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abstract: 'In this paper, we present two new ways of quantum synchronization coding based on the $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ construction and the product construction respectively, and greatly enrich the varieties of available quantum synchronizable codes. The circumstances where the maximum synchronization error tolerance can be reached are explained for both constructions. Furthermore, repeated-root cyclic codes derived from the $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ construction are shown to be able to provide better Pauli error-correcting capability than BCH codes.'
author:
- Lan Luo
- Zhi Ma
- Dongdai Lin
bibliography:
- 'bib.bib'
date: 'Received: date / Accepted: date'
title: Two New Families of Quantum Synchronizable Codes
---
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\[sec:level1\]Introduction
==========================
Block synchronization (or frame synchronization) is a critical problem in virtually any area in classical digital communications to ensure that the information transmitted can be correctly decoded by the receiver. To achieve this goal, existing classical synchronization techniques commonly require that the information receiver or processing device constantly monitors the data to exactly identify the inserted boundary signals of an information block (see Refs. [@Sklar2001Digital; @Bregni2002Synchronization] for the basics of block synchronization techniques in classical digital communications). Quantum block synchronization is also significant because the block structure is typically used in quantum information coding [@Nielsen2010Quantum; @Lidar2013Quantum] as in classical domain and procedures for manipulating it demand precise alignment [@Fujiwara2013High; @Polyanskiy2013Asynchronous; @Fujiwara2013Block]. However, since measurement of qubits usually destroys their contained quantum information, quantum analogues of above methods don’t apply.
Aiming at this problem, Fujiwara [@Fujiwara2013Block] proposed a solution—quantum synchronizable error-correcting codes, which allow us to eliminate the effects caused by block misalignment and Pauli errors. In his scheme, the construction of good quantum synchronizable codes demands a pair of nested dual-containing cyclic codes, both of which guarantee large minimum distances. Later, authors of Ref. [@Fujiwara2013Algebraic] improved the original result by widening the range of tolerable magnitude of misalignment and presented several quantum synchronizable codes from classical BCH codes and punctured Reed-Muller (RM) codes. After that, finite geometric codes [@Fujiwara2014Quantum], quadratic residue codes [@Xie2014Quantum], duadic codes [@Guenda2015Algebraic] and repeated-root codes [@xie2016Q; @Lan2018Non] etc., were shown to be applicable in synchronization coding. However, apart from the case with repeated-root cyclic codes, code parameters of other available quantum synchronizable codes are strongly limited [@Lan2018Non]. Besides, the difficulty in computing the exact minimum distances of cyclic codes keeps us away from an accurate estimate on the error-correcting capability against Pauli errors of the obtained quantum codes.
In this work, we provide two new ways of constructing quantum synchronizable codes. The first method exploits the well-known $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ construction on cyclic codes and negacyclic codes to generate new cyclic codes with twice the lengths. Two circumstances where the obtained quantum codes can achieve the maximum synchronization error tolerance are provided. In particular, repeated-root cyclic codes are shown to be able to provide better performance in correcting Pauli errors than non-primitive, narrow-sense BCH codes. The second method exploits the product construction to produce new cyclic codes from two cyclic codes with coprime lengths. With a broad range of cyclic codes as ingredients, the varieties of quantum synchronizable codes are greatly extended using cyclic product codes. Furthermore, the obtained codes can also reach the best attainable tolerance against misalignment under certain circumstances.
The rest of this paper is organized as follows: First we describe the general formalism of quantum synchronization coding in Section 2. Then we build quantum synchronizable codes based upon the $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ construction in Section 3.1. Two circumstances where the obtained quantum codes reach the maximum synchronization error tolerance are elaborated with repeated-root codes in Section 3.2. Afterwards, we discuss the minimum distances of above repeated-root codes in Section 3.3. In Section 4, we produce quantum synchronizable codes from cyclic product codes. Finally, we give concluding remarks in Section 5.
\[sec:level1\]Preliminaries
===========================
In this section, we give a brief review of quantum synchronization coding. To start with, we describe some basic facts in classical and quantum coding theory. For further details, the readers can refer to Refs. [@Nielsen2010Quantum; @Huffman2010Fundamentals].
Let $\mathbb{F}_{q}$ be a finite field where $q=p^m$ is a prime power. A classical $[n,k,d]$ linear code $C$ over $\mathbb{F}_{q}$ is a $k$-dimensional subspace of $\mathbb{F}_{q}^{n}$ such that $\text{min}\{\text{wt}(\bm{c}):\bm{c}\in C\backslash\{\bm{0}\}\}=d$, where $\text{wt}(\bm{c})$ denotes the number of non-zero coordinates of a codeword $\bm{c}$. $C$ can be determined as the null space of an $(n-k)\times k$ parity-check matrix $H$, i.e., $C=\{\bm{c}\in\mathbb{F}_{q}^{n}: H\bm{c}^{\mathsf{T}}=\bm{0}\}$. Accordingly, there exists a $k\times n$ generator matrix $G$ with its row space corresponding to $C$ such that $HG^{\mathsf{T}}=\bm{0}$. The dual code $C^{\perp}:=\{\bm{c}'\in\mathbb{F}_{q}^{n}:\bm{c}\bm{c}'^{\mathsf{T}}=0,\forall \bm{c}\in C\}$ is an $[n,n-k]$ code with a generator matrix $H$ and a parity-check matrix $G$. If $C\subset C^{\perp}$, we call $C$ a self-orthogonal code. Otherwise if $C^{\perp}\subset C$, $C$ is a dual-containing code.
A classical linear code $C$ is (nega)cyclic if it remains unchanged under a (nega)cyclic shifting of the coordinates, i.e., for a codeword $\bm{c}=(c_{0},c_{1},\dots,c_{n-1})\in C$, a cyclic shift $(c_{n-1},c_{0},\dots,c_{n-2})$ (a negacyclic shift $(-c_{n-1},c_{0},\dots,c_{n-2})$) is also a codeword of $C$. Especially, repeated-root (nega)cyclic codes [@Dinh2008On; @Chen2014Repeated; @Dinh2013Structure; @Chen2015Repeated; @Ozadam2009The; @Zeh2015Decoding] are those whose lengths are divisible by the characteristic $p$ of $\mathbb{F}_{q}$. Identify each codeword $\bm{c}$ as the coefficient vector of a polynomial $c(x)=\sum_{i=0}^{n-1}c_{i}x^{i}$. Then an $[n,k]$ (nega)cyclic code $C$ is equivalent with an ideal $\langle g(x)\rangle$ in the quotient ring $\frac{\mathbb{F}_{q}[x]}{\langle x^{n}-1\rangle}$ ($\frac{\mathbb{F}_{q}[x]}{\langle x^n+1\rangle}$). We call the monic polynomial $g(x)$ of degree $n-k$ as the generator polynomial of $C$. If the value $\frac{q-1}{\text{gcd}(n,q-1)}$ is even, there exists an isomorphism $\phi$ between the quotient rings $\frac{\mathbb{F}_{q}[x]}{\langle x^n-1\rangle}$ and $\frac{\mathbb{F}_{q}[x]}{\langle x^{n}+1\rangle}$ that maps $g(x)$ to $g(\lambda x)$ [@Chen2014Repeated], where $\lambda^{n}=-1$. Define the parity-check polynomial $h(x)$ of a (nega)cyclic code $C$ as $h(x)=\frac{x^n-1}{g(x)}$ ($h(x)=\frac{x^n+1}{g(x)}$). Accordingly, the dual (nega)cyclic code $C^{\perp}$ has a generator polynomial $g^{\perp}(x)=h^{*}(x)$, where $h^{*}(x)=h_{0}^{-1}x^{\text{deg}(h(x))}h(\frac{1}{x})$ represents the monic reciprocal polynomial of $h(x)$.
An $[[n,k,d]]$ quantum code $\mathcal{Q}$ is a $q^k$-dimensional subspace of a $q^n$-dimensional Hilbert space $\mathbb({C}^{q})^{\otimes n}$. Typically, $Q$ is designed to correct the errors caused by Pauli operators $X_{\bm{\alpha}}Z_{\bm{\beta}}$ of weight less than $\lfloor\frac{d-1}{2}\rfloor$, where $\bm{\alpha},\bm{\beta}\in\mathbb{F}_{q}^{n}$. An $(a_{l},a_{r})-[[n,k]]$ quantum synchronizable code is a quantum code that corrects not only Pauli errors, but also block misalignment to the left by $a_{l}$ qudits ($q$-ary quantum systems) and to the right by $a_{r}$ qudits for some non-negative integers $a_{l}$ and $a_{r}$. Denote the order of a polynomial $f(x)$ with $f(0)\not=0$ by $\text{ord}(f(x))$, i.e., $\text{ord}(f(x))=|\{x^a(\text{mod } f(x)):\ a\in \mathbb{N}\}|$. We give the quantum synchronization coding framework as follows.
[@xie2016Q; @Lan2018Non] Let $C$ be a dual-containing $[n,k_{C},d_{C}]$ cyclic code and $D$ be an $[n,k_{D},d_{D}]$ cyclic code such that $C\subset D$. Denote by $g_{C}(x)$ and $g_{D}(x)$ the generator polynomials of $C$ and $D$ respectively. Define the polynomial $f(x)$ of degree $k_{D}-k_{C}$ to be the quotient of $g_{C}(x)$ divided by $g_{D}(x)$. Then for any pair $a_{l},a_{r}$ of non-negative integers satisfying $a_{l}+a_{r}<\emph{ord}(f(x))$, there exists an $(a_{l},a_{r})-[[n,2k_{C}-n]]$ quantum synchronizable code that can correct up to $\lfloor\frac{d_{D}-1}{2}\rfloor$ bit errors and $\lfloor \frac{d_{C}-1}{2}\rfloor$ phase errors. \[thm1\]
We can tell from Theorem \[thm1\] that a valid construction of good quantum synchronizable codes relies on a pair of dual-containing cyclic codes, one of which is contained in the other and both guarantee large minimum distances. Furthermore, the obtained synchronizable code can correct synchronization errors (or misalignment) up to $\text{ord}(f(x))$ qubits. If $\text{ord}(f(x))=n$, then the quantum synchronizable code achieves the maximum synchronization error tolerance.
\[sec:level1\]The $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ construction
==============================================================
\[sec:level2\]Synchronization coding
------------------------------------
In this section, we describe the quantum synchronization coding based upon the $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ construction. Compared with the well-known $(\bm{u}|\bm{u}+\bm{v})$ method—an iterative way to define RM codes, the $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ technique has several advantages [@Hughes2000Constacyclic; @Ling2001On]. Apart from an estimate of minimum distances never worse than the other case, the $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ scheme enables us to obtain a $2n$-length cyclic code from an $n$-length cyclic code and an $n$-length negacyclic code.
To be specific, let $C_{1}$ and $C_{2}$ be $[n,k_{1},d_{1}]$ and $[n,k_{2},d_{2}]$ linear codes over $\mathbb{F}_{q}$ respectively, where $q=p^m$ is an odd prime power. (In this section, we leave the case with $p=2$ out of consideration.) Denote by $G_{1},G_{2}$ and $H_{1},H_{2}$ the generator matrices and parity-check matrices of $C_{1}$ and $C_{2}$, respectively. The $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ construction $C=C_{1}\curlyvee C_{2}=\{(\bm{u}+\bm{v}|\bm{u}-\bm{v}):\bm{u}\in C_{1},\bm{v}\in C_{2}\}$ [@Hughes2000Constacyclic; @Macwilliams1977The] is a $[2n, k_{1}+k_{2}, \text{min}\{2d_{1},2d_{2},\text{max}\{d_{1},d_{2}\}\}]$ code with a generator matrix $$G_{C}=\left(
\begin{array}{cc}
G_{1} & G_{1}\\
G_{2} & -G_{2}
\end{array}
\right).$$ Suppose that $C_{1}$ is cyclic with a generator polynomial $g_{1}(x)$ and $C_{2}$ is negacyclic with a generator polynomial $g_{2}(x)$, then $C$ is cyclic with a generator polynomial $g(x)=g_1(x)g_2(x)$ [@Hughes2000Constacyclic]. Clearly, $C$ is dual-containing if both $C_{1}$ and $C_{2}$ are dual-containing. Applying these properties to Theorem \[thm1\], we can build a family of quantum synchronizable codes from cyclic codes and negacyclic codes as follows.
Let $C_{i}$ be an $[n,k_{i},d_{i}]$ dual-containing code for $i\in\{1,2,3,4\}$. Suppose that $C_{1},C_{3}$ are cyclic with respective generator polynomial $g_{1}(x),g_{3}(x)$ and $C_{2},C_{4}$ are negacyclic with respective generator polynomial $g_{2}(x), g_{4}(x)$. If $C_{1}\subset C_{3}$ and $C_{2}\subset C_{4}$, define $f(x)=\frac{g_{1}(x)g_{2}(x)}{g_{3}(x)g_{4}(x)}$. Then for any pair of non-negative integers $a_{l},a_{r}$ such that $a_{l}+a_{r}<\emph{ord}(f(x))$, there exists an $(a_{l},a_{r})-[[2n,2(k_{1}+k_{2}-n)]]$ quantum synchronizable code that can correct up to $\lfloor \frac{\emph{min}\{2d_{3},2d_{4},\emph{max}\{d_{3},d_{4}\}\}-1}{2}\rfloor$ bit errors and $\lfloor \frac{\emph{min}\{2d_{1},2d_{2},\emph{max}\{d_{1},d_{2}\}\}-1}{2}\rfloor$ phase errors. \[thm2\]
It is clear that $C=C_{1}\curlyvee C_{2}$ is a $[2n,k_{1}+k_{2},\text{min}\{2d_{1},2d_{2},\text{max}\{d_{1},d_{2}\}\}]$ cyclic code with a generator polynomial $g_{C}(x)=g_{1}(x)g_{2}(x)$ and $D=C_{3}\curlyvee C_{4}$ is a $[2n,k_{3}+k_{4},\text{min}\{2d_{3},2d_{4},\text{max}\{d_{3},d_{4}\}\}]$ cyclic code with generator polynomial $g_{D}(x)=g_{3}(x)g_{4}(x)$. Furthermore, the condition $C\subset D$ holds because $C_{1}\subset C_{3}$ and $C_{2}\subset C_{4}$. By applying $C$ and $D$ to Theorem \[thm1\], we can then obtain the required quantum synchronizable codes. $\hfill\square$
Different from Theorem \[thm1\], Theorem \[thm2\] calls for two pairs of nested dual-containing classical codes in quantum synchronization coding, one of which are cyclic and the other are negacyclic. All of these codes need to guarantee large minimum distances, and are desired to make as large $\text{ord}(f(x))$ as possible to offer better synchronization recovery capability. In particular, the maximum tolerable magnitude of misalignment is $2n$. In that case, the quantum synchronizable codes from Theorem \[thm2\] can correct misalignment by up to $a_{l}$ qubits to the left and $a_{r}$ qubits to the right provided that $a_{l}+a_{r}<2n$.
Maximum synchronization error tolerance
---------------------------------------
Under two circumstances could the maximum synchronization error tolerance $2n$ be achieved, one of which is that $\text{ord}(\frac{g_{1}(x)}{g_{3}(x)})=n$ and $\text{ord}(\frac{g_{2}(x)}{g_{4}(x)})=2$ where $\text{gcd}(n,2)=1$, and the other is that $\text{ord}(\frac{g_{2}(x)}{g_{4}(x)})=2n$ whatever the value of $\text{ord}(\frac{g_{1}(x)}{g_{3}(x)})$ is.
### The first circumstance
The condition $\text{ord}(\frac{g_{1}(x)}{g_{3}(x)})=n$ in the first circumstance has been investigated on nearly all available quantum synchronizable codes, and is applicable to many cyclic codes, e.g., BCH codes [@Fujiwara2013Algebraic], punctured RM codes [@Fujiwara2013Algebraic], quadratic residue codes [@Xie2014Quantum] and repeated-root cyclic codes [@xie2016Q; @Lan2018Non]. The other condition $\text{ord}(\frac{g_{2}(x)}{g_{4}(x)})=2$, however, has limited applications subject to the dual-containing constraint. One feasible solution is to use repeated-root codes of length $p^s$, where $s$ is a positive integer.
To be concrete, let $C_{1},C_{3}$ be $p^s$-length dual-containing cyclic codes and $C_{2},C_{4}$ be $p^s$-length dual-containing negacyclic codes. Then $C_{1},C_{2},C_{3},C_{4}$ have generator polynomials [@Dinh2008On] $$\begin{array}{l}
g_{i}(x)=(x-1)^{p^s-k_{i}},\quad i=1,3,\\
g_{j}(x)=(x+1)^{p^s-k_{j}},\quad j=2,4,\\
\end{array}$$ where $\frac{p^s}{2}\leq k_{i},k_{j}\leq p^s$. With the help of these codes, we can build a family of quantum synchronizable codes that possess the maximum synchronization error tolerance.
Let $C_{i}$ be a $[p^s,k_{i}]$ cyclic code and $C_{j}$ be a $[p^s,k_{j}]$ negacyclic code, where $\frac{p^s}{2}\leq k_{i},k_{j}\leq p^s$ for $i\in\{1,3\}$ and $j\in\{2,4\}$. Suppose that $k_{3}-k_{1}>p^{s-1}$ and $k_{4}=k_{2}+1$, then for non-negative integers $a_{l}$ and $a_{r}$ such that $a_{l}+a_{r}<2p^s$, there exists an $(a_{l},a_{r})-[[2p^s,2(k_{1}+k_{3}-p^s)]]$ quantum synchronizable code.
The fact that $C_{1}\subset C_{3}$ and $C_{2}\subset C_{4}$ is evident since $k_{1}<k_{3}$ and $k_{2}<k_{4}$. Furthermore, the order of the polynomial $f(x)=\frac{g_{1}(x)g_{2}(x)}{g_{3}(x)g_{4}(x)}=(x+1)(x-1)^{k_{3}-k_{1}}$ is $2p^s$. By applying these properties to Theorem \[thm2\], we can naturally obtain the quantum synchronizable codes of desired parameters. $\hfill\square$
### The second circumstance
Assume that $\frac{q-1}{\text{gcd}(n,q-1)}$ is even, then there exists an isomorphism $\phi$ between the quotient rings $\frac{\mathbb{F}_{q}[x]}{\langle x^{n}-1\rangle}$ and $\frac{\mathbb{F}_{q}[x]}{\langle x^{n}+1\rangle}$ which maps $c(x)$ to $c(\lambda x)$, where $\lambda^{n}$=-1 [@Chen2014Repeated]. Furthermore, if the order of $c(x)$ is $n$, the order of $c(\lambda x)$ is $2n$. Therefore, the condition $\text{ord}(\frac{g_{2}(x)}{g_{4}(x)})=2n$ on negacyclic codes $C_{2},C_{4}$ can be achieved by finding two suitable cyclic codes. On that condition, most existing quantum synchronizable codes that provide the highest tolerance against synchronization errors can be generalized to quantum synchronizable codes of twice the lengths. As an example, we consider the use of $lp^s$-length repeated-root codes where $l$ is a prime distinct from $p$.
We first deal with the case $l\not=2$. Pick a primitive $l$-th root $\zeta$ of unity in the extension field $\mathbb{F}_{q^w}$ with $w=ord_{l}(q)$ indicating the order of $q$ in $\mathbb{Z}_{l}^{*}$. For $0\leq t\leq e=\frac{l-1}{w}$, denote by $M_{t}(x)$ the minimal polynomial of $\zeta^{t}$ over $\mathbb{F}_{q}$. The following lemma describes the structures of $lp^s$-length cyclic codes and negacyclic codes explicitly.
[@Chen2014Repeated] Let $C_{1}$ be an $lp^s$-length cyclic code with a generator polynomial $g_{1}(x)$ and let $C_{2}$ be an $lp^s$-length negacyclic code with a generator polynomial $g_{2}(x)$.
- If $\emph{gcd}(l,q-1)=1$, then $$\begin{array}{l}
g_{1}(x)= \prod_{t=0}^{e}(M_{t}(x))^{p^s-a_{1,t}},\\
g_{2}(x)=\prod_{t=0}^{e}(\hat{M}_{t}(-x))^{p^{s}-a_{2,t}},
\end{array}$$ where $0\leq a_{1,t},a_{2,t}\leq p^s$ for all $t$. The notation $\hat{M}_{t}(x)$ denotes the monic polynomial of $M_{t}(x)$ dividing its leading coefficient. In particular when $w=ord_{l}(q)$ is odd, the generator polynomials of $C_{1}$ and $C_{2}$ are given by $$\begin{array}{l}
g_{1}(x)=(x-1)^{p^s-a_{1,0}}\prod_{t=1}^{\frac{e}{2}}(M_{t}(x))^{p^s-a_{1,t}}(M_{-t}(x))^{p^s-a_{1,-t}},\\
g_{2}(x)=(x+1)^{p^s-a_{2,0}}\prod_{t=1}^{\frac{e}{2}}(\hat{M}_{t}(-x))^{p^{s}-a_{2,t}}(\hat{M}_{-t}(-x))^{p^s-a_{2,-t}},
\end{array}$$ where $0\leq a_{1,t},a_{2,t},a_{1,-t},a_{2,-t}\leq p^s$ for $0\leq t\leq \frac{e}{2}$.
Correspondingly, if $w$ is even, the dual codes $C_{1}^{\perp}$ and $C_{2}^{\perp}$ have generator polynomials $$\begin{array}{l}
g_{1}^{\perp}(x)=\prod_{t=0}^{e}(M_{t}(x))^{a_{1,t}},\\
g_{2}^{\perp}(x)=\prod_{t=0}^{e}(\hat{M}_{t}(-x))^{a_{2,t}},
\end{array}$$ respectively. Otherwise if $w$ is odd, the dual codes have respective generator polynomial $$\begin{array}{l}
g_{1}^{\perp}(x)=(x-1)^{a_{1,0}}\prod_{t=1}^{\frac{e}{2}}(M_{t}(x))^{a_{1,-t}}(M_{-t}(x))^{a_{1,t}},\\
g_{2}^{\perp}(x)=(x+1)^{a_{2,0}}\prod_{t=1}^{\frac{e}{2}}(\hat{M}_{t}(-x))^{a_{2,-t}}(\hat{M}_{-t}(-x))^{a_{2,t}}.
\end{array}$$
- If $\emph{gcd}(l,q-1)=l$, then we have $$\begin{array}{l}
g_{1}(x)=(x-1)^{p^s-a_{1,0}}\prod_{t=1}^{\frac{l-1}{2}}(x-\zeta^{t})^{p^s-a_{1,t}}(x-\zeta^{-t})^{p^s-a_{1,-t}},\\
g_{2}(x)=(x+1)^{p^s-a_{2,0}}\prod_{t=1}^{\frac{l-1}{2}}(x+\zeta^{t})^{p^s-a_{2,t}}(x+\zeta^{-t})^{p^s-a_{2,-t}},
\end{array}$$ where $0\leq a_{1,t},a_{2,t},a_{1,-t},a_{2,-t}\leq p^s$ for $0\leq t\leq \frac{l-1}{2}$. The dual codes $C_{1}^{\perp}$ and $C_{2}^{\perp}$ have generator polynomials $$\begin{array}{l}
g_{1}^{\perp}(x)=(x-1)^{a_{1,0}}\prod_{t=1}^{\frac{l-1}{2}}(x-\zeta^{t})^{a_{1,-t}}(x-\zeta^{-t})^{a_{1,t}},\\
g_{2}^{\perp}(x)=(x+1)^{a_{2,0}}\prod_{t=1}^{\frac{l-1}{2}}(x+\zeta^{t})^{a_{2,-t}}(x+\zeta^{-t})^{a_{2,t}},
\end{array}$$ respectively.
By applying above codes to Theorem \[thm2\], we can build quantum synchronizable codes of length $2lp^s$ as follows.
Let $l$ be an odd prime such that $\emph{gcd}(l,q-1)=1$. Suppose that $C_{1}, C_{3}$ are dual-containing cyclic codes of length $lp^s$ and $C_{2}, C_{4}$ are dual-containing negacyclic codes of length $lp^s$.
- If $w$ is even, then $C_{i},C_{j}$ have generator polynomials $$\begin{array}{l}
g_{i}(x)=\prod_{t=0}^{e}(M_{t}(x))^{p^s-a_{i,t}},\quad i\in\{1,3\},\\
g_{j}(x)=\prod_{t=0}^{e}(\hat{M}_{t}(-x))^{p^s-a_{j,t}},\quad j\in\{2,4\},
\end{array}$$ respectively, where $\frac{p^s}{2}\leq a_{i,t},a_{j,t}\leq p^s$ for $0\leq t\leq e$. Assume that $a_{1,t}\leq a_{3,t}$ and $a_{2,t}\leq a_{4,t}$ for all $t$. If there exists an integer $r$ in the range $0\leq r\leq e$ such that $\emph{gcd}(r,l)=1$ and $a_{4,r}-a_{2,r}>p^{s-1}$, then we can construct an $(a_{l},a_{r})-[[2lp^s,k]]$ quantum synchronizable code where $$k=2\left(\sum_{t=1}^{e}(a_{1,t}+a_{2,t})w+(a_{1,0}+a_{2,0})-lp^s\right).$$
- If $w$ is odd, then $C_{i}$ and $C_{j}$ have generator polynomials $$\begin{array}{l}
g_{i}(x)=(x-1)^{p^s-a_{i,0}}\prod_{t=1}^{\frac{e}{2}}(M_{t}(x))^{p^s-a_{i,t}}(M_{-t}(x))^{p^{s}-a_{i,-t}},\quad i\in\{1,3\},\\
g_{j}(x)=(x+1)^{p^s-a_{j,0}}\prod_{t=1}^{\frac{e}{2}}(\hat{M}_{t}(-x))^{p^s-a_{j,t}}(\hat{M}_{-t}(-x))^{p^s-a_{j,-t}},\quad j\in\{2,4\},
\end{array}$$ respectively, where $\frac{p^s}{2}\leq a_{i,0},a_{j,0}\leq p^s$ and $p^s\leq a_{i,t}+a_{i,-t},a_{j,t}+a_{j,-t}\leq 2p^s$ for $1\leq t\leq \frac{e}{2}$. Assume that $$\begin{array}{ll}
a_{1,0}\leq a_{3,0},& a_{2,0}\leq a_{4,0},\\
a_{1,t}\leq a_{3,t},& a_{2,t}\leq a_{4,t},\\
a_{1,-t}\leq a_{3,-t},& a_{2,-t}\leq a_{4,-t},
\end{array}$$ for $1\leq t\leq \frac{e}{2}$. If there exists an integer $r$ in the range $-\frac{e}{2} \leq r\leq \frac{e}{2}$ such that $\emph{gcd}(r,l)=1$ and $a_{4,r}-a_{2,r}>p^{s-1}$, then for any non-negative integers $a_{l},a_{r}$ satisfying $a_{l}+a_{r}<2lp^s$, we can obtain an $(a_{l},a_{r})-[[2lp^s,k]]$ quantum synchronizable code where $$k=2\left(\sum\limits_{1\leq t\leq \frac{e}{2}}(a_{1,t}+a_{1,-t}+a_{2,t}+a_{2,-t})w+(a_{1,0}+a_{2,0})-lp^s\right).$$
\[thm4\]
Given an even $w$, the dual-containing properties of $C_{i}$ and $C_{j}$, for $i\in\{1,3\}$ and $j\in\{2,4\}$, are guaranteed when the parameters $a_{i,t},a_{j,t}$ are in the range $\frac{p^s}{2}\leq a_{i,t},a_{j,t}\leq p^s$ for all $t$. Furthermore, due to the assumption that $a_{1,t}\leq a_{3,t}$ and $a_{2,t}\leq a_{4,t}$ for $0\leq t\leq e$, we have $C_{1}\subset C_{3}$ and $C_{2}\subset C_{4}$. In that case, the polynomial $f(x)$ in Theorem \[thm2\] is $$\begin{array}{ll}
f(x)&=\frac{g_{1}(x)g_{2}(x)}{g_{3}(x)g_{4}(x)}=\frac{\prod_{t=0}^{e}(M_{t}(x))^{p^s-a_{1,t}}(\hat{M}_{t}(-x))^{p^s-a_{2,t}}}{\prod_{t=0}^{e}(M_{t}(x))^{p^s-a_{3,t}}(\hat{M}_{t}(-x))^{p^s-a_{4,t}}}\\
& =\prod_{t=0}^{e}(M_{t}(x))^{a_{3,t}-a_{1,t}}(\hat{M}_{t}(-x))^{a_{4,t}-a_{2,t}}.
\end{array}$$ Pick an integer $r$ that is relatively prime to $l$ such that $a_{4,r}-a_{2,r}>p^{s-1}$, then the order of $f(x)$ has a factor $p^s\cdot\text{ord}(\hat{M}_{r}(-x))$. Note that $\text{ord}(\hat{M}_{r}(x))=\frac{l}{\text{gcd}(r,l)}=l$. Hence we have $\text{ord}(\hat{M}_{r}(-x))=2l$, indicating that $\text{ord}(f(x))\geq 2lp^s$. Due to the fact that $\text{ord}(f(x))\leq 2lp^s$, we can finally obtain that $\text{ord}(f(x))=2lp^s$. Moreover, $C_{i}$ has dimension $$k_{i}=lp^s-\left((p^s-a_{i,0})+\sum_{t=1}^{e}(p^s-a_{i,t})w\right)=a_{i,0}+\sum_{t=1}^{e}a_{i,t}\cdot w.$$ And analogously, $C_{j}$ has dimension $k_{j}=a_{j,0}+\sum_{t=1}^{e}a_{j,t}\cdot w$. Therefore, the quantum synchronizable code built on them has the desired parameters. For an odd $w$, the statements in (II) can be verified using similar arguments. $\hfill\square$
Let $l$ be an odd prime such that $\text{gcd}(l,q-1)=l$. Suppose that $C_{1},C_{3}$ are dual-containing cyclic codes of length $lp^s$ and $C_{2},C_{4}$ are dual-containing negacyclic codes of length $lp^s$. The generator polynomials of $C_{i},C_{j}$ for $i\in\{1,3\}$ and $j\in\{2,4\}$ are $$\begin{array}{l}
g_{i}(x)=(x-1)^{p^s-a_{i,0}}\prod_{t=1}^{\frac{l-1}{2}}(x-\zeta^{t})^{p^{s}-a_{i,t}}(x-\zeta^{-t})^{p^s-a_{i,-t}},\\
g_{j}(x)=(x+1)^{p^s-a_{j,0}}\prod_{t=1}^{\frac{l-1}{2}}(x+\zeta^{t})^{p^s-a_{j,t}}(x+\zeta^{-t})^{p^s-a_{j,-t}},
\end{array}$$ respectively, where $\frac{p^s}{2}\leq a_{i,0},a_{j,0}\leq p^s$ and $p^s\leq a_{i,t}+a_{i,-t},a_{j,t}+a_{j,-t}\leq 2p^s$ for $1\leq t\leq \frac{l-1}{2}$. Assume that $$\begin{array}{ll}
a_{1,0}\leq a_{3,0},& a_{2,0}\leq a_{4,0},\\
a_{1,t}\leq a_{3,t},& a_{2,t}\leq a_{4,t},\\
a_{1,-t}\leq a_{3,-t},& a_{2,-t}\leq a_{4,-t},
\end{array}$$ for $1\leq t\leq \frac{l-1}{2}$. If we can pick an integer $r$ with $-\frac{l-1}{2}\leq r\leq \frac{l-1}{2}$ such that $\text{gcd}(r,l)=1$ and $a_{4,r}-a_{2,r}>p^{s-1}$, then given a pair of non-negative integers $a_{l},a_{r}$ satisfying $a_{l}+a_{r}<2lp^s$, there exists a quantum synchronizable code of length $2lp^s$ and dimension $$k=2\left((a_{1,0}+a_{2,0})+\sum_{t=1}^{\frac{l-1}{2}}(a_{1,t}+a_{1,-t}+a_{2,t}+a_{2,-t})-lp^s\right).$$ \[thm5\]
Following a similar proof to that of Theorem \[thm4\], we can obtain the desired results. $\hfill\square$
In the case of $l=2$, quantum synchronizable codes that possess the maximum synchronization error tolerance can also be constructed from $lp^s$-length cyclic codes and negacyclic codes. The following lemma describes the structures of $2p^s$-length repeated-root codes clearly.
[@Chen2014Repeated] Let $C_{1}$ be a cyclic code of length $2p^s$. Then $C_{1}$ and its dual code $C_{1}^{\perp}$ have respective generator polynomial $$\begin{array}{l}
g_{1}(x)=(x-1)^{p^s-a_{1,0}}(x+1)^{p^s-a_{1,1}},\\
g_{1}^{\perp}(x)=(x-1)^{a_{1,0}}(x+1)^{a_{1,1}},
\end{array}$$ where $0\leq a_{1,0},a_{1,1}\leq p^s$.
Let $C_{2}$ be a negacyclic code of length $2p^s$. If $q\equiv 1(\emph{mod } 4)$, there exists an element $\eta\in\mathbb{F}_{q}^{*}$ such that $\eta^{2}=-1$. In that case, $C_{2}$ and $C_{2}^{\perp}$ have generator polynomials $$\begin{array}{l}
g_{2}(x)=(x-\eta)^{p^s-a_{2,0}}(x+\eta)^{p^s-a_{2,1}},\\
g_{2}^{\perp}(x)=(x-\eta)^{a_{2,1}}(x+\eta)^{a_{2,0}},
\end{array}$$ respectively, where $0\leq a_{2,0},a_{2,1}\leq p^s$. Otherwise if $q\equiv 3(\emph{mod } 4)$, their generator polynomials are given by $$\begin{array}{l}
g_{2}(x)=(x^2+1)^{p^s-a_{2}},\\
g_{2}^{\perp}(x)=(x^2+1)^{a_{2}},
\end{array}$$ where $0\leq a_{2}\leq p^s$. \[lem3\]
Taking similar arguments to the proof of Theorem \[thm4\], we can obtain the following results.
Suppose that $C_{1},C_{3}$ are dual-containing cyclic codes of length $2p^s$ and $C_{2},C_{4}$ are dual-containing negacyclic codes of length $2p^s$.
- If $q\equiv 1(\emph{mod }4)$, the generator polynomials of $C_{1},C_{2},C_{3},C_{4}$ are of the forms $$\begin{array}{ll}
g_{i}(x)=(x-1)^{p^s-a_{i,0}}(x+1)^{p^s-a_{i,1}}, & i\in\{1,3\},\\
g_{j}(x)=(x-\eta)^{p^s-a_{j,0}}(x+\eta)^{p^s-a_{j,1}}, & j\in\{2,4\},
\end{array}$$ where $\frac{p^s}{2}\leq a_{i,0},a_{i,1}\leq p^s$ and $p^s\leq a_{j,0}+a_{j,1}\leq 2p^s$. The notation $\eta$ denotes the element in $\mathbb{F}_{q}^{*}$ such that $\eta^{2}=-1$. Assume that $$\begin{array}{ll}
a_{1,0}\leq a_{3,0}, & a_{1,1}\leq a_{3,1},\\
a_{2,0}\leq a_{4,0}, & a_{2,1}\leq a_{4,1}.
\end{array}$$ If either $a_{4,0}-a_{2,0}>p^{s-1}$ or $a_{4,1}-a_{2,1}>p^{s-1}$ holds, then for any non-negative pair $a_{l},a_{r}$ satisfying $a_{l}+a_{r}<4p^s$, we can build an $(a_{l},a_{r})-[[4p^s,k]]$ quantum synchronizable code, where $$k=2\left(a_{1,0}+a_{1,1}+a_{2,0}+a_{2,1}-2p^s\right).$$
- If $q\equiv 3(\emph{mod } 4)$, the generator polynomials of $C_{1},C_{2},C_{3},C_{4}$ are given by $$\begin{array}{ll}
g_{i}(x)=(x-1)^{p^s-a_{i,0}}(x+1)^{p^s-a_{i,1}},& i\in\{1,3\},\\
g_{j}(x)=(x^2+1)^{p^s-a_{j}},& j\in\{2,4\},
\end{array}$$ where $\frac{p^s}{2}\leq a_{i,0},a_{i,2},a_{j}\leq p^s$. Assume that $a_{1,0}\leq a_{3,0}$, $a_{1,1}\leq a_{3,1}$ and $a_{2}\leq a_{4}$. If $a_{4}-a_{2}>p^{s-1}$, then for any non-negative integers $a_{l},a_{r}$ such that $a_{l}+a_{r}<4p^s$, there exists an $(a_{l},a_{r})-[[4p^s,k]]$ quantum synchronizable code, where $$k=2\left(a_{1,0}+a_{1,1}+2a_{2}-2p^s\right).$$
\[thm6\]
We can tell from Theorems \[thm4\], \[thm5\] and \[thm6\] that quantum synchronizable codes of length $2lp^s$ that reach the maximum synchronization error tolerance can be derived from cyclic codes and negacyclic codes of length $lp^s$. This can be seen as a generalization of results in Ref. [@Lan2018Non], where $lp^s$-length cyclic codes are exploited in the construction of $lp^s$-length quantum synchronizable codes that tolerate misalignment by $lp^s$ qubits. Similar generalizations can be applied to other existing quantum synchronizable codes of length $n$ due to the isomorphism $\phi$ between $\frac{\mathbb{F}_{q}[x]}{\langle x^n-1\rangle}$ and $\frac{\mathbb{F}_{q}[x]}{\langle x^n+1\rangle}$ [@Chen2014Repeated] that maps $c(x)$ to $c(\lambda x)$ where $\lambda^n=-1$, when $\frac{q-1}{\text{gcd}(n,q-1)}$ is even.
\[sec:level2\]The minimum distances
-----------------------------------
From Theorem \[thm2\] we can see that, quantum synchronizable codes derived from cyclic codes $C_{1}\curlyvee C_{2}$ and $C_{3}\curlyvee C_{4}$ have minimum distances no worse or up to twice larger than those from the component cyclic codes $C_{1}$ and $C_{3}$. In other words, quantum synchronizable codes based on the $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ scheme can provide good performance in correcting Pauli errors. Take the codes from Theorem \[thm4\] (I) for an example.
Suppose that $l$ is an odd prime such that $\text{gcd}(l,q-1)$. If $w$ is even, then an $lp^s$-length cyclic code $C_{i}$ and an $lp^s$-length negacyclic code $C_{j}$, for $i\in\{1,3\}$ and $j\in\{2,4\}$, have respective generator polynomial $$\begin{array}{l}
g_{i}(x)=\prod_{t=0}^{e}(M_{t}(x))^{p^s-a_{i,t}},\\
g_{j}(x)=\prod_{t=0}^{e}(\hat{M}_{t}(-x))^{p^s-a_{j,t}},
\end{array}$$ where $0\leq a_{i,t},a_{j,t}\leq p^s$ for all $t$. The minimum distance of $C_{i}$ has been thoroughly investigated in Ref. [@Lan2018Non]. Due to the isomorphism $\phi$ between $\frac{\mathbb{F}_{q}[x]}{\langle x^{lp^s}-1\rangle}$ and $\frac{\mathbb{F}_{q}[x]}{\langle x^{lp^s}+1\rangle}$ that maps $c(x)$ to $c(-x)$, the minimum distance of the negacyclic code $C_{j}$ can be computed using the same strategies.
Define a set of $l$-length negacyclic codes $\{\overline{C}_{j,v}:0\leq v\leq p^s-1\}$ with respective generator polynomial $\overline{g}_{j,v}(x)=\prod_{t=0}^{e}(\hat{M}_{t}(-x))^{f_{v,a_{j,t}}}$, where $f_{v,a_{j,t}}=\left\{\begin{array}{ll}1,& p^s-a_{j,t}>v,\\ 0, & \text{otherwise.}\end{array}\right.$ For $0\leq v\leq p^s-1$, denote by $P_{v}$ the Hamming weight of the polynomial $(x-1)^{v}$ [@Castagnoli1991On] and define the set $$V=\left\{\sum_{\mu=1}^{u-1}(p-1)p^{s-\mu}+\tau p^{s-u}:1\leq u\leq s, 1
\leq \tau\leq p-1\right\}\cup\{0\}.$$ The minimum distance of $C_{j}$ is demonstrated in Table \[tab1\].
[p[0.5cm]{}<|p[3cm]{}<|p[3cm]{}<|p[3cm]{}<]{} **Case** & $\bm{b_{\rm{min}}}^{\dagger}$ & $\bm{b_{\rm{max}}}$ & **minimum distance**\
1 & $(0, p^{s-1}]$ & $(0, p^{s-1}]$ & 2\
2 & $(0, p^{s-1}]$ & $(\beta^{\star} p^{s-1},(\beta+1)p^{s-1}]$ & $\text{min}\{2d'^{\ddagger},\beta+2\}$\
3 & $(0, p^{s-1}]$ & $(p^s-p^{s-\mu}+(\tau-1)p^{s-\mu-1}, p^s-p^{s-\mu}+\tau p^{s-\mu-1}]$ & $\text{min}\{2d',(\tau+1)p^{\mu}\}$\
4& $(\beta_{0}p^{s-1},(\beta_{0}+1)p^{s-1}]$ & $(\beta_{1}p^{s-1},(\beta_{1}+1)p^{s-1}]$ & $\text{min}\{(\beta_{0}+2)d',\beta_{1}+2\}$\
5 & $(\beta p^{s-1},(\beta+1)p^{s-1}]$ & $(p^s-p^{s-\mu}+(\tau-1)p^{s-\mu-1},p^s-p^{s-\mu}+\tau p^{s-\mu-1}]$ & $\text{min}\{(\beta+2)d',(\tau+1)p^{\mu}\}$\
6 & $(p^s-p^{s-\mu_{0}}+(\tau_{0}-1)p^{s-\mu_{0}-1},p^s-p^{s-u_{0}}+\tau_{0}p^{s-\mu_{0}-1}]$ & $(p^{s}-p^{s-\mu_{1}}+(\tau_{1}-1)p^{s-\mu_{1}-1},p^{s}-p^{s-\mu_{1}}+\tau_{1}p^{s-\mu_{1}-1}]$ & $\text{min}\{(\tau_{0}+1)p^{\mu_{0}}d',(\tau_{1}+1)p^{\mu_{1}}\}$\
7 & $(0,p^{s-1}]$ & $p^s$ & $2d'$\
8 & $(\beta p^{s-1},(\beta+1)p^{s-1}]$ & $p^s$ & $(\beta+2)d'$\
9 & $(p^s-p^{s-\mu}+(\tau-1)p^{s-\mu-1},p^s-p^{s-\mu}+\tau p^{s-\mu-1}]$ & $p^s$ & $(\tau+1)p^{\mu}d'$\
The notations $b_{\text{min}}$ and $b_{\text{max}}$ denote the minimum and maximum elements in the set $\{p^s-a_{j,t}:0\leq t\leq e\}$, respectively.
The parameters are in the ranges $1\leq\beta,\beta_{0},\beta_{1}\leq p-2$, $1\leq \mu,\mu_{0},\mu_{1}\leq s-1$ and $1\leq \tau,\tau_{0},\tau_{1}\leq p-1$.
$d'$ denotes the minimum distance of $\overline{C}_{j,v'}$ where $P_{v'}=\text{min}\{P_{v}:b_{\text{min}}\leq v< b_{\text{max}},v\in V\}$.
\[tab1\]
Furthermore, set $l=3$ and $q\equiv 2(\text{mod } 3)$. Thus a $3p^s$-length negacyclic code $C_{j}$ has a generator polynomial $$g_{j}(x)=(x+1)^{p^s-a_{j,0}}(x^2-x+1)^{p^s-a_{j,1}},$$ where $0\leq a_{j,0},a_{j,1}\leq p^s$. The $l$-length negacyclic code $\overline{C}_{j,v}$ has minimum distance $$d(\overline{C}_{j,v})=\left\{
\begin{array}{ll}
1, &\text{if } p^s-a_{j,0}\leq v, p^s-a_{j,1}\leq v,\\
2, &\text{if } p^s-a_{j,0}>v, p^s-a_{j,1}\leq v,\\
3, &\text{if } p^s-a_{j,0}\leq v, p^s-a_{j,1}>v,\\
\infty, &\text{if } p^s-a_{j,0}>v, p^s-a_{j,1}>v.
\end{array}
\right.$$ Hence if we assume that $p^s-a_{j,0}< p^s-a_{j,1}$, the minimum distance of $\overline{C}_{j,v'}$ is 3, where $P_{v'}=\text{min}\{p^s-a_{j,0}\leq v<p^s-a_{j,1} : v\in V\}$. On that condition, Table \[tab2\] lists sample parameters for $C_{j}$.
[p[0.9cm]{}<p[0.9cm]{}<p[0.9cm]{}<p[0.9cm]{}<p[0.9cm]{}<p[0.9cm]{}<p[0.9cm]{}<]{} $\bm{p}$ & $\bm{s}$ & $n_{j}$ & $\bm{a_{j,0}}$ & $\bm{a_{j,1}}$ & $k_{j}$ & $\bm{d_{j}}$\
5 & 2 & 75 & 19 & 18 & 55 & 3\
5 & 2 & 75 & 19 & 4 & 27 & 9\
5 & 2 & 75 & 14 & 3 & 20 & 12\
5 & 2 & 75 & 4 & 2 & 8 & 20\
5 & 3 & 375 & 14 & 4 & 22& 50\
5 & 3 & 375 & 9 & 3 & 15 & 75\
5 & 4 & 1875 & 19 & 4 & 27 & 225\
5 & 4 & 1875 & 9 & 3 & 15 & 375\
11& 2 & 363 & 109 & 65 & 239 & 7\
11& 2 & 363 & 21 & 10 & 41 & 33\
11 & 2 & 363 & 10 & 6 & 22 & 66\
11 & 3 & 3993 & 65 & 10 & 85 & 231\
11 & 3 & 3993 & 32 & 9 & 50 & 330\
23 & 2 & 1587 & 459 & 275 & 1009 & 13\
23 & 2 & 1587 & 229 & 22 & 273 & 45\
23 & 2 & 1587 & 45 & 22 & 89 & 69\
23 & 2 & 1587 & 21 & 16 & 53 & 184\
23 & 3 & 36501 & 68 & 21 & 110 & 1518\
23 & 3 & 36501 & 45 & 21 & 87 & 1587\
\[tab2\]
[p[0.5cm]{}<|p[0.5cm]{}<p[0.5cm]{}<p[0.5cm]{}<p[0.5cm]{}<p[0.5cm]{}<p[0.5cm]{}<p[0.5cm]{}<p[0.5cm]{}<p[0.5cm]{}<p[0.5cm]{}<p[0.5cm]{}<]{} **Case** & $\bm{p}$ & $\bm{s}$ & $\bm{n}$ & $\bm{a_{i,0}}$ & $\bm{a_{i,1}}$ & $\bm{a_{j,0}}$ & $\bm{a_{j,1}}$ & $\bm{d_{i}}^{\dagger}$ & $\bm{d_{j}}$ & $\bm{d}$ &$\bm{k}$\
1 & 5 & 2 & 150 & 19 & 18 & 19 & 4 & 3 & 9 & 6 & 82\
2 & 5 & 2 & 150 & 19 & 4 & 14 & 3 & 9 & 12 & 12 & 47\
3 & 5 & 2 & 150 & 19 & 4 & 4 & 2 & 9 & 20 & 18 & 35\
4 & 5 & 2 & 150 & 4 & 2 & 14 & 3 & 20 & 12 & 20 & 28\
5 & 5 & 3 & 750 & 14 & 4 & 9 & 3 & 50 & 75 & 75 & 37\
6 & 5 & 4 & 3750 & 19 & 4 & 9 & 3 & 225 & 375 & 375 & 42\
7 & 11& 2 & 726 & 109 & 65 & 21 & 10 & 7 & 33 & 14 & 280\
8 & 11 & 2 & 726 & 21 & 10 & 10 & 6 & 33 & 66 & 66 & 63\
9 & 11 & 3 & 7986 & 65 & 10 & 32 & 9 & 231 & 330 & 330 & 135\
10 & 23 & 2 & 3174 & 459 & 275 & 229 & 22 & 13 & 45 & 26 & 1282\
11 & 23 & 2 & 3174 & 229 & 22 & 45 & 22 & 45 & 69 & 69 & 362\
12 & 23 & 2 & 3174 & 229 & 22 & 21 & 16 & 45 & 184 & 90 & 326\
13 & 23 & 2 & 3174 & 45 & 22 & 21 & 16 & 69 & 184 & 138 & 142\
$d_{i}$ and $d_{j}$ denote the minimum distances of $C_{i}$ and $C_{j}$, respectively.
\[tab3\]
Combined with the results of Ref. [@Lan2018Non] regarding a $3p^s$-length cyclic code $C_{i}=\langle (x-1)^{p^s-a_{i,0}}(x^2+x+1)^{p^s-a_{i,1}}\rangle$ with $a_{i,1}<a_{i,0}$, the minimum distance of a $6p^s$-length cyclic code $C_{i}\curlyvee C_{j}$ on $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ construction can thus be determined. Sample parameters are provided in Table \[tab3\].
We can tell that in the cases 3, 7, 8, 10, 12 and 13 in Table \[tab3\], $C_{i}\curlyvee C_{j}$ have minimum distances twice as large as the component cyclic codes $C_{i}$, for $i\in\{1,3\}$ and $j\in \{2,4\}$. As a consequence, the quantum synchronizable codes derived from $C_{i}\curlyvee C_{j}$ can correct Pauli errors of weight twice larger than those constructed from $C_{i}$. In many instances, the former codes also have better error-correcting capability against Pauli errors than the quantum synchronizable codes derived from non-primitive narrow-sense BCH codes [@Fujiwara2013Algebraic]. Denote by $\delta$ the precise lower bound [@Lan2018Non; @Aly2007On] for the minimum distance of a dual-containing BCH code. Table \[tab4\] lists some sample parameters of dual-containing non-primitive, narrow-sense BCH codes.
**Case** $\bm{p}$ **length** $\bm{\delta}$ **dimension**
---------- ---------- ------------ --------------- ---------------
1 5 146 5 130
2 5 748 28 638
3 11 725 60 563
4 11 7985 65 7749
5 ¡¡23 3172 132 2794
: Sample parameters for $p$-ary non-primitive, narrow-sense BCH codes.
\[tab4\]
By the comparison between Tables \[tab3\] and \[tab4\] we can see that, given the same base field $\mathbb{F}_{p}$, repeated-root cyclic codes $C_{i}\curlyvee C_{j}$, with well-chosen parameters, can possess larger minimum distances than non-primitive, narrow sense BCH codes of close lengths. In particular, over the base fields $\mathbb{F}_{11}$ and $\mathbb{F}_{23}$, repeated-root cyclic codes $C_{i}\curlyvee C_{j}$ of lengths 726 and 3174, respectively, can reach a larger minimum distance provided that parameters $a_{i,0},a_{i,1},a_{j,0},a_{j,1}$ are sufficiently small. In that case, quantum sychronizable codes constructed from $C_{i}\curlyvee C_{j}$ have better performance in correcting Pauli errors than those from non-primitive, narrow-sense BCH codes.
\[sec:level1\]The product construction
======================================
Apart from the $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ method, the product construction is another useful technique of generating new cyclic codes from old ones. Without loss of generality, we restrict the following discussion to the binary case.
