array}{cc}
q & 1 \\
0 & q\\
\end{array}
\right),q,1\right), \;\;\;B_1=e_{14} , \;\;\;B_2=e_{34}
\label{f76}$$ and again $B(A)^2=0$. 4. $\epsilon=q^2$. By (\[f71\]) and (\[f72\]) we have $\gamma=0$ and equalities (\[f66\]) and (\[f67\]) imply $\beta=ce_{11}$. So the representation has the form $$A=diag\left(
\left(
\begin{array}{cc}
q^2 & 1 \\
0 & q^2\\
\end{array}
\right),q,1\right), \;\;\;B_1=e_{13} , \;\;\;B_2=e_{34}.\label{f77}$$ This representation coincides with (\[f36\]).$\Box$
Note that if $q^3=1$ or $q^4=1,$ then Theorem \[t2\] is not valid. For $q=e^{\frac{2\pi i}{3}}$ and $q=\pm i$ there exist, respectively, three dimensional and four dimensional irreducible representations (see for instance [@Mi]) while for $q^m\neq 1$ all finite dimensional representations of the $q$-spinor are one dimensional (see [@Sm], Chapter 2).
Equivalence of representations
==============================
All irreducible finite dimensional representations of $GL_q(2,C),$ $q^{m}\neq 1$ are one dimensional. This is a folklore fact which can be easily obtained from the Y.S. Soibelman work [@So] or from the FRT-duality [@RTT] and the fact that $U_q(g)$ is pointed (see [@M93b]). We need the following corollary from this fact.
. Every finite dimensional representation of $GL_q(2,C),$ $q^m\neq 1,$ is triangular; i.e. it is equivalent to a representation by triangular matrices $a_{ij}\rightarrow A_{ij}.$ \[c2\]
. For every finite dimensional representation $a_{ij}\rightarrow A_{ij}$ of $GL_q(2,C), q^m\neq 1$, the elements $A_{11}$, $A_{22}$ are invertible, while $A_{12}$, $A_{21}$ are nilpotent. \[c3\]
[*Proof*]{}. From Corollary \[c2\] we can suppose that $A_{ij}$ are triangular matrices. In this case the matrix $$(q-q^{-1})^{-1}(A_{11}A_{22}-A_{22}A_{11})$$ has only zero entries on the main diagonal. This matrix is equal to $A_{12}A_{21}.$ From this follows that the main diagonal of $A_{11}A_{22}$ and that of the invertible matrix $det_q$= $A_{11}A_{12}-qA_{12}A_{21}$ coincide. This means that $A_{11}$ and $A_{22}$ have no zero terms on the main diagonal and therefore they are invertible. $\Box$ Now we are ready to answer the question when inner actions defined by two different representations are equivalent to each other.
. Let $a_{ij}\rightarrow A_{ij}$ and $a_{ij}\rightarrow A'_{ij}$ be two representations of $GL_q(2,C)$ in ${\it C}(1,3)$. Then Hopf algebra actions $$a_{ij}\cdot v=\sum_kA_{ik}vA^*_{kj}
\label{f96}$$ and $$a_{ij}*v=\sum_kA'_{ik}vA'^*_{kj}
\label{f97}$$ are equivalent if and only if $$A'_{11}=uA_{11}u^{-1}\alpha_1,\;
A'_{12}=uA_{12}u^{-1}\alpha_2,\;$$ $$A'_{21}=uA_{21}u^{-1}\alpha_1,\;
A'_{22}=uA_{22}u^{-1}\alpha_2,
\label{f98}$$ for some nonzero complex numbers $\alpha_1$, $\alpha_2$ and invertible $u\in {\it C}(1,3)$. Formulas $({}{f96})$ and $({}{f97})$ define the same action if and only if the elements $A'_{ij}$ are connected with $A_{ij}$ by formulas $({}{f98}),$ with $u=1$. \[t4\]
[*Proof*]{}. First of all, it is easy to see that if $a_{ij}\rightarrow A_{ij}$ is a representation and $A'_{ij}$ is defined by (98), then $a_{ij}\rightarrow A'_{ij}$ is also a representation of $GL_q(2,C)$. In order to show that they define equivalent actions let us present formulas(\[f96\]),(\[f97\]) in matrix form $$\left(
\begin{array}{cc}
a_{11}\cdot v & a_{12}\cdot v\\
a_{21}\cdot v & a_{22}\cdot v
\end{array}
\right)=
A\left(
\begin{array}{cc}
v & 0 \\
0 & v
\end{array}
\right)A^{-1}
\label{f99}$$ and $$\left(
\begin{array}{cc}
a_{11}*v & a_{12}*v \\
a_{21}*v & a_{22}*v
\end{array}
\right)=
A_1\left(
\begin{array}{cc}
v & 0 \\
0 & v
\end{array}
\right)A^{-1}_1,
\label{f100}$$ where by Theorem 1 $$A=
\left(
\begin{array}{cc}
A_{11} & A_{12}\\
A_{21} & A_{22}
\end{array}
\right),\;\;\;
A^{-1}=
\left(
\begin{array}{cc}
A^*_{11} & A^*_{12}\\
A^*_{21} & A^*_{22}
\end{array}
\right)
\label{f101}$$ and $$A_1=
\left(
\begin{array}{cc}
A'_{11} & A'_{12}\\
A'_{21} & A'_{22}
\end{array}
\right)=
\left(
\begin{array}{cc}
uA_{11}u^{-1}\alpha_1 & uA_{12}u^{-1}\alpha_2\\
uA_{21}u^{-1}\alpha_1 & Au_{22}u^{-1}\alpha_2
\end{array}
\right)=
\label{f102}$$ $$\left(
\begin{array}{cc}
u & 0\\
0 & u
\end{array}
\right)A
\left(
\begin{array}{cc}
u^{-1} & 0\\
0 & u^{-1}
\end{array}
\right)
\left(
\begin{array}{cc}
\alpha_1 & 0\\
0 & \alpha_2
\end{array}
\right).$$ From this formula we have $$A^{-1}_1=
\left(
\begin{array}{cc}
\alpha^{-1}_1 & 0\\
0 & \alpha^{-1}_2
\end{array}
\right)
\left(
\begin{array}{cc}
u & 0\\
0 & u
\end{array}
\right)A^{-1}
\left(
\begin{array}{cc}
u^{-1} & 0\\
0 & u^{-1}
\end{array}
\right).
\label{f103}$$ Taking into account that $diag(v,v)$ commute with all $2\times 2$ matrix with complex coefficients (and in particular with $diag(\alpha_1,\alpha_2)$) we have $$A_1
\left(
\begin{array}{cc}
v & 0 \\
0 & v
\end{array}
\right)A^{-1}_1=
\label{f104}$$ $$\left(
\begin{array}{cc}
u & 0 \\
0 & u
\end{array}
\right)A
\left(
\begin{array}{cc}
u^{-1} & 0 \\
0 & u^{-1}
\end{array}
\right)
\left(
\begin{array}{cc}
v & 0 \\
0 & v
\end{array}
\right)
\left(
\begin{array}{cc}
u & 0 \\
0 & u
\end{array}
\right)A^{-1}
\left(
\begin{array}{cc}
u^{-1} & 0 \\
0 & u^{-1}
\end{array}
\right)=$$ $$\left(
\begin{array}{cc}
u & 0 \\
0 & u
\end{array}
\right)A
\left(
\begin{array}{cc}
u^{-1}vu & 0 \\
0 & u^{-1}vu
\end{array}
\right)A^{-1}
\left(
\begin{array}{cc}
u^{-1} & 0 \\
0 & u^{-1}
\end{array}
\right)=$$ $$\left(
\begin{array}{cc}
ua_{11}\cdot(u^{-1}vu)u^{-1} & ua_{12}\cdot(u^{-1}vu)u^{-1}\\
ua_{21}\cdot(u^{-1}vu)u^{-1} & ua_{22}\cdot(u^{-1}vu)u^{-1}
\end{array}
\right).$$ If we set $w=u^{-1}vu,$ then by(\[f100\]) and(\[f104\]) we will get $$a_{ij}*(uwu^{-1})=u(a_{ij}\cdot w)u^{-1}.
\label{f105}$$ The last formula concides with(\[fequiv\]) and therefore the actions are equivalent. If $u=1,$ then(\[f105\]) shows that the actions coincide. Inversely, let $a_{ij}\rightarrow A_{ij}$, $a_{ij}\rightarrow A'_{ij}$ be two representations which define equivalent (or equal) actions. Then we have the relations of type(\[fequiv\]) (with $u=1,$ if the actions coincide). In the matrix from for $v=uwu^{-1}$ this relations are equivalent to $$A_1
\left(
\begin{array}{cc}
v & 0 \\
0 & v
\end{array}
\right)A^{-1}_1=
\label{f106}$$ $$\left(
\begin{array}{cc}
u & 0 \\
0 & u
\end{array}
\right)A
\left(
\begin{array}{cc}
u^{-1}vu & 0 \\
0 & u^{-1}vu
\end{array}
\right)A^{-1}
\left(
\begin{array}{cc}
u^{-1} & 0 \\
0 & u^{-1}
\end{array}
\right)$$ Let us multiply this relation from the left by $$\left(
\begin{array}{cc}
u & 0 \\
0 & u
\end{array}
\right)A^{-1}
\left(
\begin{array}{cc}
u^{-1} & 0 \\
0 & u^{-1}
\end{array}
\right)$$ and from the right by $A_1$. We will get $$\left(
\begin{array}{cc}
u & 0 \\
0 & u
\end{array}
\right)A^{-1}
\left(
\begin{array}{cc}
u^{-1} & 0 \\
0 & u^{-1}
\end{array}
\right)A_1
\left(
\begin{array}{cc}
v & 0 \\
0 & v
\end{array}
\right)=
\label{f107}$$ $$\left(
\begin{array}{cc}
v & 0 \\
0 & v
\end{array}
\right)
\left(
\begin{array}{cc}
u & 0 \\
0 & u
\end{array}
\right)A^{-1}
\left(
\begin{array}{cc}
u^{-1} & 0 \\
0 & u^{-1}
\end{array}
\right)A_1.$$ If we denote by $\alpha_{ij}$ the coefficients of $\left(
\begin{array}{cc}
u & 0 \\
0 & u
\end{array}
\right)A^{-1}
\left(
\begin{array}{cc}
u^{-1} & 0 \\
0 & u^{-1}
\end{array}
\right)A_1$, then this relation reduces to $$\left(
\begin{array}{cc}
\alpha_{11}v & \alpha_{12}v \\
\alpha_{21}v & \alpha_{22}v
\end{array}
\right)=
\left(
\begin{array}{cc}
v\alpha_{11} & v\alpha_{12} \\
v\alpha_{21} & v\alpha_{22}
\end{array}
\right),
\label{f108}$$ i.e. $[\alpha_{ij},v]=0$. As $v$ is an arbitrary element from ${\it C}(1,3)$, the elements $\alpha_{ij}$ belong to the center of ${\it C}(1,3)$. The center of ${\it C}(1,3)$ coincide with the set of all complex numbers , i.e. $\alpha_{ij}\in C$. Thus, we have $$A_1=\left(
\begin{array}{cc}
u^{-1} & 0 \\
0 & u^{-1}
\end{array}
\right)A
\left(
\begin{array}{cc}
u & 0 \\
0 & u
\end{array}
\right)
\left(
\begin{array}{cc}
\alpha_{11} & \alpha_{12} \\
\alpha_{21} & \alpha_{22}
\end{array}
\right)
\label{f109}$$ or in details $$A'_{11}=u^{-1}(A_{11}\alpha_{11}+A_{12}\alpha_{21})u,\;\;
A'_{22}=u^{-1}(A_{21}\alpha_{12}+A_{22}\alpha_{22})u
\label{f110}$$ $$A'_{12}=u^{-1}(A_{11}\alpha_{12}+A_{12}\alpha_{22})u,\;\;
A'_{21}=u^{-1}(A_{21}\alpha_{11}+A_{22}\alpha_{21})u.
\label{f111}$$ We know that $a_{ij}\rightarrow A'_{ij}$ is a representation of $GL_q(2,C)$ and therefore by Corollary 3 the elements $A'_{12}$, $A'_{21}$ are nilpotent. By Corollary 2 we can suppose that $A_{ij}$ are triangular matrices of the form (90) and by Corollary 3 all entries on the main diagonal of $A_{11}$, $A_{22}$ are nonzero, while all entires on the main diagonal of $A_{12}$, $A_{21}$ are zero. Now, if $\alpha_{12}\neq 0$ then the first formula of(\[f111\]) shows that all entries on the main diagonal of $uA'_{12}u^{-1}$ are nonzero, therefore $uA'_{12}u^{-1}$ as well as $A'_{12}$ are invertible. This is impossible as $A'_{12}$ is nilpotent. So $\alpha_{12}=0$. In the same way $\alpha_{21}=0$. Thus, if we denote $\alpha_1=\alpha_{11}$, $\alpha_2=\alpha_{22}$ then(\[f110\]) and(\[f111\]) reduce to(\[f98\]).$\Box$
[**Definition**]{}. [*We call two representations $a_{ij}\rightarrow A_{ij}$, $a_{ij}\rightarrow A'_{ij}$ equivalent if they define equivalent actions of $GL_q(2,C)$; i.e., if relations $({}{f98})$ are satisfied*]{}.\
As a homomorphism $\varphi: GL_q(2,C)\rightarrow {\it C}(1,3)$ defines a four dimensional left module over the algebra $GL_q(2,C)$ and viceversa, Theorem \[t4\] shows that the equivalence of representations means that corresponding modules $V_1$, $V_2$ are related by formula $V_1\simeq V_2\otimes {\it U}$, where ${\it U}$ is any one dimensional module. Indeed, numbers $\alpha_1$, $\alpha_2$ define one dimensional representation $a_{11}\rightarrow \alpha_1$, $a_{12}\rightarrow 0$, $a_{21}\rightarrow 0$, $a_{22}\rightarrow \alpha_2$ and every one dimensional representation $a_{ij}\rightarrow \alpha_{ij}$ is defined by two nonzero numbers $\alpha_{11}$, $\alpha_{22}$, while $\alpha_{12}$=$\alpha_{21}$=$0$ ($\alpha_{11}\alpha_{12}$=$q\alpha_{12}\alpha_{11}$=$q\alpha_{11}\alpha_{12}$, $\alpha_{11}\neq 0$).
Algebra $R_q$ and its representations
=====================================
Let us consider some simplification of the algebraic structure of $GL_q(2,C).$ Let the algebra $R_q$ be generated by the elements $a_{11}, a_{12}, a_{21}, r_{22}$ with the relations between $a_{11}, a_{12}, a_{21}$ defined by (\[p1\]) and the additional relations $$a_{12}r_{22}=qr_{22}a_{12}, \ \ \ a_{21}r_{22}=qr_{22}a_{21},
\label{sp1}$$ $$a_{11}r_{22}=r_{22}a_{11}.
\label{sp2}$$ In fact this means that $R_q$ is defined by the same system of spinors with commutative diagonal: $$\begin{picture}(80,80)
\put(15.29,2.39){\makebox(0,0)[cc]{$a_{21}$}}
\put(60.0,2.59){\makebox(0,0)[cc]{$r_{22}$}}
\put(60.0,45.46){\makebox(0,0)[cc]{$a_{12}$}}
\put(15.29,45.46){\makebox(0,0)[cc]{$a_{11}$}}
\put(22,45.46){\vector(1,0){28}}
\put(22,2.39){\vector(1,0){28}}
\put(15.29,38){\vector(0,-1){28}}
\put(58.32,38){\vector(0,-1){28}}
\put(20.0,8.0){\line(1,1){30}}
\put(25.0,38.0){\line(1,-1){30}}
\end{picture}
\label{p1}$$
. If $a_{ij}\rightarrow A_{ij}$ is a finite dimensional representation of the algebra $GL_q(2,C)$, $q^m\neq 1$, then $$a_{ij}\rightarrow A_{ij}, \;\;
r_{22}\rightarrow R_{22}=A_{22}-A_{12}A^{-1}_{11}A_{21}
\label{repspi}$$ is a representation of $R_q$ with invertible $A_{11}R_{22}$. Inversely, if $a_{ij}\rightarrow A_{ij}$ $(i\neq2$ or $j\neq 2)$, $r_{22}\rightarrow R_{22}$ is a finite dimensional representation of $R_q$, such that $A_{11}$, $R_{22}$ are invertible then $$a_{ij}\rightarrow A_{ij},\;\;
a_{22}\rightarrow A_{22}=R_{22}+A_{12}A^{-1}_{11}A_{21}
\label{repglq}$$ is a representation of algebra $GL_q(2,C)$. In this case $({}{repglq})$ is a representation of $SL_q(2,C)$ iff $R_{22}=A^{-1}_{11}$. \[spin\]
[*Proof*]{}. Let $a_{ij}\rightarrow A_{ij}$ be a finite dimensional representation of $GL_q(2,C)$. Then by Corollary \[c3\] the elements $A_{11}$, $A_{22}$ are invertible, therefore formula (\[repspi\]) is correctly defined. It is easy to see that $$[A_{12}, R_{22}]_q=[A_{21}, R_{22}]_q=[A_{11},R_{22}]=0,
\label{sp3}$$ where we have denoted $[x,y]_q=xy-qyx, \ [x,y]=xy-yx.$
Besides we have $$R_{22}=A_{22}-qA^{-1}_{11}A_{12}A_{21}=
A^{-1}_{11}(A_{11}A_{22}-qA_{12}A_{21})=A^{-1}_{11}det_q,$$ which shows that $R_{22}$ is invertible. Inversely, let $r_{22}\rightarrow R_{22}$, $a_{ij}\rightarrow A_{ij}$; $i\neq 2$ or $j\neq 2$, be a finite dimensional representation of $R_q$ with invertible $A_{11}$, $R_{22}$. Straightforward calculations show that $$[A_{12}, A_{22}]_q=[A_{21}, A_{22}]_q=0,\;\;
[A_{11}, A_{22}]=(q-q^{-1})A_{12}A_{21}.
\label{fo27}$$
We have $$det_q=A_{11}A_{22}-qA_{12}A_{21}=A_{11}(R_{22}+A_{12}A^{-1}_{11}A_{21})-
qA_{12}A_{21}=$$ $$A_{11}R_{22}+qA_{12}A_{21}-
qA_{12}A_{21}=A_{11}R_{22},$$ therefore $det_q$ is invertible. $\Box$
Actions defined by $SL_q(2,C)$-representations
==============================================
For a classification of the algebra representations of $GL_q(2,C)$ in ${\it C}(1,3),$ by Theorem 1, it is enough to describe representations of $R_q$ with invertible $a_{11}$, $r_{22}$. In this case we have two $q$-spinors ($a_{11}, a_{12}$), $(a_{11}, a_{21})$ and two $q^{-1}$-spinors $(r_{22}, a_{12})$, $(a_{22}, a_{21})$ with invertible first terms, such that $a_{11}r_{22}=r_{22}a_{11}$. Note that if $(x,y)$ is a $q$-spinor then $(x^{-1},y)$ is a $q^{-1}$-spinor. Therefore for each of the two representations $(x, y)\rightarrow (A_{11}, A_{12}),$ $(x, y)\rightarrow (A_{11}, A_{21})$ with invertible $A_{11}$ and commuting $A_{12}$, $A_{21}$ we have a representation of the algebra $R_q:$ $$a_{11}\rightarrow A_{11},\;\;
a_{12}\rightarrow A_{12},\;\;
a_{21}\rightarrow A_{21},\;\;
r_{22}\rightarrow A^{-1}_{11}.
\label{repslq}$$ By Theorem \[spin\] this representation defines a representation of $SL_q(2,C)$. In this way, using $q$-spinor representations given in Theorem \[t2\] , we can write a number of $SL_q(2,C)$-representations with nontrivial “perturbation" — see all representations in the Table, which are marked by the letter $S$: $S1, S2a, S2a^{\prime },$ e.t.c.
. Every representation $a_{ij}\rightarrow A_{ij}$ of the algebra $SL_q(2,C), \ q^m\neq 1$ in $C(1,3)$ with nontrivial “perturbation" is equivalent to one of the representations marked by letter $S$ in the Table. Invariants and operator algebras of corresponding inner actions are given in the Table. \[slq\]
[*Proof*]{}. Let $a_{ij}\rightarrow A_{ij}$ be a representation of $ GL_q(2,C)$ and $a_{ij}\rightarrow A_{ij}$, $r_{22}\rightarrow R_{22}$ be the corresponding representation of $R_q$; i.e. $R_{22}$=$A_{22}-qA^{-1}_{11}A_{12}A_{21}$ (see (\[repspi\])). By Theorem 1 we have $R_{22}$=$A^{-1}_{11}$ and therefore $$A_{22}=A^{-1}_{11}(1+qA_{12}A_{21}).
\label{s}$$ We know that $a_{11}\rightarrow A_{11}$, $a_{21}\rightarrow A_{21}$ and $a_{11}\rightarrow A_{11}$, $a_{12}\rightarrow A_{12}$ are two representations of the $q$-spinor with an invertible first term. Recall that for a matrix $A$ we have denoted by $B(A)$ the linear space of all matrices $B$ such that $AB=qBA$. Thus $A_{12}, A_{21}\in B(A_{11})$ and so $B(A_{11})^2\neq 0$. By Theorem \[t2\] we can assume that $A_{11}$ is one of the seven matrices which appear in this theorem. Let us consider the seven cases separately. In this case we have $$A_{12}=\alpha e_{12}+\beta e_{23}+\gamma e_{24},\;\;\;
A_{21}=\alpha_1 e_{12}+\beta_1 e_{23}+\gamma_1 e_{24},
\label{s1.1}$$ therefore $$A_{12}A_{21}=\alpha\beta_1e_{13}+\alpha\gamma_1e_{14}=A_{21}A_{12}=
\alpha_1\beta e_{13}+\alpha_1\gamma e_{14}\neq 0.
\label{s1.2}$$ These relations imply that $$0=\alpha\beta_1-\alpha_1\beta=det
\left(
\begin{array}{cc}
\alpha & \beta \\
\alpha_1 & \beta_1
\end{array}
\right),\;\;
0=\alpha\gamma_1-\alpha_1\gamma=det
\left(
\begin{array}{cc}
\alpha & \gamma \\
\alpha_1 & \gamma_1
\end{array}
\right).
\label{s1.3}$$ The first of equalities (\[s1.3\]) shows that $(\alpha, \beta)$ and $(\alpha_1, \beta_1)$ are linearly dependent vectors and by (\[s1.2\]) we can write $(\alpha_1, \beta_1)$=$\frac{\alpha_1}{\alpha}(\alpha,\beta)$. In the same way $(\alpha_1, \gamma_1)$=$\frac{\alpha_1}{\alpha}(\alpha, \gamma)$. These two relations are equivalent to $$A_{21}=\epsilon A_{12},\;\;\epsilon=\frac{\alpha_1}{\alpha}\neq 0.
\label{s1.4}$$ Let us consider a matrix of the form $
{\it U}=diag(1, d, M),
$ where $M$ is an invertible $2\times 2$ matrix and $d$ a nonzero complex number. This matrix commutes with $A_{11}$ therefore the conjugation by it will not change $A_{11}$, while $A_{12}$ is changed in the following way $${\it U}A_{12}{\it U}^{-1}=
\left(
\begin{array}{ccc}
0 & \alpha d^{-1} & \begin{array}{cc}0\ \ &\ \ 0\end{array} \\
0 & 0 & (d\beta,\alpha\gamma) M^{-1} \\
0 & 0 & \begin{array}{cc}0\ \ &\ \ 0\end{array} \\
0 & 0 & \begin{array}{cc}0\ \ &\ \ 0\end{array}
\end{array}
\right).
\label{s1.7}$$ Let us take $d=\alpha$. Then $(d\beta, \alpha \gamma )$ = $(\alpha\beta, \alpha\gamma)\neq 0$ and there exists an invertible $2\times 2$ matrix $M$ such that $(\alpha\beta, \alpha\gamma)M^{-1}=(1,0)$. If we replace $\epsilon$ by $\alpha$, then we will get the representation $S1$ of the Table. Let us calculate $\Re $ and the invariants of the $SL_q(2,C)$-action defined by this representation. All three degrees of $A_{11}$ are linearly independent, as the Vandermond determinant does not vanish. This means that the algebra $\Re $ contains the elements $e_{11}, e_{22}, e_{33}+e_{44}$. We have $A_{12}$=$e_{12}+e_{23}\in \Re $ and $e_{11}A_{12}$=$e_{12}\in \Re $. Thus $e_{12}$, $e_{23}$, $e_{13}$= $e_{12}e_{23}\in \Re $. The linear space generated by these six elements is a subalgebra given in S1 in the Table, which evidentely is isomorphic to the algebra of triangular $3 \times 3$ matrices $T_3.$ For calculating the invariants we can use Lemma 1.It is easy to see that the centralizer of $\Re $ is equal to the algebra of matrices $\beta E+\gamma e_{44}+\delta e_{43},$ which is isomorphic to the algebra of $2 \times 2$ triangular matrices. In fact in this case we have just two basic nontrivial invariants (see \[matrix\]) $$I_1=(1-\gamma_0)(-\gamma_1+i\gamma_2)\gamma_3,\;\;
I_2=(1-\gamma_0)(1-i\gamma_{12}),
\label{s1.12}$$ while all others are linear combinations of them and of the unit element. By theorem \[t2\] we have $A_{12}$= $\alpha e_{12}+\beta e_{13}+\gamma e_{24}+\delta e_{34}$, $A_{21}$= $\alpha_1 e_{12}+\beta_1 e_{13}+\gamma_1 e_{24}+\delta_1 e_{34}$.Therefore $$A_{12}A_{21}=\left(\alpha\gamma_1+\beta\delta_1\right)e_{14}=A_{21}A_{12}=
\left(\alpha_1\gamma+\beta_1\delta\right)e_{14}\neq 0.
\label{s2.1}$$
From (\[s2.1\]) we have $$\alpha\gamma_1+\beta\delta_1=\alpha_1\gamma+\beta_1\delta\neq 0.
\label{s2.2}$$ The matrix $A_{11}$ commutes with all matrices ${\it U}$=$diag(d, M, 1)$, where $d$ is a nonzero complex number and $M$ is an invertible $2\times 2$ matrix. Conjugation by a matrix ${\it U}$ does not change $A_{11}$, while $A_{12}$, $A_{21}$ are changed by the following formulae $$A_{12}\rightarrow
\left(
\begin{array}{ccc}
0 & (d\alpha , d\beta )M^{-1} & 0 \\
\begin{array}{c} 0 \\ 0\end{array} &
\begin{array}{cc} 0 \ \ &\ \ 0 \\ 0\ \ & \ \ 0\end{array} &
M\left(\begin{array}{c} \gamma \\ \delta \end{array}\right) \\
0 & 0\ \ \ \ \, \ \ 0 & 0
\end{array}
\right);$$ $$\ \ \ A_{21}\rightarrow
\left(
\begin{array}{ccc}
0 & (d\alpha_1 , d\beta_1 )M^{-1} & 0 \\
\begin{array}{c} 0 \\ 0\end{array} &
\begin{array}{cc} 0 \ \ &\ \ 0 \\ 0\ \ &\ \ 0\end{array} &
M\left(\begin{array}{c} \gamma_1 \\ \delta_1\end{array}\right) \\
0 & 0\ \ \, \ \ \ \ 0 & 0
\end{array}
\right).
\label{Suemi}$$ In particular, the matrix $\left(
\begin{array}{cc}
\gamma & \gamma_1 \\
\delta & \delta_1
\end{array}
\right)$ under this conjugation is multiplied by $M$ from the left. For this matrix we have two possibilities. a) $det\left(\begin{array}{cc}
\gamma & \gamma_1 \\
\delta & \delta_1
\end{array}
\right)\neq 0$. In this case this matrix is invertible and we can take $M$ to be its inverse and (\[Suemi\]) reduces to the form $A_{12}\rightarrow d\alpha e_{12}+d\beta e_{13}+e_{24},$$A_{21}\rightarrow d\alpha _1e_{12}+d\beta _1e_{13}+e_{34},$with new parameters $\alpha$, $\beta$, $\alpha_1$, $\beta_1$. Relations (\[s2.2\]) become $\alpha\cdot 0+\beta\cdot 1\neq 0$; i.e. $\beta\neq 0$, and $\beta$=$\alpha_1\cdot 1+\beta_1\cdot 0$=$\alpha_1\neq 0$. If we let $d$=$\beta^{-1}$=$\alpha^{-1}_1,$ then changing $\alpha^{-1}_1\alpha\rightarrow\alpha$, $\alpha^{-1}_1\beta_1\rightarrow\beta ,$ we get the representation $S2a.$ For calculating the algebra $\Re $ we can make the same procedure as in the first case. The first three degrees of $A_{11}$ are linearly independent and so $\Re $ contains $e_{11}$, $e_{22}+e_{33}$, $e_{44}$. Multiplication of $A_{12}$ and $A_{21}$ by $e_{44}$ from the right gives two elements $e_{24}$, $e_{34}$ and also $\alpha e_{12}+e_{13}$, $e_{12}+\beta e_{13}$.
If $\alpha\beta\neq 1$ then these two elements are linearly independent and $e_{12}$, $e_{13}\in \Re $. So $\Re $ consists of matrices of the form given in the Table for representation $S2a$. If $\alpha\beta=1$ then the elements $\alpha e_{12}+e_{13}$ and $e_{12}+\beta e_{13}$ are linearly dependent and all matrices from $\Re $ have $(-_{12}, -_{13})$-entries proportional to $(\alpha , 1)=\alpha (1, \beta )$; i.e. $\Re $ has the form given in the Table for $S2a^{\prime }$. Easy calculations show that in both cases the centralizer of $\Re $ is equal to $1\cdot C$. b) $det\left(
\begin{array}{cc}
\gamma & \gamma_1 \\
\delta & \delta_1
\end{array}
\right)=0$. By the relation (\[s2.1\]) we have $\left(
\begin{array}{c}
\gamma\\
\delta
\end{array}
\right)\neq 0$, $\left(
\begin{array}{c}
\gamma_1\\
\delta_1
\end{array}
\right)\neq 0,$ therefore it is possible to find a matrix $M_1$ such that $$M_1
\left(
\begin{array}{c}
\gamma \\
\delta
\end{array}
\right)=
\left(
\begin{array}{c}
1\\
0
\end{array}
\right);\ \
M_1
\left(
\begin{array}{c}
\gamma_1 \\
\delta_1
\end{array}
\right)=
\left(
\begin{array}{c}
\gamma'_1\\
0
\end{array}
\right),\ \
\gamma'_1\neq 0.
\label{s2.6}$$ Let $M=M_2M_1$, where $M_2$ is an invertible matrix, such that $M_2\left(
\begin{array}{c}
1 \\
0
\end{array}
\right)$= $\left(
\begin{array}{c}
1 \\
0
\end{array}
\right)$; i.e. $
M_2=
\left(
\begin{array}{cc}
1 & u' \\
0 & v
\end{array}
\right),\ \
v\neq 0.
\label{s2.7}
$ In this case formula (\[Suemi\]) takes the form $$A_{12}\rightarrow
\left(
\begin{array}{ccc}
0 & (d\alpha' , d\beta' )M^{-1}_2 & 0 \\
\begin{array}{c} 0 \\ 0\end{array} &
\begin{array}{cc} 0 \ \ &\ \ 0 \\ 0\ \ & \ \ 0\end{array} &
\begin{array}{c} 1 \\ 0 \end{array} \\
0 & 0\ \ \ \ \, \ \ 0 & 0
\end{array}
\right);$$ $$\ \ \ A_{21}\rightarrow
\left(
\begin{array}{ccc}
0 & (d\alpha'_1 , d\beta'_1 )M^{-1}_2 & 0 \\
\begin{array}{c} 0 \\ 0\end{array} &
\begin{array}{cc} 0 \ \ &\ \ 0 \\ 0\ \ &\ \ 0\end{array} &
\begin{array}{c} \gamma'_1 \\ 0\end{array} \\
0 & 0\ \ \, \ \ \ \ 0 & 0
\end{array}
\right),
\label{sue}$$ where $(\alpha', \beta')$=$(\alpha, \beta)M^{-1}_1$, $(\alpha'_1, \beta'_1)$=$(\alpha_1, \beta_1)M^{-1}_1$ and $$M^{-1}_2=
\left(
\begin{array}{cc}
1 & u \\
0 & v^{-1}
\end{array}
\right),\;\; u=-u'v^{-1}.
\label{s2.8}$$ Shortly we can write (\[sue\]) in the form $$A_{12}\rightarrow d\alpha''e_{12}+d\beta''e_{13}+e_{24},\;\;
A_{21}\rightarrow d\alpha''_1e_{12}+d\beta''_1e_{13}+\gamma''_1e_{24},
\label{suem}$$ where $$(\alpha'', \beta'')=(\alpha', \beta')
\left(
\begin{array}{cc}
1 & u \\
0 & v^{-1}
\end{array}
\right)=
(\alpha', \alpha'u+\beta'v^{-1})
\label{s2.9}$$ and $$(\alpha''_1, \beta''_1)=(\alpha'_1, \beta'_1)
\left(
\begin{array}{cc}
1 & u \\
0 & v^{-1}
\end{array}
\right)=
(\alpha'_1, \alpha'_1u+\beta'_1v^{-1}).
\label{s2.10}$$ By using (\[s2.2\]), applied to primed parameters, we have $$\alpha'\gamma_1'+\beta'\cdot 0=\alpha'_1\cdot 1+\beta'_1\cdot 0\neq 0;
\mbox{ i.e.}\;\; \alpha'\neq 0,
\;\;\alpha'\gamma'_1=\alpha'_1.
\label{s2.11}$$ If we suppose $u=-(\alpha')^{-1}\beta'v^{-1}$ then we get $\beta''=0$ and $$\beta''_1=-\alpha'_1(\alpha')^{-1}\beta'v^{-1}+\beta'_1v^{-1}=
(\beta'_1-\gamma'_1\beta')v^{-1}. \label{last}$$ If $\beta'_1\neq \gamma'_1\beta'$ in formula (\[last\]), then we can take $v$= $(\beta'_1-\gamma'_1\beta')(\alpha')^{-1}$. In this case $\beta''_1$=$\alpha'$ and we can let $d$=$(\alpha')^{-1}$ in (\[suem\]) in order to obtain $S2b$ (where by $\alpha$ we mean $(\alpha')^{-1}\alpha''_1$= $(\alpha')^{-1}\alpha'_1=\gamma'_1)$. If $\beta'_1$=$\gamma_1\beta'$ in (\[last\]), then $\beta''_1$=$0$ and (\[suem\]) with $d$=$(\alpha')^{-1}$ is equal to $S2b^{\prime }.$ In this case the algebra $\Re $ contains the elements $e_{11}$, $e_{22}+e_{33}$, $e_{44}$, $e_{12}$, $e_{24}$, $e_{14}$ which form a basis of this algebra. This algebra is isomorphic to $T_3$ (see in the Table an action S2$b^{\prime }$.)
The centralizer of $\Re $ is equal to the set of diagonal matrices of the form $diag(\beta, \beta,\delta,\beta);$ therefore it is isomorphic to the direct sum $C\oplus C.$ It is interesting to note that in this case the algebra $\Re $ is abstractly isomorphic to the algebra $\Re $ for representation $S1$, but they are not conjugate in ${\it C}(1,3)$ because they have nonisomorphic centralizers. Thus, we have that the action of the quantum group $SL_q(2,C)$ defined by the representation $S2b^{\prime }$ has only one basic invariant; i.e. $$I=(1-\gamma_0)(1+i\gamma_{12}).$$
For the representation $S2b$ the algebra $\Re $ contains one more matrix, $e_{13}$, therefore it is a 7-dimensional algebra (see the Table), whose centralizer is equal to $C\cdot 1$ and the corresponding $SL_q(2,C)$-action has only trivial invariants.
[**3. $\bf A_{11}=diag(q^2, q^2, q, 1)$.**]{} By theorem \[t2\] we have $$A_{12}=\alpha e_{13}+\beta e_{23}+\gamma e_{34},\;\;
A_{21}=\alpha_1e_{13}+\beta_1e_{23}+\gamma_1e_{34}\; ;$$ thus $$A_{12}A_{21}=\alpha\gamma_1e_{14}+\beta\gamma_1e_{24}=A_{21}A_{12}=
\alpha_1\gamma e_{14}+\beta_1\gamma e_{24}\neq 0.
\label{s3.1}$$ These relations imply $$0=\alpha\gamma_1-\alpha_1\gamma=det
\left(
\begin{array}{cc}
\alpha & \gamma \\
\alpha_1 & \gamma_1
\end{array}
\right),\;\;
0=\beta\gamma_1-\gamma\beta_1=det
\left(
\begin{array}{cc}
\beta & \gamma \\
\beta_1 & \gamma_1
\end{array}
\right),
\label{s3.2}$$ $$(\alpha , \beta )\neq 0,\;\;
\gamma_1\neq 0,\;\;
(\alpha_1 , \beta_1)\neq 0,\;\;
\gamma\neq 0.
\label{s3.3}$$ From these relations we have $$A_{21}=\epsilon A_{12},\;\;
\epsilon=\gamma_1/\gamma\neq 0.
\label{s3.4}$$ Let us consider a matrix of the form $
{\it U}=diag(M, d, 1),
$ where $M$ is an invertible $2\times 2$ matrix and $d$ is a nonzero complex number. The conjugation by this matrix does not change $A_{11}$ but $A_{12}$ changes in the following way $${\it U}A_{12}{\it U}^{-1}=\left(
\begin{array}{cccc}
\begin{array}{c} 0 \\ 0 \end{array} & \begin{array}{c} 0 \\ 0 \end{array} &
M\left( \begin{array}{c} \alpha d^{-1}\\ \beta d^{-1} \end{array}\right)&
\begin{array}{c} 0 \\ 0 \end{array} \\
0 & 0 & 0 & d\gamma \\
0 & 0 & 0 & 0
\end{array}
\right).
\label{3.suemi}$$ If we take $d=\gamma^{-1}$ then $(\alpha d^{-1}, \beta d^{-1})=(\alpha\gamma , \beta\gamma )$ is a nonzero vector. Thus there exists an invertible matrix $M$ such that $M\left(
\begin{array}{c}
\alpha \gamma \\
\beta \gamma
\end{array}
\right)$=$
\left(
\begin{array}{c}
1 \\
0
\end{array}
\right)$. In this way we have obtained the representation $S3,$ where by $\alpha$ is denoted the parameter $\epsilon ,$ see (\[s3.4\]). In this case the algebra $\Re $ is generated by the elements $e_{11}+e_{22}$, $e_{33}$, $e_{44}$, $e_{13}$, $e_{34}$, $e_{14}$; i.e. it consists of matrices $$\left(
\begin{array}{cccc}
\epsilon & 0 & * & * \\
0 & \epsilon & 0 & 0 \\
0 & 0 & * & * \\
0 & 0 & 0 & *
\end{array}
\right).$$ This algebra is isomorphic to $T_3,$ while its centralizer is the algebra of matrices of the form $\beta E+\gamma e_{22}+\delta e_{12}$ this is isomorphic to the algebra of triangular $2\times 2$ matrices $T_2.$ Now formulae (\[matrix\]) show that the action defined by this representation has the following basic invariants $$I_1=(1+\gamma_1)(1-i\gamma_{12}),
I_2=(1+\gamma_0)(\gamma_1+i\gamma_2)\gamma_3.$$ [**4. $\bf A_{11}=diag(q^3, q^2, q, 1)$.**]{} By theorem \[t2\] we have $$A_{12}=\alpha e_{12}+\beta e_{23}+\gamma e_{34}, \ \
A_{21}=\alpha_1 e_{12}+\beta_1 e_{23}+\gamma_1 e_{34}\; ;$$ thus $$A_{12}A_{21}=
\alpha\beta_1e_{13}+\beta\gamma_1e_{24}=A_{21}A_{12}=\alpha_1\beta
e_{13}+\beta_1\gamma e_{24}\neq 0.
\label{s4.1}$$ These relations imply $$0=\alpha\beta_1-\alpha_1\beta=
det\left(
\begin{array}{cc}
\alpha & \beta \\
\alpha_1 & \beta_1
\end{array}
\right),\;\;
0=\beta\gamma_1-\beta_1\gamma=
det\left(
\begin{array}{cc}
\beta & \gamma \\
\beta_1 & \gamma_1
\end{array}
\right).
\label{s4.2}$$ Therefore the vectors $(\alpha , \beta )$ and $(\alpha_1 , \beta_1 )$ are linearly dependent and so are $(\beta , \gamma )$ and $(\beta_1 , \gamma_1 ).$ By (\[s4.1\]), one of the numbers $\beta , \beta_1$ is nonzero. If, for example, $\beta\neq 0$ then $(\alpha_1 , \beta_1 )=\frac{\beta_1}{\beta}
(\alpha , \beta )$ and $(\beta_1 , \gamma_1 )=\frac{\beta_1}{\beta}
(\beta , \gamma )$ so $$A_{21}=\epsilon A_{12},\;\;\epsilon=\frac{\beta_1}{\beta},
\label{s4.3}$$ where $\epsilon\neq 0$ as $A_{21}\neq 0$. In the same way, if $\beta_1\neq 0$, then we get the following relation $$A_{12}=\epsilon' A_{21},\;\;\epsilon'=\frac{\beta}{\beta_1}\neq 0.$$ Thus, in both cases we have (\[s4.3\]) with $\epsilon\neq 0$ and $\beta$, $\beta_1\neq 0$. If $\alpha$, $\gamma\neq 0$ then the conjugation by a diagonal matrix $diag(1,\alpha,\alpha\beta,\alpha\beta\gamma)$ does not change $A_{11}$, while $A_{12}$ is reduced to $e_{12}+e_{23}+e_{34}$; i.e. we obtain $S4a.$ In this case the algebra $\Re $ contains the elements $e_{11}$, $e_{22}$, $e_{33}$, $e_{44}$ ( as first four powers of $A_{11}$ are linearly independent) and $e_{12}$=$e_{11}A_{12}$, $e_{23}$=$e_{22}A_{12}$, $e_{34}$=$e_{33}A_{13}$ as well as elements $e_{13}$=$e_{12}e_{23}$, $e_{14}$=$e_{13}e_{34}$, $e_{24}$=$e_{23}e_{34}$. This means that $\Re $ contains all triangular $4\times 4$ matrices and it has the maximal possible dimension.
If $\alpha\neq 0$, $\gamma$=$0$ then, the conjugation by a diagonal matrix $diag(1,\alpha,\alpha\beta,1),$ gives us the representation $S4b.$ The algebra $\Re $ is generated by the elements $e_{11}$, $e_{22}$, $e_{33}$, $e_{44}$, $e_{12}$, $e_{23}$, $e_{13}$. So this is a 7-dimensional algebra of matrices (see the Table) which is isomorphic to a direct sum $T_3\oplus C$. The centralizer consists of diagonal matrices $diag(\beta,\beta,\beta,\delta)$ and is isomorphic to $C\oplus C.$ Formulas (\[matrix\]) show that the action defined by this representation have only one basic invariant; i.e. $$I=(1-\gamma_0)(1-i\gamma_{12}).$$
If $\gamma\neq 0$, $\alpha=0,$ then the conjugation by a diagonal matrix $diag(1,1,\beta,\beta\gamma)$ gives us the representation $S5,$ where the parameter $\alpha $ equals $q^3$ (of course the parameter $\alpha $ in the Table is not the same as in (\[s4.1\]) and (\[s4.2\]), which is now zero).
The algebra $\Re $ is generated by $e_{11}$, $e_{22}$, $e_{33}$, $e_{44}$, $e_{34}$, $e_{23}$, $e_{24}$. This is also a 7-dimensional algebra (see the Table) which is isomorphic to $T_3\oplus C$, with the centralizer $\{diag(\gamma,\delta,\delta,\delta)\}$ isomorphic to $C\oplus C$. By formulae (\[matrix\]) we have that the action defined by this representation, has only one basic invariant; i.e. $$I=(1+\gamma_0)(1+i\gamma_{12}).$$
[**5. $\bf A_{11}=diag(\alpha, q^2, q, 1)$, $\alpha\neq 0, q^{-1}, 1, q, q^2, q^3$.**]{} In this case by theorem \[t2\] we have $$A_{12}=\beta e_{23}+\gamma e_{34},\ \ \
A_{21}=\beta_1e_{23}+\gamma_1e_{34}$$ and $$A_{12}A_{21}=\beta\gamma_1e_{24}=A_{21}A_{12}=\beta_1\gamma e_{24}\neq 0.
\label{s5.1}$$ So $$0=\beta\gamma_1-\beta_1\gamma=det
\left(
\begin{array}{cc}
\beta & \gamma \\
\beta_1 & \gamma_1
\end{array}
\right),\ \
\beta,\gamma,\beta_1,\gamma_1\neq 0.
\label{s5.2}$$ Therefore $$A_{21}=\epsilon A_{12},\ \
\epsilon=\frac{\beta_1}{\beta}\neq 0.
\label{s5.3}$$ Now the conjugation by a diagonal matrix $diag(1,1,\beta,\beta\gamma)$ gives us the representation $S5$ and the action defined by this representation also has only one basic invariant; i.e. $$I=(1+\gamma_0)(1+i\gamma_{12}).$$ [**6. $\bf A_{11}=diag(q^2, q^2, q,1)+e_{12}$.**]{} By theorem \[t2\], $A_{12}$=$\alpha e_{13}+\beta e_{34}$, $A_{21}$=$\alpha_1e_{13}+\beta_1e_{34},$ thus $$A_{12}A_{21}=
\alpha\beta_1e_{14}=\alpha_1\beta e_{14}\neq 0,
\label{s6.1}$$ which implies that $$A_{21}=\epsilon A_{12},\ \ \epsilon=\frac{\alpha_1}{\alpha}\neq 0.
\label{s6.2}$$ Conjugation by a diagonal matrix $diag(1,1,\alpha,\alpha\beta)$ does not change the matrix $A_{11}$ while $A_{12}$ is reduced by this to $e_{13}+e_{34}$ and we have the representation $S6.$ For the calculation of $\Re $ let us note that $$A^k_{11}=diag(q^{2k}, q^{2k}, q^k, 1)+2^{k-1}q^{2k-2}e_{12}.
\label{s6.3}$$ Evidently a subalgebra generated by $A_{11}$ is contained in a four dimensional algebra of matrices generated by $e_{11}+e_{22}$, $e_{33}$, $e_{44}$, $e_{12}$. In order to show that these algebras are equal to each other it is enough to show that $E$, $A_{11}$, $A^2_{11}$, $A^3_{11}$ are linearly independent. If this is not the case then $A_{11}$ is a root of some polynomial $f(x)$ of degree three. This polynomial must be a divisor of the characteristic polynomial of $A_{11}$, which is equal to $$(x-q^2)^2(x-q)(x-1).
\label{s6.4}$$ The polynomial (\[s6.4\]) has just three divisors of degree three and none of them has $A_{11}$ as its root (if $q\neq \pm 1$). Thus the algebra $\Re $ contains the elements $e_{11}+e_{22}$, $e_{33}$, $e_{44}$, $e_{12}$, $e_{13}=A_{12}e_{33}$, $e_{34}$=$A_{12}e_{44}$, $e_{14}$=$e_{13}e_{34}$ and is generated by these elements like a linear space; i.e. it is the 7-dimensional algebra of matrices presented in the Table. Its centralizer is the two dimensional algebra $C+Ce_{12}\cong T'_2$ i.e. by formulae (\[matrix\]) the action defined by this representation has only one basic invariant; i.e. $$I=(1+\gamma_0)(\gamma_1+i\gamma_2)\gamma_3.$$ [**7. $\bf A_{11}=diag(q^2, q, 1, 1)+e_{34}$.**]{} By theorem \[t2\], we have $A_{12}$= $\alpha e_{12}+\beta e_{24}$, $A_{21}=\alpha_1e_{12}+\beta_1e_{24}$ and $$A_{12}A_{21}=\alpha\beta_1e_{14}=A_{21}A_{12}=\alpha_1\beta e_{14}\neq 0,
\label{s7.1}$$ which immediately implies that $$A_{21}=\epsilon A_{12},\;\;
\epsilon=\frac{\alpha_1}{\alpha}\neq 0.
\label{s7.2}$$ A conjugation by the matrix $diag(\alpha^{-1}\beta^{-1}, \beta^{-1}, 1, 1)$ gives the representation $S7.$ As in the previous case, the algebra $\Re $ is generated by elements $e_{11}$, $e_{22}$, $e_{33}+e_{44}$, $e_{34}$, $e_{12}$, $e_{24}$, $e_{14},$ i.e. this is the algebra of matrices given in the Table and its centralizer is equal to $C+e_{34}C\cong T'_2$; i.e. the action defined by this representation has only one basic invariant $$I=(1-\gamma_0)(\gamma_1+i\gamma_2)\gamma_3.$$ Thus the Theorem \[slq\] is proved. $\Box$
Actions defined by $GL_q(2,C)$-representations
==============================================
Now let us consider representations of $GL_q(2,C)$ which are not equivalent to representations of $SL_q(2,C).$ If $$a_{ij}\rightarrow A_{ij}
\label{given}$$ is such a representation then we can define a representation of $SL_q(2,C)$ setting $a_{ij}\rightarrow A_{ij}$ for $i\neq 2$ or $j\neq 2$ and (see formula (\[s\])) $$a_{22}\rightarrow A_{22}^{\prime }=A_{11}^{-1}(1+qA_{12}A_{21}).
\label{119}$$ We will call these two representation [*connected*]{} to each other. Respectively, inner actions defined by connected representations will also be called [*connected*]{}.
If we denote by $D$ the quantum determinant of the given $GL_q(2,C)$-representation $$D=A_{11}A_{22}-qA_{12}A_{21}, \label{120}$$ then we obtain $$A_{22}=A_{11}^{-1}(D+qA_{12}A_{21})=
A_{22}^{\prime }+A_{11}^{-1}(D-1). \label{121}$$ In the last formula, $D$ is an invertible matrix which commute with all $A_{ij}.$ In particular it is an invariant for the connected $SL_q(2,C)$-action (\[119\]).
Inversely, suppose that $$a_{ij}\rightarrow A_{ij}, \ \ i\neq 2 \ {\rm or}\ j\neq 2,\ \
a_{22}\rightarrow A_{22}^{\prime } \label{122}$$ is a representation of $SL_q(2,C)$ and $D$ is an invertible invariant for the action defined by this representation. In this case, formula (\[121\]) defines a representation of $GL_q(2,C)$ connected with (\[122\]).
Thus, every $GL_q(2,C)$-representation is completely defined by the connected $SL_q(2,C)$-representation (which can be taken from the Table) and by an invariant $D$ of the inner action corresponding to this connected $SL_q(2,C)$-representation (which also can be found in the Table).
If $u$ is an invertible invariant for the connected action (\[119\]), then $u$ commutes with all $A_{ij}, \ (i\neq 2$ or $j\neq 2).$ Therefore the representation (\[given\]), where $A_{22}$ is replaced by $$A_{22}=A^{\prime }_{22}+A_{11}^{-1}(uDu^{-1}-1)
\label{given1}$$ is equivalent to (\[given\]) (see formula (\[121\]) and Theorem \[t4\]. In the same way if we multiply $A_{12}$ and $D$ by a nonzero scalar $\alpha _{2}$ then we will get an equivalent representation with (probably) another connected $SL_q(2,C)$-representation and a new $D$ proportional to the old one.
These considerations show that for the classification, up to the equivalence of all representations of $GL_q(2,C)$ connected with a given $SL_q(2,C)$-representation, it is enough to take just one element in every projective class of conjugated elements of the group of invertible invariants for a connected inner $SL_q(2,C)$-action. In particular, if an inner $SL_q(2,C)$-action has no nontrivial invariants (like $S2a, S2b, S4a$) then all $GL_q(2,C)$-actions connected with this action are $SL_q(2,C)$-actions.
By Theorem \[slq\] (see representations marked by $S$ in the Table) for the algebra of invariants of an $SL_q(2,C)$-action we have just three nontrivial possibilities: $$Inv\cong \left(\matrix{*&*\cr
0&*\cr}\right)=T_2,\ \ \
Inv\cong C\oplus C, \ \ \ Inv\cong
\left(\matrix{\epsilon &*\cr
0&\epsilon \cr}\right)=T_{2}^{\prime }.$$
It is easy to see that if a triangular $2\times 2$ matrix has different nonzero elements on the diagonal, then this matrix is conjugated in $T_2$ with a diagonal one. If this matrix has on its diagonal elements equal to $\epsilon $ then this matrix is conjugated in $T_2$ with a triangular matrix whose nonzero entries are all equal to $\epsilon .$ Therefore, in the first of these three cases we have just two types of possible values for $D,$ which belong to different projective classes of conjugated elements: $$D=diag (1, \beta),\ \beta \neq 0,1;\ \ D=\left(\matrix{1&1\cr
0&1\cr}\right). \label{123}$$
In the second and third cases the algebra $Inv$ is commutative; therefore $D$ can be choosen, respectively, in the forms $$D=1 \oplus \beta ,\ \ \beta \neq 0,1 \label{124}$$ and $$D=\left(\matrix{1&\xi \cr 0&1\cr}\right),
\ \ \xi \neq 0. \label{125}$$
Thus, using Theorem \[slq\] and formula (\[121\]) with $D$ defined by (\[123\]), (\[124\]), (\[125\]) and making usual calculations for finding the algebras $\Re $ and their centralizers one can obtain the following result.
. Every representation $a_{ij}\rightarrow A_{ij}$ of the algebra $GL_q(2,C), q^m\neq 1,$ in $C(1,3)$ with nontrivial perturbation is equivalent to one of the representations given in the Table, which contains the operator algebras, the quantum determinants and the invariants of the corresponding inner actions. \[glq\]
In order to show that the Table presents a complete classification it is necessary to prove that different representations in the Table define different inner actions.
. Two representations given in the Table are equivalent if and only if they are equal to each other. \[=\]
[*Proof.*]{} Let $a_{ij}\rightarrow A_{ij}$ and $a_{ij}\rightarrow A_{ij}^{\prime }$ be equivalent representations given in the Table. By Theorem \[t2\] we have $$A_{11}^{\prime }=uA_{11}u^{-1}\alpha _1,\
A_{12}^{\prime }=uA_{12}u^{-1}\alpha _2,$$ $$A_{21}^{\prime }=uA_{21}u^{-1}\alpha _1,\
A_{22}^{\prime }=uA_{22}u^{-1}\alpha _2. \label{126}$$ The first of these equations shows that matrices $A_{11}^{\prime }\alpha_1^{-1}$ and $A_{11}$ have the same Jordan Normal Form. In particular, if one of the matrices $A_{11}, A_{11}^{\prime }$ is diagonal then so is the other one and $A_{11}^{\prime }\alpha _1^{-1}$ can be obtained from $A_{11}$ by permutation of its diagonal elements (note that all matrices $A_{11}$ in the Table have a Jordan Normal Form). It easy to see that no one pair of [*different*]{} diagonal matrices $A_{11}$ from the Table satisfies this property (here it is essential that $q^4, q^3\neq 1,$ and $\alpha \neq q^{-1}$ in $S5$). Thus, in these case $A_{11}^{\prime }=A_{11}.$
If both matrices $A_{11}, A_{11}^{\prime }$ are not diagonal and $A_{11}\neq A_{11}^{\prime }$ then one of this matrices appears in S6 while another in S7. Let us assume $A_{11}^{\prime }=diag(q^2,q,1,1)+e_{34}$ and $A_{11}=diag(q^2,q^2,q,1)+e_{12}$. The matrix $A_{11}$ has a Jordan Normal Form while the Jordan Normal Form of $A_{11}^{\prime }\alpha _1^{-1}$ is equal to $$diag(\alpha _1^{-1},\alpha _1^{-1}, q^2\alpha _1^{-1},q\alpha _1^{-1})
+e_{12}. \label{127}$$ Therefore $\alpha _1^{-1}=q^2$ and either $q^2\alpha _1^{-1}=q, q\alpha _1^{-1}=1$ or $q^2\alpha _1^{-1}=1, q\alpha _1^{-1}=q.$ Both cases are impossible since $q^4, q^3\neq 1.$ Thus, we have proved that $A_{11}^{\prime }=A_{11}$ in all cases.
Now we have $uA_{11}u^{-1}\alpha _1=A_{11}$ or $uA_{11}\alpha _1=A_{11}u,$ i.e. $x\rightarrow A_{11}, y\rightarrow u$ is a representation of the $\alpha _1$-spinor with invertible both terms. This implies that $\alpha _1$ is a root of unity, $\alpha _1^m=1.$ If $(\beta _1, \beta _2, \beta _3, \beta _4)$ is a quadruple of eigenvalues (or, more exactly, the main diagonal of the Jordan Normal Form) of $A_{11},$ then the eigenvalues of $A_{11}\alpha _1$ form a quadruple $(\beta _1\alpha _1, \beta _2\alpha _1, \beta _3\alpha _1, \beta _4\alpha _1)$ and the equality $u^{-1}A_{11}u=A_{11}\alpha _1$ shows that these two quadruples coincide up to a permutation $$(\beta _1, \beta _2, \beta _3, \beta _4)=
(\beta _{\pi (1)}\alpha _1, \beta _{\pi (2)}\alpha _1,
\beta _{\pi (3)}\alpha _1, \beta _{\pi (4)}\alpha _1), \label{128}$$ where $\pi $ is a permutation of four indices.
Now it is easy to see that no one of the matrices $A_{11}$ given in the Table satisfies (\[128\]) for $\alpha_1^m=1, \alpha_1\neq 1.$ Thus we have shown that $\alpha _1=1$ and $u$ commutes with $A_{11}.$
Let us prove that $\alpha _2=1.$ The following relation is valid for quantum determinants $$D^{\prime }=uDu^{-1}\alpha _1\alpha _2=uDu^{-1}\alpha _2. \label{129}$$ This relation shows that quadruples of eigenvalues of $D^{\prime }$ and $D$ are connected by the following relation $$(\beta _1^{\prime }, \beta _2^{\prime }, \beta _3^{\prime },
\beta _4^{\prime })=
(\beta _{\pi (1)}\alpha _2, \beta _{\pi (2)}\alpha _2,
\beta _{\pi (3)}\alpha _2, \beta _{\pi (4)}\alpha _2). \label{130}$$ It is easy to see that for every quantum determinant from the Table either all four eigenvalues are equal to 1 or three of them are equal to 1 while the fourth one is arbitrary (like in $G1a, G2, G3a, G4, G5$). In both cases the relations (\[130\]) are possible only if $\alpha _2=1.$
Assume now that no one of our two representations belongs to $S2a, S2a^\prime , S2b.$
All the representations from the Table not belonging to these groups have $A_{12}$ with no parameters. If $A_{11}\neq diag(q^3, q^2, q, 1)$ then this fact immediately implies that $A_{12}^{\prime }=A_{12}.$ For $A_{11}=diag(q^3, q^2, q, 1)$ the element $u$ commutes with this diagonal matrix, therefore it is itself a diagonal matrix. We see that no different values for $A_{12}$ (that is $e_{12}+e_{23}+e_{34}; \ e_{12}+e_{23}; \ e_{23}+e_{34}$) are conjugated by a diagonal matrix, so again $A_{12}^{\prime }=A_{12}$ and in all the cases $uA_{12}u^{-1}=A_{12},$ i.e. $u$ commutes both with $A_{11}$ and $A_{12}.$ In all of the cases under consideraton $A_{21}=\alpha A_{12},$ therefore $$A_{21}^{\prime }=uA_{21}u^{-1}=\alpha uA_{12}u^{-1}=\alpha A_{12}=A_{21}.$$ By formula (\[119\]) this implies that connected $SL_q(2,C)$-representations coincide and $u$ belongs to the algebra of invariants of the connected $SL_q(2,C)$-representation. So the relation $D^{\prime }=uDu^{-1}$ implies $D^{\prime }=D$ as by the choice of $D$ (see (\[123\]), (\[124\]), (\[125\])) different $D, D^{\prime }$ are not conjugated in the algebra of invariants of the connected $SL_q(2,C)$-representation. Now, formula (\[121\]) shows that $A_{22}^{\prime }=A_{22}$ and the representations coincide.
Finally, we have to consider representations with $A_{11}=diag(q^2, q, q, 1).$ Representations from distinct groups $S2a, S2a^{\prime },\; S2b,\;
S2b^{\prime },\; G2$ have nonisomorphic algebras $\Re $ and therefore they cannot be equivalent (recall that by (\[126\]) the algebras $\Re $ for equivalent representations are conjugated by $u$). Thus, our representations belong to the same group and we need to consider the first three groups (in $S2b^{\prime }$ and $G2$ matrices $A_{12}$ have no parameters and $A_{11}, A_{12}$ generate $\Re $ for the connected $SL_q(2,C)$-representation).
[$\bf S2a, S2a^\prime $.]{} In these cases $A_{22}$ has no parameters, therefore $A_{22}^{\prime }=A_{22}$ and the matrix $u$ commutes with $A_{11}, A_{22}$. This means that $u=diag(1, M, 1),$ where $M$ is an invertible $2\times 2$ matrix.
If $A_{12}=\alpha e_{12}+e_{13}+e_{24}$ and $A_{12}^{\prime }=\alpha ^{\prime }e_{12}+e_{13}+e_{24},$ then the relation $uA_{12}=A_{12}^{\prime }u$ implies $$M\left(\matrix{1\cr 0\cr}\right)=\left(\matrix{1\cr 0\cr}
\right), \ \ \
(\alpha ,1)=(\alpha ^{\prime },1)M. \label{131}$$ The first of these equations shows that $M=\left(\matrix{1&*\cr 0&*\cr }\right)$ and in this case $(\alpha ^{\prime }, 1)M=(\alpha ^{\prime }, *),$ so $\alpha ^{\prime }=\alpha $ by the second equation of (\[131\]).
In the same way, if $A_{21}=e_{12}+\beta e_{13}+e_{34}$ and $A_{21}^{\prime }=e_{12}+\beta ^{\prime }e_{13}+e_{34},$ then $$(1, \beta )M=(1, \beta ^{\prime }), \ \ \
M\left(\matrix{0\cr 1\cr}\right)=
\left(\matrix{0\cr 1\cr}\right) \label{132}$$ and $$\beta ^{\prime }=(1, \beta ^{\prime })\left(\matrix{0\cr 1\cr}\right)=
(1, \beta )M\left(\matrix{0\cr 1\cr}\right)=
(1, \beta )\left(\matrix{0\cr 1\cr}\right)=\beta . \label{133}$$
[$\bf S2b.$]{} In this case $A_{12}$ has no parameters, so $A_{12}^{\prime }=A_{12}$ and $u$ commutes with $A_{11}, A_{12}$. An algebra generated by $A_{11}, A_{12}$ of $S2b$ is equal to $\Re $ for $S2b^{\prime }$, thus $u$ is an invariant for $S2b^{\prime }$ i.e. $u=diag(\beta , \beta , \delta , \beta )$ (see the Table). Conjugation by $u$ of $A_{21}=\alpha e_{12}+\alpha e_{24}+e_{13}$ gives $A_{21}^{\prime }=uA_{12}u^{-1}=\alpha e_{12}+\alpha e_{24}+
\alpha \delta ^{-1}e_{13}$, while $A_{21}^{\prime }=\alpha ^{\prime }e_{12}+\alpha ^{\prime }e_{24}+e_{13}$, therefore $\alpha \delta ^{-1}=1$ and $\alpha =\alpha ^{\prime }$. $\Box$
[*NOTE*]{}. In our proofs we do not need the fact that every finite dimensional rerpresentation of $GL_q(2,C)$ is triangular. We need this fact only for four or less dimensional representations. For this we need only restrictions $q^6, q^8\neq 1$ and $q$ can be a root of unity. Our classification is correct for $q^6, q^8\neq 1$ (for example if $q=exp({2\pi i\over 5})$). It is easy to see that this classification is not valid if $q=\pm i$ or $q=\pm exp({2\pi i\over 3})$ as in this cases there exist, respectively, four and three dimensional irreducible representations of $SL_q(2,C)$ with $\dim \Re =16, 9,$ respectively. Nevertheless we do not know if this classification is correct for $q=\pm exp({\pi i\over 3})$ or $q^4=-1$.
Table and corollaries
=====================
In the Table we have denoted by $S$ (followed by some symbols) the representations of $SL_q(2,C)$ and by $G$ (followed by the same symbols) representations of $GL_q(2,C),$ connected with the corresponding representation of $SL_q(2,C).$ We have shown in the Table five ingredients for every representation: the values of $A_{ij},$ the matrix form of the operator algebra $\Re ,$ its dimension, the invariants of the inner action defined by this representation and the value of the quantum determinant. If the value of some $A_{ij}$ is not shown for a particular case then this means that it coincides with the value of the previous representation in the Table. By bold style we have denote seven representations which start groups with the same values of $A_{11}.$
**Table**
From the Table, it is easy to see that the following result is valid (roughly speaking this means that the quantum determinants are the only quantum invariants).
. If an inner action defined by a representation of $GL_q(1,C),$ $q^m\neq 1,$ in $C(1,3)$ with nontrivial perturbation is not an action of $SL_q(2,C),$ then it has only one basic invariant which can be taken to be equal to the quantum determinant. \[gli\]
Of course for $SL_q(2,C)$-actions this result is not valid (in this case the quantum determinant equals 1). Nevertheless, we have seen (formulae (\[122\], \[121\])) that for every invariant $D$ there exists a connected $GL_q(2,C)$-action defined by a representation for which the quantum determinant equals $D$ (even if the initial $SL_q(2,C)$-action is defined by a representation with trivial perturbation). So we can formulate
. Every invariant of the inner $SL_q(2,C)$-action on $C(1,3)$ is equal to a quantum determinant of a connected $GL_q(2,C)$-action. \[sli\]
Acknowledgments
===============
We wish to thank Dr. Zbigniew Oziewicz for helpful discussions. VKK wishes to thank SNI and CONACYT-México for its support under grant No. 940411-R96 and also the Russian Foundation for Fundamental Research, grant 95-01-01356. SRR wishes to thank CONACYT-México for partial support under grant No 4336-E.
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[^1]: $\hspace*{-6mm}^{\ast}$ e-mail: suemi$@$servidor.unam.mx
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abstract: 'In the past decades, it was recognized that quantum chaos, which is essential for the emergence of statistical mechanics and thermodynamics, manifests itself in the effective description of the eigenstates of chaotic Hamiltonians through random matrix ensembles and the eigenstate thermalization hypothesis. Standard measures of chaos in quantum many-body systems are level statistics and the spectral form factor. In this work, we show that the norm of the adiabatic gauge potential, the generator of adiabatic deformations between eigenstates, serves as a much more sensitive measure of quantum chaos. We are able to detect transitions from non-ergodic to ergodic behavior at perturbation strengths orders of magnitude smaller than those required for standard measures. Using this alternative probe in two generic classes of spin chains, we show that the chaotic threshold decreases exponentially with system size and that one can immediately detect integrability-breaking (chaotic) perturbations by analyzing infinitesimal perturbations even at the integrable point. In some cases, small integrability-breaking is shown to lead to anomalously slow relaxation of the system, exponentially long in system size.'
author:
- Mohit Pandey
- 'Pieter W. Claeys'
- 'David K. Campbell'
- Anatoli Polkovnikov
- Dries Sels
bibliography:
- 'ref\_general.bib'
title: Adiabatic eigenstate deformations as a sensitive probe for quantum chaos
---
Introduction
============
Finding signatures of chaos in the quantum world has been a long-standing puzzle [@haakequantum; @stockmann1999quantum; @berry1989quantum]. In the last few years exciting progress has been made on characterizing the effects of chaos on dynamical properties of quantum many-body systems, see Fig. \[fig:intro\] [@shenker_black_2014; @maldacena_bound_2016; @von_keyserlingk_operator_2018; @rakovszky_diffusive_2018; @nahum_operator_2018; @swingle_unscrambling_2018; @khemani_operator_2018; @kudler2020conformal]. Classical chaos is usually described in terms of an exponential sensitivity of trajectories to initial conditions [@vulpiani2010chaos]. However, the quantum world precludes any definition of chaos in terms of physical trajectories due to the Heisenberg uncertainty principle. Alternatively, chaos can be defined in terms of the absence of integrability. Classical Liouville-Arnold integrability is formulated in terms of independent Poisson-commuting integrals of motion. Again, although there have been many attempts to characterize quantum integrability in a similar way, no such unique definition exists [@caux2011remarks; @yuzbashyan2013quantum; @yuzbashyan2016rotationally; @ilievski2016quasilocal].
In the last two decades, Random Matrix Theory (RMT) [@brody1981random; @guhr1998random; @mehta2004random] has shown outstanding success in the understanding of quantum chaos. Following the work of Wigner [@wigner1958distribution; @wigner1993characteristic], Bohigas, Giannoni, and Schmit [@bohigas1984characterization] conjectured that the energy-level statistics of all quantum systems whose classical analogues are chaotic should show level repulsion and belong to one of three universal classes depending upon their symmetry: the Gaussian orthogonal ensemble, the Gaussian unitary ensemble, or the Gaussian symplectic ensemble. On the other hand, according to the Berry-Tabor conjecture [@berry1977level], integrable systems have uncorrelated energy levels and usually exhibit Poissonian level spacing statistics. These ideas were later extended to generic quantum systems and tested numerically under the general framework of the eigenstate thermalization hypothesis (ETH) [@deutsch1991quantum; @srednicki1994chaos; @rigol2008thermalization; @borgonovi2016quantum; @d2016quantum; @deutsch2018eigenstate]. By now, the emergence of the random matrix behavior of quantum eigenstates is an accepted definition of quantum chaos.
Numerically, two additional steps are required before one can accurately compare the statistical properties (e.g. through level statistics or the spectral form factor [@muller_semiclassical_2004; @bertini_exact_2018]) of a particular quantum system with the predictions of RMT: (1) remove any symmetries; and (2) rescale the spectrum, setting the local mean level spacing to unity (also called unfolding the spectrum). Firstly, if symmetries are not removed, energy levels in different symmetry sectors don’t have any correlations, so that spectra of chaotic systems can show Poissonian distributions [@gubin2012quantum; @kudo2003unexpected]. However, finding *all* symmetries of a many-body Hamiltonian is computationally hard without any physical intuition, since this effectively involves searching for all possible (local) operators that commute with the Hamiltonian. Secondly, there are various methods to unfold the spectrum, and it is known that statistics, especially ones measuring long-range correlations, can be sensitive to the adopted unfolding procedure [@gomez2002misleading]. Moreover, the procedure can also exhibit finite-size effects. In light of these issues, it is advisable rather to use the ratio of two consecutive level spacings [@oganesyan2007localization; @atas2013distribution] or survival probability (see Ref. [@Torres-Herrera_2017; @schiulaz2019thouless]).
Here we propose an alternative tool to detect chaos in quantum systems, based on the rate of deformations of eigenstates under infinitesimal perturbations. Mathematically, the distance between nearby eigenstates (also known as the Fubini-Study metric [@kolodrubetz2017geometry; @page1987geometrical; @kobayashi1963foundations; @provost1980riemannian] ) can be expressed as the Frobenius norm of the so-called adiabatic gauge potential (AGP) [@berry2009transitionless; @demirplak2005assisted; @demirplak2003adiabatic; @kolodrubetz2017geometry], which is exactly the operator that generates such deformations. It is straightforward to show that this norm should scale exponentially with the system size in chaotic systems satisfying ETH [@kolodrubetz2017geometry]. In this sense, quantum chaos manifests itself through an exponential sensitivity of the eigenstates to infinitesimal perturbations, which can be viewed as an analogue to classical chaos, reflected in the exponential sensitivity of trajectories to such perturbations. Moreover, unlike standard probes of RMT such as the spectral form factor (see e.g. Ref. [@vsuntajs2019quantum]) or the closely related survival probability (see Ref. [@Torres-Herrera_2017; @schiulaz2019thouless]), as well as level statistics, which only depend on the eigenvalues of the Hamiltonian, the AGP norm is sensitive to both the level spacings and the specific kind of adiabatic deformation (perturbation).
We find that the norm of the AGP shows a remarkably different, and extremely sensitive, scaling with system size for integrable and chaotic systems: polynomial versus exponential. In our method, we do not need to remove any symmetries before computing the AGP norm needed in the analysis of the level spacing distributions and do not need to average over different Hamiltonians, which is necessary to analyze the (non self-averaging) spectral form factor. We show that one can detect chaos through the sharp crossover between the polynomial and exponential scaling of the norm. The sensitivity of this norm to chaotic perturbations is orders of magnitude greater than that of the aforementioned methods. Using this approach, we find several, previously-unexpected, results for a particular but fairly generic integrable XXZ spin chain with additional small perturbations: i) The strength of the integrability-breaking perturbation scales exponentially down with the system size, much faster than in previous estimates [@modak2014finite; @Modak2014]; ii) integrability-breaking deformations immediately lead to an exponential scaling of the norm of the AGP, showing that chaotic perturbations can be already detected in the integrable regimes; and iii) in the presence of small integrability-breaking terms, the system can exhibit exponentially slow relaxation dynamics, which is similar to the slow dynamics observed in some classical nearly-integrable systems like the Fermi-Pasta-Ulam-Tsingou (FPUT) chain [@gallavotti2007fermi; @danieli2017intermittent; @pace2019behavior]. We also find that such relaxation dynamics are very different for observables conjugate (see Eq. below) to integrable and chaotic directions (perturbations) of the Hamiltonian. We find similar results for an Ising model, where the integrability is broken by introducing a longitudinal field.
The connection with relaxation is not surprising, since one representation of the AGP is in terms of the long-time evolution of a local operator conjugate to the coupling. Hence, our results relate to recent studies of information propagation through operator growth in quantum many-body systems [@eisert2015quantum; @swingle2018unscrambling; @lewis2019dynamics], where chaotic and integrable systems are again expected to exhibit qualitatively different behavior (e.g. in operator entanglement [@zhou2017operator; @PhysRevLett.122.250603] and Lanczos coefficients [@parker2018universal; @avdoshkin2019euclidean]). Whereas most of the previous works focused mainly on short-time effects, here we effectively focus on dynamics and operator growth at times that are exponentially long in the system size (Fig. \[fig:intro\]).
Adiabatic Gauge Potential
=========================
Before proceeding, let us define the adiabatic gauge potential (AGP) and discuss some of its key properties. Given a Hamiltonian $H(\lambda)$ depending on a parameter $\lambda$, the adiabatic evolution of its eigenstates as we vary this parameter is generated by the AGP as (in units with $\hbar=1$): $$\mathcal A_\lambda |n(\lambda)\rangle = i \partial_\lambda |n(\lambda)\rangle, \quad H(\lambda) |n(\lambda)\rangle=E_n(\lambda) |n(\lambda)\rangle.$$
Using the Hellmann-Feynman theorem, it is easy to see that the matrix elements of the AGP between such eigenstates are given by $$\langle m |\mathcal{A_{\lambda}} | n \rangle = -\frac{i}{\omega_{mn}}\langle m |\partial_{\lambda}H | n \rangle,
\label{AGP_orign_def}$$ where $\omega_{mn}= E_m(\lambda)-E_n(\lambda)$, $\partial_{\lambda}H $ is the operator conjugate to the coupling $\lambda$, and we have made the dependence on $\lambda$ implicit. The diagonal elements of $\mathcal A_\lambda$ can be chosen arbitrarily due to the gauge freedom in defining the phases of eigenstates. A convenient choice consists of setting all diagonal elements equal to zero. For simplicity we will assume there are no degeneracies in the spectrum, but as will be clear shortly, this assumption is not necessary and does not affect any of the results below. We define the $L_2$ (Frobenius) norm, also called Hilbert–Schmidt norm, of this operator as: $$||\mathcal A_\lambda||^2=\dfrac{1} {\mathcal{D}} \sum_n \sum_{m\neq n} |\langle n | \mathcal A_\lambda |m\rangle|^2,
\label{eq:L2_norm}$$ where $\mathcal{D}$ is the dimension of the Hilbert space.
This expression should scale exponentially with the system size in chaotic systems satisfying ETH: $||\mathcal A_\lambda||^2\sim \exp\left[S\right]$, where $S$ is the entropy of the system [@kolodrubetz2017geometry]. Within ETH, the off-diagonal matrix elements of local operators, including $\partial_{\lambda}H$, scale as $\langle m |\partial_{\lambda}H | n \rangle\propto \exp[-S/2]$ [@srednicki1994chaos; @d2016quantum] while the minimum energy gap between states, $\omega_{mn}$, scales as $\exp\left[-S\right]$. The scaling of individual matrix elements was already explored in the literature to study the crossover between chaotic and non-ergodic behavior, e.g. in the context of disordered systems [@serbyn2015criterion; @crowley2019avalanche]. As we will demonstrate, the exponential scaling of the norm of the AGP can be used to detect the emergence of chaotic behavior in the system with tremendous (exponential) precision.
However, Eq. is not particularly convenient: the norm of the exact AGP can be dominated by the smallest energy difference between eigenstates, and as such it is highly unstable and difficult to analyze, especially close to the ergodicity transition. Accidental degeneracies in the spectrum that are lifted by $\partial_{\lambda} H$ also cause the norm to formally be infinite. To resolve this issue, it is convenient to instead define a ‘regularized’ AGP as follows: $$\begin{aligned}
\langle m |\mathcal{A_{\lambda}}(\mu) | n \rangle= -i \dfrac{\omega_{mn}}{ \omega_{mn}^2 + \mu^2} \langle m | \partial_{\lambda}H | n \rangle ,
\label{AGP_mu_def}\end{aligned}$$ where $\mu$ is a small energy cutoff. For the sake of brevity, we are going to drop the argument $\mu$ and unless specified otherwise $\mathcal{A}_\lambda$ refers to the regularized AGP. This has a clear physical intuition: instead of considering transitions (matrix elements) between individual eigenstates, we now only consider transitions between energy shells with width $\mu$. For eigenstates with $|\omega_{mn}| \gg \mu$, this reproduces the exact AGP, whereas in the limit $|\omega_{mn}| \ll \mu$, the AGP no longer diverges but reduces to a constant. Alternatively, within the operator growth representation (see Eq. below), $\mu^{-1}$ has the interpretation of a cutoff time. Numerically, this regularization has the immediate advantage that it gets rid of any problem with (near-)divergences. Note that $\mu$ does not need to be system-size independent for this. Interestingly, as long as $\mu\propto \exp[-S]$, the norm of the AGP within chaotic systems should also remain proportional to $\exp[S]$. We can use this flexibility in defining $\mu$ to our advantage, choosing it to be parametrically larger than the level spacing to eliminate any effect of accidental degeneracies, but still exponentially small to minimize the deviation from the exact AGP. We find that choosing $\mu(L) \propto L \exp[-S(L)]$, where $L$ is the system size, is the most convenient choice (see Appendix \[append.muScaling\]).
From Eqs. and the norm of the regularized AGP reads $$\begin{aligned}
||\mathcal A_\lambda ||^2&={1\over \mathcal D}\sum_n \sum_{m\neq n} {\omega_{mn}^2\over (\omega_{nm}^2+\mu^2)^2} |\langle m|\partial_\lambda H |n\rangle|^2\\
& =\int_{-\infty}^\infty d\omega \dfrac{\omega^2}{ (\omega^2+\mu^2)^2}\overline{|f_\lambda(\omega)|^2},
\label{eq:norm_agp_f_omega}\end{aligned}$$ where in the second equation we replaced the summation with an integration over the energy difference $\omega_{mn}=E_m(\lambda)-E_n(\lambda)$ and also defined the response function $$\begin{aligned}
\overline{|f_\lambda(\omega)|^2}&=&{1\over \mathcal D} \sum_n \sum_{m\neq n} |\langle n | \partial_\lambda H| m\rangle|^2 \delta(\omega_{nm}-\omega) \\
&=& \dfrac{1}{\mathcal{D}} \sum_n \int_{-\infty}^{\infty} \frac{dt}{4 \pi}\, e^{i \omega t} \langle n|\{\partial_{\lambda} H (t), \partial_{\lambda} H(0)\}| n \rangle_c, \nonumber
\label{eq:f_omega_def}\end{aligned}$$ where $\{...\}$ stands for the anti-commutator and connected correlation function $\langle n|\partial_{\lambda} H (t) \partial_{\lambda} H(0)| n \rangle_c=\langle n|\partial_{\lambda} H (t) \partial_{\lambda} H(0)| n \rangle-\langle n|\partial_{\lambda} H (t)|n \rangle \langle n |\partial_{\lambda} H(0)| n \rangle$. Formally, this function represents an average over eigenstates $n$ of the sum of the squares of the off-diagonal matrix elements $|\langle n | \partial_\lambda H| m\rangle|^2$ with a fixed energy difference $\omega_{mn}=\omega$, which can also be obtained as the Fourier transform of the non-equal time correlation function of $\partial_{\lambda}H$. Within the ETH ansatz, this function exactly coincides with the (averaged over eigenstates) square of the function $f_\lambda(\omega)$ introduced by M. Srednicki [@srednicki1994chaos], according to $$\begin{aligned}
&\langle m| \partial_\lambda H |n\rangle= f_\lambda(\omega, \bar{E}) \mathrm e^{-S (\bar E)/2} \sigma_{mn}, \label{eq:fwETH}\\
&\qquad\omega=E_m-E_n,\; \bar E=(E_n+E_m)/2,\end{aligned}$$ with $\sigma_{nm}$ a random variable with zero mean and unit variance. Recently it was shown that the function $|f_\lambda(\omega)|^2$ remains well defined and smooth in generic integrable systems [@leblond2019entanglement; @brenes_2020; @brenes_2020a].
Alternatively, it is convenient to rewrite the regularized AGP as a time integral [@berry1993mv; @jarzynski1995geometric; @claeys2019floquet]: $$\mathcal{A}_{\lambda} = - \dfrac{1}{2} \int_{-\infty}^{\infty} dt \mathrm \, {\rm sgn}(t)\,e^{-\mu |t|}\,
\left(\partial_{\lambda}H\right) (t),
\label{AGP_operator_spreading}$$ where ${\rm sgn}(t)$ is the sign function and $$(\partial_{\lambda}H) (t)= \mathrm e^{i H t} (\partial_{\lambda}H) \mathrm e^{-i H t}$$ is the operator conjugate to the coupling $\lambda$ in the Heisenberg representation. The exponential factor $\exp[-\mu |t|]$ can be seen as a particular choice of a filter function in the context of quasi-adiabatic continuation [@hastings2010quasi; @Nachtergaele2012; @deroeck2017]. Notably, Eq. remains valid for classical systems [@berry1993mv; @jarzynski1995geometric] and therefore the scaling of the AGP norm can be used to detect classical chaos, which we leave for future work.
Further, Eq. , makes clear that the inverse of the parameter $\mu$ plays the role of a cutoff time, limiting the growth of $(\partial_\lambda H) (t)$ in the operator space. Note that this time is much longer than the time scales generally studied in literature (e.g, the time scale characterizing the ballistic propagation of information $t_{LR}=L/v_{LR}$, where $v_{LR}$ is the Lieb-Robinson velocity and $L$ is the system size)[@eisert2015quantum; @swingle2018unscrambling; @lewis2019dynamics]. One of the outcomes of our work is that an exponential sensitivity to detecting the onset of chaos requires access to exponentially long time scales (Fig. \[fig:intro\]).
Numerical results
=================
We can now compare with results for the AGP in integrable/non-ergodic models. Specifically, we move to the analysis of the norm of the regularized AGP for a specific integrable XXZ model with open boundary conditions [@orbach_linear_1958; @yang_one-dimensional_1966; @sutherland_beautiful_2004; @gaudin_bethe_2014; @franchini2017introduction], whose Hamiltonian is given below: $$H_{\text{XXZ}}=\sum_{i=1}^{L-1} ( \sigma_{i+1}^x \sigma_{i}^x + \sigma_{i+1}^y \sigma_{i}^y) + \Delta \sum_{i=1}^{L-1} \sigma_{i+1}^z \sigma_{i}^z.
\label{Ham.XXZ}$$ We will now consider the effects of various integrability-breaking terms. Although the thermodynamics of the above model can be solved exactly using the Bethe ansatz [@orbach_linear_1958; @yang_one-dimensional_1966; @sutherland_beautiful_2004; @gaudin_bethe_2014; @franchini2017introduction], we still don’t have access to matrix elements of general local operators $\langle n | \partial_{\lambda}H | m \rangle$ and the exact AGP remains out of reach even in the integrable limit. Consequently, there are also no results on the scaling of the AGP with increasing system size.
For reference, we also analyze an Ising model in the presence of a longitudinal field whose Hamiltonian is given below: $$H_{\text{Ising}}=\sum_{i=1}^{L-1} \sigma_{i+1}^z \sigma_{i}^z + h_z \sum_{i=1}^L \sigma^z_i +h_x \sum_{i=1}^L \sigma_i^x.
\label{Ham.chaoticIsing}$$ where open boundary conditions are chosen for the chaotic Ising model. This model has a trivially-integrable limit at zero longitudinal field $h_z=0$, which maps to a system of free fermions [@sachdev2007quantum]. In this non-interacting (free) limit, the AGP can be computed analytically [@del2012assisted; @kolodrubetz2017geometry] (see Appendix \[append.free\]). In the presence of the longitudinal field, this model shows a Wigner-Dyson type distribution of the energy level spacings, which is particularly pronounced at the parameters $h_x=(\sqrt{5}+5)/8$ and $h_z=(\sqrt{5}+1)/4$ [@kim2013ballistic]. We will use these values when computing the AGP in the chaotic regime.
In Fig. \[exact\_regim\_norm\], we show the AGP norm scaled by the system size $||\mathcal A_\lambda||^2/L$ [^1] for the interacting XXZ model and the Ising model both at the chaotic and non-interacting points. Fig. \[exact\_regim\_norm\] clearly shows the remarkably different scalings with system size $L$ for chaotic, integrable and free models. For chaotic models, the scaled AGP norm shows the exponential scaling expected from ETH. For the free model, the scaled norm is system-size independent up to exponentially small corrections away from the critical point (see Appendix \[append.free\]). For the integrable XXZ model, the scaled AGP norm shows a nontrivial polynomial scaling: $||\mathcal A_{\lambda}||^2/L\propto L^\beta$. We find that the exponent $\beta$ is non-universal and depends on the choice of the anisotropy $\Delta$ (see Appendix \[append.XXZ\]). We have chosen $\lambda=h_x$ for both the integrable and non-integrable Ising models and $\lambda=\Delta$ for the XXZ model.
While the exponential scaling of the AGP norm in the chaotic regime and the constant AGP norm in the free model are expected, the polynomial scaling of this norm of the XXZ integrable model is very interesting and leads to non-trivial conclusions. Recently, LeBlond [*et al.*]{} [@leblond2019entanglement] have shown that the matrix elements of local operators in this integrable model are not sparse (as compared to the matrix elements of non-interacting integrable models). The latter implies that Eq. for the AGP norm still applies, where $|f_\lambda(\omega)|^2$ can also be found from the Fourier transform of the symmetric correlation function (see Appendix \[append.chaotic\]). Since we have chosen $\mu$ to be exponentially small in the system size and $||\mathcal A_\lambda||^2$ is polynomially (not exponentially) large, the function $f_\lambda(\omega)$ must vanish as $\omega\to 0$. This behavior is to be contrasted with chaotic systems where at small $\omega$ this function saturates at a constant value, in agreement with the Random Matrix Theory [@d2016quantum].
Integrability breaking
======================
Having established the scaling of the AGP norm in three different regimes, we will move to the analysis of integrability breaking by small perturbations and focus on a more generic XXZ model. As an integrability-breaking term, we choose a magnetic field coupled to a single spin in the middle of the chain, acting as a single-site defect, $$V=\sigma^z_{{\lceil}(L+1)/2 {\rceil}},$$ where ${\lceil}(L+1)/2 {\rceil}$ stands for the smallest integer greater than or equal to $(L+1)/2$. Then we analyze the AGP for the total Hamiltonian $$H=H_{\rm XXZ}+\epsilon_d V,
\label{Ham_XXZ_defect}$$ as a function of the integrability-breaking parameter $\epsilon_d$. Interestingly, in Ref. [@Santos2004] it was argued based on the same model that even a single site defect is sufficient to induce chaos in the thermodynamic limit. In Appendix \[append.NNN\], we analyze an extensive integrability-breaking perturbation by considering $H=H_{\rm XXZ}+\Delta_2 V$ with $V=\sum_i \sigma^z_{i+2}\sigma^z_i$ and find the results to be consistent. The similarity between the effects of local and global perturbations on spectral properties was also found in Ref. [@torres2014local].
\
A challenging question is how quickly chaos emerges when a non-ergodic, or integrable system, is subjected to an integrability-breaking perturbation. In classical systems with few degrees of freedom, it is known from KAM theory that integrable systems are stable against small perturbations [@moser_invariant_1962; @kolmogorov_conservation_1954; @arnold_proof_1963]. It is widely believed that quantum chaos is generally induced by infinitesimal perturbations in the thermodynamic limit [@Rabson2004; @Santos2010; @modak2014finite; @Modak2014], with the potential exception of many-body localization [@nandkishore2015many; @abanin2017recent], although the precise scaling of the critical perturbation strength with the system size remains an open question. A standard limitation of numerical approaches (using e.g., level statistics or spectral form factor) addressing this question is the small system sizes amenable to simulations, where it is possible to reliably extract the data.
In Fig. \[defect\_XXZ\] a) we show the scaling of the norm of the AGP as a function of the system size for different perturbation strengths $\epsilon_d$. We choose the zero magnetization subspace of the XXZ chain with number of spins up $N_{\uparrow}={\lfloor}L/2{\rfloor}$, where ${\lfloor}L/2{\rfloor}$ stands for the largest integer less than or equal to $L/2$, and for the direction of the AGP we choose $\lambda=\Delta$, i.e. as in Fig. \[exact\_regim\_norm\]. For the cutoff, we choose $\mu=L\mathcal{D}_0^{-1}$, where $\mathcal{D}_0$ is the dimension of zero magnetization sector. From the figure, we clearly see a sharp crossover in the scaling of the norm of the AGP as a function of system size from the integrable power law behavior to the chaotic exponential behavior. The straight lines are obtained by a least squares fit, with the slope extracted for the largest $\epsilon_d$ and then used for other perturbations. After the best fitting parameters were found, the critical system sizes were obtained for a particular defect energy at which the integrable (polynomial) and chaotic (exponential) curves intersect. These values are shown in the inset of Fig. \[defect\_XXZ\] a), showing a clear exponential scaling of the critical perturbation strength with the system size. Interestingly, the slope of the exponential scaling $\beta\approx 1.28$ is almost twice the slope predicted by ETH, $\beta=\log(2)\approx 0.69$. Notably, the slope of $2\log(2)$ is the largest possible growth rate of the AGP norm (see Appendix \[append.chaotic\]). In the next section we will return to this point and relate it to the emergence of relaxation times that are exponentially long in system size.
Consistent results are obtained for the Ising model , where one can consider breaking the integrability of the transverse field Ising model ($h_z=0$) by introducing a small non-zero $h_z$-field, while probing the integrable direction $\lambda = h_x$. The results are shown in Fig. \[defect\_XXZ\] b). As in the XXZ case, we observe a sharp crossover from the the unperturbed scaling of the AGP norm (see Fig. \[exact\_regim\_norm\]) to exponential scaling with an exponent that exceeds the ETH expectation, once again having implications on the long time relaxation of the system.
To contrast the scaling of the AGP norm with more traditional approaches in Fig. \[XXZlevelstat\], we show the mean ratio of energy level statistics as a function of defect energy for system size $L=16$. Given subsequent energy level spacings $s_n = E_{n+1}-E_{n}$, this ratio is defined as $$r_n = \frac{\min(s_n,s_{n+1})}{\max(s_n,s_{n+1})}.$$ For non-ergodic systems and Poissonian level statistics, $\langle r \rangle\approx 0.386$, whereas for chaotic systems and Wigner-Dyson statistics $\langle r \rangle\approx 0.536$. In this model, the average ratio $\langle r \rangle$ shows the crossover from non-ergodic to ergodic behavior at $\epsilon_d^*\sim 0.1$ [@chavda2014poisson]. This crossover value of $\epsilon_d$ has a very weak dependence on the system size. In comparison, for the same system size $L=16$ the AGP norm shows a clear crossover to chaos for a much smaller $\epsilon_d^* \sim 10^{-3}$ (see Fig. \[defect\_XXZ\] a) ). For larger system sizes, the gap between the chaos thresholds extracted by these two methods becomes even larger. Moreover, we also estimated the critical perturbation strength using the spectral form factor for the same system size $L=16$. Since this generally doesn’t self-average [@prange1997spectral; @braun2015self], we added disorder to the $zz$-coupling in the Hamiltonian (Eq. ) which reduces the sensitivity of this probe to detect chaos. From the spectral form factor we find $\epsilon_d^*\sim 0.1$, a value where the level statistics is roughly half way between Poisson and Wigner-Dyson (see Fig. \[XXZlevelstat\]). Such a correspondence was also observed for disordered models in Ref. [@vsuntajs2019quantum].
We believe that the reason that the AGP norm is so much more sensitive is that it effectively detects the change in the differential of the norm with the system size. The absolute value of the AGP norm at the threshold is still much closer to the integrable value than to the chaotic one. Such a differential is much harder to detect using other measures, e.g. the level spacing ratio, because this crossover is much smoother, and it is harder to define a sharp threshold.
In Fig. \[defect\_XXZ\_B\] we show similar results, now choosing to deform the Hamiltonian in the direction of the integrability-breaking operator itself, i.e. $\lambda=\epsilon_d$ for the XXZ chain and $\lambda=h_z$ for the Ising chain. We choose to work in the full Hilbert space with dimension $\mathcal{D}=2^L$. We find that the AGP norm shows exponential scaling even when $\epsilon_d=0$, i.e. when the Hamiltonian is integrable. We find a good fit to the exponential scaling $||\mathcal{A}_\lambda||^2 \sim e^{\beta L}$, with now $\beta \approx \log(2)$. Again we confirm that the results remain the same if we use an extensive integrability-breaking term instead (see Appendix \[append.universalslope\]).
Long relaxation times
=====================
We already mentioned a very peculiar fact following from Fig. \[defect\_XXZ\]: namely, instead of the perhaps expected crossover of the integrable polynomial scaling of the AGP norm to the ETH exponential scaling with the slope $\log(2)$ the AGP crosses over to the exponential scaling regime with the slope $\beta=1.28$, which is almost twice as large as the slope predicted by ETH, $\beta=\log (2)\approx 0.69$. Combining this result with Eq. , which we highlight works in both integrable and nonintegrable regimes, we conclude that at small $\omega$ the function $|f_\lambda(\omega)|^2$ should scale exponentially with the system size. This implies that the system must have exponentially long relaxation times, which are known to exist in classical chaotic systems like the FPUT chain [@gallavotti2007fermi; @danieli2017intermittent; @pace2019behavior]. Although we cannot rule out the eventual relaxation to the ETH value for system sizes greater than those we have studied, our results here suggest that, while an exponentially small perturbation is sufficient to induce chaos in the system, it takes an exponentially long time for the system to relax to the steady state. In Appendix \[append.NNN\], we show that a similar behavior persists if we break the integrability by a small extensive perturbation, here chosen as the second nearest-neighbor Ising interactions. We found the same slope of $\beta\approx 1.28$, ruling out that this scaling is induced by the ultra-local nature of the perturbation in Fig. \[defect\_XXZ\] a). As the defect energy is increased further to large values (in particular, $\epsilon_d\sim 1$), we find that the slope of AGP norm’s exponential growth reduces again to the ETH value of $\beta\approx \log (2)$ (see Appendix \[append.universalslope\]).
To make the connection between the AGP norm and the relaxation time more explicit let us observe that from Eq. for sufficiently small $\mu$ one can make the following estimate: $$||\mathcal{A_\lambda}||^2 \sim \frac{|f_\lambda(\mu)|^2}{\mu}.$$ For integrable directions $\lambda$ (e.g. $\lambda=\Delta$ for the XXZ model) and $L> L^\ast$, where the AGP norm has exponential scaling, the norm becomes $$||\mathcal{A_\lambda}||^2 \sim C e^{\beta (L-L^\ast)},$$ where $C$ roughly is the value of the unperturbed AGP norm at $L^\ast$. Recall that we observed a scaling of the critical perturbation strength like $\epsilon_d \sim e^{-\alpha L^\ast}$, such that one finds $$\begin{aligned}
|f_\lambda(\mu)|^2 &\sim C\mu e^{\beta (L-L^\ast)}\sim C \epsilon_d^\eta e^{\kappa L}, \end{aligned}$$ where $\eta=\beta/\alpha$, and $\kappa=\beta-\log(2)$, and we have neglected all polynomial factors in system size. For the XXZ model, the exponents are $\eta\approx 1.6$ and $\kappa \approx 0.85 \, \log(2)$ (see caption of Fig. \[defect\_XXZ\]). Because $|f_\lambda(\omega)|^2$ is the Fourier transform of the two-point correlation function of $\partial_\lambda H$ (see ), as $\omega\to 0$ it is proportional to the relaxation time of the system. Combining these considerations, we see that for the XXZ model we have $$\tau \sim \epsilon_d^\eta e^{\kappa L},$$ with both $\kappa$ and $\eta$ of $O(1)$. Similarly, for the Ising model, $\tau \sim h_z^\eta e^{\kappa L}$ where $\eta\approx 1.8$ and $\kappa \approx 1.28 \, \log(2)$ (see caption of Fig. \[defect\_XXZ\]). We see that the relaxation time increases exponentially with the system size. For large system sizes it can saturate at some $L$-independent value, which should diverge as $\epsilon_d\to 0$. This would reflect the crossover of the scaling of the AGP norm to the ETH result: $||A_\lambda||^2\propto \exp[S(L)]=\exp[\log(2) L]$. While this scenario seems likely, we do not see any signatures for such a crossover within our numerics and thus cannot rule out more exotic scenarios for the behavior of the relaxation time with the system size. Moreover, at intermediate system sizes accessible to our numerics, we see an extremely stable exponential scaling of the AGP norm (and hence of the relaxation time), with the exponent $\beta$ independent of the strength of the integrability-breaking perturbation as long as it is sufficiently small. Interestingly, in a follow up work [@tamiro_2020] a similar exponential scaling of the AGP norm with $\beta\approx 2
\log(2)$ was observed in a disordered central spin model even in the absence of any small parameters, i.e. at large integrability-breaking perturbations. We note that in all the systems analyzed so far in this regime, $\beta$ saturates near the maximum allowed value $2\log(2)$, within numerical precision. From the point of view of operator spreading, this value is very reminiscent to the $2\log(2)$ scaling of the operator entanglement entropy in maximally chaotic dual-unitary models [@Bertini_2020]. Whether it is a simple coincidence or there is a deeper connection remains to be understood.
To illustrate these general considerations about the relaxation times we extracted the function $|f_\lambda(\omega)|^2$ directly. Usually it is very difficult to do so at exponentially small frequencies of interest, since there are very few eigenstates involved, hence leading to large fluctuations. Here we computed $|f_\lambda(\omega)|^2$ by replacing all the delta-functions in Eq. with Lorentzians of width $\mu$. In all the figures $\mu=L2^{-L}$, consistent with the AGP regularization. The total spectral weight was subsequently computed on a logarithmically spaced grid. All the figures show the average spectral weight in each bin.
In Fig. \[spectral\_XXZ\] we show the extracted spectral weight $|f_\lambda(\omega)|^2$ for the XXZ model with $\lambda=\Delta$ for a small integrability breaking perturbation $\epsilon_d=0.05$ (solid lines) and exactly at the integrable point $\epsilon_d=0$ for four different system sizes $L=12,14,16,18$. As predicted from the AGP scaling, there is a clear exponentially growing spectral weight at small frequencies with an exponentially shrinking frequency range, where it plateaus. In the integrable regime, conversely $|f_\lambda(\omega)|^2$ is exponentially decreasing with the system size, approaching zero in the thermodynamic limit.
\
To contrast this behavior of the spectral function with the other two regimes where the AGP norm shows exponential scaling with $\beta=\log(2)$, in Fig. \[spectral\_XXZ\_1\] we show $|f_\lambda(\omega)|^2$ in such regimes. The top plot shows the $|f_\lambda(\omega)|^2$ for the nonintegrable perturbation $\lambda=\epsilon_d$ at the integrable point of the XXZ model $\epsilon_d=0$. While the bottom plot corresponds to the perturbation $\lambda=\Delta$ at the strongly nonintegrable point $\epsilon_d=0.5$ where the system satisfies ETH [@gubin2012quantum; @brenes_2020].
Distinguishing between integrable and ETH regimes
==================================================
The AGP clearly depends on both the Hamiltonian $H$ and the direction along which it is deformed, i.e. $\partial_{\lambda}H$. In the previous sections, we argued that generic perturbations in chaotic systems lead to an AGP norm scaling exponentially with system size, whereas in integrable models integrability-preserving perturbations lead to an AGP norm scaling polynomially. This scaling is directly reflected in the relaxation times of $\partial_{\lambda}H$ through its probing of the zero-frequency limit of $|f_{\lambda}(\omega)|^2$. However, in specific cases, polynomial scaling of the gauge potential can also be observed in chaotic systems.
In particular, there is a special class of operators which can be represented as $K=i [H,B]$, where $B$ is a local operator or a sum of local operators. A current can, e.g., be represented in this way as $B=\sum_i i\, n_i$, where $n_i$ is the conserved charge; $n_i=\sigma_z^i$ for the XXZ model. For such operators $\mathcal A_\lambda=B$ by construction, and the AGP will have a polynomial norm irrespective of whether the system is integrable or chaotic. For such operators $|f_\lambda(\omega)|^2$ must also vanish at $\omega \to 0$, consistent with recent numerical results [@brenes_2020]. On a related note, see [@dymarsky2019new]. Physically, this non-divergence of the AGP, even in the chaotic systems satisfying ETH, simply follows from the fact that deforming the Hamiltonian with the operator $K$ is a symmetry transformation, which does not change the spectrum of the Hamiltonian, but simply transforms the eigenstates with the unitary operator $U=\exp(-i\lambda B)$. When checking for quantum chaos, such deformations can be explicitly excluded when probing the scaling of the gauge potential.
While the existence of nontrivial deformations with polynomial scaling of the AGP norm is an indicator of integrability, generic integrability-breaking perturbations give rise to exponential scaling, in which case the specific dependence on $\mu$ offers further information. Note that this also implies the existence of a family of integrable models, excluding more exotic ‘isolated’ integrable systems where every possible perturbation breaks integrability.
In the previous section, the scaling of the AGP norm was the same as one would expect from ETH, even though at $\epsilon_d=0$ the system is integrable and ETH is clearly violated. The non ETH-behavior can be seen,e.g., in large eigenstate-to-eigenstate fluctuations of the expectation value of $\sigma^z_{{\lceil}(L+1)/2 {\rceil}}$ [@brenes_2020a]. For this perturbation the scaling of the AGP with the system size simply tells us that $|f_\lambda(\omega)|^2$, which remains well-defined in such models, saturates to a nonzero constant at small $\omega$, as confirmed directly in the previous section. Similar to the usual matrix elements of observables, the information about the integrability of the system is now contained in the statistical properties of the AGP norm.
More specifically, for random matrix ensembles the statistical properties of the fidelity susceptibility (equivalent to the contributions to the AGP norm for individual eigenstates) were analyzed in Ref. [@Sierant_2019], where the distribution for different eigenstates is considered. The fidelity susceptibility $z_{n,\lambda}$ of an eigenstate $|n(\lambda)\rangle$ is equivalent to $$\begin{aligned}
z_{n,\lambda} \equiv \frac{1}{\mathcal{D}} {\langle n|\mathcal {A_\lambda}^2|n\rangle_c} \equiv \frac{1}{\mathcal{D}} \sum_{m\neq n}|{\langle n|\mathcal {A_\lambda}|m\rangle|^2},\end{aligned}$$ such that $||\mathcal{A}_{\lambda}||^2 = \sum_n z_{n,\lambda}$.
Let us briefly present a simple derivation of the tail of this distribution and then contrast the AGP distribution for integrable and ETH regimes. The tail of this distribution for typical (random) perturbations will be dominated by contributions from neighbouring energy levels, such that its distribution will be inheriting its properties from the level spacing distribution.
Recall that the exact AGP norm with $\mu=0$ is given by Eqs. and . For a typical perturbation we can replace the numerator of Eq. with a random matrix such that typical matrix elements are of order $1/\sqrt{\mathcal{D}}$ (see ). The tail of the distribution for large $z_{n,\lambda}$ is dominated by nearby energy levels and we can approximate $$\label{eq:z_l}
z_{n,\lambda} \approx \frac{C}{s_n^2},$$ where $s_n$ is the level spacing $E_{n+1}-E_{n}$ now normalized by the Hilbert space dimension (such that the mean value of $s$ is unity) and $C$ is an unimportant constant, which we can set to one. The scaling of the probability distribution at large $z_\lambda$ follows as $${\rm Pr} (z_\lambda=x) \sim \frac{1}{x^{3/2}}P\left({1\over \sqrt{ x}}\right),$$ where $P(s)$ is the normalized nearest-neighbour level spacing distribution.
For integrable systems there is no level repulsion, $P(s \to 0) \neq 0$, and we have (to dominant order) $${ \rm Pr} (z_\lambda=x) \propto {x^{-3/2}},$$ for $x\gg 1$. Note that, as a consequence of this fat tail, the mean AGP diverges without regularization. The regularization with $\mu$ in the norm of the AGP effectively introduces a cutoff to the energy denominator at the rescaled cutoff $\bar \mu= \mu \mathcal D$. Assuming that the AGP norm is dominated by the contributions $z_{n,\lambda}$ for which the derived scaling holds, we can say that the average fidelity susceptibility is given by $\left< z_\lambda \right> \propto 1/\bar \mu$, and hence $||\mathcal A_\lambda||^2=\mathcal D \left< z_\lambda \right>~\sim \mathcal D/\bar \mu$. This agrees with the observed scaling shown in Fig. \[defect\_XXZ\_B\]. On the other hand, chaotic systems satisfying ETH exhibit level repulsion and $P(s) \approx s^\beta$, resulting in ${\rm Pr} (z_\lambda=x) \propto x^{-(3+\beta)/2}$ at large values of $x$. For the considered Ising and XXZ model, the relevant random matrix ensemble is Gaussian orthogonal ensemble (GOE), for which $\beta=1$ and $${\rm Pr} (z_\lambda=x) \propto x^{-2}.$$ In contrast to the integrable model, the mean $\left< z_\lambda \right> \sim - \log( {\bar \mu}^2)$ diverges only logarithmically with the cutoff. These simple scaling arguments agree very well with numerical observations shown in Fig. \[distribution\].
From this analysis, we can conclude that choosing a fixed $\mu\sim 1/\mathcal D$ leads to the same scaling of the AGP norm with the Hilbert space dimension for the integrable model with a chaotic deformation $\lambda$ and for the ergodic ETH model. However, these two limits can still be distinguished by either the different scaling of the AGP norm with the cutoff $\mu$ or, equivalently, by the presence of an exponential-in-system-size difference between the typical and the average contributions of individual states to the AGP norm in the former (integrable) regime and the lack of such exponential difference in the latter (ETH) regime.
Conclusions
===========
We found that the properly-regularized norm of the adiabatic gauge potential, the generator of adiabatic deformations, can serve as an extremely sensitive probe of quantum chaotic behavior. Within chaotic systems, this norm scales exponentially with system size, whereas it scales polynomially in interacting integrable systems and is approximately system-size independent in non-interacting systems for adiabatic deformations preserving integrability. For adiabatic deformations breaking integrability, exponential scaling is generally observed.
Using the present method to investigate the effects of an integrability-breaking perturbation on the XXZ and Ising chains, we found that perturbations that are exponentially small in system size suffice to induce chaotic behavior. We also found that such a small integrability-breaking term leads to anomalously slow dynamics along the integrable directions, with the relaxation time scaling exponentially with system size. Such integrability-breaking perturbations can also be detected at the integrable point, where no anomalous dynamics occur. Even though typical perturbations show exponential scaling of the regularized norm of the adiabatic gauge potential, regardless of whether the system is integrable or not, one can distinguish the two cases by their dependence on the regularization parameter or by their fluctuations.
This motivates the use of the adiabatic gauge potential, which is connected with both deformations of eigenstates and operator dynamics, as a sensitive probe into either chaotic or integrable behavior of quantum many-body systems.
Acknowledgements {#acknowledgements .unnumbered}
================
Some of the numerical computations were performed using QuSpin [@weinberg2017quspin; @weinberg2019quspin]. We would like to thank Marcos Rigol and Lea Santos for detailed and very useful feedback on the manuscript. We also thank Anushya Chandran, Anatoly Dymarsky Lev Vidmar, Phil Crowley, Pranay Patil, Tamiro Villazon, Phil Weinberg and Jonathan Wurtz for useful discussions. We would also like to acknowledge technical support by Boston University’s Research Computing Services. M.P and D.K.C acknowledge support from Banco Santander Boston University-National University of Singapore grant. P.W.C gratefully acknowledges support from a Francqui Foundation Fellowship from the Belgian American Educational Foundation (BAEF), Boston University’s Condensed Matter Theory Visitors program, and EPSRC Grant No. EP/P034616/1. A.P. was supported by the NSF Grant DMR-1813499 and the AFOSR Grant FA9550-16- 1-0334. D.S acknowledges support from the FWO as post-doctoral fellow of the Research Foundation – Flanders.
Cutoff scaling with system size {#append.muScaling}
===============================
Unless stated otherwise, in all calculations we have chosen a cutoff $\mu=L \mathcal{D}^{-1}$, where $\mathcal{D}$ is the dimension of the Hilbert space. The prefactor $L$ has been chosen to remove the logarithmic correction coming from the zero-frequency contribution of $|f(\omega=0)|^2 = L$ in chaotic models (see Appendix \[append.chaotic\]). This can also be motivated by plotting the AGP norm and comparing it w.r.t. different choices of cutoff. We first study this norm close to chaotic-integrable transition point and then later describe its effect deep in the chaotic regime.
When we are close to the chaotic-integrable transition point and the cutoff is too small (e.g. $\mu=L^{-1/2}\mathcal{D}^{-1}$), then we find that the AGP norm is too sensitive to the exponentially close eigenstates, showing a non-smooth exponential scaling, which makes it hard to draw any conclusions (see Fig. \[append\_fig\_XXZ\_small\_mu\] a) ). On the other hand, if the cutoff is too large (e.g. $\mu=L^2 \mathcal{D}^{-1}$), then the AGP norm, albeit smooth, is no longer sensitive to the small strength of integrability-breaking perturbation(see Fig. \[append\_fig\_XXZ\_small\_mu\] b)). In Fig. \[append\_fig\_XXZ\_small\_mu\] c) with $\mu=L \mathcal{D}^{-1}$ , we find that the rescaled AGP norm shows an exponential scaling that is both appropriately smooth and exponentially sensitive to integrability-breaking perturbations.
Deep in the chaotic (ergodic) phase, we find that the numerically-obtained scaling for the norm of the AGP is almost the same for the different choices of cutoff scaling we studied.
Derivation of AGP for the free model {#append.free}
====================================
As shown in Refs. [@del2012assisted; @kolodrubetz2017geometry], the AGP for changing the transverse field $h_x$ in a free Ising model with periodic boundary conditions is given by $$\mathcal{A}_{h}= \sum_{l=1}^{L} \alpha_l O_l,$$ where the operators $O_l$ are given by the following Pauli string operators $$O_l= \sum_{j=1}^L ( \sigma_j^x \sigma_{j+1}^z \ldots \sigma_{j+l-1}^z \sigma_{j+l}^y + \sigma_j^y \sigma_{j+1}^z \ldots \sigma_{j+l-1}^z \sigma_{j+l}^x),$$ and the coefficients $\alpha_l$ are given by $$\alpha_l= -\dfrac{1}{4 L} \sum_{k=0}^{\pi(L-1) /L} \dfrac{\sin(k) \sin(lk)}{(\cos k - h_x)^2 + \sin^2 k}.$$ The norm of the AGP follows as $$\begin{aligned}
||\mathcal{A}_{h}||^2 = \dfrac{1}{2^L} \operatorname{Tr}\left[ \mathcal{A}_{h}^2\right] =2 L \sum_{l=1}^{L} \alpha_l ^2, \end{aligned}$$ where $\operatorname{Tr}\left[O_l O_p\right] =2^{L+1}L $ was used since all strings of Pauli matrices are trace-orthogonal. The above expression was used to compute the AGP norm for the free model in Fig. \[exact\_regim\_norm\] in the main text.
To obtain the scaling with system size, we can use the analytical expressions of $\alpha_l$ for large enough system sizes [@kolodrubetz2017geometry], i.e. $\alpha_l=h_x^{-l-1}$ in the paramagnetic phase where $h_x^2 >1$. Using this, we find that $$\begin{aligned}
||\mathcal{A}_{h}||^2 &\sim \frac{1}{h_x^2 (h_x^2-1)} L (1-e^{-2L \log h_x }).\end{aligned}$$ Recall that the correlation length in the transverse field Ising model $\sim 1/\log h_x$.
AGP bound {#append.chaotic}
=========
Recall that the norm of the AGP can be expressed as $$\begin{aligned}
||\mathcal{A}_\lambda||^2 &=& \int d \omega \, \dfrac{\omega^2}{ (\omega^2 + \mu^2)^2} \overline{|f_\lambda(\omega)|^2},
\label{eq:appC_normA}\end{aligned}$$ with $$\overline{|f_\lambda(\omega)|^2}={1\over \mathcal D} \sum_n \sum_{m\neq n} |\langle n | \partial_\lambda H| m\rangle|^2 \delta(\omega_{nm}-\omega),$$ and $\omega_{nm}=E_n-E_m$. It follows directly from eq. , and $x^2/(x^2+1)^2\leq 1/4$, that $$||\mathcal{A}_\lambda||^2\leq \frac{1}{4\mu^2} \int d \omega \, |f_\lambda(\omega)|^2 = \frac{||\partial_\lambda H||^2}{4\mu^2}
\label{eq:appCbound}$$ Consequently, for any local perturbation the norm of the regularized AGP – where we set $\mu \sim L 2^{-L}$ – can’t grow faster than $4^L$. Not only does it appear that this bound is saturated when probing integrable direction $\partial_\lambda H$ in models in which the integrability is weakly broken, it further implies that those observables $\partial_\lambda H$ take exponentially long to relax. Indeed, the above scaling can only be achieved by effectively having $|f_\lambda(\mu)|^2 \sim 2^L$. Yet, the total spectral weight, $\int d \omega \, |f_\lambda(\omega)|^2$, is only polynomially large in the system size, implying that the corresponding spectral weight must be localized in a region $\Delta \omega \sim 2^{-L}$. Combined with expression , the latter implies $\partial_\lambda H(t)$ takes exponentially long to relax to equilibrium.
For interacting integrable models we found $||\mathcal{A}||^2 \sim L^{\beta}$, where the exponent $\beta$ is non-universal. Since the norm is not exponential in system size, the function $|f_\lambda(\mu)|^2 \sim 2^{-L}$. This means that the function should vanish in the zero frequency limit, which implies oscillatory dynamics of the observable $\partial_\lambda H(t)$.
Effects of the anisotropy in the XXZ model. {#append.XXZ}
===========================================
In this Appendix, we will again consider the XXZ Hamiltonian (Eq. ):
$$H_{\text{XXZ}}=\sum_{i=1}^{L-1} ( \sigma_{i+1}^x \sigma_{i}^x + \sigma_{i+1}^y \sigma_{i}^y) + \Delta \sum_{i=1}^{L-1} \sigma_{i+1}^z \sigma_{i}^z,
\label{Ham_XXZ_append}$$
where $\Delta$ is the anisotropy, and we take $\Delta=\lambda$ as the adiabatic deformation, but now at different values of $\Delta$. We find that the slope of the AGP norm depends non-trivially on $\Delta$ (Fig. \[append\_XXZ\]).
NNN interactions in the XXZ chain {#append.NNN}
=================================
In the main text, we studied the effect of strictly local integrability-breaking operator (whose support is a single site). Looking into the effects of the locality, we here study an extensive integrability-breaking operator. We add a next-nearest-neighbor (NNN) interaction to the XXZ chain, with Hamiltonian given as: $$H_{\text{NNN}}=H_{\text{XXZ}}+ \Delta_2\sum_{i=1}^{L-2} \sigma^z_{i+2} \sigma^z_{i} ,
\label{NNN_Ham}$$ The above model is chaotic for large enough $\Delta_2$ [@gubin2012quantum]. We choose $\lambda=\Delta$ (Fig. \[global\_pertubrb\_NNN1\]) and $\lambda=\Delta_2$ (Fig. \[global\_pertubrb\_NNN2\]). In the limit $\Delta_2 \rightarrow 0$, when the above Hamiltonian (eqn. \[NNN\_Ham\]) is integrable, the former (latter) is the integrability-preserving (breaking) direction. As shown in Figs. \[global\_pertubrb\_NNN1\] and \[global\_pertubrb\_NNN2\], results are similar as for the strictly local perturbation studied in the main text. This implies our results are robust to the nature of the adiabatic deformation.
![ **Integrability breaking through NNN interaction:** Rescaled AGP norm $||\mathcal{A_\lambda}||^2/L$ with $\lambda=\Delta$ of the XXZ chain at $\Delta=1.1$ shows a sharp crossover from polynomial to exponential scaling with system size, even for very small perturbation strengths $\Delta_2$. As $\Delta_2$ decreases, the system size where this crossover happens increases. Straight lines are the exponential fits with $|A_{\lambda}||^2/L \sim e^{1.28 L}$. *Inset:* The integrability-breaking defect energy scales exponentially with system size, i.e. $\Delta_2^* \sim e^{-0.9 L}$. This is calculated for the symmetry sector with zero magnetization.[]{data-label="global_pertubrb_NNN1"}](append_Fig3.pdf){width="47.00000%"}
Universal slope of the AGP norm {#append.universalslope}
===============================
Here we study the AGP norm in the XXZ chain in the limit when the magnitude of the integrability-breaking perturbation (either the local defect energy $\epsilon_d$ or the NNN interaction strength $\Delta_2$) is of the same magnitude as the $\Delta/J$ energy scale. In this limit, we find that the AGP has an exponential scaling with system size characterized by an almost universal slope $\beta \approx \log 2$, which is close to the one predicted by ETH. Details about the model and its parameters are given in the caption of Fig. \[universal\_slope\].
[^1]: We divide the norm of the AGP by the system size for extensive perturbations, to account for the trivial extensivity of the AGP
|
---
abstract: 'The origin of the nonlocal nature of quantum mechanics is investigated in the context of Everett’s formulation of quantum mechanics. EPR phenomenon can fully be explained without introducing any kind of decoherence.'
address: |
ASCII Corporation\
Yoyogi 4-33-10, Shibuya-Ku, Tokyo, Japan
author:
- Toshifumi Sakaguchi
title: On the EPR Phenomenon
---
From the time when Einstein, Podolsky and Rosen argued [@rf:epr] the incompleteness of quantum mechanics by showing so-called EPR thought experiment, there has been heated discussions [@rf:book] on the nonlocal nature of quantum mechanics. Among them, Bell proposed [@rf:bell] an inequality which has to be satisfied if the nature has a locality, and many experimental tests have subsequently been performed. [@rf:exp1; @rf:exp2; @rf:exp3; @rf:exp4; @rf:exp5; @rf:exp6; @rf:exp7]
According to the experiments, the inequality is really violated, and so we have to conclude that the nature possesses a [*nonlocality*]{}. But how can this be possible by [*local*]{} interaction? One of the answers could be “that is quantum mechanics”. But if one takes Everett’s formulation of quantum mechanics, it can be understood from a more fundamental level. This paper is aimed at this point. I will not introduce any new interpretation to explain the phenomenon. The only assumption the formulation is based upon is that the physical law which governs microscopic phenomena is also applicable to macroscopic human beings. Those who are not familiar with Everett’s formulation of quantum mechanics should read his original paper. [@rf:everett]
We use the same notation as the previous paper [@rf:me] for describing observer state: An observer state in which an observed value $\alpha^{i}$, one of the eigenvalues of an observable $\alpha$, is recorded, is written as $| [\alpha^{i}] \rangle$. When an object system is in a superposition $| \psi \rangle = \sum_{i} C_{i} | \alpha^{i} \rangle$, the measurement process can be described as follows: $$\begin{aligned}
U(t) |\psi \rangle|[\:] \rangle = \sum_{i} C_{i}
|\alpha^{i} \rangle|[\alpha^{i}] \rangle,
\label{eq:1}\end{aligned}$$ where $U(t)$ is a time evolution operator obtained from a Hamiltonian which includes the interaction between the object and observer systems. It means that when the observer observes the object system in the state $|\psi\rangle$, the observer state $|[\:]\rangle$ branches into a number of different observer states $|[\alpha^{i}] \rangle$, each of which describes independent observer. This is a direct consequence of [*the linearity of the time evolution operator $U(t)$*]{} and of [*the relation which should be satisfied if the object system is prepared in an [*eigenstate*]{} $|\alpha^{i} \rangle$ and the observer system is prepared to observe the observable $\alpha$*]{}: $$\begin{aligned}
U(t) |\alpha^{i} \rangle|[\:] \rangle = |\alpha^{i} \rangle|[\alpha^{i}]
\rangle.
\label{eq:2}\end{aligned}$$ When the observer system comes into interaction with the object system for a second time, we can apply the time evolution operator to the branched states again and we get a superposition of states $|\alpha^{i} \rangle | [ \alpha^{i} \alpha^{i} ] \rangle$, where the same value with the one observed first is recorded in each observer’s memory. Therefore, it will [*appear*]{} to each observer, who is described by each observer state in the superposition, that the [*object*]{} system in the state $| \psi \rangle$ has [*collapsed*]{} into one of the eigenstates of the observable $\alpha$. The collapse-of-wavefunction [*phenomenon*]{} of the [*object*]{} system described above is nothing but the branching process of the [*observer*]{} state according to Schrödinger’s equation, which can never happen in classical physics. It plays an essential role in the EPR phenomenon. That is, the nonlocal nature comes from the fact that a state of a pair of observers [*branches*]{} into a number of states of mutually correlated pairs by the [*local interaction*]{} between the object and observer systems. This is shown below in detail.
We consider a two particle system in a state: $$\begin{aligned}
\sum_{ij} C_{ij} | \alpha^{i} \rangle_{1} | \alpha^{j} \rangle_{2}.
\label{eq:3}\end{aligned}$$ The two particles can be separated at any distance from each other. When the observer $1$ observes the observable $\alpha$ of the particle $1$ prepared in the state $| \alpha^{i} \rangle_{1}$, the measurement process will be described as: $$\begin{aligned}
U_{1} | \alpha^{i} \rangle_{1} | \: \rangle_{2}
| [\:] \rangle_{1} | [\:] \rangle_{2}
= | \alpha^{i} \rangle_{1} | [\alpha^{i}] \rangle_{1}
| \: \rangle_{2} | [\:] \rangle_{2}.
\label{eq:4}\end{aligned}$$ It should be noted that the measurement process is local and it does not affect the state of the system $2$. When the observer $2$ observes an observable $\beta$, which does not commute with $\alpha$, of the particle $2$ prepared in the state $|\alpha^{j} \rangle_{2}$, the measurement process will be described as: $$\begin{aligned}
&U_{2}& | \: \rangle_{1} | \alpha^{j} \rangle_{2}
| [\:] \rangle_{1} | [\:] \rangle_{2} \nonumber \\
&&= | \: \rangle_{1} | [\:] \rangle_{1}
\sum_{j^{\prime}}
{}_{2} \langle \beta^{j^{\prime}} | \alpha^{j} \rangle_{2}
|\beta^{j^{\prime}} \rangle_{2}
|[\beta^{j^{\prime}}] \rangle_{2}.
\label{eq:5}\end{aligned}$$ That is, the state of the system $1$ remains unaffected, while the state of the observer $2$ branches into the states $|[\beta^{j^{\prime}}] \rangle_{2}$. The state vector $|\Psi \rangle$ of the whole system after the two observations performed therefore becomes $$\begin{aligned}
| \Psi \rangle &=& U_{1} U_{2} \sum_{ij} C_{ij}
| \alpha^{i} \rangle_{1} | \alpha^{j} \rangle_{2}
| [\:] \rangle_{1} | [\:] \rangle_{2}
\nonumber \\
&=& \sum_{i j^{\prime}}
|\alpha^{i} \rangle_{1}|[\alpha^{i}] \rangle_{1}
\{ \sum_{j} C_{ij}
{}_{2}\langle \beta^{j^{\prime}} | \alpha^{j} \rangle_{2} \}
|\beta^{j^{\prime}} \rangle_{2} |[\beta^{j^{\prime}}] \rangle_{2}
\nonumber \\
&=& \sum_{i j^{\prime}} K_{i j^{\prime}}
| \alpha^{i} \beta^{j^{\prime}} \rangle
| [(\alpha^{i} \beta^{j^{\prime}})] \rangle,
\label{eq:6}\end{aligned}$$ where $K_{i j^{\prime}} \equiv \sum_{j} C_{ij} {}_{2}\langle
\beta^{j^{\prime}} | \alpha^{j} \rangle_{2}$, $|\alpha^{i} \beta^{j^{\prime}} \rangle
\equiv | \alpha^{i} \rangle_{1} | \beta^{j^{\prime}} \rangle_{2}$ and $| [(\alpha^{i} \beta^{j^{\prime}})] \rangle
\equiv | [\alpha^{i}] \rangle_{1} | [\beta^{j^{\prime}}] \rangle_{2}$.
Preparing $N$ identical pairs of particles in the same state (\[eq:3\]) and performing the above observation sequentially, we will get a superposition of branched states of the form $$\begin{aligned}
K_{i i^{\prime}} K_{j j^{\prime}}
&& \ldots K_{k k^{\prime}}
| \alpha^{i} \beta^{i^{\prime}} \rangle
| \alpha^{j} \beta^{j^{\prime}} \rangle
\ldots
| \alpha^{k} \beta^{k^{\prime}} \rangle
\nonumber \\
&& \otimes
| [(\alpha^{i} \beta^{i^{\prime}})
(\alpha^{j} \beta^{j^{\prime}})
\ldots
(\alpha^{k} \beta^{k^{\prime}}) ] \rangle,
\label{eq:7}\end{aligned}$$ where $$\begin{aligned}
| [(\alpha^{i} \beta^{i^{\prime}})
(\alpha^{j} \beta^{j^{\prime}}) &&\ldots
(\alpha^{k} \beta^{k^{\prime}}) ] \rangle \nonumber \\
&&\equiv | [\alpha^{i} \alpha^{j} \ldots \alpha^{k}] \rangle
| [\beta^{i^{\prime}} \beta^{j^{\prime}} \ldots
\beta^{k^{\prime}}] \rangle.
\label{eq:8}\end{aligned}$$ In the limit $N \rightarrow \infty$, we can show [@rf:me] that the pair $(\alpha^{p} \beta^{p^{\prime}})$ in $[ \ldots ]$ appears $| K_{p p^{\prime}} |^{2} N$ times in almost all branched states in the superposition. Therefore, each pair of observers in the superposition will conclude that the pair $(\alpha^{p} \beta^{p^{\prime}})$ can be obtained with probability $$\begin{aligned}
P(\alpha^{p}, \beta^{p^{\prime}}) &=& | K_{p p^{\prime}} |^{2}
\nonumber \\
&=& \sum_{qr} C^{\ast}_{pq} C_{pr}
\langle \alpha^{q} | \beta^{p^{\prime}} \rangle
\langle \beta^{p^{\prime}} | \alpha^{r} \rangle.
\label{eq:9}\end{aligned}$$ If the state (\[eq:3\]) is the spin singlet state of spin-1/2 particles; $$\begin{aligned}
\bigl( C_{ij} \bigr) =
\frac{1}{\sqrt{2}}
\left(
\begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}
\right),
\label{eq:10}\end{aligned}$$ and $$\begin{aligned}
\bigl( \langle S^{i}_{z} | S^{j}_{z^{\prime}} \rangle \bigr) =
\left(
\begin{array}{cc}
\cos \theta / 2 & - i \sin \theta / 2 \\
- i \sin \theta / 2 & \cos \theta / 2
\end{array}
\right),
\label{eq:11}\end{aligned}$$ where $\theta$ is the angle between the $z$-axis and $z^{\prime}$-axis, the probability becomes $$\begin{aligned}
\bigl( P(S^{i}_{z}, S^{j}_{z^{\prime}}) \bigr) =
\frac{1}{2}
\left(
\begin{array}{cc}
\sin^{2} \theta / 2 & \cos^{2} \theta / 2 \\
\cos^{2} \theta / 2 & \sin^{2} \theta / 2
\end{array}
\right).
\label{eq:12}\end{aligned}$$
The branching process caused by the local interaction between the object and observer systems is local and does not affect the observation performed by the other observer at a remote site as shown in Eqs. (\[eq:4\]) and (\[eq:5\]). This is why superluminal communication is impossible. The probability derived above, which is shown to violate Bell’s inequality, can be recognized only when the two observers meet each other to compare the data in their world.
A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. [**47**]{}, 777 (1935). For a review, see [*Quantum Theory and Measurement, Edited by J. A. Wheeler and W. H. Zurek*]{}, Princeton University Press (1983). J. S. Bell, Physics [**1**]{}, 195 (1965). J. F. Clauser and A. Shimony, Rep. Prog. Phys. [**41**]{}, 1881 (1971). S. J. Freedman and J. F. Clauser, Phys. Rev. Lett. [**28**]{}, 938 (1972). J. F. Clauser, Phys. Rev. Lett. [**36**]{}, 1223 (1976). E. S. Fry and R. C. Thompson, Phys. Rev. Lett. [**37**]{}, 465 (1976). A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. [**47**]{}, 460 (1981). A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. [**49**]{}, 91 (1982). A. Aspect, J. Dalibard and G. Roger, Phys. Rev. Lett. [**49**]{}, 1804 (1982). H. Everett III, Rev. Mod. Phys. [**29**]{}, 454 (1957). T. Sakaguchi, quant-ph/9506042
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---
abstract: 'Using methods from Banach space theory, we prove two new structural results on maximal regularity. The first says that there exist positive analytic semigroups on UMD-Banach lattices, namely $\ell_p(\ell_q)$ for $p \neq q \in (1, \infty)$, without maximal regularity. In the second result we show that the extrapolation problem for maximal regularity behaves in the worst possible way: for every interval $I \subset (1, \infty)$ with $2 \in I$ there exists a family of consistent bounded analytic semigroups $(T_p(z))_{z \in \Sigma_{\pi/2}}$ on $L_p({\mathbb{R}})$ such that $(T_p(z))$ has maximal regularity if and only if $p \in I$.'
address: 'Institute of Applied Analysis, University of Ulm, Helmholtzstr. 18, 89069 Ulm'
author:
- Stephan Fackler
bibliography:
- 'new\_counterexamples\_mr\_biber.bib'
title: 'Maximal Regularity: Positive Counterexamples on UMD-Banach Lattices and Exact Intervals for the Negative Solution of the Extrapolation Problem'
---
=1
[^1]
Introduction
============
Let $-A$ be the generator of a $C_0$-semigroup $(T(t))_{t \ge 0}$ on a Banach space $X$. One says that $-A$ has *maximal regularity* (for one or equivalently all choices of $T>0$ and of $p \in (1, \infty)$) if for all $f \in L_p([0,T];X)$ the mild solution $u(t) = \int_0^t T(t-s) f(s) {\mathop{}\!d}s$ of the inhomogeneous abstract Cauchy problem $$\left\{ \begin{aligned}
\dot{u}(t) + A(u(t)) & = f(t) \\
u(0) & = 0
\end{aligned} \right.$$ satisfies $u \in W_{p}^{1}([0,T];X) \cap L_p([0,T]; D(A))$. Maximal regularity is a fundamental tool in the study of non-linear partial differential equations (see [@KunWei04], [@DHP03], [@Pru02] and the references therein).
Although maximal regularity has been very successful for concrete applications, fundamental questions in the structural understanding of this concept are still open (for an explicit statement see for example [@Kal01 Section 7]). For example, until recently, no explicit example of a generator of an analytic semigroup without maximal regularity on $L_p$ for $p \in (1, \infty) \setminus \{2\}$ has been known, although the existence was shown in [@KalLan00]. In fact, the first explicit examples were given by the author in [@Fac13] and [@Fac14]. The aim of this note is to develop the techniques from [@Fac14] further in order to give new contributions to the study of the structure of maximal regularity with the help of the theory of Schauder bases. For a survey on the open questions in this area we refer to [@Fac14b].
In the first main result (Theorem \[thm:positive\_counterexamples\]) we show that there exist generators of positive analytic semigroups on UMD-Banach lattices, namely on $\ell_p(\ell_q)$ for $p \neq q \in (1, \infty)$, without maximal regularity. Hence, positivity does not imply maximal regularity. This is in contrast to the following positive result due to L. Weis [@Wei01 Remark 4.9c)]: the generator of a bounded analytic semigroup on some $L_p$-space for $p \in (1, \infty)$ that is positive and contractive on the real line has maximal regularity.
In the second main result (Theorem \[thm:extrapolation\_mr\_complete\_counterexample\]) we study the extrapolation problem for maximal regularity. In [@Fac14] it was shown that maximal regularity does in general not extrapolate from $L_2$ to the $L_p$-scale, i.e. there exists a family of consistent semigroups $(T_p(t))_{t \ge 0}$ on $L_p$ for $p \in (1, \infty)$ such that $(T_2(t))_{t \ge 0}$ has maximal regularity, but $(T_p(t))_{t \ge 0}$ fails maximal regularity for some, indeed all, $p \in (1, \infty) \setminus \{2\}$. Here we extend this negative result. Indeed, we show that the extrapolation problem behaves in the worst possible way. We now shortly explain what this means.
Suppose that one has given a family $(T_p(z))$ of consistent analytic $C_0$-semigroups on $L_p$ for $p \in (1, \infty)$ and let $M \subset (1,\infty)$ be the set all $p \in (1, \infty)$ for which the semigroup $(T_p(z))$ has maximal regularity. Since an analytic $C_0$-semigroup on a Hilbert space has maximal regularity by a result of de Simon [@Sim64 Lemma 3,1], one has $2 \in M$. Moreover, it follows from complex interpolation that $M$ is a subinterval of $(1, \infty)$. We show that apart from these obvious structural restrictions one cannot obtain any further positive results for the extrapolation problem: for every interval $I \subset (1, \infty)$ with $2 \in I$ there exists a family of consistent $C_0$-semigroups $(T_p(z))_{z \in \Sigma_{\frac{\pi}{2}}}$ on $L_p({\mathbb{R}})$ such that $(T_p(z))_{z \in \Sigma_{\frac{\pi}{2}}}$ has maximal regularity if and only if $p \in I$.
In contrast, positive results for the extrapolation problem are known under additional assumptions on the semigroups (see [@Fac13+ Section 6] and the references therein).
R-Sectorial Operators and Associated Operators
==============================================
In this section we present the necessary background on sectorial operators and maximal regularity.
A densely defined operator $A$ on a Banach space $X$ is called *sectorial* if there exists an $\omega \in (0, \pi)$ such that $$\label{sectorial}
\tag{$S_{\omega}$}
\sigma(A) \subset \overline{\Sigma_{\omega}} \qquad \text{and} \qquad \sup_{\lambda \not\in \overline{\Sigma_{\omega + {\varepsilon}}}} {\left\lVert\lambda R(\lambda, A)\right\rVert} < \infty \quad \forall {\varepsilon}> 0.$$ One defines the *sectorial angle of $A$* as $\omega(A) \coloneqq \inf \{ \omega: \text{\eqref{sectorial} holds} \}$.
Recall that on Banach spaces sectorial operators are exactly the (negative) generators of strongly continuous analytic semigroups. Maximal regularity can be characterized by a stronger boundedness property than the boundedness in operator norm. Let $r_k(t) \coloneqq \operatorname{sign}\sin (2^k \pi t)$ be the $k$-th *Rademacher function*.
A family of operators $\mathcal{T} \subset \mathcal{B}(X)$ on a Banach space $X$ is called *$\mathcal{R}$-bounded* if there exists a finite constant $C \ge 0$ such that for each finite subset $\{T_1, \ldots, T_n \}$ of $\mathcal{T}$ and arbitrary $x_1, \ldots, x_n \in X$ one has $${\biggl\lVert\sum_{k = 1}^n r_k T_k x_k\biggr\rVert}_{L_2([0,1]; X)} \le C {\biggl\lVert\sum_{k=1}^n r_k x_k\biggr\rVert}_{L_2([0,1]; X)}. \label{eq:R-ineq}$$ The best constant $C$ such that holds is denoted by $\mathcal{R}(\mathcal{T})$.
Further let $\operatorname{Rad}(X)$ denote the closed linear span in $L_2([0,1];X)$ of functions of the form $\sum_{k=1}^n r_k x_k$ for $n \in {\mathbb{N}}$ and $x_1, \ldots, x_n \in X$. We need the following strengthening of the sectoriality condition.
A sectorial operator $A$ on a Banach space $X$ is called *$\mathcal{R}$-sectorial* if for some $\omega > \omega(A)$ one has $$\mathcal{R} \{ \lambda R(\lambda,A): \lambda \not\in \overline{\Sigma_{\omega}} \} < \infty. \label{R-sectorial}\tag{$\mathcal{R}_{\omega}$}$$ One defines the *$\mathcal{R}$-sectorial angle* as $\omega_R(A) \coloneqq \inf\{ \omega: \text{\eqref{R-sectorial} holds} \}$.
One has the following connection between maximal regularity and $\mathcal{R}$-sectorial operators by L. Weis [@Wei01 Theorem 4.2] (the UMD assumption is actually only needed in one implication). For the definition and further information on UMD-spaces we refer to [@Fra86] and [@Bur01]. In the following we only need that mixed $L_p(L_q)$-spaces for $p, q \in (1, \infty)$ are UMD.
\[thm:weis\] Let $-A$ be the generator of an analytic $C_0$-semigroup $(T(z))_{z \in \Sigma}$ on a Banach space $X$. Then the following hold:
If $-A$ has maximal regularity, then $A$ is $\mathcal{R}$-sectorial with $\omega_R(A) < \frac{\pi}{2}$.
Conversely, if $X$ is UMD, $-A$ has maximal regularity if $A$ is an $\mathcal{R}$-sectorial operator with $\omega_R(A) < \frac{\pi}{2}$.
Hence, it suffices to construct sectorial operators which are not $\mathcal{R}$-sectorial to give counterexamples to maximal regularity. We now transfer the $\mathcal{R}$-sectoriality of an operator on $X$ to the boundedness of an associated operator on $\operatorname{Rad}(X)$. This variant of the transference result in [@Fac14 Theorem 3.3] (see also [@Fac14b Proposition 3.16] and [@AreBu03 Theorem 3.6]) is the central tool for the counterexamples to be given later.
\[prop:resolvent\_r\_associated\_operator\] Let $A$ be an $\mathcal{R}$-sectorial operator. Then there exists a constant $C \ge 0$ such that for all $(q_n)_{n \in {\mathbb{N}}} \subset {\mathbb{R}}_{-}$ the associated operator $$\mathcal{R}\colon \sum_{n=1}^{N} r_n x_n \mapsto \sum_{n=1}^{N} r_n q_n R(q_n,A)x_n$$ defined on the finite Rademacher sums extends to a bounded operator on $\operatorname{Rad}(X)$ with operator norm at most $C$.
Since $A$ is $\mathcal{R}$-sectorial, one has $C \coloneqq \mathcal{R}\{ \lambda R(\lambda, A): \lambda \in {\mathbb{R}}_{-} \} < \infty$. Hence, for all finite Rademacher sums we have by the definition of $\mathcal{R}$-boundedness $${\biggl\lVert\sum_{n=1}^N r_n q_n R(q_n,A)x_n\biggr\rVert} \le C {\biggl\lVert\sum_{n=1}^N r_n x_n\biggr\rVert}. \qedhere$$
Positive Analytic Semigroups without Maximal Regularity
=======================================================
In this section we construct generators of positive analytic semigroups without maximal regularity. Let $X$ be a Banach space that admits an $1$-unconditional Schauder basis $(e_m)_{m \in {\mathbb{N}}}$. Then it is well-known that $(e_m)_{m \in {\mathbb{N}}}$ induces on $X$ via $$x = \sum_{m=1}^{\infty} a_m e_m \ge 0 \quad :\Leftrightarrow \quad a_m \ge 0 \quad \text{for all } m \in {\mathbb{N}}$$ the structure of a Banach lattice. Let $\pi\colon {\mathbb{N}}\to {\mathbb{N}}$ be a permutation of the even numbers. Then by a classical perturbation result $(f_m)_{m \in {\mathbb{N}}}$ defined by $$\label{eq:basis_perturbation}
f_m =
\begin{cases}
e_m & m \text{ odd} \\
e_{m-1} + e_{\pi(m)} & m \text{ even}
\end{cases}$$ is a Schauder basis for $X$ [@Sin70 Ch. I, Proposition 4.4]. Let $A$ be the *Schauder multiplier* associated to some real positive sequence $(\gamma_m)_{m \in {\mathbb{N}}}$ with respect to $(f_m)_{m \in {\mathbb{N}}}$, that is $$\begin{aligned}
D(A) = \biggl\{ x = \sum_{m=1}^{\infty} a_m f_m: \sum_{m=1}^{\infty} \gamma_m a_m f_m \text{ exists} \biggr\} \\
A \left(\sum_{m=1}^{\infty} a_m f_m \right) = \sum_{m=1}^{\infty} \gamma_m a_m f_m.
\end{aligned}$$ Then $A$ clearly is a closed densely defined operator. Let $BV$ be the Banach space of all sequences $(x_m)_{m \in {\mathbb{N}}}$ with bounded variation, i.e. ${\left\lVert(x_m)\right\rVert}_{\mathrm{BV}} \coloneqq {\left\lVertx_1\right\rVert} + \sum_{m=1}^{\infty} {\left\lvertx_{m+1} - x_m\right\rvert} < \infty$. Concerning the boundedness of $A$, one has the following positive result [@Ven93 Lemma 2.4].
\[prop:sm\_bounded\] Let $(e_m)_{m \in {\mathbb{N}}}$ be a Schauder basis for a Banach space $X$. Then there exists a constant $K \ge 0$ such that for all $(\gamma_m)_{m \in {\mathbb{N}}} \in BV$ the Schauder multiplier $A$ associated to $(\gamma_m)_{m \in {\mathbb{N}}}$ with respect to $(e_m)_{m \in {\mathbb{N}}}$ is bounded and satisfies $${\left\lVertA\right\rVert} \le K {\left\lVert(\gamma_m)\right\rVert}_{\mathrm{BV}}.$$
We now study the positivity of the formal semigroup $(e^{-tA})_{t \ge 0}$ on $X$ generated by $-A$ as defined above with respect to the just defined lattice structure. Clearly, the semigroup is positive if and only if $e^{-tA} e_m \ge 0$ for all $m \in {\mathbb{N}}$ and all $t \ge 0$. For odd $m$ this is satisfied because of $e^{-tA}e_m = e^{-tA} f_m = e^{-\gamma_m t} e_m$. For even $m$ one has $$\label{eq:application_semigroup}
\begin{split}
e^{-tA} e_m & = e^{-tA} (f_{\pi^{-1}(m)} - e_{\pi^{-1}(m)-1}) = e^{-tA} (f_{\pi^{-1}(m)} - f_{\pi^{-1}(m)-1}) \\
& = e^{-t\gamma_{\pi^{-1}(m)}} f_{\pi^{-1}(m)} - e^{-t \gamma_{\pi^{-1}(m)-1}} f_{\pi^{-1}(m)-1} \\
& = e^{-t\gamma_{\pi^{-1}(m)}} (e_{\pi^{-1}(m)-1} + e_m) - e^{-t \gamma_{\pi^{-1}(m)-1}} e_{\pi^{-1}(m)-1} \\
& = (e^{-t\gamma_{\pi^{-1}(m)}} - e^{-t \gamma_{\pi^{-1}(m)-1}}) e_{\pi^{-1}(m)-1} + e^{-t\gamma_{\pi^{-1}(m)}} e_m.
\end{split}$$ Therefore $(e^{-tA})_{t \ge 0}$ is positive if and only if $\gamma_{m} \le \gamma_{m-1}$ for all even $m \in {\mathbb{N}}$.
Later, the following elementary observation will be useful.
\[lem:maximize\_resolvent\_sequence\] For $\gamma_m > \gamma_{m-1} > 0$ consider the function $d(t) \coloneqq t [(t+\gamma_{m-1})^{-1} - (t+\gamma_{m})^{-1}]$ on ${\mathbb{R}}_{+}$. Then $d$ has a maximum which is bigger than $\frac{1}{2} \frac{\gamma_m - \gamma_{m-1}}{\gamma_m + \gamma_{m-1}}$.
With these preliminary observations we obtain the following result.
\[thm:positive\_counterexamples\] Let $X$ be a Banach space that admits a normalized non-symmetric 1-unconditional Schauder basis $(e_m)_{m \in {\mathbb{N}}}$. We consider $X$ as a Banach lattice with the order induced by $(e_m)_{m \in {\mathbb{N}}}$. Then there exists a non-$\mathcal{R}$-sectorial operator $A$ with $\omega(A) = 0$ such that $-A$ generates a positive analytic $C_0$-semigroup on $X$.
It follows from [@Sin70 first part of the proof of Proposition 23.2] that there exists a permutation $\pi\colon {\mathbb{N}}\to {\mathbb{N}}$ of the even numbers such that $(e_{2m - 1})_{m \in {\mathbb{N}}}$ and $(e_{\pi(2m)})_{m \in {\mathbb{N}}}$ are not equivalent. Hence, there exists a sequence $(a_m)_{m \in {\mathbb{N}}}$ such that the expansion for $(a_m)_{m \in {\mathbb{N}}}$ converges with respect to $(e_{2m - 1})_{m \in {\mathbb{N}}}$ or $(e_{\pi(2m)})_{m \in {\mathbb{N}}}$ but not for both. For the rest of the proof we will assume that the expansion converges for $(e_{\pi(2m)})_{m \in {\mathbb{N}}}$. In the other case a similar argument can be applied if one replaces $(f_m)_{m \in {\mathbb{N}}}$ by $$f_m = \begin{cases}
e_m + e_{\pi(m+1)} & m \text{ odd} \\
e_{\pi(m)} & m \text{ even}.
\end{cases}$$ We use the following twisted version of the lacunary sequence $(2^m)_{m \in {\mathbb{N}}}$: $$\gamma_m = \begin{cases}
2^{m+1} & m \text{ odd} \\
2^{m-1} & m \text{ even}.
\end{cases}$$ By definition, one has $\gamma_m < \gamma_{m-1}$ for all even $m \in {\mathbb{N}}$. By the above observation this implies that the formal semigroup generated by the Schauder multiplier associated to $(-\gamma_m)_{m \in {\mathbb{N}}}$ is positive. It therefore remains to show that the multiplier $A$ associated to $(\gamma_m)_{m \in {\mathbb{N}}}$ is a sectorial operator with $\omega(A) = 0$ which is not $\mathcal{R}$-sectorial. For this let us consider the sequence $(e^{-t\gamma_m^{\alpha}})_{m \in {\mathbb{N}}}$ for $t > 0$ und $\alpha > 0$. For its total variation one obtains $$\begin{aligned}
\MoveEqLeft \sum_{m = 1}^{\infty} e^{-t 2^{(2m-1)\alpha}} - e^{-t 2^{2m\alpha}} + e^{-t 2^{(2m-1)\alpha}} - e^{-t 2^{2(m+1) \alpha}} \\
& \le t \sum_{m=1}^{\infty} (2^{2m\alpha} - 2^{(2m-1)\alpha}) e^{-t2^{(2m-1)\alpha}} + (2^{(2m+2)\alpha} - 2^{(2m-1)\alpha}) e^{-t 2^{(2m-1)\alpha}} \\
& = (2^{3\alpha} + 2^{\alpha} - 2)t \sum_{m=1}^{\infty} 2^{(2m-1)\alpha} e^{-t 2^{(2m-1)\alpha}} \\
& = \frac{2^{3\alpha} + 2^{\alpha} - 2}{2^{\alpha} - 1}t \sum_{m=1}^{\infty} \int_{2^{(2m-1)\alpha}}^{2^{2m\alpha}} e^{-t2^{(2m-1)\alpha}} \, {\mathop{}\!d}s \\
& \le \frac{2^{3\alpha} + 2^{\alpha} - 2}{2^{\alpha} - 1}t \sum_{m=1}^{\infty} \int_{2^{(2m-1)\alpha}}^{2^{2m\alpha}} e^{-ts/2^{\alpha}} \, {\mathop{}\!d}s \\
& \le \frac{2^{3\alpha} + 2^{\alpha} - 2}{2^{\alpha} - 1}t \int_{2^{\alpha}}^{\infty} e^{-ts/2^{\alpha}} \, {\mathop{}\!d}s = \frac{2^{\alpha}}{2^{\alpha} - 1} (2^{3\alpha} + 2^{\alpha} - 2) e^{-t}.
\end{aligned}$$ It follows from $\omega(A^{\alpha}) = \alpha \omega(A)$ [@KunWei04 Theorem 15.16] and Proposition \[prop:sm\_bounded\] that $A$ is sectorial with $\omega(A) = 0$.
Now assume that $A$ is $\mathcal{R}$-sectorial. Let $(q_m)_{m \in {\mathbb{N}}} \subset {\mathbb{R}}_{-}$ be a sequence to be chosen later. Then it follows from Proposition \[prop:resolvent\_r\_associated\_operator\] that the operator $\mathcal{R}\colon \operatorname{Rad}(X) \to \operatorname{Rad}(X)$ associated to the sequence $(q_m)_{m \in {\mathbb{N}}}$ is bounded. We now apply $\mathcal{R}$ to the element $x = \sum_{m=1}^{\infty} a_m e_{\pi(2m)} r_m$ of $\operatorname{Rad}(X)$. Since $R(\lambda, A)$ is the multiplier associated to the sequence $((\lambda - \gamma_m)^{-1})_{m \in {\mathbb{N}}}$ and $e_{\pi(2m)} = f_{2m} - f_{2m-1}$, we obtain $$\begin{aligned}
\mathcal{R}(x) & = \mathcal{R} \biggl( \sum_{m=1}^{\infty} a_m (f_{2m} - f_{2m-1}) r_m \biggr) \\
& = \sum_{m=1}^{\infty} r_m \frac{a_m q_m}{q_m - \gamma_{2m}} f_{2m} - r_m \frac{a_m q_m}{q_m - \gamma_{2m-1}} f_{2m-1} \\
& = \sum_{m=1}^{\infty}r_m \frac{a_m q_m}{q_m - \gamma_{2m}} (e_{\pi(2m)} + e_{2m-1}) - r_m \frac{a_m q_m}{q_m - \gamma_{2m-1}} e_{2m-1} \\
& = \sum_{m=1}^{\infty} r_m \frac{a_m q_m}{q_m - \gamma_{2m}} e_{\pi(2m)} + r_m a_m q_m \biggl( \frac{1}{q_m - \gamma_{2m}} - \frac{1}{q_m - \gamma_{2m-1}} \biggr) e_{2m-1}.
\end{aligned}$$ Now take $q_m = -2^{2m-1}$ as motivated in Lemma \[lem:maximize\_resolvent\_sequence\]. Then we see that $$\sum_{m=1}^{\infty} \frac{1}{2} r_m a_m e_{\pi(2m)} + \frac{1}{6} r_m a_m e_{2m-1}.$$ exists in $\operatorname{Rad}(X)$. By [@DJT95 Theorem 12.3] and the unconditionality of the basis, $\sum_{m=1}^{\infty} a_m e_{2m-1}$ converges. This contradicts the choice of $(a_m)_{m \in {\mathbb{N}}}$ and therefore $A$ is not $\mathcal{R}$-sectorial.
We now give some concrete examples of spaces for which the above theorem can be applied.
For $p, q \in (1, \infty)$ consider the UMD-spaces $\ell_p(\ell_q)$ with their natural lattice structure. Its ordering is induced by the standard unit vector basis $(e_m)_{m \in {\mathbb{N}}}$ of $\ell_p(\ell_q)$ for some enumeration of ${\mathbb{N}}\times {\mathbb{N}}$. Clearly, $\ell_p(\ell_q)$ contains both copies of $\ell_p$ and $\ell_q$ and therefore for $p \neq q$ the basis $(e_m)_{m \in {\mathbb{N}}}$ is 1-unconditional and non-symmetric. Hence for $p \neq q$, Theorem \[thm:positive\_counterexamples\] yields a sectorial operator $A$ on $\ell_p(\ell_q)$ with $\omega(A) = 0$ such that $-A$ generates a positive analytic $C_0$-semigroup without maximal regularity.
In the next section we see that for $p \in (1,\infty) \setminus \{2\}$ the space $\ell_p$ admits after equivalent renorming a non-symmetric $1$-unconditional basis. If one uses the ordering induced by this basis, one sees with the help of Theorem \[thm:positive\_counterexamples\] that one can give $\ell_p$ after equivalent renorming a non-standard lattice structure for which there exists a generator $-A$ of a positive analytic $C_0$-semigroup without maximal regularity satisfying $\omega(A) = 0$. Further, these arguments apply to every normalized unconditional basis of $L_p([0,1])$ for $p \in (1, \infty) \setminus \{2\}$ as such bases are automatically non-symmetric [@Sin70 Ch. II, Theorem 21.1].
Exact Control of the Extrapolation Scale
========================================
In this section we give the announced complete negative solution of the extrapolation problem for maximal regularity. For $p \in (1, \infty)$ let $(e_m)_{m \in {\mathbb{N}}}$ be the standard unit vector basis of $X_p \coloneqq (\oplus_{n=1}^{\infty} \ell_2^n)_{\ell_p}$ seen as a sequence space, the $\ell_p$-sum of finite dimensional Euclidean spaces of increasing dimension. Consider the basis $(f_m)_{m \in {\mathbb{N}}}$ given by with respect to the following permutation already used in [@Fac14]. Let $b_0, b_1, b_2, \ldots$ be the first even numbers in the blocks $B_k \coloneqq [ \frac{(k-1)k}{2} + 1, \frac{k(k+1)}{2} ]$ $(k \in {\mathbb{N}})$. Now, define $$\pi(m) = \begin{cases}
m & m \text{ odd} \\
b_k & m = 4k + 2 \\
\min 2{\mathbb{N}}\setminus (\{ b_n: n \in {\mathbb{N}}\} \cup \pi ([1, m-1])) & m = 4k.
\end{cases}$$ We need the following technical result proved in [@Fac14 Proposition 6.6].
\[prop:constructed\_basis\_unconditional\] The basis $(f_m)_{m \in {\mathbb{N}}}$ is unconditional for $p \in (1, 2]$.
We are interested in the case $p > 2$. We make frequent use of the following technical observation. For a sequence $(a_m)_{m \in {\mathbb{N}}}$ let $(b_m) = (0, \ldots, 0, a_1, 0, \ldots, 0, a_2, \ldots)$ be a sequence built from $(a_m)_{m \in {\mathbb{N}}}$ by inserting zeros. Denote by ${\varphi}\colon {\mathbb{N}}\to {\mathbb{N}}$ the mapping which sends $k$ to the position of $a_k$ in the new sequence $(b_m)_{m \in {\mathbb{N}}}$. Then the following hold (for a proof of the first implication see [@Fac14 Lemma 6.7], the second can be proved analogously).
\[lem:technical\_lemma\] Let $p \in [1,\infty)$, $(a_m)_{m \in {\mathbb{N}}}$ be a sequence, $(b_m)_{m \in {\mathbb{N}}}$ and ${\varphi}\colon {\mathbb{N}}\to {\mathbb{N}}$ be as above and suppose that $$M \coloneqq \sup_{k \in {\mathbb{N}}} {\varphi}(k+1) - {\varphi}(k) < \infty.$$ If $(a_m)_{m \in {\mathbb{N}}} \in X_p$, then $(b_m)_{m \in {\mathbb{N}}} \in X_p$ as well. Conversely, if $(b_m)_{m \in {\mathbb{N}}} \in X_p$, then $(a_m)_{m \in {\mathbb{N}}} \in X_p$.
Recall that in the proof of Theorem \[thm:positive\_counterexamples\] the fundamental property of $(\gamma_m)_{m \in {\mathbb{N}}}$ used (as clarified in Lemma \[lem:maximize\_resolvent\_sequence\]) was that the ratios $$\label{eq:ratio}
\frac{\gamma_m - \gamma_{m-1}}{\gamma_m + \gamma_{m-1}}$$ are bounded from below. We now study more precisely the Schauder multipliers associated to various sequences $(\gamma_m)_{m \in {\mathbb{N}}}$ for the basis $(f_m)_{m \in {\mathbb{N}}}$.
As a starting point we make the very elementary observation that one can find sequences $(\gamma_m)_{m \in {\mathbb{N}}}$ for which the ratio has a prescribed growth.
\[lem:existence\_c\_m\] Let $(c_m)_{m \ge 2}$ be a sequence of real numbers with $c_m \in (0,\frac{1}{2})$ for all $m \in {\mathbb{N}}$. Then there exists a unique strictly increasing sequence $(\gamma_m)_{m \in {\mathbb{N}}}$ of real numbers with $\gamma_1 = 1$ and $$\label{eq:define_c_m}
\frac{1}{2} \frac{\gamma_m - \gamma_{m-1}}{\gamma_m + \gamma_{m-1}} = c_m \qquad \text{for all } m \ge 2.$$
We now formulate a necessary condition for the sequence $(c_m)_{m \in {\mathbb{N}}}$ that implies that the Schauder multiplier associated to the sequence $(\gamma_m)_{m \in {\mathbb{N}}}$ given by with respect to the basis $(f_m)_{m \in {\mathbb{N}}}$ is $\mathcal{R}$-sectorial.
\[prop:mr\_necessary\_condition\] Let $(c_m)_{m \ge 2}$ be a sequence with $c_m \in (0, \frac{1}{2})$ for all $m \ge 2$ and $(\gamma_m)_{m \in {\mathbb{N}}}$ the sequence given by Lemma \[lem:existence\_c\_m\]. Suppose that for some $p > 2$ the sectorial operator $A$ on $X_p$ given as the Schauder multiplier $$\begin{aligned}
D(A) = \biggl\{ x = \sum_{m=1}^{\infty} a_m f_m: \sum_{m=1}^{\infty} \gamma_m a_m f_m \quad \mathrm{ exists} \biggr\} \\
A \biggl( \sum_{m=1}^{\infty} a_m f_m \biggr) = \sum_{m=1}^{\infty} \gamma_m a_m f_m
\end{aligned}$$ is $\mathcal{R}$-sectorial. Then $(a_m c_{4m+2})_{m \in {\mathbb{N}}} \in X_p$ for all $(a_m)_{m \in {\mathbb{N}}} \in \ell_p$.
Observe that the basic sequence $(e_{\pi(4m+2)})_{m \in {\mathbb{N}}}$ is isometrically equivalent to the standard unit vector basis of $\ell_p$. Let $(a_m)_{m \in {\mathbb{N}}} \in \ell_p$. Then the Rademacher series $x = \sum_{m=1}^{\infty} r_m a_m e_{\pi(4m+2)}$ lies in $\operatorname{Rad}(X_p)$. One can now argue as in the proof of Theorem \[thm:positive\_counterexamples\]:
Let $(q_m)_{m \in {\mathbb{N}}} \subset {\mathbb{R}}_{-}$ be a sequence to be chosen later. Since $A$ is $\mathcal{R}$-sectorial by assumption, it follows from Proposition \[prop:resolvent\_r\_associated\_operator\] that the operator $\mathcal{R}\colon \operatorname{Rad}(X) \to \operatorname{Rad}(X)$ associated to the sequence $(q_m)_{m \in {\mathbb{N}}}$ is bounded. We now apply $\mathcal{R}$ to $x$. Because of $e_{\pi(4m+2)} = f_{4m+2} - f_{4m+1}$ we obtain $$\begin{aligned}
\mathcal{R}(x) & = \mathcal{R} \biggl( \sum_{m=1}^{\infty} r_m a_m (f_{4m+2} - f_{4m+1}) \biggr) \\
& = \sum_{m=1}^{\infty} r_m \frac{a_m q_m}{q_m - \gamma_{4m+2}} f_{4m+2} - r_m \frac{a_m q_m}{q_m - \gamma_{4m+1}} f_{4m+1} \\
& = \sum_{m=1}^{\infty} r_m \frac{a_m q_m}{q_m - \gamma_{4m+2}} (e_{\pi(4m+2)} + e_{4m+1}) - r_m \frac{a_m q_m}{q_m - \gamma_{4m+1}} e_{4m+1} \\
& = \sum_{m=1}^{\infty} r_m \frac{a_m q_m}{q_m - \gamma_{4m+2}} e_{\pi(4m+2)} \\
& \quad + r_m a_m q_m \biggl( \frac{1}{q_m - \gamma_{4m+2}} - \frac{1}{q_m - \gamma_{4m+1}} \biggr) e_{4m+1}.
\end{aligned}$$ By Lemma \[lem:maximize\_resolvent\_sequence\] one has for $t = \gamma_{4m+2}$ $$t [ (t + \gamma_{4m+2})^{-1} - (t + \gamma_{4m+1})^{-1} ] = -\frac{1}{2} \frac{\gamma_{4m+2} - \gamma_{4m+1}}{\gamma_{4m+2} + \gamma_{4m+1}} = -c_{4m+2}.$$ Hence, for the choice $q_m = -\gamma_{4m+2}$ we obtain $$\mathcal{R}(x) = \sum_{m=1}^{\infty} \frac{1}{2} r_m a_m e_{\pi(4m+2)} - c_{4m+2} r_m a_m e_{4m+1}.$$ As in the proof of Theorem \[thm:positive\_counterexamples\] one deduces that $\sum_{m=1}^{\infty} c_{4m+2} a_m e_{4m+1}$ converges in $X_p$. By Lemma \[lem:technical\_lemma\] this implies that $(a_m c_{4m+2})_{m \in {\mathbb{N}}} \in X_p$.
In the next step we prove a sufficient criterion for maximal regularity. In fact, we will establish the boundedness of the imaginary powers. Let $A$ denote the Schauder multiplier associated to some sequence $(\gamma_m)_{m \in {\mathbb{N}}}$ as above. It then follows from formula that the imaginary powers $A^{it}$ for $t \in {\mathbb{R}}$ act formally as $$\label{eq:ait}
\sum_{m=1}^{\infty} a_m e_m \mapsto \sum_{m=1}^{\infty} \tilde{\gamma}_m^{it} a_m e_m + \sum_{m=1}^{\infty} a_{2m} (\gamma_{\pi^{-1}(2m)}^{it} - \gamma_{\pi^{-1}(2m) - 1}^{it}) e_{\pi^{-1}(2m) - 1},$$ where $$\tilde{\gamma}_m = \begin{cases}
\gamma_m, & m \text{ odd} \\
\gamma_{\pi^{-1}(m)}, & m \text{ even}
\end{cases}.$$ It is clear that the first series of the right hand side of converges for all $(a_m)_{m \in {\mathbb{N}}} \in X_p$. The crucial point is therefore the question whether the second series, which by the unconditionality of the basis $(e_m)_{m \in {\mathbb{N}}}$ can be rewritten as $$\sum_{m=1}^{\infty} a_{2m} (\gamma_{\pi^{-1}(2m)}^{it} - \gamma_{\pi^{-1}(2m) - 1}^{it}) e_{\pi^{-1}(2m) - 1} = \sum_{m=1}^{\infty} a_{\pi(2m)} (\gamma_{2m}^{it} - \gamma_{2m-1}^{it}) e_{2m-1},$$ converges in $X_p$ for all $(a_m)_{m \in {\mathbb{N}}} \in X_p$. Equivalently by Lemma \[lem:technical\_lemma\], the sequence $(a_{\pi(2m)} (\gamma_{2m}^{it} - \gamma_{2m-1}^{it}))_{m \in {\mathbb{N}}}$ must lie in $X_p$ for all $(a_m)_{m \in {\mathbb{N}}} \in X_p$. We now give a sufficient condition. Here we use the fact that if a sectorial operator $A$ on some UMD-space has bounded imaginary powers of polynomial growth, then $A$ is $\mathcal{R}$-sectorial with $\omega_{R}(A) = 0$ [@DHP03 Theorem 4.5].
\[prop:mr\_sufficient\_condition\] Let $(c_m)_{m \in {\mathbb{N}}}$ be a sequence with $c_m \in (0, \frac{1}{8})$ for all $m \ge 2$ and let $(\gamma_m)_{m \in {\mathbb{N}}}$ be the sequence given by Lemma \[lem:existence\_c\_m\]. Consider for $p > 2$ the sectorial operator $A$ on $X_p$ defined as $$\begin{aligned}
D(A) = \biggl\{ x = \sum_{m=1}^{\infty} a_m f_m: \sum_{m=1}^{\infty} \gamma_m a_m f_m \quad \mathrm{ exists} \biggr\} \\
A \biggl( \sum_{m=1}^{\infty} a_m f_m \biggr) = \sum_{m=1}^{\infty} \gamma_m a_m f_m.
\end{aligned}$$ If $(b_m c_{2m})_{m \in {\mathbb{N}}}$ lies in $X_p$ for all $(b_m)_{m \in {\mathbb{N}}} \in \ell_p$, then $A$ has bounded imaginary powers with linear growth. In particular, $A$ is $\mathcal{R}$-sectorial with $\omega_R(A) = 0$.
A short calculation shows that one has for all $m \in {\mathbb{N}}$ $$\begin{aligned}
{\lvert\gamma_{2m}^{it} - \gamma_{2m-1}^{it}\rvert}^{2} & = {\left\lvert\exp(it \log \gamma_{2m}) - \exp(it \log \gamma_{2m-1})\right\rvert}^2 \\
& = {\lvert\exp(it \log \gamma_{2m})\rvert}^2 + {\lvert\exp(it \log \gamma_{2m-1})\rvert}^2 \\
& \qquad - 2 {\operatorname{Re}}\exp(it (\log(\gamma_{2m-1} - \log \gamma_{2m}))) \\
& = 2(1 - \cos(t (\log \gamma_{2m-1} - \log \gamma_{2m}))).
\end{aligned}$$ Here we have used the identity $${\left\lvertz-w\right\rvert}^2 = (z-w)(\overline{z} - \overline{w}) = {\left\lvertz\right\rvert}^2 + {\left\lvertw\right\rvert}^2 - (z \overline{w} + \overline{z} w) = {\left\lvertz\right\rvert}^2 + {\left\lvertw\right\rvert}^2 - 2 {\operatorname{Re}}z \overline{w}.$$ Further, one has $$\begin{aligned}
{\left\lvert\log \gamma_{2m-1} - \log \gamma_{2m}\right\rvert} & = {\left\lvert\log \left( \frac{\gamma_{2m-1}}{\gamma_{2m}} \right)\right\rvert} = {\left\lvert\log \left( 1 - \frac{\gamma_{2m} - \gamma_{2m-1}}{\gamma_{2m}} \right)\right\rvert} \\
& \le {\left\lvert \log \left( 1- 2 \frac{\gamma_{2m} - \gamma_{2m-1}}{\gamma_{2m} + \gamma_{2m-1}} \right)\right\rvert} = {\left\lvert\log ( 1 - 4c_{2m})\right\rvert}.
\end{aligned}$$ It follows from elementary calculus that $1 - \cos x \le \frac{x^2}{2}$ for all $x \in {\mathbb{R}}$. In particular, we obtain the estimate $$\begin{aligned}
2(1 - \cos(t (\log \gamma_{2m-1} - \log \gamma_{2m})) \le t^{2} \log^2(1-4c_{2m}).
\end{aligned}$$ A further elementary estimate from calculus is that ${\left\lvert\log(1-4x)\right\rvert} \le 8x$ holds for all $x \in [0,\frac{1}{8}]$. Therefore we see that for all $m \in {\mathbb{N}}$ one has $$\label{eq:img_powers_fundamental_inequality}
{\lvert\gamma_{2m}^{it} - \gamma_{2m-1}^{it}\rvert} \le 8 {\left\lvertt\right\rvert} c_{2m}.$$ Now, let $(a_m)_{m \in {\mathbb{N}}} \in X_p$. Since $p > 2$, we have the inclusion $X_p \hookrightarrow \ell_p$. Hence, $(a_{\pi(2m)})_{m \in {\mathbb{N}}} \in \ell_p$. By assumption, the mapping $(b_m)_{m \in {\mathbb{N}}} \mapsto (b_m c_{2m})_{m \in {\mathbb{N}}}$ from $\ell_p$ into $X_p$ is well-defined and closed. Hence, by the closed graph theorem there exists a constant $C \ge 0$ such that ${\left\lVert(c_{2m} b_m)\right\rVert}_{X_p} \le C {\left\lVert(b_m)\right\rVert}_{\ell_p}$ for all $(b_m)_{m \in {\mathbb{N}}}$ in $\ell_p$. Hence, we obtain that $(a_{\pi(2m)} c_{2m})_{m \in {\mathbb{N}}} \in X_p$ with $${\lVert(a_{\pi(2m)} c_{2m})\rVert}_{X_p} \le C {\lVert(a_{\pi(2m)})\rVert}_{\ell_p} \le C {\left\lVert(a_m)\right\rVert}_{X_p}.$$ It is now a direct consequence of equation that $((\gamma_{2m}^{it} - \gamma_{2m-1}^{it}) a_{\pi(2m)}) \in X_p$ with $${\lVert((\gamma_{2m}^{it} - \gamma_{2m-1}^{it}) a_{\pi(2m)})\rVert}_{X_p} \le 8 C {\left\lvertt\right\rvert} {\left\lVert(a_m)\right\rVert}_{X_p}.$$ Altogether this shows that $A$ has bounded imaginary powers with ${\lVertA^{it}\rVert} \le K (1 + {\left\lvertt\right\rvert})$ for some constant $K > 0$.
The same conditions as in Proposition \[prop:mr\_sufficient\_condition\] even imply the boundedness of the $H^{\infty}$-calculus for $A$ (for the necessary background see [@KunWei04] and [@DHP03]). For this let first $g \in H^{\infty}(H_{\omega})$, where $H_{\omega} \coloneqq \{ z \in {\mathbb{C}}: {\left\lvert{\operatorname{Im}}z\right\rvert} < \omega \}$ is the strip of height $\omega > 0$. Then for $z \in {\mathbb{R}}$ it follows from Cauchy’s integral formula that for all $\tilde{\omega} \in (0, \omega)$ and all $k \ge 1$ $$\begin{aligned}
\frac{{\left\lvertg^{k}(z)\right\rvert}}{k!} & = {\left\lvert\frac{1}{2\pi i} \left( \int_{-\infty + i\tilde{\omega}}^{\infty + i\tilde{\omega}} - \int_{-\infty - i\tilde{\omega}}^{\infty - i\tilde{\omega}} \right) \frac{g(w)}{(w-z)^{k+1}} {\mathop{}\!d}w\right\rvert} \\
& \le \frac{{\left\lVertg\right\rVert}_{H_{\omega}}}{\pi} \int_{-\infty}^{\infty} \frac{1}{{\left\lverts + i\tilde{\omega} - z\right\rvert}^{k+1}} {\mathop{}\!d}s = \frac{{\left\lVertg\right\rVert}_{H_{\omega}}}{\pi} \int_{-\infty}^{\infty} \frac{1}{(s^2 + \tilde{\omega}^2)^{(k+1)/2}} {\mathop{}\!d}s \\
& = \frac{{\left\lVertg\right\rVert}_{H_{\omega}}}{\pi} \tilde{\omega}^{-k} \int_{-\infty}^{\infty} \frac{1}{(1+s^2)^{(k+1)/2}} {\mathop{}\!d}s \le \tilde{\omega}^{-k} {\left\lVertg\right\rVert}_{H_{\omega}},
\end{aligned}$$ where $H_{\omega}$ is endowed with the supremum norm. Hence, we obtain for $z_0, z \in {\mathbb{R}}$ with ${\left\lvertz-z_0\right\rvert} < \tilde{\omega}$ that $$\begin{aligned}
{\left\lvertg(z) - g(z_0)\right\rvert} \le \sum_{k=1}^{\infty} \omega^{-k} {\left\lVertg\right\rVert}_{H_{\omega}} {\left\lvertz-z_0\right\rvert}^k \le C_{\tilde{\omega}} {\left\lVertg\right\rVert}_{H_{\omega}} {\left\lvertz-z_0\right\rvert}
\end{aligned}$$ for a universal constant $C_{\tilde{\omega}} > 0$. Using the notation from Proposition \[prop:mr\_sufficient\_condition\] we obtain for $f \in H^{\infty}(\Sigma_{\theta})$ and $\theta > 0$ $$\begin{aligned}
\MoveEqLeft {\left\lvertf(\gamma_{2m}) - f(\gamma_{2m-1})\right\rvert} = {\left\lvert(f \circ \exp)(\log \gamma_{2m}) - (f \circ \exp)(\log \gamma_{2m-1})\right\rvert} \\
& \le C_{\tilde{\theta}} {\left\lVertf \circ \exp\right\rVert}_{H_{\theta}} {\left\lvert\log \gamma_{2m} - \log \gamma_{2m-1}\right\rvert} \le 8 C_{\tilde{\theta}} {\left\lVertf\right\rVert}_{\Sigma_{\theta}} c_{2m}
\end{aligned}$$ using the estimates from the proof of Proposition \[prop:mr\_sufficient\_condition\] provided $8c_{2m} < \tilde{\theta}$ for some $\tilde{\theta} \in (0, \theta)$. By the above assumptions $(c_m)_{m \in {\mathbb{N}}}$ is a zero sequence and therefore this condition is satisfied for sufficiently large $m$ and we can deduce the boundedness of the $H^{\infty}$-calculus for $A$ with $\omega_{H^{\infty}}(A) = 0$ as in the above proof.
For a special type of sequences $(c_m)_{m \in {\mathbb{N}}}$ one can use the above results to obtain a complete characterization of maximal regularity.
\[cor:characterization\_mr\_via\_sequences\] Let $(c_m)_{m \in {\mathbb{N}}}$ be an eventually decreasing sequence with $c_m \in (0, \frac{1}{8})$ for all $m \ge 2$ and $(\gamma_m)_{m \in {\mathbb{N}}}$ the sequence given by Lemma \[lem:existence\_c\_m\]. Consider for $p > 2$ the sectorial operator $A$ on $X_p$ defined by $$\begin{aligned}
D(A) = \biggl\{ x = \sum_{m=1}^{\infty} a_m f_m: \sum_{m=1}^{\infty} \gamma_m a_m f_m \quad \mathrm{ exists} \biggr\} \\
A \biggl( \sum_{m=1}^{\infty} a_m f_m \biggr) = \sum_{m=1}^{\infty} \gamma_m a_m f_m.
\end{aligned}$$ Then $A$ is $\mathcal{R}$-sectorial if and only if $(c_m)_{m \in {\mathbb{N}}} \in (\oplus_{n=1}^{\infty} \ell_q^n)_{\ell_{\infty}}$, where $\frac{1}{2} = \frac{1}{p} + \frac{1}{q}$. Moreover, in this case one has $\omega_R(A) = 0$.
Clearly, it suffices to show the corollary for decreasing sequences. As a first observation we show that both the conditions in Proposition \[prop:mr\_necessary\_condition\] and Proposition \[prop:mr\_sufficient\_condition\] are equivalent to: $(a_m c_m)_{m \in {\mathbb{N}}} \in X_p$ for all $(a_m)_{m \in {\mathbb{N}}} \in \ell_p$. We show only the non-trivial implication for the condition in Proposition \[prop:mr\_sufficient\_condition\]. Of course, for the condition in Proposition \[prop:mr\_necessary\_condition\] the proof is completely analogous. So assume that $(a_m c_{2m})_{m \in {\mathbb{N}}} \in X_p$ for all $(a_m)_{m \in {\mathbb{N}}} \in \ell_p$. Now let $(a_m)_{m \in {\mathbb{N}}} \in \ell_p$. In order to show that $(a_m c_{m})_{m \in {\mathbb{N}}} \in X_p$, by Lemma \[lem:technical\_lemma\], it suffices to show that $(a_{2m} c_{2m})_{m \in {\mathbb{N}}}$ and $(a_{2m+1} c_{2m+1})_{m \in {\mathbb{N}}}$ lie in $X_p$. For the first sequence this follows directly from the assumption and for the second this follows from the monotonicity of $(c_m)_{m \in {\mathbb{N}}}$.
Hence, we have shown that $A$ is $\mathcal{R}$-sectorial if and only if $(a_m c_m)_{m \in {\mathbb{N}}} \in X_p$ for all $(a_m)_{m \in {\mathbb{N}}} \in \ell_p$. In this case, by the closed graph theorem, there exists a constant $M \ge 0$ such that $$\label{eq:characterizing_inclusion}
{\left\lVert(a_m c_m)\right\rVert}_{X_p} \le M {\left\lVert(a_m)\right\rVert}_{\ell_p}$$ for all $(a_m)_{m \in {\mathbb{N}}} \in \ell_p$. Now, it remains to show that this condition is equivalent to $(c_m)_{m \in {\mathbb{N}}} \in (\oplus_{n=1}^{\infty} \ell_q^n)_{\ell_{\infty}}$. On the one hand it follows from Hölder’s inequality that for $(c_m)_{m \in {\mathbb{N}}} \in (\oplus_{n=1}^{\infty} \ell_q^n)_{\ell_{\infty}}$ one has $$\begin{aligned}
{\left\lVert(a_m c_m)\right\rVert}_{X_p} \le \sup_{m \in {\mathbb{N}}} \bigg( \sum_{k \in B_m} {\left\lverta_k\right\rvert}^q \bigg)^{1/q} \bigg( \sum_{m=1}^{\infty} {\left\lverta_m\right\rvert}^p \bigg)^{1/p},
\end{aligned}$$ which is . On the other hand it follows from that for all $n \in {\mathbb{N}}$ $$\sup_{{\left\lVert(a_m)\right\rVert}_{\ell_p} \le 1} \bigg( \sum_{k \in B_n} {\left\lverta_k c_k\right\rvert}^2 \bigg)^{1/2} \le M.$$ This implies that for all $n \in {\mathbb{N}}$ one has $$\bigg( \sum_{k \in B_n} {\left\lvertc_k\right\rvert}^q \bigg)^{1/q} \le M.$$ In other words one has $(c_m)_{m \in {\mathbb{N}}} \in (\oplus_{n=1}^{\infty} \ell_q^n)_{\ell_{\infty}}$. This finishes the proof.
We now give two fundamental examples for $(c_m)_{m \in {\mathbb{N}}}$. First, for $c_m = k^{-\alpha}$ for $m \in B_k$ and $\alpha \in (0, \frac{1}{2})$ one has $(c_m)_{m \in {\mathbb{N}}} \in (\oplus_{n=1}^{\infty} \ell_q^n)_{\ell_{\infty}}$ if and only if $p \le \frac{2}{1-2\alpha}$. Second, $c_m = k^{-\alpha} \log k$ for $m \in B_k$ lies in $(\oplus_{n=1}^{\infty} \ell_q^n)_{\ell_{\infty}}$ if and only if $p < \frac{2}{1-2\alpha}$. These two families of sequences can now be used to obtain a complete answer to the maximal regularity extrapolation problem.
\[thm:extrapolation\_mr\_complete\_counterexample\] Let $I \subset (1, \infty)$ be an arbitrary interval with $2 \in I$. Then there exists a family of consistent analytic $C_0$-semigroups $(T_p(z))_{z \in \Sigma_{\frac{\pi}{2}}}$ on $L_p({\mathbb{R}})$ for $p \in (1, \infty)$ such that $(T_p(z))_{z \in \Sigma_{\frac{\pi}{2}}}$ has maximal regularity (resp. bounded imaginary powers / a bounded $H^{\infty}$-calculus) if and only if $p \in I$.
Let $I$ be such an interval and let $p_0$ be the right end of $I$. We first construct a family $(T_p(z))_{z \in \Sigma_{\frac{\pi}{2}}}$ that has maximal regularity if and only if $p \in (1,2) \cup I$. For $p_0 = 2$ this has already been done in [@Fac14 Corollary 6.8]. So we may assume $p_0 > 2$. Choose $c_m = k^{-\alpha}$ for $m \in B_k$ and $\alpha = \frac{p_0 - 2}{2p_0}$ if $p_0 \in I$ or $c_m = k^{-\alpha} \log k$ for $m \in B_k$ and $\alpha = \frac{p_0 - 2}{2p_0}$ if $p_0 \not\in I$ multiplied by appropriate scaling constants such that $c_m \in (0, \frac{1}{8})$ for all $m \ge 2$. Then it follows from Corollary \[cor:characterization\_mr\_via\_sequences\] and the above calculations that the analytic semigroups on $X_p$ for $p \in (1, \infty)$ whose generators are the Schauder multipliers associated the sequence $(-\gamma_m)_{m \in {\mathbb{N}}}$ given by Lemma \[lem:existence\_c\_m\] with respect to the basis $(f_m)_{m \in {\mathbb{N}}}$ have maximal regularity for $p \in (2, \infty)$ if and only if $p \in I \cap (2, \infty)$. Moreover, it follows from Proposition \[prop:constructed\_basis\_unconditional\] and [@Fac14 Theorem 2.5] that these semigroups have maximal regularity for $p \in (1,2]$. In order to obtain consistent semigroups $(T_p(z))_{z \in \Sigma_{\frac{\pi}{2}}}$ on $L_p$ which have maximal regularity if and only if $p \in (1,2) \cup I$, one needs to transfer the just constructed example consistently in $p \in (1, \infty)$ from the $X_p$- to the $L_p$-scale. In fact, this can be done as in the proof of [@Fac14 Theorem 6.3]:
From the Khintchine inequality one obtains consistent isomorphisms $X_p = (\oplus_{n=1}^{\infty} \ell_2^n)_{\ell_p} \xrightarrow{\sim} (\oplus_{n=1}^{\infty} \operatorname{Rad}_n)_{\ell_p}$, where $\operatorname{Rad}_n$ is the span of the first $n$ Rademacher functions in $L_p([0,1])$. Hence, $(\oplus_{n=1}^{\infty} \operatorname{Rad}_n)_{\ell_p}$ can be identified with a closed subspace of $L_p([0,\infty))$. Together with the projection given by the direct sum of the consistent Rademacher projections $L_p([0,1]) \to \operatorname{Rad}_n$ we are able to transport the counterexample consistently to $L_p([0,\infty))$.
Taking dual semigroups, it follows from the first part of the proof that there exist consistent analytic $C_0$-semigroups $(S_p(z))_{z \in \Sigma_{\frac{\pi}{2}}}$ on $L_p([0,\infty))$ for $p \in (1, \infty)$ such that $(S_p(z))_{z \in \Sigma_{\frac{\pi}{2}}}$ has maximal regularity if and only if $p \in (2,\infty) \cup I$. Taking the direct sum of $(T_p(z))_{z \in \Sigma_{\frac{\pi}{2}}}$ and $(S_p(z))_{z \in \Sigma_{\frac{\pi}{2}}}$ one obtains the desired family of semigroups.
It is an open problem whether every generator of a bounded analytic $C_0$-semigroup on a uniformly convex UMD-space that is contractive on ${\mathbb{R}}_{\ge 0}$ has maximal regularity or even a bounded $H^{\infty}$-calculus.
We now comment on what we know about the contractivity of the semigroups considered above. First, it is easy to see that on $X_2 = \ell_2$ the semigroup given by a sequence $(c_m)_{m \in {\mathbb{N}}}$ is contractive for all allowed $(c_m)_{m \in {\mathbb{N}}}$. Further, on $X_{\infty} \coloneqq (\oplus_{n=1}^{\infty} \ell_2^n)_{c_0}$ one can again use the standard basis $(e_m)_{m \in {\mathbb{N}}}$ to define the Schauder basis $(f_m)_{m \in {\mathbb{N}}}$ and the sectorial operators as for example formulated in Corollary \[cor:characterization\_mr\_via\_sequences\]. Using the same notation as before, the operator $B = -A$ given by a sequence $(c_m)_{m \in {\mathbb{N}}}$ is the generator of a bounded analytic $C_0$-semigroup on $X_{\infty}$. Thus, by the Lumer–Phillips theorem, $B$ generates a contractive semigroup on $X_{\infty}$ if and only if $B$ is dissipative, i.e. if for all $x \in D(B)$ there exists an $x^* \in J(x) \coloneqq \{ x^* \in X_{\infty}^*: \langle x, x^* \rangle = {\left\lVertx\right\rVert}^2 = {\left\lVertx^*\right\rVert}^2 \}$ such that ${\operatorname{Re}}\langle Bx, x^* \rangle \le 0$. We now study this condition.
Observe that one has $Be_m = -\gamma_m e_m$ for odd $m$ and $Be_{\pi(m)} = -(\gamma_m - \gamma_{m-1})e_{m-1} - \gamma_m e_{\pi(m)}$ for even $m$. Then one has for $x \in D(B)$ and $x^* \in X_{\infty}^*$ $$\begin{aligned}
\langle Bx, x^* \rangle & = -\sum_{m \text{ odd}} \gamma_m x_m x^*_m - \sum_{m \text{ even}} \gamma_{\pi^{-1}(m)} x_{m} x^*_m \\
& - \sum_{m \text{ odd}} (\gamma_{m+1} - \gamma_m) x_{\pi(m+1)} x^*_{m}.
\end{aligned}$$ Now, choose $k \in {\mathbb{N}}$ and $x_m = \frac{\gamma_{m+1} - \gamma_m}{2 \gamma_m}$ for $m \in B_k$ and $m \equiv 1 \mod 4$ and $x_{\pi(m+1)} = -1$ for $m \in B_k$ and $m+1 \equiv 2 \mod 4$ and $x_m = 0$ otherwise. Notice that if $k$ is sufficiently large and $\sum_{m \in B_k} {\left\lvertx_m\right\rvert}^2 > 1$, then $x^* = \sum_{m \in B_k} x_m e_m$ is the unique element in $J(x)$. One therefore obtains $$\begin{aligned}
\langle Bx, x^* \rangle & = -\sum_{\substack{m \in B_k: \\m \equiv 1 \mod 4}} \gamma_m {\left\lvertx_m\right\rvert}^2 + \sum_{\substack{m \in B_k: \\m \equiv 1 \mod 4}} (\gamma_{m+1} - \gamma_m) {\left\lvertx_m\right\rvert} \\
& = \sum_{\substack{m \in B_k: \\m \equiv 1 \mod 4}} \frac{1}{4} \frac{(\gamma_{m+1} - \gamma_m)^2}{\gamma_m} > 0.
\end{aligned}$$ If $(c_m)_{m \in {\mathbb{N}}} \not\in (\oplus_{n=1}^{\infty} \ell_2^n)_{\ell_{\infty}}$, one has $\sum_{m \in B_k} {\left\lvertx_m\right\rvert}^2 > 1$ for sufficiently large $k$ because of the monotonicity of $(c_m)_{m \in {\mathbb{N}}}$ and the estimate $c_m \le \frac{\gamma_{m+1} - \gamma_m}{2 \gamma_m} \le K c_m$ for some $K > 0$ and all $m \in {\mathbb{N}}$. Hence, it follows from the above calculation that $(c_m)_{m \in {\mathbb{N}}} \not\in (\oplus_{n=1}^{\infty} \ell_2^n)_{\ell_{\infty}}$ implies the non-contractivity of the semigroup. Notice that $(c_m)_{m \in {\mathbb{N}}} \not\in (\oplus_{n=1}^{\infty} \ell_2^n)_{\ell_{\infty}}$ is exactly the limit case for $p \to \infty$ of the condition given in Corollary \[cor:characterization\_mr\_via\_sequences\].
We do not know whether analogously for $p \in (2, \infty)$ the condition $(c_m)_{m \in {\mathbb{N}}} \not\in (\oplus_{n=1}^{\infty} \ell_q^n)_{\ell_{\infty}}$ with $\frac{1}{2} = \frac{1}{p} + \frac{1}{q}$ implies that the generated semigroup is not contractive.
[^1]: The author was supported by a scholarship of the “Landesgraduiertenförderung Baden-Württemberg”.
|
---
abstract: |
Expectation for the emergence of higher functions is getting larger in the framework of end-to-end comprehensive reinforcement learning using a recurrent neural network. However, the emergence of “thinking” that is a typical higher function is difficult to realize because “thinking” needs non fixed-point, flow-type attractors with both convergence and transition dynamics. Furthermore, in order to introduce “inspiration” or “discovery” in “thinking”, not completely random but unexpected transition should be also required.
By analogy to “chaotic itinerancy”, we have hypothesized that [**“exploration” grows into “thinking” through learning by forming flow-type attractors on chaotic random-like dynamics**]{}. It is expected that if rational dynamics are learned in a chaotic neural network (ChNN), coexistence of rational state transition, inspiration-like state transition and also random-like exploration for unknown situation can be realized.
Based on the above idea, we have proposed [**new reinforcement learning using a ChNN**]{}. The positioning of exploration is completely different from the conventional one. Here, [**the chaotic dynamics inside the ChNN produces exploration factors by itself**]{}. Since external random numbers for stochastic action selection are not used, exploration factors cannot be isolated from the output. Therefore, the learning method is also completely different from the conventional one.
One variable named causality trace is put at each connection, and takes in and maintains the input through the connection according to the change in its output. Using these causality traces and TD error, the connection weights except for the feedback connections are updated in the actor ChNN.
In this paper, as the result of a recent simple task to see whether the new learning works appropriately or not, it is shown that a robot with two wheels and two visual sensors reaches a target while avoiding an obstacle after learning though there are still many rooms for improvement.
author:
- |
Katsunari Shibata[^1] & Yuki Goto\
Department of Innovative (Electrical and Electronic) Engineering\
Oita University\
700 Dannoharu, Oita 870-1192, JAPAN\
`katsunarishibata@gmail.com`\
title: |
New Reinforcement Learning Using a Chaotic Neural Network\
for Emergence of “Thinking”\
— “Exploration” Grows into “Thinking” through Learning —
---
Introduction
============
Expectation for the emergence of artificial intelligence is growing these days triggered by the recent results in reinforcement learning (RL) using a deep neural network (NN)[@DQN; @AlphaGo]. Our group has propounded for around 20 years that end-to-end RL from sensors to motors using a recurrent NN (RNN) plays an important role for the emergence[@Intech; @RLDM17]. Especially, different from “recognition” whose inputs are given as sensor signals or “control” whose outputs are given as motor commands, higher functions are very difficult to design by human hands, and the function emergence approach through end-to-end RL is highly expected. Our group has shown that not only recognition and motion, but also memory, prediction, individuality, and also similar activities to those in monkey brain at tool use emerge[@RLDM17]. We have also shown that a variety of communications emerge in the same framework[@RLDM17COM]. However, the emergence of what can be called “thinking” that is one of the typical higher functions has not been shown yet. In this paper, the difficulty of emergence of “thinking” is discussed at first. Then our hypothesis that “exploration” grows into “thinking” through learning is introduced[@IJCNN15]. To realize the hypothesis, the use of a chaotic NN (ChNN) in RL and new deterministic RL for it are introduced[@IJCNN15]. Finally, it is shown that the new RL works in a simple task[@JCSS] though that cannot be called “thinking” yet and there are still many rooms for improvement. No other works with a similar direction to ours have not been found.
Difficulty in Emergence of “Thinking”
=====================================
The definition of “Thinking” must be varied depending on the person. However, we can “think” even when we close our eyes and ears, and what we think does not change randomly, but logically or rationally. Therefore, we hope many ones can agree that in order to realize “thinking”, rational multi-stage or flow-type state transition should be formed. As a kind of dynamic functions, we have shown that a variety of memory-required functions emerge in an RNN in simple tasks[@RLDM17]. It is not so difficult to form memories as fixed-point convergence dynamics if the initial feedback connection weights are set such that the transition matrix for the connection is the identity matrix or close to it when being linearly approximated. That can also solve the vanishing gradient problem in error back propagation. However, state transition needs not only convergence dynamics for association, but also transition dynamics from one state to another.
Then, we employed a multi-room task in which an agent moves and when it pushes a button, one of the doors opens and then the agent can move to another room. The sensor signals are the inputs of the agent’s RNN, and the RNN was trained based on RL from a reward when it reached the goal. We expected that the internal state changed drastically between before and after the door open though the difference in the sensor signals was not so large. After learning, a large change in the internal state could be observed in some degree, but the learning was very difficult[@Sawatsubashi].
Furthermore, when we “think”, “inspiration” or “discovery”, which is a kind of unexpected but not completely random and rational transition, must be also essential. The convergence and transition dynamics seem contradict at a glance, and it seems very difficult to form both dynamics from scratch in a regular RNN.
[ccc]{}
![“Exploration” and “thinking” are both a kind of internal dynamics and deeply related with each other.[]{data-label="fig:ExplorationThinking"}](ForkProblem.eps){height="3.4cm"}
![“Exploration” and “thinking” are both a kind of internal dynamics and deeply related with each other.[]{data-label="fig:ExplorationThinking"}](ExplorationThinking.eps){height="3.7cm"}
Chaos and Hypothesis: Growth from “Exploration” to “Thinking” through Learning
==============================================================================
Suppose we are standing at a forked road as shown in Fig. \[fig:Fork\]. Usually, we choose one from two options: going right and going left. We do not care many other possible actions such as going straight and dancing. It is not motor(actuator)-level lower exploration, but higher exploration supported by some prior or learned knowledge[@Higher]. Furthermore, at a fork, we may wonder such that “this path looks more rough, but that way looks to go away from the destination”. This can be considered as a kind of “exploration” and also as a kind of “thinking”. The place where the wave in mind occurs is inside the process before making a decision, and learning should be reflected largely on the exploration. The author’s group has thought that exploration should be generated inside of a recurrent NN (RNN) that generates motion commands[@Exploration]. “Exploration” and “thinking” are both generated as internal dynamics as shown in Fig. \[fig:ExplorationThinking\]. “Exploration” is more random-like state transitions. On the other hand, “thinking” is more rational or logical state transition, and sometimes higher exploration or unexpected but rational state transition such as inspiration or discovery occurs in it.
![Rough schematic diagram of the combination of attractors and chaotic dynamics. See text in detail.[]{data-label="fig:AttractorsAndChaos"}](AttractorsAndChaos.eps){height="6.9cm"}
Analogy to such dynamics can be found in chaotic dynamics. In regular associative memory, fixed-point attractor (basin) is formed around each memorized or learned pattern as shown in the upper left part in Fig. \[fig:AttractorsAndChaos\]. However, when a ChNN is used in it, transition dynamics among the memorized patters called “Chaotic Itinerancy”[@Itinerancy] can be seen as the green arrows in the lower left part in Fig. \[fig:AttractorsAndChaos\]. If rational or logical state transitions are learned, it is expected that flow-type attractors are formed as the red or black arrows in the lower right part in Fig. \[fig:AttractorsAndChaos\]. It is also expected that as the green arrows, (1)inspiration or discovery emerges as irregular transitions among the attractors but reflecting the distance, (2)higher exploration emerges at branches of the flow, and (3)in unknown situations, no attractor is formed, and remaining random-like chaotic dynamics appears until return to a known situation. Skarda et al. reported that the activities on the olfactory bulb in rabbits become chaotic for unknown stimuli[@Skarda]. Osana et al. have shown the difference of chaotic property between known and unknown patterns on an associative memory using a ChNN, and also after an unknown pattern is learned, association to the pattern is formed as well as the other known patterns[@Osana]. From the above discussion, we have hypothesized that [**“exploration” grows into “thinking” through learning by forming flow-type attractors on chaotic random-like dynamics and that can be realized on reinforcement learning using a ChNN**]{}.
New Reinforcement Learning (RL) Using a Chaotic Neural Network (ChNN)
=====================================================================
In order to realize the above idea, our group has proposed new reinforcement learning using a ChNN[@IJCNN15]. The positioning of exploration in learning is completely different from the conventional one. Here, the chaotic dynamics inside the ChNN produces exploration factors by itself. Since external random numbers for stochastic action selection are not used, exploration factors cannot be isolated from the output. Then the learning method has to be completely different from the conventional one.
Assuming that the motions are continuous, actor-critic type reinforcement learning architecture is employed. Here, to isolate the chaotic dynamics from the critic, the actor is implemented in a ChNN and the critic is implemented in another regular layered NN as shown in Fig. \[fig:Task\]. The inputs are the sensor signals for the both networks. Here, only the learning of actor ChNN that is largely different from the conventional reinforcement learning is explained comparing with the conventional one using Fig. \[fig:ChaosNN\]. In our conventional works, as shown in Fig. \[fig:ChaosNN\](a), by adding a random number (noise) $rnd_{j,t}$ to each actor output, the agent explores. The actor network is trained by BPTT (Back Propagation Through Time) using the product of the random number and TD error as the error for each output of the RNN.
In the proposed method, there is no external random numbers added to the actor outputs. The network is a kind of RNN, but by setting each feedback connection to a large random value, it can produce chaotic dynamics internally. Because the learning of recurrent connections does not work well, only the connections from inputs to hidden neurons and from hidden neurons to output neurons are trained. One variable $C_{ji}$ named causality trace is put on each connection, and takes in and maintains the input through the connection according to the change in its output as $$C^{[l]}_{ji,t}=(1-|\Delta x^{[l]}_{j,t}|)C^{[l]}_{ji,t-1}+\Delta x^{[l]}_{j,t} x^{[l-1]}_{i,t}
\label{Eq:C-Trace-RL}$$ where $x^{[l]}_{j,t}$: output of the $j$-th neuron in the $l$-th layer at time $t$, $\Delta x_t = x_t - x_{t-1}$.\
Using the causality trace $C_{ji}$ and TD error ${\hat r}_t$, the weight $w^{[l]}_{ji}$ from the $i$-th neuron in $(l\!\!-\!\!1\!)$-th layer to the $j$-th neuron in the $l$-th layer is updated with a learning rate $\eta$ as $$\vspace{-1mm}
\Delta w^{[l]}_{ji,t}=\eta {\hat r_t} C^{[l]}_{ji,t}.
\label{Eq:LearnActor}$$
![Comparison of conventional RL and proposed RL\
(only actor network) [@IJCNN15][]{data-label="fig:ChaosNN"}](ChaosRL.eps){height="6.4cm"}
Learning of Obstacle Avoidance
==============================
[r]{}[80mm]{} {height="8.4cm"}
Since the learning is completely different from the conventional one, it is necessary to show whether the learning works appropriately in a variety of tasks or not. It was already applied to several tasks[@IJCNN15][@Higher]. In this paper, the result of a recent task in which a robot has two wheels and two visual sensors is shown[@JCSS]. As shown in Fig. \[fig:Task\], there is a $20 \!\!\times \!\!20$ size of field, and its center is the origin of the field. What the robot has to do is to reach a goal while avoiding an obstacle. The goal is put on the location (0, 8) with radius $r\!\!=\!\!1.0$, and a robot ($r\!\!=\!\!0.5$) and an obstacle ($r\!\!=\!\!1.5$) are put randomly at each episode. The orientation of the robot is also decided randomly. Each of the two omnidirectional visual sensors has 72 cells, and catches only goal or obstacle respectively. Total of 144 sensor signals are the inputs of both critic and actor networks, and the right and left wheels rotate according to the two actor outputs. When the robot comes in the goal area, a reward is given, and when it collides with the obstacle, a small penalty is given. One episode is defined as until arrival to the goal or 1000 steps from start. The both NNs have three layers including input layer. The number of hidden neurons is 10 for critic network and 100 for actor ChNN.
The learning results are shown in Fig. \[fig:Results\]. Fig. \[fig:Results\](a) shows learning curve and the vertical axis indicates the number of steps to the goal. The blue line shows its average over each 100 episodes. It can be seen that according to the number of episodes, the number of steps decreases. Fig. \[fig:Results\](b) and (c) show the robot trajectories before and after learning respectively. Actually, the goal location is fixed and initial robot location is varied. But to see the initial robot orientation, the robot is on the center of the field, and its orientation is for upper-side initially on the figure. Relatively, the goal location is varied. The size of the obstacle in the figure shows the area where the robot collides with the obstacle and then is larger than the actual size. Before learning, the robot explored almost randomly, but after learning, it could reach the goal while avoiding the obstacle. In conventional learning, randomness of action selection is often decreased as learning progresses, but here, it can be seen that exploration factors decreased autonomously according to learning. However, there are some strange behaviors such that the robot changes its direction suddenly. It was observed that the actor hidden neurons were likely to have a value around the maximum 0.5 or minimum -0.5. There is still a large space to improve the learning method.
Fig. \[fig:Results\](d) shows Lyapunov exponent to see the chaotic property of this system including the environment. The robot is located at one of the 8 locations in Fig. \[fig:Results\](c), and the obstacle is also located at one of 8 locations. For each of the $8 \times 8 = 64$ combinations, a small perturbation vector with the size $d_{before}$ is added to the internal states of hidden neurons, and the distance $d_{after}$ in the internal states at the next time between the cases of no perturbation and addition of the perturbation is compared. The average of $ln(d_{after}/d_{before})$ is used as the exponent here. From the figure, it can be seen that Lyapunov exponent is gradually decreased, but since the value is more than 0.0, it is considered that the chaotic property is maintained. In [@IJCNN15], it was observed that when the environment changed, Lyapunov exponent increased again.
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[^1]: http://shws.cc.oita-u.ac.jp/\~shibata/home.html
|
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Introduction
============
Saslaw and Zipoy (1967) realized the importance of gas phase molecule formation in primordial gas for the formation of proto–galactic objects. Employing this mechanism in Jeans unstable clouds, Peebles and Dicke (1968) formulated their model for the formation of primordial globular clusters. Further pioneering studies in this subject were carried out by Takeda (1969), Matsuda (1969), and Hirasawa (1969) who followed in detail the gas kinetics in collapsing objects and studied the possible formation of very massive objects (VMO’s). Hutchins (1976) then looked in greater detail at the effects of rotation and asked what minimum Jeans mass can be reached in a collapsing primordial gas cloud. In the 1980’s major contributions to this field were made by Rees and Kashlinsky (1983), Carr (1984), and Couchman and Rees (1986), who all studied the possible cosmological consequences of population III star formation. Their main conclusion was that for hierarchical structure formation scenarios the first objects might reheat and reionize the universe and thus raise the Jeans mass, influencing subsequent structure formation quite dramatically. Massive Pop III stars would also pre–enrich the intergalactic medium with metals.
It is clear that the uncertainties mentioned above arise due to the inherently multidimensional, nonlinear, nonequilibrium physics which determine the collapse and possible fragmentation of gravitationally and thermally unstable primordial gas clouds. The computational expense of solving the network of chemical rate equations forced early studies to focus on single cell calculations (cf. Hirasawa 1969; Hutchins 1976; Palla 1983; MacLow and Shull 1986; Puy 1996; Tegmark 1997) adopting simple collapse models. Bodenheimer (1986) was the first to address the hydrodynamic aspects of the problem, in spherical symmetry. Similarly, using a spherical Lagrangian hydrodynamics code and solving the kinetic rate equations simultaneously, Haiman, Thoul and Loeb (1996) studied the important question of which mass scales are able to cool efficiently enough to collapse. Unfortunately, the issue of fragmentation cannot reliably be addressed in such spherically symmetric models.
Multi–dimesional studies of first structure formation have only recently become computationally feasible (Abel 1995, Anninos & Norman 1996, Zhang 1997, Gnedin & Ostriker 1997, Abel 1998a, Abel 1998b). Specifically in the context of CDM–type structure formation models we have studied the collapse of high–$\sigma$ density fluctuations on small mass scales (Abel 1998, herafter AANZ98).
Here we present first results from three–dimensional adaptive mesh cosmological hydrodynamics following the collapse and fragmentation of the very first objects formed in hierarchical, CDM–like models of structure formation.
Simulations
===========
The three dimensional adaptive mesh refinement calculations presented here use for the hydrodynamic portion an algorithm very similar to the one described by Berger and Collela (1989). The code utilizes an adaptive hierarchy of grid patches at various levels of resolution. Each rectangular grid patch covers some region of space in its parent grid needing higher resolution, and may itself become the parent grid to an even higher resolution child grid. The general implimentation of AMR places no restriction on the number of grids at a given level of refinement, or the number of levels of refinement. Additionally the dark matter is followed with methods similar to the ones presented by Couchamn (1991). Furthermore, the algorithm of Anninos (1997) to solve the accurate time–dependent chemistry and cooling model for primordial gas of Abel (1997). Detailed description of the code are given in Bryan & Norman (1997,1998), and Norman & Bryan (1997,1998).
The simulations are initialized at redshift 100 with density perturbations of a sCDM model with $\Omega_B = 0.06$, $h=0.5$, and $\sigma_8=0.7$. The abundances of the 9 chemical species (H, , , He, , , , , e$^-$) and the temperature are initialized as discussed in Anninos and Norman (1996). After a collapsing high–$\sigma$ peaks has been identified in a low resolution simulation the simulations is reinitialized with multiple refinement levels on the Langrangian volume of the collapsing structure. The mass resolution in the inital conditions within this Langrangian volume are $0.53\Ms$ in the gas and $8.96\Ms$ for the dark matter component. The refinement criterium ensures the local Jeans length to be resolved by at least 4 grid zones as well as that no cell contains more than 4 times its initial mass of $0.53\Ms$. We limit the refinement to 12 levels within a $64^3$ top grid which translates to a maximum dynamical range of $64\times 2^{12}=262,144$. As we will show below the simulation is not resolution but physics limited.
Results
=======
In Figure \[profile\] we show mass weighted, spherically averaged quantities around the densest cell found in the simulation at redshift 19.1 of various interesting quantities. In panel c) we indicate two distinct regions. Region I) ranges from the virial radius to $r_{cool}\sim 5\pc$, the radius at which the infalling material has cooled down to $T\sim 200\K$. For most of this region the cooling time $t_{H2}$ is shorter than the free–fall time, $t_{ff}=[3\pi/(32G\rho)]^{1/2}$, as is illustrated in panel b) of Figure \[profile\]. The number fraction rises from $7\tento{-6}$ to $2\tento{-4}$ as the free electron fraction drops from $2\tento{-4}$ to $2\tento{-5}$ (panel a).
At $r_{cool}$ the free–fall time becomes smaller than the cooling time. Also at $r_{cool}$ the time it takes a sound wave to travel to the center, $t_{cross}=r/c_s=7.6\tento{6} r_{pc}/\sqrt{T_K}$yrs , becomes shorter than the cooling time. The cooling time approaches a constant value at small radii (high densities) due to the transition from non-LTE to LTE populations of the rotational/vibrational states. From the time scales one would conclude that below $r_{cool}$ the gas should evolve quasi–hydrostatically on the cooling time scale. This is, however, not strictly true. It turns out that centrigufal forces play an important role in the collapse. To illustrate this Figure \[fe\_profile\] shows the mass weighted radial profiles of various energies and force terms. From the lower panel of Figure \[fe\_profile\] it is evident that pressure + centrifugal forces dominate gravity in the range of $0.3<r/1\pc< 10$ yielding a net outward force (deceleration). Indeed the gas has zero radial velocity at $r=0.3\pc$, separating the contracting core of $\sim
100\Ms$ at smaller radii. There are multiple possible origins for a “centrifugal barrier” as the one observed at $r=0.3\pc$. The simplest explanation being angular momentum conservation. However, this would imply $v_\perp \times r =const.$ and the centrifugal force, $F_c$, would be proportional to $r^{-3}$. However, $F_c\simpr r^{-1}$ holds for more than three orders of magnitude, as can be seen from Figure \[fe\_profile\]. However, the cooling instability and the continous merging of small scale structure offer simple alternative explanations.
Discussion
==========
Multiple interesting features of the collapsing and fragmenting “primordial molecular cloud” are identified. Most notably is the fact of the existence of a contracting quasi–hydrostatic core of $\sim 100\Ms$. Within this core the number densities increase from $10^{5}$ to $10^8\cm^{-3}$. For densities $\gsim 10^6\cm^-3$, however, three–body formation of becomes the dominant formation mechanism transforming all hydrogen into its molecular form (Palla 1983). Our chemical reaction network does not include this reaction and the solution cannot be correct at $r\lsim 0.1\pc$. The most interesting effect of the tree–body reaction is that it will increase the cooling rate by a factor $\sim 10^3$ leading to a further dramatic density enhancement within the core. This will decrease the dynamical timescales to $\ll 100\yrs$ effectively “decoupling” the evolution of the fragment from the evolution of its “host primordial molecular cloud”. Silk (1983) has argued that (due the enhanced cooling from the 3–body produced ) fragmentation of this core might continue until individual fragments are opacity limited (i.e. they become opaque to their cooling radiation). Verifying this possibility will have to await yet higher resolution simulations.
Clearly, it is the evolution of the $\sim 100\Ms$ core that could participate in population III star formation. If it would form an open star cluster with 100% efficiency about $6\tento{63}$ UV photons would be liberated during the average life time of massive star ($\sim
5\tento{7}\yrs$). This is about hundred times more than the $\sim
4\tento{61}$ hydrogen atoms within the virial radius. However, the average recombination time $(nk_{rec})^{-1}\sim 5\tento{5}\yrs$ within the virial radius is a factor 100 less than the average lifetime of a massive star. Hence, one expects only very small or zero UV escape fractions for these objects.
The column density from the core to the virial radius is $\sim
10^{20}\cm^{-2}$. This is much larger than the typical $10^{14}\cm^{-2}$ required for self–shielding in stationary photodissociation regions (cf. Bertoldi and Draine 1996). Hence also many sub–Lyman limit photons will be absorbed locally and cannot reach the IGM. The host primordial molecular cloud is only transparent for photons below $\sim 11\eV$ where no Lyman Werner Band absorption can occur.
These optical depth arguments will almost certainly become incorrect once supernova explosion alter the hydrodynamics and the chemistry and cooling via collisional ionization, dissociation and shock heating. More detailed understanding of the role of local feedback will have to await new more detailed simulations.
Conclusions
===========
We have reported first results from an ongoing project that studies the physics of fragmentation and primordial star formation in the cosmological context. The discussed results clearly illustrate the advantages and power of structured adaptive mesh refinement cosmological hydrodynamic methods to cover a wide range of mass, length and timescales reliably. All findings of AANZ98 are confirmed in this study. Among other things, these are that
- a significant number fraction of hydrogen molecules is only found in structures at the intersection of filaments
- only a few percent of the gas in a virialized halo are cooled to $T\ll T_{vir}$.
The improvement of a factor $\sim 1000$ in resolution over AANZ98 has given new ininsights in the details of the fragmentation process and constraints on the possible nature of the first structures:
- Only $\lsim 1\%$ of the baryons within a virialized object can participate in population III star formation.
- The formation of super massive black holes or very massive objects in small halos seem very unlikely.
- If the gas were able to fragment further through 3–body and/or opacity limited fragmentation only a small fraction of the gas will be converted into small mass objects.
- The escape fraction of UV photons above the Lyman limit or in the Lyman Werner band should be small due to the high cloumn densities of HI ($N_{HI}\sim 10^{23}\cm^{-2}$) and ($N_{H_2}\sim
10^{20}\cm^{-2}$) in the surrounding of star forming regions.
[**Cautionary Remark:**]{} These latter conclusions are drawn from one simulation and are to be understood as preliminary results from an ongoing investigation.
Acknowledgments {#acknowledgments .unnumbered}
===============
Tom Abel acknowledges support from NASA grant NAG5-3923 and useful discussions with Karsten Jedamzik, Martin Rees, and Simon White.
[99]{}
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---
abstract: 'In this paper, we report our multiwavelength observations of the vertical oscillation of a coronal cavity on 2011 March 16. The elliptical cavity with an underlying horn-like quiescent prominence was observed by the Atmospheric Imaging Assembly (AIA) on board the *Solar Dynamics Observatory* (*SDO*). The width and height of the cavity are 150$\arcsec$ and 240$\arcsec$, and the centroid of cavity is 128$\arcsec$ above the solar surface. At $\sim$17:50 UT, a C3.8 two-ribbon flare took place in active region 11169 close to the solar western limb. Meanwhile, a partial halo coronal mass ejection (CME) erupted and propagated at a linear speed of $\sim$682 km s$^{-1}$. Associated with the eruption, a coronal extreme-ultraviolet (EUV) wave was generated and propagated in the northeast direction at a speed of $\sim$120 km s$^{-1}$. Once the EUV wave arrived at the cavity from the top, it pushed the large-scale overlying magnetic field lines downward before bouncing back. At the same time, the cavity started to oscillate coherently in the vertical direction and lasted for $\sim$2 cycles before disappearing. The amplitude, period, and damping time are 2.4$-$3.5 Mm, 29$-$37 minutes, and 26$-$78 minutes, respectively. The vertical oscillation of the cavity is explained by a global standing MHD wave of fast kink mode. To estimate the magnetic field strength of the cavity, we use two independent methods of prominence seismology. It is found that the magnetic field strength is only a few Gauss and less than 10 G.'
author:
- 'Q. M. Zhang'
- 'H. S. Ji'
title: Vertical oscillation of a coronal cavity triggered by an EUV wave
---
Introduction {#sec:intro}
============
Solar prominences are cool and dense plasma structures suspending in the corona [@lab10; @mac10; @par14]. The routine observation of prominences has a long history. They can be observed in radio, H$\alpha$, Ca[ii]{}H, He[i]{}10830 [Å]{}, extreme-ultraviolet (EUV), and soft X-ray (SXR) wavelengths [e.g., @gop03; @ji03; @ber08; @zqm15; @ste15; @wang16]. When prominences appear on the disk in the H$\alpha$ full-disk images, they look darker than the surrounding area due to the absorption of the background radiation. Therefore, they are also called filaments [@mar98]. Prominences (or filaments) can form in active regions (ARs), quiet region, and polar region [e.g., @hei08; @su15; @yan15]. It is generally believed that the gravity of prominences should be balanced by the upward magnetic tension force of the dips in sheared arcades [@jor92; @mar01; @xia12] or magnetic flux ropes [MFRs; @aul98; @hil13; @xia16; @yan18].
Prominences tend to oscillate after being disturbed [@oli92; @tri09; @arr12; @luna18 and references therein]. According to the amplitude, prominence oscillations can be classified into the small-amplitude [e.g., @oka07; @ning09] and large-amplitude [@chen08] types. According to the period, prominence oscillations can be divided into the short-period ($\leq$10 min) and long-period ($\geq$40 min) [@lin07; @zqm17a] types. A more reasonable classification is based on the direction of oscillation. For longitudinal oscillations, the prominence material oscillates along the threads with small angles of $10^{\circ}-20^{\circ}$ between the threads and spine [@jing03; @zqm12; @li12; @luna12; @luna14]. For transverse oscillations, the direction is horizontal (vertical) if the whole body of the prominence oscillates parallel (vertical) to the solar surface [@hyd66; @ram66; @kle69; @shen14a]. Generally speaking, the amplitudes and periods of longitudinal oscillations are larger than those of transverse oscillations. Occasionally, two types of oscillations coexist in a single event, showing a very complex behavior [@gil08; @shen14b; @zqm17b]. Recently, state-of-the-art magnetohydrodynamics (MHD) numerical simulations have thrown light on the triggering mechanisms, restoring forces, and damping mechanisms of prominence oscillations [@zqm13; @ter13; @ter15; @luna16; @zhou17; @zhou18].
Coronal cavities are dark structures as a result of density depletion in the white light (WL), EUV, and SXR wavelengths [@vai73; @gib06; @vas09]. The densities of cavities are 25%$-$50% lower than the adjacent streamer material [@mar04; @ful08; @ful09; @sch11]. Cavities show diverse shapes, such as semi-circular, circular, elliptical, and teardrop [@for13; @kar15a]. The lengths, heights, widths of cavities are 0.06$-$2.9 $R_{\odot}$, 0.08$-$0.5 $R_{\odot}$, and 0.09$-$0.4 $R_{\odot}$, respectively [@kar15a]. With a broader differential emission measure distribution, the average temperatures (1$-$2 MK) of cavities are slightly higher than the streamers [@hud99; @hab10; @ree12]. Both observations and numerical experiments indicate that the magnetic structure of cavities could be excellently described by a twisted/helical MFR [@gib10; @dove11; @fan12; @bak13; @rach13; @chen18]. At the bottom of cavities, there are denser filament plasmas drained down by gravity [@low95; @reg11]. Spectroscopic observations reveal that there are continuous whirling/spinning motion [@wang10] and large-scale flows with Doppler speeds of 5$-$10 km s$^{-1}$ in prominence cavities [@sch09]. The stable structures can last for days or even weeks to months [@kar15b]. After being strongly disturbed or a certain type of MHD instability is activated, a cavity may experience loss of equilibrium and erupt to drive a coronal mass ejection [CME; @ill85].
For the first time, @liu12 reported the transverse (horizontal) oscillation of a twisted MFR cavity and its embedded filament after the arrival of the leading EUV wave front at a speed of $\geq$1000 km s$^{-1}$. The average amplitude, initial velocity, period, and *e*-folding damping time are $\sim$2.3 Mm, $\sim$8.8 km s$^{-1}$, $\sim$27.6 minutes, and $\sim$119 minutes, respectively. Sometimes, MFRs carrying prominence material undergo global vertical oscillations with periods of hundreds of seconds, which are explained by the standing wave of fast kink mode [@kim14; @zhou16]. However, the vertical oscillation of a cavity as a result of the interaction between a coronal EUV wave and the cavity has rarely been observed and investigated.
In this paper, we report our multiwavelength observations of the vertical oscillation of a coronal cavity triggered by an EUV wave on 2011 March 16. The wave was caused by the eruption of a C3.8 two-ribbon flare and a partial halo CME at the remote AR 11169. The paper is structured as follows. Data analysis is described in detail in Section \[sec:data\]. Results are shown in Section \[sec:result\]. Estimation of the magnetic field strength of the cavity is arranged in Section \[sec:discuss\]. Finally, we give a brief summary in Section \[sec:summary\].
Instruments and data analysis {#sec:data}
=============================
Located north to NOAA AR 11169 (N17W74), the coronal cavity (N62W75) with an underlying prominence was continuously observed by the Global Oscillation Network Group (GONG) in H$\alpha$ line center (6562.8 [Å]{}) and by the Atmospheric Imaging Assembly [AIA; @lem12] on board the *Solar Dynamics Observatory* [*SDO*; @pes12] in UV (1600 [Å]{}) and EUV (171, 193, 211, and 304 [Å]{}) wavelengths. The photospheric line-of-sight (LOS) magnetograms were observed by the Helioseismic and Magnetic Imager [HMI; @sch12] on board *SDO*. The level\_1 data from AIA and HMI were calibrated using the standard *Solar Software* (*SSW*) programs *aia\_prep.pro* and *hmi\_prep.pro*. The full-disk H$\alpha$ and AIA 304 [Å]{} images were coaligned with a precision of $\sim$1$\farcs$2 using the cross correlation method. The CME was observed by the C2 WL coronagraph of the Large Angle Spectroscopic Coronalgraph [LASCO; @bru95] on board *SOHO*[^1]. The LASCO/C2 data were calibrated using the *SSW* program *c2\_calibrate.pro*. The CME was also observed by the COR1[^2] with a field of view (FOV) of 1.3$-$4.0 $R_{\odot}$ on board the *Solar TErrestrial RElations Observatory* [*STEREO*; @kai05]. The ahead satellite (hereafter STA) and behind satellite (hereafter STB) had separation angles of $\sim$88$^{\circ}$ and $\sim$95$^{\circ}$ with respect to the Sun-Earth direction on 2011 March 16. The coronal EUV wave associated with the CME was observed by AIA and the Extreme-Ultraviolet Imager (EUVI) in the Sun Earth Connection Coronal and Heliospheric Investigation package [SECCHI; @how08]. EUVI observes the Sun in four wavelengths (171, 195, 284, and 304 [Å]{}). Calibrations of the COR1 and EUVI data were performed using the *SSW* program *secchi\_prep.pro*. The deviation of STA north-south direction from the solar rotation axis was corrected. The EUV flux of the C3.8 flare in 1$-$70 [Å]{} was recorded by the Extreme Ultraviolet Variability Experiment [EVE; @wood12] on board *SDO*. The SXR flux of the flare in 1$-$8 [Å]{} was recorded by the *GOES* spacecraft. To investigate the hard X-ray (HXR) source of the flare, we made HXR images at different energy bands (6$-$12 keV and 12$-$25 keV) observed by the *Reuven Ramaty High-Energy Solar Spectroscopic Imager* [*RHESSI*; @lin02]. The HXR images were generated using the CLEAN method with integration time of 10 s. The observational parameters, including the instrument, wavelength, time, cadence, and pixel size are summarized in Table \[tab:para\].
[ccccc]{} LASCO & WL & 19:00$-$23:36 & 720 & 11.4\
COR1 & WL & 19:00$-$20:30 & 300 & 15.0\
HMI & 6173 & 17:30$-$23:30 & 45 & 0.6\
AIA & 1600 & 17:30$-$23:30 & 24 & 0.6\
AIA & 171$-$304 & 17:30$-$23:30 & 12 & 0.6\
EUVI & 171 & 17:30$-$23:30 & 150 & 1.6\
EUVI & 195, 284 & 17:30$-$23:30 & 300 & 1.6\
GONG & 6563 & 17:30$-$23:30 & 60 & 1.0\
*GOES* & 1$-$8 & 17:30$-$23:30 & 2.05 &\
EVE & 1$-$70 & 17:30$-$23:30 & 0.25 &\
*RHESSI* & 6$-$25 keV & 17:30$-$23:30 & 10 & 4\
Results {#sec:result}
=======
Flare and CME {#s-flare}
-------------
Figure \[fig1\] shows the H$\alpha$ and EUV images before the flare. In panels (a) and (b), the arrows point at the quiescent prominence resembling a horn structure. Above the prominence, there is a dark void with depleted EUV emissions (see panels (c-e)). This is the typical coronal cavity showing an elliptical shape. The height and width of the cavity are $\sim$240$\arcsec$ and $\sim$150$\arcsec$, and the centroid of the cavity has a height of $\sim$128$\arcsec$, respectively. In panel (f), the HMI LOS magnetogram features AR 11169 close to the western solar limb. The distance between the AR and cavity is $\sim$300$\arcsec$.
In Figure \[fig2\], the bottom panel shows the temporal evolutions of the EUV irradiance and SXR flux of the flare. It is clear that the EUV and SXR intensities of the flare started to rise gradually at $\sim$17:50 UT and reached the peak values around 21:00 UT, which was followed by a long main phase.
The flare observed in H$\alpha$, EUV, and UV wavelengths near the peak time are displayed in Figure \[fig3\]. An online animation (*20110316.mov*) is a combination of panels (c-e) observed by *SDO*/AIA. With a time cadence of 240 s, the animation starts from 17:44 UT on 2011 March 16 to 00:00 UT on 2011 March 17, featuring the vertical oscillation of the dark cavity with an embedded prominence. The flare is a typical two-ribbon flare with bright ribbons and semi-circular post flare loops in AR 11169. In panel (b), an above-the-loop-top HXR source is successfully reconstructed at 6$-$12 keV and 12$-$25 keV energy bands. The intensity contours of the source are drawn with green and yellow lines.
In Figure \[fig4\], the top and bottom panels demonstrate the WL images of the flare-related CME in the FOVs of LASCO/C2 and STA/COR1, respectively. With the central position angle (CPA) and angular width being 268$^{\circ}$ and 184$^{\circ}$, the CME appeared first in the FOV of C2 at $\sim$19:12 UT. During its forward propagation at a linear speed of $\sim$682 km s$^{-1}$, the CME underwent lateral expansion and evolved into a typical three-part structure, which consists of a bright leading edge, a dark cavity, and a bright core [@ill85]. Evolution of the CME in the FOV of STA/COR1 is similar to that in C2, except for a different CPA due to the different perspective of STA.
In Figure \[fig2\](a), we plot the temporal evolution of the CME height with green diamonds. A quadratic polynomial ($h=h_{0}+v_{0}t+at^2/2$) is applied to fit the temporal evolution of the CME height, which is represented by the green dashed line. The initial velocity $v_{0}=24.3$ km s$^{-1}$ at 17:50 UT and the constant acceleration $a=62.7$ m s$^{-2}$.
Coronal EUV Wave {#s-wave}
----------------
The fast and wide CME generated a coronal EUV wave propagating on the solar surface. To illustrate the wave more clearly, we take the EUV images at 17:30 UT as base images and obtain base-difference images at the following times. Figure \[fig5\] demonstrate eight snapshots of the base-difference images in AIA 171 [Å]{} during 18:00$-$19:24 UT. It is seen that as the CME proceeds, a bright EUV wave front forms when the lateral magnetic field lines of the CME are stretched and pressed, which is indicated by the arrow in panel (d). Behind the EUV wave front, there is a dark dimming region where the electron density decreases significantly after the impulsive expulsion of material carried by the CME [@zqm17c].
In order to investigate the evolution of the EUV wave, we select a curved slice (S1) in Figure \[fig5\](h). The long slice with a length of 1220$\arcsec$ starts from AR 11169 and passes through the cavity. Concentric with the solar limb, S1 has a height of 201$\arcsec$. Figure \[fig6\] shows the time-slice diagrams of S1 in 211, 193, and 171 [Å]{}. In panels (a) and (b), the bright inclined feature indicates the propagation of EUV wave in the northeast direction during 18:40$-$19:00 UT, which is overlaid by white dashed lines. The slopes of the dashed lines are equal to the velocities ($\sim$120 km s$^{-1}$) of EUV wave. It is obvious that the EUV wave reached and interacted with the dark cavity. In panel (c), the dashed line indicates the slow lateral expansion of the CME at a speed of $\sim$3 km s$^{-1}$ before 18:40 UT in 171 [Å]{}.
The EUV wave was also observed by STA/EUVI in various wavelengths. Like in Figure \[fig5\], we take the EUV images at 17:30 UT as base images and obtain base-difference images at the following times. Eight snapshots of the base-difference images in EUVI 195 [Å]{} during 18:20$-$18:55 UT are displayed in Figure \[fig7\]. It is clear that as the CME proceeds, the bright EUV wave front propagates in the northeast direction.
In Figure \[fig7\](b), we select a second slice (S2) with a length of 418 Mm, which starts from AR 11169 and extends in the same direction as the EUV wave. The time-slice diagrams of S2 in 171, 195, and 284 [Å]{} are plotted in Figure \[fig8\]. Like in Figure \[fig6\], the bright inclined feature indicates the propagation of EUV wave in the northeast direction during 18:30$-$18:50 UT, which is overlaid by the yellow dashed lines. The slopes of the dashed lines are equal to the velocity ($\sim$114 km s$^{-1}$) of EUV wave, which is close to the values in the FOV of AIA. The CME underwent a slow lateral expansion at a speed of $\sim$5.5 km s$^{-1}$ before $\sim$18:20 UT.
Vertical Oscillation of the Cavity {#s-osci}
----------------------------------
From the online animation (*20110316.mov*), we found clear evidence for vertical oscillation of the cavity after the arrival of the EUV wave. In Figure \[fig1\](e), we select a third slice (S3). With a length of 404$\farcs$5, S3 starts from the solar surface and passes through the cavity, covering the large-scale envelope coronal loops. The time-slice diagrams of S3 in 211, 193, and 171 [Å]{} are plotted in Figure \[fig9\]. It is obvious that after the arrival of the EUV wave at $\sim$19:00 UT, the overlying coronal loops are pressed down before bouncing back gradually to their initial states. The velocities of the downward motion are $\sim$387, $\sim$344, and $\sim$149 km s$^{-1}$ in 211, 193, and 171 [Å]{}, respectively.
Oscillation of the cavity with small amplitudes could not be distinctly displayed. However, when we magnify the regions within the white boxes of Figure \[fig9\], things are different. Close-ups of the regions with a better contrast are shown in Figure \[fig10\]. Now, the damping oscillations lasting for about 2 cycles are clear. We mark the positions of the cavity manually with white plus symbols. In order to obtain parameters of the oscillations, we performed curve fittings using the standard *SSW* program *mpfit.pro* and the same function as that described in previous literatures [@zqm12; @zqm17a; @zqm17b]: $$\label{eqn-1}
y=y_0+bt+A_0\sin(\frac{2\pi}{P}t+\phi_0)e^{-t/\tau},$$ where $y_0$, $A_0$, and $\phi_0$ represent the initial position, amplitude, and phase. $b$, $P$, and $\tau$ stand for the linear velocity of cavity, period, and damping time of the oscillations.
The results of curve fittings are shown in Figure \[fig11\], indicating that the function in Equation \[eqn-1\] can excellently describe the vertical oscillation of the cavity. The parameters in different wavelengths are labeled in each panel and listed in Table \[tab:fitting\] for a easier comparison. The amplitudes, periods, and damping times are 2.4$-$3.5 Mm, 29$-$37 minutes, and 26$-$78 minutes, respectively. The amplitudes are consistent with the values for the vertical oscillation of MFRs [@kim14; @zhou16]. The periods are close to the values for the vertical oscillation of prominences [@hyd66; @gil08; @bocc11]. Since the prominence-cavity system consists of multithermal plasma, the initial positions of oscillation in different wavelengths, which are labeled with white circles in Figure \[fig1\](c-e), agree with their formation temperatures. In other words, hot plasmas oscillate at higher altitudes and cool plasmas oscillate at lower altitudes, implying that the prominence-carrying cavity oscillates as a whole body after the arrival of the EUV wave. The positive values of $b$ in the third column of Table \[tab:fitting\] suggest that the cavity ascended slowly at a speed of 1$-$2 km s$^{-1}$ during the oscillation.
Using the simple formula $v=ds/dt$, the velocities of the vertical oscillations in 211, 193, and 171 [Å]{} are calculated and displayed in Figure \[fig12\]. The velocity amplitudes of the oscillations are less than 10 km s$^{-1}$, which are close to the values previously reported by @shen14a and @zhou16.
[ccccccc]{} 211 & 84.89 & 1.04 & 2.39 & 0.48 & 29.4 & 77.8\
193 & 82.77 & 1.43 & 3.51 & 2.34 & 36.6 & 66.2\
171 & 44.81 & 1.74 & 2.57 & 1.50 & 29.9 & 25.9\
Discussion {#sec:discuss}
==========
How is the Oscillation Triggered?
---------------------------------
Despite of substantial observations and investigations since the discovery of prominence oscillations [e.g., @hyd66; @ram66], the triggering mechanisms of prominence oscillations are far from clear. The longitudinal oscillations can be triggered by microflares or subflares [@jing03; @zqm12], shock waves [@shen14b], magnetic reconnections in the filament channels [@zqm17a], and coronal jets at the legs of filaments [@luna14] or from a remote AR [@zqm17b]. The transverse oscillations, including horizontal and vertical oscillations, are usually triggered by EUV or Moreton waves [e.g., @eto02; @liu12; @shen14a; @shen17]. @liu12 reported the transverse oscillation of a cavity triggered by the arrival of a coronal EUV wave from the remote AR 11105 on 2010 September 8-9. The oscillation is explained by the fast kink-mode wave. In our study, the oscillation of the prominence-carrying cavity is also triggered by a coronal EUV wave from AR 11169. Timeline of the whole events is drawn in Figure \[fig13\]. On the one hand, the direction of oscillation is vertical (see Figure \[fig10\]), which is different from the situation in @liu12. This is most probably due to the different location of interplay. As shown in the schematic cartoon of @liu12, the EUV wave front collides with the cavity laterally. In our case, the EUV wave collides with the cavity from the top, which is supported by the fast downward motion and bouncing back of the overlying magnetic field lines above the cavity (see Figure \[fig9\]). Horizontal oscillation is not found from the time-slice diagrams of S1 (see Figure \[fig6\]). On the other hand, the speed ($\sim$120 km s$^{-1}$) of the EUV wave from AR 11169 accounts for only 10% of that in @liu12. This is probably due to the different component of EUV wave. On 2010 September 8, the fast component of the global EUV wave interacting with the cavity is interpreted by a fast MHD wave. Considering the absence of fast component EUV wave in the time-slice diagrams of S1 and S2 (see Figure \[fig6\] and Figure \[fig8\]), the EUV wave in our case is most probably the slow component of CME-caused restructuring [@chen02; @chen05].
In Figure \[fig14\], we draw a schematic cartoon to illustrate the interaction between the cavity and the EUV wave. In AR 11169 close to the solar limb, a C3.8 two-ribbon flare (red arcades) and a CME (green line and blue circle) take place. Owing to the stretch and lateral expansion of the magnetic field lines constraining the CME, a coronal EUV wave (black lines) is generated and propagates in the northeast direction. As soon as the EUV wave reaches the cavity from top, it presses the overlying magnetic field lines (purple lines) above the cavity, leading to fast downward motion of the magnetic loops. At the same time, the cavity (dark ellipse) and the prominence at the bottom start to oscillate in the vertical direction. Due to the quick attenuation, the oscillation lasts for $\sim$2 cycles and disappears.
Magnetic Field Strength of the Cavity
-------------------------------------
The restoring force of vertical prominence oscillation is another key issue that should be addressed. Recently, @zhou18 performed three-dimensional (3D) ideal MHD simulations of prominence oscillations along a MFR in three directions. The vertical oscillation has a period of $\sim$14 minutes, which can be well explained using a slab model [@diaz01]. The magnetic tension force serves as the dominant restoring force. Considering that the periods of oscillation at different heights are approximately equal, we conclude that the cavity experienced a global vertical oscillation of fast kink mode [@zhou16]. The magnetic field strength of the cavity can be roughly estimated as follows: $$\label{eqn-2}
B_c=2L_c\sqrt{4\pi\rho_c}/P,$$ where $L_c$ and $\rho_c$ represent the total length and density of the cavity, and $P$ denotes the period of oscillation. Since the length of cavity could not be measured directly, we take $L_c$ in the range of 0.06$-$2.9 $R_{\odot}$ [@kar15a]. We adopt the typical density of $n_{e}=1.0\times10^8$ cm$^{-3}$ ($\rho_c=1.67\times10^{-16}$ g cm$^{-3}$) for the coronal cavity at a height of 1.13 $R_{\odot}$ [@ful09; @sch11]. Using the fitted period of $\sim$30 minutes (see Table \[tab:fitting\]), $B_c$ is calculated to be 0.2$-$10 G, which is consistent with the value of 6 G in @liu12.
Another way of estimating the magnetic field strength of prominence at the bottom of cavity is described by @hyd66: $$\label{eqn-3}
B_{r}^2=\pi\rho_{p} r_{0}^2(4\pi^2 P^{-2}+\tau^{-2}),$$ where $B_r$, $\rho_{p}$, and $r_0$ are the radial component of magnetic field, density, and scale height of the prominence, $P$ and $\tau$ are the period and damping time of the oscillation. Equation \[eqn-3\] can be simplified: $$\label{eqn-4}
B_{r}^2=4.8\times10^{-12}r_{0}^2(P^{-2}+0.025\tau^{-2}),$$ if we assume $\rho_p=4\times10^{-14}$ g cm$^{-3}$, which is two orders of magnitude higher than $\rho_c$, and $r_0=3\times10^9$ cm [@shen17]. Using the fitted period of $\sim$30 minutes and damping time of 25.9 minutes (see Table \[tab:fitting\]), $B_r$ is calculated to be 3.8 G, which is close to the values reported in previous literatures [@hyd66; @shen14a; @shen17]. In brief, the magnetic field strength of the prominence-carrying cavity in our case is a few Gauss and less than 10 G.
Summary {#sec:summary}
=======
In this paper, we report our multiwavelength observations of the vertical oscillation of a coronal cavity on 2011 March 16. The main results are summarized as follows:
1. [In EUV wavelengths, the width and height of the elliptical cavity are $\sim$150$\arcsec$ and $\sim$240$\arcsec$. The centroid of the cavity has a height of $\sim$128$\arcsec$ (0.13 $R_{\odot}$) above the solar surface. At the bottom of the cavity, there is a horn-like quiescent prominence.]{}
2. [At $\sim$17:50 UT, a C3.8 two-ribbon flare and a partial halo CME took place in AR 11169 close to the western limb. The EUV and SXR intensities of the flare rose gradually to the peak values at $\sim$21:00 UT. The linear velocity, central position angle, and angular width of the CME are 682 km s$^{-1}$, 268$^{\circ}$, and 184$^{\circ}$ in the LASCO/C2 FOV.]{}
3. [During the impulsive phase of the flare and acceleration phase of the CME, a coronal EUV wave was generated and propagated in the northeast direction at a speed of $\sim$120 km s$^{-1}$. The EUV wave is interpreted by the field line stretching and restructuring caused by the eruption of CME.]{}
4. [Once the EUV wave arrived at the cavity, it interacted with the large-scale overlying magnetic field lines from the top, resulting in quick downward motion and bouncing back of the overlying loops. Meanwhile, the cavity started to oscillate coherently in the vertical direction. The amplitude, period, and damping time are 2.4$-$3.5 Mm, 29$-$37 minutes, and 26$-$78 minutes, respectively. The oscillation lasted for $\sim$2 cycles before disappearing.]{}
5. [The vertical oscillation of the cavity is explained by a global standing MHD wave of fast kink mode. Using two independent methods of prominence seismology, we carry out a rough estimation of the magnetic field strength of the cavity, which is a few Gauss and less than 10 G. Additional case studies using the multiwavelength and high-resolution observations are required to investigate the interaction between a global EUV wave and a prominence-cavity system. MHD numerical simulations are worthwhile to gain an insight into the restoring forces and damping mechanisms of transverse prominence oscillations.]{}
We would like to thank Y. N. Su, D. Li, T. Li, R. S. Zheng, and Y. D. Shen for fruitful and valuable discussions. *SDO* is a mission of NASAs Living With a Star Program. AIA and HMI data are courtesy of the NASA/*SDO* science teams. This work utilizes GONG data from NSO, which is operated by AURA under a cooperative agreement with NSF and with additional financial support from NOAA, NASA, and USAF. QMZ is supported by the Youth Innovation Promotion Association CAS, NSFC (No. 11333009, 11790302, 11773079), the Fund of Jiangsu Province (BK20161618 and BK20161095), “Strategic Pilot Projects in Space Science” of CAS (XDA15052200), and CAS Key Laboratory of Solar Activity, National Astronomical Observatories (KLSA201716).
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[^1]: http://cdaw.gsfc.nasa.gov/CME\_list/
[^2]: http://cor1.gsfc.nasa.gov/catalog/cme/2011/
|
---
abstract: 'As a fundamental problem of natural language processing, it is important to measure the distance between different documents. Among the existing methods, the Word Mover’s Distance (WMD) has shown remarkable success in document semantic matching for its clear physical insight as a parameter-free model. However, WMD is essentially based on the classical Wasserstein metric, thus it often fails to robustly represent the semantic similarity between texts of different lengths. In this paper, we apply the newly developed Wasserstein-Fisher-Rao (WFR) metric from unbalanced optimal transport theory to measure the distance between different documents. The proposed WFR document distance maintains the great interpretability and simplicity as WMD. We demonstrate that the WFR document distance has significant advantages when comparing the texts of different lengths. The varying length matching and KNN classification results on eight datasets have shown its clear improvement over WMD. Furthermore, WFR could also improve WMD under other frameworks.'
author:
- |
Zihao Wang\
Department of Computer Science and Technology\
Tsinghua University\
`wzh17@mails.tsinghua.edu.cn`\
Datong Zhou\
Department of Mathematical Science\
Tsinghua University\
`zdt14@mails.tsinghua.edu.cn`\
Yong Zhang\
Department of Computer Science and Technology\
Tsinghua University\
`zhangy05@tsinghua.edu.cn`\
Hao Wu\
Department of Mathematical Science\
Tsinghua University\
`hwu@tsinghua.edu.cn`\
Chenglong Bao\
Yau Mathematical Sciences Center\
Tsinghua University, China\
clbao@math.tsinghua.edu.cn
bibliography:
- 'main.bib'
title: 'Wasserstein-Fisher-Rao Document Distance'
---
|
---
abstract: 'In this work, the control of snake robot loco-motion via economic model predictive control (MPC) is studied. Only very few examples of applications of MPC to snake robots exist and rigorous proofs for recursive feasibility and convergence are missing. We propose an economic MPC algorithm that maximizes the robot’s forward velocity and integrates the choice of the gait pattern into the closed loop. We show recursive feasibility of the MPC optimization problem, where some of the developed techniques are also applicable for the analysis of a more general class of system. Besides, we provide performance results and illustrate the achieved performance by numerical simulations. We thereby show that the economic MPC algorithm outperforms a standard lateral undulation controller and achieves constraint satisfaction. Surprisingly, a gait pattern different to lateral undulation results from the optimization.'
author:
- 'Marko Nonhoff, Philipp N. Köhler, Anna M. Kohl, Kristin Y. Pettersen, and Frank Allgöwer [^1]'
bibliography:
- 'Bibliography.bib'
title: ' **Economic model predictive control for snake robot locomotion** '
---
Introduction
============
There has been active research on the mechanisms and control of snake robot locomotion over the last decades. As opposed to robots which move in a more traditional way, for example wheeled or legged robots, snake robots carry the potential to not only move in cluttered and irregular environments, but also make use of obstacles to aid in their locomotion. Since the snake robot is a robot manipulator arm that can also locomote, it has a wide range of applications, including firefighting as well as search and rescue tasks. Correspondingly, the scope of research activities ranges from movement in tight spaces on land to subsea operations. An overview over previous literature on modelling, analysis, control and application of snake robots is found in [@Liljeback2012Overview; @Liljeback2012; @PETTERSEN2017].
The complex dynamics of snake robot locomotion, possible environment interactions, and the presence of constraints on both the input and the states make model predictive control (MPC) a promising approach to control the motion of snake robots. MPC is a control method that solves a finite-horizon optimal control problem at every sampling time instance and applies the first part of the optimal input, see for example [@Rawlings2009]. In many applications, setpoint stabilization may not be the primary control objective, but rather optimization of some general performance criterion. For this reason, so called economic MPC was developed, which allows to minimize a general performance criterion [@MUller2017_EconomicandDistributed]. Economic MPC has been studied in detail and convergence, stability, and performance results are available for the closed loop (see, e.g., [@MUller2017_EconomicandDistributed; @Ellis2014; @angeli2012; @Muller2016]). It is a well-known property of economic MPC that it can lead to periodic behavior which commonly occurs in snake robot locomotion. Nevertheless, only very few results exist for applications of MPC to the control of snake robot locomotion. In [@Marafioti2014], MPC is employed for path following of a snake robot in terms of optimizing the parameters of a predefined gait pattern. However, the gait pattern was chosen offline and no rigorous proofs for recursive feasibility of the MPC optimization problem were given for the proposed controller.
The work at hand develops a theoretical basis and provides first results on utilizing Model Predictive Control techniques for the control of snake robot locomotion without a predefined gait pattern. Therefore, we consider a simplified model and assume the snake robot moves on a flat and unbounded surface. In particular, by directly computing the input signals for each joint of the snake robot, the choice of the gait pattern is integrated into the closed loop, as opposed to existing approaches which, e.g., employ feedback controllers to track a predefined gait pattern [@Liljeback2012Overview; @Liljeback2012; @PETTERSEN2017] or central pattern generators [@IJSPEERT2008].
Thereby, we enable the snake robot to adapt its motion to, e.g., a changing environment, faults or changing performance criteria given by an altered cost function. For simplicity, we focus on maximizing the forward velocity of the snake robot throughout this work, since this yields a simple value function with an intuitive physical interpretation and is a reasonable objective of locomotion. Nevertheless, the central ideas are applicable to more stage costs as well.
This paper is structured as follows: In Section 2, we introduce the mathematical model of the snake robot used in this work and review the lateral undulation gait pattern and a corresponding controller from the literature. introduces the economic MPC scheme, which is investigated regarding recursive feasibility and performance in Section 4. Subsequently, numerical simulations are shown in Section 5, and concluding remarks are given in Section 6.
*Notation:* Let $\mathbb{I}_{[a,b]}$ denote the set of all integers in the interval $[a,b] \subset \mathbb{R}$, and let $\mathbb{I}_{>a}$ denote the set of all integers larger than $a \in \mathbb{R}$. $\lceil x \rceil$ is the ceiling function, i.e., and ${\text{sgn}}(x)$ represents the signum function.
We consider a discrete-time nonlinear system $$\begin{aligned}
\label{eq:defgensys}
x(t+1) &= f(x(t),u(t)), \hspace{1cm} x(0) = x_0,
\end{aligned}$$ where $f:\mathbb{X} \times \mathbb{U} \rightarrow \mathbb{R}^n$, $x(t) \in \mathbb{X} \subseteq \mathbb{R}^n$ and are the system dynamics, the state and the control input at time $t \in \mathbb{I}_{\geq 0}$, and $\mathbb{X}$ and $\mathbb{U}$ denote the state and input constraint sets, respectively. The solution of system for a control sequence ${\boldsymbol{u}} = ( u(0), \ldots, u(K-1) ) \in \mathbb{U}^{K}$ starting at the initial value $x_0 \in \mathbb{X}$ is denoted by ${\boldsymbol{x}}^{{\boldsymbol{u}}}(t, x_0)$, $t=0,\dots,K$, which is abbreviated as ${\boldsymbol{x}}^{{\boldsymbol{u}}}(t)$ whenever $x_0$ is clear from the context.
Problem Setup
=============
In this section, we present a mathematical model of the snake robot which will be used for controller design, analysis and simulations in the remainder of this work. Modelling a snake robot in full detail yields a dynamical system too complicated for contoller design. For this reason, we will use a simplified model throughout this work, specifically developed for controller design and analysis. A detailed derivation of this simplified model can be found in [@Liljeback2012; @Liljeback2013]. The main idea is to describe the robot by a serial connection of translational instead of revolute joints connecting the links of the snake robot, since analysis shows that it is the transversal motion of the links that is significant for forward motion. A schematic representation of the modelling approach is provided in Figure \[fig:snakemodel\].
We consider a planar snake robot consisting of $N_l$ links, all with the same mass $m$ and length $l$, which are interconnected by $N_l-1$ translational joints. The center of mass of each link is located at its center point. The robot moves on a horizontal and flat surface and is driven by $N_l -1$ actuators, one at each joint.
First, we define a set of matrices which will be used extensively in this work.
A &= \^[(N\_l-1) N\_l]{},\
D &= \^[(N\_l-1) N\_l]{},\
e &= \[1, …, 1\]\^T \^[N\_l]{}, [1.5mu1.5mu]{} = \[1, …, 1\]\^T \^[N\_l-1]{},\
[1.5mu1.5mu]{} &= D\^T ( D D\^T ) \^[-1]{} \^[N\_l (N\_l-1)]{}.
The matrices $A$ and $D$ represent an addition and subtraction, respectively, of adjacent elements of a vector. We assume that the snake robot is subject to an anisotropic viscous ground friction force. More specifically, the ground friction normal to the link is greater than the ground friction parallel to the link. Both for biological snakes and for snake robots, this is the essential feature enabling them to propel forward. We denote the friction coefficient in normal direction by and in tangential direction by Furthermore, we define a propulsion coefficient $c_p$ as $$\begin{aligned}
c_p &= \frac{c_n-c_t}{2l}.
\end{aligned}$$
We are now ready to state the complete simplified model. As opposed to [@Liljeback2012; @Liljeback2013], we give a discretized version, since our MPC scheme will be formulated in discrete-time with sampling time $T_s$.
$$\begin{aligned}
\phi(t+1) &= \phi(t) + T_s v_{\phi}(t) \label{eq:SysDynBegin} \\
\theta(t+1) &= \theta(t) + T_s v_{\theta}(t) \\
p_{x}(t+1) &= p_{x}(t) + T_s \big( v_{t}(t) \cos(\theta(t)) - v_{n}(t) \sin(\theta(t)) \big) \\
p_{y}(t+1) &= p_{y}(t) + T_s \big( v_{t}(t) \sin(\theta(t)) + v_{n}(t) \cos(\theta(t)) \big) \\
v_{\phi}(t+1) &= v_{\phi}(t) + T_s u(t) \label{eq:SysDynvphi} \\
v_{\theta} (t+1) &= v_{\theta}(t) + T_s \left( -\lambda_1 v_{\theta}(t) + \frac{\lambda_2}{N_l-1} v_{t}(t) {{\mkern 1.5mu\overline{\mkern-1.5mue\mkern-1.5mu}\mkern 1.5mu}}^T \phi(t) \right) \\
v_{t}(t+1) &= v_{t}(t) + T_s \left(-\frac{ c_t}{m} v_{t}(t) + \frac{2 c_p}{N_lm} v_{n}(t) {{\mkern 1.5mu\overline{\mkern-1.5mue\mkern-1.5mu}\mkern 1.5mu}}^T \phi(t) \right. \nonumber \\ & \quad \left. -\frac{c_p}{N_lm} \phi^T(t) A {{\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu}} v_{\phi}(t) \right) \label{eq:SysDynvt} \\
v_{n}(t+1) &= v_{n}(t) + T_s \left( - \frac{c_n}{m} v_{n}(t) + \frac{2 c_p}{N_lm} v_{t}(t) {{\mkern 1.5mu\overline{\mkern-1.5mue\mkern-1.5mu}\mkern 1.5mu}}^T \phi(t) \right), \label{eq:SysDynEnd}
\end{aligned}$$
\[eq:SysDyn\]
where $\phi(t) \in \mathbb{R}^{N_l-1}$ and $v_\phi(t) \in \mathbb{R}^{N_l-1}$ denote the joint distances and velocities, $(p_x(t),p_y(t)) \in \mathbb{R}^2$ represents the position of the snake robot’s center of mass in the global frame, $v_t(t) \in \mathbb{R}$ and $v_n(t) \in \mathbb{R}$ are the tangential and normal velocities of the center of mass in the $t-n$ frame, denotes the orientation and $v_\theta(t) \in \mathbb{R}$ the snake robot’s rotational velocity. The parameters $\lambda_1 \in \mathbb{R}$ and $\lambda_2 \in \mathbb{R}$ are empirical constants which describe the rotational dynamics. A reasonable choice of parameters for this model is presented in [@Liljeback2012].
Note that we directly consider the input $u$ obtained through an input transformation as shown in [@Liljeback2012; @Liljeback2013], leading to the simple dynamics .
Due to mechanical restrictions of the snake robot and the model only being valid for joint distances $\phi(t)$ which are sufficiently small [@Liljeback2012], state constraints on the joint distances $\phi(t)$ and the corresponding velocities $v_\phi(t)$ and input constraints need to be respected. These constraints are given as box constraints, hence, [ $$\begin{aligned}
\begin{split}\mathbb{X} = \{ &x(t) \in \mathbb{R}^{2N_l + 4} \,\big|\, \phi_i(t) \in [-\phi_{\max}, \phi_{\max}], \\ &v_{\phi,i}(t) \in [-v_{\phi,\max}, v_{\phi,\max}] \hspace{.2cm}\forall i \in \mathbb{I}_{[1,N_l-1]} \}, \end{split} \label{const:states} \\
\mathbb{U} = \{ &u(t) \in \mathbb{R}^{N_l-1} | u_i(t) \in [-u_{\max}, u_{\max}] \hspace{.2cm} \forall i \in \mathbb{I}_{[1,N_l-1]} \}, \label{const:input}
\end{aligned}$$ ]{} where $$\begin{aligned}
\label{eq:defXsnake}
x(t) &= [\phi(t), \theta(t), p_x(t), p_y(t), v_\phi(t), v_\theta(t), v_t(t), v_n(t)]^T, \end{aligned}$$ and $\phi_{\max} \in \mathbb{R}_{>0}$, $v_{\phi,\max} \in \mathbb{R}_{>0}$ and $u_{\max} \in \mathbb{R}_{>0}$. We assume that the remaining states are unconstrained.
In [@Liljeback2012; @Liljeback2013], a controller was presented which steers the joint distances $\phi(t)$ to a given reference trajectory, defined by the gait pattern lateral undulation (LU). This gait pattern propagates a body wave from head to tail of the snake robot and is defined as $$\begin{aligned}
\label{eq:DefLU}
\phi_{\mathrm{LU},i}(t) &= \alpha \sin \left( \omega t + (i-1) \delta \right),
\end{aligned}$$ where $i \in \mathbb{I}_{[1, N_l-1]}$, and $\alpha,\omega,\delta \in \mathbb{R}$ are constant parameters. This gait pattern was studied in detail, e.g., in [@Liljeback2012; @Saito2002].
The corresponding lateral undulation controller is given by $$\begin{aligned}
\begin{split}
u(t) &= u_{\mathrm{ref}}(t) + k_d ({v}_{\phi,\mathrm{ref}}(t) - {v}_\phi(t)) \\ &\quad + k_p (\phi_{\mathrm{ref}}(t) - \phi(t)),
\end{split} \label{eq:defcontLU}
\end{aligned}$$ with $$\begin{aligned}
\phi_{\mathrm{ref},i}(t) &= \phi_{\mathrm{LU},i}(t), \\
v_{\phi,\mathrm{ref},i}(t) &= \frac{\mathrm{d}}{\mathrm{d}t} \phi_{\mathrm{ref},i}(t), & u_{\mathrm{ref},i}(t) = \frac{\mathrm{d}^2}{\mathrm{d}t^2} \phi_{\mathrm{ref},i}(t).
\end{aligned}$$ This controller was proven to exponentially stabilize the reference gait pattern for the snake robot model . Furthermore, in [@Liljeback2012; @Liljeback2013] it was shown that the average forward velocity converges exponentially fast to a velocity which depends on the parameters describing the gait pattern $\alpha, \omega, \delta$ and $\phi_0$, the friction coefficients $c_n$, $c_t$ and $c_p$, and the number of links $N_l$.
Economic MPC scheme for snake robot locomotion {#sec:EMPCpreliminaries}
==============================================
As mentioned in the introduction, we aim at maximizing the forward velocity of the snake robot as a reasonable objective, and in order to arrive at a simple cost function with an intuitive physical interpretation. We therefore employ $-v_t(t)$ as the cost to be minimized by the MPC optimization problem. However, other choices are possible. For instance, in order to limit the energy consumption of the snake robot, a term $\gamma u^T(k|t) u(k|t)$ with $\gamma \in \mathbb{R}_{>0}$ could be added (cf. Section \[sec:sim\]). All our results on recursive feasibility provided in the next section are independent of the choice of the specific cost function. However, our results on the performance of the closed loop would need to be adjusted for a modified cost function.
Next, we state our proposed economic MPC algorithm for snake robot locomotion. At each time step $t$, given an initial value $x(t) \in \mathbb{X}$, the following MPC optimization problem is solved.
[(Economic MPC optimization problem)]{} \[prob:EMPC\] $$\begin{aligned}
\min_{{\boldsymbol{u}}(t) \in \mathbb{U}^{N_p}} \hspace{.3cm} &J(x(t), {\boldsymbol{u}}(t)) = -\sum_{k=0}^{N_p} v_t(k|t) \\
\text{s.t.} \hspace{.6cm}
&x(0|t) = x(t) \\
&x(k+1|t) = f(x(k|t), u(k|t))\\
&u(k|t) \in \mathbb{U} \subseteq \mathbb{R}^{N_l-1} \\
&\hspace{2cm} k = 0, \ldots, N_p-1\\
&x(k|t) \in \mathbb{X} \subseteq \mathbb{R}^{2N_l+4} \\
&\hspace{2cm} k = 0, \ldots, N_p.
\end{aligned}$$
In Problem \[prob:EMPC\], the snake robot’s dynamics $f(x(k|t), u(k|t))$ and states $x(t)$ are given by and , respectively. We denote the solution to this MPC optimization problem by and the corresponding predicted trajectory by , where the asteriks signify optimality. At each timestep, the first element of the optimal input sequence ${\boldsymbol{u}}^*(t)$ is applied to the system, hence, the control input is given by $u_{\mathrm{MPC}}(t) = u^*(0|t)$.
Analysis of the closed loop
===========================
Recursive feasibility
---------------------
Recursive feasibility is a fundamental property of MPC algorithms, which is required to apply the MPC scheme to a system. Recursive feasibility means that, if a feasible solution to the MPC optimization problem at time $t=t_0$ exists, then there is a feasible solution to the problem for every $t > t_0$. Hence, recursive feasibility establishes that the control law given by the above algorithm is defined at all times. In the following, we present a sufficient condition for recursive feasibility of Problem \[prob:EMPC\], which can be achieved without additional terminal costs or constraints, facilitating implementation and, more importantly, not requiring any a priori knowledge of, e.g., a desirable gait pattern. We note that the results of this section can be applied to a wider range of problems of similar structure, i.e., systems with box constraints on states adhering to double integrator dynamics. This is discussed in more detail below.
The main idea of the following derivations is to provide a candidate solution to the MPC optimization Problem \[prob:EMPC\] based on a feasible solution from the previous time step in order to certify that the resulting control input is always defined.
Given the current state $x(t)$ (including $v_\phi(t)$), let the input sequence ${\boldsymbol{u}}^c(t) \in \mathbb{U}^{N_p}$ and the $i$-th entry (referring to the $i$-th joint) of its $k$-th element ${\boldsymbol{u}}_i^c(k|t)$, $i \in \mathbb{I}_{[1, N_l-1]}$, be defined by the feedback law $$\begin{aligned}
\label{eq:Defuc}
u_i^c(k|t) &=
\begin{cases}
-{\text{sgn}}\left(v_{\phi,i} (k|t)\right) u_\mathrm{max} & \text{if } |v_{\phi,i}(k|t)| > T_s u_\mathrm{max} \\
-\frac{v_{\phi,i}(k|t)}{T_s} & \text{else}
\end{cases}
\end{aligned}$$ and let $$\begin{aligned}
\label{eq:defb}
b = \lceil\frac{v_{\phi,\mathrm{max}}}{T_s u_\mathrm{max}}\rceil.
\end{aligned}$$ Hence, it is always possible to steer $v_\phi(t)$ to zero in $b$ time steps. Let $$\begin{aligned}
{\boldsymbol{\phi}}^*(t) &= ( \phi^*(0|t), \ldots, \phi^*(N_p|t)), \\
{\boldsymbol{v}}_\phi^*(t) &= ( v_\phi^*(0|t), \ldots, v_\phi^*(N_p|t))
\end{aligned}$$ denote the optimal trajectories of the joint distances and velocities at time $t$, respectively. We define the candidate input sequence $\mathbf{\tilde{u}}(t)$ by shifting the previously optimal solution for the first $N_p-b-1$ time steps and extending it by ${\boldsymbol{u}}^c(t)$, i.e., $$\label{eq:Defudi}
\begin{split}
& \mathbf{\tilde{u}}(t+1) = ( u^*(1|t), \ldots, u^*(N_p-b-1|t), \\ &\qquad u^c(N_p-b-1|t+1), \ldots, u^c(N_p-1|t+1)).
\end{split}$$
We use the candidate input $\mathbf{\tilde{u}}(t)$ only for feasibility analysis, and do not intend to actually apply it to the snake robot. When applied to a snake robot, the forward velocity achieved with this input would be undesirable, since it steers the joint velocities $v_\phi(t)$ to zero. Therefore, no acceleration can be achieved by body shape changes and the robot is decelerated due to friction.
The main idea behind defining this input is that it indeed steers the joint velocities to zero, as will be shown by the next lemma. We thereby ensure that the states $v_\phi(t)$ and $\phi(t)$ remain feasible thereafter and only the transient phase of the candidate input sequence ${\boldsymbol{u}}^c(t)$ and the corresponding state trajectories remain to be analyzed. In order to prove the main result of this section, we first examine the response of the joint velocities $v_\phi^{{\boldsymbol{u}}^c(t)}(t)$ when actuated by the candidate input sequence ${\boldsymbol{u}}^c(t)$. Namely, we show that the absolute value of $v_{\phi,i}^{{\boldsymbol{u}}^c}(t)$ is decreasing and its sign does not change if $|v_{\phi,i}(t)|>T_su_{\max}$. Loosely speaking, this means that the joint velocities are steered towards zero.
\[lemma:aux\] Let ${\boldsymbol{u}}^c(t)$ be defined by (\[eq:Defuc\]). Then it holds that
- $|v_{\phi,i}(k|t)| \geq |v_{\phi,i}^{{\boldsymbol{u}}_i^c}(k+1|t)|$ for all $i \in \mathbb{I}_{[1,N_l-1]}$ and $k \in \mathbb{I}_{[0,N_p-1]}$.
- Moreover, if $|v_{\phi,i}(k|t)| > T_su_{\max}$, then ${\text{sgn}}(v_{\phi,i}(k|t)) = {\text{sgn}}\left(v_{\phi,i}^{u_i^c}(k+1|t)\right)$ for all $i \in \mathbb{I}_{[1,N_l-1]}$ and $k \in \mathbb{I}_{[0,N_p-1]}$.
\(i) Analyzing the dynamics of $v_{\phi,i}(k|t)$ for some and $k \in \mathbb{I}_{[0,N_p-1]}$ given by when applying $u_i^c(k|t)$ yields $$\begin{aligned}
|v_{\phi,i}^{u_i^c}(k+1|t)| &= |v_{\phi,i} (k|t) + T_s u^c_i(k|t)|.
\intertext{First, assume that $|v_{\phi,i}(k|t)| > T_s u_{\max}$, which gives}
|v_{\phi,i}^{u_i^c}(k+1|t)|| &= |v_{\phi,i}(k|t)-T_s {\text{sgn}}\left( v_{\phi,i}(k|t)\right) u_{\max}| \\
&< |v_{\phi,i}(k|t)|.
\intertext{Second, assume otherwise $|v_{\phi,i}(k|t)| \leq T_s u_{\max}$, which gives}
|v_{\phi,i}^{u_i^c}(k+1|t)|| &= |v_{\phi,i}(k|t)-T_s \frac{v_{\phi,i}(k|t)}{T_s}| = 0 \\
&\leq |v_{\phi,i}(k|t)|.
\end{aligned}$$ Combining these two results yields the desired inequality $|v_{\phi,i}(k|t)| \geq |v_{\phi,i}^{u_i^c}(k+1|t)|$.
\(ii) Due to $|v_{\phi,i}(k|t)| > T_su_{\max}$ and the definition of the input $u_i^c(k|t)$ in it holds that Therefore, $$\begin{aligned}
&{\text{sgn}}\left(v_{\phi,i}^{u_i^c}(k+1|t)\right) = {\text{sgn}}\left(v_{\phi,i}(k|t) + T_s u_i^c(k|t) \right) \\
= &{\text{sgn}}\left( {\text{sgn}}\left( v_{\phi,i}(k|t) \right) \left( |v_{\phi,i}(k|t)| - T_s u_{\max} \right) \right) \\
= &{\text{sgn}}\left(v_{\phi,i}(k|t) \right). \qedhere
\end{aligned}$$
Note that Lemma \[lemma:aux\](i) already implies recursive feasibility of $v_{\phi,i}(k|t)$, i.e., if $|v_{\phi,i}(k|t)| \leq v_{\phi,\max}$ then
Next, we state the main result of this section. It provides a sufficient condition for recursive feasibility of Problem \[prob:EMPC\]. The only required assumption is existence of an initially feasible input at time $t=t_0$ as is standard in MPC feasibility analysis.
\[lemma:recfeas\] Suppose that there exists an initially feasible solution ${\boldsymbol{u}}^*(t)$ to Problem \[prob:EMPC\] at time $t = t_0$. If $N_p>b$, then there exists a feasible solution to Problem \[prob:EMPC\] for all $t \in \mathbb{I}_{>t_0}$.
We assume that Problem \[prob:EMPC\] was feasible at time $t$. Then, it has to be shown that the input constraint and both state constraints can be satisfied at time $t+1$. As common in MPC, this is done by showing feasibility of a suboptimal candidate trajectory, which in our case results from application of the candidate input sequence .
First, we examine the input constraint. For the first time steps the candidate input sequence is feasible since it is the same as the shifted optimal input ${{\boldsymbol{u}}}^*(t)$ which is feasible by assumption.
For the remaining time steps, the candidate input sequence is feasible, since $u^c_i(k|t)$ is either defined by $$\begin{aligned}
|u_i^c(k|t+1)| &= |-{\text{sgn}}\left(v_{\phi,i} (k|t+1)\right) u_{\max}| \leq u_{\max}
\intertext{or by}
|u_i^c(k|t+1)| &= |-\frac{v_{\phi,i}(k|t+1)}{T_s}| \leq \frac{|T_s u_{\max}|}{T_s} = u_{\max}.
\end{aligned}$$ It follows that the candidate input trajectory satisfies the input constraints for all $k \in \mathbb{I}_{[0,N_p-1]}$.
Second, we investigate the response of the joint velocities ${\boldsymbol{v}}_\phi(k|t+1)$ to the candidate input sequence. For the first $N_p-b$ steps it holds that the constraint on $v_\phi^{\mathbf{\tilde{u}}}(k|t+1)$ is satisfied, since the solution of Problem \[prob:EMPC\] at time $t$ is feasible and, by the construction of the candidate input sequence , the resulting trajectories are the same. For the remaining time steps, constraint satisfaction is provided by Lemma \[lemma:aux\](i).
Finally, we examine the response of the joint distances $\phi(k|t+1)$ to the candidate input. For notational convenience, we define ${\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu} = N_p - b$. Again, by the construction of the candidate input sequence , the trajectory is the same as the optimal trajectory for the first $N_p-b+1$ time steps. $$\begin{aligned}
&\phi^{\mathbf{\tilde{u}}}(k|t+1) = \phi^* (k+1|t), \qquad k \in \mathbb{I}_{[0,N_p-b]}. \nonumber
\intertext{For the remaining time steps, we obtain $p \in \mathbb{I}_{[1,b]}$ and}
&\phi^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}+p|t+1) \nonumber \\
= &\phi^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-1+p|t+1) +T_s v_\phi^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-1+p|t+1), \nonumber
\intertext{and by recursively inserting the system dynamics, first \eqref{eq:SysDynBegin} and then \eqref{eq:SysDynvphi},}
= &\phi^* ({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}+1|t) + T_s \sum_{q = 1}^{p} v_\phi^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-1+q|t+1) \label{eq:Phiudidyn1} \\
\begin{split} = &\phi^*({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}+1|t) + T_s \left( \sum_{q = 1}^{p} v_\phi^* ({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}|t) \right. \\ &\qquad + T_s \left. \sum_{r=1}^{q} u^c({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-2+r|t+1) \right).
\end{split} \label{eq:Phiudidyn2}
\end{aligned}$$
Again, the first $N_p-b+1$ time steps are feasible because the candidate input sequence $\mathbf{\tilde{u}}(t+1)$ is the same as the previously feasible solution ${\boldsymbol{u}}^*(t)$. The main idea of proving feasibility of the remaining time steps is to bound the candidate joint distances by the previously feasible trajectory. In the following, we need to consider three different cases, depending on the candidate input $u^c(k|t)$. In the first two cases, we compare the candidate trajectory directly to the previously feasible trajectory of the joint distances. In the third case, we show feasibility by induction.
First, assume for some $p \in \mathbb{I}_{[1,b]}$ which implies $$\begin{aligned}
v_{\phi,i}^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-2+r|t+1) &> T_s u_{\max} > 0
\intertext{and}
u^c_i({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-2+r|t+1) &= -u_{\max} \\
&\leq u^*_i({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-1+r|t)
\end{aligned}$$ for all $r \in \mathbb{I}_{[1,p]}$ by Lemma \[lemma:aux\] and also $$\begin{aligned}
v_{\phi,i}^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-1+p|t+1)>0
\end{aligned}$$ due to Lemma \[lemma:aux\](ii). Then, inserting yields $$\begin{aligned}
&\phi_i^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}+p|t+1) \\
= &\phi_i^*({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}+1|t) + T_s \sum_{q = 1}^{p} \Bigg( v_{\phi,i}^*({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}|t) \\ &\qquad + T_s \sum_{r=1}^{q} u_i^c({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-2+r|t+1) \Bigg) \\
\leq &\phi_i^*({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}+1|t) + T_s \sum_{q = 1}^{p} \left( v_{\phi,i}^*({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}|t) + T_s \sum_{r=1}^{q} u^*_i({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-1+r|t) \right) \\
= &\phi_i^*({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}+1+p|t) \\
\leq &\phi_{\max}
\end{aligned}$$ and, by inserting , $$\begin{aligned}
&\phi_i^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}+p|t+1) \\ = &\phi_i^*({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}+1|t) + T_s \sum_{q = 1}^{p} v_{\phi,i}^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-1+q|t+1) \\
> &\phi_i^*({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}+1|t) \\
\geq &-\phi_{\max}.
\end{aligned}$$
Second, if $v_{\phi,i}^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-2+p|t+1) < -T_s u_{\max}$, the arguments are the same as above and are therefore omitted.
Third, if $|v_{\phi,i}^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-2+p|t+1)| \leq T_s u_{\max}$ holds, which implies $$\begin{aligned}
u^c_i({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-2+p|t+1) &= \linebreak -\frac{1}{T_s}v_{\phi,i}^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-2+p|t+1),
\end{aligned}$$ then the joint distance $\phi_i^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}+p|t+1)$ remains constant and, hence, feasible by induction: $$\begin{aligned}
&\phi_i^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}+p|t+1) \\
= &\phi_i^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-1+p|t+1) + T_s v_{\phi,i}^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-2+p|t+1) \\ &\qquad + T_s^2 u_i^c ({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-2+p|t+1) \\
= &\phi_i^{\mathbf{\tilde{u}}}({\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu}-1+p|t+1).
\end{aligned}$$
Combining these three results yields feasiblity for all elements of the sequence ${\boldsymbol{\phi}}^{\mathbf{\tilde{u}}}(t+1)$ except for the last one. Hence, it only remains to show that the last element $\phi^{\mathbf{\tilde{u}}}(N_p|t+1)$ of the is feasible. If $v_{\phi}^{\mathbf{\tilde{u}}} (N_p-1|t+1) = 0$ then would be feasible as shown above. In the following, we show that indeed $v_{\phi}^{\mathbf{\tilde{u}}} (N_p-1|t+1) = 0$ holds, by a contradiction argument to feasibility of the previous solution ${\boldsymbol{u}}^*(t)$.
First, assume otherwise $v_{\phi,i}^{\mathbf{\tilde{u}}}(N_p-1|t+1) \neq 0$ for some $i \in \mathbb{I}_{[1,N_l-1]}$.
If $v_{\phi,i}^{\mathbf{\tilde{u}}}(N_p-2|t+1) = 0$, then $u^c_i(N_p-2|t+1) = 0$ holds and thus $v_{\phi,i}^{\mathbf{\tilde{u}}}(N_p-1|t+1) = 0$, which is a contradiction. Hence, $v_{\phi,i}^{\mathbf{\tilde{u}}}(N_p-2|t+1) \neq 0$.
If $|v_{\phi,i}^{\mathbf{\tilde{u}}}(N_p-2|t+1)| \leq T_s u_{\max}$, then, by , which yields $v_{\phi,i}^{\mathbf{\tilde{u}}}(N_p-1|t+1) = 0$ due to . Again, this is a contradiction, so $|v_{\phi,i}^{\mathbf{\tilde{u}}}(N_p-2|t+1)| > T_s u_{\max}$.
Moreover, if $v_{\phi,i}^{\mathbf{\tilde{u}}}(N_p-2|t+1) > T_s u_{\max}$ then this is also true for every time step before where $u_i^c(k|t+1)$ was applied, due to Lemma \[lemma:aux\]. Furthermore, this implies because of Lemma \[lemma:aux\](ii). Repeatedly inserting the system dynamics into this equation yields $$\begin{aligned}
&0 < v_{\phi,i}^{\mathbf{\tilde{u}}}(N_p-1|t+1) \\
= &v_{\phi,i}^*(N_p-b-1|t) + T_s \sum_{q=1}^{b} u_i^c(N_p-b-2+q|t+1) \\
= &v_{\phi,i}^*(N_p-b-1|t) - T_s b u_{\max} \\
\leq &v_{\phi,i}^*(N_p-b-1|t) - v_{\phi,\max} \\
\Rightarrow &v_{\phi,\max} < v_{\phi,i}^*(N_p-b-1|t),
\end{aligned}$$ which is a contradiction to feasibility of the optimal solution at time $t$. Lastly, if $v_{\phi,i}^{\mathbf{\tilde{u}}}(N_p-2|t+1) < T_s u_{\max}$ the same contradiction to of the optimal solution is achieved by the same arguments. Hence, and $\phi^{{\boldsymbol{u}}_{di,i}}(N_p|k+1)= \phi^{\mathbf{\tilde{u}}}(N_p-1|t+1)$ which concludes the proof.
Note that in this proof we only investigated the constrained states $\phi(t)$, $v_\phi(t)$ and the input $u(t)$, which adhere to double integrator dynamics. Neither the additional states nor the cost function enter our analysis. This result can therefore be applied to any system with constraints on a integrator subsystem, possibly additional unconstrained states and an arbitrary cost function, which is found in many different kinds of mobile robot applications.
Performance guarantees {#sec:performance}
----------------------
In this section, we briefly discuss performance guarantees for the proposed economic MPC scheme in terms of achieving a certain benchmark velocity. However, a thorough presentation of our results is out of the scope of this paper. The analysis in this section is tailored to an economic MPC scheme which aims to maximize the snake robot’s forward velocity. In the following, the cost function in is therefore chosen as , as discussed in the previous sections. Hence, the results in this section would need to be adapted for a different cost function.
In the literature, convergence to an optimal periodic orbit for economic MPC schemes is usually shown by exploiting certain dissipativity properties of the system; see for instance [@MUller2017_EconomicandDistributed; @angeli2012; @Muller2015]. Due to the complexity of the snake robot model, showing such a dissipativity seems not to be possible. Instead, we assume existence of an auxiliary controller, which can sufficiently accelerate the snake robot, and yields a cost which is upper bounded. This is detailed below. Moreover, to properly state our result below, we additionally need to make the reasonable assumption that the economic MPC scheme will keep a certain velocity level once it has reached it.
More precisely, let $v_t(t)$ be the current velocity and let $\tilde{v}_t \in \mathbb{R}$ be a benchmark forward velocity used for performance characterization. Then, we assume that the set $\{ x(t) \in \mathbb{X} | v_t(t) \geq \tilde{v}_t \}$ is forward invariant under the proposed economic MPC scheme.
\[assum:inv\] Define $\mathcal{V} = \{ x(t) \in \mathbb{X} | v_t(t) \geq \tilde{v}_t \}$, where $\tilde{v}_t \in \mathbb{R}_{>0}$. Then, the set $\mathcal{V}$ is invariant in the economic MPC closed loop.
Moreover, for a given sampling time $T_s \in \mathbb{R}_{>0}$, denote by $\epsilon \in \mathbb{R}$ the solution of $$\begin{split}
\epsilon := \max_{x(t) \in \mathbb{X}} \hspace{.5cm} & \frac{1}{T_s} (v_t(t) - v_t(t+1)).
\end{split}$$ This constant $\epsilon$ is a measure for how much the snake robot can slow down from one time step to the next, i.e., holds for all $x(t) \in \mathbb{X}$. Note that it is independent of $u(t)$, since the input does not directly enter the dynamics of the forward velocity $v_t(t+1)$, if we investigate only one time step.
\[assum:conv\] Let $\tilde{v}_t \in \mathbb{R}_{>0}$. There exists some[^2] $\beta\in\mathcal{KL}$, such that for every $x(t) \in \mathbb{X}$ with $v_t(t) < \tilde{v}_t$ there exists an input sequence ${\boldsymbol{\bar{u}}}(t) \in \mathbb{U}^{N_p}$ which satisfies $$\begin{aligned}
\tilde{v}_t - v_t^{{\boldsymbol{\bar{u}}}}(t+k_1) \leq \beta(\tilde{v}_t - v_t(t), k_1), \\
\sum_{i=1}^{k_2} v_t^{{\boldsymbol{\bar{u}}}}(t+i) \geq mv_t(t) - T_s \epsilon,
\end{aligned}$$ for all $k_1 \in \mathbb{I}_{[0,N_p]}$ and for all $k_2 \in \mathbb{I}_{[1,N_p]}$.
Assumption \[assum:conv\] can be justified by investigating the closed-loop trajectory of the forward velocity when applying the standard lateral undulation controller given by . these two assumptions yields the desired performance result, i.e., convergence of the closed loop to the set $\mathcal{V}$. Hence, the application of the proposed economic MPC scheme leads to an acceleration sufficient to reach the velocity $\tilde{v}_t$ and, once it is achieved, remains greater than this benchmark velocity.
\[lemma:conv\] Let Assumptions \[assum:inv\] and \[assum:conv\] hold and assume that Problem \[prob:EMPC\] is initially feasible at time $t=t_0$ with $N_p > b$. Then, Problem \[prob:EMPC\] is feasible for all $t \in \mathbb{I}_{\geq t_0}$ and the closed loop converges to the set $\mathcal{V}$.
The proof of Lemma \[lemma:conv\] is omitted in this paper. All details can be found in [@Nonhoff2018]. Summarizing, we give performance guarantees for the economic MPC closed-loop snake robot system which are based on assumptions that can be justified, e.g., by simulations of existing standard controllers.
Numerical Simulations {#sec:sim}
=====================
Having established recursive feasibility, we demonstrate the effectiveness of the proposed economic MPC scheme and compare it to a standard lateral undulation controller through numerical simulations. Our proposed ecnomic MPC algorithm was implemented in CasADi [@Andersson2018]. For the sake of comparability we choose to use a similar set of parameters as in [@Liljeback2012; @Marafioti2014], and [@Liljeback2013]. The snake robot’s $N_l = 9$ links all have the same mass , the same length and friction coefficients in normal and tangential direction $c_n=3$ and $c_t=1$, respectively. We set and initialize the snake robot with , , and . The sampling time is chosen as and the prediction horizon is set to $N_p=20$ which is equivalent to . Finally, the constraint sets are given as $$\begin{aligned}
\phi_{\max} &= 0.052 \text{ m}, \\
v_{\phi,\max} &= 0.109 \text{ m/s}, \\
u_{\max} &= 0.2276 \text{ m/s\textsuperscript{2}}.
\end{aligned}$$
In order to confirm recursive feasibility we verify the conditions of Lemma \[lemma:recfeas\] and obtain $b = 10 < N_p = 20$ from . Hence, the ecnomic MPC scheme is feasible for all $t \in \mathbb{I}_{>0}$, since it is easy to see that it is initially feasible at $t=0$ for ${\boldsymbol{u}}(0) = ( 0, \ldots, 0)$.
For comparison, we consider the standard lateral undulation controller (\[eq:defcontLU\]) which is taken from [@Liljeback2012; @Liljeback2013]. The lateral undulation gait pattern parameters and the tuning parameters of the controller are adjusted such that it fully utilizes the entire constraint sets once it periodically applies the lateral undulation gait pattern and reaches a high asymptotic forward velocity. Therefore, the gait pattern parameters are chosen as , , and . The tuning parameters of the controller are taken from [@Liljeback2012; @Liljeback2013] and are set to and .
Economic MPC vs. Lateral Undulation
-----------------------------------
In the following, we first compare the velocity achieved by both controllers, i.e., the economic MPC scheme and the standard lateral undulation controller. Additionally, we highlight below the ability of our proposed economic MPC approach to incorporate arbitrary objectives in the optimization by accounting for energy consumption in the objective function, and again, compare it to the standard lateral undulation controller.
In the following, we set the cost function in the economic MPC optimization Problem \[prob:EMPC\] to $J(x(t), {\boldsymbol{u}}(t)) =\sum_{k=0}^{N_p} -v_t(k|t) + \gamma u^T(k|t) u(k|t)$, where $\gamma \in \mathbb{R}_{\geq 0}$. Here, the term $\gamma u^T(k|t) u(k|t)$ accounts for consumed energy, which is investigated below. We compare the proposed economic MPC algorithm for two different values of $\gamma$ with the standard lateral undulation controller. In the first experiment, we choose $\gamma =0$, which yields the cost function analyzed in Section \[sec:performance\], and in the second experiment, we choose $\gamma = 0.025$, which means that energy consumption is taken into account in the optimization.
Figure \[fig:velocititesEMPC\] shows the resulting closed-loop forward for the lateral undulation controller given by (\[eq:defcontLU\]) and the economic MPC scheme. One can identify a transient phase until approximately $100$ time steps and a periodic orbit afterwards. The asymptotic average velocity achieved by economic MPC for both values of $\gamma$ is clearly superior to lateral undulation.
This is achieved by a more efficient gait pattern: In , the trajectories of the joint velocities of joint $3$ are illustrated together with the corresponding constraint $\pm v_{\phi,\max}$ for both controllers. Surprisingly, a gait pattern different to lateral undulation emerges from economic MPC. Instead of a sinusoid as in lateral undulation, the proposed controller leads to an approximately piece-wise linear which results in the observed performance gain. Moreover, the economic MPC scheme achieves constraint satisfaction while the lateral undulation controller violates the constraints during the transient phase, which explains the qualitatively different closed-loop forward velocity during the first $50$ time steps.
The superior velocity comes at the price of an increased energy consumption, which is investigated next. In order to compare the asymptotic average energy consumption and forward velocity, we computed the average energy consumption and velocity over the last $200$ time steps of our simulations. We consider $E=\sum_{t = 100}^{300} \frac{u^T(t)u(t)}{200}$ as a measure for the consumed energy by the controllers, while the asymptotic average velocity is computed by $v_{av} = \sum_{t=100}^{300} \frac{v_t(t)}{200}$. The results of this comparison are given in Table \[tab\].
---------------------- ---------- ------------ ---------- ------------
Gait pattern absolute relative absolute relative
LU $0.2072$ $100.0 \%$ $0.0494$ $100 \%$
EMPC, $\gamma=0$ $0.2624$ $126.6 \%$ $0.0609$ $123.3 \%$
EMPC, $\gamma=0.025$ $0.1979$ $ 95.5 \%$ $0.0554$ $112.1 \%$
---------------------- ---------- ------------ ---------- ------------
: Comparison of the asymptotic average energy consumption and forward velocity[]{data-label="tab"}
The data demonstrates that the economic MPC algorithm reaches a higher asymptotic average forward velocity in both cases, while it simultaneously leads to a lower energy consumption if the value of $\gamma$ is tuned to take energy consumption into account. Hence, the advantage of the proposed economic MPC scheme is twofold: It achieves a higher performance in terms of forward velocity as well as in terms of energy consumption by finding a superior gait pattern while ensuring constraint satisfaction.
Actuator fault
--------------
We next investigate the case of a joint failure occuring during operation to specifically illustrate the advantage of integrating the choice of the gait pattern into the closed loop. We simulate the scenario of joint $4$ blocking after , and we therefore fix afterwards. For this experiment, we set $\gamma = 0$ in the cost function and compare an economic MPC algorithm which has access to information about the failure, i.e., where we set and in the dynamic constraints in Problem \[prob:EMPC\], with a fault-unaware economic MPC scheme.
Figure \[fig:FaultVelocities\] shows the resulting closed-loop velocity for both controllers. It can be observed, that the fault-aware controller achieves a slightly higher asymptotic average velocity after the failure has occured. the average velocity as in the previous subsection by yields for the algorithm with access to information about the failure and for the fault-unaware algorithm.
Moreover, in Figure \[fig:FaultPhi\] the corresponding joint distances for joint $i = 3$ are illustrated. It becomes apparent that the economic MPC scheme adapts the gait pattern once the failure occurs in order to reach a higher forward velocity.
Conclusion
==========
In this work, we proposed an economic MPC scheme for snake robot locomotion that integrates the choice of the gait pattern into the closed loop. We proved recursive feasibility of the economic MPC scheme in the closed loop and briefly sketched performance guarantees. We compared the proposed economic MPC algorithm to a standard lateral undulation controller from the literature and demonstrated that economic MPC is able to improve the snake robot’s performance in terms of velocity and energy consumption at the same time, while ensuring constraint satisfaction.
The theoretical foundations and insights gained by this work serve as a first step towards applying MPC to the control of snake robot locomotion with different criteria, subject to additional constraints and in the presence of . We believe that the ability to implicitly choose a gait pattern online through the economic MPC algorithm will be particularly beneficial for utilizing obstacles for locomotion, which is part of our ongoing work.
[^1]: Marko Nonhoff is with the Institute of Automatic Control, Leibniz University Hannover, 30167 Hannover, Germany, Philipp N. Köhler and Frank Allgöwer are with the Institute for Systems Theory and Automatic Control, University of Stuttgart, 70550 Stuttgart, Germany. Anna Kohl and Kristin Y. Pettersen are with Centre for Autonomous Marine Operations and Systems (NTNU AMOS), Department of Engineering Cybernetics, Norwegian University of Science and Technology, 7491 Trondheim, Norway.
[^2]: A continuous function $\beta:\mathbb{R}_{\geq0}\times\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq0}$ is said to be of class $\mathcal{KL}$ if $\beta(\cdot,t)$ is strictly increasing, $\beta(0,t)=0$, and $\beta(r,\cdot)$ is decreasing with $\lim_{k\rightarrow\infty}\beta(r,k)=0$, cf. [@Khalil2013].
|
---
abstract: 'Surface-state contributions to the dc conductivity of most homogeneous metals exposed to uniform electric fields are usually as small as the system size is large compared to the lattice constant. In this work, we show that surface states of topological metals can contribute with the same order of magnitude as the bulk even in large systems. This effect is intimately related to the intrinsic anomalous Hall effect, in which an applied voltage induces chiral surface-state currents proportional to the system size. Unlike the anomalous Hall effect, the large contribution of surface states to the dc conductivity is also present in time-reversal invariant Weyl semimetals, where the surface states come in counter-propagating time-reversed pairs. While the Hall voltage vanishes in the presence of time-reversal symmetry, the twinned chiral surface currents develop similarly as in the time-reversal broken case. For this effect to occur, the relaxation length associated with scattering between time-reversed partner states needs to be larger than the separation of contributing surfaces, which results in a characteristic size dependence of the resistivity and a highly inhomogeneous current-density profile across the sample.'
author:
- 'M. Breitkreiz'
- 'P. W. Brouwer'
bibliography:
- 'library.bib'
date: March 2019
title: Large contribution of Fermi arcs to the conductivity of topological metals
---
*Introduction.*— Weyl and Dirac semimetals have attracted enormous attention recently, especially after their experimental discovery four years ago [@Xu2015a; @Xu2015b; @Lv2015; @Borisenko2014; @Neupane2014; @Liu2014a; @Xiong2015a]. The fascination for these materials derives in large part from novel responses to electromagnetic fields [@Armitage2017; @Yan2017]. Some of the most striking phenomena are associated with magnetic Weyl metals, which in their minimal model consist of two Weyl nodes close to the Fermi level. For a strictly linear dispersion around the Weyl points and with the nodes being separated by a distance $k_0$ in the $k_z$ direction, the $xy$ part of the conductivity tensor for this system is [@Burkov2011] $$\sigma_0 = e^2 \begin{pmatrix} n_{\mathrm{b}}D
&\frac{k_0}{2\pi h} \\ -\frac{k_0}{2\pi h} & n_{\mathrm{b}}D
\end{pmatrix},
\label{cond0}$$ where $n_{\mathrm{b}}$ is the density of bulk states at the Fermi level, $D$ is the bulk diffusion constant, and $h$ is Plank’s constant. While the diagonal part follows the standard Drude relation, the off-diagonal part stems from Berry-curvature singularities at the two Weyl points, which give wave packets an anomalous contribution to their velocity perpendicular to the applied electric field and the direction of intrinsic magnetization (the $z$ direction) [@Xiao2010]. This is the anomalous Hall effect — a transverse conductivity in the absence of an external magnetic field. The ratio $\sigma_0^{xy}/\sigma_0^{xx}$ becomes large when Weyl fermions are the only mobile charge carriers and the Weyl nodes are close to the Fermi level, in which case the density of bulk states goes to zero. Experimentally, strong anomalous Hall effects with $\sigma_0^{xy}/\sigma_0^{xx}\sim 0.2$ have been observed in GdPtBi [@Suzuki2016] and Co$_3$Sn$_2$S$_2$ [@Liu2017c], demonstrating topological Weyl physics in these materials via the associated anomalous transport properties.
In this regard it is unfortunate that most existing Weyl metals are time-reversal (TR) invariant [@Jia2016]. While Berry-curvature singularities and anomalous velocities are equally present in the presence of TR symmetry, these materials lack the striking signature of the anomalous Hall voltage. One way to see this is to consider a TR-invariant Weyl metal as the sum of a TR-broken subsystem and its TR-conjugate. Denoting the conductivity tensors of the two subsystems by $\sigma_0$ and $\sigma_0^{\rm T}$, the total conductivity is then given by the sum $\sigma_0+\sigma_0^{\rm T} =
2 \sigma_0^{xx}\,\mathds{1}$, which misses the anomalous off-diagonal part.
Here we will show that the total conductivity is no longer given by $\sigma_0+\sigma_0^{\rm T}$ if the relaxation length $\bar l$ associated with scattering between the time-reversed subsystems is comparable to or larger than the transverse system size $W$. In the limit $\bar l \gg W$ of fully decoupled TR subsystems we find that the conductivity is $2 (\rho_{0}^{xx})^{-1}$, where $\rho_{0}^{xx}$ is the longitudinal resistivity of a single TR subsystem, which is the same for the two TR subsystems. Since $\rho_{0}^{xx} =
\sigma_{0}^{xx}/[(\sigma_{0}^{xx})^{2} + (\sigma_{0}^{xy})^{2}]$, the anomalous Hall conductivity $\sigma_0^{xy}$ of each subsystem does not cancel from the expression for the conductivity, but reappears in the diagonal part. The additional contribution to the longitudinal conductivity comes, as we will show, from topological surface states, which contribute to the conductivity with the same order of magnitude as bulk states if $\bar l \gtrsim W$. Before deriving this result and discussing how the conductivity depends on relevant scattering lengths, we find it useful to first reconsider the anomalous Hall effect for a finite-size system.
*Finite-size anomalous Hall effect.* — For a finite system with dimensions $-L_x/2 < x < L_x/2$, $-W_y/2 < y < W_y/2$, $- L_z/2 < z < L_z/2$, the non-trivial topology of the band structure implies the presence of surface states, which reside on “Fermi arcs”, that connect the Weyl-fermion Fermi surfaces in momentum space [@Armitage2017; @Balents2011], as illustrated in Fig. \[fig1\](a). We consider a minimal model of two Weyl nodes, for which the Fermi arcs are straight lines of length $k_0$ and the velocities $\mathbf{v}_{{\mathrm{s}}} = \hbar^{-1} \nabla_{\mathbf{k}_{{\mathrm{s}}}} \varepsilon$ are $\mathbf{k}$-independent . In particular, the surface-state velocities are $\pm v\hat{\mathbf{x}}$ and $\pm v\hat{\mathbf{y}}$ at the surfaces at $y=\mp W_y/2$ and $x=\pm L_x/2$ respectively. The density of Fermi-arc states at each of those four surfaces is $$n_\mathrm{s}= \frac{1}{(2\pi)^2}\int_\text{s} d^2 k \delta (\varepsilon-\varepsilon_F) = \frac{k_0}{2\pi vh}.
\label{ns}$$ To determine the conductivity of the finite-size system we follow the Landauer approach and interpret the applied electric field $\mathbf{E}$ as a chemical-potential difference between the system boundaries. In this picture, the bulk contribution $\mathbf{j}_{\mathrm{b}}$ to the current-density misses any anomalous transverse terms from a momentum-shifting $\mathbf{E}$-field [@Xiao2010] and instead follows the standard relation $$\begin{aligned}
\mathbf{j}_{\mathrm{b}}=& e^2 n_{\mathrm{b}}D\, \mathbf{E}.
\label{jb1}\end{aligned}$$ Opposite surfaces have counterpropagating surface states, so that the surface contribution $\mathbf{j}_{\mathrm{s}}$ to the current density is driven by the transverse potential differences $E^x L_x$ and $E^y W_y$, $$\begin{aligned}
j_{\mathrm{s}}^x =& e^2 n_{\mathrm{s}}v\, E^y,\;\;\;\;\;\; j_{\mathrm{s}}^y = -e^2 n_{\mathrm{s}}v\, E^x.
\label{js1}\end{aligned}$$ Taken together, Eqs. – reproduce the conductivity tensor . We note that the contribution of the anomalous (topological) surface states to the average current density of the three-dimensional sample is not antiproportional to the system size, unlike that of non-topological surface states. Owing to spatial separation of countermovers, the non-equilibrium occupation of chiral surface states is proportional to $L_x$ or $W_{y}$, which cancels with the same factor, by which the total current is divided to obtain the current density.
![(a) Weyl semimetal with two Weyl cones (green) and straight Fermi-arc surface states (blue) illustrated in mixed momentum/real space. (b) A Weyl semimetal in slab geometry (finite in $y$ direction) with an imposed potential gradient in the $x$ direction. The enlarged view illustrates the chemical-potential gradients of surface and bulk states and the resulting current flow in case of strong surface bulk scattering $l_{\mathrm{sb}}\ll W$ (left) and weak surface-bulk scattering $l_{\mathrm{sb}}\gg W$ (right). []{data-label="fig1"}](fig1)
In the above derivation we assumed local equilibrium between surface and bulk states, something that is only justified if the coupling between bulk and surface is sufficiently strong [@Ye2018; @Vinkler-Aviv2018]. To describe a finite coupling strength between bulk and surface states, we introduce the relaxation length $l_{\mathrm{sb}}$, corresponding to elastic isotropic scattering between surface and bulk states, neglecting any material-specific dependence of the scattering amplitude on the scattering states [@Resta2018]. We consider a slab geometry, for which $L_{x}$ and $L_z$ are taken to infinity, whereas $W_y \equiv W$ is finite, see Fig. \[fig1\](b). (Here “infinite” means $L_{x}$, $L_z \gg l_{\mathrm{sb}}$). The current is applied in the $x$ direction. The (surface) charge densities $c_{{\mathrm{s}}\pm}$ of surface states at $y=\pm W/2$ and the (bulk) charge density of the bulk states read, respectively, $$\begin{aligned}
c_{{\mathrm{s}}\pm}=-en_{\mathrm{s}}\mu_{{\mathrm{s}}\pm},\ \ c_{\mathrm{b}}=-e n_{\mathrm{b}}\mu_{\mathrm{b}}, \label{ci}\end{aligned}$$ where $\mu_{{\mathrm{s}}\pm}$, $\mu_{{\mathrm{b}}}$ is the corresponding (local) deviation of the chemical potentials from the Fermi energy. We assume that the penetration depth of surface states is much smaller than all other relevant length scales. Introducing surface and bulk current densities $\mathbf{j}_{{\mathrm{s}}\pm}=j_{{\mathrm{s}}\pm}\hat{\mathbf{x}}$ and $\mathbf{j}_{\mathrm{b}}$, the set of equations that determine the non-equilibrium steady state reads
$$\begin{aligned}
\mathbf{j}_{\mathrm{b}}=& -D\boldsymbol{\nabla} c_\mathrm{b}, \label{jb}\\
j_{{\mathrm{s}}\pm}=& \, \mp v \, c_{{\mathrm{s}}\pm},\label{js}\\
\boldsymbol{\nabla}\cdot\mathbf{j}_{\mathrm{b}}=& - \sum_{\pm}
\partial_x j_{{\mathrm{s}}\pm} \delta(y\mp W/2), \label{djb}\\
\partial_x j_{{\mathrm{s}}\pm} =& -e n_{\mathrm{s}}v \frac{\mu_b(\pm W/2)-\mu_{{\mathrm{s}}\pm}}{l_{\mathrm{sb}}}.\label{djs}\end{aligned}$$
\[e1\]
The first equation describes the diffusion of bulk particles with the diffusion constant $D$. The charge current density of chiral surface particles is directly proportional to their charge density. The change of the current densities is related to scattering between bulk and surface. This is captured by the continuity equation and by Eq. , which relates the rate of change of $c_{{\mathrm{s}}\pm}$ to the chemical potential difference $\mu_b-\mu_{{\mathrm{s}}\pm}$ and the scattering length $l_{{\mathrm{sb}}}$. The surface-state penetration depth being much smaller than $W$ excludes direct scattering between surface states of opposite surfaces.
![Resistivity as a function of the slab width for different values of $\sigma_0^{xy}/\sigma_0^{xx}$. With decreasing $W/\l_{\mathrm{sb}}$ the resistivity goes to zero since the current is increasingly conducted via surface states, which dissipation decreases. []{data-label="fig2"}](rho1)
Translation invariance in the $x$-direction allows us to assume a constant gradient of the chemical potential, corresponding to an applied electric field $E$ in the $x$ direction. To linear order in the gradients, we thus may set $$\mu_{\mathrm{b}}=eEx+\tilde{\mu}_{\mathrm{b}}(y)\;\;\;\;\;\; \mu_{{\mathrm{s}}\pm}=eEx+\tilde{\mu}_{{\mathrm{s}}\pm}. \label{mui}$$ Inserting this ansatz into gives $j_{\mathrm{b}}= \sigma_0^{xx} E$ and $$\begin{aligned}
\tilde{\mu}_{\mathrm{b}}(y)=eE_{\perp} y, \ \
\tilde{\mu}_{{\mathrm{s}}\pm}=\pm eE_{\perp} \frac{W}{2}\pm eE\, l_{\mathrm{sb}},
\label{musy}\end{aligned}$$ where $E_{\perp} = \sigma^{xy}_0 E/\sigma^{xx}_0$ and where $\sigma_0^{xx} = e^2D n_{\mathrm{b}}$ and $\sigma_0^{xy} = e^2v n_{\mathrm{s}}$ are the components of the infinite-system conductivity . The combined current density $j_{\mathrm{s}}= (j_{{\mathrm{s}}+} + j_{{\mathrm{s}}-})/ W$ coming from surface states is then $$\begin{aligned}
j_{\mathrm{s}}=& \sigma_0^{xy} \Bigg( \frac{\sigma_0^{xy}}{\sigma_0^{xx}}+\frac{2l_{\mathrm{sb}}}{W}\Bigg)E.
\label{js2}\end{aligned}$$ In the limit $W\gg l_{\mathrm{sb}}$ we recover the infinite-system conductivity by inverting the resistivity tensor $\rho^{xx} = \rho^{yy} = E/(j_{{\mathrm{s}}} + j_{\rm b})$ and $\rho^{yx} = -\rho^{xy} = E_{\perp}/(j_{{\mathrm{s}}} + j_{\rm b})$. For finite $l_{\mathrm{sb}}/W$, the transverse resistivity is not well defined due to the chemical-potential difference between surface and bulk states — it depends on to which subsystem the apparatus measuring the Hall voltage couples [@Ye2018; @Vinkler-Aviv2018]. The longitudinal resistivity remains well defined, however, see Fig. \[fig2\]. It includes the finite-size term $2 l_{\mathrm{sb}}/W$, which grows for weak surface-bulk coupling and pushes the current density profile towards the surfaces, as shown in Fig. \[fig1\](c). Decreasing the slab width below $ l_{\mathrm{sb}}/ |\sigma_0^{xy}/\sigma_0^{xx}
+\sigma_0^{xx}/\sigma_0^{xy}|$ the current flows with less and less dissipation — more and more current is carried by the surface states — and the resistivity goes to zero. Note that $\rho/\rho_0$ is an even function of $\sigma_0^{xx}/\sigma_0^{xy}$ and $\sigma_0^{xy}/\sigma_0^{xx}$, hence the resistivity has a generic minimum at $\sigma_0^{xy}/\sigma_0^{xx}=1$ for fixed $W/l_{\mathrm{sb}}$.
\] separated by $\Delta k$ and coupled by scattering (quantified by the relaxation length $l_{\mathrm{s \bar s}}$ and $l_{\mathrm{b \bar b}}$). (b) Potential gradients and current flow of the two subsystems for $\l_{\mathrm{sb}}\ll W \ll \bar l =\min (l_{\mathrm{s \bar s}},\sqrt{2Dl_{\mathrm{b \bar b}}/v})$. Both subsystems exhibit the anomalous Hall effect of a TR broken Weyl semimetal \[cf. Fig. \[fig1\](c)\]. The Hall voltages cancel each other but the surface-state currents add up.[]{data-label="fig3"}](fig2)
*Unbroken time-reversal symmetry*. — To describe TR symmetric Weyl semimetals we extend the two-node minimal model discussed above by adding its time-reversed copy and separating the copies in momentum space by $\Delta k$, as illustrated in Fig. \[fig3\](a). Using $\bar {\mathrm{s}}$ and $\bar {\mathrm{b}}$ to denote surface states and bulk states in the time-reversed copy, we introduce relaxation length $l_\mathrm{b \bar b}$ and $l_\mathrm{s \bar s}$ for scattering between bulk and surface states of different TR subsystems. As before, $l_{{\mathrm{sb}}}$ is the relaxation length for scattering between surface and bulk states in the same TR subsystem. Scattering processes of the type ${\mathrm{s}}\leftrightarrow \rm \bar b$, which connect surface and bulk states of different TR subsystems, are expected to be weaker than processes of the type ${\mathrm{s}}\leftrightarrow \rm \bar s$, because the density of bulk states at the surface is much smaller than the density of surface states, as seen, [*e.g.*]{}, in ARPES experiments [@Xu2015a; @Lv2015; @Liu2016]. For simplicity, we here neglect these scattering processes completely, although we note that our theory can be easily extended to include them.
The proper generalization of Eqs. and then reads
$$\begin{aligned}
\boldsymbol{\nabla}\cdot \mathbf{j}_{\mathrm{b}}=&\, \sum_{\pm} e n_{\mathrm{s}}v \delta(y\mp W/2)\frac{\mu_b-\mu_{{\mathrm{s}}\pm}}{l_{\mathrm{sb}}}
+v\frac{c_{\bar{{\mathrm{b}}}}-c_{\mathrm{b}}}{l_\mathrm{b \bar b}},\label{djb2}\\
\partial_x j_{{\mathrm{s}}\pm} =&\, - e n_{\mathrm{s}}v \frac{\mu_b(\pm W/2)-\mu_{{\mathrm{s}}\pm}}{l_{\mathrm{sb}}}
+ v \frac{c_{\bar{{\mathrm{s}}}}-c_{\mathrm{s}}}{l_\mathrm{s \bar s}},\label{djs2}\end{aligned}$$
\[trseq\]
plus two equations for $\boldsymbol{\nabla} \cdot \mathbf{j}_{\bar {\mathrm{b}}}$ and $\partial_x j_{\bar {\mathrm{s}}\pm}$. Solving these equations (see supplemental material for details) gives $j_{\mathrm{b}}= 2 \sigma_0^{xx}$ and $$\begin{aligned}
j_{\mathrm{s}}=& \,\frac{2 \sigma_0^{xy} l_{\mathrm{s \bar s}}^2}{W(l_{\mathrm{s \bar s}}+2 l_{\mathrm{sb}})}\label{resjs0}
\\
&\, \mbox{} \times \Bigg[ \frac{\sigma_0^{xy} {l^*}}{\sigma_0^{xy} {l^*}+\sigma_0^{xx} (l_{\mathrm{s \bar s}}+ 2 l_{\mathrm{sb}}) \coth (W/{l^*})}
+2 \frac{l_{\mathrm{sb}}}{l_{\mathrm{s \bar s}}}\Bigg]\, E, \nonumber\end{aligned}$$ where ${l^*}=\sqrt{2Dl_{\mathrm{b \bar b}}/v}$. In the limit $l_{\mathrm{sb}}\ll l_{\mathrm{s \bar s}},\, W$ Eq. simplifies to $$\begin{aligned}
j_{\mathrm{s}}=& 2\frac{(\sigma_0^{xy})^2}{\sigma_0^{xx}}\,
\bigg(\frac{W}{l_{\mathrm{s \bar s}}}\,\frac{\sigma_0^{xy}}{\sigma_0^{xx}}
+\frac{W}{{l^*}} \coth\frac{W}{{l^*}}\bigg)^{-1}
\, E. \label{resjs1}\end{aligned}$$
In the limit $W\ll \bar l \equiv \min({l^*},l_{{\mathrm{s \bar s}}})$, we obtain twice the current of the previously discussed TR-broken Weyl semimetal \[cf. \], where the presence of a large surface current was associated with the anomalous Hall effect. Here the potential and current pattern of the anomalous Hall effect appear in both TR subsystems, as illustrated in Fig. \[fig3\](b). Added together, the transverse Hall fields $\pm j_{\mathrm{s}}/(2\sigma_0^{xy})$ cancel, while the longitudinal current contributions of surface states add up.
![Resistivity $\rho$ versus slab width $W$ for different ratios $l^\ast / l_{\mathrm{sb}}$ at $\sigma_0^{xy}/\sigma_0^{xx}=1$. The other parameters were chosen as $l_{\mathrm{s \bar s}}=l_{\mathrm{b \bar b}}$ and $2D/v = l_{\mathrm{sb}}$ so that $\bar l = \min({l^*},l_{{\mathrm{s \bar s}}}) = {l^*}$. If $l_{\rm sb} < \bar l$, the anomalous contribution of surface states leads to a drop from $\rho/\rho_0=1$ for $\bar l \ll W$ to a plateau at $\rho/\rho_0=1/[1+(\sigma_0^{xy}/\sigma_0^{xx})^2]$ (green dotted line) for $l_{\mathrm{sb}}\ll W\ll \bar l$.[]{data-label="fig4"}](rho2)
The total resistivity of the slab, including the bulk contribution $j_{\mathrm{b}}=2\sigma_0^{xx}E$, shows characteristic signatures of this “twin anomalous Hall regime”. As shown in Fig. \[fig4\], in the limit $l_{\mathrm{sb}}\ll \bar l $ of weak scattering between the two TR subsystems the resistivity $\rho$ drops in two steps upon decreasing the system width $W$: First a drop from $\rho_0 = 1/2 \sigma^{xx}_0$ to $\rho_0/[1+(\sigma_0^{xy}/\sigma_0^{xx})^2]$ at $W \sim \bar l$, followed by a drop to zero at $W \lesssim l_{\mathrm{sb}}$. The first drop is due to the contribution of surface states, which sets in when the two subsystems effectively decouple and develop oppositely directed Hall voltages. The second drop then occurs for the same reasons as for the TR-broken case: The current is pushed towards the surfaces, as these become perfectly conducting channels. If the condition $l_{\mathrm{sb}}\ll \bar l$ is not met, the conductivity drops to zero in a single step at $W \sim \min(l_{{\mathrm{sb}}},l_{{\mathrm{s \bar s}}})$.
*Discussion*. — We have shown that Fermi-arc surface states in TR-invariant topological metals can contribute to transport if the system size is much larger than the lattice constant or the bulk mean free path, as long as the transverse system size $W$ is smaller than $\bar l \equiv \min(l_{\mathrm{s \bar s}},l^\ast)$, where $l_{\mathrm{s \bar s}}$ and ${l^*}= \sqrt{2Dl_{\mathrm{b \bar b}}/v}$ are the characteristic length scales describing the coupling between TR-conjugated subsystems. In this regime, the topological metal is effectively the sum of two time-reversed subsystems, which each exhibit an anomalous Hall effect. The longitudinal conductivity increases from the sum $\sigma_0 + \sigma_0^T = 2 \sigma_0^{xx}$ of subsystem conductivities in the limit $W \gg \bar l$ to twice the inverse longitudinal resistivity $2 (\rho_0^{xx})^{-1} = 2 \sigma_0^{xx} + 2 (\sigma_0^{xy})^2/\sigma_0^{xx}$ if $W \ll \bar l$, where the extra term is the contribution of the Fermi-arc surface states. That one has to average the resistivities instead of conductivities can also be understood directly from the fact that for each decoupled subsystem the transverse current must vanish, while the transverse potential gradient is set by the applied electric field. Remarkably, the surface contribution to the conductivity does not scale inversely with the width $W$ as long as $W \ll \bar l$ and is *inversely proportional* to $\sigma_0^{xx}$, making it larger for stronger scattering rates. This is in stark contrast to the opposite limit $W \gg \bar l$, where the surface contribution to the conductivity is proportional to $l_{\mathrm{sb}}/W$ (assuming $l_{\mathrm{sb}}\ll l_{\mathrm{s \bar s}}$), thus inversely proportional to both the width and surface-bulk scattering amplitude [@Gorbar2016a].
To estimate the characteristic length $\bar l$ for existing Weyl semimetals from the TaAs family, we compare it to the bulk scattering length $2D/v \sim 1\, \mu m$ given by the bulk transport lifetime and Fermi velocity and estimated from resistivity measurements [@Zhang2017d] combined with ab-initio calculations [@Lee2015]. Assuming that scattering is dominated by a Coulomb-disorder potential [@Burkov2011a; @Ominato2015], the scattering amplitude is suppressed with increasing momentum difference $q$ of scattering states $\propto 1/q^2$. Consequently, $2D/v$ is dominated by scattering processes within Weyl cones — on momentum-space distances $ k_F\sim 0.01/\mathrm{\AA}$ [@Xu2015a; @Arnold2016]. The scattering lengths $l_{\mathrm{b \bar b}}$ and $l_{\mathrm{s \bar s}}$ are instead governed by scattering on distances $\Delta k \sim 1 / \mathrm{\AA}$, which gives the estimate $\bar l\sim 100\, \mu m$. Additionally $\bar l$ can be increased if scattering between time-reversed states involves a spin flip [@Xu2016; @Inoue2016] or is dominated by long-ranged Gaussian impurities [@Ominato2015].
An experimental signature of the anomalous contribution of Fermi arcs is the decreasing resistivity with decreasing system size. The magnitude of this effect grows with increasing subsystem Hall angle $\sigma_0^{xy} / \sigma_0^{xx}$. While the estimation of this quantity for specific materials goes beyond the scope of this work, we note that a large value of $\sigma_0^{xy} / \sigma_0^{xx}$ is likely to occur, given the presence of a large Hall angle in TR-broken Weyl metals [@Suzuki2016; @Liu2017c]. We also note that the Hall angle may be increased by an enhanced long-ranged disorder, which would decrease $\sigma_0^{xx}$ without restricting the width bound $W \lesssim \bar l $.
The predicted decrease of the resistivity with decreasing sample thickness ($\sim 100\,\mu m$) is in qualitative agreement with recent measurements on NbAs nanobelts , remarkably contrasting with measurements on the Dirac semimetal Cd$_3$As$_2$ [@Liang2015; @Zhang2017b; @Schumann2018], where such a decrease has not been observed. Note that in Dirac semimetals $\Delta k \to 0$ strongly suppressing the characteristic width at which the twin anomalous Hall effect can set in. Other striking signatures and potential applications of the discussed phenomenon lie in transport devices that access the conductivity locally. The peculiar width independence of the surface contribution to the *average* conductivity implies a highly inhomogeneous *local* conductivity — enhanced by $(W/\xi)\,2 (\sigma_0^{xy})^2 / \sigma_0^{xx} $ at the surface, where $\xi$ is the penetration depth of the surface states. This factor can be large even at small Hall angles, since $W$ is only bound by $\bar l$, while $\xi$ is typically on the order of the lattice constant.
An example of how this surface current can be detected, is the Edelstein effect [@Edelstein1990], which has been predicted to be large in TaAs [@Johansson2018], due to the strong spin polarization of Fermi arcs in these materials [@Xu2016; @Inoue2016]. The current-induced magnetization will inherits the width dependence of the local Fermi-arc current density, predicted by our work. In particular, this will lead to a strong enhancement of the Edelstein effect in thin films with $W\lesssim \bar l$.
In this context, it is interesting to note that a wedge geometry, in which the width $W$ varies in the direction of current flow between the regimes $W\ll\bar l$ and $W\gg \bar l$ converts a surface current into uniform bulk current and vice versa, dependent on the current direction. Such a device could be an example how the peculiar size dependence of topological metals can be used for nanotransport circuit design.
*Acknowledgments*. We would like to thank P. Silvestrov, A. Johansson, C. Timm, and F. Xiu for valuable discussions. This research was supported by project A02 of the CRC-TR 183 “entangled states of matter” and the Grant No. 18688556 of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).