Let $C_{1}$ and $C_{2}$ be linear codes of parameters $[n_{1},k_{1},d_{1}]$ and $[n_{2},k_{2},d_{2}]$ respectively. A product code $C=C_{1}\otimes C_{2}$ [@Blahut2003Algebraic] is defined to be an $[n_{1}n_{2},k_{1}k_{2},d_{1}d_{2}]$ linear code whose codewords are all the two-dimensional arrays where each row is a codeword in $C_{1}$ and each column is a codeword in $C_{2}$. Denote by $c_{i,j}$ the element in the $(i+1)^{\text{th}}$ row and $(j+1)^{\text{th}}$ column of the array, where $0\leq i\leq n_{1}-1$ and $0\leq j\leq n_{2}-1$. Then a codeword of $C$ can be identified with a bivariate polynomial $c(y,z)=\sum_{i=0}^{n_{1}-1}\sum_{j=0}^{n_{2}-1}c_{i,j}y^{i}z^{j}\in\frac{\mathbb{F}_{2}[x,y]}{\langle (y^{n_{1}}-1)(z^{n_{2}}-1)\rangle}$. According to the Chinese remainder theorem, there exists a unique integer $\theta$ in the range $0\leq \theta\leq n_{1}n_{2}-1$ such that $\theta\equiv i(\text{mod } n_{1})$ and $\theta\equiv j (\text{mod } n_{2})$, provided that $\text{gcd}(n_{1},n_{2})=1$. In that case, ${c}(x)=\sum_{\theta=0}^{n_{1}n_{2}-1}c_{(\theta(\text{mod } n_{1}), \theta(\text{mod } n_{2}))}x^{\theta}$ is also a polynomial representation of code $C$.
Suppose that $C_{1}$ and $C_{2}$ are both cyclic. Then $C$ is also cyclic since $x{c}(x)(\text{mod } x^{n_{1}n_{2}}-1)$, which corresponds to $yzc(y,z)(\text{mod } y^{n_{1}}-1)(\text{mod } z^{n_{2}}-1)$, is a codeword of $C$. Denote by $g_{1}(x)$ and $g_{2}(x)$ the respective generator polynomial of $C_{1}$ and $C_{2}$. Then $C$ and the dual code $C^{\perp}$ have respective generator polynomial [@Lin1970Further] $$\begin{array}{l}
g_{C}(x)=\text{gcd}(g_{1}(x^{\beta n_{2}})g_{2}(x^{\alpha n_{1}}),x^{n_{1}n_{2}}-1),\\
g^{\perp}_{C}(x)=\text{gcd}(g_{1}^{\perp}(x^{\beta n_{2}}),g_{2}^{\perp}(x^{\alpha n_{1}})),
\end{array}$$ where $\alpha,\beta$ are integers satisfying $\alpha n_{1}+\beta n_{2}=1$, and $g_{1}^{\perp}(x)$ and $g_{2}^{\perp}(x)$ represent the respective generator polynomial of $C_{1}^{\perp}$ and $C_{2}^{\perp}$. Clearly, $C$ is self-orthogonal if either $C_{1}$ or $C_{2}$ is self-orthogonal. Applying cyclic product codes to Theorem \[thm1\], we can then obtain a broad family of quantum synchronizable codes as follows.
Let $C_{1}=\langle g_{1}(x)\rangle$ be a self-orthogonal $[n_{1},k_{1}]$ cyclic code and $C_{2}=\langle g_{2}(x)\rangle$ be an $[n_{2},k_{2}]$ cyclic code with $\alpha n_{1}+\beta n_{2}=1$, where $\alpha,\beta$ are integers. Suppose that $C_{3}$ is a cyclic code with a generator polynomial $g_{3}(x)=g_{2}(x)\rho(x)$, where $\rho(x)$ is a non-trivial polynomial such that $\rho(0)=1$. Assume that $\emph{gcd}(g_{3}^{\perp}(x),\rho^{*}(x))=1$ where $\rho^{*}(x)$ denotes the reciprocal polynomial of $\rho(x)$. Then for any non-negative pair $a_{l},a_{r}$ such that $a_{l}+a_{r}<\emph{ord}(\emph{gcd}(g_{1}^{\perp}(x^{\beta n_{2}}),\rho^{*}(x^{\alpha n_{1}})))$, there exists an $(a_{l},a_{r})-[[n_{1}n_{2},n_{1}n_{2}-2k_{1}k_{2}]]$ quantum synchronizable code. \[thm7\]
The cyclic product code $D=C_{1}\otimes C_{3}$ and its dual code $D^{\perp}$ have generator polynomials $$\begin{array}{l}
g_{D}(x)=\text{gcd}(g_{1}(x^{\beta n_{2}})g_{3}(x^{\alpha n_{1}}),x^{n_{1}n_{2}}-1),\\
g_{D}^{\perp}(x)=\text{gcd}(g_{1}^{\perp}(x^{\beta n_{2}}),g_{3}^{\perp}(x^{\alpha n_{1}})),
\end{array}$$ respectively. Denote by $C$ the product code of $C_{1}$ and $C_{2}$. Then the dual codes $C^{\perp}$ and $D^{\perp}$ are dual-containing cyclic codes such that $C^{\perp}\subset D^{\perp}$. Note that $$g_{2}^{\perp}(x)=\left(\frac{x^{n_{2}}-1}{g_{2}(x)}\right)^{*}=\left(\frac{x^{n_{2}}-1}{g_{3}(x)}\rho(x)\right)^{*}=g_{3}^{\perp}(x)\rho^{*}(x).$$ Hence the quotient polynomial $f(x)$ of the generator polynomials of $C^{\perp}$ and $D^{\perp}$ is given by $$f(x)=\frac{g_{C}^{\perp}(x)}{g_{D}^{\perp}(x)}=\frac{\text{gcd}(g_{1}^{\perp}(x^{\beta n_{2}}),g_{3}^{\perp}(x^{\alpha n_{1}})\rho^{*}(x^{\alpha n_{1}}))}{\text{gcd}(g_{1}^{\perp}(x^{\beta n_{2}}),g_{3}^{\perp}(x^{\alpha n_{1}}))}=\text{gcd}(g_{1}^{\perp}(x^{\beta n_{2}}),\rho^{*}(x^{\alpha n_{1}})).$$ Apply $C^{\perp}$ and $D^{\perp}$ to Theorem \[thm1\], we can then obtain the quantum synchronizable code with the desired parameters.
$\square$
Note that the constraint $\text{gcd}(g_{3}^{\perp}(x),\rho^{*}(x))=1$ is equivalent with $$\text{gcd}(h_{3}^{*}(x),\rho^{*}(x))=\text{gcd}(h_{3}(x),\rho(x))=\text{gcd}\left(\frac{h_{2}(x)}{\rho(x)},\rho(x)\right)=1,$$ where $h_{i}(x)$ denotes the parity check polynomial of $C_{i}$ for $i\in\{2,3\}$. Hence the non-trivial polynomial $\rho(x)$ can always be found, provided that $h_{2}(x)$ has at least two irreducible factors. On that condition, a broad range of $n_{2}$-length cyclic codes can be applied to the construction in Theorem \[thm7\], which, accordingly, widen the family of quantum synchronizable codes to a large extent. In particular, the range of parameters’ selection for quantum synchronization coding is greatly enlarged considering that lengths of previous quantum synchronizable codes, apart from those built on repeated-root cyclic codes, are of limited forms, e.g., $2^s-1$ [@Fujiwara2013Algebraic] and $\frac{2^{st}-1}{2^s-1}$ [@Fujiwara2014Quantum], where $s,t$ are positive integers.
Furthermore, the quantum synchronizable codes obtained from cyclic product codes can also reach the maximum synchronization error tolerance if $$\text{ord}(\text{gcd}(g_{1}^{\perp}(x^{\beta n_{2}}),\rho^{*}(x^{\alpha n_{1}})))=\text{ord}(\text{gcd}(h_{1}(x^{\beta n_{2}}),\rho(x^{\alpha n_{1}})))=n_{1}n_{2},$$ where $h_{1}(x^{\beta n_{2}})=\frac{x^{\beta n_{1}n_{2}}-1}{g_{1}(x^{\beta n_{2}})}$. For example, let $C_{1}$ be a $[7,3]$ cyclic code with a generator polynomial $$g_{1}(x)=(x+1)(x^3+x+1).$$ The dual code $C_{1}^{\perp}$ has a generator polynomial $ g_{1}^{\perp}(x)=x^3+x+1$. Clearly, $C_{1}$ is a self-orthogonal code. Let $C_{2}$ be a $[15,11]$ cyclic code with a generator polynomial $$g_{2}(x)=x^4+x^3+1.$$ Therefore, $C=C_{1}\otimes C_{2}$ is an $[105,33]$ cyclic code. Note that $$h_{2}(x)=(x+1)(x^2+x+1)(x^4+x+1)(x^4+x^3+x^2+x+1).$$ By choosing $\rho(x)$ to be $x^4+x+1$, we can then obtain a $[15,7]$ cyclic code $C_{3}$ with a generator polynomial $$g_{3}(x)=(x^4+x^3+1)(x^4+x+1).$$ In that case, the cyclic product code $D=C_{1}\otimes C_{3}$ is an $[105,21]$ code.
Since $(-2)\times 7+1\times 15=1$, we have $$\begin{array}{l}
g_{1}^{\perp}(x^{15})=x^{45}+x^{15}+1,\\
\rho^{*}(x^{-14})=x^{-56}+x^{-42}+1=x^{-56}(x^{56}+x^{14}+1).
\end{array}$$ Their greatest common divisor is $$\text{gcd}(g_{1}^{\perp}(x^{15}),f^{*}(x^{-14}))=x^{12}+x^{10}+x^9+x^7+x^6+x^4+1,$$ which is of order 105. Hence following Theorem \[thm7\], we can build an $[[105,39]]$ quantum synchronizable code that can tolerate misalignment by up to 105 qubits.
\[sec:level1\]Conclusions
=========================
In this paper, we present two families of quantum synchronizable codes from cyclic codes built on the $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ construction and the product construction. In the former case, most existing quantum synchronizable codes that provide the highest tolerance against synchronization errors can be generalized to larger cases. In particular, repeated-root codes of length $lp^s$ have been thoroughly investigated in quantum synchronization coding and can provide a better performance in correcting Pauli errors than non-primitive, narrow-sense BCH codes. In the latter case, the loose restrictions on the component cyclic codes ensure a large augmentation of available quantum synchronizable codes. Besides, their synchronization error tolerance can also reach the maximum under certain circumstances.
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---
abstract: 'The ubiquity of connecting technologies in smart vehicles and the incremental automation of its functionalities promise significant benefits, including a significant decline in congestion and road fatalities. However, increasing automation and connectedness broadens the attack surface and heightens the likelihood of a malicious entity successfully executing an attack. In this paper, we propose a Blockchain based Framework for sEcuring smaRt vehicLes (B-FERL). B-FERL uses permissioned blockchain technology to tailor information access to restricted entities in the connected vehicle ecosystem. It also uses a challenge-response data exchange between the vehicles and roadside units to monitor the internal state of the vehicle to identify cases of in-vehicle network compromise. In order to enable authentic and valid communication in the vehicular network, only vehicles with a verifiable record in the blockchain can exchange messages. Through qualitative arguments, we show that B-FERL is resilient to identified attacks. Also, quantitative evaluations in an emulated scenario show that B-FERL ensures a suitable response time and required storage size compatible with realistic scenarios. Finally, we demonstrate how B-FERL achieves various important functions relevant to the automotive ecosystem such as trust management, vehicular forensics and secure vehicular networks.'
author:
- Chuka Oham
- Regio Michellin
- 'Salil S. Kanhere'
- Raja Jurdak
- Sanjay Jha
title: 'B-FERL: Blockchain based Framework for Securing Smart Vehicles'
---
*CAVs*, ECUs, Blockchain, Merkle root, Forensics, Security, Vehicular network, Sensors, Trust management.
Introduction
============
With technological advancements in the automotive industry in recent times, modern vehicles are no longer made up of only mechanical devices but are also an assemblage of complex electronic devices called electronic control units (ECUs) which provide advanced vehicle functionality and facilitate independent decision making. ECUs receive input from sensors and runs computations for their required tasks [@Alam:2018]. These vehicles are also fitted with an increasing number of sensing and communication technologies to facilitate driving decisions and to be *self aware* [@Anupam:2018]. However, the proliferation of these technologies have been found to facilitate the remote exploitation of the vehicle \[7\]. Malicious entities could inject malware in ECUs to compromise the internal network of the vehicle [@Anupam:2018]. The internal network of a vehicle refers to the communications between the multiple ECUs in the vehicle over on-board buses such as the controller area network (CAN) [@Han:2014]. The authors in \[7\] and \[8\] demonstrated the possibility of such remote exploitation on a connected and autonomous vehicle (CAV), which allowed the malicious entity to gain full control of the driving system and bring the vehicle to a halt.\
To comprehend the extent to which smart vehicles are vulnerable, we conducted a risk analysis for connected vehicles in \[1\] and identified likely threats and their sources. Furthermore, using the Threat Vulnerability Risk Assessment (TVRA) methodology, we classified identified threats based on their impact on the vehicles and found that compromising one or more of the myriad of ECUs installed in the vehicles poses a considerable threat to the security of smart vehicles and the vehicular network. Vehicular network here refers to communication between smart vehicles and roadside units (RSUs) which are installed and managed by the transport authority. These entities exchange routine and safety messages according to the IEEE802.11p standard \[4\]. By compromising ECUs fitted in a vehicle, a malicious entity could for example, broadcast false information in the network to affect the driving decisions of other vehicles. Therefore, in this paper, we focus on monitoring the state of the in-vehicle network to enable the detection of an ECU compromise.
Previous efforts that focus on the security of in-vehicle networks have focused on intrusion and anomaly detection which enables the detection of unauthorized access to in-vehicle network \[9-11\], \[15\], \[23\] and the identification of deviation from acceptable vehicle behavior [@Wasicek:2014]. Several challenges however persist. First, proposed security solutions are based on a centralized design which relies on a Master ECU that is responsible for ensuring valid communications between in-vehicle ECUs \[9-10\] \[23\]. However, these solutions are vulnerable to a single point of failure attack where an attacker’s aim is to compromise the centralized security design. Furthermore, if the Master ECU is either compromised or faulty, the attacker could easily execute actions that undermine the security of the in-vehicle network. In-addition, efforts that focus on intrusion detection by comparing ECU firmware versions \[10\] \[11\] \[15\] are also vulnerable to a single point of exploitation whereby the previous version which is centrally stored could be altered. These works \[11\] \[15\] also rely on the vehicle manufacturer to ultimately verify the state of ECUs. However, vehicle manufacturers could be motivated to execute malicious actions for their benefits such as to evade liability \[3\].Therefore, decentralization of the ECU state verification among entities in the vehicular ecosystem is desirable for the security of smart vehicles. Finally, the solution proposed in \[24\] which focuses on observing deviations from acceptable behavior utilized data generated from a subset of ECUs. However, this present a data reliability challenge when an ECU not included in the ECU subset is compromised.\
We argue in this paper that Blockchain (BC) \[12\] technology has the potential to address the aforementioned challenges including centralization, availability and data reliability.\
**BC** is an immutable and distributed ledger technology that provides verifiable record of transactions in the form of an interconnected series of data blocks. BC can be public or permissioned \[3\] to differentiate user capabilities including who has the right to participate in the BC network. BC replaces centralization with a trustless consensus which when applied to our context can ensure that no single entity can assume full control of verifying the state of ECUs in a smart vehicle. The decentralized consensus provided by BC is well-suited for securing the internal network of smart vehicles by keeping track of historical operations executed on the vehicle’s ECUs such as firmware updates, thus easily identifying any change to the ECU and who was responsible for that change. Also, the distributed structure of BC provides robustness to a single point of failure.
Contributions and Paper Layout
------------------------------
Having identified the limitations of existing works, we propose a Blockchain based Framework for sEcuring smaRt vehicLes (B-FERL). B-FERL is an apposite countermeasure for in-vehicle network security that exposes threats in smart vehicles by ascertaining the state of the vehicle’s internal controls. Also, given that data modification depicts a successful attempt to alter the state of an ECU, B-FERL also suffices as a data reliability solution that ensures that a vehicle’s data is trustworthy. We utilize a permissioned BC to allow only trusted entities manage the record of vehicles in the BC network. This means that state changes of an ECU are summarized, stored and managed distributedly in the BC.\
*The key contributions of this paper are summarized as follows:*\
**(1)** We present B-FERL; a decentralized security framework for in-vehicle networks. B-FERL ascertains the integrity of in-vehicle ECUs and highlights the existence of threats in a smart vehicle. To achieve this, we define a two-tier blockchain-based architecture, which introduces an initialization operation used to create record vehicles for authentication purposes and a challenge-response mechanism where the integrity of a vehicle’s internal network is queried when it connects to an RSU to ensure its security.\
**(2)** We conduct a qualitative evaluation of B-FERL to evaluate its resilience to identified attacks. We also conduct a comparative evaluation with existing approaches and highlight the practical benefits of B-FERL. Finally, we characterize the performance of B-FERL via extensive simulations using the CORE simulator against key performance measures such as the time and storage overheads for smart vehicles and RSUs.\
**(3)** Our proposal is tailored to meet the integrity requirement for securing smart vehicles and the availability requirement for securing vehicular networks and we provide succinct discussion on the applicability of our proposal to achieve various critical automotive functions such as vehicular forensics, secure vehicular communication and trust management.\
This paper is an extension of our preliminary ideas presented in \[1\]. Here, we present a security framework for detecting when an in-vehicle network compromise occurs and provide evidence that reflect actions on ECUs in a vehicle. Also, we present extensive evaluations to demonstrate the efficacy of B-FERL.\
The rest of the paper is structured as follows. In section 2, we discuss related works and present an overview of our proposed framework in Section 3 where we describe our system, network and threat model. Section 4 describes the details of our proposed framework. In section 5, we discuss results of the performance evaluation. Section 6 present discussions on the potential use cases of B-FERL, comparative evaluation with closely related works, and we conclude the paper in Section 7.
Related Work
============
BC has been proposed as security solutions for vehicular networks. However, proposed solutions have not focused on the identification of compromised ECUs for securing vehicular networks. The author in [@Blackchain:2017] proposed Blackchain, a BC based message revocation and accountability system for secure vehicular communication. However, their proposal does not consider the reliability of data communicated in the vehicular network which could be threatened when an in-vehicle ECU is compromised. The author in [@Ali:2017] presents a BC based architecture for securing automotive networks. However they have not described how their architecture is secured from insider attacks where authorised entities could be motivated to execute rogue actions for their benefits. Also, their proposal does not consider the veracity of data from vehicles. The authors in [@cube:2018] proposed a security platform for autonomous vehicle based on blockchain but have not presented a description of their architecture and its applicability for practical scenarios. Also, their security is towards the prevention of unauthorized network entry using a centralized intrusion detector which is vulnerable to a single point of failure attack. Their proposal do not also consider the malicious tendencies of authorized entities as described in [@Oham:2018].
The authors in [@Coin:2018] proposed CreditCoin; a privacy preserving BlockChain based incentive announcement and reputation management scheme for smart vehicles. Their proposal is based on threshold authentication where a number of vehicles agree on a message generated by a vehicle and then the agreed message is sent to a nearby roadside unit. However, in addition to the possibility of collusion attacks, the requirement that vehicles would manage a copy of the blockchain presents a significant storage and scalability constraint for vehicles. The authors in [@BARS:2018] have proposed a Blockchain-based Anonymous Reputation System (BARS) for Trust Management in VANETs however, they have not presented details on how reputation is built for vehicles and have also not presented justifications for their choice of reputation evaluation parameters. The authors in [@Contract:2018] have proposed an enhanced Delegated Proof-of-stake (DPoS) consensus scheme with a two-stage soft security solution for secure vehicular communications. However, their proposal is directed at establishing reputation for road side infrastructures and preventing collusion attacks in the network. These authors [@Coin:2018] [@BARS:2018] [@Contract:2018] have also not considered the security of in-vehicle networks.
B-FERL Overview and Threat Model
================================
In this section, we present a brief overview of B-FERL including the roles of interacting entities, and a description of the network and threat models.
Architecture overview
---------------------
The architecture of our proposed security solution (B-FERL) is described in Figure \[fig:framework\]. Due to the need to keep track of changes to ECU states and to monitor the behaviour of a vehicle while operational, B-FERL consists of two main BC tiers namely, upper and lower tiers. Furthermore, these tiers clarify the role of interacting entities and ensure that entities are privy to only information they need to know.
The upper tier comprises vehicle manufacturers, service technicians, insurance companies, legal and road transport authorities. The integration of these entities in the upper tier makes it easy to also keep track of actions executed by vehicle manufacturers and service technicians on ECUs such as firmware updates which changes the state of an ECU and allows only trusted entities such as transport and legal authorities to verify such ECU state changes. Interactions between entities in this tier focus on vehicle registration and maintenance. The initial registration data of a vehicle is used to create a record (block) for the vehicle in the upper tier. This record stores the state of the vehicle and the hash values of all ECUs in the vehicle and is used to perform vehicle validation in the lower tier BC. This is accomplished by comparing the current state of the vehicle and the firmware hashes of each ECU in the vehicle to their values in the lower tier BC. Also, the upper tier stores scheduled maintenance or diagnostics data that reflects the actions of vehicle manufacturers and service technicians on a smart vehicle. This information is useful for the monitoring of the vehicle while operational and for making liability decisions in the multi-entity liability attribution model [@Oham:2018].\
In the following, we describe actions that trigger interactions in the upper tier. In the rest of the paper unless specifically mentioned, we refer to smart vehicles as *CAVs*.
- When a *CAV* is assembled, the vehicle manufacturer obtains the ECU Merkle root value ($SS_{ID}$) by computing hash values of all ECUs in the vehicle and forwards this value to the road transport and legal authorities to create a public record (block) for the vehicle. This record is utilized by RSUs to validate vehicles in the lower tier. We present a detailed description of this process in Section 3.
- When a maintenance occurs in the vehicle, vehicle manufacturers or service technicians follow the process of obtaining the updated $SS_{ID}$ value above and communicate this to the transport and legal authorities to update the record of the vehicle and assess the integrity of its ECUs. We present a detailed description of this process in Section 3. Maintenance here means any activity that alters the state of any of the vehicle’s ECUs.
The lower tier comprises roadside units (*RSUs*), smart vehicles, legal and road transport authorities. Interactions in this tier focus on identifying when an ECU in a vehicle has been compromised. To achieve this, a vehicle needs to prove its ECUs firmware integrity whenever it connects to an *RSU*. When a vehicle approaches the area of coverage of an *RSU*, the *RSU* sends the vehicle a challenge request to prove the state of its ECUs. To provide a response, the vehicle computes the cumulative hash value of all of its ECUs i.e. its ECU Merkle root ($SS_{ID}$). The response provided by the vehicle is then used to validate its ECUs current state in comparison to the previous state in the lower tier. Also, as a vehicle moves from one *RSU* to the other, an additional layer of verification is added by comparing the time stamps of its current response to the previous response to prevent the possibility of a replay attack. It is noteworthy, that compared to traditional BC which executes a consensus algorithm in order to insert transactions into a block, B-FERL relies on the appendable block concept (ABC) proposed in [@Michelin:2018] where transactions are added to the blocks by valid block owners represented by their public key. Therefore, no consensus algorithm is required in B-FERL to append transactions to the block. To ensure that the integrity of a block is not compromised, ABC decouples the block header from the transactions to enable network nodes store transactions off-chain without compromising block integrity. Furthermore, to ensure scalability in the lower tier, we only store two transactions (which represents the previous and current ECU’s firmware state) per vehicle and push other transactions to the cloud where historical data of the vehicle can be accessed when necessary. However, this operation could introduce additional latency for pushing the extra transaction from the RSU to the cloud storage. This further imposes an additional computing and bandwidth requirement for the RSU.\
Next, we discuss our network model which describes interacting entities in our proposed framework and their roles.
{width="90.00000%"}
Network model
-------------
To restrict the flow of information to only concerned and authorized entities, we consider a two-tiered network model as shown in Figure \[fig:framework\]. The upper tier features the road transport and legal authorities responsible for managing the vehicular network. This tier also integrates entities responsible for the maintenance of vehicles such as vehicle manufacturers and the service technicians. It could also include auto-insurance companies who could request complimentary evidence from Transport and Legal authorities for facilitating liability decisions. For simplicity we focus on single entities for each of these however, our proposal is generalizable to the case when there are several of each entity.\
The lower tier features *CAVs* as well as RSUs which are installed by the road transport authority for the management and monitoring of traffic situation in the road network. For interactions between *CAVs* and RSUs , we utilize the IEEE802.11p communication standard which has been widely used to enable vehicle-to-vehicle and vehicle-to-infrastructure communications \[4\]. However, 5G is envisaged to bring about a new vehicular communication era with higher reliability, expedited data transmissions and reduced delay \[5\]. Also, we utilise PKI to issue identifiable digital identities to entities and establish secure communication channels for permissible communication. The upper tier features a permissioned blockchain platform managed by the road transport and legal authorities. Vehicle manufacturers and service technicians participate in this BC network by sending sensor update notification transactions which are verified and validated by the BC network managers. Insurance companies on the other hand participate by sending request transactions for complimentary evidence to facilitate liability attribution and compensation payments. The lower tier also features a permissioned BC platform managed by the road transport, legal authorities and RSUs. In this tier, we maintain vehicle-specific profiles. To achieve this, once a vehicle enters the area of coverage of a roadside unit (RSU), the RSU sends a challenge request to the vehicle by which it reports the current state of its ECUs. Once a valid response is provided, the vehicle is considered trustworthy until another challenge-response activity.\
We present a full description of the entire process involved in our proposed framework in section 3.
Threat Model
------------
Given the exposure of *CAVs* to the Internet, they become susceptible to multiple security attacks which may impact the credibility of data communicated by a vehicle. In the attack model, we consider how relevant entities could execute actions to undermine the proposed framework. The considered attacks include:\
**Fake data:** A compromised vehicle could try to send misleading information in the vehicular network for its benefit. For example, it could generate false messages about a traffic incident to gain advantage on the road. Also, to avoid being liable in the case of an accident, a vehicle owner could manipulate an ECU to generate false data.\
**Code injection:** Likely liable entities such as the vehicle manufacturer and service technician could send malware to evade liability. vehicle owners on the other hand could execute such actions to for example reduce the odometer value for the vehicle to increase its resale value.\
**Sybil attack:** A vehicle could create multiple identities to manipulate vehicular network, for example by creating false alarm such as false traffic jam etc.\
**Masquerade attack (fake vehicle):** A compromised roadside unit could create a fake vehicle or an external adversary could create a fake vehicle for the purpose of causing an accident or changing the facts of an accident.\
**ECU State Reversal Attack:** A vehicle owner could extract the current firmware version of an ECU and install its malicious version and revert to the original version for verification purpose.
Blockchain based Framework for sEcuring smaRt vehicLes (B-FERL) {#sec:b-ferl}
===============================================================
This section outlines the architecture of the proposed framework. As described in Figure \[fig:framework\], entities involved in our framework include vehicle manufacturers, service technicians, insurance companies, *CAVs*, RSUs, road transport and legal authorities. Based on entity-roles described in section 2, we categorize entities as verifiers and proposers. Verifiers are entities that verify and validate data sent to the BC. Verifiers in B-FERL include RSUs, road transport and legal authorities. Proposers are entities sending data to the BC or providing a response to a challenge request. Proposers in our framework include *CAVs*, vehicle manufacturers, service technicians and insurance companies.\
In B-FERL architecture, we assume that the CAVs are producing many transactions, especially in high density smart city areas. Most of blockchains implementations are designed to group transactions, add them into a block and only after that append the new block into the blockchain, which leads to a sequential transaction insertion. To tackle this limitation, in B-FERL we adopted a blockchain framework presented by Michelin et al. [@Michelin:2018] which introduces the appendable block concept (ABC). This blockchain solution enables multiple CAVs to append transactions in different blocks at same time. The framework identifies each CAV by its public key, and for each different public key, a block is created in the blockchain data structure. The block is divided in two distinct parts: (i) block header, which contains the CAV public key, the previous block header hash, the timestamp; (ii) block payload, where all the transactions are stored. The transaction storage follows a linked list data structure, the first transaction contains the block header hash, while the coming transactions contain the previous transaction hash. This data structure allows the solution to insert new transaction into existing blocks. Each transaction must be signed by the CAV private key, once the transaction signature is validated with the block’s public key, the RSU can proceed appending the transaction into the block identified by the CAV public key. Based on the public key, the BC maps all the transactions from a specific entity to the same block.
Transactions
------------
Transactions are the basic communication primitive in BC for the exchange of information among entities in B-FERL. Having discussed the roles of entities in each tier in B-FERL, in this section, we discuss the details of communication in each tier facilitated by the different kind of transactions. Transactions generated are secured using cryptographic hash functions (SHA-256), digital signatures and asymmetric encryption.\
***Upper tier***\
Upper tier transactions include relevant information about authorized actions executed on a *CAV*. They also contain interactions that reflect the time a vehicle was assembled. Also, in this tier, insurance companies could seek complementary evidence from the road transport and legal authorities in the event of an accident hence, a request transaction is also sent in this tier.\
![Obtaining the Merkle tree root value[]{data-label="fig:merkle"}](Merkletree.PNG){width="50.00000%"}
**Genesis transaction:** This transaction is initiated by a vehicle manufacturer when a vehicle is assembled. The genesis transaction contains the initial $SS_{ID}$ value which is the Merkle tree root from the *CAV’s* ECU firmware hashes at *CAV* creation time, time stamp, firmware hashes of each ECU and associated timestamps, ($H(ECU){_1}$, $T{_1}$), ($H(ECU){_2}$, $T{_2}$), .....($H(ECU){_n}$, $T{_n}$) which reflect when an action was executed on the ECU, the public key and signature of the vehicle manufacturer. Figure \[fig:merkle\] shows how the $SS_{ID}$ of a *CAV* with 8 ECUs is derived.
Genesis = \[$SS_{ID}$, TimeStamp, ($H(ECU){_1}$, $T{_1}$), ($H(ECU){_2}$, $T{_2}$), .....($H(ECU){_n}$, $T{_n}$), PubKey, Sign\]
The genesis transaction is used by the transport and legal authorities to create a genesis block for a *CAV*. This block is a permanent record of the *CAV* and used to validate its authenticity in the lower tier. It contains the genesis transaction, public key of the *CAV*, time stamp which is the time of block creation and an external address such as an address to a cloud storage where *CAV* generated data would be stored as the block size increases.\
**Update transaction:** This transaction could be initiated by a vehicle manufacturer or a service technician. It is initiated when the firmware version of an ECU in the *CAV* is updated during scheduled maintenance or diagnostics. An update transaction leads to a change in the initial $SS_{ID}$ value and contains the updated $SS_{ID}$ value, time stamp, public key of *CAV*, public key of vehicle manufacturer or service technician and their signatures.\
When an update transaction is received in the upper tier, the update transaction updates the record (block) of the *CAV* in the lower tier. The updated *CAV* block will now be utilized by RSUs to validate the authenticity of the *CAV* in the lower tier.\
**Request transaction:** This transaction is initiated by an insurance company to facilitate liability decisions and compensation payments. It contains the signature of the insurance company, the data request and its public key.\
***Lower tier***\
Communication in the lower tier reflect how transactions generated in the upper tier for CAVs are appended to their public record (block) in the lower tier. Additionally, we describe how the block is managed by an RSU in the lower tier and the transport and legal authorities in the upper tier. Lower tier communications also feature the interactions between *CAVs* and RSUs and describes how the integrity of ECUs in a *CAV* is verified. In the following, we describe the interactions that occur in the lower tier.\
**Updating CAV block:** Updating the block of a *CAV* is either performed by the road transport and legal authorities or by an RSU. It is performed by the road transport and legal authorities after an update transaction is received in the upper tier. It is performed by an RSU after it receives a response to a challenge request sent to the vehicle. The challenge-response scenario is described in the next type of transaction. The update executed by an RSU contains a *CAV’s* response which includes the signature of the *CAV*, time stamp, response to the challenge and *CAV’s* public key. It also contains the hash of the previous transaction in the block computed by the RSU, the signature and public key of the RSU.\
**Challenge-Response transaction:** The Challenge-Response transaction is a request from an RSU to prove the integrity of its ECUs. This request is received when the *CAV* comes into the RSU’s area of coverage. When this occurs, the *CAV* receives a twofold challenge from the RSU. First is a challenge to compute its $SS_{ID}$ to ascertain the integrity of its state. Next challenge is to compute the hash value of randomly selected ECUs to prevent and detect the malicious tendencies of vehicle owners discussed in Section 3.\
The *CAV* responds by providing a digitally signed response to the request.
Operation
---------
In this section we describe key operations in our proposed framework. The proposed framework works in a permissioned mode where road transport and legal authorities have rights to manage the BC in the upper and lower tier. Service technicians as well as vehicle manufacturers generate data when they execute actions that alters the internal state of a *CAV* while *CAVs* prove the integrity of their ECUs when they connect to a RSU.\
We define 2 critical operations in our proposed framework:
### Initialization
Describes the process of creating a record for a vehicle in the vehicular network. Once a genesis transaction is generated for a *CAV* by a vehicle manufacturer, upper tier verifiers verify the transaction and upon a successful verification, a genesis block is broadcasted in the lower tier for the *CAV*.\
![*CAV* record initialization (black) and upper-tier update (blue) operations.[]{data-label="fig:operation"}](tiert.PNG){width="53.00000%"}
Figure \[fig:operation\] describes the process of block creation (assembling) for *CAVs*. It outlines the requisite steps leading to the creation of a block (record) for a *CAV*.
### Update
Describes the process of updating the record of the vehicle in the vehicular network. The update operation results in a change in the block of a *CAV* in the lower tier. The update operation occurs in the upper and lower tier. In the upper tier, an update operation occurs when a vehicle manufacturer performs a diagnostic on a *CAV* or when a scheduled maintenance is conducted by a service technician. In the lower tier, it occurs when a *CAV* provides a response to the challenge request initiated by an RSU. In the following we discuss the update operation that occurs at both tiers.\
**Upper-tier update:** Here, we describe how the earlier mentioned actions of the vehicle manufacturer or service technician alters the existing record for a *CAV* in the vehicular network.\
Figure \[fig:operation\] outlines the necessary steps to update the record of a vehicle. After completing the diagnostics or scheduled maintenance (step 1), the vehicle manufacturer or service technician retrieves the hash of all sensors in the vehicle (step 2) and computes a new ECU Merkle root value (step 3). Next, an update transaction is created to reflect the action on the vehicle (step 4). This transaction includes the computed ECU Merkle root value, time stamp to reflect when the diagnostics or maintenance was conducted, signature of the entity conducting the maintenance or diagnostics and a metadata field that describes what maintenance or diagnostics was conducted on the *CAV*. Next, the transaction is broadcasted in the upper tier (step 5) and verified by verifiers (step 6); road transport and legal authorities by validating the signature of the proposer (step 7). Upon signature validation, an update block is created by the verifiers for the *CAV* (step 8) and broadcasted in the lower tier (step 9).\
**Lower tier update:** We describe here how the update of a *CAV’s* record is executed by an RSU after the initialization steps in the lower tier.\
Figure \[fig:lowupdate\] describes the necessary steps involved in updating the record of the *CAV* in the lower tier. When a *CAV* approaches the area of coverage of an RSU, the RSU sends the *CAV* a challenge request which is to prove that it is a valid *CAV* by proving its current sensor state (Step 1). For this, the *CAV* computes its current $SS_{ID}$ value as well as the hash values of selected ECUs (Step 2) and forward it to the RSU including its signature, time stamp and public key (Step 3).\
{width="85.00000%"}
When the RSU receives the response data from the *CAV*, it first verifies that the vehicle is a valid *CAV* by using its public key ($PubKey_{CAV}$) to check that the vehicle has a block in the BC (Step 4). Only valid vehicles have a block (record) in the BC. When the RSU retrieves $PubKey_{CAV}$, it validates the signature on the response data (Step 4.1). If validation succeeds, the RSU retrieves the firmware hash value in the *CAV’s* block (Step 5) proceeds to compare the computed hash values with the value on the *CAV’s* block (Step 5.1). Otherwise, the RSU reports to the road transport and legal authorities of the presence of a malicious *CAV* or an illegal *CAV* if there is no block for such *CAV* in the BC (Step 4.2). If the comparison of hash values succeeds, the RSU updates the *CAV’s* record in the lower tier to include the $SS_{ID}$ value, the time stamp, and public key of the *CAV* (Step 6). This becomes the latest record of the *CAV* in the lower tier until another challenge-response round or another maintenance or diagnostic session. However, if the hash value differs, the RSU reports to the road transport and legal authorities of the presence of a malicious *CAV* (Step 5.2).\
When the *CAV* encounters another RSU, another challenge-response activity begins. This time, the RSU repeats the steps (1-5), in-addition, another layer of verification is executed. The RSU compares the time stamp on the response data to the immediate previous record stored on the lower tier blockchain (Step 5.1.2). The time stamp value is expected to continuously increase as the vehicle travels, if this is the case, RSU executes updates the *CAV’s* block (Step 6). Otherwise, the RSU can detect a malicious action and report this to the road transport and legal authority (Step 5.2). However, if a malicious *CAV* reports a time stamp greater than its previous time stamp, we rely on the assumption that one or more of its ECUs would have been compromised and so it would produce an $SS_{ID}$ different from its record in the lower tier. Another alternative is to comparatively evaluate its time-stamp against the time-stamp of other vehicles in the RSU area of coverage. To ensure that the blockchain in the lower tier scales efficiently, we store only two transactions per *CAV* block. In this case, after successfully executing (Step 5.1.2), the RSU removes the genesis transaction from the block and stores it in a cloud storage which can be accessed using the external address value in the *CAV’s* block.\
With the challenge-response activity, we build a behaviour profile for *CAV’s* and continuously prove the trustworthiness of a vehicle while operational. Also, by keeping track of the actions of likely liable entities such as the service technician and vehicle manufacturer and by storing vehicle’s behaviour profile in the blockchain, we obtain historical proof that could be utilised as contributing evidence for facilitating liability decisions.
Performance Evaluation
======================
The evaluation of B-FERL was performed in an emulated scenario using Common Open Research Emulator (CORE), running in a Linux Virtual Machine using six processor cores and 12 Gb of RAM. Based on the appendable blocks concept described in section \[sec:b-ferl\], B-FERL supports adding transactions of a specific *CAV* to a block. This block is used to identify the *CAV* in the lower tier and stores all of its records.\
The initial experiments aim to identify the project viability, and thus enable us to plan ahead for real world scenario experimentation. The evaluated scenario consists of multiple CAVs (varying from 10 to 200) exchanging information with a peer-to-peer network with five RSU in the lower tier. Initially, we evaluate the time it takes B-FERL to perform the system initialization. This refers to the time it takes the upper tier validators to create a record (block) for a *CAV*. Recall that in Figure \[fig:operation\], creating a record for a *CAV* is based on the successful verification of the genesis transaction sent from vehicle manufacturers. The results presented are the average of ten runs and we also show the standard deviation for each given scenario. In this first evaluation, we vary the amount of genesis transactions received by validators from 10 to 200 to identify how B-FERL responds to the increasing number of simultaneous transactions received. The results are presented in Figure \[fig:createBlock\]. Time increases in a linear progression as the number of *CAVs* increases. The time measured in milliseconds increases from 0.31 ms (standard deviation 0.12 ms) for 10 *CAVs*, to 0.49 ms (standard deviation 0.22 ms) for 200 *CAVs* which is still relatively low compared to the scenario with 10 *CAVs*.\
Once the blocks are created for the *CAVs*, the upper tier validators broadcast the blocks to the RSUs. In the next evaluation, we measure the time taken for an RSU to update its BC with the new block. The time required for this action is 0.06 ms for 200 *CAVs* which reflects the efficiency of B-FERL given the number of *CAVs*.
![Time taken to create a block[]{data-label="fig:createBlock"}](TimeToAddBlock.pdf){width="50.00000%"}
The next evaluation was the time that each RSU takes to evaluate the challenge response. This is an important measure in our proposed solution as it reflects the time taken by an RSU to verify the authenticity of a *CAV* and conduct the ECU integrity check. This process is described in steps 4 to 6 presented in Figure \[fig:lowupdate\]. Figure \[fig:validateChallenge\] presents the average time, which increases linearly from 1.37 ms (standard deviation 0.15 ms) for 10 *CAVs* to 2.02 ms (standard deviation 0.72 ms) for 200 *CAVs*. From the result, we can see that the actual values are small even for large group of *CAVs*.
![Time taken to validate a challenge from vehicles[]{data-label="fig:validateChallenge"}](TimeToValidateChallenge.pdf){width="50.00000%"}
![Time taken to calculate Merkle tree root[]{data-label="fig:merkleResult"}](TimeMerkleTree.pdf){width="50.00000%"}
In the next evaluation, we evaluate the time it takes a *CAV* to compute its merkle tree root defined as the cumulative sum of all its ECU hash. According to NXP, a semiconductor supplier for automotive industries [@NXP:2017], the number of ECUs range from 30 to 100 in a modern vehicle. In this evaluation, we assume that as vehicle functions become more automated, the number of ECUs is likely to increase. Therefore, in our experiments, we vary the number of ECUs from 10 to 1,000. Figure \[fig:merkleResult\] presents the time to compute the Merkle tree root. The results present a linear growth as the number of ECUS increases. In our result, even when the number of ECUs in a *CAV* are 1000, the time to compute the Merkle tree root is about 12 ms which is still an acceptable time for a highly complex scenario.
![Blockchain size[]{data-label="fig:blocksize"}](SizeBlockchain.pdf){width="50.00000%"}
In the final evaluation, we consider the amount of storage required by an RSU to store the BC for different number of *CAVs*. To get a realistic picture of required storage, we considered the number of vehicles in New South Wales (NSW), Australia in 2018. As presented in Figure \[fig:blocksize\], the number of blocks (which represents the number of vehicles) was changed from 100,000 representing a small city in NSW to 5,600,0000[^1]. Based on the results, an RSU must have around 5 Gb to store the BC structure in the state of New South Wales. This result show that it is feasible for an RSU to maintain the BC for all *CAVs* in NSW.
Discussion
==========
In this section, we provide a further discussion considering the security, Use cases as well a comparative evaluation of B-FERL against related work.
Security analysis
-----------------
In this section, we discuss how our proposal demonstrates resilience against attacks described in the attack model.\
**Fake data:** For this to occur, one or more data generating ECU of a *CAV* would have been compromised. We can detect this attack during the challenge-response activity between the compromised *CAV* and an RSU where the *CAV* is expected to prove the integrity of its ECU by computing its ECU Merkle tree root value.\
**Code injection:** Actions executed by service technicians and vehicle manufacturers are stored in the upper tier and could be traced back to them. Vehicle owners are not be able to alter their odometer value as such actions would make the $SS_{ID}$ value different from what is in its record in the lower tier.\
**Sybil attack:** The only entities capable of creating entities in the vehicular networks are the verifiers in the upper tier who are assumed to be trusted. A vehicle trying to create multiple entities must be able to create valid blocks for those entities which is infeasible in our approach.\
**Masquerade attack (fake vehicles):** A compromised RSU cannot create a block for a *CAV*. As such, this attack is unlikely to be undetected in B-FERL. Also, a *CAV* is considered valid only if its public key exists in the BC managed by the road transport and legal authorities.\
**ECU State Reversal Attack:** We address this attack using the random ECU integrity verification challenge. By randomly requesting the values of ECU in a *CAV*, RSUs could detect the reversal attack by comparing the timestamps ECUs against their entries in the lower tier BC. Having discussed our defense mechanism, it is noteworthy that while the utilization of a public key introduces a trade-off that compromises privacy and anonymity of a vehicle, the public key is only utilized by a RSU to identify a vehicle in the challenge-response transaction which ascertains the state of a vehicle and does not require the transmission of sensitive and privacy related information.
Use case
--------
In this section, we discuss the applicability of our proposed solution to the following use cases in the vehicular networks domain: (1) Vehicular forensics, (2) Trust management, and (3) Secure vehicular communication.\
***Vehicular forensics:*** In the liability attribution model proposed for *CAVs* in [@Oham:2018], liability in the event of an accident could be split amongst entities responsible for the day-to-day operation of the *CAVs* including the vehicle manufacturers, service technicians and vehicle owners. Also, the authors in [@Norton:2017] have identified conditions for the attribution of liability to the aforementioned entities. The consensus is to attribute liability to vehicle manufacturer and technicians for product defect and service failure respectively and to the vehicle owners for negligence. In our proposed work, we keep track of authorized actions of vehicle manufacturers and service technicians in the upper tier and so we are able to identify which entity executed the last action on the vehicle before the accident. Also, with the challenge-response between RSUs and *CAVs* in the lower tier, we are able to obtain historical proof that proves how honest or rogue a vehicle has been in the vehicular network. Consider the *CAV* in Figure 1, if before entering the coverage region of an RSU, an accident occurs, we could generate evidence before the occurrence of the accident in the lower tier that reflects the behavior of the *CAV* and such evidence could be utilized with the accident data captured by the vehicle for facilitating liability decisions.\
***Trust Management:*** Trust management in vehicular networks either assesses the veracity of data generated by a vehicle or the reputation of a vehicle \[19\]. This information is used to evaluate trust in the network. However, existing works on trust management for vehicular networks significantly relies on the presence of witness vehicles to make trust based decisions \[19-22\] and could therefore make wrong trust decisions if there are little or no witnesses available. Also, reliance on witnesses also facilitate tactical attacks like collusion and badmouthing. In our proposal, we rely solely on data generated by a *CAV* and we can confirm the veracity of data generated or communicated by the *CAV* by obtaining such evidence in the lower tier from the historical challenge-response activity between a *CAV* and RSUs as the *CAV* travels.\
***Secure vehicular communication networks:*** Given that the successful execution of a malicious action by a *CAV* reflects that at least one of the *CAV’s* ECUs has been compromised and as a result, undermines the security of the vehicular networks. We describe below how our proposal suffices as an apposite security solution for vehicular networks.\
**Identifying compromised *CAVs***: By proving the state of ECUs in *CAVs*, we can quickly identify cases of ECU tampering and quickly broadcast a notification of malicious presence in the vehicular network to prevent other *CAVs* from communicating with the compromised *CAV*.\
**Effective revocation mechanism:** Upon the identification of a malicious *CAV* during the challenge-response activity, Road transport authorities could also efficiently revoke the communication rights of such compromised *CAV* to prevent further compromise such as the propagation of false messages in the network by the compromised *CAV*.
Comparative evaluation
----------------------
In this section, we comparatively evaluate B-FERL against the works proposed in \[9-10\], \[15\], \[23\] using identified requirements for securing in-vehicle networks.\
**Adversaries**: Identified works are vulnerable to attacks executed by authorized entities (insider attacks) but in B-FERL, we address this challenge by capturing all interactions between all entities responsible for the operation of the *CAV* including the owner, manufacturer and service technician. By recording these actions in the upper tier (BC), we ensure that no entity can repudiate its actions. Furthermore, by proving the state of ECUs in a *CAV*, we are able to identify possible attacks.\
**Decentralization:** By storing vehicle related data as well as actions executed by manufacturers and service technicians in the BC, we ensure that no entity can alter or modify any of its actions. Also, by verifying the internal state of a *CAV* as it moves from one RSU to another, we preserve the security of the vehicular networks.\
**Privacy:** By restricting access to information to only authorized entities in B-FERL, we preserve the privacy of concerned entities in our proposed framework.\
**Safety:** By verifying the current state of a *CAV* against its record in the lower tier, we ensure communication occurs only between valid and honest *CAVs* which ultimately translates to secure communications in the vehicular network.
Conclusion
==========
In this paper, we have presented a Blockchain based Framework for sEcuring smaRt vehicLes (B-FERL). The purpose of B-FERL is to identify when an ECU of a smart vehicle have been compromised by querying the internal state of the vehicle and escalate identified compromise to requisite authorities such as the road transport and legal authority who takes necessary measure to prevent such compromised vehicles from causing harm to the vehicular network. Given this possibility, B-FERL doubles as a detection and reaction mechanism offering adequate security to vehicles and the vehicular network. Also, we demonstrated the practical applicability of B-FERL to critical applications in the vehicular networks domain including trust management, secure vehicular network and vehicular forensics where we discuss how B-FERL could offer non-repudiable and reliable evidence to facilitate liability attribution. Furthermore, by qualitatively evaluating the performance of B-FERL, we demonstrate how it addresses key challenges of earlier identified works. Security analysis also confirms B-FERL’s resilience to a broad range of attacks perpetuated by adversaries including those executed by supposedly benign internal entities. Simulation results reflect the practical applicability of B-FERL in realistic scenarios.\
Our current proposal provides security for smart vehicles by identifying when a vehicle becomes compromised and secures the vehicle against possible exploitations by internal adversaries. An interesting future direction would be to consider the privacy implication for a smart vehicle as it travels from one roadside unit to another.
[1]{}
C. Oham, R. Jurdak, and S. Jha, *Risk Analysis Study of Fully Autonomous Vehicle.* Also available on: https://arxiv.org/pdf/1905.10910.pdf RMS NSW, *Renew your registration.* Available on: https://www.rms.nsw.gov.au/roads/registration/renew/index.html C. Oham, R. Jurdak, Salil S. Kanhere, A. Dorri and S. Jha, *B-FICA: Blockchain based Framework for Auto-insurance Claim and Adjudication.* In proceedings of the IEEE 2018 International Conference on BlockChain, Halifax J. Chen, T. Li and J. Panneerselvam, *TMEC: A Trust Management Based on Evidence Combination on Attack-Resistant and Collaborative Internet of Vehicles.*” 2017.
L. Nkenyereye, J. Kwon and Yoon-Ho Choi, *Secure and Lightweight Cloud-Assisted Video Reporting Protocol over 5G-Enabled Vehicular Networks.* MDPI, Sensors, 2017. H. Ueda, R. Kurachi, H. Takada, T. Mizutani, M. Inoue and S. Horihata *Security Authentication System for In-Vehicle Network.* Automotive, 2015. Andy Greenberg, *Hackers Remotely Kill a Jeep on the Highway – With me in it.* Andy Greenberg Security. Available on: https://www.wired.com/2015/07/hackers-remotely-kill-jeep-highway/ 2015. S. Woo, H.J. Jo and D.H. Lee *A Practical Wireless Attack on the Connected Car and Security Protocol for In-Vehicle CAN* IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 16, NO. 2, APRIL, 2015 M. Salem, M. Mohammed, and A. Rodan, *Security Approach for In-Vehicle Networking Using Blockchain Technology* In: Barolli L., Xhafa F., Khan Z., Odhabi H. (eds) Advances in Internet, Data and Web Technologies. EIDWT 2019. Lecture Notes on Data Engineering and Communications Technologies, vol 29. Springer, Cham D. Nilsson and U. Larson, *Conducting forensic investigations of cyber attacks on automobile in-vehicle networks.* Proceedings of the 1st international conference on Forensic applications and techniques in telecommunications, information, and multimedia and workshop, 2008. CUBE, *Autonomous Car Network Security Platform based on Blockchain* 2018 S. Nakamoto, *Bitcoin: A peer-to-peer electronic cash system.* 2008. Michelin, Regio A. and Dorri, Ali and Steger, Marco and Lunardi, Roben C. and Kanhere, Salil S. and Jurdak, Raja and Zorzo, Avelino F. *SpeedyChain: A Framework for Decoupling Data from Blockchain for Smart Cities* Proceedings of the 15th EAI International Conference on Mobile and Ubiquitous Systems: Computing, Networking and Services, 2018.
Trusted Computing Group, *TCG TPM 2.0 Automotive Thin Profile.* 2016. N. Akosan, F. Brasser, A. Ibrahim, A. R. Sadeghi, M. Schunter, G. Tsudik, and C. Wachsmann, *SEDA: Scalable Embedded Device Attestation.* 2015. R. Heijden, F. Engelmann, D. Modinger, F. Schonig and F. Kargl, *Blackchain: Scalability for Resource-Constrained Accountable Vehicle-to-X Communication.* SERIAL’17: ScalablE and Resilient InfrAstructures for distributed Ledgers, December 11–15, 2017. A. Dorri, M. Steger, S. Kanhere and R. Jurdak, *BlockChain: A Distributed Solution to Automotive Security and Privacy.* IEEE Communication Magazine, 2017. Norton Rose Fulbright, *Autonomous vehicles: The legal landscape of dedicated short range communication in the US, UK and Germany.* July, 2017.
W. Li, and H. Song, *ART: An Attack-Resistant Trust Management Scheme for Securing Vehicular Ad hoc Networks.* IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 17, NO. 4, APRIL 2016. F, Dotzer, L. Fischer, and P. Magiera, *Vars: a vehicle ad-hoc network reputation system.* In: IEEE international symposium on a world of wireless mobile and multimedia networks, 2005 pp 454–456. Q. Ding, M. Jiang, X. Li and X. Zhou, *Reputation-based Trust Model in Vehicular Ad-hoc Networks.* IEEE, 2010. S. Park, B. Aslam and C. Zou, *Long-term Reputation System for Vehicular Networking based on Vehicle’s Daily Commute Routine.* In Proceeding of the 8th Annual IEEE Consumer Communications and Networking Conference – Security and Protection, 2011. H. Oguma, A. Yoshioka, and M. Nishikawa, *New Attestation-Based Security Architecture for In-vehicle Communication* In Proceedings of the IEEE “GLOBECOM”, 2008 A. Wasicek and A. Weimerskirch, *Recognizing Manipulated Electronic Control Units.* University of California, Berkeley and University of Michigan, 2014. M. Swawibe Ul Alam, *Securing Vehicle Electronic Control Unit (ECU) Communications and Stored Data* Msc. Thesis, Queens University, School of Computing, 2018 Anupam Chattopadhyay and Kwok-Yan Lam, *Autonomous Vehicle: Security by Design* School of Computer Science and Engineering, Nanyang Technological University, Singapore Kyusuk Han, André Weimerskirch, and Kang G. Shin, *Automotive Cybersecurity for In-Vehicle Communication* IQT Quarterly, Ford Motors’ OpenXC (http://openxcplatform.com/) Luc van Dijk, *Future Vehicle Networks and ECUs* Architecture and Technology considerations, NXP Semiconductors, Also available at: https://www.nxp.com/docs/en/white-paper/FVNECUA4WP.pdf Li L, Liu J, Cheng L, Qiu S, Wang W, et al. (2018) *CreditCoin: A Privacy-Preserving Blockchain-Based Incentive Announcement Network for Communications of Smart Vehicles.* IEEE Transactions on Intelligent Transportation Systems: 1–17. Zhaojun Lu, Qian Wang, Gang Qu , and Zhenglin Liu, [BARS: a Blockchain-based Anonymous Reputation System for Trust Management in VANETs.]{} July, 2018. Jiawen Kang, Zehui Xiong, Dusit Niyato, Dongdong Ye, Dong In Kim, Jun Zhao, *Towards Secure Blockchain-enabled Internet of Vehicles: Optimizing Consensus Management Using Reputation and Contract Theory* Sept, 2018.
[^1]: 5,600,0000 represents the number of cars in the state of New South Wales, according to www.abs.gov.au
|
---
abstract: |
We present new optical and infrared photometry for a statistically complete sample of seven 1.1 mm selected sources with accurate coordinates. We determine photometric redshifts for four of the seven sources of 4.64, 4.54, 1.49 and 0.18. Of the other three sources two are undetected at optical wavelengths down to the limits of very deep Subaru and Canada-France-Hawaii Telescope images ($\sim$27 mag AB, i band) and the remaining source is obscured by a bright nearby galaxy. The sources with the highest redshifts are at higher redshifts than all but one of the $\sim$200 sources taken from the largest recent 850 $\mu$m surveys, which may indicate that 1.1 mm surveys are more efficient at finding sources at very high redshifts than 850 $\mu$m surveys.
We investigate the evolution of the number density with redshift of our sample using a banded $V_{e}/V_{a}$ analysis and find no evidence for a redshift cutoff, although the number of sources is very small. We also perform the same analysis on a statistically complete sample of 38 galaxies selected at 850$\mu$m from the GOODS-N field and find evidence for a drop-off in the number density beyond $z\sim1$ and 2, confirming the earlier conclusion of Wall, Pope & Scott (2008). We also find evidence for the existence of two differently evolving sub-populations separated in luminosity, with the drop-off in density for the low-luminosity sources occurring at a lower redshift.
author:
- |
G. Raymond$^1$[^1], S. A. Eales$^1$, S. Dye$^1$, R. Carlberg$^2$ and M. Sullivan$^3$\
\
$^1$Cardiff University, School of Physics & Astronomy, Queens Buildings, The Parade, Cardiff, CF24 3AA, U.K.\
$^2$Department of Astronomy and Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada\
$^3$Department of Physics (Astrophysics), University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, U.K.
date: '27-07-09'
title: 'Is there a redshift cutoff for submillimetre galaxies?'
---
\[firstpage\]
galaxies:distances and redshifts, galaxies:evolution, galaxies:high-redshift, submillimetre, infrared:galaxies, galaxies:statistics
Introduction
============
Submillimetre (submm) galaxies (SMGs), first detected at 850 $\mu$m with the Submillimetre Common User Bolometer Array (SCUBA) [@holland:1999], are a significant population of high redshift star forming galaxies [e.g., @hughes:1998; @blain:2002]. They are believed to be dust enshrouded galaxies undergoing prodigious levels of star formation [e.g., @hughes:1998; @eales:1999] in which the optical/UV radiation emitted by the stars is absorbed by the dust and then re-emitted in the submm. Star formation rates in excess of 1000 M$_{\odot}$yr$^{-1}$ have been inferred [@scott:2002], much higher than locally. The galaxies in these samples have been found to account for up to one tenth of the total far-infrared/submm energetic background [e.g., @dye:2007] and many authors have argued that these galaxies are the progenitors for the elliptical galaxies we see in the local Universe [@eales:1999; @scott:2002; @dunne:2003]. Thus understanding the nature of these sources is of great importance for the understanding of galaxy formation and evolution as a whole.
Observations of SMGs at $\sim$1 mm benefit from a negative K-correction out to high redshifts due the shape of their spectral energy distribution (SED). As the redshift of an SMG increases, the peak of its rest-frame SED moves toward the observed waveband, offsetting the dimming caused by the increasing luminosity distance. This fact accounts for the surprising ability of SCUBA to find large numbers of high-redshift galaxies.
The large amount of dust responsible for the strong submm emission gives rise to high levels of attenuation in the optical. This in conjunction with the poor angular resolution of single dish submm facilities makes the cross identification of SMGs at different wavelengths difficult. Moreover, even when an optical counterpart can be identified, the high levels of dust attenuation makes the determination of a spectroscopic redshift difficult. As such we are currently unable to determine spectroscopic redshifts for the majority of SMGs. The strong correlation between dust emission and radio emission which appears to hold true in both the low-redshift and high-redshift universe [@vlahakis:2007] has been useful for both identifying the counterparts and estimating redshifts. Due to the low surface density of radio sources on the sky, the probability of the radio counterpart being coincidental with the submm source by chance is small. Due to the high positional accuracy of radio observations, it is then possible to identify the optical counterpart and retrieve a spectroscopic redshift. It is also possible to estimate the redshift using the ratio of radio to submm flux [e.g., @hughes:1998; @carilli:1999; @carilli:2000; @smail:2000].
@chapman:2005, using the Low Resolution Imaging Spectrograph (LRIS) [@oke:1995] on the Keck I telescope, managed to obtain spectroscopic redshifts for a total of 73 radio-identified SMGs with a median 850 $\mu$m flux of 5.3 mJy. The galaxies in this sample were found to lie at a median redshift of $z=2.2$ out to a maximum value of $z_{max}=3.6$. However, the K-correction which allows us to detect high-redshift SMGs does not similarly benefit their radio fluxes and so radio identified SMGs are subjected to a radio selection effect which limits redshifts to $z\simless3$.
@pope:2006, produced the first complete (i.e. not requiring radio IDs) sample of 850 $\mu$m selected SMGs that has close to 100% redshifts. The sample consists of 35 galaxies, 21 with secure optical counterparts and 12 with tentative optical counterparts, and its completeness means that unlike previous surveys it is not biased towards low-z sources. The median redshift determined for this sample is $z\sim 2.2$. Using this sample, @wall:2008 examined the epoch dependency of the number density of SMGs. They found an apparent redshift cutoff at $z>3$ with further evidence for two separately evolving populations, divided by luminosity. However this result was based on calculations using a single model galaxy SED. Since the predicted relationship between submm flux-density and redshift depends strongly on the assumed SED, one of the aims of this paper is to re-examine their conclusion using a range of empirical SEDs.
There have been a number of explanations for the lack of high-redshift SMGs. Given that dust is thought to form in the atmospheres of highly evolved stars, it is possible that at high redshifts simply not enough time has passed for dust to form [@morgan:2003]. Observations of high-redshift quasars have however detected high levels of dust [e.g., @priddey:2001], suggesting that this is not the explanation. Another possible explanation is that there are fewer large star-forming galaxies at high redshifts.
@eales:2003 presented evidence that SMGs typically have low values for the ratio of the 850 $\mu$m to 1200 $\mu$m fluxes compared to that expected from a low redshift galaxy. One possible explanation is that these sources are at very high redshifts. If this is true, then observations at 1.1 mm would be better at detecting SMGs at the highest redshifts than observations at 850 $\mu$m. A new complete sample of 1.1 mm selected SMGs located in the COSMOS field [@scoville:2007] has been compiled by @younger:2007. The sources were selected initially at 1.1 mm with the AzTEC camera [@scott:2008; @wilson:2008] on the JCMT. The resultant catalogue consists of 44 sources with S/N$>3.5\sigma$, 10 of which are robust with S/N$>5\sigma$. Follow up observations by @younger:2007 were then made with the Submillimetre Array (SMA) at 890 $\mu$m for the 7 highest significance AzTEC sources, allowing their positions to be determined with an accuracy of $\sim 0.2$”. The COSMOS field offers a wealth of data over a great number of wavebands including the optical and infrared. Thus the high positional accuracy allows for the identification of optical counterparts and hence the determination of photometric redshifts. Of the seven AzTEC sources imaged with the SMA six have IRAC counterparts, and one source is obscured by a nearby bright galaxy. Using deep Hubble Space Telescope (HST) imaging acquired with the Advanced Camera for Surveys (ACS), @koekemoer:2007 found optical counterpart candidates for only three of these sources.
The main aim of this paper is to carry out a deeper search for the optical counterparts for the AzTEC sources. We give photometry from deep Subaru and Canada-France-Hawaii-Telescope (CFHT) imaging and find one new possible optical ID. We estimate photometric redshifts for the AzTEC sources using the [<span style="font-variant:small-caps;">HyperZ</span>]{} photometric redshift package [@bolzonella:2000]. Throughout this work we employ a concordance cosmological model with $\Omega_{total}=1$, $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$ and $H_{0} = 75$ kms$^{-1}$Mpc$^{-1}$. All magnitudes quoted are AB.
NEW IMAGES AND PHOTOMETRY FOR THE AzTEC SAMPLE
==============================================
We searched for optical counterparts and measured new photometry using deep Subaru[^2], CFHT and IRAC images of the AzTEC sources. The IRAC and Subaru images are the publicly available COSMOS images taken by the COSMOS team [@scoville:2007]. The CFHT images are taken from the CFHT Deep Legacy Survey. The images we used were taken using the CFHT $g_{\mathrm{M}}$, $r_{\mathrm{M}}$, $i_{\mathrm{M}}$, $z_{\mathrm{M}}$, Subaru B$_{j}$, V$_{j}$, r+, i+, z+ and IRAC channel 1 and 2 filters to average 3$\sigma$ depths of approximately 28.4, 27.9, 27.6, 26.5, 29.0, 28.2, 28.3, 27.7, 26.4, 24.1 and 23.6 mags respectively.
We searched the i-band images (figure \[fig:images\]) at the SMA coordinates. We find bright i-band counterparts for AzTEC1, 3 and 7, all of which were previously known. We also find a faint i-band counterpart for AzTEC5 at the SMA coordinates. We find no objects directly at the SMA coordinates for AzTEC2, but there is a bright object offset from this position by 3”, meaning that the magnitude limits of this SMG are not useful.
We find no optical counterparts directly at the SMA coordinates for AzTEC4 and 6 in the Subaru and CFHT imaging. For the latter source, however, there is a bright i-band counterpart offset from the SMA position by $\sim$0.6” ($\sim$3$\sigma$) which could be AzTEC6’s counterpart or the true counterpart may be too faint to see. There is also a faint i-band source, offset from AzTEC4’s SMA position by $\sim$0.8” ($\sim$4$\sigma$), in the Subaru imaging. For these two sources we added the i band CFHT and Subaru images, inversely weighting the images by the square of the noise, in order to try and detect any very faint possible counterparts. The stacked i-band images for AzTEC4 and 6 are shown in figure \[fig:images\]. We still do not find counterparts at the SMA positions for AzTEC4 and 6 and given the good coincidence between the SMA and optical positions for the other AzTEC sources we tentatively conclude that the true counterparts have not yet been detected.
The typical full-width half-maximum (FWHM) of the optical and IRAC channel 1 and 2 point spread functions (PSFs) are $\sim$0.8”, 1.66” and 1.72” respectively. Magnitudes were determined manually by placing apertures onto the images, ensuring that the aperture was large enough to contain as much of the emission from the galaxy as possible without including any emission from neighboring objects. Thus the sizes of the optical apertures vary from source to source, although are constant for a given source. We used larger apertures for the IRAC sources due to the images having a larger PSF, but use the procedure outlined below to correct for this.
Due to the difference in the PSF between the optical and IRAC images as well as the difference in the aperture sizes, a small correction needed to be applied to the IRAC magnitudes before they could be used in conjunction with optical magnitudes to determine a photometric redshift. We corrected IRAC magnitudes by firstly fitting a 2D Gaussian to the IRAC source. We then scaled it to have the FWHM it would have had if observed with CFHT/Subaru. The flux was then computed using the scaled Gaussian and new aperture size. All the corrections applied in this work increase the IRAC magnitudes, and the more extended the source the greater the correction. Corrections range from 0.01 to 0.46 magnitudes.
The new photometry is summarized in table \[table:magnitudes\], and the details of the individual objects are discussed below.
\[fig:images\]
Notes on Individual Objects
---------------------------
**[AzTEC1]{}- *[J095942.86+022938.2]{}- AzTEC1 is the brightest submm source in the sample with fluxes of $F_{890\mu m} =
15.6\pm1.1$ mJy and $F_{1.1mm} = 10.7\pm1.3$ mJy. There is a bright i-band object located directly at the SMA position. Optical fluxes are measured using an aperture 1.94” in diameter and the source is detected in the Subaru i+ band at 25.11$\pm$0.03 mag which is in agreement with the HST i-band magnitude given in @younger:2007. There is some disagreement within the same wavebands between the CFHT and Subaru and photometry (table \[table:magnitudes\]), but the discrepancy is small enough that the photometric redshift is not significantly affected. IRAC magnitudes were measured using an aperture of diameter 4.45”. Only a small correction was applied to the IRAC magnitudes: +0.01 mag in both IRAC channel 1 and 2. **[AzTEC2]{}- *[J100008.05+022612.2]{}- AzTEC2 is detected in the submm with fluxes $F_{890\mu m} = (12.4\pm1.0)$ mJy and $F_{1.1mm} = (9.0\pm1.3)$ mJy. No objects are found directly at the SMA coordinates, but there is a bright object offset from SMA position by 3”. Thus the limit on the magnitude of the optical counterpart is not very useful. **[AzTEC3]{}- *[J100020.70+023520.5]{}- AzTEC3 is detected in the submm with fluxes $F_{890\mu m} = 8.7\pm1.5$ mJy and $F_{1.1mm} = 7.6\pm1.2$ mJy. There is a bright i-band object located at the SMA coordinates as well as three companion objects offset by between 1” and 2”. Since SMGs often seem to consist of multiple components [@ivison:1998] it is possible that these companion objects are also part of AzTEC3. However, since six of the seven AzTEC sources are detected in the IRAC bands, it seems likely that if the companion objects are part of the same galaxy then they should also be contributing to the IRAC emission. We attempted to determine whether this is the case by convolving the Subaru image with the IRAC beam and comparing the FWHM of the IRAC source with that of the convolved Subaru image. We find that the FWHM of the convolved image is $\sim$4.4”. The FWHM of the IRAC 3.6 $\mu$m image is $\sim$2.87”, suggesting that the 3.6 $\mu$m emission is associated only with the central object. Optical fluxes were measured using a aperture of diameter 1.26” and the source is detected in the Subaru i+ band at 26.18$\pm$0.08 mag which is in agreement with the HST i-band magnitude. The CFHT and Subaru magnitudes within the same bands are consistent with each other. IRAC magnitudes were measured using an aperture of diameter 4.80”. A correction of +0.2 mag was applied to the IRAC magnitudes in channels 1 and 2. **[AzTEC4]{}- *[J095931.72+023044.0]{}- AzTEC4 is detected in the submm with fluxes $F_{890\mu m} = 14.4\pm1.9$ mJy and $F_{1.1mm} = 6.8\pm1.3$ mJy. We find a tentative i-band counterpart, offset from the SMA position by 0.8” ($\sim$3$\sigma$), in the Subaru image with a magnitude of 27.43$\pm$0.13. In the combined image (see above), the counterpart can be seen more clearly and has a magnitude of 26.99$\pm$0.18 in a 2.57” diameter aperture. However we find it is too faint to detect in our other Subaru and CFHT bands. IRAC magnitudes were measured using an aperture of diameter 4.80”. Corrections of +0.11 and +0.04 mags were applied to the IRAC magnitudes in channels 1 and 2 respectively. Because of the good agreement between the SMA the optical positions for the other AzTEC sources we tentatively conclude that this is not the true counterpart. **[AzTEC5]{}- *[J100019.75+023204.4]{}- AzTEC5 is detected in the submm with fluxes $F_{890\mu m} = 9.3\pm1.3$ mJy and $F_{1.1mm} = 7.6\pm1.3$ mJy. @younger:2007 found no optical counterpart in ACS imaging, but we find a faint Subaru source at the SMA coordinates with a Subaru i+ band magnitude of 26.74$\pm$0.13, measured in an aperture of diameter 1.68”. The CFHT and Subaru magnitudes within the same bands are consistent with each other. IRAC magnitudes were measured using a aperture of diameter 4.80”. Corrections of +0.46 and +0.14 mag were applied to the IRAC magnitudes in channels 1 and 2 respectively. . **[AzTEC6]{}- *[J100006.50+023837.7]{}- AzTEC6 is detected in the submm with fluxes $F_{890\mu m} = 8.6\pm1.3$ mJy and $F_{1.1mm} = 7.9\pm1.2$ mJy. @younger:2007 find no optical counterpart in ACS imaging. In CFHT and Subaru imaging we find no source directly at the SMA coordinates, but we do find a source offset from the SMA position by $\sim$0.6” ($\sim$3$\sigma$). This could therefore be the optical counterpart, or the true counterpart may be too faint to detect. The source offset from the SMA position has a Subaru i+ magnitude of 25.38$\pm$0.04 magnitudes, measured in an aperture of diameter 1.59”. The CFHT and Subaru magnitudes within the same bands are consistent with each other. IRAC magnitudes were measured using an aperture of diameter 5.88”. A correction of +0.13 mag is applied to the IRAC magnitudes in channels 1 and 2. Because of the good agreement between the SMA the optical positions for the other AzTEC sources we tentatively conclude that this is not the true counterpart, although we do estimate a photometric redshift for it. **[AzTEC7]{}- *[J100018.06+024830.5]{}- AzTEC7 is detected in the submm with fluxes $F_{890\mu m} = 12.0\pm1.5$ mJy and $F_{1.1mm} = 8.3\pm1.4$ mJy. We find an optical counterpart with a disturbed morphology at the SMA coordinates which could be a system of merging galaxies. Optical fluxes were measured by placing an aperture of diameter 2.87” over the whole of the system. The source is detected in the Subaru i+ band at 24.20$\pm$0.04 mag. IRAC magnitudes were measured using an aperture of diameter 6.12”. Corrections of +0.08 and +0.02 mag were applied to the IRAC magnitudes in channels 1 and 2 respectively. .*********************
AzTEC1 AzTEC2 AzTEC3 AzTEC4 AzTEC5 AzTEC6 AzTEC 7
------------------ ---------------- ------------- ---------------- ------------------ ---------------- -------------------------- ----------------
RA 09:59:42.86 10:00:08.05 10:00:20.70 09:59:31.72 10:00:19.75 10:00:06.50 10:00:18.06
Dec +02:29:38.2 +02:26:12.2 +02:35:20.5 +02:30:44.0 +02:32:04.4 +02:38:37.7 +02:48:30.5
Optical Ap. Size 1.94” ... 1.26” 2.57” 1.68” ...(1.59”) 2.87”
$m_{B}$ $>$28.88 ... $>$29.14 $>$28.67 28.80$\pm$0.47 $>$28.98(25.80$\pm$0.30) 25.67$\pm$0.31
$m_{V}$ 27.13$\pm$0.17 ... 28.77$\pm$0.75 $>$28.21 28.76$\pm$0.62 $>$28.32(25.67$\pm$0.04) 25.10$\pm$0.06
$m_{r+}$ 26.21$\pm$0.06 ... 27.39$\pm$0.22 $>$28.02 27.07$\pm$0.13 $>$28.47(25.77$\pm$0.03) 24.97$\pm$0.05
$m_{i+}$ 25.11$\pm$0.03 ... 26.18$\pm$0.08 (27.43$\pm$0.13) 26.74$\pm$0.13 $>$27.97(25.38$\pm$0.04) 24.20$\pm$0.04
$m_{z+}$ 25.02$\pm$0.02 ... 25.58$\pm$0.15 $>$26.50 26.07$\pm$0.22 $>$26.79(24.80$\pm$0.06) 23.65$\pm$0.07
$g_{\mathrm{M}}$ $>$28.12 ... $>$28.71 $>$27.93 $>$28.41 $>$28.42(26.16$\pm$0.05) N/A
$r_{\mathrm{M}}$ 26.54$\pm$0.15 ... 27.13$\pm$0.24 $>$27.46 27.15$\pm$0.20 $>$27.96(25.61$\pm$0.05) N/A
$i_{\mathrm{M}}$ 25.25$\pm$0.05 ... 26.30$\pm$0.12 $>$26.19 26.50$\pm$0.13 $>$27.81(25.42$\pm$0.05) N/A
$z_{\mathrm{M}}$ 25.11$\pm$0.13 ... 25.69$\pm$0.22 $>$26.27 26.46$\pm$0.38 $>$26.06(24.87$\pm$0.08) N/A
IRAC Ap. Size 4.45” ... 4.80” 4.80” 4.80” 5.88” 6.12”
$m_{3.6\mu m}$ 23.40$\pm$0.07 ... 23.72$\pm$0.11 22.11$\pm$0.04 23.24$\pm$0.08 24.13$\pm$0.25 20.63$\pm$0.01
$m_{4.5\mu m}$ 23.08$\pm$0.08 ... 22.98$\pm$0.12 22.15$\pm$0.04 22.31$\pm$0.06 23.50$\pm$0.27 20.15$\pm$0.02
ESTIMATED REDSHIFTS FOR THE AzTEC SAMPLE
========================================
Photometric redshifts were determined by applying the photometric redshift package, [<span style="font-variant:small-caps;">HyperZ</span>]{} [@bolzonella:2000], to our 11 band photometry (Subaru:B, V, r+, i+, z+; CFHT: $g_{\mathrm{M}}$, $r_{\mathrm{M}}$, $i_{\mathrm{M}}$, $z_{\mathrm{M}}$; IRAC: 3.6 $\mu$m, 4.5 $\mu$m). The spectra used for fitting in this work are taken from the set compiled by @dye:2008, which is optimized for the determination of photometric redshifts when including filters in the near/mid-infrared. @dye:2008 compared the photometric redshifts determined using these spectral templates with those determined using synthetic spectra constructed from the best-fit star formation history for their sample of 60 SCUBA sources. Since these methods are completely independent and the redshifts found using both sets of templates were found to be in good agreement, we assume that our template set is adequate.
We varied the redshift in the range $z = 0$ to 10. We employed the reddening regime of @calzetti:2000, with $A_{V}$ allowed to vary in the range $A_{V}= 0$ to 5 in steps of 0.2. We used a minimum photometric error of 0.05 magnitudes for each band. For wavebands in which we have no detection we took the flux of the source to be zero with a $1\sigma$ error equal to the sensitivity of the detector in that waveband. The photometric redshifts obtained are listed in table \[table:redshift\].
ID z $\chi^{2}_{min} $ Notes
-------- ----------------- ------------------- -----------------------------------------------------------------------
AzTEC1 $4.64\pm0.06$ 1.537 ...
AzTEC2 ... ... No optical counterpart.
AzTEC3 $4.54\pm0.10$ 2.196 There is a secondary chi-squared minimum at the lower
redshift of $z\sim0.4$ with a chi-squared fit value of $\sim$3.5.
AzTEC4 ... ... Nearest counterpart only detected in one optical band.
AzTEC5 $1.49\pm0.10$ 1.488 There is a secondary chi-squared minimum at the higher
redshift of $z\sim4$ with a chi-squared fit value of $\sim$4.
AzTEC6 ($2.09\pm0.01$) (6.172) The redshift and chi-squared values are for the optical source
offset from AzTEC6’s SMA position. The chi-squared fit to
this source is much poorer compared to the others in the sample.
This may further imply that the nearby optical counterpart we have
selected is not the true counterpart to AzTEC6 and that the IRAC
emission is unassociated with the optical emission.
AzTEC7 $0.18\pm0.01$ 7.021 CFHT data not available. There are several other possible redshifts
with chi-squared fit values of $\sim10$ up to $z\sim2$. Even the best
chi-squared fit is still relatively poor however, which may be due
to the unusual nature of the source.
\[fig:zphot\]
The median redshift of the sample is 2.7 which is somewhat higher than the median redshift, 2.2, of the sample presented by @chapman:2005. The maximum redshift found is 4.64 and the minimum redshift found is 0.18. Comparing the redshift distribution of this sample to that of the samples presented in @chapman:2005, @pope:2006, @dye:2008 and @clements:2008, we note that only one of the sources in this combined sample of $\sim$200 850 $\mu$m selected sources is at a comparably high redshift as our two highest redshift sources, although this difference is not significant when analyzed with a Kolmogorov-Smirnov test. However two of the other AzTEC sources are undetected to very faint limits in the i-band, and these facts may indicate that 1.1 mm surveys find more sources at very high redshifts than 850 $\mu$m surveys.
A BANDED $V_{e}$/$V_{a}$ ANALYSIS
=================================
@wall:2008 examined a sample of 38 SMGs in the GOODS-N field and found evidence for a diminution in the space density of SMGs at redshifts $z>3$. They also found evidence for two separately evolving sub-populations separated by luminosity. In this paper we present the results of our re-examination of this result using a banded $V_{e}/V_{a}$ analysis and a range of empirical SEDs rather than the theoretical SED used by Wall et al.
The most well known method of investigating the evolution of the space density of galaxies with redshift is the $\langle V/V_{max}\rangle$ test [@schmidt:1968; @rowan-robinson:1968]. $V$ is the co-moving volume enclosed by the galaxy (that volume which the field of view traces out in moving from a redshift of $z=0$ out to the galaxy) and $V_{max}$ is the volume that would be enclosed by the galaxy were it pushed to the redshift at which its flux drops to the survey limit. This method encounters problems when a survey encloses two galaxy populations, one undergoing positive evolution, and the other negative. If we have a uniform distribution of galaxies in space, then we expect the value of $\langle V/V_{max}\rangle$ to be 0.5$\pm(12N)^{-0.5}$, where $N$ is the number of sources in the sample. A value of $\langle V/V_{max}\rangle > 0.5$ then implies a concentration of sources toward the more distant regions of their accessible volume and a value of $\langle V/V_{max}\rangle < 0.5$ implies a deficit of sources at higher redshifts. Therefore if we have in our sample separate populations undergoing high levels of positive and negative evolution, then $\langle V/V_{max}\rangle$ may still be close to 0.5, incorrectly implying zero evolution.
This problem can be solved by implementing instead a $\langle
V_{e}/V_{a} \rangle$ test [@dunlop:peacock]. This is effectively a banded version of the $\langle V/V_{max}\rangle$ test. $V_{e}$, the effective volume, is the volume enclosed between a minimum redshift $z_{low}$ and the redshift of the galaxy. $V_{a}$, the accessible volume, is the volume enclosed between $z_{low}$ and the redshift at which the galaxy’s flux drops below the sensitivity of the survey. By investigating the variation of $\langle V_{e}/V_{a}\rangle$ with $z_{low}$ we can distinguish between a positively evolving and a negatively evolving population.
We investigated the evolution of the space density of the sample with redshift through the implementation of a $\langle V_{e}/V_{a} \rangle$ test. Wall et al. based the k-correction necessary to calculate accessible volume on a single theoretical SED, whereas real galaxies have a range of SEDs. To investigate this, we carried out the $\langle V_{e}/V_{a} \rangle$ analysis using two different assumptions about SEDs. We used the two extreme two-component dust models of @dunne:eales, who provided fits to the hottest and coldest local SMGs. The cold SED, based on NGC 958, contains dust at temperatures of 20 and 44 K with a cold-to-hot dust mass ratio of 186:1. The hot SED, based on IR1525+36, contains dust at temperatures of 19 and 45 K with a cold-to-hot dust mass ratio of 15:1. Figure \[fig:f\_v\_z\], which shows the predicted flux versus redshift plot for the different models, shows the effect of using different SED templates on the flux-redshift relation. The two SED types are normalized such that they produce a flux of 1 mJy at a redshift of $z=1$.
We took the limiting flux of each source in the GOODS sample to be $3.5\sigma$ and measured $\langle V_{e}/V_{a}\rangle$ for $z_{low}=0$ to 4 in steps of 0.1. We also separated sources into two samples of equal size according to luminosity. In doing this we are able to determine whether there are differences in the evolution of the two sub-populations.
Our results for the 38 SMGs of Wall, Pope & Scott are shown in figure \[fig:goodsveva\]. We find good evidence for the existence of a redshift cutoff at $z>1$ for the hot SED, and slightly weaker evidence for a redshift cutoff at $z>2$ for the cold SED. Dividing the sample into separate populations of high and low luminosity sources shows differences in the evolution of the two populations. The low luminosity sources show much sharper redshift cutoffs whereas the evidence for redshift cutoffs in the high luminosity sources is far more marginal. Thus we find evidence to support the conclusions given in [@wall:2008]: there is a redshift cutoff for the sample and that there is evidence for two separately evolving sub-populations
An additional uncertainty about this results is that @pope:2006 claim that only 60% of their identifications are reliable. Therefore we also performed the $\langle V_{e}/V_{a}\rangle$ analysis only on sources with reliable identifications, the results of which are shown in figure \[fig:goodsvevarobust\]. Using these sources only, we still find good evidence for a redshift cutoff at $z>1$ for the hot SED, and some marginal evidence for a cutoff at $z>2$ for the cold SED. However, we are unable to find any clear evidence for two separately evolving galaxy sub-populations, separated in luminosity, as the sample size is too small.
However, by only taking into account the reliable identifications, we are probably biased towards optically brighter galaxies and therefore lower redshifts. We further investigated the effect of the unreliable identifications by putting four (roughly half) of the unreliable identifications at $z = 4$ and repeating the analysis (figure \[fig:goodsveva\_pushed\]). Doing this, we find that for hot SEDs our results are largely unaffected, with a relatively clear cutoff at redshifts higher than $z = 1$. However for the cold SEDs we find that our results are strongly affected, with no clear redshift cutoff up to a redshift of z$\sim$3.
We also performed a banded $\langle V_{e}/V_{a}\rangle$ analysis on our sample of AzTEC sources (excluding the AzTEC6 counterpart) the results of which are shown in figure \[fig:aztecveva\], but our sample is too small to find any clear evidence of a redshift cutoff.
CONCLUSIONS
===========
We give new Subaru, CFHT and IRAC photometry for a number of sources in the AzTEC / COSMOS survey with accurate coordinates from SMA imaging. We have estimated photometric redshifts for four of the seven galaxies in the sample. We find a median redshift of $z_{mean}\sim2.57$ and a maximum of $z_{max}=4.50$. Of the sources in the combined 850 $\mu$m surveys presented in @chapman:2005, @pope:2006, @dye:2008 and @clements:2008, consisting of $\sim$200 sources, only one is at a redshift greater than our two highest redshift sources. This in addition to the fact that we are unable to detect two of our sources in the optical bands down to very faint magnitudes may indicate that 1.1 mm surveys are more efficient at detecting very high-redshift sources than 850 $\mu$m surveys.
Re-investigating the space density evolution of a sample of 38 GOODS-N sources [@pope:2006; @wall:2008] with more realistic SEDs we find a redshift cutoff at $z\sim1$ if we assume a ’hot’ SED and marginal evidence for a cutoff at $z\sim2$ if we assume a ’cold’ SED (in reasonable agreement with Wall et al.). Similar to @wall:2008 we also found evidence for two differently evolving sub-populations of SMGs, separated in luminosity, with high luminosity sources showing a less negative evolution.
We performed a similar test on the AzTEC sources but were unable to draw any reliable conclusions as the sample is too small. The GOODS-N sample is also relatively small, and therefore any evidence for redshift cutoffs and differently evolving sub-populations must be treated with caution. In order to harden our conclusions in general we require larger surveys with accurate redshifts. We would also need surveys taken over larger areas of sky in order to take into account the effects of cosmic variance. Future, larger surveys (e.g. with Herschel, SCUBA2) therefore will enable us to more robustly determine the nature of the number density evolution of SMGs in the Universe.
[**Acknowledgements**]{}
G. Raymond, S. Eales and S. Dye acknowledge support from the Science and Technologies Facilities Council.
Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS.
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\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: An additional uncertainty of 0.3 mags in the Subaru B$_{j}$ band magnitudes is taken into account in this photometry due to the possibility of a red leak or a shift in the blue cutoff of this filter.
|
---
author:
- 'Alfredo T. Suzuki, Alexandre G. M. Schmidt'
title: |
Loop integrals in three outstanding gauges:\
Feynman, Light-cone and Coulomb
---
Introduction
============
Perturbative approach for Quantum Field Theory in any gauge deals with Feynman diagrams, which are expressed as $D$-dimensional integrals. The success of such approach can be understood from the comparison between the $a=\half (g-2)$ measure for the electron, && a\_[The]{} = 1159652201.2(2.1)(27.1) 10\^[-12]{}\
&& a\_[Exp]{} = 1159652188.4(4.3) 10\^[-12]{} ,see for instance [@laporta].
This is the best motivation for studying Quantum Field Theory, no physical theory can give such accuracy in any measurement. In other words, it is the very best we have.
In section 2 we discuss some 2-loop 4-point functions, namely, on-shell double boxes with 5 and 6 massless propagators; section 3 is devoted to non-covariant gauges: the light-cone and Coulomb ones. The integrals we study for the former have 7 propagators (2-loops) and the latter is 1-loop and have 4, however it is also complicated since we have to use split dimensional regularization(SDR). In the final section, 4, we present our concluding remarks.
Feynman gauge: scalar two-loop four-point massless integrals
============================================================
Of course, covariant gauges are the most popular, in what we could call “gauge market"[@leib-rmp]. Several methods were and are still developed to evaluate complicated Feynman loop integrals, being them concerned with analytic or numerical results[@laporta; @chetyrkin; @halliday], all in the context of dimensional regularization[@bolini].
Our work in concerned with the application of negative-dimensional integration method (). It is a technique which can be applied to any gauge, covariant or non-covariant alike. The results are always expressed as hypergeometric series which have definite regions of convergence allowing one to study the referred diagrams or process in specific kinematical regions of external momenta and/or masses.
On the other side, has a drawback: the amazing number of series – in the case where one is considering massless diagrams – which must be summed. When such sums are of gaussian type, it is quite easy to write a small computer program that can do the job algebraically. However, when the series are of superior order, $_{p+1} F_p$, for $p\geq 2$, there are no known formulas which can reduce it to a product of gamma functions for any value of its parameters. Despite this technical problem, proved to be an excellent method[@probing; @box; @without; @coulomb].
A question which is often arised is: what is more difficult to handle, graphs with more loops or graphs with more legs? In our point of view, i.e., in the context of , the greater the number of loops the heftier the calculations will be needed to solve it. We will consider in this section a diagram which has both (great number of legs and loops, four and two respectively), a scalar two-loop double-box integral where all the particles are massless and the external legs are on-shell.
Double box with 5 and 6 propagators
-----------------------------------
Let us consider the diagram of figure 1. Consider as the generating functional for our negative-dimensional integral the gaussian one, where all external legs are on-shell, G\_b &=& d\^D q d\^D r\
&& . -(q-r)\^2 \],\
&=& ()\^[D/2]{} , where $(s,t)$ are the usual Mandelstam variables and we use $s+t+u=0$. Observe that in the particular case where $\alpha=0$ we recover the gaussian integral for the diagram of figure 2. We also define $\Lambda = \alpha\theta + \alpha\phi + \alpha\omega
+\beta\theta+\beta\phi+ \beta\omega +\gamma\theta + \gamma\phi
+\gamma\omega+ \phi\omega +\theta\phi$.
\[double-box 6 prop\]
(600,200)(0,150) (20,200)[(1,0)[300]{}]{} (20,190)[(1,0)[30]{}]{}(20,300)[(1,0)[300]{}]{} (20,310)[(1,0)[30]{}]{} (50,200)[(0,1)[100]{}]{} (200,199)(150,299) (200,199)(240,299) (290,190)[(1,0)[30]{}]{} (290,310)[(1,0)[30]{}]{}
[ (110,190)[(0,0)\[b\][$q-p-p'$]{}]{} (310,320)[(0,0)\[b\][$p_1$]{}]{} (310,180)[(0,0)\[b\][$p_1'$]{}]{} (10,320)[(0,0)\[b\][$p$]{}]{} (10,180)[(0,0)\[b\][$p'$]{}]{} (100,310)[(0,0)\[b\][$q$]{}]{} (160,250)[(0,0)\[b\][$r$]{}]{} (70,245)[(0,0)\[b\][$q-p$]{}]{} (190,310)[(0,0)\[b\][$q-r$]{}]{} (255,250)[(0,0)\[b\][$q-r-p_1$]{}]{}]{}
The usual technique reveals that there are thirteen sums and seven equations. From the combinatorics one can solve such constraints in 1716 different ways. Of course several systems have no solution – not even in the homogeneous case – and from our previous works we know that some results are $n$-fold degenerated and others are related by . The result for the integral in question
&=& d\^D q d\^D r (q\^2)\^i (q-p)\^[2j]{} (q-p-p’)\^[2k]{} (q-r-p\_1)\^[2l]{} (r\^2)\^m (q-r)\^[2n]{},\
&=& (-)\^D i!j!k!l!m!n!(1-\_b-D/2) \_[[all]{}=0]{}\^ ,where “all” means $\{X_1,X_2,Y_1,Y_2,Y_3, Y_4,Y_5,Y_6,Y_7,Y_8, Y_9,Z_1,Z_2\}$ and $${\cal BOX} = {\cal BOX}(i,j,k,l,m,n),$$ must be understood and ${\bf delta}$ represents the system of constraints. The above expression can be expressed, in principle, as a seven-fold hypergeometric series, there are three possibilities, (...|z,z\^[-1]{},1), (...|z,1), (...|z\^[-1]{},1), where $z=-s/t$. Some series with unit argument, if they were gaussian can be summed up. However, a hypergeometric function is meaningful only if the series which defines it was convergent. Since the first possibility cannot be convergent we disregard it.
Among the 624 total solutions of the system of constraints we look for the simplest solution, namely, the one in which we can sum the great number of series. It is not difficult to find it using computer facilities, \[6prop\] [BOX]{}\^[AC]{} (i,j,k,l,m,n) = f\_1(i,j,k,l,m,n) \_3F\_2 ({1}|z) ,where the five parameters are quoted in the table, $\s_b=
i+j+k+l+m+n+D$ and f\_1(i,j,k,l,m,n) &=& \^D (t\^2)\^[\_b]{} (-j|\_b) (-l|\_b) (\_b+D/2|-2\_b-D/2)\
&& (-m|l+m+n+D/2) (i+j+k+m+D|-m-D/2)\
&& (l+m+n+D|-l-n-D/2), besides this one, we can have Appel’s, Lauricella’s and even more complicated hypergeometric functions. Moreover, $$(x|y) \equiv (x)_y = \frac{\G(x+y)}{\G(x)}.$$ If we remember that the final result should be the sum of linearly independent series[@box; @oleari], we can rightfully ask if one is not missing two other $_3F_2$ functions. According to Luke and Slater[@luke], the differential equation for $_pF_q$ has $p$ linearly independent solutions, so we should write a sum of three terms. On the other hand, according to No$\!\!\! /$rlund[@norlund], if the difference between an upper parameter and a lower one was an integer number, then some series do no exist — we used this theorem in [@box]. So, eq.(\[6prop\]) is the final result for the referred integral in the region where $|z|<1$. The expression for the same graph outside this region can be obtained making the substitutions,
s t, jk, ln, \[subst\] so we have other $_3F_2$ as the result for $|z|>1$.
[@lcr]{} Parameters & $\;_3F_2(\{1\}|z) $ & $\;_3F_2(\{2\}|z) $ a& $-k$ & $-k$b& $-n$ & $-n$c& $-\s_b$ & $-\s_b'$e& $1+j-\s_b$ & $1+j-\s_b'$f& $1+l-\s_b$ & $1+l-\s_b'$
Another solution for the Feynman integral can be written as a triple hypergeometric series, \_3 &=& \^D t\^j s\^[\_b-j]{} f\_3\_[Y\_i=0]{}\^\
&&,\
&& + (jl), where f\_3 &=& (-l|j) (l+m+n+D|i) (\_b+D/2|-j-n-D/2) (-n|l+2n+D/2)\
&& (i+j+k+m+D|-m-D/2)(-k|j+k-\_b)(-m|2m+D/2), obviously the above series converges if $|z|<1$, besides other possible condition on $z$. So there is an overlapping between the regions of convergence of ${\cal BOX}$ and ${\cal BOX}_3$, so there exists an formula which relates both. As fas as we know textbooks do not show formulas relating triple hypergeometric series with simple ones.
We have also 4-fold series, \_4 &=& \^D t\^l s\^[\_b-l]{} f\_4 \_[Y\_i=0]{}\^\
&&\
&& + (jl), where f\_4 &=& (-j|l) (l+m+n+D|i+j-l) (\_b+D/2|-k-l-D/2)(-m|2m+D/2)\
&&(-k|k+l-\_b) (-n|l+2n+D/2). Observe that the two previous results are singular when $j-l={\rm integer}$, since we have $\G(j-l)$ or $\G(l-j)$ in the numerator. However, such singularity cancels if one consider propagators exponents in the analytic regularization context, i.e., introduce[@box; @davyd] for instance $j=-1+\delta$, then expand the whole expression around $\delta=0$. Proceeding in this way the pole in $\delta$ cancels.
We have above reduction formulas which transform a hypergeometric function defined by triple and 4-fold series in a simpler function defined by a unique sum. These formulas are not in the textbooks on the subject. It is an original result.
### Double box with 5 propagators
The graph of figure 2, is a special case of the previous one. In the gaussian integral $\alpha$ must be zero, so in the final result we must merely take $i=0$,
\[double-box 5 prop\]
(600,200)(0,150) (20,200)[(1,0)[300]{}]{} (20,190)[(1,0)[30]{}]{} (20,300)[(1,0)[300]{}]{} (20,310)[(1,0)[30]{}]{} (200,199)(150,299) (200,199)(240,299) (150,299)(100,199) (290,190)[(1,0)[30]{}]{} (290,310)[(1,0)[30]{}]{}
[(140,190)[(0,0)\[b\][$q-p-p'$]{}]{} (310,320)[(0,0)\[b\][$p_1$]{}]{} (310,180)[(0,0)\[b\][$p_1'$]{}]{} (10,320)[(0,0)\[b\][$p$]{}]{} (10,180)[(0,0)\[b\][$p'$]{}]{} (160,250)[(0,0)\[b\][$r$]{}]{}(100,245)[(0,0)\[b\][$q-p$]{}]{} (190,310)[(0,0)\[b\][$q-r$]{}]{} (255,250)[(0,0)\[b\][$q-k-r$]{}]{}]{}
\[5prop\] [BOX]{}\^[AC]{} (0,j,k,l,m,n) = f(0,j,k,l,m,n) \_3F\_2 ({2}|z) ,where the parameters are listed in the table and we define $\s_b'= j+k+l+m+n+D$ and f(0,j,k,l,m,n) &=& \^D (t\^2)\^[\_b’]{} (-j|\_b’) (-l|\_b’) (\_b’+D/2|-2\_b’-D/2)\
&&(-m|l+m+n+D/2) (j+k+m+D|-m-D/2)\
&&(l+m+n+D|-l-n-D/2), we can proceed with the same substitutions (\[subst\]) to obtain the result outside the region $|z|<1$.
Finally, when all the exponents are equal to minus one, the $_3F_2$ colapses to a $_2F_1$ which can be written as an elementary function.
The results ${\cal BOX}_3$ and ${\cal BOX}_4$, in the same special case $(i=0)$, are for the integral in question. Since they are different and depend on the same variable we must sum them in order to get a triple series representation for ${\cal BOX}_3(0,j,k,l,m,n)$.
Non-covariant gauges: Light-cone and Coulomb
============================================
Recently there have been many works on non-covariant gauges, namely, light-cone[@lc; @lc2], Coulomb[@coul] and radial and axial gauges[@radial]. Despite they are not so popular as covariant ones, they have some important features which can help our study on certain physical problems.
Light-cone gauge, as far as we know, is the only one where certain supersymmetric theories can be shown to be UV finite and possess a local Nicolai map[@leib-cjp]. Moreover, ghosts decouple from physical particles and one is left with a reduced number of diagrams. On the other hand, the price to pay seemed to be so high, since the gauge boson propagator did generate spurious poles in physical amplitudes. This problem was overcame when andelstam and Leibbrandt[@mandel] introduced causal prescriptions to treat such poles (there are also other causal prescription which can be implemented, proposed by Pimentel and Suzuki[@pimentel], known as causal Cauchy principal value prescription.) However, the famous ML-prescription necessarily forces one to use partial fractioning tricks and integration over components, which turn the calculations rather involved[@leib-nyeo].
Negative-dimensional approach can avoid at all the use of prescriptions and provide physically acceptable results, i.e., causality preserving results. The calculation we will present is the very first test beyond 1-loop order without invoking ML-prescription, as we called in [@without] is a prescriptionless method. Still, integration over componenets and partial fractioning tricks can be completely abandoned as well as parametric integrals. The important point to note[@without] is that the dual light-like 4-vector $n^*_\mu$ is necessary in order to span the needed 4-dimensional space[@leib-cjp; @tetrad]. This is the very reason why our calculation for one-degree of covariance violation failed[@probing].
The second non-covariant gauge we deal with in this paper is the Coulomb gauge. Potential between quarks and studies on confinement are easily performed in this gauge[@lc2; @coul]. Besides, the ghost propagator has no pole in this gauge! As light-cone gauge, Coulomb also have problems with gauge boson propagator. In the former, loop integrals generated aditional poles; in the latter, such integrals are not even defined[@taylor] since they have the form, ,such objects are the so-called energy-integrals. Doust and Taylor[@taylor] presented a solution for this issue in a form of a interpolating gauge (between Feynman and Coulomb). Leibbrandt and co-workers[@split] presented also a solution, a procedure they called [*split dimensional regularization*]{}, (SDR), which introduces two regulating parameters, one for the energy component and another for the 3-momentum one. So, the measure becomes, d\^D q = dq\_4 d\^[D-1]{} \^[ sdr]{} d\^D q = d\^ q\_4 d\^, in Euclidean space.
can also deal with Coulomb gauge loop integrals, but it needs to make use of SDR. In this work we propose to apply to scalar integrals with four massless propagators. Our results are given in terms of hypergeometric series involving external momenta, exponents of propagators and regulating parameters $\omega$ and $\p$.
The Light-Cone Gauge
--------------------
So far, we have tested our for integrals pertaining to one-loop class. Now we apply such technology to some massless two-loop integrals. Let us consider an integral studied by Leibbrandt and Nyeo[@leib-nyeo], since they did not present the full result for it, C\_3 = d\^D q d\^D k , where in their calculation ML-prescription must be understood. On the other hand, in the context the key point is to introduce the dual vector $n_\mu^*$ in order to span the needed space[@without; @leib-cjp; @tetrad]. If we do not consider it, our result will violate causality, giving the Cauchy principal value of the integral in question, as we conclude in [@probing].
can consider lots of integrals in a single calculation. Our aim to is perform,
= d\^D q d\^D k\_1 (k\_1\^2)\^i (q\^2)\^j (q-k\_1)\^[2k]{} (k\_1-p)\^[2l]{} (k\_1n)\^m (qn)\^s (k\_1n\^\*)\^r, we will carry out this integral and then present results for special cases, including Leibbrandt and Nyeo’s $C_3$, where $i=-1$, $r=0$ and the other exponents equal to minus one. Observe that the integral must be considered as function of external momentum, exponents of propagators and dimension, = [N]{}(i,j,k,l,m,r,s;P,D), where $P$ represents $(p^2, p^+, p^-, \half (n\cdot n^*))$, and we adopt the usual notation of light-cone gauge[@leib-rmp].