Supplemental Material {#supplemental-material .unnumbered}
=====================
Current contribution of curved Fermi arcs
=========================================
Our analysis can be easily generalized to curved Fermi arcs instead of the straight Fermi arcs considered for simplicity in the main text. In the following we show that the expression for the current density used in the main text given by combination of Eqs. and , $$j_{{\mathrm{s}}}^x = \frac{e^2k_0}{2\pi\, h}\,E_y,\;\;\;\;\;\;\;
j_{{\mathrm{s}}}^y = -\frac{e^2k_0}{2\pi\, h}\,E_x,$$ continues to hold in case of curved Fermi arcs, $k_0$ being the length of a straight line connecting the end points of the arc. In case of curved Fermi arcs, the current contribution of Fermi arcs at two opposite surfaces is given by $$\mathbf{j}_{{\mathrm{s}}\pm}=-e\,\mu_{{\mathrm{s}}\pm}\,\frac{1}{(2\pi)^2}\int d^2k\,\delta (\varepsilon-\varepsilon_F) \mathbf{v}_{{\mathrm{s}}\pm},$$ where $\mu_{{\mathrm{s}}\pm}$ is the deviation of the chemical potential from the Fermi level. Executing the integration over energy, the integral reduces to the 1D integration along the Fermi arc, $$\hbar \int d^2k\,\delta (\varepsilon-\varepsilon_F) \mathbf{v}_{{\mathrm{s}}\pm}
= \int dk\, \frac{\mathbf{v}_{{\mathrm{s}}\pm}}{|\mathbf{v}_{{\mathrm{s}}\pm}|}. \label{int1}$$ The integral is solved by closing the integration contour with a straight line of length $k_0$ that connects the two end points of the Fermi arc. Let $\mathbf{n}_\pm$ be a unit vector normal to this line, such that the angle between $\mathbf{v}_{{\mathrm{s}}\pm}\big/|\mathbf{v}_{{\mathrm{s}}\pm}|$ and the integration contour is kept constant if we define $\mathbf{v}_{{\mathrm{s}}\pm}\big/|\mathbf{v}_{{\mathrm{s}}\pm}|=\mathbf{n}_\pm$ on this line. We then obtain $$\int dk\, \frac{\mathbf{v}_{{\mathrm{s}}\pm}}{|\mathbf{v}_{{\mathrm{s}}\pm}|}
= \underbrace{\oint dk\, \frac{\mathbf{v}_{{\mathrm{s}}\pm}}{|\mathbf{v}_{{\mathrm{s}}\pm}|}}_{=0} - k_0\,\mathbf{n}_\pm,$$ leading to the result $$\mathbf{j}_{{\mathrm{s}}\pm}=\frac{e\,k_0}{2\pi\, h}\,\mu_{{\mathrm{s}}\pm}\,\mathbf{n}_\pm.$$ Applying to the current contribution of the surfaces $y=\pm W_y/2$, where $\mathbf{n}_\pm = \pm \hat{\mathbf{x}}$, and the surfaces $x=\pm L_x/2$, where $\mathbf{n}_\pm = \mp \hat{\mathbf{y}}$, we obtain, respectively, $$j_{{\mathrm{s}}}^x = \frac{j^x_{{\mathrm{s}}+}+j^x_{{\mathrm{s}}-} }{W_y}= \frac{e^2k_0}{2\pi\, h}\,E_y,\;\;\;\;\;\;\;
j_{{\mathrm{s}}}^y = \frac{j^y_{{\mathrm{s}}+}+j^y_{{\mathrm{s}}-} }{L_x}= -\frac{e^2k_0}{2\pi\, h}\,E_x.$$
Note that, while the final expression for the current density of surface states remains valid for curved Fermi arcs, the expression for the density of states $n_s$ given in is no valid in case of curved Fermi arcs. The expressions for the current density used in the main text, however, remain valid in case of curved Fermi arcs if the product $n_s v$ is replaced by $k_0/2\pi h$.
Unbroken time-reversal symmetry
===============================
In the following we discuss in detail the solution for the TR-preserved case. From Eqs. , , , , , and we obtain the set of equations
$$\begin{aligned}
j_{\mathrm{b}}^y=& e n_{\mathrm{b}}D\, \partial_y\mu_{\mathrm{b}}(y),\label{jbdmu}\\
\partial_y j_{\mathrm{b}}^y =& e\frac{n_{\mathrm{s}}v}{l_{\mathrm{sb}}}\sum_\pm\big[\mu_{\mathrm{b}}(y)-\mu_{{\mathrm{s}}\pm}\big]\delta\big(y\mp\tfrac{W}{2}\big)+2e\frac{n_{\mathrm{b}}v}{l_{\mathrm{b \bar b}}}\mu_{\mathrm{b}}(y),\\
\pm eE =& - \frac{\mu_{\mathrm{b}}(\pm W/2)-\mu_{{\mathrm{s}}\pm}}{l_{\mathrm{sb}}}+2\frac{\mu_{{\mathrm{s}}\pm}}{l_{\mathrm{s \bar s}}}.\end{aligned}$$
\[deq3\]
Eliminating $j_{\mathrm{b}}^y$ in and using the symmetry relation $\mu_{\mathrm{b}}\big(\pm\tfrac{W}{2}\big)=\pm\mu_{\mathrm{b}}\big(\tfrac{W}{2}\big)$ we obtain
$$\begin{aligned}
\partial_y^2 \mu_{\mathrm{b}}=& \frac{n_{\mathrm{s}}v}{n_{\mathrm{b}}D}\sum_\pm
\frac{\mu_{\mathrm{b}}(y)-\mu_{{\mathrm{s}}\pm}}{ l_{\mathrm{sb}}}\delta\big(y\mp\tfrac{W}{2}\big)
+\frac{2v}{D\,l_{\mathrm{b \bar b}}}\mu_{\mathrm{b}}(y),\label{d2mu} \\
\mu_{{\mathrm{s}}\pm} =& \pm\frac{ eE\,l_{\mathrm{sb}}+\mu_{\mathrm{b}}\big(\tfrac{W}{2}\big)}{1+2l_{\mathrm{sb}}/l_{\mathrm{s \bar s}}}. \label{d2mus} \end{aligned}$$
\[deq5\]
Integration of Eq. with the boundary condition $\mu_{\mathrm{b}}'(-W/2)=0$, which according to corresponds to $j_{\mathrm{b}}^y(-W/2)=0$, and inserting Eq. gives $$\begin{aligned}
\mathrm{lim}_{\xi\to 0}\,\mu_{\mathrm{b}}'(-W/2+\xi)=& \mathrm{lim}_{\xi\to 0}\,\int_{-W/2}^{-W/2+\xi}dy \,\partial_y^2 \mu_{\mathrm{b}}= \frac{n_{\mathrm{s}}v}{n_{\mathrm{b}}D}\frac{\mu_{\mathrm{b}}(-W/2)-\mu_{s-}}{l_{\mathrm{sb}}} \nonumber \\
=& \frac{n_{\mathrm{s}}v}{n_{\mathrm{b}}D}\bigg[eE+\frac{2\mu_{\mathrm{b}}(-W/2)}{l_{\mathrm{s \bar s}}+2 l_{\mathrm{sb}}}\bigg].\label{BC1}\end{aligned}$$ The solution of for $-W/2<y<W/2$ with $\mu_{\mathrm{b}}(-y)=-\mu_{\mathrm{b}}( y)$ reads $$\mu_{\mathrm{b}}( y) = A\,\Big(e^{-2(y+W/2)/l^\ast}-e^{2(y-W/2)/l^\ast}\Big),\;\;\;\; l^\ast=\sqrt{2Dl_{\mathrm{b \bar b}}/v}.$$ The integration constant $A$ is determined by the boundary condition , giving the solution $$\mu_{\mathrm{b}}( y) = eE\,\frac{W}{2}\,\frac{1}{1+2l_{\mathrm{sb}}/l_{\mathrm{s \bar s}}}\,
\frac{e^{2(y-W/2)/l^\ast}-e^{-2(y+W/2)/l^\ast}}{\frac{W}{l_{\mathrm{s \bar s}}+2l_{\mathrm{sb}}}
\big(1-e^{-2 W/l^\ast}\big)+\frac{ W}{l^\ast}\big(1+e^{-2 W/l^\ast}\big)\frac{n_{\mathrm{b}}D}{n_{\mathrm{s}}v}}.$$ Using Eq. we obtain $$\begin{aligned}
\frac{\mu_{s+}-\mu_{s-}}{W} =& eE\,\frac{1}{1+2 l_{\mathrm{sb}}/l_{\mathrm{s \bar s}}}
\Bigg[ \frac{1}{1+2l_{\mathrm{sb}}/l_{\mathrm{s \bar s}}}\,
\frac{1}{\frac{W}{l_{\mathrm{s \bar s}}+2l_{\mathrm{sb}}}
+\frac{n_{\mathrm{b}}D}{n_{\mathrm{s}}v}\, \frac{W}{l^\ast} \coth\tfrac{W}{l^\ast}}
+\frac{2 l_{\mathrm{sb}}}{W}\Bigg].\end{aligned}$$ The current-density contribution of Fermi arcs is given by $j_{\mathrm{s}}=2e vn_{\mathrm{s}}(\mu_{{\mathrm{s}}+}-\mu_{{\mathrm{s}}-})/W$, where the factor $2$ accounts for the equivalent contribution of the TR subsystem, $$\begin{aligned}
j_{\mathrm{s}}=& 2 \sigma_0^{xy}\,\frac{1}{1+2 l_{\mathrm{sb}}/l_{\mathrm{s \bar s}}}
\Bigg[ \frac{1}{1+2l_{\mathrm{sb}}/l_{\mathrm{s \bar s}}}\,
\frac{1}{\frac{W}{l_{\mathrm{s \bar s}}+2l_{\mathrm{sb}}}
+\frac{\sigma_0^{xx}}{\sigma_0^{xy}}\, \frac{ W}{l^\ast} \coth\tfrac{ W}{l^\ast}}
+\frac{2 l_{\mathrm{sb}}}{W}\Bigg]\, E.\end{aligned}$$ This is Eq. (\[resjs0\]) of the main text. In the limit $l_{\mathrm{sb}}\ll l_{\mathrm{s \bar s}},\; W$ this simplifies to $$\begin{aligned}
j_{\mathrm{s}}=& 2\frac{(\sigma_0^{xy})^2}{\sigma_0^{xx}}\,
\bigg(\frac{W}{l_{\mathrm{s \bar s}}}\,\frac{\sigma_0^{xy}}{\sigma_0^{xx}}
+\frac{ W}{l^\ast} \coth\frac{ W}{l^\ast}\bigg)^{-1}
\, E.\end{aligned}$$ The $W$-dependent term goes to $1$ when $W\to 0$. Hence the twin anomalous Hall regime, in which the current-density contribution of surface states is $W$-independent, $j_{\mathrm{s}}= 2(\sigma_0^{xy})^2 / \sigma_0^{xx}\, E$, is characterized by $$\frac{W}{l_{\mathrm{s \bar s}}}\,\frac{\sigma_0^{xy}}{\sigma_0^{xx}}
+\frac{ W}{l^\ast} \coth\frac{ W}{l^\ast} \lesssim 1, \label{cond}$$ which, assuming $\sigma_0^{xy}/\sigma_0^{xx}\sim 1$, is equivalent to $W \lesssim \min(l_{\mathrm{s \bar s}},l^\ast)$.
|
---
abstract: 'Deep image completion usually fails to harmonically blend the restored image into existing content, especially in the boundary area. This paper handles with this problem from a new perspective of creating a smooth transition and proposes a concise Deep Fusion Network (DFNet). Firstly, a fusion block is introduced to generate a flexible alpha composition map for combining known and unknown regions. The fusion block not only provides a smooth fusion between restored and existing content, but also provides an attention map to make network focus more on the unknown pixels. In this way, it builds a bridge for structural and texture information, so that information can be naturally propagated from known region into completion. Furthermore, fusion blocks are embedded into several decoder layers of the network. Accompanied by the adjustable loss constraints on each layer, more accurate structure information are achieved. We qualitatively and quantitatively compare our method with other state-of-the-art methods on Places2 and CelebA datasets. The results show the superior performance of DFNet, especially in the aspects of harmonious texture transition, texture detail and semantic structural consistency. Our source code will be avaiable at: <https://github.com/hughplay/DFNet>'
author:
- |
Xin Hong[^1^]{}[^1], Pengfei Xiong[^2^]{}, Renhe Ji[^2^]{}, and Haoqiang Fan[^2^]{}\
[[^1^]{}Institute of Computing Technology, Chinese Academy of Sciences]{} [[^2^]{}Megvii Technology]{}\
[`hongxin@ict.ac.cn`]{} [`{xiongpengfei, jirenhe, fhq}@megvii.com`]{}
bibliography:
- 'main.bib'
title: Deep Fusion Network for Image Completion
---
{width="\textwidth"}
Introduction
============
Image completion, which aims to fill unknown region of an image, is a fundamental task in computer vision. It can be broadly applied to the fields of image editing, such as old photo recovering, object removal, and seamless inpainting for damaged image. For most such applications, it is a critical problem to generate perceptually plausible completion results, specifically with natural transition between known and unknown region.
Previous approaches based on deep learning have shown great progress in image completion task [@liu2018partialinpainting; @yu2018generative; @yu2018free; @IizukaSIGGRAPH2017; @nazeri2019edgeconnect; @pathakCVPR16context; @song2018spg; @Yuhang2018Contextualinpaint; @yiwang2018MultiColumn; @Huy2018Structural; @Haoran2018Progressive; @Raymond2018Semantic]. As mentioned in [@bertalmio2000image], these methods can be divided into two groups. One group of works focus on building a contextual attention architecture or applying effective loss functions to generate more realistic content in the missing area. They assume the gaps should be filled with similar content from background. A typical arrangement is applying Partial Convolutions[@liu2018partialinpainting] to concentrate on the unknown region. Other methods regard structural consistency as more important thing. Context priors such as edges are the most frequently used in these methods to ensure structural continuity. For instance, [@nazeri2019edgeconnect] proposed the Edge Connect method which can recover images with good semantic structural consistency. These approaches is dedicated to infer the unknown region with visually realistic and semantically related content. However, realizing smooth transition is more critical than restoring texture-rich images in most scenarios, as shown in Figure \[fig:teaser\].
Humans has an incredible ability to detect discontinuous transition region. Consequently, The filled region must be perceptually plausible in the transition zone with sufficiently similar texture and consistent structure. In order to achieve smooth transition, [@perez2003poisson] proposed a method to iteratively optimize the pixel gradient in edge transitional region. Given two images, the fusion quality depends on the consistency of the gradient changes of these two images, which is similar with the relationship between the restored content and the known region in image completion. This inspires us to build a network to simulate the composition process.
In this work, we design a learnable fusion block to implement pixel level fusion in the transition region. As shown in Figure \[fig:fusion-block\], the fusion block is introduced that can be embedded to an encoder-decoder structure. Different from the previous methods, we develop an extra convolutional block to generate an alpha map, which is similar to the hole mask but has smoother weights especially on the boundary region. In the process of gradient descent optimization, the alpha composition map adjusts the balance between restored image and ground truth content to make the transition smoother. Similar ideas have also been used in image matting [@ImageMatting; @NingXu_DIM]. However, The purpose of these method is to extract the smooth coefficients from background and foreground images, while the proposed fusion block is to combine them together.
In detail, we propose a Deep Fusion Network (DFNet). Firstly, a fusion block is adopted as an adaptable module to combine the restored part of image and original image. In addition to providing a smooth transition, the fusion block avoids learning unnecessary identity mapping for pixels in unknown region, and provides an attention map to make network focusing more on the missing pixels. With fusion block, structural and texture information can be naturally propagated from known region into unknown region. Secondly, we embed this module into different decoder layers. We find out that by considering the prediction of different fusion blocks with multi-scale constraints, the deep fusion network outperforms the network with only one fusion block embedded to the final layer. Furthermore, while different layers provides different feature presentations, we selectively switch on and off structure and texture loss, to recover the structural information from lower layers and refine texture details in high layers. The whole architecture of DFNet is displayed in Figure \[fig:dfn\].
The proposed DFNet is evaluated on two standard benchmarks, Places2 and CelebA. In order to better verify the proposed method, we define Boundary Pixels Error to measure the transition performance near the boundary of unknown region. Also, $\ell_{1}$ and FID are applied to verify global texture and consistency. Experiments demonstrate the superior performance of DFNet while compared with other state-of-the-art methods both in quantitative and qualitative aspects. It achieves better results in not only smooth texture transition but also structural consistency and more detailed textures. As conclusion, the main contributions can be summarized as follows:
- We investigate the image completion problem with the perspective of better transition region and propose fusion block which predicts an alpha composition map to achieve smooth transition.
- Fusion block avoids learning unnecessary identity mapping for known region and provides an attention mechanism. In this way, structure and texture information can propagate from known region to completion more naturally.
- We propose Deep Fusion Network, a U-Net architecture embedded with multiple fusion blocks to apply multi-scale constraints.
- A new measurement Boundary Pixels Error (BPE) is introduced to measure the transition performance near the boundary of missing hole.
- The results on Places2 and CelebA show that our method outperforms state-of-the-art methods in both qualitative and quantitative aspects.
Related Work
============
**Context Aware** Context aware based image completion methods imagine the semantic content can be filled based on the overall scene. Context Encoders[@pathakCVPR16context] introduces a encoder-decoder network to restore images from damaged inputs and holes. It applies a discriminator to increase the authenticity of restored images. Yang et al.[@Yang_2017_CVPR] takes its result as input and then propagates the texture information from unknown region to fill the missing area. Li et al.[@GFC-CVPR-2017] and Iizuka et al.[@IizukaSIGGRAPH2017] extends Context Encoders by defining both global and local discriminators to pay more attention on the missing areas. Iizuka et al. applies Poisson Blending[@perez2003poisson] as post-processing. Liu et al. [@liu2018partialinpainting] introduces partial convolution layers to avoid capturing too many zeros from unknown region. These methods depend entirely on the training image to generate semantically relevant structures and texture confidence.
**Texture Generation** In the field of texture generation, perceptual loss is adopted to fill in visually realistic content for missing regions. Liu et al.[@liu2018partialinpainting] applies perceptual loss[@gatys2015neural; @Johnson2016Perceptual] which uses a VGG[@simonyan2014very] network as a feature extractor. It computes loss use extracted high level features to achieve higher resolution textures in completion. Other methods usually rely on GAN[@goodfellow2014generative] loss to obtain better details. For instance, Yu et al.[@yu2018generative] replaces the post-processing with a refinement network powered by the contextual attention layers.
![Illustration of Fusion Block. A fusion block extracts raw completion from feature maps by learnable function $\mathcal{M}$, and also predicts an alpha composition map with function $\mathcal{A}$. Finally it combines the raw completion with scaled input image by blending function $\mathcal{B}$. The detail of blocks can be found in Section \[sec:fusion\_block\].[]{data-label="fig:fusion-block"}](imgs/fusion-block.pdf){width="1\linewidth"}
**Structure constraints** To better control the completing behaviour of networks, other works[@song2018spg; @yu2018free; @nazeri2019edgeconnect] explore providing extra information for inpainting. Song et al.[@song2018spg] uses a DeepLabv3+[@chen2018encoder] model to first predict a segmentation map, and then completes the unknown region with predicting segmentation map as prior. Yu et al.[@yu2018free] proposes gated convolution which generalize partial convolution and the new structure is compatible with user guides, usually strokes to indicate edges. Like Song et al.[@nazeri2019edgeconnect] uses a two staged networks for completion. It first completes edges corresponding to the input image and then use completed strokes to guide the full color images. In some extent, those methods can manually control the completion result of network by replacing the priors with custom one or giving extra edge information.
**Image Embedding** As a similar work with image completion, image embedding and matting are also studied in the past decades. [@perez2003poisson] proposes a method to iteratively optimize the pixel gradient in edge transitional region. Then poisson matting[@sun2004poisson] firstly introduces a Poisson blending method into alpha matting by solving a poisson equation, which proves the effectiveness of alpha composition. Deep Image matting[@NingXu_DIM] also generates an alpha map with a encoder-decoder network. Cho et al[@ImageMatting] takes the matting results of [@closed-form] and normalized RGB colors as inputs and learn an end-to-end deep network to predict a new alpha matte. These methods prove that alpha matting based on deep learning is more realistic for image embedding and matting.
![Corresponding results in a fusion block.[]{data-label="fig:sample-alpha"}](imgs/alpha.png){width="0.8\linewidth"}
{width="0.98\linewidth"}
Deep Fusion Network
===================
*Deep Fusion Network* is built on a U-Net[@ronneberger2015u] like architecture, which is widely used in recent image segmentation[@Long_2015_CVPR] task and image to image translation[@pix2pix2017; @wang2018pix2pixHD; @StarGAN2018; @Kupyn_2018_CVPR] tasks. The difference between our DFNet and original U-Net is that we embed fusion blocks to several layers of decoder. Fusion blocks help us to achieve smoother transition near the boundary and is the key components for our multi-scale constraints. In this section, we first introduce the fusion block and then discuss our network architecture and loss functions.
Fusion Block {#sec:fusion_block}
------------
The task of image completion is to restore the missing area with visually plausible content from a damaged image $\mathbf{I}_{in}$ and a binary mask $\mathbf{M}$ which represents the location of the unknown region.
Recently deep learning based methods usually predict the whole image $\mathbf{I}_{out}$ which even includes known region and use it to calculate loss during training. However, they take $\mathbf{I}_{comp}$ ($\mathbf{I}_{comp} = \mathbf{M} \odot \mathbf{I}_{in} +
(\mathbf{1} - \mathbf{M}) \odot \mathbf{I}_{out}$, $\odot$ denotes Hadamard product) rather than $\mathbf{I}_{out}$ for testing. The composition process replaces known region in $\mathbf{I}_{out}$ with corresponding pixels in $\mathbf{I}_{in}$. Only a few methods[@liu2018partialinpainting] use both $\mathbf{I}_{out}$ and $\mathbf{I}_{comp}$ to compute loss.
This training strategy has problems. Firstly, the mission of image completion is to complete the unknown region only. It is actually hard to complete missing hole while keeping a strict identity mapping for known area. Secondly, the inconsistent use of $\mathbf{I}_{comp}$ and $\mathbf{I}_{out}$ during training and testing, along with the rigid composition method, usually produces visible artifacts around the boundary of missing area. As shown in the first case of Figure \[fig:teaser\], the result of Edge Connect[@nazeri2019edgeconnect] has a clear edge at the boundary of completion.
To remove the artifacts around the boundary and avoid neural networks learning unnecessary identity mapping, we propose *Fusion Block*. As shown in Figure \[fig:fusion-block\], a fusion block feed with two elements, an input image with unknown region $\mathbf{I}_{in}$ and feature maps $\mathbf{F}_k$ form $k_{\text{th}}$ layer ($1_\text{st}$ layer is the last decoder layer of U-Net). The fusion block first extracts raw completion $\mathbf{C}_k$ from feature maps, and then predicts an alpha composition map $\boldsymbol{\alpha}_k$ to combine them. The final result $\mathbf{\hat{I}}_k$ is obtained by: $$\mathbf{\hat{I}}_k = \mathcal{B}(\boldsymbol{\alpha}_k, \mathbf{C}_k, \mathbf{I}_{k})
= \boldsymbol{\alpha}_k \odot \mathbf{\mathbf{C}_k}
+ (1 - \boldsymbol{\alpha}_k) \odot \mathbf{I}_{k}$$ We resize $\mathbf{I}_{in}$ to obtain $\mathbf{I}_{k}$. The raw completion $\mathbf{C}_k$ extracted from feature maps $\mathbf{F}_k$ by a learnable function $\mathcal{M}$: $$\mathbf{C}_k = \mathcal{M}(\mathbf{F}_k)$$ $\mathcal{M}(\mathbf{x})$ transforms $n$ channel feature maps $\mathbf{x}$ into a 3 channel image with the resolution unchanged which is exactly the raw completion. Actually, we use a $1\times 1$ convolutional layer following with a sigmoid function to learn $\mathcal{M}$.
The alpha composition map $\boldsymbol\alpha_k$ is produced by another learnable function $\mathcal{A}$ from raw completion and the scaled input image: $$\boldsymbol{\alpha}_k = \mathcal{A}(\mathbf{F}_k, \mathbf{I}_{k})$$ $\boldsymbol\alpha_k$ has two choices in the number of channels, either single channel for image-wise alpha composition or 3 channels for channel-wise alpha composition. In practice, we find channel-wise alpha composition performs better. As for $\mathcal{A}$, we use three convolutional layers and the kernel size of them are 1, 3, 1. First two convolutional layers are followed with a Batch Normalization[@pmlr-v37-ioffe15] layer and a leaky ReLU function. And we apply sigmoid function to the output of last convolutional layer.
The fusion block enables network to avoid learning unnecessary identity mapping while completing unknown region with soft transition near the boundary. We also give an example of corresponding images in a fusion block in Figure \[fig:sample-alpha\]. Completion performance can be further improved with multi-scale constraints by embedding fusion blocks to the last few decoder layers of U-Net.
Network Architecture
--------------------
It’s intuitive that when completing an image, constructing structures is easier in lower resolution for algorithms, while recovering texture is more feasible in higher resolution. We embed fusion blocks to the last few decoder layers of the U-Net and obtain completion results in different resolution. And then we can apply structure and texture constraints to different resolution as we want. The overview of our DFNet is shown in Figure \[fig:dfn\]. We choose U-Net[@ronneberger2015u] like the one used in [@pix2pix2017; @liu2018partialinpainting] as our backbone architecture. The difference is that the last few decoder layers are embedded with fusion blocks. Each fusion block outputs a completion result $\mathbf{C}_i$ with the same resolution as the input feature maps $\mathbf{F}_i$. According to their resolution, we can provide different constraints as we want during training. We will discuss these constraints in Section \[sec:loss\]. During testing, only the completion result $\mathbf{\hat{\mathbf{I}}_0}$ from last layer is needed.
Loss Functions {#sec:loss}
--------------
The target of image completion is to generate visually plausible results in both aspects of structure and texture. Reconstruction loss, which is mean absolute error of each pixel between prediction and ground truth, is usually used to guarantee accurate structures in completion results. However, high resolution textures is beyond the capability of reconstruction loss. Previous works use GAN loss [@goodfellow2014generative] or perceptual loss along with style loss [@Johnson2016Perceptual] to obtain vivid textures. These two loss have same drawback which is known as producing checkerboard and fish scale artifacts[@liu2018partialinpainting]. Total variation loss is usually used to counter this drawback. Results from [@liu2018partialinpainting] shows that this artifact can be reduced more obviously by increasing the weight of style loss.
**Reconstruction Loss** Reconstruction loss is defined as mean absolute error of completion result $\hat{\mathbf{I}}_k$ and target image $\mathbf{I}_k$: $$\mathcal{L}_{\ell_1}^k = \frac{1}{C_{k}H_{k}W_{k}} \lVert \mathbf{I}_k - \hat{\mathbf{I}}_k \rVert_1$$ The number of channels is $C$, the height is $H$ and the width is $W$.
**Perceptual Loss and Style Loss** Perceptual loss and style loss are first used in style transfer[@gatys2015neural; @Johnson2016Perceptual]. They use a pre-trained VGG network to extract high level features. The errors are computed between these features rather than original images. Let $\phi_j(x)$ be the features of $j$th layer in a VGG network when given image $x$. The size of $\phi_j(x)$ is $C_j\times H_j \times W_j$. Perceptual loss is defined as the error of these features: $$\mathcal{L}_{p}^k = \sum_{j\in J}\lVert \phi_j(\mathbf{I}_k) - \phi_j(\hat{\mathbf{I}}_k) \rVert_1$$ $J$ is selected VGG layers. *Gram matrix* is a $C_j \times C_j$ matrix, whose elements are defined as: $$G_{j}^{\phi}(x)_{c,c'} = \frac{1}{C_{j}H_{j}W_{j}}\sum_{h=1}^{H_j}\sum_{w=1}^{W_j}\phi_{j}(x)_{h,w,c}\phi_{j}(x)_{h,w,c'}$$ And then style loss is $L_1$ mean absolute error between corresponding Gram matrices of the output and target image: $$\mathcal{L}_{s}^k = \sum_{j\in J}\lVert G_{j}^{\phi}(\mathbf{I}_k) - G_{j}^{\phi}(\hat{\mathbf{I}}_k) \rVert_1$$ Style loss doesn’t consider the position of pixels but cares about how high level features appear simultaneously[@Johnson2016Perceptual], so that it’s better for constraining the entire style of an image.
**Total Variation Loss** Total variation loss $\mathcal{L}_{tv}$ is errors computed only use predictions. Each pixel will compute errors with top pixel and left pixel respectively. This can be implemented more easily by using a convolution layer with a fixed kernel.
**Total Loss.** We group loss functions into *Structure Loss* and *Texture Loss*. Structure Loss is represented as weighted reconstruction loss: $$\mathcal{L}_{struct}^{k} = \lambda_{\ell_1} \mathcal{L}_{\ell_1}^k$$ And texture loss is a combination of three loss: $$\mathcal{L}_{text}^{k} = \lambda_{p} \mathcal{L}_{p}
+ \lambda_{s}\mathcal{L}_{s}^k + \lambda_{tv}\mathcal{L}_{tv}^k$$ Our final loss is sum of structure loss and texture loss from different resolution completion results: $$\mathcal{L}_{total} =
\frac{1}{\lvert P \rvert}
\sum_{k\in{P}} \mathcal{L}_{struct}^{k} +
\frac{1}{\lvert Q \rvert}
\sum_{k\in{Q}} \mathcal{L}_{text}^{k}$$ $P$ is the set of layers which consider structure loss while $Q$ is for texture loss. And for brevity, we use $(p, q)$ to represent the choice of $P,Q$, which takes last $p$ layers as $P$ and last $q$ layers as $Q$. For example, $(2, 1)$ represent $P = \{1, 2\}$ and $Q = \{1\}$, which means completion results from last two layers of U-Net will be used to compute structure loss and only last one for texture loss. Corresponding part will be ignored if the total number of layers $\lvert P \rvert$ or $\lvert Q \rvert$ equal to zero. We will discuss the choice of $P$ and $Q$ in Section \[sec:analysis\].
{width="\textwidth"} \[fig:effectiveness\]
{width="\textwidth"} \[fig:multilayer\]
{width="\textwidth"} \[fig:losstune\]
{width="\linewidth"}
Experiments
===========
Experiment Details
------------------
We evaluate DFNet on two public datasets: **Places2**[@zhou2017places] and **CelebA** [@liu2015faceattributes]. For Places2, we use the original train, test, and val splits. For CelebA, we randomly partition into 27K images for training and 3Kimages for testing. Images in Places2 and CelebA are respectively resized to $512\times 512$ and $256 \times 256$ during training and testing. We randomly generate 1000 masks according to the method in [@yu2018free] and perform augmentation to these masks during training. To analysis the influence of unknown region range, these masks are categorized into five classes, including \[0 10%), \[10%, 20%), \[20%, 30%), \[30%, 40%), \[40%, 50%).
Models are separately trained on each dataset. Our proposed model is implemented in PyTorch[@paszke2017automatic] and trained in a single machine with 8 GeForce RTX 2080 Ti. We use Horovod[@alex2018horovod] as our distributed training framework. With a batch size of 6 for each GPU, it usually takes about 3 days to train a model. Forwarding is extremely fast, it only takes $8.29 ms$ to complete an image. As a common configuration, Adam[@kingma2014adam] is applied for optimization. The learning rate reduced from $2e-3$ to $2e-6$ in 20 epochs, with a decay rate of $0.1$ and step size of $5$.
Evaluation Metrics
------------------
Different from the tasks as image classification, detection and segmentation, image generation usually don’t have strict targets. The basic rule is visually plausible. For image completion, it requires the completion not only looks real but also transit naturally from known region. So we apply $\ell_1$, and *Fréchet Inception Distance* (FID) [@heusel2017gans] as evaluation metrics both in perspective of pixels and features to quantitatively analysis the performance of DFNet.
Furthermore, we observe that pixels in unknown region that near the boundary have very small variance while these pixels play the most important role in structure and texture transition. To measure the transition performance of models, we propose *Boundary Pixels Error* (BPE) which only consider pixels error near the boundary. For boundary area $\mathbf{b}$, which is $n$ pixels narrow band adjacent to the boundary of unknown region, BPE is mean absolute error of those pixels between ground truth $\mathbf{I}$ and prediction $\hat{\mathbf{I}}$: $$BPE = \frac{\lVert \mathbf{b} \odot (\mathbf{I} - \hat{\mathbf{I}}) \rVert_1}{\lVert \mathbf{b} \rVert_1}$$
=0.128cm
---------------------------------------- ---------- ---------- ---------- ---------- ----------- ---------- ---------- ---------- ---------- ---------- ----------- ----------- ----------- ------------ ------------
(lr)[2-6]{} (lr)[7-11]{} (lr)[12-16]{} 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5
**DeepFill** 2.79 6.75 10.63 15.35 28.38 1.33 1.81 2.39 2.91 5.13 24.04 56.55 98.25 173.90 324.97
**PConv** 1.51 4.22 7.01 10.52 12.83 0.17 0.37 0.62 0.87 1.60 14.98 41.21 84.60 166.72 217.48
**EdgeConnect** 1.43 3.94 **6.41** **9.64** **11.38** 0.33 0.69 1.11 1.48 2.55 19.24 35.91 68.29 131.16 147.51
**DFNet** **1.40** **3.91** 6.50 9.89 11.96 **0.15** **0.33** **0.55** **0.74** **1.42** **12.27** **34.64** **65.25** **127.58** **136.22**
---------------------------------------- ---------- ---------- ---------- ---------- ----------- ---------- ---------- ---------- ---------- ---------- ----------- ----------- ----------- ------------ ------------
Analysis of DFNet architecture {#sec:analysis}
------------------------------
In this section, we investigate the performance of the proposed modules in DFNet. First, we show the effectiveness of fusion blocks. Then we focus on the effect of multi-scale constraints by gradually increasing fusion blocks to DFNet and evaluating it. Finally, we discuss how to apply structure loss and texture loss on different resolution completion results to achieve the best results.
### Effectiveness of Fusion Block. {#sec:effectiveness}
We compare our results with predictions from a normal U-Net and predictions from a DFNet but directly using mask to replace alpha composition map. We use only one fusion block for fair comparison, which means $P = Q = \{1\}$ for $\mathcal{L}_{total}$.(Section \[sec:loss\]).
As can be seen in the 1st row of Figure \[fig:exp\], fusion block leads to the best transition near the boundary. Although most of semantic information has been restored, there exists obvious color transition inconsistent in the result of standard network without mask constraints. This is because global semantic consistency constraints can only leads to similar texture in the missing areas, but structural consistency can not be guaranteed. Based on the mask constraints, the pixel transition in filling area becomes more natural, which proves the effect of the proposed method on the propagation of structural and texture information. As mentioned above, the alpha composition map is a attention mechanism to enhance the structural consistency. Furthermore, the result of learned alpha mapping is even better in the edge transition to eliminate the visible artifacts near the boundary.
The same detailed conclusion can be seen in Figure \[fig:sample-alpha\]. Based on the proposed fusion block, the structure between the known and unknown areas are well preserved, even beyond the mask area. The sharp edge of the roof is retained into the reconstructed image with other useless part discarded.
### Multi-scale constraints. {#sec:multiscale}
We compare DFNets with different number of fusion blocks from one to six. Formally speaking, $P$ and $Q$ in Section \[sec:loss\] both increase from $\{1\}$ to $\{6\}$. In this section, $P$ and $Q$ are equal to only analyze the role of multi-scale fusion.
As can be seen in the 2nd row of Figure \[fig:exp\], the structure of building is more clear and accurate based on more fusion blocks. Also the shapes of houses are depicted in the result of 3 fusion blocks instead of the noises in the result based on 1 fusion block. While high level layers in encoder have bigger receptive field and global context, the structure information can be more easily reconstructed with more layers in decoder. Nevertheless, although the result of 6 fusion blocks retains these structural information, its texture is not very stable compare to 3 fusion blocks. We guess this is because we shouldn’t apply texture constraints for low resolution completion result. In the next section, we will go into more detail about how to choose the number of blocks layers.
We also give the quantitatively comparison in the second row of Figure \[fig:choice\]. Results are separated according to the range of mask in each evaluation metric. With fusion blocks increased, FID gets lower and lower. This means multi-layer constraints helps to capture contextual information and makes the whole image looks more real. The BPE increases slightly with fusion blocks increased. This can be explained that finer texture and smoother transition is a trade-off. However, globally visual effect is more important and the change in BPE actually is very small.
### Loss Ablation and Tuning. {#sec:loss_tune}
Firstly, the effect of structure loss $\mathcal{L}_{struct}$ and texture loss are showed $\mathcal{L}_{text}$ by respectively trained DFNet only applies only one of them. As seen in the 3rd Figure \[fig:exp\], the result without texture loss is blurry but with accurately structure consistency, while the other one completely destroys the structure, it fails to recover edges of object although they have finer textures. This provides strong evidence for loss design in this paper.
We further discuss the dynamic loss design in each layer. Based on the visualization results in \[sec:multiscale\], we make a comprehensive comparison of the loss design in different layers. As shown in Figure \[fig:choice\], the performance depicts the same trend with different ranges of hole size. We choose $P = \{1, 2, 3, 4, 5, 6\}$ to compute structure loss and $Q = \{1, 2, 3\}$ for texture loss in the final architecture. This can be explained that, although the structure information is more and more abundant with higher and higher encoder layers, the high-level features will lead to texture noise due to the loss of global semantic constraints.
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Comparisons with Other Methods
------------------------------
We quantitatively and qualitatively compare our DFNet with 3 recently methods, including DeepFill [@yu2018generative], PConv [@liu2018partialinpainting] and Edge Connect[@nazeri2019edgeconnect]. Results of DeepFill and Edge Connect are obtained by using their pre-trained models [^2] [^3]. However, we don’t find the official implementation of PConv, so we implement one with the same settings described in the original paper.
### Quantitative Comparisons.
Table \[tab:compare\_other\] shows the comparison results on Places2[@zhou2017places]. We use three metrics including $\ell_1$, BPE and *Fréchet Inception Distance* (FID) [@heusel2017gans]. Results from ours outperforms others on both boundary transition and realistic on overall image. Our predictions on BPE is significantly lower than those from Edge Connect[@nazeri2019edgeconnect] and other methods. This means completion from our methods have better transitional area near the boundary, which also proves the effectiveness of proposed fusion blocks.
Edge Connect works well on maintaining structural consistency by applying additional edge constraints. However it doesn’t pay much attention to smooth transition. The constraints on the structure of the whole image can’t lead to natural image restoration, especially in detail. Results of Edge connect shows lower $\ell_1$ than ours while the missing hole is large. But this only state results of Edge Connect is more similar to original images. Because completion can be more diverse while hole is larger.
PConv use partial convolution to progressively reduce missing region, which can be considered as providing a hidden attention map gradually enlarged from boundary area to full known region. This enhance the learning ability near the boundary, which have the similar effects with the proposed DFNet when considering transition performance. However, this architecture is not good at large hole because information can’t be transmitted effectively to inner area. When comparing PConv and Edge Connect on BPE and FID, we can find PConv has better transition near the boundary than Edge Connect and comparable FID when missing hole is small, however, when missing hole becomes larger, Edge Connect will have more realistic results.
### Qualitative Comparisons.
Figure \[fig:compare\_places2\] shows the comparison on Places2 and CelebA without any post-processing. As shown in the figure, we can see our model has the best performance in texture consistency near the boundary, and also good at keeping the structure consistency even better than Edge Connect. Results from different datasets shows the generalization ability of our methods.
There is one thing should be noticed, as shown in the 1st case of Figure \[fig:compare\_places2\], we find PConv and Edge Connect sometimes fail to complete the missing hole when the missing hole cover the border of an image. For PConv, we think this is the limit of partial convolution, which can’t transmit information into a very large hole. While for Edge Connect, it always produces clouds like completion in similar situation. We couldn’t figure it out the reason.
Conclusion
==========
In this paper, we analysis the image completion technology from a new perspective. We propose Deep Fusion Network by designing a fusion block to predict an alpha composition map for combining completion and existing content and embedding it on multi-scale layers. Results of experiments on Places2 and CelebA dataset shows our method achieves state-of-the-art results, especially in the filed of harmonious texture transition, texture detail and semantic structural consistency.
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[^1]: This work is done when Xin Hong is an intern at Megvii Technology.
[^2]: <https://github.com/JiahuiYu/generative_inpainting>
[^3]: <https://github.com/knazeri/edge-connect/>
|
---
abstract: 'An analysis of the literature shows that there are two types of non-memristive models that have been widely used in the modeling of so-called “memristive” neural networks. Here, we demonstrate that such models have nothing in common with the concept of memristive elements: they describe either [*non-linear resistors*]{} or certain [*bi-state systems*]{}, which all are devices [*without*]{} memory. Therefore, the results presented in a significant number of publications are at least questionable, if not completely irrelevant to the actual field of memristive neural networks.'
address:
- 'Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208, USA'
- 'Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA'
author:
- 'Yuriy V. Pershin'
- Massimiliano Di Ventra
bibliography:
- 'memr\_mm.bib'
title: On the validity of memristor modeling in the neural network literature
---
=6
This Letter refers to a number of publications on “memristive” neural networks (MNNs) published during the last decade [@T1_1; @T1_2; @T1_3; @T1_4; @T1_5; @T1_6; @T1_8; @T1_9; @T1_10; @T1_11; @T1_12; @T1_13; @T1_14; @T1_15; @T1_16; @T1_17; @T1_18; @T1_19; @T1_20; @T1_21; @T1_22; @T2_1; @T2_2; @T2_3; @T2_4; @T2_5; @T2_6; @T1_7; @T2_7; @T2_8; @T2_9; @T2_10; @T2_11; @T2_12; @T2_13; @T2_14; @T2_15; @T2_16; @T2_17; @T2_18; @T2_19; @T2_20; @T2_21; @T2_22; @T2_23; @T2_24; @T2_25; @T2_26; @T2_27; @T2_28; @T2_29; @T2_30; @T2_31; @T2_32; @T2_33; @M1_1; @M1_2; @M1_3; @M1_4; @M1_5] (note that this list may be incomplete, as there may be other publications that slipped through our search). We put the word “memristive” in quotes, because as we will show in the present paper, the referenced published papers refer to models that have nothing to do with resistive memories (memristive elements).
In fact, in Refs. [@T1_1; @T1_2; @T1_3; @T1_4; @T1_5; @T1_6; @T1_8; @T1_9; @T1_10; @T1_11; @T1_12; @T1_13; @T1_14; @T1_15; @T1_16; @T1_17; @T1_18; @T1_19; @T1_20; @T1_21; @T1_22; @T2_1; @T2_2; @T2_3; @T2_4; @T2_5; @T2_6; @T1_7; @T2_7; @T2_8; @T2_9; @T2_10; @T2_11; @T2_12; @T2_13; @T2_14; @T2_15; @T2_16; @T2_17; @T2_18; @T2_19; @T2_20; @T2_21; @T2_22; @T2_23; @T2_24; @T2_25; @T2_26; @T2_27; @T2_28; @T2_29; @T2_30; @T2_31; @T2_32; @T2_33; @M1_1; @M1_2; @M1_3; @M1_4; @M1_5], two types of [*non-memristive*]{} models were used in the modeling/simulation of MNNs. Our main statement in this work is that the devices considered in these publications have [*no memory*]{} of past dynamics, and as such they cannot represent memristive elements. Consequently, the results obtained with these models have no relevance to the field of [*actual*]{} memristive neural networks [@Pershin2010].
To simplify the presentation, we will refer to the aforementioned models as “type 1” and “type 2” models. The type 1 model [@T1_1; @T1_2; @T1_3; @T1_4; @T1_5; @T1_6; @T1_8; @T1_9; @T1_10; @T1_11; @T1_12; @T1_13; @T1_14; @T1_15; @T1_16; @T1_17; @T1_18; @T1_19; @T1_20; @T1_21; @T1_22] claims to approximate a “memristive element” by an expression of the type $$\label{eq:1}
R_M^{(1)}(\dot{V}_M(t))=\left\{
\begin{array}{ccc}
R_{on}, & \dot{V}_M(t)>0 & \\
R_{off}, & \dot{V}_M(t)<0 & , \\
\textnormal{unchanged}, & \dot{V}_M(t)=0 &
\end{array}
\right.$$ where $R_M^{(1)}$ is supposed to be the memristance (memory resistance), $V_M(t)$ is the voltage across the device, $R_{on}$ and $R_{off}$ are the low- and high-resistance states of the device, respectively, and the dot denotes the time derivative. To the best of our knowledge, the first use of Eq. (\[eq:1\]) was proposed in Ref. [@T1_13].
In the type 2 model [@T2_1; @T2_2; @T2_3; @T2_4; @T2_5; @T2_6; @T1_7; @T2_7; @T2_8; @T2_9; @T2_10; @T2_11; @T2_12; @T2_13; @T2_14; @T2_15; @T2_16; @T2_17; @T2_18; @T2_19; @T2_20; @T2_21; @T2_22; @T2_23; @T2_24; @T2_25; @T2_26; @T2_27; @T2_28; @T2_29; @T2_30; @T2_31; @T2_32; @T2_33], the memristance in a MNN is represented by an expression of the form $$\label{eq:2}
R_{M,ij}^{(2)}(V_j)=\left\{
\begin{array}{ccc}
\hat{R}_{ij}, & |V_j|>T_i,\\
\check{R}_{ij}, & |V_j|<T_i, \\
\end{array}
\right.$$ where $T_i$ are thresholds, $\hat{R}_{ij}$ and $\check{R}_{ij}$ are constants, and $V_j$ is the voltage at a node $j$ of the network.
Based on a literature search, the model represented by Eq. (\[eq:2\]) was pioneered by the authors of Ref. [@T2_15]. Moreover, there is a sub-set of publications [@M1_1; @M1_2; @M1_3; @M1_4; @M1_5] where both type 1 and type 2 models are mentioned. While Eqs. (\[eq:1\]) and (\[eq:2\]) look different, they have a feature in common: the devices that they describe are [*not*]{} memristive elements.
\(a) (b)
To proceed, let us first recall the definition of [*actual*]{} memristive elements [@chua76a]. These are two-terminal resistive devices with memory defined (in the voltage-controlled case [@chua76a]) by $$\begin{aligned}
I&=&R^{-1}_M\left( \bm{x}, V_M \right) V_M, \label{eq:3} \\
\dot{\bm{x}}&=&\bm{f}\left(\bm{x}, V_M \right), \label{eq:4}\end{aligned}$$ where $I$ and $V_M$ are the current through and voltage across the device, respectively, $R_M\left( \bm{x}, V_M\right)$ is the memristance (memory resistance), $\bm{x}$ is an $n$-component vector of internal state variables, and $\bm{f}\left(\bm{x}, V_M\right)$ is a vector function.
The memory feature of memristive elements is related to their internal state that evolves according to Eq. (\[eq:4\]) and is manifested in the device response (notice that $R_M$ is a function of $\bm{x}$). When subjected to time-dependent input, memristive elements typically exhibit pinched hysteresis loops. Importantly, due to the presence of memory, these loops must be strongly dependent on the input frequency (and voltage amplitude) [@chua76a; @diventra09a]. Note that this is physically necessary for [*any*]{} system with memory [@13_properties]. For instance, for high-frequency input signals the hysteresis loop closes, as there is not enough time for the internal state variables to follow the fast-varying input.
Now, a brief comparison of Eqs. (\[eq:1\]) and (\[eq:2\]) with Eqs. (\[eq:3\]) and (\[eq:4\]) is sufficient to establish the fact that the devices described by the type 1 and type 2 models are [*not*]{} memristive. While the actual memristive elements are characterized by a memory (time non-locality) of signals applied in the past, the response of type 1 and type 2 devices is effectively [*history-independent*]{}. This feature is readily evident in the case of type 2 model that simply describes a [*non-linear resistor*]{}, whose resistance is fully determined by the [*instantaneous*]{} voltage (which, in some publications [@T2_15], is not even the voltage across the device).
In the case of type 1 models, the instantaneous response is determined by the sign of the time-derivative of the voltage. Even though the time derivative implies the dependence on the voltage at an infinitesimally close preceding moment of time, this alone is not sufficient for the device to be classified as a memristive element. We emphasize that not only does the time derivative of the voltage not enter Eqs. (\[eq:3\]) and (\[eq:4\]), but also it is difficult to imagine an actual [*physical*]{} device with such a voltage differentiation capability (definitely the physical memristive elements behave differently [@pershin11a]).
Finally, consider the last line in Eq. (\[eq:1\]), which is the condition that the response of type 1 devices is unchanged when $\dot{V}_M(t)=0$. Such an isolated point condition is irrelevant since it is singular.
To further emphasize the distinction between the type 1 and type 2 devices with an actual memristive model, Fig. \[fig:1\] compares their response under the condition of periodic bias. Here, the memristive device is exemplified by a threshold-type model [@pershin18a; @pershin09b] that mimics the most common bipolar memristive elements [@pershin11a], while the response of the type 1 and 2 devices is plotted based on Eqs. (\[eq:1\]) and (\[eq:2\]), respectively.
First of all, consider the application of a simple sinusoidal voltage. This is shown in Fig. \[fig:1\](a). The response of the type 1 device seems deceptively similar to that of an actual memristive element, but close inspection shows that such a similarity is superficial. Indeed, unlike the actual memristive element, the type 1 device exhibits [*frequency-independent*]{} pinched hysteresis loops in the voltage-current plane (shown in the top left inset in Fig. \[fig:2\]) and its switching occurs [*always*]{} at voltage extrema but not at the threshold voltages defined by the physical processes responsible for memory as in actual memristive elements. Frequency-independence of the I-V curve is also evident for the type 2 device as shown in the bottom right inset in Fig. \[fig:2\]. In addition, the non-hysteretic character of these curves indicates the absence of memory in the type 2 model.
Next, consider the response to more complex waveforms. Fig. \[fig:1\](b) shows that small higher-frequency oscillations added to the main sinusoidal waveform change drastically the response of the type 1 device. Now its resistance switches at the frequency of small-amplitude signal, and has nothing in common with the behavior of an actual memristive element (whose resistance has not changed significantly compared to Fig. \[fig:1\](a)). This demonstrates that the type 1 devices are highly sensitive to small amplitude variations as opposed to the actual memristive element. In Fig. \[fig:1\](a) and (b), the resistance dynamics for the type 2 model involves a frequency doubling. According to the discussion above, the absence of memory in this model is evident.
To conclude, in this Letter we have shown that two types of “memristive” models widely used in the literature to model/simulate memristive neural networks are, in fact, [*not*]{} memristive. During the past decade, multiple studies based on these models have been reported in leading specialized journals, such as Neurocomputing, Neural Networks, etc. There are serious reasons to doubt the validity of these papers as the models adopted by their authors do not qualify as memristive, and as such have nothing to do with actual memristive neural networks.
|
---
abstract: |
A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least $2$, the neighbors of $v$ receive at least $2$ colors. The smallest integer $k$ such that $G$ has a dynamic coloring with $ k $ colors, is called the [*dynamic chromatic number*]{} of $G$ and denoted by $\chi_2(G)$. In this paper we will show that if $G$ is a regular graph, then $ \chi_{2}(G)- \chi(G) \leq 2\lfloor \log^{\alpha(G)}_{2}\rfloor +3 $ and if $G$ is a graph and $\delta(G)\geq 2$, then $ \chi_{2}(G)- \chi(G) \leq \lceil \sqrt[\delta -1]{4\Delta^{2}} $ $\rceil (\lfloor \log^{\alpha(G)}_{\frac{2\Delta(G)}{2\Delta(G)-\delta(G)}} \rfloor +1)+1 $ and in general case if $G$ is a graph, then $ \chi_{2}(G)- \chi(G) \leq 3+ \min \lbrace \alpha(G),\alpha^{\prime}(G),\frac{\alpha(G)+\omega(G)}{2}\rbrace $.
[**Key words:**]{} Dynamic chromatic number; Independent number.
[**Subject classification: 05C15, 05D40.**]{}
author:
- |
[ and [A. Dehghan]{}]{}\
\
\
bibliography:
- 'Refff.bib'
title: '**Upper Bounds for the Dynamic Chromatic Number of Graphs in Terms of the Independent Number**'
---
Introduction
============
All graphs in this paper are finite, undirected and simple. We follow the notation and terminology of [@MR1367739]. A [*proper vertex coloring*]{} of $G$ by $k$ colors is a function $c: V(G)\longrightarrow \lbrace 1, \ldots ,k\rbrace$, with this property: if $u,v\in V(G)$ are adjacent, then $c(u)$ and $c(v)$ are different. A [*vertex $k$-coloring*]{} is a proper vertex coloring by $k$ colors. We denote a bipartite graph $G$ with bipartition $(X,Y)$ by $G[X,Y]$. Let $G$ be a graph with a proper vertex coloring $c$. For every $v\in V (G)$, we denote the degree of $v$ in $G$, the neighbor set of $v$ and the color of $v$ by $d(v)$, $N(v)$, and $c(v)$, respectively. For any $S\subseteq V(G)$, $N(S)$ denote the set of vertices of $G$, such that each of them has at least one neighbour in $S$.
There are many ways to color the vertices of graphs, an interesting way for vertex coloring was recently introduced by Montgomery et al. in [@Mont]. A proper vertex $k$-coloring of a graph $G$ is called [*dynamic*]{} if for every vertex $v$ with degree at least $2$, the neighbors of $v$ receive at least two different colors. The smallest integer $k$ such that $G$ has a dynamic $k$-coloring is called the [*dynamic chromatic number*]{} of $G$ and denoted by $\chi_2(G)$.
There exists a generalization for the dynamic coloring of graphs [@Mont]. For an integer $r > 0$, a conditional $(k, r)-$coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to vertices with at least $\min \lbrace r, d(v) \rbrace$ different colors. The smallest integer $k$ for which a graph $G$ has a conditional $(k, r)-$coloring is called the $r$th order conditional chromatic number, denoted by $\chi_r(G)$. Conditional coloring is a generalization of the traditional vertex coloring for which $r = 1$. From [@MR2251583] we know that if $\Delta(G)\leq 2$, for any $r$ we can easily have an algorithm of polynomial time to give the graph $G$ a $(k, r)-$coloring but for any $k \geq 3$ and $r \geq 2$ it is $NP-$complete to check if a graph is $(k, r)-$colorable [@MR2483491].
The other concept that has a relationship with the dynamic coloring is the hypergraph coloring. A hypergraph $ H $, is a pair $ (X,Y) $, where $ X $ is the set of vertices and $ Y $ is a set of non-empty subsets of $ X $, called edges. The proper coloring of $ H $ is a coloring of $ X $ such that for every edge $ e $ with $ \vert e \vert >1 $, there exist $ v,u\in X $ such that $ c(u)\neq c(v) $. For the hypergraph $ H=(X,Y) $, consider the bipartite graph $ \widehat{H} $ with two parts $ X $ and $ Y $, that $ v\in X $ is adjacent to $ e\in Y $ if and only if $ v\in e $ in $ H $. Now consider a dynamic coloring $ c $ of $ \widehat{H} $, clearly by inducing $ c $ on $ X $, we obtain a proper coloring of $ H $.
The graph $G^{\frac{1}{2}}$ is said to be the $2$-subdivision of a graph $G$ if $G^{\frac{1}{2}}$ is obtained from $G$ by replacing each edge with a path with exactly one inner vertices [@MR2519165]. There exists a relationship between $ \chi(G)$ and $ \chi_{2}(G^{\frac{1}{2}}) $. We have $\chi(G) \leq \chi_{2}(G^{\frac{1}{2}}) $ and $ \chi(G^{\frac{1}{2}})=2 $. For example it was shown in [@Mont] if $ G\cong K_{n} $ then $ \chi_{2}(K_{n}^{\frac{1}{2}}) \geq n $.
In the previous example and Proposition $1$ and $2$, we present some graphs such that the difference between the chromatic number and the dynamic chromatic number can be arbitrarily large. It seems that when $ \Delta (G) $ is close to $ \delta(G) $, then $ \chi_{2}(G) $ is also close to $ \chi(G) $. Montgomery conjectured that for regular graphs the difference is at most $2$.
[**\[Montgomery [@Mont]**\]****]{} For any $r$-regular graph $G$, $\chi_2(G)-\chi(G)\leq 2$.
In [@strongly] it was proved that if $G$ is a strongly regular graph and $G \neq C_{4},C_{5},K_{r,r}$, then $\chi_2(G)-\chi(G)\leq 1$. Also in [@Akbari] it was proved that if $G$ is an $r$-regular graph, $\chi_2(G)-\chi(G)\leq \chi(G)$. Recently the dynamic coloring of Cartesian product of graphs has been studied in [@product].
We will prove some inequalities for the difference between the chromatic number and the dynamic chromatic number of regular graphs. For example, it was shown [@Akbari] that, if $G$ is a regular graph, then $\chi_{2}(G)- \chi(G) \leq
\lceil\frac{\alpha(G)}{2} \rceil +1 $, we prove that if $G$ is a regular graph, then $ \chi_{2}(G)- \chi(G) \leq 2\lfloor\log^{\alpha(G)}_{2}\rfloor +3 $.
In general case, finding the optimal upper bound for $ \chi_{2}(G)-\chi(G) $ seems to be an intriguing problem. In this paper we will prove various inequalities relating it to other graph parameters. At the end we will introduce a generalization of the Montgomery’s Conjecture for the dynamic coloring of regular graphs.
Let $c$ be a proper vertex coloring of a graph $G$, define $B_{c}$ as the set of vertices such that for each vertex $v\in B_{c}$, $d(v)\geq 2$ and the neighbors of $v$ receives a same color and let $A_{c}=V(G)\backslash B_{c}$.
Now we state some lemmas and theorems without proof.
[[@MR1991048]]{} For a connected graph $G$ if $\Delta(G)\leq 3$, then $\chi_{2}(G) \leq 4$ unless $G = C_{5}$, in which case $\chi_{2}(C_{5}) = 5$ and if $\Delta(G) \geq 4$, then $\chi_{2}(G) \leq \Delta(G)+ 1$.
So if $\Delta(G)\leq 3$, then $\chi_{2}(G) \leq 5$.
[**\[The Lovasz Local Lemma [@MR1885388]**\]****]{} Suppose ${A_{1},\ldots ,A_{n}}$ be a set of random events such that for each $i$, $Pr(A_{i})\leq p$ and $A_{i}$ is mutually independent of the set of all but at most $d$ other events. If $4pd\leq 1$, then with positive probability, none of the events occur.
[[@Akbari]]{} Let $r \geq 4$ be a natural number. Suppose that $G [A,B]$ is a bipartite graph such that all vertices of Part $A$ have degree $r$ and all vertices of Part $B$ have degree at most $r$. Then one can color the vertices of Part $B$ with two colors such that the dynamic property holds for all vertices of Part $A$.
[[@MR1367739]]{} A set of vertices in a graph is an independent dominating set if and only if it is a maximal independent set.
So if $G$ is a graph, then $G$ has an independent dominating set $T$.
[**\[Dirac [@MR1367739]**\]****]{} If $G$ is a simple graph with at least three vertices and $ \delta(G) \geq \frac{n(G)}{2} $, then $G$ is Hamiltonian.
Main Results
============
Before proving the theorems we need to prove some lemmas.
If $G$ is a graph, $G\neq \overline{K_{n}}$ and $T_{1}$ is an independent set of $G$, then there exists $T_{2}$ such that, $T_{2}$ is an independent dominating set for $T_{1}$ and $ \vert T_{1} \cap T_{2}\vert \leq \frac{2\Delta(G) -\delta(G) }{2\Delta(G)} \vert T_{1}\vert$.
The proof is constructive. For each $u\in N(T_{1})$, define the variable $ f(u) $ as the number of vertices which are adjacent to $u$ and are in $T_{1}$. $\sum_{u\in N(T_{1})} f(u)$ is the number of edges of $G[T_{1},N(T_{1})]$, so $\sum_{u\in N(T_{1})} f(u)\geq \vert T_{1}\vert\delta(G)$.
Let $T_{3}=T_{1}$, $T_{4}=\emptyset $, $s=0$, $i=1$ and repeat the following procedure until $\sum_{u\in N(T_{1})} f(u)=0$.
[**Step 1.**]{} Select a vertex $u$ such that $f(u)$ is maximum among $ \lbrace f(v) \vert v\in N(T_{1})\rbrace$ and add $u$ to the set $T_{4}$ and let $t_{i}=f(u)$.
[**Step 2.**]{} For each $v\in N(T_{1})$ that is adjacent to $u$, change the value of $ f(v) $ to $0$. Change the value of $ f(u) $ to $0$.
[**Step 3.**]{} For each $v\in N(T_{1})$ that is adjacent to at least one vertex of $N(u)\cap T_{3}$ and it is not adjacent to $u$, decrease $ f(v)$ by the number of common neighbours of $v$ and $u$ in $T_{3}$.
[**Step 4.**]{} Remove the elements of $N(u)$ from $T_{3}$. Increase $s$ by $t_{i}$ and $i$ by $1$.
When the above procedure terminates, because of the steps $1$ and $2$, $ T_{4} $ is an independent set and because of the steps $1$ and $4$, $ T_{4} $ is a dominating set for $ T_{1} \backslash T_{3} $. Now let $T_{2}=T_{4}\cup T_{3}$, because of the step $4$, $ T_{2} $ is an independent dominating set for $T_{1}$.
Assume that the above procedure has $l$ iterations. Because of the step $4$, we have $ s=\sum_{i=1}^{l} t_{i} $. Each vertex in $N(T_{1})$ has at most $\Delta(G)-1 $ neighbours in $N(T_{1}) $, so in the step $2$ of the $i$th iteration, $\sum_{u\in N(T_{1})} f(u)$ is decreased at most $t_{i} \Delta(G)$ and in the step $3$ of the $i$th iteration, $\sum_{u\in N(T_{1})} f(u)$ is decreased at most $ t_{i} \Delta(G)$, so in the $i$th iteration, $\sum_{u\in N(T_{1})} f(u)$ is decreased at most $2 t_{i} \Delta(G)$. When the procedure terminates, $\sum_{ u\in N(T_{1})} f(u)=0 $, so:
$ \delta(G)\vert T_{1}\vert - \sum_{ i=1 }^{ i=l } (2t_{i}\Delta(G)) \leq 0$,
$ \delta(G)\vert T_{1}\vert - 2s\Delta(G) \leq 0$,
$ s\geq \frac{\delta(G)}{2\Delta(G)} \vert T_{1}\vert$ ,
$\vert T_{1} \cap T_{2}\vert =\vert T_{3}\vert = \vert T_{1}\vert - s \leq \frac{2\Delta(G) -\delta(G) }{2\Delta(G)} \vert T_{1}\vert$.
If $G$ is a graph, $ \delta \geq 2 $ and $T$ is an independent set of $G$, then we can color the vertices of $T$ with $ \lceil (4\Delta^{2})^{\frac{1}{\delta-1}} \rceil $ colors such that for each $u\in \lbrace v \vert v\in V(G), N(v)\subseteq T\rbrace$, $N(u)$ has at least two different colors.
Let $ \eta = \lceil (4\Delta^{2})^{\frac{1}{\delta-1}} \rceil $. Color every vertex of $T$ randomly and independently by one color from $ \lbrace 1,\cdots,\eta \rbrace $, with the same probability. For each $u\in \lbrace v \vert v\in V(G), N(v)\subseteq T\rbrace$, let $A_{u}$ be the event that all of the neighbors of $ u $ have the same color.
Each $A_{u}$ is mutually independent of a set of all $A_{v}$ events but at most $\Delta ^{2}$ of them. Clearly, $Pr(A_{u})\leq \frac{1}{\eta^{\delta-1}}$. We have:
$4pd= 4(\frac{1}{\eta})^{\delta-1} \Delta ^{2}\leq 1$.
So by Local Lemma there exists a coloring with our condition for $T$ with positive probability.
Let $c$ be a vertex $k$-coloring of a graph $G$, then there exists a dynamic coloring of $G$ with at most $k+\vert B_{c}\vert$ colors.
Suppose that $B_{c}=\lbrace v_{1},\ldots ,v_{\vert B_{c}\vert}\rbrace$. For each $1\leq i\leq \vert B_{c}\vert$ select $u_{i}\in N(v_{i})$ and recolor $u_{i}$ by the color $k+i$, respectively. The result is a dynamic coloring of $G$ with at most $k+\vert B_{c} \vert$ colors.
\[t1\] If $G$ is a graph and $\delta(G) \geq 2$, then $ \chi_{2}(G)- \chi(G) \leq \lceil \sqrt[\delta -1]{4\Delta^{2}} \rceil (\lfloor \log^{\alpha(G)}_{\frac{2\Delta(G)}{2\Delta(G)-\delta(G)}} \rfloor +1)+1 $.
Let $ \eta = \lceil \sqrt[\delta -1]{4\Delta^{2}} \rceil $ and $k=\lfloor \log^{\alpha(G)}_{\frac{2\Delta(G)}{2\Delta(G)-\delta(G)}}\rfloor +1$. By Lemma $ 3 $, let $T_{1}$ be an independent dominating set for $G$. Consider the vertex $\chi(G)$-coloring of $G$, by Lemma $ 5 $, recolor the vertices of $T_{1}$ by the colors $\chi +1,\ldots,\chi+\eta$ such that for each $u\in \lbrace v \vert v\in V(G), N(v)\subseteq T_{1}\rbrace$, $N(u)$ has at least two different colors. Therefore we obtain a coloring $c_{1}$ such that $B_{c_{1}}\subseteq T_{1}$.
Let $T'_{1}=T_{1}$. For $i=2$ to $i=k$ repeat the following procedure:
[**Step 1.**]{} By Lemma $ 4 $, find an independent set $T_{i}$ such that, $T_{i}$ is an independent dominating set for $T'_{i-1}$ and $\vert T_{i} \cap T'_{i-1} \vert\leq \frac{2\Delta(G) -\delta(G)} {2\Delta(G) } \vert T'_{i-1}\vert$.
[**Step 2.**]{} By Lemma $ 5 $, recolor the vertices of $T_{i}$ with the colors $\chi+\eta i-(\eta -1),\ldots, \chi+\eta i $ such that for each $u\in \lbrace v \vert v\in V(G), N(v)\subseteq T_{i}\rbrace$, $N(u)$ has at least two different colors.
[**Step 3.**]{} Let $ T'_{i}=T_{i} \cap T'_{i-1} $.
After each iteration of above procedure we obtain a proper coloring $c_{i}$ such that $B_{c_{i}}\subseteq T'_{i}$, so when the procedure terminates, we have a coloring $c_{k}$ with at most $\chi(G) +\eta k $ colors, such that $B_{c_{k}}\subseteq T'_{k}$ and $\vert T'_{k}\vert \leq 1$, so by Lemma $ 6 $ we have a dynamic coloring with at most $\chi(G) +\lceil \sqrt[\delta -1]{4\Delta^{2}} \rceil (\lfloor \log^{\alpha(G)}_{\frac{2\Delta(G)}{2\Delta(G)-\delta(G)}} \rfloor +1)+1 $ colors.
if $G$ is a graph and $\Delta(G)\leq 2^{\frac{\delta(G) -3}{2}}$, then $ \chi_{2}(G)- \chi(G) \leq 2\lfloor \log^{\alpha(G)}_{\frac{2\Delta(G)}{2\Delta(G)-\delta(G)}} \rfloor +3 $.
If $G$ is an $r$-regular graph , then $ \chi_{2}(G)- \chi(G) \leq 2\lfloor \log^{\alpha(G)}_{2} \rfloor +3$.
If $r=0$, then the theorem is obvious. For $1\leq r\leq 3$, we have $\chi(G)\geq 2$, by Theorem $1$, $ \chi_{2}(G)\leq 5 $ so $\chi_{2}(G)\leq \chi(G) +3$.
So assume that $ r \geq 4 $, we use a proof similar to the proof of Theorem $3$, in the proof of Theorem $3$, for each $i$, $ 1\leq i \leq k $ we used Lemma $ 5 $, to recolor the vertices of $T_{i}$ with the colors $\chi+\eta i-(\eta -1),\ldots, \chi+\eta i $ such that for each $u\in \lbrace v \vert v\in V(G), N(v)\subseteq T_{i}\rbrace$, $N(u)$ has at least two different colors. In the new proof, for each $i$, $ 1\leq i \leq k $, let $ A= \lbrace v \vert v\in V(G), N(v)\subseteq T_{i}\rbrace$ and $B=T_{i}$ and by Lemma $2$, recolor the vertices of $T_{i}$ with the colors $ \chi+2i-1 $ and $ \chi+2i $ such that for each $u\in \lbrace v \vert v\in V(G), N(v)\subseteq T_{i}\rbrace$, $N(u)$ has at least two different colors. The other parts of the proof are similar. This completes the proof.
Suppose that $c$ is a proper vertex coloring of $G$, let $H_{c}=G [B_{c}] $, define $X_{c}$ as the set of isolated vertices in $ H_{c} $ and let $ Y_{c}=G[B_{c}\backslash X_{c}] $.
If $c$ is a proper vertex coloring of $G$, then $ H_{c} $ is a bipartite graph.
Since the vertices of $ H_{c} $ are in $ B_{c} $, so $ H_{c} $ has at most $2$ different colors in each of it’s connectivity components, so $ H_{c} $ is a bipartite graph.
If $G$ is a simple graph, then there exists a vertex $(\chi(G)+3 )$-coloring $ c^{\prime} $ of $G$ such that $ B_{c^{\prime}} $ is an independent set.
If $ \chi(G) \leq 1 $, then the theorem is obvious, so suppose that $ \chi(G) \geq 2 $. Let $c$ be a vertex $(\chi(G)+1)$-coloring of $G$, such that $\vert B_{c}\vert$ is minimum. Every vertex in $B_{c}$ has a neighbour in $A_{c}$, otherwise if there exists $v\in B_{c}$ such that $ N(v)\subseteq B_{c} $, then because $\chi(G)+1\geq 3$, so we can change the color of $v$ such that the size of $B_{c}$ decreases, but it is a contradiction.
Let $T_{c}=\lbrace v\in A_{c} \mid N(v)\subseteq Y_{c} \rbrace$. Clearly $ T_{c}\cup X_{c} $ is an independent set. Since $ Y_{c}\subseteq H_{c} $ and by Lemma $7$, $ Y_{c} $ is a bipartite graph. Properly recolor the vertices of $ Y_{c} $ by the colors $ \chi+2 $, $ \chi+3 $, retain the color of other vertices and name this coloring $c'$. Clearly $ Y_{c'}\subseteq Y_{c} $. Since every vertex of $ Y_{c} $ has at least one neighbour in $ A_{c} $ and one neighbour in $ Y_{c} $, so $ Y_{c} $ is a subset of $A_{c'}$. So $Y_{c'}=\emptyset$ and $X_{c'}\subseteq T_{c}\cup X_{c}$. Therefore $ c^{\prime} $ is a vertex $(\chi(G)+3 )$-coloring of $G$ such that $ B_{c^{\prime}} $ is an independent set.
If $G$ is a simple graph, then $ \chi_{2}(G)-\chi(G)\leq \alpha(G)+3 $.
[ By Lemma $ 8 $ and Lemma $ 6 $ the proof is easy. ]{}
If $G$ is a simple graph, then $ \chi_{2}(G)-\chi(G)\leq \alpha(G)+1 $.
For $\chi(G)=1 $ the theorem is obvious and if $\chi(G)=2 $, then $ \Delta(G)\leq\alpha(G) $ so by Theorem $1$, $ \chi_{2}(G)\leq \Delta(G) +2 \leq \alpha(G)+ \chi(G)$. If $ \alpha(G)=1 $, then $ \chi_{2}(G)= \chi(G) $.
Suppose that $G$ is the connected graph, otherwise we apply the following proof for each of it’s connectivity components and also suppose $\chi(G)\geq 3$ and $ \alpha(G)\geq 2 $. Let $c$ be a vertex $\chi(G)$-coloring of $G$, such that $\vert B_{c}\vert$ is minimum. Every vertex in $B_{c}$ has a neighbour in $A_{c}$, otherwise there exists $v\in B_{c}$ such that $ N(v)\subseteq B_{c} $, then because $\chi(G)\geq 3$, by changing the color of $v$ decrease the size of $B_{c}$.
Now, two cases can be considered:
Case 1. $Y_{c}=\emptyset$. In this case we have $B_{c}=X_{c}$. Since the vertices of $X_{c}$ is an independent set, so $\vert X_{c} \vert \leq \alpha(G)$. By Lemma $ 6 $, there exists a dynamic coloring with $\chi(G)+\alpha(G) $ colors.
Case 2. $Y_{c}\neq\emptyset$. Let $T_{c}=\lbrace v\in A_{c} \mid N(v)\subseteq Y_{c} \rbrace$. Clearly $ T_{c}\cup X_{c} $ is an independent set. Since $ Y_{c}\subseteq H_{c} $ and by Lemma $7$, $ Y_{c} $ is a bipartite graph. Properly color the vertices of $ Y_{c} $ by the colors $ \chi+1 $, $ \chi+2 $, retain the color of other vertices and name this coloring $c'$. Clearly $ Y_{c'}\subseteq Y_{c} $. Since every vertex of $ Y_{c} $ has at least one neighbour in $ A_{c} $ and one neighbour in $ Y_{c} $, so $ Y_{c} $ are the subset of $A_{c'}$. So $Y_{c'}=\emptyset$ and $X_{c'}\subseteq T_{c}\cup X_{c}$. If $\vert X_{c'}\vert =1 $, then the theorem is obvious, so assume $ \vert X_{c'}\vert \geq 1 $. Let $ v\in X_{c'} $. Now we have two situations:
Case 2A. $ N(v) \cap N(X_{c'}\setminus\lbrace v\rbrace) =\emptyset$. So $ N(v) \cup X_{c'}\setminus \lbrace v \rbrace $ is the independent set. So $ \vert N(v) \cup X_{c'}\setminus \lbrace v \rbrace \vert \leq \alpha(G)$. Since $ \vert N(v) \vert \geq 2 $, $ \vert X_{c'} \vert \leq \alpha(G)-1$. By Lemma $ 6 $, there exists a dynamic coloring with ${\displaystyle{\chi(G)+2+(\alpha(G)-1)}} $ colors.
Case 2B. $ N(v) \cap N(X_{c'}\setminus\lbrace v\rbrace) \neq\emptyset$. Therefore there exists a vertex $ u\in X_{c'} $ such that $ N(v) \cap N(u)\neq \emptyset $. Color one of the common neighbours of $ v $ and $ u $, with the color $ \chi+3 $, retain the color of other vertices and name this coloring $c''$. Clearly $ B_{c''}\subseteq X_{c'} \setminus \lbrace v,u\rbrace $. $ \vert B_{c''} \vert \leq \vert X_{c'}\setminus \lbrace v,u\rbrace \vert \leq \alpha(G)-2 $. By Lemma $ 6 $, there exists a dynamic coloring with $\chi(G)+3+(\alpha(G)-2) $ colors.
For $G=C_{4},C_{5}$, we have $ \chi_{2}(G)= \chi(G)+\alpha(G) $.
For any two numbers $a$, $b$ $ (a\geq b \geq 3) $, there exists a graph $G$, such that $ \chi(G)=a $, $ \alpha(G) =b $ and $ \chi_{2}(G)-\chi(G)=\alpha(G) -1 $.
[ For every two number $a$, $b$ $ (a\geq b \geq 3) $, consider the graph $G$ derived from $K_{a+b-1}$ by replacing the edges of a matching of size $ b-1 $ with $ P_{3} $. It is easy to see that $ G $ satisfies the conditions of the proposition. ]{}
If $G$ is a simple graph, then $ \chi_{2}(G)-\chi(G) \leq \frac{\alpha(G)+\omega(G) }{2} +3 $.
For $\chi(G)=1 $ the theorem is obvious. Suppose that $G$ is the connected graph with $ \chi(G) \geq 2 $, otherwise we apply the following proof for each of it’s connectivity components. By Lemma $8$, suppose that $ c $ is a vertex $(\chi(G)+3 )$-coloring of $G$ such that $ B_{c} $ is an independent set. Also suppose that $ T_{1} $ is a maximal independent set that contains $ B_{c} $. Consider a partition $ \lbrace \lbrace v_{1},v_{2} \rbrace ,\ldots \lbrace v_{2s-1},v_{2s}\rbrace ,T_{2}=\lbrace v_{2s+1},\ldots ,v_{l} \rbrace \rbrace $ for the vertices of $T_{1} $ such that for $ 1\leq i \leq s $, $ N(v_{2i-1})\cap N(v_{2i})\neq \emptyset$ and for $ 2s < i <j \leq l $, $ N(v_{i})\cap N(v_{j})=\emptyset $. For $ 1\leq i \leq s $, let $ w_{i}\in N(v_{2i-1})\cap N(v_{2i})$, recolor $ w_{i} $ by the color $ \chi+3+i $ and name it $c^{\prime}$.
Consider a partition $ \lbrace\lbrace v_{2s+1},v_{2s+2} \rbrace , \ldots ,\lbrace v_{2t-1},v_{2t} \rbrace , T_{3}=\lbrace v_{2t+1},\ldots ,v_{l} \rbrace \rbrace $ for the vertices of $ T_{2} $ such that for $ s < i \leq t $ there exist $ u_{2i-1}\in N(v_{2i-1}) $ and $ u_{2i}\in N(v_{2i}) $, such that $ u_{2i-1}$ and $u_{2i} $ are not adjacent and for $ 2t < i <j \leq l $, every neighbour of $v_{i} $ is adjacent to every neighbour of $v_{j} $. For $ s < i \leq t $, suppose that $ u_{2i-1}\in N(v_{2i-1}) $, $ u_{2i}\in N(v_{2i}) $ such that $ u_{2i-1}$ and $u_{2i} $ are not adjacent. Now if $ c(u_{2i-1}) \neq c^{\prime}(u_{2i-1}) $, then recolor $u_{2i-1}$ by the color $ \chi+3+i $ and also if $ c(u_{2i}) \neq c^{\prime}(u_{2i}) $, then recolor $u_{2i}$ by the color $ \chi+3+i $.
After above procedure we obtain a coloring, name it $ c^{\prime\prime} $. We state that $c^{\prime\prime}$ maintains the condition of the dynamic coloring, since if $ z $ is a vertex with $ N(z)=\lbrace u_{2i-1},u_{2i} \rbrace $ for some $ s < i \leq t $ and $ c^{\prime\prime}(u_{2i-1})=c^{\prime\prime}(u_{2i} )$, therefore $ c^{\prime\prime}(u_{2i-1})=c^{\prime\prime}(u_{2i} )=\chi+3+i$. It means that $ c(u_{2i})=c^{\prime}(u_{2i} )$, so $u_{2i} $ is the common neighbour of $ v_{2i} $ and $ z $, so $ z\in T_{1} $. Therefore $\lbrace z , v_{2i-1} \rbrace \in T_{1}\setminus T_{2}$. It is a contradiction.
For $ v_{i}\in T_{3} $ let $ x_{i}\in N(v_{i}) $. Suppose that $X= \lbrace x_{i} \vert v_{i}\in T_{3} \rbrace $. The vertices of $X $ make a clique, recolor each of them by a new color. We have $ \vert X \vert =l-2t\leq \omega(G) $. Therefore:
$\chi_{2}(G)-\chi(G) \leq s+(t-s)+(l-2t)+3 \leq \frac{\alpha(G) +\omega(G)}{2}+3$.
If $G$ is a triangle-free graph, then $ \chi_{2}(G)-\chi(G) \leq \frac{\alpha(G) }{2} +4 $
If $G$ is an $r$-regular graph and $ r > \frac{n}{2} $, then every vertex $ v\in V(G) $ appears in some triangles, therefore $ \chi_{2}(G)=\chi(G)$. In the next theorem, we present an upper bound for the dynamic chromatic number of $r$-regular graph $G$ with $ r \geq \frac{n}{k} $ in terms of $ n $ and $ r $.
If $G$ is an $r$-regular graph with $ n $ vertices, then $ \chi_{2}(G)-\chi(G) \leq 2 \lceil \frac{n}{r} \rceil -2$.
If $ r\leq 2 $, then the theorem is obvious. For $ r=3 $, if $ n\geq 8 $, for every vertex $v$, we have $ d_{\overline{G}}(v) \geq \frac{n}{2}$, so by Theorem $ 2 $, $\overline{G} $ is Hamiltonian, so $\overline{G} $ has a perfect matching. Therefore $G$ has a vertex $ ( \frac{n}{2} )$-coloring $ c $, such that every color used in exactly two vertices. $ c $ is a dynamic coloring and we have: $ \chi_{2}(G) \leq \frac{n}{2} \leq \chi(G) + 2 \lceil \frac{n}{r} \rceil -2 $. Also for graphs with $ r=3 $ and $ n \leq 7 $, the theorem is obvious.
Therefore suppose that, $ r\geq 4 $ and $ c $ is a vertex $ \chi(G) $-coloring of $G$. For every $ 1 \leq k \leq \lceil \frac{n}{r} \rceil -1 $, let $ T_{k} $ be a maximum independent set of $ G \setminus \cup _{i=1}^{k-1}T_{i}$. By Lemma $ 2 $, recolor the vertices of $T_{1}$ with two new colors, such that for each $u\in \lbrace v \vert v\in V(G), N(v)\subseteq T_{1}\rbrace$, $u$ has two different colors in $N(u)$. Therefore $G$ has the coloring $ c^{\prime} $ by $ \chi(G) +2 $ colors such that $ B_{c^{\prime}}\subseteq T_{1} $.
Also by Lemma $ 3 $, by recoloring every $ T_{k} $ $( 2 \leq k \leq \lceil \frac{n}{r} \rceil -1 ) $, by two new colors, $ G $ has a coloring $ c^{\prime\prime} $ such that for every vertex $ v\in V(G) $ with $ N(v)\subseteq T_{k} $, for some $ k $, $ v $ has at least two different colors in its neighbours. We state that $ c^{\prime\prime} $ is a dynamic coloring, otherwise suppose that $ u\in B_{c^{\prime\prime}} $, we have $ u\in T_{1} $ so $ N(u) $ is an independent set and $ N(u) \cap ( \cup_{i=1}^{\lceil \frac{n}{r} \rceil -1} T_{k})= \emptyset$. Considering the definitions of $ T_{k} $ $ ( 1 \leq k \leq \lceil \frac{n}{r} \rceil -1) $, we have:
$ r= \vert N(u) \vert \leq \vert T_{\lceil \frac{n}{r} \rceil-1}\vert \leq \vert T_{\lceil \frac{n}{r} \rceil-2}\vert \leq \ldots \leq \vert T_{2}\vert $,
and $ \vert T_{2}\vert < \vert T_{1}\vert $, since otherwise, $ T_{2} \cup \lbrace u\rbrace $ is an independent set and $ \vert T_{2} \cup \lbrace u \rbrace \vert > \vert T_{1} \vert$.
Therefore $ n \geq r \lceil \frac{n}{r} \rceil +1 $, but it is a contradiction. This completes the proof.
For any number $a \geq 3$, there is a graph $G$ with $ \alpha^{\prime}(G) =a $ such that $\chi_2(G)-\chi(G) \geq \alpha^{\prime}(G)-2$. To see this for a given number $a$ consider a bipartite graph $G$. Assume that every vertex in $A$ is corresponding to a $2$-subset of the set $\{1,\ldots ,a\}$ and $B=\{1, \ldots, a\}$. If $A=\{i,j\}$, then we join this vertex to $i$ and $j$ in part $B$ $ \cite{akbari2}$.
If $G$ is a simple graph, then $ \chi_{2}(G)-\chi(G) \leq \alpha^{\prime}(G) +3 $.
Let $G$ be a simple graph, by lemma $8$, suppose that $c$ is a vertex $ (\chi(G)+3 )$-coloring of $G$ such that $ B_{c} $ is an independent set. Let $ M=\lbrace v_{1}u_{1},\ldots,v_{\alpha^{\prime}}u_{\alpha^{\prime}} \rbrace $ be a maximum matching of $G$ and $ W=\lbrace v_{1},u_{1},\ldots,v_{\alpha^{\prime}},u_{\alpha^{\prime}} \rbrace $. Let $ X=B_{c} \cap W $ and $ Y= \lbrace v_{i}\vert u_{i}\in X \rbrace \cup \lbrace u_{i}\vert v_{i}\in X \rbrace$. Recolor the vertices of $Y$ by a different new color. Also recolor every vertex in $ N(B_{c}\setminus X) \cap W $, by a different new color. Name this coloring $ c^{\prime} $. We state that $ c^{\prime} $ is a dynamic coloring of $G$. In order to complete the proof, it is enough to show that we used at most $\alpha^{\prime}(G)$ new colors in $ c^{\prime} $.
If $ e=v_{i}u_{i} $ $ (1 \leq i \leq \alpha^{\prime}(G) ) $ is an edge of $ M $ such that $ c(v_{i})\neq c^{\prime}(v_{i}) $ and $ c(u_{i})\neq c^{\prime}(u_{i}) $, then three cases can be considered:
Case A: $ v_{i},u_{i}\in Y $. It means that $ \lbrace v_{i},u_{i} \rbrace \subseteq B_{c} $. Therefore $ v_{i} $ and $ u_{i} $ are adjacent, but $ B_{c} $ is an independent set.
Case B: $( v_{i}\in Y$ and $u_{i}\notin Y )$ or $( u_{i}\in Y$ and $v_{i}\notin Y )$. Without loss of generality suppose that $v_{i}\in Y$ and $u_{i}\notin Y$. So $ u_{i}\in X $ and $ v_{i}\notin X $, therefore there exists $ u^{\prime} \in B_{c} $ such that $u^{\prime} u_{i} \in E(G) $, but $ B_{c} $ is an independent set.
Case C: $ v_{i},u_{i}\notin Y $. It means that $ v_{i},u_{i}\notin X $ and there exist $ v^{\prime},u^{\prime}\in B_{c} $ such that $ v^{\prime}v_{i},u^{\prime}u_{i}\in E(G)$. Now $ M^{\prime}= (M \setminus \lbrace v_{i}u_{i}\rbrace ) \cup \lbrace v^{\prime}v_{i},u^{\prime}u_{i}\rbrace$ is a matching that is greater than $ M $.
Therefore we recolor at most one of the $ v_{i}$ and $u_{i} $ for each $ 1\leq i \leq \alpha^{\prime}(G) $, this completes the proof.
A Generalization of the Montgomery’s Conjecture
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For every two numbers $ a $, $ b $ $ (a\geq b \geq 2) $, there exists a graph $ G_{a,b} $ such that $ \delta(G_{a,b})=b $, $ \Delta(G_{a,b})=a+1 $ and $ \chi_{2}(G_{a,b} ) - \chi(G_{a,b}) = \lceil \frac{\Delta(G_{a,b})}{\delta(G_{a,b})} \rceil $.
To see this for given two numbers $ a $, $ b $ $ (a\geq b \geq 2) $, consider a graph $ G_{a,b} $ by the following definition. Suppose that $ a=bc+d $ such that $ 0\leq d < b $.
$ V( G_{a,b} )=\lbrace v_{i,j} \vert 1\leq i \leq b, 1 \leq j\leq c+1 \rbrace$
$\cup \lbrace v_{i,c+2} \vert 1 \leq i\leq d \rbrace \cup \lbrace w_{k} \vert 1\leq k \leq c+1\rbrace $,
$ E(G_{a,b})=\lbrace v_{i_{1},j_{1}}v_{i_{2},j_{2}}\vert j_{1}\neq j_{2},\lbrace j_{1},j_{2}\rbrace \neq \lbrace c+1 ,c+2 \rbrace \rbrace \cup \lbrace v_{i,j}w_{k} \vert j=k \rbrace $.
We have $ \chi(G_{a,b})=c+1$, $ \chi_{2}(G_{a,b} )=2(c+1)$, it is easy to see that $G_{a,b}$ has the conditions of the proposition.
(154.95,152.6)(0,0) (48,92)(47.98,92.85)(47.41,93.41) (47.41,93.41)(46.85,93.98)(46,94) (46,94)(45.15,93.98)(44.59,93.41) (44.59,93.41)(44.02,92.85)(44,92) (44,92)(44.02,91.15)(44.59,90.59) (44.59,90.59)(45.15,90.02)(46,90) (46,90)(46.85,90.02)(47.41,90.59) (47.41,90.59)(47.98,91.15)(48,92) (48,20)(47.98,20.85)(47.41,21.41) (47.41,21.41)(46.85,21.98)(46,22) (46,22)(45.15,21.98)(44.59,21.41) (44.59,21.41)(44.02,20.85)(44,20) (44,20)(44.02,19.15)(44.59,18.59) (44.59,18.59)(45.15,18.02)(46,18) (46,18)(46.85,18.02)(47.41,18.59) (47.41,18.59)(47.98,19.15)(48,20) (38,58)(37.98,58.85)(37.41,59.41) (37.41,59.41)(36.85,59.98)(36,60) (36,60)(35.15,59.98)(34.59,59.41) (34.59,59.41)(34.02,58.85)(34,58) (34,58)(34.02,57.15)(34.59,56.59) (34.59,56.59)(35.15,56.02)(36,56) (36,56)(36.85,56.02)(37.41,56.59) (37.41,56.59)(37.98,57.15)(38,58) (102,92)(101.98,92.85)(101.41,93.41) (101.41,93.41)(100.85,93.98)(100,94) (100,94)(99.15,93.98)(98.59,93.41) (98.59,93.41)(98.02,92.85)(98,92) (98,92)(98.02,91.15)(98.59,90.59) (98.59,90.59)(99.15,90.02)(100,90) (100,90)(100.85,90.02)(101.41,90.59) (101.41,90.59)(101.98,91.15)(102,92) (102,20)(101.98,20.85)(101.41,21.41) (101.41,21.41)(100.85,21.98)(100,22) (100,22)(99.15,21.98)(98.59,21.41) (98.59,21.41)(98.02,20.85)(98,20) (98,20)(98.02,19.15)(98.59,18.59) (98.59,18.59)(99.15,18.02)(100,18) (100,18)(100.85,18.02)(101.41,18.59) (101.41,18.59)(101.98,19.15)(102,20) (92,58)(91.98,58.85)(91.41,59.41) (91.41,59.41)(90.85,59.98)(90,60) (90,60)(89.15,59.98)(88.59,59.41) (88.59,59.41)(88.02,58.85)(88,58) (88,58)(88.02,57.15)(88.59,56.59) (88.59,56.59)(89.15,56.02)(90,56) (90,56)(90.85,56.02)(91.41,56.59) (91.41,56.59)(91.98,57.15)(92,58) (154,92)(153.98,92.85)(153.41,93.41) (153.41,93.41)(152.85,93.98)(152,94) (152,94)(151.15,93.98)(150.59,93.41) (150.59,93.41)(150.02,92.85)(150,92) (150,92)(150.02,91.15)(150.59,90.59) (150.59,90.59)(151.15,90.02)(152,90) (152,90)(152.85,90.02)(153.41,90.59) (153.41,90.59)(153.98,91.15)(154,92) (48,92)[(1,0)[50]{}]{} (48,20)[(1,0)[50]{}]{} (46,22)(0.12,0.15)[450]{}[(0,1)[0.15]{}]{} (46,90)(0.12,-0.15)[450]{}[(0,-1)[0.15]{}]{} (90,56)(0.12,-0.41)[83]{}[(0,-1)[0.41]{}]{} (90,60)(0.12,0.36)[83]{}[(0,1)[0.36]{}]{} (38,58)[(1,0)[50]{}]{} (36,56)(0.12,-0.41)[83]{}[(0,-1)[0.41]{}]{} (36,60)(0.12,0.45)[67]{}[(0,1)[0.45]{}]{} (152,94)(127.34,108.67)(99.71,109.01) (99.71,109.01)(72.09,109.36)(48,94) (46,18)(0.22,-0.13)[2]{}[(1,0)[0.22]{}]{} (46.43,17.75)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (46.87,17.5)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (47.3,17.25)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (47.74,17.01)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (48.18,16.77)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (48.62,16.53)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (49.06,16.3)(0.22,-0.12)[2]{}[(1,0)[0.22]{}]{} (49.51,16.07)(0.22,-0.11)[2]{}[(1,0)[0.22]{}]{} (49.95,15.84)(0.22,-0.11)[2]{}[(1,0)[0.22]{}]{} (50.4,15.61)(0.22,-0.11)[2]{}[(1,0)[0.22]{}]{} (50.85,15.39)(0.23,-0.11)[2]{}[(1,0)[0.23]{}]{} (51.3,15.17)(0.23,-0.11)[2]{}[(1,0)[0.23]{}]{} (51.75,14.96)(0.23,-0.11)[2]{}[(1,0)[0.23]{}]{} (52.21,14.74)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (52.66,14.53)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (53.12,14.33)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (53.57,14.12)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (54.03,13.92)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (54.49,13.73)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (54.96,13.53)(0.23,-0.1)[2]{}[(1,0)[0.23]{}]{} (55.42,13.34)(0.23,-0.09)[2]{}[(1,0)[0.23]{}]{} (55.88,13.16)(0.23,-0.09)[2]{}[(1,0)[0.23]{}]{} (56.35,12.97)(0.23,-0.09)[2]{}[(1,0)[0.23]{}]{} (56.82,12.79)(0.47,-0.18)[1]{}[(1,0)[0.47]{}]{} (57.28,12.61)(0.47,-0.17)[1]{}[(1,0)[0.47]{}]{} (57.75,12.44)(0.47,-0.17)[1]{}[(1,0)[0.47]{}]{} (58.22,12.27)(0.47,-0.17)[1]{}[(1,0)[0.47]{}]{} (58.7,12.1)(0.47,-0.16)[1]{}[(1,0)[0.47]{}]{} (59.17,11.94)(0.47,-0.16)[1]{}[(1,0)[0.47]{}]{} (59.64,11.78)(0.48,-0.16)[1]{}[(1,0)[0.48]{}]{} (60.12,11.62)(0.48,-0.15)[1]{}[(1,0)[0.48]{}]{} (60.6,11.46)(0.48,-0.15)[1]{}[(1,0)[0.48]{}]{} (61.07,11.31)(0.48,-0.15)[1]{}[(1,0)[0.48]{}]{} (61.55,11.17)(0.48,-0.14)[1]{}[(1,0)[0.48]{}]{} (62.03,11.02)(0.48,-0.14)[1]{}[(1,0)[0.48]{}]{} (62.51,10.88)(0.48,-0.14)[1]{}[(1,0)[0.48]{}]{} (62.99,10.74)(0.48,-0.13)[1]{}[(1,0)[0.48]{}]{} (63.47,10.61)(0.48,-0.13)[1]{}[(1,0)[0.48]{}]{} (63.96,10.48)(0.48,-0.13)[1]{}[(1,0)[0.48]{}]{} (64.44,10.35)(0.49,-0.12)[1]{}[(1,0)[0.49]{}]{} (64.93,10.22)(0.49,-0.12)[1]{}[(1,0)[0.49]{}]{} (65.41,10.1)(0.49,-0.12)[1]{}[(1,0)[0.49]{}]{} (65.9,9.98)(0.49,-0.11)[1]{}[(1,0)[0.49]{}]{} (66.39,9.87)(0.49,-0.11)[1]{}[(1,0)[0.49]{}]{} (66.88,9.76)(0.49,-0.11)[1]{}[(1,0)[0.49]{}]{} (67.36,9.65)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (67.85,9.55)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (68.34,9.45)(0.49,-0.1)[1]{}[(1,0)[0.49]{}]{} (68.84,9.35)(0.49,-0.09)[1]{}[(1,0)[0.49]{}]{} (69.33,9.25)(0.49,-0.09)[1]{}[(1,0)[0.49]{}]{} (69.82,9.16)(0.49,-0.09)[1]{}[(1,0)[0.49]{}]{} (70.31,9.08)(0.49,-0.08)[1]{}[(1,0)[0.49]{}]{} (70.81,8.99)(0.49,-0.08)[1]{}[(1,0)[0.49]{}]{} (71.3,8.91)(0.49,-0.08)[1]{}[(1,0)[0.49]{}]{} (71.8,8.84)(0.5,-0.07)[1]{}[(1,0)[0.5]{}]{} (72.29,8.76)(0.5,-0.07)[1]{}[(1,0)[0.5]{}]{} (72.79,8.69)(0.5,-0.07)[1]{}[(1,0)[0.5]{}]{} (73.28,8.63)(0.5,-0.06)[1]{}[(1,0)[0.5]{}]{} (73.78,8.56)(0.5,-0.06)[1]{}[(1,0)[0.5]{}]{} (74.28,8.5)(0.5,-0.06)[1]{}[(1,0)[0.5]{}]{} (74.77,8.45)(0.5,-0.05)[1]{}[(1,0)[0.5]{}]{} (75.27,8.39)(0.5,-0.05)[1]{}[(1,0)[0.5]{}]{} (75.77,8.34)(0.5,-0.05)[1]{}[(1,0)[0.5]{}]{} (76.27,8.3)(0.5,-0.04)[1]{}[(1,0)[0.5]{}]{} (76.77,8.26)(0.5,-0.04)[1]{}[(1,0)[0.5]{}]{} (77.27,8.22)(0.5,-0.04)[1]{}[(1,0)[0.5]{}]{} (77.77,8.18)(0.5,-0.03)[1]{}[(1,0)[0.5]{}]{} (78.27,8.15)(0.5,-0.03)[1]{}[(1,0)[0.5]{}]{} (78.77,8.12)(0.5,-0.03)[1]{}[(1,0)[0.5]{}]{} (79.27,8.09)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (79.77,8.07)(0.5,-0.02)[1]{}[(1,0)[0.5]{}]{} (80.27,8.05)(0.5,-0.01)[1]{}[(1,0)[0.5]{}]{} (80.77,8.04)(0.5,-0.01)[1]{}[(1,0)[0.5]{}]{} (81.27,8.03)(0.5,-0.01)[1]{}[(1,0)[0.5]{}]{} (81.77,8.02)(0.5,-0)[1]{}[(1,0)[0.5]{}]{} (82.27,8.02)(0.5,-0)[1]{}[(1,0)[0.5]{}]{} (82.77,8.01)(0.5,0)[1]{}[(1,0)[0.5]{}]{} (83.27,8.02)(0.5,0.01)[1]{}[(1,0)[0.5]{}]{} (83.77,8.02)(0.5,0.01)[1]{}[(1,0)[0.5]{}]{} (84.27,8.03)(0.5,0.01)[1]{}[(1,0)[0.5]{}]{} (84.77,8.05)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (85.27,8.06)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (85.77,8.08)(0.5,0.02)[1]{}[(1,0)[0.5]{}]{} (86.28,8.11)(0.5,0.03)[1]{}[(1,0)[0.5]{}]{} (86.78,8.13)(0.5,0.03)[1]{}[(1,0)[0.5]{}]{} (87.28,8.16)(0.5,0.03)[1]{}[(1,0)[0.5]{}]{} (87.77,8.2)(0.5,0.04)[1]{}[(1,0)[0.5]{}]{} (88.27,8.23)(0.5,0.04)[1]{}[(1,0)[0.5]{}]{} (88.77,8.27)(0.5,0.04)[1]{}[(1,0)[0.5]{}]{} (89.27,8.32)(0.5,0.05)[1]{}[(1,0)[0.5]{}]{} (89.77,8.37)(0.5,0.05)[1]{}[(1,0)[0.5]{}]{} (90.27,8.42)(0.5,0.05)[1]{}[(1,0)[0.5]{}]{} (90.77,8.47)(0.5,0.06)[1]{}[(1,0)[0.5]{}]{} (91.26,8.53)(0.5,0.06)[1]{}[(1,0)[0.5]{}]{} (91.76,8.59)(0.5,0.06)[1]{}[(1,0)[0.5]{}]{} (92.26,8.65)(0.5,0.07)[1]{}[(1,0)[0.5]{}]{} (92.75,8.72)(0.5,0.07)[1]{}[(1,0)[0.5]{}]{} (93.25,8.79)(0.5,0.08)[1]{}[(1,0)[0.5]{}]{} (93.74,8.87)(0.49,0.08)[1]{}[(1,0)[0.49]{}]{} (94.24,8.95)(0.49,0.08)[1]{}[(1,0)[0.49]{}]{} (94.73,9.03)(0.49,0.09)[1]{}[(1,0)[0.49]{}]{} (95.23,9.12)(0.49,0.09)[1]{}[(1,0)[0.49]{}]{} (95.72,9.2)(0.49,0.09)[1]{}[(1,0)[0.49]{}]{} (96.21,9.3)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (96.7,9.39)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (97.19,9.49)(0.49,0.1)[1]{}[(1,0)[0.49]{}]{} (97.68,9.59)(0.49,0.11)[1]{}[(1,0)[0.49]{}]{} (98.17,9.7)(0.49,0.11)[1]{}[(1,0)[0.49]{}]{} (98.66,9.81)(0.49,0.11)[1]{}[(1,0)[0.49]{}]{} (99.15,9.92)(0.49,0.12)[1]{}[(1,0)[0.49]{}]{} (99.64,10.04)(0.49,0.12)[1]{}[(1,0)[0.49]{}]{} (100.12,10.16)(0.49,0.12)[1]{}[(1,0)[0.49]{}]{} (100.61,10.28)(0.48,0.13)[1]{}[(1,0)[0.48]{}]{} (101.09,10.4)(0.48,0.13)[1]{}[(1,0)[0.48]{}]{} (101.58,10.53)(0.48,0.13)[1]{}[(1,0)[0.48]{}]{} (102.06,10.67)(0.48,0.14)[1]{}[(1,0)[0.48]{}]{} (102.54,10.8)(0.48,0.14)[1]{}[(1,0)[0.48]{}]{} (103.02,10.94)(0.48,0.14)[1]{}[(1,0)[0.48]{}]{} (103.5,11.08)(0.48,0.15)[1]{}[(1,0)[0.48]{}]{} (103.98,11.23)(0.48,0.15)[1]{}[(1,0)[0.48]{}]{} (104.46,11.38)(0.48,0.15)[1]{}[(1,0)[0.48]{}]{} (104.94,11.53)(0.48,0.16)[1]{}[(1,0)[0.48]{}]{} (105.41,11.69)(0.47,0.16)[1]{}[(1,0)[0.47]{}]{} (105.89,11.85)(0.47,0.16)[1]{}[(1,0)[0.47]{}]{} (106.36,12.01)(0.47,0.17)[1]{}[(1,0)[0.47]{}]{} (106.83,12.18)(0.47,0.17)[1]{}[(1,0)[0.47]{}]{} (107.31,12.35)(0.47,0.17)[1]{}[(1,0)[0.47]{}]{} (107.78,12.52)(0.47,0.18)[1]{}[(1,0)[0.47]{}]{} (108.24,12.69)(0.47,0.18)[1]{}[(1,0)[0.47]{}]{} (108.71,12.87)(0.23,0.09)[2]{}[(1,0)[0.23]{}]{} (109.18,13.05)(0.23,0.09)[2]{}[(1,0)[0.23]{}]{} (109.64,13.24)(0.23,0.09)[2]{}[(1,0)[0.23]{}]{} (110.11,13.43)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (110.57,13.62)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (111.03,13.81)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (111.49,14.01)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (111.95,14.21)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (112.41,14.42)(0.23,0.1)[2]{}[(1,0)[0.23]{}]{} (112.86,14.63)(0.23,0.11)[2]{}[(1,0)[0.23]{}]{} (113.32,14.84)(0.23,0.11)[2]{}[(1,0)[0.23]{}]{} (113.77,15.05)(0.23,0.11)[2]{}[(1,0)[0.23]{}]{} (114.22,15.27)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (114.67,15.49)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (115.12,15.71)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (115.57,15.94)(0.22,0.11)[2]{}[(1,0)[0.22]{}]{} (116.01,16.17)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (116.45,16.4)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (116.9,16.64)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (117.34,16.87)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (117.77,17.12)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (118.21,17.36)(0.22,0.12)[2]{}[(1,0)[0.22]{}]{} (118.65,17.61)(0.22,0.13)[2]{}[(1,0)[0.22]{}]{} (119.08,17.86)(0.22,0.13)[2]{}[(1,0)[0.22]{}]{} (119.51,18.11)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (119.94,18.37)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (120.37,18.63)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (120.8,18.89)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (121.22,19.16)(0.21,0.13)[2]{}[(1,0)[0.21]{}]{} (121.64,19.43)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (122.06,19.7)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (122.48,19.97)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (122.9,20.25)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (123.32,20.53)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (123.73,20.81)(0.21,0.14)[2]{}[(1,0)[0.21]{}]{} (124.14,21.1)(0.2,0.14)[2]{}[(1,0)[0.2]{}]{} (124.55,21.39)(0.2,0.15)[2]{}[(1,0)[0.2]{}]{} (124.96,21.68)(0.2,0.15)[2]{}[(1,0)[0.2]{}]{} (125.36,21.97)(0.2,0.15)[2]{}[(1,0)[0.2]{}]{} (125.76,22.27)(0.2,0.15)[2]{}[(1,0)[0.2]{}]{} (126.16,22.57)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (126.56,22.87)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (126.96,23.18)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (127.36,23.49)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (127.75,23.8)(0.13,0.1)[3]{}[(1,0)[0.13]{}]{} (128.14,24.11)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (128.53,24.43)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (128.91,24.75)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (129.3,25.07)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (129.68,25.39)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (130.06,25.72)(0.13,0.11)[3]{}[(1,0)[0.13]{}]{} (130.43,26.05)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (130.81,26.38)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (131.18,26.72)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (131.55,27.05)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (131.92,27.39)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (132.28,27.74)(0.12,0.11)[3]{}[(1,0)[0.12]{}]{} (132.65,28.08)(0.12,0.12)[3]{}[(1,0)[0.12]{}]{} (133.01,28.43)(0.12,0.12)[3]{}[(1,0)[0.12]{}]{} (133.37,28.78)(0.12,0.12)[3]{}[(1,0)[0.12]{}]{} (133.72,29.13)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (134.07,29.49)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (134.43,29.84)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (134.77,30.2)(0.12,0.12)[3]{}[(0,1)[0.12]{}]{} (135.12,30.57)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (135.46,30.93)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (135.8,31.3)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (136.14,31.67)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (136.48,32.04)(0.11,0.12)[3]{}[(0,1)[0.12]{}]{} (136.81,32.41)(0.11,0.13)[3]{}[(0,1)[0.13]{}]{} (137.14,32.79)(0.11,0.13)[3]{}[(0,1)[0.13]{}]{} (137.47,33.17)(0.11,0.13)[3]{}[(0,1)[0.13]{}]{} (137.79,33.55)(0.11,0.13)[3]{}[(0,1)[0.13]{}]{} (138.12,33.93)(0.11,0.13)[3]{}[(0,1)[0.13]{}]{} (138.44,34.32)(0.11,0.13)[3]{}[(0,1)[0.13]{}]{} (138.75,34.7)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (139.07,35.09)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (139.38,35.48)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (139.69,35.88)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (140,36.27)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (140.3,36.67)(0.1,0.13)[3]{}[(0,1)[0.13]{}]{} (140.6,37.07)(0.15,0.2)[2]{}[(0,1)[0.2]{}]{} (140.9,37.48)(0.15,0.2)[2]{}[(0,1)[0.2]{}]{} (141.19,37.88)(0.15,0.2)[2]{}[(0,1)[0.2]{}]{} (141.49,38.29)(0.14,0.2)[2]{}[(0,1)[0.2]{}]{} (141.78,38.69)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (142.06,39.11)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (142.35,39.52)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (142.63,39.93)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (142.91,40.35)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (143.18,40.77)(0.14,0.21)[2]{}[(0,1)[0.21]{}]{} (143.45,41.19)(0.13,0.21)[2]{}[(0,1)[0.21]{}]{} (143.72,41.61)(0.13,0.21)[2]{}[(0,1)[0.21]{}]{} (143.99,42.03)(0.13,0.21)[2]{}[(0,1)[0.21]{}]{} (144.25,42.46)(0.13,0.21)[2]{}[(0,1)[0.21]{}]{} (144.51,42.89)(0.13,0.21)[2]{}[(0,1)[0.21]{}]{} (144.77,43.32)(0.13,0.22)[2]{}[(0,1)[0.22]{}]{} (145.03,43.75)(0.13,0.22)[2]{}[(0,1)[0.22]{}]{} (145.28,44.18)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (145.53,44.61)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (145.77,45.05)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (146.01,45.49)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (146.25,45.93)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (146.49,46.37)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (146.72,46.81)(0.12,0.22)[2]{}[(0,1)[0.22]{}]{} (146.96,47.26)(0.11,0.22)[2]{}[(0,1)[0.22]{}]{} (147.18,47.7)(0.11,0.22)[2]{}[(0,1)[0.22]{}]{} (147.41,48.15)(0.11,0.22)[2]{}[(0,1)[0.22]{}]{} (147.63,48.6)(0.11,0.23)[2]{}[(0,1)[0.23]{}]{} (147.85,49.05)(0.11,0.23)[2]{}[(0,1)[0.23]{}]{} (148.06,49.5)(0.11,0.23)[2]{}[(0,1)[0.23]{}]{} (148.27,49.96)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (148.48,50.41)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (148.69,50.87)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (148.89,51.33)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (149.09,51.79)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (149.28,52.25)(0.1,0.23)[2]{}[(0,1)[0.23]{}]{} (149.48,52.71)(0.09,0.23)[2]{}[(0,1)[0.23]{}]{} (149.67,53.17)(0.09,0.23)[2]{}[(0,1)[0.23]{}]{} (149.85,53.64)(0.09,0.23)[2]{}[(0,1)[0.23]{}]{} (150.04,54.1)(0.18,0.47)[1]{}[(0,1)[0.47]{}]{} (150.22,54.57)(0.18,0.47)[1]{}[(0,1)[0.47]{}]{} (150.39,55.04)(0.17,0.47)[1]{}[(0,1)[0.47]{}]{} (150.57,55.51)(0.17,0.47)[1]{}[(0,1)[0.47]{}]{} (150.74,55.98)(0.17,0.47)[1]{}[(0,1)[0.47]{}]{} (150.9,56.45)(0.16,0.47)[1]{}[(0,1)[0.47]{}]{} (151.07,56.93)(0.16,0.47)[1]{}[(0,1)[0.47]{}]{} (151.23,57.4)(0.16,0.48)[1]{}[(0,1)[0.48]{}]{} (151.38,57.88)(0.15,0.48)[1]{}[(0,1)[0.48]{}]{} (151.54,58.35)(0.15,0.48)[1]{}[(0,1)[0.48]{}]{} (151.69,58.83)(0.15,0.48)[1]{}[(0,1)[0.48]{}]{} (151.84,59.31)(0.14,0.48)[1]{}[(0,1)[0.48]{}]{} (151.98,59.79)(0.14,0.48)[1]{}[(0,1)[0.48]{}]{} (152.12,60.27)(0.14,0.48)[1]{}[(0,1)[0.48]{}]{} (152.26,60.75)(0.13,0.48)[1]{}[(0,1)[0.48]{}]{} (152.39,61.23)(0.13,0.48)[1]{}[(0,1)[0.48]{}]{} (152.52,61.72)(0.13,0.48)[1]{}[(0,1)[0.48]{}]{} (152.65,62.2)(0.12,0.49)[1]{}[(0,1)[0.49]{}]{} (152.77,62.69)(0.12,0.49)[1]{}[(0,1)[0.49]{}]{} (152.89,63.17)(0.12,0.49)[1]{}[(0,1)[0.49]{}]{} (153.01,63.66)(0.11,0.49)[1]{}[(0,1)[0.49]{}]{} (153.12,64.15)(0.11,0.49)[1]{}[(0,1)[0.49]{}]{} (153.23,64.64)(0.11,0.49)[1]{}[(0,1)[0.49]{}]{} (153.34,65.12)(0.1,0.49)[1]{}[(0,1)[0.49]{}]{} (153.44,65.61)(0.1,0.49)[1]{}[(0,1)[0.49]{}]{} (153.54,66.11)(0.1,0.49)[1]{}[(0,1)[0.49]{}]{} (153.64,66.6)(0.09,0.49)[1]{}[(0,1)[0.49]{}]{} (153.73,67.09)(0.09,0.49)[1]{}[(0,1)[0.49]{}]{} (153.82,67.58)(0.09,0.49)[1]{}[(0,1)[0.49]{}]{} (153.91,68.07)(0.08,0.49)[1]{}[(0,1)[0.49]{}]{} (153.99,68.57)(0.08,0.49)[1]{}[(0,1)[0.49]{}]{} (154.07,69.06)(0.08,0.49)[1]{}[(0,1)[0.49]{}]{} (154.15,69.56)(0.07,0.5)[1]{}[(0,1)[0.5]{}]{} (154.22,70.05)(0.07,0.5)[1]{}[(0,1)[0.5]{}]{} (154.29,70.55)(0.07,0.5)[1]{}[(0,1)[0.5]{}]{} (154.36,71.05)(0.06,0.5)[1]{}[(0,1)[0.5]{}]{} (154.42,71.54)(0.06,0.5)[1]{}[(0,1)[0.5]{}]{} (154.48,72.04)(0.06,0.5)[1]{}[(0,1)[0.5]{}]{} (154.53,72.54)(0.05,0.5)[1]{}[(0,1)[0.5]{}]{} (154.59,73.04)(0.05,0.5)[1]{}[(0,1)[0.5]{}]{} (154.63,73.53)(0.05,0.5)[1]{}[(0,1)[0.5]{}]{} (154.68,74.03)(0.04,0.5)[1]{}[(0,1)[0.5]{}]{} (154.72,74.53)(0.04,0.5)[1]{}[(0,1)[0.5]{}]{} (154.76,75.03)(0.03,0.5)[1]{}[(0,1)[0.5]{}]{} (154.79,75.53)(0.03,0.5)[1]{}[(0,1)[0.5]{}]{} (154.83,76.03)(0.03,0.5)[1]{}[(0,1)[0.5]{}]{} (154.85,76.53)(0.02,0.5)[1]{}[(0,1)[0.5]{}]{} (154.88,77.03)(0.02,0.5)[1]{}[(0,1)[0.5]{}]{} (154.9,77.53)(0.02,0.5)[1]{}[(0,1)[0.5]{}]{} (154.92,78.03)(0.01,0.5)[1]{}[(0,1)[0.5]{}]{} (154.93,78.53)(0.01,0.5)[1]{}[(0,1)[0.5]{}]{} (154.94,79.03)(0.01,0.5)[1]{}[(0,1)[0.5]{}]{} (154.95,79.53)(0,0.5)[1]{}[(0,1)[0.5]{}]{} (154.95,80.03)[(0,1)[0.5]{}]{} (154.95,81.03)(0,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.94,81.54)(0.01,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.93,82.04)(0.01,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.92,82.54)(0.01,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.9,83.04)(0.02,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.88,83.54)(0.02,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.86,84.04)(0.02,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.83,84.54)(0.03,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.8,85.04)(0.03,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.76,85.54)(0.03,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.72,86.04)(0.04,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.68,86.54)(0.04,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.64,87.03)(0.04,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.59,87.53)(0.05,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.54,88.03)(0.05,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.48,88.53)(0.06,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.42,89.03)(0.06,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.36,89.52)(0.06,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.3,90.02)(0.07,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.23,90.52)(0.07,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.16,91.01)(0.07,-0.5)[1]{}[(0,-1)[0.5]{}]{} (154.08,91.51)(0.08,-0.49)[1]{}[(0,-1)[0.49]{}]{} (154,92)(0.08,-0.49)[1]{}[(0,-1)[0.49]{}]{}
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(102,100)[(0,0)\[cc\][$�V_{12}� �$]{}]{}
(100,58)[(0,0)\[cc\][$W_{2}� �$]{}]{}
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(34,22)[(0,0)\[cc\][$�V_{21}� �$]{}]{}
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Now, we state a conjecture that is a generalization of the Montgomery’s Conjecture.
For any nontrivial connected graph $G$, $\chi_2(G)-\chi(G)\leq 2\lceil \frac{\Delta(G)}{\delta(G)} \rceil$.
By Theorem $1$, if $G$ is a graph and $ 1 \leq \delta(G) \leq 2$, then Conjecture $2$ is true.
If $G$ is a graph and $ \delta\geq 2 $, then $ \chi_{2}(G)\leq \lceil(4\Delta^{2})^{\frac{1}{\delta-1}} \rceil\chi(G) $.
Suppose that $G$ is a graph, $ \delta\geq 2 $ and let $ k = \lceil(4\Delta^{2})^{\frac{1}{\delta-1}} \rceil $. Consider a vertex $\chi(G)$-coloring of $G$ and for each $1\leq i \leq \chi(G)$, recolor every vertex colored by $i$ randomly and independently by one of the colors $\lbrace ki-(k-1),ki-(k-2),\cdots,ki \rbrace$, with the same probability. Obviously no two adjacent vertices have the same color. For each vertex $v$ let $E_{v}$ be the event that all of the neighbors of $v$ have the same color. We have $P(E_{v})\leq (\frac{1}{k})^{\delta-1}$. $E_{v}$ is dependent to $N[v]\cup N[N[v]]$, so depends to at most $\Delta^{2}$ events. We have $4pd\leq 4\frac{1}{4\Delta^{2}}\Delta^{2} \leq 1$, so by Local Lemma there exists a dynamic coloring by $\lceil(4\Delta^{2})^{\frac{1}{\delta-1}} \rceil\chi(G)$ colors with positive probability.
There exists $ \delta_{0} $ such that every bipartite graph $G$ with $ \delta(G)\geq \delta_{0} $, has $\chi_2(G)-\chi(G)\leq 2\lceil\frac{\Delta(G)}{\delta(G)}\rceil$.
If $G$ is an $ r $-regular graph and $ r\geq 4 $ then it was proved by [@akbari2], that $\chi_2(G)-\chi(G)\leq 2$, so suppose that $ \delta(G) \neq \Delta(G) $. Now two cases can be considered:
Case A. $ \frac{\Delta }{\delta } \geq \delta^{\frac{3}{\delta -4} } $. We have:
$\Delta \geq \delta^{\frac{\delta-1}{\delta-4}} $,
$ \Delta^{\delta-4} \geq \delta^{\delta-1}$,
$ (\frac{\Delta}{\delta})^{\delta-1} \geq \Delta^{3} $,
$ (\frac{\Delta}{\delta})^{\delta-1} \geq 4\Delta^{2} $,
$ \frac{\Delta}{\delta} \geq (4\Delta^{2})^{\frac{1}{\delta-1}} $.
So by Lemma $ 9$, $\chi_2(G)-\chi(G)\leq 2\lceil\frac{\Delta(G)}{\delta(G)}\rceil$.
Case B. $ \frac{\Delta }{\delta } \leq \delta^{\frac{3}{\delta -4} } $.
There exists $ \delta_{0} $ such that: $(4\Delta^{2})^{\frac{1}{\delta-1}} \leq (4(\delta)^{\frac{2\delta-2}{\delta-4}})^{\frac{1}{\delta-1}} \leq (4(\delta_{0})^{\frac{2\delta_{0}-2}{\delta_{0}-4}})^{\frac{1}{\delta_{0}-1}} \leq 3$. So by Lemma $ 9 $ and since $ \delta \neq \Delta $ we have $\chi_2(G)-\chi(G)\leq 2\lceil\frac{\Delta(G)}{\delta(G)}\rceil$.
Concluding Remarks About the Montgomery’s Conjecture
====================================================
In Lemma $ 4 $ we proved if $T_{1}$ is an independent set for a graph $G$, then there exists $T_{2}$ such that, $T_{2}$ is an independent dominating set for $T_{1}$ and $ \vert T_{1} \cap T_{2}\vert \leq \frac{2\Delta(G) -\delta(G) }{2\Delta(G)} \vert T_{1}\vert$. Finding the optimal upper bound for $ \vert T_{1} \cap T_{2}\vert$ seems to be an intriguing open problem, we do conjecture the following:
There is a constant $C$ such that for every $r$-regular graph with $ r\neq 0 $, if $T_{1}$ is an independent set, then there exists an independent dominating set $T_{2}$ for $T_{1}$ such that $ \vert T_{1} \cap T_{2}\vert \leq C $.
If Conjecture $ 3 $ is true, then there is a constant $C_{1}$ such that for every $r$-regular graph $G$, $\chi_{2}(G)-\chi(G) \leq C_{1}$.
If $r\leq 3$, then by Theorem $1$, $\chi_{2}(G)-\chi(G) \leq \chi_{2}(G)-1 \leq 4$.
So, we can assume that $ r\geq 4 $. Consider a vertex $\chi(G)$-coloring of $G$, by Lemma $ 3 $, let $T_{1}$ be an independent dominating set for $G$. Suppose that $ A= \lbrace v \vert v\in V(G), N(v)\subseteq T_{1}\rbrace$ and $B=T_{1}$ and by Lemma $2$, recolor the vertices of $T_{1}$ with the colors $ \chi+1 $ and $ \chi+2 $ such that for each $u\in \lbrace v \vert v\in V(G), N(v)\subseteq T_{1}\rbrace$, $N(u)$ has at least two different colors, retain the color of other vertices and name this coloring $c$. $B_{c}$ is the independent set, by Conjecture $ 3 $, there is a constant $C$ such that there exists an independent dominating set $T_{2}$ for $T_{1}$ and $\vert T_{1} \cap T_{2}\vert \leq C$, now suppose that $ A= \lbrace v \vert v\in V(G), N(v)\subseteq T_{2}\rbrace$ and $B=T_{2}$ and by Lemma $2$, recolor the vertices of $T_{2}$ with the colors $ \chi+3 $ and $ \chi+4 $ such that for each $u\in \lbrace v \vert v\in V(G), N(v)\subseteq T_{2}\rbrace$, $N(u)$ has at least two different colors, retain the color of other vertices and name it $c'$. $ \vert B_{c'}\vert \leq C $, so by Lemma $ 6 $, there exists a dynamic coloring with at most $ \chi(G)+4 +C $ colors. Let $C_{1}=C+4$, this completes the proof.
Acknowledgment
==============
The authors would like to thank Professor Saieed Akbari for his invaluable comments.
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abstract: 'Measurements of the decay strength of superallowed $0^+$$\rightarrow 0^+$nuclear $\beta$ transitions shed light on the fundamental properties of weak interactions. Because of their impact, such measurements were first reported 60 years ago in the early 1950s and have continued unabated ever since, always taking advantage of improvements in experimental techniques to achieve ever higher precision. The results helped first to shape the Electroweak Standard Model but more recently have evolved into sensitively testing that model’s predictions. Today they provide the most demanding test of vector-current conservation and of the unitarity of the Cabibbo-Kobayashi-Maskawa matrix. Here, we review the experimental and theoretical methods that have been, and are being, used to characterize superallowed $0^+$$\rightarrow 0^+$$\beta$ transitions and to extract fundamentally important parameters from their analysis.'
address: 'Cyclotron Institute, Texas A&M University, College Station, TX 77843-3366, U.S.A.'
author:
- J C Hardy and I S Towner
---
\[intro\] Introduction
======================
In 1953, Sherr and Gerhart published a paper [@Sh53] on “Experimental evidence for the Fermi interaction in the $\beta$ decay of $^{14}$O and $^{10}$C." It was less than five years since Sherr had first discovered these two nuclei [@Sh49], yet already the two authors were using the decays to probe for the first time the fundamental nature of $\beta$ decay. They were able to identify superallowed transitions in both decays – they called them “allowed favoured transitions" – and recognized that the Fermi theory of $\beta$ decay predicted that the comparative half-lives, or $ft$ values, for the two transitions should be the same, a prediction they could test. The $ft$ value for a transition depends on the energy released by the transition as well as its branching ratio and the half-life of the initial state. Using degraders to determine the energy of the emitted positrons, a NaI(Tl) scintillation spectrometer to establish the branching ratios and Geiger counters to measure the half lives, Sherr and Gerhard were able to conclude that the two $ft$ values were indeed the same, albeit with large error bars: They obtained 3300$\pm$750 s for the superallowed $ft$ value of $^{14}$O and 5900$\pm$2400 s for that of $^{10}$C.
By 1960, when Bardin [*et al.*]{} reported a much improved result for $^{14}$O [@Ba60], experimental techniques had advanced considerably. The decay energy was no longer dependent on positron range measurements, but rather on measured Q values for the reactions $^{12}$C($^3$He,$n$)$^{14}$O and $^{12}$C($^3$He,$p$)$^{14}$N\*, which populated the superallowed transition’s parent and daughter states, respectively, from a common target. Also, the branching ratio could be corrected for the weak non-superallowed $\beta$ branch to the ground state of $^{14}$N, which had by then been measured with the help of a magnetic lens spectrometer [@Sh55]. They reported an $ft$ value of 3060$\pm$13 s, which for its time is a remarkably precise result and stands only 1.4 standard deviations away from the currently accepted value of 3042.3$\pm$2.7 s.
The precision obtained by Bardin [*et al.*]{} was sufficient for them to compare the vector coupling constant $\GV$, derived from their $ft$-value, with $\GF$, the weak-interaction constant derived from the purely leptonic decay of the muon. The Conserved Vector Current (CVC) hypothesis had been proposed by Feynman and Gell-Mann two years earlier [@Fe58] but the role of strangeness was not yet understood, so the authors expected that $\GV$ should equal $\GF$. Although their measurement actually showed a small difference between the two, the authors noted that radiative and Coulomb corrections could account for the discrepancy and concluded that it was too soon to tell if this constituted a failure of the universality hypothesis.
The discrepancy persisted, however, and was joined by other discrepancies observed between weak decays that changed strangeness $S$ and those that did not, the most glaring example being the decays of the $K^+$ ($S$=1) and the $\pi^+$ ($S$=0) mesons to the same final state, $\mu^+$ + $\nu$. All these apparent conflicts with vector-current universality were resolved by Cabibbo in 1963 [@Ca63] when he recognized that the universality of the weak interaction was manifest only if one considered the total strength of both the strangeness non-changing and the strangeness changing decays. In modern terminology we would say that he was the first to realize that there is mixing between the first two generations of quarks, and to express that mixing in terms of a unitary rotation.
As it turned out, that was only part of the story. A year after Cabibbo introduced his rotation angle, another symmetry – that of CP – was observed to be violated in the weak decay of the long-lived neutral kaon. This result, which remained a puzzle for nearly a decade, ultimately led Kobayashi and Maskawa [@Ko73] in 1973 to postulate the existence of a third generation of quarks – subsequently confirmed by experiment – and to replace Cabibbo’s single rotation angle $\theta_C$, which was in effect a 2$\times$2 rotation matrix, by the now familiar 3$\times$3 unitary rotation matrix referred to as the Cabibbo-Kobayashi-Maskawa, or CKM, matrix. It became one of the pillars of the Standard Model.
In the conclusion to his 1963 paper, Cabibbo remarked that $\GV$ should no longer be expected to be equal to $\GF$ but rather to $\GF$cos$\theta_C$. However, he noted that although this change was “in the right direction to eliminate the discrepancy between $^{14}$O and muon lifetimes," in fact the correction was “too large, leaving about 2% to be explained" [@Ca63]. That observation was enough to stimulate a great deal of activity in superallowed $\beta$ decay during the decade between Cabibbo’s insight and that of Kobayashi and Maskawa. By 1973, the number of well-measured superallowed transitions had grown from one to seven; the 2% discrepancy was being explained in terms of radiative corrections and charge-dependent nuclear corrections [@To73]; and plausible values for the Cabibbo angle had been extracted from the results. Once the CKM matrix took center stage though, it was not long before the focus of superallowed $\beta$ decay had shifted to a determination of the upper left element of the CKM matrix, $V_{ud}$, the value of which is in fact closely related to $\cos{\theta_C}$.
Today, 40 years on, although the Standard Model is by now well established, the limits of its applicability are still being probed aggressively. One such experimental probe is to test the unitarity of the CKM matrix. Although the Standard Model does not prescribe values for the nine elements of the matrix – they must all be determined from experiment – it does require the matrix itself to be unitary. Superallowed $\beta$ decay is currently the source of the most precise value for $V_{ud}$ [@Ha09; @PDG12], which is the largest element in the matrix, and is therefore a crucial contributor to the most sensitive available test of CKM unitarity: the sum of squares of the three top-row elements [@TH10]. The current value for that unitarity sum is 1.00008$\pm$0.00056 [@Ha13], a remarkably precise result that agrees with unitarity and significantly limits the scope for new physics beyond the Standard Model. However, even this precision, $\pm$0.06%, will likely be improved before long by upgraded measurements of superallowed transitions, which are already underway.
In what follows we review particularly the experimental methods that have been, and are being, used to characterize superallowed $0^+$$\rightarrow 0^+$$\beta$ transitions. Although these are all nuclear-physics measurements, the extraordinary demands for precision in the Standard-Model test have motivated the development of highly refined techniques not commonly employed in other nuclear-physics applications. In addition, since theory also plays a vital role in the extraction of $V_{ud}$, we also outline the methods used for calculating the small radiative and isospin-symmetry-breaking corrections that must be applied to the experimental data.
\[exp\] Experiments, past and present
=====================================
Since the axial current cannot contribute to transitions between spin-0 states, superallowed $0^+$$\rightarrow 0^+$ $\beta$ decay between $T$=1 analogue states depends uniquely on the vector part of the weak interaction. As already noted, the CVC principle indicates that the experimental $ft$ value for such a transition should be related to the vector coupling constant, $\GV$, which must be common to all nuclear vector transitions. In turn, $\GV$ itself is related to the fundamental weak-interaction coupling constant, $\GF$, via the relation = V\_[ud]{}, \[GVtoVud\] where $\GF /(\hbar c )^3 = (1.1663787 \pm 0.0000006) \times 10^{-5}$ GeV$^{-2}$, as obtained from the measured muon lifetime [@PDG12]
In practice, the expression for $ft$ includes several small correction terms. It is convenient to combine some of these terms with the $ft$ value and define a “corrected" $\F t$ value. Thus, we write [@Ha09] t ft (1 + \_R\^) (1 + \_[NS]{} - \_C ) = , \[Ftconst\] where $K/(\hbar c )^6 = 2 \pi^3 \hbar \ln 2 / (m_e c^2)^5 = 8120.2787(11) \times 10^{-10}$ GeV$^{-4}$s, $\delta_C$ is the isospin-symmetry-breaking correction, and $\DRV$ is the transition-independent part of the radiative correction. The terms $\delta_R^{\prime}$ and $\delta_{NS}$ comprise the transition-dependent part of the radiative correction, the former being a function only of the electron’s energy and the $Z$ of the daughter nucleus, while the latter, like $\delta_C$, depends in its evaluation on the details of the nuclear structure of the parent and daughter states. All these correction terms are of order 1% or less, with uncertainties at least an order of magnitude smaller than that, so equation (\[Ftconst\]) provides an experimental method for determining $\GV$ – and thus $V_{ud}$ – to better than a part in a thousand.
Experimentally, the $ft$ value that characterizes a superallowed transition – or any $\beta$ transition for that matter – is determined from three measured quantities: the total transition energy, $Q_{EC}$, the half-life, $t_{1/2}$, of the parent state, and the branching ratio, $R$, for the particular transition of interest. The $Q_{EC}$-value is required to determine the phase-space integral, $f$, while the half-life and branching ratio combine to yield the partial half-life, $t$. Since the $ft$ value incorporates three experimental quantities, each one of those quantities must be measured to substantially better than 0.1% precision in order to achieve that precision on the combination. This is particularly true for $Q_{EC}$ since it enters to the fifth power in the calculation of $f$.
![\[fig:chart\] Partial chart of nuclides showing parents of the precisely measured superallowed transitions as solid black squares. The stable nuclei appear as grey squares. The two labeled diagonal lines mark the loci of the $T_Z$=$-1$ and $T_Z$=0 parents. From left to right, the former are [$^{10}$C]{}, [$^{14}$O]{}, [$^{22}$Mg]{} and [$^{34}$Ar]{}; and the latter are [$^{26m}$Al]{}, [$^{34}$Cl]{}, [$^{38m}$K]{}, [$^{42}$Sc]{}, [$^{46}$V]{}, [$^{50}$Mn]{}, [$^{54}$Co]{}, [$^{62}$Ga]{} and [$^{74}$Rb]{}.](fig1.eps){width="10cm"}
To date, the $ft$ values for ten $0^+$$\rightarrow 0^+$ transitions – with parents [$^{14}$O]{}, [$^{26m}$Al]{}, [$^{34}$Cl]{}, [$^{38m}$K]{}, [$^{42}$Sc]{}, [$^{46}$V]{}, [$^{50}$Mn]{}, [$^{54}$Co]{}, [$^{62}$Ga]{} and [$^{74}$Rb]{} – are known to 0.1% relative precision or better; and three more – [$^{10}$C]{}, [$^{22}$Mg]{} and [$^{34}$Ar]{} – are known to $<$0.3%. The 13 cases are shown on the chart of nuclides in Fig.\[fig:chart\]. How has this level of precision been achieved in all these cases for the combination of three experimental quantities? In answering this question in the following sections, we frequently reference examples of various measurement techniques. A complete referenced list of [*all*]{} measurements that currently contribute to world data for superallowed $0^+$$\rightarrow 0^+$ decays appears in the 2009 survey by Hardy and Towner [@Ha09].
\[QEC\] $Q_{EC}$-value measurements
-----------------------------------
Already in 1960, Barden [*et al.*]{} [@Ba60] appreciated that a determination of the end-point energy from a measured $\beta$ spectrum, even if that spectrum were obtained with a magnetic spectrometer, could not possibly match the precision possible with a nuclear reaction. From that time on, until the advent of on-line Penning traps less than a decade ago, nuclear reactions were the only method used to determine precise $Q_{EC}$ values. The favoured ones were ($p,n$) and ($^3$He,$t$) reactions on the $\beta$-decay daughter nuclei, which are stable for all $0^+$$\rightarrow 0^+$ superallowed decays that have been studied until recently. Since these reactions connect the same nuclei as the corresponding $\beta$ decay, their $Q$ values are directly related to the $\beta$-decay $Q_{EC}$ value.
![\[fig:history\]The relative precision, $\Delta Q/Q$, for $Q_{EC}$-value measurements of superallowed transitions is plotted against their publication date, where $Q$ is the measured $Q_{EC}$ value and $\Delta Q$ is its quoted uncertainty. The data encompass the superallowed transitions from [$^{10}$C]{}, [$^{14}$O]{}, [$^{26m}$Al]{}, [$^{34}$Cl]{}, [$^{38m}$K]{}, [$^{42}$Sc]{}, [$^{46}$V]{}, [$^{50}$Mn]{} and [$^{54}$Co]{}, and are taken from a series of survey articles [@Ha09; @Ha75; @Ha90; @Ha05] plus two more-recent publications [@Er09; @Er11]. Each point is identified by the experimental method used in the corresponding measurement. The line simply illustrates the decreasing trend. (Adapted from ref.[@Er12].)](fig2.eps){width="13cm"}
The relative precision, $\Delta Q/Q$, obtained for various measurements and techniques is plotted in Fig.\[fig:history\] as a function of publication date, starting in 1960. Evidently the relative precision improved steadily over the years, from $\sim$5$\times$$10^{-4}$ in the early 1960s to $\sim$3$\times$$10^{-5}$ by 1990, but at that point a limit seemed to have been reached: for the next fifteen years there were no further improvements in $Q_{EC}$-value precision, not even with the appearance of the first few Penning-trap measurements. However, the figure shows that in the space of only a few years after their first contribution, Penning traps had improved the relative precision for measured $Q_{EC}$ values by a factor of 5, to as low as $\sim$7$\times$$10^{-6}$. To put this last number into perspective, it applies to the decay of [$^{38m}$K]{} and corresponds to an uncertainty of $\pm$40 eV on a total $Q_{EC}$ value of 6044.22 keV [@Er09].
Although Penning traps are now outstripping nuclear reactions in the precision of their $Q_{EC}$-value results, for each transition it is the average of all measurements with uncertainties within a factor of ten of the best measurement that is used in the determination of $V_{ud}$. Many reaction measurements therefore still make significant contributions to the world averages for $Q_{EC}$ values. We will describe several examples of important reaction measurements before doing the same for Penning-trap measurements.
### \[pnreact\] The ($p$,$n$) reaction
It can easily be seen in Fig.\[fig:history\] that before 2005 the dominant choice for making $Q_{EC}$-value measurements of $0^+$$\rightarrow 0^+$ transitions was the ($p$,$n$) reaction. There are a total of 21 such cases documented in the figure, with all but one – the first – being threshold measurements. Between 1962 and 1976, all but two of the measurements were made by Freeman and her collaborators using the 12 MeV Tandem accelerator at Harwell in England (e.g. [@Fr66; @Sq76]). From 1977 on, all but two were made by Barker [*et al.*]{} using the folded Tandem accelerator AURA2 at the University of Auckland in New Zealand (e.g. [@Ba77; @To03]).
The Freeman measurements and the early Barker measurements all used the same general approach: For a superallowed decay P$\rightarrow$D they bombarded a thin target of the daughter material D with a proton beam for a well-determined time, and then interrupted the beam while they determined the amount of the parent P that had been produced by recording the characteristic activity of P, usually its emitted positrons. This beam on-off cycle was repeated until sufficient statistics had been accumulated. Then the process was repeated at a succession of different beam energies until a threshold curve for the production of P had been obtained. They calibrated the proton beam energy near threshold by scattering the beam at 90$^\circ$ from a thin gold foil into a broad-range magnetic spectrograph, where it was compared with a known $\alpha$-particle group from the decay of a standard source such as [$^{212}$Po]{}. This, of course, meant that the accuracy of their threshold energy – and the resultant $Q_{EC}$ value – relied upon the accuracy of the $\alpha$-particle energy as it was known at the time but, by clearly indicating the $\alpha$ energy they used, they ensured that their result could be upgraded in future whenever that calibration energy was improved.
Barker’s group refined this technique by passing the primary proton beam through an Enge split-pole magnetic spectrograph, which was set at 0$^\circ$. The beam path and width were constrained by a set of slots and apertures, resulting in an energy profile with FWHM (full width half maximum) of 50-100 ppm (parts per million) at the focal plane of the spectrograph [@To03]. It was this prepared beam that bombarded the target, which was located 50 cm beyond the focal plane. To determine the energy of the proton beam, the magnetic rigidity of the spectrograph orbit was calibrated by leaving the field strength unchanged and passing a monoenergetic heavy-ion beam of cesium or potassium around the same constrained orbit. That heavy-ion beam was produced by surface ionization and then accelerated through a potential $V$, which was adjusted until the ions were observed just upstream from the target. Finally, the optimized voltage $V$ was compared with a 1-volt standard via two successive stages of resistive division. The threshold determined from a yield curve was thus based firmly on a primary calibration standard – the volt – independent of any secondary reaction $Q$ values. However it did require corrections for finite energy spread of the beam, for non-uniform proton energy loss and for atomic excitation. It was also subject to near-threshold resonances, which, if present, could misguide the analysis [@Br94; @Er12]. Nevertheless, at its apotheosis, this technique achieved a precision of $\sim$2$\times$$10^{-5}$ on several measured $Q_{EC}$ values.
### \[pgreact\] Combined ($p$,$\gamma$) and ($n$,$\gamma$) reactions
A second reaction-based approach that ultimately led to very precise results was to measure both ($p$,$\gamma$) and ($n$,$\gamma$) reactions on a common target, which had been chosen so that the ($p$,$\gamma$) reaction would produce the parent of a $0^+$$\rightarrow 0^+$ $\beta$ decay, and the ($n$,$\gamma$) reaction would produce the daughter: for example, [$^{25}$Mg]{}($p$,$\gamma$)[$^{26m}$Al]{} and [$^{25}$Mg]{}($n$,$\gamma$)[$^{26}$Mg]{} [@Ki91]. In such cases, the reaction $Q$ values yield $Q_{EC}$ through the relation Q\_[EC]{} = Q\_[n]{} - Q\_[p]{} - 782.347 [keV,]{} \[Qpgpn\] which is independent of the mass of the target nucleus.
Continuing with the same example [@Ki91], the [$^{25}$Mg]{}($n$,$\gamma$)[$^{26}$Mg]{} reaction was studied with thermal neutrons from the Los Alamos Omega West reactor. Gamma rays were detected with a Compton-suppressed Ge(Li) detector, and their energies precisely determined from a calibration based on well known lines observed in the [$^{1}$H]{}($n$,$\gamma$), [$^{12}$C]{}($n$,$\gamma$) and [$^{14}$N]{}($n$,$\gamma$) reactions. The value of $Q_{n\gamma}$ was then obtained from the average summed energy of a number of $\gamma$-ray cascades de-exciting the capture state in [$^{26}$Mg]{}. For the [$^{25}$Mg]{}($p$,$\gamma$)[$^{26m}$Al]{} reaction a dual Mg-Al target (upper half Mg, lower half Al) was employed, over which a proton beam from the Utrecht 3 MV Van der Graaff accelerator was wobbled up and down. The proton energies at four [$^{25}$Mg]{}($p$,$\gamma$)[$^{26m}$Al]{} resonances were compared with four accurately known resonances in the [$^{27}$Al]{}($p$,$\gamma$)[$^{28}$Si]{} reaction. The proton beam was scanned in 200-eV steps over each [$^{25}$Mg]{} resonance and its nearest neighbour [$^{27}$Al]{} resonance, which in all four cases was only a few keV away. Thus each of the proton energy differences could be determined precisely and an average $Q_{p\gamma}$-value obtained. Obviously this method for determining $Q_{EC}$ depended on secondary calibration standards but it was less dependent on experimental corrections than was the ($p$,$n$) reaction described in Sec. \[pnreact\]. At its best, it achieved a relative precision of 3-5$\times$$10^{-5}$.
### \[3het\] Relative ($^3\mathrm{He},t$) reactions
Like ($p$,$n$), a ($^3$He,$t$) reaction acting on the daughter of a superallowed $0^+$$\rightarrow 0^+$ decay produces the parent of the decay, so the reaction $Q$ value is directly related to $Q_{EC}$. In contrast with ($p$,$n$) though, the energy of the outgoing particle – a tritium ion – can be measured conveniently. Even so, to determine the $Q$ value, the energies of both the tritons and the $^3$He projectiles must still be calibrated at high precision against an established energy standard, not an easy task at the energies involved. Koslowsky, Hardy and collaborators at Chalk River dealt with this difficulty by developing a novel system that allowed them to measure the $Q$-[*value differences*]{} between two ($^3$He,$t$) reactions produced concurrently from a target containing two components, each the daughter of a superallowed $\beta$ emitter [@Ko87].
![\[fig:doublet\]Schematic diagram illustrating the principle of biasing a two-component target in order to peak-match the reaction-product group $g_1$ from one component with group $g_2$ from the other component. (Adapted from ref.[@Ko87].)](fig3.eps){width="11cm"}
Their method is illustrated in Fig.\[fig:doublet\]. A two-component target was bombarded by a doubly ionized $^3$He beam of 20-30 MeV from an MP Tandem accelerator. The ejected tritons were analyzed at 0$^\circ$ by a high-resolution Q3D magnetic spectrometer, which transformed their energy spectrum into a distribution in position along its focal plane. The target assembly was constructed so that it could be intermittently biased at $+V$ volts relative to its grounded surroundings, with $V$ being adjustable up to 150 kV. With the target at voltage $+V$, the $^3$He beam, being doubly ionized, was retarded by 2$V$ eV, while the singly-charged tritons were re-accelerated by only $V$ eV. As a result, the net effect of the imposed voltage was to reduce the triton energy by $V$ eV relative to its value with no voltage on the target, thus shifting its position on the focal plane. The target voltage could then be adjusted until the shifted position of triton group $g_2$ exactly coincided with the unshifted position of $g_1$ (see Fig.\[fig:doublet\]). That adjusted voltage, which was related by resistive division to the standard volt, corresponded to the energy difference between groups $g_1$ and $g_2$. Furthermore, since the shifted tritons ($g_2$) followed the same path as the unshifted ones ($g_1$), the result was independent of that path. With $g_1$ chosen to be a triton group from one target component and $g_2$ being from the other, the difference between reaction $Q$ values could be precisely determined with reference to the standard volt.
Of course, the superallowed $Q_{EC}$-value differences themselves were not 150 keV or less. In practice, for each doublet, the experimental team determined the $Q$-value difference for the population of an excited state in each of the two $\beta$-decay parents, these states being chosen so that their $Q$-value difference was indeed within 150 keV. Their excitation energies were either known or determined separately via $\gamma$-ray spectroscopy, so the measured reaction $Q$-value differences could be related quite precisely to the superallowed $Q_{EC}$-value differences. Four pairs of superallowed decay energies were studied in this way [@Ko87]. Being differences, the results do not appear in Fig.\[fig:history\], but their precision was comparable to the best reaction results that do appear there, and they continue to figure prominently in the world averages for $Q_{EC}$ values [@Ha09].
### \[Penning\] Penning-trap mass measurements
The $Q_{EC}$ value for $\beta$ decay is simply the atomic mass difference between the parent state and its daughter: It could be derived from those masses if they were known precisely enough. So far, though, we have just described methods for measuring the difference directly since, until recently, this approach yielded the only precise results. As explanation, consider the superallowed $0^+$$\rightarrow 0^+$ decay, $^{26m}$Al$\rightarrow$$^{26}$Mg. The masses of the nuclear states in this case are $\sim$$2.4\times10^{10}$ eV and the difference between them ($Q_{EC}$) is $\sim$$4.2\times10^6$ eV. Reaction measurements yield $Q_{EC}$-values with relative precision of $\sim$$3\times10^{-5}$ (see Fig.\[fig:history\]), which is $\sim$120 eV in this case. To achieve the same precision with a pair of mass measurements would require them each to have a relative precision of $\sim$$4\times10^{-9}$. This was beyond the capability of conventional mass spectrometry, which in any case was limited to effectively stable nuclei.
The balance has shifted with the appearance of Penning traps coupled on-line to an accelerator. The Penning trap itself can confine charged particles to a small volume by means of static magnetic and electric fields, the former being homogeneous and the latter quadrupolar. The trapped ions exhibit three eigenmotions: one along the axis of the magnetic field, and the other two in the radial plane perpendicular to that axis. By combining the frequencies of these three eigenmotions, one can obtain the cyclotron frequency, $\nu_c$, of the trapped ions and, from it, the mass of the ion itself since \_c = , \[cycf\] where $q$ and $m$ are the charge and mass of the ion, and $B$ is the magnetic field. For stable nuclei, the frequency can be determined via external circuitry, without the trapped ions being released [@Br86]. A relative precision of 2-3$\times10^{-12}$ has been achieved this way for the stable nuclei, $^{32}$S and $^{31}$P, for example [@Re08].
![\[fig:Penning\]Time of flight (TOF) resonance measured with an on-line Penning trap for the superallowed emitter $^{26m}$Al, which has a half-life of 6.3 s. The solid curve is a fitted function. (Adapted from ref.[@Er06].)](fig4.eps){width="10cm"}
However, the parent nuclei of the superallowed decays – and sometimes the daughters as well – have short half-lives, from a few seconds down to a few tens of milliseconds. This requires a number of additional steps in the experimental procedures. First the ions of interest must be produced by an accelerator; next they are cooled, bunched and if necessary purified; and then they are injected into the Penning trap. Because of the short half-life of the ions, this cycle is repeated continuously with fresh ion bunches being delivered every few seconds. In each cycle, once the ions are trapped, the cyclotron frequency in the trap is probed with an applied radiofrequency electric field, after which the ions are released and their time of flight measured to a microchannel-plate detector located outside the high-field region. As the applied frequency is scanned through the cyclotron frequency, the ions’ time-of-flight passes through a distinct minimum, as is shown in Fig.\[fig:Penning\]. To this basic technique numerous refinements have been applied, the most significant of which is to excite the ion motion with Ramsey’s method of time-separated oscillatory fields [@Ra90].
On-line Penning trap measurements for superallowed $\beta$ decay have been, and are being performed at four different facilities: ISOLTRAP ([*e.g.*]{} Ref.[@Mu04]), CPT ([*e.g.*]{} Ref.[@Sa05]), LEBIT ([*e.g.*]{} Ref.[@Ri07]) and JYFLTRAP ([*e.g.*]{} Refs. [@Er09; @Er11]). An early high-impact measurement made with the CPT Penning trap was of the $Q_{EC}$ value for the decay of $^{46}$V [@Sa05]. The result, which was determined with an uncertainty of $\pm$400 eV, differed by 2.5 keV from a long-trusted 1977 ($^3$He,$t$) reaction measurement that had claimed a similar precision. The latter measurement was based on a “precision time-of-flight measuring system" with the Q3D spectrograph at the Munich Tandem Laboratory; it was one of seven superallowed $Q_{EC}$ values that appeared in a single publication [@Vo77] and had stood unchallenged for nearly 30 years. However, the $^{46}$V discrepancy was soon confirmed by a second Penning trap, JYFLTRAP, which also identified similar disagreements with the Munich measurements for three other cases, $^{42}$Sc, $^{50}$Mn and $^{54}$Co [@Er06; @Er08]. When a repeat ($^3$He,$t$) measurement [@Fa09], made at Munich in 2009 with much of the same equipment, agreed with the Penning trap results, it was decided to eliminate Ref.[@Vo77] from surveyed world data [@Ha09]. Fortunately, this was the only significant disagreement found between Penning-trap results and those from earlier reaction measurements. It has since been demonstrated that one can safely combine the results of both types of measurement without including any additional systematic uncertainties [@Er12].
To date, the most precise Penning-trap measurements of superallowed $Q_{EC}$ values have been done by Eronen and collaborators with the JYFLTRAP trap at the University of Jyvaskyla, where it is coupled to a cyclotron through their Ion Guide Isotope Separator On Line (IGISOL) [@Er12]. In addition to all the refinements to achieve high trap precision, this facility has an added advantage: It can produce both parent and daughter nuclei with the same beam. For the case illustrated in Fig.\[fig:Penning\], the superallowed emitter $^{26m}$Al was produced by the ($p$,$n$) reaction at 15MeV on a target of $^{26}$Mg. Ions of $^{26}$Mg, which is the $\beta$-decay daughter, were also released by elastic scattering of the proton beam. The $Q_{EC}$ value is then given by Q\_ = M\_p - M\_d = ( -1 ) (M\_d - m\_e ) + \_[p,d]{}, \[qec\] where $M_p$ and $M_d$ are the parent and daughter masses, and $\nu_{c,\mathrm{p}}$ and $\nu_{c,\mathrm{d}}$ are their respective measured cyclotron frequencies; $m_e$ is the electron rest mass; and the term $\Delta_{p,d}$ arises from atomic-electron binding-energy differences between the parent and daughter, known to sub-eV accuracy. Since the term $(\nu_{c,\mathrm{d}}/\nu_{c,\mathrm{p}} - 1)$ in (\[qec\]) is always $\ll$$10^{-3}$ for the superallowed parent-daughter pairs, $M_d$ needs only to be known to few-keV precision in order for its uncertainty to have a negligible impact on the $Q_{EC}$-value precision.
Recent $Q_{EC}$ measurements with JYFLTRAP interleave parent and daughter frequency measurements by switching automatically back and forth between parent and daughter ions after each complete frequency scan, typically every minute or so. This effectively eliminates any systematic differences that might occur from drifts in the magnetic field for example. As a result, a relative precision of $7\times10^{-6}$ has been achieved in several cases, including the $Q_{EC}$ value for the superallowed branch from $^{38m}$K, which was determined to be 6044.223(41)keV [@Er09].
\[t\] Half-lives
----------------
Precise half-life measurements are deceptively difficult. Problems such as impurity activities, rate-dependent thresholds, dead-time and pile-up effects, as well as statistically flawed analyses, offer no obvious signals of their magnitude, or even of their presence. It is not surprising that many half-life measurements of superallowed emitters have had to be rejected from surveys of world data (see table VII in Ref. [@Ha09]).
The superallowed $0^+$$\rightarrow 0^+$ transitions we are considering here take place between $T$=1 analogue states. As shown in Fig.\[fig:chart\], the parents are of two types: either odd-$Z$-odd-$N$ nuclei with $T_Z$=0, or even-even ones with $T_Z$=$-1$. The two types exhibit quite different decay patterns, as is shown in Fig.\[fig:38decays\], where $^{38m}$K$\rightarrow ^{38}$Ar is an example of the first type and $^{38}$Ca$\rightarrow ^{38}$K is an example of the second. Not surprisingly, most of the best-measured decays in the past have been of $T_Z$=0 nuclei, where the superallowed branch is overwhelmingly predominant. For such cases the only way to measure the half-life is to detect the emitted positrons (or possibly the 511-keV annihilation radiation). However, for the decays of $T_Z$=$-1$ nuclei – two of which are the classic cases of $^{10}$C and $^{14}$O, discussed in the Introduction – the total $\beta$-decay strength is spread over a number of branches and ample $\beta$-delayed $\gamma$ rays are produced. In these cases half-life measurements based on $\gamma$-ray detection have been reported as well as those in which only positrons were recorded. We will briefly describe both measurement techniques with their advantages and disadvantages.
![\[fig:38decays\]Partial decay schemes of $^{38}$Ca and $^{38m}$K.](fig5.eps){width="10cm"}
### \[beta\] Beta detection methods
Direct detection of decay positrons can be accomplished with high efficiency, and the signals from the detector – either a plastic scintillator or a gas counter – can be processed safely at quite high rates. Against these advantages must be balanced the disadvantage presented by the positrons’ broad energy distribution, which cannot in general be used to distinguish one decaying nuclide from another. Without some external means of ensuring source purity, a decay measurement can easily be invalidated by the presence of an undetected impurity. Typically, before 1983, activities were produced by low-energy proton beams on enriched targets, a combination that minimized contaminant activities but could not eliminate them entirely. In a few cases, where impurities were identified, their contribution was corrected for ([ *e.g.*]{} Refs.[@Ry73a; @Az75]) but, in most cases, purity was simply a fervent belief. Since 1983 however, with rare exceptions all measurements have employed sources produced via either an on-line isotope separator – first at Chalk River ([*e.g.*]{} Ref.[@Ko83]) and later at TRIUMF ([*e.g.*]{} Ref.[@Fi11]) – or the magnetic recoil spectrometer at the Texas A&M cyclotron ([*e.g.*]{} Ref.[@Pa12]). These devices eliminate or at least minimize impurities, with the recoil spectrometer also being capable of identifying and quantifying any weak impurities that remain.
Although half-life measurements in the past have frequently used plastic scintillators to detect $\beta$ particles, in all but one of the thirteen cases of superallowed decays that currently contribute to world data [@Ha09], the most precise half-life measurements have all been made with 4$\pi$ gas proportional counters, all built from the design first developed for this purpose by Koslowsky, Hagberg and Hardy at Chalk River 30 years ago [@Ko83]. It was modeled after the “pill box" detectors long used by radiation metrologists [@NC85], chosen because they have low background and are nearly 100% efficient for $\beta$ particles, while being insensitive to $\gamma$ rays. The detector consists of two separate gas cells machined from copper (as pictured in Fig.\[fig:TAMUexp\]), each containing an anode of gold-plated tungsten wire 13 $\mu$m in diameter, and each hermetically sealed by a Havar window 3.7cm in diameter and 1.5$\mu$m thick. Methane at just above atmospheric pressure is continuously flushed through both cells. When assembled together, there is a 25-mm slot between the two cell windows, through which a thin tape can slide. The assembled detector is easily held in the palm of one hand.
![\[fig:TAMUexp\]Simplified experimental arrangement used, for example, in measuring the half-lives of [$^{38}$Ca]{} [@Pa11] and [$^{46}$V]{} [@Pa12]. The two halves of the 4$\pi$ gas proportional counter are pictured at the bottom right. In operation, each half is sealed with a Havar window (as shown on the right half only) and the two halves are bolted together with the Havar windows facing one another. There is a thin slot that remains so the tape can pass between the windows.](fig6.eps){width="12cm"}
Though the detector itself is small, the equipment required to deliver a clean source into the detector is not. In the experimental configuration employed at Texas A&M (see top and left side of Fig.\[fig:TAMUexp\]) the activity of interest is first produced by bombardment of a cooled hydrogen gas target. Taking the superallowed parent [$^{46}$V]{} ($t_{1/2}$=423 ms) as an example [@Pa12], the activity was produced from the [$^{1}$H]{}([$^{47}$Ti]{},2$n$)[$^{46}$V]{} reaction initiated by a 1.5-GeV beam of [$^{47}$Ti]{} from the K500 cyclotron. The fully stripped reaction products exiting the gas cell entered the Momentum Achromat Recoil Spectrometer (MARS) [@Tr89], where they were separated according to their charge-to-mass ratio $q/m$, with [$^{46}$V]{} being selected by slits in the focal plane. A position-sensitive silicon detector was periodically inserted at the focal plane to identify and monitor any weak contaminants that were also passing through the slits.
The purified [$^{46}$V]{} beam was extracted into air, degraded and implanted into the 76-$\mu$m-thick aluminized Mylar tape of a fast tape-transport system. The combination of $q/m$ selectivity in MARS and range sensitivity in the degraders led to collected samples of radioactive [$^{46}$V]{}, in which the only interfering activity was determined to contribute less than 0.012% to the total. After a sample had been collected for 0.5s, the beam was turned off and the tape moved the sample 90 cm into the center of the shielded 4$\pi$ gas detector, where it stopped less than 200ms later. Signals from the detector were then multiscaled for 10s, which is more than 20 half-lives of [$^{46}$V]{}. This cycle was repeated many thousands of times until the desired statistics had been reached.
In all measurements of this type, the counting electronics impose a well-defined non-extendable dead time. This dead time as well as other measurement parameters, such as detector bias and discriminator levels, are altered from time to time in order to test for possible systematic effects. Careful analysis and fitting procedures are applied and these procedures are checked with hypothetical data, computer-generated by Monte Carlo techniques to simulate closely the experimental counting conditions.
In recent years this system has also been used to measure the half-lives of $T_Z$=$-1$ superallowed parents such as [$^{38}$Ca]{} (see Fig.\[fig:38decays\]). There a complication arises: The $\beta$ decay of the $T_Z$=$-1$ parent feeds a second superallowed decay, from the $T_Z$=0 daughter to the $T_Z$=+1 granddaughter. Of course, the positrons from both decays are recorded simultaneously in the $\beta$ detector. The time-decay spectrum is therefore the sum of the decay of the parent and the growth-and-decay of the daughter. Typically, the parent’s half-life is about a factor of two shorter than the daughter’s, so the decay of the daughter almost completely masks the decay of the parent ([*e.g.*]{} see Refs.[@Pa11; @Ia10]). However the composite decay can still be used to determine the parent’s half-life if three conditions are met: 1) the half-life of the daughter is known precisely from an independent measurement; 2) the $T_Z$=$-1$ parent activity deposited on the transport tape is pure; and 3) the rate at which it is deposited is known. The first condition can easily be met by a measurement of the type already described for [$^{46}$V]{}; the second is assured by the combination of electromagnetic separation (MARS) and range selectivity; and the third was met by insertion of a thin scintillator into the degrader stack (see Fig.\[fig:TAMUexp\]), from which the number of ions were recorded as a function of time during sample collection.
With these methods, half-lives with a relative precision of $\sim$0.03% for $T_Z$=$-1$ parents ([*e.g.*]{} [$^{26}$Si]{} [@Ia10]) and $\sim$0.01% for $T_Z$=0 parents ([*e.g.*]{} [$^{46}$V]{} [@Pa12]) have been obtained in the best cases. A result of comparable precision was also obtained for [$^{26m}$Al]{} at TRIUMF [@Fi11], where the same type of gas counter was used but with the sample produced by their isotope separator ISAC. Very recently another measurement of the [$^{26m}$Al]{} half-life with similar precision has been reported [@Ch13], for which digital pulse-analysis was used to process signals from the 4$\pi$ gas detector instead of analogue electronics. This meant that parameters like dead-time and discriminator level could be investigated, after the fact, on the saved pulse shapes, thus improving the efficiency of data-taking. The relative precision quoted for this measurement was also at the 0.01% precision level, with some promise for future improvement.
### \[gamma\] Gamma detection methods
For a half-life measurement, high-resolution $\gamma$-ray detection makes source purity a much less critical requirement, since analysis can focus on the photopeak of a $\gamma$ ray that is characteristic of the activity of interest. This is the method’s principal advantage. On the other side of the ledger must be placed the relatively low efficiency of germanium detectors and the slow signals that are derived from them, with the consequently long time required to process those signals. The latter introduces uncertainties in accounting for dead time and especially for pulse pile-up, which is of course rate dependent. A further disadvantage is that the method can only be applied to $T_Z$=$-1$ parents of superallowed decays since they are the only ones that produce $\beta$-delayed $\gamma$ rays of sufficient intensity (see Fig.\[fig:38decays\]).
A recent measurement at TRIUMF illustrates the inherent difficulties with this method. Using a separated beam from the ISAC facility, Laffoley [*et al.*]{} [@La13] measured the half-life of [$^{14}$O]{}. This nucleus $\beta$-decays to [$^{14}$N]{} with a 99.4% branch to the 2313-keV excited state, which then de-excites by emitting a $\gamma$ ray to the ground state. The [$^{14}$O]{} half-life can then be measured through detecting either $\gamma$ rays or $\beta$ particles. Laffoley [*et al.*]{} did both simultaneously. They implanted [$^{14}$O]{} from ISAC into thin aluminum at the center of the “8$\pi$" $\gamma$-ray spectrometer, a spherically symmetric array of 20 HPGe detectors. A fast plastic scintillating detector was placed immediately behind the implantation location to detect $\beta$ particles. The $\gamma$-ray data (for the 2313-keV transition) were carefully analyzed with well worked-out techniques [@Gr07] to account for pile-up and other time-dependent effects. The $\beta$ signals were handled in a very similar way to that used with a gas proportional counter. The results are revealing: The half-life obtained from $\gamma$ counting was $70.632 \pm 0.086_{stat} \pm 0.037_{syst}$s, while from $\beta$ counting it was $70.610 \pm 0.020_{stat} \pm 0.023_{syst}$s. Though the two measurements were made simultaneously on the same collected sources, both the statistical and systematic uncertainties were larger when $\gamma$-rays were employed. Happily though the two results agreed with one another well within their quoted uncertainties.
\[R\] Branching ratios
----------------------
Of the three experimental quantities – $Q_{EC}$ values, half-lives and branching ratios – needed to obtain an $ft$ value, the most difficult to measure precisely is the branching-ratio. Since the continuous energy distribution of emitted positrons leaves little opportunity to distinguish one transition from another, such measurements must be based on detection of the $\beta$-delayed $\gamma$ rays emitted from levels populated in the $\beta$-decay daughter. To make matters more difficult, in most cases one of the $\beta$ transitions populates the ground state or an isomeric state, from which no $\gamma$-ray signal is forthcoming. Thus it is not enough to measure the relative branching to excited states. What is needed is the absolute branching for each transition: [*i.e.*]{} the fraction each accounts for out of the total decays of the parent nucleus.
What are the competing transitions? Since the parents of our decays of interest have spin-parity $0^+$, they can populate $1^+$ states in the daughter by allowed Gamow-Teller decay, in addition to populating the analogue $0^+$ state by the superallowed (Fermi) branch. Furthermore, weak Fermi branches are also possible to excited $0^+$ states via charge-driven mixing with the analogue state. The nuclear structure of the daughter nucleus and the energy available for the parent’s $\beta$ decay together determine how many such states can be populated.
Here again the challenges are different for the decays of $T_Z$=0 superallowed parents compared to the $T_Z$=$-1$ cases. The superallowed branch from each $T_Z$=0 parent carries $>$99% of the decay strength and populates the ground state of its daughter (see Fig.\[fig:38decays\]). To determine its exact branching ratio, all that is required is to measure the weak competing branches, if any, with modest precision and subtract their total from 100%. Since the absolute value of the uncertainty on the total of the weak branches becomes the uncertainty on the superallowed branch, a poor relative precision on the former becomes a very good relative precision on the latter. For example, the decay of [$^{42}$Sc]{} includes a single competing Gamow-Teller $\beta$-decay branch to the 1.84-MeV state in [$^{42}$Ca]{}. Its branching ratio, based on four separate measurements, is 0.0059(14)% [@Ha09], a result with $\pm$24% relative precision. The superallowed branching ratio obtained from this result is 99.9941(14), which has a relative precision of $\pm$0.0014%! Obviously, high precision is not the issue with these measurements. Rather, the difficulty lies in even observing branches with relative intensities that are less than 100 parts per million of the total decay.
The decays of $T_Z$=$-1$ superallowed parents are quite different. In general they are characterized by much stronger competing Gamow-Teller transitions (see Fig.\[fig:38decays\]) and, in a few cases, the superallowed branch is not even the strongest transition: In [$^{10}$C]{} decay, for example, the superallowed branch only accounts for 1.4646(19)% of the total decay strength [@Ha09]. In addition, with two exceptions – [$^{10}$C]{} and [$^{22}$Mg]{} – all the known cases include decay branches that do not produce a subsequent $\gamma$ ray. Consequently, absolute branching ratios must be determined, but without nearly the precision improvement factor just described for decays like that of [$^{42}$Sc]{}. These ratios must therefore be directly determined with $\sim$0.1% precision. Only very recently has it become possible to do so.
### \[TZ0\] $T_Z$=0 parent decays
Because the daughter of a $T_Z$=0 superallowed parent is an even-$Z$-even-$N$ nucleus, its excited $1^+$ and $0^+$ states are at a relatively high excitation energy above the $0^+$ ground state, which is strongly populated by the superallowed transition. For the lightest nuclei, with $A$$\leq$38, the $\beta$-decay energy window is such that none of these excited states is populated, the limit of observation being at $\sim$10 parts per million. However, as $A$ increases, the $\beta$-decay energy of the parent grows and the level density in the daughter increases; so competing branches become greater in number and in strength. For [$^{42}$Sc]{}, the lightest emitter for which a non-superallowed $\beta$ branch has been observed, that branching ratio is 0.0059%; while for [$^{74}$Rb]{}, the heaviest well-measured case, there are a number of competing branches, which total to a 0.50% branching ratio.
There have been relatively few measurements of these weak non-superallowed branching ratios, since parts-per-million sensitivity is not easy to achieve. One example is the work of Hagberg and collaborators [@Ha94] at Chalk River, who investigated four emitters, [$^{38m}$K]{}, [$^{46}$V]{}, [$^{50}$Mn]{} and [$^{54}$Co]{}. The first was produced by an ($\alpha$,$n$) reaction, the other three by ($p$,$n$) reactions, in all cases on isotopically enriched targets. A helium-jet gas-transfer system was used to convey each activity to a low-background counting location where the activity-loaded NaCl aerosol clusters in the helium were deposited onto the aluminized Mylar tape of a fast tape-transfer system. After a short collection period, typically $\sim$0.5s, a paddle was inserted between the helium-jet nozzle and tape, and the tape moved the sample in sequence to two different detector stations, stopping at each; then the cycle was repeated until adequate statistics had been acquired. To achieve the required sensitivity, MBq-level sources were required for each cycle.
The first detector station consisted of two thin plastic scintillators located on either side of the tape, with an HPGe detector in close geometry behind one of them. The latter was passively shielded against the high flux of energetic positrons from the dominant ground-state branch. Gamma-ray signals from the HPGe detector were only recorded if they were in coincidence with $\beta$ signals from the opposite-side scintillator and in anti-coincidence with those from the same-side scintillator. This singled out true $\beta$-delayed $\gamma$ rays while eliminating bremsstrahlung radiation in the HPGe detector caused by $\beta$ particles backscattering from the opposite-side plastic.
At the second detector station the tape stopped in the center of a 4$\pi$ gas proportional counter (see Sec.\[beta\]) with nearly 100% efficiency for $\beta$ particles. The multiscaled data from this detector were used to determine the strength of the source in each cycle. With the total strength known, the branching ratio corresponding to any $\gamma$-ray peak observed in the HPGe spectrum could be obtained. The HPGe detector efficiency was calibrated with standard sources and there were also significant dead-time and other corrections to be applied so only $\sim$10% relative precision could be quoted on the result but, because the transitions were so weak in the first place, that was more than sufficient. Portions of the $\gamma$-ray spectra they recorded from [$^{38m}$K]{} and [$^{46}$V]{} are shown in Fig.\[fig:gspect\]. The arrows indicate where $\gamma$ rays from known excited $0^+$ states would appear if those states were populated by $\beta$ decay. An upper limit of 19 ppm was obtained for this possible non-superallowed branch from [$^{38m}$K]{}, and a value of 39(4) ppm was derived from the clearly observed peak in the case of [$^{46}$V]{}. Results were also obtained for [$^{50}$Mn]{} and [$^{54}$Co]{}.
![\[fig:gspect\]Portions of gated $\gamma$-ray spectra obtained following $\beta$ decays of [$^{38m}$K]{} and [$^{46}$V]{} [@Ha94]. The position of the possible $0^+_1$$\rightarrow$$2^+$ $\gamma$ ray is indicated with an arrow in both cases. The strong peak in the [$^{38m}$K]{} spectrum is the double-escape peak from the 2168-keV $\gamma$ ray from the $\beta$ decay of the [$^{38}$K]{} ground state. (Adapted from ref.[@Ha94].)](fig7.eps){width="12cm"}
The situation becomes much more complex for $T_Z$=0 parents with $A$$\geq$62. This is well illustrated by the work of Finlay [*et al.*]{} [@Fi08] at TRIUMF, who studied the $\beta$ decay of [$^{62}$Ga]{}. They identified 30 $\beta$-coincident $\gamma$ rays, which they attributed to non-superallowed $\beta$ transitions from [$^{62}$Ga]{} to 10 excited $0^+$ or $1^+$ states in its daughter [$^{62}$Zn]{}. To obtain this result they deposited [$^{62}$Ga]{} ions from the ISAC separator onto aluminized Mylar tape at the mutual centers of an array of 20 thin plastic scintillators with $\sim$80% efficiency for $\beta$ detection, and the “8$\pi$" $\gamma$-ray spectrometer, an array of 20 HPGe detectors operated in Compton-suppressed mode. Although data were recorded continuously, the beam was cycled on and off, with an implantation period lasting 30s sandwiched between two shorter periods: one before for background counting, and one after for decay counting. At the end of each cycle, a tape-transport system moved the collected sample to a shielded location, leaving a fresh portion of tape for the next cycle. More than 5000 cycles were recorded in all.
While this thorough experiment might seem to have quantified all possible decay branches, actually it cannot. For these higher $A$ values, hundreds or even thousands of $1^+$ states in the daughter can become accessible to $\beta^+$/EC decay. Although most of these transitions are undoubtedly very weak, their aggregate can be quite significant at the level of precision required in these measurements. This is the “Pandemonium" effect, which was first described in 1977 in a more general context [@Ha77] and, more recently, has been applied specifically to superallowed $\beta$ decay [@Ha02]. In the case of [$^{62}$Ga]{} decay, shell-model calculations predict that over a hundred $1^+$ states in [$^{62}$Zn]{} can be populated by $\beta$ decay [@Fi08]. With only 10 identified, Finlay [*et al.*]{} had to make a correction to their result to account for transitions that they could not observe individually. Their approach hinged on two low-lying $2^+$ states in [$^{62}$Zn]{}, which cannot be populated by allowed $\beta$ decay yet were seen to have more $\gamma$-ray intensity de-populating them than feeding them. The missing feeding could only be attributed to the presence of a large number of $\gamma$-ray transitions, each too weak to observe, from excited states fed by correspondingly weak Gamow-Teller $\beta$ transitions. After a further adjustment from a comparison with theory, they concluded that their detailed spectroscopy had only identified $\sim$94% of the non-superallowed $\beta$ intensity, and they corrected their result accordingly, arriving at a final superallowed branching ratio of 99.858(8)%.
### \[TZ-1\] $T_Z$=$-$1 parent decays
![\[fig:10C\] Left: Decay scheme of [$^{10}$C]{}, which is produced by the [$^{10}$B]{}($p$,$n$) reaction. Right: Main de-excitation route for the 2.154-MeV level in [$^{10}$B]{} populated by inelastic proton scattering on [$^{10}$B]{}. The level energies are given in MeV; the $\gamma$-ray energies are in keV.](fig8.eps){width="8cm"}
As described in the Introduction, the first $T_Z$=$-1$ parent decays to be studied were those of [$^{10}$C]{} and [$^{14}$O]{}. Each has its own unique problem that even today limits the precision with which its branching ratio can be measured. The decay scheme of [$^{10}$C]{} is shown on the left side of Fig.\[fig:10C\]. Since $\beta$ decay to the $3^+$ ground state is second forbidden and decay to the $1^+$ state at 2.154 MeV is energetically disfavoured, the superallowed branching ratio is simply given by the ratio of the number of $\gamma$ rays emitted at 1022 keV relative to the number at 718 keV. There are two problems with this: the superallowed branch is weak and the energy of its characteristic $\gamma$ ray is exactly twice that of the 511-keV annihilation radiation, which appears in abundance from the decay positrons. Precision requires high statistics together with confidence that the pile up of annihilation radiation has not contaminated the peak of interest. These two conditions tend to work against one another.
Savard [*et al.*]{} at Chalk River [@Sa95] dealt with these conflicting demands by using the array of 20 HPGe detectors that constituted the 8$\pi$ spectrometer. This yielded a twentyfold reduction in 511-511 pile-up compared with a single detector at the same total counting rate. The experiment itself comprised two interleaved measurements. One was a repeated cycle in which the [$^{10}$C]{} was first produced by the ($p$,$n$) reaction on a [$^{10}$B]{} target mounted at the center of the spectrometer; then the beam was interrupted and the $\beta$-delayed $\gamma$ rays from the decay were observed in singles mode. The second measurement was performed in beam with $\gamma$-$\gamma$ coincidences recorded from the deexcitation of the 2.154-MeV level in [$^{10}$B]{}, which was populated by the ($p$,$p^{\prime}$) reaction (see right side of Fig.\[fig:10C\]). The ratio of the number of $\gamma_{414}$-$\gamma_{718}$ coincidences to $\gamma_{414}$-$\gamma_{1022}$ coincidences in the second measurement determines the relative counting efficiencies for the 718- and 1022-keV $\gamma$ rays, which can then be used to determine the relative $\gamma$-ray intensities in the first measurement. In this way the superallowed branching ratio was determined to be 1.4625(25)%. Subsequently, a similar measurement was made with the Gammasphere detector array [@Fu99], which gave a slightly less precise, but nevertheless consistent result.
The decay of [$^{14}$O]{} has an even more challenging feature. The superallowed branch carries more than 99% of the decay strength and populates the analogue state at 2.31 MeV in [$^{14}$N]{}. However, its strongest competition comes from a 0.6% Gamow-Teller branch to the ground state, which emits no subsequent $\gamma$ ray to signal its appearance. The only way to determine the precise strength of this ground state $\beta$-decay branch is to measure the energy spectrum of all the emitted positrons and tease out the contribution of the ground state branch. The last time this measurement was made was in 1966 [@Si66] (though the analysis was updated more recently [@To05]). This measurement begs to be repeated.
There are, of course, other $T_Z$=$-1$ superallowed parents with $A$$\geq$$18$ but, until very recently, their branching ratios have defied $\pm$0.1% measurements since in all cases but one, [$^{22}$Mg]{}, not every $\beta$-decay branch feeds excited states that subsequently emit $\gamma$-rays. With the ground state – or a low-lying isomeric state – also populated, the only way to arrive at correct branching ratios is to measure the intensity of the $\gamma$-ray peaks relative to the total number of decays of the parent nucleus. A method to achieve this has now been developed at Texas A&M University by Hardy, Iacob and Park [@Pa14] using the same source-production and delivery system as illustrated in Fig,\[fig:TAMUexp\]. For these measurements, though, the gas proportional detector shown in that figure is replaced at the counting station by a thin plastic scintillator on one side of the tape and an HPGe detector on the other side, as shown on the left side of Fig.\[fig:BR\]. They record $\beta$ singles and $\beta$-$\gamma$ coincidences.
![\[fig:BR\] Left: Arrangement of the $\beta$ and $\gamma$-ray detectors, between which the source is placed by the tape transport system illustrated in Fig.\[fig:TAMUexp\]. Right: Spectrum of $\beta$-coincident $\gamma$ rays recorded from 61,000 cycles, each having a 1.54-s counting period [@Pa14].](fig9.eps){width="12cm"}
If the $\gamma$ ray de-exciting state $i$ in the daughter is denoted by $\gamma_i$, then the $\beta$-branching ratio, $R_i$ for the $\beta$ transition populating that state can be written: R\_i = k, where $N_{\beta\gamma_i}$ is the total number of $\beta$-$\gamma$ coincidences measured in the $\gamma_i$ peak, $N_{\beta}$ is the total number of $\beta$ singles, $\epsilon_{\gamma_i}$ is the HPGe efficiency for detecting $\gamma_i$, and $k$ is a small correction factor ([*i.e.*]{} $k$$\sim$1) that accounts for dead time and pile-up, coincident summing, and the small changes in $\beta$-detector efficiency for the different energy transitions participating in the decay. The equation highlights the importance of having a pure sample – so that $N_{\beta}$ can be relied upon – as well as having a precise absolute efficiency calibration for the $\gamma$-ray detector, and a reasonable knowledge of relative efficiencies in the beta detector. The key ingredient that the Texas A&M team has painstakingly developed is an HPGe detector whose absolute efficiency has been accurately determined (to $\pm$0.2% for 50-1400 keV $\gamma$ rays and to $\pm$0.4% up to 3500 keV) from source measurements and Monte Carlo calculations [@He03; @He04].
This method yields the branching ratios of all transitions except the one to the ground (or isomeric) state which has no subsequent $\gamma$ ray. However, the branching ratio of the latter transition can be obtained by subtracting the sum of the former branching ratios from 100%. In the case of the [$^{38}$Ca]{} decay, which is shown in Fig.\[fig:38decays\], the “missing" transition is the superallowed one, and the subtraction from 100% actually has a very salutary effect on the relative precision, reducing it by more than a factor of 3 (= 22.7/77.3). Very recently with this system, the branching ratio for the [$^{38}$Ca]{} superallowed transition has been determined to $\pm$0.2% [@Pa14]. This completes the information needed to obtain a precise $ft$ value for this transition and will add [$^{38}$Ca]{} to the current list of well known superallowed decays (see Fig.\[fig:chart\]), the first to be added in nearly a decade. There will likely be several more additions of this type in the next few years.
\[survey\] Survey of world data
-------------------------------
Many independent measurements contribute to the determination of superallowed $ft$ values, so for the past four decades we have periodically produced critical surveys of relevant world data available at the time of writing. All published measurements are carefully considered, with some being rejected but only if a specific fault has been identified. Of the surviving results, only those with uncertainties that are within a factor of 10 of the most precise measurement for each quantity are retained. They are then averaged by the same procedures as those adopted by the Particle Data Group [@PDG12]. In columns two, three and four of table \[Ftdata\] we present the results from the most recent survey [@Ha09]. In almost all cases, the tabulated values are averages of several – sometimes many – experimental measurements with comparable uncertainties.
For each superallowed transition, the three measured quantities, $Q_{EC}$, $t_{1/2}$ and $R$, together with a small correction ($\sim$0.1%) [@Ha09] to account for the contribution of electron capture, are combined to obtain an $ft$ value. These $ft$ values appear in column five of the table. The next step is to use (\[Ftconst\]) to obtain the corrected $\F t$ value for each transition. Before doing this though, we need to examine the theoretical correction terms used to account for radiative and isospin-symmetry-breaking effects.
\[s:theo\] Theoretical corrections
==================================
\[ss:radc\] Radiative corrections
---------------------------------
As described in the Introduction, the historical determination of the vector coupling constant $\GV$ in the semi-leptonic decays of [$^{14}$O]{} and [$^{10}$C]{} differed by a few percent from $\GF$, which was obtained from the purely leptonic decay of the muon – seemingly violating universality as espoused by Cabibbo. At the time though it was understood that radiative corrections could largely explain the discrepancy. Simply put, the lifetimes measured in the decays of [$^{14}$O]{} and [$^{10}$C]{} include both the bare $\beta$-decay process and the radiative process, in which the emitted electron releases a bremsstrahlung photon that is undetected. Since $\GV$ is determined from the bare $\beta$-decay process, the contribution from radiative effects needs to be computed and subtracted from the measured result. Calculations of this contribution, however, have to consider not just the bremsstrahlung process alone but also the loop diagram in which a photon is exchanged between charged particles. The key point is that the bremsstrahlung contribution diverges in the limit when the photon energy goes to zero; the loop graph likewise diverges but with the opposite sign, so the combination of the two remains finite in the low-energy limit. That this happens is a consequence of the renormalizability of quantum electrodynamics.
There was another problem for the calculations performed in the 1950s [@Be58; @KS59]: the loop graph also diverged in the limit of infinite photon energy, so some form of arbitrary cut-off had to be imposed. The difficulty at the time was what to choose for the cut-off: As the cut-off increased so did the discrepancy between $\GV$ and $\GF$. In this era the $\beta$-decay process was treated as a four-fermion contact interaction and the loop diagram was of triangular geometry.
During the 1960s, the Standard Model began to be formulated and the four-fermion contact interaction was replaced by one in which an intermediate vector boson – the $W$ boson – mediated between the leptons and hadrons in semi-leptonic decay. With the $W$ boson, the loop diagram became of rectangular geometry and the mass of the boson then provided a natural cut-off [@Si67]. The Standard Model also introduced a neutral $Z$ boson that mixed with the photon, so a further class of loop diagrams involving $Z$-boson exchange had to be included in the radiative-correction calculation. The main effect of this addition was to increase the effective cut-off in the loop diagram from the $W$-boson mass to the $Z$-boson mass [@Si74; @Si78].
It was realized all along that the purely leptonic muon decay was also subject to radiative corrections and that many of the potential contributions were identical in both pure- and semi-leptonic decays. Such identical contributions are called ‘universal’. Thus, in the Standard Model, $\GF$ came to be defined as the weak-interaction coupling constant, with the understanding that it included within it all universal radiative corrections. Thus, the radiative correction applied to the semi-leptonic decays only needed to include terms that were non-universal. A longer discussion of this point is given by Sirlin [@Si74].
So far, the radiative correction was only calculated to lowest order in the fine-structure constant $\alpha$. In the 1970s, Jaus and Rasche [@JR70; @Ja72] gave the first estimate of the order-$Z \alpha^2$ contributions, where $Z$ is the charge number of the daughter nucleus. This correction is defined as the contribution at this order not already contained in the product $F(Z,E) (1 + \delta_1)$, where $F(Z,E)$ is the Fermi function, $E$ the electon total energy and $\delta_1$ the order-$\alpha$ correction. Also in the 1970s, experimental results of superallowed beta decay began to emerge in the higher-$Z$ elements of [$^{42}$Sc]{}, [$^{46}$V]{}, [$^{50}$Mn]{} and [$^{54}$Co]{}. These results raised a problem [@TH84]: the high-$Z$ cases seemed to be failing the required Cabibbo universality. This prompted a reexamination of the order-$Z \alpha^2$ radiative correction by Sirlin and Zucchini [@SZ86; @Si87] and Jaus and Rasche [@JR87]. An error was discovered in the earlier work and Cabibbo universality was duly recovered.
In the loop graph there is another interesting wrinkle. If the nucleus is treated as a collection of nucleons, and if the weak and the electromagnetic interactions occurring in the radiative correction interact with the same nucleon, then the graph evaluation leads to a result that is proportional to the expectation value of the isospin ladder operator. Since the bare $\beta$-decay process also proceeds via the isospin ladder operator, the radiative correction simply scales with the bare $\beta$-decay value. This implies that the calculation is independent of nuclear structure, the result depending only on the electron energy and the charge $Z$ of the daughter nucleus. If, however, the weak and electromagnetic interactions occur with different nucleons, then the scaling property fails and a full nuclear-structure-dependent calculation is required. The first such calculation was provided by Jaus and Rasche [@JR90; @BBJR92] and later refined by Towner [@To92; @To94]. These results constitute the $\delta_{NS}$ correction, which appears in (\[Ftconst\]).
Another property of the loop graph is that one can usefully employ different approximations for the hadronic structure depending upon whether the photon energy is small or large. At low photon energy, it is sufficient to treat the nucleus as a collection of nucleons and use nucleon weak and electromagnetic form factors at the vertices. At high photon energy, the hadronic structure is essentially that of a soup of quarks. In this limit, the calculation becomes independent of the details of hadronic structure and leads to the term $\DRV$ in (\[Ftconst\]). Marciano and Sirlin [@MS06] have critically examined this separation into low- and high-energy contributions and the linkages between them. Their recommended value for $\DRV$ is in current use: = (2.361 0.038) % . \[DRV\] Lastly, a correction to order $\alpha^2$ was recently considered by Czarnecki, Marciano and Sirlin [@CMS04] and is now included in current computations.
All contributions to the radiative correction considered to date are collated into $\delta_R^{\prime}$, $\delta_{NS}$ and $\DRV$ as displayed in (\[Ftconst\]). The currently accepted values [@TH08] for $\delta_R^{\prime}$ and $\delta_{NS}$ are given in columns six and seven of table \[Ftdata\].
\[ss:deltc\] Isospin symmetry-breaking corrections
--------------------------------------------------
The Conserved Vector Current (CVC) hypothesis asserts that the vector coupling constant $\GV$ is not renormalized in the nuclear medium but is the same for all nuclei. This assertion can be tested by the measurement of superallowed $\beta$ decays, which cover a wide range of nuclei spanning from [$^{10}$C]{} to [$^{74}$Rb]{}. However, the CVC hypothesis is only valid in the isospin-symmetry limit, so if we are to determine $\GV$ from a wide range of nuclei then a correction has first to be applied that removes the effects of isospin-symmetry breaking. Isospin symmetry is naturally broken in nuclei because the protons are subject to the Coulomb interaction, which is not felt by the neutrons. Thus, the wave function of a proton in a given quantum state differs slightly from a neutron in the mirror nucleus in the same quantum state. This is reflected in the nuclear matrix element describing the beta-decay transition between mirror states. The isospin symmetry-breaking correction $\delta_C$ is defined by the relation |M\_F|\^2 = |M\_F\^0|\^2 (1 - \_C ) , \[MFdc\] where $M_F$ is the exact Fermi matrix element and $M_F^0$ its isospin-symmetry-limit value, being just the expectation value of the isospin ladder operator.
One of the earliest estimates of $\delta_C$ was provided by Damgaard [@Da69] in 1969. He suggested that the radial function of the proton in beta decay could be expanded in a complete set of neutron radial functions of the same angular momentum; the terms in the set differing in the number of radial nodes in the function. The expansion coefficients were determined in perturbation theory, with the Coulomb force being the perturbing interaction. With the basis states taken as harmonic oscillator functions, Damgaard obtained the expression \_C = (n + 1)(n + + ) . \[dam1\] If one adopts the relationships for the characteristic oscillator energy, $\hbar \omega = 41 A^{-1/3}$ MeV, and the nuclear radius, $r = 1.2 A^{1/3}$ fm, this expression becomes \_C = 0.2645 Z\^2 A\^[-2/3]{} (n+1) (n + +) , \[dam2\] which, for the relatively light nuclei we are interested in here, exhibits the general behaviour $\delta_C \propto Z^{4/3}$ with some shell structure superimposed through the choice of oscillator quantum numbers $n$ and $\ell$. In particular, a proton radial function with one radial node gets a factor of 2 enhancement in its $\delta_C$ value over one that has no radial nodes simply from the factor $(n+1)$ in (\[dam2\]). This enhancement is clearly evident in table \[Ftdata\] for the upper $pf$ shell, where there is a significant increase in experimental $ft$ value when going from [$^{54}$Co]{} ($ft$=3051s) to [$^{62}$Ga]{} ($ft$=3074s).
Beginning in 1973, Damgaard’s approach has been improved upon by Towner and Hardy [@To73; @TH08; @To77; @To02] in two significant ways. The oscillator basis states have been replaced by eigenfunctions of a Woods-Saxon potential and the extreme non-interacting single-particle model has been replaced by the interacting shell model. The perturbing Coulomb force is long range and influences a huge number of configurations in the shell model. Thus it has become expedient to divide $\delta_C$ into two components \_C = \_[C1]{} + \_[C2]{} , \[dc1dc2\] where $\delta_{C1}$ is a contribution from a finite-sized shell-model calculation, typically containing all configurations within one major shell, while $\delta_{C2}$ steps outside that model space to states that typically have a different number of radial nodes influencing the proton and neutron radial functions. For superallowed transitions between the $0^+$ states of interest here, Towner and Hardy [@TH08] find that the $\delta_{C1}$ contributions are much smaller than the $\delta_{C2}$ ones. Their results are shown in table \[t:tab2\], where they are labelled SM-WS. We note that these results are considered to be semi-phenomenological since a number of isospin-specific nuclear properties have been fitted in their derivation: $viz.$ the measured proton and neutron separation energies in the parents and daughters, respectively, were used in the calculation of $\delta_{C2}$; and each calculation of $\delta_{C1}$ was tuned to fit the $b$- and $c$-coefficients in the isobaric multiplet mass equation corresponding to the specific $T=1$ multiplet that included the parent and daughter state.
From 1985 to 1995, Ormand and Brown [@OB85; @OB89; @OB95] adopted the same general procedure as the one just described but used eigenfunctions of a Hartree-Fock mean field rather than a Woods-Saxon potential. Their results were systematically smaller than the SM-WS values and this difference was used to assess a systematic error in the analysis of superallowed beta-decay data. This systematic difference, however, was much reduced by a 2009 calculation of Hardy and Towner [@Ha09], in which the protocol for the Hartree-Fock calculation was altered to ensure that the Coulomb part of the proton mean field obtained from the Hartree-Fock calculation had the appropriate asymptotic form. The results from this calculation are labelled SM-HF in table \[t:tab2\].
Besides Damgaard’s model and the two shell-model approaches to $\delta_C$, which apply to the full range of measured superallowed transitions, there have been many other less-complete computations by various authors [@Sa86; @Li09; @Au09; @MS08; @MS09; @Gr10; @Sa12] using a diverse set of nuclear models. There are too many to discuss all these cases here. We have selected two of them, one based on the random phase approximation (RPA) and the other on density functional theories (DFT) to include in table \[t:tab2\]. In the former, the RPA work of Sagawa [*et al*]{} [@Sa86], improved upon by Liang [*et al*]{} [@Li09], treats the even-even nucleus of the parent-daughter pair as a core, and the analogue odd-odd nucleus as a particle-hole excitation built on that core. The particle-hole calculation is carried out in the charge-exchange version of the RPA. The more recent of the two works [@Li09] replaces zero-range interactions with finite-range meson-exchange potentials, and a relativistic rather than nonrelativistic treatment (RHF-RPA) is used. In a variation of this approach, density-dependent meson-exchange vertices were introduced in a Hartree (only) computation with nonlocal interactions (RH-RPA). Both these sets of results are listed in table \[t:tab2\].
Most recently, Satula [*et al*]{} [@Sa12] used an isospin- and angular-momentum-projected density functional theory (DFT). This method accounts for spontaneous symmetry breaking, configuration mixing and long-range Coulomb polarization effects. The results are also listed in table \[t:tab2\].
[@lllllll]{} &\
&\
Nucleus & Damgaard & DFT & RHF-RPA & RH-RPA & SM-HF & SM-WS\
[$^{10}$C]{} & 0.046 & 0.462(65) & 0.082 & 0.150 & 0.225(36) & 0.175(18)\
[$^{14}$O]{} & 0.111 & 0.480(48) & 0.114 & 0.197 & 0.310(36) & 0.330(25)\
[$^{22}$Mg]{} & 0.153 & 0.432(49) & & & 0.260(56) & 0.380(22)\
[$^{26m}$Al]{}& 0.182 & 0.307(62) & 0.139 & 0.198 & 0.440(51) & 0.310(18)\
[$^{34}$Cl]{} & 0.326 & & 0.234 & 0.307 & 0.695(56) & 0.650(46)\
[$^{34}$Ar]{} & 0.285 & 1.08(42) & 0.268 & 0.376 & 0.540(61) & 0.665(56)\
[$^{38m}$K]{} & 0.370 & & 0.278 & 0.371 & 0.745(63) & 0.655(59)\
[$^{42}$Sc]{} & 0.414 & 0.70(32) & 0.333 & 0.448 & 0.640(56) & 0.665(56)\
[$^{46}$V]{} & 0.524 & 0.375(96) & & & 0.600(63) & 0.620(63)\
[$^{50}$Mn]{} & 0.550 & 0.39(13) & & & 0.620(59) & 0.655(54)\
[$^{54}$Co]{} & 0.613 & 0.51(20) & 0.319 & 0.393 & 0.685(63) & 0.770(67)\
[$^{62}$Ga]{} & 1.34 & & & & 1.21(17) & 1.48(21)\
[$^{74}$Rb]{} & 1.42 & 0.90(22) & 1.088 & 1.258 & 1.42(17) & 1.63(31)\
$\chi^2 / n_d$ & 1.7 & 1.9 & 2.7 & 2.1 & 2.2 & 0.4\
The six sets of $\delta_C$ values in table \[t:tab2\] show a wide variation. Some yardstick is required to distinguish the quality of one set relative to another, so Towner and Hardy [@TH10] proposed such a test using the premise that the CVC hypothesis is valid. The requirement is that a calculated set of $\delta_C$ values should produce a statistically consistent set of $\F t$ values, the average of which we can write as $\overline{\F t}$. Then (\[Ftconst\]) can be written for each individual transition in the set as \_C = 1 + \_[NS]{} - .\[dctest\] \[deltaCtest\] For any set of corrections to be acceptable, the calculated value of $\delta_C$ for each superallowed transition must satisfy this equation, where $ft$ is the measured result for the transition and $\overline{\F t}$ has the same value for all of them. Thus, to test a set of $\delta_C$ values for $n$ superallowed transitions, one can treat $\overline{\F t}$ as a single adjustable parameter and use it to bring the $n$ results for the right-hand side of (\[dctest\]), which are based predominantly on experiment, into the best possible agreement with the corresponding $n$ calculated values of $\delta_C$ on the left-hand side of the equation. The normalized $\chi^2$, minimized by this process then provides a figure of merit for that set of calculations. The $\chi^2$ for each fit, expressed as $\chi^2/n_d$, where $n_d = n-1$ is the number of degrees of freedom, is given in the last row of table \[t:tab2\].
The most obvious outcome of this analysis is that the model, SM-WS, has a $\chi^2$ smaller by a factor of five than the other five cases cited. For this reason, the SM-WS $\delta_C$ values are the ones used in the calculation of the $\F t$ values, and it is these $\delta_C$ values that appear in the eighth column of table \[Ftdata\]. However, the other $\delta_C$ calculations can be used to help establish a systematic-uncertainty assignment on this analysis.
\[s:impact\] Impact on weak-interaction physics
===============================================
\[ss:Vud\] The value of $V_{ud}$
--------------------------------
The $\F t$ values obtained with the Woods-Saxon isospin-symmetry breaking correction SM-WS appear in the ninth column of table \[Ftdata\], where all are seen to be mutually consistent. Thus, we are justified in averaging the 13 entries in table \[Ftdata\] and using the result, $\overline{\F t} = 3072.08 \pm
0.79$ s, to determine the CKM matrix element $V_{ud}$. However, before doing so we must consider the impact that a different set of isospin-symmetry breaking corrections might have on the result. In the past [@Ha05], we compared the SM-WS calculations with the Hartree-Fock calculations of Ormand and Brown [@OB85; @OB89; @OB95], whose $\delta_C$ corrections covered all measured transitions and were consistently smaller than those obtained with Woods-Saxon functions. We considered that this provided a valid estimate of the systematic (theoretical) uncertainty, and so incorporated it into the overall result by deriving two average $\overline{\F t}$ values, one for each set of $\delta_C$ calculations, then adopting the average of the two and assigning a systematic uncertainty equal to half the spread between them.
This procedure was continued in the most recent 2009 survey [@Ha09], except that the Ormand and Brown calculations were replaced by the Towner-Hardy Hartree-Fock values, SM-HF in table \[t:tab2\]. With the SM-HF $\delta_C$ values, the average $\overline{\F t}$ value became $3071.55 \pm 0.89$ s but with a substantially increased chi-square of $\chi^2/\nu = 0.93$. This normalized chi-square is three times what was obtained in table \[Ftdata\] with Woods-Saxon corrections, which arguably could justify a rejection of the Hartree-Fock results outright. However, to be safe, we proceeded as in the 2005 survey and take the average of the SM-WS and SM-HF results, adding a systematic uncertainty equal to half the spread between the two results. Thus, the 2009 survey [@Ha09] arrived at & = & 3071.81 0.79\_[stat]{} 0.27\_[syst]{} [s]{}\
& = & 3071.81 0.83 [s]{} , \[Ftfinal\] where on the second line the two uncertainties have been added in quadrature.
With $\overline{\F t}$ thus obtained, the CKM matrix element $V_{ud}$ is derived from a rearrangement of (\[GVtoVud\]) and (\[Ftconst\]): |V\_[ud]{}|\^2 = = , \[Vud2\] where the overall weak-interaction coupling constant from muon decay is $\GF /(\hbar c)^3 = 1.1663787(6) \times 10^{-5}$ GeV$^{-2}$ from Ref.[@PDG12] and $\DRV$ is taken from (\[DRV\]). On applying our recommended value of $\overline{\F t}$ from (\[Ftfinal\]) we arrive at |V\_[ud]{}| = 0.97425 0.00022 , \[Vud\] a value with $0.02 \%$ precision.
This result is certainly the most precise current determination of $V_{ud}$, but superallowed $0^+ \rightarrow 0^+$ $\beta$ decay is not the only experimental approach to $V_{ud}$. Neutron decay, nuclear $T = 1/2$ mirror decays and pion beta decay have all been used for this purpose. For now, these other methods cannot compete with $0^+ \rightarrow 0^+$ decays for precision, although they yield statistically consistent results [@TH10]. We need not consider them farther in this context.
\[ss:unitarity\] Unitarity of the CKM matrix
--------------------------------------------
That the sum of the squares of the top-row elements should add to one is the most stringent test of the CKM matrix’s unitarity. Here $V_{ud}$ plays the dominant role with by far the largest magnitude and the smallest relative uncertainty; but, when it comes to the unitarity sum, $V_{us}$ and $V_{ud}$ contribute equally to the uncertainty because the terms themselves have such different magnitudes. For $V_{us}$ we will use the value reported at the recent CIPANP 2012 conference [@Mo12] (which updates the 2012 Particle Data Group value [@PDG12]): |V\_[us]{}| = 0.2256 0.0008 . \[Vus\] This value is an average of results obtained from kaon semi-leptonic decays, $K_{\ell 3}$, of both charged and neutral kaons, and from the purely leptonic decay of the kaon, $K_{\ell 2}$. Both methods rely on lattice QCD calculations for values of the hadronic form factors. Other determinations from hyperon decays and hadronic tau decay do not have the precision at the present time to challenge the results from kaon decays.
The third element of the top row of the CKM matrix, $V_{ub}$, is very small and hardly impacts on the unitarity test at all. Its value from the 2012 PDG compilation [@PDG12] is |V\_[ub]{}| = (4.15 0.49) 10\^[-3]{} . \[Vub\]
Combining the values given in (\[Vud\]), (\[Vus\]) and (\[Vub\]), the sum of the squares of the top-row elements of the CKM matrix becomes |V\_[ud]{}|\^2 + |V\_[us]{}|\^2 + |V\_[ub]{}|\^2 & = & 1.00008 0.00043\_[V\_[ud]{}]{} 0.00036\_[V\_[us]{}]{} ,\
& = & 1.00008 0.00056 , \[Usum\] a result that shows unitarity to be fully satisfied at the $0.06 \%$ level. In the first line of (\[Usum\]), the two errors shown are firstly from the uncertainty in $V_{ud}$ and secondly from $V_{us}$. They are combined in quadrature in the second line. Observe that kaon decays contribute slightly less than nuclear decays to the error budget.
No other row or column approaches this precision on a unitarity test. The first column comes closest, with $|V_{ud}|^2+|V_{cd}|^2+|V_{td}|^2 = 1.0021 \pm 0.0051$ [@CLS12], but this is a factor of ten less precise than the top-row sum. The corresponding sums for the second row and second column are $1.067 \pm 0.047$ and $1.065 \pm 0.046$ respectively [@CLS12], another order of magnitude less precise. Without question the top-row sum provides the most demanding test of CKM unitarity, $V_{ud}$ is its dominant contributor, and superallowed $\beta$ decay is effectively the sole experimental source for the value of $V_{ud}$.
The excellent experimental agreement with unitarity provides strong confirmation of the Standard-Model radiative corrections that enter both nuclear and kaon decays at the 3 to $4 \%$ level and, to a lesser extent, confirmation of isospin-symmetry breaking estimates, again in both nuclear and kaon decays. In addition it implies constraints on new physics beyond the Standard Model. New physics can enter in one of two ways: directly, via a new semi-leptonic interaction ([*e.g.*]{} scalar currents or right-hand currents), or indirectly, via loop-graph contributions to the radiative corrections ([*e.g.*]{} extra $Z$ bosons). Discussions of these issues can be found in [@TH10; @BM12; @NG13] as well as in other contributions to this volume.
\[s:future\] Future Prospects
=============================
Because the unitary CKM matrix is an essential pillar of the Standard Model, the uncertainty limits on the unitarity sum in (\[Usum\]) constrain the scope of whatever new physics may be anticipated to lie beyond that model. Consequently, there is ample motivation to search for improvements in both theory and experiment that can reduce the uncertainty on the sum, and potentially expose – or rule out – some classes of new physics.
Currently the uncertainty in the value of $V_{us}$ is dominated by the lattice QCD estimate for the form factor used to extract it from the experimental measurements on $K_{\ell 3}$ decays. It is predicted, though, that these lattice calculations will improve considerably over the next decade, with reasonable prospects of reducing the $V_{us}$ uncertainty by a factor of 2 [@Mo12; @Ha13]. With this improvement in sight, it is evident from (\[Usum\]) that any future improvements in $V_{ud}$ will thus have a significant impact on the overall uncertainty of the unitarity sum. Are such improvements foreseeable?
It is the uncertainties on the calculated correction terms, particularly $\DRV$, $\delta_C$ and $\delta_{NS}$, that have the greatest influence on the uncertainty of $V_{ud}$. The largest contributor, $\DRV$, may be open to some improvement, with a 30% reduction in uncertainty having been suggested as possible [@Ma13]. This is a challenge exclusively for theorists. However, improvements in the nuclear-structure-dependent corrections, $\delta_C$ and $\delta_{NS}$, can actually be achieved with the help of experiments. As described in Section \[ss:deltc\], these terms are subject to a test: They can be applied to the uncorrected experimental $ft$ values to obtain a set of $\F t$ values, which can then be evaluated for the consistency required by CVC – see (\[deltaCtest\]) and table \[t:tab2\]. The more precisely the $ft$ values have been measured, the more demanding this test can be; and if new superallowed transitions with larger predicted nuclear corrections can be measured, the test will be improved still more.
Of particular importance in this context are superallowed $0^+$$\rightarrow 0^+$decays of the $T_Z$=$-1$ nuclei $^{26}$Si, $^{34}$Ar, $^{38}$Ca and $^{42}$Ti. As noted in Section \[TZ-1\], these decays are more complex than the currently well studied superallowed transitions, which mostly have $T_Z$=0 parents. Only very recently have $T_Z$=$-1$ decays in this mass region become amenable to precise $ft$-value measurements [@Pa14] and it is anticipated that all four will be fully characterized within the next few years. These cases will be influential not only because they have relatively large nuclear corrections but also because each of the four transitions is mirror to another well known transition, $^{26m}$Al, $^{34}$Cl, $^{38m}$K and $^{42}$Sc, respectively. It turns out that the ratio of the $ft$-values for mirror superallowed transitions is extremely sensitive to the model used to calculate $\delta_C$ and $\delta_{NS}$ [@Pa14]. There is good reason to expect that these new cases will tighten the model constraints enough to shrink or even remove entirely the systematic uncertainty now applied to the structure-dependent correction terms.
The potential improvements in all the correction terms should act to reduce the uncertainty in $V_{ud}$ by about 25% and, together with the improvements expected in $V_{us}$, can be expected to reduce the uncertainty in the CKM unitarity sum – see (\[Usum\]) – from $\pm$0.00056 to $\pm$0.00037. This is likely to be the extent of possible improvements in the foreseeable future.
This work was supported by the U.S. Department of Energy under Grant No.DE-FG03-93ER40773 and by the Robert A. Welch Foundation under Grant No.A-1397.
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|
---
abstract: 'Given a combinatorial optimization problem $\Pi$ and an increasing finite sequence $c$ of natural numbers, we obtain a cardinality constrained version $\Pi_c$ of $\Pi$ by permitting only those feasible solutions of $\Pi$ whose cardinalities are members of $c$. We are interested in polyhedra associated with those problems, in particular in inequalities that cut off solutions of forbidden cardinality. Maurras [@Maurras77] and Camion and Maurras [@CM82] introduced a family of inequalities, that we call [*forbidden set inequalities*]{}, which can be used to cut off those solutions. However, these inequalities are in general not facet defining for the polyhedron associated with $\Pi_c$. In [@KS] it was shown how one can combine integer characterizations for cycle and path polytopes and a modified form of forbidden set inequalities to give facet defining integer representations for the cardinality restricted versions of these polytopes. Motivated by this work, we apply the same approach on the matroid polytope. It is well known that the so-called rank inequalities together with the nonnegativity constraints provide a complete linear description of the matroid polytope (see Edmonds [@Edmonds1971]). By essentially adding the forbidden set inequalities in an appropriate form, we obtain a complete linear description of the cardinality constrained matroid polytope which is the convex hull of the incidence vectors of those independent sets that have a feasible cardinality. Moreover, we show how the separation problem for the forbidden set inequalities can be reduced to that for the rank inequalities. We also give necessary and sufficient conditions for a forbidden set inequality to be facet defining.'
author:
- 'Jean François Maurras[^1] and Rüdiger Stephan[^2]'
title: On the cardinality constrained matroid polytope
---
Introduction
============
Let $E$ be a finite set and $\mathcal{I}$ a subset of the power set of $E$. The pair $(E,\mathcal{I})$ is called an *independence system* if (i) $\emptyset \in \mathcal{I}$ and (ii) whenever $I \in
\mathcal{I}$ then $J \in \mathcal{I}$ for all $J \subset I$. If $I
\subseteq E$ is in $\mathcal{I}$, then $I$ is called an *independent set*, otherwise it is called a *dependent set*. Dependent sets $\{e\}$ with $e\in E$ are called *loops*. For any set $F \subseteq E$, $B \subseteq F$ is called a *basis* of $F$ if $B \in \mathcal{I}$ and $B \cup \{e\}$ is dependent for all $e \in F \setminus B$. The *rank* of $F$ is defined by ${r_{\mathcal{I}}}(F):=
\max \{|B| : B \mbox{ basis of } F\}$. The set of all bases $B$ of $E$ is called a *basis system*. There are many different ways to characterize when an independence system is a matroid. For our purposes the following definition will be most comfortable. $(E,\mathcal{I})$ is called a *matroid*, and then it will be denoted by $M=(E,\mathcal{I})$, if $$\label{def-matroid}
\mbox{(iii)} \hspace{1cm}I,J \in \mathcal{I}, |I|<|J| \; \Rightarrow \;
\exists \,K \subseteq J
\setminus I : |I \cup K|=|J|, \; K \cup I \in \mathcal{I}.$$ Equivalent to (iii) is the requirement that for each $F \subseteq E$ all its bases have the same cardinality. Throughout the paper we deal only with loopless matroids. The results of the paper can be easily brought forward to matroids containing loops.
Let $M=(E,\mathcal{I})$ be a matroid. A set $F \subseteq E$ is said to be *closed* if ${r_{\mathcal{I}}}(F)<{r_{\mathcal{I}}}(F \cup \{e\})$ for all $e \in E
\setminus F$ and *inseparable* if there are no sets $F_1 \neq \emptyset
\neq F_2$ with $F_1 \,\dot{\cup} \,F_2=F$ such that ${r_{\mathcal{I}}}(F_1)+
{r_{\mathcal{I}}}(F_2) \leq {r_{\mathcal{I}}}(F)$.
Given any independence system $(E,\mathcal{I})$ and any weights $w_e
\in \mathbb{R}$ on the elements $e \in E$, the combinatorial optimization problem $\max w(I) , I \in \mathcal{I}$, where $w(I):=\sum_{e \in I} w_e$, is called the *maximum weight independent set problem*. The convex hull of the incidence vectors of the feasible solutions $I \in \mathcal{I}$ is called the *independent set polytope* and will be denoted by $P_{\mathcal{I}}(E)$. If $(E,\mathcal{I})$ is a matroid, then $P_{\mathcal{I}}(E)$ is also called the *matroid polytope*.
As it is well known, the maximum weight independent set problem on a matroid can be solved to optimality with the greedy algorithm. Moreover, the matroid polytope $P_{\mathcal{I}}(E)$ is determined by the rank inequalities and the nonnegativity constraints (see Edmonds [@Edmonds1971]), i.e., $P_{\mathcal{I}}(E)$ is the set of all points $x \in \mathbb{R}^E$ satisfying $$\label{matroid}
\begin{array}{rcll}
\sum\limits_{e \in F} x_e & \leq & {r_{\mathcal{I}}}(F) & \mbox{for all }
\emptyset \neq F \subseteq E,\\
x_e & \geq & 0 & \mbox{for all } e \in E.
\end{array}$$ The rank inequality associated with $F$ is facet defining for $P_{\mathcal{I}}(E)$ if and only if $F$ is closed and inseparable (see Edmonds [@Edmonds1971]).
Let $c=(c_1,\dots,c_m)$ be a finite sequence of integers with $0 \leq c_1 < c_2 < \ldots < c_m$. Then, the *cardinality constrained independent set polytope* ${P_{\mathcal{I}}^c(E)}$ is defined to be the convex hull of the incidence vectors of the independent sets $I \in \mathcal{I}$ with $|I|=c_p$ for some $p \in \{1,\dots,m\}$, that is, ${P_{\mathcal{I}}^c(E)}= \mbox{conv}
\{\chi^I \in \mathbb{R}^E : I \in \mathcal{I}, \, |I|=c_p \mbox{ for
some } p \in \{1,\dots,m\}\}$. If $(E,\mathcal{I})$ is a matroid, then ${P_{\mathcal{I}}^c(E)}$ is called the *cardinality constrained matroid polytope*. In the next section we will see that, if $(E,\mathcal{I})$ is a matroid, then the associated combinatorial optimization problem $\max
w^T x,\; x \in {P_{\mathcal{I}}^c(E)}$ can be solved in polynomial time. Since $c=(c_1,\dots,c_m)$ is linked to a cardinality constrained optimization problem, it is called a *cardinality sequence.* Throughout the paper we assume that $m \geq 2$.
The underlying basic problem of cardinality restrictions can be completely described in terms of linear inequalities. Given a finite set $B$ and a cardinality sequence $c=(c_1,\dots,c_m)$, the set ${\mbox{CHS}}^{c}(B):=\{F \subseteq B
: |F|=c_p \mbox{ for some } p\}$ is called a *cardinality homogenous set system*. The polytope associated with ${\mbox{CHS}}^c(B)$, namely the convex hull of the incidence vectors of elements of ${\mbox{CHS}}^c(B)$, is completely described by the *trivial inequalities* $0 \leq z_e \leq 1$, $e \in B$, the *cardinality bounds* $c_1 \leq \sum_{e \in B} z_e \leq c_m$, and the *forbidden set inequalities* $$\label{FS}
\begin{array}{l}
(c_{p+1} - |F|) \sum\limits_{e \in F} z_e\;-\;(|F| - c_p) \sum\limits_{e \in B
\setminus F} z_e \leq
c_p(c_{p+1}-|F|) \\
\hspace{0.5cm} \mbox{for all } F \subseteq B \mbox{ with } c_p < |F| <
c_{p+1} \mbox{ for some } p \in \{1,\dots,m-1\}.
\end{array}$$ This result is due to Maurras [@Maurras77] and Camion and Maurras [@CM82]. Grötschel [@Groetschel] rediscovered inequalities independently and proved the same result.
In [@KS] the authors investigated cardinality constrained cycle and path problems. They observed that inequalities define very low dimensional faces of the associated polyhedra. However, with a modified version of the cardinality forcing inequalities they were able to provide characterizations of the integer points of cardinality constrained cycle and path polytopes by facet defining inequalities.
In our context “modified version” means to replace $|F|$ by ${r_{\mathcal{I}}}(F)$. To this end, consider, for instance, the cardinality constrained graphic matroid. The independence system is the collection of all forests. Figure 1 illustrates the support graph of an ordinary forbidden set inequality. The set of bold edges, denoted by $F$, is of forbidden cardinality, since $9$ is not in the cardinality sequence $c=(3,5,12,14)$. The forbidden set inequality associated with $F$ has coefficients $3$ on the bold edges and $-4$ on the dashed edges. The right hand side is $15$. As it is not hard to see, none of the incidence vectors of forests of feasible cardinality satisfies the inequality at equality. However, if we fill up $F$ with further edges such that we obtain an edge set, say $F'$, of rank $9$, then the resulting inequality, which is illustrated in Figure 2, remains valid. Moreover, there are forests of cardinality $5$ and $12$ whose incidence vectors satisfy the resulting inequality at equality.
{width="45.00000%"} {width="45.00000%"}
With respect to $M=(E,\mathcal{I})$, ${P_{\mathcal{I}}^c(E)}= \mbox{conv}
\{\chi^I \in \mathbb{R}^E : I \in \mathcal{I} \cap {\mbox{CHS}}^c(E)\}$. By default, we assume that $c_m \leq {r_{\mathcal{I}}}(E)$. Our main result is that the system $$\begin{gathered}
\nonumber {\mbox{FS}}_F(x) := (c_{p+1}-{r_{\mathcal{I}}}(F)) x(F)-
({r_{\mathcal{I}}}(F)-c_p)x(E \setminus F) \leq
c_p(c_{p+1}-{r_{\mathcal{I}}}(F))\\
\hspace{0.5cm}\mbox{for all $F \subseteq E$ with $c_p <
{r_{\mathcal{I}}}(F) < c_{p+1}$ for some $p \in \{0,\dots,m-1\}$,} \label{eq_FS}\end{gathered}$$
$$\begin{aligned}
x(E) & \geq c_1,& \label{eq_lowerBound}\\
x(E) & \leq c_m, &\label{eq_upperBound}\\
x(F) & \leq {r_{\mathcal{I}}}(F) &\hspace{2cm}\mbox{for all } \emptyset \neq F
\subseteq E,\label{eq_rankInequalities}\\
x_e & \geq 0 &\hspace{2cm}\mbox{for all } e \in E\label{eq_nn}\end{aligned}$$ completely describes ${P_{\mathcal{I}}^c(E)}$. Here, for any $I \subseteq E$ we set $x(I):=\sum_{e \in I} x_e$. Of course, each $x \in {P_{\mathcal{I}}^c(E)}$ satisfies $c_1 \leq x(E) \leq c_m$. Inequalities are called [*rank induced forbidden set inequalities*]{}. The inequality ${\mbox{FS}}_F(x)
\leq c_p(c_{p+1}-{r_{\mathcal{I}}}(F))$ associated with $F$, where $c_p<{r_{\mathcal{I}}}(F)<c_{p+1}$, is valid as can be seen as follows. The incidence vector of any $I \in
\mathcal{I}$ of cardinality at most $c_p$ satisfies the inequality, since $|I \cap F| = {r_{\mathcal{I}}}(I \cap F) \leq c_p$: $$\begin{aligned}
(c_{p+1}-{r_{\mathcal{I}}}(F)) \chi^I(F)-
({r_{\mathcal{I}}}(F)-c_p)\chi^I(E \setminus F) & \leq &
(c_{p+1}-{r_{\mathcal{I}}}(F)) \chi^I(F)\\
& \leq & (c_{p+1}-{r_{\mathcal{I}}}(F)) c_p.\end{aligned}$$ The incidence vector of any $I \in
\mathcal{I}$ of cardinality at least $c_{p+1}$ satisfies also the inequality, since ${r_{\mathcal{I}}}(I \cap F) \leq {r_{\mathcal{I}}}(F)$ and thus ${r_{\mathcal{I}}}(I \cap (E \setminus F)) \geq
c_{p+1}-{r_{\mathcal{I}}}(F)$: $$\begin{aligned}
& & (c_{p+1}-{r_{\mathcal{I}}}(F)) \chi^I(F)-
({r_{\mathcal{I}}}(F)-c_p)\chi^I(E \setminus F)\\
& \leq &
(c_{p+1}-{r_{\mathcal{I}}}(F)) {r_{\mathcal{I}}}(F)
-({r_{\mathcal{I}}}(F)-c_p)\chi^I(E \setminus F)\\
& \leq &
(c_{p+1}-{r_{\mathcal{I}}}(F)) {r_{\mathcal{I}}}(F)
-({r_{\mathcal{I}}}(F)-c_p) (c_{p+1}- {r_{\mathcal{I}}}(F))\\
& = & c_p(c_{p+1}-{r_{\mathcal{I}}}(F)).\end{aligned}$$ However, it is not hard to see that some incidence vectors of independent sets $I$ with $c_p < |I| < c_{p+1}$ violate the inequality.
When $M=(E,\mathcal{I})$ is the trivial matroid, i.e., all $F
\subseteq E$ are independent sets, then $\mathcal{I} \cap {\mbox{CHS}}^c(E) =
{\mbox{CHS}}^c(E)$. Thus, cardinality constrained matroids are a generalization of cardinality homogenous set systems.
The paper is organized as follows. In Section 2 we prove that the system - provides a complete linear description of the cardinality constrained matroid polytope. Next, we will give sufficient conditions for the rank induced forbidden set inequalities to be facet defining. Finally, we show that the separation problem for the rank induced forbidden set inequalities can be reduced to that for the rank inequalities. This results in a polynomial time separation routine based on Cunningham’s separation algorithm for the rank inequalities. In Section 3 we briefly discuss some consequences for cardinality constrained combinatorial optimization problems and in particular for the intersection of two cardinality constrained matroid polytopes.
Polyhedral analysis of ${P_{\mathcal{I}}^c(E)}$
===============================================
Let $M=(E,\mathcal{I})$ be a matroid. As already mentioned, $P_{\mathcal{I}}(E)$ is determined by . For any natural number $k$, the independence system $M' :=(E,
\mathcal{I'})$ defined by $\mathcal{I'} := \{I \in \mathcal{I} : |I|
\leq k\}$ is again a matroid and is called the $k$-*truncation* of $M$. Therefore, the matroid polytope $P_{\mathcal{I'}}^c(E)$ associated with the $k$-truncation of $M$ is defined by system , where the rank inequalities are indexed with $\mathcal{I'}$ instead of $\mathcal{I}$. Following an argument of Gamble and Pulleyblank [@GP], the only set of the $k$-truncation which might be closed and inseparable with respect to the truncation, but not with respect to the original matroid $M$ is $E$ itself, and the rank inequality associated with $E$ is the cardinality bound $x(E) \leq k$. Hence, in context of the original matroid $M$, $P_{\mathcal{I'}}^c(E)$ is described by $$\label{<=k}
\begin{array}{rcll}
x(F) & \leq & {r_{\mathcal{I}}}(F) & \mbox{for all } \emptyset \neq F
\subseteq E,\\
x(E) & \leq & k,\\
x_e & \geq & 0 & \mbox{for all } e \in E.
\end{array}$$ Of course, the connection to cardinality constraints is obvious, since $P_{\mathcal{I'}}^c(E)= P_{\mathcal{I}}^{(0,\dots,k)}(E)$. The basis system of $M'$ is the set of all bases $B$ of $E$ (with respect to $M'$) and in case of ${r_{\mathcal{I}}}(E) \geq r_{\mathcal{I'}}(E)$ the bases are all of cardinality $k$. Assuming ${r_{\mathcal{I}}}(E) \geq
r_{\mathcal{I'}}(E)$, the associated polytope $$\mbox{conv}\{\chi^B \in
\mathbb{R}^E : B \mbox{ basis of $E$ with respect to $M'$}\}$$ is determined by $$\label{=k}
\begin{array}{rcll}
x(F) & \leq & {r_{\mathcal{I}}}(F) & \mbox{for all } \emptyset \neq F
\subseteq E,\\
x(E) & = & k,\\
x_e & \geq & 0 & \mbox{for all } e \in E.
\end{array}$$ On a basis system of a matroid one can optimize in polynomial time by application of the greedy algorithm. Thus, for each member $c_p$ of a cardinality sequence $c=(c_1,\dots,c_m)$ an optimal solution $I^p$ of the linear optimization problem $\max w(I),\; I \in \mathcal{I}, |I|=c_p$ can be found in polynomial time. The best of the solutions $I^p, p=1,\dots,m$ with respect to the linear objective $w$ is then the optimal solution of $\max w(I),\; I \in \mathcal{I} \cap {\mbox{CHS}}^c(E)$. Since $0 \leq c_1< \dots < c_m \leq {r_{\mathcal{I}}}(E) \leq |E|$ and thus $m \leq
|E|$, it can be found by at most $|E|+1$ applications of the greedy algorithm.
These preliminary remarks are sufficient to present our main theorem. In the sequel, we denote the rank function by $r$ instead of $r_{I}$. Given a valid inequality $ax \leq a_0$ with $a \in \mathbb{R}^E$, $F \subseteq E$ is said to be *tight* if $a
\chi^F = a_0$. A valid inequality $ax \leq a_0$ is *dominated* by another valid inequality $bx \leq b_0$, if $\{x \in {P_{\mathcal{I}}^c(E)}: ax = a_0\}
\subseteq \{x \in {P_{\mathcal{I}}^c(E)}: bx = b_0\}$. It is said to be *strictly dominated* by $bx \leq b_0$, if $\{x \in {P_{\mathcal{I}}^c(E)}: ax = a_0\}
\subsetneq \{x \in {P_{\mathcal{I}}^c(E)}: bx = b_0\}$.
A complete linear description
-----------------------------
The cardinality constrained matroid polytope ${P_{\mathcal{I}}^c(E)}$ is completely described by system -.
Since all inequalities of system - are valid, $P_{\mathcal{I}}^c(M)$ is contained in the polyhedron defined by -. To show the converse, we consider any valid inequality $bx \leq b_0$ for $P_{\mathcal{I}}^c(M)$ and associate with the inequality the following subsets of $E$: $$\begin{aligned}
P &:=& \{e \in E : b_e > 0\},\\
Z &:=& \{e \in E : b_e = 0\},\\
N &:=& \{e \in E : b_e < 0\}.\\\end{aligned}$$ We will show by case by case enumeration that the inequality $bx \leq
b_0$ is dominated by some inequality of the system -. By definition, $E = P \dot{\cup}
Z \dot{\cup} N$, and hence, if $P=Z=N=\emptyset$, then $E=\emptyset$, and it is nothing to show. By a scaling argument we may assume that either $b_0=1$, $b_0=0$, or $b_0=-1$.
1. $b_0=-1$.
1. $c_1=0$. Then $0 \in {P_{\mathcal{I}}^c(E)}$, and hence $0=b \cdot 0 \leq -1$, a contradiction.
2. $c_1>0$.
1. $P=Z=\emptyset, \,N \neq \emptyset$. Assume that there is some tight $I \in \mathcal{I}$ with $|I|=c_p$, $p \geq 2$. Then, for any $J \subset I$ with $|J|=c_1$ holds: $\chi^J \in {P_{\mathcal{I}}^c(E)}$ and $b
\chi^J > b \chi^I = -1$, a contradiction. Therefore, if any $I \in
\mathcal{I} \cap {\mbox{CHS}}^c(E)$ is tight, then $|I|=c_1$. Thus, $bx
\leq -1$ is dominated by the cardinality bound $x(E) \geq c_1$.
2. $P \cup Z \neq \emptyset, \,N = \emptyset$. Then, $by \geq 0$ for all $y \in {P_{\mathcal{I}}^c(E)}$, a contradiction.
3. $P \cup Z \neq \emptyset, \,N \neq \emptyset$. If $c_1 \leq r(P \cup Z)$, then there is some independent set $I
\subseteq P \cup Z$ of cardinality $c_1$, and hence, $b \chi^I \geq
0$, a contradiction. Thus, $c_1 > r(P \cup Z)$. Assume, for the sake of contradiction, that there is some tight independent set $J$ of cardinality $c_p$ with $p \geq 2$. If $J \subseteq N$, then the incidence vector of any $K \subset J$ with $|K|=c_1$ violates $bx
\leq -1$. Hence, $J \cap (P \cup Z) \neq \emptyset$. On the other hand, $J \cap N
\neq \emptyset$ due to $c_p > c_1 > r(P \cup Z)$. However, by removing any $(c_p-c_1)$ elements in $N \cap J$, we obtain some independent set $K$ of cardinality $c_1$ whose incidence vector violates the inequality $bx \leq -1$, a contradiction. Therefore, if any $T
\in \mathcal{I} \cap {\mbox{CHS}}^c(E)$ is tight, then $|T|=c_1$. Thus, $bx
\leq -1$ is dominated by the bound $x(E) \geq c_1$.
2. $b_0=0$.
1. $P \cup Z \neq \emptyset, \, N= \emptyset$. Then, either $bx
\leq 0$ is not valid or $b=0$.
2. $P=\emptyset, \, Z \cup N \neq \emptyset$. Then, $bx \leq 0$ is dominated by the nonnegativity constraints $x_e \geq 0$ for $e \in
N$ or $b=0$.
3. $P \neq \emptyset, \, N \neq \emptyset$.
1. $c_1 >0$. If $c_1 \leq r(P \cup Z)$, then there is some independent set $I \subseteq P \cup Z$ with $I \cap P \neq
\emptyset$ of cardinality $c_1$, and hence, $b \chi^I > 0$, a contradiction. Thus, $c_1 > r(P \cup Z)$. Assume, for the sake of contradiction, that there is some tight independent set $J$ of cardinality $c_p$ with $p \geq 2$. Since $c_p>c_1>r(P \cup Z)$ and $J$ is tight, $J \cap (P \cup Z) \neq \emptyset \neq J \cap N$. From here, the proof for this case can be finished as the proof for the case (1.2.3) with $b_0=0$ instead of $b_0=-1$ in order to show that $bx \leq 0$ is dominated by the cardinality bound $x(E) \geq c_1$.
2. $c_1 =0$. As in case (2.3.1), it follows immediately that $c_2 >r(P \cup Z)$, and if $I \in \mathcal{I} \cap {\mbox{CHS}}^c(E)$ is tight, then $|I|=c_1=0$, that is, $I=\emptyset$, or $|I|=c_2$. Moreover, if $I \in \mathcal{I}$ with $|I|=c_2$ is tight, then follows $|I \cap (P
\cup Z)|= r(P \cup Z)$. Hence, $bx \leq b_0$ is dominated by the rank induced forbidden set inequality ${\mbox{FS}}_F(x) \leq 0$ with $F=P \cup Z$.
3. $b_0=1$.
1. $P= \emptyset, \, Z \cup N \neq \emptyset$. Then, $b \leq
0$, and hence $bx \leq 1$ is dominated by any nonnegativity constraint $x_e \geq 0$, $e \in
E$.
2. $P \cup Z \neq \emptyset, \, N =
\emptyset$. Assume that there is some $I \in \mathcal{I}, I \notin {\mbox{CHS}}^c(E)$ with $|I| < c_m$ that violates $bx \leq 1$. Then, of course, all independent sets $J \supset I$ violate $bx \leq 1$, in particular, those $J$ with $|J|=c_m$, a contradiction. Hence, $bx \leq 1$ is not only a valid inequality for ${P_{\mathcal{I}}^c(E)}$ but also for $P_{\mathcal{I}}^{(0,1,\ldots, c_m)}(E)$, that is, $bx \leq 1$ is dominated by some inequality of the system with $k=c_m$.
3. $P \neq \emptyset, \, N \neq \emptyset$. Let $p \in
\{1,\dots,m\}$ be minimal such that there is a tight independent set $I^*$ of cardinality $c_p$. Of course, $c_p>0$, because otherwise $I^*$ could not be tight. If $p=m$, then $bx \leq 1$ is dominated by the cardinality bound $x(E) \leq c_m$, because then all tight $J \in
\mathcal{I} \cap {\mbox{CHS}}^c(E)$ have to be of cardinality $c_p=c_m$. So, let $0< c_p<c_m$. We distinguish 2 subcases.
1. $c_p \geq r(P \cup Z)$. Suppose, for the sake of contradiction, that there is some tight independent set $I$ of cardinality $c_p$ such that $|I \cap
(P \cup Z)| < r(P \cup Z)$. Then, $I \cap (P \cup Z)$ can be completed to a basis $B$ of $P \cup Z$, and since $|B|
\leq |I|$, there is some $K \subseteq I \setminus B$ such that $I' := B \cup K \in \mathcal{I}$ and $|I'|=|I|$. $K$ is maybe the empty set. Anyway, by construction, $I'$ is of cardinality $c_p$ and violates the inequality $bx \leq
1$. Thus, $|I \cap (P \cup Z)| = r(P \cup Z)$. For the same reason, any tight $J \in \mathcal{I} \cap {\mbox{CHS}}^c(E)$ satisfies $|J \cap (P \cup Z)| = r(P \cup Z)$, and since $p$ is minimal, $|J| \geq c_p$. Now, with similar arguments as in case (1.2.3) one can show that if $T \in \mathcal{I} \cap {\mbox{CHS}}^c(E)$ is tight, then $|T|=c_p$. Thus, $c_p=c_1 >0$ and $bx \leq 1$ is dominated by the cardinality bound $x(E) \geq c_1$.
2. $c_p < r(P \cup Z)$. Following the argumentation line in (3.3.1), we see that $I \subseteq P \cup Z$ and $|I
\cap P|$ has to be maximal for any tight independent set $I$ of cardinality $c_p$. Assume that $c_{p+1} \leq r(P \cup
Z)$. Then, from any tight independent set $I$ with $|I|=c_p$ we can construct a tight independent set $J$ with $|J|=c_{p+1}$ by adding some elements $e \in Z$. However, it is not hard to see that there is no tight $K \in \mathcal{I} \cap {\mbox{CHS}}^c(E)$ that contains some $e \in N$. Thus, when $c_{p+1} \leq r(P \cup Z)$, $bx \leq 1$ is dominated by the nonnegativity constraints $y_e
\geq 0$, $e \in N$. Therefore, $c_{p+1} > r(P \cup Z)$. The following is now immediate: If $I \in \mathcal{I} \cap
{\mbox{CHS}}^c(E)$ is tight, then $|I|=c_p$ or $|I|=c_{p+1}$; if $|I|=c_p$, then $I \subset P \cup Z$, and if $|I|=c_{p+1}$, then $|I \cap (P \cup Z)|= r(P \cup Z)$ and $c_{p+1} > r(P \cup
Z)$. Thus, $bx \leq 1$ is dominated by the rank induced forbidden set inequality ${\mbox{FS}}_{P \cup Z}(x) \leq c_p(c_{p+1}-r(P \cup
Z))$.
Facets {#Sec:Facets}
------
We first study the facial structure of a single cardinality constrained matroid polytope $P_{\mathcal{I}}^{(k)}(E)$. All points of $P_{\mathcal{I}}^{(k)}(E)$ satisfy the equation $x(E)=k$, and hence, any inequality $x(F) \leq r(F)$ is equivalent to the inequality $x(E \setminus F) \geq k - r(F)$. Motivated by this observation, we introduce the following definitions. For any $F \subseteq E$, the number $r^k(F) := k - r(E \setminus F)$ is called the $k$-*rank* of $F$. Due to the submodularity of $r$ we have $r^k(F_1) + r^k(F_2) \leq r^k (F)$ for all $F_1,F_2$ with $F = F_1 \dot{\cup} F_2$, and $F$ is said to be $k$-*separable* if equality holds for some $F_1 \neq \emptyset \neq F_2$, otherwise $k$-*inseparable*. Due to the equation $x(E)=k$, $\dim P_{\mathcal{I}}^{(k)}(E) \leq |E|-1$, and in fact, in the most cases we have equality. However, if $\dim P_{\mathcal{I}}^{(k)}(E) < |E|-1$, then at least one rank inequality $x(F) \leq r(F)$ with $\emptyset \neq F \subsetneq E$ is an implicit equation. As is easily seen, this implies that an inequality $x(F') \leq r(F')$ (or $x(F') \geq r^k(F')$) does not necessarily induce a facet of $P_{\mathcal{I}}^{(k)}(E)$, although $F$ is inseparable ($k$-inseparable). To avoid the challenges involved, we only characterize the polytopes $P_{\mathcal{I}}^{(k)}(E)$ of dimension $|E|-1$.
\[L:facets1\] Let $M=(E,\mathcal{I})$ be a matroid and for any $k \in \mathbb{N}$, $0<k<r(E)$, $M_k=(E,\mathcal{I}_k)$ the $k$-truncation of $M$ with rank function $r_k$. Then, $E$ is inseparable with respect to $r_k$.
Let $E = F_1 \dot{\cup} F_2$ with $F_1 \neq \emptyset \neq F_2$ be any partition of $E$. We have to show that $r_k(F_1) + r_k(F_2) > r_k(E)$. By definition, $r_k(E)=k$. First, let $r(F_i) \leq k$ for $i=1,2$. Then, $r_k(F_i) = r(F_i)$ and consequently, $r_k(F_1) + r_k(F_2) = r(F_1)+r(F_2) \geq r(E)>k$ due to the submodularity of $r$. Next, let w.l.o.g. $r(F_1)>k$. Then, $r_k(F_1) = k$ and, since $F_2 \neq \emptyset$, $r_k(F_2)>0$. Thus, $r_k(F_1) + r_k(F_2)= k + r_k(F_2) >k$.
\[L:facets2\] Let $M=(E,\mathcal{I})$ be a matroid, $M_k=(E,\mathcal{I}_k)$ its $k$-truncation with rank function $r_k$, $\emptyset \neq F \subseteq E$, and $\bar{F} = E \setminus F$ be closed with $r(\bar{F})<k<r(E)$. Then, $F$ is $k$-inseparable with respect to $r_k$.
$r(\bar{F})<k$ implies $r_k(\bar{F})=r(\bar{F})$, and since beyond it $\bar{F}$ is closed with respect to $r$, it is also closed with respect to $r_k$. Let $F = F_1 \dot{\cup} F_2$ be a proper partition of $F$. We have to show that $r^k_k(F_1)+r^k_k(F_2) < r^k_k(F)$. First, suppose that $I \in \mathcal{I}$ with $|I|=k$ and $|I \cap \bar{F}|= r_k(\bar{F})$ implies $I \cap F_1 = \emptyset$ or $I \cap F_2 = \emptyset$. Since $\bar{F}$ is closed with respect to $r_k$, it follows that $r_k^k(F_1)=r^k_k(F_2)=0$, while $r^k_k(F)= k - r_k(\bar{F}) >0$. So assume that there is some independent set $I'$ of cardinality $k$ such that $|I' \cap \bar{F}|=r_k(\bar{F})$ and $I' \cap F_i \neq \emptyset$ for $i=1,2$. Since $k<r(E)$, there is some element $e$ such that $I := I' \cup \{e\}$ is independent with respect to $r$. Set $I_1 := I \setminus \{f_1\}$ and $I_2 := I \setminus \{f_2\}$ for $f_1 \in I \cap F_1, f_2 \in I \cap F_2$. Then, $r^k_k(F_1) \leq |I_1 \cap F_1|$ and $r^k_k(F_2) \leq |I_2 \cap F_2|$. Hence, $r^k_k(F_1)+r^k_k(F_2) \leq |I_1 \cap F_1|+|I_2 \cap F_2| < |I_1 \cap F_1|+|I_1 \cap F_2| = |I_1 \cap F| = r^k_k(F)$.
\[L:facets3\] Let $M=(E,\mathcal{I})$ be a matroid, $\emptyset \neq F \subseteq E$, and $A$ the matrix whose rows are the incidence vectors of $I \in \mathcal{I}$ with $|I|=k$ that satisfy the inequality $x(F) \geq r^k(F)$ at equality. Moreover, denote by $A_F$ the submatrix of $A$ restricted to $F$. Then, rank$(A_F) = |F|$ if and only if $r^k(F) \geq 1$, $\bar{F} := E \setminus F$ is closed, and (i) $F$ is $k$-inseparable or (ii) $k<r(E)$.
*Necessity.* The inequality $x(F) \geq r^k(F)$ is valid for $P_{\mathcal{I}}^{(k)}(E)$. As is easily seen, if $r^k(F) \leq 0$, then $\mbox{rank}(A_F) < |F|$. Next, assume that $\bar{F}$ is not closed. Then, there is some $e \in F$ such that $r(\bar{F} \cup \{e\})=r(\bar{F})$ which is equivalent to $r^k(F)=r^k(F \setminus \{e\})$. Thus, $x(F) \geq r^k(F)$ is the sum of the inequalities $x(F \setminus \{e\}) \geq r^k(F \setminus \{e\})$ and $x_e \geq 0$. This implies $\chi^I_e=0$ for all incidence vectors of independent sets $I$ with $|I|=k$ satisfying $x(F) \geq r^k(F)$ at equality. Again, it follows $\mbox{rank}(A_F) < |F|$. Finally, suppose that neither $k<r(E)$ nor $F$ is $k$-inseparable. Then, $k=r(E)$ and $F$ is $r(E)$-separable. Thus, the inequality $x(F) \geq r^{r(E)}(F)$ is the sum of the valid inequalities $x(F_1) \geq r^{r(E)}(F_1)$ and $x(F_2) \geq r^{r(E)}(F_2)$ for some $F_1 \neq \emptyset \neq F_2$ with $F = F_1 \dot{\cup} F_2$. Setting $\lambda := r^{r(E)}(F_2) \chi_F^{F_1}-r^{r(E)}(F_1) \chi_F^{F_2}$, we see that for any $|F| \times |F|$ submatrix $\tilde{A}_F$ of $A_F$ we have $\tilde{A}_F \lambda = 0$, that is, the columns of $\tilde{A}_F$ are linearly dependent which implies $\mbox{rank}(A_F) < |F|$.
*Suffiency.* First, let $k=r(E)$. Suppose $\mbox{rank}(A_F)<|F|$. Then, $A_F \lambda = 0$ for some $\lambda \in \mathbb{R}^F, \, \lambda \neq 0$. Since $\bar{F}$ is closed and $r^k(F) \geq 1$ (that is, $r(\bar{F})<k$), for each $e \in F$ there is an independent set $I$ with $|I|=k$ that contains $e$ and whose incidence vector satisfies $x(F) \geq r^k(F)$ at equality. Thus, $A_F$ does not contain a zero-column. Moreover, $A_F \geq 0$, and hence, $F_1 := \{e \in F : \lambda_e > 0\}$ and $F_2 := \{e
\in F : \lambda_e \leq 0\}$ defines a proper partition of $F$. Let $J \subseteq \bar{F}$ with $|J|=r(\bar{F})$ be an independent set. For $i=1,2$, let $B_i \subseteq F$ be an independent set such that $J \cup B_i$ is a basis of $E$ and $J \cup (B_i \cap F_i)$ is a basis of $\bar{F} \cup F_i$. Set $S_i :=B_i \cap F_i$ and $T_i := B_i \setminus S_i$ ($i=1,2$). By construction, $T_1 \subseteq F_2$ and $T_2 \subseteq
F_1$. By matroid axiom (iii), to $J \cup S_1$ there is some $U_1 \subseteq J \cup B_2$ such that $K := J \cup S_1 \cup U_1$ is a basis of $F$. Clearly, $U_1 \subseteq (B_2 \cap F_2) = S_2$. Since the incidence vectors of $J \cup B_1$ and $K$ are rows of $A$, it follows immediately $\lambda(T_1) = \lambda(U_1)$. With an analogous construction one can show that there is some $U_2 \subseteq S_1$ such that $\lambda(U_2)=\lambda(T_2)$. It follows, $\lambda(T_2) = -\lambda(S_2) \geq -
\lambda(U_1) = -\lambda(T_1) = \lambda(S_1) \geq
\lambda(U_2) = \lambda(T_2)$. Thus, between all terms we have equality implying $\lambda(S_1) = \lambda(U_2)$. Moreover, since $U_2
\subseteq S_1$ and $\lambda_e >0$ for all $e \in S_1$, it follows $S_1
= U_2$. Hence, $K = J \cup S_1 \cup S_2$. This, in turn, implies that $F$ is $k$-separable, a contradiction.
It remains to show that the statement is true if $k<r(E)$. Let $M_k = (E,\mathcal{I}_k)$ be the $k$-truncation of $M$ with rank function $r_k$. By hypothesis, all conditions of Lemma \[L:facets2\] hold. Hence, $F$ is $k$-inseparable with respect to $r_k$. Thus, all conditions of the lemma hold for $r_k$ instead of $r$ and hence, $\mbox{rank}(A_F) = |F|$.
\[T:facets4\] Let $M=(E,\mathcal{I})$ be a matroid and $k \in \mathbb{N}$, $0 < k \leq r(E)$.
1. $P_{\mathcal{I}}^{(k)}(E)$ has dimension $|E|-1$ if and only if $E$ is inseparable or $k < r(E)$.
2. Let $\dim P_{\mathcal{I}}^{(k)}(E) = |E|-1$ and $\emptyset \neq F \subsetneq E$. The inequality $x(F) \leq {r}(F)$ defines a facet of $P_{\mathcal{I}}^{(k)}(E)$ if and only if $F$ is closed and inseparable, $r(F) < k$, and (i) $\bar{F} := E \setminus F$ is $k$-inseparable or (ii) $k < r(E)$.
\(a) First, let $k=r(E)$. For any $\emptyset \neq F \subseteq E$, the rank inequality $x(F) \leq r(F)$ defines a facet of $P_{\mathcal{I}}(E)$ if and only if $F$ is closed and inseparable. Consequently, the polytope $P_{\mathcal{I}}^{(r(E))}(E)$, which is a face of $P_{\mathcal{I}}(E)$, has dimension $|E|-1$ if and only if $E$ is inseparable. Next, let $0 < k < r(E)$. By Lemma \[L:facets1\], $E$ is inseparable with respect to the rank function $r_k$ of the $k$-truncation $M_k=(E,\mathcal{I}_k)$. Consequently, $x(E) \leq r_k(E)=k$ defines a facet of $P_{\mathcal{I}_k}(E)$ and hence, $\dim P_{\mathcal{I}}^{(k)}(E) = |E|-1$.
\(b) Clearly, $x(F) \leq r(F)$ does not induce a facet of $P_{\mathcal{I}}^{(k)}(E)$ if $F$ is separable or not closed, since $\dim P_{\mathcal{I}}^{(k)}(E) = |E|-1$, and hence, any inequality that is not facet defining for $P_{\mathcal{I}}(E)$ is also not facet defining for $P_{\mathcal{I}}^{(k)}(E)$. Next, if $r(F) \geq k$, then holds obviously $x(F) \leq x(E)=k \leq r(F)$, that is, either $F$ is not closed, $x(F) \leq r(F)$ is an implicit equation, or the face induced by $x(F) \leq r(F)$ is the emptyset. Finally, assume that $F$ is closed but neither (i) nor (ii) holds. Then, $k=r(E)$ and $\bar{F}$ is $k$-separable. Thus, there are nonempty subsets $\bar{F}_1, \bar{F}_2$ of $\bar{F}$ with $\bar{F} = \bar{F}_1 \dot{\cup} \bar{F}_2$ such that $r^k(\bar{F})= r^k(\bar{F}_1)+ r^k(\bar{F}_2)$. Now, the inequality $x(\bar{F}) \geq r^{k}(\bar{F})$, which is equivalent to $x(F) \leq r(F)$, is the sum of the valid inequalities $x(\bar{F}_i) \geq r^k(\bar{F}_i)$, $i=1,2$, both not being implicit equations.
To show the converse, let $F$ satisfy all conditions mentioned in Theorem \[T:facets4\] (b). The restriction of $M=(E,\mathcal{I})$ to $F$ is again a matroid. Denote it by $M'=(F, \mathcal{I}')$ and its rank function by $r'$. $F$ remains inseparable with respect to $r'$. Thus, the restriction of $x(F) \leq r(F)$ to $F$, denoted by $x_F(F) \leq r(F)=r'(F)$, induces a facet of $P_{\mathcal{I}'}(F)$. A set of affinely independent vectors whose sum of components is equal to some $\ell$, is also linearly independent. Thus, there are $|F|$ linearly independent vectors $\chi^{I'_j}$ of independent sets $I'_j \in \mathcal{I'}$ of cardinality $r'(F)$ ($j=1,\ldots,|F|$). The sets $I'_j$ are also independent sets with respect to $\mathcal{I}$. Due to the matroid axiom (iii), $P:=I'_1$ can be completed to an independent set $I_1$ of cardinality $k$. Since $P \subseteq F$ and $|P|=r(F)$, $Q := I_1 \setminus P \subseteq \bar{F}$. Now, $I'_j, I_1 \in \mathcal{I}$, $I'_j \subseteq F$, and $r(F) = |I'_j| < |I_1|=k$. Hence, $I_j := I'_j \cup Q \in \mathcal{I}$ for all $j$. Consequently, we have $|F|$ linearly independent vectors $\chi^{I_j} \in P_{\mathcal{I}}^{(k)}(E)$ satisfying $x(F) \leq r(F)$ at equality.
Next, let $A$ be the matrix whose rows are the incidence vectors of tight independent sets and $A_{\bar{F}}$ its restriction to $\bar{F}$. By Lemma \[L:facets3\], $A_{\bar{F}}$ contains a $|\bar{F}| \times |\bar{F}|$ submatrix $B$ of full rank. By construction, each row $B_i$ of $B$ is an incidence vector of an independent set $J'_i \subseteq \bar{F}$ with $|J'_i|=r^k(\bar{F})$. W.l.o.g. we may assume that $B_1 = \chi^{Q}$, that is, $Q = J'_1$. By a similar argument as above, the independent sets $J_i := J'_i \cup P$ are tight and its incidence vectors are linearly independent.
Alltogether we have $|F|$ linearly independent vectors $\chi^{I_j}$ with $I_j \cap \bar{F} = Q$ and $|\bar{F}|$ linearly independent vectors $\chi^{J_i}$ with $J_i \cap F = P$, where $J_1=I_1$. As is easily seen, this yields a system of $|F|+|\bar{F}|-1 = |E|-1$ linearly independent vectors satisfying $x(F) \leq r(F)$ at equality.
\[T:facets5\] ${P_{\mathcal{I}}^c(E)}$ is fulldimensional unless $c=(0,r(E))$ and $E$ is separable.
Clearly, $\dim {P_{\mathcal{I}}^c(E)}\geq \dim P_{\mathcal{I}}^{(c_p)}(E) + 1$ for all $p$, since the equation $x(E) = c_p$ is satisfied by all $y \in P_{\mathcal{I}}^{(c_p)}(E)$ but violated by at least one vector $z \in {P_{\mathcal{I}}^c(E)}$.
If $0 < c_p < r(E)$ for some $p$, then, by Theorem \[T:facets4\], $\dim P_{\mathcal{I}}^{(c_p)}(E) = |E|-1$, and consequently $\dim {P_{\mathcal{I}}^c(E)}= |E|$. If there is no such $p$, then $c = (0,r(E))$. Again by Theorem \[T:facets4\], $\dim P_{\mathcal{I}}^{(r(E))}(E) = |E|-1$ if and only if $E$ is inseparable. Since $\dim P_{\mathcal{I}}^{(0,r(E))}(E) = \dim P_{\mathcal{I}}^{(r(E))}(E) + 1$, it follows the claim.
\[T:facets6\] For any $\emptyset \neq F \subseteq E$, the rank inequality $x(F) \leq r(F)$ defines a facet of ${P_{\mathcal{I}}^c(E)}$ if and only if one of the following conditions holds.
1. $0<r(F)<c_{m-1}$ and $F$ is closed and inseparable.
2. $0<c_{m-1}=r(F)<c_m<r(E)$, and $F$ is closed and inseparable.
3. $0<c_{m-1}=r(F)<c_m=r(E)$, $F$ is closed and inseparable, $\bar{F}$ is $c_m$-inseparable, and $E$ is inseparable.
4. $0<c_{m-1}<c_m=r(F)$, $F=E$, and $c_m<r(E)$ or $E$ inseparable.
5. $c_{m-1}=c_1=0$, $c_m=r(E)$, and $r(F)+r(E \setminus F)=r(E)$.
We prove the theorem by case by case enumeration.
\(a) Let $0<r(F)<c_{m-1}$. It is not hard to see that if $F$ is separable or not closed, then $x(F) \leq r(F)$ does not define a facet of ${P_{\mathcal{I}}^c(E)}$. So, let $F$ be closed and inseparable. By Theorem \[T:facets4\], $x(F) \leq r(F)$ defines a facet of $P_{\mathcal{I}}^{(c_{m-1})}(E)$ and $\dim P_{\mathcal{I}}^{(c_{m-1})}(E)= |E|-1$. Thus, it defines also a facet of ${P_{\mathcal{I}}^c(E)}$.
\(b) Let $0<c_{m-1}=r(F)<c_m<r(E)$. Clear by interchanging $c_{m-1}$ and $c_m$ in item (a).
\(c) Let $0<c_{m-1}=r(F)<c_m=r(E)$. The conditions mentioned in (iii) are equivalent to the postulation that $x(F) \leq r(F)$ defines a facet of $P_{\mathcal{I}}^{(c_m)}(E)$ and $\dim P_{\mathcal{I}}^{(c_m)}(E)=|E|-1$. If, indeed, the latter is true, then $x(F) \leq r(F)$ induces a facet also of ${P_{\mathcal{I}}^c(E)}$. To show the converse, suppose, for the sake of contradiction, that $x(F) \leq r(F)$ does not induce a facet of $P_{\mathcal{I}}^{(c_m)}(E)$ or $\dim P_{\mathcal{I}}^{(c_m)}(E)<|E|-1$. Let $\mathcal{B} := \{\chi^{I_j} : I_j \in \mathcal{I}, |I_j|=c_m, j=1,\ldots, z,\}$ be an affine basis of the face of $P_{\mathcal{I}}^{(c_m)}(E)$ induced by $x(F) \leq r(F)$. By hypothesis, $z \leq |E|-2$. Moreover, set $J := I_1 \cap F$ and $K:= I_1 \setminus J$. Then, any incidence vector of an independent set $L \subseteq F$ with $|L|=c_{m-1}$ can be obtained as an affine combination of the set $\mathcal{B}' := \mathcal{B} \cup \{\chi^J\}$, which can be seen as follows: $L,I_1 \in \mathcal{I}$, and $|L|=r(F)$ implies $L \cup {K} \in \mathcal{I}$. Consequently, $\chi^L = \chi^{L \cup K} - \chi^K$. Now, $\chi^K = \chi^{I_1} - \chi^J$ and $\chi^{L \cup K}= \sum_{j=1}^z \lambda_j \chi^{I_j}$ with $\sum_{j=1}^z \lambda_j =1$, since $L \cup K$ is tight. Thus, $\chi^L= \sum_{j=1}^z \lambda_j \chi^{I_j}-\chi^{I_1} + \chi^J$, that is, $\chi^L$ is in the affine hull of $\mathcal{B}'$. Since $|\mathcal{B}'| \leq |E|-1$, $x(F) \leq r(F)$ is not facet defining for ${P_{\mathcal{I}}^c(E)}$, a contradiction.
\(d) Let $0<c_{m-1} < r(F)<c_m$. Since none of the independent sets $I$ with $|I|=c_p$ is tight for $p=1,\ldots,m-1$, $x(F) \leq r(F)$ defines a facet of ${P_{\mathcal{I}}^c(E)}$ if and only if it is an implicit equation for $P_{\mathcal{I}}^{(c_m)}(E)$ and $\dim P_{\mathcal{I}}^{(c_m)}(E)=|E|-1$. However, $\dim P_{\mathcal{I}}^{(c_m)}(E)=|E|-1$ implies $c_m<r(E)$ or $E$ is inseparable. In either case, it follows that $x(F) \leq r(F)$ is an implicit equation for $P_{\mathcal{I}}^{(c_m)}(E)$ if and only if $F=E$. Thus, $r(F)=c_m$, a contradiction.
\(e) Let $0<c_{m-1}<c_m=r(F)$. Clearly, if $F \subset E$, then $x(F) \leq r(F)$ is strictly dominated by the cardinality bound $x(E) \leq c_m$. Consequently, $F=E$ and $x(F) \leq r(F)$ is an implicit equation for $P_{\mathcal{I}}^{(c_m)}(E)$. For the same reasons as in (d), $\dim P_{\mathcal{I}}^{(c_m)}(E)=|E|-1$. Hence, $c_m<r(E)$ or $E$ is inseparable.
\(f) Let $c_{m-1}=c_1=0$. Again, $x(F) \leq r(F)$ defines a facet of ${P_{\mathcal{I}}^c(E)}$ if and only if it is an implicit equation for $P_{\mathcal{I}}^{(c_m)}(E)$. This is the case if and only if $c_m=r(E)$ and $r(F)+r(E \setminus F)=r(E)$.
\(g) Let $r(F)>c_m$. Then, $x(F) \leq x(E)\leq c_m<r(F)$, that is, the face induced by $x(F) \leq r(F)$ is the empty set.
\[T:facets7\] Let $F \subseteq E$ with $c_p< r(F) < c_{p+1}$ for some $p \in
\{1,\dots,m-1\}$. Then, the rank induced forbidden set inequality ${\mbox{FS}}_F(x)
\leq c_p (c_{p+1}-r(F))$ defines a facet of ${P_{\mathcal{I}}^c(E)}$ if and only if
1. $c_p=c_1=0$ and the inequality $x(F) \leq r(F)$ defines a facet of $P_{\mathcal{I}}^{(c_{p+1})}(E)$, or
2. $c_p>0$, $F$ is closed and (i) $\bar{F} := E \setminus F$ is $c_{p+1}$-inseparable or (ii) $c_{p+1}<r(E)$.
For $P_{\mathcal{I}}^{(c_{p+1})}(E)$, the inequality ${\mbox{FS}}_F(x) \leq c_p (c_{p+1}-r(F))$ is equivalent to $x(F) \leq r(F)$, while for $P_{\mathcal{I}}^{(c_{p})}(E)$, it is equivalent to $x(F) \leq c_p$. Hence, in case $c_p=c_1=0$, ${\mbox{FS}}_F(x) \leq c_p (c_{p+1}-r(F))$ induces a facet of ${P_{\mathcal{I}}^c(E)}$ if and only if it induces a facet of $P_{\mathcal{I}}^{(c_{p+1})}(E)$. When $\dim P_{\mathcal{I}}^{(c_{p+1})}(E) = |E|-1$, this is the case if and only if $F$ is closed and inseparable and (i) $\bar{F}$ is $c_{p+1}$-inseparable or (ii) $c_{p+1} < r(E)$, see Theorem \[T:facets4\] (b).
In the following, let $c_p>0$. Let $A$ be the matrix whose rows are the incidence vectors of $I \in \mathcal{I}$ with $|I|=c_p$ or $|I|=c_{p+1}$ that satisfy the inequality ${\mbox{FS}}_F(x) \leq c_p (c_{p+1}-r(F))$ at equality. Denote by $A_F$ and $A_{\bar{F}}$ the restriction of $A$ to $F$ and $\bar{F}$, respectively. By Theorem \[T:facets5\], ${P_{\mathcal{I}}^c(E)}$ is fulldimensional. Hence, ${\mbox{FS}}_F(x) \leq c_p (c_{p+1}-r(F))$ is facet defining if and only if the affine rank of $A$ is equal to $|E|$.
If $F$ is not closed, then there is some $e \in \bar{F}$ with $r(F \cup \{e\}) = r(F)$. Thus, ${\mbox{FS}}_{F'}(x) \leq c_p (c_{p+1}-r(F'))$ is a valid inequality for ${P_{\mathcal{I}}^c(E)}$, where $F' := F \cup \{e\}$, and ${\mbox{FS}}_F(x) \leq c_p (c_{p+1}-r(F))$ is the sum of this inequality and $- (c_{p+1} - c_p) x_e \leq 0$. Next, assume that neither (i) nor (ii) holds. Then, $c_{p+1}=r(E)$ and $\bar{F}$ is $r(E)$-separable. Thus, there is a proper partition $\bar{F} = \bar{F}_1 \dot{\cup} \bar{F}_2$ of $\bar{F}$ with $r^{r(E)}(\bar{F}_1) + r^{r(E)}(\bar{F}_2) = r^{r(E)}(\bar{F})$. Since $F$ is closed, it is not hard to see that $r^{r(E)}(\bar{F}_i) >0$ which implies $c_p < r(F \cup \bar{F}_i) < r(E)$ for $i=1,2$, and hence, the inequalities ${\mbox{FS}}_{F \cup \bar{F}_1}(x) \leq c_p (c_{p+1}-r(F \cup \bar{F}_1))$ and ${\mbox{FS}}_{F \cup \bar{F}_2}(x) \leq c_p (c_{p+1}-r(F \cup \bar{F}_2))$ are valid. One can check again that then ${\mbox{FS}}_F(x) \leq c_p (c_{p+1}-r(F))$ is the sum of these both rank induced forbidden set inequalities.
To show the converse, let $M^F = (F, \mathcal{I}^F)$ with $\mathcal{I}^F := \{I \cap F : I \in \mathcal{I}\}$ be the restriction of $M$ to $F$ and $M^F_{c_p} = (F,\mathcal{I}^F_{c_p})$ the $c_p$-truncation of $M^F$. Since $0 < c_p < r(F)$, Lemma \[L:facets1\] implies that $F$ is inseparable with respect to the rank function of $M^F_{c_p}$. Consequently, the restriction of $x(F) \leq c_p$ to $F$ defines a facet of $P_{\mathcal{I}^F_{c_p}}(F)$. Hence, $A$ contains a $|F| \times |E|$ submatrix $B$ such that $B_F$ is nonsingular and $B_{\bar{F}} = 0$. Next, since $F$ is closed, $r^{c_{p+1}}(\bar{F}) \geq 1$, and (i) $\bar{F}$ is $c_{p+1}$-inseparable or (ii) $c_{p+1}<r(E)$, Lemma \[L:facets3\] implies that $A$ contains a $|\bar{F}| \times |E|$ submatrix $C$ such that $C_{\bar{F}}$ is nonsingular. Thus, $$D :=
\begin{pmatrix}
B_F & 0 \\
C_F & C_{\bar{F}}
\end{pmatrix}$$ is a nonsingular $|E| \times |E|$ submatrix of $A$ (or a row permutation of $A$).
Separation problem
------------------
Given any ${P_{\mathcal{I}}^c(E)}$ and any $x^* \in \mathbb{R}^E$, the separation problem consists of finding an inequality among - violated by $x^*$ if there is any. This problem should be solvable efficiently, due to the polynomial time equivalence of optimization and separation (see Grötschel, Lovász, and Schrijver [@GLS]). By default, we may assume that $x^*$ satisfies the cardinality bounds , and the nonnegativity constraints . A violated rank inequality among (if there is any) can be found by a polynomial time algorithm proposed by Cunningham [@Cunningham]. So, we are actually interested only in finding an efficient algorithm that solves the separation problem for the class of rank induced forbidden set inequalities . If ${r}(F)=|F|$ for all $F \subseteq E$, then the separation routine proposed by Grötschel [@Groetschel] can be applied: For each forbidden cardinality $k$ one just needs to take the first $k$ greatest weights, say $x^*_{e_1},\ldots, x^*_{e_k},$ and check whether the forbidden set inequality associated with $F:= \{e_1,\ldots,e_k\}$ is violated by $x^*$. Otherwise we shall see that the separation problem for the rank induced forbidden set inequalities can be transformed to that for the rank inequalities.
The separation problem for the class of rank induced forbidden set inequalities consists of checking whether or not $$\begin{array}{rcll}
\multicolumn{4}{l}{
(c_{p+1}-{r}(F)) x^*(F)-
({r}(F)-c_p)x^*(E \setminus F) \; \leq \;
c_p(c_{p+1}-{r}(F))}\\
\multicolumn{4}{r}{\mbox{for all $F \subseteq E$ with $c_p <
{r}(F) <
c_{p+1}$} \mbox{ for some $p \in \{0,\dots,m-1\}$}.}
\end{array}$$ For any $F \subseteq E$, $$\begin{array}{cl}
&(c_{p+1}-{r}(F)) x^*(F)-({r}(F)-c_p)x^*(E \setminus F) \leq
c_p(c_{p+1}-{r}(F))\\[0.2cm]
\Leftrightarrow & (c_{p+1}-c_p) x^*(F)-({r}(F)-c_p)x^*(E) \leq
c_p(c_{p+1}-{r}(F))\\[0.2cm]
\Leftrightarrow & x^*(F) \leq \frac{
c_p(c_{p+1}-{r}(F))+({r}(F)-c_p)x^*(E)}{ (c_{p+1}-c_p)} =: \gamma_F.
\end{array}$$ Moreover, for any $k \in \{1,\dots,{r}(E)\}$, the right hand sides of the inequalities $x^*(F) \leq \gamma_F$ for $F \subseteq E$ with ${r}(F)=k$ are equal and differ only by a constant to the right hand sides of the corresponding rank inequalities $x(F) \leq {r}(F)=k$. Thus, both the separation problem for the rank inequalities and rank induced forbidden set inequalities could be solved by finding, for each $k \in
\{1,\dots,|E|\}$, a set $F^* \subseteq E$ of rank $k$ that maximizes $x^*(F)$. If $x^*(F^*)>k$, then the inequality $x(F^*)\leq {r}(F^*)$ is violated by $x^*$. If, in addition, $c_p<k<c_{p+1}$ for some $p \in
\{1,\dots,m-1\}$ and $x^*(F^*)> \gamma_{F^*}$, then $x^*$ violates the rank induced forbidden set inequality associated with $F^*$.
This natural generalization of Grötschel’s separation algorithm, however, seems usually not to result in an efficient separation routine. In order to mark the difficulties, we investigate the above approach for the class of rank inequalities, when $M=(E,\mathcal{I})$ is the graphic matroid defined on some graph $G=(V,E)$. It is well known that the closed and inseparable rank inequalities for the graphic matroid are of the form $x(E(W)) \leq
|W|-1$ for $\emptyset \neq W \subseteq V$. If we would tackle the separation problem for this class of inequalities by finding, for each $k \in \{1,\dots, |W|\}$ separately, a set $W^*_k$ that maximizes $x^*(E(W))$ such that $|W|=k$, then we would run into trouble, since for each $k$, such a problem is the weighted version of the densest $k$-subgraph problem which is known to be NP-hard (see Feige and Seltser [@FS]).
The last line of argument indicates that it is probably not a good idea to split the separation problem for the rank induced forbidden set inequalities into separation problems for the subclasses ${\mbox{FS}}_F(x) \leq c_p(c_{p+1}-{r}(F))$ with ${r}(F)=k, \, k
\in \{c_1+1,\dots, c_m-1\} \setminus \{c_2,c_3,\dots,c_{m-1}\}$. It would be rather better to approach it as “non-cardinality constrained” problem. And this is exactly what Cunningham did for the rank inequalities.
In the sequel, we firstly remind of some important facts regarding Cunningham’s algorithm for the separation of the rank inequalities. Afterwards, we show how the separation problem for the rank induced forbidden set inequalities can be reduced to that for the rank inequalities.
The theoretical background of Cunningham’s separation routine is the following min-max relation.
\[Tmm\] For any $x^* \in \mathbb{R}^E_+$, $\max \{y(E) : y \in P_M(E), y \leq x^*\}=
\min \{{r}(F)+x^*(E \setminus F) : F \subseteq E\}$. $\Box$
Indeed, for any $y \in P_M(E)$ with $y \leq x^*$, $y(E)=y(F)+y(E \setminus
F) \leq {r}(F)+x^*(E \setminus F)$, and equality will be attained if only if $y(F)={r}(F)$ and $y(E \setminus F)= x^*(E \setminus
F)$. Theorem \[Tmm\] guarantees that any $F$ minimizing ${r}(F)+x^*(E \setminus F)$ maximizes $x^*(F)-{r}(F)$. For any matroid $M=(E,\mathcal{I})$ given by an independence testing oracle and any $x^* \in \mathbb{R}^E_+$, Cunningham’s algorithm finds a $y
\in P_M(E)$ with $y \leq x^*$ maximizing $y(E)$, a decomposition of $y$ as convex combination of incidence vectors of independent sets, and a set $F^* \subseteq E$ with ${r}(F^*)+x^*(E \setminus
F^*)=y(E)$ in strongly polynomial time. The vector $y$ will be constructed by path augmentations along shortest paths in an auxiliary digraph.
Next, we return to the separation problem for the rank induced forbidden set inequalities . In the sequel, we suppose that $x^*$ satisfies the rank inequalities .
\[L\_SEP1\] Let $x^* \in \mathbb{R}^E_+$ satisfying all rank inequalities . If a rank induced forbidden set inequality ${\mbox{FS}}_F(x) \leq
c_p(c_{p+1}-{r}(F))$ with $c_p<{r}(F)<c_{p+1}$ is violated by $x^*$, then $c_p < x^*(E) < c_{p+1}$.
First, assume that $x^*(E) \leq c_p$. Then $x^*(F) \leq c_p$, and hence, $$\begin{array}{cl}
& (c_{p+1}-{r}(F)) x^*(F) - ({r}(F)-c_p)x^*(E \setminus F)\\
\leq & (c_{p+1}-{r}(F)) c_p - ({r}(F)-c_p)x^*(E \setminus F)\\
\leq & c_p(c_{p+1}-{r}(F)).
\end{array}$$
Next, assume that $x^*(E) \geq c_{p+1}$. By hypothesis, $x^*$ satisfies all rank inequalities , in particular, $x(F) \leq {r}(F)$. Thus, $$\begin{array}{cl}
&(c_{p+1}-{r}(F)) x^*(F) - ({r}(F)-c_p)x^*(E \setminus F)\\
= & (c_{p+1}-c_p) x^*(F) - ({r}(F)-c_p)x^*(E)\\
\leq & (c_{p+1}-c_p) {r}(F) - ({r}(F)-c_p)x^*(E)\\
\leq & (c_{p+1}-c_p) {r}(F) - ({r}(F)-c_p)c_{p+1}\\
= & c_p(c_{p+1}-{r}(F)).
\end{array}$$
\[L\_SEP2\] Let $x^* \in \mathbb{R}^E_+$ satisfying all rank inequalities , and let $c_p<x^*(E)<c_{p+1}$ for some $p \in \{1,\dots,m-1\}$. Then for any $F \subseteq E$ we have: If $(c_{p+1}-{r}(F)) x^*(F) -
({r}(F)-c_p)x^*(E \setminus F) > c_p(c_{p+1}-{r}(F))$, then $c_p<{r}(F)<c_{p+1}$.
Let $F \subseteq E$, and assume that ${r}(F) \leq c_p$. Then, $$\begin{array}{cl}
&(c_{p+1}-{r}(F)) x^*(F) - ({r}(F)-c_p)x^*(E \setminus F)-
c_p(c_{p+1}-{r}(F))\\
=& (c_{p+1}-c_p) x^*(F) - ({r}(F)-c_p)x^*(E)-
c_p(c_{p+1}-{r}(F))\\
\leq & (c_{p+1}-c_p) {r}(F) - ({r}(F)-c_p)x^*(E)-
c_p(c_{p+1}-{r}(F))\\
= & \underbrace{(c_{p+1}-x^*(E))}_{>
0}\underbrace{({r}(F)-c_p)}_{\leq 0}\; \leq \; 0.
\end{array}$$
Next, if ${r}(F) \geq c_{p+1}$, then $$\begin{array}{cl}
&(c_{p+1}-{r}(F)) x^*(F) - ({r}(F)-c_p)x^*(E \setminus F)-
c_p(c_{p+1}-{r}(F))\\
=& (c_{p+1}-c_p) x^*(F) - ({r}(F)-c_p)x^*(E)-
c_p(c_{p+1}-{r}(F))\\
\leq & (c_{p+1}-c_p) x^*(E) - ({r}(F)-c_p)x^*(E)-
c_p(c_{p+1}-{r}(F))\\
= & \underbrace{(c_{p+1}-{r}(F))}_{\leq
0}\underbrace{(x^*(E)-c_p)}_{> 0}\; \leq \; 0.
\end{array}$$
Thus, $(c_{p+1}-{r}(F)) x^*(F) -
({r}(F)-c_p)x^*(E \setminus F) > c_p(c_{p+1}-{r}(F))$ at most if $c_p<{r}(F)<c_{p+1}$.
\[T\_SEP\] Given a matroid $M=(E,\mathcal{I})$ by an independence testing oracle, a cardinality sequence $c$, and a vector $x^\star \in \mathbb{R}^E_+$ satisfying all rank inequalities , the separation problem for $x^*$ and the rank induced forbidden set inequalities can be solved in strongly polynomial time.
By Lemmas \[L\_SEP1\] and \[L\_SEP2\] we know that $x^*$ violates a rank induced forbidden set inequality at most if $c_p<x^*(E)<c_{p+1}$ for some $p \in \{1,\dots,m-1\}$. Thus, if $x^*(E)=c_q$ for some $q \in
\{1,\dots,m\}$, then $x^* \in {P_{\mathcal{I}}^c(E)}$.
Suppose that $c_p<x^*(E)<c_{p+1}$ for some $p \in \{1,\dots,m-1\}$. We would like to find some $F' \subseteq E$ such that $$(c_{p+1}-{r}(F')) x^*(F') - ({r}(F')-c_p)x^*(E \setminus F') -
c_p(c_{p+1}-{r}(F')) >0$$ if there is any. Lemma \[L\_SEP2\] says that $c_p<{r}(F')<c_{p+1}$, and thus, the inequality ${\mbox{FS}}_{F'}(x)
\leq c_p(c_{p+1}-{r}(F'))$ is indeed a rank induced forbidden set inequality among violated by $x^*$. If there is no such $F'$, then for all $F \subseteq E$ with $c_p<{r}(F)<c_{p+1}$ the associated rank induced forbidden set inequality with $F$ is satisfied by $x^*$, and by Lemma \[L\_SEP1\], all other rank induced forbidden set inequalities among are also satisfied by $x^*$.
To find such a subset $F'$ of $E$, set $\delta := \frac{x^*(E)-c_p}{c_{p+1}-c_p}$. Since $c_p<x^*(E)<c_{p+1}$, $0<\delta<1$. Moreover, $
\frac{c_{p+1}-x^*(E)}{c_{p+1}-c_p}=1-\delta$. For any $F
\subseteq E$ it now follows: $$\begin{array}{crcl}
& (c_{p+1}-c_p) x^*(F) - ({r}(F)-c_p)x^*(E)-
c_p(c_{p+1}-{r}(F)) & > & 0\\
\Leftrightarrow & x^*(F) - \frac{{r}(F)x^*(E)+c_px^*(E)-
c_p c_{p+1}+c_p{r}(F)}{ c_{p+1}-c_p} & > & 0\\
\Leftrightarrow & x^*(F) - {r}(F)\frac{x^*(E)-c_p}{c_{p+1}-c_p} -
c_p\frac{c_{p+1}-x^*(E)}{ c_{p+1}-c_p} & > & 0\\
\Leftrightarrow & x^*(F) - {r}(F)\delta & > & c_p(1-\delta)\\
\Leftrightarrow & \frac{x^*(F)}{\delta} - {r}(F) & > &
c_p\frac{(1-\delta)}{\delta}.\\
\end{array}$$ Setting $x' := \frac{1}{\delta} x^*$, we see that the last inequality is equivalent to $x'(F) - {r}(F) > c_p\frac{(1-\delta)}{\delta}$. Thus, we can apply Cunningham’s algorithm to find some $F \subseteq E$ that maximizes $x'(F)-{r}(F)$. If $x'(F)-{r}(F)>
c_p\frac{(1-\delta)}{\delta}$, then $c_p<{r}(F)<c_{p+1}$ and the rank induced forbidden set inequality associated with $F$ is violated by $x^*$.
Consequently, we suggest a separation routine that works as follows. Assume that the fractional point $x^*$ satisfies the nonnegativity constraints and the cardinality bounds. First, compute with Cunningham’s algorithm a subset $F$ of $E$ maximizing $x^*(F)-{r}(F)$. If $x^*(F)-{r}(F)>0$, then the associated rank inequality $x(F) \leq {r}(F)$ is violated by $x^*$. If $x^*(F)-{r}(F) \leq 0$, then $x^*$ satisfies all rank inequalities , and if, in addition, $x^*(E)=c_p$ for some $p$, then we know that $x^* \in
{P_{\mathcal{I}}^c(E)}$. Otherwise, i.e., if $c_p<x^*(E)<c_{p+1}$ for some $p \in
\{1,\dots,m-1\}$, then we check whether or not there is a violated rank induced forbidden set inequality among by applying Cunningham’s algorithm on $M=(E,\mathcal{I})$ and $x' =
\frac{1}{\delta}x^*$.
\[C:SEP2\] Given a matroid $M=(E,\mathcal{I})$ by an independence testing oracle, a cardinality sequence $c$, and a vector $x^\star \in \mathbb{R}^E_+$, the separation problem for $x^\star$ and ${P_{\mathcal{I}}^c(E)}$ can be solved in strongly polynomial time. $\Box$
Concluding remarks {#Sec:remarks}
==================
The cardinality constrained matroid polytope turns out to be a useful object to enhance the theory of polyhedra associated with cardinality constrained combinatorial optimization problems. Imposing cardinality constraints on a combinatorial optimization problem does not necessarily turn it into a harder problem: The cardinality constrained version of the maximum weight independent set problem in a matroid is manageable on the algorithmic as well as on the polyhedral side without any difficulties. Facets related to cardinality restrictions (rank induced forbidden set inequalities) are linked to well known notions of matroid theory (closed subsets of $E$). The analysis of the separation problem for the rank induced forbidden set inequalities discloses that it is sometimes better not to split a cardinality constrained problem into “simpler” cardinality constrained problems but to transform it into one or more non-cardinality restricted problems.
It stands to reason to investigate the intersection of two matroids with regard to cardinality restrictions. As it is well known, if an independence system $\mathcal{I}$ defined on some ground set $E$ can be described as the intersection of two matroids $M_1=(E,\mathcal{I}_1)$ and $M_2=(E,\mathcal{I}_2)$, then the optimization problem $\max w(I),
\, I \in \mathcal{I}$ can be solved in polynomial time, for instance with Lawler’s weighted matroid intersection algorithm [@Lawler]. This algorithm solves also the cardinality constrained version $\max w(I), \, I \in
\mathcal{I} \cap {\mbox{CHS}}^c(E)$, since for each cardinality $p \leq
{r}(E)$ it generates an independent set $I$ of cardinality $p$ which is optimal among all independent sets $J$ of cardinality $p$. Thus, from an algorithmic point of view the problem is well studied. However, there is an open question regarding the associated polytope. As it is well known, $P_{\mathcal{I}}(E)
=P_{\mathcal{I}_1}(E) \cap P_{\mathcal{I}_2}(E)$, that is, the non-cardinality constrained independent set polytope $P_{\mathcal{I}}(E)$ is determined by the nonnegativity constraints $x_e \geq 0$, $e \in E$, and the rank inequalities $x(F) \leq r_j(F)$, $\emptyset \neq F \subseteq E$, $j=1,2$, where $r_j$ is the rank function with respect to $\mathcal{I}_j$. We do not know, however, whether or not $P_{\mathcal{I}}^c(E)=P_{\mathcal{I}_1}^c(E) \cap
P_{\mathcal{I}_2}^c(E)$ holds. So far, we have not found any counterexample contradicting the hypothesis that equality holds.
[10]{}
P. Camion and J. F. Maurras. Mélanges, hommage á P. Gillis, Cahiers du Centre d’Études de Recherche Opérationnelle, Bruxelles, vol 24 (1982), 107–120.
W. H. Cunningham, [*[Testing membership in matroid polyhedra.]{}*]{} J. Comb. Theory, Ser. B 36 [(1984)]{}, 161-188.
J. Edmonds.
J. Edmonds, [*[Matroids and the greedy algorithm.]{}*]{} Math. Program. 1 [(1971)]{}, 127-136.
U. Feige and M. Seltser. . Technical report, Department of Applied Mathematics and Computer Science, The Weizmann Institute, Rehobot, 1997.
A. B. Gamble and W. R. Pulleyblank, [*[Forest covers and a polyhedral intersection theorem.]{}*]{} Math. Program., Ser. B 45 [(1989)]{}, 49-58.
M. Grötschel.
M. Grötschel, L. Lovász, and A. Schrijver. Springer, Berlin, 1988.
V. Kaibel and R. Stephan. . Technical report, Zuse Institute Berlin, 2007. Available at http://opus.kobv.de/zib/volltexte/2007/1024/.
E. L. Lawler, [*[Matroid intersection algorithms.]{}*]{} Math. Program. 9 [(1975)]{}, 31-56.
J. F. Maurras.
[^1]: Laboratoire d’Informatique Fondamentale, UMR 6166, Université de la Mediterranée, Faculté des sciences de Luminy, 163 Avenue de Luminy, 13288 Marseille, France, e-mail : jean-francois.maurras@lif.univ-mrs.fr
[^2]: Institut für Mathematik, Technische Universität Berlin, Stra[ß]{}e des 17. Juni 136, 10623 Berlin, e-mail : stephan@math.tu-berlin.de
|
---
abstract: 'The best measured rotation curve for any galaxy is that of the dwarf spiral DDO 154, which extends out to about 20 disk scale lengths. It provides an ideal laboratory for testing the universal density profile prediction from high resolution numerical simulations of hierarchical clustering in cold dark matter-dominated cosmological models. We find that the observed rotation curve cannot be fit either at small radii, as previously noted, or at large radii. We advocate a resolution of this dilemma by postulating the existence of a dark spheroid of baryons amounting to several times the mass of the observed disk component and comparable to that of the cold dark matter halo component. Such an additional mass component provides an excellent fit to the rotation curve provided that the outer halo is still cold dark matter-dominated with a density profile and mass-radius scaling relation as predicted by standard CDM-dominated models. The universal existence of such dark baryonic spheroidal components provides a natural explanation of [ the universal rotation curves observed in spiral galaxies]{}, may have a similar origin and composition to the local counterpart that has been detected as MACHOs in our own galactic halo via gravitational microlensing, and is consistent with, and even motivated by, primordial nucleosynthesis estimates of the baryon fraction.'
author:
- |
Andreas Burkert$^{1}$ and Joseph Silk$^{2,3}$\
[$^1$Max-Planck-Institut für Astronomie]{}\
[Königstuhl 17, D-69117 Heidelberg, Germany]{}\
[$^2$Institut d’Astrophysique, 75014 Paris, France]{}\
[$^3$Department of Astronomy and Physics, and Center for Particle Astrophysics]{}\
[University of California, Berkeley CA 97420, USA]{}
title: Dark Baryons and Rotation Curves
---
Introduction
============
Cosmological cold dark matter theories of structure formation in the universe via hierarchical merging are in some difficulty. Recent high-resolution cosmological N-body simulations (Navarro et al. 1996a, 1997) have substantially improved our understanding of the equilibrium density profiles which dark matter halos achieve when formed through hierarchical clustering. It has been shown that the violent, collisionless dynamical relaxation processes during the formation phase of dark matter halos lead to equilibrium profiles that have similar shapes, independent of halo mass, initial density fluctuation spectrum, and adopted cosmological model. All dark matter profiles can be well fit by the simple formulae for density and mass: $$\begin{aligned}
\rho_{DM}(r) & = & \frac{4 \rho_0}{(r/R_s)(1 + r/R_s)^2} \nonumber \\
M_{DM}(r) & = & M_0 \times [ \ln (1+r/R_s) - (r/R_s)/(1+r/R_s) ] ,\end{aligned}$$ where $\rho_0$ is the density of the dark matter halo evaluated at the scale radius $R_s$ and $M_0$ is the characteristic mass; $R_s$ and $M_0$ (or $\rho_0$) are free parameters. $M_0$ is a function of the total virial mass $M_{200}$ inside the virial radius $R_{200}$ $$M_0 = \frac{M_{200}}{\ln (1+R_{200}/R_s) - (R_{200}/R_s)/
(1 + R_{200}/R_s)}$$ where $R_{200}$ denotes the radius inside which the averaged overdensity of dark matter is 200 times the critical density of the universe. The simulations also show that, for any particular cosmology, $R_s$ and $M_0$ are strongly correlated: $R_s = 1.63 \times 10^{-2-c} (M_{200}/M_{\odot})^{1/3} h^{-2/3}\rm \, kpc$ where $c = \log (R_{200}/R_s)$ is approximately 1.4 for low-mass dark matter halos as considered in this paper. Low-mass halos are denser than more massive systems. This results from the fact that lower mass halos form earlier, at times when the universe is significantly denser. Dark matter halos therefore represent a 1-parameter family, being completely described by equation 1 and their virial mass $M_{200}$ or virial radius $R_{200}$. The universal profile has been verified by simulations to halo masses as small as $M_{200} \approx 10^{11}
M_{\odot}$ but there is no reason to believe that these results would not be valid for halos which are even one order of magnitude lower in mass.
Substantial progress has also been made [ within the past decade ]{} on the observational front. The observations show that the rotation curves of low-luminosity disk galaxies and low-surface brightness galaxies are strongly dark matter-dominated (Persic & Salucci 1988, 1990, Puche & Carignan 1991, Broeils 1992, Persic et al. 1996, de Blok et al. 1996). The spherically-averaged mass distributions $M(r)=V_c^2 r/G$ can be derived from the measured circular velocity distributions $V_c(r)$ of gaseous galactic disks. Subtraction of the contributions by the visible components gives the dark matter mass profiles $M_{DM}(r)$ or the corresponding dark matter rotation curves $V_{c,DM}(r) =
(G M_{DM}/r)^{1/2}$. Given these theoretical and observational developments, an important question arises as to whether the theoretically predicted rotation curves, determined by equation 1, are in agreement with the observations.
The dark matter halo of DDO 154
===============================
The low-surface brightness dwarf galaxy DDO 154 has one of the most extended and best-studied dark matter halo rotation curves (Carignan & Freeman 1988, Carignan & Beaulieu 1989, Carignan & Purton 1997), with a precise decomposition into contributions from stars, gas and dark matter. It is also one of the most gas-rich systems known with an inner stellar disk component of mass $5(\pm 2.5) \times 10^7 M_{\odot}$ and an extended HI disk with scale radius of $0.4(\pm 0.05)$ kpc and mass of approximately $3(\pm 1) \times 10^8 M_{\odot}$ [ (Carignan & Beaulieu 1989). ]{} The shape of the rotation curve, even in the innermost regions, is completely dominated by the dark matter halo with a total mass [ within $R_{200}$]{} of about $5 \times 10^9 M_{\odot}$ [ (see section 3).]{}
DDO 154 is a [ good]{} candidate with which to test cosmological models. Fig. 1 (upper panel) shows the dark matter rotation curve of DDO 154 and compares it with the profiles predicted from cosmological models. The stellar disk and the HI contribution (Carignan & Freeman 1988) have been subtracted, adopting a stellar mass-to-light ratio of (M/L$\rm_B$)$_*$ = 1. The error bars indicate the uncertainties in the observations. Note that the data extends out to 21 disk scale lengths and that the rotation curve clearly decreases beyond 5 kpc. The thick dashed and dotted curves fit the theoretically predicted rotation curves as determined by equation 1 to the innermost and outermost regions, respectively. Fitting the inner regions (dotted curve, labeled A) is well known to pose a problem (Flores & Primack 1994, Moore 1994, Burkert 1995) as the theoretical models predict a central $r^{-1}$ density cusp, whereas the observed velocity profile indicates a large isothermal core with a constant density. As a result, the theoretical models lead to far more mass in the innermost region than is seen. It has been suggested that this discrepancy could be solved by assuming secular processes (e.g. violent galactic winds) in the baryonic component (Navarro et al. 1996b) which could also affect the innermost parts of dark matter halos.
However a similar problem exists in the outermost regions. Whereas the observed dark matter rotation curve clearly decreases beyond 5 kpc, the theoretically predicted universal profile fit leads to a very massive and extended halo ($M_{200}=1.8 \times 10^{13} h^{-1} M_{\odot};
R_{200} = 863 h^{-1}$ kpc, where $h$ is the Hubble constant in units of 100 km s$^{-1}$ Mpc$^{-1}$) with a rotation curve that increases beyond 8 kpc. The outer regions certainly cannot be affected by secular mass loss involving a baryonic component, and a different explanation must be sought. The dashed line in the upper panel of Fig. 1 (labeled B) shows a fit to the outer rotation curve of DDO 154. In this case the dark matter excess in the inner regions is unacceptably large.Approximately $10^9 M_{\odot}$ in dark matter would have to be moved from the inner 2 kpc into the region between 2 kpc and 4 kpc to explain the discrepancy between the observed inner rotation curve and the predicted one. This is far more than would be expected as a result of secular processes.
Figure 2 compares the values of $M_0$ and $R_s$ (see equation 1), derived from fitting the inner or outer rotation curve of DDO 154 with the standard, cluster-normalized CDM predictions (Navarro et al 1996a, 1997). Note that both fits require pairs of parameter values which are not in agreement with theory. In order to lie within the expected parameter range one would have to choose values for $M_0$ and $R_s$ which do not fit the rotation curve, either in the inner or in the outer regime.
The dark baryonic component of DDO 154
======================================
What is wrong with either CDM or with the failure to fit the rotation curve of DDO 154? There is accumulating evidence that the CDM models can account for many aspects of large-scale structure and therefore should not be dismissed. Errors in determining the rotation curve of DDO 154 are equally unlikely due to its very regular velocity field from which $V_c(r)$ can be derived and unambiguously corrected for warping. As described above, secular processes cannot explain the discrepancy. There also cannot exist much more mass in the HI disk, e.g. in $H_2$, as such a massive disk would be gravitationally unstable and would efficiently form molecular clouds and stars, in contradiction to the observations.
We propose, that in addition to the CDM dark matter halo which can be represented by the one-parameter family of universal profiles, DDO 154 contains a second dark but compact baryonic component whose presence we infer in order to explain the observed rotation curve. Because of the arguments mentioned above, this new component cannot be gaseous and located in the disk. We identify it with a [*spheroidal distribution of massive compact baryonic objects, MACHOS*]{}, located in the inner regions of the dark matter halo.
There is in fact ample room for such a subdominant baryonic component, relative to the massive dark halos of the standard CDM theory in the form of MACHOS. Primordial nucleosynthesis requires a baryonic component of [ $\Omega_b h^2 \approx 0.015 \pm 0.008$ (Kurki-Suonio, Jedamzik & Mathews 1997, Copi, Schramm & Turner 1995) whereas the observed value for stellar and gaseous components in disks lies in the range of $\Omega_d \approx 0.004 \pm 0.002$ (Persic & Salucci 1992, note that $\Omega_d \approx 0.006$]{} for DDO 154, assuming $\Omega_{halo} = 0.1$). Where are all these additional baryons? Most of them should be found in galaxies unless winds have strongly depleted all the initial baryons during the protogalaxy phase. Even if protogalactic winds have occurred, as is suggested by theory and observation, it is unlikely that all of the baryons that are not in the disk would have been lost. The recent MACHO experiments (Alcock et al. 1996) demonstrate that there indeed exists a substantial baryonic component in a more extended, spheroidal distribution in the Milky Way, at least between the Sun and the LMC, in addition to the observed stellar and gaseous baryonic components. It is likely that a similar, yet hitherto unobserved, component also exists in DDO 154.
The mass distribution $M_{sph}(r)$ of the proposed spheroidal dark baryonic component is determined by $$M_{sph}(r) = V_{c,DM}^2 r/G - M_{DM}^*(r)$$ which is the difference in total mass inside a radius $r$ as expected from the dark matter rotation curve $V_{c,DM}$ and the dark matter mass $M_{DM}^*$ inside $r$. We consider two extreme possibilities for determining $M_{DM}^*(r)$. One possibility is just to adopt the universal profile predicted by hierarchical infall models (equation 1): $M_{DM}^*(r) = M_{DM}(r)$. In this case the baryonic dark component formed at the same time or even earlier than the extended dark halo. In the opposite limit we imagine that the MACHO spheroid formed after the extended dark halo profile was established. In this case, if the dark baryonic component is of comparable mass inside a certain radius, the dark halo within this radius will undergo adiabatic contraction. For simplicity we will assume spherical symmetry and a formation timescale of the MACHO halo which is long compared to the dynamical timescale of the dark halo. In this case the corrected dark matter mass profile is determined by (Binney & Tremaine 1987) $M_{DM}^*(r) = M_{DM}(r^*)
$ where $M_{DM}(r)$ is determined by equation 1 and $$r^* = r \times \frac{M_{DM}(r^*) + M_{sph}(r)}{M_{DM}(r^*)}.$$ Inserting $M_{sph}$ (equation 3) into equation 4 and choosing values for $M_0$ and $R_s$, one can determine iteratively the adiabatically contracted dark matter mass profile $M_{DM}^*$ and $M_{sph}(r)$. Our approach seems at first sight a very contrived solution to the problem of the rotation curve, as we attribute the discrepancy between observations and theory to a third, invisible component which introduces an extra degree of freedom. How can this idea be tested? There exist in fact very strong constraints on the shape of $M_{sph}(r)$, which must be fulfilled in order for our model to be physically correct. First of all, the total mass distribution of the dark spheroid has to be positive everywhere and increase monotonically with increasing radius, up to a maximum radius $R_{sph}$ beyond which it has to remain constant. Second, the density of the spheroid $\rho_{sph}(r) = (dM_{sph}/dr) / (4 \pi r^2)$ must decrease monotonically with increasing radius.
These constraints require that the dark matter rotation curve lies everywhere below the observed rotation curve. Dark matter fits like the curves labeled A and B in the upper panel of Fig. 1 are ruled out. $R_{sph}$ could in principle lie outside the observed radius regime. However for DDO 154 the measured rotation curve extends out to 21 disk scale lengths. We identify the dark spheroid with an intermediate baryonic component that presumably formed during the dissipative protogalactic collapse phase, and which should therefore be more centrally concentrated than the dark matter halo. The rotation curve of DDO 154 clearly decreases outside 5 kpc, and this indicates that a large fraction of its mass lies at smaller radii. As the dark matter halo dominates the total mass in the outer regions, we expect that the mass distribution of the baryonic spheroid becomes constant and equal to its total mass $M_{sph}^{tot}$ at a radius $R_{sph} \leq 5 $ kpc.
Note that the radial dependence of $V_c$ and $M_{DM}$ is given by the observations and equation 1. It is therefore not trivial that the difference of the terms on the right hand side of equation 3 gives a profile which meets all of the constraints mentioned above, even if the dark matter scale parameters $M_0$ and $R_s$ are treated as free parameters. We have indeed found such a solution, however only for very special values of $M_0$ and $R_s$, values which for the models with and without adiabatic contraction lie within the small shaded and dark regions of figure 2, respectively. These sets of solutions uniquely determine the total mass of the spheroid $M_{sph}^{tot}$, and hence, via equation 2, the total mass $M_{200}$ of the dark matter halo.
The solid lines in the upper and lower panels of figure 1 provide an excellent fit to the rotation curve, adopting our Macho model with the non-baryonic dark matter halo parameters fixed at $M_0 = 4 \times 10^9 M_{\odot}$ and $R_s = 4.5$ kpc for the case without adiabatic contraction and $M_0 = 3 \times 10^9 M_{\odot}$ and $R_s = 5.5$ kpc if adiabatic contraction is included. The inserts within Figure 2 show the mass and density profiles of the dark baryonic spheroid and the dark matter halo without and with adiabatic contraction. Note that in the former case without adiabatic contraction, the MACHO component would dominate the inner mass and density distribution, whereas in the latter case the baryonic and non-baryonic halos have comparable masses in the inner region. For all acceptable models we find $R_{sph} \approx 5$ kpc, $M_{sph}^{tot} \approx 1.5 \times 10^9 M_{\odot}$ and $M_{200} \approx 5 \times 10^9 M_{\odot}$. The density profiles of the MACHO component can be well approximated by an isothermal sphere with a constant velocity dispersion of $\sigma = 40$ km/s.
There exists an additional independent constraint in order for our model to be acceptable within the framework of standard cosmology: $M_0$ and $R_s$ must follow the tight relationship, predicted by cosmological models. Indeed, figure 2 shows that the values which have to be adopted in order to give a physically correct mass distribution overlap with the parameter space of values expected from standard cosmology, in contrast to the one-component dark matter fits (starred points). Moreover, as we identify the dark spheroid with the missing baryonic component, early universe nucleosynthesis requires that $\Omega_b/\Omega_d = (M_{sph}^{tot}+M_{visible}^{tot})/
M_{visible}^{tot} \approx 5 (\pm 3)\times h^{-2}$, where $M_{visible}^{tot} = 3 \times 10^8 M_{\odot}$ is the total visible mass of DDO 154. For the areas of possible solutions we find a dark-to-luminous mass fraction of approximately 6, in agreement with the expectations.
Note that an arbitrary choice of $M_0$, $R_s$ and $M_{sph}$ would almost certainly fail to meet all of these constraints. The excellent agreement of our model with the predictions from cosmological models of structure formation and primordial nucleosynthesis provides additional evidence for the presence of a dark baryonic component in DDO 154.
Discussion and Conclusions
==========================
Our three component model of luminous baryons in a disk configuration, and MACHOS and cold dark matter in a spheroidal distribution, can reconcile the most detailed observations of a rotation curve to date with the hierarchical clustering theory of galaxy formation. This might be the first (indirect) detection of a MACHO component in another galaxy. It also has allowed us to study in detail for the first time the internal density distribution of a dark baryonic spheroid due to the excellent high-resolution data of DDO 154’s rotation curve.
The structure of the MACHO spheroid in DDO 154 is surprisingly similar to the MACHO halo of the Milky Way, the existence and mass of which has been inferred from a completely different method: gravitational microlensing events of stars in the Large Magellanic Cloud. Alcock et al. (1996, 1997) find for the Galaxy that within 50 kpc (14 disk scale lengths) the total masses of MACHOS and dark matter are comparable and of order $2.5 \times 10^{11} M_{\odot}$. This is 4 to 5 times the mass of the galactic disk ($M_{d} \approx 6 \times 10^{10} M_{\odot}$). The dark baryonic spheroid of DDO 154 also extends out to 14 disk scale lengths at which radius the mass of the MACHO halo is again similar to the mass of the dark matter halo, with, in this case, a mass of order $1.5 \times 10^9 M_{\odot}$. The inferred MACHO mass is also of order 5 times the mass of the HI disk. This agreement in the relative mixture of dark matter, MACHOS and disk material indicates that these components formed in both galaxies from a similar continuous dynamical process, with the MACHO spheroid representing a presumably dissipative component intermediate between the collisionless non-baryonic dark halo and the strongly collisional, dissipation-dominated, rotationally-supported disk. This might provide an explanation [ for the puzzling observational result that disk galaxies have universal rotation curves (Casertano & van Gorkom 1991, Rubin et al. 1985, Persic et al. 1996), requiring a connection between their galactic disks and their dark spheroidal components. Universal rotation curves would be expected]{} for galaxies of any mass where the relative mass and radius ratios between the dark matter halo, the MACHO halo and the disk are universal numbers.
Our results indicate that the baryonic component of DDO154 and probably also of other disk galaxies consists of two components, a spheroidal MACHO component which represents about 22% of the total mass and a visible disk component with only 4% of the total mass. The total baryon fraction in galaxies is then of the order of a quarter of the total mass, a value which is higher than expected from the primordial nucleosynthesis predictions if the relative mixture of baryonic and non-baryonic matter is universal. This baryon segregation could result from dissipative processes during the formation of halos which concentrate the baryons relative to the nondissipative dark matter.
The origin of two separate baryonic components, namely a dominant dark spheroidal component and a disk component, is an interesting and yet unsolved theoretical puzzle which could provide important information on the dissipative formation history and evolution of galaxies.
We thank Dr. C. Carignan for making his new data of DDO 154’s rotation curve available prior to publication, Dr. J. Navarro for sending us a subroutine that generates the scaling relations as predicted from cosmological models and Dr. S. White and the referee for helpful suggestions. The research of J.S. has been supported in part by grants from NASA and NSF, and he also acknowledges with gratitude the hospitality of the Institut d’Astrophysique de Paris as a Blaise-Pascale Visiting Professor, and the Institute of Astronomy at Cambridge as a Sackler Visiting Astronomer.
Alcock, C. et al 1996, , 461, 84 Alcock, C. et al 1997, submitted to [*Astrophys. J.*]{}, astro-ph/9606165 Bahcall, J. N. & Casertano, S. 1985, , 293, L7 Binney, J, & Tremaine, S. 1987, Galactic Dynamics (Princeton Univ. Press: Princeton) Broeils, A.H. 1992, PhD thesis Burkert, A. 1995, , 447, L25 Carignan, C. & Freeman, K.C. 1988, , 332, L33 Carignan, C. & Beaulieu, S. 1989, , 347, 760 Carignan, C. & Purton, C. 1997, submitted to [*Astrophys. J.*]{} Casertano, S. & van Gorkom, J.H. 1991, , 101, 1231 Copi, C.J., Schramm, D.N. & Turner, M.S. 1995, Science, 267, 192 de Blok, W.J.G., McGaugh, S.S. & van der Hulst, J.M. 1997, , 283, 18 Flores, R.A. & Primack, J.R. 1994, , 427, L1 Kurki-Suonia, H., Jedamzik, K, & Mathews, G.J. 1997, , 479, 31 Moore, B. 1994, Nature, 370, 629 Navarro, J.F., Frenk, C.S., & White, S.D.M. 1996a, , 462, 563 Navarro, J.F., Eke, V.R. & Frenk, C.S. 1996b, , 283, L72 Navarro, J.F., Frenk, C.S., & White, S.D.M. 1997, submitted to [*Mon. Not. R. Astron. Soc.*]{}, astro-ph/9611107 Persic, M., Salucci 1988, , 234, 131 Persic, M., Salucci 1990, , 245, 577 Persic, M., Salucci 1992, , 258, 14 Persic, M., Salucci, P. & Stel, F. 1996, , 281, 27 Puche, D. & Carignan, C. 1991, , 378, 487 Rubin, V.C., Burstein, D., Ford, W.K. & Thonnard, N. 1985, , 289, 81
[FIGURE CAPTIONS]{}
[F[IG]{}.]{} 1. [*Upper panel*]{}: The dark matter rotation curve of DDO 154 is shown with error bars. The dotted line (A) shows a fit to the inner parts of the rotation curve, adopting the dark matter halo structure as predicted by cosmological models. The dashed line (B) shows a dark matter halo fit to the outer part of the rotation curve. The solid line shows the fit achieved with the 2-component MACHO model without adiabatic contraction, assuming dark matter halo parameters $M_0 = 4 \times 10^9 M_{\odot}$ and $R_s = 4.5$ kpc. The lower dashed and dot-dashed curves show the contribution to the rotation curve of the MACHO spheroid and the dark matter halo (C), respectively.
[*Lower panel*]{}: The 2-component MACHO model with adiabatic contraction is shown, adopting halo parameters $M_0 = 3 \times 10^9 M_{\odot}$ and $R_s = 5.5$ kpc. The dashed curve shows the contribution by the MACHO spheroid. The dot-dashed curve (D) and the dotted curve (C$^*$) show the contribution of the dark matter halo after and before adiabatic contraction, respectively.
[F[IG]{}.]{} 2.— Standard cluster normalized cold dark matter models predict that the dark matter halo scale radii $R_s$ and scale masses $M_0$ should lie within the narrow band enclosed by the two parallel solid lines. The parallel dashed lines enclose the region of scale parameters, expected for the less favoured COBE-normalized cold dark matter model. One-component dark matter fits to the rotation curve of DDO 154 would result in scale parameters as shown by the two stars for the inner (A) and outer (B) fits. The two-component MACHO model without adiabatic contraction and with adiabatic contraction requires the scale parameters to lie within the dark area (labeled C) and the shaded area (labeled C$^*$), respectively. The upper inserts show the mass and density distribution ($\rho$ in units of $M_{\odot}$ pc$^{-3}$) for the standard model ($M_0 = 4 \times 10^9 M_{\odot}, R_s = 4.5$ kpc, star inside dark area) without adiabatic contraction, with the solid and dashed lines representing the dark baryonic spheroid and the dark matter halo, respectively. The lower insert shows the mass distribution of the standard model ($M_0 = 3 \times 10^9 M_{\odot}, R_s = 5.5$ kpc, star inside shaded area) with adiabatic contraction. The dots represent the total dark matter mass profile as predicted from the rotation curve. The lower solid line shows the mass profile of the MACHO halo. The dot-dashed and the dotted curves show the mass distribution of the non-baryonic dark matter halo after and before adiabatic contraction, respectively.
|
---
author:
- 'L. Pagani'
- 'A. Bacmann'
- 'S. Cabrit'
- 'C. Vastel'
date: 'Received 31/10/2006; accepted 26/01/2007'
title: 'Depletion and low gas temperature in the L183 prestellar core: the N$_2$H$^+$ - N$_2$D$^+$ tool[^1]'
---
[The study of pre-stellar cores (PSCs) suffers from a lack of undepleted species to trace the gas physical properties in their very dense inner parts.]{} [We want to carry out detailed modelling of N$_2$H$^+$ and N$_2$D$^+$ cuts across the L183 main core to evaluate the depletion of these species and their usefulness as a probe of physical conditions in PSCs.]{} [We have developed a non-LTE (NLTE) Monte-Carlo code treating the 1D radiative transfer of both N$_2$H$^+$ and N$_2$D$^+$, making use of recently published collisional coefficients with He between individual hyperfine levels. The code includes line overlap between hyperfine transitions. An extensive set of core models is calculated and compared with observations. Special attention is paid to the issue of source coupling to the antenna beam.]{} [The best fitting models indicate that i) gas in the core center is very cold (7$\pm$ 1 K) and thermalized with dust, ii) depletion of N$_2$H$^+$ does occur, starting at densities 5–7$\times$10$^5$ cm$^{-3}$ and reaching a factor of 6$^{+13}_{-3}$ in abundance, iii) deuterium fractionation reaches $\sim$70% at the core center, and iv) the density profile is proportional to r$^{-1}$ out to $\sim$4000 AU, and to r$^{-2}$ beyond.]{} [Our NLTE code could be used to (re-)interpret recent and upcoming observations of N$_2$H$^+$ and N$_2$D$^+$ in many pre-stellar cores of interest, to obtain better temperature and abundance profiles.]{}
Introduction
============
Understanding star formation is critically dependent upon the characterisation of the initial conditions of gravitational collapse, which remain poorly known. It is therefore of prime importance to study the properties of pre-collapse objects, the so-called pre-stellar cores (hereafter PSCs). As bolometers and infrared extinction maps have been unveiling PSCs through their dust component, it has become clear that depletion of molecules onto ice mantles is taking place inside these cores, preventing their study with the usual spectroscopic tools. In most PSCs, very few observable species seem to survive in the gas phase in the dense and cold inner parts, namely N$_2$H$^+$, NH$_3$, H$_2$D$^+$ and their isotopologues (e.g. Tafalla et al. [@Tafalla02]). In a few cases, it is advocated that even N-bearing species also deplete (e.g. B68: Bergin et al. [@Bergin02]; L1544: Walmsley et al. [@Walmsley04]; L183: Pagani et al. [@Pagani05], hereafter PPABC).
Among the above three species, H$_2$D$^+$ is thus the only one not to deplete. However, it has only one (ortho) transition observable from the ground, that moreover requires excellent sky conditions. The para form ground transition at 1.4 THz should not be detectable in emission in cores with T$_\mathrm{kin} \leq$ 10 K. Therefore H$_2$D$^+$ is useful for chemical and dynamical studies, but brings little information on gas physical conditions. NH$_3$ inversion lines at 23 GHz can provide kinetic temperature measurements as long as higher lying non-metastable levels are not significantly populated (Walsmley & Ungerechts [@Walmsley83]). However, because the critical density of the NH$_3$ (1,1) inversion line is only $\sim$2000 cm$^{-3}$, this tracer may have substantial contribution from external, warmer layers, not representative of the densest parts of PSCs. The third species, N$_2$H$^+$, appears very promising: it has the strong advantage of having mm transitions with critical densities in the range 0.5–70$\times$10$^6$ cm$^{-3}$ and intense hyperfine components for the (J:1–0) line in typical PSC conditions. The ratio of hyperfine components gives an estimate of the opacity and excitation temperature of the line. Fitting both the hyperfine ratios and the (J:1–0) to (J:3–2) ratio with a rigorous NLTE model should thus bring strong constraints on both T$_\mathrm{kin}$ and n(H$_2$). Collisional coefficients (with He) between individual hyperfine sublevels have become available recently (Daniel et al. [@Daniel05]). However, current excitation models (Daniel et al. [@Daniel06a]) do not take into account line overlap, which limits their accuracy. Another question that remains is whether N$_2$H$^+$ is indeed able to probe the central core regions, despite the depletion effects which have been reported.
In this paper, we introduce a new NLTE Monte-Carlo 1D code[^2] treating N$_2$H$^+$ and N$_2$D$^+$ radiative transfer with line overlap, and apply it to detailed analysis of the main PSC in L183, a clear-cut case of N$_2$H$^+$ depletion (cf. PPABC). We demonstrate the capability of the model to constrain physical conditions inside the PSC (temperature, density profile, abundance and depletion, deuterium fractionation). In particular we show that the gas is very cold at the core center and thermalized with the dust, and that N$_2$D$^+$ appears a very useful tracer of physical conditions in the innermost core regions.
Observations and analysis {#sec:obs}
=========================
[cccccccccccccccccc]{} &&&&&&&\
&&&&&& &\
&&&&&&&\
&&&&&&&\
&&&&&&&\
& V\_ & [T\_R\^\*]{} & Area & & V\_ & [T\_R\^\*]{} & Area & & V\_ & [T\_R\^\*]{} & Area\
&[(kms\^[-1]{})]{}&[(K)]{}&[(Kkms\^[-1]{})]{}&[obs]{}&[weights]{}&[(kms\^ [-1]{})]{}&[(K)]{}&[(Kkms\^[-1]{})]{}&&[(kms\^[-1]{})]{}&[(K)]{}&[(Kkms\^[-1]{})]{}\
1 &-8.009(7)&2.05(3)&0.47(3)&0.932 & [*0.428*]{}&-2.58(1)&0.17(6)&0.013(5)&&-2.63(2)&0.20(6)&0.025 (8)\
2 &-0.611(8)&1.81(3)&0.42(3)&0.822 & [*0.428*]{}&0.00(1)&0.50(6)&0.29(2)&&0.00(1)&0.67(6)&0.37(2)\
3 &0.000(7)&2.20(3)&0.60(3)&1 &[*1*]{}&4.70(8)&2.25(6)&0.014(5)&&4.68(3)&2.12(6)&0.028(9)\
4 &0.956(7)&2.08(3)&0.54(3)&0.945 & [*0.714*]{}\
5 &5.546(7)&2.12(3)&0.47(3)&0.964 & [*0.428*]{}\
6 &5.983(8)&2.04(3)&0.53(3)&0.927& [*0.714*]{}\
7 &6.94(1)&1.26(3)&0.24(3)&0.573 & [*0.143*]{}\
&&&&&&&\
&&&&&& &\
&&&&&&&\
&&&&&&&\
&&&&&&&\
& V\_ & [T\_R\^\*]{} & Area & & V\_ & [T\_R\^\*]{} & Area & & V\_ & [T\_R\^\*]{} & Area\
&[(kms\^[-1]{})]{}&[(K)]{}&[(Kkms\^[-1]{})]{}&[obs]{}&[weights]{}&[(kms\^[-1]{})]{}&[(K)]{}&[(Kkms\^[-1]{})]{}&&[(kms\^[-1]{})]{}&[(K)]{}&[(Kkms\^[-1]{})]{}\
1 &-9.697 (8)&0.77(3)&0.15(1)&0.7&[*0.428*]{}&-5.35(2)&0.16(3)&0.137(5)&&-3.26(1)&<0.12\
2 &-0.763 (8)&0.76(3)&0.15(1)&0.691&[*0.428*]{}&-0.759 (8)&0.21(3)&0.080 (3)&&0.000(5)&0.72(4) &0.186(8)\
3 &0.000 (7)&1.10(3)&0.33(2)&1&[*1*]{}&0.000 (1)&1.37(3)&0.513 (3)&&0.46(2)&0.19(4)&0.06(1)\
4 &1.146 (8)&0.92(3)&0.25(2)&0.836&[*0.714*]{}&0.646 (4)&0.40(3)&0.139 (3)&&2.47(3)&0.11(4)&0.028 (7)\
5 &6.65 (2)&0.55(3)&0.16(2)&0.5&[*0.428*]{}&2.5(1)&0.08(3)&0.02(2)&&\
6 &7.19 (1)&0.90(3)&0.23(2)&0.818&[*0.714*]{}&2.97(2)&0.43(3)&0.08(2)&&\
7 &8.34 (2)&0.34(3)&0.05(2)&0.309&[*0.143*]{}&3.57(5)&0.26(3)&0.16(3)&&\
&&&&\
&&&&&\
&&&&\
&&&&&\
&&&&\
& V\_ & [T\_R\^\*]{} & Area & & V\_ & [T\_R\^\*]{} & Area\
&[(kms\^[-1]{})]{}&[(K)]{}&[(Kkms\^[-1]{})]{}&[obs]{}&[weights]{}&[(kms\^[-1]{})]{}&[(K)]{}&[(Kkms\^[-1]{})]{}\
1&-19.504(6)&2.62(7)&0.89(4)&0.922&[*0.363*]{}&-0.005(5)&0.73(8)&0.152(7)\
2& -7.814(5)&2.68(7)&0.69(3)&0.944&[*0.273*]{}\
3& -7.255(8)&1.95(7)&0.57(3)&0.687&[*0.182*]{}\
4& -0.17(1)&2.84(7)&1.15(9)&1&[*1*]{}\
5& 0.30(2)&2.78(7)&1.0(1)&0.979&[*0.636*]{}\
6& 7.465(9)&2.60(7)&0.72(5)&0.915&[*0.303*]{}\
7& 7.89(1)&1.91(7)&0.48(5)&0.673&[*0.152*]{}\
8& 19.318(9)&2.48(7)&0.62(4)&0.873&[*0.242*]{}\
9& 19.85(1)&1.72(7)&0.42(5)&0.606&[*0.121*]{}\
CSO observations are done with a resolution around 0.1 kms$^{-1}$ (or more) which explains this larger value
The hyperfine structure is too complex to be detailed completely. Only the main groups of hyperfine transitions are given
The main dust core in (Pagani et al. [@Pagani04]) was observed at the IRAM 30-m telescope in November 2003, July 2004 and October 2006. Spectra were taken on a 12 grid in an East-West strip across the core (centered at $\alpha$(2000) = 15h54m08.5s, $\delta$(2000)= $-$25248). Symmetric eastern and western spectra were averaged out to $\pm$ 48 to give a more representative radial profile. A small anti-symmetric velocity shift of a few tens of m s$^{-1}$ was noticed on either sides of the PSC center at distances beyond $\pm$ 30, possibly indicative of rotation (see Section \[sec:coremodel\]). This shift was compensated for when averaging eastern and western spectra, in order not to artificially broaden the lines. Beyond 48, only eastern positions are considered, as the western side is contaminated by a separate core (Peak 3; Pagani et al. 2004). Lines of N$_2$H$^+$ (J:1–0), (J:3–2) and of N$_2$D$^+$ (J:1–0), (J:2–1) and (J:3–2) were observed in Frequency switching mode, with velocity sampling 30–50 ms$^{-1}$ and T$_\mathrm{sys}$ ranging from 100 K at 3mm up to 1000 K at 1mm. The N$_2$H$^+$ (J:2–1) line at 186 GHz was not observed, as it lies only 3 GHz away from the telluric water line at 183 GHz, hence its usefulness to constrain excitation conditions would be limited by calibration sensitivity to even tiny sky fluctuations. The problem does not apply to N$_2$D$^+$ which lies a factor of 1.2 lower in frequency. Spatial resolution ranges from 33 at 77 GHz to 9 at 279 GHz. Additional CSO 10-m observations of N$_2$H$^+$ (J:3–2) were obtained in June 2004 at selected positions. Observations were done in Position Switch mode with the high resolution AOS (48 kHz sampling) and a T$_\mathrm{sys}$ around 600 K. A major interest to observe the (J:3–2) line with the CSO is that the beam size and efficiency is very similar to the 30-m values for the (J:1–0) line, thus almost canceling out beam correction errors in the comparison between the two. It is thus a useful constraint for radiative transfer modelling, even though the higher resolution (9) 30-m (J:3–2) observations will be essential to constrain abundances and physical conditions in the innermost core regions. Standard reduction techniques were applied using the CLASS reduction package[^3].
Complementary observations of NH$_3$ (1,1) and (2,2) inversion lines towards the PSC center were also obtained at the new Green Bank 100-m telescope (GBT) in November 2006 with velocity sampling of 20 ms$^{-1}$ and a typical T$_\mathrm{sys}$ of 50 K, in Frequency Switch mode. The angular resolution ($\sim$35) is close to that of the 30-m in the low frequency (J:1-0) N$_2$D$^+$line. The main beam efficiency of the GBT at 23 GHz is between 0.95 and 1, hence no correction was applied to put the spectra on T$_\mathrm{R}^*$ scale.
Table \[data\] summarizes the main observational parameters of the lines observed towards the core center: noise level of the spectrum (rms), rotational line total opacity and intrinsic FWHM width of individual hyperfine transitions (as fitted by the CLASS HFS routine, which assumes equal excitation temperature for all sublevels), and relative velocity centroids and intensities of the main hyperfine groups in our spectra (derived from gaussian fits). Note that the hyperfine groups are always broader than the intrinsic linewidth derived by CLASS, due to optical depth effects and/or to the presence of adjacent hyperfine components too close to be spectroscopically separated.