Our starting point is the generating function for our negative-dimensional integrals, G\_N &=& d\^D q d\^D k\
&& . -(kn\^\*)\], then after a little bit of algebra we integrate it, G\_N &=& ()\^[D/2]{} , where $$g_1 = (\alpha\beta + \alpha\gamma +\beta\gamma)\theta, \quad g_2 = (\beta\phi+ \gamma\omega +\gamma\phi)\theta,\quad g_3 = (\beta+\gamma)\eta\theta, \quad g_4 = \eta\frac{g_2}{\theta},$$ and $ \lambda = \alpha\beta+ \alpha\gamma + \beta\gamma + \beta\theta + \gamma\theta.$
Taylor expanding the exponentials one obtain, &=& (-\^D) i!j!k!l!m!r!s!(1-\_n-D/2) \_[[all]{}=0]{}\^\
&&( -)\^[Y\_[123]{}]{}, where $\s_n=i+j+k+l+m+r+s+D$ and $\delta$ represents the system of constraints $(8\times 16)$ for the negative-dimensional integral. In the end of the day we have 12870 possible solutions for such system! Most of them, 9142, have no solution while 3728 present solutions which can be written as hypergeometric series. Of course several of these will provide the same series representation, these solutions we call degenerate.
We present a result for the referred integral as a double hypergeometric series, &=& \^D f\_n P\_n \_[Z\_j=0]{}\^\
&& ()\^[Z\_[45]{}]{}, where f\_n &=& (-m|-s)(-i-j-k-D/2|-\_n-D/2) (j+k+s+D|i-s+r)\
&&\
&&(-j|-i-k-m-r-s-D),are the Pochhammer symbols and P\_n = (p\^2)\^[\_n+i+j+k+D]{} (p\^+)\^[l+m+s-\_n]{}(p\^-)\^[l+r-\_n]{} ( )\^[\_n-l]{}.
Now we can consider the special case ($i=1, j=k=l=m=s=-1, r=0$), studied in [@leib-nyeo],
\[caso-especial\] [N]{}\_[SC]{} &=& \^D (p\^2)\^[2D-5]{}\
&&(p\^+)\^[1-D]{} (p\^-)\^[3-D]{} ( ) \^[D-3]{} \_[Z\_4,Z\_5=0]{}\^\
&& ( )\^[Z\_[45]{}]{} , observe that it exibits a double pole, as stated by Leibbrandt and Nyeo[@leib-nyeo].
The Coulomb Gauge
-----------------
We will present the full calculation of an integral which has four propagators,J(i,j,k,m) = d\^D q (q\^2)\^i(q-p)\^[2j]{}[**q**]{}\^[2k]{} ([**q+p**]{})\^[2m]{}, in order to regulate the possible divergences originated by the energy component, SDR must be understood, namely, d\^D q = d\^ q\_4 d\^, where $D=\p+\omega$.
The generating functional for our negative-dimensional integrals is the gaussian-like integral, G\_c = d\^D q , which can be easily integrated, G\_c = . There are results given by double, triple, 4-fold and 5-fold hypergeometric series in the variable $\pp^2/p_4^2$ or its inverse.
We will present two of such , the first one is a 4-fold series, \[4-sum\] J\_4(i,j,k,m) &=& C\_4(i,j,k,m) \_[X\_i=0]{}\^ ()\^[X\_[1234]{}]{}\
&&\
&&\
&& + (ij, km), where C\_4(i,j,k,m) &=& \^[D/2]{} (p\_4\^2)\^i (\^2)\^[\_c-i]{} (-j|-/2) (-j-m-/2|-k-/2)\
&&(-k|2k+/2)(j+k+m+D/2+/2|-k-/2), where $\s_c=i+j+k+m+D/2$ and the second a 5-fold hypergeometric series, \[5-sum\] J\_5(i,j,k,m) &=& C\_5(i,j,k,m) \_[X\_i=0]{}\^\
&&()\^[2X\_1+X\_[2345]{}]{}\
&& , where C\_5(i,j,k,m) &=& \^[D/2]{}(p\_4\^2)\^[i+j+/2]{} (\^2)\^[k+m+/2]{} (-i|i+j+/2) (-j|i+j+/2)\
&&(-k|k+m+/2)(-m|k+m+/2) (i+j+|-\_c-/2)\
&& (k+m+|-\_c-/2), observe that the above result is also symmetric in $(i\leftrightarrow j, k\leftrightarrow m)$, which means in the loop integral, $ q^\mu \rightarrow q^\mu+p^\mu$.
Another important point to observe is that the final result must a sum of linearly independent hypergeometric series[@probing; @box]. The above 5-fold series, $J_5$, appears only one time whereas $J_4$ is degenerate since several systems give its two hypergeometric functions. This must be considered if one wants to apply to more complicated diagrams which can in principle generate even more involved.
Moreover, the above expressions, $J_4$ and $J_5$, are related by direct , since both are convergent for $|\pp^2/p_4^2|<1$. When one is considering simple , several formulas are known; on the other hand, for rather complicated hypergeometric series, as we obtained — four and five-fold series —, there are very few of such formulas. can fill this gap, since it is the only method which provides for Feynman loop integrals, in different kinematical regions, and related by direct or indirect alike.
The above hypergeometric series (only the series, not the factors!), $J_4$ and $J_5$, can be written as generalized s [@luke] of four and five variables, & & [F]{} \^[6:0;0;0;0]{}\_[2:0;0;0;0]{} , and & & [F]{} \^[3:0;0;0;0;0]{}\_[3:0;0;0;0;0]{} ,where $x=\pp^2/p_4^2$.
Conclusion
==========
The technique of Feynman parametrization can of course be used to perform loop integrals in different gauges but it is very difficult to perform the parametric integrals for arbitrary exponents of propagators. Not so with , carry loop integrals out with particular exponents is as easy as dealing with arbitrary ones – besides, one can come across with singularities which depend on them and not on dimension $D$ – this fact is very important when we are studying light-cone gauge Feynman integrals, because one could have to handle products like $(q^+)^a
\left[(q-p)^+\right]^b$, being $a$ and $b$ negative. can calculate all of them simultaneously, but if one chooses partial fractioning tricks then he/she will be forced to carry out each integral separately. Besides usual covariant integrals and the trickier light-cone gauge ones, was probed in the Coulomb gauge, where a procedure – introduced by Leibbrandt and co-workers – called [*split dimensional regularization*]{} is needed in order to render the energy integrals well-defined.
In this paper, we studied Feynman loop integrals pertaining to three outstanding gauges: the usual, and more popular, covariant Feynman gauge and two of the trickiest non-covariant gauges, the light-cone and the Coulomb ones. Our results are given in terms of hypergeometric functions and in the dimensional regularization context.
AGMS gratefully acknowledges FAPESP (Fundação de Amparo à Pesquisa de São Paulo) for financial support.
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abstract: 'The complex impedance of a semiconductor superlattice biased into the regime of negative differential conductivity and driven by an additional GHz ac voltage is computed. From a simulation of the nonlinear spatio-temporal dynamics of traveling field domains we obtain strong variations of the amplitude and phase of the impedance with increasing driving frequency. These serve as fingerprints of the underlying quasiperiodic or frequency locking behavior. An anomalous phase shift appears as a result of phase synchronization of the traveling domains. If the imaginary part of the impedance is compensated by an external inductor, both the frequency and the intensity of the oscillations strongly increase.'
address: |
$^{a}$Institute for Theoretical Physics, Technical University of Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany\
$^{b}$ Institute for Applied Physics, Universit[ä]{}t Regensburg, D-93040 Regensburg, Germany
author:
- 'A.-K. Jappsen$^{a}$, A. Amann$^{a}$, A. Wacker$^{a}$, E. Schomburg$^{b}$, E. Sch[ö]{}ll$^{a}$'
title: 'High-Frequency Impedance of Driven Superlattices'
---
=10000
[2]{}
Semiconductor superlattices (SL) show pronounced negative differential conductivity (NDC) [@ESA70]. If the total bias is chosen such that the average electric field is in the NDC region, stable inhomogeneous field distributions (field domains)[@ESA74] or self-sustained oscillations [@KAS95] with frequencies up to 150 GHz at room temperature[@SCH99h] appear. Which of these scenarios occurs depends on bias, doping, temperature, and the properties of the injecting contact [@WAC97a; @PAT98; @SAN99; @WAN99e; @STE00a; @AMA01]. For a recent overview see Ref. [@WAC02].
In order to apply the self-sustained oscillations as a high–frequency generator in an electronic device, it is crucial to know the response of the SL in an external circuit. A key ingredient for the analysis is the complex impedance of the SL in the respective frequency range. This is the subject of this paper where the complex impedance is evaluated numerically by imposing an additional ac bias to the SL. The interplay of the self-sustained oscillations and the external frequency causes a variety of interesting phenomena such as frequency locking, quasi-periodic and chaotic behavior which has been extensively studied both theoretically [@BUL95; @CAO99a; @SAN01] and experimentally [@ZHA96; @SCH02b]. In contrast to those studies we focus on the response to the circuit and concurrent phase synchronization phenomena in this work.
We describe the dynamical evolution of the SL by rate equations for the electron densities in the quantum wells, together with Poisson’s equation for the electric fields. The current densities $J_{j\to j+1}$ between adjacent quantum wells are evaluated within the model of sequential tunneling, and Ohmic boundary conditions with a contact conductivity $\sigma$ are used. For details see Refs. [@AMA01; @WAC02]. Here we apply a periodic bias signal $U(t)=U_{\rm dc}+U_{\rm ac}\sin(2\pi\nu_1 t)$ and study the total current $I(t)=A\sum_{j=0}^NJ_{j\to j+1}/(N+1)$. In particular we consider the Fourier component $I_{\rm ac}(\nu_1) \sin(2\pi\nu_1t-\phi)$ which gives the complex impedance $$Z(\nu_1)=\frac{U_{\rm ac}}{I_{\rm ac}(\nu_1)}e^{i\phi}\, .
\label{EqZdef}$$
As an example we consider the SL structure studied in Ref. [@SCH02b]. It consists of $N=120$ GaAs wells of width 4.9 nm separated by 1.3 nm AlAs barriers. The sample is n-doped with a density of $8.7\times 10^{10}{\rm cm}^{-2}$ per period and the sample cross section is $A=64\, (\mu {\rm m})^2$. We estimate the energy broadening $\Gamma= 15$ meV, which effectively gives the sum of phonon, impurity and interface roughness scattering rates. All calculations are performed at room temperature.
In Fig. \[FigKenn\]a we display the calculated current–voltage characteristic for $U_{\rm ac}=0$. While for large values of the contact conductivity $\sigma$ stationary field domains occur, self-sustained current oscillations (where pairs of accumulation and depletion fronts travel through the sample) are found for lower $\sigma$ (Fig. \[FigKenn\]b). This scenario is quite general for moderately doped samples [@SAN99; @WAC02]. In the following we use $\sigma =20$ A/Vm giving best agreement with the measured current-voltage characteristic.
At $U_{\rm dc}=2$ V, the self-sustained oscillations exhibit a frequency of $\nu_0=2.1$ GHz. Now we study the change in the current signal by imposing an additional ac-bias amplitude $U_{\rm ac}$ with frequency $\nu_1$. Fig. \[FigZges\] displays the absolute value and the phase of the complex impedance given by Eq. (\[EqZdef\]). For both strengths of the ac amplitude one observes strong variations in the amplitude and phase of $Z$. In particular one observes pronounced minima in $|Z|$ and a corresponding monotonic increase in $\phi$ around frequencies $\nu_1$ which are approximately integer multiples of $\nu_0$. This is due to frequency locking [@SCH88a], where the external frequency $\nu_1$ modifies the main oscillation frequency to $\nu_0^*$ such that $\nu_1/\nu_0^*$ is a rational number. While locking occurs in a rather wide frequency range for $\nu_1/\nu_0^*\in \mathbb{N}$, locking into rational numbers with larger denominators are less easy to observe (The corresponding widths of the locking intervals roughly follow the position in the Farey-tree of the rational numbers, see [@SCH88a]. The width of the locking ranges also increases with $U_{\rm ac}$.) A more detailed examination indicates that the local minima in $|Z|$ correspond to the locking regions, and $\phi$ increases with $\nu_1$ in those intervals, while a decrease is frequently observed outside these ranges where quasi-periodic or possibly chaotic behavior occurs. Thus the presence of several locking intervals[@SCH02b] explains the variety of peaks.
For high frequencies $\nu_1\gg \nu_0$ the variations of $Z$ become less pronounced and $|Z|\sim 300\Omega$ and $\phi\approx -\pi/2$ is observed. This corresponds to a capacitive current with an effective frequency-dependent capacitance $C_{\rm eff}(\nu_1)\approx 1/(2\pi \nu_1|Z| )$. At $\nu_1=10$ GHz this gives $C_{\rm eff}\approx 50$ fF. This capacitance results from the interaction with the internal front dynamics in the SL structure; the intrinsic sample capacitance $\epsilon_r\epsilon_0A/(Nd)=10$ fF is significantly smaller.
While we have only shown results for $U_{\rm dc}=2$ V and $\sigma=20$ A/Vm here, we checked that the features discussed above neither change for different biases nor different contact conductivities, albeit the main oscillation frequency $\nu_0$ and the locking ranges change slightly.
Now we focus on the 1/1 locking region which is of interest if the device is used as an oscillator. In Fig. \[FigTimeresolved\] we show the current and bias signal for different frequencies $\nu_1$. In parts (a) ($\nu_1=1.95$ GHz) and (f) ($\nu_1=2.3$ GHz) no complete locking occurs and the current does not exhibit a periodic signal. Between these frequencies the current signal is periodic with a frequency $\nu_0^*=\nu_1$ imposed by the external bias. While the current peaks occur around the minima of the external bias for $\nu_1=2$ GHz (corresponding to $\phi\approx -\pi$), the delay between the current peaks and the bias maxima decreases with frequency until they are approximately in phase at the upper boundary of the locking range at $\nu_1=2.25$ GHz (corresponding to $\phi\approx 0$). It is intriguing to note that this behavior is opposite to the response of a damped linear oscillator, where the phase between the driving signal and the response shifts from $0$ to $\pi$ with increasing driving frequency. The current peaks correspond to the formation of a domain at the emitter, while during the current minima the domain traverses the sample. The domain transit velocity is larger for smaller voltage. The phase shift between current and voltage adjusts such that the domain velocity increases with increasing driving frequency during the whole locking interval. This is a phase synchronization effect induced by the domain dynamics.
Let us analyze the behavior close to the onset of the locking interval in detail, see Fig. \[FigDetail\]. The main frequency component of the current signal is given by $\nu_0^*$. In the locking region it is equal to the frequency of the driving bias $\nu_1$ such that $\nu_0^*-\nu_1$ vanishes in a finite range of $\nu_1$. In contrast, far away from the locking $\nu_0^*$ is essentially given by the free oscillation frequency $\nu_0$, thus $\nu_0^*-\nu_1$ exhibits a linear relation. Close to the boundaries of the locking region a square root behavior $\nu_0^*-\nu_1\propto
\sqrt{|\nu_1-\nu_1^{\rm crit}|}$ can be detected, which is a general feature of frequency locking, see, e.g., [@PEI92].
Fig. \[FigDetail\]b shows the amplitude and the monotonically increasing phase of the impedance over the whole locking interval. We extract an impedance $Z\approx -i 150 \Omega$ for $\nu_0=2.1 GHz$. Now we compensate this impedance by an external circuit with an inductor of $L=10$ nH and a resistor of 25 $\Omega$ in series with the SL. (The resistor may result both from connecting wires or due to radiation damping in a real device.) Fig. \[FigInductor\] shows that the oscillation mode is strongly affected: The frequency is increased to $\nu=5.6$ GHz, the oscillation is more sinusoidal, and the current amplitude increases. Thus the device performance of the SL is strongly improved. This effect is due to a different oscillation mode, where the domains are quenched such that they only traverse a small part of the SL. As the domain velocity is almost constant, the frequency increases.
We have shown that the amplitude and the phase of the complex impedance $Z$ exhibit strong variations with the driving frequency $\nu_1$. The distinct minima of the amplitude of the impedance correspond directly to the frequency locking intervals. In these regions the frequency components of the current at $\nu_1$ are particularly large, leading to small $|Z|$. The phase of the impedance is monotonically increasing with frequency from $-\pi$ to zero in the locking intervals, which constitutes a phase shift of $\pi$ compared to the standard behavior of a driven oscillator. These effects are caused by phase synchronization of the spatio-temporal dynamics of the traveling field domains in the NDC regime. Compensating the imaginary part of the impedance in an external circuit strongly improves the device performance of the SL.
Partial support from Sfb 555 is acknowledged.
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![(a) Calculated current-voltage characteristics for different contact conductivities $\sigma$, and $U_{\rm ac}=0$. For $\sigma=20$ A/Vm and 40 A/Vm self-sustained current oscillations are found in the indicated bias ranges. In this case we display the time-averaged current. (b) Self-sustained oscillations for $U_{\rm dc}=2$ V, $\sigma=20$ A/Vm. The time series of the current $I(t)$ and the space-time plot of the electron densities $n_i(t)$ in the quantum wells is shown. Black indicates low electron density (depletion front), white indicates high electron density (accumulation front). The emitter is at the bottom, the collector at the top. []{data-label="FigKenn"}](fig1.eps){width="8.5cm"}
![Amplitude $|Z|$ (full line) and phase $\phi$ (dashed line) of the complex impedance $Z$ as a function of the driving frequency for two different values of the ac bias: (a) $U_{ac}=0.2$V, (b) $U_{ac}=0.6$V. The shaded areas indicate locking intervals marked by $\nu_1/\nu_0^*$ ($U_{dc}=2$V, $\sigma=20$ A/Vm). []{data-label="FigZges"}](fig2.eps){width="6.cm"}
![Current $I$ (full line) and driving bias $U$ (dashed line) versus time for different driving frequencies $\nu_1$ ($U_{\rm ac}=0.4 V$, $U_{dc}=2$V). []{data-label="FigTimeresolved"}](fig3.eps){width="8.5cm"}
![(a) Shift of the fundamental frequency $\nu_0^*$ of the current signal and (b) amplitude and phase of the complex impedance as a function of the driving frequency $\nu_1$ in the vicinity of the 1/1 locking region ($U_{\rm ac}=0.2$ V, $U_{dc}=2$V). []{data-label="FigDetail"}](fig4.eps){width="8.5cm"}
![Operation of the SL device with an inductor of 10 nH and a resistor of 25 $\Omega$ in series ($U_{dc}=2$V, $U_{\rm ac}=0$) (a) Current versus time and space-time plot of the electron density in the SL (as in Fig.1b). (b) Electric field versus position for different times (as indicated in a). []{data-label="FigInductor"}](fig5.eps){width="8.5cm"}
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---
abstract: 'Graph matrices are a type of matrix which appears when analyzing the sum of squares hierarchy and other methods using higher moments. However, except for rough norm bounds, little is known about graph matrices. In this paper, we take a step towards better understanding graph matrices by determining the spectrum of the singular values of Z-shaped graph matrices.'
author:
- Wenjun Cai and Aaron Potechin
bibliography:
- 's0\_main.bib'
title: 'The Spectrum of the Singular Values of Z-Shaped Graph Matrices'
---
Introduction
============
Techniques {#section prem}
==========
Trace Powers of the Z-shaped Graph Matrix {#section z-graph matrix}
=========================================
The Spectrum of the Z-shaped Graph Matrix {#section spect z}
=========================================
Trace Powers of Multi-Z-shaped Graph Matrices {#section mz-graph matrix}
=============================================
The Spectrum of a Multi-Z-shaped Graph Matrix
=============================================
Dominant Constraint Graphs on $H(\a_{Z_m},2q)$ are Well-Behaved
===============================================================
|
---
abstract: 'We demonstrate that bistability of the nuclear spin polarization in optically pumped semiconductor quantum dots is a general phenomenon possible in dots with a wide range of parameters. In experiment, this bistability manifests itself via the hysteresis behavior of the electron Zeeman splitting as a function of either pump power or external magnetic field. In addition, our theory predicts that the nuclear polarization can strongly influence the charge dynamics in the dot leading to bistability in the average dot charge.'
author:
- 'A. Russell'
- 'Vladimir I. Fal’ko'
- 'A. I. Tartakovskii'
- 'M. S. Skolnick'
title: 'Bistability of optically-induced nuclear spin orientation in quantum dots'
---
The hyperfine interaction in solids between the electron and nuclear spins [@overhauser] leads to the Overhauser energy shift $\delta$ of the electron spin states, produced by the magnetic dipole moments of orientated nuclear spins, often described in terms of the effective nuclear magnetic field $B_{N}=2\delta/g_{e}\mu_{B}$. The hyperfine interaction is also responsible for the transfer of spin from electrons to nuclei and may lead to a significant nuclear spin polarization $S$, if the system is pumped with highly polarized electrons [@pines; @paget; @braun]. Recently, nuclear spin effects have been observed in optically pumped semiconductor quantum dots (QDs) [@gammon; @yokoi; @lai; @akimov; @brauntwo; @tartakovskii; @maletinsky; @eble]. In these experiments circularly polarized light generated electron-hole pairs which, then, relax into the ground state of the dot, with electrons exhibiting a longer spin memory than the holes (which lose their spin polarization due to a stronger spin-orbit coupling [@uenoyama; @ebbens]).
Recently, the nuclear spin orientation in optically pumped dots has been found to display a pronounced bistability in externally applied magnetic fields [@brauntwo; @tartakovskii; @maletinsky]. This appears as a threshold-like switching of the nuclear magnetic field $B_N \sim 2-3T$ and a characteristic hysteresis behavior observed in the dependence of the nuclear polarization on either the intensity of the polarized light [@brauntwo; @tartakovskii] or external magnetic field [@maletinsky; @ono].
In this Letter we propose a theory of the nuclear polarization bistability in optically pumped QDs. We study the dynamics of nuclear spins in a dot populated by electrons (el) and holes (h) which arrive into its ground state with the independent rates $w$ (el) and $\tilde{w}$ (h) and polarization degrees $\sigma$ (el) and $\tilde{\sigma}$ (h) (see Fig. \[schem\](a)). It has been recently noticed that nuclear polarization bistability may occur in the regime when light generates 100% spin-polarized excitons on the dot. Here, we demonstrate that bistability is a general phenomenon possible in a wide range of experimental conditions, including the non-resonant excitation conditions and in the regime when a dot often appears in a positively charged (trion) state. We also predict a new phenomenon caused by the bistable behavior of the nuclear spin orientation: the bistability of the dot average charge.
![\[schem\] (a)Diagram of optical excitation of the dot. The case of strongly polarized electrons and weakly polarized holes is shown. (b) Spin-flip-assisted el-h recombination accompanied by flipping one of the nuclear spins.](fig1.eps){width="8cm"}
In optically pumped dots, nuclear spins become orientated due to the spin flip-flops in which a single electron and one of the nuclei exchange spins via the hyperfine interaction. The process leading to spin transfer consists of an el-nucleus spin flip-flop accompanying the recombination of a polarized electron with a spin $\pm
\tfrac{3}{2}$ heavy hole in a QD carrying a neutral or positively charged exciton (see Fig. \[schem\](b)). In such a process, the electron occupies the intermediate inverted-spin state on the dot virtually, since in a magnetic field a real single-electron spin-flip is prohibited by energy conservation and the Zeeman splitting. The rate of the electron spin-flip recombination involving a single nucleus [@erlingsson] is, $$w_{s}=|u|^2w_{r} / ( \epsilon_{eZ}^2 + \tfrac{1}{4}\gamma^2 )
\label{w_s} .$$ Here $u$ is a typical energy of the hyperfine interaction with a single nucleus, $\gamma$ is broadening of the electron energy level, and $w_{r}$ is the rate at which the bright exciton recombines on the dot. The electron Zeeman splitting, modified by the Overhauser field $B_N$, is $\epsilon_{eZ}=g_{e}\mu_{B}(B-B_{N})$. The form of the Eq. (\[w\_s\]) implies a feedback due to the dependence of $w_s$ on $B_N$, which is key to the nuclear spin bistability.
The kinetic model describing the carrier population in the ground state of the dot is formulated in terms of the probabilities of its $16$ allowed configurations based upon the two electron and two hole spin states corresponding to the lowest el/h orbitals in the QD. We solve the rate equations for the populations of these states and for the nuclear orientation, and then find the steady-state magnitude of the nuclear spin polarization $S$. Here, we denote the probability that the dot is empty by $n$, and use $n_{\mu}$ ($n^{\mu}$) for the probabilities of the dot occupation by a single electron (hole), with the index $\mu=+/-$ representing the spin state of the particle. We refer to these states as $D, D_{\mu}$ and $D^{\mu}$, respectively. The probabilities for the dot to be occupied with two electrons or two holes (states $D_{+-}$ and $D^{+-}$) are $n_{+-}$ and $n^{+-}$. The probability to find the dot in a dark exciton state $X_{\mu}^{\mu}$ is $n_{\mu}^{\mu}$, and in a bright exciton state $X_{\mu}^{-\mu}$ is $n_{\mu}^{-\mu}$. The probability to find the dot in a negative (positive) trion state labelled as $X_{+-}^{\mu}$ ($X_{\mu}^{+-}$) is $n_{+-}^{\mu}$ ($n_{\mu}^{+-}$) and, finally, $n_{+-}^{+-}$ represents the dot in the biexciton state, $X_{+-}^{+-}$.
Below we list the balance equations for the dot population. The first two equations describe the probability of the dot occupation by a single carrier. $$\begin{aligned}
\dot{n}_{\mu} = && \tfrac{1}{2} (1 + \mu \sigma) w n + w_r
n_{+-}^{\mu} - \left[ \tilde{w} + \tfrac{1}{2} (1 - \mu\sigma)w
\right]n_{\mu};\nonumber
\\
\dot{n}^{\mu}= && \tfrac{1}{2} (1 + \mu \tilde{\sigma}) \tilde{w} n
+ w_r n_{\mu}^{+-} + \tfrac{1}{2} (1+\mu S) Nw_{s} n_{-\mu}^{+-}
\nonumber
\\* && - \left[ w + \tfrac{1}{2} (1 -\mu \tilde{\sigma})\tilde{w} \right]n^{\mu}.
\label{kinset1}\end{aligned}$$ Both include “gains” due to the arrivals of an electron/hole into the empty dot (Fig. \[schem\](a)) and the recombination of a charged bright exciton, and “losses” due to the arrival of an electron or a hole. The second equation also has a gain due to a possible spin-flip-assisted recombination from a positive trion $X_{\mu}^{+-} \rightarrow D^{-\mu}$ in which the spin is transferred to a nucleus [@virtual], Fig. \[schem\](b). This process is impossible for a negative trion since in the lowest orbital state the flip-flop is blocked by the presence of the second electron [@virtual]. The probability for an el-h pair to recombine via spin-flip depends on the number of nuclei available, which leads to the term $(1+\mu S) Nw_{s} n_{-\mu}^{+-}$ in Eq. (\[kinset1\]), where $S$ is the degree of nuclear polarization and $N$ is the total number of nuclei covered by the electron wave function ($N\sim 10^4
\div 10^5$ in a typical InGaAs/GaAs dot).
Equations describing the QD states $D_{+-}$ and $D^{+-}$ are: $$\begin{aligned}
\dot{n}_{+-}= \tfrac{1}{2}\sum_{\mu}( 1-\mu\sigma) w n_{\mu} -
\tilde{w} n_{+-};\nonumber
\\
\dot{n}^{+-}= \tfrac{1}{2}\sum_{\mu}( 1 - \mu \tilde{\sigma})
\tilde{w} n^{\mu} - w n^{+-}. \label{kinset2}\end{aligned}$$
Kinetics of the the neutral bright and dark excitons $X_{\mu}^{\mu}$ and $X_{\mu}^{-\mu}$ are described by $$\begin{aligned}
\dot{n}_{\mu}^{\mu}=&& \tfrac{1}{2}(1 + \mu\tilde{\sigma} )
\tilde{w} n_{\mu} + \tfrac{1}{2}(1 + \mu\sigma ) w n^{\mu} \nonumber
\\* &&- \tfrac{1}{2}\left[(1 - \mu S)Nw_{s} + ( 1 - \mu \sigma
) w + ( 1 - \mu \tilde{\sigma} )\tilde{w} \right]
n_{\mu}^{\mu};\nonumber
\\
\dot{n}_{\mu}^{-\mu}= && \tfrac{1}{2}(1 -
\mu\tilde{\sigma})\tilde{w} n_{\mu} + \tfrac{1}{2}(1 + \mu\sigma )w
n^{-\mu} + w_r n_{+-} ^{+-} \nonumber
\\* && - \left[ w_r + \tfrac{1}{2}(1 - \mu\sigma)w + \tfrac{1}{2}( 1 +
\mu\tilde{\sigma})\tilde{w} \right]n_{\mu}^{-\mu}. \label{kinset3}\end{aligned}$$ Both neutral bright and dark exciton populations decrease when more carriers arrive onto the dot. The neutral bright exciton can also be created and removed due to the el-h pair recombination in the processes $X^{+-}_{+-}\rightarrow X_{\mu}^{-\mu}$ and $X_{\mu}^{-\mu}\rightarrow D$, respectively. The dark exciton can decay due to the spin-flip-assisted recombination ($X_{\mu}^{\mu}
\rightarrow D$) leading to spin transfer to nuclei [@virtual].
Kinetics of the trions $X_{+-}^{\mu}$ and $X_{\mu}^{+-}$ are described by $$\begin{aligned}
\dot{n}_{+-}^{\mu} =&& \tfrac{1}{2}(1 +\mu \tilde{\sigma})
\tilde{w} n_{+-} + \tfrac{1}{2}\sum_{\nu=\pm} (1-\nu
\sigma)wn_{\nu}^{\mu} \nonumber
\\* && - \left[ w_r + \tfrac{1}{2}(1 - \mu \tilde{\sigma})\tilde{w}
\right] n_{+-}^{\mu} ; \label{kinset4}
\\
\dot{n}_{\mu}^{+-} =&& \tfrac{1}{2}(1+ \mu \sigma)w n^{+-} +
\tfrac{1}{2}\sum_{\nu=\pm}(1-\nu\sigma)wn_{\mu}^{\nu} \nonumber
\\* && - \left[ w_r + \tfrac{1}{2}(1-\mu S)Nw_{s} +
\tfrac{1}{2}(1-\mu\sigma) w\right] n_{\mu}^{+-}. \nonumber\end{aligned}$$ Both trion populations change due to the recombinations $X_{\mu}^{+-} \rightarrow D^{\mu}, X_{+-}^{\mu} \rightarrow D_{\mu}$ and arrival of a single additional charge (the ground states of the dot permit maximum four carriers). A positive trion can also recombine in the spin-flip-assisted process $X_{\mu}^{+-}
\rightarrow X^{-\mu}$, forbidden for the negative trions [@virtual].
Finally, the biexciton state $X_{+-}^{+-}$ cannot contribute to the nuclear spin pumping as it decays without the spin-flip, $X_{+-}^{+-}\rightarrow X_{+}^{-},X_{-}^{+}$, $$\begin{aligned}
\dot{n}_{+-}^{+-} = && \tfrac{1}{2}\sum_{\mu} \left[ (1 -
\mu\tilde{\sigma})\tilde{w} n_{+-}^{\mu} + (1 -\mu\sigma) w
n_{\mu}^{+-} \right]\nonumber
\\* && - 2w_r n_{+-}^{+-}. \label{kinset5}\end{aligned}$$
The probabilities for the dot with a given nuclear polarization to be in each of the $16$ configurations are found using the normalization condition $1=n + n_{+-} + n^{+-} + n_{+-}^{+-}+
\sum_{\mu}n_{\mu}+n^{\mu}+ n_{\mu}^{\mu} +
n_{\mu}^{-\mu}+n_{+-}^{\mu}+n_{\mu}^{+-}$ and the steady state condition for Eqs. (\[kinset1\]-\[kinset5\]). We formally write these equations in the form $\hat{M} \vec{n}=\left(
1,0,...,0\right)^T$, where the components of $\vec{n}$ are the occupation numbers and $\hat{M}$ is a $16\times 16$ matrix with elements determined by the coefficients in Eqs. (\[kinset1\]-\[kinset5\]) and the normalization condition. The formal solutions for components of $\vec{n}$ are given by $C_{i,1} / detM$ where $C_{i,1}$ is the relevant cofactor of $\hat{M}$.
A steady-state value for the nuclear polarization S (defined as $S=f_{\Uparrow}-f_{\Downarrow}$) can be obtained by substituting formal steady-state solutions of Eqs. (\[kinset1\]-\[kinset5\]) for a given $S$ into the balance equation for the occupation numbers of spin up ($f_{\Uparrow}$) and down ($f_{\Downarrow}$) nuclei [@spinhalf], $$\dot{S}=I\equiv \sum_{\mu}\mu\left( 1-\mu S\right)\left(
n_{\mu}^{\mu} + n_{\mu}^{+-} \right)w_{s} - 2Sw_{d}. \label{dotS}$$ It summarizes the processes leading to the nuclear spin pumping: $S$ is increased as a result of the spin-flip-assisted recombination of $X^{+}_{+}$ and $X^{+-}_{+}$ and reduced due to a similar recombination process involving $X^{-}_{-}$ and $X^{+-}_{-}$. Thus the balance between the populations of $X^{+}_{+}$ and $X^{+-}_{+}$ on one hand and $X^{-}_{-}$ and $X^{+-}_{-}$ on the other will eventually define the sign of the net nuclear polarization [@virtual]. However, an additional important contribution to the depolarization of the nuclei has to be taken into account. It arises from their mutual dipole-dipole interaction effectively leading to the nuclear spin diffusion from the dot into the bulk semiconductor [@pagettwo], described in our model by the rate $w_d$ [@dep].
To present the analysis of the above equations, we employ the following parameters: $$x=\frac{B}{B_{N}^{max}},\qquad z=2N^2\frac{w_{d}}{w_r},\qquad P=
\frac{\tilde{w}}{zw_r}.\label{P}$$ Here $B_N^{max}$ is defined through the Overhauser field $B_{N}$ as $B_N=B_{N}^{max}S$ and, after introducing $\alpha = \gamma/g_e
\mu_{B}B_{N}^{max}$, Eq. (\[w\_s\]) can be represented in the form [@values]: $$w_{s}\equiv \frac{w_{r}}{ N^2\left( x - S\right)^2 +
\tfrac{1}{4}\alpha^2}. \label{w_s2}$$ The steady-state values of $S$ determined by the feedback built into Eqs. (\[w\_s\]-\[w\_s2\]) are given by the solutions of the equation $I\left(S\right)=0$, satisfying the condition $\frac{dI}{dS}<0$ (solutions with $\frac{dI}{dS}>0$ are unstable). Figure \[I\] demonstrates that for a fixed external magnetic field the number of stable solutions for the nuclear spin polarization varies: it can be one or two depending on the incident power and other experimental parameters such as $w_{d},\sigma,\tilde{\sigma}$ and the ratio $w/\tilde{w}$. At small powers only a single low value of $S$ is possible. At high powers when two stable solutions appear, including one with a large $S>x$, the dot enters the regime of the nuclear spin bistability. This result strongly depends on the depolarization parameter $z$, defined in Eq. (\[P\]), so that in the following discussion we specify the range of $z$ where a bistability occurs.
![\[I\] The function $I(S)$ for the situation where $w=\tilde{w}, \sigma=0.9, \tilde{\sigma}=-0.2 ,x=0.6$ and $z=8$ for three different powers: $P=0.0001,0.0003,0.0005$. Stable roots correspond to the solutions of $I(S)=0$ where $\frac{dI}{dS}<0$.](fig2.eps){width="8cm"}
![\[hyst\] Evolution of nuclear polarization ($S$) and the average charging state of the dot ($Q$) for $w=\tilde{w},\sigma=0.9$ and $\tilde{\sigma}=-0.2$: (a) as a function of power for $x=0.6$ and various values of $z$, with the arrows indicating a forwards or backwards sweep. Although not shown in the figure, at high powers $P \approx 1$, both $Q$ and $S$ start to decrease due to the dot being dominated by the biexciton (for which the spin-flip process is blocked); (b) as a function of magnetic field for $z=8$ and various power values.](fig3.eps){width="8cm"}
The bottom parts of Figs. \[hyst\](a) and (b) show the calculated evolution of the nuclear polarization in a dot for realistic magnitudes of the depolarization parameter $z$ in the regime where electrons have a high degree of spin memory and arrive with the same rate as the depolarized holes. Fig. \[hyst\](a) contains a large hysteresis loop in the power dependence of $S$ for a fixed magnetic field (here, $x=0.6$), similar to those observed in Refs. 10,11. The bistable behavior occurs for a wide range of the depolarization parameter $z$: $5\lesssim z\lesssim 14$. Experimentally, the evolution of $S$ can be detected in polarization-resolved PL experiments on individual self-assembled InGaAS/GaAs quantum dots, by deducing it from the measured exciton Zeeman splitting.
We also find that the bistability in $S$ leads to a novel phenomenon: a hysteresis in the average dot charge, $Q$ (see top parts of Fig. \[hyst\](a,b)). This occurs when the electrons arriving to the dot have a high degree of spin polarization, permitting their recombination with only one spin orientation of holes. Thus, an extra hole with the opposite spin is likely to remain on the dot, leading to, on average, a positive dot charge. The enhancement of the spin-flip-assisted recombination for large $S$, removing such holes, will result in reduction of the charge. Therefore the hysteresis in $S$ will be reflected as a hysteresis in the average dot charge. A similar bistable behaviour in both $S$ and $Q$ can also be found if the external magnetic field is varied [@maletinsky] at a fixed optical pump power, as shown in Fig. \[hyst\](b).
![\[diffpol\]Evolution of $S$ with $P$ in the regime where $w=0.1\tilde{w}$ for $z=0.16,0.4$ and different polarizations of arriving electrons/holes. (a) $\sigma=0.9,\tilde{\sigma}=-0.2$. The inset shows the evolution of the charging state of the dot for $z=0.4$ (with a very small hysteresis loop). (b) Same for $\sigma=0.45,\tilde{\sigma}=-0.1$.](fig4.eps){width="8cm"}
Figure \[diffpol\] illustrates that the range of parameters for which the bistability can occur strongly depends on the ratio between the arrival rates of electrons and holes, $w$ and $\tilde{w}$, as well as on their polarizations, $\sigma$ and $\tilde{\sigma}$. In experiment the ratio $w/\tilde{w}$ can be varied by applying an electric field in a diode containing QDs in the intrinsic region [@braun; @lai; @tartakovskii; @maletinsky]: because of a light effective mass, electrons can tunnel out before relaxing to the dot ground state, which in effect reduces their arrival rate as compared to that of the holes. Fig. \[diffpol\](a) shows the evolution of $S(P)$ for $w=0.1\tilde{w}$. The dot is mainly in the state $D^{+-}$ so that its average charge is $Q\approx
+1.8$ and exhibits a weak power-dependence with a negligible hysteresis loop (see inset), despite a pronounced hysteresis loop in the nuclear polarization. As seen from the figure, for such low values of $w/\tilde{w}$ higher powers are required to pump a significant nuclear polarization, and the bistability in $S$ is moved towards smaller values of the depolarization parameter ($0.2
\lesssim z \lesssim 0.5$). Figure \[diffpol\](b) illustrates that when the polarizations of both electrons and holes is reduced by $50\%$ the bistability can still be observed, but only for $0.1\lesssim z \lesssim 0.2$.
To summarize, we have shown that for a wide range of dot parameters (including the number of nuclei, el-h radiative recombination time and nuclear spin diffusion rate) the polarization of nuclei in a non-resonantly optically pumped semiconductor quantum dot can exhibit a bistable behavior. Thus, we conclude that the nuclear spin bistability is a general phenomenon for dots pumped with circularly polarized light. In addition, we find that the nuclear spin polarization can also strongly influence the charge dynamics in the dot leading to the bistability of the average dot charge.
We thank A. Imamoglu, O. Tsyplyatyev and A. Yacobi for discussions. This work has been supported by the Lancaster EPSRC Portfolio Partnership No. EP/C511743, the Sheffield EPSRC Programme Grant No. GR/S76076, the EPSRC IRC for Quantum Information Processing, ESF-EPSRC network Spico EP/D062918, EPSRC Advanced Research Fellowship EP/C54563X/1.
A. W. Overhauser, Phys. Rev. [**92**]{}, 411 (1953). D. Pines, J. Bardeen, and C. P. Slichter, Phys. Rev. [**106**]{}, 489 (1957) D. Paget [*et al.*]{}, Phys. Rev. [**B 15**]{}, 5780 (1977) P.-F. Braun [*et al.*]{}, Phys. Rev. Lett. [**94**]{}, 116601 (2005). D. Gammon [*et al.*]{}, Phys. Rev. Lett. [**86**]{}, 5176 (2001); A. S. Bracker [*et al.*]{}, [*ibid.*]{} [**94**]{}, 047402 (2005). C. W. Lai [*et al.*]{}, Phys. Rev. Lett. [**96**]{}, 167403 (2006).
T. Yokoi [*et al.*]{}, Phys. Rev. [**B 71**]{}, 041307 (2005).
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Hints of bistability in nuclear orientation have been seen in the spin-polarized tunneling experiments: K. Ono and S. Tarucha, Phys. Rev. Lett. [ **92**]{}, 256803 (2004); S. Tarucha [*et al.*]{}, Phsy. Stat. Sol. [**B 243**]{}, 3673 (2006). T. Uenoyama and L. J. Sham, Phys. Rev. Lett. [**64**]{}, 25 (1989); T. C. Damen [*et al.*]{}, [*ibid.*]{} [**67**]{}, 24 (1991).
A. Ebbens [*et al.*]{}, Phys. Rev. [**B 72**]{}, 073307 (2005).
In the opposite polarization of light, the Overhauser field enhances the Zeeman splitting so that it cannot cause a bistability. Also, virtual spin-flip processes via excited orbital states have a much smaller probability, and are thus neglected. Here we also neglect the effect of the electron-hole exchange on the electron spin splitting.
S. I. Erlingsson, Y. V. Nazarov, and V. I. Fal’ko, Phys. Rev. [**B 64**]{}, 195306 (2001). For simplicity, we consider spin $\tfrac{1}{2}$ nuclei. Higher spins will result only in the re-parameterization of $N$.
D. Paget, Phys. Rev. [**B 25**]{}, 4444 (1982). For a dot with radius $r\approx 5nm$, we approximate $w_{dep}\approx D_{N}/r^{2}\approx 1-10 s^{-1}$, where $D_{N}\approx
\mu_{n}^{2}/\hbar a$ is the coefficient of polarization diffusion due to the dipole-dipole interaction between magnetic moments $\mu_{n}$ of neighbouring nuclei and $a=0.56nm$ is the lattice constant.
For InGaAs quantum dots used in [@tartakovskii], $w_{r}\approx 10^{9}s^{-1}, N\approx 10^4,
B_{N}^{max}\approx 2-3T, \alpha \approx 0.01$.
|
---
abstract: 'We present a new millimeter CO-line observation toward supernova remnant (SNR) [CTB87]{}, which was regarded purely as a pulsar wind nebula (PWN), and an optical investigation of a coincident surrounding superbubble. The CO observation shows that the SNR delineated by the radio emission is projectively covered by a molecular cloud (MC) complex at ${V_{\rm LSR}}=-60$ to $-54{\,{\rm km}}{\,{\rm s}^{-1}}$. Both the symmetric axis of the radio emission and the trailing X-ray PWN appear projectively to be along a gap between two molecular gas patches at $-58$ to $-57{\,{\rm km}}{\,{\rm s}^{-1}}$. Asymmetric broad profiles of [$^{12}$CO]{} lines peaked at $-58{\,{\rm km}}{\,{\rm s}^{-1}}$ are found at the eastern and southwestern edges of the radio emission. This represents a kinematic signature consistent with an SNR-MC interaction. We also find that a superbubble, $\sim37''$ in radius, appears to surround the SNR from HI 21cm (${V_{\rm LSR}}\sim-61$ to $-68{\,{\rm km}}{\,{\rm s}^{-1}}$), *WISE* mid-IR, and optical extinction data. We build a multi-band photometric stellar sample of stars within the superbubble region and find 82 OB star candidates. The likely peak distance in the stars’ distribution seems consistent with the distance previously suggested for [CTB87]{}. We suggest the arc-like radio emission is mainly a relic of the part of blastwave that propagates into the MC complex and is now in a radiative stage while the other part of blastwave has been expanding into the low-density region in the superbubble. This scenario naturally explains the lack of the X-ray emission related to the ejecta and blastwave. The SNR-MC interaction also favors a hadronic contribution to the $\gamma$-ray emission from the [CTB87]{} region.'
author:
- 'Qian-Cheng Liu$^1$; Yang Chen$^{1,2,8}$; Bing-Qiu Chen$^3$; Ping Zhou$^{1,4}$; Xiao-Tao Wang$^{1,5}$; Yang Su$^{6,7}$'
title: 'An Investigation of the Interstellar Environment of Supernova Remnant [CTB87]{}'
---
Introduction
============
The progenitors of core-collapse (CC) supernovae (SNe) are massive stars, which are expected to be born mostly in OB associations, and their environments are altered by energy feedback processes dominated by energetic stellar winds and SN explosions as well as strong ionizing radiation. Such processes can produce superbubbles filled with low-density material and significantly affect the evolution of supernova remnants (SNRs) therein. On the other hand, remnants of CC SNe, are often close (a few pc) to molecular clouds (MCs), the birthplace of the progenitor stars that end their evolution in short lifetimes. So far about 70 SNRs are confirmed or suggested to be associated with MCs [@2010ApJ...712.1147J; @2014IAUS..296..170C] among the known $\sim300$ SNRs in the Milky Way [@2012AdSpR..49.1313F; @2014BASI...42...47G]. Six types of observational evidence for SNR-MC interaction are summarized, including the 1720 MHz OH maser, molecular line broadening, morphological agreement of molecular features with SNR features, etc.[@2010ApJ...712.1147J; @2014IAUS..296..170C]. In this paper, we investigate a perplexing SNR, [CTB87]{}, that may interact with an adjacent MC but may be located near the inner edge of a superbubble.
[CTB87]{} has been classified as a filled-center type SNR, with a radio size of about $8'\times6'$, centered at R.A.$= 20^{\rm h}16^{\rm m}02^{\rm s}$, decl.$= 37^{\circ}12'00''$. It was first cataloged in a radio survey at 960MHz of the Galactic plane conducted by the Owens Valley Radio Observatory [@1960PASP...72..331W]. The distance to it was suggested to be $12{\,{\rm kpc}}$ according to HI absorption measurements . Based on the extinction-distance relation [@2002ASPC..276..123F], a new distance of $6.1\pm0.9{\,{\rm kpc}}$ was established, which implies that [CTB87]{} is located in the Perseus spiral arm [@2003ApJ...588..852K]. In X-rays, [CTB87]{} is centrally brightened with a size of about $5'$ as observed by the *Einstein* satellite observatory [@1980ApJ...241L..19W]. A detailed *Chandra* ACIS analysis shows that the pulsar wind nebula (PWN) harbours a point source, which may putatively be a pulsar, $100''$ southeast away from the radio emission peak [@2013ApJ...774...33M]. In $\gamma$-rays, the TeV point source VER J2016+371, spatially coincident with [CTB87]{}, is resolved by the VERITAS telescope system [@2014ApJ...788...78A]. The GeV *Fermi*-LAT source 3FGL J2015.6+3709 is spatially close to VER J2016+371, with its origin still under debate [@2012ApJ...746..159K; @2015ApJS..218...23A; @2016ApJS..224....8A; @2016MNRAS.460.3563S]. Recently, the multiwavelength spectrum obtained from the [CTB87]{} region has been interpreted to be comprised of contributions from a Maxwellian distribution of electrons and a broken power-law distribution of electrons [@2016MNRAS.460.3563S].
[CTB87]{} has been proposed to be associated with molecular gas at a systemic local standard of rest (LSR) velocity ${V_{\rm LSR}}$ of about $-58{\,{\rm km}}{\,{\rm s}^{-1}}$ [@1986ApJ...309..804H; @1994AJ....108..634C; @2003ApJ...588..852K], while it has also been argued to lie at the inner boundary of an expanding HI superbubble at a ${V_{\rm LSR}}$ of about $-70{\,{\rm km}}{\,{\rm s}^{-1}}$ . The molecular gas was resolved into several subclumps, with [CTB87]{} seemingly located between two of them [@1994AJ....108..634C]. However, no kinematic evidence has yet been found for the SNR-MC interaction. Also, no 1720MHz OH maser, a reliable signpost of interaction between SNR and MC, has been detected for [CTB87]{} [@1996AJ....111.1651F]. About 30% of the SNRs that are confirmed to be in physical contact with MCs do not show OH masers [@2010ApJ...712.1147J], most probably because the shocked molecular gas does not satisfy the appropriate physical conditions, namely density of order $\sim 10^5 {\rm cm}^{-3}$ and temperature in the range of $50$–$125$K [@1999ApJ...511..235L].
In previous work, [@1994AJ....108..634C] mapped the [CTB87]{} region in CO lines with a half-power beamwidth of $2.7'$ and a grid spacing of $3'$, and [@2003ApJ...588..852K] presented a mapping in CO ([$J$=1–0]{}) with a beamwidth of $2'$ and a grid spacing of $1'$, as well as a partial mapping in CO ($J$=3–2). In this paper, we present a new CO-line observation toward the [CTB87]{} region and an optical investigation of the coincident superbubble, aiming to explore the environment of [CTB87]{}and the interaction of the SNR with the adjacent MC. We first describe the CO observations and data reduction process in §\[data\]; we analyze the CO-line data in detail in §\[result\] and discuss the main result in §\[discussion\]. Finally, we present our summary in §\[summarize\].
Observations and data reduction {#data}
===============================
Our observations of millimeter emission toward SNR [CTB87]{} were made simultaneously in [$^{12}$CO]{} ([$J$=1–0]{}), [$^{13}$CO]{} ([$J$=1–0]{}), and [C$^{18}$O]{}($J=$1–0) lines during 2013 May 11–18 using the 13.7m millimeter-wavelength telescope of the Purple Mountain Observatory at Delingha. We used the new $3\times3$ pixel Superconducting Spectro-scopic Array Receiver as the front end, which is constructed with Superconductor-Insulator-Superconductor mixers using the sideband separating scheme [@2011AcASn..52..152Z; @2012ITTST...2..593S]. We did on-the-fly mapping toward a $34'\times60'$ field that covers the full angular extent of SNR [CTB87]{}with a grid spacing of $30''$. The half-power beamwidth of the telescope is about $52''$ and the pointing accuracy of the telescope is better than $4''$. The typical system temperature is about 220K for 115.2GHz and 130K for 110.2GHz. The bandwidth of the spectrometer is 1GHz, with 16,384 channels and a spectral resolution of 61KHz. Thus the velocity resolution is $0.16{\,{\rm km}}{\,{\rm s}^{-1}}$ for [$^{12}$CO]{}and $0.17{\,{\rm km}}{\,{\rm s}^{-1}}$ for [$^{13}$CO]{} and [C$^{18}$O]{}. All the CO-line data were reduced using the GILDAS/CLASS package developed by the IRAM observatory[^1].
For a multiwavelength investigation of the environment of this SNR, we also used *Chandra* X-ray (ObsID: 11092, PI: Safi-Harb; which we reduced with CIAO ver. 4.7), and *WISE* 22 $\mu$m mid-infrared (IR) (*WISE* Science Data Center, IPAC, Caltech) data. The HI line emission data were obtained from the Canadian Galactic Plane Survey [CGPS; @2003AJ....125.3145T], and the 1.4GHz radio continuum emission data were from the NRAO VLA Sky Survey [NVSS; @1998AJ....115.1693C]. Optical and near-IR data from the Panoramic Survey Telescope and Rapid Response System , the Isaac Newton Telescope (INT) Photometric ${\rm H}\alpha$ Survey of the Northern Galactic Plane [IPHAS; @2005MNRAS.362..753D], and the Two Micron All Sky Survey [2MASS; @2006AJ....131.1163S] were used to investigate the OB star candidates within the coincident superbubble.
Results {#result}
=======
Spatial distribution of the ambient clouds {#spatial}
------------------------------------------
Figure \[overallspec\] shows the averaged CO spectra from a region covering SNR [CTB87]{}. There are two prominent [$^{12}$CO]{} ([$J$=1–0]{}) emission peaks, at around $-58 {\,{\rm km}}{\,{\rm s}^{-1}}$ and $-38 {\,{\rm km}}{\,{\rm s}^{-1}}$. The [$^{13}$CO]{} ([$J$=1–0]{}) emission is only prominent at $\sim-58{\,{\rm km}}{\,{\rm s}^{-1}}$, and the [C$^{18}$O]{} ([$J$=1–0]{}) emission is not significant across the velocity range.
We made [$^{12}$CO]{} emission channel maps around the two peaks with velocity interval $0.5{\,{\rm km}}{\,{\rm s}^{-1}}$ (see Figure \[channelmap1\] and Figure \[channelmap2\]) to examine the spatial distribution of the two molecular components. The $\sim-38{\,{\rm km}}{\,{\rm s}^{-1}}$ [$^{12}$CO]{} component seems to overlap the SNR at its southeastern “apex" by projection in interval $-39$ to $-37.5{\,{\rm km}}{\,{\rm s}^{-1}}$, but there is no systematic morphological feature correspondence between the molecular gas and the SNR (Figure \[channelmap1\]).
The $\sim-58{\,{\rm km}}{\,{\rm s}^{-1}}$ component appears to spread over a large area covering the SNR, and a few LSR-velocity dependent structures are noteworthy. (1) In velocity interval $-55.5$ to $-54.5{\,{\rm km}}{\,{\rm s}^{-1}}$, there is a linear structure of [$^{12}$CO]{} ([$J$=1–0]{}) emission along the eastern edge of the SNR (Figure \[channelmap2\]), somewhat similar to that described in [@2003ApJ...588..852K] (in passing, note that a Galactic coordinate system was used there while an equatorial coordinate system is used here, and the shell-like structure shown there is not complete due to the limited field of view). The structure can also be discerned in the [$^{13}$CO]{} ([$J$=1–0]{}) channel map at $-55{\,{\rm km}}{\,{\rm s}^{-1}}$ (Figure \[channelmapl\]). (2) In velocity interval $-56.5$ to $-55.5{\,{\rm km}}{\,{\rm s}^{-1}}$, a bar-like molecular structure appears to pass through the SNR along the northeast-southwest orientation and, in particular, through the brightness peak of the radio emission. (3) In interval $-58$ to $-57 {\,{\rm km}}{\,{\rm s}^{-1}}$, both [$^{12}$CO]{} and [$^{13}$CO]{}emissions show two patches of molecular material in the eastern and southwestern edges of the SNR, similar to the sub-clumps described in [@1994AJ....108..634C]. The integrated [$^{12}$CO]{} ([$J$=1–0]{}) intensity map in the main velocity range of the $\sim-58{\,{\rm km}}{\,{\rm s}^{-1}}$ component, $-60$ to $-54 {\,{\rm km}}{\,{\rm s}^{-1}}$, is presented in Figure \[overall\] (in green). It is overlaid with *Chandra* 0.5–7keV X-ray image of [CTB87]{}(in blue) and the radio continuum image (in red). The X-ray emitting part of the PWN, with a southeast-northwest oriented elongation, appears to be located in a gap of molecular gas, where CO emission is weak, between the two patches at $-58$ to $-57{\,{\rm km}}{\,{\rm s}^{-1}}$.
Kinematic evidence for SNR-MC interaction ? {#kinematic}
-------------------------------------------
We have inspected the [$^{12}$CO]{} ([$J$=1–0]{}) line profiles of the two molecular components toward [CTB87]{}. For the $\sim-38{\,{\rm km}}{\,{\rm s}^{-1}}$ component, we do not find any asymmetric broad profiles of the [$^{12}$CO]{} line in the grid of CO spectra. (This component, as mentioned in §\[spatial\], has neither morphological feature corresponding to the SNR). For the other component, around $-58{\,{\rm km}}{\,{\rm s}^{-1}}$, we show a grid of [$^{12}$CO]{} ([$J$=1–0]{}) and [$^{13}$CO]{} ([$J$=1–0]{}) spectra in the velocity range of $-61$ to $-53 {\,{\rm km}}{\,{\rm s}^{-1}}$ (Figure \[linegrid60\]) and find asymmetric broad profiles of the [$^{12}$CO]{} line in the eastern and southwestern edges. Figure \[region\] shows the average line profiles of some of the pixels at the edges of the SNR (regions “E" and “SW” marked in Figure \[linegrid60\]). There is a secondary peak at $-56$ to $-55{\,{\rm km}}{\,{\rm s}^{-1}}$ in region “E" (also seen in most of the pixels therein) besides the main peak at $-58{\,{\rm km}}{\,{\rm s}^{-1}}$, with little [$^{13}$CO]{} counterpart ($\la3\sigma$). Although the broad red (right) wing may include a contribution from real line broadening due to shock disturbation, we cannot exclude contamination by line-of-sight emission in a wide region. We note that there is similar secondary peak on the east of the radio boundary and there is non-negligible intensity at $-56$ to $-55{\,{\rm km}}{\,{\rm s}^{-1}}$ on the northeast of the boundary. However, in the three bottom pixels of region “E", there are unique plateaus in the blue (left) wings ($\sim-61$ to $-59{\,{\rm km}}{\,{\rm s}^{-1}}$), which are also clearly ($>4\sigma$) reflected in the average profile for region “E" (top panel, Figure \[region\]). This plateau-like spectral feature is somewhat similar to that detected in the [$^{12}$CO]{} spectra in the western edge of SNR 3C397 [@2010ApJ...712.1147J]. In the spectra of the pixels in the southwestern edge, as typified by region “SW”, the line profiles are characterized by strong peaks at $-58{\,{\rm km}}{\,{\rm s}^{-1}}$ for both [$^{12}$CO]{} and [$^{13}$CO]{} and a broad red wing of [$^{12}$CO]{} extending to $\sim-54{\,{\rm km}}{\,{\rm s}^{-1}}$without a significant [$^{13}$CO]{} counterpart (bottom panel, Figure \[region\]). The surrounding pixels essentially display different shapes of line profiles, and the emission outside the southern edge becomes weak. Since [$^{13}$CO]{} emission, which is usually optically thin (like that with $\tau$([$^{13}$CO]{})$\ll1$ given in Tables \[parameter\] and \[parameter2\]), is yielded in quiescent, intrinsically high-column-density molecular gas, a broad [$^{12}$CO]{}-line wing without a [$^{13}$CO]{} counterpart is very likely to represent a perturbed gas deviating from the systemic LSR velocity. Therefore, the asymmetric [$^{12}$CO]{} wings from the edges of [CTB87]{} (especially the blue wings in region “E" and the red wings in region “SW") very likely result from Doppler broadening of the $-58{\,{\rm km}}{\,{\rm s}^{-1}}$ line and thus might provide a kinematic evidence for interaction between the $\sim-58{\,{\rm km}}{\,{\rm s}^{-1}}$ MC complex and the SNR.
We fit the CO emissions with Gaussian lines for the $\sim-58 {\,{\rm km}}{\,{\rm s}^{-1}}$ molecular gas in regions “R" and “L" (as defined in Figure \[overall\]), and the derived parameters, molecular column density $N(\mbox{H}_2)$, excitation temperature $T_{\rm ex}$, and optical depth of [$^{13}$CO]{}([$J$=1–0]{}) $\tau$([$^{13}$CO]{}) are summarized in Table \[parameter\] and Table \[parameter2\]. The distance to the MC/SNR is taken to be 6.1 kpc, as suggested by [@2003ApJ...588..852K]. The column density of H$_2$ and the mass of the molecular gas are estimated using two methods. In the first method, the conversion relation for the molecular column densities, $N$(H$_2$)$\approx 7\times 10^5 N$ ([$^{13}$CO]{}) [@1982ApJ...262..590F], is used under the assumption of local thermodynamic equilibrium for the molecular gas and optically thick condition for the [$^{12}$CO]{} ([$J$=1–0]{}) line. In the second method, a value of the CO-to-H$_2$ mass conversion factor, $N$(H$_2$)/$W$([$^{12}$CO]{}) (known as the “X-factor"), $1.8\times 10^{20}$cm$^{-2}$K$^{-1}$km$^{-1}$s [@2001ApJ...547..792D] is adopted.
The superbubble at ${V_{\rm LSR}}\sim-64{\,{\rm km}}{\,{\rm s}^{-1}}$ {#superbubble}
---------------------------------------------------------------------
By projection, SNR [CTB87]{} is located within a superbubble, centered at R.A.=$20^{\rm h}14^{\rm m}03^{\rm s}$, decl.=$37^{\circ}13'27''$, with a circular boundary (brightened in the south and north) of angular radius $\sim37'$, and nears its eastern edge (shown in Figure \[bubble\]). We investigate the *WISE* mid-IR observation toward SNR [CTB87]{} and find the superbubble is bright at both 12 and $22\mu$m. The HI emission in this sky region also shows a cavity at the LSR velocity around $-64 {\,{\rm km}}{\,{\rm s}^{-1}}$ ($-61$to$-68{\,{\rm km}}{\,{\rm s}^{-1}}$), which happens to be spatially coincident with the mid-IR superbubble and to have the same angular size. Furthermore, such a bubble-like structure has a counterpart in the optical extinction map (also see Figure \[bubble\]).
As superbubbles are usually the products of energy and material feedback from massive OB stars into the interstellar medium, we searched for possible OB stars in the direction toward this superbubble.
We built a multi-band photometric stellar sample of over 0.18 million stars within the projected region of the superbubble (i.e., the circle shown in Figure \[bubble\]) by cross-matching the photometric cataloges of the Pan-Starrs, IPHAS, and 2MASS. We used a spectral energy distribution (SED) fitting algorithm, similar to those employed by [@2014MNRAS.443.1192C] and @2015MNRAS.450.3855M, to obtain the effective temperature $T_{\rm eff}$, optical extinction $A_V$, and distance modulus $\mu$ of the individual stars. We compared the observed SED of each star to the stellar models from the Padova isochrone data base [CMD v3.0; @2012MNRAS.427..127B; @2017ApJ...835...77M]. We considered only the main-sequence (surface gravity $\log g> 3.8$dex), and Galactic thin disk metallicity (\[Fe/H\] = $-0.2$dex) models. The observed magnitudes of a star can be modeled by $$m^{\rm obs}_i=m^{\rm mod}_i+A_i+\mu,$$ where $m^{\rm obs}_i$ and $m^{\rm mod}_i$ are the observed and Padova magnitudes in the $i$th band ($i$=1, 2, ..., 11 corresponding to Pan-Starrs $g_{\rm P},~ r_{\rm P},~i_{\rm P},~z_{\rm P},~y_{\rm P}$, IPHAS $r',~i'$, H$\alpha$, and 2MASS $J$, $H$, and $K_{\rm S}$ bands, respectively), $A_i$ is the extinction in the $i$th band, and $\mu$ is the distance modulus. We adopt the $R_V=3.1$ extinction law from [@1989ApJ...345..245C] to convert the optical extinction $A_V$ to extinction in each individual band $A_i$. A simple Bayesian scheme based on the Markov Chain Monte Carlo sampling is adopted to obtain the parameters $\log T_{\rm eff}$, $A_V$, and $\mu$ for the individual stars. We adopt the likelihood $$L_r=\Pi \frac{1}{\sqrt{2\pi \sigma _i}}
\exp{\frac{-(m^{\rm obs}_i-m^{\rm mod}_i)^2}{2\sigma^2 _i}},$$ where $\sigma_i$ is the uncertainty of the $i$th band observed magnitude, and the priors $$\begin{gathered}
P(T_{\rm eff}, A_V, \mu) = \begin{cases} 1 \quad \text{if} \,\,\begin{cases} 4.3 \,\, \leq \,\, {\rm log}\,(T_{\rm eff}) \,\, \leq \,\, 4.7 \\
0 \,\, \leq \,\, A_V \,\, \leq \,\, 20 \\
0 \,\, \leq \,\, \mu \,\, \leq \,\, 20
\end{cases}\\
0 \quad \text{else}.
\end{cases}\end{gathered}$$ We adopt the results for the stars with high photometric precisions ($\sigma_i$ $\le$ 0.1mag for each band) and high posterior probabilities ($\log P\ge 5$, where 5 is around the median value of the logarithmic posterior probabilities). We note that, due to the lack of information in blue optical wavelengths (i.e., $u$ band), there could be degeneracies between the effective temperatures and the amount of extinction for the individual stars.
As a result, we found 82 OB star candidates. Figure \[obstar\] shows the distribution of distance and optical extinction for all these objects. There seems to be a peak of the distribution of distance of these OB candidates at $d$=7.3kpc, with a dispersion of 1.9kpc, although the selected stars may be an incomplete and contaminated sample of the candidates owing to the lack of the $u$ band data. Notably, this distance range covers that of the Perseus spiral arm in the direction of [CTB87]{}[@2003ApJ...588..852K Figure8 therein]. As some massive stars should be responsible for the superbubble, it is most likely that they are located in the dispersion range of the peak. The distance range of the massive stars or the superbubble also seems to be consistent with the distance to SNR [CTB87]{}, either $6.1 \pm 0.9$kpc, that is derived from the intervening neutral hydrogen column density [@2003ApJ...588..852K] or $\sim6$–8kpc estimated from the column density of the line-of-sight X-ray absorbing hydrogen atoms and molecules (see § \[snrenvironment\]).
Discussion
==========
SNR Environment {#snrenvironment}
---------------
Our new CO-line observation shows spatial correspondence of the SNR [CTB87]{} with an MC complex at LSR velocities around $-58{\,{\rm km}}{\,{\rm s}^{-1}}$ ($-54$ to $-60{\,{\rm km}}{\,{\rm s}^{-1}}$). We have also demonstrated a probable kinematic evidence of the SNR shocking against molecular patches at ${V_{\rm LSR}}\sim-58{\,{\rm km}}{\,{\rm s}^{-1}}$. Hereafter we parameterize the distance to [CTB87]{} and the associated MC as $d=6.1{d_{6.1}}{\,{\rm kpc}}$. We can make a rough estimate of the density of the associated molecular gas. We adopt angular sizes of the two patches of molecular gas in regions “R" and “L" as $\sim 7.5'\times 7.5'$ and $\sim 3.9'\times 3.5'$ (i.e., $\sim 13.3{d_{6.1}}{\,{\rm pc}}\times 13.3{d_{6.1}}{\,{\rm pc}}$ and $\sim 6.9{d_{6.1}}{\,{\rm pc}}\times 6.2{d_{6.1}}{\,{\rm pc}}$), respectively, and assume line-of-sight sizes 13.3pc and 6.6pc for them. The estimated gas density, $n(\mbox{H}_2)$, and gas mass, $M$, for the two patches are given in Tables \[parameter\] and \[parameter2\], respectively.
The multiwavelength spectrum from radio to $\gamma$-rays for [CTB87]{}, which incorporates the TeV emission of VERJ2016+371 [@2014ApJ...788...78A] and the GeV emission from 3FGL J2015.6+3709 [@2012ApJ...746..159K; @2015ApJS..218...23A; @2016ApJS..224....8A], has been explained using a leptonic scenario by [@2016MNRAS.460.3563S], with a combination of a broken power law and a Maxwellian distribution of electrons. Nonetheless, it is also mentioned that the observed GeV–TeV data can be explained by a hadronic scenario as long as there is a dense ambient target medium with a mean hydrogen atom density $\sim 20\,{\rm cm}^{-3}$ or higher (depending on detailed hadronic interaction mechanisms) [@2016MNRAS.460.3563S]. Actually, it is notable that the centroid of the [*VERITAS*]{} very high energy $\gamma$-ray emission [@2014ApJ...788...78A] is essentially coincident with both the radio emission of the SNR and the $\sim-58{\,{\rm km}}{\,{\rm s}^{-1}}$ MC complex. The possible SNR-MC interaction revealed here provides a likely hotbed for the production of hadronic $\gamma$-rays. The gas density of the two molecular patches, a few tens cm$^{-3}$, estimated here, appears to satisfy the hadronic scenario.
[CTB87]{} is very likely to be located within (but near the eastern boundary of) the revealed superbubble, in view of the following issues. (1) The LSR velocity of the [$^{12}$CO]{} emission of the associated MC, $-54$ to $-60{\,{\rm km}}{\,{\rm s}^{-1}}$, is very similar to that of the HI emission for the superbubble, $-61$ to $-68{\,{\rm km}}{\,{\rm s}^{-1}}$. (2) The distance to the SNR, derived both from the foreground intervening neutral hydrogen column density, $6.1\pm0.9{\,{\rm kpc}}$ [@2003ApJ...588..852K], and from the X-ray-absorbing hydrogen column density, $6$ to $8{\,{\rm kpc}}$ (see below), is consistent with the likely peak distance of the OB star candidates that are projected inside the superbubble, 7.3$\pm$1.9kpc, which can be regarded as the distance to the superbubble and the OB star candidates inside. The hydrogen (including atoms and molecules) column density for the entire diffuse X-ray emission region of [CTB87]{}is $N_{\rm H}=1.21$–$1.57{\times 10^{22}}{\,{\rm cm}}^{-2}$ [@2013ApJ...774...33M]. The optical extinction can then be estimated to be $A_V=4.8\pm 0.7$ from the empirical relation $N_{\rm H}=(2.87\pm0.12){\times 10^{21}}A_V{\,{\rm cm}}^{-2}$ [@foight16], which indicates a distance of $\sim6$–8kpc to [CTB87]{}using the $A_V$-distance relation toward this SNR given in [@2003ApJ...588..852K]. (3) There is little detection of thermal X-rays related to the SN ejecta and blast wave, which sets an upper limit $0.2{\,{\rm cm}}^{-3}$ to the density of the medium [@2013ApJ...774...33M]. This can be readily explained if the SN exploded material is blown into the tenuous and hot gas within the superbubble (in [CTB87]{}, another part of the gas may have shocked into the MC; see §\[configuration\]). In the superbubble, the SN ejecta can freely move away at a very high speed, and the blast shock will propagate with a small Mach number so that it can be expected to be weak and become thermalized, with most of the SN kinetic energy carried by this part of the material being deposited in the thermal energy of the hot medium [@2005ApJ...628..205T].
The MC with which the SNR probably interacts may survive the strong radiation of the OB stars in the superbubble. The photodissociation timescale of the MC can be comparable to, or even larger than, the age of the superbubble. For simplicity, the dissociating far ultraviolet (FUV) radiation of the OB stars (of number $N_{\rm OB}$), to which the MC is exposed, is approximated as being from the bubble center, and the molecular column density along the bubble radial is approximated to the line-of-sight column density ${N({\rm H}_2)}$. The distance from the MC to the center, $R_{\rm MC}$, is not less than the projected distance to the center ($\sim22.5'$ or $\sim40{d_{6.1}}$pc). Each dissociating photon is absorbed by a hydrogen molecule with a photodissociation probability, $p_{\rm D}$ $\sim0.15$ (@2011piim.book.....D). Thus, the photodissociation timescale of the MC is given by $\tau_{\rm MC}\ga {N({\rm H}_2)}(4\pi R_{\rm MC}^2)/
(p_{\rm D}N_{\rm OB}S_{\rm D})$, where $S_{\rm D}$ is the mean production rate of Lyman band dissociating photons of an OB star ($\sim10^{47.5}{\,{\rm s}^{-1}}$, @1998ApJ...501..192D). It is estimated to be $\tau_{\rm MC}\ga
6.4{\times 10^{6}}({N({\rm H}_2)}/1{\times 10^{21}}{\,{\rm cm}}^{-2})
(N_{\rm OB}/20)^{-1}
(S_{\rm D}/10^{47.5}{\,{\rm s}^{-1}})^{-1}
{d_{6.1}}^2 {\,{\rm yr}}$, where the reference value $1{\times 10^{21}}{\,{\rm cm}}^{-2}$ is adopted for ${N({\rm H}_2)}$ according to Table \[parameter\] and Table \[parameter2\], and $N_{\rm OB}\sim20$ is adopted as a reference value considering this approximate number of stars are possibly inside and responsible for the superbubble (based on Figure \[obstar\]). On the other hand, the age of the superbubble is estimated to be [@1977ApJ...218..377W] $t_{\rm SB}\sim 5.8{\times 10^{6}} d_{6.1}^{5/3} (N_{\rm OB}/20)^{-1/3}
\times (L_w/10^{34} {\,{\rm erg}}{\,{\rm s}^{-1}})^{-1/3} (n_{0}/0.6
{\rm cm}^{-3})^{1/3} {\rm yr}$, where $L_w$ is the power of the wind of an OB star, and $n_0$ is the atomic number density of the environment medium into which the superbubble expands. Here we use a density of $0.6{\,{\rm cm}}^{-3}$ typical for a warm interstellar HI gas (which takes the largest volume fraction next to the coronal gas in the Galactic disk) to be a reference value for $n_0$ [@2011piim.book.....D]. Thus their ratio is $\tau_{\rm MC}/t_{\rm SB}\ga
1.1\times (N({\rm H}_2)/1{\times 10^{21}}{\rm cm}^{-2})
(N_{\rm OB}/20)^{-2/3}d_{6.1}^{1/3}
(S_{\rm D}/10^{47.5} {\rm s}^{-1})^{-1}
(L_w/10^{34}{\,{\rm erg}}{\,{\rm s}^{-1}})^{1/3} (n_0/0.6 {\rm cm}^{-3})^{-1/3} $. Here we have ignored the formation of new molecules and the extinction of the dissociating photons by dust grains in the molecular gas. This estimate indicates that, with some typical or proper values of parameters adopted, the photodissociation timescale appears not less than the age of the superbubble, and thus the survival of the MC in the superbubble could be reasonable.
It has been suggested that [CTB87]{} is inside another superbubble found in HI emission at ${V_{\rm LSR}}\sim -70 {\,{\rm km}}{\,{\rm s}^{-1}}$, with a radius in the range $\sim38'$–$70'$, centered at approximately R.A.$=20^{\rm h}16^{\rm m}15^{\rm s}$, decl.=$36^\circ 40'$ (J2000) . This superbubble, however, is not as favored as the $\sim-64{\,{\rm km}}{\,{\rm s}^{-1}}$ one in view of the bigger offset of the LSR velocity $\sim-70{\,{\rm km}}{\,{\rm s}^{-1}}$ from the CO-line velocity of the associated MC ($\sim-58{\,{\rm km}}{\,{\rm s}^{-1}}$) than that of $\sim-64{\,{\rm km}}{\,{\rm s}^{-1}}$. Incidentally, we do not find the counterpart of this HI superbubble in the *WISE* near- and mid-IR observations.
Radio and X-ray Configuration {#configuration}
-----------------------------
The X-ray observation of [CTB87]{} shows a cometary-like trailing structure, $\sim 200''\times 300''$ or $\sim 5.9{d_{6.1}}{\,{\rm pc}}\times 8.9{d_{6.1}}{\,{\rm pc}}$ in size at 0.3–7keV [@2013ApJ...774...33M]. The radio emission takes a blow-out, arc-like shape, with the X-ray trail at its symmetric axis, and has a remarkably larger size ($\sim6'\times8'$ or $10.6{d_{6.1}}{\,{\rm pc}}\times14.2 {d_{6.1}}{\,{\rm pc}}$, even up to $16'$) than the X-ray emission and a different brightness peak location from the X-ray one [@2003ApJ...588..852K; @2013ApJ...774...33M]. The apparent one-sided confinement of the X-ray emission represents a typical structure for a relative oriented motion between the ambient gas and the PWN. But why is the radio emission much more extended than the X-ray nebula, with different locations for the brightness peaks in the two bands?
[@2013ApJ...774...33M] suggest that the radio emission of [CTB87]{}is a relic PWN that was crushed by the reverse shock propagating back from somewhere external, and the X-ray nebula is the new trailing PWN after the passage of the reverse shock and the subsequent oscillation of the nebula. The pulsar has moved for about 10 kyr southeastward from the explosion site, assumed to be at the location of the radio brightness peak. However, as the authors point out, no sign of a forward shock (especially in the nearby southeastern region) has been found in this scenario. Also, according to the simulation in [@2001ApJ...563..806B], , and [@kolb17], relic radio PWN are apparently swept/left behind the head of the new, small X-ray nebula. In particular, given the physical contact of the eastern and southwestern edges of the extended radio emission with the MC, this seems to leave no room for the reverse shock that shocks against and crushes the suggested radio relic PWN.
With the probable interaction of [CTB87]{} with the ambient MC revealed here, we suggest instead that the radio emission is mainly a remnant of the blastwave that propagates into the MC at ${V_{\rm LSR}}\sim-58{\,{\rm km}}{\,{\rm s}^{-1}}$, although it may include a contribution from the PWN. This scenario also allows for a reverse shock to have moved backward from the remnant’s edge and crushed the PWN, forming the trailing morphology of X-ray emission, as in the case of other, similar PWNe (e.g. N157B, ). This scenario naturally addresses the question, as noted in [@2013ApJ...774...33M], of the lack of the X-ray emission related to the ejecta and blastwave.
The progenitor exploded somewhere along the symmetric axis of the radio emission or the X-ray trail, with a part of the blastwave shocking against the MC and the other part expanding, and quickly becoming sufficiently faint in a low-density region, possibly a portion of the superbubble. The shock in the MC decelerated and entered the intense radiative stage shortly thereafter, with the shocked gas at that point at too low a temperature to emit X-rays. Actually, the cooling time scale of the shock propagating into the molecular gas is $ t_{\rm cool}=2.8{\times 10^{3}} (E/10^{51}{\,{\rm erg}})^{0.24}
(n_{\rm a}/80{\,{\rm cm}}^{-3})^{-0.52}
$yr [@1981MNRAS.195.1011F], where $E$ is the SNR’s explosion energy and $n_{\rm a}$ is the preshock H atom density. For an average molecular density $n({\rm H}_2)\sim40{\,{\rm cm}}^{-3}$ (see Tables \[parameter\] and \[parameter2\]), the cooling time is much shorter than the spin down time $\sim1{\times 10^{4}}{\,{\rm yr}}$ [@2013ApJ...774...33M]. Although the SNR may have a part blown out, for simplicity we crudely estimate the evolution of the radiative part in the MC as a complete sphere with the radius as $r_s\sim5(\epsilon/0.24)^{5/21}(E/10^{51}{\,{\rm erg}})^{5/21}
(n_{\rm a}/80{\,{\rm cm}}^{-3})^{-5/21}(t/10^{4}{\,{\rm yr}})^{2/7}{\,{\rm pc}},
$ where $\epsilon\sim0.24$ is the energy fraction [@mckee77; @blinnikov82]. If we approximate the pulsar’s spindown time of $\sim10^{4}{\,{\rm yr}}$ as the age of the remnant and adopt an explosion energy $\sim10^{51}{\,{\rm erg}}$, then the radius is $\sim5{\,{\rm pc}}$, very similar to the size of the radio emission. The shock velocity is $ v_s \sim (2/7)r_s/t \sim140(\epsilon/0.24)^{5/21}
(E/10^{51}{\,{\rm erg}})^{5/21}(n_{\rm a}/80{\,{\rm cm}}^{-3})^{-5/21}(t/10^{4}{\,{\rm yr}})^{-5/7}
{\,{\rm km}}{\,{\rm s}^{-1}}.
$ Since MCs are usually highly clumpy, the observed shocked molecular gas showing broad CO-line wings should be in dense clumps, in which the shock velocity can be well below $50{\,{\rm km}}{\,{\rm s}^{-1}}$ [@draine93] so that the molecules are not dissociated. The radio emission as a relic of the blastwave in this scenario indicates a composite nature for SNR [CTB87]{}.
Summary {#summarize}
=======
We have performed a new millimeter CO-line observation toward the region of [CTB87]{}, which was thought to be a filled-center type SNR, and an optical investigation of the coincident superbubble. The CO observation shows that the SNR delineated by the radio emission is projectively covered by a bar-like molecular structure at ${V_{\rm LSR}}=-56.5$ to $-55.5{\,{\rm km}}{\,{\rm s}^{-1}}$. Both the symmetric axis of the radio emission and the trailing X-ray PWN appear projectively to be at a gap between two molecular gas patches at $-58$ to $-57{\,{\rm km}}{\,{\rm s}^{-1}}$. Asymmetric broad line profiles of the $\sim-58{\,{\rm km}}{\,{\rm s}^{-1}}$ [$^{12}$CO]{}-line are obtained from the molecular gas at the eastern and southwestern boundary of the radio emission. This could well be a kinematic evidence of the physical contact between [CTB87]{} and the $\sim-58{\,{\rm km}}{\,{\rm s}^{-1}}$ ($-60$ to $-54{\,{\rm km}}{\,{\rm s}^{-1}}$) MC complex. A superbubble, $\sim37'$ in angular radius and centered at ($20^{\rm h}14^{\rm m}03^{\rm s}$, $37^{\circ}13'27''$, J2000), seemingly surrounding the SNR, is found in HI 21cm (${V_{\rm LSR}}\sim-61$ to $-68{\,{\rm km}}{\,{\rm s}^{-1}}$), *WISE* mid-IR, and optical extinction observations. We built a multi-band photometric stellar sample of over 0.18 million stars within the superbubble region and found 82 OB star candidates. The distribution of the stars’ distances is likely peaked at 7.3kpc (with a dispersion of 1.9kpc) and seems consistent with the previously suggested distance of $6.1\pm0.9{\,{\rm kpc}}$ for [CTB87]{}. We suggest the arc-like radio emission is mainly a relic of the part of the blastwave that is driven into the MC complex, and is now in a radiative stage, while the other part of the blastwave has been expanding into the low-density region, very likely in the superbubble. This scenario naturally explains the lack of X-ray emission related to the ejecta and blastwave. The SNR-MC interaction also favors a hadronic contribution to the $\gamma$-ray emission from the [CTB87]{} region.
We are grateful to the staff of Qinghai Radio Observing Station at Delingha for their help during the observation. We highly appreciate Xin Zhou and Gao-Yuan Zhang for the advice on the data analysis, and Xiao Zhang for the discussion about PWN physics. This work is supported by the 973 Program grants 2015CB857100 and 2017YFA0402600 and NSFC grants 11233001, 11633007, 11773014, 11503008, and 11590781. This research has made use of the NVSS data and NASA’s Astrophysics Data System[^2].
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[ccccc]{} [$^{12}$CO]{}([$J$=1–0]{}) & $-57.5$ & 2.6 & 3.6 & 9.9\
[$^{13}$CO]{}([$J$=1–0]{}) & $-57.1$ & 2.0 & 0.6 & 1.3\
$N$(H$_2$)($10^{21}$cm$^{-2}$) & $n(\mbox{H}_2){d_{6.1}}({\,{\rm cm}}^{-3})$ & $M{d_{6.1}}^{-2}(10^3M_\odot)$ & $T_{\rm ex}$(K) & $\tau$([$^{13}$CO]{})\
\
1.4/1.8 & 34/43 & 4.1/5.1 & 9.5 & 0.18
[ccccc]{} [$^{12}$CO]{}([$J$=1–0]{}) & $-57.5$ & 2.1 & 2.9 & 6.4\
[$^{13}$CO]{}([$J$=1–0]{}) & $-57.4$ & 0.8 & 0.6 & 0.5\
$N$(H$_2$)($10^{21}$cm$^{-2}$) & $n(\mbox{H}_2){d_{6.1}}({\,{\rm cm}}^{-3})$ & $M{d_{6.1}}^{-2}(10^3M_\odot)$ & $T_{\rm ex}$(K) & $\tau$([$^{13}$CO]{})\
\
0.4/1.1 & 21/57 & 0.3/0.8 & 7.5 & 0.21
![Average CO spectra from a $6.5'\times6.5'$ region (centered at R.A.=$20^{\rm h}16^{\rm m}06^{\rm s}.7$, decl.=$37^{\circ}12'32''$) covering the pulsar wind nebula, in the velocity range of $-80 {\,{\rm km}}{\,{\rm s}^{-1}}$– $20 {\,{\rm km}}{\,{\rm s}^{-1}}$. The black line is for [$^{12}$CO]{} ([$J$=1–0]{}), the green line for [$^{13}$CO]{} ([$J$=1–0]{}), and the red line for [C$^{18}$O]{}([$J$=1–0]{}). []{data-label="overallspec"}](overallspec.eps)
![[$^{12}$CO]{} intensity maps integrated each $0.5 {\,{\rm km}}{\,{\rm s}^{-1}}$ in the velocity range of $-40.5$ to $-36.5 {\,{\rm km}}{\,{\rm s}^{-1}}$, overlaid by NVSS 1.4 GHz radio continuum emission in gray contours with levels of 4, 45, 86, 127, 168, 209, and 250 mJy beam$^{-1}$. The lowest contour level is larger than the 5$\sigma$ value of the background. \[channelmap1\]](chanel_U42.eps)
![[$^{12}$CO]{} intensity maps integrated each $0.5 {\,{\rm km}}{\,{\rm s}^{-1}}$ in the velocity range $-61$ to $-54 {\,{\rm km}}{\,{\rm s}^{-1}}$. The contours are the same as those in Figure \[channelmap1\]. \[channelmap2\]](chanel_U60.eps)
![[$^{13}$CO]{} intensity maps integrated each $0.5 {\,{\rm km}}{\,{\rm s}^{-1}}$ in the velocity range of $-61$ to $-54 {\,{\rm km}}{\,{\rm s}^{-1}}$. The contours are the same as those in Figure \[channelmap1\]. \[channelmapl\]](chanel_L.eps)
![Multiwave map of the SNR [CTB87]{}: NVSS 1.4 GHz radio continuum emission in red, [$^{12}$CO]{} ([$J$=1–0]{}) intensity map in velocity interval $-60$ to $-54 {\,{\rm km}}{\,{\rm s}^{-1}}$ in green, *Chandra* X-ray image in energy band 0.5–7 keV in blue. We have also overlaid the contours of the NVSS 1.4 GHz radio continuum emission with levels the same as in Figure \[channelmap1\]. The white cross indicates the point source CXOUJ201609.2+371110 reported in [@2013ApJ...774...33M]. \[overall\]](overall.eps)
![Grid of [$^{12}$CO]{} ([$J$=1–0]{}) and [$^{13}$CO]{} ([$J$=1–0]{}) spectra in the velocity range of $-61$ to $-53 {\,{\rm km}}{\,{\rm s}^{-1}}$. Black lines denote [$^{12}$CO]{} spectra, red lines denote [$^{13}$CO]{}, and dashed lines denote the 0K main-beam temperature. The size of each pixel is $30''\times30''$. The radio contours are the same as those in Figure \[channelmap1\]. Two regions delineated by blue rectangles (labelled as “E” and “SW”) are used to extract CO spectra, in which redward broadened wings are shown for the [$^{12}$CO]{} emission at systemic velocity $\sim-58 {\,{\rm km}}{\,{\rm s}^{-1}}$ (see Figure \[region\]). \[linegrid60\]](linegrid60.eps)
![Averaged spectra of regions “E" (top panel) and “SW" (bottom panel) in the velocity range $-65$–$-50{\,{\rm km}}{\,{\rm s}^{-1}}$. The two regions are defined in Figure \[linegrid60\]. The black line denotes the [$^{12}$CO]{} spectra, the green line represents [$^{13}$CO]{}, and the dotted line represents the 0K main-beam temperature. \[region\]](region.eps)
![Multiwave band morphology of the superbubble in the direction of SNR [CTB87]{}; HI emission around ${V_{\rm LSR}}=-64 {\,{\rm km}}{\,{\rm s}^{-1}}$ in red, *WISE* 22.194 $\mu$m mid-IR image in green, optical extinction map in blue. The radio contours in white are the same as those in Figure \[channelmap1\]. The large yellow circle outlines the superbubble region, and the cyan crosses mark the project positions of the OB star candidates. \[bubble\] ](bubblenewx.eps)
![ Distribution of distance (top panel) and optical extinction (bottom panel) for the OB star candidates ($\log P\ge 5$ and $\log T_{\rm eff}\ge 4.3$) within the projected region of the superbubble (i.e., the circle shown in Figure \[bubble\]). The red curve in the top panel is the Gaussian fitting result of the peak-like component. \[obstar\] ](obstarnew.eps)
\[lastpage\]
[^1]: http://www.iram.fr/IRAMFR/GILDAS
[^2]: http://adswww.harvard.edu/
|
---
abstract: 'Making available and archiving scientific results is for the most part still considered the task of classical publishing companies, despite the fact that classical forms of publishing centered around printed narrative articles no longer seem well-suited in the digital age. In particular, there exist currently no efficient, reliable, and agreed-upon methods for publishing scientific datasets, which have become increasingly important for science. Here we propose to design scientific data publishing as a Web-based bottom-up process, without top-down control of central authorities such as publishing companies. Based on a novel combination of existing concepts and technologies, we present a server network to decentrally store and archive data in the form of nanopublications, an RDF-based format to represent scientific data. We show how this approach allows researchers to publish, retrieve, verify, and recombine datasets of nanopublications in a reliable and trustworthy manner, and we argue that this architecture could be used for the Semantic Web in general. Evaluation of the current small network shows that this system is efficient and reliable.'
author:
- |
Tobias Kuhn, Christine Chichester, Michael Krauthammer, and\
Michel Dumontier
bibliography:
- 'trustypublishing.bib'
title: |
Publishing without Publishers:\
a Decentralized Approach to Dissemination,\
Retrieval, and Archiving of Data
---
Introduction
============
Modern science increasingly depends on datasets, which however are left out in the classical way of publishing, i.e. through narrative (printed or online) articles in journals or conference proceedings. This means that the publications that describe scientific findings get disconnected from the data they are based on, which can seriously impair the verifiability and reproducibility of their results. Addressing this issue raises a number of practical problems: How should one publish scientific datasets and how can one refer to them in the respective scientific publications? How can we be sure that the data will remain available in the future and how can we be sure that data we find on the Web have not been corrupted or tampered with? Moreover, how can we refer to specific entries or subsets from large datasets?
To address some of these problems, a number of scientific data repositories have appeared, such as Figshare and Dryad.[^1] Furthermore, Digital Object Identifiers (DOI) have been advocated to be used not only for articles but also for scientific data [@paskin2005digital]. While these services certainly improve the situation of scientific data, in particular when combined with Semantic Web techniques, they have nevertheless a number of drawbacks: They have centralized architectures, they give us no possibility to check whether the data have been (deliberately or accidentally) modified, and they do not support access or referencing on a more granular level than entire datasets (such as individual data entries). Even if we put aside worst-case scenarios of organizations going bankrupt or becoming uninterested in sustaining their services, their websites have typically not a perfect uptime and might be down for a few minutes or even hours every once in a while. This is certainly acceptable for most use cases involving a human user accessing the data, but it can quickly become a problem in the case of automated access embedded in a larger service. Furthermore, it is possible that somebody gains access to their database and silently modifies part of the data, or that the data get corrupted during the transfer from the server to the client. Below we present an approach to tackle these problems, building upon existing Semantic Web technologies, in particular RDF and nanopublications, and adhering to accepted Web principles, such as decentralization and REST APIs. Specifically, our research question is: Can we create a decentralized, reliable, trustworthy, and scalable system for publishing, retrieving, and archiving datasets in the form of sets of nanopublications based on existing Web standards and infrastructure?
Background {#sec:background}
==========
Nanopublications [@groth2010isu] are a relatively recent proposal for improving the efficiency of finding, connecting, and curating scientific findings in a manner that takes attribution, quality levels, and provenance into account. While narrative articles would still have their place in the academic landscape, small formal data snippets in the form of nanopublications should take their central position in scholarly communication [@mons2011naturegen]. Most importantly, nanopublications can be automatically interpreted and aggregated and they allow for fine-grained citation metrics on the level of individual claims. On the technical level, nanopublications use the RDF language with named graphs [@carroll2005www] to represent assertions, as well as their provenance and metadata. Conceptually, the approach boils down to the ideas of subdividing scientific results into atomic assertions, representing these assertions in RDF, attaching provenance information in RDF on the level of individual assertions, and treating each of these tiny entities as an individual publication. Nanopublications have been applied to a number of domains, so far mostly from the life sciences including pharmacology [@williams2012open], genomics [@patrinos2012humanmutation], and proteomics [@chichester2014sw]. An increasing number of datasets formatted as nanopublications are openly available, including neXtProt [@chichester2014querying] and DisGeNET [@queralt2014semanticweb], and the nanopublication concept has been combined with and integrated into existing frameworks for data discovery and integration, such as CKAN [@mccusker2013next]. Research Objects are a related proposal to establish “self-contained units of knowledge” [@belhajjame2012sepublica], and they constitute in a sense the antipode approach to nanopublications. We could call them “megapublications,” as they contain much more than a typical narrative publication, namely resources like input and output data, workflow definitions, log files, and presentation slides. We demonstrate in this paper, however, that bundling all resources of scientific studies in large packages is not a necessity to ensure reproducibility and trust, but we can achieve these properties also with strong identifiers and a decentralized server network.
SPARQL endpoints, i.e. query APIs to RDF triple stores, are a widely used technique for making linked data available on the Web in a flexible manner. While off-the-shelf triple stores can nowadays handle billions of triples or more, they require a significant amount of resources in the form of memory and processor time to do so, at least if the full expressive power of the SPARQL language is supported. A recent study found that more than half of the publicly accessible SPARQL endpoints are available less than 95% of the time [@builaranda2013iswc], posing a major problem to services depending on them, in particular to those that depend on several endpoints at the same time. To solve these problems, alternative approaches and platforms — such as Linked Data Fragments [@verborgh2014ldow], the Linked Data Platform [@speicher2015ldp], and CumulusRDF [@ladwig2011ssws] — have been proposed, providing less powerful query interfaces and thereby shifting the workload from the server to the client. Fully reliable services, however, can only be achieved with distributed architectures, which have been proposed by a number of existing approaches related to data publishing. For example, distributed file systems that are based on cryptographic methods have been designed for data that are public [@fu2002acm] or private [@clarke2001freenet]. In contrast to the design principles of the Semantic Web, these approaches implement their own internet protocols and follow the hierarchical organization of file systems. Other approaches build upon the existing BitTorrent protocol and apply it to data publishing [@markman2014dlib; @cohen2014xsede], and there is interesting work on repurposing the proof-of-work tasks of Bitcoin for data preservation [@miller2014sp]. There exist furthermore a number of approaches to applying peer-to-peer networks for RDF data [@filali2011lsdkcs], but they do not allow for the kind of permanent and provenance-aware publishing that we propose below. Moreover, only for the centralized and closed-world setting of database systems, approaches exist that allow for robust and granular references to subsets of dynamic datasets [@proell2014data].
Our approach is based on previous work, in which we proposed *trusty URIs* to make nanopublications and their entire reference trees verifiable and immutable by the use of cryptographic hash values [@kuhn2014eswc; @kuhn2015tkde]. This is an example of such a trusty URI:
``` {basicstyle="\small\ttfamily"}
http://example.org/r1.RA5AbXdpz5DcaYXCh9l3eI9ruBosiL5XDU3rxBbBaUO70
```
The last 45 characters of this URI (i.e. everything after “[`.`]{}”) is what we call the *artifact code*. It contains a hash value that is calculated on the RDF content it represents, such as the RDF graphs of a nanopublication. Because this hash is part of the URI, any link to such an artifact comes with the possibility to verify its content, including other trusty URI links it might contain. In this way, the range of verifiability extends to the entire reference tree.
Furthermore, we argued in previous work that the assertion of a nanopublication need not be fully formalized, but we can allow for informal or underspecified assertions [@kuhn2013eswc]. We also sketched how “science bots” could autonomously produce and publish nanopublications, and how algorithms could thereby be tightly linked to their generated data [@kuhn2015savesd], which requires the existence of a reliable and trustworthy publishing system, such as the one we present here.
Approach
========
Our approach builds upon the existing concept of nanopublications and our previously introduced method of trusty URIs. It is a proposal of a reliable implementation of accepted Semantic Web principles, in particular of what has become known as the *follow-your-nose* principle: Looking up a URI should return relevant data and links to other URIs, which allows one (i.e. humans as well as machines) to discover things by navigating through this data space [@berners2006linked]. We argue that approaches following this principle can only be reliable and efficient if we have some sort of guarantee that the resolution and processing of any single identifier will succeed in one way or another and only takes up a small amount of time and resources. This requires (1) that RDF representations are made available on several distributed servers, so the chance that they all happen to be inaccessible at the same time is negligible, and that (2) these representations are reasonably small, so that downloading them is a matter of fractions of a second, and so that one has to process only a reasonable amount of data to decide which links to follow. We address the first requirement by proposing a distributed server network and the second one by building upon the concept of nanopublications. Below we explain the general architecture, the functioning and the interaction of the nanopublication servers, and the concept of nanopublication indexes.
Architecture {#sec:architecture}
------------
![Illustration of current architectures of Semantic Web applications and our proposed approach[]{data-label="fig:swarch"}](swarch.pdf){width="\textwidth"}
There are currently at least three possible architectures for Semantic Web applications (and mixtures thereof), as shown in a simplified manner in Figure \[fig:swarch\]. The first option is the use of plain HTTP GET requests. Applying the follow-your-nose principle, resolvable URIs provide the data based on which the application performs the tasks of finding relevant resources, running queries, analyzing and aggregating the results, and using them for the purpose of the application. If SPARQL endpoints are used, as a second option, most of the workload is shifted from the application to the server via the expressive power of the SPARQL query language. A more reasonable approach, in our view, is the third option of Linked Data Fragments, where servers provide only limited query features and where the tasks are distributed between servers and applications in more balanced fashion. However, all these current solutions are based on two-layer architectures, and have moreover no inherent replication mechanisms. A single point of failure can cause applications to be unable to complete their tasks: A single URI that does not resolve or a single server that does not respond can break the entire process.
We argue here that we need distributed and decentralized services to allow for robust and reliable applications that consume linked data. At the same time, the most low-level task of providing linked data is essential for all other tasks at higher levels, and therefore needs to be the most stable and robust one. We argue that this can be best achieved if we free this lowest layer from all tasks except the provision and archiving of data entries (nanopublications in our case) and decouple it from the tasks of providing services for finding, querying, or analyzing the data. This makes us advocate a multi-layer architecture, a possible realization of which is shown at the bottom of Figure \[fig:swarch\].
Below we present a concrete proposal of such a low-level data provision infrastructure in the form of a nanopublication server network. Based on such an infrastructure, one can then build different kinds of services operating on a subset of the nanopublications they find in the underlying network. “Core services” could involve things like resolving backwards references (i.e. “which nanopublications refer to the given one?”) and the retrieval of the nanopublications published by a given person or containing a particular URI. Based on such core services for finding nanopublications, one could then provide “advanced services” that allow us to run queries on subsets of the data and ask for aggregated output. (These higher layers could of course make use of existing techniques such as SPARQL endpoints and Linked Data Fragments.) While the lowest layer would necessarily be accessible to everybody, some of the services on the higher level could be private or limited to a small (possibly paying) user group. We have in particular scientific data in mind, but we think that an architecture of this kind could also be used for Semantic Web content in general.
Nanopublication Servers
-----------------------
As a concrete proposal of a low-level data provision layer, as explained above, we present here a decentralized nanopublication server network with a REST API to provide and propagate nanopublications identified by trusty URIs.[^2] The nanopublication servers of such a network connect to each other to retrieve and replicate their nanopublications, and they allow users to upload new nanopublications, which are then automatically distributed through the network. Basing the content of this network on nanopublications with trusty URIs has a number of positive consequences for its design: The first benefit is that the fact that nanopublications are all similar in size and always small makes it easy to estimate how much time is needed to process an entity (such as validating its hash) and how much space to store it (e.g. as a serialized RDF string in a database). Moreover it ensures that these processing times remain mostly in the fraction-of-a-second range, guaranteeing quick responses, and that these entities are never too large to be analyzed in memory. The second benefit is that servers do not have to deal with identifier management, as the nanopublications already come with trusty URIs, which are guaranteed to be unique and universal. The third and possibly most important benefit is that nanopublications with trusty URIs are immutable and verifiable. This means that servers only have to deal with *adding* new entries but not with *updating* or *correcting* any of them, which eliminates the hard problems of concurrency control and data integrity in distributed systems. Together, these aspects significantly simplify the design of such a network and its synchronization protocol, and make it reliable and efficient even with limited resources.
Specifically, a nanopublication server of the current network has the following components:
- A **key-value store** of its nanopublications (with the trusty URI as the key)
- A **journal** consisting of a journal identifier and a list of the identifiers of all loaded nanopublications, subdivided into pages of a fixed size.
- Optionally, a **cache of gzipped packages** containing all nanopublications for a given journal page (but they can also be generated on the fly)
- A **list of known peers**, i.e. the URLs of other nanopublication servers
- **Information about each known peer**, including the journal identifier and the total number of nanopublications at the time it was last visited
Based on these components, the servers respond to the following request (in the form of HTTP GET):
- Each server needs to return general **server information**, including the journal identifier and the number of stored nanopublications
- Given an artifact code (i.e. the final part of a trusty URI) of a known nanopublication, the server returns the given **nanopublication** in a format like TriG, TriX, or N-Quads (depending on content negotiation).
- A **journal page** can be requested by page number as a list of trusty URIs.
- For every journal page (except for incomplete last pages), a gzipped **package** can be requested containing the respective nanopublications.
- The **list of known peers** can be requested as a list of URLs.
In addition, a server can optionally support the following two actions (in the form of HTTP POST requests):
- A server may accept requests to **add a given individual nanopublication** to its database.
- A server may also accept requests to **add the URL of a new nanopublication server** to its peer list.
Server administrators have the additional possibility to load nanopublications from the local file system. Together, these server components and their possible interactions allow for efficient decentralized distribution of published nanopublications.
The current system can be seen as an unstructured peer-to-peer network, where each node can freely decide which other nodes to connect to and which nanopublications to replicate. As the network is still very small, the present five nodes connect to all other nodes and replicate all nanopublications they can find. The current implementation is furthermore designed to be run on normal Web servers alongside with other applications, with economic use of the server’s resources in terms of memory and processing time. In order to avoid overload of the server or the network connection, we restrict outgoing connections to other servers to one at a time. The current system and its protocol are not set in stone but, if successful, will have to evolve in the future — in particular with respect to network topology and partial replication — to accommodate a network of possibly thousands of servers and billions of nanopublications.