To compare with models, beam efficiency correction is an important issue. Since L183 (as well as other PSCs) is very extended compared to the primary beam of 10–30, the T$_\mathrm{mb}$ intensity scale is inadequate, as it will overcorrect for source-antenna coupling and thus overestimate the true source surface brightness[^4] (cf. PPABC). To avoid this problem, observed spectra are corrected only for moon efficiency ($\sim$ T$_\mathrm{R}^*$ scale), while models are convolved by the full beam pattern of the 30m telescope (cf. Greve et al. [@Greve98])[^5], yielding intensities on the same T$_\mathrm{R}^*$ scale. For the CSO, we also correct the data for moon efficiency ($\eta_\mathrm{moon}$ $\approx$ 0.8 at 279 GHz) and as the details of the beam pattern are not known, the model output is convolved with a simple gaussian beam. Because the CSO main beam coupling is very good at 1.1 mm ([$\eta_\mathrm{MB} \geq$ 0.7]{}), the uncertainty of the correction should be limited.
In order to treat correctly the beam coupling to the source, we also take into account the fact that the L183 core is located within a larger-scale N$_2$H$^+$ filament oriented roughly north-south, as revealed by previous low resolution maps (PPABC). This elongation is mostly constant in intensity over $\simeq$ 6 arcminutes. Therefore we approximate the brightness distribution as a cylinder of length 6, with its axis in the plane of the sky. The intensity distribution along the east-west section of the cylinder is taken as the emergent signal along the equator of our spherical Monte-Carlo model. We then replicate these values along the (north-south) cylinder axis before convolving by the antenna beam. For N$_2$D$^+$, the (unpublished) large-scale map shows a smaller north-south extent ($\simeq$ 2) hence we use a cylinder length of 2 for convolving our model.
In Figs. \[n2hp\] and \[n2dp\], we plot the observations in T$_\mathrm{R}^*$ compared to our best model (see Section \[sec:bestfit\]), convolved by the antenna beam as explained above.
Modelling and discussion
========================
Radiative transfer code {#sec:montecarlo}
-----------------------
Our spherical Monte-Carlo code is derived from Bernes’ code (Bernes [@Bernes]) and has been extensively tested on models of CO emission from dense cores (e.g. Pagani [@Pagani98]). It was recently updated to take into account overlap between hyperfine transitions occuring at close or identical frequencies. This is done by treating simultaneously the photon transfer for all hyperfine transitions of the same rotational line (see also Appendix in Gónzalez-Alfonso & Cernicharo [@Gonzalez93]). Instead of choosing randomly the frequency of the emitted photon, a binned frequency vector is defined for each rotational transition inside which all hyperfine transitions are positioned. All bins are filled with their share of spontaneously emitted photons (or 2.7K continuum background photons) and, during photon propagation, absorption is calculated for each bin by summing all the hyperfine transition opacities at that frequency
$$\tau(\nu) = \int\kappa(\nu)\mathrm{d}s = \sum_{i}\tau_{i}\phi_{i}(\nu)
=\int\left(\sum_{i}\kappa_i\phi_{i}(\nu)\right)\mathrm{d}s$$
where $\kappa_i$ is the absorption coefficient, $\tau_{i}$ the opacity and $\phi_{i}(\nu)$ the local frequency profile of the i$^{th}$ hyperfine transition. The total number of absorbed photons I$_o(\nu)e^{-\tau(\nu)}$ is then redistributed among hyperfine levels according to their relative absorption coefficients $$\mathrm{d}I_{i}(\nu) = \frac{\kappa_{i}\phi_{i}(\nu)}{\kappa(\nu)}
\times I_o(\nu)e^{-\tau(\nu)}$$
Overlap is treated for all rotational transitions. Statistical equilibrium, including collisions, is then solved separately for each hyperfine energy level.
Collisional coefficients are available for transitions between all individual hyperfine levels, but were calculated with He as a collisioner instead of H$_2$ (Daniel et al. [@Daniel05]). We scaled them up by 1.37 to correct for the difference in reduced mass, but note that this correction is only approximate (see Willey et al. [@Willey02] and Daniel et al. [@Daniel06a]). The code works for both isotopologues. As one can probably neglect the variation of the electric dipole moment (Gerin et al. [@Gerin01]), the N$_2$D$^+$ Einstein spontaneous emission coefficients are derived from those of N$_2$H$^+$ by simply scaling them down by 1.2$^3$. Deexcitation collisional coefficients are kept the same as for N$_2$H$^+$ (following Daniel et al. [@Daniel06b]).
The local line profile in our 1D code takes into account both thermal and turbulent broadening, as well as any radial and rotational velocity fields. In principle, rotation introduces a deviation from spherical symmetry. However, for small rotational velocities (less than half the linewidth, typically), the excitation conditions do not noticeably change within a given radial shell, as shown by a previous 2–D version of this code (Pagani & Bréart de Boisanger, [@Pagani96]). Therefore the (much faster) 1D version remains sufficiently accurate.
Our Monte-Carlo model shows that excitation temperatures among individual hyperfine transitions within a given rotational line differ by up to 15 %, hence the usual assumption of a single excitation temperature to estimate the line opacity (e.g. in the CLASS HFS routine) is not fully accurate as already noticed by Caselli et al. ([@Caselli95]). Neglecting line overlap affects differentially the excitation temperature of hyperfine components when the opacity is high enough, mostly decreasing it, but also increasing it in a few cases. For example, in our best model for the L183 PSC, the excitation temperatures of individual hyperfine components change by up to 10%. As shown in the last column of Fig. \[n2hp\], a noticeable difference between models with and without overlap appears in the (J:3–2) line shape.
Grid of models {#sec:coremodel}
--------------
[cccccccccccccccccc]{} &&&&&&&&\
[Radius]{} & [V. rot.]{}&[Density]{} &T\_ & [Abundance]{}&[Abundance]{}& &[Density]{} &T\_ & [Abundance]{}& &[Density]{} &T\_ & [Abundance]{}& & [Density]{} &T\_ & [Abundance]{}\
[A.U.]{} &[(kms\^[-1]{})]{} & [(cm\^[-3]{})]{} &[(K)]{} & [N\_2H\^+]{} & [N\_2D\^+]{} & & [(cm\^[-3]{})]{} & [(K)]{} & [N\_2H\^+]{} & & [(cm\^[-3]{})]{}& [(K)]{} & [N\_2H\^+]{} & & [(cm\^[-3]{})]{}& [(K)]{} & [N\_2H\^+]{}\
330& 0 & 2.34(6)\^& 7 &2.4(-11) & 2(-11) & & 2.34(6)\^ & 7 & 7(-11) & & 3.30(6)\^& 7 & 1(-11) & & 3.46(6)\^& 9 & 1(-11)\
660& 0 & 2.05(6)\^& 7 &2.4(-11) & 2(-11) & & 2.05(6)\^ & 7 & 7(-11) & & 2.89(6)\^& 7 & 1(-11) & & 3.03(6)\^& 9 & 1(-11)\
990& 0 & 1.55(6)\^& 7 &8.5(-11) & 4(-11) & & 1.55(6)\^ & 7 & 7(-11) & & 2.19(6)\^& 7 & 7(-11) & & 2.29(6)\^& 9 & 3(-11)\
1320& 0 & 1.16(6) & 7 &8.5(-11) & 4(-11) & & 1.16(6) & 7 & 7(-11) & & 1.19(6) & 7 & 7(-11) & & 1.46(6)& 9 & 3(-11)\
1650& 0 & 9.27(5) & 7 &8.5(-11) & 4(-11) & & 9.27(5) & 7 & 7(-11) & & 7.74(5) & 7 & 7(-11) & & 9.09(5)& 9 & 3(-11)\
1980& 0 & 7.73(5) & 7 &8.5(-11) & 4(-11) & & 7.73(5) & 7 & 7(-11) & & 5.55(5) & 7 & 7(-11) & & 6.10(5)& 9 & 3(-11)\
2310& 0 & 6.62(5) & 7 &1.1(-10) & 3(-11) & & 6.62(5) & 7 & 1(-10) & & 4.20(5) & 7 & 4(-10) & & 4.39(5)& 9 & 1(-10)\
2640& 0 & 5.80(5) & 7 &1.1(-10) & 3(-11) & & 5.80(5) & 7 & 1(-10) & & 3.34(5) & 7 & 4(-10) & & 3.54(5)& 9 & 1(-10)\
2970& 0 & 5.15(5) & 7 &1.1(-10) & 3(-11) & & 5.15(5) & 7 & 1(-10) & & 2.74(5) & 7 & 4(-10) & & 2.68(5)& 9 & 1(-10)\
3300& 0.01 & 4.64(5) & 7 &1.1(-10) & 3(-11) & & 4.64(5) & 7 & 1(-10) & & 2.29(5) & 7 & 4(-10) & & 2.20(5)& 9 & 1(-10)\
3630& 0.01 & 4.21(5) & 7 &1.5(-10) & 3(-11) & & 4.21(5) & 7 & 1.5(-10)& & 1.95(5) & 7 & 3(-10) & & 1.83(5)& 9 & 6(-10)\
3960& 0.01 & 3.54(5) & 7 &1.5(-10) & 3(-11) & & 3.86(5) & 7 & 1.5(-10)& & 1.70(5) & 7 & 3(-10) & & 1.46(5)& 9 & 6(-10)\
4290& 0.02 & 3.01(5) & 7 &1.5(-10) & 3(-11) & & 3.57(5) & 7 & 1.5(-10)& & 1.49(5) & 7 & 3(-10) & & 1.22(5)& 9 & 6(-10)\
4620& 0.02 & 2.60(5) & 7 &1.5(-10) & 3(-11) & & 3.31(5) & 7 & 1.5(-10)& & 1.32(5) & 7 & 3(-10) & & 1.07(5)& 9 & 6(-10)\
4950& 0.02 & 2.26(5) & 7 &1.3(-10) & 5(-12) & & 3.09(5) & 7 & 5(-11) & & 1.18(5) & 7 & 3(-10) & & 9.27(4)& 9 & 4(-10)\
5280& 0.05 & 1.99(5) & 7 &1.3(-10) & 5(-12) & & 2.90(5) & 7 & 5(-11) & & 1.07(5) & 7 & 3(-10) & & 8.5(4) & 9 & 4(-10)\
5610& 0.05 & 1.76(5) & 7 &1.3(-10) & 5(-12) & & 2.73(5) & 7 & 5(-11) & & 9.67(4) & 7 & 3(-10) & & 7.20(4)& 9 & 4(-10)\
5940& 0.05 & 1.57(5) & 8 &1.3(-10) & 5(-12) & & 2.58(5) & 8 & 5(-11) & & 8.82(4) & 7 & 3(-10) & & 6.34(4)& 9 & 4(-10)\
6667& 0.05 & 1.27(5) & 8 &1(-10) & 4(-12) & & 2.02(5) & 8 & 2.5(-11)& & 8.10(4) & 7 & 1.5(-10) & & 5.25(4)& 9 & 4(-10)\
7733& 0.05 & 8.50(4) & 9 &1(-10) & 4(-12) & & 1.57(5) & 9 & 2.5(-11)& & 6.77(4) & 7 & 1.5(-10) & & 4.55(4)& 9 & 4(-10)\
8867& 0.05 & 6.50(4) & 10 &1(-10) & 4(-12) & & 8.50(4)\^ & 10& 2.5(-11)& & 5.42(4) & 7 & 1.5(-10) & & 3.46(4)& 9 & 4(-10)\
10000& 0.02 & 5.20(4) & 11 &1(-10) & 4(-12) & & 5.20(4)\^ & 11& 2.5(-11)& & 4.41(4) & 7 & 1.5(-10) & & 2.68(4)& 9 & 4(-10)\
13333& 0 & 3.45(4) & 12 &1(-10) & 4(-12) & & 3.45(4)\^ & 12& 2.5(-11)& & 3.68(4) & 7 & 1.5(-10) & & 1.70(4)& 9 & 4(-10)\
$\rho\propto$ r$^{-1}$ (r $<$ 4000 AU), $\rho\propto$ r$^{-2}$ (r $>$ 4000 AU)
The first three layer densities do not follow a power law but the density profile derived from the ISOCAM absorption map (Pagani et al. [@Pagani04])
Same as note (b) above, but rescaled to maintain a constant total column density of 1.4$\times$ 10$^{23}$ cm$^{-2}$ (see text).
For the pure $\rho\propto$ r$^{-1}$ case, density is not falling off fast enough to reach ambient cloud density and we correct for this in the last 3 layers.
\[tab:model\]
Our core model has an outer radius of 13333 AU $\simeq$ 2, fixed by the maximum width $\simeq$ 4of detectable N$_2$H$^+$ emission in the east-west direction across the PSC, as seen in large scale maps (PPABC). We assume that N$_2$H$^+$ abundance is zero outside of this radius (due to chemical destruction by CO).
For all models, the microscopic turbulence is set to $\Delta$v$_\mathrm{turb}$(FWHM) = 0.136kms$^{-1}$. This was found sufficient to reproduce the width of individual hyperfine groups, and is comparable to the thermal width contribution. A small rotational velocity field of a few tens of ms$^{-1}$ was further imposed beyond 3000 AU (see Table \[tab:model\]) to reproduce the small anti-symmetric velocity shift from PSC center observed at distances beyond $\pm$ 30 (cf. Section \[sec:obs\]). The radial velocity was kept to 0 in all layers. As seen in Figs. 1 and 2, this approximation already gives a remarkable fit to observed line profiles, and is therefore sufficient to derive the overall temperature and abundance structure.
We considered a variety of density profiles: single power law profiles $\rho \propto$ r$^{-1}$, $\rho \propto$ r$^{-1.5}$, $\rho \propto$ r$^{-2}$, as well as broken power laws with $\rho \propto$ r$^{-1}$ out to 4000 AU followed by r$^{-2}$, or by r$^{-3.5}$. For good accuracy, the density profile was sampled in 330 AU = 3 thick shells, i.e. 3 to 10 times smaller than our observational beam sizes. In all cases, divergence at $r = 0$ was avoided by adopting inside r = 990 AU the density slope derived from the ISOCAM data (Pagani et al. 2004). The density profiles were then scaled to give the same total column density N(H$_2$) = 1.4$\times$10$^{23}$ cm$^{-2}$ towards the PSC center, as estimated from the MAMBO emission map (Pagani et al. 2004). The effect of changing the total N(H$_2$) on the fit results will be discussed in Section \[sec:bestdenstemp\].
We then considered a variety of temperatures spanning the range 6–10 K in the inner core regions. We also tested non-constant temperature laws, increasing up to 12 K in the outermost layers and/or in the innermost layers.
For each combination of density and temperature laws, a range of N$_2$H$^+$ abundance profiles was investigated. To avoid a prohibitive number of cases, abundances were sampled in 6 concentric layers only, corresponding to the spatial sampling of our spectra (except the last, wider layer which encompasses the outermost two offsets), and multiple maxima/minima were forbidden. This lead to a total of about 40,000 calculated models with 8 free parameters (6 abundances, 1 temperature profile, 1 density profile). Table \[tab:model\] lists the values of H$_2$ densities for 4 density laws, together with the temperature and abundance profiles that best fitted the data in each case (cf. next section).
Selection criteria {#sec:selection}
------------------
Our best fit selection criteria are based on the use of several $\chi^2$ and (reduced) $\chi^2_{\nu}$ estimates. Because the N$_2$H$^+$ (J:1–0) data have much more overall weight than the (J:3–2) data (more isolated hyperfine groups per spectrum, more observed positions, higher signal-to-noise ratio), computing a single $\chi^2_\nu$ per model is hardly sensitive to the quality of the (J:3–2) line fit. However, the (J:1–0) spectra being optically thick, they give little constraints on the temperature and N$_2$H$^+$ abundance of the innermost layers (within a 6 radius). Those are only constrained by the IRAM (J:3–2) observations. The CSO (J:3–2) spectrum at (24,0) also adds interesting information on the density profile.
To constrain the models we thus evaluate the goodness of fit independently for the (J:1–0) and (J:3–2) data, by computing $\chi^2$ values on 4 types of measurements : 1) full area of each of the 7 (J:1–0) hyperfine components, at each of the 7 offset positions (49 values). This measurement is sensitive to the global temperature and abundance profiles. Fitting separately each hyperfine component allows to discard solutions with overall identical area but different hyperfine line ratios; 2) area of the 5 central channels (= 0.15 kms$^{-1}$) of each hyperfine component for the same lines (49 values), in order to reject self-absorbed profiles, i.e. line profiles which have the same total area but are wider and self-absorbed; 3) total area of the (J:3–2) CSO spectrum at (24,0) (1 value). This single measurement helps to constrain the density profile; 4) total area of the (J:3-2) 30-m spectrum (1 value) which constrains the abundance (and the temperature to some extent) at the core center. These four measurements are of course not completely independent from each other, and trying to improve one fit by changing a model parameter can degrade one or several of the others. We used $\chi^2_\nu$ for the first two (with 41 degrees of freedom) and $\chi^2$ for the last two (only one measurement) so that in all cases $\Delta\chi^2 \approx$ 1 is equivalent to a 1$\sigma$ deviation. With this choice, a 3$\sigma$ variation is equivalent to $\Delta\chi^2_\nu \approx$ 1.5 for the first two and $\Delta\chi^2$ = 9 for the last two.
After having run all of the models in our grid, we first used a single, global $\chi^2_\nu$ combining the four types of measurements to visualize the global fit quality of the various models. Fig. \[fig:chi2\] plots the global $\chi^2_\nu$ as a function of temperature, for the various density laws that we investigated. Symbols with arrows indicate models where temperature rises in the outermost layers. For each density-temperature pair, we plot only the smallest $\chi^2_\nu$ obtained by adjusting the abundance profile.
It is readily seen that the best fits are found for relatively low overall core temperatures $\simeq$ 6.5–8 K. Hence, the (J:1–0) data alone (which dominate the global $\chi^2_\nu$) set a strong constraint on the core temperature: indeed, the combination of strong opacity and relatively low intensity in this line requires low excitation temperature. Because the (J:1–0) line should be thermalized in the dense center, this rules out kinetic temperatures above 8 K (for our adopted total column density of 1.4$\times$ 10$^{23}$ cm$^{-2}$).
To further discriminate among models, we then looked individually at the four quality indicators described above, in particular those related to the (J:3-2) lines. Table \[tab:chi2\] lists these 4 values as well as the global $\chi^2_\nu$ for various models selected from Fig. \[fig:chi2\], namely: the best model fits for a given constant core temperature (from 6 to 10 K), and the best model fits for a given density law (with corresponding temperature and abundance distributions given in Table \[tab:model\]).
[llcccccc]{} &&&&\
& Total area $\chi^2_{\nu}$& 5 channels area $\chi^2_{\nu}$&& CSO Total area $\chi^2$& IRAM Total area $\chi^2$&Global $\chi^2_{\nu}$\
\
T = 6 K &($\rho\propto$ r$^{-1}$,r$^{-2}$)&4.7 &3.8 && 1.5 & 1.0& 3.9\
T = 7 K &($\rho\propto$ r$^{-1}$,r$^{-2}$) &1.9 & 2.4 && 12.6 & 4.1& 2.1\
T = 8 K &($\rho\propto$ r$^{-1.5}$)&3.1 & 3.2 && 0.2& 3.4& 2.9\
T = 9 K &($\rho\propto$ r$^{-2}$)& 4.5 & 5.6&& 7.5& 6.8& 4.7\
T = 10 K &($\rho\propto$ r$^{-2}$)& 4.8& 6.3&& 16.9& 28.7& 5.5\
\
&[**(T = 7 $\rightarrow$ 12 K)**]{}&[**2.4**]{}&[**2.7**]{}&&[**0.3**]{}&[**0.0**]{}&[ **2.3**]{}\
$\rho\propto$ r$^{-1}$&(T = 7 $\rightarrow$ 12 K)& 3.7&3.8&&1.2&2.9&3.4\
$\rho\propto$ r$^{-1.5}$&(T = 7 K)&3.4&3.2&&0.1&0.1&2.9\
$\rho\propto$ r$^{-2}$&(T = 9 K)&4.5&5.6&& 7.5&6.8&4.7\
\
X(N$_2$H$^+$) = &10$^{-12}$&2.6 & 2.8&& 0.3& 2.0& 2.5\
X(N$_2$H$^+$) =&8$\times$10$^{-12}$&2.6 & 2.8&& 0.3& 1.0& 2.5\
[**X(N$_2$H$^+$) =**]{}&[**2.4$\times$10$^{-11}$**]{}&[**2.4**]{}&[**2.7**]{}&&[**0.3**]{}&[**0.0**]{}&[**2.3**]{}\
X(N$_2$H$^+$) =&8$\times$10$^{-11}$&2.1 & 2.4 && 0.5 & 9.2 &2.1\
\[tab:chi2\]
It can be seen in Table \[tab:chi2\] that for T$_\mathrm{kin}$ = 10 K the (J:3–2) lines are not well fitted (4 and 5 $\sigma$ deviations for the CSO and IRAM lines respectively), hence such a high temperature seems ruled out here. Overall, it is rather difficult to have low $\chi^2$ values in both the (J:1-0) and the (J:3-2) indicators. Only one density-temperature combination (indicated by bold faces in Table \[tab:chi2\]) reaches low values in all 4 individual $\chi^2_\nu$ and $\chi^2$, each within 1$\sigma$ of their minimum value, among all the models in Table \[tab:chi2\]. This combination is referred to as our “best model” in the following, and its inferred physical parameters are discussed in the next section. It can also be noticed in Table \[tab:chi2\] that the model with T$_\mathrm{kin}$ = 7 K has a slightly better global $\chi^2_\nu$ than the best model (2.1 compared to 2.3) but is to be eliminated as the (J:3–2) lines are 2 and 3.5 $\sigma$ off the mark.
Physical conditions in the L183 main core {#sec:bestfit}
-----------------------------------------
### Density and temperature structure {#sec:bestdenstemp}
Our best density-temperature combination (circled 5-branch star symbol in Fig. \[fig:chi2\]) has a density law of r$^{-1}$ out to r $\approx$ 4000 A.U., consistent with the ISOCAM profile (Pagani et al. 2004) and a slope of r$^{-2}$ beyond. The temperature is constant at 7 K out to 5600 A.U. and increases up to 12 K at the core edge (see Table \[tab:model\]). Further increasing the number of warmer layers failed, as well as introducing warmer layers in the center. Keeping the temperature constant at 7 K everywhere significantly worsens the fit of the (J:3–2) lines (see Table \[tab:chi2\]).
The main uncertainty in our temperature determination stems from the assumed total gas column density through the core. Decreasing/increasing it by a factor 1.4 (the typical uncertainty found by Pagani et al. 2004) changes all volume densities by the same factor. To recover the same N$_2$H$^+$ emission, we find that the kinetic temperature in our models must be increased/decreased by $\simeq$1 K (respectively, while the N$_2$H$^+$ abundance must be scaled accordingly to keep the same column density). Since the dust temperature in the L183 core is 7.5 $\pm$ 0.5 K (Pagani et al. 2004), gas and dust are thermalized inside this core within the uncertainties.
A different result was found by Bergin et al. ([@Bergin06]) in the B68 PSC, where outer layers emitting in CO are consistent with a kinetic temperature of 7–8K while NH$_3$ measurements indicate a higher temperature of 10–11K in the inner 40. This inward increase in gas temperature was attributed to the lack of efficient CO cooling in the depleted core center, and requires an order of magnitude reduction in the gas to dust coupling, possibly due to grain coagulation (Bergin et al. [@Bergin06]). Such an effect is not seen in L183, despite strong CO depletion within the inner 1= 6600 AU radius. Our best fit temperature law is more consistent with the thermo-chemical evolution of slowly contracting prestellar cores with standard gas-dust coupling coefficients, which predicts low gas temperatures $\simeq$ 6 K near the core center, increasing to $\simeq$ 14 K near the core surface (Lesaffre et al. [@lesaffre]).
### N$_2$H$^+$ abundance profile {#sec:bestabun}
With our best density-temperature model, several N$_2$H$^+$ abundance profiles give equally good fits (within $<$1$\sigma$ on all 4 quality indicators). From all these acceptable abundance profiles, we determined a median profile (plotted as a solid histogram in Fig. \[ratio\]a), with error bars representing the range of acceptable values in each layer (though not any combination of these values does fit the observations). This median profile, whose values are listed in Table \[tab:model\], is used to compute the fit displayed in Fig. \[n2hp\]. The total N$_2$H$^+$ column density is 1.2$\pm 0.1$ $\times$ 10$^{13}$ cm$^{-2}$, comparable with the value reported by Dickens et al. ([@Dickens]) and a factor of $\sim$2 below that in Crapsi et al. ([@Crapsi05]).
The maximum N$_2$H$^+$ abundance is 1.5$_{-0.3}^{+0.4}$ $\times$ 10$^{-10}$, the same as found by Tafalla et al. ([@Tafalla04]) in L1498 and L1517. However, there is a definite drop in abundance in the inner layers. This drop is imposed by the fit to the (J:3–2) IRAM central spectrum (9 beam), which is quite sensitive to variations in the central abundance of N$_2$H$^+$, as illustrated in the last rows of Table \[tab:chi2\]. In contrast, the (J:1–0) line is optically thick at the core center and the $\chi_\nu^2$ is thus not sensitive to this parameter (cf. Table \[tab:chi2\]).
Exploring a broad range of abundance profiles, we find only a limited range of possible N$_2$H$^+$ abundances in the inner layer of 6 radius (r $<$ 660 AU): values above 4.5$\times$10$^{-11}$ always produce too strong (J:3–2) emission in the central IRAM beam ($\chi^2 > 1$), while abundances below 10$^{-12}$ give a (J:3–2) line marginally too weak. We thus derive a range of 2.4 $^{+2.1}_{-2.3}$$\times$10$^{-11}$ for the central N$_2$H$^+$ abundance. However, we note that our results for N$_2$D$^+$ would favor a value of at least 10$^{-11}$ (see next section).
Compared with the maximum abundance, this gives a volume depletion factor of $\simeq$ 6$^{+13}_{-3}$ at the core center. The median abundance drop is less than (but marginally consistent with) that expected from simple geometrical arguments in PPABC, but it confirms that the leveling of N$_2$H$^+$ intensity seen across the dust peak is not due to pure opacity effects. As can be seen from Fig. \[ratio\]a-b, depletion starts at a density of 5–7$\times$10$^5$ cm$^{-3}$ and increases as density goes up, in agreement with the conclusions of PPABC. The abundance also drops slightly in the outermost regions, possibly due to partial destruction by CO. Indeed, the outermost layers of the model have densities of a few 10$^4$ cm$^{-3}$ which is the limit above which CO starts to deplete in this source (PPABC).
### N$_2$D$^+$ abundance profile {#sec:bestn2dp}
As seen in Fig.\[n2dp\], good fits to the N$_2$D$^+$ data may be obtained with the same density and temperature profile as our best model for N$_2$H$^+$, although it is difficult to reproduce simultaneously the intensities of both (J:1-0) and (J:3-2) lines. We plot in Fig. \[ratio\]a the N$_2$D$^+$ abundance profiles giving the best fit either for (J:1-0) and (J:2-1) (dashed histogram) or for (J:2-1) and (J:3-2) (dotted histogram). The latter was used to produce Fig. \[n2dp\]. A better fit could be obtained with a temperature of 8K instead of 7K. This might indicate that the collisional coefficients with He are somewhat inaccurate to represent those with H$_2$. Indeed, it has been shown for NH$_3$ that collisional coefficients with He could differ by a factor up to 4 with respect to those computed with para-H$_2$ (Willey et al. [@Willey02]). A similar problem may occur here (see also the discussion in Daniel et al. [@Daniel06a]). Therefore we consider that the temperatures are compatible within the uncertainties on the collisional coefficients.
The N$_2$D$^+$ abundance profile is quite different from that of N$_2$H$^+$ (cf. Fig. \[ratio\]a). Its abundance is essentially an upper limit beyond 6000 A.U. It increases sharply by about an order of magnitude in the region between 600 and 4000 A.U., then slightly drops by a factor of 2–2.5 in the core center, reaching an abundance between 1.3 and 2$\times$10$^{-11}$. The low optical depth of the line ($\tau$ = 0.84 for the strongest hyperfine component, J$_{FF\arcmin}$:1$_{23}$–0$_{12}$) allows to measure the contribution of all layers to the emission and to determine with relatively little uncertainty the abundance profile. In particular, for the chosen density and temperature profiles, we find that the abundance of N$_2$D$^+$ in the center of the cloud cannot be below 10$^{-11}$. Hence, we can set tighter constraints on the N$_2$D$^+$ abundance at the core center than was possible for N$_2$H$^+$.
The N$_2$D$^+$/N$_2$H$^+$ ratio obtained by comparing the N$_2$D$^+$ range of abundances to the median N$_2$H$^+$ abundance profile is plotted in Fig. \[ratio\]b. The deuteration ratio varies from an upper limit $\leq$0.05–0.1 away from the core to a very high factor of 0.7$\pm$0.12 in the depletion region. This is larger than what Tine et al. ([@Tine]) reported 48 further north in the same source, and 3–4 times larger than what Crapsi et al. ([@Crapsi05]) report towards the PSC. Deuterium enrichment is thus very strong and can certainly be linked to the strong and extended H$_2$D$^+$ line detected towards this source (Vastel et al. [@Vastel06]). If we consider the full range of possible values for N$_2$H$^+$ itself, the deuteration ratio in the inner layer ranges from $\sim$0.3 to $\geq$ 20. However, as D$_2$H$^+$ has not been detected in this source despite its strong H$_2$D$^+$ line (Vastel et al. [@Vastel06]), an enrichment above 1 seems unprobable, suggesting that the central abundance of N$_2$H$^+$ is probably at least 10$^{-11}$.
Comparison with NH$_3$ temperature estimates
--------------------------------------------
The low kinetic temperatures $\le$8 K inferred from our N$_2$H$^+$ and N$_2$D$^+$ modelling are somewhat lower than previous temperature estimates in L183 from NH$_3$ inversion lines, which gave values in the range 9–10 K (Ungerechts et al. [@Ungerechts80], Dickens et al. [@Dickens]) up to 12 K (Swade et al. [@Swade89]). However, these NH$_3$ spectra were not obtained towards the PSC center itself, and had relatively low angular resolution for the last two. We thus briefly reconsider this issue using our NH$_3$ spectra obtained at the PSC center position, which also benefit from the higher resolution and beam coupling of the new GBT. The NH$_3$ spectra were analyzed in the standard way, as discussed by Ho & Townes ([@Ho83]) and Walmsley & Ungerechts ([@Walmsley83]). Using the CLASS NH3 fitting procedure, we found a total opacity of 24 for the (1,1) inversion line, and an excitation temperature of 5.5 K assuming a beam filling-factor of 1 (Figs. \[nh311\] & \[nh322\]). Making the usual assumption of constant temperature on the line of sight and of negligible population in the non-metastable levels, the intensity ratio of the (2,2) to (1,1) main lines indicates a rotation temperature of 8.4$\pm$0.3K which should correspond to a kinetic temperature of 8.6$\pm$0.3K. Hence, NH$_3$ emission towards the PSC center indicates an only slightly higher kinetic temperature than N$_2$H$^+$ (8.6$\pm$0.3K instead of 7$\pm$1K), almost equal to that obtained with N$_2$D$^+$ ($\sim$8K). The discrepancy between NH$_3$ and N$_2$H$^+$ is thus not as large as originally thought. Both tracers point to very cold gas in the core, close to thermal equilibrium with the dust.
Reasons for a possible difference between NH$_3$ and N$_2$H$^+$ temperature determinations in L183 include the following: 1) the higher sensitivity of NH$_3$ to the warmer outer layers of the core, since NH$_3$ inversion lines are much easier to thermalize (n$_\mathrm{crit} \approx$ 2000 cm$^{-3}$) than N$_2$H$^+$ lines. 2) a slight overestimate in the total column density towards the PSC (reducing it to 10$^{23}$cm$^{-2}$, the temperature has to be raised to 8 K to compensate for the density decrease and recover the same N$_2$H$^+$ emission). 3) systematic errors introduced by the use of collisional coefficients with He instead of H$_2$ for N$_2$H$^+$ (see Sect. \[sec:bestn2dp\]). 4) concerning NH$_3$, standard hypotheses leads to a puzzling discrepancy between T$_\mathrm{ex}$ = 5.5 K and T$_\mathrm{rot}$ = 8.4 K towards the L183 PSC center, where NH$_3$ inversion lines should be fully thermalized. Beam dilution has been invoked for giant molecular clouds (thus increasing T$_\mathrm{ex}$), but it seems unreasonable to extend this to dense cloud cores (Swade [@Swade89], and references therein). A Monte-Carlo code (or equivalent) would be needed to model NH$_3$ taking into account the strong density gradients present in PSCs, and the possible population of non-metastable levels at very high densities.
Conclusions
===========
1. We have presented a new Monte-Carlo code (available upon request to the author) to compute more realistically the NLTE emission of N$_2$H$^+$ and N$_2$D$^+$, taking into account both line overlap and hyperfine structure. This code may be used to infer valuable information on physical conditions in PSCs.
2. The best kinetic temperature to explain N$_2$H$^+$ observations of the L183 main core is 7$\pm$1 K (and $\sim$ 8K for N$_2$D$^+$) inside 5600 AU, therefore gas appears thermalized with dust in this source.
3. There is no major discrepancy with NH$_3$ measurements which also indicate very cold gas (8.6$\pm$0.3K) towards the PSC.
4. We have found a noticeable depletion of N$_2$H$^+$ by a factor of 6$^{+13}_{-3}$, and of N$_2$D$^+$ by a smaller factor of 2–2.5. This smaller depletion is probably due to a strong (0.7$\pm$0.12) deuterium fractionation, consistent with the detection of H$_2$D$^+$ in this core.
5. N$_2$D$^+$ should be a useful probe of the innermost core regions, thanks to its low optical depth combined with its strong enhancement.
We thank the IRAM direction and staff for their support, S. Léon for his dedicated assistance during pool observing, and F. Daniel for providing routines to compute the frequencies and A$_{ul}$ coefficients of N$_2$H$^+$. We also thank an anonymous referee and C.M. Walmsley for suggestions which helped to improve this paper.
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Pagani, L., Pardo, J.-R., Apponi, A.J., Bacmann, A., and Cabrit, S., 2005, A&A 429, 181 (PPABC) Swade, D.A., 1989, ApJ, 345, 828 Tafalla, M., Myers, P.C., Caselli, P., Walmsley, C.M., Comito, C., 2002, ApJ 569, 815 Tafalla, M., Myers, P.C., Caselli, P., Walmsley, C.M., 2004, A&A 416, 191 Teyssier, D., Hennebelle, P., Pérault, M., 2002, A&A, 382, 624 Tiné, S., Roueff, E., Falgarone, E., Gerin, M., Pineau des Forêts, G., 2000, A&A 356, 1039 Ungerechts, H, Walmsley, C.M., Winnewisser, G., 1980 A&A, 88, 259 Vastel, C., Phillips, T. G., Caselli, P., Ceccarelli, C., Pagani, L. 2006, proceedings of the Royal Society meeting Physics, Chemistry, and Astronomy of H+ \[arXiv:astro-ph/0605126\] Walmsley, C. M., & Ungerechts, H., 1983, A&A 122, 164 Walmsley, C. M., Flower, D. R., Pineau des Forêts, G. 2004, A&A, 418, 1035 Willey, D.R., Timlin, R.E., Jr, Merlin, J.M., Sowa, M.M., Wesolek, W.M., 2002 ApJS, 139, 191
[^1]: Based on observations made with the IRAM 30-m and the CSO 10-m. IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain).
[^2]: The Fortran code is available upon request to the author
[^3]: http://www.iram.fr/IRAMFR/GILDAS/
[^4]: Teyssier et al. ([@Teyssier02]) provide appropriate corrections for 30-m data, but only for uniform circular disk sources, and for data taken before the 1998 surface readjustment.
[^5]: The last surface readjustment of the 30-m in 1998 has not been characterized in detail, therefore we have scaled down the error beam coupling coefficients given in Greve et al. ([@Greve98]) to retrieve the new, improved beam efficiencies
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abstract: 'Melanoma is the deadliest form of skin cancer. Automated skin lesion analysis plays an important role for early detection. Nowadays, the ISIC Archive and the Atlas of Dermoscopy dataset are the most employed skin lesion sources to benchmark deep-learning based tools. However, all datasets contain biases, often unintentional, due to how they were acquired and annotated. Those biases distort the performance of machine-learning models, creating spurious correlations that the models can unfairly exploit, or, contrarily destroying cogent correlations that the models could learn. In this paper, we propose a set of experiments that reveal both types of biases, positive and negative, in existing skin lesion datasets. Our results show that models can correctly classify skin lesion images without clinically-meaningful information: disturbingly, the machine-learning model learned over images where no information about the lesion remains, presents an accuracy above the AI benchmark curated with dermatologists’ performances. That strongly suggests spurious correlations guiding the models. We fed models with additional clinically meaningful information, which failed to improve the results even slightly, suggesting the destruction of cogent correlations. Our main findings raise awareness of the limitations of models trained and evaluated in small datasets such as the ones we evaluated, and may suggest future guidelines for models intended for real-world deployment.'
author:
- |
Alceu Bissoto^1^ Michel Fornaciali^2^ Eduardo Valle^2^ Sandra Avila^1^\
^1^Institute of Computing (IC) ^2^School of Electrical and Computing Engineering (FEEC)\
RECOD Lab., University of Campinas (UNICAMP), Brazil\
bibliography:
- 'egbib.bib'
title: '(De)Constructing Bias on Skin Lesion Datasets'
---
Introduction {#sec:intro}
============
The amount of people diagnosed with melanoma is rapidly increasing in the past decades. Today, it is already treated as a public health challenge, especially in high sun exposition areas with Caucasian populations[^1]. Melanoma is the deadliest form of skin cancer, and early detection is crucial [@survival-rates-skin-cancer] for good prognosis, creating a need for efficient early-detection techniques, and thus an incentive for research on automated detection.
Deep learning methods are the state-of-the-art on skin cancer classification [@esteva2017; @haenssle2018man]. That task is challenging due to the vast visual variability of skin lesions, and the subtlety of the cues that differentiate benign and malignant cases. To compound the difficulty, datasets to train the data-hungry models are small, when compared with general-purpose image datasets (, ImageNet, MSCOCO, LabelMe).
Due to the scarcity of good-quality, annotated skin lesion images, two datasets dominate research on automated skin lesion analysis: the Interactive Atlas of Dermoscopy [@argenziano2002dermoscopy] and the ISIC Archive [@isicarchive]. The Atlas is an educational medical resource, with many standardized metadata over the cases it contains, while the ISIC Archive is a much larger, but also less controlled dataset, with images of different sources. Nowadays nearly every reproducible work in the field refer to these datasets for training, evaluating or comparing its models [@celebi2015state; @valle2018data; @bissoto2018skin; @brinker2019comparing], and the ISIC Archive deserves special mention as the source of the images used in the ISIC Challenge [@isic2016; @isic2017; @isic2018], an annual event where different teams compare the performance of their algorithms under the controlled supervision of the organizers.
The problem of having so few, relatively small datasets dominating much of research in automated skin analysis, is the risk of datasets biases. Indeed, the (re)use of relatively small datasets by a research community poses certain risks for research on Machine Learning [@peng2011reproducible]. Dataset biases may both inflate the performance of models (presenting them features that are not truthful to real-world data), or play down their performance (by destroying correlations that occur in real-world data, and thus preventing models from exploiting them). If we think of general datasets, there can be bias over the scenes (rural or urban), acquisition methods (professional or amateur), amount of objects in the scene, angles of views, among other factors [@torralba2011unbiased].
If bias is present even in bigger and more diverse datasets [@torralba2011unbiased] like ImageNet [@ILSVRC15], it is naive to think it is not present in the smaller and harder to obtain skin cancer datasets, where we lack works identifying the possible sources of dataset bias. We know, however, that there are visible artifacts introduced during the image acquisition process (, dark corners, marker ink, gel bubbles, color charts, ruler marks, skin hair) [@mishra2016overview] that could inflate models performances due to spurious correlations.
Despite being impossible to wholly eliminate, it is important to understand bias and its sources to further improve our image acquisition processes and deep learning models.
A useful way to measure the first possible effect of a dataset bias (undue inflation of performances due to spurious correlations in the dataset), is a counterfactual experiment, which destroys the cogent information in the data, and measures how much the performance of models drops. Therefore, our first set of experiments follows that procedure, gradually removing information from skin lesion images, and measuring the network performance. We perform single- (training and testing on the same dataset) and cross-dataset (training on ISIC and testing on Atlas) experiments, and find that in both cases, the networks are able to maintain a surprising amount of accuracy, even after almost all cogent information has been destroyed.
Measuring the second possible effect (inability to provide useful correlations for learning) is much harder, since we cannot, *a priori* prove those correlations exist in the real-world, neither that the machine-learning model would learn from them if they were correctly represented in the dataset. The best we can do is to provide additional evidence for the models that we expect would be useful for a human, and measure if that makes any difference.
Thus, in our second experiment set, we add progressively more features, based upon fine-grained dermoscopic attributes (pigment network, negative network, streaks, milia-like cysts, and globules) spatially located on the lesions. In order to provide those features, we employ the annotations available for the Task 2 (Lesion Attribute Detection) of the ISIC Challenge. We expected that such clinically-meaningful skin lesion information would improve the network learning process, but in fact, the performance fails to improve in all scenarios we tested, even when we feed the network with all the image’s pixels with an additional channel containing extra clinically-meaningful information.
Summarizing, the main contributions in this work are:
- We assess whether the models’ performance are inflated due to dataset bias by performing a counterfactual experiment, where we gradually destroy meaningful information in the data, and measure the performance of our models.
- We evaluate whether the dataset is providing useful correlations for learning, by gradually feeding the network with extra clinically-meaningful information.
- We provide a discussion to raise awareness of bias in the automated skin lesion analysis community to improve the next generation of solutions for classifying skin lesions in the real world.
We organized the text as follows. We introduce our motivation and related works in Section \[sec:intro\]. We present our methodology, materials and goals in Section \[sec:matmeth\]. We detail our experiment to gradually destroy clinically-meaningful information in skin lesion images and evaluate the network’s responses in Section \[sec:destruct\]. We detail our experiment where we try to guide the network’s learning process through additional clinical information in Section \[sec:construct\]. Finally, we review and discuss our findings in Section \[sec:conclusion\].
Materials and Methods {#sec:matmeth}
=====================
Datasets
--------
We employ two of the most important skin lesion datasets: the Interactive Atlas of Dermoscopy (Atlas) [@argenziano2002dermoscopy] and the International Skin Imaging Collaboration (ISIC) Archive [@isicarchive]. While Atlas excels for having rich metadata associated with each lesion image, ISIC excels for being diverse. Since both are publicly available, most of the recent works in skin lesion analysis rely on these datasets. When exploring reproducible works on lesion segmentation [@celebi2015state; @xue2018adversarial], dermoscopic attribute segmentation [@kawahara2018fully], skin lesion classification [@valle2018data; @brinker2019comparing; @perez2018data], or skin lesion synthesis [@bissoto2018skin] those two datasets are almost certain to be included. Next we describe their individual characteristics, and discuss how they differ.
The **Atlas** [@argenziano2002dermoscopy] is a medical educational dataset composed of $+1,000$ cases of pigmented skin lesions. Each case is associated with clinical and dermoscopic images. Each skin lesion has clinical data (, location, diameter, elevation), histopathological results, diagnosis, and the presence or absence of dermoscopic attributes. The presence of those rich metadata correspond to the pedagogical objectives of the Atlas of teaching dermoscopy through reliable and understandable medical algorithms (, the 7-point checklist). The Atlas also groups the lesions according to their level of diagnostic difficulty (low, medium or high), which indicates how difficult it is to identify the medical attributes (, networks, dots-and-globules, etc.) in the lesions. The difficulty relies on the morphological variability of a given criterion, which explains the sometimes low intra- and interobserver agreement of such medical algorithm.
Since it is a dataset for medical education purposes, the statistics of its metadata do not necessarily reflect their occurrences in any real-world population. In this work, we are especially interested in the dermoscopic attributes annotation. Lesions’ dermoscopic attributes analysis (through pattern-based medical algorithms) is crucial for dermatologists to diagnose skin cancer. This information enable us to verify bias by comparing the medical algorithm performance, the network performance, and an Artificial Intelligence benchmark for melanoma classification [@brinker2019comparing]. Segmentation masks, which is also relevant to this discussion are not available, but we employ computational methods to obtain them.
For our experiments, we select only the dermoscopic samples, remove “duplicates” (some medical cases have multiple images), and include only the classes present in the dataset of task $2$ of $2018$ ISIC Challenge (melanoma, nevus, and seborrheic keratosis). Those alterations result in a dataset containing $872$ images.
The **ISIC Archive (ISIC)** dataset [@isicarchive] is a bigger and more generic dataset, composed of more than $23,000$ images collected from different leading clinical centers internationally, using a variety of devices for acquisition. Since the first ISIC Challenge in 2016 [@isic2016], this dataset is increasing in size and in the amount of information available for each lesion. Segmentation masks and maps over five dermoscopic attributes (pigment network, negative network, streaks, milia-like cysts, and globules) are available for smaller subsets of the dataset.
It is import to note that the dermoscopic attributes annotations in ISIC and Atlas differ in two ways. First, in ISIC the annotation is a mask that maps the dermoscopic attributes in the original images. In the Atlas dataset, we only have the information about the presence or absence of each dermoscopic attribute. Second, the two datasets annotated information about different dermoscopic attributes, with different levels of detail. Unfortunately, only the patterns present in the Atlas dataset allow to apply (and evaluate) the medical pattern-based algorithms .
For all of this work’s experiments, we use only the data from the second task of the ISIC 2018 Challenge [@isic2018] for dermoscopic attribute detection. This subset contain $2,594$ lesions’ dermoscopic attributes information.
For **both** datasets, the class frequencies (types of skin lesions, , melanomas, nevi, keratoses) do not reflect any real-world population. That, however, is a necessity for training and evaluating machine-learning models, since in real-world populations, the proportion of melanomas to nevi, for example, is *extremely* small, generating huge imbalances that most models would not tolerate. The need to “rebalance” the classes for machine-learning, however, can generate models biased towards some classes, and inflate the rate of false positive for melanomas, for example.
Methodology
-----------
To evaluate the presence and effect of dataset bias in Atlas and ISIC, we propose to:
- Perform destructive actions (see Figure \[fig:atlasmodifications\]) in the dataset to analyze if the network can still learn patterns to correctly classify skin lesions, even without clinically-meaningful information available.
- Apply the 7-point checklist algorithm [@7points] to the Atlas dataset, and analyze the result comparing it with the recent melanoma classification benchmark for AI [@brinker2019comparing] to verify how biased it is due to its educational purposes and acquisition methods.
- Perform constructive actions (see Figure \[fig:isicmodified\]) in the dataset, building from clinically-meaningful information to guide the network’s learning, to analyze if the result improves.
To accomplish our goals, we propose destructive and constructive actions in the target datasets. We present the details of each destructive action and the result data in Section \[sec:destruct\], and the details of each constructive action and the result data in Section \[sec:construct\]. Next, we introduce our ideas to exploit the deep neural network learning capabilities.
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\[fig:atlasrgb\]
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\[fig:atlasbg\]
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\[fig:atlasbbox\]
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**Destructing Atlas-dataset**: We employ the Atlas dataset with our disruptive actions for both training and testing the network in the destruction of information approach. We use $10$ splits that we keep the same throughout all sets of images (*Traditional, Only Skin, Bounding Box, Bounding Box 70%*) to make comparisons fair. To compose each training split, we randomly select $70\%$ of the images of each diagnostic difficulty present in the Atlas dataset (low, medium and high). We compose the corresponding test split using the $30\%$ that is left. Following this procedure, we reduce the possibility of biasing our results with a split that is especially good for a given set of images. Since the training and test sets come from the same data distribution (same dataset), we expect these results to be optimistic, and that motivates our three next designs. **Destructing ISIC-dataset**: We also apply the destruction of information approach to the ISIC dataset. We do that to confirm the behavior verified in Atlas in a more generic dataset, with fewer effects of human bias. We apply the same $10$ split generation procedure we described for this experiment, except for the diagnostic difficulty stratification (the information is not present for the ISIC dataset).
**Destructing Cross-dataset**: We increase the difficulty by experimenting with a cross-dataset fashion. We train with all $2,594$ samples from the ISIC dataset and evaluate on the complete $872$ images set from Atlas. The differences between the statistics between those two datasets make this task harder, and better reflect a real-world setting [@torralba2011unbiased]. We repeat that experiment $10$ times, for statistical significance.
**Constructing ISIC-dataset**: We attempt to guide the network’s learning using the dermoscopic attribute information available for the ISIC dataset. We create three sets of images (, *Grayscale Attributes, RGB Attributes, Traditional $+$ Grayscale Attributes*), where the amount of information is gradually increased (see Section \[sec:construct\]). We keep the same training procedure and splits from *Destructing ISIC-dataset*.
Training and Evaluation Setup
-----------------------------
We use the same network architecture and hyperparameters for all experiments. We employ an Inceptionv4 network [@szegedy2017inception], widely used for computer vision, and well-established for skin lesion analysis. To train each network, we use Stochastic Gradient Descent (SGD) with momentum $0.9$, weight decay $0.001$ and learning rate $1$e-$3$, which we reduce to $1$e-$4$ after epoch $25$. We use a batch size of $32$, shuffling the data before each epoch.
We fine-tune the ImageNet [@ILSVRC15] pre-trained network to the target dataset. We resize the input images to $299\times299$ to fit the input size of Inceptionv4. To augment the dataset [@perez2018data], we apply random horizontal and vertical flips, random resized crops that contain from $75\%$ to $100\%$ of the original image, random rotations between $-45$ and $45$ degrees, and random hue changes between $-20\%$ to $20\%$. We apply the same augmentations on both train and test. For the evaluation, we average the predictions over $50$ augmented versions of each image. We normalize the input using the z-score, computed on ImageNet’s training set mean and standard deviation. For all experiments, we report the Area Under the ROC Curve (AUC).
We limit our datasets to contain lesions’ dermoscopic attributes for every sample, shrinking ISIC considerably. Since both our datasets are relatively small, we choose not to use a validation set, using the weights after the $60$th epoch for test evaluation[^2].
Information Destruction Experiments {#sec:destruct}
===================================
In this section, we detail our information destruction experiments. We intend to investigate the presence of dataset bias by gradually removing cogent information. First, we introduce the disrupted datasets used and proceed to show and discuss our results.
Data
----
Next, we present the different datasets modifications made for our first experiments and our motivations behind each one. In Figure \[fig:atlasmodifications\] we show examples of each variation. We point that we keep the same modifications for both training and testing our networks.
**Traditional**: This dataset contains the usual information used for training and evaluating skin lesion analysis networks. The images contain all pixels’ information and we expect it to have the highest scores in our tests, being our upper bound baseline.
**Only Skin**: To create this dataset, we take advantage of segmentation masks. We apply the mask in the samples from the *Traditional* dataset, removing the pixels’ information (they turn black) inside the actual lesion. We keep only the silhouette of the lesion and the skin of the image. Our intention when creating this dataset is to destroy the lesion information while verifying if the network could still make sense of the remained pixels to classify the samples correctly. Unlike the ISIC dataset, the Atlas dataset does not provide the lesions’ ground truth segmentation masks. To obtain them, we choose to use the SeGAN model [@xue2018adversarial], which placed $4$th on the segmentation task at the 2018 ISIC Challenge making use of a generative approach for skin lesion segmentation.
**Bounding Box (Bbox)**: The lesion border is an essential feature to diagnose skin lesions. The classic ABCD medical algorithm [@abcd] consider this feature, which accounts border symmetry and border regularity. To destroy this information from the dataset, we cover the silhouette of the lesion with a black bounding box. At this point, we already removed the lesion and its borders information. Only healthy skin and artifacts reminiscent from the image acquisition process are available for the network to learn.
**Bounding Box $70$% (Bbox$70$)**: The diameter (size) of the lesion is considered by dermatologists to diagnose skin lesions since melanomas are usually bigger (start with a diameter of more than $6$mm [@friedman1985early] than benign lesions. The diameter is the last clinical feature we attempt to remove from the network’s learning possibilities. For this purpose, we define that every bounding box must at least have the size of a $250\times250$ square (note that images are $299\times299$). We keep intact bounding boxes that need to be bigger to cover the lesion. The $250\times250$ square is sufficient to cover $70\%$ of the pixels. We place this square at the center of the lesion. If the lesion is not in the center of the image, part of the box is not visible. In these specific cases, it is possible that the bounding boxes cover less than $70\%$ of the pixels. At this point, there is no information left to apply any of the factors from the ABCD [@friedman1985early; @abcd], ABCDE [@abcde] or any pattern-based algorithm [@7points].
Results and Discussion
----------------------
We employ the melanoma classification benchmark [@brinker2019comparing] to measure the expected performance for dermatologists, in an unbiased scenario. This benchmark is the result of a study with $157$ German dermatologists to be a reliable benchmark for artificial intelligence algorithms. Brinker ’s procedure were to send an electronic questionnaire to dermatologists containing $100$ dermoscopic images ($80$ nevi and $20$ biopsy-verified melanoma) randomly chosen from the ISIC Archive, asking for their evaluation. The AUC achieved by dermatologists for dermoscopic images (which is the case for our Atlas set) is $67\%$.
We employ 7-point checklist [@7points], a score-based medical algorithm, to verify bias in the Atlas dataset. This way we can isolate the neural network’s learning capabilities. Dermatologists use attribution pattern analysis to diagnose malignant cases. The 7-point medical algorithm assigns a score to each of the dermoscopic attributes. The medical practitioner needs to accumulate the scores over the detected present attributes. If this score surpasses a threshold, the lesion is assigned as a melanoma. Dermatologists use this information in addition to clinical information (if the lesion is growing, if it itches, if it bleeds, if it hurts, its location and patient’s age and sex), to diagnose skin lesions. We use the 7-points checklist score available as metadata of the Atlas dataset[^3]. It achieves $91.7\%$ AUC over all selected Atlas samples (see Figure \[fig:7pointsresult\]).
The huge gap between the 7-point checklist performance with the melanoma classification benchmark reveals it is biased due to the characteristics and educational objectives of the Atlas dataset. Low and medium difficulty cases selected to compose the dataset are probably hand-picked to be good examples to teach new medical practitioners to identify and classify dermoscopic attributes, while hard cases are exceptions to the pattern-based analysis.
Next, we try to find the source of bias, by gradually destructing clinical-meaningful information from the images, and assessing the network’s performance on them. Figures \[fig:deconstructionall\] and \[fig:deconstructiondiffs\] show the network’s performance for the different sets in the Atlas, Cross-dataset, and ISIC experiments respectively.
High difficulty lesions classification seem to be a very hard and specific task to the network, as it is for dermatologists. It could not learn clinical patterns properly with the training set, and destroying information do not influence the results. We understand that the network is probably exploiting image acquisition artifacts and dataset bias.
When experimenting in a cross-dataset fashion, the performance drops as expected, because of the differences between the statistics of Atlas and ISIC. The behavior of the network is similar in all experiments, and the following analysis can be generalized.
*Traditional* has the best overall performance, as expected. The network results follow the annotation of difficulty to diagnose by dermatologists. The results start to drop in *Only Skin*, where we start to deconstruct the information. When we remove the pixel information inside the lesion, we are removing all the information about dermoscopic attributes. The only clinically-meaningful information present is the border of the lesion, that could be used to verify its symmetry and irregularity, and skin features, such as vascularization.
When we remove the information of the borders, on *Bbox*, the performance lower, even more, revealing that we removed an essential feature for classification. An explanation, referring to medical algorithms like ABCD [@abcd], is that the diameter of the box contains the information on the size of the lesion, which is also relevant information when diagnosing skin lesions.
At *Bbox$70$*, we remove $70\%$ of all pixels in the image and all medical relevant features that could aid the classification. Still, surprisingly, the network can make sense of visual features to make decisions that are much better than chance. There is a pattern within the available pixels that contain information that leads to the correct label. This is shocking. The numbers achieved by the network at this point even surpass the AUC achieved by dermatologists on the melanoma classification benchmark. As sanity check, we performed an experiment hiding all image information, feeding the network (for training and testing) only zero-filled images. We achieved an AUC of 50%, which is expected since AUC is insensitive to class balance.
We believe that dataset bias is the culprit for inflating the network’s performance in our destructive experiments, introducing artifacts [@mishra2016overview] that undesirably can deviate the network’s attention from more critical features. We also verify that bias is not only present in the smaller educational purpose Atlas dataset, but also the most diversified ISIC dataset. Even performing the experiments in a cross-dataset fashion (the network is trained on ISIC, and tested on Atlas), the unnatural behavior persists, attesting to the fact that these two datasets may also share the same bias. We will address the exact causes and artifacts in future works.
Another possibility is that there is meaningful information at the borders of the images (parts that were not affected by the destruction procedures). This is unlikely because according to medical algorithms [@7points; @abcd; @abcde], there is no information left to account.
Information Construction Experiments {#sec:construct}
====================================
Since we have masks that maps the dermoscopic attributes in the lesion, we want to verify if we can simplify and guide the learning process by feeding the network with that detailed clinically-meaningful information. We gradually increase the amount of information fed to the network, building from only the attributes information. We describe each set of data in the next subsection.
Data
----
We introduce further modifications that are only possible with the dermoscopic attributes masks available on the ISIC dataset. Please refer to examples in Figure \[fig:isicmodified\].
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**Grayscale Attributes**: To compose each image in this set, we use a lesion’s masks from ISIC that show the location of five dermoscopic attributes and the same lesions’ segmentation masks. The skin without lesion, the lesion without markers, and each dermoscopic attribute are assigned a different value, equally spaced from each other. Dermatologists look for this information to diagnose skin lesions, and it is the basis for different medical algorithms, therefore being one of the most critical parts of the image.
**RGB Attributes**: This dataset only shows the RGB values of the regions of the image that belongs to an annotated dermoscopic attribute, and mask the others. This way, the network does not know in principle what are the skin patterns in the image or how many of them are present, but it gain access to their RGB values. We keep the segmentation mask information from *Grayscale Attributes* in this set to display some information for cases that do not present any skin patterns. ISIC’s annotation over the dermoscopic attributes is not as detailed as Atlas’. By letting the network analyze the RGB pixels that belong to a dermoscopic attribute, we are forcing the network to focus on the attributes, to discover more details about them (typical or atypical, regular or irregular, etc.), and to rely the classification on this information.
**Traditional$+$Grayscale Attributes**: Here we aim to guide the learning process by giving to the network extra information that is very relevant to dermatologists. We concatenate a fourth channel to the *Traditional* image, containing the information described in the *Grayscale Attributes*. We need to adapt the network to receive the extra channel in the input. We add an extra convolutional layer at the beginning of the network, initialized to prioritize receiving information from the RGB channels, and progressively learn to make use of the mask provided. We expected the results to be better than *Traditional* since we are adding clinically-meaningful information to guide the network to a better understanding of the process according to human knowledge.
Results and Discussion
----------------------
We show in Figure \[fig:constructionisic\] our results evaluating all different sets on the ISIC dataset.
Our attempt to guide the network’s learning process did not yield better results. Starting from *Grayscale Attributes*, we are feeding the network with enough information to verify global patterns present in the lesion, and location of some local features (pigment network, globules, streaks, negative network, and milia-like cysts). We note that the dermoscopic attributes information is not as detailed as the one present in Atlas, and this may affect the capability of the network to make correct predictions.
In *RGB Attributes*, we add pixel information to the images. That enables the network to learn details about each different dermoscopic attribute and improve classification. However, we did not observe that behavior. The extra information did not help the network to improve its understanding of the problem.
In *Traditional$+$Grayscale Attributes*, where we are adding clinical relevant information to the usual classification procedure to guide the learning process, the result did not improve as well in comparison to the *Traditional* baseline.
Conclusion {#sec:conclusion}
==========
If we hide the same lesion information from the networks, can it still learn patterns that help differentiate benign from malignant lesions? We believe that when a model learns to classify malignant lesions by analyzing only the skin —without information on the borders, biological markers or lesions’ diameter— it strongly relies on patterns introduced during image acquisition and general dataset bias.
Surprisingly, the result when feeding the network with clinically-meaningful information from the dermoscopic attribute maps (*Grayscale Attributes* and *RGB Attributes* sets) is worse than feeding it only with healthy skin information (*Only Skin* and *Bounding Box* sets). That leads us to believe that also our networks’ results towards both datasets is optimistic, not only the performance of 7-points over Atlas (which is expected).
That problem is critical for deploying automated skin lesion analysis. When performing in the real world, we want the network to be as unbiased as possible to make decisions based on clinical features. Therefore, it is urgent to understand the current bias in the datasets used to train and evaluate our works.
Acknowledgments {#acknowledgments .unnumbered}
===============
A. Bissoto and S. Avila are partially funded Google LARA 2018. A. Bissoto is also partially funded by CNPq. E. Valle is partially funded by a CNPq PQ-2 grant (311905/2017-0). This work was funded by grants from CNPq (424958/2016-3), FAPESP (2017/16246-0) and FAEPEX (3125/17). The RECOD Lab receives addition funds from FAPESP, CNPq, and CAPES. We gratefully acknowledge NVIDIA for the donation of GPU hardware.
[^1]: <http://www.cancer.net/cancer-types/melanoma/statistics>
[^2]: All our source code is readily available on <https://github.com/alceubissoto/deconstructing-bias-skin-lesion>.
[^3]: <http://derm.cs.sfu.ca>
|
---
author:
- Colin Goldblatt
title: 'The Inhabitance Paradox: how habitability and inhabitancy are inseparable'
---
Introduction {#introduction .unnumbered}
============
The dominant paradigm in assigning “habitability” to terrestrial planets is to define a circumstellar habitable zone: the locus of orbital radii in which the planet is neither too hot nor too cold for life as we know it. One dimensional climate models have identified theoretically impressive boundaries for this zone: a runaway greenhouse or water loss at the inner edge (Venus), and low-latitude glaciation followed by formation of clouds at the outer edge. A cottage industry now exists to “refine” the definition of these boundaries each year to the third decimal place of an AU. Using the same class of climate model, I show that the different climate states can overlap very substantially and that “snowball Earth”, moist temperate climate, hot moist climate and a post-runaway dry climate can all be stable under the same solar flux. The radial extent of the temperate climate band is very narrow for pure water atmospheres, but can be widened with di-nitrogen and carbon dioxide. The width of the habitable zone is thus determined by the atmospheric inventories of these gases. Yet Earth teaches us that these abundances are very heavily influenced (perhaps even controlled) by biology. This is paradoxical: the habitable zone seeks to define the region a planet should be capable of harbouring life; yet whether the planet is inhabited will determine whether the climate may be habitable at any given distance from the star. This matters, because future life detection missions may use habitable zone boundaries in mission design.
The habitable zone and physical climate {#the-habitable-zone-and-physical-climate .unnumbered}
=======================================
A typical description of the “habitable zone” is the region of circumstellar space in which a planet might be habitable. Rather by convention, the practical definition has become the existence of liquid water, for life as we know it (that is: Earth) relies on this.
As our conceptualization relies on the existence of on a certain phase of a certain chemical, our problem becomes finding the conditions of the physical climate of a planet which would put the surface and atmosphere in a desirable region of the pressure–temperature space of the phase diagram. Necessarily, we need to consider all three phases (all exist on Earth). This becomes a rather rich problem, for the different phases interact differently with electromagnetic radiation. There are, therefore, various climate feedbacks which express through water.
The framework of dynamical systems theory becomes fundamental, as climate state depends on history; our task is to look for stable steady states of climate. To make progress, we can separately consider the fate of solar photons incident on Earth which may transfer energy to us, and the escape of thermally emitted photons which transfer energy away. Where energy fluxes are equal, there exists a steady state. Steady states which are stable to a small perturbation are habitable zone candidates. Thus, in Figures 1 and 2, we derive bifurcation diagrams for Earth’s climate, mapping climate states as a function of circumstellar distance. There exist ice-covered, temperate moist, hot moist and hot dry climate states. The moist states both enjoy liquid water surfaces, but the sea-level temperature conditions are only favourable for life-as-we-know-it in the temperate moist state.
![Graphical derivation of steady states of climate with varying background gas inventories: none / pure water (blue) with orange through brown representing increasing p. (a) Planetary co-albedo (b) Outgoing thermal flux (c) Example case of finding steady states by balancing outgoing thermal flux with the product of incident solar flux and co-albedo (d) Consequent bifurcation diagram, with steady states as a function of distance from the Sun, stable in solid lines and unstable dotted. These figures are schematics, sketched from my previous numerical results[@Goldblatt2013; @Goldblatt2015]](schematic_nitrogen){width="\columnwidth"}
{width="\columnwidth"}
For the fundamentals of how energy fluxes change as a response to water inventory (controlled by surface temperature via the Clausius-Claperon equation), I defer to descriptions in my previous papers[@Goldblatt2012a; @Goldblatt2013; @Goldblatt2015]. Herein, the focus is on how variation in atmospheric composition modifies these.
Gases may be separated into *radiatively active* and *background* gases. The former absorb photons directly, and the most important subset of these, *greenhouse gases*, does so in the infrared; carbon dioxide is the prototype for Earth. The latter absorb little radiation directly but will broaden the absorption of the radiative gases and scatter solar photons; di-nitrogen is the prototype for Earth.
More background gas means less energy absorbed from the Sun because of higher albedo, and less or more thermal energy emitted depending on temperature (less at low temperatures where pressure-broadening is dominant, more at intermediate temperature where dilution of water aloft lowers the emission level). Consequent is the wet temperate climate state moving to higher solar constants and broadening: the habitable zone is wider and further from the Sun.
More greenhouse gas simply reduces outgoing thermal radiation, and only does so significantly at lower temperatures. Thus the wet temperate climate state is moved to lower solar constants and is broader: the habitable zone is wider and nearer to the Sun.
Controls on atmospheric composition {#controls-on-atmospheric-composition .unnumbered}
===================================
[[*Is there an atmosphere?*]{}]{}
There are two kinds of planets, those with atmospheres and those without. The planets with atmospheres are simply those which have not lost them. This can be seen clearly on the “Zahnle diagram”: with axis of escape velocity (planet mass) and solar heating, a simple power law empirically separates the two classes[@Zahnle2008; @Zahnle2013].
[[*Carbon Dioxide*]{}]{}
There are two end member cases of how a planet may store a carbon dioxide reservoir, represented in our solar system by Earth and Venus. On Earth, there is a small atmospheric reservoir and a larger ocean reservoir, but the vast majority of oxidized carbon is stored in carbonate rocks. On Venus, the atmospheric carbon dioxide inventory is roughly equivalent to the contents of Earth’s carbonate rocks, but there is neither an ocean inventory (as there is no ocean) nor any evidence of carbonate rocks.
Let us consider how oxidized carbon is partitioned between reservoirs on an Earth-like planet. The partial pressure of atmospheric and the dissolved concentration in the ocean are in direct proportion, described by Henry’s Law. But dissolved is not the major reservoir; in aqueous solution, the dissolution products of carbonic acid, bicarbonate ion and carbonate ion, will often be larger reservoirs. Conservation of mass is described via Dissolved Inorganic Carbon (). Conservation of change of weak acids occurs through alkalinity, which balances net positive change from salts (). For some DIC, alkalinity will control the partitioning between different reservoirs; low alkalinity requires high and low , high alkalinity the opposite. At intermediate alkalinity dominates. Thus, for given DIC, alkalinity determines the atmospheric and thus climate (Figure 3).
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{width="5cm"} {width="5cm"}
Plant roots increase soil and secrete organic acids which will dissolve rock.
In the ocean, high alkalinity means that will increase to the point where precipitation of is thermodynamically favoured, and precipitation removes both DIC and alkalinity. Thus, the consequence of an alkalinity flux from weathering is removal of inorganic carbon from the atmosphere-ocean system and deposition in rock. The saturation product describes when precipitation is thermodynamically favoured, but this is kinetically inhibited. Authigenic (abiological) precipitation requires $\Omega = 30$, but on Earth biology rules the roost yet again as organisms that build calcium carbonate shells are able to do so at $\Omega = 3$. Just as biology enhances weathering, it enhances carbonate deposition, and living Earth has lower than her sterile equivalent would.
Carbonate rocks will ultimate be destroyed through metamorphism, either of during subduction of ocean crust or in regional metamorphism of carbonate rocks uplifted to the continents. This recycled carbon contributes a larger part of the oft-referred to “volcanic carbon” source than juvenile carbon. Geological processes are thus involved in the partitioning between atmosphere-ocean and solid planet reservoirs.
Now, the transition to Venus: first, the deposition of carbonates is aqueous, so with the evaporation and loss of the oceans none can be deposited; second, at high temperatures, calcium carbonate will break down . This reaction is operated industrially on Earth in lime kilns. The required temperature of around 1200K is quite reasonable for the surface of Venus in a runaway greenhouse (the hot dry state) before water is lost to space.
In summary, to understand the first order controls of atmospheric , one must understand planetary climate, ocean chemistry, surface weathering processes and the action of life.
[[*Di-nitrogen*]{}]{}
The geological cycle of nitrogen, hence the possible variation of atmospheric di-nitrogen inventory, have long been under-appreciated. However, modern whole-Earth nitrogen budgets suggest that Earth’s mantle contains more nitrogen than the atmosphere and that mantle nitrogen is of subduction origin. An atmospheric mass worth of nitrogen can be subducted in around a billion years, so variations in the atmospheric budget have to be considered [@Goldblatt2013; @Johnson2015].
This mantle nitrogen is biological in origin. The only way that large amounts of di-nitrogen can be fixed is through biological nitrogen fixation, to make ammonium (). Ammonium has a similar ionic radius to potassium ion, so will readily substitute into K-rich minerals (particularly clays). Stable in a geological setting, it may then be subducted. Noble gas isotopes indicate that the mantle nitrogen (one to a few times the amount in the atmosphere) is indeed of subducted origin, and not primordial.
As with carbon dioxide, there is a prima facia case that a mix of biological, geochemical and geological processes control the atmospheric inventory.
[[*Water*]{}]{}
Water may be somewhat unique amongst the atmospheric gases in that its controls do appear to be physical, dominated by evaporation-condensation: warmer temperatures leads directly to higher partial pressure. Feedback processes in interaction with the radiation field give rise to the number of climate states discussed above.
There are other considerations though. Existence of an ocean requires: initial delivery of water to a planet inside the snowline; failure to lose that ocean via hydrogen escape; failure to subduct all water to the mantle through hydration and subduction of rocks (which does happen to an extent on Earth).
Conclusion {#conclusion .unnumbered}
==========
Were Earth to have only water in its atmosphere, it would today exist perilously close to both the inner and outer bounds of the conventionally defined habitable zone. In dynamical systems terms, there are several overlapping stable steady states of climate near present solar constant. The state with liquid water and modest temperature is narrow.
Expanding the habitable zone requires other gases in the atmosphere: conceptually, a greenhouse gas when there is little incoming sunlight and a background gas to either scatter sunlight when that is abundant, or broaden the absorption of greenhouse gasses when sunlight is scarce. The boundaries of the climate states are determined by atmospheric physics with abundances of these gases as a free parameter. On Earth, appeals to physical feedbacks to control carbon dioxide or di-nitrogen inventories fail: those free parameters are controlled by geology, geochemistry and life itself.
Life, we see, has entered the business of controlling the boundaries of its own habitat in space and time, niche construction in ecological parlance. As we view Gaia through the atmosphere, we see that habitability and inhabitance are inseparable.
[[*Acknowledgements*]{}]{} This work was funded by an NSERC discovery grant. Thanks to Ben Johnson for comments on the manuscript.
[1]{}
C. Goldblatt, Tyler D. Robinson, K. J. Zahnle, and David Crisp. . , 6(8):661–667, July 2013.
C. Goldblatt. , 15(5):362–70, May 2015.
C. Goldblatt and A. J. Watson. , 370(1974):4197–4216, 2012.
. . In [*AGU Fall Meeting Abstracts*]{}, 2008.
K. J. Zahnle. . In [*44th Lunar and Planetary Science Conference*]{}, page 2787, 2013.
R. E. Zeebe and D. Wolf-Gladrow. . Elsevier, 2001.
B. Johnson and C. Goldblatt. . , 148:150–173, 2015.
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abstract: 'We explore the Gaia DR2 proper motions of six young, main-sequence stars, members of the Large Magellanic Cloud (LMC) reported by @mb17. These stars are located in the outskirts of the disk, between $7\arcdeg$ and $13\arcdeg$ from the LMC’s center where there is very low H I content. Gaia DR2 proper motions confirm that four stars formed locally, in situ, while two are consistent with being expelled via dynamical interactions from inner, more gas-rich regions of the LMC. This finding establishes that recent star formation occurred in the periphery of the LMC, where thus far only old populations were known.'
author:
- 'Dana I. Casetti-Dinescu'
- 'Terrence M. Girard'
- Christian Moni Bidin
- Lan Zhang
- 'Rene A. Mendez'
- Katherine Vieira
- 'Vladimir I. Korchagin'
- 'William F. van Altena'
title: 'In-Situ Star Formation in the Outskirts of the Large Magellanic Cloud: Gaia DR2 Confirmation'
---
Introduction {#sec:intro}
============
In a recent contribution, @mb17 [hereafter MB17] presented a spectroscopic analysis of a set of candidate young, B-type, main-sequence stars in the outskirts of the LMC. The candidates were selected via a large-area study that combined UV, optical and IR photometry to specifically look for young stars far from known regions of star formation [@cas12]. found six stars with distances and radial velocities consistent with LMC membership. They argued for in-situ star formation based on small line-of-sight velocity residuals from a disk model of the LMC. Lacking proper-motion measures and the remaining two velocity components, this result could not be conclusive. Indeed, @boub17 proposed that these stars are runaways from the inner regions of the LMC. We revisit this issue using Gaia DR2 proper motions . Besides proper motions for our target stars, Gaia DR2 provides proper motions across the full field of the LMC to an unprecedented combination of precision and density. This allows us to work differentially, i.e., by obtaining proper-motion differences with respect to the local LMC motion within each subfield, without the need for a disk model. Moreover, we have a test case to interpret the kinematics, since one of our six stars is a known member of a young stellar association.
Analysis {#sec:ana}
========
We extract the proper motions for our six young stars from the Gaia DR2 catalog . Our stars have $G$ magnitudes between $\sim 15.1$ and 16.4, and thus are well measured by Gaia. The stars’ IDs (Gaia DR2 and ), proper motions and uncertainties are listed in Table \[tab1\], columns 1, 2, 7 and 8[^1] respectively. In addition, Gaia DR2 parallaxes confirm that these are distant stars, in agreement with the spectroscopic distances determined by . Specifically, all six stars have parallaxes compatible with zero at the $\pm 1\sigma_{\pi}$ level.
The bulk motion of the LMC, as well as its rotation — nicely evidenced by @helmi18 — must be taken into account when analyzing the proper motions of our target stars. We first determine the mean motion of LMC stars in the area around each of our target stars. This local field motion is then subtracted from the proper motion of the target star to obtain the star’s motion with respect to its LMC neighbors. We extract from Gaia DR2 subfields of radius $1\arcdeg$ centered on each target star. We then identify LMC members within each subfield. Our first attempt was to select members from the Gaia DR2 color-magnitude diagram (CMD) in $B_P$ and $R_P$. We illustrate this selection in Figure \[fig1\], left panel, for the field of star 292. This field is the richest, and in addition to the well-represented giant branch and red clump, we also see a young population of stars at blue colors ($B_P-R_P \sim 0$). These are stars in the known young stellar association ICA76 located toward the Bridge; noted that star 292 belongs to this association. Unfortunately, the remaining five regions are in much less populated areas of the Cloud and as such, foreground Milky Way stars overwhelm any selection made in the $B_{P}R_{P}$ CMD. Compounding the problem, the Gaia DR2 proper-motion errors increase rapidly with magnitude. For these two reasons we have opted to use 2MASS @skr06 to photometrically select LMC members.
Gaia DR2 coordinates in each field were matched with 2MASS, using a tolerance of $0\farcs5$. The $J,H$ CMD in the field of 292 is shown in Fig.\[fig1\], right panel. Our CMD cut is a box with $12.5 \le J \le 15.5$ and $0.65 \le J-H \le 1.0$. The giant branch represented by M giants of intermediate age and metallicity is distinct from the foreground field in this $JH$ CMD (as evidenced early on by e.g., @maj03). These stars are further trimmed by distance, i.e., keeping those with parallaxes compatibile with zero at $2\sigma_{\pi}$ level. Such stars are highlighted in Fig.\[fig1\] (red symbols), with the parallax criterion effectively cleaning up the remaining foreground stars within the CMD box cut.
In Figure \[fig2\] we show the proper-motion diagrams within each subfield. Left panels show the $B_{P}R_{P}$-plus-parallax selected samples, while the right panels show the $JH$-plus-parallax selected samples. We purposely use the same plot limits for each subfield to illustrate the change in the mean motion from subfield to subfield, reflecting the rotation of the LMC disk. In all panels, the blue symbol with error bars shows the proper motion of the target, young star. The $JH$-selected samples exhibit much tighter proper-motion clumps compared to the $B_{P}R_{P}$-selected samples, confirming these are better-measured samples.
The mean proper motion within each subfield is determined from the $JH$-plus-parallax selected sample, after trimming proper-motion outliers by eye. Those stars used in each determination are highlighted in red in Fig. \[fig2\]. The resulting mean motion is represented with a black square. The values of these means, and the number of stars used in their determination, are listed in Tab.\[tab1\], columns 4, 5 and 3 respectively. Finally, the last two columns of Tab.\[tab1\] show the proper-motion difference between the target star and the mean of the field. It is clear that stars 390 and 403 have proper motions significantly different from those of the local field, while stars 292, 307, 405, and 406 have proper motions within the dispersion of the local field.
Gaia DR2 ID ID N$_{f}$
--------------------- ----- --------- ---------------- ----------------- ---------------- ----------------- ------------------ -----------------
4629325831365569408 292 620 $1.971(0.006)$ $-0.272(0.007)$ $2.275(0.086)$ $-0.187(0.088)$ $0.304(0.086) $ $0.085(0.088)$
5263888695091870976 307 82 $1.418(0.020)$ $1.288(0.020)$ $1.429(0.070)$ $1.412(0.077)$ $0.011(0.073) $ $0.124(0.080)$
5281835782874813952 390 81 $1.329(0.013)$ $1.231(0.014)$ $2.182(0.151)$ $1.129(0.180)$ $0.853(0.152) $ $-0.102(0.181)$
4774221707057337088 403 13 $1.890(0.056)$ $-0.236(0.049)$ $1.603(0.096)$ $0.864(0.095)$ $-0.287(0.111) $ $1.127(0.107)$
4764998110170179328 405 36 $1.589(0.052)$ $0.515(0.041)$ $1.614(0.068)$ $0.798(0.085)$ $0.025(0.086) $ $0.283(0.094)$
5495816929075206912 406 15 $1.584(0.036)$ $0.516(0.044)$ $1.507(0.091)$ $0.890(0.081)$ $-0.077(0.098) $ $0.374(0.092)$
Proper-motion differences are multiplied by a fixed 50-kpc distance, in agreement with the LMC’s distance modulus of 18.49 [@pie13], yielding velocities in the plane of the sky. The total tangential-velocity differences, and their uncertainties[^2] are listed in Table \[tab2\] as $\Delta V_{T}$. Ages, as derived by , are also listed in Tab. \[tab2\] along with their estimated uncertainties. These are combined to calculate lifetime tangential-travel distances that are also given in Tab. \[tab2\], in both degrees and kpc. For comparison, we also list in the last column of Tab. \[tab2\], the line-of-sight velocity difference $\Delta V_{los}$ between the target star and the prediction of an LMC disk model as determined in . $\Delta V_{los}$ is not used in the determination of the travel distance.
There are several implicit assumptions to our analysis that deserve discussion. First, in order to convert proper-motion differences to velocity differences, it is assumed the young target stars and the (angularly) nearby LMC stars are at the same mean distance. This is reasonable given that the young star and the M giants are likely to both belong to the disk of the LMC. (Regardless, we are attempting to ascertain if the two are comoving and it is highly unlikely that a combination of discordant distances and tangential velocities would conspire to yield such a small proper-motion difference as is seen in four of the six cases.) Second, we assume that the mean motion of M giants within each field is representative of the local LMC disk motion. Third, we assume that while velocity gradients across each $2\arcdeg$-diameter area may distort the proper-motion distribution, the location of the target star at the spatial center of the field ensures that such gradients should not affect the mean motion.
Only stars 390 and 403 show velocity differences in excess of 200 km s$^{-1}$, with corresponding large travel distances of $\sim 6-7$ kpc. The remaining stars have velocity differences $< 100$ km s$^{-1}$, and travel distances of the order of 1 kpc. A marginal exception is star 406, with a travel distance of 4 kpc, due to its having the largest age in our sample. However, its near proximity to star 405 and their similar proper-motion differences (i.e., velocity vectors) hint that they were possibly formed in the same local area. We note star 292 is a known member of a nearby young association. As such, this star serves as an example of an in-situ formation referred to the local velocity field of an intermediate-age M-giant population.
----- ----------- ---------- ---------- ------------ ----------
ID
292 $75(21)$ $11(06)$ 1.0(0.6) $0.8(0.5)$ $-52(9)$
307 $29(19)$ $18(05)$ 0.6(0.2) $0.5(0.4)$ $14(8)$
390 $204(36)$ $35(08)$ 8.4(2.4) $7.1(2.1)$ $-28(7)$
403 $276(25)$ $20(05)$ 6.5(1.7) $5.5(1.5)$ $-48(7)$
405 $67(22)$ $20(05)$ 1.6(0.5) $1.3(0.6)$ $-21(5)$
406 $90(22)$ $45(10)$ 4.8(1.3) $4.1(1.3)$ $40(8)$
----- ----------- ---------- ---------- ------------ ----------
: Velocity and travel distance[]{data-label="tab2"}
In Figure \[fig4\] we show the distribution of our target stars in the plane of the sky. We use Magellanic coordinates [@nide08] in a gnomonic projection centered on the LMC. The top panel shows the H I column density map from the GASS survey [@mcc09; @kal15] with $V_{lsr} = 100 - 450$ km s$^{-1}$. For each star, the velocity-difference vectors are indicated. The two stars with velocity differences in excess of 200 km s$^{-1}$ are moving away from the inner regions of the LMC disk. Thus, their likely origin is in the denser parts of the disk, having been ejected toward the outskirts by dynamical interactions of the type described by @boub17. Their velocities in excess of 200 km s$^{-1}$ at current radii indicate that they escape the LMC [see e.g., Fig. 4 in @boub17]. The remaining four stars appear to have formed locally within a few degrees of their present location.
We illustrate the likely origin of the six stars in the bottom panel of Figure \[fig4\]. The difference in proper motion between each star and the mean motion of its nearby LMC members is combined with the star’s age estimate to predict the location of the star when it formed by simply propagating backward in time by the star’s age. In the figure, error ellipses are drawn at 1 Myr intervals around the star’s estimated age, $\pm 1\sigma$ the age uncertainty. Thus, for each star, the complex of error ellipses represents its likely place of origin. For reference, the region with H I column density exceeding $10^{20}$cm$^{-2}$ is shown in gray. Evidently, stars 390 and 403 formed in regions with higher gas density, compared to their present location, while stars 307, 405, and 406 likely formed in low-density regions. Of course, this simple approach does not explicitly include the LMC’s potential when tracing back in time each star’s position. Nonetheless, for the slow moving stars this differential approach is appropriate and allows us to reach a definite conclusion regarding their origin. For the two fast moving stars the approach is admittedly simplified, but should still indicate the rough direction and amount of offset of each star’s origin location relative to its nearby LMC neighbors. A more rigorous orbit integration might possibly improve the estimate, but considering the size of the uncertainties in age and transverse velocity, we do not think it is warranted at this time.
It is necessary to explore the possibility that stars 390 and 403 were expelled from the Milky Way. To do so, we integrate back in time the orbits of these two stars in an analytic 3-component Galactic potential [@jsh95], using the distance moduli derived in , and ignoring the LMC. We find that the pericenter of star 390 is 49 kpc, reached some 17 Myr ago, i.e., within its current age range, compared to its current Galactocentric distance of 50 kpc. For star 403, the predicted pericenter is 52 kpc some 39 Myr ago, i.e., beyond its age range, while its current Galactocentric distance is 54 kpc. Young stars with pericenters of $\sim 50$ kpc are unlikely to have originated in our Galaxy. They more likely escaped from a more gas-rich region of the LMC, and are analagous to high-velocity star HVS3 that was recently confirmed to have a Magellanic origin [@er18].
Summary {#sec:sum}
=======
We use Gaia DR2 data to confirm the origin of six young stars located in the outskirts of the LMC. We find that four stars have low velocities. Combining this with age estimates derived in an earlier study , the four stars must have been born within $1\arcdeg$ to $\sim 5\arcdeg$ of their current location. Three of these stars do not belong to any known young association and have formed in very low H I density regions, in the periphery of the Cloud. It is conceivable that the recent ($\sim 200$ Myr) collision between the Small Magellanic Cloud and the LMC could have triggered star formation in the far outskirts of the LMC’s disk . The remaining two stars have velocity differences in excess of 200 km s$^{-1}$ and in directions roughly outward from the LMC, indicating their origin is consistent with being runaways from the inner LMC.\
\
DIC and TMG were supported, in part, by NASA grant 80NSSC18K0422. DIC is grateful to the Vatican Observatory Summer School where she was supported as a lecturer during revision of this manuscript, (and where her lab students were given the opportunity to replicate the results of this study!). LZ acknowledges support by the National Science Foundation of China grants 11773033 and 11390371/2. VIK acknowledges support by grant 3.858.2017/4.6 of the Ministry of Education and Science (Russia). This work has made use of data from the European Space Agency (ESA) mission [*Gaia*]{} (<https://www.cosmos.esa.int/gaia>), processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the NASA and the NSF.
Boubert, D., Erkal, D., Evans, N. W., and Izzard, R. G. 2017, , 469, 2151 Casetti-Dinescu, D. I., Vieira, K., Girard, T. M., and van Altena, W. F. 2012, , 753, 123 Erkal. D., Boubert, D., Gualandris, A., Evans, N. W., and Antonini, F. 2018, astro-ph 1804.10197 Gaia Collaboration, Prusti, T. et al., 2016, , 595, 1 Gaia Collaboration, Brown, A. G. A. et al., 2018, , special issue for Gaia DR2 Gaia Collaboration, Helmi, A. et al. 2018, , special issue for Gaia DR2 Gaia Collaboration, Lindegren, L. et al. 2018, , special issue for Gaia DR2 Johnston, K. V., Spergel, D. N., and Hernquist, L. 1995, , 451, 598 Kalberla, P. M. W., and Haud, U., 2015, , 578, A78 Mackey, A. D., Koposov, S. E., Erkal, D., Belokurov, V., Da Costa, G. S., and G[ó]{}mez, F. A. 2016, , 459, 239 Majewski, S. R, Skrutskie, M. F., Weinberg, M. D., and Ostheimer, J. C. 2003, , 599, 1082 McClure-Griffiths, N. M., et al. 2009, , 181, 398 Moni Bidin, C., Casetti-Dinescu, D. I., Girard, T. M., Zhang, L., M[é]{}ndez, R. A., Vieira, K., Korchagin, V. I., and van Altena, W. F. 2017, , 466, 3077 Murray, C. E., Stanimir, S., McClure-Griffiths, N. M., Putman, M. E., Listz, H. S., Wong, T., Richter, P., Dawson, J. R., Dickey, J. M., Linder, R. R., Babler, B. L., and Allison, J. R. 2015, , 808, 41 Nidever, D., L., Majewski, S. R., and Butler, B. W. 2008, , 679, 432 Pietrzy[ń]{}ski, Graczyk, D., Gieren, W., et al. 2013, Nature, 495, 76 Skrutskie, M. F. et al. 2006, , 131,1163
[^1]: Throughout the paper, proper-motion units are mas yr$^{-1}$, and $\mu_{\alpha}$ is actually $\mu_{\alpha}$cos$\delta$.
[^2]: As calculated, the uncertainty in $\Delta V_{T}$ is dominated by the Gaia DR2 proper-motion uncertainty of each target star. While the known correlations between the $\alpha$ and $\delta$ components of the Gaia $\mu$ measures [@lind18] will affect the final uncertainty in $\Delta V_{T}$, in actuality the effect of the correlation amounts to $\sim 1$ km s$^{-1}$ or less for these stars. This is negligible relative to the overall uncertainty values of $\sim 20-30$ km s$^{-1}$ (see Table \[tab2\]) and has been ignored.
|
---
author:
- |
$^1$, for the Pierre Auger Collaboration$^2$ [^1]\
$^1$ Laboratoire de Physique Subatomique et de Cosmologie (LPSC), UJF-INPG, CNRS/IN2P3, Grenoble, France.\
$^2$ Observatorio Pierre Auger, Av. San Martín Norte 304, 5613 Malargüe, Argentina.\
E-mail:
title: 'Radio-detection of extensive air showers at the Pierre Auger Observatory – Results and enhancements'
---
Introduction {#sec:intro}
============
The Pierre Auger Observatory is the largest operating cosmic ray observatory ever built [@OPA]. It is designed to measure the flux, arrival directions and mass composition of cosmic rays from above $3\times 10^{17}~$eV to the very highest energies. The observatory is composed of 1660 water Cherenkov detectors covering 3000 km$^2$, and 27 fluorescence telescopes surrounding the array on four sites. Both of these techniques have some limitations: whereas the interpretation of the Cherenkov detector data is very dependent on hadronic interaction models which are not yet tested in laboratories at the highest energies, the optical telescopes are operated only during dark nights (duty cycle close to $13\%$). It is in this context that the Pierre Auger Collaboration is pursuing R&D projects to enhance the detection capabilities and to test new techniques for the next generation of ground-based detectors. Currently, the most promising technique is probably the radio-detection with a duty cycle reaching almost $100\%$ and no atmospheric attenuation. This technique allows one to record the whole development of the extensive air shower with a quasi-calorimetric measurement of the shower energy. Two different frequency domains are probed at the Pierre Auger Observatory: the MHz band and, more recently, the GHz band. In the following, we give a status on both activities and their next challenges.
The Auger Engineering Radio Array (AERA) – MHz frequencies {#sec:mhz}
==========================================================
The Auger Engineering Radio Array (AERA) records radio signals produced by extensive air showers in the MHz frequency domain, from 30 to 80 MHz [@AugerRadio_FAL]. After a first phase, started in April 2011 with 24 stations distributed over an area of 0.5 km$^2$, the second phase is operating since April 2013 with 100 additional stations covering an area of around 6 km$^2$. It is planned to deploy an additional 36 stations in 2014 to cover, finally, approximately 20 km$^2$. Although AERA can be operated in a self-trigger mode, a portion of the radio stations can get an external trigger from the water Cherenkov detectors or from scintillators integrated directly into the stations. In this MHz frequency range, there is a polarised coherent emission resulting from two relevant mechanisms: [*the geomagnetic effect*]{} corresponding to the transversal separation of charged particles due to the geomagnetic field (linear polarisation), and the [*Askaryan effect*]{} related to the variation of charge excess along the longitudinal development of the air shower (radial polarisation). Both of them have been confirmed by the Pierre Auger Collaboration using events recorded during the first phase of AERA [@Melissas; @HuegePolar]. Using multi-hybrid events and Monte Carlo simulation codes dedicated to radio emission, we are improving our understanding of radio emission mechanisms. Current studies are focused on evaluating the precision of AERA for estimating the arrival direction, energy and composition of air showers in the energy range from $10^{17}$ to $10^{19}~$eV.
Microwave radiation (AMBER, EASIER, MIDAS) – GHz frequencies {#sec:ghz}
============================================================
The observation of a microwave continuum emission from air shower plasmas has raised the idea to measure extensive air showers in the GHz frequency range [@Gorham]. This recorded signal was interpreted as molecular bremsstrahlung radiation (MBR), [*i.e.*]{} an emission produced by low-energy electrons (coming from ionisation of air molecules by primary electrons of the air shower) scattering into the electromagnetic field of neutral atmospheric molecules. Radio emission produced by this mechanism presents interesting features: it is isotropic and unpolarised, with a very low natural background. Three prototypes, AMBER, EASIER and MIDAS are now operating at the Pierre Auger Observatory to measure microwave radiation coming from extensive air showers [@RGaior]. The main goal is to characterise this microwave emission and to test the feasibility of such a technique to detect ultra-high energy cosmic rays. AMBER and MIDAS are imaging telescopes composed of horn antennas placed at the focus of a parabolic dish. This design allows one to record the shower longitudinal profile by exploiting the isotropic nature of the MBR. EASIER instruments a set of 61 water Cherenkov tanks with a microwave receiver observing the shower with a wide field-of-view (around $100^\circ$) pointing to the zenith. In this set-up, the very small effective area of the antennas is compensated by time compression of the signal and closer distances to the shower. All of them record signals in the C-band ($3.4-4.2~$GHz) and AMBER has additional horn antennas in the Ku-band ($10.9-14.5~$GHz). The MIDAS detector, during a first phase of data-taking at the University of Chicago, sets a limit on air shower microwave emission and excludes a quadratic scaling with the air shower energy (for an isotropic emission) [@MIDAS]. Up to now, only the EASIER project recorded microwave signals in coincidence with an extensive air shower, especially with antennas close to the shower core and with an E-W polarisation [@RGaior]. Several analyses to understand the origin of these signals are on-going.
Conclusion and outlook {#sec:conclusion}
======================
The Pierre Auger Observatory is a great facility for studying alternative air shower detection techniques. Whereas the collaboration is now working on the MHz radio emission to estimate properties of cosmic rays, we are still in an exploratory phase to identify the main mechanism(s) producing signals in the microwave band.
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T. Huege, for the Pierre Auger Collaboration, *Probing the radio emission from cosmic-ray induced air showers by polarization measurements*, Proc. 33rd ICRC, Rio de Janeiro (2013) \[[astro-ph/1307.5059]{}\].
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[^1]: http://www.auger.org/archive/authors\_2013\_06.html
|
---
author:
- 'С.С. Галкин, П.П. Попов'
title: '****'
---
Заметка посвящена вопросу стабильной бирациональной классификации произведений симметрических степеней фиксированного алгебраического многообразия $Y,$ неприводимого над полем $k$. Говорят, что $Y,Y'$ стабильно бирациональны (над $k$), если существуют числа $n,n'$ и изоморфизм полей функций $k(Y)(z_1,...,z_n) \simeq k(Y')(z'_1,...,z'_{n'})$. Пусть $\Sigma_n$ это группа перестановок $n$-элементного множества, $Y^{(n)} := Y^n/\Sigma_n$ — $n$-ая симметрическая степень $Y$, $Y^{[n]}$ — схема Гильберта $n$ точек на $Y$ (если $Y$ это гладкая поверхность, то $Y^{[n]}\to Y^{(n)}$ это разрешение особенностей по теореме Фогарти), $X\subset\P^3_k$ — геометрически целая кубическая поверхность над $k$.
[**Теорема.**]{} Многообразия $X\times X^{(3)}$ и $X\times X^{(4)}$ стабильно бирациональны над $k$.
[**Лемма 1.**]{} Многообразия $X^{(2)}\times X^{(4)}$ и $X^{(2)}\times X^{(3)}\times\P^2$ бирациональны над $k$.
[**Лемма 2.**]{} Многообразия $X^{(2)}$ и $X\times\P^2$ бирациональны над $k$.
Заметим, что существуют $k$ и $X/k$, такие что $X^{(4)}$ стабильно не бирационально $X^{(3)}$.
[**Пример 1.**]{} Поверхность $F = \operatorname{Proj}k[X_0,X_1,X_2,X_3]/(X_0^3 + 7 X_1^3 + 7^2 X_2^3 - 2 X_3^3)$. Следуя [@CM04], $F$ гладкая над $\Q$ и $F(K)=\emptyset$ для всех расширений $K/\Q_7$ степеней 1,2,4. Поэтому $F^{(4)}(\Q_7)=\emptyset$. Но $F^{(3)}$ имеет точки над любым полем. По теореме Нишимуры [@Nis55] многообразия $F^{(4)}$ и $F^{(3)}$ стабильно не бирациональны над $\Q_7$.
[**Пример 2.**]{} Поверхности Севери–Брауэра ($Y/k$ такие что $Y_{\bar{k}} \simeq \P^2_{\bar{k}}$). Согласно [@Kol16] стабильная бирациональная классификация описывается булевой функцией <<пусто ли множество $Y(k)$>>. Поверхности Севери–Брауэра бирациональны кубическим поверхностям, имеющим $\operatorname{Gal}_k$-инвариантную шестёрку попарно дизьюнктных прямых.
[**Пример 3.**]{} Кубические кривые. Пусть $C\subset\P^2_k$ — гладкая кубическая кривая с якобианом $E := \operatorname{Pic}^0 C$. Отображение Абеля показывает, что $C^{(n)}$ стабильно бирационально $\operatorname{Pic}^n C$, причём $\operatorname{Pic}^{3n\pm1} C \simeq C$ и $\operatorname{Pic}^{3n} C \simeq E$. Если $C(k)=\emptyset$, то $C^{(3)}$ и $C^{(4)}$ стабильно не бирациональны. Но $C\times C$ изоморфно отображается в $C\times E$: $(P,Q)\to (P,\mathcal{O}(P-Q))$. Поэтому $C\times C^{(4)}$ стабильно бирационально $C\times C^{(3)}$. Поскольку конус стабильно бирационален своему основанию, все результаты о кубических кривых эквивалентны аналогичным результатам о кубических поверхностях-конусах.
[**Скрученные кубики.**]{} Говорят, что шестёрка точек в пространстве находится в общем положении, если никакие четыре из них не лежат в одной плоскости. Через такую шестёрку можно провести единственную скрученную кубику (см. например Теорему 1.18 в [@Ha92ru]), рациональную нормальную гладкую кривую степени $3$. Также любая шестёрка точек на любой скрученной кубике является шестёркой точек в общем положении. Множество $M_3$ всех скрученных кубик это $12$-мерное связное гладкое многообразие, изоморфное $PGL(4)/PGL(2)$. Будем говорить, что кривая трансверсальна к поверхности, если их схемное пересечение это приведённая артинова схема. Для каждой приведённой кубической поверхности $X\subset\P^3$ подмножество $U\subset M_3$ трансверсальных к $X$ скрученных кубик непусто и открыто по Зарисскому. Пусть $\tilde{U} \to U$ это нормальное этальное накрытие с накрывающей группой $\Sigma_9$: точка $\tilde{U}$ это трансверсальная к $X$ скрученная кубика $T$ плюс полный порядок на множестве $T\cap X$. Стандартно вложим $\Sigma_2\times\Sigma_3\times\Sigma_4$ в $\Sigma_9$, и обозначим через $U_{2,3,4} = \tilde{U}/(\Sigma_2\times\Sigma_3\times\Sigma_4)$ промежуточный фактор. Точка $(T;B,C,D)\in U_{2,3,4}$ это трансверсальная к $X$ скрученная кубика $T$ плюс разбиение множества $T\cap X$ на три группы $B=\{B_1,B_2\}, C=\{C_1,C_2,C_3\}, D=\{D_1,D_2,D_3,D_4\}$. Определим отображение $\pi_{2,4}:U_{2,3,4}\to X^{(2)}\times X^{(4)}$ так: $\pi_{2,4}(T;B,C,D) = (B,D)$. Если $(B,D)$ лежит в образе $\pi_{2,4}$, то шестёрка точек $B\cup D$ лежит хотя бы на одной скрученной кубике $T$, а следовательно находится в общем положении и лежит на единственной скрученной кубике; поэтому $C = (T\cap X)\backslash(B\cup D)$ и $\pi_{2,4}$ это открытое вложение. Зафиксируем плоскость $\Pi\subset\P^3$ и пару точек $O,O'\in\P^3(k)\backslash\Pi$. Определим подмножество $V\subset U_{2,3,4}$ условием, что точки $O,O',B_1,B_2$ находятся в общем положении. Пусть $W = \{(T;A,B,C,D) | (T;B,C,D)\in V, A\in(T\cap\overline{OB_1B_2})\backslash(B\cup C)\}$. Морфизм $\pi_{2,3,1'}: W\to X^{(2)}\times X^{(3)}\times\Pi$ задан так: $\pi_{2,3,1'}(T;A,B,C,D) = (B,C,A')$, где $A' := \overline{O'A}\cap\Pi$. Если $(B,C,A')$ лежит в образе $\pi_{2,3,1'}$, то шестёрка точек $B\cup C\cup A$ (где $A = \overline{O'A'}\cap\overline{OB_1B_2}$) лежит на скрученной кубике $T$, следовательно находится в общем положении и лежит на единственной скрученной кубике, поэтому $D = (T\cap X)\backslash(B\cup C)$ и $\pi_{2,3,1'}$ это открытое вложение. Таким образом, и $X^{(2)}\times X^{(4)}$, и $X^{(2)}\times X^{(3)}\times\Pi$ имеют открытые по Зарисскому подмножества, изоморфные $W$. Лемма 1 доказана. Лемма 2 доказывается аналогично, с использованием прямых вместо скрученных кубик (также см. [@Popov18]). Теорема является мгновенным следствием комбинации двух Лемм.
[**Следствия.**]{} Определим $\Z[SB_k]$ как свободную абелеву группу, порождённую классами $\{Y\}$ стабильной бирациональной эквивалентности неприводимых над $k$ многообразий $Y$. Её можно снабдить операциями умножения и симметрических степеней. Теорема и примеры влекут, что для всех $X$ выполнено тождество $\{X\}\cdot\{X^{(4)}\}=\{X\}\cdot\{X^{(3)}\}\in\Z[SB_k]$, но существуют $X$ для которых $\{X^{(4)}\}\neq\{X^{(3)}\}\in\Z[SB_k]$. Такие кубические поверхности доставляют примеры делителей нуля в кольце $\Z[SB_k]$. Кольцо Гротендика $k$-многообразий $K(Vars/k)$ порождено классами $[Y]$ схем $Y$ конечного типа над $k$ и соотношениями $[Y] = [W] + [Y\backslash W]$ для замкнутых подсхем $W\subset Y$. Пусть $\L := [\A^1]$ это класс аффинной прямой. Определим фактор-кольцо $LL(k) := K(Vars/k)/(\L)$. Согласно [@LL03] в характеристике $0$ существует изоморфизм $\mu:LL(k)\to\Z[SB_k]$, такой что $\mu([Y])=\{Y\}$ для гладких проективных связных $Y/k$. Поверхности из примеров 1 и 2 также являются делителями нуля в кольце $LL(k)$. В [@Popov18] доказано, что есть лишь одно соотношение степени $4$ между $\operatorname{Gal}_k$-представлениями $H(X^{[n]})$ и $H(M_3(X))$, где $M_3(X)$ это многообразие обобщённых скрученных кубик на гладкой $X$. Это соотношение не выполнено в $K(Vars/k)$, потому что по модулю $(\L)$ оно имеет вид $[X^{(4)}] = [X^{(3)}] \in LL(k)$, что противоречит примерам 1 и 2. Мы ожидаем, что модификация формулы из [@Popov18], получающаяся домножением на $X^{[2]}$ уже может быть выполнена и в $K(Vars/k)$. Теорема влечёт, что в характеристике $0$ для гладких $X$ существует пара гладких $10$-мерных многообразий $N_{\pm}$, таких что $[X^{[2]}]\cdot([X^{[4]}] - [X^{[3]}\cdot\P^2]) = \L\cdot([N_+]-[N_-]) \in K(Vars/k)$. Чтобы найти $N_\pm$ нужно разложить бирациональный морфизм $\pi_{2,4}\circ{(\pi_{2,3,1'})}^{-1}$ в последовательность раздутий и стягиваний с гладкими центрами, и описать эти центры.
Мы благодарим К. Шрамова за ссылку [@Nis55], а также К.Вуазен, К.Пескина, Ф.Хана за проявленный интерес к нашей работе.
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H. Nishimura. Some remarks on rational points. , 29:189–192, 1955.
J. [Koll[á]{}r]{}. . 2016, [[arXiv:1603.02104](http://arxiv.org/abs/1603.02104)]{}.
Дж. Харрис. . МЦНМО, 2006. 2-е издание, стереотипное.
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M. Larsen, V. Lunts. Motivic measures and stable birational geometry. , 3(1):85–95, 259, 2003.
[**С.С. Галкин (S.S. Galkin)**]{}
Национальный исследовательский университет <<Высшая школа экономики>>
[**П.П. Попов (P.P. Popov)**]{}
Национальный исследовательский университет <<Высшая школа экономики>>
[*E-mail*]{}: lightmor@gmail.com
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---
abstract: 'We explore the stress-energy tensor arising from the interaction of $U(1)$ symmetric quantum and gravitational fields. Using scalar-tensor theories of gravity, a conformal factor $\Omega^2$ is defined as the rest mass corrected by the quantum potential. The quantum potential, derived from the Klein-Gordon equation, allows for matter’s intrinsic interaction with spacetime. A Lagrange multiplier $\lambda$ is used as a constraint to properly couple matter with gravity. The Heisenberg uncertainty principle appears as a natural artifact of $\lambda$. Unlike the classical limit, $\lambda$ in the quantum regime strongly influence the stress-energy tensor and it is therefore suggested that it is characteristic of the quantum vacuum. Additionally, the cosmological constant $\Lambda$, defined from the modified Einstein’s equation, is formulated for any particle of mass $m$. The mysterious variation in $\Lambda$ is properly evaluated from its cosmological value to that of an electron, from which we obtained a 77 order difference.'
author:
- Dor Gabay
- 'Sijo K. Joseph'
title: On the Cosmological Constant in a Conformally Transformed Einstein Equation
---
Extended versions of Einstein’s theory of gravity are getting more attractive in recent years [@BeyondEinsteinGravity; @Shojai_Article]. There are many different extension of the general theory of relativity [@WeylReview2017; @Bergmann1968; @fR_HamFormlation; @Shojai2008; @fofT1; @*fofT2; @Bekenstein2011_TeVeS; @*BekensteinPRD_TeVeS; @Moffat_STVG; @*MOG_Moffat]. Weyl conformal gravity is especially interesting due to the fact it is a higher derivative theory of gravity with many advantages over Einstein’s theory [@Mannheim2012; @*MannheimGalRot2013]. Weyl conformal theory shines light at many theoretical issues in General Relativity, but many conceptual problems have yet to be tackled. The main problem facing conformal gravity is that it is non-unitary, making it incompatible with our current understanding of quantum mechanics. As a resolution, Manheim and Bender proposed that the Hermitian nature of quantum mechanics can be generalized to account for space-time reflection, or PT-Symmetry [@Bender1; @*Bender2; @Manheim_PTSym]. Assuming PT-Symmetry is a proper extension of Hermiticity, conformal gravity can play a fundamental role in unifying quantum and gravitational fields.
Many of the difficulties in quantum-gravity can be circumvented if one were to accept the duality between the two theories [@QRGM_Susskind; @EReqEPR]. The classical field theories of Quantum Mechanics and General Relativity are governed by different fundamental assumptions. Quantum theory defines a set of field variables over a flat Minkowski space-time. General Relativity, on the other hand, displaces the gravitational forces onto a manifold structure using a space-time metric $g_{\mu\nu}$. Given the presumed flat space-time geometry of quantum theory, properly incorporating gravity can be a difficult task. Additionally, unlike the deterministic interpretation of General Relativity, quantum mechanics is inherently defined as a probabilistic theory with complex quantities. All of these difficulties can be circumvented by reformulating the U(1) symmetry of quantum mechanics into its geometrical form [@BohmI].
In this letter, we show that one could define the required conformal frames necessary to allow matter and spacetime to coexist. Geometrically, this can be achieved by identifying the conformal factor with quantum potential emerging from the quantum mechanical scalar matter field (e.g. Klein-Gordon field). Initially, we define the stress energy tensors related to matter, quantum, and vacuum contributions within the framework of a scalar-tensor theory. In our framework, the gravitational scalar field appearing in the theory is just the Klien-Gordon quantum potential. An interpretation is given for the less obvious vacuum contribution arising from coupling of matter with gravity via the Lagrange multiplier $\lambda$. As a special case, $\lambda$ is studied for a static Gaussian distribution to validate our general approach. $\lambda$ is shown to decay from the quantum to classical regime, as might be expected from the quantum vacuum. The standard deviation $s$ associated to the Gaussian distribution is confined to the Planck length in order to conform with cosmological observables. Finally, an analytical expression is defined for the cosmological constant $\Lambda$ and the alleged 60-120 order difference in $\Lambda$ is exemplified for an arbitrary mass $m$ ranging from the universe to a single electron.
Interpreting classical trajectories using the de-Broglie-Bohm picture of quantum-gravity is one possible way for integrating the classical and quantum theories consistently. Classically, one must ensure the particle trajectories contain the fluctuations arising from quantum mechanics. One possible way of incorporating such fluctuations is through a conformal factor. Nalikar and Padmanabhan studied a quantized version of conformal fluctuations associated to the spacetime geometry [@Narlikar1981; @Tanu_PLA; @*Tanu_Nature]. Later on, Santamato derived a modified Schrödinger equation by considering the scale-invariant Weyl theory [@Santamato_Schrodinger]. Thereafter, Sidharth attempted to provide a geometrical interpretation of quantum mechanics [@Sidharth_QM]. Then Shojai et. al [@Shojai_Article], inspired by their work, defined a conformally transformed action along with a Lagrange multiplier. Here we define a more generalized form of the action to fully incorporate the general exponential constraint condition using the Lagrange multiplier $\lambda$, taking $c=1$ $$\begin{aligned}
& & A[g_{\mu\nu},{\Omega}, S, \rho, \lambda]=\nonumber \\
& & \frac{1}{2\kappa}\int{d^4x\sqrt{-g}\left(R\Omega^2-6\nabla_{\mu}\Omega\nabla^{\mu}\Omega\right)} \nonumber \\
& & +\int{d^4x\sqrt{-g} \left(\frac{\rho}{m}\Omega^2 \nabla_{\mu}S \nabla^{\mu}S-m\rho\Omega^4\right)} \nonumber \\
& & +\int{d^4x\sqrt{-g}\lambda\left[\ln{\Omega^2}-\left(\frac{\hbar^2}{m^2}\frac{\nabla_{\mu}\nabla^{\mu}\sqrt{\rho}}{\sqrt{\rho}}
\right)\right]} \label{Actioneq} \end{aligned}$$ Here, $\lambda$ is employed to constraint the conformal factor $\Omega^2$ to the quantum nature of the Klein-Gordon field. The constraint effectively bypasses the fine-tuning problem articulated by Weinberg [@WeinbegCosmo89] (see *Appendix A* on how Lagrange multiplier bypasses Weinberg’s no-go theorem). The proposed action (Eq. \[Actioneq\]) reduces to its classical form in the $\hbar\to 0$ limit (e.g. $\Omega^2\to 1$). By minimizing the action with respect to $S, \rho, {\Omega}, g_{\mu\nu}$ and $\lambda$, here we derive the equations of motion for a relativistic matter field. In taking the variation of $A$ with respect to $\rho$, the equation of motion for the particle (e.g. matter field) is obtained $$\begin{aligned}
(\nabla_{\mu}S \nabla^{\mu}S- m^2\Omega^2)\Omega^2 \sqrt{\rho}+f(\lambda,\rho)=0 \label{EqMotion}\end{aligned}$$ Here, $f(\lambda,\rho)=\frac{\hbar^2}{2m}[\Box({\frac{\lambda}{\sqrt{\rho}})}-\lambda\frac{\Box\sqrt{\rho}}{\rho}]$ is the coupling contribution arising from the Lagrange multiplier. The equation of motion is fully defined in terms of the density $\rho$, Hamilton’s principal function $S$, and the Lagrange multiplier $\lambda$. In taking the variation with respect to the classical action $S$, one can similarly arrive at the corresponding continuity equation $$\begin{aligned}
\nabla_{\nu}(\rho\Omega^2\nabla^{\nu}S) =0. \label{ContinuityEq}\end{aligned}$$ The quantum mechanical behavior of the particle can therefore fully be described by two real fields $\rho$ and $S$, along with the yet to be defined coupling contribution. For $f(\lambda,\rho)=0$ one gets the usual Klein-Gordon equation. It is therefore apparent that the equations of motion arising from the conformal factor are more general, with an additional contribution given by $f(\lambda,\rho)$. Unfortunately, Shojai assumed $\lambda$ to be zero giving her no coupling contributions. The physical meaning of $\lambda$ is worthwhile to explore in the quantum mechanical context. In this article we interpret $\lambda$ as a quantity associated to the energy density of the quantum vacuum. There are many reasons to suspect $\lambda$ is a vacuum contribution (as will be discussed in later sections). Even in a flat spacetime, the $\lambda$ contribution seems to generalize the governing quantum mechanical equations of motion. It can be seen that, in the absence of matter’s interaction with gravity, Eq. \[EqMotion\] yields $f(\lambda,\rho)=0$, which results in a wave equation for $\lambda$. In addition to the equation of motion (Eq. \[EqMotion\]) and the continuity equation (Eq. \[ContinuityEq\]), the equation associated to the scalar curvature $R$ can be determined by varying the action with respect to $\Omega$ $$\begin{aligned}
R\Omega+6 \Box\Omega +\frac{2\kappa}{m}\rho \Omega (\nabla_{\mu}S \nabla^{\mu}S-2m^2\Omega^2)+\frac{2\kappa\lambda}{\Omega}=0.
\label{TraceEq}\end{aligned}$$ Here, the conformal factor $\Omega^2$ is defined as $\exp{(Q)}$, where $Q$ is a quantum mechanical quantity known as the quantum potential [@BohmI; @BohmII; @Carroll2005; @*RCarroll2007]. Here $Q$ is contained within the constraint equation, which can be obtained by similarly varying the action (Eq.1) with respect to $\lambda$ $$\begin{aligned}
\Omega^2=\exp{\left(\frac{\hbar^2}{m^2}\frac{\nabla_{\mu}\nabla^{\mu}\sqrt{\rho}}{\sqrt{\rho}}\right)}. \label{CnstrEq}\end{aligned}$$ This constraint equation is particularly interesting since we are identifying a purely geometrical quantity $\Omega$ with a quantum mechanical descriptor. Similarly, the variation of the action with respect to the metric tensor ${g}_{\mu \nu}$ generates the modified Einstein equation $$\begin{aligned}
\mathcal{G}_{\mu \nu}=T^{\bf{matter}}_{\mu\nu}(S,\rho)+T^{\bf{qm}}_{\mu\nu}(\Omega)+T^{\bf{vac}}_{\mu\nu}(\lambda,\rho).
\label{EinsteinEq}\end{aligned}$$ Here the stress-energy tensors are given by, $$\begin{aligned}
T^{\bf{matter}}_{\mu\nu}(S,\rho)&=& -\frac{2\kappa}{m}\rho \nabla_{\mu}S \nabla_{\nu}S
+\frac{\kappa}{m} \rho\,g_{\mu\nu}\nabla_{\sigma}S \nabla^{\sigma}S \nonumber \\
& &-\kappa m \rho \Omega^2 g_{\mu\nu}. \end{aligned}$$ and $$\begin{aligned}
T^{\bf{qm}}_{\mu\nu}(\Omega) &=&\frac{(g_{\mu \nu}\Box{\Omega^2}-
\nabla_{\mu}\nabla_{\nu}{\Omega^2})}{\Omega^2}+6\frac{\nabla_{\mu}\Omega
\nabla_{\nu}\Omega}{\Omega^2} \nonumber \\
& & -3 g_{\mu \nu} \frac{\nabla_{\sigma}\Omega \nabla^{\sigma}\Omega}{\Omega^2}. \end{aligned}$$ The remaining components associated to $\lambda$ are defined as the vacuum energy contributions for reasons to later be clarified $$\begin{aligned}
%\begin{align}
T^{\bf{vac}}_{\mu\nu}(\lambda,\rho)&=&
-\frac{\kappa\hbar^2}{m^2\Omega^2}[\nabla_{\mu}\sqrt{\rho}\nabla_{\nu}(\frac{\lambda}{\sqrt{\rho}})+\nabla_{\nu}\sqrt{\rho}\nabla_{\mu
}(\frac { \lambda}{\sqrt{\rho}})] \nonumber \\
& & +\frac{\kappa\hbar^2}{m^2\Omega^2}g_{\mu\nu}\nabla_{\sigma}(\lambda\frac{\nabla^{\sigma}{\sqrt{\rho}}}{\sqrt{\rho}}).
\label{TVacuum}\end{aligned}$$ The modified equation contains a stress-energy tensor related to the matter $T^{\bf{matter}}_{\mu\nu}(S,\rho)$, quantum $T^{\bf{qm}}_{\mu\nu}(\Omega)$ and coupling $T^{\bf{vac}}_{\mu\nu}(\lambda,\rho)$ contributions. The coupling contribution $T^{\bf{vac}}_{\mu\nu}(\lambda,\rho)$ is usually ignored by applying perturbative schemes to the Lagrangian multiplier $\lambda$ (typically assumed to be a parameter). It is worthwhile to explore the physical meaning of $\lambda$ as a density-dependent field, arising from the coupling of quantum matter with gravity. The physical interpretation of the coupling contribution and its relevance to the Planck scale is of important physical consequence. As will be seen, an expression of $\lambda$ for a single-boson can be easily defined for a static density, otherwise $\lambda$ must be interpreted dynamically. In Eq. \[EinsteinEq\], the Lagrange multiplier $\lambda$ appears to mediate the interaction between Klein-Gordon and Gravitational fields within the stress-energy tensor $T^{\bf{vac}}_{\mu\nu}(\lambda,\rho)$. We therefore seek to introduce a new physical meaning to the Lagrange multiplier $\lambda$ appearing in the theory.
Additionally, it is apparent that $T^{\bf{vac}}_{\mu\nu}(\lambda,\rho)$ brings about negative energies within the modified Einstein equation. General Relativists are particularly interested in negative energies because of their correspondence to expansion behavior [@Guth_Inflation]. On the other hand, to particle physicists, negative energies are simply a consequence of the particle’s interaction with the quantum vacuum. Negative energies are well known to contribute to spacetime expansion. In the Brans-Dicke theory, scalar fields contribute to the negative energies of gravitational fields [@BransDicke]. Similarly $f(R)$ theories also predict a negative energy contribution [@fofr_agravity; @EnrgyFRGravity; @EnerfofrandBransDicke]. Hence, it is fair to assume that the source of negative energy appearing in the currently proposed theory is simply a geometrical manifestation of the quantum vacuum. A connection to the vacuum energy is difficult to interpret geometrically since there is no ’geometrodynamic’ theory of quantum mechanics, rather just a probabilistic one. Nonetheless, in the Bohmian framework, one can more easily conceive the physical meaning of the coupling of matter to gravity via the conformal factor $\Omega^2$. Instead of assuming $\lambda$ to be spatially and temporally uniform, one can assume a spacetime dependence. As will be speculated later, the reason for the spacetime dependence can be attributed to the nontrivial nature of the vacuum energy. Assuming a near-zero velocity and mass, the scalar curvature $R$ of the trace equation (Eq. \[TraceEq\]) contains an intimate relationship to $\lambda$ $$\begin{aligned}
R \approx -2\kappa\frac{\lambda}{\Omega^2}\end{aligned}$$ Hence $\lambda$ gives negative scalar curvature $R$ in the near-zero velocity classical regime (when $\Omega\approx 1$). This is just a heuristic argument concerning the relationship between $\lambda$ and scalar curvature $R$. A complete mathematical relationship can be obtained by combining Eq. \[EqMotion\] and Eq. \[TraceEq\], $$R= +2\kappa\rho m {\Omega}^2 -6\frac{\Box\Omega}{\Omega} -2\frac{\kappa \lambda}{\Omega^2}
+\frac{\kappa \hbar^2
\sqrt{\rho}}{m^2\Omega^2}{\Bigl(\Box{(\frac{\lambda}{\sqrt{\rho}})}-\lambda\frac{\Box{\sqrt{\rho}}}{\rho}\Bigr)} \label{Req}$$ It is to be noted that the conformal factor $\Omega^2$ gives a negative contribution to the scalar curvature $R$ which is independent of the gravitational constant $\kappa$. The last three terms in Eq. \[Req\] give the quantum mechanical correction to the scalar curvature due to $\lambda$. There is a positive contribution to $R$ from rest mass $m$ of the quantum particle while the contribution from $\lambda$ can be negative. Spatial and temporal dynamics of $\lambda$ play an important role in the limit of small mass. As $m \to 0$ the vacuum coupling contributions within the scalar curvature dominate and therefore become critical in the quantum regime. Additionally, the vacuum coupling contribution vanishes in the classical limit as $m \to \infty$. It is then easy to conceive that the rest mass, characterizing a positive curvature, can be accompanied with negative curvature as a consequence of quantum mechanical effects. Given negative curvature plays a central role in the quantum regime, we claim $\lambda$ to be a vacuum contribution. In order to determine the quantum vacuum contributions, the first task is to find a way to express $\lambda$ analytically. Once $\lambda$ is known, it is possible to compute all the observable quantities appearing in the theory, which can be compared to experimental observations.
The spacetime behavior of $\lambda$ can be obtained by substituting the scalar curvature Eq. \[TraceEq\] into the contracted Einstein equation (Eq. \[EinsteinEq\]) $$\begin{aligned}
2\nabla_{\alpha}\Bigl(\lambda\frac{\nabla^{\alpha}\sqrt{\rho}}{\sqrt{\rho}}\Bigr)
-\nabla_{\mu}\sqrt{\rho}\,\nabla^{\mu}\Bigl(\frac{\lambda}{\sqrt{\rho}}\Bigr)=\frac{\lambda\,m^2}{\hbar^2}.\end{aligned}$$ Simplifying the L.H.S using Eq.\[CnstrEq\], followed by rearranging the terms involving $\lambda$ and $\rho$, one can arrive at a complete equation for $\lambda$
$$\begin{aligned}
\frac{\nabla_{\mu}\sqrt{\rho}}{\sqrt{\rho}}\frac{\nabla^{\mu}\lambda}{\lambda}=\Bigl(\frac{m^2(1-Q)}{\hbar^2}
-\frac{\Box\sqrt{\rho}}{\sqrt{\rho}}+\frac{\nabla_{\mu}\sqrt{\rho}\,\nabla^{\mu}\sqrt{\rho}}{\rho} \Bigr)\nonumber\\.
\label{LambdaEq}\end{aligned}$$
Assuming a spherically symmetric form of the density $\rho$ and ignoring variations in $\theta$ and $\phi$, one arrives at a more appealing form of the alleged vacuum energy equation $$\begin{aligned}
\nabla_{t}\lambda &=& \Bigl(\frac{m^2(1-Q)\sqrt{\rho}}{\hbar^2\nabla^{t} \rho}
-\frac{\Box{\sqrt{\rho}}}{\nabla^{t}\sqrt{\rho}}+\frac{\nabla_{\alpha}\sqrt{\rho}\,\nabla^{\alpha}\sqrt{\rho}}
{\sqrt{\rho}\nabla^{t}\sqrt{\rho}}\Bigr) \lambda \nonumber\\
& & +(\frac{\nabla^{r}\sqrt{\rho}}{\nabla^{t}\sqrt{\rho}}) \nabla_{r}\lambda.\end{aligned}$$ Here, $\lambda$ needs to be solved dynamically. In the static case, with only a radial contribution, $\lambda$ can be analytically represented in a much simpler exponential form $$\begin{aligned}
\lambda(r)&=& \exp{\Bigl(-\int{dr\,\beta(r)+ {C}_{\beta} }\Bigr)} \label{LambdaRadEq} \\
\beta(r)&=&\frac{\sqrt{\rho}}{\nabla_{r}\sqrt{\rho}}\Bigl(\frac{m^2(1-Q)}{\hbar^2}
+\frac{\Box_{r}\sqrt{\rho}}{\sqrt{\rho}}-\frac{\nabla_{r}\sqrt{\rho}\,\nabla_{r}\sqrt{\rho}}{\rho}\Bigr).\nonumber\\ \label{Betaeq}\end{aligned}$$ The subscript $r$ denotes differentiation in the radial component and $\lambda_{0}=\exp({C}_{\beta})$ is the resulting integration constant. As of now, the constraint for $\lambda_{0}$ is unknown and open for debate. It is speculated that, like the vacuum energy, $\lambda_{0}$ could be defined by a group theory characterizing the particles allowed in nature. In the case of a separable density, it can also be shown that $\lambda$ is of the form $\lambda =\lambda_{0}
\exp{(-\int{dt\,\alpha_{1}(\rho(t))}+\int{dr \beta_{2}(\rho(r)})}$. Integrating $\beta(r)$ within the exponential and computing ${C}_{\beta}$ results in the expression of $\lambda$. Once $\lambda$ is known, interpreting the negative energies arising from the quantum mechanical nature of matter can become a trivial task. Therefore, finding an analytical expression for $\lambda$ is critical to understanding the physical implications of the coupling contribution.
To better understand this, we define the density $\sqrt{\rho}$ of a single quantum mechanical particle as a Gaussian wavepacket in spherical coordinates $$\begin{aligned}
\sqrt{\rho(r,s)} = {\bigl(\frac{1}{\pi\,s^2}\bigr)}^{3/4} \exp{\Biggl(\frac{-r^2}{2s^2}\Biggr)}. \label{dnstyExp}\end{aligned}$$ Here, $N(s)={(\frac{1}{\pi\,s^2})}^{3/4}$ is the normalization constant and $s=\sigma+\sqrt{2}\ell_{p}$ is the spatial variation of the particle. We assume that an external potential, necessary to allow for a Gaussian density, is added to the equation of motion (Eq. \[EqMotion\]). Although $\sigma$ can be chosen freely, a fundamental Planck length limit $\ell_{p}$ is considered so as to conform with cosmological observables. As will be seen, the Gaussian standard deviation $s$ cannot arbitrarily decrease to zero, rather must obey the minimum allowable standard deviation $\sqrt{2}\ell_{p}$. Here, $\sqrt{2}$ is taken to eliminate the singularity which will later be shown to appear in the cosmological constant expression. The partial differential equation of $\lambda$ (Eq. \[LambdaEq\]) can be simplified for a general quantum mechanical density $\rho$ $$\begin{aligned}
\lambda=\frac{1}{(1-Q)}\frac{\hbar^2}{m^2}\nabla_{\sigma}(\lambda\frac{\nabla^{\sigma}{\sqrt{\rho}}}{\sqrt{\rho}}) \label{lambda_deq}.\end{aligned}$$ By ansatz, one can easily verify that this simple function obeys Eq. \[lambda\_deq\]. Similarly, the total vacuum energy density in the static case also simplifies to $T^{\bf{vac}}_{00}=\kappa\lambda/\Omega^{2}$. Using Eq. \[LambdaRadEq\] and Eq. \[Betaeq\] along with the predefined Gaussian density, $\lambda$ appears to take a fascinating form $$\begin{aligned}
\lambda=\lambda_{0}\exp{\Biggl(\frac{-r^2}{2s^2}\Biggr)} {r}^{(s/l_{c})^2} \label{eternal_eq}.\end{aligned}$$ As can be seen, a natural expansion and decay behavior is characterized by the polynomial and Gaussian, respectively. The form of Eq. \[lambda\_deq\] for a linear constraint $$\begin{aligned}
\lambda_{lin}=\frac{\hbar^2}{m^2}\nabla_{\sigma}(\lambda_{lin}\frac{\nabla^{\sigma}{\sqrt{\rho}}}{\sqrt{\rho}})
\label{lambda_deq_linear}\end{aligned}$$ Unlike Eq. \[eternal\_eq\], it can be seen that the expansion behavior of $\lambda_{lin}$ ceases to exist $$\begin{aligned}
\lambda_{lin}=\lambda_{0}\, r^{-(D-1)+ (s/{l_{c})^2}} \label{lambda_resultd}.\end{aligned}$$ Here, $D$ is representative of the number of dimensions and $l_{c}=\hbar/m$ is the Compton wavelength of a particle of mass $m$. It can be seen that $\lambda_{lin}$ contains a singularity when $s=l_{c}$. Therefore, $s\geq \sqrt{(D-1)}\,l_{c}$ must be satisfied to avoid the singularity. For a 1D-Gaussian ($1+1$ spacetime ($D=2$)), it can be shown that, once the free parameter $s$ is identified as $s=\sqrt{2} \Delta{x}$ and the maximum uncertainty in momentum as $\Delta p \propto m \implies l_{c}=\hbar/\sqrt{2} \Delta p $ (where $c=1$), avoiding the singularity in Eq. \[lambda\_resultd\] implies a new but familiar relation $$\begin{aligned}
\Delta x \Delta p \geq \,\hbar/2.\end{aligned}$$ This is just the uncertainty principle in $1+1$-dimension. Here, the uncertainty principle emerges from a more fundamental condition; that is avoiding the singularity of the quantum vacuum. This singularity can be eliminated by taking into account a minimal length $\ell_{p}$ requirement, enforcing a fundamental threshold. Note that, the singularity in $\lambda_{lin}$ appears for even $r\geq 0$ in the linear order theory $(\Omega^2=1+Q)$. Since we go beyond linear order theory, this problem doesn’t appear for $r>0$ and $s=0$ in Eq. \[eternal\_eq\]. But $\lambda$ in Eq. \[eternal\_eq\] is undefined when $r=0$ and $s=0$, again this can be solved by taking into account a minimal length $\ell_{p}$ requirement.
To better understand the coupling contributions defined in Eq. \[TVacuum\], one must determine whether observables are properly reproduced in the quantum and universal domains. One such observable is the cosmological constant $\Lambda$. The alleged 60-120 order difference in the transition of $\Lambda$ from the classical to quantum regime is a long-lived problem yet to be solved. The difficulty arises due to the lack of: 1) the proper characterization of quantum mechanical matter within the framework of General Relativity; 2) the lack of geometrical interpretation in quantum mechanics, via conformal frame. The cosmological constant can be identified as the negative term of the stress-energy tensor containing $g_{\mu\nu}$ $$\begin{aligned}
\Lambda_{v+q+g}&=& 3 \frac{\nabla_{\sigma}\Omega \nabla^{\sigma}\Omega}{\Omega^2} -\frac{\Box\Omega^2}{\Omega^2}
- \frac{\kappa}{m} \rho\,\nabla_{\sigma}S \nabla^{\sigma}S \nonumber \\
& &
+ \kappa m \rho \Omega^2 - \frac{\kappa \lambda }{\Omega^2}. \label{Lambda1}\end{aligned}$$ Substituting the equation of motion (Eq. \[EqMotion\]) into Eq. \[Lambda1\], one gets a quantity completely analogues to the vacuum and quantum contributions $$\begin{aligned}
\Lambda_{v+q+g}&=& 3 \frac{\nabla_{\sigma}\Omega \nabla^{\sigma}\Omega}{\Omega^2} -\frac{\Box\Omega^2}{\Omega^2} - \frac{\kappa
\lambda (1-Q)}{\Omega^2} \nonumber \\
& &
+\frac{\kappa\hbar^2\sqrt{\rho}}{2m^2\Omega^2}\Bigl[\Box({\frac{\lambda}{\sqrt{\rho}})}-\lambda\frac{\Box\sqrt{\rho}}{\rho}\Bigr].
\label{Cosmo}\end{aligned}$$ For quantum mechanical particles ($m\to0$), the vacuum contributions in Eq. \[Cosmo\] can play a significant role in characterizing $\Lambda$. Terms containing $\Omega$ will dominate in Eq. \[Cosmo\], deeming the conformal factor, arising from the quantum potential $Q$, an important contribution. Once $m$ is of the order of the Planck mass $\Bigl(\sqrt{\frac{\hbar\,c}{G}}\,\Bigr)$, the gravitational contribution suppresses the vacuum in the short distance. For smaller particles (i.e. mass of an electron), the vacuum tends to dominate the gravitational contribution. There is also an important point to take into consideration: the cosmological constant $\Lambda_{v+q+g}$ in Eq. \[Cosmo\] contains a spatial dependence, via $r$. For the more generalized vacuum contribution, a temporal dependence also naturally arises. By evaluating $\Lambda_{q+v+g}$ in Eq. \[Cosmo\] for the presumed density (Eq. \[dnstyExp\]), using vacuum contribution $\lambda$ (Eq. \[cosmo\_secret\]), and defined conformal factor $\Omega^2$ (Eq. \[CnstrEq\]), one arrives at an expression for the cosmological constant $$\begin{split}
& \Lambda_{v+q+g}= -6 \Bigl(\frac{l_{c}^2}{s^4}\Bigr)-\Bigl(\frac{l_{c}^2}{s^4}\Bigr)^2 r^2 \\
& -\kappa\,r^{s^2/l_{c}^2}\exp{\Biggl(\frac{-r^2}{2s^2}-\frac{l_{c}^2}{s^4} (r^2 -3
s^2)\Biggr)}\Biggl(1+\frac{l_{c}^2}{s^4}(3s^2-r^2)\Biggr) \\
& +\frac{\hbar^2 \kappa}{2 m^2 l_{c}^4 s^4} r^{-2 +s^2/l_{c}^2}\exp{\Biggl(\frac{-r^2}{2s^2}-\frac{l_{c}^2}{s^4} (r^2 -3
s^2)\Biggr)}\\
& \times \bigl[s^6 (l_{c}^2 + s^2)- l_{c}^4 (r^4 - 3 r^2 s^2) \bigr]. \label{cosmo_secret}
\end{split}$$ Equation \[cosmo\_secret\] contains the standard deviation of the Gaussian density $s$, allowing for a smooth quantum to classical transition. By considering the dominant contributions in Eq. \[cosmo\_secret\], one arrives at a simplified expression $$\begin{aligned}
\Lambda= 6 \Bigl(\frac{l_{c}^2}{s^4}\Bigr)+\Bigl(\frac{l_{c}^2}{s^4}\Bigr)^2 r^2 \label{CosmoSimpEq1}\end{aligned}$$ Here, $r$ defines the scale of observation and can vary from the Planck length to the observable universe. To properly conform to cosmology, one must fix $s=\sigma+\sqrt{2}l_{p}$. Here, $s=\sigma+\sqrt{2}l_{p}$ comes from the condition $s\geq\sqrt{2}\,l$ to avoid $r^{-2}$ singularity in the third term of the expression $\Lambda_{v+q+g}$ (See Eq. \[cosmo\_secret\]). The minimum-length element $l_{p}$ is chosen to avoid illogical mathematical scenario $(\frac{1}{0})^0$ appearing in $\Lambda_{v+q+g}$ (See Eq. \[cosmo\_secret\]) when $r=0$ and $s=0$. Fixing the minimum length $l_{p}$ transforms $(\frac{1}{0})^0\to{(\frac{1}{l_{p}})}^{\sqrt{2}l_{p}}$. Interestingly enough, the number $6$ in the conformal transformation is a result of the dimensionality of our spacetime $(D=4)
\implies
(D-1)(D-2)=6$. In the cosmological scale $(s \approx \sqrt{2}l_{p}\gg l_{c})$, Eq. \[CosmoSimpEq1\] can be written entirely in terms of the Schwartzschild radius $r_{s}=2GM/c^2$, $$\begin{aligned}
\Lambda \approx 6 \Bigl(\frac{1}{r_{s}^2}\Bigr)+\Bigl(\frac{1}{r_{s}^4}\Bigr) r^2 \label{CosmoAstroEq1}\end{aligned}$$ For $r\ll{r_{s}}$, the static term dominates and can be shown to play a fundamental role in defining its astronomical value $$\begin{aligned}
\Lambda_{\bf{astr}} \approx 6 \Bigl(\frac{1}{r_{s}^2}\Bigr) \label{CosmoAstroEq2}\end{aligned}$$ The cosmological constant can be determined theoretically once we know the analytical expression for the mass of the universe. It is well known that the Hoyle-Carvalho relation [@Carvalho1995] gives a theoretical estimation for the mass of the universe, where they had shown that the mass of the universe can be determined using only microscopic quantities. In addition, D. Valev pointed out that, using dimensionality arguments only, the Hoyle-Carvalho relation can be derived [@DValev2014] $$\begin{aligned}
M_{u} \propto \frac{c^3}{G\,H_{0}} \label{Massuni}.\end{aligned}$$ Given the recently measured Hubble constant ($H_{0} = 73.52 \pm 1.62\, km\, s^{-1}\, {Mpc}^{-1}$) [@ExperiHubble2016; @ExperiHubble2018], the mass of the universe can be estimated $M_{u} \approx 1.6\times 10^{53} kg$. Using Eq. \[CosmoAstroEq2\] and Eq. \[Massuni\], the cosmological constant can be written in terms of Hubble’s constant $H_{0}$. $$\begin{aligned}
\Lambda_{\bf{astr}} \approx \frac{3}{2} \Bigl(\frac{H_{0}^2}{c^2}\Bigr) \label{CosmoAstroEq3}\end{aligned}$$ This is just like the standard result of the cosmological constant arising in the Friedmann equation for a flat universe when the gravitational mass density contribution is ignored [@Friedman1999]. Note that, we started with a pure quantum mechanical problem and arrived at a standard result in Einstein’s gravity differing only by a factor of two. Approximately taking the mass of the universe $M \approx 1.6\times{10}^{53}\,kg$, and using Eq. \[CosmoAstroEq2\], one gets a static contribution of $\Lambda_{\bf{astr}}=1.06\times{10}^{-52} m^{-2}$. This is close to the value measured in a recent experiment [@PlanckMission]. The radially dependent term in Eq. \[CosmoAstroEq1\] plays a fundamental role in characterizing the expansion of our universe. At $r=r_s$ the rate of expansion naturally increases $\Lambda=1.24\times{10}^{-52} m^{-2}$. The quadratic trend in $r$ is the result of considering the exponential form of the conformal factor $\Omega^2=e^{Q}$ to ensure non-tachyonic behavior (e.g. beyond the linear form of $\Omega^2$). In the quantum regime, similar conclusions can be made for the standard deviation in Eq. \[CosmoSimpEq1\] when $s\approx l_{c}\gg \sqrt{2}l_{p}$, $$\begin{aligned}
\Lambda_{\bf{qm}}\approx 6 \Bigl(\frac{1}{l_{c}^2}\Bigr)+\Bigl(\frac{1}{l_{c}^2}\Bigr)^2 r^2 \label{LCosmoQM}\end{aligned}$$ Similarly, ignoring the spatially dependent contribution, one arrives at the static value of the cosmological constant for an electron mass $\Lambda_{el}=4.02363\times{10}^{25}{m}^{-2}$. At the Compton wavelength of an electron $r=l_{c}$, the spatial dependence of the cosmological constant once again begins to dominate $\Lambda_{el}=4.69424\times{10}^{25}{m}^{-2}$. The large discrepancy from the astronomical value has been pointed out by physicists for decades [@SeanCarrol_Cosmo; @Zeldovich_Cosmo]. Here, a natural variation in the value of $\Lambda$ is present simply by the consideration of the conformal factor. For a more accurate expression of the cosmological constant, particularly in the quantum regime (where the quantum vacuum plays a more fundamental role), one can use the generalized expression in Eq. \[cosmo\_secret\].
In this paper, we have explored the vacuum energy contributions resulting from the Bohmian framework of Quantum-Gravity. The conformal factor was defined by the quantum potential associated to the Klein-Gordon equation. By a geometrical means and an inherently probabilistic interpretation, we were able to couple the U(1) symmetry of quantum mechanics (beyond linear order) to arbitrary gravitational fields. Weinberg no-go theorem is bypassed using a Lagrange multiplier within the action, resulting in a vacuum density field $\lambda$. The identified vacuum contribution plays a significant role in the quantum regime (in the form of a correction), and naturally decays in the classical regime. After rigorous analysis, the proposed theory potentially serves as a solution to the cosmological constant problem. $\Lambda$ naturally varies by 77 orders from the cosmological to quantum scale and almost perfectly conforms to the measured astronomical value. Further experiments are needed to confirm the legitimacy of the order of expansion, via the cosmological constant, identified in the quantum regime. We hope that these results will shine light at the unification of quantum and gravitational fields. We plan to further explore the defined vacuum energy contribution $\lambda$ and its consequence to a geometrical realization of quantum mechanics.
Appendix A - Bypassing Weinberg’s no-go
=======================================
Weinberg proves his no-go theorem based on the usual understanding of coupling scalar fields to gravity, but our action couples the scalar field, $\Omega^2=e^{Q}$, in an entirely different manner. The key difference lies in our constraint field $\lambda$, which we identify as the vacuum density. $\lambda$ does not simply fix the conformal factor to the exponential of the quantum potential, rather allows one to overcome the no-go theorem by enforcing the scalar field to conform with the contracted stress-energy tensor. In this section, We make it apparent that Weinberg’s no-go theorem can be bypassed using a Lagrangian constraint.
Weinberg starts with the Euler Lagrange Equation (See Equation 6.2 and 6.3 in his article [@WeinbegCosmo89]). Looking for stationary solutions of the scalar and tensor equations, he finds that, for mathematical consistency, fine tuning is needed.
We identify that the aforementioned fine tuning problem arises within any conformally transformed Lagrangian $\mathcal{L}$ (e.g. Brans-Dicke theories) and suggest to overcome it by considering a yet unexplored field $\lambda$. Here $\lambda$ acts as a Lagrange multiplier enforcing the metric and scalar field to conform with the particle’s background potential, $Q$. A more thorough physical interpretation of $\lambda$ is given in this article. To simplify matters, we articulate a simplified action (of little physical meaning) instead of Weinberg’s proposed action to show the inconsistency can be eliminated $$\begin{aligned}
\mathcal{L}&=&e^{4\phi}\sqrt{-g}\mathcal{L}_{0}(\sigma)\nonumber \\
& & +\sqrt{-g}\lambda\Bigl(\phi-\mathcal{Q}(\rho,\nabla_{\mu}\rho,\Box\rho,...)\Bigr) \label{weinbergLag}\end{aligned}$$ By varying the action with respect to ${g_{\mu\nu},\phi,\lambda}$ we get a stress-energy tensor, scalar field equation, constraint equation, and, the yet unexplored, $\lambda$ equation $$\begin{aligned}
T_{\mu\nu}=T_{\mu\nu}^{WB}-2\frac{\delta}{\delta g_{\mu\nu}}\Bigl(\lambda\, \mathcal{Q}(\rho,\nabla_{\mu}\rho,\Box\rho,...)\Bigr) \label{einsteinEq}\end{aligned}$$ $$\begin{aligned}
\frac{\partial\mathcal{L}}{\partial \phi}=0 \implies (T^{\mu}_{\mu})^{WB} +\lambda=0 \label{scalarEq}\end{aligned}$$ The constraint equation is given by, $$\begin{aligned}
\frac{\partial\mathcal{L}}{\partial \lambda}=0 \implies \phi =\mathcal{Q}(\rho,\nabla_{\mu}\rho,\Box\rho,...) \label{CnstrEq0}\end{aligned}$$
According to Weinberg, consistency of the above equations requires that the trace of the energy-momentum tensor $T_{\mu\nu}$ ($g^{\mu\nu}{T_{\mu\nu}}={T^{\mu}_{\mu}}$) be equivalent to $\frac{\partial\mathcal{L}}{\partial \phi}=0$. The added field variable $\lambda$ allows for the scalar field $\phi$ to properly conform within Einstein’s equation by implying the following constraint field equation (Obtained using Eq.\[einsteinEq\] and Eq.\[scalarEq\])
$$\begin{aligned}
\lambda=-2g^{\mu\nu}\frac{\delta}{\delta g_{\mu\nu}}\Bigl(\lambda \, \mathcal{Q}(\rho,\nabla_{\mu}\rho,\Box\rho,...)\Bigr) \label{CnstrFieldEq1}\end{aligned}$$
With this field equation (See Eq. \[CnstrFieldEq1\]) satisfied, the scalar field and Einstein’s equations are consistent. The conditions imposed by Weinberg are satisfied: $$\begin{aligned}
\frac{\partial\mathcal{L}}{\partial g_{\mu\nu}}=0\\
\frac{\partial\mathcal{L}}{\partial\phi}=0 \label{ScalarEq1}\end{aligned}$$
The Lagrange multiplier imposes a condition on the scalar field, allowing one to overcome the no-go theorem. In the simplified scenario proposed by Weinberg, $\lambda=0$ leads to the fine tuning problem, suggesting a nonzero $\lambda$ should play an essential role in properly balancing the scalar (Eq. \[scalarEq\]) and contracted Einstein (Eq. \[einsteinEq\]) equations. When $\mathcal{Q}(\rho,\nabla_{\mu}\rho,\Box\rho,....)=K$, where $K$ is a constant, the $\lambda$ equation results in $\phi=K$. In such a scenario, the proposed field $\lambda=0$ and the Lagrangian in Eq. \[weinbergLag\] simplifies to Weinberg’s proposed Lagrangian (see Ref.[@WeinbegCosmo89]). Hence, the fine tuning problem can be superposed by an unexplored, nonzero field $\lambda$. With the presence of the constraint field equation (Eq. \[CnstrFieldEq1\]), the scalar field and Einstein’s equations can be made to be consistent. This justifies our reasoning for enforcing the scalar field $\phi$ to contain a metric dependence, via a higher order-derivative of the quantum density $\sqrt{\rho}$ (see the action in Eq. \[Actioneq\]).
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abstract: 'Understanding the wave transport and localisation is a major goal in the study of lattices of different nature. In general, inhibiting the energy transport on a perfectly periodic and disorder-free system is challenging, however, some specific lattice geometries allow localisation due to the presence of dispersionless (flat) bands in the energy spectrum. Here, we report on the experimental realisation of a quasi-one-dimensional photonic graphene ribbon supporting four flat-bands. We study the dynamics of fundamental and dipolar modes, which are analogous to the $s$ and $p$ orbitals, respectively. In the experiment, both modes (orbitals) are effectively decoupled from each other, implying two sets of six bands, where two of them are completely flat. Using an image generator setup, we excite the $s$ and $p$ flat band modes and demonstrate their non-diffracting propagation for the first time. Our results open an exciting route towards photonic emulation of higher orbital dynamics.'
author:
- 'Camilo Cantillano[^1]'
- 'Sebabrata Mukherjee$^{\dagger,}$'
- 'Luis Morales-Inostroza'
- Bastián Real
- 'Gabriel Cáceres-Aravena'
- 'Carla Hermann-Avigliano'
- 'Robert R. Thomson'
- 'Rodrigo A. Vicencio'
title: Observation of Ground and Excited Flat Band States in Graphene Photonic Ribbons
---
The physics of graphene is an intense field of research primarily focused on their unique electronics and magnetic properties. In particular, graphene nanoribbons can exhibit edge states [@gr1] as well as the transition from semiconductors to semi-metals, depending on the number of coupled ribbons [@gr2; @gr3]. Several attempts to fabricate and characterize these graphene-like structures have been reported due to their fundamental relevance for future applications in nanoelectronics [@gr4; @gr5]. This includes room-temperature ballistic transport [@gr6], well-controlled atomic configurations [@gr7], photonics and optoelectronic applications [@gr8]. In the photonic platform, graphene lattices have already been induced in photorefractive crystals at [ the micrometer]{} scale, where conical diffraction and nonlinear localisation were experimentally observed [@hcmoti]. Additionally, the observation of unconventional edge states [@hcedge], photonic floquet topological insulators [@htopo], and pseudospin-mediated vortex generation [@hcpseudo] have been reported in graphene optical lattices. The ability of directly imaging the wavefunction gives an important experimental advantage for photonic setups [@rep1; @rep2; @Garanovich2012], in comparison to solid-state physics.
Recent advancement in experimental physics enabled us to emulate various semi-classical and quantum phenomena in a highly controllable environment. Ultracold atoms in optical lattices [@Bloch2005; @Jaksch2005] and periodic arrays of coupled optical waveguides (photonic lattices) [@rep1; @rep2; @Garanovich2012] are two parallel experimental platforms which were extensively used to observe and probe various intriguing solid-state phenomena. This includes the localisation effects induced by external fields [@Dreisow2008; @WSL], disorder [@Schwartz2007; @Billy2008] and particle interactions [@Greiner2002; @Szameit2006]. Indeed, localisation is a major goal in diverse areas of physics, where the trapping and control of excitations of different nature become crucial [@rep2]. During several years, photonics has taken a central role on this problem, being particularly intense in the context of photonic lattices. Different fabrication techniques have been developed, being the femtosecond-laser technique probably the most flexible one in order to fabricate arbitrary three-dimensional configurations [@fslt; @femtoseba]. Most of the known methods to localise energy rely on the modification of the lattice using linear or nonlinear defects, or by destroying the periodicity of the system. However, localised states in a photonic Lieb lattice [@lieb2; @liebseba] were recently observed in the linear optical regime, due to the existence of a completely flat-band (FB). The states residing on this non-diffractive band occupy only a few sites and can be considered as localised states in the continuum [@bic; @lieb2]. Unfortunately, the flatness of a given band can be modified if extra interactions are also considered in the model [@desa1]. This is a frequent problem on several FB systems which diminishes the chances for an experimental excitation of FB localised states. However, by inspecting the discrete properties of a given system, it is possible to identify some lattices where next or even next-next nearest neighbour (NN) interactions preserve the flatness of the band. This requires a high degree of symmetry in order to effectively cancel the transport at different connector sites [@luis].
Almost all the experimental research devoted to the study of periodical systems has focused on the excitation of fundamental modes on different lattice sites. This is essentially due to experimental complications of exciting higher order modes, which in the case of cold atoms have been solved indirectly by selectively populating p-band states [@becdipole]. However, a precise excitation of dipolar states has only been possible very recently on optical waveguide lattices using an image generator setup [@dipole1], where a [well-defined]{} contrast between the transport of fundamental and dipolar states has been shown. The possibility to experimentally excite and control higher bands excitations, in optical lattice systems, paves the venue in which the study of remarkable properties of correlated systems such as superfluidity, superconductivity, antiferromagnetic ordering, among others, becomes possible [@wu; @li1; @yin; @li2].
In this Article, we study theoretically and experimentally a graphene-like ribbon [ where each lattice site supports two non-degenerate modes, the fundamental and dipolar modes.]{} This system is particularly interesting because it [ can possess]{} two flat bands per mode, and these bands are robust against higher-order coupling interactions. This implies that the excitation of FB states is quite stable in realistic experimental conditions, as we show below. As these modes possess a large propagation constant detuning, the interaction between them is effectively absent in the dynamics. To the best of our knowledge, this is the first experimental [ realisation]{} of a periodical lattice possessing [ multiple flat-bands,]{} corroborated by the [observation of the spatially localised flat-band states.]{}
[*Model.*]{} The unit cell of a graphene ribbon consists of a sequence of six-sites as sketched in Fig. \[f1\](a), where each waveguide is separated from its nearest neighbour by a centre-to-centre distance “$a$”. The interaction between lattices sites is governed by the evanescent coupling which decreases exponentially with the distance between waveguides [@fslt; @femtoseba]. We define the nearest and next-nearest neighbour coupling coefficients in Fig. \[f1\](b), where the horizontal coupling is $V_1$, the short-diagonal one is $V_2$, the vertical coefficient is $V_3$, and the long-diagonal one is $V_4$. The contribution of all other long range couplings can be safely neglected for the maximum propagation distance considered here. For our laser inscribed photonic lattice (PL), each waveguide supports elliptically oriented modes with the major axis along the vertical, implying that $V_2>V_1$ and $V_3>V_4$.
![[****]{} (a) A graphene-ribbon lattice. (b) Couplings interactions represented by lines. Linear spectrum for (c) $V_1=1$, $V_2=2$, $V_3=V_4=0$ and (d) $V_1=1$, $V_2=2$, $V_3=V_4=0.5$. \[f1\]](fig1.pdf){width="0.9\linewidth"}
In the scalar-paraxial approximation, the evolution of light waves across a graphene ribbon is governed by the following discrete linear Schrödinger-like equations [@rep1; @rep2; @Garanovich2012] $$-i\frac{\partial\psi_{\vec n}^j}{\partial z}=\beta_j\psi_{\vec n}^j+\sum_{\vec m\neq\vec n}V_{\vec n,\vec m}^j\psi_{\vec m}^j\ .
\label{eq1}$$ Here, $\psi_{\vec n}^j$ describes the field amplitude of a given mode, $j\!=\!\{s, p$}, at the $\vec n$-th site, with propagation constant $\beta_j$, $z$ corresponds to the propagation coordinate (dynamical variable) along the waveguides, and $V_{\vec n,\vec m}^j$ represents the coupling interactions between sites $\vec n$ and $\vec m$ for mode $j$. In model (\[eq1\]), it was assumed that the $s$ and $p$ modes are effectively decoupled. First, we consider that each waveguide supports only a single mode, for [example,]{} the fundamental $s$ mode. In order to find the linear spectrum of this lattice, we first define the unitary cell composed of sites $A$, $B$, $C$, $D$, $E$, and $F$ as shown in Fig. \[f1\](b), and insert a plane wave ansatz $\vec{\Psi}_n(z)=\vec{\Psi}_0\exp(i k_x a n) \exp(i k_z z)$, with $\vec{\Psi}_l\equiv \{A_l,B_l,C_l,D_l,E_l,F_l\}$. Here, $k_x$ and $k_z$ [correspond]{} to the transverse and longitudinal propagation constants, respectively. By solving the eigenvalue problem, we identify two flat bands, $k_z^{\pm}(k_x)=\pm V_1-V_3$, with degenerate eigenmodes, as indicated by the red horizontal lines in Figs. \[f1\](c) and (d). It should be highlighted that the flatness of these two bands is independent of the next-nearest neighbour interactions due to the symmetry of the lattice geometry. Only for a reduced set of parameters, the rest four linear bands can be expressed in a closed form. Therefore, for generality, we show the band structure in Fig. \[f1\] for two different cases considering (c) only NN and (d) NN plus next NN interactions. In both cases, one can observe two perfectly flat bands, demonstrating the robustness of FB phenomena against the next-nearest neighbour interactions in this lattice geometry. The compact localised states occupy only four sites ($B$, $C$, $E$, and $F$) of a unit cell, with equal intensity and the following phase distributions: [ $\{+,+,-,-\}$]{} for the upper and [ $\{+,-,+,-\}$]{} lower flat bands, [respectively]{}. We can easily identify the destructive interference at sites $A_n$ and $D_n$, as expected considering the properties of mini-arrays [@luis]. When exciting these localised FB states, the transport is absolutely cancelled across the lattice due to the perfectly zero amplitude at the connector sites.
![[****]{} Composed linear spectrum of a graphene-ribbon with $V_1^s\!=\!0.5$, $V_2^s\!=\!1$ and $V_1^p\!=\!1$, $V_2^p\!=\!-2$. (a)–(d) Intensity and phase profiles of the FB modes. Here, red (green) colour represents a positive (negative) phase. \[f2\]](fig2.pdf){width="0.95\linewidth"}
Now, we consider that each waveguide in the lattice supports two modes, the fundamental ($s$) and the vertically-oriented dipolar ($p$) modes. It should be mentioned that the propagation constant (or the analogous energy) of the supported modes, and hence, the excitation of [higher-order]{} modes, can be efficiently controlled by tuning the wavelength $\lambda$ of incident light. The coupling between the two modes at the same lattice site is forbidden due to orthogonality. The large mismatch in propagation constants (defined as $\Delta \beta\equiv |\beta_s-\beta_p|$) [@dipole1] causes a negligible effective coupling interaction between the $s$ and $p$ modes at adjacent waveguides, as we confirmed experimentally below. The dynamical excitation of an orthogonal mode on a neighbour waveguide is proportional to the ratio $V_{sp}/\Delta \beta$, where $V_{sp}$ is the NN coupling interaction between the s and p modes. For standard elliptical waveguides [@dipole1], $V_{sp}/\Delta \beta\sim1/30$. (In atomic systems, this is related to the energy difference between different energy levels. Note that the coupling interaction between the $s$ and $p$ modes on adjacent sites can induce interesting phenomena, such as topological edge modes [@li1]; however, its experimental atomic implementation is still a challenge.) By following these considerations, now we can write the dynamical equations for both modes just by identifying $j=s$ or $p$ in Eq. (\[eq1\]) and by writing $V_i$ as $V_i^j$, to distinguish the coupling constants for different modes (in general, as the wave-function of the fundamental mode has a shorter evanescent tail [@li1; @dipole1], $|V_i^p|> |V_i^s|$). To simplify the description, we will consider only NN coupling such that $V_1,V_2\gg V_3,V_4$, and a detuning $\Delta\beta\equiv\beta_s-\beta_p\approx30$ cm$^{-1}$ [@dipole1]. In Fig. \[f2\], we present an example of the composed linear spectrum for this two-mode-system. We observe four flat bands located [at]{} $\pm V_1^p$ and $\Delta\beta\pm V_1^s$, and also the corresponding FB mode profiles. These states satisfy a destructive interference condition at connector sites (white zero amplitudes at the central row), depending on the sign of coupling constants. The relative sign of the coupling coefficients is determined by the parity symmetry of the $s$ and $p$ modes, considering the profiles sketched in Figs. \[f2\]. Whereas the fundamental coupling constants are always positive ($V_1^s,V_2^s>0$), the dipolar ones are determined by the specific geometry: $V_1^p>0$ and $V_2^p<0$. In Fig. \[f2\] we observe that the simplest fundamental FB mode ($a$) possesses the larger longitudinal propagation constant $k_z$, while the more complex dipolar one ($d$) has a shorter value, for this two-modes system.
![[****]{} (a) White-light-micrograph of the output facet of a graphene ribbon (PL). (b) Experimental setup to excite and measure the dynamics of a given input state. Here, L: lens; P: polariser; SLM-t (SLM-r): transmission (reflection) SLM; M: Mirror; BS: Beam Splitter; CCD: Camera; PL: Photonic Lattice. The inset shows a dipolar input state launched at an $A$ site. (c) and (d) show the output intensity profiles of the dipolar single-site excitations injected separately at the $A$ and $B$ sites, respectively, as indicated with the white circles. To highlight the low intensities, a nonlinear colour-map was used. \[f3\]](fig3.pdf){width="0.8\linewidth"}
[*Experiments.*]{} Photonic graphene ribbons are directly fabricated inside a borosilicate substrate (Corning Eagle$^{2000}$) using ultrafast laser inscription [@ol96; @ol17sm]. Our fabrication method produces waveguides which are elongated along the vertical direction, therefore, the dipolar ($p$) modes are constrained to exist in that direction too. In Fig. \[f3\](a), a white-light transmission micrograph of the output facet is presented, showing the vertically oriented waveguides. The laser-writing parameters are optimised to produce single-mode waveguides with low propagation losses at a $780$ nm wavelength. The final lattices are inscribed in a $3$-cm-long substrate, with $a\!=\!17\ \mu$m waveguide spacing. In order to study the dynamics of the $s$ and $p$ modes, we reduced the wavelength to perform the experiment at $\lambda=640$ nm. We implement an *image generator setup* [@dipole1] as shown in Fig. \[f3\](b), which enables us to generate an arbitrary input state that can be launched on the photonic lattice PL (this is mounted on a $5$-axis-stage, which is not shown in the figure). The key element of this setup is a sequence of two spatial light modulators (SLM), that modulate the amplitude (SLM-t) and the phase (SLM-r) of an incident laser beam. Using this configuration, we launch a desired input state (with specific spatial profile and intensity and/or phase distribution) at a given lattice site. For example, the inset in Fig \[f3\](b) shows a dipolar input state generated by the image generator setup. Figs. \[f3\](c,d) present the output intensity distributions for single-site dipolar excitations at $A$ and $B$ sites, respectively. We observe how the energy diffracts in the lattice, due to the excitation of dispersive bands in the spectrum.
![[****]{} Output profiles for different input conditions: fundamental FB profiles at (a) $k_z=\Delta\beta+V_1^s$ and (b) $k_z=\Delta\beta-V_1^s$, and (c) four in-phase sites. Left (right)-insets: interferogram of a tilted plane wave with input (output) profiles. To highlight the low intensities, a nonlinear colour-map was used. \[f4\]](fig4.pdf){width="0.8\linewidth"}
In order to observe the dynamics of flat-band states, we use our image generator setup to excite only four desired sites of a unit cell with different spatial and phase profiles. First, to excite the fundamental FB modes, we generate two input states as sketched in Figs. \[f2\](a) and (b). The observed outputs are presented in Figs. \[f4\](a) and (b), respectively. We see that both FB input states propagate along the crystal without exhibiting any significant diffraction across the lattice, with an evident zero background. These states remain quite localised in space and occupy only four sites of the lattice, constituting two completely independent orthogonal states. To measure the phase profile of the input as well as of the output states, we implement an interferogram setup \[this is not shown in Fig. \[f3\](b) and simply consists on superposing the output profile with an extended tilted plane wave\]. The left and right insets in Figs. \[f4\](a) and (b) show the input and output phase structure, respectively. As the intensity and phase profiles are preserved in the dynamics, we can confirm the first excitation of the two fundamental FB modes. Additionally, we inject an in-phase four-sites excitation pattern and observe that the energy starts to spread to the rest of the lattice by the excitation of $A$ and $D$ connector sites \[see Fig. \[f4\](c)\]. This input condition [excites]{} most of the linear spectrum and, therefore, for a longer propagation distance or a shorter waveguide separation, the energy would spread faster and would cover a larger transverse area, as we have confirmed numerically.
In the next step, we excite the dipolar ($p$) flat-band modes, which is [considerably]{} more challenging due to the complexity of the required spatial and phase profiles. \[Note that the single-site excitations of the dipolar mode [were]{} presented in Fig. \[f3\](c,d).\] Precise control of the input [state,]{} as well as its accurate overlap with the dipolar modes of the lattice [sites (waveguides),]{} is required. We generated two dipolar FB modes sketched in Figs. \[f2\](c) and (d) and measured outputs are shown in Figs. \[f5\](a) and (b). In both cases, we observe a spatially localised state which occupies only four sites of the lattice, with a zero background. The interferograms show that the input and output phase profiles are preserved during the propagation, confirming the excitation of p-FB modes. We probe the relevance of the phase structure, on the cancellation of the transport through connector sites [@luis], by injecting an input pattern composed of four in-phase dipolar waveguide modes. In Fig. \[f5\](c) we show a complete destruction of the input profile, as a consequence of exciting the dispersive part of the spectrum.
![[****]{} Output profiles for different input conditions: dipolar FB profiles at (a) $k_z=V_1^p$ and (b) $k_z=-V_1^p$, and (c) four in-phase sites. Left (right)-insets: interferogram of a tilted plane wave with input (output) profiles. To highlight the low intensities, a nonlinear colour-map was used. \[f5\]](fig5.pdf){width="0.8\linewidth"}
[*Discussions*]{} We studied a graphene-ribbon lattice and showed the existence of $s$ and $p$ flat-band modes in the linear optical regime. Due to the symmetry of this lattice geometry, FB states can exist even in the presence of next NN interactions, as we have theoretically demonstrated by inspecting the linear properties of this system. Our lattice model possesses two FB per mode which correspond to bulk FB states, something that is particularly different to the already predicted FB edge modes in graphene-like lattices [@gr2]. In our homogeneous lattice, fundamental and dipolar modes are effectively decoupled, showing no interaction between these modes. We carefully prepared several input states and experimentally observed a stable propagation of the four FB modes, what is confirmed by the analysis of the corresponding phase structure. This is the first experimental evidence of a controlled excitation of a system possessing two flat-bands per mode and, also, this is the first observation ever of a p-FB mode in any physical system. The ability to precisely control the input states in photonic lattices gives us a unique access to investigate more complex phenomena, as it has been suggested in different areas of physics [@rep1; @rep2; @Garanovich2012; @Bloch2005; @Jaksch2005; @li1; @yin; @li2]. It should also be highlighted that the effective coupling between the spatial modes (orbitals) can be controlled by tuning their energy (propagation constant) mismatch. Experimental realisation of such engineered photonic lattices with interacting spatial modes will enable us to investigate intriguing phenomena [@yin; @li1] with more complex dynamics.
[*Fabrication.*]{} The photonic devices, used in the experiments, were fabricated using the ultrafast laser inscription [@ol96]. By focusing sub-picosecond laser pulses (Menlo System BlueCut) inside a borosilicate substrate (Corning Eagle$^{2000}$), the refractive index of the substrate is modified permanently. Each optical waveguide is fabricated by translating the substrate (at 8 mm/s speed) once through the focus of the laser beam.
[*Image generation.*]{} We generate specific input conditions by using an *image generator setup*. This setup consists of a sequence of several optical elements described as follows: We first expand a laser beam in order to cover completely the screen of a transmission spatial light modulator (SLM). By setting two polarisers, we optimize the amplitude modulation response of this SLM and create a given light pattern. In our experiments, this pattern consists of several light spots located on specific positions depending on the studied lattice. Then, we modulate only in phase this pattern using a reflective SLM. After this point, we obtain a light pattern which is already modulated in amplitude and phase. Finally, by using different optics and a $10\times$ microscope objective (MO), we decrease the size of the pattern in order to match the specific dimensions of the lattice. To calibrate this, we install a beam splitter before the MO and take an image on a CCD camera using the reflected light on this facet. By launching white light on the output facet, we are able to observe the waveguide positions and check the dimensions of the generated image in the input facet. Finally, we take several images of the output patterns by installing after the sample another $10\times$ MO and a second CCD camera.\
[*Acknowledgment.*]{} The authors sincerely thank financial support from Programa ICM RC130001, FONDECYT Grant No. 1151444, UK Science and Technology Facilities Council (STFC) through ST/N000625/1.
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[^1]: These authors contributed equally to this work.
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abstract: 'The conventional slow-roll approximation is broken in the so-called “ultra slow-roll” models of inflation, for which the inflaton potential is exactly (or extremely) flat. The interesting nature of (canonical) ultra slow-roll inflation is that the curvature perturbation grows on superhorizon scales, but has a scale-invariant power spectrum. We study the ultra slow-roll inflationary dynamics in the presence of noncanonical kinetic terms of the scalar field, namely ultra slow-roll G-inflation. We compute the evolution of the curvature perturbation and show that the primordial power spectrum follows a broken power law with an oscillation feature. It is demonstrated that this could explain the lack of large-scale power in the cosmic microwave background temperature anisotropies. We also point out that the violation of the null energy condition is prohibited in ultra slow-roll G-inflation and hence a blue tensor tilt is impossible as long as inflation is driven by the potential. This statement is, however, not true if the energy density is dominated by the kinetic energy of the scalar field.'
author:
- 'Shin’ichi Hirano'
- Tsutomu Kobayashi
- Shuichiro Yokoyama
title: 'Ultra slow-roll G-inflation'
---
Introduction
============
Inflation is a very successful scenario of the early universe that resolves the problems in the standard big bang cosmology [@Guth:1981; @Starobinsky:1980; @Sato:1981]. Conventional inflation models employ a scalar field $\phi$ (the inflaton) rolling slowly on a nearly flat potential, $V(\phi)$. The energy density $\rho$ and the pressure $p$ of the scalar field are then given by $\rho\simeq V$ and $p\simeq -V$, leading to a quasi-de Sitter expansion. Quantum fluctuations in $\phi$ are generated during inflation [@Mukhanov:1981xt], which nicely provide the seeds for the large-scale structure of the Universe. Standard single-field slow-roll inflation indeed predicts nearly scale-invariant, adiabatic, and Gaussian density perturbations consistent with the observed cosmic microwave background (CMB) anisotropies [@Planck:2013jfk; @Planck2], though the agreement is not perfect and some anomalies have been reported in CMB data such as the lack of large-scale power [@Planck2].
Slow-roll inflation driven by a canonical scalar field is no doubt very attractive, but it is not the only option. For example, the inflaton sector of the Lagrangian could contain non-canonical kinetic terms and/or multiple fields. It is also possible that the inflaton dynamics is away from slow roll. Those different possibilities yield different predictions for CMB power spectrum features and thus can be tested against observations. It would also be interesting if non-standard models of inflation could explain the observed CMB anomalies.
Since the deviation from the exactly scale-invariant power spectrum is characterized by the slow-roll parameters, the violation of the slow-roll approximation would imply a large tilt as well as a non-de Sitter expansion. This common wisdom, however, is not true. If the inflaton is subject to the so-called “ultra slow-roll” (or “nonattractor”) dynamics [@Kinney:2005vj; @Inoue:2001zt], one of the slow-roll parameters is of ${\cal O}(1)$, but the Universe undergoes a quasi-de Sitter phase. In the ultra slow-roll phase, the curvature perturbation grows on superhorizon scales, and this growing mode has a scale-invariant spectrum. Ultra slow-roll inflation is unique in that large local non-Gaussianity is produced due to the superhorizon growth despite a single field model [@Namjoo:2012aa; @Martin:2012pe; @Huang:2013lda; @Motohashi:2014ppa; @Mooij:2015yka]. The same behavior of the curvature perturbation is also found in the other backgrounds such as matter bounce [@Finelli:2001sr; @Wands:1998yp; @Allen:2004vz; @Cai:2009fn] and a variant of Galilean genesis [@Liu:2011ns; @Piao:2010bi; @Nishi:2015pta].
In this paper, we explore the consequences of non-canonical kinetic terms on the ultra slow-roll dynamics of inflation in order to construct a phenomenological model that can explain the CMB anomalies. In Refs. [@Chen:2013aj; @Chen:2013eea] ultra slow-roll models have already been generalized to the Lagrangian containing higher power kinetic terms, $(\partial_\mu\phi\partial^\mu\phi)^n$, [*i.e.,*]{} the k-essence Lagrangian [@ArmendarizPicon:1999rj], with a particular emphasis on the violation of Maldacena’s consistency relation for the bispectrum in the squeezed limit [@Maldacena:2002vr]. In this paper, we allow the inflaton Lagrangian to depend on second derivatives of the scalar field [@Deffayet:2010qz; @Kobayashi:2010cm]. That is, we consider [*ultra slow-roll G-inflation*]{}. Provided that the duration of inflation is “just enough,” the primordial power spectrum is of the broken power-law form having a blue tilt on large scales in ultra slow-roll k/G-inflation. We would therefore point out that this could explain the large-scale suppression of the CMB power.
G-inflation is an interesting class of models that admits in principle a blue spectrum of primordial tensor modes by violating the null energy condition stably [@Kobayashi:2010cm] (see, however, Ref. [@Cai:2014uka]). Therefore, it should be clarified in which concrete models this is indeed possible. Under the usual slow-roll conditions, it was shown in Ref. [@Kamada:2010qe] that stable potential-driven models cannot produce a blue tensor spectrum. This result leads to the question whether stable violation of the null energy condition is possible under the ultra slow-roll conditions, which also motivates the study of the combined scenario of ultra slow-roll inflation and G-inflation.
The paper is organized as follows. In the next section we introduce ultra slow-roll G-inflation and study the evolution of the curvature perturbation. The suppressed CMB power due to the broken power-law primordial spectrum is demonstrated. We then consider the inflationary universe approaching a kinetically driven de Sitter attractor and show that a blue spectrum of primordial tensor modes is possible in Sec. III. We draw our conclusions in Sec. IV.
Ultra slow-roll G-inflation
===========================
The background equations
------------------------
Let us consider the action of the form $$\begin{aligned}
&S=\int{{\rm d}}^4x\sqrt{-g}
\left[
\frac{{M_{\rm Pl}}^2}{2}R+{\cal L}_\phi
\right],
\\
&{\cal L}_\phi=-V(\phi)+K(X)-G(X)\Box\phi,\label{Lphi1}\end{aligned}$$ where $K$ and $G$ are arbitrary functions of $X:=-(1/2)g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$. This action was obtained by generalizing the Galileon [@Nicolis:2008in], and the first application to dark energy/inflation was done in Refs. [@Deffayet:2010qz; @Kobayashi:2010cm]. A further generalization yields generalized G-inflation [@Kobayashi:2011nu], [*i.e.,*]{} inflation in the Horndeski theory [@Horndeski:1974wa; @Deffayet:2011gz; @Charmousis:2011bf], but just for simplicity we focus on the above subclass; G-inflation captures important aspects of generalized G-inflation and in the following analysis extending the former to the latter would be more or less straightforward.
For a flat Friedmann universe, ${{\rm d}}s^2=-{{\rm d}}t^2+a^2(t)\delta_{ij}{{\rm d}}x^i{{\rm d}}x^j$, the field equations read $$\begin{aligned}
&3{M_{\rm Pl}}^2 H^2=V-K+2XF,\label{bg1}
\\
&{M_{\rm Pl}}^2\dot H=-XF+XG_X\ddot\phi,\label{bg2}
\\
&\frac{{{\rm d}}}{{{\rm d}}t}\left(\dot\phi F\right)+3H\dot\phi F+\frac{{{\rm d}}V}{{{\rm d}}\phi}=0,\label{bg3}\end{aligned}$$ where $$\begin{aligned}
F:=K_X+3G_X H \dot\phi,\end{aligned}$$ and a subscript $X$ denotes differentiation in terms of $X$. For a canonical model of inflation, we have $K=X$, $G=0$, so that $F=1$, and in this case Eqs. (\[bg1\])–(\[bg3\]) reproduce the standard equations.
[*Slow-roll*]{} G-inflation presumes that the energy density is dominated by the potential in Eq. (\[bg1\]) and the friction term balances the slope of the potential in Eq. (\[bg3\]), [*i.e.*]{}, $$\begin{aligned}
3{M_{\rm Pl}}^2H^2\simeq V,
\quad
3H\dot\phi F+\frac{{{\rm d}}V}{{{\rm d}}\phi}\simeq 0.\end{aligned}$$ This class of G-inflation models has been investigated in [@Kamada:2010qe] with an emphasis on the consequence of a non-canonical kinetic term with $F\gg 1$.
In this paper, we propose [*ultra slow-roll*]{} G-inflation for which the potential is [*exactly*]{} flat, $V=V_0=$ const. This is an extension of the notion of ultra slow-roll inflation [@Kinney:2005vj; @Inoue:2001zt] to the models with noncanonical kinetic terms. Ultra slow-roll k-inflation has been previously discussed in Refs. [@Huang:2013lda; @Chen:2013aj; @Chen:2013eea]. When $V=$ const, the friction term cannot balance the potential slope because ${{\rm d}}V/{{\rm d}}\phi =0$, and as a result the scalar field equation of motion yields $$\begin{aligned}
\frac{{{\rm d}}}{{{\rm d}}t}\left(\dot\phi F\right)+3H\dot\phi F=0
\quad\Rightarrow\quad \dot\phi F\propto a^{-3}.\label{beki}\end{aligned}$$ It is still assumed that $V_0\gg |K|,\;|XF|$, and hence $3{M_{\rm Pl}}^2H^2\simeq V_0$. Note that even in the case of an extremely flat potential ultra slow-roll inflation is possible. However, in the present paper we focus on the exactly flat potential just for simplicity and clarity.
To be more specific, let us consider the case with $F\propto X^{p-1}$. Here, $p\,(\ge 1)$ is not necessarily an integer. It follows from Eq. (\[beki\]) that $$\begin{aligned}
\dot\phi\propto a^{-3/(2p-1)},
\quad
\ddot\phi=-\frac{3H\dot\phi}{2p-1}\propto a^{-3/(2p-1)}.\end{aligned}$$ Then, Eq. (\[bg2\]) reduces to $$\begin{aligned}
-{M_{\rm Pl}}^2\dot H =X\left(F+\frac{3}{2p-1}G_XH\dot\phi\right).\end{aligned}$$ Since $|F|\gtrsim|G_XH\dot\phi|$ (barring from accidental cancellation of the two terms in $F$), we have $$\begin{aligned}
\epsilon:=-\frac{\dot H}{H^2}={\cal O}(|XF|/V_0)\ll 1,\end{aligned}$$ and $$\begin{aligned}
\epsilon\propto X^p\propto a^{-6p/(2p-1)}.\end{aligned}$$ The second slow-roll parameter, $\eta:=\dot\epsilon/H\epsilon$, is given by $$\begin{aligned}
\eta=-\frac{6p}{2p-1}={\rm const}<0,\label{eta-formula}\end{aligned}$$ where $-6\le \eta < -3$ for $p\ge 0$ and hence $|\eta|$ is not small. For example, the k-inflation model with $K\propto X^{n}$ and $G=0$ gives $p=n$, while the model with $K=0$ and $G\propto X^{n}$ leads to $p=n+1/2$. In the latter case $p$ is a half-integer if $n$ is an integer. Substituting $p=1$ to Eq. (\[eta-formula\]), one can recover the previous result of ultra slow-roll inflation, $\eta=-6$.
![The evolution of the slow-roll parameters, $\epsilon$ (a) and $\eta$ (b), in the two-phase model of ultra slow-roll G-inflation as functions of $e$-folding number $N$. []{data-label="fig:plt1.eps"}](plt1.eps){height="95mm"}
Bearing in mind the above ultra slow-roll dynamics with a noncanonical kinetic term, let us suppose that ${\cal L}_\phi$ is composed of the standard kinetic term plus some higher-power term, as $$\begin{aligned}
\label{k-type}
{\cal L}_\phi=-V_0+X+\frac{X^n}{M^{4(n-1)}}\quad (n\ge2),\end{aligned}$$ or $$\begin{aligned}
{\cal L}_\phi=-V_0+X\mp \frac{X^n}{M^{4n-1}}\Box\phi\quad(n\ge1),\end{aligned}$$ and in the early stage of inflation the higher-power term is much larger than $X$ (but smaller than $V_0$). During this early stage, the Universe undergoes ultra slow-roll G-inflation (or ultra slow-roll k-inflation) with $\eta\neq -6$ and $\dot\phi$ decreases as $\dot\phi\propto a^{3+\eta}$. The higher-power term immediately becomes smaller than the standard kinetic energy, $X$, and then the Universe switches to the conventional ultra slow-roll phase with $\eta=-6$. It is thus easy to incorporate the automatic transition between the two phases of inflation in this setup. In this paper, we study such two-phase models of ultra slow-roll inflation. We dub the early phase as USRK1$_\eta$ if it is governed by $K\sim X^n$ and as USRG1$_\eta$ if governed by $G\sim X^n$, while we call the second ultra slow-roll phase simply USR2. A numerical example of the background evolution for $$\begin{aligned}
\label{g-type}
{\cal L}_\phi=-V_0+X-\frac{X}{M^3}\Box\phi,\end{aligned}$$ [*i.e.,*]{} the transition from the USRG1$_{-9/2}$ phase to the USR2 phase, is shown in Fig. \[fig:plt1.eps\].
Cosmological perturbations
--------------------------
We assume that the duration of the second phase is just enough, so that the first phase can be probed by the cosmological perturbations on the largest scales in the CMB observations. It has been known that in the previous models of ultra slow-roll inflation the curvature perturbation grows, contrary to folklore, on superhorizon scales. As we will see below, this also happens in the case of ultra slow-roll G-inflation, though in the two-phase model we have to be more careful about the evolution of the curvature perturbation across the transition between the two phases.
The quadratic action for the curvature perturbation $\zeta$ on uniform $\phi$ hypersurfaces is given by [@Deffayet:2010qz; @Kobayashi:2010cm; @Kobayashi:2011nu] $$\begin{aligned}
S^{(2)}_\zeta=\int {{\rm d}}t{{\rm d}}^3x a^3\left[{\cal G}_S\dot\zeta^2-\frac{{\cal F}_S}{a^2}(\partial\zeta)^2\right],
\label{qaczeta}\end{aligned}$$ where $$\begin{aligned}
{\cal F}_S&=\frac{{M_{\rm Pl}}^4}{a}\frac{{{\rm d}}}{{{\rm d}}t}\left(\frac{a}{\Theta}\right)-{M_{\rm Pl}}^2,
\\
{\cal G}_S&=\frac{{M_{\rm Pl}}^4\Sigma }{\Theta^2}+3{M_{\rm Pl}}^2,\end{aligned}$$ with $$\begin{aligned}
\Sigma &:=XK_X+2X^2K_{XX}+12H\dot\phi XG_X
\notag \\ &\quad
+6H\dot\phi X^2G_{XX}-3{M_{\rm Pl}}^2H^2,
\\
\Theta&:={M_{\rm Pl}}^2H-\dot\phi XG_X.\end{aligned}$$ By using these expressions, the sound speed of the curvature perturbation can be written as $c_s=\sqrt{{\cal F}_S/{\cal G}_S}$.
Let us first consider the USRK1$_\eta$ phase. In this phase, we have $$\begin{aligned}
{\cal F}_S={M_{\rm Pl}}^2\epsilon,
\quad
{\cal G}_S=\frac{{M_{\rm Pl}}^2\epsilon}{-1-\eta/3},\label{usrk1-fg}\end{aligned}$$ so that $c_s^2=-1-\eta/3$. Since $\eta<-3$, the ghost and gradient instabilities can be avoided if $\epsilon >0$. In the USRG1$_\eta$ phase, we find $$\begin{aligned}
{\cal F}_S=\frac{{M_{\rm Pl}}^2\epsilon}{-\eta},
\quad
{\cal G}_S=\frac{9{M_{\rm Pl}}^2\epsilon}{\eta(3+\eta)},\label{usrg1-fg}\end{aligned}$$ and therefore $c_s^2=(-3-\eta)/9$. The ghost and gradient instabilities can thus be avoided provided that $\epsilon > 0$. Finally, the result in the USR2 phase can be obtained by substituting $\eta=-6$ to Eq. (\[usrk1-fg\]), $$\begin{aligned}
{\cal F}_S={\cal G}_S={M_{\rm Pl}}^2\epsilon.\label{usr2-fg}\end{aligned}$$
When the galileon-like $G\Box\phi$ term comes into play, the stability conditions at the level of linear perturbations are in general uncorrelated with the sign of $\epsilon$ or, equivalently, the sign of $\dot H$ [@Deffayet:2010qz; @Kobayashi:2010cm; @Kobayashi:2011nu]. Therefore, the null energy condition can in principle be violated stably. However, in the potential-driven inflation models it has been shown, [*under the slow-roll approximation*]{}, that the stability conditions require $\epsilon>0$ [@Kamada:2010qe]. Here, we have generalized the previous statement and shown that the stability of the potential-driven models amounts to $\epsilon>0$ even in the case of [*ultra slow-roll*]{}, where the different approximation is made. Note that nonlinear stability is yet unclear [@Sawicki:2012pz].
In all three cases described by Eqs. (\[usrk1-fg\])–(\[usr2-fg\]), the functions ${\cal F}_S$ and ${\cal G}_S$ are of the form $$\begin{aligned}
{\cal F}_S={M_{\rm Pl}}^2 f(\eta)\epsilon,
\quad
{\cal G}_S={M_{\rm Pl}}^2 g(\eta)\epsilon.\end{aligned}$$ Let us look at the solutions to the equation of motion for $\zeta$ in each $\eta=\;$const phase of the ultra slow-roll background, $\epsilon \propto a^{\eta}$. The equation of motion in the Fourier space reduces to $$\begin{aligned}
\zeta_k''-\frac{2+\eta}{\tau}\zeta_k'+c_s^2k^2\zeta_k=0,\label{perteq}\end{aligned}$$ where a dash stands for differentiation with respect to the conformal time $\tau=-1/aH$ and $c_s^2=f/g=\;$const. The linearly independent solutions are given by $\psi_k$ and its complex conjugate, $\psi_k^\ast$, where $$\begin{aligned}
\psi_k&=\frac{1}{2}\sqrt{\frac{\pi}{2}}\frac{H}{{M_{\rm Pl}}k^{3/2}}
\frac{g^{1/4}}{f^{3/4}\epsilon^{1/2}}
\notag \\&\quad\times
(-c_sk\tau)^{3/2}
H_{(3+\eta)/2}^{(1)}(-c_sk\tau),\end{aligned}$$ and $H_\nu^{(1)}$ is the Hankel function of the first kind. Note the normalization of $\psi_k$, $$\begin{aligned}
\psi_k\approx
\frac{1}{a\left(2{\cal G}_S\right)^{1/2}}\cdot \frac{e^{-ic_sk\tau}}{\sqrt{2c_sk}}
\quad (|c_sk\tau|\to\infty),\label{advac}\end{aligned}$$ up to a phase factor. In particular, in the USR2 phase the linearly independent solutions are given by $y_k$ and $y_k^\ast$, where $$\begin{aligned}
y_k=\frac{1}{2}\sqrt{\frac{\pi}{2}}\frac{H}{{M_{\rm Pl}}k^{3/2}\epsilon^{1/2}}(-k\tau)^{3/2}H_{-3/2}^{(1)}(-k\tau).\end{aligned}$$
On superhorizon scales ($|c_sk\tau|\ll 1$), we find, for $3+\eta<0$, $$\begin{aligned}
\psi_k\simeq
\frac{A}{2}\frac{H}{{M_{\rm Pl}}\epsilon^{1/2}}\frac{g^{1/4}}{f^{3/4}}
|c_s\tau|^{3+\eta/2}k^{(3+\eta)/2},\label{assympsi}\end{aligned}$$ where $$\begin{aligned}
A:=2^{-\nu}\sqrt{\frac{\pi}{2}}\left[
\frac{1}{\Gamma(1+\nu)}+\frac{\cos(\pi\nu)\Gamma(-\nu)}{\pi i}\right],\end{aligned}$$ with $$\begin{aligned}
\nu:=\frac{3+\eta}{2}.\end{aligned}$$ From Eq. (\[assympsi\]) we see that $$\begin{aligned}
\psi_k\propto \frac{|\tau|^{3+\eta/2}}{\epsilon^{1/2}}\propto |\tau|^{3+\eta},\end{aligned}$$ and hence the curvature perturbation grows on superhorizon scales.
In the two-phase model of ultra slow-roll inflation, Eq. (\[advac\]) shows that the appropriate initial condition deep in the USRK1$_\eta$/USRG1$_\eta$ phase is given by $$\begin{aligned}
\zeta_k\to \psi_k.\end{aligned}$$ One can then solve Eq. (\[perteq\]) numerically to obtain the power spectrum of $\zeta_k$ at the end of inflation. Here we provide a complementary argument for evaluating analytically the late-time amplitude of $\zeta_k$. To do so, let us assume that the transition between the two phases occurs instantaneously at some $\tau=\tau_\ast$. The solution in the USR2 phase ($\tau>\tau_\ast$) can be written as $$\begin{aligned}
\zeta_k=\alpha_ky_k+\beta_ky_k^\ast,\end{aligned}$$ where the coefficients $\alpha_k$ and $\beta_k$ are fixed by matching the curvature perturbation at $\tau=\tau_\ast$. Noting that ${\cal G}_S={M_{\rm Pl}}^2g(\eta)\epsilon$ and $\epsilon$ is continuous while $g(\eta)$ undergoes a sudden change $g\to 1$ across the transition, the matching conditions are summarized as [@Deruelle:1995kd; @Nishi:2014bsa] $$\begin{aligned}
\zeta_k|_{\tau_\ast-0}=\zeta_k|_{\tau_\ast+0},
\quad
g(\eta)\zeta_k'|_{\tau_\ast-0}
=\zeta_k'|_{\tau_\ast+0},\end{aligned}$$ where $g=1/(-1-\eta/3)$ for USRK1$_\eta$ and $g=9/\eta(3+\eta)$ for USRG1$_\eta$, as we have mentioned. We thus obtain $$\begin{aligned}
\alpha_k&=\frac{\pi i}{4}\frac{|k\tau_\ast|}{g^{1/2}}\bigl[
H_\nu^{(1)}(-c_sk\tau_\ast)H_{-5/2}^{(2)}(-k\tau_\ast)
\notag \\ &\qquad\qquad\quad
-c_sgH_{\nu-1}^{(1)}(-c_sk\tau_\ast)H_{-3/2}^{(2)}(-k\tau_\ast)
\bigr],\label{calpha}
\\
\beta_k&=-\frac{\pi i}{4}\frac{|k\tau_\ast|}{g^{1/2}}\bigl[
H_\nu^{(1)}(-c_sk\tau_\ast)H_{-5/2}^{(1)}(-k\tau_\ast)
\notag \\ &\qquad\qquad\quad
-c_sgH_{\nu-1}^{(1)}(-c_sk\tau_\ast)H_{-3/2}^{(1)}(-k\tau_\ast)
\bigr],\label{cbeta}\end{aligned}$$ and using those coefficients the power spectrum is obtained as $$\begin{aligned}
{\cal P}_\zeta(k)=\frac{|\alpha_k+\beta_k|^2}{8\pi^2}\frac{H^2}{{M_{\rm Pl}}^2\epsilon}.\label{psan}\end{aligned}$$
In the $|k\tau_\ast|\ll 1$ limit, we find $$\begin{aligned}
|\alpha_k+\beta_k|^2\simeq
\frac{2^{-2-\eta}\Gamma^2(1-\nu)}{9\pi}c_s^{3+\eta}g|k\tau_\ast|^{6+\eta}.\label{amp1}\end{aligned}$$ This corresponds to the modes that exit the horizon during the USRK1$_\eta$/USRG1$_\eta$ phase. Equation (\[amp1\]) shows that for $|k\tau_\ast|\ll 1$ $$\begin{aligned}
n_s-1=6+\eta,\end{aligned}$$ and thus the power spectrum is blue on large scales.
In the opposite limit, $|k\tau_\ast|\gg 1$, we have $$\begin{aligned}
\label{amp2}
|\alpha_k+\beta_k|^2\approx
c_s g+\frac{1-c_s^2 g^2}{c_s g}\sin^2(k\tau_\ast).\end{aligned}$$ It can be seen that the power spectrum oscillates even at large $k$ [@Nakashima:2010sa]. However, the oscillation is due to the artifact of the sudden change approximation enforcing infinitely large $\dot \eta/H\eta$ and $\dot c_s/Hc_s$ at the transition, which leads to the nonadiabatic evolution of all the modes.[^1] From the numerical example in Fig. \[fig:plt1.eps\] we see that the actual transition occurs smoothly on a time scale of $N={\cal O}( 1)$. In this case, the modes with sufficiently large $k$ remain in their adiabatic vacuum, yielding $$\begin{aligned}
|\alpha_k+\beta_k|^2\approx 1.\end{aligned}$$
Full numerical solutions confirm the above argument. As examples, we consider two simple models where the Lagrangians are respectively given by Eq. (\[g-type\]), that is, USRK1$_{-9/2}$, and Eq. (\[k-type\]) with $n = 2$, that is, USRG1$_{-4}$. We plot the power spectrum of the primordial curvature perturbation computed numerically for each model (Fig. \[fig:USRG\_power.eps\] for the two-phase model composed of USRG1$_{-9/2}$ and USR2, and Fig. \[fig:USRK\_power.eps\] for that composed of USRK1$_{-4}$ and USR2). In both figures, the red lines represent the numerical results, while the blue dashed lines correspond to the analytic results for $|k \tau_*| \ll 1$ under the sudden transition approximation. The black dotted lines are given by ${\mathcal P}_\zeta \simeq H^2 / (8 \pi^2 M_{\rm Pl}^2 \epsilon)$, [*i.e.*]{}, $|\alpha_k+\beta_k|\approx 1$. The amplitude and the scales are taken so as to be consistent with the CMB observation, by choosing the inflationary Hubble scale and the scale $M$ appropriately. From these figures, we find that suppression of the power on large scales can be realized due to the fact that the USRG1$_{-9/2}$/USRK1$_{-4}$ phase exists prior to the USR2 phase which produces the scale-invariant curvature perturbation. Furthermore, we can see that our analytic formula (\[psan\]) with the coefficients (\[calpha\]) and (\[cbeta\]) reproduces the numerical result except that the oscillating feature for large $k$ appears in Eq. (\[amp2\]). This is because the actual transition occurs smoothly on a time scale of $N={\cal O}( 1)$, as we have discussed. Note that since $\dot c_s/Hc_s, \dot\eta/H\eta\sim 1$ at the transition, a few oscillations are still found around the break in the spectrum.
In the present setup, the equation of motion for the tensor perturbations remains the same as the standard one. Therefore, the power spectrum of the primordial tensor modes is given by $$\begin{aligned}
{\cal P}_h=\frac{8}{{M_{\rm Pl}}^2}\left.\left(\frac{H}{2\pi}\right)^2\right|_{k=aH},
\quad
n_t=-2\epsilon.\label{tensorsp}\end{aligned}$$ Since $\epsilon>0$ is required for stability, the tensor power spectrum is always red tilted.
Before closing this subsection, it is appropriate to give a short comment on potential drawbacks that have not been discussed so far. First, the present model does not have a mechanism to end the period of inflation. Second, the curvature perturbation generated in the second phase has an exactly scale-invariant spectrum, while observations imply that $n_s\simeq 0.96$. Both of the drawbacks stem from the exactly flat potential. Introducing a mild slope of the potential [@Chen:2013aj; @Chen:2013eea], this issue is expected to be evaded.
![The power spectrum of the primordial curvature perturbations for the model given by Eq. (\[g-type\]). The amplitude and the scales are taken so as to be consistent with the CMB observation. []{data-label="fig:USRG_power.eps"}](USRG_power.eps){height="50mm"}
![The power spectrum of the primordial curvature perturbations for the model given by Eq. (\[k-type\]) with $n = 2$. The amplitude and the scales are taken so as to be consistent with the CMB observation. []{data-label="fig:USRK_power.eps"}](USRK_power.eps){height="50mm"}
Suppression of CMB power on the largest scales
----------------------------------------------
![CMB angular power spectrum calculated from the primordial power spectrum shown in Fig. \[fig:USRK\_power.eps\] (red solid line) with Planck low-$\ell$ data [@Planck2] (purple dots with error bars) and the power spectrum for the best fit power-law $\Lambda$CDM model (blue dotted line). []{data-label="fig:cmb_temp.eps"}](cmb_temp.eps){height="60mm"}
Given that the power spectrum of the primordial curvature perturbation has the broken power-law form, let us briefly discuss the CMB temperature power spectrum in ultra slow-roll G-inflation.
Our model relies on the assumption that there are two phases during inflation and the duration of the second phase is just enough. A number of attempts in this direction have been done so far to explain the lack of large-scale power in the CMB, e.g., by changing the inflaton potential or considering the pre-inflationary era [@Contaldi:2003zv; @Kawasaki:2003dd; @Wang:2007ws; @Scardigli:2010gm; @BouhmadiLopez:2012by; @Kouwn:2014aia; @Jain:2008dw; @Dudas:2012vv; @Namjoo:2012xs; @White:2014aua; @Biswas:2013dry; @Chen:2015gla]. Comparing with these possibilities, it should be emphasized that in our model the first phase too is quasi-de Sitter. Nevertheless, the primordial spectrum is tilted on the largest scales.
As an example, Fig. \[fig:cmb\_temp.eps\] shows the CMB angular power spectrum calculated from the primordial power spectrum shown in Fig. \[fig:USRK\_power.eps\] (red solid line). In this figure, we also plot the Planck low-$\ell$ data [@Planck2] (purple dots with error bars) and the power spectrum for the best fit power-law $\Lambda$CDM model (blue dotted line). The angular power spectrum is calculated by using the CLASS code [@Blas:2011rf]. From this figure, we find that the suppression of CMB power on the largest scales can indeed be realized in our model. Using $H_{\rm inf} = 2.7 \times 10^{-39} {M_{\rm Pl}}$ and $M = 6.8 \times 10^{-23} {M_{\rm Pl}}$, we find that this model roughly improves the effective $\chi^2$ by $\Delta \chi^2_{\mathrm{eff}} \approx -3.4$.
Approaching de Sitter Universe dominated by kinetic energy {#sec:bluetensor}
==========================================================
So far we have studied ultra slow-roll inflation driven by the constant potential energy of a scalar field. Its kinetic energy is much smaller than the potential energy and dilutes as the Universe expands with a nearly constant Hubble rate. Here one would notice that quasi-de Sitter inflation can be driven as well by nearly constant kinetic energy of a noncanonical scalar field. In this section, we study inflationary expansion that is similar to ultra slow-roll inflation in the sense that the Universe approaches the de Sitter attractor, but in the present case the Universe is dominated by the kinetic energy rather than the potential.
The scalar-field Lagrangian we consider is $$\begin{aligned}
{\cal L}_\phi=K(X)-G(X)\Box\phi,\end{aligned}$$ where the potential term is removed from Eq. (\[Lphi1\]). The background equations are thus given by $$\begin{aligned}
3{M_{\rm Pl}}^2H^2&=-K+\dot\phi J,
\\
{M_{\rm Pl}}^2\dot H&=-\frac{1}{2}\dot\phi J+X G_X\ddot\phi,
\\
\dot J+3H J&=0,\end{aligned}$$ where we write $$\begin{aligned}
J:=\dot\phi F=\dot\phi K_X+6XG_XH.\end{aligned}$$
We introduce a small parameter $\xi$ and assume the following ansatz: $$\begin{aligned}
H=H_0+\delta H(t),
\quad
\dot\phi=\dot\phi_0+\dot{\delta\phi}(t),\end{aligned}$$ where $H_0$ and $\dot\phi_0$ are constants and $\{\delta H,\dot{\delta\phi}\}={\cal O}(\xi)\times\{H_0,\dot\phi_0\}$. At leading order, $$\begin{aligned}
3{M_{\rm Pl}}^2H^2=-K,
\quad
J=0.\end{aligned}$$ Those two equations algebraically determine $H_0$ and $\dot\phi_0$, yielding de Sitter inflation, $a\propto e^{H_0 t}$. Then, at ${\cal O}(\xi)$ we have $$\begin{aligned}
6{M_{\rm Pl}}^2H_0\delta H&=-\dot\phi_0K_X\dot{\delta\phi}+\dot\phi_0\delta J,\label{bgeq21}
\\
{M_{\rm Pl}}^2\dot{\delta H}&=-\frac{1}{2}\dot\phi_0\delta J+X_0G_X\ddot{\delta\phi},\label{bgeq22}
\\
\dot{\delta J}+3H_0 \delta J&=0,\label{bgeq23}\end{aligned}$$ where $X_0:=\dot\phi_0^2/2$, $$\begin{aligned}
\delta J &=\left[K_{X}+2XK_{XX}+6H\dot\phi\left(G_{X}+XG_{XX}\right)\right]\dot{\delta \phi}
\notag \\ &\quad
+\left[6XG_{X}\right]\delta H,\label{def-deltaJ}\end{aligned}$$ and the quantities in the square brackets in Eq. (\[def-deltaJ\]) are evaluated at $H=H_0$ and $\dot\phi=\dot\phi_0$. Note that Eq. (\[bgeq22\]) \[Eq. (\[bgeq23\])\] can be derived from Eq. (\[bgeq23\]) \[Eq. (\[bgeq22\])\] and Eq. (\[bgeq21\]).
We have consistent solutions to Eqs. (\[bgeq21\])–(\[bgeq23\]) satisfying $$\begin{aligned}
&\delta H\propto a^{-3},
&&\dot{\delta\phi}\propto a^{-3},
\\
&\dot{\delta H}=-3H_0\delta H \propto a^{-3},
&&\ddot{\delta\phi}=-3H_0\dot{\delta\phi}\propto a^{-3},\end{aligned}$$ characterized by one integration constant. The slow-roll parameter is given by $$\begin{aligned}
\epsilon=-\frac{\dot H}{H^2}\simeq -\frac{\dot{\delta H}}{H_0^2}
=\frac{3\delta H}{H_0}\propto a^{-3},\end{aligned}$$ and hence $\epsilon$ is of ${\cal O}(\xi)$. However, the second slow-roll parameter is not small: $\eta=-3$. Introducing a constant ${\cal C}$ \[$={\cal O}(\xi)$\], we write $\epsilon={\cal C}/a^3$, and then $\delta H$ and $\dot{\delta \phi}$ can be written in terms of ${\cal C}$ as $$\begin{aligned}
\delta H=\frac{H_0}{3}\frac{{\cal C}}{a^3},\end{aligned}$$ and $$\begin{aligned}
\dot{\delta\phi}=
\frac{H_0\Theta_0}{X_0\Gamma_0}\frac{{\cal C}}{a^3}.\end{aligned}$$ where $$\begin{aligned}
\Theta_0&:={M_{\rm Pl}}^2H_0-\dot\phi_0X_0G_X,
\\
\Gamma_0&:=\dot\phi_0K_{XX}+6H_0(G_X+X_0G_{XX}).\end{aligned}$$ Note that, at this stage, there is no privileged choice of the sign of ${\cal C}$ , and hence $\epsilon$ can be negative, because ${\cal C}$ is just an integration constant.
Let us move to discuss the behavior of cosmological perturbations around this quasi-de Sitter background. Also in this case the tensor perturbations obey the standard formula, so that the primordial power spectrum is given by Eq. (\[tensorsp\]). Therefore, the tensor amplitude is determined at leading order as ${\cal P}_h\simeq 2H_0^2/\pi^2{M_{\rm Pl}}^2$, and its tilt is given in terms of the ${\cal O}(\xi)$ quantity as $n_t=-2\epsilon$.
The quadratic action for the curvature perturbation is of the form (\[qaczeta\]) with $$\begin{aligned}
{\cal F}_S&={\cal F}_{S0}+f_S\frac{{\cal C}}{a^3},
\\
{\cal G}_S&={\cal G}_{S0}+g_S\frac{{\cal C}}{a^3}.\end{aligned}$$ Here, ${\cal F}_{S0}$ and ${\cal G}_{S0}$ are leading-order quantities and are given explicitly by $$\begin{aligned}
{\cal F}_{S0}&=\frac{{M_{\rm Pl}}^2\dot\phi_0 X_0G_X}{\Theta_0},
\\
{\cal G}_{S0}&=\frac{{M_{\rm Pl}}^4X_0}{\Theta^2}\left(K_X+\dot\phi_0\Gamma_0+\frac{6X_0^2G_X^2}{{M_{\rm Pl}}^2}\right).\end{aligned}$$ The coefficients of the ${\cal O}(\xi)$ terms are also constant and $$\begin{aligned}
f_S&=\frac{{M_{\rm Pl}}^4H_0\dot\phi_0}{3\Theta^2_0}\left[
2X_0G_X+\frac{3\Theta_0(K_{XX}+H_0\dot\phi_0 G_{XX})}{\Gamma_0}
\right],\end{aligned}$$ while the form of $g_S$ is messy and involves $G_{XXX}$ as well as $K_{XXX}$. The stability conditions are given at leading order by $$\begin{aligned}
{\cal F}_{S0}>0,\quad{\cal G}_{S0}>0.\end{aligned}$$ It should be emphasized here that the sign of ${\cal C}$ is not important for stability, as its contribution is subleading. Therefore, the tensor power spectrum can be blue on a healthy background, depending on the initial conditions.
The primordial power spectrum reduces to $$\begin{aligned}
{\cal P}_\zeta=\left.\frac{{\cal G}_S^{1/2}}{{\cal F}_S^{3/2}}\frac{H^2}{4\pi^2}\right|_{k=aH/c_s},\end{aligned}$$ and therefore the amplitude is determined by the leading order terms. The spectral index is $$\begin{aligned}
n_s-1=\left(-2
-\frac{3g_S}{2{\cal G}_{S0}}+\frac{9f_S}{2{\cal F}_{S0}}
\right)\epsilon.\end{aligned}$$ It can be seen that we have a nearly scale-invariant spectrum even though the second slow-roll parameter is as large as $-3$. While the stability conditions are given by the first and second derivatives of $K(X)$ and $G(X)$, $n_s$ depends on the third derivatives of those functions through $g_S$. Therefore, using the functional degrees of freedom one can realize $n_s\simeq 0.96$ on a stable background.
Summary
=======
Under the motivation of constructing a phenomenological model to explain the CMB anomalies, we have studied ultra slow-roll models of G-inflation with a constant potential. We have considered an earlier phase of inflation where a higher-power term in $X=-(\partial_\mu\phi\partial^\mu\phi)/2$ governs the scalar-field dynamics prior to the usual ultra slow-roll phase. The primordial power spectrum of the curvature perturbation in such a two-phase model has been evaluated analytically and numerically to show that the growing curvature perturbation has a broken power-law spectrum. The transition between the two phases occurs on a time scale of $N={\cal O}(1)$, and a few oscillations appear around the break of the spectrum due to the change of the sound speed $c_s$. As a consequence of such a feature, our model could explain the suppression of CMB power on the largest scales which has been reported in the CMB observations [@Planck2].
In the presence of the Galileon-like interaction $G(\phi, X)\Box\phi$, the null energy condition can in principle be violated stably, which opens up the interesting possibility of a blue primordial tensor spectrum. However, in Ref. [@Kamada:2010qe] it was shown under the slow-roll approximation that this is not possible in general potential-driven inflation models. Motivated by this fact, in this paper we have extended the previous argument [@Kamada:2010qe] and showed that also in ultra slow-roll models the stability conditions require the null energy condition. To realize the stable violation of the null energy condition, we have to give up potential-driven inflation and consider kinetically driven models. We have studied the inflationary Universe approaching the kinetically driven de Sitter attractor in the way similar to potential-driven ultra slow-roll, and found that in this case the null energy condition can be violated stably.
This work was supported in part by YITP-W-15-16 in workshop JGRG25 and the JSPS, Grant-in-Aid for Scientific Research No. 24740161 (T.K.), No. 15K17659(S.Y.), and No. 15H05888 (T.K. and S.Y.). We thank Kazuhiro Yamamoto and Kiyotomo Ichiki for useful discussions.
The sudden change approximation and the oscillating power spectrum {#append}
==================================================================
![The power spectrum of $\zeta$ for a sudden transition with $\lambda=100$ (solid curve). For comparison the analytic estimate is also shown as a dotted curve. For sufficiently large $k$ the spectrum is given by ${\cal P}_\zeta\simeq H^2/(8\pi^2{M_{\rm Pl}}^2\epsilon)$ (dashed line), though such behavior is not shown explicitly in the figure. []{data-label="fig:TML100.eps"}](TML100.eps){height="45mm"}
![The power spectrum of $\zeta$ for a mild transition with $\lambda=10$ (solid curve). For comparison the analytic estimate is also shown as a dotted curve. For sufficiently large $k$ the spectrum is given by ${\cal P}_\zeta\simeq H^2/(8\pi^2{M_{\rm Pl}}^2\epsilon)$ (dashed line). []{data-label="fig:TML10.eps"}](TML10.eps){height="45mm"}
To see how the sudden change approximation affects the power spectrum of $\zeta$ at large $k$, let us study the toy model of two-phase inflation in which the “sharpness” of the transition is controllable. The model is characterized by the slow-roll parameter $$\begin{aligned}
\epsilon=\epsilon_\ast
\left\{\cosh[\lambda(N-N_\ast)]\right\}^{(\eta_1-\eta_0)/2\lambda}e^{(\eta_1+\eta_0)(N-N_\ast)/2},\end{aligned}$$ which yields $$\begin{aligned}
\eta = \frac{1}{2}\left\{
(\eta_1-\eta_0)\tanh[\lambda(N-N_0)]+\eta_1+\eta_0
\right\},\end{aligned}$$ with $\epsilon_\ast \ll 1$. This describes a change from $\eta_0$ to $\eta_1$ at around $N=N_\ast$ on a quasi-de Sitter background, and thus mimics the background evolution of the two-phase model of ultra slow-roll inflation presented in Fig. \[fig:plt1.eps\]. However, in this toy model the “sharpness” of the transition can be controlled by the dimensionless parameter $\lambda$.
The above background can be realized by a suitable choice of the ultra slow-roll k-inflation Lagrangian. The evolution of the curvature perturbation follows from the quadratic action with ${\cal F}_S={M_{\rm Pl}}^2\epsilon$ and ${\cal G}_S={M_{\rm Pl}}^2/(-1-\eta/3)$. Numerical solutions for the power spectrum of $\zeta$ evaluated at some later time are shown in Figs. \[fig:TML100.eps\] and \[fig:TML10.eps\] in the case of the transition from $\eta_0=-4$ to $\eta_1=-6$. The solid curve in Fig. \[fig:TML100.eps\] corresponds to a sudden transition with $\lambda = 100$, while that in Fig. \[fig:TML10.eps\] corresponds to a milder transition with $\lambda = 10$, showing that in the case of the sudden transition the oscillation in the power spectrum persists down to small scales. For a mild transition the analytic formula obtained in the main text cannot be used on small scales, but instead one may have ${\cal P}_\zeta\simeq H^2/(8\pi^2{M_{\rm Pl}}^2\epsilon)$.
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[^1]: We further discuss the relation between the sudden change approximation and the oscillating spectrum at large $k$ in Appendix \[append\].
|
---
author:
- |
Csaba Balázs\
HEP Division, Argonne National Laboratory,\
9700 Cass Ave., Argonne, IL 60439, USA\
E-mail:
- |
István Szapudi\
Institute for Astronomy, University of Hawaii,\
2680 Woodlawn Dr., Honolulu, HI 96822, USA\
E-mail:
bibliography:
- 'ms.bib'
title: |
Holographic Quantum Statistics\
from Dual Thermodynamics
---
Introduction
============
The quantum estimate of the cosmological vacuum energy density is about 123 orders of magnitude larger than its measured value [@Weinberg:1988cp; @Riess:1998cb]. This discrepancy might be resolved by an upper limit on the entropy of the universe, since such a bound implies an upper limit on its total energy density. Along the lines of this argument, recently we conjectured that after imposing a holographic entropy limit, the resulting quantum theory leads exactly to the measured value of the total energy density [@Balazs:2006kc]. In this work we construct an example, where imposing the holographic entropy limit on a quantum system indeed leads to an upper limit on its energy density. We choose a black hole as a representative holographic quantum system, since it is easy to relate the entropy and energy using black hole mechanics.
Black hole mechanics is one of the most tantalizing features of general relativity. Its formal analogy with thermodynamics was noticed early on [@Bekenstein:1973ur; @Bekenstein:1974ax; @Bardeen:1973gs]. The quantum nature of the underlying physics was clarified by the discovery of the Hawking radiation [@Hawking:1974rv; @Hawking:1974sw]. While the quantum theory of black holes struggles with unsolved problems such as the information paradox [@Susskind:1993if], it successfully motivated the holographic conjecture [@Bekenstein:1972tm; @'tHooft:1993gx; @Susskind:1994vu]: the entropy in a spherical volume cannot exceed the quarter of its surface area in Planck units (see [@Bousso:1999xy; @Bousso:2002ju] for alternative and generalized definitions). Despite the general belief that the explanation of the holographic conjecture must be of quantum nature, it is non-trivial to find simple quantum systems representing black hole mechanics. In this paper, we set out to construct a quantum statistical model displaying the holographic behavior of a Schwarzschild black hole.
Following [@Wald:1999vt], we summarize the laws of black hole mechanics for an isolated Schwarzschild black hole (i.e. one without electric charge or angular momentum) characterized with its mass $M$, surface gravity $\kappa = 1/(4M)$, and surface area $A$.
- [*Zeroth Law:*]{} $\kappa$ is the same everywhere on the horizon in a time independent black hole.
- [*First Law:*]{} $$\delta M = \frac{\kappa}{8\pi}\delta A
\label{Eq:BHD1}$$
- [*Second Law:*]{} $$\delta A \ge 0$$
Based on the analogy with thermodynamics one associates the temperature with the surface gravity $T \to \kappa/2\pi$, the energy with the mass $E
\to M$, and the entropy with the horizon area $S \to A/4$. Holography in this context amounts to the conjecture that a black hole maximizes the entropy [@'tHooft:1993gx]. The traditional assignment of variables is further motivated by the Hawking effect: a black hole emits black body radiation with temperature $\kappa/2\pi$. A black hole weakly interacting with its surrounding appears to obey the generalized version of the second law where the sum of the entropies of matter and the black hole always increases [@Bekenstein:1974ax].
On the other hand, black hole evaporation due to Hawking radiation raises questions: is the unitary nature of quantum mechanics broken? It appears that an initial pure quantum state will evolve into a mixed thermal state through the process of black hole formation and evaporation; this amounts to information loss. In addition, it is not clear precisely what quantum degrees of freedom are responsible for the entropy of the black hole.
The resolution of the information paradox is still open. Either quantum gravity allows correlations to be restored through the horizon, in contrast with classical gravity, thus information could be restored during evaporation; or quantum mechanics needs to be modified to allow non-unitary evolution [@Banks:1983by; @Wald:1995yp; @Hartle:1996rp]. The degrees of freedom responsible for the entropy have stimulated vigorous research. Competing models involve weakly coupled string states [@Strominger:1996sh; @Peet:1997es], conformal symmetry [@Aharony:1999ti; @Carlip:2000nv; @Carlip:2006fm], string fuzzballs [@Mathur:2005zp] spin network states at [@Ashtekar:1997yu] or inside [@Livine:2005mw] the horizon, “heavy” degrees of freedom in induced gravity [@Frolov:1997up], or non-local topological properties of black hole spacetime [@Hawking:1998jf]. In each of these models the entropy is proportional to the horizon area, yet none of them offer a clear and comprehensive picture associated with black hole thermodynamics.
Another peculiarity of black hole mechanics is rarely discussed: it violates the (strong version of the) third law of thermodynamics [@Wald:1997qp]. The entropy is in fact singular (tends to infinity rather than zero) as the temperature approaches zero. Since virtually every known quantum mechanical system in nature obeys the third law, we argue that this singularity expresses yet another way the difficulty of quantizing gravity.
In order to construct quantum models representing black hole mechanics, we propose dual thermodynamics with the following assignments $E' \to A/4, S'
\to M$, and $T' \to 2\pi/\kappa$ in Planck units. In the dual description, the entropy and energy exchange roles, and the inverse of the Hawking temperature plays the role of the temperature. By definition, the dual variables obey the first law while the validity of the second law follows from the fact that the mass of a Schwarzschild black hole can only increase if we neglect Hawking radiation. Note that at this point we do not aim to model the generalized second law which includes the sum of matter and black hole entropies. The advantage of our proposal is that the dual entropy of a black hole is proportional to its dual temperature $S' \propto T'$, i.e. the third law of dual thermodynamics is satisfied as well.
In the next section we formalize the duality transformation between two thermodynamical systems, and in Section \[Sec:DualStat\] we construct a simple quantum model which reproduces the dual thermodynamics. In the last section we summarize our results after transforming back from the dual variables and compare the quantum corrections to the entropy with those available in the literature.
Energy-entropy duality {#Sec:TheDua}
======================
We define the energy-entropy duality by assuming that the first law in entropy representation corresponds to the first law in dual energy representation. We allow for pressure and chemical potential for full generality, although we exclude for the moment other entities, such as electric charge, magnetic susceptibility, etc. The first law in both representations reads $$\begin{aligned}
dS &=& \frac{1}{T}dE+\frac{p}{T}dV-\frac{\mu}{T}dN, \\
dE' &=& T'dS' - p'dV' + \mu' dN',\end{aligned}$$ where $S, E, V, N, T, p, \mu$ are the thermodynamic entropy, energy, volume, species number, temperature, pressure, and chemical potential, respectively, and primed symbols denote dual thermodynamical variables. Comparison of the two equations motivates the definition of the entropy-energy duality $$\begin{aligned}
S \to E', ~~~~~
E \to S', ~~~~~
T \to \frac{1}{T'}, ~~~~~
\frac{\mu}{T} \to -\mu'.
\label{eq:eduality}\end{aligned}$$ We assumed that the degrees of freedom $N$ is the same in both spaces, which fixes the transformation of $\mu$. From these transformation rules it follows that $\frac{p dV}{T} \to -p' dV'$ from the first law.
While we presented a definition which might be more generally applicable, we aim specifically at application to black hole mechanics. As shown at the end of the previous section, for vanishing chemical potential, and $p dV = 0$, all three dual thermodynamic laws will be satisfied as a consequence of black hole mechanics. For the more general case with non vanishing chemical potential, we can show that it is sufficient to have $p' dV'=0$, $\mu > 0$ and $N \sim
M$ for the dual second law to hold. In this case $$dS' = \frac{1}{T'}(dE' + p'dV' - \mu'dN') \ge 0,$$ since $dE' = dS \ge 0$ and $- \mu'dN' \ge 0$ separately. We will show that the specific quantum statistics we propose to represent the dual thermodynamics satisfies these sufficient conditions. The above arguments show that it is reasonable to assume that the proposed entropy-energy duality is meaningful for the specific case of black hole mechanics, as all three laws of thermodynamics hold in the dual variables. The precise necessary conditions for the applicability of the entropy-energy duality in a more general case are less clear, and are beyond the scope of the present paper.
Our duality proposal was motivated by the third law of thermodynamics. The behaviour $S' \propto T'$ and $E' \propto T'^2$ in dual variables is now easily reproducible by ordinary quantum statistics. In fact, probably multiple quantum systems could satisfy these sensible relations. We show in the next section, that one dimensional quantum gases (whether Fermi or Bose) provide a fully consistent dual quantum statistical model for the dual black hole thermodynamics.
Dual quantum statistics: one dimensional quantum gas {#Sec:DualStat}
====================================================
In this section, we summarize the main results for the statistical mechanics of one dimensional quantum gases. As we show in the next section, they provide a representation of dual black hole thermodynamics in the limit where $|\mu'| \ll T'$, with $\mu'$ corresponding to quantum corrections. All equation in this section are written in dual variables, therefore we omit the primes for convenience.
The energy, pressure and particle number of one dimensional quantum gases are calculated as $$\begin{aligned}
E = pL &=&\frac{g L}{2\pi }\int_0^\infty f(\epsilon) \epsilon \; d\epsilon,
\label{Eq:EpL} \\
N &=& \frac{g L}{2\pi }\int_0^\infty f(\epsilon) \; d\epsilon,\end{aligned}$$ where $f(\epsilon) = 1/(e^{(\epsilon-\mu)/T}\pm 1)$. Hereafter the upper (lower) sign represents the Fermi-Dirac (Bose-Einstein) distribution. The coefficient $g$ gives the internal degrees of freedom of the gas particles, and $L$ is the size of the system. The entropy is obtained from the integral form of the first thermodynamic law $$S = \frac{1}{T}(E+p L - \mu N).$$
The above integrals can be expressed in terms of the polylogarithm function $\rm{Li}_n(z) \equiv \sum_{k=1}^{\infty} z^k/k^n$: $$\begin{aligned}
E = p L &=& \mp \frac{g L}{2 \pi} \; {\rm Li}_2\left(\mp e^{\frac{\mu}{T}}\right) T^2, \\
N &=& \pm \frac{g L}{2 \pi} \log \left(1 \pm e^{\frac{\mu }{T}}\right) T.\end{aligned}$$ We will use the expansions of these formulae in the limit of $|\mu| \ll T$. For a Fermi gas $$\begin{aligned}
E = p L &=& \frac{1}{24} g L \pi T^2
+\frac{g L}{2 \pi} \log (2) T \mu
+\frac{g L}{8 \pi } \mu ^2
+O\left(\mu ^3\right), \label{Eq:EpF} \\
N &=& \frac{g L}{2 \pi } \log (2) T
+\frac{g L}{4 \pi } \mu
+\frac{g L}{16 \pi T} \mu ^2
+O\left(\mu ^3\right), \\
S &=& \frac{1}{12} g L \pi {T}
+\frac{g L}{2 \pi } \log(2) {{\mu }}
+ O(\mu _{D}^{3}),
\label{Eq:SpF}\end{aligned}$$ and for a Bose gas $$\begin{aligned}
E = p L &=&\frac{1}{12} g L \pi T^2
- \frac{ g L}{2 \pi } \left(\log \left(-\mu/T\right)-1\right) T \mu
- \frac{g L}{8\pi } \mu ^2
+ O\left(\mu ^3\right),
\label{Eq:EpB} \\
N &=&-\frac{g L}{2 \pi } \log \left(-\mu/T\right) T
- \frac{g L}{4 \pi } \mu
- \frac{g L}{48} \frac{\mu ^2}{\pi T}
+ O\left(\mu^3\right), \\
S &=&\frac{1}{6} g L \pi T
- \frac{g L}{2 \pi } (\log (-\mu/T ) - 1) \mu
+ O\left(\mu ^3\right).
\label{Eq:SpB}\end{aligned}$$
Quantum statistical model of a black hole
=========================================
In this section we use the previous results to show that dual black hole thermodynamics is represented by one dimensional quantum gases. Note that we return to primed notation for dual variables. In addition, we derive quantum corrections to the black hole mechanics based on our toy model. When the dual temperature, which is proportional to the mass of the black hole, approaches the Planck scale, we expect quantum corrections. In the dual space quantum effects will manifest themselves when $T'$ approaches $\mu'$, therefore we can assume that $\mu'$ is at most of the order of the Planck mass or smaller. In that case, for a large black hole, $\mu N$ will be completely negligible.
We can set $g L$ of our dual quantum theory to ensure consistency of the energy with that of the black hole: $$g L = \frac{3}{2 \pi^2}, \, \, \frac{3}{4 \pi^2},
\label{Eq:gL}$$ for Fermi and Bose systems, respectively. This means that the one dimensional dual quantum system lives in a space of size order of Planck length.
Equation (\[Eq:gL\]) fixes the parametric freedom in our quantum models. Transforming back from the dual variables, the leading terms of Eqs. (\[Eq:SpF\]) and (\[Eq:SpB\]) reproduce the black hole energy (assuming $\mu$ is small). Meanwhile, the leading terms of Eqs. (\[Eq:EpF\]) and (\[Eq:EpB\]) reproduce the black hole entropy as demonstrated by the leading terms of Eqs. (\[Eq:SFexp\]) and (\[Eq:SBexp\]). This itself is a non-trivial result. After fixing $g L$ to obtain the correct coefficient of the black hole energy, the dual system does not have any freely adjustable parameters. Yet, not only the temperature (or mass) dependences of the black hole energy and entropy are reproduced, but also the coefficient of the entropy term is given correctly. Finally, from $dL = 0$ follows $p dV = 0$, which means that black hole mechanics with its holographic aspects is perfectly represented by our dual model.
For the Fermi model, we also obtain that the number of degrees of freedom $$N = \frac{6 \log(2)}{\pi^2} M,$$ is essentially given by the black hole mass in Planck mass units. This is an entirely sensible prediction: the number of degrees of freedom grow extensively with the black hole mass, while our model faithfully reproduces the holographic growth of entropy.
For the Bose model, $N$ has a logarithmic sensitivity for $\mu'$, but for a wide range of values $N \simeq M\log M$ will hold, including the value we suggest from consistency with previous quantum gravity calculations for the entropy correction. Note that we do not discuss the possible effects Bose condensation on $N$, as we assumed that $\mu' \ll T'$.
Next, we calculate quantum corrections to the black hole entropy considering corrections to the energy of the dual quantum gas. The entropy according to our Fermi model is $$\begin{aligned}
S &=& -\frac{48}{\pi}{\rm Li}_2(-e^{-\mu}) M^2 = \\
& & 4 \pi M^2
- \frac{48 }{\pi} \log(2) M^2 \mu
+ \frac{12 }{\pi} M^2 \mu^2
- \frac{ 2 }{\pi} M^2 \mu^3
+ O(\mu^4),
\label{Eq:SFexp}\end{aligned}$$ while the Bose model gives $$\begin{aligned}
S &=& \frac{24}{\pi} \, {\rm Li}_2(e^{-\mu }) \, M^2 = \\
& & 4 \pi M^2
+ \frac{24}{\pi } \mu M^2 (\log (\mu ) - 1)
- \frac{6 }{\pi } \mu^2 M^2
+ O(\mu^4).
\label{Eq:SBexp}\end{aligned}$$
Quantum corrections to the entropy of various black holes were calculated using the Cardy formula [@Cardy:1986ie; @Bloete:1986qm]. These corrections allow us to fix the exact value of $\mu$. Comparing our expression with Refs. [@Kaul:2000kf; @Carlip:2000nv; @Das:2000bx], we find that the Bose case easily lends itself for obtaining a logarithmic correction to the black hole entropy. Setting $$\begin{aligned}
\mu = \frac{\pi}{16 M^2},\end{aligned}$$ the black hole entropy derived from the dual Bose gas becomes $$\begin{aligned}
S = 4\,\pi \,M^2
- \frac{3}{2} \, \log (4\, \pi M^2)
- \frac{3}{2} \left( 1+ \log \left( \frac{4}{\pi^2}\right) \right)
- \frac{3\,\pi } {128\,M^2}
+ O(1/M^3).\end{aligned}$$ We find it remarkable that our simplest toy model, namely a dual one dimensional Bose gas, is able to reproduce not only the correct semi classical black hole energy and entropy but also their highly non-trivial and minute logarithmic quantum corrections. It is possible to obtain a value of $\mu$ for the Fermi gas as well which would be consistent with logarithmic correction to the entropy.
Conclusions
===========
We proposed a dual thermodynamics for an isolated Schwarzschild black hole based on our conjectured energy-entropy duality. Our new formulation has the major advantage of obeying all laws of thermodynamics. This means that ordinary quantum statistics can represent the otherwise mysterious holography encoded in a black hole. While it is conceivable that that there are many dual quantum systems which would be equally suitable for this purpose, we present toy models based on one dimensional Fermi or Bose gases. We show that these exactly reproduce the holographic thermodynamics of black holes. In particular, since energy and entropy play dual roles, the holographic entropy cut off in real space corresponds to the energy cut off in dual space from the Fermi-Dirac or Bose-Einstein distribution and vice versa, as it was conjectured in Ref. [@Balazs:2006kc].
The energy-entropy duality transforms a gravitating, strongly interacting, low temperature system into a weakly interacting, high temperature dual quantum system. This weakly interacting system obeys the laws of ordinary quantum mechanics. The duality transformation allows the interpretation of results obtained in dual space where calculations are straightforward. We have demonstrated this by presenting quantum corrections to the black hole mechanics in the framework of our toy model. We find it remarkable that our simple model not only reproduces black hole mechanics, but it is also consistent with previously obtained logarithmic corrections to the entropy.
The most conservative interpretation of our result $p dV = 0$ would be that the pressure vanishes, $p = 0$. This is consistent with the integral form $E
= 2 TS$, as can be easily checked from the explicit expressions of energy, entropy, and temperature. Our quantum holographic model reproduces all these quantities as well as their relationship. However, in extensive thermodynamics $E = TS - pV $ should hold if the pressure is non-zero. A literal interpretation of our dual model suggests a non-zero negative pressure $p = -TS/V$, yielding $E = TS-pV = 2 TS$, consistently with black hole mechanics. This pressure term, however, does not enter into the (differential) first law of black hole mechanics due to $dL=0$ in our underlying quantum model. Thus, taken at face value, this picture predicts that a black hole, which classically has zero temperature and pressure, has both a non-zero temperature and pressure associated with it. Although this pressure does not manifest itself in the differential black hole mechanics due to a constraint ($S$, and $V$ are not independent variables), it appears in the integral of the first law. Just as the temperature associated with black hole mechanics, this pressure is entirely of quantum nature and has an equation of state $w = -\frac{1}{2}$, the hallmark of dark energy.
The similarity of Friedmann universes to black holes raises the possibility of applying similar ideas toward the development of quantum cosmology. In particular, it is intriguing that negative pressure is entirely natural in this context. Further research into this area could shed light on the dark energy component of the universe, and thereby touch base with observations. In [@Balazs:2006kc], based on simple thermodynamical considerations, we found that the present value of the cosmological constant is natural in the holographic context. We conjecture that, possibly when “bulk viscosity” effects corresponding to holographic entropy production are taken into account, this negative quantum gravitational pressure might account for the apparent acceleration of the universe [@Balazs:2006xx].
The energy-entropy duality as defined in Eq. \[eq:eduality\] establishes a transformation among partition functions as well. For example, the transformation of the grand canonical partition function should be $$q \equiv \frac{pV}{T} \to -q' T'.$$ Exploring the partition functions should shed more light on the density of states for a black hole. Further investigation of this issue is left for future research.
It is clear that our considerations can be generalized for other types of black holes; e.g., angular momentum and charge add further terms into the dual thermodynamics. At the moment general relativity is taken into account only in a fairly implicit way, through the phenomenology of black hole mechanics. However, derivation of Einsteins’s equation exists directly from on holographic thermodynamics [@Jacobson:1995ab]. This raises the possibility of covariant formulation of these ideas, possibly based on the work in Ref. [@Bousso:1999xy]. We speculate that this could lead to an energy-entropy dual of Einstein’s equation.
It is also clear that a range of dual models could reproduce the same large $M$ asymptotic thermodynamics, and possibly different models could produce different quantum effects. The present range of models can be characterized by $\mu$ and the choice of Fermi or Bose statistics, however, it is likely that other weakly interacting models could be producing similar results. Some of them, such as Ising type spin network models, could have more direct connection with previous work in this area, although one could hardly miss the striking similarity of one dimensional quantum gases with string excitations. Indeed, string theory appears to have a thermodynamic duality $T \to 1/T$ according to Ref. [@Dienes:2003sq; @Dienes:2003dv] and our dual description is similar to the well known AdS/CFT duality [@Maldacena:1997re]. We will explore the range possible models and connections with existing formulations in the future.
Ultimately, it would be desirable to extend these ideas to the interaction of a black hole with its surrounding. It is might be possible to relate the entropy current of the dual system to the energy current of the surrounding of the original system, and vice versa. Such calculations could shed more light on the information paradox associated with the Hawking radiation, and are left for future research.
We thank Nick Kaiser, Arjun Menon and Robert Wald for stimulating discussions. IS was supported by NASA through AISR NAG5-11996, and ATP NASA NAG5-12101 as well as by NSF grants AST02-06243, AST-0434413 and ITR 1120201-128440. Research at the HEP Division of ANL is supported in part by the US DOE, Division of HEP, Contract W-31-109-ENG-38. CB also thanks the Aspen Center for Physics for its hospitality and financial support.
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---
address: 'Institut für Festkörper- und Werkstofforschung Dresden, P.O. Box 270016, D-01171 Dresden, Germany'
author:
- 'H. Rosner, S.-L. Drechsler, K. Koepernik, R. Hayn, and H. Eschrig'
title: ' The electronic structure of CuSiO$_3$ – a possible candidate for a new inorganic spin-Peierls compound ?'
---
Low-dimensional spin systems such as chains or ladders are of fundamental interest for contemporary solid state physics due to their peculiar electronic and magnetic properties. During the last years, many related materials have been found within the cuprate family, famous for the high temperature superconductivity. All cuprates contain CuO$_4$ plaquettes. In most cases it is energetically favorable to connect these plaquettes by the formation of chains or planes. According to the number ($n=1,2$) of oxygen atoms shared by adjacent plaquettes, these compounds can be classified as so-called edge-shared ($n=2$) or corner-shared ($n=1$) compounds.
Obviously, the type of sharing affects strongly the physical properties of the compounds under consideration. For example, corner sharing leads to strong antiferromagnetic coupling between neighboring plaquettes compared with the weak inter-chain interactions.[@rosner97] As a result, the straight CuO$_3$ chain in Sr$_2$CuO$_3$ is the best known realization of the one-dimensional spin-1/2 Heisenberg model,[@tsvelik95] with an in-chain exchange coupling of about 2200 K, but with a Néel temperature of only 5 K and with an extremely small ordered magnetic moment of about 0.06 $\mu
_{\text{B}} $,[@kojima97] both due to a small residual interchain exchange coupling. Spin-charge separation in the excitation spectra could be observed for Sr$_2$CuO$_3$ and for the double chain compound SrCuO$_2$.[@kim96]
Somewhat surprisingly, in contrast to the similarity between different corner-shared chain compounds, the magnetic properties in the edge-shared chain family exhibit a remarkable variance. Thus, the edge-shared CuO$_2$ plaquettes in Li$_2$CuO$_2$ order antiferromagnetically with a ferromagnetic arrangement along those chains and with a large ordered moment of 0.9 $\mu
_{\text{B}}$,[@sapina90] whereas the same chain in CuGeO$_3$ shows a spin-Peierls transition at low temperatures.[@hase93] Antiferromagnetically ordered chains were observed in Cu$_{1-x}$Zn$_x$GeO$_3$ for small concentrations of Zn impurities.[@hase93a] It is noteworthy that, even for the intensively studied CuGeO$_3$, a consensus with respect to the quantitative description of competing or complementary interactions such as the inter-chain coupling, frustration and spin-phonon coupling has not been reached so -.75
far,[@uhrig97; @bouzeras99] despite the achieved qualitative understanding of their influence on different magnetically ordered states.
Naturally, the magnetic properties depend very sensitively on the electronic interactions in these systems. Therefore, a comparative study of the electronic properties of closely related systems can shed light on the interactions responsible for the magnetically ordered states mentioned above. In this context, the recent discovery and first investigations of the long searched for compound CuSiO$_3$,[@otto99] which is isostructural to the prototypical inorganic spin-Peierls system CuGeO$_3$ is of great scientific interest. The crystal structure of CuSiO$_3$ is shown in Fig. \[struct\]. The most important feature for the magnetic properties are the planar edge-shared CuO$_2$ chains running along [**c**]{}-direction. These chains are very similar to those of CuGeO$_3$. The Cu-O(2) bond length in CuSiO$_3$ (CuGeO$_3$) is 1.941 Å (1.942 Å), the Cu-O(2)-Cu bonding angle is 94$^\circ$ (99$^\circ$).
Thus, the question arises, whether the very recently observed phase transition [@baenitz00] near 8 K does point to a new inorganic spin-Peierls system or to another ordered state realized at low temperature. To get theoretical insight into possible scenarios, we present here comparative band-structure calculations and tight-binding examinations for CuSiO$_3$ and CuGeO$_3$. In this context we note that for the latter compound several (non full-potential) bandstructure calculation have been reported (e.g. in Ref. ), but to our knowledge the inter-chain interaction has not been analyzed in detail.
The relevant electronic structure of these materials is very sensitive to details of hybridization and charge balance. In order to obtain a realistic and reliable hopping part of a tight binding Hamiltonian, band-structure calculations were performed using the full-potential nonorthogonal local-orbital minimum-basis scheme [@koepernik99] within the local density approximation (LDA). In the scalar relativistic calculations we used the exchange and correlation potential of Perdew and Zunger.[@perdew81] Cu($4s$, $4p$, 3$d$), O(2$s$, 2$p$, 3$d$), Ge(3$d$, 4$s$, 4$p$, 4$d$) and Si(2$p$, 3$s$, 3$p$, 3$d$) states, respectively, were chosen as minimum basis set. All lower lying states were treated as core states. The inclusion of Ge 3$d$ and Si 2$p$ states in the valence states was necessary to account for non-negligible core-core overlaps. The O and Si 3$d$ as well as the Ge 4$d$ states were taken into account to increase the completeness of the basis set. The spatial extension of the basis orbitals, controlled by a confining potential [@eschrig89] $(r/r_0)^4$, was optimized to minimize the total energy.
The results of the paramagnetic calculation[@remark3] for CuSiO$_3$ (see Fig. \[band\] (a)) and CuGeO$_3$ (see Fig. \[band\] (b); we find similar results as the non full-potential calculation of Ref. ) show a valence band complex of about 10 eV width with two bands crossing the Fermi level in both cases. These two bands are well separated from the rest of the valence band complex and show mainly Cu 3$d$ and O(2) 2$p$ character in the analysis of the corresponding partial densities of states (not shown). We note that the occupancy of the two O(2) 2$p$ orbitals along and perpendicular to the chain (lying in the plaquette-planes) is rather different, but it is almost identical for the corresponding orbitals in both compounds. Therein, we found only a small admixture of O(1) 2$p$ and Ge 4$s$ and 4$p$ states, respectively, with a total amount of few percent. The examination of the eigenstates of the latter bands at high symmetry points yields an antibonding character typical for cuprates. Here these relatively narrow antibonding bands are half-filled. Therefore, strong correlation effects can be expected which explain the experimentally observed insulating groundstate. Despite almost perfect qualitative one to one correspondence of all valence bands and main peak structures in the densities of states (DOS) (compare right panels in Fig. \[band\]), the most important differences between both compounds -.25
occur for the antibonding bands (shown in detail in Fig. \[band\](c)). Therefore, we restrict ourselves to the extended tight-binding analysis and the discussion of these antibonding bands.
The dispersion of these bands has been analyzed in terms of nearest neighbor transfer (NN), next nearest neighbor transfer (NNN) and higher neighbor terms in chain direction, but only NN hopping and a diagonal transition term between the CuO$_2$-chains have been considered (see Fig. \[skizze\]). Then, the corresponding dispersion relation takes the form $$\begin{aligned}
E(\vec{k})&=&-2\big(\sum_{m=1,4}t_{mz}\cos (mz)
+\cos(x)\left[t_{x} +2t_{xz}\cos(z)\right]\nonumber \\
&&+\cos(y/2)\left[t_y+
2t_{yz}\cos(z)+2t_{xy}\cos(x)\right]\big),
%&&+2t_{xz}\cos(k_xa)\cos(k_zc)\big).\nonumber \\\end{aligned}$$ where $x=k_za$, $y=k_yb$, $z=k_zc$. Notice that in our effective one-band description the upper band (see Fig. \[band\](c)) e.g.along $\Gamma$–X corresponds to $k_y$ = 0, whereas the lower one corresponds to $k_y$ = $2\pi /b$. The assignment of the parameters has been achieved by two numerically independent procedures: By straightforward least square fitting of the whole antibonding band in all directions and by using the bandwidths, the slopes and the curvatures at special selected high symmetry points. The latter procedure has the advantage to be less affected by hybridization effects from lower lying bands near the bottom of the antibonding band (being of some relevance near the Z-point in Fig. \[band\]).
The results are shown in Tab. \[tabel1\]. The errors can be estimated between 1% for the large and 10% for the small parameters from the difference of both mentioned above fitting procedures. The analyzed antibonding bands of both compounds exhibit a rather similar shape except near the Z-points, where the hybridization with lower lying bands produces an additional band-crossing for CuGeO$_3$ (see Fig. \[band\](c)). Recall that the main difference to the corner-shared chains as e.g. in Sr$_2$CuO$_3$ is a much smaller in-chain NN transfer due to the different geometry.
In spite of the qualitative similarity, the calculated values for the transfer integrals are quite different. The in-chain dispersion is nearly twice as large for CuGeO$_3$ in comparison to CuSiO$_3$. This can be attributed mainly to the larger Cu-O-Cu bond angle in CuGeO$_3$ (99$^\circ$ and 94$^\circ$, respectively). However, this geometrical effect is somewhat reduced by the different on-site energies of the oxygen orbitals along and perpendicular to the chain (lying in the plaquettes planes). The latter difference is reflected by the larger separation of the corresponding bands at the Z-point in CuSiO$_3$ (see Fig. \[band\]).
The inter-chain dispersions in $b$ direction are comparable. For both compounds, we find also rather significant diagonal hopping terms $t_{yz}$ which are reflected by different dispersions along the X–S and the T–Z directions. Somewhat surprisingly, we found a sizeable dispersion in $x$-direction for CuGeO$_3$ but only a very weak one for the CuSiO$_3$ counterpart.
From the transfer integrals discussed above, we conclude that both compounds are not so well-defined quasi one-dimensional systems as compared to the corner-shared CuO$_3$ chain compounds [@rosner97; @rosner99]. The inter-chain coupling is rather significant for CuGeO$_3$, and CuSiO$_3$ can even be regarded as an anisotropic two-dimensional system. Since increasing inter-chain coupling tends to destabilize the spin-Peierls state[@inagaki83b], a Néel ordered antiferromagnetic ground state might be expected for CuSiO$_3$ in contrast to the spin-Peierls state realized in CuGeO$_3$.
The obtained transfer integrals enables us to estimate the relevant exchange integrals $J$. This knowledge is crucial for the derivation and examination of magnetic model Hamiltonians of the spin-1/2 Heisenberg type frequently used in the literature: $$H_{spin}={\sum_{ij}}^{\prime}J_{ij}\vec{S_i}\vec{S_j} \, .$$ In general, the total exchange $J$ can be divided into an antiferromagnetic and a ferromagnetic contribution $J$ = $J^{AFM} +
J^{FM}$. In the strongly correlated limit, valid for typical cuprates, the former can be calculated in terms of the one-band extended Hubbard model $J^{AFM}_{ij}$ = $4t^2_{ij}/(U-V_{ij})$. The indices $i$ and $j$ correspond to nearest and next nearest neighbors, $U$ is the on-site Coulomb repulsion and $V_{ij}$ is the inter-site Coulomb interaction. From experimental data [@parmigiani96] mapped from the standard $pd$-model onto the one-band description, one estimates $U - V$ $\sim$ 4.2 eV. For the sake of simplicity, we neglect the difference in the quantity $U - V$ in the compounds. The calculated values for the exchange integrals are given in Tab. \[tabel2\].
The value of the NN exchange integral $J^{AFM}_{1}$ $\sim$ 30 meV in CuGeO$_3$ exceeds the experimental values of about 11 meV from inelastic neutron scattering data [@regnault96], about 14 meV from magnetic susceptibility[@fabricius98] and about 22 meV from Raman scattering [@kuroe97]. This points to a significant ferromagnetic contribution due to the Goodenough-Kanamori-Anderson-type interaction[@anderson59]. In the following, we shall adopt 15 meV for the resulting total exchange coupling $J_1$ as a representative value, suggested by the average of the above mentioned experimental data. Owing to the lack of experimental data we assume the same ratio $J_1/J_1^{AFM}$ in CuSiO$_3$ as in CuGeO$_3$, suggested by the quite similar O(2) 2$p$ orbital occupancies mentioned above. For the latter compound, we note the reasonable agreement with the available experimental data and most of our calculated antiferromagnetic values for the remaining exchange parameters. Hence, further possible ferromagnetic contributions seem to be less relevant and are neglected in the following considerations.
Further simplification can be obtained mapping J$_1$ and the frustrated NN term J$_2$ onto an effective intra-chain coupling $J_{\parallel}=J_1 - 1.12J_2$.[@fledderjohann97] The calculated values for $J_{\parallel}$ are 12.2 meV for CuGeO$_3$ and 2.8 meV for CuSiO$_3$, respectively. The latter value is close to the value of 2 meV reported by Baenitz [*et al.*]{} from a one-dimensional fit of magnetic susceptibility data.[@baenitz00] We find also a considerable inter-chain frustration $J_{yz}=\beta J_y$ with $\beta$=0.36 (0.34) for the Ge- (Si-) compound. This is in good agreement with the suggestions of Uhrig [@uhrig97] $\beta \approx
0.5$ for CuGeO$_3$.
Transfering the above mentioned idea to map frustrating terms onto one effective coupling,[@fledderjohann97] we adopt $J_{\perp}=J_y-2J_{yz}$ for the effective inter-chain exchange parameters in $b$-direction. The factor of two is introduced to account approximately the twice as large number of second neighbors. The effective anisotropy ratio $R = J_{\perp}/J_{\parallel}$ measures approximately the magnitude of quantum fluctuations. In the crossover region between one and two dimensions, quantum fluctuations do strongly affect the magnitude of the staggered magnetization $m$ and the local Cu moment $\mu=g_{L}n_dm$ at $T=0$ for a Néel ground state, where $g_L$=2.06 to 2.26 [@honda96] denotes the (anisotropic) Landé-factor (tensor) for Cu$^{2+}$ in CuGeO$_3$ and $n_d\approx $ 0.8 is the hole occupation number of the related Cu 3$d$ plaquette orbital. Using the expression $$m=0.39\sqrt{R}(1+0.095R)\ln^{1/3}(1.3/R) ,$$ taken from Ref. , we arrive at 0.17$\mu_B$ in reasonable agreement [@remark2] with the neutron data 0.22$\pm 0.02$ [@hase96] and 0.2 [@sasago96] for the disorder induced Néel state achieved below 4.5K in Zn-doped CuGeO$_3$. The same approach predicts a significantly larger value of about 0.35$\mu_B$ for CuSiO$_3$ realized in a possible Néel state.
To summarize, our LDA-FPLO calculation reveals valuable insight into the relevant couplings of CuGeO$_3$ and CuSiO$_3$. We can classify CuGeO$_3$ as a quasi one-dimensional compound with significant inter-chain interaction, whereas CuSiO$_3$ is closer to an anisotropic two-dimensional compound. The significantly reduced energy scale of the in-chain exchange interactions and the large inter-chain interaction in CuSiO$_3$ are less favorable for a spin-Peierls state than for a Néel order. However, due to the large frustrations other states such as a spin-Peierls state cannot be excluded. Further investigations are required to elucidate the unknown ground state.
We acknowledge fruitful discussions with M. Baenitz, C. Geibel, W. Pickett and G. Uhrig. This work was supported by individual grants of the DAAD (H.R.) and the DFG (S.D.).\
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abstract: 'We give a detailed and refined proof of the Dobrushin-Pechersky uniqueness criterion extended to the case of Gibbs fields on general graphs and single-spin spaces, which in particular need not be locally compact. The exponential decay of correlations under the uniqueness condition has also been established.'
address:
- 'Fakutät für Mathematik, Universität Bielefeld, Bielefeld D-33615, Germany'
- 'Fakutät für Mathematik, Universität Bielefeld, Bielefeld D-33615, Germany'
- 'Instytut Matematyki, Uniwersytet Marii Curie-Sk[ł]{}odowskiej, 20-031 Lublin, Poland'
- 'Fakutät für Mathematik, Universität Bielefeld, Bielefeld D-33615, Germany'
author:
- Diana Conache
- Yuri Kondratiev
- Yuri Kozitsky
- Tanja Pasurek
title: 'Gibbs Fields: Uniqueness and Decay of Correlations. Revisiting Dobrushin and Pechersky'
---
Introduction
============
A random field on a countable set ${\sf L}$ is a collection of random variables – [*spins*]{}, indexed by $\ell \in {\sf L}$. Each variable is defined on some probability space and takes values in the corresponding [*single-spin*]{} space $\Xi_\ell$. Typically, it is assumed that each $\Xi_\ell$ is a copy of a Polish space $\Xi$. In a ‘canonical version’, the probability space is $(\Xi^{\sf L},
\mathcal{B}(\Xi^{\sf L}), \mu)$, where $\mu$ is a probability measure on the Borel $\sigma$-field $\mathcal{B}(\Xi^{\sf L})$. Then also $\mu$ is referred to as random field. A particular case of such a field is the product measure of some single-spin probability measures $\sigma_\ell$. [*Gibbs random fields*]{} with pair interactions are constructed as ‘perturbations’ of the product measure $\otimes_{\ell \in {\sf L}} \sigma_\ell$ by the ‘densities’ $$\exp\left(\sum W_{\ell \ell'} (\xi_\ell, \xi_{\ell'} ) \right)$$ where $W_{\ell \ell'}: \Xi\times \Xi \to \mathbb{R}$ are measurable functions – [*interaction potentials*]{}, whereas the sum is taken over the set ${\sf E} \subset {\sf L}\times {\sf L}$ such that $W_{\ell\ell'}\neq 0$ for $(\ell,\ell')\in {\sf E}$. The latter condition defines the underlying graph ${\sf G} = ({\sf L}, {\sf
E})$. For bounded potentials, the perturbed measures usually exist. Moreover, there is only one such measure if the potentials are small enough and the underlying graph is enough ‘regular’. If the potentials are unbounded, both the existence and uniqueness issues turn into serious problems of the theory. Starting from the first results in constructing Gibbs fields with ‘unbounded spins’ [@Lebow], attempts to elaborating tools for proving their uniqueness were being undertaken [@COPP; @DobP; @Mal]. However, except for the results of [@Mal] obtained for the potentials and single-spin measures of a special type, and also for methods applicable to ‘attractive’ potentials, see [@KP; @Pasurek; @Diana], there is only one work presenting a kind of general approach to this problem. This work is due to R. L. Dobrushin and E. A. Pechersky [@DobP], which was first published in Russian and later on translated to English. In spite of its great importance, the work remains almost unknown (it has been cited only few times) presumably for the following reasons: (i) the English translation in [@DobP] was made with numerous typos and errors of mathematical nature, whereas the Russian version is inaccessible for the most of the readers; (ii) most of the proofs in [@DobP] are very involved and complex, and essential parts of them are only sketched or even missing. In the present publication, we give a refined and complete proof of the Dobrushin-Pechersky result extended in the following directions: (a) we do not employ the compactness arguments crucially used in [@DobP]; (b) we settle (in Proposition \[TVpn\] below) the measurability issues not even discussed in [@DobP]; (c) instead of the cubic lattice $\mathbb{Z}^d$ we consider general graphs as underlying sets of the Gibbs fields. The refinement consists, among others, in explicitly calculating the threshold value of $K$ in (\[K\]) and the constants in the basic estimates in Lemma \[R3lm\]. Due to (a), as the single-spin spaces $\Xi$ one can consider just standard Borel spaces, e.g., infinite dimensional spaces which are not locally compact, see [@KP; @Pasurek]. Due to (c), one can apply the criterion to varios models employing graphs as underlying sets. One can also apply the criterion to the equilibrium states of continuum particle systems, see [@Diana Chapter 4] and Section \[222\] below.
The structure of this paper is as follows. In Section \[2SEC\], we give necessary preliminaries and formulate the results in Theorems \[1tm\] and \[2tm\]. Section \[3SEC\] contains the proof of these theorems based on the estimates obtained in Lemmas \[R3lm\] and \[dclm\], respectively, as well as on a number of other facts proved thein. In Section \[4SEC\], we perform detailed constructions yielding the proof of the mentioned lemmas.
Setup and the Result {#2SEC}
====================
Notations and preliminaries
---------------------------
The underlying set for the spin configurations of our model is a countable simple connected graph $({\sf L}, {\sf E})$. For a vertex $\ell \in {\sf L}$, by $\partial \ell$ we denote the neighborhood of $\ell$, i.e., the set of vertices adjacent to $\ell$. The vertex [*degree*]{} $\varDelta_\ell$ is then the cardinality of $\partial
\ell$. The only assumption regarding the graph is that $$\label{1}
\sup_{\ell\in {\sf L}}\varDelta_\ell=: \varDelta < \infty,$$ i.e., the vertex degree is globally bounded. A given ${\sf V}\subset {\sf L}$ is said to be an [*independent set*]{} of vertices if $$\label{2}
\forall \ell \in {\sf V} \quad \partial l \cap {\sf V} = \emptyset.$$ The [*chromatic number*]{} $\chi\in \mathbb{N}$ is the smallest number such that $$\label{3}
{\sf L} = \bigcup_{j=0}^{\chi-1} {\sf V}_j, \qquad {\sf V}_j \ - \ {\rm independent}, \ j=0, \dots , \chi-1.$$ Obviously, $\chi \leq \varDelta+1$. However, by Brook’s theorem, see, e.g., [@Lovasz], for our graph we have that $\chi \leq
\varDelta$.
For a measuarble space $(E, \mathcal{E})$, by $\mathcal{P}(E)$ we denote the set of all probability measures on $\mathcal{E}$. All measurable spaces we deal with in this article are standard Borel spaces. The prototype example is a Polish space endowed with the corresponding Borel $\sigma$-field. For $\sigma \in \mathcal{P}(E)$ and a suitable function $f:E \to \mathbb{R}$, we write $$%\label{4}
\sigma(f) = \int_{E} f d \sigma.$$ For our model, the single-spin spaces $(\Xi_\ell,
\mathcal{B}(\Xi_\ell))$, $\ell \in {\sf L}$, are copies of a standard Borel space $(\Xi, \mathcal{B}(\Xi))$. Then the configuration space $X = \Xi^{\sf L}$ equipped with the product $\sigma$-field $\mathcal{B}(X)=\mathcal{B}(\Xi^{\sf L})$ is also a standard Borel space. Likewise, for a nonempty ${\sf D} \subset
{\sf L}$, $\Xi^{\sf D}$ is the product of $\Xi_\ell$, $\ell \in {\sf
D}$. Its elements are denoted by $x_{\sf D} = (x_\ell)_{\ell \in
{\sf D}}$, whereas the elements of $X$ are written simply as $x=
(x_\ell)_{\ell \in {\sf L}}$. For $y,z\in X$, by $y_{\sf D} \times
z_{{\sf D}^c}$ we denote the configuration $x\in X$ such that $x_{\sf D} = y_{\sf D}$ and $x_{{\sf D}^c} = z_{{\sf D}^c}$, ${\sf
D}^c:= {\sf L} \setminus {\sf D}$. For ${\sf D}\subsetneq {\sf L}$, we denote $\mathcal{F}_{\sf D} = \mathcal{B}(\Xi^{{\sf D}^c})$ and write $\mathcal{F}_\ell$ if ${\sf D}=\{\ell\}$.
\[0df\] Given $\ell \in {\sf L}$, let $ \pi_\ell:=\{\pi_\ell^x : x \in
X\}\subset \mathcal{P}(\Xi_\ell)$ be such that the map $X\ni x \mapsto
\pi^x_\ell (A)\in \mathbb{R}$ is $\mathcal{F}_\ell$-measurable for each $A\in \mathcal{B}(\Xi_\ell)$. A family $\pi=\{\pi_\ell\}_{\ell \in
{\sf L}}$ of the maps of this kind is said to be a [*one-site*]{} specification.
\[1df\] A given $\mu \in \mathcal{P}(X)$ is said to be [*consistent*]{} with a one-site specification $\pi$ in a given ${\sf D}\subseteq {\sf L}$ if $\mu(\cdot|\mathcal{F}_\ell)(x) = \pi_\ell^x$ for $\mu$-almost all $x$ and each $\ell \in {\sf D}$. By $\mathcal{M}_{\sf D}(\pi)$ we denote the set of all $\mu\in \mathcal{P}(X)$ consistent with $\pi$ in ${\sf D}$. We say that $\mu$ is consistent with $\pi$ if it is consistent in ${\sf L}$, and write just $\mathcal{M}(\pi)$ in this case.
Obviously, $\mu\in \mathcal{M}(\pi)$ if and only if it satisfies the following equation $$\begin{aligned}
\label{5}
\mu(A)& = & \int_X \int_X \mathbb{I}_A (x) \pi_\ell^y (dx_\ell) \prod_{\ell'\neq \ell} \delta_{y_{\ell'}} (d x_{\ell'}) \mu (dy)\\[.2cm]
& = & \int_X \left(\int_\Xi \mathbb{I}_A (\xi \times y_{\{\ell\}^c}) \pi_\ell^{y} (d\xi) \right) \mu (dy),\nonumber\end{aligned}$$ which holds for every $\ell\in {\sf L}$ and $A\in \mathcal{B}(X)$. Here, for $\eta\in \Xi$, $\delta_\eta\in \mathcal{P}(\Xi)$ is the Dirac measure centered at $\eta$ and $\mathbb{I}_A$ stands for the indicator of $A$.
For a standard Borel space $(E, \mathcal{E})$, let $(E^2,
\mathcal{E}^2)$ be the product space. For $\sigma, \varsigma \in
\mathcal{P}(E)$, let $\varrho\in \mathcal{P}(E^2)$ be such that $\varrho(A\times E)= \sigma(A)$ and $\varrho(E\times A) = \varsigma
(A)$ for all $A\in \mathcal{B}(E)$. Then we say that $\varrho$ is a [*coupling*]{} of $\sigma$ and $\varsigma$. By $\mathcal{C}(\sigma,
\varsigma)$ we denote the set of all such couplings.
For $\xi, \eta \in \Xi$, we set $$%\label{up}
\upsilon(\xi, \eta) = \left\{ \begin{array}{ll} 0, \quad {\rm if} \ \ \xi = \eta;\\[.3cm]
1, \quad {\rm otherwise},
\end{array} \right.$$ which is a measurable function on $\Xi^2$ since $\Xi$ is a standard Borel space. Then we equip $\mathcal{P}(\Xi)$ with the [*total variation distance*]{} $$\label{TV}
d(\sigma, \varsigma) = \sup_{A \in \mathcal{B}(\Xi)} |\sigma (A) -
\varsigma (A)|,$$ that, by duality, can also be written in the form $$% \label{6}
d(\sigma, \varsigma) = \inf_{\varrho\in \mathcal{C}(\sigma, \varsigma)} \int_{\Xi^2} \upsilon (\xi, \eta) \varrho( d \xi , d \eta).$$
\[TVpn\] For each $\ell \in {\sf L}$ and $(x,y)\in X^2$, there exists $\varrho^{x,y}_\ell \in \mathcal{C}(\pi_\ell^x, \pi_\ell^y)$ such that: (a) for each $B\in \mathcal{B}(\Xi^2_\ell)$, the map $X^2\ni
(x,y)\mapsto \varrho^{x,y}_\ell(B)$ is $\mathcal{F}_\ell^2$-measurable; (b) the following holds $$\label{7}
d(\pi_\ell^x, \pi_\ell^y) = \int_{\Xi^2} \upsilon (\xi, \eta) \varrho_\ell^{x,y}( d \xi , d \eta).$$
Set $$%\label{7TV}
(\pi_\ell^x \wedge \pi_\ell^y) (A) = \min\{\pi_\ell^x (A);
\pi_\ell^y(A) \}, \qquad A\in \mathcal{B}(\Xi_\ell).$$ In view of the measurability as in Definition \[0df\], the map $X^2\ni (x,y) \mapsto (\pi_\ell^x \wedge \pi_\ell^y) (A)$ is $\mathcal{F}_\ell^2$-measurable since, given $a\in [0,1]$, we have that $$\begin{aligned}
\{ (x,y): a \leq (\pi_\ell^x \wedge \pi_\ell^y) (A) \} = \{ x:
a\leq \pi_\ell^x (A)\}^2.\end{aligned}$$ Then both maps $(x,y) \mapsto (\pi^x_\ell - \pi_\ell^x \wedge
\pi_\ell^y)(A)$ and $(x,y) \mapsto (\pi^y_\ell - \pi_\ell^x \wedge
\pi_\ell^y)(A)$ are $\mathcal{F}_\ell^2$-measurable. By (\[TV\]) also $(x,y) \mapsto d
(\pi_\ell^x, \pi_\ell^y)$ is measurable in the same sense.
Set $D_\ell = \{(\xi, \xi): \xi \in \Xi_\ell\}$. Since $\Xi_\ell$ is a standard Borel space, the map $\xi \mapsto
\psi (\xi) = (\xi, \xi)\in D_\ell$ is measurable. Then, for each $B\in
\mathcal{B}(\Xi^2_\ell)$, we have that $\psi^{-1} (B\cap D_\ell) \in
\mathcal{B}(\Xi_\ell)$, which allows us to define $\omega_\ell^{x,y} \in
\mathcal{P}(\Xi^2_\ell)$ by setting $$\omega_\ell^{x,y} (B) = (\pi_\ell^x \wedge \pi_\ell^y) \left(
\psi^{-1} (B\cap D_\ell) \right).$$ The coupling for which (\[7\]) holds has the form, see [@Lind Eq. (5.3), page 19], $$% \label{Lind1}
\varrho_\ell^{x,y} = \omega_\ell^{x,y} + (\pi^x_\ell - \pi_\ell^x
\wedge \pi_\ell^y) \otimes (\pi^y_\ell - \pi_\ell^x \wedge
\pi_\ell^y)/ d (\pi_\ell^x, \pi_\ell^y).$$ Then the $\mathcal{F}_\ell^2$-measurability of the maps $(x,y) \mapsto \varrho^{x,y}_\ell (A_1 \times A_2)$, $A_1,A_2 \in \mathcal{B}(\Xi_\ell)$, follows by the arguments given above. This yields the proof of claim (a) as $\mathcal{B}(\Xi^2_\ell)$ is a product $\sigma$-field.
Let $\varpi$ be the family of $\varpi_\ell = \{\varpi_\ell^{x,y}:
(x,y) \in X^2\}$, $\ell \in {\sf L}$, such that each $\varpi_\ell^{x,y}$ is in $\mathcal{P}(\Xi^2_\ell)$ and, for any $B\in
\mathcal{B}(\Xi^2_\ell)$, the map $(x,y) \mapsto \varpi_\ell^{x,y} (B)$ is $\mathcal{F}_\ell^2$-measurable. Then $\varpi$ is a one-point specification in the sense of Definition \[1df\], which determines the set $\mathcal{M}(\varpi)$ of $\nu\in
\mathcal{P}(X^2)$ consistent with $\varpi$. Like in (\[5\]), $\nu \in
\mathcal{M}(\varpi)$ if and only if it satisfies $$\begin{aligned}
\label{8}
\nu(B) & = & \int_{X^2} \int_{X^2} \mathbb{I}_B(x,y) \varpi_\ell^{y,\tilde{y}}(dx_\ell, d\tilde{x}_\ell)\\[.2cm]
& \times & \prod_{\ell'\neq \ell} \delta_{y_{\ell'}} (d x_{\ell'})\delta_{\tilde{y}_{\ell'}} (d \tilde{x}_{\ell'})\nu(dy, d\tilde{y}),\nonumber\end{aligned}$$ which holds for all $\ell\in {\sf L}$ and $B\in \mathcal{B}(X^2)$.
\[1pn\] Suppose that $\varpi^{x, \tilde{x}}_\ell\in \mathcal{C}(\pi_\ell^x, \pi_\ell^{\tilde{x}})$ for all $\ell\in {\sf L}$ and $x, \tilde{x}\in X$. Then each $\nu\in \mathcal{M}(\varpi)$ is a coupling of some $\mu_1 , \mu_2 \in \mathcal{M}(\pi)$.
The equality $\mu_1(A)= \nu(A\times X)$, $A\in \mathcal{B}(X)$, determines a probability measure on $X$. Thus, for $A\in \mathcal{B}(X)$, by (\[8\]) we get $$\begin{aligned}
\mu_1(A) & = & \int_{X^2} \int_{X^2}\mathbb{I}_A (x) \varpi_\ell^{y,\tilde{y}}(dx_\ell, d\tilde{x}_\ell)
\prod_{\ell'\neq \ell} \delta_{y_{\ell'}} (d x_{\ell'})\delta_{\tilde{y}_{\ell'}} (d \tilde{x}_{\ell'})\nu(dy, d\tilde{y})\\[.2cm]
& = & \int_{X^2} \int_{X}\mathbb{I}_A (x) \pi_\ell^y (dx_\ell) \prod_{\ell'\neq \ell} \delta_{y_{\ell'}}(dx_{\ell'})\nu(dy, d\tilde{y})\\[.2cm]
& = & \int_{X} \int_{X}\mathbb{I}_A (x) \pi_\ell^y (dx_\ell) \prod_{\ell'\neq \ell} \delta_{y_{\ell'}}(dx_{\ell'})
\int_X \nu(dy, d\tilde{y})\nonumber \\[.2cm]
& = & \int_{X} \int_{X}\mathbb{I}_A (x) \pi_\ell^y (dx_\ell) \prod_{\ell'\neq \ell} \delta_{y_{\ell'}}(dx_{\ell'}) \mu_1(dy).\end{aligned}$$ Therefore, $\mu_1$ solves (\[5\]) and hence $\mu_1\in \mathcal{M}(\pi)$. The same is true for the second marginal measure $\mu_2$.
The results
-----------
Our main concern is under which conditions imposed on the family $\pi$ the set $\mathcal{M}(\pi)$ contains one element at most. If each $\pi_\ell^x$ is independent of $x$, the unique element of $\mathcal{M}(\pi)$ is the product measure $\otimes_{\ell \in {\sf
L}} \pi_\ell$, which readily follows from (\[5\]). Therefore, one may try to relate the uniqueness in question to the weak dependence of $\pi_\ell^x$ on $x$, formulated in terms of the metric defined in (\[7\]). Thus, let us take $x,y\in X$ such that $x=y$ off some $\ell' \in \partial \ell$, and consider $d(\pi_\ell^x,
\pi_{\ell}^y)$. If this quantity were bounded by a certain $\kappa_{\ell\ell'}$, uniformly in $x$ and $y$, this bound (Dobrushin’s estimator, cf. [@Beth pp. 20, 21]) could be used to formulate the celebrated Dobrushin uniqueness condition in the form $$\label{9}
\sup_{\ell \in {\sf L}} \sum_{\ell'\in \partial \ell} \kappa_{\ell \ell'} =: \bar{\kappa} < 1.$$ However, in a number of applications, especially where $\Xi$ is a noncompact topological space, the mentioned boundedness does not hold. The way of treating such cases suggested in [@DobP] may be outlined as follows. Assume that there exists a matrix $(\kappa_{\ell \ell'})$ with the property as in (\[9\]) such that, for each $\ell \in {\sf L}$, the following holds $$\label{10}
d(\pi_\ell^x, \pi_{\ell}^y) \leq \sum_{\ell' \in \partial \ell} \kappa_{\ell\ell'} \upsilon(x_{\ell'}, y_{\ell'}),$$ for $x$ and $y$ belonging to the set $$\label{11}
X_\ell (h,K) := \{ x \in X: h(x_{\ell'}) \leq K \ \ {\rm for} \ {\rm all} \ \ell' \in \partial \ell\}.$$ Here $K>0$ is a parameter and $h:\Xi \to [0,+\infty)$ is a given measurable function. Clearly, if $h$ is bounded, then $X_\ell
(h,K)=X$ for big enough $K$, and hence (\[10\]) turns into the mentioned Dobrushin condition. Thus, in order to cover the case of interest we have to take $h$ unbounded and $\pi_\ell^x$-integrable, with an appropriate control of the dependence of $\pi_\ell^x(h)$ on $x$. Namely, we shall assume that, for each $\ell\in {\sf L}$ and $x
\in X$, the following holds $$\label{12}
\pi_\ell^x (h) \leq 1 + \sum_{\ell'
\in \partial \ell} c_{\ell \ell'} h(x_{\ell'}),$$ for some matrix $c=(c_{\ell \ell'})$, which satisfies $$\label{13}
(a) \quad c_{\ell\ell'} \geq 0; \qquad \quad (b) \quad \sup_{\ell
\in {\sf L}} \sum_{\ell' \in \partial \ell} c_{\ell \ell'} =:\bar{c}
< 1/ \varDelta^\chi.$$ In the original work [@DobP], the first summand on the right-hand side of (\[12\]) is a constant $C>0$, the value of which determines the scale of $K$, see (\[11\]). We thus take it as above for the sake of convenience.
\[2df\] Let $h$, $K$, $\kappa$, and $c$ be as in (\[9\]) – (\[13\]). Then by $\Pi(h,K,\kappa,c)$ we denote the set of one-site specifications $\pi$ for which both estimates (\[10\]), (\[11\]) and (\[12\]) hold true for each $\ell \in {\sf L}$.
Given $\mu\in \mathcal{M}(\pi)$, the integrability assumed in (\[12\]) does not yet imply that $h$ is $\mu$-integrable. For $\pi$ satisfying (\[12\]), by $\mathcal{M}(\pi, h)$ we denote the subset of $\mathcal{M}(\pi)$ consisting of those measures for which the following holds $$\label{14}
\mu(h):=\sup_{\ell \in {\sf L}} \int_{X} h(x_\ell) \mu(d x) < \infty.$$ In a similar way, we introduce the set $\mathcal{M}_{\sf D}(\pi, h)$ for a given ${\sf D}\subset {\sf L}$, cf. Definition \[1df\].
From now on we fix the graph, the function $h$, and the matrices $c$ and $\kappa$. Thereafter, we set $$\label{K}
K_* = \max\left\{\frac{4 \varDelta^{\chi+1}}{\bar{c}(1 - \bar{\kappa})} ; \ \
\frac{2 \varDelta^{\chi+1} (2 \varDelta^{\chi-1} +1-\bar{c} \varDelta^\chi)}{(1-\bar{\kappa})^2( 1- \bar{c} \varDelta^\chi)}\right\}.$$
\[1tm\] For each $K>K_*$ and $\pi \in \Pi(h,K,\kappa,c)$, the set $\mathcal{M}(\pi, h)$ contains at most one element.
An important characteristic of the states $\mu\in \mathcal{M}(\pi)$ is the decay of correlations. Fix two distinct vertices $\ell_1,
\ell_2 \in {\sf L}$ and consider bounded functions $f, g:X\to
\mathbb{R}_{+}$, such that $f$ is $\mathcal{B}(\Xi_{\ell_1})$-measurable and $g$ is $\mathcal{B}(\Xi_{\ell_2})$-measurable. Set $${\rm Cov}_\mu (f;g) = \mu(fg) - \mu(f)\mu(g),$$ and let $\delta$ denote the path distance on the underlying graph.
\[2tm\] Let $\pi$ and $K$ be as in Theorem \[1tm\], and $\mathcal{M}(\pi, h)$ be nonempty and hence contain a single state $\mu$. Let also $f$ and $g$ be as just described and $\|\cdot \|_\infty$ denote the sup-norm on $X$. Then there exist positive $C_K$ and $\alpha_K$, dependent on $K$ only, such that $$\label{dc}
| {\rm Cov}_\mu (f;g)| \leq C_K \|f\|_\infty \|g\|_\infty \exp\left[- \alpha_K \delta(\ell_1 , \ell_2) \right].$$
Comments and applications {#222}
-------------------------
Let us make some comments to the above results. For further comments related to the proof of these results see the end of Section \[3SEC\].
- According to [@Preston Section 8], the elements of $\mathcal{M}(\pi)$ as in Definition \[1df\] are one-site Gibbs states. In [@Ge Theorem 1.33, page 23] and [@Preston Section 8], there are given conditions under which the elements of $\mathcal{M}(\pi)$ are ‘usual’ Gibbs states, e.g., in the sense of [@Ge Definition 1.23, page 16]. This, in particular, holds if $\pi$ is a subset of the set of all local kernels $\Pi_{\sf D}$ defined for all finite ${\sf D}\subset {\sf
L}$, which determine the states. In this case, Theorem \[1tm\] yields the existence and uniqueness of the usual states, see [@Diana].
- The condition in (\[14\]) is usually satisfied for [*tempered*]{} measures, i.e., for those elements of $\mathcal{M}(\pi)$ which are supported on [*tempered*]{} configurations, cf., e.g., [@Lebow].
- As mentioned above, we do not require that $h$ be [*compact*]{} in the sense of [@DobP]. This our extension gets important if one deals with single-spin spaces which are not locally compact, e.g., with spaces of Hölder continuous functions as in [@Mon; @KP; @Pasurek].
- In contrast to [@DobP Theorem 1], in (\[K\]) we give an explicit expression for the threshold value $K_*$, which depends only on the parameters of the underlying graph and on the norms $\bar{c}$ and $\bar{\kappa}$.
- The novelty of Theorem \[2tm\] consists in the following. The decay of correlations under the uniqueness condition was proven only for compact single-spin spaces, see [@Ku], where the classical Dobrushin criterion can be applied. For ‘unbounded spins’, the corresponding results are usually obtained by cluster expansions, see, e.g., [@PS], where the correlations are shown to decay due to ‘weak enough’ interactions’ and no information on the number of states is available.
- The parameters $C_K$ and $\alpha_K$ in (\[dc\]) are also given explicitly, see (\[constan\]) below.
Now we turn to briefly outlining possible applications of Theorems \[1tm\] and \[2tm\]. A more detailed discussion of this issue can be found in [@Diana], see also the related parts of [@Pasurek]. Further results in these directions will be published in forthcoming articles.
By means of Theorems \[1tm\] and \[2tm\] the uniqueness of equilibrium states and the decay of correlations can be established in the following models:
- Systems of classical $N$-dimensional anharmonic oscillators described by the energy functional $$H(x) = \sum_{\ell \in {\sf L}} V(\xi_\ell) + \sum_{(\ell, \ell')\in
{\sf E}} W_{\ell \ell'} (\xi_\ell, \xi_{\ell'}), \qquad \xi_\ell \in
\mathbb{R}^N, \ N \in \mathbb{N}$$
- Systems of quantum $N$-dimensional anharmonic oscillators described by the Hamiltonian $$H = \sum_{\ell \in {\sf L}} H_\ell + \sum_{(\ell, \ell')\in {\sf E}}
W_{\ell \ell'} (q_\ell, q_{\ell'}),$$ where $q_\ell= (q^{(1)}_\ell, \dots , q^{(N)}_\ell)$ is the position operator and $H_\ell$ is the one-particle Hamiltonian defined on the corresponding physical Hilbert space. States of such models are constructed in a path integral approach as probability measures on the products of continuous periodic functions, which are not locally compact, see [@Mon; @KP; @Pasurek].
- Systems of interacting particles in the continuum (e.g. $\mathbb{R}^d$), including the Lebowitz-Mazel-Presutti model [@LMP], and systems of ‘particles’ lying on the cone of discrete measures introduced in [@Hagedorn]. Note that to continuum systems the original version [@DobP] of the Dobrushin-Pechersky criterion was used in [@BP; @Pechersky].
The Proof of Theorems \[1tm\] and \[2tm\] {#3SEC}
=========================================
The ingredients of the proof
----------------------------
First we introduce the notion of [*locality*]{}. By writing ${\sf D}\Subset {\sf L}$ we mean that $\sf D$ is a nonempty finite subset of $\sf L$. For such $\sf D$, elements of $\mathcal{B}(\Xi^{\sf D})\subset \mathcal{B}(X)$ are called local sets. A function $f:X \to \mathbb{R}$ is called local if it is $\mathcal{B}(\Xi^{\sf D})$-measurable for some ${\sf D}\Subset {\sf L}$. Likewise, $B\in \mathcal{B}(X^2)$ is local if $B\in\mathcal{B}((\Xi\times \Xi)^{\sf D})$ for such ${\sf D}$. Locality of functions $f:X^2 \to \mathbb{R}$ is defined in the same way.
\[1lm\] Given a one-site specification $\pi$ and $\mu_1, \mu_2 \in \mathcal{M}(\pi)$, suppose there exists $\nu_*\in \mathcal{C}(\mu_1, \mu_2)$ such that $$\label{15}
\int_{X^2} \upsilon (x_\ell , y_\ell) \nu_* (d x, dy) = 0,$$ holding for all $\ell \in {\sf L}$. Then $\mu_1 = \mu_2$.
Local sets $A\subset X$ are measure defining, that is, $\mu_1 , \mu_2 \in \mathcal{P}(X)$ coincide if they coincide on local sets. For $A\in \mathcal{B}(\Xi^{\sf D})$ and the indicator $\mathbb{I}_A$, we have $$|\mathbb{I}_A (x) - \mathbb{I}_A (y)| \leq \sum_{\ell \in {\sf D}} \upsilon (x_\ell, y_\ell),$$ and then $$|\mu_1(A) - \mu_2 (A) | \leq \sum_{\ell \in {\sf D}} \int_{X^2} \upsilon (x_\ell, y_\ell) \nu_* (d x, dy) = 0,$$ which yields the proof.
The proof of Theorem \[1tm\] will be done by showing that, for each $\mu_1 , \mu_2 \in \mathcal{M}(\pi,h)$, the set $\mathcal{C}(\mu_1 , \mu_2)$ contains a certain $\nu_*$ such that (\[15\]) holds. This coupling $\nu_*$ will be obtained by taking the limit in the topology of local setwise convergence, cf. [@Ge], which we introduce as follows.
\[3df\] A net $\{\nu_\alpha\}_{\alpha \in I} \subset \mathcal{P}(X^2)$ is said to be convergent to a $\nu_*\in \mathcal{P}(X^2)$ in the topology of local setwise convergence ($\mathfrak{L}$-topology, for short), if $\nu_\alpha (B) \to \nu_* (B)$ for all local $B\in \mathcal{B}(X^2)$. Or, equivalently, $\nu_\alpha (f) \to \nu_*(f)$ for all bounded local functions. The same definition applies also to nets $\{\mu_\alpha\}_{\alpha \in I} \subset \mathcal{P}(X)$.
Note that the $\mathfrak{L}$-topology is Hausdorff, but not metrizable if $\Xi$ is not a compact topological space.
\[2lm\] Given $\mu_1 , \mu_2 \in \mathcal{P}(X)$, let $\{\nu_\alpha \}_{\alpha \in I} \subset \mathcal{C}(\mu_1 , \mu_2)$ be convergent to a certain $\nu\in \mathcal{P}(X^2)$ in the $\mathfrak{L}$-topology. Then $\nu\in \mathcal{C}(\mu_1 , \mu_2)$.
The proof of this lemma is rather obvious. The coupling in question $\nu_*$ will be constructed within a step-by-step procedure based on the mapping $$\label{16}
(R_\ell \nu)(f) = \int_{X^2} \left( \int_{\Xi^2} f( \xi\times x_{\{\ell\}^c}, \eta\times y_{\{\ell\}^c })
\varrho^{x,y}_\ell (d \xi, d\eta)\right) \nu( d x, d y),$$ where $\ell \in {\sf L}$, $\varrho_\ell^{x,y}$ is as in (\[7\]), and $f:X^2 \to \mathbb{R}$ is a function such that both $\nu(f)$ and the integral on the right-hand side of (\[16\]) exist.
\[Rlm\] For each $\ell \in {\sf L}$, the mapping (\[16\]) has the following properties: (a) if $\nu\in \mathcal{C}(\mu_1, \mu_2)$ for some $\mu_1, \mu_2\in \mathcal{M}(\pi)$, then also $R_\ell \nu \in \mathcal{C}(\mu_1, \mu_2)$; (b) if $f$ is $\mathcal{F}_\ell (X^2)$-measurable and $\nu$-integrable, then $(R_\ell \nu)(f) = \nu(f)$.
Claim (a) is true since $\varrho_\ell^{x,y} \in \mathcal{C}(\pi_\ell^x, \pi_\ell^y)$ for all $x, y \in X$. Claim (b) follows by the fact that the considered $f$ in (\[16\]) is independent of $\xi$ and $\eta$, and that $\varrho_\ell^{x,y}$ is a probability measure.
Given $\ell \in {\sf L}$, we set $$% \label{17}
Y_\ell = \{(x^1, x^2)\in X^2: \upsilon (x_\ell^1, x_\ell^2) \leq \sum_{\ell'\in \partial \ell} \upsilon (x_{\ell'}^1, x_{\ell'}^2) \}.$$
\[R1lm\] For each $\nu\in \mathcal{P}(X^2)$ and $\ell \in {\sf L}$, it follows that $(R_\ell \nu)(Y_\ell) =1$.
If $(x^1, x^2)$ is in $Y_\ell^c$, then $\upsilon (x_\ell^1,
x_\ell^2) =1$ and $\upsilon (x_{\ell'}^1, x_{\ell'}^2)=0$ for all $\ell'\in \partial \ell$, which follows by the fact that $\upsilon$ takes values in $\{0,1\}$. This means that $x_\ell^1 \neq x_\ell^2$ and $x_{\ell'}^1 = x_{\ell'}^2$ for all $\ell'\in \partial \ell$. For such $(x^1, x^2)$, the definition of $\pi$ implies that $\pi_\ell^{x^1} = \pi_\ell^{x^2}$, and hence $$\int_{\Xi^2} \upsilon (\xi, \eta) \varrho^{x^1,x^2}_\ell (d\xi,
d\eta) = d(\pi^{x_1}_\ell, \pi^{x_2}_\ell) = 0,$$ which by (\[16\]) yields $(R_\ell \nu)(Y_\ell^c) = 0$.
The proof of Theorem \[1tm\] will be done by showing that, for each $\mu_1, \mu_2 \in \mathcal{M}(\pi, h)$, there exists $\nu_* \in \mathcal{C}(\mu_1, \mu_2)$, for wich (\[15\]) holds. To this end we construct a sequence $\{\hat{\nu}_n\}_{n\in \mathbb{N}_0} \subset
\mathcal{C}(\mu_1, \mu_2)$ such that $$\label{18}
\gamma(\hat{\nu}_n):= \sup_{\ell \in {\sf L}} \int_{X^2} \upsilon (x^1_\ell , x_\ell^2) \hat{\nu}_n (d x^1 , d x^2) \to 0, \qquad n \to +\infty.$$ This sequence will be obtained by a procedure based on the mapping (\[16\]) and the estimates which we derive in the next subsection. The proof of Theorem \[2tm\] will be obtained as a byproduct.
The main estimates
------------------
In the sequel, we use the following functions indexed by $\ell \in {\sf L}$ $$\label{19}
I_\ell (x^1, x^2) = \upsilon (x^1_\ell , x_\ell^2), \qquad H^i_\ell (x^1, x^2) = h(x_\ell^i), \ \ i = 1,2.$$ By claim (b) of Lemma \[Rlm\], we have that $$%\label{20}
(R_\ell \nu)(I_{\ell_1}) = \nu (I_{\ell_1}), \quad (R_\ell \nu)(I_{\ell_1}H^i_{\ell_2}) =
\nu(I_{\ell_1}H^i_{\ell_2}) \quad {\rm for} \ \ell \neq \ell_1, \ \ell \neq \ell_2,$$ whenever $H^i_\ell$ is $\nu$-integrable. We recall that $\varrho_\ell^{x,y}$ in (\[16\]) is a coupling of $\pi_\ell^x$ and $\pi_\ell^y$, for which (\[10\]) and (\[12\]) hold true.
\[R2lm\] Let $\nu \in \mathcal{P}(X^2)$ be such that the integrals on both sides of (\[16\]) exist for $f=H^i_\ell$, $\ell \in {\sf L}$ and $i=1,2$. Then the following estimates hold $$\begin{gathered}
\label{21}
(R_\ell \nu)(I_{\ell}) \leq \sum_{\ell' \in\partial \ell} \kappa_{\ell \ell'} \nu(I_{\ell'}) +
K^{-1} \sum_{i=1,2} \sum_{\ell_1 , \ell_2 \in \partial \ell} \nu(I_{\ell_2}H^i_{\ell_1}), \\[.2cm]
\label{22}
(R_\ell \nu)(I_{\ell_1}H^i_{\ell}) \leq \nu (I_{\ell_1}) + \sum_{\ell_2 \in \partial \ell}c_{\ell\ell_2}
\nu(I_{\ell_1}H^i_{\ell_2}),\\[.2cm]
\label{23}
(R_\ell \nu)(I_{\ell}H^i_{\ell_1}) \leq \sum_{\ell_2 \in \partial \ell} \nu(I_{\ell_2}H^i_{\ell_1}), \qquad \ell_1 \neq \ell, \\[.3cm]
\label{24}
(R_\ell \nu)(I_{\ell}H^i_{\ell}) \leq \sum_{\ell_1 \in \partial \ell} \nu(I_{\ell_1}) +
\sum_{\ell_1 , \ell_2 \in \partial \ell}c_{\ell \ell_2} \nu(I_{\ell_1}H^i_{\ell_2}).\end{gathered}$$
The proof of (\[23\]) readily follows by Lemma \[R1lm\]. Let us prove (\[21\]). By (\[7\]) and (\[16\]), we have $$\begin{aligned}
%\label{25}
(R_\ell \nu)(I_{\ell}) & = & \int_{X^2} d(\pi_\ell^{x^1}, \pi_\ell^{x^2}) \nu(dx^1 , dx^2)\\[.2cm]
& = & \int_{X^2} \mathbf{1}_\ell (x^1)\mathbf{1}_\ell (x^2) d(\pi_\ell^{x^1}, \pi_\ell^{x^2}) \nu(dx^1 , dx^2)\nonumber \\[.2cm]
& + & \int_{X^2} \left[ 1 - \mathbf{1}_\ell (x^1)\mathbf{1}_\ell (x^2) \right] d(\pi_\ell^{x^1}, \pi_\ell^{x^2}) \nu(dx^1 , dx^2) ,\nonumber\end{aligned}$$ where $\mathbf{1}_\ell$ is the indicator of the set defined in (\[11\]). By (\[10\]), we have $$\int_{X^2} \mathbf{1}_\ell (x^1)\mathbf{1}_\ell (x^2) d(\pi_\ell^{x^1}, \pi_\ell^{x^2}) \nu(dx^1 , dx^2) \leq
\sum_{\ell' \in\partial \ell} \kappa_{\ell \ell'} \nu(I_{\ell'}),$$ which yields the first term of the right-hand side of (\[21\]). By (\[11\]), we have $$\left[ 1 - \mathbf{1}_\ell (x^1)\mathbf{1}_\ell (x^2) \right] \leq \sum_{i=1,2} \sum_{\ell_1 \in \partial \ell}
\left[1 - \mathbb{I}_{h\leq K} (x_{\ell_1}^i)\right],$$ where $\mathbb{I}_{h\leq K}$ is the indicator of $\{\xi \in \Xi: h(\xi) \leq K\}$. Then the second term of the right-hand side of (\[21\]) cannot exceed the following $$\begin{aligned}
& & \sum_{i=1,2} \sum_{\ell_1 \in \partial \ell} \int_{X^2}
\left[1 - \mathbb{I}_{h\leq K} (x_{\ell_1}^i)\right] d(\pi_\ell^{x^1}, \pi_\ell^{x^2}) \nu(dx^1 , dx^2)\\[.2cm]
& & \qquad \leq K^{-1} \sum_{i=1,2} \sum_{\ell_1 \in \partial \ell} \int_{X^2} h(x^i_{\ell_1})
d(\pi_\ell^{x^1}, \pi_\ell^{x^2}) \nu(dx^1 , dx^2)\\[.2cm]
& & \qquad \leq K^{-1} \sum_{i=1,2} \sum_{\ell_1 , \ell_2\in \partial \ell} \nu(I_{\ell_2} H^i_{\ell_1}).\end{aligned}$$ The latter line has been obtained by (\[23\]).
Let us prove now (\[22\]). By (\[16\]) and the fact that $\varrho_\ell^{x,y} \in \mathcal{C}(\pi_\ell^x, \pi_\ell^y)$, we have $$\begin{aligned}
(R_\ell \nu)(I_{\ell_1}H^i_{\ell}) & = & \int_{X^2} \left( \int_{\Xi} h (\xi) \pi^{x^i} (d\xi) \right)
\upsilon(x^{1}_{\ell_1} , x^{2}_{\ell_1}) \nu(dx^1, dx^2)\\[.2cm]
& \leq & {\rm RHS}(\ref{22}),\end{aligned}$$ where we have used (\[12\]). To prove (\[24\]) we employ Lemma \[R1lm\], by which we get $${\rm LHS}(\ref{24}) \leq \sum_{\ell_1 \in \partial \ell} (R_\ell \nu) (I_{\ell_1} H^i_\ell) \leq {\rm RHS}(\ref{24}) ,$$ where the latter estimate follows by (\[22\]).
From the lemma just proven it follows that along with the parameter $\gamma(\nu)$ defined in (\[18\]) one has to control also the following $$\label{26}
\lambda (\nu) = \max_{i=1,2} \sup_{\ell, \ell' \in {\sf L}} \nu(I_\ell H^i_{\ell'}),$$ where $\nu\in \mathcal{C}(\mu_1 , \mu_2)$, $\mu_1, \mu_2 \in \mathcal{M}_h(\pi)$, and $\pi \in \Pi(h,K,\kappa, c)$, see Definition \[2df\].
The proof of Theorem \[1tm\]
----------------------------
The proof is based on constructing a sequence with the property (\[18\]). Given $\mu_1, \mu_2 \in \mathcal{M}(\pi,h)$ with $\pi \in \Pi(h,K,\kappa, c)$, we take an arbitrary $\nu_0\in \mathcal{C}(\mu_1, \mu_2 )$ and construct $\nu \in \mathcal{C}(\mu_1, \mu_2 )$ by applying the mapping defined in (\[16\]) to the initial $\nu_0$ with $\ell$ running over the set ${\sf L}$. Each time we use the estimates derived in Lemma \[R2lm\]. Then the first two elements of the sequence in question are set $\hat{\nu}_0 = \nu_0$ and $\hat{\nu}_1 = \nu$. Afterwards, we produce $\hat{\nu}_2$ from $\hat{\nu}_1$, etc.
Recall that the underlying graph is supposed to have the property defined in (\[1\]) and $\chi\leq \varDelta$ is its chromatic number. Set $$\label{26a}
A = \frac{2 \varDelta^{\chi +1}}{1 - \bar{\kappa}}.$$ Then, for $K>K_*$, see (\[K\]), the following holds $$\label{26b}
K^{-1} < \frac{\bar{c}(1 - \bar{\kappa})}{4 \varDelta^{\chi+1}}, \qquad A K^{-1} < \bar{c}/2.$$
\[R3lm\] For $K> K_*$, take $\pi \in \Pi(h,K,\kappa, c)$ and $\mu_1 , \mu_2
\in \mathcal{M}(\pi, h)$. Then for each $\nu_0 \in
\mathcal{C}(\mu_1 , \mu_2)$ there exists $\nu \in \mathcal{C}(\mu_1
, \mu_2)$ for which the following estimates hold $$\begin{gathered}
\label{27}
\gamma(\nu) \leq \left[\bar{\kappa} + AK^{-1} \right] \gamma(\nu_0) + 2 A K^{-1} \lambda (\nu_0),\\[.3cm]
\label{28}
\lambda (\nu) \leq \Delta^{\chi-1} \gamma(\nu_0) + \bar{c} \varDelta^\chi \lambda(\nu_0).\end{gathered}$$
The proof of the lemma will be given in the subsequent parts of the paper. .1cm [*Proof Theorem \[1tm\]:*]{} As already mentioned, we let $\hat{\nu}_1\in \mathcal{C}(\mu_1, \mu_2)$ and $\hat{\nu}_0\in \mathcal{C}(\mu_1, \mu_2)$ be the measures on the left-hand sides and right-hand sides of (\[27\]) and (\[28\]), respectively. Then we apply to $\hat{\nu}_1$ the same reconstruction procedure and obtain $\hat{\nu}_2 \in \mathcal{C}(\mu_1, \mu_2)$, for which both estimates (\[27\]), (\[28\]) hold with $\hat{\nu}_1$ on the right-hand sides. We repeat this due times and obtain $\hat{\nu}_n \in \mathcal{C}(\mu_1, \mu_2)$ such that $$\label{28a}
\left(\begin{array}{ll} \gamma(\hat{\nu}_n)\\[.3cm] \lambda (\hat{\nu}_n) \end{array}
\right) \leq\left[ M(K) \right]^n \left(\begin{array}{ll} \gamma(\nu_0)\\[.3cm] \lambda (\nu_0) \end{array}
\right),$$ where $M(K)$ is the matrix defined by the right-hand sides of (\[27\]) and (\[28\]). Its spectral radius is $$\label{srM}
r_K = \frac{1}{2}\left[\bar{\kappa} + A K^{-1} + \bar{c}\varDelta^\chi + \sqrt{(\bar{\kappa} + A K^{-1} - \bar{c}\varDelta^\chi)^2 +
8 \varDelta^{\chi} AK^{-1}} \right].$$ For $K>K_*$, see (\[K\]), we have $r_K < 1$, which by (\[28a\]) yields (\[18\]) and thereby completes the proof.
The proof of Theorem \[2tm\]
----------------------------
The proof of this theorem is based on the version of the estimates in Lemma \[R3lm\] obtained in a subset ${\sf D}\subset {\sf L}$. For such $\sf D$, we define $$%\label{dc1}
\partial {\sf D} = \{\ell' \in {\sf D}^c: \partial \ell' \cap {\sf D}\neq \emptyset\},$$ which is the external boundary of ${\sf D}$. For $\nu \in \mathcal{P}(X^2)$ such that all $H^{i}_\ell$, $i=1,2$, $\ell \in {\sf D}\cup \partial {\sf D}$ are $\nu$-integrable, see (\[19\]), we set, cf. (\[18\]) and (\[26\]), $$\label{dc2}
\gamma_{\sf D} (\nu) = \sup_{\ell \in {\sf D}} \nu(I_\ell), \qquad \lambda_{\sf D}(\nu) = \max_{i=1,2}\sup_{\ell_1, \ell_2 \in {\sf D}}
\nu(I_{\ell_1} H^{i}_{\ell_2}).$$ Next, for $\ell_1$ as in (\[dc\]) and $N= \delta (\ell_1 , \ell_2)$, we set $$%\label{dc4}
{\sf D}_0 = \{\ell_1\}, \quad {\sf D}_{k} = {\sf D}_{k-1} \cup \partial {\sf D}_{k-1}, \quad k=1, \dots , N-1.$$ Let $\mu^x(\cdot)$ denote the conditional measure $\mu
(\cdot|\mathcal{F}_{{\sf D}_{N-1}})(x)$. For brevity, we say that $\nu^x \in \mathcal{P}(X^2)$ is $\mathcal{F}_{{\sf
D}_{N-1}}$-measurable if the maps $x\mapsto \nu^x(B)$ are $\mathcal{F}_{{\sf D}_{N-1}}$-measurable for all $B\in
\mathcal{B}(X^2)$. Clearly, $\nu_0^x = \mu^x \otimes \mu$ possesses this property. The version of Lemma \[R3lm\] which we need is the following statement.
\[dclm\] Let $\pi$, $K$, and $\mu$ be as in Theorem \[2tm\] and $\nu_0^x =
\mu^x \otimes \mu$. Then there exist $\nu_1^x, \dots , \nu_{N-1}^x
\in \mathcal{C}(\mu^x, \mu)$, all $\mathcal{F}_{{\sf
D}_{N-1}}$-measurable, such that for the parameters defined in (\[dc2\]) the following estimates hold $$\label{dc3}
\left(\begin{array}{ll} \gamma_{{\sf D}_{N-s-1}}(\nu^x_{s})\\[.3cm] \lambda_{{\sf D}_{N-s-1}} (\nu^x_s) \end{array}
\right) \leq M(K) \left(\begin{array}{ll} \gamma_{{\sf D}_{N-s}}(\nu^x_{s-1})\\[.3cm]
\lambda_{{\sf D}_{N-s}} (\nu^x_{s-1}) \end{array}
\right),$$for all $s=1, \dots , N-1$ and $\mu$-almost all $x\in X$.
.1cm [*Proof of Theorem \[2tm\]:*]{} Since $g$ is $\mathcal{F}_{{\sf D}_{N-1}}$-measurable, we have $$%\label{dc5}
\int_{X} f(x) g(x) \mu(dx) = \int_{X} g(x)\left(\int_X f(y) \mu^x(d y) \right)\mu(dx),$$ which yields $$\label{dc6}
{\rm Cov}_\mu (f;g) = \int_X g(x) \Phi(x)\mu(dx),$$ where $$\begin{aligned}
\label{dc7}
\Phi (x) = \int_{X^2} \left( f(y)-f(z)\right)\mu^x (dy) \mu(dz).\end{aligned}$$ For each $\nu^x_s$, $s=0, \dots , N-1$, as in Lemma \[dclm\], we then have $$\label{dc8}
\Phi(x) = \int_{X^2} \left( f(y)-f(z)\right) \nu^x_s(d y, dz),$$ and hence $$\label{dc9}
\left\vert \Phi(x) \right\vert \leq 2 \|f\|_{\infty} \nu^x_{N-1}(I_{\ell_1}) = 2 \|f\|_{\infty} \gamma_{{\sf D}_0} (\nu^x_{N-1}).$$ Note that the function defined in (\[dc7\]), (\[dc8\]) is related to the quantity which characterizes mixing in state $\mu$, cf. [@Ku Proposition 2.5].
Let $v_s$ and $v_{s-1}$ denote the column vector on the left-hand and right-hand sides of (\[dc3\]), respectively. Set $$%\label{dc10}
\xi = \frac{\varDelta^{\chi-1}}{r_K - \bar{c}\varDelta^\chi} = \frac{r_K - \bar{\kappa} - AK^{-1}}{2 AK^{-1}} >0,$$ and let $T$ be the $2\times 2$ diagonal matrix with $T_{11} = \xi$ and $T_{22} =1$. Then the matrix $$\label{dc11}
\widetilde{M}(K) := T M(K) T^{-1},$$ cf. [@BL Corollary 2.9.4, page 102], is positive and such that both its rows sum up to $r_K$. Set $\tilde{v}_s = T v_s$ and let $\tilde{v}_s^{i}$, $i=1,2$, be the entries of $\tilde{v}_s$. By (\[dc3\]) we then get $$\|\tilde{v}_s\|:= \max\{\tilde{v}_s^{1}; \tilde{v}_s^{2}\} \leq \|\widetilde{M}(K)\| \|\tilde{v}_{s-1}\| = r_K
\max\{\tilde{v}_{s-1}^{1}; \tilde{v}_{s-1}^{2}\},$$ which yields $$\label{dc12}
\gamma_{{\sf D}_0} (\nu^x_{N-1}) \leq r_K^{N-1} \max\{ \gamma_{{\sf D}_{N-1}}(\nu^x_0); \xi^{-1} \lambda_{{\sf D}_{N-1}}(\nu^x_0)\}.$$ Applying this estimate in (\[dc9\]) and then in (\[dc6\]) we arrive at (\[dc\]) with, cf. (\[srM\]) and (\[14\]), $$\label{constan}
\alpha_K = - \log r_K, \qquad C_K = 2 r_K^{-1} \max\{1; \xi^{-1} \mu(h)\}.$$ Let us make now further comments on the above results and their proof.
- The mapping in (\[16\]), which is the main reconstruction tool, see Section \[4SEC\] below, was first introduced in another seminal paper by R. L. Dobrushin [@Dob]. In a rather general context, it was used in [@dl]. The main feature of this mapping, which was not pointed out in [@DobP], is the measurability of the coupling $\varrho_\ell^{x,y}$ in $(x,y)\in X^2$. A similar property of the couplings in Lemma \[dclm\] was crucial for the proof of Theorem \[2tm\].
- We avoid using ‘compactness’ of $h$, and hence the related topological properties of the single-spin space $\Xi$, by employing the $\mathfrak{L}$-topology, see Definition \[3df\].
- In contrast to the estimates obtained in [@DobP Lemma 5], our estimate in (\[28\]) is independent of $K$. The only constant in (\[27\]) is given explicitly in (\[26a\]). This allowed us to calculate explicitly the spectral radius (\[srM\]), which was then used to obtain the decay parameter $\alpha_K$, see (\[constan\]).
- The proof of Lemma \[dclm\] was performed in the spirit of the proof of Proposition 2.5 of [@Ku]. Our $\Phi(x)$ in (\[dc7\]), (\[dc8\]) can be used to prove a kind of mixing in state $\mu$. However, here we cannot estimate this function uniformly in $x$, and hence employ its measurable estimate (\[dc9\]) which is then integrated in (\[dc6\]).
- The transformation used in (\[dc11\]) allowed us to find explicitly the operator norm of $M(K)$ equal to its spectral radius $r_K$. This then was used to find in (\[dc12\]) the exact rate of the decay of correlations in $\mu$.
Proof of Lemmas \[R3lm\] and \[dclm\] {#4SEC}
=====================================
For the partition (\[3\]) of the set of vertices ${\sf L}$, which has the property (\[2\]), we set $$\label{29}
{\sf U}_j = \bigcup_{i=0}^j {\sf V}_i, \qquad {\sf W}_j = {\sf L}\setminus {\sf U}_j, \quad j=0, \dots , \chi-1.$$ The measure $\nu$ in (\[27\]), (\[28\]) will be obtained in the course of consecutive reconstructions with $\ell\in {\sf V}_j$. The first step is .1cm
Reconstruction over ${\sf V}_0$ {#sec:1}
-------------------------------
Let $\{\ell_1, \ell_2, \dots , \}$ be any numbering of the elements of ${\sf V}_0$. Set $$\label{30}
{\sf V}_0^{(n)} = \{\ell_1, \dots , \ell_n\}, \qquad \nu_0^{(n)} = R_{\ell_n} R_{\ell_{n-1}}\cdots R_{\ell_1} \nu_0, \quad n \in \mathbb{N}.$$ Our first task is to estimate $\nu_0^{(n)} (I_\ell)$. By claim (b) of Lemma \[Rlm\] we have that $$\label{31}
\nu_0^{(n)} (I_\ell) = \nu_0 (I_\ell), \qquad {\rm for} \ \ \ell \notin {\sf V}_0^{(n)}.$$ For $k\leq n$, by (\[2\]) and claim (b) of Lemma \[Rlm\], and then by (\[21\]) and (\[31\]), we have $$\begin{aligned}
\label{32}
\nu_0^{(n)} (I_{\ell_k}) = \nu_0^{(k)} (I_{\ell_k}) & \leq & \sum_{\ell \in \partial \ell_k} \kappa_{\ell_k\ell} \nu_0(I_{\ell}) +
K^{-1} \sum_{i=1,2} \sum_{\ell, \ell' \in \partial \ell_k} \nu_0 (I_\ell H^i_{\ell'})\nonumber \\[.2cm]
& \leq & \bar{\kappa} \gamma(\nu_0) + 2 \varDelta^2 K^{-1} \lambda (\nu_0),\end{aligned}$$ see also (\[9\]), (\[18\]), and (\[26\]).
Next we turn to estimating $\nu_0^{(n)} (I_\ell H^i_{\ell'})$. As in (\[31\]) we have $$%\label{33}
\nu_0^{(n)} (I_\ell H^i_{\ell'}) = \nu_0 (I_\ell H^i_{\ell'}) \qquad {\rm for} \ \ \ell, \ell' \notin {\sf V}_0^{(n)}.$$ For $k<m \leq n$, by claim (b) of Lemma \[Rlm\], and then by (\[22\]), (\[21\]), (\[32\]), and (\[23\]), we have $$\begin{aligned}
%\label{34}
& & \nu_0^{(n)} (I_{\ell_k} H^i_{\ell_{m}}) = \nu_0^{(m)} (I_{\ell_k} H^i_{\ell_{m}})
\leq \nu_0^{(k)}(I_{\ell_k}) + \sum_{\ell \in \partial \ell_m} c_{\ell_m \ell} \nu^{(k)}_0(I_{\ell_k}H^i_{\ell}) \nonumber \\[.2cm]
& & \qquad \leq \bar{\kappa} \gamma(\nu_0) + 2 \varDelta^2 K^{-1} \lambda (\nu_0) + \sum_{\ell \in \partial \ell_m} c_{\ell_m \ell}
\sum_{\ell' \in \partial \ell_k} \nu_0 (I_{\ell'} H^i_{\ell}) \nonumber \\[.2cm]
& & \qquad \leq \bar{\kappa} \gamma(\nu_0) + \left[\varDelta \bar{c} + 2 \varDelta^2 K^{-1} \right] \lambda (\nu_0).
\end{aligned}$$ For $k\leq n$, by (\[24\]) we have $$\begin{aligned}
%\label{35}
\nu_0^{(n)} (I_{\ell_k} H^i_{\ell_{k}}) = \nu_0^{(k)} (I_{\ell_k} H^i_{\ell_{k}}) & \leq &
\sum_{\ell \in \partial \ell_k} \nu_0(I_\ell) +
\sum_{\ell, \ell' \in \partial \ell_k} c_{\ell_k \ell'} \nu_0( I_\ell H^i_{\ell'}) \nonumber \\[.2cm]
& \leq & \varDelta \gamma(\nu_0) + \varDelta \bar{c} \lambda (\nu_0).
\end{aligned}$$ Next, for $m < k \leq n$, by (\[23\]) and (\[22\]) we have $$\begin{aligned}
%\label{36}
\nu_0^{(n)} (I_{\ell_k} H^i_{\ell_{m}})& = & \nu_0^{(k)} (I_{\ell_k} H^i_{\ell_{m}})
\leq \sum_{\ell \in \partial \ell_k} \nu_0^{(m)} (I_\ell H^i_{\ell_m}) \nonumber \\[.2cm]
& \leq & \sum_{\ell \in \partial \ell_k} \left( \nu_0 (I_\ell) + \sum_{\ell' \in \partial \ell_m}
c_{\ell_m \ell'} \nu_0 (I_{\ell} H^i_{\ell'}) \right) \nonumber \\[.2cm]
& \leq & \varDelta \gamma(\nu_0) + \varDelta \bar{c} \lambda (\nu_0).\end{aligned}$$ Now we consider the case where $k\leq n$ and $\ell \notin {\sf V}_0^{(n)}$. Then by (\[23\]) we have $$%\label{37}
\nu_0^{(n)} (I_{\ell_k} H^i_{\ell}) = \nu_0^{(k)} (I_{\ell_k} H^i_{\ell})
\leq \sum_{\ell' \in \partial \ell_k} \nu_0 (I_{\ell'} H^i_\ell) \leq \varDelta \lambda (\nu_0).$$ For $k\leq n$ and $\ell \notin {\sf V}_0^{(n)}$, we also have by (\[22\]) that $$\begin{aligned}
\label{38}
\nu_0^{(n)} (I_{\ell} H^i_{\ell_{k}}) = \nu_0^{(k)} (I_{\ell} H^i_{\ell_{k}}) & \leq & \nu_0 (I_\ell) + \sum_{\ell' \in \partial \ell_k}
c_{\ell_k \ell'} \nu_0 (I_\ell H^i_{\ell'}) \nonumber \\[.2cm]
& \leq & \gamma(\nu_0) + \bar{c} \lambda (\nu_0).\end{aligned}$$ Now let us consider the sequence $\{\nu_0^{(n)}\}_{n\in \mathbb{N}_0}$ defined in (\[30\]). By claim (b) of Lemma \[Rlm\] it stabilizes on local sets $B\in \mathcal{B}(X^2)$, and hence is convergent in the $\mathfrak{L}$-topology. Let $\nu_1$ be its limit. By Lemma \[2lm\] we have that $\nu_1 \in \mathcal{C}(\mu_1, \mu_2)$. At the same time, by (\[29\]), (\[31\]), and (\[32\]) it follows that $$\label{39}
\nu_1 (I_\ell) \leq \left\{\begin{array}{ll}
\bar{\kappa} \gamma(\nu_0) + 2 \varDelta^2 K^{-1} \lambda (\nu_0),& \qquad {\rm for} \ \ \ell\in {\sf V}_0 ; \\[.3cm]
\gamma(\nu_0), & \qquad {\rm for} \ \ \ell\in {\sf W}_0 .
\end{array} \right.$$ Similarly, by (\[32\]) – (\[38\]) we obtain $$\label{40}
\nu_1 (I_\ell H^{i}_{\ell'}) \leq \left\{\begin{array}{ll}
\varDelta \gamma(\nu_0) + \left[ \varDelta \bar{c} + 2 \varDelta^2 K^{-1} \right] \lambda(\nu_0), & \
\ell, \ell' \in {\sf V}_0;\\[.3cm]
\varDelta \lambda (\nu_0), & \
\ell \in {\sf V}_0, \ell' \in {\sf W}_0 ;\\[.3cm]
\gamma(\nu_0) + \bar{c} \lambda(\nu_0), & \ \ell\in {\sf W}_0, \ell' \in {\sf V}_0;\\[.3cm]
\lambda (\nu_0), & \
\ell, \ell' \in {\sf W}_0.
\end{array} \right.$$ These estimates complete the reconstruction over ${\sf V}_0$. .1cm
Reconstruction over ${\sf V}_j$: Proof of Lemma \[R3lm\] {#sec:2}
--------------------------------------------------------
Here we assume that $\nu_j$ satisfies the following estimates, cf. (\[39\]), where $A$ is as in (\[26a\]): $$\label{41}
\nu_j (I_\ell) \leq \left\{\begin{array}{ll}
\left[\bar{\kappa} + A K^{-1}\right] \gamma(\nu_0) + 2 A K^{-1} \lambda (\nu_0),& \qquad {\rm for} \ \ \ell\in {\sf U}_{j-1} ; \\[.3cm]
\gamma(\nu_0), & \qquad {\rm for} \ \ \ell\in {\sf W}_{j-1} .
\end{array} \right.$$ And also, cf. (\[40\]), $$\label{42}
\nu_j (I_\ell H^{i}_{\ell'}) \leq \left\{\begin{array}{ll}
\varDelta^j
\gamma(\nu_0) + \bar{c}\varDelta^{j+1} \lambda(\nu_0), & \
\ell, \ell' \in {\sf U}_{j-1};\\[.3cm]
\varDelta^j \lambda (\nu_0), & \
\ell \in {\sf U}_{j-1}, \ell' \in {\sf W}_{j-1} ;\\[.3cm]
j \gamma(\nu_0) + \bar{c} \lambda(\nu_0), & \ \ell\in {\sf W}_{j-1}, \ell' \in {\sf V}_{j-1};\\[.3cm]
\lambda (\nu_0), & \
\ell, \ell' \in {\sf W}_{j-1}.
\end{array} \right.$$ Since ${\sf W}_{\varDelta-1} = \emptyset$, see (\[29\]), for $j= \varDelta -1$ we have just the first lines in (\[41\]) and (\[42\]), which yields (\[27\]) and (\[28\]), respectively, and thus the proof of Lemma \[R3lm\]. Note that (\[39\]) agrees with (\[41\]) as $\varDelta^2 < A$, see (\[26a\]). Also (\[40\]) agrees with (\[42\]), which follows from the fact that $$\bar{c} \varDelta + 2 \varDelta^2 K^{-1} < \bar{c} \varDelta + A K^{-1} \leq \bar{c} \varDelta + \bar{c}/2 < \bar{c} \varDelta^2
\leq \bar{c} \varDelta^{j+1}, \quad j = 1, \dots \chi-1,$$ see (\[26a\]) and (\[26b\]).
Thus, our aim now is to prove that the estimates as in (\[41\]) and (\[42\]) hold also for $j+1$. Note that the last lines in these estimates follow by claim (b) of Lemma \[Rlm\]. As above, we enumerate ${\sf V}_j = \{\ell_1 , \ell_2 , \cdots \}$ and set $$\nu_j^{(n)} = R_{\ell_n}R_{\ell_{n-1}} \cdots R_{\ell_1}\nu_j.$$ For $k\leq n$, by (\[21\]) we have, cf. (\[32\]), $$\begin{aligned}
%\label{44}
\nu_j^{(n)} (I_{\ell_k}) & = & \nu_j^{(k)} (I_{\ell_k}) \leq
\sum_{\ell \in \partial \ell_k \cap {\sf U}_{j-1}}\kappa_{\ell_k\ell} \nu_j(I_\ell)
+ \sum_{\ell \in \partial \ell_k \cap {\sf W}_{j}}\kappa_{\ell_k\ell} \nu_j(I_\ell)\qquad \nonumber \\[.2cm]
& + & K^{-1} \sum_{i=1,2} \sum_{\ell, \ell' \in \partial \ell_k \cap {\sf U}_{j-1}} \nu_j (I_\ell H^i_{\ell'}) \nonumber \\[.2cm]
& + & K^{-1} \sum_{i=1,2} \sum_{\ell \in \partial \ell_k \cap {\sf U}_{j-1}} \sum_{ \ell' \in \partial \ell_k \cap {\sf W}_{j} } \nu_j (I_\ell H^i_{\ell'}) \nonumber \\[.2cm]
& + & K^{-1} \sum_{i=1,2} \sum_{\ell \in \partial \ell_k \cap {\sf W}_{j} }\sum_{ \ell' \in \partial \ell_k \cap {\sf U}_{j-1} } \nu_j (I_\ell H^i_{\ell'}) \nonumber \\[.2cm]
& + & K^{-1} \sum_{i=1,2} \sum_{\ell, \ell' \in \partial \ell_k \cap {\sf W}_{j}} \nu_j (I_\ell H^i_{\ell'}).\end{aligned}$$ Now we use the assumptions in (\[41\]) and (\[42\]) and obtain herefrom $$\begin{aligned}
\label{45}
\nu_j^{(n)} (I_{\ell_k}) & \leq & \left[\bar{\kappa} + K^{-1} \left( \bar{\kappa} A + 2 \varDelta^j \varDelta^2_j + 2 j \varDelta_j
\widetilde{\varDelta}_j \right) \right]\gamma(\nu_0) \\[.2cm]
& + &2 K^{-1} \left[\bar{\kappa} A + \bar{c} \varDelta^{j+1} \varDelta_j^2 + \varDelta^j \varDelta_j\widetilde{ \varDelta}_j \right. \nonumber \\[.2cm]
& + & \left. \bar{c} \varDelta_j\widetilde{ \varDelta}_j + \widetilde{ \varDelta}^2_j \right]\lambda (\nu_0) ,\nonumber\end{aligned}$$ where $$\varDelta_j := | \partial \ell_k \cap {\sf U}_{j-1}| , \qquad \widetilde{\varDelta}_j:= | \partial \ell_k \cap {\sf W}_{j}|.$$ To prove that $$\bar{\kappa} A + 2 \varDelta^j \varDelta^2_j + 2 j \varDelta_j
\widetilde{\varDelta}_j \leq A$$ see the first line in (\[41\]), we use (\[26a\]), take into account that $\varDelta \geq 2$ (hence, $j \leq \varDelta^j$, $j=1, 2 , \dots \chi -1$) and obtain $$2 \varDelta^j \varDelta^2_j + 2 j \varDelta_j
\widetilde{\varDelta}_j \leq 2 \varDelta^j \varDelta_j\left( \varDelta_j + \widetilde{\varDelta}_j (j/\varDelta^j)\right)\leq 2 \varDelta^{j+2}
\leq A (1 - \bar{\kappa}),$$ where we have taken into account that $j+2 \leq \chi+1$, see (\[26a\]). To prove that the coefficient at $\lambda (\nu_0)$ in (\[45\]) agrees with that in (\[41\]) we use the following estimates $$\begin{aligned}
& & \bar{c} \varDelta^{j+1} \varDelta_j^2 + \varDelta^j \varDelta_j\widetilde{ \varDelta}_j +
\bar{c} \varDelta_j\widetilde{ \varDelta}_j + \widetilde{ \varDelta}^2_j \\[.2cm]
& & \quad = \bar{c} \varDelta^{j+1} \varDelta_j \left( \varDelta_j + \widetilde{\varDelta}_j \varDelta^{-j}\right) +
\varDelta^{j}\widetilde{\varDelta}_j \left( \varDelta_j + \widetilde{\varDelta}_j \varDelta^{-(j+1)}\right)\\[.2cm]
& & \quad \leq \varDelta^2 + \varDelta^{j+2} \leq 2\varDelta^{j+2}
\leq A (1 - \bar{\kappa}).\end{aligned}$$ For $\ell \in {\sf U}_{j-1}$, $\nu_j^{(n)}(I_\ell) = \nu_j (I_\ell)$ and hence obeys the first line of (\[41\]). For $\ell \in {\sf W}_{j}$, again $\nu_j^{(n)}(I_\ell) = \nu_j (I_\ell)$ and hence obeys the second line of (\[41\]). Here we also used that $\bar{c} < 1/\varDelta^\chi$ and $j+1\leq \chi$, see (\[13\]). Thus, (\[41\]) with $j+1$ holds true.
Now we turn to estimating $\nu_j^{(n)} (I_{\ell} H^i_{\ell'})$. In the situation where $\ell, \ell' \in {\sf U}_{j-1}\cup {\sf W}_j$, we have that $\nu_j^{(n)} (I_{\ell} H^i_{\ell'}) = \nu_j (I_{\ell} H^i_{\ell'})$ and hence obeys (\[42\]). Let us consider first the cases where only one vertex of $\ell, \ell'$ lies in ${\sf V}_j$.
For $\ell' \in {\sf U}_{j-1}$ and $k \leq n$, by (\[23\]) and the first and third lines in (\[42\]) we obtain $$\begin{aligned}
%\label{46}
& & \nu_j^{(n)} (I_{\ell_k} H^i_{\ell'}) = \nu_j^{(k)} (I_{\ell_k} H^i_{\ell'})
\leq \sum_{\ell \in \partial \ell_k \cap {\sf U}_{j-1}}\nu_j (I_{\ell} H^i_{\ell'})
+ \sum_{\ell \in \partial \ell_k \cap {\sf W}_{j}}\nu_j (I_{\ell} H^i_{\ell'}) \nonumber \\[.2cm]
& & \quad \leq \left[\varDelta^j \varDelta_j + j \widetilde{ \varDelta}_j \right] \gamma(\nu_0) +
\left[\bar{c}\varDelta^{j+1} \varDelta_j + \bar{c} \widetilde{ \varDelta}_j \right] \lambda(\nu_0)\nonumber \\[.2cm]
& & \quad \leq \varDelta^{j+1} \gamma(\nu_0) + \bar{c} \varDelta^{j+2}\lambda(\nu_0),
\end{aligned}$$ which yields the first line in (\[42\]) with $j+1$.
For $\ell' \in {\sf W}_{j}$ and $k \leq n$, by (\[23\]) and the second and fourth lines in (\[42\]) it follows that $$\begin{gathered}
%\label{47}
\nu_j^{(n)} (I_{\ell_k} H^i_{\ell'}) = \nu_j^{(k)} (I_{\ell_k} H^i_{\ell'}) \leq \sum_{\ell \in \partial \ell_k \cap {\sf U}_{j-1}}\nu_j (I_{\ell} H^i_{\ell'}) + \sum_{\ell \in \partial \ell_k \cap {\sf W}_{j}}\nu_j (I_{\ell} H^i_{\ell'}) \nonumber \\[.2cm]
\leq
\left(\varDelta^j \varDelta_j + \widetilde{ \varDelta}_j \right) \lambda(\nu_0) \leq \varDelta^{j+1} \lambda(\nu_0),
\end{gathered}$$ which agrees with the second line in (\[42\]).
For $\ell \in {\sf U}_{j-1}$ and $k \leq n$, by (\[22\]) and the first and second lines in (\[42\]) we get $$\begin{aligned}
%\label{48}
& & \nu_j^{(n)} (I_{\ell} H^i_{\ell_k}) = \nu_j^{(k)} (I_{\ell} H^i_{\ell_k}) \leq \nu_j (I_\ell) +
\sum_{\ell' \in \partial \ell_k \cap {\sf U}_{j-1}}c_{\ell_k\ell'}\nu_j (I_{\ell} H^i_{\ell'}) \nonumber \\[.2cm]
& & \quad + \sum_{\ell' \in \partial \ell_k \cap {\sf W}_{j}} c_{\ell_k\ell'}\nu_j (I_{\ell} H^i_{\ell'}) \leq
\left[ \bar{\kappa} + A K^{-1} \right] \gamma(\nu_0) \\[.2cm]
& & \quad + 2A K^{-1} \lambda (\nu_0) +
\left[\varDelta^j \gamma(\nu_0) + \bar{c} \varDelta^{j+1} \lambda (\nu_0) \right]
\sum_{\ell' \in \partial \ell_k \cap {\sf U}_{j-1}}c_{\ell_k\ell'} \nonumber \\[.2cm] & & \quad +
\varDelta^j \lambda (\nu_0) \sum_{\ell' \in \partial \ell_k \cap {\sf W}_{j}} c_{\ell_k\ell'}. \nonumber
\end{aligned}$$ In order for this to agree with the first line in (\[42\]), it is enough that the following holds $$\begin{aligned}
\label{48a}
& &\qquad \bar{\kappa} + AK^{-1} + \varDelta^{j} \sum_{\ell' \in \partial \ell_k \cap {\sf U}_{j-1}}c_{\ell_k\ell'} \leq \varDelta^{j+1}, \\[.2cm]
& & 2 AK^{-1} + \bar{c} \varDelta^{j+1} \sum_{\ell' \in \partial \ell_k \cap {\sf U}_{j-1}}c_{\ell_k\ell'} +
\varDelta^{j} \sum_{\ell' \in \partial \ell_k \cap {\sf W}_{j}}c_{\ell_k\ell'} \leq \bar{c} \varDelta^{j+2}. \nonumber
\end{aligned}$$ Recall that we assume $\varDelta \geq 2$. By (\[26b\]) and (\[13\]) we get that the left-hand side of the first line in (\[48a\]) does not exceed $$\bar{\kappa} + \bar{c}/2 + \varDelta^{-1} < 2 < \varDelta^{j+1}, \qquad {\rm for} \ \ j=1, \dots , \chi-1.$$ Likewise, the left-hand side of the second line in (\[48a\]) does not exceed $$\bar{c} + \bar{c} + \bar{c} \varDelta^{j} \leq \bar{c}( 2 + \varDelta^{j} ) < \bar{c} \varDelta^{j+2} \qquad {\rm for} \ \ j=1, \dots , \chi-1.$$ For $\ell \in {\sf W}_{j}$ and $k \leq n$, by (\[22\]) and the third and fourth lines in (\[42\]) we get $$\begin{aligned}
\label{49}
& & \nu_j^{(n)} (I_{\ell} H^i_{\ell_k}) = \nu_j^{(k)} (I_{\ell} H^i_{\ell_k})
\leq \nu_j (I_\ell) \nonumber \\[.2cm] & & + \sum_{\ell' \in \partial \ell_k \cap {\sf U}_{j-1}}c_{\ell_k\ell'}\nu_j (I_{\ell} H^i_{\ell'})
+ \sum_{\ell' \in \partial \ell_k \cap {\sf W}_{j}} c_{\ell_k\ell'}\nu_j (I_{\ell} H^i_{\ell'}) \\[.2cm]
& & \leq \gamma(\nu_0) + \left[j \gamma(\nu_0) +
\bar{c} \lambda (\nu_0)\right] \sum_{\ell' \in \partial \ell_k \cap {\sf U}_{j-1}}c_{\ell_k\ell'}
+ \lambda (\nu_0) \sum_{\ell' \in \partial \ell_k \cap {\sf W}_{j}} c_{\ell_k\ell'} \nonumber \\[.2cm]
& & \leq (1 + j \bar{c}) \gamma(\nu_0) + \left( \bar{c}\sum_{\ell' \in \partial \ell_k \cap {\sf U}_{j-1}}c_{\ell_k\ell'} +
\sum_{\ell' \in \partial \ell_k \cap {\sf W}_{j}}c_{\ell_k\ell'} \right) \lambda(\nu_0), \nonumber
\end{aligned}$$ which clearly agrees with the third line in (\[42\]).
Now we consider the cases where both $\ell, \ell'$ lie in ${\sf V}_j$. For $k< m\leq n$, by first (\[22\]) and (\[23\]), and then by (\[21\]), we have $$\begin{aligned}
\label{50}
\nu_j^{(n)} (I_{\ell_k} H^i_{\ell_m}) & = & \nu_j^{(m)} (I_{\ell_k} H^i_{\ell_m}) \leq \nu_j^{(k)} (I_{\ell_k}) + \sum_{\ell'\in \partial \ell_m}
c_{\ell_m\ell'} \nu_j^{(k)}(I_{\ell_k}H^i_{\ell'}) \nonumber \\[.2cm]
& \leq &\sum_{\ell \in \partial \ell_k} \kappa_{\ell_k\ell}\nu_j (I_\ell) +
K^{-1} \sum_{s=1,2} \sum_{\ell, \ell'\in \partial \ell_k} \nu_j (I_\ell H^s_{\ell'}) \nonumber \\[.2cm]
& + & \sum_{\ell'\in \partial \ell_m}
c_{\ell_m\ell'} \sum_{\ell\in \partial \ell_k} \nu_j(I_{\ell}H^i_{\ell'}).\end{aligned}$$ The next step is to split the sums in (\[50\]) as it has been done in, e.g., (\[49\]), and then use (\[41\]) and (\[42\]). By doing so we get $$\begin{aligned}
%\label{51}
& & \nu_j^{(n)} (I_{\ell_k} H^i_{\ell_m})
\leq \left[ (\bar{\kappa} + A K^{-1}) \gamma (\nu_0) + 2A K^{-1} \lambda (\nu_0)\right]
\sum_{\ell \in \partial \ell_k \cap {\sf U}_{j-1}} \kappa_{\ell_k\ell} \nonumber \\[.2cm]
& & \quad + \gamma(\nu_0) \sum_{\ell \in \partial \ell_k \cap {\sf W}_{j}} \kappa_{\ell_k\ell} +
2 K^{-1} \varDelta_j^2 \left[\varDelta^j \gamma(\nu_0) + \bar{c}\varDelta^{j+1} \lambda (\nu_0) \right] \nonumber \\[.2cm]
& & \quad + 2K^{-1} \varDelta_j \widetilde{\varDelta}_j \left[ \varDelta^j \lambda (\nu_0) + j \gamma(\nu_0) + \bar{c}\lambda( \nu_0) \right]
+ 2 K^{-1} \widetilde{\varDelta}_j^2 \lambda (\nu_0)
\nonumber \\[.2cm]
& & \quad + \varDelta_j \left[\varDelta^j \gamma(\nu_0) +
\bar{c}\varDelta^{j+1} \lambda (\nu_0) \right] \sum_{\ell' \in \partial \ell_m\cap {\sf U}_{j-1}} c_{\ell_m\ell'} \nonumber \\[.2cm]
& & \quad +
\varDelta^j \varDelta_j \lambda (\nu_0) \sum_{\ell' \in \partial \ell_m\cap {\sf W}_{j}} c_{\ell_m\ell'} +
\widetilde{\varDelta}_j (j \gamma(\nu_0) + \bar{c} \lambda (\nu_0) ) \sum_{\ell' \in \partial \ell_m\cap {\sf U}_{j-1}}
c_{\ell_m\ell'} \nonumber \\[.2cm]
& & \quad + \widetilde{\varDelta}_j \lambda (\nu_0) \sum_{\ell' \in \partial \ell_m\cap {\sf W}_{j}} c_{\ell_m\ell'}.
\end{aligned}$$ In order for this to agree with the first line in (\[42\]), it is enough that the following two estimate hold $$\begin{aligned}
\label{51a}
& & (\bar{\kappa} + A K^{-1}) \sum_{\ell \in \partial \ell_k \cap {\sf U}_{j-1}} \kappa_{\ell_k\ell}
+ \sum_{\ell \in \partial \ell_k \cap {\sf W}_{j}} \kappa_{\ell_k\ell}
+ 2K^{-1} \varDelta_j^2 \varDelta^j \\[.2cm]
& & \quad +
2K^{-1} j \varDelta_j \widetilde{ \varDelta}_j + \varDelta_j \varDelta^j \sum_{\ell' \in \partial \ell_m\cap {\sf U}_{j-1}} c_{\ell_m\ell'} +
\widetilde{ \varDelta}_j \sum_{\ell' \in \partial \ell_m\cap {\sf U}_{j-1}} c_{\ell_m\ell'} \nonumber \\[.2cm] & & \quad \leq \varDelta^{j+1}\nonumber \\[.3cm]
\label{51b}
& & 2 A K^{-1} \sum_{\ell \in \partial \ell_k \cap {\sf U}_{j-1}} \kappa_{\ell_k\ell} + 2K^{-1} \varDelta_j^2 \bar{c} \varDelta^{j+1} +
2K^{-1} \varDelta_j \widetilde{\varDelta}_j \varDelta^{j} \\[.2cm] & & \quad +
2K^{-1} \bar{c} \varDelta_j \widetilde{\varDelta}_j + 2K^{-1} \widetilde{\varDelta}_j^2 + \bar{c} \varDelta_j \varDelta^{j+1}
\sum_{\ell' \in \partial \ell_m\cap {\sf U}_{j-1}} c_{\ell_m\ell'} \nonumber \\[.2cm]
& & \quad + \varDelta_j \varDelta^{j} \sum_{\ell' \in \partial \ell_m\cap {\sf W}_{j}} c_{\ell_m\ell'} + \bar{c} \widetilde{\varDelta}_j
\sum_{\ell' \in \partial \ell_m\cap {\sf U}_{j-1}} c_{\ell_m\ell'} \nonumber \\[.2cm]
& & + \quad \widetilde{\varDelta}_j
\sum_{\ell' \in \partial \ell_m\cap {\sf W}_{j}} c_{\ell_m\ell'} \leq \bar{c} \varDelta^{j+2}. \nonumber\end{aligned}$$ Taking into account that $\bar{\kappa} < 1$ and (\[26b\]) one can show that the left-hand side of (\[51a\]) does not exceed $$\begin{gathered}
1 + \bar{c}/2 + 2 K^{-1} \varDelta^j \varDelta_j \left( \varDelta_j + \widetilde{\varDelta}_j (j/\varDelta^j)\right) + \bar{c} \varDelta^{j+1}
\\[.2cm]\leq 1 + \bar{c}/2 + \bar{c}/2 + \bar{c} \varDelta^{j+1} < 2 + \frac{1}{\varDelta^{\chi}} < \varDelta^{j+1}.\end{gathered}$$ To prove (\[51b\]) we use (\[26b\]), (\[13\]), the inequality $\varDelta_j \widetilde{\varDelta}_j \leq \varDelta^2/4$, and perform the following calculations $$\begin{gathered}
{\rm LHS(\ref{51b})} \leq 2AK^{-1} \bar{\kappa} + \frac{1}{2} K^{-1} \varDelta^{j+2} + 2K^{-1} \left(\varDelta_j^2 +
\bar{c} \varDelta_j \widetilde{\varDelta}_j + \widetilde{\varDelta}^2_j\right) \\[.2cm]
+ \varDelta_j \sum_{\ell' \in \partial \ell_m\cap {\sf U}_{j-1}} c_{\ell_m\ell'} +
\varDelta^j \varDelta_j \sum_{\ell' \in \partial \ell_m\cap {\sf W}_{j}} c_{\ell_m\ell'} \\[.2cm]
+ \bar{c} \widetilde{\varDelta}_j \sum_{\ell' \in \partial \ell_m\cap {\sf U}_{j-1}} c_{\ell_m\ell'} +
\widetilde{\varDelta}_j \sum_{\ell' \in \partial \ell_m\cap {\sf W}_{j}} c_{\ell_m\ell'}\\[.2cm]
\leq \bar{c} + \frac{\bar{c} \varDelta^{j+2}}{8 \varDelta^{\chi+1}} + \frac{\bar{c} \varDelta^{2}}{2 \varDelta^{\chi+1}} + \bar{c} \varDelta^{j+1}
+ \bar{c} \varDelta < \bar{c} \varDelta^{j+2},\end{gathered}$$ which holds even for $j=1$, $\chi = 2$, and $\varDelta = 2$.
Next, for $k\leq n$, by (\[24\]) we have $$\label{52}
\nu_j^{(n)} (I_{\ell_k} H^i_{\ell_k})= \nu_j^{(k)} (I_{\ell_k} H^i_{\ell_k}) \leq
\sum_{\ell\in \partial \ell_k} \nu_j (I_\ell) + \sum_{\ell, \ell'\in \partial \ell_k} c_{\ell_k \ell'} \nu_j(I_\ell H^i_{\ell'})$$ As above, we split the sums in (\[52\]) and then use (\[41\]) and (\[42\]), and obtain $$\begin{aligned}
\label{53}
& & \nu_j^{(n)} (I_{\ell_k} H^i_{\ell_k}) \leq \varDelta_j \left[\bar{\kappa} + A K^{-1} \right] \gamma(\nu_0) +
\varDelta_j 2A K^{-1} \lambda (\nu_0) \nonumber\\[.2cm] & & \quad
+ \widetilde{\varDelta}_j \gamma(\nu_0) + \varDelta_j \left[ \varDelta^j \gamma(\nu_0) +
\bar{c} \varDelta^{j+1} \lambda(\nu_0) \right]\sum_{\ell' \in \partial \ell_k \cap {\sf U}_{j-1}} c_{\ell_k \ell'} \nonumber \\[.2cm]
& & \quad
+ \varDelta_j \varDelta^j\lambda (\nu_0) \sum_{\ell' \in \partial \ell_k \cap {\sf W}_{j}} c_{\ell_k \ell'} +
\widetilde{\varDelta}_j (j \gamma(\nu_0)
+ \bar{c} \lambda (\nu_0)) \sum_{\ell' \in \partial \ell_k \cap {\sf U}_{j-1}} c_{\ell_k \ell'} \nonumber \\[.2cm]
& & \quad
+ \widetilde{\varDelta}_j \lambda(\nu_0) \sum_{\ell' \in \partial \ell_k \cap {\sf W}_{j}} c_{\ell_k \ell'}.\end{aligned}$$ In order for this to agree with the first line in (\[42\]), it is sufficient that the following two inequalities hold $$\begin{aligned}
\label{53a}
& & \varDelta_j \left[\bar{\kappa} + A K^{-1} \right] + \widetilde{\varDelta}_j + \left( \varDelta^j \varDelta_j + j \widetilde{\varDelta}^j
\right) \sum_{\ell' \in \partial \ell_k \cap {\sf U}_{j-1}} c_{\ell_k \ell'} \leq \varDelta^{j+1}, \qquad \quad
\\[.3cm]
\label{53b}
& & 2 A K^{-1} \varDelta_j + \bar{c} \varDelta^{j+1} \varDelta_j \sum_{\ell' \in \partial \ell_k \cap {\sf U}_{j-1}} c_{\ell_k \ell'}
+ \varDelta^{j} \varDelta_j \sum_{\ell' \in \partial \ell_k \cap {\sf W}_{j}} c_{\ell_k \ell'}\\[.2cm]
& & \quad + \bar{c} \widetilde{\varDelta_j} \sum_{\ell' \in \partial \ell_k \cap {\sf U}_{j-1}} c_{\ell_k \ell'} +
\widetilde{\varDelta}_j \sum_{\ell' \in \partial \ell_k \cap {\sf W}_{j}} c_{\ell_k \ell'} \leq \bar{c} \varDelta^{j+2}. \nonumber\end{aligned}$$ By means of (\[26b\]) we get $$\begin{gathered}
{\rm LHS(\ref{53a})} \leq \varDelta + \varDelta AK^{-1} + \bar{c} \varDelta^{j+1} < \varDelta + \frac{1}{2 \varDelta^{\chi-1}} + 1 < \varDelta^{j+1}.\end{gathered}$$ Similarly, $$\begin{gathered}
{\rm LHS(\ref{53b})} \leq \bar{c} \varDelta_j + \varDelta_j \sum_{\ell' \in \partial \ell_k \cap {\sf U}_{j-1}} c_{\ell_k \ell'} +
\varDelta^{j} \varDelta_j \sum_{\ell' \in \partial \ell_k \cap {\sf W}_{j}} c_{\ell_k \ell'} \\[.2cm]
+ \bar{c}\widetilde{\varDelta}_j \sum_{\ell' \in \partial \ell_k \cap {\sf U}_{j-1}} c_{\ell_k \ell'}
+\widetilde{\varDelta}_j \sum_{\ell' \in \partial \ell_k \cap {\sf W}_{j}} c_{\ell_k \ell'} \\[.2cm]
\leq \bar{c} \varDelta + \bar{c} \varDelta^j \varDelta_j + \bar{c} \widetilde{\varDelta}_j < \bar{c} \varDelta + \bar{c} \varDelta^{j+1} \leq
\bar{c} \varDelta^{j+2}.\end{gathered}$$ Now we consider the case where $ m <k \leq n$. By (\[23\]), and then by (\[22\]), we have $$\begin{aligned}
\label{54}
\nu_j^{(n)} (I_{\ell_k} H^i_{\ell_m})& = & \nu_j^{(k)} (I_{\ell_k} H^i_{\ell_m}) \leq \sum_{\ell \in \partial \ell_k} \nu_j^{(m)} (I_\ell H^i_{\ell_m}) \\[.2cm]
& \leq & \sum_{\ell \in \partial \ell_k} \nu_j(I_\ell) + \sum_{\ell \in \partial \ell_k} \sum_{\ell' \in \partial \ell_m}c_{\ell_m\ell'}
\nu_j (I_{\ell} H^i_{\ell'}). \nonumber\end{aligned}$$ Again we split the sums in (\[54\]) and then use (\[41\]) and (\[42\]), and obtain that $$\nu_j^{(n)} (I_{\ell_k} H^i_{\ell_m}) \leq {\rm RHS}(\ref{53}).$$ Thus, we have that (\[42\]) with $j+1$ holds also in this case. The proof is complete.
The proof of Lemma \[dclm\]
---------------------------
Assume that we have given $\nu^x_{s-1}\in \mathcal{C}(\mu^x, \mu)$ with the properties in question. Then we split ${\sf D}_{N-s-1}$ into independent subsets by taking intersections with the sets ${\sf V}_j$, as in (\[3\]). Let $\ell^1 , \dots , \ell^m$ be a numbering of ${\sf D}_{N-s-1}\cap {\sf V}_0$. Set $$\tilde{\nu}^x_0 = \nu^x_{s-1} \quad {\rm and} \quad \tilde{\nu}^x_k = R_{\ell^k} \tilde{\nu}^x_{k-1}, \qquad k=1, \dots , m,$$ where $R_\ell$ is defined in (\[16\]). Thus, $\tilde{\nu}_m^x$ is $\mathcal{F}_{{\sf D}_{N-1}}$-measurable, and $\tilde{\nu}_m^x (I_\ell)$ and $\tilde{\nu}_m^x (I_\ell H^i_{\ell'})$, $\ell, \ell' \in {\sf D}_{N-s-1}$, satisfy the inequalities in (\[39\]) and (\[40\]), respectively, in which the right-hand sides contain $\gamma_{{\sf D}_{N-s}} (\nu^x_{s-1})$ and $\lambda_{{\sf D}_{N-s}} (\nu^x_{s-1})$. Then we perform the reconstruction over the remaining independent subsets of ${\sf D}_{N-s-1}$ and obtain an element of $\mathcal{C}(\mu^x, \mu)$, which we denote by $\nu^x_s$. Its $\mathcal{F}_{{\sf D}_{N-1}}$-measurability is then guarateed by construction, and the parameters $\gamma_{{\sf D}_{N-s-1}} (\nu^x_{s})$ and $\lambda_{{\sf D}_{N-s-1}} (\nu^x_{s})$ satisfy the first-line inequalities in (\[41\]) and (\[42\]), respectively, and hence (\[dc3\]) with $\gamma_{{\sf D}_{N-s}} (\nu^x_{s-1})$ and $\lambda_{{\sf D}_{N-s}} (\nu^x_{s-1})$ on the right-hand side. The $\mathcal{F}_{{\sf D}_{N-1}}$-measurability of $\nu_0^x$ is straightforward.
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abstract: 'We propose and experimentally demonstrate a new spectroscopic method, image-charge detection, for the Rydberg states of surface electrons on liquid helium. The excitation of the Rydberg states of the electrons induces an image current in the circuit to which the electrons are capacitively coupled. In contrast to the conventional microwave absorption measurement, this method makes it possible to resolve the transitions to high-lying Rydberg states of the surface electrons. We also show that this method can potentially be used to detect quantum states of a single electron, which paves a way to utilize the quantum states of the surface electrons on liquid helium for quantum computing.'
author:
- Erika Kawakami
- Asem Elarabi
- Denis Konstantinov
title: 'Image-charge detection of the Rydberg states of surface electrons on liquid helium'
---
Surface electrons (SE) above liquid helium constitute an exquisitely pure quantum system which for a long time served as a unique experimental platform to discover interesting many-electron phenomena [@Andrei-book]. The quantized (Rydberg) states of SE with a hydrogen-like energy spectrum $E_n=-R_e/n^2$, where $R_e\sim 10^{-3}$ eV and $n$ is a positive integer number, are formed due to the attractive interaction between an electron and its image charge inside the liquid, as well as a strong repulsive barrier experienced by the electron at the vapor-liquid interface. For typical experimental temperatures below 1 K, electrons occupy the ground state and are localized about 10 nm above the surface. The higher-energy Rydberg states can be excited by millimeter-waves ($\sim 100$ GHz) radiation. Grimes and Brown first measured the transitions from the ground ($n=1$) to the low-lying excited states ($n=2\sim 6$) by detecting the change in the microwave (MW) absorption caused by the excitation of the Rydberg states of SE using a cryogenic bolometer [@GrimesPRL1974]. A renewed interest in the Rydberg states of SE has emerged from their potential as qubit states [@PlatzDykm1999; @DykmPRB2003], which was followed by several other proposals to use either orbital or spin states of SE as qubit states [@LyonPRA2006; @SchuPRL2010]. A crucial point for successful qubit implementation is the ability to manipulate and detect quantum states of a single electron. For such an application, the conventional MW absorption measurement is inappropriate as a detection of the quantum state because it is applicable only for a sufficiently large number of electrons [@GrimesPRL1974; @LambPRL1980; @CollPRL2002], and so is the indirect detection of the Rydberg transitions via the measurement of SE conductivity [@VolJETP1981; @KonsPRL2007]. Originally, a destructive readout of the Rydberg states was proposed, in which the electrons leave the liquid surface depending on the occupied Rydberg states [@PlatzDykm1999; @WillPRL1971]. An interesting and promising idea is to use the strong coupling of a single electron to a superconducting resonator to realize a non-destructive readout of electron quantum states [@SchuPRL2010; @GePRX2016]. However, this method is limited to a low transition frequency which should match the frequency of the coplanar resonator ($\sim 5$ GHz), thus it is not applicable for the detection of the excitation of the Rydberg states.
Here, we propose and demonstrate a new spectroscopic method, image-charge detection, for the Rydberg states of SE on liquid helium. Our method makes use of the fact that, as an electron is excited from the ground state to a higher excited state, the electron wave function spreads farther away from the liquid surface and its average distance from the liquid surface increases. Thus, when the system is placed near an electrode aligned parallel to the liquid surface, the excitation of SE causes a change in the image charge induced in the electrode by SE. In our experiment, the method is demonstrated with MW-excited SE placed in a parallel-plate capacitor. By measuring the image current in the capacitor, we managed to detect the excitation to the high-lying Rydberg states which were unable to be measured with the above-mentioned conventional methods. We also discuss an alternative detection scheme where the electric susceptibility of the MW-excited SE is detected as a relative change in the capacitance, a technique which can be scaled down to detection of the excitation of a single electron.
![(color online) Principle diagram of the excited-state-population detection of the electrons trapped on the surface of liquid helium (light blue circles) placed between top plates and bottom plates of a capacitor C. The electrons that are excited to the first excited state (green circles) are elevated above a charged layer of the ground-state SE by a distance $\Delta z_{2} \sim 35$ nm and forms an electric dipole $p=e\Delta z_{2} $. In the experiments shown here, an induced image current $i(t)$ was detected using a lock-in current amplifier by periodically varying $i$ by means of the modulated MW excitation. A suitable bias-tee is placed at room temperature for each capacitor plate (see text for more details).[]{data-label="fig:principle_diagram"}](fig1_modified.eps){width="8.5cm"}
The principle diagram of the image-charge detection is shown in Figure \[fig:principle\_diagram\]. SE are formed on the surface of liquid helium which are placed between two plates of a capacitor C. Here we use a parallel-plate capacitor (the area of each plate $S$ and the distance between the plates $D$). SE are confined on the surface of liquid helium by applying a positive dc bias $V_\textrm{dc}$ to the bottom capacitor plate. Owing to the linear Stark shift of the Rydberg levels caused by a dc electric field $E_\perp$ applied perpendicular to the surface, the transition frequency of SE, $\omega_{1n}(E_\perp)=(E_n - E_1)/\hbar$, can be adjusted to match with the microwave frequency $\omega_0$ by varying the value of $V_\textrm{dc}$. The MW-excited electrons are elevated above the charged layer of the ground-state electrons by a distance $\Delta z_{n} = z_{nn} - z_{11}$, where $z_{nn}$ is the average value of $z$-coordinate (counted from the surface) of an electron occupying the Rydberg state of index $n$. For the first excited state, $n=2$, this distance is $\Delta z_{2} \approx 35$ nm, and it increases with $n$. As a fraction $\rho_{nn}$ of SE is excited to the $n$-th Rydberg state, the image charge induced by SE in the top (bottom) capacitance plate changes by $\Delta q$ ($-
\Delta q$):
$$\Delta q = \frac{\Delta z_{n} }{D} e n_s \rho_{nn} S =\frac{\Delta z_{n} }{D} P_e S ,
\label{eq:deltaq}$$
where $e(>0)$ is the elementary charge and $n_s$ is the areal density of SE. Here, for later discussion, we introduce the quantity $P_e =n_s S \rho_{nn} p/ (S \Delta z_{n} )=en_s \rho_{nn}$, with $p=e \Delta z_n$ being the electric dipole moment of one electron. This quantity can be viewed as the electric dipole moment per unit volume of the electron system induced by the excitation of SE to the $n$-th excited state. The change in the image charge induces a current $i_{t,b}$ in the top (bottom) capacitor plate:
$$i_{t,b}(t)=\pm \frac{d\Delta q}{dt}.
\label{eq:i}$$
This current can be readily detected using, for example, a lock-in amplifier by periodically varying the fractional occupancy $\rho_{nn}$. The periodic variation of $\rho_{nn}$ can be realized by MW modulation or by the modulation of the voltage applied to the capacitor plates. In the latter case, $\rho_{nn}$ is determined by the detuning which is related to the electric field $E_\perp$ applied to SE by $\delta \omega = \omega_{1n}(E_\perp) - \omega_0$. This case is discussed later as a future experiment and the former is realized by applying a pulse-modulated MW excitation to the system in the experiment described here. Assuming the harmonic time-dependence of $\rho_{nn}$ at the pulse modulation frequency $\omega_m$: $\rho_{nn}=\rho_{nn}^{(0)} e^{i \omega_m t}$, the current amplitude can be estimated as
$$|i_{t,b}(t)| = \left| \frac{d\Delta q}{dt} \right| = \frac{ e n_s C_0\omega_m \Delta z_{n} \rho_{nn}^{(0)}}{ \varepsilon_0},
\label{eq:i_PM}$$
where $C_0=\varepsilon_0 S/D$ is the the capacitance of the parallel plate capacitor and $\varepsilon_0$ is the vacuum permittivity. For typical values of $n_s=10^8$ cm$^{-2}$, $C_0\sim 1$ pF, the modulation frequency $\omega_m/2\pi=100$ kHz, and $\rho_{nn}^{(0)}=10\%$, we obtain $|i_{t,b}(t)|\sim 100$ pA.
The experiment is carried out in a leak-tight copper cell cooled below 1 K in a dilution refrigerator and filled with condensed He gas. In the experiments shown here, we used liquid $^3$He. Similar results were obtained using liquid $^4$He. The helium liquid surface is set approximately midway between two plates of a parallel-plate capacitor formed by two round (diameter 24 mm) conducting plates separated by four quartz spacers of height $2$ mm. This height sets the gap $D$ between the capacitor plates. Using a parallel-plate capacitor with the gap smaller than used here is not practical because the helium liquid, which has the capillary length $\sim 0.5$ mm, would wet the top plate by the capillary rise and form a meniscus [@CollPRB2017]. Each plate of the capacitor consists of three concentric electrodes separated by two gaps of diameters 14 and 20 mm and width 0.2 mm. Electrons are produced by the thermionic emission from a tungsten filament placed near the capacitor and above the liquid surface, and electrons are attracted towards the liquid surface by applying a positive bias to all the three concentric electrodes of the bottom plate. After the SE system is formed on the surface above the bottom plate, the areal density of SE is determined by the voltage bias from the condition of complete screening of the electric field above the surface. After the system is formed, the voltage of the outer concentric electrode (guard) is set to a large negative value (typically -30 V) to confine SE above the two inner concentric electrodes and ensure that electrons do not escape to the grounded walls of the cell. The two inner electrodes of each plate constitute a Corbino disk which allows detection of SE by driving one of the electrodes with a small ac voltage and measuring the current induced in the other plate by the lateral motion of SE [@SommPRL1971]. MW radiation at a fixed frequency $\omega_0$ is introduced into the cell through a sealed rectangular single-mode waveguide with inner dimensions $1.6\times 0.8$ mm. In order to increase the area illuminated by MW, the waveguide gradually transforms to an overmoded size ($3.8\times 1.9$ mm) inside the cell (shown schematically in Fig. \[fig:principle\_diagram\]).
A resonant $1\rightarrow n$ transition between the Rydberg states of SE is excited by adjusting voltage $V_\textrm{dc}$ applied to the Corbino disk of the bottom plate to match $\omega_{1n}$ with $\omega_0$ via Stark shift. In the experiment, we employ on/off MW pulse modulation at frequency $\omega_m$ and measure the corresponding ac currents at the Corbino disk of either the bottom or top capacitor plates using a lock-in amplifier, while sweeping $V_\textrm{dc}$ through the resonance. The typical result of such measurements is shown in Fig. \[fig:2\] where we plot the in-phase component of the current measured by the lock-in amplifier at the modulation frequency $\omega_m/2\pi=250$ kHz. The sharp enhancement of current corresponds to the resonant $1\rightarrow 2$ transition of SE excited by the applied MW at frequency $\omega_0=140$ GHz. The resonance value of $V_\textrm{dc}$ corresponds to the electric field acting on the electrons $E_\perp=V_\textrm{dc}/D\approx 110$ V/cm, which agrees reasonably well with the calculated value $116$ V/cm for the Stark-shifted Rydberg levels assuming an infinitely-large barrier for the electrons at the vapor-liquid interface. Possible causes for this deviation, as well as for the deviation seen in Fig. \[fig:4\] discussed later, come from the approximate model used to calculate the Rydberg spectrum and the contribution to $E_\perp$ from the image charges induced by SE in the capacitor plates. The full width of the peak at its half-maximum is about $400$ MHz, which is significantly larger than the expected intrinsic linewidth $\gamma\approx 1$ MHz due to the elastic scattering of SE from ripplons [@IsshJLTP2007]. This width is determined by the inhomogeneous broadening of the transition energy due to the nonuniformity of $E_\perp$, which most likely arises from the misalignment between the capacitor plates and liquid surface.
An important feature of the result shown in Fig. \[fig:2\] is that the current signals measured at the top and bottom plates are nearly equal in magnitude but opposite in sign as expected from Eq. (\[eq:i\]). The inset of Fig. \[fig:2\] shows a nearly linear dependence of the magnitude of $i$ at the resonance for different values of $\omega_m$, as expected from Eq. (\[eq:i\_PM\]).
![(color online) The current signal measured at the bottom (red line) and top (blue line) capacitor plate for SE irradiated with pulse-modulated MW at frequency $\omega_0=140$ GHz. (Inset) The current signal at the resonance measured for different values of the pulse-modulation frequency $\omega_m$.[]{data-label="fig:2"}](fig2_old.eps){width="7.5cm"}
![(color online) The current signal measured at the top capacitor plate for SE irradiated with pulse-modulated MW at frequency $\omega_0/2\pi=140$ GHz for different MW power. Inset: The current signal at the resonance versus the normalized MW power.[]{data-label="fig:3"}](fig3.eps){width="7.5cm"}
Figure \[fig:3\] shows the dependence on MW power of the current signal measured at the top plate for the resonant $1 \rightarrow 2$ transition. The magnitude of the current at the resonance versus normalized power is shown in the inset of this figure. For a two-level system, the fractional occupancy $\rho_{22}$ is expected to increase linearly for small powers and saturate at the value $0.5$ at high powers. The measured dependence shows a deviation from such a dependence. It is known that in SE the two-level approximation fails for a high MW power due to the electron heating by MW radiation, which results in the thermal population of the Rydberg states which are not directly excited by MW [@KonsPRL2007]. Thus, the expression $\Delta z_{n} \rho_{nn}^{(0)}$ in Eq. (\[eq:i\_PM\]) becomes reformulated as $\sum_{k \geq n} \Delta z_{k} \rho_{kk}^{(0)}$. The data in Fig. \[fig:3\] shows that the resonance shifts towards lower values of $V_\textrm{dc}$ with increasing power. This is due to the Coulomb shift of the Rydberg levels caused by the electron-electron interaction [@KonsPRL2009]. Using the experimentally determined $\omega_{12}(V_\textrm{dc})\sim 2.1$ GHz/V, the estimated Coulomb shift is about 0.2 GHz for the highest applied MW power (red line in Fig. \[fig:3\]). Assuming the Boltzmann population for all the Rydberg states, the corresponding estimated electron temperature is $T_e\approx 5$ K [@KonsPRL2007]. The thermal population of the excited states with higher quantum numbers should further increase the image current $i$, thus a stronger rise of $i$ with $P$ should be expected at higher $P$.
The fact that the image current induced by the excitation of SE strongly increases with quantum number $n$ of the excited state suggests one of the important advantages of our method comparing with the conventional MW absorption measurement. The MW absorption due to the resonant $1\rightarrow n$ transition is proportional to the oscillator strength $f_{1n} \propto\hbar \omega_{1n} |\braket{1|z|n}|^2 $. The Thomas-Reiche-Kuhn sum rule, $\sum\limits_n f_{1n}=1$, states that the oscillator strength, and therefore the absorbed power, must decrease rapidly with increasing $n$ in order to ensure a convergence of the sum. Indeed, in a typical power absorption measurement, the measured signal decreases rapidly with increasing $n$ and for large $n$ the transitions become practically unobservable [@GrimPRB1976]. In our method, the rapid decrease of the $1\rightarrow n$ transition rate $|\braket{1|z|n}|^2$ with increasing $n$ is compensated by an increase of $\Delta z_n$. Figure \[fig:4\] shows the current signal measured at a fixed MW power as $V_\textrm{dc}$ is swept to zero, i.e., tuning transitions to higher-$n$ states to the resonance with the applied MW. We can clearly observe the transitions up to $n=14$. At lower $V_\textrm{dc}$, since the Stark shift of the Rydberg levels is smaller, the overlap between different transitions becomes larger and produces a smooth background, which makes it difficult to resolve each transition. However, this is not a fundamental limitation. High-lying Rydberg states can be resolved by increasing MW frequency $\omega_0$, thus shifting resonances to higher $V_\textrm{dc}$.
![(color online) The current signal measured at the bottom capacitor plate for SE irradiated with pulse-modulated MW at frequency $\omega_0/2\pi=200$ GHz while bias $V_\textrm{dc}$ applied to the bottom electrode is swept down to zero. The arrows indicate theoretical predictions for the transitions between the ground states and excited states. []{data-label="fig:4"}](fig4_modified.eps){width="7.5cm"}
In order to increase the sensitivity and the bandwidth of the image-charge detection, a number of improvements can be readily done, such as an employment of a cryogenic high-electron-mobility transistor (HEMT) amplifier [@VinkAPL2007; @AschRSI2002]. Another possibility, as mentioned earlier, is to apply a small ac voltage $u(t)$ to a capacitor plate in addition to the dc bias $V_\textrm{dc}$ to induce the periodic variation of the fractional occupancy $\rho_{nn}$. In this case, we can draw an analogy between the capacitance change induced by inserting a dielectric slab into a capacitor and that by the MW-excited SE. We introduce the electric field $\mathcal{E}=u/D$ due to the ac voltage applied to the capacitor and, for the sake of simplicity, ignore the effect of the electric field from the image charges induced by SE in the capacitor plates. By making expansion of $\rho_{nn}(u)=\rho_{nn}|_{u=0}+\frac{d\rho_{nn}}{d \mathcal{E}}|_{u=0}\mathcal{E}+\frac{d^2\rho_{nn}}{d \mathcal{E}^2}|_{u=0}\frac{\mathcal{E}^2}{2} +..$ , the electric dipole moment per unit volume of the electron system can be cast in the form $P_e = P_e^{(0)}+\epsilon_0 \chi_e \mathcal{E}$, where $\chi_e =\chi_e^{(1)}+\chi_e^{(2)}\mathcal{E}+..$ is the nonlinear ac electric susceptibility of the electron system. By inserting this form into Eq. (\[eq:deltaq\]), a current in the capacitor plate can be written as $$\begin{aligned}
i(t)& =&\frac{d(C_0 u(t) + \Delta q)}{dt} \label{eq:i_Vac_0} \nonumber \\
& =& C_0 \left(1+\eta \chi_e^{(1)}+ \eta \chi_e^{(2)} \frac{2 u(t)}{D}+.. \right)\frac{du(t) }{dt}
\label{eq:i_Vac}\end{aligned}$$ with $\eta =\Delta z_n/D$. The image current is caused by the electric susceptibility of the electron system induced by the MW-induced population of the excited states. The linear susceptibility is given by $\chi_e^{(1)}=-\frac{ en_s }{\varepsilon_0} \frac{d\rho_{nn}}{d \mathcal{E}}|_{u=0} $ and can be rewritten as $$\chi_e^{(1)}=-\frac{\alpha en_s }{\varepsilon_0} \left( \frac{\partial\rho_{nn}}{\partial\omega_{1n}} \right),
\label{eq:chi}$$ using an approximated linear dependence $\alpha=d\omega_{1n}/dE_\perp$. Eq. (\[eq:i\_Vac\]) states that the relative change in capacitance $\Delta C /C_0 =\eta \chi_e^{(1)} $ due to the linear part of $\chi_e$ is equivalent to inserting a dielectric slab with thickness $\Delta z_n $ and susceptibility $\chi_e^{(1)} $ and the nonlinear part of $\chi_e$ produces higher harmonic time-dependence. Note that, unlike the current $i$ which is proportional to the total number of electrons $N_e=n_s S$ in the capacitor, the relative change $\Delta C/C_0$ is proportional to the electron density $n_s$. Thus, the required number of the electrons for detection scales down with the size of the capacitor C employed for the detection. The sensitivity of the measurement can be improved, for example, by including the capacitor $C$ in an $LC$-circuit [@Gonzalez-ZalbaNatCom2015]. As mentioned earlier, employment of a parallel-plate capacitor of a much smaller size than used here is not practical due to the capillary action of liquid helium [@CollPRB2017]. It would be preferable to use, for example, a coplanar-plate capacitor covered by a thin helium film. For the detection of the quantum states of a single electron, the linear dimensions of such a capacitor can be reduced to $\mu$m-size. This would also give about three-order of magnitude enhancement in the value of $\eta$, which is inversely proportional to the capacitor gap $D$. Considering a capacitor with $S=10\times 10$ $\mu$m$^2$ and $D=1$ $\mu$m, and using experimentally determined $\alpha\sim 5$ MHz/(V/m), for a single electron we obtain $\Delta C/C_0\sim 30 (\partial\rho_{22}/\partial \omega_{1n})$, where $\omega_{1n}$ is in MHz. For a single electron localized in a dc electrostatic trap we can assume an intrinsic broadening of the transition line $\gamma\sim 1$ MHz, which gives us $\Delta C/C_0>> 1$. This crude estimate ignores experimental effects, such as stray capacitance, lateral motion of an electron in the trap, and inhomogenious broadening of the transition line. However, bearing in mind that employment of $LC$-circuits allows for determination of relative changes $\Delta C/C_0$ to part-per-million [@Gonzalez-ZalbaNatCom2015], the proposed scheme of detection is promising.
Once the detection of the excitation of the Rydberg state is realized for a single electron, we could transform it to a non-destructive readout of the spin state of a single electron with the help of a magnetic field gradient. A current running through a superconducting wire in the vicinity of the trapped electrons [@SchuPRL2010] or a ferromagnet under an external magnetic field [@TokuraPRL2006] can create a local magnetic field gradient. Thanks to the field gradient, the electron feels a different magnetic field depending on the Rydberg state, which allows spin-selective excitation of the Rydberg state. The detailed schematics for the detection of the spin state of a single electron and its manipulation are beyond the scope of this paper and will be discussed elsewhere.
In summary, we have proposed a new method, the image-charge detection, for spectroscopic study of the Rydberg states of surface electrons on liquid helium. The method is demonstrated by measuring the image current induced in the capacitor circuit by a pulse-modulated MW excitation. The method is simple, does not require expensive devices such as a cryogenic hot-electron (InSb) bolometer, and can be readily employed using a conventional lock-in amplifier. Moreover, it is demonstrated that the image-charge detection provides other advantages over the conventional methods, such as the ability to do spectroscopic studies of the high-lying Rydberg states. In addition to SE, this method can be potentially useful to study radiation-induced intersubband transitions in 2D electron gas in semiconductor heterostructures, where energy structure of electrons trapped at the interface of two solid materials has many similarities with that of SE on liquid helium [@AndoRevModPhys1982]. We also discussed the possibility of using this method to detect the spin state of a single electron, which opens a new pathway for using spin states of SE on liquid helium for quantum computing.
This work was supported by JST-PRESTO (Grant No. JPMJPR1762) and an internal grant from Okinawa Institute of Science and Technology (OIST) Graduate University. We are grateful to V. P. Dvornichenko for providing technical support.
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---
abstract: 'We studied the pairing instabilities in K$_x$Fe$_{2-y}$Se$_2$ using a two stage functional renormalization group (FRG) method. Our results suggest the leading and subleading pairing symmetries are nodeless $d_{x^2-y^2}$ and nodal extended $s$ respectively. In addition, despite having no Fermi surfaces we find the buried hole bands make important contributions to the final effective interaction. From the bandstructure, spin susceptibility and the FRG results we conclude that the low energy effective interaction in K$_x$Fe$_{2-y}$Se$_2$ is well described by a $J_1-J_2$ model with dominant nearest-neighbor antiferromagnetic interaction $J_1$ (at least as far as the superconducting pairing is concerned). In the end we briefly mention several obvious experiments to test whether the pairing symmetry is indeed $d_{x^2-y^2}$.'
author:
- |
Fa Wang$^{1}$, Fan Yang$^{2}$, Miao Gao$^{3}$, Zhong-Yi Lu$^{3}$,Tao Xiang$^{4,5}$, and Dung-Hai Lee$^{6,7}$\
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title: 'The Electron Pairing of K$_x$Fe$_{2-y}$Se$_2$'
---
Introduction
============
Very recently a new wave of excitements occurred in the field of iron-based superconductors. This is stirred up by the discovery of K$_x$Fe$_{2-y}$Se$_2$[@Kfese]. These compounds are isostructural to the “122”-family of iron pnictides, [e.g.]{} BaFe$_2$As$_2$, with the highest transition temperature $T_c\approx$ 33K among iron chalcogenides. The reason these compounds attracted considerable interests is not their “high $T_c$”. Rather it is because they have a very different electronic structure from all other iron-based superconductors. In particular, the hole pockets near the Brillouin zone center in all other iron-based superconductors are found to be absent in the highest $T_c$ K$_x$Fe$_{2-y}$Se$_2$[@dl; @ding]. The result of ref. [@dl] suggests a very uniform ($\approx 10$meV) superconducting gap around the electron pockets.
The absence of hole pockets is expected from valence count: KFe$_2$Se$_2$ should have $0.5$ doped electron per Fe relative to the parent iron pnictides. In view of the wide spread belief that the scattering between the hole and electron pockets is crucial for the high pairing scale in the iron-based superconductors[@mazin; @kuroki; @rg1; @seo; @rg2; @rg3; @rg4; @rg5; @rg6; @rg7; @rg8; @flex1; @flex2; @rpa1; @rpa2], it is surprising that a material without hole pocket can support such a high $T_c$. It is fair to say the relatively high pairing scale in the absence of hole pockets calls for a re-evaluation of the spin fluctuation pairing mechanism. This is so because in the other extreme KFe$_2$As$_2$, which has $0.5$ doped holes per Fe and only hole pockets, is a very low $T_c$ ($\approx 3K$)[@kfeas1; @kfeas2] superconductor with experimental evidences of gap nodes [@nodes].
In this paper we apply the functional renormalization group (FRG) method[@rg1; @rg3; @rg8] to study the pairing instability of K$_x$Fe$_2$Se$_2$. When the electron-electron interaction is weak compared with the bandwidth, this method is unbiased. It sums all virtual one-loop scattering processes including particle-hole, particle-particle and vertex corrections. As other iron-based superconductors, the strength of electron-electron interaction in K$_x$Fe$_{2-y}$Se$_2$ is uncertain. Ideally, we should combine FRG with the variational Monte-Carlo calculation[@fan], which is underway.
Model
=====
According to an earlier DFT result[@shein], KFe$_2$Se$_2$ has cylindrical electron pockets around $(\pi,0,k_z)$ and $(0,\pi,k_z)$ as well as a 3D electron pocket centered around $(0,0,\pi)$. In our study we uses a two dimensional(2D) five-band tight binding model to describe the $k_z=0$ plane of the bandstructure. It is obtained via the maximally localized Wannier function fit[@wannier] to our own DFT calculation. In our DFT calculation the plane wave basis method was used [@pwscf]. We adopted the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof formula [@pbe] for the exchange-correlation potentials. The ultrasoft pseudopotentials [@vanderbilt] were used to model the electron-ion interactions. After the full convergence test, the kinetic energy cut-off and the charge density cut-off of the plane wave basis were chosen to be 800eV and 6400 eV, respectively. The Gaussian broadening technique was used and a mesh of $18\times 18\times 9$ k-points were sampled for the Brillouin-zone integration. In the calculations, the experimental tetragonal lattice parameters were adopted, and the internal atomic coordinates within the cell were determined by the energy minimization.
Our bandstructure and the associated Fermi surfaces are shown in fig. \[band\].
![(Color online) The band structure and Fermi surfaces. Note that all the Fermi surfaces are electron like. []{data-label="band"}](fig1.eps)
Because experimentally there is uncertainty about the Fe and K content, we have adjusted the chemical potential so that the distance from the top of hole bands at $\Gamma$ to the Fermi energy is about $0.1$eV. It mimics the ARPES finding of ref. [@ding]. The parameters of the tight-binding model involves the nearest- and second-neighbor hopping among the Fe 3d orbitals: $$H_{\mathrm{band}}=\sum_{ij}\sum_{\alpha\beta}\sum_s
t_{ij}^{\alpha\beta} c^\dagger_{i\alpha s}c_{j\beta s}^{\vphantom{\dagger}}.$$ Here $i,j$ labels the Fe sites, $\alpha,\beta$ label the five different Fe orbitals ($\alpha=1,..5$ denotes $Z^2$, $XZ$, $YZ$, $X^2-Y^2$, $XY$, respectively), $s$ labels spin. The label $X,Y$ refer to the diagonal directions, while $x,y$ refer to the nearest neighbor Fe-Fe directions.
In unit of eV the tight-binding parameter $t_{i,i}^{\alpha\beta}$ are given by $$\begin{aligned}
&& t_{i,i}^{\alpha\beta}=\left(
\begin{array}{ccccc}
7.13365 & 0 & 0 & 0 & 0 \\
0 & 7.38168 & 0 & 0 & 0 \\
0 & 0 & 7.38168 & 0 & 0 \\
0 & 0 & 0 & 7.31272 & 0 \\
0 & 0 & 0 & 0 & 7.06574
\end{array}
\right)\equiv K_0^{\alpha\beta},
\\
&&t_{i,i+\hat{y}}^{\alpha\beta}=\left(
\begin{array}{ccccc}
0.00376 & 0.07892 & 0.07892 & 0 & -0.25696 \\
0.07892 & -0.1406 & -0.05034 & 0.18873 & 0.20995 \\
0.07892 & -0.05034 & -0.1406 & -0.18873 & 0.20995 \\
0 & 0.18873 & -0.18873 & -0.10725 & 0 \\
-0.25696 & 0.20995 & 0.20995 & 0 & -0.35102
\end{array}
\right)\equiv K_1^{\alpha\beta},
\\
&&t_{i,i+\hat{x}+\hat{y}}^{\alpha\beta}=\left(
\begin{array}{ccccc}
-0.00559 & 0 & -0.13684 & 0.11502 & 0 \\
0 & 0.07104 & 0 & 0 & 0.05964 \\
0.13684 & 0 & 0.23042 & 0.00961 & 0 \\
0.11502 & 0 & -0.00961 & 0.08569 & 0 \\
0 & -0.05964 & 0 & 0 & -0.11843
\end{array}
\right)\equiv K_2^{\alpha\beta}.\end{aligned}$$
In terms of these parameters the $5\times 5$ Bloch Hamiltonian is given by $$H({\mathbf{k}})= M_{\mathrm{2nd}}({\mathbf{k}})+{\frac{1}{2}}\left[M_{\mathrm{1st}}({\mathbf{k}})R_z+R_zM_{\mathrm{1st}}^T({\mathbf{k}})\right],$$ where $$\begin{split}
M_{\mathrm{1st}}({\mathbf{k}})= & K_1 e^{i k_y}+e^{i k_x}R K_1R^{-1}+e^{-i k_x}R^{-1}K_1R
\\ &
+e^{-i k_y}R^{-2}K_1R^2,
\end{split}$$ and $$\begin{split}
M_{\mathrm{2nd}}({\mathbf{k}})= & K_0+e^{i(k_x+k_y)}K_2+e^{i(k_y-k_x)}R^{-1}K_2R
\\ &
+e^{-i(k_x+k_y)}R^{-2}KR^2+e^{-i(k_y-k_x)}RK_2R^{-1}.
\end{split}$$ In the above $$\begin{array}{l}
R=\left(
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & -1
\end{array}
\right),
\\
R_z=-\left(
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1
\end{array}
\right),
\end{array}$$ are the matrices representing the combined $90^\circ$ rotation and $z$ reflection and the combined $z$ reflection and a gauge transformation on one Fe sublattice, respectively.
The bare magnetic susceptibility computed using the above bandstructure is shown in fig. \[sus\].
![(Color online) The real(left) and imaginary(right) part of the bare spin susceptibility. []{data-label="sus"}](fig2.eps)
A broad maximum around ${\mathbf{Q}}=(\pi,\pi)$ is found in Re($\chi$). If we make an analogy with the pnictide superconductors, where ${\mathbf{Q}}=(\pi,0)$ or $(0,\pi)$ and the predicted pairing form factor is $s_\pm$, it is natural to guess the pairing form factor here to be $d_{x^2-y^2}$[@ding; @thomale]. Note that this gives rise to fully gapped superconducting state with the order parameter changing sign between the electron Fermi surfaces. In the following we check whether this is true via a FRG calculation.
As in ref. [@rg3] we model the electronic correlations by the Hubbard and Hunds types of local interactions. The Hamiltonian is given as follows: $$\begin{split}
H= &
H_{\rm band}+
U_{1}\sum_{i\alpha}n_{i\alpha\uparrow}n_{i\alpha\downarrow}+
U_{2}\sum_{i,\alpha<\beta}n_{i\alpha}n_{i\beta}
\\ &
+J_{\mathrm{H}}
\Big[
\sum_{\alpha<\beta}
\sum_{\sigma\sigma^{\prime}}
c^{+}_{i\alpha\sigma}c^{+}_{i\beta\sigma^{\prime}}
c_{i\alpha\sigma^{\prime}}c_{i\beta\sigma}
\\ &
+c^{+}_{i\alpha\uparrow}c^{+}_{i\alpha\downarrow}
c_{i\beta\downarrow}c_{i\beta\uparrow}+h.c.\Big].
\end{split}
\label{H-H-model}$$ In the rest of the paper we use $(U_1,U_2,J_{\mathrm{H}})$=(4, 2, 0.7) eV respectively. These value are suggested by early dynamical mean-field theory results for the pnictides[@DMFT]. The sole reason for using this set of parameter is they represent intermediate couplings, which we believe K$_x$Fe$_{2-y}$Se$_2$ is likely to be.
FRG Method
==========
Details of the FRG method are discussed in ref. [@rg3; @rg8]. We made one important modification to include the buried hole bands in the FRG calculation. In our previous treatment bands without Fermi surface were ignored completely. In the case of K$_x$Fe$_2$Se$_2$ (and possibly in heavily hole-doped KFe$_2$As$_2$ as well) we believe this is no longer legitimate. The reason is two-fold. First consider the three different energy scales in the iron-based superconductors, the largest energy scale is the bare bandwidth and bare Coulomb interactions of order eV, the second energy scale is the effective spin exchange interactions about $50$meV as indicated by neutron studies[@spinwave], the pairing scale(gap) is the smallest about $\sim 10$meV. The distance between the top of the hole bands and the Fermi level in K$_x$Fe$_2$Se$_2$ is smaller than the first but comparable to the second energy scale. Therefore for a wide range of cutoff energies virtual excitations from the hole band to above the Fermi level should contribute to the renormalization of the effective interaction. The second reason is that a preliminary FRG which only include the single band that intersects the Fermi energy did not produce any strong pairing tendency, except very weak Kohn-Luttinger-type pairing in high angular momentum channel.
Physically when the FRG cutoff energy $\Lambda$ is much larger than the distance $\Delta E_0$ between the top of the hole bands and the Fermi energy, the fact that there are no hole pockets should not be important. Only when $\Lambda\lesssim\Delta E_0$ the lack of hole pockets will play an important role. Consequently, our FRG scheme consists of two stages. In the first stage the FRG scheme is the same as the case where the hole pockets are present. Thus we discretize the electron pocket band into patches represented by Fermi surfaces points as before. For the hole bands we put fictitious “Fermi surfaces”, small circles, around $(0,0)$ and $(\pi,\pi)$ and discretize the Brillouin zone as if hole pockets exist. Each (fictitious) Fermi surface is discretized into $16$ patches as in ref. [@rg1; @rg3; @rg8]. In the calculation of RG flow these hole band patches represented by the fictitious Fermi surface points are treated in the same way as electron band patches. In the second stage where $\Lambda\lesssim\Delta E_0$ the hole bands no longer contribute virtual states in the one-loop diagrams. When we compute pairing the fictitious Fermi surface are dropped. Thus in the second stage of the FRG the flow of pairing channel is essentially the marginally relevant Cooper channel flow discussed by Shankar[@shankar].
In our previous FRG study for systems involving both electron and hole pockets, the final effective interaction is captured reasonably well by a $J_1-J_2$ model[@rg8] (as far as the antiferromagnetism and pairing are concerned). In addition, in those cases, the final effective interaction is a result of the FRG flow over the entire bandwidth, not just in slivers around the Fermi surfaces. Thus to gain a qualitative understanding of the pairing in K$_x$Fe$_{2-y}$Se$_2$ one can use an effective $J_1-J_2$ interaction[@seo] to replace the FRG flow for cutoff $\Lambda >\Delta E_0$. We have analyzed the pairing form factor using such a model and find that for $J_1,J_2$ that are both antiferromagnetic and weak, the nodeless $d_{x^2-y^2}$ is the dominant pairing channel for the K$_x$Fe$_2$Se$_2$ bandstructure as long as $J_1/J_2 > 0.213$. In contrast the $s_{\pm}$ is dominant in the LaFeAsO band structure from $J_1/J_2=0$ up to $J_1/J_2\sim 1$. Of course, in general what affects the final pairing strength and form factor involves both the strength and form of the effective interaction as well as the geometry of the Fermi surfaces.
FRG Results
===========
The presentation of our results follows the format in our previous publications[@rg1; @rg3; @rg8]. For this FRG calculation with the parameters and RG scheme described above, we obtained divergent flow of several pairing channels, the flow of the two leading channels are plotted in fig. \[fig:flow\]. The strongest pairing channel has the $d_{x^2-y^2}$ symmetry. The associated divergence energy scale is $\sim 4$meV. The second strongest pairing is a extended $s$-wave with form factor approximately described by $\cos(k_x)+\cos(k_y)$. It has four nodes on each electron Fermi surface. The pairing form factors of both channels are depicted in fig. \[fig:formfactor\]. Interestingly both pairing channels are favored by the nearest-neighbor antiferromagnetic interactions $J_1$. In fact as long as $J_1/J_2 > 0.2$ the $J_1-J_2$ effective interaction predicts a leading $d_{x^2-y^2}$ pairing whose form factor is almost identical to our FRG result. The overlap between the form factor predicted by the $J_1-J_2$ model and that determined by FRG is shown in fig. \[fig:overlap\].
![(Color online) FRG flow of two leading pairing channels for $(U_1,U_2,J_{\mathrm{H}})$=(4, 2, 0.7) eV. Horizontal axis is the negative of the decadic logarithm of RG cutoff $\Lambda$. Vertical axis is the effective pairing strength (c.f. ref. [@rg8]). []{data-label="fig:flow"}](fig3.eps)
![(Color online) Gap form factors (c.f. ref. [@rg8]) of the two leading pairing channel for $(U_1,U_2,J_{\mathrm{H}})$=(4, 2, 0.7) eV. Horizontal axis $\theta$ is the polar angle on each Fermi surface with respect to its center. $\theta=0$ indicates the $+k_x$ direction. Upward(red) and downward(green) triangles label the two electron Fermi surfaces at $(0,\pi)$ and $(\pi,0)$ respectively. Top: The Leading $d_{x^2-y^2}$ gap. Bottom: The subleading extended $s$-wave gap. []{data-label="fig:formfactor"}](fig4a.eps "fig:") ![(Color online) Gap form factors (c.f. ref. [@rg8]) of the two leading pairing channel for $(U_1,U_2,J_{\mathrm{H}})$=(4, 2, 0.7) eV. Horizontal axis $\theta$ is the polar angle on each Fermi surface with respect to its center. $\theta=0$ indicates the $+k_x$ direction. Upward(red) and downward(green) triangles label the two electron Fermi surfaces at $(0,\pi)$ and $(\pi,0)$ respectively. Top: The Leading $d_{x^2-y^2}$ gap. Bottom: The subleading extended $s$-wave gap. []{data-label="fig:formfactor"}](fig4b.eps "fig:")
![(Color online) The overlap between the gap form factors of leading pairing channels obtained by FRG \[$(U_1,U_2,J_{\mathrm{H}})$=(4, 2, 0.7) eV\] and $J_1-J_2$ mean field theory versus the ratio $J_1/J_2$ ($J_2>0$ is antiferromagnetic). []{data-label="fig:overlap"}](fig5.eps)
Discussions
===========
There are evidences from NMR that the antiferromagnetic fluctuation is weak in K$_x$Fe$_{2-y}$Se$_2$[@gf]. On the surface this seems to be at odds with the notion that spin fluctuation is the main source of pairing. However, NMR probes spin fluctuations at very low energy. It is the high energy spin fluctuations that are good for pairing.
Obvious implications of the nodeless $d_{x^2-y^2}$ pairing is the presence of $(\pi,\pi)$ neutron resonance in the superconducting state. In addition, phase sensitive measurement utilizing the relative orientation of crystallographic grains, or the detection of randomly trapped half flux quanta in polycrystalline materials, can be done similar to the cuprates. Fourier transform scanning tunneling spectroscopy should in principle reveal the enhancement of the $(\pi,\pi)$ scattering by the magnetic field. Whether any of these is true is remained to be studied.
Finally we would like to list a few caveats. First, there is no [*a priori*]{} justification for ignoring the $(0,0,\pi)$ electron pocket found in the DFT calculation. It is important to study the effect of the three dimensional electron pocket on the pairing symmetry. Second, although the particular parameter set (including the chemical potential) we used in this study yielded $d_{x^2-y^2}$ pairing, it is important to investigate the robustness of this pairing symmetry against chemical potential shift and/or the changing of interaction parameters. Presently we have studied a different set of interaction parameters $(U_1,U_2,J_{\mathrm{H}})$=(4, 4, 0) eV and found a weaker pairing divergence (fig. \[fig:flow440\]). More importantly the leading pairing symmetry becomes a fully gapped $s$-wave, and the subleading one becomes $d_{xy}$ (fig. \[fig:formfactor440\]). Both these channels may be driven by strong $J_2$. The fully gapped $s$-wave can be viewed as the remnant of the $s_{\pm}$ pairing after the central hole pockets are removed. This result clearly calls for a more systematic study of the dependence of pairing symmetry on the parameters in our model. Third, the real materials have Fe vacancies, and even possible partial Fe vacancy ordering. This is obviously ignored in the current study. Last, the effects of impurity scattering on destabilizing the $d_{x^2-y^2}$ pairing clearly need to be addressed.
![(Color online) FRG flow of two leading pairing channels for $(U_1,U_2,J_{\mathrm{H}})$=(4, 4, 0) eV. Horizontal axis is the negative of the decadic logarithm of RG cutoff $\Lambda$. Vertical axis is the effective pairing strength (c.f. ref. [@rg8]). []{data-label="fig:flow440"}](fig6.eps)
![(Color online) Gap form factors (c.f. ref. [@rg8]) of the two leading pairing channel for $(U_1,U_2,J_{\mathrm{H}})$=(4, 4, 0) eV. Horizontal axis $\theta$ is the polar angle on each Fermi surface with respect to its center. $\theta=0$ indicates the $+k_x$ direction. Upward(red) and downward(green) triangles label the two electron Fermi surfaces at $(0,\pi)$ and $(\pi,0)$ respectively. Top: The Leading $s$-wave gap. Bottom: The subleading $d_{xy}$ gap. []{data-label="fig:formfactor440"}](fig7a.eps "fig:") ![(Color online) Gap form factors (c.f. ref. [@rg8]) of the two leading pairing channel for $(U_1,U_2,J_{\mathrm{H}})$=(4, 4, 0) eV. Horizontal axis $\theta$ is the polar angle on each Fermi surface with respect to its center. $\theta=0$ indicates the $+k_x$ direction. Upward(red) and downward(green) triangles label the two electron Fermi surfaces at $(0,\pi)$ and $(\pi,0)$ respectively. Top: The Leading $s$-wave gap. Bottom: The subleading $d_{xy}$ gap. []{data-label="fig:formfactor440"}](fig7b.eps "fig:")
In summary we have studied the pairing instabilities in K$_x$Fe$_{2-y}$Se$_2$ using a two stage functional renormalization group method. Our calculation suggests the leading pairing channel is $d_{x^2-y^2}$ and the subleading one is extended $s$. We find the buried hole bands make important contributions to the final effective interaction because of the closeness of their maxima to the Fermi energy. The leading $d_{x^2-y^2}$ pairing form factor is well captured by an effective $J_1-J_2$ interaction with dominant nearest neighbor antiferromagnetic exchange. At the end of the paper we briefly mentioned several obvious experiments that can test the $d_{x^2-y^2}$ symmetry. We also list some caveats of the present study.
DHL is supported by DOE grant number DE-AC02-05CH11231. FY is supported by the NSFC Grant No.10704008. ZYL and TX are supported by National Natural Science Foundation of China and by National Program for Basic Research of MOST, China.
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---
abstract: 'The extraordinary properties of nonlinear optical propagation processes in double-domain positive/negative index metamaterials are reviewed. These processes include second harmonic generation, three- and four-wave frequency mixing, and optical parametric amplification. Striking contrasts with the properties of the counterparts in ordinary materials are shown. We also discuss the possibilities for compensating strong losses inherent to plasmonic metamaterials, which present a major obstacle in numerous exciting applications, and the possibilities for creation of unique ultracompact photonic devices such as data processing chips and nonlinear-optical sensors. Finally, we propose similar extraordinary three-wave mixing processes in crystals based on optical phonons with negative dispersion.'
author:
- |
Alexander K. Popov and Vladimir M. Shalaev Department of Physics and Astronomy, University of Wisconsin-Stevens Point,\
Stevens Point, WI 54481, U. S. A.\
Birck Nanotechnology Center and School of Electrical and Computer Engineering,\
Purdue University, West Lafayette, IN 47907, U. S. A.
title: 'Merging Nonlinear Optics and Negative-Index Metamaterials'
---
INTRODUCTION {#sec:intro}
============
In the late 1960s, V. G. Veselago considered the propagation of electromagnetic waves in an fictitious, isotropic medium with simultaneously negative dielectric permittivity $\epsilon$ and magnetic permeability $\mu$ and showed that it would exhibit very unusual properties [@Vesel1; @Vesel2]. Specifically, the simultaneously negative dielectric permittivity, $\epsilon<0$, and magnetic permeability, $\mu<0$, would lead to a negative refraction index and to a left-handed triplet of the electric field, magnetic field and the wavevector. The energy flow (Poynting vector) in this case appears counter-directed with respect to the wavevector. This is rather counter-intuitive and in a sharp contrast with normal, positive-index materials (PIMs), also referred to as right-handed materials (RHMs). Negative-index materials (NIMs), also referred to as left-handed materials (LHMs), do not exist naturally. It was also generally accepted in physical optics that the magnetization at optical frequencies is negligible and, hence, did not play any essential role [@Land]. In accordance with this, the magnetic permeability $\mu$ was normally set to be equal to one in the basic Maxwell’s equations describing linear and nonlinear optical processes.
Metamaterials, i.e., artificially designed and engineered materials, can have properties unattainable in nature, including a negative refractive index. As outlined above, the *backwardness* of electromagnetic waves, i.e., the phenomenon of counter-directed energy flow and phase velocity for electromagnetic waves, does not exist in naturally occurring materials. Its realization has become feasible owing to the advent of nanotechnologies. Significant progress in the design of bulk metamaterials has been achieved during the last few years (see, e.g., [@Jas; @SM; @Zh]). This dictates a revision of many concepts concerning electromagnetic propagation processes in man-made materials, which hold the promise of revolutionary breakthroughs in microwave and photonic device technologies [@Sh].
Negative refraction can be implemented to develop a wide variety of devices with enhanced and uncommon functions, such as hyper-resolution lenses and optical cloaking devices. Nanostructured metamaterials are expected to play a key role in the development of all-optical data processing chips. The opposite directions of the energy flow and wavevector (phase velocity) in NIMs determines unique linear and nonlinear optical processes. Initially, metamaterials with a magnetic response and a negative refractive index were fabricated for the microwave and terahertz frequency ranges [@Smith1; @Smith11; @Smith2]. The optical frequency range imposes increasing difficulties and challenges for metamaterials. A negative magnetic permeability in the optical range, which is a precursor for a negative refraction, has been demonstrated in Refs. [@mu1; @mu2; @mu3] Finally, simultaneously negative magnetic permeability and electric permittivity in the optical telecommunicationfrequency range, and a hence negative refractive index, were experimentally demonstrated for metal-dielectric metamaterials [@NIMExp1; @NIMExp2; @NIMExp21]. For review, see Ref. [@Sh] Great progress was recently achieved in fabricating plasmonic NIMs, including chiral NIMs, and is described in Refs. [@Jas; @SM; @Zh; @WeLi; @SZh; @Pl]
To satisfy the casuality principle, a NIM must be lossy [@caus1; @caus2]. The majority of NIMs realized to date contain metal nanostructures. These structures introduce strong losses inherent to metals that are difficult to avoid, especially in the visible range of frequencies. Irrespective of their origin, losses constitute a major hurdle to the practical realization of the unique optical applications of these structures. Therefore, developing efficient loss-compensating techniques is of a paramount importance. So far, the most common approach to compensating losses in NIMs is associated with the possibility of embedding amplifying centers in the host matrix. The amplification is supposed to be provided through a population inversion between the energy levels of the embedded centers. Such a method to remove absorption losses and dissipation by optical amplification, e.g. by introducing a lasing medium instead of a passive dielectric into a negative-positive index composite and to utilize alternating amplifying layers instead of a bulk NIM, was suggested in [@amp1]. Although there are problems with this approach (for example, regarding amplified spontaneous noise, the influence of surface plasmons and the extreme localization of fields in optical amplification), it appears that it could furnish improved results [@amp2; @amp3; @amp4; @amp5; @Sh]. Raman amplification can be also used for removing the loss obstacle [@ampr]. Exciting progress was achieved during the last year in producing population-inversion-based amplification in plasmonic metamaterials [@Siv; @Nog], including recent a breakthrough with compensating losses in NIMs [@Ampl].
The main emphasis in the studies of NIMs has been placed so far on linear optical effects. Nonlinear optics in NIMs, especially frequency mixing, still remains a less-developed branch of photonics. It has been shown that NIMs, which include structural elements with non-symmetric current-voltage characteristics – its “meta-atoms,” can possess a nonlinear magnetic response at high frequencies [@Lap1; @Zhar; @Lap2; @Gorkopa; @Shadr]. Such meta-atoms have highly controllable magnetic and electric responses. Hence, NIMs can combine unprecedented linear and nonlinear electromagnetic properties. The feasibility of crafting NIMs with nonlinear optical (NLO) responses in the optical wavelength range, such as second-harmonic generation (SHG) and third-harmonic generation in magnetic metamaterials, has been experimentally demonstrated [@shg1e; @shg2e; @Kl]. On a fundamental level, the NLO response of nanostructured metamaterials is not yet completely understood or characterized. Nevertheless, it is well established that local-field-enhanced nonlinearities can be attributed to plasmonic nanostructures. Most negative-index (NI) plasmonic metamaterials are metal-dielectric composites made of metallic nanostructures shaped as split rings, horseshoes, fishnets, or chiral structures that are plunged into a dielectric host. This makes *ab initio* calculations of nonlinear propagation processes a complex task. Such works fall into a special, rapidly expanding research category. Recent progress in studying NLO responses of plasmonic metamaterials, both in theory and proof-of-principle experiments, is described in Refs. [@Shad; @Pet; @Ze; @Nie] and references therein.
The important properties of SHG in NIMs in the constant-pump approximation were discussed in Ref.[@Agr] for loss-less, semi-infinite materials and in Refs. [@Lens; @Lens1] for a slab of a finite thickness. Unlike ordinary NLO materials, the latter appeared to bring major qualitative effects. The possibility of exact phase-matching for waves with counter-propagating energy-flows has been shown in Ref. [@KivSHG] for the case when the fundamental wave falls in the negative-index frequency domain and the second-harmonic (SH) wave lies in the positive-index domain. The possibility of the existence of multistable nonlinear effects in SHG was also predicted in Ref. [@KivSHG] As outlined here, absorption is one of the most challenging problems that needs to be addressed for the realization of practical applications of NIMs. A transfer of the near-field image into aSH frequency domain, where absorption may be lower, was proposed in REfs. [@Agr; @Lens] as a possible means to overcome dissipative losses and thus enable the superlens. The propagation of microwave radiation in nonlinear transmission lines, which are the one-dimensional analog of NIMs, was investigated in Ref. [@Kozyr] The possibility to achieve conversion efficiency up to 12% was shown in Ref.[@Sc] for phase-matched SHG in a strongly absorbing, bulk NIM. In that case, the input picosecond-pulse peak intensity was assumed to be about 500 MW/cm$^2$, the nonlinear susceptibility of the metamaterial was assumed to be $\chi^2 \sim$ 80 pm/V, and the attenuation depth of the NIM was about 50 $\mu$m. The possibility of achieving phase- and group-velocity matching was shown for the above-indicated pulses based on the Drude dispersion model for metals. Nonlinear-optical processes in NIMs possess unusual, sometimes counter-intuitive, properties. They are described below with the example of SHG in loss-less NIMs of a finite thickness taken from our recent paper [@SHG]. The extraordinary NLO properties of such propagation processes in NIMs, including three-wave mixing (TWM), four-wave mixing (FWM) and optical parametric amplification (OPA), also have been predicted and are in stark contrast with their counterparts in natural materials [@SHG; @APB; @met; @OL; @OLM; @APL; @LSh; @APB09; @OL09; @SPI; @JOA; @WAS; @EPJD; @Ch]. Striking changes in the properties of nonlinear pulse propagation and temporal solitons [@Laz], spatial solitons in systems with bistability [@Tas; @Kos; @Boa], gap solitons [@Agu], and optical bistability in layered structures including NIMs [@Lit; @LitOL] have been revealed. A review of some of the corresponding theoretical approaches is given in Refs. [@Gab; @El] Frequency-degenerate multi-wave mixing and self-oscillations of counter-propagating waves in ordinary materials have been extensively studied earlier because of their easily achievable phase matching. Phase matching for TWM and four-wave mixing (FWM) of contra-propagating waves that are [far from degeneracy]{} seem impossible in ordinary materials and presents a technical challenge in the metamaterials. It has become achievable only recently due to the advances in nanotechnology [@Pas; @Kh]. The possibility and extraordinary properties of TWM, including mirrorless self-oscillations, was proposed in Ref. [@Har] (and references therein) more than 40 years ago (see also Refs. [@Vol; @Yar]) for two co-propagating waves with nearly-degenerate frequencies that fall within an anomalous dispersion frequency domain and gives rise to generation of a far-infrared difference-frequency counter-propagating wave in an anisotropic crystal. However, far-infrared radiation is typically strongly absorbed in crystals, which presents an unavoidable and strong detrimental factor. The proposal was not realized until recently. For the first time, a TWM backward-wave (BW) mirrorless optical parametrical oscillator (BWMOPO) with all three significantly different optical wavelengths was realized in Ref [@Pas]. Phase-matching of counter-propagating waves has been achieved in a periodically poled NLO crystal with the period on a submicrometer scale. Both in the proposal [@Har] and in the experiment [@Pas], the *opposite* orientation of wavevectors was required to establish a distributed feedback and to produce mirrorless OPO. This was due to the fact that ordinary, positive-index crystals were proposed for the implementation. As outlined, a major technical problem in creating BWMOPO stems from the requirement of phase matching for the opposite orientation of wavevectors in PIMs. This paper reviews the counter-intuitive effects and unusual features in the energy exchange between coupled ordinary and contra-propagating backward electromagnetic waves associated with three types of nonlinear optical processes: second harmonic generation beyond the constant fundamental wave approximation, three-wave mixing, and four-wave mixing.
The extraordinary properties of second-harmonic generation in a frequency double-domain positive/negative matamaterial are studied in Section \[shg\]. Both semi-infinite and finite-length NIM slabs are considered and compared with each other and with ordinary PIMs for SHG. The feasibility of a nonlinear-optical mirror converting the incident radiation into a reflected SH is shown and specific features of the process are investigated both for continuous-wave and pulsed regimes. The paper is organized as follows. The “backward” features of the electromagnetic waves in NIMs are discussed in Section \[pv\]. The basic equations for the negative-index fundamental and positive-index SH waves are derived and the relevant Manley-Rowe relations are discussed and compared with those in PIMs in Section \[bemr\]. The unusual spatial distributions of the field intensities for SHG in a NIM slab of a finite thickness are described in Section \[sl\] for the model of a loss-free, double-domain metamaterial and a continuous-wave regime. These distributions are compared with their counterparts in ordinary PIMs (Section \[shgpim\]) and semi-infinite NIM slabs (Section \[sis\]). Section \[sis\] also analyzes the properties of a nonlinear-optical mirror with a semi-infinite thickness that converts an incident fundamental beam into a reflected SH one. The role of absorption and phase mismatch is discussed in Section \[abs\]. The extraordinary features associated with SHG in the pulsed regime, which include significant differences in the pulse shapes for the fundamental and SH radiations dependent on the intensity and on the pulse duration of the incident pulses, are considered in Section \[pul\].
The extraordinary features of coherent, NLO energy transfer from the control optical field(s) to the coupled contra-propagating negative-phase (negative index, NI) and positive-phase (positive index, PI) waves through three- and four-wave mixing and related optical parametric amplification are described in Sections \[twm\] and \[fwm\]. The uncommon phenomenon of generating a contra-propagating wave at an appreciably different difference frequency in the direction of reflection is investigated. The feasibility and unparalleled features of such energy transfer are shown, which stem from the unusual fact that the energy flow of one of the coupled electromagnetic waves is *contra-directed* relative to the others, whereas the wavevectors for all coupled waves remain *parallel*. Such an opportunity makes phase matching of counter-propagating waves much easier and is offered by the backwardness of electromagnetic waves that is natural to NIMs. Consequently, distributed-feedback features become possible, while the antiparallel orientation of the wavevectors of the coupled waves is no longer required. The different properties attributed to such an all-optically tailored nonlinear optical mirror-amplifier for the case of a negative-index control field and for the alternative option of a negative-index idler are discussed and compared in Sections \[mcon\] and \[mid\]. The characteristic magnitudes of the required intensity of the control field and of the slab thickness are given in Section \[est1\].
The feasibility of independently crafting of a negative index and a resonantly enhanced four-wave mixing nonlinearity through embedded four-level centers is considered in Section \[fwm\]. The characteristic values of the required density of the embedded resonant nonlinear-optical centers, their relaxation properties, and the intensity of the control field are given in Section \[est2\]. The possibility of substituting a nanostructured negative-index metal-dielectric composites, which requires sophisticated fubrication techniques, with extensively used and studied ordinary crystals in order to simulate the unparalleled properties of coherent NLO energy exchange between the ordinary and backward waves is described in Section \[ph\]. Finally, the possibilities of implementing originally strongly absorbing microscopic samples of plasmonic, nanostructured, metal-dielectric composites for remote, all-optical tailoring of the transparency and reflectivity of metamaterial films as well as the concepts of several options of unique, ultracompact photonic devices based on the outlined three- and four-wave processes are summarized in the concluding Section \[con\].
Second-harmonic generation in a lossy, dispersive double-domain slab {#shg}
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Wavevectors and Poynting vectors for the fundamental and second-harmonic waves in a double-domain metamaterial {#pv}
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We consider a loss-free material that is left-handed at the fundamental frequency $\omega_1$ ($\epsilon_1<0$, $\mu_1<0$), whereas it is right-handed at the SH frequency $\omega_2=2\omega_1$ ($\epsilon_2>0$, $\mu_2>0$). The relations between the vectors of the electric, $\mathbf{E}$, and magnetic, $\mathbf{H}$, field components and the wavevector $\mathbf{k}$ for an electromagnetic wave written as $$\begin{aligned}
\mathbf{E}(\mathbf{r},t)&=&\mathbf{E}_0(\mathbf{r})\exp[-i(\omega t-\mathbf{k\cdot r})]+ c.c., \label{EM} \\
\mathbf{H}(\mathbf{r},t)&=&\mathbf{H}_0(\mathbf{r})\exp[-i(\omega t -\mathbf{k\cdot r})]+ c.c., \label{HM}\end{aligned}$$ traveling in a loss-free medium with dielectric permittivity $\epsilon$ and magnetic permeability $\mu$ are given by the equations $$\begin{aligned}
&\mathbf{k}\times\mathbf{E} = ({\omega}/{c})\mu\mathbf{H},\quad\mathbf{k}\times\mathbf{H} =- ({\omega}/{c}) \epsilon\mathbf{E},& \label{kh} \\
&\sqrt{\epsilon}{E}(\mathbf{r},t)=-\sqrt{\mu}{H}(\mathbf{r},t),&
\label{eh}\end{aligned}$$ which follow from Maxwell’s equations. Equations (\[kh\]) show that the vector triplet $\mathbf{E}$, $\mathbf{H}$ and $\mathbf{k}$ forms a right-handed system for the SH wave and a left-handed system for the fundamental beam. Simultaneously negative material parameters ($\epsilon<0$ and $\mu<0$) result in a negative refractive index $n= -
\sqrt{\mu\epsilon}$. As seen from Eqs. (\[EM\]) and (\[HM\]), the phase velocity $\mathbf{v}_{ph}$ is co-directed with $\mathbf{k}$ and is given by $\mathbf{v}_{ph}=({\mathbf{k}}/{k})({\omega}/{k})=({\mathbf{k}}/{k})({c}/{|n|})$, where ${k}^{2}=n^{2}(\omega/{c})^{2}$. In contrast, the direction of the energy flow (Poynting vector) $\mathbf{S}$ with respect to $\mathbf{k}$ depends on the signs of $\epsilon $ and $\mu $: $$\mathbf{S}(\mathbf{r},t) =\frac{c}{4\pi}[\mathbf{E}\times\mathbf{H}] =\frac{c^{2}}{4\pi\omega\epsilon}[\mathbf{H}\times\mathbf{k}\times\mathbf{H}]
= \frac{c^{2}\mathbf{k}}{4\pi\omega\epsilon}H^{2} =\frac{c^{2}\mathbf{k}}{4\pi\omega\mu}E^{2}. \label{s}$$
Since we have taken the material to be loss-free, we assume here that all indices of $\epsilon$, $\mu$ and $n$ are real numbers. Thus, the energy-flow $\mathbf{S}_1$ at $\omega_1$ is directed opposite to $\mathbf{k}_1$, whereas $\mathbf{S}_2$ is co-directed with $\mathbf{k}_2$.
Basic equations and the Manley-Rowe relations for SHG process in NIMs and PIMs {#bemr}
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![\[fig1\] The difference in the phase-matching geometry and in the intensity distribution for the fundamental and the second-harmonic waves, $h_1^2$ and $h_2^2$, between a slab of the left-handed (negative-index) metamaterial, (a), and a right-handed material (ordinary, positive-index) material, (b). ](fig1.eps){width=".6\columnwidth"}
### Basic equations {#be}
We assume that an incident flow of fundamental radiation $\mathbf{
S}_{1}$ at $\omega _{1}$ propagates along the z-axis, which is normal to the surface of a metamaterial. According to (\[s\]), the phase of the wave at $\omega _{1}$ travels in the reverse direction inside the NIM \[Fig.\[fig1\](a)\]. Because of the phase-matching requirement, the generated SH radiation also travels backward with energy flow in the same backward direction. This is in contrast with the standard coupling geometry in a PIM \[Fig.\[fig1\](b)\]. As shown below, the basic features of the process associated with electric or magnetic nonlinearities are similar. Following Ref. [@KivSHG], we assume that a nonlinear response is primarily associated with the magnetic component of the waves. Then the equations for the coupled fields inside a NIM in the approximation of slow-varying amplitudes acquire the form: $$\begin{aligned}
{dA_{2}}/{dz} &=&i\sigma_2A_{1}^{2}\exp (-i\Delta kz), \label{A2} \\
{dA_{1}}/{dz} &=&i\sigma_1A_{2}A_{1}^{\ast }\exp (i\Delta kz).
\label{A1}\end{aligned}$$ Here, $\Delta k=k_{2}-2k_{1}$; $\sigma_1 =({\epsilon _{1}\omega
_{1}^{2}}/{k_{1}c^{2}})8\pi \chi ^{(2)}$, $\sigma_2=({\epsilon
_{2}\omega _{2}^{2}}/{k_{2}c^{2}})4\pi \chi^{(2)}$; $\chi^{(2)}$ is the effective nonlinear susceptibility; $A_{2}$ and $A_{1}$ are the slowly varying amplitudes of the waves with the phases traveling against the z-axis, $${H}_{j}(z,t)={A}_{j}\exp [-i(\omega _{j}t+k_{j}z)]+c.c., \label{Az}$$ $j=\{1,2\}$; $\omega _{2}=2\omega _{1}$; and $k_{1,2}>0$ are the moduli of the wavevectors directed against the z-axis. We note that according to Eq. (\[eh\]), the corresponding equations for the electric components can be written in a similar form, with $\epsilon
_{j}$ substituted by $\mu _{j}$ and vice versa. The factors $\mu _{j}$ were usually assumed to be equal to one in similar equations for PIMs. However, this assumption does not hold for the case of NIMs, and this fact dramatically changes many conventional electromagnetic relations.
### The Manley-Rowe relations {#mrr}
The Manley-Rowe relations [@MR56; @MR] for the field intensities and for the energy flows follow from Eqs. (\[s\]) - (\[A1\]): $$\frac{k_{1}}{\epsilon
_{1}}\dfrac{d|A_{1}|^{2}}{dz}+\frac{k_{2}}{2\epsilon
_{2}}\dfrac{d|A_{2}|^{2}}{dz}=0,\quad
\dfrac{d|S_{1}|^{2}}{dz}-\dfrac{d|S_{2}|^{2}}{dz}=0. \label{MR1}$$ The latter equation accounts for the difference in the signs of $\epsilon _{1}$ and $\epsilon _{2}$, which brings radical changes to the spatial dependence of the field intensities, as discussed below.
In order to outline the basic difference between the SHG process in NIMs and PIMs, we assume in our further consideration that the phase-matching condition $k_{2}=2k_{1}$ is fulfilled. The spatially-invariant form of the Manley-Rowe relations follows from equation (\[MR1\]): $$|A_{1}|^{2}/\epsilon _{1}+|A_{2}|^{2}/\epsilon _{2}=C, \label{I}$$ where $C$ is an integration constant. With $\epsilon _{1}=-\epsilon
_{2}$ Equation (\[I\]) takes the form: $$|A_{1}|^{2}-|A_{2}|^{2}=C, \label{D}$$
Equations (\[I\]) and (\[D\]) predict a *concurrent decrease* of the amplitudes of both waves along the z-axis so that the *difference* between the squared amplitudes remains constant through the sample, as schematically depicted in Fig. \[fig1\](a). Hence, Eqs. (\[I\]) and (\[D\]) also predict that the *difference* between the number of pairs of photons $\hbar\omega_1$ and the number of photons $\hbar\omega_2$ remains constant through the sample. This is in striking contrast with the requirement that the *sum* of the squared amplitudes and the *sum* of the corresponding photon numbers is constant in the analogous case in a PIM, as schematically shown in Fig. \[fig1\](b) and described in textbooks on nonlinear optics.
Ultimately, the described extraordinary features of SHG in a double-domain NIM and their dependence on the slab geometry stem from the backwardness of electromagnetic waves in NIMs and the opposite directions of the energy flows in fundamental-harmonic (FH) and SH waves. As shown in the next sections for three-wave mixing processes, different depletion rates caused by different absorption indices at frequencies of the contra-propagating wave may qualitatively change their propagation properties, the distribution across the slab, and the output characteristics depending on the specific NLO propagation process.
We now introduce the real phases and amplitudes of the fields as $A_{1,2}=h_{1,2}\exp (i\phi_{1,2})$. Then the equations for the real amplitudes and phases $$\begin{aligned}
\label{psi}
&&dh_{2}/dz=\sigma_2h^2_{1}sin\Psi, \nonumber\\&&dh_{1}/dz=-\sigma_1h_1h_2sin\Psi,\nonumber\\&&d\Psi/dz=-(2\sigma_1h_2-\sigma_2h^2_1/h_2)cos\Psi,\\&&\Psi\equiv\phi_2(z)-2\phi_1(z).\nonumber,\end{aligned}$$ which follow from Eqs. (\[A2\]) and (\[A1\]), show that if any of the fields becomes zero at any point, the integral (\[I\]) corresponds to the solution with the constant phase difference $2\phi_{1}-\phi_{2}=\pi /2$ over the entire sample.
SHG in PIMs {#shgpim}
-----------
The equations for the slowly varying amplitudes corresponding to the ordinary coupling scheme in a PIM, shown in Fig. \[fig1\](b), are readily obtained from Eqs. (\[A2\]) - (\[Az\]) by changing the signs of $k_{1}$ and $k_{2}$. This does not change the integral (\[I\]); more importantly, the relation between $\epsilon_{1}$ and $\epsilon_{2}$ required by phase matching now changes to $\epsilon_{1}=\epsilon_{2}$, where both constants are positive. The phase difference remains the same. Because of the boundary conditions $h_{1}(0)=h_{10}$ and $h_{2}(0)=h_{20}=0$, the integration constant becomes $C=h_{10}^{2}$. Thus, the equations for the real amplitudes in the case of a PIM acquire the form: $$\begin{aligned}
&&h_{1}(z)=\sqrt{h_{10}^{2}-h_{2}(z)^{2}}, \label{D3} \\
&&dh_{2}/{dz}=\kappa \lbrack h_{10}^{2}-h_{2}(z)^{2}], \label{h24}\end{aligned}$$ with the known solution $$\begin{aligned}
h_{2}(z) &=&h_{10}\tanh (z/z_{0}), \label{rhm2} \\
h_{1}(z) &=&h_{10}/\cosh (z/z_{0}),\,z_{0}=[\kappa h_{10}]^{-1}.
\label{rhm1}\end{aligned}$$ Here, $\kappa =({\epsilon_{2}\omega_{2}^{2}}/{k_{2}c^{2}})4\pi
\chi_{eff}^{(2)}$. *The solution has the same form for an arbitrary slab thickness* with decreasing fundamental and increasing SH squared amplitudes along the z-axis, as shown schematically in Fig. \[fig1\](b).
SHG in a NIM slab {#sl}
-----------------
Now consider phase-matched SHG in a loss-less NIM slab of a finite length L. Equations (\[A2\]) and (\[D\]) take the form: $$\begin{aligned}
h_{1}(z)^{2} &=&C+h_{2}(z)^{2}, \label{D1} \\
dh_{2}/{dz} &=&-\kappa \lbrack C+h_{2}(z)^{2}]. \label{h12}\end{aligned}$$ Taking into account the *different boundary conditions in a NIM as compared to a PIM*, $h_{1}(0)=h_{10}$ and $h_{2}(L)=0$, the solution to these equations is $$\begin{aligned}
h_{2} &=&\sqrt{C}\tan [\sqrt{C}\kappa (L-z)], \label{h22} \\
h_{1} &=&\sqrt{C}/\cos [\sqrt{C}\kappa (L-z)], \label{h11}\end{aligned}$$ where the integration parameter $C$ *depends on the slab thickness $L$* and on the amplitude of the incident fundamental radiation as $$\sqrt{C}\kappa L=\cos ^{-1}(\sqrt{C}/h_{10}). \label{C}$$ Thus, the spatially invariant field intensity difference between the fundamental and SH waves in NIMs depends on the slab thickness, which is in *strict contrast* with the case in PIMs. As seen from Equation (\[D1\]), the integration parameter $C=h_{1}(z)^{2}-h_{2}(z)^{2}$ now represents the deviation of the conversion efficiency $\eta =h_{20}^{2}/h_{10}^{2}$ from unity: $(C/h_{10}^{2})=1-\eta $. Figure \[f2\] shows the dependence of this parameter on the conversion length $z_{0}=(\kappa h_{10})^{-1}$.
![The normalized integration constant $C/h_{10}^{2}$ and the energy conversion efficiency $\protect\eta $ vs. the normalized length of a NIM slab.[]{data-label="f2"}](f2.eps){width=".6\textwidth"}
The figure shows that for a conversion length of 2.5, the NIM slab, which acts as nonlinear mirror, provides about 80% conversion of the fundamental beam into a reflected SH wave. Figure \[f3\] depicts the field distribution along the slab length. One can see from the figure, with an increase in slab length (or intensity of the fundamental wave), the gap between the two plots decreases while the conversion efficiency increases (comparing the main plot and the inset).
![The squared amplitudes for the fundamental wave (dashed line) and SHG (solid line) in a loss-less NIM slab of a finite length. Inset: the slab has a length equal to one conversion length. Main plot: the slab has a length equal to five conversion lengths. The dash-dot lines show the energy conversion for a semi-infinite NIM.[]{data-label="f3"}](f3.eps){height=".6\textwidth"}
SHG in a semi-infinite NIM {#sis}
--------------------------
Now we consider the case of a semi-infinite NIM at $z>0$. Since both waves disappear at $z\rightarrow \infty $ due to the entire conversion of the fundamental beam into SH, $C=0$. Then equations (\[D1\]) and (\[h12\]) for the amplitudes take the simple form $$\begin{aligned}
&&h_{2}(z)=h_{1}(z), \label{D2} \\
&&dh_{2}/{dz}=-\kappa h_{2}^{2}. \label{h23}\end{aligned}$$ Equation (\[D2\]) indicates 100% conversion of the incident fundamental wave into the reflected second harmonic at $z=0$ in a loss-less, semi-infinite medium provided that the phase-matching condition $\Delta k=0$ is fulfilled. The integration of (\[h23\]) with the boundary condition $h_{1}(0)=h_{10}$ yields $$h_{2}(z)=\dfrac{h_{10}}{(z/z_{0})+1},\,z_{0}=(\kappa h_{10})^{-1}.
\label{si}$$ Equation (\[si\]) describes a *concurrent decrease* of both waves of *equal amplitudes* along the z-axis; this is shown by the dash-dot plots in Fig. \[f3\]. For $z\gg z_{0}$, the dependence is inversely proportional to $z$. These spatial dependencies, shown in Fig. \[f3\], are also in striking contrast with those for the conventional process of SHG in a PIM, which are shown in various textbooks \[compare, for example, with Fig.\[fig1\](b)\].
SHG in a lossy dispersive NIM slab {#abs}
----------------------------------
It is convenient to introduce effective amplitudes, $a_{j}$, and nonlinear coupling parameters, $X_{j}$, which are defined as $$\begin{aligned}
a_{j}=\sqrt{|\mu_j|/k_j}A_j,\quad
X=\sqrt{k_1^2k_2/|\mu_1^2\mu_2|} 4\pi\chi^{(2)}_{\rm eff}.
\label{ec}\end{aligned}$$ Here, $\chi^{(2)}_{\rm eff}$ is the effective nonlinear susceptibility, and the quantities $|a_j|^2$ are proportional to the photon numbers in the energy fluxes.
In the general case of SHG in a dispersive double-domain material with $\alpha(\omega_1)\neq\alpha(\omega_2)$ and $\Delta k=k_{2}-2k_{1}~\neq0$, the equations for the amplitudes $a_j$ take the form: $$\begin{aligned}
&&da_1/dz=-i2Xa^*_1a_2\exp(i\Delta k z)-(\alpha_1/2)a_1,\label{am1}\\
&&da_2/dz=iXa_1^2\exp(-i\Delta k z)+(\alpha_2/2)a_2.\label{am2}\end{aligned}$$ Here, $\alpha_{1,2}$ are the absorption indices at the corresponding frequencies. The equations account for opposite signs of $\epsilon_1$, $\mu_1$ and $\epsilon_2$, $\mu_2$. Three differences distinguish these equations from their counterparts in ordinary PIMs – opposite signs with the NLO coupling factors, opposite signs with the absorption indices, and boundary conditions to be applied to the opposite sides of the slab. These differences stem from the backwardness of the fundamental wave in the NIM in this case. Note that, for a NIM with an electric nonlinearity, the corresponding equations for the electric components can be written in a similar form, with $\epsilon
_{j}$ substituted by $\mu _{j}$ and vice versa.
\
(a)(b)
\
(a)(b)
Equations (\[am1\])-(\[am2\]) do not have analytical solutions. Figures \[fig2\] and \[fig3\] depict the results of a numerical analysis of the chief features of SHG in a dispersive, absorptive slab for boundary conditions $a_1(z=0)=a_{10}$, $a_2(z=L)=0$. The quantity $T_1=|a_1(z)|^2/|a_{10}|^2$ represents the distribution of the fundamental beam inside the slab determined by both absorption and energy conversion as well as the ultimate slab transparency, $T_1(z=L)$. The quantity $\eta_2=2|a_2(z)|^2/|a_{10}|^2$ represents the quantum conversion efficiency of the process, $g=Xa_{10}$, and $g^{-1}$ is the nonlinear conversion length, i.e., the characteristic slab thickness that is required for significant energy conversion and is dependent on the input intensity of the fundamental field. The plot for $\alpha_{1}=\alpha_{2}=0$ in Fig. \[fig2\](a) displays two curves with a constant gap, which determines the output value of about 80% for the reflected beam ($z=0$) at $2\omega_1$ for the given input intensity of the fundamental field at $\omega_1$. By comparing this plot with the others, we see that absorption causes variable a gap that increases in the output area $z\geq0$, so that the output value of SHG drops to about 40% for $\alpha_{1}L=2.3$ and $\alpha_{2}L=1$ and the same intensity of the input fundamental beam. For $\alpha_{1}L=2.3$ and $a_{10}\rightarrow0$, the transparency of the slab at $\omega_1$ reaches up to 10%, which is characteristic for plasmonic samples. Figure \[fig2\](b) shows that the conversion efficiency decreases with an increase in phase mismatch, however it is less than it would be for the ordinary coupling schemes in PIMs. Figure \[fig3\](a) shows that, for the given parameters, the SH field concentrates in the area close to the exit facet of the slab at $z=0$. The dependence on $g$ at $z=0$ indicates that a significant part of the incident beam can be converted into SH for $gL\sim1$ even for the given phase mismatch and absorption rates. This is shown in Fig. \[fig3\](b), which depicts the significant depletion of the FH field with an increase of its input intensity and the slab length (parameter $gL$).
Pulsed SHG in a transparent NIM slab {#pul}
------------------------------------
The equations for the quantities $a_1$ and $a_2$ introduced above are $$\begin{aligned}
\label{eq1}
\frac1{v_1}\frac{{\partial}a_1}{{\partial}t}+ \frac{{\partial}a_1}{{\partial}z}&=& -i2ga_1^*a_2\exp{(i\Delta kz)}-\frac{\alpha_1}2a_1 \nonumber\\
-\frac1{v_2}\frac{{\partial}a_2}{{\partial}t}+ \frac{{\partial}a_2}{{\partial}z}&=& iga_1^2\exp{(-i\Delta kz)}+\frac{\alpha_2}2a_2.\end{aligned}$$ Here, $|a_{1,2}|^2$ are the above-introduced slowly varying amplitudes that are proportional to the photon numbers in the energy fluxes corresponding to the pulse maximum, and $v_i$ are the group velocities for the corresponding pulses. After introducing $\xi=z/L$ and $\tau=t/\Delta\tau$, $d=L/v_1\Delta\tau$, where $d$ is the slub thickness $L$ reduced by the input pulse length and $\Delta\tau$ is the duration of the input fundamental pulse, Eqs. (\[eq1\]) take the form: $$\begin{aligned}
\label{eq2}
d\frac{{\partial}a_1}{{\partial}\tau}+ \frac{{\partial}a_1}{{\partial}\xi}&=& -i2\tilde{g}a_1^*a_2\exp{(i\Delta \tilde{k}\xi)}-\frac{\tilde{\alpha}_1}2a_1 \nonumber\\
-d\frac{{\partial}a_2}{{\partial}\tau}+ \frac{{\partial}a_2}{{\partial}\xi}&=& i\tilde{g}a_1^2\exp{(-i\Delta \tilde{k}\xi)}+\frac{\tilde{\alpha}_2}2a_1\end{aligned}$$ The input pulse shape is taken as being close to a rectangular form $$F(\tau)=0.5\left(\tanh\frac{\tau_0+1-\tau}{\delta\tau}-\tanh\frac{\tau_0-\tau}{\delta\tau}\right),$$ where $\delta\tau$ is the duration of the pulse front and tail, and $\tau_0$ is the shift of the front relative to $t=0$. The magnitudes $\delta\tau=0.01$ and $\tau_0=0.5$ have been selected for numerical simulations.
![\[fi1\] Input $T_1(z=0)$ and output $T_1(z=L)$ pulses of fundamental and negative-index SH radiation $\eta_2(z=0)$ for relatively short pulses of fundamental radiation. (a) Input pulse area $S_{10}=0.9750$; output pulse areas $S_{1L}= 0.5031$, $S_{20}= 0.2392$. (c) Input pulse area $S_{10}=0.9750$; output pulse areas $S_{1L}= 0.0396$, $S_{20}= 0.4742$.](fi1a.eps "fig:"){width=".48\textwidth"} ![\[fi1\] Input $T_1(z=0)$ and output $T_1(z=L)$ pulses of fundamental and negative-index SH radiation $\eta_2(z=0)$ for relatively short pulses of fundamental radiation. (a) Input pulse area $S_{10}=0.9750$; output pulse areas $S_{1L}= 0.5031$, $S_{20}= 0.2392$. (c) Input pulse area $S_{10}=0.9750$; output pulse areas $S_{1L}= 0.0396$, $S_{20}= 0.4742$.](fi1b.eps "fig:"){width=".48\textwidth"}\
(a) (b)\
![\[fi1\] Input $T_1(z=0)$ and output $T_1(z=L)$ pulses of fundamental and negative-index SH radiation $\eta_2(z=0)$ for relatively short pulses of fundamental radiation. (a) Input pulse area $S_{10}=0.9750$; output pulse areas $S_{1L}= 0.5031$, $S_{20}= 0.2392$. (c) Input pulse area $S_{10}=0.9750$; output pulse areas $S_{1L}= 0.0396$, $S_{20}= 0.4742$.](fi1c.eps "fig:"){width=".48\textwidth"} ![\[fi1\] Input $T_1(z=0)$ and output $T_1(z=L)$ pulses of fundamental and negative-index SH radiation $\eta_2(z=0)$ for relatively short pulses of fundamental radiation. (a) Input pulse area $S_{10}=0.9750$; output pulse areas $S_{1L}= 0.5031$, $S_{20}= 0.2392$. (c) Input pulse area $S_{10}=0.9750$; output pulse areas $S_{1L}= 0.0396$, $S_{20}= 0.4742$.](fi1d.eps "fig:"){width=".48\textwidth"}\
(c) (d)\
![\[fi5\] Input $T_1(z=0)$ and output $T_1(z=L)$ pulses of fundamental and negative-index SH radiation $\eta_2(z=0)$ for relatively long pulses of fundamental radiation. (a) Input pulse area $S_{10}= 0.9900$; output pulse areas $S_{1L}= 0.2516$, $S_{20}= 0.3692$. (c) Input pulse area $S_{10}= 0.9900$; output pulse areas $S_{1L}=0.0161$, $S_{20}= 0.4870$.](fi5a.eps "fig:"){width=".48\textwidth"} ![\[fi5\] Input $T_1(z=0)$ and output $T_1(z=L)$ pulses of fundamental and negative-index SH radiation $\eta_2(z=0)$ for relatively long pulses of fundamental radiation. (a) Input pulse area $S_{10}= 0.9900$; output pulse areas $S_{1L}= 0.2516$, $S_{20}= 0.3692$. (c) Input pulse area $S_{10}= 0.9900$; output pulse areas $S_{1L}=0.0161$, $S_{20}= 0.4870$.](fi5b.eps "fig:"){width=".48\textwidth"}\
(a) (b)\
![\[fi5\] Input $T_1(z=0)$ and output $T_1(z=L)$ pulses of fundamental and negative-index SH radiation $\eta_2(z=0)$ for relatively long pulses of fundamental radiation. (a) Input pulse area $S_{10}= 0.9900$; output pulse areas $S_{1L}= 0.2516$, $S_{20}= 0.3692$. (c) Input pulse area $S_{10}= 0.9900$; output pulse areas $S_{1L}=0.0161$, $S_{20}= 0.4870$.](fi5c.eps "fig:"){width=".48\textwidth"} ![\[fi5\] Input $T_1(z=0)$ and output $T_1(z=L)$ pulses of fundamental and negative-index SH radiation $\eta_2(z=0)$ for relatively long pulses of fundamental radiation. (a) Input pulse area $S_{10}= 0.9900$; output pulse areas $S_{1L}= 0.2516$, $S_{20}= 0.3692$. (c) Input pulse area $S_{10}= 0.9900$; output pulse areas $S_{1L}=0.0161$, $S_{20}= 0.4870$.](fi5d.eps "fig:"){width=".48\textwidth"}\
(c) (d)\
![\[fi9\] Effect of phase mismatch on SHG in a NIM slab for the case of strong, relatively long pulses. ](fi9a.eps "fig:"){width=".49\textwidth"} ![\[fi9\] Effect of phase mismatch on SHG in a NIM slab for the case of strong, relatively long pulses. ](fi9b.eps "fig:"){width=".49\textwidth"}\
(a) (b)\
![\[fi11\] Effect of difference in the group velocities on SHG in a NIM slab for the case of strong, relatively long pulses. ](fi11.eps){width=".6\textwidth"}
The unusual properties of SHG in NIMs in the pulsed regime stem from the fact that it occurs only inside the pulse of fundamental radiation. Generation begins on the leading edge of the pulse, grows towards its trailing edge, and then exits the pulse with no further changes. Since the fundamental pulse propagates across the slab, the duration of the SH pulse is longer than the fundamental one. Depletion of the fundamental radiation along its pulse length and the conversion efficiency depend on its initial maximum intensity and phase matching. Ultimately, the overall properties of SHG, the pulse length, and the photon conversion efficiency depend on the ratio of the fundamental pulse and slab length. This extraordinary behavior is illustrated in Figs. \[fi1\]-\[fi11\].
Figures \[fi1\]-\[fi11\] display the input shape of the fundamental pulse $T_1=|a_1(z)|^2/|a_{10}|^2$ at $z=0$ when its leading front enters the medium and the results of numerical simulations for the output fundamental pulse when its tail reaches the slab boundary at $z=L$ as well as the shape and the conversion efficiency of the output second harmonic pulse $\eta_2=|a_2(z)|^2/|a_{10}|^2$ traveling against the z-axis, when its tail passes the slab’s edge at $z=0$. For clarity, here, the medium is assumed loss-less and group velocities of the fundamental and SH pulses assumed equal for Figs. \[fi1\]-\[fi9\].
Figure \[fi1\] corresponds to the fundamental pulse four time shorter than the slab thickness. It shows increase of the conversion efficiency with increase the intensity of the input pulse. It is followed by the shortening of the SH pulse. Phase mismatch causes changes in the depletion and shape of the output fundamental pulse. However overall conversion rate does not change significantly. The outlined properties satisfy to the conservation law: the number of annihilated pair of photons of fundamental radiation ($S_{10}-S_{1L})/2$ is equal to the number of output SH photons $S_{20}$. Figure \[fi5\] displays the corresponding changes for a longer input pulse length equal to 0.5 of the slab thickness. Here, conversion efficiency increases at a lower input intensity because of the longer conversion length. The changes in the SH pulse length and conversion efficiency with increasing input intensity appear less significant.
Figure \[fi9\] shows the effect of phase mismatch on the shape of the SH pulse for relatively long pulses and a high-intensity input pulse of fundamental radiation.
Figure \[fi11\] shows that the properties of the output SH pulse do not change significantly with an increase of the group velocity difference up to 20% for long pulses and a high-intensity fundamental field.
Three-wave mixing of contra-propagating electromagnetic waves: tailored transparency and reflectivity of a plasmonic metaslab, and applications to ultracompact nonlinear-optical data processing chips and sensors {#twm}
====================================================================================================================================================================================================================
An extraordinary electromagnetic property of NIMs stems from the fact that the energy flow and phase velocity of electromagnetic waves become counter-directed inside the NIM slab. Usually, a negative refractive index exists only inside a certain frequency band. The metamaterial remains ordinary, PI, outside that band. Consider a slab of thickness $L$ that possesses a quadratic nonlinearity, which enables three-wave mixing processes $\omega_3=\omega_1+\omega_2$. The amount of coherent energy transfer between the coupled light waves increases with the decrease of phase mismatch $\Delta kL$, where $\Delta k=k_3-k_2-k_1$. This dictates the requirement of co-directed wavevectors for all coupled waves. Consequently, the energy flux for the wave that falls in the negative-index frequency domain appears contra-directed relative two others. Figure \[fig4\] depicts three corresponding possible options for the phase-matched TWM NLO coupling of ordinary and backward waves.
\
(a)(b)(c)\
For the continuous-wave regime, the equations for the slowly varying amplitudes of the coupled waves can be written in the form $$\begin{aligned}
da_1/dz&=&\pm[iX_1a^*_{2}a_3\exp(i\Delta k z)-({\alpha_1}/{2})a_{1}],\label{a1}\\
da_2/dz&=&iX_2a^*_{1}a_3\exp(i\Delta k z)-({\alpha_2}/{2})a_{2},\label{a2}\\
da_3/dz&=&\mp[iX_{3}a_1a_2\exp(-i\Delta k z)-({\alpha_3}/{2})a_{3}] \label{a3}.
$$ The minus and plus signs refer to the negative- and positive-index waves, correspondingly. Here, $\Delta k=k_3-k_2-k_1$, and all three waves experience strong dissipation described by absorption indices $\alpha_{1,2,3}$. The slowly varying effective amplitudes of the waves, $a_{e,m,j}$, (j={1,2,3}) and nonlinear coupling parameters, $g_{e,m}$, for the [electric (*e*)]{} and magnetic (*m*) types of quadratic nonlinearity can be conveniently introduced as $$\begin{aligned}
a_{ej}=\sqrt{|\epsilon_j/k_j|}E_j,
X_{e}=\sqrt{|k_1k_2/\epsilon_1\epsilon_2|} 2\pi\chi^{(2)}_{ej};
\nonumber\\a_{mj}=\sqrt{|\mu_j/k_j|}H_j,
X_{m}=\sqrt{|k_1k_2/\mu_1\mu_2|} 2\pi
\chi^{(2)}_{mj}.\nonumber$$ The quantities $|a_j|^2$ are proportional to the photon numbers in the energy fluxes. Equations for the amplitudes $a_j$ are identical for the both types of the nonlinearities.
We note the following three fundamental differences in Equations (\[a1\]) and (\[a3\]) as compared with their counterparts in ordinary, PI materials \[Eq. (\[a2\])\]. First, the signs with $g_{1}$ or $g_{3}$ are opposite to that with $g_{2}$. This is because the corresponding $\epsilon_{j}<0$ and $\mu_{j}<0$. Second, the opposite sign appears with $\alpha_{j}$ because the corresponding energy flow $\mathbf{S_{j}}$ is against the $z$-axis. Third, the boundary conditions for the negative-index wave are defined at the opposite side of the sample as compared to the ordinary waves because their energy flows are counter-directed. These modifications lead to dramatic changes in the equation solution as compared with the exponential dependence on $z$ that is characteristic for the ordinary, naturally occurring, PI materials.
Consider the example depicted in Fig. \[fig4\](a). We assume that the wave at $\omega_{1}$ with the wavevector $\mathbf{k}_1$ directed along the $z$-axis is a PI ($n_{1}>0$) signal. Usually it experiences strong absorption caused by metal inclusions. The medium is assumed to possess a quadratic nonlinearity $\chi^{(2)}$ and is illuminated by the strong, higher-frequency control field at $\omega_{3}$, which falls into the NI domain. Due to the TWM interaction, the control and signal fields generate a difference-frequency idler at $\omega_{2}=\omega_{3}-\omega_{1}$, which is also assumed to be a PI wave ($n_{2}>0$). The idler, in cooperation with the control field, contributes back into the wave at $\omega_{1}$ through the same type of TWM interaction and thus enables OPA at $\omega_{1}$ by converting the energy of the control fields into the signal. In order to ensure effective energy conversion the induced traveling wave of nonlinear polarization in the medium and the coupled electromagnetic wave at the same frequency must be phase matched. Hence, all phase velocities (wavevectors) must be co-directed. Since $n(\omega_3)<0$, the control field is a backward wave, i.e., its energy flow $\mathbf{S}_{3}
=(c/4\pi)[\mathbf{E_3}\times \mathbf{H_3}]$ is directed against the $z$-axis. Such a device can be employed as NLO sensor. It can be conveniently and remotely interrogated to actuate frequency up-conversion and amplification of a signal directed towards the remote detector. Such a signal could be incoming far-infrared thermal radiation emitted by the object of interest or signal that carries important spectral information about the chemical composition of the environment, for example. Two other schemes depicted in Fig. \[fig4\](b),(c) offer different advantages and operational properties for NLO NIM-based devices. The research challenge is that such an unprecedented NLO coupling scheme leads to changes in the set of coupled nonlinear propagation equations and boundary conditions compared to the standard ones known from published literature. This in turn results in dramatic changes in their solutions and in the multi-parameter dependencies of the operational properties of the proposed devices.
Unusual propagation and energy-conversion properties in double-domain NIM/PIM slabs are readily seen, for example, in the coupling scheme of Fig. \[fig4\](a) and with a loss-free medium. Then the following Manley-Rowe relations can be derived from Maxwell’s equations for the slowly varying amplitudes: $$\begin{aligned}
\label{manley}
&{d}(|a_1|^2-|a_2|^2)/{dz}=0,& \label{a12}\\
&{d}(|a_3|^2-|a_1|^2)/{dz}=0, \quad{d}(|a_3|^2-|a_2|^2)/{dz}=0.&\label{a23}\end{aligned}$$ Here, $|a_j|^2$ present photon numbers. Equation (\[a12\]) predicts that the difference of the numbers of photons $\hbar\omega_1$ and $\hbar\omega_1$ remains constant through the sample, which indicates their creation in pairs due to the split of photons $\hbar\omega_3$. However, Eqs. (\[a23\]) predict that the differences of the numbers of photons $\hbar\omega_1$ and $\hbar\omega_3$ as well as of $\hbar\omega_2$ and $\hbar\omega_3$ *also remain constant* through the sample. This looks like a breaking of the energy conservation law and is in seemingly striking difference with the fact that the *sum* of the corresponding photon numbers is constant in the analogous case in a PIM. Actually, such unusual dependencies stem from the fact that the waves propagate in the opposite direction. Consequently, *extraordinary distributions* of these fields across the slab and the dependence of their output values on the linear and nonlinear optical properties of the given NIM and on the input intensities of the coupled fields are expected, especially when the conversion efficiency becomes large. Particularly, the conversion rate is expected to grow across the slab with a different rate than in an ordinary medium with a standard coupling geometry. Equations (\[a23\]) indicate unusual feedback, which provides correlated depletion of the control field on one hand and growth of the signal and the idler on the other hand, so that the difference must remain constant along the metaslab. Absorption would change this behavior, which may strongly depend on the absorption dispersion and on the phase mismatch. Investigation of the indicated dependencies is important for optimizing the operational properties of the proposed sensor. Consequently, accounting for the boundary conditions that must be defined at the opposite facets of the slab, this suggests extraordinary distributions of these fields across the slab and extraordinary dependencies of their output values on the linear and nonlinear optical properties of the given NIM and on the input magnitudes of the fields. Such dependencies are essentially different for the coupling schemes depicted in Fig. \[fig4\].
![\[opa\_weak\_w\_abs\] Distribution of the coupled fields across the slab and output characteristics of the amplified transmitted, generated, and depleted control fields for the case of a weak input signal $a_{10}=10^{-4}a_{3L}$. ](fi12T1.eps "fig:"){width=".45\columnwidth"} ![\[opa\_weak\_w\_abs\] Distribution of the coupled fields across the slab and output characteristics of the amplified transmitted, generated, and depleted control fields for the case of a weak input signal $a_{10}=10^{-4}a_{3L}$. ](fi12T2.eps "fig:"){width=".45\columnwidth"}\
![\[opa\_weak\_w\_abs\] Distribution of the coupled fields across the slab and output characteristics of the amplified transmitted, generated, and depleted control fields for the case of a weak input signal $a_{10}=10^{-4}a_{3L}$. ](fi12T3.eps "fig:"){width=".45\columnwidth"}
![\[opa\_strong\_w\_abs\] Distribution of the coupled fields across the slab and output characteristics of the amplified transmitted, generated, and depleted control fields for the case of a strong input signal $a_{10}=0.1 a_{3L}$.](fi13T1.eps "fig:"){width=".45\columnwidth"} ![\[opa\_strong\_w\_abs\] Distribution of the coupled fields across the slab and output characteristics of the amplified transmitted, generated, and depleted control fields for the case of a strong input signal $a_{10}=0.1 a_{3L}$.](fi13T2.eps "fig:"){width=".45\columnwidth"}\
![\[opa\_strong\_w\_abs\] Distribution of the coupled fields across the slab and output characteristics of the amplified transmitted, generated, and depleted control fields for the case of a strong input signal $a_{10}=0.1 a_{3L}$.](fi13T3.eps "fig:"){width=".45\columnwidth"}
Properties of a nonlinear optical reflector and amplifier tailored by the negative-index control field {#mcon}
------------------------------------------------------------------------------------------------------
Consider the scheme depicted in Fig. \[fig4\](a). An analytical solution is not possible for the problem that underlies the concept of the proposed sensor. Figures \[opa\_weak\_w\_abs\] and \[opa\_strong\_w\_abs\] display the results of numerical simulations for this case. Here, z is the length across the slab of thickness L, $T_3(z)=|a_3(z)/a_{3L}|^2$ and $T_1(z)=|a_1(z)/a_{3L}|^2$ are transparencies, $\eta_2(z)=|a_2(z)/a_{3L}|^2$ is the photon conversion efficiency, $g=Xa_3(L)$, $X_{ej}=\sqrt{|k_1k_2/\epsilon_1\epsilon_2|} 2\pi\chi^{(2)}_{ej}$, and $a_{3L}=a_3(L)$ is the amplitude of the input control field. Absorption indices, $\alpha_j$, for the coupled field are indicated, and $\Delta k=0$. Figure \[opa\_weak\_w\_abs\] illustrates the case of a weak input signal so that the depletion of the control field due to the conversion becomes significant only in the vicinity of z=L. Figures \[opa\_weak\_w\_abs\](a) and (b) show the possibility to achieve many orders of amplification of the signal traveling against the control beam and its conversion to the frequency-shifted wave for the intensity of the incoming control field corresponding to $gL\approx 15...20$. Then the numbers of the output photons $\hbar\omega_1$ and $\hbar\omega_2$ make about 10% of that of the input contra-directed control field, which means amplification on the order of $10^7$. Figure \[opa\_strong\_w\_abs\] demonstrates a stronger energy-conversion effect due to a higher input intensity of the signal which, however, leads to lower overall amplification.
Tunable NLO mirror employing a negative-index idler: weak signal and idler approximation {#mid}
-----------------------------------------------------------------------------------------
The physical principles of the proposed nonlinear-optical micromirror, which also can be viewed as an optical data-processing chip, are based on difference-frequency generation of a backward, NI wave in a strongly absorbing double-domain NIM slab \[Fig. \[fig5\](a), which is equivalent to Fig. \[fig4\](c)\].
\
(a)(b)\
Simultaneously, the incident PI signal propagating through the slab experiences OPA. Only the reflected-wave frequency is assumed to fall in the negative-index frequency domain. Here, two incident co-propagating PI waves would produce an all-optically controlled reflectivity accompanied by a tunable frequency shift of the reflected beam. A strong control field at $\omega_3$ and a strong incident wave at $\omega_2$ are both assumed to be PI. The generated difference-frequency wave at $\omega_1=\omega_3-\omega_2$ is NI and, therefore, a backward wave. All three waves experience strong dissipation described by absorption indices $\alpha_{1,2,3}$.
First, consider a simplified model that neglects depletion in the control field due to the TWM conversion and absorption. Hence, the field is assumed to be uniform across the slab. This model allows one to understand the basic properties of TWM coupling of backward waves and difference-frequency generation of a backward NI wave. For the case of a spatially homogeneous, constant control field and a real, nonlinear susceptibility, an analytical solution to Eqs. (\[a1\])-(\[a2\]) can be found, and then the reflectivity, $r_{1}=|a_1(0)/a_{20}^{\ast}|^2$, is given by the equation $$r_{1}=\left|\frac{(g/R)\sin RL}{\cos RL+\left(s/R\right) \sin RL}\right|^2,\quad \label{gen11}$$ whereas the corresponding transmission factor for the PI incident wave, $T_{2}=\left\vert
{a_{2}(L)}/{a_{20}}\right\vert ^{2}$, is $$T_{2}=\left|\frac{\exp \left\{-\left[ \left(
\alpha_{2}/2\right)-s\right] L\right\}}{\cos RL+\left(s/R\right) \sin RL}\right|^2.
\label{T2}$$ Here, $L$ is the thickness of the slab, $g=g_0=Xa_{30}$, $R=\sqrt{g^{2}-s^{2}}$, and $s=[({\alpha_{1}+\alpha_{2}})/{4}]-[i{\Delta
k}/{2}]$. It is seen that the reflectivity and transmittance experience a *set of “geometrical” resonances* [@Yar]. For example, at $s=0$, $r_{1}$=tan$^2(gL)$, $r_{1}=1/cos^2(gL)$ and they tend to infinity at $g L \rightarrow (2j+1)\pi/2$, (j=0, 1, 2, ...), which indicates the possibility of *mirrorless self-oscillations*. A similar behavior is characteristic for distributed-feedback lasers and is equivalent to a great extension of the NLO coupling length. It is known that even weak amplification per unit length may lead to lasing provided that the corresponding frequency coincides with high-quality cavity or feedback resonances. Such a *giant enhancement* of the conversion efficiency in the vicinity of “geometrical” resonances is in *striking contrast* with the exponential dependencies $ T_2\propto \exp(2gL)$ known for the counterparts in ordinary nonlinear materials. Figure \[fig5\](b) displays such resonances in an initially strongly opaque slab. It shows the dependence of the reflectivity $r_1$ and transmittance $T_2$ of the slab on the parameter $gL$ for specific absorption indices at the frequencies of the coupled waves indicated in the figure and $\Delta k=k_{3}-k_{2}-k_{1}=0$. It is seen that the modulation properties of the reflected and transmitted beams differ greatly. The transmitted beam experiences amplification for any magnitude of the parameter $gL$ above a certain level. On the contrary, the intensity of the reflected beam at $\omega_1$ can be varied over a wide range by changing the intensity of the control field. It can be modulated from zero to magnified values *exceeding generation threshold* for both coupled waves. The Manley-Rowe equations for the given coupling scheme predict *the sum* of $|a_1|^2$ and $|a_2|^2$ to be conserved across the slab. Note that in a transparent PI medium, the *difference* of these values is invariant across the slab. It appears that absorption of the control field causes broadening the resonances but does not destroy the resonance behavior. The transmission minima are dependent on the ratio of absorption rates; this is in contrast with the reflectivity minima, which remain robust. Ultimately, the NLO coupling under consideration may provide for significantly sharper growth of the entangled counter-propagating photons in the vicinity of the “geometrical” resonances than the conventional schemes for the same parameters of the slab and the control field. The described extraordinary properties can be utilized for sensing, filtering and conversion of weak light signals.
(a)(b)\
(c)(d)
Alternatively, the phase mismatch causes a decrease of the reflectivity maxima and an increase of the minima. Reflectivity becomes relatively robust against phase mismatch with an increase of the intensity of the control field. It drops dramatically in the range of small phase mismatch and then remains relatively robust at the lower level within the range of greater phase mismatch, as seen in Fig. \[f10\]. The outlined properties of the NLO mirror are determined by the interplay of several processes that have a strong effect on the NLO coupling of contra-propagating waves and, consequently, on their distributions inside the slab.
The depletion of the control field due to energy conversion to the reflected and transmitted beams may have a strong effect on the processes under investigation. Particularly, it limits the amplification in the maxima and may even change the resonance behavior. Basically, the reflected wave has a different frequency than the incident beam. The simulations predict that the quantum conversion efficiency with respect to the control field may be up to several tens of percent, which indicates great enhancement of the reflected and transmitted beams with respect to the incident, positive-index, weak field and the reflectivity that may significantly exceed 100%. Ultimately, the simulations show the possibility of *all-optical tailoring and switching of the reflectivity and transparency* of such a mirror over a wide range by changing the intensity of the control field. More details regarding the given coupling scheme and the one displayed in Fig. \[fig4\](b) can be found in Refs. [@OL; @APB; @EPJD; @WAS; @Ch]
Estimates {#est1}
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The characteristic magnitude of the parameter $gL$, which is required to realize the effects predicted above, is on the order of 1. Assuming $\chi^{(2)}\sim 10^{-6}$ ESU ($\sim10^3$ pm/V), which is on the order of that for CdGeAs$_2$ crystals, and a control field of $I\sim~100$ kW focused on a spot of $D\sim 50$ $\mu$m in diameter, one can estimate that the typical required value of the parameter $gL\sim1$ can be achieved for a slab thickness in the *[microscopic]{}* range of $L\sim1$$ \mu$m, which is comparable with that of the multilayer NIM samples fabricated to date.
Quantum engineering of nonlinearity and coherent quantum control {#fwm}
================================================================
Figure \[fig6\] depicts the basic principles of independent quantum engineering of nonlinearity and its coherent quantum control.
\
(a) (b) (c) (d)
Here, a resonantly enhanced, higher-order, electrical $ \chi ^{(3)}$ NLO response and four-wave mixing are employed in a composite metamaterial with embedded NLO centers (quantum dots, ions, or molecules). Two different options correspondingly depicted in panels (a), (b) and (c), (d) offer different means of quantum control by the driving fields at frequencies $\omega_3$ and $\omega_4$. Only the wave at $\omega_1$ is presumed to be a negative-index one, and all others are PI waves. Four-wave mixing $\omega_1=\omega_3+\omega_4-\omega_2$ replaces the three-wave mixing process described above, so that in this case two strong fields are used for controlling the process. Resonant processes at the depicted quantum transitions driven by the strong control fields make all linear and nonlinear characteristics of the composite intensity-dependent. Among the specific features attributed to the resonant coupling is the fact that the nonlinear susceptibility becomes complex in the vicinity of the introduced resonances, which is followed by a phase shift between the fields and the polarizations. Due to the different contributions of the populations of the coupled energy levels to $\chi^{(3)}_1$ and to $\chi^{(3)}_2$, the latter becomes different and intensity-dependent [@GPRA]. The ratio of real and imaginary parts of $\chi^{(3)}_j$ varies with the change of the resonance offsets. It is also different for schemes depicted in Figs. \[fig6\](b) and (d). It appears that the detrimental role of even a large phase mismatch can be diminished or eliminated through the appropriate adjustment of the real and imaginary parts of $\chi^{(3)}_j$. Such a possibility is not available for off-resonant wave mixing. The properties of the coherent energy transfer from the control fields to the negative-phase signal depend strongly on the ratio of the absorption rates and NLO susceptibilities for the signal and the idler. Figures \[fig6\](b) and (d) present two alternative options for quantum control of NLO propagation processes. Figure \[fig6\](b) depicts the scheme that offers the possibility of incoherent amplification (population inversion or Raman-type) for the idler [@OLM; @APB09; @OL09]. This possibility depends on the set of quantum relaxation rates inherent to the specific doping agent. Figure \[fig6\](d) presents a scheme that offers an alternative possibility. Here, the idler frequency is close to a higher-frequency transition from the ground state, and the signal corresponds to a lower-frequency transition between the excited states. Hence, absorption for the idler is typically larger than for the signal. This situation appears advantageous for robust transparency tailoring. No incoherent amplification is possible here for the idler, whereas incoherent absorption for the signal can be controlled and turned to amplification. Such a possibility is also contingent on the appropriate partial population and coherence relaxation rates attributed to the embedded centers. In all of the schemes outlined above, the linear and nonlinear local parameters can be tailored through quantum control by varying the intensities and frequency-resonance offsets for combinations of the two control driving fields.
Depending on the partial transition relaxation rates associated with levels *m* and *g*, the numerical model relevant to the coupling scheme depicted in Fig. \[fig6\](d) offers neither the possibility of population inversion nor Raman-type amplification. The fact that all involved optical transitions are absorptive determines essentially different features of the overall resonant loss-compensation technique compared to that proposed in Refs. [@OLM; @APB09; @OL09] The following model, which is characteristic of ions and some molecules embedded in a solid host, has been adopted: energy level relaxation rates $\Gamma_n=20$, $\Gamma_g=\Gamma_m=120$ (all in $ 10^6$ s$^{-1}$); partial transition probabilities $\gamma_{gn}=50$, $\gamma_{mn}=90$, (all in $ 10^6$ s$^{-1}$); homogeneous transition half-widths $\Gamma_{gl}$=1.8, $\Gamma_{mn}$=1.9, $\Gamma_{gn}$=1, $\Gamma_{ml}$=1.5, $\Gamma_{mg}=5\times10^{-2}$, $\Gamma_{nl}=5\times10^{-3}$ (all in $10^{11}$ s$^{-1}$); $\lambda_1=756$ nm and $\lambda_2=480$ nm. The density-matrix method [@GPRA] has been used for calculating intensity-dependent local parameters while accounting for the quantum nonlinear interference effects. This allows one to investigate the changes in absorption, amplification, and refractive indices as well as in the magnitudes and signs of the NLO susceptibilities caused by the control fields. These changes depend on the population redistribution over the coupled levels, which in turn strongly depends on the ratio of the partial transition probabilities. Numerical simulations have led to promising conclusions regarding the enhanced reflectivity and amplification of the incident beam as well as possibilities of modulating these characteristics of the proposed microscopic metadevices [@OLM; @OL09; @APB09; @JOA; @WAS; @EPJD; @Ch].
Estimates {#est2}
---------
For transition properties that are characteristic of molecules embedded in a solid environment, estimates show that the required intensity of the control fields at $\omega_3$ and $\omega_4$ are on the order of $I\sim$ 1W/(0.1mm)$^2$. With a resonant absorption cross-section $\sigma_{40}~\sim~10^{-16}$ cm$^2$, which is typical for transitions with an oscillator strength of about one, and a concentration of embedded centers $ N~\sim~10^{19}$ cm$^{-3}$, one estimates $\alpha_{10}~\sim 10^3$ cm$^{-1}$ and the required slab thickness in the [microscopic]{} range $L~\sim (1 - 100) \mu$m. The contribution to the refractive index by the impurities then is estimated as $ \Delta n< 0.5(\lambda/4\pi)\alpha_{40}\sim 10^{-3}$, which essentially does not change the negative refractive index.
Backward-wave optical phonons and distributed nonlinear-optical feedback {#ph}
========================================================================
A different scheme of TWM for ordinary and backward waves has been proposed previously [@ph]. It builds on stimulated Raman scattering (SRS) where two ordinary electromagnetic waves excite a backward, elastic, vibrational wave in a crystal and thus initiate TWM. The possibility of elastic BWs was predicted by L. I. Mandelstam in 1945 [@Ma], who also pointed out that negative refraction is a general property of BWs. The basic idea of replacing the negative-index composites with ordinary crystals and mimicing the uncommon properties of coherent NLO energy exchange between the ordinary and backward waves was shown in Ref. [@ph] and is reviewed below.
![\[fig7\] Negative dispersion of optical phonons and two phase-matching options: (a) co-propagating, and (b) contra-propagating fundamental, control, and Stokes signal waves. Insets: relative directions of the energy flows and the wavevectors.](fig7.eps){width=".7\columnwidth"}
The dispersion curve $\omega(k)$ of phonons in crystals containing more than one atom per unit cell has two branches: acoustic and optical. For the optical branch, the dispersion is negative in the range from zero to the boundary of the first Brillouin zone (Fig. \[fig7\]). Hence, the group velocity of optical phonons, $\mathbf{v}_{v}^{gr}$, is antiparallel with respect to its wavevector, $\mathbf{k}_{v}^{ph}$, and phase velocity, $\mathbf{v}_{v}^{ph}$. This is because $v_{gr}=\partial \omega(k)/\partial k < 0$. Optical vibrations can be excited by light waves through two-photon (Raman) scattering. This gives us the grounds to consider these crystals as analogues to a medium with a negative refractive index at the phonon frequency and to examine the processes of parametric interaction of three waves. Two of these waves are ordinary electromagnetic waves. The third wave is the backward wave of elastic vibrations with directions of its energy flow and wavevector that are opposite to each other. We investigat the extraordinary nonlinear propagation and output properties of the Stokes electromagnetic wave in one of two different coupling geometries, depicted in Fig. \[fig7\], with both utilizing the backward elastic waves. Such unusual properties are in contrast with those attributed to the counterparts in the standard schemes that build on the coupling of co-propagating photons and phonons. They are also different from the properties of the phase-matched mixing of optical and acoustic waves for the case when the latter has its energy flux and wavevector directed against those of one of the optical waves. The revealed properties can be utilized for the creation of optical switches, filters, amplifiers and cavity-free OPOs based on ordinary NLO crystals without the requirement of periodically poling at the nanoscale [@Kh]. Therefore, Fig. \[fig7\] depicts the concept of substituting a sophisticated, nanostructured, negative-index metal-dielectric composite by extensively used and studied ordinary crystals in order to mimic the unparalleled properties of coherent NLO energy exchange between ordinary and backward waves.
Conclusion {#con}
==========
In conclusion, we have described the extraordinary properties of coherent, nonlinear-optical processes in negative-index metamaterials, such as second-harmonic generation, difference-frequency generation and optical parametric amplification. The feasibility of compensating the strong losses inherent to nanostructured, negative-index, plasmonic, metal-dielectric composites is outlined and illustrated with the aid of numerical simulations. The elimination of the detrimental role of optical losses in negative-index metamaterials is the key problem that limits the numerous revolutionary applications of this novel class of electromagnetic metamaterials. All-optical tailoring of transparency and reflectivity is shown to be possible through coherent, nonlinear-optical energy transfer from the ordinary control wave to a negative-index, backward-wave field. This property is intrinsic to negative-index metamaterials. It is shown that besides the nonlinearity attributed to the building blocks of the negative-index host, a strong nonlinear optical response in the composite can be independently provided through embedded, resonant, four-level nonlinear-optical centers. This opens the way to independent nanoengineering and adjustment of the negative index and nonlinearities of metamaterials. In addition, we have described the opportunity for quantum control over the local optical parameters of the metamaterial in this case, which employs constructive and destructive quantum interference tailored by two auxiliary driving control fields. Such a possibility is proven with the aid of a realistic numerical model. The possibility of mimicing the extraordinary properties of coherent energy transfer between ordinary and the contra-propagating, backward waves inherent to negative-index, plasmonic metamaterials is described by making use of an easily available class of crystals with negative dispersion for optical phonons. Among the possible applications of the described nonlinear-optical processes are a novel class of miniature, all-optically tailored sensors, mirrors, frequency-tunable narrow-band filters, quantum switches, amplifiers, cavity-free microscopic optical parametric oscillators that allow generation of entangled counter-propagating left- and right-handed photons, and all-optical data-processing chips. The unique, unparalleled features of the underlying processes are outlined and include the strongly resonant behavior with respect to the material thickness, the density of the embedded resonant centers and the intensities of the control fields, the feasibility of negating the linear phase-mismatch introduced by the host material, and the role of absorption or, conversely, the supplementary nonparametric amplification of the idler.
This work was supported in part by the U.S. National Science Foundation under Grant No. ECCS-1028353, by the U.S. Air Force Office of Scientific Research, by the U.S. Army Research Office under grants W911NF-0710261 and ARO-MURI grant W911NF-0910539, and by the U.S. Office of Naval Research under ONR MURI grant N00014-010942. The authors thank S. A. Myslivets for help with numerical simulations, thoughtful comments and useful discussions and Mark Thoreson for help with preparing the manuscript.
[10]{}
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|
---
author:
- '[Tristan Bice]{}'
title: ' ****\'
---
#### Abstract
Two sets are said to be almost disjoint if their intersection is finite. Almost disjoint subsets of $[\omega]^\omega$ and $\omega^\omega$ have been studied for quite some time. In particular, the cardinal invariants $\mathfrak{a}$ and $\mathfrak{a}_e$, defined to be the minimum cardinality of a maximal infinite almost disjoint family of $[\omega]^\omega$ and $\omega^\omega$ respectively, are known to be consistently less than $\mathfrak{c}$. Here we examine analogs for functions in $\mathbb{R}^\omega$ and projections on $l^2$, showing that they too can be consistently less than $\mathfrak{c}$.
Introduction
============
Projections
-----------
A cardinal invariant is, broadly speaking, any cardinal with some combinatorial definition whose value can not be determined in ZFC. For example, as mentioned above, there is the cardinal invariant $\mathfrak{a}$ whose value must lie within $\aleph_1$ and $\mathfrak{c}$ but (if CH fails) can consistently take almost any value in between. Indeed, we can define a general $R_\kappa$-disjointness number as follows.
Let $\kappa$ be a cardinal, $S$ a set and $R$ a (symmetric) binary relation on $S$. We define $$\mathfrak{a}_\kappa((S,R))=\inf\{|A|:|A|\geq\kappa\wedge A\subseteq S\wedge\forall a,b\in A(aRb)\wedge\forall s\in S\exists a\in A\neg(sRa)\}.$$
So $\mathfrak{a}=\mathfrak{a}_{\aleph_0}(([\omega]^\omega,\not|^*))$, where $A \not|^*\ B\Leftrightarrow|A\cap B|<\infty$. Note here that, as there are trivial examples of finite maximal almost disjoint families of subsets of $\omega$, we need to eliminate these from consideration to define the non-trivial cardinal invariant $\mathfrak{a}$. However, with the relations $R$ we will be concerned with, there will be no such finite maximal $R$-disjoint families, in which case we may omit the restriction that $|A|\geq\aleph_0$ and accordingly omit the subscript, i.e. $\mathfrak{a}((S,R))=\mathfrak{a}_0((S,R))$. Also, as is common practice with such definitions, we shall often abbreviate $\mathfrak{a}_\kappa((S,R))$ by $\mathfrak{a}_\kappa(S)$ or $\mathfrak{a}_\kappa(R)$ when the context makes it clear so, for example, $\mathfrak{a}=\mathfrak{a}_{\aleph_0}(\not|^*)$.
These kinds of cardinal invariants are the focus of this paper. However, they are certainly not the only kinds of cardinal invariants that can be defined see [@a] for a comprehensive introduction to cardinal invariants defined from subsets of $[\omega]^\omega$, for example. In the past few years, analogous cardinal invariants have been defined from subsets of $\mathcal{P}_\infty(H)$, the collection of infinite rank projections (idempotent self-adjoint operators $P\in\mathcal{B}(H)$, i.e. satisfying $P^2=P$ and $P^*=P$) on a separable infinite dimensional Hilbert space $H$ (which, in fact, is unique up to an isometric isomorphism, $l^2$ being the typical example of such a Hilbert space). For example in [@b] the cardinal invariant $\mathfrak{a}^*$ is defined to be the minimum cardinality of a maximal infinite almost orthogonal subset of $\mathcal{P}_\infty(H)$, according to the following definition.
$P,Q\in\mathcal{P}(H)$ are *almost orthogonal*, written $P\bot^*Q$, if $PQ$ is a compact operator.
So, in our notation, $\mathfrak{a}^*=\mathfrak{a}_{\aleph_0}(\bot^*)=\mathfrak{a}_{\aleph_0}((\mathcal{P}_\infty(H),\bot^*))$.
Here we study a slightly different analog of $\mathfrak{a}$ from subsets of $\mathcal{P}_\infty(H)$, namely $\mathfrak{a}(\top^*)$. Indeed, it is a general fact that cardinal invariants defined using infinite subsets of $\omega$ have (at least) two different natural analogs for infinite rank projections, depending on whether we reinterpret $\not|^*$ as $\bot^*$ or $\top^*$. To define $\top^*$, recall that the compact operators $\mathcal{K}(H)$ on $H$ form a closed ideal and so we can form the factor C$^*$-algebra, known as the Calkin algebra of $H$, denoted by $\mathcal{C}(H)$, with canonical homomorphism $\pi:\mathcal{B}(H)\mapsto\mathcal{C}(H)$.
$P,Q\in\mathcal{P}(H)$ are *almost disjoint*, written $P\top^*Q$, if $||\pi(PQ)||<1$.
Note that $P,Q\in\mathcal{P}(H)$ are almost orthogonal if and only if $||\pi(PQ)||=0$, so $P\bot^*Q$ implies $P\top^*Q$. In fact, almost orthogonality is equivalent to almost disjointness together with commutivity (modulo compact operators).
For $P,Q\in\mathcal{P}_\infty(H)$, $$P\bot^*Q\quad\Leftrightarrow\quad\pi(PQ)=\pi(QP)\ \wedge\ P\top^*Q.\nonumber$$
#### Proof:
If $P\bot^*Q$ then $\pi(QP)=\pi((PQ)^*)=\pi(PQ)^*=0^*=0=\pi(PQ)$ and, as mentioned above, $P\top^*Q$. For the other direction note that $\pi(PQ)=\pi(QP)$ implies that $\pi(PQ)$ is a projection and hence $\pi(PQ)=0$, because $||p||=1$ for all non-zero projections $p\in\mathcal{C}(H)$. $\Box$\
Indeed, most research on cardinal invariants defined from projections up till now has focused on (modulo compact) commutative subsets of $\mathcal{P}_\infty(H)$. This is probably because even basic questions about non-commutative projections had not been answered, such as the question of when two non-commutative $P,Q\in\mathcal{P}(H)$ have a greatest $\leq^*$-lower bound, where $$P\leq^*Q\quad\Leftrightarrow\quad\pi(PQ)=\pi(P)\quad\Leftrightarrow\quad PQ-P\textrm{ is compact.}$$
However, recent progress in this direction has been made in [@c], where it is also shown that $||\pi(PQ)||<1$ is equivalent to saying $P$ and $Q$ have no non-trivial (i.e. infinite rank) lower bound with respect to $\leq^*$. So $\mathfrak{a}(\top^*)$ is the minimum cardinality of a maximal antichain of infinite rank projections with respect to $\leq^*$. As $\mathfrak{a}$ is the minimum cardinality of a maximal infinite antichain of infinite subsets of $\omega$ with respect to $\subseteq^*$ (inclusion modulo finite subsets), $\mathfrak{a}(\top^*)$ is perhaps the more natural purely order-theoretic analog of $\mathfrak{a}$ and thus more deserving of the notation $\mathfrak{a}^*$. However, to avoid confusion, we shall avoid this notation entirely and refer only to $\mathfrak{a}(\bot^*)$ and $\mathfrak{a}(\top^*)$.
Functions on $\omega$
---------------------
There are also structures that lie somewhere in between $([\omega]^\omega,\not|^*)$ and $(\mathcal{P}_\infty(H),\top^*)$. To see this, consider $H=\bigoplus_\omega\mathbb{F}^2$ (here $\mathbb{F}$ denotes the scalar field of $H$, either the reals $\mathbb{R}$ or the complex numbers $\mathbb{C}$). Given $(\theta_n)\subseteq[0,\pi/2]$ set $\mathbf{v}_n(m)=\delta_{m,n}(\cos\theta_n,\sin\theta_n)\in H$ and let $P_{(\theta_n)}\in\mathcal{P}(H)$ be such that $\mathcal{R}(P_{(\theta_n)})=V_{(\theta_n)}=\overline{\mathrm{span}}(\mathbf{v}_n)$. Then, for $(\theta_n),(\phi_n)\subseteq[0,\pi/2]$, $$||\pi(P_{(\theta_n)}P_{(\phi_n)})||=\lim\inf_n\cos(\theta_n-\phi_n),$$ so $P_{(\theta_n)}\top^*P_{(\phi_n)}\Leftrightarrow\lim\inf|\theta_n-\phi_n|>0$. This inspires the following definition. $$\label{f|g}
f\not\approx g\Leftrightarrow\lim\inf|f(n)-g(n)|>0.$$ Of course, $\mathfrak{a}([0,\pi/2]^\omega,\not\approx)$ will only equal the minimum cardinality of a maximal antichain of a certain subcollection of projections, but still provides a good intuitive basis for studying antichains of arbitrary projections. In fact, more generally, we define the following.
$f,g\in\mathscr{P}(\mathbb{R})^\omega$ are *lim-inf disjoint*, written $f\not\approx g$, if there exists $m\in\omega$ and $\epsilon>0$ such that $\forall k>m\forall s\in f(k)\forall t\in g(k)(|s-t|>\epsilon)$.
For a function $f$ on $\omega$ define $f'$ on $\omega$ by $f'(n)=\{f(n)\}$, for all $n\in\omega$. So, for $f,g\in\mathbb{R}^\omega$, we have $f\not\approx g$ according to (\[f|g\]) if and only $f'\not\approx g'$ according to . Context should make it clear which definition is being used. If $\mathcal{F}$ is a collection of functions on $\omega$, let $\mathcal{F}_\infty$ be the subset of all $f\in\mathcal{F}$ such that $\{n:f(n)\neq0\}$ is infinite. We are primarily interested in $\mathfrak{a}(\mathbb{R}^\omega)$, $\mathfrak{a}(([\mathbb{R}]^{<\omega}\backslash\{0\})^\omega)$, $\mathfrak{a}(([\mathbb{R}]^{\leq1})^\omega_\infty)$, and $\mathfrak{a}(([\mathbb{R}]^{<\omega})^\omega_\infty)$, all with respect to the relation $\not\approx $ (strictly speaking, the first w.r.t. (\[f|g\]) and the latter three w.r.t. ), both for their intrinsic interest and their relation to $\mathfrak{a}(\top^*)$.
The first thing to note is that it makes no difference if we replace $\mathbb{R}$ in each of these invariants with $\mathbb{Q}$ because every real can be approximated arbitrarily closely by rationals. From a forcing point of view, it is nicer to deal with $\mathbb{Q}$ as rationals are countable and absolute w.r.t. transitive models of ZFC, in contrast to reals.
The next thing to note is that all these invariants are at least as large as $\mathfrak{b}$, the minimum cardinality of a $\leq^*$-unbounded subset of $\omega^\omega$ (where $f\leq^*g\Leftrightarrow\forall^\infty n\in\omega(f(n)\leq^*g(n))$). We also immediately obtain the inequalities $\mathfrak{a}(([\mathbb{R}]^{<\omega}\backslash\{0\})^\omega)\leq\mathfrak{a}(\mathbb{R}^\omega)$ and $\mathfrak{a}(([\mathbb{R}]^{<\omega})^\omega_\infty)\leq\mathfrak{a}(([\mathbb{R}]^{\leq1})^\omega_\infty)$. We show later that these inequalities can consistently be strict, specifically we show in that they differ in the random model.
It is not difficult to see that $\mathfrak{a}=\mathfrak{a}(([\omega]^{<\omega})^\omega_\infty)$. It seems possible that also $\mathfrak{a}=\mathfrak{a}(([\mathbb{R}]^{<\omega})^\omega_\infty)$, although this is not investigated in this paper.
Main Results and Open Questions
-------------------------------
Our main result is that the cardinal invariants just defined can be consistently less than $\mathfrak{c}$, specifically that this holds in the Sacks model. This is done for functions in and for projections in . However, it remains open whether we can actually prove $\mathfrak{a}(\top^*)=\aleph_1$ in ZFC. Indeed, while we have $\mathfrak{a}(\mathbb{R}^\omega)\geq\mathfrak{b}$, for example, and hence consistently $\mathfrak{a}(\mathbb{R}^\omega)(\geq\mathfrak{b})>\aleph_1$, it is not clear that the same is true for bounded functions, i.e. whether we consistently have $\mathfrak{a}([0,1]^\omega)>\aleph_1$. There is even evidence to the contrary. Specifically, we can show that a weaker (i.e. smaller) version of $\mathfrak{a}([0,1]^\omega)$ is indeed equal to $\aleph_1$ in ZFC.
Let $\kappa$ be a cardinal, $S$ a set and $R$ a (symmetric) binary relation on $S$. We define $$\mathfrak{p}_\kappa((S,R))=\inf\{|A|:A\subseteq S\wedge\forall s\in S\exists a\in A\neg(sRa)\wedge\forall B\in[A]^{<\kappa}\exists s\in S\forall b\in B(sRb)\}.$$
Note that we always have $\mathfrak{p}_\kappa((S,R))\leq\mathfrak{a}_\kappa((S,R))$. Also $\mathfrak{p}=\mathfrak{p}_{\aleph_0}(([\omega]^\omega,\not|^*))$, where $\mathfrak{p}$ is the standard pseudointersection number.
In ZFC we have $\mathfrak{p}([0,1]^\omega)=\aleph_1$.
#### Proof:
First we claim that we have a net $(X_x)_{x\in[\aleph_1]^{<\omega}}\subseteq[\omega]^\omega$ satisfying the following properties $$\begin{aligned}
X_x &\subseteq& X_{x\backslash\{\max(x)\}},\textrm{ for all }x\in[\aleph_1]^{<\omega},\label{X1}\\
\textrm{and }X_x\cap X_y &=& 0,\textrm{ for all }x,y\in[\aleph_1]^{<\omega}\textrm{ with }\max(x)=\max(y)\textrm{ and }x\neq y.\label{X2}\end{aligned}$$ To see this, recursively define $(X_x)_{x\in[\aleph_1]^{<\omega}}$ as follows. Set $X_0=\omega$ and, once $(X_x)_{x\in[\xi]^{<\omega}}$ has been defined for some $\xi<\aleph_1$, note that $[\xi]^{<\omega}$ is countable and hence we have a sequence $(x_n)\subseteq[\xi]^{<\omega}$ such that, for each $x\in[\xi]^{<\omega}$, there exists infinitely many $n\in\omega$ such that $x=x_n$. For each $n\in\omega$, recursively pick $m_n\in X_{x_n}\backslash\{m_k:k<n\}$. Once this is done set $X_{x\cup\{\xi\}}=\{m_n:x=x_n\}$, for each $x\in[\xi]^{<\omega}$.
Define $(f_\xi)_{\xi<\aleph_1}\subseteq\mathbb{R}^{\omega\times\omega}$ as follows. For each $x\in[\aleph_1]^{<\omega}$, $n\in X_x$ and $m\in\omega$ let us set $f_{\max(x)}(n,m)=|x|/(m+1)$, noting that (\[X2\]) ensures that each $f_\xi$ is a well defined function on a subset of $\omega$ (we may define $f_\xi$ arbitrarily for the other values).
Now take $f\in[0,1]^{\omega\times\omega}$ and assume that $f\not\approx f_\xi$ for all $\xi\in\aleph_1$. By the pigeonhole principle, we must therefore have $k\in\omega$, $F\in[\omega\times\omega]^{<\omega}$ and $X\in[\aleph_1]^{\aleph_1}$ such that $|f(n,m)-f_\xi(n,m)|>1/k$, for all $\xi\in X$ and $(n,m)\in\omega\times\omega\backslash F$. But then, letting $x$ be any $k$-element subset of $X$ and letting $n\in X_x$ be large enough that $(n,k)\notin F$, we see that $f_{\xi_j}(n,k)=(j+1)/(k+1)$, for all $j\in k$, where $(\xi_j)_{j\in k}$ is the increasing enumeration of $x$. As $f(n,k)\in[0,1]$, we must therefore have $|f(n,k)-f_\xi(n,k)|\leq1/(k+1)$ for some $\xi\in x\subseteq X$, a contradiction. $\Box$\
If we go in between bounded and unbounded functions, i.e. if we look at $\prod I_n$ for intervals $(I_n)\subseteq\mathbb{R}$ which increase in size sufficiently fast, then we see that consistently $\mathfrak{p}(\prod I_n)>\aleph_1$ by the same argument used to show $\mathfrak{a}(\mathbb{R}^\omega)=\mathfrak{c}$ in . On the other hand, it is open whether a slight variant of $\mathfrak{p}([0,1]^\omega)$, namely $\mathfrak{p}(([0,1]^{\leq1})^\omega_\infty)$, is still $\aleph_1$ in ZFC. It is not clear which versions of $\mathfrak{p}$-like and $\mathfrak{a}$-like invariants for functions are closest to those for projections, but an answer to this last question for $\mathfrak{p}(([0,1]^{\leq1})^\omega_\infty)$ will most likely shed light on whether $\mathfrak{p}(\top^*)=\aleph_1$ in ZFC. If $\mathfrak{p}(\top^*)$ is consisently $>\aleph_1$ then the same is true for $\mathfrak{a}(\top^*)$, while a proof that $\mathfrak{p}(\top^*)=\aleph_1$ in ZFC may well provide a way of also proving the stronger statement that $\mathfrak{a}(\top^*)=\aleph_1$ in ZFC.
Consistency Results
===================
Functions on $\omega$
---------------------
A forcing notion $\mathbb{P}$ is $\omega^\omega$-bounding if, for all $p\in\mathbb{P}$ and names $\dot{f}$ for elements of $\omega^\omega$, there exists $q\leq p$ and $g\in\omega^\omega$ such that $q\Vdash\forall n\in\omega(\dot{f}(n)\leq g(n))$.
A Cohen indestructible infinite MAD family of $[\omega]^\omega$ is constructed in [@e] VIII Theorem 2.3. Our approach, using instead the $\omega^\omega$-bounding property (which does not apply to Cohen forcing), is more similar to [@f] Lemma III.1.
If CH holds in the ground model $V$ and $\mathbb{P}$ is an $\omega^\omega$-bounding proper forcing notion with $(|\mathbb{P}|=\aleph_1)^V$, we have lim-inf disjoint $\mathcal{F}\subseteq([\mathbb{Q}]^{<\omega}\backslash\{0\})^\omega$ in $V$ such that $\mathds{1}\Vdash_\mathbb{P}\mathcal{F}$ is maximal.
#### Proof:
Thanks to CH and properness, there are only $\aleph_1$ many nice names for elements of $([\mathbb{Q}]^{\leq1})^\omega_\infty$. Let $(p_\xi,\tau_\xi)_{\xi\in\omega_1}$ enumerate all pairs of elements of $\mathbb{P}$ and such nice names. We construct $(f_\xi)_{\xi\in\omega_1}\subseteq([\mathbb{Q}]^{<\omega}\backslash\{0\})^\omega$ by recursion as follows. Say we have constructed $(f_\xi)_{\xi\in\gamma}$. If $$p_\gamma\not\Vdash\forall\xi\in\gamma(f_\xi\not\approx \tau_\gamma)$$ then set $f_\gamma$ to be, say, $f_0$ (or simply leave $f_\gamma$ undefined). Otherwise, let $(\xi_n)_{n\in\omega}$ be an enumeration of all $\xi\in\gamma$. Thanks to the $\omega^\omega$-bounding property, there exists $(N_n),(m_n)\subseteq\omega$ in the ground model and $p'_\gamma\leq p_\gamma$ such that $$\label{b1}
p'_\gamma\Vdash\forall n\in\omega\forall k\geq m_n\forall s\in f_{\xi_n}(k)\forall t\in\tau_\gamma(k)(|s-t|>1/N_n).$$ Again, by the $\omega^\omega$-bounding property, we may find $p''_\gamma\leq p'_\gamma$ and $f\in([\mathbb{Q}]^{<\omega})^\omega$ such that we have $p''_\gamma\Vdash\forall n\in\omega(\tau_\gamma(n)\subseteq f(n))$. For all $n\in\omega$ let $$f_\gamma(n)=\{r\in\mathbb{Q}:\exists p\leq p''_\gamma(p\Vdash r\in\tau_\gamma(n))\}\subseteq f(n).$$ Note that (\[b1\]) implies that $f_\xi\not\approx f_\gamma$, for all $\xi<\gamma$. As $\gamma$ is a countable ordinal, we may also recursively make $f_\gamma(n)\neq0$, for all $n\in\omega$, while still having $f_\xi\not\approx f_\gamma$, for all $\xi<\gamma$. This completes the recursion.
Let $\mathcal{F}=\{f_\xi:\xi\in\omega_1\}$. If we had $\mathds{1}\not\Vdash_\mathbb{P}\mathcal{F}$ is maximal, then there would exist $p\in\mathbb{P}$ and a nice name $\tau$ for an element of $([\mathbb{Q}]^{\leq1})^\omega_\infty$ such that $p\Vdash\forall\xi\in\omega_1(f_\xi\not\approx \tau)$. Taking $\gamma\in\omega_1$ such that $p=p_\gamma$ and $\tau=\tau_\gamma$ we have $$\begin{aligned}
p_\gamma &\Vdash& \forall\xi\in\gamma(f_\xi\not\approx \tau)\qquad\textrm{and hence}\\
p''_\gamma &\Vdash& \forall n\in\omega(\tau_\gamma(n)\subseteq f_\gamma(n)),\end{aligned}$$ by our recursive construction. But we also have $p_\gamma\Vdash(f_\gamma\not\approx \tau_\gamma)$ and thus $p''_\gamma\leq p_\gamma$ forces contradictory statements, a contradiction. $\Box$\
Two reals $f,g\in\omega^\omega$ are *eventually different* if $\{n\in\omega:f(n)=g(n)\}$ is finite.
If $V$ is a transitive model of ZFC and $f\in\omega^\omega$ is eventually different from every element of $\omega^\omega\cap V$ then $f$ is lim-inf disjoint from every element of $\mathbb{R}^\omega\cap V$.
#### Proof:
Simply note that, for any $g\in\mathbb{R}^\omega\cap V$, there exists $h\in\omega^\omega\cap V$ such that $|g(n)-h(n)|\leq1/2$, for all $n\in\omega$, and hence $\{n\in\omega:|f(n)-g(n)|<1/2\}\subseteq\{n\in\omega:|f(n)-h(n)|<1\}$ is finite. $\Box$\
For any index set $I$, let $\mu$ be the standard product measure on $2^I$ and let $\mathbb{B}_I\in V$ be the collection of (equivalence classes of) Baire subsets of $2^I$ ordered by inclusion modulo null subsets.
If $V$ is a c.t.m. of ZFC+CH and $(\kappa=\kappa^\omega)^V$ then in $V^{\mathbb{B}_I}$ we have $$\begin{aligned}
\aleph_1 &=& \mathfrak{a}(([\mathbb{R}]^{<\omega}\backslash\{0\})^\omega)=\mathfrak{a}(([\mathbb{R}]^{<\omega})^\omega_\infty)\quad\textrm{and}\label{r1}\\
\mathfrak{c} &=& \mathfrak{a}(\mathbb{R}^\omega)=\mathfrak{a}(([\mathbb{R}]^{\leq1})^\omega_\infty)=\kappa.\label{r2}\end{aligned}$$
#### Proof:
Any new real in $V^{\mathbb{B}_I}$ will be in $V^{\mathbb{B}_J}$ for some countable $J\subseteq I$. But $V$ and $\mathbb{B}_J$ satisfy the hypotheses of so (\[r1\]) follows from this. On the other hand, any collection of $<\kappa$ reals will be in a proper submodel of $V^{\mathbb{B}_I}$, over which $V^{\mathbb{B}_I}$ will contain an eventually different real, so (\[r2\]) follows from . $\Box$\
A forcing notion $\mathbb{P}$ has the Sacks property if, for all $p\in\mathbb{P}$ and names $\dot{f}$ for elements of $\omega^\omega$, there exists $q\leq p$ and $F\in\mathscr{P}(\omega)^\omega$ such that $|F(n)|=2^n$, for all $n\in\omega$, and $q\Vdash\forall n\in\omega(\dot{f}(n)\in F(n))$.
If CH holds in the ground model $V$ and $\mathbb{P}$ is a proper forcing notion with the Sacks property and $(|\mathbb{P}|=\aleph_1)^V$ then we have lim-inf disjoint $\mathcal{F}\subseteq\mathbb{Q}^\omega$ in $V$ such that $\mathds{1}\Vdash_\mathbb{P}\mathcal{F}$ is maximal.
#### Proof:
It is immediate that any forcing notion with the Sacks property is $\omega^\omega$-bounding so we may proceed as in the proof of up to and including (\[b1\]). Yet again, by $\omega^\omega$-bounding, we may decrease $p'_\gamma$ and increase $(m_n)$ if necessary so that $$p'_\gamma\Vdash\forall n\in\omega(|\{k\in m_{n+1}\backslash m_n:\tau_\gamma(k)\neq0\}|\geq2^n).$$ Then, by the Sacks property, there exists $p''_\gamma\leq p'_\gamma$ and a function $F$ on $\omega$ such that, for each $n\in\omega$, $F(n)$ is a non-empty collection of functions from $m_{n+1}\backslash m_n$ to $[\mathbb{Q}]^{\leq1}$ of size at most $2^n$ satisfying $$\begin{aligned}
&& p''_\gamma\Vdash\forall n(\tau_\gamma\upharpoonright m_{n+1}\backslash m_n\in F(n))\qquad\textrm{and}\label{cond2}\\
&& \forall n\forall f\in F(n)\exists p\leq p''_\gamma(p\Vdash\tau_\gamma\upharpoonright m_{n+1}\backslash m_n=f).\label{cond3}\end{aligned}$$ Let $f_\gamma\in([\mathbb{Q}]^{\leq1})^\omega$ satisfy $$\begin{aligned}
&& \forall n\forall f\in F(n)\exists k\in m_{n+1}\backslash m_n({f_\gamma(k)}={f(k)}\neq0)\qquad\textrm{and}\label{cond4}\\
&& \forall n\forall k\in m_{n+1}\backslash m_n\exists f\in F(n)(f_\gamma(k)=f(k)).\label{cond5}\end{aligned}$$ (\[b1\]), (\[cond3\]) and (\[cond5\]) imply that $f_\xi\not\approx f_\gamma$, for all $\xi<\gamma$. As $\gamma$ is a countable ordinal, we may again recursively make $|f_\gamma(n)|=1$, for all $n\in\omega$, while still having $f_\xi\not\approx f_\gamma$, for all $\xi<\gamma$. On the other hand, (\[cond2\]) and (\[cond4\]) imply that $$\begin{aligned}
p''_\gamma &\Vdash& \forall n\exists k\in m_{n+1}\backslash m_n(\tau_\gamma(k)=f_\gamma(k)\neq0)\\
&\Vdash& \exists^\infty n(\tau_\gamma(n)=f_\gamma(n)\neq0),\end{aligned}$$ leading to essentially the same contradiction as in the proof of . $\Box$\
Let $\mathbb{S}$ be Sacks forcing and, for any ordinal $\xi$, let $\mathbb{S}_\xi$ its $\xi$-step countable support iteration.
If CH holds in the ground model $V$ then in $V^{\mathbb{S}_{\omega_2}}$ we have $$\mathfrak{a}(\mathbb{R}^\omega)=\mathfrak{a}(([\mathbb{R}]^{\leq1})^\omega_\infty)=\aleph_1(<\aleph_2=\mathfrak{c}).$$
#### Proof:
As shown in [@f] Theorem III.2, it suffices to find $\mathcal{F}\in V\cap\mathbb{Q}^\omega$ such that $\mathds{1}\Vdash_{\mathbb{S}_{\omega_1}}\mathcal{F}$ is maximal. As $\mathbb{S}$ is proper and has the Sacks property, the same applies to its iterations and hence, as $\mathbb{S}_{\omega_1}$ has a dense subset of size $\aleph_1$, this follows from . $\Box$\
Projections
-----------
As before, let $H$ be $l^2$, a separable infinite dimensional Hilbert space with (Hilbert) basis $(e_n)$.
Say we have orthogonal unit vectors $v_0,\ldots,v_{n-1}$ in $H$ and a projection $P$ on $H$ such that $v_i\perp Pv_j$, for all distinct $i,j\in n$. Then $||PP_{\mathrm{span}_{i\in n}(v_i)}||=\max_{i\in n}||Pv_i||$.
#### Proof:
For all distinct $i,j\in n$, we have $v_i\perp Pv_j$ and hence $\langle Pv_i, Pv_j\rangle=\langle v_i, Pv_j\rangle=0$, i.e. $Pv_i\perp Pv_j$. Thus, for all $\alpha_0,\ldots,\alpha_{n-1}\in\mathbb{F}$ such that $|\alpha_0|^2+\ldots+|\alpha_{n-1}|^2=1$, $$\begin{aligned}
||P(\alpha_0v_0+\ldots+\alpha_{n-1}v_{n-1})||^2 &=& ||\alpha_0Pv_0+\ldots+\alpha_{n-1}Pv_{n-1}||^2\\
&=& |\alpha_0|^2||Pv_0||^2+\ldots+|\alpha_{n-1}|^2||Pv_{n-1}||^2\\
&\leq& \max_{i\in n}||Pv_i||^2.\end{aligned}$$ i.e. $||PP_{\mathrm{span}_{i\in n}(v_i)}||\leq\max_{i\in n}||Pv_i||$. The reverse inequality is immediate. $\Box$\
Let $(I_n)$ be a sequence of finite intervals of $\omega$, i.e. of sets of the form $m\backslash n$ for some $n\in m\in\omega$. We say $(I_n)$ is an *interval subpartition* (of $\omega$) if $\max(I_n)<\min(I_{n+1})$, for all $n\in\omega$. $(I_n)$ is an *interval partition* if $\min(I_0)=0$ and $\max(I_n)+1=\min(I_{n+1})$, for all $n\in\omega$.
$V\subseteq H$ is a *block subspace* (w.r.t. $(e_n)$) if there exists an interval (sub)partition $(I_n)$ and (unit) vectors $(v_n)\subseteq H$ such that $v_n\in\mathrm{span}_{i\in I_n}(e_i)$, for all $n\in\omega$, and $V=\overline{\mathrm{span}}(v_n)$. $V\subseteq H$ is a *generalized block subspace* (w.r.t. $(e_n)$) if there exists an interval (sub)partition $(I_n)$ and finite subspaces $(F_n)\subseteq H$ such that $F_n\subseteq\mathrm{span}_{i\in I_n}(e_i)$, for all $n\in\omega$, and $V=\bigoplus F_n(=\overline{\sum F_n}$, i.e. the closure of the subspace of finite linear combinations of elements of $\bigcup F_n)$.
Let $\mathbb{G}$ be an absolute (w.r.t. countable transitive models of set theory) countable dense subfield of the scalar field $\mathbb{F}$ that is closed under taking square roots (eg. the algebraic numbers). Then we define $\mathbb{G}$-block subspaces analogously, with the extra requirement that each $v_n$ is a $\mathbb{G}$-vector, i.e. in the $\mathbb{G}$-span of the basis vectors $(e_i)$.
Note that the projections onto infinite dimensional $\mathbb{G}$-block subspaces are $\leq^*$-dense among all infinite rank projections. For any $P,Q,R\in\mathcal{P}(H)$ we have $$R\leq^*Q\Rightarrow||\pi(PR)||=||\pi(PQR)||\leq||\pi(PQ)||$$ and hence, to verify that some almost disjoint family $\mathcal{P}$ of infinite rank projections is maximal we need only verify that, for all projections $R$ onto infinite dimensional $\mathbb{G}$-block subspaces of $H$, there exists $P\in\mathcal{P}$ such that $||\pi(PR)||=1$.
If CH holds in the ground model $V$ and $\mathbb{P}$ is a proper forcing notion with the Sacks property such that $(|\mathbb{P}|=\aleph_1)^V$ then there exists an almost disjoint family $\mathcal{P}\subseteq\mathcal{P}_\infty(H)$ in $V$ such that $\mathds{1}\Vdash_\mathbb{P}\mathcal{P}$ is maximal.
#### Proof:
Thanks to CH and properness, there are only $\aleph_1$ many nice names for projections onto infinite dimensional $\mathbb{G}$-block subspaces of $H$. In fact, all bounded linear operators can be considered as reals by looking at their matrix representations. Indeed, if we are being truly formal, we have to deal with these matrices as coding the projections we are talking about, seeing as $l^2$, the domain of the projections, becomes larger in any extension adding reals. That aside, let $(p_\xi,\tau_\xi)_{\xi\in\omega_1}$ enumerate all pairs of elements of $\mathbb{P}$ and such nice names. We construct $(P_\xi)_{\xi\in\omega_1}\subseteq\mathcal{P}$ and interval subpartitions $((K^\xi_n))_{\xi\in\omega_1}$ (each $(K^\xi_n)$ corresponding to the generalized blocks of $P_\xi$) by recursion as follows. Say we have constructed $(P_\xi)_{\xi\in\gamma}$. If $$p_\gamma\not\Vdash\forall\xi\in\gamma||\pi(\tau_\gamma P_\xi)||<1$$ then set $P_\gamma$ to be, say, $P_0$ (or simply leave $P_\gamma$ undefined). Otherwise, let $(\xi_n)_{n\in\omega}$ be an enumeration of all $\xi\in\gamma$ and let $(\dot{I}_n)$ and $(\dot{v}_n)$ be names for the interval partition and corresponding unit vectors that define the block subspace $\mathcal{R}(\tau_\gamma)$. Thanks to the Sacks property which in turn implies the $\omega^\omega$-bounding property, there exists $(N_n)\subseteq\omega$ in the ground model and $p'_\gamma\leq p_\gamma$ such that $$p'_\gamma\Vdash\forall n||\pi(\tau_\gamma P_{\xi_n})||<N_n/(N_n+1).$$ Then, again thanks to the $\omega^\omega$-bounding property, there exists increasing $(m_n)\subseteq\omega$, with $m_0=0$, in the ground model $V$ and $p''_\gamma\leq p'_\gamma$ such that $$\label{cond1p}
p''_\gamma\Vdash\forall l<n||P_{\xi_l}P_{\mathrm{span}\{\dot
{v}_k:k\geq m_n\}}||<N_l/(N_l+1).$$ Yet again thanks to the $\omega^\omega$-bounding property, there exists increasing an interval partition $(J_n)$ of $\omega$ in the ground model $V$ and $p'''_\gamma\leq p''_\gamma$ such that $$p'''_\gamma\Vdash\forall n\exists m(\dot{I}_m\subseteq J_n).$$ Let $(K^\gamma_n)$ be an interval subpartition such that, for all $n\in\omega$, $$\max\{k:K^\gamma_n\cap K^{\xi_l}_k\neq0\}<\min\{k:K^\gamma_{n+1}\cap K^{\xi_l}_k\neq0\},$$ for all $l<n$, $K^\gamma_n$ contains only intervals $J_k$ such that $k\geq m_n$ and contains more than $(n+1)(2^n-1)$ such intervals, and hence $$p'''_\gamma\Vdash\forall n((\dot{I}_k\subseteq K^\gamma_n\Rightarrow k\geq m_n)\wedge|\{k:\dot{I}_k\subseteq K^\gamma_n\}|>(n+1)(2^n-1)).$$ By the Sacks property, there exists $p''''_\gamma\leq p'''_\gamma$ and function $F$ on $\omega$ such that, for each $n\in\omega$, $|F(n)|\leq2^n$ and each element of $F(n)$ is a collection of $\mathbb{G}$-vectors such that $$\begin{aligned}
&& p''''_\gamma\Vdash\forall n(\{\dot{v}_k:\dot{I}_k\subseteq K^\gamma_n\}\in F(n))\label{cond2p}\\
&\textrm{and}& \forall n\forall \mathcal{V}\in F(n)\exists p\leq p''_\gamma(p\Vdash\{\dot{v}_k:\dot{I}_k\subseteq K^\gamma_n\}=\mathcal{V}).\label{cond3p}\end{aligned}$$
For each $n$, let $\mathcal{V}_0,\ldots,\mathcal{V}_{|F(n)|-1}$ enumerate the elements of $F(n)$ and define vectors $u_n^0,\ldots,u_n^{|F(n)|-1}$ recursively as follows. Let $u_n^0$ be any element of $\mathcal{V}_0$. Once $u_n^j$ has been defined, for $j<i$, let $u_n^i$ be any unit ($\mathbb{G}$-)vector in $\mathrm{span}(\mathcal{V}_i)$ that is perpendicular to $u_0,\ldots,u_{i-1}$ and $P_ku_j$, for all $k\in n$ and $j\in i$. As $\dim\mathrm{span}(\mathcal{V})=|\mathcal{V}|>(n+1)(2^n-1)$, for all $\mathcal{V}\in F(n)$, and $|F(n)|\leq2^n$ we may continue this recursion until we have defined $u_n^{|F(n)|-1}$. Let $U_n=\mathrm{span}(u_n^i)_{i\in|F(n)|}$ and let $P_\gamma$ be the projection onto $\bigoplus U_n$. , (\[cond1p\]) and (\[cond3p\]) imply that $$\forall l\in\omega(||P_{\xi_l}P_{\bigoplus_{n>l}U_n}||<N_l/(N_l+1))$$ and hence that $\forall n(||\pi(P_{\xi_n}P_\gamma)||<N_n/(N_n+1))$, as $\pi(P_{\bigoplus_{n>l}U_n})=\pi(P_\gamma)$, which, in particular, means that $P_\gamma$ is almost disjoint from $(P_\xi)_{\xi\in\gamma}$. On the other hand, (\[cond2p\]) implies that $$p''''_\gamma\Vdash\forall n(\mathcal{R}(\tau_\gamma)\cap\mathcal{R}(P_\gamma)\cap\mathrm{span}_{i\in K_n}(e_i)\neq0)$$ which, in particular, means that $p''''_\gamma\Vdash\mathcal{R}(\tau_\gamma)\cap\mathcal{R}(P_\gamma)$ is infinite dimensional, so $$p''''_\gamma\Vdash||\pi(\tau_\gamma P_\gamma)||=1.$$ This completes the recursive construction.
If we had $\mathds{1}\not\Vdash_\mathbb{P}\mathcal{P}$ is maximal, then there would exist $p\in\mathbb{P}$ and a nice name $\tau$ for a projection onto an infinite dimensional $\mathbb{G}$-block subspace of $H$ such that $$p\Vdash\forall\xi\in\omega_1(||\pi(\tau_\xi P_\xi)||<1).$$ Taking $\gamma\in\omega_1$ such that $p=p_\gamma$ and $\tau=\tau_\gamma$ we have $$\begin{aligned}
p_\gamma &\Vdash& \forall\xi\in\gamma(||\pi(\tau_\gamma P_\xi)||<1)\qquad\textrm{and hence}\\
p''''_\gamma &\Vdash& (||\pi(\tau_\gamma P_\gamma)||=1),\end{aligned}$$ by our recursive construction. But we also have $$p_\gamma\Vdash||\pi(\tau_\gamma P_\gamma)||<1$$ and thus $p''''_\gamma\leq p_\gamma$ forces contradictory statements, a contradiction. $\Box$\
If CH holds in the ground model $V$ then there exists almost disjoint $\mathcal{P}\subseteq\mathcal{P}_\infty(H)$ in $V$ such that $\mathds{1}\Vdash_{\mathbb{S}_{\omega_2}}\mathcal{P}$ is maximal.
#### Proof:
As shown in [@f] Theorem III.2, it suffices to find $\mathcal{P}$ such that $\mathds{1}\Vdash_{\mathbb{S}_{\omega_1}}\mathcal{P}$ is maximal. As $\mathbb{S}$ is proper and has the Sacks property, the same applies to its iterations and hence, as $\mathbb{S}_{\omega_1}$ has a dense subset of size $\aleph_1$, this follows from . $\Box$\
If CH holds in the ground model $V$ then $V^{\mathbb{S}_{\omega_2}}$ satsifies $\mathfrak{a}(\top^*)=\aleph_1(<\aleph_2=\mathfrak{c})$.
[6]{} Tristan Bice, *The Order on Projections in C$^*$-Algebras of Real Rank Zero*, Bull. Polish Acad. Sci. Math. 60 (2012), 37-58. Tomek Bartoszynski and Haim Judah, *Set Theory: On the Structure of the Real Line*, A K Peters (1995). Andreas Blass, *Combinatorial Cardinal Characteristics of the Continuum*, Handbook of Set Theory Volume I p395 (2010). Michael Hrusak, *Life in the Sacks Model*, Jaroslav Tiser and Bohuslav Balcar (eds.): Proceedings of the 29th Winter School on Abstract Analysis. Charles University, Praha (2001), Acta Universitatis Carolinae - Mathematica et Physica, Vol. 42, No. 2. 43-58, http://dml.cz/dmlcz/702077. Kenneth Kunen, *Set Theory: An Introduction to Independence Proofs*, Studies in Logic and the Foundations of Mathematics Volume 102 (1980), Elsevier. Eric Wofsey, *$P(\omega)/\mathrm{Fin}$ and Projections in the Calkin Algebra*, Proc. Amer. Math. Soc. 136 (2008), no. 2, 719-726.
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---
abstract: 'One important question in relativistic heavy ion collisions is if hadrons, specifically anti-hyperons, are in equilibrium before thermal freezeout because strangeness enhancement has long been pointed to as a signature for Quark Gluon Plasma. Because anti-baryons have long equilibration times in the hadron gas phase it has been suggested that they are “born" into equilibrium. However, Hagedorn states, massive resonances, which are thought to appear near the critical temperature, contribute to fast chemical equilibration times for a hadron gas by providing extra degrees of freedom. Here we use master equations to describe the interplay between Hagedorn resonances, pions, and baryon anti-baryon pairs as they equilibrate over time and observe if the baryons and anti-baryons are fully equilibrated within the fireball.'
author:
- 'J. Noronha-Hostler'
- 'C. Greiner'
- 'I. Shovkovy'
title: Chemical Equilibration at the Hagedorn Temperature
---
Introduction
============
It has been suggested that one of the main signatures for the Quark Gluon Plasma (QGP), which is a state of matter that consists of deconfined quarks and gluons, could be the equilibration of anti-hyperons and multi-strange baryons [@Koch:1986ud]. The idea was that within a hadron gas there would not be enough time for these particles to reach chemical equilibrium and thus they should be “born" into equilibrium after the QGP phase transition [@Stock:1999hm; @Heinz:2006ur]. Indeed, experimentally we know that the particle abundancies reach chemical equilibration close to the phase transition [@Braun-Munzinger]. Multi-mesonic reactions within the standard hadron gas model, which can explain abundancies at SPS energies [@Rapp:2000gy; @Greiner], cannot account for the short chemical equilibration times and high abundancies of anti-baryons at RHIC [@Kapusta; @Huovinen:2003sa] (but also see [@BSW]). In this paper we focus on Hagedorn resonances (heavy resonances that have an exponential mass spectrum that appear near the critical temperature) that drive multi-pionic reactions and also produce baryon anti-baryon pairs as was suggested in Ref. [@Greiner:2004vm]. Once we include the Hagedorn resonances we see that it is possible for the anti-baryons (future work will discuss anti-hyperons and multi-strange baryons) to quickly equilibrate within the fireball just “below" the phase transition.
Originally, (anti-)strangeness enhancement at CERN-SPS energies in comparison to $pp$-data, which was primarily observed in anti-hyperons and multi-strange baryons, was thought of as a signature for QGP. Using binary strangeness production reactions such as $$\label{eqn:strangeprod}
\pi+\bar{p}\leftrightarrow \bar{K}+\bar{\Lambda}$$ or binary strangeness exchange reactions $$\label{eqn:strangeex}
K+\bar{p}\leftrightarrow \pi+\bar{\Lambda}$$ it was concluded that it took far too long for chemical equilibrium to be reached within the hadron gas phase [@Koch:1986ud]. Thus, it was proposed that the QGP was already observed at SPS because strange quarks can be produced more abundantly by gluon fusion, which would then account for strangeness enhancement in hadrons following hadronization and rescattering of strange quarks [@Koch:1986ud].
On the other hand, to explain secondary production of anti-hyperons the idea was suggested that strangeness enhancement could be explained using multi-mesonic reactions such as $$\label{eqn:antiproton}
\bar{p}+N\leftrightarrow n\pi$$ for the anti-protons [@Rapp:2000gy] and for the anti-hyperons $$\begin{aligned}
\label{eqn:antihypall}
\bar{\Sigma},\bar{\Lambda}+N&\leftrightarrow &n\pi+K\nonumber\\
\bar{\Xi}+N&\leftrightarrow &n\pi+2K\nonumber\\
\bar{\Omega}+N&\leftrightarrow &n\pi+3K\end{aligned}$$ found in Ref. [@Greiner]. The anti-hyperons can be rewritten into the general equation $$\label{eqn:antihyp}
\bar{Y}+N\leftrightarrow n\pi+n_{Y}K.$$ The arrows in Eqs. (\[eqn:antiproton\]), (\[eqn:antihypall\]) signify the equal probability that a decay can occur in either direction otherwise known as detailed balance.
The time scale of a standard hadron gas at SPS can be then estimated using the multi mesonic reactions in Eq. (\[eqn:antiproton\]), (\[eqn:antihypall\]). Using $$\label{eqn:crosssection}
\sigma_{N\bar{Y}}\approx\sigma_{N\bar{p}}\approx 50\;\mathrm{mb}$$ the chemical equilibration time is then $$\label{eqn:prochemtime}
\tau_{\bar{Y}}=\left(\Gamma_{\bar{Y}}\right)^{-1}=
\frac{1}{\langle\langle\sigma_{N\bar{Y}}v_{N\bar{Y}}\rangle\rangle
\rho_{B}}\approx 1-3\; \frac{\mathrm{fm}}{c},$$ where $\rho_{B}\approx \rho_{0}\;\mathrm{to}\;2\rho_{0}$, which is typical for SPS [@Greiner; @Rapp:2000gy]. Therefore, the time scale given in Eq. (\[eqn:prochemtime\]) is short enough to account for chemical equilibration within the cooling hadronic fireball at SPS.
If we apply our same understanding of the hadron gas phase to RHIC temperatures our time scales are much longer. The equilibrium rate of $\Omega$ at RHIC at $T=170$ MeV is $\Gamma_{\Omega}^{chem}\approx\langle\sigma_{\Omega
\bar{B}}v_{\Omega \bar{B}}\rangle N_{\bar{B}}$ where the cross section is $\sigma\approx
30\;\mathrm{mb}$ and the baryon density is $N_{B}^{eq}=N_{\bar{B}}^{eq}\approx0.04\;\mathrm{fm}^{-3}$ , which leads to a time scale of $\tau_{\Omega}=\frac{1}{\Gamma_{\Omega}}\approx
10\;\frac{\mathrm{fm}}{\mathrm{c}}$. However, considering that the fireball’s time scale in the hadronic stage is $\tau<
4\;\frac{\mathrm{fm}}{\mathrm{c}}$, a standard hadron gas could not explain the apparent chemical equilibration observed in baryons within the fireball. Moreover, these results were also backed up in Ref. [@Kapusta] where using a fluctuation-dissipation theorem it was found that the equilibration time of baryons and anti-baryons, $\tau\approx
10\left[\frac{\mathrm{fm}}{\mathrm{c}}\right]$, at RHIC temperatures. In the $5\%$ most central Au-Au collisions the baryon anti-baryon production is roughly three times lower than the measured experimental values if it starts out of equilibrium (specifically at zero for reactions of type (\[eqn:antiproton\]) and (\[eqn:antihypall\])) shown in Fig. (\[fig:alambda\]), taken from Ref. [@Huovinen:2003sa], for $\bar{\Lambda}$ production.
![Anti-lambda production in the most central collisions at RHIC. The bottom three dashed lines start with no anti-lambdas at $T=180$ MeV whereas the solid lines assume that the anti-lambdas begin in equilibrium. The shaded area shows the experimental results taken from Ref. [@Huovinen:2003sa]. The set of lines shows the difference for varying coupling coefficients.[]{data-label="fig:alambda"}](alambda.eps){width="6cm"}
Because of the apparent differences in the equilibration times some have suggested that the hadrons are “born" into equilibrium i.e. the system is already in a chemically frozen out state at the end of the phase transition [@Heinz:2006ur].
In this paper we use Hagedorn states to provide the extra degrees of freedom needed to match experimental results when the anti-baryons begin out of equilibrium. Baryon anti-baryon production develops according to the possible reaction $$\label{eqn:bbproduction}
n\pi\leftrightarrow HS\leftrightarrow n\pi+B\bar{B},$$ which is initially discussed in Ref. [@Greiner:2004vm]. The Hagedorn states arethought as massive resonances with short time scales and only contribute near the critical temperature. They can then catalyze rapid equilibration of baryons and anti-baryons near $T_{c} $. The results obtained here lead us to believe that the baryon anti-baryon pairs have sufficient time to equilibrate within the fireball close to the phase boundary.
Hagedorn States
===============
In the 1960’s Hagedorn found a fit for an experimentally growing mass spectrum [@Hagedorn:1968jf], $\rho$, which is described as $$\begin{aligned}
\label{eqn:Hagedorn}
\rho&=&\int_{m_{0}}^{M}F(m)e^{\frac{m}{T_{H}}}dm,\nonumber\\
F(m)&=&\frac{A}{\left(m^2 +m^2 _{0}\right)^{\frac{5}{4}}}.\end{aligned}$$ where the minimum mass is $m_{0}=500$ MeV, the maximum mass is $M=7$ GeV, the Hagedorn temperature is $T_{H}=180$ MeV, which fits within Lattice QCD predictions [@Karsch], and $A=0.5$ is a free parameter. Because of the exponential growth, near the Hagedorn temperature the Hagedorn states can account for the extra degrees of freedom needed to match experimental values. $A$ is then chosen by looking at the energy density and trying take into account the extra degrees of freedom needed. Here we are considering only mesonic, non-strange Hagedorn states with masses between $2\;\mathrm{GeV\; and}\;7$ GeV.
To describe the dynamical behaviour of the Hagedorn states we use rate equations. Rate equations have both loss and gain terms and have the basic form $$\label{eqn:rateex}
\frac{dn}{dt}=-loss+gain.$$ We assume a system with no net baryon density i.e. $N_{B}=N_{\bar{B}}=N_{B\bar{B}}$. This should approximately be the case at RHIC at midrapidity. For the reaction $HS\leftrightarrow n\pi+B\bar{B}$ the behaviour of the density of the $i^{th}$ Hagedorn resonance $N_{R(i)}$, pions $N_{\pi }$, and baryon anti-baryon pairs $N_{B\bar{B}}$ is described by the following set of equation: $$\begin{aligned}
\label{eqn:networkbab}
\frac{dN_{R(i)}}{dt}&=&-\Gamma_{i}^{tot}
N_{R(i)}+\sum_{n}\Gamma^{tot}_{i,\pi} \Re _{i,n}(T) (N_{\pi
})^{n}B_{i\rightarrow n\pi}\nonumber\\
&+&\Gamma_{i,B\bar{B}}^{tot} \Re_{i,\langle n\rangle}^{\langle n\rangle\pi B\bar{B}} (T)
(N_{\pi })^{\langle n\rangle} N_{B\bar{B}}^2 {}\nonumber\\
\frac{dN_{\pi }}{dt} &=&\sum_{i} \sum_{n}\Gamma_{i,\pi}^{tot} n
B_{i\rightarrow n\pi}\left(N_{R(i)}-\Re (T) (N_{\pi })^{n}
\right)\nonumber\\
&+&\sum_{i} \Gamma_{i,B\bar{B}}^{tot} \langle n\rangle\left(N_{R(i)}-
\Re_{i,\langle n\rangle}^{{\langle n\rangle}\pi B\bar{B}} (T)
(N_{\pi })^{\langle n\rangle}N_{B\bar{B}}^2\right){} \nonumber\\
\frac{dN_{B\bar{B}}}{dt}&=&-\sum_{i}\Gamma_{i,B\bar{B}}^{tot}\left(
N_{B\bar{B}}^2 N_{\pi}^{\langle n\rangle} \Re _{i,\langle n\rangle}(T)- N_{R(i)}\right)\end{aligned}$$ where $\Gamma$ is the decay width, $B_{i\rightarrow n\pi}$ represents the branching ratios, and $\langle n\rangle$ is the average number of pions that each Hagedorn state decays into when a baryon anti-baryon pair is included. We also have two separate detailed balance factors, $\Re (T)=\frac{N_{R(i)}^{eq}}{\left(N_{\pi}^{eq}\right)^{n}}$ for the decay $HS\leftrightarrow n\pi$ and $\Re_{i,\langle n\rangle}^{{\langle n\rangle}\pi B\bar{B}} (T)=\frac{N_{R(i)}^{eq}}{N_{B\bar{B}}^2 \left(N_{\pi}^{eq}\right)^{n}}$ for the decay $HS\leftrightarrow n\pi+B\bar{B}$. The detailed balance factors ensure that detailed balance is maintained in equilibrium. They are also temperature dependent because the equilibrium values of the density are dependent on the temperature.
The branching ratios are described by a gaussian distribution $$\begin{aligned}
\label{eqn:branchingratio}
B_{i\rightarrow n\pi}\approx
\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(n-\langle n\rangle)^{2}}{2\sigma ^{2}}}\end{aligned}$$ where $\langle n\rangle=0.6+0.3\frac{m_{i}}{m_{\pi}}\approx 5-16$ is the average pion number that each Hagedorn state decays into and $\sigma=0.26\sqrt{\frac{m_{i}}{m_{\pi}}}$ is the width of the distribution [@Hamer:1972wz]. The decay width $\Gamma_{i}=0.17m_{i}-88$, which has the range $\Gamma_{i}=250\;\mathrm{MeV\;to}\;1100$ MeV, is a linear fit extrapolated from the data from the particle data group [@Eidelman:2004wy]. In the future we will also use a microcanonical model to find the branching ratios as shown in Ref. [@Liu].
Results
=======
We have solved the rate equations in Eq. (\[eqn:networkbab\]) considering several different initial conditions for a statistical system. At first we take the simplest example and observe only the decay $HS\leftrightarrow n\pi$ when the pions are held in equilibrium. We then consider a resonance bath and allow the pions to equilibrate. We also allow both the pions and the resonances to develop until they reach equilibrium. Then we include baryon anti-baryon pairs into our decay $HS\rightarrow n\pi+B\bar{B}$ where at first the pions are held in equilibrium, then the Hagedorn states, and also both the Hagedorn resonances and the pions are held in equilibrium while the baryon anti-baryon pairs are allowed to equilibrate. Finally, we consider the case when all the constituents are allowed to equilibrate simultaneously.
Case 1 $HS\leftrightarrow n\pi$: Pions held in Equilibrium
----------------------------------------------------------
Initially, when the Hagedorn states decay only into multiple pions, $HS\leftrightarrow n\pi$ we assume that the pions start in equilibrium and the Hagedorn resonances start at zero, which could be an approximation for a physical system immediately following hadronization when the correlation lengths are very short. Then we can make an estimate for the time scale with the inverse of the decay width $\Gamma=\frac{1}{\tau}$. The equilibration time estimate is then between $\tau\approx0.2\;\frac{\mathrm{fm}}{c}\mathrm{\;and}\;0.5\;\frac{\mathrm{fm}}{c}$.
![Hagedorn resonances for various masses at $T=170$ MeV.[]{data-label="fig:pionsineq"}](piineq.eps){width="6cm"}
In Fig. (\[fig:pionsineq\]) the result of the rate equations is shown, which matches the time estimates well. From Fig. (\[fig:pionsineq\]) we see that the Hagedorn states equilibrate quickly (in comparison to typical expansion times). Hence, they should be at chemical equilibrium as long as the pions are at chemical equilibrium.
Case 2 $HS\leftrightarrow n\pi$: Hagedorn Resonances held in Equilibrium
------------------------------------------------------------------------
On the other hand, if the Hagedorn resonances are held in equilibrium and treated like a resonance bath then we can define a new effective production rate for pions: $$\label{eqn:gamHSinEq}
\Gamma_{\pi}^{eff}=\sum_{i}\Gamma_{i}^{tot}\langle n_{\pi}^{i}\rangle\frac{N_{R(i)}^{eq}}{N_{\pi}^{eq}},$$ which gives the range for the equilibration time $\tau\approx0.02\;\frac{\mathrm{fm}}{c}\mathrm{\;to}\;0.25\;\frac{\mathrm{fm}}{c}$. Here the pions start at zero, which could be an approximation for a physical system immediately following hadronization when the correlation lengths are very long. Again the time scale can be compared with the results from the rate equations, which is shown in Fig. (\[fig:reineq\]).
![Number density of pions as they go to equilibrium while the Hagedorn resonances are held in equilibrium. []{data-label="fig:reineq"}](reineq.eps){width="6cm"}
Indeed, the pion become populated very quickly towards chemical equilibrium for all temperatures.
Case 3 $HS\leftrightarrow n\pi$: Hagedorn Resonances and Pions are both out of Equilibrium
------------------------------------------------------------------------------------------
The most interesting case is when both the Hagedorn states and the pions are out of equilibrium. The coupled network of rate equations is $$\begin{aligned}
\label{eqn:setresonances}
\frac{dN_{R(i)}}{dt}&=&\Gamma^{tot}_{i} \left( \sum_{n=2}
\Re (T) (N_{\pi })^{n}B_{i\rightarrow n\pi}-N_{R(i)}\right) {} \nonumber\\
\frac{dN_{\pi }}{dt} &=&\sum _{i}^{\infty} \sum _{n=2}^{\infty}
\Gamma_{i}^{tot} n B_{i\rightarrow n\pi}(N_{R(i)}-
\Re (T) (N_{\pi })^{n} ).\end{aligned}$$ The right-hand side of the rate equations goes to zero when the particles are in equilibrium. In Eq. (\[eqn:setresonances\]) the resonances and pions reach a quasi-equilibrium configuration before full equilibrium is reached. In the quasi-equilibrium state the right-hand side nears zero and thus slows down the equilibration time, thus, $N_{R(i)}\approx N_{R(i)}^{eq}\left(\frac{N_{\pi}}{N_{\pi}^{eq}}\right)^{\langle n\rangle}$ and $N_{\pi}\approx N_{\pi}^{eq}\left(\frac{N_{R(i)}}{N_{R(i)}^{eq}}\right)^{\frac{1}{\langle n\rangle}}$.
-- -- --
-- -- --
In Fig. (\[fig:2re0pi\]) quasi-equilibrium is quickly reached on the time scales from case 1 and case 2.
The quasi-equilibrium can be understood by looking at the number of effective pions in the system i.e. $$\label{eqn:effpionspre}
\tilde{N}_{\pi}=N_{\pi}+\sum_{R}\langle n\rangle N_{R}$$ where $\langle n\rangle$ is the average number of pions that the Hagedorn resonance $N_{R}$ will decay into. Essentially in this particular example, there are too many effectives pions in the system and they must be killed off, which can account for the longer time scales. The direct pions are practically already in equilibrium after $\tau>0.5\;\frac{\mathrm{fm}}{c}$, especially at higher temperatures, however, the Hagedorn states are clearly overpopulated, which implies that the effective number of pions are also overpopulated. Thus, the long time scale consists of reactions such as such as $n\pi\rightarrow HS\rightarrow m\pi$ where $n>m$, which kills off the effective pions in the system. Moreover, the heavier Hagedorn resonances are more overpopulated because they have a higher probability to decay into more pions than lighter resonances. A more in depth look into the time scales of the effective pion number will be discussed in an upcoming paper.
$HS\leftrightarrow n\pi+B\bar{B}$
---------------------------------
Finally, we want to determine the production of baryon anti-baryon pairs close to the phase transition. The baryon anti-baryon pairs are produced through the reaction found in Eq. (\[eqn:bbproduction\]) and these particular branching ratios are determined using a microcanonical model. The average pion number for each corresponding mass is calculated by making a fit to the multiplicities in the microcanonical model found in Ref.[@Liu].
![Various baryon multiplicities, which are used to determine the total baryon density in our system, for the decay $HS\leftrightarrow n\pi +B\bar{B}$ [@Greiner:2004vm; @Liu].[]{data-label="fig:micro"}](baryonsmulti.eps){width="8cm"}
For baryon anti-baryon pairs we need to also reconfigure the decay widths, which is also done with the baryon multiplicities from Ref.[@Liu]. We assume that for every baryon there is a corresponding anti-baryon and then we add up the multiplicities for the three dominat baryons: the proton, the neutron and lambda, to determine the total baryon multiplicity $\langle
B\rangle$ as shown in Fig. (\[fig:micro\]) We can then estimate the relative decay width for the baryon anti-baryon decay, $\Gamma_{i,B\bar{B}}^{tot}=\langle
B\rangle\Gamma_{i,\pi}^{tot}$. The total baryon multiplicity varies between $0.2\mathrm{\;and}\;0.4$, which then gives a decay width between $\Gamma_{i,B\bar{B}}^{tot}=50\;\mathrm{MeV\;and}\;450$ MeV [@Greiner:2004vm].
We start with case 1 when both the Hagedorn resonances and the pions can be held in equilibrium, while the baryon anti-baryon pairs are driven to equilibrium as shown in case 1 in Fig. (\[fig:INeqBaB\]).
-- -- --
-- -- --
Case 2 is again when the pions are held in equilibrium (and the resonances begin at zero) and, case 3 is when the Hagedorn resonances are held in equilibrium (and the pions are started at zero). As before we can make a time scale estimate, this time for the baryon anti-baryon pairs. Our effective chemical equilibration for the baryon anti-baryon pairs is then $$\label{eqn:babdw}
\Gamma_{i,B\bar{B}}^{eff}=-\sum_{i}\Gamma_{i,B\bar{B}}^{tot}
\left(\frac{ N_{R(i)}^{eq}}{N_{B\bar{B}}^{eq}}\right),$$ which gives equilibration times $\tau=0.2\;\frac{\mathrm{fm}}{c}\mathrm{\;to}\;1\;\frac{\mathrm{fm}}{c}$. The results of the pions being held in equilibrium are shown in case 2 in Fig. (\[fig:INeqBaB\]) and the results for the Hagedorn states held in equilibrium are shown in case 3 in Fig. (\[fig:INeqBaB\]).
For case 2 the baryon anti-baryon pairs take slightly longer to equilibrate than their time scale estimate because quasi-equilibrium is reached with the Hagedorn states, which occurs around $\tau\approx 1\;\frac{fm}{c}$. Case 3 is not as affected by a quasi-equilibrium state because the pions equilibrate so quickly that they do not affect the equilibration time of the baryon anti-baryon pairs. Comparing the graph of baryon anti-baryon pairs in case 1 to that in case 2 in Fig. (\[fig:INeqBaB\]) we clearly see that they are still almost identical. The reason is that the baryon anti-baryon pairs are not affected by the pions because the pions equilibrate almost immediately and thus the approximation that the pions are held in equilibrium can be made.
When everything is allowed to develop out of equilibrium, quasi-equilibrium is reached on a time scale of the previous estimated equilibration times. The results are given in Fig.(\[fig:HS2pi0bab0\])
-- --
-- --
Although we only see a small deviation in the pions from equilibrium in Fig. (\[fig:HS2pi0bab0\]) that minor deviation drastically affects the resonances because even the lightest resonance, whose mass is $M=2$ GeV, decays on average into $\langle n\rangle\approx 5$ pions. In Fig. (\[fig:HS2pi0bab0\]) we clearly see that quasi-equilibrium is reached quickly i.e. $\tau_{quasi-eq}<1\;\frac{\mathrm{fm}}{c}$. After quasi-equilibrium is reached the remaining constituents (especially the resonances) slowly reach equilibrium. What we can get from Fig. (\[fig:HS2pi0bab0\]) is that the pions and the baryon anti-baryon pairs quickly equilibrate, especially for higher temperatures. In the graph of the baryon anti-baron pairs we see that at $T=180$ MeV the baryon anti-baryon pairs quickly populate near chemical equilibrium and while they are not in complete equilibrium are quite close to it. Hence, for temperatures between $T=180\;\mathrm{MeV\;and}\;160$ MeV, the baryon anti-baryon pairs are populated with equilibration times faster or equal to $2\;\frac{\mathrm{fm}}{c}$. This constitutes a very promising finding.
Conclusions
===========
Our preliminary results and time scale estimates indicate that baryon anti-baryon pairs can be “born" out of equilibrium after hadronization and then equilibrated by the subsequent population and decay of Hagedorn states. Cases 1-3 for the decay $HS\leftrightarrow n\pi+B\bar{B}$ clearly show quick equilibration times for baryon anti-baryon pairs between temperatures of $180$ MeV and $160$ MeV i.e. slightly below the phase transition. When all the particles started out of equilibration the baryon anti-baryon pairs quickly neared equilibrium although a minor deviation was still observed from chemical equilibration. Afterwards, due to affects from the need to kill of the number of effective pions in the system, longer time scales were observed when the pions, Hagedorn states and baryon anti-baryon pairs were out of equilibrium. However, the pions and baryon anti-baryon pairs were quickly populated near $T_{c}$ and remained close to their equilibrium values even when it took longer for the Hagedorn states to reach chemical equilibrium. Since the Hagedorn states only contribute near $T_{c}$ these appear to be acceptable results.
In an upcoming publication we will delve more thoroughly into the effects of the quasi-equilibrated state seen in both Fig. (\[fig:2re0pi\]), (\[fig:HS2pi0bab0\]) and we also will discuss its effects on the fireball as it cools over time due to a Bjorken expansion. Thus, we expect to see the baryon anti-baryon pairs quickly reach equilibrium and then they will not be produced further when the system is cooled. Moreover, we will vary our choices of initial conditions and discuss their implications. Finally, a non-zero strangeness will be included specifically in the baryon anti-baryon pairs so that we can study the equilibration times of anti-hyperons and multi-strange baryons. The continuation of our work is promising in order to explain results for strangeness in baryons and anti-baryons found at RHIC.
ACKNOWLEDGEMENTS
================
This work was supported by FIGSS. J.N-H. would also like to thank Jorge Noronha for thoughtful discussions on this work. Furthermore, it is a pleasure to thank the organizers of the XLV International Winter Meeting on Nuclear Physics, Bormio 2007 for allowing us to present our results.
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abstract: 'Highly polarized components of pulse profiles are investigated by analyzing observational data and simulating the emission processes. The highly polarized components appear at the leading or trailing part of a pulse profile, which preferably have a flat spectrum and a flat polarization angle curve compared with the low polarized components. By considering the emission processes and propagation effects, we simulate the distributions of wave modes and fractional linear polarization within the entire pulsar emission beam. We show that the highly polarized components can appear at the leading, central, and/or trailing parts of pulse profiles in the models, depending on pulsar geometry. The depolarization is caused by orthogonal modes or scattering. When a sight line cuts across pulsar emission beam with a small impact angle, the detected highly polarized component will be of the O mode, and have a flat polarization angle curve and/or a flat spectrum as observed. Otherwise, the highly polarized component will be of the X mode and have a steep polarization angle curve.'
author:
- |
P. F. Wang[^1] and J. L. Han\
National Astronomical Observatories, Chinese Academy of Sciences. A20 Datun Road, Chaoyang District, Beijing 100012, China\
bibliography:
- 'HiP.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: Highly polarized components of integrated pulse profiles
---
\[firstpage\]
pulsars: general – polarization – acceleration of particles
INTRODUCTION
============
Integrated pulse profiles are obtained by integrating tens of thousands of individual pulses. Features of pulse profiles have been investigated to understand the geometry and physical processes within pulsar magnetosphere [e.g. @ran83; @lm88; @kwj+94; @nsk+15]. Integrated pulse profiles generally comprise several components, and are characterized by diverse polarization features, including prominent linear polarization, ‘S’-shaped polarization angle curves, single sign or sign reversals of circular polarization. Some integrated pulse profiles are highly polarized for the whole pulse, even 100% polarized such as PSR B1259-63 and B1823-13. These pulsars are generally young and have very high spin-down luminosity $\dot{E}$ and flat spectrum [@qml+95; @vkk98; @cmk01; @wj08]. Some pulsars have highly linearly polarized leading or trailing components, for example, the leading components of PSRs B0355+54 and B0450+55 [@lm88; @vx97; @gl98], and the trailing components of PSRs B1650-38 and B1931+24 [@kjm05; @hdv+09]. @vkk98 noticed that the highly polarized leading component of PSR B0355+54 has a flat spectrum and becomes increasingly prominent at higher frequencies. The highly polarized trailing component of PSR B2224+65 has a flat polarization angle curve [@mr11].
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Theoretical efforts have been made to understand various polarization features. In general, pulsar polarizations are closely related to the emission processes of the relativistic particles streaming along the curved magnetic field lines [e.g. @bcw91; @wwh12], the propagation effects within pulsar magnetosphere [e.g. @ba86; @wlh10; @bp12], and the scattering within the interstellar medium [@lh03]. However, these investigations have been conducted separately on each aspect, rarely done jointly. Curvature radiation which serves as one of the most probable mechanisms for pulsar emission can produce highly polarized emission [@gan10; @wwh12]. Propagation effects are succeeded in demonstrating the interaction of the ordinary (O) and extra-ordinary (X) modes within pulsar magnetosphere and can lead to diverse depolarization features [@cr79; @ba86; @wlh10; @bp12], though initial ratios for both modes are uncertain. Propagation effects within the interstellar medium need to be investigated further. Recently, we investigated the emission processes jointly with propagation effects [@wwh14; @wwh15], which provides us a new opportunity to understand the highly polarized components, because distributions of the X-mode and O-mode within pulsar magnetosphere are related to the depolarization across pulsar emission beam by considering the refraction and corotation effects. Emission can be highly depolarized in some beam regions where both modes have comparable intensities, but dominated by one mode in other regions and hence the resulting profile can be highly polarized.
In this paper, we summarize observations for highly polarized components of integrated pulse profiles in literature and then theoretically explain them by modeling emission and propagation processes. In Section 2, we analyze various features for highly polarized components of observed pulsar profiles. In Section 3, we simulate polarized pulsar beams and pulse profiles by considering the emission processes and propagation effects. Discussions and conclusions are given in Section 4.
-------------- ---------- --------- -------------------- ----------------------------------- ----------------------------------------------------
PSR Jname Bname Period DM Polarization Features References
(s) ($\rm cm^{-3} pc$)
J0014+4746 B0011+47 1.24069 30.8 Leading 31, 70, 86
J0358+5413 B0355+54 0.15638 57.1 Leading, Flat Spec., Orth. Modes 8, 9, 16, 20, 24, 26, 30, 31, 34, 77, 81
J0454+5543 B0450+55 0.34072 14.5 Leading, Flat Spec., Flat PA 16, 20, 30, 31, 81
J0814+7429 B0809+74 1.29224 5.7 Leading, Orth. Modes 1, 9, 30, 31, 44, 58, 78, 88
J0942-5657 B0941-56 0.80812 159.7 Leading 23, 33, 69, 90
J0954-5430 0.47283 200.3 Leading 69, 90
J1048-5832 B1046-58 0.12367 129.1 Leading, Flat Spec. 23, 56, 59, 69, 76, 89, 90
J1057-5226IP B1055-52 0.19710 30.1 Leading, Flat Spec. 4, 6, 12, 16, 29, 69, 72, 77, 89, 90
J1112-6103 0.06496 599.1 Leading, Scattering 69, 89, 90
J1341-6220 B1338-62 0.19333 717.3 Leading, Scattering 23, 39, 59, 60, 69, 90
J1410-6132 0.05005 960.0 Leading, Scattering 68, 69, 89, 90
J1453-6413 B1449-64 0.17948 71.0 Leading, Flat PA, Orth. Modes 5, 6, 7, 29, 54, 59, 69, 71, 81, 90
J1730-3350 B1727-33 0.13946 259.0 Leading, Scattering 31, 39, 59, 69, 89, 90
J1805+0306 B1802+03 0.21871 80.8 Leading, Flat PA 31, 38
J1823-3106 B1820-31 0.28405 50.2 Leading 21, 31
J1825-0935MP B1822-09 0.76900 19.3 Leading, Flat Spec., Orth. Modes 7, 9, 24, 29, 30, 31, 49, 61, 63, 65, 73, 77
J1844+1454 B1842+14 0.37546 41.4 Leading, Flat Spec., Flat PA 17, 31, 38, 54, 65, 74, 77, 81, 90
J1849+2423 0.27564 62.2 Leading 70
J1937+2544 B1935+25 0.20098 53.2 Leading, Flat PA 31, 38, 63, 70, 81, 90
J2008+2513 0.58919 60.5 Leading, Orth. Modes 70
J0601-0527 B0559-05 0.39596 80.5 Trailing, Flat Spec., Flat PA 20, 23, 31, 34, 54, 69, 90
J0922+0638 B0919+06 0.43062 27.2 Trailing, Flat Spec., Orth. Modes 13, 19, 24, 29–31, 38, 49, 54, 59
62, 65, 74, 81, 90
J1401-6357 B1358-63 0.84278 98.0 Trailing 21, 23, 29
J1539-5626 B1535-56 0.24339 175.8 Trailing, Flat Spec., Flat PA 23, 56, 59, 69, 90
J1548-5607 0.17093 315.5 Trailing 69, 90
J1653-3838 B1650-38 0.30503 207.2 Trailing 56, 69, 90
J1739-1313 1.21569 58.2 Trailing 69, 90
J1808-3249 0.36491 147.3 Trailing 56, 69, 90
J1933+2421 B1931+24 0.81369 106.0 Trailing 31, 70
J2013+3845 B2011+38 0.23019 238.2 Trailing, Flat PA 31, 81
J2225+6535 B2224+65 0.68254 36.0 Trailing, Flat Spec., Flat PA 9, 16, 31, 77, 86, 88
J0737-3039A 0.02269 48.9 MSP, Leading&Trailing 47, 55, 57, 82
Flat PA, Orth. Mod
J1012+5307 0.00525 9.0 MSP, Trailing, Flat PA 35, 37, 70, 88
J1022+1001 0.01645 10.2 MSP, Trailing 35–37, 46, 48, 52, 80, 84, 85, 87, 88
J1300+1240 B1257+12 0.00621 10.1 MSP, Leading 35, 70
J0108-1431 0.80756 2.3 Whole 33, 69, 90
J0134-2937 0.13696 21.8 Whole 33, 65, 69, 71, 90
J0139+5814 B0136+57 0.27245 73.7 Whole 16, 20, 30, 31, 81, 86, 88
J0538+2817 0.14315 39.5 Whole 30, 81
J0543+2329 B0540+23 0.24597 77.7 Whole, Pol. dec. with freq. 9, 10, 17, 19, 24, 30, 31, 38, 49
53, 63, 65, 69, 74, 77, 81, 90
J0614+2229 B0611+22 0.33495 96.9 Whole, Strong CP 10, 16, 19, 31, 34, 38, 53, 63, 65, 69, 74
J0630-2834 B0628-28 1.24441 34.4 Whole 5, 6, 7, 9, 16, 29, 31, 34, 54, 59, 65, 69, 81, 90
J0631+1036 0.28780 125.3 Whole 27, 69, 75, 89, 90
J0659+1414 B0656+14 0.38489 13.9 Whole 17, 31, 38, 40, 53, 59, 63, 64, 69, 74
75, 81, 89, 90
J0742-2822 B0740-28 0.16676 73.7 Whole 6, 7, 9, 16, 24, 29–31, 33, 49, 54, 59
61, 69, 71, 75, 77, 81, 83, 89, 90
J0835-4510 B0833-45 0.08932 67.9 Whole, Strong CP 2, 3, 5–7, 11, 29, 43, 54, 59, 61
69, 71, 76, 81, 89, 90
J0901-4624 0.44199 198.8 Whole, Strong CP 69, 90
J0905-5127 0.34628 196.4 Whole 69, 90
J0908-4913 B0906-49 0.10675 180.3 Whole, Inter Pulse 21, 23, 32, 59, 66, 69, 76, 89, 90
J1015-5719 0.13988 278.7 Whole 60, 69, 90
J1028-5819 0.09140 96.5 Whole 67, 69, 89
J1057-5226MP B1055-52 0.19710 30.1 Whole 4, 6, 12, 16, 29, 69, 72,77, 89, 90
J1105-6107 0.06319 271.0 Whole 39, 60, 69, 89, 90
-------------- ---------- --------- -------------------- ----------------------------------- ----------------------------------------------------
-------------- ---------- --------- ------- ------------------------------- ------------------------------------------------
J1119-6127 0.40796 707.4 Whole 45, 60, 69, 79, 89, 90
J1302-6350 B1259-63 0.04776 146.7 Whole, Strong CP 22, 25, 42, 56, 59, 69, 90
J1321+8323 B1322+83 0.67003 13.3 Whole 31, 70, 86
J1359-6038 B1356-60 0.12750 293.7 Whole, Strong CP 21, 29, 33, 59, 61, 69, 90
J1420-6048 0.06817 358.8 Whole, Strong CP 41, 60, 69, 75, 89, 90
J1614-5048 B1610-50 0.23169 582.8 Whole, Scattering, Strong CP 23, 56, 69, 90
J1637-4553 B1634-45 0.11877 193.2 Whole 56, 69, 90
J1702-4128 0.18213 367.1 Whole 69, 89, 90
J1705-1906IP B1702-19 0.29898 22.9 Whole, Strong CP, Inter-pulse 15, 16, 29, 31, 34, 49, 65, 69, 90
J1705-3950 0.31894 207.1 Whole, Strong CP 69, 90
J1709-4429 B1706-44 0.10245 75.6 Whole, Strong CP 23, 54, 56, 59, 69, 89, 90
J1718-3825 0.07466 247.4 Whole 69, 75, 89, 90
J1733-3716 B1730-37 0.33758 153.5 Whole 56, 69, 90
J1740-3015 B1737-30 0.60688 152.1 Whole, Strong CP 21, 23, 31, 34, 59, 69, 76, 90
J1801-2451 B1757-24 0.12491 289.0 Whole, Strong CP 31, 63, 69, 81, 86, 89, 90
J1803-2137 B1800-21 0.13366 233.9 Whole, Strong CP 21, 31, 34, 69, 86, 90
J1809-1917 0.08274 197.1 Whole, Strong CP 69, 90
J1826-1334 B1823-13 0.10148 231.0 Whole, Strong CP 31, 34, 69, 90
J1830-1059 B1828-11 0.40504 161.5 Whole, Strong CP 31, 69, 90
J1841-0345 0.20406 194.3 Whole 69, 90
J1841-0425 B1838-04 0.18614 325.4 Whole 31, 63, 69, 90
J1850+1335 B1848+13 0.34558 60.1 Whole 31, 38, 63, 65, 81, 90
J1915+1009 B1913+10 0.40454 241.6 Whole, Strong CP 17, 31, 34, 38, 63, 81, 90
J1926+1648 B1924+16 0.57982 176.8 Whole 10, 17, 19, 31, 38, 77
J1932+1059 B1929+10 0.22651 3.1 Whole 2, 9, 10, 13, 14, 16–20, 24, 26, 28–31, 37, 38
40, 49–51, 53, 54, 70, 74, 81, 86, 88, 90
-------------- ---------- --------- ------- ------------------------------- ------------------------------------------------
Observational features for highly polarized pulse components
============================================================
The highly polarized components of integrated pulse profiles exhibit diverse polarization features. To demonstrate the properties, a sample of 78 pulsars is collected from literatures, as listed in Table \[high\_L\]. Among them, 20 pulsars have highly polarized leading components, 11 pulsars have highly polarized trailing components, four millisecond pulsars have both highly polarized leading and/or trailing components, and 43 pulsars are highly polarized for the whole pulse profile. The fractional linear polarization is larger than 70% for highly polarized components or the whole profile for these pulsars at more than one frequency.
Flat spectra of highly polarized components
-------------------------------------------
Multi-frequency observations demonstrate that pulsar flux density generally decreases with frequency, following a power-law spectrum [e.g. @sie73]. Different components for a given pulsar could evolve differently with frequency. For example, the relative spectra for the leading and trailing components are diverse for the conal double pulsars [@whq01]. The highly polarized components also show frequency evolution. Fig. \[fig:profiles\] shows the polarized pulse profiles at three frequencies for two pulsars, J1048-5832 and J2225+6535. PSR J1048-5832 exhibits highly polarized leading component with polarization degree approaching 100%. At 692 MHz, the peak intensity of the highly polarized leading component is weaker than the low polarized trailing component. As observation frequency increases, the highly polarized leading component gradually dominates, as shown by the profiles of 1369 and 3068 MHz. Similar features have been seen from PSRs J0358+5413, J0454+5543, J1057-5226IP, J1825-0935MP and J1844+1454. In contrast, PSR J2225+6535 is an example for highly polarized trailing component, which becomes dominant as observation frequency increases. Similar cases can be found from PSRs J0601-0527, J0922+0638 and J1539-5626.
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![The frequency evolution for the peak intensity ratio of the highly polarized leading components for six pulsars with respect to the low polarized trailing ones. The intensity ratios are listed in Table \[table:leading\_ratio\], which can be described by a power-law as $\rm I_{HiP}/I_{LowP} \sim a\nu^k$. []{data-label="fig:leading_ratio"}](J0358+5413.ps "fig:"){width="22.20000%"} ![The frequency evolution for the peak intensity ratio of the highly polarized leading components for six pulsars with respect to the low polarized trailing ones. The intensity ratios are listed in Table \[table:leading\_ratio\], which can be described by a power-law as $\rm I_{HiP}/I_{LowP} \sim a\nu^k$. []{data-label="fig:leading_ratio"}](J0454+5543.ps "fig:"){width="22.20000%"}
![The frequency evolution for the peak intensity ratio of the highly polarized leading components for six pulsars with respect to the low polarized trailing ones. The intensity ratios are listed in Table \[table:leading\_ratio\], which can be described by a power-law as $\rm I_{HiP}/I_{LowP} \sim a\nu^k$. []{data-label="fig:leading_ratio"}](J1048-5832.ps "fig:"){width="22.20000%"} ![The frequency evolution for the peak intensity ratio of the highly polarized leading components for six pulsars with respect to the low polarized trailing ones. The intensity ratios are listed in Table \[table:leading\_ratio\], which can be described by a power-law as $\rm I_{HiP}/I_{LowP} \sim a\nu^k$. []{data-label="fig:leading_ratio"}](J1057-5226IP.ps "fig:"){width="22.20000%"}
![The frequency evolution for the peak intensity ratio of the highly polarized leading components for six pulsars with respect to the low polarized trailing ones. The intensity ratios are listed in Table \[table:leading\_ratio\], which can be described by a power-law as $\rm I_{HiP}/I_{LowP} \sim a\nu^k$. []{data-label="fig:leading_ratio"}](J1825-0935MP.ps "fig:"){width="22.20000%"} ![The frequency evolution for the peak intensity ratio of the highly polarized leading components for six pulsars with respect to the low polarized trailing ones. The intensity ratios are listed in Table \[table:leading\_ratio\], which can be described by a power-law as $\rm I_{HiP}/I_{LowP} \sim a\nu^k$. []{data-label="fig:leading_ratio"}](J1844+1454.ps "fig:"){width="22.20000%"}
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![Same as Fig. \[fig:leading\_ratio\] but for the highly polarized trailing components for four pulsars. Data are listed in Table \[table:trailing\_ratio\]. []{data-label="fig:trailing_ratio"}](J0601-0527.ps "fig:"){width="22.20000%"} ![Same as Fig. \[fig:leading\_ratio\] but for the highly polarized trailing components for four pulsars. Data are listed in Table \[table:trailing\_ratio\]. []{data-label="fig:trailing_ratio"}](J0922+0638.ps "fig:"){width="22.20000%"}
![Same as Fig. \[fig:leading\_ratio\] but for the highly polarized trailing components for four pulsars. Data are listed in Table \[table:trailing\_ratio\]. []{data-label="fig:trailing_ratio"}](J1539-5626.ps "fig:"){width="22.20000%"} ![Same as Fig. \[fig:leading\_ratio\] but for the highly polarized trailing components for four pulsars. Data are listed in Table \[table:trailing\_ratio\]. []{data-label="fig:trailing_ratio"}](J2225+6535.ps "fig:"){width="22.20000%"}
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Figs. \[fig:leading\_ratio\] and \[fig:trailing\_ratio\] quantitatively demonstrate the frequency evolution of the peak intensity ratios, $\rm I_{HiP}/I_{LowP}$, of the highly polarized components with respect to the low polarized ones at a series of frequencies, see data in Tables \[table:leading\_ratio\] and \[table:trailing\_ratio\] in the Appendix. Clearly, $\rm
I_{HiP}/I_{LowP}$ generally increases with frequency for highly polarized leading or trailing components, and can be described by a power-law, though the power-law indices vary from 0.35 to 1.79 for different pulsars. We conclude that the highly polarized components exhibit a flatter spectrum than the low polarized components, regardless of its location at the leading or trailing phase.
Polarization angle curves of highly polarized components
--------------------------------------------------------
1.5mm
-------------- ---------------- ------- ------------------------ ------------------------- ------
PSR Hi. Pol. Comp. Freq. $\Delta PA/\Delta\phi$ $\Delta PA/\Delta \phi$ Ref.
(GHz) High Pol. Comp. Low Pol. Comp.
J0454+5543 Leading 1.41 $-0.6\pm0.2$ $-7.8\pm0.3$ 30
J1453-6413 Leading 1.4 $0.0\pm0.2$ $8.0\pm0.3$ 81
J1805+0306 Leading 1.4 $-1.2\pm0.5$ $13.1\pm0.6$ 38
J1844+1454 Leading 1.4 $2.4\pm1.1$ $17.0\pm2.1$ 54
J1937+2544 Leading 0.774 $-1.5\pm0.3$ $-7.6\pm0.6$ 70
J0601-0527 Trailing 0.692 $2.6\pm0.6$ $5.6\pm1.0$ 90
J2225+6535 Trailing 0.325 $-0.1\pm0.8$ $-3.7\pm0.7$ 77
J0358+5413 Leading 1.408 $-1.4\pm0.2$ Orth. Modes 31
J0814+7429 Leading 1.41 $-0.6\pm0.2$ Orth. Modes 30
J1825-0935 Leading 0.691 $3.7\pm0.2$ Orth. Modes 63
J2008+2513 Leading 0.774 $1.4\pm0.7$ Orth. Modes 70
J0922+0638 Trailing 0.692 $5.2\pm0.4$ Orth. Modes 90
J1112-6103 Leading 1.5 $-0.6\pm0.1$ Scattering 69
3.0 $-5.0\pm0.1$ - 69
J1341-6220 Leading 1.5 $0.6\pm0.1$ Scattering 60
3.0 $7.1\pm0.2$ - 60
J1410-6132 Leading 1.5 $0.0\pm0.1$ Scattering 69
3.1 $4.0\pm0.2$ - 68
J1730-3350 Leading 1.5 $-1.5\pm0.1$ Scattering 69
3.0 $-5.4\pm0.3$ - 69
J0014+4746 Leading 0.774 $-1.0\pm0.1$ $-1.4\pm0.1$ 70
J0954-5430 Leading 1.4 $12.1\pm2.1$ $10.8\pm1.7$ 90
J1057-5226IP Leading 1.377 $0.6\pm0.5$ $1.2\pm1.3$ 90
J0942-5657 Leading 1.5 $14.2\pm0.4$ Mixed 69
J1048-5832 Leading 1.369 $4.3\pm0.2$ Mixed 90
J1823-3106 Leading 1.4 $-4.5\pm0.3$ Mixed 31
J1401-6357 Trailing 0.955 $8.1\pm1.4$ Mixed 29
J1739-1313 Trailing 1.377 $8.9\pm1.5$ Mixed 90
J2013+3845 Trailing 1.408 $-1.0\pm0.2$ Mixed 31
J1849+2423 Leading 0.774 $-1.2\pm0.2$ Weak Pol. 70
J1539-5626 Trailing 1.5 $0.0\pm0.1$ Weak Pol. 69
J1548-5607 Trailing 1.4 $2.4\pm0.3$ Weak Pol. 90
J1653-3838 Trailing 1.377 $3.4\pm1.2$ Weak Pol. 90
J1808-3249 Trailing 1.377 $-8.2\pm1.6$ Weak Pol. 90
J1933+2421 Trailing 0.774 $9.4\pm0.3$ Weak Pol. 70
J0737-3039A MSP-Leading 1.4 $0.0\pm0.2$ Orth. Modes 82
MSP-Trailing 1.4 $0.0\pm0.3$ - 82
J1012+5307 MSP-Trailing 0.774 $-0.4\pm0.1$ Mixed 70
J1022+1001 MSP-Trailing 1.3 $3.3\pm0.2$ $4.4\pm0.1$ 87
J1300+1240 MSP-Leading 0.774 $0.4\pm0.3$ Weak Pol. 70
-------------- ---------------- ------- ------------------------ ------------------------- ------
: Gradients of polarization curves for the highly polarized and low polarized components. References are numbered in Table A1.[]{data-label="table:PA_gradient"}
Highly polarized components differ from low polarized components also in polarization angle curves. Table \[table:PA\_gradient\] summarizes the gradients of polarization angle curves for 35 pulsars extracted from Table \[high\_L\]. The highly polarized components generally have flat polarization angle curves. For example, the gradient of polarization angle curve for the highly polarized trailing component of PSR J2225+6535 in Fig. \[fig:profiles\] approximates to be -0.1 at 325MHz, but it is -3.7 for the low polarized leading component [@mr11]. The gradient is 2.4 for the highly polarized leading component of PSR J1844+1454 at 1.4GHz, but 17.0 for low polarized trailing component [@jhv+05]. The large difference for the gradients can also be found for PSRs J0454+5543, J1453-6413, J1805+0306, J1937+2544 and J0601-0527, as listed in Table \[table:PA\_gradient\]. It implies that the highly polarized emission of these pulsars might be generated from the beam regions well away from the magnetic meridional plane.
However, the highly polarized emission of some pulsars might also be produced near the meridional plane, e.g. J0942-5657 and J1933+2421. Both of them have very steep polarization angle curves with gradients of 14.2 and 9.4 for the highly polarized components. Gradients for low polarized components of many pulsars are hard to determine due to various reasons as noted in the fifth column of Table \[table:PA\_gradient\].
![Histograms for absolute values of gradients of the polarization angle curves for highly polarized and low polarized components. The gradient values are listed in Table \[table:PA\_gradient\]. []{data-label="fig:PA_gradient"}](PA-Gradient.ps){width="40.00000%"}
As shown in Fig. \[fig:PA\_gradient\], the gradients of polarization angle curves for the highly polarized components are concentrated near 0.0. The gradients for the low polarized components have fewer data but are widely distributed. We therefore conclude that the highly polarized emission tends to have a flat polarization angle curve.
Depolarization and other properties
-----------------------------------
There are two mechanisms for depolarization of pulsar profiles: orthogonally polarized radiation and scattering within the interstellar medium. Single pulse observations [e.g. @scr+84; @scw+84] show the orthogonal modes of pulsar emission, and highly polarized components of integrated profiles are generally of one mode. The orthogonal modes often depolarize the integrated profiles and lead to low polarized components, as shown for PSRs J0814+7429 and J0922+0638 by @scr+84 and @rrs+02. PSRs J0358+5413, J1825-0935 and J2008+2513 also show orthogonal modes and have depolarized trailing components, as listed in Table \[table:PA\_gradient\].
Scattering during the propagation of pulsed emission in the interstellar medium can also cause depolarization at the trailing parts of profiles and result in a flat polarization angle curve [@lh03]. For example, PSR J1112-6103 has a dispersion measure of 599.1 and has two highly polarized components at 3.1GHz [@wj08]. But at frequencies below 1.5GHz, the effect of scattering becomes very significant and causes depolarization in the trailing part. The polarization angle curves are also flattened, as indicated by the gradient values in Table \[table:PA\_gradient\]. The other three pulsars, PSRs J1341-6220, J1410-6132 and J1730-3350 show similar polarization profiles due to scattering.
Millisecond pulsars exhibit highly polarized components as the normal pulsars. PSR J0737-3039A is an orthogonal rotator and has an inter-pulse. The leading part of the main pulse and the trailing part of the interpulse are highly linearly polarized with a nearly constant position angle. The gradient of polarization angle curve is near 0.0 as listed in Table \[table:PA\_gradient\]. Orthogonal modes might happen at the trailing part of the main pulse and the leading part of the interpulse [@gkj+13]. PSR J1012+5307 is an aligned rotator and has emission at almost all rotation phases. The trailing part of the brightest component and all the other components are highly linearly polarized [@stc99; @hdv+09]. The swing of polarization angle is nearly flat at all these phases. PSRs J1022+1001 and J1300+1240 show similar polarization features.
Theoretical explanations of highly polarized components
=======================================================
It can be summarized from observations that the highly polarized components preferably have a flat spectrum and a flat polarization angle curve. Orthogonal modes and scattering could cause depolarization. Millisecond pulsars exhibit similarly highly polarized components as the normal pulsars. After we analyze literature data to uncover these features for highly polarized components, we here carry out numerical simulations of emission processes and propagation effects to understand the polarization.
A theoretical model for emission processes and propagation effects
------------------------------------------------------------------
![The distributions of wave modes and fractional linear polarization within simulated pulsar emission beam. The upper panels are plotted for the X-mode and O-mode intensities, $I_X$ and $I_O$. The bottom panel shows the degree of linear polarization. Seven density patches labeled as $\it a$, $\it b$, $\it c$, $\it d$, $\it e$, $\it f$ and $\it g$ are shown in the figure and their locations are listed in Table \[table:patch\_loc\]. Example sight lines at $\zeta=31^o$ and $34^o$ from the rotation axis of a neutron star are indicated by the dashed lines. Other pulsar parameters used for simulations are the inclination angle of the magnetic axis from the rotation axis $\alpha=30^o$ and pulsar period $P=1s$. Relativistic particles are assumed to have a Lorentz factor of $\gamma=500$, and emit at 1.4GHz.[]{data-label="fig:patch_dis"}](Patch_distribution.ps){width="40.00000%"}
In general, pulsar magnetosphere is assumed to be an dipole, $${\bmath B}=B_{\star}(\frac{R_{\star}}{r})^3[3{\hat{\bmath r}}({\hat{\bmath r}}\cdot
{\hat{\bmath m}})-{\hat{\bmath m}}],
\label{eq:staticb}$$ here $R_{\star}$ and $B_{\star}$ represent neutron star radius and the magnetic field on its surface, ${\hat{\bmath r}}$ and ${\hat{\bmath m}}$ are the unit vectors along ${\bmath r}$ and the magnetic dipole moment. The magnetic axis inclines to the rotation axis by an inclination angle $\alpha$. It rotates freely in space. Relativistic particles with a Lorentz factor of $\gamma$ are produced by the sparking processes above the polar cap. They stream out along the curved magnetic field lines and co-rotate with pulsar magnetosphere. As influenced by the perpendicular acceleration, relativistic particles will produce curvature radiation. The radiation field $\bmath E(t)$ and its Fourier components $\bmath E(\omega)$ can be calculated by using circular path approximation [@wwh12]. Curvature radiation at a given position of pulsar magnetosphere actually contains the contributions from all the nearby field lines within a $1/\gamma$ cone around the tangential direction. The polarization patterns of emission cones are further distorted by rotation effects, as demonstrated by @wwh12.
In general, there are four wave modes (two transverse and two longitudinal) in the plasma of pulsar magnetosphere [@bp12]. Two modes are damped at large distances from the neutron star in the magnetosphere. Only the X-mode and superluminous O-mode, hereafter the O-mode, can escape from the magnetosphere to be observed. Immediately after the waves are generated in the emission region, they are coupled to the local X-mode and O-mode to propagate outwards. Within the $1/\gamma$ emission cone, both components have comparable intensities and propagate separately. The X-mode component propagates in a straight line, while the O-mode component suffers refraction [@ba86]. Hence, both mode components are separated [@wwh14]. The detectable emission at a given position consists of incoherent superposition of X-mode and O-mode components coming from discrete emission regions. Both mode components experience ‘adiabatic walking’, wave mode coupling, and cyclotron absorption [@wlh10; @bp12].
These emission processes and propagation effects have been considered jointly by @wwh14 for four particle density models in the form of uniformity, cone, core and patches. We demonstrated that refraction and co-rotation significantly affect pulsar polarizations. Refraction bends O-mode emission towards the outer part of pulsar emission beam, and causes the separation of both modes. Co-rotation will lead to different ratios for both modes at different parts of pulsar emission beam. Investigations on the influences of both effects have been extended to a wide range of frequencies, and succeeded in demonstrating the frequency dependence of pulsar linear polarization [@wwh15].
Index $\theta_i$ $\phi_i(^o)$ $\sigma_{\theta}$ $\sigma_{\phi}(^o)$
--------- ------------ -------------- ------------------- ---------------------
$\it a$ 0.8 85 0.06 5
$\it b$ 0.5 40 0.08 12
$\it c$ 0.5 -55 0.09 12
$\it d$ 0.85 -85 0.06 5
$\it e$ 0.8 40 0.06 5
$\it f$ 0.8 -10 0.06 5
$\it g$ 0.85 -55 0.06 5
: Assumed seven density patches within a pulsar emission beam. Here, $\theta_i$ and $\phi_i$ represent the peak positions for the Gaussian density patches in the magnetic colatitude $\theta$ and azimuth $\phi$ directions within ranges of $0<\theta_i<1$ and $-180^o<\phi_i<180^o$. $\sigma_\theta$ and $\sigma_\phi$ represent the width of Gaussian distribution of the density distribution of particles. []{data-label="table:patch_loc"}
Based on our previous studies [@wwh12; @wwh14; @wwh15], we here simulate the curvature radiation processes and propagation effects, but focus mainly on the distribution of highly polarized emission regions within pulsar emission beam. Fig. \[fig:patch\_dis\] represents a very typical case for the distributions of wave modes and fractional linear polarization, based on an uniform density model demonstrated in @wwh14. It shows that the intensity distributions for both modes are quite different. The X-mode components, $I_X$, are stronger at the two sides of pulsar beam in the $\zeta$ direction, as shown in the top left panel of Fig. \[fig:patch\_dis\], while $I_O$ are stronger at the two sides of pulsar beam in the $\varphi$ direction, as shown in the top right panel of Fig. \[fig:patch\_dis\]. Here, $\zeta$ is the sight line angle, i.e., the angle between sight line and the rotation axis, $\varphi$ represents the rotation phase. Depolarization is caused by two modes. Some regions in emission beam is dominated by one mode that can be highly polarized. The depolarization leads the distribution of fractional linear polarization $|I_X-I_O|/|I_X+I_O|$ to be quadruple. It implies that the highly polarized emission could be produced at four parts of pulsar emission beam, i.e., the leading (O-mode), trailing (O-mode), top (X-mode) and bottom (X-mode) parts of the beam.
In order to demonstrate the formation of highly polarized components, seven density patches ($\it a$, $\it b$, $\it c$, $\it d$, $\it e$, $\it f$ and $\it g$) are simulated as listed in Table \[table:patch\_loc\]. As shown in the bottom panel of Fig. \[fig:patch\_dis\], patches $\it a$ and $\it d$ are dominated by the O-mode emission, while patch $\it f$ by the X-mode. The emission from these regions should have a large fraction of linear polarization. However, emission from density patches $\it b$, $\it c$ and $\it e$ have both the X and O modes with comparable intensity, hence the observed emission from these regions is depolarized.
Polarized pulse profiles obtained by a small impact angle
---------------------------------------------------------
![Pulse profiles resulting from the cut of density patches ($\it a$, $\it b$) and ($\it c$, $\it d$) to explain the highly polarized leading and trailing components, depending on the available density patches in the emission region. The sold lines represent the total intensity, the dashed and dotted lines are for the linear polarization and polarization angle curves. The wave modes are marked near the polarization angle curves.[]{data-label="fig:patch_prof_s"}](Profile_stack_ab.ps "fig:"){width="23.50000%"} ![Pulse profiles resulting from the cut of density patches ($\it a$, $\it b$) and ($\it c$, $\it d$) to explain the highly polarized leading and trailing components, depending on the available density patches in the emission region. The sold lines represent the total intensity, the dashed and dotted lines are for the linear polarization and polarization angle curves. The wave modes are marked near the polarization angle curves.[]{data-label="fig:patch_prof_s"}](Profile_stack_cd.ps "fig:"){width="23.50000%"}
When a sight line has a small impact angle, $\beta$, i.e., $\zeta-\alpha$, cutting across pulsar emission beam, it will detect emissions from the density patches $\it a$, $\it b$, $\it c$ and $\it
d$ in Fig. \[fig:patch\_dis\]. The resulting pulse profiles are shown in Fig. \[fig:patch\_prof\_s\], depending on the available density patch combinations, for example ($\it a$, $\it b$) or ($\it c$, $\it
d$). We can conclude from simulations that:
1\) The highly polarized components can be generated from the leading (patch $\it a$) and the trailing (patch $\it d$) parts of pulsar emission beam, both of which are dominated by the O-mode.
2\) The highly polarized components have a flat polarization angle curve, because density patches $\it a$ and $\it d$ are away from the meridional plan of $\varphi=0^o$.
3\) The low polarized components exhibit orthogonal modes, and the emission from the X and O modes has comparable intensity. Orthogonal mode jump happens when one mode dominates over the other, as shown by the polarization angle curves.
In addition, the simulations predict that the highly polarized components are more likely to be generated at the leading parts of pulse profiles, because the highly polarized leading part of pulsar emission beam is broader than the trailing one (see Fig.\[fig:patch\_dis\]) due to rotation-induced asymmetry. Highly polarized components would have a flatter spectrum than the low polarized components, because the beam regions further away from the magnetic axis tend to have a flat spectrum according to @lm88, while the detailed spectrum behavior is not modeled in our simulations.
Polarized pulse profiles obtained by a large impact angle
---------------------------------------------------------
![Same as Fig. \[fig:patch\_prof\_s\] but for density patch combinations of ($\it e$, $\it f$), ($\it f$, $\it g$) and ($\it
e$, $\it f$, $\it g$).[]{data-label="fig:patch_prof_l"}](Profile_stack_ef.ps "fig:"){width="23.50000%"} ![Same as Fig. \[fig:patch\_prof\_s\] but for density patch combinations of ($\it e$, $\it f$), ($\it f$, $\it g$) and ($\it
e$, $\it f$, $\it g$).[]{data-label="fig:patch_prof_l"}](Profile_stack_fg.ps "fig:"){width="23.50000%"} ![Same as Fig. \[fig:patch\_prof\_s\] but for density patch combinations of ($\it e$, $\it f$), ($\it f$, $\it g$) and ($\it
e$, $\it f$, $\it g$).[]{data-label="fig:patch_prof_l"}](Profile_stack_efg.ps "fig:"){width="23.50000%"}
When a sight line has a large impact angle to cut across the pulsar emission beam, it will detect emission from density patches $\it e$, $\it f$ and $\it g$ in Fig. \[fig:patch\_dis\]. The resulting pulse profiles are shown in Fig. \[fig:patch\_prof\_l\] for different patch combinations ($\it e$, $\it f$), ($\it f$, $\it g$) or ($\it e$, $\it
f$, $\it g$) for available density distributions of particles. Highly polarized components can appear at the leading, central or trailing part of pulse profiles. These profiles have similar features as those in Fig. \[fig:patch\_prof\_s\], but differences are as following.
1\) The highly polarized component from the bottom part, i.e., density patch $\it f$, of pulsar emission beam is dominated by the X-mode, rather than the O-mode.
2\) Highly polarized component has a steeper polarization angle curve, because the component is generated near the meridional plane, where the polarization angle has the maximum rate of change approximating $(d PA/d \varphi)_{\rm max}=\sin \alpha/\sin \beta$. The gradient is inversely proportional to the impact angle $\beta$.
3\) Highly polarized component may have a similar spectrum to the low polarized components, since all components are generated at comparable distances from the magnetic axis.
In summary, joint simulations of emission processes and propagation effects demonstrate that highly polarized components can be produced at the leading, central and trailing parts of pulse profiles. The properties of emission components, polarization angle curve, mode characteristic and spectrum, depend on pulsar geometry and the density patches of radiating particles.
Discussions and Conclusions
===========================
In this paper, we have investigated the highly polarized components of integrated pulse profiles observationally and theoretically. We found from observational data that:
\(i) Highly polarized components of pulsar profiles have a flatter spectrum than the low polarized components, regardless of their locations at the leading or trailing phase;
\(ii) Highly polarized components tend to have a flat polarization angle curve, though a small fraction of pulsars have very steep polarization angle curves;
\(iii) Highly polarized components generally have one mode, while the low polarized components often show orthogonal modes;
\(iv) Significant scattering will cause depolarization at the trailing parts of pulse profiles and result in flat polarization angle curves;
\(v) Millisecond pulsars can have highly polarized components as normal pulsars.
We simulated emission processes and propagation effects within pulsar magnetosphere, and found that highly polarized emission could be produced at the leading (O-mode), trailing (O-mode), top (X-mode) and bottom (X-mode) parts of pulsar emission beam. When a sight line cuts across the beam with different impact angles, the detected highly polarized components have different properties, depending on the specific geometry and available density patches of the radiating particles:
\(i) Highly polarized component generated from the leading or trailing part of pulsar emission beam is of the O-mode, and has a flat polarization angle curve;
\(ii) Highly polarized component generated from the top or bottom part of pulsar emission beam is of the X-mode, and has a steep polarization angle curve.
In the observational aspect, polarization observations at multiple frequencies are important to reveal the frequency dependencies of intensities and polarization degrees of the components. The polarization observations should have higher signal to noise ratio and time resolution. For example, PSR J1048-5832 appeared to have one component at 1.44GHz due to limited time resolution [@qml+95], but it is clearly resolved to two components by the recent polarization observations at 1.5GHz [@wj08], which show clearly the gradient differences of polarization angle curves between the highly polarized and low polarized components. In addition, single pulse observations can help to identify the wave modes and depolarization processes [@scr+84; @scw+84].
In the theoretical aspect, our simulations here represent the further development of joint researches on the emission processes and propagation effects [@wwh14] and focus mainly on the properties for the highly polarized components within the wave mode separated magnetosphere. Note, however that in our current calculations, the magnetic field is assumed to be a rotating dipole for an empty magnetosphere. Radiation correction is neglected, and the effect of loaded plasma on magnetic fields is not yet incorporated. Furthermore, the energy and density distributions of relativistic particles are assumed to follow a simple model. Therefore, the conclusions and predictions under these assumptions may be altered if more complicated pulsar magnetosphere is considered.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been supported by the National Natural Science Foundation of China (11403043, 11473034 and 11273029), and the Strategic Priority Research Programme “The Emergence of Cosmological Structures” of the Chinese Academy of Sciences (Grant No. XDB09010200).
Peak intensity ratios for highly and low polarized components
=============================================================
Tables \[table:leading\_ratio\] and \[table:trailing\_ratio\] list the peak intensity ratios, $\rm I_{HiP}/I_{LowP}$, of the highly polarized components at the leading and trailing part of profiles with respect to the low polarized components at a series of frequencies.
1.5mm
PSR Freq.(GHz) $\rm I_{HiP}/I_{LowP}$ References
-------------- ------------ ------------------------ ----------------
J0358+5413 0.234 $<0.05$ 31
0.325 $<0.03$ 77
0.408 $0.05\pm0.02$ 16, 31
0.610 $0.17\pm0.02$ 31
0.925 $0.36\pm0.02$ 31
1.408 $1.00\pm0.04$ 31
1.642 $1.33\pm0.06$ 31
1.71 $1.77\pm0.08$ 30
1.72 $2.13\pm0.07$ 9, 20
2.65 $2.40\pm0.26$ 8, 9
4.85 $9.00\pm2.26$ 30, 34, 81
8.7 $11.00\pm5.52$ 9
10.55 $10.00\pm2.01$ 24, 30, 34
32.0 $>8.25$ 26
J0454+5543 0.234 $0.32\pm0.05$ 31
0.408 $0.36\pm0.02$ 16, 31
0.610 $0.48\pm0.03$ 31
0.91 $0.57\pm0.05$ 31
1.408 $0.74\pm0.04$ 30, 31
1.642 $0.70\pm0.09$ 31
1.72 $0.81\pm0.04$ 20
4.85 $0.83\pm0.05$ 30, 81
J1048-5832 0.692 $0.70\pm0.06$ 90
1.369 $1.50\pm0.09$ 90
1.5 $1.71\pm0.06$ 69
3.0 $3.71\pm0.27$ 69
3.1 $4.75\pm0.91$ 56, 90
6.387 $7.33\pm3.70$ 90
8.4 $11.50\pm2.89$ 59, 90
J1057-5226IP 0.17 $0.81\pm0.25$ 12
0.325 $1.00\pm0.07$ 77
0.631 $1.60\pm0.28$ 6, 12
0.64 $1.88\pm0.40$ 16
0.692 $1.50\pm0.27$ 90
0.95 $2.60\pm1.95$ 29
1.369 $2.33\pm0.21$ 4, 72, 90
1.5 $2.15\pm0.18$ 69
3.0 $2.78\pm0.82$ 69, 90
J1825-0935MP 0.243 $0.07\pm0.06$ 65
0.325 $0.07\pm0.02$ 65, 73, 77
0.408 $0.09\pm0.02$ 31
0.61 $0.14\pm0.02$ 31
0.69 $0.13\pm0.02$ 63, 65
0.925 $0.17\pm0.02$ 31
0.95 $0.19\pm0.02$ 29
1.4 $0.20\pm0.02$ 30, 49, 61, 65
1.612 $0.24\pm0.02$ 7
3.1 $0.26\pm0.02$ 61, 63, 65
4.85 $0.24\pm0.02$ 30
10.45 $0.42\pm0.18$ 30
J1844+1454 0.243 $<0.05$ 65, 90
0.325 $<0.02$ 65, 77, 90
0.408 $<0.05$ 31
0.61 $0.25\pm0.03$ 31
0.69 $0.33\pm0.08$ 65, 90
0.925 $1.00\pm0.21$ 31
1.4 $1.15\pm0.09$ 17, 31, 38
54, 65, 74
1.642 $1.18\pm0.23$ 31
3.1 $2.00\pm0.67$ 65, 90
: The ratios for peak intensities of highly polarized leading components, $\rm I_{HiP}$, with respect to low polarized ones, $\rm
I_{LowP}$, at a series of frequencies. References are numbered in Table A1.[]{data-label="table:leading_ratio"}
1.5mm
PSR Freq.(GHz) $\rm I_{HiP}/I_{LowP}$ References
------------ ------------ ------------------------ ------------
J0601-0527 0.243 $0.34\pm0.14$ 90
0.325 $0.42\pm0.05$ 90
0.408 $0.45\pm0.08$ 31
0.61 $0.50\pm0.03$ 31
0.692 $0.66\pm0.09$ 90
0.925 $0.63\pm0.19$ 31
1.408 $0.95\pm0.07$ 31, 54, 90
1.5 $1.02\pm0.05$ 69
1.642 $1.05\pm0.19$ 31
1.72 $1.30\pm0.15$ 20
3.068 $1.79\pm0.26$ 90
4.85 $2.12\pm0.52$ 34
6.2 $2.86\pm1.41$ 90
J0922+0638 0.243 $0.77\pm0.05$ 65, 90
0.322 $1.0\pm0.07$ 62, 65, 90
0.43 $1.1\pm0.05$ 19
0.690 $1.59\pm0.13$ 65, 90
J1539-5626 0.692 $0.45\pm0.37$ 90
1.377 $0.60\pm0.08$ 90
1.5 $0.58\pm0.03$ 69
3.1 $0.78\pm0.06$ 56, 69, 90
6.2 $1.77\pm0.65$ 90
8.356 $1.95\pm0.58$ 59, 90
J2225+6535 0.15 $0.07\pm0.06$ 88
0.234 $0.20\pm0.10$ 31
0.325 $0.28\pm0.10$ 77
0.408 $0.37\pm0.14$ 16, 31
0.61 $0.57\pm0.05$ 31
0.925 $1.11\pm0.17$ 31
1.408 $1.33\pm0.07$ 31
1.5 $1.39\pm0.07$ 86
1.642 $1.43\pm0.19$ 31
1.7 $1.38\pm0.21$ 9
: Same as Table \[table:leading\_ratio\] but for highly polarized trailing components. []{data-label="table:trailing_ratio"}
\[lastpage\]
[^1]: E-mail: pfwang@nao.cas.cn
|
---
abstract: 'Disks in binary systems can cause exotic eclipsing events. [MWC 882]{} (BD-22 4376, EPIC 225300403) is such a disk-eclipsing system identified from observations during Campaign 11 of the [*K2*]{} mission. We propose that [MWC 882]{} is a post-Algol system with a B7 donor star of mass ${\ensuremath{0.542 \pm 0.053}}\,M_\odot$ in a 72day period orbit around an A0 accreting star of mass ${\ensuremath{3.24 \pm 0.29}}\,M_\odot$. The ${\ensuremath{59.9 \pm 6.2}}\,R_\odot$ disk around the accreting star occults the donor star once every orbit, inducing 19day long, 7% deep eclipses identified by [*K2*]{}, and subsequently found in pre-discovery ASAS and ASAS-SN observations. We coordinated a campaign of photometric and spectroscopic observations for [MWC 882]{} to measure the dynamical masses of the components and to monitor the system during eclipse. We found the photometric eclipse to be gray to $\approx 1$%. We found the primary star exhibits spectroscopic signatures of active accretion, and observed gas absorption features from the disk during eclipse. We suggest [MWC 882]{} initially consisted of a $\approx 3.6\,M_\odot$ donor star transferring mass via Roche lobe overflow to a $\approx 2.1\,M_\odot$ accretor in a $\approx 7$day initial orbit. Through angular momentum conservation, the donor star is pushed outward during mass transfer to its current orbit of 72days. The observed state of the system corresponds with the donor star having left the Red Giant Branch $\sim 0.3$Myr ago, terminating active mass transfer. The present disk is expected to be short-lived ($10^2$ years) without an active feeding mechanism, presenting a challenge to this model.'
author:
- 'G. Zhou , S. Rappaport , L. Nelson , C.X. Huang , A. Senhadji , J.E. Rodriguez , A. Vanderburg , S. Quinn , C.I. Johnson , D.W. Latham , G. Torres , B.L. Gary , T.G. Tan , M.C. Johnson , J. Burt , M.H. Kristiansen , T.L. Jacobs , D. LaCourse , H. M. Schwengeler , I. Terentev , A. Bieryla , G.A. Esquerdo , P. Berlind , M.L. Calkins , J. Bento , W.D. Cochran , M. Karjalainen , A.P. Hatzes , R. Karjalainen , B. Holden , R.P. Butler'
bibliography:
- 'mybib.bib'
title: 'Occultations from an active accretion disk in a 72day detached post-Algol system detected by [*K2*]{}'
---
Introduction {#sec:introduction}
============
Approximately 70% of intermediate mass stars reside in multi-stellar systems , and perhaps only a half of them can go through life without experiencing the influence of their partners. Binary stars are important astrophysical laboratories, where the primordial stars share a common age, and these are excellent test-beds of binary stellar evolution, mass-transfer, and accretion processes. The final states of many binaries are shaped by mass transfer events during their lifetimes.
Algols are binary stars that experience mass transfer when the primary star in the primordial binary evolves. In these binaries [@1989SSRv...50....1B; @1989SSRv...50.....B; @2001ASSL..264...79P], the more massive companion evolves off the main sequence first, and expands to fill its Roche lobe. Overflowing matter from the more massive donor star accretes onto the mass-gaining star (hereafter ‘accretor’), leading to an inverted mass ratio for the resultant system. The actual pathway for each Algol depends on the initial mass ratio of the binary, the initial orbital period, how conservative the mass transfer is, and how much specific angular momentum is carried away with the lost mater . Some Algols are stable mass-transferring systems and leave remnant thermally bloated white dwarfs [e.g. @2015ApJ...803...82R] or subdwarf B and O stars [e.g. @2002MNRAS.336..449H]. However, shortly after the mass-transfer phase ends, the tenuous residual atmosphere of the mass losing giant can remain bloated for up to a Myr until the remaining hydrogen is consumed, while the envelope shrinks and gets hotter, to the point where the underlying white dwarf or subdwarf is revealed. This might be called a ‘transitional phase’.
It may be possible for the accretion disks to persist in systems that have ended active mass transfer. Occultations induced by persistent disks have been detected in a handful of long period binaries. Photometric occultations by disks are often distinct from other events, as they exhibit long duration, deep eclipse-like signals that can be identified in photometric surveys. $\epsilon$ Aurigae is the classical example of such a system, with a post-AGB F0 supergiant with a B-star companion embedded in a dusty disk with an orbital period of 27 years . The occultation is observed in photometry [e.g. @1970VA.....12..199G; @1986ApJ...300L..11K; @1991ApJ...367..278C], interferometry [@2010Natur.464..870K; @2015ApJS..220...14K], and spectroscopy , and the system’s orbit and masses are constrained by the radial velocities [@2010AJ....139.1254S]. The two year long eclipse is inferred to be from a $\sim 4$AU diameter disk consisting of both dust and gas [e.g. @1996ApJ...465..371L]. The dusty component is required to explain the strong IR excess in the system [@2010ApJ...714..549H], while the gaseous component in the vertically extended disk shell induces a series of spectroscopic features that map out the Keplerian disk during occultation . A central brightening is seen during eclipse, leading to a flared disk geometry interpretation . Similar long period, long duration disk occultations in other potential mass-transfer systems have also been identified: the 96 day period V383 Sco , the 5.6 year period EE Cep [@1999MNRAS.303..521M], the 468 day periodiocally recurring eclipses of OGLE-LMC-ECL-11893 [@2014ApJ...788...41D; @2014ApJ...797....6S], and 1277 day periodic eclipses of OGLE-BLG182.1.162852 [@2015MNRAS.447L..31R]. The recent advent of wide field photometric surveys is now also enabling numerous new discoveries, including the 69 year eclipsing disk system TYC 2505-672-1 [@2016AJ....151..123R].
[MWC 882]{} ($V_\mathrm{mag} = 10.8$) was originally identified in the Mount Wilson Catalog (MWC) of A and B stars due to its Balmer line emissions [@1949ApJ...110..387M]. Here we report that [MWC 882]{} is a 72-day period binary involving a B7 post Red Giant Branch (RGB) star and an A0 accretor, exhibiting disk occultations on each orbit. Occultations of the donor by the accretion disk around the accretor were detected by the [*K2*]{} mission (during Campaign 11), and subsequently identified in pre-discovery observations by ground-based photometric surveys. The basic geometry of the system is illustrated in Figure \[fig:disk\_illustration\]. As we detail below, the disk signatures of [MWC 882]{} are eerily reminiscent of $\epsilon$ Aurigae. The eclipse light curve exhibits a central brightening that we also argue to be from a flared-disk geometry. Spectroscopic signatures of the disk were detected, via a similar set of absorption lines, as that from the recent $\epsilon$ Aurigae eclipse. Unlike $\epsilon$ Aurigae, both members of the system are directly detected, and their dynamical masses can be measured. Our interpretation of [MWC 882]{}, as a post-Algol with a donor star that relatively recently ended its mass transfer phase, will aid the understanding of similar long period disk occultation systems.
![ In the [MWC 882]{} system, the $3.2\,M_\odot$ A0 accreting star is enshrouded in an accretion disk $\approx 60\,R_\odot$ in radius, orbited by a post-RGB $0.5\,M_\odot$ B7 donor star with an orbital period of 72days at a separation of $114\, R_\odot$. The donor star is occulted by the accretion disk every orbit, causing periodic eclipses $\approx 7$% deep. Our observations suggest that the donor star is much smaller than its Roche lobe, and is not sustaining active mass transfer. Illustration shows the geometry of the system from the observer, and is not to scale. \[fig:disk\_illustration\]](disk_illustration2.png){width="\linewidth"}
Detection and Photometric follow-up observations {#sec:Photobs}
================================================
The photometric and astrometric properties of [MWC 882]{} are listed in Table \[tab:litparams\]. A total of 38 photometric occultations were recorded in multiple datasets, including the [*K2*]{} discovery light curves, pre-discovery datasets from ground-based wide-field photometric surveys, and our multi-wavelength follow-up observations. They are summarized below, listed in Table \[tab:photobs\] for clarity, and plotted in Figure \[fig:lc\] in full.
\
[llrr]{} ASAS & 2001 Jan 31 – 2009 Oct 25 & 662 & $V$\
ASAS-SN & 2015 Feb 16 – 2017 Jul 2 & 418 & $V$\
[*K2*]{} & 2016 Sep 24 – 2016 Oct 18 & 1062 & $Kp$\
[*K2*]{} & 2016 Oct 21 – 2016 Dec 07 & 2174 & $Kp$\
HAO & 2017 Aug 30 – 2017 Sep 20 & 1000 & $g'$\
HAO & 2017 Aug 30 – 2017 Sep 20 & 870 & $z'$\
PEST & 2017 Sep 2 – 2017 Sep 12 & 79 & $B$\
PEST & 2017 Sep 2 – 2017 Sep 12 & 79 & $V$\
PEST & 2017 Sep 2 – 2017 Sep 12 & 83 & $I$\
Identification from [*K2*]{} photometry {#sec:K2}
---------------------------------------
[MWC 882]{} was observed during Campaign 11 of the [*K2*]{} mission with the [*Kepler*]{} spacecraft [@2014PASP..126..398H] under the designation EPIC 225300403. [*K2*]{} provides photometric coverage of selected stars in fields distributed across the ecliptic plane over a temporal baseline of $\approx 72$ days per field. [MWC 882]{} was observed with a cadence of $30$min by [*K2*]{}. The target pixels were downloaded upon public release of the campaign data, and reduced as per @2014PASP..126..948V[^1].
We conducted a visual examination of Campaign 11 light curves for astrophysical features not usually identified by automated algorithms and periodic signal analyses. The inspections are aided by the [*LCTOOLS*]{} software and [*LCViewer*]{} packages, and was conducted in the fashion described in @2017arXiv170806069R. The light curve of [MWC 882]{} revealed a transit-like feature, with a depth of $\approx 7$%, and a full duration of $\approx 19.4$ days (see Figure \[fig:lc\]). The occultation is slightly asymmetric, with ‘ingress’ spanning over 2 days, and the egress extending over 8 days. A $\approx 3$% brightening is seen at the center of the eclipse.
Pre-covery with ASAS and ASAS-SN {#sec:precovery}
--------------------------------
Once the eclipse of [MWC 882]{} was identified in the [*K2*]{} data, we searched for additional events in archival photometric observations from the ASAS and ASAS-SN surveys. Here we present a brief overview of each survey and the observations available on [MWC 882]{}.
Designed to survey the entire sky and catalog all variable stars brighter than $V_\mathrm{mag}=14$, the All-Sky Automated Survey [ASAS, @1997AcA....47..467P; @2001ASPC..246...53P] accomplished this goal by obtaining simultaneous $V$ and $I$ band photometry. The survey has two units, one located in Las Campanas, Chile and the other in Haleakala, Maui. Each unit has two telescopes equipped with wide-field Minolta 200/2.8 APO-G telephoto lenses and a 2K$\times$2K Apogee CCD. Each telescope setup has an 8.8$^{\circ}\times8.8^{\circ}$ field of view. ASAS observed [MWC 882]{} in the $V$ band at 662 epochs at a median cadence of 2.04 days from UT 2001 January 31 to UT 2009 October 25.
The eclipse signals were also recovered from observations by the All Sky Automated Survey for SuperNovae (ASAS-SN). Using two separate telescope units at Mount Haleakala in Hawaii and Cerro Tololo Observatory in Chile, ASAS-SN is monitoring the entire sky down to $V_\mathrm{mag} \approx$17 to detect new supernovae and transients [@2014ApJ...788...48S; @2017PASP..129j4502K]. Each unit has four 14cm aperture Nikon telephoto lens with 2K $\times$ 2K thinned CCDs, and can survey over 20,000 deg$^2$ each night, allowing the entire visible sky to be observed every 2 days. The telescope setup results in a 4.5$^{\circ}\times4.5^{\circ}$ field of view. The reduction pipeline is described in @2017PASP..129j4502K. ASAS-SN observed [MWC 882]{} 418 times at a mean cadence of 2.08 days from 2015 February 16 – 2017 July 2 UT.
We used the Box Least Squares algorithm to search for the period of the eclipse signal in the combined ASAS, ASAS-SN, and K2 data sets with the eclipse epoch fixed to be the center of the K2 event. The BLS spectrum has a strong detection at $72.416 \pm 0.016$ days. Uncertainties in this period estimation stem mainly from the difficulty in accurately determining the eclipse centroid. We use this period estimation to guide further follow up observations. The period is also independently confirmed via a summed harmonics Fast Fourier Transform on the ensemble photometry, yielding a period of $72.428\pm0.030$ days, as well as via a Stellingwerf transform [@1978ApJ...224..953S] at a period of $72.417\pm0.030$ days.
To check for possible period changes, we fitted for a set of eclipse times over 1 year segments of the ASAS and ASAS-SN light curves, using the [*K2*]{} eclipse as a template. We find no eclipse timing variations, with a $2\sigma$ upper limit of $|\dot P| < 0.016$ dayyear$^{-1}$ over the 16 year baseline. From the description of the binary system evolution set out in Section \[sec:evolution\], we expect no detectable period changes in the current system configuration, even if mass transfer is occurring at a rate of $10^{-8}\,M_\odot \, \mathrm{yr}^{-1}$ as inferred from the spectroscopy (Section \[sec:sig\_accretion\]). Based on the simplest dimensional arguments using conservation of orbital angular momentum, we expect period changes of $\dot P/P \simeq 3 |\dot M|/M_{\rm don}$. From that, we find that $\dot P$ is likely to be $\lesssim 4 \times 10^{-6}\,\mathrm{days}\,\mathrm{yr}^{-1}$, orders of magnitude smaller than our measurements could possibly detect.
Ground-based photometric follow-up {#sec:PHFU}
----------------------------------
Following the period determination for the occultation of [MWC 882]{}, we targeted the predicted occultation in early 2017 September with a series of ground-based photometric observations. This occultation occurred five orbits after that observed by the [*K2*]{} mission, and was the first observable occultation post [*K2*]{} data release.
Observations were obtained at the Hereford Arizona Observatory (HAO), using a 0.36m Meade LX200 GPS telescope with a Santa Barbara Instrument Group (SBIG) ST-10XME CCD camera. We made observations during 16 nights between 2017 August 30 to 2017 September 19, successfully recording the ingress, full eclipse, egress, and three days of post-occultation baseline photometry. The observations and reductions procedure follows that described in @2016MNRAS.458.3904R [@2017arXiv170908195R]. To check for color dependencies in the occultation depth, the observations were made in the $g'$ and $z'$ bands. On each night, continuous observations were performed over a two-hour span, with integration times of 20–40s. The per night average magnitudes are listed in Table \[tab:lc\_table\] and plotted in Figure \[fig:lc\].
We also obtained multi-band follow-up observations of the 2017 September occultation with the Perth Exoplanet Survey Telescope (PEST) located in Perth, Australia. PEST operates a fully automated 0.3m Meade LX200 telescope, coupled with a SBIG ST-8XME CCD camera. The observations covered parts of the full eclipse and egress, and were made over 2017 September 2 – 2017 September 12 period. To check for colour dependencies in the occultation, observations were made in the $B$, $V$, and $Ic$ bands. As with the HAO observations, the nightly averages of the light curves are shown in Figure \[fig:lc\].
[rrrrr]{}
2451940.8811800&0.9959&0.0290&ASAS&V\
2451949.8930800&0.9922&0.0420&ASAS&V\
2451953.8868900&0.9786&0.0310&ASAS&V\
2451954.8783100&1.0144&0.0350&ASAS&V\
2451961.8914600&0.9786&0.0330&ASAS&V\
2451963.8718300&0.9886&0.0290&ASAS&V\
2451965.8651100&1.0257&0.0300&ASAS&V\
2451967.8676900&1.0125&0.0340&ASAS&V\
2451978.8217600&0.9895&0.0400&ASAS&V\
...&...&...&...&...\
Spectroscopic observations {#sec:spec}
==========================
We obtained spectroscopic observations of [MWC 882]{} with a series of facilities over 2017 August and September to measure the masses of the system, estimate spectral classifications, and to monitor for temporal changes in the spectroscopic features. These observations are summarized in Table \[tab:specobssummary\].
A total of 11 observations were obtained using the Tillinghast Reflector Echelle Spectrograph (TRES) on the 1.5m telescope at Fred Lawrence Whipple Observatory, Mt. Hopkins, Arizona, USA. TRES is a fibre fed spectrograph with a spectral resolution of $R \equiv \lambda / \Delta \lambda = 44000$ over the wavelength range 3900–9100Å via 51 echelle orders. Each observation is made of three sequential exposures, combined to minimize the impact of cosmic rays, and were reduced as per @2010ApJ...720.1118B. Our observations spanned the period of 2017 July 31 – 2017 September 25, over orbital phases between 0.25 and 0.75. Of the 11 observations, 5 were obtained during the occultation at phase 0.5.
We observed [MWC 882]{} six times between 2017 August 28 and 2017 September 15 with the 2.7 m Harlan J. Smith Telescope and its Robert G. Tull Coudé Spectrograph [@1995PASP..107..251T] at McDonald Observatory, Mt. Locke, Texas, USA. We used the TS23 spectrograph configuration, providing $R=60,000$ over 58 echelle orders. The wavelength coverage is 3570-10200 Å, and is complete below 5691 Å with increasingly large inter-order gaps red-ward of this point; notably, H$\alpha$ falls into one of these gaps and is not captured. We obtained one spectrum per epoch. The data were reduced and the spectrum extracted and wavelength calibrated using standard IRAF tasks.
We also obtained an observation of [MWC 882]{} using the 2.4m Automated Planet Finder (APF) and the Levy spectrograph, located at Lick observatory, Mt Hamilton, California, USA. The APF is coupled with a high resolution, slit-fed, spectrograph that works at a typical resolution of R $\approx$ 110,000. The telescope is operated by a dynamic scheduler, has a peak overall system throughput of 15%, and its data reduction pipeline extracts spectra from 3750-7600Å [@Vogt2014; @Burt2015]. The APF spectrum used in this work was extracted from a single, 20 minute exposure taken with the 1x3" slit on 2017 July 26.
[llrrrr]{} APF 2.4m & 2017 Jul 26 & 1 & 110000\
FLWO 1.5m/TRES & 2017 Jul 31 – 2017 Sep 25 & 11 & 44000\
McDonald 2.7m & 2017 Aug 28 – 2017 Sep 15 & 6 & 60000\
Radial velocities of the binary {#sec:RV}
-------------------------------
The spectra of [MWC 882]{} show obvious signs of blending by the two stellar components in the system. Luckily, only the donor star is hot enough to exhibit strong He I absorption lines. As such, we make use of the unblended He I lines at 5876Å and 6678Å to measure the radial velocity of the donor star (Figure \[fig:He\]). To measure the velocity of the accretor, we perform a least-squares deconvolution of the spectrum to derive the broadening kernel of the spectral lines. The broadening kernel contains the radial velocity information of the system, since the observed spectrum is the convolution of the kernel of the two stars and a synthetic non-rotating single star spectral template. Following the technique laid out in @1997MNRAS.291..658D, a deconvolution is performed between the observed spectrum and a single-star non-rotating synthetic spectral template. The template is generated with the SPECTRUM code[^2] [@1994AJ....107..742G] with the ATLAS9 model atmospheres [@2004astro.ph..5087C], over the wavelength range of 4000-6100Å, and has the atmospheric properties of a $T_\mathrm{eff}=12000$K, ${\ensuremath{\log{g}}}= 4.5$ main-sequence star. We find that broadening profiles from least-squares deconvolutions often resolve blended lines better than classical cross correlation functions, due to the sharp-edge nature of rotational broadening kernels.
Temporal variations of the broadening profile, as a function of the orbital phase, are shown in Figure \[fig:lsd\]. These temporal variations capture the radial velocity orbit of the system, and have the potential of revealing line width and asymmetry variations if they are present. We fit the broadening profiles with a two-star model, and each kernel of each star is modeled with the convolution of a rotational kernel and a macroturbulence kernel, along with the parameters for velocity centroid and height ratio. The velocity of the broadening kernel of the donor is fixed to that measured from the He I line centroids, while their broadening values are fitted for simultaneously with the fitting of the donor’s profile. The fitting is performed via a Markov chain Monte Carlo (MCMC) exercise, using the [*emcee*]{} affine invariant ensemble sampler [@2013PASP..125..306F]. The derived radial velocities are presented in Table \[tab:rv\_table\], and plotted in Figure \[fig:rv\].
Keplerian fits to the radial velocities yield the dynamical masses of the system. The Keplerian orbit parameters and associated uncertainties are determined via an MCMC analysis. To yield realistic uncertainty estimates and to account for underestimation of per-point velocity measurement errors, the per-point velocity uncertainties were inflated such that the reduced $\chi^2$ for the best fit solution after the MCMC burn-in chain is at unity. We did not attempt to correct the velocities from different facilities for systematic offsets. Instead we chose to increase the formal uncertainties to account for this effect. The per-point uncertainties were assigned to be at least 0.5[$\rm km\,s^{-1}$]{} to account for these possible systematic uncertainties. We find, for assumed circular orbits, that the radial velocity amplitudes are $K_1 = {\ensuremath{14.3 _{-1.3}^{+1.4}}}$[$\rm km\,s^{-1}$]{} and $K_2 = {\ensuremath{85.5 _{-2.2}^{+1.9}}}$[$\rm km\,s^{-1}$]{}, resulting in minimum mass measurements of $M_\mathrm{acc} = {\ensuremath{3.24 \pm 0.29}}\,M_\odot$ and $M_\mathrm{don} = {\ensuremath{0.542 \pm 0.053}}\,M_\odot$. The residuals from the best fit model have scatters of 3.0[$\rm km\,s^{-1}$]{} and 1.4[$\rm km\,s^{-1}$]{} for the accretor and donor velocities, respectively. To test the effect of combining velocities from multiple instruments in our analysis, we recomputed the masses from the TRES velocities alone, finding $M_\mathrm{acc} = 3.14\pm0.14 \, M_\odot$ and $M_\mathrm{don} = 0.54\pm0.02 \, M_\odot$, consistent with the combined analysis. We also performed an independent radial velocity analysis of the TRES spectra via the two-dimensional cross correlation technique [*TODCOR*]{} [@1994ApJ...420..806Z], finding $M_\mathrm{acc} = 3.01 \pm 0.17 \, M_\odot$, and $M_\mathrm{don} = 0.508 \pm 0.017 \, M_\odot$, consistent with the masses quoted above to within $1\sigma$.
We checked that the orbit is indeed consistent with being nearly circular, with eccentricity formally constrained to be $e = 0.021 \pm 0.010$. While we cannot rule out a small, but non-zero eccentricity for the system, significantly more radial velocities are required to avoid biases at near-zero eccentricities [e.g. @1971AJ.....76..544L; @1978Obs....98..122B]. We note, however, that non-zero eccentricities have been detected for other Algols [e.g. TT Hydrae @2007ApJ...656.1075M], so such follow-up observations are worthwhile.
[lrrrrr]{} 2457960.7368 & -24.62 & 0.21 & -61.67 & 0.50 & APF\
2457965.6830$^d$ & ... & ... & -26.84 & 1.79 & TRES\
2457990.6635$^d$ & ... & ... & 53.73 & 0.55 & TRES\
2457993.6303 & -31.83 & 0.15 & 38.44 & 0.50 & McDonald\
2457994.6378 & -34.05 & 0.31 & 34.32 & 0.50 & McDonald\
2457994.6673 & -33.38 & 0.23 & 34.94 & 0.50 & TRES\
2458002.6414 & -21.70 & 0.33 & -21.58 & 2.00 & TRES\
2458006.6275$^e$ & ... & ... & -47.81 & 1.31 & TRES\
2458007.6279$^e$ & ... & ... & -56.62 & 1.02 & McDonald\
2458008.6041$^e$ & ... & ... & -63.44 & 0.50 & McDonald\
2458008.6290$^e$ & ... & ... & -62.36 & 0.50 & TRES\
2458009.5981$^e$ & ... & ... & -71.15 & 0.50 & McDonald\
2458009.6244$^e$ & ... & ... & -69.11 & 0.50 & TRES\
2458011.6228 & -14.25 & 0.16 & -79.53 & 0.50 & McDonald\
2458019.6026 & -7.77 & 0.16 & -106.26 & 0.50 & TRES\
2458020.6046 & -8.17 & 0.30 & -107.93 & 0.96 & TRES\
2458021.6009 & -8.74 & 0.10 & -106.84 & 0.50 & TRES\
\[ht\]
\[ht\]
\[ht\]
Spectral classification {#sec:spec_class}
-----------------------
Stellar classification was particularly difficult given the complexity of the system: the spectra are blended, both stars exhibit signatures of chemical peculiarity (see Section \[sec:chemical\_peculiarity\]), active accretion prevents the use of Balmer line strengths for temperature estimates, and both stars are severely reddened by interstellar and circumstellar material. As such, no single set of synthetic template spectra and fluxes fit the observed spectra well.
Since the stars exhibit chemical peculiarity and accretion signatures, we did not perform a global spectral matching of observed and synthetic spectra. Instead, we make use of temperature sensitive spectral features. For example, strong and broad near-UV/optical He I absorption lines are prominent features in B-star spectra, but decrease in strength toward cooler temperatures. As a result, He I lines can be useful temperature indicators. Figure \[fig:He\] shows the radial velocity orbital phase variations of the donor star by its He I absorption feature. Note that we see He I only at the spectrum of the donor star, not the accretor, suggesting that the donor star is hotter than $\approx 13000\,\mathrm{K}$. The Ca II H & K lines grow weaker and narrower for earlier type A and B stars, while C II lines grow in strength for B stars. Figure \[fig:spec\_class\] shows the observed spectra of [MWC 882]{} for the Ca II K and C II lines, compared to synthetic spectra from ATLAS12 models [@2005MSAIS...8..189K] taken from the POLLUX spectral database . Synthetic spectra with metallicity of $\mathrm{[Fe/H]} = +0.5$ were chosen, since the stellar surface of both stars appears metal enriched and chemically peculiar (Section \[sec:chemical\_peculiarity\]). We also adopt models with a surface gravity of ${\ensuremath{\log{g}}}= 4.5$ for the accretor, corresponding to its expected properties after mass transfer (see Section \[sec:evolution\]), and a gravity of ${\ensuremath{\log{g}}}= 3.5$ for the donor, the lowest gravity template available in the ATLAS12 grid. We take as the ‘observed spectrum’ to be evaluated an average of the spectra taken over 2017 September 23 – 25, at orbital phase 0.75–0.77, where the radial velocity separation between the accretor and donor were at its greatest. We find the accretor star to be consistent with an A0 spectral type with an effective temperature of $12000\pm1000$K, while the donor is likely a B7 star, with an effective temperature of $15000\pm1000$K. The similar temperatures between the accretor and donor star are consistent with the grey eclipse observed during our follow-up observations (Section \[sec:PHFU\]).
Broadening of the spectral lines informs us of the projected rotational velocity of the stars. We fit for rotation profiles [defined in @2005oasp.book.....G] to the least-squares deconvolution kernels taken at phase 0.25 and 0.75, where the velocities of the two stars are well separated. We find both stars exhibit similar line broadenings, with the accretor rotating at $v\sin i = 25.6\pm4.5\,{\ensuremath{\rm km\,s^{-1}}}$, and the donor star with $v\sin i = 31\pm11\,{\ensuremath{\rm km\,s^{-1}}}$. We note, however, that our quoted rotational broadening velocities do not account for the effects of macroturbulence. The broadening profile of both stars in the [MWC 882]{} system deviate significantly from the standard rotational profiles, and the fits that incorporate both [$v \sin{I_\star}$]{} and macroturbulence are highly degenerate. A similar issue was encountered when analysing the spectra of $\epsilon$ Aurigae, with multiple [$v \sin{I_\star}$]{} values quoted, ranging from $5\,{\ensuremath{\rm km\,s^{-1}}}$ to 41[$\rm km\,s^{-1}$]{} [@2010AJ....139.1254S]. Regardless of the assumptions on non-rotational line broadenings, the stars in both [MWC 882]{} and $\epsilon$ Aurigae are rotating significantly slower than the average A and B stars , and certainly not near breakup velocity as expected for actively accreting systems. @2010MNRAS.406.1071D suggest a strong stellar wind as a mechanism for angular momentum dumping. They expect that strong differential rotation, induced by accretion, can generate strong magnetic fields that enhance the stellar wind and mass loss, spinning down the stars.
\[ht\]
----------------------------------------------- ---------------------------------------------
{width="0.5\linewidth"} {width="0.5\linewidth"}
----------------------------------------------- ---------------------------------------------
The relative line strengths of the accretor and donor spectra also inform us of the flux ratio of the system. To measure the flux ratio, we fit for the contributions of the donor and accretor stars over the entire TRES spectrum, order by order, using the ATLAS12 models. Only a few orders have high signal-to-noise and enough lines from both stars to constrain the flux ratio. We find the spectral region over 48 orders between 4000–6000Å satisfy this requirement well, yielding $F_\mathrm{acc} / (F_\mathrm{don} + F_\mathrm{acc}) = 0.52 \pm 0.18$, with the uncertainty being the scatter in the flux ratios measured. An illustrative section of the spectral fitting is shown in Figure \[fig:fluxratio\].
\[ht\] {width="0.95\linewidth"}
Reddening, UV, and IR excess are constrained by the spectral energy distribution (SED, Figure \[fig:sed\]). We model the SED with 12000K and 15000K synthetic templates from @2004astro.ph..5087C, and fit for a black body component to constrain any IR excess, and an extinction coefficient to account for interstellar and circumstellar reddening. Since both stellar components have very similar temperatures, the flux ratio is impossible to constrain using the SED only, so we fix the flux ratio to 0.52 as determined from spectral line fitting. We recover significant reddening of $E(B-V) = 1.07$, and is better fit with an extinction law of $R(V) = A(V) / E(B - V) = 2.5$. Minimal IR excess is detected in the SED, and can be rectified by a black body of 1200K. The IR excess may be due to contaminating M-dwarfs along the line of sight, or low levels of a dusty envelope around either of the stars.
\[ht\]
Chemical peculiarity {#sec:chemical_peculiarity}
--------------------
Some A and B stars exhibit chemical and magnetic peculiarity in their atmospheres. The spectra of both stars in the [MWC 882]{} system are consistent with classification as chemically peculiar stars. In particular, Figure \[fig:SiSrCr\] shows that both the donor and accretor star exhibit signatures of strong Si enhancement. Enhancement in the Cr lines are also seen in the accretor star, but not the donor star. We did not see enhancement in Sr, which is common amongst stars that are enriched in Si and Cr. We also searched for enhancements in Y, Hg, and Mn (Figure \[fig:HgMn\]), seen in the HgMn class of Bp/Ap stars, but did not detect their presence. The slow rotational ($\la$ 30 km s$^{\rm -1}$) velocities of both stars in the [MWC 882]{} system are consistent with the presence of strong magnetic fields, and both stars reside within the roughly B6-F4 spectral range of known Bp/Ap stars . Therefore, we have classified the donor and accretor stars in [MWC 882]{} as BpSi and ApSiCr stars, respectively.
\[ht\] {width="0.8\linewidth"}
\[ht\] {width="0.6\linewidth"}
Spectroscopic signatures of accretion {#sec:sig_accretion}
-------------------------------------
Evidence for active accretion is apparent in the spectra of [MWC 882]{}. H$\alpha$ is seen strongly in emission during all observations, with an equivalent width of $12\pm2$Å measured from the out-of-occultation spectra. The feature is double peaked, and is representative of those seen in longer period $(P>6\,\mathrm{d})$ Algols [e.g. @1999ApJS..123..537R; @2005ApJ...623..411B]. The wings of the H$\alpha$ feature extend to $\approx 400\,\mathrm{km\,s}^{-1}$, corresponding to an inferred inner disk radius of $\approx 3.7 \,R_\odot$[^3]. In comparison, the isochrone-fitted radius of the $3.24 M_\odot$ accreting star is ${\ensuremath{3.09 \pm 0.59}}\,R_\odot$ and, as such, the inner edge of the accretion disk extends close to the stellar surface. The peak emission of both H$\alpha$ and H$\beta$ are located at $\approx 150\,\mathrm{km\,s}^{-1}$, corresponding to a disk radius of $\approx 28 \,R_\odot$. The Ca II triplet (CaT) lines exhibit inverted P-Cygni profiles, again indicative of active accretion. The line profiles and phase variations of the accretion features are shown in Figure \[fig:accretion\_lineprof\]. We see no He features in the accreting star, nor any signatures of wind and outflows in the He I or O I lines.
The mass-transfer rate onto the accreting star can be estimated via the H$\alpha$ luminosity. By comparing the H$\alpha$ flux against the bolometric luminosity from our SED fit, we measure $L_{\mathrm{H}\alpha} / L_\mathrm{bol,acc} = 0.0023$, implying $L_{\mathrm{H}\alpha} = 0.32_{-0.07}^{+0.15}\,L_\odot$. Assuming complete conversion of potential energy, the mass transfer rate estimated from the H$\alpha$ flux is $1.3\pm0.3\times10^{-8}\,M_\odot \, \mathrm{yr}^{-1}$. For comparison, typical accretion rates measured for Algols in 4 to 13 day period orbits range from $10^{-9}$ to $10^{-5}\,M_\odot\,\mathrm{yr}^{-1}$ . We note that an accurate estimate of the accretion rate should model the H$\alpha$ profile to extract the accretion luminosity, and account for flux from other accretion signatures. Future observations in the UV and more sophisticated modeling of the line profiles will yield more accurate accretion rates for this system.
\[ht\]
-------------------------------------------- --------------------------------------------- ------------------------------------------
{width="0.3\linewidth"} {width="0.3\linewidth"} {width="0.3\linewidth"}
-------------------------------------------- --------------------------------------------- ------------------------------------------
Gas disk absorption during occultation {#sec:diskspecfeatures}
--------------------------------------
\[ht\]
Occultations like those exhibited by [MWC 882]{} present opportunities to study the gas material within the accretion disk. During occultation, light from the donor star passes through the accretion disk, and spectroscopic absorption features from the optically thin parts of the gaseous disk are imprinted on the observed spectrum. These absorption lines trace the velocities of disk gas along the line of sight between the donor star and the observer.
Such occultation disk spectroscopy was obtained during eclipses of $\epsilon$ Aurigae in 1982-1984 [@1986PASP...98..389L] and the more recent 2009-2011 event [@2012JAVSO..40..729L; @2013PASP..125..775G; @2014MNRAS.445.2884M; @2014AN....335..904S]. The thin dusty and gaseous disk is seen to transit in front of an F0 supergiant, which we view almost edge on (near $90^\circ$ inclination). Disk lines from the optically thin disk ‘shell’ around $\epsilon$ Aurigae were observed at varying widths, depths, and velocities through the two year eclipse. In particular, lines of low excitation potential species were strongly detected, such as Na I, K I, Mg I, H$\alpha$, and H$\beta$. These lines traced the Keplerian velocity of the disk, and helped to constrain the morphology, homogeneity, and potential eccentricity of the disk.
Our spectra obtained during the occultation of [MWC 882]{} revealed a series of broad absorption features that are offset in velocity from both the donor and accretor. These features are obvious in the Balmer lines (Figure \[fig:accretion\_lineprof\]), as well as many neutral and ionised metal lines, such as the Ca II triplet (Figure \[fig:accretion\_lineprof\]), Si II (Figure \[fig:Si6300\]), Fe I, II, Na I D, K I, Mg I. The disk features are also present in the ensemble line profiles derived from the least-squares deconvolution analysis, shown in Figure \[fig:lsd\]. The velocities derived from selected unblended lines are shown in Figure \[fig:blobrv\].
Assuming that the greatest contribution in flux to the disk lines come from the inner-most regions of the disk that actually occult the donor star, we can use these disk line velocities to constrain the occultation geometry and the system architecture (Section \[sec:system\_architecture\]).
\[ht\]
System architecture {#sec:system_architecture}
===================
\[ht\]
\[ht\]
\[ht\]
Algols undergoing active accretion are rarely detected at periods as long as 72 days [e.g. @1999ApJS..123..537R]. Unravelling the true system architecture is crucial to understanding its evolutionary state.
From the light curves and radial velocities, we know that the disk around the accreting star is occulting the donor star, the two orbiting each other with a period of 72 days. The dynamical masses indicate the accretor is now ${\ensuremath{3.24 \pm 0.29}}\,M_\odot$ and the donor is ${\ensuremath{0.542 \pm 0.053}}\,M_\odot$, separated by a distance of $114.0 \pm 3.1\,R_\odot$. Examinations of the spectroscopic lines find that the donor is slightly hotter than the accretor, at 15000K and 12000K respectively, both contributing approximately equally to the flux of the system. Properties of the accretor can be further estimated by fitting its mass and effective temperature to standard stellar isochrones. Using the Geneva isochrones for high mass stars , at solar metallicity, and no rotation, the radius and luminosity of the accretor is estimated to be ${\ensuremath{3.09 \pm 0.59}}\,R_\odot$ and $142_{-33}^{+67}\,L_\odot$. Assuming Stefan-Boltzmann’s law and a flux ratio of $F_\mathrm{don} / (F_\mathrm{don} + F_\mathrm{acc}) = 0.52 \pm 0.18$, we calculate the radius of the donor to be $1.79 \pm 0.72\,R_\odot$. We note that the Geneva isochrones are single star evolution tracks, and begin at zero-age main-sequence. The accretor star, however, is a lower mass star that is ‘rejuvenated’ by the mass transfer episode, with different interior structures. Section \[sec:evolution\] discusses binary evolution tracks that account for the co-evolution of the two stars. However, the Geneva grid presents a reasonable estimate of the properties of the accretor star, and an interpolation of the grid allows us to estimate uncertainties on the properties of both stars, which is more difficult with the scenario-specific binary evolution tracks.
Adopting these masses and radii, neither star is close to filling its Roche lobe (Figure \[fig:roche\]). The estimated sizes of the Roche lobes are $61.1\pm2.5\,R_\odot$ for the accretor and $27.3\pm 0.9 \,R_\odot$ for the donor [following the approximation in @1983ApJ...268..368E]. In fact, a stable disk can only be as large as 80% of the Roche lobe [@1977ApJ...216..822P], with a maximum radius of $49.4\pm3.2\,R_\odot$.
We create a toy disk model to simultaneously fit the eclipse light curve and spectroscopic disk velocities. In this model, the eclipse duration constrains the diameter of the disk. The ingress and egress timescales constrain the radius of the donor star, if we assume the disk has a sharp edge and uniform optical depth. The central brightening is modeled by including a flaring to the disk, such that a smaller area of the donor star is covered during mid-eclipse. The velocities of the disk absorption lines seen during occultation are also fitted for simultaneously. At each phase, we take the disk line velocity to be described by the Keplerian orbital velocity of the innermost disk annulus that is occulting the limb of the donor star.
We fit for a disk of radius $R_\mathrm{disk}$ using the following prior. The probability is not penalized if $R_\mathrm{disk} < 49.4$, but follows a Gaussian distribution with $\sigma = 3.2 ~R_\odot$ (see above) if the disk is larger. We also fit for the donor star radius, $R_\mathrm{don}$, constrained by a Gaussian prior about $1.79 \pm 0.72\,R_\odot$, its inferred radius being taken from spectroscopic analyses. The system is inclined to our line of sight by angle $i$, and the disk exhibits a flare angle of $\beta$, with thickness $h$ at the inner edge, and has uniform optical depth of $\tau$. We make use of all available light curves, and fit for a reference eclipse time $T_c$ and period $P$. We estimated the best-fit values and uncertainties for model parameters with an MCMC exercise. Since the star exhibits photometric jitter only at the 7mmag level, the per-point uncertainties are inflated to be the scatter of the out-of-occultation light curve. We find a best fit radius of the donor star of ${\ensuremath{2.01 \pm 0.52}}\,R_\odot$, being occulted by a disk of $R_\mathrm{disk} = {\ensuremath{59.9 \pm 6.2}}\,R_\odot$, with a flaring angle of $1.63 \pm 0.47 ^\circ$ and inclined to our line of sight by $1.87 \pm 0.58 ^\circ$. The best fit light curve model is illustrated in Figure \[fig:lcmodel\], and the best fit parameters summarized in Table \[tab:paramtable\].
Some of the parameters in our model are highly degenerate, and the model is overly simplistic for this system. From the MCMC analysis, the flare angle is highly correlated with the donor star radius and line of sight inclination. The true disk will also not have a sharp edge, and the ingress and egress timescales will be degenerate with the optical depth of the disk and the donor star radius. For future reference, we also note that our adopted priors on disk and donor star radii significantly influence the goodness-of-fit of the light curve. We found that an unconstrained model fits the light curve much better, but tends to adopt a significantly larger disk radius of $\approx 65 \,R_\odot$ and larger donor star of $\approx 15 \,R_\odot$, neither are physical solutions given our known constraints on the system. Even with the Gaussian prior on the disk radius, our derived value of ${\ensuremath{59.9 \pm 6.2}}\,R_\odot$ is in tension with that expected from the Roche lobe constraint at the $1.5\,\sigma$ level. The disk size in our model depends on the eclipse impact factor through a perfectly elliptical projected disk. However, we know from the ingress and egress light curve that the disk is irregular. In this simplistic modeling, we are forcing the disk to be well aligned and symmetric. Asymmetries in the disk can be better explored by future observations that provide better spectroscopic coverage of the disk absorption lines during occultation (Section \[sec:diskspecfeatures\].
It is also possible that the disk contains significant amounts of dust. [MWC 882]{} exhibits levels of infrared excess consistent with a 1500K dusty disk. Similarly, dust $\gtrsim 5$$\mu\mathrm{m}$ are thought to be the source of infrared excess in $\epsilon$ Aurigae [@2010ApJ...714..549H]. The dust in the disk of $\epsilon$ Aurigae may be partially responsible for the mid-eclipse brightening via forward scattering .
Forward scattering can also effectively reproduce the mid-eclipse brightening seen in the occultations of [MWC 882]{}. During the eclipse, light from the donor star is passing through, or skimming along the surface of the disk. There are three components to the forward scattering pattern: (1) the angular size of the light source as seen by a dust grain; (2) the scattering pattern (i.e., ‘phase function’) vs. scattering angle for individual dust grains; and (3) the angular size of the dust region as seen from the donor star. These three angular components would be convolved together. The first of these is just a few degrees due to the small size of the donor. The second component is given approximately from Mie scattering theory by $\sim 20^\circ/s$ where $s$ is the grain size in $\mu$m. For 5–10 $\mu$m particles, this angle is only $2-4^\circ$. Finally, the size of the whole dust cloud is essentially $60^\circ$ across, and which dominates the system scattering pattern. As such, at orbital phase $\phi$, the magnitude of the forward scattering effect is roughly proportional to the length of the chord through the disk at each angle $\phi$: $$B(\phi) \propto \sqrt{ \left(R_\mathrm{disk}/a \right)^2 - \sin^2\phi} \, .$$
Figure \[fig:scatter\_lcmodel\] demonstrates the effect of forward scattering on a non-flared disk occultation model. The disk is chosen to be $60\,R_\odot$ wide, inclined to our line of sight at an angle of $1.5^\circ$ with an intrinsic width of $0.5\,R_\odot$. In this demonstration, the magnitude of the scattering effect is arbitrarily normalized to replicate the central brightening seen in the photometry.
System Evolution {#sec:evolution}
================
We propose that [MWC 882]{} is a post-Algol binary system. Although there are a number of different configurations for the progenitor binary that could have evolved to match the currently observed system, we believe that the progenitor binary likely consisted of a 3.6 M$ _\odot$ primary (the donor star) and a 2.1 M$ _\odot$ secondary star (accretor) with an orbital period of about 7.05 days. Some of the difficulties in determining the properties of the progenitor binary are related to: (i) uncertainties in the physics describing the amount of mass lost from the binary as it evolves; and, (ii) the magnitude of orbital angular momentum loss due to systemic mass loss, stellar winds, and tidal friction [@2000NewAR..44..111E]. It is very difficult to quantify these phenomena but previous theoretical calculations based on reasonable physical assumptions have matched the properties of many observed post-Algol systems [e.g. @1989SSRv...50....1B and references therein]. Moreover, the uncertainties can be parameterized and constrained within certain physical limits [e.g. @2000NewAR..44..111E].
According to our favored scenario, a 3.6 M$ _\odot$ primary evolves off of the Main Sequence and burns sufficient hydrogen to form a 0.35 M$ _\odot$ helium ash core that is surrounded by a thin hydrogen-burning shell just as it first begins to overflow its Roche lobe. At this point (see Figure \[fig:evolutionary\_state\]) the luminosity of the donor is about 230 L$ _\odot$ and its radius is $\sim 10$ R$ _\odot$. Mass transfer from the donor star to the 2.1 M$ _\odot$ accreting companion proceeds stably as the donor evolves up the RGB[^4]. Mass-loss rates from the donor can approach 10$^{-5.5}$ M$ _\odot$/yr because of its short nuclear time scale and because of angular momentum carried away by the (systemic) mass-loss from the binary. After the donor has lost about 85[%]{} of its mass, it is largely composed of a compact, degenerate helium core with a mass of approximately 0.5 M$ _\odot$, and a very tenuous envelope composed mostly of hydrogen that has a mass of approximately 0.05 M$ _\odot$ and extends over a cross-sectional radius of 25 R$ _\odot$. Continued mass loss cannot be sustained, as this causes this envelope to collapse once a certain threshold in pressure is reached due to the ever decreasing gas densities in the envelope. Thus the donor contracts within its Roche lobe on its thermal (Kelvin-Helmholtz) timescale with the concomitant cessation of mass transfer [see @2004ApJ...616.1124N and references therein for a detailed discussion].
Once mass transfer stops, nuclear burning near the surface of the helium core persists for several million years. This causes the surface luminosity to remain high (more than 100 $L_\odot)$ and approximately constant during this phase. As a result, the effective temperature continues to rise as the donor star evolves thermally. We find that the donor star evolves through this post-RGB “horizontal branch” phase for approximately 0.35 Myr before attaining a surface temperature and luminosity close to what are inferred for [MWC 882]{}. At this juncture the donor has a mass of 0.495$M_\odot$ and a luminosity of approximately $300\, L_\odot$. The orbital period of the binary is 72.4 days and the mass of the companion is $3.03\, M_\odot$.
In order to reproduce the observed properties of [MWC 882]{} a grid of models was computed using the MESA stellar evolution code [@2011ApJS..192....3P; @2015ApJS..220...15P]. The evolution of the donor star was followed according to the Roche lobe overflow model. The evolution of the accretor was calculated simultaneously, and the computation was stopped after the accretor evolved to become a giant, thereby filling its Roche lobe and leading to a ‘mass-transfer reversal’. The chemical composition of the progenitor binary was assumed to be solar (Z$=$0.02) and magnetic braking was assumed to operate in low-mass stars ($\lesssim 1.5 ~M_\odot$) with convective envelopes and radiative cores.
Although highly uncertain, we set the systemic mass-loss and angular momentum-loss parameters such that $\alpha$=0.4, $\beta$=0.3, $\gamma$=0, and $\delta$=0 [for details see @2006csxs.book..623T]. Here, $\alpha$ and $\beta$ are fractions of the mass lost by the donor star that get ejected from the binary and carry away the specific angular momentum of the donor star and accreting star, respectively. $\gamma$ and $\delta$ are parameters that are associated with a circumbinary disk and are not used here. Note that these values imply that: (i) mass-transfer is quite non-conservative with an accretion efficiency of 30% (i.e., with 70% of the mass being lost via a fast Jeans’ ejection); and, (ii) none of the mass lost from the binary forms a circumbinary torus that can extract additional orbital angular momentum.
It is important to note that the track presented in Figure \[fig:evolutionary\_state\] is only one of several possible tracks that could reproduce the observed properties of the system. This implies that the properties of the progenitor binary (e.g., the initial masses and separation) and the physics of systemic mass and angular-momentum loss (e.g., $\alpha$ and $\beta$) do not have to be fine-tuned in order to match the observations. This increases the probability that the post-Algol hypothesis is correct.
The donor star in [MWC 882]{} is currently evolving through the Slowly Pulsating B-star (‘SPB’) instability strip[^5] and will contract further until substantial helium burning is ignited in its core after $\simeq 3$ Myr has elapsed. During the next $\simeq 50$ Myr the donor star resembles a subwarf B (sdB) star and begins the process of alpha-capture to produce oxygen in the core. This corresponds to the loop that is seen in Figure \[fig:evolutionary\_state\] for a temperature of 29,000K and a luminosity of 25$L_\odot$. After an additional $\simeq 35$ Myr, burning at the center of the donor will be complete and the chemical composition at the center is approximately 68% oxygen and 30% carbon. After passing through the sdO (subdwarf O star) region of the HR diagram, the donor reaches its maximum effective temperature (nuclear burning has been quenched) and subsequently enters the degenerate dwarf cooling phase. During this final phase its radius is approximately constant as it evolves to a completely electron degenerate configuration.
We also predict that the accretor will evolve off the main sequence approximately 300 Myr from now and fill its Roche lobe—resulting in a ‘mass-transfer reversal’ episode. The orange track in Figure \[fig:evolutionary\_state\] illustrates the future evolution of the accretor. At that time, the 3$M_\odot$ accretor will have evolved up the AGB and have a 0.54$M_\odot$ CO core with a thin helium-burning layer surrounding it. Once mass transfer onto the 0.495$M_\odot$ “donor” is initiated, it will be dynamically unstable with the “donor” spiraling in towards the center of the accretor [@1984ApJ...277..355W]. This common envelope (CE) phase of evolution will last for $\sim$100 to 1000 years, after which the envelope of the accretor will be ejected into the interstellar medium leaving a nascent double-degenerate CO-CO white dwarf binary of nearly equal mass in a tight orbit. In order to determine the likely orbital period of the double-degenerate binary we follow the approach used by @2017MNRAS.471..948R (see their Eqn. 10) and set their CE efficiency parameter to 0.5. This implies that the remnant binary will have an orbital period of about 1 hour and will be composed of a 0.54$M_\odot$ CO white dwarf and a 0.50$M_\odot$ CO white dwarf in a nearly circular orbit. The merger of these two white dwarfs as a result of gravitational radiation losses, which will occur within 40 Myrs of the CE phase, may even result in a Type Ia supernova if sub-Chandrasekhar mergers are admissible [@2010ApJ...722L.157V]. Because of the fine-tuning required, it is more likely that the merger product with be a very massive single white dwarf such as inferred for J0317-853 and GD362as (see @2013ApJ...773..136J for a discussion of mergers). If the $0.50\,M_\odot$ white dwarf has a slightly lower mass than our models predict, it is also possible that the merger could lead to the creation of an R CrB star (see, for example, @2006ApJ...638..454P).
Finally, it is important to note that the current-epoch donor star will be a member of the predicted population of subdwarf-B stars in wide binaries around A star accretors while it is in the helium-burning phase of evolution. Our model produces a very natural evolutionary pathway that reproduces the properties of members of this population. Binary population synthesis by @2003MNRAS.341..669H predicts a population of donor stars with masses sharply peaked at about $0.5\,M_\odot$, and with orbital periods extending up to $\lesssim 1000$ days, from the first stable Roche lobe overflow channel. The resultant accretors in these systems can have a wide range of spectral types based on their masses and degree of nuclear evolution. We could reasonably expect that most unevolved accretors would fall within the F to B V spectral classes; the evolved systems could have much cooler temperatures but are less likely to be observed because of their relatively short lifetimes. They note that most of these systems are probably beyond the detection thresholds of typical radial velocity surveys [e.g. @2001MNRAS.326.1391M] due to the small orbital semi-amplitudes of the luminous accretor stars.
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Why is the disk still around? {#sec:disk_puzzle}
-----------------------------
In the scenario outlined above, active mass transfer ended $\sim 0.3$Myr ago when the donor star evolved off the RGB, and under-filled its Roche lobe. The occultation we observe, however, indicates that a substantial disk of size ${\ensuremath{59.9 \pm 6.2}}\,R_\odot$ still remains around the accretor. We explore a few simple arguments for the lifetimes of gas and dusty disks below, to show that without the disk being actively fed, it is difficult for the disk to remain in its current state.
To estimate the lifetime of a gaseous disk, we adopt the $\alpha$-disk model for an approximation of the disk filling timescale $\tau_\mathrm{fill}$: $$\begin{aligned}
\tau_\mathrm{fill} &= \frac{M_\mathrm{disk}} {\dot{M}} \nonumber{} \\
& \simeq 240 \left(\frac{\alpha}{0.01}\right)^{-4/5} \left( \frac{\dot{M}}{10^{-8} M_\odot / \mathrm{yr}} \right)^{-3/10} \left( \frac{r_\mathrm{max}}{60\, R_\odot} \right)^{5/4} \,\mathrm{yr}\,, \end{aligned}$$ where $\dot{M}$ is the accretion rate, and $r_\mathrm{max}$ is the outer radius of the disk, and the parameter $\alpha$ specifies the efficiency of viscous angular momentum transport in the disk. Adopting our H$\alpha$ derived accretion rate of $10^{-8}\,M_\odot\,\mathrm{yr}^{-1}$, and an $\alpha$ parameter of 0.01, the disk should have dissipated in just 200 years. In fact, while our accretion rate may be somewhat underestimated, it needs to be smaller by 10 orders of magnitude to achieve a disk lifetime of 0.2 Myr. Alternatively, $\alpha$ would have to be $2 \times 10^{-6}$ for the disk to last for 0.25 Myr. We also note that for the disk to maintain its current accretion rate of $1.3 \times 10^{-8}\,M_\odot\,\mathrm{yr}^{-1}$ over 0.2Myr, the disk needs to be more massive than $0.003 \, M_\odot$. However, given that the currently inferred accretion rate is similar in magnitude to the theoretically expected mass transfer rate during the Roche-lobe-filling phase, it would be difficult to build up such a massive disk.
We can also explore the possibility that the disk contains significant dust. Dust particles in the disk are subjected to the Poynting-Robertson drag, by which dust grains lose orbital angular momentum. This results from the fact that in the rest frame of the dust there is a small component of the momentum flux of the photons that is in the direction opposed to the orbital motion. Figure \[fig:dust\_timescales\] shows the expected orbital decay timescales from Poynting-Robertson drag on dust particles in the disk at $49\,R_\odot$, for dust having different imaginary indices of refraction $k$. As can be seen from Figure \[fig:dust\_timescales\], such dust particles have orbital lifetimes of no more than $\sim$1000 years, insufficient to sustain the disk in its current state. Furthermore, most dust particles smaller than $\sim$10 $\mu$m have ratios of radiation-pressure forces to gravity $\gtrsim 0.5$ and therefore they become unbound from the system on a dynamical timescale.
\[ht\] ![A dusty disk is subjected to Poynting-Robertson orbital decay if the dust is relatively optically thin. We plot the orbital decay timescale for dust particles of various grain sizes and imaginary refraction indices $k$ when placed in an $49\,R_\odot$ orbit around the accretor. These P-R timescales have been computed using only the dust [*absorption*]{} cross sections, thereby providing an upper limit to the orbital decay timescales. Dust particles smaller than $100\,\mu$m experience decay on time scales no more than $\sim 10^3$ years, and are not sufficient in sustaining the disk seen around [MWC 882]{} today. The vertical line at 10 $\mu$m indicates the dividing line where, for smaller particles, radiation pressure forces will directly unbind them from the system. \[fig:dust\_timescales\]](new_fig_18.pdf "fig:"){width="1\linewidth"}
Summary
=======
[MWC 882]{} is a post-Algol binary with a B7 post-RGB donor star that, until $\sim 0.3$Myr ago, was transferring mass to its A0 companion. The donor star, now pushed outward to an orbital separation of ${\ensuremath{114.0 \pm 3.1}}\,R_\odot$, is occulted by the remnant accretion disk around the accretor once every 72 days. The occultations were observed during Campaign 11 of the [*K2*]{} mission, and subsequently identified in pre-discovery light curves from the ASAS and ASAS-SN surveys. The dynamical masses of the system were measured to be ${\ensuremath{3.24 \pm 0.29}}\,M_\odot$ and ${\ensuremath{0.542 \pm 0.053}}\,M_\odot$. The strengths of temperature sensitive lines yielded a spectral type estimate of A0 for the accretor and B7 for the donor star, with the two stars exhibiting approximately equal luminosities. We estimate, via isochrone fitting and light curve modeling that the radii of the two stars are ${\ensuremath{3.09 \pm 0.59}}\,R_\odot$ and ${\ensuremath{2.01 \pm 0.52}}\,R_\odot$.
We coordinated a campaign of multi-band photometric and spectroscopic observations over the occultation event centered on September 2017. Our eclipse light curves agreed well with those observed by [*K2*]{} and pre-discovery surveys. The eclipse was found to be colorless to within detection limits. We obtained a series of spectroscopic observations over the second half of the eclipse, and detected a series of absorption lines from the disk around the accretor, reminiscent of those seen during the eclipse of $\epsilon$ Aurigae . We used a toy disk model to simultaneously fit for the photometric and spectroscopic eclipses, finding a disk ${\ensuremath{59.9 \pm 6.2}}\,R_\odot$ in radius. The central brightening seen during eclipse is explained by including a flaring geometry to the disk, such that a smaller area of the donor star is covered during mid-eclipse, or by a dusty disk inducing significant forward scattering along the line of sight .
Our interpretation of the system cannot account for the persistence of the accretion disk. We expect the donor star to have begun contraction $\sim0.3$Myr ago, terminating mass transfer, while the disk is nominally expected to survive for only hundreds of years without active feeding, regardless of its dust to gas composition. We suggest future observations could search for signatures of active mass transfer in the system. Full spectroscopic coverage of the orbital phase can reconstruct a Doppler tomographic image of the inner disk [e.g. @1988MNRAS.235..269M], and help map out the ongoing accretion mechanics. Sporadic hot spots in the disk are also signs of mass transfer, and may be identified by frequent monitoring of the system at all phases. In particular, higher signal-to-noise spectra during the eclipse, covering ingress and egress, can help us constrain the optical depths at the edges of the disk, and allow us to resolve degeneracies that plague the light curve modeling. The disk of $\epsilon$ Aurigae is thought to consist of large dust grains [e.g. @2010ApJ...714..549H], which may help extend the disk lifetime. Similar mid- and far-infrared observations of [MWC 882]{} may constrain the disk gas to dust ratio, and help better understand why the disk is still present. Continuous spectroscopic and spectro-polarimetric monitoring of the system over the rotation period of the inner disk and the stars may reveal connections between accretion and the chemical peculiarity of the stars. Ap/Bp stars are known to exhibit strong magnetic fields, spots, and inhomogeneities in the abundances across their surfaces . We can search for links between the inner disk structure and the potential spot distribution in the stars of [MWC 882]{}.
Work by G.Z. is provided by NASA through Hubble Fellowship grant HST-HF2-51402.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. L.N. thanks the Natural Sciences and Engineering Research Council (Canada) for financial support provided through a Discovery grant. C.I.J. gratefully acknowledges support from the Clay Fellowship, administered by the Smithsonian Astrophysical Observatory. M.H.K would like to acknowledge Allan Schmitt for his LcTools software. Work by C.H. is supported by the Juan Carlos Torres Fellowship. Work by A.V is performed in part under contract with the California Institute of Technology/Jet Propulsion Laboratory funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. This work makes use of the Smithsonian Institution High Performance Cluster (SI/HPC) We also thank Calcul Québec, the Canada Foundation for Innovation (CFI), NanoQuébec, RMGA, and the Fonds de recherche du Québec - Nature et technologies (FRQNT) for computational facilites. We would also like to thank J. Cannizzo for insightful discussions. This paper includes data taken at The McDonald Observatory of The University of Texas at Austin.
Facilities:
[^1]: After identifying [MWC 882]{} as a target of interest, we re-reduced the light curve allowing low-frequency variations to be modeled with a faster basis spline, which yielded a light curve with fewer systematics.
[^2]: `http://www1.appstate.edu/dept/physics/spectrum/spectrum.html`
[^3]: The exact extent of the H$\alpha$ wings depends on the spectral normalization and blaze function removal, and is not well constrained.
[^4]: Mass transfer that occurs after a helium ash core has formed inside the donor star is known as Case B evolution
[^5]: However, the internal structure of this star is radically different from SPB stars that are thought to be close to the main sequence.
|
---
abstract: 'This article presents a coupling approach for the approximation of iterated stochastic integrals of length three. The generation of such integrals is the central problem of higher-order pathwise approximations for SDEs, which still lacks a satisfactory answer due to the restriction of dimensionality and computational load. Here we start from the Fourier representation of the triple stochastic integral and investigate the global behaviour of the joint density of the representation. Finally in the main result we give a coupling in the quadratic Vaserstein distance.'
author:
- 'Xīlíng Zhāng [^1]'
bibliography:
- 'xiling.bib'
title: '<span style="font-variant:small-caps;">A Coupling for Triple Stochastic Integrals</span>'
---
Introduction
============
Let[^2] $d,q\in\ZZ^+$ and $(\Omega,\mathscr{F},\prb)$ be a complete probability space equipped with a right-continuous filtration $\mathbb{F}=\{\mathscr{F}_t\}_{t\gq0}$. Consider an $\RR^d$-valued autonomous stochastic differential equation driven by a $q$-dimensional $\mathbb{F}$-Wiener martingale $W$: $$\label{sde}
x_t=x_0+\int_0^tb(x_s)\td s+\int_0^t\sigma(x_s)\td W_s.$$ Assume that the coefficients $b:\RR^d\to\RR^d$ and $\sigma:\RR^d\to\RR^{d\times q}$ are sufficiently smooth. It is well-known that one can derive numerical schemes that converge in the strong $L^p$ sense of order greater than $1/2$ from stochastic Taylor expansions, as is shown in [@kloeden1995NumSolStoDifEqu]. For example, by applying Itô’s formula to $b$ and $\sigma$, one obtains the Itô-Taylor expansion of length $2$: for each component $i=1,\cdots,d$ on the interval $[s,t]$, $$\begin{aligned}
x_t^i=&x_s^i+b_i(x_s)(t-s)+\sum_{j=1}^q\sigma_{ij}(x_s)(W_t^j-W_s^j)\nonumber\\
&+\int_s^t\int_s^r\mathcal{L}b_i(x_u)\td u\td r+\sum_{j=1}^q\int_s^t\int_s^r\sum_{k=1}^d\sigma_{kj}(x_u)\partial_kb_i(x_u)\td W_u^j\td r\nonumber\\
&+\sum_{j=1}^q\int_s^t\int_s^r\mathcal{L}\sigma_{ij}(x_u)\td u\td W_r^j+\sum_{j,k=1}^q\int_s^t\int_s^r\sum_{l=1}^d\sigma_{lk}(x_u)\partial_l\sigma_{ij}(x_u)\td W_u^k\td W_r^j,\label{ito_taylor_2}\end{aligned}$$ where $\partial_k$ is the partial derivative w.r.t. the $k$-th coordinate. The last term in involves an iterated stochastic integral, and it gives rise to Milstein’s method: for each component $i=1,\cdots,d$, $$\label{milstein}
X_{k+1}^i=X_k^i+b_i(X_k)h+\lb\sum_{j=1}^q\sigma_{ij}(X_k)\Delta W_{k+1}^j+\sum_{j,l=1}^q\varsigma_{ijl}(X_k)A_k(j,l)\rb,$$ where $h\in(0,1)$ is the step size, $\Delta W^j_{k+1}=W^j_{t_{k+1}}-W^j_{t_k},~\varsigma_{ijl}(x):=\sum_{m=1}^d\sigma_{mj}(x)\partial_m\sigma_{il}(x)$ and $$A_k(j,l):=\int_{t_k}^{t_{k+1}}(W_t^j-W_{t_k}^j)\td W_t^l.$$ The scheme has strong-$L^2$ convergence rate $O(h)$ according to Kloeden and Platen [@kloeden1995NumSolStoDifEqu] (Section 10.3), but the problem lies in the generation of the double integral $I_{jl}=\int_0^hW_t^j\td W_t^l$, which is non-trivial for $q\gq2$.