Nanopublication Indexes
-----------------------
To make the infrastructure described above practically useful, we have to introduce the concept of indexes. One of the core ideas behind nanopublications is that each of them is a tiny atomic piece of data. This implies that analyses will mostly involve more than just one nanopublication and typically a large number of them. Similarly, most processes will generate more than just one nanopublication, possibly thousands or even millions of them. Therefore, we need to be able to group nanopublications and to identify and use large collections of them.
Given the versatility of the nanopublication standard, it seems straightforward to represent such collections as nanopublications themselves. However, if we let such “collection nanopublications” contain other nanopublications, then the former would become very large for large collections and would quickly lose their property of being *nano*. We can solve part of that problem by applying a principle that we can call *reference instead of containment*: nanopublications cannot contain but only refer to other nanopublications, and trusty URIs allow us to make these reference links almost as strong as containment links. To emphasize this principle, we call them *indexes* and not collections.
![Schematic example of nanopublication indexes []{data-label="fig:indexes"}](indexes2.pdf){width="85.00000%"}
However, even by only containing references and not the complete nanopublications, these indexes can still become quite large. To ensure that all such index nanopublications remain *nano* in size, we need to put some limit on the number of references, and to support sets of arbitrary size, we can allow indexes to be appended by other indexes. We set 1000 nanopublication references as the upper limit any single index can directly contain. This limit is admittedly arbitrary, but it seems to be a reasonable compromise between ensuring that nanopublications remain small on the one hand and limiting the number of nanopublications needed to define large indexes on the other. A set of 100,000 nanopublications, for example, can therefore be defined by a sequence of 100 indexes, where the first one stands for the first 1000 nanopublications, the second one appends to the first and adds another 1000 nanopublications (thereby representing 2000 of them), and so on up to the last index, which appends to the second to last and thereby stands for the entire set. In addition, to allow datasets to be organized in hierarchies, we define that the references of an index can also point to sub-indexes. In this way we end up with three types of relations: an index can *append to* another index, it can contain other indexes as *sub-indexes*, and it can contain nanopublications as *elements*. These relations defining the structure of nanopublication indexes are shown schematically in Figure \[fig:indexes\]. Index (a) in the shown example contains five nanopublications, three of them via sub-index (c). The latter is also part of index (b), which additionally contains eight nanopublications via sub-index (f). Two of these eight nanopublications belong directly to (f), whereas the remaining six come from appending to index (e). Index (e) in turn gets half of its nanopublications by appending to index (d). We see that some nanopublications may not be referenced by any index at all, while others may belong to several indexes at the same time. Below we show how this general concept of indexes can be used to define sets of new or existing nanopublications, and how such index nanopublications can be published and their nanopublications retrieved.
Trusty Publishing
-----------------
Let us consider two simple exemplary scenarios to illustrate and motivate the general concepts, using the [`np`]{} command from the [`nanopub-java`]{} library[^3]. Given, for example, a file [`nanopubs.trig`]{} with three nanopublications, we have to assign them trusty URIs before they can be published:
{c}$ np mktrusty -v nanopubs.trig
{o}Nanopub URI: http://example.org/np1#RAQoZlp22LHIvtYqHCosPbUtX8yeGs1Y5AfqcjMneLQ2I
{o}Nanopub URI: http://example.org/np2#RAT5swlSLyMbuD03KzJsYHVV2oM1wRhluRxMrvpkZCDUQ
{o}Nanopub URI: http://example.org/np3#RAkvUpysi9Ql3itlc6-iIJMG7YSt3-PI8dAJXcmafU71s{end}
This gives us the file [`trusty.nanopubs.trig`]{}, which contains transformed versions of the three nanopublications, now having trusty URIs as identifiers. We can now publish these nanopublications to the network:
{c}$ np publish trusty.nanopubs.trig
{o}3 nanopubs published at http://np.inn.ac/{end}
We can check the publication status of the given nanopublications:
{c}$ np status -a http://example.org/np1#RAQoZlp22LHIvtYqHCosPbUtX8yeGs1Y5AfqcjMneLQ2I
{o}URL: http://np.inn.ac/RAQoZlp22LHIvtYqHCosPbUtX8yeGs1Y5AfqcjMneLQ2I
{o}Found on 1 nanopub server.{end}
This is what we see immediately after publication, but only a few minutes later the given nanopublication is found on several servers:
{c}$ np status -a http://example.org/np1#RAQoZlp22LHIvtYqHCosPbUtX8yeGs1Y5AfqcjMneLQ2I
{o}URL: http://np.inn.ac/RAQoZlp22LHIvtYqHCosPbUtX8yeGs1Y5AfqcjMneLQ2I
{o}URL: http://ristretto.med.yale.edu:8080/nanopub-server/RAQoZlp22LHIvtYqHCosPbUtX8yeGs{k}...
{o}URL: http://nanopub-server.ops.labs.vu.nl/RAQoZlp22LHIvtYqHCosPbUtX8yeGs1Y5AfqcjMneLQ2I
{o}URL: http://nanopubs.stanford.edu/nanopub-server/RAQoZlp22LHIvtYqHCosPbUtX8yeGs1Y5Afq{k}...
{o}URL: http://nanopubs.semanticscience.org/RAQoZlp22LHIvtYqHCosPbUtX8yeGs1Y5AfqcjMneLQ2I
{o}Found on 5 nanopub servers.{end}
Next, we can make an index pointing to these three nanopublications:
{c}$ np mkindex -o index.nanopubs.trig trusty.nanopubs.trig
{o}Index URI: http://np.inn.ac/RAXsXUhY8iDbfDdY6sm64hRFPr7eAwYXRlSsqQAz1LE14{end}
This creates a local file [`index.nanopubs.trig`]{} containing the index, identified by the URI shown above. As this index is itself a nanopublication, we can publish it in the same way as described above, and then everybody can conveniently and reliably retrieve the given set of nanopublications:
{c}$ np get -c http://np.inn.ac/RAXsXUhY8iDbfDdY6sm64hRFPr7eAwYXRlSsqQAz1LE14{end}
This command downloads the content of the given index, i.e. the three nanopublications we just created and published.
As another exemplary scenario, let us imagine a researcher in the biomedical domain who is interested in the protein CDKN2A and who has derived some conclusion based on the data found in existing nanopublications. Specifically, let us suppose this researcher analyzed five nanopublications from different sources, specified by the following artifact codes (they can be viewed online by appending the artifact code to the URL ):
{o}RAEoxLTy4pEJYbZwA9FuBJ6ogSquJobFitoFMbUmkBJh0
{o}RAoMW0xMemwKEjCNWLFt8CgRmg_TGjfVSsh15hGfEmcz4
{o}RA3BH_GncwEK_UXFGTvHcMVZ1hW775eupAccDdho5Tiow
{o}RA3HvJ69nO0mD5d4m4u-Oc4bpXlxIWYN6L3wvB9jntTXk
{o}RASx-fnzWJzluqRDe6GVMWFEyWLok8S6nTNkyElwapwno{end}
These nanopublications can be downloaded from the network with the [`np get`]{} command and stored in a file, which we name here [`cdkn2a-nanopubs.trig`]{}. In order to be able to refer to such a collection of nanopublications with a single identifier, a new index is needed that refers to just these five nanopublications. This time we give the index a title (which is optional):
{c}$ np mkindex -t "Data about CDKN2A from BEL2nanopub & neXtProt" \
{c} -o index.cdkn2a-nanopubs.trig cdkn2a-nanopubs.trig
{o}Index URI: http://np.inn.ac/RA6jrrPL2NxxFWlo6HFWas1ufp0OdZzS_XKwQDXpJg3CY{end}
The generated index is stored in the file [`index.cdkn2a-nanopubs.trig`]{}, and our exemplary researcher can now publish this index to let others know about it:
{c}$ np publish index.cdkn2a-nanopubs.trig
{o}1 nanopub published at http://np.inn.ac/{end}
There is no need to publish the five nanopublications this index is referring to, because they are already public (this is how we got them in the first place). The index URI can be used to refer to this new collection of existing nanopublications in an unambiguous and reliable manner, for example as a reference in a paper, as we do it for the datasets of this article [@nanopubindex2015aidagenerif; @nanopubindex2015openbel1; @nanopubindex2015openbel2; @nanopubindex2015disgenet; @nanopubindex2015nextprot].
Evaluation {#sec:eval}
==========
{width="95.00000%"}
To evaluate our approach, we want to find out whether a small server network run on normal Web servers, without dedicated infrastructure, is able to handle the amount of nanopublications we can expect to become publicly available in the next few years. At the time the evaluation was performed, the server network consisted of three servers in Zurich, New Haven, and Ottawa. Two new servers in Amsterdam and Stanford have joined the network since. The current network of five servers is shown in Figure \[fig:map\], which is a screenshot of a nanopublication monitor that we have implemented. Such monitors regularly check the nanopublication server network, register changes (currently once per minute), and test the response times and the correct operation of the servers by requesting a random nanopublication and verifying the returned data.
Evaluation Design
-----------------
[@\*[6]{}[l|rr|rr|l]{}@]{} & & & initial location\
dataset & *index* & *content* & *index* & *content* & for evaluation\
GeneRIF/AIDA [@nanopubindex2015aidagenerif] & 157 & 156,026 & 157,909 & 2,340,390 & New Haven\
OpenBEL 1.0 [@nanopubindex2015openbel1] & 53 & 50,707 & 51,448 & 1,502,574 & New Haven\
OpenBEL 20131211 [@nanopubindex2015openbel2] & 76 & 74,173 & 75,236 & 2,186,874 & New Haven\
DisGeNET v2.1.0.0 [@nanopubindex2015disgenet] & 941 & 940,034 & 951,325 & 31,961,156 & Zurich\
neXtProt [@nanopubindex2015nextprot] & 4,026 & 4,025,981 & 4,078,318 & 156,263,513 & Ottawa\
total & 5,253 & 5,246,921 & 5,314,236 & 194,254,507 &\
Table \[tab:datasets\] shows the existing datasets that we use for the first part of the evaluation. This includes all datasets we are aware of that use trusty URIs, with a total of more than 5 million nanopublications and close to 200 million RDF triples, including nanopublication indexes that we generated for each dataset. The total size of these datasets when stored as uncompressed TriG files amounts to 15.6 GB. Each of the datasets is assigned to one of the three servers, where it is loaded from the local file systems. The first nanopublications start spreading to the other servers, while others are still being loaded from the file system. We therefore test the reliability and capacity of the network under constant streams of new nanopublications coming from different servers, and we use two nanopublication monitors (in Zurich and Ottawa) to evaluate the responsiveness of the network. In the second part of the evaluation we expose a server to heavy load from clients to test its retrieval capacity. For this we use a service called Load Impact[^4] to let up to 100 clients access a nanopublication server in parallel. We test the server in Zurich over a time of five minutes under the load from a linearly increasing number of clients (from 0 to 100) located in Dublin. These clients are programmed to request a randomly chosen journal page, then to go though the entries of that page one by one, requesting the respective nanopublication with a probability of 10%, and starting over again with a different page. As a comparison, we run a second session, for which we load the same data into a Virtuoso SPARQL endpoint on the same server in Zurich (with 16 GB of memory given to Virtuoso and two 2.40 GHz Intel Xeon processors). Then, we perform exactly the same stress test on the SPARQL endpoint, requesting the nanopublications in the form of SPARQL queries instead of requests to the nanopublication server interface. This comparison is admittedly not a fair one, as SPARQL endpoints are much more powerful and are not tailor-made for the retrieval of nanopublications, but they provide nevertheless a valuable and well-established reference point to evaluate the performance of our system.
Evaluation Results
------------------
The first part of the evaluation lasted 13 hours and 21 minutes, at which point all nanopublications were replicated on all three servers, and therefore the nanopublication traffic came to an end. Figure \[fig:dataflow\] shows the type and intensity of the data flow (i.e. the transfer of nanopublications) between the three servers over the time of the evaluation. The network was able to handle an average of about 400,000 new nanopublications per hour, which corresponds to more than 100 new nanopublications per second. This includes the time needed for loading each nanopublication once from the local file system (at the first server), transferring it through the network two times (to the other two servers), and for verifying it three times (once when loaded and twice when received by the other two servers).
{width="\textwidth"}\
{width="\textwidth"}
Figure \[fig:responsetimes\] shows the response times of the three servers as measured by the two nanopublication monitors in Zurich (top) and Ottawa (bottom) from the start of the evaluation until 24 hours later, therefore covering the entire evaluation plus an additional 10 hours and 39 minutes after its end. We see that the observed latency is mostly due to the geographical distance between the servers and the monitors. The response time was always less than 0.25 seconds when the server was on the same continent as the measuring monitor. In 99.86% of all cases (including those across continents) the response time was below 0.5 seconds, and it was always below 1.1 seconds. Not a single one of the 8636 individual HTTP requests timed out, led to an error, or received a nanopublication that could not be successfully verified. We see that the load put onto the network did not have much of an impact on the response times. Except for a handful of spikes, one barely notices the difference between the heavy-load and zero-load situations.
{width="90.00000%"}
Figure \[fig:loadimpact\] shows the result of the second part of the evaluation. The nanopublication server was able to handle 113,178 requests in total (i.e. an average of 377 requests per second) with an average response time of 0.12 seconds. In contrast, the SPARQL endpoint answering the same kind of requests needed 100 times longer to process them (13 seconds on average), consequently handled about 100 times fewer requests (1267), and started to hit the timeout of 60 seconds for some requests when more than 40 client accessed it in parallel. In the case of the nanopublication server, the majority of the requests were answered within less than 0.1 seconds for up to around 50 parallel clients, and this value remained below 0.17 seconds all the way up to 100 clients. As the round-trip network latency alone between Ireland and Zurich amounts to around 0.03 to 0.04 seconds, further improvements can be achieved for a denser network due to the reduced distance to the nearest server.
The first part of the evaluation shows that the overall replication capacity of the current server network is around 9.4 million new nanopublications per day or 3.4 billion per year. The results of the second part show that the load on a server when measured as response times is barely noticeable for up to 50 parallel clients, and therefore the network can easily handle $50 \cdot x$ parallel client connections or more, where $x$ is the number of servers in the network (currently $x = 5$). The second part thereby also shows that the restriction of avoiding parallel outgoing connections for the replication between servers is actually a very conservative measure that could be relaxed, if needed, to allow for a higher replication capacity.
Discussion and Conclusion
=========================
We have presented here a low-level infrastructure for data sharing, which is just one piece of a bigger ecosystem to be established. The implementation of components that rely on this low-level data sharing infrastructure is ongoing and future work. This includes the development of “core services” (see Section \[sec:architecture\]) on top of the server network to allow people to find nanopublications and “advanced services” to query and analyze the content of nanopublications. In addition, we need to establish standards and best practices of how to use existing ontologies (and to define new ones where necessary) to describe properties and relations of nanopublications, such as referring to earlier versions, marking nanopublications as retracted, and reviewing of nanopublications.
Apart from that, we also have to scale up the current small network. As our protocol only allows for simple key-based lookup, the time complexity for all types of requests is sublinear and therefore scales up well. The main limiting factor is disk space, which is relatively cheap and easy to add. Still, the servers will have to specialize, i.e. replicate only a part of all nanopublications, in order to handle really large amounts of data, which can be done in a number of ways: Servers can restrict themselves to nanopublications from a certain internet domain, or to particular types of nanopublications, e.g. to specific topics or authors, and communicate this to the network; inspired by the Bitcoin system, certain servers could only accept nanopublications whose hash starts with a given number of zero bits, which makes it costly to publish; and some servers could be specialized to new nanopublications, providing fast access but only for a restricted time, while others could take care of archiving old nanopublications, possibly on tape and with considerable delays between request and delivery. Lastly, there could also emerge interesting synergies with novel approaches to internet networking, such as Content-Centric Networking [@jacobson2012acm], with which — consistent with our proposal — requests are based on content rather than hosts.
We argue that data publishing and archiving can and should be done in a decentralized manner. We believe that the presented server network can serve as a solid basis for semantic publishing, and possibly also for the Semantic Web in general. It could contribute to improve the availability and reproducibility of scientific results and put a reliable and trustworthy layer underneath the Semantic Web.
[^1]: <http://figshare.com>, <http://datadryad.org>
[^2]: <https://github.com/tkuhn/nanopub-server>
[^3]: <https://github.com/Nanopublication/nanopub-java>
[^4]: <https://loadimpact.com>
|
---
abstract: 'New experimental information has been recently obtained on the odd-odd nucleus $^{134}$I. We interpret the five observed excited states up to the energy of $\sim$3 MeV on the basis of a realistic shell-model calculation, and make spin-parity assignments accordingly. A very good agreement is found between the experimental and calculated energies.'
author:
- 'L. Coraggio$^1$'
- 'A. Covello$^{1,2}$'
- 'A. Gargano$^1$'
- 'N. Itaco$^{1,2}$'
title: 'Shell-model interpretation of high-spin states in $^{134}$I'
---
In a recent paper [@Liu09], excited levels up to an energy of about 3 MeV were identified for the first time in $^{134}$I through the measurements of prompt $\gamma$ rays from the spontaneous fission of $^{252}$Cf. A five transition cascade was observed, but the measured angular correlations were not sufficient to assign spins and parities. In this connection, it may be mentioned that preliminary results were also reported in Ref. [@Mason09] on some new $\gamma$ transitions in $^{134}$I populated in the reaction $^{136}$Xe + $^{208}$Pb. It is the aim of the present paper to give a shell-model interpretation of the observed levels.
The study of nuclei in the vicinity of doubly magic $^{132}$Sn is indeed a subject of great current experimental and theoretical interest. Experimental information on these nuclei, which have been long inaccessible to spectroscopic studies, is now becoming available offering the opportunity to test shell-model calculations in regions of shell closures off stability.
The odd-odd nucleus $^{134}$I with three protons and one neutron hole away from $^{132}$Sn represents an important source of information on the matrix elements of the proton-neutron hole interaction. Actually, the most appropriate system to study this interaction is $^{132}$Sb with only one proton valence particle and one neutron valence hole. Experimental information on this nucleus was provided by the studies of Refs. [@Stone89; @Mach95; @Bhattacharyya01] and there have also been various shell-model studies employing realistic effective interactions [@Andreozzi99; @Coraggio02; @Brown05]. Both the calculations of Refs. [@Coraggio02] and [@Brown05] start from the CD-Bonn nucleon-nucleon ($NN$) potential and derive the effective proton-neutron interaction within the particle-hole formalism. In the former paper, however, the short-range repulsion of the $NN$ potential is renormalized by means of the $V_{\rm low-k}$ approach [@Coraggio09] while in [@Brown05] use is made of the traditional Brueckner $G$-matrix method.
The nucleus $^{134}$I, with an additional pair of protons with respect to $^{132}$Sb, may certainly contribute to improve our knowledge of the two-body effective interaction, since it offers the opportunity to investigate its effects when moving away from the one proton-one neutron hole system. This was also the motivation for extending, in Ref. [@Coraggio02], our shell model calculations to the nucleus $^{130}$Sb with three neutron holes.
In our calculations we consider $^{132}$Sn as a closed core and let the valence protons and neutron hole occupy the five levels $0g_{7/2}$, $1d_{5/2}$, $1d_{3/2}$, $2s_{1/2}$, and $0h_{11/2}$ of the 50-82 shell. The single-particle and single-hole energies have been taken from the experimental spectra [@XUNDL] of $^{133}$Sb and $^{131}$Sn, respectively. The only exception is the proton $\epsilon _{s_{1/2}}$ which has been taken from Ref. [@Andreozzi97], since the corresponding single-particle level is still missing in the spectrum of $^{133}$Sb. Our adopted values for the proton single-particle energies are (in MeV): $\epsilon_{g_{7/2}} = 0.0$, $\epsilon_{d_{5/2}} = 0.962$, $\epsilon_{d_{3/2}} = 2.439$, $\epsilon_{h_{11/2}} = 2.793$, and $\epsilon _{s_{1/2}}= 2.800$, and for the neutron single-hole energies: $\epsilon^{-1}_{d_{3/2}} = 0.0,$ $\epsilon^{-1}_{h_{11/2}} = 0.065$, $\epsilon^{-1}_{s_{1/2}} =0.332$, $\epsilon^{-1}_{d_{5/2}} = 1.655$, and $\epsilon^{-1}_{g_{7/2}} = 2.434$. Note that for the $h_{11/2}$ level we have taken the position suggested in Ref. [@Fogelberg04].
As in our recent studies [@Coraggio05; @Coraggio06; @Covello07; @Gargano09] in the $^{132}$Sn region, we start from the CD-Bonn $NN$ potential [@Machleidt01] and derive the $V_{\rm low-k}$ with a value of the cutoff parameter $\Lambda=2.2$ fm$^{-1}$. This low-momentum potential, with the addition of the Coulomb force for protons, is then used to derive the effective interaction $V_ {\rm eff}$ within the framework of the $\hat Q$-box folded diagram expansion [@Coraggio09] including diagrams up to second order in $V_{\rm low-k}$. The computation of these diagrams is performed within the harmonic-oscillator basis, using intermediate states composed of all possible hole states and particle states restricted to the five shells above the Fermi surface. This guarantees stability of the results when increasing the number of intermediate states. The oscillator parameter is $\hbar \omega = 7.88$ MeV.
As mentioned above, the effective proton-neutron interaction is derived directly in the particle-hole representation, while for the proton-proton interaction we use the particle-particle formalism. The shell-model calculations have been performed using the OXBASH computer code [@Oxbash].
![\[LevelScheme\] The five-transition cascade observed in $^{134}$I is compared with the calculated level scheme for negative-parity yrast states.](Fignew.eps){width="40.00000%"}
Let us now start to present our results by comparing in Table \[134Ilow\] the calculated low-energy spectrum of $^{134}$I with the experimental one [@ENSDF]. The levels shown in this table, which have all been identified in the $^{134}$Te $\beta$ decay, were also observed in the experiment of Ref. [@Liu09]. We see that the experimental energies are very well reproduced by theory, the largest discrepancy, 160 keV, occurring for the $8^-$ state. For the two positive-parity states at 0.181 and 0.210 MeV excitation energy our calculation speaks in favour of a $J=2$ and 3 assignment, respectively.
As regards the wave functions, it turns out that the six considered states are dominated by a single configuration, as is shown by the percentages reported in Table \[134Ilow\]. It is interesting to note that these states may be viewed as the evolution of the six lowest-lying states in $^{132}$Sb, the $8^-$ state being tentatively identified [@ENSDF] with the 0.200 MeV level. As discussed in Ref. [@Coraggio02], the first four positive-parity states in $^{132}$Sb are interpreted as members of the $\pi g_{7/2} \nu d^{-1}_{3/2}$ multiplet , while the $3^+$ and $8^-$ states as the next to the highest $J$ member of the $\pi g_{7/2} \nu s^{-1}_{1/2}$and $\pi g_{7/2} \nu h^{-1}_{11/2}$ multiplets, respectively . The corresponding states in $^{134}$I are dominated by the same proton-neutron hole configuration with the remaining two protons forming a zero-coupled pair. The main feature of the proton-neutron hole multiplets, namely the lowest position of the state with next to the highest $J$, seems to be preserved when adding two valence protons. We find, however, that in $^{134}$I the members of a given multiplet lie in a smaller energy interval with respect to $^{132}$Sb, as is experimentally confirmed in the case of the $\pi g_{7/2} \nu d^{-1}_{3/2}$ multiplet.
We now come to discuss the level scheme identified in Ref. [@Liu09], which is reported in Fig. \[LevelScheme\] together with our shell-model interpretation. The observed $\gamma$ cascade is composed of five transitions and was supposed to be built on the 0.316 MeV $8^-$ isomeric state. With this assumption, we find that the excitation energies of the experimental levels are well reproduced by the calculated yrast sequence shown in the figure. More quantitatively, the $\it {rms}$ deviation between the calculated and experimental energies is about 100 keV.
We have verified that a different assumption for the lowest-lying populated level does not lead to any theoretical sequence that matches well with the experimental energies and is consistent with the observed transitions. It should be mentioned that at about 0.250 MeV below the $10^-$ state we find the yrast $9^-$ state, whose probability to be populated from the $10^-$ state is, however, about 30 times smaller than that relative to the $8^-$ state.
------------------------------------------------------------------------------------------------------------------------
J$_{Exp}^{\pi}$ E$_{Exp}$ J$_{Th}^{\pi}$ E$_{Th}$ Configuration Probability
----------------- ----------- ---------------- ---------- ------------------------------------------------ -------------
(4)$^{+}$ 0 4$^{+}$ 0 $\pi(g_{7/2})^{3} 78
\nu (d_{3/2})^{-1}$
(5)$^{+}$ 44 3$^{+}$ 51 $\pi(g_{7/2})^{3} 77
\nu (d_{3/2})^{-1}$
(3)$^{+}$ 79 5$^{+}$ 122 $\pi(g_{7/2})^{3} 81
\nu (d_{3/2})^{-1}$
(2,3)$^{+}$ 181 2$^{+}$ 158 $\pi(g_{7/2})^{3} 78
\nu (d_{3/2})^{-1}$
(2,3)$^{+}$ 210 3$^{+}$ 312 $\pi(g_{7/2})^{3} 60
\nu (s_{1/2})^{-1}$
$\pi g_{7/2})^{3} \nu (d_{3/2})^{-1}$ 24
(8)$^{-}$ 316 8$^{-}$ 475 $\pi(g_{7/2})^{2} 75
\nu (h_{11/2})^{-1}$
$\pi g_7/2 (d_{3/2})^{2} \nu (h_{11/2})^{-1}$ 12
------------------------------------------------------------------------------------------------------------------------
: \[134Ilow\] Experimental and calculated energies (in keV) of the lowest lying states in $^{134}$I. Wave-function components with a percentage $\ge$ 10% are reported.
In Table \[134Ihigh\], we report the percentages of configurations larger than 10% for the $J^ \pi =$ $10^-$, $11^-$, $12^-$, $13^-$, and $14^-$ states. As in the case of the low-lying states (see Table \[134Ilow\]), each of these high-spin states is dominated by a single configuration. We see that in all five states the neutron hole occupies the $h_{11/2}$ level while the three protons are in the $(g_{7/2})^3$ or $(g_{7/2})^2
d_{5/2}$ configurations, with two protons coupled to $J \ne 0$. In particular, the two highest-lying levels arise from the maximum spin alignment of the corresponding configurations.
J$^{\pi}$ Component Probability
----------- ----------------------------------------------- -------------
10$^{-}$ $\pi (g_{7/2})^3 \nu h_{11/2}^{-1}$ 89
11$^{-}$ $\pi (g_{7/2})^{2} d_{5/2} \nu h_{11/2}^{-1}$ 92
12$^{-}$ $\pi (g_{7/2})^{2} d_{5/2} \nu h_{11/2}^{-1}$ 93
13$^{-}$ $\pi (g_{7/2})^3 \nu h_{11/2}^{-1}$ 66
$\pi (g_{7/2})^{2} d_{5/2} \nu h_{11/2}^{-1}$ 32
14$^{-}$ $\pi (g_{7/2})^{2} d_{5/2} \nu h_{11/2}^{-1}$ 97
: \[134Ihigh\] Wave functions of negative-parity yrast states of $^{134}$I (components $\ge$ 10 % are reported).
In summary, we have given here a shell-model description of $^{134}$I, focusing attention on the energy levels recently identified from the spontaneous fission of $^{252}$Cf [@Liu09]. We have assigned spins and parity to the new observed levels and obtained a very good agreement between experimental and calculated energies. Our shell-model effective interaction has been derived from the CD-Bonn $NN$ potential without using any adjustable parameter, in line with our previous studies of other nuclei in the $^{132}$Sn region. The accurate description obtained for all the investigated nuclei makes us confident in the results of the present work.
[99]{}
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---
abstract: 'We have studied the possible isospin corrections on the skewness and kurtosis of net-baryon and net-charge fluctuations in the isospin asymmetric matter formed in Au+Au collisions at RHIC-BES energies, based on a 3-flavor polyakov-looped Nambu-Jona-Lasinio model. With typical scalar-isovector and vector-isovector couplings leading to the splitting of $u$ and $d$ quark chiral phase transition boundaries and critical points, we have observed dramatic isospin effects on the susceptibilities, especially those of net-charge fluctuations. Reliable experimental measurements at even lower collision energies are encouraged to confirm the observed isospin effects.'
author:
- He Liu
- 'Jun Xu[^1]'
title: Isospin effect on baryon and charge fluctuations from the pNJL model
---
Studying the hadron-quark phase transition and exploring the phase structure of quantum chromodynamics (QCD) matter are the fundamental goals of relativistic heavy-ion collision experiments. Lattice QCD simulations predict that the transition between the hadronic phase and the partonic phase is a smooth crossover at nearly zero baryon chemical potential ($\mu_B \sim 0$) [@Ber05; @Aok06; @Baz12a]. At larger $\mu_B$, the transition can be a first-order one based on investigations from theoretical models, see, e.g., studies from the Nambu-Jona-Lasinio (NJL) model and its extensions [@Asa89; @Fuk08; @Car10; @Bra13]. The knowledge of the QCD critical point (CP) in-between the smooth crossover and the first-order phase transition boundaries is important in mapping out the whole QCD phase diagram [@Ste06]. In order to find the signature of the QCD CP at finite baryon chemical potentials, the Beam Energy Scan (BES) program at the Relativistic Heavy-Ion Collider (RHIC) [@MMA10] has been carrying out $``$low-energy$"$ relativistic heavy-ion collisions and obtained many interesting results. However, heavy-ion collisions with neutron-rich beams produce isospin asymmetric quark matter consisting of different net numbers of $u$ and $d$ quarks, and the isospin degree of freedom is expected to be increasingly important at lower collision energies with larger net baryon densities, related to the QCD phase structure at finite isospin chemical potentials $\mu_I$ [@Son01]. Although lattice QCD can explore the phase transition at finite $\mu_I$ [@Kog02; @Det12; @Bra16], it suffers from the fermion sign problem at finite $\mu_B$ [@Bar86; @Kar02; @Mur03]. Based on the NJL model studies, the isovector couplings may lead to the isospin splittings of chiral phase transition boundaries and critical points for $u$ and $d$ quarks in isospin asymmetric quark matter [@Fra03; @Tou03; @Zha14; @Liu16]. This may influence the search of QCD CP signals in heavy-ion collisions at RHIC-BES energies.
It was proposed [@Ste09] that the non-Gaussian fluctuations of observables in relativistic heavy-ion collisions can be a signal of the phase transition critical point, and the moment of these fluctuations can be used to measure the magnitude of the correlation length. Typically, the susceptibilities of conserved quantities carry information of the QCD phase boundary as well as the position of the critical point [@Asa09]. The above findings have stimulated measurements of the kurtosis and skewness of net-baryon, net-charge, and net-strangeness fluctuations from $\sqrt{s_{NN}}=7.7$ to $200$ GeV at RHIC [@Agg10; @Ada14; @Ada14c; @Ada16; @Ada17], as well as lattice QCD calculations [@Baz12; @Bor13]. For recent reviews on this topic, we refer the reader to Refs. [@Asa16; @Luo17]. However, to the best of our knowledge, the isospin effect on these fluctuations was assumed to be small and has never been addressed. Actually, based on studies from the statistic model [@And09; @Sta14; @Hat16], the isospin chemical potential can be as larger as 10 MeV at the chemical freeze-out stage of heavy-ion collisions at RHIC-BES energies. Taking this isospin effect into consideration, it is expected that the isospin splitting of the chiral phase transition boundaries and the critical points may affect the moments of fluctuations, especially for the net-charge fluctuations. In the present study, we investigate the isospin effects based on the 3-flavor polyakov-looped NJL (pNJL) model with scalar-isovector and vector-isovector couplings incorporated in Ref. [@Liu16].
The thermodynamic potential of the 3-flavor pNJL model with scalar-isovector and vector-isovector couplings at temperature $T$ can be expressed as [@Liu16] $$\begin{aligned}
\label{eq1}
\Omega_{\textrm{pNJL}} &=& \mathcal{U}(\Phi,\bar{\Phi},T)-2N_c\sum_{i=u,d,s}\int_0^\Lambda\frac{d^3p}{(2\pi)^3}E_i
\notag\\
&-&2T\sum_{i=u,d,s}\int\frac{d^3p}{(2\pi)^3}[\ln(1+e^{-3\beta(E_i-\tilde{\mu}_i)}
\notag\\
&+&3\Phi e^{-\beta(E_i-\tilde{\mu}_i)}
+3\bar{\Phi}e^{-2\beta(E_i-\tilde{\mu}_i)})
\notag\\
&+&\ln(1+e^{-3\beta(E_i+\tilde{\mu}_i)}
+3\bar{\Phi} e^{-\beta(E_i+\tilde{\mu}_i)}
\notag\\
&+&3\Phi e^{-2\beta(E_i+\tilde{\mu}_i)})]
+G_S(\sigma_u^2+\sigma_d^2+\sigma_s^2)
\notag\\
&-&4K\sigma_u\sigma_d\sigma_s
+G_V(\rho_u^2+\rho_d^2+\rho_s^3)
\notag\\
&+&G_{IS}(\sigma_u-\sigma_d)^2
+G_{IV}(\rho_u-\rho_d)^2.\end{aligned}$$ In the above, the temperature-dependent effective potential $\mathcal{U}(\Phi, \bar{\Phi}, T)$ as a function of the polyakov loop $\Phi$ and $\bar{\Phi}$ is expressed as [@Fuk08] $$\begin{aligned}
\mathcal{U}(\Phi,\bar{\Phi},T) &=& -b \cdot T\{54e^{-a/T\Phi\bar{\Phi}}
+\ln[1-6\Phi\bar{\Phi}
\notag\\
&-&3(\Phi\bar{\Phi})^2+4(\Phi^3+\bar{\Phi}^3)]\},\end{aligned}$$ with $a = 664$ MeV and $b$ = 0.015$\Lambda^3$ [@Fuk08], where $\Lambda = 750$ MeV is the cutoff value in the momentum integral of the second term in Eg. (\[eq1\]). The factor $2N_c$ with $N_c=3$ represents the spin and color degeneracy, and $\beta = 1/T$ represents the temperature. $G_S$ and $G_V$ are respectively the scalar and vector coupling constants, $K$ is the coupling constant of the six-point Kobayashi-Maskawa-t¡¯ Hooft (KMT) interaction, and $G_{IS}$ and $G_{IV}$ are respectively the strength of the scalar-isovector and vector-isovector couplings that break the SU(3) symmetry while keeping the isospin symmetry. For the ease of discussions, we define $R_{IS} = G_{IS}/G_S$ and $R_{IV} = G_{IV}/G_S$ as the reduced strength of the scalar-isovector and vector-isovector coupling. The energy $E_i$ of quarks with flavor $i$ is expressed as $E_i(p)=\sqrt{p^2 +M_i^2}$, where $M_i$ is the constituent quark mass. In the mean-field approximation, quarks can be considered as quasiparticles with constituent masses $M_i$ determined by the gap equation $$\begin{aligned}
\label{m}
M_i &=& m_i-2G_S\sigma_i+2K\sigma_j\sigma_k-2G_{IS}\tau_{3i}(\sigma_u-\sigma_d),\end{aligned}$$ where $m_i$ is the current quark mass, $\sigma_i$ stands for quark condensate, ($i$, $j$, $k$) is any permutation of ($u$, $d$, $s$), and $\tau_{3i}$ is the isospin quantum number of quark flavor $i$, i.e., $\tau_{3u} = 1$, $\tau_{3d} = -1$, and $\tau_{3s} = 0$. As shown in Eq. , $\sigma_d$ and $\sigma_s$ contribute to the constituent quark mass of $u$ quarks as a result of the six-point interaction and the scalar-isovector coupling. Similarly, the effective chemical potential expressed as $$\begin{aligned}
\tilde{\mu}_i&=&\mu_i+2G_V\rho_i+2G_{IV}\tau_{3i}(\rho_u-\rho_d)\end{aligned}$$ has also the contribution from quarks of other isospin states through the vector-isovector coupling. The net quark number density of flavor $i$ can be calculated from $$\begin{aligned}
\rho_i=2N_c\int(f_i-\bar{f_i})\frac{d^3p}{(2\pi)^3}\end{aligned}$$ where $$\begin{aligned}
f_i=\frac{1+2\bar{\Phi}\xi_i+\Phi\xi_i^2}{1+3\bar{\Phi}\xi_i+3\Phi\xi_i^2+\xi_i^3}\end{aligned}$$ and $$\begin{aligned}
\bar{f_i}=\frac{1+2\Phi{\xi}'_i+\bar{\Phi}{{\xi}'_i}^2}{1+3\Phi{\xi}'_i+3\bar{\Phi}{{\xi}'_i}^2+{{\xi}'_i}^3}\end{aligned}$$ are the effective phase-space distribution functions for quarks and antiquarks in the pNJL model with $\xi_i= e^{(E_i-\tilde{\mu_i})/T}$ and ${\xi}'_i= e^{(E_i+\tilde{\mu_i})/T}$.
In the present study, we adopt the values of parameters [@Bra13; @Lut92] as $m_u = m_d = 3.6$ MeV, $m_s = 87$ MeV, $G_S\Lambda^2 = 3.6$, and $K\Lambda^5 = 8.9$. Since the purpose is not to study the effect of $G_V$ on the structure of phase diagram [@Asa89; @Fuk08; @Car10; @Bra13], it is set to 0 in the present study. The following equations $$\begin{aligned}
\frac{\partial\Omega_{\textrm{pNJL}}}{\partial\sigma_u}
=\frac{\partial\Omega_{\textrm{pNJL}}}{\partial\sigma_d}
=\frac{\partial\Omega_{\textrm{pNJL}}}{\partial\sigma_s}
=\frac{\partial\Omega_{\textrm{pNJL}}}{\partial\Phi}
=\frac{\partial\Omega_{\textrm{pNJL}}}{\partial\bar{\Phi}}
=0
\notag\\\end{aligned}$$ are used to obtain the values of $\sigma_u$, $\sigma_d$, $\sigma_s$, $\Phi$, and $\bar{\Phi}$ at the minimum thermodynamic potential in the pNJL model.
In the following, we consider fluctuation moments of conserved quantities, such as net-baryon and net-charge fluctuations, from the above 3-flavor pNJL model. The $n$-order susceptibility representing the cumulant of a given conserved quantity in the grand ensemble can be expressed as the derivative of the thermodynamic potential as $$\begin{aligned}
\chi^{(n)}_X = \frac{\partial^n(-\Omega/T)}{\partial(\mu_X/T)^n},\end{aligned}$$ where $\mu_X$ represents the baryon ($\mu_B$) or the charge ($\mu_Q$) chemical potential. Numerically, the isospin chemical potential $\mu_I$ and the charge chemical potential $\mu_Q$ are equal to each other [@Baz12; @Che15]. In the present study, we use the empirical relation between the isospin chemical potential and the baryon chemical potential, i.e., $\mu_I = -0.293 - 0.0264\mu_B $ (MeV), determined from the statistical model fits of the experimental data from Au+Au collisions from center-of-mass energy $\sqrt{s_{NN}}$ = 7.7 GeV to 200 GeV [@And09; @Sta14; @Hat16]. For the strangeness chemical potential, we take the empirical relation as $\mu_S = 1.032 + 0.232\mu_B $ (MeV) [@And09; @Sta14; @Hat16]. Note that the chemical potentials of $u$, $d$, and $s$ quarks in the 3-flavor pNJL model can be expressed in terms of $\mu_B$, $\mu_I$, and $\mu_S$. The higher-order susceptibilities are related to the skewness $S$ and kurtosis $\kappa$ measured experimentally in relativistic heavy-ion collisions through the relations $$\begin{aligned}
S\sigma = \frac{\chi^{(3)}}{\chi^{(2)}},
\quad \kappa\sigma^2 = \frac{\chi^{(4)}}{\chi^{(2)}},\end{aligned}$$ where $\sigma$ is the variance of the conserved quantity. The subscript of the net baryon (B) or the net charge (Q) is omitted in the above equation.
We begin our discussion with the higher-order fluctuations of net baryon and net charge from the 3-flavor pNJL model in the $\mu_B - T$ plane with various isovector coupling constants, as shown in Fig. \[phase\]. As shown in Fig. 3 of Ref. [@Liu16], $R_{IS}=0.14$ and $R_{IV}=0.5$ lead to the splittings of $u$ and $d$ quark chiral phase transition boundaries as well as their critical points, which are plotted in all panels of Fig. \[phase\] for references. We note that the empirical relation $\mu_I = -0.293 - 0.0264\mu_B$ (MeV) is only well valid near the phase boundary, and this is exactly the region where we are interested. As seen in Fig. \[phase\], chiral phase transition boundaries separate the red and blue areas for skewness results, representing respectively the positive and negative values of $S\sigma$. For kurtosis results, however, the chiral phase transition boundaries go through the blue areas, and the critical points stand at the ends of the blue areas. It is also interesting to see that the skewness of net-charge fluctuations gives the different orders of the red and blue areas compared to that of net-baryon fluctuations. Using the empirical relation between $\mu_I$ and $\mu_B$, the splitting of the chiral phase transition boundaries from $R_{IV}=0.5$ is much weaker compared to that observed in Ref. [@Liu16].
 
In relativistic heavy-ion collision experiments, the net-baryon and net-charge fluctuations are measured at chemical freeze-out. It is well-known that the phase transition temperature from lattice QCD calculations at zero baryon density is the same as that extracted from the statistical model at top RHIC energy or LHC energy, leading to the conclusion that the chemical freeze-out happens right after hadron-quark phase transition in extreme relativistic heavy-ion collisions. In collisions at RHIC-BES energies, when the chemical freeze-out happens is not known. There are several empirical criteria for chemical freeze-out in relativistic heavy-ion collisions, such as fixed energy per particle, baryon+antibaryon density, normalized entropy density, as well as percolation model and so on (see Ref. [@Cle06] and references therein). In order to compare qualitatively the higher-order susceptibilities from the pNJL model with experimental results, we obtain the hypothetical chemical freeze-out lines by rescaling $\mu_B$ of the averaged chiral phase transition boundaries of $u$ and $d$ quarks with factors of $0.98$, $0.95$, and $0.90$, corresponding respectively to the dash-dotted, dashed, and dotted curves in Fig. \[phase-boundary\]. We note that a similar assumption of the chemical freeze-out lines was made in Ref. [@Che17], and the present hypothetical chemical freeze-out lines are always below the chiral phase transition boundaries of both $u$ and $d$ quarks. For the net-baryon susceptibility shown in the upper panels of Fig. \[phase-boundary\], it is seen that $S\sigma(B)$ has one positive peak while $\kappa\sigma^2(B)$ has a positive and a negative peak along the chemical freeze-out lines, and the peaks are sharper if the hypothetical chemical freeze-out lines are closer to the chiral phase transition boundary. The critical point of the $d$ quark chiral phase transition is always at the low-temperature or low-energy side of the positive peaks for both $S\sigma(B)$ and $\kappa\sigma^2(B)$, and the distance between the positive peaks and the critical point is related to that between the chemical freeze-out line and the chiral phase transition boundary. For the net-charge susceptibility shown in the lower panels of Fig. \[phase-boundary\], we have also observed sharper peaks if the hypothetical chemical freeze-out line is closer to the chiral phase transition boundary. It is seen that $S\sigma(Q)$ has a negative peak at lower temperatures/energies and a positive peak at higher temperatures/energies, and the broad positive peak of $\kappa\sigma^2(Q)$ turns to two positive peaks and a negative one if the hypothetical chemical freeze-out line is very close to the chiral phase transition boundary. Again, the critical point of the $d$ quark chiral phase transition is at the low-temperature or low-energy side of the negative peak for $S\sigma(Q)$ or the positive peak for $\kappa\sigma^2(Q)$.
In order to compare the higher-order susceptibilities with and without the isospin effect, we display in Fig. \[boundary\] the skewness and kurtosis of net-baryon and net-charge fluctuations with four different scenarios of isospin chemical potentials and isovector couplings: $\mu_I = 0$, $R_{IS} = 0$, $R_{IV} = 0$; $\mu_I = -0.293 - 0.0264\mu_B$ (MeV), $R_{IS} = 0$, $R_{IV} = 0$; $\mu_I = -0.293 - 0.0264\mu_B$ (MeV), $R_{IS} = 0.14$, $R_{IV} = 0$; $\mu_I = -0.293 - 0.0264\mu_B$ (MeV), $R_{IS} = 0$, $R_{IV} = 0.5$. In order to illustrate the largest possible isospin effect, the susceptibilities are calculated along the closest hypothetical chemical freeze-out line to the phase boundary, i.e., using the rescaling factor of 0.98. It is seen that even without isovector couplings, the skewness and kurtosis can be slightly different for $\mu_I=0$ and $\mu_I \neq 0$, especially for net-charge susceptibilities. With finite isovector coupling constants, the isospin effect is largely enhanced. The peaks of the net-baryon susceptibilities move to the low-temperature or low-energy side, especially for $R_{IS}=0.14$. The general shape of the net-baryon susceptibility is qualitatively consistent with the experimental data [@Ada14; @Luo]. The net-baryon susceptibility is not a unique probe of the isospin effect, since it is largely affected by other effects as well, such as the vector coupling. On the other hand, the isospin couplings affect dramatically the net-charge susceptibilities. It is seen that the isovector couplings lead to a negative peak at lower temperatures/energies for $S\sigma(Q)$. For $\kappa\sigma^2(Q)$, the isovector couplings lead to two peaks and $R_{IS}=0.14$ is the only scenario that leads to negative values. The experimental results from STAR and PHENIX Collaborations for net-charge susceptibilities are not consistent with each other yet [@Ada14c; @Ada16]. So far the experimental results for $S\sigma(Q)$ seem to be positive above $\sqrt{s_{NN}}=7.7$ GeV from both STAR and PHENIX measurements, and it is of great interest to distinguish different scenarios if reliable measurements are done at even lower collision energies. For the $\kappa\sigma^2(Q)$ results, the STAR results lead to negative values at lower $\sqrt{s_{NN}}$ [@Ada14c] while the PHENIX results remain positive at all energies [@Ada16]. It is again of great interest to check experimentally whether another peak appears at even lower collision energies, as seen from the $\kappa\sigma^2(Q)$ results with $R_{IS}=0.14$.
To summarize, based on the 3-flavor polyakov-looped Nambu-Jona-Lasinio model with the scalar-isovector and vector-isovector couplings, we have studied the higher-order susceptibilities of net-baryon and net-charge fluctuations in isospin asymmetric matter formed in Au+Au collisions at RHIC-BES energies. Although the general features remain the same, the isovector couplings move the peaks of the net-baryon susceptibilities to the low-temperature or low-energy side. On the other hand, the shape of the net-charge susceptibilities is largely changed with the isovector couplings, i.e., an additional negative peak appear in the skewness results and two positive peaks peaks appear in the kurtosis results along the hypothetical chemical freeze-out line, if it is very close to the phase boundary. It is of great interest to confirm our findings by measuring the net-charge susceptibility at even lower collision energies, such as that in the future FAIR-CBM program. Such analysis may also be helpful in extracting the strength of the isovector couplings or even the information of the isospin dependence of the QCD phase diagram.
We acknowledge helpful communications with Xiao-Feng Luo. This work was supported by the Major State Basic Research Development Program (973 Program) of China under Contract Nos. 2015CB856904 and 2014CB845401, the National Natural Science Foundation of China under Grant Nos. 11475243 and 11421505, the “100-talent plan” of Shanghai Institute of Applied Physics under Grant Nos. Y290061011 and Y526011011 from the Chinese Academy of Sciences, and the Shanghai Key Laboratory of Particle Physics and Cosmology under Grant No. 15DZ2272100.
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[^1]: correspond author: [email protected]
|
---
abstract: 'In this paper we state a full classification for Coxeter polytopes in $\mathbb{H}^{n}$ with $n+3$ facets which are non-compact and have precisely one non-simple vertex.'
author:
- |
<span style="font-variant:small-caps;">Mike Roberts</span>\
Liverpool University\
<[email protected]>
bibliography:
- 'biblio.bib'
title: '**A Classification of Non-Compact Coxeter Polytopes with $n+3$ Facets and One Non-Simple Vertex**'
---
=1
Introduction
============
A polytope $P$ in hyperbolic $n$-space $\mathbb{H}^{n}$ is called a Coxeter polytope if bounded by hyperplanes which intersect at angle $\frac{\pi}{m_{ij}}$, $m_{ij}\in\mathbb{N}, m_{ij}\ge 2$ for hyperplanes $H_{i}$ and $H_{j}$. They have the interesting property that they tessellate the space $\mathbb{H}^{n}$ through reflections in the bounding hyperplanes.
We can define $P$ as $P=\cap_{i\in J} H_{i}^{-}$, $J$ an arbitrary index set and $H_{i}^{-}$ the half-space containing $P$. The intersection of a Coxeter polytope with a bounding hyperplane is called a facet of the polytope.
If the Coxeter polytope $P$ has a vertex at infinity, i.e. the hyperplanes intersect at infinity to form a vertex, then $P$ is said to be non-compact, otherwise we say $P$ is compact. Also, if all vertices of $P\subset\mathbb{H}^{n}$ are formed by the intersection of precisely $n$ hyperplanes then $P$ is said to be simple, otherwise we say $P$ is non-simple.
All Coxeter polytopes with $n+1$ facets (simplices) have been fully classified [@chein1969; @coxeter1950world; @lannr1950complexes]. Coxeter polytopes with $n+2$ facets have been fully classified by Kaplinskaya [@kaplinskaya1974discrete] for simplicial prisms, Esselmann [@Esselmann:1996aa] for compact Coxeter polytopes and Tumarkin [@Tumarkin:2004ab] for non-compact Coxeter polytopes.
In the case of Coxeter polytopes with $n+3$ facets Esselmann [@esselmann1994kompakte] began and Tumarkin [@Tumarkin:2007aa] completed the classification in the compact case. The non-compact case remains without classification currently, however there are some results.
In Tumarkin [@Tumarkin:2004aa] there is the following theorem:
There are no Coxeter polytopes of finite volume with $n+3$ facets in $\mathbb{H}^{n}$ of dimension $n\ge 17$. There is just one such polytope in $\mathbb{H}^{16}$ and it has the following Coxeter diagram:
Andreev [@andreev1970convex] proved that in dimension $3$ there are infinitely many Coxeter polytopes with $n+3$ facets.
Non-simple pyramids have been classified by Tumarkin [@Tumarkin:2004aa] and will not be considered in this text. Therefore a classification is desired for dimensions $4$ to $16$ for non-compact non-pyramidal Coxeter polytopes with $n+3$ facets.
This paper focusses on those polytopes with precisely one non-simple vertex, i.e. the Coxeter polytopes in $\mathbb{H}^{n}$ which have one vertex formed by the intersection of $n+1$ hyperplanes. A full classification of these is displayed in Appendix \[sec:newpoly\].
The majority of this text is copied directly from my Masters dissertation in which a near classification of the non-compact non-pyramidal Coxeter polytopes with $n+3$ facets and one non-simple vertex was obtained.