As mentioned by Wiktorsson [@wiktorsson2001JoiChaFunSimSimIteItIntMulIndBroMot] and Davie [@davie2014Patappstodifequusicou] (Section 2), if the diffusion matrix satisfies the commutativity condition $\varsigma_{ijl}(x)=\varsigma_{ilj}(x)$ for all $x\in\RR^d$ and all $i=1,\cdots,d,~j,l=1,\cdots,q$, one only needs to generate the Wiener increments $\Delta W_{k+1}$ to achieve the order-$1$ convergence. But this is not always the case: using only the Wiener increments $\Delta W_{k+1}$ to implement a numerical method will, in general, result in a convergence rate no more than $O(h^{1/2})$, according to [@clark1978MaxRatConDisAppStoDifEqu].
One attempt to generate the double integral $I_{jl}$ was made by Lyons and Gaines [@lyons1994RanGenStoAreInt], but their method only works for $q=2$. Recently a strong result for any dimension has been proved by Davie [@davie2014Patappstodifequusicou] (Theorem 4) under the condition that the diffusion matrix $\sigma$ has rank $d$ everywhere, and it provides a way to approximate the SDE up to an arbitrary order. This is a significant improvement concerning higher-order approximations. The idea is that, rather than generating the double integrals at each step $k$, one approximates the quantity inside the big parentheses in as a whole. This is a completely different approach than the usual ones, as Davie’s arguments are based on the coupling method, quantifying the strong-$L^p$ convergence in terms of the Vaserstein[^3] metrics.
#### The coupling method.
For probability measures $\prb,\qrb$ on $\RR^q$ and $p\gq1$, the *Vaserstein $p$-distance* is defined by $$\wass_p(\prb,\qrb):=\inf_{\pi\in\Pi(\prb,\qrb)}\lb\int_{\RR^q\times\RR^q}|x-y|^p\pi(\td x,\td y)\rb^{1/p},$$ where $\Pi(\prb,\qrb)$ is the set of all joint probability measures on $\RR^q\times\RR^q$ with marginal laws $\prb$ and $\qrb$. In general $\prb$ and $\qrb$ need not be defined on the same probability space, but this definition is enough for the purpose of this article. The notation $\wass_p(X,Y)$ will not cause any confusion for random variables $X$ and $Y$ having laws $\prb$ and $\qrb$, respectively. If one can show a bound for the distance between the two laws, we then say there is a *coupling* between $X$ and $Y$ (or $\prb$ and $\qrb$).
The significance of using the Vaserstein distances instead of other ones is that, when generating numerical schemes for an SDE, the convergence in the Vaserstein-type distance $\wass_{p,\infty}$ (replacing $|x-y|^p$ in the definition above by $\max_k|x_k-y_k|^p$) is equivalent to the usual strong $L^p$-convergence, for the purpose of simulation at least. To see this, suppose we have found a coupling between the grid points of the solution $x=\{x_{t_k}\}_k$ and a numerical scheme $X=\{X_k\}_k$ with $\wass_{p,\infty}(x,X)\lq Ch^\gamma$ for some $\gamma>0$. Then by definition, $\forall\eps>0$ there is a random vector $Y^\eps$ on the same probability space as the solution $x$, having the same distribution as $X$, s.t. $(\ex\max_k|x_{t_k}-Y_k|^p)^{1/p}\lq\wass_{p,\infty}(x,X)+\eps$. Choose $\eps=h^\gamma$ and in practice one generates $Y$ instead of $X$ to approximate $x$. The reader is referred to Section 12 in [@davie2014Patappstodifequusicou] for a detailed discussion on the contexts where such a substitution holds or fails.
Although there is no general formulas for the quantity $\wass_p(\prb,\qrb)$, if $\prb$ and $\qrb$ have densities $f$ and $g$, respectively, then there is the elementary and yet important inequality $$\label{crude_bound_1}
\wass_p(\prb,\mathds{Q})\lq C_p\lb\int_{\RR^q}|x|^p|f(x)-g(x)|\td x\rb^{1/p},$$ for all $p\gq1$, as a variant of Proposition 7.10 in [@villani2003TopOptTra]. This inequality serves as a main tool to give an $\wass_2$-estimate in [@davie2014Patappstodifequusicou] and [@davie2015Polpernordis], and will be used for all the main result in this article.
The more difficult situation is that $\sigma$ has rank less than $d$, which could well happen. In Section 9 in [@davie2014Patappstodifequusicou] a different approach based on the Fourier expansion introduced in Section 5.8 in [@kloeden1995NumSolStoDifEqu] is proposed, giving a coupling for the double integral $I_{jl}$. The motivation of this article is to provide a feasible approximation for SDEs of a higher order. For the equation on the interval $[0,T]$, by applying Itô’s formula again to the term $\sigma_{kl}(X_u)\partial_k\sigma_{ij}(X_u)$ in , one obtains, for each component $i=1,\cdots,d$ on the interval $[s,t]$, $$\begin{aligned}
X_t^i=&X_s^i+b_i(X_s)(t-s)+\sigma_{ij}(X_s)(W_t^j-W_s^j)+\sigma_{kl}(X_s)\partial_k\sigma_{ij}(X_s)\int_s^t\int_s^r\td W_u^l\td W_r^j\\
&+\int_s^t\int_s^r\mathcal{L}b_i(X_u)\td u\td r+\int_s^t\int_s^r\sigma_{kl}(X_u)\partial_kb_i(X_u)\td W_u^l\td r\\
&+\int_s^t\int_s^r\mathcal{L}\sigma_{ij}(X_u)\td u\td W_r^j+\int_s^t\int_s^r\int_s^u\mathcal{L}\lb\sigma_{kl}(X_v)\partial_k\sigma_{ij}(X_v)\rb\td v\td W_u^l\td W_r^j\\
&+\int_s^t\int_s^r\int_s^u\partial_m\lb\sigma_{kl}(X_v)\partial_k\sigma_{ij}(X_v)\rb\sigma_{mn}(X_v)\td W_v^n\td W_u^l\td W_r^j,\end{aligned}$$ where the summation signs over repeated indices are omitted. From this expression one can obtain a suitable numerical scheme (formula (10.4.6) in [@kloeden1995NumSolStoDifEqu]) with strong convergence order $O(h^{3/2})$. Just as the Milstein scheme, the crucial ingredient to achieve such a higher-order convergence is the generation of the triple integrals $$I_{jkl}(s,t):=\int_s^t\int_s^r\int_s^u\td W_v^j\td W_u^k\td W_r^l,$$ for indices $(j,k,l)\in\{1,\cdots,q\}^3$.
Similar to the way the double stochastic integral is treated in [@davie2014Patappstodifequusicou], one would expect the same method to be extended to treat triple integrals. For the simplicity of formulation, the Stratonovich triple integral $I^\circ_{jkl}(s,t):=\int_s^t\int_s^r\int_s^u\td W_v^j\circ\td W_u^k\circ\td W_r^l$ will be considered instead of the Itô version, since the Fourier representation of the former has a relatively simpler form. This is due to the fact that the product of two Stratonovich integrals is a shuffle product - see Proposition 2.2 in [@gaines1994AlgIteStoInt]. In other words, an iterated Stratonovich integral of longer length can be represented by shorter ones in a much simpler way compared its Itô counterpart.
#### The double integral case.
The goal of this paper is to find a random variable $\bar{I}_{jkl}$ whose law is close to that of $I^\circ_{jkl}$ in the Vaserstein distance, which in turn gives a feasible $O(h^{3/2})$-approximation for the SDE . In order to have a better understanding of the method let us briefly review Davie’s Fourier method (Section 9 in [@davie2014Patappstodifequusicou]). Consider the interval $[0,1]$ for simplicity. According to [@kloeden1995NumSolStoDifEqu] (Section 5.8), the Brownian bridge process $W_t-tW_1$ has Fourier expansion $$\label{bridge}
W_t^j-tW_1^j=\frac{1}{2\sqrt{2}\pi}x_{j0}+\frac{1}{\sqrt{2}\pi}\sum_{r=1}^\infty x_{jr}\cos(2\pi rt)+\frac{1}{\sqrt{2}\pi}\sum_{r=1}^\infty y_{jr}\sin(2\pi rt),$$ where $x_{jr},~y_{jr}$ are $\N(0,1)$-random variables mutually independent for different values of $j=1,\cdots,q$ or $r\in\NN$, all independent of $W_1$. Then the double integral $I^\circ_{jk}=\int_0^1W_s^j\circ\td W_s^k$ has Fourier representation $$\label{double_int}
I^\circ_{jk}=\frac{1}{2}W_1^jW_1^k+\frac{1}{\sqrt{2}\pi}\lb W_1^jz_k-W_1^kz_k\rb+\frac{1}{2\pi}\lambda_{jk},$$ where $\lambda_{jk}=\sum_{r\gq1}r^{-1}(x_{jr}y_{kr}-y_{jr}x_{kr})$ and $z_j=\sum_{r\gq1}r^{-1}x_{jr}$. One then needs to approximate each $\lambda_{jk}$ and $z_j$ by their partial sums $\lambda_{jk}=\sum_{r=1}^pr^{-1}(x_{jr}y_{kr}-y_{jr}k_{jr})$ and $z_j=\sum_{r=1}^pr^{-1}x_{jr}$. Denote $\wt\lambda_{jk}^{(p)}=\lambda_{jk}-\lambda_{jk}^{(p)},~\wt z_j^{(p)}=z_j-z_j^{(p)}$ and $U:=(\lambda,z),~U_p:=(\lambda^{(p)},z^{(p)}),~\wt U_p:=(\wt\lambda^{(p)},\wt z^{(p)})$.
Davie’s result states that if there is a random variable $\bar{U}_p$, independent of $U_p$, having the same moments as $\wt U_p$ up to order $m-1$ and satisfying $\ex\exp(a\sqrt{p}|\bar{U}_p|)\lq b$ for some positive constants $a,b$ for all $p$, then $\wass_2(U,~U_p+\bar{U}_p)=O(p^{-m/2})$ for $p$ sufficiently large. The idea is to estimate the densities $g(\zeta)$ of $U$ and $h(\zeta)$ of $U_p+\bar{U}_p$. If $f_p$ is the density of $U_p$, then $g(\zeta)=\ex f_p(\zeta-\wt U_p)$ and $h(\zeta)=\ex f_p(\zeta-\bar{U}_p)$. By Taylor’s theorem, for all $\zeta,w\in\RR^d$, $$\begin{aligned}
f_p(\zeta-w)=&\sum_{|\beta|=0}^{m-1}\frac{(-1)^{|\beta|}}{\beta!}w^\beta\partial^\beta f_p(\zeta)\nonumber\\
&+\sum_{|\beta|=m}\frac{|\beta|(-1)^{|\beta|}}{\beta!}\int_0^1(1-\theta)^{|\beta|-1}w^\beta\partial^\beta f_p(\zeta-\theta w)\td\theta.\label{taylor_density}\end{aligned}$$ Since up to the $(m-1)$-th moments of $\wt U_p$ and $\bar{U}$ match, when taking the difference $g(\zeta)-h(\zeta)$ the first summation vanishes, and hence $\forall\zeta\in\RR^d$, $$\label{different_g_h}
g(\zeta)-h(\zeta)=\sum_{|\beta|=m}\int_0^1C_{\beta,\theta}\ex\lb\wt U_p^\beta\partial^\beta f_p(\zeta-\theta\wt U_p)-\bar{U}^\beta\partial^\beta f_p(\zeta-\theta\bar{U}_p)\rb\td\theta,$$ where $C_{\beta,\theta}=|\beta|(-1)^{|\beta|}(1-\theta)^{|\beta|-1}/\beta!$. If one can give a uniform bound for some higher derivatives of $f_p$ in terms of $p$, then using an interpolation argument one can show a reasonable decay for the $m$-th derivative of $f_p$, and finally one finds a coupling between $U$ and $U_p+\bar U_p$ by the inequality .
The main advantage of the double integral $I^\circ_{jk}$ compared to the triple one is the fact that its Fourier representation only involves $\lambda$ and $z$, whose summands are independent. This ensures that $U$ has a smooth density (as the convolution of the density $f_p$ of $U_p$ and the law of $\wt U_p$), which significantly simplifies the analysis. More importantly, the characteristic function of $U_p$ can be explicitly calculated - see formula (32) in the proof of Lemma 11 in [@davie2014Patappstodifequusicou]. This provides some convenience for investigating the global and local behaviour of the density $f_p$ (Lemma 12, 13 and 14). In particular, Lemma 14 therein gives a lower bound for $f_p$, which is essential for achieving a coupling for $U$ of the optimal order $O(p^{-m/2})$ in the $\wass_2$ distance.
Without Lemma 14, one can still achieve a suboptimal $\wass_2$-rate $O(p^{-m/4})$ by directly showing a decay of the difference $|g(\zeta)-h(\zeta)|$ - this is the goal of the present paper, but the treatment of the densities is quite different from the double integral case.
The latter is much more straighforward to see. For $p$ sufficiently large, the vector $\tD^{2m}f_p$ of partial derivatives of order $2m$ is uniformly bounded everywhere due to part (1) of Lemma 11 in [@davie2014Patappstodifequusicou]. Also by Lemma 12 therein, one has $f_p(\zeta)\lq e^{-c_q|\zeta|}$ for $|\zeta|$ sufficiently large. Then one can apply Lemma 9 therein to get a rapid decay for $\tD^mf_p(\zeta)$. To see this, consider $|\zeta|>p$ sufficiently large and the ball $B(\zeta,1)$ that is disjoint with $B(0,p)$. Then $\sup_{y\in B(\zeta,1)}f_p(y)\lq e^{-c_q(|\zeta|-1)}$, and by applying Lemma 9 to the ball $B(\zeta,1)$ one sees the following bound for (the Euclidean norm of) the $m$-th derivatives: $$\label{interpolation}
|\tD^mf_p(\zeta)|\lq C_{q,m}\max\left\{\sup_{y\in B(\zeta,1)}\sqrt{f_p(y)}\sup_{y\in B(\zeta,1)}\sqrt{|\tD^{2m}f_p(y)|},~\sup_{y\in B(\zeta,1)}f_p(y)\right\}.$$ This yields $|\tD^mf_p(\zeta)|\lq C_{q,m}e^{-c_q|\zeta|}$. Therefore from and part (2) of Lemma 11 in [@davie2014Patappstodifequusicou] one has, by the Cauchy-Schwartz inequality, that for all $\zeta\in\RR^{q(q+1)/2}$, $$\begin{aligned}
|g(\zeta)-h(\zeta)|\lq&C_{d,m}\sum_{|\beta|=m}\lb\ex|\wt U_p^\beta\partial^\beta f_p(\zeta-\wt U_p)|+\ex|\bar{U}^\beta\partial^\beta f_p(\zeta-\bar{U}_p)|\rb\\
\lq&C_{d,m}p^{-m/2}\lb\sqrt{\ex|\tD^mf_p(\zeta-\wt U_p)|^2}+\sqrt{\ex|\tD^mf_p(\zeta-\bar{U}_p)|^2}\rb.\end{aligned}$$ Notice that, on the set $\{\omega:~|\wt U_p|\lq1\}$ one has $\|\tD^mf_p(\zeta-\wt U_p)\|^2\lq C_qe^{-c_q|\zeta|}$ by the rapid decay of $\tD^mf_p$; on the complement $\{\omega:~|\wt U_p|>1\}$, part (2) of Lemma 11 and Chebyshev’s inequality imply that $\prb(|\wt U_p|>1)\lq C_Mp^{-M}$ for any $M>0$. The same argument works for the second term above involving $\bar{U}$, and so by the inequality for the quadratic distance, $$\wass_2(U,~\wt U_p+\bar{U}_p)\lq C\lb\int_{\RR^{q(q+1)/2}}|\zeta|^2|g(\zeta)-h(\zeta)|\td\zeta\rb^{1/2}\lq C_{q,m}p^{-m/4}.$$
From this calculation one sees that the key step towards a good coupling result depends on how well the behaviour of $f_p$ is understood. Davie’s result is a significant improvement to the existing rate of approximation - see the discussion following the proof of Theorem 15 therein. This is due to some careful estimates (Lemma 12, 13 and 14 in [@davie2014Patappstodifequusicou]) for the density $f_p$. For the triple integral $I^\circ_{jkl}$, however, showing similar estimates becomes much more complicated as the Fourier coefficients for $I^\circ_{jkl}$ have summands that are not independent of each other - see the definition of the random variable $\Delta_{jkl}$ below.
#### Notation.
Throughout this paper we will denote by $\phi$ the standard normal density of dimension $1$, by $B(x,r)$ the open ball of radius $r$ centred at $x$, and by $\Lambda^d$ the Lebesgue measure on $\RR^d$. The notation $C_0^\infty$ stands for the set of $C^\infty$-functions with compact support. Unless specified otherwise, the single bars $|\cdot|$ stand for the Euclidean norm, modulus of a complex number, or the cardinality of a set, and the double bars $\|\cdot\|$ stand for the operator norm, which in the context of matrices is equivalent to any other matrix norm. The letter $C$ will be used for a generic constant that may change value from line to line, with subscripts specifying its dependence on the parameters. The symbol $\lesssim_\alpha$ ($\gtrsim_\alpha$) means that the inequality $\lq$ ($\gq$) holds up to a multiplicative constant $C_\alpha$, and $\simeq_\alpha$ is used when both inequalities hold. For a function $f(x,y)$ of two variables, we also write $f(x;y)$ when, especially differentiating, $y$ is treated as a fixed parameter. For example, $\tD f(x;y)=\partial_xf(x,y)$.
#### Acknowledgement.
This work was completed under the patient guidance of my Ph.D. advisor, Prof. Alexander M. Davie, who suggested the problem and gave many crucial advices on the main steps of the arguments as well as technical details.
The Fourier Representation
==========================
For the simplicity of presentation let us consider the triple integral on the unit interval $[0,1]$. Following Section 5.8 in [@kloeden1995NumSolStoDifEqu], from the Fourier expansion the triple Stratonovich integral $$I^\circ_{jkl}=\int_0^1\int_0^tW_s^j\circ\td W^k_s\circ\td W^l_t,$$ for each $(j,k,l)\in\{1,\cdots,q\}^3$ has the following representation: $$\begin{aligned}
I^\circ_{jkl}=&\frac{1}{6}W_1^jW_1^kW_1^l-\frac{1}{2\sqrt{2}\pi}W_1^jW_1^k\lb z_l-\frac{1}{\pi}u_l\rb-\frac{1}{2\sqrt{2}\pi}W_1^kW_1^l\lb z_j-\frac{1}{\pi}u_j\rb\\
&-\frac{1}{\sqrt{2}\pi^2}W_1^jW_1^lu_k-\frac{1}{2\pi^2}z_j\lb W_1^kz_l-W_1^lz_k\rb+\frac{1}{2\pi}W_1^l\lb\frac{1}{2}\lambda_{jk}+\frac{1}{\pi}\nu_{kj}\rb\\
&+\frac{1}{2\pi}W_1^j\lb\frac{1}{2}\lambda_{kl}-\frac{1}{\pi}\nu_{kl}\rb+\frac{1}{4\pi^2}\lb W_1^j\mu_{kl}-W_1^k\mu_{jl}\rb-\frac{1}{2\sqrt{2}\pi^2}z_j\lambda_{kl}\\
&+\frac{1}{4\sqrt{2}\pi}\Delta_{jkl},\end{aligned}$$ where the coefficients $z,u,\lambda,\mu,\nu$ are defined as $$\begin{aligned}
z_j=&\sum_{r=1}^\infty\frac{1}{r}x_{jr},~u_j=\sum_{r=1}^\infty\frac{1}{r^2}y_{jr},\\
\lambda_{jk}=&\sum_{r=1}^\infty\frac{1}{r}\lb x_{jr}y_{kr}-y_{jr}x_{kr}\rb,~\mu_{jk}=\sum_{r=1}^\infty\frac{1}{r^2}\lb x_{jr}x_{kr}+y_{jr}y_{kr}\rb,\\
\nu_{jk}=&\sum_{\substack{r,s=1\\r\neq s}}^\infty\frac{1}{r^2-s^2}\lb\frac{r}{s}x_{jr}x_{ks}-y_{jr}y_{ks}\rb,\end{aligned}$$ with $x_{jr},y_{jr}$, again, being $\N(0,1)$-random variables independent for different indices $j=1,\cdots,q,~r\in\ZZ^+$ and all independent of $W_1^j$, and the last coefficient $\Delta$ is given by $$\begin{aligned}
\Delta_{jkl}=\sum_{r,s=1}^\infty\left\{-\frac{1}{r(r+s)}\left[(x_{jr}y_{ks}+y_{jr}x_{ks})x_{l,r+s}+(-x_{jr}x_{ks}+y_{jr}y_{ks})y_{l,r+s}\right]\right.&\\
+\frac{1}{rs}\left[(x_{jr}y_{ls}+y_{jr}x_{ls})x_{k,r+s}+(-x_{jr}x_{ls}+y_{jr}y_{ls})y_{k,r+s}\right]&\\
\left.+\frac{1}{s(r+s)}\left[(-x_{kr}y_{ls}+y_{kr}x_{ls})x_{j,r+s}+(x_{kr}x_{ls}+y_{kr}y_{ls})y_{j,r+s}\right]\right\}&\end{aligned}$$ For a positive integer $p$, write $z^{(p)}$ as the $p$-th partial sum of $z$ and $\wt z^{(p)}=z-z^{(p)}$. Similar notations are applied to $u,\lambda$ and $\mu$. Let $\nu^{(p)}$ be the partial sum of $\nu$ over $r,s\lq p,~r\neq s$ and $\wt\nu^{(p)}=\nu-\nu^{(p)}$, whilst $\Delta^{(p)}$ denotes the partial sum of $\Delta$ up to $r+s\lq p$ and $\wt\Delta^{(p)}=\Delta-\Delta^{(p)}$.
From the definition of $\nu_{jk}^{(p)}$ one observes that, by splitting each variable $\mu_{jk}^{(p)}$ into two parts: $$\mu_{jk}^{(1,p)}:=\sum_{r=1}^p\frac{1}{r^2}x_{jr}x_{kr},~\mu_{jk}^{(2,p)}:=\sum_{r=1}^p\frac{1}{r^2}y_{jr}y_{kr},$$ one need only generate $\nu_{jk}^{(p)}$ for $j<k$, since $$\nu_{jk}^{(p)}+\nu_{kj}^{(p)}=z_j^{(p)}z_k^{(p)}-\mu_{jk}^{(1,p)}.$$ Equivalent notations for the infinite sums are used by omitting the superscript $(p)$ and the identity still holds. Therefore one need only consider $\nu_{jk}$ for $j<k$.
Another observation is that one need not consider all possible choices of the $3$-tuple $(j,k,l)\in\{1,\cdots,q\}^3$ for $\Delta$; it suffices to focus on those terms with $(j,k,l)$ being a **Lyndon word** - a word that is strictly less than all of its proper right factors in the lexicographic order. This is due to the fact that all triple Stratonovich integrals $I^\circ_{jkl}$ can be expressed by the Lyndon words of length at most $3$ - see Corollary 3.3 in [@gaines1994AlgIteStoInt].
For a word $w$ in a totally ordered set $A$, if it is the concatenation of two non-empty words $u,v\in A$, i.e. $w=uv$, then $v$ is called a proper right factor of $w$. For example, $(1,1,2)$ and $(1,3,2)$ are both Lyndon words but $(1,2,1)$ is not. By definition, a triple $(j,k,l)$ is a Lyndon word if and only if $j<k\wedge l$ or $j=k<l$. Denote by $\mathfrak{L}_{3,q}\subset\{1,\cdots,q\}^3$ the set of Lyndon words of length $3$, then according to [@gaines1994AlgIteStoInt] $|\mathfrak{L}_{3,q}|=(q^3-q)/3$.
As an analogue of the work by Davie [@davie2014Patappstodifequusicou] (Section 9), one seeks to approximate the variable $V=(z,u,\lambda,\mu,\nu,\Delta)$ by studying the distribution of the partial sums $$V_p=(z^{(p)},u^{(p)},\lambda^{(p)},\mu^{(p)},\nu^{(p)},\Delta^{(p)}),$$ and that of the remainder $\wt{V}_p:=(\wt z^{(p)},\wt u^{(p)},\wt\lambda^{(p)},\wt\mu^{(p)},\wt\nu^{(p)},\wt\Delta^{(p)})$. Note that for an $O(h^{3/2})$-approximation of the SDE , one also needs to simulate the double integrals along with the triple ones. But they are determined by the variables $(z,\lambda)$, which are already included in $V$.
To develop an analogue of Davie’s results in [@davie2014Patappstodifequusicou], it is necessary to give some suitable moment estimates for the remainder $\wt V_p$. For simplicity denote the dimension of $V$ by $$d=2q^2+2q+(q^3-q)/3,$$ and denote by $v_p$ the $\RR^{2qp}$-vector consisting of $x_{jr},y_{ks}$ for $j,k=1,\cdots,q$ and $r,s=1,\cdots,p$.
For vectors $\omega:=(\alpha,\beta^{(1)},\beta^{(2)},\gamma,a,b,\rho)\in\RR^d$ and $v=(x_{jr},y_{jr})_{j,r}\in\RR^{2qp}$, define the cubic **phase function** $\Phi_p:\RR^{2qp}\times\RR^d\to\RR$ by $$\begin{aligned}
\Phi_p(v;\omega)=&\sum_{j<k}\lb\alpha_{jk}\lambda_{jk}^{(p)}+\gamma_{jk}\nu_{jk}^{(p)}\rb+\sum_{j\lq k}\lb\beta_{jk}^{(1)}\mu_{jk}^{(1,p)}+\beta_{jk}^{(2)}\mu_{jk}^{(2,p)}\rb\nonumber\\
&+\sum_{j=1}^q\lb a_jz_j^{(p)}+b_ju_j^{(p)}\rb+ \sum_{(j,k,l)\in\mathfrak{L}_{3,q}}^q\rho_{jkl}\Delta_{jkl}^{(p)}.\label{phase}\end{aligned}$$ Then by definition the characteristic function $\psi_p(\xi)$ of $V_p$ is given by $$\begin{aligned}
\psi_p(\xi)=&\int_{\RR^{2qp}}\exp\{i|\xi|\Phi_p(x,y;\omega_0)\}\prod_{j=1}^q\prod_{r=1}^p\phi(x_{jr})\phi(y_{jr})\td x \td y\\
=:&\int_{\RR^{2qp}}\exp\{i|\xi|\Phi_p(v;\omega_0)\}\phi_p(v)\td v,\end{aligned}$$ where $\phi$ is the density function of $\N(0,1)$ and $\omega_0=\xi/|\xi|\in\mathbb{S}^{d-1}$. Observe that the matrices $\lambda$ and $\mu$ are skew-symmetric and symmetric, respectively, so it would be convenient to extend the values of the coefficients $\alpha,~\beta:=(\beta^{(1)},~\beta^{(2)})$ to their lower-triangles by setting $\alpha_{kj}=-\alpha_{jk},~\beta_{kj}^{(i)}=\beta_{jk}^{(i)}$ for all $i=1,2,~j,k=1,\cdots,q$. Set $\gamma_{jk}=0$ for all $j\gq k$ and $\rho_{jkl}=0$ if $(j,k,l)$ is not a Lyndon word.
Throughout this article we will be frequently dealing with oscillatory integrals of the form $\psi_p(\xi)$, and we will conveniently call the function $\phi_p$ the **amplitude**. In order to give a good estimate for magnitude of $\psi_p(\xi)$ one resorts to the method of stationary phase, and for that one needs to study the derivatives of the phase function $\Phi_p$.
To find the gradient $\nabla\Phi_p(v;\omega)$, one can make use the extended definitions of $\alpha,\beta,\gamma$ and write down the partial derivatives. For each $j=1,\cdots,q$ and $r=1,\cdots,p$, differentiating w.r.t. $x_{jr}$ and $y_{jr}$ gives $$\begin{aligned}
\partial_{x_{jr}}&\Phi_p(v;\omega)=\frac{1}{r}\alpha_{jk}y_{kr}+\frac{1}{r^2}(1+\delta_{kj})\beta_{jk}^{(1)}x_{kr}+\sum_{\substack{s=1\\s\neq r}}^p\frac{1}{r^2-s^2}\lb\frac{r}{s}\gamma_{jk}-\frac{s}{r}\gamma_{kj}\rb x_{ks}+\frac{1}{r}a_j\\
&+\sum_{s=1}^{p-r}\left[\lb\frac{-\rho_{jkl}+\rho_{lkj}}{r(r+s)}-\frac{\rho_{kjl}}{s(r+s)}\rb y_{ks}x_{l,r+s}+\lb\frac{\rho_{jkl}+\rho_{lkj}}{r(r+s)}+\frac{\rho_{kjl}}{s(r+s)}\rb x_{ks}y_{l,r+s}\right.\\
&\qquad\quad\left.+\lb\frac{\rho_{jkl}+\rho_{lkj}}{rs}-\frac{\rho_{kjl}}{s(r+s)}\rb (y_{ls}x_{k,r+s}-x_{ls}y_{k,r+s})\right]\\
&+\sum_{s=1}^{r-1}\left[\lb-\frac{\rho_{jkl}}{rs}+\frac{\rho_{kjl}}{(r-s)s}\rb x_{k,r-s}y_{ls}+\lb\frac{\rho_{jkl}}{rs}+\frac{\rho_{kjl}}{(r-s)s}\rb y_{k,r-s}x_{ls}\right.\\
&\qquad\quad\left.-\frac{\rho_{lkj}}{(r-s)r}\lb x_{l,r-s}y_{ks}+y_{l,r-s}x_{ks}\rb\right],{\stepcounter{equation}\tag{{{\arabic{section}}.\arabic{equation}}}}\label{1st_der_x}\\
\partial_{y_{jr}}&\Phi_p(v;\omega)=-\frac{1}{r}\alpha_{jk}x_{kr}+\frac{1}{r^2}(1+\delta_{kj})\beta_{jk}^{(2)}y_{kr}-\sum_{\substack{s=1\\s\neq r}}^p\frac{1}{r^2-s^2}(\gamma_{jk}-\gamma_{kj})y_{ks}+\frac{1}{r^2}b_j\\
&+\sum_{s=1}^{p-r}\left[\lb-\frac{\rho_{jkl}+\rho_{lkj}}{r(r+s)}-\frac{\rho_{kjl}}{s(r+s)}\rb x_{ks}x_{l,r+s}+\lb\frac{-\rho_{jkl}+\rho_{lkj}}{r(r+s)}-\frac{\rho_{kjl}}{s(r+s)}\rb y_{ks}y_{l,r+s}\right.\\
&\qquad\quad\left.+\lb\frac{\rho_{jkl}+\rho_{lkj}}{rs}+\frac{\rho_{kjl}}{s(r+s)}\rb \lb x_{ls}x_{k,r+s}+y_{ls}y_{k,r+s}\rb\right]\\
&+\sum_{s=1}^{r-1}\left[\lb\frac{\rho_{jkl}}{(r-s)r}-\frac{\rho_{kjl}}{(r-s)s}\rb x_{k,r-s}x_{ls}+\lb\frac{\rho_{jkl}}{rs}+\frac{\rho_{kjl}}{(r-s)s}\rb y_{k,r-s}y_{ls}\right.\\
&\qquad\quad\left.+\frac{\rho_{lkj}}{(r-s)r}\lb x_{l,r-s}x_{ks}-y_{l,r-s}y_{ks}\rb\right],{\stepcounter{equation}\tag{{{\arabic{section}}.\arabic{equation}}}}\label{1st_der_y}\end{aligned}$$ where $\delta_{jk}$ is the Krönecker delta, the summation signs over the repeated indices $k,l=1,\cdots,q$ are omitted, and all $x$ and $y$-terms with second subscripts outwith the interval $[1,p]$ are assumed to vanish. The Hessian matrix of $\Phi_p$ takes the form $$\label{hessian}
\tD^2\Phi_p(v;\omega)=\begin{pmatrix}
H_{xx}(1,1) & \cdots & H_{xx}(1,q) & H_{xy}(1,1) & \cdots & H_{xy}(1,q)\\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\
H_{xx}(q,1) & \cdots & H_{xx}(q,q) & H_{xy}(q,1) & \cdots & H_{xy}(q,q)\\
H_{yx}(1,1) & \cdots & H_{yx}(1,q) & H_{yy}(1,1) & \cdots & H_{yy}(1,q)\\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\
H_{yx}(q,1) & \cdots & H_{yx}(q,q) & H_{yy}(q,1) & \cdots & H_{yy}(q,q)
\end{pmatrix},$$ where for each pair $(j,k)\in\{1,\cdots,q\}^2$ the blocks $H_{xx}(j,k),~H_{xy}(j,k),~H_{yy}(j,k)$ are $p\times p$ matrices, e.g., $$\label{block}
H_{xx}(j,k)=\begin{pmatrix}
\partial^2_{x_{j1}x_{k1}} & \partial^2_{x_{j1}x_{k2}} & \cdots & \partial^2_{x_{j1}x_{kp}}\\
\partial^2_{x_{j2}x_{k1}} & \partial^2_{x_{j2}x_{k2}} & \cdots & \partial^2_{x_{j2}x_{kp}}\\
\vdots & \vdots & \ddots & \vdots\\
\partial^2_{x_{jp}x_{k1}} & \partial^2_{x_{jp}x_{k2}} & \cdots & \partial^2_{x_{jp}x_{kp}}
\end{pmatrix}\Phi_p(v,\omega),$$ and the rest are similarly defined. From the gradient of $\Phi_p$ in $v$ one can compute the second derivative $\tD^2\Phi_p$ by finding the mixed derivatives for each pair $(j,k)$ and $(r,s)$. The $(r,s)$-th entries of the blocks $H_{xx}(j,k),~H_{yy}(j,k)$ and $H_{xy}(j,k)$ are given by $$\begin{aligned}
\partial^2_{x_{jr}x_{ks}}\Phi_p(v;\omega)=&\frac{1}{r^2}(1+\delta_{jk})\beta_{jk}^{(1)}\delta_{rs}+\frac{1}{r^2-s^2}\lb\frac{r}{s}\gamma_{jk}-\frac{s}{r}\gamma_{kj}\rb(1-\delta_{rs})\\
&+\lb\frac{\rho_{jkl}+\rho_{lkj}}{r(r+s)}+\frac{\rho_{kjl}+\rho_{ljk}}{s(r+s)}-\frac{\rho_{jlk}+\rho_{klj}}{rs}\rb y_{l,r+s}\\
&+\lb\frac{-\rho_{jlk}+\rho_{klj}}{rs}-\frac{\rho_{ljk}+\rho_{kjl}}{(s-r)s}+\frac{\rho_{jkl}+\rho_{lkj}}{r(s-r)}\rb y_{l,s-r}\\
&+\lb\frac{-\rho_{klj}+\rho_{jlk}}{rs}+\frac{\rho_{kjl}+\rho_{ljk}}{(r-s)s}-\frac{\rho_{jkl}+\rho_{lkj}}{(r-s)r}\rb y_{l,r-s},{\stepcounter{equation}\tag{{{\arabic{section}}.\arabic{equation}}}}\label{2nd_der_xx}\\
\partial^2_{y_{jr}y_{ks}}\Phi_p(v;\omega)=&\frac{1}{r^2}(1+\delta_{jk})\beta_{jk}^{(2)}\delta_{rs}-\frac{1}{r^2-s^2}(\gamma_{jk}-\gamma_{kj})(1-\delta_{rs})\\
&+\lb\frac{-\rho_{jkl}+\rho_{lkj}}{r(r+s)}+\frac{-\rho_{kjl}+\rho_{ljk}}{s(r+s)}+\frac{\rho_{jlk}+\rho_{klj}}{rs}\rb y_{l,r+s}\\
&+\lb\frac{-\rho_{jlk}+\rho_{klj}}{rs}+\frac{-\rho_{ljk}+\rho_{kjl}}{(s-r)s}+\frac{\rho_{jkl}+\rho_{lkj}}{r(s-r)}\rb y_{l,s-r}\\
&+\lb\frac{\rho_{jlk}-\rho_{klj}}{rs}+\frac{\rho_{ljk}+\rho_{kjl}}{(r-s)s}+\frac{\rho_{jkl}-\rho_{lkj}}{(r-s)r}\rb y_{l,r-s},{\stepcounter{equation}\tag{{{\arabic{section}}.\arabic{equation}}}}\label{2nd_der_yy}\\
\partial^2_{x_{jr}y_{ks}}\Phi_p(v;\omega)=&\frac{1}{r}\alpha_{jk}\delta_{rs}+\lb\frac{-\rho_{jkl}+\rho_{lkj}}{r(r+s)}-\frac{\rho_{kjl}+\rho_{ljk}}{s(r+s)}+\frac{\rho_{jlk}+\rho_{klj}}{rs}\rb x_{l,r+s}\\
&+\lb\frac{\rho_{jlk}+\rho_{klj}}{rs}+\frac{\rho_{ljk}+\rho_{kjl}}{(s-r)s}-\frac{\rho_{jkl}+\rho_{lkj}}{r(s-r)}\rb x_{l,s-r}\\
&+\lb-\frac{\rho_{jlk}+\rho_{klj}}{rs}+\frac{\rho_{ljk}+\rho_{kjl}}{(r-s)s}+\frac{-\rho_{lkj}+\rho_{jkl}}{(r-s)r}\rb x_{l,r-s},{\stepcounter{equation}\tag{{{\arabic{section}}.\arabic{equation}}}}\label{2nd_der_xy}\end{aligned}$$ where, again, the summation sign over the repeated index $l=1,\cdots,q$ is omitted, and all $x$ and $y$-terms with second subscripts outwith the interval $[1,p]$ are assumed to vanish.
The Joint Characteristic Function of the Partial Sums
=====================================================
With the gradient and the Hessian matrix of the phase function $\Phi_p(v;\omega)$ in $v$ given above, one can apply the method of stationary phase to study the asymptotic behaviour of the oscillatory integral $\psi_p(\xi)$. A useful tool for this is provided in [@sogge1993FouIntClaAna] (Lemma 0.4.7), and the first estimate given in the following lemma is a more quantitative version of it.
Before stating the lemma let us introduce the norm $$|\varphi|_{K,\Omega}:=\max_{0\lq n\lq K}\sup_{x\in\Omega}|\tD^n\varphi(x)|$$ for any smooth function $\varphi$ on a bounded domain $\Omega\subset\RR^d$ and any natural number $K$.
\[stationary\_phase\] Let $\Psi,\varphi\in C^\infty(\RR^k)$ with $\supp\varphi=\Omega$ bounded. Then for all $\delta,R>0$ and $K\in\NN$, $$\lv\int_\Omega e^{iR\Psi(x)}\varphi(x)\td x\rv\lq C|\varphi|_{K,\Omega}\lb|\Psi|_{K,\Omega}^K\vee1\rb\delta^{-2K}R^{-K}+2|\varphi|_{0,\Omega}\Lambda^k(\Omega\setminus\Omega_\delta),$$ where $\Omega_\delta:=\{x\in\Omega:|\nabla\Psi(x)|>\delta\}$ and the constant $C$ depends on $k,K$ and $\Lambda^k(\Omega)$.
It suffices to bound the integral on $\Omega_\delta$. For any fixed $K>0$ write $M=|\Psi|_{K,\Omega}\vee1$ and split the set $\Omega_\delta$ into the level sets of the gradient of the phase function: $$\Omega_r:=\{x\in\Omega_\delta:~2^{-r}M<|\nabla\Psi(x)|\lq2^{-r+1}M\},$$ for $r=1,\cdots,r_0:=[\log_2(M/\delta)]$; there are at most $[\log_2(M/\delta)]+1$ non-empty $\Omega_r$’s. On each level set $\Omega_r$, choose $\eps_r=2^{-r}M/(M+1)$ and let $N_r=N_r(d,\eps_r)$ be the maximum number s.t. there are $x_1,\cdots,x_{N_r}\in\Omega_r$ so that the balls $B(x_j,\eps_r/2)$ are all disjoint. Then the balls $\{B(x_j,\eps_r)\}_j$ must cover $\Omega_r$: if there is $x_\ast\in\Omega_r$ s.t. $|x_\ast-x_j|>\eps_r$ for all $j$, then $B(x_\ast,\eps_r/2)$ is disjoint from all other balls $B(x_j,\eps_r)$ or those with half of the radius, which contradicts the maximality of $N_r$. Note that $\bigcup_{j=1}^{N_r}B(x_j,\frac{\eps_r}{2})\subset\Omega_r^\frac{\eps_r}{2}$, the $\frac{\eps_r}{2}$-neighbourhood of $\Omega_r$, therefore $$N_r\lq\frac{\Lambda^k\lb\Omega_r^\frac{\eps_r}{2}\rb}{\Lambda^k\lb B(x_j,\frac{\eps_r}{2})\rb}\lq C2^k\eps_r^{-k}\Lambda^k\lb\Omega^\frac{1}{4}\rb\lq C\eps_r^{-k},$$ where $C$ is a constant depending on $k$ and the Lebesgue measure of $\Omega$.
These balls altogether provide a finite open cover for the entire $\Omega_\delta$, on which there exist non-negative functions $\alpha_{j,r}\in C_0^\infty(B(x_j,\eps_r))$ that give a partition of unity: $\forall x\in\Omega_\delta$: $$\sum_{r=1}^{r_0}\sum_{j=1}^{N_r}\alpha_{j,r}(x)=1.$$ For each $r$, set further a smaller value for $\eps_r$ s.t. the balls covering $\Omega_s,~s\lq r$, do not intersect those of $\Omega_{r+2}$. Then one may choose, by Theorem 1.4.1 and 1.4.4 in [@hormander1990AnaLinParDifOpeI:DisTheFouAna], such functions $\alpha_{j,r}$ that satisfy $|\alpha_{j,r}|_{K,B(x_j,\eps_r)}\lq C_{d,K}\eps_r^{-K}$ for each $j,r$ and any $K>0$. For each $j$ and $r$ define $\wt{\Psi}_{j,r}(y):=M^{-1}\eps_r^{-2}(\Psi(\eps_ry+x_j)-\Psi(x_j))$. Then for each $y\in B(0,1)$, the point $\eps_ry+x_j\in B(x_j,\eps_r)$, and by Taylor’s theorem, there is some $x'\in B(x_j,\eps_r)$ s.t. $$\begin{aligned}
\lv\nabla\wt{\Psi}_{j,r}(y)\rv=&M^{-1}\eps_r^{-1}|\nabla\Psi(\eps_ry+x_j)|\\
\gq&M^{-1}\eps_r^{-1}|\nabla\Psi(x_j)|-\frac{1}{2}M^{-1}|\tD^2\Psi(x')|\\
\gq&\eps_r^{-1}2^{-r}-\frac{1}{2}>\frac{1}{2}.\end{aligned}$$ Since each $x_j\in\Omega_r$, by Taylor’s theorem again, for all $y\in B(0,1)$ and some $x''\in B(x_j,\eps_r)$, $$\begin{aligned}
\lv\wt{\Psi}_{j,r}(y)\rv\lq&M^{-1}\eps_r^{-1}\lv\nabla\Psi(x_j)\rv+\frac{1}{2}M^{-1}|\tD^2\Psi(x'')|\\
\lq&\eps_r^{-1}2^{-r+1}+\frac{1}{2}\lq\frac{9}{2};\end{aligned}$$ the same argument gives the same upper bound for $|\nabla\wt\Psi_{j,r}(y)|$. For all $n\gq2$, one also has the Euclidean norm $|\tD^n\wt{\Psi}_{j,r}(y)|\lq M^{-1}\eps_r^{n-2}|\tD^n\Psi(x_j)|\lq1$. Therefore $\wt\Psi_{j,r}$ is in a (uniformly) bounded subset of $C^\infty(B(0,1))$.
Now that each function $\varphi_{j,r}:=\alpha_{j,r}\varphi$ is supported on the ball $B(x_j,\eps_r)$, the function $\psi_{j,r}(y):=\varphi_{j,r}(\eps_ry+x_j)$ is then supported on $B(0,1)$, satisfying $|\psi_{j,r}|_{K,B(0,1)}\lq C_{k,K}|\varphi|_{K,\Omega}$ for all $K,j,r$. Hence using the same arguments as in the proof of Lemma 0.4.7 in [@sogge1993FouIntClaAna] one sees: $$\begin{aligned}
\lv\int_{B(x_j,\eps_r)}e^{iR\Psi(x)}\varphi_{j,r}(x)\td x\rv=&\eps_r^k\lv\int_{B(0,1)}e^{iM\eps_r^2R\wt{\Psi}_{j,r}(y)}\varphi_{j,r}(\eps_ry+x_j)\td y\rv\\
\lq&C_{k,K}|\varphi|_{K,\Omega}M^{-K}\eps_r^{k-2K}R^{-K}.\end{aligned}$$ Finally, since $\supp\varphi=\Omega$, by the triangle inequality one deduces that $$\begin{aligned}
\lv\int_\Omega e^{iR\Psi(x)}\varphi(x)\td x\rv\lq&\sum_r\sum_{j=1}^{N_r}\lv\int_{B(x_j,\eps_r)}e^{iR\Psi(x)}\varphi_{j,r}(x)\td x\rv+|\varphi|_{0,\Omega}\Lambda^k\lb\bigcup_{r,j}B(x_j,\eps_r)\setminus\Omega_\delta\rb\\
\lq&C|\varphi|_{K,\Omega}M^{-K}\sum_rN_r\eps_r^{k-2K}R^{-K}+|\varphi|_{0,\Omega}\Lambda^k(\Omega\setminus\Omega_\delta)\\
\lq&C|\varphi|_{K,\Omega}M^{-K}R^{-K}\sum_r\eps_r^{-2K}+|\varphi|_{0,\Omega}\Lambda^k(\Omega\setminus\Omega_\delta)\\
\lq&C|\varphi|_{K,\Omega}M^K\delta^{-2K}R^{-K}+|\varphi|_{0,\Omega}\Lambda^k(\Omega\setminus\Omega_\delta),\end{aligned}$$ where $C$ is a constant depending on $k,K$ and $\Lambda^k(\Omega)$.
This lemma is to be applied to $\Psi(v)=\Phi_p(v;\omega_0)$ and $\Omega_\delta=\{v\in\Omega,~|\nabla\Phi_p(v;\omega_0)|>\delta\}$ for some bounded domain $\Omega\subset\RR^{2qp}$ and any $\delta>0$; in this case the phase function $\Psi$ also depends on the parameter $\omega_0=\xi/|\xi|$. Instead of a unit vector consider $\omega\in\RR^d$ s.t. $|\omega|\gq c$ for some $c>0$: if the $v$-derivatives of $\Psi(v;\omega)$ have no singularity in $\omega$, then the result holds with $|\Psi|_{K,\Omega}$ replaced by $\sup_{|\omega|\gq c}|\Psi(\cdot;\omega)|_{K,\Omega}$. If the amplitude $\varphi$ also depends on $\omega$, then $|\varphi|_{K,\Omega}$ should be replaced by $\sup_{|\omega|\gq c}|\varphi(\cdot;\omega)|_{K,\Omega}$.
It then remains to estimate the Lebesgue measure of the exceptional set $\Omega\setminus\Omega_\delta$, which would also depend on $\omega$ if $\Psi=\Psi(v;\omega)$. The next three lemmas are devoted to this; the idea is to study the degeneracy of the Hessian matrix $\tD^2\Phi_p(v;\omega)$ described by , and . We start with the following general fact.
\[factorisation\] Let $\Omega\subset\RR^k$ be open and bounded, $f:\Omega\to\RR^l$ be a $C^1$ function. For each $x$, let $\sigma_1(x)\gq\sigma_2(x)\gq\cdots\gq\sigma_{k\wedge l}(x)$ be the singular values of its derivative $\tD f(x)$. For any $n\in[1,k\wedge l]\cap\NN$ and $\eta>0$, define $$G_{n,\eta}(f):=\left\{x\in\Omega:\sigma_n(x)>\eta\right\}.$$ If $\tD f$ is Lipschitz continuous with Lipschitz constant $L$, then $\forall\delta>0$, $$\Lambda^d\lb G_{n,\eta}(f)\cap\{|f|\lq\delta\}\rb\lq CL^n\eta^{-2n}\delta^n,$$ where the constant $C$ depends on $k,l$ and $\Lambda^k(\Omega)$.
For fixed $n,\eta$ and any $z\in G_{n,\eta}(f)$, by definition the matrix $\tD f(z)$ has rank $n$. This implies that for each $z$ there are $n$-dimensional subspaces $E_z$ of $\RR^k$ and $F_z$ of $\RR^l$ s.t., with $g_z(\cdot):=\pi_{F_z}\circ f|_{E_z}(\cdot)$ and $\pi_\cdot$ being the orthogonal projection, the linear map $\tD g_z(z)$ is invertible. Denote by $E_z^\perp$ the orthogonal complement of $E_z$ for each $z$.
By the continuity of $\tD f$ the set $G_{n,\eta}(f)$ is open, and the inverse function theorem implies that $g_z$ is a diffeomorphism in some neighbourhood[^4] $B^{(n)}(z,\eps)\subset E_z$. Moreover, in the proof of the inverse function theorem (see, e.g., Theorem 9.24 in [@rudin1976PriMatAna] or Theorem 1.1.7 in [@hormander1990AnaLinParDifOpeI:DisTheFouAna]), the ball $B^{(n)}(z,\eps)$ can be typically constructed with radius $$\eps\lq\frac{1}{2L|(\tD g_z(z))^{-1}|}\lq\frac{|\tD f(z)|}{2L}.$$ As $z\in G_{n,\eta}(f)$, one can choose e.g. $\eps=\eta/(4L)\wedge1$.
Since $G_{n,\eta}(f)$ is bounded, similar to the proof of Lemma \[stationary\_phase\] there are finitely many points $z_1,\cdots,z_{N_\eps}\in G_{n,\eta}(f)$ s.t. $G_{n,\eta}(f)\subset\bigcup_{j=1}^{N_\eps}B(z_j,\eps)$, with the number of balls satisfying $$N_\eps\lq\frac{\Lambda^k\lb G_{n,\eta}^{\eps/2}(f)\rb}{\Lambda^k\lb B(z_j,\eps/2)\rb}\lq C2^k\eps^{-k}\Lambda^k\lb\Omega^{1/2}\rb\lq C\eps^{-k},$$ for some constant $C$ depending on $k$ and $\Lambda^k(\Omega)$.
Write $\Gamma_j=B(z_j,\eps)\cap G_{n,\eta}(f)$ and $P_{j,\delta}=\pi_{E_{z_j}^\perp}(\Gamma_j\cap\{|f|\lq\delta\})$ for $\delta>0$. For each $(k-n)$-dimensional vector $(x_{n+1},\cdots,x_k)\in P_{j,\delta}$ let $S_{j,\delta}=S_{j,\delta}(x_{n+1},\cdots,x_k)$ be the corresponding ‘slice’ of the set $\Gamma_j\cap\{|f|\lq\delta\}$ parallel to $E_{z_j}$. Then $$\Gamma_j\cap\{|f|\lq\delta\}=\bigcup_{(x_{n+1},\cdots,x_k)\in P_{j,\delta}}S_{j,\delta}.$$ Notice that all the singular values of $\tD g_z$ are greater than $\eta$ on $\Gamma_j$. Then by a change of coordinates and variables, one has that $$\begin{aligned}
\Lambda^k&\lb G_{n,\eta}(f)\cap\{|f|\lq\delta\}\rb\lq\sum_{j=1}^{N_\eps}\int_{\Gamma_j\cap\{|f|\lq\delta\}}\td x_1\cdots\td x_k\\
&=\sum_{j=1}^{N_\eps}\int_{P_{j,\delta}}\td x_{n+1}\cdots\td x_k\int_{S_{j,\delta}}\td x_1\cdots\td x_n\\
&=\sum_{j=1}^{N_\eps}\int_{P_{j,\delta}}\td x_{n+1}\cdots\td x_k\int_{g_{z_j}(\Gamma_j)\cap\{|y|\lq\delta\}}\lv\det(\tD g_{z_j})^{-1}(y)\rv\td y_1\cdots\td y_n\\
&\lq C\lb\min_{j}\inf_{x\in\Gamma_j}\lv\det\tD g_{z_j}(x)\rv\rb^{-1}\delta^n\sum_{j=1}^{N_\eps}\Lambda^{k-n}\lb B^{(k-n)}(z_j,\eps)\rb\\
&\lq C\eta^{-n}\delta^nN_\eps\eps^{k-n},\end{aligned}$$ where the constant $C$ depends on $k,l$ and $\Lambda^k(\Omega)$. Then the result follows from the bound for $N_\eps$ and the choice of $\eps$.
Now write $G_{n,\eta}=G_{n,\eta}(\nabla\Phi_p(\cdot;\omega))$ as defined in Lemma \[factorisation\] with $k=l=2qp$. One then needs to estimate the measure of the complement $\Omega\setminus G_{n,\eta}$ for suitable values of $\eta$ and $n\lq2qp$. From the expressions , and one sees that the behaviour of the second derivatives depends on the magnitude of the parameter $\rho$. Since the differentiation is done w.r.t. the variable $v$, the measure $\Lambda^{2qp}(\Omega\setminus G_{n,\eta})$ may depend on $\omega$, which for now we do not assume to be a unit vector.
The following result gives an estimate for the case where $\rho$ is not too small.
\[rho\_big\] Let $\Omega\subset\RR^{2qp}$ be bounded and $n\lq\sqrt{2p}/4$ be an integer. If $|\rho|>\eps$ for some fixed $\eps\in(0,|\omega|)$, then one has $\Lambda^{2qp}(\Omega\setminus G_{n,\eta})\lq C\eps^{1-2n}\eta^n$, where $C$ is a constant depending on $q,p,n$ and $\diam(\Omega)$.
It suffices to focus on a submatrix of $\tD^2\Phi_p(v;\omega)$ since $\wh{G}_{n,\eta}\subset G_{n,\eta}$ where $\wh{G}_{n,\eta}$ is similarly defined by the singular values of the submatrix. Since $|\rho|>\eps$, locate the (Lyndon) word $(j,k,l^\ast)$ that gives the maximum entry $|\rho_{jkl^\ast}|\gq\eps\sqrt{3/(q^3-q)}$. Then for the fixed pair $(j,k)$ we will focus on the submatrix $H_{xx}(j,k)$.
For a particular pair $(r,s)$, observe from that $\partial^2_{x_{jr}x_{ks}}\Phi_p(v;\omega)$ contains all the permutations of the word $(j,k,l)$ for each index $l$. Recall that all non-Lyndon entries of $\rho$ are defined to be $0$, and that if $(j,k,l)$ is a Lyndon word, we have either $j<k\wedge l$ or $j=k<l$. Thus for every Lyndon word $(j,k,l)$, out of the rest five permutations only one of $\rho_{jlk}$ and $\rho_{kjl}$ may not vanish, corresponding to those two cases respectively. If $j<k$ one has that $$\begin{aligned}
\partial^2_{x_{jr}x_{ks}}\Phi_p(v;\omega)=&\frac{1}{r^2}\beta_{jk}^{(1)}\delta_{rs}+\frac{1}{r^2-s^2}\lb\frac{r}{s}\gamma_{jk}-\frac{s}{r}\gamma_{kj}\rb(1-\delta_{rs})\\
&+\sum_{l>j}\lb\frac{\rho_{jkl}}{r(r+s)}-\frac{\rho_{jlk}}{rs}\rb y_{l,r+s}+\sum_{l>j}\lb\frac{\rho_{jkl}}{r|s-r|}-\frac{\rho_{jlk}}{rs}\rb\sign(s-r)y_{l,|s-r|}.\end{aligned}$$ Clearly, when $r\neq s$ the coefficients of $y_{l^\ast,r+s}$ and $y_{l^\ast,|s-r|}$ cannot vanish simultaneously. This is trivial if $j=k$, for one has instead $$\partial^2_{x_{jr}x_{js}}\Phi_p(v;\omega)=\frac{2}{r^2}\beta_{jk}^{(1)}\delta_{rs}+\sum_{l>j}\frac{\rho_{jjl}}{rs}(y_{l,r+s}+y_{l,|s-r|}).$$ This means that for fixed $r\neq s$ the entries of the submatrix $H_{xx}(j,k)$ involve different components $y_{l,r+s}$ and $y_{l,|s-r|}$ of the vector $y$. Let us combine these two cases and write $$\label{entry}
\partial^2_{x_{jr}x_{ks}}\Phi_p(v;\omega)=\kappa_{rs}+w_{rs}\cdot y$$ with a constant term $\kappa_{rs}=\kappa_{rs}(\gamma_{jk},r,s)$ and coefficient $w_{rs}=w_{rs}(\rho_{jk\cdot},r,s)\in\RR^{qp}$.
For integers $n\lq m\lq\sqrt{p/2}-1$, one can choose $r_1,\cdots,r_m,s_1,\cdots,s_m\lq p$ s.t. $r_a\neq s_b$ and the integers $r_a+s_b,|r_c-s_d|$ are all different from one another for all choices of $a,b,c,d=1,\cdots,m$. For example, one may choose $r_a=a,~s_a=a(2m+1)$. In this case, the only choice of $(a,b,c,d)$ s.t. $r_a+s_b=r_c+s_d$, i.e. $(c-a)+(d-b)(2m+1)=0$, is that $a=c$ and $b=d$; the same for $r_a-s_b=r_c-s_d$. There is no choice of $(a,b,c,d)$ for the equation $(a+c)+(b-d)(2m+1)=0$ to hold so $r_a+s_b=s_c-r_d$ is never satisfied. Since we also require that all of them are no greater than $p$, it is necessary that $\max_{a,b}(r_a+s_b)=2m(m+1)\lq p$.
With this particular choice of $r_1,\cdots,r_m,s_1,\cdots,s_m$, one obtains an $m\times m$ submatrix $Q_m(y)=Q_m(y;\rho,\gamma)$ of $H_{xx}(j,k)$, of which each entry takes the form ; write $\kappa_{ab}=\kappa_{r_as_b},~w_{ab}=w_{r_as_b},~a,b=1,\cdots,m$ for short. Then $|w_{ab}|\gq c_{q,p}\eps$ for each $(a,b)$, since for the particular case $l=l^\ast$ we have that $|\rho_{jjl^\ast}/(rs)|\gq c_q\eps/p^2$ and, by the maximality of $|\rho_{jkl^\ast}|$, that $$\lv\frac{\rho_{jkl^\ast}}{r(s-r)}-\frac{\rho_{jl^\ast k}}{rs}\rv\gq\frac{|\rho_{jkl^\ast}|}{r(s-r)}-\frac{|\rho_{jl^\ast k}|}{rs}\gq\frac{|\rho_{jkl^\ast}|}{s(s-r)}\gq c_q\frac{\eps}{p^2}.$$ Secondly, this particular choice of $\{r_a,s_b\}_{a,b}$ ensures that each entry of the submatrix $Q_m(y)$, translated by the constant $\kappa_{ab}$, is a linear combination of different components of $y$ that are *all distinct* from those appearing in other entries; in other words, the $m^2$ vectors $\{w_{ab}\}_{a,b}$ are *mutually orthogonal*. Denote the rows of $Q_m(y)$ by $q_1(y),\cdots,q_m(y)$, then each $q_a^\top(y)=\kappa_a+W_ay$ where $\kappa_a=(\kappa_{a1},\cdots,\kappa_{am})^\top$ and $W_a$ is the $m\times qp$ matrix consisting of the rows $w_{a1},\cdots,w_{am}$.
Now define for $a=1,\cdots,n$ the set $$\label{set_diff}
F_a:=\{(x,y)\in\Omega:~\dist(q_a(y),~\spa\{q_b(y):~b=1,\cdots,n,~b\neq a\})>\sqrt{n}\eta\},$$ then $Q_m(y)$ has rank at least $n$ for $(x,y)\in\bigcap_{a=1}^nF_a$. Every point $(x,y)\in F_a$ satisfies $$\inf_{c_1,\cdots,c_n\in\RR}\lv \kappa_a-\sum_{b\neq a}c_b\kappa_b+\lb W_a-\sum_{b\neq a}c_bW_b\rb y\rv>\sqrt{n}\eta.$$ The mutual orthogonality of the vectors $\{w_{ab}\}_{a,b}$ implies the mutual orthogonality of the $m$ rows of the matrix $U_a:=W_a-\sum_{b\neq a}c_bW_b$, which thereby has a right inverse on an $m$-dimensional subspace $E_m$ of $\RR^{qp}$. Note also that each $|w_{ab}|\gq c_{q,p}\eps$ implies that $U_a$ restricted on $E_m$ has norm at least $c_{q,p}\eps$. Hence by the translation-invariance of the Lebesgue measure and the boundedness of $\Omega$, that for each $a$, $$\begin{aligned}
\Lambda^{2qp}(\Omega\setminus F_a)\lq&C|\det(U_a|_{E_m})|^{-1}(\sqrt{n}\eta)^{m-n+1}\\
\lq&C\|(U_a|_{E_m})\|^{-m}(\sqrt{n}\eta)^{m-n+1}\lq C\eps^{-m}(\sqrt{n}\eta)^{m-n+1},\end{aligned}$$ where the constant $C=C(q,p,m,\diam(\Omega))$ grows at most exponentially in $m$.
For each point $(x,y)\in\bigcap_{a=1}^nF_a$ and any unit vector $e=(e_1,\cdots,e_n)$, consider the linear combination $e\cdot(q_1(y),\cdots,q_n(y))$ of the $n$ rows. Choose $a$ s.t. $|e_a|=\max\{|e_1|,\cdots,|e_n|\}\gq1/\sqrt{n}$, then $$|e_1q_1(y)+\cdots+e_nq_n(y)|=|e_a|\lv q_a(y)+\sum_{b\neq a}e_a^{-1}e_bq_b(y)\rv\gq\eta.$$ Thus, the $n\times m$ submatrix $\wh Q_n(y):=(q_1(y)^\top,\cdots,q_n(y)^\top)^\top$ has a right inverse $R_n(y)$ on an $n$-dimensional subspace $E_n$ of $\RR^m$, and $$\lvv R_n(y)\rvv=\sup_{|e|=1}\lv R_n(y)e^\top\rv\lq\lb\inf_{|e|=1}\lv e\wh Q_n(y)\rv\rb^{-1}\lq \eta^{-1}.$$ It then follows from the singular-value decomposition that the singular values of the matrix $\wh Q_n(y)$ are all bounded from below by $\|R_n(y)\|^{-1}\gq\eta$, which in turn gives an estimate for the measure of the exceptional set: $$\Lambda^{2qp}(\Omega\setminus G_{n,\eta})\lq\Lambda^{2qp}\lb\Omega\setminus\wh{G}_{n,\eta}\rb\lq\Lambda^{2qp}\lb\bigcup_{a=1}^n(\Omega\setminus F_a)\rb\lq Cn\eps^{-m}(\sqrt{n}\eta)^{m-n+1},$$ and the result follows by taking $m=2n-1$.
The result of Lemma \[rho\_big\] is meaningful for small values of $\eps$ and $\eta$. It remains to show that the measure $\Lambda^{2qp}(\Omega\setminus G_{n,\eta})$ is also small when $\rho$ is small.
\[rho\_small\] Let $\Omega\subset\RR^{2qp}$ be bounded and $n\lq p$ be an even integer s.t. $n+1$ is prime. Then, depending on $q,p,n$ and $\diam(\Omega)$, one can choose $\delta,\eta,\eps\lesssim_{q,p,n}|\omega|$ sufficiently small s.t. for $|\rho|\lq\eps$, either $\Omega_\delta=\Omega$ or $G_{n,\eta}=\Omega$.
For $\eps\in(0,|\omega|/\sqrt{2})$ define $\bar\eps=\sqrt{|\omega|^2-\eps^2}\in(|\omega|/\sqrt{2},|\omega|)$, and assume $\diam(\Omega)=1$ w.l.o.g., otherwise replace $\eps$ with $\eps/(1\vee\diam(\Omega))$. First of all that $|\rho|\lq\eps$ implies that the vector $(\alpha,\beta,\gamma,a,b)$ has modulus no less than $\bar\eps$. The proof is divided into several cases depending on which components of this vector are dominant in modulus or norm.
Let us first consider the case where the coefficients $(a,b)$ are ‘dominant’ in the sense that $|(a,b)|>\bar\eps\sqrt{1-\theta^2}\gq|\omega|/2$ for some $\theta\in(0,1/\sqrt{2})$ to be chosen later. In this case $|(\alpha,\beta,\gamma)|\lq\bar\eps\theta$. From the expression for the first derivatives one has the following bound: $$\label{1st_der_bound}
\lv\partial_{x_{jr}}\Phi_p(v;\omega)\rv^2\gq\frac{1}{r^2}a_j^2-\frac{2}{r}|a_j||Q_{x_{jr}}(v;\rho)|-2\lb\frac{1}{r}|a_j|+|Q_{x_{jr}}(v;\rho)|\rb|L_{x_{jr}}(v;\alpha,\beta,\gamma)|,$$ where $L_{x_{jr}}(v;\alpha,\beta,\gamma)$ and $Q_{x_{jr}}(v;\rho)$ denote the linear and quadratic parts for $v$ in . Since $x$ and $y$ are bounded, one has that $$\label{1st_der_quadratic}
\sup_{v\in\Omega}|Q_{x_{jr}}(v;\rho)|\lq C_q|\rho|\frac{1}{r}\lb\sum_{s=1}^{p-r}\frac{1}{s}+\sum_{s=1}^{r-1}\frac{1}{s}\rb\lq C_q\frac{\eps}{r}\log p,$$ and that, omitting the summation in $k$, $$\begin{aligned}
\sup_{v\in\Omega}|L_{x_{jr}}(v;\alpha,\beta,\gamma)|\lq&\sup_{v\in\Omega}\lb\frac{1}{r}|\alpha_{jk}||y_{kr}|+\frac{2}{r^2}|\beta_{jk}^{(1)}||x_{kr}|+\frac{1}{r}(|\gamma_{jk}|+|\gamma_{kj}|)\sum_{s\neq r}\frac{1}{s}|x_{ks}|\rb\nonumber\\
\lq&C_q\frac{\log p}{r}\bar\eps\theta.\label{1st_der_linear}\end{aligned}$$ Hence one derives that $$\inf_{v\in\Omega}|\partial_{x_{jr}}\Phi_p(v;\omega)|^2\gq\frac{1}{r^2}a_j^2-C_q\frac{\eps\log p}{r^2}|a_j|-C_q\bar\eps\theta\frac{\log p}{r}\lb\frac{1}{r}|a_j|+\frac{\eps}{r}\log p\rb,$$ and a similar inequality for $|\partial_{y_{jr}}\Phi_p|^2$ with $a_j/r$ replaced with $b_j/r^2$ as per . Thus, summing up $j$ and $r$ one has that $$\begin{aligned}
\inf_{v\in\Omega}|\nabla\Phi_p(v;\omega)|^2\gq&C|(a,b)|^2-C_q|(a,b)|\eps\log p-C_q\bar\eps\theta(\log p)^2(|(a,b)|+\eps\log p)\nonumber\\
\gq&C|\omega|^2-C_q|\omega|\eps\log p-C_q|\omega|\theta(\log p)^2(|\omega|+\eps\log p),\label{grad_bound}\end{aligned}$$ which has a fixed lower bound for $\eps<|\omega|(\log p)^{-1}$ and $\theta<(\log p)^{-2}$ sufficiently small. Then for small values of $\delta<C|\omega|$ we have $\Omega=\Omega_\delta$.
Now suppose that $|(a,b)|\lq\bar\eps\sqrt{1-\theta^2}$, then $|(\alpha,\beta,\gamma)|\gq\bar\eps\theta$. The latter corresponds to the constant terms in the second derivative $\tD^2\Phi_p(v;\omega)$. Write $$\label{2nd_der_mat}
\tD^2\Phi_p(v;\omega)=A_p+L_p(v;\rho)$$ according to the expressions , and , where $A_p=A_p(\alpha,\beta,\gamma)$ and $L_p(v;\rho)=\{(L_{x_{jr}x_{ks}},L_{y_{jr}y_{ks}},L_{x_{jr}y_{ks}})(v;\rho)\}_{j,k,r,s}$ are the constant and linear parts in $v$, respectively. Then for each $(j,k)$ and $(r,s)$, $$\sup_{v\in\Omega}|L_{x_{jr}x_{ks}}(v;\rho)|\lq C_q|\rho|\lb\frac{1}{rs}+\frac{\delta_{rs}}{r|r-s|}+\frac{\delta_{rs}}{s|r-s|}\rb\lq C_q\frac{\eps}{r\wedge s},$$ and the same bound holds for $L_{y_{jr}y_{ks}}$ and $L_{x_{jr}y_{ks}}$. Let $H_n(v;\omega)=A_n+L_n(v;\rho)$ be an $n\times n$ submatrix of $\tD^2\Phi_p(v;\omega)$ with ‘constant’ part $A_n=A_n(\alpha,\beta,\gamma)$ and linear part $L_n(v;\rho)$. Then the estimate above implies that $$\label{Ln_bound}
\sup_{v\in\Omega}\|L_n(v;\rho)\|\lq C_{q,p}\eps.$$ Thus if $A_n$ is invertible, one can choose $\eps\lesssim_{q,p}\|A_n\|$ sufficiently small s.t. $H_n$ is also invertible for all $v\in\Omega$. Furthermore, choose $\eps<\|A_n^{-1}\|^{-1}$ small enough so that $$\label{Hn_lower_bound}
\|H_n^{-1}\|\lq\|A_n^{-1}\|\|(I+A_n^{-1}L_n)^{-1}\|\lq\|A_n^{-1}\|/(1-\|A_n^{-1}\|\|L_n\|)\lq2\eps^{-1},$$ for all $v\in\Omega$. This follows from the fact that $\|(I+B)^{-1}\|=\|\sum_{k\gq0}(-B)^k\|\lq\sum_{k\gq0}\|B\|^k=1/(1-\|B\|)$ for any square matrix $B$ s.t. $\|B\|<1$. Henceforth by the singular-value decomposition the estimate will imply that $H_n(v;\omega)$ - as an $n\times n$ submatrix of $\tD^2\Phi_p(v;\omega)$ - has singular values no less than $\eps/2$ for all $v\in\Omega$, in other words, $\Omega\setminus G_{n,\tau/2}=\varnothing$. In particular, for any $\eta\lq\eps/2$ we have $\Omega\setminus G_{n,\eta}=\varnothing$, too. Henceforth, one looks for an invertible $n\times n$ submatrix $A_n$ of $A_p$ with an appropriate bound for $\|A_n^{-1}\|$, and the result will follow by choosing sufficiently small values of $\eps$ and $\eta$.
Write $D_n=\diag(1,1/2,\cdots,1/n),~n\lq p$ for simplicity. If the component $\alpha$ is ‘dominant’ amongst $\alpha,\beta,\gamma$ in the sense that, for example, $\|\alpha\|>\bar\eps\theta/\sqrt{3}>|\omega|\theta/\sqrt{6}$, choose the largest entry $|\alpha_{jk}|\gq c_q|\omega|\theta$. Then by the constant part of the $n$-th principle submatrix of the block $H_{xy}(j,k)$ is $A_n=\alpha_{jk}D_n$, and $\|A_n^{-1}\|\lq|\alpha_{jk}|^{-1}n$. Thus the result holds for $\eps\lesssim_{q,p}|\omega|\theta/n$.
On the other hand we need to consider the case where $|(\beta,\gamma)|\gq\bar\eps\theta\sqrt{2/3}$. If the largest entry of $(\beta,\gamma)$ is located on the diagonal, i.e. $|\beta_{jj}^{(i)}|\gq c_q\bar\eps\theta$ (recall that $\gamma_{jj}=0$) for $i=1$ or $2$ and some $j$, then the constant part of the $n$-th principle submatrix of the block $H_{xx}(j,j)$ or the block $H_{yy}(j,j)$ is $A_n^{(i)}=2\beta_{jj}^{(i)}D_n^2$ by and . Hence we have that $\|A_n^{-1}\|\lq|2\beta_{jj}^{(i)}|^{-1}n^2$ and we need $\eps\lesssim_{q,p}|\omega|\theta/n^2$.
The situation is trickier when the largest entry is found off the diagonal, i.e. for some pair $(j,k)$ (assuming $j<k$ w.l.o.g.). Consider the constant part $A_n^{(2)}$ of the $n$-th principle submatrix of the block $H_{yy}(j,k)$. By it takes the form $$A_n^{(2)}=\begin{pmatrix}
\beta_{jk}^{(2)} & \frac{1}{3}\gamma_{jk} & \frac{1}{8}\gamma_{jk} & \cdots & \frac{1}{n^2-1}\gamma_{jk} \\
-\frac{1}{3}\gamma_{jk} & \frac{1}{4}\beta_{jk}^{(2)} & \frac{1}{5}\gamma_{jk} & \cdots & \frac{1}{n^2-4}\gamma_{jk} \\
-\frac{1}{8}\gamma_{jk} & -\frac{1}{5}\gamma_{jk} & \frac{1}{9}\beta_{jk}^{(2)} & \cdots & \frac{1}{n^2-9}\gamma_{jk} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
-\frac{1}{n^2-1}\gamma_{jk} & -\frac{1}{n^2-4}\gamma_{jk} & -\frac{1}{n^2-9}\gamma_{jk} & \cdots & \frac{1}{n^2}\beta_{jk}^{(2)}
\end{pmatrix}=\beta_{jk}^{(2)}D_n^2+\gamma_{jk}S_n,$$ where $S_n$ is the skew-symmetric matrix with $(r,s)$-th entry $(s^2-r^2)^{-1},~r\neq s$ and $0$ on the diagonal. If $|\beta_{jk}^{(2)}|\gq c_q\bar\eps\theta$, then the matrix $A_n^{(2)}$ has full rank. To see this, notice that the matrix $\bar{S}_n:=D_n^{-1}S_nD_n^{-1}$ is also skew-symmetric and has purely imaginary eigenvalues only. Then all the eigenvalues of the scaled matrix $\bar{A}_n^{(2)}:=I+\gamma_{jk}\bar{S}_n/\beta_{jk}^{(2)}$ have real parts $1$, which serves as a lower bound for the operator norm of $\bar A_n^{(2)}$ as it is in fact a normal matrix, and so $\|(\bar{A}_n^{(2)})^{-1}\|\lq1$. Therefore $\|(A_n^{(2)})^{-1}\|=\|(\beta_{jk}^{(2)}D_n\bar{A}_n^{(2)}D_n)^{-1}\|\lq|\beta_{jk}^{(2)}|^{-1}n^2$ and again we need $\eps\lesssim_{q,p}|\omega|\theta/n^2$.
The same applies to the case where $|\beta_{jk}^{(1)}|\gq c_q\bar\eps\theta$: instead of $A_n^{(2)}$ consider the constant part $A_n^{(1)}$ of the $n$-th principle submatrix of the block $H_{xx}(j,k)$, which by takes the form $A_n^{(1)}=\beta_{jk}^{(1)}D_n^2+\gamma_{jk}S'_n$ where $S'_n$ is the matrix with $(r,s)$-th entry $(r^2-s^2)^{-1}r/s$. Then it suffices to observe that $S'_n=-D_n^{-1}S_nD_n$ and $A_n^{(1)}=\beta_{jk}^{(1)}(I-\gamma_{jk}\bar{S}_n/\beta_{jk}^{(1)})D_n^2$.
Finally, if $|\gamma_{jk}|\gq c_q\bar\eps\theta$ is the largest entry of $(\beta,\gamma)$, we return to the matrix $A_n^{(2)}$. Since $S_n$ is skew-symmetric, $\det S_n=0$ for all odd $n$. If $n$ is even, by definition the determinant of $S_n$ is given by the expansion $$\det S_n=\sum_{\sigma\in\Pi_n^\ast}\sign(\sigma)\frac{1}{1-\sigma_1^2}\frac{1}{4-\sigma_2^2}\cdots\frac{1}{n^2-\sigma_n^2},$$ where $\Pi_n^\ast$ is the set of permutations of $(1,\cdots,n)$ with no fixed points. Notice that this summation includes the product of all the entries along the reflected diagonal $r+s=n+1$, each of which has denominator divisible by $n+1$. Clearly, out of all the permutations this product is the only term in the above expansion whose denominator is divisible by $(n+1)^n$ if $n+1$ is prime. Then it follows from the fundamental theorem of arithmetic that $\text{\thorn}_n:=\det S_n\neq0$. It is rather difficult to compute the the value $\text{\thorn}_n$ explicitly; computer results for large values of $n$ up to $400$ shows that it decays roughly exponentially. Notice that $A_n^{(2)} =D_n(I+\beta_{jk}^{(2)}\gamma_{jk}^{-1}\bar{S}_n^{-1})D_n^{-1}\gamma_{jk}S_n$ and that the matrix $\bar{S}_n^{-1}$ is still skew-symmetric, the same argument used in the previous cases still applies. Therefore $\|(A_n^{(2)})^{-1}\|\lq|\gamma_{jk}|^{-1}\|S_n^{-1}\|n$, and we need $\eps\lesssim_{q,p}|\omega|\theta\text{\thorn}_n^{1/n}/n$. Combining all the criteria above, for an even integer $n$ s.t. $n+1$ is prime one can choose $\eps\lesssim_{q,p}|\omega|\text{\thorn}_n^{1/n}n^{-2}$ s.t. the result holds true for $\delta\lesssim_q|\omega|/4$ and $\eta<\eps/2$ sufficiently small.
These lemmas altogether give an estimate for oscillatory integrals of the type $$T(R,\omega)=\int_\Omega e^{iR\Phi_p(v;\omega)}\varphi(v)\td v,$$ for a bounded domain $\Omega$ and a smooth amplitude $\varphi$ supported on $\Omega$. In order to study the global behaviour of it, in particular, the characteristic function $\psi_p(\xi)$ of $V_p$, some cut-off arguments will be needed to derive a similar estimate as in Lemma \[stationary\_phase\] on the whole space $\RR^{2qp}$.
But as the reader will realise later, to find a desired coupling for $V_p$ it is necessary to estimate oscillatory integrals with amplitudes other than just $\phi_p$. For a Schwartz function $\varphi\in\mathscr{S}(\RR^q)$ and $k,l\in\NN$, introduce the norm $$\|\varphi\|_{j,k}=\max_{|\theta|\lq j,|\tau|\lq k}\sup_{x\in\RR^q}|x^\theta\partial^\tau\varphi(x)|,$$ where $\theta,\tau\in\NN^q$ are multi-indices. Then for $\varphi\in C_0^\infty(\Omega)$ it holds that $|\varphi|_{k,\Omega}\simeq_q\|\varphi\|_{0,k}$.
\[global\_estimate\] For any $K>0$, let $p_0>8K^2$ be an even integer s.t. $[\sqrt{2p_0}/4]+1$ is a prime number and let $\varkappa=(2q+1/2)p_0+K+1$. For any $p\gq p_0,~\xi\in\RR^d,~\omega_0:=\xi/|\xi|$ and a Schwartz function $\varphi\in\mathscr{S}(\RR^{2qp})$, define $$I_p(\xi)=\int_{\RR^{2qp}}e^{i|\xi|\Phi_p(v;\omega_0)}\varphi(v)\td v,$$ and separate the phase function $\Phi_p$ in terms of distinct $v_0$-monomials: $$\label{separate_phase}
\Phi_p(v;\omega_0)=\sum_{|\beta|\lq3}v_0^{\beta}P_\beta(v'),$$ where for each multi-index $\beta$ the polynomial $P_\beta$ has degree $3-|\beta|$. If $\varphi$ can be factorised as the product of two further Schwartz functions $\varphi_0\in\mathscr{S}(\RR^{2qp_0})$ and $\varphi_1\in\mathscr{S}(\RR^{2qp'}),~p':=p-p_0$, then $I_p\in C^\infty(\RR^d)$, and for any $k\in\NN$ and $|\xi|$ sufficiently large it holds that $$\label{DI_estimate}
|\tD^kI_p(\xi)|\lq C_{q,p_0,k,K}|\xi|^{-\frac{K}{16}}\|\varphi_0\|_{\varkappa+3k,K}\int_{\RR^{2qp'}}\lb1+\sum_{|\beta|\lq3}|P_\beta(v')|^{\sqrt{2p_0}+k-2}\rb\varphi_1(v')\td v',$$ provided that the last integral is finite.
As we shall see later, this result is only useful when the integral in is indepdent of $p$.
Let us prove the rapid decay of $|I_p(\xi)|$ first. Instead of $I_p(\xi)$ let us consider for now the oscillatory integral $$I_{p_0}(\xi)=\int_{\RR^{2qp_0}}e^{i|\xi|\Phi_{p_0}(v_0;\omega_0)}\varphi_0(v_0)\td v_0$$ for a fixed positive integer $p_0$, and write $\omega_0=(a,b,\alpha,\beta,\gamma,\rho)\in\mathbb{S}^{d-1}$. First choose a non-negative, smooth cut-off function $\zeta_0\in C_0^\infty(B(0,2))$ s.t. $\zeta_0\equiv1$ on $B(0,1)$ and all its derivatives are bounded on $B(0,2)\setminus B(0,1)$. Divide the rest of $\RR^{2qp_0}$ by the annuli $$A_r:=\{v_0\in\RR^{2qp_0}:~2^{r-1}\lq|v_0|<2^r\},~r\in\NN,$$ and define the fattened annuli $A'_r:=\{2^{r-2}\lq|v_0|<2^{r+1}\}$. Choose another non-negative, smooth cut-off $\zeta_1\in C_0^\infty(A'_1)$ taking value $1$ on $A_1$ and bounded derivatives on $A'_1\setminus A_1$, and define $\zeta_r(v_0):=\zeta_1(2^{-r+1}v_0),~\forall r\gq2$. Then for each $r\gq1$, the smooth function $\zeta_r$ is supported on the fattened annulus $A'_r$ with value $1$ on $A_r$ and bounded derivatives on $A'_r\setminus A_r$; the sum $\sigma(v_0):=\sum_{r=0}^\infty\zeta_r(v_0)$ is then supported on the whole of $\RR^{2qp_0}$.
If one further sets $\wt{\zeta}_r(v_0):=\zeta_r(v_0)/\sigma(v_0)$, then each $\wt{\zeta}_r$ has the same properties as those of $\zeta_r$, and $\sum_{r=0}^\infty\wt{\zeta}_r\equiv1$ trivially. Then one can write $$\begin{aligned}
I_{p_0}(\xi)&=\int_{B(0,2)}e^{i|\xi|\Phi_{p_0}(v_0;\omega_0)}\varphi_0(v_0)\wt{\zeta}_0(v_0)\td v_0+\sum_{r=1}^\infty\int_{A'_r}e^{i|\xi|\Phi_{p_0}(v_0;\omega_0)}\varphi_0(v_0)\wt{\zeta}_r(v_0)\td v_0\\
&=:T_0(\xi)+\sum_{r=1}^\infty T_r(\xi),\end{aligned}$$ where the first integral can be readily estimated by the lemmas above. Since the function $\Phi_{p_0}(v_0;\omega_0)$ is a cubic polynomial and the vector $\omega_0=(a,b,\alpha,\beta,\gamma,\rho)$ is normalised, all the derivatives of $\Phi_{p_0}$ are uniformly bounded on $B(0,2)$; so do all the derivatives of $\wt\zeta_0$ by its construction.
Thus, applying Lemma \[stationary\_phase\] we have, $\forall K,\delta_0>0,~\xi\in\RR^d$, $$|T_0(\xi)|\lq C_{q,p_0,K}|\varphi_0|_{K,B(0,2)}\delta_0^{-2K}|\xi|^{-K}+2|\varphi_0|_{0,B(0,2)}\Lambda^{2qp_0}(\Gamma_{\delta_0}),$$ where $\Gamma_{\delta_0}=\{v_0\in B(0,2):|\nabla\Phi_{p_0}(v_0,\omega_0)|\lq\delta_0\}$. The set $\Gamma_{\delta_0}$ can be further split by the set $G_{n,\eta_0}=G_{n,\eta_0}(\nabla\Phi_{p_0})$ as defined in Lemma \[factorisation\] and its complement for some $\eta_0>0$ and some integer $n$. Note that the Lipschitz constant of $\tD^2\Phi_{p_0}$ is at most $|\rho|\lq1$. Then by Lemma \[factorisation\], \[rho\_big\] and \[rho\_small\] one sees that for any $\delta_0,\eta_0,\eps_0>0$ sufficiently small and any even integer $n\lq\sqrt{2p_0}/4$ s.t. $n+1$ is prime, one has that $$\begin{aligned}
\Lambda^{2qp_0}(\Gamma_{\delta_0})&\lq\Lambda^{2qp_0}(\Gamma_{\delta_0}\cap G_{n,\eta_0})+\Lambda^{2qp_0}(\Gamma_{\delta_0}\setminus G_{n,\eta_0})\nonumber\\
&\lesssim_{q,p_0,n}\eta_0^{-2n}\delta_0^n+\eps_0^{1-2n}\eta_0^n.\label{est_excep}\end{aligned}$$ Thus, choosing $\eta_0=\delta_0^{1/4},~\delta_0=|\xi|^{-1/4}$ and $\eps_0\ll_{q,p_0}1$ one has that $$\begin{aligned}
|T_0(\xi)|\lq&C_{q,p_0,K}|\varphi_0|_{K,B(0,2)}|\xi|^{-\frac{1}{2}K}+C_{q,p_0,n}|\varphi_0|_{0,B(0,2)}\lb|\xi|^{-\frac{1}{8}n}+|\xi|^{-\frac{1}{16}n}\rb\\
\lq&C_{q,p_0,n,K}\|\varphi_0\|_{0,K}|\xi|^{-\frac{1}{16}K}\end{aligned}$$ for $|\xi|$ sufficiently large and $n>K$. Hence we choose $p_0>8K^2$ s.t. $[\sqrt{2p_0}/4]+1$ is prime and set $n=[\sqrt{2p_0}/4]$.
For each $r\gq1$, let $\omega_r:=(\rho,2^{-r}\alpha,2^{-r}\beta,2^{-r}\gamma,2^{-2r}a,2^{-2r}b)$ and consider the scaled phase function $\Phi_{p_0}(u_0;\omega_r)=2^{-3r}\Phi_{p_0}(2^ru_0;\omega_0)$ for all $u_0\in A'_0,~\omega_0\in\mathbb{S}^{d-1}$. This is again a cubic polynomial with bounded coefficients and so $|\Phi_{p_0}(\cdot;\omega_r)|_{K,A'_0}\lq C_{q,p_0,K}$ for any $K>0$. Scaling each annulus $A'_r$ down to $A'_0\subset B(0,2)$ one has that $$T_r(\xi)=\int_{A_0'}e^{i2^{3r}|\xi|\Phi_{p_0}(u_0;\omega_r)}\chi_r(u_0)\td u_0,$$ where $\chi_r(u_0)=2^{2qp_0r}\varphi_0(2^ru_0)\zeta_1(2u_0)/\sigma(2^ru_0)\in C_0^\infty(A'_0)$. Applying Lemma \[stationary\_phase\] again to this new expression of $T_r$ on $A'_0$ one sees that $\forall\delta_r,K>0$, $$|T_r(\xi)|\lq C_{q,p_0,K}|\chi_r|_{K,A'_0}\delta_r^{-2K}(2^{3r}|\xi|)^{-K}+2|\chi_r|_{0,A'_0}\Lambda^{2qp_0}(\wt\Gamma_{\delta_r}),$$ where $\wt\Gamma_{\delta_r}=\{v_0\in A'_0:|\nabla\Phi_{p_0}(v_0;\omega_r)|\lq\delta_r\}$. Splitting $\wt\Gamma_{\delta_r}$ according to the set $\wt G_{n,\eta_r}:=G_{n,\eta_r}(\nabla\Phi_{p_0}(\cdot;\omega_r))$ one obtains the estimate for the measure of the set $\wt\Gamma_{\delta_r}$ again from Lemma \[factorisation\], \[rho\_big\] and \[rho\_small\]. Note that here instead of unit frequency we have $1>|\omega_r|\gq2^{-2r}$, and so applying Lemma \[rho\_small\] w.r.t. $\omega_r$ one may choose $\delta_r=2^{-2r}\delta_0,\eta_r=2^{-2r}\eta_0,~\eps_r=2^{-2rn}\eps_0$, with the same values of $n$ and $p_0$, so that for $|\xi|$ sufficiently large, $$|\xi|^{K/16}|T_r(\xi)|\lesssim_{q,p_0,n,K}|\chi_r|_{K,A'_0}2^{rK}+|\chi_r|_{0,A'_0}2^{4rn(n-1)}\lesssim_{q,p_0,K}|\chi_r|_{K,A'_0}2^{rp_0/2}.$$ Notice that $|u_0|\gq1/4$ for any $u_0\in A'_0$, and so for any multi-indices $\theta,\tau\in\NN^{2qp_0}$ and $r\gq1$, $$2^{r|\theta|}\sup_{u_0\in A'_0}|\partial^\tau\varphi_0(2^ru_0)|\lq4^{|\theta|}\sup_{u_0\in A'_0}\lv(2^ru_0)^\theta\partial^\tau\varphi_0(2^ru_0)\rv\lq4^{|\theta|}\|\varphi_0\|_{|\theta|,|\tau|}.$$ Therefore for any $k,l\gq0$, differentiating $\chi_r$ up to $k$ times by Leibniz’s rule one sees that $2^{rl}|\chi_r|_{k,A'_0}\lesssim_{q,p_0,k,l}\|\varphi_0\|_{2qp_0+l+k,k}$ from the boundedness of the derivatives of $\zeta_1$ and $\sigma$. This in turn implies that $|T_r(\xi)|\lesssim_{q,p_0,K}2^{-r}|\xi|^{-K/16}\|\varphi_0\|_{\varkappa,K}$, where $\varkappa=2qp_0+p_0/2+K+1$. Summing up in $r$ and one achieves the bound $$|I_{p_0}(\xi)|\lq C_{q,p_0,K}|\xi|^{-K/16}\|\varphi_0\|_{\varkappa,K}.$$ It remains to bound the original integral $I_p(\xi)$ in question for all $p\gq p_0$.
Write $v'_p:=\{(x_{jr},y_{ks}):~j,k=1,\cdots,q;~r,s=p_0+1,\cdots,p\}$, then conditional on $v'_p$ the integral $I_p(\xi)$ can be written as, by the factorisation assumption, $$\begin{aligned}
I_p(\xi)&=\int_{\RR^{2qp'}}\varphi_1(v')\td v'\int_{\RR^{2qp_0}}e^{i|\xi|\Phi_p(v_0;v',\omega_0)}\varphi_0(v_0)\td v_0\nonumber\\
&=:\int_{\RR^{2qp'}}J_p(\xi,v')\varphi_1(v')\td v'.\label{I_p}\end{aligned}$$ If one can show that $|J_p(\xi,v')|$ has a global decay in $|\xi|$ uniformly in $p$ and at most polynomial growth in $v'$, then such a decay should be passed on to $|I_p(\xi)|$ by the finite moments of $\varphi_1$. The idea is that for a fixed value of $v'$ (equivalently, conditional on the random variable $v'_p$) the oscillatory integral $J_p(\xi,v')$ has the same global behaviour in $\xi$ as $I_{p_0}(\xi)$.
Using the same cut-off arguments, it suffices to focus on the case where the amplitude $\varphi_0$ is compactly supported on $B(0,2)\subset\RR^{2qp_0}$. Conditional on $v_p'$ (fixing $v'$), Lemma \[stationary\_phase\] can be readily applied w.r.t. $v_0$ for any $K>0$ and the given choice of $\delta_0$, $$\label{J_bound}
|J_p(\xi,v')|\lq C_{q,p_0,K}|\varphi_0|_{K,B(0,2)}|\Phi_p(\cdot;v')|_{K,B(0,2)}^K\delta_0^{-2K}|\xi|^{-K}+2|\varphi_0|_{0,B(0,2)}\Lambda^{2qp_0}(\Gamma'_{\delta_0}),$$ where $\Gamma'_{\delta_0}=\{v_0\in B(0,2):|\nabla\Phi_p(v_0;v')|\lq\delta_0\}$. To estimate the measure of the exceptional set $\Gamma'_{\delta_0}$, which may depend on $v'$, one divides it by the set $G'_{n,\eta_0}:=G_{n,\eta_0}(\nabla\Phi_p(\cdot;v'))$ just like before. The key is then to recognise the $v_0$-derivatives of $\Phi_p(v;\omega_0)$.
Separating the monomials that involve $v_0$ in the phase function $\Phi_p(v;\omega_0)$, we write $$\label{phase_p}
\Phi_p(v;\omega_0)=\Phi_{p_0}(v_0;\omega_0)+\Theta_p(v_0;v',\gamma,\rho)+\Upsilon_p(v';\omega_0),$$ where the second term is given by, omitting the summation signs over the repeated indices $(j,k)$ and $(j,k,l)$ like before, $$\begin{aligned}
\Theta_p(v_0;v',\gamma,\rho)=&\gamma_{jk}\lb\sum_{\substack{r\lq p_0\\p_0<s\lq p}}+\sum_{\substack{s\lq p_0\\p_0<r\lq p}}\rb\frac{1}{r^2-s^2}\lb\frac{r}{s}x_{jr}x_{ks}-y_{jr}y_{ks}\rb\\
&+\rho_{jkl}\lb\sum_{\substack{r\lq p_0\\p_0<s\lq p-r}}+\sum_{\substack{s\lq p_0\\p_0<r\lq p-s}}+\sum_{\substack{r\lq p_0\\p_0-r<s\lq p_0}}\rb\\
&\qquad\quad\left\{-\frac{1}{r(r+s)}\left[(x_{jr}y_{ks}+y_{jr}x_{ks})x_{l,r+s}+(-x_{jr}x_{ks}+y_{jr}y_{ks})y_{l,r+s}\right]\right.\\
&\qquad\qquad+\frac{1}{rs}\left[(x_{jr}y_{ls}+y_{jr}x_{ls})x_{k,r+s}+(-x_{jr}x_{ls}+y_{jr}y_{ls})y_{k,r+s}\right]\\
&\qquad\qquad\left.+\frac{1}{s(r+s)}\left[(-x_{kr}y_{ls}+y_{kr}x_{ls})x_{j,r+s}+(x_{kr}x_{ls}+y_{kr}y_{ls})y_{j,r+s}\right]\right\},\end{aligned}$$ and $\Upsilon_p(v';\omega_0)$ is the sum of remaining monomials that do not involve $v_0$. Then it is clear that the function $\Phi_p(v;\omega_0)$ has the same derivatives in $v_0$ as the function $$\Psi_p(v_0;v',\omega_0):=\Phi_{p_0}(v_0;\omega_0)+\Theta_p(v_0;v',\gamma,\rho).$$ An important observation is that the polynomial $\Theta_p(v_0;v',\gamma,\rho)$ is *at most quadratic* in $v_0$. The summations over the indices $(r,s)$ are plotted in the shaded areas in Figure \[shade1\] and \[shade2\], corresponding to $\nu^{(p)}-\nu^{(p_0)}$ and $\Delta^{(p)}-\Delta^{(p_0)}$, respectively.
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(0,1)–(1,1)–(1,2.5)–(0,2.5); (1,0)–(2.5,0)–(2.5,1)–(1,1); (0,0)–(3,0); (0,0)–(0,3); (0,0)–(3,3); (0,1)–(3,1); (1,0)–(1,3); (0,2.5)–(3,2.5); (2.5,0)–(2.5,3); (3,0) node\[anchor=west\][$r$]{}; (0,3) node\[anchor=south\][$s$]{}; (1,0) node\[anchor=north\][$p_0$]{}; (0,1) node\[anchor=east\][$p_0$]{}; (2.5,0) node\[anchor=north\][$p$]{}; (0,2.5) node\[anchor=east\][$p$]{};
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(0,1)–(0,2.5)–(1,1.5)–(1,1); (1,0)–(2.5,0)–(1.5,1)–(1,1); (1,0)–(1,1)–(0,1); (0,0)–(3,0); (0,0)–(0,3); (0,1)–(3,1); (1,0)–(1,3); (0,1)–(1,0); (0,2.5)–(2.5,0); (3,0) node\[anchor=west\][$r$]{}; (0,3) node\[anchor=south\][$s$]{}; (1,0) node\[anchor=north\][$p_0$]{}; (0,1) node\[anchor=east\][$p_0$]{}; (2.5,0) node\[anchor=north\][$p$]{}; (0,2.5) node\[anchor=east\][$p$]{};
Considering the variable $v_0$ only, the Lipschitz constant of the linear function $\tD^2\Psi_p(v_0;v')$ is identical to that of $\tD^2\Phi_{p_0}(v_0)$ since they only differ by a *constant* matrix $\tD^2\Theta_p(v_0;v',\gamma,\rho)$. Hence Lemma \[factorisation\] applies directly and gives $\Lambda^{2qp_0}(\Gamma'_{\delta_0}\cap G'_{n,\eta_0})\lq C_{q,p_0}\eta_0^{-2n}\delta_0^n$ uniformly in $p$. This uniformity also holds when applying Lemma \[rho\_big\]: the difference here is that, in its proof, a constant vector $\theta_a(v')$ from the matrix $\tD^2\Theta_p$ is added to each row $q_a(y)$ of the submatrix $Q_m(y)$ therein, and the sets $F_a$ in are replaced by $$F'_a=\{(x,y):~\dist(q_a(y)+\theta_a(v'),~\spa\{q_b(y)+\theta_b(v'):~a\neq b=1,\cdots,n\})>\sqrt{n}\eta_0\}.$$ Then under the image of the same matrix $U_a$, each $F'_a$ is just a translated copy of $F_a$, whose Lebesgue measure remains unchanged. Therefore $\Lambda^{2qp_0}(\Gamma'_{\delta_0}\setminus G'_{n,\eta_0})\lq C_{q,p_0,n}(\eps'_0)^{1-2n}\eta_0^n$ for $|\rho|>\eps'_0$ for any $\eps'_0\in(0,1)$ by Lemma \[rho\_big\].
However, in order to handle the case where $|\rho|\lq\eps'_0$ the choice of $\eps'_0$ will depend on $v'$, because previously the choice of $\eps_0$ was obtained by studying the specific form of the derivatives of the phase function $\Phi_{p_0}(v_0)$ in Lemma \[rho\_small\], and now the new phase function $\Psi_p(v_0;v',\omega_0)$ does contain the extra parameter $v'$ in its $v_0$-derivatives. Thankfully, only a slight change (introducing the parameter $v'$) of the proof is needed for Lemma \[rho\_small\] to hold for $\Psi_p$.
For the case where the coefficients $(a,b)$ are dominant, since the first $v_0$-derivatives of $\Theta_p$ are only linear (in $v_0$), the $v_0$-quadratic part of $\nabla_{v_0}\Psi_p$ is the same as that of $\nabla\Phi_{p_0}(v_0;\omega_0)$, and so the estimate still holds for $\eps=\eps_0$. For a fixed value of $v'$, the $v_0$-linear part of $\nabla\Psi_p(v_0;v',\omega_0)$ equals that of $\nabla\Phi_{p_0}(v_0;\omega_0)$ plus $\nabla\Theta_p(v_0;v',\gamma,\rho)$. One then sees that, instead of , we have a bound $C_{q,p_0}(1+\theta'\sup_{v_0\in B(0,2)}|\nabla_{v_0}\Theta_p|)$ for the linear part, and we choose $\theta'=\theta/(1+\sup_{v_0\in B(0,2)}|\nabla_{v_0}\Theta_p|)$ with the original choice of $\theta$ in the proof.
In the other case where the coefficients $(\alpha,\beta,\gamma)$ are dominant, the constant matrix $A_{p_0}$ in is perturbed by the ‘constant’ matrix $\tD^2\Theta_p(v_0;v',\gamma,\rho)$. Instead of looking for a non-singular $n\times n$ submatrix $H_n(v_0;\omega_0)$ of $\tD^2\Phi_{p_0}(v_0;\omega_0)$ one looks for such a submatrix $H_n(v_0;v',\omega_0)$ of $\tD^2_{v_0}\Psi_p(v_0,v';\omega_0)$. From the expression $$H_n(v_0;v',\omega_0)=A_{p_0}+\tD^2\Theta_p(v_0;v',\rho)+L_n(v_0;\rho)$$ with the linear part $L_n$ still satisfying , we arrive at the choice $\eps'_0=\eps_0\theta'/(1+|\tD^2_{v_0}\Theta_p|)\lq\eps_0/(1+|\Theta_p(\cdot;v',\gamma,\rho)|^2_{2,B(0,2)})$, as the rest of the arguments in the proof of Lemma \[rho\_small\] remain the same.
Therefore, recalling the choices for $\eps_0,\delta_0,\eta_0$ altogether we have that, for $|\xi|$ sufficiently large, $$\begin{aligned}
\Lambda^{2qp_0}(\Gamma'_{\delta'})\lesssim_{q,p_0}&\eta_0^{-2n}\delta_0^n+\eps_0^{1-2n}(1+|\Theta_p(\cdot;v',\gamma,\rho)|^2_{2,B(0,2)})^{2n-1}\eta_0^n\\
\lesssim_{q,p_0,n}&|\xi|^{-n/16}\lb1+|\Theta_p(\cdot;v',\gamma,\rho)|^{4n-2}_{2,B(0,2)}\rb.\end{aligned}$$ Returning to , with the same values of $n=[\sqrt{2p_0}/4]>K$ and $p_0$ as before we have that, for any $p\gq p_0$ and $\xi$ sufficiently large, $$|J_p(\xi,v')|\lesssim_{q,p_0,K}\|\varphi_0\|_{0,K}\lb1+|\Phi_p(\cdot;v',\omega_0)|_{K,B(0,2)}^K+|\Theta_p(\cdot;v',\gamma,\rho)|^{\sqrt{2p_0}-2}_{2,B(0,2)}\rb|\xi|^{-K/16}.$$ Now recalling the separation of the $v_0$-monomials in $\Phi_p$ and the fact that the latter is a cubic polynomial, this bound is reduced to $|J_p(\xi,v')|\lesssim_{q,p_0,K}\|\varphi_0\|_{0,K}(1+\sum_\beta|P_\beta(v')|^{\sqrt{2p_0}-2})$. Thus, by the same scaling argument for a general Schwartz function $\varphi_0$ we have the above result with $\|\varphi_0\|_{0,K}$ replaced by the norm $\|\varphi_0\|_{\varkappa,K}$, and the desired bound for $|I_p(\xi)|$ follows from the relation .
The function $I_p(\xi)$ is obviously smooth. Since $|\xi|\Phi_p(v;\omega_0)=\xi\cdot V_p(v)$, with a slight abuse of notation $V_p=V_p(v_p)$, we have, for any multi-index $\alpha\in\NN^d,~|\alpha|=k$, that $$\partial^\alpha I_p(\xi)=i^k\int_{\RR^{2qp}}e^{i|\xi|\Phi_p(v;\omega_0)}V_p^\alpha(v)\varphi(v)\td v.$$ Similar to the equality separate the $v_0$-monomials in $V_p^\alpha(v)$, and one sees that the derivative $\partial^\alpha I_p(\xi)$ is a sum of oscillatory integrals of the same type as $I_p(\xi)$ itself (with Schwarz amplitudes). The result then follows from the observation that the number of terms in depends only on $q,p_0,k$ and that $$\max_{|\theta|\lq j,|\sigma|\lq l,|\tau|\lq m}\sup_{v_0\in\RR^{2qp_0}}|v_0^{\theta}\partial^\tau(v_0^{\sigma}\varphi_0(v_0))|\lq\|\varphi_0\|_{j+l,m},$$ for any $j,l,m\in\NN$ and multi-indices $\theta,\sigma,\tau\in\NN^{2qp_0}$.
As we will encounter standard Fourier integrals as well later on in the next section, perhaps it is a good time now to remark that much less restriction on the amplitude is needed to show the rapid decay for such special cases.
\[fourier\_bound\] For arbitrary $K\in\NN$ and $\epsilon>0$, let $h:\RR^d\times\RR^d\to\RR$ be sufficiently smooth w.r.t the first variable s.t. $\sup_z\|h(\cdot,z)\|_{d+\epsilon,K}<\infty$. Then it holds that $\forall z\in\RR^d$, $$\lv\int_{\RR^d}e^{\pm iz\cdot u}h(u,z)\td u\rv\lq C_{d,\epsilon}\sup_{z\in\RR^d}\|h(\cdot,z)\|_{d+\epsilon,K}|z|^{-K}.$$
By assumption $h$ vanishes as $|u|\to\infty$. Then, for each fixed $z$ and any $\alpha\in\NN^d$ s.t. $|\alpha|=K$, by integration by parts we have that $$|z|^K\lv\int_{\RR^d}e^{\pm iz\cdot u}h(u;z)\td u\rv=\lv\int_{\RR^d}e^{\pm iz\cdot u}\partial_u^\alpha h(u,z)\td u\rv\lq\int_{\RR^d}|\partial_u^\alpha h(u,z)|\td u.$$ Rewrite the right-hand-side integral as a sum of integrals over the unit ball and the annuli $\{u:2^r\lq|u|<2^{r+1}\}_{r\gq0}$, and the result follows.
Returning to Theorem \[global\_estimate\], as a special case the characteristic function $\psi_p(\xi)$ of $V_p$ has Gaussian amplitude $\varphi(v)=\phi_p(v)$, which can be factorised as the product of $\varphi_0(v_0)=\phi_{p_0}(v_0)$ and $\varphi_1(v')=\phi_{p'}(v'):=\phi_p(v)/\phi_{p_0}(v_0)=\prod_{j=1}^q\prod_{r=p_0+1}^p\phi(x_{jr})\phi(y_{jr})$. As the reader will see later, in this case the integral in against $\varphi_1$ is not only finite but *independent of* $p$ as well. This will follow from the moment estimates in the next section.
Moment Estimates and Density Decay
==================================
Recall the notations $d=2q^2+2q+(q^3-q)/3$ and $v_p=\{(x_{jr},y_{ks}):j,k=1,\cdots,q,r,s=1,\cdots,p\}\in\RR^{2qp}$. It is not quite clear yet how the method described in the introduction for the double integral can be applied to the triple integral case. In fact, the expression no longer holds as $g$ is no longer the convolution of $f_p$ and the law of $\wt V_p$ - the latter is not independent of $V_p$. Instead, let $\kappa_y,\chi_y$ be the densities of $\wt V_p$ and $\bar V_p$ conditional on that $V_p=y$, respectively. Then one has that $$g(z)=\int_{\RR^d}f_p(z-w)\kappa_{z-w}(w)\td w,~h(z)=\int_{\RR^d}f_p(z-w)\chi_{z-w}(w)\td w,$$ and by , for all $z\in\RR^d$ one arrives at $$\begin{aligned}
|g(z)-h(z)|\lq&C_{d,m}\sum_{|\beta|=0}^{m-1}\lv\int_{\RR^d}\lb w^\beta\kappa_{z-w}(w)-w^\beta\chi_{z-w}(w)\rb\td w\rv\\
&+C_{d,m}\sum_{|\beta|=m}\int_{\RR^d}\lv w^\beta\kappa_{z-w}(w)-w^\beta\chi_{z-w}(w)\rv\td w.\end{aligned}$$ One then sees the complication of estimating the integrands above, compared to the proof of Theorem 15 in [@davie2014Patappstodifequusicou]: in the double integral case, due to the independence between $U_p$ and $\wt U_p$ the first integral above will just be $\ex\wt U_p^\beta-\ex\bar{U}_p^\beta$, which vanishes by assumption, and the rest is of order $O(p^{-m/2})$ by Lemma \[moments\] below (or Lemma 11 in [@davie2014Patappstodifequusicou]). However, here $\int_{\RR^d}w^\beta\kappa_{z-w}\td w$ is not even the conditional moment of $\wt V_p$ due to the appearance of $w$ in the subscript of $\kappa$. One may apply Taylor’s theorem about $z$ in the subscript and impose certain smoothness condition on $\chi_a$, but whether $\kappa_a$ is smooth in $a$ is not clear.
Instead of this approach we follow a somewhat more primitive way of deriving the coupling bound via the Fourier inversion formula, for which we need much detailed moment estimates for $\wt V_p$.
\[moments\] For fixed $p_0\in\ZZ^+$ and any $p,N\in \ZZ^+,~N>p\gq2p_0$, define the notation $v'_N:=\{(x_{jr},y_{jr}):j=1,\cdots,q;r=p_0+1,\cdots,N\}$ and let $\wt V_{p,N}=V_N-V_p=\wt V_p-\wt V_N$. Then for any $2\lq m\in\ZZ^+$ and $\alpha\in\NN^d$ s.t. $|\alpha|=m$, the following hold:
(i)
: We have that $\wt V_{p,N}^\alpha=\sum_\beta v_{p_0}^\beta\wt P_\beta(v'_N)$, where each $\wt P_\beta$, depending on $p$ and $\alpha$, is a polynomial of degree at most $3m-|\beta|$ and the summation depends on $q,p_0,\alpha$ with $|\beta|\lq m$;
(ii)
: For each $\beta$ we have that $\ex|\wt P_\beta(v'_N)|\lesssim_{q,\alpha}p^{-m/2}$ uniformly in $N$.
One can write $\alpha=(\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5,\alpha_6)$ s.t. $\sum_{i=1}^6|\alpha_i|=m$ and $$\label{power_split}
\wt V_{p,N}^\alpha=\lb\wt z^{(p,N)}\rb^{\alpha_1}\lb\wt u^{(p,N)}\rb^{\alpha_2}\lb\wt\lambda^{(p,N)}\rb^{\alpha_3} \lb\wt\mu^{(p,N)}\rb^{\alpha_4}\lb\wt\nu^{(p,N)}\rb^{\alpha_5} \lb\wt\Delta^{(p,N)}\rb^{\alpha_6},$$ where the terms on the right-hand side are similarly defined as the components of $\wt V_p$ and each multi-index $\alpha_i$ is of corresponding dimension. It is then easier to work with powers of each component.
Clearly the contribution of $v_{p_0}$ comes from $\wt\nu^{(p,N)}$ and $\wt\Delta^{(p,N)}$ only, as the rest are all independent of $v_p$ (and thus of $v_{p_0}$). For each admissible $j,k,l$, the truncated sum $\wt\nu_{jk}^{(p,N)}$ of $\wt\nu_{jk}^{(p)}$ over $p<r\vee s\lq N,~r\neq s$ is at most linear in $v_{p_0}$, and the truncated sum $\wt\Delta_{jkl}^{(p,N)}$ of $\wt\Delta_{jkl}^{(p)}$ over $p<r+s\lq N$ is also at most linear in $v_{p_0}$ as the assumption $p\gq2p_0$ implies that at least one of $r$ and $s$ must be greater than $p_0$. Thus by multiplying out the powers in and re-grouping the monomials involving $v_{p_0}$ one see that the power $\wt V_{p,N}^\alpha$ has degree at most $|\alpha_5|+|\alpha_6|$ in $v_{p_0}$, and the representation (i) follows.
To give a bound for each $\ex|\wt P_\beta(v'_N)|=\ex(|\wt P_\beta(v'_N)||v_{p_0})$, observe that, as per , each $\wt P_\beta(v'_N)$ is a mixed product of the random variables $\wt z^{(p,N)},\wt u^{(p,N)},\wt\lambda^{(p,N)},\wt\mu^{(p,N)},\wt\nu^{(p,N)},\wt\Delta^{(p,N)}$. Separating them by Young’s inequality, it suffices to bound the $m$-th moment of each component of $\wt V_{p,N}$. In particular, it suffices to bound the $m$-th conditional (on $v_{p_0}$) moments of $\wt\nu^{(p,N)}$ and $\wt\Delta^{(p,N)}$, since one can then evaluate $v_{p_0}=(1,\cdots,1)$, for example.
It is easy to estimate the moments of the random variables $\wt z^{(p,N)},\wt u^{(p,N)},\wt\lambda^{(p,N)},\wt\mu^{(p,N)}$, since they are all sums of independent random variables. For $\wt u^{(p,N)}=\wt u^{(p)}-\wt u^{(N)}$, each component $\wt u_j^{(p,N)}$ follows $\N(0,\sum_{r=p+1}^Nr^{-4})$ and one sees that $\ex|\wt u^{(p,N)}|^m\lq C_{q,m}p^{-3m/2}$. For $\wt\mu^{(p,N)}$, consider $\wt\mu_{jk}^{(1,p,N)}:=\wt\mu_{jk}^{(1,p)}-\wt\mu_{jk}^{(1,N)}$ for instance: by Rosenthal’s inequality (see Theorem 3 in [@rosenthal1970SubLppSpabySeqIndRanVar] or Theorem 2.1 in [@deacosta1981IneB-vRanVecApptoStrLawLarNum]), for any $N>p$, $$\begin{aligned}
\ex\lv\sum_{r=p+1}^N\frac{1}{r^2}x_{jr}x_{kr}\rv^m\lesssim_m&\sum_{r=p+1}^N\frac{1}{r^{2m}}\ex|x_{jr}|^m\ex|x_{kr}|^m+\lb\sum_{r=p+1}^N\frac{1}{r^4}\ex|x_{jr}|^2\ex|x_{kr}|^2\rb^{m/2}\\
\lesssim_m&(p+1)^{1-2m}+(p+1)^{-3m/2}.\end{aligned}$$ Obviously the same bound holds for the $m$-th moment of $\wt{\mu}_{jk}^{(2,p,N)}$, too. One also sees a bound $C_{q,m}p^{-m/2}$ for the $m$-th moments of $\wt z^{(p,N)}$ and $\wt\lambda^{(p,N)}$ for the same reason, but this is in fact implied by part (2) of Lemma 11 in [@davie2014Patappstodifequusicou] where a stronger estimate is given.
It is much less straightforward to compute the conditional moments of $\wt\nu^{(p,N)}$ and $\wt\Delta^{(p,N)}$ as they are not sums of independent random variables. For the rest of the proof use the shorthand notation $\ex_{p_0}:=\ex(\cdot|v_{p_0})$. For each pair $(j,k)$ write $\wt\nu_{jk}^{(p,N)}=A_{jk}-B_{jk}$ where $A_{jk}$ is the corresponding sum of $(r^2-s^2)^{-1}rs^{-1}x_{jr}x_{ks}$ and $B_{jk}$ of $(r^2-s^2)^{-1}y_{jr}y_{ks}$. One can further write (see Figure \[split1\] below) $$A_{jk}=\lb\sum_{\substack{s\lq p_0\\p<r\lq N}}+\sum_{\substack{r\lq p_0\\p<s\lq N}}+\sum_{\substack{p_0<s<r\\p<r\lq N}}+\sum_{\substack{p_0<r<s\\p<s\lq N}}\rb\frac{1}{r^2-s^2}\frac{r}{s}x_{jr}x_{ks}=:T_1-T_2+T_3-T_4,$$ and $B_{jk}$ can be similarly split into four smaller sums. Hence it suffices to bound the $m$-th conditional moment of each of those smaller sums. Moreover, it suffices to consider the case where $m$ is even, as the odd moments can be derived from the Cauchy-Schwartz inequality.
Multiplying out the power one obtains that $$T_1^m=\sum_{\substack{s.\lq p_0\\p<r.\lq N}}\frac{1}{(r_1^2-s_1^2)\cdots(r_m^2-s_m^2)}\frac{r_1\cdots r_m}{s_1\cdots s_m}x_{jr_1}\cdots x_{jr_m}x_{ks_1}\cdots x_{ks_m}$$ and similar expressions for $T_2^m,T_3^m$ and $T_4^m$, where the summations are in $s_\alpha,r_\alpha$ accordingly for all $\alpha=1,\cdots,m$. Note that the random variables $x_{jr_1},\cdots,x_{jr_m}$ are independent of $x_{ks_1},\cdots,x_{ks_m}$ as $j<k$. For the conditional expectation not to vanish, the indices $r_1,\cdots,r_m$ must match in pairs; meanwhile since $p\gq2p_0$, one has that $r_\alpha>2s_\alpha$ and $r_\alpha/(r_\alpha-s_\alpha)\lq2$ for each $\alpha$. Thus $$\ex_{p_0}T_1^m\lesssim_m\lb\sum_{p<r\lq N}\frac{1}{r^2}\rb^{m/2}\lb\sum_{s\lq p_0}\frac{1}{s}x_{ks}\rb^m,$$ and by symmetry $\ex_{p_0}T_2^m$ has the same bound with $k$ replaced by $j$.
As for $T_3$ the indices $s_1,\cdots,s_m$ must also match in pairs. If $r_\alpha>2s_\alpha$ then one immediately obtains a bound $C_mp^{-m/2}$; if $r_\alpha\lq2s_\alpha$, then the corresponding expected sum is bounded by (up to the number of matchings) $$\lb\sum_{r>p}\sum_{s<r}\frac{1}{(r-s)^2s^2}\rb^{m/2}\lesssim_m\lb\sum_{r>p}\frac{1}{r^2}\sum_{s<r/2}\frac{1}{s^2}\rb^{m/2}\lesssim_mp^{-m/2}.$$ Thus $\ex_{p_0}T_3^m\lq C_mp^{m/2}$, and the same holds for $\ex_{p_0}T_4^m$ by symmetry. So altogether we have that $\ex_{p_0}|A_{jk}|^m\lq C_m(1+(\sum_{r\lq p_0}r^{-1}|x_{jr}|)^m+(\sum_{r\lq p_0}r^{-1}|x_{kr}|)^m)p^{-m/2}$, and it is easy to see that the same bound with $x$ replaced by $y$ holds for $\ex_{p_0}|B_{jk}|^m$.
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(0,0)–(2,0)–(2,2)–(0,2); (0,0)–(4,0); (0,0)–(0,4); (0,1)–(4,1); (1,0)–(1,4); (0,2)–(2,2); (2,0)–(2,2); (2,2)–(4,4); (4,0) node\[anchor=west\][$r$]{}; (0,4) node\[anchor=south\][$s$]{}; (1,0) node\[anchor=north\][$p_0$]{}; (0,1) node\[anchor=east\][$p_0$]{}; (2,0) node\[anchor=north\][$p$]{}; (0,2) node\[anchor=east\][$p$]{}; (3,0.5) node[$T_1$]{}; (0.5,3) node[$T_2$]{}; (3,2) node[$T_3$]{}; (2,3) node[$T_4$]{};
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(0,0)–(2,0)–(0,2); (0,0)–(4,0); (0,0)–(0,4); (0,0.5)–(4,0.5); (0.5,0)–(0.5,4); (0,2)–(2,2); (2,0)–(2,4); (2,0)–(0,2); (4,0) node\[anchor=west\][$r$]{}; (0,4) node\[anchor=south\][$s$]{}; (0.5,0) node\[anchor=north\][$p_0$]{}; (0,0.5) node\[anchor=east\][$p_0$]{}; (2,0) node\[anchor=north\][$p$]{}; (0,2) node\[anchor=east\][$p$]{}; (1.5,1.5) node[$S_1$]{}; (1,3) node[$S_2$]{}; (3,2) node[$S_3$]{};
It remains to estimate the conditional moments of $\wt\Delta_{jkl}^{(p,N)}$ for each Lyndon word $(j,k,l)$. Take, for instance, the sum $$\begin{aligned}
\Sigma_{jkl}:=\sum_{p<r+s\lq N}\frac{1}{rs}x_{jr}y_{ls}x_{k,r+s},\end{aligned}$$ for a certain Lyndon word $(j,k,l)$. Write $\Sigma_{jkl}=S_1+S_2+S_3$ where, as is illustrated by Figure \[split2\], $S_1$ is the sum over $p-s<r\lq p,s\lq p$, $S_2$ is the sum over $p<s\lq N-r,~r\lq p$, and $S_3$ is the sum over $p<r\lq N-s,~s<N-p$. Further write $t=r+s$ for simplicity. Then the $m$-th power of $\Sigma_{jkl}$ can be expressed as $$\Sigma_{jkl}^m=\sum_{s.\lq N}\frac{1}{s_1\cdots s_m}y_{ls_1}\cdots y_{ls_m}\lb\sum_{\substack{r.\lq N\\p<t.\lq N}}\frac{1}{r_1\cdots r_m}x_{jr_1}\cdots x_{jr_m}x_{kt_1}\cdots x_{kt_m}\rb,$$ and $S_1,S_2$ and $S_3$ can also be written in this form. We also write $r.$ as the $m$-tuple $(r_1,\cdots,r_m)$ and $s.,t.$ likewise. Denote by $\Pi_m$ the set of all pair-matching patterns for an $m$-tuple.
Notice that in $S_1$ the random variables $x_{jr_1},\cdots,x_{jr_m}$ are all independent of $x_{kt_1},\cdots,x_{kt_m}$. For the conditional expectation not to vanish, the indices $t.$ must match in pairs. Thus $$\ex_{p_0}S_1^m=\sum_{s.\lq p}\frac{1}{s_1\cdots s_m}\ex_{p_0}y_{ls_1}\cdots y_{ls_m}\lb\sum_{t.\in \Pi_m}C_{t.}\sum_{r.}\frac{1}{r_1\cdots r_m}\ex_{p_0}x_{jr_1}\cdots x_{jr_m}\rb,$$ where the last summation is over $p-s<r_\alpha\lq p$ for all $\alpha=1,\cdots,m$ subject to a fixed pair-matching pattern of $t.$, and the constant $C_{t.}$ is the product of the corresponding even moments of $x_{kt_\alpha}$. Note that the indices $r.$ and $s.$ cannot be both less than or equal to $p_0$ as $p\gq2p_0$, so they must match in pairs, respectively, too. Hence $\ex_{p_0}S_1^m$ is a polynomial in $x_{j1},\cdots,x_{jp_0},y_{l1},\cdots,y_{lp_0}$ of degree $m$. Distributing out the summation above and using the restriction $p-p_0\gq p/2$ one sees that $$\begin{aligned}
\ex_{p_0}S_1^m\lesssim_m&\lb\sum_{\substack{p-s<r\lq p\\s\lq p_0}}\frac{1}{s^2r^2}y_{ls}^2\rb^{m/2}+\lb\sum_{\substack{p-s<r\lq p_0\\p-p_0<s\lq p}}\frac{1}{s^2r^2}x_{jr}^2\rb^{m/2}+\lb\sum_{\substack{(p-s)\vee p_0<r\lq p\\p_0<s\lq p}}\frac{1}{s^2r^2}\rb^{m/2}\\
\lesssim_m&\lb\frac{1}{p}\sum_{s\lq p_0}\frac{1}{s^2}(x_{js}^2+y_{ls}^2)\rb^{m/2}+\lb\frac{1}{p}\sum_{s\lq p}\frac{1}{s(p-s+1)}\rb^{m/2}.\end{aligned}$$ The last summation is bounded by $2\sum_{s\lq p/2}s^{-2}$, and therefore $\ex_{p_0}S_1^m\lq C_m(1+|v_{p_0}|^m)p^{-m/2}$.
For $S_2$, the random variables $x_{jr.}$ and $x_{kt.}$ are still independent as $r.\lq p$. In addition to $t.$, under conditional expectation the indices $s.$ must match in pairs. This means that the indices $r.$ must also match in pairs, and hence $$\ex_{p_0}S_2^m=\lb\sum_{p<s.\lq N}\frac{C_{s.}}{s_1^2\cdots s_{m/2}^2}\rb\lb\sum_{r.\lq p_0}\frac{C_{t.}}{r_1^2\cdots r_{m/2}^2}x_{jr_1}^2\cdots x_{jr_{m/2}}^2+\sum_{p<r.\lq N}\frac{C_{t.,r.}}{r_1^2\cdots r_{m/2}^2}\rb,$$ where the constants reflect the number of pair-matching patterns for $s.,t.$ and $r.$ and the even moments of $y_{ls_.},x_{kt.}$ and $x_{jr.}$. This is a polynomial in $x_{j1},\cdots,x_{jp_0}$ of degree $m$, and it has the same bound as $\ex_{p_0}S_1^m$ uniformly in $N$.
In $S_3$ some index $r_\alpha$ may match some other $t_\beta$, thus the we need to consider some more specific cases. Recall that there are only two types of Lyndon words for $3$-tuple $(j,k,l)$. For the case where $j<k\wedge l$, we still have the independence between the random variables $x_{jr_1},\cdots,x_{jr_m}$ and $x_{kt_1},\cdots,x_{kt_m}$. Thus, after taking conditional expectation, only those with pair-matching indices $r.$ and $t.$, respectively, remain non-vanishing. The indices $s.$ must also match in pairs, and therefore similar to $S_2$ one has that $$\ex_{p_0}S_3^m=\lb\sum_{p<r.\lq N}\frac{C_{r.,t.}}{r_1^2\cdots r_{m/2}^2}\rb\lb\sum_{s.\lq p_0}\frac{1}{s_1^2\cdots s_{m/2}^2}y_{ls_1}^2\cdots y_{ls_{m/2}}^2+\sum_{p_0<s.\lq N}\frac{C_{s.}}{s_1^2\cdots s_{m/2}^2}\rb,$$ which again leads to the same bound.
It is more intricate to deal with the case where $j=k<l$, as the independence between $x_{jr_1},\cdots,x_{jr_m}$ and $x_{jt_1},\cdots,x_{jt_m}$ is no longer true. For the conditional expectation not to vanish, the $2m$-tuple $\tau:=(r_1,\cdots,r_m,t_1,\cdots t_m)$ must match in pairs. So it suffices to consider the following terms in $S_3$: $$\label{S_3}
\sum_{s.\lq N}\frac{1}{s_1\cdots s_m}y_{ls_1}\cdots y_{ls_m}\lb\sum_{\tau\in \Pi_{2m}}\sum_{r.}\frac{1}{r_1\cdots r_m}x_{jr_1}\cdots x_{jr_m}x_{jt_1}\cdots x_{jt_m}\rb,$$ where the last summation is over $p<r.\lq N$ subject to a pair-matching pattern of $\tau$. For fixed values of $s_1,\cdots,s_m$ and a pattern $\tau$, define $\alpha$ and $\beta$ as equivalent on $\{1,\cdots,m\}$ if $r_\alpha$ or $t_\alpha$ is equal to $r_\beta$ or $t_\beta$. Consider the equivalence relation generated by this relation. If $\alpha$ and $\beta$ are in an equivalent class $E\subset\{1,\cdots,m\}$, then the difference $r_\alpha-r_\beta$ is determined by the fixed choice of $s_\alpha,s_\beta$ and the matching constraint of $r_\alpha,t_\alpha,r_\beta,t_\beta$. In effect, for any $\alpha\in E$ the value of $r_\alpha$ determines the values of $r_\beta$ for all $\beta\in E$. Thus, one can choose $r_{\alpha_\ast}=\min\{r_\alpha:\alpha\in E\}$ and rewrite the last summation above as $$\sum_{r.}\frac{1}{r_1\cdots r_m}x_{jr_1}\cdots x_{jr_m}x_{kt_1}\cdots x_{kt_m}=\prod_E\sum_{p<r_{\alpha_\ast}\lq N}\frac{1}{r_{\alpha_1}\cdots r_{\alpha_{|E|}}}x_{jr_{\alpha_1}}\cdots x_{jr_{\alpha_{|E|}}}x_{jt_{\alpha_1}}\cdots x_{jt_{\alpha_{|E|}}}$$ where the product is taken over the equivalent class partition of the set $\{1,\cdots,m\}$. Then the expectation of the terms in the big parentheses in is bounded by $$C_m\prod_E\sum_{r_{\alpha_\ast}>p}r_{\alpha_\ast}^{-|E|}\lq C_m\prod_Ep^{1-|E|}=C_mp^{n_m-m},$$ where $n_m$ is the number of equivalent classes, which is at most $m/2$, giving an upper bound $C_mp^{-m/2}$ for the quantity above. Thus $\ex_{p_0}S_3^m$ is again a polynomial in $y_{l1},\cdots,y_{lp_0}$ of degree $m$, and one has that $$\ex_{p_0}S_3^m\lesssim_mp^{-m/2}\lb\sum_{s\lq p_0}\frac{1}{s}y_{ls}\rb^m+p^{-m/2}\lb\sum_{p_0<s\lq N}\frac{1}{s^2}\rb^{m/2},$$ which again gives the same bound as previous cases.
It then follows from the triangle inequality that $\ex_{p_0}\Sigma_{jkl}^m\lq C_mp^{-m/2}(1+|v_{p_0}|_\ast^m)$. The same arguments apply to all other terms in $\wt \Delta^{(p)}$ (as for the terms $y_{jr}y_{ls}y_{k,r+s}$, the indices $j,k,l$ are never all the same), and the result is proved.
\[moment2\] From the proof one sees that $\ex(|\wt V_p|^m|v_{p_0})\lesssim_{q,m}(1+|v_{p_0}|_\ast^m)p^{-m/2}$, where $|v_{p_0}|_\ast:=\sum_{j=1}^q\sum_{r\lq p_0}r^{-1}(|x_{jr}|+|y_{jr}|)$. Taking expectation again one gets $\ex|\wt V_p|^m\lq C_{q,m}p^{-m/2}$, and by setting $p_0=1,~p=2$ one obtains from the triangle inequality (obviously $V_2$ has bounded moments) that $\sup_{p\gq1}\ex|V_p|^m\lq C_{q,m}$.
Returning to the applicaiton of Theorem \[global\_estimate\] to the characteristic function $\psi_p(\xi)$ of $V_p$, observe that the phase function can be written as $\Phi_p(v;\omega_0)=\omega_0\cdot V_p$, and so the polynomials $P_\beta$ in correspond to the $\wt P_\beta$ in (i) of Lemma \[moments\] with $|\alpha|=1,N=p$. Then in this case Lemma \[moments\] (ii) implies that the integral against $\varphi_1=\phi_{p'}$ in is bounded for all $p\gq2p_0$, and thereby Theorem \[global\_estimate\] immediately implies:
\[density\] The density $f_p$ of $V_p$ has continuous and uniformly bounded derivatives up to order $N$ if $p\gq2p_0$, where $p_0>2048(N+d)^2$ is an even integer s.t. $[\sqrt{2p_0}/4]+1$ is prime.
This is an analogue of part (1) of Lemma 11 in [@davie2014Patappstodifequusicou]; it is not clear whether part (2) of that lemma holds in the triple integral case. But at least we can conclude that the density $f_p$ converges uniformly in $p$ and we have the following:
The random variable $V$ has a density with continuous and bounded derivatives up to any order.
By the mean value theorem for $p>p_0$ all the derivatives of $f_p$ up to order $N$ are Lipschitz. Combining this fact with the last comment in Remark \[moment2\] one can show the rapid decay of the density $f_p$ of $V_p$ for large $p$, due to the following observation.
\[density\_decay\] Let $f:\RR^d\to\RR$ be a Lipschitz function with Lipschitz constant $L$. If the moments $\mu_m:=\int_{\RR^d}|x|^m|f(x)|\td x<\infty$ for all $m>0$, then $|f(x)|\lesssim_{d,L}\mu_{r(d+1)}^{1/(d+1)}|x|^{-r}$ for any $r>0$ and $|x|$ sufficiently large.
If $f\equiv0$ outside a large ball then the claim is trivial. Otherwise take an $x\notin B(0,|f(0)|/L)$ s.t. $\alpha:=|f(x)|>0$. By the Lipschitz condition $|f(x)|<2L|x|$. Moreover, for any $y\in B(x,\frac{3\alpha}{4L})$ one has that $|f(y)|\gq|f(x)|-L|x-y|\gq\alpha/4$. Thus, by assumption for any $m>0$ we have that $$\mu_m\gq\int_{B(x,\frac{3\alpha}{4L})\cap\{|y|\gq|x|\}}|y|^m|f(y)|\td y\gq C_d\lb\frac{3\alpha}{4L}\rb^d|x|^m\frac{\alpha}{4}=C_{d,L}|x|^m\alpha^{d+1},$$ which in turn implies that $|f(x)|\lq C_{d,L}(\mu_m|x|^{-m})^{1/(d+1)}$. For any $r>0$ let $m=r(d+1)$, and the result follows from the arbitrary choice of $x$.
Henceforth we have that $|f_p(y)|\lq C_{d,p_0,r}|y|^{-r}$ for any $r>0,~p>p_0$ and $y\in\RR^d$ sufficiently far away from $0$. Moreover, the argument for the interpolation inequality also applies here, with $m$ replace by any $k\gq0$ and the decay rate $C_{q,m}e^{-c_q|y|}$ replaced by $C_{d,p_0,k}|y|^{-r}$ for the derivative $\tD^kf_p$. Therefore we arrive at the following conclusion.
\[density\_smooth\] For any $N\gq1$ let $p_0$ be defined as in Theorem \[density\]. Then for all $p\gq2p_0$, all the derivatives of $f_p$ up to order $N$ have rapid decay.
Main Result and Further Discussion
==================================
We are now finally ready to proceed to the coupling result, by which we wish to characterise a candidate random variable $\bar V_p$ s.t. the distance $\wass_2(V,V_p+\bar V_p)$ is small.
\[main\] Let $2\lq m\in\ZZ^+$ and $p_0>2048(m+3d+3)^2$ be an even integer s.t. $[\sqrt{2p_0}/4]+1$ is prime. For any $p\gq2p_0$, suppose there exists an $\RR^d$-random variable $\bar{V}_p$ having the same conditional moments given $v_p$ as those of $\wt V_p$ up to order $m-1$ and having density $\chi_y$ conditional on that $V_p=y$. If the function $y\mapsto\chi_y(w)$ is at least $C^{m+d+1}(\RR^d)$ and for $n,k\gq0$ there is a constant $M(n,k)\gq0$ s.t. $$\int_{\RR^d}|w|^n|\tD_y^k\chi_y(w)|\td w\lq C_{d,n,k}(1+|y|^{M(n,k)})p^{-n/2},$$ then there exists a constant $C$ depending on $d,m,p_0$ and the density $f_p$ of $V_p$ s.t. $\forall p\gq2p_0$, $$\wass_2(V,V_p+\bar{V}_p)\lq Cp^{-m/4}.$$
Let $g$ and $h$ be the densities of $V$ and $\bar{V}:=V_p+\bar{V}_p$, respectively. Then by the inversion formula and the tower property, for all $z\in\RR^d$, $$\begin{aligned}
g(z)-h(z)=&\frac{1}{(2\pi)^d}\int_{\RR^d}e^{-iz\cdot\xi}\lb\ex e^{i\xi\cdot V}-\ex e^{i\xi\cdot\bar{V}}\rb\td\xi\\
=&\frac{1}{(2\pi)^d}\int_{\RR^d}e^{-iz\cdot\xi}\ex\lb e^{i\xi\cdot V_p}\ex_p\lb e^{i\xi\cdot\wt{V}_p}-e^{i\xi\cdot\bar{V}_p}\rb\rb\td\xi,\end{aligned}$$ where $\ex_p:=\ex(\cdot|v_p)$. Applying Taylor’s theorem to $\exp(i\xi\cdot\wt V_p)$ and $\exp(i\xi\cdot\bar V_p)$ inside the conditional expectation up to order $m$, one sees that the first $m-1$ differences vanish due to the moment matching assumption and hence $$\ex_p\lb e^{i\xi\cdot\wt{V}_p}-e^{i\xi\cdot\bar{V}_p}\rb=i^m\sum_{|\alpha|=m}\frac{m}{\alpha!}\xi^\alpha\int_0^1(1-\theta)^{m-1}\ex_p\lb\wt V_p^\alpha e^{i\theta\xi\cdot\wt V_p}-\bar V_p^\alpha e^{i\theta\xi\cdot\bar V_p}\rb\td\theta.$$ Thus we have the identity $$g(z)-h(z)=\frac{1}{(2\pi)^d}\sum_{|\alpha|=m}i^m\frac{m}{\alpha!}\int_0^1(1-\theta)^{m-1}\int_{\RR^d}e^{-iz\cdot\xi}\xi^\alpha\lb\rho(\xi)-\eta(\xi)\rb\td\xi\td\theta,$$ where $$\rho(\xi)=\rho(\xi;p,\alpha,\theta):=\ex\lb e^{i\xi\cdot(V_p+\theta\wt V_p)}\wt V_p^\alpha\rb,$$ and $\eta(\xi)$ is similarly defined by replacing $\wt V_p$ with $\bar V_p$. The goal is then to show the rapid decay in $|z|$ of the $\td\xi$ integral above with an appropriate rate of decay in $p$. This should follow from the rapid decay in $|\xi|$ of the derivatives of $\rho(\xi)$ and $\eta(\xi)$.
To this end it is easier to work with, instead of $\wt V_p$, the ‘truncated remainder’ $\wt V_{p,N}:=V_N-V_p$ up to some integer $N\gg p$. Replace $\wt V_p$ with $\wt V_{p,N}$ in $\rho(\xi)$ and denote it by $\rho_N(\xi)$. Recall the notations $v'_N=\{(x_{jr},y_{jr}):j=1,\cdots,q;r=p_0+1,\cdots,N\}$ and $N'=N-p_0$. For $p\gq2p_0$, recall from Lemma \[moments\] (i) the power $\wt V_{p,N}^\alpha$, as a function of $v_{p_0}$ and $v'_N$, takes the form $$\wt V_{p,N}^\alpha=\sum_\beta v_{p_0}^\beta\wt P_\beta(v'_N),$$ where the number of summands depends only on $p_0$ and $\alpha$. By the tower property again, $$\begin{aligned}
\rho_N(\xi):=&\ex\lb e^{i\xi\cdot(V_p+\theta\wt V_{p,N})}\wt V_{p,N}^\alpha\rb=\ex\sum_\beta\wt P_\beta(v'_N)\ex\lb\left.e^{i\xi\cdot(V_p+\theta\wt V_{p,N})}v_{p_0}^\beta\rv v'_N\rb\\
=&\sum_\beta\int_{\RR^{2qN'}}\wt P_\beta(v')\phi_{N'}(v')\td v'\int_{\RR^{2qp_0}}e^{i|\xi|\wh\Phi_N(v_0,v';\theta,\omega_0)}v_0^\beta\phi_{p_0}(v_0)\td v_0,\end{aligned}$$ where $\omega_0=\xi/|\xi|$ and the phase function $\wh\Phi_N(v_0,v';\theta,\omega_0)$ is similar to $\Phi_N(v;\omega_0)$ in the sense that some of the terms in $\wh\Phi_N(v;\theta,\omega_0)-\Phi_{p_0}(v_0)$ have dilated frequency $\theta\omega_0$ - recall the decomposition with $p$ replaced by $N$.
The functions $\varphi_0(v_0):=v_0^\beta\phi_{p_0}(v_0)$ and $\varphi_1(v'):=\wt P_\beta(v')\phi_{N'}(v')$ are both Schwartz, and for any $K,k\in\NN,~\|\varphi_0\|_{\varkappa+3k,K}\lq C_{q,p_0,k,K}$. Similar to write $\wh\Phi_N(v;\theta,\omega_0)=\sum_\gamma v_0^\gamma\wh P_\gamma(v')$. Then Lemma \[moments\] (ii), together with the Cauchy-Schwartz inequality, gives $$\begin{aligned}
\int_{\RR^{2qN'}}\lb1+\sum_\gamma|\wh P_\gamma(v')|^{\sqrt{2p_0}+k-2}\rb\varphi_1(v')\td v'=&\ex\lb1+\sum_\gamma|\wh P_\gamma(v'_N)|^{\sqrt{2p_0}+k-2}\rb|\wt P_\beta(v'_N)|\\
\lq&C_{q,p_0,\alpha,k}p^{-m/2},\end{aligned}$$ for all multi-indices $\beta$ and all $k\in\NN$. Moreover, for fixed $\theta$ and $v'$ the function $\wh\Phi_N(v;\theta,\omega_0)$ only differs from $\Phi_{p_0}(v_0;\omega)$ by a quadratic polynomial in $v_0$ with no singularity in $\theta$, which is insignificant according to the last part of the proof of Theorem \[global\_estimate\] (an additional power in $v'$ may appear in the integral against $\varphi_1$, but this integral will again be independent of $p$ by Lemma \[moments\]). Thus, by Theorem \[global\_estimate\] (for $K=m+d+1$) the integral $\rho_N(\xi)$ is a smooth function s.t. for all $k\gq0,~N\gg p,~\theta\in(0,1)$ and $|\xi|$ sufficiently large, $$\label{rho_derivatives}
|\tD^k\rho_N(\xi)|\lq C_{q,p_0,m,k}|\xi|^{-m-d-1}p^{-m/2}.$$ Thereby for all $N$ and $\theta$ the function $G_N(\xi):=\xi^\alpha\rho_N(\xi)$ has uniformly bounded derivatives, and in particular $\|G_N\|_{d+1,d+3}\lq\|\rho_N\|_{m+d+1,d+3}\lq C_{d,p_0,m}p^{-m/2}$. Applying Lemma \[fourier\_bound\] with ($k\lq$)$K=d+3$ one deduces that $$\label{rho_decay}
\lv\int_{\RR^d}e^{-iz\cdot\xi}\xi^\alpha\rho_N(\xi)\td\xi\rv\lq C_{d,p_0,m}|z|^{-d-3}p^{-m/2}.$$ This estimate is uniform in $N$, and therefore by taking the limit $N\to\infty$ the same bound holds for the integral $\int_{\RR^d}e^{-iz\cdot\xi}\xi^\alpha\rho(\xi)\td\xi$.
For the other integral $\int_{\RR^d}e^{-iz\cdot\xi}\xi^\alpha\eta(\xi)\td\xi$, from the same arguments above it suffices to show that $\forall0\lq k\lq d+3,~N\gg p,~\theta\in(0,1)$ and $|\xi|$ sufficiently large the estimate also holds for $\eta$. By conditioning on the value of $V_p$ one finds the identity $$\eta(\xi)=\ex e^{i\xi\cdot V_p}\ex\lb e^{i\theta\xi\cdot\bar V_p}\bar V_p^\alpha\left|V_p\right.\rb=\int_{\RR^{d}}e^{i\xi\cdot y}f_p(y)\int_{\RR^d}e^{i\theta\xi\cdot w}w^\alpha\chi_y(w)\td w\td y.$$ Differentiating $\eta(\xi)$ by $k$ times by Leibniz’s rule, one has that $\forall\tau\in\NN^d,~|\tau|=k$, $$\begin{aligned}
\partial^\tau\eta(\xi)&=\int_{\RR^d}e^{i\xi\cdot y}f_p(y)\sum_{\sigma\lq\tau}\binom{\tau}{\sigma}(iy)^{\tau-\sigma}\partial_\xi^\sigma\lb\int_{\RR^d}e^{i\theta\xi\cdot w}w^\alpha\chi_y(w)\td w\rb\td y\\
&=i^k\sum_{\sigma\lq\tau}\binom{\tau}{\sigma}\theta^{|\sigma|}\int_{\RR^d}e^{i\xi\cdot y}\underbrace{y^{\tau-\sigma}f_p(y)\int_{\RR^d}e^{i\theta\xi\cdot w}w^{\sigma+\alpha}\chi_y(w)\td w}_{=:H_{\tau,\sigma}(y;\xi,\theta)}\td y.\end{aligned}$$ Then by Leibniz’s rule again for any $\beta\in\NN^d,~|\beta|\lq m+d+1$, $$\begin{aligned}
\partial_y^\beta H_{\tau,\sigma}(y;\xi,\theta)=&\sum_{\nu\lq\beta}\binom{\beta}{\nu}\partial_y^{\beta-\nu}(y^{\tau-\sigma}f_p(y))\int_{\RR^d}e^{i\theta\xi\cdot w}w^{\sigma+\alpha}\partial_y^\nu\chi_y(w)\td w,\end{aligned}$$ which is bounded in $\xi$ and $\theta$. Thus, by the moment assumption on the conditional density of $\chi_y$ and counting the maximum power of $y$, one sees from Theorem \[density\_smooth\] that $$\sup_{\xi\in\RR^d,\theta\in(0,1)}\|H_{\tau,\sigma}(\cdot;\xi,\theta)\|_{d+1,m+d+1}\lq C_{m,d,k}p^{-m/2}\|f_p\|_{k+d+1+M(m+k,m+d+1),m+d+1},$$ for all $\theta\in(0,1)$ and $\xi\in\RR^d$. Thus the sought-after estimate for $|\tD^k\eta(\xi)|$ follows from Lemma \[fourier\_bound\] for $K=m+d+1$, with the Schwarz norm of $f_p$ above as a multiplicative factor. Apply Lemma \[fourier\_bound\] again for ($k\lq$)$K=d+3$ we obtain with $\rho_N$ replaced by $\eta$ and a multiplicative factor $\|f_p\|_{2d+4+M(m+d+3,m+d+1),m+d+1}$.
The result then follows from the inequality for $p=2$.
To finish off this article I shall make the following remarks regarding the remaining difficulties of the coupling problem for the triple stochastic integral.
#### Rate of convergence.
As opposed to the rate $O(p^{-m/2})$ obtained in [@davie2014Patappstodifequusicou] for the double integral, the rate $O(p^{-m/4})$ is probably the best one can expect simply from Theorem \[global\_estimate\] and Theorem \[density\] alone. This is because the particular form of the phase function $\Phi_p$ and its derivatives are not fully exploited. In fact we have only used the fact that the phase function $\Phi_p$ is a cubic polynomial in Lemma \[rho\_big\] and Lemma \[rho\_small\]. Moreoever, the inequality itself is not very sharp, especially for the quadratic distance $\wass_2$.
Despite this limitation, to my best knowledge what is proved so far is the first attempt to find a coupling for triple stochastic integrals. I believe that Davie’s rate $O(p^{-m/2})$ could still be achieved if analogues of Lemma 12, 13 and especially 14 in [@davie2014Patappstodifequusicou] can be proved.
#### Generation of $\bar V_p$.
The requirements for the candidate $\bar V_p$ in Theorem \[main\] are not straightforward. While the moment-matching condition is relatively easy to meet (similar to the discussion after Theorem 15 in [@davie2014Patappstodifequusicou]), it is not clear how to generate $\bar V_p$ s.t. its conditional density satisfies the smoothness conditions.
#### Application to SDE approximation.
The numerical scheme based on a coupling of order $O(p^{-m/4})$ is computationally equivalent to the Milstein scheme based on Wiktorsson’s result [@wiktorsson2001JoiChaFunSimSimIteItIntMulIndBroMot] with step size $h^{3/2}$- see the discussion following the proof of Theorem 15 in [@davie2014Patappstodifequusicou]. To achieve a genuine improvement one needs a better rate than Theorem \[main\], which requires careful estimates for the density $f_p$ of $V_p$ as mentioned above. Moreover, it is also not clear how one can combine Davie’s coupling for the double integral and the coupling for the triple integral together to form a genuinedly improved numerical scheme. These questions are left open for future investigation.
[^1]: School of Mathematics, The University of Edinburgh. e-mail: xiling.zhang@ed.ac.uk
[^2]: Throughout the paper $\ZZ^+$ denotes the set of positive integers and $\NN$ denotes the set of natural numbers $\ZZ^+\cup\{0\}$.
[^3]: Also spelt as “Wasserstein".
[^4]: The superscript $(n)$ signifies that it is a ball in $\RR^n$. Balls without superscripts lie in the whole space $\RR^k$.
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**Quark masses in QCD: a progress report** [[^1]]{}
C. A. Dominguez
Centre for Theoretical and Mathematical Physics, University of Cape Town, Rondebosch 7700, South Africa, and Department of Physics, Stellenbosch University, Stellenbosch 7600, South Africa
**Abstract**
Recent progress on QCD sum rule determinations of the light and heavy quark masses is reported. In the light quark sector a major breakthrough has been made recently in connection with the historical systematic uncertainties due to a lack of experimental information on the pseudoscalar resonance spectral functions. It is now possible to suppress this contribution to the $1 \%$ level by using suitable integration kernels in Finite Energy QCD sum rules. This allows to determine the up-, down-, and strange-quark masses with an unprecedented precision of some $8 - 10\, \%$. Further reduction of this uncertainty will be possible with improved accuracy in the strong coupling, now the main source of error. In the heavy quark sector, the availability of experimental data in the vector channel, and the use of suitable multipurpose integration kernels allows to increase the accuracy of the charm- and bottom-quarks masses to the $1 \%$ level.
Introduction
============
Quark and gluon confinement in Quantum Chromodynamics (QCD) precludes direct experimental measurements of the fundamental QCD parameters, i.e. the strong interaction coupling and the quark masses. Hence, in order to determine these parameters analytically one needs to relate them to experimentally measurable quantities. Alternatively, simulations of QCD on a lattice provide increasingly accurate numerical values for these parameters, but little if any insight into their origin. The first approach relies on the intimate relation between QCD Green functions, in particular their Operator Product Expansion (OPE) beyond perturbation theory, and their hadronic counterparts. This relation follows from Cauchy’s theorem in the complex energy plane, and is known as the QCD sum rule technique [@QCDSR]. In addition to producing numerical values for the QCD parameters, this method provides a detailed breakdown of the relative impact of the various dynamical contributions. For instance, the strong coupling at the scale of the $\tau$-lepton mass essentially follows from the relation between the experimentally measured $\tau$ ratio, $R_\tau$, and a contour integral involving the perturbative QCD (PQCD) expression of the $V+A$ correlator. This is the cleanest, most transparent, and model independent determination of the strong coupling [@PICH1]-[@PICH2]. It also allows to gauge the impact of each individual term in PQCD, up to the currently known five-loop order. Similarly, in the case of the quark masses one considers a QCD correlation function which on the one hand involves the quark masses and other QCD parameters, and on the other hand it involves a measurable (hadronic) spectral function. Using Cauchy’s theorem to relate both representations, the quark masses become a function of QCD parameters, e.g. the strong coupling, some vacuum condensates reflecting confinement, etc., and measurable hadronic parameters. The virtue of this approach is that it provides a breakdown of each contribution to the final value of the quark masses. More importantly, it allows to tune the relative weight of each of these contributions by introducing suitable integration kernels. This last feature has been used recently to solve the historical problem of the systematic uncertainties affecting light quark mass determinations, to be discussed in this report.\
In the case of the light quark masses the ideal Green function is the pseudoscalar current correlator. This contains the square of the quark masses as an overall factor multiplying the PQCD expansion, and the leading power corrections in the OPE. Unfortunately, this correlator is not realistically accessible experimentally beyond the pseudoscalar meson pole. While the existence of at least two radial excitations of the pion and the kaon are known from hadronic interaction data, this information is hardly enough to reconstruct the full spectral functions. In spite of many attempts over the years to model them, there remains an unknown systematic uncertainty that has plagued light quark mass determinations from QCD sum rules. The use of the vector current correlator, for which there is plenty of experimental data from $\tau$ decays and $e^+ e^-$ annihilation, is not a realistic option for the light quarks as their masses enter as sub-leading terms in the OPE. The scalar correlator, involving the square of quark mass differences, at some stage offered some promise for determining the strange quark mass with reduced systematic uncertainties. This was due to the availability of data on $K-\pi$ phase shifts. Unfortunately, these data do not fully determine the hadronic spectral function. The latter can be reconstructed from phase shift data only after substantial theoretical manipulations, implying a large unknown systematic uncertainty. A breakthrough has been made recently by introducing an integration kernel in the contour integral in the complex energy plane. This allows to suppress substantially the unknown hadronic resonance contribution to the pseudoscalar current correlator. As it follows from Cauchy’s theorem, this suppression implies that the quarks masses are determined essentially from the well known pseudoscalar meson pole and PQCD (well known up to five-loop level). In this way it has been possible to reduce the hadronic resonance contribution to the 1% level, allowing for an unprecedented accuracy of some $8 - 10\, \%$ in the values of the up-, down-, and strange-quark masses. Further improvement on this accuracy will be possible with further reduction of the uncertainty in the strong coupling, now the main source of error.\
The determination of the charm- and bottom-quark masses has been free of systematic uncertainties due to the hadronic resonance sector, as there is plenty of experimental information in the vector channel from $e^+ e^-$ annihilation into hadrons. One problem, though, is that the massive vector current correlator is not known in PQCD to the same level as the light pseudoscalar correlation function. Nevertheless, substantial theoretical progress has been made over the years leading to extremely accurate charm- and bottom-quark masses. The novel idea of introducing suitable integration kernels in Cauchy’s contour integrals, as described above, has also been used recently as a way of improving accuracy in the heavy-quark sector. For instance, kernels can be used to suppress regions where the data is either not as accurate, or simply unavailable. This will also be reported here.\
The paper is organized as follows. First, determinations of quark-mass ratios from various hadronic data, as well as from chiral perturbation theory, will be reviewed in Section 2. These ratios are quite useful as consistency checks for results from QCD sum rules. Section 3 describes the OPE beyond perturbation theory, one of the two pillars of QCD sum rules. Section 4 discusses quark-hadron duality and finite energy sum rules. These sum rules weighted by suitable integration kernels will be analyzed in the light quark sector in Section 5. In particular it will be shown how this technique unveils the subjacent hadronic systematic uncertainty plaguing light quark mass determinations for the past thirty years. In Section 6 recent progress on charm- and bottom-quark mass determinations will be reported. Comparison with lattice QCD results for all quark masses will also be made. Finally, Section 7 provides a very short summary of this report.\
As an important disclaimer, this paper is not a comprehensive review of past quark mass determinations from QCD sum rules. It is, rather, a report on recent progress on the subject. Given that past determinations of light quark masses were affected by unknown systematic uncertainties, essentially from the hadronic resonance sector, it makes little or no sense to review them once the main uncertainty has been exposed. Any agreement between values affected by this uncertainty and current results, free of it, would only be fortuitous. In fact, once this hadronic resonance uncertainty is removed, the values of all three light quark masses get reduced by some 15 - 20 %, a clear sign of a systematic uncertainty acting in only one direction. Last but not least, light quark masses from QCD sum rules before 2006 employed correlators up to at most four-loop level in PQCD, together with superseded values of the strong coupling.
Quark mass ratios
=================
Quark masses actually precede QCD by a number of years, albeit under the guise of [*current algebra quark masses*]{}, which clearly lacked today’s detailed understanding of quark-mass renormalization. In fact, the study of global ${\mbox{SU}}(3) \times {\mbox{SU}}(3)$ chiral symmetry realized á la Nambu-Goldstone, and its breaking down to ${\mbox{SU}}(2) \times {\mbox{SU}}(2)$, followed by a breaking down to ${\mbox{SU}}(2)$, and finally to ${\mbox{U}}(1)$ was first done using the strong interaction Hamiltonian [@CA1]-[@HLR] $$H(x)\, =\, H_0(x)\, + \,\epsilon_0\; u_0(x)\, + \,\epsilon_3\; u_3(x)\, + \,\epsilon_8\; u_8(x) \;.$$ The term $H_0(x)$ above is $\mbox{SU}(3) \times \mbox{SU}(3)$ invariant, the $\epsilon_{0,3,8}$ are symmetry breaking parameters, and the scalar densities $u_{0,3,8}(x)$ transform according to the $3 \,\overline{3} \;+\;\overline{3} \,3$ representation of $\mbox{SU}(3) \times \mbox{SU}(3)$. In modern language, $\epsilon_8$ is related to the strange quark mass $m_s$, and $\epsilon_3$ to the difference between the down- and the up-quark masses $m_d-m_u$, while the scalar densities are related to products of quark-anti-quark field operators. For instance, the ratio of $\mbox{SU}(3)$ breaking to $\mbox{SU}(2)$ breaking is given by $$R \equiv \frac{m_s - m_{ud}}{m_d - m_u} = \frac{\sqrt{3}}{2} \; \frac{\epsilon_8}{\epsilon_3} \;,$$ where $m_{ud} \equiv (m_u + m_d)/2$. In the pre-QCD era many relations for quark-mass ratios were obtained from hadron mass ratios, as well as from other hadronic information, e.g. $\eta \rightarrow 3 \pi$, $K_{l 3}$ decay, etc. [@HLR]. To mention a pioneering determination of the ratio $R$ above, from a solution to the $\eta \rightarrow 3 \pi$ puzzle proposed in [@CAD1] it followed [@CAD2] $R^{-1} = 0.020 \pm 0.002$, in remarkable agreement with a later determination based on baryon mass splittings [@MZ] $R^{-1} = 0.021 \pm 0.003$. With the advent of chiral perturbation theory (CHPT) [@HP]-[@HLR], [@CHPT1]-[@CHPT2], certain quark mass ratios turned out to be renormalization scale independent to leading order, and could be expressed in terms of pseudoscalar meson mass ratios [@HLR],[@CHPT1]-[@SW], e.g. $$\frac{m_u}{m_d} = \frac{M_{K^+}^2 - M_{K^0}^2 + 2 M_{\pi^0}^2 - M_{\pi^+}^2}{M_{K^0}^2 - M_{K^+}^2 + M_{\pi^+}^2} = 0.56$$ $$\frac{m_s}{m_d} = \frac{M_{K^+}^2 + M_{K^0}^2 - M_{\pi^+}^2}{M_{K^0}^2 - M_{K^+}^2 + M_{\pi^+}^2} = 20.2 \;,$$ where the numerical results follow after some subtle corrections due to electromagnetic self energies [@CHPT2]. Beyond leading order in CHPT things become complicated. At next to leading order (NLO) the only parameter free relation is $$Q^2 \equiv \frac{m_s^2 - m_{ud}^2}{m_d^2 - m_u^2} =
\frac{M_{K}^2 - M_{\pi}^2}{M_{K^0}^2 - M_{K^+}^2}\; \frac{M_K^2}{M_{\pi}^2} \;.$$ Other quark mass ratios at NLO and beyond depend on the renormalization scale, as well as on some CHPT low energy constants which need to be determined independently [@CHPT1]-[@CHPT2]. After taking into account electromagnetic self energies, Eq.(5) gives [@CHPT1] $Q = 24.3$, while a recent analysis of $\eta \rightarrow 3 \pi$ [@CHPT1], [@GC] gives $$Q = 22.3 \pm 0.8 \;.$$ The ratios $R$, Eq.(2), and $Q$, Eq.(5), together with the leading order ratios Eqs.(3)-(4), will prove useful for comparisons with QCD sum rule results. An additional useful quark mass ratio involving the ratios Eqs.(3)-(4) is $$r_s \equiv \frac{m_s}{m_{ud}} = \frac{2\; m_s/m_d}{1 + m_u/m_d} = 28.1 \pm 1.3 \;,$$ where the numerical value follows from the NLO CHPT relation [@CHPT1], to be compared with the LO result from Eqs.(3)-(4), $r_s = 25.9$, and a large $N_c$ estimate [@eta] $r_s = 26.6 \pm 1.6$.
Operator product expansion beyond perturbation theory
=====================================================
The OPE beyond perturbation theory in QCD, one of the two pillars of the sum rule technique, is an effective tool to introduce quark-gluon confinement dynamics. It is not a model, but rather a parametrization of quark and gluon propagator corrections due to confinement, done in a rigorous renormalizable quantum field theory framework. Let us consider a typical object in QCD in the form of the two-point function, or current correlator $$\Pi(q^2)\,=\,i\; \int \,d^4 x \; e^{iqx} \; <0|\,T(J(x)\,J(0))\,|0 >,$$ where the local current $J(x)$ is built from the quark and gluon fields entering the QCD Lagrangian. Equivalently, this current can also be written in terms of hadronic fields with the same quantum numbers. A relation between the two representations follows from Cauchy’s theorem in the complex energy (squared) plane. This is often referred to as quark-hadron duality, the second pillar of the QCD sum rules method to be discussed in the next section. The QCD correlator, Eq.(8), contains a perturbative piece (PQCD), and a non perturbative one mostly reflecting quark-gluon confinement. The leading order in PQCD is shown in Fig.1. Since confinement has not been proven analytically in QCD, its effects can only be introduced effectively, e.g. by parameterizing quark and gluon propagator corrections in terms of vacuum condensates. This is done as follows. In the case of the quark propagator $$S_F (p) = \frac{i}{\not{p} - m}\;\;\Longrightarrow \;\;\frac{i}{\not{p} - m + \Sigma(p^2)} \;,$$ the propagator correction $\Sigma(p^2)$ contains the information on confinement, a purely non-perturbative effect. One expects this correction to peak at and near the quark mass-shell, e.g. for $p \simeq 0$ in the case of light quarks. Effectively, this can be viewed as in Fig. 2, where the (infrared) quarks in the loop have zero momentum and interact strongly with the physical QCD vacuum. This effect is then parameterized in terms of the quark condensate $\langle 0| \bar{q}(0) q(0) | 0 \rangle$.
![[]{data-label="fig:figure1"}](loop.eps){height=".1\textheight"}
Similarly, in the case of the gluon propagator $$D_F (k) = \frac{i}{k^2}\;\;\Longrightarrow \;\;\frac{i}{k^2 + \Lambda(k^2)} \;,$$ the propagator correction $\Lambda(k^2)$ will peak at $k\simeq 0$, and the effect of confinement in this case can be parameterized by the gluon condensate $\langle 0| \alpha_s\; \vec{G}^{\mu\nu} \,\cdot\, \vec{G}_{\mu\nu}|0\rangle$ (see Fig.3). In addition to the quark and the gluon condensate there is a plethora of higher order condensates entering the OPE of the current correlator at short distances, i.e. $$\Pi(q^2)|_{QCD}\,=\, C_0\,\hat{I} \,+\,\sum_{N=0}\;C_{2N+2}(q^2,\mu^2)\;\langle0|\hat{O}_{2N+2}(\mu^2)|0\rangle \;,$$ where $\mu^2$ is the renormalization scale, and where the Wilson coefficients in this expansion, $ C_{2N+2}(q^2,\mu^2)$, depend on the Lorentz indices and quantum numbers of $J(x)$ and of the local gauge invariant operators $\hat{O}_N$ built from the quark and gluon fields. These operators are ordered by increasing dimensionality and the Wilson coefficients, calculable in PQCD, fall off by corresponding powers of $-q^2$. In other words, this OPE achieves a factorization of short distance effects encapsulated in the Wilson coefficients, and long distance dynamics present in the vacuum condensates. Since there are no gauge invariant operators of dimension $d=2$ involving the quark and gluon fields in QCD, it is normally assumed that the OPE starts at dimension $d=4$. This is supported by results from QCD sum rule analyses of $\tau$-lepton decay data, which show no evidence of $d=2$ operators [@C2a]-[@C2b].
![[]{data-label="fig:figure2"}](qcond2.eps){height=".06\textheight"}
The unit operator $\hat{I}$ in Eq.(11) has dimension $d=0$ and $C_0 \hat{I}$ stands for the purely perturbative contribution. The Wilson coefficients as well as the vacuum condensates depend on the renormalization scale. For light quarks, and for the leading $d=4$ terms in Eq.(11), the $\mu^2$ dependence of the quark mass cancels the corresponding dependence of the quark condensate, so that this contribution is a renormalization group (RG) invariant. Similarly, the gluon condensate is also a RG invariant, hence once determined in some channel these condensates can be used throughout. The numerical values of the vacuum condensates cannot be calculated analytically from first principles as this would be tantamount to solving QCD exactly. One exception is that of the quark condensate which enters in the Gell-Mann-Oakes-Renner relation, a QCD low energy theorem following from the global chiral symmetry of the QCD Lagrangian [@GMOR]. Otherwise, it is possible to extract values for the leading vacuum condensates using QCD sum rules together with experimental data, e.g. $e^+ e^-$ annihilation into hadrons, and hadronic decays of the $\tau$-lepton. Alternatively, as lattice QCD improves in accuracy it should become a valuable source of information on these condensates.\
![[]{data-label="fig:figure3"}](gcond2.eps){height=".12\textheight"}
Quark-hadron duality and finite energy QCD sum rules
====================================================
Turning to the hadronic sector, bound states and resonances appear in the complex energy (squared) plane (s-plane) as poles on the real axis, and singularities in the second Riemann sheet, respectively. All these singularities lead to a discontinuity across the positive real axis. Choosing an integration contour as shown in Fig. 4, and given that there are no other singularities in the complex s-plane, Cauchy’s theorem leads to the finite energy sum rule (FESR)
![[]{data-label="fig:figure4"}](FESR.eps){height=".25\textheight"}
$$\int_{\mathrm{sth}}^{s_0} ds\; \frac{1}{\pi}\; f(s) \;Im \,\Pi(s)|_{HAD} \; = \; -\, \frac{1}{2 \pi i} \; \oint_{C(|s_0|) }\, ds \;f(s) \;\Pi(s)|_{QCD} \;,$$
where $f(s)$ is an arbitrary (analytic) function, $s_{th}$ is the hadronic threshold, and the finite radius of the circle, $s_0$, is large enough for QCD and the OPE to be used on the circle. Physical observables determined from FESR should be independent of $s_0$. In practice, though, this is not exact, and there is usually a region of stability where observables are fairly independent of $s_0$, typically somewhere inside the range $s_0 \simeq 1 - 4 \; \mbox{GeV}^2$. Equation (12) is the mathematical statement of what is usually referred to as quark-hadron duality. Since QCD is not valid in the time-like region ($s \geq 0$), in principle there is a possibility of problems on the circle near the real axis (duality violations), to be discussed shortly (this issue was identified very early in [@Shankar] long before the present formulation of QCD sum rules). The right hand side of this FESR involves the QCD correlator which is expressed in terms of the OPE as in Eq.(11). The left hand side involves the hadronic spectral function which is written as $$Im \,\Pi(s)|_{HAD}\,=\, Im \,\Pi(s)|_{POLE}\,+\, Im \,\Pi(s)|_{RES} \,\theta(s_0-s)\,+\, Im\, \Pi(s)|_{PQCD}\,\theta(s-s_0) \;,$$ where the ground state pole is followed by the resonances which merge smoothly into the hadronic continuum above some threshold $s_0$. This continuum is expected to be well represented by PQCD if $s_0$ is large enough. Hence, if one were to consider an integration contour in Eq.(12) extending to infinity, the cancellation between the hadronic continuum on the left hand side and the PQCD contribution on the right hand side, would render the sum rule a FESR.
![ []{data-label="fig:figure5"}](FPI1.eps){height=".28\textheight"}
The integration in the complex s-plane of the QCD correlator is usually carried out in two different ways, Fixed Order Perturbation Theory (FOPT) and Contour Improved Perturbation Theory (CIPT). The first method treats running quark masses and the strong coupling as fixed at a given value of $s_0$. After integrating all logarithmic terms ($\ln(-s/\mu^2)$) the RG improvement is achieved by setting the renormalization scale to $\mu^2 = - s_0$. In CIPT the RG improvement is performed before integration, thus eliminating logarithmic terms, and the running quark masses and strong coupling are integrated around the circle. This requires solving numerically the RGE for the quark masses and the coupling at each point on the circle. The FESR Eq.(12) with $f(s)=1$ and in FOPT can be written as $$(-)^N \, C_{2N+2} \, \langle0| \hat{O}_{2N+2}|0 \rangle = \int_0^{s_0} \,ds\, s^N \, \frac{1}{\pi}\, Im \,\Pi(s)|_{HAD} \,-\, s_0^{N+1} \; M_{2N+2}(s_0) \, ,$$ where the dimensionless PQCD moments $M_{2N+2}(s_0)$ are given by $$M_{2N+2}(s_0) = \frac{1}{s_0^{(N+1)}} \, \int_0^{s_0}\, ds\,s^N \, \frac{1}{\pi} \, Im \, \Pi(s)|_{PQCD}\;.$$
![[]{data-label="fig:figure6"}](FPI2.eps){height=".35\textheight"}
If the hadronic spectral function is known in some channel from experiment, e.g. from $\tau$-decay into hadrons, then $Im \,\Pi(s)|_{HAD} \equiv Im \,\Pi(s)|_{DATA}$, and Eq.(14) can be used to determine the values of the vacuum condensates. Subsequently, Eq.(14) can be used in a different channel for a different application. It is important to mention that the correlator $\Pi(q^2)$ is generally not a physical observable. However, this has no effect in FOPT as the unphysical quantities (polynomials) in the correlator do not contribute to the integrals. In the case of CIPT, though, this requires modified sum rules involving as many derivatives of the correlator as necessary to render it physical.\
Next, let us consider an application where the integration kernel $f(s)$ in Eq.(12) is of great importance [@FPI]. In the axial-vector channel, the FESR Eq.(14) with $f(s)=1$ and $N=0$ can be confronted with data from $\tau$-decay. The hadronic spectral function is then written as the sum of the pion pole and the resonance data known up to the kinematical end point $s_0 = M_\tau^2$. The moment $M_2(s_0)$ is known up to five-loop order in PQCD, so that the FESR can be used to confront the resonance data plus PQCD with e.g. $f_\pi$. As seen from Fig. 5 the agreement is rather poor, except possibly near the end point. At first sight, this may be interpreted as a signal for quark-hadron duality violations near the real axis, even at this high enough energy. In fact, it has been known for quite some time that e.g. the Weinberg (chiral) sum rules are not saturated by the $\tau$ decay data unless one introduces [*pinched*]{} integration kernels, e.g. $f(s) = [1 - (s/s_0)]^{(N+1)}$ [@PINCH1]-[@PINCH2]. Unfortunately, the $\tau$-lepton is not massive enough to probe higher energy regions. In spite of this it is still possible to explore a wider energy range by introducing as integration kernel a polynomial $f(s) \equiv P(s, s_0, s_1)$ tuned to eliminate the (unknown) hadronic contribution to the integral between $s_1$ and $s_0 \geq s_1$, where $s_1$ is at or near the end point of the data. It has been shown [@FPI] that the optimal degree of $P(s)$ is the simplest, i.e. the linear function $$P(s,s_0,s_1)=1-\frac{2s}{s_{0}+s_{1}} \,,$$ so that $$\mbox{constant} \times \int_{s_1}^{s_0} P(s,s_0,s_1) ds = 0\,.$$ In this case the complete FESR becomes a linear combination of a dimension-two and a dimension-four FESR, which from Eqs.(14) and (16) is given by $$\begin{aligned}
2 \, f_\pi^2 &=& - \int_{0}^{s_{1}} ds \, P(s) \, \frac{1}{\pi}\, Im \,\Pi(s)|_{DATA}
+ \frac{s_0}{4 \pi^2} \left[ M_2(s_0) - \frac{2 s_0}{s_0+s_1} M_4 (s_0) \right] \nonumber \\[.3cm]
&+& \frac{1}{4 \pi^2} \left[ C_2 \langle \hat{O}_2 \rangle +\frac{2}{s_0+s_1} C_4 \langle \hat{O}_4 \rangle \right] \, + \Delta(s_0)\,,\end{aligned}$$ where the pion pole has been separated from the data, and the chiral limit is understood. The term $\Delta(s_0)$ is the error being made by assuming that the data is constant in the interval $s_1 - s_0$. It is possible to estimate this error which turns out to be two to three orders of magnitude smaller than $2 f_\pi^2$ on the left hand side of Eq.(18) [@FPI]. As can be seen from Fig. 6 the FESR Eq.(18) shows an excellent consistency between QCD and the $\tau$ data in the axial-vector channel in a remarkably wide region $s_0 \simeq 4 \, -\,10\, \mbox{GeV}^2$. A similar consistency is also found in the vector channel, where QCD is now confronted with zero (there is no pole in this channel). This result shows either no evidence for quark-hadron duality violations in these channels, or if they are present it indicates that they are suppressed by the integration kernel (some model dependent analyses claim the existence of duality violations [@PICH3]).\
Light quark masses
==================
Traditionally, the light quark masses have been determined using the correlator, Eq.(8), involving the pseudoscalar currents $J(x) \equiv \partial_\mu A^\mu(x)|^i_j = [\overline{m}_i(\mu) + \overline{m}_j(\mu)]:\overline{q}_j(x) i \gamma_5 q_i(x):$, where $A_\mu(x)$ is the axial vector current of flavours $i$ and $j$, $\overline{m}_i(\mu)$ the quark mass in the $\overline{MS}$ scheme, $\mu$ the renormalization scale and $q_i(x)$ are the quark fields. An issue of major concern in the past was the presence of logarithmic quark-mass singularities in these correlators. This problem has been satisfactorily resolved some time ago in [@LMS1]-[@LMS2]. These correlators are now known to five-loop order in PQCD [@CHET5], and free of logarithmic quark mass singularities. The Wilson coefficients of the leading power corrections, i.e. the gluon and the quark condensates, are also known up to two-loop level [@LMS1]-[@LMS2]. Higher dimensional condensates, as well as quark mass corrections of order ${\cal{O}}$$(m_i^4)$ (with respect to the one-loop term) and higher turn out to be negligible. From Cauchy’s theorem, Eq.(12), the FESR to determine the quark masses can be written as $$\begin{aligned}
- \frac{1}{2\pi i}
\oint_{C(|s_0|)}
ds \;\psi_{5}^{QCD}(s)\; \Delta_5(s) &=& 2\; f_P^2 \; M_P^4\; \Delta_5(M_P^2) \nonumber \\ [.3cm]
&+&
\int_{s_{th}}^{s_0}
ds \;\frac{1}{\pi} \;Im \;\psi_{5}(s)|_{RES}\;\Delta_5(s) \, , \end{aligned}$$ where $\Delta_5(s)$ is an (analytic) integration kernel to be introduced shortly, the first term on the right hand side is the pseudoscalar meson pole contribution ($P = \pi, K$), $s_{th}$ is the hadronic threshold, and $Im\, \psi_5(s)|_{RES}$ is the hadronic resonance spectral function. The radius of integration $s_0$ is assumed to be large enough for QCD to be valid on the circle. For later convenience this FESR can be rewritten as $$\delta_5(s_0)|_{QCD} \,=\, \delta_5|_{POLE}\, + \, \delta_5(s_0)|_{RES}\;,$$ where the meaning of each term is self evident. Historically, the problem with the pseudoscalar correlator has been the lack of direct experimental information on the hadronic resonance spectral functions. Two radial excitations of the pion and of the kaon, with known masses and widths, have been observed in hadronic interactions [@PDG]. However, this information is hardly enough to reconstruct the full spectral function. In fact, inelasticity, non-resonant background and resonance interference are impossible to guess, leaving no choice but to model these functions. This introduces an unknown systematic uncertainty which has been present in all past QCD sum rule determinations of the light quark masses. Since the FESR Eq.(19) is valid for any analytic $\Delta_5(s)$ one can choose this kernel in such a way as to suppress $\delta_5(s_0)|_{RES}$ as much as possible. An example of such a function is the second degree polynomial [@ss]-[@IJMPA] $$\Delta_5(s)|_{RES} \,=\, 1 \, - a_0 \,s - a_1\, s^2\;,$$ where $a_0$ and $a_1$ are constants fixed by the requirement $\Delta_5(M_1^2) = \Delta_5(M_2^2) =0$, where $M_{1,2}$ are the masses of the first two radial excitations of the pion or kaon. This simple kernel suppresses enormously the resonance contribution, which becomes only a couple of a percent of the pole contribution, and well below the current uncertainty due to the strong coupling. This welcome feature is essentially independent of the model chosen to parametrize the resonances. A practical parametrization consists of two Breit-Wigner forms normalized at threshold according to chiral perturbation theory, as first proposed in [@CADCHPT1] for the pionic channel, and in [@CADCHPT2] for the kaonic channel. Detailed results for $\delta_5(s_0)|_{QCD}$, to five-loop order in PQCD and up to dimension $d=4$ in the OPE, after integrating in FOPT may be found in [@ms]. In the case of CIPT the FESR must be written in terms of the second derivative of the current correlator. This is in order to eliminate the unphysical first degree polynomial present in $\psi_5(s)$, which unlike the case of FOPT would otherwise contribute to the FESR which then becomes $$\begin{aligned}
- \frac{1}{2\pi i}
\oint_{C(|s_0|)}
&ds& \psi_{5}^{'' QCD}(s)\,[F(s) - F(s_0)] = 2\; f_P^2 \; M_P^4\; \Delta_5(M_P^2) \nonumber \\ [.3cm]
&+&
\frac{1}{\pi} \; \int_{s_{th}}^{s_0}
ds \; Im \;\psi_{5}(s)|_{RES}\;\Delta_5(s) \, , \end{aligned}$$ where $$F(s) = - s \left(s_0 - a_0\,\frac{s_0^2}{2} - a_1\, \frac{s_0^3}{3} \right) + \frac{s^2}{2} - a_0\, \frac{s^3}{6} - a_1\, \frac{s^4}{12} \;,$$ and $$F(s_0) = - \frac{s_0^2}{2} + a_0\, \frac{s_0^3}{3} + a_1\, \frac{s_0^4}{4} \;.$$
![[]{data-label="fig:figure7"}](BOUNDMS.eps){height=".35\textheight"}
The RG improvement is used before integration, so that all logarithmic terms vanish. The running coupling as well as the running quark masses are no longer frozen as in FOPT, but must be integrated. This can be done by solving numerically the respective RG equations at each point on the integration circle in the complex s-plane. Detailed expressions are given in [@ms]-[@mq].\
The parameters of the integration kernel, Eq.(21), are $a_0 = 0.897 \;\mbox{GeV}^{-2}$, and $a_1 = -\, 0.1806 \;\mbox{GeV}^{-4}$ for the pionic channel, and $a_0 = 0.768 \;\mbox{GeV}^{-2}$, and $a_1 = -\, 0.140 \;\mbox{GeV}^{-4}$ for the kaonic channel. These values correspond to the radial excitations $\pi(1300)$, $\pi(1800)$, $K(1460)$ and $K(1830)$. The pion and kaon decay constants are [@PDG] $f_\pi = 92.21 \,\pm 0.14 \;\mbox{MeV}$, and $f_K = (1.22 \pm 0.01) f_\pi$. In the QCD sector it is best to use the value of the strong coupling determined at the scale of the $\tau$-mass, as this is close to the scale in current use for the light quark masses, i.e. $\mu= 2 \;\mbox{GeV}$. The extraction of $\alpha_s(M_\tau)$ from the $R_\tau$ ratio involves an integral with a natural kinematical integration kernel that eliminates the contribution of the $d=4$ term in the OPE. This welcome feature improves the accuracy of the determination, and it makes little sense to introduce additional spurious integration kernels which would artificially recover this $d=4$ contribution. The different values obtained from $\tau$ decay using CIPT are all in agreement with each other, i.e. $\alpha_s(M_\tau) = 0.338 \pm 0.012$ [@PICH2], $\alpha_s(M_\tau) = 0.341 \pm 0.008$ [@CVETIC], $\alpha_s(M_\tau) = 0.344 \pm 0.009$ [@DAVIER], and $\alpha_s(M_\tau) = 0.332 \pm 0.016$ [@BAIKOV]. These determinations are model independent and extremely transparent, with $\alpha_s$ obtained essentially by confronting PQCD with the single experimental number $R_\tau$. The method of FOPT is known to give rise to a pathological non-convergent perturbative series in this application [@PICH1], so it will not be considered here. The $d=4$ gluon condensate has been extracted from $\tau$ decays [@C2b], but one can conservatively consider the wide range$<\alpha_s G^2> = 0.01 - 0.12 \;\mbox{GeV}^4$. The impact of the light quark condensate is at the level of $1 \%$ in the quark masses. A $\pm \;30 \%$ uncertainty in the resonance contribution $\delta_5(s_0)|_{RES}$ in Eq.(20) translates into a safe $1 \%$ change in the quark masses. Finally, it has been assumed that the unknown six-loop PQCD contribution is equal to the five-loop result, an extreme but very conservative estimate of higher orders in PQCD.\
[ccccccccc]{}\
Source & $\overline{m}_u$ & $\overline{m}_d$& $\overline{m}_s$& $\overline{m}_{ud}$ & $\overline{m}_u/\overline{m}_d$ & $\overline{m}_s/\overline{m}_{ud}$ & $R$ & $Q$\
\
QCDSR [@mq] & $2.6 \pm 0.3$ & $5.6 \pm 0.4$ & - & $4.1 \pm 0.3$ & $0.46 \pm 0.06$ & - &- & -\
\
QCDSR [@ms] & - & - & $ 102 \pm 8$ & - & - & $24.9 \pm 2.7$ &$33 \pm 6$ & $21 \pm 3$\
\
FLAG [@CHPT2] & $2.2 \pm 0.3$ & $4.6 \pm 0.6$ & $ 95 \pm 10$ & $3.4 \pm 0.4$ & $0.47 \pm 0.04$ & $27.8 \pm 1.0$ & $37.2 \pm 4.1$ & $23.1 \pm 1.5$\
Beginning with the strange quark mass, Fig. 1 shows the results for $\overline{m}_s (2\; \mbox{GeV})|_{\overline{MS}}$ with no integration kernel, $\Delta_5(s) = 1$, and taking into account only the kaon pole, curve (a), and the kaon pole plus a two Breit-Wigner resonance model with a threshold constraint from CHPT [@CADCHPT2], curve (b) (a misprint in the formula for the spectral function in [@CADCHPT2] has been corrected in [@CADCHPT3]). These curves are for the central value of $\alpha_s(M_\tau)$ whose uncertainties will be considered afterwards. The latter result is reasonably stable in the wide region $s_0 = 2 - 4 \; \mbox{GeV}^2$, so that it could lead us to conclude that $\overline{m}_s (2\; \mbox{GeV})|_{\overline{MS}} \simeq 100 - 120 \; \mbox{MeV}$, albeit with a yet unknown systematic uncertainty arising from the resonance sector. Introducing the kernel, Eq.(21), leads to curve (c) and to a dramatic unveiling of this systematic uncertainty. In fact, the [*real*]{} value of the quark mass is $\overline{m}_s (2\; \mbox{GeV})|_{\overline{MS}} = 102 \pm 8 \; \mbox{MeV}$, or some $20 \%$ below the former result (this error now includes the uncertainty in $\alpha_s$). In addition, and as a bonus the systematic uncertainty-free result is remarkably stable in the unusually wide region $s_0 \simeq 1 - 4 \; \mbox{GeV}^2$ (typical stability regions are only half as wide).\
It must be recalled that the pseudoscalar correlator involves the overal factor $(m_s + m_{ud})^2$. Hence, in order to determine $m_s$ an input value for the ratio $m_s/m_{ud}$ is needed in the result from the sum rule, which is $$\overline{m}_s (2\; \mbox{GeV})|_{\overline{MS}} = \frac{105.5 \pm 8.2 \; \mbox{MeV}}{1 + m_{ud}/m_s} \;.$$ Using the wide range $m_s/m_{ud} = 24 - 29$ leads to $\overline{m}_s (2\; \mbox{GeV})|_{\overline{MS}} = 102 \pm 8 \; \mbox{MeV}$. In this case the impact of the uncertainty in the quark mass ratio is small. However, in the case of the up- and down-quark masses the corresponding ratio $m_u/m_d$ plays a more important role in the result from the sum rule, which is $$\overline{m}_d (2\; \mbox{GeV})|_{\overline{MS}} = \frac{8.2 \pm 0.6 \; \mbox{MeV}}{1 + m_u/m_d} \;.$$ The input used in Eq.(26) for the ratio $m_u/m_d = 0.47 \pm 0.04$ is from the overall lattice QCD analysis of the FLAG group [@CHPT2]. Once $m_d$ is determined from Eq.(26), $m_u$ follows. Using these results for the individual masses one obtains the ratios $m_u/m_d$ and $m_s/m_{ud}$ shown in Table 1. The quark masses $\overline{m}_u (2\; \mbox{GeV})|_{\overline{MS}}$ and $\overline{m}_d (2\; \mbox{GeV})|_{\overline{MS}}$ also exhibit a remarkably wide stability region $s_0 \simeq 1 - 4 \; \mbox{GeV}^2$ [@mq]-[@IJMPA]. In Table 1 one finds a summary of the results for the light quark masses, and the ratios $R$ and $Q$ defined on the left hand side of Eqs. (2) and (5), together with the results of the Flag group [@CHPT2]. The values of the up- and down-quark masses and their ratios in Table 1 are slightly different from those in [@mq]-[@IJMPA] due to the input value for the ratio $m_u/m_d$ (in [@mq]-[@IJMPA] the value $m_u/m_d = 0.55$ was used). The various sources of errors in the quark masses discussed earlier combine into the final values given in Table 1. Having all but eliminated the systematic uncertainty from the hadronic resonance sector, the main source of error is now due to the strong coupling. Improved accuracy in the determination of $\alpha_s$ would then allow for a reduction of the uncertainties in the light quark masses.
Heavy quark masses
==================
Determinations of the charm- and bottom-quark masses are not affected by a lack of data, as there is plenty of experimental information from $e^+ e^-$ annihilation into hadrons [@PDG], except for a gap in the region $25 \;\mbox{GeV}^2 \lesssim s \lesssim 50\; \mbox{GeV}^2$ . On the theoretical side there has been very good progress on PQCD up to four-loop level [@QCD1]-[@QCD14]. The leading power correction in the OPE is due to the gluon condensate with its Wilson coefficient known at the two-loop level [@BROAD]. The correlator, Eq.(8), involves the vector current $J(x)\equiv V_\mu(x) = \bar{Q}(x) \gamma_\mu Q(x)$, where $Q(x)$ is the charm- or bottom-quark field. The experimental data is in the form of the $R_Q$-ratio for charm (bottom) production, which determines the hadronic spectral function. Modern determinations of the heavy-quark masses have been based on inverse moment (Hilbert-type) QCD sum rules, e.g. Eq.(12) with $f(s) = 1/s^n$. These sum rules require QCD knowledge of the vector correlator in the low energy region, around the open charm (bottom) threshold, as well as in the high energy region. A recent update [@KUHN10] of earlier determinations [@QCD1]-[@QCD3], [@QCD5]-[@QCD7] reports a charm-quark mass in the $\overline{MS}$ scheme accurate to $1 \%$, and half this uncertainty for the bottom-quark mass. However, the analysis of [@QCD14] claims an error a factor two larger for the charm-quark mass. It appears that the discrepancy arises from the treatment of PQCD. In fact, in [@QCD14] two different renormalization scales were used, one for the strong coupling and another one for the quark mass. This unconventional choice results in a much larger error in the charm-quark mass obtained from inverse (Hilbert) moment QCD sum rules. It does not affect, though, sum rules involving positive powers of $s$. In any case, the philosophy in current use is to choose the result from the method leading to the smallest uncertainty.\
Beginning with the charm-quark mass, an alternative procedure was proposed some years ago based only on the high energy expansion of the heavy-quark vector correlator [@KS1]-[@KS2]. This method was followed recently [@SB1], but with updated PQCD information and the inclusion of integration kernels in the FESR, Eq.(12), tuned to enhance/suppress contributions from data in certain regions. The first such kernel is the so-called [*pinched*]{} kernel [@PINCH1]-[@PINCH2] $$f(s) = 1 - \frac{s}{s_0} \;,$$ which is supposed to suppress potential duality violations close to the real s-axis in the complex s-plane. In connection with the charm-quark mass application, this kernel enhances the contribution from the first two narrow resonances, $J/\psi$ and $\psi(2S)$, and reduces the weight of the broad resonance region, particularly near the onset of the continuum. The latter feature is better achieved with the alternative kernel [@SB2] $$f(s) = 1 - \left(\frac{s_0}{s}\right)^2 \;,$$ which produces an obvious larger enhancement of the narrow resonances, and a larger quenching of the broad resonance region. This kernel has been used together with both the high and the low energy expansion of the vector correlator in [@SB2]. Since there are no data in the region $25 \;\mbox{GeV}^2 \lesssim s \lesssim 50\; \mbox{GeV}^2$, while the data for $s\gtrsim 50\;\mbox{GeV}^2$ agrees with PQCD, it is useful to introduce a kernel that will allow a suppression in the former region, e.g. $$f(s) = \mathcal{P}_n [x(s)] \;,$$ where $$x(s) = \frac{2 s - (s_0 + s_1)}{s_0 - s_1}\;,$$ with $s_0 > s_1$, and $\mathcal{P}_n(x)$ are the standard Legendre polynomials, i.e. $\mathcal{P}_1(x) = x$, $\mathcal{P}_2(x) =(5 x^3 - 3 x)/2$, etc., which satisfy the constraint $$\int_{s_1}^{s_0} s^k \;\mathcal{P}_n[x(s)]\;ds =0 \;,$$ where $s_1 \simeq 24 \; \mbox{GeV}^2$, and $s_0$ varies in the region where there is no data. These Legendre-type kernels provide extra weight to the well known resonance region on account of their rapid growth for $s < s_1$. The resulting charm-quark masses are essentially insensitive to the order of these polynomials, with $n=3-6$ giving answers differing at the $0.1 \%$ level.\
[cccccccc]{}\
\
Kernel & $\overline{m}_c(3\,\mbox{GeV})$ & EXP & $\Delta \alpha_s$ & $\Delta \mu$ & NP & $s_0$ & Total\
\
$1 - s/s_0$ & 983 & 9 & 1 & - & 1 & 16 &25\
\
$\mathcal{P}_{5}[x(s)]$ & 1007 & 22 & 1 & 8 & 2 & $< 1$ & 23\
\
$s^{-2}$ & 995 & 9 &3 &1 &1 &14 &17\
\
$1-(s_0/s)^2$ & 987 & 7 &4 &1 &1 &4 &9\
\
The results for $\overline{m}_c(3\,\mbox{GeV})$ in the $\overline{MS}$ scheme using various integration kernels are listed in Table 2, together with the various uncertainties. The merits of each kernel may be judged by its ability to minimize these uncertainties, in particular those that might be most affected by systematic errors, such as e.g. the experimental data. The kernel $f(s) = 1 - (s_0/s)^2$ appears to be optimal as it produces the smallest uncertainty due to the data, and is very stable against changes in $s_0$. Some recent determinations of the charm-quark mass are based on Hilbert moments with no $s_0$-dependent kernel, corresponding to the third row in Table 2. While there is no explicit $s_0$-dependence in Hilbert moments (the integrals extend to infinity), there is definitely a residual dependence when choosing the threshold for the onset of PQCD. From a FESR perspective, the major drawback of the kernel $1/s^2$ is clearly the poor stability against changes in $s_0$. In this regard, the Legendre polynomial kernels would be optimal, except that they have the largest uncertainty due to the data. An important remark is in order concerning the uncertainty due to changes in $s_0$. From current data it is not totally clear where does PQCD start. This problem not only affects FESR, with their explicitly obvious $s_0$-dependence, but also Hilbert moments with an implicit $s_0$-dependence, as there is no data all the way up to infinity. The range of variation in $s_0$ and its contribution to the uncertainty in the quark mass, as appearing in Table 2, is as follows. For rows 1, 3 and 4 $s_0 \simeq 15 - 23 \; \mbox{GeV}^2$, and for the Legendre polynomial kernels, row 2, it is $s_0 \simeq 100 - 200 \; \mbox{GeV}^2$. Two of the most recent results for $\overline{m}_c(3\,\mbox{GeV})$ [@KUHN10], [@QCD14], together with the weighted FESR value [@SB2] (last row in Table 2) are $$\begin{aligned}
\overline{m}_c(3\,\mbox{GeV}) \; = \;\Bigg\{
\begin{array}{lcl}
986 \; \pm 13 \;\mbox{MeV} \; \cite{KUHN10}\\
998 \; \pm 29 \;\mbox{MeV} \; \cite{QCD14} \\
987 \; \pm \;\; 9 \;\mbox{MeV} \; \cite{SB2}\;,
\end{array}\end{aligned}$$ in very good agreement with each other, except for the errors. The small uncertainty from [@SB2] is due in part to improved quenching of the data in the broad resonance region, but mostly due to a strong reduction in the sensitivity to $s_0$, i.e. the onset of PQCD. For comparison, a recent lattice QCD determination gives [@LATT1] $$\overline{m}_c(3\,\mbox{GeV}) \; = \; 986 \; \pm\; 6 \;\mbox{MeV}\;.$$ Turning to the bottom-quark mass, the most recent determination from Hilbert moment QCD sum rules [@KUHN10] is $$\overline{m}_b(10\,\mbox{GeV}) \; = \; 3610 \; \pm\; 16 \;\mbox{MeV}\;.$$ The error above is due to uncertainties in the data, the strong coupling and the renormalization scale. It does not include, though, an uncertainty due to the onset of PQCD. There is an appreciable difference if PQCD were to start at the end point of the (BABAR) data ($\sqrt{s_0} \simeq 11.2\; \mbox{GeV}$), or rather at higher energies ($\sqrt{s_0} \simeq 13.0 \;\mbox{GeV}$). If one were to adopt the procedure of [@SB1], i.e. add in quadrature the first three errors, but only linearly the one due to $s_0$, then using the information in [@KUHN10] the result would change to $\overline{m}_b(10\;\mbox{GeV}) \; = \; 3600 \; \pm\; 47 \;\mbox{MeV}$. Presumably, the error would be larger if a broader region in $s_0$ were to be considered. One of the virtues of weighted sum rules is to reduce this uncertainty, as may appreciated from Table 2 for the case of the charm-quark mass. For the bottom-quark mass work on weighted FESR is in progress [@bquark]. In any case, it should be kept in mind that by choosing different kernels it is possible to generate a large number of predictions for the quark masses with a variety of different errors. According to current philosophy one chooses the determination having the smallest error.
Conclusions
===========
After a short review of quark mass ratios the method of QCD sum rules was discussed, in connection with determinations of individual values of the quark masses. The historical (unknown) hadronic systematic uncertainty affecting light quark mass determinations was highlighted. Details of the recent breakthrough in eliminating this uncertainty were provided. Future improvement in accuracy is now possible, and depends essentially on more accurate determinations of the strong coupling, now the main source of error. The new values of the light quark masses and their ratios, free from this systematic uncertainty, agree well with lattice QCD results. In the heavy quark sector recent more accurate determinations of the charm- and bottom-quark masses were reported. While these values are all in agreement, there is some disagreement on the size of the errors. The use of suitable multipurpose integration kernels in FESR allows to tune the weight of the various contributions to the quark masses. This in turn allows to minimize the error due to the data, as well as to the uncertainty in the onset of PQCD. The latter uncertainty impacts FESR as well as Hilbert moment sum rules, as there is no data all the way up to infinity. If no kernel, other than inverse powers of $s$ is used then this uncertainty would be much larger than normally reported, as may be appreciated from Table 2. However, according to current philosophy one chooses the determination having the smallest error. Marginal improvement of the current total error in this framework should be possible with improved accuracy in the data and in the strong coupling.
Acknowledgements
================
The author wishes to thank his collaborators in the various projects on quark masses reported here: S. Bodenstein, J. Bordes, N. Nasrallah, J. Peñarrocha, R. Röntsch and K. Schilcher. Enlightening correspondence with H. Leutwyler and P. Minkowski is greatly appreciated. This work was supported in part by NRF (South Africa).
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---
abstract: 'We present the first interferometric observation of a zebra-pattern radio burst with simultaneous high spectral ($\approx 1$ MHz) and high time (20 ms) resolution. The Frequency-Agile Solar Radiotelescope (FASR) Subsystem Testbed (FST) and the Owens Valley Solar Array (OVSA) were used in parallel to observe the X1.5 flare on 14 December 2006. By using OVSA to calibrate the FST the source position of the zebra pattern can be located on the solar disk. With the help of multi-wavelength observations and a nonlinear force-free field (NLFFF) extrapolation, the zebra source is explored in relation to the magnetic field configuration. New constraints are placed on the source size and position as a function of frequency and time. We conclude that the zebra burst is consistent with a double-plasma resonance (DPR) model in which the radio emission occurs in resonance layers where the upper hybrid frequency is harmonically related to the electron cyclotron frequency in a coronal magnetic loop.'
author:
- 'Bin Chen$^{1,2}$, T. S. Bastian$^2$, D. E. Gary$^3$, and Ju Jing$^4$'
bibliography:
- 'apj-jour.bib'
- 'FSTpaper.bib'
title: Spatially and Spectrally Resolved Observations of a Zebra Pattern in Solar Decimetric Radio Burst
---
INTRODUCTION
============
Fine structures in the solar radio bursts - in both the time and frequency domains - have been studied for many years. They are believed to embody important information about charged particle acceleration processes, particle dynamics, and emission mechanisms [@1994SoPh..153..403F]. Many such fine structures - type III bursts and their variants, spike bursts, pulsations, fiber bursts - are believed to be the result of non-equilibrium processes in the coronal plasma. Zebra-pattern radio bursts (Slottje 1972; hereafter “zebra pattern” will be abbreviated to “ZP”) are one of the most striking examples of such fine structures.
The observed properties of ZP radio bursts have been presented in detail in the review by @2006SSRv..127..195C and are reiterated briefly here. ZP bursts appear in radio dynamic spectra as closely-spaced, quasi-parallel bands of emission, typically ranging from $\sim\!5-20$ in number but sometimes showing as many as 70. They have been observed at meter wavelengths for decades [@1959Natur.184..887E; @1972SoPh...25..210S; @1975PhDT.........1K]; more recently, similar structures have been reported at decimeter and centimeter wavelengths. For the purposes of discussion we denote the instantaneous frequency of a single ZP emission band or stripe by $f_e$, the frequency bandwidth of a ZP emission band by $\Delta f_e$, the separation between adjacent ZP emission bands by $\Delta f_{s}$, the mean frequency of two adjacent emission bands as $f_m$, the mean frequency of the ZP emission bands as a whole by $\langle f_e\rangle$, and the overall frequency bandwidth occupied by ZP emission bands by $\Delta f_{tot}$. Generally, $\Delta f_{tot}/\langle f_e\rangle$ decreases with frequency whereas $\Delta f_s/\langle f_e\rangle$ increases with frequency; the relative bandwidth of individual ZP emission bands $\Delta f_e/f_e$ shows no obvious trend with frequency and is typically $\lesssim\!1\%$ [@2006SSRv..127..195C]. There are few reports of the brightness temperature $T_B$ of ZP bursts. In those cases where such constraints are available, the brightness temperature is typically very high: @1994SoPh..155..373C estimated the $T_B$ of a metric ZP to be $\approx 10^{10}$ K with the source size constrained by the Nançay Radioheliograph (NRH); In , a decimetric ZP that consisted of spiky superfine structures was estimated to have $T_B \gtrsim 10^{13}$ K by assuming the burst had the same source size as a spike burst; used the Siberian Solar Radio Telescope (SSRT) to observe a ZP burst at $\approx 5.7$ GHz, the highest frequency ever reported for ZP emission, which yielded a lower limit of $T_B\approx2\times10^7$ K, the source size being $\lesssim 10''$. ZP bursts are typically observed during the impulsive and/or decay phases of the flares. They are typically polarized in the sense of the ordinary wave mode and the degree of polarization can be very high. The durations of ZP bursts can vary from a few minutes down to a few seconds at meter wavelengths to decimeter/centimeter wavelengths, respectively. The narrow-band features, high degree of circular polarization, and indications of high brightness temperatures suggest that the corresponding emission mechanism is coherent.
ZP bursts often appear with the presence of type IV continuum emission (hereafter “continuum”). Many other fine structures, including type III bursts, broad band pulsations (BBP), fiber bursts, and spikes accompany, or are associated with, ZP emission. There are also some rare examples where ZPs consist of pulsating superfine structures [@2007SoPh..241..127K; @2007SoPh..246..431C; @2008SoPh..253..103K], and they appear in fast-drift, type-III-like, absorption features [@2009SoPh..255..273Z], which could be related to fast electron beam injections into the magnetic trap.
There is no broadly accepted interpretation for ZP emission. Several types of models purport to explain the ZP phenomenon [see @2006SSRv..127..195C; @2009CEAB...33..281Z]. Most involve the growth and conversion of electrostatic wave modes to transverse modes. One type of model suggests that all the ZP stripes originate from the same discrete source, the dimensions of which are assumed to be small enough for the inhomogeneity of the plasma density and the magnetic field to be neglected. In these models, zebra stripes are assumed to be simultaneously generated at several harmonics of the local electron cyclotron frequency, due to nonlinear couplings of Bernstein waves with each other or with upper-hybrid waves (hereafter “Bernstein models”) [e.g. @1972SoPh...25..188R; @1975SoPh...43..431Z; @1975SoPh...44..461Z; @1983SoPh...88..297Z].
A second type of ZP model is based on trapping upper hybrid Z mode waves in density inhomogeneities [@2003ApJ...593.1195L]. The trap results in a discrete spectrum of eigenfrequencies. The model depends on the emission by many such discrete traps distributed over a larger volume.
Models based on propagation phenomena have also been proposed. and @2006PlPhR..32..866L suggest that coronal fine structure can behave as an optical filter or produce Bragg-like reflections, resulting in regular emission bands. Alternatively, interference between direct and reflected rays from a coherent source have been suggested .
Another class of models argues that ZPs are related to an extended source filled with energetic electrons. The different zebra stripes originate from different locations in the extended source, where resonance conditions are fulfilled. The most popular model of this kind is the so-called double plasma resonance (DPR) model, first proposed by @1975SoPh...43..431Z [@1975SoPh...44..461Z], and subsequently developed by several authors [e.g. @1986ApJ...307..808W; @2007SoPh..241..127K]. In this class of models, upper-hybrid waves are generated most efficiently at locations where the double plasma resonance occurs: $$f_{UH}=(f^{2}_{pe}+f^{2}_{ce})^{1/2}=sf_{ce}$$ where $f_{UH}$ is the upper-hybrid frequency, $f_{pe}$ is the electron plasma frequency, $f_{ce}$ is the electron cyclotron frequency, and $s$ is the harmonic number. The distribution of the DPR levels in the flare loop is determined by spatial gradients in the plasma density and magnetic field.
Finally, models are based on propagation of whistler wave packets [@1976SvA....20..582C; @1990SoPh..130...75C] across, or along, the magnetic trap where the energetic electrons generate Langmuir waves (hereafter “whistler model”). ZPs are produced by coalescence of the Langmuir waves (*l*) and whistlers (*w*) through the process $l+w\longrightarrow t$, where *t* stands for transverse waves that can be observed as emission near the local plasma frequency. They propose that ZPs as a whole are the manifestation of the ensemble of periodically generated whistler wave packets propagating in the magnetic trap, in which each zebra stripe corresponds to one propagating whistler wave packet. In this way, zebra stripes can be separated regularly from each other in height (and emit at different frequencies) by a distance determined by the whistler propagation velocity and time interval of generating the whistlers.
Both @2006SSRv..127..195C and @2009CEAB...33..281Z discuss each of these models and summarize their strengths and weaknesses. Many display significant theoretical shortcomings. In light of these shortcomings, Chernov favors whistler models. In contrast, Zlotnik favors DPR models. We will therefore direct most of our attention toward the last two classes of models - DPR and whistler models - in subsequent discussion.
As for the type IV continuum emission associated with ZP bursts, it is assumed by most models that it arises from fast electrons trapped magnetically in the coronal loops. However, the relationship between the continuum and ZP emission varies from model to model. In the Bernstein and DPR models, it is suggested that the only difference between their formation is whether these trapped electrons favor conditions for creating the zebra pattern or not, which is most likely the forms of anisotropic momentum distributions. In the whistler model, the ZP is formed through interactions of the plasma waves that can result in the type IV continuum. The propagation models, however, suggest that either the ZP sources reside in the continuum source [@2006SoPh..233..129L], or ZP emission as well as the “background” continuum as a whole are just type IV continuum emission modulated by the inhomogeneous medium . However, because of the lack of spatial information, the physical relationship between the emission sources of ZPs and type IV continuum is still not known.
The means of placing meaningful constraints on models for ZP and continuum emission has been limited by the unavailability of spatial resolution at each of the frequencies and times recorded by the dynamic spectrum. Interferometric observations of ZP emission at even a single frequency are relatively few in number. Several examples are provided by fixed-frequency observations of ZP by the NRH in combination with observations made by a spectrometer . Other examples have been provided by and who combined interferometric observations of ZP made by the SSRT at 5.7 GHz with spectroscopic observations obtained by the Chinese Solar Broadband Radio Spectrometer (SBRS/Huairou).
In the present paper we describe the first interferometric observations of ZP emission for which both high-time resolution and high-spectral resolution observations are simultaneously available over a significant frequency bandwidth. The relevant instrumentation is described in §2. The ZP event and its analysis are described in §3. The event is placed in a physical context in §4, where we argue that the data are consistent with a DPR model. We conclude in §5.
Instrumentation {#sect:instr}
===============
The observations were obtained by the Frequency-Agile Solar Radiotelescope (FASR) Subsystem Testbed (FST) [@2007PASP..119..303L]. FASR is a next generation solar radio telescope [@2004ASSL..314.....G] designed to provide simultaneous imaging and spectroscopic observations over a large bandwidth, with high angular, time, and spectral resolutions. The FST is a prototype and testbed system for FASR. As such, it is the first system with the ability to combine Nyquist-limited high time and frequency resolution with interferometric ability to locate sources [@2007PASP..119..303L].
The FST uses the existing antenna system of the Owens Valley Solar Array (OVSA). OVSA [@1994ApJ...420..903G] is a solar-dedicated interferometric array that is composed of two 27-meter antennas and five 1.8-meter antennas. OVSA can observe the Sun at up to 86 frequencies in the range 1-18 GHz. The FST employs three of the 1.8-meter OVSA antennas (antenna number 5, 6 and 7) as shown in Figure \[fig:ants\]. The longest baseline, between antennas 6 and 7, is nearly 280 meters which yields a minimum fringe spacing of 221$''$ at 1 GHz. @2007PASP..119..303L describe the FST system configuration in detail, and we briefly reiterate it here. A radio frequency (RF) splitter divides the output signals of the three OVSA/FST antennas to simultaneously feed the OVSA receivers and the FST. Thus, the FST can be used in parallel with OVSA, observing the same source as OVSA, without affecting OVSA’s normal operation. FST signals are amplified and transmitted by a broadband optical fiber to a block down-converter. The 1-5 GHz or 5-9 GHz band of RF signal from each antenna is selected and down-converted to 1-5 GHz as necessary. The spectral-line down-converter is used to select a single-sideband, 500 MHz band which is tunable anywhere within the 1-5 GHz band. The selected 500 MHz band is further down-converted it to an intermediate frequency (IF) band of 500-1000 MHz. The 500 MHz IF band is sampled at 1 Gsps and the data are written to disk. The full-resolution time-domain data are then correlated offline using a software correlator (written in IDL) to produce amplitude and phase spectra on the three interferometric baselines. For daily solar observing, the system employs a time resolution of $\approx\!20$ ms and a frequency resolution of $0.98$ MHz.
The small size of the FST antennas precludes calibration of the FST against sidereal radio sources. However, since the event was observed by both FST and OVSA in parallel, we can make use of the OVSA observations to cross-calibrate FST. OVSA is calibrated against sidereal standards in order to determine the complex gain of each antenna. However, OVSA only samples the RF spectrum at 1.2 and 1.4 GHz. Moreover, the 1.2 GHz OVSA data were found to be corrupted and were therefore unusable. Therefore, the FST 1.4 GHz data were averaged in frequency and time to match the OVSA data. The calibrated OVSA amplitudes and phases of antenna baselines 5-6, 6-7, and 5-7 were then compared to those measured by FST, and the FST amplitudes and phases were corrected accordingly. As Figure \[fig:comp\] shows, they agree with each other quite well after the cross-calibration. However, this process doesn’t directly allow bandpass calibration of FST. Based on an examination of broadband continuum emissions that occupy the entire frequency band, we conclude that the FST has approximately linear bandpass patterns in phase with good stability in time on all three baselines. We therefore applied linear fits as the bandpass correction to the phases.
Observations
============
The FST observed the powerful GOES class X1.5 flare that occurred on 2006 Dec 14 in NOAA/USAF active region 10930 at S06W46, the site of a X3.4 flare the previous day [see, e.g., @2007PASJ...59S.785S]. The flare on Dec 14 was accompanied by a fast halo CME and a solar energetic particle event. Figure \[fig:goes\_rstn\]a shows the GOES SXR light curve; the flare started at 21:07 UT, peaked at 22:15 UT, and ended at around 04:00 UT on December 15. The radio time profiles at 1.415 GHz, 2.695 GHz, and 8.8 GHz obtained by the Radio Solar Telescope Network (RSTN, operated by the U.S. Air Force) are shown in Figure \[fig:goes\_rstn\]b$-$d. Note the difference in scale between the intense 1.415 GHz emission and that at 2.695 and 8.8 GHz.
The two X-class flares on 2006 Dec 13 and 2006 Dec 14 were observed by the X-ray Telescope (XRT) [@2007SoPh..243...63G] and the Solar Optical Telescope (SOT) [@2008SoPh..249..167T] aboard [*Hinode*]{} [@2007SoPh..243....3K]. Most of the XRT images were taken with a $512''\times 512''$ field of view (FOV) and a cadence of 60 s. Vector photospheric magnetograms were obtained by the SOT SpectroPolarimeter (SP). The slit length and width of SOT/SP are $164''$ and $0.16''$, respectively. For each magnetogram with a $297''\times 164''$ FOV, the slit scanned from east to west for about one hour, with a resolution of $0.297''$ and $0.320''$ in the east-west and north-south directions, respectively. The SOT Ca II H band samples the chromospheric structure at a very high spatial resolution ($0.1”$ per pixel). Like H$\alpha$ this band is sensitive to plasma heating by precipitating electrons and/or thermal conduction. The images were taken with a $218''\times 109''$ FOV with a cadence of 120 s. Figure \[fig:SOTSP\]a and b give the longitudinal magnetogram at the photospheric level $B_z(0)$ observed from 22:00:05 to 23:03:16 UT, and an example of a Ca II H image at 22:37:35 UT.
Figure \[fig:ca2h\_n\_xrt\] shows the Ca II H (Figure \[fig:ca2h\_n\_xrt\]a$-$c) and XRT (Figure \[fig:ca2h\_n\_xrt\]d$-$f) images at times prior to the flare (a and d), during the flare maximum (b and e), and during the decay phase (c and f) at the time of the ZP. The core flare region (marked as region “1” in Figure \[fig:SOTSP\]b) is located between the two sunspots with opposite polarity. According to @2007PASJ...59S.785S, the flare loops inside this region evolve from highly sheared in the pre-flare phase into less sheared in the post-flare phase, based on the XRT observations. Except for the flare core region, there are two other major bright regions in the Ca II H images, where the brightness of soft X-ray loops is enhanced at the same time, as shown in the XRT images. One is located about $50''$ to the west of the core region (region “2” in Figure \[fig:SOTSP\]b), and another is located to the north-west near the larger sunspot with negative polarity (region “3” in Figure \[fig:SOTSP\]b).
A more subtle Ca II H brightening is observed at the time of the ZP (at about 22:40 UT), indicated by a white arrow in Figure \[fig:ca2h\_n\_xrt\]c). This brightening appeared in Ca II H images at around 22:10 UT (near the flare maximum) and persisted for more than an hour.
FST ZP Observations {#obs:fst}
-------------------
The FST observed the 2006 Dec 14 flare over a frequency range of 1.0$-$1.5 GHz. The instrument observed both right- and left-circularly polarized radiation, switching between the two polarizations every 4 s. A spectrum of 511 channels across the 500 MHz bandwidth was produced every $\approx\!20$ ms. The 2006 Dec 14 flare produced a high level of type IV burst activity in the post-flare phase for more than one hour. A rich variety of fine structures was observed in the 1.0$-$1.5 GHz band, including ZP, fiber bursts, pulsations, and others. The time of the ZP event is marked by the vertical line in Figure \[fig:goes\_rstn\]b, at about 22:40 UT, during the decay phase.
### Total Power Dynamic Spectrum
Figure \[fig:zebra\]a shows a dynamic spectrum of roughly 1 s of total power data near 22:40:07 UT. A striking ZP radio burst is present. The data are right-circularly polarized (RCP); the left-circularly polarized (LCP) observations made just prior to those presented here showed no ZP emission. We conclude that the ZP burst is highly right-circularly polarized. A precise measure of the degree of polarization is not possible, however, because the noise in the LCP observations is dominated by polarization leakage from the RCP channel. The ZP burst shows as many as 12 distinct bands or stripes superposed on a type IV-like continuum background centered on $\langle f_e \rangle \approx 1.32$ GHz and with a frequency bandwidth $\Delta f_{tot}$ of up to 150 MHz, or $\Delta f_{tot}/\langle f_e\rangle \gtrsim 10\%$. All of the zebra stripes drift in frequency together irregularly with time. Figure \[fig:zebra\]c shows a histogram of the drift rates of the zebra stripes in the dashed box marked in Figure \[fig:zebra\]a. An average drift rate of about $-$50 MHz s$^{-1}$ is indicated by the vertical line. The overall trend is to drift from higher to lower frequencies. The drift rates of the zebra stripes are mainly between $-100$ MHz s$^{-1}$ to 100 MHz s$^{-1}$, comparable to, but generally slower than, those of the so-called “fiber burst” in the same frequency range [@1982ITABO..53.....E].
The frequency profile of the ZP emission (averaged in the time denoted by the small solid box in Figure \[fig:zebra\]a) is shown in Figure \[fig:zebra\]b. The contrast of each stripe is defined by $(I_{on}-I_{off})/I_{off}$ (where $I_{on}$ and $I_{off}$ are the intensities “on” and “off” a zebra stripe, respectively) and is shown in Figure \[fig:zebra\]d. The relative frequency separations between adjacent zebra stripes $\Delta f_s/f_m$ in the dashed box of Figure \[fig:zebra\]a are shown in Figure \[fig:zb\_spacing\]a as diamonds. They are color-coded in time from blue to red. One can clearly see that $\Delta f_s/f_m$ increases with frequency. This phenomenon is commonly seen in zebra patterns reported by other authors in both the meter and decimeter wavelength range .
### Apparent ZP Source Size {#obs:srcsize}
With only three antenna baselines it is not possible to image the ZP source. However, we are able to constrain the source size using the visibility amplitudes as a function of antenna baseline. We assume that we can characterize the source brightness distribution as a symmetrical Gaussian. The visibility function is then likewise a Gaussian and a simple model, characterized by a single spatial scale, can be fit to the normalized visibility amplitudes as a function of spatial frequency. We have fit the source at frequencies “on” the bright zebra stripes (hereafter “on-stripe source”) and at frequencies “off” the zebra stripes; that is, in between the zebra stripes (hereafter “off-stripe source”). As is shown in Figure \[fig:srcsize\]a, a simple Gaussian model adequately fits both the on- and off-stripe sources. The visibility amplitudes of the three baselines have been normalized by the total power measured by each antenna. Figure \[fig:srcsize\]b shows the corresponding Gaussian FWHMs of the on- and off-stripe sources in the spatial domain for the six zebra stripes in Figure \[fig:zebra\]a. They both have values near $50''$, but the sizes of the off-stripe sources are systematically larger than those of the on-stripe sources by $\approx\!10''$, except for the stripes at lower frequencies where the contrast is low (see Figure \[fig:zebra\]d), a point to which we return in §3.1.3.
The uncertainty of the source size estimation depends on the accuracy of normalized amplitude measurements. Within the context of the model a difference of 20% in the inferred FWHM of the on- and off-stripe sources implies that the normalized amplitude of the off-stripe source on the longest antenna baseline (antennas 6-7) is lower than that of the on-stripe source by $\gtrsim 10\%$. The normalized amplitude measurements used in our fitting are averaged values along each zebra stripe. Therefore, the systematic errors can be represented by the statistical standard deviation of the mean of the sample, which are respectively $\lesssim~1\%$ and $\lesssim~2\%$ for the on- and off-stripe sources. We therefore regard errors as large as 10% as unlikely and conclude that the on-stripe source is marginally more compact than the off-stripe source. Qualitatively similar results were obtained by @1994SoPh..155..373C using one-dimensional NRH observations at a much lower frequency of 164 MHz.
Note that both sources are likely strongly affected by scattering. @1994ApJ...426..774B showed that scattering by the overlying inhomogeneous corona can play an important role in modifying the angular structure of the emission source at wavelengths longer than a few centimeters. The estimated source sizes of $\approx\!50''$ are consistent with angular scattering and the intrinsic source sizes could be significantly smaller. Given an apparent source size of $50''$, the lower limit of the brightness temperature of ZP source can be estimated to be $\approx\!10^9$ K.
### Relative Locations of the ZP and Continuum Sources
We now ask whether the on- and off-stripe emissions originate from two different source locations. If they are indeed separated from each other spatially, how are they related to each other? Second, how do the source locations vary with time? Finally, do the on- and off-stripe source locations have any significant dependence on frequency? In other words, does the radiation at different frequencies come from different spatial locations or not?
In interferometry, the interferometric phase is a direct response to the spatial information of the radiation source. We first consider the “dynamic phase spectrum” of the ZP emission for the three baselines to gain a qualitative impression (Figure \[fig:zb\_phz\]). For baselines 5-6 and 6-7, zebra stripes can be distinguished as darker colors compared with the background continuum. This means that the on- and off-stripe phases are measurably different. In addition, the phase difference seems larger in the upper-left region (higher frequencies and earlier times) than later in the event. For baseline 7-5, however, no phase differences are evident between on-stripe and off-stripe emissions.
In order to characterize on- and off-stripe phases quantitatively, we employ “phasor diagrams”. In a phasor diagram, the amplitude and phase of each measurement are displayed in a polar plot. We have made phase measurements for both the on- and off-stripe emissions of the spectral fragment indicated by the dashed box in Figure \[fig:zebra\]a, which includes six zebra stripes in an interval of 0.48 s with 24 consecutive observations (from 22:40:06.86 UT to 22:40:07.34 UT). The rms phase noise of a single data point (duration 20 ms and bandwidth roughly 1 MHz), is about 5 degrees, too large for tracking the phase variations in time and frequency among individual measurements. To increase the signal-to-noise ratio we averaged the data as follows. First, we obtained the on- and off-stripe phases by averaging the three frequencies about the flux maximum/minimum at each time. Second, we considered averages of the phases “along” the on- and off-stripe positions (in time) and “across” the six on- and off-stripe positions (in frequency). By averaging along the zebra stripes, we constrain the phase variations of the on- and off-stripe sources as a function of frequency, so that the spatial distribution of the ZP source over frequency can be revealed. Implicit in this treatment is the assumption that the ZP source is likely to maintain the same structure as a function of frequency during the brief averaging time. Similarly, by averaging the on- and off-stripe phases in frequency, we track the phase variation of the ZP source with time, so that the evolution of the source centroid location with time can be seen, considering that different stripes of the ZP show the same drift motion according to the dynamic spectra. This approach allowed us to reduce the statistical phase error by a factor of several, and to show systematic variations in the on- and off-stripe phases in the phasor diagrams with increased signal-to-noise ratio.
Figure \[fig:phasor\_t\] shows the results for the on-stripe (pluses) and off-stripe (triangles) emission after averaging across the six zebra stripes. The pluses and triangles are colored from dark blue to red to indicate the variation of amplitudes and phases in time. The dashed lines in each panel represent increments of $5^\circ$ in phase. It can be seen that the on-stripe phases of baseline 5-6 and 6-7 drift by $\approx\!9^\circ$ within 0.48 s towards the off-stripe phases, while the off-stripe phases show no evident drift. No significant phase drift is seen for baseline 7-5, thus the direction of the spatial drift is nearly along the 7-5 interferometric fringe, which is NE-SW at the time of observation. By applying the fringe spacings (which are respectively 274$''$, 176$''$, and 439$''$ for baselines 5-6, 6-7 and 7-5 at the time of observation) and orientations of the three FST baselines (see Figure \[fig:SOTSP\]b), this amount of phase drift in time can be translated into a spatial drift of $15.6\pm6.5''$ from NE to SW on the solar disk, corresponding to a projected drift velocity of $2.5\pm1.0\times10^9$ cm$~s^{-1}$ ($\approx\!0.1c$). The estimated spatial error is based on the rms error of the FST phase measurements ($\approx\!5^\circ$), the total number of data points averaged, and triangulation of the three fringes. The fringe orientations of the three baselines yield the error to be orientationally dependent, which is larger in the NE-SW direction than the SE-NW direction.
In addition, there is an evident difference in mean phase of about $6^\circ$ between the on- and off-stripe emissions for baseline 5-6 and 6-7, shown by the arrows in Figure \[fig:phasor\_t\]. That means the on- and off-stripe emission are separated in space by an average of $8\pm0.9''$ in the NE-SW direction.
In the two bottom panels of Figure \[fig:phasor\_t\] the ZP phases shown in the top-left and top-middle panels (baselines 5-6 and 6-7) are plotted as a function of time. The solid line, repeated in each panel, represents the ZP frequency as a function of time (note that the frequency scale on the right axis is reversed - with frequency decreasing from high to low values). Although the variations in time are irregular, there is a rather good correlation between the phase and the frequency; stated another way, there is a good correlation between the ZP source centroid position and the mean ZP frequency.
Figure \[fig:phasor\_f\] shows similar phasor diagrams. Here, however, the data are averaged along the zebra stripes, instead of across the zebra stripes as above, and the amplitude and phase variations are shown as a function of frequency. The data are color-coded from black to red for stripes with decreasing frequencies. The pluses and triangles denote the on- and off-stripes sources, respectively. The amplitudes of the off-stripe source barely change, but those of the on-stripe source drop with decreasing frequency. There is still little change in phase for baseline 7-5, but both the on- and off-stripe phases shift by several degrees monotonically across the six stripes. The shifts of on-stripe phases seem to be slightly larger than those of the off-stripe phases. Therefore, it appears that the source centroid positions of both the on- and off-stripe sources show a systematic displacement with frequency, which is respectively $14.6\pm2.8''$ and $8.3\pm2.8''$ across the six zebra stripes from NE to SW.
The spatial difference between the on-stripe and off-stripe emissions demonstrates that there are indeed two spatially separated sources that contribute to the event, namely, the zebra-stripe emission and the background continuum source. This, too, has been noted in previous examples of ZP events using one-dimensional NRH observations at much lower frequencies [@2006SSRv..127..195C and references therein]. Care must be taken in further interpretation of the phasor diagrams. The off-stripe emission is dominated by the continuum source. However, the on-stripe emission has contributions from both the continuum and ZP sources. Therefore, a change of relative intensities of the zebra and continuum sources can result in an apparent position shift of the emission centroid measured at on-stripe frequencies. The phase drift of the on-stripe emission in time (Figure \[fig:phasor\_t\]) is not in question in this respect because the amplitudes barely change in time along a given stripe. However, there is an obvious change of on-stripe amplitude with frequency at a given time - the amplitude decreases from high to low frequencies and the phase tends toward the value of the off-stripe emission (Figure \[fig:phasor\_f\]), which could be caused by the intensity modulation itself. In order to correct for this effect, we vector-subtract the complex amplitudes of the off-stripe source from those of the on-stripe source in the phasor diagram, and plot the actual phases of the zebra source as “X” symbols in the first and second panels of the bottom row in Figure \[fig:phasor\_f\]. After the correction, the ZP phases of the baseline 5-6 and 6-7 still show monotonic displacements as a function of frequency, but now they become comparable to those of the continuum (off-stripe) phases. Again converting to angular displacements, the corrected ZP source centroid has a displacement of $8.5\pm2.8''$ across the six zebra stripes with a direction from NE to SW, which is smaller than the value we obtained previously for the on-stripe source ($14.6\pm2.8''$) but comparable to that of the continuum (off-stripe) source ($8.3\pm2.8''$).
The variation in the on-stripe source sizes seen in Figure \[fig:srcsize\]b can also be explained by the variation of intensity contrast. The off-stripe source sizes are nearly unaffected by the contrast variation because the continuum source continuously dominates the emission at off-stripe frequencies. But since the on-stripe emission is a mixture of zebra and continuum emission, the variation in contrast plays an important role. At high contrasts (e.g., $P\geqslant 1.5$ at the Nos. 1$-$4 stripes), the zebra source dominates the on-stripe emission, and the source size measurements primarily reflect the property of the zebra source. At lower contrasts (e.g. $P\leqslant 1$ at the Nos. 5$-$6 stripes), the contribution from the continuum source becomes increasingly important and the source size measurements start to reflect the property of the continuum source. Therefore, the on-stripe source sizes start to increase at lower-frequency zebra stripes with decreasing intensity contrasts, and reach the size of the off-stripe source, as shown in Figure \[fig:srcsize\]b. If we do the vector subtraction on the on-stripe source as above, the corrected relative intensities of zebra source can be obtained, hence their sizes can be fitted accordingly and plotted in Figure \[fig:srcsize\]b as “X” symbols. An up to $20''$ difference between the zebra and continuum (off-stripe) source sizes can be seen after the correction (note the fitted value of the lowest frequency stripe is unreliable, because its relative intensities have large errors that comes from the subtraction of two sources with comparable intensities). We suggest that the size difference is real and indicates the zebra source is yet more compact ($35-40''$) than the source size deduced in §3.1.2 ($45-50''$) and the lower limit to the ZP brightness temperature should therefore be increased by $\approx 60\%$ to $1.6 \times 10^9$ K.
### Absolute ZP Source Location
Once calibration of the FST was achieved, as described in §\[sect:instr\], we could locate the absolute source positions by using the three interferometric fringes corresponding to the three antenna baselines to triangulate. The averaged location (in time and frequency) of the ZP source from 22:40:06.86 to 22:40:07.34 UT on the solar disk is shown in Figure \[fig:SOTSP\] as the intersection of the three fringes, with a dashed circle representing the apparent source size of $\approx\!50''$. The error of the source location is between 1.3$''$ and 3.5$''$ depending on direction. The ZP source is located $\approx\!100''$ to the west of the main flaring emission. For comparison, the OVSA 4.6$-$6.2 GHz map at the same time is over-plotted as contours (the levels have an increment of 10% of the maximum), showing that the higher-frequency emission is well-correlated with the three Ca II H bright regions (Figure \[fig:SOTSP\]b). The 4.6$-$6.2 GHz source is far less intense than the ZP emission (cf. Figure \[fig:goes\_rstn\]) and is due to incoherent, non-thermal gyrosynchrotron emission.
The variation of relative source centroid locations in time and frequency, as was discussed in §3.1.3, can be now seen from the change of absolute locations on the solar disk. The variation of source location along the zebra stripes – that is, from 22:40:06.86 to 22:40:07.34 UT – is shown in Figure \[fig:zbpos\]a. The pluses denote the on-stripe source locations, with the connecting arrow representing the projected spatial drift of $15.6''$ from NE to SW in 0.48 s. The error bar in the lower-left corner gives the estimated error of the on-stripe location. The off-stripe continuum source location is denoted by a single triangle because it does not display a significant drift in time, with the error bar plotted in the lower-right corner. Figure \[fig:zbpos\]b shows the source locations as a function of frequency across six zebra stripes. The on- and off-stripe source locations from stripes numbered 1 to 6 (decreasing in frequency) in Figure \[fig:zebra\] are marked by the pluses and triangles. The “X” symbols show the “actual” locations of the zebra source after removing the effect of relative intensity variations. The error is denoted by the error bar in the lower-left corner. The connecting arrows show the variation of the source locations with decreasing frequency. We can see that as the frequency decreases, the zebra and continuum (off-stripe) source locations both shift from NE to SW by about $8-9''$, and they are separated from each other by $\approx 11''$.
Magnetic field configuration {#sec:nlfff}
----------------------------
We are interested in constraining the location of the ZP source within the 3D magnetic field configuration. We performed a nonlinear force-free field (NLFFF) extrapolation based on the photospheric vector magnetogram obtained by the SpectroPolarimeter (SP) of the Solar Optical Telescope on board [*Hinode*]{}. The SOT/SP measures Stokes profiles of two magnetically sensitive Fe lines at 630.15 and 630.25 nm. We started from the SOT/SP level 2 data that are inverted using the “MERLIN" inversion code from the polarization spectra. The 180$^{\circ}$ azimuthal ambiguity in the transverse magnetogram was resolved using the “minimum energy” method [@2006SoPh..237..267M]. The observed vector magnetogram in the image plane (in heliocentric coordinates) was then transformed to heliographic Cartesian coordinates. Because the photospheric magnetic field is not necessarily force-free, an assumption inherent to NLFFF extrapolation as a boundary condition, and since the measurements contain inconsistencies and noise, the measured photospheric magnetic field was preprocessed to mitigate these effects. Here we used the preprocessing method developed by @2006SoPh..233..215W. Finally, we performed the NLFFF extrapolation using the resulting magnetogram in the heliographic Cartesian coordinates using the weighted optimization method [@2004SoPh..219...87W], which is an implementation of the original method of @2000ApJ...540.1150W. Best results for a NLFFF extrapolation are achieved when positive and negative magnetic flux are balanced; the FOV we selected to perform the extrapolation is therefore $244''\times 163''$, centered on the active region. The extrapolation was then calculated on a 240$\times$160$\times$160 grid with a resolution of $1.02''$ in the x-, y- and z-directions (corresponding to the directions of west, north and normal to the tangent plane centered on the active region).
In Figure \[fig:SOTSP\], we have already seen that the location of the ZP source is nearly $100''$ away from the active region center in the image plane. Since the active region is at S06W46, the ZP source is apparently located relatively high in the solar corona above the active region. The projected ZP source location is known but the location along the line of sight is unknown. We obtained a series of possible 3D locations of the radio source, consistent with its projected location, and plot them in Figure \[fig:nlfff\]. The zebra and continuum source locations (averaged in time and frequency) are marked as “X” symbols and triangles respectively, and color-coded in magnetic field strength (from light to dark blue, the magnetic field changes from $\approx\!90$ G to $\approx\!30$ G at coronal heights from $\approx\!40$ Mm to $\approx\!80$ Mm). It can be seen that as the radio source locations are placed higher in the corona, the magnetic field strength decreases, and they move to a position more nearly over the large sunspot with negative polarity. A group of extrapolated field lines in blue colors is drawn passing these possible source locations, showing a post-flare loop system that is connected with the large sunspot with negative polarity. The polarity of these field lines suggest that since the ZP is RCP, it is polarized in the sense of the ordinary mode. We note, too, that a Ca II H brightening pointed out at the beginning of §3 (in Figure \[fig:ca2h\_n\_xrt\]c) is near the footpoints of the post-flare loops in which the ZP source may be located. These field lines have orientations from NE to SW, and extend to large coronal heights. For completeness, we also show the extrapolated field lines of the three major Ca II H bright regions (see Figure \[fig:SOTSP\]b), which are grouped and colored in red and green to represent increasing coronal heights.
Discussion {#sec:discussion}
==========
The key findings of the previous section may be summarized as follows:
- [An intense X1.5 flare was observed on 2006 Dec 14. The flare was accompanied by intense, prolonged, and variable decimeter wavelength emission as well as gyrosynchrotron emission at centimeter wavelengths. A striking ZP radio burst was observed during the decay phase, centered near 1.32 GHz, with an overall frequency bandwidth of up to 150 MHz. It is completely right-circularly polarized and consists of up to 12 zebra stripes superposed on broadband continuum emission. The stripes drift in the spectrum irregularly as an entity with an average drift rate of $-$50 MHz s$^{-1}$. The relative frequency separations $\Delta f_s/f_m$ between adjacent stripes are $\approx\!1.3\%$, with $\Delta f_s$ increasing with frequency. ]{}
- [Two radio sources with a spatial separation of $\approx\!11''$ - zebra and background continuum - contribute to the ZP emission. The apparent zebra source size is around $35''$, which is systematically smaller than that of the continuum source by as much as $20''$, after the correction for the relative intensity effect. The source size of both the ZP and continuum sources are likely strongly affected by scattering. The lower limit of the brightness temperature of ZP is estimated to be $1.6\times 10^9$ K.]{}
- [The zebra source centroid drifts irregularly in time with an average drift of $\approx\!16''$ in 0.48 s from NE to SW (corresponding to a projected average velocity of $2.5\times 10^9$ cm s$^{-1}$), while the continuum locations show no evident drift with time. ]{}
- [Both the ZP on-stripe and off-stripe source locations are frequency dependent. After correction of the source centroid positions for relative intensity variations, the centroid position shifts $8-9''$ for both the zebra and continuum sources in the direction from NE to SW across the six stripes from high to low frequency at a fixed time. ]{}
- [The zebra and continuum sources are possibly located in a post-flare loop system with an orientation from NE to SW, which connects to the large sunspot with negative polarity. The polarity of the magnetic field and the observed sense of circular polarization of the ZP imply that it is o-mode. ]{}
Of these findings, perhaps the most significant are that the ZP (on-stripe) and continuum (off-stripe) emissions show an angular separation as well as a difference in source size; and that the ZP and continuum source locations are frequency dependent. In other words, the radiation at different frequencies originates from different spatial locations. We are able to conclude that the ZP and continuum sources are not spatially coincident and that both the ZP and continuum sources are spatially extended.
We should point out that the spatial displacement of the on- and off-stripe emissions suggests that the previous measurements of ZP spatial drift in time based on interferometric observations at a single frequency may be misleading. Since the ZP shows frequency drifts with time as an entity, a record of the flux density with time at a single frequency cuts through different zebra stripes as well as the off-stripe continuum source. Apparent spatial shifts can appear as a result of the switch between different sources with different locations. We note that reported a microwave zebra-pattern structure near 5.7 GHz that was observed on 2003 January 05 by the SBRS and the SSRT. The apparent source size, measured at a frequency far less influenced by scattering, was measured to be $<10''$. The source positions of two successive ZP stripes coincide spatially in the east-west direction. For these reasons, the authors concluded that the emission is due to the nonlinear coupling of Bernstein waves in an unresolved source. Yet in the spatial coincidence of the two ZP stripes was based on measurements in one dimension at a fixed frequency at different times. It is possible that the ZP source location drifting with time can result in the apparent “coincidence” of the two stripes. Moreover, even if the two stripes do coincide spatially in the east-west direction, there could be an unknown separation in the north-south direction that is not revealed by the one-dimensional measurements.
In the case of the 14 December 2006 ZP event reported here, and in contrast to the observations of , there is clear evidence that the source location is frequency-dependent. Taken together with the theoretical difficulties raised by various authors [e.g. @2009CEAB...33..281Z], we conclude that Bernstein models are unlikely to be relevant to this ZP emission and we turn our attention to whistler and DPR models.
The observed projected average spatial drift velocity of the ZP source along the zebra stripes ($2.5\times 10^9$ cm s$^{-1}$ in the NE to SW direction) is about 40 times larger than the local Alfvén velocity of $\approx\!700$ km s$^{-1}$ calculated as $V_A=B/(4\pi n_i m_i)^{1/2}\approx 2\times 10^{11} B n_e^{-1/2}$, where $B$ is the magnetic strength in the radio source, assumed here to be $\sim\!50$ G, and $n_e^{1/2}=f/(e^2/\pi m_e)^{1/2}\approx f/9000$ if one assumes the ZP emission at frequency $f$ is near the local plasma frequency. This velocity is too high for most physical movements of the radio source reacting to the magnetic field variation. But it could be explained in terms of the whistler model by propagation of low-frequency whistler wave packets with group velocities ($v_{gr}$) given by [@1976SvA....20..582C] $$v_{gr}=2c\frac{f_{ce}}{f_{pe}}\sqrt{\frac{f_w}{f_{ce}}(1-\frac{f_w}{f_{ce}})^3}$$ in the quasi-longitudinal case, where $c$ is the speed of light and $f_w$ is the whistler wave frequency. For $f_w/f_{ce}\approx 0.25$, the ratio at which the growth of the whistler wave is preferred in the decimetric range [@2006SSRv..127..195C], the group velocities can reach the observed value of $2.5\times 10^9$ cm s$^{-1}$ for $f_{pe}/f_{ce}\lesssim 10$ [see Fig. 2 of @1976SvA....20..582C].
Is the whistler model able to explain the spatial drift of the ZP source in frequency and time simultaneously? In the whistler model, periodically generated whistler wave packets propagate along a trajectory in the magnetic trap at the whistler group velocity $v_{gr}$. They separate regularly from each other in space so that they can emit at discrete frequencies, thereby producing zebra stripes. The frequency drift of each zebra stripe in the spectrum is determined by the motion of the corresponding whistler wave packet along its trajectory, the density gradient along its trajectory, and wave-wave interactions. We suppose the observed spatial drift with time corresponds to $v_{gr}$. If the coronal number density decreases exponentially with height $$\label{eq:n_prof}
n_e=n_{e0}e^{-\Delta h/L_n},$$ where $n_{e0}$ is the plasma density at the reference height $h_0$, $\Delta h$ is the height from $h_0$ and $L_n=dh/(-dn_e/n_e)$ is the plasma density scale height, then the plasma frequency $f_{pe}$ can be written as $$\label{eq:f_prof}
f_{pe}=f_{pe0}e^{-\Delta h/2L_n}.$$ The radiation is near the plasma frequency, so the frequency drift rate is $$\dfrac{df}{dt}=-f\dfrac{v_h}{2L_n},$$ where $v_h$ is the vertical component of the whistler group velocity (normal to the solar surface). Given the observed average frequency drift rate $df/dt\approx -50$ MHz s$^{-1}$ and the projected velocity of $v_{proj}=2.5\times 10^9$ cm s$^{-1}$ (which, given its nearly N-S orientation, is nearly parallel to the solar surface), the tangent angle of the trajectory is then $\tan\alpha_1=v_h/v_{proj}=3\times 10^{-11}L_n$. On the other hand, we observe the projected spatial displacement from zebra stripes Nos. 1$-$6 to be $\Delta l_{1-6}=8.5''$, or $6.2\times 10^8$ cm, and their frequencies differ by $\Delta f_{1-6}\approx\!85$ MHz. It’s easy to see from Eq. \[eq:f\_prof\] that the height difference between zebra stripes Nos. 1$-$6 $\Delta h_{1-6}\approx\Delta f_{1-6}/f\cdot 2L_n$ for small variations of $f$ and $L_n$. Therefore, we have another estimate of the tangent angle of the whistler wave packet trajectory $\tan\alpha_2=\Delta h_{1-6}/\Delta l_{1-6}\approx 2\times 10^{-10}L_n$, which is an order of magnitude larger than $\tan\alpha_1$ that we obtained based on the assumption that the observed source drift with time corresponded to the whistler group velocity $v_{gr}$. In other words, the trajectory of whistler wave packets cannot be reconciled with both the observed spatial displacement of the ZP source with frequency and the observed frequency drift with time.
It is also worth noting that the whistler model does not account for the systematic increase of the relative spacing ${\Delta f_s}/{f_m}$ with increasing ZP frequencies as seen in Figure \[fig:zb\_spacing\]a, because the spatial separation between adjacent whistler wave packets is assumed to be result from the periodic injection of whistler wave packets and their frequency separation is therefore essentially constant.
We now consider whether the DPR model is consistent with the observed features of the ZP and continuum source, including both the total power spectral features and their apparent shift in spatial location as a function of time and frequency. As the ZP and continuum sources emit near the local upper hybrid frequency $f_{UH}$ which, with $f_{ce}\ll f_{pe}$, is near the local electron plasma frequency $f_{pe}$, lower frequency emission should come from greater coronal heights for both the ZP and continuum sources. As a result, a spatial shift with frequency is expected for both the ZP and continuum sources, in accordance with the observations. Furthermore, The NE-SW direction of the spatial drift is generally consistent with the NE-SW orientation of the post-flare loop system in which the emission sources are located. However, the absolute height of the emission source is not yet known. The ZP stripes are those locations where $f_{UH}\approx f_{pe}=sf_{ce}$ in the DPR model. We again assume an exponential dependence of coronal number density as in Eq. \[eq:n\_prof\], with a scale height $L_n$. The coronal magnetic field is likewise assumed to decrease exponentially with height: $$B=B_0e^{-\Delta h/L_B},$$ where $B_0$ is the magnetic field at $h_0$ and $L_B$ is the magnetic field scale height. The electron cyclotron frequency $f_{ce}\propto B$ and so $$f_{ce}=f_{ce0}e^{-\Delta h/L_B}.$$ It can be shown that the relative frequency spacing between the adjacent zebra stripes in the DPR model is then given by [@2007SoPh..241..127K] $$\label{eq:DPR}
\left|\frac{\Delta f_s}{f_m}\right|\simeq \frac{1}{s}\frac{1}{1-(2L_n/L_B)},$$ where $s$ is the harmonic number with $f=sf_{ce}$. Given the continuity of each zebra stripe in the dynamic spectrum for the $\approx 0.5$ s duration of the spectral fragment in our analysis (the dashed box in Figure \[fig:zebra\]a), we assume that each stripe corresponds to a single integer harmonic $s$ of the electron cyclotron frequency $f_{ce}$. Moreover, successive zebra stripes emitting at discrete frequencies should have a one-to-one correspondence with successive integer values of $s$, i.e. $s_0$, $s_0+1$, ..., where $s_0$ is the reference harmonic number at $h_0$. Thus the frequency spacing $\Delta f_s/f_m$ is a function of ($s_0+i$), i=0, 1, 2 ... .
By using the observed values of $\Delta f_s/f_m$ for each ZP frequency at a fixed time, we obtain a pair of best-fit values of $s_0$ and $L_n/L_B$. In particular, for a given time, we start with a fixed $s_0$ and perform a least-squares fit to find $L_n/L_B$. Then we increment the value of $s_0$ by integer values until a minimum in the standard deviation of the fit to $\Delta f_s/f_m$ is reached. We did such fits of $\Delta f_s/f_m$ as a function of ($s_0+i$) for the six zebra stripes for the 24 consecutive integrations shown by the dashed box in Figure \[fig:zebra\]a. Figure \[fig:zb\_spacing\]a shows examples of these fits at five near-equally spaced times spanning the entire fitting time range (22:40:06.86 UT, 22:40:06.96 UT, 22:40:07.04 UT, 22:40:07.16 UT, and 22:40:07.26 UT), where square symbols and solid lines represent respectively the measured and best-fit values, and are color-coded in time from blue to red. From the distribution of best-fit values of $s_0$, after excluding a few outliers, we conclude that $s_0=8$ is the most probable value, with a scatter of $\sigma_{s_0}\approx\!1.4$. Therefore we assign harmonic numbers of $s=8, 9, 10, 11, 12, 13$ to the six successive zebra stripes numbered from 1 to 6 in Figure \[fig:zebra\]a, with a corresponding $L_n/L_B\approx\!4.4\pm 0.5$. Note that a stripe with lower frequency corresponds to a higher harmonic number, which is consistent with the previous results based on DPR models . Such a ratio of the scale heights means the magnetic field changes faster than the plasma density with height, which is usually the case in the solar corona. The magnetic field strengths $B$ from stripes 1 to 6 (from low to high in height) can be estimated to be from 62 G to 35 G, by using the harmonic values $s$ and assuming $f\simeq f_{pe}\simeq sf_{ce}$. The uncertainty in $B$ can be estimated from the scatter in the distribution of $s_0$. We find that it ranges from 17% at $s_0=8$ to 11% for $s=s_0+5=13$. Using the magnetic fields derived from the NLFFF extrapolation results in §\[sec:nlfff\] for guidance, we suggest the zebra stripes 1 to 6 of the ZP source are consistent with a location in the the post-flare loop system at coronal heights from 57 to 75 Mm above the photosphere. The magnetic scale height $L_B$ can be estimated to be $L_B=dh/(-dB/B)\approx 3.2\times 10^9$ cm. On the other hand, an estimation of the density scale height $L_n$ is also available by $L_n=dh/(-2df/f)\approx 1.4\times 10^{10}$ cm, from the known frequencies of the zebra stripes (an equivalent result of $L_n$ can be obtained by using $L_B$ and the fitted value of $L_n/L_B\approx 4.4$).
DPR levels are formed as a result of plasma density and magnetic field variations in height, as demonstrated by Figure \[fig:zb\_spacing\]b. By applying the values of $L_n$, $L_B$, and $s$, the dependencies of $sf_{ce}$ and $f_{pe}$ as a function of $\Delta h$ are given for the time denoted as the solid box in Figure \[fig:zebra\]a. The reference height $h_0$ is set to be that of the lowest DPR level (the No. 1 stripe with $s_0=8$). The intersections of the curves $sf_{ce}(\Delta h)$ and the plasma frequency distribution $f_{pe}(\Delta h)$ (marked by diamonds) are the DPR levels, which coincide with the observed frequencies of zebra stripes (denoted by the horizontal lines). In this event, the zebra stripes drift irregularly and rapidly in the dynamic spectrum, indicating that the DPR levels do not remain at precisely the same height in the flare loop - that is, $h_0$ varies by a few percent of $L_n$, or a few thousand kilometers - which also accounts for the scatter and temporal variation seen in Figure \[fig:zb\_spacing\]a.
In the whistler model, formation of the regularly spaced stripes of ZP in frequency is based on propagation of the periodically generated whistler wave packets along a given trajectory - that is, the trajectory of each whistler wave packet determines the spatial drift of each zebra stripe in time (with tangent angle $\tan\alpha_1$), should follow the same orientation of the spatial extent of the entire ZP source, represented by the spatial displacement of zebra stripes at different frequencies (with tangent angle $\tan\alpha_2$), i.e. $\tan\alpha_1 \approx \tan\alpha_2$. We have already demonstrated that they can not be reconciled in the context of the whistler model. For the DPR model the two “trajectories” need not coincide. The DPR levels, corresponding to the observed zebra stripes, are distributed along a coronal loop at locations where the resonant conditions are matched. At the same time, the locations of each DPR level can change with time in response to the variation of source conditions. Therefore, the “trajectory” of each DPR level is not required to follow the same orientation of the spatial distribution of the resonance layes in the loop. In fact, given the density scale height of $L_n\approx 1.4\times 10^{10}$ cm, we have $\alpha_1\approx 23^{\circ}$ and $\alpha_2\approx 70^{\circ}$. That means the DPR levels are distributed near vertically in the coronal loop while their apparent motion is nearly horizontal in time.
Both the ZP and continuum sources probably reside on the same post-flare/post-CME loops that extend field lines from in or near the large sunspot with negative polarity to well up into the corona with a NE to SW orientation. The radio emission could be powered by an energy release site high up in the corona above the radio source. This site could be related to magnetic reconnections induced by the fast halo CME associated with this flare, which may also account for the intense and prolonged type IV burst activity in the post-flare phase, from 22:07-23:15 UT. As mentioned in §\[sec:nlfff\], the radio source may be magnetically associated with the Ca II H brightening feature that persists for the duration of the type IV emission. The brightening may indicate the magnetic footpoints of the field lines on which the ZP and type IV emission originates.
In the DPR model, ZPs are thought to arise from energetic electrons in a magnetic trap with a large gradient in electron momentum distribution function perpendicular to the magnetic field $\partial f/\partial p_\bot$, as might arise from a distribution function sharply peaked perpendicular to the magnetic field (e.g., a Dory-Guest-Harris, or ring, distribution), or a loss-cone distribution with sufficiently narrow momentum dispersion [e.g. @1986ApJ...307..808W; @2007SoPh..241..127K; @2009SoPh..255..273Z]. At the same time, the continuum emission can also arise from an anisotropic electron distribution in the magnetic trap, most likely, a common loss-cone distribution [e.g. @2007SoPh..241..127K; @2009SoPh..255..273Z]. Therefore, it is plausible that the ZP and continuum observed by the FST is related to the injection of fast electrons that originate from the energy release site above the radio source, perhaps as the result of magnetic reconnection behind the fast halo CME associated with the flare. These downward-propagating electrons establish the anisotropic distribution that drives the zebra and continuum emission. It is highly unlikely that the ambient electron number density or the magnetic field change significantly during the $\sim 1$ s duration of the ZP event reported here and therefore cannot be the reason for variations in the ZP frequency drift with time. More likely, variations in the number and/or degree of anisotropy of the injected electrons lead to variations in the height where wave growth is favored and the DPR condition is met. As a result, the heights of the established anisotropic distribution and/or the DPR levels can be modulated, and form the irregular ZP frequency drift in the dynamic spectrum. The observed irregular drift of ZP source centroid in time can also be attributed to this effect.
As has been proposed by many authors [e.g. @1986ApJ...307..808W; @2007SoPh..241..127K; @2009SoPh..255..273Z], the conditions required to produce a ZP signature through the DPR instability are more stringent than those for producing a continuum, e.g., a low-momentum electron deficit, a narrower momentum dispersion, a higher overall momentum, etc. In addition, for achieving the observed large intensity contrast of the ZP to the continuum in this event, the growth rate of the ZP emission should be sufficiently higher than that of the continuum, which may require a higher non-equilibrium electron density. All the peculiarities mentioned above could possibly explain the observations of the relatively short-lived and fast-varying ZP and the more persistent and stable continuum in this event.
Conclusion
==========
We present FST observations of a striking zebra pattern radio burst that occurred during the 14 December 2006 X3.4 flare. This is the first observation of zebra pattern emission that combines simultaneous high spectral and time resolution data with interferometric observations over the entire bandwidth. After calibrating the FST against OVSA we can obtain the absolute locations of radio fine structures on the solar disk, and study their spatial and spectral features. We conclude that the DPR model is the most favorable model since it can fit most spectral and spatial features of this ZP event.
The cartoon in Figure \[fig:cartoon\] summarizes our interpretation of the ZP event within the framework of DPR model: the zebra and continuum source are located at a height of $60-80$ Mm in a post-flare/post-CME loop system that connects the large sunspot with negative polarity with a NE to SW orientation. Both the zebra and continuum are extended sources occupying a total height range of $\approx\!20$ Mm in the post-flare loops, which explains the spatial drifts of ZP and continuum in frequency. Within the zebra source, individual stripes correspond to emissions near the local plasma frequencies at the DPR levels. We suggest that the fast electrons responsible for the continuum and ZP emission originate in an energy release site high up in the corona above the radio source, perhaps the result of magnetic reconnections induced by the fast halo CME. The fast and irregular spatial drift of the ZP source centroid in time and the irregular frequency drift of ZP likely result from time variations in properties of the fast electrons injected into the field well above the ZP source. The continuum source is comparatively more extended in size, with its emission centroid separated from that of the zebra source in the NE$-$SW direction. The continuum emission requires less stringent conditions for the anisotropic distribution of the injected electrons and can therefore have a different size and a different source centroid location. This may also explain why the continuum emission is comparatively more stable in time and frequency.
The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. [*Hinode*]{} is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as a domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). OVSA and FST are supported by NSF grant AST-0908344 to New Jersey Institute of Technology. Bin Chen is supported by NSF grant AGS-1010652 and Ju Jing is supported by NSF grant ATM 09-36665.
![The nearly right triangle antenna configuration of FASR Subsystem Testbed, which consists of three 1.8-m antennas numbered 5, 6, and 7 of the Owens Valley Solar Array [adapted from @2007PhDT....Liu].[]{data-label="fig:ants"}](f1.eps){width="80.00000%"}
![Comparison of OVSA and FST amplitudes and phases (OVSA: plus; FST: solid line). From top to bottom: baselines of antenna 5-6, 6-7, 7-5, respectively.[]{data-label="fig:comp"}](f2.eps){width="85.00000%"}
![GOES and RSTN time profiles of the 2006 December 14 flare. Note the difference in scale between the intense 1.415 GHz emission and that at 2.695 and 8.8 GHz. The time of the ZP event is marked by the vertical line in (b), at about 22:40 UT, during the decay phase.[]{data-label="fig:goes_rstn"}](f3.eps){width="75.00000%"}
![Longitudinal photometric magnetogram $B_z(0)$ (a) and an example of a Ca II H image at 22:37:35 UT (b) by SOT. North is up and west is to the right. The magnetogram was observed from 22:00:05 to 23:03:16 for about one hour. The intersection of the three interferometric fringes of FST denote the location of the ZP emission centroid (observed around 22:40 UT), with a dashed circle representing the $\approx 50''$ apparent source size obtained in §\[obs:srcsize\]. The error of the source location is between 1.3$''$ and 3.5$''$ depending on direction. The contours are the OVSA 4.6$-$6.2 GHz map at the same time of the ZP (the levels have an increment of 10% of the maximum). The numbers “1”, “2”, and “3” in (b) mark the locations of the three major Ca II H bright regions seen in Figure \[fig:ca2h\_n\_xrt\]a-c.[]{data-label="fig:SOTSP"}](f4.eps){width="85.00000%"}
![Hinode SOT Ca II H and XRT observations of the 2004 December 14 flare showing the chromospheric and coronal evolution. The left column: Hinode Ca II H images at 19:43:37, 22:09:39, and 22:37:35 UT. The right column: Hinode XRT images at 19:43:52, 22:09:29, and 22:40:42 UT. They correspond to the times prior to the flare, during the flare maximum, and of the occurrence of ZP in the post-flare phase. The white arrow in (c) shows the subtle Ca II H brightening that may be magnetically associated with the ZP source. []{data-label="fig:ca2h_n_xrt"}](f5.eps){width="85.00000%"}
![(a): A zebra-pattern structure observed at around 22:40 UT on 2006 December 14. Six successive strong stripes with decreasing frequencies are marked by numbers 1 to 6. (b): Frequency profile of zebra-pattern structure, averaged in the time denoted by the small solid box (between 22:40:07.18 UT and 22:40:07.26 UT). (c): Histogram of drift rates of zebra stripes in the large dashed box. Vertical line indicates the mean drift rate is about -50 MHz/s. (d): Intensity contrasts $P=(I_{on}-I_{off})/I_{off}$ at the six zebra stripes. Pluses, stars, and triangles denote respective contrasts of baseline 5-6, 6-7, and 7-5. []{data-label="fig:zebra"}](f6.eps "fig:"){width="85.00000%"}\
![(a): An example of Gaussian fits of the relative visibility amplitudes on one zebra stripe. Pluses and triangles denote the on- and off-stripe sources respectively. The seven data points for each fit are the relative visibility amplitudes for the three FST baselines (both positive and negative, plus one total power/zero-spacing amplitude). A narrower Gaussian in this visibility plot implies a spatially larger source. (b): Source size estimations on the six zebra stripes in Figure \[fig:zebra\]a. The “X” symbols denote the zebra source sizes after removing the contribution from the continuum (off-stripe) source. The average value of the on- and off-stripe source sizes is given by the horizontal dashed line. []{data-label="fig:srcsize"}](f7.eps){width="12cm"}
![The total power dynamic spectrum as well as the “dynamic phase spectrum” of the three baselines. For baseline 5-6 and 6-7 the phases at the zebra stripes (darker colors) and background continuum are evidently different, but there is no notable difference for baseline 7-5.[]{data-label="fig:zb_phz"}](f8.eps){width="85.00000%"}
![The first row: phasor diagrams showing the visibility amplitude and phase variations of ZP for all the three baselines. The visibilities are averaged across the zebra stripes, showing the variations in time (colored from blue to red). The amplitude and phase for each visibility data are represented by its absolute distance from the origin and the direction, respectively. Pluses and triangles denote the on- and off-stripe sources. Arrows give the average amplitudes and phases. The second row: the ZP phases of baselines 5-6 and 6-7 as a function of time. The solid line, repeated in each panel, represents the ZP frequency as a function of time (note that the frequency scale on the right axis is reversed - with frequency decreasing from high to low values). []{data-label="fig:phasor_t"}](f9.eps){width="85.00000%"}
![The first row: phasor diagrams showing the visibility amplitude and phase variations of ZP for all the three baselines. The visibilities are averaged along the zebra stripes, showing the variations in frequency (colored from black to red for the six stripes with decreasing frequency). Pluses and triangles denote the on- and off-stripe sources. Arrows give the average amplitudes and phases. The second and third rows give the visibility amplitudes and phases of the six stripes. In the first and second panels of the third row, the “X” symbols denote the corrected phases of the zebra source. []{data-label="fig:phasor_f"}](f10.eps){width="85.00000%"}
![(a): The variation of source centroid location along the zebra stripes (with time). The pluses denote the on-stripe source locations along the zebra stripes from 22:40:06.86 to 22:40:07.34 UT (marked in the dashed box in Figure \[fig:zebra\]), and the arrow shows the position drift. The error is represented by the error bar in the lower-left corner. The triangle denote the off-stripe (continuum) source location, which shows no evident drift, with the error bar plotted in the lower-right corner. (b): the variation of source centroid location across the six zebra stripes (with frequency).The pluses, triangles, and the “X” symbol denote respectively the on-stripe, off-stripe and corrected zebra source locations from stripe Nos. 1 to 6 in Figure \[fig:zebra\] (decreasing in frequency). The arrows give their corresponding position displacements. The error bar in the lower-left corner gives the error. []{data-label="fig:zbpos"}](f11.eps){width="85.00000%"}
![The deprojected zebra and continuum source locations with the NLFFF extrapolation field lines. The left image is the view from top, while the two images on the right are for side views from south and east (the x- and y-axis points at west and north, and the z-axis is perpendicular to the tangent plane centered at the active region). The “X” symbols and triangles denote a series of possible 3D locations of continuum and zebra sources consistent with their projected locations (averaged in time and frequency) on the image plane. The extrapolated field lines in blue colors passing the possible radio source locations are probably the post-flare loops in which the radio source is located. The colors from light to dark blue denote the magnetic field strengths from 90 to 30 Gauss at heights from 40 to 80 Mm. The magnetic configurations of the three major Ca II H bright regions (see Figure \[fig:SOTSP\]b) are also shown by the extrapolated field lines, which are grouped and colored in red and green to represent increasing coronal heights.[]{data-label="fig:nlfff"}](f12.eps){width="90.00000%"}
![(a): Fitting examples of ${\Delta f_s}/{f_m}$ as a function of $f_m$ for the six zebra stripes in the dashed box of Figure \[fig:zebra\]a (which has 24 consecutive measurements in time from 22:40:06.86 UT to 22:40:07.34 UT). The square symbols and solid lines represent respectively the measured and best-fit values at five near-equally spaced times spanning the entire fitting time range (22:40:06.86 UT, 22:40:06.96 UT, 22:40:07.04 UT, 22:40:07.16 UT, and 22:40:07.26 UT). They are color-coded in time from blue to red. The most probable fitting values of the hamonic numbers are $s=8, 9, 10, 11, 12, 13$ for the six zebra stripes numbered from 1 to 6 in Figure \[fig:zebra\]a, and the corresponding $L_n/L_B=4.4$. (b) Horizontal lines are the averaged peak frequencies of the six zebra stripes in the solid box of Figure \[fig:zebra\]a (around 22:40:07.21 UT). The curves of gyroharmonics $sf_{ce}$ with $s=8-13$ are plotted as a function of coronal height $\Delta h$ relative to the height of the $s=8$ layer. The intersections of the curves $sf_{ce}(\Delta h)$ and the plasma frequency distribution $f_{pe}(\Delta h)$ are the DPR levels.[]{data-label="fig:zb_spacing"}](f13.eps){width="90.00000%"}
![A simplified source model: the zebra and continuum source are located at a height of $60-80$ Mm in a post-flare/post-CME loop system that connects the large sunspot with negative polarity with a NE to SW orientation. Both the zebra and continuum are extended sources occupying a total height range of $\approx\!20$ Mm in the post-flare loops. Within the zebra source, individual stripes correspond to emissions near the local plasma frequencies at the DPR levels (horizontal dashed lines, of which the lowest one corresponds to the $s_0 = f_{pe}/f_{ce} = 8$ layer). An energy release site is located high up in the corona above the radio source. Electron beams generated from this site propagate downwards along the magnetic field lines into the magnetic trap and give rise to the instability for emission. The fast and irregular spatial drift of the ZP source centroid in time ($\approx 0.1c$, indicated by the arrow showing the general drift direction of NE$-$SW) and the irregular frequency drift of ZP likely result from time variations in properties of the fast electrons injected into the field well above the ZP source. The continuum source is comparatively more extended in size, with its emission centroid separated from that of the zebra source in the NE$-$SW direction.[]{data-label="fig:cartoon"}](f14.eps){width="80.00000%"}
|
---
abstract: |
Recent studies have shown that the primordial non-Gaussianity affects clustering of dark matter halos through a scale-dependent bias and various constraints on the non-Gaussianity through this scale-dependent bias have been placed. Here we introduce the cross-correlation between the CMB lensing potential and the galaxy angular distribution to effectively extract information about the bias from the galaxy distribution. Then, we estimate the error of non-linear parameter, $f_{\rm NL}$, for the on-going CMB experiments and galaxy surveys, such as Planck and Hyper Suprime-Cam (HSC). We found that for the constraint on $f_{\rm NL}$ with Planck and HSC, the wide field galaxy survey is preferable to the deep one, and the expected error on $f_{\rm NL}$ can be as small as: $\Delta f_{\rm
NL} \sim 20$ for $b_0 = 2$ and $\Delta f_{\rm NL} \sim 10$ for $b_0 = 4$, where $b_0$ is the linear bias parameter. It is also found that future wide field galaxy survey could achieve $\Delta
f_{\rm NL} \sim 5$ with CMB prior from Planck if one could observe highly biased objects at higher redshift ($z\sim 2$).
author:
- 'Yoshitaka Takeuchi$^1$, Kiyotomo Ichiki$^1$ and Takahiko Matsubara$^2$'
bibliography:
- 'CMB-Galaxy.bib'
title: 'Constraints on primordial non-Gaussianity from Galaxy-CMB lensing cross-correlation'
---
Introduction {#sec:Intro}
============
The cosmic microwave background (CMB) temperature anisotropy is a quite useful probe for cosmology. The contribution to the anisotropy is dominated by fluctuations at the last scattering surface. The CMB photon, however, encounters the large-scale structure along the line of sight and some additional effects are imprinted on the temperature and polarization as secondary anisotropies. The deflection of the CMB photon due to gravitational potential produced by the large-scale structure is one of them. The effect of the gravitational lensing on the CMB photon through the large-scale structure is known as the CMB lensing [@Lewis-06]. An on-going CMB observation by Planck [@Planck] or various ground-based experiments are expected to detect this signal, while the effect of the CMB lensing are imprinted on small scales which the Wilkinson Microwave Anisotropy Probe (WMAP) satellite could not resolve. The lensing effect reflects the late time evolution of the universe at relatively low-redshifts. Therefore, the lensing information plays an important role to determine the cosmological parameters, such as the neutrino mass, the cosmological constant, the equation of state parameter of dark energy and so on.
The large-scale structures are formed at relatively late time and they become the source of the gravitational potential. They are correlated with the CMB temperature anisotropy through the Integrated Sachs-Wolfe (ISW) effect, which generates the secondary anisotropies due to the time variation of the potential [@Sachs-67]. The cross-correlation has an advantage for observations of ISW effect whose signal is weak. Cross-correlations with complementary probes are expected to provide additional information on top of their respective auto-correlations. Similarly, we expect that lensing potential should correlate with the large-scale structure and the information from their cross-correlation may precisely determine the cosmological parameters.\
Recently, the deviations from Gaussian initial conditions (primordial non-Gaussianity) are intensively focused on and discussed. Inspection of them offers an important window into the very early Universe because non-standard models of inflation allow for a large non-Gaussianity while standard single-field slow-roll models predict the small deviations from Gaussianity. The most popular method to detect the primordial non-Gaussianity is to measure higher-order correlations of CMB anisotropies and distributions of galaxies, for example, the bispectrum or the three-point correlation function of CMB [@Komatsu-01; @Bartolo-04] or the large-scale structure bispectrum [@Scoccimarro-00; @Verde-00; @Scoccimarro-04; @Sefusatti-07].
Some studies have shown that the primordial non-Gaussianity affects clustering of dark matter halos through a scale-dependent bias, both by analytic calculations and by N-body simulations [@Afshordi-08; @Carbone-08; @Dalal-08; @Pillepich-10; @Slosar-08]. By considering the scale-dependent bias, one can constrain on the non-Gaussianity by the power spectrum. Various constraints on the non-Gaussianity through the scale-dependent bias have already been placed [@Slosar-08; @Oguri-09; @Carbone-10; @Cunha-10; @DeBernardis-10; @Xia-10]. One may expect, however, that a constraint on non-linear parameter $f_{\rm
NL}$, which describes the primordial non-Gaussianity, is degenerated with other cosmological parameters and has a large error. Because of such degeneracy, especially with the linear bias $b_0$, it is important to combine unbiased observations which are sensitive to the matter power spectrum in the near universe, like CMB lensing, shear and so on. Here we use the cross-correlation between the CMB lensing potential and the large-scale galaxy distribution and estimate expected errors of non-linear parameter $f_{\rm NL}$ for future CMB experiments and galaxy surveys. We expect that this cross-correlation may play a role to break degeneracy and give us more stringent constraints.
This paper is organized as follows. We review the scale-dependent bias due to the non-Gaussianity in section \[sec:SBias\] and the theory behind the cross-correlation between CMB lensing potential and galaxy distribution in section \[sec:APS\]. In section \[sec:Modeling\] we describe the survey model for the galaxy distribution in the Hyper Suprime-Cam (HSC) survey, which is a fully funded imaging survey at Subaru telescope. In section \[sec:Fisher\] and section \[sec:Forecasts\] we explain the method of our analysis. Finally, in section \[sec:Result\] and \[sec:Summary\] we discuss the results and summarize our conclusions. Throughout this paper we assume a spatially flat universe for simplicity.
Scale-Dependent Bias {#sec:SBias}
====================
Deviations from Gaussian initial conditions are commonly parameterized in term of the dimensionless $f_{\rm NL}$ parameter and primordial non-Gaussianity of the local-type is defined as [@Komatsu-01] $$\Phi = \phi +
f_{\rm NL} (\phi^2 - \langle \phi^2 \rangle) ,$$ where $\Phi$ denotes Bardeen’s gauge-invariant potential and $\phi$ denotes a Gaussian random field. On subhorizon scale, $\Phi = - \Psi$, where $\Psi$ denotes the usual Newtonian gravitational potential related to density fluctuations via Poisson’s equation. For example, simple slow-role inflation gives a parameter $f_{\rm NL}$ of the order of $10^{-2} - 1$ [@Salopek-90; @Gangui-94; @Falk-93]. On the other hand, large values of $f_{\rm NL}$ can be expected in models of multifield inflation, tachyonic preheating in hybrid inflation [@Barnaby-06] or ghost inflation [@Arkani-Hamed-04], for instance. Thus, the information about the inflation physics is closely related to the parameter $f_{\rm NL}$.
Recent studies show that the effect of the primordial non-Gaussianity of the local-type is seen in the clustering of halos through a scale-dependent bias, $$P_g(k)
= b_0^2 P(k) \rightarrow [b_0 + \Delta b(k)]^2 P(k) ,$$ where $P_g (k)$ and $P (k)$ are the power spectrum of galaxy and matter density fluctuations as a function of the wave number $k$, respectively. $b_0$ is the Gaussian-case bias which relates the galaxy density fluctuations with the matter density fluctuations, and $\Delta
b(k)$ represents the scale-dependence due to the non-Gaussianity [@Afshordi-08; @Carbone-08; @Dalal-08; @Pillepich-10; @Slosar-08], $$\Delta b(k) =
\dfrac{3(b_0 -1) f_{\rm NL} \Omega_m H_0^2 \delta_c}{D(z) k^2 T(k)} ,$$ where $D(z)$ and $T(k)$ are the growth rate and the transfer function for linear matter density fluctuations, respectively. $\delta_c
\simeq 1.68$ is the threshold linear density contrast for a spherical collapse of an overdensity region. Primordial non-Gaussianity of the local-type gives rise to a strong scale-dependent bias on large scales, while the bias is roughly constant on large scales in the Gaussian case. However it is necessary to emphasize that the constraint through the scale-dependent bias is sensitive only to the local-type non-Gaussianity. For the constraints on the other non-Gaussianity models we must consider higher-order correlation such as bispectrum, trispectrum and so on.
The Angular Power Spectrum {#sec:APS}
==========================
The cross-correlations, for example, between CMB and galaxy, are well known as providing additional information other than their respective auto-correlation. In Ref. [@Jeong-09], they investigated the cross-correlation between the shear of CMB lensing and halos. In this paper, we introduce the cross-correlation between the CMB lensing and galaxy angular distribution to estimate errors in constraining cosmological parameters.
Galaxy Distribution
-------------------
Probably the most obvious tracers of the large-scale density field in the linear regime are luminous sources such as galaxies at optical wavelengths and AGNs at x-rays and/or radio wavelengths. The projected density contrast of the tracers can be written as $$\delta_g (\hat{\bm n}) =
\int dz \dfrac{dN}{dz} \delta_g (\chi \hat{\bm n}, z),$$ where $\delta_g$ represents the density contrast of tracers, $\hat{\bm
n}$ is the direction to the line of sight, $dN/dz$ is a normalized distribution function of tracers in redshift such that $\int dz dN/dz
= 1$ and $\chi (z)$ is the comoving distance to the redshift $z$. We assume the following analytic form of the normalized galaxy distribution function, $$\dfrac{dN}{dz} =
\dfrac{\beta z^{\alpha}}{\Gamma
\left[(\alpha + 1)/\beta \right] z_0^{\alpha + 1}} \exp
\left[ -\left(\dfrac{z}{z_0} \right)^{\beta}\right] ,
\label{Gdist}$$ where $\alpha$, $\beta$ and $z_0$ are the free parameters. In this parameterization $\alpha$ and $\beta$ denote the slope of the distribution at low and high-redshifts, respectively, and $z_0$ determines the peak of the distribution. We assume that the tracer density field is related to the underlying matter density field via a scale- and redshift-dependent bias factor, so that $\delta_g({\bm k},
z) = b(k, z) \delta({\bm k}, z)$. On large scales, where the mass fluctuations are small $\delta \ll 1$, the perturbations grow according to the linear growth rate, $\delta ({\bm k}, z) = \delta
({\bm k}) T(k) D(z)$, where $\delta({\bm k})$ is the primordial value of matter density and $T(k)$ is the transfer function. The linear angular power spectrum of the galaxy distribution for a flat universe is given by $$\cgg = \dfrac{2}{\pi} \int k^2 dk P(k) {\Delta_l^g (k)}^2 ,$$ where $$\Delta_l^g (k) = \int dz \dfrac{dN}{dz} b(k, z) T(k) D(z) j_l ({k}\chi) .
\label{eq:delg}$$ and $P(k)$ is the linear power spectrum as a function of the wave number $k$ and $j_l(k\chi)$ is a spherical Bessel function.
In order to estimate errors in parameters and signal-to-noise ratios, we need to describe the noise contribution due to the finiteness in numbers of sources associated with source catalogs. We can write the shot noise contribution as $$N_l^{gg} = \dfrac{1}{n_{\rm L}} ,$$ where $n_{\rm L}$ is the surface density of sources per steradian and related to the total number of available samples $N_{\rm g}$ as $n_{\rm L} = N_{\rm g} / 4\pi f_{\rm sky}$. We show the angular power spectrum of the galaxy distribution, $\cgg$, and the noise spectrum, $N_l^{gg}$, in the left panel of Fig. \[fig:CMBNoise\].
CMB Lensing Potential
---------------------
We consider the potential that deflects CMB photons. The relationship between the lensed temperature anisotropy, $\tilde{T}(\hat{\bm n})$, and unlensed one, $T(\hat{\bm n})$, is related by $\tilde{T}(\hat{\bm
n}) = T(\hat{\bm n}+{\bm d})$ and the deflection angle ${\bm d}(
\hat{\bm n})$ is related to the line of sight projection of the gravitational potential $\Psi (\chi \hat{\bm n}, \eta)$ as ${\bm d}(
\hat{\bm n}) = \nabla \psi ( \hat{\bm n})$, where $$\psi(\hat{\bm n}) = -2 \int d\chi \dfrac{f_K(\chi_{*}) -
f_K(\chi)}{f_K(\chi_{*}) f_K(\chi)} \Psi (\chi \hat{\bm n} ; \eta_0 - \chi) .$$ Here $\psi(\hat{\bm n})$ is the lensing potential, $f_K(\chi)$ is the angular diameter distance, $\chi$ is the radial comoving distance along the line of sight, and $\chi_{*}$ denotes the distance to the last scattering surface. For a flat universe angular diameter distance is related to the comoving distance as $f_K(\chi) = \chi$. The angular power spectrum of the lensing potential for a flat universe can be written as $$\cpp = \dfrac{2}{\pi} \int k^2 dk P(k) \Delta_l^{\psi} (k) ^2,$$ where $$\Delta_l^{\psi} (k) = -2 \int_0^{\chi_*} d\chi T_{\Psi}(
k ; \eta_0 - \chi) \left(\dfrac{\chi_{*} -
\chi}{\chi_{*}\chi}\right) j_l (k\chi) .
\label{eq:delp}$$ In the linear theory, we define a transfer function for the gravitational potential $T_{\Psi}(k; \eta)$ so that $P_{\Psi}({k}; \eta) = T_{\Psi}^{2}(k; \eta) P({k})$.
The lensing potential can be reconstructed using quadratic statistics in the temperature and polarization data that are optimized to extract the lensing signal. To reconstruct the lensing potential $\psi$, one needs to use the non-Gaussian information imprinted into the CMB. Lensing conserves surface brightness, so that the probability distribution function of the temperatures remains unchanged. Therefore the lowest order nonzero estimator of the lensing potential is quadratic. This quadratic estimator has been investigated by [@Hu-02; @Okamoto-03] and the minimum variance estimator was given by [@Hirata-03a]. A quadratic estimator in the flat-sky approximation generally has the form [@Hu-02] $$\hat{\psi}({\bm L}) =
N(\bm{L}) \int \dfrac{d^2{\bm l}}{(2 \pi)^2}\tilde{\Theta} ({\bm l})
\tilde{\Theta}^{'} ({\bm L}-{\bm l}) {g}({\bm l}, {\bm L}),
\label{eq:lp}$$ where $\tilde{\Theta}$ and $\tilde{\Theta}^{'}$ are lensed temperature and/or polarization modes on the sky, $i.e.,$ $\tilde{\Theta}$, $\tilde{\Theta}^{'} = \tilde{T}$, $\tilde{E}$, $\tilde{B}$. The optimal weight ${g}({\bm l}, {\bm L})$ and normalization $N(\bm{L})$ for each mode are found using the fact that the deflection position can be written as a first order expansion of the displacement around the undeflected position, $\tilde{\Theta} (\hat{\bm n}) = \Theta
(\hat{\bm n} + {\bm d}) = \Theta (\hat{\bm n}) + \nabla^i \psi
(\hat{\bm n}) \nabla_i \Theta (\hat{\bm n})$. Requiring the estimator to be unbiased and minimizing the variance, the optimal weight for $TT$ estimator is $${g}({\bm l}, {\bm L}) =
\dfrac{({\bm L} - {\bm l})\cdot{\bm L}C_{|{\bm L}-{\bm l}|}+{\bm l}
\cdot {\bm L}C_l}{2 \tilde{C}_l^{\rm tot}
\tilde{C}_{|{\bm L}-{\bm l}|}^{\rm tot}} ,$$ where $C_l$ ($\tilde{C}_l$) is the unlensed (lensed) temperature power spectrum. For other estimators, $C_l$ ($\tilde{C}_l$) represents the temperature or polarization one. The superscript “tot” originates from the fact that the lensed CMB and the noise enter in the variance, $\tilde{C}_l^{tot} = \tilde{C}_l + N_l$.
With the definition in Eq. (\[eq:lp\]), the lowest order noise of the lensing reconstruction equals to the normalization which is determined by $$\delta({\bm 0})\langle |\hat{\psi}({\bm L})|^2 \rangle =
N(\bm{L}) = \left[ \int \dfrac{d^2 {\bm l}}{(2\pi)^2}
\left[ ({\bm L} - {\bm l})\cdot{\bm L}C_{|{\bm L}-{\bm l}|}+{\bm l}
\cdot {\bm L}C_l \right]
\times {g}({\bm l}, {\bm L}) \right]^{-1} .$$ Physically the variance is a combination of the noise introduced by primary anisotropies themselves and the instrumental noise. The all-sky generalization is presented in Ref. [@Okamoto-03].
Here, the noise power spectrum of the CMB experiment reads $$N_{l,\, \nu}^{XX} = (\theta_{\rm FWHM} \Delta_X)^2 \exp
\left[ l(l + 1)\theta_{\rm FWHM}^2 /8 \ln 2 \right],$$ with $X \in \{ T, E, B\}$, where $\Delta_{X}$ is the temperature and polarization sensitivities per pixel of the combined detectors and $\theta_{\rm FWHM}$ describes the spatial resolution of the beam. These values are given for each frequency bands $\nu$ and we show the values for some CMB experiments in Table \[tb:CMBNoisePar\]. When there are multiple frequency bands or $channels$, the global noise of the experiment is given by $$N_l^{XX} = \left[ \sum_{\nu} (N_{l,\, \nu}^{XX} )^{-1} \right]^{-1},$$ where the sum is over the individual channels. We show the angular pawer spectrum of the CMB lensing potential, $\cpp$, and its noise spectrum, $N_l^{\psi \psi}$, for various CMB experiments in the right panel of Fig. \[fig:CMBNoise\]. As Planck does not have much sensitivity to reconstruct the lensing potential from the polarization components, $TT$ provides the best estimator for the Planck. For the reference experiment like the CMBPol, the lensing potential, however, is reconstructed from polarization components and $EB$ provides the best estimator.
Experiments $f_{\rm sky}$ $\nu$ \[[GHz]{}\] $\theta_{\rm FWHM}$ $\Delta_T$ $\Delta_P$
------------------------ --------------- ------------------- --------------------- ------------ ------------
Planck [@Planck] 0.65 100 9.5$^{'}$ 6.8 10.9
143 7.1$^{'}$ 6.0 11.4
217 5.0$^{'}$ 13.1 26.7
PolarBear [@PolarBear] 0.03 90 6.7$^{'}$ 1.13 1.6
150 4.0$^{'}$ 1.70 2.4
220 2.7$^{'}$ 8.00 11.3
CMBPol [@CMBPol] 0.65 100 4.2$^{'}$ 0.87 1.18
150 2.8$^{'}$ 1.26 1.76
220 1.9$^{'}$ 1.84 2.60
: The current designs of CMB experiments. $\theta_{\rm FWHM}$ is the Gaussian beam width at FWHM, $\Delta_T$ and $\Delta_P$ are the temperature and polarization noises, respectively. Planck and CMBPol are the satellite experiments and PolarBear is the ground based experiment. \[tb:CMBNoisePar\]
Cross-Correlation: Galaxy & Lensing Potential
---------------------------------------------
We focus on the linear cross-spectrum of the galaxy with the CMB lensing potential, $$C_l^{\psi g} = \dfrac{2}{\pi} \int
k^2 dk P(k) \Delta_l^{\psi} (k) \Delta_l^g (k) .$$The most important assumption we have made so far is that the galaxy distribution and the lensing potential is linear and Gaussian. On small scales this will not be quite correct due to non-linear evolution. For simple models, fits to numerical simulation like the HALOFIT code of Ref. [@Smith-03] can be used to compute an approximate non-linear power spectrum. A good approximation is simply to scale the transfer functions $T(k)$ of Eq. (\[eq:delg\]), (\[eq:delp\]) so that the power spectrum has the correction from the non-linear effect $$T(k) ~ \longrightarrow ~ T(k) \sqrt{\dfrac{P^{\rm non-linear}(k) }{P(k)}} .$$
We also include the other cross-correlation components, $\cte$, $\ctp$ and $\ctg$, for the estimation of the parameter errors. However, we assume that there is no cross-correlation between the polarization and the lensing potential or the galaxy distribution, $\cep = \ceg =
0$. This is because the polarization is mainly produced by the Thomson scattering at the last scattering surface while the lensing potential and the galaxy distribution exist in the late-time universe. We show the angular power spectrum $\cpg$ in Fig. \[fig:Clpg\]. The redshift dependence and the effect of the primordial non-Gaussianity through a scale-dependent bias are clearly seen in that figure.
Modeling Galaxy Sample {#sec:Modeling}
======================
We showed the analytic form of the normalized galaxy distribution function in Eq. (\[Gdist\]). The mean redshift $z_{\rm m}$ is related to the peak redshift $z_0$ and determined by $$z_{\rm m} = \int dz z \dfrac{dN}{dz} = \dfrac{z_0 \Gamma
\left[(\alpha + 2)/\beta \right]}{\Gamma \left[(\alpha + 1)/\beta \right]} .
\label{meanredshift}$$ The relation between $z_0$ and $z_{\rm m}$ is, for example, $z_0 =
{z_{\rm m}} /0.64$ for ($\alpha$, $\beta$) = (0.5, 3.0) and $z_0 =
{z_{\rm m}} /1.41$ for ($\alpha$, $\beta$) = (2.0, 1.5). In this paper, we consider a wide field survey such as the on-going Hyper Suprime-Cam (HSC) project. This is a fully funded imaging survey at Subaru telescope. The surface density $n_{\rm L}$ and the mean redshift ${z_{\rm m}}$ are related to the exposure time $t_{\exp}$ as [@Amara-06; @Yamamoto-07], &=& 0.9 ( )\^[0.067]{} ,\
\[eq:model-zm\] n\_[L]{} &=& 35 ( )\^[0.44]{} . \[eq:model-nl\] In Ref. [@Amara-06] and [@Yamamoto-07], ($\alpha$, $\beta$) = (0.5, 3.0) and (2.0, 1.5) are adopted, respectively. In this paper, we adopt both cases and compare the differences between the survey models.
The validity of the above form of the galaxy distribution is shown in Ref. [@Yamamoto-07]. They compared it with the Canada-France-Hawaii telescope (CFHT) photometric redshift data [@Ilbert-06]. The relationship between magnitude limit and exposure time was scaled for the published Subaru Suprime-Cam specification [@Miyazaki-02], and these data are shown in Table \[tb:Exptime\] for the $i, g, r, z$ passbands.
The total survey area can be expressed as [@Yamamoto-07] $${\rm area} = \pi \left( \dfrac{\rm field \,of\, view}{2} \right)^2
\dfrac{T_{\rm total}}{1.1 \times t_{\rm \exp} + t_{\rm op}} ,$$ where we assume that the field of view is 1.5$^\circ $, the total observation time $T_{\rm total}$ is fixed as 800 hours, and the overhead time is modeled by constant, $t_{\rm op} = 5$ min, plus a fraction (10%) of the exposure time $t_{\rm \exp}$ for one field of view.
$i_{\rm AB~limit}$ $i(S/N=10)$ $g(S/N=5)$ $r(S/N=5)$ $z(S/N=5)$
-------------------- ------------- ------------- ------------- -------------
$22.97$ $1$ mins. $3$ mins. $1.1$ mins. $0.3$ mins.
$23.84$ $5$ mins. $15$ mins. $7$ mins. $1.4$ mins.
$24.22$ $10$ mins. $30$ mins. $12$ mins. $3.5$ mins.
$24.81$ $30$ mins. $90$ mins. $34$ mins. $8.1$ mins.
$25.04$ $45$ mins. $130$ mins. $50$ mins. $13$ mins.
: Exposure time for the bands, $i, g, r, z$. The relation between magnitude limit and exposure time was scaled for the published Subaru Suprime-Cam specification [@Miyazaki-02; @Yamamoto-07]. \[tb:Exptime\]
The Fisher matrix analysis {#sec:Fisher}
==========================
For our Fisher matrix analysis, we refer to the method of Ref. [@Perotto-06] and expand it to take into account the cross-correlation between the lensing potential and the galaxy distribution. In Ref. [@Perotto-06], the $5 \times 5$ covariance matrix is calculated for primary CMB and CMB lensing. In our case, we expand it into $8 \times 8$ covariance matrix for the cross-correlation between CMB lensing and galaxy distribution.
Likelihood Function
-------------------
Each data points have contributions from both signal and noise. If we assume both contributions are Gaussian distributed, we can write the likelihood function of the data given the theoretical model as $${\cal L}({\bm d} |\Theta) \propto \dfrac{1}{\sqrt{{\rm det}
\bar{C}(\Theta)}}\exp \left( -\frac{1}{2} \bm{d}^\dagger
[\bar{C}(\Theta)^{-1}] \bm{d} \right),
\label{LikeF}$$ where $\bm{d} = (a^T_{lm},a^E_{lm},a^{\psi}_{lm}, a^g_{lm} )$ is the data vector, $\Theta = (\theta_1, \theta_2, \ldots)$ is a vector describing the theoretical model parameters, and $\bar{C}(\Theta)$ is the theoretical data covariance matrix represented by both signal and noise. For it to be a good estimate, we would like it to be unbiased, $i.e.$, $\langle \Theta \rangle = \Theta_0$ , where $\Theta_0$ indicates the true parameter vector of the underlying cosmological model, $\Theta$ is the one constructed by the data vector $\bm{d}$ they minimizing the likelihood function ${\cal L}({\bm d} |\Theta)$ ($i.e.$, the so-called best fit model) and $\langle \ldots \rangle$ denotes an average over many independent realizations.
We can derive the effective chi-square, $\chi_{\rm eff}^2
\equiv \sum_{XY} \sum_{lm}(- 2) \ln {\cal L}$, from (\[LikeF\]) as $$\chi^2_{\rm eff} = \sum_{l} (2l+1) \left( \frac{D}{|\bar{C}|} + \ln
|\bar{C}| \right),
\label{effchi}$$ where $\sum_{XY}$ represents the summation for each modes, $X, Y =
\{T, E, \phi, g\}$, and $|\bar{C}|$ denotes the determinant of the theoretical data covariance matrix, ||[C]{}| &=& ()\^2()\^2 -()\^2-()\^2\
&& -()\^2 -()\^2 ++2 .\
Here $D$ is defined as D &=& +2 ()\^2 +2 ()\^2\
&& -()\^2 -()\^2 -2\
&& -()\^2 -()\^2 -2\
&& -()\^2 -()\^2 -2\
&& -()\^2 -()\^2 -2\
&& ++++\
&& +2 +2 +2 +2 . In the above expression, we have assumed that the polarization component does not correlate with the lensing potential and galaxy distribution, so we put $C_l^{E\psi} = C_l^{E g} = 0$.
On the other hand, the mock data covariance matrix $\hat{C}$ is given from the simulations and defined as $\hat{C} \equiv \langle \bm{d}\,
\bm{d}^\dagger \rangle$. We can estimate the power spectrum of the mock data through the following definition, $$\sum_m a_{lm}^{X*}a_{lm}^{Y} = (2l+1)\hat{C}_l^{XY} ~.$$ From Bayes’ theorem, we assume (\[effchi\]) to be the distribution of theoretical data covariance matrix $\bar{C}(\Theta)$ when mock covariance matrix $\hat{C}$ is given. Then, we can account $\bar{C}$ to be a variable and $\hat{C}$ to be a constant. All expressions introduced so far assume a full sky coverage survey. However, real experiments can only see a fraction of the sky. We introduce a factor $f_{\rm sky}$, where $f_{\rm sky}$ denotes the observed fraction of the sky in the effective $\chi^2$. We are interested only in the confidence levels, so the normalization factor in front of the likelihood function (\[LikeF\]) is irrelevant. We normalize as $\chi_{\rm eff}^2 = 0$ if $\bar{C} = \hat{C}$ by adding arbitrary constant and redefine $\chi_{\rm eff}^2$ from (\[effchi\]) as $$\chi^2_{\rm eff} = \sum_{l} (2l+1) f_{\rm sky} \left(
\frac{D}{|\bar{C}|} + \ln{\frac{|\bar{C}|}{|\hat{C}|}} - 4 \right) ,
\label{nchieff}$$ and $|\hat{C}|$ denotes the determinant of the mock (observed) data covariance matrix, || &=& ()\^2 ()\^2 -()\^2 -()\^2\
&& -()\^2 -()\^2 ++2 .
Fisher Information Matrix {#sec:FIM}
-------------------------
The Fisher matrix formalism can be used to understand how accurately we can estimate the values of vector of parameters ${\bm \Theta}$ for a given model from one or more data sets [@Tegmark-97]. The Fisher matrix approximates the curvature of the likelihood function $\L$ around its maximum in a space spanned by the parameters $\theta$. The usual formula requires a slight generalization to account for the possibility that different surveys may only partially overlap in sky coverage as we shall show below. The likelihood function should peak at $\Theta \simeq \Theta_0$, and can be Taylor expanded to second order around this value. The relevant term at second order is the Fisher information matrix, defined as F\_[ij]{}. - |\_[ \_0]{} . \[fisher\] From the Cramer-Rao inequality, the marginalized error on a given parameter $\theta_i$ is given by $\sigma(\theta_i) = \sqrt{(F^{-1})_{ii}}$ for an optimal unbiased estimator such as the maximum likelihood.
Substituting equations (\[LikeF\]) and (\[nchieff\]) into the above expression, the Fisher information matrix is written by $$F_{ij} = \sum_{l=2}^{l_{\rm max}}\sum_{XX',YY'}\frac{\partial
C_l^{XX'}}{\partial
\theta_i}({\rm Cov}_l^{-1})_{XX'YY'}\frac{\partial C_l^{YY'}}{\partial \theta_j},
\label{cov}$$ where $i, j$ run over the cosmological parameters, $l_{\rm max}$ is the maximum multipole available given the angular resolution of the considered experiment, and $XX', YY' \in \{ TT, EE, TE, \psi \psi, T\psi, gg, Tg, \psi g\}$. The matrix ${\rm Cov}_l$ is the power spectrum covariance matrix at the $l$-multipole, \_l=(
[cccccccc]{} \_[TTTT]{} & \_[TTEE]{} & \_[TTTE]{} & \_[TT ]{} & \_[TTgg]{} & \_[TT T]{} & \_[TT Tg]{} & \_[TT g]{}\
\_[TTEE]{} & \_[EEEE]{} & \_[TEEE]{} & 0 & 0 & 0 & 0 & 0\
\_[TTTE]{} & \_[TEEE]{} & \_[TETE]{} & 0 & 0 & 0 & 0 & 0\
\_[TT]{} & 0 & 0 & \_ & \_[gg]{} & \_[T]{} & \_[Tg ]{} & \_[g]{}\
\_[TTgg]{} & 0 & 0 & \_[gg]{} & \_[gggg]{} & \_[Tgg]{} & \_[Tggg]{} & \_[g gg]{}\
\_[TTT]{} & 0 & 0 & \_[T]{} & \_[Tgg]{} & \_[TT]{} & \_[TTg]{} & \_[Tg]{}\
\_[TTTg]{} & 0 & 0 & \_[Tg ]{} & \_[Tg Tg]{} & \_[TTg]{} & \_[Tg g]{} & \_[Tg g]{}\
\_[TTg]{} & 0 & 0 & \_[g]{} & \_[g gg]{} & \_[Tg]{} & \_[Tg g]{} & \_[g g]{}
) , \[eq:Cov\] where the auto correlation coefficients are given by
$$\begin{aligned}
\Xi_{TTTT}
&=& (\ctt)^2 + \dfrac{2 (\cte)^2 \left[ \cpp (\ctg)^2 + \cgg (\ctp)^2 -2 \ctp \ctg \cpg \right]}{\left[ (\cpg)^2 - \cpp \cgg \right]} , \\
\Xi_{EEEE}
&=& (\cee)^2 , \\
\Xi_{TETE}
&=& \dfrac{1}{2}\left[ (\cte)^2 + \ctt \cee \right] + \dfrac{\cee \left[ \cpp (\ctg)^2 + \cgg (\ctp)^2 -2 \ctp \ctg \cpg \right]}{2 \left[ (\cpg)^2 - \cpp \cgg \right]} , \\
\Xi_{\psi \psi \psi \psi}
&=& (\cpp)^2 , \\ \nn \\
\Xi_{gggg}
&=& (\cgg)^2 , \\
\Xi_{T\psi T\psi}
&=& \dfrac{1}{2}\left[ (\ctp)^2 + \ctt \cpp \right] - \dfrac{\cpp (\cte)^2}{2 \cee} , \\
\Xi_{TgTg}
&=& \dfrac{1}{2}\left[ (\ctg)^2 + \ctt \cgg \right] - \dfrac{\cgg (\cte)^2}{2 \cee} , \\
\Xi_{\psi g \psi g}
&=& \dfrac{1}{2}\left[ (\cpg)^2 + \cpp \cgg \right] ,\end{aligned}$$
while the cross-correlation ones are
$$\begin{aligned}
\Xi_{TTEE}
&=& (\cte)^2 , \\
\Xi_{TTTE}
&=& \ctt \cte + \dfrac{\cte \left[ \cpp (\ctg)^2 + \cgg (\ctp)^2 -2 \ctp \ctg \cpg \right]}{\left[ (\cpg)^2 - \cpp \cgg \right]} , \\
\Xi_{TT\psi \psi}
&=& (\ctp)^2 , \\
\Xi_{TTT\psi}
&=& \ctp \left[ \ctt - \dfrac{(\cte)^2}{\cee} \right] , \\
\Xi_{TTgg}
&=& (\ctg)^2 , \\
\Xi_{TTTg}
&=& \ctg \left[ \ctt - \dfrac{(\cte)^2}{\cee} \right] , \\
\Xi_{TT\psi g}
&=& \ctp \ctg , \\ \nn \\
\Xi_{TEEE}
&=& \cte \cee , \\ \nn \\
\Xi_{T\psi \psi \psi}
&=& \ctp \cpp , \\ \nn \\
\Xi_{T\psi gg}
&=& \ctg \cpg , \\
\Xi_{T\psi Tg}
&=& \dfrac{1}{2}\left( \ctp \ctg + \ctt \cpg \right) - \dfrac{(\cte)^2 \cpg}{2 \cee} , \\
\Xi_{T\psi \psi g}
&=& \dfrac{1}{2}\left( \ctp \cpg + \ctg \cpp \right) , \\ \nn \\
\Xi_{Tg \psi \psi}
&=& \ctp \cpg , \\ \nn \\
\Xi_{Tg gg}
&=& \ctg \cgg , \\ \nn \\
\Xi_{Tg \psi g}
&=& \dfrac{1}{2}\left( \ctg \cpg + \cgg \ctp \right) , \\ \nn \\
\Xi_{\psi \psi gg}
&=& (\cpg)^2 , \\ \nn \\
\Xi_{\psi \psi \psi g}
&=& \cpp \cpg , \\ \nn \\
\Xi_{\psi g gg}
&=& \cgg \cpg .\end{aligned}$$
Forecasts {#sec:Forecasts}
=========
We estimate the parameter errors for Planck satellite using Fisher analysis following the method introduced in Sec. \[sec:Fisher\]. Our fiducial cosmology is based on the WMAP 7-year result [@Komatsu-10] within a flat, $\Lambda$CDM model framework.
The fiducial model parameters we consider are given as : &=& { \_b h\^2, \_c h\^2, \_, , f\_, Y\_[He]{}, n\_s, \_[R]{}\^[2]{}(k\_0) 10\^[9]{}, w\_0, N\_[eff]{}, \_s, f\_[NL]{}, b\_0 }\
&=& { 0.02258, 0.1109, 0.734, 0.088, 0.02, 0.24, 0.963, 2.45, -1.0, 3.0, 0, 0, 2.0 } , where $\Omega_b$, $\Omega_c$ and $\Omega_{\Lambda}$ are the density parameters for baryon, cold dark matter and dark energy, respectively, $h$ is the Hubble constant, $\tau$ is the Thomson scattering optical depth to the last scattering surface, $f_\nu$ is the mass density of the massive neutrino relative to the total matter density: $f_\nu \equiv \Omega_\nu /\Omega_m$, $Y_{\rm He}$ is the primordial helium fraction, $n_s$ is spectral index of the primordial power spectrum, $\Delta_{\rm R}^{2}(k_0)$ is the amplitude of the primordial power spectrum normalized at $k_0 = 0.002 ~{\rm Mpc^{-1}}$, $w$ is the equation of state parameter of dark energy, $N_{\rm eff}$ is the effective number of neutrinos, $\alpha_s$ is the running index, $f_{\rm NL}$ is the non-linear parameter which represents the primordial non-Gaussianity and $b_0$ is the linear bias parameter. Because we assume a flat universe the Hubble parameter is adjusted to keep our universe flat when we vary $\Omega_\Lambda$. For neutrino parameters, we assume the standard three neutrino species. In our analysis, the non-linear parameter $f_{\rm NL}$ and the linear bias parameter $b_0$ are determined by the galaxy surveys only, and the CMB experiment plays a role in breaking the parameter degeneracies. We use CAMB code [@Lewis-00] and HALOFIT code [@Smith-03] to calculate the angular power spectrum $C_l^{XY}$ and the non-linear region of the angular power spectrum of the galaxy distribution and lensing potential.
[@c@|p[2pt]{}ccccp[2pt]{}cp[2pt]{}ccc@]{} && && galaxy &&\
\
$C_l^{XY}$ && $\ctt$ & $\cee$ & $\cte$ & $\cpp$ && $\cgg$ && $\ctp$ & $\ctg$ & $\cpg$\
\
$l_{\rm max}$ && 2500 & 2500 & 2500 & 1000 && 1000 && 1000 & 1000 & 1000\
$f_{\rm sky}$ && 0.65 & 0.65 & 0.65 & 0.65 && 0.10 && 0.10 & 0.10 & 0.10\
In our estimation, we include the information from temperature anisotropies, E-mode polarization and reconstructed lensing potential. The range of multipoles are $2 \leq l \leq 2500$ for $\ctt$ and $\cee$ and $2 \leq l \leq 1000$ for $\cpp$, respectively, and survey area is taken as $f_{\rm sky}^{\rm CMB} = 0.65$ for CMB survey. On the other hand, we include the information from galaxy survey, $\cgg$, where the range of multipoles are $2 \leq l \leq 1000$ and survey area is $f_{\rm sky}^{\rm galaxy} = 0.10$. We assume that there is no correlation between different patches, so that the area where there is a correlation between CMB and galaxy survey corresponds to the galaxy survey area $f_{\rm galaxy}$. We summarize the values we used mainly in the following calculation in Table \[tb:survey\].
The non-linear effect on the angular spectrum of the galaxy distribution begins to appear at $l \geq 100$. As the calculation of Fisher matrix assumes that all fields are random, the non-linear region of the galaxy distribution is inadequate for this calculation. However, because the auto correlation signal of the galaxy distribution is dominated by the noise term in this region as seen in Fig. \[fig:CMBNoise\] ($Left$), little information of the galaxy distribution from this region can be expected. Therefore, we neglect the non-linear effect of the angular power spectrum of the galaxy distribution in this paper.
Usually, the full Fisher matrix for joint experiment of galaxy survey and CMB is obtained simply by adding each Fisher matrices: $F_{ij} =
F_{ij}^{CMB} + F_{ij}^{\rm galaxy}$. This method, however, does not include all the available information because it does not account for the cross-correlation of temperature-galaxy $\ctg$ and lensing potential-galaxy $\cpg$. To use the angular power spectrum $\ctg$ and $\cpg$ for the estimation, and make the most of the available information, we consider all of the conceivable cross-correlations and calculate the full covariance matrix, which in this case is $8 {\rm
\times} 8$ matrix while we assume that $\cep$ and $\ceg$ are not correlated. Here, in order to account for the difference of the each survey area, and to investigate the significance of the cross-correlation signal, we calculate the full Fisher matrix of the following form,
$$\begin{aligned}
{\rm Case~(i)}~:~F_{ij}
&=& \sum_{l = 2}^{1000}F_{ij} \{
f_{\rm sky}^{\rm CMB}:\ctt, \cee, \cte, \cpp, \ctp \} \nn
+ \sum_{l = 2}^{1000}F_{ij} \{ f_{\rm sky}^{\rm galaxy} :\cgg \} \\
&+& \sum_{l = 1001}^{2500}F_{ij} \{ f_{\rm sky}^{\rm CMB}:\ctt, \cee, \cte \} , \label{eq:case1} \\
{\rm Case~(ii)}~:~F_{ij}
&=& \sum_{l = 2}^{2500}F_{ij} \{ f_{\rm
sky}^{\rm CMB}:\ctt, \cee, \cte \} \nn
+ \sum_{l = 2}^{1000}F_{ij} \{ f_{\rm sky}^{\rm galaxy} :\cpp, \cgg, \cpg \} \label{eq:case2} \\
&+& \sum_{l = 2}^{1000}F_{ij} \{ f_{\rm sky}^{\rm CMB} - f_{\rm sky}^{\rm galaxy} :\cpp \} , \\
{\rm Case~(iii)}~:~F_{ij}
&=& \sum_{l = 2}^{1000}F_{ij} \{ f_{\rm sky}^{\rm galaxy}:\ctt, \cee, \cte, \cpp, \ctp, \cgg, \ctg, \cpg \} \nn \\
&+& \sum_{l = 2}^{1000}F_{ij} \{f_{\rm sky}^{\rm CMB} - f_{\rm
sky}^{\rm galaxy}:\ctt, \cee, \cte \} +
\sum_{l = 1001}^{2500}F_{ij} \{ f_{\rm sky}^{\rm CMB}:\ctt, \cee,
\cte \} , \label{eq:case3} \end{aligned}$$
where “$f_{\rm sky}^{(\rm Survey)}$ ” represents the available sky fraction in the Fisher matrix with $({\rm Survey}) = \{{\rm galaxy},
{\rm CMB}\}$, and “$C_l^{XY} , \cdot \cdot \cdot$ ” denotes the angular power spectra included in the Fisher matrix with $X, Y = \{ T,
~E, ~\psi, ~g \}$. Case (i) does not include the cross-correlations between CMB and galaxy survey, $\ctg$ and $\cpg$, and Case (ii) does not include the cross-correlations between temperature and galaxy or lensing potential, $\ctg$ and $\ctp$, while Case (iii) takes account of all cross-correlations. We shall compare the difference in these three cases in the following section.
Result {#sec:Result}
======
Signal-to-Noise
---------------
The signal-to-noise ratio ($S/N$) for a cross-correlation between $X$ and $Y$ can be estimated by [@Peiris-00] $$\left( \dfrac{S}{N}
\right)^2 = f_{\rm sky} \sum_{2}^{l_{\rm max}} (2 l+1)
\dfrac{(C_l^{XY})^2}{(C_l^{XY})^2 + (C_l^{XX} + N_l^{XX})(C_l^{YY} +
N_l^{YY})} .$$ We show the $S/N$ value in Fig. \[fig:SN\] where we fixed the survey area as $f_{\rm sky} = 0.10$ and total number of the galaxies as $N_{\rm g} = 10^6$, respectively. The cross-correlations between temperature and lensing potential $\ctp$ and temperature and galaxy $\ctg$ are through the Integrated Sachs-Wolfe (ISW) effect imprinted in the CMB and the distribution of matter at late time. The ISW effect arise from the time-variation of the scalar metric perturbations and it is usually divided into an early ISW effect and a late ISW effect. The origin of the late ISW effect is from the time variation of the gravitational potential by the dark energy component and its effect emerges at low-multipoles. Therefore, the $S/N$ both from $\ctp$ and $\ctg$ saturate around $l_{\rm max} \simeq 40$, although their amplitudes are different.
On the other hand, the cross-correlation between lensing potential and galaxy has another feature. The survey which can explore the small scale region with large $l_{\rm max}$, can get large $S/N$ from $\cpg$ more than those from $\ctp$ and $\ctg$ while their amplitude is very small for the low resolution survey with small $l_{\rm max}$. The saturation of $S/N$ from $\cpg$ at large $l_{\rm max}$ for Planck in Fig. \[fig:SN\] (top two panels) is due to the noise contribution of the surveys. This justifies our omitting the proper modeling on small scales where non-linear evolutions are important. The signal-to-noise may be improved for high-precision future CMB survey, such as CMBPol, because it will be obtain much more information about small scale region than Planck. Since the cross-correlation between lensing potential and galaxy $\cpg$ has larger $S/N$ than other cross-correlation components, $\cpg$ would be more powerful tool when small-scale powers are observed. However, it should be noted that the correct non-linear model will be necessary on these scales.
Parameter errors
----------------
We show the 1$\sigma$ marginalized error of each parameter in Table \[tb:ParError\] and error contours in Fig. \[fig:ErrorCont\]. First, we compare the analysis methods, $i.e.$, the Cases (i) - (iii) defined in Eqs. (\[eq:case1\]) - (\[eq:case3\]). From the figure we find that the cosmological parameters, such as $\Omega_{\Lambda}$, $f_{\nu}$, and $w$ are tightly constrained in Case (i), while the constraint on the non-Gaussianity parameter $f_{\rm NL}$ is tighter in Case (ii) than Case (i). Because the main difference between Case (i) and (ii) is whether one includes $\ctp$ or $\cpg$, respectively, we can conclude that the cross-correlation between lensing potential and galaxy $\cpg$ is important to determine $f_{\rm NL}$ more precisely. Assuming more high-precision CMB survey, CMBPol, the significance of $\cpg$ for constraints on $f_{\rm NL}$ are seen more clearly.
Next, we investigate how the different galaxy sampling models affect the determination of $f_{\rm NL}$. For this purpose we consider two cases, the cases ($\alpha$, $\beta$) = (0.5, 3.0) and ($\alpha$, $\beta$) = (2.0, 1.5), and the results are shown in Fig. \[fig:ParErrorComp\]. The features of the two models are that the former has a gradual distribution and the latter has a sharp one around the peak redshift $z_0$. Generally, the expected errors rapidly decrease with redshift $z_0$ for $z_0 \lesssim 1$. As for the Case (iii) the error is almost independent of $z_0$ for the case with ($\alpha$, $\beta$) = (2.0, 1.5), while it has strong dependence for the case with ($\alpha$, $\beta$) = (0.5, 3.0). However, translating the peak redshift $z_0$ to the mean redshift ${z_{\rm m}}$, the tendency of the two models is similar to each other, although the constraint from the sharp distribution model is somewhat stronger than the gradual one, where the mean redshift ${z_{\rm m}}$ is determined by Eq. (\[meanredshift\]) and the relation between ${z_{\rm m}}$ and $z_0$ depends on the galaxy sampling model. For the same value of the peak redshift $z_0$, the gradual model represents relatively low mean redshift observation and the sharp model represents high redshift one. Therefore, the mean redshift of galaxies, rather than their distribution, determines how tight constraint one can obtain.
The difference of the constraints due to sampling models can be attributed to the degree of the correlation between the lensing potential and the galaxy distribution. Because the sharp one has narrow peak around the peak redshift, it has much large correlation with the lensing potential in this narrow range. The degree of the correlation takes maximum value at certain redshift, and it gradually decreases above the redshift. This fact reflects that $\Delta f_{\rm
NL}$ slowly increases with increasing of the peak redshift above the certain redshift. On the other hand, because the gradual distribution model has a wide peak, the galaxy distribution gas the lower degree of correlation with the lensing potential, even though the correlation exists over the wider range in $k$-space. In other words, to constrain the primordial non-Gaussianity from galaxy-CMB lensing cross-correlation one should select the galaxies whose correlation with the CMB lensing potential becomes maximum.
![ 1$\sigma$ confidence limits on the pair ($f_{\rm NL}$, $\theta_i$) in our 13-dimensional model. We show for Case (i) (dot-dashed line), Case (ii) (dashed line) and Case (iii) (solid line), respectively. We assume Planck (thick line) and CMBPol (thin line) for CMB survey, and for the galaxy survey the sky coverage and total number of galaxy sampling of $f_{\rm sky} = 0.1$ and $N_{\rm
g} = 10^6$. The parameters of galaxy sampling model are fixed as $\alpha = 0.5$, $\beta = 3.0$ and $z_0 = 1.8$. \[fig:ErrorCont\]](fig/contour.eps){width="100.00000%"}
[c|p[0.5pt]{}cp[0.5pt]{}cp[0.5pt]{}cccp[0.5pt]{}cccp[0.5pt]{}ccc]{} && && && && &&\
\
&& No lensing && Lensing && $z_0 = 0.6$ & $z_0 = 1.2$ & $z_0 = 1.8$ && $z_0 = 0.6$ & $z_0 = 1.2$ & $z_0 = 1.8$ && $z_0 = 0.6$ & $z_0 = 1.2$ & $z_0 = 1.8$\
$100 \Omega_b h^2$ && 0.0243 && 0.0225 && 0.0225 & 0.0225 & 0.0225 && 0.0224 & 0.0225 & 0.0225 && 0.0223 & 0.0223 & 0.0224\
$\Omega_c h^2$ && 0.00222 && 0.00214 && 0.00213 & 0.00214 & 0.00214 && 0.00212 & 0.00215 & 0.00216 && 0.00211 & 0.00213 & 0.00214\
$\Omega_{\Lambda}$ && 0.1922 && 0.0530 && 0.0519 & 0.0527 & 0.0528 && 0.0635 & 0.0677 & 0.0680 && 0.0504 & 0.0526 & 0.0527\
$\tau$ && 0.00553 && 0.00457 && 0.00457 & 0.00457 & 0.00457 && 0.00463 & 0.00463 & 0.00463 && 0.00456 & 0.00456 & 0.00456\
$f_\nu$ && 0.0384 && 0.0107 && 0.0107 & 0.0106 & 0.0107 && 0.0120 & 0.0120 & 0.0121 && 0.0107 & 0.0106 & 0.0106\
$Y_{\rm He}$ && 0.0159 && 0.0152 && 0.0152 & 0.0152 & 0.0152 && 0.0153 & 0.0154 & 0.0154 && 0.0151 & 0.0152 & 0.0152\
$n_s$ && 0.01016 && 0.00937 && 0.00933 & 0.00934 & 0.00936 && 0.00926 & 0.00932 & 0.00936 && 0.00924 & 0.00929 & 0.00933\
$\Delta_{\rm R}^{2}(k_0)\times 10^9$ && 0.0295 && 0.0237 && 0.0237 & 0.0237 & 0.0237 && 0.0242 & 0.0243 & 0.0243 && 0.0237 & 0.0237 & 0.0237\
$w$ && 0.651 && 0.197 && 0.191 & 0.195 & 0.195 && 0.253 & 0.269 & 0.270 && 0.187 & 0.195 & 0.195\
$N_{\rm eff}$ && 0.135 && 0.116 && 0.116 & 0.116 & 0.116 && 0.116 & 0.117 & 0.117 && 0.116 & 0.116 & 0.116\
$\alpha$ && 0.00841 && 0.00755 && 0.00749 & 0.00752 & 0.00754 && 0.00761 & 0.00767 & 0.00769 && 0.00745 & 0.00751 & 0.00753\
$f_{\rm NL}$ && —– && —– &&105.7 & 39.9 & 27.0 &&104.9 & 38.3 & 25.9 &&100.8 & 36.9 & 25.1\
$b_0$ && —– && —– && 0.0682 & 0.0718 & 0.0931 && 0.0703 & 0.0826 & 0.1076 && 0.0662 & 0.0707 & 0.0911
Finally, we focus on the observing redshift dependence for the constraints on the primordial non-Gaussianit $f_{\rm NL}$. We found that the constraints on $f_{\rm NL}$ considerably depends on the mean redshift of the observations ${z_{\rm m}}$, which is related with model parameter $z_{0}$ by Eq. (\[meanredshift\]). We show its redshift dependence in Fig. \[fig:ParErrorComp\]. The errors of $f_{\rm NL}$ rapidly decrease with redshift at low redshift, $z_0 \leq
1$, while it increases slowly at high redshift, $z_0 \geq 1$. The effect of the primordial non-Gaussianity through the scale-dependent bias becomes large at high redshift, so that this is the reason why the smaller error of $f_{\rm NL}$ can be obtained when the higher redshift is probed. On the other hand, the auto-correlation signal of the galaxy distribution becomes small with increasing redshift. This is an opposite effect to that from the scale-dependent bias for constraint on $f_{\rm NL}$ and this is the reason why the error of $f_{\rm NL}$ gradually increases at higher redshift, in particular, in Case (i) There are two reasons for increasing of the error of $f_{\rm
NL} $ at high-redshift. One is due to galaxy sampling model defined by Eq. (\[Gdist\]). This analytic form may drop the information of the low-redshift galaxies at high $z_0$. The other is that the cross-correlation between lensing potential and galaxy becomes weak at high redshift $z_0$, as seen in Fig. \[fig:Clpg\] for $f_{\rm NL} =
0$ (solid line).
Moreover, we compare the constraints from various galaxy survey conditions and linear bias parameters $b_0$ in Fig. \[fig:dfNL-FS\]. From this figure, we clearly see that both “depth” and “width” for galaxy survey are important for constraint on $f_{\rm NL}$ and what should be stressed is that the case of large bias $b_0$ constrains more strictly than the case of low bias. The objects with large bias are affected by the primordial non-Gaussianity more strongly than the objects with low bias, so one of the key points to put constraint on the primordial non-Gaussianity is to explore the highly biased objects. This results indicate that some galaxy survey exploring the highly biased objects could constrain $f_{\rm NL}$ in less than 10 even with Planck, for example, $\Delta f_{\rm NL} \sim ~ 5$ at $z_{\rm m} = 2.0$.
The estimations given above do not take into account conditions of realistic observations, because we vary only the peak redshift $z_0$ fixing the survey area $f_{\rm sky}$ and the available galaxy samples $N_{\rm g}$. In fact, in the real galaxy survey the survey area and the available galaxy samples will also vary due to the change of the observed mean redshift because the total observation time is finite as explained in Sec. \[sec:Modeling\]. Accounting the realistic galaxy survey condition, what strategy should we develop for constraining the primordial non-Gaussianity, $e.g.$ deep survey or wide survey ? In the next section, we search for the best condition for constraining $f_{\rm NL}$.
For the Galaxy Survey with Modeling
-----------------------------------
Assuming observations like HSC, we estimate a more realistic constraint on $f_{\rm NL}$ using the survey model introduced in Sec.\[sec:Modeling\]. We show the results in Fig.\[fig:ParErrorM\] and the relations between various survey parameters, namely, peak redshift $z_0$, survey area $f_{\rm sky}$, number density of sampling galaxy $n_{\rm L}$ and exposure time $t_{\rm exp}$ with a fixed observation time in Table \[tb:ParRelation\].
In Fig. \[fig:ParErrorM\], we find that there is a minimal point for the error of $f_{\rm NL}$ around ${z_{\rm m}} \simeq 0.7$ and in this point, the constraints on $f_{\rm NL}$ are $\Delta f_{\rm NL} \sim
~20$ for $b_0 = 2$ and $\Delta f_{\rm NL} \sim ~10$ for $b_0 = 4$. From this result, the target redshift we should observe is not deep enough, in which case the surface number density of the sample galaxies is relatively small, and the observation area can be wide, $n_{\rm L} \simeq 7.0$ $[{\rm arcmin}^{-2}]$ and $f_{\rm sky} \simeq
0.35$ (at $z_{\rm m} = 0.7$). (see Table \[tb:ParRelation\].) For the question whether we should make wide or deep survey, the answer is that we should select the wide survey. The signal of the primordial non-Gaussianity in the scale-dependent bias $b (z, k)$ is more significant in relatively large-scale regions, as seen in Fig. \[fig:Clpg\]. The noise contributions in large-scale regions are dominant by the cosmic variance due to the finiteness of survey area. On the other hand, small scale region is dominated by shot noise related to the surface number density of the galaxy samples $n_{\rm L}$, although the signal of the primordial non-Gaussianity is not sensitive there. Therefore, we conclude that the better constraining the primordial non-Gaussianity, $f_{\rm NL}$, prefers wide area survey to the deep survey. However, note that this conclusion is derived by assuming Planck and HSC experiments with a fixed observation time. Accounting for the redshift dependence of the effect of the primordial non-Gaussianity through scale-dependent bias, we should keep it in mind that the deep survey also becomes important to put a tighter constraint on $f_{\rm NL}$.
Summary and Discussion {#sec:Summary}
======================
In this paper we have estimated the constraints on the cosmological parameters newly taking into account the cross-correlation between CMB lensing and galaxy angular distributions. In particular, we have focused on the constraint on the primordial non-Gaussianity through the scale-dependent bias $b (z, k)$ and estimated how much the cross-correlation between CMB lensing and galaxy would improve the constraint by the Fisher matrix analysis. In order to make the most general Fisher matrix analysis with CMB and galaxy survey experiments, we have taken into account the all the auto- and cross-correlations available which is expressed by the 8$\times$8 covariance matrix as Eq. (\[eq:Cov\]). We have paid particular attention to the CMB and galaxy survey just coming up now, namely Planck satellite and HSC, and also to the future experiments like CMBPol, LSST and so on. Our estimations are mainly based on the Planck and HSC surveys, however we also show some cases for comparison when CMBPol or ambitious survey conditions are assumed.
As for the constraints on the conventional cosmological parameters, the improvement can not be expected very much from the simple estimate in which the Fisher matrices for the CMB and galaxy surveys are combined properly even if the cross-correlations are properly taken into account. However, focusing on the determination of $f_{\rm NL}$, we found that the role of the cross-correlation between CMB and galaxy is important, especially the one between CMB lensing potential and galaxy $\cpg$ contributes the determination of $f_{\rm NL}$.
We have estimated the constraints on the $f_{\rm NL}$ for the coming experiments. First, we gave the rough estimation in cases where the survey area $f_{\rm sky}$ and galaxy samples $N_{\rm g}$ are fixed and only peak redshift $z_0$ is varied. It was found that the keys for more strictly constraining the primordial non-Gaussianity are observing higher redshift and larger biased objects. Second, considering the realistic observations with fixed observation time, we have estimated the constraint on $f_{\rm NL}$, adapting the galaxy survey model in Ref. [@Yamamoto-07], which is scaled to the HSC survey. As a result we found on optimized target redshift to be $z_0 \simeq
1$, which brings us to a conclusion that we should make a wide survey rather than a deep survey for constraining the primordial non-Gaussianity with HSC-like observation. This is because the effect of the primordial non-Gaussianity through the scale-dependent bias is significant on large scales rather than small scales. The large-scale region is dominated by the cosmic variance related to the survey area while the small-scale region is dominated by the shot noise term related to the number of the galaxy samples. Therefore, we should explore the wide survey area rather than observing a lot of galaxies. However, we should keep it in mind that high-redshift and highly-biased objects are much affected by primordial non-Gaussianity, so that the deep survey will be also essential, in the future.
The constraints on the primordial non-Gaussianity expected from HSC survey with Planck are : $\Delta f_{\rm NL} \sim ~20$ for $b_0 = 2$ and $\Delta f_{\rm NL} \sim ~10$ for $b_0 = 4$. Slosar et al. [@Slosar-08] obtained constraints on $f_{\rm NL}$ for highly biased tracers using available luminous red galaxy (LRG) or quasar (QSO) data from Sloan Digital Sky Survey (SDSS) [@Ho-08] and CMB data from WMAP 5, with the error of $\Delta f_{\rm NL} \sim 30$. The reason why the combination of HSC and Planck observations does not make significant improvement over the current constraints is explained as below. We have considered only 800 hours for HSC and normal galaxies. The SDSS data is obtained over longer period of time, and QSOs seem to be observed more than normal galaxies at high-redshift. These constraints are weaker than those expected with the CMB bispectrum constraints achievable with an ideal CMB experiment, $\Delta f_{\rm NL} \sim ~{\rm few}$ (Ref. [@Yadav-07; @Liguori-08]). However, the constraint on $f_{\rm NL}$ presented in this paper depends highly on the galaxy survey condition, $i.e.,$ survey area $f_{\rm sky}$, number density of sample galaxies $n_{\rm
L}$ and observing peak redshift $z_0$. It is found that with some galaxy survey with Planck $f_{\rm NL}$ could achieve $\Delta f_{\rm
NL} \sim ~ 5$ at ${z_{\rm m}} = 2.0$ (Fig. \[fig:dfNL-FS\]). Recently it is reported that an ambitious future galaxy survey (like the LSST survey), which provides large survey area of 30,000 ${\rm
deg^2}$ and highly biased galaxy samples, can measure the primordial non-Gaussianity with the order $\Delta f_{\rm NL} \sim ~2-5$ [@Carbone-10]. The other method using the full covariance of cluster counts for Dark Energy Survey (DES) can yield $\Delta f_{\rm
NL} \sim 1-5$ [@Cunha-10]. In Ref. [@Carbone-10; @Cunha-10], they add the Planck and galaxy survey Fisher matrices, $i.e.,$ $F_{ij}
= F_{ij}^{\rm galaxy} + F_{ij}^{\rm CMB} $, and do not include the cross-correlation between CMB and galaxy survey, $i.e.,$ $\ctg$ an $\cpg$, so that more tight constraint on $f_{\rm NL}$ may be expected with these cross-correlations. In any case, it is worth pursuing how well we can put a constraint on non-Gaussianity of the local-type from the large-scale structure because it contains information on non-Gaussianity at different epoch from CMB and thus the constraint through the scale-dependent bias will be an important cross check against the CMB bispectrum.
![ The 1$\sigma$ error of $f_{\rm NL}$ in the case with fixed sky coverage $f_{\rm sky} = 0.1$ and total number of galaxy $N_{\rm g} = 10^6$ (Left), and in the case with modeling for HSC survey (Right). For the CMB experiment we consider two cases. The thick lines show the errors against peak redshift $z_0$ and the thin lines are against mean redshift $z_{\rm m}$. \[fig:ParErrorM\]](fig/dfNL-0530-40.eps "fig:"){width="45.00000%"} ![ The 1$\sigma$ error of $f_{\rm NL}$ in the case with fixed sky coverage $f_{\rm sky} = 0.1$ and total number of galaxy $N_{\rm g} = 10^6$ (Left), and in the case with modeling for HSC survey (Right). For the CMB experiment we consider two cases. The thick lines show the errors against peak redshift $z_0$ and the thin lines are against mean redshift $z_{\rm m}$. \[fig:ParErrorM\]](fig/fnl-HSC_0530.eps "fig:"){width="45.00000%"}
$z_0$ $z_{\rm m}$ $\Delta f_{\rm NL}$ $n_{\rm L} {\rm [arcmin^{-2}]}$ $f_{\rm sky}$
------- ------------- --------------------- --------------------------------- ---------------
0.8 0.51 30.1 0.9 0.42
0.9 0.58 24.5 1.9 0.41
1.0 0.64 20.7 3.7 0.40
1.1 0.70 18.5 7.0 0.35
1.2 0.77 18.6 12.4 0.25
1.3 0.83 21.9 20.9 0.14
1.4 0.90 29.1 34.0 0.06
1.5 0.96 40.5 53.5 0.02
: The relation of parameters with fixed observation time, between the peak redshift $z_0$, the mean redshift $z_{\rm m}$, the $1\sigma$ error of $f_{\rm NL}$, the number density of sampling galaxy $n_{\rm L}$ and the survey area $f_{\rm sky}$, for the HSC-like survey ($b_0=2$). \[tb:ParRelation\]
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abstract: 'We study an abstract model for the coevolution between mutating viruses and the adaptive immune system. In sequence space, these two populations are localized around transiently dominant strains. Delocalization or error thresholds exhibit a novel interdependence because immune response is conditional on the viral attack. An evolutionary chase is induced by stochastic fluctuations and can occur via periodic or intermittent cycles. Using simulations and stochastic analysis, we show how the transition between these two dynamic regimes depends on mutation rate, immune response, and population size.'
author:
- Alexander Seeholzer
- Erwin Frey
- Benedikt Obermayer
title: Periodic versus Intermittent Adaptive Cycles in Quasispecies Coevolution
---
Evolution is commonly pictured as a dynamic process on a fitness landscape in sequence space. In general, this landscape depends not only on the genotype but varies dynamically as a function of the environment and coevolving interaction partners [@Wright1932]. Prominent biological examples are the coevolutionary dynamics between the adaptive immune system and virus populations such as HIV [@Nowak1990; @Woo2012] or influenza [@Bedford2011], or between bacteria and their phages [@Levin2013]. Continuous evolutionary innovations allow the virus to transiently escape immune suppression, triggering subsequent adaptations of the immune system. These dynamics can lead to coevolutionary cycles, which have been generally described in two different forms [@Woolhouse2002; @Bedford2011]: either as an intermittent series of quasistationary states connected by stochastic jumps, or as periodic and largely deterministic oscillations. From a modeling perspective, this highly complex process is determined by three main features [@Perelson2002]. First, mutation rates are high and populations are large, which implies large genetic heterogeneity within the populations [@Domingo1985]. This has often been pictured in terms of broad quasispecies distributions around peaks in the fitness landscape [@Wang2006; @Eigen1977]. At the same time, continuous adaption and coevolutionary arms races are driven by strong ecological interactions [@Woolhouse2002; @Tellier2013]. These modulate effective fitness landscapes [@Gavrilets1998; @Nowak1990; @Arnoldt2012] and lead to nontrivial nonlinear population dynamics. Finally, stochastic effects in finite populations become especially pronounced whenever the first two issues are relevant at the same time [@Tsimring1996; @Rouzine2001; @Tellier2013; @Gokhale2014].
![(a) Schematic model for the coevolutionary dynamics of virus (V) and immune system (IS). The two populations are subject to mutation and selection (left), but also to ecological interactions (right). (b) Exemplary trajectories of the relative frequencies of virulent strains $x_i$ (top) and corresponding antibodies $y_i$ (bottom). Regular oscillatory dynamics involving three strains turn into simpler two-strain oscillations at $T_{3\to2}$ and finally transition into intermittency at $T_\text{int}$. Genetic variability within the populations is calculated from the average pairwise Hamming distance and indicated in gray dashed lines. (c) Sketch of the dynamics of the full population distributions in sequence space as described in the text.[]{data-label="fig1"}](fig1){width="8.6cm"}
Here, we offer a synthetic perspective on these processes. In our model \[see Fig. \], we consider a population of $N$ viruses represented by their genotypes (binary sequences of length $L$ and frequency $x_i$) and replication rates $r_i=1$. A small number $n$ of these genotypes corresponding to particularly virulent strains have a fitness advantage $\alpha$ over the unit baseline, giving $r_i=1+\alpha$ for $i=p,q,\ldots$. Offspring sequences undergo mutations with per-base rate $\mu_x$. In the absence of immune suppression, and in the stationary state, the viral population localizes as so-called quasispecies around any of the fittest genotypes, provided the mutation rate is smaller than Eigen’s error threshold $\mu_\text{c}\approx \ln(\alpha+1)/L$ [@Eigen1977]. This simple picture is considerably complicated by the host’s adaptive immune system, which produces antibodies that recognize and neutralize viruses with matching epitopes [@Alberts2002]. Antibody production is specifically increased and variability in the binding affinity is introduced when viruses with matching genotype are encountered [@Rajewsky1996], in a process that can be modeled in terms of mutation and selection. Similar concepts can be used for bacterial immune systems, where spacer sequences in the host genome complementary to genetic elements of a phage take antibodylike functions [@Levin2013]. Hence, for the immune system we introduce a second population of $N$ binary sequences with frequencies $y_i$, mutation rate $\mu_y$, unit replication rate for unstimulated production, and stimulated antibody production in the presence of perfectly matching viruses [@Kamp2002; @Wang2006]. Ecological interactions then introduce frequency-dependent fitness terms $\propto x_i y_i$ for such matched virus and antibody pairs. Including these terms leads to a reduction of the viral load and stimulation of antibody production \[Fig. , right\]. In the deterministic limit ($N\to\infty$), our model is described by [@supplement] $$\label{eq:1}
\begin{split}
\dot x_i &= \textstyle{\sum_j} m^x_{ij} r_j x_j - \alpha x_i y_i - x_i\phi_x,\\
\dot y_i &= \textstyle{\sum_j} m^y_{ij} (1+\gamma x_j) y_j -y_i\phi_y,
\end{split}$$ where $i$ and $j$ run over all $2^L$ sequences. The fitness advantage $\alpha$ of virulent strains can be suppressed to background levels by a perfectly adapted immune system, which undergoes stimulated production at rate $\gamma$ when encountering matching viral epitopes. Further, $m^{x}_{ij}=\mu_{x}^{d_{ij}}(1-\mu_{x})^{L-d_{ij}}$ is the probability of having $d_{ij}$ simultaneous mutations, where $d_{ij}$ is the Hamming distance between $x_i$ and $x_j$. The dilution terms $\phi_{x/y}$ are obtained from the conditions $\sum_i \dot x_i=0$ and $\sum_i \dot y_i=0$, respectively, and keep the sizes of the two populations fluctuating around constant values. This constraint applies to the stationary phase of the adaptive race, while we ignore some of the effects of a changing viral load [@Nowak1990; @Woo2012; @Wang2006; @Bull2007] and also neglect immune system memory [@Bianconi2010] and unspecific recognition [@Alberts2002].
To facilitate a systematic study of the effects of demographic noise by means of simulations and theoretical analysis, our starting point is the underlying stochastic master equation [@supplement] which has rarely been used in this context. Its deterministic limit leads to Eq. and connects to established quasispecies theory [@Eigen1977; @Kamp2002]. Exemplary simulation results obtained with the Gillespie algorithm [@Gillespie1992] are shown in Fig. , with parameters in the coexistence regime discussed below. We readily identify characteristics of the intermittent coevolutionary dynamics. First, a particularly virulent strain with its associated quasispecies “cloud” of mutants triggers a specific immune response (a), leading to a corresponding localization in the antibody sequence space (b). This gives alternative viral strains that are not under immune attack a fitness advantage, and after a brief “search” period during which the viral population becomes delocalized, this new fitness peak is colonized in a “growth” phase (c), awaiting the adaptive immune response (d). The delocalization and relocalization dynamics of each population in sequence space are clearly visible as transient increases in their respective mean pairwise Hamming distances \[Fig. \]. Intriguingly, this sequence of events can occur both in the form of regular oscillations as well as by means of stochastically intermittent cycles [@Woolhouse2002]. The former occurs when the large genetic diversity within the population extends across the valleys between different fitness peaks and signifies periodic shifts in the extent to which these peaks are populated [@Gavrilets1998]. The latter case indicates that adaptation proceeds stochastically via the random discovery of previously unpopulated fitness peaks by relatively tightly localized populations.
#### Steady-state regimes: coexistence for mutation rates below interdependent error thresholds.
We use a reduced deterministic version of the model to determine stationary states and the associated error thresholds. We restrict the analysis to the populations of the $n$ virulent strains $x_{p,q,\ldots}$ and their respective antibodies $y_{p,q,\ldots}$, and lump all mutant sequences together in the so-called error tail [@Schuster1988]. The high-dimensional system is then reduced to [@supplement] $$\label{eq:2}
\begin{split}
\dot x_p &= \left[Q_{x}\left(1+\alpha\right)-\alpha y_{p}-\bar{\phi}_{x}\right]x_{p},\\
\dot y_p &= \left[Q_{y}\left(1+\gamma x_{p}\right)-\bar{\phi}_{y}\right]y_{p},
\end{split}$$ where $p$ runs over the $n$ strains which are coupled by the corresponding dilution terms $\bar\phi_{x/y}$. $Q_{x/y}=(1-\mu_{x/y})^L$ are the quality factors. A straightforward stability analysis of fixed points in this system with respect to $\mu_{x/y}$ as bifurcation parameters yields the phase diagrams of Fig. .
![(a) Regimes of coevolution. High mutation rates $\mu_x$ of the virus lead to population delocalization, while for lower mutation rates a regime of coexistence emerges. Intermediate values lead to a degenerate localization regime for the virus (see text). (c) Steady-state values for relative frequencies $x_p=x_q$ and $y_p=y_q$ as a function of $\mu_y$ with $\mu_x=\mu_y$ (above) or $\mu_x=0.05$ (below). Solid lines are solutions of Eq. and dots are simulation results. Panels (a) and (c) are for $\gamma=\alpha$, while (b) and (d) show analogous result for $\gamma=10\alpha$, where $L=8$, $n=2$, and $\alpha=10$.[]{data-label="fig2"}](fig2){width="8.6cm"}
As expected, we recover the classical result that the viral population localizes around a fitness peak only if $Q_{x}>Q_{c}\equiv\left(\alpha+1\right)^{-1}$, with increasing genetic variability (i.e., the width of the population distribution) for larger mutation rate $\mu_x$. However, antibodies are localized only (1) if their mutation rate $\mu_y$ is small enough, (2) if their production rate $\gamma$ is high enough and (3) if the virus attack is specific enough (i.e., tightly localized). These interdependent requirements are an inevitable consequence of ecological interactions, and they translate into the condition $Q_y = \left\{\left(\gamma/\alpha n \right) \left[ Q_x(\alpha +1)-1\right]+1 \right\}^{-1}$ as the analytical limit for the coexistence regime (blue dashed lines in Fig. ). Only in this regime do we find the intriguing oscillatory dynamics shown in Fig. that will be discussed below. Finally, in a somewhat model-specific “degenerate” regime bounded by $Q_y = \left\{\left(\gamma/\alpha\right) \left[ Q_x(\alpha +1)-1\right]+1 \right\}^{-1}$, the virus population can stably localize about several fitness peaks simultaneously such that none of these quasispecies is sufficiently tight to trigger a specific response of the immune system, which thus remains delocalized. The fixed points of the approximate system coincide closely with the mean steady-state concentrations obtained by stochastic simulation of the full system \[see Figs. and \]. Interestingly, for $\alpha=\gamma$ and symmetric mutation rates $\mu_{x,y} \equiv \mu$, the critical condition of coexistence can be approximated by $\mu \approx \left(1/2L\right)\ln\left(\alpha/2\right)$ for large $\alpha$ and $L$, which generalizes a comparable result for mutualistic frequency-dependent fitness [@Obermayer2009] to the case of antagonistic interactions. This correspondence also suggests that the error thresholds derived here should be largely unchanged if recognition between the two population tolerates some mismatches [@Obermayer2010].
#### Noise-driven oscillations in the coexistence regime.
Performing a linear stability analysis in the coexistence regime reveals that the oscillations seen in the simulations are caused by $n-1$ pairs of purely imaginary eigenvalues. Numerical solutions of the deterministic Eqs. show complex but regular oscillations involving all $n$ strains with slow amplitude variations controlled by higher-order nonlinearities (see Fig. S1 in the Supplemental Material [@supplement]). Results from stochastic simulations, however, suggest that such complex patterns quickly give rise to simpler oscillations involving only two strains, which at a later time transition into intermittency (cf. Fig. ). Investigating the case $n > 2$ by simulations below, we restrict further analysis to $n=2$. Also, here we only display more compact analytical results for the case $\gamma=\alpha$ (see the Supplemental Material [@supplement] for general results). Our analysis exploits that in the coexistence regime mutation rates $\mu_{x/y} \lesssim \ln\alpha/L$ are small compared to the error thresholds and can be used as expansion parameters. To obtain the nonlinearities that control oscillation amplitudes, we expand Eq. to first order and transform to polar normal form on the two-dimensional stable manifold [@supplement]:
\[eq:3\] $$\begin{aligned}
\dot u &= -\frac{4}{5}L\left[\mu_{x}\left(\alpha+1\right)+\mu_{y}\right] u^2,\label{eq:3a}\\
\dot \varphi &= \frac{\alpha}{2} - \frac{L}{2}(\alpha + 1) (\mu_x + \mu_y)+\mathcal{O}(u),\label{eq:3b}\end{aligned}$$
where $u$ is a squared radial coordinate indicating deviations from the coexistence fixed point and $\varphi$ measures the phase of the oscillations. Equation exhibits a weak geometric decay of the oscillation amplitude $\mathcal{O}(u^2 \mu_x)\ll1$, which makes the fixed point only marginally stable and thus vulnerable to stochastic fluctuations [@Reichenbach2006; @Cremer2009; @Bladon2010]. Notably, the oscillation frequency of Eq. depends mainly on the fitness advantage $\alpha$, and is only weakly slowed down by mutations. In this deterministic regime, the quasispecies distribution in sequence space is broad enough that the time required to shift to a new fitness peak is dominated by the *growth* of the subpopulation already on the new peak (with a rate $\alpha$) rather than the *search* for this new peak in the first place (via mutations). We note that this effect is even stronger if the two fitness peaks are close in sequence space, i.e., if direct mutations between them are not ignored as in Eq. . In contrast, when the coexistence regime displays intermittent dynamics, because the relevant sequence space is not already inhabited by the virus population, the dynamics are inherently stochastic and mutation rates can be *too small* for the virus to explore enough sequence space to escape immune suppression in time. This would correspond to an adaptation threshold as found in a previous study [@Kamp2002]. However, as shown more formally below, this situation is incompatible with the presence of deterministic dynamics, which is an assumption of standard quasispecies theory. Instead, population genetics models should be used [@Tellier2013; @Arnoldt2012].
#### Noise determines if dynamics are periodic or intermittent.
We can characterize how stochastic noise controls the transition between periodic and intermittent adaptive dynamics by means of stochastic averaging. This technique enables a systematic derivation of effective one-dimensional Fokker-Planck equations in relevant subspaces of more complex high-dimensional nonlinear dynamics such as those arising in evolutionary game theory [@Dobrinevski2012]. It is based on the time scale separation between slow radial and fast azimuthal dynamics in Eq. : $\varphi$ evolves on fast time scales ($\dot\varphi\propto \alpha$), while $u$ changes much more slowly ($\dot u\propto \mu L u^2$). Using this observation, we can derive effective coefficients governing the evolution of the probability distribution $P(u,t)$ of the radial variable by averaging the angular dynamics over one oscillation period [@supplement]. To leading order, we get $$\label{eq:4}
\partial_{t}P= -\partial_{u}\left[\left(-a_{1}u^{2}+\frac{a_{2}}{N}\right)P \right] +\frac{1}{N}\partial_{u}^{2}\left(a_{2}u\,P\right),$$ with $a_1=\tfrac{4}{5}L[\mu_{x}\left(\alpha+1\right)+\mu_{y}]$ and $a_2=\frac{1}{16}\big\{4+3 \alpha - \mu_x L\left[\left(4/\alpha\right)+11+7 \alpha \right]- \mu_y L\left[\left(4/\alpha\right)+7+3 \alpha\right]\big\}$. Note that in the deterministic limit $N\to\infty$ we recover Eq. . For a finite population, we now find the deterministic decay $\propto a_1 u^2$ towards the coexistence fixed point in competition with a stochastic outward drift $\propto \left( a_2 / N \right)$, which destabilizes the fixed point and leads to a finite oscillation amplitude $\langle u\rangle= \sqrt{\left(2/\pi\right)\left(a_2/a_1 N\right)}$ . As mutation rates get small, expected oscillation amplitudes grow as $[(\alpha+1) \mu_x + 4 \mu_y]^{-1/2}$, eventually hitting the borders of the concentration simplex. This indicates the transition from regular oscillations to intermittent behavior: during large-amplitude oscillations the fittest virus genotypes are temporarily lost from the population and are only much later recovered through spontaneous mutations.
A more detailed understanding of this transition is obtained by estimating the lifetime of the regular oscillations. To this end, we use the bounds on the radial variable $u_\text{max} = \tfrac{1}{8}-\mathcal{O}(\mu L)$, where the populations are fully localized about only one peak. The chances of observing a transition to intermittent behavior are estimated from the mean first passage time (MFPT) $T_\text{int}$ from $u=0$ to $u=u_\text{max}$ under Eq. . Using standard methods [@Kampen1992], we find the result [@supplement] $$\label{eq:5}
T_\text{int} = N \frac{u_\text{max}}{a_2} \tilde{F}\left(\frac{N a_1 u^2_\text{max}}{2 a_2}\right),$$ where ${\tilde F}(x)$ is the generalized hypergeometric function ${_2 F_2} (\nicefrac{1}{2},1;\nicefrac{3}{2},\nicefrac{3}{2}; x)$. Equation can be brought into scaling form by defining $N^*=\left(2 a_2/a_1 u^2_\text{max} \right)$ and $T^*=N^* \left(u_\text{max} / a_2 \right)$. To compare this result to simulations, we plot the rescaled MFPT $\left(T_\text{int}/T^* \right) = \left(N/N^*\right) {\tilde F} (N/N^*)$ (see Fig. ). The nearly perfect data collapse for different parameter choices validates our analytical approach.
While $N^*$ measures the population size at the crossover from periodic to intermittent dynamics, $T^*$ denotes the corresponding typical duration of the transition. For large populations ($N > N^*$), we find $T_\text{int}\sim N^{-1/2} e^{N/N^*}$; this almost exponential growth of the MFPT indicates that the dynamics are effectively deterministic and intermittent behavior extremely unlikely. For $N < N^*$, we find $T_\text{int}\sim N$ and the dynamics thus easily transition into intermittency. This distinction based on the scaling of $T_\text{int}$ with $N$ has recently been suggested in the context of game theory [@Cremer2009]. In our case, however, finite mutation rates prevent permanent extinction of subpopulations and stabilize regular oscillatory behavior even in small populations, because the deterministic decay in Eq. is strengthened and the critical population size $N^* \propto (\mu_x L)^{-1}$ is reduced. Thus, even for small populations, mutations can act as a driving force for the stabilization of regular oscillations, which *a posteriori* justifies assumptions underlying quasispecies theory and generalizes previous observations [@Gokhale2014]. In contrast, from results for general $\gamma$ (see Fig. S2 in the Supplemental Material [@supplement]), we find that a strong immune response (i.e., $\gamma > \alpha$) promotes early transitions into intermittency \[cf. Fig. \], since both $N^*$ and $T^*$ increase with $\gamma$. However, these parameters are insensitive against the precise value of $\mu_y$ \[cf. Fig. \]; this suggests that effective immune suppression is achieved via a strong stimulated response rather than high adaptive flexibility. Indeed, extreme antibody secretion rates have been reported in the literature [@Hibi1986]. Finally, we support our choice of limiting the analysis to $n=2$ strains by simulating a system with $n=3$ strains, measuring the time $T_{3\to 2}$ until one strain is lost as well as the subsequent $T_\text{int}$ until the remaining two strains transition to intermittency. As shown in Fig. , the state with all three strains present is short lived compared to the two-strain oscillations, especially in the relevant deterministic regime of larger population size. Hence, apart from numerical prefactors the general trend captured in Eq. also describes systems with larger $n$.
![Mean time until transition from regular oscillations to intermittency. Dashed lines are from the analytical result . (a) Rescaled simulation data for $L=8$, $\alpha=\gamma=10$ and different choices of $\mu_{x/y}$ collapse onto a universal curve (unscaled data shown in the inset). (b) Transition times decrease for increasing $\gamma$ \[parameters otherwise as in (a)\]. (c) For $n=3$ virulent strains, the transition time $T_{3\to 2}$ until one strain is lost is much shorter than $T_\text{int}$, especially for large populations.[]{data-label="fig3"}](fig3){width="8.6cm"}
#### Conclusions.
We have analyzed a model for the coevolutionary dynamics of virus and immune system, combining simulations with nonlinear deterministic and stochastic analysis. Starting from the established quasispecies treatment of this problem, we explicitly introduced interactions between the populations. These lead to interdependent error thresholds, because a focused immune defense against a specific viral strain is impaired for large genetic variability in the virus population. Further, we performed a rigorous analysis of stochastic effects in the coexistence regime: regular yet noise-induced oscillatory behavior for large populations, large mutation rates, and weak immune response turn into stochastic intermittent cycles for smaller populations, smaller mutation rates and strong immune response. Our simulations indicate that the reverse transition from intermittency towards regular oscillations is a rare event occurring on time scales well beyond $T_\text{int}$. It cannot easily be analyzed within our reduced two-dimensional model as it will depend on the entire population structure. Finally, we note that our abstract model based on quasispecies theory focuses on the dynamics of genetic variability within populations of constant size. This assumption is of course violated for some biological scenarios, where immune response modulates the viral load [@Nowak1990; @Wang2006; @Bull2007] and may well lead to extinction of the virus [@Tejero2010; @Woo2012]. We expect that more detailed models including these and other effects relevant in biological situations [@Bianconi2010; @Alberts2002] will also be amenable to theoretical analysis based on the stochastic averaging techniques used here.
We acknowledge helpful comments by anonymous reviewers and financial support by the Deutsche Forschungsgemeinschaft in the framework of the SFB/TR12 – Symmetries and Universality in Mesoscopic Systems.
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abstract: 'Convolutional neural networks (CNNs) have been widely and successfully used for medical image segmentation. However, CNNs are typically considered to require large numbers of dedicated expert-segmented training volumes, which may be limiting in practice. This work investigates whether clinically obtained segmentations which are readily available in picture archiving and communication systems (PACS) could provide a possible source of data to train a CNN for segmentation of organs-at-risk (OARs) in radiotherapy treatment planning. In such data, delineations of structures deemed irrelevant to the target clinical use may be lacking. To overcome this issue, we use multi-label instead of multi-class segmentation. We empirically assess how many clinical delineations would be sufficient to train a CNN for the segmentation of OARs and find that increasing the training set size beyond a limited number of images leads to sharply diminishing returns. Moreover, we find that by using multi-label segmentation, missing structures in the reference standard do not have a negative effect on overall segmentation accuracy. These results indicate that segmentations obtained in a clinical workflow can be used to train an accurate OAR segmentation model.'
author:
- 'Louis D. van Harten'
- 'Jelmer M. Wolterink'
- 'Joost J.C. Verhoeff'
- Ivana Išgum
bibliography:
- 'spie\_seg.bib'
title: 'Exploiting Clinically Available Delineations for CNN-based Segmentation in Radiotherapy Treatment Planning'
---
Introduction
============
Recent years have seen substantial progress in automatic medical image segmentation. These advances can be primarily attributed to the emergence of convolutional neural networks (CNNs) [@ciresan2012deep; @Litj17]. CNN-based segmentation is typically performed in a fully supervised setting, where a network is trained based on available segmentation maps in a training set. In such settings, the performance varies based on the network architecture and hyper-parameters, the optimization procedure during training, and on the size and quality of the training set. A number of recent benchmarks have shown that most state-of-the-art CNN-based methods achieve comparable results when applied to the same medical image data set. For example, a recent challenge in cardiac cine MR images found that for left ventricle segmentation, there were no significant differences between the top 8 methods, even though all used different architectures, hyper parameters, and optimization schemes.[@Bern18]. All methods used the same training data, suggesting that the properties of the available training data used to train a convolutional neural network may be among the most important factors for the network performance.
For medical image segmentation, creating a labeled training set typically entails time-consuming and costly annotation by medical experts. As a consequence, the number of available labeled examples in medical image training sets is generally much lower than the number of labeled examples in training sets for natural images. This issue is exacerbated by the large variety of medical imaging modalities and sequences, which generally means a completely new data set is required for every medical segmentation problem. A possible solution is to use data that is produced and manually labeled as part of a clinical workflow. For example, in radiotherapy, manual segmentations of organs-at-risk (OARs) are routinely made for treatment planning. In this work, we investigate whether – instead of obtaining a data set of dedicated segmentations by a clinical expert – these readily available clinical segmentations could be used to train a CNN for automatic segmentation of OARs. One challenge to overcome is that this data often lacks delineations of structures deemed irrelevant for the clinical task. For example, organs that are far away from a tumor, and hence not at risk of irradiation, are often not segmented.
The use of partially segmented training volumes raises interesting methodological questions, as there is no unambiguous definition of “background” in such volumes. This problem has previously been addressed using conventional machine learning techniques [@moeskops_evaluation_2015]. Recently, CNN training strategies have been adapted for training with missing annotations by considering segmentation to be a *multi-label* instead of a *multi-class* problem [@petit_handling_2018]. While multi-class segmentation requires all voxels to exclusively belong to the background class or to one of the foreground classes, a multi-label segmentation model produces a result for each of the foreground classes independently from the others. This property can be exploited to train the network with images for which not all classes have a ground truth label.
Here, we perform an empirical study in which we systematically investigate the number of reference delineations that is necessary to achieve adequate model performance for OAR segmentation, and whether similarly adequate results can be obtained using reference delineations with missing structures. By training and evaluating the performance of 96 CNNs trained on different subsets of our data set, we assess the feasibility of developing a successful OAR segmentation model using varying amounts of clinically available segmentations with varying levels of completeness.
![A schematic overview of the multi-label training process. Function $G$ represents a convolutional neural segmentation network with $N$ outputs, corresponding to each class. For each class $n$, a sigmoid function generates a probability map that is compared to a binary reference image through binary cross-entropy, resulting in a loss $\lambda_n$. The total loss is the sum over all $\lambda_n$, weighted by $c_n\in\{0,1\}$ depending on the presence of class $n$ in the ground truth (GT) of the training sample.[]{data-label="fig:pipeline_architecture"}](pipeline_architecture_isolated_equifontsize_raster_small.pdf){width="99.00000%"}
Data {#sec:data}
====
With permission of the local medical ethics board, we included brain MRI studies of 52 patients undergoing radiotherapy treatment planning. All patients received a T~1~-weighted MR scan at the University Medical Center Utrecht (Utrecht, the Netherlands). Volumes were acquired using a Philips Ingenia 1.5T MR system with a voxel size of $1.1\times1.1\times1.0$ mm$^3$, 8 flip angle, 7 ms repetition time, and 3.1 ms echo time. Scans were reconstructed to a voxel size of $0.9\times0.9\times1.0$ mm$^3$. Patients were scanned in an immobilising mask, ensuring a similar orientation of the head for all patients.
This data set includes annotations of the brain stem, pituitary gland and optic chiasm, and the left and right optic nerves, eyes, cochleas, and lenses acquired as part of RT treatment planning. OARs were typically segmented only if they were in clinically relevant proximity to the clinical target volume for the RT treatment. On average 9.4$\pm$1.8 out of 11 possible OARs were annotated in each patient. For 15 out of 52 patients, all OARs were available. Delineations were made on CT images and propagated to corresponding MR images. This regularly led to over- and under-segmentation when visualised in the MR image. Given that these are representative of clinical segmentations, we used these potentially suboptimal segmentations for training. However, as the results must be evaluated on a ground truth, such errors would interfere with the evaluation. Hence, a clinical expert corrected all manual segmentations in a subset of 20 volumes, which was used as a test set.
![An illustration of the two sampling methods used in this work, visualised for subsets of size $M$=2. The full data set contains some MR images for which not all class labels are available (indicated by strike-through). *Concentrated label* subsets contain $M$ volumes for which all labels are available. *Distributed label* subsets contain more MR volumes, but labels for each class are only available in $M$ volumes.[]{data-label="fig:sampling_diagram"}](sampling_diagram_wide_noxenos.pdf){width="70.00000%"}
![A detailed view of the 3D CNN architecture used in this work, indicated as function $G$ in Figure \[fig:pipeline\_architecture\]. Each unit in this figure represents a 3D convolution layer with a ReLu activation, with numbers in each unit indicating the kernel size and number of filter maps. The down- and upsampling paths are implemented using strided convolutions (indicated by ‘/2’) and transposed convolutions (indicated by ‘\*2’) respectively. $N_c$ indicates the amount of classes and is equal to 11 in this work.[]{data-label="fig:detailed_architecture"}](detail_architecture.pdf){width="99.00000%"}
Methods {#sec:methods}
=======
We perform a series of experiments to address two separate but related research questions. We investigate whether incomplete segmentations obtained from a clinical workflow are an appropriate substitute for a dedicated training set for training a segmentation network for various brain structures. Additionally, we investigate the impact of the number of segmented training volumes on the performance of such a network, and whether the required number of segmented volumes increases when only part of the target structures are segmented in each training image.
To address these questions, we perform 96 experiments in which we train CNNs with sampled subsets of the available training data. A subset of size $M$ is defined as a data set in which each structure has been segmented $M$ times. Subsets are randomly sampled in two distinct ways, as illustrated in Fig. \[fig:sampling\_diagram\]. In the *concentrated labels* setting, we sample $M$ volumes with full reference segmentations, which is equivalent to selecting the MR volumes of $M$ patients. In the *distributed labels* setting, a subset includes all training volumes, but for each class, labels are only included in $M$ randomly selected volumes. In this setting, a training subset contains more volumes of different patients, but labels for only part of the structures in each volume are available. Subsets in the distributed labels setting were pseudo-randomly sampled to evenly spread the labels over as many volumes as possible. In both settings, the trained networks see the same number of labeled structures and – assuming similarly sized structures among patients – a similar number of training voxels. This equates to an approximately equal amount of work required by a clinical expert to create the training sets, which allows us to compare the results fairly. As smaller subsets can be sampled in many ways, we repeat experiments multiple times with different subsets of the same size.
In all experiments, we use the same 3D fully convolutional network with residual connections, adapted from an existing 2D network[@he_deep_2015]. This network was recently shown to exhibit competitive performance in a challenge on OAR segmentation in thoracic CT images[@vanharten2019automatic]. Its architecture is shown in Fig. \[fig:detailed\_architecture\]. The network contains two strided convolutional downsampling layers, followed by 16 residual blocks and two transposed convolutional upsampling layers. The residual blocks are implemented using the updated residual configuration proposed in He et al. [@he2016identity]. Instead of a softmax activation function, which is typically used in multi-class segmentation problems, the output layer contains one sigmoid activation function per class (as shown in Fig. \[fig:pipeline\_architecture\]). By using a sigmoid instead of a softmax output activation function, any $N$-class segmentation problem can be modelled as a combination of $N$ binary label segmentation problems[@petit_handling_2018]. A class loss $\lambda_n$ can be calculated for each binary label separately; the total loss is calculated as the sum over all class losses, weighted by an optional weighting factor $c_n$. If during training class $n$ is not present in the reference segmentation of the current training volume, $c_n$ is set to 0. This amounts to ignoring the loss component corresponding to this class for the current training volume. In this work, $c_n$ is otherwise set to 1 for all classes.
![Network performance in terms of Dice similarity coefficient as a function of the number of delineated ground truth references per class, when trained in the concentrated (i.e.: fully segmented) training configuration. Data points on the graph are the average Dice score for each OAR on the test set, averaged over the results from each network trained on the same training subset size. Vertical bars indicate the 95% confidence interval. []{data-label="fig:results_mrct_concentrated"}](mean_perf_perclass_mrct_hor_wide_linx.pdf){height="\detgraphheight"}
Experiments and Results
=======================
From the full data set, 30 volumes were used for training, 2 volumes were used for validation, and the remaining 20 volumes were used for testing. We trained 96 networks on the OAR set: 48 on concentrated subsets and 48 on distributed subsets, as described in Sec. \[sec:methods\]. In the training set, 15 volumes included reference delineations for all 11 target OARs and could be used to sample concentrated subsets. No data augmentation was used in any of the experiments. All networks were trained with the Adam optimizer (learning rate: 0.001) for 15000 iterations with batches of four cubic 64^3^-voxel patches per iteration. Training was done on a shared computing cluster containing various consumer-grade NVIDIA GPUs; training times ranged between 4 and 12 hours per network, depending on the load on the cluster.
[0.48]{} {height="\stripheight"}
[0.48]{} {height="\stripheight"}
Fig. \[fig:results\_mrct\_concentrated\] shows the average Dice similarity coefficients attained by networks trained on concentrated (i.e. fully segmented) training sets. The results show that, as may be expected, the performance increases when more training volumes are used. The improvement of the average performance is most pronounced up until five reference segmentations per structure. The results for most classes still slightly improve when increasing the training set size further, albeit with sharply diminishing returns.
We assessed whether the same number of reference segmentations can also be provided to the CNN in a distributed fashion, i.e. using volumes in which only a subset of structures has been segmented. We compare the results for the concentrated and distributed settings for the right eye and the brain stem in Fig. \[fig:results\_all\_concentrated\]. These figures show that for the larger training sets, the performance of networks trained in both settings is similar, although networks trained in the distributed setting produce a smaller number of worst-case outliers.
Interestingly, the results show that the networks trained on small sets of distributed data perform substantially worse on the eye, whereas they are mostly comparable with the networks trained on concentrated data for the brain stem. This discrepancy could be explained by the presence of visual ambiguity in the classes that have contralateral equivalents. Because of our pseudo-random sampling in the distributed experiment setting, it was highly unlikely that reference segmentations for both versions of a symmetric OAR would be included for the same patient. As a result, these networks perform worse at distinguishing symmetric OARs from their contralateral equivalents. Intuitively, a right eye is difficult to distinguish from a left eye based on local geometry: the networks may fail to learn the distinction unless labels for a matching set of eyes is available.
Discussion
==========
In this study, we aimed to answer two research questions. First, we evaluated the number of segmented volumes required to train an adequate OAR segmentation network using clinically obtained data. We observed a saturation point between five and seven labeled examples in the training sets (depending on the class) after which an increased data set size showed sharply diminishing returns on the network performance. This challenges the common assumptions concerning the large amounts of data required for training a deep neural network. Although these results are promising, they were acquired using a relatively small test set in a single segmentation task; further research is needed to investigate the extent to which our findings apply to different segmentation tasks.
Second, we evaluated whether a similarly performing network could be trained using a data set in which only some classes were segmented in each volume, as is generally the case with clinically performed segmentations. In this setting, we observed a large discrepancy in performance on the symmetric OARs with contralateral equivalents in networks trained on small training sets. Our results imply that the networks have trouble learning to discriminate between visually similar structures unless both are segmented in the same training volume.
It should be noted that the problem illustrated above is only present when training on subsets where the majority of segmented volumes only contain one version of the symmetric OARs. In the full data set, segmentations of contralaterally equivalent organs are usually both present if a segmentation for at least one of them is available. This discrepancy could be considered an artifact of our sampling method. However, the presence of unsegmented visually ambiguous structures is not unthinkable in other clinically obtained data sets. For example, similar problems could emerge with partially segmented vertebral columns, or abdominal images where only part of the large intestine is delineated. Future work could investigate the merit of cropping such unsegmented visual ambiguities out of the training images before training the network.
Conclusion
==========
In this work, we have addressed the common assumption that segmentation CNNs require large amounts of training data and we have investigated whether routinely acquired clinical segmentations can be used to train an OAR segmentation model instead of a dedicated training set. We found that training such networks on a small number of incomplete clinical segmentations is feasible, as long as there are no clear ambiguities between classes. We have shown that this limitation can be overcome by increasing the size of the training set.
New or breakthrough work to be presented
========================================
We have shown that it is possible to train an accurate OAR segmentation network with a small training set of clinically acquired delineations, without any data augmentation. Our results show that as long as there is little ambiguity in the class definitions, it is possible to train such a network even if part of the target class delineations is missing in each of the training volumes.
|
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abstract: ' data are used to obtain two-dimensional gas temperature maps of the hot non-cooling flow clusters A2256, A2319, A2163 and A665. In all four clusters, the temperature decreases significantly at off-center distances of $\sim 1\,h^{-1}$ Mpc ($H_0\equiv 100\,h$). Central regions of the two nearer clusters A2256 and A2319 are resolved by and appear largely isothermal except for the cooler spots coincident with the subunits in their X-ray surface brightness. Although the existence of this substructure may suggest ongoing merger activity, no asymmetric features in the temperature distribution resembling those in the hydrodynamic merger simulations (e.g., Schindler & 1993) are apparent. In the outer parts of the clusters, the temperature declines symmetrically with radius. In A2256 and A2319, it follows a polytropic slope with $\gamma\simeq 1.3-1.5$. This is somewhat steeper than the simulations predict for a flat CDM universe and is closer to the open universe predictions (Evrard et al. 1996). The temperature drop is more prominent in the outer regions of A2163 and A665 and appears even steeper than adiabatic (although not inconsistent with it). If the gas in the outskirts of these two clusters is indeed as cool as we measure, the cluster atmospheres should be convectively unstable and transient. Also, such a steep temperature profile could not possibly emerge if the gas was heated only via the release of its own gravitational energy during infall. This may indicate the presence of an additional heat source in the inner cluster, such as merger shocks transferring energy from the dark matter to the gas. The results suggest that A2256 and A2319 are pre-merger systems and A2163 and A665 are ongoing or post-mergers.'
author:
- Maxim Markevitch
title: TEMPERATURE STRUCTURE OF FOUR HOT CLUSTERS WITH
---
Accepted for [*The ApJ Letters*]{}
INTRODUCTION
============
Spatially-resolved measurements of the cluster gas temperature are necessary for such an extensive and important problem as determining cluster masses (e.g., Fabricant et al. 1984; White et al. 1993). Beyond that, the cluster temperature structure can provide information on the dynamical history of these systems. Rich and massive clusters should be just forming now in the hierarchical clustering scenarios with a high matter density parameter $\Omega$ (e.g., Blumenthal et al. 1984). On the other hand, in an open universe most present-day clusters should be old, because their formation is inhibited after $t\sim \Omega t_0$, $t_0$ denoting the present epoch (White & Rees 1978). Hydrodynamic simulations of cluster growth (e.g., Navarro et al. 1995; Evrard et al. 1996, hereafter EMN) predict a largely constant temperature profile in the inner part and its decline in the outer regions for the relaxed clusters, with a steeper decline in the open universe models. Young clusters which have recently undergone a merger should retain a complex temperature structure (e.g., Schindler & 1993). However, while there is a wealth of cluster simulations in various cosmological scenarios, until recently, direct spatially-resolved temperature measurements have been possible with only a limited accuracy, especially for the hotter, more massive systems (e.g., Hughes 1991; Eyles et al. 1991; Miyaji et al. 1993; Briel & Henry 1994, hereafter BH; Henry & Briel 1995). with its broad energy coverage combined with imaging capability (Tanaka et al. 1994) is set to significantly improve the situation. Some results have already appeared (e.g., Arnaud et al. 1994; Markevitch et al. 1994, 1996, hereafter M94 and M96; Ikebe et al. 1996).
In this [*Letter*]{}, we use data to derive temperature maps of nearby A2256 ($z=0.058$), A2319 ($z=0.056$), and distant A2163 ($z=0.201$) and A665 ($z=0.18$). All four are hot, lack cooling flows and are probably not fully relaxed, which is suggested by either the substructure in their X-ray images or by galaxy velocities (e.g., Briel et al. 1991; Elbaz et al. 1995). For A2256, a temperature map was earlier presented by BH who used PSPC, and our results are compared with theirs. results on the temperature structure near the center of A2163 were reported in M94. In M96, a steep radial temperature decrease was found in this cluster. Interestingly, a recent measurement of the Sunyaev-Zeldovich effect toward A2163 by Holzapfel et al. (1996) independently suggests a similar decrease, although with marginal significance. Below, a less model-dependent, two-dimensional approach to the data is employed to confirm the result of M96 and find similar phenomena in other three clusters.
DATA AND METHOD
===============
For A2256, data from the two pointings were used which total 63 ks of useful time. The pointings are offset by 6 from one another, have the “roll angles” between the cluster and mirror axes which differ by about 100, and are carried out with different GIS onboard background rejection modes (high background during the early 28 ks and normal background during the subsequent observation.) All of this facilitated useful internal consistency checks of the results. A2319 has two pointings of 28 ks in total with a relative offset of 12. For these two bigger clusters not entirely covered by the SIS, only the GIS data were used for computational simplicity. Both SIS and GIS were used for A2163, which has a single 27 ks pointing. A665 was observed in one pointing for 33 ks. The detector plus sky background was modeled using the blank field observations normalized according to their exposures. A $1\sigma$ relative error of 20% was assigned to these normalizations (5% for the A2163 GIS data, see M96). A problem was encountered while modeling the SIS background for A665, for which the normalization calculated this way was obviously too high. We chose to use only the GIS data for this cluster.
To reconstruct the cluster temperature maps, the scheme described in M96 was used, which consists of simultaneous fitting of the spectra from all chosen image regions, all detectors and all pointings, taking into account the PSF scattering. The projected temperature was assumed constant within each region of interest. The iron abundance and $N_H$ were fixed at the same values for all regions. To minimize $\chi^2$ as a function of many free parameters and avoid false minima, the annealing method from Press et al.(1992) was used. The image regions we use are all larger than 5–6across, which is sufficiently large compared to the $3'$ half-power diameter of the PSF. The PSF was modeled by interpolation between the GIS images of Cyg X-1 (Takahashi et al. 1995). Only data above 2.5 keV were used due to the uncertainty of our PSF model below that energy. Relative PSF uncertainty of 15% ($1\sigma$) was included in the confidence intervals calculation. For the purpose of this work, PSPC images were used to model the cluster projected emission measure. They were corrected for the gas non-isothermality in each iteration by dividing the brightness by the plasma emissivity in the PSPC band for a given temperature (which doesn’t significantly change the results; M96). The images were obtained using the Snowden et al. (1994) procedure. statistical and background errors were included into the confidence intervals of our temperature values. To correct for inaccuracy of the attitude solution, and images were aligned by eye after convolving the former with the response. Uncertainty of this operation (about $\pm 0.3'$ at 90%) was included in the confidence intervals for the two big clusters.
Given the relative complexity of the method, it is useful to present some consistency checks which were performed. Our results may be affected by the following:
[*Errors in the code.*]{} Currently there are at least three other independent techniques for the cluster analysis, which give results consistent with this method. An isothermal cluster was simulated by Ikebe (1996) and it was possible to extract its input temperatures with the code used here. R. Mushotzky communicated that the temperature profile of A2163 derived using the code of K. Arnaud is similar to that in M96. Using another independent technique, Churazov et al. (1996) derived temperatures for A2256 and A2319 similar to those presented here.
[*Use of ROSAT brightness map.*]{} An incorrect emission measure model may result in distorted temperatures. For A2163, the temperatures and a $\beta$-model density profile were fitted together using only, and the obtained density profile was in agreement with a better-constrained profile (M96). For the remaining three clusters, an analogous test was performed, in which relative normalizations between the model annuli, set by the image, were freed and fitted together with the temperatures. Their resulting values were consistent with 1, as is expected if the emission measure model is adequate. Cooling flows would require a different approach, which is the reason of our choice of non-cooling flow clusters for the present work.
[*PSF and effective area miscalibration.*]{} All analysis methods are currently using essentially the same PSF model. This model has been tested by comparison with the point sources at different focal plane positions (Takahashi et al. 1995; Ueda 1996) and found adequate to the accuracy level which is used here. Ikebe (1996) checked the effective area calibration and found that it is adequate after certain correction. We also note that there are no bright sources in the vicinity of these clusters that may produce any significant stray light contamination. For the bigger clusters A2256 and A2319, the PSF-scattered contribution to the flux of a particular image region, was less than a half for most of the regions, thus a PSF error does not propagate strongly to the measured temperatures. However, images of these clusters span the whole GIS field of view and the effective area miscalibrations may in principle affect the results, especially for the hotter A2319. Different pointings to these clusters help reduce the chances of both errors. For the two smaller clusters, the PSF-scattered contributions were about 2/3 of the outer flux and even greater for some of the A665 regions, making the results for these clusters dependent on the reliability of the PSF model.
It is not inconceivable that other instrumental effects exist that are not understood at the moment, so the best check of the results would be that by another telescope. PSPC results for A2256 and A665 are included below for this purpose. More on this comparison is presented in Markevitch & Vikhlinin (1996, hereafter MV).
RESULTS
=======
For A2256, we obtained average values over the inner $r=15'$ of $T_e=7.5\pm0.4$ keV and an iron abundance relative to Allen (1973) of $0.23\pm0.05$[^1], in agreement with (Hatsukade 1989). For A2319, our average $T_e=10.0\pm0.7$ keV is in agreement with MPC (David et al.1993), and our abundance is $0.30\pm0.08$. For A665, an average temperature of $T_e=8.0\pm1.0$ keV was found in the inner $r=10'$, in agreement with (Hughes & Tanaka 1992). Results for the whole of A2163 were reported in M94. For all clusters, the average spectra can be acceptably fit with a single-temperature model. However, spatially resolved fits reveal significant deviations from isothermality for all four clusters. Their two-dimensional temperature maps overlaid on the brightness contours are presented in Fig. 1 (Plate 1), and Fig. 2 shows projected temperatures averaged over the concentric annuli (the A2163 profile is presented in M96). The best-fit $\chi^2$ values for the maps in Fig. 1 are 301/408–17 d.o.f., 220/266–14 d.o.f., 55/100–10 d.o.f. and 39/60–7 d.o.f. for A2256, A2319, A2163 and A665, respectively, suggesting that our conservative compound errors are, if anything, slightly overestimated. When fit separately, different pointings to A2256 and A2319 give consistent results, with best-fit values within the 90% intervals for simultaneous fit in all but a couple of regions, which is even fewer than expected from statistical scatter.
A2256 and A2319
---------------
Cooler spots are observed near the centers of these two clusters, coincident with the substructures in their X-ray images. In A2256, the approximate subgroup region has best-fit $T_e=6.2\pm0.8$ keV, compared to an average of 8.7 keV for the immediate surrounding. It is higher than the value of $3.6^{+0.9}_{-0.5}$ keV (J. P. Henry, personal comm.) and than the value (Miyaji et al. 1993). Because the subgroup is projected on the hotter main cluster, one expects that the value would be higher. [^2] In A2319, the subgroup region has best-fit $T_e=8.4\pm1.2$ keV, compared to an average of 10 keV at that radius. Otherwise, the central $r=0.5\,h^{-1}$ Mpc regions of these two clusters are roughly isothermal at the present accuracy (the center of A2256 appears hotter than the second ring, but only with a marginal significance). We could not confirm existence of the two hot spots in A2256 reported in BH and ascribed to the effect of a merger. The temperatures in our approximately correspondent regions 6 and 9 are consistent with the average temperature at this radius within the much smaller errors (Fig. 1[*a*]{}). A reanalysis by MV of the pointings using a scheme less sensitive to calibration uncertainties, has shown that those hot spots are probably artifacts.
Beyond $r=0.5\,h^{-1}$ Mpc, the temperature in both clusters is found to decline with radius, with no significant deviations from symmetry. The A2256 temperature profile is in good agreement with that from obtained by MV (Fig. 2[*a*]{}), while having a much better accuracy. The measurement of Miyaji et al. for $r<10'$ is in agreement as well. BH derived a different temperature profile (although insignificantly so), which is addressed by MV. Using BH and Jones & Forman (1984) fits of the density profiles, the temperatures correspond to a polytropic index of $\gamma\simeq 1.55$ (1.4–1.7 90% interval) and $\gamma\simeq 1.25$ (1.08–1.40) for outer A2256 and A2319, respectively. For A2319, a tentative temperature estimate is also obtained for $r\sim1\,h^{-1}$ Mpc, suggesting that the profile steepens with radius. The temperature profiles in these clusters are to some extent similar to that of Coma (Hughes 1991).
A2163 and A665
--------------
For A2163 and A665, greater off-center distances are within the field of view. A2163 was reported earlier (M96) to have a largely isothermal inner $r=0.7\,h^{-1}$ Mpc profile at around 11 keV, and beyond this, a sharp drop of the temperature down to 4 keV at $1.5\,h^{-1}$ Mpc. An azimuthally-resolved measurement (Fig. 1[*c*]{}) shows the temperature dropping in all four directions off the cluster center, although the individual constraints are poor.
Averaged over the central $r=0.7\,h^{-1}$ Mpc ($=6'$) region, A665 has a temperature of $8.3\pm1.5$ keV. The temperature drops to $2.2^{+2.2}$ keV on average in the 6–12 annulus (because of our restricted energy band, only the upper bound and not the best-fits value is meaningful.) It decreases in all four off-center directions as well (Fig. 1[*d*]{}). Region 6 has a point source with an apparently non-thermal spectrum. It was fitted simultaneously with other spectra although its influence on the other regions was small. Since the outer cluster temperature appears to be within the energy band, we have undertaken to analyze the archival 40 ks PSPC observation of A665 to confirm the result. The PSPC and GIS temperatures obtained in two concentric annuli are shown in Fig. 2[*c*]{}. The two instruments are in good agreement, and a temperature drop is suggested by the PSPC data as well, although marginally significantly. The absorption column was kept fixed at its Galactic value in the fit since varying it was not required by the $F$-test, while freeing it makes the best-fit outer temperature still lower.
The observed temperature decline in the outer part of A2163 corresponds to a polytropic index of $\gamma\simeq 1.9$ ($\gamma>1.7$ at 90% confidence), while for A665, $\gamma\simeq 1.7$ ($\gamma>1.3$), adopting the density profiles from Elbaz et al. (1995) and the image, respectively.
DISCUSSION
==========
Although A2256 and A2319 clearly exhibit substructure in their X-ray images, no large-scale merger signatures, such as those predicted by hydrodynamic simulations (e.g., Schindler & 1993; EMN), are seen in the temperature maps of their central parts. This may indicate that the current mergers have not proceeded far enough to disturb the bulk of gas. For example, Roettiger et al. (1995) specifically simulated A2256 and found that the cluster image, galactic velocities and absence of the cluster-scale temperature variations are consistent with an epoch of about 0.2 Gyr before core passage of an infalling subunit. The observed relative symmetry of the temperatures in A2256 and probably A2319 suggests that their outer parts have been undisturbed by major mergers for the past few Gyr, making these clusters good candidates for an accurate mass measurement, which will be made in a future paper.
Apart from the subgroups, the temperature profiles of these two clusters are qualitatively similar to those predicted by the simulations of Navarro et al. and EMN for clusters in equilibrium. Interestingly, the observed temperature decline starts at smaller radii than EMN predict for the flat universe models without galactic winds, and is closer to their open universe model, in which clusters are expected to have steeper density and temperature profiles (Hoffman & Shaham 1985; Crone et al. 1994; Jing et al. 1995). However, the published simulations including gas are limited to the CDM initial perturbations spectrum, and our sample is limited to just a couple of rather specific clusters (e.g., lacking cooling flows unlike most of the clusters). A study of several other clusters is underway with , which will show how common this phenomenon is.
The temperature falls even steeper in A2163 and perhaps in A665. However, as was noted in M96, the low outer values may in fact not be representative of the mean gas temperature at those radii. For example, the measured electron temperature may be lower than that of ions heated by shock waves, because the timescale of electron-ion equipartition via collisions becomes non-negligible at such low plasma densities. Other possibilities involve cold gas clumps or point sources which cannot be localized by either or but significantly contribute to the flux. On the other hand, if the outer gas temperatures are indeed as low as measured, the observed steep profiles would have interesting implications for the physical conditions of the gas (although it may be premature to speculate using such poor data constraints.) Firstly, the outer cluster parts with a steeper than adiabatic temperature decline should be convectively unstable, and convection should develop on a timescale of the order of the free-fall time (a few Gyr) and erase the gradient. Thus, existence of a steep gradient implies that the cluster cannot have remained in its present state for a longer time than this. Another interesting problem is how such a temperature distribution may have emerged. Early simulations of infall of the cold gas into the cluster and its heating via the release of its potential energy (e.g., Bertschinger 1985) predict that such a process should form shallower temperature distributions. A steeper slope may therefore indicate that the gas in the central part has accumulated additional energy from another source. Hydrodynamic merger simulations predict (Pearce et al. 1994) that during a merger, energy is transferred from the dark matter to the gas, increasing its entropy in the center. Thus the observed profiles may independently indicate that these clusters have experienced major mergers. There is some evidence of the asymmetric temperature variations in the central part of A2163 (M94) and recent weak lensing analysis reveals two mass peaks near its center (Squires et al. 1996). A665 has a markedly asymmetric X-ray image which may remain from a merger. Schindler & (1993) predict that a merger shock wave would manifest itself at certain stages as a sharp projected temperature gradient in the cluster outer part, not accompanied by a similarly noticeable feature in the wide-band surface brightness, which is what we may be observing in the two more distant clusters.
The author thanks team for continuous support. He is grateful to C. L. Sarazin, A. C. Fabian, R. F. Mushotzky and the referees for useful discussions, and to A. Vikhlinin for help with the data analysis. He thanks ISAS, where most of this work was done, for its hospitality and support. Further support was provided by the Smithsonian Institution, and by NASA grants NAG5-2526 and NAG5-1891.
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(0,0)(20,20) (0,-1)
[to ]{} [Fig.]{} 2.—Projected temperatures of A2256, A2319 and A665 in concentric annuli. Errors are 90%. Panels [*(a)*]{} and [*(c)*]{} also show PSPC results from MV and this work, respectively. Parts of the cluster affected by cooler structures in A2256 (regions 4, 9 and 10 in Fig. 1[*a*]{}) and A2319 (region 2) are excluded and for A2319 shown separately. The outer measurement in A2319, shown by dashed cross, is less reliable because in different pointings this annulus is always close to the edge of the FOV and covers non-overlapping parts of the cluster.
(0,0)(17,23) (-2,24.5)
=10.5cm
(7.5,24.5)
=10.5cm
(-2,10)
=6.8cm
(7.5,10)
=6.8cm
(-1.5,4)
[Fig.]{} 1.— temperature maps (color) of A2256, A2319, A2163 and A665, overlaid on the PSPC brightness contours. Regions in which the temperature is measured are sectors of the concentric annuli. For A2256, the regions are centered on the main subcluster peak, and for other clusters, on the brightness peak. The regions are numbered in the maps and their temperatures are shown in the accompanying panels with 90% errors. Dotted horizontal lines in those panels correspond to the average temperature within the annulus. Regions 1 and 14 of A2319 and region 1 of A2163 and A665 are whole rings. The outer region of A2319 is not fully covered by the two pointings. The linear scale shown is for $h=1$.
[^1]: Errors are 90% one-parameter intervals throughout.
[^2]: For example, for a mixture of two components, an 8 keV underlying cluster contributing 2/3 of the emission measure and a 1.5 keV projected group contributing the rest, PSPC gives a 3.7 keV single-temperature fit and GIS gives 6.2 keV in the 2.5–11 keV band we use. This indicates that our results are insensitive to the cool substructure unless it is very prominent.
|
---
abstract: 'We investigate the reconstruction capabilities of Dark Matter mass and spin-independent cross-section from future ton-scale direct detection experiments using germanium, xenon or argon as targets. Adopting realistic values for the exposure, energy threshold and resolution of Dark Matter experiments which will come online within 5 to 10 years, the degree of complementarity between different targets is quantified. We investigate how the uncertainty in the astrophysical parameters controlling the local Dark Matter density and velocity distribution affects the reconstruction. For a 50 GeV WIMP, astrophysical uncertainties degrade the accuracy in the mass reconstruction by up to a factor of $\sim 4$ for xenon and germanium, compared to the case when astrophysical quantities are fixed. However, combination of argon, germanium and xenon data increases the constraining power by a factor of $\sim 2$ compared to germanium or xenon alone. We show that future direct detection experiments can achieve self-calibration of some astrophysical parameters, and they will be able to constrain the WIMP mass with only very weak external astrophysical constraints.'
author:
- Miguel Pato
- Laura Baudis
- Gianfranco Bertone
- Roberto Ruiz de Austri
- 'Louis E. Strigari'
- Roberto Trotta
title: Complementarity of Dark Matter Direct Detection Targets
---
=1
Introduction {#secintro}
============
Many experiments are currently searching for Dark Matter (DM) in the form of Weakly Interacting Massive Particles (WIMPs), by looking for rare scattering events off nuclei in the detectors, and many others are planned for the next decade [@book; @Lewin; @Bergstrom; @Munoz; @Bertone; @CerdenoGreen]. This direct DM detection strategy has brought over the last year several interesting observations and upper limits. The results of the DAMA/LIBRA [@dama] and, more recently, the CoGeNT [@cogent] collaborations have been tentatively interpreted as due to DM particles. It appears however that these results cannot be fully reconciled with other experimental findings, in particular with the null searches from XENON100 [@xe100; @gelmini1; @Bezrukov] or CDMS [@cdms10], and are also in tension with ZEPLIN-III [@zeplin]. In this context, the next generation of low-background, underground detectors is eagerly awaited and will hopefully confirm or rule out a DM interpretation.
If convincing evidence is obtained for DM particles with direct detection experiments, the obvious next step will be to attempt a reconstruction of the physical parameters of the DM particle, namely its mass and scattering cross-section (see e.g. Refs. [@Goudelis; @Green1; @ST]). This is a non-trivial task, hindered by the different uncertainties associated with the computation of WIMP-induced recoil spectra. In particular, Galactic model uncertainties – i.e. uncertainties pertaining to the density and velocity distribution of WIMPs in our neighbourhood – play a crucial role. In attempting reconstruction, the simplest assumption to make is a fixed local DM density $\rho_0=0.3 \textrm{ GeV/cm}^3$ and a “standard halo model”, i.e. an isotropic isothermal sphere density profile and a Maxwell-Boltzmann distribution of velocities with a given galactic escape velocity $v_{esc}$ and one-dimensional dispersion $\sigma^2\equiv v_0^2/2 = v_{lsr}^2/2$ ($v_0$ being the most probable velocity and $v_{lsr}$ the local circular velocity, see below). However, the Galactic model parameters are only estimated to varying degrees of accuracy, so that the true local population of DM likely deviates from the highly idealised standard halo model.
Several attempts have been made to improve on the standard approach [@ST; @Ling; @McCabe; @Green2]. In the case of a detected signal at one experiment, recent analyses have studied how complementary detectors can extract dark matter properties, independent of our knowledge of the Galactic model [@fox1]. Certain properties of dark matter may also be extracted under assumptions about the nature of the nuclear recoil events [@fox2]. Furthermore, eventual multiple signals at different targets have been shown to be useful in constraining both dark matter and astrophysical properties [@APeter] and in extracting spin-dependent and spin-independent couplings [@Vergados; @BertoneCerdeno]. Here, using a Bayesian approach, we study how uncertainties on Galactic model parameters affect the determination of the DM mass $m_{\chi}$ and spin-independent WIMP-proton scattering cross-section $\sigma_{SI}^p$. In particular we focus on realistic experimental capabilities for the future generation of ton-scale detectors – to be reached within the next 10 years – with noble liquids (argon, xenon) and cryogenic (germanium) technologies.
The main focus of this paper is the complementarity between different detection targets. It is well-known (see e.g. [@Lewin]) that different targets are sensitive to different directions in the $m_{\chi}-\sigma_{SI}^p$ plane, which is very useful to achieve improved reconstruction capabilities – or more stringent bounds in the case of null results. This problem has often been addressed without taking proper account of Galactic model uncertainties. Using xenon (Xe), argon (Ar) and germanium (Ge) as case-studies, we ascertain to what extent unknowns in Galactic model parameters limit target complementarity. A thorough understanding of complementarity will be crucial in the near future since it provides us with a sound handle to compare experiments and, if needed, decide upon the best target to bet on future detectors. Our results also have important consequences for the combination of collider observables and direct detection results (for a recent work see [@Fornasa]).
Besides degrading the extraction of physical properties like $m_{\chi}$ and $\sigma_{SI}^p$, uncertainties in the Galactic model will challenge our ability to distinguish between different particle physics frameworks in case of a positive signal. Other relevant unknowns are hadronic uncertainties, related essentially to the content of nucleons [@Ellis]. Here, we undertake a model-independent approach without specifying an underlying WIMP theory and using $m_{\chi}$ and $\sigma_{SI}^p$ as our phenomenological parameters – for this reason we shall not address hadronic uncertainties (hidden in $\sigma_{SI}^p$). A comprehensive work complementary to ours and done in the supersymmetric framework has been presented recently [@Akrami1; @Akrami2].
The paper is organised as follows. In the next section, we give some basic formulae for WIMP-nucleus recoil rates in direct detection experiments. In Section \[exp\] the upcoming experimental capabilities are detailed, while Section \[statsection\] describes our Bayesian approach. We outline the relevant Galactic model uncertainties and our modelling of the velocity distribution function in Section \[astrounc\] and present our results in Section \[results\] before concluding in Section \[secconc\].
Basics of direct dark matter detection {#DD}
======================================
Several thorough reviews on direct dark matter searches exist in the literature [@book; @Lewin; @Bergstrom; @Munoz; @Bertone; @CerdenoGreen]. In this section, we simply recall the relevant formulae, emphasizing the impact of target properties and unknown quantities.
The elastic recoil spectrum produced by WIMPs of mass $m_{\chi}$ and local density $\rho_0$ on target nuclei $N(A,Z)$ of mass $m_N$ is $$\label{recoil1}
\frac{dR}{dE_R}(E_R)=\frac{\rho_0}{m_{\chi}m_{N}} \, \int_{\cal V}
%{v>v_{min}(E_R)}
{ d^3\vec{v} \, \, v f\left(\vec{v}+\vec{v_e}\right) \frac{d\sigma_{\chi-N}}{dE_R}(v,E_R) } ,$$ where $\vec{v}$ is the WIMP velocity in the detector rest frame, $\vec{v}_e$ is the Earth velocity in the Galactic rest frame, $f(\vec{w})$ is the WIMP velocity distribution in the galactic rest frame and $\sigma_{\chi-N}$ is the WIMP-nucleus cross-section. The integral is performed over ${\cal V}: v>v_{min}(E_R)$, where $v_{min}$ is the minimum WIMP velocity that produces a nuclear recoil of energy $E_R$. Eq. simply states that the recoil rate is the flux of WIMPs $\rho_0 v / m_{\chi}$, averaged over the velocity distribution $f(\vec{w})$, times the probability of interaction with one target nucleus $\sigma_{\chi-N}$. Anticipating the scale of future detectors, we will think of measuring $dR/dE_R$ in units of counts/ton/yr/keV. For non-relativistic (elastic) collisions – as appropriate for halo WIMPs, presenting $v/c\sim 10^{-3}$ – the kinematics fixes the recoil energy $$E_R(m_{\chi},v,A,\theta')=\frac{\mu_N^2 v^2 (1-\cos\theta')}{m_N} \quad ,$$ and the minimum velocity $$v_{min}(m_{\chi},E_R,A)=\sqrt{\frac{m_N E_R}{2 \mu_N^2}}$$ in which $\theta'$ is the scattering angle in the centre of mass and $\mu_N=\frac{m_{\chi}m_N}{m_{\chi}+m_N}$ is the WIMP-nucleus reduced mass.
In principle, all WIMP-nucleus couplings enter in the cross-section $\sigma_{\chi-N}$. However, we shall focus solely on spin-independent (SI) scalar interactions so that $$\frac{d\sigma_{\chi-N}}{dE_R}=\frac{m_N}{2 \mu_N^2 v^2} \sigma_{SI}^N F^2(A,E_R) \quad ,$$ where $\sigma_{SI}^N = \frac{4 \mu_N^2}{\pi} \left[ Z f_p + (A-Z) f_n \right]^2$ is the WIMP-nucleus spin-independent cross-section at zero momentum transfer and $F(A,E_R)$ is the so-called form factor that accounts for the exchange of momentum. Assuming that the WIMP couplings to protons and neutrons are similar, $f_p\sim f_n$, and defining $\sigma_{SI}^p\equiv \frac{4\mu_p^2}{\pi}f_p^2$ ($\mu_p$ being the WIMP-proton reduced mass), one gets $$\label{sigma}
\frac{d\sigma_{\chi-N}}{dE_R} = \frac{m_N}{2\mu_p^2 v^2} \sigma_{SI}^p A^2 F^2(A,E_R) \quad .$$ For the form factor, we use the parameterisation in [@Lewin] appropriate for spin-independent couplings, namely $$F(A,E_R)=3\frac{\textrm{sin}(q r_n)-(q r_n)\textrm{cos}(q r_n)}{(q r_n)^3} \textrm{exp}(-(q s)^2/2) \quad ,$$ with $qr=6.92\times 10^{-3} A^{1/2} (E/\textrm{keV})^{1/2} r/\textrm{fm}$, $s\simeq 0.9$ fm, $r_n^2=c^2+\frac{7}{3}\pi^2 a^2-5 s^2$, $c/\textrm{fm}=1.23A^{1/3}-0.6$ and $a\simeq0.52$ fm.
As noticed above, in Eq. $\vec{v}_e$ is the Earth velocity with respect to the galactic rest frame and amounts to $\vec{v}_e=\vec{v}_{lsr}+\vec{v}_{pec}+\vec{v}_{orb}$, where $v_{lsr}\sim\mathcal{O}(250)$ km/s is the local circular velocity, $v_{pec}\sim\mathcal{O}(10)$ km/s is the peculiar velocity of the Sun (with respect to $\vec{v}_{lsr}$) and $v_{orb}\sim\mathcal{O}(30)$ km/s is the Earth velocity with respect to the Sun (i.e. the Earth orbit). Here, we are not interested in the annual modulation signal nor directional signatures but rather in the average recoil rate – therefore we shall neglect $\vec{v}_{pec}$ and $\vec{v}_{orb}$ and take $\vec{v}_e\simeq \vec{v}_{lsr} = \text{const}$.
Under these assumptions, Eq. may be recast in a very convenient way: $$\begin{aligned}
\nonumber
\frac{dR}{dE_R}(E_R)&=&\frac{\rho_0 \sigma_{SI}^p}{2 \mu_p^2 m_{\chi}} \times A^2 F^2(A,E_R) \times \\ \label{recoil2}
& & \mathcal{F}\left(v_{min}(m_{\chi},E_R,A),\vec{v}_e;v_0,v_{esc}\right) \, ,\end{aligned}$$ where we have used Eq. , defined $$\label{eq:F}
\mathcal{F}\equiv\int_{\cal V}{ d^3\vec{v} \frac{f\left(\vec{v}+\vec{v_e}\right)}{v}}$$ and made explicit the dependence of $\mathcal{F}$ on the velocity distribution parameters $v_0$ and $v_{esc}$. Below we discuss in more detail the connection between the parameters $v_0$ and $v_{lsr}$. The distribution of DM is encoded in the factor $\mathcal{F}$ (and $\rho_0$), whereas the detector-related quantities appear in $A^2 F^2(A,E_R)$ (and $v_{min}$). The apparent degeneracy along the direction $\rho_0 \sigma_{SI}^p/m_{\chi}=\text{const}$ may be broken by using different recoil energies and/or different targets since $\mathcal{F}$ is sensitive to a non-trivial combination of $m_{\chi}$, $E_{R}$ and $A$. Nevertheless, for very massive WIMPs $m_{\chi}\gg m_{N}\sim \mathcal{O}(100) \textrm{ GeV} \gg m_p$, the minimum velocity becomes independent of $m_{\chi}$, $v_{min}\simeq \sqrt{E_R/(2m_N)}$, and the degeneracy $\rho_0 \sigma_{SI}^p/m_{\chi}$ cannot be broken. Depending on the target being used, this usually happens for WIMP masses above a few hundred GeV.
Ultimately, the observable we will be interested in is the number of recoil events in a given energy bin $E_1 < E_R < E_2$: $$\label{eq:NR}
N_R(E_1,E_2) = \int_{E_1}^{E_2}{ dE_R \, \, \epsilon_{eff} \, \frac{d\tilde{R}}{dE_R} } \quad ,$$ $\epsilon_{eff}$ being the effective exposure (usually expressed in ton$\times$yr) and $d\tilde{R}/dE_R$ the recoil rate smeared according to the energy resolution of the detector $\sigma(E)$, $$\frac{d\tilde{R}}{dE_R} = \int{ dE' \, \frac{dR}{dE_R}(E') \, \frac{1}{\sqrt{2\pi}\sigma(E')} \textrm{exp}\left(-\frac{(E-E')^2}{2\sigma^2(E')}\right) } \quad .$$
Three fiducial WIMP models will be used to assess the capabilities of future direct detection experiments: $m_{\chi}=$25, 50 and 250 GeV, all with $\sigma_{SI}^p=10^{-9}$ pb. These models are representative of well-motivated candidates such as neutralinos in supersymmetric theories [@ruiz].
Upcoming experimental capabilities {#exp}
==================================
target $\epsilon$ \[ton$\times$yr\] $\eta_{cut}$ $A_{NR}$ $\epsilon_{eff}$ \[ton$\times$yr\] $E_{thr}$ \[keV\] $\sigma(E)$ \[keV\] background events/$\epsilon_{eff}$
-------- ------------------------------ -------------- ---------- ------------------------------------ ------------------- --------------------- ------------------------------------
Xe 5.0 0.8 0.5 2.00 10 Eq. $<1$
Ge 3.0 0.8 0.9 2.16 10 Eq. $<1$
Ar 10.0 0.8 0.8 6.40 30 Eq. $<1$
Currently, the most stringent constraints on the SI WIMP-nucleon coupling are those obtained by the CDMS [@cdms] and XENON [@xe100] collaborations. While XENON100 should probe the cross-section region down to $5\times10^{-45}$ cm$^2$ with data already in hand, the XENON1T [@xenon1T] detector, whose construction is scheduled to start by mid 2011, is expected to reach another order of magnitude in sensitivity improvement. To test the $\sigma_{SI}^p$ region down to $10^{-47}$ cm$^2\equiv 10^{-11}$ pb and below, a new generation of detectors with larger WIMP target masses and ultra-low backgrounds is needed. Since we are interested in the prospects for detection in the next 5 to 10 years, we discuss new projects that can realistically be built on this time scale, adopting the most promising detection techniques, namely noble liquid time projection chambers (TPCs) and cryogenic detectors operated at mK temperatures.
In Europe, two large consortia, DARWIN [@darwin] and EURECA [@eureca], gathering the expertise of several groups working on existing DM experiments are funded for R&D and design studies to push noble liquid and cryogenic experiments to the multi-ton and ton scale, respectively. DARWIN is devoted to noble liquids, having as main goal the construction of a multi-ton liquid Xe (LXe) and/or liquid Ar (LAr) instrument [@darwin_IDM2010], with data taking to start around 2016. The XENON, ArDM and WARP collaborations participate actively in the DARWIN project. EURECA is a design study dedicated to cryogenic dark matter detectors operated at mK temperatures. The proposed roadmap is to improve upon CRESST [@cresst] and EDELWEISS [@edel] technologies and build a ton-scale detector by 2018, with a SI sensitivity of about $10^{-46}$ cm$^2\equiv 10^{-10}$ pb. The complementarity between DARWIN and EURECA is of utmost importance for dark matter direct searches since a solid, uncontroversial discovery requires signals in distinct targets and preferentially distinct technologies. In an international context, two engineering studies (MAX [@max] and LZS [@lzs]) are funded in the US for ton to multi-ton scale LXe and LAr TPCs and the SuperCDMS/GEODM collaboration [@geodm] plans to operate an 1.5ton Ge cryogenic experiment at DUSEL [@dusel]. In Japan, the XMASS experiment [@xmass], using a total of 800kg of liquid xenon in a single-phase detector, is under commissioning at the Kamioka underground laboratory [@kamioka], while a large single-phase liquid argon detector, DEAP-3600 [@deap3600], using 3.6tons of LAr is under construction at SNOLab [@snolab].
Given these developments, we will focus on the three most promising targets: Xe and Ar as examples of noble liquid detectors, and Ge as a case-study for the cryogenic technique. In the case of a Ge target, we assume an 1.5 ton detector (1ton as fiducial target mass), 3 years of operation, an energy threshold for nuclear recoils of $E_{thr,Ge}=10$ keV and an energy resolution given by $$\label{resGe}
\sigma_{Ge}(E)=\sqrt{(0.3)^2+(0.06)^2 E/\textrm{keV}} \textrm{ keV} \quad .$$
For a liquid Xe detector, we assume a total mass of 8tons (5tons in the fiducial region), 1 year of operation, an energy threshold for nuclear recoils of $E_{thr,Xe}=10$ keV and an energy resolution of $$\label{resXe}
\sigma_{Xe}(E) = 0.6 \textrm{ keV} \sqrt{E/\textrm{keV}} \quad .$$
Finally, for a liquid Ar detector, we assume a total mass of 20tons (10tons in the fiducial region), 1 year of operation, an energy threshold for nuclear recoils of $E_{thr,Ar}=30$ keV and an energy resolution of [@privRegenfuss] $$\label{resAr}
\sigma_{Ar}(E) = 0.7 \textrm{ keV} \sqrt{E/\textrm{keV}} \quad .$$
To calculate realistic exposures, we make the following assumptions: nuclear recoils acceptances $A_{NR}$ of 90%, 80% and 50% for Ge, Ar and Xe, respectively, and an additional, overall cut efficiency $\eta_{cut}$ of 80% in all cases, which for simplicity we consider to be constant in energy. We hypothesise less than one background event per given effective exposure $\epsilon_{eff}$, which amounts to 2.16 ton$\times$yr in Ge, 6.4 ton$\times$yr in Ar and 2 ton$\times$yr in Xe, after allowing for all cuts. Such an ultra-low background will be achieved by a combination of background rejection using the ratio of charge-to-light in Ar and Xe, and charge-to-phonon in Ge, the timing characteristics of raw signals, the self-shielding properties and extreme radio-purity of detector materials, as well as minimisation of exposure to cosmic rays above ground.
The described characteristics are summarised in Table \[tabExp\]. We note that in the following we shall consider recoil energies below 100 keV only; to increase this maximal value may add some information but the effect is likely small given the exponential nature of WIMP-induced recoiling spectra.
Statistical methodology {#statsection}
=======================
We take a Bayesian approach to parameter inference. We begin by briefly summarizing the basics, and we refer the reader to [@Trotta:2008qt] for further details. Bayesian inference rests on Bayes theorem, which reads $$\label{eq:bayes}
p({\Theta}| {d}) = \frac{p({d}| {\Theta}) p({\Theta})}{p({d})},$$ where $p({\Theta}| {d})$ is the posterior probability density function (pdf) for the parameters of interest, ${\Theta}$, given data ${d}$, $p({d}| {\Theta}) = {{\mathcal L}}({\Theta})$ is the likelihood function (when viewed as a function of ${\Theta}$ for fixed data ${d}$) and $p({\Theta})$ is the prior. Bayes theorem thus updates our prior knowledge about the parameters to the posterior by accounting for the information contained in the likelihood. The normalization constant on the r.h.s. of Eq. is the Bayesian evidence and it is given by the average likelihood under the prior: $$p({d}) = \int d{\Theta}p({d}| {\Theta}) p({\Theta}).$$ The evidence is the central quantity for Bayesian model comparison [@Trotta:2005ar], but it is just a normalisation constant in the context of the present paper.
Parameter Prior range Prior constraint
------------------------------------------------------------- --------------- -------------------------
$\textrm{log}_{10} \left( m_{\chi}/\textrm{GeV} \right)$ $(0.1,3.0)$ Uniform prior
$\textrm{log}_{10} \left( \sigma_{SI}^p/\textrm{pb}\right)$ $(-10,-6)$ Uniform prior
$\rho_0/(\textrm{GeV/cm}^3)$ $(0.001,0.9)$ Gaussian: $0.4 \pm 0.1$
$v_0/(\textrm{km/s})$ $(80,380)$ Gaussian: $230 \pm 30$
$v_{esc}/(\textrm{km/s})$ $(379,709)$ Gaussian: $544 \pm 33$
$k$ $(0.5,3.5)$ Uniform prior
The parameter set ${\Theta}$ contains the DM quantities we are interested in (mass and scattering cross-section), and also the Galactic model parameters, which we regard as nuisance parameters, entering the calculation of direct detection signals, namely $\rho_0$, $v_0$, $v_{esc}$, $k$, see Eq. and Section \[astrounc\]. We further need to define priors $p({\Theta})$ for all of our parameters. For the DM parameters, we adopt flat priors on the log of the mass and cross-section, reflecting ignorance on their scale. For the Galactic model parameters, we choose priors that reflect our state of knowledge about their plausible values, as specified in the next section. Those priors are informed by available observational constraints as well as plausible estimations of underlying systematical errors, for example for $\rho_0$. Finally, the likelihood function for each of the direct detection experiments is given by a product of independent Poisson likelihoods over the energy bins: $${{\mathcal L}}({\Theta}) = \prod_{b} \frac{N_R^{\hat{N}_b}}{\hat{N}_b!}\exp\left(-N_R\right),$$ where $\hat{N}_b$ is the number of counts in each bin (generated from the true model with no shot noise, as explained below) and $N_R = N_R(E_b^\text{min}, E_b^\text{max})$ is the number of counts in the $b$-th bin (in the energy range $E_b^\text{min} \leq E \leq E_b^\text{max}$) when the parameters take on the value ${\Theta}$, and it is given by Eq. . We use 10 bins for each experiment, uniformly spaced on a linear scale between the threshold energy and 100 keV. We have checked that our results are robust if we double the number of assumed energy bins. Using the experimental capabilities outlined in Section \[exp\], we compute the counts $N_R$ that the benchmark WIMPs would generate, and include no background events since the expected background level in the fiducial mass region is negligible (cf. Table \[tabExp\]). The mock counts are generated from the true model, i.e. without Poisson scatter. This is because we want to test the reconstruction capabilities without having to worry about realization noise (such a data set has been called “Asimov data” in the particle physics context [@Cowan:2010js]).
To sample the posterior distribution we employ the MultiNest code [@Feroz:2007kg; @Feroz:2008xx; @Trotta:2008bp], an extremely efficient sampler of the posterior distribution even for likelihood functions defined over a parameter space of large dimensionality with a very complex structure. In our case, the likelihood function is unimodal and well-behaved, so Monte Carlo Markov Chain (MCMC) techniques would be sufficient to explore it. However, MultiNest also computes the Bayesian evidence (which MCMC methods do not return), as it is an implementation of the nested sampling algorithm [@Skilling:2006]. In this work, we run MultiNest with 2000 live points, an efficiency parameter of 1.0 and a tolerance of 0.8 (see [@Feroz:2007kg; @Feroz:2008xx] for details).
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Velocity distribution and Galactic model parameters {#astrounc}
===================================================
We now move onto discussing our modeling of the velocity distribution function and the Galactic model parameters that are input for Eq. . We model only the smooth component of the velocity distribution – recent results from numerical simulations indicate that the velocity distribution component arising from localised streams and substructures is likely sub-dominant in the calculation of direct dark matter detection signals [@Vogelsberger:2008qb; @Kuhlen:2009vh].
We model the velocity distribution function as spherical and isotropic, and parameterise it as [@Lisanti:2010qx], $$\label{fv}
f(w)=\left\{\begin{array}{ll}
\frac{1}{N_f}\left[\textrm{exp}\left(\frac{v_{esc}^2-w^2}{k v_0^2}\right)-1\right]^k & \text{if } w \leq v_{esc}\\
0 & \text{if } w > v_{esc}
\end{array} \right. \, .$$ This velocity distribution function was found to be flexible enough to describe the range of dark matter halo profiles found in cosmological simulations [@Lisanti:2010qx]. Boosting into the rest frame of the Earth implies the transformation $w^2=v^2+v_e^2+2v v_e \textrm{cos}\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{v}_e\sim \vec{v}_{lsr}$. The shape parameter that determines the power law tail of the velocity distribution is $k$, the escape velocity is $v_{esc}$, while $v_0$ is a fit parameter that we discuss in detail below, and $N_f$ is the appropriate normalisation constant. The special case $k=1$ represents the standard halo model with a truncated Maxwellian distribution, and the corresponding expressions for $N_f$ and $\mathcal{F}$ have been derived analytically in the literature – see for instance [@McCabe]. Note as well that, for any value of $k$, this distribution matches a Maxwellian distribution for sufficiently small velocities $w$ and if $v_{esc}>v_0$.
The high-velocity tail of the distributions found in numerical simulations of pure dark matter galactic halos are well modelled by $1.5 < k < 3.5$ [@Lisanti:2010qx]. In our analysis we will expand this range to also include models that behave similar to pure Maxwellian distributions near the tail of the distribution, so that in our analysis we vary $k$ in the range $$k = 0.5-3.5 \quad (\textrm{flat}) \quad.$$ We adopt an uniform (i.e., flat) prior within the above range for $k$.
The range we take for the $v_{esc}$ is motivated by the results of Ref. [@vescref], where a sample of high-velocity stars is used to derive a median likelihood local escape velocity of $\bar{v}_{esc}=544$ km/s and a 90% confidence level interval $498 \textrm{ km/s} < v_{esc} < 608$ km/s. Assuming Gaussian errors this translates into a 1$\sigma$ uncertainty of 33 km/s. It is important to note that this constraint on the escape velocity is derived assuming a range in the power law tail for the distribution of stars in the local neighbourhood, which is then related to the power law tail in the dark matter distribution [@vescref]. Motivated by obtaining conservative limits on the reconstructed mass and cross-section of the dark matter, in our modelling we will not include such correlations between the escape velocity and the power law index $k$, so that in the end we take a Gaussian prior on $v_{esc}$ with mean and standard deviation given by $$v_{esc}= 544 \pm 33 \textrm{ km/s} \quad (1\sigma) \quad .$$
Having specified ranges for $v_{esc}$ and $k$, it remains to consider a range for $v_0$ in Eq. . As defined in that equation, the quantity $v_0$ does not directly correspond to the local circular velocity, $v_{lsr}$, but rather is primarily set by $v_{lsr}$ and the dark matter profile. Following a procedure similar to that discussed in Ref. [@Lisanti:2010qx], we find the range of values $v_0$ compatible with a given a dark matter halo profile, $\rho_0$ and a range for $v_{lsr}$. For the above range in $v_{lsr}$ and the values $\rho_{0}$ in Eq. below, we find that the parameter $v_0$ can take values in the range $200-300$ km/s for pure Navarro-Frenk-White (NFW) dark matter halos with outer density slopes $\rho \propto r^{-3}$. Larger values of $v_0$ are allowed for steeper outer density slopes, though the range is found to not expand significantly if we restrict ourselves to models with outer slopes similar to the NFW case. With these caveats in mind regarding the mapping between $v_0$ and $v_{lsr}$ for steeper outer slopes, for simplicity and transparency in our analysis, we will consider a similar range for $v_0$ as for the local circular velocity, so we take $v_0 = v_{lsr}$ (that holds in the case of the standard halo model).
For the local circular velocity and its uncertainty, a variety of measurements presents a broad range of central values and uncertainties [@v0refs]. To again remain conservative we use an interval bracketing recent determinations: $$v_0 = v_{lsr} = 230 \pm 30 \textrm{ km/s} \quad (1\sigma) \quad ,$$ where we take a Gaussian prior with the above mean and standard deviation. To account for the variation of the local density of dark matter in our modeling, we will take a mean value and error given by [@CatenaUllio; @localDM] $$\rho_0 = 0.4 \pm 0.1 \textrm{ GeV/cm}^3 \quad (1\sigma) \quad ,
\label{eq:rho0}$$ There are several other recent results that determine $\rho_0$, both consistent [@Salucci] and somewhat discrepant [@Weber:2009pt] with our adopted value. Even in light of these uncertainties, we take Eq. to represent a conservative range for the purposes of our study.
For completeness Table \[tabPars\] summarises the information on the parameters used in our analysis.
Results
=======
Complementarity of targets
--------------------------
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We start by assuming the three dark matter benchmark models described in Section \[DD\] ($m_{\chi}=25,50,250$ GeV with $\sigma_{SI}^p=10^{-9}$ pb) and fix the Galactic model parameters to their fiducial values, $\rho_0=0.4 \textrm{ GeV}/\textrm{cm}^3$, $v_0=230$ km/s, $v_{esc}=544$ km/s, $k=1$. With the experimental capabilities outlined in Section \[exp\], we generate mock data that in turn are used to reconstruct the posterior for the DM parameters $m_{\chi}$ and $\sigma_{SI}^p$. The left frame of Fig. \[figfixed\] presents the results for the three benchmarks and for Xe, Ge and Ar separately. Contours in the figure delimit regions of joint 68% and 95% posterior probability. Several comments are in order here. First, it is evident that the Ar configuration is less constraining than Xe or Ge ones, which can be traced back to its smaller $A$ and larger $E_{thr}$. Moreover, it is also apparent that, while Ge is the most effective target for the benchmarks with $m_{\chi}=25,250$ GeV, Xe appears the best for a WIMP with $m_{\chi}=50$ GeV (see below for a detailed discussion). Let us stress as well that the 250 GeV WIMP proves very difficult to constrain in terms of mass and cross-section due to the high-mass degeneracy explained in Section \[DD\]. Taking into account the differences in adopted values and procedures, our results are in qualitative agreement with Ref. [@Akrami1], where a study on the supersymmetrical framework was performed. However, it is worth noticing that the contours in Ref. [@Akrami1] do not extend to high masses as ours for the 250 GeV benchmark – this is likely because the volume at high masses in a supersymmetrical parameter space is small.
---------- ------------------- -------------------
$m_{\chi}=25$ GeV $m_{\chi}=50$ GeV
Xe 6.5% (14.3%) 8.1% (20.4%)
Ge 5.5% (16.0%) 7.0% (29.6%)
Ar 12.3% (23.4%) 14.7% (86.5%)
Xe+Ge 3.9% (10.9%) 5.2% (15.2%)
Xe+Ge+Ar 3.6% (9.0%) 4.5% (10.7%)
---------- ------------------- -------------------
In the right frame of Fig. \[figfixed\] we show the reconstruction capabilities attained if one combines Xe and Ge data, or Xe, Ge and Ar together, [*again for when the Galactic model parameters are kept fixed*]{}. In this case, for $m_{\chi}=25,50$ GeV, the configuration Xe+Ar+Ge allows the extraction of the correct mass to better than $\mathcal{O}(10)$ GeV accuracy. For reference, the (marginalised) mass accuracy for different mock data sets is listed in Table \[tabAccuracy\]. For $m_{\chi}=250$ GeV, it is only possible to obtain a lower limit on $m_{\chi}$.
Fig. \[figastro\] shows the results of a more realistic analysis, that keeps into account the large uncertainties associated with Galactic model parameters, as discussed in Section \[astrounc\]. The left frame of Fig. \[figastro\] shows the effect of varying only $\rho_0$ (dashed lines, blue surfaces), only $v_0$ (solid lines, red surfaces) and all Galactic model parameters (dotted lines, yellow surfaces) for Xe and $m_{\chi}=50$ GeV. The Galactic model uncertainties are dominated by $\rho_0$ and $v_0$, and, once marginalised over, they blow up the constraints obtained with fixed Galactic model parameters. This amounts to a very significant degradation of mass (cf. Table \[tabAccuracy\]) and scattering cross-section reconstruction. Inevitably, the complementarity between different targets is affected – see the right frame of Fig. \[figastro\]. Still, for the 50 GeV benchmark, combining Xe, Ge and Ar data improves the mass reconstruction accuracy with respect to the Xe only case, essentially by constraining the high-mass tail.
![Figure of merit quantifying the relative information gain on Dark Matter parameters for different targets and combinations thereof. The values of the figure of merit are normalised to the Ar case at $m_{\chi}=250$ GeV with fixed astrophysical parameters. Empty (filled) bars are for fixed astrophysical parameters (including astrophysical uncertainties).[]{data-label="figinfo"}](infoarea.pdf){width="8.9cm" height="7.6cm"}
In order to be more quantitative in assessing the usefulness of different targets and their complementarity, we use as [*figure of merit*]{} the inverse area enclosed by the 95% marginalised contour in the $\log_{10}(m_{\chi})-\log_{10}(\sigma_{SI}^p)$ plane inside the prior range. Notice that for the 250 GeV benchmark the degeneracy between mass and cross-section is not broken – this does not lead to a vanishing figure of merit (i.e. infinite area under the contour) because we are restricting ourselves to the prior range. Fig. \[figinfo\] displays this figure of merit for several cases, where we have normalised to the Ar target at $m_{\chi}=250$ GeV with fixed Galactic model. Analyses with fixed Galactic model parameters are represented by empty bars, while the cases where all Galactic model parameters are marginalised over with priors as in Table \[tabPars\] are represented by filled bars. Firstly, one can see that all three targets perform better for WIMP masses around 50 GeV than 25 or 250 GeV if the Galactic model is fixed. When astrophysical uncertainties are marginalised over, the constraining power of the experiments becomes very similar for benchmark WIMP masses of 25 and 50 GeV. Secondly, Fig. \[figinfo\] also confirms what was already apparent from Fig. \[figfixed\]: Ge is the best target for $m_{\chi}=25,250$ GeV (although by a narrow margin), whereas Xe appears the most effective for a 50 GeV WIMP (again, by a narrow margin). Furthermore, the inclusion of uncertainties drastically reduces the amount of information one can extract from the data: the filled bars are systematically below the empty ones. Now, astrophysical uncertainties affect the complementarity between different targets in a non-trivial way. To understand this point, let us focus on the two rightmost bars for each benchmark in Fig. \[figinfo\], corresponding to the data sets Xe+Ge and Xe+Ge+Ar. For instance, in the case of a 250 GeV WIMP, astrophysical uncertainties seem to reduce target complementarity: adding Ar to Xe+Ge leads to a significant increase in the figure of merit for analyses with fixed astrophysics (empty bars) but has a negligible effect for analyses with varying astrophysical parameters (filled bars). For low mass benchmarks, the effect of combining two (Xe+Ge) or three targets (Xe+Ge+Ar) is to increase the figure of merit by about a factor of 2 compared to Xe alone or Ge alone, almost independently of whether the astrophysical parameters are fixed or marginalised over. However, the overall information gain on the Dark Matter parameters (for light WIMPs) is reduced by a factor $\sim 10$ if astrophysical uncertainties are taken into account, compared to the case where the Galactic model is fixed.
----------------- ------------ ----------------- ---------- -------- ----------- -------- ------------ ----------------- ---------- -------- ----------- -------- ------------ ----------------- ---------- -------- ----------- --------
$m_{\chi}$ $\sigma_{SI}^p$ $\rho_0$ $v_0$ $v_{esc}$ $k$ $m_{\chi}$ $\sigma_{SI}^p$ $\rho_0$ $v_0$ $v_{esc}$ $k$ $m_{\chi}$ $\sigma_{SI}^p$ $\rho_0$ $v_0$ $v_{esc}$ $k$
$m_{\chi}$ $-$ 0.039 -0.006 -0.850 -0.238 -0.002 $-$ 0.098 -0.006 -0.870 -0.079 -0.004 $-$ 0.874 -0.011 -0.615 -0.027 0.022
$\sigma_{SI}^p$ $-$ $-$ -0.887 -0.237 0.116 0.010 $-$ $-$ -0.957 -0.175 0.026 -0.031 $-$ $-$ -0.452 -0.525 -0.024 0.015
$\rho_0$ $-$ $-$ $-$ 0.013 -0.005 0.005 $-$ $-$ $-$ 0.014 -0.010 0.030 $-$ $-$ $-$ 0.002 0.015 0.010
$v_0$ $-$ $-$ $-$ $-$ -0.087 -0.004 $-$ $-$ $-$ $-$ -0.151 0.011 $-$ $-$ $-$ $-$ -0.049 -0.008
$v_{esc}$ $-$ $-$ $-$ $-$ $-$ 0.000 $-$ $-$ $-$ $-$ $-$ -0.009 $-$ $-$ $-$ $-$ $-$ 0.001
----------------- ------------ ----------------- ---------- -------- ----------- -------- ------------ ----------------- ---------- -------- ----------- -------- ------------ ----------------- ---------- -------- ----------- --------
Reduction in uncertainties and self-calibration
-----------------------------------------------
The uncertainties used thus far and outlined in Section \[astrounc\] are a reasonable representation of the current knowledge. For illustration it is also interesting to consider the effect of tighter constraints on Galactic model parameters in the reconstruction of WIMP properties. We start by computing the correlation coefficient between the parameters ($m_{\chi}$, $\sigma_{SI}^p$, $\rho_0$, $v_0$, $v_{esc}$, $k$) when they are constrained by the combined data set Xe+Ge+Ar – see Table \[tabCorr\]. Clearly, for all benchmark models, $\sigma_{SI}^p$ and $\rho_0$ as well as $m_{\chi}$ and $v_0$ are strongly anti-correlated. The anti-correlation between $\sigma_{SI}^p$ and $\rho_0$ is obvious since $dR/dE_R\propto \sigma_{SI}^p\rho_0$. As for the degeneracy between $m_{\chi}$ and $v_0$, it is easy to verify that, for $v_{min}\ll v_e \sim v_0 \ll v_{esc}$, $\mathcal{F}$ defined in Eq. goes approximately as $1/v_0$ and thus $dR/dE_R\propto 1/(m_{\chi}v_0)$. Table \[tabCorr\] also shows a small (anti-)correlation between $\sigma_{SI}^p$ and $v_0$; all other correlations are negligible. Therefore, $\rho_0$ and $v_0$ are the dominant sources of uncertainty and their more accurate determination will lead to a significant improvement on the reconstruction of $m_{\chi}$ and $\sigma_{SI}^p$. To illustrate this point we follow [@CatenaUllio] and apply a 7% (4.2%) uncertainty on $\rho_0$ ($v_0$), while maintaining the same central values as before, thus reducing the realistic error bars used above by a factor $\sim 3.0-3.5$ for both parameters. The results are shown in Fig. \[figredastro\] where we consider the combination Xe+Ge+Ar. A future more constrained astrophysical setup may indeed lead to a better reconstruction of the WIMP mass and scattering cross-section.
![The effect of reducing the uncertainty on the astrophysical parameters $\rho_0$ and $v_0$. The red surfaces refer to the scan using the fiducial astrophysical setup; the yellow surfaces (and dotted lines) indicate the effect of marginalising over the uncertainties in Table \[tabPars\]; the blue surfaces (and solid lines) correspond to the reduced uncertainties $\rho_0 = 0.4 \pm 0.028 \textrm{ GeV/cm}^3$, $v_0 = 230 \pm 9.76 \textrm{ km/s}$, $v_{esc}= 544 \pm 33 \textrm{ km/s}$, $k = 0.5-3.5$.[]{data-label="figredastro"}](astrounc_red.pdf){width="8.9cm" height="7.6cm"}
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To this point we have studied the impact of Galactic model uncertainties on the extraction of DM properties from direct detection data. However, once a positive signal is well-established, it may be used to determine some of the Galactic parameters directly from direct detection data (see e.g. [@APeter]), without relying on external priors. This would amount to achieving a self-calibration of the astrophysical uncertainties affecting direct detection rates. In order to explore such possibility we re-ran our analysis but dropping the Gaussian priors on $\rho_0$, $v_0$ and $v_{esc}$ described in Section \[astrounc\]. Instead, we used uniform, non-informative priors on $\rho_0$, $v_0$, $v_{esc}$ and $k$ in the ranges indicated in the middle column of Table \[tabPars\]. We focus on the 50 GeV benchmark and use the data sets Xe, Xe+Ge and Xe+Ge+Ar. With this large freedom on the astrophysical side, it turns out that direct detection data alone leave $\rho_0$, $v_{esc}$ and $k$ unconstrained within their ranges while $\sigma_{SI}^p$ is pinpointed within approximately one order of magnitude. Only the DM mass $m_{\chi}$ and the circular velocity $v_0$ can be constrained by direct detection, as shown in Fig. \[figflatastro\]. This figure stresses two interesting results. First, if $m_{\chi}=50$ GeV (and $\sigma_{SI}^p=10^{-9}$ pb), the next generation of experiments will be able to determine the WIMP mass within a few tens of GeV (percent 1$\sigma$ accuracy of 11.8%) even with very loose assumptions on the local DM distribution. Second, the right frame in Fig. \[figflatastro\] shows that the combination of Xe, Ge and Ar targets is very powerful in constraining $v_0$ on its own without external priors. In particular, the data set Xe+Ge+Ar (solid blue line) is sufficient to infer at 1$\sigma$ $v_0=238\pm 22$ km/s (compared to the top-hat prior in the range 80$-$380 km/s). This represents already a smaller uncertainty than the present-day constraint that we have taken, $v_0=230\pm 30$ km/s – in case of a positive signal, a combination of direct detection experiments will probe in an effective way the local circular velocity. Repeating the same exercise for the 25 GeV benchmark we find good mass reconstruction but a weaker constraint: $v_0=253\pm 39$ km/s. Again, we stress that the quoted $v_0$ uncertainties in this paragraph do not take into account possible systematic deviations from the parameterisation in Eq. .
Conclusions {#secconc}
===========
We have discussed the reconstruction of the key phenomenological parameters of WIMPs, namely mass and scattering cross-section off nuclei, in case of positive detection with one or more direct DM experiments planned for the next decade. We have in particular studied the complementarity of ton scale experiments with Xe, Ar and Ge targets, adopting experimental configurations that may realistically become available over this time scale.
To quantify the degree of complementarity of different targets we have introduced a figure of merit measuring the inverse of the area enclosed by the 95% marginalised contours in the plane $\log_{10}(m_\chi)-\log_{10}(\sigma_{SI}^p)$. There is a high degree of complementarity of different targets: for our benchmark with $m_\chi=50$ GeV and our fiducial set of Galactic model parameters, the relative error on the reconstructed mass goes from 8.1% for an analysis based on a xenon experiment only, to 5.2% for a combined analysis with germanium, to 4.5% adding also argon. Allowing the parameters to vary within the observational uncertainties significantly degrades the reconstruction of the mass, increasing the relative error by up to a factor of $\sim$4 for xenon and germanium, especially due to the uncertainty on $\rho_0$ and $v_0$. However, we found that combining data from Ar, Ge and Xe should allow to reconstruct a 50 GeV WIMP mass to 11.8% accuracy even under weaker astrophysical constraints than currently available.
Although the mass reconstruction accuracy may appear modest, any improvement of this reconstruction is important, in particular in view of the possible measurement of the same quantity at the Large Hadron Collider at CERN. The existence of a particle with a mass compatible, within the respective uncertainties, with that deduced from direct detection experiments would provide a convincing proof that the particles produced in accelerators are stable over cosmological time scales. Although this is not sufficient to claim discovery of DM [@Fornasa], it would certainly be reassuring.
Despite the strong dependence of direct detection experiments on the Galactic model degrades the reconstruction of DM properties, it does open up the possibility to potentially constrain the local distribution of DM, in case of detection with multiple targets. For example in the case of a low mass 50 GeV WIMP, we have shown that the local circular velocity can be determined from direct detection data alone more accurately than it is presently measured using the local distribution of stars and gas clouds. Additionally, directly detecting DM provides the most realistic way of measuring the local DM velocity distribution. This will in principle provide invaluable information on the structure and formation of the Milky Way halo.
[*Acknowledgements:*]{} G.B., R.T. and M.P. would like to thank the organisers of the workshop “Dark Matter all around” for a stimulating meeting. We wish to thank the authors of the paper [@Akrami1] for providing their preliminary results, as well as Henrique Araujo and Alastair Currie for useful discussions. We also acknowledge support from the SNF grant 20AS21-29329 and the University of Zurich. M.P. is supported by Fundação para a Ciência e Tecnologia (Ministério da Ciência, Tecnologia e Ensino Superior).
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---
abstract: 'Information processing with light is ubiquitous, from communication, metrology and imaging to computing. When we consider light as a quantum mechanical object, new ways of information processing become possible. In this review I give an overview how quantum information processing can be implemented with single photons, and what hurdles still need to be overcome to implement the various applications in practice. I will place special emphasis on the quantum mechanical properties of light that make it different from classical light, and how these properties relate to quantum information processing tasks.'
author:
- |
Pieter Kok\
Department of Physics & Astronomy, The University of Sheffield
---
Information processing with light
=================================
From gestures and smoke signals to books, paintings, telecom and high-capacity optical fibres connecting continents, light is one of the main information carriers that has been driving human civilisation since antiquity. Aside from the cultural aspect of communicating ideas with pictures, optical information processing is an important economic engine. In 2020, the market volume of the photonics industry will be 650 billion euro and is expected to continue to outperform GDP [@photonics13]. Light is the ideal medium for fast, reliable and high-bandwidth communication. The amount of data that can be transmitted through optical fibres continues to grow, and we are approaching the limit for the capacity of single-mode fibres. To increase the capacity, multi-mode fibres can be employed to achieve a data transmission rate of 255 Terabits per second over a distance of one kilometre [@vanuden14]. For comparison, this is similar to streaming one hundred thousand full length HD movies *each second*.
Similarly, imaging is fundamentally an optical task, and it has been known for a long time that the wave nature of light places a limit on how well we can make out small details using microscopes and telescopes. The resolution limit for imaging is called the Abbe limit, and depends on the opening angle of the aperture of the imaging device, called the *numerical aperture*. The bigger the numerical aperture, the higher the resolution. This is why telescopes are made in larger and larger sizes. In microscopy, where one can have more control over the object that is to be imaged, several techniques have been developed that can beat the Abbe limit (e.g., STED [@hell94], STORM [@rust06], PALM [@betzig06; @hess06]). These techniques use prior information about the source and selective activation of photo-emitters to achieve so-called super-resolution.
Finally, conventional computing faces some important barriers, such as heat generation and bandwidth limitations. Classical optics can help heat reduction by using passive elements and reversible computing, and optics also allows us to handle complex calculations by using fan-in and fan-out, in addition to parallelisation [@shlomi10]. These techniques are expected to become more prevalent in the near future.
All of the above examples are using classical optics. However, light is not classical. From a fundamental physics point of view, optical information processing must be extended to take quantum effects into account. These effects do not just add noise to existing techniques, but enable dramatic improvements in information processing with light, from communication, metrology and imaging to full-scale quantum computing.
In this review, I will sketch the physical principles and phenomena that lie at the heart of optical quantum information processing. I place special emphasis on the quantum mechanical properties of light that make it different from classical light, and how these properties relate to information processing tasks. In section \[sec:transition\] I introduce the concepts of coherence, anti-bunching and the Hong-Ou-Mandel effect, and in section \[sec:photons\] I give a brief introduction how photons as quantum information carriers are described mathematically. Section \[sec:comm\] is devoted to quantum communication, covering the no-cloning theorem, quantum key distribution, teleportation and repeaters. In section \[sec:metro\] I introduce the idea of precision metrology with classical and quantum light and show how similar techniques can be used for high resolution imaging. Section \[sec:comp\] starts with a discussion about optical entanglement and introduces the famous *KLM protocol* for quantum computing. I also sketch the most recent ideas for creating a quantum computer architecture based on linear optics. Finally, in section \[sec:future\] I give an overview of the practical challenges that still remain in implementing the ideas presented in this review.
From classical light to quantum light {#sec:transition}
=====================================
As we continue to improve the information processing capabilities of our computer processors and networks, everything is being made smaller and more efficient. In miniaturizing and squeezing all the information out of every last bit of light, we will be coming up against a fundamental limit of nature, namely the fact that light comes in discrete quanta called *photons*. Entirely new laws of physics come into play that have a profound effect on our capabilities for information processing. Before we can explore these new capabilities, however, we need to establish in some more detail what makes quantum light different from classical light. To this end we will briefly describe the idea of coherence, anti-bunching, and two-photon interference.
Coherence {#sub:coherence}
---------
Classical light exists in roughly two categories, namely *thermal* and *coherent* light. Thermal light is the type that is emitted by sources like stars, light bulbs and LEDs, while coherent light is typically associated with the output of a laser. The two types should not be seen as completely distinct, but rather as the extremes of a whole spectrum that spans from coherent, via partially coherent, to thermal (or incoherent) light. A key concept in this regard is the *coherence length* of a beam of light.
![Coherence: A wave of constant frequency over a long time (left figure) will have a well-defined phase relationship both at short times (between point $A$ and point $B$), as well as long times (between point $A$ and point $C$). Alternatively, if the frequency fluctuates over time (right figure), there will still be an appreciable phase correlation at short times (between point $A$ and point $B$), but at long times the phase relationship fluctuates (between point $A$ and point $C$). The characteristic time at which the coherence between the phases drops below a certain level is called the coherence time. Sometimes this is also referred to as the coherence length.[]{data-label="fig:coherence"}](coherent.pdf "fig:"){height="33mm"} ![Coherence: A wave of constant frequency over a long time (left figure) will have a well-defined phase relationship both at short times (between point $A$ and point $B$), as well as long times (between point $A$ and point $C$). Alternatively, if the frequency fluctuates over time (right figure), there will still be an appreciable phase correlation at short times (between point $A$ and point $B$), but at long times the phase relationship fluctuates (between point $A$ and point $C$). The characteristic time at which the coherence between the phases drops below a certain level is called the coherence time. Sometimes this is also referred to as the coherence length.[]{data-label="fig:coherence"}](coherent2.pdf "fig:"){height="33mm"}
The coherence length is most easily explained by first looking at the coherence *time*. Consider a wave of constant frequency $\omega$, as shown in figure \[fig:coherence\] on the left. The phase of the wave at time $t$ will be given by $\phi$. We can calculate the difference between the phases at difference times. For example, $\phi_B - \phi_A$ is the difference in the phase of the wave for the time interval $\Delta t = t_B - t_A$. Since the frequency of the wave is constant over time, the phase difference $\phi_B - \phi_A$ between two points in time separated by $\Delta t$ will be constant as well. In addition, given a third phase $\phi_C$ at time $t_C$ much later than $t_A$ we have again a constant phase difference $ \phi_C - \phi_A$ for a time interval $\Delta T = t_C - t_A$: $$\begin{aligned}
\phi_B - \phi_A = \text{constant} \qquad\text{and}\qquad \phi_C - \phi_A= \text{constant.} \end{aligned}$$ Next, consider a wave whose frequency fluctuates over time, as shown in figure \[fig:coherence\] on the right (the effect is quite subtle). If the time interval $\Delta t = t_B - t_A$ is short compared to the (inverse) rate of change of $\omega$, the difference between the phases at $A$ and $B$ will still be nearly constant. However, for longer times between two points on the wave this will no longer be the case. The fluctuations in the frequency will cause fluctuations in the phase relationship between $A$ and $C$, and we have $$\begin{aligned}
\phi_B - \phi_A \simeq \text{constant} \qquad\text{and}\qquad \phi_C - \phi_A = f(t)\, ,\end{aligned}$$ where $f(t)$ is a strongly fluctuating function over time taking values in the entire interval $[0,2\pi)$. There will be a characteristic time $\tau$ where the behaviour of the phase difference changes from nearly constant to strongly fluctuating. This is the coherence time of the wave. Multiplying the coherence time by the velocity of the wave in the medium gives us the coherence length. A very similar argument can be made for the phase coherence of two emitters a distance $d$ apart, leading to the concept of *transverse* coherence length.
The coherence length of light plays a fundamental role in classical interference. Coherent sources interfere, while incoherent sources do not. These properties carry over to the quantum mechanical treatment of light, and we will see that coherence is a crucial property for optical quantum information tasks.
Anti-bunching
-------------
Classical waves can have any amplitude, no matter how small. However, this is not the case for quantum mechanical light. When the power of a light source is reduced, at some point a detector will no longer record a constant signal, but rather we find that the light arrives in bursts. These bursts are called photons.
We can easily imagine how such bursts come about: the atom that emits the photon does so by having one of its electrons drop down to a lower energy state (the difference in energy escapes in the form of our photon). Creating a second burst of light requires the electron to be loaded back up into the excited state, which takes time. This leads to the phenomenon of *anti-bunching*: it is relatively unlikely that two photons are detected in much shorter succession than the time it takes the electron to reoccupy the excited state.
To make this description a bit more mathematically precise, we can consider the probability distribution of the number of photons that are recorded in a time interval $\Delta t$. If the photons arrive completely randomly, this distribution will be Poissonian: $$\begin{aligned}
\label{eq:poisson}
\pr{n} = \frac{\lambda^n {\rm e}^{-\lambda}}{n!}\, ,\end{aligned}$$ where $n$ is the number of photons detected in the interval $\Delta t$ and $\lambda$ is the (dimensionless) intensity of the light, which corresponds to the average number of photons in $\Delta t$. When a source exhibits anti-bunching at a time scale $\Delta t$, the probabilities of finding two, three, four, etc., photons will be suppressed compared to the Poisson distribution in Eq (\[eq:poisson\]).
![Second-order correlations: By counting the number of coincidence detections in detectors 1 and 2 we can determine the second order correlation function $g^{(2)}$. We vary the path length of one of the detectors. For a Poissonian distribution (blue curve) the $g^{(2)}$ function remains constant at 1, while at very short times (i.e., equal distances of the detectors to the beam splitter) the probability of getting detector coincidences drops to zero for anti-bunched light (green curve). Note that thermal light is bunched (orange curve), in the sense that it is *more* likely that two photons are emitted at the same time.[]{data-label="fig:hbt"}](HBT.pdf "fig:"){height="41mm"} ![Second-order correlations: By counting the number of coincidence detections in detectors 1 and 2 we can determine the second order correlation function $g^{(2)}$. We vary the path length of one of the detectors. For a Poissonian distribution (blue curve) the $g^{(2)}$ function remains constant at 1, while at very short times (i.e., equal distances of the detectors to the beam splitter) the probability of getting detector coincidences drops to zero for anti-bunched light (green curve). Note that thermal light is bunched (orange curve), in the sense that it is *more* likely that two photons are emitted at the same time.[]{data-label="fig:hbt"}](g2.pdf "fig:"){height="41mm"}
Experimentally, we can demonstrate this by putting a beam splitter in the light beam and count the number of coincidences in the two photodetectors, as shown in figure \[fig:hbt\]. The number operator for detector 1 is given by $\hat{n}_1$ and that of detector 2 by $\hat{n}_2$. The average photon number in detector $j=1,2$ at time $t$ is then calculated via the quantum mechanical expectation value $\braket{\hat{n}_j(t)}$. The second-order correlation is measured by the $g^{(2)}$ function defined according to $$\begin{aligned}
g^{(2)}(\tau) = \frac{\Braket{\hat{n}_1(t)\;\hat{n}_2(t+\tau)}}{\Braket{\hat{n}_1(t)}\Braket{\hat{n}_2(t+\tau)}}\, ,\end{aligned}$$ which does not depend on $t$ for stationary processes, such as the situation we consider here. A typical $g^{(2)}(\tau)$ function is shown in figure \[fig:hbt\]. For long time scales the intensities for anti-bunched light in both detectors become independent, and the $g^{(2)}$ function for anti-bunched light (the green curve) tends towards that of Poissonian light (the blue curve). For completeness we also plotted the $g^{(2)}$ function for thermal light (the orange curve), which shows *bunching*: it is more likely that the two detectors fire in unison at very short timescales compared to Poissonian light. This is in fact also a quantum mechanical effect: a thermal “gas” of photons obeys the Bose-Einstein distribution, rather than the classical Maxwell-Boltzmann distribution.
The equations of motion for a photon
------------------------------------
The question is now what makes light quintessentially quantum-mechanical, and which features of light are well-explained by the classical theory. Maxwell’s equations do not provide a description of quantised energy, but they do very accurately describe the shapes of the wave packets. In particular, the coherence lengths (transverse and longitudinal) that are determined by the classical theory determine the interference properties of photons. To illustrate this, we consider Young’s double-slit experiment with photons.
Suppose that a source of single photons illuminates a screen with two narrow slits of width $a$ placed at a distance $d$ apart. The classical theory predicts that the intensity pattern at a screen a distance $L$ from the slits is given by $$\begin{aligned}
I(x) = I_0 \cos\left(\frac{xd\pi}{L\lambda}\right) \operatorname{sinc}\left(\frac{xa\pi}{L\lambda}\right) \qquad\text{with}\qquad \operatorname{sinc}\beta \equiv \frac{\sin\beta}{\beta}\, ,\end{aligned}$$ where $x$ is the position along the screen and $\lambda$ is the wavelength of the light. When the source emits single photons one at a time, after collecting many photons the average intensity on the screen will be exactly the same. Therefore the spatial behaviour of single photons is identical to that of classical waves. However, this presupposes that we do not know which slit the photon travels through. This is equivalent to saying that the amplitudes of the photon wave packet at the left slit and the right slit add coherently. In other words, the transverse coherence of the photon wave packet must be larger than the distance between the slits.
Next, we consider the quantum mechanical description of light. Classically, we can write the electric field $E$ as a solution to the wave equation: $$\begin{aligned}
\label{eq:classicalfieldop}
E(\mathbf{x},t) = \sum_{\mathbf{k}} A_ {\mathbf{k}}\, u_ {\mathbf{k}} (\mathbf{x},t) + A_ {\mathbf{k}}^*\, u_ {\mathbf{k}}^* (\mathbf{x},t)\, ,\end{aligned}$$ where $A$ is the complex amplitude of a wave in mode $\mathbf{k}$ (commonly referred to as the wave vector, but in principle we can have more exotic labelings), and the sum over $\mathbf{k}$ indicates that the waves can be superposed. The functions $u_ {\mathbf{k}} (\mathbf{x},t)$ in Eq. (\[eq:fieldop\]) are the so-called *mode functions*, and they form a complete set of solutions to the classical Maxwell equations. Often a plane wave expansion for ${E}(\mathbf{x},t)$ is given, in which $$\begin{aligned}
u_ {\mathbf{k}} (\mathbf{x},t) \propto \bm{\epsilon}_\mathbf{k}\, {\rm e}^{i\mathbf{k}\cdot\mathbf{x} - i\omega_{\mathbf{k}}t} ,\end{aligned}$$ and the sum in Eq. (\[eq:fieldop\]) becomes an integral over $\mathbf{k}$ with $\omega_\mathbf{k}$ the frequency of the wave vector $\mathbf{k}$. The vector $ \bm{\epsilon}_\mathbf{k}$ determines the polarisation of the mode. Other expansions are also possible, and may be more convenient depending on the application (for example, a plane wave expansion is not very suited to describe a wave in an optical fibre).
In the full quantum mechanical description of optics, the electric field becomes an operator and can be written as [@koklovett10]: $$\begin{aligned}
\label{eq:fieldop}
\hat{E}(\mathbf{x},t) = \sum_{\mathbf{k}} \hat{a}_ {\mathbf{k}}\, u_ {\mathbf{k}} (\mathbf{x},t) + \hat{a}_ {\mathbf{k}}^\dagger\, u_ {\mathbf{k}}^* (\mathbf{x},t)\, ,\end{aligned}$$ where $\hat{a}_ {\mathbf{k}}$ and $\hat{a}_ {\mathbf{k}}^\dagger$ are the annihilation and creation operator for the optical mode indicated by $\mathbf{k}$. These operators replace the complex amplitudes in the classical theory and obey the commutation relations $$\begin{aligned}
\left[ \hat{a}_ {\mathbf{k}}, \hat{a}_ {\mathbf{k}'} \right] = \left[ \hat{a}_ {\mathbf{k}}^\dagger, \hat{a}_ {\mathbf{k}'}^\dagger \right] = 0 \qquad\text{and}\qquad \left[ \hat{a}_ {\mathbf{k}}, \hat{a}_ {\mathbf{k}'}^\dagger \right] = \delta_{\mathbf{k}\, \mathbf{k}'}\, ,\end{aligned}$$ where $\delta_{\mathbf{k}\, \mathbf{k}'}$ is the Kronecker delta symbol[^1]. The creation and annihilation operators act on photon number states $\ket{n}_\mathbf{k}$ according to the rules $$\begin{aligned}
\hat{a}_{\mathbf{k}} \ket{n}_{\mathbf{k}} = \sqrt{n} \ket{n-1}_{\mathbf{k}} \qquad\text{and}\qquad \hat{a}_{\mathbf{k}}^\dagger \ket{n}_{\mathbf{k}} = \sqrt{n+1} \ket{n+1}_{\mathbf{k}}\, .\end{aligned}$$ The number operator for mode $\mathbf{k}$ is then given by $\hat{n}_{\mathbf{k}} = \hat{a}_{\mathbf{k}}^\dagger \hat{a}_{\mathbf{k}}$. The algebra of these operators is identical to that of the simple harmonic oscillator.
Creating a photon in mode $\mathbf{k}$ means that the photon will behave according to the spatio-temporal description provided by $u_ {\mathbf{k}} (\mathbf{x},t)$. Since $u_ {\mathbf{k}} (\mathbf{x},t)$ is determined by Maxwell’s equations, we can say that the classical Maxwell equations are the equations of motion for the photon. The quantum mechanical behaviour of light is restricted to the photon *statistics*.
The Hong-Ou-Mandel effect
-------------------------
While the mode shapes of propagating photons are determined by the classical theory of electrodynamics, the quantum behaviour of light is most apparent in multi-photon effects. For the purposes of quantum information processing, the most important example of this is the Hong-Ou-Mandel effect, which is a two-photon intensity interference effect. Specifically, the Hong-Ou-Mandel effect occurs when two photons with identical frequency, polarisation and shape of the wave packet enter a 50:50 beam splitter on either side. If we place detectors in the two outgoing modes of the beam splitter, each photon has two ways of triggering the detectors. Either the photon triggers the upper detector, or it triggers the lower detector. The resulting four possible paths for the two photons are shown in figure \[fig:hom\]. In (a) the top input photon is transmitted while the bottom photon is reflected. In that case, both photons end up in the bottom detector. And vice versa, both photons may end up in the top detector (d). Whenever a photon is reflected off the topside of the beam splitter it experiences a $\pi$ phase shift, which results in a factor $e^{i\pi} = -1$ in the state of the photon.
![The Hong-Ou-Mandel effect: If we send two photons into the two input beams of a 50:50 beam splitter there are four possible outcomes, as shown in the figure. In quantum mechanics, these four possibilities are superposed. Due to a $\pi$ phase shift upon reflection of the top surface, the case where both photons are reflected has a relative minus sign compared to the case where both photons are transmitted. When the input photons are identical, we cannot distinguish between the two middle outcomes, and they cancel. So the identical photons will always pair off towards the same output beam, and never leave the beam splitter in different beams.[]{data-label="fig:hom"}](hom.pdf){width="110mm"}
An interesting effect happens when we consider situations (b) and (c) in figure \[fig:hom\]. In case (b), both photons are reflected by the beam splitter, while in case (c), both photons are transmitted. Since the photons are identical, the two processes (b) and (c) are *indistinguishable* from each other. No physical process can tell whether the photons were reflected or transmitted, and the beam splitter itself holds no memory of the process. According to Feynman, this means that the two contributions must be superposed coherently and are allowed to interfere [@feynman]. However, since the top photon in process (b) picks up a phase factor $e^{i\pi}$, the two states must be subtracted. This leads to destructive interference, and as a result the two identical photons will *never* end up in separate detectors. This effect was first observed by Hong, Ou, and Mandel in 1987 [@hong87]. The effect lies at the heart of the protocols that enable quantum computing with single photons and linear optics. A recent experimental realisation of the Hong-Ou-Mandel effect is shown in figure \[fig:homexp\] [@prtljago16]. The depth of the dip indicates the level of indistinguishability between the two photons. Together with a single-mode $g^{(2)}$ measurement to verify the presence of only a single photon, the Hong-Ou-Mandel dip gives a good indication for the quality of single-photon sources.
![Experimental data for the Hong-Ou-Mandel effect using one dot and one laser input [@prtljago16]. The dip is not as deep as one would expect for two indistinguishable single photon sources, because the laser is not a single-photon source (it obeys Poissonian statistics, rather than anti-bunching). Nevertheless, the dip (black curve) is lower than expected for classical light (red curve).[]{data-label="fig:homexp"}](HOM_exp.pdf){width="80mm"}
Photons as quantum information carriers {#sec:photons}
=======================================
As we encounter the quantum limits of light, we may ask what we can do in terms of information processing if we embrace this natural behaviour. Imagine that we send classical bits using the polarisation degree of freedom. In other words, an optical pulse of horizontally polarised light ($H$) is defined as the bit value 0, and a vertically polarised pulse ($V$) is the bit value 1. At the single photon level the polarisation is still a well-defined physical property, since it is determined by the mode functions (and therefore obey Maxwell’s equations). The polarised photons and their bit values can be described quantum mechanically with the quantum states $$\begin{aligned}
\ket{H} \equiv \ket{0} \qquad\text{and}\qquad \ket{V} \equiv \ket{1}\, .\end{aligned}$$ A fundamental property of classical light is that two pulses can be prepared in superposition. For our example, this means that we can make a superposition of vertical and horizontal light. The result is a new pulse with a polarisation in a different direction depending on the relative phase between the $H$ and $V$ pulses. The new polarisation can be linear, circular, or elliptical.
This property carries over to photons. For example, left- and right-handed circularly polarised photons have quantum states $\ket{L}$ and $\ket{R}$, respectively: $$\begin{aligned}
\ket{L} = \frac{\ket{H}+i\ket{V}}{\sqrt{2}} \qquad\text{and}\qquad \ket{R} = \frac{\ket{H}-i\ket{V}}{\sqrt{2}} \, .\end{aligned}$$ If we treat the horizontal and vertical polarisation of a photon as bit values 0 and 1, we see that we now have two *new* bit values that are superpositions of $\ket{0}$ and $\ket{1}$: $$\begin{aligned}
\ket{\circlearrowleft} = \frac{\ket{0}+i\ket{1}}{\sqrt{2}} \qquad\text{and}\qquad \ket{\circlearrowright} = \frac{\ket{0}-i\ket{1}}{\sqrt{2}} \, .\end{aligned}$$ This is not possible for a classical bit, and we therefore call the polarised photon a quantum bit, or *qubit* for short. Every classical polarisation has a corresponding qubit when we bring the optical pulse down to a single photon. This requires that the two pulses have a well-defined phase relationship, and are therefore *coherent* in the sense of the discussion in section \[sub:coherence\]. The extra states in a qubit over a classical bit suggest that information processing with qubits can be more powerful than information processing with classical bits, because—loosely speaking—more available states means more room for the information to play in. Instead of polarisation, we can use other degrees of freedom of light [@koklovett10], but for simplicity we will restrict our discussion to polarised photons in this article.
Since the qubit structure is directly inherited from the classical superposition principle, the next question is: what makes the photonic qubit a fundamentally quantum mechanical object? The answer is given by anti-bunching. There is only one indivisible photon (see figure \[fig:hbt\]) that triggers either detector 1 or detector 2. Classically, both detectors could register a non-zero signal simultaneously. The fact that this is not possible for single photons means that the photon ends up in either one or the other detector in a *probabilistic* manner. If we wish to measure the polarisation of the photon, we must first choose which *basis* we want to measure ($\ket{H}$ and $\ket{V}$, or $\ket{L}$ and $\ket{R}$). If we measure the photon in the state $\ket{L}$ in the $H/V$ basis, we will obtain the outcome “H” or “V” with 50:50 probability.
The superposition principle—together with the concept of the photon as a particle—further leads to the phenomenon of *entanglement*. Two photons can be prepared in the state $$\begin{aligned}
\label{eq:phiplus}
\ket{\Phi^+} = \frac{\ket{HH} + \ket{VV}}{\sqrt{2}}\, ,\end{aligned}$$ where $\ket{HH}\equiv\ket{H}_1\ket{H}_2$ and $\ket{VV}\equiv\ket{V}_1\ket{V}_2$ are short-hand for the polarisation states of the two photons. It is easy to see that the state in Eq. (\[eq:phiplus\]) cannot be written as the product of two separate photon states: $$\begin{aligned}
\ket{\Phi^+} \neq \left( \alpha_H \ket{H} + \alpha_V \ket{V} \right) \left( \beta_H \ket{H} + \beta_V \ket{V} \right)\, ,\end{aligned}$$ where $\alpha$ and $\beta$ are complex numbers obeying $\abs{\alpha_H}^2 + \abs{\alpha_V}^2 = 1$ and $\abs{\beta_H}^2 + \abs{\beta_V}^2 = 1$. The effect of entanglement is that the two photons are more strongly correlated than is possible classically: $$\begin{aligned}
\ket{\Phi^+} & = \frac{\ket{HH} + \ket{VV}}{\sqrt{2}} = \frac{1}{2\sqrt{2}} \left[ (\ket{L}+\ket{R}) (\ket{L}+\ket{R}) + (-i)^2 (\ket{L}-\ket{R}) (\ket{L}-\ket{R}) \right] \cr & = \frac{\ket{LR} + \ket{RL}}{\sqrt{2}}\, .\end{aligned}$$ There is not only a correlation in $H$ and $V$, but also in $L$ and $R$. This is not possible in classical systems, and these stronger quantum correlations can be utilised in information processing. For future use, we define a basis of four entangled states, called the *Bell states*: $$\begin{aligned}
\ket{\Phi^+} = \frac{\ket{HH} + \ket{VV}}{\sqrt{2}} & \qquad\phantom{\text{and}}\qquad \ket{\Psi^+} = \frac{\ket{HV} + \ket{VH}}{\sqrt{2}} \, ,\cr
\ket{\Phi^-} = \frac{\ket{HH} - \ket{VV}}{\sqrt{2}} & \qquad\text{and}\qquad \ket{\Psi^-} = \frac{\ket{HV} - \ket{VH}}{\sqrt{2}} \, .\end{aligned}$$ A measurement in this basis is called a *Bell measurement*, and plays a crucial role in quantum information processing.
Now that we have a photonic qubit, what exactly can we do with it? We would expect that all the classical information processing tasks with light will in some way carry over to quantum light with some enhancements due to qubit superpositions and entanglement. Indeed, we can construct new communication protocols and create a quantum internet [@kimble08], we can use photons as measurement probes to achieve a much higher precision in parameter estimation [@giovannetti11] and imaging [@boto00; @kolobov07], and we can use photons as the fundamental information carriers in quantum computing [@kok07].
However, photons are not equally good at all these things. While they are clearly very good data carriers over long distances, it is rather hard to slow them down significantly or even stop them completely. Typical classical and quantum information processing tasks require feed-forward operations in which the state of a qubit is modified in a way that depends on the measurement outcome of another process. If the photon flies away at the speed of light while the measurement is being made, we cannot perform feed-forward because we cannot catch up with the photon. We therefore need to store the photon in some kind of photon memory. This is a complicated process that will likely introduce a lot of noise.
Another complication is that while the polarisation of a single photon is very easy to manipulate using half wave plates and quarter wave plates, the creation of entanglement between two photons is extremely hard. This is due to the complete absence of direct photon-photon interactions. An operation that entangles two photons must therefore be an inherently nonlinear process, either involving nonlinear materials or a clever arrangement of projective measurements. In this review we will consider the latter.
Quantum communication {#sec:comm}
=====================
Classical light makes for an excellent information carrier over long distances. This is also true for quantum light. Moreover, we can use the quantum mechanical properties of photons to accomplish new communication tasks that are more difficult or impossible with classical light. As an example, we will consider secure communication using quantum key distribution, and explore what requirements are necessary to extend these techniques beyond a few hundred kilometres.
The no-cloning theorem and quantum key distribution
---------------------------------------------------
Consider a photon with horizontal and vertical polarisation states $\ket{H}$ and $\ket{V}$, respectively. As we have seen, we can make quantum superpositions of these states to obtain different polarisation states. The no-cloning theorem says that it is impossible to create a machine that copies an unknown quantum state perfectly [@wootters82; @dieks82]. To see this, suppose that we have a photon in some unknown polarisation state $\ket{\psi}$ and a second photon in a known initial polarisation state, e.g., $\ket{H}$. A proper copying machine would have to produce the following effect on *any* state $\ket{\psi}$: $$\begin{aligned}
\ket{\psi}\ket{H} \underset{\rm copy}{\longrightarrow} \ket{\psi}\ket{\psi}\, .\end{aligned}$$ In particular, the machine must act on the states $\ket{H}$ and $\ket{V}$ according to $$\begin{aligned}
\ket{H}\ket{H} \underset{\rm copy}{\longrightarrow} \ket{H}\ket{H} \qquad\text{and}\qquad \ket{V}\ket{H} \underset{\rm copy}{\longrightarrow} \ket{V}\ket{V} \, .\end{aligned}$$ This completely determines how the copying machine will handle the unknown polarisation state, because any polarisation state can be written as a quantum superposition of $\ket{H}$ and $\ket{V}$. For example, suppose that the state $\ket{\psi}$ is in fact the left-handed circularly polarised state $\ket{L}$. Then $$\begin{aligned}
\ket{\psi} = \frac{\ket{H}+i \ket{V} }{\sqrt{2}} \, ,\end{aligned}$$ and the copying machine will produce $$\begin{aligned}
\ket{\psi}\ket{H} = \frac{\ket{H}+i\ket{V}}{\sqrt{2}} \ket{H} \underset{\rm copy}{\longrightarrow} \frac{1}{\sqrt{2}}\ket{H}\ket{H}+\frac{i}{\sqrt{2}}\ket{V}\ket{V} \, .\end{aligned}$$ However, this is *not* the same as $\ket{\psi}\ket{\psi}$, as you can tell when we write it out in the $H/V$ basis: $$\begin{aligned}
\ket{\psi}\ket{\psi} & = \left( \frac{\ket{H}+i\ket{V}}{\sqrt{2}}\right) \left( \frac{\ket{H}+i\ket{V}}{\sqrt{2}} \right) \cr & = \frac12 \ket{H}\ket{H} + \frac{i}{2}\ket{H}\ket{V} + \frac{i}{2}\ket{V}\ket{H} - \frac12 \ket{V}\ket{V} \, .\end{aligned}$$ This means that a copying machine that works for $\ket{H}$ and $\ket{V}$ will not faithfully copy $\ket{L}$ and $\ket{R}$, and vice versa. Practically, this means that the badly copied photon behaves differently from a photon in the original state. The no-cloning theorem is a fundamental result in quantum mechanics and applies to all physical systems.
Next, consider the situation in which two agents, Alice and Bob, wish to communicate in private. One way they can accomplish this if they share a secret string of random zeros and ones called a *key*: Alice adds this string to her binary message, creating the encrypted message. Bob decrypts the message by subtracting the key from the encrypted message. Since no-one else has the secret key, nobody can decrypt the message but Alice and Bob. The question is how to generate such a secret key.
Sending the secret key over a public channel will invite eavesdroppers to copy it and gain access to the private message between Alice and Bob. If Alice and Bob can detect the eavesdropper, they know the channel is compromised and move to a different channel. This is what the no-cloning theorem allows them to do: Let’s suppose that $\ket{H}$ and $\ket{L}$ denote a bit value of zero, and $\ket{V}$ and $\ket{R}$ denote a bit value of one. Alice sends a random string of photons in polarisation states $\ket{H}$, $\ket{V}$, $\ket{L}$, and $\ket{R}$. Bob measures randomly in the $H/V$ basis or the $L/R$ basis. About half the time Alice will have created a photon in the same basis as Bob’s measurement, and in those cases both Alice and Bob will know whether they shared a zero or a one. In the rest of the cases there is no correlation between the bit value sent by Alice and the bit value measured by Bob. Alice and Bob then publicly compare their preparation and measurement bases ($H/V$ or $L/R$), and keep only those bits for which the preparation and measurement bases coincide. Note that they do not reveal the actual bit values, only the bases.
To see whether there is an eavesdropper on the line, Alice and Bob sacrifice a small part of their secret key. They publicly compare this part of the key and see if the bit values match up. If they do, there was no eavesdropper, but if there is a sizeable amount of errors there may have been an eavesdropper. Anyone trying to copy the secret key as it was being established must copy the information of the photon polarisation. However, since the photons were sent in two different bases (unknown to anyone but Alice), a copying machine that works perfectly in the $H/V$ basis will create imperfect copies and cause incorrect measurement results for Bob. These incorrect measurement results will show up when Alice and Bob compare part of their secret key. The secrecy of the key is therefore guaranteed by the no-cloning theorem.
The comparison between Alice and Bob of a fraction of their secret key is what guarantees the privacy of the key. There is a trade-off between the amount of information Eve can gain, and the level of privacy attained by Alice and Bob. When this protocol is implemented with real devices, additional noise will appear in the system, and Alice and Bob must be able to account for that also. To this end, they can further sacrifice part of the key to increase their privacy. This is called *privacy amplification* [@koashi07], and is a crucial part of any practical implementation of quantum key distribution. The general trade-off in the communication between Alice and Bob is then privacy versus bit rate.
A final observation about this quantum key distribution protocol is that it relies critically on the quantum mechanical possibility of anti-bunching of light. If light did not come in discrete packages (photons), the eavesdropper could siphon off a small part of the signal with a beam splitter and measure the polarisation of this very weak field. The fact that there is only one photon in each successive pulse from Alice means that it either shows up in Bob’s detector, or it arrives in the eavesdropper’s detector, in which case Bob will register a failed transmission. In either case that photon will not be used for the secret key and is useless to the eavesdropper. If there was a possibility of more than one photon in each pulse (a Poissonian or thermal distribution), the eavesdropper could measure one photon while an identical photon makes its way to Bob. Note that this does not violate the no-cloning theorem, since Alice can create however many photons she chooses in a state of her choice.
Quantum repeaters and quantum memories
--------------------------------------
When a photon travels in an optical fibre, it has a certain probability of being scattered by impurities in the fibre. In this case the photon does not make it to the end of the fibre, and we call this *photon loss*. A fibre can be characterised by an attenuation length $\ell$ at which the original signal is reduced by a factor $1/e$. The attenuation for a fibre of length $L$ is then given by $\exp(-L/\ell)$. This is an exponential decay, which means that we cannot lay arbitrarily long fibre-optic cables and still expect a sizeable bit rate from end to end. In practice, the cable length can be a few hundred kilometres at most.
If we want to extend the reach of quantum communication protocols, we have to add some active devices in the communications channel. Classically, this is accomplished by repeater stations, which amplify the signal and transmit it to the next repeater station. However, amplification is a form of copying, and we have just seen that the no-cloning theorem prevents such devices from working properly on general qubit states. At first this looks like quantum communication will remain viable only for short distances. However, another fundamental protocol in quantum mechanics comes to the rescue here, namely quantum *teleportation* [@bennet93].
![Quantum teleportation. Alice takes a photon in an arbitrary quantum state (qubit 1) and one half of an entangled photon pair (qubit 2) and performs a Bell measurement. The outcome of the measurement determines the operation that Bob needs to perform on his photon (qubit 3). After teleportation is complete, the quantum state of qubit 1 has been transferred to qubit 3.[]{data-label="fig:teleportation"}](teleportation.pdf){height="40mm"}
In quantum teleportation, shown schematically in figure \[fig:teleportation\], Alice wishes to send an arbitrary quantum state of a photon to Bob. Rather than sending the state directly (which would be subject to photon loss), they first establish entangled photons pairs between each other. We denote the arbitrary quantum state by $\ket{\psi}_1$ and the shared entanglement is in the Bell state $\ket{\Phi^+}_{23}$, where the labels 1, 2, and 3 indicate the photons. Photons 1 and 2 are held by Alice, and photon 3 is held by Bob. The total state of the three photons is then given by $$\begin{aligned}
\ket{\psi}_1 \ket{\Phi^+}_{23} = \left( \alpha\ket{H}_1 + \beta\ket{V}_1 \right) \frac{\ket{HH}_{23}+\ket{VV}_{23}}{\sqrt{2}} \equiv \ket{\chi}_{123}\, ,\end{aligned}$$ where $\alpha$ and $\beta$ are complex numbers obeying $\abs{\alpha}^2 + \abs{\beta}^2 = 1$. We can write this as $$\begin{aligned}
\ket{\chi}_{123} = \frac{1}{\sqrt{2}} \left( \alpha\ket{HHH} + \alpha\ket{HVV} + \beta\ket{VHH} + \beta\ket{VVV} \right) .\end{aligned}$$ We suppressed the photon labels for brevity.
Next, Alice performs a Bell measurement, in which her two photons are projected onto the Bell states. Writing the states $\ket{HH}$, $\ket{HV}$, $\ket{VH}$, and $\ket{VV}$ in the Bell basis and rearranging the terms, the state just before the Bell measurement is given by $$\begin{aligned}
\ket{\chi}_{123} = & \frac12 \left[ \ket{\Phi^+} \left( \alpha \ket{H} + \beta\ket{V}\right) + \ket{\Phi^-} \left( \alpha \ket{H} - \beta\ket{V}\right) \right. \cr & + \left. \ket{\Psi^+} \left( \beta \ket{H} + \alpha\ket{V}\right) + \ket{\Psi^-} \left( \beta \ket{H} - \alpha\ket{V}\right) \right] .\end{aligned}$$ The outcomes of Alice’s Bell measurement indicate which state Bob’s photon is in: $$\begin{aligned}
{\Phi^+}: & \quad \ket{\psi}_3 = \alpha \ket{H} + \beta\ket{V} \qquad {\Psi^+}: \quad \ket{\psi}_3 = \beta \ket{H} + \alpha\ket{V} , \cr
{\Phi^-}: & \quad \ket{\psi}_3 = \alpha \ket{H} - \beta\ket{V} \qquad {\Psi^-}: \quad \ket{\psi}_3 = \beta \ket{H} - \alpha\ket{V} . \end{aligned}$$ Bob does not know which of these states his photon is in until he receives a message from Alice telling him her measurement outcome. After receiving the message, Bob can apply a corrective operation (using half wave plates and quarter wave plates) to bring the quantum state of his photon back to the original $\alpha \ket{H} + \beta\ket{V}$. This completes the teleportation protocol. Quantum teleportation was demonstrated in 1997 and 1998 in various optical implementations [@bouwmeester97; @boschi98; @furusawa98], and to various degrees of completeness [@kok00].
A quantum repeater based on quantum teleportation works as follows (see figure \[fig:repeater\]) [@briegel98]: Alice needs to send a polarised photon to Bob, who is too far away to send directly. Instead, she sends it to a repeater station somewhere between her and Bob. For now, let’s assume that this station is close enough to Alice. The repeater station establishes shared entangled pairs with Bob, or another repeater (more on that later). This allows the repeater station to receive the incoming photon from Alice and teleport it to Bob using the shared entanglement (the “Swap” gate in figure \[fig:repeater\]). For this to work, the Bell measurement must reveal whether the photon sent by Alice made it to the repeater station, and the entanglement between the repeater station and Bob must be (near) perfect. The repeater station then informs Bob what the corrective operation on his part of the entangled pair must be, and after making this correction Bob can measure his photon in the basis of his choice. Alice and Bob can now be far away from each other and still establish a secret key with a sufficiently high bit rate.
![A series of quantum repeaters can be used to extend the range of quantum communication. A quantum repeater teleports an input photon to a photon in the next repeater via the “Swap” operation. The entanglement between the repeaters is established beforehand and made near perfect using entanglement distillation. Bob receives a cumulative corrective instruction that is determined by all the Swap operations in the repeaters, and includes information which of his photons caries the teleported state.[]{data-label="fig:repeater"}](repeater.pdf){width="110mm"}
In the description above we have, of course, cheated! We magically assumed that the repeater station and Bob share near-perfect entangled photon pairs. This is far from trivial to establish. The photons must be created together, and at least one of them must travel the distance between the repeater station and Bob. That photon will inevitably incur losses of the same magnitude as the photon sent by Alice. The repeater station and Bob can share several pairs and try to find out which of the photons made it through. This is difficult, because it requires detecting a photon without destroying it (after all, we still need to use it in the teleportation protocol). Alternatively, we can perform entanglement distillation, which takes several imperfect entangled pairs and extracts one perfectly entangled pair. This is not easy either, because it requires entangling gates between photons. Finally, while all this processing is going on, the photons don’t just sit there in the repeater station. They need to be actively stored in either an optical delay line or a quantum memory. If the distillation protocol requires communication between the repeater station and Bob, the length of time for which the photon needs to be stored is comparable to the time it takes light to travel from Bob to the repeater station. Any delay line memory would then incur the same amount of photon loss as the channel between the repeater station and Bob, and we’re back to square one.
There are several architectures that attempt to circumvent these various difficulties, and one particular question of interest is what are the minimal requirements for a repeater to work? Does it need two-way communication between stations or can we construct a protocol that requires only one-way communication as shown in figure \[fig:repeater\]? Does the repeater require memories that last as long (or longer) than the flight time of the photons between repeater stations? A lot of progress is being made on these questions, and this is currently an active area of research [@briegel98; @kok02; @jiang09; @zwerger12; @azuma15; @reiserer15].
Finally, we note that while anti-bunching and the no-cloning theorem are sufficient for the design of quantum key distribution, the implementation of this protocol over long distances will most likely require the use of polarisation entanglement. This lifts the construction of repeaters into a new realm of difficulty over direct quantum communication.
Quantum metrology and imaging {#sec:metro}
=============================
Light is also extremely useful in metrological applications. As a simple example, consider the measurement of the thickness of a thin piece of foil. A traditional mechanical micrometer has a precision of about 0.05 mm, which is not good enough to measure foils that are much thinner[^2]. To obtain the required precision, we can use an optical interferometric micrometer shown in figure \[fig:umm\], the principle of which is identical to that of Newton’s rings. The foil is placed at the end of a mirror and holds up a plate of glass. The reflected light will consist of two contributions, namely the light that is reflected off the inside surface of the glass plate, and the light that is reflected off the mirror. Whether constructive or destructive interference occurs at the outgoing light depends on the path difference $t$ between the two contributions. In figure \[fig:umm\], we show this effect at three different positions along the mirror.
![Light can be used to measure the thickness $d$ of a piece of foil by counting the number of interference fringes along a distance $L$ as seen from above. The precision in the measurement of $d$ can be estimated as $\Delta d \sim \frac12 \lambda$, where $\lambda$ is the wavelength of the light. This is much more precise than a mechanical micrometer, which has a precision of about 0.05 mm.[]{data-label="fig:umm"}](micrometer.pdf){width="90mm"}
We find destructive interference between the two reflected light waves when the path length $t$ is a half-integer multiple of the wavelength $\lambda$: $$\begin{aligned}
t = \left( m + \frac12 \right) \lambda = x \tan\theta = \frac{xd}{L}\, ,\end{aligned}$$ where $m$ is an integer, $\theta$ is the angle between the glass plate and the mirror, $L$ is the distance between the pivot point and the foil, and $x$ is the distance from the pivot point to the dark fringe. Adjacent fringes are a distance $\Delta x$ apart, with $$\begin{aligned}
\Delta x = \frac{\lambda L}{2d} \qquad\text{or}\qquad d = \frac{\lambda L}{2\Delta x} \equiv \frac12 \lambda L \sigma \, ,\end{aligned}$$ where $\sigma$ is the number of fringes per unit length. If we know $\lambda$ and $L$, we can count the fringes to obtain $\sigma$, which in turn gives us a value for the foil thickness $d$.
The precision of this measurement can be estimated by noting that the number of fringes per unit length $\sigma$ can be counted very accurately, as long as there is at least one fringe across the length $L$, or $\Delta\sigma \sim 1/L$. Using the error propagation formula we calculate $\Delta d$ from this estimate as follows: $$\begin{aligned}
\Delta d = \frac{\Delta\sigma}{|{\rm d}\sigma/{\rm d}d|} = \frac{\lambda}{2} L\Delta\sigma \sim \frac{\lambda}{2}\, .\end{aligned}$$ We see that this device can reach a precision of a few hundred nanometers, which is two to three orders of magnitude better than the mechanical micrometer. The above example is using only classical light. Can we improve on this technique by using quantum light? In order to answer this question we will have to go back to what makes quantum light different from classical light.
We can in principle improve the precision of the optical micrometer by using a lot of light: If we measure the intensity along the $x$-direction with very high precision we can detect any variation in intensity, even if that amounts to much fewer than one fringe per length $L$. There is a catch, however, since the intensity itself is noisy. A (transversely) coherent state of light with on average $\braket{n}$ photons has intensity fluctuations that are proportional to $\sqrt{\braket{n}}$. The signal to noise ratio (SNR) is then $\braket{n}/\sqrt{\braket{n}} = \sqrt{\braket{n}}$. The more photons, the higher the SNR, and the higher the SNR, the more precisely the intensity curve in the $x$-direction can be measured (inversely proportional to the SNR). The number of photons is therefore a *resource* for measuring the foil thickness $d$: the more you have of them, the better you can estimate $d$. The precision of $d$ will scale according to $$\begin{aligned}
\label{eq:sql}
\Delta d \propto \frac{1}{\sqrt{\braket{n}}}\, .\end{aligned}$$ This is called the *shot noise limit*, or the *standard quantum limit* (SQL). It originates in the natural intensity fluctuations of light. The $1/\sqrt{\braket{n}}$ behaviour is specific to the type of light, which in this case is classical coherent light. We therefore sometimes refer to this precision as the classical precision limit.
If there is a way to suppress these fluctuations in the intensity we may be able to increase the precision $\Delta d$. How would this work? To answer this, note that the photons arrive in the detector completely independently of each other, which means that they will cluster randomly at each pixel in the detector according to the Poisson distribution in Eq. (\[eq:poisson\]). To remove this randomness we need a “conspiracy” between the photons in the form of transverse anti-bunching (e.g., see figure \[fig:hbt\], where $\tau$ may now denote the distance between the detected positions). If the photons arrive nicely spaced out, the intensity fluctuations at each pixel will be suppressed. The periodic structure of the fringes will then become clear much quicker, and an accurate count of the fringes can be performed with fewer photons [@boyer15].
However, there is a limit to the precision gain that can be obtained this way. No matter how evenly spaced, we still need an appreciable number of photons to reveal the fringes. The ultimate precision in $d$ can be calculated as $$\begin{aligned}
\Delta d \propto \frac{1}{\braket{n}}\, .\end{aligned}$$ This is called the *Heisenberg limit* [@holland93]. Since reaching this limit requires anti-bunching (which in this case will require some form of entanglement between the photons) this is a truly quantum mechanical precision scaling without a classical implementation.
More generally, the ultimate precision allowed by quantum mechanics of the measurement of a parameter $\theta$ is given by the expectation value of the operator $K$ that drives the changes in that parameter. These operators are called *generators*. For example, the generator for changes in time is the Hamiltonian, the generator for translations in space is the momentum operator, and the generator for phase changes is the number operator. The unitary evolution that imparts the parameter $\theta$ onto the quantum state is given by $U(\theta) = \exp(-i K \theta/\hbar)$. The ultimate precision is then written as a *lower bound* on the root mean square error of $\theta$ [@braunstein96]: $$\begin{aligned}
\label{eq:hl}
\Delta\theta \geq \frac{\hbar}{2} \frac{1}{\Delta{K}}\, .\end{aligned}$$ For optimal states, the quantum mechanical operator variance $(\Delta K)^2$ is bounded by the (squared) expectation value $\braket{K}^2$, and $\braket{K}$ is the proper definition of the resource (e.g., the amount) that allows us to increase the precision of the measurement of $\theta$ [@zwierz10]. The expectation value $\braket{K}$ is often much easier to estimate than the variance $(\Delta K)^2$.
From Eq. (\[eq:hl\]) it is clear why this limit is called the Heisenberg limit: if $\theta$ is the position $x$ of a particle and $K$ is its momentum $p$, then the inequality becomes $$\begin{aligned}
\Delta x\, \Delta p \geq \frac{\hbar}{2} \, ,\end{aligned}$$ which you will recognise as Heisenberg’s uncertainty relation for position and momentum. Eq. (\[eq:hl\]) is more general than the traditional Heisenberg-Robertson relation that is derived only for (non-commuting) observables, since it is valid also for physical quantities that do not have an associated quantum operator, such as time, phase and rotation angle.
It is often argued that entanglement is a prerequisite for reaching the Heisenberg limit [@giovannetti06; @giovannetti11]. While this is certainly true in the context of estimation procedures involving many distinguishable particles, the situation in optics—where photons may be indistinguishable—is a little more subtle. Since the Heisenberg limit in Eq. (\[eq:hl\]) depends on $\Delta{K}$, we can in principle construct a quantum state on a single optical mode that maximises $\Delta{K}$. For example, if we wish to measure an optical phase $\varphi$, the relevant generator is the number operator. The state with a maximal variance (and bounded maximum number states) is given by $$\begin{aligned}
\label{eq:0N}
\ket{\psi(0)} = \frac{\ket{0} + \ket{N}}{\sqrt{2}}\, ,\end{aligned}$$ with $\ket{0}$ the state of no photons, and $\ket{N}$ the state of $N$ photons in the optical mode. A phase shift on the optical mode then leads to the transformation $$\begin{aligned}
\ket{\psi(0)} = \frac{\ket{0} + \ket{N}}{\sqrt{2}} \quad\to\quad \frac{\ket{0} + {\rm e}^{iN\varphi}\ket{N}}{\sqrt{2}} = \ket{\psi(\varphi)}\, .\end{aligned}$$ Measuring the observable $X_N = \ket{N}\bra{0} - \ket{0}\bra{N}$ will then give a precision [@koklovett10] $$\begin{aligned}
\Delta\varphi = \frac{\pi}{2} \frac{1}{N}\, .\end{aligned}$$ There is no entanglement in a single optical mode, but we still attain the Heisenberg limit. In any real estimation procedure, however, the use of entanglement can help overcome practical difficulties such as creating the superposition in Eq. (\[eq:0N\]) or implementing the observable $X_N$. For example, instead of the state in Eq. (\[eq:0N\]), we may want to create the so-called [noon]{} state [@bollinger96; @lee02] $$\begin{aligned}
\label{eq:noon}
\ket{\psi(0)} = \frac{\ket{N,0} + \ket{0,N}}{\sqrt{2}}\, ,\end{aligned}$$ which is a two-mode state in which all the photons are in one mode (but it is undetermined which mode). A phase shift in one of the modes will then induce the same relative phase factor ${\rm e}^{iN\varphi}$ as in Eq. (\[eq:0N\]). However, since this is not a superposition of different photon numbers but rather a superposition of the distribution of $N$ photons, it is conceptually easier to see how this can be made in practice [@mitchell04; @walther04].
Still, creating [noon]{} states is extraordinarily difficult, and they are extremely sensitive to decoherence. More promising is the use of *sqeezed* light. This is another type of quantum mechanical light that has no classical analog. Instead of considering the photon number, which are the eigenvalues of the operator $\hat{n} = \hat{a}^\dagger \hat{a}$, we may look at the *quadrature* operators $$\begin{aligned}
\hat{X} = \frac{\hat{a}+\hat{a}^\dagger}{2} \qquad\text{and}\qquad \hat{Y} = -i \frac{\hat{a}-\hat{a}^\dagger}{2}\, .\end{aligned}$$ The commutation relation between $\hat{X}$ and $\hat{Y}$ is $[\hat{X},\hat{Y}] = i$, which means that they obey the uncertainty relation $$\begin{aligned}
\label{eq:quad}
\Delta X \, \Delta Y \geq \frac12\, .\end{aligned}$$ To get an idea what these operators mean, remember that the creation and annihilation operators are mathematically identical to the ladder operators of the simple harmonic oscillator. By analogy, $\hat{X}$ and $\hat{Y}$ behave as the position and momentum of the simple harmonic oscillator. Measuring $\hat{X}$ would then be equivalent to measuring the amplitude of the oscillator, while measuring $\hat{Y}$ would be equivalent to measuring the momentum of the oscillator. In the language of waves these are called quadratures.
A classical coherent state of light is a minimum uncertainty state, in the sense that Eq. (\[eq:quad\]) becomes an inequality. Not only that, the two variances are also equal: $$\begin{aligned}
(\Delta X)^2 = (\Delta Y)^2 = \frac12\, .\end{aligned}$$ Quantum mechanically, we can reduce the variance in one quadrature at the expense of the other while still obeying Eq. (\[eq:quad\]). We can write this as $$\begin{aligned}
(\Delta X)^2 = \frac{{\rm e}^{-2r}}{2} \qquad\text{and}\qquad (\Delta Y)^2 = \frac{{\rm e}^{2r}}{2} \, ,\end{aligned}$$ where $r$ is called the squeezing parameter. Using these types of light allows us to achieve a measurement precision in $\varphi$ of $$\begin{aligned}
\Delta\varphi \geq \frac{{\rm e}^{-r}}{\sqrt{M \braket{n}}}\, ,\end{aligned}$$ where $\braket{n}$ is the average number of photons in the light probe, and $M$ is the number of times the experiment is repeated [@pezze08]. The advantage of this approach is that it does not require too exotic quantum states such as the [noon]{} state, and the measurement can be achieved by ordinary homodyne detection. This method is proposed as part of Advanced LIGO for the measurements of gravitational waves [@caves81; @aligo13]
![The image of a small circular source exhibits diffraction fringes, and has a smeared out character that makes it difficult to find the exact position and radius of the source. If the image was perfectly smooth we would be able to characterise its parameters from Fourier analysis, but the intensity profile has a fair amount of noise that leads to an error in the estimation of the source dimensions.[]{data-label="fig:aperture"}](aperture.jpg){height="60mm"}
Finally, the same transverse anti-bunching effect that was used to increase the precision of the optical micrometer can be employed to improve the resolution in imaging. Suppose that we wish to image an object that we know is circular (for example a star or a small aperture). We can use a telescope or a microscope and obtain an image like the one shown in figure \[fig:aperture\]. We may infer the radius (or, more precisely, the opening angle) of the sources, as well as the position of the source, by matching the intensity of the light in the imaging plane with a theoretical model of the image (related to the Fourier transform of the source geometry). Given a perfectly smooth intensity profile in the imaging plane we can find the position and radius with arbitrarily high precision. However, as before the intensity of the light in the image plane fluctuates, and this will create a degree of uncertainty in the fitting of the theoretical curve to the data [@perezdelgado12; @pearce15]. If the fluctuations can be suppressed via transverse anti-bunching, the precision of the estimates of the position and radius can achieve the Heisenberg limit [@boyer15].
Quantum computing {#sec:comp}
=================
The last, and arguably most challenging information processing task with single photon qubits is quantum computing. Quantum computers have very stringent noise requirements, which linear optics can in principle meet. The major challenge, however, is in the generation of entanglement. To make matters worse, not all entanglement in created equal.
Entanglement generation between photons
---------------------------------------
One of the key ingredients in quantum information processing is quantum entanglement. For example, in Section \[sec:comm\] we used the maximally entangled Bell states as resources for quantum teleportation. For quantum computing, entanglement is also a key resource. No exponential speed-up can be achieved without it. However, when we deal with photons as our information carriers, we must distinguish between different types of entanglement.
Consider a single photon impinging on a 50:50 beam splitter. There are two input and two output modes for a simple beam splitter, and we can write the mode transformations as $$\begin{aligned}
\hat{a}_1 \to \frac{\hat{b}_1+\hat{b}_2}{\sqrt{2}} \qquad\text{and}\qquad \hat{a}_2 \to \frac{\hat{b}_1-\hat{b}_2}{\sqrt{2}} \, ,\end{aligned}$$ where $\hat{a}_1$ and $\hat{a}_2$ are the annihilation operators for the input modes, and $\hat{b}_1$ and $\hat{b}_2$ are the annihilation operators for the output modes. A single photon entering the beam splitter in mode 1 can then be written as a quantum state transformation $$\begin{aligned}
\ket{1,0}_{12} \to \frac{ \ket{1,0}_{12}+ \ket{0,1}_{12}}{\sqrt{2}}\, .\end{aligned}$$ This state is entangled. It can in principle be used to violate a Bell inequality (even though it would be difficult to implement in practice). The entanglement is between the spatial degree of freedom (mode 1 or 2), and the photon number degree of freedom (0 or 1 photons). In general, quantum states that are not thermal or classical coherent states become entangled when they interact with beam splitters [@koklovett10].
Unfortunately, this type of entanglement is of limited use for quantum computation. To see this, consider a Bell state required for quantum computing: $$\begin{aligned}
\ket{\Phi^+}_{12} = \frac{ \ket{H,H}_{12} + \ket{V,V}_{12}}{\sqrt{2}}\, .\end{aligned}$$ This is a state of two photons with polarisation degree of freedom ($H$ and $V$) in two spatial modes (1 and 2). We can write this in terms of creation operators acting on the vacuum as $$\begin{aligned}
\ket{\Phi^+}_{12} = \frac{1}{\sqrt{2}}\left( \hat{a}_{1,H}^\dagger \hat{a}_{2,H}^\dagger + \hat{a}_{1,V}^\dagger \hat{a}_{2,V}^\dagger \right) \ket{0}\, .\end{aligned}$$ Suppose that we wish to create this state from the separable input state $\ket{H,H}$. The mode transformation that must be implemented is then $$\begin{aligned}
\label{eq:insep}
\hat{a}_{1,H}^\dagger \hat{a}_{2,H}^\dagger \to \frac{1}{\sqrt{2}}\left( \hat{b}_{1,H}^\dagger \hat{b}_{2,H}^\dagger + \hat{b}_{1,V}^\dagger \hat{b}_{2,V}^\dagger \right)\, .\end{aligned}$$ Linear optics is *linear* in the mode transformations, which means that each input mode operator transforms into a linear combination of the output mode operators. In other words, $$\begin{aligned}
\label{eq:sep}
\hat{a}_{1,H}^\dagger \to \sum_{j,s} U_{1j,Hs} \hat{b}_{j,s}\, ,\end{aligned}$$ where $U_{1j,Hs}$ are the elements of a unitary matrix[^3]. Each mode operator is replaced with a sum over mode operators. However, the substitution rule of Eq. (\[eq:sep\]) applied to the left-hand side of Eq. (\[eq:insep\]) can never produce the right-hand side of Eq. (\[eq:insep\]) because the left-hand side is separable into a product of two mode operators, whereas the right-hand side is not. Therefore, linear optics alone cannot be used to create the necessary entanglement for quantum computing.
One potential way around this problem is to use an induced photon-photon interaction, for example using a Kerr nonlinearity. Such a nonlinearity imparts a phase shift on one optical mode that is proportional to the intensity in another mode. At the single-photon level this can act as a coherent switch. Unfortunately, Kerr nonlinearities are inherently noisy and cannot be used for single-photon quantum gates [@shapiro06]. The question is then whether we can use photonic qubits for quantum computing.
The Knill-Laflamme-Milburn protocol
-----------------------------------
The problem was solved in 2000 by Knill, Laflamme and Milburn, in what was to become one of the classic papers in quantum information processing [@klm01]. Instead of a medium-induced photon-photon interaction, the required nonlinearity of the mode transformations is provided by projective measurements. In addition to the photons that are part of a computation, we may send extra *ancilla* photons through a linear optical network of beam splitters and phase shifters. This gives us the freedom to detect photons in very specific output modes, as shown in the example of the nonlinear phase shift circuit in figure \[fig:ns\], also known as the NS gate. Since the number of added ancilla photons is the same as the number of detected photons, the photon number in the input mode does not change once it has passed through the network.
![A linear optical network that implements a nonlinear phase shift (an NS gate) using only beam splitters and projective measurements. Instead of each photon in the input mode accumulating a $-1$ phase shift, only the two-photon component picks up the $-1$ phase shift, leaving the one-photon component unaffected. The implementation requires one ancilla photon in the middle mode, and a detection signature of one photon and no photons in the two detectors in the output. The beam splitters BS1, BS2 and BS3 are not 50:50, but have specially chosen transmission coefficients.[]{data-label="fig:ns"}](ns.pdf){height="25mm"}
Of course, it is not guaranteed that the two detectors will detect one and zero photons, respectively. If it was, there would be no need for detection. This implies that the circuit in figure \[fig:ns\] succeeds only part of the time (in this case, the success probability of the gate is one quarter). This is no good for quantum computation, in which all the circuits must be successful simultaneously. To overcome this problem, Knill, Laflamme and Milburn employed quantum teleportation: Instead of trying to apply the probabilistic gate directly to the quantum information carrying qubits (which cannot be copied and must therefore be handled with care), the gate is applied to one half of an entangled pair. If the gate is successful, the now modified entangled pair is used as the entanglement resource in quantum teleportation of the information carrying qubit. The teleported qubit emerges with the gate applied to it. Knill, Laflamme and Milburn found a way to make the teleportation procedure nearly deterministic, which means that the probabilistic gate can now be applied deterministically to the qubit, and linear optical quantum computing was in principle possible.
Once we can create an NS gate, we can use the Hong-Ou-Mandel effect to create controlled Pauli $\sigma_z$ gates, or CZ gate. These are the two-qubit gates that can create the entanglement necessary for quantum computing. In terms of qubits, the CZ gate operates as follows on the two-qubit states: $$\begin{aligned}
\label{eq:cz}
U_{CZ} \ket{00} = \ket{00}\, , & \quad
U_{CZ} \ket{01} = \ket{01}\, , \cr
U_{CZ} \ket{10} = \ket{10}\, , & \quad
U_{CZ} \ket{11} = -\ket{11}\, .\end{aligned}$$ In other words, when both qubits are in the $\ket{1}$ state, the gate applies a phase shift ${\rm e}^{i\pi} = -1$. To see how this gate can be implemented with two NS gates and the Hong-Ou-Mandel effect, consider the circuit in figure \[fig:cz\]. We can arrange the two incoming qubits Q1 and Q2 in such a way that the $\ket{0}$ states for each qubit—i.e., horizontal polarisation—are mapped onto the top and bottom modes that propagate freely. The $\ket{1}$ states for each qubit are the vertically polarised photons, and will be reflected in the polarising beam splitters (PBS). The photons will combine in the first 50:50 beam splitter. If an input state $\ket{V,V}$ enters this circuit, both photons will meet at the first beam splitter and experience to Hong-Ou-Mandel effect. This means that both photons will exit the beam splitter in a quantum superposition of both photons in the top mode and both photons in the bottom mode. The NS gate, if successful, will then impart a $-1$ phase shift on the two-photon state. The second beam splitter will apply the Hong-Ou-Mandel effect in reverse, such that $$\begin{aligned}
\ket{V,V} ~\to~ \frac{\ket{2V,0}-\ket{0,2V}}{\sqrt{2}} ~\to~ \frac{-\ket{2V,0}+\ket{0,2V}}{\sqrt{2}} ~\to~ -\ket{V,V}\, .\end{aligned}$$ If only one photon enters the first beam splitter, for example because Q1 is in the logical state $\ket{1}$ and Q2 is in the state $\ket{0}$, there will only be one photon going through the NS gates, and there will be no phase shift. Similarly, when both photons are in the top and bottom mode, no photons travel through the NS gates and no phase shift is imported on the quantum state. The result is that the circuit in figure \[fig:cz\] implements the gate in Eq. (\[eq:cz\]). The gate is successful when both NS gates are successful, and the total success probability is therefore $p_{\rm CZ} = (\frac14)^2 = 1/16$. The gate can be applied to qubits in the computation using the teleportation trick described above.
![The controlled-$\sigma_z$ (CZ) gate. Two photonic qubits, Q1 and Q2, enter the interferometer. The modes corresponding to qubit value $\ket{1}$ are sent into a 50:50 beam splitter, the output of which are subject to a nonlinear phase shift (NS). If there is a photon in each input mode of the beam splitter, the Hong-Ou-Mandel effect guarantees that the two photons will either go both through the top NS gate or through the bottom NS gate. Consequently, these photons will pick up a $-1$ phase. The second 50:50 beam splitter will separate the two photons again into one photon in each output mode of the beam splitter.[]{data-label="fig:cz"}](cz.pdf){height="30mm"}
It is important for the operation of the CZ gate that the Hong-Ou-Mandel effect works perfectly. This means that the two photons must be indistinguishable in every respect, including frequency, polarisation and mode shape. Imperfections in the photon source, the beam splitters, or the NS gate will create faulty gates that can ruin the computation.
Measurement-based quantum computing
-----------------------------------
The Knill-Laflamme-Milburn protocol is a type of measurement-based quantum computing, in which the computations are induced by measurements and feed-forward processing of the measurement outcomes. As a practical scheme, however, it has many downsides: a single entangling gate needs tens to hundreds of thousands of ancilla photons; the detectors must be nearly perfectly efficient and be able to tell the difference between 0, 1 and 2 photons; all the photons must be identical to an extremely high degree; and the feed-forward procedure requires high-quality, low-loss optical switches. In addition, while the feed-forward takes place, the photons must be stored in a quantum memory.
The first problem, the resource count, can be mitigated if instead of single photon ancilla states we use entangled photons from the start. This requires a reliable source of photons in one of the Bell states (it does not really matter which one). Traditionally, entangled photon pairs have been generated using a process called Spontaneous Parametric Down-Conversion (SPDC), in which a high energy laser pumps a nonlinear crystal. The photons of the laser have a very small probability of “breaking up” in the crystal into two photons of lower energy. Depending on the arrangement, these two photons can be created in an entangled polarisation state $(\ket{H,H}+\ket{V,V})/\sqrt{2}$ [@kwiat95]. Alternatively, we can engineer quantum dot structures that create entangled photons on demand [@stevenson06] as shown in figure \[fig:spdc\]. The dot can be placed in a Bragg stack that sends photons in the vertical direction, and a prism separates the different frequency components. While SPDC is clean and straightforward to implement, the rate of photon pair production is extremely low, and occasionally two or more pairs are created. The quantum dot approach would therefore be preferable, but it is currently still in the research stage.
![Creating two-photon entangled states. a) Spontaneous parametric down-conversion creates photon pairs probabilistically when one or more pump photons break down into two entangled photons whose frequencies sum to the pump frequency. This can be achieved for example in a BBO crystal. Two BBO crystals back-to-back rotated ninety degrees with respect to each other will create polarisation-entangled photon pairs. b) A single quantum dot in a Bragg stack can be excited and decay along two different paths, creating polarisation-entangled photons on demand. However, while this gives in principle superior performance over SPDC, the fabrication challenges are significant and this approach is currently still in the research stage.[]{data-label="fig:spdc"}](pairs.pdf){width="110mm"}
Assuming that we have a reliable two-photon source we can design a new architecture for linear optical quantum computing that requires significantly fewer resources. The key is still to use gate teleportation, but instead of the complicated states required by the Knill-Laflamme-Milburn protocol we create conceptually (and practically) simpler cluster states. Consider two polarised photons in the (unnormalised) entangled state $$\begin{aligned}
\nonumber
\ket{H,H} + \ket{H,V} + \ket{V,H} - \ket{V,V} .\end{aligned}$$ We can measure the first photon in a special basis “$\pm\alpha$” with eigenstates $$\begin{aligned}
\ket{+\alpha} \equiv \frac{\ket{H} + {\rm e}^{i\alpha}\ket{V}}{\sqrt{2}} \qquad\text{and}\qquad \ket{-\alpha} \equiv \frac{\ket{H} - {\rm e}^{i\alpha}\ket{V}}{\sqrt{2}}\, .\end{aligned}$$ After finding, say the measurement outcome $+\alpha$ the quantum state of the remaining photon is $$\begin{aligned}
\ket{\psi_{\rm out}} = \frac{1+{\rm e}^{-i\alpha}}{2} \ket{H} + \frac{1-{\rm e}^{-i\alpha}}{2} \ket{V} = H U_Z(\alpha) \left( \frac{\ket{H}+\ket{V}}{\sqrt{2}} \right)\, ,\end{aligned}$$ where the last equality is true up to an unobservable global phase, and $U_Z(\alpha) = \exp(-i\alpha\sigma_z/2)$ is a rotation generated by the Pauli $\sigma_z$ operator. In other words, measuring the first photon in the special basis “$\pm\alpha$” produces a unitary gate $H U_Z(\alpha)$ on the second photon. We can daisy-chain this process by using a four-photon entangled state and measuring the first three photons in special bases defined by successive angles $\alpha$, $\beta$, and $\gamma$. The resulting operation on the final photon is the unitary gate $$\begin{aligned}
U(\alpha,\beta,\gamma) = H U_Z(\gamma) H U_Z(\beta) H U_Z(\alpha) = H U_Z(\gamma) U_X(\beta) U_Z(\alpha)\, ,\end{aligned}$$ where we have used that $H U_Z H = U_X$, a rotation generated by the Pauli $\sigma_x$ operator. Such a gate can implement any single qubit operation given judiciously chosen values of $\alpha$, $\beta$, and $\gamma$. Depending on the (probabilistic) measurement outcome ($\pm\alpha$), the subsequent measurement angle must be chosen as $\pm\beta$, and the measurement outcome $\pm\beta$ determines the angle $\pm\gamma$. This forward dependence of the measurement angles creates a definite direction of the computation.
The four-photon state can be graphically represented as a linear (one-dimensional) graph, in which the nodes denote the photons and the edges denote entanglement created by CZ gates between the photons: $$\begin{aligned}
\label{eq:1Dcluster}
\ket{\psi_{\rm 1D}} = \vcenter{\hbox{\includegraphics[width=40mm]{linear_cluster.pdf}}}\end{aligned}$$ We can create two-dimensional graphs in which the vertical entanglement connections represent entangling gates: $$\begin{aligned}
\label{eq:2Dcluster}
\ket{\psi_{\rm 2D}} = \vcenter{\hbox{\includegraphics[width=40mm]{2D_cluster.pdf}}}\end{aligned}$$ These structures can be mapped onto any quantum computational circuit and are therefore universal for quantum computation. The entangled states in Eqs. (\[eq:1Dcluster\]) and (\[eq:2Dcluster\]) are called *cluster states*, and the method is called *one-way* quantum computation [@raussendorf01]. Since photon measurements can in principle be carried out efficiently, the challenge is to create the required cluster states.
![Fusion gates for creating cluster states. a) Type-I fusion allows us to take two Bell pairs and create an entangled three-photon state by measuring a single photon in detector D after mixing on a linear polarising beam splitter (PBS1) and a half wave plate (HWP). b) Type-II fusion is a form of entanglement swapping that uses a circular polarising beam splitter (PBS2) and two detection events at D1 and D2. Here, the fusion gate is applied to two Bell pairs, which results in another pair. In larger systems the type-II fusion gate can be used to create bigger cluster states without the noise drawbacks of the type-I fusion gate.[]{data-label="fig:fusion"}](fusion.pdf){width="110mm"}
A particularly promising way to create large cluster states is to use so-called *fusion gates*, shown in figure \[fig:fusion\] [@browne05]. The entanglement is created by a variation of a probabilistic Bell measurement for polarised photons, and can be implemented with linear optical elements such as half wave plates and polarising beam splitters. Since the creation of the cluster state occurs before we introduce the quantum computation via measurements, we are at liberty to create the cluster state in a probabilistic manner and purify the result until we have the desired fidelity.
There are two types of fusion gates, type-I and type-II. The first type purports to detect a single photon, leaving the three remaining photons in an entangled linear cluster state. This fusion gate can be described mathematically by the operator $$\begin{aligned}
\label{eq:type1}
\mathcal{F}_{{\rm I},\pm} = \ket{H}\bra{H,H} \pm \ket{V}\bra{V,V}\, ,\end{aligned}$$ where the sign $\pm$ is determined by the polarisation of the photon measured in detector D. Similarly, the type-II fusion gate can be described by the operator $$\begin{aligned}
\label{eq:type2}
\mathcal{F}_{\rm II} =
\begin{cases}
\bra{H,V} + \bra{V,H} & \text{for outcome $(H,H)$ or $(V,V)$ in D1 and D2,}\cr
\bra{H,H} + \bra{V,V} & \text{for outcome $(H,V)$ or $(V,H)$ in D1 and D2.}
\end{cases} \end{aligned}$$ Starting with Bell pairs, the type-II fusion gates clearly cannot grow large clusters on their own since they remove two photons from the entangled state. However, type-II has a much more beneficial behaviour that type-I when the fusion gate fails. The best strategy is therefore to create three-photon entangled states using type-I gates, and subsequently create larger cluster states using only type-II fusion gates. Improvements in the architecture of linear optical quantum computers continue to be made, and in a recent proposal the need for quantum memories is reduced by using a percolation-based *ballistic* approach [@gimeno15].
The future of optical quantum information processing {#sec:future}
====================================================
One of the aims of this review is to show that different quantum information processing tasks have different technological requirements. For quantum computing, we need sources that produce Bell pairs on demand with high efficiency and, perhaps more importantly, identical mode shapes to accommodate the Hong-Ou-Mandel effect. Moreover, the Bell pairs must be very close to pure states. The photodetectors must have a high detection efficiency, so that photon loss in the course of the computation remains low. The linear optical components must similarly be low-loss and accurate. The polarising beam splitters must have nearly perfect transmission or reflection for the polarised photons, and beam splitters must have carefully calibrated transmission coefficients. The exact allowed tolerances of the components of a linear optical quantum computer will be determined by the error correction mechanism that is employed. Finally, the feed-forward nature of linear optical quantum computing means that we require fast, low-loss optical switches. This is currently a major challenge.
An actual implementation of a linear optical quantum computer will not use bulk optical elements, but rather have a chip-based architecture in which microscopic waveguides are wired into programmable circuits. Beam splitters can then be constructed from evanescently coupled waveguides. By adjusting the distance between the waveguides, the transmission coefficient can in principle be carefully calibrated. Recently, photon sources have been placed in or on top of waveguides, which allows for directional coupling of the photon into the waveguide depending on the spin of the photon source [@zayats13; @lodahl15; @coles15]. This new technology can be employed for alternative Bell pair generation methods based on photon which-path erasure and spin readout.
Quantum metrology is similarly challenging to implement. It is known that in the limit of large photon numbers the Heisenberg limit is extremely sensitive to noise [@rafal12]. This means that some type of quantum error correction must be employed in order to achieve the Heisenberg limit, and this places the practical challenge on a par with the construction of a full-scale quantum computer. On the other hand, quantum metrology techniques that do not achieve the Heisenberg limit but that nonetheless improve on the shot-noise limit by a constant factor will still be very welcome. Squeezed (quantum) light will be used in the next generation gravitational wave detectors [@aligo13], especially now that gravitational waves have been observed directly [@abbott16].
Quantum communication is arguably the least challenging task to implement in practice. Quantum key distribution requires single-photon sources that may not be fully indistinguishable from each other. However, much care must be taken in the prevention of side-channel detection, in which an eavesdropper can infer or influence the polarisation of a photon via classical methods (e.g., monitoring the photon source for tell-tale signals, etc.). These can be difficult engineering questions that must be solved. Extending quantum communication over longer distances will require quantum repeaters. These devices are much more challenging to build, and require multi-photon entanglement, high-efficiency photodetectors, and generally rather large optical circuits. While repeaters do not have strict fault-tolerance requirements, the techniques that will make them work will likely be similar to those of full-scale quantum computers (indistinguishable photons, fast low-loss switches, etc.).
To conclude, optical quantum information processing presents various physical and engineering challenges for different tasks. Some processes, such as quantum key distribution are currently being implemented in commercial products, while others are still very much in the research stage. I have shown that different tasks have very similar physical requirements at different stages of development, which makes it more likely that as our understanding and mastery of Nature continues, even the more exotic applications will find their way into working devices.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank Nikola Prtljaga for providing me with the data that was used in figure 4.
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[^1]: We can also define the electric field operator over a continuum $\mathbf{k}$, in which case the sum over $\mathbf{k}$ becomes an integral and the Kronecker delta becomes a Dirac delta $\delta(\mathbf{k} - \mathbf{k}')$.
[^2]: Metal foils a few micrometres thick are readily available.
[^3]: There is a one-to-one relation between unitary matrices and linear optical networks that consist of beam splitters, phase shifters and polarisation rotations [@reck94].
|
---
abstract: 'We present an approach for learning simple algorithms such as copying, multi-digit addition and single digit multiplication directly from examples. Our framework consists of a set of *interfaces*, accessed by a *controller*. Typical interfaces are 1-D tapes or 2-D grids that hold the input and output data. For the controller, we explore a range of neural network-based models which vary in their ability to abstract the underlying algorithm from training instances and generalize to test examples with many thousands of digits. The controller is trained using $Q$-learning with several enhancements and we show that the bottleneck is in the capabilities of the controller rather than in the search incurred by $Q$-learning.'
author:
- |
Wojciech Zaremba$^{\S}$\
New York University\
`woj.zaremba@gmail.com`\
Tomas Mikolov Armand Joulin Rob Fergus\
Facebook AI Research\
`{tmikolov,ajoulin,robfergus}@fb.com`\
bibliography:
- 'bibliography.bib'
title: Learning Simple Algorithms from Examples
---
Introduction
============
Many every day tasks require a multi-step interaction with the world. For example, picking an apple from a tree requires visual localization of the apple; extending the arm and then fine muscle control, guided by visual feedback, to pluck it from the tree. While each individual procedure is not complex, the task nevertheless requires careful sequencing of operations across both visual and motor systems.
This paper explores how machines can learn algorithms involving a similar compositional structure. Since our emphasis is on learning the correct sequence of operations, we consider the domain of arithmetic where the operations themselves are very simple. For example, although learning to add two digits is straightforward, solving addition of two multi-digit numbers requires precise coordination of this operation with movement over the sequence and recording of the carry. We explore a variety of algorithms in this domain, including complex tasks involving addition and multiplication. Our approach formalizes the notion of a central controller that interacts with the world via a set of interfaces, appropriate to the task at hand. The controller is a neural network model which must learn to control the interfaces, via a set of discrete actions (e.g. “move input tape left”, “read”, “write symbol to output tape”, “write nothing this time step” ) to produce the correct output for given input patterns. Specifically, we train the controller from large sets of examples of input and output patterns using reinforcement learning. Our reward signal is sparse, only being received when the model emits the correct symbol on the output tape.
We consider two separate settings. In the first, we provide supervision in the form of ground truth actions. In the second, we train only with input-output pairs (i.e. no supervision over actions). While we are able to solve all the tasks in the latter case, the supervised setting provides insights about the model limitations and an upper bound on the performance. We evaluate our model on sequences far longer than those present during training. Surprisingly, we find that controllers with even modest capacity to recall previous states can easily overfit the short training sequences and not generalize to the test examples, even if the correct actions are provided. Even with an appropriate controller, off-the-shelf $Q$-learning fails on the majority of our tasks. We therefore introduce a series of modifications that dramatically improve performance. These include: (i) a novel dynamic discount term that makes the reward invariant to the sequence length; (ii) an extra penalty that aids generalization and (iii) the deployment of Watkins Q-lambda [@sutton1998reinforcement].
We would like to direct the reader to the video accompanying this paper ([](https://youtu.be/GVe6kfJnRAw)). This gives a concise overview of our approach and complements the following explanations. Full source code for this work can be found at [](https://github.com/wojzaremba/algorithm-learning).
Model
=====
Our model consists of an RNN-based controller that accesses the environment through a series of pre-defined interfaces. Each interface has a specific structure and set of actions it can perform. The interfaces are manually selected according to the task (see [Section \[sec:tasks\]]{}). The controller is the only part of the system that learns and has no prior knowledge of how the interfaces operate. Thus the controller must learn the sequence of actions over the various interfaces that allow it to solve a task. We make use of three different interfaces:
[**Input Tape:**]{} This provides access to the input data symbols stored on an “infinite” 1-D tape. A read head accesses a single character at a time through the *read* action. The head can be moved via the *left* and *right* actions.
[**Input Grid:**]{} This is a 2D version of the input tape where the read head can now be moved by actions *up*, *down*, *left* and *right*.
[**Output Tape:**]{} This is similar to the input tape, except that the head now writes a single symbol at a time to the tape, as provided the controller. The vocabulary includes a no-operation symbol (NOP) enabling the controller to defer output if it desires. During training, the written and target symbols are compared using a cross-entropy loss. This provides a differentiable learning signal that is used in addition to the sparse reward signal provided by the $Q$-learning.
[Fig. \[fig:interfaces\]]{}(a) shows examples of the input tape and grid interfaces. [Fig. \[fig:interfaces\]]{}(b) gives an overview of our controller–interface abstraction and [Fig. \[fig:interfaces\]]{}(c) shows an example of this on the addition task (for two time steps).
![(a): The input tape and grid interfaces. Both have a single head (gray box) that reads one character at a time, in response to a *read* action from the controller. It can also move the location of the head with the *left* and *right* (and *up*, *down*) actions. (b) An overview of the model, showing the abstraction of controller and a set of interfaces (in our experiments the memory interface is not used). (c) An example of the model applied to the addition task. At time step $t_1$, the controller, a form of RNN, reads the symbol $4$ from the input grid and outputs a no-operation symbol ($\oslash$) on the output tape and a *down* action on the input interface, as well as passing the hidden state to the next timestep.[]{data-label="fig:interfaces"}](figs/tape_grid_1.png "fig:"){width="\linewidth"}\
(a)
\
(b)
![(a): The input tape and grid interfaces. Both have a single head (gray box) that reads one character at a time, in response to a *read* action from the controller. It can also move the location of the head with the *left* and *right* (and *up*, *down*) actions. (b) An overview of the model, showing the abstraction of controller and a set of interfaces (in our experiments the memory interface is not used). (c) An example of the model applied to the addition task. At time step $t_1$, the controller, a form of RNN, reads the symbol $4$ from the input grid and outputs a no-operation symbol ($\oslash$) on the output tape and a *down* action on the input interface, as well as passing the hidden state to the next timestep.[]{data-label="fig:interfaces"}](figs/full2.png "fig:"){width="\linewidth"}\
(c)
For the controller, we explore several recurrent neural network architectures: two different sizes of 1-layer LSTM [@hochreiter1997long], a gated-recurrent unit (GRU)[@cho2014learning] and a vanilla feed-forward network. Note that RNN-based models are able to remember previous network state, unlike the the feed-forward network. This is important because some tasks explicitly require some form of memory, e.g. the carry in addition.
The simple algorithms we consider (see [Section \[sec:tasks\]]{}) have deterministic solutions that can be expressed as a finite state automata. Thus during training we hope the controller will implicitly learn the correct automata from the training samples, since this would ensure generalization to sequences of arbitrary length. On some tasks like reverse, we observe a higher-order form of over-fitting: the model learns to solve the training tasks correctly and generalizes successfully to test sequences of the same length (thus is not over-fitting in the standard sense). However, when presented with longer test sequences the model fails completely. This suggests that the model has converged to an incorrect local minima, one corresponding to an alternate automata which have an implicit awareness of the sequence length of which they were trained. See [Fig. \[fig:automata\]]{} for an example of this on the reverse task. Note that this behavior results from the controller, not the learning scheme, since it is present in both the supervised ([Section \[sec:supervised\]]{}) and $Q$-learning settings ([Section \[sec:qlearn\]]{}). These experiments show the need to carefully adjust the controller capacity to prevent it learning any dependencies on the length of training sequences, yet ensuring it has enough state to implement the algorithm in question.
As illustrated in [Fig. \[fig:interfaces\]]{}(c), the controller passes two signals to the output tape: a discrete action (move left, move right, write something) and a symbol from the vocabulary. This symbol is produced by taking the max from the softmax output on the top of the controller. In training, two different signals are computed from this: (i) a cross-entropy loss is used to compare the softmax output to the target symbol and (ii) a discrete 1/0 reward if the symbol is correct/incorrect. The first signal gives a continuous gradient to update the controller parameters via backpropagation. Leveraging the reward requires reinforcement learning, since many actions might occur before a symbol is written to the output tape. Thus the action output of the controller is trained with reinforcement learning and the symbol output is trained by backpropagation.
Tasks {#sec:tasks}
=====
We consider six different tasks: copy, reverse, walk, multi-digit addition, 3 number addition and single digit multiplication. The input interface for copy and reverse is an input tape, but an input grid for the others. All tasks use an output tape interface. Unless otherwise stated, all arithmetic operations use base 10. Examples of the six tasks are shown in [Fig. \[fig:tasks\]]{}. [**Copy:**]{} This task involves copying the symbols from the input tape to the output tape. Although simple, the model still has to learn the correspondence between input and output symbols, as well as executing the move right action on the input tape.
[**Reverse:**]{} Here the goal is to reverse a sequence of symbols on the input tape. We provide a special character “r” to indicate the end of the sequence. The model must learn to move right multiple times until it hits the “r” symbol, then move to the left, copying the symbols to the output tape. [**Walk:**]{} The goal is to copy symbols, according to the directions given by an arrow symbol. The controller starts by moving to the right (suppressing prediction) until reaching one of the symbols $\uparrow, \downarrow, \leftarrow$. Then it should change it’s direction accordingly, and copy all symbols encountered to the output tape.
[**Addition:**]{} The goal is to add two multi-digit sequences, provided on an input grid. The sequences are provided in two adjacent rows, with their right edges aligned. The initial position of the read head is the last digit of the top number (i.e. upper-right corner). The model has to: (i) memorize an addition table for pairs of digits; (ii) learn how to move over the input grid and (iii) discover the concept of a carry.
[**3 Number Addition:**]{} As for the addition task, but now three numbers are to be added. This is more challenging as the reward signal is less frequent (since more correct actions must be completed before a correct output digit can be produced). Also the carry now can take on three states (0, 1 and 2), compared with two for the 2 number addition task.
[**Single Digit Multiplication:**]{} This involves multiplying a single digit with a long multi-digit number. It is of similar complexity to the 2 number addition task, except that the carry can take on more values $\in [0, 8]$.
Related Work {#sec:related}
============
A variety of recent work has explored the learning of simple algorithms. Many of them are different embodiments of the controller-interface abstraction formalized in our model. The Neural Turing Machine (NTM) [@graves2014neural] uses a modified LSTM [@hochreiter1997long] as the controller, and has three inferences: sequential input, delayed output and a differentiable memory. The model is able to learn simple algorithms including copying and sorting. The Stack RNN [@joulin2015inferring] has an RNN controller and three interfaces: sequential input, a stack memory and sequential output. The learning of simple binary patterns and regular expressions is demonstrated. A closely related work to this is [@Das92], which was recently extended in the Neural DeQue [@grefenstette2015learning] to use a list instead. End-to-End Memory Networks [@sukhbaatar2015weakly] use a feed-forward network as the controller and interfaces consisting of a soft-attention input, plus a delayed output (by a fixed number of “hops”). The model is applied to simple Q&A tasks, some of which involve logical reasoning. In contrast, our model automatically determines when to produce output and uses more general interfaces.
However, most of these approaches use continuous interfaces that permit training via back-propagation of gradients. Our approach differs in that it uses discrete interfaces thus is more challenging to train since as we must rely on reinforcement learning instead. A notable exception is the Reinforcement Learning Neural Turing Machine (RLNTM) [@zaremba2015reinforcement] which is a version of the NTM with discrete interfaces. The Stack-RNN [@joulin2015inferring] also uses a discrete search procedure for its interfaces but it is unclear how this would scale to larger problems. The problem of learning algorithms has its origins in the field of program induction [@nordin1997evolutionary; @liang2013learning; @wineberg1994representation; @solomonoff1964formal]. In this domain, the model has to infer the source code of a program that solves a given problem. This is a similar goal to ours, but in quite a different setting. I.e. we do not produce a computer program, but rather a neural net that can operate with interfaces such as tapes and so implements the program without being human-readable. A more relevant work is [@schmidhuber2004optimal] which learns an algorithms for the Hanoi tower problem, using a simple form of program induction and incremental learning components. Genetic algorithms [@Holland; @Goldberg] also can be considered a form of program induction, but are mostly based on a random search strategy rather than a learned one.
Similar to [@mnih2013playing], we train the controller to approximate the $Q$-function. However, we introduce several modifications on top of the classical $Q$-learning. First, we use Watkins $Q(\lambda)$ [@watkinsq; @sutton1998reinforcement]. This helps to overcome a *non-stationary* environment. We are unaware of any prior work that uses Watkins $Q(\lambda)$ for this purpose. Second, we reparametrized $Q$ function, to become invariant to the sequence length. Finally, we penalize $||Q(s, \bullet)||$, which might help to remove positive bias [@hasselt2010double].
Supervised Experiments {#sec:supervised}
======================
To understand the behavior of our model and to provide an upper bound on performance, we train our model in a supervised setting, i.e. where the ground truth actions are provided. Note that the controller must still learn which symbol to output. But this now can be done purely with backpropagation since the actions are known.
To facilitate comparisons of difficulty between tasks, we use a common measure of *complexity*, corresponding to the number of time steps required to solve each task (using the ground truth actions[^1]). For instance, a reserve task involving a sequence of length $10$ requires $20$ time-steps ($10$ steps to move to the “r” and $10$ steps to move back to the start). The conversion factors between sequence lengths and complexity are as follows: copy=1; reverse=2; walk=1; addition=2; 3 row addition=3 and single digit multiplication=1.
For each task, we train a separate model, starting with sequences of complexity 6 and incrementing by 4 once it achieves 100% accuracy on held-out examples of the current length. Training stops once the model successfully generalizes to examples of complexity 1000. Three different cores for the controllers are explored: (i) a 200 unit, 1-layer LSTM; (iii) a 200 unit, 1-layer GRU model and (iii) a 200 unit, 1-layer feed-forward network. An additional linear layer is placed on top of these model that maps the hidden state to either action for a given interface, or the target symbol.
In [Fig. \[fig:supervised\]]{} we show the accuracy of the different controllers on the six tasks for test instances of increasing complexity, up to $20,000$ time-steps. The simple feed-forward controller generalizes perfectly on the copy, reverse and walk tasks but completely fails on the remaining ones, due to a lack of required memory[^2]. The RNN-based controllers succeed to varying degrees, although some variability in performance is observed. Further insight can be obtained by examining the internal state of the controller. To do this, we compute the autocorrelation matrix[^3] $A$ of the network state over time when the model is processing a reverse task example of length $35$, having been trained on sequences of length $10$ or shorter. For this problem there should be two distinct states: move right until “r” is reached and then move left to the start. [Fig. \[fig:autocorrelation\]]{} plots $A$ for models with three different controllers. The larger the controller capacity, the less similar the states are within the two phases of execution, showing how it has not captured the correct algorithm. The figure also shows the confidence in the two actions over time. In the case of the high capacity models, the initial confidence in the move left action is high, but this drops off after moving along the sequence. This is because the controller has learned during training that it should change direction after at most $10$ steps. Consequently, the unexpectedly long test sequence makes it unsure of what the correct action is. By contrast, the simple feed-forward controller does not show this behavior since it is stateless, thus has no capacity to know where it is within a sequence. The equivalent automata is shown in [Fig. \[fig:automata\]]{}(a), while [Fig. \[fig:automata\]]{}(b) shows the incorrect time-dependent automata learned by the over-expressive RNN-based controllers. We note that this argument is empirically supported by our results in [Table \[tab:simple\]]{}, as well as related work such as [@graves2014neural] and [@joulin2015inferring] which found limited capacity controllers to be most effective. For example, in the latter case, the counting and memorization tasks used controllers with just $40$ and $100$ units respectively.
![Test accuracy for all tasks with **supervised** actions over $10$ runs for feed-forward (green), GRU (red) and LSTM (yellow) controllers. In this setting the optimal policy is provided. Complexity is the number of time steps required to compute the solution. Every task has slightly different conversion factor between complexity and the sequence length: a complexity of $10^4$ for copy and walk would mean $10^4$ input symbols; for reverse would correspond to $\frac{10^4}{2}$ input symbols; for addition would involve two $\frac{10^4}{2}$ long numbers; for 3 row addition would involve three $\frac{10^4}{3}$ long numbers and for single digit multiplication would involve a single $10^4$ long number.[]{data-label="fig:supervised"}](figs/combined_sup.pdf){width="1\linewidth"}
{width="\linewidth"}\
\(A) [[right]{}]{}; (B) \[right of=A\] [[left]{}]{}; (A) edge \[loop above\] node (A) edge node (B) (B) edge \[loop above\] node (B);
\
(a)
(A0) [[right[$_1$]{}]{}]{}; (A1) \[right of=A0\] [[right[$_2$]{}]{}]{}; (A2) \[right of=A1\] [[right[$_3$]{}]{}]{}; (A3) \[right of=A2\] [[right[$_4$]{}]{}]{}; (B) \[right of=A3\] [[left]{}]{}; (A0) edge \[out=60,in=120,looseness=1\] node (B) edge node (A1) (A1) edge \[out=50,in=130,looseness=1\] node (B) edge node (A2) (A2) edge \[out=40,in=140,looseness=1\] node (B) edge node (A3) (A3) edge \[out=30,in=150,looseness=1\] node (B) (B) edge \[loop above\] node (B);
\
(b)
\[fig:automata\]
Q-Learning {#sec:qlearn}
==========
In the previous section, we assumed that the optimal controller actions were given during training. This meant only the output symbols need to be predicted and these could be learned via backpropagation. We now consider the setting where the actions are also learned, to test the true capabilities of the models to learn simple algorithms from pairs of input and output sequences.
We use $Q$-learning, a standard reinforcement learning algorithm to learn a sequence of discrete actions that solves a problem. A function $Q$, the estimated sum of future rewards, is updated during training according to: $$\begin{aligned}
\label{eqn:q} Q_{t + 1}(s, a) = Q_t(s, a) - \alpha \big[Q_t(s, a) - \big(R(s') +
\gamma \max_{a}Q_n(s', a)\big)\big]\end{aligned}$$ Taking the action $a$ in state $s$ causes a transition to state $s'$, which in our case is deterministic. $R(s')$ is the reward experienced in the state $s'$. The discount factor is $\gamma$ and $\alpha$ is the learning rate. The another commonly considered quantity is $V(s) = \max_a Q(s, a)$. $V$ is called the value function, and $V(s)$ is the expected sum of future rewards starting from the state $s$. Moreover, $Q^*$ and $V^*$ are function values for the optimal policy.
Our controller receives a reward of $1$ every time it correctly predicts a digit (and $0$ otherwise). Since the overall solution to the task requires all digits to be correct, we terminate a training episode as soon as an incorrect prediction is made. This learning environment is [*non-stationary*]{}, since even if the model initially picks the right actions, the symbol prediction is unlikely to be correct, so the model receives no reward. But further on in training, when the symbol prediction is more reliable, the correct action will be rewarded[^4]. This is important because reinforcement learning algorithms assume stationarity of the environment, which is not true in our case. Learning in non-stationary environments is not well understood and there are no definitive methods to deal with it. However, empirically we find that this non-stationarity can be partially addressed by the use of Watkins $Q(\lambda)$ [@watkinsq], as detailed in [Section \[sec:watkins\]]{}.
Dynamic Discount {#sec:time}
----------------
The purpose of the reinforcement learning is to learn a policy that yields the highest sum of the future rewards. $Q$-learning does it indirectly by learning a $Q$-function. The optimal policy can be extracted by taking ${\arg\!\max}$ over $Q(s, \bullet)$. Note that shifting or scaling $Q$ induces the same policy. We propose to dynamically rescale $Q$ so (i) it is independent of the length of the episode and (ii) $Q$ is within a small range, making it easier to predict.
We define $\hat{Q}$ to be our reparametrization. $\hat{Q}(s, a)$ should be roughly in range $[0, 1]$, and it should correspond to how close we are to $V^*(s)$. $Q$ could be decomposed multiplicatively as ${Q(s, a) = \hat{Q}(s,a) V^*(s)}$. However, in practice, we do not have access to $V^*(s)$, thus instead we use an estimate of future rewards based on the total number of digits left in the sequence. Since every correct prediction yields a reward of $1$, the optimal policy should achieve sum of future rewards equal to the number of remaining symbols to predict. The number of remaining symbols to predict is known and we denote it by $\hat{V}(s)$. Note that this is a form of supervision, albeit a weak one.
Therefore, we normalize the $Q$-function by the remaining sum of rewards left in the task: $$\begin{aligned}
\hat{Q}(s,a) := \frac{Q(s,a)}{\hat{V}(s)}\end{aligned}$$ We assume that $s$ transitions to $s'$, and we re-write the $Q$-learning update equations: $$\begin{aligned}
\hat{Q}(s, a) &= \frac{R(s')}{\hat{V}(s)} + \gamma \max_{a}\frac{\hat{V}(s')}{\hat{V}(s)} \hat{Q}(s', a) \\
\hat{Q}_{t + 1}(s, a) &= \hat{Q}_{t}(s, a) -
\alpha\big[\hat{Q}_{t}(s, a) -
\big(\frac{R(s')}{\hat{V}(s)} + \gamma \max_{a}\frac{\hat{V}(s')}{\hat{V}(s)} \hat{Q}_t(s', a)\big)\big]\end{aligned}$$ Note that $\hat{V}(s) \geq \hat{V}(s')$, with equality if no digit was predicted at the current time-step. As the episode progresses, the discount factor $\frac{\hat{V}(s')}{\hat{V}(s)}$ decreases, making the model greedier. At the end of the sequence, the discount drops to $\frac{1}{2}$.
Watkins $Q(\lambda)$ {#sec:watkins}
--------------------
The update to $Q(s,a)$ in [Eqn. \[eqn:q\]]{} comes from two parts: the observed reward $R(s')$ and the estimated future reward $Q(s',a)$. In our setting, there are two factors that make the former far more reliable than the latter: (i) rewards are deterministic and (ii) the non-stationarity (induced by the ongoing learning of the symbol output by backpropagation) means that estimates of $Q(s,a)$ are unreliable as environment evolves. Consequently, the single action recurrence used in [Eqn. \[eqn:q\]]{} can be improved upon when on-policy actions are chosen. More precisely, let $a_t, a_{t+1}, \dots, a_{t+T}$ be consecutive actions induced by $Q$: $$\begin{aligned}
a_{t+i} = {\arg\!\max}_a Q(s_{t+i},a) \\
s_{t+i} \xrightarrow{a_{t + i}} s_{t + i + 1} \end{aligned}$$ Then the optimal $Q^*$ follows the following recursive equation: $$Q^*(s_t, a_t) = \sum_{i=1}^T \gamma^{i - 1} R(s_{t + i}) + \gamma^T \max_{a}Q^*(s_{t + n + 1}, a)$$ and the update rule corresponding to [Eqn. \[eqn:q\]]{} becomes: $$Q_{t + 1}(s_t, a_t) = Q_t(s_t, a_t) - \alpha\big[Q_t(s_t, a_t) -
\big(\sum_{i=1}^T \gamma^{i-1} R(s_{t + i}) + \gamma^T \max_{a}Q_t(s_{t + n + 1}, a)\big)
\big]$$ This is a special form of Watkins $Q(\lambda)$ [@watkinsq] where $\lambda=1$. The classical applications of Watkins $Q(\lambda)$ suggest choosing a small $\lambda$, which trades-off estimates based on various numbers of future rewards. $\lambda=0$ rolls back to the classical $Q$-learning. Due to reliability of our rewards, we found $\lambda = 1$ to be better than $\lambda < 1$, however this needs further study.
Note that this unrolling of rewards can only take place until a non-greedy action is taken. When using an $\epsilon$-greedy policy, this means we would expect to be able to unroll $\epsilon^{-1}$ steps, on average. For the value of $\epsilon=0.05$ used in our experiments, this corresponds to $20$ steps on average.
Penalty on $Q$-function {#sec:penalty}
-----------------------
After reparameterizing the $Q$-function to $\hat{Q}$ ([Section \[sec:time\]]{}), the optimal $\hat{Q}^*(s,a)$ should be 1 for the correct action and zero otherwise. To encourage our estimate $\hat{Q}(s,a)$ to converge to this, we introduce a penalty that “pushes down” on incorrect actions: $\kappa \| \sum_a \hat{Q}(s,a) - 1 \|^2$. This has the effect of introducing a margin between correct and incorrect actions, greatly improving generalization. We commence training with $\kappa=0$ and make it non-zero once good accuracy is reached on short samples (introducing it from the outset hurts learning).
Reinforcement Learning Experiments
----------------------------------
We apply our enhancements to the six tasks in a series of experiments designed to examine the contribution of each of them. Unless otherwise specified, the controller is a 1-layer GRU model with 200 units. This was selected on the basis of its mean performance across the six tasks in the supervised setting (see [Section \[sec:supervised\]]{}). As the performance of reinforcement learning methods tend to be highly stochastic, we repeat each experiment $10$ times with a different random seed. Each model is trained using $3
\times 10^7$ characters which takes $\sim4$ hrs. A model is considered to have successfully solved the task if it able to give a perfect answer to $50$ test instances, each $100$ digits in length. The GRU model is trained with a batch size of $20$, a learning rate of $\alpha=0.1$, using the same initialization as [@glorot2010understanding] but multiplied by 2. All tasks are trained with the same curriculum used in the supervised experiments (and in [@joulin2015inferring]), whereby the sequences are initially of complexity $6$ (corresponding to 2 or 3 digits, depending on the task) and once 100% accuracy is achieved, increased by $4$ until the model is able to solve validation sequences of length $100$.
For 3-row addition, a more elaborate curriculum was needed which started with examples that did not involve a carry and contained many zero. The test distribution was unaffected. Some examples: ${ \tikz[anchor=base,baseline]{
\node[inner sep=1pt,](h){$\displaystyle\begin{matrix}1\\ 2\\ 2\end{matrix}\mathstrut$};
\draw(h.south east)--(h.south west)--(h.north west)
--(h.north east)--(h.south east);
}}$ ; ${ \tikz[anchor=base,baseline]{
\node[inner sep=1pt,](h){$\displaystyle\begin{matrix}2\\ 0\\ 2\end{matrix}\mathstrut$};
\draw(h.south east)--(h.south west)--(h.north west)
--(h.north east)--(h.south east);
}}$ ; ${ \tikz[anchor=base,baseline]{
\node[inner sep=1pt,](h){$\displaystyle\begin{matrix}8 & 3\\ 3 & 3\\ 3 & 7\end{matrix}\mathstrut$};
\draw(h.south east)--(h.south west)--(h.north west)
--(h.north east)--(h.south east);
}}$ ; ${ \tikz[anchor=base,baseline]{
\node[inner sep=1pt,](h){$\displaystyle\begin{matrix}3 & 2 & 0 & 6 & 9\\ 1 & 3 & 1 & 3 & 1\\ 2 & 8 & 0 & 8 & 3\end{matrix}\mathstrut$};
\draw(h.south east)--(h.south west)--(h.north west)
--(h.north east)--(h.south east);
}}$ ; ${ \tikz[anchor=base,baseline]{
\node[inner sep=1pt,](h){$\displaystyle\begin{matrix}8 & 0 & 1 & 8 & 5 & 2 & 0 & 2 & 1\\ 1 & 3 & 1 & 4 & 0 & 7 & 0 & 5 & 4\\ 3 & 1 & 3 & 2 & 7 & 5 & 0 & 7 & 1\end{matrix}\mathstrut$};
\draw(h.south east)--(h.south west)--(h.north west)
--(h.north east)--(h.south east);
}}$.
We show results for various combinations of terms in [Table \[tab:simple\]]{}. The experiments demonstrate that standard $Q$-learning fails on most of our tasks (first six columns). Each of our additions (dynamic discount, Watkins $Q(\lambda)$ and penalty term) give significant improvements. When all three are used our model is able to succeed at all tasks, providing the appropriate curriculum and controller are used. For the reverse and walk tasks, the default GRU controller failed completely. However, using a feed-forward controller instead enabled the model to succeed, when dynamic discount and Watkins $Q(\lambda)$ was used. As noted above, the 3-row addition required a more careful curriculum before the model was able to learn successfully. Increasing the capacity of the controller (columns 2-4) hurts performance, echoing [Fig. \[fig:autocorrelation\]]{}. The last two columns of [Table \[tab:simple\]]{} show results on test sequences of length 1000. Except for multiplication, the models still generalized successfully.
[|ll||P[0.6cm]{}|P[0.6cm]{}||P[0.6cm]{}|P[0.6cm]{}|P[0.6cm]{}|P[0.6cm]{}|P[0.6cm]{}|P[0.6cm]{}||P[0.6cm]{}|P[0.6cm]{}|]{} & Test length & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 1000 & 1000\
& \#Units & 600 & 400 & 200 & 200 & 200 & 200 & 200 & 200 & 200 & 200\
& Discount $\gamma$ & 1 & 1 & 1 & 0.99 & 0.95 & D & D & D & D & D\
& Watkins $Q(\lambda)$ & [ $\color{Maroon}\times$ ]{}& [ $\color{Maroon}\times$ ]{}& [ $\color{Maroon}\times$ ]{}& [ $\color{Maroon}\times$ ]{}& [ $\color{Maroon}\times$ ]{}& [ $\color{Maroon}\times$ ]{}& [ $\color{LimeGreen}\checkmark$ ]{}& [ $\color{LimeGreen}\checkmark$ ]{}& [ $\color{LimeGreen}\checkmark$ ]{}& [ $\color{LimeGreen}\checkmark$ ]{}\
Task & Penalty & [ $\color{Maroon}\times$ ]{}& [ $\color{Maroon}\times$ ]{}& [ $\color{Maroon}\times$ ]{}& [ $\color{Maroon}\times$ ]{}& [ $\color{Maroon}\times$ ]{}& [ $\color{Maroon}\times$ ]{}& [ $\color{Maroon}\times$ ]{}& [ $\color{LimeGreen}\checkmark$ ]{}& [ $\color{Maroon}\times$ ]{}& [ $\color{LimeGreen}\checkmark$ ]{}\
Copying & & 30% & 60% & 90% & 50% & 70% & 90% & 100% & 100% & 100% & 100%\
& 0% & 0% & 0% & 0% & 0% & 0% & 0% & 0%& 0% & 0%\
& 0% & 0% & 0% & 0% & 0% & 0% & 100% & 90% & 100% & 90%\
& 0% & 0% & 0% & 0% & 0% & 0% & 10% & 90% & 10% & 80%\
& 0% & 0% & 0% & 0% & 0% & 0% & 100% & 100% & 100% & 100%\
2-row Addition & & 10% & 70% & 70% & 70% & 80% & 60% & 60% & 100% & 40% & 100%\
& 0% & 0% & 0% & 0% & 0% & 0% & 0% & 0%& 0% & 0%\
& 0% & 50% & 80% & 40% & 50% & 50% & 80% & 80% & 10% & 60%\
& 0% & 0% & 0% & 0% & 0% & 100% & 100% & 100% & 0% & 0%\
[Fig. \[fig:accuracy\]]{} shows accuracy as a function of test example complexity for standard $Q$-learning and our enhanced version. The difference is performance is clear. At very high complexity, corresponding to $1000$’s of digits, the accuracy starts to drop on the more complicated tasks. We note that these trends are essentially the same as those observed in the supervised setting ([Fig. \[fig:supervised\]]{}), suggesting that $Q$-learning is not to blame. Instead, the inability of the controller to learn an automata seems to be the cause. Potential solutions to this might include (i) noise injection, (ii) discretization of state, (iii) a state error correction mechanism or (iv) regularizing the learned automata using MDL principles. However, this issue, the inability of RNN to perfectly represent an automata can be examined separately from the setting where actions have to be learnt (i.e. in the supervised domain).
![Test accuracy as a function of task complexity ($10$ runs) for standard $Q$-learning (blue) and our enhanced version (dynamic discount, Watkins $Q(\lambda)$ and penalty term). Accuracy corresponds to the fraction of correct test cases (all digits must be predicted correctly for the instance to be considered correct). []{data-label="fig:accuracy"}](figs/combined.pdf){width="\linewidth"}
Further results can be found in the appendices. For the addition task, our model was able to discover multiple correct solutions, each with a different movement pattern over the input tape (see Appendix A). [Table \[tab:base\]]{} in Appendix B sheds light on the trade-off between errors in actions and errors in symbol prediction by varying the base used in the arithmetic operations and hence the size of the target vocabulary. Appendix C explores the use of non-integer rewards. Surprisingly, this slows down training, relative to the 0/1 reward structure.
Discussion
==========
We have explored the ability of neural network models to learn algorithms for simple arithmetic operations. Through experiments with supervision and reinforcement learning, we have shown that they are able to do this successfully, albeit with caveats. $Q$-learning was shown to work as well as the supervised case. But, disappointingly, we were not able to find a single controller that could solve all tasks. We found that for some tasks, generalization ability was sensitive to the memory capacity of the controller: too little and it would be unable to solve more complex tasks that rely on carrying state across time; too much and the resulting model would overfit the length of the training sequences. Finding automatic methods to control model capacity would seem to be important in developing robust models for this type of learning problem.
### Acknowledgments {#acknowledgments .unnumbered}
We wish to thank Jason Weston, Marc’Aurelio Ranzato and Przemysław Mazur for useful discussions, and comments. We also thank Christopher Olah for LSTM figures that have been used in the paper and the accompanying video.
Appendix A: Different Solutions to Addition Task {#appendix-a-different-solutions-to-addition-task .unnumbered}
=================================================
On examination of the models learned on the addition task, we notice that three different solutions were discovered. While they all give the correct answer, they differ in their actions over the input grid, as shown in [Fig. \[fig:addition\_solutions\]]{}.
(280, 100) (0,50)(20,0)[3]{}
(0, 0)[(20, 20)]{} (0, 20)[(20, 20)]{} (20, 0)[(20, 20)]{} (20, 20)[(20, 20)]{}
(34, 30)[(-0.2, -1)[4]{}]{} (29, 10)[(-0.2, 1)[4]{}]{} (25, 30)[(-1, 0)[12]{}]{}
(100, 50)(20,0)[3]{}
(0, 0)[(20, 20)]{} (0, 20)[(20, 20)]{} (20, 0)[(20, 20)]{} (20, 20)[(20, 20)]{}
(29, 30)[(-0.2, -1)[4]{}]{} (14, 10)[(-0.2, 1)[4]{}]{} (25, 10)[(-1, 0)[12]{}]{}
(200, 50)(40,0)[2]{}
(0, 0)[(20, 20)]{} (0, 20)[(20, 20)]{} (20, 0)[(20, 20)]{} (20, 20)[(20, 20)]{}
(30, 30)[(0, -1)[20]{}]{} (30, 10)[(-1, 0)[20]{}]{} (10, 10)[(0, 1)[20]{}]{} (10, 30)[(-1, 0)[20]{}]{}
Appendix B:Reward Frequency vs Reward Reliability {#appendix-breward-frequency-vs-reward-reliability .unnumbered}
=================================================
We explore how learning time varies as the size of the target vocabulary is varied. This trades off reward frequency and reliability. For small vocabularies, the reward occurs more often but is less reliable since the chance of the wrong action sequence yielding the correct result is relatively high (and vice-versa for for larger vocabularies). For copying and reverse tasks, altering the vocabulary size just alters the variety of symbols on the tape. However, for the arithmetic operations this involves a change of base, which influences the task in a more complex way. For instance, addition in base $4$ requires the memorization of digit-to-digit addition table of size $16$ instead of $100$ for the base $10$. [Table \[tab:base\]]{} shows the median training time as a function of vocabulary size. The results suggest that an infrequent but reliable reward is preferred to a frequent but noisy one.
-------------------------------------- --------- ------ ------- ------- ------- ------- ------ ------ ------
Coping 1.4 0.6 0.6 0.6 0.6 0.5 0.6 0.6 0.6
Reverse (FF controller) 6.5 23.8 3.1 3.6 3.8 2.5 2.8 2.0 3.1
Walk (FF controller) 8.7 6.9 6.8 4.0 6.2 5.3 4.4 3.9 11.1
Addition 250.0 30.9 14.5 26.1 21.8 21.9 25.0 23.4 21.1
3-number Addition (extra curriculum) 250.0 61.5 250.0 250.0 112.2 178.2 93.8 79.1 81.9
Single Digit Multiplication invalid 6.2 17.8 20.9 21.4 21.5 22.3 23.3 24.7
-------------------------------------- --------- ------ ------- ------- ------- ------- ------ ------ ------
: Median training time (minutes) over $10$ runs as we vary the base used (hence vocabulary size) on different problems. Training stops when the model successfully generalizes to test sequences of length $100$. The results show the relative importance of reward frequency versus reliability, with the latter being more important. []{data-label="tab:base"}
Appendix C: Reward Structure {#appendix-c-reward-structure .unnumbered}
============================
Reward in reinforcement learning systems drives the learning process. In our setting we control the rewards, deciding when, and how much to give. We now examine various kinds of rewards and their influence on the learning time of our system.
Our vanilla setting gives a reward of $1$ for every correct prediction, and reward $0$ for every incorrect one. We refer to this setting as “0/1 reward”. We consider two other settings in addition to this, both of which rely on the probabilities of the correct prediction. Let $y$ be the target symbol and $p_i = p(y = i), i \in [0, 9]$ be the probability of predicting label $i$.
In setting “Discretized reward”, we sort $p_i$. That gives us an order on indices $a_1, a_2, \dots, a_{10}$, i.e. $p_{a_1} \geq p_{a_2} \geq p_{a_3} \dots \geq p_{a_{10}}$. “Discretized reward” yields reward $1$ iff $a_1 \equiv y$, reward $\frac{1}{2}$ iff $a_2 \equiv y$, and reward $\frac{1}{3}$ iff $a_3 \equiv y$. Otherwise, environment gives a reward $0$. In the “Continuous reward” setting, a reward of $p_y$ is given for every prediction. One could also consider reward $\log(p_y)$, however this quantity is unbounded, and further processing might be necessary to make it work.
Table \[tab:more\_reward\] gives results for the three different reward structures, showing training time for the five tasks (training is stopped once the model generalizes to test sequences of length $100$). One might expect that a continuous reward would convey more information than a discrete one, thus result in faster training. However, the results do not support this hypothesis, as training seems harder with continuous reward than a discrete one. We hypothesize, that the continuous reward makes environment less stationary, which might make $Q$-learning less efficient, although this needs further verification.
-------------------------------------- ------ ------ -------
Coping 0.6 0.6 0.8
Reverse (FF controller) 3.1 3.1 59.7
Walk (FF controller) 11.1 9.5 250.0
Addition 21.1 21.6 24.2
3-number Addition (extra curriculum) 81.9 77.9 131.9
Single Digit Multiplication 24.7 26.5 26.6
-------------------------------------- ------ ------ -------
: Median training time (minutes) for the five tasks for the three different reward structures. “0/1 reward”: the model gets a reward of $1$ for every correct prediction, and $0$ otherwise. “Discretized reward” provides a few more values of reward prediction, if sufficiently close to the correct one. “Continuous reward” gives a probability of correct answer as the reward. See text for details.[]{data-label="tab:more_reward"}
[^1]: In practice, multiple solutions can exist (see Appendix A), thus the measure is approximate.
[^2]: Ammending the interfaces to allow both reading and writing on the same interface would provide a mechanism for long-term memory, even with a feed-forward controller. But then the same lack of generalization issues (encountered with more powerful controllers) would become an issue.
[^3]: Let $h_i$ be the controller state at time $i$, then the autocorrelation $A_{i,j}$ between time-steps $i$ and $j$ is given by $A_{i,j} = \frac{\langle h_i - E, h_j - E \rangle}{\sigma^2}, i, j =
1, \dots, T$ where $E = \frac{\sum_{k=1}^T h_k}{T}, \sigma^2 = \frac{\sum_{k=1}^T
\langle h_k - E, h_k - E\rangle}{T} $. $T$ is the number of time steps (i.e. complexity).
[^4]: If we were to use reinforcement to train the symbol output as well as the actions, then the environment would be stationary. However, this would mean ignoring the reliable signal available from direct backpropagation of the symbol output.
|
---
abstract: 'The electronic structure of lanthanide and actinide compounds is often characterized by orbital ordering of localized $f$-electrons. Density-functional theory (DFT) studies of such systems using the currently available LDA+$U$ method are plagued by significant orbital-dependent self-interaction, leading to erroneous orbital ground states. An alternative scheme that modifies the exchange, not Hartree, energy is proposed as a remedy. We show that our LDA+$U$ approach reproduces the expected degeneracy of $f^1$ and certain $f^2$ states in free ions and the correct ground states in solid PrO$_2$. We expect our method to be useful in studying electronic excitations and entropies in $f$- and heavy-$d$ elements.'
author:
- Fei Zhou
- 'V. Ozolinš'
title: 'Obtaining correct orbital ground states in $f$ electron systems using a nonspherical self-interaction corrected LDA+$U$ method'
---
Introduction
============
Interesting physical phenomena associated with the strongly correlated $f$-electrons in lanthanide and actinide compounds continue to attract lively interest [@Pepper1991CR719; @Dolg1996]. Strong on-site interactions between the $f$-electrons in these materials present serious challenges to modern density-functional theory (DFT) based electronic-structure techniques, causing most approximate functionals, such as the local density (LDA) or generalized gradient approximation (GGA), to fail qualitatively. To overcome the deficiencies of the LDA/GGA in studying $f$-element compounds, several recent studies have employed the self-interaction-corrected LDA [@Perdew1981PRB5048] e.g. in Refs. , the hybrid functional method [@Becke1993JCP1372; @Becke1993JCP5648] in Refs. , or the dynamical mean-field theory (DMFT) [@Metzner1989PRL324] in Refs. . The LDA+$U$ method [@Anisimov1991PRB943] has emerged as a well-established model to deal with strong electron correlations in $d$- and $f$-systems, combining high efficiency with an explicit treatment of correlation within a Hubbard-like model for the localized electrons. This method has been very successful in transition metal oxides (for a review see Ref. ) and has yielded promising results for band gaps in $f$ systems [@Larson2007PRB45114; @DaSilva2007PRB45121; @Tran2008PRB85123]. However, systematic studies of its effectiveness remain inconclusive, with issues of orbital ordering [@Hotta2006RPP2061] and multiple self-consistent solutions attracting heightened attention [@Shick2001JES753; @Larson2007PRB45114; @Jomard2008PRB75125; @Amadon2008PRB155104; @Ylvisaker2009PRB35103]. Here, we show that the currently popular versions of LDA+$U$, by Liechtenstein and co-workers [@Liechtenstein1995PRB5467] and by Dudarev and co-workers [@Dudarev1998PRB1505], respectively, encounter serious difficulties in $f$ systems due to large orbital-dependent self-interaction (SI) effects, which result in an unphysical splitting of up to 0.4 eV between degenerate $f^1$ multiplets. Since the SI errors (SIE) are typically larger than the crystal field (CF) splitting energies, and comparable to the strength of the spin-orbit coupling (SOC), they lead to qualitatively incorrect electronic ground states in solids. We propose a new, orbital SI free form of the LDA+$U$ method that leaves the LDA Hartree term intact and only replaces the LDA exchange with the Hartree-Fock exchange. In our method, the Hartree-Fock exchange term cancels the LDA self-interaction energy to a high degree of accuracy, ensuring near-degeneracy of real- and complex-valued orbitals in free ions and correctly reproducing the $\Gamma_8$ ground state and $\Gamma_8 \rightarrow \Gamma_7$ excitation energies in the PrO$_2$ solid. The accuracy of this functional is sufficient for evaluating high-temperature electronic entropies of $f$ electron systems.
Method and computational details
================================
All DFT calculations were carried out using the VASP package [@Kresse1996PRB11169; @Kresse1999PRB1758] with projected augmented wave (PAW) potentials [@Blochl1994PRB17953], energy cutoff of 450 eV, and without any constraint symmetry or ionic relaxation. For free ions, a 12 Åcubic cell containing one ion and uniform compensating background charge were used. For the PrO$_{2}$ solid, we consider a primitive cell of the fcc supercell (lattice constant of 5.386 Å[@Gardiner2004PRB24415]). The term “LDA+$U$” is used irrespective of the $xc$ functional since the LDA and GGA results are found similar. Each calculation was initialized in a specific atomic orbital and self-consistently converged to either states very close to the initial orbital with the results reported, or distinctly different states with lower energy. SOC was excluded from the calculations unless its inclusion is stated explicitly to make realistic comparison with experiment. Finally, we fix the $U$ parameter in the LDA+$U$ method to 6 eV and leave the discussions of this choice to the end.
Aspherical self-interaction error of LDA+$U$ for $f$-electrons
--------------------------------------------------------------
We begin by showing that the conventional LDA+$U$ approach fails to reproduce the degeneracy of different $| m \rangle$ orbitals of $f^{1}$ ions. First consider real orbitals with angular dependence of real $y^R_{3m}=\sqrt{2} \Re Y_{3m}$ without spin-orbit effects to simplify the presentation of our method. Complex orbitals and SOC are discussed later. Fig. \[fig:E-ion\] shows the energies of different $y^{R}_{3m}$ orbitals (with the exception of $y^R_{31}$, which converges to $y^R_{32}$) in several lanthanide and actinide ions calculated using the LDA+$U$ scheme of Liechtenstein [*et al.*]{} [@Liechtenstein1995PRB5467] with $J=0.5$ and $U=6$ eV. Contrary to the expected degeneracy, the energies of the different $y^R_{3m}$ orbitals differ substantially, up to 0.4 eV, and the $y^R_{31}$ orbital was found unstable and converged to $y^R_{32}$. Varying $J$ between 0 (i.e. the Dudarev scheme [@Dudarev1998PRB1505]) $\sim 1$ eV changes the results by only a few meV.
The above results demonstrate that the conventional LDA+$U$ approach commits large errors of up to 0.4 eV/electron in the predicted relative orbital energies of $f$ electrons. To understand the reasons for the unphysical splitting of the $f^1$ states, we examine the conventional LDA+$U$ total energy functional[@Anisimov1991PRB943]: $$\begin{aligned}
E ^{\mathrm{LDA}+U} = E ^{\mathrm{LDA}} + E_U - E_{\mathrm{dc}},
\label{eq:lda+u} \end{aligned}$$ where the LDA description of the on-site interaction, approximately represented by the so-called double-counting term $E_{\mathrm{dc}}$, is replaced with a Hubbard-like $E_U$. The latter is essentially the Hartree-Fock energy, expressed in a rotationally invariant form by Liechtenstein [*et al.*]{} [@Liechtenstein1995PRB5467] as a sum of the Hartree (H) and exchange (X) terms, $E_U = E_{\mathrm{H}} + E_{\mathrm{X}}$, where $$\begin{aligned}
E_{\mathrm{H}} &=& \frac12 \sum_{ \{m\} }
\langle m, m'' | V_{\mathrm{ee}} | m' ,m''' \rangle n_{m m'} n_{m'' m'''},
\label{eq:EHartree} \\
E_{\mathrm{X}} &=& - \frac12 \sum_{ \{m\} , \sigma}
\langle m, m'' | V_{\mathrm{ee}} | m''' , m' \rangle
n_{m m'}^\sigma n_{m'' m'''}^{\sigma}.
\label{eq:EFock} \end{aligned}$$ The on-site density matrix $n^\sigma_{m m'}$ is obtained by projecting the Kohn-Sham orbitals $\psi_{\alpha}^{\sigma}$ of occupancy $f_{\alpha}^{\sigma}$ onto atomic states $| nlm(m') \rangle$ $$\begin{aligned}
n^\sigma_{m m'} &=& \sum_{\alpha} f_\alpha^\sigma \langle \psi_\alpha^\sigma | nlm' \rangle
\langle nlm | \psi^\sigma_\alpha \rangle,\end{aligned}$$ while the Slater integrals $\langle m m'| V_{\mathrm{ee}}| m''m'''\rangle$ are evaluated in terms of the Gaunt coefficients and the screened Coulomb $U$ and exchange $J$ parameters (the diagonal $m=m'=m''=m'''$ terms are given in Table \[tab:exchange-analytical-LSD\]). A simplified version by Dudarev [*et al.*]{} [@Dudarev1998PRB1505] adopts the $J=0$ limit. States with real $n^\sigma_{m m'}$ are referred to as “real”.
$y^R_{l3}$ $y^R_{l2}$ $y^R_{l1}$ $Y_{l0}$ $Y_{l1}$ $Y_{l2}$ $Y_{l3}$
----- ------------ ------------ ------------ ---------- ---------- ---------- ----------
$l$
1 0.4 0.4 0.1
2 0.571 0.571 0.571 0.186 0.358
3 0.880 0.422 0.807 0.716 0.332 0.194 0.696
$l$
1 0.409 0.409 0.364
2 0.364 0.364 0.356 0.324 0.324
3 0.339 0.328 0.335 0.323 0.298 0.292 0.302
: Hartree energy $E _{\mathrm{H}}$, Eq. \[eq:EHartree\], and LSD exchange energy $E_x^{\mathrm{LSD}}$, Eq. (\[eq:exchange-LSD\]), for one $l$-electron in orbitals with real ($y^R_{lm}$,$Y_{l0}$) and complex ($Y_{lm}$ for $m>0$) angular wavefunctions. \[tab:exchange-analytical-LSD\]
For a free $f^{1}$ ion, the Hartree-Fock energy $E_U$ in Eq. \[eq:lda+u\] naturally vanishes, while $E_{\mathrm{dc}}$ in the Liechtenstein and Dudarev schemes depends only on the number of electrons, $N^\sigma=\sum_m n_{mm}^\sigma$, and not on the type of the occupied orbital. Therefore, Eq. \[eq:lda+u\] becomes $$\begin{aligned}
E ^{\mathrm{LDA}+U}
%= E _{\mathrm{H}} + E ^{\mathrm{LDA}}_{x} + \mathrm{const}
%\approx E _{\mathrm{H}} [\{ n \}] + \mathrm{const} .
= E ^{\mathrm{LDA}} + \mathrm{const} \approx E _{\mathrm{H}} + \mathrm{const}.
\label{eq:E-one-electron}\end{aligned}$$ In the above approximation we assumed 1) the LDA exchange is not sensitive to orbital filling (more on this later) and 2) the Hartree energy difference comes mainly from the on-site Hartree term $E _{\mathrm{H}}$ of eq. \[eq:EHartree\]. The resulting error in the relative orbital energies is then entirely due to the orbital-dependence of the SIE of the LDA, which is reflected in $E _{\mathrm{H}}$. To see the validity of our argument, we list in Table \[tab:exchange-analytical-LSD\] the on-site $E _{\mathrm{H}}$ calculated from eq. \[eq:EHartree\] for atomic orbitals; these expressions are expected to closely approximate the SI for localized orbitals in the LDA+$U$. Even though $E_{\mathrm{H}}$ is identical for all real $p$ or $d$ orbitals, it is orbital-dependent for $f$ multiplets, and in all cases splits the SI energies of real [*vs.*]{} complex orbitals. The predicted ordering of $E_{\mathrm{H}}$ is $y_{32}<
y_{30} < y_{31} < y_{33}$, in agreement with the LDA+$U$ results shown in Fig. \[fig:E-ion\], demonstrating that the unphysical splitting of $f^1$ states in conventional LDA+$U$ is due to orbital-dependent SIE. Note that with real orbitals the problem of orbital-dependent SIE does not affect $p$ or $d$ electrons. We will show later that complex $p$ and $d$ orbitals are affected. According to Table \[tab:exchange-analytical-LSD\], the SIE is proportional to $J$; for typical values of $J$ in the range of $0.1$ to $1$ eV, it is comparable to or even larger than other important on-site effects, such as CF and SOC, which can lead to qualitatively incorrect predictions of electronic ground states in solids by the current LDA+$U$ methods. These deficiencies of the conventional LDA+$U$ approach can be traced back to its treatment of the Hartree and exchange energies. The LDA+$U$ approach replaces the LDA Hartree energy with an on-site model expression $E_{\mathrm{H}}$ given by Eq. \[eq:EHartree\]. Even though the $E_{\mathrm{H}}$ term is capable of reproducing the correct orbital energetics, the LDA+$U$ double-counting energy $E_{\mathrm{dc}}$ is [*orbital-indepedent*]{} and fails to properly account for the orbital-dependence of the LDA SIE in open-shell systems. Similar considerations hold for the orbital-dependence of the LDA exchange energy, which is mainly sensitive to the choice of real [*vs.*]{} complex orbitals (see Table \[tab:exchange-analytical-LSD\]); this factor acquires importance in systems with strong SOC, when the orbitals with a definite value of the total angular momentum $J$ are necessarily complex.
Reformulated LDA+$U$
--------------------
To correct the orbital-dependent SIE in the Hartree and exchange terms, we propose a new formulation of the LDA+$U$ method by modifying only the exchange term of the LDA: $$\begin{aligned}
E ^{\mathrm{LDA}+U} = E ^{\mathrm{LDA}} + E_{\mathrm{X}} - E_{\mathrm{dcX}},
% =E ^{\mathrm{LDA}} +\Delta E .
\label{eq:newlda+u}\end{aligned}$$ where the orbital-dependent Hartree-Fock exchange $E_{\mathrm{X}}$ of Eq. (\[eq:EFock\]) contains a term that approximately cancels the SIE in the LDA Hartree energy; the remainder of the LDA Hartree energy is exact by definition and therefore left unmodified in our approach. The exchange double-counting term $E_{\mathrm{dcX}}$ accounts for the LDA exchange energy and is given by a linear combination of the exchange double-counting in the Liechtenstein scheme and the on-site local-spin-density (LSD) exchange: $$\begin{aligned}
%E_{\mathrm{dcX}} &=& (1-c) E_{\mathrm{dcX}}^{\mathrm{FLL}} + c E_\mathrm{X}^{\mathrm{LSD}} \\
%E_{\mathrm{dcX}} ^{\mathrm{FLL}} &=& \frac12 \sum_{ \sigma} [ U N^\sigma + J N^{\sigma} (N^{\sigma} -1)] \\
E_{\mathrm{dcX}} &=& -\frac{1-c}{2} \sum_{ \sigma} [ U N^\sigma + J N^{\sigma} (N^{\sigma} -1)] + cE_\mathrm{X}^{\mathrm{LSD}}, \label{eq:dcX}\\
% VO Please check - I introduced parentheses around rho^sigma
E_\mathrm{X}^{\mathrm{LSD}} &=& - \frac{3}{2} \left(\frac{3}{4 \pi}\right)^{1/3} \sum_\sigma \int d^3r (\rho^{\sigma})^{4/3} \nonumber \\
%&=& - \frac{3}{2} \left(\frac{3}{4 \pi}\right)^{1/3} \sum_\sigma \int d^3r \left[n_{mm'}^{\sigma} \bar{\psi}_m(r) \psi_{m'} \right] ^{4/3} \nonumber \\
&=& - \frac{3}{2} \left(\frac{3}{4 \pi}\right)^{1/3} \sum_\sigma \int R_{l}^{8/3}(r) r^2 dr d\Omega \left[ n_{mm'}^{\sigma} \bar{Y}_{lm} (\Omega) Y_{lm'} (\Omega) \right] ^{4/3} \nonumber \\
&=& - \left( \frac{ 4 \pi}{2l +1} \right ) ^{1/3} \frac{K}{2} \sum_\sigma \int d\Omega \left[ n_{mm'}^{\sigma} \bar{Y}_{lm} Y_{lm'} \right] ^{4/3} , \nonumber\\
&=& - \left( \frac{ 4 \pi}{2l +1} \right ) ^{1/3} \frac{K}{2} \sum_\sigma \int d\Omega \left[\tilde{\rho}^\sigma(\Omega)\right] ^{4/3}
\label{eq:exchange-LSD}\end{aligned}$$ where $c$ is the interpolation coefficient, $\rho^{\sigma}$ is the charge density of spin component $\sigma$, which can be obtained from the on-site occupation matrix $n_{mm'}^{\sigma}$ as well as radial function $R_{l}(r)$ and spherical $Y_{lm}(\Omega)$, $K$ is the LSD exchange strength parameter, and $\tilde{\rho}$ represents the angular part of $\rho$. Only the $E_\mathrm{X}^{\mathrm{LSD}}$ term in Eq. (\[eq:dcX\]) is orbital-dependent. The linear interpolation is conceptually similar to hybrid functional approaches and serves the purpose of subtracting the orbital-dependence of the LDA exchange energy. The potential corresponding to the correction energy $E_{\mathrm{X}} - E_{\mathrm{dcX}}$, obtained by differentiating with respect to the on-site density matrix $n_{mm'}$, is then $$\begin{aligned}
\Delta V^\sigma_{m m'} &=& \frac{2c}{3} \left( \frac{ 4 \pi}{2l +1} \right ) ^{1/3} K \int d\Omega \left[\tilde{\rho}^\sigma(\Omega)\right] ^{1/3} \bar{Y}_{lm} Y_{lm'} \nonumber \\
&+&(1-c) (\frac{U-J} {2} + n^\sigma J) \delta_{m m'} - \langle m, m'' | V_{ee} | m''' m' \rangle n_{m'' m'''}^{\sigma} \end{aligned}$$ It is possible to reduce the number of independent parameters by requiring that $E_{\mathrm{X}}-E_{\mathrm{dcX}}$ vanishes for full $l$-shells ($n^{\uparrow}_{mm'}=n^{\downarrow}_{mm'}=\delta_{mm'}$), $$E_{\mathrm{X}}-E_{\mathrm{dcX}}= -c(2l+1)(U+2lJ)+ c(2l+1)K =0,$$ which gives $$\begin{aligned}
K=U + 2lJ.\end{aligned}$$ The main advantage of Eqs. (\[eq:newlda+u\])-(\[eq:exchange-LSD\]) is that the LDA self-interaction energy is canceled by the corresponding exchange term in $E_{\mathrm{X}}$. As a result, the proposed method is self-interaction free to high accuracy.
Results and discussions
=======================
In this section, we analyze the parameter dependence of the proposed method and then presents results for the example of PrO$_{2}$ solid.
Determination of parameters to remove aspherical SIE
----------------------------------------------------
We demonstrate orbital degeneracy for free Pr ions with one and two $f$-electrons. Figure \[fig:Pr-EvsJ-U6\]a displays the energy of Pr$^{4+}$ in real atomic orbitals calculated with our method (assuming $c=0$) as a function of the exchange parameter $J$. At $J=0$, a splitting of more than 0.3 eV is found, similar to the behavior of the original LDA+$U$ in Fig. \[fig:E-ion\]. The splitting is reduced by increasing $J$ and at the optimal value of $J^{\mathrm{o}}=0.783$ eV, it is less than 40 meV, i.e., the four real orbitals $y^R_{3m}$ are almost degenerate. The $y^R_{31}$ orbital can only be stabilized in the vicinity of $J^{\mathrm{o}}$, relaxing otherwise to the more stable $y^R_{32}$ or $y^R_{33}$. Hence, just one point for $y^R_{31}$ is shown in the inset of Fig. \[fig:Pr-EvsJ-U6\]a.
The energy of the Pr$^{3+}$ ion ($f^{2}$) is shown in Fig. \[fig:Pr-EvsJ-U6\]b (also at $c=0$). Consider three distinct $f^2$ states with $S=1$ and degenerate Hartree-Fock energy $E_U$. Using the basis defined by real-valued spherical harmonics, $\{ y^I_{l|m|}=\sqrt{2} \Im Y_{l|m|} (-l \le m < 0$), $Y_{l0}$, $y^R_{lm} (0 < m \le l)\}$ (shown for $l=3$ in Fig. \[fig:energy-levels-WF\]b), the first of these states, designated by $\phi_{13}$, has electrons in orbitals $y^I_{31}$ and $y^I_{33}$, or $n^\sigma_{mm'}=0$ except $n^\uparrow_{11}=n^\uparrow_{33}=1$, while the other two $f^{2}$ states, designated by $\phi_{14}$ and $\phi_{15}$, correspond to $n^\uparrow_{11}=n^\uparrow_{44}=1$ and $n^\uparrow_{11}=n^\uparrow_{55}=1$, respectively. Their angular wavefunctions are shown in Fig. \[fig:Pr-EvsJ-U6\]b. Similar to the $f^{1}$ case, the energy splitting is large at $J=0$ and gets reduced to less than 30 meV at the optimal value $J^{\mathrm{o}}$. Note that $\phi_{15}$ can be stabilized only for $J \gtrsim J^{\mathrm{o}}$.
So far, we have used $c=0$, assuming that the LSD exchange functional is insensitive to the orbital and can be ignored. The lower part of Table \[tab:exchange-analytical-LSD\] proves this assumption for the real orbitals: $E_{\mathrm{X}}^{\mathrm{LSD}}$ varies by less than $0.02
K$. However, Table \[tab:exchange-analytical-LSD\] also shows that $E_{\mathrm{X}}^{\mathrm{LSD}}$ of complex orbitals is substantially lower (by $\sim 0.3 K$), indicating a large lowering of the exchange energy in states with nonzero orbital current. Since $E_{\mathrm{X}}^{\mathrm{LSD}} \sim - \rho^{4/3}$ is concave, it favors inhomogeneous charge distributions (such as real orbitals compared to complex ones) and therefore the LDA exchange energies in Table \[tab:exchange-analytical-LSD\] of real orbitals are lower than those for complex orbitals. The difference may play an important role in systems with strong SOC, when the resulting electronic states are complex combinations of real $y_{lm}$’s with the orbital angular momentum unsuppressed. In Fig. \[fig:Evsc\], we show the dependence of the energies of real- and complex-valued orbitals for Pr$^{4+}$ on the mixing coefficient $c$ in Eq. (\[eq:exchange-LSD\]), using the optimal value of the exchange parameter, $J^{\mathrm{o}}$. It is seen that at $c=0$, the energies of real and complex orbitals differ by more than $0.2$ eV due to their different LSD exchange, and the spurious splitting is minimized to approximately 70 meV at the optimal $c\approx 0.6$.
In our approach, the $J$ and $c$ parameters are [*a priori*]{} determined by the physical requirement of degeneracy once the $U$ parameter is given (6 eV in this work). They hardly change when $U=4$ eV is used, suggesting that our method is relatively insensitive to the choice of $U$.
Eigenstates of PrO$_{2}$ without SOC
------------------------------------
[|c|c|c|c|]{} & $Y_{30} \ (t_{1u})$ &$y^R_{32} \ (t_{2u})$ &$y^I_{32}\ (a_{2u})$\
\
& **[-3]{} & 1 & 6\
\
Liechtenstein & **[-23.848]{} & **[-23.843]{} &-23.488\
Dudarev &-23.693 & **[-23.877]{} &-23.458\
This work & **[-24.260]{} &-24.128 &-23.834**********
Finally, we demonstrate the advantages of our method for extended solids by considering PrO$_2$ in the cubic fluorite structure. The Pr$^{4+}$ ion is coordinated by eight oxygen atoms in a cube. Figure \[fig:energy-levels-WF\] shows the $f^1$ energy level splitting scheme in the presence of cubic CF and SOC. Without SOC, the cubic CF splits the $f^1$ states into the $t_{1u}$ ground state and $t_{2u}$, $a_{2u}$ excited states (see Fig. \[fig:energy-levels-WF\]a,c). Table \[tab:PrO2\] lists the CF eigenvalues of these states (small 6th-order CF ignored), and the calculated LDA+$U$ energies using the conventional approaches and our new scheme at the optimal values of $J=0.783$ eV and $c=0.6$. The conventional schemes predict orbital enegies that deviate dramatically from the expected CF order: the Liechtenstein approach predicts almost degenerate $t_{1u}$ and $t_{2u}$, while $t_{2u}$ is the ground state in the Dudarev method. In contrast, our new method successfully finds the correct $t_{1u}$ ground state.
Eigenstates of PrO$_{2}$ with SOC
---------------------------------
The physics of orbital ordering in $f$ systems is affected by strong relativistic effects [@Hotta2006RPP2061], necessitating the inclusion of SOC to make direct comparisons with experiment. Including SOC, our method predicts that the energies of the CF-degenerate $\Gamma_8^a$ and $\Gamma_8^b$, and the excited $\Gamma_7$ states in PrO$_2$ (Fig. \[fig:energy-levels-WF\]d) are 0 (reference), 69 and 142 meV, respectively. The spurious 69 meV splitting between the two degenerate $\Gamma_8$ states is consistent with the accuracy shown in Fig. \[fig:Evsc\]. Neglecting Jan-Teller lattice distortions and magnetic ordering effects, we estimate that the $\Gamma_7/\Gamma_8$ CF splitting is between 73 and 142 meV, in good agreement with the measured value of 131 meV from neutron diffraction [@Boothroyd2001PRL2082].
Aspherical SIE in other methods
-------------------------------
Our method bears some likeness to the hybrid functional approach. The difference in the latter is that the exchange interactions are calculated directly from the wavefunctions, with the amount of exact or Fock exchange ($U/2+a J$ for one localized electron in terms of LDA+$U$) as well the replaced LDA/GGA exchange controlled by a fixed parameter $a_{\mathrm{EXX}}$. However, $a_{\mathrm{EXX}}$ in the hybrid functional method is often system-dependent and fitted to experimental data, just like $U$ in LDA+$U$. For instance, Ref. found that in $f$-ystems good results were obtained using $40-70$% Fock exchange, while $d$-systems typically require $20-50$% [@Cora2004SB171]. However, such an $a_{\mathrm{EXX}}$ may not necessarily lead to accurate removal of the aspherical SIE. Fig. \[fig:EvsaEXX\] shows the energy of Pr$^{4+}$ ion as a function of $a_{\mathrm{EXX}}$ calculated with the hybrid functional (HSE06) [@Heyd2006JCP219906]. Nearest degeneracy is obtained at $a_{\mathrm{EXX}} \approx 85\%$. Given the sensitive orbital dependence of SI demonstrated in this work, in general the accuracy of hybrid functional calculations for $f$-electron systems may still suffer from incomplete removal of aspherical SIE. After the first submission of this manuscript, we became aware that the idea of removing on-site $E_{\mathrm{H}}$ from LDA+$U$ was previously proposed from a different perspective in Ref. , in which the correction energy is independent of the orbital filling, an important different from our approach. Therefore, the method of Ref. is not expected to give accurate removal of the orbital-dependent SIE.
Summary
=======
In summary, we have identified a serious problem in applying the LDA+$U$ method to $f$-electron systems: the degeneracy of atomic orbitals is lifted, resulting in qualitatively incorrect electronic ground states and orbital excitation spectra. Aspherical orbital-dependent self-interaction is identified as the main source of error. To correct it, a new LDA+$U$ scheme is proposed, which leaves the Hartree intact and only replaces the LDA exchange with the Hartree-Fock exchange. Our method has one adjustable parameter $U$, with the other two ($J$ and $c$) being determined from the condition of orbital degeneracy in free ions. The computational expense is approximately the same as in the conventional LDA+$U$, and very competitive compared to hybrid functional approaches [@Novak2006PSSB563]. We expect that our method will scale to large systems and will significantly improve the accuracy of first-principles studies of $f$- as well as heavy $d$-systems with significant relativistic effects. Additionally, more advanced methods such as GW and DMFT could benefit from the correct input ground state orbitals generated by our method.
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---
author:
- 'Xintao Duan, Haoxian Song'
title: Coverless information hiding based on Generative Model
---
of the current information hiding technologies embed the secret information into the carrier with the slight modification of the carrier data (digital image, video and audio), and hide the dense carrier as a disguise of the secret information. The popularization of personal computers and the proliferation of multimedia data on the internet have provided convenient conditions for implementing information hiding and have made information hiding rapid development. However, the pace of development in recent years has slowed down. The main reason is that at the same time as the development of information hiding, the detection technology for hidden information, also called steganalysis, has also been rapidly developed. The technology is based on the statistical anomaly of the carrier data caused by information embedding to determine whether the secret information exists, and it has posed a serious threat to information hiding. According to the different hiding methods, the commonly used steganography methods are divided into two types: space domain hiding method and transform domain hiding method. The space domain hiding method has the adaptive LSB hiding method[@yang2008adaptive], the spatial adaptive steganography algorithm S-UNIWARD [@Holub2014Universal], HUGO[@Pevn2010Using], WOW[@Holub2012Designing] and so on. The transform domain method is to modify the host image data to change some statistical features to achieve data hiding, such as the hidden method in DFT(discrete Fourier transform) domain[@Ruanaidh1996Phase], DCT (discrete cosine transform) domain[@Cox1999Secure], and DWT (discrete wavelet transform) domain[@Lin2008An]. These methods inevitably leave some modifications to the carrier. In order to fundamentally resist the detection of various detection algorithms, this paper presents a new coverless image information hiding method based on generative model. As shown in Fig.1, we only need to deliver a meaning-normal image which is not related to the secret image to the receiver, so that the receiver can generate an image visually the same as the secret image without worrying about the analysis of the steganography, even less attack.
![[]{data-label="fig_1"}](1){width="3.5in"}
As mentioned above, we proposed a new approach to hide image information, it can generate visually the same image as the secret image by sending a generated image that is not related to the secret image. The transmitted image is only a normal-meaningful image rather than the image embedded any information of the secret image, and also achieve the same effect as transferring the secret image. This method can effectively resist steganalysis tools, and greatly improves the security of the image. To summarize, the major contributions of our work as below:
- [强调]{}We do not need to pass the secret image, on the contrary, we transmit a meaning-normal image which is completely unrelated to the secret image. This method has high security.
<!-- -->
- [强调]{}The image we transmit does not embed any secret information, it is a normal image, and the image steganography analysis does not work.
<!-- -->
- [强调]{}As long as the training is enough, this effect can be achieved and the capacity is large.
Related work
============
Restricted Boltzmann Machines (RBMs) [@Smolensky1986Information], deep Boltzmann machines (DBMs) [@Salakhutdinov2009Deep] and their numerous variants are undirected graphical models with latent variables. The interactions within such models are represented as the product of unnormalized potential functions, normalized by a global summation/integration over all states of the random variables. This quantity and its gradient are intractable for all but the most trivial instances, although they can be estimated by Markov chain Monte Carlo (MCMC) methods. Mixing poses a significant problem for learning algorithms that rely on MCMC [@Bengio2012Better],[@Bengio2014Deep]. Deep belief networks (DBNs) [@Hinton2006A] are hybrid models containing a single undirected layer and several directed layers. While a fast approximate layer-wise training criterion exists, DBNs incur the computational difficulties associated with both undirected and directed models. Variational Auto-Encoders (VAEs) [@glorot2011deep] and Generative Adversarial Networks (GANs) [@Bengio2013Generalized] are well known to us. VAEs focus on the approximate likelihood of the examples, and they share the limitation of the standard models and need to fiddle with additional noise terms. Ian Goodfellow put forward GAN [@Goodfellow2014Generative] in 2014. Goodfellow theoretically proved the convergence of the algorithm, and when the model converges, the generated data has the same distribution as the real data. GAN provides a new training idea for many generative models and has hastened many subsequent works. GAN takes a random variable (it can be Gauss distribution, or uniform distribution between 0 and 1) to carry on inverse transformation sampling of the probability distribution through the parameterized probability generative model (it is usually parameterized by a neural network model), then a generative probability distribution is obtained. The GAN model includes a generative model G and a discriminative model D. The training objective of the discriminative model D is to maximize the accuracy of its own discriminator, and the training objective of generative model G is to minimize the discriminator accuracy of the discriminative model D. The objective function of GAN is a zero-sum game between D and G and also a minimum - maximization problem. GAN adopts a very direct way of alternate optimization, and it can be divided into two stages. In the first stage, the discriminative model D is fixed, the generative model G is optimized to minimize the accuracy of the discriminative model. In the second stage, the generative model G is the fixed in order to improve the accuracy of the discriminative model D. As a generative model, GAN is directly applied to modeling of the real data distribution, including generating images, videos, music and natural sentences, etc. Because of the mechanism of internal confrontation training, GAN can solve the problem of insufficient data in some traditional machine learning. GANs offer much more flexibility in the definition of the objective function, including Jensen-Shannon, and all f-divergences [@Hinton2012Improving] as well as some exotic combinations. Therefore, it can be used in semi-supervised learning, unsupervised learning, multi-view learning and multi-tasking learning. In addition, it has been successfully used in reinforcement learning to improve its learning efficiency. Although GAN is applied widely, there are some problems with GAN, difficulty in training, lack of diversity. Besides, generator and discriminator cannot indicate the training process. On the other hand, training GANs is well known for being delicate and unstable. The better discriminator is trained, the more serious gradient of the generator disappears, leading to gradient instability and insufficient diversity. WGAN (Wasserstein Generative Adversarial Networks [@Arjovsky2017Towards], [@Arjovsky2017Wasserstein]) is an improvement to GAN, and it applies Wasserstein distance instead of JS divergence in the GAN. Compared to KL divergence and JS divergence, the advantage of Wasserstein distance is that it can still reflect their distance even if there is no overlap between the two distributions. At the same time, the problem of training stability and process indicating are solved.
Therefore, this paper chooses Wasserstein GAN so as to guarantee training stability instead of GAN. It is no longer necessary to carefully balance the training extent between generator and discriminator. It basically solves the problem of collapse mode and ensures the diversity of samples.
Approach
========
The WGAN model is applied to generate the handwritten word by feeding the random noise z, but when the random noise z is changed to a secret image img, the model can still generate the meaning-normal and independent image IMG’ which is not related to the secret image we want to transmit. These several images taken from the standard set of images were evaluated in the paper, they are Lena, baboon, cameraman and peppers, and they have the same size as 256 by 256. The feed is the secret image, and we train the generative model database through the WGAN, then it can generate a meaning-normal and independent image which is not related to the secret image. So we transmit the meaning-normal image to the receiver, and this generated image is fed to the generative model database to generate another generated image visually the same as the secret image. The flow charts of the whole experiment are shown in Fig. \[fig\_2\] and Fig. \[fig\_3\].
![[]{data-label="fig_2"}](2){width="3.8in"}
![[]{data-label="fig_3"}](3){width="3.5in"}
D and G play the following two-player minimax game with value function $$V(G;D)$$ in WGAN:
$$\mathop {\min }\limits_G \mathop {\max }\limits_D V(D,G) = {E_{x \sim pdata(x)}}[logD(x)] +
{E_{z \sim {p_z}(z)}}[log(1 - D(G(z)))]$$
D(x) represents the probability that x came from the data rather than pg. We train D to maximize the probability of assigning the correct label to both training examples and samples from G. We simultaneously train G to minimize $$log(1 - D(G(z)))$$. We first make a gradient ascent step on D and then a gradient descent step on G, then the update rules are:
Keeping the G fixed, update the model D by$${\theta _D} \leftarrow {\theta _D} + {\gamma _D}{\nabla _D}L$$with
$${\nabla _D}L = \frac{\partial }{{\partial {\theta _D}}}\{ {E_{x \sim Pdata(x)}}[logD(x,{\theta _D})] +
{E_{z \sim Pnoise(z)}}[log(1 - D(G(z,{\theta _G}),{\theta _D}))]\}$$ Keeping D fixed, update G by$${\theta _G} \leftarrow {\theta _G} - {\gamma _G}{\nabla _G}L$$where
$${\nabla _G}L = \frac{\partial }{{\partial {\theta _G}}}{E_{z \sim Pdata(z)}}[log(1 - D(G(z,{\theta _G}),{\theta _D}))]$$
Wasserstein distance is also called the EM(Earth-Mover) distance $$W({P_r},{P_g}) = \mathop {\inf }\limits_{\gamma \sim \prod ({p_{r,}}{p_g})} {E_{(x,y) \sim \gamma }}[\parallel x - y\parallel ]$$
Where $\prod {({P_r},{P_g})} $ denotes the set of all joint distributions $\gamma (x,y)$ whose marginal are respectively ${P_r}$ and ${P_g}$ . Intuitively, $\gamma (x,y)$ indicates how much “mass” must be transported from x to y in order to transform the distributions ${P_r}$ into the distribution ${P_g}$ . The EM distance then is the “cost” of the optimal transport plan.
Experiments
===========
In this paper, 5,000 images are randomly selected from the CelebA dataset to experiment, and the results show that the coverless image information hiding based on generative model method can be implemented well. The sender and receiver share the same dataset and the same parameters. As shown in Fig. \[fig\_4\] and Fig. \[fig\_5\], we feed the secret image img into the generative model, generating the meaning-normal and independent IMG’ which is not related to the secret image we want to transmit.
![[]{data-label="fig_4"}](4){width="3.0in"}
![[]{data-label="fig_5"}](5){width="3.0in"}
As shown above, we choose Lena as the secret image img, it can generate the IMG’ visually the same as Baboon we want to transmit. In the meantime, we also trained Baboon to generate the IMG’ visually the same as Lena through the WGAN. We save the corresponding generative model G1 and G2 of generating visually the same as Baboon and Lena respectively. Using the same method, we take the cameraman and peppers as the secret image to experiment respectively, and they can generate corresponding peppers and cameraman. We also save the corresponding generative model G3 and G4 of generating visually the same as peppers and cameraman respectively, and apply them to the next experiment, instead of the WGAN. We put the generative model G1, G2, G3 and G4 of generating visually the same as Baboon, Lena, peppers and cameraman in a database respectively, so that the generative model database is built. Since both the sender and the receiver train well the generative model database, we perform experiments as shown in Fig. \[fig\_6\] and Fig. \[fig\_7\].
![[]{data-label="fig_6"}](6){width="3.0in"}
![[]{data-label="fig_7"}](7){width="3.0in"}
As shown above, when the sender wants to transmit the secret image Lena, the generated image Baboon can be transmitted to the receiver to generate a generated image visually the same as the secret image Lena, similarly, if you want to transmit Baboon, you can transmit the generated image Lena, if you want to transmit cameraman, you can transmit the generated image peppers, if you want to transmit peppers, you can transmit the generated image cameraman. In this experiment, we have successfully achieved the effect of coverless image information hiding based on generative model method by feeding a secret image to generate a meaning-normal and independent image which is not related to the secret image we want to transmit, and when the secret image is given, the transmitted image is unique and specific. Consequently, the image information hiding method proposed in this paper is feasible. In practical application, we are more concerned with the content of the image rather than the pixels in addition to professional image workers, this method can produce a meaning-normal and independent image which is not related to the secret image we want to transmit, which can satisfy most requirements, thereby, we suppose that if you want to send a secret image, you only need to transmit a meaning-normal and independent image to the receiver, the receiver only need to feed transmitted image to the generative model database, generate an image visually the same as the secret one, no needing direct transmission of the secret image. Besides, the transmitted image does not embed any information of the secret image, so it does not give visual cues to attackers, and the image steganography analysis does not work. This method can resist detection of all the existing steganalysis tools, and improve the security of the image.
Conclusion
==========
To sum up, the paper proposed the coverless image information hiding based on generative model method. An image visually the same as the secret image is generated by transmitting a normal-meaningful image to the receiver. A fed image corresponds uniquely to a secret image. This method is practical. Therefore, it can be applied to image hiding and image protection.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by the National Natural Science Foundation of China under Grant NO.U1204606, U1404603, the Science and Technology Foundation of Henan Province under Grant No.172102210335.
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---
abstract: 'For a wide class of monotonic functions $f$, we develop a Chernoff-style concentration inequality for quadratic forms $Q_f \sim \sum\limits_{i=1}^n f(\eta_i) (Z_i + \delta_i)^2$, where $Z_i \sim N(0,1)$. The inequality is expressed in terms of traces that are rapid to compute, making it useful for bounding p-values in high-dimensional screening applications. The bounds we obtain are significantly tighter than those that have been previously developed, which we illustrate with numerical examples.'
address:
- 'Department of Mathematical Sciences, Durham University, United Kingdom'
- 'The Alan Turing Institute, London, United Kingdom'
- 'Department of Statistics, Oxford University, United Kingdom'
- 'McDonnell Genome Institute, Washington University in St. Louis, United States'
author:
- 'Robert E. Gallagher'
- 'Louis J. M. Aslett'
- David Steinsaltz
- 'Ryan R. Christ'
bibliography:
- 'references2.bib'
title: Improved Concentration Bounds for Gaussian Quadratic Forms
---
quadratic form ,generalized non-central chi-square distribution ,concentration inequality ,Hilbert-Schmidt Information Criteria ,tail bound
Introduction and Background
===========================
We consider the problem of finding an upper bound for the cumulative distribution function (cdf) of random variables of the form $Q_f \sim \sum\limits_{i=1}^n f(\eta_i) (Z_i + \delta_i)^2$, where $Z_i \sim N(0,1)$, $f: \mathbb{R} \rightarrow \mathbb{R}$, and $\delta_i$ and $\eta_i$ are deterministic scalars. Many applications lead to this form with $\{\eta_i\}_{i=1}^n$ being the eigenvalues of a symmetric matrix $M \in \mathbb{R}^{n \times n}$; for example, a quadratic form $X^\top f(M) X$ where $X \sim N(\mu,I)$ and $f(M)$ represents $f$ applied to the eigenvalues of $M$. As described in @Christ:2017aa and @Christ2020, results of this kind can be generalized to cases where $M$ is asymmetric with careful treatment of $f$.
$Q_f$ arises as the limiting distribution of test statistics used in a wide range of applications. These statistics include the Hilbert-Schmidt Information Criterion used for high-dimensional independence testing [@gretton2005; @Zhang2018], score statistics for linear and genearlized linear mixed models commonly used in genomics [@lin1997variance; @wu2011rare], and the goodness-of-fit statistic proposed by @pena2002powerful for ARMA models in time series analysis. It is easy to see that $Q_f$ has mean $$\mathbb{E}(Q_f) = \sum\limits_{i=1}^n f(\eta_i) \delta_i^2 + \sum\limits_{i=1}^n f(\eta_i)$$ and variance $$\operatorname{\mathrm{Var}}(Q_f) = 2 \left( \sum\limits_{i=1}^n f(\eta_i)^2 \delta_i^2 + 2 \sum\limits_{i=1}^n f(\eta_i)^2\right).$$ Work in [@Christ2020] established a concentration inequality to bound the tails of $Q_f$, which yield a set of bounds for different functions. The results of [@Christ2020] show that it is possible to find polynomial bounds, but these are not constructed explicitly. We provide here explicit optimal coefficients for bounds of this form in the single-spectrum case. This earlier work yielded the following bound on $Q$ (by which we designate the base version of $Q_f$, where $f$ is the identity function):
\[theorem1\] Let $X \sim N(\mu,I)$ and $M$ be a real, symmetric matrix. Let $Q=X^\top M X$. Let $\nu = 2 \left( 4 \left| \left| M \mu \right| \right|^2_{2} + 2 \left| \left| M \right| \right|^2_{HS} \right)$ and let $b=\underset{i}{\max}\left| \lambda_i\right|$, where $\{\lambda_i\}_{i=1}^n$ are the eigenvalues of $M$. Then, for all $q > \mathbb{E}\left[ Q \right]$,
$$\mathbb{P}\left( Q > q \right) \leq \left\{ \begin{array}{lcc}
\exp\left( -\frac{1}{2} \frac{(q-\mathbb{E}\left[ Q \right])^2}{\nu}\right) & & \mathbb{E}\left[ Q \right] < q \leq \frac{\nu}{4 b}+ \mathbb{E}\left[ Q \right] \\
\exp\left( \frac{1}{2} \frac{\nu}{\left(4b\right)^2 } \right) \exp\left( -\frac{q-\mathbb{E}\left[ Q \right]}{4b}\right) & & q > \frac{\nu}{4 b }+\mathbb{E}\left[ Q \right].
\end{array}\right.$$
Similarly, for all $q < \mathbb{E}\left[ Q \right]$, $$\mathbb{P}\left( Q < q \right) \leq \left\{ \begin{array}{lcc}
\exp\left( -\frac{1}{2} \frac{(\mathbb{E}\left[ Q \right]-q)^2}{\nu}\right) & & \mathbb{E}\left[ Q \right] -\frac{\nu}{4 b} \leq q < \mathbb{E}\left[ Q \right] \\
\exp\left( \frac{1}{2} \frac{\nu}{\left(4b\right)^2 } \right) \exp\left( -\frac{\mathbb{E}\left[ Q \right]-q}{4b}\right) & & q < \mathbb{E}\left[ Q \right]-\frac{\nu}{4 b }.
\end{array}\right.$$
The proof of this result relies on a Chernoff-style bound involving the cumulant generating function (cgf) of $Q$, which has two main types of terms: $$\begin{aligned}
{\mathcal{L}}_1(x)&=-\log(1-2x)/2 \quad\text{ and}\\
{\mathcal{L}}_2(x)&=\frac{x}{1-2x}.\end{aligned}$$ Each of these is bounded by a quadratic function, leading to an overall bound in terms of easily computable coefficients. We improve on this previous work by constructing a family of quadratics that yield pointwise tighter bounds on ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$. We then show how these can be incorporated into an optimisation step to yield tighter bounds on the tails of $Q_f$.
In Section 2 we present our main results. First we present Lemma \[lemma1\], which tightens the quadratic bounds above from [@Christ:2017aa]. From this lemma, we derive the corresponding improved bounds on the tails of $Q_f$ in Theorem \[Thm1\]. Specialisation of these results to some particular functions $f$ then follow in corollaries. In Section 3, we empirically demonstrate the improvement provided by these bounds with an application to a simulatd matrix with a exponentially decaying spectrum. Section 4 concludes with discussion of potential future improvements. Proofs for the main results are presented in Section 5.
Main Results
============
Our results depend upon elementary upper bounds on ${\mathcal{L}}_1(x)$ and ${\mathcal{L}}_2(x)$ in the form of parabolas passing through the origin. We describe the coefficients of these parabolas in terms of the width of the (symmetric) interval on which the bounds are to be applied, and on the parameter $t$ that arises from the cgf. We exploit two openings for improvement: optimising the coefficients of the parabola and optimising the width of the scaled domain over which it bounds ${\mathcal{L}}_1 \left(tf(x)\right)$ and ${\mathcal{L}}_2 \left(tf(x)\right)$.
\[lemma1\] Let $f(x)$ be a monotonic increasing function such that $f(0)=0$. Let $L$ be a fixed positive real number, and $t \in [0,t^{\star})$, where $$t^{\star} = \min\bigl\{|1/2f(L)|,|1/2f(-L)| \bigr\}.$$ Furthermore, suppose that over the region $x \in (0,L], t \in [0, t^{\star})$ the following inequalities are satisfied for both ${\mathcal{L}}_1\left(tf(x)\right)$ and ${\mathcal{L}}_2 \left(tf(x)\right)$: $$\begin{aligned}
x\bigl(\partial_x {\mathcal{L}}\left(tf(x)\right)/2 + t f'(0) \bigr) &\geq {\mathcal{L}}\left(tf(x)\right), \label{E:lemcond1}\\
\frac{{\mathcal{L}}\left(tf(x)\right) - {\mathcal{L}}\left(tf(x)\right)}{2x} &\geq t f'(0). \label{E:lemcond2}\end{aligned}$$ For each $t \in [0,t^{\star})$ define $$\begin{aligned}
\alpha_f (L,t) &= {\mathcal{L}}_1\left(tf(x)\right)/ L^2 - t f'(0)/L, \\
\beta_f (L,t) &= {\mathcal{L}}_2 \left(tf(x)\right) / L^2 - t f'(0)/L,\\
\gamma_f(t) &= t f'(0) .\end{aligned}$$
Then for each $t \in [0,t^{\star})$, among all quadratic function $x \mapsto ax^2 + b x$ that maintain $g^1_t(x)\le 0$ over the whole region $x \in [-L,L]$, where $$g^1_t(x) := {\mathcal{L}}_1 \left(tf(x)\right) - \bigl( a x^2 + b x \bigr),$$ the difference $|g^1_t(x)|$ is minimised at every point $x$ by the choice $a= \alpha_f(L,t)$ and $b=\gamma_f(t)$; and among those that maintain $g^2_t(x)\le 0$ over the whole region $x \in [-L,L]$, where $$g^2_t(x) := {\mathcal{L}}_2 \left(tf(x)\right) - \bigl( a x^2 + b x \bigr),$$ the difference $|g^2_t(x)|$ is minimised at every point $x$ by the choice $a= \beta_f(L,t)$ and $b=\gamma_f(t)$.
This lemma will allow us to build on the existing result from [@Christ:2017aa]. In the original form of this theorem $t$ was restricted so that $tf(x) < 1/4$, avoiding the asymptote at $1/2$. We remove this boundary at 1/4 and allow $tf(x)$ to get arbitrarily close to $1/2$. We also reinterpret $L$, so that it now defines the domain of $x$ rather than that of $tf(x)$. It also means that for every endpoint along the interval $[-L,L]$ we can obtain optimal coefficients on our quadratic bounds. This yields a new bound on the tails of $Q_f$ as follows.
\[Thm1\] Let $\xi = c \left(\sum\limits_{i=1}^n \eta_i \delta_i^2 + \sum\limits_{i=1}^n \eta_i\right)$ where $c=f'(0)$, and let $L$ be set to $\underset{i}{\max}\left| \eta_i\right|$. Suppose $f$ satisfies the conditions in Lemma \[lemma1\]. Then for all $q > \xi$, $$\label{eq:PQf}
\operatorname{\mathbb{P}}(Q_f > q) \leq \min_{t \in (0, 1/2d)} \bigl[ \exp(\nu_f(t)/2 -\left(q -\xi\right) t) \bigr],$$ where $d = \underset{i}{\max}\left| f(\eta_i)\right|$, and $$\label{eq:nuft}
\nu_f(t)=2 \left( \beta_f(L,t) \sum\limits_{i=1}^n \eta_i^2 \delta_i^2 + \alpha_f(L,t) \sum\limits_{i=1}^n \eta_i^2\right).$$ Furthermore, for all $q < \xi$, $$\label{eq:PQf2}
\operatorname{\mathbb{P}}(Q_f < q) \leq \min_{t \in (0, 1/2d)} \bigl[\exp(\nu_f(t)/2 -\left(\xi -q\right) t) \bigr].$$
In the central use case, where $Q_f$ arises as $X^\top f(M) X$, we can apply Theorem \[Thm1\], where $\xi = c \, (\mu^\top M \mu + \mathrm{tr}\left(M\right))$ and $$\nu_f(t) = 2 \left( \beta_f(L,t) \left| \left| M \mu \right| \right|^2_{2} + \alpha_f(L,t) \left| \left| M \right| \right|^2_{HS}\right).$$ This allows us to quickly compute tight tail bounds on $X^\top f(M) X$. In the following corollaries we address special cases of $f$.
\[identity\_corollary\] Let $f(x)=x$. Then the cdf of $Q_f$ is bounded as in equations and where in equation we set ${\alpha_f(L,t) = {\mathcal{L}}_1\left(tL\right)/L^2 - t/L}$ and ${\beta_f(L,t) = tL/(L^2(1-2 tL)) - t/L}$.
*Proof:* Since $|f(L)| = |f(-L)|$, the $t^{\star}$ from Lemma 1 is equal to $1/2f(L)=1/2L$. The conditions and may be written in terms of the variable $z = tx$, and these inequalities then need to hold for $z \in [0, 1/2)$. The two conditions for ${\mathcal{L}}_1$ become $$\begin{aligned}
\frac{z}{(1-2z)} + 2z &\geq -\log(1-2z) \quad \text{and} \\
-\log(1-2z) + \log(1+2z) &\geq 4z ,\end{aligned}$$ while the two conditions for ${\mathcal{L}}_2$ become $$\begin{aligned}
\frac{z}{(1-2z)^2} + 2z &\geq \frac{2z}{1-2z} \quad \text{and} \\
\frac{z}{1-2z} + \frac{z}{1+2z} &\geq 2z.\end{aligned}$$ All of these inequalities hold for $z \in (0,1/2)$, and so Lemma 1 holds where $f$ is the identity function. The result follows by application of Theorem 1.
\[C:xp\] Let $f(x)=x^p$ for some positive integer $p \ge 2$. Then the cdf of $Q_f$ is bounded as in equations and where in equation , $${\alpha_f(L,t) = {\mathcal{L}}_1\left(tL^p\right)/L^2} \quad\text{and}\quad \beta_f(L,t) = tL^{p-2}/(1-2 tL^p).$$
*Proof:* Since $|f(L)|=|f(-L)|$, the $t^{\star}$ from Lemma 1 is equal to $1/2L^p$. We introduce the variable $z = tx^p$ and note that our original region, $x \in [0,L]$ and $t \in [0, 1/2L^p)$, corresponds to $z \in [0, 1/2)$.\
Substituting the definitions of $z, {\mathcal{L}}_1, {\mathcal{L}}_2$ into condition yields $$\begin{aligned}
\frac{pz}{1-2z} &\geq -\log(1-2z), \\
\frac{pz}{(1-2z)^2} &\geq \frac{2z}{1-2z}.\end{aligned}$$ The condition is trivial for even $p$, while for odd $p$ it becomes $$\begin{aligned}
-\log(1-2z) + \log(1+2z) &\geq 0, \\
\frac{z}{1-2z} + \frac{z}{1+2z} &\geq 2z.\end{aligned}$$ All of these inequalities hold for $z \in [0,1/2)$ and $p\ge 2$, so Lemma \[lemma1\] holds for $f(x) = x^p$. The result follows by application of Theorem \[Thm1\].
With essentially the same proof used for Corollary \[C:xp\], we can formulate the result of Theorem \[Thm1\] for matrix powers. Note that in following case, $\xi = 0$.
\[C:xpMat\] For any positive integer $p \ge 2$, for each $q>0$ $$\operatorname{\mathbb{P}}(X^\top M^p X > q) \leq \min_{t \in (0, 1/2d)}{\mathrm{e}}^{-qt + \nu_f(t)/2},$$ and for $q < 0$ $$\operatorname{\mathbb{P}}(X^\top M^p X < q) \leq \min_{t \in (0, 1/2d)}{\mathrm{e}}^{qt + \nu_f(t)/2},$$ where $\nu_f(t)$ is defined in and $\alpha_f(L,t) = -\log(1-2t L^p)/2L^2$, $\beta_f(L,t) = tL^{p-2}/(1-2 tL^p)$.
Examples
========
Here we compare the bounds in Corollary \[identity\_corollary\] and Corollary \[C:xpMat\] to the bounds provided in @Christ:2017aa and @Christ2020 for different matrix powers $p =1,2,3,4$. For this comparison, we simluated a matrix with an exponentially decaying spectrum of eigenvalues, a case which is relatively common in applications. See Figure \[fig:modqq\].
![Modified $Q-Q$ plot showing the difference between the negative base 10 logarithm of the true right tail probabilities estimated by our simulations and those estimated by $U_p(q)$. In other words, we plot $z - \Omega(z)$, where $\Omega\left(z\right) = -\log_{10}\left(1-U_p\left(\hat{F}^{-1}_p\left(1-10^{-z}\right)\right)\right).$ []{data-label="fig:modqq"}](expdecay_powers_compared.pdf){width="\linewidth"}
For this comparison, we simluated a matrix with an exponentially decaying spectrum of eigenvalues, a case which is relatively common in applications. See Figure \[fig:modqq\]. Note that we have plotted the logarithm (base 10) of the true probability on the $x$ axis, and the error in the bounds on the $y$ axis. Thus, using the solid red line in Figure \[fig:modqq\], if the true tail probability of $Q_f$ is $10^{-4}$ ($z=4$), then our new bound for $p=1$ would be approximately of the order $10^{-2}$.
Particularly of note is that while our bounds show an improvement for all functions satisfying the assumptions of Lemma \[lemma1\], the improvement is much greater for even functions. This is because our bounds are quadratic, so they must yield the same error bound on both sides of the real line for even functions; however, when bounding an odd function, our bounds will be tight by construction for $x>0$ but may be much looser for $x<0$. As expected, our bounds perform worse for higher powers $p$, which is effectively a result of attempting to control the higher-order behavior of the matrix given traces that measure the empirical mean and variance of the matrix elements.
Conclusions
===========
We have placed tighter bounds than were previously available on the tails of $Q_f$. Although our bounds are not available in an explicit form, since we optimise over two parameters that previous results set arbitrarily, our bounds are at least as good, which is seen in practice. We further observe that they tend to be significantly tighter and improve relative to the old bounds as we go further out into the tails.
Although our results do give a significantly tighter bound on the tails of $Q_f$, they only work for a specific class of $f$ satisfying the conditions of Lemma 1, which notably excludes functions such as $\exp(x)$. Future developments could improve on this; one possible way would be to introduce an intercept into our quadratic bounds for ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$, which would maintain the ease of computability while extending it to a wider range of $f$. A further source of improvement may be achieved by modifying Lemma \[lemma1\] to account for the asymmetry on $x \leq 0$ vs. $x \geq 0$. Treating each side of the real line separately could enable one to use both the smallest and largest eigenvalue, rather than just $\underset{i}{\max}\left| \lambda_i\right|$.
Though outside the scope of this paper, it would be possible to achieve similar bounds for sub-Gaussian random variables. This would provide tighter results than currently exist in those cases if the Hanson–Wright inequality argument [@rudelson2013hanson] were reworked in terms of explicit constants.
Proofs of Main Results
======================
**Proof of Lemma \[lemma1\]**
In the special case $t=0$ we simply have that ${\mathcal{L}}_1\left(0\right)$, ${\mathcal{L}}_2(0)$, $\alpha_f(L,0)$, $\beta_f(L,0)$, and $\gamma_f(0)$ are all $0$, so the Lemma clearly holds. We assume now $t \neq 0$.
Since $g_t^1(0)=g_t^2(0)=0$, the choice of $\gamma_f(t)$ is fixed by the need to make 0 a critical point for both of these functions. It remains only to consider the choice of $a$.
Consider ${\mathcal{L}}$ being either ${\mathcal{L}}_1$ or ${\mathcal{L}}_2$. Write $g(x,a)={\mathcal{L}}\left(tf(x)\right) - (ax^2 + bx)$, where $b=\gamma_f(t)$. Since $b$ is fixed, the quadratic functions are strictly increasing in $a$ at every point. For $x\in (0,L]$ define $$a_x := \frac{{\mathcal{L}}\left(tf(x)\right)}{x^2} - \frac{{\mathcal{L}}(tf)'(0)}{x}.$$ Then $a_x$ is the minimum $a$ such that $g(x,a) \le 0$, and the optimum $a$ that we are looking for is $\sup_{x\in (0,L]} a_x$. We have $$\begin{aligned}
\frac{\dif a_x}{\dif x} & = \frac{{\mathcal{L}}' \left(h(x)\right)}{x^2} - \frac{2{\mathcal{L}}\left(tf(x)\right)}{x^3} + \frac{2 {\mathcal{L}}(tf)'(0) }{x^2} \\
&= 2x^{-3} \left( x\left(\frac{\partial_x {\mathcal{L}}\left( t f(x)\right)}{2} + t f'(0) \right) - {\mathcal{L}}\left(t f(x)\right) \right) \\
&\ge 0\end{aligned}$$ by assumption \[E:lemcond1\]. Thus $a_x$ is non-decreasing in $x$, and so has its maximum at $L$. This shows that taking $a=a_L$ makes $g(x,a)\le 0$ for any $x\in (0,L]$, and it is the smallest such $a$. Note that $a_L= \alpha_f(L)$ when ${\mathcal{L}}={\mathcal{L}}_1$, and $a_L=\beta_f(L)$ when ${\mathcal{L}}={\mathcal{L}}_2$.
Assumption \[E:lemcond2\] tells us that for $x\in (0,L]$ we have $\frac{{\mathcal{L}}\left(tf(x)\right) - {\mathcal{L}}\left(tf(-x)\right)}{2x} \geq tf'(0)$. This implies that $g(-x,a_L) \le g(x,a_L)\le 0$, so the same choice of $a=a_L$ provides a bound — that is, $g(x,a_L) \le 0$ — over the whole interval $[x,L]$.
**Proof of Theorem \[Thm1\]**
We credit [@Pollard:website] for the proof technique used below.
Using Lemma 3.1.3 in [@Christ:2017aa p.75], for $t < 1/2 d$ $$\begin{aligned}
\mathbb{E}\left[e^{t Q_f }\right] &= \prod\limits_{i=1}^n \left(1-2 t f(\eta_i) \right)^{-1/2} \exp\left(\delta_i^2 t f(\eta_i) /(1-2 t f(\eta_i)) \right) \\
&= \exp\left(\sum\limits_{i=1}^n \delta_i^2 t f(\eta_i)/(1-2 t f(\eta_i)) - \log\left( 1- 2 t f(\eta_i)\right)/2 \right).\end{aligned}$$ By Lemma \[lemma1\] we know, setting $L=\underset{i}{\max}\left| \eta_i\right|$, that for $x\in \left[-L,L\right]$, $$\begin{aligned}
{\mathcal{L}}_1\left(tf(x)\right) &\leq \alpha_f(L,t)x^2 + tf'(0)x, \\
{\mathcal{L}}_2\left(tf(x)\right) & \leq \beta_f(L,t)x^2 + tf'(0)x.\end{aligned}$$ We claim that this is the optimal choice of $L$. Smaller $L$ will void the inequalities for some $\eta_i$ and so cannot be considered. On the other hand, we know that both $\alpha_f(L,t)$ and $\beta_f(L,t)$ are increasing in $L$ so any larger $L$ would simultaneously weaken the quadratic bound and shrink the range of values $t$ to which it can be applied, since $1/2f(L)$ is decreasing in $L$.
Therefore, $$\begin{aligned}
\mathbb{E}\left[e^{tQ_f}\right] &\leq \exp\left(\sum\limits_{i=1}^n \delta_i^2 \left( \beta_f(L,t) \eta_i^2 + c \eta_i t \right) + \alpha_f(L,t) \eta_i^2 + c \eta_i t \right) \\
&\leq \exp\left( \beta_f(L,t) \sum\limits_{i=1}^n \eta_i^2 \delta_i^2 + c t \sum\limits_{i=1}^n \eta_i \delta_i^2 + \alpha_f(L,t) \sum\limits_{i=1}^n \eta_i^2 + ct \sum\limits_{i=1}^n \eta_i \right).\end{aligned}$$ Applying the definitions of $\xi$ and $\nu_f(t)$ we have $$\mathbb{E}\left[{\mathrm{e}}^{t \left( Q_f - \xi \right) }\right] \leq {\mathrm{e}}^{\nu_f(t)/2} .$$ By Markov’s Inequality, for any $q \in \mathbb{R}$, $$\begin{gathered}
\mathbb{P}\left( Q_f > q \right) =
\mathbb{P}\left( Q_f - \xi > q -\xi \right) = \mathbb{P}\left( {\mathrm{e}}^{Q_f - \xi} > {\mathrm{e}}^{q -\xi} \right) \\
\leq {\mathrm{e}}^{-(q -\xi ) t + \nu_f(t)/2 } \ \text{ for all } t \in \left( 0, 1/2d \right).\end{gathered}$$
For $q \leq \xi$, since $\nu_f(t)$ is positive we have the trivial bound $\mathbb{P}\left( Q_f > q \right) \leq 1$.
The bound for $\mathbb{P}\left( Q_f < q \right)$ is derived identically.
Acknowledgements {#acknowledgements .unnumbered}
================
The first two authors would like to acknowledge the support of the Engineering and Physical Sciences Research Council \[grant number EP/M507854/1\]. The last author would like to acknowledge the support of the Summer Opportunities Abroad Program (SOAP) - WUSM Global Health & Medicine and the WUSM Dean’s Fellowship, both from the Washington University School of Medicine in St. Louis.
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abstract: 'Visual attention mechanisms have proven to be integrally important constituent components of many modern deep neural architectures. They provide an efficient and effective way to utilize visual information selectively, which has shown to be especially valuable in multi-modal learning tasks. However, all prior attention frameworks lack the ability to explicitly model structural dependencies among attention variables, making it difficult to predict consistent attention masks. In this paper we develop a novel [*structured spatial*]{} attention mechanism which is end-to-end trainable and can be integrated with any feed-forward convolutional neural network. This proposed AttentionRNN layer explicitly enforces structure over the spatial attention variables by sequentially predicting attention values in the spatial mask in a bi-directional raster-scan and inverse raster-scan order. As a result, each attention value depends not only on local image or contextual information, but also on the previously predicted attention values. Our experiments show consistent quantitative and qualitative improvements on a variety of recognition tasks and datasets; including image categorization, question answering and image generation.'
author:
- |
Siddhesh Khandelwal\
University of British Columbia\
[skhandel@cs.ubc.ca]{}
- |
Leonid Sigal\
University of British Columbia\
[lsigal@cs.ubc.ca]{}
bibliography:
- 'egbib.bib'
title: '[AttentionRNN]{}: A Structured Spatial Attention Mechanism'
---
=1
Introduction
============
In recent years, computer vision has made tremendous progress across many complex recognition tasks, including image classification [@krizhevsky2012nips; @zheng2017learning], image captioning [@chen2017sca; @johnson2016cvpr; @xu2015show; @you2016image], image generation [@tang2014nips; @zhang2018arxiv; @zhao2018modular] and visual question answering (VQA) [@antol2015iccv; @fukui2016multimodal; @johnson2017cvpr; @lu2016hierarchical; @seo2017visual; @tommasi2014bmvc; @xu2016ask; @yang2016stacked]. Arguably, much of this success can be attributed to the use of visual attention mechanisms which, similar to the human perception, identify the important regions of an image. Attention mechanisms typically produce a spatial mask for the given image feature tensor. In an ideal scenario, the mask is expected to have higher activation values over the features corresponding to the regions of interest, and lower activation values everywhere else. For tasks that are multi-modal in nature, like VQA, a query (, a question) can additionally be used as an input to generate the mask. In such cases, the attention activation is usually a function of similarity between the corresponding encoding of the image region and the question in a pre-defined or learned embedding space.
![[**AttentionRNN.**]{} Illustration of the proposed structured attention network as a module for down stream task.[]{data-label="fig:model"}](model){width="0.8\linewidth"}
Existing visual attention mechanisms can be broadly characterized into two categories: [*global*]{} or [*local*]{}; see Figure \[subfig1:att\] and \[subfig2:att\] respectively for illustration. Global mechanisms predict all the attention variables jointly, typically based on a dense representation of the image feature map. Such mechanisms are prone to overfitting and are only computationally feasible for low-resolution image features. Therefore, typically, these are only applied at the last convolutional layer of a CNN [@lu2016hierarchical; @zhu2016visual7w]. The local mechanisms generate attention values for each spatial attention variable independently based on corresponding image region [@fukui2016multimodal; @seo2017visual; @seo2016progressive] (, feature column) or with the help of local context [@seo2016progressive; @woo2018cbam; @zhao2018modular]. As such, local attention mechanisms can be applied at arbitrary resolution and can be used at various places within a CNN network (, in [@seo2016progressive] authors use them before each sub-sampling layer and in [@woo2018cbam] as part of each residual block). However, all the aforementioned models lack explicit structure in the generated attention masks. This is often exhibited by lack of coherence or sharp discontinuities in the generated attention activation values [@seo2016progressive].
[.2]{} {width="\linewidth"}
[.2]{} {width="\linewidth"}
[.2]{} {width="\linewidth"}
Consider a VQA model attending to regions required to answer the question, “Do the two spheres next to each other have the same color?”. Intuitively, attention mechanisms should focus on the two spheres in question. Furthermore, attention region corresponding to one sphere should inform the estimates for attention region for the other, both in terms of shape and size. However, most traditional attention mechanisms have no ability to encode such dependencies. Recent modularized architectures [@andreas2016cvpr; @hu2018eccv] are able to address some of these issues with attentive [*reasoning*]{}, but they are relevant only for a narrow class of VQA problems. Such models are inapplicable to scenarios involving self-attention [@woo2018cbam] or generative architectures, where granular shape-coherent attention masks are typically needed [@zhao2018modular].
In this paper, we argue that these challenges can be addressed by [*structured*]{} spatial attention. Such class of attention models can potentially encode arbitrary constraints between attention variables, be it top-down structured knowledge or local/global consistency and dependencies. To enforce this structure, we propose a novel attention mechanism which we refer to as AttentionRNN (see Figure \[subfig3:att\] for illustration). We draw inspiration from the Diagonal BiLSTM architecture proposed in [@van2016pixel]. As such, AttentionRNN generates a spatial attention mask by traversing the image diagonally, starting from a corner at the top and going to the opposite corner at the bottom. When predicting the attention value for a particular image feature location, structure is enforced by taking into account: (i) local image context around the corresponding image feature location and, more importantly, (ii) information about previously generated attention values.
One of the key benefits of our model is that it can be used agnostically in any existing feed-forward neural network at one or multiple convolutional feature levels (see Figure \[fig:model\]). To support this claim, we evaluate our method on different tasks and with different backbone architectures. For VQA, we consider the Progressive Attention Network (PAN) [@seo2016progressive] and Multimodal Compact Bilinear Pooling (MCB) [@fukui2016multimodal]. For image generation, we consider the Modular Generative Adversarial Networks (MGAN) [@zhao2018modular]. For image categorization, we consider the Convolutional Block Attention Module (CBAM) [@woo2018cbam]. When we replace the existing attention mechanisms in these models with our proposed AttentionRNN, we observe higher overall performance along with better spatial attention masks.
[**Contributions:**]{} Our contributions can be summarized as follows: (1) We propose a novel spatial attention mechanism which explicitly encodes structure over the spatial attention variables by sequentially predicting values. As a consequence, each attention value in a spatial mask depends not only on local image or contextual information, but also on previously predicted attention values. (2) We illustrate that this general attention mechanism can work with any existing model that relies on, or can benefit from, spatial attention; showing its effectiveness on a variety of different tasks and datasets. (3) Through experimental evaluation, we observe improved performance and better attention masks on VQA, image generation and image categorization tasks.
Related Work
============
Visual Attention
----------------
Visual attention mechanisms have been widely adopted in the computer vision community owing to their ability to focus on important regions in an image. Even though there is a large variety of methods that deploy visual attention, they can be categorized based on key properties of the underlying attention mechanisms. For ease of understanding, we segregate related research using these properties.
**Placement of attention in a network**. Visual attention mechanisms are typically applied on features extracted by a convolutional neural network (CNN). Visual attention can either be applied: (1) at the end of a CNN network, or (2) iteratively at different layers within a CNN network.
Applying visual attention at the end of a CNN network is the most straightforward way of incorporating visual attention in deep models. This has led to an improvement in model performance across a variety of computer vision tasks, including image captioning [@chen2017sca; @xu2015show; @you2016image], image recognition [@zheng2017learning], VQA [@lu2016hierarchical; @xu2016ask; @yang2016stacked; @zhu2016visual7w], and visual dialog [@seo2017visual].
On the other hand, there have been several approaches that iteratively apply visual attention, operating over multiple CNN feature layers [@jaderberg2015spatial; @seo2016progressive; @woo2018cbam]. Seo [@seo2016progressive] progressively apply attention after each pooling layer of a CNN network to accurately attend over target objects of various scales and shape. Woo [@woo2018cbam] use a similar approach, but instead apply two different types of attention - one that attends over feature channels and the other that attends over the spatial domain.
**Context used to compute attention**. Attention mechanisms differ on how much information they use to compute the attention mask. They can be *global*, that is use all the available image context to jointly predict the values in an attention mask [@lu2016hierarchical; @xu2015show]. As an example, [@lu2016hierarchical] propose an attention mechanism for VQA where the attention mask is computed by projecting the image features into some latent space and then computing its similarity with the question.
Attention mechanisms can also be *local*, where-in attention for each variable is generated independently or using a corresponding local image region [@fukui2016multimodal; @seo2017visual; @seo2016progressive; @woo2018cbam; @zhao2018modular]. For example, [@seo2016progressive; @woo2018cbam; @zhao2018modular] use a $k\times k$ convolutional kernel to compute a particular attention value, allowing them to capture local information around the corresponding location.
None of the aforementioned works enforce structure over the generated attention masks. Structure over the values of an image, however, has been exploited in many autoregressive models trained to generate images. The next section briefly describes the relevant work in this area.
Autoregressive Models for Image Generation
------------------------------------------
Generative image modelling is a key problem in computer vision. In recent years, there has been significant work in this area [@goodfellow2014generative; @kingma2014auto; @rezende2014stochastic; @salimans2017pixelcnn++; @van2016pixel; @zhang2017stackgan; @zhao2018modular]. Although most work uses stochastic latent variable models like VAEs [@kingma2014auto; @rezende2014stochastic] or GANs [@goodfellow2014generative; @zhang2017stackgan; @zhao2018modular], autoregressive models [@salimans2017pixelcnn++; @van2016conditional; @van2016pixel] provide a more tractable approach to model the joint distribution over the pixels. These models leverage the inherent structure over the image, which enables them to express the joint distribution as a product of conditional distributions - where the value of the next pixel is dependent on all the previously generated pixels.
Van [@van2016pixel] propose a PixelRNN network that uses LSTMs [@hochreiter1997long] to model this sequential dependency between the pixels. They also introduce a variant, called PixelCNN, that uses CNNs instead of LSTMs to allow for faster computations. They later extend PixelCNN to allow the model to be conditioned on some query [@van2016conditional]. Finally, [@salimans2017pixelcnn++] propose further simplifications to the PixelCNN architecture to improve performance.
Our work draws inspiration from the PixelRNN architecture proposed in [@van2016pixel]. We extend PixelRNN to model structural dependencies within attention masks.
Approach
========
Given an input image feature $\mathbf{X} \in \mathbb{R}^{h \times m \times n}$, our goal is to predict a spatial attention mask $\mathbf{A} \in \mathbb{R}^{m\times n}$, where $h$ represents the number of channels, and $m$ and $n$ are the number of rows and the columns of $\mathbf{X}$ respectively. Let $\mathbf{X} = \{\mathbf{x}_{1,1}, \dots, \mathbf{x}_{m,n}\}$, where $\mathbf{x}_{i,j} \in \mathbb{R}^h$ be a column feature corresponding to the spatial location $(i,j)$. Similarly, let $\mathbf{A} = \{a_{1,1}, \dots, a_{m,n}\}$, where $a_{i,j} \in \mathbb{R}$ be the attention value corresponding to $\mathbf{x}_{i,j}$. Formally, we want to model the conditional distribution $p(\mathbf{A}~|~\mathbf{X})$. In certain problems, we may also want to condition on other auxiliary information in addition to $\mathbf{X}$, in VQA on a question. While in this paper we formulate and model attention probabilistically, most traditional attention models directly predict the attention values, which can be regarded as a point estimate (or expected value) of $\mathbf{A}$ under our formulation.
Global attention mechanisms [@lu2016hierarchical; @zhu2016visual7w] predict $\mathbf{A}$ directly from $\mathbf{X}$ using a fully connected layer. Although this makes no assumptions on the factorization of $p(\mathbf{A}~|~\mathbf{X})$, it becomes intractable as the size of $\mathbf{X}$ increases. This is mainly due to the large number of parameters in the fully connected layer.
Local attention mechanisms [@seo2017visual; @seo2016progressive; @woo2018cbam; @zhao2018modular], on the other hand, make strong independence assumptions on the interactions between the attention variables $a_{i,j}$. Particularly, they assume each attention variable $a_{i,j}$ to be independent of other variables given some local spatial context $\delta(\mathbf{x}_{i, j})$. More formally, for local attention mechanisms, $$\begin{aligned}
\begin{split}
p\left(\mathbf{A}~|~\mathbf{X}\right) &\approx \prod_{i=1, j=1}^{i=m, j=n} p\left(a_{i,j}~|~ \delta(\mathbf{x}_{i,j})\right)
\end{split}\end{aligned}$$ Even though such a factorization improves tractability, the strong independence assumption often leads to attention masks that lack coherence and contain sharp discontinuities.
Contrary to local attention mechanisms, our proposed *AttentionRNN* tries to capture some of the structural dependencies between the attention variables $a_{i,j}$. We assume $$\begin{aligned}
\label{eq:chainrule}
p(\mathbf{A}~|~\mathbf{X}) &= \prod_{i=1, j=1}^{i=m, j=n} p\left(a_{i,j} ~|~ \mathbf{a}_{<i,j},~ \mathbf{X})\right)\\
\label{eq:arrnprob}
&\approx \prod_{i=1, j=1}^{i=m, j=n} p\left(a_{i,j} ~|~ \mathbf{a}_{<i,j},~ \delta(\mathbf{x}_{i,j})\right)\end{aligned}$$ where $\mathbf{a}_{<i,j} = \{a_{1,1}, \dots, a_{i-1,j}\}$ (The blue and green region in Figure \[fig:skewing\]). That is, each attention variable $a_{i,j}$ is now dependent on : (i) the local spatial context $\delta(\mathbf{x}_{i,j})$, and (ii) all the previous attention variables $\mathbf{a}_{<i,j}$. Note that Equation \[eq:chainrule\] is just a direct application of the chain rule. Similar to local attention mechanisms, and to reduce the computation overhead, we assume that a local spatial context $\delta(\mathbf{x}_{i,j})$ is a sufficient proxy for the image features $\mathbf{X}$ when computing $a_{i,j}$. Equation \[eq:arrnprob\] describes the final factorization we assume.
One of the key challenges in estimating $\mathbf{A}$ based on Equation \[eq:arrnprob\] is to efficiently compute the term $\mathbf{a}_{<i,j}$. A straightforward solution is to use a recurrent neural network (LSTMs) to summarize the previously predicted attention values $\mathbf{a}_{<i,j}$ into its hidden state. This is a common approach employed in many sequence prediction methods [@bahdanau2014neural; @shankar2018posterior; @venugopalan2015sequence]. However, while sequences have a well defined ordering, image features can be traversed in multiple ways due to their spatial nature. Naively parsing the image along its rows using an LSTM, though provides an estimate for $\mathbf{a}_{<i,j}$, fails to correctly encode the necessary information required to predict $a_{i,j}$. As an example, the LSTM will tend to forget information from the neighbouring variable $a_{i-1,j}$ as it was processed $n$ time steps ago.
To alleviate this issue, *AttentionRNN* instead parses the image in a diagonal fashion, starting from a corner at the top and going to the opposite corner in the bottom. It builds upon the Diagonal BiLSTM layer proposed by [@van2016pixel] to efficiently perform this traversal. The next section describes our proposed *AttentionRNN* mechanism in detail.
AttentionRNN {#sec:attention}
------------
Our proposed structured attention mechanism builds upon the Diagonal BiLSTM layer proposed by [@van2016pixel]. We employ two LSTMs, one going from the top-left to bottom-right corner ($\mathcal{L}^{l}$) and the other from the top-right to the bottom-left corner ($\mathcal{L}^r$).
As mentioned in Equation \[eq:arrnprob\], for each $a_{i,j}$, our objective is to estimate $p\left(a_{i,j} ~|~ \mathbf{a}_{<i,j}, \delta(\mathbf{x}_{i,j})\right)$. We assume that this can be approximated via a combination of two distributions. $$\begin{aligned}
\label{eq:decompose}
\begin{split}
p\left(a_{i,j}|\mathbf{a}_{<i,j}\right) &= \Gamma\left<p\left(a_{i,j}| \mathbf{a}_{<i,<j}\right),~p\left(a_{i,j}|\mathbf{a}_{<i,>j}\right) \right>
\end{split}\end{aligned}$$ where $\mathbf{a}_{<i,<j}$ is the set of attention variables to the top and left (blue region in Figure \[fig:skewing\]) of $a_{i,j}$, $\mathbf{a}_{<i,>j}$ is the set of attention variables to the top and right of $a_{i,j}$ (green region in Figure \[fig:skewing\]), and $\Gamma$ is some combination function. For brevity, we omit explicitly writing $\delta(\mathbf{x}_{i,j})$. Equation \[eq:decompose\] is further simplified by assuming that all distributions are Gaussian. $$\begin{aligned}
\label{eq:gaussian}
\begin{split}
p\left(a_{i,j}|\mathbf{a}_{<i,<j}\right) &\approx \mathcal{N}\left(\mu_{i,j}^l, {\sigma_{i,j}^l} \right)\\
p\left(a_{i,j}|\mathbf{a}_{<i,>j}\right) &\approx \mathcal{N}\left(\mu_{i,j}^r, {\sigma_{i,j}^r} \right)\\
p\left(a_{i,j}|\mathbf{a}_{<i,j}\right) &\approx \mathcal{N}\left(\mu_{i,j}, {\sigma_{i,j}} \right)
\end{split}\end{aligned}$$ where, $$\begin{aligned}
\label{eq:gaussparams}
\begin{split}
(\mu_{i,j}^l, \sigma_{i,j}^l)& = f_l\left(\mathbf{a}_{<i,<j}\right);~~ (\mu_{i,j}^r, \sigma_{i,j}^r) = f_r\left(\mathbf{a}_{<i,>j}\right)\\
&(\mu_{i,j}, \sigma_{i,j}) = \Gamma\left(\mu_{i,j}^l, \sigma_{i,j}^l, \mu_{i,j}^r, \sigma_{i,j}^r \right)
\end{split}\end{aligned}$$ $f_l$ and $f_r$ are fully connected layers. Our choice for the combination function $\Gamma$ is explained in Section \[sec:combination\]. For each $a_{i,j}$, $\mathcal{L}^l$ is trained to estimate $(\mu_{i,j}^l, \sigma_{i,j}^l)$, and $\mathcal{L}^r$ is trained to estimate $(\mu_{i,j}^r, \sigma_{i,j}^r)$. We now explain the computation for $\mathcal{L}^l$. $\mathcal{L}^r$ is analogous and has the same formulation.
$\mathcal{L}^l$ needs to correctly approximate $\mathbf{a}_{<i,<j}$ in order to obtain a good estimate of $(\mu_{i,j}^l, \sigma_{i,j}^l)$. As we are parsing the image diagonally, from Figure \[fig:skewing\] it can be seen that the following recursive relation holds, $$\begin{aligned}
\label{eq:recursion}
\mathbf{a}_{<i,<j} = f(\mathbf{a}_{<i-1,<j}~~,~~ \mathbf{a}_{<i,<j-1})\end{aligned}$$ That is, for each location $(i,j)$, $\mathcal{L}^l$ only needs to consider two attention variables- one above and the other to the left; [@van2016pixel] show that this is sufficient for it to be able to obtain information from all the previous attention variables.
![[**Skewing operation.**]{} This makes it easier to compute convolutions along the diagonal. The arrows indicate dependencies between attention values. To obtain the image on the right, each row of the left image is offset by one position with respect to its previous row.[]{data-label="fig:skewing"}](skew){width="48.00000%"}
To make computations along the diagonal easier, similar to [@van2016pixel], we first skew $\mathbf{X}$ into a new image feature $\mathbf{\widehat{X}}$. Figure \[fig:skewing\] illustrates the skewing procedure. Each row of $\mathbf{X}$ is offset by one position with respect to the previous row. $\mathbf{\widehat{X}}$ is now an image feature of size $h\times m \times (2n - 1)$. Traversing $\mathbf{X}$ in a diagonal fashion from top left to bottom right is now equivalent to traversing $\mathbf{\widehat{X}}$ along its columns from left to right. As spatial locations $(i-1,j)$ and $(i,j-1)$ in $\mathbf{X}$ are now in the same column in $\widehat{\mathbf{X}}$, we can implement the recursion described in Equation \[eq:recursion\] efficiently by performing computations on an entire column of $\widehat{\mathbf{X}}$ at once.
Let $\mathbf{\widehat{X}}_j$ denote the $j^{th}$ column of $\mathbf{\widehat{X}}$. Also, let $\mathbf{\widehat{h}}_{j-1}^l$ and $\mathbf{\widehat{c}}_{j-1}^l$ respectively denote the hidden and memory state of $\mathcal{L}^l$ before processing $\mathbf{\widehat{X}}_j$. Both $\mathbf{\widehat{h}}_{j-1}^l$ and $\mathbf{\widehat{c}}_{j-1}^l$ are tensors of size $t\times m$, where $t$ is the number of latent features. The new hidden and memory state is computed as follows. $$\begin{aligned}
\label{eq:gates}
\begin{split}
[\mathbf{o}_j, \mathbf{f}_j, \mathbf{i}_j, \mathbf{g}_j] &= \sigma\left(\mathbf{K}^{h} \circledast \mathbf{\widehat{h}}_{j-1}^l + \mathbf{K}^{x} \circledast \mathbf{\widehat{X}}_{j}^c\right)\\
\mathbf{\widehat{c}}_{j}^l &= \mathbf{f}_j \odot \mathbf{\widehat{c}}_{j-1}^l + \mathbf{i}_j \odot \mathbf{g}_j\\
\mathbf{\widehat{h}}_{j}^l &= \mathbf{o}_j \odot \text{tanh}(\mathbf{c}_{j}^l)
\end{split}\end{aligned}$$ Here $\circledast$ represents the convolution operation and $\odot$ represents element-wise multiplication. $\mathbf{K}^h$ is a $2 \times 1$ convolution kernel which effectively implements the recursive relation described in Equation \[eq:recursion\], and $\mathbf{K}^x$ is a $1 \times 1$ convolution kernel. Both $\mathbf{K}^h$ and $\mathbf{K}^u$ produce a tensor of size $4t \times m$. $\mathbf{\widehat{X}}_{j}^c$ is the $j^{th}$ column of the skewed local context $\mathbf{\widehat{X}}^c$, which is obtained as follows. $$\begin{aligned}
\label{eq:localcontext}
\begin{split}
\mathbf{\widehat{X}}^c = \text{skew}\left(\mathbf{K}^c \circledast \mathbf{X}\right)
\end{split}\end{aligned}$$ where $\mathbf{K}^c$ is a convolutional kernel that captures a $\delta$-size context. For tasks that are multi-modal in nature, a query $\mathbf{Q}$ can additionally be used to condition the generation of $a_{i,j}$. This allows the model to generate different attention mask for the same image features depending on $\mathbf{Q}$. For example, in tasks like VQA, the relevant regions of an image will depend on the question asked. The nature of $\mathbf{Q}$ will also dictate the encoding procedure. As an example, if $\mathbf{Q}$ is a natural language question, it can be encoded using a LSTM layer. $\mathbf{Q}$ can be easily incorporated into *AttentionRNN* by concatenating it with $\mathbf{\widehat{X}}^c$ before passing it to Equation \[eq:gates\].
Let $\mathbf{\widehat{h}}^l = \{\mathbf{\widehat{h}}_{1}^l,\dots, \mathbf{\widehat{h}}_{2n-1}^l\}$ be the set of all hidden states obtained from $\mathcal{L}^l$, and $\mathbf{h}^l$ be the set obtained by applying the reverse skewing operation on $\mathbf{\widehat{h}}^l$. For each $a_{i,j}$, $\mathbf{a}_{<i,<j}$ is then simply the $(i,j)$ spatial element of $\mathbf{h}^l$. $\mathbf{a}_{<i,>j}$ can be obtained by repeating the aforementioned process for $\mathcal{L}^r$, which traverses $\mathbf{X}$ from top-right to bottom-left. Note that this is equivalent to running $\mathcal{L}^r$ from top-left to bottom-right after mirroring $\mathbf{X}$ along the column dimension, and then mirroring the output hidden states $\mathbf{h}^r$ again. Similar to [@van2016pixel], $\mathbf{h}^r$ is shifted down by one row to prevent $\mathbf{a}_{<i,>j}$ from incorporating future attention values.
Once $\mathbf{a}_{<i,<j}$ and $\mathbf{a}_{<i,>j}$ are computed (as discussed above), we can obtain the Gaussian distribution for the attention variable $\mathcal{N}\left(\mu_{i,j}, \sigma_{i,j} \right)$ by following Equation \[eq:gaussparams\]. The attention $a_{i,j}$ could then be obtained by either sampling a value from $\mathcal{N}\left(\mu_{i,j}, \sigma_{i,j} \right)$ or simply by taking the expectation and setting $a_{i,j} = \mu_{i,j}$. For most problems, as we will see in the experiment section, taking the expectation is going to be most efficient and effective. However, sampling maybe useful in cases where attention is inherently multi-modal. Focusing on different modes using coherent masks might be more beneficial in such situations.
![[**Block AttentionRNN**]{} for $\gamma=2$. The input is first down-sized using a $\gamma \times \gamma$ convolutional kernel. Attention is computed on this smaller map.[]{data-label="fig:upscale"}](upscale){width="\linewidth"}
Combination Function {#sec:combination}
--------------------
The choice of the combination function $\Gamma$ implicitly imposes some constraints on the interaction between the distributions $\mathcal{N}\left(\mu_{i,j}^l, \sigma_{i,j}^l\right)$ and $\mathcal{N}\left(\mu_{i,j}^r, \sigma_{i,j}^r\right)$. For example, assumption of independence would dictate a simple product for $\Gamma$, with resulting operations to produce $(\mu_{i,j}, \sigma_{i,j})$ being expressed in closed form. However, it is clear that independence is unlikely to hold due to image correlations. To allow for a more flexible interaction between variables and combination function, we instead use a fully connected layer to learn the appropriate $\Gamma$ for a particular task. $$\begin{aligned}
\label{eq:combination}
\begin{split}
\mu_{i,j}, \sigma_{i,j} = f_{comb}\left(\mu_{i,j}^l, \sigma_{i,j}^l, \mu_{i,j}^r, \sigma_{i,j}^r\right)
\end{split}\end{aligned}$$
Block AttentionRNN {#sec:upscale}
------------------
Due to the poor performance of LSTMs over large sequences, the AttentionRNN layer doesn’t scale well to large image feature maps. We introduce a simple modification to the method described in Section \[sec:attention\] to alleviate this problem, which we refer to as Block AttentionRNN (BRNN).
BRNN reduces the size of the input feature map $\mathbf{X}$ before computing the attention mask. This is done by splitting $\mathbf{X}$ into smaller blocks, each of size $\gamma \times \gamma$. This is equivalent to down-sampling the original image $\mathbf{X}$ to $\mathbf{X}^{ds}$ as follows. $$\begin{aligned}
\mathbf{X}^{ds} = \mathbf{K}^{ds} \circledast \mathbf{X}\end{aligned}$$ where $\mathbf{K}^{ds}$ is a convolution kernel of size $\gamma \times \gamma$ applied with stride $\gamma$. In essence, each value in $\mathbf{X}^{ds}$ now corresponds to a $\gamma \times \gamma$ region in $\mathbf{X}$.