Gram matrices {#sec:grammatrices}
=============
Gram matrices are an elegant way to display and encode the structural information about a Coxeter polytope in matrix form. They also provide the opportunity to question the structure of the Coxeter polytope from a Linear Algebra approach by, for example, considering the eigenvalues and determinant of the Gram matrix.
\[defn:grammatrix\] The Gram matrix $G=(g_{ij})$ of a Coxeter polytope $P=\bigcap_{i\in J}H_{i}^{-}\subset\mathbb{H}^{n}$ bounded by $|J|$ hyperplanes is a $|J|\times |J|$ matrix where:
1. [$g_{ij}=1$ if $i=j$.]{}
2. [$g_{ij}=-cos(\frac{\pi}{m_{ij}})$ if $H_{i},H_{j}$ intersect.]{}
3. [$g_{ij}=-1$ if $H_{i},H_{j}$ are parallel.]{}
4. [$g_{ij}=-cosh(\rho(H_{i},H_{j}))$ if $H_{i}$ and $H_{j}$ are ultraparallel, where $\rho(H_{i},H_{j})$ is the distance between $H_{i}$ and $H_{j}$.]{}
Constructing a Gram matrix in this way encodes the angles or distances between the hyperplanes that bound $P$ into a matrix. The matrix is square and from its construction it can be seen to be symmetric, with the number of columns and rows equal to the number of hyperplanes bounding $P$.
\[rmk:detzero\] Consider the columns of the Gram matrix as vectors. $\mathbb{H}^{n}$ can be represented as $\mathbb{E}^{n,1}$ ($n+1$ dimensional Euclidean space with Lorentzian metric). If there are more than $n+1$ columns (vectors) there is a linear dependence between some of the rows or columns and so the Gram matrix has determinant zero.
This text will consider Coxeter polytopes in $\mathbb{H}^{n}$ with $n+3$ facets and so the Gram matrix of the Coxeter polytope will have determinant zero. A minor formed from $n+2$ rows and columns of this Gram matrix will similarly have determinant zero.
Coxeter diagrams \[sec:coxdiagrams\]
====================================
Coxeter diagrams encode into a two-dimensional diagram the structure of the bounding hyperplanes of the Coxeter polytope. A Coxeter polytope has a unique Coxeter diagram.
\[def:vincox\]For a Coxeter polytope $P$, construct a graph where the vertices $a_{i}$ represent the hyperplanes $H_{i}$ and each node is joined to every other by either a solid, bold or dashed line. The edge between $a_{i}, a_{j}$ is one of the following:
- [$(m_{ij}-2)-$fold line which is unlabelled (or single line labelled $m_{ij}$) where the dihedral angle between $H_{i}$ and $H_{j}$ is $\frac{\pi}{m_{ij}}$ (note that when $m_{ij}=2$ this is represented with no line),]{}
- [a thick line (or single line labelled $\infty$) when the hyperplanes $H_{i}$ and $H_{j}$ are parallel; or]{}
- [a dashed line labelled $cosh(\rho(H_{i},H_{j}))$ when the hyperplanes $H_{i}$ and $H_{j}$ are ultraparallel, where $\rho(H_{i},H_{j})$ is the distance between $H_{i}$ and $H_{j}$.]{}
By reference to Definition \[defn:grammatrix\] we see that the Coxeter diagram for a given polytope has the same information contained in it as in the Gram matrix for that polytope, i.e. each Coxeter diagram corresponds to a unique Gram matrix. Therefore the Coxeter diagram defines the combinatorics of the polytope by Vinberg [@Vinberg:1985aa] §3.
Gale diagrams \[sec:galediagrams\]
==================================
A Gale diagram is obtained by Gale Transform (linear algebra operation) on the vertices of the polytope. Gale transforms will not be discussed here but for more information see, for example, Grünbaum [@Grunbaum:1967aa].
A Gale diagram is of the form $\mathbb{S}^{p-n-2}$ where $p$ is the number of bounding hyperplanes and $n$ is the dimension of the space. There is a node (vertex) labelled $1$ on the boundary of the $(p-n-2)$-sphere for each of the $k$ hyperplanes.
This text focusses on Coxeter polytopes with $n+3$ bounding hyperplanes and hence the Gale diagram is $\mathbb{S}^{1}$, i.e. a circle, and the nodes for the hyperplanes are on the circumference of it.
We use a reformulation of the definition of the Gale diagram taken from Tumarkin [@Tumarkin:2004aa] p1 which will give a standard (contracted) form for Gale diagrams.
\[defn:galediagrams\] Every combinatorial type of an $n$-dimensional polytope with $n + 3$ facets can be represented by a standard two-dimensional Gale diagram. This consists of vertices of regular $2k-$gon in $\mathbb{E}^{2}$ centered at the origin which are labelled according to the following rules:
1. [Each label is a non-negative integer, the sum of labels equals $n + 3$.]{}
2. [Labels of neighbouring vertices cannot be equal to zero simultaneously.]{}
3. [Labels of opposite vertices can not be equal to zero simultaneously.]{}
4. [The points that lie in any open halfspace bounded by a hyperplane through the origin have labels whose sum is at least two.]{}
\[rmk:galefaces\] The combinatorial type of a convex polytope can be read off from the Gale diagram in the following way. Each vertex $a_{i}, i = 1, . . . , 2k$, with label $\mu_{i}$ corresponds to $\mu_{i}$ facets $f_{i,1},\ldots,f_{i,\mu_{i}}$ of $P$. For any subset $I$ of the set of facets of $P$ the intersection of facets $\{ f_{j,\gamma} | (j,\gamma )\in I \}$ is a face of $P$ if and only if the origin is contained in the set $conv \{ a_{j} | ( j , \gamma ) \not\in I \}$.
Methodical approach for constructing Coxeter polytopes
======================================================
Introduction
------------
In this section we will detail the rigorous approach taken to classify the non-compact non-pyramidal Coxeter polytopes with $n+3$ facets and one non-simple vertex. We start by introducing some notation and then two important lemmas.
If $G$ is the Gale diagram of a polytope $P$, denote by $S_{m,l}$ the subdiagram of the Coxeter diagram $S(P)$ corresponding to the $l-m+1$ (mod $2k$) consecutive vertices $a_{m},\ldots, a_{l}$ of $G$. For $l=m$, denote this subdiagram by $S_{m}$. The weight of the vertex is denoted by $\mu(a_{i})$.
\[lem:tum1\] Let $G$ be the Gale diagram of a polytope $P$. Suppose that the weights of $a_{i},a_{k+i}$ are non-zero. Then:
1. [the vertices $a_{i}$ and $a_{k+i}$ have weight $1$ and the Coxeter diagrams $S_{i+1,k+i-1}$ and $S_{k+i+1,i-1}$ are connected and parabolic.]{}
2. [if $a_{i+1}$ and $a_{k+i+1}$ have non-zero weights, the Coxeter diagram $S_{i+1,k+i}$ is quasi-Lannér.]{}
3. [if $a_{i+1}$ has weight zero, the Coxeter diagram $S_{i+2,k+i}$ is quasi-Lannér.]{}
\[lem:tum2\] Let $G$ be the Gale diagram of a polytope $P$, and suppose that the weights of the vertices $a_{i}$ and $a_{k+i-1}$ are zero. Then the diagram $S_{i+1,k+i-2}$ is Lannér.
We see that a non-simple Coxeter polytope is represented by a Gale diagram which has pairs of nodes opposite one another around the circumference and whose labels must be $1$ (Lemma \[lem:tum1\].1). In this text we focus on those Gale diagrams which are a circle with precisely one pair of nodes opposite one another.
Notice that there are only a finite number of Gale diagrams in each dimension which have one pair of nodes opposite each other, this can be seen as the task of splitting integer $n+1$ over the nodes which are not opposite one another.
As the final point before detailing the methodical approach taken, we quote the following important theorem:
\[thm:vin21\] Let $G=(g_{ij})$ be an indecomposable symmetric matrix of signature $(n,1)$ with $1's$ along the diagonal and non-positive entries off it. Then there is a convex polytope $P$ in $\mathbb{H}^{n}$ whose Gram matrix is $G$. The polytope $P$ is uniquely determined up to isometry in $\mathbb{H}^{n}$.
The method explained
--------------------
Suppose we look to find all non-compact non-pyramidal Coxeter polytopes in $\mathbb{H}^{n}$ which have $n+3$ facets and one non-simple vertex, then applying the following method will give all such Coxeter polytopes.
1. [Determine all possible Gale diagrams up to congruence.]{}
2. [Apply Tumarkin’s Lemma’s \[lem:tum1\] and \[lem:tum2\]. These will restrict Coxeter subdiagrams which can be formed.]{}
3. [Apply the restrictions which can be read off from the Gale diagram such as those subdiagrams which are necessarily elliptic, etc.]{}
4. [Form a Gram matrix $G$ for the candidate Coxeter diagrams and check that the determinant of $G$ and of any $(n+2)\times(n+2)$ minor is zero (Remark \[rmk:detzero\])]{}
5. [Apply Vinberg’s Theorem \[thm:vin21\] and check the signature of the Gram matrix $G$. If the signature is $(n,1,2)$ then the Gram matrix $G$ corresponds to a Coxeter polytope.]{}
Results and Further Work
========================
The full classification for the non-compact non-pyramidal Coxeter polytopes with $n+3$ facets and precisely one non-simple vertex is shown in Appendix \[sec:newpoly\]. It is interesting to note that there are no examples in dimension 11 and above.
To obtain a full classification of the non-compact Coxeter polytopes with $n+3$ facets it remains to classify such simple Coxeter polytopes and non-simple non-pyramidal Coxeter polytopes with more than one non-simple vertex.
Once this is completed then along with along with [@Tumarkin:2007aa] and [@Tumarkin:2004aa] this will complete the classification of Coxeter polytopes with $n+3$ facets.
The majority of this work was completed at Durham University as a part of my Masters thesis. I am extremely grateful to my supervisor Dr Pavel Tumarkin for all his help and advice whilst I was performing the work and for his helpful comments on this paper. I’m also thankful to Rafael Guglielmetti for his efforts in verifying the Coxeter polytopes found and providing me with corrections.
New Coxeter Polytopes \[sec:newpoly\]
=====================================
This section details the new Coxeter polytopes found which have $n+3$ facets, are non-compact non-pyramidal and have precisely one non-simple vertex. They are split over the following subsections by the dimension in which they exist.
The nodes are labelled by the location of that node in its Gale diagram. For example, a node labelled $j$ will appear at location $a_{j}$ in the Gale diagram with label $1$, a node with label $i,j$ (with $j\in\{ 1,\ldots s\}$) will be one hyperplane from node $a_{i}$ in the Gale diagram (which has label $s$). This is purely to provide an easy trail back to the Gale diagram for the Coxeter polytope and these node labels may be ignored.
Dimension 4
-----------
Dimension 5
-----------
Dimension 6
-----------
Dimension 7
-----------
Dimension 8
-----------
Dimension 9
-----------
Dimension 10
------------
|
---
abstract: 'Potential effects of sublimation of water ice from very slowly moving millimeter-sized and larger grains. a product of activity at $\sim$10 AU or farther from the Sun driven presumably by annealing of amorphous water ice, are investigated by comparing 2I/Borisov with a nominal Oort Cloud comet of equal perihelion distance of 2 AU. This comparison suggests that the strongly hyperbolic motion of 2I mitigates the integrated sublimation effect. The population of these grains near the nucleus of 2I is likely to have been responsible for the comet appearing excessively bright in pre-discovery images at 5–6 AU from the Sun, when the sublimation rate was exceedingly low, as well as for the prominent nuclear condensation more recently. All grains smaller than 2-3 cm across had been devolatilized by mid-October 2019 and some subjected to rapid disintegration. This left only larger chunks of the initial icy-dust halo contributing to the comet’s strongly suspected hyperactivity. Sublimation of water ice from the nucleus has been increasing since the time of discovery, but the rate has not been high enough to exert a measurable nongravitational acceleration on the orbital motion of 2I/Borisov.'
author:
- Zdenek Sekanina
title: |
Sublimation of Water Ice from a Population of Large, Long-Lasting Grains\
Near the Nucleus of 2I/Borisov?
---
Introduction
============
McKay et al.’s (2019) detection of the red line of atomic oxygen in 2I/Borisov on 2019 October 11 (when the comet was 2.38 AU from the Sun) and interpretation in terms of the production of water resulted in their determination of a production rate of 0.63$\times$10$^{27}$molecules s$^{-1}$ with an error of $\pm$24 percent. On the other hand, from a 15 hour integration time of their radio observations of the OH line at 1667 MHz on October 2–25, Crovisier et al. (2019) estimated an OH production rate of 3.3$
\times $10$^{27}$molecules s$^{-1}$ with an uncertainty of $\pm$27 percent. This result is virtually identical with Bolin et al.’s (2019) water production rate estimate of 100 kg s$^{-1}$ based on their approximate scaling of a typical CN/H$_2$O abundance ratio and an available CN production rate (Fitzsimmons et al. 2019).
Combining Crovisier et al.’s production rate result with Bolin et al.’s estimated upper limit of 1.4 km on the comet’s nuclear diameter and with McKay et al.’s estimate of an averaged water sublimation rate of 0.37$\times
$10$^{27}$molecules km$^{-2}$ s$^{-1}$ at the given heliocentric distance, one finds an active area of more than 140 percent of the upper limit of the total surface area of the nucleus. It appears that the suggestion of the comet’s potential hyperactivity, expressed independently by McKay et al., is well taken. In the following, I present a scenario not considered by McKay et al. or Bolin et al. that is in line with a high abundance of water, without the need of introducing carbon monoxide as the driver of activity in the pre-discovery observations.
Halo of Large Icy-Dust Grains and Chunks
========================================
In a recent short communication (Sekanina 2019a) I noted that the inner coma of 2I/Borisov, resolved in the widely disseminated image taken by a camera on board the Hubble Space Telescope one day after McKay et al. (2019) made their observation, shows a feature extending over a few arcseconds from the nuclear condensation in a general direction of the negative orbital-velocity vector. Orbital considerations suggest that this projected side of the inner coma could be occupied by a halo of nearly-stationary, centimeter-sized and larger icy-dust grains or pebbles released from the nucleus at times when the comet was still far (on the order of 10 AU or more) from the Sun. This type of debris (in a range from a millimeter across up) is commonplace in the characteristic tails of Oort Cloud comets with perihelia near or beyond the snow line (Sekanina 1975). The activity that releases this debris is believed to be driven by annealing of amorphous water ice in comets just arriving from the Oort Cloud (Meech et al.2009).
I now propose that 2I/Borisov has undergone similar evolution as do Oort Cloud comets on their way to perihelion and that sublimating centimeter-sized and larger grains and chunks in its atmosphere could contribute substantially to the reported water production, thereby explaining the comet’s hyperactivity even if only a relatively small fraction of a relatively small nucleus is actually active. A very similar scenario was considered by A’Hearn et al. (1984) for an Oort Cloud comet C/1980 E1 (Bowell; old designation 1980 I = 1980b); they attributed the observed high production rates of water to a cloud of pre-existing sizable grains proposed in my earlier paper (Sekanina 1982).
To examine the plausibility of applying this hypothesis to 2I/Borisov, I assume that dark grains, consisting of dirty water ice (i.e., contaminated by refractory material), are in thermal equilibrium, having a fairly uniform temperature dictated at a given heliocentric distance by the energy balance between the radiation input from the Sun and the losses by the thermal reradiation and water ice sublimation. Somewhat arbitrarily I assume the Bond albedo of 0.04 and the emissivity of 0.9. Because this sublimation regime differs from the one assumed for the nucleus by McKay et al. (2019), the sublimation area of the putative source of detected water is also different, equaling 5.8 km$^2$ instead of 1.7 km$^2$.
Let the sublimation rate of water ice implied by the energy balance at time $t$, when the comet is at heliocentric distance $r(t)$, be $\dot{Z}(t)$, expressed in molecules cm$^{-2}$ s$^{-1}$. Next, I introduce a columnar sublimation rate, $d\Lambda(t)/dt$ or $\dot{\Lambda}(t)$, at which the thickness of a layer or the length of a column of water ice contracts (in cm s$^{-1}$, for example) due to its sublimation, by $$\dot{\Lambda}(t) = \frac{\mu \dot{Z}(t)}{\delta},{\vspace{-0.1cm}}$$ where $\mu$ is the mass of a water molecule and $\delta$ is the bulk density of water ice in the grain. Integrating (1) from an early time before the sublimation began, the length of a column of water ice that has sublimated away by a reference time, $t_{\rm ref}$, is $$\Lambda(t_{\rm ref}) = \!\! \int_{-\infty}^{t_{\rm ref}} \! \dot{\Lambda}(t) \, dt
= \frac{\mu}{\delta} \! \int_{\infty}^{r_{\rm ref}} \! \dot{Z}(r) \, \dot{r}^{-1}
dr ,{\vspace{-0.1cm}}$$ where $t_{\rm ref}$ is reckoned from perihelion, $t_\pi$, and $r_{\rm ref}$ is the heliocentric distance at time $t_{\rm ref}$. In general, there are no constraints on $t_{\rm ref}$, but in the following I only consider the contraction of a column of water ice by its sublimation along the preperihelion leg of the orbit, in which case and .
To facilitate the integration, I introduce a new, dimensionless variable, , by substituting $$z = \exp \!\left( \frac{\tau}{\tau_0} \! \right),{\vspace{-0.2cm}}$$ where and is a constant. The length of a column of water ice sublimated by time $t_{\rm ref}$ is then $$\Lambda(t_{\rm ref}) = \tau_0 \int_{0}^{\exp(\tau_{\rm ref}/\tau_0)} \!
{\cal F}(t) \, dz,$$ where and $${\cal F}(t) = \frac{\dot{\Lambda}(t)}{z} = \dot{\Lambda}(t) \, \exp \! \left(
\! -\frac{\tau}{\tau_0} \! \right).$$
The columnar rates $\dot{\Lambda}(t)$ of water ice sublimation for 2I/Borisov and an Oort Cloud comet of the perihelion distance of 2.006 AU, are compared in Figure 1 as a function of time reckoned from perihelion. The lower rates for 2I/Borisov stem from the higher orbital velocity in its strongly hyperbolic path, i.e., at an equal time from perihelion 2I is farther from the Sun than the Oort Cloud comet of equal perihelion distance.
The length of a column of water ice sublimated by the reference time, defined by Equation (2), is obviously also shorter for 2I than for the nominal Oort Cloud comet. To determine this quantitatively, the integration of the expression on the right-hand side of Equation (2) or (4) requires decisions on two issues: (i) to devise an appropriate method of integration; and (ii) to choose the constant $\tau_0$ separately for 2I and the nominal comet in the parabolic orbit once Equation (4) is used.
The choice of $\tau_0$ is not at all critical, the only concern being that, for the purpose of smooth integration, ${\cal F}$ is approximately flat near perihelion, where $\dot{\Lambda}$ reaches its maximum. This condition is satisfied by for 2I/Borisov and by for the nominal Oort Cloud comet in the parabolic orbit.
An obvious choice for approximating a function defined by a sequence of data pairs is a polynomial $y(x)$, whose coefficients are determined by least squares. The advantage of this approach is twofold:first, one can optimize the polynomial’s degree by searching for the lowest possible degree that secures the requested quality of fit; and second, the integration is straightforward. To follow this procedure, I write\
$${\cal F}_j(t) = \sum_{k=0}^{n} a_{k,j} z^k,\\[-0.1cm]$$ where for 2I/Borisov and for the Oort Cloud comet. In accordance with this notation, I refer from now on to $\Lambda$ and $\tau_0$ for 2I/Borisov as $\Lambda_1$ and $\tau_{0,1}$, for the Oort Cloud comet as $\Lambda_2$ and $\tau_{0.2}$, respectively. Experimentation with a number of reference points ($z$, ${\cal F}$) has shown that a cubic polynomial fits both 2I/Borisov and the Oort Cloud comet quite adequately, if a typical error of approximately $\pm$0.001 cm day$^{-1}$ for ${\cal F}_j$ is acceptable, but [*only*]{} over the range of heliocentric distances not exceeding , equivalent to about and for 2I/Borisov and to and for the nominal Oort Cloud comet. The failure of any fit of this kind for all values of $z$ smaller (and times from perihelion larger) than these limits has two major implications. One is the need to devise an integration approach other than via Equation (4) for all reference times preceding $t^{\displaystyle \ast}$ (i.e., for heliocentric distances larger than $r^{\displaystyle \ast}$). And two, all reference times between $t^{\displaystyle \ast}$ and the perihelion time $t_\pi$ call obviously for a revision of the integration limits in Equation (4).
The remedy of the first problem is greatly facilitated by the fact that at heliocentric distances exceeding 3.4 AU the incident solar radiation is spent overwhelmingly on reradiation, so that the columnar rate of water ice sublimation follows closely a relation $$\dot{\Lambda} = A \exp \!\left( \! -B \sqrt{r} \,\right),$$ where and . One bypasses Equation (4) by integrating directly Equation (2) to obtain for the length of the sublimated column of water ice by time $t^{\displaystyle \ast\!}$ an expression $$\Lambda_j(t^{\displaystyle \ast}) = A \!\! \int_{-\infty}^{t^{\scriptstyle \ast}}
\!\!\! \exp \! \left( \! -B \sqrt{r} \, \right) dt .$$ Next one substitutes with $$\dot{r} = \frac{k_0 e \sin u}{\sqrt{p}},$$ where is the Gaussian gravitational constant, $e$ the orbit eccentricity, the semi-latus rectum (in AU), $q$ the perihelion distance (in AU), and $u$ the true anomaly. At a heliocentric distance of 3.4 AU preperihelion one obtains for 2I/Borisov, but $-$14.63 km s$^{-1}$ for the nominal Oort Cloud comet. Examining $\Lambda_2(t_2^{\displaystyle \ast})$ first, I note that for a parabolic orbit Equation (9) becomes before perihelion $$\dot{r} = -\frac{k_0 \sqrt{2}}{r} \sqrt{r \!-\! q},$$ which for satisfies a condition $$|\dot{r}|^{-1} < \frac{\sqrt{r}}{k_0 \sqrt{2}} \sqrt{\frac{f}{f \!-\! 1}}.$$ Inserting Equation (11) into Equation (8) I find $$\begin{aligned}
\Lambda_2(t_2^{\displaystyle \ast}) & < & \frac{A}{k_0 \sqrt{2}} \, \sqrt{\frac{f}
{f \!-\! 1}} \int_{r^{\scriptstyle \ast}}^{\infty} \!\! \sqrt{r} \exp \! \left(
-B \sqrt{r} \, \right) dr \nonumber \\[-0.2cm]
& & \\[-0.2cm]
& = & \frac{2A \sqrt{2}}{k_0 B^3} \sqrt{\frac{f}{f
\!-\! 1}} \exp \! \left( \!-B \sqrt{r^{\displaystyle \ast}} \right)
\!\! \left( \! 1 \!+\! B \sqrt{r^{\displaystyle \ast}} \!+\! {\textstyle
\frac{1}{2}} B^2 r^{\displaystyle \ast} \! \right)\! . \nonumber\end{aligned}$$ For I find $$\sqrt{\frac{f}{f \!-\! 1}} = \sqrt{\frac{r^{\displaystyle \ast}}
{r^{\displaystyle \ast\!} \!-\! q}} = 1.562.$$ Inserting the numerical values into Equation (12), one obtains for the nominal Oort Cloud comet and from the hyperbolic-to-parabolic ratio of the radial velocities for 2I/Borisov, both implying negligible sublimation losses along either trajectory for centimeter-sized grains by the time the object had reached 3.4 AU.
The second remedy, required by the limits of the polynomial approximation (6) to the columnar sublimation rate, is predictably a relatively minor modification of Equation (4). Marking now with primes the lengths of the sublimated columns computed by integrating ${\cal F}_j$ via polynomials, $\Lambda_j^\prime$, I find (dropping the subscript [*ref*]{}) $$\begin{aligned}
\Lambda_j(t) & = & \tau_{0,j} \!\!\int_{\exp(\tau_j^{\scriptstyle
\ast\!}/\tau_0)}^{\exp(\tau_j/\tau_0)} \!\! {\cal F}_j \, dz +
\Lambda_j(t_j^{\displaystyle \ast}) \nonumber \\[-0.25cm]
& & \\[-0.15cm]
& = & \tau_{0,j} \!\left[ \Lambda_j^\prime(t)\!-\!
\Lambda_j^\prime(t_j^{\displaystyle \ast\!}) ] + o[\Lambda_j(t)
\right], \;\;\; j = 1, 2, \nonumber\end{aligned}$$ where , $o[x]$ means a negligible contribution to $x$, and again for 2I/Borisov and for the nominal Oort Cloud comet.
I already remarked that cubic polynomials represent adequate approximations for both objects; the columns $\Lambda_j^\prime$ sublimated by time $t$ (), employed in Equation (14), equal: $$\Lambda_j^\prime(t) = \tau_{0,j} z \sum_{k=0}^{3} \frac{b_{k,j} z^{k}}{k+1},
\;\;\; j = 1, 2,$$ where $$b_{k,j} = \frac{a_{k,j}}{k+1}, \;\;\; k\!=\!0, \ldots 3; \;\; j \!=\! 1, 2,$$ and $$\begin{aligned}
& & \;\;\;\; j = 1 \,\;\;\;\;\;\;\;\;\;\; j = 2 \nonumber \\
b_{0,j} & = & +0.78485, \;\;\; +0.96249, \nonumber \\
b_{1,j} & = & -2.06856, \;\;\; -2.41993, \nonumber \\
b_{2,j} & = & +2.26169, \;\;\; +2.53858, \nonumber \\
b_{3,j} & = & -0.81706, \;\;\; -0.89328.\end{aligned}$$
Table 1 compares 2I/Borisov with the nominal Oort Cloud comet in terms of both the columnar rate of water ice sublimation and the total length of sublimated column of ice at various times before perihelion. Figures 2 and 3 show the sublimated columnar lengths as a function of time and heliocentric distance, respectively.
Large differences between the two objects are plainly apparent, especially when plotted against time. For example, by the time of the Hubble Space Telescope’s observation, 57 days before perihelion, the column of water ice that sublimated away should have been about 1 cm, so that ice would have been gone from all grains of up to about 2 cm across. According to Table 1 of Sekanina (2019a), such chunks would be at 6300 km from the nucleus if they had been released $\sim$1 year before perihelion, at $\sim$8 AU from the Sun. By contrast, an Oort Cloud comet of equal perihelion distance would have lost by that time a column of water ice more than 6 cm thick, so only very massive chunks, much larger than 10 cm across would still possess some remaining ice. Thus, the substantially higher orbital velocity and the hyperbolic shape of the trajectory have helped 2I/Borisov lose water ice at a significantly lower rate than does an Oort Cloud of equal perihelion distance.
Population of Large Grains and the Hyperactivity
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The term [*hyperactive comets*]{} is usually employed to refer to a small group of short-period comets, including 21P/Giacobini-Zinner, 45P/Honda-Mrkos-Pajdušáková, 46P/Wirtanen, and 103P/Hartley, whose water production is anomalously high, often exceeding the level that implies the entire surface area of the nucleus is active. The best examined member of this group is 103P, the target of the EPOXI mission, whose hyperactivity was proposed by A’Hearn et al.(2011) to be driven primarily by gaseous CO$_2$ that blasts off chunks of sublimating water ice. Yet, Harker et al.(2018) have concluded that the underlying cause of the hyperactivity is still unknown.
Independently of the outcome of this dispute, I believe that 103P or any other short-period comet is not an appropriate standard for investigating the hyperactivity of 2I/Borisov because of their very different histories. A much better benchmark is C/1980 E1 (Bowell), an Oort Cloud comet with perihelion at 3.36 AU, which turned out to be extremely hyperactive around 5 AU preperihelion, although this term was not yet in use in those days. A’Hearn et al.(1984) reported in their paper a water production rate of 0.74$\times$10$^{29}$ molecules s$^{-1}$ at 5.25 AU and 1.55$\times$10$ ^{29}$ molecules s$^{-1}$ at 4.63 AU from the Sun. These rates imply the surface area of a nonrotating nucleus of, respectively, 460 km and 190 km in diameter (!) and even a larger rapidly rotating nucleus. Although the nuclear size of C/1980 E1 is unknown, it has been estimated at not more than 10 km by A’Hearn et al.(1984), a discrepancy of more than one order of magnitude.
A’Hearn et al. resolved this disparity by assuming that the production of water near 5 AU preperihelion came from a source other than the nucleus, referring to my work on a population of large grains ($>$0.5 mm across) in the coma and tail of C/1980 E1 released from the nucleus probably near AU from the Sun (Sekanina 1982), a product of the comet’s activity on its first journey to the inner Solar System. Estimates of the total mass of this population of grains ranged from more than 10$^{13}$ g to 5$\times$10$
^{15}$ g, corresponding to a layer of between $\sim$1 cm and $\sim$1 m on a 10 km nucleus.
Grains observed in the tail of an Oort Cloud comet, whose perihelion is near or beyond the snow line, are released from the nucleus over a period of time, but the tail’s narrow width and the characteristic major gap between the tail and the antisolar direction demonstrate that the grain release stopped long before observation. The time it began is unfortunately not well determined because of poor temporal resolution among early emissions in the tail orientation. Yet, I showed on an example of comet C/1954 O1 (Baade) that measuring the position angle of the tail’s axis (as is commonly done) instead of the position angle of its maximum extent has a tendency to significantly underestimate both the minimum grain size in the tail and the heliocentric distance at release (Sekanina 1975). For any comet displaying such a tail before perihelion, the heliocentric distance at release must obviously exceed markedly the perihelion distance. While Oort Cloud comets discovered years ago had perihelia at distances of up to at most 5 AU, more recently the limit has moved up by several AU, yet some of these objects still display the tails, some even before perihelion.[^1] This is a very strong argument for the onset of release of grains from Oort Cloud comets at heliocentric distances of at least $\sim$10 AU from the Sun, corresponding to temperatures of not more than 80K for a rapidly rotating comet and lower than 100K for a nonrotating comet.
A paper by Meech et al. (2009) addresses the problem of the potential mechanism of large-grain release at very low temperatures; they prefer annealing of amorphous water ice, which is stable up to at least 120–130K, over sublimation of carbon monoxide as the driver of activity far from the Sun in Oort Cloud comets. They show, in fact, that in laboratory experiments the annealing process begins at a temperature as low as 37K, corresponding to a heliocentric distance of 59 AU for a rapidly rotating (isothermal) nucleus and still farther from the Sun for a nonrotating nucleus. Ice physics thus provides only modest constraints in terms of heliocentric distance at the time of release of grains.
Additional issues relate to the grains. Relative fractions of water ice, other ices, and refractory material in the grains upon arrival from the Oort Cloud are unknown. Nor is the grains’ degree of coherence and their fate after the evacuation of water ice: does the skeleton hold together or does it disintegrate? And if it holds together, how does the loss of water ice affect its optical properties, such as the albedo? On the last issue, there is evidence that might indicate either the grains’ failure to survive without the ice or a major drop in their reflectivity because the tails have a tendency to disappear in comets with perihelion distances below the snow line, being replaced near and after perihelion with tails of lesser age. Yet, these tails also lack grains substantially smaller than $\sim$1 mm and no post-perihelion emission of dust is typically detected.
Population of Large Grains and Chunks in 2I/Borisov and Absence of Nongravitational Acceleration in Its Orbital Motion
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Bolin et al. (2019) reported Ye et al.’s (2019) detection of pre-discovery $r$-filter images of 2I/Borisov on 2019 March 17–18 and May 2 and 5 (when the comet was between 6.03 AU and 5.09 AU from the Sun), taken with the Zwicky Transient Facility’s (ZTF) wide-field camera mounted on the 122-cm Schmidt telescope at Palomar, thus extending the observed arc of the orbit by 5.5 months. The authors pointed out that the comet was at the time of magnitude 20.5 to 21.0, much brighter than predicted by a water ice sublimation model, and argued that the brightness was consistent with activity driven by sublimation of carbon monoxide. The plot in their Figure 6 shows, however, that between 6 AU and 2.5 AU from the Sun the scattering cross-sectional area then varied with heliocentric distance $r$ as $r^{-\frac{2}{3}}$ rather than the expected $\sim \! r^{-2}$ or steeper. And because activity at 2.5 AU was driven by sublimation of water ice and other volatiles from the nucleus, the exponent for the CO model should be still closer to zero. In addition, missing in Bolin et al.’s Figure 6 is evidence for a definite increase in the scattering cross section of dust from March to May, which would be expected in the case of CO driven activity.
On the other hand, if the comet’s excessive brightness observed in the ZTF images of 2I/Borisov was due to a population of sizable, extremely slowly moving grains, the product of activity at very large heliocentric distance driven presumably by annealing of amorphous water ice (Meech et al. 2009), one would expect the cross-sectional area to stay constant between 2019 March and May. For an assumed albedo of 0.04, Bolin et al.’s results suggest the total cross-sectional area of the comet to equal $\sim$200 km$^2$ at AU before perihelion. Assuming this refers to the grain halo and a typical grain diameter of $\sim$2 mm, the volume is 3$\times$10$^{11}$cm$^{3}$, which makes a layer of more than 10 cm thick on the nucleus smaller than 1.4 km across, in the same range as the result for the population of grains in C/1980 E1 (Section 3).
Because the sublimation rate of water ice from grains in the atmosphere of 2I/Borisov increases exponentially with time at heliocentric distances exceeding $\sim$3 AU, reaching an integrated columnar length of 0.1 cm near 2.77 AU, 1 cm near 2.39 AU, 2 cm near 2.23 AU, and 4 cm near 2.1 AU (Figure 3), grains of the respective sizes become devolatilized at the respective heliocentric distances and subject to potential disintegration. This could explain three additional points:(i) the presumed presence, in the Hubble Space Telescope’s image taken on October 12, of a population of centimeter-sized and larger chunks in the inner coma (Sekanina 2019a) and, simultaneously, the absence of a tail-like extension, composed of millimeter-sized grains of the same population and pointing along the direction of the negative orbital-velocity vector; (ii) the comet’s hyperactivity over a broad range of heliocentric distances around 2.5 AU preperihelion, with a major contribution by the same centimeter-sized and larger grains because of their water sublimation; and (iii) the bumps in the comet’s light curve in the course of September and early October (Bolin et al. 2019), which could be triggered by clouds of microscopic-sized debris of large devolatilized grains disintegrating at temporally variable rates.
Bolin et al. (2019) appear to admit the presence of a nongravitational acceleration in the orbital motion of 2I/Borisov by noting that “\[m\]oderate non-gravitational force parameters have been measured for the orbit of 2I in pre-discovery data when the comet’s activity was weaker (Ye et al. 2019).” However, consideration of the reported outgassing rates of 2I suggests that after normalization to 1 AU from the Sun, the nongravitational acceleration on this comet should at best be two orders of magnitude lower than the observed anomalous effect on 1I/‘Oumuamua. In fact, Nakano (2019) linked the pre-discovery astrometry from the ZTF database for 2018 December 13 and 2019 February 24, March 17, April 9, and May 2 and 5 with more than 1600 observations from August 30 through November 6 with no need to incorporate nongravitational terms into the equations of motion. His purely gravitational solution provides an excellent fit to the dataset with a mean residual of $\pm$0$^{\prime\prime\!}$.67 and a very satisfactory distribution of individual residuals (which are explicitly tabulated by Nakano), free from any systematic trends.
Conclusions
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While I by no means rule out sublimation of monoxide (or other volatiles not directly tied to sublimation of water ice) from 2I/Borisov, I see no compelling evidence from the pre-discovery observations for CO driving the comet’s activity before water ice sublimation began to gradually dominate, as documented by the data acquired since discovery. I find that the observations reported thus far are consistent with a pre-existing population of (initially) millimeter-sized and larger icy-dust grains and chunks in the coma, presumably a product of the process of annealing of amorphous water ice (Meech et al. 2009) that proceeded in the conglomerate nucleus when the comet was, say, 10 AU or farther from the Sun. The presence of such a massive grain population made the comet bright in the 2019 March–May images, at 5–6 AU from the Sun. Devolatilization and subsequent disintegration of the smallest, millimeter-sized grains from this population was being completed in the course of September of 2019, and their microscopic-sized relics were gradually but temporally unevenly eliminated from the comet’s proximity by solar radiation pressure.
By mid-October, eight weeks before perihelion, sublimation of water continued only from the halo’s chunks $>$2–3 cm in diameter, the grain population’s sublimation cross section dropping from the initial $\sim$200 km$^2$ down to $<$6 km$^2$ or $<$30 km$^2$, depending on whether one uses McKay et al.’s (2019) or Crovisier et al.’s (2019 data. Before the comet reaches perihelion, water sublimation from the nucleus should begin to dominate and the hyperactivity vanish, if 2I evolves as do Oort Cloud comets. As of mid-November 2019, the activity of 2I/Borisov has closely replicated — except for the reduced sublimation effect — that of Oort Cloud comets of similar perihelion distance, as previously suspected (Sekanina 2019b); it remains to be seen whether this affinity is going to be maintained throughout 2I’s journey about the Sun.\
This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.\
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Bolin, B. T., Lisse, C. M., Kasliwal, M. M., et al. 2019, eprint arXiv:1910.14004\
Crovisier, J., Colom, P., Biver, N., et al. 2019, CBET 4691\
Fitzsimmons, A., Hainaut, O., Meech, K. J., et al. 2019, ApJ, 885, L9\
Harker, D. E., Woodward, C. E., Kelley, M. S. P., et al. 2018, AJ, 155, 199\
McKay, A. J., Cochran, A. L., Dello Russo, N., et al. 2019, eprint arXiv:1910.12785\
Meech, K. J., Pittichová, J., Bar-Nun, A., et al. 2009, Icarus, 201, 719\
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Sekanina, Z. 1975, Icarus, 25, 218\
Sekanina, Z. 1982, AJ, 87, 161\
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Ye, Q.-Z., Kelley, M. S. P., Bolin, B. T., et al. 2019, in preparation
[**ADDENDUM, dated November 18, 2019**]{}
Now that the results of the pre-discovery observations investigated by Ye et al. (2019, eprint arXiv:1911.05902) have been published, two of the highlights of their work stand out as the most diagnostic: (i) a ballooning of the cross-sectional area of the comet from an undetected level (with a $3\sigma$ upper limit of 140 km$^2$) in November 2018 to about 330 km$^2$ a month later at heliocentric distances $r$ near 8 AU; and (ii) the comet’s intrinsic brightness, which varies approximately as $r^{-2}$ from 8 AU down all the way to 2.4 AU, implies an essentially constant cross-sectional area over this range of heliocentric distances, until at least early October 2019, a 10-month period.
The most straightforward interpretation of the findings — which should likewise accommodate the high production rate of water in October — is liberation, over a period of several weeks to a few months and at extremely low velocities, of a large amount of icy-dust debris from the nucleus into the atmosphere, where it has been lingering for the 10 months seemingly unaltered. The process of annealing as a driver of this activity, proposed by Meech et al. (2009), was based on Bar-Nun et al.’s laboratory experiments, but independent and/or more recent work on annealing \[e.g., S. A. Sandford & L. J. Allamandola (1988, Icarus, 76, 201), B. Schmitt et al. (1989, ESA SP-302, 65), O.’ Ó. Gálvez et al. (2008, Icarus, 197, 599), L. J. Karssemeijer et al.(2013, ApJ, 781, 16), R. Martín-Doménech et al. (2014, A&A, 564, A8), A. N. Greenberg et al. (2017, MNRAS, 469, S517)\] does in general support this suggestion. It is possible that the process was already in progress in November 2018 or perhaps even earlier, lasting for a few months. Some of the experiments indicate that especially CO$_2$/H$_2$O ice mixtures exhibit a strong annealing effect above 90K, equivalent to the relevant range of heliocentric distances beyond 8 AU. Crystallization of amorphous ice, which also could trigger activity, should start near the subsolar point at about the same time or soon afterward. One has to keep in mind a potentially large temperature variations over the surface of the nucleus.
The requirement of extremely low velocities implies large size of the released debris. An order-of-magnitude estimate for the velocities is derived from an extreme condition that after 10 months the chunks should stay confined to within a fairly small distance, say, 5000 to 10000 km, of the nucleus, even if they keep expanding; this suggests 0.2 to 0.4 m s$^{-1}$, comparable to the velocity of escape from a small comet. If an average piece of debris is between 0.1 cm and 1 cm across and of a bulk density of 0.5 g cm$^{-3}$, the estimated mass of fragments with the cross-sectional area of 330 km$^2$ is ()$\times$10$^{11}$g. On a nucleus of [*less*]{} than 1.4 km across (Bolin et al. 2019) and equal density, this mass would be distributed in a layer of [*more*]{} than thick, an estimate whose upper bound is in order-of-magnitude agreement with the thickness of a layer of grains released from the nucleus of C/1980 E1, estimated by A’Hearn et al. (1984) from their determination of the comet’s sublimation rate of water ice near 5 AU preperihelion.
The pre-discovery observations refer to a total [*scattering*]{} cross section of the debris, offering no information of the [*sublimation*]{} cross section, which is assumed to be the same near 8 AU, but dropping with decreasing heliocentric distance because of progressive depletion of water ice from smaller fragments by increasing sublimation nearer perihelion, as shown in Table 1. It is still unclear whether, or at what rate, does the scattering cross section of the devolatilized debris drop with time. Potential fragmentation temporarily increases the cross section but has eventually the opposite effect as microscopic debris is removed by solar radiation pressure. In addition, the size of the contribution from increasing activity of the nucleus itself at is yet to be determined. On the other hand, the total sublimation cross section in mid-October seems to have been at least crudely established to range between 6 km$^2$ and 30 km$^2$. Interestingly, the size of the largest water-ice depleted pieces of debris just about equals the size of the debris that, released near 8 AU from the Sun, was in mid-October, at the Hubble Space Telescope’s imaging time, at the boundary of the inner coma. Thus, the inner coma, within $\sim$6000 km or so of the nucleus, still contains sublimating chunks released near 8 AU from the Sun.
[^1]: An excellent example of what can be learned from high-quality imaging obtained at an appropriate time is Meech et al.’s (2009) image of C/1999 J2, taken on 2000 February 24, or 42 days preperihelion, at 7.11 AU. Compared to the other five images of this comet in the paper, the great advantage of this one is the large angle between the radius vector (p.a. 279$^\circ$) and the negative orbital velocity vector (p.a. 20$^\circ$). For the left, sharper, and perfectly rectilinear boundary of the tail I measure a position angle of 13$^\circ$, which implies a release time of 1600 days before perihelion at 12 AU from the Sun. Assuming the tail extends to the edge of the frame, the smallest grains are subjected to a radiation pressure acceleration of 0.0014 the Sun’s gravitational acceleration and are about 1.6 mm across. Their temperature is about 80K. The right, much shorter, and less sharp boundary of the tail extends to a position angle of about 2–3$^\circ$, implying a terminal release time 600 days before perihelion and a heliocentric distance of $\sim$8 AU. The grains’ implied temperature is about 100K. The process extended for approximately 1000 days with an activating temperature range of $\sim$20$^\circ$.
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abstract: 'In this paper, an expression for the asymptotic growth rate of the number of small linear-weight codewords of irregular doubly-generalized LDPC (D-GLDPC) codes is derived. The expression is compact and generalizes existing results for LDPC and generalized LDPC (GLDPC) codes. Assuming that there exist check and variable nodes with minimum distance $2$, it is shown that the growth rate depends only on these nodes. An important connection between this new result and the stability condition of D-GLDPC codes over the BEC is highlighted. Such a connection, previously observed for LDPC and GLDPC codes, is now extended to the case of D-GLDPC codes.'
author:
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title: 'On the Growth Rate of the Weight Distribution of Irregular Doubly-Generalized LDPC Codes'
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Doubly-generalized LDPC codes, irregular code ensembles, weight distribution.
Introduction
============
Recently, low-density parity-check (LDPC) codes have been intensively studied due to their near-Shannon-limit performance under iterative belief-propagation decoding. Binary regular LDPC codes were first proposed by Gallager in 1963 [@Gallager]. In the last decade the capability of irregular LDPC codes to outperform regular ones in the waterfall region of the performance curve and to asymptotically approach (or even achieve) the communication channel capacity has been recognized and deeply investigated (see for instance [@luby01:improved; @luby01:efficient; @richardson01:design; @richardson01:dB; @pfister05:capacity-achieving; @pfister07:ara]).
It is usual to represent an LDPC code as a bipartite graph, i.e., as a graph where the nodes are grouped into two disjoint sets, namely, the variable nodes (VNs) and the check nodes (CNs), such that each edge may only connect a VN to a CN. The bipartite graph is also known as a Tanner graph [@Tanner_GLDPC]. In the Tanner graph of an LDPC code, a generic degree-$q$ VN can be interpreted as a length-$q$ repetition code, as it repeats $q$ times its single information bit towards the CNs. Similarly, a degree-$s$ CN of an LDPC code can be interpreted as a length-$s$ single parity-check (SPC) code, as it checks the parity of the $s$ VNs connected to it.
The growth rate of the weight distribution of Gallager’s regular LDPC codes was investigated in [@Gallager]. The analysis demonstrated that, provided that the smallest VN degree is at least 3, for large enough codeword length $N$, the expected minimum distance of a randomly chosen code in the ensemble is a linear function of $N$.
More recently, the study of the weight distribution of binary LDPC codes has been extended to irregular ensembles. Important works in this area are [@litsyn02:ensembles; @Burshtein_Miller; @Di_Richardson_Urbanke]. In [@Di_Richardson_Urbanke] a complete solution for the growth rate of the weight distribution of binary irregular LDPC codes was developed. One of the main results of [@Di_Richardson_Urbanke] is a connection between the expected behavior of the weight distribution of a code randomly chosen from the ensemble and the parameter $\lambda'(0)\rho'(1)$, $\lambda(x)$ and $\rho(x)$ being the edge-perspective VN and CN degree distributions, respectively. More specifically, it was shown that for a code randomly chosen from the ensemble, one can expect an exponentially small number of small linear-weight codewords if $0 \leq
\lambda'(0)\rho'(1)<1$, and an exponentially large number of small linear-weight codewords if $\lambda'(0)\rho'(1)>1$.
This result establishes a connection between the statistical properties of the weight distribution of binary irregular LDPC codes and the stability condition of binary irregular LDPC codes over the binary erasure channel (BEC) [@luby01:efficient; @richardson01:design]. If $q^*$ denotes the LDPC asymptotic iterative decoding threshold over the BEC, the stability condition states that we always have $$\begin{aligned}
\label{eq:stability_LDPC}
q^* \leq \left[ \lambda'(0)\rho'(1) \right]^{-1}.\end{aligned}$$
Prior to the rediscovery of LDPC codes, binary generalized LDPC (GLDPC) codes were introduced by Tanner in 1981 [@Tanner_GLDPC]. A GLDPC code generalizes the concept of an LDPC code in that a degree-$s$ CN may in principle be any $(s,h)$ linear block code, $s$ being the code length and $h$ the code dimension. Such a CN accounts for $s-h$ linearly independent parity-check equations. A CN associated with a linear block code which is not a SPC code is said to be a *generalized CN*. In [@Tanner_GLDPC] *regular* GLDPC codes (also known as Tanner codes) were investigated, these being GLDPC codes where the VNs are all repetition codes of the same length and the CNs are all linear block codes of the same type.
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The growth rate of the weight distribution of binary GLDPC codes was investigated in [@boutros99:generalized; @lentmaier99:generalized; @Tillich04:weight; @paolini08:weight]. In [@boutros99:generalized] the growth rate is calculated for Tanner codes with BCH check component codes and length-2 repetition VNs, leading to an asymptotic lower bound on the minimum distance. The same lower bound is developed in [@lentmaier99:generalized] assuming Hamming CNs and length-2 repetition VNs. Both works extend the approach developed by Gallager in [@Gallager Chapter 2] to show that, for sufficiently large $N$, the minimum distance is a linear function of $N$. The growth rate of the number of small weight codewords for GLDPC codes with a uniform CN set (all CN of the same type) and an irregular VN set (repetition VNs with different lengths) is investigated in [@Tillich04:weight]. It is shown that for sufficiently large $N$, a minimum distance increasing linearly with $N$ is expected when either the uniform CN set is composed of linear block codes with minimum distance at least $3$, or the minimum length of the repetition VNs is 3. On the other hand, if the minimum distance of the CNs and the minimum length of the repetition VNs are both equal to 2, then for a randomly selected GLDPC code in the ensemble we expect a minimum distance growing as a linear or sublinear function of $N$ (for large $N$) depending on the sign of the first order coefficient in the growth rate Taylor series expansion. The results developed in [@Tillich04:weight] were further extended in [@paolini08:weight] to GLDPC ensembles with an irregular CN set (CNs of different types). It was there proved that, provided that there exist CNs with minimum distance $2$, a parameter $\lambda'(0)C$, generalizing the parameter $\lambda'(0)\rho'(1)$ of LDPC code ensembles, plays in the context of the weight distribution of GLDPC codes the same role played by $\lambda'(0)\rho'(1)$ in the context of the weight distribution of LDPC codes. The parameter $C$ is defined in Section \[section:further\_def\_&\_notation\].
Interestingly, this latter results extends to binary GLDPC codes the same connection between the statistical properties of the weight distribution of irregular codes and the stability condition over the BEC. In fact, it was shown in [@paolini08:stability] that the stability condition of binary irregular GLDPC codes over the BEC is given by $$\begin{aligned}
\label{eq:stability_GLDPC}
q^* \leq \left[ \lambda'(0)C \right]^{-1}.\end{aligned}$$
Generalized LDPC codes represent a promising solution for low-rate channel coding schemes, due to an overall rate loss introduced by the generalized CNs [@miladinovic08:generalized]. Doubly-generalized LDPC (D-GLDPC) codes generalize the concept of GLDPC codes while facilitating much greater design flexibility in terms of code rate [@Wang_Fossorier_DG_LDPC] (an analogous idea may be found in the previous work [@Dolinar]). In a D-GLDPC code, the VNs as well as the CNs may be of any generic linear block code types. A degree-$q$ VN may in principle be any $(q,k)$ linear block code, $q$ being the code length and $k$ the code dimension. Such a VN is associated with $k$ D-GLDPC code bits. It interprets these bits as its local information bits and interfaces to the CN set through its $q$ local code bits. A VN which corresponds to a linear block code which is not a repetition code is said to be a *generalized VN*. A D-GLDPC code is said to be *regular* if all of its VNs are of the same type and all of its CNs are of the same type and is said to be *irregular* otherwise. The structure of a D-GLDPC code is depicted in Fig. \[fig:DGLDPC\].
In this paper the growth rate of the weight distribution of binary irregular D-GLDPC codes is analyzed for small weight codewords. It is shown that, provided there exist both VNs and CNs with minimum distance $2$, a parameter $1/P^{-1}(1/C)$ discriminates between an asymptotically small and an asymptotically large expected number of small linear-weight codewords in a D-GLDPC code randomly drawn from a given irregular ensemble (the function $P(x)$ is defined in Section \[section:further\_def\_&\_notation\]). The parameter $1/P^{-1}(1/C)$ generalizes the above mentioned parameters $\lambda'(0)\rho'(1)$ and $\lambda'(0)C$ to the case where both generalized VNs and generalized CNs are present. The obtained result also represents the extension to the D-GLDPC case of the previously recalled connection with the stability condition over the BEC. In fact, it was proved in [@paolini08:stability] that the stability condition of D-GLDPC codes over the BEC is given by $$\begin{aligned}
\label{eq:stability_D-GLDPC}
q^* \leq P^{-1}(1/C)\, .\end{aligned}$$ The paper is organized as follows. Section \[section:irregular\_D\_GLDPC\] defines the D-GLDPC ensemble of interest, and introduces some definitions and notation pertaining to this ensemble. Section \[section:further\_def\_&\_notation\] defines further terms regarding the VNs and CNs which compose the D-GLDPC codes in the ensemble. Finally, Section \[section:growth\_rate\] states and proves the main result of the paper regarding the growth rate of the weight distribution.