Instead of predicting a different attention probability for each individual spatial location $(i,j)$ in $\mathbf{X}$, BRNN predicts a single probability for each $\gamma \times \gamma$ region. This is done by first computing the attention mask $\mathbf{A}^{ds}$ for the down-sampled image $\mathbf{X}^{ds}$ using AttentionRNN (Section \[sec:attention\]), and then $\mathbf{A}^{ds}$ is then scaled up using a transposed convolutional layer to obtain the attention mask $\mathbf{A}$ for the original image feature $\mathbf{X}$. Figure \[fig:upscale\] illustrates the BRNN procedure.
BRNN essentially computes a coarse attention mask for $\mathbf{X}$. Intuitively, this coarse attention can be used in the first few layers of a deep CNN network to identify the key region blocks in the image. The later layers can use this coarse information to generate a more granular attention mask.
Experiments
===========
To show the efficacy and generality of our approach, we conduct experiments over four different tasks: visual attribute prediction, image classification, visual question answering (VQA) and image generation. We highlight that our goal is not to necessarily obtain the absolute highest raw performance (although we do in many of our experiments), but to show improvements from integrating AttentionRNN into existing state-of-the-art models across a variety of tasks and architectures. Due to space limitations, all model architectures and additional visualizations are described in the supplementary material.
[.15]{} ![[**Synthetic Dataset Samples.**]{} Example images taken from the three synthetic datasets proposed in [@seo2017visual].[]{data-label="fig:synthetic"}](mref "fig:"){width="0.9\linewidth"}
[.15]{} ![[**Synthetic Dataset Samples.**]{} Example images taken from the three synthetic datasets proposed in [@seo2017visual].[]{data-label="fig:synthetic"}](mdist "fig:"){width="0.9\linewidth"}
[.15]{} ![[**Synthetic Dataset Samples.**]{} Example images taken from the three synthetic datasets proposed in [@seo2017visual].[]{data-label="fig:synthetic"}](mbg "fig:"){width="0.9\linewidth"}
Visual Attribute Prediction {#subsec:vap}
---------------------------
**Datasets.** We experiment on the synthetic MREF, MDIST and MBG datasets proposed in [@seo2016progressive]. Figure \[fig:synthetic\] shows example images from the datasets. The images in the datasets are created from MNIST [@lecun1998gradient] by sampling five to nine distinct digits with different colors (green, yellow, white, red, or blue) and varying scales (between 0.5 and 3.0). The datasets have images of size 100 x 100 and only differ in how the background is generated. MREF has a black background, MDIST has a black background with some Gaussian noise, and MBG has real images sampled from the SUN Database [@xiao2016sun] as background. The training, validation and test sets contain 30,000, 10,000 and 10,000 images respectively.
[l|ccc|c]{} Attention & MREF & MDIST & MBG &
---------
Rel.
Runtime
---------
: [**Color prediction accuracy.**]{} Results are in % on MREF, MDIST and MBG datasets. Our AttentionRNN-based model, CNN+ARNN, outperforms all the baselines.[]{data-label="tab:vapresultspred"}
\
SAN [@xu2015show] & 83.42 & 80.06 & 58.07 & 1x\
$\lnot \text{CTX}$ [@seo2016progressive] & 95.69 & 89.92 & 69.33 & 1.08x\
$\text{CTX}$ [@seo2016progressive] & 98.00 & 95.37 & 79.00 & 1.10x\
$\text{ARNN}_{ind}^{\sim}$ & 98.72 & 96.70 & 83.68 &\
$\text{ARNN}_{ind}$ & 98.58 & 96.29 & 84.27 &\
$\text{ARNN}^{\sim}$ & 98.65 & 96.82 & 83.74 &\
$\text{ARNN}$ & **98.93** & **96.91** & **85.84** &\
**Experimental Setup.** The performance of AttentionRNN (ARNN) is compared against two [*local*]{} attention mechanisms proposed in [@seo2016progressive], which are referred as $\lnot {\text{CTX}}$ and CTX. ARNN assumes $a_{i,j}=\mu_{i,j}, \delta=3$, where $\mu_{i,j}$ is defined in Equation \[eq:combination\]. To compute the attention for a particular spatial location $(i,j)$, CTX uses a $\delta = 3$ local context around $(i,j)$, whereas $\lnot {\text{CTX}}$ only uses the information from location $(i,j)$. We additionally define three variants of ARNN: i) $\text{ARNN}^\sim$ where each $a_{i,j}$ is sampled from $\mathcal{N}\left(\mu_{i,j}, \sigma_{i,j} \right)$, ii) $\text{ARNN}_{ind}$ where the combination function $\Gamma$ assumes the input distributions are independent, and iii) $\text{ARNN}_{ind}^{\sim}$ where $\Gamma$ assumes independence and $a_{i,j}$ is sampled. The soft attention mechanism (SAN) proposed by [@xu2015show] is used as an additional baseline. The same base CNN architecture is used for all the attention mechanisms for fair comparison. The CNN is composed of four stacks of $3 \times 3$ convolutions with 32 channels followed by $2 \times 2$ max pooling layer. SAN computes attention only on the output of the last convolution layer, while $\lnot {\text{CTX}}$, CTX and all variants of ARNN are applied after each pooling layer. Given an image, the models are trained to predict the color of the number specified by a query. Chance performance is 20%.
[0.5]{} \[subfig1:viz\] {width="0.9\linewidth"}
[0.5]{} \[subfig2:viz\] {width="0.9\linewidth"}
**Results.** Table \[tab:vapresultspred\] shows the color prediction accuracy of various models on MREF, MDIST and MBG datasets. It can be seen that ARNN and all its variants clearly outperform the other baseline methods. The difference in performance is amplified for the more noisy MBG dataset, where ARNN is 6.8% better than the closest baseline. $\text{ARNN}_{ind}$ performs poorly compared to $\text{ARNN}$, which furthers the reasoning of using a neural network to model $\Gamma$ instead of assuming independence. Similar to [@seo2016progressive], we also evaluate the models on their sensitivity to the size of the target. The test set is divided into five uniform scale intervals for which model accuracy is computed. Table \[tab:vapresultsscale\] shows the results on the MBG dataset. ARNN is robust to scale variations and performs consistently well on small and large targets. We also test the correctness of the mask generated using the metric proposed by [@liu2017attention], which computes the percentage of attention values in the region of interest. For models that apply attention after each pooling layer, the masks from different layers are combined by upsampling and taking a product over corresponding pixel values. The results are shown for the MBG dataset in Table \[tab:vapresultsscale\]. ARNN is able to more accurately attend to the correct regions, which is evident from the high correctness score.
[@l|c|ccccc@]{} & &\
& & 0.5-1.0 & 1.0-1.5 & 1.5-2.0 & 2.0-2.5 & 2.5-3.0\
SAN [@xu2015show] & 0.15 & 53.05 & 74.85 & 72.18 & 59.52 & 54.91\
$\lnot\text{CTX}$ [@seo2016progressive] & 0.28 & 68.20 & 76.37 & 73.30 & 61.60 & 57.28\
CTX [@seo2016progressive] & 0.31 & 77.39 & 87.13 & 84.96 & 75.59 & 63.72\
$\text{ARNN}_{ind}^{\sim}$ & 0.36 & 82.23 & 89.41 & 86.46 & 84.52 & 81.35\
$\text{ARNN}_{ind}$ & 0.34 & 82.89 & 89.47 & 88.34 & 84.22 & 80.00\
$\text{ARNN}^{\sim}$ & 0.39 & 82.23 & 89.41 & 86.46 & 84.52 & 81.35\
$\text{ARNN}$ & **0.42** & **84.45** & **91.40** & **86.84** & **88.39** & **82.37**\
From Tables \[tab:vapresultspred\] and \[tab:vapresultsscale\], it can be seen that $\text{ARNN}^{\sim}$ provides no significant advantage over its deterministic counterpart. This can be attributed to the datasets encouraging point estimates, as each input query can only have one correct answer. As a consequence, for each $a_{i,j}$, $\sigma_{i,j}$ was observed to underestimate variance. However, in situations where an input query can have multiple correct answers, $\text{ARNN}^{\sim}$ can be used to generate diverse attention masks. To corroborate this claim, we test the pre-trained $\text{ARNN}^{\sim}$ on images that are similar to the MBG dataset but have the same digit in multiple colors. Figure \[subfig1:viz\] shows the individual layer attended feature maps for three different samples from $\text{ARNN}^{\sim}$ for a fixed image and query. For the query “9”, $\text{ARNN}^{\sim}$ is able to identify the three modes. Note that since $\sigma_{i,j}$’s were underestimated due to aforementioned reasons, they were scaled up before generating the samples. Despite being underestimated $\sigma_{i,j}$’s still capture crucial information.
**Inverse Attribute Prediction.** Figure \[subfig1:viz\] leads to an interesting observation regarding the nature of the task. Even though $\text{ARNN}^{\sim}$ is able to identify the correct number, it only needs to focus on a tiny part of the target region to be able to accurately classify the color. To further demonstrate the ARNN’s ability to model longer dependencies, we test the performance of ARNN, CTX and $\lnot\text{CTX}$ on the $\text{MBG}^{inv}$ dataset, which defines the inverse attribute prediction problem - given a color identify the number corresponding to that color. The base CNN architecture is identical to the one used in the previous experiment. ARNN, CTX and $\lnot\text{CTX}$ achieve an accuracy of 72.77%, 66.37% and 40.15% and a correctness score[@liu2017attention] of 0.39, 0.24 and 0.20 respectively. Figure \[subfig2:viz\] shows layer-wise attended feature maps for the three models. ARNN is able to capture the entire number structure, whereas the other two methods only focus on a part of the target region. Even though CTX uses some local context to compute the attention masks, it fails to identify the complete structure for the number “0”. A plausible reason for this is that a $3 \times 3$ local context is too small to capture the entire target region. As a consequence, the attention mask is computed in patches. CTX maintains no information about the previously computed attention values, and therefore is unable to assign correlated attention scores for all the different target region patches. ARNN, on the other hand, captures constraints between attention variables, making it much more effective in this situation.
[@c|c|ccccc@]{} & &\
& Total & 0.5-1.0 & 1.0-1.5 & 1.5-2.0 & 2.0-2.5 & 2.5-3.0\
NONE & 91.43 & 85.63 & 92.57 & 94.96 & 94.77 & 93.59\
ARNN & 91.09 & 84.89 & 92.25 & 94.24 & 94.70 & **94.52**\
$\text{BRNN}^{\gamma=3}$ & 91.67 & 85.97 & 93.46 & 94.81 & 94.35 & 93.68\
$\text{BRNN}^{\gamma=2}$ & **92.68** & **88.10** & **94.23** & **95.32** & **94.80** & 94.01\
**Scalability of ARNN.** The results shown in Table \[tab:vapresultspred\] correspond to models trained on $100\times100$ input images, where the first attention layer is applied on an image feature of size $50\times50$. To analyze the performance of ARNN on comparatively larger image features, we create a new dataset of $224\times224$ images which we refer to as $\text{MBG}^\text{b}$. The data generation process for $\text{MBG}^\text{b}$ is identical to MBG. We perform an ablation study analyzing the effect of using Block AttentionRNN (BRNN) (Section \[sec:upscale\]) instead of ARNN on larger image features. For the base architecture, the ARNN model from the previous experiment is augmented with an additional stack of convolutional and max pooling layer. The detailed architecture is mentioned in the supplementary material. Table \[tab:ablation\] shows the color prediction accuracy on different scale intervals for the $\text{MBG}^\text{b}$ dataset. As the first attention layer is now applied on a feature map of size $112\times 112$, ARNN performs worse than the case when no attention (NONE) is applied due to the poor tractability of LSTMs over large sequences. BRNN$^{\gamma=2}$, on the other hand, is able to perform better as it reduces the image feature size before applying attention. However, there is a considerable difference in the performance of BRNN when $\gamma=2$ and $\gamma=3$. When $\gamma=3$, BRNN applies a $3 \times 3$ convolution with stride $3$. This aggressive size reduction causes loss of information.
Image Classification
--------------------
**Dataset.** We use the CIFAR-100 [@krizhevsky2009learning] to verify the performance of AttentionRNN on the task of image classification. The dataset consists of 60,000 $32\times32$ images from 100 classes. The training/test set contain 50,000/10,000 images. **Experimental Setup.** We augment ARNN to the convolution block attention module (CBAM) proposed by [@woo2018cbam]. For a given feature map, CBAM computes two different types of attentions: 1) channel attention that exploits the inter-channel dependencies in a feature map, and 2) spatial attention that uses local context to identify relationships in the spatial domain. We replace *only* the spatial attention in CBAM with ARNN. This modified module is referred to as CBAM+ARNN. ResNet18 [@he2016deep] is used as the base model for our experiments. ResNet18+CBAM is the model obtained by using CBAM in the Resnet18 model, as described in [@woo2018cbam]. Resnet18+CBAM+ARNN is defined analogously. We use a local context of $3 \times 3$ to compute the spatial attention for both CBAM and CBAM+ARNN.
**Results.** Top-1 and top-5 error is used to evaluate the performance of the models. The results are summarized in Table \[tab:cbam\]. CBAM+ARNN provides an improvement of 0.89% on top-1 error over the closest baseline. Note that this gain, though seems marginal, is larger than what CBAM obtains over ResNet18 with no attention (0.49% on top-1 error).
[@l|cc|c@]{} &
-----------
Top-1
Error (%)
-----------
: **Performance on Image Classification.** The Top-1 and Top-5 error % are shown for all the models. The ARNN based model outperforms all other baselines.[]{data-label="tab:cbam"}
&
-----------
Top-5
Error (%)
-----------
: **Performance on Image Classification.** The Top-1 and Top-5 error % are shown for all the models. The ARNN based model outperforms all other baselines.[]{data-label="tab:cbam"}
&
---------
Rel.
Runtime
---------
: **Performance on Image Classification.** The Top-1 and Top-5 error % are shown for all the models. The ARNN based model outperforms all other baselines.[]{data-label="tab:cbam"}
\
ResNet18 [@he2016deep] & 25.56 & 6.87 & 1x\
ResNet18 + CBAM [@woo2018cbam] & 25.07 & 6.57 & 1.43x\
ResNet18 + CBAM + ARNN & **24.18** & **6.42** & 4.81x\
[l|cccc|c]{} & Yes/No & Number & Other & Total &
---------
Rel.
Runtime
---------
: **Performance on VQA.** In % accuracy.[]{data-label="tab:vqa"}
\
MCB [@fukui2016multimodal] & 76.06 & 35.32 & 43.87 & 54.84 & 1x\
MCB+ATT [@fukui2016multimodal] & 76.12 & 35.84 & 47.84 & 56.89 & 1.66x\
MCB+ARNN & **77.13** & **36.75** & **48.23** & **57.58** & 2.46x\
Visual Question Answering
-------------------------
**Dataset.** We evaluate the performance of ARNN on the task of VQA [@antol2015iccv]. The experiments are done on the VQA 2.0 dataset [@goyal2017making], which contains images from MSCOCO [@lin2014microsoft] and corresponding questions. As the test set is not publicly available, we evaluate performance on the validation set.
**Experimental Setup.** We augment ARNN to the Multimodal Compact Bilinear Pooling (MCB) architecture proposed by [@fukui2016multimodal]. This is referred to as MCB+ARNN. Note that even though MCB doesn’t give state-of-the-art performance on this task, it is a competitive baseline that allows for easy ARNN integration. MCB+ATT is a variant to MCB that uses a local attention mechanism with $\delta=1$ from [@fukui2016multimodal]. For fair comparison, MCB+ARNN also uses a $\delta=1$ context.
**Results.** The models are evaluated using the accuracy measure defined in [@antol2015iccv]. The results are summarized in Table \[tab:vqa\]. MCB+ARNN achieves a 0.69% improvement over the closest baseline. We believe this marginal improvement is because all the models, for each spatial location $(i,j)$, use no context from neighbouring locations (as $\delta=1$).
Image Generation
----------------
**Dataset.** We analyze the effect of using ARNN on the task of image generation. Experiments are performed on the CelebA dataset [@liu2015faceattributes], which contains 202,599 face images of celebrities, with 40 binary attributes. The data pre-processing is identical to [@zhao2018modular]. The models are evaluated on three attributes: hair color = [*{black, blond, brown}*]{}, gender = [*{male, female*}]{}, and smile = [*{smile, nosmile}*]{}.
**Experimental Setup.** We compare ARNN to a local attention mechanism used in the ModularGAN (MGAN) framework [@zhao2018modular]. MGAN uses a $3 \times 3$ local context to obtain attention values. We define MGAN+ARNN as the network obtained by replacing the local attention with ARNN. The models are trained to transform an image given an attribute.
**Results.** To evaluate the performance of the models, similar to [@zhao2018modular], we train a ResNet18 [@he2016deep] model that classifies the hair color, facial expression and gender on the CelebA dataset. The trained classifier achieves an accuracy of 93.9%, 99.0% and 93.7% on hair color, gender and smile respectively. For each transformation, we pass the generated images through this classifier and compute the classification error (shown in Table \[tab:modulargan\]). MGAN+ARNN outperforms the baseline on all categories except *hair color*. To analyse this further, we look at the attention masks generated for the *hair color* transformation by both models. As shown in Figure \[fig:gan\], we observe that the attention masks generated by MGAN lack coherence over the target region due to discontinuities. MGAN+ARNN, though has a slightly higher classification error, generates uniform activation values over the target region by encoding structural dependencies.
[0.45]{} ![[**Qualitative Results on ModularGAN.**]{} Attention masks generated by original ModularGAN [@zhao2018modular] and ModularGAN augmented with ARNN are shown. Notice that the hair mask is more uniform for MGAN+ARNN as it is able to encode structural dependencies in the attention mask. Additional results shown in supplementary material.[]{data-label="fig:gan"}](gan1_1 "fig:"){width="\linewidth"}
[l|ccc|c]{} & Hair & Gender & Smile &
---------
Rel.
Runtime
---------
: **Performance on Image Generation.** ResNet18 [@he2016deep] Classification Errors (%) for each attribute transformation. ARNN achieves better performance on two tasks.[]{data-label="tab:modulargan"}
\
MGAN [@zhao2018modular] & **2.5** & 3.2 & 12.6 &1x\
MGAN+ARNN & 3.0 & **1.4** & **11.4** &1.96x\
Conclusion
==========
In this paper, we developed a novel [*structured*]{} spatial attention mechanism which is end-to-end trainable and can be integrated with any feed-forward convolutional neural network. The proposed AttentionRNN layer explicitly enforces structure over the spatial attention variables by sequentially predicting attention values in the spatial mask. Experiments show consistent quantitative and qualitative improvements on a large variety of recognition tasks, datasets and backbone architectures.
Section 1 explains the architectures for the models used in the experiments (Section 4 in the main paper). Section 2 provides additional visualizations for the task of Visual Attribute Prediction (Section 4.1 in the main paper) and Image Generation (Section 4.4 in the main paper). These further show the effectiveness of our proposed structured attention mechanism.
Model Architectures
===================
Visual Attribute Prediction {#visual-attribute-prediction}
---------------------------
Please refer to Section 4.1 of the main paper for the task definition. Similar to [@seo2016progressive], the base CNN architecture is composed of four stacks of $3 \times 3$ convolutions with 32 channels followed by $2 \times 2$ max pooling layer. SAN computes attention only on the output of the last convolution layer, while $\lnot {\text{CTX}}$, CTX and all variants of ARNN are applied after each pooling layer. Table \[tab:modelarc\] illustrates the model architectures for each network. {$\lnot\text{CTX}$, CTX, ARNN}$_{sigmoid}$ refers to using sigmoid non-linearity on the generated attention mask before applying it to the image features. Similarly, {$\lnot\text{CTX}$, CTX, ARNN}$_{softmax}$ refers to using softmax non-linearity on the generated attention mask. We use the same hyper-parameters and training procedure for all models, which is identical to [@seo2016progressive].
For the scalability experiment described in Section 4.1, we add an additional stack of $3 \times 3$ convolution layer followed by a $2 \times 2$ max pooling layer to the ARNN architecture described in Table \[tab:modelarc\]. This is used as the base architecture. Table \[tab:ablationsupp\] illustrates the differences between the models used to obtain results mentioned in Table 3 of the main paper.
[|c|c|c|c|]{} **SAN** & **$\lnot \text{CTX}$** & **CTX** & **ARNN**\
\
\
$\downarrow$ & $\lnot \text{CTX}_{sigmoid}$ & CTX$_{sigmoid}$ & ARNN$_{sigmoid}$\
\
\
$\downarrow$ & $\lnot \text{CTX}_{sigmoid}$ & CTX$_{sigmoid}$ & ARNN$_{sigmoid}$\
\
\
$\downarrow$ & $\lnot \text{CTX}_{sigmoid}$ & CTX$_{sigmoid}$ & ARNN$_{sigmoid}$\
\
\
SAN & $\lnot \text{CTX}_{softmax}$ & CTX$_{softmax}$ & ARNN$_{softmax}$\
[|c|c|c|c|]{} **NONE** & **ARNN** & **BRNN**\
\
\
$\downarrow$ & $\text{ARNN}_{sigmoid}$ & $\text{BRNN}^{\delta}_{sigmoid}$\
\
Image Classification
--------------------
Please refer to Section 4.2 of the main paper for the task definition. We augment ARNN to the convolution block attention module (CBAM) proposed by [@woo2018cbam]. For a given feature map, CBAM computes two different types of attentions: 1) channel attention that exploits the inter channel dependencies in a feature map, and 2) spatial attention that uses local context to identify relationships in the spatial domain. Figure \[subfig1:cbam\] shows the CBAM module integrated with a ResNet [@he2016deep] block. We replace only the *spatial attention* in CBAM with ARNN. This modified module is referred to as CBAM+ARNN. Figure \[subfig2:cbam\] better illustrates this modification. Both CBAM and CBAM+ARNN use a local context of $3 \times 3$ to compute attention. We use the same hyper-parameters and training procedure for both CBAM and CBAM+ARNN, which is identical to [@woo2018cbam].
[0.85]{} ![[**Difference between CBAM and CBAM+ARNN.**]{} (a) CBAM[@woo2018cbam] module integrated with a ResNet[@he2016deep] block. (b) CBAM+ARNN replaces the spatial attention in CBAM with ARNN. It is applied similar to (a) after each ResNet[@he2016deep] block. Refer to Section 4.2 of the main paper for more details.[]{data-label="fig:cbam"}](cbam1 "fig:"){width="\linewidth"}
[0.85]{} ![[**Difference between CBAM and CBAM+ARNN.**]{} (a) CBAM[@woo2018cbam] module integrated with a ResNet[@he2016deep] block. (b) CBAM+ARNN replaces the spatial attention in CBAM with ARNN. It is applied similar to (a) after each ResNet[@he2016deep] block. Refer to Section 4.2 of the main paper for more details.[]{data-label="fig:cbam"}](cbam+arnn "fig:"){width="\linewidth"}
Visual Question Answering
-------------------------
Please refer to Section 4.3 of the main paper for task definition. We use the Multimodal Compact Bilinear Pooling with Attention (MCB+ATT) architecture proposed by [@fukui2016multimodal] as a baseline for our experiment. To compute attention, MCB+ATT uses two $1 \times 1$ convolutions over the features obtained after using the compact bilinear pooling operation. Figure \[subfig1:mcb\] illustrates the architecture for MCB+ATT. We replace this attention with ARNN to obtain MCB+ARNN. MCB+ARNN also uses a $1 \times 1$ local context to compute attention. Figure \[subfig2:mcb\] better illustrates this modification. We use the same hyper-parameters and training procedure for MCB, MCB+ATT and MCB+ARNN, which is identical to [@fukui2016multimodal].
![[**Difference between MCB+ATT and MCB+ARNN.**]{} (a) MCB+ATT model architecture proposed by [@fukui2016multimodal]. It uses a $1\times 1$ context to compute attention over the image features. (b) MCB+ARNN replaces the attention mechanism in MCB+ATT with ARNN. It is applied in the same location as (a) with $1 \times 1$ context. Refer to Section 4.3 of the main paper for more details.[]{data-label="fig:mcb"}](mcb){width="\linewidth"}
![[**Difference between MCB+ATT and MCB+ARNN.**]{} (a) MCB+ATT model architecture proposed by [@fukui2016multimodal]. It uses a $1\times 1$ context to compute attention over the image features. (b) MCB+ARNN replaces the attention mechanism in MCB+ATT with ARNN. It is applied in the same location as (a) with $1 \times 1$ context. Refer to Section 4.3 of the main paper for more details.[]{data-label="fig:mcb"}](mcb+arnn){width="\linewidth"}
Image Generation
----------------
Please refer to Section 4.4 of the main paper for task definitions. We compare ARNN to a local attention mechanism used in the ModularGAN (MGAN) framework [@zhao2018modular]. MGAN consists of three modules: 1) encoder module that encodes an input image into an intermediate feature representation, 2) generator module that generates an image given an intermediate feature representation as input, and 3) transformer module that transforms a given intermediate representation to a new intermediate representation according to some input condition. The transformer module uses a $3×3$ local context to compute attention over the feature representations. Figure \[subfig1:mgan\] illustrates the transformer module proposed by [@zhao2018modular]. We define MGAN+ARNN as the network obtained by replacing this local attention mechanism in the transformer module with ARNN. Note that the generator and encoder modules are unchanged. MGAN+ARNN also uses a $3 \times 3$ local context to compute attention. Figure \[subfig2:mgan\] better illustrates this modification to the transformer module. We use the same hyper-parameters and training procedure for both MGAN and MGAN+ARNN, which is identical to [@zhao2018modular].
[0.85]{} ![[**Difference between MGAN and MGAN+ARNN.**]{} (a) The transformer module for the ModularGAN (MGAN) architecture proposed by [@zhao2018modular]. It uses a $3\times 3$ local context to compute attention over the intermediate features. (b) MGAN+ARNN replaces the attention mechanism in MGAN with ARNN. It is applied in the same location as (a) with $3 \times 3$ local context. Note that the generator and encoder modules in MGAN and MGAN+ARNN are identical. Refer to Section 4.4 of the main paper for more details.[]{data-label="fig:mgan"}](mgan "fig:"){width="\linewidth"}
[0.85]{} ![[**Difference between MGAN and MGAN+ARNN.**]{} (a) The transformer module for the ModularGAN (MGAN) architecture proposed by [@zhao2018modular]. It uses a $3\times 3$ local context to compute attention over the intermediate features. (b) MGAN+ARNN replaces the attention mechanism in MGAN with ARNN. It is applied in the same location as (a) with $3 \times 3$ local context. Note that the generator and encoder modules in MGAN and MGAN+ARNN are identical. Refer to Section 4.4 of the main paper for more details.[]{data-label="fig:mgan"}](mgan+arnn "fig:"){width="\linewidth"}
Additional Visualizations
=========================
Visual Attribute Prediction {#visual-attribute-prediction-1}
---------------------------
Please refer to Section 4.1 of the main paper for task definition. Figures \[fig:sample0\] - \[fig:sample2\] show the individual layer attended feature maps for three different samples from $\text{ARNN}^{\sim}$ for a fixed image and query. It can be seen that $\text{ARNN}^{\sim}$ is able to identify the different modes in each of the images.
![[**Qualitative Analysis of Attention Masks sampled from $\text{ARNN}^{\sim}$.**]{} Layer-wise attended feature maps sampled from $\text{ARNN}^{\sim}$ for a fixed image and query. The masks are able to span the different modes in the image. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:sample0"}](sample_7 "fig:"){width="0.7\linewidth"} ![[**Qualitative Analysis of Attention Masks sampled from $\text{ARNN}^{\sim}$.**]{} Layer-wise attended feature maps sampled from $\text{ARNN}^{\sim}$ for a fixed image and query. The masks are able to span the different modes in the image. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:sample0"}](sample_8 "fig:"){width="0.7\linewidth"}
![[**Qualitative Analysis of Attention Masks sampled from $\text{ARNN}^{\sim}$.**]{} Layer-wise attended feature maps sampled from $\text{ARNN}^{\sim}$ for a fixed image and query. The masks are able to span the different modes in the image. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:sample1"}](sample_1 "fig:"){width="0.7\linewidth"} ![[**Qualitative Analysis of Attention Masks sampled from $\text{ARNN}^{\sim}$.**]{} Layer-wise attended feature maps sampled from $\text{ARNN}^{\sim}$ for a fixed image and query. The masks are able to span the different modes in the image. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:sample1"}](sample_2 "fig:"){width="0.7\linewidth"} ![[**Qualitative Analysis of Attention Masks sampled from $\text{ARNN}^{\sim}$.**]{} Layer-wise attended feature maps sampled from $\text{ARNN}^{\sim}$ for a fixed image and query. The masks are able to span the different modes in the image. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:sample1"}](sample_3 "fig:"){width="0.7\linewidth"}
![[**Qualitative Analysis of Attention Masks sampled from $\text{ARNN}^{\sim}$.**]{} Layer-wise attended feature maps sampled from $\text{ARNN}^{\sim}$ for a fixed image and query. The masks are able to span the different modes in the image. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:sample2"}](sample_4 "fig:"){width="0.7\linewidth"} ![[**Qualitative Analysis of Attention Masks sampled from $\text{ARNN}^{\sim}$.**]{} Layer-wise attended feature maps sampled from $\text{ARNN}^{\sim}$ for a fixed image and query. The masks are able to span the different modes in the image. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:sample2"}](sample_5 "fig:"){width="0.7\linewidth"} ![[**Qualitative Analysis of Attention Masks sampled from $\text{ARNN}^{\sim}$.**]{} Layer-wise attended feature maps sampled from $\text{ARNN}^{\sim}$ for a fixed image and query. The masks are able to span the different modes in the image. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:sample2"}](sample_6 "fig:"){width="0.7\linewidth"}
Inverse Attribute Prediction
----------------------------
Please refer to Section 4.1 of the main paper for task definition. Figures \[fig:vap0\] - \[fig:vap2\] show the individual layer attended feature maps comparing the different attention mechanisms on the $\text{MBG}^{inv}$ dataset. It can be seen that ARNN captures the entire number structure, whereas the other two methods only focus on a part of the target region or on some background region with the same color as the number, leading to incorrect predictions.
![[**Qualitative Analysis of Attention Masks on $\text{MBG}^{inv}$.**]{} Layer-wise attended feature maps generated by different mechanisms visualized on images from $\text{MBG}^{inv}$ dataset. ARNN is able to capture the entire number structure, whereas the other two methods only focus on a part of the target region or on some background region with the same color as the target number. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:vap0"}](vap_1 "fig:"){width="0.7\linewidth"} ![[**Qualitative Analysis of Attention Masks on $\text{MBG}^{inv}$.**]{} Layer-wise attended feature maps generated by different mechanisms visualized on images from $\text{MBG}^{inv}$ dataset. ARNN is able to capture the entire number structure, whereas the other two methods only focus on a part of the target region or on some background region with the same color as the target number. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:vap0"}](vap_6 "fig:"){width="0.7\linewidth"}
![[**Qualitative Analysis of Attention Masks on $\text{MBG}^{inv}$.**]{} Layer-wise attended feature maps generated by different mechanisms visualized on images from $\text{MBG}^{inv}$ dataset. ARNN is able to capture the entire number structure, whereas the other two methods only focus on a part of the target region or on some background region with the same color as the target number. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:vap1"}](vis_9 "fig:"){width="0.7\linewidth"} ![[**Qualitative Analysis of Attention Masks on $\text{MBG}^{inv}$.**]{} Layer-wise attended feature maps generated by different mechanisms visualized on images from $\text{MBG}^{inv}$ dataset. ARNN is able to capture the entire number structure, whereas the other two methods only focus on a part of the target region or on some background region with the same color as the target number. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:vap1"}](vap_2 "fig:"){width="0.7\linewidth"} ![[**Qualitative Analysis of Attention Masks on $\text{MBG}^{inv}$.**]{} Layer-wise attended feature maps generated by different mechanisms visualized on images from $\text{MBG}^{inv}$ dataset. ARNN is able to capture the entire number structure, whereas the other two methods only focus on a part of the target region or on some background region with the same color as the target number. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:vap1"}](vap_7 "fig:"){width="0.7\linewidth"}
![[**Qualitative Analysis of Attention Masks on $\text{MBG}^{inv}$.**]{} Layer-wise attended feature maps generated by different mechanisms visualized on images from $\text{MBG}^{inv}$ dataset. ARNN is able to capture the entire number structure, whereas the other two methods only focus on a part of the target region or on some background region with the same color as the target number. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:vap2"}](vap_5 "fig:"){width="0.7\linewidth"} ![[**Qualitative Analysis of Attention Masks on $\text{MBG}^{inv}$.**]{} Layer-wise attended feature maps generated by different mechanisms visualized on images from $\text{MBG}^{inv}$ dataset. ARNN is able to capture the entire number structure, whereas the other two methods only focus on a part of the target region or on some background region with the same color as the target number. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:vap2"}](vap_3 "fig:"){width="0.7\linewidth"} ![[**Qualitative Analysis of Attention Masks on $\text{MBG}^{inv}$.**]{} Layer-wise attended feature maps generated by different mechanisms visualized on images from $\text{MBG}^{inv}$ dataset. ARNN is able to capture the entire number structure, whereas the other two methods only focus on a part of the target region or on some background region with the same color as the target number. For detailed explanation see Section 4.1 of the main paper.[]{data-label="fig:vap2"}](vap_10 "fig:"){width="0.7\linewidth"}
Image Generation
----------------
Please refer to Section 4.4 of the main paper for task definition. Figures \[fig:gan1\] and \[fig:gan2\] show the attention masks generated by MGAN and MGAN+ARNN for the task of *hair color* transformation. MGAN+ARNN encodes structural dependencies in the attention values, which is evident from the more uniform and continuous attention masks. MGAN, on the other hand, has sharp discontinuities which, in some cases, leads to less accurate hair color transformations.
![[**Qualitative Results for Image Generation.**]{} Attention masks generated by MGAN and MGAN+ARNN are shown. Notice that the hair mask is more uniform for MGAN+ARNN as it is able to encode structural dependencies in the attention mask. For detailed explanation see Section 4.4 of the main paper.[]{data-label="fig:gan1"}](gan1_2 "fig:"){width="0.8\linewidth"} ![[**Qualitative Results for Image Generation.**]{} Attention masks generated by MGAN and MGAN+ARNN are shown. Notice that the hair mask is more uniform for MGAN+ARNN as it is able to encode structural dependencies in the attention mask. For detailed explanation see Section 4.4 of the main paper.[]{data-label="fig:gan1"}](gan1_8 "fig:"){width="0.8\linewidth"} ![[**Qualitative Results for Image Generation.**]{} Attention masks generated by MGAN and MGAN+ARNN are shown. Notice that the hair mask is more uniform for MGAN+ARNN as it is able to encode structural dependencies in the attention mask. For detailed explanation see Section 4.4 of the main paper.[]{data-label="fig:gan1"}](gan1_3 "fig:"){width="0.8\linewidth"} ![[**Qualitative Results for Image Generation.**]{} Attention masks generated by MGAN and MGAN+ARNN are shown. Notice that the hair mask is more uniform for MGAN+ARNN as it is able to encode structural dependencies in the attention mask. For detailed explanation see Section 4.4 of the main paper.[]{data-label="fig:gan1"}](gan1_4 "fig:"){width="0.8\linewidth"}
![[**Qualitative Results for Image Generation.**]{} Attention masks generated by MGAN and MGAN+ARNN are shown. Notice that the hair mask is more uniform for MGAN+ARNN as it is able to encode structural dependencies in the attention mask. For detailed explanation see Section 4.4 of the main paper.[]{data-label="fig:gan2"}](gan1_9 "fig:"){width="0.8\linewidth"} ![[**Qualitative Results for Image Generation.**]{} Attention masks generated by MGAN and MGAN+ARNN are shown. Notice that the hair mask is more uniform for MGAN+ARNN as it is able to encode structural dependencies in the attention mask. For detailed explanation see Section 4.4 of the main paper.[]{data-label="fig:gan2"}](gan1_5 "fig:"){width="0.8\linewidth"} ![[**Qualitative Results for Image Generation.**]{} Attention masks generated by MGAN and MGAN+ARNN are shown. Notice that the hair mask is more uniform for MGAN+ARNN as it is able to encode structural dependencies in the attention mask. For detailed explanation see Section 4.4 of the main paper.[]{data-label="fig:gan2"}](gan1_6 "fig:"){width="0.8\linewidth"} ![[**Qualitative Results for Image Generation.**]{} Attention masks generated by MGAN and MGAN+ARNN are shown. Notice that the hair mask is more uniform for MGAN+ARNN as it is able to encode structural dependencies in the attention mask. For detailed explanation see Section 4.4 of the main paper.[]{data-label="fig:gan2"}](gan1_7 "fig:"){width="0.8\linewidth"}
|
---
abstract: 'We consider a class of simple, non-trivial models of evolving weighted scale-free networks. The network evolution in these models is determined by attachment of new vertices to ends of preferentially chosen weighted edges and by updating the weights of these edges. Resulting networks have scale-free distributions of the edge weight, of the vertex degree, and of the vertex strength. We discuss situations where this mechanism operates. Apart of stochastic models of weighted networks, we introduce a wide class of deterministic, scale-free, weighted graphs with the small-world effect. We show also how one can easily construct an equilibrium weighted network by using a generalization of the configuration model.'
author:
- 'S. N. Dorogovtsev'
- 'J. F. F. Mendes'
title: 'Minimal models of weighted scale-free networks'
---
Introduction {#s-introduction}
============
All real-world networks are weighted. Let us explain this strong claim in more detail. Usual objects of interest in the science of networks are relatively simple nets where all edges have equal “weights” [@ab02; @dm02; @n03; @dmbook03; @pvbook04]. The elements of the adjacency matrices of these networks are ones and zeros. We stress that these simple graphs are only parodies of real networks where connections between vertices are not equal. Edge weights are introduced to describe this diversity. In the simplest case, which we discuss here, weights are positive numbers [@yjbt01; @n01; @lm03; @ztzh03; @bbpv04; @bbv04; @bbv04a; @lc03; @flc04; @mab04; @lc04; @r04; @nr02; @bbchs03; @p04; @yzwfw04; @foc04] (see Fig. \[f1\]). So, the elements of the resulting adjacency matrices are zeros and positive numbers, $w_{ij}$. However, in principle, in more complex situation, edges may be described by a set of variables or operators. Note that an extra set of variables may be introduced to describe an individual properties of vertices (so called hidden variables [@gkk02; @s02; @cl02; @ccrm02; @bp03; @mmk04]).
Edge weights allow one to better describe reality. For example, in the simplest, unweighted version of a network of social contacts, each edge connects a pair of individuals who had one or two or more contacts. In a far more informative weighted network of social contacts, the weight of an edge shows the frequency of recent contacts between the corresponding pair of persons [@g73; @g83]. Another example is a network of coauthorships, where edge weights show the number of joint papers of two coauthors [@lc03; @flc04]. (This network may be treated as a one-mode projection of a bipartite graph.) Note that networks with multiple connections can be considered as weighted ones with integer edge weights.
Weighted networks provide an informative, usable picture of reality. Our first example is collaboration networks. Collaborations are adequately represented by bipartite collaboration graphs, where one kind of vertices is collaborators and the other one is acts of collaboration \[see Fig. \[f2\], graph A\]. Usually, empirical researchers consider one-mode projections of these graphs, where a pair of vertices-collaborators is interconnected by an edge if there has been at least one act of collaboration between them \[see Fig. \[f2\], graph B\]. This net is more simple than the original bipartite graph, but an essential information is lost. In particular, one cannot recover the bipartite original by using this projection. The weighted one-mode projection shown in Fig. \[f2\], graph C, is more informative than unweighted projection B but, nonetheless, is less informative than original bipartite graph A. One still cannot recover bipartite graph A if weighted projection C is known.
The second example is energy landscape networks, where each vertex is a local minimum of the potential energy landscape, and an edge connects two minima with a saddle point between them [@sab01; @d02]. An unweighted version of this network reflects only a general structure of connections in the configuration space of a system. In contrast, directed weighted graphs, where edge weights indicate transition rates provide basic necessary information about the relaxation and kinetics of a system. (One may also ascribe the energies of the potential minima to the vertices, and the energies of the saddle points to the edges.)
The third example is various chemical reaction networks [@sm04] (e.g., networks of metabolic reactions [@akvob04]). In a one-partite version of these networks, weights show the chemical reaction weights. Networks of corporate ownerships are also directed weighted graphs (edge weights indicate the fractions of company’s shares in hands of other companies). The links of spacial networks [@gn04] has a natural characteristic—their length. So, in this case an edge weight may be introduced as some function of the length of the corresponding edge.
The simplest local characteristics of weighted networks are (i) the weight of an edge, $w_{ij}$, (ii) the degree of a vertex, and (iii) “the strength” of a vertex,
$$s_i \equiv \sum_{j \in i} w_{ij}
\, ,
\label{e1}$$
Here the sum is over the nearest neighbors of the vertex $i$. (One can introduce directed weighted networks, where vertices have “in-strength” and “out-strength”.) The distributions of these local characteristics in random networks are a weight distribution $Q(w)$, a degree distribution $P(k)$, and a strength distribution $R(s)$.
A number of models, which provide complex, in particular, scale-free weighted networks, were proposed (see, e.g., Refs. [@yjbt01; @lc04; @ztzh03; @bbpv04; @bbv04; @bbv04a; @foc04]). Obviously, one may easily construct a complex weighted network independently ascribing degrees to vertices and weights to edges (see the next section). For this, (i) build an unweighted network by any of existing algorithms and afterwords (ii) arrange weights of edges. These are trivial constructions. A much more interesting scale-free network model, where the evolutions of degrees and weights are coupled, was proposed in Refs. [@bbpv04; @bbv04; @bbv04a].
In this network, in addition to a standard preferential attachment rule [@ba99; @baj99] (see also [@p76]), a specific redistribution of edge weights was introduced. In more detail, (i) each new vertex is attached to a preferentially selected existing vertex (the probability of the attachment is proportional to the vertex strength), and (ii) weights of connections of the latter vertex are updated in a specific way depending on the strength of the vertex. That is, in this model, a preferential attachment to a vertex induces changes in weights of its edges. This evolution results in (a) a power-law weight distribution, (b) a linear dependence of the strength of a vertex on its degree, and (c) power-law strength and degree distributions. Note that the specific form of rule (ii) is necessary to obtain scale-free networks.
In the present paper we show that quite similar results \[(a), (b), and (c)\] can be obtained by using a more simple construction. Actually we use the same idea as in our network evolving due to attachment to the ends (or to an end) of a randomly chosen edge [@dms01]. In the present network,
the weight of a preferentially chosen edge is increased by a constant, and
its end receives a new connection of a unit weight.
One can see that this model is so simple, that it may be considered as a minimal one.
The paper is organized as follows. In Sec. \[s-equilibrium\] we, for comparison, demonstrate a “trivial” construction of an equilibrium weighted network, which is a direct generalization of the configuration model. In Sec. \[s-model\] we define our stochastic model and present its solution. In Sec. \[s-how\_to\_choose\] we explain how one can choose edges at random or accounting for their weights. In Sec. \[s-generalizations\] we describe simple generalizations of the model. In Sec. \[s-deterministic\] we introduce deterministic scale-free weighted networks. In Sec. \[s-applications\] we discuss possible applications of our models.
Equilibrium weighted networks {#s-equilibrium}
=============================
As a starting point, we consider a “trivial” equilibrium, weighted network—the simplest generalization of the configuration model of a random network. The standard configuration model is, in simple terms, a maximally random graph with a given degree distribution, $P(k)$, see Ref. [@b80], where the model was introduced in this form, and also Refs. [@bbk72; @bc78; @w81] with very similar constructions. In addition, we assume that the weights of the edges in the random graph ensemble of the configuration model are random, independent, and distributed according to a given weight distribution $Q(w)$. The question is: what is the resulting distribution of the vertex strength, $R(s)$, in this network?
For brevity, in this section, we consider only networks with integer edge weights. However, the final result for the asymptotics of distributions is also valid without this assumption.
Actually, we must calculate the distribution of the sum of independent equally distributed random variables. The resulting relation is especially simple in a $Z$-transformed form. Let us introduce the $Z$-transforms (generating functions) of the distributions $P(k)$, $Q(w)$, and $R(s)$: $$\begin{aligned}
& &
\Phi(z) \equiv \sum_k z^k P(k)
\nonumber
\\[5pt]
& &
\Sigma(z) \equiv \sum_w z^w Q(w)
\nonumber
\\[5pt]
& &
\Psi(z) \equiv \sum_s z^s R(s)
\, .
\label{e2}\end{aligned}$$ Then, accounting for the definition of the strength (\[e1\]) and for the mutual independence of edge weights (and vertex degrees), we immediately have
$$\Psi(z) = \sum_k P(k)\Sigma^k(z) = \Phi(\Sigma(z))
\, .
\label{e3}$$
Here we used a standard relation for the $Z$-transform of the distribution of the sum of independent random variables (see, e.g., Ref. [@nsw01]).
The asymptotic behavior of the original distribution at large values of a random variable is related to the analytical structure of the corresponding $Z$-transformation near $z=1$. For example, for a power-law original, we have the following correspondence: $$\begin{aligned}
& &
\!\!\!\!\!P(k\gg1)\sim k^{-\gamma_k} \ \Longleftrightarrow
\nonumber
\\[5pt]
\!\!\!\!\!\!\Psi(z\ \text{near}\ 1) &\cong&
1 + \sum_{i>0}a_i (1-z)^i + b(1-z)^{\gamma_k-1}
\, .
\label{e4}\end{aligned}$$ On the other hand, the $Z$-transformation of a rapidly decreasing distribution (i.e., with the finite moments), is analytical at $z=1$. So, accounting for correspondence (\[e4\]), relation (\[e3\]) leads to the following conclusions for this network:
if both the distributions $P(k)$ and $Q(w)$ are rapidly decreasing, then the strength distribution $R(s)$ is rapidly decreasing too;
if $P(k)$ is rapidly decreasing, and $Q(w)$ is, e.g., a power law, $Q(w) \sim w^{-\gamma_w}$, then the resulting strength distribution is also power-law, with the same exponent, $R(s) \sim s^{-\gamma_w}$, i.e., $\gamma_s=\gamma_w$;
if $P(k)$ is, e.g., scale-free, $P(k)\sim k^{-\gamma_k}$, and $Q(w)$ rapidly decreasing (i.e., $\Sigma(z)$ analytic at $z=1$), then $R(s) \sim s^{-\gamma_k}$, i.e., $\gamma_s=\gamma_k$;
if both $P(k)$ and $Q(w)$ are scale-free, then the resulting distribution $R(s)$ is also scale-free with exponent $\gamma_s = \min(\gamma_k,\gamma_w)$.
These are important, although particular, cases. However, relation (\[e3\]) allows one to easily obtain, after an inverse $Z$-transform, a strength distribution for the weighted graph with arbitrary degree and weight distributions.
Note that, also, one may consider a more “serious”, “non-trivial” generalization of the configuration model. These are labeled random graphs with a given sequence of “generalized degrees”. The generalized degree of a vertex is a complete set of numbers, where each number shows how many edges of a given weight are attached to this vertex. This is quite similar to the constructions of graphs with hidden color of Refs. [@s03].
Attachment to weighted edges {#s-model}
============================
Let us introduce the model of a growing scale-free network in more precise terms than in the Introduction. We first discuss the simplest case.
We assume that the growth starts from an arbitrary configuration of vertices and edges, e.g., from a single edge of weight $1$. At each successive time step,
choose an edge with probability proportional to its weight and increase this weight by a constant $\Delta \geq 0$,
attach a new vertex to both the ends of this edge by edges of weight $1$
(see Fig. \[f3\]).
In the particular case of $\Delta=0$, this network is reduced to our model of Ref. [@dms01]. Rule (i) may be interpreted in the following way. Suppose that $\Delta$ and the weights of the edges are integer numbers. In this case, an edge of weight $w$ may be treated as a $w$-multiple edge. If we choose each “elementary” edge in this network with multiple edges at random, with equal probability, we just arrive at the proportional preferential choice of edges in the corresponding weighted network.
Rule (ii) may be easily modified: attach a new vertex to one of the ends of the selected edge and not to the both of them—see below. In the present form, rule (ii) results in the following rigid coupling of the degree $k_i$ of a vertex and its strength $s_i$:
$$s_i = k_i(1+\Delta) - 2\Delta
\, .
\label{e5}$$
Indeed, in this model, each attachment to a vertex increases its degree by $1$ and its strength by $1+\Delta$ ($1$—due to the link of weight $1$ to a new vertex and $\Delta$ due to the modification of the selected edge). Taking into account that new vertices have degree and strength equal to $2$ results in relation (\[e5\]). So the degree and strength distributions have quite similar asymptotic behaviors.
We will show below that the weight distribution, the strength distribution, and the degree distribution of this network have power-law asymptotics:
$$Q(w) \sim w^{-\gamma_w}, \ \ \
R(s) \sim s^{-\gamma_s}, \ \ \
P(k) \sim k^{-\gamma_k}
%%\, .
\label{e5a}$$
with exponents
$$\begin{aligned}
& &
\gamma_w = 1 + \frac{2+\Delta}{\Delta} = 2 + \frac{2}{\Delta}
\, ,
\label{e5b}
%%\nonumber
\\[5pt]
& &
\gamma_s = \gamma_k = 1 + \frac{2+\Delta}{1+\Delta} = 2 + \frac{1}{1+\Delta}
\, .
\label{e5c}\end{aligned}$$
Formula (\[e5b\]) is valid at $\Delta>0$, and formula (\[e5c\]) is valid at $\Delta \geq 0$. Note that formulas (\[e5b\]), (\[e5c\]) give $\gamma_s=\gamma_k<\gamma_w$. Recall the relation $\gamma_s=\text{min}(\gamma_k,\gamma_w)$, which we have obtained for scale-free equilibrium networks (the “configuration model of a weighted network”). So, formulas (\[e5b\]), (\[e5c\]) satisfy that relation. One can easily check that the same is valid for the other scale-free weighted networks in this paper \[see formulas (\[e14\]) and (\[e15\]), (\[e19\]) and (\[e20\]), (\[e21\]) and (\[e22\])\].
Note that the evolution of edge weights in this model is quite similar to that in the Simon model [@s55]. One must take into account that in our network the total weight of edges growth proportionally to $t$: $W(t) \cong t(2+\Delta)$, which is asymptotic expression at large $t$. This gives the following evolution equation for the mean number $\overline{N}(w,t)$ of edges of weight $w$ at time $t$:
$$\begin{aligned}
& &
\overline{N}(w,t+1) = \overline{N}(w,t) + 2\delta_{w,1}
\nonumber
\\[5pt]
& &
+
\frac{w-\Delta}{t(2+\Delta)}\overline{N}(w-\Delta,t) - \frac{w}{t(2+\Delta)}\overline{N}(w,t)
\, .
\label{e6}\end{aligned}$$
(For similar equations for degree distributions in growing scale-free networks, see Refs. [@krl00] and [@dms00].) At each time step two new edges of weight $1$ emerge. This produces the term $2\delta_{w,1}$ on the right-hand side of the equation. The third term describes increase in the number of edges of weight $w$ due to the modification of edges of weight $w-\Delta$. The last term on the right-hand side describes the reduction of the number of edges of weight $w$ due to their modification.
We pass to the continuum $t$ limit and assume a stationary form of the weight distribution, $\overline{N}(w,t) = 2tQ(w,t) \to 2tQ(w)$. Then, passing to the continuum limit of weight allows us to obtain a power-law form of the weight distribution $Q(w) \sim w^{-\gamma_w}$ with the $\gamma_w$ exponent given by formula (\[e5b\]).
The exact solution of the stationary limit of Eq. (\[e8\]),
$$Q(w) \propto B(w/\Delta,\gamma_w)
\, ,
\label{e8}$$
is quite similar to the solution of the Simon model (we have omitted a normalization factor on the right-hand side of this formula). Here $B(\ ,\ )$ is the beta-function. As is natural, at large $w$, this expression has a power-law form with the same exponent $\gamma_w$ given by relation (\[e5b\]).
The total strength of the vertices in the network is $S(t)=2W(t) \cong 2t(2+\Delta)$. \[The total degree of the network evolves in the following way: $K(t) \cong 4t$.\] The attachment of a new vertex to the end vertices of an edge increases their strengths by $1+\Delta$. Furthermore, the rules of the model actually result in preferential (proportional) attachment to vertices of higher strength. So, the evolution equation for the mean number $\overline{N}(s,t)$ of vertices of strength $s$ at time $t$ looks as follows:
$$\begin{aligned}
& &
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\overline{N}(s,t+1) = \overline{N}(s,t) + \delta_{s,2}
\nonumber
\\[5pt]
& &
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+
2\frac{s\!-\!(1\!+\!\Delta)}{2t(2+\Delta)}\overline{N}(s\!-\!(1\!+\!\Delta),t) - 2\frac{s}{2t(2+\Delta)}\overline{N}(s,t)
%%\,
.
\label{e9}\end{aligned}$$
The factors $2$ of the third and the fourth terms on the right-hand side of this equation are due to the attachment of each new vertex to the two ends of a selected edge. The resulting stationary strength distribution $R(s)$, $\overline{N}(s,t) = tR(s,t) \to R(s)$, is of the form
$$R(s) \propto B(s/(1\!+\!\Delta),\gamma_s)
\,
\label{e10}$$
with $\gamma_s$ given by formula (\[e5c\]). Consequently, due to linear relation (\[e5\]) between strength and degree, the resulting degree distribution is $$P(k) \sim B(k,\gamma_k)
\,
\label{e12}$$ with $\gamma_k$ given by formula (\[e5c\]). One can see that the resulting expressions of exponents, Eqs. (\[e5a\]) and (\[e5b\]), are similar to those obtained for the model of Refs. [@bbpv04; @bbv04; @bbv04a]. The ranges of variation with the parameter of a problem—$(2,\infty)$ for $\gamma_w$ and $(3,\infty)$ for $\gamma_s$ and $\gamma_k$—are the same in both these networks.
By construction, our networks have numerous loops of length three, which indicates high clustering (see Ref. [@dms01]). However, one should be careful with this claim. There are two distinct integrated clustering characteristics. One can show that the average clustering (the average clustering coefficient of a vertex) in this case is finite in the infinite network limit. On the other hand, the clustering coefficient (in simple terms, the density of triangles in the network) approaches zero in the infinite network.
How to choose an edge at random {#s-how_to_choose}
===============================
A meticulous reader may say: “Your model, as it was formulated, is indeed looks more simple than that of Refs. [@bbpv04; @bbv04; @bbv04a]. However, your model is based on the selection of connections, which may be, computationally, more complex task than the selection of vertices. This may depreciate this model.”
To convince this reader, we indicate two easy (and quick) ways to select connections. (i) Make a list of edges (this list is sufficiently short in sparse networks) and choose from it. (ii) Choose edges, starting the procedure from random vertices. Below we describe the second way in more detail.
Let us first demonstrate how one can select at random edges in a network. The simple procedure looks as follows:
choose a random vertex and then
subsequently choose each of its edges with some sufficiently small probability $p$,
then, repeat. Here, the probability $p$ must be small: $p < 1/k_{\text{max}}$, or at least $p \ll 1/\overline{k}$, where $k_{\text{max}}$ is the maximal degree of a vertex in the network, and $\overline{k}$ is the mean degree. This condition is necessary to subsequently select single edges but not bunches of edges—“hedgehogs”. (Note that, unlike the above procedure, if we simply select a random edge of a random vertex, we will not choose a random edge.)
Similarly, in a weighted network, one can choose an edge with probability proportional to its weight. For this,
Choose a random vertex and then
subsequently choose each of its (weighted) edges with some sufficiently small probability proportional to its weight.
then, repeat. Rule (ii) means that, after a vertex (say vertex $i$) is selected, each of its weighted edges is selected with probability $p w_{ij}/s_i$. Here, $s_i$ is the strength of this vertex \[see definition (\[e1\])\] and $w_{ij}$ is the strength of an edge attached to this vertex. $p$ is an arbitrary parameter, so small that there is a small chance to subsequently select two or more edges attached to one vertex (see the above procedure).
Simple generalizations {#s-generalizations}
======================
As we promised, let us consider the variation of the model with attachment to one end vertex of a selected edge. At each successive time step,
choose an edge with probability proportional to its weight and increase this weight by a constant $\Delta \geq 0$,
attach a new vertex to one the ends of this edge by edges of weight $1$
(see Fig. \[f4\]).
The total weight, the total degree, and the total strength of the vertices of this network grow in the following way: $W(t) \cong t(1+\Delta)$, $K(t) \cong 2t$, $S(t) \cong 2t(1+\Delta)$, respectively. We will show that the network has power-law distributions (\[e5a\]) with exponents
$$\begin{aligned}
& &
\gamma_w = 1 + \frac{1+\Delta}{\Delta} = 2 + \frac{1}{\Delta}
\, ,
\label{e14}
%%\nonumber
\\[5pt]
& &
\gamma_s = \gamma_k = 1 + 2\,\frac{1+\Delta}{1+2\Delta} = 2 + \frac{1}{1+2\Delta}
\, .
\label{e15}\end{aligned}$$
Instead of rigid coupling (\[e5\]) between the degree and strength of an individual vertex in the network of Sec. \[s-model\], now we have only the asymptotic relation between the mean values of the strength and degree of an individual vertex: $$\overline{s}_i \cong (1 + 2\Delta)\overline{k}_i
\, .
\label{e16}$$ This relation is valid for highly connected vertices (with large degrees). It shows that $\gamma_s = \gamma_k$ in this network.
Asymptotic equality (\[e16\]) follows from the following simple considerations. At each modification of vertex $i$, its degree and strength are modified in the following way: (i) with probability $1/2$, $k_i \to k_i+1$ and $s_i \to s_i+\Delta+1$, and (ii) with probability $1/2$, $k_i \to k_i$ and $s_i \to s_i+\Delta$. This directly leads to relation (\[e16\]).
One can see that for this network, the equation for the mean number of edges of weight $w$ at time $t$ is
$$\begin{aligned}
& &
\!\!\!\!\!\!\!\!\!\!
\overline{N}(w,t+1) = \overline{N}(w,t) + \delta_{w,1}
\nonumber
\\[5pt]
& &
\!\!\!\!\!\!\!\!\!\!
+
\frac{w-\Delta}{t(1+\Delta)}\overline{N}(w-\Delta,t) - \frac{w}{t(1+\Delta)}\overline{N}(w,t)
\,
%%.
\label{e17}\end{aligned}$$
\[compare with Eq. (\[e6\])\]. This immediately leads to formula (\[e14\]) for the $\gamma_w$ exponent of the weight distribution.
For the mean number of vertices of strength $s$ in this network, we have the following equation:
$$\begin{aligned}
& &
\!\!\!\!\!\!\!\!\!\!
\overline{N}(s,t+1) = \overline{N}(s,t) + \delta_{s,1}
\nonumber
\\[5pt]
& &
\!\!\!\!\!\!\!\!\!\!
+
\left[\frac{s-(1\!+\!\Delta)}{2t(1+\Delta)}\overline{N}(s-(1\!+\!\Delta),t) - \frac{s}{2t(1+\Delta)}\overline{N}(s,t)\right]
\nonumber
\\[5pt]
& &
\!\!\!\!\!\!\!\!\!\!
+
\left[\frac{s-\Delta}{2t(1+\Delta)}\overline{N}(s-\Delta,t) - \frac{s}{2t(1+\Delta)}\overline{N}(s,t)\right]
%%\, .
\label{e18}\end{aligned}$$
\[compare with Eq. (\[e9\])\]. This, together with relation (\[e16\]) leads to formula (\[e15\]) for the $\gamma_s$ exponent of the strength distribution and the $\gamma_k$ exponent of the degree distribution. One can see that the only difference between formulas (\[e5b\]), (\[e5c\]) and (\[e14\]), (\[e15\]) is that the parameter $\Delta$ in the results of Sec. \[s-model\] is now substituted by $2\delta$.
In the same way as above, one can consider a combination of the models that we have already discussed. For example, the following combination of the evolution channels may be introduced. At each time step: (i) Choose preferentially an edge and increase its weight. (ii) (a) With probability $p_2$, attach a new vertex to both the ends of this edges; (b) with probability $p_1$, attach a new vertex to one of the end vertices of this edge; (c) with probability $p_0$, do not add a new vertex. Here, $p_0+p_1+p_2=1$. One can also add the processes of linking of existing vertices, rewiring of connections, and introduce variation of $\Delta$.
Up to now we discussed the proportional preferential choice of edges, that is, edges were chosen with probability proportional to their weights. In general, one may choose edges with probability proportional to some function $f(w)$ of their weight (preference function). Furthermore, one may consider an inhomogeneous situation, where a preference function depends on an edge $f_{ij}(w)$. Here the preference functions $f_{ij}(w)$ (and their parameters) may be randomly distributed similarly to what was considered for more traditional preferential attachments to vertices in inhomogeneous networks (see Refs. [@bb01a; @bb01b]). $\Delta$ also may be made random variable. One may also, at each time step, attach a new vertex to ends of several edges but not to a single edge as above (see Fig. \[f5\]). By using these generalizations, we can easily obtain various complex weighted network architectures with fat-tailed and rapidly decreasing distributions, with condensation, gelation, etc., previously studied in unweighted networks [@krl00; @dms00; @bb01a; @bb01b].
Let us suppose, for example, that (i) each new vertex becomes attached to all the ends of $m\geq 1$ preferentially selected edges, (ii) the preference function is a linear function: $f(w)=w+a$ where $a>-1$, (iii) weights of the selected edges are increased by a constant $\Delta$. In this case, formulas (\[e5b\]) and (\[e5c\]) take the forms:
$$\begin{aligned}
& &
\gamma_w =
%%1 + \frac{2+\Delta+2a}{\Delta} =
2 + 2\,\frac{1+a}{\Delta}
\, ,
\label{e19}
%%\nonumber
\\[5pt]
& &
\gamma_s = \gamma_k =
%%1 + \frac{2+\Delta+2a}{1+\Delta+a} =
2 + \frac{1+a}{1+\Delta+a}
\, .
\label{e20}\end{aligned}$$
The derivation of these expressions is very similar to that in Sec. \[s-model\] \[one must take into account relation (\[e5\])\]. If $a \to -1$, then all the three exponents $\gamma_w, \gamma_s, \gamma_k \to 2$. Note that if $a \to \infty$, then $\gamma_s, \gamma_k \to 3$. One can easily understand this limit. The infinite $a$ actually means the absence of preference, and edges are chosen at random, without accounting for their weights, exactly as in our network of Ref. [@dms01]. So, we arrive at the same value of exponents as in that model. (Recall that $\gamma_k=3$ in the Barabási-Albert model [@ba99].)
Another natural generalization of the process in Fig. \[f3\] is shown in Fig. \[f6m\]. In this process, new edges also can have different weights. Furthermore, all three numbers, $\Delta_1$, $\Delta_2$, and $\Delta_3$, may be random.
Deterministic weighted networks (pseudofractals) {#s-deterministic}
================================================
Compact growing networks with the small-world effect may be produced in a deterministic way . These graphs have a discrete spectrum of degrees. These spectra may have a variety of shapes: they may be fat-tailed, in particular, scale-free, they may be rapidly decreasing, e.g., exponential. The key and necessary feature of these deterministic graphs is the small-world effect: their mean intervertex distances grow slower than any (positive) power of the numbers of vertices. So, while these graphs visually look very similar to fractals, they are only parodies of fractals. The difference is crucial. Fractals have a finite dimension (i.e., their intervertex distances grow as a power of the numbers of their vertices). In contrast, the networks, which we discuss, have infinite dimension (i.e., their intervertex distances grow slower than any power of the numbers of vertices, e.g., logarithmically). This is why we call deterministic graphs of this kind [*pseudofractals*]{} [@dm02; @dgm02].
One should stress that a scale-free architecture (a power-law degree spectrum) is not not a definitive, necessary feature of these graphs. Many fractals have scale-free spectra of degrees (see discussion in Ref. [@dmbook03]).
How may these deterministic compact networks with a self-similar structure be constructed? There are two ways to obtain a compact deterministic network:
The first was realized in Ref. [@brv01]. At each step of the evolution, generate a number of the copies of a graph and connect them together, e.g., by adding new edges, which decreases intervertex distances.
One can use the following approach [@dm02; @dgm02]. At each step, in a regular manner, transform given basic subgraphs of the network into other, larger but compact, configurations consisting of the same basic clusters. These basic clusters may be vertices, edges, triangles, squares, etc. The words “transform in a regular manner”, in the simplest, very particular case, mean “transform all the given basic subgraphs of the graph”. The words “larger but compact configurations” are illustrated by an example below.
(Note that in some specific situations, these approaches are equivalent.)
Let us demonstrate how to construct a deterministic, scale-free, weighted, compact network by using the second approach. We will exploit the equivalent representation of integer-weighted edges as multiple connections (see Fig. \[f6\]).
As an example, we use the transformation that is a deterministic variation of the transformation shown in Fig. \[f3\] with an integer $\Delta$. This transformation is shown in Fig. \[f7\]. The edge of integer weight $w$ is transformed into the edge of weight $w(1+\Delta)$ with $w$ triangles of $1$-weight edges attached. The growth starts from the triangle of edges of weight $1$. At each successive step, each edge of the graph is transformed in the way shown in Fig. \[f7\]. The result is a weighted pseudofractal network, shown in Fig. \[f8\] in particular case of $\Delta=1$. If $\Delta=0$, we get the deterministic graph of Refs. [@dm02; @dgm02].
We will show below that the weight, strength, and degree distributions of this graph are power-law with exponents
$$\begin{aligned}
& &
\gamma_w = 1 + \frac{\ln(3+\Delta)}{\ln(1+\Delta)}
\, ,
\label{e21}
%%\nonumber
\\[5pt]
& &
\gamma_s = \gamma_k = 1 + \frac{\ln(3+\Delta)}{\ln(2+\Delta)}
\, ,
\label{e22}\end{aligned}$$
respectively. Note that $\gamma_w,\gamma_s,\gamma_k \to 2$ as $\Delta \to \infty$.
We emphasize that this is only a particular example. The readers can easily consider numerous variations: use other transformations of this kind or their combinations, transform different clusters, transform not each of given elements of a graph but only some of them, selected in a regular way, consider networks embedded in a finite dimension space, consider trees, and so on. These variations produce a wide range of network architectures and a wide range of the exponents values in the range $(2,\infty)$.
Let us consider the growth of the graph. One can see that at each step of the evolution, a weight of each individual edge of the graph, a strength of each individual vertex and its degree transform in the following way:
$$\begin{aligned}
& &
w\ \to\ w' = (1+\Delta)w
\, ,
%%\label{e22}
\nonumber
\\[5pt]
& &
s\ \to\ s' = (2+\Delta)s
\, ,
%%\label{e23}
\nonumber
\\[5pt]
& &
k\ \to\ k' = k+s
\, ,
\label{e22a}\end{aligned}$$
respectively (the indices of edges and vertices are not shown).
We introduce the following notations for basic numbers: $N_n$, $L_n$, and $W_n$, which are the total numbers of vertices and edges and the total weight of the edges in this deterministic graph in an $n$th generation, respectively. The total degree is $K_n=2L_n$. The total strength of vertices is $S_n = 2W_n$.
In the initial state, $N_0=L_0=W_0=3$. Then, one can obtain
$$W_n = 3(3+\Delta)^n
\, .
\label{e23}$$
One can see that
$$\begin{aligned}
& &
N_{n+1} = N_n + W_n
\, ,
\nonumber
\\[5pt]
& &
L_{n+1} = L_n + 2W_n
\, .
\label{e24}\end{aligned}$$
This gives
$$\begin{aligned}
& &
N_n = \frac{3}{2+\Delta}[(3+\Delta)^n + 1+\Delta]
\, ,
\nonumber
\\[5pt]
& &
L_n = \frac{3}{2+\Delta}[2(3+\Delta)^n + \Delta]
\, .
\label{e25}\end{aligned}$$
To find the weight, strength, and degree spectra we take into account the following circumstances. New edges have weight $1$, new vertices have degree and strength equal to $2$. All the edges that emerge simultaneously have the same weight. All the vertices that emerge simultaneously have the same strength and the same degree. The weight $w_m$ of edges emerged $m \leq n$ generation ago and the strength $s_m$ and the degree $k_m$ of vertices emerged at that moment are
$$\begin{aligned}
& &
w_m = (1+\Delta)^{m-1}
\, ,
%%\label{e22}
\nonumber
\\[5pt]
& &
s_m = 2(2+\Delta)^{m-1}
\, ,
%%\label{e23}
\nonumber
\\[5pt]
& &
k_m = \frac{2}{1+\Delta}[(2+\Delta)^{m-1}+\Delta]
\, .
\label{e26}\end{aligned}$$
The $n$th generation graph contains $W_{n-1}=3(3+\Delta)^{n-1}$ new vertices and $2W_{n-1}=6(3+\Delta)^{n-1}$ new edges. Consequently, the $n$th generation graph contains $6(3+\Delta)^{n-m-1}$ edges of weight $w_m$ and $3(3+\Delta)^{n-m-1}$ vertices of strength $s_m$ and degree $k_m$. Here $w_m$, $s_m$, and $k_m$ are given by formula (\[e26\]).
As a result, we obtain a power-law behavior of the spectrum of the weight: the number of the edges of weight $w_m$ is $N(w_m) \propto w_m^{-\ln(3+\Delta)/\ln(1+\Delta)}$. This is a discrete spectrum with gaps between discrete weights growing with $w$. To present our result in a form which can be compared with results for stochastic models, where distributions are continuous and spectrum gaps are absent, we pass to the cumulative weight distribution: $\sum_{w_l\geq w_m}N(w_l) \propto w_m^{-\ln(3+\Delta)/\ln(1+\Delta)}$. This distribution already has no gaps like the corresponding cumulative distribution of the edge weight of stochastic networks, $N_{\text{cum}}(w) \propto w^{-(\gamma_w-1)}$. So we arrive at formula (\[e21\]) for the $\gamma_w$ exponent of the edge-weight distribution.
Similarly, we obtain the power strength spectrum: the number of vertices of strength $s_m$ is $N(s_m) \propto s_m^{-\ln(3+\Delta)/\ln(2+\Delta)}$. The cumulative distribution of the vertex strength is $\sum_{s_l\geq _m}N(s_l) \propto s_m^{-\ln(3+\Delta)/\ln(2+\Delta)}$, and the corresponding cumulative continuum distribution of the vertex strength in stochastic models is $N_{\text{cum}}(s) \propto s^{-(\gamma_s-1)}$. One can see that the degree spectrum of the deterministic graph is quite similar to the strength spectrum. So, we finally obtain formula (\[e22\]) for the exponents of the strength and degree distributions.
Other structural characteristics of this graph also can be easily found. We present here only an expression for the standard local degree-dependent clustering, which is defined as
$$C(k)\!\equiv\! \frac
{\text{mean\! \#\! of triangles attached\! to\! a\! vertex of deg.}\,k}
{k(k-1)/2}
%\,
.
\label{e28}$$
By construction, a vertex of degree $k$ in this graph has $k-1$ triangles attached, so we readily find
$$C(k)=2/k
\,
\label{e28}$$
Note that this expression is independent of $\Delta$, and we have the same result as for the network of Ref. [@dm02; @dgm02].
The rationale behind these models {#s-applications}
=================================
We discussed networks where edges of high weight attract new connections. Does this occur in the real world? Here we present only one illustration.
Let us consider the evolution of a weighted network of scientific coauthorships, taken in a one-mode representation. So, the vertices are authors, and the weighted edges are (pair-wise) coauthorships. Intensive pair-wise collaborations have a greater chance to attract new collaborators than occasional connections. Speaking in simple terms, if two coauthors have only a single paper, e.g., written 20 years ago, there is a small chance that somebody suddenly decide to write a paper with them. In contrast, fruitful joint efforts lead to new coauthorships much more frequently. One may say, it is the papers—scientific results, but not their authors that attract new coauthors. This is precisely the mechanism that is discussed in the present paper.
Discussion and summary {#s-summary}
======================
One point should be stressed. We have shown that our approach leads to results which are close to those obtained in Refs. [@bbpv04; @bbv04; @bbv04a]. So that, these two approaches are related. In the model of Refs. [@bbpv04; @bbv04; @bbv04a], “strong” vertices attract new connections and afterwards the weights of the edges of these vertices are specifically modified. In contrast, in our case, links of high weight increase their weights and attract new connections. That is, in one approach, the attachment is to (“strong”) vertices and in the other approach, the attachment is to (“heavy”) edges. This is a principal difference which is related to distinct real situations. We have shown that the introduced mechanism operates in real-world networks.
We have demonstrated that our approach allows one to formulate minimal, non-trivial, solvable models of evolving scale-free, weighted networks. We have found a number of basic structural characteristics of these networks. We indicated some possible generalizations. In particular, we have introduced a wide class of weighted deterministic graphs of a pseudofractal type and have considered the simplest scale-free, weighted, deterministic graph in detail.
In summary, we have developed an effective, simple approach to evolving weighted networks. This approach allows one to extend numerous results for well studied unweighted networks to a wide class of weighted networks.
[*Note added.*]{} After this paper had been prepared, works [@hjw04; @hjd04; @ak04], where the linking process also is determined by the edge weights, have appeared in the cond-mat electronic archive.
This work was partially supported by projects POCTI/FAT/46241/2002andPOCTI/MAT/46176/2002. A part of this work was made when one of the authors (SD) attended the Exystence Thematic Institute on Networks and Risks (Collegium Budapest, June 2004). SD thanks A. Barrat and A. Vespignani for useful discussions in Budapest.
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|
---
abstract: 'Using special polynomials related to the Andrews-Gordon identities and the colored Jones polynomial of torus knots, we construct classes of $q$-hypergeometric series lying in the Habiro ring. These give rise to new families of quantum modular forms, and their Fourier coefficients encode distinguished Maass cusp forms. The cuspidality of these Maass waveforms is proven by making use of the Habiro ring representations of the associated quantum modular forms. Thus, we provide an example of how the $q$-hypergeometric structure of the associated series to can be used to establish modularity properties which are otherwise non-obvious. We conclude the paper with a number of motivating questions and possible connections with Hecke characters, combinatorics, and still mysterious relations between $q$-hypergeometric series and the passage from positive to negative coefficients of Maass waveforms.'
address:
- 'Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany'
- 'CNRS LIAFA Universite Denis Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, France'
- 'Hamilton Mathematics Institute & School of Mathematics, Trinity College, Dublin 2, Ireland'
author:
- Kathrin Bringmann
- Jeremy Lovejoy
- Larry Rolen
title: 'On some special families of $q$-hypergeometric Maass forms'
---
[^1]
Introduction and Statement of Results {#Intro}
=====================================
We begin by introducing the special polynomials $H_n(k,\ell;b;q)$ which play a key role in our constructions. To do so, we recall the *$q$-rising factorial*, defined by $$(a)_n = (a;q)_n := \prod_{k=0}^{n-1} \big(1-aq^{k}\big),$$ along with the *Gaussian polynomials*, given by $$\begin{bmatrix} n \\ k \end{bmatrix}_q :=
\begin{cases}
\frac{(q)_n}{(q)_{n-k}(q)_k} & \text{if $0 \leq k \leq n$}, \\
0 & \text{otherwise}. \notag
\end{cases}$$ Then for $k\in\mathbb{N}$, $1 \leq \ell \leq k$, and $b\in\{0,1\}$, we define the polynomials $H_{n}(k,\ell;b;q)$ by $$\label{Hdef}
H_{n}(k,\ell;b;q) := \sum_{n = n_k \geq n_{k-1} \geq \ldots \geq n_1 \geq 0} \prod_{j=1}^{k-1} q^{n_j^2+(1-b)n_j} \begin{bmatrix} n_{j+1}-n_j - bj + \sum_{r=1}^j (2n_r + \chi_{\ell > r}) \\ n_{j+1}-n_j \end{bmatrix}_q.$$ Here we use the usual charactersitic function $\chi_{A}$, defined to be $1$ if $A$ is true and $0$ otherwise.
These polynomials occurred explicitly (in the case $b=1$) in recent work on torus knots [@Hi-Lo1], and they can also be related to generating functions for the partitions occurring in Gordon’s generalization of the Rogers-Ramanujan identities [@Wa1]. To describe the latter, let $G_{k,i,i',L}(q)$ be the generating function for partitions of the form $$\label{A-Grelation}
\sum_{j=1}^{L-1} j f_j,$$ with $f_1 \leq i-1$, $f_{L-1} \leq i'-1$, and $f_i + f_{i+1} \leq k$ for $1 \leq k \leq L-2$. Using the fact that $$\begin{bmatrix} n \\ k \end{bmatrix}_{q^{-1}} = q^{-k(n-k)}\begin{bmatrix} n \\ k \end{bmatrix}_{q},$$ making some judicious changes of variable and comparing with Theorem 5 of [@Wa1], it can be shown that $$\label{A-Grelationbis}
H_n\big(k,\ell;b,q^{-1}\big) = q^{(k-1)bn - 2(k-1)\binom{n+1}{2}} G_{k-1,\ell,k,2n-b+1}(q).$$
In the context of torus knots, the $n$-th coefficient in Habiro’s cyclotomic expansion of the colored Jones polynomial of the left-handed torus knot $T(2,2k+1)$ was shown in [@Hi-Lo1] to be $q^{n+1-k}H_{n+1}(k,1;1;q)$, and the general $H_{n}(k,\ell;1;q)$ were used to construct a class of $q$-hypergeometric series with interesting behavior both at roots of unity and inside the unit circle. As we shall see shortly, this is the heart of the quantum modular phenomenon; the reader is also referred to [@Hi-Lo1] for more details.
In this paper, we consider classes of $q$-hypergeometric Maass cusp forms constructed from the polynomials $H_{n}(k,\ell;b;q)$. These functions, denoted [by]{} $F_j(k,\ell;q)$ $\big(j\in\{1,2,3,4\}\big)$, are defined as follows:[^2] $$\label{FFnsDefn}\begin{aligned}
F_1(k,\ell;q) &:= \sum_{n \geq 0} (q)_{n}(-1)^{n}q^{\binom{n+1}{2}} {H}_{n}(k,\ell;0;q), \\ F_2(k,\ell;q) &:= \sum_{n \geq 0} \big(q^2;q^2\big)_{n}(-1)^{n} {H}_{n}(k,\ell;0;q), \\F_3(k,\ell;q) &:= \sum_{\substack{n \geq 1}} (q)_{n-1}(-1)^{n}q^{\binom{n+1}{2}} H_{n}(k,\ell;1;q), \\ F_4(k,\ell;q) &:= \sum_{\substack{n \geq 1}} (-1)_{n}(q)_{n-1}(-q)^{n} H_{n}(k,\ell;1;q). \end{aligned}$$ Note that when $k=1$ the polynomials in are identically $1$, and so the above contain two celebrated $q$-series of Andrews, Dyson, and Hickerson [@An-Dy-Hi1] as special cases. Namely, we have $$\label{ConnectionOurFamilySigma}
2F_2(1,1;q) = \sigma\big(q^2\big)$$ and $$\label{ConnectionOurFamilySigmaStar}
F_4(1,1;q) = -\sigma^*(-q),$$ where $$\begin{aligned}
\label{sigmadef}
\sigma(q) :=& \sum_{n \geq 0} \frac{q^{\binom{n+1}{2}}}{(-q)_n} \\
=&\ 1 + \sum_{n \geq 0} (-1)^nq^{n+1}(q)_n \label{id1}\\
=&\ 2\sum_{n \geq 0}(-1)^n(q)_n , \label{id2}\\
\sigma^*(q) :=&\ 2\sum_{n \geq 1} \frac{(-1)^nq^{n^2}}{(q;q^2)_n} \label{sigma*def}\\
=&\ -2\sum_{n \geq 0} q^{n+1}\big(q^2;q^2\big)_n. \label{id3}\end{aligned}$$ The definitions in and are the original definitions of Andrews, Dyson, and Hickerson, while the identities and were established by Cohen [@Co1], and follows easily. The function $\sigma$ was first considered in Ramanujan’s “Lost” notebook (see [@AndrewsLostNotebookV]). Andrews, Dyson, and Hickerson showed [@An-Dy-Hi1] that this series satisfies several striking and beautiful properties, and in particular that if $
\sigma(q)=\sum_{n\geq0}S(n)q^n
,$ then $\lim \sup |S(n)|=\infty$ but $S(n)=0$ for infinitely many $n$. Their proof is closely related to indefinite theta series representations of $\sigma$, such as: $$\sigma(q)=\sum_{\substack{n\geq0\\ |\nu|\leq n}}(-1)^{n+\nu}q^{\frac{n(3n+1)}2-\nu^2}\big(1-q^{2n+1}\big)
.$$ The coefficients of $\sigma^*(q)$ have the same properties.
Subsequently Cohen [@Co1] showed how to nicely package the $q$-series of Andrews, Dyson, and Hickerson within a single modular object. Namely, he proved that if coefficients $\{T(n)\}_{n\in1+24{\mathbb Z}}$ are defined by $$\label{Tofndef}
\sigma\big(q^{24}\big)=\sum_{n\geq0}T(n)q^{n-1}
,
\quad\quad\quad\quad
\sigma^*\big(q^{24}\big)=\sum_{n<0}T(n)q^{1-n},$$ then the $T(n)$ are the Fourier coefficients of a Maass waveform. The definitions of Maass waveforms and the details of this construction are reviewed in Section \[MaassFormsSctn\]. In this paper, we show that the functions $F_j(k,\ell;q)$ have a similar connection to Maass waveforms, and by and may thus be considered as a $q$-hypergeometric framework containing the examples of Andrews, Dyson, and Hickerson and Cohen.\
In what follows, we let $f$ be a Maass waveform with eigenvalue $1/4$ (under the hyperbolic Laplacian $\Delta$) on a congruence subgroup of $\SL_2(\mathbb{Z})$ (and with a possible multiplier), which is cuspidal at $i\infty$. If the Fourier expansion of $f$ is given, as in Lemma \[Maass0Fourier\], by $(\tau=u+iv)$ $$f(\tau)
=
v^{\frac{1}{2}}\sum_{n\neq0}A(n)K_{0}\bigg(\frac{2\pi |n|v}{N}\bigg)e\bigg(\frac{n u}{N}\bigg)
,$$ where $e(w):=e^{2\pi i w}$, then the $q$-series associated to the positive coefficients of $f$ is defined by $$\label{pluspart}
f^+(\tau):=\sum_{n>0}A(n)q^{\frac{n}{N}}.$$ We remark in passing that such a map from Maass forms to $q$-series was studied extensively by Lewis and Zagier [@LewisZagier1; @LewisZagier2], and, as we shall see, was used by Zagier [@Za1] to show that such functions are quantum modular forms. Such a construction is also closely related to the study of automorphic distributions in [@MS].
\[mainthm1\] For any $k, \ell\in\mathbb{N}$, with $1 \leq \ell \leq k$, and $j\in\{1,2,3,4\}$, there exists a Maass cusp form $G_{j,k,\ell}$ with eigenvalue $1/4$ for some congruence subgroup of $\operatorname{SL}_2({\mathbb Z})$, such that $$G_{j,k,\ell}^+(\tau)
=
q^{\alpha}
F_j\big(k,\ell;q^d\big)$$ for some $\alpha\in{{\mathbb Q}}$, where $d=1$ if $j\in\{1,3\}$ and $d=2$ if $j\in\{2,4\}$ .
The cuspidality of the Maass waveform $G_{j,k,\ell}$ is far from obvious. Indeed, the authors are aware of only two approaches to prove such a result: either to explicitly write down representations for the $G_{j,k,\ell}$ in terms of Hecke characters (which we suspect exist, but which we were unable to identify), or, as we show below, to use the $q$-hypergeometric representations of the $F_j$ directly. This connection, which was hinted at for certain examples in [@RobMaass], utilizes $q$-hypergeometric series to deduce modularity properties in an essential way.
The proof of Theorem \[mainthm1\] relies on the Bailey pair machinery and important results of Zwegers [@ZwegersMockMaass] giving modular completions for indefinite theta functions of a general shape. In particular, this family of indefinite theta functions naturally describes the behavior of functions studied by many others in the literature, as described in a recent proof of Krauel, Woodbury, and the second author [@KRW] of unifying conjectures of Li, Ngo, and Rhoades [@RobMaass]. The indefinite theta functions considered here are given for $M\in{\mathbb N}_{\ge2}$ and vectors $a=(a_1,a_2)\in{{\mathbb Q}}^2$ and $b=(b_1,b_2)\in{{\mathbb Q}}^2$ such that $a_1 \pm a_2 \not \in \mathbb{Z}$: $$\label{Sdef}
\begin{aligned}
&
S_{a,b;M}(\tau)
:=
\\
&
\Bigg(\sum_{n\pm \nu\geq-\lfloor a_1\pm a_2\rfloor}+\sum_{n\pm\nu<-\lfloor a_1\pm a_2\rfloor}\Bigg)e\big((M+1)b_1n-(M-1)b_2\nu\big)q^{\frac12\big((M+1)(n+a_1)^2-(M-1)(\nu+a_2)^2\big)}.
\end{aligned}$$ The next theorem states conditions under which $S_{a,b;M}$ is the image of a Maass waveform under the map defined in . The definitions of $\gamma_M$, the equivalence relation $\sim$, and the operation $^*$ are given in Section \[ZwegersWorkSection\].
\[mainthm2\] Suppose that $a,b\in{{\mathbb Q}}^2$ with $a\neq0$, $a_1\pm a_2\not\in{\mathbb Z}$, $M\in{\mathbb N}_{\geq2}$, and $(\gamma_M a,\gamma_M b)\sim(a,b)$ or both $(\gamma_M a,\gamma_M b)\sim(a^*,b^*)$ and $(\gamma_M a^*,\gamma_M b^*)\sim(a,b)$ hold. Then $S_{a,b;M}=F^+$ for a Maass waveform $F$ of eigenvalue $1/4$ on a congruence subgroup of $\operatorname{SL}_2({\mathbb Z})$.
The family of $q$-series $F_j$ in Theorem \[mainthm1\] are all specializations of the series $S_{a,b;M}$. We note that although all of the functions $F_j$ are cusp forms, for a general Maass form $F$ in Theorem \[mainthm2\] this is not always true. For example, the function $W_1$ considered in Theorem 2.1 of [@RobMaass] is shown not to be a cusp form, and using Theorem 4.2 and (4.4) one can easily check that the function $W_1$ fits into the family $S_{a,b;M}$.
In addition to the relations of the $q$-series $F_j$ to Maass forms, following Zagier’s work, we find that these functions are instances of so-called quantum modular forms [@Za1]. These new types of modular objects, which are reviewed in Section \[MaassFormsHolomorphization\], are connected to many important combinatorial generating functions, knot and $3$-manifold invariants, and are intimately tied to the important volume conjecture for hyperbolic knots. Roughly speaking, a *quantum modular form* is a function which is defined on a subset of $\mathbb Q$ and whose failure to transform modularly is described by a particularly “nice” function. (See Definition \[cocycle\].) Viewed from a general modularity framework, the generating function of the set of positive coefficients of a Maass form automatically has quantum modular transformations when considered as (possibly divergent) asymptotic expansions. The situation becomes much nicer when, as happens for $\sigma$ and $\sigma^*$, $q$-hypergeometric representations can be furnished which show convergence at various roots of unity (as the series specialize to finite sums of roots of unity). This quantum modularity result, as well as the relation of the associated quantum modular forms to the cuspidality of the Maass form is described in Theorem \[MaassQMFThm\]. In particular, if such a $q$-series is an element of the Habiro ring, which essentially means that it can be written as $$\sum_{n\geq0}a_n(q)(q)_n$$ for polynomials $a_n(q)\in{\mathbb Z}[q]$, then it is apparent that it converges at all roots of unity $q$, and hence the associated Maass form is cuspidal. This observation, combined with Zagier’s ideas, yields the following corollary (the definitions of quantum modular forms and related terms are given in Section \[MaassFormsHolomorphization\]).
\[mainthm3\] For any choice of $j,k,\ell$ as in Theorem \[mainthm1\], the functions $F_{j,k,\ell}$ are quantum modular forms of weight $1$ on a congruence subgroup with quantum set $\mathbb P^1({{\mathbb Q}})$. Moreover, the cocycles $r_{\gamma}$, defined in , are real-analytic on ${\mathbb R}\setminus\{\gamma^{-1}i\infty\}$.
The paper is organized as follows. In Section \[PrelimSection\], we recall the basic preliminaries and definitions needed for the proofs and explicit formulations of the main theorems, which are then proven in Section \[ProofsSection\]. As mentioned above, the main tools are the Bailey pair method, work of Zwegers in [@ZwegersMockMaass], and ideas from Zagier’s seminal paper on quantum modular forms [@Za1]. We conclude in Section \[QuestionsSection\] with further commentary on related questions and possible future work.
Preliminaries {#PrelimSection}
=============
Bailey pairs {#BaileyPairsSection}
------------
In this subsection, we briefly recall the Bailey pair machinery, which is a powerful tool for connecting $q$-hypergeometric series with series such as indefinite theta functions. The basic input of this method is a *Bailey pair* relative to $a$, which is a pair of sequences $(\alpha_n,\beta_n)_{n \geq 0}$ satisfying $$\beta_n = \sum_{k=0}^n \frac{\alpha_k}{(q)_{n-k}(aq)_{n+k}}. \notag$$ Bailey’s lemma then provides a framework for proving many $q$-series identities. For our purposes, we need only a limiting form, which says that if $(\alpha_n,\beta_n)$ is a Bailey pair relative to $a$, then, provided both sums converge, we have the identity
$$\label{limitBailey}
\sum_{n \geq 0} (\rho_1)_n(\rho_2)_n \bigg(\frac{aq}{\rho_1 \rho_2}\bigg)^n \beta_n = \frac{\Big(\frac{aq}{\rho_1}\Big)_{\infty}\Big(\frac{aq}{\rho_2}\Big)_{\infty}}{(aq)_{\infty}\Big(\frac{aq}{\rho_1 \rho_2}\Big)_{\infty}} \sum_{n \geq 0} \frac{(\rho_1)_n(\rho_2)_n\Big(\frac{aq}{\rho_1 \rho_2}\Big)^n }{\Big(\frac{aq}{\rho_1}\Big)_n\Big(\frac{aq}{\rho_2}\Big)_n}\alpha_n.$$
For more on Bailey pairs and Bailey’s lemma, see [@An1; @An2; @war].
We record four special cases of for later use.
\[Baileylemmaspecial\] The following are identities are true, provided that both sides converge. If $(\alpha_n,\beta_n)$ is a Bailey pair relative to $1$, then $$\begin{aligned}
\sum_{n \geq 1} (-1)^n(q)_{n-1}q^{\binom{n+1}{2}}\beta_n &= \sum_{n \geq 1} \frac{(-1)^nq^{\binom{n+1}{2}}}{1-q^n}\alpha_n, \label{Baileya=1eq1} \\
\sum_{n \geq 1} \big(q^2;q^2\big)_{n-1} (-q)^n\beta_n &= \sum_{n \geq 1} \frac{(-q)^n}{1-q^{2n}}\alpha_n, \label{Baileya=1eq2}\end{aligned}$$ and if $(\alpha_n,\beta_n)$ is a Bailey pair relative to $q$, then $$\begin{aligned}
\sum_{n \geq 0} (-1)^n(q)_{n}q^{\binom{n+1}{2}}\beta_n &= (1-q)\sum_{n \geq 0} (-1)^nq^{\binom{n+1}{2}}\alpha_n, \label{Baileya=qeq1} \\
\sum_{n \geq 0} \big(q^2;q^2\big)_{n} (-1)^n\beta_n &= \frac{1-q}{2}\sum_{n \geq 0} (-1)^n\alpha_n. \label{Baileya=qeq2}\end{aligned}$$
For the first two we set $a=1$ in , take the derivative $\frac{d}{d\rho_1} \big | _{\rho_1=1}$, and let $\rho_2 \to \infty$ or $\rho_2 = -1$. For the second two we set $a=q$, $\rho_1=q$, and let $\rho_2 \to \infty$ and $\rho_2 = -q$, respectively.
Maass waveforms and Cohen’s example {#MaassFormsSctn}
-----------------------------------
We now recall the basic definitions and facts from the theory of Maass waveforms. The interested reader is also referred to [@Bump; @Iwaniec02] for more details. Maass waveforms, or simply Maass forms, are functions on $\mathbb H$ which transform like modular functions but instead of being meromorphic are eigenfunctions of the hyperbolic Laplacian. For $\tau = u+i v \in \mathbb H$ (with $u, v\in {\mathbb R}$), this operator is defined by $$\Delta := -v^2 \bigg(\frac{\partial^2}{\partial u^2} + \frac{\partial^2}{\partial v^2}\bigg).$$ We also require the fact that any translation invariant function $f$, namely a function satisfying $f(\tau+1)=f(\tau)$, has a Fourier expansion at infinity of the form $$\begin{aligned}
\label{Fexpgeneral} f(\tau) = \sum_{n\in\mathbb Z} a_f(v;n) e(nu),
\end{aligned}$$ where $$a_f(v;n) := \int_{0}^1 f(t+iv)e(-nt)dt.$$ Similarly, such an $f$ has Fourier expansions at any cusp $\mathfrak a$ of a congruence subgroup $\Gamma \subseteq \SL_2(\mathbb{Z})$. We denote these Fourier coefficients by $a_{f,\mathfrak a}(v;n)$.
\[MaassForm0Def\] Let $\Gamma \subseteq \textnormal{SL}_2(\mathbb Z)$ be a congruence subgroup. A [*[Maass waveform]{}*]{} $f$ on $\Gamma$ with eigenvalue $\lambda=s(1-s) \in \mathbb C$ is a smooth function $f:\mathbb H \to \mathbb C$ satisfying
1. $f(\gamma \tau) = f(\tau)$ for all $\gamma \in \Gamma;$
2. $f$ grows at most polynomially at the cusps;
3. $\Delta (f)=\lambda f$.
If, moreover, $a_{f,\mathfrak a}(0;0) = 0$ for each cusp $\mathfrak a$ of $\Gamma$, then $f$ is a [*[Maass cusp form]{}*]{}.
We also require the general shape of Fourier expansions of such Maass forms. As all of our forms are cusp forms, the following is sufficient for our purposes. The proof may be found in any standard text on Maass forms (such as those listed above), but we note that it follows from the differential equation (iii), growth condition (ii), and the periodicity of $f$.
\[Maass0Fourier\] Let [**$f$**]{} be a Maass cusp form with eigenvalue $\lambda=s(1-s)$. Then there exist $\kappa_1, \kappa_2, a_f(n) \in \mathbb C$, $n\neq 0$, such that $$f(\tau) = \kappa_1 v^s + \kappa_2 v^{1-s}\delta_s(v) + v^{\frac12} \sum_{n\neq 0} a_f(n) K_{s-\frac12} (2\pi |n| v)e(nu),$$ where $K_\nu$ is the modified Bessel function of the second kind, and $\delta_s(v)$ is equal to $\log(v)$ or $1$, depending on whether $s=1/2$ or $s\neq 1/2$, respectively. Such an expansion also exists at all cusps.
Cohen proved (in the notation of ) that the function $$\notag f(\tau)
:=
v^{\frac{1}{2}}\sum_{n\in1+24{\mathbb Z}}T(n)K_0\bigg(\frac{2\pi |n|{v}}{24}\bigg)e\bigg(\frac {nu}{24}\bigg)$$ is a Maass form on the congruence subgroup $\Gamma_0(2)$ with a multiplier. Namely, $u$ satisfies the transformations $$f\bigg(\!-\frac1{2\tau}\bigg)=\overline{f(\tau)}, \qquad
f(\tau+1)=e\bigg(\frac1{24}\bigg)f(\tau),$$ and is an eigenfunction of $\Delta$ with eigenvalue $1/4$. Put another way, Cohen showed that $$f^+(\tau)=\sigma(q),$$ (in the notation of ) and that $\sigma^*$ similarly interprets the negative Fourier coefficients of $u$. Cohen’s proof relies on connections between $\sigma,\sigma^*$ and the arithmetic of a quadratic field, which also forms the basis of investigations by many authors of the series discussed by Li, Ngo, and Rhoades in [@RobMaass]. However, as noted above, computing the Hecke characters related to such $q$-series using Cohen’s methods quickly becomes computationally difficult. Instead, we use work of Zwegers which provides a convenient framework for giving examples of Maass forms and allows us to circumvent these problems.
Work of Zwegers and related notation {#ZwegersWorkSection}
------------------------------------
In this section we summarize the important recent work of Zwegers [@ZwegersMockMaass], which allows us to study the relation between indefinite theta functions and Maass forms. Effectively, Zwegers showed for a large class of indefinite theta functions how to define eigenfunctions of $\Delta$, along with special completion terms which correct their (non)-modularity. What is especially useful in our case, is the theory which Zwegers provides for describing when these completion terms vanish. To describe the setup, suppose that $A$ is a symmetric $2\times 2$ matrix with integral coefficients such that the quadratic form $Q$ defined by $Q({r}) := \frac12 {r}^T A {r}$ is indefinite of signature $(1,1)$, where ${r}^T$ denotes the transpose of ${r}$. Let $B({r},\mu)$ be the associated bilinear form given by $$\notag B({r},\mu) := {r}^T A \mu = Q({r}+\mu)-Q(r)-Q(\mu),$$ and take vectors $c_1,c_2\in {\mathbb R}^2$ with $Q(c_j)=-1$ and $B(c_1,c_2)<0$. In other words, we are assuming that $c_1$ and $c_2$ belong to the same one of the two components of the space of vectors $c$ satisfying $Q(c)=-1$. We denote this choice of component by $$C_Q:=\{x \in {\mathbb R}^2\mid Q(x)=-1,B(x,c_1)<0\}.$$ It is easily seen that $Q$ splits over $\mathbb{R}$ as a product of linear factors $Q(r)=Q_0(Pr)$ for some (non-unique) $P\in \GL_2({\mathbb R})$, where $Q_0(r):=r_1r_2$ with $r=(r_1,r_2)$. Note that $P$ satisfies $A=P^T(\begin{smallmatrix} 0&1\\1&0\end{smallmatrix})P$. The choice of $P$ is not unique, however we fix a $P$ with sign chosen so that $P^{-1}\binom{ \hspace{2mm}1}{-1}\in C_Q$. Then, for each $c\in C_Q$, there is a unique $t \in \mathbb{R}$ such that $$\label{eq:c(t)}
c = c(t):= P^{-1}\begin{pmatrix} e^t \\ -e^{-t} \end{pmatrix}.$$ Additionally, for $c\in C_Q$ we let $c^\perp =c^\perp(t):=P^{-1}{\left(\begin{smallmatrix} e^t \\ e^{-t} \end{smallmatrix}\right)}$. Note that $B(c,c^\perp)=0$, and $Q(c^\perp)=1$. It is easily seen that these two conditions determine $c^\perp$ up to sign.
Set $$\begin{aligned}
\rho_A(r):=\rho_{A}^{c_1,c_2}(r):=&\frac{1}{2} \Big(1-{\operatorname{sgn}}\big(B(r,c_1)B(r,c_2)\big) \Big)
,\end{aligned}$$ and for convenience, let $\rho_A^{\perp}:=\rho_A^{c_1^{\perp},c_2^{\perp}}$. Then, for $c_j=c(t_j)\in C_Q$, Zwegers defined the function $$\label{Phidef}
\begin{aligned}
\Phi_{a,b}(\tau)=\Phi_{a,b}^{c_1,c_2}(\tau) :&= {\operatorname{sgn}}(t_2-t_1) v^{\frac{1}{2}} \sum_{r \in a+{\mathbb Z}^2} \rho_A(r) e( Q(r)u+ B(r,b))K_0(2\pi Q(r)v) \\
& \quad + {\operatorname{sgn}}(t_2-t_1) v^{\frac{1}{2}} \sum_{r \in a+{\mathbb Z}^2} \rho_A^\perp (r) e( Q(r)u+B(r,b))K_0(-2\pi Q(r)v)
.
\end{aligned}$$ Note in particular that $$\label{plus}
\Phi_{a, b}^+(\tau)={\operatorname{sgn}}(t_2-t_1) \sum_{r \in a+{\mathbb Z}^2} \rho_A(r) e(B(r,b))q^{Q(r)}.$$ Here we used that in the proof of convergence in [@ZwegersMockMaass] it is shown that for the first sum in the definition of $\Phi_{a,b}$, $Q$ is positive definite whereas in the second sum $Q$ is negative definite from. Given convergence of this series, it is immediate from the differential equation satisfied by $K_0$ that $\Phi_{a,b}$ is an eigenfunction of the Laplace operator $\Delta$ with eigenvalue $1/4$.
Zwegers then found a completion of $\Phi_{a,b}$. Moreover, he gave useful conditions to determine when the extra completion term vanishes. To describe this, we first consider for $c\in C_Q$ the $q$-series $$\varphi_{a,b}^c(\tau) := {v}^{\frac{1}{2}}\sum_{{r}\in a+{\mathbb Z}^2} \alpha_{t} \big({r} {v}^{\frac{1}{2}} \big) q^{Q({r})}e(B({r},b))
,$$ $t$ as defined in and $$\alpha_{t}({r}):=
\begin{cases} \displaystyle{\int_{t}^\infty} e^{-\pi B({r},c(x))^2}dx & \mbox{ if }B({r},c)B\big({r},c^\perp\big)>0, \\[2ex]
-\displaystyle{\int_{-\infty}^{t}} e^{-\pi B({r},c(x))^2}dx & \mbox{ if }B({r},c)B\big({r},c^\perp \big)<0, \\
0 & \mbox{ otherwise, }
\end{cases}
$$ These functions satisfy the following transformation properties.
\[Zlem\] For $c\in C_Q$ and $a,b\in{\mathbb R}^2$, we have that $$\begin{aligned}
\varphi_{a+\lambda,b+\mu}^c &= e(B(a,\mu))\varphi_{a,b}^c\quad\mbox{for all $\lambda\in {\mathbb Z}^2$ and }\mu\in A^{-1}{\mathbb Z}^2, \\
\varphi_{-a,-b}^c &= \varphi_{a,b}^c, \\
\varphi_{\gamma a,\gamma b}^{\gamma c} &= \varphi_{a,b}^c \quad \mbox{for all }\gamma\in \mathrm{Aut}^+(Q,{\mathbb Z}),\end{aligned}$$ where $$\mathrm{Aut}^+(Q,{\mathbb Z}):=\big\{\gamma \in \operatorname{GL}_2({\mathbb R})\big\vert \gamma\circ Q=Q,\gamma{\mathbb Z}^2={\mathbb Z}^2, \gamma(C_Q)=C_Q, \det(\gamma)=1\big\}.$$
Zwegers’ main result is as follows, where $$\label{ZwegersPhiHatDefn}
\widehat\Phi_{a,b}(\tau)=\widehat\Phi_{a,b}^{c_1,c_2}(\tau):=v^{\frac12}\sum_{r\in a+{\mathbb Z}^2}q^{Q(r)}e(B(r,b))\int_{t_1}^{t_2}e^{-\pi vB(r,c(x))^2}dx
.$$
\[Zthm\] The function $\Phi_{a,b}$ converges absolutely for any choice of parameters $a,b$, and $Q$ such that $Q$ is non-zero on $a+{\mathbb Z}^2$. Moreover, the function $\widehat \Phi_{a,b}$ converges absolutely and can be decomposed as $$\notag \widehat{\Phi}^{c_1,c_2}_{a,b} = \Phi^{c_1,c_2}_{a,b}+\varphi_{a,b}^{c_1}-\varphi_{a,b}^{c_2}
.$$ Moreover, it satisfies the elliptic transformations $$\begin{aligned}
\widehat{\Phi}^{c_1,c_2}_{a+\lambda,b+\mu} &= e(B(a,\mu))\widehat{\Phi}^{c_1,c_2}_{a,b}\quad \mbox{for all $\lambda\in {\mathbb Z}^2$ and }\mu\in A^{-1}{\mathbb Z}^2,\\
\widehat{\Phi}^{c_1,c_2}_{-a,-b} &= \widehat{\Phi}^{c_1,c_2}_{a,b},\end{aligned}$$ and the modular relations $$\begin{aligned}
\widehat{\Phi}^{c_1,c_2}_{a,b}(\tau+1) & = e\bigg(-Q(a)-\frac12 B\big(A^{-1}A^*,a\big)\bigg)\widehat{\Phi}^{c_1,c_2}_{a,a+b+\frac12 A^{-1}A^*}(\tau), \\
\widehat{\Phi}^{c_1,c_2}_{a,b}\bigg(-\frac{1}{\tau}\bigg) & = \frac{e(B(a,b))}{\sqrt{-\det{(A)}}} \sum_{p\in A^{-1}{\mathbb Z}^2\pmod{{\mathbb Z}^2}} \widehat{\Phi}^{c_1,c_2}_{-b+p,a}(\tau),\end{aligned}$$ where $A^*:=(A_{11},\ldots,A_{rr})^{\mathrm T}$.
These results can be conveniently repackaged in the language of Theorem \[mainthm2\] as follows. We note that the addition of the transformation results involving $(a^*,b^*)$ is based on the discussion of the proof of (14) in [@ZwegersMockMaass]. For future reference, we also define the equivalence relation on ${\mathbb R}^2$ $$(a,b)\sim(\alpha,\beta)$$ if $a\pm \alpha\in{\mathbb Z}^2$ and $b\pm \beta=:\mu\in{\mathbb Z}^2$ with $B(a,\mu)\in{\mathbb Z}$ (note that the two $\pm$ are required to have the same sign).
\[MainThm2ZwegersResult\] If $a,b$ are chosen so that $(\gamma a,\gamma b)\sim(a,b)$ for some $\gamma\in \mathrm{Aut}^+(Q,{\mathbb Z})$ with $\gamma c_1=c_2$, then $$\widehat{\Phi}^{c_1,c_2}_{a,b} = \Phi^{c_1,c_2}_{a,b}
.$$ In particular, it is a Maass form (with a multiplier) on a congruence subgroup.
The connection of the series $S_{a,b;M}$ with Maass forms (once these series are decorated with the proper modified Bessel functions of the second kind) follows from Zwegers’ work, given certain special conditions. To describe these, we define an equivalence relation on the set of pairs $(a,b)$. We also set $$\label{gm}
\gamma_M:=\bigg(\begin{matrix}M&M-1\\ M+1&M\end{matrix}\bigg),$$ which is useful for our purposes as it lies in $\mathrm{Aut}^+(Q,{\mathbb Z})$ and satisfies $\gamma_M c=c'$.
Finally, for a generic vector $x=(x_1,x_2)$, we let $$x^*:=(-x_1,x_2).$$
Quantum modular forms and the map $F\mapsto F^+$ {#MaassFormsHolomorphization}
------------------------------------------------
In this section, we review Lewis and Zagier’s construction [@LewisZagier2] of period functions for Maass waveforms, and following Zagier [@Za1] indicate how so-called quantum modular forms may be formed using them. We also use this construction in the proof of Theorem \[mainthm1\], as we shall see that the $q$-hypergeometric forms of the associated quantum modular forms are essential for showing cuspidality of the Maass waveforms.
We begin by recalling the definition of quantum modular forms (see [@Za1] for a general survey).
\[cocycle\] For any subset $X\subseteq\mathbb P^1({{\mathbb Q}})$, a function $f\colon X\rightarrow{\mathbb C}$ is a [*quantum modular form*]{} with [*quantum set $X$*]{} of weight $k\in\frac12{\mathbb Z}$ on a congruence subgroup $\Gamma$ if for all $\gamma\in\Gamma$, the cocycle ($|_k$ the usual slash operator) $$r_{\gamma}(x):=f|_{k}(1-\gamma)(x)$$
extends to an open subset of ${\mathbb R}$ and is real-analytic.
Zagier left his definition of quantum modular forms more open only requiring for $r_\gamma$ to be “nice”. In general, one knows one is dealing with a quantum modular form if it has a certain feel, which Zagier brilliantly explained in his several motivating examples.
The first main example Zagier gave, and the one most relevant for us here, is that of quantum modular forms attached to the positive (and negative) coefficients of Maass forms. Although Zagier only worked out this example explicitly in one case, and the work of Lewis and Zagier only studied Maass cusp forms of level one, for our purposes it is important to consider a more general situation. This is described in the following result, which extends observations of Lewis and Zagier for Maass Eisenstein series of Li, Ngo, and Rhoades for special examples in [@RobMaass], and where, for a Maass form $F$ on a congruence subgroup $\Gamma$, we set $$\Gamma_F:=\Gamma\cap\big\{\gamma\in\Gamma : F \text{ is cuspidal at } \gamma^{-1}i\infty\big\}
.$$
\[MaassQMFThm\] Let $F$ be a Maass waveform on a congruence subgroup $\Gamma$ with eigenvalue $1/4$ under $\Delta$ which is cuspidal at $i\infty$. Then $F^+$ defines a quantum modular form of weight one on a subset $X\subseteq\mathbb P^1({{\mathbb Q}})$ on $\Gamma_F$. Moreover, $F$ is cuspidal exactly at those cusps which lie in the maximal such set $X$.
1. The quantum modular form defined by $F^+$ may formally be given on all of $\mathbb P^1({{\mathbb Q}})$. This is done by considering asymptotic expansions of $F^+$ near the cusps, instead of simply values. This consideration leads to Zagier’s notion of a [*strong quantum modular form*]{}.
2. There is also a quantum modular form associated to the negative coefficients of $F$, which is also a part of the object corresponding to $F$ under the Lewis-Zagier correspondence of [@LewisZagier2].
The key idea, already present in [@LewisZagier2], is to realize $F^+$ as an integral transform of $F$ defined in . To describe this, we require the real-analytic function $R_{\tau}$ given by ($z=x+iy$ with $x,y\in{\mathbb R}$) $$R_{\tau}(z):=\frac{y^{\frac12}}{\sqrt{(x-\tau)^2+y^2}}
.$$ This function is an eigenfunction of $\Delta$ with eigenvalue $1/4$. For two real-analytic functions $f,g$ defined on $\mathbb H$, we also consider their Green’s form $$[f,g]:=\frac{\partial f}{\partial z}gdz+\frac{\partial g}{\partial \overline{z}}fd\overline{z}.$$
Then Lewis and Zagier showed (see also Proposition 3.5 of [@RobMaass] for a direct statement and a detailed proof) that $$F^+(\tau)=-\frac 2{\pi}\int_{\tau}^{i\infty}\big[F(z),R_{\tau}(z)\big]
.$$ This formula, which may also be thought of as an Abel transform, can also be rephrased as in the proposition of Chapter II, Section 2 of [@LewisZagier2] in the following convenient form: $$\label{FPlusAltInt}
F^+(\tau)\
=
\mathcal C\int_{\tau}^{i\infty}\Bigg(
\frac{\partial F(z)}{\partial z}\frac{y^{\frac12}}{(z-\tau)^{\frac12}(\overline z-\tau)^{\frac12}}dz
+\frac i4 F(z)\frac{(z-\tau)^{\frac12}}{y^{\frac12}(\overline z-\tau)^{\frac32}}d\overline{z}
\Bigg)
,$$ where $\mathcal C$ is a constant. Now for general functions $f,g$ which are eigenfunctions of $\Delta$ with eigenvalue $1/4$, the quantity $[f,g]$ is actually a closed one-form. This fact, combined with the modularity transformations of $F$ and the equivariance property $$R_{\gamma \tau}(\gamma z)=(c\tau+d)R_{\tau}(z)$$ for $\gamma=\big(\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\big)\in\operatorname{SL}_2({\mathbb R})$ directly shows (as in (14) of [@Za1]) that $$\label{MaassQuantumCocycle}
F^+(\tau)-(c\tau+d)^{-1}F^+(\gamma \tau)=-\int_{\gamma^{-1}i\infty}^{i\infty}\big[F(z),R_{\tau}(z)\big]$$ for all $\gamma\in\Gamma_F$. This last integral converges since, by assumption, $F$ is cuspidal at $\gamma^{-1}i\infty$. As the integral on the right hand side of the last formula is real-analytic on ${\mathbb R}\setminus\{\gamma^{-1}i\infty\}$, this establishes the first claim, if we note that the values of the quantum modular form, if they converge, are given as the limits towards rational points from above. That is, the value of the quantum modular form at $\alpha\in{{\mathbb Q}}$ equals $$\label{LimitEquationFPlus}
F^+(\alpha):=\lim_{t\rightarrow0^+} F^+(\alpha+it).$$
We next establish the second claim, which states that exists precisely for those $\alpha$ for which $F$ is cuspidal. By the existence of a Fourier expansion at all cusps in Lemma \[Maass0Fourier\], and using the exponential decay of $K_0(x)$ as $x\to\infty$, we find that, for $t>0$, $$\label{FourierExpansionCuspAsympExp}
F(\alpha+it)\approx\frac{\kappa_1}{|c|\sqrt{t}}-\frac{\kappa_2}{|c|\sqrt{t}}\log\big(c^2t\big)
,$$
where $\gamma=\big(\begin{smallmatrix}a & b\\ c& d\end{smallmatrix}\big)\in\operatorname{SL}_2({\mathbb Z})$ is chosen such that $\gamma \alpha=i\infty$ and we write $f(t) \approx g(t)$ if $f-g$ decays faster than any polynomial in $t$, as $t\rightarrow0^+$. We have also used the fact that $c\alpha+d=0$ to note that the imaginary part of $\gamma(\alpha+it)$ is $t/|c\alpha+cit+d|^2=1/(c^2t)$. Our goal is to show that converges if and only if $\kappa_1=\kappa_2=0$.
For this, we also require an estimate on $\frac{\partial F}{\partial z}(\alpha+it)$. To compute this, we note that $$\frac{\partial}{\partial z}[F(z)]_{z=\alpha+it}
=\frac{\partial}{\partial z}\bigg[F(\gamma z)\bigg]_{z=\alpha+it}=\frac{1}{j(\gamma, \alpha+it)^2}F'\big(\gamma(\alpha+it)\big),$$ where $j(\gamma,z):=(cz+d)$. Now using Lemma \[Maass0Fourier\], we obtain $$F'(z)\approx\frac{\partial}{\partial z}\Big(\kappa_1y^{\frac{1}{2}}+\kappa_2y^{\frac{1}{2}}\log(y)\Big)=\frac{i}{4}\Big((\kappa_1+2\kappa_2)y^{-\frac{1}{2}}+\kappa_2y^{-\frac{1}{2}}\log(y)\Big).$$ Using that $
j(\gamma, \alpha+it)=cit \text{ and } \mathrm{Im}(\gamma(\alpha+it))=1/(c^2t),
$ we obtain that $$\frac{\partial F}{\partial z}(\alpha+it)\approx
\frac{-i}{4|c|t^{\frac{3}{2}}}\Big(\kappa_1+2\kappa_2-\kappa_2\log\big(c^2t\big)\Big).$$
To determine when $\lim_{t\to 0^+}F^+(\alpha+it)$ exists, we need the following to converge: $$\int_\alpha^{i\infty}\Bigg(\frac{\partial F(z)}{\partial z}\frac{y^{\frac12}}{(z-\alpha)^{\frac12}(\overline{z}-\alpha)^{\frac12}}dz+\frac{i}{4}F(z)\frac{(z-\alpha)^{\frac12}}{y^{\frac12}(\overline{z}-\alpha)^{\frac32}}d\overline{z}\Bigg).$$ Making the change of variables $z=\alpha+it$ (note that we need to conjugate the second term) gives $$i\int_0^\infty\Bigg( \frac{\partial}{\partial z}\big[F(z)\big]_{z=\alpha+it}(-it)^{-\frac12}+\frac{i}{4}
\overline{F(\alpha+it)}(it)^{-\frac12}\Bigg).$$ The top part of this integral, say from $1$ to $\infty$, is always convergent, since we assumed that $F$ is cuspidal at $i\infty$. Towards $0$, the integrand behaves like $$-\frac{i}{4|c|t^{\frac32}}\Big(\kappa_1+2\kappa_2-\kappa_2\log\big(c^2t\big)\Big)(-it)^{-\frac12}+\frac{i}{4}\Bigg(\frac1{|c|t^{\frac12}}\Big(\overline{\kappa}_1-\overline{\kappa}_2\log\big(c^2 t\big)\Big)(it)^{-\frac12}\Bigg).$$ Comparing alike powers then gives that the integral only converges for $\kappa_1=\kappa_2=0$, i.e., if $F$ is cuspidal.
Proofs of the main results {#ProofsSection}
==========================
Proof of Theorem \[mainthm2\]
-----------------------------
For any $M\in{\mathbb N}_{\geq2}$, consider the quadratic form $Q(x,y):=\frac12\big((M+1)x^2-(M-1)y^2\big)$ associated to the symmetric matrix $
A:=\big(\begin{smallmatrix}
M+1 & 0
\\
0 & 1-M
\end{smallmatrix} \big)
$ and for $\ell\in\{1,2\}$ the vectors $$\begin{aligned}
c_\ell:=\frac{1}{\sqrt{M^2-1}}\big((-1)^\ell(M-1), M+1\big)^T.\end{aligned}$$ It is easily checked that $Q(c_\ell)=-1$ and $B(c_1,c_2)=-2M<0$, so that these two vectors lie in the same component $C_Q$. Choose $a=(a_1,a_2)\in{{\mathbb Q}}^2$ and $b=(b_1,b_2)\in{{\mathbb Q}}^2$. Then, for any vector $r=(n,\nu)^{\mathrm T}$, we find that $$B(r,c_1)B(r,c_2)=(M^2-1)(\nu - n)(\nu + n),$$ and thus $$\rho_A(a+r)
=
\frac12\Big(1+{\operatorname{sgn}}\big((a_1 - a_2 -\nu + n)(a_1 + a_2 + \nu + n)\big)\Big)
.$$ Given these choices, we find that the family of indefinite theta functions $S_{a,b;M}$ may be understood in Zwegers’ notation via the relation $$\Phi_{a,b}^+=\operatorname{sgn}(t_2-t_1) e\big((M+1)a_1b_1-(M-1)a_2b_2\big)S_{a,b;M}.$$ Since $\gamma_M$, defined in , can easily be verified to lie in $\operatorname{Aut}^+(Q,{\mathbb Z})$ and $\gamma_Mc_1=c_2$, the theorem then follows from Proposition \[MainThm2ZwegersResult\] if $(\gamma_M a,\gamma_M b)\sim(a,b)$. We next prove the theorem if $(\gamma_M a,\gamma_M b)\sim(a^*,b^*)$ and $(\gamma_M a^*,\gamma_M b^*)\sim(a,b)$. The key step is to show that the involution $(a,b)\mapsto(a^*,b^*)$ fixes $\widehat \Phi_{a,b}^{c,c'}$. For this, we compute a parameterization $c(t)$ of $C_Q$. We find that a suitable choice for $P$ is given by $
P=\frac1{\sqrt2}
\Big(
\begin{smallmatrix}
\sqrt{M+1} & \sqrt{M-1}
\\
\sqrt{M+1} & -\sqrt{M-1}
\end{smallmatrix}
\Big)
.
$ Then we obtain that $
c(t)
=
\bigg(\begin{smallmatrix}
\sqrt{\frac{2}{M+1}}\sinh(t)
\\
\sqrt{\frac{2}{M-1}}\cosh(t)
\end{smallmatrix}\bigg)
.
$ In this parameterization, we have $t_\ell=(-1)^{\ell+1}\operatorname{arcsinh}(-\sqrt{(M-1)/2})$ for $\ell\in\{1,2\}$, and $c(-t)=c^*(t)$. Hence, by sending $x\mapsto-x$ and $r\mapsto r^*$ in , we find that $\widehat \Phi_{a^*,b^*}^{c_1,c_2}=\widehat \Phi_{a,b}^{c_1,c_2}$. We then obtain, using Lemma \[Zlem\], that $$\begin{aligned}
2\widehat \Phi_{a,b}^{c_1,c_2}
&
=
\widehat \Phi_{a,b}^{c_1,c_2}+\widehat \Phi_{a^*,b^*}^{c_1,c_2}
\\
&
=
\Phi_{a,b}^{c_1,c_2}+\Phi_{a^*,b^*}^{c_1,c_2}+\varphi_{a,b}^{c_1}-\varphi_{a,b}^{c_2}+\varphi_{a^*,b^*}^{c_1}-\varphi_{a^*,b^*}^{c_2}
\\
&
=
\Phi_{a,b}^{c_1,c_2}+\Phi_{a^*,b^*}^{c_1,c_2}+\varphi_{a^*,b^*}^{c_2}-\varphi_{a,b}^{c_2}+\varphi_{a,b}^{c_2}-\varphi_{a^*,b^*}^{c_2}
\\
&
=
\Phi_{a,b}^{c_1,c_2}+\Phi_{a^*,b^*}^{c_1,c_2}
,
\end{aligned}$$ which shows that the completion terms in $\Phi_{a,b}^{c_1,c_2}$ cancel out, as desired. Finally, we note that since $M\in{\mathbb N}_{\ge2}$, $Q(x,y)$ cannot vanish at rational values $x,y$ unless $x=y=0$ since $M-1$ and $M+1$ are coprime and cannot both be squares. As we have supposed that $a\in{{\mathbb Q}}^2\setminus\{0\}$, it automatically follows that the quadratic form above cannot vanish on $a+\mathbb{Z}^2$, and hence our choice satisfies the convergence requirement in Theorem \[Zthm\].
Proof of Theorem \[mainthm1\]
-----------------------------
We begin by showing the connection of the relevant $q$-series to indefinite theta functions. To do so, we use the Bailey pairs in the following lemma. These pairs have the rare and important feature that the $\beta_n$ are polynomials.
\[twopairslemma\] Let $k,\ell \in \mathbb{N}$ with $1 \leq \ell \leq k$. We have the Bailey pair relative to $1$, $$\begin{aligned}
\alpha_n &= -q^{(k+1)n^2-n}\big(1-q^{2n}\big)\sum_{\nu=-n}^{n-1}(-1)^\nu q^{-\frac{1}{2}(2k+1)\nu^2 - \frac{1}{2}(2k-(2\ell-1))\nu} \label{firstalpha} \\
\intertext{and}
\beta_n &= H_n(k,\ell;1;q) \cdot \chi_{n \neq 0} \label{firstbeta},\end{aligned}$$ and the Bailey pair relative to $q$, $$\begin{aligned}
\alpha_n &= \frac{1-q^{2n+1}}{1-q}q^{(k+1)n^2+kn}\sum_{\nu=-n}^{n}(-1)^\nu q^{-\frac{1}{2}(2k+1)\nu^2 - \frac{1}{2}(2k - (2\ell-1))\nu} \label{secondalpha}\\
\intertext{and}
\beta_n &= H_n(k,\ell;0;q). \label{secondbeta}\end{aligned}$$
The Bailey pair relative to $1$ was established in [@Hi-Lo1 Section 5]. The proof of the Bailey pair relative to $q$ follows by using a similar argument. We begin by replacing $K$ by $k$ and $\ell$ by $k-\ell$ in part (i) of Theorem 1.1 of [@Lo1]. This gives that $(\alpha_n,\beta_n)$ is a Bailey pair relative to $q$, where $\alpha_n$ is given in and $\beta_n$ is the $z=1$ instance of $$\label{zcase}
\beta_n(z) = \sum_{n \geq m_{2k-1} \geq \ldots \geq m_1 \geq 0} \frac{q^{\sum_{\nu=1}^{k-1} (m_{k+\nu}^2+m_{k+\nu}) + \binom{m_k+1}{2} - \sum_{\nu=1}^{k-1} m_\nu m_{\nu+1} - \sum_{\nu=1}^{k-\ell} m_\nu}(-z)^{m_k}}{(q)_{m_{2k}-m_{2k-1}}(q)_{m_{2k-1}-m_{2k-2}}\cdot\ldots\cdot (q)_{m_2-m_1}(q)_{m_1}},$$ where $m_{2k} := n$. To transform the above into , we argue as in Sections 3 and 5 of [@Hi-Lo1]. We replace $m_1,\dots,m_{2k-1}$ by the new summation variables $n_1,\dots,n_{k-1}$ and $u_1,\dots,u_k$ as follows: $$\label{replacement}
m_{\nu} \mapsto
\begin{cases}
u_{k-\nu+1} + \cdots + u_k & \text{for $1 \leq \nu \leq k$}, \\
n_{\nu-k} + u_{\nu-k+1} + \cdots + u_k & \text{for $k+1 \leq \nu \leq 2k-1$}.
\end{cases}$$ With $m_0 = n_0 = 0$ and $n_{k} = n$, the inequalities $m_{i+1} - m_i \geq 0$ in for $0 \leq i \leq k-1$ give $u_i \geq 0$ and the inequalities $m_{k+i+1} - m_{k+i} \geq 0$ for $0 \leq i \leq k-1$ then give $0 \leq u_i \leq n_{i+1} - n_i$. Thus after a calculation to determine the image of the summand of under the transformations in , we find that $$\beta_n(z) = \sum_{n \geq n_{k-1} \geq \cdots \geq n_1 \geq 0} \prod_{\nu = 1}^k \sum_{u_{\nu} = 0}^{n_{\nu} - n_{\nu-1}} \frac{\big(-zq^{\min\{\nu,\ell\} + 2\sum_{\mu = 1}^{\nu-1}n_{\mu}}\big)^{u_{\nu}}q^{\binom{u_{\nu}}{2} + 2\binom{n_{\nu-1} + 1}{2}}}{(q)_{n_{\nu} - n_{\nu-1}}} \begin{bmatrix} n_{\nu} - n_{\nu-1} \\ u_{\nu} \end{bmatrix}.$$ By the $q$-binomial theorem $$\label{qbin}
\sum_{u=0}^n (-z)^uq^{\binom{u}{2}}\begin{bmatrix} n \\ u \end{bmatrix} = (z)_n,$$ each of the sums over $u_{\nu}$ may be carried out, giving $$\beta_n(z) = \sum_{n \geq n_{k-1} \geq \cdots \geq n_1 \geq 0} \prod_{\nu = 1}^k q^{2\binom{n_{\nu-1} + 1}{2}} \frac{\big(zq^{\min\{\nu,\ell\} + 2\sum_{\mu = 1}^{\nu-1} n_{\mu}}\big)_{n_{\nu} - n_{\nu-1}}}{(q)_{n_{\nu} - n_{\nu-1}}}.$$ Using the fact that $$\begin{bmatrix} n \\ k \end{bmatrix}_q = \frac{(q^{k+1})_{n-k}}{(q)_{n-k}},$$ we then have $$\begin{aligned}
\beta_n(1) &= \sum_{n \geq n_{k-1} \geq \cdots \geq n_1 \geq 0} \prod_{\nu = 1}^k q^{2\binom{n_{\nu-1} + 1}{2}} \begin{bmatrix} \min\{\nu,\ell\} - 1 + n_{\nu} - n_{\nu-1} + 2\sum_{\mu=1}^{\nu-1} n_{\mu} \\ n_{\nu} - n_{\nu-1} \end{bmatrix} \\
&=\sum_{n \geq n_{k-1} \geq \cdots \geq n_1 \geq 0} \prod_{\nu = 1}^{k-1} q^{2\binom{n_{\nu} + 1}{2}} \begin{bmatrix} \min\{\nu,\ell - 1\} + n_{\nu+1} - n_{\nu} + 2\sum_{\mu=1}^{\nu} n_{\mu} \\ n_{\nu+1} - n_{\nu} \end{bmatrix},\end{aligned}$$ in agreement with the $H_n(k,\ell;0;q)$, defined in .
The referee has observed that Lemma \[twopairslemma\] could also be proved by using together with ideas from [@Wa2].
With these Bailey pairs we prove the following key proposition.
\[IndefThetaFj\] We have $$\begin{aligned}
F_1(k,\ell;q) & = \sum_{n \geq 0} \sum_{|\nu | \leq n} (-1)^{n+\nu }q^{(k+1)n^2+kn+\binom{n+1}{2} - \frac12\big((2k+1)\nu ^2 + (2k - (2\ell-1))\nu \big)}\big(1-q^{2n+1}\big), \label{F1identity} \\
F_2(k,\ell;q) & = \frac{1}{2}\sum_{n \geq 0} \sum_{|\nu | \leq n} (-1)^{n+\nu }q^{(k+1)n^2+kn - \frac12\big((2k+1)\nu ^2 + (2k - (2\ell-1))\nu \big)}\big(1-q^{2n+1}\big), \notag \\ F_3(k,\ell;q) & = -\sum_{n \geq 1} \sum_{\nu = -n}^{n-1} (-1)^{n+\nu }q^{(k+1)n^2+ \binom{n}{2} - \frac12\big((2k+1)\nu ^2 + (2k - (2\ell-1)) \nu \big)}\big(1+q^n\big), \notag \\ F_4(k,\ell;q) & = -2\sum_{n \geq 1} \sum_{\nu = -n}^{n-1} (-1)^{n+\nu }q^{(k+1)n^2 - \frac12\big((2k+1)\nu ^2 + (2k - (2\ell-1)) \nu \big)}. \notag\end{aligned}$$
The first two identities follow upon using the Bailey pair in and in equations and , while the second two use and in equations and .
We are now ready to prove our main result.
We first apply Theorem \[mainthm2\] to the indefinite theta function representations of the $F_\nu $, given in Proposition \[IndefThetaFj\]. We begin with $F_1$. Using the term $(1-q^{2n+1})$ to split the right-hand side into two sums and then replacing $n$ by $-n-1$ in the second sum, we obtain $$F_1(k,\ell;q) = \Bigg(\sum_{n\pm \nu \geq 0} + \sum_{n\pm\nu < 0 }\Bigg) (-1)^{n+\nu }q^{(k+1)n^2+kn+\binom{n+1}{2} - \frac12\big((2k+1)\nu ^2 + (2k - (2\ell-1))\nu \big)}.$$ By completing the square, we directly compute that $$q^{\frac{(2k+1)^2}{8(2k+3)} - \frac{(2k-2\ell+1)^2}{8(2k+1)}}F_1(k,\ell;q) = \Bigg(\sum_{n\pm\nu \geq 0 } + \sum_{n\pm\nu < 0 }\Bigg) (-1)^{n+\nu } q^{\frac{1}{2}(2k+3)\big(n+\frac{2k+1}{2(2k+3)}\big)^2 - \frac{1}{2}(2k+1)\big(\nu +\frac{2k-2\ell+1}{2(2k+1)}\big)^2}.$$ We claim that the right-hand side is equal to $S_{a,b;M}$ defined in with $ M=2k + 2, a=(\frac{2k+1}{2(2k+3)},\frac{2k-2\ell+1}{2(2k+1)})^{\mathrm T}$, and $b=(\frac1{2(2k+3)},\frac1{2(2k+1)})^{\mathrm T}.$ The summand is directly seen to match that of . To show that the summation bounds are correct, we use the restrictions on $k$ and $\ell$ to verify the inequalities $0<a_1\pm a_2<1$. For example, to see this for $a_1-a_2$, we note that $$a_1-a_2=\frac{{(2k+3)\ell-2k-1}}{(2k+3)(2k+1)}$$ is positive exactly if $\ell>(2k+1)/(2k+3)$. As $0<(2k+1)/(2k+3)<1$ and $\ell\geq1$, this inequality automatically holds. To check the upper bound, note that $a_1-a_2<1$ exactly if $\ell<\frac{2(k+2)(2k+1)}{2k+3}$. This last expression is always bigger than $k$, and $\ell$ is, by assumption, bounded by $k$, so this inequality holds. The inequalities on $a_1+a_2$ may be checked in a similar manner. We then show that $$\gamma_M a+(\ell-2k-1)\begin{pmatrix}1\\ 1\end{pmatrix}=a^*,\quad \gamma_M a^*+\ell\begin{pmatrix}1\\ 1\end{pmatrix}=a, \quad \gamma_M b-\begin{pmatrix}1\\ 1\end{pmatrix}=b^*,\quad\text{and}\quad
\gamma_M b^*=b$$ and also that $B(a,(-1,-1)^{\mathrm T})=-\ell\in{\mathbb Z}$ Theorem \[mainthm2\] yields the first claim in Theorem \[mainthm1\] for $F_1$, namely that it is the generating function for the positive coefficients of a Maass waveform. We return to the question of cuspidality of this Maass form below, after indicating the related calculations which must be performed on the other $F_j$.
In the case of $F_2$, we find in the same manner that $F_2(k,\ell;q^2)$ is equal (up to a rational power of $q$) to $\frac12S_{a,b;M}$, where $M=4k+3, a=(\frac{k}{2(k+1)},\frac{2k-2\ell+1}{2(2k+1)})^{\mathrm T},\text{ and }b=(\frac1{8(k+1)},\frac1{4(2k+1)})^{\mathrm T}.$ As above, we check that $$\gamma_M a+(2\ell-4k-1)\begin{pmatrix}1\\ 1\end{pmatrix}=a^*, \quad\gamma_M a^*+(2\ell-1)\begin{pmatrix}1\\ 1\end{pmatrix}=a, \quad\gamma_M b-\begin{pmatrix}1\\ 1\end{pmatrix}=b^*, \quad\text{and}\quad\gamma_M b^*=b.$$ Here, we also have $0<a_1\pm a_2<1$, and we compute $B(a,(-1,-1)^{\mathrm T})=-2\ell+1\in{\mathbb Z}$, which establishes the theorem for $F_2$.
For $F_3$, we use the specializations $M=2k+2, a=(-\frac1{2(2k+3)},\frac{2k-2\ell+1}{2(2k+1)})^\mathrm{T}, b=(\frac1{2(2k+3)}, \frac1{2(2k+1)})^\mathrm{T},$ and find that $0<a_1+ a_2<1, -1<a_1- a_2<0$ $$\gamma_M a+(\ell-k)\begin{pmatrix} 1\\ 1\end{pmatrix}=a^*, \quad \gamma_M a^*+(\ell-k-1)\begin{pmatrix} 1\\ 1\end{pmatrix}=a,
\quad\gamma_M b-\begin{pmatrix}1\\ 1\end{pmatrix}=b^*, \quad\gamma_M b^*=b,$$ and $B(a,(-1,-1)^{\mathrm T})=k-\ell+1$.
Finally, for $F_4$, we have $M=4k+3, a=(0, \frac{2k-2\ell+1}{2(2k+1)})^\mathrm{T}, b=(\frac1{8(k+1)}, \frac1{4(2k+1)})^\mathrm{T},$ and calculate that $0<a_1+ a_2<1, -1<a_1- a_2<0$, while $$\gamma_M a+(2\ell-2k-1)\begin{pmatrix} 1\\ 1\end{pmatrix}=a,
\quad\gamma_M b^*=b, \quad\gamma_M b-\begin{pmatrix}1\\ 1\end{pmatrix}=b^*
,$$ and $B(a,(-1,-1)^{\mathrm T})=2k-2\ell+1$.
Thus, we have shown that the $q$-series in Theorem \[mainthm1\] are indeed the positive parts of Maass forms. By the construction of Zwegers’ Maass forms via the Fourier expansions in , we see that the Maass forms here are all cuspidal at $i\infty$ whenever they converge.
Theorems \[MaassQMFThm\] and \[mainthm3\] then imply that in fact each of the Maass forms in Theorem \[mainthm1\] are indeed cusp forms.
Proof of Theorem \[mainthm3\]
------------------------------
Theorem \[mainthm3\] follows directly from Theorem \[mainthm1\] and Theorem \[MaassQMFThm\], together with the observation that the $q$-series in converge (as they are finite sums) at all roots of unity, which implies that their radial limits exist and equal these values by Abel’s theorem. Although Theorem \[MaassQMFThm\] is only stated for Maass forms with trivial multiplier for simplicity, a review of the proof shows that the method applies equally well to our Maass waveforms with multipliers.
Further questions and outlook {#QuestionsSection}
=============================
There are several outstanding questions which naturally arise from the main results considered here. In what follows, we outline five interesting directions for future investigation.
[**1).**]{} As the example of $\sigma,\sigma^*$ indicates, it is worthwhile to look at the negative coefficients of the related Maass form. Thus, it is natural to ask: are there nice hypergeometric representations for the $q$-series formed by the negative coefficients of the Maass forms $G_{j ,k,\ell}$ in Theorem \[mainthm2\]? For example, one such series has the shape $$\sum_{n, \nu \in {\mathbb Z}\atop |(M+1)n+M-1|<2|(M-1)\nu+M-1-2 \ell|}(-1)^{n+\nu}q^{-\frac{1}{8(M+1)(M-1)}\big((M-1)(2(M+1)n+M-1)^2-(M+1)(2(M-1)\nu+M-1-2\ell)^2\big)}.$$ If so, do they have relations to the $q$-hypergeometric series defining the $F_j$-functions, as $\sigma$ and $\sigma^*$ satisfy? Such a connection could help explain relationships between passing from positive to negative coefficients of Maass waveforms and letting $q\mapsto q^{-1}$ in $q$-hypergeometric series. Examples of such relationships were observed by Li, Ngo, and Rhoades [@RobMaass], and further commented on in [@KRW]. However, the authors were unable to identify suitable Bailey pairs to make this idea work in our case.
[**2).**]{} As in Theorem \[Zthm\], we may also think of the Maass forms corresponding to the $F_j$-functions as components of vector-valued Maass waveforms (as discussed in detail for the $\sigma,\sigma^*$ case in [@ZwegersMockMaass]). Is it possible to find nice $q$-hypergeometric interpretations for the corresponding positive (or negative) coefficients of the other components of such vectors as well? That is, are the $q$-series associated to the expansions of the Maass waveforms at other cusps than $i\infty$ interesting from a $q$-series or combinatorial point of view?
[**3).**]{} Define the $q$-series $$\notag \mathcal{U}_k^{(\ell)}(x;q) := \sum_{n \geq 0}q^{n} (-x)_{n}\bigg(\frac{-q}{x}\bigg)_{n}H_{n}(k,\ell;0;q).$$ These are analogous to the series $U_k^{(\ell)}(x;q)$, defined by Hikami and the second author [@Hi-Lo1] by $$\notag U_k^{(\ell)}(x;q) := q^{-k}\sum_{n \geq 1} q^{n}(-xq)_{n-1}\bigg(\frac{-q}{x}\bigg)_{n-1} H_{n}(k,\ell;1;q).$$ At roots of unity the functions $U_k^{(\ell)}(-1;q)$ are (vector-valued) quantum modular forms which are “dual" to the generalized Kontsevich-Zagier functions $$\notag F_k^{(\ell)}(q)
:=
q^k
\sum_{n_1, \dots, n_k\geq 0}
(q)_{n_k} \,
q^{n_1^{2} + \cdots + n_{k-1}^{2} + n_{\ell} + \cdots + n_{k-1}} \,
\prod_{j=1}^{k-1}
\begin{bmatrix}
n_{j+1} + \delta_{j,\ell-1} \\
n_j
\end{bmatrix},$$ in the sense that $$\notag F_k^{(\ell)}(\zeta_N) = U_k^{(\ell)}\big(-1;\zeta_N^{-1}\big),$$ where $\zeta_N:=e^{2\pi i /N}$. Are the $\mathcal{U}_k^{(\ell)}(-1;q)$ also quantum modular forms like the $U_k^{(\ell)}(-1;q)$? Are they related at roots of unity to some sort of Kontsevich-Zagier type series?
[**4).**]{} Using Bailey pair methods one can show that $$\begin{aligned}
\mathcal{U}_k^{(\ell)}(-x;q)
&= \frac{(x)_{\infty} \big(\frac{q}{x}\big)_\infty}{
(q)_\infty ^2} \notag \\
&\times
{\bBigg@{4}}(
\sum_{\substack{r,s,t \geq 0 \\ r \equiv s \pmod{2}}} +
\sum_{\substack{r,s,t < 0 \\ r \equiv s \pmod{2}}}
{\bBigg@{4}})
(-1)^{\frac{r-s}{2}}x^t
q^{\frac{r^2}{8}+
\frac{4k+3}{4} r s + \frac{s^2}{8}+\frac{4k+3-2\ell}{4} r
+
\frac{1+2\ell}{4} s + t\frac{r+s}{2}} . \nonumber
\end{aligned}$$ This is analogous to [@Hi-Lo1] $$\begin{aligned}
U_k^{(\ell)}(-x;q)
&=
-q^{-\frac{k}{2}-\frac{\ell}{2}+\frac{3}{8}}
\frac{(x q)_{\infty} \big(\frac{q}{x}\big)_\infty}{
(q)_\infty ^2} \notag \\
&\times
{\bBigg@{4}}(
\sum_{\substack{r,s,t \geq 0 \\ r \not \equiv s \pmod{2}}} +
\sum_{\substack{r,s,t < 0 \\ r \not \equiv s \pmod{2}}}
{\bBigg@{4}})
(-1)^{\frac{r-s-1}{2}}x^t
q^{\frac{r^2}{8}+
\frac{4k+3}{4} r s + \frac{s^2}{8}+\frac{1+\ell+k}{2} r
+
\frac{1-\ell+k}{2} s + t\frac{r+s+1}{2}} . \nonumber
\end{aligned}$$ What sort of modular behavior is implied by these expansions?
[**5).**]{} As per the discussion in [@RobMaass], there is hope that the Maass forms in Theorem \[mainthm1\] are related to Hecke characters or multiplicative $q$-series. In fact, such connections were related to all related examples of $q$-hypergeometric examples found in the literature, although finding a general formulation seems intractable at the moment since as the discriminants of the quadratic fields grow, explicitly identifying such characters becomes computationally difficult.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to the referee for many helpful comments, especially the observation that the polynomials $H_n(k,\ell;b;q)$ can be related to the Andrews-Gordon identities and for simplifying the proof of Lemma \[twopairslemma\].
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[^1]: The research of the first author is supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results receives funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant agreement n. 335220 - AQSER. The third author thanks the University of Cologne and the DFG for their generous support via the University of Cologne postdoc grant DFG Grant D-72133-G-403-151001011, funded under the Institutional Strategy of the University of Cologne within the German Excellence Initiative.
[^2]: Note that $F_2(k,\ell;q)$ has a convergence issue, which we overcome by averaging over the even and odd partial sums with respect to $n$.
|
---
abstract: |
We consider the graph classes [<span style="font-variant:small-caps;">Grounded</span>-[[]{.nodecor}]{}]{} and [<span style="font-variant:small-caps;">Grounded</span>-$\{{\textnormal{\textsf{L}}},\JJ\}$]{} corresponding to graphs that admit an intersection representation by [[]{.nodecor}]{}-shaped curves (or [[]{.nodecor}]{}-shaped and -shaped curves, respectively), where additionally the topmost points of each curve are assumed to belong to a common horizontal line. We prove that [<span style="font-variant:small-caps;">Grounded</span>-[[]{.nodecor}]{}]{} graphs admit an equivalent characterisation in terms of vertex ordering with forbidden patterns.
We also compare these classes to related intersection classes, such as the grounded segment graphs, the monotone [[]{.nodecor}]{}-graphs (a.k.a. max point-tolerance graphs), or the outer-1-string graphs. We give constructions showing that these classes are all distinct and satisfy only trivial or previously known inclusions.
author:
- |
Vít Jelínek[^1]\
Computer Science Institute, Faculty of Mathematics and Physics,\
Charles University, Prague, Czechia\
jelinek@iuuk.mff.cuni.cz\
- |
Martin Töpfer[^2]\
Institute of Science and Technology, Klosterneuburg, Austria\
mtopfer@gmail.com
bibliography:
- 'outerl.bib'
title: 'On grounded L-graphs and their relatives'
---
Introduction
============
An *intersection representation* of a graph $G=(V,E)$ is a map that assigns to every vertex $x\in V$ a set $s_x$ in such a way that two vertices $x$ and $y$ are adjacent if and only if the two corresponding sets $s_x$ and $s_y$ intersect. The graph $G$ is then the *intersection graph* of the set system $\{s_x;\; x\in V\}$. Many natural graph classes can be defined as intersection graphs of sets of a special type.
One of the most general classes of this type is the class of *string graphs*, denoted [<span style="font-variant:small-caps;">String</span>]{}. A string graph is an intersection graph of *strings*, which are bounded continuous curves in the plane. All the graph classes we consider in this paper are subclasses of string graphs.
A natural way of restricting a string representation is to impose geometric restrictions on the strings we consider. This leads, for instance, to *segment graphs*, which are intersection graphs of straight line segments, or to *[[]{.nodecor}]{}-graphs*, which are intersection graphs of [[]{.nodecor}]{}-shapes, where an [[]{.nodecor}]{}-shape is a union of a vertical segment and a horizontal segment, in which the bottom endpoint of the vertical segment coincides with the left endpoint of the horizontal one. Apart from [[]{.nodecor}]{}-shapes, we shall also consider -shapes, which are obtained by reflecting an [[]{.nodecor}]{}-shape along a vertical axis.
Apart from restricting the geometry of the strings, one may also restrict a string representation by imposing conditions on the placement of their endpoints. Following the terminology of Cardinal et al. [@Cardinal], we will say that a representation is *grounded* if all the strings have one endpoint on a common line (called *grounding line*) and the remaining points of the strings are confined to a single open halfplane with respect to the grounding line. We will usually assume that the grounding line is the $x$-axis, and the strings extend below the line. The endpoint belonging to the grounding line is the *anchor* of the string.
Similarly, a string representation is an *outer* representation, if all the strings are confined to a disk, and each string has one endpoint on the boundary of the disk. The endpoint on the boundary is again called the *anchor* of the string. One may easily see that a graph admits a grounded string representation if and only if it admits an outer-string representation. Such graphs are known as *outer-string* graphs, and we denote their class [<span style="font-variant:small-caps;">Outer-string</span>]{}.
![Graph classes considered in this paper. Arrows indicate inclusions. We will see in Section \[sec-sep\] that there are no other inclusions among these classes apart from those implied by the depicted arrows. In particular, the classes are all distinct.[]{data-label="fig-classes"}](classes){width="90.00000%"}
Our first main result, Theorem \[thm-patterns\] in Section \[sec-ord\], is a characterisation of the class of grounded [[]{.nodecor}]{}-graphs by vertex orderings avoiding a pair of forbidden patterns. Our next main result, presented in Section \[sec-sep\], is a collection of constructions providing separations between the classes in Figure \[fig-classes\], showing that there are no nontrivial previously unknown inclusions among them.
Let us now formally introduce the graph classes we are interested in, and briefly review some relevant previously known results.
*1-string graphs* are the graphs that admit a string representation in which any two distinct strings intersect at most once. The class of 1-string graphs is denoted [<span style="font-variant:small-caps;">1-String</span>]{}.
*Outer-1-string graphs* (denoted [<span style="font-variant:small-caps;">Outer-1-string</span>]{}) are the graphs that have a string intersection representation which is simultaneously a 1-string representation and an outer-string representation. Note that not every graph from [<span style="font-variant:small-caps;">1-string</span>]{}$\,\cap\,$[<span style="font-variant:small-caps;">Outer-string</span>]{} is necessarily in [<span style="font-variant:small-caps;">Outer-1-string</span>]{}, as we shall see in Section \[sec-sep\].
*[[]{.nodecor}]{}-graphs* ([[]{.nodecor}]{}) are the intersection graphs of [[]{.nodecor}]{}-shapes. This type of representation has received significant amount of interest lately. A notable recent result is a theorem of Gonc calves, Isenmann and Pennarun [@Goncalves] showing that every planar graph is an [[]{.nodecor}]{}-graph. Since it is known that [[]{.nodecor}]{}-graphs are a subclass of segment graphs [@Middendorf], this result strengthens an earlier result of Chalopin and Gonçalves [@CG] showing that all planar graphs are segment graphs.
*Max point-tolerance graphs* ([<span style="font-variant:small-caps;">Mpt</span>]{}), also known as *monotone [[]{.nodecor}]{}-graphs*, are the graphs with an [[]{.nodecor}]{}-representation in which all the bends of the [[]{.nodecor}]{}-shapes belong to a common downward-sloping line. This class was independently introduced by Soto and Thraves Caro [@Soto], by Catanzaro et al. [@Catanzaro] and by Ahmed et al. [@Ahmed]. Apart from the above intersection representation by [[]{.nodecor}]{}-shapes, it admits several other equivalent characterisations. Notably, [<span style="font-variant:small-caps;">Mpt</span>]{} graphs can be characterised as graphs that admit a vertex ordering that avoids a certain forbidden pattern [@Ahmed; @Catanzaro; @Soto]. This graph class is known to be a superclass of several important graph classes, such as outerplanar graphs and interval graphs, among others [@Ahmed; @Catanzaro; @Soto].
*Grounded segment graphs* ([<span style="font-variant:small-caps;">Grounded-seg</span>]{}) are the intersection graphs admitting a grounded segment representation. Cardinal et al. [@Cardinal] proved that these are also precisely the intersection graphs of downward rays in the plane. Note that any grounded segment graph also admits an outer-segment representation, but the converse does not hold, as shown by Cardinal et al. [@Cardinal]. Cardinal et al. also showed that outer-segment graphs are a proper subclass of outer-1-string graphs. This strengthens an earlier result of Cabello and Jejčič [@CabelloJejcic], who showed that outer-segment graphs are a proper subclass of outer-string graphs.
*Grounded [[]{.nodecor}]{}-graphs* ([<span style="font-variant:small-caps;">Grounded</span>-[[]{.nodecor}]{}]{}) are the intersection graphs of grounded [[]{.nodecor}]{}-shapes, that is, [[]{.nodecor}]{}-shapes with top endpoint on the $x$-axis. This class of graphs has been first considered by McGuiness [@McGuiness], who represented them as intersections of upward-infinite [[]{.nodecor}]{}-shapes. These graphs can also equivalently be represented as intersections of [[]{.nodecor}]{}-shapes inside a disk, with the top endpoint of each [[]{.nodecor}]{}-shape anchored to the boundary of the disk. McGuiness has shown that this class is $\chi$-bounded, i.e., these graphs have chromatic number bounded from above by a function of their clique number. The $\chi$-boundedness result has been later generalized to all outer-string graphs by Rok and Walczak [@Rok].
*Grounded $\{{\textnormal{\textsf{L}}},\JJ\}$-graphs* ([<span style="font-variant:small-caps;">Grounded</span>-$\{{\textnormal{\textsf{L}}},\JJ\}$]{}) are analogous to grounded [[]{.nodecor}]{}-graphs, but their representation may use both [[]{.nodecor}]{}-shapes and -shapes. An argument of Middendorf and Pfeiffer [@Middendorf] shows that [<span style="font-variant:small-caps;">Grounded</span>-$\{{\textnormal{\textsf{L}}},\JJ\}$]{} is a subclass of [<span style="font-variant:small-caps;">Grounded-Seg</span>]{}.
*Circle graphs* ([<span style="font-variant:small-caps;">Circle</span>]{}) are the intersection graphs of chords inside a circle, or equivalently, the intersection graphs of [[]{.nodecor}]{}-shapes drawn inside a circle, so that both endpoints of each [[]{.nodecor}]{}-shape touch the circle. Circle graphs include all outerplanar graphs [@Wessel].
*Interval graphs* ([<span style="font-variant:small-caps;">Int</span>]{}) are the intersection graphs of intervals on the real line. Equivalently, we may easily observe that these are exactly the graphs with an intersection representation which is simultaneously an [<span style="font-variant:small-caps;">Mpt</span>]{}-representation and a [<span style="font-variant:small-caps;">Grounded</span>-[[]{.nodecor}]{}]{}-representation. But note that not every graph from the intersection of [<span style="font-variant:small-caps;">Mpt</span>]{} and [<span style="font-variant:small-caps;">Grounded</span>-[[]{.nodecor}]{}]{} is an interval graph, as witnessed, e.g., by any cycle $C_n$ of length $n\ge 4$.
*Permutation graphs* ([<span style="font-variant:small-caps;">Per</span>]{}) are the intersection graphs of segments between a pair of parallel lines, with each segment having one endpoint on each of the two lines. Equivalently, we may observe that these are exactly the graphs admitting an [[]{.nodecor}]{}-representation in which the top endpoints of all the [[]{.nodecor}]{}-shapes are on a common horizontal line and the right endpoints are on a common vertical line.
We will always assume implicitly that the intersection representations we deal with satisfy certain non-degeneracy assumptions. In particular, we will assume that the strings have no self-intersections, that any two strings intersect in at most finitely many points (except for interval representations), and that any intersection of two strings is a proper crossing. In particular, an endpoint of a string does not belong to another string. Moreover, we will assume that every segment in a segment representation is non-degenerate, i.e., it has distinct endpoints. This also applies to horizontal and vertical segments forming an [[]{.nodecor}]{}-shape or -shape. These assumptions imply, in particular, that in any $\{{\textnormal{\textsf{L}}},\JJ\}$-representation, each intersection is realized as a crossing of a horizontal segment with a vertical one.
Note that in any grounded representation with a horizontal grounding line, the left-to-right ordering of the anchors on the grounding line defines a linear order on the vertex set of the represented graph. We say that this linear order is *induced* by the representation. Similarly, for an [<span style="font-variant:small-caps;">Mpt</span>]{} representation, we can define the induced order by following the top-left to bottom-right order of the bends along their common supporting line. Induced vertex orders play an important part both in characterising graphs in a given class and in separating different classes.
Vertex orders with forbidden patterns {#sec-ord}
=====================================
Our main result is a characterisation of grounded [[]{.nodecor}]{}-graphs as graphs that admit vertex orderings avoiding a pair of four-vertex patterns. Let us begin by formalising the key notions.
An *ordered graph* is a pair $(G,<)$, where $G=(V,E)$ is a graph and $<$ is a linear order on $V$. A *pattern of order $k$* is a triple $P=(W,C,F)$ where $W$ is the set $\{1,2,\dotsc,k\}$ while $C$ and $F$ are two disjoint subsets of $\binom{W}{2}$. The set $W$ is the *vertex set* of the pattern $P$, $C$ is the set of *compulsory edges* of $P$, and $F$ is the set of *forbidden edges*.
For an ordered graph $(G,<)$ with $G=(V,E)$, we say that $(G,<)$ *contains* a pattern $P=(W,C,F)$ of order $k$ if $G$ contains $k$ distinct vertices $x_1<x_2<\dotsb<x_k$ such that for every $\{i,j\}\in C$ the vertices $x_i$ and $x_j$ are adjacent in $G$, while for every $\{i,j\}\in F$, $x_i$ and $x_j$ are non-adjacent in $G$. If $(G,<)$ does not contain $P$, we say that it *avoids* $P$. For simplicity, we will often write an edge $\{i,j\}$ as $ij$.
![Forbidden order patterns for various graph classes [@Brandstadt; @Catanzaro; @Damaschke]. The solid arcs denote compulsory edges and the dotted arcs are forbidden edges.[]{data-label="fig-pats"}](pats){width="\textwidth"}
Many important graph classes can be characterised in terms of vertex orderings with forbidden patterns, that is, for a class ${\mathcal{C}}$ there is a pattern $P_{\mathcal{C}}$ such that a graph $G=(V,E)$ is in ${\mathcal{C}}$ if and only if it admits a linear order $<$ such that $(G,<)$ avoids $P_{\mathcal{C}}$; see Figure \[fig-pats\] for examples of classes with their forbidden patterns. The forbidden pattern characterisation of [<span style="font-variant:small-caps;">Mpt</span>]{} was found independently by at least three groups of authors [@Ahmed; @Catanzaro; @Soto].
As our first main result, we show that [<span style="font-variant:small-caps;">Grounded</span>-[[]{.nodecor}]{}]{} is characterised by a pair of forbidden patterns.
![The two forbidden ordering patterns for the class [<span style="font-variant:small-caps;">Grounded</span>-[[]{.nodecor}]{}]{}.[]{data-label="fig-grlpats"}](grlpats){width="80.00000%"}
\[thm-patterns\] Consider the patterns ${P_1}=(\{1,2,3,4\},\{13,24\},\{12,23\})$ and ${P_2}=(\{1,2,3,4\},\{12,14,23\},\{13\})$; see Figure \[fig-grlpats\]. A graph $G=(V,E)$ is a grounded [[]{.nodecor}]{}-graph if and only if it has a vertex ordering that avoids both ${P_1}$ and ${P_2}$. In fact, a linear order $<$ on $V$ avoids the two patterns ${P_1}$ and ${P_2}$ if and only if $G$ has a grounded [[]{.nodecor}]{}-representation which induces the linear order $<$.
Suppose first that $G$ has a grounded [[]{.nodecor}]{}-representation. Let $\ell_1,\ell_2,\dotsc,\ell_n$ be the [[]{.nodecor}]{}-shapes used in the representation, ordered left to right according to the positions of their anchors. Let $h_i$ and $v_i$ denote, respectively, the horizontal and vertical segment of $\ell_i$. Let $x_i$ be the vertex represented by $\ell_i$. We will show that the vertex ordering $x_1<x_2<\dotsb<x_n$ avoids the two patterns ${P_1}$ and ${P_2}$.
Assume that $(G,<)$ contains ${P_1}$, and let $x_p<x_q<x_r<x_s$ be the four vertices forming a copy of ${P_1}$. Since $x_q x_s$ is an edge, the two [[]{.nodecor}]{}-shapes $\ell_q$ and $\ell_s$ intersect. Let $R$ be the rectangle whose vertices are the anchors of $\ell_q$ and $\ell_s$, the bend of $\ell_q$ and the intersection of $\ell_q$ and $\ell_s$. Since neither $x_p$ nor $x_r$ is adjacent to $x_q$, we see that $\ell_p$ is completely outside of $R$, while $v_r$ is inside $R$. It follows that $\ell_p$ and $\ell_r$ are disjoint, and fail to represent the compulsory edge $13$ of ${P_1}$.
Suppose now that $(G,<)$ contains ${P_2}$, and let $x_p<x_q<x_r<x_s$ now be the four vertices forming a copy ${P_2}$. Since $x_p x_s$ is an edge, the segment $h_p$ intersects $v_s$. Moreover, $v_q$ intersects $h_p$, while $v_r$ does not intersect $h_p$, and in particular, $\ell_q$ and $\ell_r$ fail to represent the compulsory edge $23$ of ${P_2}$. We conclude that any grounded [[]{.nodecor}]{}-representation of $G$ induces a vertex order that avoids ${P_1}$ and ${P_2}$.
To prove the converse, assume that $G$ is a graph with a vertex ordering $x_1<x_2<\dotsb<x_n$ which avoids both ${P_1}$ and ${P_2}$. We will construct a grounded [[]{.nodecor}]{}-representation $\ell_1,\ell_2,\dotsc,\ell_n$ of $G$ inducing the order $<$, with $\ell_i$ being the [[]{.nodecor}]{}-shape representing the vertex $x_i$.
We fix the anchor of $\ell_i$ to be the point $(i,0)$ on the horizontal axis. Next, we process the vertices left to right, and for a vertex $x_i$ we define the representing shape $\ell_i$, assuming $\ell_1,\ell_2,\dotsc,\ell_{i-1}$ have already been defined, and assuming further that for any $j<i$ such that $x_j x_i$ is an edge of $G$, the horizontal segment $h_j$ of $\ell_j$ reaches to the right of the point $(i,0)$.
To define $\ell_i$, we first describe its vertical segment $v_i$. Let $N^-_i$ be the set of vertices $x_j$ such that $j<i$ and $x_j x_i\in E$. If $N^-_i$ is empty, choose the vertical segment $v_i$ to be shorter than any of $v_1,\dotsc,v_{i-1}$. In particular, $v_i$ will not intersect any of the [[]{.nodecor}]{}-shapes constructed in previous steps. If $N^-_i$ is nonempty, let $x_p$ be a vertex from $N^-_i$ chosen so that $v_p$ is as long as possible (and therefore $h_p$ is as low as possible). Then define $v_i$ to be slightly longer than $v_p$, so that $v_i$ intersects $h_p$ (recall that $h_p$ reaches to the right of $(i,0)$) but does not intersect any [[]{.nodecor}]{}-shape whose horizontal segment is below $h_p$. This choice of $v_i$ guarantees that $v_i$ intersects $h_j$ for any $x_j\in N^-_i$.
It remains to define the segment $h_i$. Let $j$ be the largest index such that $j>i$ and $x_i x_j\in E$. If no such $j$ exists, set $j=i$. The horizontal segment $h_i$ then has length $j-i+\frac12$, and in particular, its right endpoint has horizontal coordinate $j+\frac12$.
Having defined the [[]{.nodecor}]{}-shapes $\ell_1,\dotsc,\ell_n$ as above, let us verify that their intersection graph is $G$. If $x_j x_i$ is an edge of $G$ with $j<i$, then the definition of $v_i$ guarantees that $v_i$ intersects $h_j$, and therefore the two [[]{.nodecor}]{}-shapes $\ell_j$ and $\ell_i$ intersect.
To prove the converse, suppose for contradiction that for some $j<i$ the two [[]{.nodecor}]{}-shapes $\ell_j$ and $\ell_i$ intersect while $x_j x_i$ is not an edge of $G$. Choose such a pair $i,j$ so that $i$ is the smallest possible. There must be an index $k>i$ such that $x_j x_k$ is an edge of $G$, otherwise $h_j$ would be too short to intersect $v_i$. Similarly, there must be an index $m<i$ such that $x_m x_i$ is an edge of $G$, and $v_m$ is longer than $v_j$, otherwise $v_i$ would not be long enough to intersect $h_j$.
We now distinguish two cases depending on the relative position of $m$ and $j$. If $m<j$, then $\ell_m$ and $\ell_j$ are disjoint (recall that $v_m$ is longer than $v_j$) and hence $x_m x_j$ is not an edge of $G$. It follows that the four vertices $x_m<x_j<x_i<x_k$ form the pattern ${P_1}$, a contradiction. Suppose now that $j<m$. It follows that $\ell_j$ intersects $\ell_m$, and therefore $x_j x_m$ is an edge of $G$, by the minimality of $i$. Thus, the four vertices $x_j<x_m<x_i<x_k$ form the pattern ${P_2}$, which is again a contradiction.
Separations between classes {#sec-sep}
===========================
Consider again the classes in Figure \[fig-classes\]. The inclusions indicated by arrows are either easy to observe or follow from known results that we have pointed out in the introduction. Our goal now is to argue that there are no other inclusions among these classes except those that follow by transitivity from the depicted arrows. In particular, the classes are all distinct.
![An ordered graph $G$, and an example of its cycle extension $H$.[]{data-label="fig-cyclex"}](cyclex){width="90.00000%"}
As our main tool, we will use a lemma which is a slight modification of the ‘Cycle Lemma’ of Cardinal et al. [@Cardinal]. The lemma allows to prescribe the cyclic order of a subset of vertices in an outer-1-string representation of a graph. Let $G=(V_G,E_G)$ be a graph on $n$ vertices $x_1, x_2,\dotsc,x_n$, and let $<$ be the linear order $x_1<x_2<\dotsb<x_n$. The *cyclic shift* of $<$ is the linear order $<_s$ defined as $x_n<_s x_1<_s x_2<_s\dotsb<_s x_{n-1}$. The *reversal* of $<$, denoted $<_r$, is defined as with $x_n <_r x_{n-1}<_r \dotsb<_r x_1$. We say that two linear orders of $V$ are *equivalent* if one can be obtained from the other by a sequence of cyclic shifts and reversals.
A *cycle extension* of the ordered graph $(G,<)$ is an (unordered) graph $H=(V_H,E_H)$ with these properties (see Figure \[fig-cyclex\]):
- $V_H$ is the disjoint union of the sets $V_G=\{x_1,\dotsc,x_n\}$ and $V_C=\{y_1,\dotsc,y_{5n}\}$. The vertices $V_G$ induce the graph $G$ (in particular, $E_G\subseteq E_H$), and $V_C$ induce a cycle of length $5n$ with edges $y_1 y_2,\allowbreak y_2
y_3,\dotsc,\allowbreak
y_{5n-1}y_{5n},\allowbreak y_{5n} y_1$.
- For each vertex $x_i\in V_G$, either $x_i$ is adjacent to $y_{5i}$ and has no other neighbors in $V_C$, or $x_i$ is adjacent to $y_{5i-1}$ and $y_{5i}$ and has no other neighbors in $V_C$.
For the classes of graphs we consider, an intersection representation of a graph $G$ inducing an order $<$ can always be extended into a representation of a cycle extension of $G$, without modifying the curves representing $G$. This is formalised by the next lemma.
\[lem-extend\] Given a graph class ${\mathcal{C}}\in\{\text{{\textsc{Grounded}-{\textnormal{\textsf{L}}}}},\text{{\textsc{Grounded}-\ensuremath{\{{\textnormal{\textsf{L}}},\JJ\}}}},\allowbreak\text{{\textsc{Mpt}}},\allowbreak
\text{{\textsc{Grounded-seg}}}, \allowbreak \text{{\textsc{Outer-1-string}}}\}$, for every ${\mathcal{C}}$-representation of a graph $G$ inducing an order $<$ on $V_G$ there is a cycle extension $H$ of $(G,<)$ such that a ${\mathcal{C}}$-representation of $H$ can be constructed by adding into the given ${\mathcal{C}}$-representation of $G$ the curves representing the vertices of $V_H\setminus V_G$.
Suppose we are given a ${\mathcal{C}}$-representation of $G$. It is easy to see that we can add the curves representing the cycle $V_C$ close enough to the grounding line; see Figure \[fig-extend\]. Note that for [<span style="font-variant:small-caps;">Mpt</span>]{}-representations, each original [[]{.nodecor}]{}-shape may have to be intersected by two consecutive [[]{.nodecor}]{}-shapes from the added cycle. In all the other types of representations, each vertex $x_i$ of $G$ will have a unique neighbor $y_{5i}$ among the $V_C$.
![Extending the representation of $G$ into a representation of a cycle extension for grounded [[]{.nodecor}]{}-representations (left) and [<span style="font-variant:small-caps;">Mpt</span>]{} representations (right).[]{data-label="fig-extend"}](extend){width="90.00000%"}
Recall that two linear orders are equivalent if one can be obtained from the other by a sequence of cyclic shifts and reversals. The key property of cycle extensions of $(G,<)$ is that they restrict the possible vertex orders of the $G$-part to an order equivalent to $<$, as shown by the next lemma.
\[lem-cycle2\] If $(G,<)$ is an ordered graph with a cycle extension $H$, then in every grounded 1-string representation of $H$, the order of the vertices of $G$ induced by the representation is equivalent to the order $<$.
The proof follows the same ideas as the proof of the Cycle Lemma of Cardinal et al. [@Cardinal_arx Lemma 6].
Suppose $(G,<)$ is an ordered graph with vertices $V_G=\{x_1<x_2<\dotsb<x_n\}$ and edge-set $E_G$, and $H$ is its cycle extension, with vertices $V_H=V_G\cup V_C$ as in the definition of cycle extension and $V_C=\{y_1,\dotsc,y_{5n}\}$. When working with the indices of the vertices in $V_C$, we will assume that arithmetic operations are performed modulo $5n$, so $5n+1=1$, etc.
Suppose that $H$ has a grounded 1-string representation. We may transform this representation into an outer-1-string representation, while preserving the induced vertex order up to equivalence. Suppose then that an outer-1-string representation of $H$ is given, inside a disk whose boundary is a circle $B$. Let $c_j$ be the string representing $y_j$, and let $p_{j,j+1}$ be the intersection point of $c_j$ and $c_{j+1}$. The subcurve of $c_j$ between the two intersection points $p_{j-1,j}$ and $p_{j,j+1}$ is the *central part* of $c_j$, denoted ${\text{center}}(j)$. The part of $c_j$ between the anchor and the first point of ${\text{center}}(j)$ is the *initial part* of $c_j$, denoted ${\text{start}}(j)$. Let $p_j$ be the common endpoint of ${\text{start}}(j)$ and ${\text{center}}(j)$. Note that $p_j$ is equal to $p_{j-1,j}$ or to $p_{j,j+1}$. The sequence of curves ${\text{center}}(1),{\text{center}}(2),\dotsc,{\text{center}}({5n})$ forms a closed Jordan curve, denoted by $C$. Note that $C$ contains all the points $\{p_{k,k+1};\;k=1,\dotsc,5n\}$. Let $R_C$ be the interior region of $C$.
Consider now a vertex $x_i$, represented by a string $s_i$. Note that $s_i$ can only intersect the curve $C$ in a point of ${\text{center}}({5i})$ or possibly ${\text{center}}({5i-1})$. Let $R_i$ be the planar region bounded by the union of the following four curves: ${\text{start}}(5i-3)$, ${\text{start}}(5i+2)$, the arc of $C$ between $p_{5i-3}$ and $p_{5i+2}$ that contains ${\text{center}}(5i-1)\cup{\text{center}}(5i)$, and the arc of $B$ between the anchors of $c_{5i-3}$ and $c_{5i+2}$ that contains the anchors of $c_{5i-1}$ and $c_{5i}$.
Note that $s_i$ is the only string among the strings representing $V_G$ that can intersect the boundary of $R_i$. Note also that the string $c_{5i}$ cannot intersect the boundary of $R_k$ for $k\neq i$, and therefore $c_{5i}$ is contained in $R_i\cup R_C$. Since $s_i$ intersects $c_{5i}$, and since $s_i$ also cannot cross the boundary of $R_k$ for $k\neq i$, it follows that $s_i$ is also contained in $R_C\cup R_i$, and in particular, the anchor of $s_i$ is in $R_i\cap B$. Therefore, the anchors of $s_1,\dotsc,s_n$ appear on $B$ in the order which, up to equivalence, corresponds to the order $<$ on $V_G$.
We will now use Lemmas \[lem-extend\] and \[lem-cycle2\] to construct graphs that have no representation in a given intersection class. Our goal is to show that there are no inclusions missing in Figure \[fig-classes\]. The classes [<span style="font-variant:small-caps;">Int</span>]{}, [<span style="font-variant:small-caps;">Circle</span>]{}, [<span style="font-variant:small-caps;">Outerplanar</span>]{} and [<span style="font-variant:small-caps;">Per</span>]{} are well studied [@Brandstadt], and simple examples show that there are no inclusions among them other than those depicted in Figure \[fig-classes\].
Catanzaro et al. [@Catanzaro Observation 6.9] observed that the graph $K_{2,2,2}$ (the octahedron) is a permutation graph not in [<span style="font-variant:small-caps;">Mpt</span>]{}, and therefore neither [<span style="font-variant:small-caps;">Per</span>]{} nor any superclass of [<span style="font-variant:small-caps;">Per</span>]{} is contained in [<span style="font-variant:small-caps;">Mpt</span>]{}. Cardinal et al. [@Cardinal] showed that [<span style="font-variant:small-caps;">Grounded-seg</span>]{} is a proper subclass of [<span style="font-variant:small-caps;">Outer-1-string</span>]{}. To complete the hierarchy, we only need the following separations.
![The three intersection representations used to prove Theorem \[thm-separate\]. In each case, a representation cannot be replaced by a representation from a smaller class while preserving the induced vertex order. Left: a [<span style="font-variant:small-caps;">Grounded</span>-$\{{\textnormal{\textsf{L}}},\JJ\}$]{} representation which cannot be replaced by a [<span style="font-variant:small-caps;">Grounded</span>-[[]{.nodecor}]{}]{} one. Middle: a [<span style="font-variant:small-caps;">Grounded-seg</span>]{} representation which cannot be replaced by a [<span style="font-variant:small-caps;">Grounded</span>-$\{{\textnormal{\textsf{L}}},\JJ\}$]{} one. Right: an [<span style="font-variant:small-caps;">Mpt</span>]{} representation which cannot be replaced by an [<span style="font-variant:small-caps;">Outer-1-string</span>]{} one.[]{data-label="fig-sep"}](sep){width="90.00000%"}
\[thm-separate\] The following properties hold.
- The class [<span style="font-variant:small-caps;">Grounded</span>-$\{{\textnormal{\textsf{L}}},\JJ\}$]{} is not a subclass of [<span style="font-variant:small-caps;">Grounded</span>-[[]{.nodecor}]{}]{}.
- The class [<span style="font-variant:small-caps;">Grounded-seg</span>]{} is not a subclass of [<span style="font-variant:small-caps;">Grounded</span>-$\{{\textnormal{\textsf{L}}},\JJ\}$]{}.
- The class [<span style="font-variant:small-caps;">Mpt</span>]{} is not a subclass of [<span style="font-variant:small-caps;">Outer-1-string</span>]{}.
We first prove part (i) of the theorem. Consider the graph $G=(V,E)$ with $V=\{x_1,x_2,x_3,x_4\}$ and $E=\{x_1x_2,\allowbreak x_2x_3,\allowbreak x_1x_4\}$. Figure \[fig-sep\] (left) shows a grounded $\{{\textnormal{\textsf{L}}},\JJ\}$-representation of $G$ which induces the order $<$ defined as $x_1< x_2< x_3< x_4$ on $V$. Note that there is no grounded [[]{.nodecor}]{}-representation of $G$ that would induce the vertex order $<$, because $(G,<)$ contains the pattern ${P_2}$ of Theorem \[thm-patterns\].
Let $(G', <')$ be the ordered graph obtained by putting $(G,<)$ and the mirror image of $(G,<)$ side by side. Formally, $(G',<')$ has vertex set $V'=\{x_1, x_2,x_3,\allowbreak x_4,y_1,y_2,\allowbreak
y_3,y_4\}$, edge set $E'=\{ x_1x_2, x_2x_3, x_1x_4, y_1y_2, y_2y_3,y_1y_4\}$ and vertex order $x_1<'
x_2<' x_3<' x_4<'y_4<'y_3<' y_2<' y_1$. Finally, let $(G'',<'')$ be the ordered graph obtained by placing two disjoint copies of $(G',<')$ side by side. Clearly $G''$ has a grounded $\{{\textnormal{\textsf{L}}},\JJ\}$-representation which induces the vertex order $<''$. However, $G''$ has no grounded [[]{.nodecor}]{}-representation inducing a vertex order equivalent with $<''$, since in any vertex order equivalent with $<''$ there are four consecutive vertices forming a copy of ${P_2}$.
By Lemma \[lem-extend\], the ordered graph $(G'',<'')$ has a cycle extension $H$ that admits a grounded $\{{\textnormal{\textsf{L}}},\JJ\}$-representation. By Lemma \[lem-cycle2\], any grounded 1-string representation (and therefore any grounded [[]{.nodecor}]{}-representation) of $H$ induces on $V''$ an order which is equivalent with $<''$. It follows that $H$ has no grounded [[]{.nodecor}]{}-representation, and therefore [<span style="font-variant:small-caps;">Grounded</span>-$\{{\textnormal{\textsf{L}}},\JJ\}$]{} is not a subclass of [<span style="font-variant:small-caps;">Grounded</span>-[[]{.nodecor}]{}]{}, as claimed.
For the other two parts of the theorem, the argument is analogous, the main difference is in the choice of the initial ordered graph $(G,<)$. To prove part (ii), consider the graph $G$ on six vertices whose [<span style="font-variant:small-caps;">Grounded-seg</span>]{} representation is in the middle of Figure \[fig-sep\], and let $<$ be the vertex order induced by the depicted representation.
Let us argue that $G$ has no grounded $\{{\textnormal{\textsf{L}}},\JJ\}$-representation inducing the vertex order $<$. For contradiction, suppose that such a representation exists, and let $\ell_i$ denote the [[]{.nodecor}]{}-shape or -shape representing $x_i$ in this representation. Let $h_i$ and $v_i$ be the horizontal and vertical segment of $\ell_i$, respectively. Suppose, without loss of generality, that $v_1$ is longer than $v_6$. Since $\ell_1$ and $\ell_6$ intersect, $h_6$ must intersect $v_1$, and $\ell_6$ is a -shape. Since $\ell_2$ intersects both $\ell_1$ and $\ell_6$, $v_2$ must be longer than $v_6$, and $v_2$ intersects $h_6$. But this means that $\ell_3$ must intersect either $\ell_2$ or $\ell_6$ in order to intersect $\ell_1$, a contradiction.
Note that the graph $(G,<)$ is isomorphic to its reversal. Consider the ordered graph $(G',<')$ obtained by placing two copies of $(G,<)$ side by side: note that in any vertex order equivalent to $<'$, $G'$ contains a copy of $(G,<)$, and therefore there is no grounded $\{{\textnormal{\textsf{L}}},\JJ\}$-representation of $G'$ inducing a vertex order equivalent to $<'$. We apply Lemmas \[lem-extend\] and \[lem-cycle2\] to $(G',<')$ and obtain its cycle extension $H$, which is in [<span style="font-variant:small-caps;">Grounded-seg</span>]{} but not in [<span style="font-variant:small-caps;">Grounded</span>-$\{{\textnormal{\textsf{L}}},\JJ\}$]{}.
To prove part (iii), consider the graph $G$ whose [<span style="font-variant:small-caps;">Mpt</span>]{}-representation is depicted in the right part of Figure \[fig-sep\], and let $<$ be the vertex order induced by the representation. We claim that there is no grounded 1-string representation of $G$ inducing the order $<$. For contradiction, suppose that such a representation exists, and let $s_i$ be the string representing the vertex $x_i$. Additionally, let $a_i$ denote the anchor of $s_i$, and for a pair of intersecting strings $s_i$, $s_j$ let $p_{ij}$ denote their intersection.
Suppose, without loss of generality, that when we follow $s_4$ starting in $a_4$, we encounter $p_{24}$ before we encounter $p_{46}$. Let $C$ be the closed Jordan curve obtained as the union of the subcurve of $s_1$ between $a_1$ and $p_{17}$, the subcurve of $s_7$ between $p_{17}$ and $p_{37}$, the subcurve of $s_3$ between $p_{37}$ and $a_3$, and the segment $a_1a_3$ of the grounding line. Note that $s_2$ is inside $C$ (except $a_2$, which lies on $C$), and both $a_4$ and $s_6$ are outside $C$. Therefore, $s_4$ must intersect $C$ at least twice: once between $a_4$ and $p_{24}$, and once between $p_{24}$ and $p_{46}$. However, $s_4$ can only intersect $C$ in the point $p_{34}$, a contradiction.
To complete the proof, we first observe that $G$ has no grounded 1-string representation inducing a vertex order equivalent with $<$, since such a representation could be trivially transformed into a grounded 1-string representation inducing $<$. We apply Lemmas \[lem-extend\] and \[lem-cycle2\] to $G$, to obtain a graph $H$ which is in ${\textsc{Mpt}}$ but not in [<span style="font-variant:small-caps;">Outer-1-string</span>]{}.
Note that these results imply that [<span style="font-variant:small-caps;">Outer-string</span>]{} is a proper superclass of both [<span style="font-variant:small-caps;">Mpt</span>]{} and [<span style="font-variant:small-caps;">Outer-1-string</span>]{}.
We remark that [<span style="font-variant:small-caps;">Mpt</span>]{} is clearly a subclass of [<span style="font-variant:small-caps;">1-string</span>]{} and of [<span style="font-variant:small-caps;">Outer-string</span>]{}, but it is not a subclass of [<span style="font-variant:small-caps;">Outer-1-string</span>]{}, as we just saw.
Concluding remarks
==================
We have seen that the vertex orders induced by grounded [[]{.nodecor}]{}-representations can be characterised by a pair of forbidden patterns. Previously, a characterisation by a single forbidden pattern has been found for vertex orders induced by [<span style="font-variant:small-caps;">Mpt</span>]{} representations [@Ahmed; @Catanzaro; @Soto]. It is an open problem whether such a characterisation can be obtained for other similar grounded intersection classes, such as the class [<span style="font-variant:small-caps;">Grounded</span>-$\{{\textnormal{\textsf{L}}},\JJ\}$]{}.
Another problem concerns the recognition complexity of the graph classes we considered. Recognition of max point-tolerance graphs is mentioned as a prominent open problem by Ahmed et al. [@Ahmed], by Catanzaro et al. [@Catanzaro], as well as by Soto and Thraves Caro [@Soto]. For the classes [<span style="font-variant:small-caps;">Grounded</span>-[[]{.nodecor}]{}]{} and [<span style="font-variant:small-caps;">Grounded</span>-$\{{\textnormal{\textsf{L}}},\JJ\}$]{}, recognition is open as well. On the other hand, the recognition problem for [<span style="font-variant:small-caps;">Grounded-seg</span>]{} is known to be $\exists \mathbb{R}$-complete, as shown by Cardinal et al. [@Cardinal]. In particular, [<span style="font-variant:small-caps;">Grounded-seg</span>]{} cannot be characterised by finitely many forbidden vertex order patterns, unless $\exists\mathbb{R}$ is equal to NP.
The characterisation of [<span style="font-variant:small-caps;">Grounded</span>-[[]{.nodecor}]{}]{} by forbidden vertex order patterns might conceivably be helpful in designing a polynomial recognition algorithm, but note that even a graph class characterised by a forbidden vertex order pattern may have NP-hard recognition [@Duffus], although it is known that recognition is polynomial for all classes described by a set of forbidden patterns of order at most three [@HMR].
[^1]: Supported by the project 16-01602Y of the Czech Science Foundation and by project Neuron Impuls of the Neuron Fund for Support of Science.
[^2]: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665385.
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abstract: 'We calculate numerically the logarithmic contribution to the entanglement entropy of a cylindrical region in three spatial dimensions for both, free scalar and Dirac fields. The coefficient is universal and proportional to the type $c$ conformal anomaly in agreement with recent analytical predictions. We also calculate the mass corrections to the entanglement entropy for scalar and Dirac fields in a disk. These apparently unrelated problems make contact through the dimensional reduction procedure valid for free fields whereby the entanglement entropy for the cylinder can be calculated as an integral over masses of the disk entanglement entropies. Coming from the same numerical evaluation in the lattice, each coefficient is cross checked by the other, testing in this way the two results simultaneously.'
author:
- |
Marina Huerta[^1]\
[*Centro Atómico Bariloche, 8400-S.C. de Bariloche, Río Negro, Argentina*]{}
title: Numerical determination of the entanglement entropy for free fields in the cylinder
---
Introduction
============
The entanglement entropy, being a measure of the correlations between two subsystems, depends on the geometry of the separating surface, becoming for this reason a quantity with a strong geometric character. This has inspired a holographic interpretation within the AdS-CFT correspondence framework [@ryu_taka; @ryu_taka2], which provides a purely geometric way of calculating the entropy of conformal field theories. There are several results in the literature where the geometrical properties for different sets, dimensions and theories are explored both, from the quantum field theory and holographic approaches. Among these, the coefficient $s$ of the logarithmically divergent term in the entanglement entropy $S_{\log}=s\,\log\epsilon$, where $\epsilon$ is the ultraviolet cut-off, for general conformal field theories in four dimensions has been found to be proportional to the type $a$ and $c$ conformal anomaly coefficients [@solo]
$$s=\frac{a}{180} \,\chi(\partial V)+\frac{c}{240 \pi} \int_{\partial V}(k_i^{\mu \nu}k^i_{\nu \mu} -\frac{1}{2} k_i^{\mu \mu}k^i_{\mu \mu})\,.\label{general}$$
Here $\chi(\partial V)$ is the Euler number of the surface, $k^i_{\mu\nu}=-\gamma^\alpha_\mu \gamma^\beta_\nu \partial_\alpha n^i_\beta$ is the second fundamental form, $n^\mu_i$ with $i=1,2$ are a pair of unit vectors orthogonal to $\partial V$, and $\gamma_{\mu\nu}=\delta_{\mu\nu}-n^i_\mu n^i_\nu$ is the induced metric on the surface. From (\[general\]), it can be seen that the sphere and the cylinder are sensitive only to one type of anomaly coefficient, $a$ or $c$, respectively. This is [@solo] $$s=\frac{a}{90} \,,$$ for the sphere, while for a cylinder of length $L$ and radius $R$ it is $$s= \frac{c}{240}\frac{L}{R}\,.
\label{coefs}$$ For spherical sets, this result was later recalculated, numerically and analytically [@num; @dowker; @cashue; @solo2; @dowker1], using different technics, and recently extended to any theory [@cashuemye]. Instead, the validity of the result (\[coefs\]) for the cylinder is more subtle. It has been also studied from the holographic point of view in [@Myers2] where $s$ is found to be proportional to the $c$ conformal anomaly in four dimensions as in (\[coefs\]), but it cannot be generalized to higher dimensions where other conformally invariant terms can be added to (\[general\]).
In this paper, we study numerically the logarithmic contributions to the entropy for a cylinder in $(3+1)$ dimensions for free massless scalar and Dirac fields. The method we use consists first, in dimensionally reducing the problem of the three dimensional cylinder to the one of an infinite set of massive fields living in a two dimensional spatial disk. The entanglement entropy is then calculated numerically by the real time approach [@review] where the reduced density matrix is written in terms of free correlators. This is based, on one hand, on the method developed by Sredniki for calculating the entropy for scalars in spherical sets [@sred] where the field is discretized in the radial direction in polar coordinates, and on the other, on the work by Peschel, whereby the reduced density matrix can be written in terms of correlators for both, bosonic and fermionic, solvable lattice systems [@peschel]. Thus, for the numerical evaluation of the entropy of a Dirac field in the disk, we extend the method described in [@sred], originally applicable only to scalar fields.
Then, from the expansion of the entropy in powers of $(mR)$ valid in the large $mR$ limit, we find the coefficient $s$ for the cylinder (\[coefs\]). This is directly related to the coefficient of the term $(mR)^{-1}$. In the same expansion, we also identify the coefficient of the linear term $mR$. This last, has been recently calculated analytically in [@wilczek], where the same type of expansion is considered for general smooth geometries and massive scalar theories. From the holographic point of view, new contributions to the entanglement entropy were also found when considering theories deformed by a relevant operator [@Myers3], consistently to that reported in [@wilczek].
We find, for the cylinder, within a porcentual error $\sim 0.15 \%$ for scalars and $\sim 1\%$ for Dirac fields, $$\begin{aligned}
s_{s}&=&\frac{L}{240R}\,, \nonumber \\
s_{f}&=&\frac{6L}{240R}\,, \label{cyl}\end{aligned}$$ and for the disk, within a porcentual error $\sim 0.05\%$, $$\Delta S_m=-\frac{2 \pi }{12}R~ m\,, \label{disk}$$ in agreement with [@solo] and [@wilczek]. Coming from the same expansion, each coefficient is cross checked by the other testing both at the same time. This concordance sums support to the general validity of the method used in [@solo]. We will discuss this issue in more detail, later in the Discussion.
The paper is organized as follows: in the second Section, we discuss the dimensional reduction procedure and show the problem of massless fields in the cylinder can be reduced to the one of massive fields in a disk. In the third Section, we discuss the approach we use for the numerical evaluation of the entanglement entropy in the disk, for both, scalar and Dirac massive fields. In Section 4 we present our results and finally, its interpretation in the Discussion.
The cylinder by dimensional reduction
=====================================
For free fields, some universal terms in the entanglement entropy in high dimensions can be obtained via dimensional reduction technics from results calculated in lower dimensions [@review]. Let us consider a set in three spatial dimensions of the form $V=D\times X$, where $X$ is a line on the first coordinate $x_1$, of length $L$, and $D$ is a sphere in two dimensions (disk). We are interested in the entropy of $V$ in the limit of large $L$. The direction $x_1$ can be compactified by imposing periodic boundary conditions $x_1\equiv x_1+L$, without changing the result of the leading extensive term. For a free massless field we Fourier decompose it into the corresponding field modes in the compact direction $$\phi_n(x_1,x_2,x_3,t)=\tilde\phi_n(x_2,x_3,t)e^{i\frac{ 2 \pi x_1 n}{L}}\,.$$
The problem then reduces to a two dimensional one with an infinite tower of massive fields. For example, for a free scalar we obtain the tower of fields $\tilde\phi$, $$\Box_4 \phi_n=\Box_3 \tilde\phi_n+(\frac{2 \pi n}{L})^2 \tilde\phi_n\,.$$ From the point of view of the non compact $x_{2}, x_{3}$ directions, these fields have masses given by $$m_{n}^2=\left(\frac{2\pi}{L}n \right) ^2\,.$$ Summing over the contributions of all the decoupled $2$ dimensional fields we have $$S(V)=\sum_{n=-{\infty}}^\infty S(D,m_{n})\,.$$ In the limit of large $L$ we can convert this sum into integral $$S(V)= \frac{L}{\pi}\int_0^\infty dm\, S(D,m)\,.\label{ala}$$ The universal terms in $S(V)$ will then come from the ones of $S(D)$ after integrating over the mass. For a Dirac field, the spin multiplicity factor $2^{[(d+1)/2]}$ has to be incorporated. In the present case, it is
$$S(V)= 2 \frac{L}{\pi}\int_0^\infty dm\, S(D,m)\,\label{ala1}\,.$$
Expanding the entanglement entropy $S(D,m)$ in powers of $mR$ for large $mR$ $$S(D,m)= c_1 mR +c_0+c_{(-1)}\frac{1}{mR}+...\label{expansion}\,,$$ and inserting (\[expansion\]) in (\[ala\]) and (\[ala1\]) for scalar and Dirac fields respectively, we obtain for the $(3+1)$ dimensional theory that the logarithmic coefficient $s$ in $S(V)$ is directly related to $c_{(-1)}$ by $$\begin{aligned}
s_{s}&=& -c^{s}_{(-1)}\frac{L}{\pi R}\,,\\
s_{f}&=&-2 c^{f}_{(-1)}\frac{L}{\pi R} \,.\label{sm}\end{aligned}$$ On the other hand, for the dimensionally reduced theory, the contribution in the entropy proportional to the mass is given by the term proportional to the coefficient $c_1$. This contribution was calculated in [@wilczek] for massive scalar fields in any dimension for a waveguide geometry with specified boundary conditions using heat kernel methods. The terms extensive in the area $A_{d-1}$ in even spatial dimensions are given by $$S=\frac{A_{d-1}}{12}\int_{\epsilon^2}^{\infty}\frac{dt}{t}\frac{1}{(4\pi t)^{\frac{d-1}{2}}}e^{-tm^2} \,.\label{heatkernel}$$ From (\[heatkernel\]) it follows, $$\Delta S_m=\gamma_d~ m^{d-1}~A_{d-1}\,,\\$$ where $\gamma_d\equiv(-1)^{(d/2)}[12 ~(2\pi)^{(d-2)/2}(d-1)!!]^{-1}$ for $d$ even. For Dirac fermions with $2^{[(d+1)/2]}$ components, the same calculation can be done taking into account the extra factor $2^{[(d+1)/2]-1}$ [@kabat], which relates the scalar and Dirac $\gamma$ coefficients, $$\gamma_d^{f}\equiv 2^{[(d+1)/2]-1}\gamma_d^{s}.$$ In our case, $d=2$ and $$\Delta S_m=c_1 R~ m\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\, c_1=2\pi \gamma_2=-\frac{2 \pi }{12},$$ both for Dirac and scalar fields.
The disk: Numerical evaluation for massive scalar and Dirac fields
==================================================================
We evaluate numerically the entanglement entropy for a two dimensional disk. In this section we describe the numerical method and models in the lattice for scalar and Dirac massive fields.
Massive scalars in a disk
-------------------------
Consider the quadratic Hamiltonian for a massive scalar in $(2+1)$ dimensions $$H=\frac{1}{2}\int dV ((\partial_t \phi)^2+ (\bigtriangledown{\phi})^2 +m^2 \phi^2)\,.$$ Following [@sred], we separate variables in polar coordinates and introduce new ones $$\begin{aligned}
\tilde\phi_n(r)&=&\sqrt{\frac{r}{2\pi}}\int d\theta ~e^{i n\theta} \phi(\theta,r)\,,\\
\tilde\pi_n(r)&=&\sqrt{\frac{r}{2\pi}}\int d\theta~ e^{i n\theta} \pi(\theta,r)\,,\\\end{aligned}$$ such that $$[\tilde\phi_n(r),\tilde\pi_{n^{\prime}}(r^{\prime})]=i \delta_{nn^{\prime}}\delta(r-r^{\prime} )\,.$$ Then, the Hamiltonian $H=\sum_n H_n$ takes the form $$H_n=\frac{1}{2}\int_0^{\infty} dr [\tilde\pi_n^2+r\partial_r(\frac{\tilde\phi_n}{\sqrt{r}})^2+m^2\tilde\phi_n^2+\frac{n^2}{r^2}\tilde\phi_n^2]\,.
\label{hscalar}$$
For a general quadratic discrete Hamiltonian $H=\frac{1}{2}\sum \pi _{i}^{2}+\frac{1}{2}\sum_{ij}\phi _{i}K_{ij}\phi
_{j}$, (dropping the $n$ index temporarily), the vacuum (ground state) two point correlators $X_{ij}$ and $P_{ij}$ $$\begin{aligned}
\left\langle \phi _{i}\phi _{j}\right\rangle &=& X_{ij} \,,\label{atre} \\
\left\langle \pi _{i}\pi _{j}\right\rangle &=&P_{ij} \,, \label{atro}\,\label{need}\end{aligned}$$ are given by [@review] $$\begin{aligned}
X_{ij} &=&\left\langle \phi _{i}\phi _{j}\right\rangle =\frac{1}{2}(K^{-
\frac{1}{2}})_{ij}\,, \label{x} \\
P_{ij} &=&\left\langle \pi _{i}\pi _{j}\right\rangle =\frac{1}{2}(K^{\frac{1
}{2}})_{ij}\,, \label{p}\end{aligned}$$ in terms of the $K$ matrix. Here, $1 \leq i,j\leq N$, where $N$ is the size of the lattice and acts as an infrared regulator.
Then, the entropy of the disk is given by $$S^D\,=S_0+\sum_{n=1}^{\infty} 2 S_n\,,
\label{sscalar}$$ with [@review] $$S_n\,= \textrm{tr}\left(( \sqrt{X_n^D P_n^D}+\frac{1}{2})\log (\sqrt{X_n^D P_n^D}+\frac{1}{2})-(\sqrt{X_n^D P_n^D}-\frac{1}{2})\log (\sqrt{X_n^D P_n^D}-\frac{1}{2})\right)\,. \label{for}$$ The superscript $D$ in (\[for\]) means the indices of the matrices are restricted to the region $D$: $1\leq i,j \leq r$, with $r$ the radius of the disk.
In this case, after discretization we find the matrix $K_n$ corresponding to (\[hscalar\]) is $$\begin{aligned}
K_n^{11}&=&\frac{3}{2}+n^2+m^2\,,\\
K_n^{ii}&=&2+\frac{n^2}{i^2}+m^2\,,\\
K_n^{i,i+1}&=&-\frac{i+1/2}{\sqrt{i(i+1)}}=K_n^{i+1,i}\,.\end{aligned}$$
Summarizing, the numerical evaluation of the entropy for massive scalar fields in a disk starts with the calculation of the $(N\times N)$ matrix $K_n$ for a given mass $m$, angular momentum $n$, and infrared regulator $N$ which gives the size of the unidimensional lattice. From (\[x\]) and (\[p\]) we calculate the two point correlators $X$ and $P$. Then, we reduce them to the disk and calculate the contribution $S_n$ (\[for\]). Finally, the entropy $S^D$ is given by the sum (\[sscalar\]).
Massive fermions in a disk.
---------------------------
The Hamiltonian for a massive Dirac field in $(2+1)$ dimensions can be written as $$H=\int dV \psi^{\dagger}(x,y) H^{(1)} \psi(x,y)\,,$$ with $H^{(1)}$ the one particle Hamiltonian $$H^{(1)}\equiv i \frac{\partial}{\partial_t}=\frac{1}{i}(\alpha_x\partial_x+\alpha_y\partial_y)+\beta m\,.$$ Choosing $$\begin{aligned}
\alpha_x&=&\sigma_1\,,\\
\alpha_y&=&\sigma_2\,,\\
\beta&=&\sigma_3\,,\end{aligned}$$ with $\sigma_i$ the Pauli matrices, the relations $\{\alpha_i,\alpha_j\}=\{\alpha_i,\beta\}=0$ and $\alpha_i^2=\beta^2=1$ are satisfied. In polar coordinates $r,\theta$ the Hamiltonian takes the form
$$H^{(1)}=\frac{1}{i} \left(\frac{1}{r}h_{\theta}\partial_{\theta}+h_{r}\partial_r \right)+m h_m\,,$$
where $h_{\theta}$, $h_r$ and $h_m$ are $$\begin{aligned}
h_{\theta}&=& \left(
\begin{array}{cc}
0 & ie^{i\theta} \\
-ie^{-i\theta} & 0 \end{array} \right)\,, \\
h_{r}&=& \left(
\begin{array}{cc}
0 & e^{i\theta} \\
e^{-i\theta} & 0 \end{array} \right) \,,\\
h_m&=&\left(
\begin{array}{cc}
1 & 0 \\
0 & -1 \end{array} \right) \,.\end{aligned}$$ Separating variables, we propose the following two component Dirac spinor $$\psi_n=\left(\begin{array}{c}
u_n(r)~ \phi^1_n(\theta) \\
v_n(r) ~\phi^2_n(\theta) \end{array} \right) \,,$$ where $\phi^1_n=\frac{1}{\sqrt {2\pi}}e^{i\theta(n+\frac{1}{2})}$ and $\phi^2_n=\frac{1}{\sqrt {2\pi}}e^{i\theta(n-\frac{1}{2})}$ are the eigenvector components of the angular momentum operator $J=\frac{1}{i}\partial_{\theta}-\frac{1}{2}\sigma_3 $ with half integer eigenvalue $n$ $$J\left(\begin{array}{c}
\phi_n^1 \\
\phi_n^2 \end{array}\right)=n\left(\begin{array}{c}
\phi_n^1 \\
\phi_n^2 \end{array}\right)\,.$$ Note that $J$ commutes with $H$. Then, we can express the Hamiltonian as a sum over $n$ such that $$H= \sum_n H_n=\sum_n \int dr~ r~\alpha^{\dagger}_n(r)H^1_n\alpha_n(r)$$ where $$H^{(1)}_n=-\frac{1}{r}(i\sigma_1/2+n\sigma_2)-i\sigma_1\partial_r+m\sigma_3\,,
\label{hfer}$$ and $$\alpha_n(r)=\left(\begin{array}{c}
u_n(r) \\
v_n(r) \end{array} \right)\,.
\label{hfer}$$
In general, in the discrete free case, the reduced density matrix can be written in terms of a Hermitian operator ${\cal H}$ $$\rho=K e^{-{\cal H}}=Ke^{\sum_V \psi_i^{\dagger}{\hat{\cal H}}_{ij}\psi_j}\,.$$ This can be expressed in terms of the correlators $C_{ij}$ $$\langle\psi_i\psi_j^{\dagger}\rangle=C_{ij}\,,
\label{corf}$$ via the identification [@review] $$\hat {\cal H}=-\log(C^{-1}-1)\,.$$ Here, the $\psi_i$ are fermion operators canonically normalized. Then, the entropy can be written in terms of $C$ as $S= -\text{tr}((1-C)\log(1-C)+C \log C)$.
For a general quadratic case, with discrete Hamiltonian $H=\sum_{ij}M_{ij}\psi_i^{\dagger}\psi_j$, the correlator is directly related to $M_{ij}$ by $$C=\Theta(-M) \,.
\label{cfe}$$ In (\[cfe\]), the indices run over the complete space. Then, we introduce an infrared regulator $N$ which is the size of the lattice.
Therefore, the entropy $S$ for the disk $D$, is given by a sum over the angular momentum $n$ as $$S(D)=\sum_n -\text{tr}((1-C^D_n)\log(1-C^D_n)+C^D_n \log C^D_n)\,.
\label{entd}$$ The correlators $C^D_n$ for fixed angular momentum $n$ are restricted to the disk region $D$.
In order to identify the $M$ matrix in the case of the disk, we first introduce the operators $\tilde{\alpha}(r)=r^{(1/2)}\alpha(r)$. These are normalized such that they satisfy the canonical anticonmutation relations required for the application of (\[entd\]). The discrete Hamiltonian in these variables for a $N$ lattice, takes the form (dropping the angular momentum index temporarily) $$H=\sum_{i,j=1}^NH_{i,j}=\sum_{i,j=1}^N(\tilde u^{*}_i,\tilde v^{*}_i)M_{i,j}\left(\begin{array}{c} \tilde u_j \\
\tilde v_j
\end{array} \right)\,,$$ where the indices $i,j$ are the discrete variables corresponding to the continuum radial coordinate $r$. $M_{i,j}$, for fixed $i,j$, is a $(2 \times 2)$ matrix such that $H_{i,j}=
M^{1,1}_{i,j}\tilde u_i^{*}\tilde u_j+M^{1,2}_{i,j}\tilde u_i^{*}\tilde v_j+M^{2,1}_{i,j}\tilde v_i^{*}\tilde u_j+M^{2,2}_{i,j}\tilde v_i^{*}\tilde v_j$ with $i,j=1,...,N$.
Finally, we define a $2N\times2N$ matrix $\tilde{M}_n^{2k+\alpha-2,2l+\beta-2}=M_{k,l}^{\alpha,\beta}$, for $k,l=1,...,N$ and $\alpha,\beta=1,2$. The non zero entries of $\tilde{M}$ (for each angular momentum $n$) are $$\begin{aligned}
\tilde{M}_n^{kk}&=&(-1)^{k+1} m,\\
\tilde{M}_n^{1,2}=i(n+\frac{1}{2}) &,& \tilde{M}_n^{2,1}=-i(n+\frac{1}{2}),\\
\tilde{M}_n^{2k-1,2k}=i~\frac{n}{k} &,& \tilde{M}_n^{2k,2k-1}=-i~\frac{n}{k},\\
\tilde{M}_n^{2k-1,2k+2}=\frac{-i}{2}&,& \tilde{M}_n^{2k,2k-3}=\frac{i}{2},\\
\tilde{M}_n^{2k-1,2k-2}=\frac{i}{2}&,& \tilde{M}_n^{2k,2k+1}=\frac{-i}{2}.\end{aligned}$$ The $\tilde{M_n}$ matrix satisfies $\tilde{M_n}^{\dagger}=\tilde{M_n}$ and it has real eigenvalues symmetric with respect to the origin. There is also a symmetry related to the mapping $n \rightarrow -n$ which can be seen also in the continuum limit.
\[tp\]
Summarizing, the numerical evaluation of the entropy for massive Dirac fields in a disk starts with the calculation of the $(2N\times 2N)$ matrix $\tilde M_n$ for a given mass $m$, angular momentum $n$, and $N$, the size of the unidimensional lattice. From (\[cfe\]) we calculate the two point correlator $C$ (\[corf\]). Then, we reduce it to the disk $C^D_n$ and calculate the contribution $S_n$ in (\[entd\]). Finally, the entropy $S^D$ is given by the sum (\[entd\]).
Results
=======
We calculate the entanglement entropy for a disk in a lattice of $200$ points. We consider regions of radii $r$ going from $50$ to $80$ in lattice units for the scalar and from $30$ to $50$ for the fermionic field. The sum over the angular momentum $n$ is done exactly up to $n_{max}=3000$ and the corrections coming from contributions $n>n_{max}$ are added by fitting the exact entropy for some large values of $n$. The fit we use is of the form $$s_n=a_2 \frac{1}{n^2}+b_2 \frac{\log n}{n^2}+a_4 \frac{1}{n^4}+b_4 \frac{\log n}{n^4}+a_6 \frac{1}{n^6}+b_6 \frac{\log n}{n^6}$$ since for large $n$ ($n \gg N$) both $C$ and $\sqrt{XP}$ defined in (\[cfe\]), (\[x\]) and (\[p\]) have expansions in even inverse powers of $n$ [@sred].
This is done for different masses in the range $1/20 < m < 1/2$ such that $m^{-1}>\epsilon$ (we fix $\epsilon=1$) and $m r > 1$. For each mass, we fit the entropy in terms of the size $r$ of the disk, $$S=c_1(m)r+c_0(m)+c_{-1}(m)\frac{1}{r}+...\,.$$
Once the coefficients $c_1(m)$ and $c_{-1}(m)$ are identified, we expand in powers of $m$, $$\begin{aligned}
c_1(m)&=&c_1 m+c^0_1+c^{-1}_1\frac{1}{m}\,,\label{c1m}\\
c_{-1}(m)&=&c^1_{-1} m+c^0_{-1}+c_{-1}\frac{1}{m}\,.\label{c-1m}\end{aligned}$$
We obtain $c_1$ and $c_{-1}$ of (\[expansion\]) as the $m$ and $\frac{1}{m}$ coefficients in (\[c1m\]) and (\[c-1m\]) respectively. For example, in the scalar case, for $m=1/10$ we calculate the entanglement entropy for disks with radii from $60$ to $80$ (in lattice units). From the best fit, we extract the coefficients of the terms proportional to $r$ and $1/r$. In this example, this gives $ 0.20744 - 0.13303/r + 0.41493 r$. Then, $c_1(m=1/10)=0.41493$ and $c_{-1}(m=1/10)=-0.13303$. The values of $c_1(m)$ and $c_{-1}(m)$ are shown in Figures (1),(2) for scalars and Figures (3),(4) for the Dirac field. In order to compare the lattice results with the continuum expectations for fermions, a factor $1/2$ has to be incorporated in the entropy formula due to the fermion doubling in the unidimensional radial lattice. We obtain $$\begin{aligned}
c^{s}_{(-1)}&=&-0.01342\sim-\frac{\pi}{240} \,\,,\,\text{within a porcentual error of $0.15$}\,,\label{ana1}\\
c^{f}_{(-1)}&=&-0.07754\sim -\frac{6 \pi}{240}\,\,,\,\text{within a porcentual error of $1.2$} \,,\label{ana2}\end{aligned}$$ and $$\begin{aligned}
c^{s}_1&=&-0.52359\sim -\frac{2 \pi }{12}\,\,\,,\,\text{within a porcentual error of $0.02$}\,,\\ \label{ana3}
c^{f}_1&=&-0.52329\sim-\frac{2 \pi }{12}\,\,\,,\,\text{within a porcentual error of $0.05$}\,,\label{ana3}\end{aligned}$$ in agreement to the analytical results.
Discussion
==========
Our numerical results agree with the ones predicted analytically within porcentual errors $\sim 1.2\%$ or less. Since both results, the one for the cylinder and the one for the disk, come from the identification of different coefficients in the same expansion (\[expansion\]), they cross check each other giving a solid basement to our numerical test.
On the other hand, by the analytical side within the QFT approach, the general grounds the result for the cylinder is based on, rely first on the application of the replication method, then on demanding conformal invariance of the logarithmic contribution in the entropy derived from the effective action of the replicated manifold, and finally on calibrating a free parameter by using the holographic correspondence [@ryu_taka]. The replica trick is a standard approach to calculate the entanglement entropy [@hlw]. In this construction, the trace of the integers powers of the reduced density matrix corresponds to the partition function of $n$ copies of the system connected consistently through the cuts along the boundary. The replicated manifold is non trivial due to the conical singularities of deficit angle $\alpha=2 \pi (n-1)$ placed on the boundary. From this, the entropy associated to the set $V$ can be calculated as $$S=-\lim_{n \rightarrow 1} \frac{\partial}{\partial n}\textrm{tr}[\rho ^n _V]=-\lim_{n \rightarrow 1}(\frac{\partial}{\partial n}-1)\log Z_n\,.
\label{ee}$$ In the above formula we assume that not only the partition function $Z_n$ is calculable, but it is also possible to analytically continue it to non integer $n$. The method presented in [@solo] differs subtly from the above: instead of continuing analytically in $n$ the partition function $Z_n$, they first analytically continue the cover manifold assuming the corresponding geometry can be defined for an arbitrary and small angular deficit in the limit $n \rightarrow 1$ and then calculate the corresponding $Z_n$ [@solo; @solo1]. Taking into account this is not possible in general, then, the identification of the logarithmic coefficient with linear combinations of the stress tensor anomalies is not fully justified [@Myers2; @Myers4; @Schwimmer; @Soloreview]. In the case of spherical sets, the objections to the applicability of the replication method disappear since there is an extra rotational symmetry in the transverse space about the entangling surface. In fact, the result for the sphere in four dimensions can be extended to any dimensions and any field theory [@cashuemye]. These results complete a solid proof for the term proportional to the type $a$ anomaly in (\[general\]).
On the other hand, for the cylinder, the situation is more obscure. Very recently, it was found in [@Myers2] for holographic models, that in four dimensions the logarithmic coefficient is proportional to the anomaly as expected, but that the same is not true for higher dimensions, where new conformally invariant terms could be added to (\[general\]). In this scenario, we conclude our results give support to the validity of the Solodukhin’s calculation for the cylinder in four dimensions.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank H.Casini for useful discussions and comments on the manuscript. I also acknowledge support from the EPLANET programme grant and Dr. G. Mussardo for his hospitality at SISSA where the initial stages of this work were done. This work was partially supported by CONICET, ANPCyT and Universidad Nacional de Cuyo, Argentina.
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[^1]: e-mail: marina.huerta@cab.cnea.gov.ar
|
---
abstract: |
Let ${\cal F}$ be a set of blocks of a $t$-set $X$. $(X,{\cal F})$ is called $(w,r)$-cover-free family ($(w,r)-$CFF) provided that, the intersection of any $w$ blocks in ${\cal F}$ is not contained in the union of any other $r$ blocks in ${\cal F}$.
We give new asymptotic lower bounds for the number of minimum points $t$ in a $(w,r)$-CFF when $w\le r=|{\cal F}|^\epsilon$ for some constant $\epsilon\ge 1/2$.
author:
- |
Ali Z. Abdi\
Convent of Nazareth High School\
Grade 12, Abas 7, Haifa
- |
Nader H. Bshouty\
Dept. of Computer Science\
Technion, Haifa, 32000
title: 'Lower Bounds for Cover-Free Families'
---
[**Keywords:**]{} Cover-Free Family, Lower Bound.
Introduction
============
Let ${\cal F}$ be a set of blocks (subsets) of a $t$-set $X$. $(X,{\cal F})$ is called $(w,r)$-cover-free family ($(w,r)-$CFF) provided that, for any $w$ blocks $A_1,A_2,\ldots,A_w\in {\cal F}$ and any other $r$ blocks $B_1,B_2,\ldots,B_r\in {\cal F}$ we have $$\bigcap_{i=1}^w A_i\not\subseteq \bigcup_{j=1}^r B_j.$$ Since using De Morgan a $(w,r)-$CFF can be turned into $(r,w)-$CFF, throughout the paper we assume that $w\le r$. Cover-free families were first introduced in 1964 by Kautz and Singleton [@KS64].
Let $N(n,(w,r))$ denote the minimum number of points $|X|$ in any $(w,r)$-CFF having $|{\cal F}|=n$ blocks. The best known lower bound for $N(n,(1,r))$ is [@DR82; @F96; @R94] $$\begin{aligned}
\label{bound1}
N(n,(1,r))=\Omega\left(\frac{r^2}{\log r}\log n\right)\end{aligned}$$ when $r\le \sqrt{n}$ and $\Omega(n)$ when $r>\sqrt{n}$. The constant of the $\Omega()$ is asymptotically $1/2$, $1/4$ and $1/8$, respectively. Stinson et. al, [@SWZ00], proved that $$\begin{aligned}
\label{base1}
N(n,(w,r))\ge N(n-1,(w-1,r))+N(n-1,(w,r-1)).\end{aligned}$$ They then use it with (\[bound1\]) to prove two bounds. The first bound is $$\begin{aligned}
\label{fb}
N(n,(w,r))\ge \Omega\left( \frac{{w+r\choose w}(w+r)}{\log {w+r\choose w}}\log n\right)
\end{aligned}$$ when $r\le \sqrt{n}$, [@SWZ00; @MW04], and $$\begin{aligned}
\label{bst}
N(n,(w,r))\ge \Omega\left( \frac{{w+r\choose w}}{\log {(w+r)}}\log n\right)\end{aligned}$$ for any $r\le n$, [@SWZ00]. To the best of our knowledge (\[bst\]) is the best bound known when $\sqrt{n}\le r\le n$. D’yachkov et. al. breakthrough result, [@DVPS14], implies that for $r\le \sqrt{n}$ and $r,n\to\infty$ $$\begin{aligned}
\label{fb2}
N(n,(w,r))= \Theta\left( \frac{{w+r\choose w}(w+r)}{\log {w+r\choose w}}\log n\right)\end{aligned}$$ and for $r\ge \sqrt{n}$ and $r,n\to\infty$ $$\begin{aligned}
\label{bst2}
N(n,(w,r))\le O\left( \frac{r}{w}\cdot\frac{{w+r\choose w}}{\log {(w+r)}}\log n\right).\end{aligned}$$
In this paper we give a new lower bound for $(w,r)$-CFF when $r>\sqrt{n}$. We combine the two techniques used in [@SWZ00; @MW04] and [@AA05] to give the following asymptotic lower bound.
\[T12\] For any $2\le k\le w<r\le n/2$ and $$(n+k-1-w)^{\frac {k-1}{k}} \leq r \leq (n+k-w)^{\frac {k}{k+1}}$$ $$N(n,(w,r)) \ge \frac {k^kk!}{2(k+1)^{2k}} \frac {r^{w+1}}{(w+1)!\ln^k r} = \Omega \left(\frac{\sqrt{k}}{e^k}\cdot\frac {r^{w+1}}{(w+1)!\ln^{k+1} r}\log n\right)$$ and for $$r=\Omega\left(({n\log n})^{\frac{w}{w+1}}\right)$$ $$N(n,(w,r)) = \Theta\left( {n\choose w}\right).$$
Our bound is $$\Theta\left(\frac{\sqrt{k}\cdot r}{w(e\ln r)^k}\right)$$ times greater than the previous bound in (\[bst\]). In particular, when $k$ is constant, our lower bound improves the bound in (\[bst\]) to $$\begin{aligned}
\label{bst3}
N(n,(w,r))\ge \Omega\left( \frac{r}{w\log^k r}\cdot \frac{{w+r\choose w}}{\log {(w+r)}}\log n\right).\end{aligned}$$
A slightly better bound can be achieved when $(n+k-w)^{\frac{k}{k+1}}\le r\le (n+k-w)^{\frac{k}{k+1}}\ln^{1/(k+1)}n$.
For example, let $w=4$. The table in Figure \[exa\] compares our results with the previous results (asymptotic values)
---------------------------- ------------------------------ ---------------------------- --------------------------------
Previous Lower Upper Our Lower
$r$ Bounds (\[fb\]), (\[bst\]) Bound [@DVPS14] Bound
$r\le n^{1/2}$ ${r^5}\frac{\log n}{\log r}$ $r^5\frac{\log n}{\log r}$ —–
$n^{1/2}\le r\le n^{2/3}$ ${r^4}\frac{\log n}{\log r}$ $r^5\frac{\log n}{\log r}$ ${r^5}\frac{\log n}{\log^3 r}$
$n^{2/3}\le r\le n^{3/4}$ ${r^4}\frac{\log n}{\log r}$ $r^5\frac{\log n}{\log r}$ ${r^5}\frac{\log n}{\log^4 r}$
$n^{3/4}\le r\le n^{4/5}$ ${r^4}\frac{\log n}{\log r}$ $r^5\frac{\log n}{\log r}$ ${r^5}\frac{\log n}{\log^5 r}$
$ n> r\ge (n\log n)^{4/5}$ $r^4$ $n^4$ $n^4$
---------------------------- ------------------------------ ---------------------------- --------------------------------
First Lower Bound
=================
In this section we prove
\[L11\] Let $w\le r\le n/2$. If $$r=\Omega\left(\left(n\log n\right)^{\frac{w}{w+1}}\right)$$ then $$\begin{aligned}
N(n,(w,r))=\Theta\left( {n\choose w}\right).\end{aligned}$$ Otherwise, $$\begin{aligned}
N(n,(w,r))\ge \Omega\left(\left(\frac{r}{(w+1)\ln r}\right)^{w+1}\log n\right).\end{aligned}$$
Lemma \[L11\] follows from the following
\[T1\] Let $\epsilon<1$ be any constant. For $w\le r\le n/2$ we have $$\begin{aligned}
N(n,(w,r))\ge \min\left((1-\epsilon)\frac{w^w}{(w+1)^{2w+1}}\cdot \frac{r^{w+1}}{\ln^wr}\ \ \ ,
\ \ \ \epsilon{n\choose w}\right)\end{aligned}$$
Let $(X,{\cal F})$ be an optimal $(w,r)$-CFF. Let ${\cal F}=\{F_1,\ldots,F_n\}$, $|X|=N=N(n,(w,r))$ and assume without loss of generality that $X=[N]:=\{1,\ldots,N\}$. Define $v^{(i)}\in\{0,1\}^n$, $i=1,\ldots,N$ where $v^{(i)}_j=1$ if and only if $i\in F_j$. Let $V=\{v^{(i)}|i=1,\ldots,N\}$. Let $V_0$ be the set of $v^{(i)}$ of weight $wt(v^{(i)})$ (i.e., $\sum_jv^{(i)}_j$) equal to $w$. Let $$m=\frac{(w+1)^2n\ln r}{wr}$$ and consider the two sets $V_1=\{v^{(i)}\ |\ w<wt(v^{(i)})< m\}$ and $V_2=\{v^{(i)}\ |\ wt(v^{(i)})\ge m\}$. Obviously, $V=V_0\cup V_1\cup V_2$ is a partition of $V$. Suppose $$|V_0|\le \epsilon{n\choose w}$$ and $$\max(|V_1|,|V_2|)\le (1-\epsilon)\frac{w^w}{(w+1)^{2w+1}}\cdot \frac{r^{w+1}}{\ln^wr}.$$
Consider $W=\{(j_1,\ldots,j_w)\ |\ 1\le j_1<\cdots<j_w\le n\}$ and $W'\subset W$ the set of all $(j_1,\ldots,j_w)$ where no $v^{(i)}\in V_0$, $i=1,\ldots,N$, satisfies $v_{j_1}^{(i)}=\cdots=v^{(i)}_{j_w}=1$. Obviously, $$|W'|={n\choose w}-|V_0|\ge (1-\epsilon){n\choose w}.$$ Fix an element $v\in V_1$ and randomly and uniformly choose $j=(j_1,\ldots,j_w)\in W'$. We have $$\begin{aligned}
{{\bf Pr}}_{j\in W'}[v_{j_1}=\cdots=v_{j_w}=1] &\le& \frac{{wt(v)\choose w}}{|W'|}
\le\frac{{m\choose w}}{(1-\epsilon){n\choose w}}.\end{aligned}$$ Therefore, the expectation of the number of $v\in V_1$ for which $v_{j_1}=\cdots=v_{j_w}=1$ is at most $$\begin{aligned}
\frac{{m\choose w}|V_1|}{(1-\epsilon){n\choose w}}&\le & \frac{1}{1-\epsilon}\left(\frac{m}{n}\right)^w|V_1|\\
&\le& \frac{1}{1-\epsilon}\frac{(w+1)^{2w}\ln^wr}{w^wr^w}\cdot (1-\epsilon)\frac{w^w}{(w+1)^{2w+1}}\cdot \frac{r^{w+1}}{\ln^wr}\\
&=& \frac{r}{w+1}.\end{aligned}$$ Therefore, there is $j'=(j_1',\ldots,j_w')\in W'$ such that the number of $v\in V_1$ that satisfies $v_{j_1'}=\cdots=v_{j_w'}=1$ is $r_1\le r/(w+1)$. Since the weight of every $v\in V_1$ is greater than $w$, we can choose $r_1$ new entries $j_1'',\ldots,j_{r_1}''\not\in \{j_1',\ldots,j_w'\}$ such that for every $v\in V_1$ where $v_{j_1'}=\cdots=v_{j_w'}=1$ there is $j_\ell''$ such that $v_{j_\ell''}=1$.
Now randomly and uniformly choose $$r_2:=\left\lceil \frac{wr}{w+1}\right\rceil$$ distinct $k_1,\ldots,k_{r_2}\in [n]$. Let $A$ be the event that $\{k_1,\ldots,k_{r_2}\}\cap \{j_1',\ldots,j_w'\}\not=\O$. The probability that $A$ does not happen is $$\begin{aligned}
\frac{{n-w\choose r_2}}{{n\choose r_2}}\ge \frac{{n-w \choose r_2}}{2^w {n-w\choose r_2}}=\frac{1}{2^w}\end{aligned}$$
Then $$\begin{aligned}
{{\bf Pr}}[A\vee(\exists v\in V_2)\ v_{k_1}=\cdots=v_{k_{r_2}}=0]&\le& 1-\frac{1}{2^w}+|V_2|\frac{{n-m\choose r_2}}{{n\choose r_2}}\\
&\le& 1-\frac{1}{2^w}+|V_2| \left(\frac{n-m}{n}\right)^{r_2}\\
&\le& 1-\frac{1}{2^w}+|V_2|e^{-\frac{mr_2}{n}}\end{aligned}$$ and $$\begin{aligned}
|V_2|e^{-\frac{mr_2}{n}}&\le&(1-\epsilon)\frac{w^w}{(w+1)^{2w+1}}\cdot \frac{r^{w+1}}{\ln^wr}\cdot e^{-\frac{(w+1)^{2}\ln r}{wr}r_2 }\\
&\le & (1-\epsilon)\frac{w^w}{(w+1)^{2w+1}}\cdot \frac{r^{w+1}}{\ln^wr}\cdot e^{-(w+1)\ln r}\\
&= & (1-\epsilon)\frac{w^w}{(w+1)^{2w+1}}\cdot \frac{1}{\ln^wr}\\
&< & \frac{1}{2^w}\end{aligned}$$ Therefore, $$\begin{aligned}
{{\bf Pr}}[A\vee(\exists v\in V_2)\ v_{k_1}=\cdots=v_{k_{r_2}}=0]<1.\end{aligned}$$
Therefore, there is $\{k_1,\ldots,k_{r_2}\}$ such that $\{k_1,\ldots,k_{r_2}\}\cap \{j_1',\ldots,j_w'\}=\O$ and for every $v\in V_2$ there is $k_\ell \in \{k_1,\ldots,k_{r_2}\}$ where $v_{k_\ell}=1$.
Now it is easy to see that there is no $v\in V$ where $v_{j_1'}=\cdots=v_{j_w'}=1$, $v_{j_1''}=\cdots=v_{j_{r_1}''}=0$ and $v_{k_1}=\cdots=v_{k_{r_2}}=0$. This implies that $$\bigcap_{i=1}^w F_{j_i'}\subseteq \bigcup_{i=1}^{r_1} F_{j_i''}\cup\bigcup_{i=1}^{r_2} F_{k_i}$$ which is a contradiction.
The Second Bound {#s2}
================
In this section we prove Theorem \[T12\].
For any $2\le k\le w\le r\le n/2$ and $$2 \leq r \leq (n+k-w)^{\frac {k}{k+1}}$$ $$N(n,(w,r)) \ge \frac {k^kk!}{2(k+1)^{2k}} \frac {r^{w+1}}{(w+1)!\ln^k r} = \Omega \left(\frac {r^{w+1}}{(w+1)!\ln^k r}\right).$$
We prove the lemma by induction on $w$.
From Lemma \[T1\] the lemma holds for $w=k$. Now assume the bound holds for some $w$ and every $r$ that satisfies $r \leq (n+k-w)^\frac{k}{k+1}$. We now prove the bound for $w+1$ and $r\le (n+k-w-1)^\frac{k}{k+1}$ $$\begin{aligned}
N(n,(w+1,r)) &\ge & N(n-1,(w,r))) + N(n-1,(w+1,r-1))\label{I5} \\
&\ge & \sum\limits_{j=1}^{r} N(n-r+j-1,(w,j))\label{I6} \\
&\ge & N(n-r,(w,1)) +\nonumber \\ & &\ \ \ \ \ \ \sum\limits_{j=2}^{r} \frac {k^kk!}{2(k+1)^{2k}} \frac {j^{w+1}}{(w+1)!\ln^k j} \label{I7}\\
&\ge & \frac {k^kk!}{2(k+1)^{2k}(w+1)!\ln^k r} \sum\limits_{j=1}^{r} j^{w+1} \nonumber \\
&\ge & \frac {k^kk!}{2(k+1)^{2k}(w+1)!\ln^k r} \int_0^r \! x^{w+1} \, \mathrm{d}x \nonumber \\
&\ge& \frac {2k^kk!}{(k+1)^{2k}} \frac {r^{w+2}}{(w+2)!\ln^k r} \nonumber\end{aligned}$$
Here, inequality (\[I5\]) comes from [@SWZ00]. Inequality (\[I6\]) follows from the fact that $N(n-r+1,(w+1,1))\ge N(n-r,(w,1))$. Inequality (\[I7\]) follows from the induction hypothesis since $$\begin{aligned}
j&=&r-(r-j)\\
&\le& (n+k-w-1)^\frac{k}{k+1}-(r-j)\\
&\le& (n+k-w-1-(r-j))^\frac{k}{k+1}\\
&=& ((n-r+j-1)+k-w)^\frac{k}{k+1}.\end{aligned}$$
N. Alon, V. Asodi. Learning a Hidden Subgraph. [*SIAM J. Discrete Math.*]{} 18(4). pp. 697–712 (2005).
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A. G. D’yachkov, I. V. Vorob’ev, N. A. Polyansky, V. Yu. Shchukin. Bounds on the rate of disjunctive codes. Problems of Information Transmission. 50(1), pp. 27–56. (2014).
Z. Füredi. On $r$-Cover-free Families. [*J. Comb. Theory, Ser. A*]{}, 73(1). pp. 172–173. (1996).
W. H. Kautz and R. C. Singleton. Nonrandom binary superimposed codes, [*IEEE Trans. Inform. Theory*]{}. 10, pp. 363–377. (1964).
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D. R. Stinson, R. Wei and L. Zhu. Some New Bounds for Cover-Free Families. [*Journal of Combinatorial Theory, Series A*]{}. 90, pp. 224–234 (2000).
|
---
author:
- |
Subhadeep Mondal,$^a$ Sanjoy Biswas,$^{b}$ Pradipta Ghosh$^c$ and Sourov Roy$^a$\
$^a$Department of Theoretical Physics,\
Indian Association for the Cultivation of Science,\
2A $\&$ 2B Raja S.C. Mullick Road, Kolkata 700032, India\
$^b$INFN, Sezione di Roma, Dipartimento di Fisica, Università di Roma La Sapienza\
Piazzale Aldo Moro 2; I-00185 Rome, Italy\
$^c$Departamento de Física Teórica UAM and Instituto de Física Teórica UAM/CSIC,\
Universidad Autónoma de Madrid (UAM), Cantoblanco, 28049 Madrid, Spain\
E-mails:
bibliography:
- 'invseesaw\_LHC\_v11.bib'
title: Exploring novel correlations in trilepton channels at the LHC for the minimal supersymmetric inverse seesaw model
---
Model
=====
In the MSISM the particle content of the MSSM is extended by a pair of SM singlet fields, $\widehat\nu^c$ and $\widehat S$ having lepton numbers $-1$ and $+1$, respectively. The model superpotential following refs. [@Arina:2008bb; @Hirsch:2009ra] is written as W= W\_[MSSM]{}+\_[ab]{}h\^i\_\^a\_i \^c\_u\^b+M\_R\^c+ \_S, \[Inverse-seesaw-superpot\] where $W_{MSSM}$ is the MSSM superpotential. In eq. (\[Inverse-seesaw-superpot\]) $\hat {L}_i$s are the $SU(2)_L$ doublet lepton superfields and $\hat H_u$ represents a up-type Higgs superfield. $\hat \nu^c$ represents a right handed neutrino superfield whereas $\hat S$ is another SM gauge-singlet superfield, but with non-zero lepton number. In eq. (\[Inverse-seesaw-superpot\]) coefficient of the lepton number violating term is given by $\mu_S$. In the limit $\mu_S$ $\to\ 0$, MSISM superpotential (see eq. (\[Inverse-seesaw-superpot\])) restores lepton number conservation, which is consistent with the t’Hooft naturalness criteria [@'tHooft:1979bh].
The corresponding soft supersymmetry breaking Lagrangian is given by -[L]{}\_[soft]{} &=&-[L]{}\^[MSSM]{}\_[soft]{} + m\^2\_[\^c]{} \^[c]{} \^c +m\^2\_S S\^ S\
&+& (\_[ab]{} A\^i\_[h\_]{} L\^a\_i \^c H\_u\^b + B\_[M\_R]{} \^c S +B\_[\_S]{}S S +[h.c.]{}), \[Inverse-seesaw-soft\] with $-{\mathcal L}^{\rm MSSM}_{\rm soft}$ representing soft terms of the MSSM. Just like the coefficient $\mu_S$ appearing in the superpotential (eq. (\[Inverse-seesaw-superpot\])), the $B_{\mu_S}$ parameter in the soft terms (eq. (\[Inverse-seesaw-soft\])) violates lepton number by two units. The MSISM thus includes two lepton number violating parameters. Both of these will contribute to the Majorana neutrino mass matrix [@Hirsch:2009ra].
Tree level neutrino mass matrix in the MSISM is a $5\times 5$ matrix in the basis $(\nu_l,\nu^c,S)$, where, $l\equiv e, \mu, \tau$. This is given by (
[ccccc]{} 0 & 0 & 0 & m\_[D\_1]{} & 0\
0 & 0 & 0 & m\_[D\_2]{} & 0\
0 & 0 & 0 & m\_[D\_3]{} & 0\
m\_[D\_1]{} & m\_[D\_2]{} & m\_[D\_3]{} & 0 & M\_R\
0 & 0 & 0 & M\_R & \_s
), \[5by5numat\] where $m_{D_i} \equiv h^i_\nu v_u$ ($i = 1,2,3$) are the three light neutrino Dirac masses with $v_u$ as the vacuum expectation value of the up-type Higgs field, $\langle H^0_u \rangle$. The quantities $h^i_\nu$ are the neutrino Yukawa couplings. The structure of the matrix shown in eq. (\[5by5numat\]) can be readily understood from eq. (\[Inverse-seesaw-superpot\]) by looking at the mixing between different doublet and singlet neutral fermions. The effective 3 $\times$ 3 mass matrix for the light neutrinos can be obtained as (M\^\_[tree]{})\_[ij]{}= m\_[D\_i]{}m\_[D\_j]{}. \[tree-neut-mass-matrix\] In the seesaw approximation $(m_{D_i}<<M_R)$, the denominator of eq. (\[tree-neut-mass-matrix\]) becomes only $M^2_R$. The five mass eigenstates of eq. (\[5by5numat\]) are denoted by $\widetilde n_i$, $i=1,\dots,5$, out of which $\widetilde n_{1,2,3}$ are nothing but three light neutrinos.
Structure of eq. (\[tree-neut-mass-matrix\]) tells us that only one neutrino is massive at the tree level with mass: m\_[\_3]{} = . \[tree-level-neut-mass-matrix\] The smallness of the neutrino mass is ascribed to the smallness of the $\mu_s$ parameter, rather than the largeness of the Majorana-type mass, $M_R$, as required for the standard seesaw mechanism [@Nunokawa:2007qh]. With the choice of normal hierarchy in the light neutrino masses this tree level mass will attribute to the atmospheric scale $\sim 10^{-11}$ GeV. In the regime of seesaw approximation, for a typical Dirac mass $m_{D_i}\sim 10^{2}$ GeV (assuming neutrino Yukawa couplings, $h_\nu^i \sim 10^{-1}$) and TeV scale $M_R$, the value of the parameter $\mu_s$ comes out to be $\sim 10^{-9}$ GeV. On the contrary, when $m_{D_i}\sim M_R$ (see eq. (\[tree-neut-mass-matrix\])) then the atmospheric neutrino scale ($\sim 10^{-11}$ GeV) is determined by $\mu_s$ only.
The neutrino mass matrix shown in eq. (\[tree-neut-mass-matrix\]) is diagonalizable using a $3\times3$ unitary matrix as U\^[[tr]{} T]{} M\^\_[tree]{} U\^[tr]{}= [diag]{}(0,0,m\_[\_3]{}). \[tree-level-diag-neut\] The matrix $U^{tr}$ contains information about the tree level neutrino mixing angles. In order to satisfy the neutrino experimental data [@Schwetz:2008er; @GonzalezGarcia:2010er; @Schwetz:2011qt] one requires a second non-zero neutrino mass eigenvalue which can arise by including the one-loop corrections in the neutrino mass matrix [@Hirsch:2009ra].
In this model the doublet and singlet sneutrinos mix after the electroweak symmetry breaking. Thus the sneutrino mass squared matrix, $M^2_{\tilde\nu}$ is now a $10\times10$ matrix for MSISM, and assuming [*CP*]{} conservation this matrix can be decomposed into two $5\times5$ block matrices corresponding to [*CP*]{}-even and [*CP*]{}-odd sneutrino fields. The sneutrino mass term in the Lagrangian, then looks like, \_= (\^R,\^I) (
[cc]{} M\^2\_+ & 0\
0 & M\^2\_-\
)(
[c]{} \^R\
\^I
), \[Lagrangian-neutral-scalar\] where, $\phi^R=(\widetilde\nu^R_i,\widetilde\nu^{cR},\widetilde S^R)$, $\phi^I=(\widetilde\nu^I_i,\widetilde\nu^{cI},\widetilde S^I)$. The two mass squared matrices $M^2_\pm$ are given by [@Arina:2008bb] M\^2\_= (
[ccc]{} (M\^2\_[L\_[i]{}]{}+M\_Z\^2 2+m\^2\_[D\_i]{})\_[ij]{} & (A\^j\_[h\_]{}v\_u- m\_[D\_j]{} ) & m\_[D\_j]{} M\_R\
(A\^i\_[h\_]{}v\_u- m\_[D\_i]{} ) & m\^2\_[\^c]{}+M\_R\^2+m\^2\_[D\_k]{} & \_S M\_RB\_[M\_R]{}\
m\_[D\_i]{} M\_R & \_S M\_RB\_[M\_R]{} &m\_S\^2+\^2\_S+M\_R\^2B\_[\_S]{}
), \[neutral-scalar-massmatrix\] where $M^2_{\widetilde L_{i}}$ denote soft supersymmetry breaking mass squared terms for $SU(2)_L$ doublet sleptons and $M_Z$ is the Z-boson mass. The ratio of the two Higgs VEVs is defined as $\tan\beta = \frac{v_u}{v_d}$, where $v_d$ is the vacuum expectation value of the down-type Higgs field $H_d$. The real symmetric mass matrix of eq. (\[Lagrangian-neutral-scalar\]) can be diagonalized by a 10$\times$10 orthogonal matrix as follows M\^2\_ \^T= [diag]{}(m\^2\_[N\_1]{},…, m\^2\_[N\_[10]{}]{}), \[scalar-mass-matrix-diag\] with $m^2_{{\widetilde N}_1} < \dots < m^2_{{\widetilde N}_{10}}.$ Diagonalizing the [*CP*]{}-even and [*CP*]{}-odd mass matrices $M^2_\pm$ separately by M\^2\_ \^T=[diag]{}(m\^2\_[[N]{}\_[i]{}]{}), i=1,…,5, \[CP-state-scalar-mass-matrix-diag\] where ${\widetilde N}_{i+}$ and ${\widetilde N}_{i-}$ denote the $i$-th [*CP*]{}-even and [*CP*]{}-odd sneutrino mass eigenstates, respectively, leads to a different parameterization which can be used in some cases. In this notation, for the set of chosen parameters (shown later) ${\widetilde N}_{1+}$ = ${\widetilde N}_2$ and ${\widetilde N}_{1-}$ = ${\widetilde N}_1$ and so on (see, eq. (\[scalar-mass-matrix-diag\])).
Decays of chargino and neutralino {#decays}
=================================
In this section we discuss the decays of charginos to charged leptons and singlet sneutrinos as well as the decays of the lighter neutralinos. We shall also show how these decays can lead to the final states, that we have proposed to study in this paper. Our choices of the four benchmark points for a detailed collider study will also be presented here.
Chargino decay {#Chargino-decay}
--------------
For the discussion of chargino decays we shall concentrate on a part of the parameter space where one of the singlet scalars of MSISM is the LSP. Hence this scalar singlet will appear at the end of the supersymmetric cascade decay chains. For the present discussion let us assume that the dominant decay mode of the lighter chargino is in the two body mode \^\_1 \_a + l\^\_i, a = 1,2, l\_i = e, , , with ${\widetilde N}_1$ being the [*CP*]{} conjugated state to ${\widetilde N}_2$. The relevant piece of the Lagrangian for the calculation of this decay width is \_[\^-]{}=(C\^L\_[ija]{}P\_L + C\^R\_[ija]{}P\_R)l\_i N\_a + h.c. , \[chargino-decay-Lagrangian\] where C\^L\_[ija]{}&=&- \[g\^\*\_[j1]{}(\_[ai]{} - i\_[a,i+5]{}) - h\^i\_\^\*\_[j2]{}(\_[a4]{} - i\_[a9]{})\],\
C\^R\_[ija]{}&=& Y\_[\_i]{} \_[j2]{}(\_[ai]{} - i\_[a,i+5]{}). \[chargino-couplings\] The $Y_{\ell_i}$s are the charged lepton Yukawa couplings and $\bU$, $\bV$ are two unitary $2\times2$ chargino mixing matrices such that $\bU^* m_{2\times2} \bV^{-1}=diag(m_{{\widetilde{\chi}^{\pm}_1}},m_{{\widetilde{\chi}^{\pm}_2}})$, where $m_{{\widetilde{\chi}^{\pm}_1}},m_{{\widetilde{\chi}^{\pm}_2}}$ are the two physical chargino masses. The $2\times2$ mass matrix $m_{2\times2}$ in the charged gaugino-higgsino basis $\psi^{+^T}=-i\widetilde{\lambda}^+_2,
\widetilde{H}^+_u,~\psi^{-^T}=-i\widetilde{\lambda}^-_2, \widetilde{H}^-_d$ is given by \[chargino\_mass\_matrix\] m\_[22]{} = (
[ccccc]{} M\_2 & [g]{}[v\_u]{}\
\
[g]{}[v\_d]{} &
). Here $g$ is the $SU(2)_L$ gauge coupling.
The corresponding decay widths are given as (\^\_1N\_a + l\^\_i)= ([C\^L\_[i1a]{}]{}\^2 + [C\^R\_[i1a]{}]{}\^2). \[chargino-decay-width\] The members of [*CP*]{} conjugated pair of sneutrinos being nearly mass degenerate ($m_{{\widetilde N}_1} \approx m_{{\widetilde N}_2}$) they are unlikely to be distinguished experimentally. Hence we sum over the [*CP*]{}-even and [*CP*]{}-odd sneutrino states of the [*CP*]{} conjugated pair. Thus (\^\_1N\_[1+2]{} + l\^\_i) \^2\_[=1]{} (\^\_1N\_+ l\^\_i). \[decay-width-sum\] One can adjust the parameters $\mu_S$ and $B_{\mu_S}$ in such a way that the tree-level neutrino mass matrix contribution determines the atmospheric mass scale, while the one-loop corrections control the solar mass scale [@Hirsch:2009ra]. In such a situation it can be shown that in order to have small reactor neutrino mixing angle and maximal atmospheric neutrino mixing angle, the parameter $m_{D_1}$ has to be considerably smaller than other two Dirac masses and simultaneously, $m_{D_2} \sim m_{D_3}$. The solar neutrino mixing angle can be kept large by keeping the parameters $\delta_i \equiv A^i_{h_\nu} v_u - \mu m_{D_i} \cot\beta$ to be of the same order for all the three flavors, $i = e, ~\mu, ~\tau$. In this case, one can show that the decay width of the lighter chargino, $\Gamma (\widetilde\chi^\pm_1\to\widetilde N_{1+2} + l^\pm_i)$ correlates with the corresponding parameter $m^2_{D_i}$. The atmospheric neutrino mixing angle at the same time also behaves as $\tan^2 \theta_{23} \sim \frac{m^2_{D_2}}{m^2_{D_3}}$. Hence, one would expect that the ratio of the branching ratios $\frac{Br(\widetilde\chi^\pm_1\to\widetilde N_{1+2}+\mu^\pm)}
{Br(\widetilde\chi^\pm_1\to\widetilde N_{1+2}+\tau^\pm)}$ must correlate with the ratio $\frac{m_{D_2}^2}{m_{D_3}^2}$. This has been shown in figure \[branching\_frac\_ratio\].
Neutralino decay
----------------
In our chosen benchmark points (defined later in this section) the lightest neutralino is the next-to-lightest supersymmetric particle (NLSP) and, decays dominantly through the two body decay channels $\widetilde\chi^0_1\to\nu_{l_i} +\widetilde N_{1,2},~l_i = e, \mu, \tau$. The relevant interaction term of the Lagrangian is: \_[\^0]{}=|\^0\_j (A\^L\_[mjb]{}P\_L + A\^R\_[mjb]{}P\_R)\_m N\_b + h.c., \[neutralino-decay-Lagrangian\] where A\^L\_[mjb]{}&=&(\^\*\_[j2]{} - \_W \^\*\_[j1]{}) (\_[bi]{} - i\_[b(i+5)]{})U\^[tr]{}\_[im]{},\
A\^R\_[mjb]{}&=&-h\^i\_U\^[tr]{}\_[im]{}\_[j4]{}(\_[b4]{} - i\_[b9]{}). \[neutralino coupling\] Here $g$ is the $SU(2)_L$ gauge coupling, $\theta_W $ is the weak mixing angle, and $\bN$ is the unitary 4x4 neutralino mixing matrix. Although the second lightest neutralino (${\widetilde \chi}^0_2$) decays mostly through the standard MSSM two-body charged lepton-slepton channel (${\widetilde \chi}^0_2 \to {\widetilde l_i}^\pm + l_i^\mp$), some of its branching fraction goes into the decay channels arising from the coupling given in eq. (\[neutralino-decay-Lagrangian\]). Here, we have neglected the charged lepton flavor violating decay of ${\widetilde \chi}^0_2$. The decay width of a neutralino decaying into neutrino-sneutrino two-body mode is given as (\^0\_jN\_b + \_m)= ([A\^L\_[mjb]{}]{}\^2 + [A\^R\_[mjb]{}]{}\^2). \[neutralino-decay-width\]
Trilepton signal and the benchmark points {#trilepton-benchmark}
-----------------------------------------
In order to illustrate the trilepton signal we simulate $\widetilde\chi^0_2
\widetilde\chi^\pm_1$ production followed by their two-body decays to produce $3\ell +~\mET$ or $2\ell + \tau-{\rm jet} + \mET$ final states, where $\ell = e, \mu$.
As discussed above the production process and the decay cascades leading to these final states are as follows &&pp\^0\_2 + \^\_1,\
&&\^\_1N\_[1,2]{} +\^/\^,\
&&\^0\_2 + \^,\
&&\^\^+ \^0\_1,\
&&\^0\_1\_[l]{} +N\_[1,2]{}. \[signal-choice\] The Feynman diagram for the above mentioned final states is shown in figure \[feyn-diag\]. In the presence of heavy squarks ($\sim 1 ~TeV$), this is the leading process for the chosen signal.
Because of the presence of the massive singlet sneutrino LSPs, $\widetilde N_{1,2}$, (quasi-degenerate in masses), we have, for this model, a large amount of missing energy in the final states. In order to have an appreciable signal rate one must have significant production cross section of ${\widetilde \chi}^0_2 - {\widetilde \chi}_1^\pm$ pair and large branching ratios for the above-mentioned decays. To achieve these we have chosen four benchmark points (BPs) in the parameter space where the detailed collider simulation has been performed. We scanned the whole parameter space to check for charged lepton flavour violating (LFV) decay widths and we found points both above and below the experimental limits in different region of the parameter space. In all of the benchmark points, constraints from LFV decays [@Nakamura] are satisfied as well as the atmospheric neutrino mixing is near maximal. The input parameters for different benchmark points are given in table \[input1\]. The choices of the parameters $m^2_{D_i}$ will be shown later. The mass splittings between the second lightest neutralino, the charged sleptons and the lightest neutralino are maintained in a way, that the second lightest neutralino decays only through charged lepton-slepton two body modes and the charged sleptons further decay into the lightest neutralino and charged lepton states. With these considerations we generated the sparticle spectrum using (version 2.41)[@Djouadi:2002ze]. Masses of the neutrino and sneutrino states are computed using a self developed code in [FORTRAN]{}. Relevant mass spectra for these benchmark points are shown in table \[mass-spectrum\].
The choice of model parameters for different benchmark points are chosen to yield statistically significant final states. As an illustrative example, production cross sections for the ${\widetilde{\chi}^0_2}{\widetilde{\chi}^{\pm}_1}$ pair with $7$ TeV center of mass energy at LHC are in the range of $200-300$ fb for the first and third benchmark points. For the fourth benchmark point with relatively heavy ${\widetilde{\chi}^0_2}{\widetilde{\chi}^{\pm}_1}$ pair (see table \[mass-spectrum\]) the production cross section is reduced by a factor of $4(3)$ compared to the first(third) benchmark point. On the contrary, a higgsino like ${\widetilde{\chi}^0_2}{\widetilde{\chi}^{\pm}_1}$ pair (BP2) yield a similar production cross-section like BP4, in spite of having a lighter ${\widetilde{\chi}^0_2}{\widetilde{\chi}^{\pm}_1}$ pair. Thus, the region of parameter space with higgsino like ${\widetilde{\chi}^0_2}{\widetilde{\chi}^{\pm}_1}$ pair is unfavorable for this analysis.
Note that hadronically quiet trilepton signal ($3 \ell + \mET$ ) will get very little contribution from squark-squark, squark-gluino and gluino-gluino pair production. On the other hand, when we have $2\ell + \tau$-jet + $\mET~$ signal, then one should consider all other sources of dilepton + 1-jet + $\mET$ events where one jet can be faked as a $\tau$-jet. For example, one can have a jet out of a squark decay (${\tilde q} \to q^\prime + {\widetilde \chi}_1^\pm$) from one side of the cascade. However, since in this model the squarks are much heavier ($\sim 1~\rm{TeV}$) and after incorporating the probability of any jet faking as a $\tau$-jet, the event rate comes out to be negligibly small compared to the one generated from chargino-neutralino production. Hence the main contribution to $2\ell+\tau$-jet + $\mET$ signal comes from ${\widetilde \chi}^0_2{\widetilde \chi}^\pm_1$ production only.
Event generation and\
background analysis {#ev-gen}
=====================
On the basis of the discussion presented in the previous section, let us now provide a detailed description of event generation and subsequently, the background analysis. The decay widths corresponding to the two-body modes shown in eq. (\[signal-choice\]) have been used to modify the branching fractions of the charginos and neutralinos obtained from . These input files are then fed to (version 6.409) [@Pythia6.4] for event generation and showering. Initial and final state radiation, decay, hadronization, fragmentation and jet formation are implemented following the standard procedures in . Factorization and renormalization scales are set at $\sqrt{\widehat s}~$ (i.e $\mu_R =\mu_F =\sqrt{\widehat s}$ ), where $\sqrt{\widehat s}~$ is the parton level centre of mass energy. We have used the leading order CTEQ5L parton distribution functions [@Lai:1999wy; @Pumplin:2002vw] for the colliding protons. Some of the background events are generated using (version 2.14) [@Mangano:2002ea] with default factorization and renormalization scales. The jets are constructed using cone algorithm in PYCELL. Only those jets are constructed which have $p_T > 20~{\rm GeV}$ and $\mid\eta\mid < 2.5$. To simulate detector effects we have taken into account smearing of jet energies by a Gaussian probability density function of width [@Barr:2009wu] $\sigma (E)/E_j = (0.6/\sqrt{E_j[GeV]}) + 0.03$ where $E_j$ is the unsmeared jet energy.
In order to find three isolated leptons in the final states we impose following cuts and isolation criteria:\
I. Leptonic events are selected only if $p^\ell_T>8 ~{\rm{GeV}}$ and $\mid\eta^\ell\mid<2.4$.\
II. Lepton-lepton separation $\Delta R(\ell, \ell)$ set to be $> 0.2$, where $\Delta R=
\sqrt{(\Delta\eta)^2 + (\Delta\phi)^2}$.\
III. Lepton-jet separation $\Delta R(\ell, j)$ chosen to be $> 0.5$.\
IV. The sum of $E_T$ deposits of the hadrons which fall within a cone of $\Delta R\le 0.2$ around a lepton, must be less than $10~ {\rm GeV}$.
A $p_T$ cut of $10~{\rm GeV}$ and $17~{\rm GeV}$ [@Bayatian:2006zz] is applied on final state muons and electrons respectively, for the analysis at $7~{\rm TeV}$ and $14~{\rm TeV}$ center of mass energies at the LHC. The $\tau$-jets are counted with $p_T\ge 20 ~{\rm GeV}$ and $\mid\eta^\tau\mid<2.4$. The $\tau$’s are then counted according to the visible energy bins. A $\tau$-jet is treated as tagged or untagged according to the efficiency ($\epsilon_\tau$) of the most efficient algorithm given in [@Tau-eff]. In reference [@Tau-eff], $\tau$ identification efficiency obtained from actual collision data at $7~{\rm TeV}$ center of mass energy has also been quoted. The efficiencies obtained from Monte-Carlo simulation and from the data agrees very well. However, for higher luminosity with $14~{\rm TeV}$ center of mass energy, a lot of underlying events are expected to be there, which can perhaps bring down the detection efficiency. In this case also we have used the same efficiency as in $7~{\rm TeV}$ case hoping the experimentalists can maintain the efficiency as we have now. Unlike $\tau$, detection efficiencies of $e$ and $\mu$ are assumed to be $100\%$.
We have analysed the SM backgrounds in some detail. The dominant background events arise from $t\bar t$ and $WZ$ production at the LHC. Apart from these, contributions from $ZZ$, $WW$, $Zb\bar b$, $Wb\bar b$, $Z+jets$, $Wt$, $tb$, $WWW$, $Wt\bar t$ events have also been studied at the leading order. We also studied $QCD$ di-jet events. But after putting the cuts to reduce backgrounds as mentioned below we found no trilepton events for $1~{\rm fb^{-1}}$ integrated luminosity from these particular $QCD$ events. We use for an estimation of $Zb\bar b$, $Wb\bar b$, $Wt$, $tb$, $Z+jets$, $WWW$, $Wt\bar t$ backgrounds. We generate these events at the parton level using and fed those partonic events to for showering, hadronization, fragmentation, decay, etc. The other events are generated and analysed using . It should be mentioned that the importance of these processes have already been emphasized in the literature[@Sullivan:2008ki; @Bhattacharyya:2008zi].
The trilepton signal in our model arising out of chargino-neutralino production is accompanied by large missing transverse energy ($\mET$ ), because of a pair of singlet sneutrino LSPs and a neutrino. As an example, the $\mET~$ spectrum of background events as well as the signal events ($3\mu + \mET$ ) for the first benchmark point (BP1) are shown in figure \[missing\_pt\].
These distributions are obtained without applying any cuts to reduce background events. It is evident from the plot in figure \[missing\_pt\] that a strong $\mET$ cut will affect the signal cross-section very mildly, but it reduces significantly background events coming from some processes. Therefore, a cut $\mET > 25~{\rm GeV}$ is applied for background rejection. For some other channels; $t\bar t, WZ, Wt\bar t, WWW$ the $\mET$ distributions do not peak before $25~ {\rm GeV}$ as shown in figure \[missing\_pt1\]. Hence, the above mentioned $\mET~$ cut does not seriously affect these background events. To reduce these events we have further applied two more cuts. An invariant mass cut on the opposite sign dilepton pair, $80~{\rm GeV} > M^{\ell\ell}_{inv} > 100~{\rm GeV}$ removes backgrounds coming from $Z$-bosons. To manifest this idea we show invariant mass distribution in figure \[inv\_mass\] constructed from opposite sign muon pairs for signal events and $WZ$ background events.
On the other hand, rejection of tagged $b$-jet events significantly reduces backgrounds coming from $t\bar t$ events. A jet (with $\mid\eta\mid<2.5$) is reconstructed as a $b$-jet if the $\Delta R$ separation between the jet and the $b$-quark (with $p_T > 5~{\rm GeV}$) is less than $0.2$. The b-jet identification efficiency is taken to be $50 \%$.
In order to perform the collider analysis we have randomly generated $m^2_{D_i}$ and $\delta^2_i$ within certain range: $(\sum_i m^2_{D_i})^{1/2} \in 10^{[-4,2.6]}$ and $(\sum_i \delta^2_i)^{1/4} \in 10^{[-4,3]}$ [@Hirsch:2009ra]. Moreover, we also consider $(\sum m^2_{D_i})^{1/2}>(\sum_i \delta^2_i)^{1/4}$, such that Dirac neutrino masses give the dominant contribution to the chargino decay [@Hirsch:2009ra]. Around each of the four benchmark points we select a set of six to seven points of these randomly generated parameters. These points will be useful for the correlation study discussed later in section \[results\]. Remember that these parameters control the neutrino masses and the mixing angles and our choices of benchmark points are such that the atmospheric neutrino mass scale is determined by the tree level neutrino mass matrix contribution. Before showering in , as mentioned earlier, the ratio $\frac{Br(\widetilde\chi^\pm_1\to \widetilde N_{1+2} + \mu^\pm)}{Br(\widetilde\chi^\pm_1\to
\widetilde N_{1+2} +\tau^\pm)}$ shows a very nice sharp correlation when plotted against $\frac{m^2_{D_2}}{m^2_{D_3}}$ which is a measure of $\tan^2 \theta_{23}$. We have done the showering for four benchmark points introduced in table \[input1\] and table \[mass-spectrum\] to look for the ratio $\frac{\sigma(pp \to \mu \sum \ell \ell + \mET~)}{\sigma(pp \to \tau \sum \ell \ell + \mET~)}$ with $\ell=e,\mu$. Since one $\mu$ and one $\tau$ in these final states always come from the decay of ${\widetilde \chi}_1^\pm$, we would expect that this ratio will also go as $\sim \tan^2\theta_{\rm 23}$. Hence, by measuring this ratio from the trilepton signals one can obtain information about the atmospheric neutrino mixing angle at the LHC. On the other hand, a precise measurement of the atmospheric neutrino mixing angle at the oscillation experiments can be used to predict the allowed range of the above ratio at the LHC. In the following section we give a quantitative estimate of this ratio for our choices of benchmark points (along with randomly selected values of $m^2_{D_i}$) and show that for each of these points the various signal events included in the calculation of this ratio can be statistically significant.
Results
=======
In order to study the correlation between the atmospheric neutrino mixing angle ($\theta_{23}$) and the final states with trilepton + $\mET$ at the LHC, we look at the ratio of cross sections $\frac{\sigma(pp \to \mu \sum \ell \ell + \mET~)}
{\sigma(pp \to \tau \sum \ell \ell + \mET~)}$, $\ell = e,\mu$. As mentioned in the introduction, in the denominator the $\tau$ must always come from the decay of ${\tilde \chi}^\pm_1$ because we are considering final states with only one $\tau$-jet and neglecting lepton flavor violating decays of ${\tilde \chi}^0_2$ and ${\tilde \ell}^\pm$. For the same reason, in the numerator one $\mu$ must always also come from the decay of ${\tilde \chi}^\pm_1$. Hence, naively we would expect that this ratio of cross sections will also show nice correlation with the atmospheric neutrino mixing angle $\theta_{23}$.
After applying different cuts to reduce backgrounds and taking into account the $\tau$-tagging efficiency, we find that the ratio of trilepton signal cross section again shows a nice correlation with the atmospheric neutrino mixing angle $\tan^2\theta_{\rm 23}$. However, in this case the numbers change from the ratio of branching ratios, discussed earlier and the straight lines obtained are steeper than the one shown in figure \[branching\_frac\_ratio\]. This happens because in our simulation we take the detection efficiency of $\mu$ to be $100 \%$ as opposed to the $\tau$ detection efficiency, which is smaller [@Bayatian:2006zz]. Since the branching fractions of $\tau$ events are in the denominator of the ratio, the numbers naturally go up.
The cross-sections and the corresponding statistical significance ($\frac{S_x}{\sqrt{B_x+S_x}}$ with $x=e,\mu,\tau$) obtained from our simulation for LHC are shown in this section. Here $S_x$ is defined as the number of $x\sum\ell\ell$ signal events and $B_x$ is defined as the number of corresponding background events. In more simple form significance for the $\mu\sum\ell\ell+\mET~$ channel is defined as $\frac{S_{\mu e e} + S_{\mu \mu\mu}}{\sqrt{S_{\mu e e} + S_{\mu \mu\mu}
+B_{\mu e e} + B_{\mu \mu\mu}}}$. In a similar fashion significance for the $\tau\sum\ell\ell+\mET~$ channel can be obtained.
We quote the results below for an integrated luminosity of 25 ${\rm fb}^{-1}$ for the LHC with $7~{\rm TeV}$ and $14~{\rm TeV}$ center-of-mass energies. The results are obtained with the cuts mentioned in section \[ev-gen\]. Throughout this analysis we have used leading order cross sections for the signals as well as all the backgrounds at the LHC. However, if next-to-leading order (NLO) corrections are included the statistical significance will not change much. For example, if NLO corrections are included the signal cross section at 14 TeV LHC is expected to increase by 1.25 to 1.35 [@Beenakker:1999xh]. As discussed above, a large contribution to the background comes from the $t {\bar t}$ events. The NLO cross section for $t {\bar t}$ production at 14 TeV LHC is about 800 pb [@Kidonakis:2008mu; @Cacciari:2003fi] which is about a factor of two larger than the leading order cross section that we have used in our analysis. Thus taking into account the NLO contribution of all the major background events along with the signal event, the significance $S_x/\sqrt{B_x+S_x}$ estimated for our signal, will not change much and remains conservative in comparison to the uncertainties in the production cross sections.
Values of the randomly generated parameters $m^2_{D_i}$, for four chosen benchmark points are presented in table \[input2\]. For the numerical analysis we choose to vary $m^2_{D_1}$ in the range of $10^{-4}-10^{-2}$ GeV$^2$, whereas $m^2_{D_{2,3}}$ are varied within $10^{-2}$ to $10^{2}$ GeV$^2$. The $\delta^2_i$ are also varied accordingly, but keeping the constraints $(\sum m^2_{D_i})^{1/2}
>(\sum \delta^2_{i})^{1/4}$. The scale of $m^2_{D_i}$ has a strong influence on the decay processes ${\widetilde{\chi}^{\pm}_1}\to\widetilde N_{1+2}+\mu^\pm/\tau^\pm$ and $\widetilde\chi^0_j\to \widetilde N_b+\nu_m$. In order to achieve a statistically significant trilepton final state originating from ${\widetilde{\chi}^0_2}{\widetilde{\chi}^{\pm}_1}$ pair, we would like to have $Br({\widetilde{\chi}^{\pm}_1}\to\widetilde N_{1+2}+\mu^\pm/\tau^\pm)$ to be large and $Br({\widetilde{\chi}^0_2}\to \widetilde N_b+\nu_m)$ to be small, simultaneously. However, in the limit $m_{D_i}\sim M_R\sim$ $\cal{O}$ $(10^2~\rm{GeV})$, the neutrino Yukawa couplings $h^i_\nu$ are $\sim$ $\cal{O}$ $(1)$. Then as can be seen from eqs. (\[chargino-decay-width\]) and (\[neutralino-decay-width\]) both of these decay widths are large and consequently, yields a smaller branching ratio for ${\widetilde{\chi}^0_2}\to \widetilde{\ell}^\pm+\ell^\mp$. We observe that in this case it is rather difficult to achieve a statistically significant final state particularly for the $\tau\sum\ell\ell+\mET~$ mode.
In the trilepton signals studied in this work, one lepton comes from the lighter chargino (${\widetilde \chi}^\pm_1$) decay and the other two same flavour opposite sign leptons come from the second lightest neutralino (${\widetilde \chi}^0_2$) decay. Since the probability of getting electrons from the chargino decay is suppressed compared to muons or taus, events with odd number of electrons ($e e e$ and $e\mu\mu$) should have smaller cross-sections compared to others, which is clearly reflected in the signal cross-sections. This feature is intrinsically related with the small but non-zero reactor neutrino angle [@Abe:2011sj], which will be discussed again later. In table \[cross-section-lhc7\] and \[cross-section-lhc14\] chosen trilepton $+\mET~$ cross sections are shown along with the total standard model background cross section for the LHC at center-of-mass energy, $\sqrt{s}=7$ and $14$ TeV, respectively. The corresponding statistical significance of the signals are shown respectively in table \[significance-lhc7\] and table \[significance-lhc14\]. We can see from table \[significance-lhc7\] that, at the LHC even with $\sqrt{s}=7$ TeV, the lowest signal significance for $\tau\sum\ell\ell+\mET~$ final state, that we have obtained, is greater than $3\sigma$ for an integrated luminosity of $25$ fb$^{-1}$. Hence, the trilepton $+\mET~$ data for $25$ fb$^{-1}$ integrated luminosity at $7$ TeV LHC should be able to constrain the theoretical parameter space of this model. These numbers (significance) are much higher for LHC with $\sqrt{s}=14$ TeV and are shown in table \[significance-lhc14\]. It is once again evident from these tables that a higgsino like ${\widetilde{\chi}^0_2}{\widetilde{\chi}^{\pm}_1}$ pair (BP2) yields statistically less significant specific trilepton final state. In other words for such benchmark points, the significance of the final state trilepton signal is less promising. This situation is comparable to a heavy gaugino like ${\widetilde{\chi}^0_2}{\widetilde{\chi}^{\pm}_1}$ pair as represented by BP4.
We present the correlation plots, obtained with different randomly generated values of $m^2_{D_i}$ and $\delta^2_i$ around each of the four benchmark points. These are shown in figure \[correlation\_plot1\] and figure \[correlation\_plot2\] for the LHC with $\sqrt{s}=7$ TeV and $14$ TeV, respectively. We present these correlations with best fit lines. It can be seen from these figures that the $3\sigma$ allowed value of $\tan^2\theta_{23}$ [@Schwetz:2008er] from atmospheric neutrino oscillation experiments predict a value of the ratio of cross sections $\frac{\sigma(pp \to \mu \sum\ell\ell + \mET~)}
{\sigma(pp \to \tau \sum\ell\ell + \mET~)}$, ($\ell = e,\mu$) to be approximately in the range $1.0-6$. These predictions can be verified at the LHC or the measured value of this ratio can give an alternative estimate of $\tan^2\theta_{23}$. On the other hand, if this ratio comes out to be very much different from the ones predicted here then one can perhaps conclude that MSISM is not the correct model for explaining neutrino masses and mixing.
Nevertheless, as we can see, from the correlation plots, that there is a different linear relationship for each different kind of benchmark points. In general then, from neutrino oscillation data we cannot give a unique prediction for the ratio of the cross-sections that can be verified at the LHC and help us in constraining the model parameters. In other words, measuring the cross section ratio at the LHC would not allow a prediction of $\theta_{23}$ that could be tested against oscillation results. This means that we need other measurements at the LHC to allow such predictions. As an example, to distinguish among the four benchmark points we plot the ratio, $m_{\widetilde\chi_1^\pm}/m_{\widetilde N_{1,2}}$ with the ratio of cross-sections of $\mu$ and $\tau$ channels which gives four separate parallel lines for the four benchmark points (figure \[bp\_separation\]). One can see from figure \[bp\_separation\] that the ratio (${m_{\widetilde\chi_1^\pm}}/{{m_{\widetilde N_{1,2}}}}$) increases as the slope of the straight lines in the correlation plot corresponding to different benchmark points decreases. This pattern can easily be understood. Increase in the mass ratio indicates greater splitting between the chargino and sneutrino masses. As the splitting increases, the leptons coming from this chargino decay become more energetic (eq.(\[signal-choice\])). This affects the $\tau$ count in the final state more than the $\mu$ count as the detection efficiency for the taus increases with the increase of visible energy of $\tau$ decay products[@Tau-eff]. Hence more $\tau$ events are expected in the final state for those benchmark points which has greater lighter chargino - LSP mass ratio for a given set of $m_{D_i}$’s. With the increase of $\tau$ events the ratio of the cross-sections plotted in the correlation plots decreases and as a consequence gives smaller slope compared to the previous benchmark point. Now it is clearly understood that if we can determine the lighter chargino-LSP mass ratio, we can distinguish among the four benchmark points.
Mass determination techniques in the context of LHC have been studied extensively. Transverse mass variable ($m_{T_2}$) [@Lester:1999tx; @Barr:2003rg] is very useful for this purpose. $m_{T_2}$ has also been generalized for the cases where the parent and daughter particles in the two decay chains are not identical [@Barr:2009jv; @Konar:2009qr]. Moreover, final state with more than two invisible particles has also been addressed in ref.[@Agashe:2010tu]. In our case, we observe the following:\
- The lightest neutralino $\widetilde\chi_1^0$ is also invisible, as mentioned earlier in the text.
- One lepton is produced from one side of the cascade and remaining two leptons from the other side of the cascade (figure \[feyn-diag\]).
- $\widetilde\chi_1^\pm$ and $\widetilde\chi_2^0$ are not mass degenerate but the difference is quite small in the context of mass measurement.
So, we see that daughters of different masses are produced here from nearly identical parents. A mass determination technique similar to refs. [@Barr:2009jv; @Konar:2009qr; @Agashe:2010tu] can be applied here too to determine the masses of the lighter chargino and the sneutrino LSP. However, a detailed analysis in this direction is beyond the scope of the present paper. Thus we see that measuring the mass ratio ($m_{\widetilde\chi_1^\pm}/{m_{\widetilde N_{1,2}}}$), along with the ratio of the trilepton cross-section, can help us pick the correct benchmark point and hence predict the correct value of $\theta_{23}$ that could be tested against the oscillation results. On the other hand, a precise determination of $\tan^2 \theta_{23}$ from oscillation experiments as well as a measurement of the cross section ratio at the LHC can give a unique prediction of the mass spectrum of the model, that can be verified by mass measurements at the LHC.
In support of our explanation for obtaining different slopes, we present the following analysis. Since this difference among the four benchmark points appears because of taking different $\tau$ identification efficiencies for different energy range and for taking separate $p_T$ cuts for $\mu$’s and $\tau$’s, we can remove this by the following strategy:
- A $p_T$ cut of $20~{\rm GeV}$ taken for both $\mu$ and $\tau$.
- A uniform $\tau$ identification efficiency of $50\%$ applied over the whole energy range.
We have presented the result in fig.\[unique\_slope\]. This shows the correlation plot for $7~{\rm TeV}$ center of mass energy under the above mentioned conditions. All the benchmark points now lie almost on one straight line. A few points still looks a little bit scattered because of the different isolation criteria used for $\mu$ and $\tau$.
Finally, in table \[ratio\_distinct-feature\] we show the ratios $\frac{\sigma(pp \to e \mu\mu + \mET~)}{\sigma(pp \to 3 \ell + \mET~)}$ and $\frac{\sigma(pp \to e e e + \mET~)}{\sigma(pp \to 3 \ell + \mET~)}$ for the four benchmark points. The smallness of these ratios is also a distinct feature of this model and arises due to the smallness of the neutrino reactor angle imposed by neutrino data [@Abe:2011sj]. In the usual MSSM scenario these ratios are expected to be much higher as there is no suppression of charginos decaying into electrons as we have in this model.
A discussion of these specific trilepton signals remains incomplete without a note on the Tevatron analysis of the considered model. For the four chosen benchmark points we observed no points with significance $\ge3\sigma$ for the $\tau\sum\ell\ell + \mET~$ final state and simultaneously consistent with the atmospheric neutrino mixing at the $3\sigma$ limit. This is a well expected result considering that the Tevatron center-of-mass energy is $1.96$ TeV with $12~{\rm fb^{-1}}$ of integrated luminosity [@tev-lumi]. For example, the statistical significance for $\tau\sum\ell\ell$ mode for BP1 with $m^2_{D_i}$s given in table \[input2\] is computed to be $1.64$.
Conclusion {#conclusions}
==========
We consider the minimal supersymmetric inverse seesaw model and study its characteristic signatures at the LHC. This model, with only one pair of singlet superfields explains existing neutrino oscillation data. The model is rich from phenomenological point of view and can lead to potentially testable signatures at the hadron colliders. In this R-parity conserving model, one of the singlet sneutrino (with a small admixture of the doublet sneutrino) is the lightest supersymmetric particle (LSP) and as a result shows up in the collider as missing energy. Charginos can decay to charged leptons plus singlet sneutrino LSP. The decay patterns of the chargino are controlled by the same parameters which generate the neutrino mixing angles.
In order to study this correlation of the chargino decays and the neutrino mixing angles, we look at specific trilepton $+~\mET~$ signatures at the LHC. We show that the ratios of cross sections of this studied trilepton $+~\mET~$ final states in certain flavour specific channels ($\mu ee + \mET$, $\mu \mu \mu + \mET$, $\tau ee + \mET$, $\tau \mu \mu + \mET~$) nicely correlate with the atmospheric neutrino mixing angle. We explore different points in the parameter space to study this correlation. A measurement of these cross sections thus provide an interesting test of the minimal supersymmetric inverse seesaw model. The hard missing $E_T$ spectrum makes this trilepton final state statistically significant by reducing certain standard model background events significantly. We adhere to different cuts to reduce the backgrounds coming from some other channels. Motivated by the recent results from the ATLAS and the CMS experiments, we work in a scenario with heavy squarks and gluinos and a relatively light electroweak sector. The results of our analysis suggest that the theoretical parameter space of this model can be constrained by the data collected at the LHC with center-of-mass energy $7$ TeV and for an integrated luminosity of $25$ fb$^{-1}$. On the other hand, a measured value of this ratio at the LHC can give us an alternative estimate of $\rm{tan}^2 \theta_{23}$ and confirm (or rule out) this minimal supersymmetric inverse seesaw model as a possible explanation of neutrino masses and mixing. We also show, as a distinct feature of this model, the cross sections of $pp \to e \mu \mu + \mET$ and $pp
\to eee + \mET$ are suppressed compared to the total chosen trilepton + $\mET~$ cross section because of the restrictions on the neutrino reactor angle imposed by neutrino data.
Acknowledgments {#acknowledgments .unnumbered}
===============
SM wishes to thank the Department of Science and Technology, Government of India for a Senior Research Fellowship. SB would like to thank the Department of Theoretical Physics, IACS for the hospitality while this work was being proposed and initiated. SB would also like to thank PhD program of HRI and RECAPP, HRI for hospitality and financial support during the initial phase of this work. PG would like to thank the Council of Scientific and Industrial Research, Government of India for the financial assistance during the initial phase of this work. PG’s work is supported by the Spanish MICINN under grant FPA2009-08958. PG also thank the support of the MICINN under the Consolider-Ingenio 2010 Programme with grant MultiDark CSD2009-00064, the Community of Madrid under grant HEPHACOS S2009/ESP-1473, and the European Union under the Marie Curie-ITN program PITN-GA-2009-237920. PG and SM wish to thank RECAPP, HRI for hospitality during a part of the investigation. We are very grateful to Asesh Krishna Datta for some very helpful discussions. We also thank Nabanita Bhattacharyya, Anindya Datta, Martin Hirsch, Partha Konar, Satyanarayan Mukhopadhyay and Sujoy Poddar for many useful comments and suggestions. PG is grateful to Chan Beom Park for his insightful suggestions. SR acknowledges the hospitality provided by the Helsinki Institute of Physics and the CERN Theory Group during the final phase of this work.
|
---
abstract: |
Consider an undirected graph $G = (VG, EG)$ and a set of six *terminals* $T = {\left \{ s_1, s_2, s_3, t_1, t_2, t_3 \right \}} \subseteq VG$. The goal is to find a collection ${\mathcal{P}}$ of three edge-disjoint paths $P_1$, $P_2$, and $P_3$, where $P_i$ connects nodes $s_i$ and $t_i$ ($i = 1, 2, 3$).
Results obtained by Robertson and Seymour by graph minor techniques imply a polynomial time solvability of this problem. The time bound of their algorithm is $O(m^3)$ (hereinafter we assume $n := {\left| VG \right|}$, $m := {\left| EG \right|}$, $n = O(m)$).
In this paper we consider a special, *Eulerian* case of $G$ and $T$. Namely, construct the *demand graph* $H = (VG, {\left \{ s_1t_1, s_2t_2, s_3t_3 \right \}})$. The edges of $H$ correspond to the desired paths in ${\mathcal{P}}$. In the Eulerian case the degrees of all nodes in the (multi-) graph $G + H$ ($ = (VG, EG \cup EH)$) are even.
Schrijver showed that, under the assumption of Eulerianess, cut conditions provide a criterion for the existence of ${\mathcal{P}}$. This, in particular, implies that checking for existence of ${\mathcal{P}}$ can be done in $O(m)$ time. Our result is a combinatorial $O(m)$-time algorithm that constructs ${\mathcal{P}}$ (if the latter exists).
author:
- 'Maxim A. Babenko [^1], Ignat I. Kolesnichenko [^2], Ilya P. Razenshteyn [^3]'
bibliography:
- 'main.bib'
nocite: '[@*]'
title: |
A Linear Time Algorithm for\
Finding Three Edge-Disjoint Paths\
in Eulerian Networks
---
Introduction {#sec:intro}
============
We shall use some standard graph-theoretic notation through the paper. For an undirected graph $G$, we denote its sets of nodes and edges by $VG$ and $EG$, respectively. For a directed graph, we speak of arcs rather than edges and denote the arc set of $G$ by $AG$. A similar notation is used for paths, trees, and etc. We allow parallel edges and arcs but not loops in graphs.
For an undirected graph $G$ and $U \subseteq VG$, we write $\delta_G(U)$ to denote the set of edges with exactly one endpoint in $U$. If $G$ is a digraph then the set of arcs entering (resp. leaving) $U$ is denoted by ${\delta^{\rm in}}_G(U)$ and ${\delta^{\rm out}}_G(U)$. For a graph $G$ and a subset $U \subseteq VG$, we write $G[U]$ to denote the subgraph of $G$ induced by $U$.
Let $G$ be an undirected graph. Consider six nodes $s_1, s_2, s_3, t_1, t_2, t_3$ in $G$. These nodes need not be distinct and will be called *terminals*. Our main problem is as follows:
\[eq:three\_pairs\] Find a collection of three edge-disjoint paths $P_1$, $P_2$, $P_3$, where $P_i$ goes from $s_i$ to $t_i$ (for $i = 1, 2, 3$).
Robertson and Seymour [@RS-95] developed sophisticated graph minor techniques that, in particular, imply a polynomial time solvability of the above problem. More specifically, they deal with the general case where $k$ pairs of terminals ${\left \{ s_1, t_1 \right \}}, \ldots, {\left \{ s_k, t_k \right \}}$ are given and are requested to be connected by paths. These paths are required to be *node-disjoint*. The edge-disjoint case, however, can be easily simulated by considering the line graph of $G$. For fixed $k$, the running time of the algorithm of Robertson and Seymour is cubic in the number of nodes (with a constant heavily depending on $k$). Since after reducing the edge-disjoint case to the node-disjoint one the number of nodes becomes $\Theta(m)$, one gets an algorithm of time complexity $O(m^3)$ (where, throughout the paper, $n := {\left| VG \right|}$, $m := {\left| EG \right|}$; moreover, it is assumed that $n = O(m)$). If $k$ is a part of input, then it was shown by Marx [@marx-04] that finding $k$ edge-disjoint paths is NP-complete even in the Eulerian case.
We may also consider a general *integer multiflow problem*. To this aim, consider an arbitrary (multi-)graph $G$ and also an arbitrary (multi-)graph $H$ obeying $VH = VG$. The latter graph $H$ is called the *demand graph*. The task is to find a function $f$ assigning each edge $uv \in EH$ a $u$–$v$ path $f(uv)$ in $G$ such that the resulting paths ${\left \{ f(e) \mid e \in EH \right \}}$ are edge-disjoint. Hence, the edges of $H$ correspond to the paths in the desired solution. By a *problem instance* we mean the pair $(G, H)$. An instance is *feasible* if the desired collection of edge-disjoint paths exists; *infeasible* otherwise.
In case of three terminal pairs one has $H = (VG, {\left \{ s_1t_1, s_2t_2, s_3t_3 \right \}})$. We can simplify the problem and get better complexity results by introducing some additional assumptions regarding the degrees of nodes in $G$. Put $G + H := (VG, EG \cup EH)$. Suppose the degrees of all nodes in $G + H$ are even; the corresponding instances are called *Eulerian*.
As observed by Schrijver, for the Eulerian case there exists a simple feasibility criterion. For a subset $U \subseteq VG$ let $d_G(U)$ (resp. $d_H(U)$) denote ${\left| \delta_G(U) \right|}$ (resp. ${\left| \delta_H(U) \right|}$).
\[th:feasibility\_criterion\] An Eulerian instance $(G,H)$ with three pairs of terminals is feasible if and only if $d_G(U) \ge d_H(U)$ holds for each $U \subseteq VG$.
The inequalities figured in the above theorem are called *cut conditions*. In a general problem (where demand graph $H$ is arbitrary) these inequalities provide necessary (but not always sufficient) feasibility requirements.
For the Eulerian case, the problem is essentially equivalent to constructing two paths (out of three requested by the demand graph). Indeed, if edge-disjoint paths $P_1$ and $P_2$ (where, as earlier, $P_i$ connects $s_i$ and $t_i$, $i = 1, 2$) are found, the remaining path $P_3$ always exists. Indeed, remove the edges of $P_1$ and $P_2$ from $G$. Assuming $s_3 \ne t_3$, the remaining graph has exactly two odd vertices, namely $s_3$ and $t_3$. Hence, these vertices are in the same connected component. However, once we no longer regard $s_3$ and $t_3$ as terminals and try to solve the four terminal instance, we lose the Eulerianess property. There are some efficient algorithms (e.g. [@SP-78; @shi-80; @tho-04; @tho-09]) for the case of two pairs of terminals (without Eulerianess assumption) but no linear time bound seems to be known.
The proof of [Theorem \[th:feasibility\_criterion\]]{} presented in [@sch-03] is rather simple but non-constructive. Our main result is as follows:
\[th:main\] An Eulerian instance of the problem [(\[eq:three\_pairs\])]{} can be checked for feasibility in $O(m)$ time. If the check turns out positive, the desired path collection can be constructed in $O(m)$ time.
The Algorithm
=============
Preliminaries {#subsec:prelim}
-------------
This subsection describes some basic techniques for working with edge-disjoint paths. If the reader is familiar with network flow theory, this subsection may be omitted.
Suppose we are given an undirected graph $G$ and a pair of distinct nodes $s$ (*source*) and $t$ (*sink*) from $VG$. An *$s$–$t$ cut* is a subset $U \subseteq VG$ such that $s \in U$, $t \notin U$.
Edge-disjoint path collections can be described in flow-like terms as follows. Let ${\overrightarrow{G}}$ denote the digraph obtained from $G$ by replacing each edge with a pair of oppositely directed arcs. A subset $F \subseteq A{\overrightarrow{G}}$ is called *balanced* if ${\left| F \cap {\delta^{\rm in}}(v) \right|} = {\left| F \cap {\delta^{\rm out}}(v) \right|}$ holds for each $v \in VG - {\left \{ s,t \right \}}$. Consider the *value* of $F$ defined as follows: $${\mathop{\rm val}\nolimits ( F )} := {\left| F \cap {\delta^{\rm out}}(s) \right|} - {\left| F \cap {\delta^{\rm in}}(s) \right|}.$$ Proofs of upcoming [Lemma \[lm:decomp\]]{}, [Lemma \[lm:augment\]]{}, and [Lemma \[lm:min\_cut\]]{} are quite standard and hence omitted (see, e.g. [@FF-62; @CLRS-01]).
\[lm:decomp\] Each balanced arc set decomposes into a collection arc-disjoint $s$–$t$ paths ${\mathcal{P}}_{st}$, a collection of $t$–$s$ paths ${\mathcal{P}}_{ts}$, and a collection of cycles ${\mathcal{C}}$. Each such decomposition obeys ${\left| {\mathcal{P}}_{st} \right|} - {\left| {\mathcal{P}}_{ts} \right|} = {\mathop{\rm val}\nolimits ( F )}$. Also, such a decomposition can be carried out in $O(m)$ time.
Obviously, for each collection ${\mathcal{P}}$ of edge-disjoint $s$–$t$ paths in $G$ there exists a balanced arc set of value ${\left| {\mathcal{P}}\right|}$. Vice versa, each balanced arc set $F$ in ${\overrightarrow{G}}$ generates at least ${\mathop{\rm val}\nolimits ( F )}$ edge-disjoint $s$–$t$ paths in $G$. Hence, finding a maximum cardinality collection of edge-disjoint $s$–$t$ paths in $G$ amounts to maximizing the value of a balanced arc set.
Given a balanced set $F$, consider the *residual digraph* ${\overrightarrow{G}}_F := (VG, (A{\overrightarrow{G}} - F) \cup F^{-1})$, where $F^{-1} := {\left \{ a^{-1} \mid a \in F \right \}}$ and $a^{-1}$ denotes the arc reverse to $a$.
\[lm:augment\] Let $P$ be an arc-simple $s$–$t$ path in ${\overrightarrow{G}}_F$. Construct the arc set $F'$ as follows: (i) take set $F$; (ii) add all arcs $a \in AP$ such that $a^{-1} \notin F$; (iii) remove all arcs $a \in F$ such that $a^{-1} \in AP$. Then, $F'$ is balanced and obeys ${\mathop{\rm val}\nolimits ( F' )} = {\mathop{\rm val}\nolimits ( F )} + 1$.
\[lm:min\_cut\] Suppose there is no $s$–$t$ path in ${\overrightarrow{G}}_F$. Then $F$ is of maximum value. Moreover, the set $U$ of nodes that are reachable from $s$ in ${\overrightarrow{G}}_F$ obeys $d_G(U) = {\mathop{\rm val}\nolimits ( F )}$. Additionally, $U$ is an inclusion-wise minimum such set.
Hence, to find a collection of $r$ edge-disjoint $s$–$t$ paths one needs to run a reachability algorithm in a digraph at most $r$ times. Totally, this takes $O(rm)$ time and is known as the *method of Ford and Fulkerson* [@FF-62].
Checking for Feasibility {#subsec:feasibility}
------------------------
We start with a number of easy observations. Firstly, there are some simpler versions of [(\[eq:three\_pairs\])]{}. Suppose only one pair of terminals is given, i.e. $H = (VG, {\left \{ s_1t_1 \right \}})$. Then the problem consists in checking if $s_1$ and $t_1$ are in the same connected component of $G$. Note that if the instance $(G, H)$ is Eulerian then it is always feasible since a connected component cannot contain a single odd vertex. An $s_1$–$t_1$ path $P_1$ can be found in $O(m)$ time.
Next, consider the case of two pairs of terminals, i.e. $H = (VG, {\left \{ s_1t_1, s_2t_2 \right \}})$. Connected components of $G$ not containing any of the terminals may be ignored. Hence, one may assume that $G$ is connected since otherwise the problem reduces to a pair of instances each having a single pair of terminals.
\[lm:two\_pairs\] Let $(G, H)$ be an Eulerian instance with two pairs of terminals. If $G$ is connected then $(G, H)$ is feasible. Also, the desired path collection ${\left \{ P_1, P_2 \right \}}$ can be found in $O(m)$ time.
The argument is the same as in [Section \[sec:intro\]]{}. Consider an arbitrary $s_1$–$t_1$ path $P_1$ and remove it from $G$. The resulting graph $G'$ may lose connectivity, however, $s_2$ and $t_2$ are the only odd vertices in it (assuming $s_2 \ne t_2$). Hence, $s_2$ and $t_2$ are in the same connected component of $G'$, we can trace an $s_2$–$t_2$ path $P_2$ and, hence, solve the problem. The time complexity of this procedure is obviously $O(m)$.
Now we explain how the feasibility of an Eulerian instance $(G, H)$ having three pairs of terminals can be checked in linear time. Put $T := {\left \{ s_1, s_2, s_3, t_1, t_2, t_3 \right \}}$. There are exponentially many subsets $U \subseteq VG$. For each subset $U$ consider its *signature* $U^* := U \cap T$. Fix an arbitrary signature $U^* \subseteq T$ and assume w.l.o.g. that $\delta_H(U^*) = {\left \{ s_1t_1, \ldots, s_kt_k \right \}}$. Construct a new undirected graph $G(U^*)$ as follows: add source $s^*$, sink $t^*$, and $2k$ auxiliary edges $s^*s_1, \ldots, s^*s_k, t_1t^*, \ldots, t_kt^*$ to $G$.
Let $\nu(U^*)$ be the maximum cardinality of a collection of edge-disjoint $s^*$–$t^*$ paths in $G(U^*)$. We restate [Theorem \[th:feasibility\_criterion\]]{} as follows:
\[lm:feasibility\_criterion\_restated\] An Eulerian instance $(G,H)$ with three pairs of terminals is feasible if and only if $\nu(U^*) \ge d_H(U^*)$ for each $U^* \subseteq T$,
Necessity being obvious, we show sufficiency. Let $(G,H)$ be infeasible, then by [Theorem \[th:feasibility\_criterion\]]{} $d_G(U) < d_H(U)$ for some $U \subseteq VG$. Consider the corresponding signature $U^* := U \cap T$. One has $d_H(U) = d_H(U^*)$, hence there is a collection of $d_H(U)$ edge-disjoint $s^*$–$t^*$ paths in $G(U^*)$. Each of these paths crosses the cut $\delta_G(U)$ by a unique edge, hence $d_G(U) \ge d_H(U)$ — a contradiction.
By the above lemma, to check $(G,H)$ for feasibility one has to validate the inequalities $\nu(U^*) \ge d_H(U^*)$ for all $U^* \subseteq T$. For each fixed signature $U^*$ we consider graph $G(U^*)$, start with an empty balanced arc set and perform up to three augmentations, as explained in [Subsection \[subsec:prelim\]]{}. Therefore, the corresponding inequality is checked in $O(m)$ time. The number of signatures is $O(1)$, which gives the linear time for the whole feasibility check.
We now present our first $O(m^2)$ time algorithm for finding the required path collection. It will not be used in the sequel but gives some initial insight on the problem. Consider an instance $(G, H)$ and let $s_1, t_1 \in T$ be a pair of terminals ($s_1t_1 \in EH$). If $s_1 = t_1$ then the problem reduces to four terminals and is solvable in linear time, as discussed earlier. Otherwise, let $e$ be an edge incident to $s_1$, and put $s_1'$ to be the other endpoint of $e$. We construct a new problem instance $(G_e, H_e)$, where $G_e = G - e$, $H_e = H - s_1t_1 + s_1't_1$ (i.e. we remove edge $e$ and choose $s_1'$ instead of $s_1$ as a terminal). Switching from $(G,H)$ to $(G_e,H_e)$ is called *a local move*. Local moves preserve Eulerianess. If a local move generates a feasible instance then it is called *feasible*, *infeasible* otherwise.
If $(G_e,H_e)$ is feasible (say, ${\mathcal{P}}_e$ is a solution) then so is $(G,H)$ as we can append the edge $e$ to the $s_1'$–$t_1$ path in ${\mathcal{P}}_e$ and obtain a solution ${\mathcal{P}}$ to $(G,H)$.
Since $(G,H)$ is feasible, there must be a feasible local move (e.g. along the first edge of an $s_1$–$t_1$ path in a solution). We find this move by enumerating all edges $e$ incident to $s_1$ and checking $(G_e,H_e)$ for feasibility. Once $e$ is found, we recurse to the instance $(G_e, H_e)$ having one less edge. This way, a solution to the initial problem is constructed.
To estimate the time complexity, note that if a move along some edge $e$ is discovered to be infeasible at some stage then it remains infeasible for the rest of the algorithm (since the edge set of $G$ can only decrease). Hence, each edge in $G$ can be checked for feasibility at most once. Each such check costs $O(m)$ time, thus yielding the total bound of $O(m^2)$. This is already an improvement over the algorithm of Robertson and Seymour. However, we can do much better.
Reduction to a Critical Instance
--------------------------------
To solve an Eulerian instance of [(\[eq:three\_pairs\])]{} in linear time we start by constructing an arbitrary node-simple $s_1$–$t_1$ path $P_1$. Let $e_1, \ldots, e_k$ be the sequence of edges of $P_1$. For each $i = 0, \ldots, k$ let $(G_i, H_i)$ be the instance obtained from the initial one $(G,H)$ by a sequence of local moves along the edges $e_1, \ldots, e_i$. In particular, $(G_0, H_0) = (G,H)$.
If $(G_k,H_k)$ is feasible (which can be checked in $O(m)$ time) then the problem reduces to four terminals and can be solved in linear time by [Lemma \[lm:two\_pairs\]]{}. Otherwise we find an index $j$ such that $(G_j,H_j)$ is a feasible instance whereas $(G_{j+1},H_{j+1})$ is not feasible.
This is done by walking in the reverse direction along $P_1$ and considering the sequence of instances $(G_k,H_k)$, …, $(G_0, H_0)$. Fix an arbitrary signature $U^*$ in $(G_k, H_k)$. As we go back along $P_1$, terminal $s_1$ is moving. We apply these moves to $U^*$ and construct a sequence of signatures $U_i^*$ in $(G_i, H_i)$. ($i = 1, \ldots, k$; in particular, $U_k^* = U^*$). Let $\nu_i(U^*)$ be the maximum cardinality of an edge-disjoint collection of $s^*$–$t^*$ paths in $G_i(U^*_i)$.
Consider a consequent pair $G_{i+1}(U_{i+1}^*)$ and $G_i(U_i^*)$. When $s_1$ is moved from node $v$ back to $v'$, edge $s^*v$ is removed and edges $s^*v'$ and $v'v$ are inserted. Note, that this cannot decrease the maximum cardinality of an edge-disjoint $s^*$–$t^*$ paths collection (if the dropped edge $s^*v$ was used by some path in a collection, then we may replace it by a sequence of edges $s^*v'$ and $v'v$). Hence, $$\nu_0(U_0^*) \ge \nu_1(U_1^*) \ge \ldots \ge \nu_k(U_k^*).$$
Our goal is to find, for each choice of $U^*$, the largest index $i$ (denote it by $j(U^*)$) such that $$\nu_i(U_i^*) \ge d_H(U_i^*).$$ Then, taking $$j := \min_{U^* \subseteq T} j(U^*)$$ we get the desired feasible instance $(G_j,H_j)$ such that $(G_{j+1},H_{j+1})$ is infeasible.
To compute the values $\nu_i(U_i^*)$ consider the following dynamic problem. Suppose one is given an undirected graph $\Gamma$ with distinguished source $s$ and sink $t$, and also an integer $r \ge 1$. We need the following operations:
> <span style="font-variant:small-caps;">Query</span>: Report $\min(r,c)$, where $c$ is the maximum cardinality of a collection of edge-disjoint $s$–$t$ paths in $\Gamma$.
> <span style="font-variant:small-caps;">Move</span>$(v,v')$: Let $v$, $v'$ be a pair nodes in $VG$, $v \ne s$, $v' \ne s$, $sv \in E\Gamma$. Remove the edge $sv$ from $\Gamma$ and add edges $sv'$ and $v'v$ to $\Gamma$.
\[lm:dyn\_flows\] There exists a data structure that can execute any sequence of <span style="font-variant:small-caps;">Query</span> and <span style="font-variant:small-caps;">Move</span> requests in $O(rm)$ time.
We use a version of a folklore incremental reachability data structure. When graph $\Gamma$ is given to us, we start computing a balanced arc set $F$ in ${\overrightarrow{\Gamma}}$ of maximum value ${\mathop{\rm val}\nolimits ( F )}$ but stop if ${\mathop{\rm val}\nolimits ( F )}$ becomes equal to $r$. This takes $O(rm)$ time. During the usage of the required data structure, the number of edge-disjoint $s$–$t$ paths (hence, ${\mathop{\rm val}\nolimits ( F )}$) cannot decrease (it can be shown using arguments similar to the described earlier). Therefore, if ${\mathop{\rm val}\nolimits ( F )} = r$ we may stop any maintenance and report the value of $r$ on each <span style="font-variant:small-caps;">Query</span>.
Otherwise, as ${\mathop{\rm val}\nolimits ( F )}$ is maximum, there is no $s$–$t$ path in ${\overrightarrow{\Gamma}}_F$ by [Lemma \[lm:min\_cut\]]{}. As long as no $r$ edge-disjoint $s$–$t$ paths in $\Gamma$ exist, the following objects are maintained:
- a balanced subset $F \subseteq A\Gamma$ of maximum value ${\mathop{\rm val}\nolimits ( F )}$ (which is less than $r$);
- an inclusionwise maximal directed tree ${\mathcal{T}}$ rooted at $t$ and consisting of arcs from ${\overrightarrow{\Gamma}}_F$ (oriented towards to $t$).
In particular, ${\mathcal{T}}$ covers exactly the set of nodes that can reach $t$ by a path in ${\overrightarrow{\Gamma}}_F$. Hence, $s$ is not covered by ${\mathcal{T}}$ (by [Lemma \[lm:augment\]]{}).
Consider a <span style="font-variant:small-caps;">Move</span>$(v,v')$ request. We update $F$ as follows. If $sv \notin F$, then no change is necessary. Otherwise, we remove $sv$ from $F$ and also add arcs $sv'$ and $v'v$ to $F$. This way, $F$ remains balanced and ${\mathop{\rm val}\nolimits ( F )}$ is preserved.
Next, we describe the maintenance of ${\mathcal{T}}$. Adding an arbitrary edge $e$ to $\Gamma$ is simple. Recall that each edge in $\Gamma$ generates a pair of oppositely directed arcs in ${\overrightarrow{\Gamma}}$. Let $a = pq$ be one of these two arcs generated by $e$. Suppose $a$ is present in ${\overrightarrow{\Gamma}}_F$. If $a \in {\delta^{\rm in}}(V{\mathcal{T}})$ (i.e., $p$ is not reachable and $q$ is reachable) then add $a$ to ${\mathcal{T}}$. Next, continue growing ${\mathcal{T}}$ incrementally from $p$ by executing a depth-first search and stopping at nodes already covered by ${\mathcal{T}}$. This way, ${\mathcal{T}}$ is extended to a maximum directed tree rooted at $t$. In other cases ($a \notin {\delta^{\rm in}}(V{\mathcal{T}})$) arc $a$ is ignored.
Next consider deleting edge $sv$ from $G$. We argue that its removal cannot invalidate ${\mathcal{T}}$, that is, $sv$ does not generate an arc from ${\mathcal{T}}$. This is true since $t$ is not reachable from $s$ and, hence, arcs incident to $s$ may not appear in ${\mathcal{T}}$.
Note that a *breakthrough* may occur during the above incremental procedure, i.e. node $t$ may become reachable from $s$ at some point. When this happens, we trace the corresponding $s$–$t$ path in ${\mathcal{T}}$, augment $F$ according to [Lemma \[lm:augment\]]{}, and recompute ${\mathcal{T}}$ from scratch. Again, any further activity stops once ${\mathop{\rm val}\nolimits ( F )}$ reaches $r$.
To estimate the complexity, note that between breakthroughs we are actually dealing with a single suspendable depth-first traversal of ${\overrightarrow{\Gamma}}_F$. Each such traversal costs $O(m)$ time and there are at most $r$ breakthroughs. Hence, the total bound of $O(rm)$ follows.
We apply the above data structure to graph $G_k(U_k^*)$ for $r = d_H(U^*)$ and make the moves in the reverse order, as explained earlier. Once <span style="font-variant:small-caps;">Query</span> reports the existence of $r$ edge-disjoint $s^*$–$t^*$ paths in $G_i(U_i^*)$, put $j(U^*) := i$ and proceed to the next signature. This way, each value $j(U^*)$ can be computed in $O(m)$ time. There are $O(1)$ signatures and $r = O(1)$, hence computing $j$ takes linear time as well.
Dealing with a Critical Instance
--------------------------------
The final stage deals with problem instance $(G_j, H_j)$. For brevity, we reset $G := G_j$, $H := H_j$ and also denote $G' := G_{j+1}$, $H' := H_{j+1}$. Consider the connected components of $G$. Components not containing any terminals may be dropped. If $G$ contains at least two components with terminals, the problem reduces to simpler cases described in [Subsection \[subsec:feasibility\]]{}. Hence, one can assume that $G$ is connected. We prove that $(G,H)$ is, in a sense, *critical*, that is, it admits a cut of a very special structure.
\[lm:critical\_cut\] There exists a subset $U \subseteq VG$ such that $d_G(U) = d_H(U) = 2$, $G[U]$ is connected and ${\left| U \cap T \right|} = 2$ (see [Fig. \[fig:critical\]]{}(b)).
The following is true:
\[eq:step\_cond\] For problem instances $(G,H)$ and $(G',H')$
- $(G,H)$ is feasible,
- $(G',H')$ is obtained from $(G,H)$ by a single local move,
- $(G',H')$ is infeasible,
- let $s \in VG'$ be the new location of the moved terminal, $st \in EH'$, then $s$ and $t$ are in the same connected component of $G'$.
Properties (i)–(iii) are ensured by the choice of $j$. Property (iv) holds since there exists a remaining (untraversed) part of the initial $s_1$–$t_1$ path in the original graph $G$.
We apply a number of contractions to $(G,H)$ and $(G',H')$ that preserve condition [(\[eq:step\_cond\])]{}. Suppose the following:
\[eq:bad\_bridge\] there is a subset $U \subseteq VG$ such that $d_G(U) = d_H(U) = 1$ and ${\left| U \cap T \right|} = 1$.
In other words, there is a *bridge* $e = uv \in EG$, $u \in U$, $v \in VG - U$ (an edge whose removal increases the number of connected components) that separates $G$ into parts $G[U]$ and $G[VG - U]$ and the former part contains a single terminal, say $s$.
We argue that the local move, which produced $(G',H')$, was carried out in the subgraph $G[VG - U]$ (but not in $G[U]$ or along the edge $e$).
Firstly, the move could not have been applied to $s$. Suppose the contrary. Terminal $s$ is connected to node $v$ by some path in $G[U \cup {\left \{ v \right \}}]$ and this property remains true even if apply a local move to $s$. (Nodes $v$ and $s$ are the only odd vertices in $G[U \cup {\left \{ v \right \}}]$, hence, these nodes cannot fall into distinct connected components after the move.) Therefore, $(G,H)$ and $(G',H')$ are simultaneously feasible or infeasible.
Next, suppose that $v$ is a terminal and the move is carried out along the bridge $e$. Then, $vs \notin EH'$ (otherwise, $(G',H')$ remains feasible). Therefore, $vw \in EH'$ for some $w \in VG - U$. Then $v$ and $w$ belong to different connected components of $G'$ after the move, which is impossible by [(\[eq:step\_cond\])]{}(iv).
Contract the set $U \cup {\left \{ v \right \}}$ in instances $(G, H)$ and $(G', H')$ thus producing instances $({\overline}G, {\overline}H)$ and $({\overline}G', {\overline}H')$, respectively. The above contraction preserves feasibility, hence $({\overline}G, {\overline}H)$ is feasible and $({\overline}G', {\overline}H')$ is infeasible. Moreover, the latter instance is obtained from the former one by a local move. Property [(\[eq:step\_cond\])]{} is preserved.
We proceed with these contractions until no subset $U$ obeying [(\[eq:bad\_bridge\])]{} remains. Next, since $(G',H')$ is infeasible by [Theorem \[th:feasibility\_criterion\]]{} there exists a subset $U \subseteq VG$ such that $d_{G'}(U) < d_{H'}(U)$. Eulerianess of $G' + H'$ implies that each cut in $G' + H'$ is crossed by an even number of edges, hence $d_{G'}(U) \equiv d_{H'}(U) \pmod{2}$. Therefore, $$\label{eq:bad_cut}
d_{G'}(U) \le d_{H'}(U) - 2.$$ At the same time, $(G,H)$ is feasible and hence $$\label{eq:good_cut}
d_G(U) \ge d_H(U).$$ Graph $G'$ is obtained from $G$ by removing a single edge. Similarly, $H'$ is obtained from $H$ by one edge removal and one edge insertion. Hence, $d_G(U)$ and $d_H(U)$ differ from $d_{G'}(U)$ and $d_{H'}(U)$ (respectively) by at most 1. Combining this with [(\[eq:bad\_cut\])]{} and [(\[eq:good\_cut\])]{}, one has $$d_{G'}(U) + 1 = d_G(U) = d_H(U) = d_{H'}(U) - 1.$$ So $d_H(U) \in {\left \{ 1,2 \right \}}$.
Suppose $d_H(U) = 1$. Subgraphs $G[U]$ and $G[VG - U]$ are connected (since otherwise $G$ is not connected). Also, ${\left| U \cap T \right|} = 3$ (otherwise, ${\left| U \cap T \right|} = 1$ or ${\left| (VG - U) \cap T \right|} = 1$ and [(\[eq:bad\_bridge\])]{} still holds). Therefore, Case 1 from [Fig. \[fig:critical\]]{}(a) applies (note that terminals $s_i$ and $t_i$ depicted there are appropriately renumbered). Let us explain, why this case is impossible. Graph $G'$ is obtained from $G$ by removing edge $uv$. Let, as in [(\[eq:step\_cond\])]{}(iv), $s$ denote the terminal in $(G,H)$ that is being moved and $t$ denote its “mate” terminal (i.e. $st \in EH$). We can assume by symmetry that $u = s$. Hence, $v$ is the new location of $s$ in $(G',H')$. By [(\[eq:step\_cond\])]{}(iv), $v$ and $t$ are in the same connected component of $G'$. The latter is only possible if $s = u = s_1$ and $t = t_1$. But then feasibility of $(G,H)$ implies that of $(G',H')$.
Finally, let $d_H(U) = 2$. Replacing $U$ by $VG - U$, if necessary, we may assume that ${\left| U \cap T \right|} = 2$, see [Fig. \[fig:critical\]]{}(b). It remains to prove that $G[U]$ is connected. Let us assume the contrary. Then, $U = U_1 \cup U_2$, $U_1 \cap U_2 = \emptyset$, $d_G(U_1) = d_H(U_1) = 1$, $d_G(U_2) = d_H(U_2) = 1$ (due to feasibility of $(G,H)$ and connectivity of $G$). Therefore, [(\[eq:bad\_bridge\])]{} still holds (both for $U := U_1$ and $U := U_2$) — a contradiction.
Once set $U$ is found, we undo the contractions described in the beginning and obtain a set $U$ for the original instance $(G,H)$. Clearly, these uncontractions preserve the required properties of $U$.
\[lm:building\_u\] Set $U$ figured in [Lemma \[lm:critical\_cut\]]{} can be constructed in $O(m)$ time.
We enumerate pairs of terminals $p, q \in T$ that might qualify for $U^* := U \cap T = {\left \{ p,q \right \}}$. Take all such pairs $U^* = {\left \{ p,q \right \}}$ except those forming an edge in $H$ ($pq \in EH$). Contract $U^*$ and $T - U^*$ into $s^*$ and $t^*$, respectively, in the graphs $G$ and $H$. The resulting graphs are denoted by $G^*$ and $H^*$. If a subset obeying [Lemma \[lm:critical\_cut\]]{} and having the signature $U^*$ exists then there must be an $s^*$–$t^*$ cut $U$ in $G^*$ such that $d_{G^*}(U) = 2$.
We try to construct $U$ by applying three iterations of the max-flow algorithm of Ford and Fulkerson, see [Subsection \[subsec:prelim\]]{}. If the third iteration succeeds, i.e. three edge-disjoint $s^*$–$t^*$ paths are found, then no desired cut $U$ having signature $U^*$ exists; we continue with the next choice of $U^*$. Otherwise, a subset $U \subseteq VG^*$ obeying $d_{G^*}(U) \le 2$ is constructed. Case $d_{G^*}(U) < 2 = d_{H^*}(U)$ is not possible due to feasibility of $(G,H)$.
Set $U$ is constructed for graph $G^*$ but may also be regarded as a subset of $VG$. We keep notation $U$ when referring to this subset.
Connectivity of $G[U]$ is the only remaining property we need to ensure. This is achieved by selecting an inclusion-wise maximal set $U$ among minimum-capacity cuts that separate ${\left \{ p,q \right \}}$ and $T - {\left \{ p,q \right \}}$. Such maximality can achieved in a standard way, i.e. by traversing the residual network in backward direction from the sink $t^*$, see [Lemma \[lm:min\_cut\]]{}.
To see that $G[U]$ is connected suppose the contrary. Then, as in the proof of [Lemma \[lm:critical\_cut\]]{}, let $U_1$ and $U_2$ be the node sets of the connected components of $G[U]$. Edges in $\delta_G(U) = {\left \{ e_1, e_2 \right \}}$ are bridges connecting $G[U_i]$ to the remaining part of graph $G$ (for $i = 1, 2$). Also, ${\left| U_1 \cap T \right|} = {\left| U_2 \cap T \right|} = 1$ (recall that $G$ is connected). Denote $U_1 \cap T = {\left \{ q_1 \right \}}$ and $U_2 \cap T = {\left \{ q_2 \right \}}$. Terminals $q_1$ and $q_2$ are not connected in $G[U]$. Since set $U$ is inclusion-wise maximal, any subset $U'$ satisfying [Lemma \[lm:critical\_cut\]]{} also obeys $U' \subseteq U$. But then $q_1$ and $q_2$ are also disconnected in $G[U']$, which is a contradiction. Therefore, no valid subset $U$ of signature $U^*$ obeying [Lemma \[lm:critical\_cut\]]{} exists.
In the algorithm, we check $G[U]$ for connectivity in $O(m)$ time. If the graph is not connected, then we proceed with the next signature $U^*$.
Now everything is in place to complete the proof of [Theorem \[th:main\]]{}. By [Lemma \[lm:building\_u\]]{}, finding set $U$ takes $O(m)$ time. It remains to construct a solution to $(G,H)$. Put $\delta_G(U) = {\left \{ e_1, e_2 \right \}}$, $e_i = u_iv_i$, $u_i \in U$, $v_i \in VG - U$, $i = 1, 2$. Again, after renaming some terminals we may assume that $s_1, s_2 \in U$, $t_1, t_2, s_3, t_3 \in VG - U$. Augment $G$ by adding two new nodes $s^*$ and $t^*$ and *auxiliary* edges $s^*u_1$, $s^*u_2$, $t_1t^*$, and $t_2t^*$. Due to feasibility of $(G,H)$, there exists (and can be constructed in $O(m)$ time) a collection of two edge-disjoint $s^*$–$t^*$ paths. After removing auxiliary edges we either obtain a $u_1$–$t_1$ path and a $u_2$–$t_2$ path (Case A) or a $u_1$–$t_2$ path and a $u_2$–$t_1$ path (Case B). To extend these paths to an $s_1$–$t_1$ path and an $s_2$–$t_2$ path we consider a four terminal instance in the subgraph $G[U]$. The demand graph is $(U, {\left \{ s_1u_1, s_2u_2 \right \}})$ in Case A and $(U, {\left \{ s_1u_2, s_2u_1 \right \}})$ in Case B. As $G[U]$ is connected, the latter instance is feasible by [Lemma \[lm:two\_pairs\]]{}. Therefore, we obtain edge-disjoint $s_1$–$t_1$ and $s_2$–$t_2$ paths $P_1$ and $P_2$, respectively. As explained earlier in [Section \[sec:intro\]]{}, the remaining path $P_3$ always exists and can be found in $O(m)$ time. Therefore, the proof of [Theorem \[th:main\]]{} is complete.
[^1]: Email: `max@adde.math.msu.su`. Supported by RFBR grant 09-01-00709-a.
[^2]: Email: `ignat1990@gmail.com`
[^3]: Email: `ilyaraz@gmail.com`
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abstract: 'Consequences of different discretizations of the two-dimensional Dirac operator on low energy properties (e.g., the number of nodes) and their relations to gauge properties are discussed. Breaking of the gauge invariance was suggested in a recent work by M. Bocquet, D. Serban, and M.R. Zirnbauer \[cond-mat/9910480\] in order to destroy an intermediate metallic phase of lattice Dirac fermions with random mass. It is explained that such a procedure is inconsistent with the underlying lattice physics. Previous results point out that the logarithmic growth of the slope of the average density of states with the system size, obtained in the field-theoretical calculation of M. Bocquet et al., could be a precursor for the appearence of an intermediate metallic phase.'
author:
- |
K. Ziegler\
Institut für Physik, Universität Augsburg\
D-86135 Augsburg, Germany\
title: '**Dirac Fermions on a Two-Dimensional Lattice and the Intermediate Metallic Phase**'
---
psbox.tex
Introduction
============
Two-dimensional (2D) Dirac (Majorana) fermions can be derived in statistical physics (2D Ising model [@dotsenko]) and in models of condensed matter systems (quasiparticles in $d$-wave superconductors [@hatsugai; @nersesyan], in the resonant valence bond state of the two-dimensional Heisenberg model [@pepin], or in the quantum Hall effect [@semenoff; @haldane; @hatsugai1; @ludwig]) always from lattice models. The relation of all these lattice models and 2D Dirac fermions is based on the agreement of their low-energy properties on certain length scales. Using continuous Dirac fermions, which have a linear dispersion, their $k=0$ node is identified with one of the nodes of the original lattice model. However, there are technical reasons to study a discrete (lattice) version of the Dirac operator because of problems related to the unrestricted spectrum of the continuum model. In the presence of randomness there is also the particular problem of correlations on a characteristic length scale which restricts the use of the unrealistic white noise randomness. In the renormalization group approach [@dotsenko] or in the saddle-point approximation [@ziegler2] it is sufficient to specify the lattice by a cutoff of the wavevector, assuming that only the $k=0$ node is important. For a more detailed discussion of the model one must define the lattice structure explicitly. The simplest case would be a nearest neighbor (NN) discretization of the lattice difference operator $$\nabla_jf(r)={1\over2}[f(r+e_j)-f(r-e_j)],
\label{nabla}$$ where $e_j$ is the lattice unit vector in direction $j$ on a square lattice. This choice, which was used in several papers by the author, exhibits an “intermediate phase”, characterized by a non-vanishing density of states (DOS), in the presence of a random mass for $|\langle m\rangle |\le m_c$. It turned out that this phase is metallic with finite conductivity [@ziegler4].
The discretization (\[nabla\]) was criticized in a recent work by Bocquet et al. [@bocquet] because of the associated sublattice symmetry. It was argued that the non-vanishing average DOS of this model [@ziegler4] is a sole consequence of the fact that it belongs to a special universality class, and that a different discretization would have a vanishing average DOS. Their argument is based on the renormalization group calculation of Ref. [@dotsenko] which gives a [*linearly vanishing*]{} DOS at $E=0$. (However, problems with this calculation exist because the RG has a strange behavior in which the slope of the DOS [*diverges*]{} logarithmically with the system size [@bocquet].) Since the non-vanishing DOS represents an intermediate metallic phase, observed and discussed for several models in Refs. [@ziegler2; @ziegler4; @kagalovsky; @senthil/marston; @senthil; @read], the change of the discretization would result in a destruction of this phase.
In the following, an interpretation of the 2D Dirac fermions with discretization (\[nabla\]) is given in terms of a tight-binding model with flux $\pi$. It will be shown that the additional sublattice symmetry is a special gauge symmetry. The destruction of the gauge symmetry by an additional next-nearest neighbor term in (\[nabla\]) leads to a lattice model with an inconsistent gauge field representation. More general, a change of the discretization affects physical quantities like the Hall conductivity which depends on the number of nodes. Finally, it will be shown, using a result from a previous calculation, that breaking of the sublattice symmetry does not neccessarily result in a vanishing average DOS at $E=0$. This indicates that even though the properties of the intermediate metallic phase vary with a change of the discretization, its very existence is rather stable.
Discussion of the Model
=======================
The 2D Dirac Hamiltonian is given as $$H=H_0+m\sigma_3,\ \ H_0=
i\nabla_1\sigma_1
+i\nabla_2\sigma_2$$ with Pauli matrices $\sigma_j$ and a random mass with $\langle m_r\rangle=0$. $\nabla_j$ can be any discrete lattice difference operator. It must be antisymmetric in order to have a Hermitean matrix $H$. For instance, it could be a NN difference operator (\[nabla\]) or a combination of a NN and a NNN difference operator. The former will be called “$\pi$ flux discretization” and the latter “flux breaking (FB) discretization” for reasons which are explained subsequently. The additional NNN difference operator adds new nodes to the dispersion of the pure system. To understand the difference of the discretizations one can consider the pure model, i.e. the Hamiltonian without the random mass. Then the dispersion for the $\pi$ flux discretization is $$E_\pi=\pm\sqrt{\sin^2 k_1+\sin^2 k_2}$$ which has nodes at $k_j=0,\pm\pi$. For the FB discretization the dispersion is $$E_{FB}=\pm\sqrt{[\sin k_1+\sin (2k_1)]^2+[\sin k_2+\sin (2k_2)]^2}.$$ with nodes at $k_j=0,\pm\pi,\arccos(-1/2)$. Additional nodes of the pure sytem can increase the DOS of the disordered system. This gives rise to the question whether all nodes contribute to the DOS at low energy or only some. On the other hand, the nodes with non-zero $k$ can be removed by adding a diagonal hopping term to the Hamiltonian. As it was shown in a previous study [@ziegler3] (and will be discussed in Sect. III), the $k=0$ node is sufficient for a non-zero average DOS at $E=0$. Thus the states with wavevector $k\approx 0$ appear with non-vanishing density.
Two sublattices can be constructed by considering two layers of the original square lattice, where the layers refer to the Dirac index $\alpha=1,2$. Then the sublattices are defined through the condition that for the site $(r,\alpha)$ either $r_1+r_2+\alpha$ is odd (sublattice $\Lambda_1$) or even (sublattice $\Lambda_2$). The Hamiltonian with $\pi$ flux discretization acts on these two sublattices separately: $$H=P_1HP_1+P_2HP_2,
\label{projection}$$ where $P_j$ is the projector on $\Lambda_j$. Adding next-nearest neighbor (NNN) terms to (\[nabla\]) results in the Hamiltonian $H_{FB}$ which couples both sublattices.
The transformation $H\to H S\sigma_3$ can be applied to the Hamiltonian with $$S_{rr'}=(-1)^{r_1+r_2}\delta_{rr'}.$$ $H$ commutes with $S\sigma_3$ [@ziegler3; @bocquet]. It changes the sign of $P_2HP_2$ but leaves $P_1HP_1$ unchanged. This means it is a gauge transformation on $\Lambda_2$.
It is important to notice that property (\[projection\]) does not mean that one can project on one of these sublattices and ignore the other projection because both sublattices are still statistically coupled through the same random mass. This can be seen in the product of the Green’s function with its complex conjugate $|(H+i\epsilon)^{-1}_{r\alpha,r'\alpha'}|^2$. It was shown [@ziegler4] that for $\alpha'=\alpha+1\ (mod\ 2)$ the symmetry of $H$ implies $$|(H+i\epsilon)^{-1}_{r\alpha,r'\alpha}|^2
=-(H+i\epsilon)^{-1}_{r\alpha,r'\alpha}
(H+i\epsilon)^{-1}_{r\alpha',r'\alpha'}$$ and $$|(H+i\epsilon)^{-1}_{r\alpha,r'\alpha'}|^2
=-(H+i\epsilon)^{-1}_{r\alpha,r'\alpha'}
(H+i\epsilon)^{-1}_{r\alpha',r'\alpha}.$$ Thus $|(H+i\epsilon)^{-1}_{r\alpha,r'\alpha'}|^2$ on one sublattice is the product of Green’s functions from [*both*]{} sublattices. The non-trivial properties of the two-particle Green’s function are a consequence of the Green’s functions on the two sublattices which share a common random mass.
The Hamiltonian $H_j=P_jHP_j$ describes a tight-binding model on the sublattice $\Lambda_j$ with hopping elements $$H_{r1,r\pm e_12}=H_{r2,r\pm e_11}=\pm i$$ $$H_{r1,r\pm e_22}=H_{r2,r\mp e_21}=\pm 1$$ which represents a Hamiltonian with flux $\pi$. Adding NNN terms to $\nabla_j$ changes this flux because the new terms have the same phase factors as the NN terms but on links which are twice as large as those of the NN hopping terms. This leads to an inconsistency in the model because particles would experience a magnetic field whose strength depends on the length of the hops. Moreover, the invariance property $[H,S\sigma_3]_-=0$, which is the gauge invariance on sublattice $\Lambda_2$, is violated by the NNN hopping term.
It is possible to add NNN hopping terms to $H$ which are consistent with the node structure of the $\pi$ flux discretization. The simplest is $$h_{nnn}=H_0^2=(i\nabla_1\sigma_1+i\nabla_2\sigma_2)^2.$$ It preserves the gauge field of the NN term. Another one is $$h_{nnn}'=H_0^2\sigma_3=(i\nabla_1\sigma_1+i\nabla_2\sigma_2)^2\sigma_3$$ which does not preserve the gauge field because it contributes a positive hopping element for $\alpha=1$ but a negative hopping element for $\alpha=2$. Nevertheless, both NNN terms obey the condition $$[h_{nnn},S\sigma_3]_-=[h_{nnn}',S\sigma_3]_-=0,
\label{gaugeinv}$$ but only $h_{nnn}'$ satisfies the defining symmetry of class D of Altland and Zirnbauer [@altland] $$h_{nnn}'=-\sigma_1{h_{nnn}'}^T\sigma_1.$$ Property (\[gaugeinv\]) can be used to estimate the non-vanishing average DOS at $E=0$, as demonstrated in Refs. [@ziegler1; @ziegler3].
The discussion of the DOS requires only the diagonal part of the one-particle Green’s function $G_{rr,aa}=(H\pm i\epsilon\sigma_0)^{-1}_{rr,aa}$, which can be represented by a sum over closed random walks [@glimm] beginning at $r$ and returning to it with the same Dirac index. Formally, the random walk representation can be obtained from the hopping expansion of the Green’s function $$(H_0+m\sigma_3+i\epsilon\sigma_0)^{-1}=(m\sigma_3+i\epsilon\sigma_0)^{-1}
\sum_{l\ge0}\Big[H_0(m\sigma_3+i\epsilon\sigma_0)^{-1}\Big]^l.$$ This expansion is convergent for sufficiently large $\epsilon$. After averaging over the random mass one can perform an analytic continuation to arbitrarily small $\epsilon>0$. The loops experience an effective gauge field because of complex hopping elements. An example is a simple plaquette (Fig. 1). In general the flux per plaquette is $\phi=\pi$, in units of the flux quantum $\phi_0$. An important property of the random walks follows from $$G(-m,i\epsilon)=\sigma_2G^T(m,i\epsilon)\sigma_2$$ because this means that the random walks are reversed by the change of the sign of the mass. An interpretation is that the currents in the model can be reversed by reversing the sign of the mass, which is related to the change of the sign of the Hall conductivity $\sigma_{xy}$ with $m$ [@semenoff].
Hall Conductivity
-----------------
A quantity which is sensitive to the type of discretization but robust against disorder is the Hall conductivity $\sigma_{xy}$. It can be measured as the linear response to an external gauge field [@semenoff]. In a pure system ($m=const.$) with Hamiltonian $H=m\sigma_3+h_1\sigma_1+h_2\sigma_2$ it reads [@ludwig] $$\sigma_{xy}={m\over2}
\int{1\over(m^2+h_1^2+h_2^2)^{3/2}}{d^2k\over(2\pi)^2}$$ in units of $e^2/\hbar$. For $m\sim 0$ only the nodes contribute significantly to the intergral: $$\sigma_{xy}(m)={m\over |m|}f(|m|),$$ where $f(0)\ne0$ is proportional to the number of nodes. In the case of the $\pi$ flux Dirac operator this gives just a Hall step for [*each*]{} sublattice which means a single Hall step for the corresponding $\pi$ flux tight-binding model. For the modified Hamiltonian $H_0+m\sigma_3+h_{nnn}'$ the Hall conductivity is $${1\over2}\int{m+\sin^2k_1+\sin^2 k_2
\over[(m+\sin^2k_1+\sin^2 k_2)^2+\sin^2k_1+\sin^2 k_2]^{3/2}}
{d^2k\over(2\pi)^2}$$ which gives $$\sigma_{xy}\sim const.+{1\over2\pi}{m\over |m|}\ \ \ (m\sim0).$$ Thus, the additional term $h_{nnn}'$ contributes a constant Hall conductivity at small values of $m$. These results indicate that approximations of the original model should not change the number of nodes nor violate the structure of the Hamiltonian by adding new terms of the type $h_{nnn}'$ in order to get the correct Hall conductivity.
A general rule is that all nodes of the lattice model should be included in the calculation of the low energy properties. This has severe consequence for the renormaliztion group (RG) calculation which usually deals only with one large length scale. The various nodes (different length scales) will create additional couplings under the RG transformation, leading eventually to a strong coupling behavior.\
Model with single node
======================
To study the contribution of the $k=0$ node to the average DOS, all nodes of the $\pi$ flux discretization must be removed except for that at $k=0$. This can be achieved by adding a diagonal term to $H$ $$H_1=H+(\Delta-2)\sigma_3$$ with $2\Delta f(r)=f(r+e_1)+f(r-e_1)+f(r+e_2)+f(r-e_2)$. Obviously, this Hamiltonian violates the sublattice gauge-invariance condition: $$\Delta\sigma_3S=-\sigma_3S\Delta.$$ Without randomness (i.e. $m=0$) the dispersion is $$E_1(k_1,k_2)=\pm\sqrt{(\cos k_1+\cos k_2-2)^2+\sin^2k_1+\sin^2k_2}$$ which is non-zero except for the node at $k_j=0$. Now this Hamiltonian can be coupled by a random field $m'$ to the Hamiltonian with nodes at $\pm\pi$ $$H_2=H-(\Delta+2)\sigma_3$$ as $${\hat H}=\pmatrix{
H_1 & m'\sigma_3 \cr
m'\sigma_3 & H_2 \cr
}.$$ Both Hamiltonians $H_1$ and $H_2$ violate the sublattice gauge-invariance condition (\[gaugeinv\]) but preserve the symmetry of class D $${\hat H}=-\sigma_1{\hat H}^T\sigma_1.$$ The Green’s function can be projected on the subspace of $H_1$ $$[H_1+i\epsilon-m'\sigma_3(H_2+i\epsilon)^{-1}\sigma_3m']^{-1}$$ It was shown that this expression leads to a non-vanishing average DOS at $E=0$ [@ziegler3], indicating that the projection on one node produces already a non-vanishing average DOS.
As a remark it should be noted that the pure 2D Ising model at the critical point is governed by the Hamiltonian [@ziegler2] $$H_{2DIM}=
a(1+a^2+2a\cos k_1-2\cos k_2)\sigma_3+2a(\sin k_1\sigma_1+\sin k_2\sigma_2)$$ with $a=\sqrt2 -1$. This Hamiltonian is similar to $H_1$ as it has only one node at $k=0$.\
Conclusions
===========
The previous discussion is based on the one-particle Green’s function without referring to an effective field theory. This approach is motivated by the aim to avoid additional symmetries which appear by the introduction of new fields and which are not related to the original Hamiltonian or the one-particle Green’s function. In the case of the Hamiltonian $H$, regardless of the discretization of $\nabla_j$, there are two [*discrete*]{} symmetries: $$H\to -\sigma_3 H \sigma_3
\label{symm1}$$ for the ensemble and, therefore, for the [*average*]{} one-particle Green’s function. Moreover, there is a symmetry under the discrete transformation $$H\to -\sigma_1 H^T \sigma_1,
\label{symm2}$$ which defines class D of Ref. [@altland]. It holds for each realization of the model with Hamiltonian $H$. $h_{nnn}$ and $h_{nnn}'$, both break symmetry (\[symm1\]) whereas only $h_{nnn}$ breaks symmetry (\[symm2\]).
From the symmetry point of view alone it is not entirely clear under which conditions a vanishing average DOS at $E=0$ exist. The competition of different nodes (i.e. different length scales) cannot be described only by global symmetries but requires more detailed knowledge. However, there is a simple argument in terms of the hopping expansion which indicates that the number of [*independent*]{} random terms in the Dirac Hamiltonian $H$ is crucial for the behavior of the average DOS around $E=0$. Taking the zero-dimensional limit of $H$, i.e. the leading order of the hopping expansion, there is a power law $$\langle \rho(E)\rangle\sim \rho_0|E|^\alpha,$$ where $\alpha=0$ (random mass) and $\alpha=1$ (two-component random vector potential). The latter, of course, violates also the symmetry (\[symm2\]).
The non-vanishing DOS for the class D model in $d=2$ with a diffusive behavior was recently discussed by Senthil and Fisher [@senthil] and Read and Green [@read] in terms of the RG flow of a non-linear sigma model. Although this approach was criticized by Bocquet et al. [@bocquet], its result is in agreement with that obtained with a different approach [@ziegler4]. The appearence of the intermediate phase is quite natural for the 2D Dirac fermions, since the pure model has a singular “metallic” phase at $E=m=0$ with conductivity $e^2/h\pi$. This phase is robust against a random vector potential, where the value of the conductivity remains unchanged [@ludwig]. A random mass has apparently a stronger effect because it reduces the conductivity at $m=0$ by a factor $1/(1+g/\pi)$, where $g$ is the variance of the random mass [@ziegler4]. Moreover, it broadens the singular phase at $m=0$ to an interval $-m_c<m<m_c$ with non-vanishing DOS. In terms of the random bond Ising model the non-vanishing DOS reflects the existence of the Griffiths-McCoy-Wu phase, which cannot be seen in the perturbative RG approach [@mccoy].
The occurence of vortex-like excitations might be an important effect, as suggested in Refs. [@bocquet; @senthil; @read]. Using the $\pi$ flux discretization, these vortices can be created by local edge currents in areas where the sign of the random mass changes: An area with a positive mass has a positive Hall conductivity whereas the surrounding area with negative mass has a Hall conductivity with opposite sign. The resulting edge currents can have a long-range interaction. An effective model for this behavior can be found in terms of the $Q$ matrix field theory of Ref. [@ziegler4], in which the random mass is replaced by a matrix field. Details will be published separately.
In conclusion, the change of the discretization has a strong effect on the node structure of the Dirac Hamiltonian. The correct discretization is determined by the effective gauge field which the Dirac fermions experience. In the case of Dirac fermions with random mass, however, the average DOS at low energies is relatively robust against the change of the discretization. In particular, the $k\approx 0$ modes have a substantial contribution to the average DOS. This indicates that also the intermediate metallic phase of the Dirac fermions with random mass and $\pi$ flux discretization should be robust under a change of the discretization.\
Acknowledgement
The author would like to acknowledge a communication with M.R. Zirnbauer about his work in Ref. [@bocquet]. He is grateful to D. Braak for interesting discussions. This work was supported by the Sonderforschungsbereich 484.
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---
abstract: '[**The ability to modify light-matter coupling in time (e.g. using external pulses) opens up the exciting possibility of generating and probing new aspects of quantum correlations in many-body light-matter systems. Here we study the impact of such a pulsed coupling on the light-matter entanglement in the Dicke model as well as the respective subsystem quantum dynamics. Our dynamical many-body analysis exploits the natural partition between the radiation and matter degrees of freedom, allowing us to explore time-dependent intra-subsystem quantum correlations by means of squeezing parameters, and the inter-subsystem Schmidt gap for different pulse duration (i.e. ramping velocity) regimes – from the near adiabatic to the sudden quench limits. Our results reveal that both types of quantities indicate the emergence of the superradiant phase when crossing the quantum critical point. In addition, at the end of the pulse light and matter remain entangled even though they become uncoupled, which could be exploited to generate entangled states in non-interacting systems.**]{}'
address:
- '$^1$ Departamento de F[í]{}sica, Universidad de los Andes, A.A. 4976, Bogot[á]{} D. C., Colombia'
- '$^2$ Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom'
- '$^3$ JILA, University of Colorado, Boulder, CO 80309, U.S.A.'
- '$^4$ Department of Physics, University of Miami, Coral Gables, FL 33124, U.S.A.'
author:
- 'F. J. Gómez-Ruiz$^1$, J. J. Mendoza-Arenas$^{1,2}$, O. L. Acevedo$^3$, F. J. Rodríguez$^1$, L. Quiroga$^1$ and N. F. Johnson$^4$'
bibliography:
- '\\jobname.bib'
title: 'Dynamics of Entanglement and the Schmidt Gap in a Driven Light-Matter System'
---
[*Keywords*]{}: Non-Equilibrium Dicke Model, Schmidt Gap, Entanglement, Squeezing.\
\
Introduction
============
The understanding, characterization and manipulation of non-equilibrium correlated many-body systems has benefitted from several remarkable experimental and theoretical advances in recent years [@Nori_RMP2014; @Zoller_NJP2011]. Although, by definition, any laboratory sample will necessarily interact with its laboratory environment [@breuer], modern technologies have succeeded in isolating quantum systems to a significant degree within a large variety of experimental settings [@Lloyd_SC96; @Schneider_RP2012; @Houck_Nat2012]. Many of these realizations can be regarded as particular cases of an interaction between matter and radiation, or some other form of bosonic excitation field. From a theoretical point of view, many of these systems can be modeled to a reasonable approximation by considering the matter subsystem as two-level systems (qubits) and the radiation subsystem as a set of independent harmonic oscillators. Examples of such modeling include cavity Quantum Electrodynamics (QED) [@nori2011nature; @xiang2013rmp] and circuit QED [@niemczyk2010nature; @peterson2012nature], impurities immersed in Bose-Einstein condensates (BECs) [@ng2008pra; @haikka2011pra; @sabin2014scirep], and artificial atoms of semiconductor heterostructures interacting with light [@Sarah_IOP2012] or with plasmonic excitations [@Dzsotjan2010prb]. Since these systems contain various degrees of freedom, their theoretical study has been traditionally approached using approximate perturbative methods [@breuer].
Most of the theoretical treatments to date rely on the assumption that the matter-radiation interaction is static, and either very weak or very strong. However from an empirical perspective, these regimes do not represent any technological boundary – indeed, the coupling strength in real systems is quite likely to be in between these limits. The potential richness of effects in this intermediate case [*and*]{} in the regime of non-static coupling, is therefore of significant interest for temporal quantum control in practical quantum information processing and quantum computation. On a more fundamental level, an open-dynamics quantum simulator would be invaluable for shedding new light on core issues at the foundations of physics, ranging from the quantum-to-classical transition and quantum measurement theory [@Zurek1] to the characterization of Markovian and non-Markovian systems [@Bruer_PRL; @PRBluis; @Cosco2017arxiv].
In our work we explore this dynamical regime which is opened up by manipulating the strength of the light-matter coupling in time – for example using external pulses that generate a coupling that cycles from weak to strong and back again. Specifically, we use a general, time-dependent many-body Hamiltonian, namely the Dicke model, to study the impact of a single pulse in the light-matter coupling, on the quantum correlations at the collective and subsystem levels. Exploiting the natural partition between the radiation and matter degrees of freedom, we explore the time-dependent squeezing parameters of each subsystem, and the entanglement spectrum through the Schmidt gap, for different pulse duration (i.e. ramping velocity) regimes, ranging from the near-adiabatic to the sudden quench limits. The results show that both the inter-subsystem [*and*]{} and intra-subsystem quantum correlations signal the emergence of the superradiant phase. In addition, in the intermediate ramping regimen, both subsystems remain entangled at the end of the applied pulse, which should be of interest for quantum control schemes.
The paper is organized as follows. In Section \[sect\_DM\] we describe the time-dependent light-matter model that we analyze. In Section \[sect\_results\] we present our main results for the driven dynamics, starting with the coherence and squeezing of the light and matter subsystems, and then considering the Schmidt gap for the light-matter bipartition. We provide a discussion of our results in Section \[sect\_discussion\]. Finally we present our conclusions in Section \[sect\_conclu\].
Model and methods {#sect_DM}
=================
The time-dependent light-matter system that we choose for our study is the non-equilibrium Dicke Model (NE-DM) [@Acevedo2014PRL], which consists of a set of qubits coupled to a single radiation mode. Its Hamiltonian is given by $\left(\hbar=1\right)$ $$\label{Hdicke}
\hat{H}{\left( {\lambda}{\left( t \right)} \right)}=\epsilon \hat{J}_{z} + \omega \hat{a}^{\dagger}\hat{a} +\frac{2\lambda(t)}{\sqrt{N}}\hat{J}_{x}\left(\hat{a}^{\dagger}+\hat{a}\right).$$ Here $N$ is the number of qubits, $\hat{J}_{\alpha}=\frac{1}{2}\sum_{i=1}^{N}{{\sigma}}_{\alpha}^{{\left( i \right)}}$ $\left(\alpha=x,z\right)$ denote collective angular momentum operators of the qubits acting on the totally symmetric manifold (known as the [*Dicke manifold*]{}), $\hat{a}^{\dag} {\left( \hat{a} \right)}$ is the creation (annihilation) operator of the radiation field, $\epsilon$ and $\omega$ represent the qubit and field transition frequencies respectively, and $\lambda{\left( t \right)}$ is the time-dependent coupling between matter and light subsystems. For all the numerical results in this paper, we consider the resonant case between the qubits and the radiation frequency, and set the energy scale by taking $\epsilon=\omega=1$. Though we focus on light-matter systems, other realizations are possible – including those where the bosonic mode corresponds to vibrational degrees of freedom.
The static properties of Dicke Model have been widely studied and characterized in the last two decades [@Nagy2010prl; @Das2016njp]. It is well known that, in the thermodynamic limit $N\to\infty$, it exhibits a second-order quantum phase transition (QPT) [@Hioe_PRA1973] at ${\lambda}_{c}=\sqrt{{\epsilon}{\omega}}/2$ with order parameter $\hat{a}^{\dag}\hat{a}/J$, separating the normal phase at ${\lambda}_{c}<\sqrt{{\epsilon}{\omega}}/2$ from the superradiant phase in which there is a finite value of the macroscopic order parameter, e.g. finite boson expectation number [@Acevedo2014PRL]. Now we discuss its dynamical properties under the time-dependent model in Eq. \[Hdicke\]. We obtain the full NE-DM instantaneous state ${\left\vert \psi{\left( t \right)} \right\rangle}$ by numerically solving the time-dependent Schrödinger equation. Our numerical solution of the NE-DM profits from the fact that the operator $\hat{{\bf J}}^{2}=\sum_{\alpha}\hat{J}_{\alpha}^{2}$ is a constant of motion with eigenvalue $J\left(J+1\right)$, and that the parity operator $\hat{\mathcal{P}}=\exp\left(\imath \pi\left[\hat{a}^{\dag}\hat{a} +{{J}}_{z}+J\right]\right)$ is also conserved and commutes with $\hat{{\bf J}}^{2}$. Since we are seeking results that have general validity, we avoid making the rotating-wave approximation that is commonly used to solve the static version of the NE-DM and which makes it Bethe ansatz integrable [@Gaudin_JPF1976]. The general structure of the state ${\left\vert \psi{\left( t \right)} \right\rangle}$ at any time $t$ is given by $$\label{state}
{\left\vert \psi{\left( t \right)} \right\rangle}=\sum_{m_{z}=-N/2}^{N/2}\sum_{n=0}^{\chi}C_{n,m_{z}}{\left( t \right)}{\left\vert m_{z},n \right\rangle}.$$ Here $\chi$ is the truncation parameter of the size of the bosonic Fock space, whose value we choose to be large enough to ensure that the numerical results converge [@Acevedo2014PRL]. The basis states ${\left\vert m_{z},n \right\rangle}={\left\vert m_{z} \right\rangle}\otimes{\left\vert n \right\rangle}$ are defined such that ${\left\vert m_{z} \right\rangle}$ is an eigenvector of $J_{z}$ in the subspace of even parity with eigenvalue $m_z$, and ${\left\vert n \right\rangle}$ is a bosonic Fock state with occupation $n$. The initial state of the dynamics at $t=0$, with negligible light-matter coupling $\lambda(t)=0$, is the non-interacting ground state ${\left\vert \psi(0) \right\rangle}=\bigotimes_{i=1}^{N}{\left\vert \downarrow \right\rangle}\otimes{\left\vert n=0 \right\rangle}={\left\vert -\frac{N}{2},0 \right\rangle}$, where both the matter and light subsystems have zero excitations. All qubits are polarized in the state with $\langle\sigma_z\rangle=-1$, and the field is in the Fock state of zero photons. Since the total angular momentum and parity are conserved quantities, we can without loss of generality restrict our study to the maximum angular momentum sector $J=N/2$ and $\mathcal{P}=1$.
We obtain the time-evolution of the system in response to an up-down pulse in $\lambda(t)$, which is chosen so that the system dynamically crosses the QPT on both the up and down portion of the $\lambda(t)$ pulse cycle. For simplicity and considering the successful and actual experimental platform for driving light-matter interaction in Dicke Model [@ExpDicke1; @ExpDicke2; @ExpDicke3], we consider $\lambda(t)$ to rise and fall linearly in time $t$ during the pulse, i.e. it has the form $\upsilon t$, and thus establishes a triangular ramping of the light-matter interaction. The rate $\upsilon=\frac{d \lambda}{d t}$ acts as a control parameter and is known as the [*annealing velocity*]{}, which is characterized by a finite time $\tau$ such that $\upsilon=1/\tau$. The particular choice of $\lambda(t)$, implies that the quantum critical point is crossed twice during the cycle, first when $t = \tau/2$, and second when $t = 3\tau/2.$
Several experimental realizations of the DM have been discussed during the past few decades, with many being built around implementations in circuit QED where superconducting qubits play the role of the matter subsystem [@ExpDicke1; @ExpDicke2; @ExpDicke3]. Additionally to date, important experimental scenarios have shown efficient and effective ways to simulate radiation-matter interaction systems with time-varying couplings. Some of the most promising experimental possibilities are thermal gases of atoms [@BlackPLR], and BECs using momentum [@ExpDicke4] and hyperfine states [@BaumannPRL]. However, we believe that the branch of experiments deserving most attention is that demonstrating DM superradiance in ultra-cold atom optical traps, especially $^{87}$Rb Bose-Einstein condensates [@Klinder_PNA2015; @ExpDicke4; @ExpDicke5; @ExpDicke6]. Indeed, these atom-trap experiments are so promising that a brief description of them is pertinent here, following Ref. [@Klinder_PNA2015]. Figure \[fig\_1\] depicts a schematic of the main components of the atom-trap DM realization. It corresponds to an ultracold cloud of $N\sim 10^{5}$ $^{87}$Rb atoms confined by a magneto-optical trap inside a high finesse Fabri-Perot cavity. The cloud is driven by a transverse pump laser whose wavelength is the same as that of the fundamental mode of the cavity. The combined cavity and pump laser setting produces an optical lattice potential that affects the motion of the atoms in the cloud through coupling with far-detuned atomic resonances. This coupling causes the atoms to interact with each other through the mediating presence of the radiation mode. At low intensity pump power $\epsilon_{p}$, the BEC remains in its (almost spatially uniform) translational ground state. However, when a critical value of $\epsilon_p$ is reached, the ground state becomes a grid-like matter wave as the one shown in Fig \[fig\_1\]. This change of configuration constitutes the QPT whose spontaneous symmetry breaking is caused by the fact that two matter wave configurations, which are distinguishable only by a phase difference of $\pi$, have the same lowest possible energy. The effective two-level (qubit) system is composed of the ground BEC translational state and the fundamental grid-like matter wave state for each atom. There are several ways to monitor the system. The two most fundamental are (1) addressing the radiation field by coupling one of the (unavoidably) leaky walls of the cavity to a detector, and (2) using time-of-flight methods to measure the matter wave modes. Note that each technique measures the state of one of the two main components of the DM, i.e. light and matter respectively.
![(Color online) Schematic of recent successful realizations of the Dicke Model in a BEC of $^{87}$Rb atoms confined by a magneto-optical trap. The atoms (qubits) are made to interact with the field mode of a high finesse cavity by means of a pumped transverse field. [@Klinder_PNA2015] []{data-label="fig_1"}](Figure1.pdf)
Results {#sect_results}
=======
We now proceed to characterize the complete dynamical QPT profile by focusing on properties of each subsystem, namely the matter subsystem composed by the all-to-all (qubit) spin network, and the radiation mode subsystem. We analyze a wide range of annealing velocities $v$, and use a logarithmic scale for showing these values of the velocity, defined by $\Gamma=\log_2(v)$. This range varies from the slow near adiabatic regime, through the intermediate regime, to the fast sudden-quench regime. In previous papers, we showed how the values of the velocity for which the change of regime is manifested depend on system size [@Gomez_Ent2016; @Acevedo2014PRL; @Acevedo2015NJP; @AcevedoPRA2015]. Here, on the other hand, we take a fixed size of the qubit subsystem, namely $N=81$, for all the numerical calculations.
Coherence and Squeezing
-----------------------
We begin our discussion of the driven dynamics of the Dicke model by considering the diagonal elements of the reduced density matrices of the bipartition. Figure \[fig\_2\] shows the instantaneous projection of the reduced density matrix for the matter and radiation subsystems over the $J_z$ and Fock basis respectively, for several values of the annealing velocity. For the slowest ramping ($\Gamma=-7.76$), both the radiation mode and qubits remain entirely unexcited before crossing the quantum critical point $\lambda_c$ when increasing $\lambda$, with only the respective states of $n=0$ and $m_z=-N/2$ being populated. After crossing $\lambda_c$ the population is transferred to states of larger $n$ and $m_z$, a process that continues up to the time where $\lambda$ starts decreasing. With the reversal of $\lambda$ the population of large values of $m_z$ and $n$ is transferred back to lower values, in a highly-symmetrical form with respect to the turning point. When $\lambda_c$ is crossed again, both the radiation field and the set of qubits become almost completely unexcited, with only the lowest values of $m_z$ and $n$ being populated. Since the reversal of the matter and light dynamics is not completely achieved, this corresponds to a near-adiabatic regime instead of a true adiabatic one (e.g. see results on Section \[gap\_section\] for lower velocities).
![(Color online) Projection of reduced density matrix for each subsystem. Upper panels: $\langle n | \rho_{B}| n\rangle$. Lower panels: $\langle m_z | \rho_{Q}| m_z\rangle$. The values of velocities are, from left to right: $\Gamma=-7.76,\, -5.76,\, -4.76,\, -2.76$. The color scale was adjusted to improve visualization of the results.[]{data-label="fig_2"}](Figure2.pdf)
For larger annealing velocities, the dynamic population of states with $m_z>-N/2$ and $n>0$ remains qualitatively similar to that of the near-adiabatic limit during the linear increase of $\lambda$. However two main qualitative differences are observed. First, this population transfer occurs further and further away from $\lambda_c$ as $v$ increases, indicating that the ground-state QPT is not being immediately captured. Second, larger values of $n$ and $m_z$ are reached, since a faster ramping velocity provides a stronger excitation to the system. On the other hand, the population dynamics of the $\lambda$ reversal regime is very different to that close to adiabaticity. Even though the population is also transferred back to states of lower quantum numbers, the symmetry with respect to the turning point is lost, and at the end of the dynamics, when the matter and radiation become uncoupled, they are still highly excited. This already indicates that for large annealing velocities, the system gets so excited that it does not simply follow the decrease of $\lambda$, which is of course only expected in the adiabatic limit. Similar asymmetric results are found in the squeezing and entanglement spectrum results shown below.
Now we describe the squeezing parameter for both subsystems, starting with the light degrees of freedom. The squeezing of light states has widely been studied in the literature. A squeezed state of light arises in a simple quantum model comprising non-linear optical processes such as optical parametric oscillation and four-wave mixing. The fundamental importance of the squeezed state is characterized by the property that the variance of the quadrature operator $\hat{x}$ is less than the value $1/2$ associated with the vacuum and coherent state. The squeezing parameter in the field mode $\xi_{B}^{2}$ is expressed in terms of the variance (Var) and covariance (Cov) of the field quadratures as [@Walls] $$\label{Sbos}
\xi_{B}^{2} = {\rm Var}{\left( \hat{x} \right)} + {\rm Var}{\left( \hat{p} \right)} - \sqrt{{\left( {\rm Var}{\left( \hat{x} \right)}-{\rm Var}{\left( \hat{p} \right)} \right)}^2 + 4 {\rm Cov}{\left( \hat{x},\hat{p} \right)}^2}.$$ In the left panel of Figure \[fig\_3\], we present a novel way to generate a photon squeezed state. At $t=0$ the quantum cavity starts in the vacuum state $|n=0\rangle$. As before, the radiation-matter parameter varies as a simple linear up-down pulse, forming a triangular ramping. Our results show the existence of a specific regime of annealing velocities such that while the pulse is applied, the photon squeezing tends to increase (besides small oscillations) even after the reversal ramping of $\lambda(t)$ has started. Furthermore, we note that for this velocity regime, the final state of light has high squeezing when the final radiation-matter parameter is zero.
![(Color online) Dynamic profiles of the matter subsystem in which, for fixed annealing velocities, time varies according to the direction of arrows. Left panel: Evolution $1-\xi_{B}^{2}$, as defined in Eq. \[Sbos\], whenever it is greater that zero (squeezed radiation). Right: Two qubit concurrence $c_{w}{\left( N-1 \right)}=1-\xi_{Q}^{2}$. []{data-label="fig_3"}](Figure3.pdf)
Now we discuss the dynamics of spin squeezing, which has also been the object of intense research in the past few decades. For example, the natural idea of transferring squeezing from light to atoms has been attracting attention both theoretically and experimentally. The notion of spin squeezing has arisen mainly from two considerations: the study of particle correlations and entanglement [@Wang_PRA2003; @Zoller; @Nick], as well as the improvement of measurement precision in experiments [@Wineland]. The experimental proposals for transferring squeezing from light to atoms include placing the latter in a high-Q cavity so they interact repeatedly with a single-field (not squeezed) mode [@Ueda], and illuminating bichromatic light on atoms in a bad cavity [@Klaus]. The intrinsic spin squeezing in a large atomic radiating system was studied in Ref. [@yukalov], where spin-squeezed states were generated by means of strong interatomic correlations induced by photon-exchange. Spin squeezing can also be produced via a squeezing exchange between motional and internal degrees of freedom of atoms [@Saito]. For a detailed review, we refer to Ref. [@Jian].
The definition of spin-squeezing is not unique [@Jian]. For our propose we use the definition given in Ref. [@Wang_PRA2003], in which a relation between entanglement for a two-qubit subsystem as measured by the Wootters concurrence $c_{w}$ [@wootters] and the spin squeezing parameter $\xi_{Q}$ was established, namely $$\label{squee_concu}
\xi_{Q}^{2}=1-{\left( N-1 \right)}c_{w}.$$ Since each qubit is equally entangled with each other, the monogamic character of entanglement is manifested in Eq. \[squee\_concu\] by the $N-1$ factor.
In the right panel of Fig. \[fig\_3\] we show the spin squeezing for a wide range of velocities. We find a regime of intermediate annealing velocities for which the squeezing is large at the end of the pulse, which coincides with the velocity regime for which the photonic squeezing is magnified. In previous works by some of us, we showed that the intermediate velocity regime allows for the generation of entanglement [@Gomez_Ent2016; @Acevedo2015NJP]; this is manifested in the generation of squeezing in both light and matter. A fundamental and novel feature of our results is that there is no need of ultra-strong coupling to have squeezing in both light and matter. In addition, we note that the squeezing after the pulse widely exceeds the values that would be achieved through a near-adiabatic evolution.
Schmidt gap {#gap_section}
-----------
The observation several years ago of the fundamental role of entanglement on quantum criticality led to intense research on characterizing QPTs by means of different measures such as entanglement entropy and concurrence [@amico2002nature; @gu2003pra; @latorre2003prl; @wu2004prl; @laflorencie2006prl; @zanardi2006njp; @buonsante2007prl; @amico2008rmp; @jjma2010pra; @pino2012pra; @hofmann2014prb]. Shortly after, it was shown that the entanglement spectrum, i.e. the set of eigenvalues of the reduced density matrix of one subsystem resulting from a bipartition, provides valuable information on the properties of topological phases [@haldane2008prl], and remarkably even more than the entanglement entropy. Since then, several works have analyzed the behavior of the entanglement spectrum, and in particular of the Schmidt gap (the difference between the two largest eigenvalues) close to criticality for different scenarios. These include zero-temperature QPTs [@chiara2012prl; @lepori2013prb; @bayat2013nat], where the Schmidt gap has been suggested as an order parameter, many-body localization [@Gray2017arxiv], and dynamical crossings of QPTs at different speeds [@canovi2014prb; @torlai2014jstat; @Qijun2015prb; @francuz2016prb; @coulamy2017pre]. The latter situation, corresponding to our point of interest in the present work, has been mostly studied for quantum spin chains. Now we discuss the dynamics of the Schmidt gap of the non-equilibrium Dicke model.
In contrast to several condensed-matter systems, the Dicke model immediately suggests a bipartition which allows for a direct study of the physical properties of subsystems of different nature, i.e. the set of qubits and the radiation field. Thus we calculate the entanglement spectrum for this bipartition. In general, the dynamical state of the total system ${\left\vert \psi{\left( t \right)} \right\rangle}$ is represented by the bipartite form in Eq. \[state\]. A standard singular-value-decomposition therefore allows us to rewrite this state as $$\label{schmidt}
{\left\vert \psi{\left( t \right)} \right\rangle}=\sum_{\alpha=1}^{\Xi}S_{\alpha}{\left( t \right)}{\left\vert \Phi_{\alpha}^{{\left[m_{z} \right]}}{\left( t \right)} \right\rangle}\otimes {\left\vert \Phi_{\alpha}^{{\left[n \right]}}{\left( t \right)} \right\rangle},$$ with $$\begin{aligned}
{\left\vert \Phi_{\alpha}^{{\left[m_{z} \right]}}{\left( t \right)} \right\rangle}&=\sum_{m=-N/2}^{N/2} U_{m,\alpha}{\left( t \right)} {\left\vert m_{z} \right\rangle},\qquad{\left\vert \Phi_{\alpha}^{{\left[n \right]}}{\left( t \right)} \right\rangle}&=\sum_{n=0}^{\chi} V_{\alpha,n}{\left( t \right)} {\left\vert n \right\rangle},\end{aligned}$$ and where the unitary matrices ${\bf U}$ and ${\bf V}$ are defined on the corresponding subspaces $\mathcal{H}_{m_z}$ and $\mathcal{H}_{n}$ of the set of qubits and radiation field respectively. The new orthonormal states ${ \left\{ {\left\vert \Phi_{\alpha}^{{\left[m_{z} \right]}}{\left( t \right)} \right\rangle} \right\} }$ and ${ \left\{ {\left\vert \Phi_{\alpha}^{{\left[n \right]}}{\left( t \right)} \right\rangle} \right\} }$ are known as Schmidt states. The diagonal elements $S_{\alpha}\geq 0$ in the expansion of Eq. \[schmidt\] are the Schmidt coefficients, which satisfy $\sum_{\alpha}S_{\alpha}^{2}=1$ due to the normalization of the state and are assumed to be arranged in descending order with $\alpha$. Finally $\Xi=\min(N+1,\chi+1)$ is the Schmidt rank, which corresponds to the total number of coefficients in the decomposition.\
The reduced density matrices for the two subsystems, $\rho_{m_z}{\left( t \right)}=tr_{n}{\left( {\left\vert \psi{\left( t \right)} \right\rangle}{\left\langle \psi{\left( t \right)} \right\vert} \right)}$ and $\rho_{n}{\left( t \right)}=tr_{m_z}{\left( {\left\vert \psi{\left( t \right)} \right\rangle}{\left\langle \psi{\left( t \right)} \right\vert} \right)}$, follow directly from the Schmidt decomposition of Eq. \[schmidt\] and are given by $$\begin{aligned}
\rho_{m_z}{\left( t \right)}&=\sum_{\alpha=1}^{\Xi} S_{\alpha}^{2}{\left( t \right)}{\left\vert \Phi_{\alpha}^{{\left[m_{z} \right]}}{\left( t \right)} \right\rangle}{\left\langle \Phi_{\alpha}^{{\left[m_{z} \right]}}{\left( t \right)} \right\vert}\\
\rho_{n}{\left( t \right)}&=\sum_{\alpha=1}^{\Xi} S_{\alpha}^{2}{\left( t \right)}{\left\vert \Phi_{\alpha}^{{\left[n \right]}}{\left( t \right)} \right\rangle}{\left\langle \Phi_{\alpha}^{{\left[n \right]}}{\left( t \right)} \right\vert},
\end{aligned}$$ which immediately shows that both $\rho_{m_z}{\left( t \right)}$ and $\rho_{n}{\left( t \right)}$ are diagonal in their respective Schmidt basis and have identical spectra. As a result, the Schmidt gap $\Delta_S$ is defined as $$\Delta_S \equiv \left| S^2_{2}-S^2_{1}\right|,$$ corresponding to the difference between the two largest eigenvalues of the reduced density matrix of any of the two subsystems, and is thus a property shared by both. In the following we describe the behavior of the Schmidt gap as the QPT of the Dicke model is crossed with the triangular ramping at different annealing velocities $v$.\
![(Color online) Left panel. Schmidt gap $\Delta_S$ as a function of the annealing velocity $\Gamma=\log_2(v)$ and the time-dependent light-matter interaction $\lambda(t)$. Right panel. Comparison of the $\lambda$ values where the Schmidt gap vanishes, and where the maximal squeezing of both qubits and photons takes place, for the same annealing velocities of the left panel and the ramping of increase of $\lambda$.[]{data-label="fig_4"}](Figure4.pdf)
We first consider the crossing of the quantum critical point $\lambda_c=1/2$ during the linear increase of $\lambda(t)$, corresponding to the $0\rightarrow1$ regime in the left panel of Fig \[fig\_4\]. Since the initial state ${\left\vert \psi(t=0) \right\rangle}$ is simply a product, only $S_1(t=0)=1$ is finite, while the other Schmidt coefficients are zero; thus $\Delta_S(t=0)=1$. During the subsequent dynamics $\Delta_S$ monotonically decreases, at a $\Gamma-$dependent rate. In the near adiabatic regime ($\Gamma\lesssim-10$) $S_1$ and $S_2$ cross and the Schmidt gap closes slightly above $\lambda_c$, which suggests that it actually captures the QPT between normal and superradiant states. This is similar to previous results of adiabatic dynamical crossings of QPTs in spin chains [@canovi2014prb; @torlai2014jstat; @Qijun2015prb], where the gap closes near to the corresponding transition. However, in contrast to these cases where the Schmidt coefficients separate and continue crossing during the subsequent dynamics, here $\Delta_S$ remains being zero. As the annealing velocity increases up to the intermediate regime, the Schmidt gap maintains the same qualitative decay with $\lambda$, closes further away from $\lambda_c$ similarly to dynamical crossings on spin chains, and remains zero afterwards. However for even faster ramping processes, in the sudden quench regime ($-3\lesssim\Gamma$), the decay of the gap is so slow that it remains finite when the reversal of $\lambda$ begins.
Now we discuss the dynamical crossing when $\lambda$ is reversed, depicted in the $1\rightarrow0$ regime of the left panel of Fig \[fig\_4\]. The main feature of the near adiabatic ramping is that slightly above $\lambda_c$ the Schmidt gap becomes finite again, signaling the return of the system to the normal phase. Moreover, the system actually goes back to the initial product state ${\left\vert \psi(0) \right\rangle}$, since the Schmidt gap reaches the value $\Delta_S=1$ when $\lambda=0$. For higher annealing velocities ($-10\lesssim\Gamma\lesssim-7$) the gap shows an initial fast non-monotonic growth, after which it tends to saturate to a finite value following an oscillatory dynamics. This indicates that even though the qubit and radiation subsystems become disconnected at the end of the pulse, the total state is not just a simple product but an entangled configuration. Thus this intermediate far-from-adiabatic triangular ramping could be exploited as a protocol for preparing entangled states of non-interacting subsystems. For larger annealing velocities but before the sudden quench regime, where the Schmidt gap became zero before starting the light-matter coupling reversal ($-7\lesssim\Gamma\lesssim-5$), it emerges again before crossing $\lambda_c$ but exhibits complex dynamics including more points of closure. For somewhat higher velocities we observe a scenario where the gap remains finite during the first stage of the driving, but since the dynamics is not so slow it still becomes zero shortly after the start of the reversal stage, before crossing $\lambda_c$ for the second time ($-5\lesssim\Gamma\lesssim-3$). This no longer occurs in the sudden quench regime, where due to the very slow dynamics the Schmidt gap never closes.
Discussion {#sect_discussion}
==========
The results presented in Section \[sect\_results\], in particular the similar qualitative profiles of the squeezing parameters and the Schmidt gap as a function of $\Gamma$ and $\lambda$, suggest that both might serve as indicators of the same non-equilibrium phenomenon. Now we briefly discuss this connection, along with a simple approach to the problem, and a possible future application.
Squeezing functions and Schmidt gap
-----------------------------------
In the right panel of Fig. \[fig\_4\] we show, for each annealing velocity considered, the value of $\lambda$ at which the Schmidt gap becomes zero during its increase ramping. As previously discussed, the gap vanishes at higher values of $\lambda>\lambda_c$ as the velocity increases, moving away from the near adiabatic limit. Close to the sudden quench regime ($-5\lesssim\Gamma\lesssim-2$) this general trend continuous, even though the increase is non-monotonic as the dynamics (and thus determining the exact closing point) becomes more involved. In spite of this behavior we find that remarkably, the closure of the gap coincides (quite well for low velocities, approximately for high velocities) with the points in which the maximal squeezing parameters of both qubits and photons take place. This is also shown in the right panel of Fig. \[fig\_4\], where the different scenarios are plotted simultaneously.
Furthermore this also agrees with the values of $\lambda$ in which the qubit and radiation order parameters become finite (see Ref. [@Acevedo2015NJP]). Thus the Schmidt gap can be considered as a complementary quantity to the order parameters of the Dicke model [@Qijun2015prb], as the former is finite when the latter are zero and vice versa [@Acevedo2015NJP]. These results suggest that both the Schmidt gap and the squeezing parameters are indicators of the emergence of the superradiant state when dynamically crossing the QPT, even at high velocity.
The behavior of these quantities is far more complex during the reversal stage. Due to the strongly-oscillating behavior at low velocities and the more erratic dynamics at high velocities, determining correctly the vanishing point of the Schmidt gap is much more complicated. However the qualitative form of the squeezing parameters depicted in Fig. \[fig\_3\] suggests that the connection between both types of quantities remains valid.
Landau-Zener-Stuckelberg approach
---------------------------------
A common theme running through our results is the appearance of large quantum correlations in the regime of intermediate pulse duration in the variation in $\lambda(t)$, or equivalently intermediate ramping velocity. A full many-body theory of this dynamical generation of quantum correlations is not possible at the present, and would likely require a novel theoretical technique for treating Eq. \[Hdicke\] in a non-perturbative way. However as a first step towards understanding the complex dynamics discussed here, we consider the simplest version of what happens to a quantum system when it crosses a quantum critical point driven by a time-dependent Hamiltonian. Specifically we provide a heuristic treatment by appealing to the phenomenon of Landau-Zener-Stuckelberg interferometry, by means of which possible trajectories of a quantum system interfere with each other when a transition between energy levels at an avoided crossing (a Landau-Zener transition) is crossed. As discussed in detail in Ref. [@Nori_RMP2014], when a two-level system is subject to periodic driving with sufficiently large amplitude, a sequence of transitions occurs. The phase accumulated between transitions (commonly known as the Stuckelberg phase) may result in constructive or destructive interference.
Following this heuristic approach, we imagine that we can approximate the complex energy-level diagram of this many-body light-matter system as simply a ground state and a excited-state manifold, separated by some minimum energy gap $\Delta$ during the driving process. During the up-sweep alone, there is a single pass through the avoided crossing (i.e. remnant of the critical point) and so the probability that the system then ends up in this excited state manifold is given by $P_+=P_{\rm LZ}={\rm exp}(-2\pi \Delta^2/4v)$ [@Nori_RMP2014]. A similar result follows for the down-sweep alone. However since a pulse involves the double-passage through the avoided crossing region, the resulting probability is given by $P_+=4P_{\rm LZ}(1-P_{\rm LZ}) {\rm sin}^2 \Phi$, where $\Phi$ is the sum of two separate phase contributions: one through the quasi-adiabatic portion and one through the non-adiabatic portion. Averaging over these phases, and hence averaging over the fine-scale oscillations seen in our results, the probability that the system ends up in the excited state manifold following the pulse is given by ${\overline P_+}=2P_{\rm LZ}(1-P_{\rm LZ})$. As a crude approximate energy scale we set $\Delta=0.5$, which is the value of $\lambda$ at which a purely static QPT occurs in Eq. \[Hdicke\]. As $v$ increases, ${\overline P_+}$ rises from zero to a maximum and then decays back to zero. Its maximum value is $0.5$ which corresponds to the maximum entropy scenario in a simple two-level system. We then obtain numerically that ${\overline P_+}$ starts decaying from its maximum when $v\approx 1$. This suggests that the correlation features that we observe should also fall off for $v\rightarrow 1$ ($\Gamma\rightarrow0$), as observed.
Future application: system-environment entanglement
---------------------------------------------------
Our findings are also relevant in an entirely different way: if we consider the matter subsystem as the system of interest, and the radiation subsystem as the environment, then our results provide new insight into how a system and its environment become entangled over time, as the system-environment interaction varies. To explore this in the future, instead of considering a single pulse as we do here, the system-environment interaction could be chosen to be a sequence of such pulses which may arrive randomly (e.g. following a Poisson distribution) or become correlated in terms of their arrival times. As such, our model and analysis can provide a first step toward a better understanding of environmental decoherence – and its flip side, quantum control – over time. This is important since a primary goal of quantum control is to reliably manipulate quantum systems while preserving advantageous properties such as coherence, entanglement, and purity. Instead of the complex interaction between the system (e.g. matter) and its surroundings (e.g. radiation) being assumed to hamper the system’s evolution, it is possible that a suitable sequence of corrective pulses might be used to provide positive feedback to the system and hence maintain its quantum coherences. We leave this for future work.\
Conclusions {#sect_conclu}
===========
We have presented theoretical results for the quantum correlations that develop in a many-body light-matter system, as a result of dynamically manipulating the strength of the light-matter coupling – specifically, in the form of a single pulse. Our approach was to solve numerically a general, time-dependent many-body Hamiltonian, and exploit the natural partition between radiation and matter degrees of freedom. Specifically, we presented results on intra-subsystem quantum correlations, namely the time-dependent matter and radiation squeezing parameters, and the inter-subsystem Schmidt gap for different pulse duration (i.e. ramping velocity) regimes, from the near adiabatic to the sudden quench limits. The results reveal that both types of quantities signal the emergence of the superradiant state when the quantum critical point is dynamically crossed, by the maximal value of the squeezing parameters and the vanishing of the gap. It is also observed that beyond the near adiabatic limit, the light and matter subsystems remain entangled even when they become uncoupled at the end of the pulse, which could be exploited as a protocol to engineer entangled states of non-interacting systems. Thus our results should also be of interest for temporal control schemes in practical quantum information processing and quantum computation. On a more fundamental level, our results may be helpful for the development of an open-dynamics quantum simulator, for shedding new light on core issues at the foundations of physics, including the quantum-to-classical transition and quantum measurement theory [@Zurek1], and characterization of Markovianity in quantum systems [@Bruer_PRL; @PRBluis; @Cosco2017arxiv]. Our findings could also help shed light on system-environment entanglement, if we view the matter subsystem as the system of interest and the radiation subsystem as the environment, and if the system-environment interaction is chosen to be a sequence of pulses with different correlation properties.
Acknowledgments {#acknowledgments .unnumbered}
===============
F.J.G-R., J.J.M-A, F.J.R., and L.Q. acknowledge financial support from Facultad de Ciencias through UniAndes-2015 project *Quantum control of nonequilibrium hybrid systems-Part II*. F.J.G.R and F.J.R acknowledges financial support from P17.160322.011/01-FISI05 Proyectos Semilla-Facultad de Ciencias at Universidad de los Andes (2017-II). O.L.A. acknowledges support from NSF-PHY-1521080, JILA-NSF-PFC-1125844, ARO, AFOSR, and MURI-AFOSR. N.F.J. acknowledges support from the National Science Foundation (NSF) under grant CNS 1522693 and the Air Force under AFOSR grant FA9550-16-1-0247. The views and conclusions contained herein are solely those of the authors and do not represent official policies or endorsements by any of the entities named in this paper.
References {#references .unnumbered}
==========
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abstract: 'After a proof of concept using Dropbox, a free storage and synchronization, showed that an evolutionary algorithm using several dissimilar computers connected via WiFi or Ethernet had a good scaling behavior in terms of evaluations per second, it remains to be proved whether that effect also translates to the algorithmic performance of the algorithm. In this paper we will check several different, and difficult, problems, and see what effects the automatic load-balancing and asynchrony have on the speed of resolution of problems.'
author:
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bibliography:
- 'dropbox.bib'
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- 'fluid.bib'
- 'pool.bib'
title: 'Cloud-based Evolutionary Algorithms: An algorithmic study'
---
Introduction
============
The main objective of this research is to find easily available means to either use or connect computing nodes in a distributed evolutionary computation experiment, and this often means resorting to free and readily available services. Dropboxis one of these services: it is commercialized as a [*cloud storage*]{} service, which is free up to a certain level of use (measured in traffic and usage). There are many other services like this one; however Dropbox was chosen due to its popularity, which also implies having many more potential volunteer users of a massive evolutionary computation experiment. there are also other features that make it the right tool for these experiments. Some other cloud storage services, like Wuala, provide a client program on which one must add explicitly the files that will be stored, which does not allow a seamless integration with the filesystem; others, like ZumoDrive, use remotely-mounted filesystems whose access is not so fast. Dropbox monitors local filesystems, and uploads them asynchronously, which makes it faster from the local point of view[^1].
In the experiments we are performing, we are interested in its use as a file [*synchronization*]{} service. When one file in one of the folders that is monitored by Dropbox is changed, it is uploaded to Dropbox servers and then distributed to all the clients that share the same folder. It is interesting, however, to note that from the programming point of view, all folders are written and read as local one, which makes its use quite easy, and also seamless.
In previous experiments [@dropbox:cec] we measured whether adding several computers to an experiment of this kind resulted in an increase in the number of simultaneous evaluations. In this paper we will measure whether besides an increase in speed, the algorithm profits from the distribution and asynchrony of the particular instance we have implemented, or on the contrary it suffers from it. In order to do that, we chose two optimization problems with a different degree of difficulty, and measured the time needed to find the solution, along with the number of evaluations.
The rest of the paper is organized as follows: after a brief section presenting the state of the art in voluntary and pool-based evolutionary computation, we describe the algorithm, the experimental setup and the implementation in Section \[sec:desc\]; results of these experiments will be briefly presented in Section \[sec:res\], to be followed by the conclusion, discussion and future lines of work in Section \[sec:conc\].
State of the Art
================
Cloud computing [@Armbrust:2010:VCC:1721654.1721672; @springerlink:10.1007/978-1-4419-6524-0_14] is an emergent technology, and as such research related to it is just recently emerging. Research addressing cloud storage is mainly related to content delivery [@Broberg20091012] or designing data redundancy schemes to ensure information integrity [@PamiesJuarez2010]. However, its use in distributed computing has not been addressed in such depth. Even if it is related to data grids [@chervenak2000data], in this paper we address the use of free cloud storage as a medium for doing distributed evolutionary computation, in a more or less parasitic way [@Barabasi2001Parasitic], since we use the infrastructure laid by the provider as part of an immigration scheme in an island-based evolutionary algorithm [@GA_Island_Model].
Thus we will have to look at pool-based distributed evolutionary algorithms for the closest methods to the one presented here. In these methods, several nodes or [*islands*]{} share a [*pool*]{} where the common information is written and read. To work against a single pool of solutions is an idea that has been considered almost from the beginning of research in distributed evolutionary algorithms. Asynchronous Teams or A-Teams [@de1991genetic; @talukdar1998asynchronous; @talukdar2003asynchronous] were proposed in the early nineties as a cooperative scheme for autonomous agents. The basic idea is to create a work-flow on a set of solutions and apply several heuristic techniques to improve them, possibly including humans working on them. This technique is not constrained to evolutionary algorithms, since it can be applied to any population based technique, but in the context of EAs, it would mean creating different single-generation algorithms, with possibly several techniques, that would create a new generation from the existing pool.
The A-Team method does not rely on a single implementation, focusing on the algorithmic and data-flow aspects, in the same way as the Meandre [@llora51meandre] system, which creates a data flow framework, with its own language (called ZigZag), which can be applied, in particular, to evolutionary algorithms.
While algorithm design is extremely important, implementation issues always matter, and some recent papers have concentrated on dealing with pool architectures in a single environment: G. Roy et al. [@pool:ga] propose a shared memory multi-threaded architecture, in which several threads work independently on a single shared memory, having read access to the whole pool, but write access to just a part of it. That way, interlock problems can be avoided, and, taking advantage of the multiple thread-optimized architecture of today’s processors, they can obtain very efficient, running time-wise, solutions, with the added algorithmic advantage of working on a distributed environment. Although they do not publish scaling results, they discuss the trade off of working with a pool whose size will have a bigger effect on performance than the population size on single-processor or distributed EAs. The same issues are considered by Bollini and Piastra in [@bollini1999distributed], who present a design pattern for persistent and distributed evolutionary algorithms; although their emphasis is on persistence, and not performance, they try to present several alternatives to decouple population storage from evolution itself ([*traditional*]{} evolutionary algorithms are applied directly on storage) and achieve that kind of persistence, for which they propose an object-oriented database management system accessed from a Java client. In this sense, our former take on browser-based evolutionary computation [@agajaj] is also similar, using for persistence a small database accessed through a web interface, but only for the purpose of interchanging individuals among the different nodes, not as storage for the whole population.
In fact, the efforts mentioned above have not had much continuity, probably due to the fact that there have been, until now, few (if any) publicly accessible online databases. However, given the rise of cloud computing platforms over the last few years, interest in this kind of algorithms has bounced back, with implementations using the public FluidDB platform [@merelo2010fluid] having been recently published.
Description of the algorithm and implementation {#sec:desc}
===============================================
A pool based evolutionary algorithm can be described as an island model [@whitley1997island] without topology; in fact, it is closer to the [*island*]{} metaphor since migrants are sent to the [*sea*]{} (pool), and come also from it, that is, the evolutionary algorithm is a canonical one with binary codification, except for two steps within the cycle that (conditionally) emit or receive immigrants. A minimum number of evaluations for the whole algorithm is set from the beginning; we will see later on how to control when this minimum number of evaluations is reached.
During the evolutionary loop, new individuals are selected using 3-tournament selection and generated using bit-flip mutation and uniform crossover. Migration is introduced in the algorithm as follows: after the population has been evaluated, migration might take place if the number of generations selected to do it is reached. The best individual is sent to the pool, and the best individual in the pool (chosen among those emitted by the other nodes) is incorporated into the population; if there has been no change in the best individual since the last migration, a random individual is added to the pool, which adds diversity to the population even if the individual fitness is not the highest. For the time being this has no influence on the result, but will have it later on when algorithmic tests are run.
Migrants, if any, are incorporated into the population substituting the worst individual, along with the offspring of the previous generation using generational replacement with a 1-elite. Population was set to 1000 individuals for all problems, and the minimum number of evaluations has been four million. Several migration rates were tested to assess its impact on performance. Besides, we introduced a 1-second delay after migration so that workload was reduced and the Dropbox daemon had enough time to propagate files to the rest of the computers. This delay makes 1-computer experiments faster when less migration is made, and will probably have to be fine-tuned in the future. The results are updated at the end of the loop to check if the algorithm has finished, that is, found the (single) solution to the problem.
One of the advantages of this topology-less arrangement is the independence from the number of computers participating in the experiment, and also the lack of need from a [*central*]{} server, although it can be arranged so that one of the computers starts first, and the others start running when some file is present. Adding a new computer, then, does not imply to arrange new connections to the current set of computers; the only thing that needs to be done is to locate the directory with the migrated individuals that is shared.
Two representative functions have been chosen to perform the tests; the main idea is that they took a long enough time to make sense in a distributed environment, but at the same tame a short enough time that experiments did not take a long time. One of them is [P-Peaks]{}, a multimodal problem generator proposed by De Jong in [@dejong97using]; a [P-Peaks]{} instance is created by generating $P$ random $N$-bit strings where the fitness value of a string $\vec x$ is the number of bits that $\vec x$ has in common with the nearest peak divided by $N$. $$\label{eq:ppeaks}
f_{P-PEAKS}(\vec x)= \frac{1}{N} \max_{1 \leq i \leq p}\{ N -
H(\vec x,Peak_i)\}$$ where $H(\vec x, \vec y)$ is the Hamming distance between binary strings $\vec x$ and $\vec y$. In the experiments made in this paper we will consider $P=100$ and $N=64$. Note that the optimum fitness is 1.0. The second function is [MMDP]{} [@goldberg92massive], which is a deceptive problem composed of $k$ subproblems of 6 bits each one ($s_i$). Depending of the number of ones (unitation) $s_i$ takes the values depicted next:\
------------------------------- -------------------------------
$fitness_{s_i}(0) = 1.0$ $fitness_{s_i}(1) = 0.0$
$fitness_{s_i}(2) = 0.360384$ $fitness_{s_i}(3) = 0.640576$
$fitness_{s_i}(4) = 0.360384$ $fitness_{s_i}(5) = 0.0$
$fitness_{s_i}(6) = 1.0$
------------------------------- -------------------------------
The fitness value is defined as the sum of the $s_i$ subproblems with an optimum of $k$ (equation \[eq:mmdp\]). Figure \[fgraf\_mmdp\] represents one of the variables of the function. The number of local optima is quite large ($22^k$), while there are only $2^k$ global solutions[^2]. $$\label{eq:mmdp}
f_{MMDP}(\vec s)= \sum_{i=1}^{k} fitness_{s_i}$$
In this paper we have used $k=20$, in order to make it difficult enough to need a parallel solution.
Experiments and results {#sec:res}
=======================
In this occasion the experiments were done in several different computers connected also in different ways; however, computers were added to the experiment in the same order; the problems were solved first in a single computer, then on two and finally with four computers. Total time, as well as the number of evaluations, were measured. Since the end of the experiment is propagated also via Dropbox, the number of evaluations is not exactly the one reached when the solution is found. This number also increases with the number of nodes.
The only parameter that was changed during experiments was migration rate. We were interested in doing this, since network performance will be impacted negatively with migration rate: bandwith usage (and maybe latency) increases with the inverse of the migration rate. On the other hand, evolutionary performance will increase in the opposite direction: the bigger the migration rate, the more similar to a panmictic population will be, which might make finding the solution easier; on the other hand, it will also decrease diversity, making the relationship between migration rate and evolutionary/runtime performance quite complex and worth studying.
To keep (the rest of) the conditions uniform for one and two machines, all parameters were fixed but for the population, which was distributed among the machines in equal proportions: all computers maintained a population of 1000, so that initial diversity was roughly the same. Further experiments will have to be made keeping population constant, but this is left for further study.
Finally, Dropbox itself was used to check for termination conditions: a file was written on the folder indicating the experiment had finished; when the other node read that file, it finished too; all nodes were kept running until the solution was found or until a maximum number of generations were reached. That is why, in some cases, solution is not found; the number of generations was computed so that it was possible in a high number of cases to find out the solution.
The computers used in this experiment were laptops connected via the University of Granada WiFi, they were different models, and were running different operating systems and versions of them. The most powerful computer was the first one; then \#2 was the second-best, and finally numbers 3 and 4 were the least powerful ones. Since computers run independently without synchronization checkpoints, load balancing is automatic, with more powerful computers contributing more evaluations to the mix, and less powerful ones contributing fewer.
The first thing that was checked with the two problems examined (P-Peaks and MMDP) was whether adding more computers affected the solution rate. For P-Peaks there was no difference, independently of migration rate and number of computers, all experiments found the solution. However, there was a difference for MMDP, shown in Table \[tab:mmdp:sols\].
--------- --------------------- ----------------
*Nodes* *Generations* *Success rate*
*between migration*
1 0.83
2 0.95
4 1
1 0.70
2 0.88
4 1
1 0.80
2 0.90
4 1
--------- --------------------- ----------------
: Success rate for the MMDP problem with different number of nodes and migration rates. \[tab:mmdp:sols\]
The evolution with the migration rate can also be observed in figure \[fig:mmdp:sols\]; as was advanced in the introduction, the relationship is quite complex and decrease or increase do not lead to a monotonic change of the success rate. In fact, the best success rate corresponds to the highest migration rate (migration after 100 generations), but the second best corresponds to the lowest one (migration after 400), which is almost akin to no migration, since taking into account that generations run asynchronously, this might mean that in fact on migrant from other nodes is incorporated into the population. This result is in accordance with the [*intermediate disturbance hypothesis*]{}, proved by us previously [@jj:2008:PPSN]. However, it is not clear in this case that migration in 100 generations can be actually considered intermediate and in 200 too high, so more experiments will have to be performed to ascertain the optimum migration rate.
Thus, having proved that success rate increases with the number of nodes, we will have to study how performance varies with it. Does the algorithm really finds the solution faster when more nodes are added? We have computed time only for the experiments that actually found the solution, and plotted the results in figures \[fig:time:ppeaks\] and \[fig:time:mmdp\].
As seen above in the case of MMDP, there is not a straightforward relationship between the migration rate and the time to solution; in this case, the relationship between the number of computers and time solution is also complex. If we look first at the P-Peaks experiment in \[fig:time:ppeaks\] we see that we obtain little time improvement when adding more nodes to the mix. Since success rate is already 100% with a single computer, and the solution takes around two minutes, the delay imposed by Dropbox implies that it is not very likely that the migrated solutions are transmitted to the other nodes. In this case it is the intermediate migration rate (every 40 generations) the only one that obtains a steady decrease of time to solution. The best time is obtained for a single node and a migration gap of 60; in general, the best times are for the highest migration gap since the total delay induced by migration is also the least. These results probably imply that there must be a certain degree of complexity in the problems to take advantage of the features in this environment. For relatively [ *easy*]{} problems, which need few generations, there is nothing to gain.
The situation varies substantially for the MMDP, as seen in figure \[fig:time:mmdp\]. In this case, the best result is obtained for four nodes and the smallest mutation gap (every 100 generations, dashed black line). However, it is interesting to observe that trend change for two nodes in all cases, either the solution takes more time than for a single node or it tales less than for four nodes; the conclusion is, anyways, that the increased number of simultaneous evaluations brought by the number of nodes eventually makes solution faster. However, a fine-tuning of the migration gap is needed in order to take full advantage of the parallel evaluation in the Dropbox-based system.
Conclusions and future work {#sec:conc}
===========================
In general, and for complex problems like the MMDP, a Dropbox-based system can be configured to take advantage of the paralellization of the evolutionary algorithm and obtain reliably (in a 100% of the cases) solutions in less time than a single computer would. Besides, it has been proved that it does not matter whether the new computers added to the set are more or less powerful than the first one. In general, however, adding more computers to a set synchronized via Dropbox has more influence in the success rate than in the time needed to find the solution, which seems roughly linked to the population size, although this hypothesis will have to be tested experimentally. On the other hand, using relatively simple problems like P-Peaks yields no sensible improvement, due to the delay in migration imposed by Dropbox, which implies that this kind of technique would be better left only for problems that are at the same time difficult from the evolutionary point of view and also slow to evaluate.
However, several issues remain to be studied. First, more accurate performance measures must be taken to measure how the time needed to find the solution in all occasions scales when new machines are added. We will have to investigate how parameter settings such as population size and migration gap (time passed between two migrations) influence these measures. This paper proves that this influence is important, but it is not clear what is the influence on the final result. It would be also interesting to test different migration policies affect final result, as done in [@Araujo2010], where it was found out that migrating the best one might not be the best policy.
An important issue too is how to interact with Dropbox so that information is distributed optimally and with a minimal latency. In this case we had to stop each node for a certain time (which was heuristically found to be 1 second) to leave time for the Dropbox daemon to distribute files. In an experiment that lasts for less than two minutes, this can take up 25% of the total time (per node), resulting in an obvious drag in performance that can take many additional nodes to compensate. A deeper examination of the Dropbox API and a fine-tuning of these parameters will be done in order to fix that.
Finally, this framework opens many new possibilities for distributed evolutionary computation: meta-evolutionary computation, artificial life simulations, and big-scale simulation using hundreds or even thousands of clients. The type of problems suitable for this, as well as the design and implementation issues, will have to be explored. Other cloud storage solutions, preferably including open source implementations, will be also tested. Since they have different models (synchronization daemon or user-mounted filesystems, mainly) latency and other features will be completely different, so we expect that performance will be affected by this.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been supported in part by the CEI BioTIC GENIL (CEB09-0010) MICINN CEI Program (PYR-2010-13) project and the Andalusian Regional Government P08-TIC-03903 and P08-TIC-03928 projects.
[^1]: The characteristics of these and others online backup services can be seen in <http://en.wikipedia.org/wiki/Comparison_of_online_backup_services>
[^2]: The local optima occur when there are 3 ones; off all the 64 possible combinations of six zeros and ones, there are 22 with exactly three ones
|
---
abstract: 'Low-temperature-resistivity plateau observed in $\rm SmB_6$ single crystal,which is due to surface, not bulk, conduction has been confirmed from electrical transport measurements. Recently, the correlation between bulk thermodynamic measurements and the low-temperature-resistance plateau in $\rm SmB_6$ have been investigated and a change in Sm valence at the surface has been obtained from x-ray absorption spectroscopy and x-ray magnetic circular dichroism. Here we show that the statement of the report are not supported by the results from x-ray absorption spectroscopy and x-ray magnetic circular dichroism.'
address: '$^1$Synchrotron SOLEIL, L’Orme des Merisiers,Saint-Aubin-BP48, 91192 GIF-sur-YVETTE CEDEX,France'
author:
- 'Kai Chen$^{1,*}$, Jean-Paul Kappler$^1$'
title: 'Comment on “Correlation between Bulk Thermodynamic Measurements and the Low-Temperature-Resistance Plateau in $\bold {SmB_6}$”'
---
In a recent article, W. A. Phelan and co-workers[@Phelan2014] report data on the correlation between bulk thermodynamic measurements and the low-temperature-resistance plateau in $\rm SmB_6$. They found surface conductivity of $\rm SmB_6$ increases systematically with bulk carbon content and addition of carbon is linked to an increase in n-type carriers, larger low-temperature electronic contributions to the specific heat and a broadening of the crossover to the insulating state. A change in Sm valence at the surface has been obtained from x-ray absorption spectroscopy(XAS) and x-ray magnetic circular dichroism(XMCD), which is claimed to be the definitive proof of changes in the electronic structure at the surface of $\rm SmB_6$. This statement is true while the data from XAS and XMCD are problematic which may misleading the further investigations.
In their report, the XAS and XMCD of surface and bulk are obtained from total electron yield (TEY) and fluorescence yield(FY), which are believed to be sensitive to the surface and bulk of the sample, respectively. However, the “surface” is not well defined here, which should be the electron escaping lengh in the order of $\rm \sim 2nm$ while the “bulk” is related to the thickness of $\sim10$ times higher. Besides, nothing is reported for the $\rm SmB_6$ surface treatment which is important for the XAS measurement since naturally oxidized or cleaved surface are quite different. Furthermore, at the $\rm M_5$ edge of Sm, XAS from FY may be quite different from that obtained from TEY and transmission[@Pompa1997], due to the 3d core hole lifetime broadening $\rm \Gamma$ dominated by the auger decay. However, such a deviation between TEY and FY is neglected in $\rm SmB_6$ shown in Fig.8a in[@Phelan2014], in which the peaks from $\rm Sm^{2+}$ and $\rm Sm^{3+}$ are well distinguished.
![ []{data-label="fig:1"}](Fig1.pdf){width="\linewidth"}
As they claimed in Fig.8 in [@Phelan2014], “the bulk spectra measured from FY (red curve) are consistent with a mixture of $\rm Sm^{2+}$ and $\rm Sm^{3+}$, with no appreciable magnetization, as previously reported[@Tarascon1980; @Mizumaki2009]". Nothing related to magentism is reported in the ref[@Tarascon1980; @Mizumaki2009]. It is also claimed: “in contrast, the surface spectra from TEY (black curve) consists of almost entirely $\rm Sm^{3+}$ and shows a discernible XMCD signal characteristic of a net magnetic moment, approximately 1/10 of that observed in ferromagnetic $\rm Sm_{0.974}Gd_{0.02}Al_2$ [@Dhesi2010]". Here the spectra of $\rm Sm^{2+}$ and $\rm Sm^{3+}$ are mistaken in the report. To clarify, the XAS of $\rm Sm^{2+}$ and $\rm Sm^{3+}$ obtained from atomic multiplet calculation using CTM4XAS[@Stavitski2010] are shown in Fig.1. The electrostatic and exchange parameters were scaled down to 80% of the atomic Hartree-Fock value. In the case of $\rm Sm^{3+}$ ions, because the two first excited, J =7/2 and J=9/2, multiplets are relatively close in energy to the fundamental J=5/2 multiplet [@Kramida2014], it is necessary to account for the crystalline electric field effects, not only on the fundamental, but also on these excited multiplets. The mixing of these higher multiplets into the fundamental leads to a reduction of the magnetic moment [@Buschow1973]. Such an effect is not considered in the calculation since only a slightly change of the XAS shape will be observed [@Dhesi2010]. The XAS of $\rm Sm^{2+}$ is left shifted compared to that of $\rm Sm^{3+}$ in Fig.1, which is normal and attributed to the chemical shift, and similar to the previous results from experimental results [@Kaindl1984] and theoretical calculation[@Thole1985]. However it is opposite in Fig.8 in[@Phelan2014], where the surface state of $\rm Sm^{3+}$ is left shifted. This needs to be corrected at least by an erratum.
According to the experimental data of XAS, it is not possible that the surface is in pure $\rm Sm^{2+}$ state while the bulk is mixed with $\rm Sm^{2+}$ and $\rm Sm^{3+}$. For Sm metals, which in the bulk is a trivalent of $\rm Sm^{3+}$ at the surface was turned into divalent configuration of $\rm Sm^{2+}$ [@Wertheim1978; @Allen1978; @Johansson1979]. For $\rm SmB_6$, surface valence between 2.5 and 2.6 was determined from X-ray photoemission spectroscopy(XPS)[@Heming2014]. Interestingly, as we calculated the shape of XAS for the surface is more like a $\rm 4f^5$ ground state with $\rm Sm^{3+}$, not a $\rm 4f^6$ ground state with $\rm Sm^{2+}$ (Fig.1). The XMCD results are also puzzling. Indeed, a magnetic TEY signal is observed in the “surface" case ($\rm Sm^{3+}$ like), with the same shape as $\rm Sm^{3+}$ in[@Dhesi2010], whereas no magnetic signal has been detected for the bulk. As the bulk is a $\rm Sm^{2+}$-$\rm Sm^{3+}$ mixing, at least the Sm atoms in the 3+ state should give a magnetic response.
There still remains the problem with the shape of XAS from FY. The intensive peak at higher energy cannot be understood. We doubt there is the energy shift in the FY XAS since the shape canbe well fitted with the XAS of $\rm Sm^{3+}$ and $\rm Sm^{2+}$, as shown in Fig.1. In this case, the XAS and XMCD data canbe well understood and supports their statements very well. However, we have no idea if there exists the energy shift between the XAS from TEY and FY shown in Fig.8a in[@Phelan2014] and needs to be checked by the authors.
We conclude that the XAS and XMCD spectra of Sm in[@Phelan2014] are problematic. Several possible mistakes have been considered to understand the results, among which the energy shift between the XAS from TEY and FY may be the explanation. We also doubt if chemical states of Sm canbe determined from the comparison from the XAS of $\rm M_{4,5}$ edge measured from TEY and FY[@Pompa1997].
[99]{}
|
---
abstract: 'The existence of a quantum butterfly effect in the form of exponential sensitivity to small perturbations has been under debate for a long time. Lately, this question gained increased interest due to the proposal to probe chaotic dynamics and scrambling using out-of-time-order correlators. In this work we study echo dynamics in the Sachdev-Ye-Kitaev model under effective time reversal in a semiclassical approach using the truncated Wigner approximation, which accounts for non-vanishing quantum fluctuations that are essential for the dynamics. We demonstrate that small imperfections introduced in the time-reversal procedure result in an exponential divergence from the perfect echo, which allows to identify a Lyapunov exponent $\lambda_L$. In particular, we find that $\lambda_L$ is twice the Lyapunov exponent of the semiclassical equations of motion. This behavior is attributed to the growth of an out-of-time-order double commutator that resembles an out-of-time-order correlator.'
author:
- Markus Schmitt
- Dries Sels
- Stefan Kehrein
- Anatoli Polkovnikov
bibliography:
- 'refs.bib'
title: 'Semiclassical echo dynamics in the Sachdev-Ye-Kitaev model'
---
Introduction
============
The question of chaos and the possibility of a butterfly effect in quantum systems is a long-standing problem that received increased attention in recent years. In studies addressing the information paradox of black holes so-called out-of-time-order correlators (OTOCs) of the form $$\begin{aligned}
\braket{\hat V(0)^\dagger\hat W(t)^\dagger\hat V(0)\hat W(t)}_\beta
\label{eq:otoc}\end{aligned}$$ were introduced to probe the sensitivity of the dynamics to small perturbations and scrambling, i.e., the delocalization of initially local information [@Hayden2007; @Sekino2008; @Shenker2014; @Kitaev2014]. A semiclassical analysis of the OTOC motivates that it can exhibit exponential growth, allowing to identify a Lyapunov exponent [@Larkin1969]. In fact, it was found that in a black hole theory OTOCs grow exponentially with the maximal possible rate $\lambda_L=\frac{2\pi}{\beta}$. [@Maldacena2016] Remarkably, there exists a solvable model of interacting fermions, which also saturates this bound at low temperatures, namely the Sachdev-Ye-Kitaev (SYK) model [@Kitaev2015; @Maldacena2016a], which is a variant of a model originally introduced by Sachdev and Ye [@Sachdev1993; @Sachdev2010].
OTOCs as dynamical probe of chaos and scrambling are also of interest in condensed matter systems beyond the AdS/CFT paradigm [@Bohrdt2017; @Huang2017; @Chen2017; @Iyoda2017; @Swingle2017; @Shen2017; @Patel2017; @Fan2017]. Particularly intriguing is the connection to the question how and in what sense closed quantum many-body systems thermalize when initially prepared far from equilibrium, which has been studied with great efforts in recent years [@DAlessio2016; @Gogolin2016]. The corresponding statistical description of the stationary state is only justified if the information about the initial state cannot be recovered in practice, i.e., the dynamics is irreversible.
To assess the irreversibility of the dynamics a common approach is to study imperfect effective time reversal. In classical systems it is understood that the exponential sensitivity of the dynamics to small perturbations prohibits recovery of the initial state, because perfect time reversal is impossible in practice [@Thompson1874; @Boltzmann1872; @Loschmidt1876; @Boltzmann1877]. Under chaotic dynamics any imperfection occurring in the time reversal operation leads to an exponential divergence from accurate recovery of the initial state with a rate that is largely independent of the perturbation, namely the Lyapunov exponent. This renders the improvement of the protocol prohibitively expensive.
Analogous approaches have been explored considering quantum systems. In few-body systems the decay characteristics of the Loschmidt echo $\mathcal L(\tau)=|\braket{\psi_0|\hat U_E^\epsilon(\tau)|\psi_0}|^2$ with the echo operator $\hat U_E^\epsilon(\tau)=e^{{\mathrm{i}}(\hat H+\epsilon\hat V)\tau}e^{-{\mathrm{i}}\hat H\tau}$, where $\epsilon\hat V$ constitutes a small perturbation to the Hamiltonian, were used as indicator of chaos and irreversibility [@Peres1984; @Gorin2006; @Jacquod2009]. For many-body systems, however, overlaps lack experimental significance. Instead, the decay of observable echos under imperfect effective time reversal was studied to investigate irreversibility [@Fine2014; @Elsayed2015; @Zangara2015; @Schmitt2016; @Schmitt2017].
In the works mentioned above the focus was on decay laws occurring in the echo dynamics at late times. By contrast, imperfect effective time reversal in classical systems features initial dynamics that is governed by the butterfly effect. The possibility of a butterfly effect that occurs analogously in quantum systems is currently under debate [@Fine2014; @Fine2017; @Rozenbaum2017; @Bohrdt2017; @Scaffidi2017; @Patel2017; @Khemani2018; @Swingle2018]. Moreover, the realization of effective time reversal was recently reported from an experiment with trapped ions, where OTOCs were measured in the form of echo dynamics [@Gaerttner2017].
In this work we study the dynamics of the SYK model under imperfect effective time reversal in a semiclassical approach using the truncated Wigner approximation. We demonstrate that the small imperfection leads to an exponential divergence from the perfect echo. This divergence can be attributed to the exponential growth of an out-of-time-order double commutator similar to an OTOC and it allows to identify a Lyapunov exponent based on the echo dynamics.
The structure of the paper is as follows: First, in Sections \[sec:time\_reversal\] and \[sec:hamiltonian\] we introduce the echo protocol under consideration and the model of interest. Section \[sec:semiclassical\_dynamics\] comprises an introduction to the truncated Wigner approximation and a discussion of its applicability to the SYK model. In Section \[section:echoes\] we present our results for the echo dynamics in the semiclassical limit. Before the final discussion in Section \[sec:discussion\] we include in Section \[sec:TWAvsMF\] an extended elaboration on the distinction between mean field dynamics and the truncated Wigner approximation in the context of the SYK model.
Imperfect effective time reversal {#sec:time_reversal}
=================================
In the following we will investigate the echo dynamics of an observable $\hat O$ under imperfect effective time reversal. The perturbation is introduced by the action of a perturbation operator $\hat P_\epsilon$ on the time-evolved state at the point of time reversal. Here $\epsilon$ denotes a parameter for the smallness of the perturbation. A natural choice for the perturbation operator is unitary time evolution for a short interval $\delta t$ with a perturbation Hamiltonian $\hat H_p$, i.e., $\hat P_{\delta t}=e^{-{\mathrm{i}}\hat H_p\delta t}$. The quantity of interest is the echo signal $$\begin{aligned}
E_{\hat O}(\tau)=\braket{\psi_0|\hat U_E^{\delta t}(\tau)^\dagger \hat O\hat U_E^{\delta t}(\tau)|\psi_0}
\label{eq:echo_definition}\end{aligned}$$ with the echo operator $\hat U_E^{\delta t}(\tau)=e^{{\mathrm{i}}\hat H\tau}\hat P_{\delta t}e^{-{\mathrm{i}}\hat H\tau}$. This constitutes an OTOC in the case that the initial state is an eigenstate of the observable, $(\hat O-\mu)\ket{\psi_0}=0$ [@Gaerttner2017; @Schmitt2017]. Moreover, expanding the echo operator $\hat U_E^{\delta t}(\tau)$ in orders of $\delta t$ yields $$\begin{aligned}
\Delta &E_{\hat O}(\tau)=\braket{\psi_0|\hat O|\psi_0}-E_{\hat O}(\tau)\nonumber\\
&={\mathrm{i}}\delta t\braket{\psi_0|[\hat H_p(\tau),\hat O]|\psi_0}\nonumber\\
&\quad
+\frac{\delta t^2}{2}\braket{\psi_0|[\hat H_p(\tau),[\hat H_p(\tau),\hat O]]|\psi_0}
+\mathcal O(\delta t^3)
\label{eq:echo_divergence}\end{aligned}$$ for the divergence from the perfect echo. In this expression the linear term corresponds to linear response and it vanishes in the case that the initial state is an eigenstate of the observable. Hence, the quadratic term constitutes the leading contribution to the divergence from the perfect echo, accounting for the sensitivity of the dynamics to small perturbations. Using the example of the SYK Hamiltonian we will demonstrate in the following that this double commutator in fact determines the initial decay of the echo and that the corresponding divergence grows exponentially in time, which allows to identify a Lyapunov exponent.
Model Hamiltonian {#sec:hamiltonian}
=================
The Hamiltonian of the fermionic SYK model is given by $$\begin{aligned}
\hat H=\frac{1}{(2N)^{3/2}}\sum_{ijkl}J_{ij;kl}\hat c_i^\dagger\hat c_j^\dagger\hat c_k\hat c_l\ ,\end{aligned}$$ where the $J_{ij;kl}$ are complex-valued Gaussian random couplings with vanishing mean and variance $\sigma^2=\overline{|J_{ij;kl}|^2}$. $N$ denotes the number of fermionic modes. The SYK model has a number of remarkable properties. Although strongly interacting, it is exactly solvable in the limit of large $N$. At low temperatures it exhibits an emergent conformal symmetry indicating the existence of a holographic dual [@Maldacena2016a]. In this regime it is maximally chaotic in the sense that the Lyapunov exponent occurring in OTOCs saturates the bound that was derived for AdS black holes [@Maldacena2016].
Semiclassical dynamics in the SYK model {#sec:semiclassical_dynamics}
=======================================
We will analyze echo dynamics using the fermionic version of the truncated Wigner approximation (TWA), which was recently developed in Refs. [@Davidson2017; @Davidson2017a].
On the applicability of the truncated Wigner approximation {#sec:onTWA}
----------------------------------------------------------
The TWA is the saddle point approximation for the Keldysh action describing the Heisenberg evolution of the observables. It can be generally derived using standard path integral methods.[@Polkovnikov2010] Within the TWA time evolution of phase space variables is governed by the classical Hamiltonian equations of motion, which have to be supplemented by fluctuating initial conditions. In turn those are encoded in the Wigner function describing the initial state. Within the accuracy of the TWA one can generally approximate this Wigner function by a Gaussian capturing means and fluctuations of the phase space variables. Note that while formally classical equations of motion are identical to the Dirac mean field equations of motion (see Section \[sec:TWAvsMF\]), the TWA reduces to the mean field approximation only if fluctuations in initial conditions are asymptotically vanishing with the saddle point parameter. This is, e.g., the case for initial coherent states or for polarized quantum spins in the large S-limit. But it is not the case, e.g., for stationary states of a high energy particle in a confining potential where the Wigner function approaches the broad in space micro canonical distribution rather than a single phase space point. In many instances, in particular when we deal with Fermions or spin one half degrees of freedom fluctuations are always large such that the mean field approximation is generally incorrect and moreover is not approached as the saddle point parameter increases (see, e.g., Refs. [@Altland2009; @Orioli2017]). In the SYK model the large N limit ensures validity of the saddle point approximation[@Maldacena2016a; @Eberlein2017] and, therefore, it is natural that the fermionic version of TWA will be asymptotically exact in the large N limit, which as we show in Section \[sec:accuracy\] is indeed the case. Hence, $N$ serves as effective $\hbar^{-1}$.
Phase space approach for Fermions {#subsec:eom}
---------------------------------
Within the fermionic TWA a phase space representation is constructed for the fermionic bilinears, which satisfy the commutation relations of $so(2N)$ [@Davidson2017; @Davidson2017a]; see also Ref. [@Yaffe1982] for a general picture of classical representations of quantum models. The Weyl symbols of the fermionic bilinears are $\tau_{\alpha\beta}=\big(\hat c_\alpha\hat c_\beta\big)_W=-\big(\hat c_\alpha^\dagger\hat c_\beta^\dagger\big)_W^*$ and $\rho_{\alpha\beta}=\frac12\big(\hat c_\alpha^\dagger\hat c_\beta-\hat c_\beta\hat c_\alpha^\dagger\big)_W$. The corresponding Weyl symbol of the SYK Hamiltonian expressed in terms of pairing operators is $$\begin{aligned}
\mathcal H&=\frac{1}{(2N)^{3/2}}\sum_{ijkl}J_{ij;kl}
\Big(\tau_{ji}^*\tau_{kl}
+\rho_{jk}\delta_{il}+\rho_{il}\delta_{kj}\Big)\ .\end{aligned}$$
Generally, for phase space variables $X_\alpha$ of operators $\hat X_\alpha$, which obey some algebra $$\begin{aligned}
[\hat X_\alpha,\hat X_\beta]={\mathrm{i}}f_{\alpha\beta\gamma}\hat X_\gamma\end{aligned}$$ with structure constants $f_{\alpha\beta\gamma}$, the classical equations of motion are determined by $$\begin{aligned}
\frac{dX_\alpha}{dt}=f_{\alpha\beta\gamma}\frac{\partial (\hat H)_W}{\partial X_\beta}X_\gamma\ ,\end{aligned}$$ where $(\hat H)_W\equiv\mathcal H$ is the Weyl symbol of the Hamiltonian.[@Davidson2017]
For the phase space variables of the fermionic bilinears and the Hamiltonian of the SYK model this yields
$$\begin{aligned}
{\mathrm{i}}\frac{d}{dt}\rho_{\alpha\beta}&=
\Bigg(
-\frac{\partial\mathcal H}{\partial\rho_{\gamma\alpha}}\rho_{\gamma\beta}
+\frac{\partial\mathcal H}{\partial\tau_{\gamma\alpha}}\tau_{\beta\gamma}
-\frac{\partial\mathcal H}{\partial\tau_{\alpha\gamma}}\tau_{\beta\gamma}
\Bigg)
-\Bigg(\alpha\leftrightarrow\beta\Bigg)^*\ ,
\nonumber\\
&=
\frac{2}{N^{3/2}}\sum_{ijkl}J_{ijkl}\delta_{\alpha l}\Big(\tau_{ji}^*\tau_{\beta k}+\delta_{ik}\rho_{j\beta}\Big)-(\alpha\leftrightarrow\beta)^*
\nonumber\\
{\mathrm{i}}\frac{d}{dt}\tau_{\alpha\beta}&=
\Bigg(
\frac{\partial\mathcal H}{\partial\rho_{\alpha\gamma}}\tau_{\gamma\beta}
+\frac{\partial\mathcal H}{\partial\tau_{\gamma\alpha}^*}\rho_{\gamma\beta}
-\frac{\partial\mathcal H}{\partial\tau_{\alpha\gamma}^*}\tau_{\gamma\beta}
\Bigg)
-\Bigg(\alpha\leftrightarrow\beta\Bigg)
\nonumber\\
&=
\frac{2}{N^{3/2}}\sum_{ijkl}J_{ijkl}\delta_{\alpha j}\Big(\delta_{il}\tau_{k\beta}-\tau_{kl}\rho_{i\beta}\Big)-(\alpha\leftrightarrow\beta)
\ .
\label{eq:twa_eom}\end{aligned}$$
In the following we will consider uncorrelated initial states that are fully characterized by orbital occupation numbers $n_\alpha=\braket{\hat c_\alpha^\dagger\hat c_\alpha}$. In that case the Wigner function is well approximated by a multivariate Gaussian fixed by the first and second moments [@Davidson2017a]. We will be interested in the expansion dynamics starting from an initially imbalanced occupation similar to the situations studied in different recent cold atom experiments [@Schneider2012; @Choi2016; @Bordia2017]. Given this initial state a suited observable to consider in the view of echo dynamics is the occupation imbalance $$\begin{aligned}
\hat M=\frac{1}{N}\sum_{\alpha=1}^N(2n_{\alpha}^0-1)(2\hat c_\alpha^\dagger\hat c_\alpha-1)\end{aligned}$$ with Weyl symbol $\mathcal M=\frac{2}{N}\sum_{\alpha=1}^N(2n_\alpha^0-1)\rho_{\alpha\alpha}$, where $n_\alpha^0$ is the initial value of $n_\alpha$.
Accuracy of the TWA
-------------------
![Comparison of TWA results to the exact dynamics. The top panel shows the time evolution of the occupation imbalance with $N=20$ modes, whereas in the bottom panel the individual mode occupation numbers are shown. In the bottom panel dashed lines correspond to the exact result. The inset shows the system size dependence of the time-averaged squared deviation of the TWA from the exact result.[]{data-label="fig:twa_precision"}](twa_precision_rev.pdf){width=".9\columnwidth"}
\[sec:accuracy\] In order to assess the accuracy of the TWA we compare the result for expansion dynamics from an uncorrelated initial state, where one quarter of the modes is occupied and the rest is empty, with exact dynamics. Fig. \[fig:twa\_precision\] displays the corresponding time evolution of the occupation imbalance $M(t)$ and the individual mode occupations $n_i(t)=\braket{\psi(t)|\hat c_i^\dagger\hat c_i|\psi(t)}$ for $N=20$ and a disorder average involving 20 realizations. The dynamics computed using TWA is in good agreement with the exact dynamics. We find empirically that the accuracy of TWA improves as $N$ is increased. As demonstrated in the inset of Fig. \[fig:twa\_precision\] the deviations from the exact result are compatible with a power law scaling; $N^{-2}$ is shown as orientation.
Semiclassical echo dynamics {#section:echoes}
===========================
For our purposes we choose the perturbation Hamiltonian $\hat H_p=\sum_\alpha J_\alpha \big(\hat c_\alpha^\dagger\hat c_{\alpha+1}+h.c.\big)$ with normally distributed random couplings $J_\alpha$ (variance $J^2=\overline{J_\alpha^2}$) and corresponding Weyl symbol $\mathcal H_p=2\sum_\alpha J_\alpha\rho_{\alpha,\alpha+1}$. Note that the dynamics under this Hamiltonian is captured exactly by the TWA, because it is quadratic.
Echoes in finite systems
------------------------
In Fig. \[fig:cmp\_echo\] we compare the result for $\Delta E_M(\tau)$ given in Eq. obtained from TWA with the exact dynamics. The presented data includes a disorder average over $80$ realizations of both the SYK and the perturbation Hamiltonian. In the initial uncorrelated state one quarter of the sites is filled and the rest is empty. We find with both methods that the echo deviates increasingly from the initial value as the waiting time $\tau$ is increased and the results are in good agreement at short times. At long times, however, there is a clear discrepancy. In the result obtained from TWA the echo signal ultimately vanishes completely, meaning that $\Delta M(\tau\to\infty)=3/4$. By contrast, the exact result saturates much earlier. The reason for this is that for finite $N$ the overlap $\braket{\psi(\tau)|\hat P_{\delta t}|\psi(\tau)}$ is non-zero, resulting in an ever-persisting revival at the echo time [@Schmitt2017]. The corresponding saturation value can be determined in the exact simulation and it is indicated in Fig. \[fig:cmp\_echo\] by the dashed line; see Appendix \[app:finite\_size\] for details. This persisting echo, however, vanishes for $N\to\infty$, because, typically, the Loschmidt echo is exponentially suppressed by the system size, $|\braket{\psi(\tau)|\hat P_{\delta t}|\psi(\tau)}|^2\sim e^{-Nr(\delta t)}$ with an intensive rate function $r(t)$. Hence, the limits $N\to\infty$ and $\delta t\to0$ do not commute. Correspondingly, the normalized difference between exact and TWA data at the echo time, Diff($\Delta E_{\hat M}(\tau)$), is reduced when the system size is increased, as indicated in the inset of Fig. \[fig:cmp\_echo\]. In Appendix \[app:finite\_size\] we include further data supporting the anticipated vanishing of the persistent echo in the exact dynamics for $N\to\infty$. This disappearing of an intrinsic difference between TWA and exact echo dynamics goes along with a generally improved accuracy of the TWA as discussed above. Therefore, we expect that in the limit $N\to\infty$ results from TWA and exact dynamics will converge. Since we find in addition that the TWA result for the echo dynamics is independent of $N$ (see Appendix \[app:finite\_size\]), we conclude that the TWA results obtained for large but finite $N$ constitute a good approximation of the behavior in the large $N$ limit.
![Echo dynamics computed with TWA in comparison with exact results for $J\delta t=0.1$ and $N=16$ at quarter filling. The dashed line indicates the corresponding persistent echo peak height derived in Appendix \[app:finite\_size\]. The inset demonstrates that the normalized difference at the echo time is reduced as the system size is increased; the black line corresponds to an exponential fit.[]{data-label="fig:cmp_echo"}](cmp_twa_edv3.pdf){width=".95\columnwidth"}
With our resources for the exact dynamics, however, $N=20$ is the largest value we can reach due to the large number of nonvanishing matrix elements in the SYK Hamiltonian and the disorder average necessary to perform a meaningful finite size analysis. For these finite systems the persisting echo can be considered to be a genuine quantum characteristic. The TWA, applicable in the semiclassical limit, does not capture this feature, because its origin is the non-vanishing overlap between the quantum states before and after application of the perturbation operator in combination with the unitarity of quantum time evolution.
![TWA results for the divergence from the perfect echo computed for $N=20$ modes. As the perturbation strength $\delta t$ is decreased the regime of exponential growth is extended, allowing for the identification of a Lyapunov exponent. The inset shows exact results for system sizes $N=8,12,16,20$ for $J\delta t=0.1$. These exact data are compatible with convergence towards the TWA result as $N\to\infty$.[]{data-label="fig:echo_decay"}](cmp_echo_decayv3.pdf){width=".95\columnwidth"}
{width="\textwidth"}
Signature of a butterfly effect in echo dynamics
------------------------------------------------
In Fig. \[fig:echo\_decay\] we show TWA results for the divergence from the perfect echo as defined in Eq. . After the short time period the data exhibit a clear exponential growth of the difference to the perfect echo although the observable is bounded. We find that the parameter $\delta t$ that determines the smallness of the perturbation controls the extent of the regime, where the exponential law is observed. In direct analogy to classical chaos the exponential divergence of the perturbed echo from the perfect echo allows to identify a Lyapunov exponent $\lambda_L$. A fit to the data in Fig. \[fig:echo\_decay\] yields $\lambda_L\approx0.87\sigma$, which is in good quantitative agreement with a result for the Lyapunov exponent in the limit of high temperature obtained via a diagrammatic large-$N$ expansion and exact numerics.[@Kobrin2018; @Roberts2018] Note that our convention for the coupling constants differs from Ref. [@Roberts2018] by a factor $\sqrt{2}$. In Appendix \[app:corr\_echo\] we include results for another observable, namely density-density correlations, showing exponential divergence with the same rate. In the following we will discuss the origin of this exponential divergence in more detail.
Role of the double commutator
-----------------------------
In the exact echo dynamics we observe that the quadratic term of Eq. , in fact, is the only relevant contribution for a large range of perturbation strengths and irrespective of the waiting time. Fig. \[fig:shorttime\]a shows exact data for $\Delta E_{\hat M}(\tau)$ in comparison with the quadratic term $\frac12\braket{\psi_0|[\hat H_p(\tau),[\hat H_p(\tau),\hat M]]|\psi_0}\delta t^2$ as function of the perturbation strength $\delta t$ for different waiting times $\tau$. Both coincide perfectly for $J\delta t<0.5$.
Even though the TWA does not capture the persistent echo, Fig. \[fig:shorttime\]b demonstrates that the semiclassical echo dynamics exhibit the same quadratic dependence on the perturbation strength $\delta t$ in the regime of exponential growth. Deviations from the quadratic scaling only occur when $\Delta E_{\hat M}(\tau)$ begins to saturate. This supports the assertion that in Eq. the second order term is the single contribution responsible for the exponential sensitivity to the imperfection in the time reversal protocol.
Similar to the OTOC , which is related to the square of the commutator of both operators, $|[V,W(t)]|^2$, expanding the double commutator reveals an out-of-time-order structure: $$\begin{aligned}
[\hat H_p(\tau),[\hat H_p(\tau),\hat M]]
&=\hat H_p(\tau)^2\hat M+\hat M\hat H_p(\tau)^2
\nonumber\\&\quad
-2\hat H_p(\tau)\hat M\hat H_p(\tau)
\label{eq:double_commutator}\end{aligned}$$ In this expression the last term accounts for the butterfly effect. For an extensive perturbation Hamiltonian $\hat H_p$ the double commutator becomes extensive at late times. In the thermodynamic limit the double commutator can grow indefinetely such that it can govern the exponential divergence from the perfect echo irrespective of the higher order terms as long as $\braket{\psi_0|[\hat H_p(\tau),[\hat H_p(\tau),\hat M]]|\psi_0}\delta t^2\ll1$. We deduce that only at late times higher order terms become important resulting in the approach to a constant. The inference that the double commutator governs the exponential divergence in the echo dynamics is supported by the relation to the Lyapunov exponent of the classical TWA equations, which is discussed next.
Classical Lyapunov exponent of the TWA equations
------------------------------------------------
The Lyapunov exponent occurring in the semiclassical echo dynamics can be related to the largest Lyapunov exponent of the dynamical system defined by the TWA equations of motion. The largest classical Lyapunov exponent is defined as $$\begin{aligned}
\lambda_{\text{cl}}=\Big\langle\lim_{t\to\infty}\lim_{d(\vec x(0),\vec x'(0))\to0}\frac{1}{t}\ln\Big|\frac{d(\vec x(t),\vec x'(t))}{d(\vec x(0),\vec x'(0))}\Big|\Big\rangle
\label{eq:class_lyapunov}\end{aligned}$$ with coordinate vectors $\vec x(t)$ and $\vec x'(t)$ and $d(\vec x,\vec x')=\sqrt{\sum_{i}(x_i-x_i')^2}$ the Euclidian distance. The time-dependence of the coordinate vectors is given by the equations of motion and $\braket{\cdot}$ indicates the classical average over an ensemble of trajectories.
To estimate the Lyapunov exponent of the TWA equations of motion we average the divergence of an ensemble of initially close-by trajectories on a fixed time interval; details are given in Appendix \[app:lyapunov\]. Fig. \[fig:shorttime\]c displays the resulting average $\braket{\ln\big|d(\vec x(t),\vec x'(t))/d(\vec x(0),\vec x'(0))\big|}$, which we computed for half and quarter filling with $d_0=10^{-8}$. We find a clear linear dependence on time and a fit yields the classical Lyapunov exponent $\lambda_{\text{cl}}\approx0.34$. The result varies only weakly as the filling is changed.
This value of $\lambda_\text{cl}$ is slightly less than half of $\lambda_L$, which we extracted before from the echo dynamics. In the following we will argue that a factor of two between both is to be expected. We attribute the slight discrepancy to the different orders of averaging and taking the logarithm, resulting in a slightly smaller classical Lyapunov exponent as reported in Ref. [@Rozenbaum2017].
The Weyl symbol of the double commutator in Eq. can be written in the form $$\begin{aligned}
&\big([\hat H_p(\tau),[\hat H_p(\tau),\hat M]]\big)_W=
\nonumber\\&
A^i_j\frac{\partial x_i(t)}{\partial x_j(0)}+B^i_j\frac{\partial x_i(0)}{\partial x_j(t)}
+C_{ij}^{kl}\frac{\partial x_k(t)}{\partial x_i(0)}\frac{\partial x_l(0)}{\partial x_j(t)}
\label{eq:dc_weyl}\end{aligned}$$ with $\vec x$ the vector of $\rho$ and $\tau$ coordinates of the TWA equations (cf. Appendix \[app:dc\]). The modulus of all derivatives occurring in this expression grows with the classical Lyapunov exponent. However, the sums of the single derivatives in the first two terms will cancel, because they correspond to linear response. Hence, if the terms in the quadratic contribution do not cancel, at late times $$\begin{aligned}
\big([\hat H_p(\tau),[\hat H_p(\tau),\hat M]]\big)_W\sim e^{2\lambda_\text{cl}t}\ .\end{aligned}$$ The Weyl symbols of higher order commutators would contain growth rates that are higher multiples of $\lambda_\text{cl}$. Since we only observe the factor of two in the echo dynamics, we conclude that the quadratic term in Eq. is in fact the one that is relevant for the butterfly effect.
On the importance of including quantum fluctuations {#sec:TWAvsMF}
===================================================
It is worthwhile elaborating more on the importance of quantum fluctuations for the dynamics of the SYK model in the semiclassical limit. In the following we will contrast mean field dynamics, which captures only fluctuations on the Gaussian level of single Fermion operators, against dynamics, which includes fluctuations that are Gaussian on the level of Fermionic bilinears.
The TWA equations of motion presented in Section \[subsec:eom\] are essentially mean field equations of motion. In the following we aim to outline the key difference between TWA and the mean field approximation, namely the fact that TWA captures fluctuations which are essential for the dynamics of the SYK model. The importance of fluctuations is due to the fact that the microscopic degrees of freedom are fermions, which are always strongly fluctuating. This is, for example, in contrast to the semiclassical limit of large spins, where the fluctuations vanish in the limit of large spin.
The most general equations of motion for a mean field approximation are
$$\begin{aligned}
{\mathrm{i}}\frac{d}{dt}\rho_{\alpha\beta}&=
\frac{2}{N^{3/2}}\sum_{ijkl}J_{ijkl}\delta_{\alpha l}\Big(
\tau_{ji}^*\tau_{\beta k}+\delta_{ik}\rho_{j\beta}+2\rho_{j\beta}\rho_{ik}
\Big)
-(\alpha\leftrightarrow\beta)^*
\nonumber\\
{\mathrm{i}}\frac{d}{dt}\tau_{\alpha\beta}
&=\frac{2}{N^{3/2}}\sum_{ijkl}J_{ijkl}\delta_{j\alpha}
\Big(\delta_{il}\tau_{k\beta}-\rho_{i\beta}\tau_{kl}-2\rho_{il}\tau_{\beta k}\Big)
-(\alpha\leftrightarrow\beta)
\label{eq:mean_field_eom}\end{aligned}$$
These equations are obtained under the assumption that the quantum state remains Gaussian for all times. In that case a Wick theorem can be used to split all higher order correlations into products of two-point functions, which correspond to the resulting phase space variables.
The mean field Hamiltonian corresponding to Eq. encorporates all possibilities, meaning that there are classical fields coupling to pairing terms as well as hopping and local potentials. It turns out (see Fig. \[fig:red\_eom\] below) that to approximate the SYK dynamics it is sufficient to consider a much simpler mean field Hamiltonian including only pairing operators, given that the quantum fluctuations in the initial state are taken into account. This simpler mean field Hamiltonian takes the form $$\begin{aligned}
\hat H=\frac{1}{\sqrt{2N}}\sum_{ij}(\Delta_{ij}(t)\hat c_i^\dagger\hat c_j^\dagger+h.c.)\end{aligned}$$ where the classical field $$\begin{aligned}
\Delta_{ij}(t)=\frac{1}{2N}\sum_{kl}J_{ijkl}\braket{\hat c_k\hat c_l}_t=\frac{1}{2N}\sum_{kl}J_{ijkl}\tau_{kl}(t)\end{aligned}$$ is determined self-consistently. The resulting equations of motion constitute a reduction of Eq. : $$\begin{aligned}
{\mathrm{i}}\frac{d\rho_{\alpha\beta}}{dt}&=-\frac{2}{\sqrt{2N}}\sum_k\Delta_{k\alpha}(t)^*\tau_{\beta k}-(\alpha\leftrightarrow\beta)^*
\nonumber\\
{\mathrm{i}}\frac{d\tau_{\alpha\beta}}{dt}&=\frac{2}{\sqrt{2N}}\sum_j\Delta_{\alpha j}(t)\rho_{j\beta}-(\alpha\leftrightarrow\beta)
\label{eq:rand_sc_eom}\end{aligned}$$
In the mean field approximation the initial condition of the phase space variables is fixed by the expectation values in the initial state, $$\begin{aligned}
\rho_{\alpha\beta}(0)&=\braket{\hat c_\alpha^\dagger\hat c_\beta}_{t=0}-\frac{\delta_{\alpha\beta}}{2}\nonumber\\
\tau_{\alpha\beta}(0)&=\braket{\hat c_\alpha\hat c_\beta}_{t=0}=0\end{aligned}$$ This means, however, that mean field dynamics with Eq. is trivial, because $\braket{\hat c_i\hat c_j}_{t=0}=0$; in the mean field approximation the system remains stationary at all times. Non-trivial dynamics is only initiated by fluctuations of the fermionic bilinears in the initial state. These fluctuations can be included by stochastic sampling of the initial condition as we will discuss below.
Within the mean field approximation non-trivial dynamics is obtained when considering the equations of motion given in Eq. . These equations account for Gaussian fluctuations on the level of single fermion operators. Fig. \[fig:mf\] shows a result for relaxation dynamics obtained in the mean field approximation using Eq. starting with an uncorrelated state with occupation imbalance just as in the main text. Although the general shape of the decay is captured quite well, the decay time scale differs from the corresponding exact result. In the mean field approximation the relaxation turns out to be too slow. This discrepancy between mean field dynamics and exact dynamics was already observed in Ref. [@Davidson2017], where, however, different mean field approximations were considered.
![Expansion dynamics from the uncorrelated initial stated as observed in the single mode occupation numbers $n_i(t)=\braket{\hat c_i^\dagger\hat c_i}_t$. The solid lines correspond to mean field dynamics based on the most general equations of motion, Eq. . The dashed lines were obtained by computing the full quantum dynamics. The data shown are for $N=20$ at quarter filling.[]{data-label="fig:mf"}](mf.pdf){width=".9\columnwidth"}
![Expansion dynamics from the uncorrelated initial stated as observed in the single mode occupation numbers $n_i(t)=\braket{\hat c_i^\dagger\hat c_i}_t$. The solid lines correspond to dynamics obtained based on the reduced mean field equations of motion, Eq. , including fluctuations in the initial state by stochastic sampling of the initial conditions. The dashed lines were obtained by computing the full quantum dynamics. The data shown are for $N=20$ at quarter filling.[]{data-label="fig:red_eom"}](self_consistent_fluct.pdf){width=".9\columnwidth"}
In order to accurately describe the relaxation dynamics it is essential to capture fluctuations of the fermionic bilinears correctly. This can be achieved by including Gaussian fluctuations of the phase space variables[^1] by stochastic sampling and an averaging of the resulting trajectories. This approach is essentially equivalent to stochastic sampling from the Wigner function of the initial state as it is done in the TWA. Fig. \[fig:red\_eom\] displays the result for relaxation dynamics obtained in this approximation using the equations of motion of the simple mean field Hamiltonian, Eq. , supplemented with fluctuations of the initial conditions. The comparison with the exact result shows very good agreement. Hence, we conclude that the relaxation dynamics is mainly driven by two-particle fluctuations, which are included in the TWA, but not in the mean field approximation.
A similar approach to incorporate quantum fluctuations in phase space dynamics has already been introduced in Ref. [@Damle1996]. However, as it is evident from the discussion above, there are various ambiguities, for which there is no a-priori resolution. Nevertheless, the corresponding choices might affect the resulting physical quantities. For example, the additional terms occurring in Eq. , which are irrelevant for the dynamics in our case, might be important under different circumstances. [@Polkovnikov2010] The TWA provides a consistent mathematical framework to set up the equations of motion and to incorporate fluctuations. The remaining ambiguity in choosing the bilinears based on which the phase space is constructed corresponds to finding the decoupling scheme where the saddle point approximation becomes asymptotically exact (cf. Section \[sec:onTWA\]).
As a final remark we would like to mention that the shortcomings of the mean field approximation are also reflected in the fact that with mean field only a sub-extensive part of the spectrum can be captured[@Scaffidi2017] and only fluctuations as included in TWA render the energies extensive.
Discussion {#sec:discussion}
==========
We found that the exponential divergence from the perfect echo in the semiclassical dynamics is due to the growth of an out-of-time-order double commutator of the form $[\hat V(\tau),[\hat V(\tau),\hat W(0)]]$. This assertion is based on the small perturbation expansion in Eq. , which does not rely on any semiclassical approximation. In future work the structure and characteristic behavior of these objects should be further explored, in particular with regard to the sensitivity of genuine quantum dynamics far from a classical limit to small perturbations. Regarding irreversibility our result implies that the dynamics of the SYK model is irreversible in the same sense as a chaotic classical system: Any imperfection in the time reversal procedure leads to an exponential divergence from the perfect echo and substantial improvement is prohibitively expensive, because the Lyapunov exponent is perturbation-independent.
The authors acknowledge helpful discussions with S. Davidson. This work was supported through SFB 1073 (project B03) of the Deutsche Forschungsgemeinschaft (DFG). M.S. acknowledges support by the Studienstiftung des Deutschen Volkes and through the Leopoldina Fellowship Programme of the German National Academy of Sciences Leopoldina (LPDS 2018-07) with additional support from the Simons Foundation. D.S. acknowledges support from the FWO as post-doctoral fellow of the Research Foundation – Flanders and CMTV. A.P. acknowledges support by NSFGrants No. DMR-1506340, No. DMR-1813499 and AFOSR Grand No. FA9550-16-1-0334. For the numerical computations Armadillo [@armadillo] and ITensor [@itensor] were used.
![Full time evolution under imperfect effective time reversal as obtained by TWA in comparison with exact dynamics for different forward times $\tau$ with system size $N=20$ at quarter filling and $J\delta t=0.25$. The exact dynamics show a persistent echo signal, whereas the TWA echo vanishes at long forward times. The dashed line indicates the persistent peak height as given by Eq. .[]{data-label="fig:full_echo_dynamics"}](syk_cmp_echo_rev.pdf)
Finite size analysis {#app:finite_size}
====================
For finite mode number $N$ the perturbed state will always have a nonvanishing overlap with the unperturbed state, $|\braket{\psi(\tau)|\hat P_{\delta t}|\psi(\tau)}|>0$. Accordingly, we can decompose $\hat P_{\delta t}\ket{\psi(\tau)}=\cos(\alpha_{\delta t})\ket{\psi(\tau)}+\sin(\alpha_{\delta t})\ket{\phi}$ by introducing the “orthogonal component" $\ket{\phi}$ with $\braket{\psi(\tau)|\phi}=0$. Considering this decomposition it becomes evident that the remaining “parallel component" of the perturbed state leads to an ever persisting echo at time $t=2\tau$: $$\begin{aligned}
&\braket{\psi(\tau)|\hat P_{\delta t}^\dagger e^{-{\mathrm{i}}\hat H\tau}\hat Me^{{\mathrm{i}}\hat H\tau}\hat P_{\delta t}|\psi(\tau)}\nonumber\\
&=\cos^2(\alpha_{\delta t})\braket{\psi_0|\hat M|\psi_0}+\sin^2(\alpha_{\delta t})\braket{\phi|e^{-{\mathrm{i}}\hat H\tau}\hat Me^{{\mathrm{i}}\hat H\tau}|\phi}
\nonumber\\&\quad
+\sin(2\alpha_{\delta t})\text{Re}\big(\braket{\psi_0|\hat Me^{{\mathrm{i}}\hat H\tau}|\phi}\big)\end{aligned}$$ For finite $N$ there is a time-independent contribution proportional to the initial value of the observable, $\braket{\psi_0|\hat M|\psi_0}$, and the overlap of the perturbed and unperturbed state, $\cos^2(\alpha_{\delta t})=|\braket{\psi(\tau)|\hat P_{\delta t}|\psi(\tau)}|^2$. At late times the expectation value in the second term will attain an equilibrium value $M_\phi^\infty=\lim_{\tau\to\infty}\braket{\phi|e^{-{\mathrm{i}}\hat H\tau}\hat Me^{{\mathrm{i}}\hat H\tau}|\phi}$ and the overlap in the third term will vanish. Therefore, the persistent echo peak height at large $\tau$ is given by $$\begin{aligned}
\lim_{\tau\to\infty}E_{\hat M}(\tau)=\cos^2(\alpha_{\delta t})\braket{\psi_0|\hat M|\psi_0}+\sin^2(\alpha_{\delta t})M_\phi^\infty\ .
\label{eq:persistent_echo_height}\end{aligned}$$ Exemplary results for the dynamics including effective time reversal are shown in Fig. \[fig:full\_echo\_dynamics\]. In the thermodynamic limit, $N\to\infty$, we will have $\alpha_{\delta t}=\pi/2$, i.e. the contribution given by the initial expectation value of $\hat M$ vanishes and we obtain $$\begin{aligned}
\lim_{N\to\infty}\lim_{\tau\to\infty}E_{\hat M}(\tau)=M_\phi^\infty\ .\end{aligned}$$
![Finite size analysis of exact results for the echo dynamics. The dashed lines indicate the saturation values obtained from the overlap of the perturbed with the unperturbed state. Here, $J\delta t=0.1$.[]{data-label="fig:ed_finite_size"}](ed_echo_system_sizev2.pdf)
Moreover, the window for possible exponential divergence from the perfect echo has a fixed size for a given finite $N$. This window cannot be increased by reducing $\delta t$, which is evident from Eq. of the main text. In a finite system the expectation value of the double commutator is bounded for all times $\tau$, $|\braket{\psi_0|[\hat H_p(\tau),[\hat H_p(\tau),\hat O]]|\psi_0}|<C(N)$. Therefore, in the limit of small $\delta t$ $$\begin{aligned}
1\leq\Big|\frac{\Delta E_{\hat O}(\tau)}{\Delta E_{\hat O}(0)}\Big|<\frac{C(N)}{|\braket{\psi_0|[\hat H_p,[\hat H_p,\hat O]]|\psi_0}|}\ .\end{aligned}$$
In the following we present data for the variation of the echo signal $E_{\hat M}(\tau)$ with changing system sizes, which supports our assertion that the persistent echo vanishes in exact quantum dynamics. For a faithful investigation of finite size effects disorder averaging is essential, because fluctuations introduced by adding new randomly coupled degrees of freedom can otherwise spoil the analysis.
In Fig. \[fig:ed\_finite\_size\] we show exact results for the divergence from the perfect echo for different system sizes, including a disorder average over 80 realizations. The dashed lines indicate the saturation value of the persistent echo computed directly according to Eq. , where at quarter filling $M_\phi^\infty=1/4$. We find very good agreement of the echo at late times with this value. As discussed in the main text and earlier in this section the saturation value increases as the system size is increased. This corresponds to the vanishing of the persistent echo in the thermodynamic limit.
Fig. \[fig:twa\_finite\_size\] displays TWA results for the divergence from the perfect echo for different system sizes. In this case we find that the results are almost identical despite a doubling of the system size.
Combining both results with the expectation that TWA becomes exact in the thermodynamic limit we conclude that the TWA result gives already at finite system sizes a good approximation of the result in the thermodynamic limit and with increasing $N$ the exact results will converge to this.
![Finite size analysis of results for the echo dynamics obtained using TWA.[]{data-label="fig:twa_finite_size"}](twa_echo_system_size_rev.pdf)
![Echo observed in the correlation average $\mathcal C$ (cf. Eq. ) as a function of waiting time $\tau$. The exponential rate is the same as in the case of the occupation imbalance.[]{data-label="fig:corr_echo"}](twa_echo_corr.pdf)
Echo in density-density correlation {#app:corr_echo}
===================================
In addition to the occupation imbalance $\hat M$ presented in the main text we investigated echos in density-density correlations. We consider the average correlation $$\begin{aligned}
\mathcal C(t)=\frac{2}{N(N-1)}\sum_{i<j}\big|\braket{\hat n_i\hat n_j}_t-\braket{\hat n_i}_t\braket{\hat n_j}_t\big|
\label{eq:corr_avg}\end{aligned}$$ with $\hat n_i=\hat c_i^\dagger\hat c_i$.
Fig. \[fig:corr\_echo\] shows the dynamics of the echo $E_{\mathcal C}(\tau)$ as defined in Eq. . With increasing waiting time $\tau$ we find also for the correlation average $\mathcal C$ an exponential divergence from the perfect echo. The exponential rate is the same as in the case of the occupation imbalance.
Approach to determine the classical Lyapunov exponent {#app:lyapunov}
=====================================================
A common numerical method to determine the largest classical Lyapunov exponent $$\begin{aligned}
\lambda_{\text{cl}}=\Big\langle\lim_{t\to\infty}\lim_{d(\vec x(0),\vec x'(0))\to0}\frac{1}{t}\ln\Big|\frac{d(\vec x(t),\vec x'(t))}{d(\vec x(0),\vec x'(0))}\Big|\Big\rangle
\label{eq:class_lyapunov}\end{aligned}$$ is to integrate the equations of motion of two close-by initial conditions $\vec x(0)$ and $\vec x'(0)$ with a small fixed $d(\vec x(0),\vec x'(0))=d_0$ and evaluate the ratio $d(\vec x(t),\vec x'(t))/d_0$ at a fixed time $t$. Then $\vec x'$ is reinitialized such that $d(\vec x(t),\vec x'(t))=d_0$ and the equations of motion are integrated for another interval $t$, before the ratio of initial and final distances is evaluated again. This procedure is iterated and the samples of $t^{-1}\ln\big|d(\vec x(t),\vec x'(t))/d(\vec x(0),\vec x'(0))\big|$ are averaged to obtain an estimate of the classical Lyapunov exponent .
To estimate the Lyapunov exponent of the TWA equations of motion we employed a similar approach. In this case $\vec x\equiv(\rho_{\alpha,\beta},\tau_{\alpha,\beta})$. During a sequence of integration and reinitialization in turns we average $\ln\big|d(\vec x(t),\vec x'(t))/d(\vec x(0),\vec x'(0))\big|$ on the whole interval $0<t<t_\text{max}$. Additionally, we average over many such sequences with initial conditions drawn from the Wigner function of the initial state under consideration. In this way we obtained the result shown in Fig. \[fig:shorttime\]c in the main text.
Structure of the Weyl symbol of the double commutator {#app:dc}
=====================================================
Let us denote the set of phase space variables by $\vec x$. In our case both $H_p$ and $M$ are linear in TWA variables, which means that the Bopp operators take the form $$\begin{aligned}
\vec H_p=h(\vec x)+\sum_i h_i(\vec x)\frac{\partial}{\partial x_i}\end{aligned}$$ and $$\begin{aligned}
\vec M=m(\vec x)+\sum_i m_i(\vec x)\frac{\partial}{\partial x_i}\end{aligned}$$ where $h(\vec x), h_i(\vec x), m(\vec x), m_i(\vec x)$ are some functions of the coordinates.
Plugging this into the double commutator yields the Weyl symbol $$\begin{aligned}
&\big([\hat H_p(t),[\hat H_p(t),\hat M]]\big)_W\nonumber\\
&=
\Big(h(\vec x(t))+\sum_i h_i(\vec x(t))\frac{\partial}{\partial x_i(t)}\Big)\Big(h(\vec x(t))+\sum_j h_j(\vec x(t))\frac{\partial}{\partial x_j(t)}\Big)m(\vec x)
\nonumber\\&\quad
+\Big(m(\vec x)+\sum_i m_i(\vec x)\frac{\partial}{\partial x_i}\Big)
\Big(h(\vec x(t))+\sum_j h_j(\vec x(t))\frac{\partial}{\partial x_j(t)}\Big)h(\vec x(t))
\nonumber\\&\quad
-2\Big(h(\vec x(t))+\sum_i h_i(\vec x(t))\frac{\partial}{\partial x_i(t)}\Big)
\Big(m(\vec x)+\sum_j m_j(\vec x)\frac{\partial}{\partial x_j}\Big)h(\vec x(t))
\nonumber\\
&=\sum_{ij}h_i(\vec x(t))h_j(\vec x(t))\frac{\partial}{\partial x_i(t)}\frac{\partial}{\partial x_j(t)}m(\vec x)
+\sum_{ij}h_i(\vec x(t))\frac{\partial h_j(\vec x(t))}{\partial x_i(t)}\frac{\partial}{\partial x_j(t)}m(\vec x)
\nonumber\\&\quad
-2\sum_{ij}h_i(\vec x(t))\frac{\partial m_j(\vec x)}{\partial x_i(t)}\frac{\partial h(\vec x(t))}{\partial x_j}
+\sum_{ij} m_i(\vec x)\frac{\partial h(\vec x(t))}{\partial x_j(t)}\frac{\partial}{\partial x_i}h_j(\vec x(t))
\nonumber\\&\quad
-\sum_{ij}m_i(\vec x)h_j(\vec x(t))\frac{\partial}{\partial x_i}\frac{\partial h(\vec x(t))}{\partial x_j(t)}\end{aligned}$$ Now we use the chain rule $\frac{\partial f(x_i(t_1))}{\partial x_j(t_2)}=\sum_k\frac{f(x_i(t_1))}{\partial x_k(t_1)}\frac{\partial x_k(t_1)}{\partial x_j(t_2)}$ wherever applicable, yielding $$\begin{aligned}
&\big([\hat H_p(t),[\hat H_p(t),\hat M]]\big)_W
\nonumber\\
&=
\sum_{ijkl}\Big(h_i(\vec x(t))h_j(\vec x(t))\frac{\partial^2 m(\vec x)}{\partial x_k\partial x_l}\Big)\frac{\partial x_k}{\partial x_i(t)}\frac{\partial x_l}{\partial x_j(t)}
+\sum_{jk}\Big(\sum_ih_i(\vec x(t))\frac{\partial h_j(\vec x(t))}{\partial x_i(t)}\frac{\partial m(\vec x)}{\partial x_k}\Big)\frac{\partial x_k}{\partial x_j(t)}
\nonumber\\&\quad
-2\sum_{ijkl}\Big(h_i(\vec x(t))\frac{\partial m_j(\vec x)}{\partial x_k}\frac{\partial h(\vec x(t))}{\partial x_l(t)}\Big)\frac{\partial x_k}{\partial x_i(t)}\frac{\partial x_l(t)}{\partial x_j}
+\sum_{ik}\Big(\sum_j m_i(\vec x)\frac{\partial h(\vec x(t))}{\partial x_j(t)}\frac{\partial h_j(\vec x(t))}{\partial x_k(t)}\Big)\frac{\partial x_k(t)}{\partial x_i}
\nonumber\\&\quad
-\sum_{ik}\Big(\sum_jm_i(\vec x)h_j(\vec x(t))\frac{\partial^2 h(\vec x(t))}{\partial x_k(t)\partial x_j(t)}\Big)\frac{\partial x_k(t)}{\partial x_i}\ .\end{aligned}$$ Since in our case $h(\vec x)$ and $m(\vec x)$ are linear in $\vec x$, the expression can be simplified to $$\begin{aligned}
&\big([\hat H_p(t),[\hat H_p(t),\hat M]]\big)_W
\nonumber\\
&=
\sum_{jk}\Big(\sum_ih_i(\vec x(t))\frac{\partial h_j}{\partial x_i}\frac{\partial m}{\partial x_k}\Big)\frac{\partial x_k}{\partial x_j(t)}
-2\sum_{ijkl}\Big(h_i(\vec x(t))\frac{\partial m_j}{\partial x_k}\frac{\partial h}{\partial x_l}\Big)\frac{\partial x_k}{\partial x_i(t)}\frac{\partial x_l(t)}{\partial x_j}
+\sum_{ik}\Big(\sum_j m_i(\vec x)\frac{\partial h}{\partial x_j}\frac{\partial h_j}{\partial x_k}\Big)\frac{\partial x_k(t)}{\partial x_i}\ .\end{aligned}$$
In this form the Weyl symbol corresponds to Eq. in the main text. This expression involves linear response type terms, which are linear in $\frac{\partial x_i(t)}{\partial x_j(0)}$, and terms that are quadratic in these derivatives. The linear terms should cancel such that they do not contribute to exponential growth; otherwise, also the response of the form $\{\hat H_p(\tau)^2,\hat M\}=\hat H_p(\tau)^2\hat M+\hat M\hat H_p(\tau)^2$ would grow exponentially.
[^1]: Note that Gaussian fluctuations on the level of the phase space variables (fermionic bilinears) includes the connected parts of the fermionic four point function.
|
---
abstract: 'Milne cosmology has recently been shown to be in broad agreement with most cosmological data while being free of the problematic notions of standard cosmology such as the dark sector. In this paper a broken symmetric unified theory of gravity and electromagnetism is introduced which has a Milne metric under a certain geometric condition. Strikingly, particles (dyons) emerge as topological charges in this theory provided the torsion vector $\Gamma_i$ is curl-less.'
author:
- |
Partha Ghose[^1]\
The National Academy of Sciences, India,\
5 Lajpatrai Road, Allahabad 211002, India
title: ' A Cosmology Inspired Unified Theory of Gravity and Electromagnetism: Classical and Quantum Aspects'
---
Introduction
============
Einstein’s famous equation $$R_{ik} - \frac{1}{2}g_{ik}R = \kappa T_{ik}\label{E}$$ of General Relativity has been extremely successful in explaining and predicting various weak field phenomena such as the precession of the perihelion of Mercury, the bending of light by stars, the Shapiro delay time and the frame-dragging precession of gyroscopes measured by the Gravity Probe B experiment [@will]. In the strong field sector the observation of gravitational waves originating in the merger of binary black holes [@gwaves] has also come as a reassurance of the general correctness of the theory.
The successful weak field predictions are all solutions of the field equations $R_{ik} = 0$, i.e. with $T_{ik} =0$ ‘outside’ a spherically symmetric body having mass and angular momentum, such as the Schwarzschild and Kerr metrics. Hence, $R_{ik} = 0$ are not necessarily ‘vacuum equations’ in the sense of a completely empty universe. On the other hand, attempts to use solutions of eqn (\[E\]) with $T_{ik} \neq 0$ in FLRW cosmology have led to many intractable problems such as the hypothetical non-baryonic dark matter [@dm], dark energy and the cosmological constant problem [@peebles], the horizon problem [@hor] and the flatness problem [@flat] which show no signs of going away.
In this context the fact that the Milne model [@milne; @milne2], which has no horizon problem, happens to be in broad agreement with most cosmological data without requiring the dark sector of the concordance $\Lambda$CDM cosmology [@vish; @vish2; @mac] comes as a surprise and points to a possible alternative paradigm. This model is, however, based on Special Relativity but is [*formally*]{} identical with the so-called ‘vacuum’ FLRW cosmology. It is therefore worth exploring if there exist some conditions under which it can be derived from a generalization of GR (a unified theory) without implying an empty universe.
Einstein himself was very unhappy with the role that the stress-energy tensor $T_{ik}$ played in GR. He had repeatedly emphasized that it was only a phenomenological representation of matter, to be regarded with caution. In 1936 he wrote [@Ein]:
> “\[General Relativity\] is sufficient–as far as we know–for the representation of the observed facts of celestial mechanics. But it is similar to a building, one wing of which is made of fine marble (left part of the equation), but the other wing of which is built of low-grade wood (right side of the equation). The phenomenological representation of matter is, in fact, only a crude substitute for a representation which would do justice to all known properties of matter.
Also, in a letter to Michele Besso he wrote [@einarch]:
> “But it is questionable whether the equation $R_{ik} - \frac{1}{2} g_{ik} R = T_{ik}$ has any reality left within it in the face of quanta. I vigorously doubt it. In contrast, the left-hand side of the equation surely contains a deeper truth. If the equation $R_{ik} = 0$ really determines the behavior of the singularities, then a law describing this behavior would be justified far more deeply than the aforementioned equation, which is not unified and only phenomenologically justified.”
These quotes show that the stress-energy tensor was unsatisfactory to Einstein for two reasons. First, it is not geometrical in nature like the left side of eqn (\[E\]) and hence not unified with it, and second, it does not reflect the quantum nature of matter and radiation. It is merely phenomenological, a placeholder for a more satisfactory theory of matter. This is why later on Einstein preferred to work with the equation $R_{ik} = 0$ in which matter appears as singularities and follow geodesics, though even this was a placeholder for a more satisfactory future theory of matter [@leh; @Ein2].
It was therefore natural for him to try and construct a unified geometrical theory of all fields with the hope that the quantum features would emerge as consequences of the nonlinearity of the theory. The only known long range interactions being electromagnetic and gravity, he sought to bring them under one umbrella. Now, in General Relativity the number of independent variables is ten (the ten components of the metric tensor $g_{\mu\nu}$). Hence, in order to incorporate electromagnetism into a unified theory, one needed additional variables. There were many options for this. After Weyl’s and Kaluza’s attempts at unification, it was Eddington [@ed] who first proposed to replace the metric as a fundamental concept by a non-symmetric affine connection $\Gamma$ which could then be split into a symmetric and an anti-symmetric part. However, Einstein [@ein0], supported by Schrödinger [@sch], extended the idea to include also a non-symmetric metric $g$. Just as passing beyond Euclidean geometry gravitation makes its appearance, so going beyond Riemannian geometry electromagnetism appears [*naturally*]{} as the anti-symmetric part of the metric [*without requiring any higher dimensional space*]{}.
A major problem with such unified theories is that the symmetry between gravity and electromagnetism is actually badly broken in the universe, the electromagnetic interaction being enormously ($\sim 10^{36}$) stronger than gravity, but this problem is not usually addressed. The purpose of this paper is to consider a broken-symmetric unified theory with the aim of exploring if, under certain conditions, it leads to the Milne metric. To lead up to it, a minimal unified theory will be presented in the next section to set the scene for the broken-symmetric theory.
A Minimal Unified Theory of Gravity and Electromagnetism
========================================================
Let $U_4$ be a smooth manifold with signature $(-,+,+,+)$ and endowed with a non-symmetric affine connection $\Gamma$ and a non-symmetric metric $g$. Let $$R_{ik} = \Gamma^\alpha_{ik,\,\alpha} - \Gamma^\alpha_{i\alpha,\,k} + \Gamma^\xi_{ik}\Gamma^\lambda_{\xi\lambda} - \Gamma^\xi_{i\lambda}\Gamma^\lambda_{\xi k} \label{R}$$ be the non-symmetric curvature tensor. Let us also define $$\begin{aligned}
\Gamma^\lambda_{(ik)} &=& \frac{1}{2}\left(\Gamma^\lambda_{ik} + \Gamma^\lambda_{ki}\right),\\
\Gamma^{\lambda}_{[ik]} &=& \frac{1}{2}\left(\Gamma^\lambda_{ik} - \Gamma^\lambda_{ki}\right),\\
\Gamma_l &=& \frac{1}{2}(\Gamma^\lambda_{l\lambda} - \Gamma^\lambda_{\lambda\ l}).\end{aligned}$$ $\Gamma^\lambda_{[ik]}$ is called the Cartan torsion tensor. Following Einstein [@ein], let us put $\Gamma_l = 0$ since it is not determined by any equation in the theory. Similarly, let $$\begin{aligned}
\bar{g}^{(ik)} &=& \frac{1}{2}\sqrt{-g}\left(g^{ik} + g^{ki}\right),\\
\bar{g}^{[ik]} &=& \frac{1}{2}\sqrt{-g}\left(g^{ik} - g^{ki}\right). \end{aligned}$$ To restrict the number of possible covariant terms in a non-symmetric theory, Einstein and Kaufman [@ein] imposed [*transposition invariance*]{} and [*$\Lambda$-transformation invariance*]{} on the theory. Let $\tilde{\Gamma}^\lambda_{ik} = \Gamma^\lambda_{ki}$ and $\tilde{g}_{ik} = g_{ki}$. Then terms that are invariant under the simultaneous replacements of $\Gamma^\lambda_{ik}$ and $g_{ik}$ by $\tilde{\Gamma}^\lambda_{ik}$ and $\tilde{g}_{ik}$ are called transposition invariant. For example, the tensor $R_{ik}$ (\[R\]) is not transposition invariant because it is transposed to $$\tilde{R}_{ik} = \Gamma^\alpha_{ki,\,\alpha} - \Gamma^\alpha_{\alpha i,\,k} + \Gamma^\xi_{ki}\Gamma^\lambda_{\xi\lambda} - \Gamma^\xi_{\lambda i}\Gamma^\lambda_{k\xi}. \label{R2}$$ Next, define the transformations $$\begin{aligned}
\Gamma^{i\prime}_{kl} &=& \Gamma^{i}_{kl} + \delta^i_k \lambda_{,\,l},\nonumber\\
g^{ik\prime} &=& g^{ik}\label{proj},\end{aligned}$$ where $\lambda$ is an arbitrary function of the coordinates. Then $R_{ik}$ (eqn \[R\]) is $\lambda$-transformation invariant (or projective invariant). What this means is that a theory characterized by $R_{ik}$ cannot determine the $\Gamma$-field completely but only up to an arbitrary function $\lambda$. Hence, in such a theory, $\Gamma$ and $\Gamma^\prime$ represent the same field. Further, this [*$\lambda$-transformation*]{} produces a non-symmetric $\Gamma^\prime$ from a $\Gamma$ that is symmetric or anti-symmetric in the lower indices. Hence, the symmetry condition for $\Gamma$ loses objective significance. This sets the ground for a genuine unification of gravity and electromagnetism, the former determined by the symmetric part and the latter by the antisymmetric part of the action.
Let us write the simplest transposition invariant and $\lambda$-transformation invariant Lagrangian [@bose] $$\begin{aligned}
{\cal{L}} &=& \frac{1}{2}\sqrt{-g}\left(g^{ik}R_{ik} + \tilde{g}^{ik}\tilde{R}_{ik}\right)\label{L},\end{aligned}$$ which can be expressed as (vide Appendix) $$\begin{aligned}
{\cal{L}} &=& \bar{g}^{(ik)}\left(R_{(ik)} - \Gamma^\lambda_{[i\xi]}\Gamma^\xi_{[\lambda k]}\right) + \bar{g}^{[ik]}\Gamma^\lambda_{[ik];\,\lambda}\\
&:=& \bar{g}^{(ik)}R^\prime_{(ik)} + \bar{g}^{[ik]}\Gamma^\lambda_{[ik];\,\lambda}. \end{aligned}$$ Variation of the action $\int {\cal{L}} d^4 x$ holding $\bar{g}^{(ik)}$ and $\bar{g}^{[ik]}$ constant at once implies $$\begin{aligned}
R^\prime_{(ik)} &=& R_{(ik)}- \Gamma^\lambda_{[i\xi]}\Gamma^\xi_{[\lambda k]} = 0,\label{1}\\
\Gamma^\lambda_{[ik];\,\lambda} &=& 0\label{2}.\end{aligned}$$ It can also be shown (vide Appendix) using the variational principle that the equation connecting $g$ and $\Gamma$ in this non-symmetric theory is $$\begin{aligned}
\bar{g}^{ik}_{\,\,\,\,\,,\lambda} &+& \bar{g}^{i\alpha}\Gamma^{\prime\, k}_{\lambda\alpha} + \bar{g}^{\alpha k}\Gamma^{\prime\, i}_{\alpha\lambda} - \bar{g}^{ik}\Gamma^\alpha_{(\lambda\alpha)} = 0,\label{3}\end{aligned}$$ where $$\begin{aligned}
\Gamma^{\prime\, k}_{\lambda\alpha} &=& \Gamma^{k}_{(\lambda\alpha)} + \Gamma^{k}_{[\lambda\alpha]},\nonumber\\
\Gamma^{\prime\, i}_{\alpha\lambda} &=& \Gamma^{i}_{(\alpha\lambda)} + \Gamma^{i}_{[\alpha\lambda]},\nonumber\end{aligned}$$ and further that $$\bar{g}^{[i\alpha]}_{\,\,\,\,\,\,\,\,\,\,,\,\alpha} = 0.\label{4}$$ The last equation can be interpreted as Maxwell’s equations for electrodynamics by identifying $\bar{g}^{[ik]}$ with the dual electromagnetic field $\tilde{F}^{ik}$. The electric current is given by $$\begin{aligned}
j^i &=& \frac{1}{3!}\zeta\epsilon^{i\nu\lambda\rho} \left(\bar{g}_{[\nu\lambda],\,\rho} + \bar{g}_{[\lambda\rho],\,\nu} + \bar{g}_{[\rho\nu],\,\lambda}\right)\nonumber\\
&=&\frac{1}{3!}\epsilon^{i\nu\lambda\rho}\left(\tilde{F}_{\nu\lambda,\,\rho} + \tilde{F}_{\lambda\rho,\,\nu} + \tilde{F}_{\rho\nu,\,\lambda}\right)\nonumber\\
&=& F^{i\nu}_{\,\,\,\,,\,\nu} = 0,\label{current}\end{aligned}$$ where $\zeta$ is a suitable dimensional constant and the current vanishes because of the Bianchi identity in the first line. Hence, this theory describes free electromagnetic fields.
The equation set (\[1\]), (\[2\]), (\[3\]), (\[4\]) are the fundamental equations of the theory.
Notice that that the Ricci tensor $R^\prime_{(ik)}$ in the theory, which is flat, has an additional term compared to the GR Ricci tensor $R_{(ik)}$. Clearly, the additional curvature has its origin in the torsion in the manifold $U_4$.
Broken Symmetric Unified Theory
===============================
The theory outlined so far unifies gravity and electromagnetism fully, but in nature, as we have seen, these two interactions are distinguished by their widely different strengths, signalling a broken symmetry. One way to break the symmetry explicitly is to admit terms into the Lagrangian that are not $\lambda$-transformation or projective invariant. To have such a theory one needs to relax the Einstein condition $\Gamma_l = 0$. A general Lagrangian of this kind is [@bose] $$\begin{aligned}
{\cal{L}} &=& \sqrt{-g}\left[g^{ik}R_{ik} + \tilde{g}^{ik}\tilde{R}_{ik}\right] + a\bar{g}^{(ik)}\Gamma_i\Gamma_k + b \bar{g}^{[ik]}\left(\Gamma_{i,\,k} - \Gamma_{k,\,i}\right) \label{L2}\end{aligned}$$ where $a$ and $b$ are two arbitrary dimensionless parameters that are not fixed by any symmetry. Projective invariance requires $a$ to vanish but not $b$. Then eqns (\[1\]) and (\[2\]) are modified to $$\begin{aligned}
{\cal{R}}_{ik} = R_{(ik)}- Q^\lambda_{i\xi}Q^\xi_{\lambda k} + x \Gamma_i\Gamma_k\nonumber\\ =
R_{(ik)}- \Gamma^\lambda_{[i\xi]}\Gamma^\xi_{[\lambda k]} + a \Gamma_i\Gamma_k &=& 0,\label{mod1}\\
Q^\lambda_{[ik];\,\lambda} - y\left(\Gamma_{i,\,k} - \Gamma_{k,\,i}\right) &=& 0,\end{aligned}$$ where $$\begin{aligned}
Q^\lambda_{ik} &=& \Gamma^{\,\,\,\,\,\,\,\,\,\,\lambda}_{[ik]} + \frac{1}{3}\delta^\lambda_i \Gamma_k - \frac{1}{3}\delta^\lambda_k \Gamma_i,\\
Q^\lambda_{ik;\,\lambda} &=& Q^\lambda_{ik,\,\lambda} - Q^\lambda_{i\xi}\Gamma^\xi_{(\lambda k)} - Q^\lambda_{\xi k}\Gamma^\xi_{(i\lambda)} + Q^\xi_{ik}\Gamma^\lambda_{(\xi\lambda)},\end{aligned}$$ and $x = a + \frac{1}{3}$, $y = \frac{1}{6} - b$. Hence, the new Ricci tensor ${\cal{R}}_{ik}$ is also flat though the universe has other fields than gravity, and is hence not empty.
Eqn (\[mod1\]) will lead to corrections to the Schwarzschild and Kerr metrics that are analogous to the Reissner-Nordström and Kerr-Newman metrics.
The variational principle is a little more complex because $Q^\lambda_{i\lambda} = 0$ and all the 24 components of $Q^\lambda_{ik}$ are not independent, and consequently one has to use a Lagrange multiplier $k^i$ [@bose]. The upshot is that the equations connecting the $g$’s and $\Gamma$’s are of the form $$\begin{aligned}
\bar{g}^{ik}_{\,\,\,\,\,,\lambda} + \bar{g}^{i\alpha}\Gamma^{\prime\prime\, k}_{\lambda\alpha} + \bar{g}^{\alpha k}\Gamma^{\prime\prime\, i}_{\alpha\lambda} &=& 3 \bar{g}^{ik}\Phi_\lambda,\label{eqcon}\\
\Phi_\lambda = g_{[\lambda\beta]}k^\beta &=& -\frac{1}{3}\left(\frac{x}{y}\right) g_{[\lambda\beta]}g^{(\beta\alpha)}\Gamma_\alpha,\end{aligned}$$ where $$\begin{aligned}
\Gamma^{\prime\prime\, k}_{\lambda\alpha} &=& \Gamma^{k}_{(\lambda\alpha)} + Q^{k}_{\lambda\alpha} + \frac{1}{\sqrt{\vert g\vert}}(g_{\lambda\beta}k^\beta \delta^k_\alpha - g_{\beta\alpha}k^\beta \delta^k_\lambda).\nonumber\\\end{aligned}$$ Equation (\[4\]) is modified to $$\bar{g}^{[i\alpha]}_{\,\,\,\,\,\,\,\,,\alpha} = 3k^i = - \frac{x}{y}\bar{g}^{(i\alpha)}\Gamma_\alpha := \theta \Gamma^i,\,\,y\neq 0.\label{curr}$$ Thus, $\Gamma^i$ turns out to be the source of the dual electromagnetic field unless $x = 0$, i.e. $a = -\frac{1}{3}$ and $k^i = 0$. If $\Gamma_l = k_l = 0$, one gets back the minimally unified theory. By multiplying the equation by the dimensional parameter $\zeta$, we can write it in the form $$\tilde{F}^{i\alpha}_{\,\,\,\,\,\,\,\,,\alpha} = j^i_m \label{jm}$$ where $j^i_m = \zeta\theta\Gamma^i$ is the magnetic source current which is automatically conserved because $\tilde{F}^{i\alpha} = -\tilde{F}^{\alpha i}$. Defining $F_{kl} = \epsilon_{kli\alpha}\tilde{F}^{i\alpha}$, we have $$\partial^k F_{kl} = \partial^k\epsilon_{kli\alpha}\tilde{F}^{i\alpha} := j_l \label{j}$$ where $j_l$ is the electric source current which is also automatically conserved. These two equations ((\[jm\]), (\[j\])) constitute the complete set of Maxwell equations in the presence of the two source currents: $$\begin{aligned}
\vec{\nabla} \times \vec{B} - \frac{\partial \vec{E}}{\partial t} &=& \vec{j},\,\,\,\,\,\,\vec{\nabla}. \vec{E} = \rho_e,\label{ma1}\\
\vec{\nabla} \times \vec{E} + \frac{\partial \vec{B}}{\partial t} &=& - \vec{j}_m,\,\,\,\,\,\,\vec{\nabla}. \vec{B} = \rho_m.\label{ma2}\end{aligned}$$ The two Bianchi identities that must be satisfied are $$\begin{aligned}
F_{\mu\nu,\,\lambda} + F_{\nu\lambda,\,\mu} + F_{\lambda\mu,\,\nu} &=& 0, \label{B1}\\
\tilde{F}_{\mu\nu,\,\lambda} + \tilde{F}_{\nu\lambda,\,\mu} + \tilde{F}_{\lambda\mu,\,\nu} &=& 0. \label{B2} \end{aligned}$$ These identities are consistent with the inhomogeneous Maxwell equations provided one makes the following identifications: $$\begin{aligned}
F^{0i} &=& -(E^i - E^{\prime\,i}),\,\,\,\,\,\, F^{ij} = - \epsilon^{ijk} (B_k - B_k^\prime),\\
\tilde{F}^{0i} &=& - (B^i - B^{\prime\,i}),\,\,\,\,\,\,\tilde{F}^{ij} = \epsilon^{ijk}(E_k - E^\prime_k), \end{aligned}$$ where $(\vec{E}^\prime, \vec{B}^\prime)$ are auxiliary fields that satisfy the conditions $$\begin{aligned}
\vec{\nabla}.\vec{E}^{\prime} &=& \rho_e,\,\,\,\,\vec{\nabla}\times\vec{E}^{\prime} = -\vec{j}_m,\\
\vec{\nabla}.\vec{B}^{\prime} &=& \rho_m,\,\,\,\,\vec{\nabla}\times\vec{B}^{\prime} = \vec{j},\\
\frac{\partial \vec{E}^{\prime}}{\partial t} &=& \frac{\partial \vec{B}^{\prime}}{\partial t} = 0. \end{aligned}$$ One can now define potentials $A^\mu = (\phi, \vec{A}), \tilde{A}^\mu = (\tilde{\phi}, \vec{\tilde{A}})$ through the relations $$\begin{aligned}
\vec{E} &=& -\frac{\partial \vec{A}}{\partial t} - \vec{\nabla}\phi,\\
\vec{B} &=& \vec{\nabla}\times \vec{A},\\
\vec{E}^{\prime} &=& \vec{\nabla}\times \vec{\tilde{A}} + \vec{\nabla}\phi, \\
\vec{B}^{\prime} &=& \vec{\nabla}\times \vec{A} + \vec{\nabla}\tilde{\phi}\end{aligned}$$ in the Lorentz gauge $\vec{\nabla}.\vec{A} = \vec{\nabla}.\vec{\tilde{A}} = 0$ and with $\nabla^2\tilde{\phi} = \rho_m,\,\,\nabla^2\phi = \rho_e, \nabla^2\vec{A} = -\vec{j}, \nabla^2\vec{\tilde{A}} = \vec{j}_m$. Hence, [*the theory allows magnetic charges and currents without Dirac strings*]{} [@dir].
An interesting feature of the presence of $\Gamma_i$ is that it makes electrodynamics invariant under continuous transformations [@heav; @lar] $$\begin{aligned}
\vec{E}\rightarrow\vec{E}^\prime &=& \vec{E}{\rm cos}\theta - \vec{B}{\rm sin}\theta,\\
\vec{B}\rightarrow \vec{B}^\prime &=& \vec{E}{\rm sin}\theta + \vec{B}{\rm cos}\theta,\end{aligned}$$ where $0\leq\theta\leq \pi/2$. Hence, $$\begin{aligned}
\vec{j}^\prime &=& \vec{j}{\rm cos}\theta - \vec{j}_m{\rm sin}\theta,\\
\vec{j}_m^\prime &=& \vec{j}{\rm sin}\theta + \vec{j}_m{\rm cos}\theta,\\
\rho_e^\prime &=& \rho_e{\rm cos}\theta - \rho_m{\rm sin}\theta,\\
\rho_m^\prime &=& \rho_e{\rm sin}\theta + \rho_m{\rm cos}\theta.\end{aligned}$$ For $\theta = \pi/2$ one has $\vec{E} \rightarrow -\vec{B}, \vec{B} \rightarrow \vec{E},
(\rho_e, \vec{j}) \rightarrow (-\rho_m, -\vec{j}_m),
(\rho_m, \vec{j}_m) \rightarrow (\rho_e, \vec{j})$. This shows that there is complete equivalence and continuous freedom in the choice of electric and magnetic quantities.
Quantization
------------
Note that $(\rho_m, \rho_e)$ are time components of the corresponding 4-currents which are determined in terms of continuous fields (vide eqns (\[current\]) and (\[curr\])). Also, note that no condition has been imposed so far on $\Gamma_i$. Instead of imposing the Einstein condition $\Gamma_i = 0$, if one imposes the weaker condition $$\Gamma_{i,\,k} - \Gamma_{k,\,i} = 0,\label{cond}$$ then like ${\cal{R}}_{ik} = 0$, $Q^\lambda_{[ik];\,\lambda} = 0$, and one immediately gets a very interesting result, namely that $\Gamma_i$, and hence the magnetic source current $j_{m\,i} = \zeta \Gamma_i$, is an [*irrotational*]{} or curl-less axial vector. Let $S = \mathbb{R}^3\backslash \left\{(0,0,z\leq 0)|z \in \mathbb{R}\right\}$ be the usual 3-dimensional space with the negative $z$-axis, along which $\vec{\Gamma} \neq 0$, removed. Then the curl-less vector $\vec{\Gamma} = -\vec{\nabla}\tilde{\Phi},\,\nabla^2 \tilde{\Phi} = 0$ has vortex solutions $\vec{\Gamma} = \vec{e}_\phi/r$ where $\vec{e}_\phi$ is a unit vector, and the integral over a unit counterclockwise circular path $C$ in the $xy$ plane enclosing the origin is $$\frac{1}{2\pi}\oint_C \vec{\Gamma}.\vec{e}_\phi\, d\phi = n,\,\,n\in \mathbb{Z},$$ where $n$ is a winding number which can be interpreted as the number of magnetic charges enclosed by the unit circle. For $n = 1$ one has a single magnetic charge, i.e. a magnetic monopole carrying some charge $g$. It is a straightforward consequence of condition (\[cond\]). Because of the Larmor-Heaviside symmetry, there is also an electric monopole, i.e. a particle carrying electrical charge $e$. Thus, particles emerge as topological charges. The product $eg$ has the dimension of action, and the fundamental unit of action in nature being the Planck constant $h$, it follows that $eg/h =$ constant, which is essentially the topological basis of Dirac quantization.
Cosmological Implications
=========================
We have seen that in the broken symmetric unified theory the Ricci tensor ${\cal{R}}_{ik}$ is flat. This will have non-trivial implications for FLRW cosmology whose metric is $$ds^2 = dt^2 - a^2(t)\left[\frac{dr^2}{1 - kr^2} + r^2 d\theta^2 + r^2 {\rm sin}^2\theta d\phi^2 \right].$$ In GR the spatial curvature $k$ is related to the Ricci scalar $R = g^{(ik)}R_{ik}$ by the relation $k = -a^2R/6$. In the broken symmetric unified theory this relation is modified to the form $$k = - \frac{a^2}{6}(R - \Gamma_T + a\Gamma)$$ where $$\begin{aligned}
\Gamma_T &=& g^{(ik)}\Gamma^\lambda_{[i\xi]}\Gamma^\xi_{[\lambda k]},\,\,\Gamma = g^{(ik)}\Gamma_i\Gamma_k.\end{aligned}$$ Hence $k = -1$ provided $$R -\Gamma_T + a\Gamma = 6/a^2.\label{X}$$ This is therefore the required condition to have the Milne metric in a broken symmetric unified theory which reduces to GR in the absence of torsion. Notice that the additional torsional terms in ${\cal{R}}_{ik}$ (\[mod1\]) are due to the presence of electromagnetic fields and charged particles in the universe which is therefore not empty. In principle it is possible to express them in terms of electromagnetic fields and currents by using solutions of the equation (\[eqcon\]) which give the connections in terms of the metrics, but they turn out to be very complex [@bose2; @ton; @hlav].
Concluding Remarks
==================
A remarkable feature of the broken symmetric unified theory with $\Gamma_i \neq 0$ is the occurrence of particles as topological charges carrying electric and magnetic charge provided $\Gamma_i$ is curl-less. Thus, Einstein’s dream of deriving particles from fields is at least partially realized.
Furthermore, the theory implies Milne cosmology if condition (\[X\]) holds. Milne cosmology is consistent with with most cosmological data without requiring the dark sector of the concordance $\Lambda$CDM cosmology.
Because of the presence of the magnetic current $\Gamma_l$ there is also a natural classical explanation of magnetism in the universe. Magnetism is ubiquitous in the universe, and primordial magnetic fields are specially important for probing the physics of the early universe [@kanda].
Acknowledgement
===============
The author thanks the National Academy of Sciences, India for a grant.
Appendix
========
Straightforward algebra gives $$\begin{aligned}
{\cal{L}} &=& \frac{1}{2}\sqrt{-g}\left(g^{ik}R_{ik} + \tilde{g}^{ik}\tilde{R}_{ik}\right)\nonumber\\
&=& \left[\bar{g}^{(ik)}(R_{ik} + \tilde{R}_{ki}) + \bar{g}^{[ik]}(R_{ik} - \tilde{R}_{ki})\right]\nonumber\\
&=& \bar{g}^{(ik)}\left[R_{ik} - \Gamma^{\,\,\,\,\,\,\,\,\,\,\lambda}_{[i\xi]}\Gamma^{\,\,\,\,\,\,\,\,\,\,\xi}_{[\lambda k]}\right]\nonumber\\ &+& \bar{g}^{[ik]}\left(\Gamma^\lambda_{[ik],\,\lambda} -\Gamma^\lambda_{[i\lambda],\,k} +\Gamma^\xi_{ik}\Gamma^\lambda_{\xi\lambda} - \Gamma^\xi_{i\lambda}\Gamma^\lambda_{\xi k} - \Gamma^\xi_{ki}\Gamma^\lambda_{\lambda\xi} + \Gamma^\xi_{\lambda i}\Gamma^\lambda_{k\xi}\right)\nonumber\\
&=& \bar{g}^{(ik)}\left[R_{ik} - \Gamma^{\,\,\,\,\,\,\,\,\,\,\lambda}_{[i\xi]}\Gamma^{\,\,\,\,\,\,\,\,\,\,\xi}_{[\lambda k]}\right] + \bar{g}^{[ik]}\left[\Gamma^\lambda_{ik;\,\lambda})\right]\end{aligned}$$ Following [@bose] but using $\Gamma_\mu = 0$, let $${\cal{L}} = H + \frac{d X^\lambda}{dx^\lambda}\label{H}$$ with $$\begin{aligned}
X^\lambda &=& \bar{g}^{(ik)}\Gamma_{(ik)}^\lambda - \bar{g}^{(i\lambda)}\Gamma^k_{(ik)} + \bar{g}^{[ik]}\Gamma^\lambda_{[ik]}\nonumber\\
H &=& - \bar{g}^{(ik)}_{\,\,\,\,\,\,,\lambda}\Gamma^\lambda_{(ik)} + \bar{g}^{(i\lambda)}_{\,\,\,\,\,,\lambda}\Gamma^k_{{(ik)}} + \bar{g}^{(ik)}\left(\Gamma^\xi_{(ik)}\Gamma^\lambda_{(\xi\lambda)} - \Gamma^\xi_{(i\lambda)}\Gamma^\lambda_{(\xi k)} - \Gamma^\lambda_{[i\xi]}\Gamma^\xi_{[\lambda k]}\right)\nonumber\\ &-& \bar{g}^{[ik]}_{\,\,\,\,\,\,,\lambda}\Gamma^\lambda_{[ik]} + \bar{g}^{[ik]}\left[- \Gamma^\lambda_{[i\xi]}\Gamma^\xi_{(\lambda k)} - \Gamma^\lambda_{[\xi k]}\Gamma^\xi_{(i\lambda)} + \Gamma^\xi_{[ik]}\Gamma^\lambda_{(\xi\lambda)}\right] \nonumber\end{aligned}$$ Thus, $H$ is free of the partial derivatives of $\Gamma^\lambda_{(ik)}$ and $\Gamma^\lambda_{[ik]}$, and the four-divergence term in the action integral is equal to a surface integral at infinity on which all arbitrary variations are taken to vanish.
Variations of $H$ w.r.t $\Gamma^\lambda_{(ik)}$ and $\Gamma^\lambda_{[ik]}$ give $$\begin{aligned}
\bar{g}^{(ik)}_{\,\,\,\,\,,\lambda} + \bar{g}^{(i\alpha)}\Gamma^k_{(\lambda\alpha)} + \bar{g}^{(\alpha k)}\Gamma^i_{(\alpha\lambda)} - \bar{g}^{(ik)}\Gamma^\alpha_{(\lambda\alpha)} = -[\bar{g}^{[i\alpha]}\Gamma^{\,\,\,\,\,\,k}_{[\lambda\alpha]} + \bar{g}^{[\alpha k]}\Gamma^{\,\,\,\,i}_{[\alpha\,\,\,\,\lambda]}]\label{1a} \\
\bar{g}^{[ik]}_{\,\,\,\,\,,\lambda} + \bar{g}^{[i\alpha]}\Gamma^{k}_{(\lambda\alpha)} + \bar{g}^{[\alpha k]}\Gamma^{i}_{(\alpha\lambda)} - \bar{g}^{[ik]}\Gamma^\alpha_{(\lambda\alpha)} = - [\bar{g}^{(i\alpha)}\Gamma^{\,\,\,\,\,\,k}_{[\lambda\alpha]} + \bar{g}^{(\alpha k)}\Gamma^{\,\,\,\,i}_{[\alpha\,\,\,\,\lambda]}]\label{2a}\end{aligned}$$ Adding (\[1a\]) and (\[2a\]), we get $$\begin{aligned}
\bar{g}^{ik}_{\,\,\,\,,\lambda} &+& \bar{g}^{i\alpha}\left(\Gamma^k_{(\lambda\alpha)} + \Gamma^{k}_{[\lambda\alpha]}\right) + \bar{g}^{\alpha k}\left(\Gamma^i_{(\alpha\lambda)} + \Gamma^{i}_{[\alpha\lambda]}\right) - \bar{g}^{ik}\Gamma^\alpha_{(\lambda\alpha)} = 0.\end{aligned}$$ This can be written as $$\begin{aligned}
\bar{g}^{ik}_{\,\,\,\,\,,\lambda} &+& \bar{g}^{i\alpha}\Gamma^{\prime\, k}_{\lambda\alpha} + \bar{g}^{\alpha k}\Gamma^{\prime\, i}_{\alpha\lambda} - \bar{g}^{ik}\Gamma^\alpha_{(\lambda\alpha)} = 0,\label{ggamma}\\
\Gamma^{\prime\, k}_{\lambda\alpha} &=& \Gamma^{k}_{(\lambda\alpha)} + \Gamma^{k}_{[\lambda\alpha]},\nonumber\\
\Gamma^{\prime\, i}_{\alpha\lambda} &=& \Gamma^{i}_{(\alpha\lambda)} + \Gamma^{i}_{[\alpha\lambda]}.\nonumber\end{aligned}$$ This is eqn (\[3\]).
By contracting (\[ggamma\]) once with respect to $(k, \lambda)$, then with respect to $(i, \lambda)$, and subtracting the equations term by term, one gets eqn (\[4\]).
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A. Einstein, [*Autobiographical Notes*]{}, The Library of Living Philosophers, Harper & Row Publishers, 1949, p. 93. A. S. Eddington, [*Proc. R. Soc. London, Ser. A*]{} [**99**]{}, 104–122 (1921). A. Einstein, [*Sitzungsber. Preuss. Akad. Wiss.*]{} [**5**]{}, 32-38, (1923), [*ibid*]{} [**22**]{}, 414-419, (1925). E. Schrödinger, [*Space-Time Structure*]{}, Cambridge University Press, 1950. A. Einstein, [*The Meaning of Relativity*]{}, Methuen & Co., London, Sixth Edition, 1956. S. N. Bose, [*Le Jour de Phys et le Radium*]{} (Paris) [**14**]{}, 641-644 (1952). G. Heaviside, [*Phil. Trans. Roy. Soc.(London) A*]{} [**183**]{}, 423 (1893). I. Larmor, [*Collected Papers*]{}, London(1928). P. A. M. Dirac, [*Proc. R. Soc. London A*]{} [**133**]{}, 60 (1931). S. N. Bose, [*Ann. of Math.*]{} [**59**]{} (1), 171-176 (1954).\
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[^1]: partha.ghose@gmail.com
|
---
abstract: 'Let $M$ be a closed oriented 3-manifold such that $S^1\times M$ admits a symplectic structure $\omega$. The goal of this paper is to show that $M$ is a fiber bundle over $S^1$. The basic idea is to use the obvious $S^1$-action on $S^1\times M$ by rotating the first factor of $S^1\times M$, and one of the key steps is to show that the $S^1$-action on $S^1\times M$ is actually symplectic with respect to a symplectic form cohomologous to $\omega$. We achieve it by crucially using the recent result or its relative version of Giroux about one-to-one correspondence between open book decompositions of $M$ up to positive stabilization and co-oriented contact structures on $M$ up to contact isotopy. As a consequence, we can give an answer to a question of Kronheimer concerning the relation between symplectic structures on $S^1\times M_K$ and fibered knots $K$, where $M_K$ denotes the result of $0$-surgery on $S^3$ along a knot $K$ in $S^3$. Moreover, a complete picture of the various intriguing implications between symplectic structures on $S^1\times M_K$ and fibered knots can be provided as in Table I below, and thus we fill in the missing links in the circle of ideas around this topic.'
author:
- Jin Hong Kim
title: '**Giroux correspondence, confoliations, and symplectic structures on $S^1\times M$**'
---
Introduction and statements of results {#sec1}
======================================
It was Thurston in [@Th] who first proved that any closed oriented smooth 4-manifold $X$ which fibers over a Riemann surface admits a symplectic structure, unless the fiber class is torsion in $H_2(X, {\bf Z})$. Thus, if the genus of the fiber of such a closed oriented 4-manifold $X$ is greater than or equal to 2, then the manifold $X$ always admits a symplectic structure. Moreover, any fibration of a closed oriented 3-manifold $M$ over a circle $S^1$ induces a symplectic structure on the 4-manifold $S^1\times M$. (See [@Mc-S] and [@G-S].) Furthermore, it seems to have been widely believed that the converse also holds (see [@Ta]).
\[conj1.1\] Let $M$ be a closed oriented $3$-manifold such that $S^1\times M$ admits a symplectic structure. Then $M$ fibers over $S^1$.
Indeed, there have been several attempts towards the conjecture, and it turns out that the conjecture is true for many important classes of 4-manifolds. For instance, see [@McC], [@C-M], [@Et], [@F-V], [@F-V2], and [@F-V3].
The goal of this paper is that by taking a completely different but elementary approach from the previous work, we give a short and affirmative proof to the Conjecture \[conj1.1\]. To do so, we shall use the $S^1$-action on the symplectic 4-manifold $S^1\times M$ obtained by rotating the first factor of $S^1\times M$. One crucial observation of the proof is that every symplectic class on $S^1\times M$ can always be represented by a symplectic form invariant under the $S^1$-action. We show this important fact by a recent result and its relative version of Giroux about one-to-one correspondence between contact structures up to contact isotopy and open book decompositions up to positive stabilizations (see [@Gi], [@HKM], [@EO], and [@Etn] for more details).
To the author’s knowledge, it is still unknown whether or not any action of a compact connected Lie group $G$ on a symplectic $2n$-manifold $X$ always induces a $G$-invariant symplectic form on $X$, in general. On the other hand, in Riemannian geometry one can always obtain a $G$-invariant Riemannian metric by taking the average of a Riemannian metric over the Lie group $G$. As $S^2\times S^2$ with the product symplectic form shows, simply taking the average of a symplectic form over the Lie group $G$ does not yield a $G$-invariant symplectic form [@Ono].
Once we show that there exists a symplectic form invariant under an $S^1$-action on $S^1\times M$, it is an easy and well-known procedure to complete the proof of the conjecture. In Section \[sec4\], for the sake of the reader we provide a proof of Theorem \[thm4.1\], using an argument of D. Tischler in [@Ti] about fibering certain foliated manifolds over $S^1$. We remark that Theorem 18 in [@F-G-M] gives an alternative argument of the second half of the proof of our main Theorem \[thm4.1\].
As a generalization of the Theorem \[thm4.1\], Baldridge asked in [@Ba] whether or not for every closed symplectic 4-manifold admitting a free $S^1$-action whose orbit space is $M$ the quotient manifold $M$ fibers over $S^1$. We think that our method of the present paper can be adapted to answer the following Conjecture \[conj1.2\]. But we do not pursue it in this paper, for the sake of simplicity.
\[conj1.2\] Let $X$ be a closed symplectic $4$-manifold admitting a free $S^1$-action whose orbit space is $M$. Then the quotient manifold $M$ fibers over $S^1$.
Let us denote by $M_K$ a 3-manifold $M_K$ obtained by 0-surgery on a knot $K$ in $S^3$. As an interesting consequence, we can give an answer to the question in [@K] of Kronheimer about symplectic structures on $S^1\times
M_K$. To be more precise, Fintushel and Stern proved in [@F-S] that if $S^1\times M_K$ admits a symplectic structure then the symmetrized Alexander polynomial $\Delta_K(t)$ is monic. On the other hand, Kronheimer proved in [@K] that if the knot has a genus $g(K)$ of two or more a necessary condition for $S^1\times
M_K$ to admit a symplectic structure is that its genus $g(K)$ be equal to the degree of its symmetrized Alexander polynomial. It is a well-known fact ([@B-Z] or [@Ro]) that if a knot $K$ is fibered then its symmetrized Alexander polynomial is monic and its genus $g(K)$ is equal to the degree of is symmetrized Alexander polynomial. Since $S^1\times M_K$ is symplectic for fibered knots, Kronheimer raised a question whether or not $S^1\times M_K$ admits a symplectic structure for *non-fibered* knots such as the pretzel knot $P(5, -3, 5)$. The symmetrized Alexander polynomial of the pretzel knot $P(5, -3, 5)$ is $t-3+t^{-1}$ and thus monic with its degree equal to the genus $1$ of the knot. Our another main result is to give a negative answer to the question of Kronheimer as follows:
\[thm1.1\] The product $4$-manifold $S^1\times M_K$ admits a symplectic structure if and only if the knot $K$ is always fibered.
According to the recent paper [@F-V] of S. Friedl and S. Vidussi, the product of $S^1$ with the $0$-surgery of $S^3$ along the pretzel knot $P(5, -3, 5)$ does not admit a symplectic structure, which fits well with our result. We give the proof of Theorem \[thm1.1\] at the end of Section 4. As a corollary, as the pretzel knot $P(5, -3, 5)$ shows, the statement that the symmetrized Alexander polynomial of a knot $K$ is monic and its genus $g(K)$ is equal to the degree of its symmetrized Alexander polynomial does not imply that $S^1\times M_K$ admits a symplectic structure. In summary, when we set the statements [**(A)**]{}, [**(B)**]{}, and [**(C)**]{} as follows,
- $S^1\times M_K$ admits a symplectic structure,
- The symmetrized Alexander polynomial of a knot $K$ with genus $\ge 2$ is monic and its knot genus $g(K)$ is equal to the degree of its symmetrized Alexander polynomial,
- The knot $K$ is fibered,
we can establish the following table for the various implications:
[Table I]{} .3cm
Implication True/False Reason
------------------------------------------------ ------------ ------------------------------------------
[**(A)**]{} $\rightarrow$ [**(B)**]{} True Proved by Kronheimer and Fintushel-Stern
[**(B)**]{} $\rightarrow$ [**(A)**]{} False e.g., Pretzel knot $P(5,-3, 5)$
[**(B)**]{} $\rightarrow$ [**(C)**]{} False e.g., Pretzel knot $P(5,-3, 5)$
[**(C)**]{} $\rightarrow$ [**(B)**]{}$\dagger$ True Proposition 8.16 in [@B-Z] (Neuwirth)
[**(A)**]{} $\rightarrow$ [**(C)**]{} True Theorem \[thm1.1\]
[**(C)**]{} $\rightarrow$ [**(A)**]{} True Proved by Thurston
$\dagger$[For this direction, we do not need the restriction on the genus of a knot.]{}
Finally a few remarks are in order. During the preparation of our paper, two papers related to the Conjecture \[conj1.1\] have appeared. In their paper [@Ku-Ta], Kutluhan and Taubes studied the Seiberg-Witten Floer homology of $M$, assuming that $S^1\times M$ admits a symplectic form. As a consequence, by combining their results with Theorem 1 of Y. Ni in [@Ni], they gave a different proof that $M$ fibers over $S^1$, in case that $M$ has the first Betti number equal to 1 and the first Chern class of the canonical line bundle is not torsion. Friedl and Vidussi also posted a preprint [@Fr-Vi08] asserting the proof of Conjecture \[conj1.1\] modulo some technical step regarding the residually finite solvability of $\pi_1(M)$ which allegedly depends on a work under preparation by M. Aschenbrenner and S. Friedl. Among other things, the twisted Alexander polynomials, algebraic group theory, and Stallings’ characterization ([@St]) for the fibration of a $3$-manifold over a circle play crucial roles in their proof.
We organize this paper as follows. In Section \[sec2\], we give some basic facts about open book decompositions for closed contact 3-manifolds, partial open book decompositions for compact contact $3$-manifolds with convex boundary, and confoliations. Section \[sec3\] is one of the key sections for this paper. In that section, we show that every symplectic class on $S^1\times M$ can always be represented by a symplectic form invariant under the naturally defined $S^1$-action. Finally Section \[sec4\] is devoted to the proofs of the main Theorems \[thm4.1\] and \[thm4.2\].
Giroux correspondence and confoliations {#sec2}
=======================================
The aim of this section is to review some basic facts about open book decompositions for contact 3-manifolds, partial open book decompositions for compact contact $3$-manifolds with convex boundary, and confoliations.
First we briefly review the definition of an open book decomposition of a closed 3-manifold $M$, and its extension to compact contact 3-manifolds with convex boundary can be easily obtained with an obvious modification (see the recent papers [@HKM] and [@EO]). Let $(F, h)$ be a pair consisting of an oriented surface $F$ and a diffeomorphism $h:F\to F$ which is the identity on $\partial F$, and $K$ be a link in $M$. An open book decomposition for $M$ with binding $K$ is the quotient space $$((F\times [0,1])/\sim_h, (\partial F\times [0,1])/\sim_h))$$ which is homeomorphic to $M$. Here the equivalence relation $\sim_h$ is given by $$\begin{split}
&(x,1)\sim_h (h(x), 0) \ \text{for}\ x\in F\ \text{and}\\
&(x, t)\sim_h(x, t')\ \text{for}\ x\in \partial F\ \text{and}\ \text{all}\ t, t'\in [0,1].
\end{split}$$ We will call $F\times \{ t \}$ for $t\in [0,1]$ a *page* of the open book decomposition. Two open book decomposition is *equivalent* if there is an ambient isotopy between them taking binding to binding and pages to pages. We can obtain a new open book decomposition $(F,
h')$ from $(F, h)$ by a *positive* (resp. *negative*) *stabilization*. Namely, $F'$ is obtained from $F$ by attaching a 1-handle $B$ along $\partial F$ and $h'$ is obtained by extending $h$ by the identity map on the 1-handle $B$ and taking the composition $R_\gamma\circ h$ (resp. $R_\gamma^{-1}\circ h$) with the right-handed Dehn twist $R_\gamma$ along a simple closed curve $\gamma$ in $F'$ dual to the core of the 1-handle $B$.
It is known that every closed 3-manifold has an open book decomposition, but it is not unique. A contact structure $\tau$ is said to be *supported* (or *adapted*) by the open book decomposition $(F, h, K)$ if there is a contact 1-form $\lambda$ satisfying the following properties:
- $\lambda$ induces a symplectic form $d\lambda$ on each fiber $F$.
- $K$ is transverse to $\tau$, and the orientation on $K$ given by $\lambda$ is the same as the boundary orientation induced from $F$ coming from the symplectic structure.
Thurston and Winkelnkemper showed in [@TW75] that any open book decomposition $(F, h, K)$ supports a contact structure. The contact planes constructed by them can be made arbitrary close to the tangent planes of the pages away from the binding. Recently E. Giroux showed in [@Gi] that the converse also holds. To be more precise, the following theorem holds:
\[thm2.1\] Every contact structures $\tau$ on a closed $3$-manifold $M$ is supported by some open book decomposition $(F,h,K)$. Moreover, two open book decompositions $(F, h,K)$ and $(F', h', K')$ which support the same contact structure $(M,\tau)$ become equivalent after applying a sequence of positive stabilizations to each.
In our situation, we do not use the full version of this theorem. Rather we will need the result of Giroux to choose a coordinate chart on $S^1\times M$ with which we can easily calculate the Lie derivative of the symplectic form for our purposes (e.g., see Lemma \[lem3.3\] for more details). Even if the above Theorem \[thm2.1\] is stated for closed contact 3-manifold, the construction of Giroux shows that the same result holds for contact 3-manifolds with contact boundary, as the papers [@HKM] and [@EO] of Honda-Kazez-Mati' c and Etg" u-Ozbagci show.
For the sake of reader’s convenience, we briefly review the relative Giroux correspondence for compact contact $3$-manifolds with convex boundary, although in the present paper we do not need the full strength of this correspondence. For more details, see [@HKM] and [@EO], and most of what is presented here can be found in those two papers.
We first begin with the abstract version of a *partial open book decomposition* which is a triple $(S, P, h)$ satisfying the following three properties:
- $S$ is a compact oriented connected surface with non-empty boundary $\partial S$,
- $P=P_1\cup P_2\cup\cdots \cup P_r$ where $P_1$, $P_2$, $\ldots$, $P_r$ are $1$-handles is a proper, but not necessarily connected, subsurface of $S$ such that $S$ is obtained from the closure of $S\backslash P$ by attaching 1-handles $P_1$, $P_2$, $\ldots$, $P_r$ successively,
- $h: P\to S$ is an embedding such that $h|_{\partial P\cup\partial S}=$identity.
Given a partial open book decomposition $(S, P, h)$, we can construct a compact $3$-manifold with boundary as follows. Let $H=(S\times [-1,0])/\sim$, where $(x, t)\sim (x, t')$ for all $x\in \partial S$ and $t, t'\in [-1,0]$, which is a solid handlebody with $S\times \{ 0 \}\cup -S \times \{-1\}$ as the boundary under the obvious relation $(x, 0)\sim (x, -1)$ for all $x\in \partial S$. We also let $N=(P\times [0,1])/\sim$, where $(x,t)\sim (x,t')$ for all $x\in \partial P\cap \partial S$ and $t,t'\in [0,1]$. Again each component of $N$ is a solid handlebody whose boundary can be described by the connected arcs of the closure of $\partial P\backslash \partial S$. In other words, let $c_1, c_2, \cdots, c_n$ denote such connected arcs. Then each disk $D_i= (c_i\times [0,1])/\sim$ is contained in the boundary of $N$. Thus the boundary of $N$ consists of the union of the disjoint disks $D_i$’s and the surface $P\times \{ 1\}\cup -P\times \{0 \}$ with the relation $(x, 0)\sim (x, 1)$ for all $x\in \partial P\cap \partial S$.
Now let $M=N\cup H$ with the identification of $P\times \{ 0\}\subset \partial N$ (resp. $P\times \{ 1\}\subset \partial N$) with $P\times \{ 0\}\subset \partial H$ (resp. $h(P)\times \{ -1\}\subset \partial H$). Then $M$ is an oriented compact $3$-manifold with oriented boundary $$\label{eq2.1}
\partial M= (S\backslash P)\times \{ 0 \} \cup -(S\backslash h(P))\times \{ -1 \} \cup (\overline{\partial P\backslash \partial S})\times [0,1]$$ with the suitable identifications. If a compact $3$-manifold $M$ with boundary is obtained from the abstract partial open book decomposition $(S, P, h)$ as above, then the triple $(S, P, h)$ is called a *partial open book decomposition* of $M$. The notions such as compatibility of a contact structure with respect to a partial open book decomposition, the isomorphism class of two partial open book decompositions, and the definition of a positive stabilization of a partial open book decomposition can also be interpreted suitably for this relative version (e.g., see Definitions 1.10, 1.11, and1.13 in [@EO]).
Recall that a closed oriented embedded surface $\Sigma$ in a contact manifold $(M,\xi)$ is called *convex* if there is vector field tansverse to $\Sigma$ which preserves the contact structure $\xi$. A generic surface $\Sigma$ inside a contact manifold can be made convex (cf. [@Gi91] and Section 2.2 of [@Honda99]). So the assumption that the boundary be convex can be imposed without loss of generality.
In [@HKM], Honda-Kazez-Mati' c associated the isomorphism classes of compact contact $3$-manifolds with convex boundary to the isomorphism classes of partial open book decompositions modulo positive stabilizations. Conversely, in [@EO] Etg" u and Ozbagci constructed its inverse by describing a compact contact $3$-manifold with convex boundary compatible with a given partial open book decomposition. As in the proof of Proposition 1.9 in [@EO], such a construction is essentially given by the explicit construction of Thurston and Winkelnkemper. Hence the property, as well as others, that for closed contact $3$-manifolds the contact planes constructed by them can be made arbitrary close to the tangent planes of the pages away from the binding can also be used for compact $3$-manifolds with convex boundary.
Now we can state a relative version of Giroux correspondence as follows, which is a relative version of Giroux correspondence for closed contact $3$-manifolds (see Theorem 0.1 in [@EO]).
\[thm2.2\] There is a one-to-one correspondence between isomorphism classes of partial open book decompositions modulo positive stabilization and the isomorphism classes of compact contact $3$-manifolds with convex boundary.
In what follows, we will also need to use the result of Eliashberg and Thurston (Theorem 2.4.1 in [@E-T97]). A plane field $\eta=\ker \theta$ on an oriented 3-manifold is called *positive* (resp. *negative*) *confoliation* if $\theta\wedge d\theta\ge 0$ (resp. $\theta\wedge d\theta\le 0$). Let us denote by $\zeta$ the product foliation of the manifold $S^2\times S^1$ by the spheres $S^2\times \{ z \}$ for $z\in S^1$.
\[thm2.3\] Suppose that a $C^2$-confoliation $\eta$ on an oriented 3-manifold is different from the foliation $\zeta$ on $S^2\times S^1$. Then $\eta$ can be $C^0$-approximated by contact structure. When $\eta$ is a foliation it can be $C^0$-approximated both by positive and negative contact structure.
Existence of $S^1$-invariant symplectic structures {#sec3}
==================================================
Recall that there exists an obvious circle action on the 4-manifold $S^1\times M$ obtained by rotating the first factor of $S^1\times M$. The aim of this section is to show that every symplectic class on $S^1\times M$ can be represented by a symplectic form which is invariant under the obvious action of $S^1$.
In what follows, we assume that $S^1\times M$ admits a symplectic structure $\omega$. If $M$ is $S^2\times S^1$, then clearly $M$ fibers over $S^1$. So from now on we also assume that $M$ is not $S^1\times S^2$, unless stated otherwise. Let $X$ be the fundamental vector field associated to the action of $S^1$, and let $\alpha=\iota_X \omega$. Since $\omega$ is a symplectic 2-form, $\alpha$ is clearly a nowhere vanishing 1-form on $S^1\times M$. Now choose an arbitrary point $t$ in $S^1$. Let $j_t$ denote the inclusion from $M$ into $S^1\times M$ given by $x\mapsto (t, x)$. Then we obtain a nowhere vanishing 1-form $\beta_t$ by the pull-back of the 1-form $\alpha$ restricted to $\{ t \}\times M$ via the inclusion $j_t$. In other words, $\beta_t=j_t^\ast (\alpha|_{\{ t \}\times M})$. Then we have the following proposition.
\[prop3.1\] The differential 3-form $\beta_{t}\wedge d_M\beta_{t}$ should vanish identically for all $t\in S^1$.
We divide the proof into the following three cases:
[**(Case 1)**]{} First of all, assume that $\beta_t\wedge d_M
\beta_t$ is non-zero for all $t$. Thus the 2-plane field $\xi_t=\ker
\beta_t$ is a family of contact structures on $M$. It is obvious that $d_M \beta_t$ is a nowhere vanishing 2-form on $M$. Moreover, the following holds.
\[lem3.1\] Under our assumption, the Lie derivative ${\mathcal L}_X \omega$ is nowhere vanishing on $S^1 \times M$.
The proof follows from the Cartan’s formula. Indeed, it suffices to note that $$\label{eq3.1}
\begin{split}
0 &\ne d_M\beta_t =d_M j_t^\ast (\iota_X \omega)=j_t^\ast d \iota_X
\omega=j_t^\ast (d\iota_X \omega+ \iota_X d\omega)\\
&=j_t^\ast ({\mathcal L}_X \omega).
\end{split}$$ This completes the proof of Lemma \[lem3.1\].
Next we can show the following
\[lem3.2\] The symplectic $2$-form $\omega$ restricted to the contact structure $\xi_t=\ker \beta_t$ along $\{ t \} \times M$ is non-zero.
To see it, first note that there exists a Reeb vector field $Z_t$ on $M$, depending on the parameter $t$, such that $$\label{eq3.2}
1=\beta_t(Z_t)=j_t^\ast (\iota_X \omega)(Z_t)=\omega(X,
(j_t)_\ast(Z_t)).$$ Since, along each point $(t, x)$ in $S^1\times M$, the vector space spanned by $X$ and $(j_t)_\ast(Z_t)$ is transversal to the contact plane $\xi_t$ and the equation is satisfied, we conclude that the restriction $\omega|_{\xi_t}$ of the symplectic form $\omega$ on $S^1\times M$ is non-zero.
Now, we apply the result of Giroux in [@Gi] concerning the open book decomposition of a contact 3-manifold (Theorem \[thm2.2\]). In our situation, we can choose a family of open book decompositions along a connected binding $B_t$ associated to the contact structure $\beta_t$ on $M$, so that the parameter $s_t$ for the base manifold $S^1$ for the fibration associated to the open book decomposition is given by the Reeb vector field $Z_t$. We recall that by the way of the construction of the open book decomposition the contact plane $\xi_t$ can be made arbitrarily close to the pages $F_t$. Thus it follows from Lemma \[lem3.2\] that, along $\{ t \} \times M$, $\omega$ restricted to the pages $F_t$ of the open book decomposition is also non-zero. That is, we see that, along $\{ t \}
\times M$, $\omega$ restricted to the pages $F_t$ of the open book decomposition is a volume form away from the binding $B_t$. Recall also that $d_M\beta_t$ is a volume form on the pages $F_t$ away from the binding $B_t$ by the construction of the open book decomposition.
Let $M_{B_t}$ be the result of 0-surgery along $B_t$. Then we have a fibration $$\pi: S^1\times M_{B_t} \to S^1\times S^1,$$ and $t$ and $s_t$ will denote the first and second angular coordinates on the base manifold $S^1\times S^1$ of the fibration $\pi$, respectively. Let $N(B_t)$ denote a tubular neighborhood of the binding $B_t$. Since, along $\{ t \} \times M$, $d_M\beta_t$ and $\omega$ are both nowhere vanishing 2-forms restricted to the contact plane $\xi_t$, we can choose a smooth function $f$ defined over $S^1\times (M_B\backslash N(B))$ satisfying the following two properties:
- $f$ is nowhere vanishing over $S^1\times (M_{B_t}\backslash N(B_t))$ and
- $f\cdot d_M\beta_t$ coincides with $\omega$, when restricted to the contact plane $\xi_t$.
Then the following lemma holds.
\[lem3.3\] On the manifold $S^1\times (M_{B_t}\backslash N(B_t))$ which can be identified with $S^1\times (M\backslash N(B_t))$, the symplectic $2$-form $\omega$ can be written locally as the form $$\label{eq3.3}
\omega=\pi^\ast(dt\wedge ds_t)+ f(t,x) d_M\beta_t + ds_t\wedge
\delta,$$ where $\delta$ is a 1-form on $S^1\times (M\backslash N(B_t))$ and $x$ denotes a local coordinate on $M$.
To see it, notice first that $\omega(Z_t, W_t)$ may be non-zero for the Reeb vector field $Z_t$ and $W_t\in\xi_t$, while $\omega(X,
W_t)$ should be zero for $W_t\in\xi_t$. Thus in local coordinates the symplectic form $\omega$ should have only the terms involving $\pi^\ast(dt\wedge ds_t)$, $d_M\beta_t$ and $ds_t\wedge \delta$. Due to the equation , the coefficient of $\pi^\ast(dt\wedge
ds_t)$ should be 1, as stated. Thus we are done.
Finally, over $S^1\times (M_{B_t}\backslash N(B_t))$ we compute the Lie derivative ${\mathcal L}_X \omega$ explicitly. To do so, note that we have $$\label{eq3.4}
\begin{split}
{\mathcal L}_X \omega &= d \iota_X(\pi^\ast(dt\wedge
ds_t)+f(t,x) d_M\beta_t+ ds_t\wedge \delta)\\
&=d (-\iota_X\delta)\wedge ds_t.
\end{split}$$ However, since we have $$1=\omega(X,(j_t)_\ast(Z_t))=1-\iota_X\delta$$ by the equations and , $\iota_X\delta$ should be zero. Thus it follows from that ${\mathcal
L}_X \omega=0$. This clearly contradicts to Lemma \[lem3.1\]. That is, this case does not occur.
[**(Case 2)**]{} We next assume that, for some $t=t_0$ in $S^1$, $\beta_{t}\wedge d_M\beta_{t}$ is non-zero for all $x\in M$. By the continuity of smooth differential forms, $\beta_{t}\wedge d_M\beta_{t}$ should be non-zero for all $t$ in some sufficiently small open interval $I$ of $t_0$ and all $x\in M$.
Then apply the arguments in (Case 1) to the manifold $I\times M$ instead of $S^1\times M$. Then we can also derive a contradiction in this case. In more detail, the manifold $I \times M$ admits a symplectic structure, denoted $\omega$, by the restriction of the symplectic form $\omega$ on $S^1\times M$ to $I\times M$. There still exists the fundamental vector field $X$ on $I\times M$ associated to the natural action of $S^1$ on $S^1\times M$, since the interval is regarded as an open submanifold of $S^1$. However, clearly there is no $S^1$-action on $I$. Using this fundamental vector field $X$ on $I\times M$ and the fact that $\beta_{t}\wedge d_M\beta_{t}$ is non-zero on $I\times M$, as in Lemma \[lem3.1\] we can show that the Lie derivative ${\mathcal L}_X \omega$ is nowhere vanishing on $I\times M$. Furthermore, one can check that other arguments as well as Lemmas \[lem3.2\] and \[lem3.3\] go through without any modification. So, we can conclude that this case does not occur, either.
[**(Case 3)**]{} In this case we assume that, for some $t=t_0$ in $S^1$ and some $x_0\in M$, $\beta_{t}\wedge d_M\beta_{t}$ is non-zero. Once again it follows from the continuity of smooth differential forms that $\beta_{t}\wedge d_M\beta_{t}$ should be non-zero for all $t$ in some sufficiently small open interval $I$ of $t_0$ and some $x_0\in M$.
In order to apply the arguments of the previous cases, we need to take the contact part $$V(\beta_t)=\{ x\in M \ | \ \beta_{t}\wedge d_M\beta_{t}\ne 0\
\text{for}\ t\in I \}.$$ For simplicity, for each $t\in I$ we shall denote by $W_t$ the closure of the connected component of the contact part $V(\beta_t)$ which contains $x_0$. Then for each $t\in I$, $W_t$ is a compact contact submanifold of $M$ of codimension 0 with (possibly empty) boundary, since $\beta_{t}\wedge d_M\beta_{t}$ is a nowhere vanishing $3$-form on $V(\beta_t)$ and so is a volume form there. Fortunately, in the present paper we do not need to know the precise information of $W_t$, unlike to the case in the paper [@H-T].
Assume now that the boundary is non-empty. (Otherwise, we are reduced to (Case 2).) Then, as already mentioned in Section \[sec2\], we may assume without loss of generality that the boundary is convex. Thus for each $t\in I$ we obtain a compact contact $3$-manifold $W_t$ with convex boundary. It is also true that as in (Case 2) above there still exists the fundamental vector field $X$ on $\cup_{t\in I} \{ t \}\times W_t\subset S^1\times M$ associated to the natural action of $S^1$ on $S^1\times M$, since the interval $I$ is again regarded as an open submanifold of $S^1$. But this case is slightly different from (Case 2) in that $W_t$ is a compact contact $3$-manifold with convex boundary. So we need to use the relative Giroux correspondence (Theorem \[thm2.2\]) instead of the Giroux correspondence (Theorem \[thm2.1\]). In other words, for each $t\in I$ apply Theorem \[thm2.2\] to $W_t$ in order to obtain its partial open book decomposition $(S_t, P_t, h_t)$. Thus $W_t$ can now be described as the gluing of two handlebodies $H_t$ and $N_t$ by the map $h_t$ whose boundary is given as in . However, since all the arguments in (Case 1) and (Case 2) are essentially local, those arguments applied to the compact contact $3$-manifold $W_t$ with convex boundary and symplectic $4$-manifold $\cup_{t\in I} \{ t \} \times W_t$ equipped with the symplectic form induced from $S^1\times M$ will again go through without any modification. This in turn gives rise to a contradiction for this case, which means that this case does not occur, either.
This completes the proof of Proposition \[prop3.1\].
With this understood, the following theorem will play a crucial role in the proof of Theorem \[thm4.1\].
\[thm3.1\] The symplectic class $[\omega]$ on $S^1\times M$ can be represented by a symplectic form which is invariant under the obvious action of $S^1$.
We prove this theorem by contradiction. That is, suppose that the cohomology class $[\omega]$ cannot be represented by any $S^1$-invariant symplectic form $\omega$ under the obvious $S^1$-action. Then we would have $$\label{eq3.5}
{\mathcal L}_X\omega\ne 0.$$ Note also that by Proposition \[prop3.1\] the differential 3-form $\beta_{t}\wedge d_M\beta_{t}$ vanishes identically for all $t\in S^1$. Then there are two possibilities we have to consider: $d_M\beta_t$ vanishes identically on $\{ t \}\times M$ for all $t\in S^1$ or not.
So, suppose first that $d_M\beta_t$ vanishes identically on $M$ as well. Then it can be shown that the Lie derivative ${\mathcal L}_X \omega$ vanishes identically. To see it, notice that it follows from the identity that we have $j_t^\ast({\mathcal L}_X \omega)=0$. Thus we have ${\mathcal L}_X\omega(Z_t, W_t)=0$ for any vector fields $Z_t$ and $W_t$ on $\{ t \}\times M$ for each $t\in S^1$. Moreover, since $\omega$ is a symplectic form on $S^1\times M$, we can choose a Darboux chart in a neighborhood of a point $(t, x)$ whose coordinate vectors are given by non-zero vector fields $X_0=X$ and $X_i$ $(i=1,2,3)$. With this coordinate chart, we have $${\mathcal L}_X \omega(X, X_i)=d\iota_X\omega(X, X_i)=X(\iota_X \omega(X_i))=X(1)=0,$$ which implies that ${\mathcal L}_X\omega(X, Y_t)=0$ for all vector field $Y_t$ of $\{ t \}\times M$. Therefore, we can conclude that in this case ${\mathcal L}_X\omega$ vanishes identically. But this clearly contradicts to the assumption .
On the other hand, if $d_M\beta_t$ does not vanish on $\{ t \}\times M$, we need to use the result of Eliashberg and Thurston about perturbing a confoliation into a contact structure. If the 3-manifold $M$ is $S^2\times S^1$, clearly $M$ fibers over $S^1$, as mentioned earlier. Thus we may assume that our foliation $\xi_t$ is different from the foliation $\zeta$ on $S^2\times S^1$. Now if we apply Theorem \[thm2.3\] to $\xi_t$ then we have a contact structure $\tilde\xi_t=\ker \tilde
\beta_t$ which is a $C^0$-approximation to $\xi_t$. Since $\tilde\xi_t$ is a $C^0$-approximation of $\xi_t$, the symplectic 2-form $\omega$ can also be $C^0$-approximated by a symplectic 2-form $\tilde\omega$ on $S^1\times M$ so that $\tilde\beta_t=j^\ast_t(\iota_X \tilde\omega|_{ \{ t \} \times M})$. So we are essentially led to the (Case 1), (Case 2), or (Case 3) of Proposition \[prop3.1\], which has already shown not to occur.
Therefore, for either case we have derived a contradiction under our assumption . This completes the proof of Theorem \[thm3.1\].
Note that Theorem \[thm3.1\] does not imply that any arbitrary symplectic form $\omega$ on $S^1\times M$ is always $S^1$-invariant under the $S^1$-action on the first factor of $S^1\times M$. This can be easily seen by taking $M$ to be the $3$-dimensional torus $T^3$. That is, if the theorem implies that any symplectic form $\omega$ on $S^1\times T^3$ is always $S^1$-invariant under the obvious $S^1$-action, the symplectic form on $S^1\times T^3=T^4$ should also be invariant under the obvious $S^1$-action of the last three $S^1$-factors of $T^4$. So we can conclude that every symplectic form on $T^4$ should be invariant under the componentwise $T^4$-action on $T^4$. But obviously this is not the case for $T^4$.
Proofs of Theorems \[thm4.1\] and \[thm4.2\] {#sec4}
============================================
In this section we present the proofs of main theorems of the present paper. Once we have established the existence of an $S^1$-invariant symplectic structure on $S^1\times M$, it is a fairly standard procedure to complete the proof of Conjecture \[conj1.1\]. For the sake of reader’s convenience, we give its proof relatively in detail.
To do so, we begin with the following well-known lemma which says that when the cohomology class $[\iota_X\omega]$ is not integral and non-zero, by some suitable perturbation we can always make it integral.
\[lem4.1\] Let $\omega$ be an $S^1$-invariant symplectic form on a closed oriented $4$-manifold $N$ such that $[\iota_X \omega]$ is non-zero. Then $N$ admits an $S^1$-invariant symplectic form $\hat \omega$ such that $[\iota_X \hat\omega]$ is non-zero and integral.
If $N$ admits an $S^1$-invariant symplectic form $\omega'$ such that $[\omega']$ is rational, then we can easily obtain an $S^1$-invariant symplectic form $\hat\omega$ such that $[\hat\omega]$ is integral by multiplying some suitable integer to $\omega'$. Note also that the class $[\iota_X \omega']$ is rational if the class $[\omega']$ is.
So assume now that the class $[\omega]$ is not rational. It is clear that there exists an arbitrary small closed 2-form $\eta$ such that $\omega+\eta$ represent a rational cohomology class. Let $\hat \eta$ be the average of $\eta$ over the $S^1$-action. Since $S^1$ is connected, for $\nu \in S^1$ $\nu^\ast\eta$ is a closed 2-form representing the same cohomology class as $\eta$. Thus $\omega+
\hat\eta$ and $\omega+\eta$ have the same rational cohomology class. Note also that $\omega'=\omega+\hat\eta$ is symplectic, provided that $\eta$ is sufficiently small. By the openness of symplectic condition again, we can further choose $\omega'$ in such a way that the class $[\iota_X
\omega']$ is non-zero. This completes the proof.
Finally we are ready to prove the main theorems.
\[thm4.1\] Let $M$ be a closed oriented $3$-manifold such that $S^1\times M$ admits a symplectic structure $\omega$. Then $M$ fibers over $S^1$.
By Theorem \[thm3.1\], we may assume that the symplectic structure $\omega$ is $S^1$-invariant. Further, we may also assume that the class $[\iota_X \omega]$ on $S^1\times M$ is integral by Lemma \[lem4.1\].
In order to prove the theorem, we first consider the case where the class $[\iota_X \omega]$ is zero. In this case, there exists a function, called the moment map $\mu:
S^1\times M\to {\bf R}$ such that $\iota_X \omega=d\mu$. Thus the $S^1$-action is Hamiltonian. But it is clear that the $S^1$-action on $S^1\times M$ does not have any fixed points that are critical points of $\mu$. This gives rise to a contradiction to the fact that any Hamiltonian function on a closed symplectic manifold should have at least two critical points (e.g., extremal points). Therefore we can conclude that the class $[\iota_X \omega]$ is actually non-zero. Under this condition, McDuff proved in [@Mc] that by using an argument of D. Tischler in [@Ti], there exists a generalized moment map $\mu:
S^1\times M\to S^1$ satisfying $\iota_X \omega=\mu^\ast(dt)$. Thus by restricting the map $\mu$ to $\{\text{a point}\}\times M$, we easily obtain a fibration of $M$ over $S^1$. This completes the proof of Theorem \[thm4.1\].
Now we close this section with a proof of Theorem \[thm1.1\] as follows.
\[thm4.2\] If $S^1\times M_K$ admits a symplectic structure, then $K$ is always a fibered knot.
Suppose that $S^1\times M_K$ admits a symplectic structure. Then it follows from Theorem \[thm4.1\] that the 3-manifold $M_K$ is a fibration of $S^1$. Moreover, by the construction in the proof of Theorem \[thm4.1\], we have a closed 1-form $\iota_X\omega$ whose class is integral and is pointwise non-zero. Now let $j:
S^3-N(K)\to M_K$ be the natural inclusion, where $N(K)$ is a tubular neighborhood of $K$. Now observe that by the pullback we have a closed 1-form $j^\ast(\iota_X\omega)$ on $S^3-N(K)$ whose class is still integral and is non-zero pointwise on $S^3-N(K)$. Then $j^\ast(\iota_X\omega)$ defines a measured foliation ${\mathcal F}$ of $S^3-N(K)$ transverse to the boundary $\partial (S^3-N(K))$ with $T{\mathcal F}=\ker j^\ast(\iota_X\omega)$. Since $j^\ast(\iota_X\omega)$ is integral, we can also write $j^\ast(\iota_X\omega)=d\pi$ for a fibration $\pi: S^3-N(K)\to S^1$ whose fibers are the leaves of ${\mathcal F}$ (e.g., see p.2 of [@Mc-Ta] for more details). Thus the knot $K$ is indeed fibered. This completes the proof.
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|
**Nonperturbative enhancement of superloop at strong coupling**
A.V. Belitsky
*Department of Physics, Arizona State University*
*Tempe, AZ 85287-1504, USA*
**Abstract**
We address the near-collinear expansion of NMHV six-particle scattering amplitudes at strong value of the ’t Hooft coupling in planar maximally supersymmetric Yang-Mills theory. We complement recent studies of this observable within the context of the Pentagon Operator Product Expansion, via the dual superWilson loop description, by studying effects of multiple scalar exchanges that accompany (or not) massive flux-tube excitations. Due to the fact that holes have a very small, nonperturbatively generated mass $m_{\rm h}$ which is exponentially suppressed in the ’t Hooft coupling, their exchanges must be resummed in the ultraviolet limit, $\tau \ll m_{\rm h}$. This procedure yields a contribution to the expectation value of the superloop which enters on equal footing with the classical area, — a phenomenon which was earlier observed for MHV amplitudes. In all components, the near-massless scalar exchanges factorize from the ones of massive particles, at leading order in strong coupling.
0
Introduction
============
The equivalence between $N$-gluon maximally helicity-violating (MHV) scattering amplitudes in planar maximally supersymmetric gauge theory and the expectation value of the Wilson loop on a null polygonal contour $C_N$ was first established at strong coupling via the analysis of the minimal area in anti-de Sitter space ending on $C_N$ [@Alday:2007hr] through the lens of gauge/string correspondence [@Maldacena:1997re; @Witten:1998qj; @Gubser:1998bc]. This was further solidified through the Thermodynamic Bethe Ansatz [@Alday:2009dv; @Alday:2010vh]. Simultaneously, extensive perturbative checks verified this duality at weak coupling as well for the MHV case [@Drummond:2007cf; @Brandhuber:2007yx]. The language suitable for analysis in both regimes of weak and strong coupling was recently suggested through the Pentagon Operator Product Expansion [@Basso:2013vsa] based on an earlier version [@Alday:2010ku]. All-order expressions in ’t Hooft coupling for the main ingredients of the framework, i.e., the pentagon transitions for all single-particle excitations, including “flavor” changing ones, were constructed in a series of papers [@Basso:2013aha; @Belitsky:2014rba; @Basso:2014koa; @Belitsky:2014sla; @Basso:2014nra; @Belitsky:2014lta; @Belitsky:2015efa; @Basso:2014hfa; @Basso:2015rta; @Belitsky:2015qla] and confronted with “data” acccumulated in other frameworks to scattering amplitudes at several loop orders [@Bern:2008ap; @DelDuca:2009au; @Goncharov:2010jf; @Dixon:2013eka; @Dixon:2011nj; @Dixon:2014iba; @Golden:2014xqa; @Golden:2014xqf; @Golden:2014pua; @Drummond:2014ffa; @Dixon:2015iva]. While the MHV amplitude at strong coupling was addressed in this Operator Product Expansion framework[^1] in Refs. [@Basso:2013vsa; @Basso:2014koa; @Basso:2014jfa; @Fioravanti:2015dma] and went beyond the area paradigm in Ref. [@Basso:2014jfa], quantitatively not much is known to date about the strong coupling behavior of amplitudes at non-MHV level. The latter are dual to a supersymmetric Wilson loop on a null polygonal contour [@CaronHuot:2010ek; @Mason:2010yk; @Belitsky:2011zm]. In a recent publication [@Belitsky:2015qla], we had a first glimpse into certain components of NMHV hexagon by deriving the inverse-coupling expansion for the pentagons involving gauge fields and fermions. However, we have ignored completely contributions due to scalars accompanying any given tree-level exchange that encodes quantum numbers of the transition under study. In the present study we will lift this limitation and address the fate of scalar exchanges in NMHV amplitudes. Echoing an earlier work on MHV scattering [@Basso:2014jfa], we will observe nonperturbative enhancement of various components due to the nonperturbatively generated hole mass. In fact, we will find that at leading order in the inverse coupling expansion, any given component factorizes into the product of terms, one corresponding to the exchange of a massive excitation and the other one due to an infinite number of hole exchanges. Of course, the purely scalar components do not admit this factorization. In the current paper, we will focus on the hexagon superloop.
Our subsequent presentation is organized as follows. In the next section, we address the phases of the direct and mirror hole-hole S-matrices and recover their recursive structure in the non-perturbative scale that allows one to fix the form of the leading contribution to the even and odd parity flux-tube functions. In Sect. \[NPcorrectionsSection\], we turn to the calculation of the first nonperturbative corrections to the latter. Using the hole flux-tube functions, we determine pentagon transitions involving at least one scalar in Sect. \[MixedPentagonsSection\]. Then we shift our attention to the application of these results to components of the NMHV hexagon that can accommodate scalars, as the only or an accompanying excitation of some transitions. We start with fermionic exchanges and demonstrate the factorizability alluded to above. The same is applicable to the gluonic NMHV exchange as well. We perform a resummation of scalar exchanges using numerical studies and a form governed by the interpretation in terms of correlation function of twist operators in O(6) sigma model as was done in Ref. [@Basso:2014jfa] for MHV amplitudes. Along these lines, we find a contribution of the same order as the classical area. Finally we conclude. A couple of appendices contain results used in the main text.
Strong-coupling expansion of hole phases
========================================
Let us start our discussion of the strong-coupling regime recalling that the leading contribution of the hole excitation to the expectation value of the Wilson loop arises from its nonperturbative regime, i.e., when its rapidity scales as $u \sim O (g^0)$. The solution to the corresponding flux-tube equations, which are quoted for completeness in Appendix \[FluxTubeEqsAppendix\], were found in Ref. [@Basso:2013pxa]. Here we present an indirect way of deducing them. The flux-tube function will not enjoy correct properties,— it will possess an infinite number of poles rather than being an entire function,— however, the terms which restore its proper analytical structure turn out to be exponentially suppressed in the ’t Hooft coupling. The first correction in this infinite series will be recovered in the following section.
The indirect method of finding the flux-tube function is based on an iterative structure of scattering phases. It was previously applied in Ref. [@Basso:2008tx] to the problem of nonperturbative corrections to the cusp anomalous dimension. The latter is the vacuum of the flux tube so it should not be surprising that the same formalism is applicable in the current circumstances of a hole excitation created on top of the vacuum.
The direct $S_{\rm hh}$ and mirror $S_{\ast \rm hh}$ hole-hole S-matrices, building up the corresponding pentagon transition $P_{\rm h|h}$ [@Basso:2013aha], are determined by the dynamical scattering phases $f^{(i)}_{\rm hh}$ [@Basso:2013pxa; @Belitsky:2014sla] which are integrals of flux-tube hole functions, $$\begin{aligned}
S_{\rm hh} (u_1, u_2)
&= \exp \left( 2 i \sigma_{\rm hh} (u_1, u_2) - 2i f^{(1)}_{\rm hh} (u_1, u_2) + 2i f^{(2)}_{\rm hh} (u_1, u_2) \right)
\, , \\
S_{\ast \rm hh} (u_1, u_2)
&= \exp \left( 2 \widehat\sigma_{\rm hh} (u_1, u_2) + 2 f^{(3)}_{\rm hh} (u_1, u_2) - 2 f^{(4)}_{\rm hh} (u_1, u_2) \right)
\, ,\end{aligned}$$ and explicit phases $\sigma_{\rm hh}$ and $\widehat\sigma_{\rm hh}$ that are quoted below in Eqs. [(\[ExplcitiSigmas\])]{}. The strong coupling expansion of $f^{(i)}_{\rm hh}$ will allow us to kill two birds with one stone: we will determine the sought after nonperturbative expansion as well as find the leading order flux-tube functions.
Let us start with $f_{\rm hh}^{(1)}$ that can be cast in the form $$\begin{aligned}
\label{PhasefHH1}
f_{\rm hh}^{(1)} (u_1, u_2)
&= \frac{1}{2} \int_0^\infty \frac{dt}{t} \frac{\sin(u_1 t)}{\sinh \frac{t}{2}} \gamma^{\rm h}_{u_2} (2 g t)
\\
&
=
\frac{1}{2}
\int_0^\infty \frac{dt}{t} \sin (u_1 t) \frac{\cosh \frac{t}{2}}{\cosh t} \left[ \Gamma^{\rm h}_{u_2, -} (2 g t) - \Gamma^{\rm h}_{u_2, +} (2 g t) \right]
\nonumber\\
&
+
\frac{1}{2}
\int_0^\infty \frac{dt}{t} \sin (u_1 t) \frac{\sinh \frac{t}{2}}{\cosh t} \left[ \Gamma^{\rm h}_{u_2, -} (2 g t) + \Gamma^{\rm h}_{u_2, +} (2 g t) \right]
\, , \nonumber\end{aligned}$$ making use of a functional transformation [@Basso:2008tx; @Basso:2008tx1], see Eq. [(\[gammaToGamma\])]{}, that eliminates explicit dependence on the coupling constant from the flux-tube equations. Performing the inverse Fourier transformation for the product of hyperbolic and trigonometric functions, $$\begin{aligned}
\label{FT1}
\sin (u t) \frac{\cosh \frac{t}{2}}{\cosh t}
&
=
- \sqrt{2} g \int_{- \infty}^{\infty} dw \, \sin (2 g t w) \frac{\cosh (g \pi w + u \pi/2)}{\cosh (2 g \pi w + u \pi)}
\, , \\
\label{FT2}
\sin (u t) \frac{\sinh \frac{t}{2}}{\cosh t}
&
=
+ \sqrt{2} g \int_{- \infty}^{\infty} dw \, \cos (2 g t w) \frac{\sinh (g \pi w + u \pi/2)}{\cosh (2 g \pi w + u \pi)}
\, ,\end{aligned}$$ we can rewrite the phase in the form $$\begin{aligned}
f_{\rm hh}^{(1)} (u_1, u_2)
=
&
- \frac{g}{\sqrt{2}}
\int_{- \infty}^{\infty} dw \, \frac{\cosh (g \pi w + u_1 \pi/2)}{\cosh (2 g \pi w + u \pi)}
\int_0^\infty \frac{dt}{t}
\sin (2 g t w)
\left[ \Gamma^{\rm h}_{u_2, -} (2 g t) - \Gamma^{\rm h}_{u_2, +} (2 g t) \right]
\nonumber\\
&+
\frac{g}{\sqrt{2}}
\int_{- \infty}^{\infty} dw \, \frac{\sinh (g \pi w + u_1 \pi/2)}{\cosh (2 g \pi w + u \pi)}
\int_0^\infty \frac{dt}{t}
[ \cos (2 g t w) - 1 ]
\left[ \Gamma^{\rm h}_{u_2, -} (2 g t) - \Gamma^{\rm h}_{u_2, +} (2 g t) \right]
\, .\end{aligned}$$ Here in the second line, we subtracted 1 without any consequences by virtue of Eq. [(\[FT2\])]{} for $t = 0$. Next, splitting the integration range for $w$ into the interval $[-1, 1]$ and the rest, we can use the flux-tube equations [(\[GammaH1sin\])]{} and [(\[GammaH2cos\])]{} for the former interval, while safely expand integrands at large coupling in the latter. After the flux-tube equations had been applied, it remains to evaluate the integrals over the region $[-1,1]$ of $w$, $$\begin{aligned}
\int_{- 1}^{1} dw \sin (2 g t w) \frac{\cosh (g \pi w + \pi u/2)}{\cosh (2 g \pi w + \pi u)}
&
=
-
\frac{1}{g \sqrt{2}}
\sin (u t)
\frac{\cosh \frac{t}{2}}{\cosh t}
+
\frac{{\rm e}^{- \pi g}}{g} \sinh \frac{u \pi}{2} \, \Re{\rm e} \left[ \frac{{\rm e}^{2 i g t}}{t + i \pi/2} \right]
\nonumber\\
&
+
O ({\rm e}^{- 3 \pi g})
\, , \nonumber\\
\int_{- 1}^{1} dw [ \cos (2 g t w) - 1] \frac{\sinh (g \pi w + \pi u/2)}{\cosh (2 g \pi w + \pi u)}
&
=
\frac{1}{g \sqrt{2}}
\sin (u t)
\frac{\sinh \frac{t}{2}}{\cosh t}
+
\frac{{\rm e}^{- \pi g}}{g} \sinh \frac{u \pi}{2} \, \Re{\rm e} \left[ \frac{i {\rm e}^{2 i g t}}{t + i \pi/2} - \frac{2}{\pi} \right]
\nonumber\\
&
+
O ({\rm e}^{- 3 \pi g})
\, . \nonumber\end{aligned}$$ Adding all of these contributions together, we get $$\begin{aligned}
f_{\rm hh}^{(1)} (u_1, u_2)
&
=
\frac{1}{2}
\int_0^\infty \frac{dt}{t} \frac{\sin (u_1 t)}{\sinh \frac{t}{2}} \left( J_0 (2 g t) - \frac{{\rm e}^{t/2} \cos (u_2 t)}{\cosh t} \right)
\\
&
-
{\rm e}^{- \pi g}
\sinh \frac{u_1 \pi}{2}
\,
\Re{\rm e}
\Bigg\{
{\rm e}^{i \pi/4}
\int_0^\infty \frac{dt}{t}
\left[
\frac{{\rm e}^{2 i g t}}{t + i \pi/2} + \frac{2i}{\pi}
\right]
\bigg[
i \frac{\cos (u_2 t)}{\sinh \frac{t}{2}}
\nonumber\\
&
\qquad\qquad\qquad\quad
+
\left( 1 + i \coth\frac{t}{2} \right)
\left(
\gamma^{\rm h}_{+, u_2} (2 g t) + i \gamma^{\rm h}_{-, u_2} (2 g t)
-
J_0 (2 g t)
\right)
\bigg]
\Bigg\}
+
O ({\rm e}^{-3 g \pi})
\, . \nonumber\end{aligned}$$ Comparing this result with Eq. [(\[PhasefHH1\])]{}, we can immediately extract the parity-even flux-tube function of the hole $$\begin{aligned}
\label{EvengammaHole}
\gamma^{\rm h}_u (2 g t)
=
J_0 (2 g t) - \frac{{\rm e}^{t/2} \cos (u t)}{\cosh t}
+
O ({\rm e}^{-\pi g})
\, ,\end{aligned}$$ up to exponentially-suppressed contributions. Substituting this expression into the $O ({\rm e}^{- \pi g})$ term in the above equation, we find that it vanishes at this order. So the first nontrivial correction to the scattering phase will come at order ${\rm e}^{- 2\pi g}$ from the the first nonperturbative term to the flux-tube function. As we pointed out earlier and as it is obvious from Eq. [(\[EvengammaHole\])]{}, $\gamma^{\rm h}_u (2 g t)$ possesses an infinite number of fixed poles on the imaginary axis. These are cancelled against the ones in nonperturbative terms that we have just mentioned. The first one in this infinite series will be determined in the following section.
To determine $\widetilde\gamma_u^{\rm h}$, we will analyze $f^{(3)}_{\rm hh}$ in the same fashion as above by first changing the basis functions [(\[gammaToGammaTilde\])]{}, $$\begin{aligned}
\label{PhasefHH3}
f_{\rm hh}^{(3)} (u_1, u_2)
&= \frac{1}{2} \int_0^\infty \frac{dt}{t} \frac{\sin(u_1 t)}{\sinh \frac{t}{2}} \widetilde\gamma^{\rm h}_{u_2} (- 2 g t)
\\
&
=
\frac{1}{2}
\int_0^\infty \frac{dt}{t} \sin (u_1 t) \frac{\sinh \frac{t}{2}}{\cosh t} \left[ \widetilde\Gamma^{\rm h}_{u_2, +} (2 g t) - \widetilde\Gamma^{\rm h}_{u_2, -} (2 g t) \right]
\nonumber\\
&
-
\frac{1}{2}
\int_0^\infty \frac{dt}{t} \sin (u_1 t) \frac{\cosh \frac{t}{2}}{\cosh t} \left[ \widetilde\Gamma^{\rm h}_{u_2, +} (2 g t) + \widetilde\Gamma^{\rm h}_{u_2, -} (2 g t) \right]
\, , \nonumber\end{aligned}$$ and then applying the Fourier transforms [(\[FT1\])]{}, [(\[FT2\])]{} with subsequent use of the flux-tube equations [(\[GammaHtilde1\])]{} and [(\[GammaHtilde2\])]{}. Then we obtain $$\begin{aligned}
f_{\rm hh}^{(3)} (u_1, u_2)
&
=
\frac{1}{2}
\int_0^\infty \frac{dt}{t} \frac{\sin (u_1 t)}{\sinh \frac{t}{2}} \left( \frac{{\rm e}^{- t/2} \sin (u_2 t)}{\cosh t} \right)
\\
&
-
{\rm e}^{- \pi g}
\sinh \frac{u_1 \pi}{2}
\,
\Re{\rm e}
\Bigg\{
{\rm e}^{i \pi/4}
\int_0^\infty \frac{dt}{t}
\left[
\frac{{\rm e}^{2 i g t}}{t + i \pi/2} + \frac{2i}{\pi}
\right]
\bigg[
- i \frac{\sin (u_2 t)}{\sinh \frac{t}{2}}
\nonumber\\
&
\qquad\qquad\qquad\quad
+
\left( 1 + i \coth\frac{t}{2} \right)
\left(
\widetilde\gamma^{\rm h}_{+, u_2} (2 g t) - i \widetilde\gamma^{\rm h}_{-, u_2} (2 g t)
\right)
\bigg]
\Bigg\}
+
O ({\rm e}^{-3 g \pi})
\, . \nonumber\end{aligned}$$ Comparing its right-hand side with Eq. [(\[PhasefHH3\])]{}, we immediately see that this equation defines an iteration for $\widetilde\gamma^{\rm h}_u$ in the perturbative parameter set by ${\rm e}^{- g \pi}$. Therefore, we find at leading order $$\begin{aligned}
\label{OddgammaHole}
\widetilde\gamma^{\rm h}_{u} (2 g t) = - \frac{\sin (u t) {\rm e}^{t/2}}{\cosh t}
+
O ({\rm e}^{-g \pi})
\, .\end{aligned}$$ Both results for $\gamma^{\rm h}_{u}$ and $\widetilde\gamma^{\rm h}_{u}$ were announced before in Ref. [@Basso:2013pxa]. Here we obtained them in a rather indirect way as well as fixed the form of the first nonperturbative correction to scattering phases. To complete the list of contributing phases, we have to find $f^{(2)}_{\rm hh}$, $$\begin{aligned}
\label{Initialf2hh}
f_{\rm hh}^{(2)} (u_1, u_2)
=
\int_0^\infty \frac{dt}{t} ({\rm e}^{t/2} \cos (u_1 t) - J_0 (2gt)) \widetilde\gamma^{\rm h}_{u_2} (2 g t)
\, ,\end{aligned}$$ and $f^{(4)}_{\rm hh}$, $$\begin{aligned}
\label{Initialf4hh}
f_{\rm hh}^{(4)} (u_1, u_2)
=
\int_0^\infty \frac{dt}{t} ({\rm e}^{t/2} \cos (u_1 t) - J_0 (2gt)) \gamma^{\rm h}_{u_2} (- 2 g t)
\, .\end{aligned}$$ The derivation follows the same footsteps. The only difference from the above calculation is the form of the Fourier transform for the integrands, namely, we need $$\begin{aligned}
\label{Fourier1}
\cos (u t)
\frac{\cosh \frac{t}{2}}{\cosh t}
&
=
\sqrt{2} g \int_{- \infty}^{\infty} dw \cos (2 g w t) \frac{\cosh (g \pi w + u \pi/2)}{\cosh (2 g \pi w + u \pi)}
\, , \\
\label{Fourier2}
\cos (u t)
\frac{\sinh \frac{t}{2}}{\cosh t}
&
=
\sqrt{2} g \int_{- \infty}^{\infty} dw \sin (2 g w t) \frac{\sinh (g \pi w + u \pi/2)}{\cosh (2 g \pi w + u \pi)}
\, .\end{aligned}$$ Repeating the analysis, we deduce for $$\begin{aligned}
f^{(2)}_{\rm hh} (u_1, u_2)
&
+
\int_0^\infty \frac{dt}{t} (1 - J_0 (2 g t)) \frac{{\rm e}^{t/2} \sin (u_2 t)}{{\rm e}^t - 1}
=
-
\int_0^\infty \frac{dt}{t} \frac{{\rm e}^{t/2} \sin (u_2 t)}{{\rm e}^t - 1}
\left(
\frac{\cos (u_1 t) {\rm e}^{t/2}}{\cosh t}
-
1
\right)
\nonumber\\
&
+
{\rm e}^{- \pi g}
\cosh \frac{u_1 \pi}{2}
\,
\Re{\rm e}
\Bigg\{
{\rm e}^{i \pi/4}
\int_0^\infty \frac{dt}{t}
\left[
\frac{{\rm e}^{2 i g t}}{t + i \pi/2} + \frac{2i}{\pi}
\right]
\bigg[
\frac{\sin (u_2 t)}{\sinh \frac{t}{2}}
\\
&\qquad\qquad\qquad\qquad\qquad\quad
+
\left( 1 + i \coth\frac{t}{2} \right)
\left(
\widetilde\gamma^{\rm h}_{+, u_2} (2 g t) - i \widetilde\gamma^{\rm h}_{-, u_2} (2 g t)
\right)
\bigg]
\Bigg\}
+
O ({\rm e}^{-3 g \pi})
\, , \nonumber\end{aligned}$$ and $$\begin{aligned}
f^{(4)}_{\rm hh} (u_1,u_2)
&
+
\int_0^\infty \frac{dt}{t} (1 - J_0 (2 g t)) \frac{{\rm e}^{t/2} \cos(u_2 t) - J_0 (2 g t)}{{\rm e}^t - 1}
\\
&
=
\int_0^\infty \frac{dt}{t ({\rm e}^t - 1)}
\left[
{\rm e}^{t/2} \cos (u_1 t) \left( J_0 (2 g t) - \frac{{\rm e}^{-t/2} \cos (u_2 t)}{\cosh t} \right)
+
\left( {\rm e}^{t/2} \cos (u_2 t) - {\rm e}^t \right) J_0 (2 g t)
\right]
\nonumber\\
&+
{\rm e}^{- \pi g}
\cosh \frac{u_1 \pi}{2}
\,
\Re{\rm e}
\Bigg\{
{\rm e}^{i \pi/4}
\int_0^\infty \frac{dt}{t}
\left[
\frac{{\rm e}^{2 i g t}}{t + i \pi/2} + \frac{2i}{\pi}
\right]
\bigg[
i \frac{\cos (u_2 t)}{\sinh \frac{t}{2}}
\nonumber\\
&
\qquad\qquad\qquad\quad
+
\left( 1 + i \coth\frac{t}{2} \right)
\left(
\gamma^{\rm h}_{+, u_2} (2 g t) + i \gamma^{\rm h}_{-, u_2} (2 g t)
-
J_0 (2 g t)
\right)
\bigg]
\Bigg\}
+
O ({\rm e}^{-3 g \pi})
\, , \nonumber\end{aligned}$$ respectively.
Taking the leading order solutions, and adding the explicit phases $\sigma_{\rm hh}$ and $\widehat\sigma_{\rm hh}$, $$\begin{aligned}
\label{ExplcitiSigmas}
\sigma_{\rm hh} (u_1, u_2)
&
=
\int_0^\infty \frac{dt}{t ({\rm e}^t - 1)}
\left[
{\rm e}^{t/2} J_0 (2gt) \sin(u_1 t) - {\rm e}^{t/2} J_0 (2gt) \sin(u_2 t) - {\rm e}^t \sin((u_1 - u_2)t)
\right]
\, , \\
\widehat\sigma_{\rm hh} (u_1, u_2)
&
=
\int_0^\infty \frac{dt}{t ({\rm e}^t - 1)}
\left[
{\rm e}^{t/2} \left( \cos(u_1 t) + \cos(u_2 t) \right) J_0 (2gt) - \cos((u_1 - u_2)t) - {\rm e}^t J_0^2 (2gt)
\right]
\, ,\end{aligned}$$ the first term in the strong coupling expansion of the hole-hole pentagon reads[^2] $$\begin{aligned}
P_{\rm h|h} (u_1|u_2)
=
\frac{\Gamma \left({{\textstyle\frac{1}{4}}} - {{\textstyle\frac{i}{4}}} (u_1 - u_2) \right) \Gamma \left( {{\textstyle\frac{i}{4}}} (u_1 - u_2) \right)}{4 \Gamma \left({{\textstyle\frac{3}{4}}} - {{\textstyle\frac{i}{4}}} (u_1 - u_2) \right) \Gamma \left( {{\textstyle\frac{1}{2}}} + {{\textstyle\frac{i}{4}}} (u_1 - u_2) \right)}
+ \dots\end{aligned}$$ while the measure $$\begin{aligned}
\mu_{\rm h} = \frac{\sqrt{2 \pi^3}}{\Gamma^2 ({{\textstyle\frac{1}{4}}})}
+
\dots \end{aligned}$$ is a transcendental constant, with the ellipsis standing for nonperturbative corrections in coupling. The latter can be evaluated with results obtained in the next section. The above expressions coincide with the ones derived in Ref. [@Basso:2014koa].
Nonperturbative corrections {#NPcorrectionsSection}
===========================
Let us now turn to the determination of the exponentially suppressed effects in the flux-tube functions of the hole. As we established in the previous section, the leading order solutions yielded functions with incorrect analytical properties. From the point of view of the flux-tube equations with hole inhomogeneities, these generate their particular solutions. We can always add homogeneous solutions to the above functions in order to restore analyticity and thus produce an entire function of $t$. As we will find below, these addenda are actually exponentially suppressed in the ’t Hooft coupling. Below we will provide a recipe for their calculation and construct an explicit first correction to both even and odd parity functions.
Even parity
-----------
We start with even parity. Let us add a solution of the homogeneous equation to Eq. [(\[EvengammaHole\])]{}, such that the resulting flux-tube function becomes an entire function in the complex $t$-plane, $$\begin{aligned}
\label{PhysicalHgamma}
\gamma^{\rm h}_{u,+} (2 g t) + i \gamma^{\rm h}_{u,-} (2 g t)
=
J_0 (2 g t) + \frac{\sinh \frac{t}{2}}{\sqrt{2} \sinh \left( \frac{t}{2} + i \frac{\pi}{4} \right)}
\left[
\Gamma^{\rm h, \, hom}_{u, +} (2 g t) + i \Gamma^{\rm h, \, hom}_{u, -} (2 g t) - \frac{i \cos (u t)}{\sinh \frac{t}{2}}
\right]
\, .\end{aligned}$$ Presently, we will focus on the cancellation of the leading singularity at $t = - i \pi/2$, however, our consideration can be easily extended to subleading terms as well. This will produce solutions to the homogeneous flux-tube equation which induce leading exponential corrections. That is, we impose the following quantization conditions $$\begin{aligned}
\Gamma^{\rm h, \, hom}_{u, +} (4 \pi i x_\ell) + i \Gamma^{\rm h, \, hom}_{u, -} (4 \pi i x_\ell)
=
- \delta_{\ell, 0} \sqrt{2} \cosh \frac{u \pi}{2}\, ,\end{aligned}$$ where $x_\ell \equiv \ell - {{\textstyle\frac{1}{4}}}$. A general solution to the homogeneous flux-tube equations was constructed in studies of the flux-tube vacuum [@Basso:2008tx1] and reads $$\begin{aligned}
\label{GeneralHomogeneousHGamma}
\Gamma_{u, +}^{\rm h, hom} (\tau) + i \Gamma_{u, -}^{\rm h, hom} (\tau)
&
=
\sum_{n \geq 1} \frac{c_u^{-} (n, g)}{4 \pi g n - i \tau}
\left[
- i \tau V_0 (- i \tau) U_1^- (4 \pi g n) + 4 \pi g n V_1 (- i \tau) U_0^- (4 \pi g n)
\right]
\nonumber\\
&
+
\sum_{n \geq 1} \frac{c_u^{+} (n, g)}{4 \pi g n + i \tau}
\left[
- i \tau V_0 (- i \tau) U_1^+ (4 \pi g n) + 4 \pi g n V_1 (- i \tau) U_0^+ (4 \pi g n)
\right]
\, ,\end{aligned}$$ where the special functions involved admit the following integral representation $$\begin{aligned}
V_n (z)
&
= \frac{\sqrt{2}}{\pi} \int_{-1}^1 dk \left( \frac{1 + k}{1 - k} \right)^{1/4} \frac{{\rm e}^{k z}}{(1 + k)^n}
\, , \\
U^\pm_n (z)
&
= \frac{1}{2} \int_{1}^\infty dk \left( \frac{k + 1}{k - 1} \right)^{\mp1/4} \frac{{\rm e}^{- k (z - 1)}}{(k \mp 1)^n}
\, ,\end{aligned}$$ and can be related to the confluent hypergemetric function. Substituting these into the quantization conditions and taking the limit $g \to \infty$, making use of their asymptotic expansions, which can be found in Refs. [@Basso:2008tx1; @Belitsky:2015qla], the above quantization conditions can be solved with the result $$\begin{aligned}
c_u^+ (n, g) = - \frac{\Lambda (u, g)}{(8 \pi g n)^{3/4}} \frac{2 \Gamma (n + {{\textstyle\frac{1}{4}}})}{\Gamma^2 ({{\textstyle\frac{1}{4}}}) \Gamma (n)}
\, , \qquad
c_u^- (n, g) = \frac{\Lambda (u, g)}{(8 \pi g n)^{1/4}} \frac{\Gamma (n - {{\textstyle\frac{1}{4}}})}{2 \Gamma^2 ({{\textstyle\frac{3}{4}}}) \Gamma (n)}
\, ,\end{aligned}$$ at leading order in the inverse coupling. Here, we introduced a nonperturbative scale $$\begin{aligned}
\Lambda (u, g) = - \sqrt{2} \cosh \frac{u \pi}{2} \, \frac{{\rm e}^{-\pi g} (2 \pi g)^{5/4}}{\Gamma ({{\textstyle\frac{5}{4}}})}
\, .\end{aligned}$$ Substituting these results into Eq. [(\[GeneralHomogeneousHGamma\])]{}, we deduce, after summing the infinite series up, the leading order contribution to the homogeneous solution of the parity-even flux-tube equation $$\begin{aligned}
\Gamma_{u, +}^{\rm h, hom} (2 g t) + i \Gamma_{u, -}^{\rm h, hom} (2 g t)
=
\frac{\Lambda (u, g)}{8 \pi g}
\bigg[
&
V_0 (- 2 i g t)
\frac{\Gamma ({{\textstyle\frac{3}{4}}}) \Gamma (1 - {{\textstyle\frac{i t}{2 \pi}}})}{ \Gamma ({{\textstyle\frac{3}{4}}} - {{\textstyle\frac{i t}{2 \pi}}})}
\\
+
&
\left( 2 V_1 (- 2 i g t) - V_0 (- 2 i g t) \right) \frac{\Gamma ({{\textstyle\frac{5}{4}}}) \Gamma (1 - {{\textstyle\frac{i t}{2 \pi}}})}{ \Gamma ({{\textstyle\frac{5}{4}}} - {{\textstyle\frac{i t}{2 \pi}}})}
\bigg]
+
O ({\rm e}^{- 2 \pi g})
\, . \nonumber\end{aligned}$$ This expression can be also recovered from the analysis of Ref. [@Basso:2008tx1] after appropriate rescaling of nonperturbative vacuum solution.
Odd parity
----------
Let us now turn to odd parity, where the flux-tube function with restored analytical properties is built from [(\[OddgammaHole\])]{} by adding again a homogeneous solution to it, $$\begin{aligned}
\label{PhysicalHgammaTilde}
\widetilde\gamma^{\rm h}_{u,+} (2 g t) - i \widetilde\gamma^{\rm h}_{u,-} (2 g t)
=
\frac{\sinh \frac{t}{2}}{\sqrt{2} \sinh \left( \frac{t}{2} + i \frac{\pi}{4} \right)}
\left[
\widetilde\Gamma^{\rm h, \, hom}_{u, +} (2 g t) - i \widetilde\Gamma^{\rm h, \, hom}_{u, -} (2 g t) - \frac{\sin (u t)}{\sinh \frac{t}{2}}
\right]
\, .\end{aligned}$$ As above, we will discuss the cancellation of the leading singularity at $t = - i \pi/2$ only. In other words, we impose the following quantization conditions $$\begin{aligned}
\widetilde\Gamma^{\rm h, \, hom}_{u, +} (4 \pi i x_\ell) - i \widetilde\Gamma^{\rm h, \, hom}_{u, -} (4 \pi i x_\ell)
=
\delta_{\ell, 0} \sqrt{2} \sinh \frac{u \pi}{2}\, ,\end{aligned}$$ where $x_\ell \equiv \ell - {{\textstyle\frac{1}{4}}}$. Since the homogeneous $\widetilde\Gamma$’s admit the same representation in terms of the infinite series [(\[GeneralHomogeneousHGamma\])]{}, deviating in minor details like certain relative signs, and differ only by the form of the quantization condition, there is no need to redo the analysis anew. We can simply obtain the final expression by replacing $\cosh \to - \sinh$ in the even parity solution constructed earlier. The result reads $$\begin{aligned}
\widetilde\Gamma_{u, +}^{\rm h, hom} (2 g t) - i \widetilde\Gamma_{u, -}^{\rm h, hom} (2 g t)
=
\frac{\widetilde\Lambda (u, g)}{8 \pi g}
\bigg[
&
V_0 (- 2 i g t)
\frac{\Gamma ({{\textstyle\frac{3}{4}}}) \Gamma (1 - {{\textstyle\frac{i t}{2 \pi}}})}{ \Gamma ({{\textstyle\frac{3}{4}}} - {{\textstyle\frac{i t}{2 \pi}}})}
\\
+
&
\left( 2 V_1 (- 2 i g t) - V_0 (- 2 i g t) \right) \frac{\Gamma ({{\textstyle\frac{5}{4}}}) \Gamma (1 - {{\textstyle\frac{i t}{2 \pi}}})}{ \Gamma ({{\textstyle\frac{5}{4}}} - {{\textstyle\frac{i t}{2 \pi}}})}
\bigg]
+
O ({\rm e}^{- 2 \pi g})
\, , \nonumber\end{aligned}$$ with $$\begin{aligned}
\widetilde\Lambda (u, g) = \sqrt{2} \sinh \frac{u \pi}{2} \, \frac{{\rm e}^{-\pi g} (2 \pi g)^{5/4}}{\Gamma ({{\textstyle\frac{5}{4}}})}
\, .\end{aligned}$$
We can verify the correctness of these expressions by substituting them into Eq. [(\[PhysicalHgamma\])]{} and calculating the energy and momentum of the hole excitation [(\[HoleEandPfromGammaH\])]{}. We find $$\begin{aligned}
E_{\rm h} (u)
=
- \frac{\Lambda (u, g)}{2 \pi g} = m_{\rm h} \cosh \frac{u \pi}{2}
\, , \qquad
p_{\rm h} (u)
=
\frac{\widetilde\Lambda (u, g)}{2 \pi g} = m_{\rm h} \sinh \frac{u \pi}{2}
\, , \end{aligned}$$ which is in agreement with the well-known leading order result [@Basso:2010in] when expressed in terms of the nonperturbatively generated mass of the O(6) sigma model [@Alday:2007mf; @Basso:2008tx] $$\begin{aligned}
m_{\rm h} = {\rm e}^{- \pi g} \frac{(8 \pi g)^{1/4}}{\Gamma ({{\textstyle\frac{5}{4}}})} + O ({\rm e}^{-2 \pi g})
\, .\end{aligned}$$ The above consideration can be extended to higher orders without facing conceptual difficulties.
Mixed pentagons {#MixedPentagonsSection}
===============
With the found explicit expressions for the hole flux-tube functions, we can determine the mixed hole-fermion and hole-gluon scattering phases at strong coupling. The only integral that one needs for the leading order solution is the following one $$\begin{aligned}
\int_0^\infty \frac{dt}{t} \frac{{\rm e}^{i \alpha t} - 1}{{\rm e}^t + 1} = \ln \frac{\Gamma \left( {{\textstyle\frac{1}{2}}} - i {{\textstyle\frac{\alpha}{2}}} \right)}{\sqrt{\pi} \Gamma (1 - i {{\textstyle\frac{\alpha}{2}}})}
\, .\end{aligned}$$ Notice that while hole’s rapidity will stay in the nonpertubative domain $u_{\rm h} \sim O (g^0)$, the ones for the fermion and gauge field should belong to the perturbative strong coupling scaling regime, where their energy and momentum are of order $g^0$, $$\begin{aligned}
E_\star = m_\star \cosh\theta
\, , \qquad
p_\star = m_\star \sinh\theta\end{aligned}$$ with $m_{\rm f} = 1$ and $m_{\rm g} = \sqrt{2}$ [@Alday:2007mf], to bestow amplitudes with leading contributions. Thus, the fermion belongs to the small fermion sheet with rapidity $u_{\rm f} = 2 g \widehat{u}_{\rm f}$ where $|\widehat{u}_{\rm f}| = |\coth( 2 \theta) | > 1$, while the gluon one scales as $u_{\rm g} = 2 g \widehat{u}_{\rm g}$ with $|\widehat{u}_{\rm g}| = |\tanh (2 \theta)| < 1$.
The hole-small fermion phases are $$\begin{aligned}
&
f_{\rm hf}^{(1)} (u, v) = - \frac{1}{16 g^2} \frac{u}{\widehat{v}^2} + \dots
\, , \quad
f_{\rm hf}^{(2)} (u, v) = \frac{1}{8 g} \frac{1}{\widehat{v}} + \dots
\, , \\
&
f_{\rm hf}^{(3)} (u, v) = - \frac{1}{4 g} \frac{u}{\widehat{v}} + \dots
\, , \qquad \
f_{\rm hf}^{(4)} (u, v) = - \frac{1}{2} \ln \big( 2 \widehat{v} \widehat{x}_{\rm f}[v] \big) + \frac{1}{4 g^2} \left( \frac{3}{16} + \frac{u^2}{4} \right) \frac{1}{\widehat{v}^2} + \dots
\, , \nonumber\end{aligned}$$ where the rescaled small-fermion Zhukowski variable in the hyperbolic parametrization reads $\widehat{x}_{\rm f} = \tanh (\theta)$. Such that the the hole-small fermion pentagon in the regime in question takes the form at leading order $$\begin{aligned}
\label{PhfLO}
P_{\rm h|f} (u|v) = \frac{1}{\sqrt{2 g \widehat{v}}} \exp\left( \frac{u^-}{2 g \widehat{v}}+ \dots \right)
\, .\end{aligned}$$ We can immediately test this form by employing constraints stemming from the $\bar{Q}$-equation [@CaronHuot:2011kk; @Bullimore:2011kg], namely, as demonstrated in Ref. [@Belitsky:2015kda], it enters the following integral equation $$\begin{aligned}
\int d v \, \mu_{\rm f} (v) {\rm e}^{- \tau (E_{\rm f} (v) - 1)} x_{\rm f}^{3/2} [v] \delta \left( p_{\rm f} (v) \right) P_{\rm \bar{f}|f} (- u + {{\textstyle\frac{3i}{2}}} | v) P_{\rm h|f} (- u | v) = \frac{2 g^3}{\Gamma (g)}
\, ,\end{aligned}$$ where $\Gamma (g)$ is the cusp anomalous dimension. Rescaling the small-fermion rapidity and re-expressing it in terms of the Zhukowski variable $\widehat{u}_{\rm f} = \widehat{x}_{\rm f}
+ 1/\widehat{x}_{\rm f}$, one can immediately confirm the leading order expression for the hole-small fermion pentagon [(\[PhfLO\])]{} making use of the known expressions for the small fermion measure and fermion-antifermion pentagon [@Basso:2014koa] $$\begin{aligned}
\mu_{\rm f} (u) = - \left( 1 - \widehat{x}_{\rm f}^2[u] \right)^{-1/2} + \dots
\, , \qquad
P_{\rm \bar{f}|f} (u| v) = \left( 1 - \widehat{x}_{\rm f} [u] \widehat{x}_{\rm f}[v] \right)^{-1/2} + \dots
\, .\end{aligned}$$
For the gluon-hole case, it is more instructive to discuss the entire direct and mirror S-matrices rather than individual phases. Then one finds by substituting the leading order hole solutions [(\[EvengammaHole\])]{} and [(\[OddgammaHole\])]{} to the dynamical phases that they cancel exactly the $\sigma$-phases on the level of integrands and thus both S-matrices are trivial $$\begin{aligned}
S_{\rm hg} = 1
\, , \qquad
S_{\rm \ast hg} = 1
\, ,\end{aligned}$$ up to nonperturbative effects in coupling. Consequently, the hole-gluon pentagon is $$\begin{aligned}
\label{HoleGluonPentagon}
P_{\rm h|g} (u| v) = 1 + O ({\rm e}^{- \pi g})
\, .\end{aligned}$$ With these results at our disposal, we are now ready to move onto explicit analyses of different components of the NMHV hexagon at strong coupling.
Hexagon superloop
=================
Let us decompose the hexagon superloop at the NMHV level in terms of Grassmann components that receive leading contribution from single particle exchanges, $$\begin{aligned}
\label{NMHVhexagon}
\mathcal{W}_6 (\tau, \sigma, \phi)
&
=
\chi_1^4 {\rm e}^{i \phi} \mathcal{W}_{(4,0)} (\tau, \sigma)
\\
&
+
\chi_1^3 \chi_4
\left(
{\rm e}^{i \phi/2} \mathcal{W}^{\rm odd}_{(3,1)} (\tau, \sigma)
+
{\rm e}^{- i \phi/2} \mathcal{W}^{\rm even}_{(3,1)} (\tau, \sigma)
\right)
+
\chi_1^2 \chi_4^2 \mathcal{W}_{(2,2)} (\tau, \sigma)
+
\dots \, . \nonumber\end{aligned}$$ Here we adopted a conventional twistor parametrization via the three variables $\tau$, $\sigma$ and $\phi$ which are equivalent to the three conformal cross-ratios $u$, $v$ and $w$ of the six-point remainder function. We will start below with the second contribution $\mathcal{W}_{(3,1)}$ in the Grassmann series, the one that is induced by the (anti)fermion production on the bottom and absorption at the top along with an infinite number of scalars. We divided their effect in the above sum in two classes, an antifermion along with an even number of scalars and a fermion with an odd number of scalars. As a consequence, $\mathcal{W}^{\rm odd}_{(3,1)}$ starts with two-particle exchanges compared to $\mathcal{W}^{\rm even}_{(3,1)}$. Both of them transform in the ${\bf \bar 4}$ of SU(4), however, possess different helicities as exhibited by accompanying phases in the above equation.
Antifermion-scalars
-------------------
The leading contribution to $\mathcal{W}^{\rm even}_{(3,1)}$ comes from the single-particle exchange with the quantum numbers of the antifermion $$\begin{aligned}
\label{Wf-bar}
\mathcal{W}_{\rm \bar{f}} = \int d \mu_{\rm f} (v) x_{\rm f} [v]
\, ,\end{aligned}$$ with the NMHV helicity form factor determined by the small-fermion Zhukowski variable $x_{\rm f} [v] = {{\textstyle\frac{1}{2}}} (v - \sqrt{v^2 - (2 g)^2})$. Here and below the single particle measure includes the propagating phases $$\begin{aligned}
d \mu_\star (v) = \frac{d v}{2 \pi} \mu_\star (v) {\rm e}^{- \tau E_\star (v) + i \sigma p_\star (v)}\end{aligned}$$ that are determined by all-order energies $E_\star$ and momenta $p_\star$ [@Basso:2010in]. Next in the infinite series comes the antifermion accompanied by two scalars $$\begin{aligned}
\mathcal{W}^{\rm\bf \bar{4}}_{\rm hh\bar{f}}
=
\frac{1}{2!}
\int d \mu_{\rm h} (u_1) d \mu_{\rm h} (u_2)
\int d \mu_{\rm f} (v) x_{\rm f} [v] \frac{1}{|P_{\rm h|f} (u_1| v) P_{\rm h|f} (u_2| v)|^2}
\frac{\Pi^{\rm\bf \bar{4}}_{\rm hhf} (u_1, u_2, v)}{|P_{\rm h|h} (u_1| u_2)|^2}
\, ,\end{aligned}$$ where the matrix part reads $$\begin{aligned}
\Pi^{\rm\bf \bar{4}}_{\rm hhf} (u_1, u_2, v)
=
\frac{3}{2}
\frac{45 + 6 u_1^2 - 8 u_1 u_2 + 6 u_2^2 - 4 (u_1 + u_2) v + 4 v^2
}{
[1 + (u_1 - u_2)^2] [4 + (u_1 - u_2)^2] [{{\textstyle\frac{9}{4}}} + (u_1 - v)^2] [{{\textstyle\frac{9}{4}}} + (u_2 - v)^2]
}
\, .\end{aligned}$$ In the scaling limit $v = 2 g \widehat{v}$ with $g \to \infty$ and $\widehat{v} = {\rm fixed}$, we get $$\begin{aligned}
\Pi^{\rm\bf \bar{4}}_{\rm hhf} (u_1, u_2, v) = \frac{1}{v^2} \Pi^{\rm\bf 1}_{\rm hh} (u_1, u_2) + O (1/v)
\, ,\end{aligned}$$ with $\Pi^{\rm\bf 1}_{\rm hh}$ being the singlet two-scalar matrix part (which defines one of the twist-two contributions in the MHV amplitude [@Basso:2014koa]) $$\begin{aligned}
\Pi^{\rm\bf 1}_{\rm hh} (u_1, u_2)
=
\frac{6
}{
[1 + (u_1 - u_2)^2] [4 + (u_1 - u_2)^2]
}
\, .\end{aligned}$$ This immediately yields a factorized form of the three-particle contribution in terms of the fermion, on the one hand, and the two-scalar pair in the singlet representation, on the other, i.e., $$\begin{aligned}
\mathcal{W}^{\rm\bf \bar{4}}_{\rm hh\bar{f}} = \mathcal{W}_{\rm \bar{f}} \mathcal{W}^{\rm\bf 1}_{\rm hh}\end{aligned}$$ where $$\begin{aligned}
\mathcal{W}^{\rm\bf 1}_{\rm hh}
=
\frac{1}{2!}
\int d \mu_{\rm h} (u_1) d \mu_{\rm h} (u_2)
\frac{\Pi^{\rm\bf 1}_{\rm hh} (u_1, u_2)}{|P_{\rm h|h} (u_1| u_2)|^2}
\, .\end{aligned}$$ A simple counting of the powers of the ’t Hooft coupling demonstrates that $\mathcal{W}^{\rm\bf 1}_{\rm hh}$ is of order $g^0$ and contributes on equal footing with Eq. [(\[Wf-bar\])]{}. The same phenomenon persists for all multi-scalar exchanges such that all scalar pairs have to be accounted for, $$\begin{aligned}
\label{WevenAllOrder}
\mathcal{W}^{\rm even}_{(3,1)}
=
\mathcal{W}_{\rm \bar{f}}
\sum_{n = 0}^\infty \, \mathcal{W}^{\rm\bf 1}_{({\rm hh})^n}
\, ,\end{aligned}$$ with $\mathcal{W}^{\rm\bf 1}_{({\rm hh})^0} = 1$ and $\mathcal{W}^{\rm\bf 1}_{({\rm hh})^n}$ having the form analogous to the two-scalar contribution $$\begin{aligned}
\label{EvenMultiScalars}
\mathcal{W}^{\rm\bf 1}_{({\rm hh})^n}
=
\frac{1}{(2 n)!}
\int d \mu_{\rm h} (u_1) \dots d \mu_{\rm h} (u_{2n})
\frac{\Pi^{\rm\bf 1}_{\rm h \dots h} (u_1, \dots , u_{2n})}{ \prod_{i<j}^{2n} |P_{\rm h|h} (u_i| u_j)|^2}\end{aligned}$$ with the matrix part $\Pi^{\rm\bf 1}_{\rm h \dots h} (u_1, \dots , u_{2n})$ that can be read off from the integral representation given in Refs. [@Basso:2014jfa; @Basso:2015uxa].
Singlet multi-scalar exchanges
------------------------------
In spite of the fact that the singlet multi-scalar resummation was analyzed in Ref. [@Basso:2014jfa], in preparation for the sextet case that is addressed next, we will repeat numerical computations here and confront them against twist-operator correlation functions. We start with the two-particle contribution. In the ultraviolet regime, it has the following asymptotic form $$\begin{aligned}
\mathcal{W}^{\rm\bf 1}_{\rm hh} |_{m_{\rm h} \xi \ll 1}
=
\mu_{\rm h}^2
\left[
\alpha^{\rm\bf 1}_{\rm hh} \ln \frac{1}{m_{\rm h} \xi} + \beta^{\rm\bf 1}_{\rm hh} \ln \ln \frac{1}{m_{\rm h} \xi} + \gamma^{\rm\bf 1}_{\rm hh}
\right]
+
O \left( (m_{\rm h} \xi)^0 \right)
\, ,\end{aligned}$$ where the relativistic invariance of the contributions is exhibited through the dependence on a single variable $\xi = \sqrt{\tau^2 + \sigma^2}$. The coefficients accompanying functional dependence on $\xi$ can be partially determined analytically $$\begin{aligned}
\alpha^{\rm\bf 1}_{\rm hh}
&
=
\frac{\Gamma^4 ({{\textstyle\frac{1}{4}}})}{84 \pi^5}
\left[
56 + 5 \, {_4{F}_3} \left.\left( {1, 1, {{\textstyle\frac{5}{4}}}, {{\textstyle\frac{9}{4}}} \atop 2, {{\textstyle\frac{7}{4}}}, {{\textstyle\frac{11}{4}}}} \right| 1 \right)
\right]
-
\frac{1}{5 \pi^3}
\left[
40
+ 3 \pi \,
{_3{F}_2} \left.\left( {{{\textstyle\frac{1}{2}}}, {{\textstyle\frac{5}{4}}}, {{\textstyle\frac{3}{2}}} \atop 2, {{\textstyle\frac{9}{4}}}} \right| 1 \right)
\right]
\nonumber\\
&
\simeq
0.087
\, , \\
\beta^{\rm\bf 1}_{\rm hh}
&
\simeq
- 0.137 \pm 0.001
\, ,\\
\gamma^{\rm\bf 1}_{\rm hh}
&
\simeq
- 0.044 \pm 0.014
\, .\end{aligned}$$ The effects of subleading terms in the expansion were analyzed numerically. In Fig. \[Hseries\] (a), we demonstrate the result of successive additions of more and more scalar exchanges in Eq. [(\[WevenAllOrder\])]{}. In these estimates, we computed the multifold integrals [(\[EvenMultiScalars\])]{} making use of an adaptive Monte Carlo method and then averaged over multiple samples. The standard deviation from the mean is shown in above formulas, while it is stripped off the graphs as not to obscure effects from multi-hole exchanges. We confirmed quick convergence of the Operator Product Expansion in the range of $m_{\rm h} \xi < 10^{-18}$ in this channel with the resulting functional fit inspired by the two-point correlation function of twist operators $\phi_{\pentagon}$, whose matrix elements correspond to the pentagon transitions as was pointed out in [@Basso:2014jfa], $$\begin{aligned}
\label{SingletSsm}
\mathcal{W}^{\rm\bf 1}_\infty
\equiv
\sum_{n = 0}^\infty \, \mathcal{W}^{\rm\bf 1}_{({\rm hh})^n} |_{m_{\rm h} \xi \ll 1}
=
C^{\rm\bf 1}_{\rm h} \, (m_{\rm h} \xi)^{-1/36} \ln^{-1/24} \frac{1}{m_{\rm h} \xi}
\, ,\end{aligned}$$ and $C^{\rm\bf 1}_{\rm h} \simeq 0.99$. This is indeed the result of Ref. [@Basso:2014jfa]. We will take it below as a basis for our analysis of the sextet component in the hexagonal Wilson loop.
Fermion-scalars
---------------
The even-scalar exchanges do not have ${\bf 6}$ in their product, e.g., ${\bf 6} \times {\bf 6} = {\bf 1} + {\bf 15} + {\bf 20}$, so they do not contribute to the component $\mathcal{W}^{\rm odd}_{(3,1)}$ in question. However, any odd number of holes does contribute. The first term in the fermion-scalar series is the one with a fermion and a hole in the ${\bf \bar 4}$ of SU(4), emerging from the product ${\bf 4} \times {\bf 6} = {\bf \bar 4} + {\bf 20}$, $$\begin{aligned}
\mathcal{W}^{\bf \bar{4}}_{\rm hf} = \int d \mu_{\rm h} (u) \int d \mu_{\rm f} (v)
\frac{\Pi^{\bf \bar{4}}_{\rm hf} (u, v) }{|P_{\rm h|f} (u | v)|^2}
\, ,\end{aligned}$$ with $$\begin{aligned}
\Pi^{\bf \bar{4}}_{\rm hf} (u, v) = \frac{3}{ [(u - v)^2 + {{\textstyle\frac{9}{4}}}] }
\, .\end{aligned}$$ The leading contribution at strong coupling emerges from the scaling limit of the small fermion rapidity $v = 2 g \widehat{v}$ with $\widehat{v} \sim O (g^0)$ and nonperturbative one for the hole, i.e., $u \sim O (g^0)$. Then the matrix part immediately simplifies and we get $$\begin{aligned}
\mathcal{W}^{\bf \bar{4}}_{\rm f h} = 3 W_{\rm f} W^{\bf 6}_{\rm h}\end{aligned}$$ with $$\begin{aligned}
W_{\rm f} = \int d \mu_{\rm f} (\widehat{v}) \frac{\widehat{x}_{\rm f}[v]}{(1 + \widehat{x}^2_{\rm f}[v])}
\, , \qquad
W^{\bf 6}_{\rm h} = g \int d \mu_{\rm h} (u)
\, .\end{aligned}$$ We pulled out the factor of 3 stemming from SU(4) tensor contraction as an overall coefficient. The next term arises from the four-particle ${\rm hhhf}$-state $$\begin{aligned}
\mathcal{W}^{\bf \bar{4}}_{\rm hhh\bar{f}}
=
\frac{1}{3!}
\int d \mu_{\rm h} (u_1) d \mu_{\rm h} (u_2) d \mu_{\rm h} (u_3)
&
\int d \mu_{\rm f} (v) \frac{1}{|P_{\rm h|f} (u_1| v) P_{\rm h|f} (u_2| v) P_{\rm h|f} (u_3| v)|^2}
\nonumber\\
&\times
\frac{\Pi^{\bf \bar{4}}_{\rm hhhf} (u_1, u_2, u_3, v)}{|P_{\rm h|h} (u_1| u_2) P_{\rm h|h} (u_1| u_3) P_{\rm h|h} (u_2| u_3)|^2}
\, ,\end{aligned}$$ where $\Pi^{\bf \bar{4}}_{\rm hhhf}$ is too cumbersome to be displayed here. However, in the scaling limit, it reduces to $$\begin{aligned}
\label{FactorizationPiSix}
\Pi^{\bf \bar{4}}_{\rm hhhf} (u_1, u_2, u_3, v)
=
\frac{3}{v^4} \Pi^{\rm\bf 6}_{\rm hhh} (u_1, u_2, u_3) + O (v^{-5})
\, , \end{aligned}$$ where the matrix part of the three-hole state in the ${\bf 6}$ of SU(4) reads $$\begin{aligned}
&
\Pi^{\rm\bf 6}_{\rm hhh} (u_1, u_2, u_3)
\\
&
=
6
\frac{
[7 + u_1^2 + u_2^2 + u_3^2 - (u_1 u_2 + u_1 u_3 + u_2 u_3)]
[12 + u_1^2 + u_2^2 + u_3^2 - (u_1 u_2 + u_1 u_3 + u_2 u_3)]
}
{
[1 + (u_1 - u_2)^2] [4 + (u_1 - u_2)^2] [1 + (u_1 - u_3)^2] [4 + (u_1 - u_3)^2] [1 + (u_2 - u_3)^2] [4 + (u_2 - u_3)^2]
}
\, . \nonumber\end{aligned}$$ Therefore, the expression factorizes again yielding $$\begin{aligned}
\mathcal{W}^{\bf \bar{4}}_{\rm hhh\bar{f}}
=
3 W_{\rm f} W^{\bf 6}_{\rm hhh}\end{aligned}$$ with $$\begin{aligned}
W^{\bf 6}_{\rm hhh}
=
\frac{g}{3!}
\int d \mu_{\rm h} (u_1) d \mu_{\rm h} (u_2) d \mu_{\rm h} (u_3)
\frac{\Pi^{\rm\bf 6}_{\rm hhh} (u_1, u_2, u_3)}{|P_{\rm h|h} (u_1| u_2) P_{\rm h|h} (u_1| u_3) P_{\rm h|h} (u_2| u_3)|^2}
\, .\end{aligned}$$ Using the integral representation of the matrix part of the pentagon transitions [@Basso:2015uxa], one can convince oneself that the above property [(\[FactorizationPiSix\])]{} persists for any odd number of hole excitations, such that any number of scalars accompanying the fermion needs to be resumed $$\begin{aligned}
\mathcal{W}^{\rm odd}_{(3,1)}
=
3 \mathcal{W}_{\rm f}
\sum_{n = 0}^\infty \, \mathcal{W}^{\rm\bf 6}_{{\rm h}({\rm hh})^n}
\, ,\end{aligned}$$ where similarly to Eq. [(\[EvenMultiScalars\])]{} $$\begin{aligned}
\label{SextetSsm}
\mathcal{W}^{\rm\bf 6}_{{\rm h} ({\rm hh})^n}
=
\frac{g}{(2 n + 1)!}
\int d \mu_{\rm h} (u_1) \dots d \mu_{\rm h} (u_{2n + 1})
\frac{\Pi^{\rm\bf 6}_{\rm h \dots h} (u_1, \dots , u_{2n + 1})}{ \prod_{i<j}^{2n + 1} |P_{\rm h|h} (u_i| u_j)|^2}\end{aligned}$$ with the matrix part of sextet hole exchanges that can be read off from the integral representation given in Refs. [@Basso:2014jfa; @Basso:2015uxa].
Sextet multi-scalar exchanges
-----------------------------
The one-particle contribution with sextet quantum numbers arises from the single hole exchange $$\begin{aligned}
\mathcal{W}^{\rm\bf 6}_{\rm h}
&
= g \int d \mu_{\rm h} (u)
= g \frac{2 \mu_{\rm h}}{\pi^2} K_0 (m_{\rm h} \xi)
\, , \nonumber\end{aligned}$$ where after the second equality sign we displayed its leading behavior from nonperturbative domain of rapidities. In the infrared regime $m_{\rm h} \xi \gg 1$, it displays the expected exponentially suppressed behavior $$\begin{aligned}
\mathcal{W}^{\rm\bf 6}_{\rm h} |_{m_{\rm h} \xi \gg 1}
=
g \frac{\sqrt{2}\mu_{\rm h}}{\pi^{3/2}}
\frac{{\rm e}^{- m_{\rm h} \xi}}{\sqrt{m_{\rm h} \xi}} \, \left( 1 + O \big( 1/(m_{\rm h} \xi) \big) \right)
\, ,\end{aligned}$$ while in the ultraviolet region $m_{\rm h} \xi \ll 1$, it shows logarithmic enhancement, $$\begin{aligned}
\mathcal{W}^{\rm\bf 6}_{\rm h} |_{m_{\rm h} \xi \ll 1} = g \frac{2 \mu_{\rm h}}{\pi^2} \ln \frac{1}{m_{\rm h} \xi} + O \big( (m_{\rm h} \xi)^0 \big)
\, .\end{aligned}$$
The enhancement of the ultraviolet regime persists and amplifies in multi-hole exchanges. For instance, in the three-particle term that reads $$\begin{aligned}
W^{\rm\bf 6}_{\rm hhh}
=
\frac{g}{3!}
\int d \mu_{\rm h} (u_1) d \mu_{\rm h} (u_2) d \mu_{\rm h} (u_3)
\frac{\Pi^{\rm\bf 6}_{\rm hhh} (u_1, u_2, u_3)}{|P_{\rm h|h} (u_1| u_2) P_{\rm h|h} (u_1| u_3) P_{\rm h|h} (u_2| u_3)|^2}
\, ,\end{aligned}$$ the analysis of the $z \to 0$ limit unravels the following behavior $$\begin{aligned}
\mathcal{W}^{\rm\bf 6}_{\rm hhh} |_{m_{\rm h} \xi \ll 1}
=
g \mu_{\rm h}^3
\ln \frac{1}{m_{\rm h} \xi}
\left[
\alpha^{\rm\bf 6}_{\rm hhh} \ln \frac{1}{m_{\rm h} \xi} + \beta^{\rm\bf 6}_{\rm hhh} \ln \ln \frac{1}{m_{\rm h} \xi} + \gamma^{\rm\bf 6}_{\rm hhh}
\right]
+ O \left( (m_{\rm h} \xi)^0 \right)
\, ,\end{aligned}$$ where $$\begin{aligned}
\alpha^{\rm\bf 6}_{\rm hhh} \simeq 0.0173 \pm 0.0001
\, , \qquad
\beta^{\rm\bf 6}_{\rm hhh} \simeq - 0.0453 \pm 0.0081
\, , \qquad
\gamma^{\rm\bf 6}_{\rm hhh} \simeq - 0.0141 \pm 0.0202
\, .\end{aligned}$$ We observe that as compared to the singlet case, there is an overall power of the logarithm accompanying the familiar $\xi$-dependence. Thus we anticipate the resummation to produce the same functional dependence on $\xi$ up to an extra logarithmic factor which stems from the anomalous dimension[^3] of the two-dimensional bosonic fields $X^i$ ($i = 1, \dots, 6$) on the five-sphere which build up the sextet pentagon twist operator $\phi^i_{\pentagon} \sim X^i \phi_{\pentagon}$ which in turn defines the scalar component of the hexagon Wilson loop in question $\delta^{ij} \mathcal{W}^{\rm 6} \sim {\langle{\phi^i_{\pentagon} (\xi) \phi^j_{\pentagon} (0)}\rangle}$. However, the overall normalization will be different and its proper extraction requires resummation. Due to complexity of the asymptotic analysis of multifold integrals and, as a consequence, the lack of explicit analytical expressions, we performed it numerically. The result of successive additions of multi-scalar exchanges up to five holes in shown in Fig. \[Hseries\] (b). The result is fitted by the following formula $$\begin{aligned}
\label{ResummedSextet}
\mathcal{W}^{\rm\bf 6}_\infty
\equiv
\sum_{n = 0}^\infty \, \mathcal{W}^{\rm\bf 6}_{{\rm h} ({\rm hh})^n} |_{m_{\rm h} \xi \ll 1}
=
g C^{\rm\bf 6}_{\rm h} \, (m_{\rm h} \xi)^{-1/36} \ln^{23/24} \frac{1}{m_{\rm h} \xi}
\, .\end{aligned}$$ with $C^{\rm\bf 6}_{\rm h} \simeq 0.11$. This analysis is in agreement[^4] with results announced in Ref. [@BSVtalk].
Notice that $\mathcal{W}_{(2,2)}$ in the superloop [(\[NMHVhexagon\])]{} does not require a dedicated study since it is determined by the sextet multi-scalar exchanges we have just discussed, $$\begin{aligned}
\mathcal{W}_{(2,2)} = \sum_{n = 0}^\infty \, \mathcal{W}^{\rm\bf 6}_{{\rm h} ({\rm hh})^n}
\, .\end{aligned}$$
Gluon-scalars
-------------
Finally, we address the component $\mathcal{W}_{(4,0)}$. Making use of Eq. [(\[HoleGluonPentagon\])]{}, it becomes obvious that as in the previous cases of heavy excitations accompanying an infinite tower of scalar exchanges, the contribution in question falls into the product of two terms $$\begin{aligned}
\mathcal{W}_{(4,0)} = \mathcal{W}_{\rm g} \sum_{n = 0}^\infty \, \mathcal{W}^{\rm\bf 1}_{({\rm hh})^n} \end{aligned}$$ with $$\begin{aligned}
\mathcal{W}_{\rm g} = \int d\mu_{\rm g} (u) \frac{x^+[u] x^- [u]}{g^2}
\, ,\end{aligned}$$ where the NMHV gluon helicity form factor is given by the product of shifted $x^\pm [u] = x [u \pm {{\textstyle\frac{i}{2}}}]$ Zhukowski variables $x [u] = {{\textstyle\frac{1}{2}}} (u + \sqrt{u^2 - (2 g)^2})$ and the infinite sum governed in the ultraviolet regime by the right-hand side of Eq. [(\[SingletSsm\])]{}.
Asymptotic form of heavy-particle exchanges
-------------------------------------------
Let us wrap up our discussion by determining the functional form of the heavy flux-tube exchanges at asymptotic values of $\tau$. Starting with the antifermion integral, we can use the saddle point approximation that immediately yields $$\begin{aligned}
\mathcal{W}_{\rm \bar{f}}
&
= - g^2 \int_{\mathbb{R} + i 0} \frac{d \theta}{\pi \sinh (\theta) \sinh (2 \theta)} {\rm e}^{- \tau \cosh (\theta) + i \sigma \sinh (\theta)}
\nonumber\\
&
\simeq
g^2
{\rm e}^{- \tau}
\sqrt{\frac{\tau}{2 \pi}} {\rm e}^{- \sigma^2/(2 \tau)}
\left[
1 + \frac{\sigma}{\sqrt{2 \tau}} {\rm e}^{\sigma^2/(2 \tau)} \left( {\rm erf} \left(\frac{\sigma}{\sqrt{2 \tau}} \right) - 1 \right)
+
\frac{5}{6} \frac{1}{\tau} + O \left(\frac{1}{\tau^{3/2}} \right)
\right]
\, .\end{aligned}$$ The leading behavior for the fermion contribution, that arises along with the odd number of accompanying scalars, differs from the above at subleading order in $\tau$ only, namely, $$\begin{aligned}
\mathcal{W}_{\rm f}
&
= - \int_{\mathbb{R} + i 0} \frac{d \theta}{2 \pi \sinh (\theta) \sinh (2 \theta)} {\rm e}^{- \tau \cosh (\theta) + i \sigma \sinh (\theta)} \frac{1}{(1 + \tanh^2(\theta))}
\nonumber\\
&
\simeq
{\rm e}^{- \tau}
\sqrt{\frac{\tau}{8 \pi}} {\rm e}^{- \sigma^2/(2 \tau)}
\left[
1 + \frac{\sigma}{\sqrt{2 \tau}} {\rm e}^{\sigma^2/(2 \tau)} \left( {\rm erf} \left(\frac{\sigma}{\sqrt{2 \tau}} \right) - 1 \right)
+
\frac{11}{6} \frac{1}{\tau} + O \left(\frac{1}{\tau^{3/2}} \right)
\right]
\, .\end{aligned}$$ Finally, as explained in Ref. [@Belitsky:2015qla], to properly take the strong coupling limit of gluons, first one has to pass to the Goldstone sheet $u \to u^{\rm G} + i/2 \to u$ with $\Im{\rm m} [u] \geq 1/2$ and then, after rescaling the rapidity $u =2 g \widehat{u}$, send $g \to \infty$, $$\begin{aligned}
\mathcal{W}_{\rm G}
&
= \int d\mu_{\rm G} (u) \frac{x^+[u]}{x^- [u]}
\simeq
\int d\mu_{\rm G} (u)
=
- 2g \int \frac{d \theta}{\pi \cosh^2 (2 \theta)} {\rm e}^{- \sqrt{2} \tau \cosh (\theta) + i \sqrt{2} \sigma \sinh (\theta)}
\nonumber\\
&\simeq
2 g {\rm e}^{- \sqrt{2} \tau}
\sqrt{\frac{\sqrt{2} \tau}{\pi}} {\rm e}^{- \sigma^2/(\sqrt{2} \tau)}
\left[
1 - \frac{\sqrt{8}}{\tau} + \frac{4 (\sigma^2 + 4)}{\tau^2} + O (\tau^{-3})
\right]
\, .\end{aligned}$$ Here the first line exhibits the helicity-independence of the gauge transition at strong coupling as the NMHV helicity form factor is $1$ to leading order in $1/g$ expansion. The same applies to bound states of $\ell$ gauge fields, whose contribution differs from the above consideration by the introducing the shifts $\pm i \ell/2$ in Zhukowski variables compared to $\ell = 1$ for a single gluon. The leading order expression is unaffected by these.
Conclusions
===========
In this work we extended the strong coupling analysis of NMHV hexagon to include hole excitations. The latter develop a nonperturbative regime compared to all other excitations with their mass gap being exponentially suppressed in strong coupling. Each individual contribution develops logarithmic dependence on the dimensionless scale $m_{\rm h} z$ which calls for an all-order resummation of all multi-hole exchanges. In all NMHV components the latter factorize into a multiplier that can be addressed separately from the accompanying heavy flux-tube excitation at leading order in strong coupling. There are two of these with either singlet or sextet quantum numbers with respect to the internal symmetry group of the parent theory. While the one corresponding to the singlet was addressed before, presently we added the latter to complete the consideration. The resumed expression was inspired by the reinterpretation of the pentagon form factor series in terms of the correlation functions of twist operators in the O(6) sigma model. Like in the MHV case [@Basso:2014jfa], we observed a nonperturbative enhancement of the classical area prediction $\exp (- 2 g A_6)$ by multiplicative factors $$\begin{aligned}
\mathcal{W}_\infty^{\bf r} =\left[ 8^{-1/4} \Gamma ({{\textstyle\frac{5}{4}}}) \right]^{1/36} \xi^{- 1/36} {\rm e}^{\pi g/36}
\left[
C_{\rm h}^{\bf 1} (\pi g)^{- 7/144}\delta_{\bf r, 1}
+
\frac{1}{\pi} C_{\rm h}^{\bf 6} (\pi g)^{281/144}\delta_{\bf r, 6}
\right]
\, , \end{aligned}$$ depending on the representation of the exchanged scalars.
It is important to find a way to predict the normalization constants in the ultraviolet limit analytically. The fact that the coefficients of the $\xi$-dependence in individual multi-hole exchanges are given by transcendental numbers suggests that direct resummation is presumably not the right way to approach this problem and therefore begs for more efficient techniques. It would be interesting to rephrase these results in a form of Thermodynamic Bethe Ansatz equations similar to the ones developed for the MHV amplitudes in Refs. [@Alday:2009dv; @Alday:2010vh]. Apart from that, the overall normalization receives corrections from heavy modes and inverse coupling expansion. One can extend our current considerations to higher polygons and establish constraints that follow from the Descent Equation [@CaronHuot:2011kk; @Bullimore:2011kg] along the lines of Ref. [@Belitsky:2015kda] for subleading corrections.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are indebted to Hank Lamm for help with numerical calculations. We would like to thank Benjamin Basso for informing us about the ongoing analysis being done in collaboration with Amit Sever and Pedro Vieira [@BSVtalk], which overlaps with the current consideration, as well as discussions at the final stage of this work completed during the visit at ENS (Paris) and IPhT (Saclay). We would like to thank Benjamin Basso and Gregory Korchemsky and for the warm hospitality at respective institutions. This research was supported by the U.S. National Science Foundation under the grants PHY-1068286 and PHY-1403891.
Flux-tube equations {#FluxTubeEqsAppendix}
===================
Let us rewrite the flux-tube equations in a form suitable for analysis at strong coupling. Their generic representation for the parity-even and parity-odd cases read [@Basso:2010in; @Basso:2013pxa; @Belitsky:2014sla] $$\begin{aligned}
\label{EvenEven}
\int_0^\infty
\frac{dt}{t} J_{2n} (2 g t) \left[ \frac{\gamma^{\star}_{u, +} (2 g t)}{1 - {\rm e}^{- t}} - \frac{\gamma^{\star}_{u, -} (2 g t)}{{\rm e}^{t} - 1} \right]
&
=
\kappa^{\star}_{2 n} (u)
\, , \\
\label{EvenOdd}
\int_0^\infty
\frac{dt}{t} J_{2n - 1} (2 g t) \left[ \frac{\gamma^{\star}_{u, -} (2 g t)}{1 - {\rm e}^{- t}} + \frac{\gamma^{\star}_{u, +} (2 g t)}{{\rm e}^{t} - 1} \right]
&
=
\kappa^{\star}_{2 n - 1} (u)
\, ,\end{aligned}$$ and $$\begin{aligned}
\label{OddEven}
\int_0^\infty
\frac{dt}{t} J_{2n} (2 g t) \left[ \frac{\widetilde\gamma^{\star}_{u, +} (2 g t)}{1 - {\rm e}^{- t}} + \frac{\widetilde\gamma^{\star}_{u, -} (2 g t)}{{\rm e}^{t} - 1} \right]
&
=
\widetilde\kappa^{\star}_{2 n} (u)
\, , \\
\label{OddOdd}
\int_0^\infty
\frac{dt}{t} J_{2n - 1} (2 g t) \left[ \frac{\widetilde\gamma^{\star}_{u, -} (2 g t)}{1 - {\rm e}^{- t}} - \frac{\widetilde\gamma^{\star}_{u, +} (2 g t)}{{\rm e}^{t} - 1} \right]
&
=
\widetilde\kappa^{\star}_{2 n - 1} (u)
\, ,\end{aligned}$$ respectively. Here the sources depend on the $\star$-type of excitations under consideration. In what follows, we only need the ones corresponding to scalars. However, since they will be defined implicitly in our subsequent formulas, we will not display the explicit form in order to save space. In addition, for future reference, we recall the form of inhomogeneities for the flux-tube vacuum which read $$\begin{aligned}
\label{VacSource}
\kappa^{\o}_{n} = 2 g \delta_{n,0}
\, , \qquad
\widetilde\kappa^{\o}_{n} = 0
\, .\end{aligned}$$
Following [@Basso:2008tx1], we introduce a functional transformation $$\begin{aligned}
\label{gammaToGamma}
\Gamma^{\rm f}_u (\tau)
&
\equiv
\Gamma^{\rm f}_{+,u} (\tau) + i \Gamma^{\rm f}_{-,u} (\tau)
=
\left(
1 + i \coth \frac{\tau}{4 g}
\right)
\gamma^{\rm f}_{u} (\tau)
\, ,
\\
\label{gammaToGammaTilde}
\widetilde\Gamma^{\rm f}_u (\tau)
&
\equiv
\widetilde\Gamma^{\rm f}_{+,u} (\tau) - i \widetilde\Gamma^{\rm f}_{-,u} (\tau)
=
\left(
1 + i \coth \frac{\tau}{4 g}
\right)
\widetilde\gamma^{\rm f}_{u} (\tau)
\, ,\end{aligned}$$ that has the advantage of removing the explicit dependence on the coupling constant from the Eqs. [(\[EvenEven\])]{} – [(\[OddOdd\])]{}. Further, using the Jacobi-Anger summation formula and the identity $$\begin{aligned}
\int_0^\infty \frac{dt}{t} J_0 (2 g t) (\cos (u_1 t) - 1) = 0
\, ,\end{aligned}$$ ($|u_1| < 2 g$) for $\Gamma$, we can cast flux-tube equations for the hole into the form $$\begin{aligned}
\label{GammaH2cos}
\int_0^\infty \frac{dt}{t} ( \cos(u_1 t) - 1 )
\left[
\Gamma^{\rm h}_{-, u_2} (2 g t)
+
\Gamma^{\rm h}_{+, u_2} (2 g t)
\right]
&
=
-
\int_0^\infty \frac{dt}{t} ( \cos(u_1 t) - 1 )
\frac{\cos(u_2 t) - {\rm e}^{t/2} J_0 (2 g t)}{\sinh \frac{t}{2}}
\, , \\
\label{GammaH1sin}
\int_0^\infty \frac{dt}{t} \sin(u_1 t)
\left[
\Gamma^{\rm h}_{-, u_2} (2 g t)
-
\Gamma^{\rm h}_{+, u_2} (2 g t)
\right]
&
=
-
\int_0^\infty \frac{dt}{t} \sin(u_1 t)
\frac{\cos(u_2 t) - {\rm e}^{- t/2} J_0 (2 g t)}{\sinh \frac{t}{2}}
\, ,\end{aligned}$$ and $$\begin{aligned}
\label{GammaHtilde2}
\int_0^\infty \frac{dt}{t} ( \cos(u_1 t) - 1 )
\left[
\widetilde\Gamma^{\rm h}_{+, u_2} (2 g t)
-
\widetilde\Gamma^{\rm h}_{-, u_2} (2 g t)
\right]
&
=
-
\int_0^\infty \frac{dt}{t} ( \cos(u_1 t) - 1 )
\frac{\sin(u_2 t)}{\sinh \frac{t}{2}}
\, , \\
\label{GammaHtilde1}
\int_0^\infty \frac{dt}{t} \sin(u_1 t)
\left[
\widetilde\Gamma^{\rm h}_{+, u_2} (2 g t)
+
\widetilde\Gamma^{\rm h}_{-, u_2} (2 g t)
\right]
&
=
-
\int_0^\infty \frac{dt}{t} \sin(u_1 t)
\frac{\sin(u_2 t)}{\sinh \frac{t}{2}}
\, .\end{aligned}$$ These results are used in the main text.
Exchange relations
==================
To partially verify our findings for nonperturbative corrections derived in the main text, we will rewrite the energy and momentum of the hole, which are are conventionally expressed via the vacuum flux-tube function $\gamma^{\o}(t)$ [@Basso:2010in] $$\begin{aligned}
E_{\rm h} (u)
&=
1 + \int_0^\infty \frac{dt}{t} \frac{\gamma^{\o} (- 2 g t)}{{\rm e}^t - 1} \left( {\rm e}^{t/2} \cos (u t) - J_0 (2 g t) \right)
\, , \\
p_{\rm h} (u)
&=
2 u - \int_0^\infty \frac{dt}{t} \frac{\gamma^{\o} (2 g t)}{{\rm e}^t - 1} {\rm e}^{t/2} \sin (u t)
\, ,\end{aligned}$$ in terms of the hole flux-tube functions $\gamma^{\rm h}_u$ and $\widetilde\gamma^{\rm h}_u$. Let us demonstrate it for the momentum and just quote the final answer for the energy.
To start with, let us recall that the even and odd components of the flux-tube functions are entire functions and admit convergent Neumann expansions in terms of Bessel functions. Then one can write the above formula in the form of an infinite series representation $$\begin{aligned}
p_{\rm h} (u)
=
2 u
+
2 \sum_{n \geq 1} (2n) \gamma^{\o}_{2n} \widetilde\kappa^{\rm h}_{2n} (u)
+
2 \sum_{n \geq 1} (2n - 1) \gamma^{\o}_{2n - 1} \widetilde\kappa^{\rm h}_{2n - 1} (u)
\, ,\end{aligned}$$ making use of the sources defining inhomogeneities in the flux-tube equations for scalars. Multiplying Eqs. [(\[OddEven\])]{} and [(\[OddOdd\])]{} by $2 (2n) J_{2n} (2 g t)$ and $2 (2n - 1) J_{2n - 1} (2 g t)$, respectively, and summing over positive values of $n$, we find for their sum $$\begin{aligned}
2 \sum_{n \geq 1} (2n) \gamma^{\o}_{2n} \widetilde\kappa^{\rm h}_{2n} (u)
+
2 \sum_{n \geq 1} (2n - 1) \gamma^{\o}_{2n - 1} \widetilde\kappa^{\rm h}_{2n - 1} (u)
&
=
\int_0^\infty \frac{dt}{t}
\bigg[
\frac{\gamma^{\o}_+ (2 g t) \widetilde\gamma^{\rm h}_{+, u} (2 g t) + \gamma^{\o}_- (2 g t) \widetilde\gamma^{\rm h}_{-, u} (2 g t) }{1 - {\rm e}^{- t}}
\nonumber\\
&\qquad\quad
+
\frac{\gamma^{\o}_+ (2 g t) \widetilde\gamma^{\rm h}_{-, u} (2 g t) - \gamma^{\o}_- (2 g t) \widetilde\gamma^{\rm h}_{+, u} (2 g t) }{{\rm e}^{t} - 1}
\bigg]
. \end{aligned}$$ Now, expanding the hole flux-tube functions in the Neumann series provides gives a very concise representation of the right-hand side $$\begin{aligned}
2 \sum_{n \geq 1} (2n) \widetilde\gamma^{\rm h}_{2n} (u) \kappa^{\o}_{2n}
+
2 \sum_{n \geq 1} (2n - 1) \widetilde\gamma^{\rm h}_{2n - 1} (u) \kappa^{\o}_{2n - 1}
=
4 g \widetilde\gamma^{\rm h}_1 (u)
\, ,\end{aligned}$$ where we employed the explicit form of the sources for the vacuum [(\[VacSource\])]{}.
Analogous consideration can be done for the energy such that one can rewrite the dispersion relation in the form $$\begin{aligned}
\label{HoleEandPfromGammaH}
E_{\rm h} (u)
=
1 + 2 \lim_{t \to 0} \frac{\gamma_u^{\rm h} (2 g t)}{t}
\, , \qquad
p_{\rm h} (u)
=
2 u + 2 \lim_{t \to 0} \frac{\widetilde\gamma_u^{\rm h} (2 g t)}{t}
\, .\end{aligned}$$ Here we relied on the fact that only the leading term in the Neumann expansion induces a nontrivial contribution (with subleading ones scaling as powers of $t$ which vanish in the limit in question). These agree with Ref. [@Basso:2013pxa].
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L.J. Dixon, M. von Hippel, A.J. McLeod, “The four-loop six-gluon NMHV ratio function,” arXiv:1509.08127 \[hep-th\]. B. Basso, A. Sever, P. Vieira, “Collinear limit of scattering amplitudes at strong coupling,” Phys. Rev. Lett. [**113**]{} (2014) 26, 261604 \[arXiv:1405.6350 \[hep-th\]\]. D. Fioravanti, S. Piscaglia, M. Rossi, “Asymptotic Bethe Ansatz on the GKP vacuum as a defect spin chain: scattering, particles and minimal area Wilson loops,” Nucl. Phys. B [**898**]{} (2015) 301 \[arXiv:1503.08795 \[hep-th\]\]. S. Caron-Huot, “Notes on the scattering amplitude/Wilson loop duality,” JHEP [**1107**]{} (2011) 058 \[arXiv:1010.1167 \[hep-th\]\]. L.J. Mason, D. Skinner, “The complete planar S-matrix of N=4 SYM as a Wilson loop in twistor space,” JHEP [**1012**]{} (2010) 018 \[arXiv:1009.2225 \[hep-th\]\]. A.V. Belitsky, G.P. Korchemsky, E. Sokatchev, “Are scattering amplitudes dual to super Wilson loops?,” Nucl. Phys. B [**855**]{} (2012) 333 \[arXiv:1103.3008 \[hep-th\]\]. L. Bianchi, M.S. Bianchi, “Worldsheet scattering for the GKP string,” arXiv:1508.07331 \[hep-th\]; “On the scattering of gluons in the GKP string,” arXiv:1511.01091 \[hep-th\]. B. Basso, A. Rej, “Bethe Ansatze for GKP strings,” Nucl. Phys. B [**879**]{} (2014) 162 \[arXiv:1306.1741 \[hep-th\]\]. B. Basso, G.P. Korchemsky, “Embedding nonlinear O(6) sigma model into N=4 super-Yang-Mills theory,” Nucl. Phys. B [**807**]{} (2009) 397 \[arXiv:0805.4194 \[hep-th\]\]. B. Basso, G.P. Korchemsky, “Nonperturbative scales in AdS/CFT,” J. Phys. A [**42**]{} (2009) 254005 \[arXiv:0901.4945 \[hep-th\]\]. B. Basso, “Exciting the GKP string at any coupling,” Nucl. Phys. B [**857**]{} (2012) 254 \[arXiv:1010.5237 \[hep-th\]\]. L.F. Alday, J.M. Maldacena, “Comments on operators with large spin,” JHEP [**0711**]{} (2007) 019 \[arXiv:0708.0672 \[hep-th\]\]. S. Caron-Huot, S. He, “Jumpstarting the all-loop S-Matrix of planar N=4 super Yang-Mills,” JHEP [**1207**]{} (2012) 174 \[arXiv:1112.1060 \[hep-th\]\]. 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[^1]: Recently scattering matrices that define pentagon transitions were independently computed at strong coupling from the perspective of the two-dimensional world-sheet sigma-model in Refs. [@Bianchi:2015vgw].
[^2]: To avoid cluttering the formulas which follow with powers of the ’t Hooft coupling, we normalized the hole-hole pentagon transition, and as a consequence the measure, to coupling independent function at leading order at strong coupling.
[^3]: Notice that $X^i$ has a vanishing canonical dimension and therefore does not affect the power-law behavior of the amplitude.
[^4]: We would like to thank Benjamin Basso for bringing the talk [@BSVtalk] to our attention and useful discussion.
|
---
abstract: 'Have our fundamental theories got time right? Does size really matter? Or is physics all in the eyes of the beholder? In this essay, we question the origin of time and scale by reevaluating the nature of measurement. We then argue for a radical scenario, supported by a suggestive calculation, where the flow of time is inseparable from the measurement process. Our scenario breaks the bond of time and space and builds a new one: the marriage of time and scale.'
author:
- 'Sean Gryb[^1]'
- 'Flavio Mercati[^2]'
bibliography:
- 'mach.bib'
- 'BibEssay.bib'
title: '**Right about time?**'
---
Introduction
============
Near the end of the 19$^\text{th}$ century, physics appeared to be slowing down. The mechanics of Newton and others rested on solid ground, statistical mechanics explained the link between the microscopic and the macroscopic, Maxwell’s equations unified electricity, magnetism, and light, and the steam engine had transformed society. But the blade of progress is double edged and, as more problems were sliced through, fewer legitimate fundamental issues remained. Physics, it seemed, was nearing an end.
Or was it? Among the few remaining unsolved issues were two experimental anomalies. As Lord Kelvin allegedly announced:“The beauty and clearness of the dynamical theory \[...\] is at present obscured by two clouds.”[@Kelvin:dark_clouds] One of these clouds was the ultra–violet catastrophe: an embarrassing prediction that hot objects like the sun should emit *infinite* energy. The other anomaly was an experiment by Michelson and Morley that measured the speed of light to be independent of how an observer was moving. Given the tremendous success of physics at that time, it would have been a safe bet that, soon, even these clouds would pass.
Never bet on a sure thing. The ultra–violet catastrophe led to the development of quantum mechanics and the Michelson–Morley experiment led to the development of relativity. These discoveries completely overturned our understanding of space, time, measurement, and the perception of reality. Physics was not over, it was just getting started.
Fast–forward a hundred years or so. Quantum mechanics and relativity rest on solid ground. The microchip and GPS have transformed society. These frameworks have led to an understanding that spans from the microscopic constituents of the nucleus to the large scale structure of the Universe. The corresponding models have become so widely accepted and successful that they have been dubbed *standard models* of particle physics and cosmology. Resultantly, the number of truly interesting questions appears to be slowly disappearing. In well over 30 years, there have been no experimental results in particle physics that can’t be explained within the basic framework laid out by the standard model. With the ever increasing cost of particle physics experiments, it seems that the data is drying up. But without input from experiment, how can physics proceed? It would appear that physics is, again, in danger of slowing down.
Or is it? Although the *number* of interesting fundamental questions appears to be decreasing, the *importance* of the remaining questions is growing. Consider two of the more disturbing experimental anomalies. The first is the *naturalness problem*, i.e., the presence of unnaturally large and small numbers in Nature. The most embarrassing of these numbers – and arguably the worst prediction of science – is the accelerated expansion of the Universe, which is some 120 orders of magnitude smaller than its natural value. The second is the *dark matter problem* that just under 85–90 percent of the matter content of our Universe is of an exotic nature that we have not yet seen in the lab. It would seem that we actually understand very little of what is happening in our Universe!
The problem is not that we don’t have enough data. The problem is that the data we do have does not seem to be amenable to explanation through incremental theoretical progress. The belief that physics is slowing down or, worse, that we are close to a final theory is just as as unimaginative now as it would have been before 1900. The lesson from that period is that the way forward is to question the fundamental assumptions of our physical theories in a radical way. This is easier said than done: one must not throw out the baby with the bath water. What is needed is a careful examination of our physical principles in the context of real experimental facts to explain *more* data using *less* assumptions.
The purpose of this work is to point out three specific assumptions made by our physical theories that might be wrong. We will not offer a definite solution to these problems but suggest a new scenario, supported by a suggestive calculation, that puts these assumptions into a new light and unifies them. The three assumptions we will question are
1. Time and space are unified.
2. Scale is physical.
3. Physical laws are independent of the measurement process.
We will argue that these three assumptions inadvertently violate the same principle: the requirement that the laws of physics depend only on what is knowable through direct measurement. They fall into a unique category of assumptions that are challenged when we ask how to adapt the scientific method, developed for understanding processes in the lab, to the cosmological setting. In other words, how can we do science on the Universe *as a whole*?
We will not directly answer this question but, rather, suggest that this difficult issue may require a radical answer that questions the very origin of time. The flow of time, we will argue, may be fundamentally linked to the process of measurement. We will then support this argument with an intriguing calculation that recovers the black hole entropy law from a simple toy model. Before getting to this, let us explain the three questionable assumptions.
Three questionable assumptions
==============================
Many of our most basic physical assumptions are made in the first week of physics education. A good example is one of the first equations we are taught: the definition of velocity, $$v = \frac {\Delta x} {\Delta t}.$$ To give this equation precise operational meaning has been an outstanding issue in physics for its entire history. This is because, to understand this equation, one has to have an operational definition of both $x$, $t$, and $\Delta$. Great minds have pondered this question and their insights have led to scientific revolutions. This includes the development of Newtonian mechanics, relativity, and quantum mechanics.[^3] Recently, the meaning of $x$ and, in particular, $t$, have been the subject of a new debate whose origin is in a theory of quantum gravity. This brings us to our first questionable assumption.
Time and space are unified {#sec:time and space}
--------------------------
The theory of relativity changed our perception of time. As Minkowski put it in 1908 [@Minkowski:seminal_address], “space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” Nowhere is this more apparent than in the main equation physicists use to construct the solutions of general relativity (GR): $$\label{eq:EH}
S_\text{Einstein-Hilbert} = \int d^4x \left( R + \mathcal L_\text{matter} \right) \sqrt{-g} \;.$$ Can you spot the $t$? It’s hidden in the $4$ of $d^4x$. But there are important structures hidden by this compact notation.
We will start by pointing out an invisible minus sign in equation [(\[eq:EH\])]{}. When calculating spacetime distances, one needs to use $$x^2 + y^2 + z^2 - t^2,$$ which has a $-$ in front of the $t^2$ instead of Pythagoras’ $+$. The minus sign looks innocent but has important consequences for the solutions of equation [(\[eq:EH\])]{}. Importantly, the minus sign implies *causality*, which means that only events in the past can effect what is going on now. This, in turn, implies that generic solutions of GR can only be solved by specifying information at a particular *time* and then seeing how this information propagates into the future. Doing the converse, i.e., specifying information at a particular *place* and seeing how that information propagates to another place, is, in general, not consistent.[^4] Thus, the minus sign already tells you that you have to use the theory in a way that treats time and space differently.
There are other ways to see how time and space are treated differently in gravity. In Julian Barbour’s 2009 essay, *The Nature of Time* [@Barbour:nature_of_time], he points out that Newton’s “absolute” time is not “absolute” at all. Indeed, the Newtonian notion of *duration* – that is, how much time has ticked by – can be *inferred* by the total change in the *spatial* separations of particles in the Universe. He derives the equation $$\label{eq:ET}
\Delta t^2 \propto \sum_i \Delta d^2_i,$$ where the $d_i$ are inter–particle separations in units where the masses of the particles are 1. The factor of proportionality is important, but not for our argument. What is important is that changes in time can be inferred by changes in distances so that absolute duration is not an input of the classical theory. This equation can be generalized to gravity where it must be solved at every point in space. The implications for the quantum theory are severe: time completely drops out of the formalism.
Expert readers will recognize this as one of the facets of the *Problem of Time* [@Isham:pot_review]. The fact that there is no equivalent *Problem of Space* can be easily traced back to the points just made: time is singled out in gravity as the variable in terms of which the evolution equations are solved. This in turn implies that local duration should be treated as an *inferred* quantity rather than something fundamental. Clearly, time and space are *not* treated on the same footing in the formalism of GR despite the rather misleading form of equation [(\[eq:EH\])]{}. Nevertheless, it is still true that the spacetime framework is incredibly useful and, as far as we know, correct. How can one reconcile this fact with the space–time asymmetry in the formalism itself? We will investigate this in section (\[sec:time from RG\]).
Scale is physical {#sec:scale}
-----------------
Before even learning the definition of velocity, the novice physicist is typically introduced to an even more primary concept that usually makes up one’s first physics lesson: *units*. Despite the rudimentary nature of units, they are probably the most commonly misunderstood concept in all of physics. If you ask ten different physicists for the physical meaning of a unit, you will likely get ten different answers. To avoid confusion, most theoreticians set all dimensionful constants equal to 1. However, one can’t predict anything until one has painfully reinserted these dimensionful quantities into the final result.
And yet, no one has *ever* directly observed a dimensionful quantity. This is because all measurements are comparisons. A meter has no intrinsic operational meaning, only the ratio of two lengths does. One can call object A a meter and measure that object B is twice its length. Then, object B has a length of 2 meters but that tells you nothing about the intrinsic length of object A. If a demon doubled the intrinsic size of the Universe, the result of the experiment would be exactly the same. So, where do units come from?
Some units, like the unit of pressure, are the result of emergent physics. We understand how they are related to more “fundamental” units like meters and seconds. However, even our most fundamental theories of Nature have dimensionful quantities in them. The standard model of particle physics and classical GR require only a singe unit: *mass*. Scale or, more technically, *conformal invariance* is then broken by the Higgs mass, which is related to all the masses of the particles in the standard model, and the Plank mass, which sets the scale of quantum gravity. As already discussed, there is a naturalness problem associated with writing all other constants of nature as dimensionless quantities.
The presence of dimensionful quantities is an indication that our “fundamental” theories are not fundamental at all. Instead, scale independence should be a basic principle of a fundamental theory. As we will see in section (\[sec:time from RG\]), there is a formulation of gravity that is *nearly* scale invariant. The “nearly” will be addressed by the considerations of the next section.
Physical laws are independent of the measurement process {#sec:measurement}
--------------------------------------------------------
There is one assumption that is so fundamental it doesn’t even enter the physics curriculum: the general applicability of the scientific method. We know that the scientific method can be applied in the laboratory where external agents (i.e., scientists) carefully control the inputs of some subsystem of the Universe and observe the subsystem’s response to these inputs. We don’t know, however, whether it is possible to apply these techniques to the Universe as a whole. On the other hand, when it comes to quantum mechanics, we *do* know whether our formalism can be consistently applied to the Universe. The answer is ‘NO’. The reasons are well understood – if not disappointingly under appreciated – and the problem even has a name: *the measurement problem*.
The measurement problem results from the fact that quantum mechanics is a framework more like statistical physics than classical mechanics. In statistical physics, one has *practical* limitations on one’s knowledge of a system so one takes an educated guess at the results of a specific experiment by calculating a probability distribution for the outcome using one’s current knowledge of the system. In quantum mechanics, one has *fundamental* limitations on one’s knowledge of the system – essentially because of the uncertainty principle – so one can only make an educated guess at the outcome of a specific experiment by calculating a probability distribution for the outcome using one’s current knowledge of the system. However, it would be strange to apply statistical mechanics to the whole Universe[^5] because the Universe itself is only given once. It is difficult to imagine an ensemble of Universes for which one can calculate a probability distribution. The same is true in quantum mechanics, but the problem is worse. The framework itself is designed to give you a probability distribution for the outcome of some measurement but how does one even define a measurement when the observer itself is taken to be part of the system? The answer is not found in any interpretation of quantum mechanics, although the problem itself takes a different form in a given interpretation. The truth is that quantum mechanics requires some additional structure, which can be thought of as the observer, in order for it to make sense. In other words, quantum mechanics can never be a theory of the whole Universe.
As a consequence of this, any approach to quantum gravity that uses quantum mechanics unmodified – including all *major* approaches to quantum gravity – is not, and *can never be* a theory of the whole Universe. It could still be used for describing quantum gravity effects on isolated subsystems of the Universe, but that is not the ambition of a full fledged quantum gravity theory. Given such a glaring foundational issue at the core of every major approach to quantum gravity, we believe that the attitude that we are nearing the end of physics is unjustified. The “shut–up and calculate” era is over. It is time for the quantum gravity community to return to these fundamental issues.
One approach is to change the ambitions of science. This is the safest and most practical option but it would mean that science is inherently a restricted framework. The other possibility is to try to address the measurement problem directly. In the next section, we will give a radical proposal that embraces the role of the observer in our fundamental description of Nature. To understand how this comes about, we need one last ingredient: *renormalization*, or the art of averaging.
A way forward
=============
The art of averaging
--------------------
It is somewhat unfortunate that the great discoveries of the first half of the $20^\text{th}$ century have overshadowed those of the second half of the century. One of these, the theory of *renormalization*, is potentially the uncelebrated triumph of $20^\text{th}$ century physics. Renormalization was born as rather ugly set of rules for removing some undesirable features of quantum field theories. From these humble beginnings, it has grown into one of the gems of physics. In its modern form due to Wilson [@Wilson:RG_review], renormalization has become a powerful tool for understanding what happens in a general system when one lacks information about the details of its fine behavior. Renormalization’s reach extends far beyond particle physics and explains, among other things, what happens during phase transitions. But, the theory of renormalization does even more: it helps us understand why physics is possible at all.
Imagine what it would be like if, to calculate everyday physics like the trajectory of Newton’s apple, one would have to compute the motions of every quark, gluon, and electron in the apple and use quantum gravity to determine the trajectory. This would be completely impractical. Fortunately, one doesn’t have to resort to this. High–school physics is sufficient to determine the motion of what is, fundamentally, an incredibly complicated system. This is possible because one can average, or *coarse grain*, over the detailed behavior of the microscopic components of the apple. Remarkably, the average motion is simple. This fact is the reason why Newtonian mechanics is expressible in terms of simple differential equations and why the standard model is made up of only a couple of interactions. In short, it is why physics is possible at all. The theory of renormalization provides a framework for understanding this.
The main idea behind renormalization is to be able to predict how the laws of physics will change when a coarse graining is performed. This is similar to what happens when one changes the magnification of a telescope. With a large magnification, one might be able to see the moons of Jupiter and some details of the structure of their atmospheres. But, if the magnification, or the *renormalization scale*, is steadily decreased, the resolution is no longer good enough to make out individual moons and the lens averages over these structures. The whole of Jupiter and its moons becomes a single dot. As we vary the renormalization scale, the laws of physics that govern the structures of the system change from the hydrodynamic laws of the atmospheres to Newton’s law of gravity.
The theory of renormalization produces precise equations that say how the laws of physics will change, or *flow*, as we change the renormalization scale. In what follows, we will propose that flow under changes of scale may be related to the flow of time.
Time from coarse graining {#sec:time from RG}
-------------------------
We are now prepared to discuss an idea that puts our three questionable assumptions into a new light by highlighting a way in which they are connected. First, we point out that there is a way to *trade* a spacetime symmetry for conformal symmetry without altering the physical structures of GR. This approach, called *Shape Dynamics* (SD), was initially advocated by Barbour [@barbour:bm_review] and was developed in [@gryb:shape_dyn; @Gomes:linking_paper]. Symmetry trading is allowed because symmetries don’t affect the physical content of a theory. In SD, the irrelevance of duration in GR is traded for local scale invariance (we will come to the word “local” in a moment). This can be done without altering the physical predictions of the theory but at the cost of having to treat time and space on a different footing. In fact, the local scale invariance is only an invariance of *space*, so that local rods – not clocks – can be rescaled arbitrarily. Time, on the other hand, is treated differently. It is a global notion that depends on the total change in the Universe.
In 2 spatial dimensions, we know that this trading is possible because of an accidental mathematical relationship between the structure of conformal symmetry in 2 dimensions and the symmetries of 3 dimensional spacetime [@gryb:2_plus_1].[^6] We are investigating whether this result will remain true in 3 spatial dimensions. If it does, it would mean that the spacetime picture and the conformal picture can coexist because of a mere mathematical accident.
We now come to a key point: in order for any time evolution to survive in SD, one cannot eliminate all of the scale. The *global* scale of the Universe cannot be traded since, then, no time would flow. Only a *redistribution* of scale from point to point is allowed (this is the significance of the word “local”) but the overall size of the Universe cannot be traded. In other words, *global scale must remain for change to be possible.* How can we understand this global scale?
Consider a world with no scale and no time. In this world, only 3 dimensional Platonic shapes exist. This kind of world has a technical name, it is a *fixed point* of renormalization – “fixed” because such a world does not flow since the renormalization scale is meaningless. This cannot yet be our world because nothing happens in this world. Now, allow for something to happen and call this “something” a *measurement*. One thing we know about measurements is that they can never be perfect. We can only compare the smallest objects of our device to larger objects and coarse grain the rest. Try as we may, we can never fully resolve the Platonic shapes of the fixed point. Thus, coarse graining by real measurements produces flow away from the fixed point. But, what about time? How can a measurement happen if no time has gone by? The scenario that we are suggesting is that the flow under the renormalization scale is exchangeable with the flow of time. Using the trading procedure of SD, the flow of time might be relatable to renormalization away from a theory of pure shape.
In this picture, time and measurement are inseparable. Like a diamond with many faces, scale and time are different reflections of a single entity. This scenario requires a radical reevaluation of our notions of time, scale, and measurement.
To be sure, a lot of thought is still needed to turn this into a coherent picture. A couple of comments are in order. Firstly, some authors [@Strominger:holo_cosmo; @Skenderis:holo_uni] have investigated a similar scenario, called *holographic cosmology* using something called *gauge/gravity duality*. However, our approach suggests that one may not have to *assume* gauge/gravity duality for this scenario but, instead, can make use of symmetry trading in SD. Furthermore, our motivation and our method of implementation is more concrete. Secondly, why should we expect that there is enough structure in a coarse graining of pure shapes to recover the rich structure of spacetime? A simple answer is the subject of the next section.[^7]
The size that matters
=====================
In this section, we perform a simple calculation suggesting that the coarse graining of shapes described in the last section could lead to gravity. This section is more technical than the others but this is necessary to set up our final result. Brave souls can find the details of the calculations in the Technical Appendix (\[TechnicalAppendix\]).
We will consider a simple “toy model” that, remarkably, recovers a key feature of gravity. Our model will be a set of $N$ free Newtonian point particles. To describe the calculation we will need to talk about two spaces: *Shape Space* and *Extended Configuration Space* (ECS). Shape Space is the space of all the shapes of the system. If $N=3$, this is the space of all triangles. ECS is the space of all Cartesian coordinates of the particles. That is, the space of all ways you can put a shape into a Cartesian coordinate system. The ECS is larger than Shape Space because it has information about the position, orientation, and size of the shapes. Although this information is unphysical, it is convenient to work with it anyway because the math is simpler. This is called a *gauge theory*. We can work with gauge theories provided we remove, or *quotient*, out the unphysical information. To understand how this is done, examine Figure (\[pfb\]) which shows schematically the relation between the ECS and Shape Space. Each point on Shape Space is a different shape of the system, like a triangle.
![Each point in Shape Space is a different shape (represented by triangles). These correspond to an equivalence class (represented by arrows) of points of the Extended Configuration Space describing the same shape with a different position, orientation, and size.[]{data-label="pfb"}](Flavio_FiberBundleNew.pdf){width="75.00000%"}
All the points along the arrows represent the same shape with a different position, orientation, or size. By picking a representative point along each arrow, we get a 1–to–1 correspondence between ECS and Shape Space. This is called *picking a gauge*. Mathematically, this is done by imposing constraints on the ECS. In our case, we need to specify a constraint that will select a triangle with a certain center of mass, orientation, and size. For technical reasons, we will assume that all particles are confined to a line so that we don’t have to worry about orientation. To specify the size of the system, we can take the “length” of the system, $R$, on ECS. This is the *moment of inertia*. By fixing the center of mass and moment of inertia in ECS, we can work indirectly with Shape Space. The main advantage of doing this is that there is a natural notion of distance in ECS. This can be used to define the distance between two shapes, which is a key input of our calculations.
To describe the calculation, we need to specify a notion of *entropy* in Shape Space. Entropy can be thought of as the amount of information needed to specify a particular macroscopic state of the system. To make this precise, we can use the notion of distance on ECS to calculate a “volume” on Shape Space. This volume roughly corresponds to the number of shapes that satisfy a particular property describing the state. The more shapes that have this property, the more information is needed to specify the state. The entropy of that state is then related to its volume, $\Omega_m$, divided by the total volume of Shape Space, $\Omega_\text{tot}$. Explicitly, $$S = -k_\text{B} \log \frac{\Omega_m}{\Omega_\text{tot}},$$ where $k_\text{B}$ is Boltzmann’s constant.
We will be interested in states described by a subsystem of $n<N$ particles that have a certain center of mass ${\bm x}_0$ and moment of inertia, $r$. To make sense of the volume, we need a familiar concept: coarse graining. We can approximate the volume of the state by chopping up the ECS into a grid of size $\ell$. Physically, the coarse graining means that we have a measuring device with a finite resolution given by $\ell$. Consider a state that is represented by some surface in ECS. This is illustrated in Figure (\[LatticeApprox\]) by a line.
![*Left:* Approximation of a line using a grid. *Right:* Further approximation of the line as a strip of thickness equal to the grid spacing.[]{data-label="LatticeApprox"}](Grid.png){width="75.00000%"}
The volume of the state is well approximated by counting the number of dark squares intersected by the line. In the Technical Appendix (\[TechnicalAppendix\]), we calculate this volume explicitly. The result is $$\Omega_\text{m} \propto \ell^2 \; r^{n-2} \; \left( R^2 - r^2 - \left(1 + \frac m {M-m} \right)\frac m M \; x_0^2\right)^{\frac{N-n-2}{2}} \;,$$ where $M$ and $R$ are the total mass and moment of inertia of the whole system and $m$ is the mass of the subsystem. We can then compare this volume to the total volume of Shape Space, which goes like the volume of an $N-1$ dimensional sphere (the $-1$ is because of the center of mass gauge fixing). Thus, $$\Omega_\text{tot} \propto R^{N-1}.$$ The resulting entropy is $$\label{eq:entropy}
S = \frac 1 2 \, k_B \, \frac N n \, \left( \frac r R \right)^2 - \, k_B \, \log \frac r R + \dots.$$ Remarkably, the first term is exactly the entropy of a black hole calculated by Bekenstein and Hawking [@BCH; @Bekenstein]. More remarkably, the second term is exactly the first correction to the Bekenstein–Hawking result calculated in field theory [@BlackHole1; @BlackHole2]. Erik Verlinde [@Verlinde:entropic_gravity] discovered a way to interpret Newtonian gravity as an *entropic* force for systems whose entropy behaves in this way. It would appear that this simple model of a coarse graining of pure shapes has the right structure to reproduce Newtonian gravity.
Conclusions
===========
We have questioned the basic assumptions that: i) time and space should be treated on the same footing, ii) scale should enter our fundamental theories of Nature, and iii) the evolution of the Universe is independent of the measurement process. This has led us to a radical proposal: that time and scale emerge from a coarse graining of a theory of pure shape. The possibility that gravity could come out of this formalism was suggested by a simple toy model. The results of this model are non–trivial. The key result was that the entropy [(\[eq:entropy\])]{} scales like $r^2$, which, dimensionally, is an area. In three dimensions, this is the signature of *holography*. Thus, in this simple model, Shape Space is holographic. If this is a generic feature of Shape Space, it would be an important observation for quantum gravity.
Moreover, the toy model may shed light on the nature of the Plank length. In this model, the Plank length is the emergent length arising in ECS given by $$L_\text{Planck}^2 = G \, \hbar \propto \frac {R^2} N \;.$$ This dimensionful quantity, however, is not observable in this model. What is physical, instead, it the dimensionless ratio $r/R$. This illustrates how a dimensionful quantity can emerge from a scale independent framework. Size doesn’t matter – but a ratio of sizes does. The proof could be gravity.
Technical Appendix {#TechnicalAppendix}
==================
The extended configuration space is $\mathbbm R^N$: the space coordinates, $r_i$, ($i= 1 ,\dots , N$) of $N$ particles in 1 dimension. To represent the reduced configuration space, or Shape Space, we can use a gauge fixing surface. To fix the translations, we can fix the center of mass to be at the origin of the coordinate system: $$\sum_{i=1}^{N} \, m_i \; r_i = 0 \;. \qquad \text{\it(center of mass at the origin)} \label{TotCenterOfMass}$$ The equation above gives three constraints selecting three orthogonal planes through the origin whose orientation is determined by the masses $m_i$. A natural gauge–fixing for the generators of dilatations is to set the moment of inertia with respect to the center of mass to a constant[^8] (the weak equation holds when the gauge–fixing (\[TotCenterOfMass\]) is applied): $$\label{DilatationGaugeFixing}
\sum_{i<j} \, \frac{m_i m_j}{M^2} \,| r_i - r_j|^2 \approx
\sum_{i=1}^{N} \, \frac{m_i}{M} \; | r_i |^2= R^2\;. \qquad \text{\it(fixed moment of inertia)}$$ The last relation defines a sphere in $\mathbbm R^N$ centered at the origin. Thus, Shape Space is the intersection of the $N-1$-dimensional sphere (\[DilatationGaugeFixing\]) with the three orthogonal planes (\[TotCenterOfMass\]).
The flat Euclidean metric, $ ds^2 = m_i \; \delta_{ij} \; \delta_{ab} \; d r_i^a \; d r_j^b $, is the natural metric on the extended configuration space $Q$. This metric induces the non–flat metric $$ds^2_\text{induced} = \left. m_i \; \delta_{ij} \; \delta_{ab} \; d r_i^a \; d r_j^b \right|_{Q_S} \;.
\label{InducedMetric}$$ on Shape Space.
Description of a macrostate in Shape Space
------------------------------------------
Consider an $N$–particle toy Universe with an $n$–particle subsystem, $n<N$. The particles in the subsystem have coordinates $x_i = r_i$, ($i = 1,\dots, n$), while the coordinates of all the other particles will be called $y_i = r_{n+i}$ , ($i= 1,\dots,N-n$). It is useful to define the coordinates of the center of mass of the subsystem and of the rest of the Universe:[^9] $$\label{CentersOfMass}
x_0 = \sum_{i=1}^n \frac{m_i}{m} \, x_i \;, \qquad y_0 = \sum_{i=1}^{N-n} \frac{m_{n+i}}{M-m} \, y_i \;, \qquad ~~~ m = \sum_{i=1}^n m_i\;,$$ and the center–of–mass moment of inertia of the two subsystems $$\label{MomentsOfInertia}
r = \sum_{i=1}^n \frac{m_i}{M} \, | x_i - x_0 |^2 \;, \qquad r' = \sum_{i=1}^{N-n} \frac{m_{n+i}}{M} \,| y_i - y_0 |^2 \;.$$ The relation between the moments of inertia of the total system and those of the two subsystems is $$R^2 = r^2 + (r')^2 + \left( 1 + \frac m {M-m} \right) \frac m M \; x_0^2 \;. \label{TotalMomentOfInertia}$$
We define a macrostate as a state in which the moment of inertia of the subsystem, $r$, and its center of mass, ${\bm x}_0$, are constant. To calculate the Shape Space volume of such a macrostate, we must integrate over all Shape Space coordinates ${\bm x}_i$ and ${\bm y}_i$ that respect the conditions (\[CentersOfMass\]), (\[MomentsOfInertia\]), and (\[TotalMomentOfInertia\]) using the measure provided by the induced metric (\[InducedMetric\]). Let’s make the following change of variables: $${\tilde x}_i = \sqrt{m_i} \left( x_i - x_0 \right) \;,
\qquad
{\tilde y}_i = \sqrt{m_{n+i}} \left( y_i - y_0 \right) \;.$$ Our equations become $$\begin{array}{c}
\frac 1 m \sum_{i=1}^n \sqrt{m_i} \; {\tilde x}_i = 0 \;, ~~~ \frac 1 {M-m} \sum_{i=1}^n \sqrt{m_{n+i}} \; {\tilde y}_i = 0
\;,\\\\
r = \frac{1}{M} \sum_{i=1}^n {\tilde x}_i^2 \;, ~~~ r' = \frac 1 {M} \sum_{i=1}^{N-n} {\tilde y}_i^2 \;, ~~~
R^2 = r^2 + (r')^2 + \left( 1 + \frac m {M-m} \right) \frac m M \; x_0^2 \;.
\end{array}$$ In the new coordinates, the metric is the identity matrix (it loses the $m_i$ factors on the diagonal). The integral is over the direct product of an $(n-2)$–dimensional sphere of radius $Mr$ and an $(N-n-2)$–dimensional sphere of radius $Mr' = M \sqrt{R^2 - r^2 - \left( 1 + \frac m {M-m} \right) \frac m M \; x_0^2}$ whose volume (calculated with a coarse–graining of size $\ell$) is: $$\Omega_\text{m} = \ell^2 \frac{ 4 \; \pi^{(N-n-1)/2} \pi^{(n-1)/2} }{ \Gamma((N-n-1)/2) \Gamma((n-1)/2)} M^{N-4} r^{n-2} \left(R^2 - r^2 - \left( 1 + \frac m {M-m} \right) \frac m M \; x_0^2 \right)^{\frac {N-n-2}{2}}\;.$$ The total volume of Shape Space is that of an $(N-1)$–dimensional sphere of radius $M R$ $$\Omega_\text{tot} =\frac{ 2 \pi^{N/2} }{ \Gamma(N/2)} \; M^{N-1} \, R^{N-1} \;.$$ Thus, the Shape Space volume per particle, in the limit $1 \ll n \ll N$, $r \ll r$, $m \ll M$ reduces to $$\omega \propto \left( \frac \ell r \right)^{2/n} \; \frac r R \left(1 - \left( \frac r R \right)^2 - \left( 1 + \frac m {M-m} \right) \frac m M \; \left( \frac{x_0} R \right)^2 \right)^{\frac {N}{2n}} \;,$$ and its logarithm has the expansion (remember that $x_0 < R$) $$S = \frac 1 2 k_B \frac {N}{n} \left( \frac r R \right)^2 - k_B \log \frac r R - \frac 2 n k_B \; \log \frac \ell r + \dots \;.$$ Notice that the numerical factors change in the 3 dimensions. In that case, they are $$S = \frac 3 2 \, k_B \, \frac N n \, \left( \frac r R \right)^2 - 3 \, k_B \, \log \frac r R - \frac 4 n \, k_B \log \frac \ell r \dots \;.$$
[^1]: s.gryb@hef.ru.nl
[^2]: fmercati@perimeterinstitute.ca
[^3]: A lot to digest in the first week!
[^4]: Technically, the difference is in the elliptic versus hyperbolic nature of the evolution equations.
[^5]: Believers in the Multiverse could substitute “Universe” for “Multiverse” in this argument.
[^6]: Technically, this is the isomorphism between the conformal group in $d$ spatial dimensions and the deSitter group in $d+1$ dimensions.
[^7]: FM and M. Lostaglio are exploring a related approach [@Matteo:thesis].
[^8]: We are using here the notion of moment of inertia with respect to a point, which we rescaled by the total mass $M = \sum_i m_i$ to give it the dimensions of a squared length.
[^9]: Notice that the two sets of coordinates must satisfy the relation $ m \; x_0 + (M-m) y_0 = 0$ in order to keep the total center of mass at the origin.
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