Irregular Doubly-Generalized LDPC Code Ensemble {#section:irregular_D_GLDPC}
===============================================
We define a D-GLDPC code ensemble ${{\mathcal{M}}}_n$ as follows, where $n$ denotes the number of VNs. There are $n_c$ different CN types $t \in I_c = \{ 1,2,\cdots, n_c\}$, and $n_v$ different VN types $t \in I_v = \{ 1,2,\cdots, n_v\}$. For each CN type $t \in I_c$, we denote by $h_t$, $s_t$ and $r_t$ the CN dimension, length and minimum distance, respectively. For each VN type $t \in I_v$, we denote by $k_t$, $q_t$ and $p_t$ the VN dimension, length and minimum distance, respectively. For $t \in I_c$, $\rho_t$ denotes the fraction of edges connected to CNs of type $t$. Similarly, for $t \in I_v$, $\lambda_t$ denotes the fraction of edges connected to VNs of type $t$. Note that all of these variables are independent of $n$.
The polynomials $\rho(x)$ and $\lambda(x)$ are defined by $$\rho(x) = \sum_{t\in I_c} \rho_t x^{s_t - 1}$$ and $$\lambda(x) = \sum_{t \in I_v} \lambda_t x^{q_t - 1} \; .$$ If $E$ denotes the number of edges in the Tanner graph, the number of CNs of type $t\in I_c$ is then given by $E \rho_t / s_t$, and the number of VNs of type $t\in I_v$ is then given by $E \lambda_t / q_t$. Denoting as usual $\int_0^1 \rho(x) \, {\rm d} x$ and $\int_0^1 \lambda(x) \, {\rm d} x$ by $\int \rho$ and $\int \lambda$ respectively, we see that the number of edges in the Tanner graph is given by $$E = \frac{n}{\int \lambda}$$ and the number of CNs is given by $m = E \int \rho$. Therefore, the fraction of CNs of type $t \in I_c$ is given by $$\gamma_t = \frac{\rho_t}{s_t \int \rho}
\label{eq:gamma_t_definition}$$ and the fraction of VNs of type $t \in I_v$ is given by $$\delta_t = \frac{\lambda_t}{q_t \int \lambda}
\label{eq:delta_t_definition}$$ Also the length of any D-GLDPC codeword in the ensemble is given by $$N = \sum_{t \in I_v} \left( \frac{E \lambda_t}{q_t} \right) k_t = \frac{n}{\int \lambda} \sum_{t \in I_v} \frac{\lambda_t k_t}{q_t} \; .
\label{eq:DG_LDPC_codeword_length}$$ Note that this is a linear function of $n$. Similarly, the total number of parity-check equations for any D-GLDPC code in the ensemble is given by $$M = \frac{m}{\int \rho} \sum_{t \in I_c} \frac{\rho_t h_t}{s_t} \; .$$ A member of the ensemble then corresponds to a permutation on the $E$ edges connecting CNs to VNs.
The growth rate of the weight distribution of the irregular D-GLDPC ensemble sequence $\{ {{\mathcal{M}}}_n \}$ is defined by $$G(\alpha) = \lim_{n\rightarrow \infty} \frac{1}{n} \log \mathbb{E}_{{{\mathcal{M}}}_n} \left[ N_{\alpha n} \right]
\label{eq:growth_rate_result}$$ where $\mathbb{E}_{{{\mathcal{M}}}_n}$ denotes the expectation operator over the ensemble ${{\mathcal{M}}}_n$, and $N_{w}$ denotes the number of codewords of weight $w$ of a randomly chosen D-GLDPC code in the ensemble ${{\mathcal{M}}}_n$. The limit in (\[eq:growth\_rate\_result\]) assumes the inclusion of only those positive integers $n$ for which $\alpha n \in \mathbb{Z}$ and $\mathbb{E}_{{{\mathcal{M}}}_n} [ N_{\alpha n} ]$ is positive (i.e., where the expression whose limit we seek is well defined). Note that the argument of the growth rate function $G(\alpha)$ is equal to the ratio of D-GLDPC codeword length to the number of VNs; by (\[eq:DG\_LDPC\_codeword\_length\]), this captures the behaviour of codewords linear in the block length, as in [@Di_Richardson_Urbanke] for the LDPC case.
An *assignment* is a subset of the edges of the Tanner graph. An assignment is said to have *weight* $k$ if it has $k$ elements. An assignment is said to be *check-valid* if the following condition holds: supposing that each edge of the assignment carries a $1$ and each of the other edges carries a $0$, each CN recognizes a valid codeword.
A *split assignment* is an assignment, together with a subset of the D-GLDPC code bits (called a *codeword assignment*). A split assignment is said to have *split weight* $(u, v)$ if its assignment has weight $v$ and its codeword assignment has $u$ elements. A split assignment is said to be *check-valid* if its assignment is check-valid. A split assignment is said to be *variable-valid* if the following condition holds: supposing that each edge of its assignment carries a $1$ and each of the other edges carries a $0$, and supposing that each D-GLDPC code bit in the codeword assigment is set to $1$ and each of the other code bits is set to $0$, each VN recognizes an input word and the corresponding valid codeword.
Note that for any D-GLDPC code, there is a bijective correspondence between the set of D-GLDPC codewords and the set of split assignments which are both variable-valid and check-valid.
Further Definitions and Notation {#section:further_def_&_notation}
================================
The weight enumerating polynomial for CN type $t \in I_c$ is given by $$\begin{aligned}
A^{(t)}(x) & = & \sum_{u=0}^{s_t} A_u^{(t)} x^u \\
& = & 1 + \sum_{u=r_t}^{s_t} A_u^{(t)} x^u \; .\end{aligned}$$ Here $A_u^{(t)} \ge 0$ denotes the number of weight-$u$ codewords for CNs of type $t$. Note that $A_{r_t}^{(t)} > 0$ for all $t \in I_c$. The bivariate weight enumerating polynomial for VN type $t \in I_v$ is given by $$\begin{aligned}
B^{(t)}(x,y) & = & \sum_{u=0}^{k_t} \sum_{v=0}^{q_t} B_{u,v}^{(t)} x^u y^v \\
& = & 1 + \sum_{u=1}^{k_t} \sum_{v=p_t}^{q_t} B_{u,v}^{(t)} x^u y^v \; .\end{aligned}$$ Here $B_{u,v}^{(t)} \ge 0$ denotes the number of weight-$v$ codewords generated by input words of weight $u$, for VNs of type $t$. Also, for each $t \in I_v$, corresponding to the polynomial $B^{(t)}(x,y)$ we denote the sets $$S_t = \{ (i,j) \in \mathbb{Z}^2 \; : \; B^{(t)}_{i,j} > 0 \}
\label{eq:St}$$ and $$S_t^{-} = S_t \backslash \{ (0,0) \} \; .
\label{eq:St-}$$
We denote the smallest minimum distance over all CN types by $$r = \min \{ r_t \; : \; t \in I_c \}$$ and the set of CN types with this minimum distance by $$X_c = \{ t \in I_c \; : \; r_t = r \} \; .$$ We also define $$C_t = \frac{r_t A^{(t)}_{r_t}}{s_t}$$ for each $t \in I_c$, and $$C = \sum_{t \in X_c} \rho_t C_t \; .
\label{eq:C_definition}$$ We note that $C_t>0$ for all $t \in I_c$, so $C > 0$.
Similarly, we denote the smallest minimum distance over all VN types by $$p = \min \{ p_t \; : \; t \in I_v \}$$ and the set of VN types with this minimum distance by $$X_v = \{ t \in I_v \; : \; p_t = p \} \; .$$ In the specific case where $p = 2$, we also introduce the following definitions. For each $t \in X_v$, define the set $L_t = \{ i \in \mathbb{Z} \; : \; B^{(t)}_{i,2} > 0 \}$ – note that these sets are nonempty. Also define the polynomial $P(x)$ by $$P(x) = \sum_{t \in X_v} \lambda_t \sum_{i \in L_t} \frac{2 B^{(t)}_{i,2}}{q_t} x^i
\label{eq:Px_definition}$$ and denote its inverse by $P^{-1}(x)$. Since all the coefficients of $P(x)$ are positive, $P(x)$ is monotonically increasing and therefore its inverse is well-defined and unique. Note that in the case $r=p=2$, both $C$ and the polynomial $P(x)$ depend only on the CNs and VNs with minimum distance equal to $2$.
Finally, note that throughout this paper, the notation $e = \exp(1)$ denotes Napier’s number.
Growth Rate for Doubly-Generalized LDPC Code Ensemble {#section:growth_rate}
=====================================================
The following theorem constitutes the main result of the paper.
Consider an irregular D-GLDPC code ensemble sequence ${{\mathcal{M}}}_n$ satisfying $r=p=2$. For sufficiently small $\alpha$, the growth rate of the weight distribution is given by $$G(\alpha) = \alpha \log \left[ \frac{1}{P^{-1}(1/C)} \right] + O(\alpha^2) \; .
\label{eq:growth_rate_case_1}$$ \[thm:growth\_rate\]
The theorem is proved next. For ease of presentation, the proof is broken into four parts.
Number of check-valid assignments of weight $\epsilon m$ over $\gamma m$ CNs of type $t \in I_c$
------------------------------------------------------------------------------------------------
Consider $\gamma m$ CNs of the same type $t \in I_c$. Using generating functions [^1] , the number of check-valid assignments (over these CNs) of weight $\epsilon m$ is given by $$N_{c,t}^{(\gamma m)}(\epsilon m) = {{\mbox{Coeff }}}\left[ \left( A^{(t)}(x) \right) ^{\gamma m}, x^{\epsilon m} \right]$$ where ${{\mbox{Coeff }}}[ p(x), x^c ]$ denotes the coefficient of $x^c$ in the polynomial $p(x)$. We now use the following result, which appears as Lemma 19 in [@Di_Richardson_Urbanke]:
Let $A(x) = 1 + \sum_{u=c}^{d} A_u x^u$, where $1 \le c \le d$, be a polynomial satisfying $A_c > 0$ and $A_u \ge 0$ for all $c < u \le d$. Then, for sufficiently small $\xi$, $$\begin{gathered}
\lim_{\ell\rightarrow \infty} \frac{1}{\ell} \log {{\mbox{Coeff }}}\left[ \left( A(x) \right) ^{\ell}, x^{\xi \ell} \right] \\
= \frac{\xi}{c} \log \left( \frac{e c A_c}{\xi} \right) + O(\xi^2)
\label{lemma_42_formula}\end{gathered}$$ \[lemma:optimization\_dominant\_term\_1D\]
The limit in (\[lemma\_42\_formula\]) assumes the inclusion of only those positive integers $\ell$ for which $\xi \ell \in \mathbb{Z}$ and ${{\mbox{Coeff }}}[ ( A(x) ) ^{\ell}, x^{\xi \ell} ]$ is positive (i.e., where the expression whose limit we seek is well defined). A proof of this lemma may be found in the Appendix; our proof is based on arguments from [@Burshtein_Miller] and Lagrange multipliers, and constitutes a different approach to that taken in [@Di_Richardson_Urbanke].
Applying this lemma by substituting $A(x) = A^{(t)}(x)$, $\ell=\gamma m$ and $\xi = \epsilon/\gamma$, we obtain that with $\gamma$ fixed, as $m \rightarrow \infty$ we have, for sufficiently small $\epsilon$, $$\begin{gathered}
N_{c,t}^{(\gamma m)}(\epsilon m) = {{\mbox{Coeff }}}\left[ \left( A^{(t)}(x) \right) ^{\gamma m}, x^{\epsilon m} \right] \rightarrow \\
\exp \left\{ m \left[ \frac{\epsilon}{r_t} \log \left( \frac{e r_t A^{(t)}_{r_t} \gamma}{\epsilon} \right) + O(\epsilon^2) \right] \right\}
\label{eq:check_node_type_t}\end{gathered}$$
Number of check-valid assignments of weight $\delta m$ {#sub:no_check_valid_assign}
------------------------------------------------------
Next we derive an expression, valid asymptotically, for the number of check-valid assignments of weight $\delta m$. For each $t \in I_c$, let $\epsilon_t m$ denote the portion of the total weight $\delta m$ apportioned to CNs of type $t$. Then $\epsilon_t \ge 0$ for each $t \in I_c$, and $\sum_{t \in I_c} \epsilon_t = \delta$. Also denote ${{\mbox{\boldmath $\epsilon$}}}= (\epsilon_1 \; \epsilon_2 \; \cdots \; \epsilon_{n_c})$. The number of check-valid assignments of weight $\delta m$ satisfying the constraint ${{\mbox{\boldmath $\epsilon$}}}$ is obtained by multiplying the numbers of check-valid assignments of weight $\epsilon_t m$ over $\gamma_t m$ CNs of type $t$, for each $t \in I_c$, $$N_c^{({{\mbox{\boldmath $\epsilon$}}})}(\delta m) = \prod_{t \in I_c} N_{c,t}^{(\gamma_t m)}(\epsilon_t m)$$ where the fraction $\gamma_t$ of CNs of type $t \in I_c$ is given by (\[eq:gamma\_t\_definition\]).
As $n\rightarrow \infty$, we have $m\rightarrow \infty$ and so we obtain using (\[eq:check\_node\_type\_t\]) that for sufficiently small $\delta$, $$\begin{gathered}
N_c^{({{\mbox{\boldmath $\epsilon$}}})}(\delta m) \rightarrow \\
\prod_{t \in I_c} \exp \left\{ m \left[ \frac{\epsilon_t}{r_t} \log \left( \frac{e r_t A^{(t)}_{r_t} \gamma_t}{\epsilon_t} \right) + O(\epsilon_t^2) \right] \right\} \\
= \exp \left\{ m \left[ \sum_{t \in I_c} \left( \frac{\epsilon_t}{r_t} \log \left( \frac{e \rho_t C_t}{\epsilon_t \int \rho} \right) \right) + O(\delta^2) \right] \right\}
\label{eq:number_of_assign_distribution}\end{gathered}$$
The number of check-valid assignments of weight $\delta m$, which we denote $N_c(\delta m)$, is equal to the sum of $N_c^{({{\mbox{\boldmath $\epsilon$}}})}(\delta m)$ over all admissible vectors ${{\mbox{\boldmath $\epsilon$}}}$. However, the asymptotic expression as $n\rightarrow \infty$ will be dominated by the distribution ${{\mbox{\boldmath $\epsilon$}}}$ which maximizes the argument of the exponential [^2] . Therefore, our next step is to maximize the function $$f({{\mbox{\boldmath $\epsilon$}}}) = \sum_{t \in I_c} \frac{\epsilon_t}{r_t} \log \left( \frac{e \rho_t C_t}{\epsilon_t \int \rho} \right)$$ subject to the constraints $$g({{\mbox{\boldmath $\epsilon$}}}) = \sum_{t \in I_c} \epsilon_t = \delta
\label{eq:sum_epsilon_equals_1}$$ and $$\epsilon_t \ge 0 \quad \forall t \in I_c \; .
\label{eq:nonnegative_constraint_epsilon}$$ We solve this optimization problem using Lagrange multipliers, ignoring for the moment the final constraint. Since $$\frac{\partial f}{\partial\epsilon_{t}} = \frac{1}{r_t} \log \left( \frac{\rho_t C_t}{\epsilon_t \int \rho} \right) \quad ; \quad \frac{\partial g}{\partial\epsilon_t} = 1$$ for all $t \in I_c$, we have to solve the $n_c$ equations (where $\mu$ is the Lagrange multiplier) $$\frac{1}{r_t} \log \left( \frac{\rho_t C_t}{\epsilon_t \int \rho} \right) + \mu = 0 \quad \forall t \in I_c
\label{eq:Lagrange_epsilons}$$ together with (\[eq:sum\_epsilon\_equals\_1\]), for the $(n_c+1)$ unknowns $\{ \mu, {{\mbox{\boldmath $\epsilon$}}}\}$. First, (\[eq:Lagrange\_epsilons\]) yields $$\epsilon_t = \frac{\rho_t C_t}{\int \rho} z^{r_t} \quad \forall t \in I_c$$ where $z = e^{\mu}$. Now using (\[eq:sum\_epsilon\_equals\_1\]), we obtain $$\frac{1}{\int \rho} \sum_{t \in I_c} C_t \rho_t z^{r_t} = \delta \; .$$ The left-hand side of this equation involves a sum of positive terms. For $\delta$ sufficiently small, we may approximate $$\begin{aligned}
\frac{1}{\int \rho} \sum_{t \in I_c} \rho_t C_t z^{r_t} & \approx & \frac{1}{\int \rho} \sum_{t \in X_c} \rho_t C_t z^{r_t} \\
& = & \frac{C}{\int \rho} z^r \; .\end{aligned}$$ Applying this approximation, we obtain $\epsilon_t = 0$ if $t \notin X_c$, and $\epsilon_t = K \rho_t C_t$ if $t \in X_c$, where $K$ is independent of $t$. Then (\[eq:sum\_epsilon\_equals\_1\]) yields $K = \delta / C$, and we obtain the solution $$\epsilon_t = \left\{ \begin{array}{cl}
\rho_t C_t \delta / C & \textrm{if } t \in X_c \\
0 & \textrm{otherwise.}\end{array}\right.$$ This solution satisfies (\[eq:nonnegative\_constraint\_epsilon\]). When substituted into (\[eq:number\_of\_assign\_distribution\]), it yields the following result: as $n\rightarrow \infty$ $$N_c(\delta m) \rightarrow \exp \left\{ m \left[ \frac{\delta}{r} \log \left( \frac{e C}{\delta \int \rho} \right) + O(\delta^2) \right] \right\}
\label{eq:number_of_check_valid_assign}$$
Number of variable-valid split assignments of split weight $(\tau n, \sigma n)$ over $\gamma n$ VNs of type $t \in I_v$
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Consider $\gamma n$ VNs of the same type $t \in I_v$. We now evaluate the number of variable-valid split assignments (over these VNs) of split weight $(\tau n, \sigma n)$. Using generating functions [^3] , this is given by $$N_{v,t}^{(\gamma n)}(\tau n, \sigma n) = {{\mbox{Coeff }}}\left[ \left( B^{(t)}(x,y) \right) ^{\gamma n}, x^{\tau n} y^{\sigma n} \right]$$ where ${{\mbox{Coeff }}}[p(x,y), x^c y^d ]$ denotes the coefficient of $x^c y^d$ in the bivariate polynomial $p(x,y)$. We make use of the following lemma from [@Burshtein_Miller Theorem 2].
Let $$B(x,y) = 1 + \sum_{u=1}^{k} \sum_{v=c}^{d} B_{u,v} x^u y^v$$ where $k \ge 1$ and $1 \le c \le d$, be a bivariate polynomial satisfying $B_{u,v} \ge 0$ for all $1 \le u \le k$, $c \le v \le d$. For fixed positive rational numbers $\xi$ and $\theta$, consider the set of positive integers $\ell$ such that $\xi \ell \in \mathbb{Z}$, $\theta \ell \in \mathbb{Z}$ and ${{\mbox{Coeff }}}[ (B(x,y) ) ^{\ell}, x^{\xi \ell}y^{\theta \ell} ] > 0$. Then either this set is empty, or has infinite cardinality; if $t$ is one such $\ell$, then so is $jt$ for every positive integer $j$. Assuming the latter case, the following limit is well defined and exists: $$\begin{gathered}
\lim_{\ell\rightarrow \infty} \frac{1}{\ell} \log {{\mbox{Coeff }}}\left[ \left( B(x,y) \right) ^{\ell}, x^{\xi \ell}y^{\theta \ell} \right] \\
= \max_{{{\mbox{\scriptsize \boldmath $\eta$}}}} \sum_{(i,j) \in S} \eta_{i,j} \log \left( \frac{B_{i,j}}{\eta_{i,j}} \right)\end{gathered}$$ where $S = \{ (i,j) \in \mathbb{Z}^2 \; : \; B_{i,j} > 0 \}$, ${{\mbox{\boldmath $\eta$}}}= ( \eta_{i,j} )_{(i,j) \in S}$, and the maximization is subject to the constraints $\sum_{(i,j) \in S} \eta_{i,j} = 1$, $\sum_{(i,j) \in S} i \eta_{i,j} = \xi$, $\sum_{(i,j) \in S} j \eta_{i,j} = \theta$ and $\eta_{i,j} \ge 0$ for all $(i,j) \in S$. \[lemma:optimization\_2D\]
Applying this lemma by substituting $B(x,y) = B^{(t)}(x,y)$, $l = \gamma n$, $\xi = \tau/\gamma$ and $\theta = \sigma/\gamma$, we obtain that with $\gamma$ fixed, as $n \rightarrow \infty$ $$\begin{aligned}
N_{v,t}^{(\gamma n)}(\tau n, \sigma n) = {{\mbox{Coeff }}}\left[ \left( B^{(t)}(x,y) \right) ^{\gamma n}, x^{\tau n} y^{\sigma n} \right]
\label{eq:Nvt_tau_sigma_start} \\
\rightarrow \exp \left\{ n \gamma \max_{{{\mbox{\scriptsize \boldmath $\eta$}}}^{(t)}} \sum_{(i,j) \in S_t} \eta^{(t)}_{i,j} \log \left( \frac{B^{(t)}_{i,j}}{\eta^{(t)}_{i,j}} \right) \right\}
\label{eq:Nvt_tau_sigma_mid} \\
\triangleq \exp \left\{ n X^{(\gamma)}_t(\tau, \sigma) \right\}
\label{eq:Nvt_tau_sigma_end}\end{aligned}$$ where the maximization over ${{\mbox{\boldmath $\eta$}}}^{(t)} = ( \eta^{(t)}_{i,j} )_{(i,j) \in S_t}$ is subject to the constraints $\sum_{(i,j) \in S_t} \eta^{(t)}_{i,j} = 1$, $\sum_{(i,j) \in S_t^{-}} i \eta^{(t)}_{i,j} = \tau / \gamma$, $\sum_{(i,j) \in S_t^{-}} j \eta^{(t)}_{i,j} = \sigma / \gamma$ and $\eta^{(t)}_{i,j} \ge 0$ for all $(i,j) \in S_t$ (recall that the sets $S_t$ and $S_t^{-}$ are given by (\[eq:St\]) and (\[eq:St-\])).
Growth rate of the weight distribution of the irregular D-GLDPC code ensemble sequence
--------------------------------------------------------------------------------------
Recall that the number of check-valid assignments of weight $\delta m$ is $N_c(\delta m)$; also, the total number of assignments of weight $\delta m$ is $\binom{E}{\delta m}$. Therefore, the probability that a randomly chosen assignment of weight $\delta m$ is check-valid is given by $$P_{\mbox{\scriptsize valid}}(\delta m) = N_c(\delta m) \Big/ \binom{E}{\delta m} \; .$$ Here we adopt the notation $\delta m = \beta n$; also we have $E = m / \int \rho = n / \int \lambda$. The binomial coefficient may be asymptotically approximated using the fact, based on Stirling’s approximation, that as $n \rightarrow \infty$ [@Di_Richardson_Urbanke] $$\binom{\tau n}{\sigma n} \rightarrow \exp \left\{ n \left[ \sigma \log \left( \frac{e \tau}{\sigma} \right) +
O(\sigma^2) \right] \right\}$$ (valid for $0 < \sigma < \tau < 1$) which yields, in this case, $$\binom{n / \int \lambda}{\beta n} \rightarrow \exp \left\{ n \left[ \beta \log \left( \frac{e}{\beta \int \lambda} \right) +
O(\beta^2) \right] \right\}$$ as $n \rightarrow \infty$. Applying this together with the asymptotic expression (\[eq:number\_of\_check\_valid\_assign\]), and assuming sufficiently small $\beta$, we find that as $n \rightarrow \infty$ (exploiting the fact that $\delta \int \rho = \beta \int \lambda$) $$P_{\mbox{\scriptsize valid}}(\beta n) \rightarrow \exp \{ n Y(\beta)\}
\label{eq:Pvalid_limit}$$ where $$Y(\beta) = \frac{\beta}{r} \log \left( \frac{e C}{\beta \int \lambda} \right) - \beta \log \left( \frac{e}{\beta \int \lambda} \right) + O(\beta^2) \; .$$
Next, we note that the expected number of D-GLDPC codewords of weight $\alpha n$ in the ensemble ${{\mathcal{M}}}_n$ is equal to the sum over $\beta$ of the expected numbers of split assignments of split weight $(\alpha n, \beta n)$ which are both check-valid and variable-valid, denoted $N^{v,c}_{\alpha n, \beta n}$: $$\mathbb{E}_{{{\mathcal{M}}}_n} \left[ N_{\alpha n} \right] = \sum_{\beta} \mathbb{E}_{{{\mathcal{M}}}_n} [ N^{v,c}_{\alpha n, \beta n} ] \; .$$ This may then be expressed as $$\begin{gathered}
\mathbb{E}_{{{\mathcal{M}}}_n} \left[ N_{\alpha n} \right] = \\
\sum_{\beta} P_{\mbox{\scriptsize valid}}(\beta n) \sum_{\substack{\sum \alpha_t = \alpha \\ \sum \beta_t = \beta}} \left[ \prod_{t \in I_v} N_{v,t}^{(\delta_t n)}(\alpha_t n, \beta_t n) \right]\end{gathered}$$ where the fraction $\delta_t$ of VNs of type $t \in I_v$ is given by (\[eq:delta\_t\_definition\]) and the second sum is over all partitions of $\alpha$ and $\beta$ into $n_v$ elements, i.e., we have $\alpha_t, \beta_t \ge 0$ for all $t \in I_v$, and $\sum_{t \in I_v} \alpha_t = \alpha$, $\sum_{t \in I_v} \beta_t = \beta$.
Now, using (\[eq:Nvt\_tau\_sigma\_start\])-(\[eq:Nvt\_tau\_sigma\_end\]), as $n \rightarrow \infty$ we have for each $t \in I_v$ $$N_{v,t}^{(\delta_t n)}(\alpha_t n, \beta_t n) \rightarrow \exp \left\{ n X^{(\delta_t)}_t(\alpha_t, \beta_t) \right\} \; ,$$ where, for each $t \in I_v$, $$X^{(\delta_t)}_t(\alpha_t, \beta_t) = \delta_t \max_{{{\mbox{\scriptsize \boldmath $\eta$}}}^{(t)}} \sum_{(i,j) \in S_t} \eta^{(t)}_{i,j} \log \left( \frac{B^{(t)}_{i,j}}{\eta^{(t)}_{i,j}} \right)
\label{eq:X_function}$$ and the maximization over ${{\mbox{\boldmath $\eta$}}}^{(t)} = ( \eta^{(t)}_{i,j} )_{(i,j) \in S_t}$ is subject to the constraints $$\sum_{(i,j) \in S_t} \eta^{(t)}_{i,j} = 1
\label{eq:eta_sum_constraint}$$ $$\sum_{(i,j) \in S_t^{-}} i \eta^{(t)}_{i,j} = \alpha_t / \delta_t
\label{eq:sum_xi_constraint}$$ $$\sum_{(i,j) \in S_t^{-}} j \eta^{(t)}_{i,j} = \beta_t / \delta_t
\label{eq:sum_theta_constraint}$$ and $$\eta^{(t)}_{i,j} \ge 0 \quad \forall (i,j) \in S_t \; .
\label{eq:nonnegative_eta_constraint}$$ Therefore, recalling (\[eq:Pvalid\_limit\]), we have that as $n \rightarrow \infty$, $$\begin{gathered}
\mathbb{E}_{{{\mathcal{M}}}_n} \left[ N_{\alpha n} \right] \rightarrow \\
\sum_{\beta} \sum_{\substack{\sum \alpha_t = \alpha \\ \sum \beta_t = \beta}}
\exp \left\{ n \left[ \sum_{t \in I_v} X^{(\delta_t)}_t(\alpha_t, \beta_t) + Y(\beta) \right] \right\} \; .
\label{eq:sum_of_exp}\end{gathered}$$ Next, for each $t \in I_v$ we define $$F_t({{\mbox{\boldmath $\eta$}}}^{(t)}) = \eta^{(t)}_{0,0} \log \left( \frac{1}{\eta^{(t)}_{0,0}} \right) - \sum_{(i,j) \in S_t^{-}} \eta^{(t)}_{i,j} \; .$$ Note that the expression (\[eq:sum\_of\_exp\]) is dominated as $n \rightarrow \infty$ by the term which maximizes the argument of the exponential. Thus we may write $$\begin{gathered}
\!\!\!\!\!\! G(\alpha) = \max_{\beta} \max_{\substack{\sum \alpha_t = \alpha \\ \sum \beta_t = \beta}} \Bigg\{ \sum_{t \in I_v} \delta_t \max_{{{\mbox{\scriptsize \boldmath $\eta$}}}^{(t)}} \Bigg[ \sum_{(i,j) \in S_t^{-}} \eta^{(t)}_{i,j} \log \left( \frac{e B^{(t)}_{i,j}}{\eta^{(t)}_{i,j}} \right) \\
+ F_t({{\mbox{\boldmath $\eta$}}}^{(t)}) \Bigg] + \frac{\beta}{r} \log \left( \frac{e C}{\beta \int \lambda} \right) \\ - \beta \log \left( \frac{e}{\beta \int \lambda} \right) + O(\beta^2) \Bigg\}
\label{eq:growth_rate_with_O_notation2}\end{gathered}$$ where the maximization over ${{\mbox{\boldmath $\eta$}}}^{(t)} = ( \eta^{(t)}_{i,j} )_{(i,j) \in S_t^{-}}$ (for each $t \in I_v$) is subject to constraints (\[eq:sum\_xi\_constraint\]) and (\[eq:sum\_theta\_constraint\]) together with $\eta^{(t)}_{i,j} \ge 0$ for all $(i,j) \in S_t^{-}$.
We next have the following lemma.
The expression $\sum_{t \in I_v} \delta_t F_t({{\mbox{\boldmath $\eta$}}}^{(t)})$ is $O(\alpha^2)$ for any ${{\mbox{\boldmath $\eta$}}}^{(t)}$ satisfying the optimization constraints (\[eq:eta\_sum\_constraint\])-(\[eq:nonnegative\_eta\_constraint\])[^4]. \[lemma:O2\_term\]
A proof of this lemma is given in the Appendix. It follows from Lemma \[lemma:O2\_term\] that the expression $\sum_{t \in I_v} \delta_t F_t({{\mbox{\boldmath $\eta$}}}^{(t)})$ is $O(\alpha^2)$ for the maximizing ${{\mbox{\boldmath $\eta$}}}^{(t)}$. Also, since $\beta/\alpha$ is bounded between two positive constants, any expression which is $O(\beta^2)$ must necessarily also be $O(\alpha^2)$. Therefore $$\begin{gathered}
\!\! G(\alpha) = \max_{\beta} \max_{\substack{\sum \alpha_t = \alpha \\ \sum \beta_t = \beta}} \Bigg[ \sum_{t \in I_v} \delta_t \max_{{{\mbox{\scriptsize \boldmath $\eta$}}}^{(t)}} \sum_{(i,j) \in S_t^{-}} \eta^{(t)}_{i,j} \log \left( \frac{e B^{(t)}_{i,j}}{\eta^{(t)}_{i,j}} \right) \\
+ \frac{\beta}{r} \log \left( \frac{e C}{\beta \int \lambda} \right) - \beta \log \left( \frac{e}{\beta \int \lambda} \right) \Bigg] + O(\alpha^2)\end{gathered}$$ where the optimization is (as before) subject to the constraints (\[eq:sum\_xi\_constraint\]) and (\[eq:sum\_theta\_constraint\]) together with $\eta^{(t)}_{i,j} \ge 0$ for all $(i,j) \in S_t^{-}$. In what follows, for convenience of presentation we shall temporarily omit the $O(\alpha^2)$ term in the expression for growth rate.
Next we make the substitution $\gamma^{(t)}_{i,j} = \delta_t \eta^{(t)}_{i,j}$ for all $t \in I_v$, $(i,j) \in S_t^{-}$. This yields $$\begin{gathered}
\!\! G(\alpha) = \max_{\beta} \max_{\substack{\sum \alpha_t = \alpha \\ \sum \beta_t = \beta}} \Bigg[ \sum_{t \in I_v} \max_{{{\mbox{\scriptsize \boldmath $\gamma$}}}^{(t)}} \sum_{(i,j) \in S_t^{-}} \gamma^{(t)}_{i,j} \log \left( \frac{e B^{(t)}_{i,j} \delta_t}{\gamma^{(t)}_{i,j}} \right) \\
+ \frac{\beta}{r} \log \left( \frac{e C}{\beta \int \lambda} \right) - \beta \log \left( \frac{e}{\beta \int \lambda} \right) \Bigg]\end{gathered}$$ where the maximization over ${{\mbox{\boldmath $\gamma$}}}^{(t)} = ( \gamma^{(t)}_{i,j} )_{(i,j) \in S_t^{-}}$ (for each $t \in I_v$) is subject to the constraints $\sum_{(i,j) \in S_t^{-}} i \gamma^{(t)}_{i,j} = \alpha_t$, $\sum_{(i,j) \in S_t^{-}} j \gamma^{(t)}_{i,j} = \beta_t$, and $\gamma^{(t)}_{i,j} \ge 0$ for all $(i,j) \in S_t^{-}$. We observe that this maximization may be recast as $$\begin{gathered}
G(\alpha) = \max_{{{\mbox{\scriptsize \boldmath $\gamma$}}}} \Bigg[ \sum_{t \in I_v} \sum_{(i,j) \in S_t^{-}} \gamma^{(t)}_{i,j} \log \left( \frac{e B^{(t)}_{i,j} \delta_t}{\gamma^{(t)}_{i,j}} \right) \\
+ \frac{\beta({{\mbox{\boldmath $\gamma$}}})}{r} \log \left( \frac{e C}{\beta({{\mbox{\boldmath $\gamma$}}}) \int \lambda} \right) - \beta({{\mbox{\boldmath $\gamma$}}}) \log \left( \frac{e}{\beta({{\mbox{\boldmath $\gamma$}}}) \int \lambda} \right) \Bigg]\end{gathered}$$ where the maximization, which is now over ${{\mbox{\boldmath $\gamma$}}}= (\gamma^{(t)}_{i,j})_{t \in I_v, (i,j) \in S_t^{-}}$, is subject to the constraints $$\sum_{t \in I_v} \sum_{(i,j) \in S_t^{-}} i \gamma^{(t)}_{i,j} = \alpha$$ and $\gamma^{(t)}_{i,j} \ge 0$ for all $t \in I_v$, $(i,j) \in S_t^{-}$, and where $$\beta({{\mbox{\boldmath $\gamma$}}}) = \sum_{t \in I_v} \sum_{(i,j) \in S_t^{-}} j \gamma^{(t)}_{i,j} \; .$$ Making the substitution $\nu^{(t)}_{i,j} = \gamma^{(t)}_{i,j} / \alpha$ for all $t \in I_v$, $(i,j) \in S_t^{-}$, we obtain $$\begin{gathered}
G(\alpha) = \alpha \max_{{{\mbox{\scriptsize \boldmath $\nu$}}}} \Bigg[ \sum_{t \in I_v} \sum_{(i,j) \in S_t^{-}} \nu^{(t)}_{i,j} \log \left( \frac{e B^{(t)}_{i,j} \delta_t}{\alpha \nu^{(t)}_{i,j}} \right) \\
+ \frac{z({{\mbox{\boldmath $\nu$}}})}{r} \log \left( \frac{e C}{\alpha z({{\mbox{\boldmath $\nu$}}}) \int \lambda} \right) - z({{\mbox{\boldmath $\nu$}}}) \log \left( \frac{e}{\alpha z({{\mbox{\boldmath $\nu$}}}) \int \lambda} \right) \Bigg]\end{gathered}$$ where the maximization over ${{\mbox{\boldmath $\nu$}}}= (\nu^{(t)}_{i,j})_{t \in I_v, (i,j) \in S_t^{-}}$ is subject to the constraints $\sum_{t \in I_v} \sum_{(i,j) \in S_t^{-}} i \nu^{(t)}_{i,j} = 1$ and $\nu^{(t)}_{i,j} \ge 0$ for all $t \in I_v$, $(i,j) \in S_t^{-}$, and where $$z({{\mbox{\boldmath $\nu$}}}) = \sum_{t \in I_v} \sum_{(i,j) \in S_t^{-}} j \nu^{(t)}_{i,j} \; .$$ Assuming the condition $r=2$, we may write $$G(\alpha) = \alpha \max_{{{\mbox{\scriptsize \boldmath $\nu$}}}} \left( K_1({{\mbox{\boldmath $\nu$}}}) + K_2({{\mbox{\boldmath $\nu$}}}) \log \alpha \right)$$ where $$K_2({{\mbox{\boldmath $\nu$}}}) = \frac{z({{\mbox{\boldmath $\nu$}}})}{2} - \sum_{t \in I_v} \sum_{(i,j) \in S_t^{-}} \nu^{(t)}_{i,j} \; .$$ Next, assuming the condition $p=2$, we make the observation that $$z({{\mbox{\boldmath $\nu$}}}) = \sum_{t \in I_v} \sum_{(i,j) \in S_t^{-}} j \nu^{(t)}_{i,j} \ge 2 \sum_{t \in I_v} \sum_{(i,j) \in S_t^{-}} \nu^{(t)}_{i,j}$$ with equality if and only if $\nu^{(t)}_{i,j} = 0 $ for all $t \in I_v, (i,j) \in S_t^{-}$ with $j>2$. Therefore $K_2({{\mbox{\boldmath $\nu$}}}) \ge 0$ with equality if and only if $\nu^{(t)}_{i,j} = 0 $ for all $t \in I_v, (i,j) \in S_t^{-}$ with $j>2$. Let ${{\mbox{\boldmath $\nu$}}}_1$ and ${{\mbox{\boldmath $\nu$}}}_2$ be two distributions satisfying the optimization constraints, and suppose that $K_2({{\mbox{\boldmath $\nu$}}}_1) > 0$ and $K_2({{\mbox{\boldmath $\nu$}}}_2) = 0$. Then for sufficiently small $\alpha$, we must have $$K_1({{\mbox{\boldmath $\nu$}}}_1) + K_2({{\mbox{\boldmath $\nu$}}}_1) \log \alpha < K_1({{\mbox{\boldmath $\nu$}}}_2) + K_2({{\mbox{\boldmath $\nu$}}}_2) \log \alpha \; .$$ This follows from the fact that the inequality $$K_2({{\mbox{\boldmath $\nu$}}}_1) \log \alpha < K_1({{\mbox{\boldmath $\nu$}}}_2) - K_1({{\mbox{\boldmath $\nu$}}}_1)$$ will always be satisfied for $\alpha$ sufficiently small (since $\log \alpha \rightarrow~-\infty$ as $\alpha \rightarrow 0$, and recalling that $K_2({{\mbox{\boldmath $\nu$}}}_1) > 0$).
Therefore, for sufficiently small $\alpha$, the vector ${{\mbox{\boldmath $\nu$}}}$ which maximizes $K_1({{\mbox{\boldmath $\nu$}}}) + K_2({{\mbox{\boldmath $\nu$}}}) \log \alpha$ must satisfy $\nu^{(t)}_{i,j} = 0 $ for all $t \in I_v, (i,j) \in S_t^{-}$ with $j>2$. Note that this implies that the maximum, and hence the growth rate, depends only on the check and VNs with minimum distance equal to $2$. Also, recall that for each $t \in X_v$, the set $L_t = \{ i \in \mathbb{Z} \; : \; B^{(t)}_{i,2} > 0 \}$; we contract the vector ${{\mbox{\boldmath $\nu$}}}$ to include only variables $\nu^{(t)}_{i,2}$ where $t \in X_v$ and $i \in L_t$ (since only these may assume positive values).
The growth rate may be written as $$\begin{gathered}
G(\alpha) = \alpha \max_{{{\mbox{\scriptsize \boldmath $\nu$}}}} \Big[ \sum_{t \in X_v} \sum_{i \in L_t} \nu^{(t)}_{i,2} \log \left( \frac{B^{(t)}_{i,2} \delta_t}{\nu^{(t)}_{i,2}} \right) \\
+ s({{\mbox{\boldmath $\nu$}}}) \log \frac{s({{\mbox{\boldmath $\nu$}}})}{\phi} \Big] \triangleq \alpha \max_{{{\mbox{\scriptsize \boldmath $\nu$}}}} \left( \log R({{\mbox{\boldmath $\nu$}}}) \right)
\label{eq:log_R_definition}\end{gathered}$$ where $$\phi \triangleq \frac{1}{2 C \int \lambda}
\label{eq:phi_definition}$$ and where the function $s({{\mbox{\boldmath $\nu$}}})$ is given by $$s({{\mbox{\boldmath $\nu$}}}) = \sum_{t \in X_v} \sum_{i \in L_t} \nu^{(t)}_{i,2} \; .
\label{eq:s_nu_definition}$$ The maximization over ${{\mbox{\boldmath $\nu$}}}= (\nu^{(t)}_{i,2})_{t \in I_v, i \in L_t}$ in (\[eq:log\_R\_definition\]) is subject to the constraints $$h({{\mbox{\boldmath $\nu$}}}) = \sum_{t \in I_v} \sum_{i \in L_t} i \nu^{(t)}_{i,2} = 1
\label{eq:final_sum_constraint}$$ and $$\nu^{(t)}_{i,2} \ge 0 \quad \forall t \in X_v, i \in L_t \; .
\label{eq:final_inequality_constraint}$$ Let the vector ${{\mbox{\boldmath $\nu$}}}$ which maximizes (\[eq:log\_R\_definition\]) be denoted by $\tilde{{{\mbox{\boldmath $\nu$}}}}$. Then, our task is to show that $$R(\tilde{{{\mbox{\boldmath $\nu$}}}}) \triangleq \tilde{R} = \frac{1}{P^{-1}(1/C)}$$ i.e., that $$P \left( \frac{1}{\tilde{R}} \right) = \frac{1}{C} \; ,
\label{eq:polynomial_match}$$ where the parameter $C$ and the polynomial $P(x)$ are defined in (\[eq:C\_definition\]) and (\[eq:Px\_definition\]), respectively. We show this using Lagrange multipliers, ignoring for the moment the constraint (\[eq:final\_inequality\_constraint\]). We have $$\frac{\partial \log R({{\mbox{\boldmath $\nu$}}})}{\partial \nu^{(t)}_{i,2}} = \log \left( \frac{B^{(t)}_{i,2} \delta_t}{\nu^{(t)}_{i,2}} \right) + \log \left( \frac{s({{\mbox{\boldmath $\nu$}}})}{\phi} \right) \:\: ; \:\: \frac{\partial h({{\mbox{\boldmath $\nu$}}})}{\partial \nu^{(t)}_{i,2}} = i$$ so that, at the maximum, $$\log \left( \frac{B_{i,2} \delta_t}{\tilde{\nu}^{(t)}_{i,2}} \right) + \log \left( \frac{s(\tilde{{{\mbox{\boldmath $\nu$}}}})}{\phi} \right) = \lambda i$$ for all $t \in X_v$, $i \in L_t$, and for some Lagrange multiplier $\lambda \in \mathbb{R}$. Substituting back into (\[eq:log\_R\_definition\]) and using (\[eq:final\_sum\_constraint\]) yields $$\begin{gathered}
\log \tilde{R} = \sum_{t \in X_v} \sum_{i \in L_t} \nu^{(t)}_{i,2} \left( \lambda i - \log \left(\frac{s(\tilde{{{\mbox{\boldmath $\nu$}}}})}{\phi} \right) \right) \\ + s(\tilde{{{\mbox{\boldmath $\nu$}}}}) \log \left(\frac{s(\tilde{{{\mbox{\boldmath $\nu$}}}})}{\phi}\right) = \lambda\end{gathered}$$ i.e., the maximum value of the function $\log R({{\mbox{\boldmath $\nu$}}})$ is equal to the Lagrange multiplier. Thus we have $$\frac{B^{(t)}_{i,2}}{(\tilde{R})^i} = \left( \frac{\phi}{s(\tilde{{{\mbox{\boldmath $\nu$}}}}) \delta_t} \right) \tilde{\nu}^{(t)}_{i,2}$$ for all $t \in X_v$, $i \in L_t$. Substituting this into the LHS of (\[eq:polynomial\_match\]) and recalling the definition (\[eq:Px\_definition\]), we obtain $$\begin{aligned}
P \left( \frac{1}{\tilde{R}} \right) & = & \sum_{t \in X_v} \sum_{i \in L_t} \frac{2 \lambda_t}{q_t} \frac{B^{(t)}_{i,2}}{(\tilde{R})^i} \\
& = & \sum_{t \in X_v} \sum_{i \in L_t} \frac{\tilde{\nu}^{(t)}_{i,j}}{C s(\tilde{{{\mbox{\boldmath $\nu$}}}})}
= \frac{1}{C}\end{aligned}$$ where we have used (\[eq:delta\_t\_definition\]), (\[eq:phi\_definition\]) and (\[eq:s\_nu\_definition\]). This completes the proof of the theorem. Note that (\[eq:growth\_rate\_case\_1\]) is a first-order Taylor series expansion around $\alpha = 0$ which directly generalizes the results of [@Di_Richardson_Urbanke] and [@paolini08:weight] (for irregular LDPC and GLDPC codes respectively) to the case of irregular D-GLDPC codes. Our result indicates that for this case also, the parameter $1 / P^{-1}(1/C)$ plays an analagous role to the parameter $\lambda'(0) \rho'(1)$ for irregular LDPC codes, and to the parameter $\lambda'(0)C$ for irregular GLDPC codes.
Conclusion
==========
An expression for the asymptotic growth rate of the weight distribution of D-GLDPC codes for small linear-weight codewords has been derived. The expression assumes the existence of minimum distance $2$ check and variable nodes, and involves the evaluation of a polynomial inverse, derived from the minimum distance $2$ variable nodes, at a point derived from the minimum distance $2$ check nodes. This generalizes known results for LDPC codes and GLDPC codes, and also generalizes the corresponding connection with the stability condition over the BEC.
Proof of Lemma \[lemma:optimization\_dominant\_term\_1D\] {#proof-of-lemmalemmaoptimization_dominant_term_1d .unnumbered}
=========================================================
First consider the set of positive rational numbers $\ell$ such that $\xi \ell \in \mathbb{Z}$ and ${{\mbox{Coeff }}}( \{ A(x) \} ^{\ell}, x^{\xi \ell}) > 0$. Then it is easy to see that either this set is empty, or it has infinite cardinality; if $t$ is one such $\ell$, then so is $jt$ for every positive integer $j$ (proof routine by induction). The former case is not of interest to us here. In the latter case, the following limit is well defined and exists [@Burshtein_Miller Theorem 1]: $$\begin{gathered}
\lim_{\ell\rightarrow \infty} \frac{1}{\ell} \log {{\mbox{Coeff }}}\left[ \left( A(x) \right) ^{\ell}, x^{\xi \ell} \right] \\
= \max_{{{\mbox{\scriptsize \boldmath $\beta$}}}} \sum_{i \in S} \beta_i \log \left( \frac{A_i}{\beta_i} \right)\end{gathered}$$ where $S = \{ i \in \mathbb{Z} \; : \; A_i > 0 \}$, ${{\mbox{\boldmath $\beta$}}}= ( \beta_i )_{i \in S}$, and the maximization is subject to the constraints $$g({{\mbox{\boldmath $\beta$}}}) = \sum_{i \in S} \beta_i = 1
\label{eq:beta_sum_constraint}$$ $$h({{\mbox{\boldmath $\beta$}}}) = \sum_{i \in S} i \beta_i = \xi
\label{eq:sum_xi_constraint_1D}$$ and $$\beta_i \ge 0 \quad \forall i \in S \; .
\label{eq:nonnegative_beta_constraint}$$
We solve this optimization problem using Lagrange multipliers, ignoring for the moment the final constraint. Defining $$f({{\mbox{\boldmath $\beta$}}}) = \sum_{i \in S} \beta_i \log \left( \frac{A_i}{\beta_i} \right)
\label{eq:f_beta}$$ we have $$\frac{\partial f}{\partial\beta_i} = \log \left(\frac{A_i}{e \beta_i}\right) \:\: ; \:\: \frac{\partial g}{\partial\beta_i} = 1 \:\: ; \:\:\: \frac{\partial h}{\partial\beta_i} = i$$ for all $i \in S$. Therefore we obtain $$\log \left(\frac{A_i}{e \beta_i}\right) + \lambda + \mu i = 0
\label{eq:Lagrange_1D}$$ for all $i \in S$, where $\lambda$ and $\mu$ are Lagrange multipliers. These equations, together with (\[eq:beta\_sum\_constraint\]) and (\[eq:sum\_xi\_constraint\_1D\]), yield $(|S|+2)$ equations in the $(|S|+2)$ unknowns $\{ \lambda, \mu, {{\mbox{\boldmath $\beta$}}}\}$. Setting $i = 0$ in (\[eq:Lagrange\_1D\]) yields $$\beta_0 = e^{\lambda-1}$$ and substituting this back into (\[eq:Lagrange\_1D\]) gives $$\beta_i = \beta_0 A_i z^i$$ for all $i \in S$, where $z = e^{\mu}$. So from (\[eq:sum\_xi\_constraint\_1D\]) $$\beta_0 \sum_{i \in S} i A_i z^i = \xi$$ Now for sufficiently small $\xi$, we may approximate $$\beta_0 \sum_{i \in S} i A_i z^i \approx \beta_0 c A_c z^c \; .$$ Applying this approximation, $\beta_i$ is nonzero only for $i \in \{ 0, c \}$. Therefore, from (\[eq:beta\_sum\_constraint\]) and (\[eq:sum\_xi\_constraint\_1D\]) we obtain the solution $$\beta_i = \left\{ \begin{array}{cl}
1 - \xi/c & \textrm{if } i = 0 \\
\xi/c & \textrm{if } i = c \\
0 & \textrm{otherwise. }\end{array}\right.
\label{eq:beta_solution}$$ It is easy to see that this solution satisfies (\[eq:nonnegative\_beta\_constraint\]). Finally, substituting the solution (\[eq:beta\_solution\]) into (\[eq:f\_beta\]) gives $$\begin{aligned}
\max_{{{\mbox{\scriptsize \boldmath $\beta$}}}} f({{\mbox{\boldmath $\beta$}}}) & = & \left(\frac{\xi}{c} - 1 \right) \log \left( 1 - \frac{\xi}{c} \right) + \frac{\xi}{c} \log \left( \frac{c A_c}{\xi} \right) \\
& = & \left(\frac{\xi}{c} - 1 \right) \left( - \frac{\xi}{c} + O(\xi^2) \right) + \frac{\xi}{c} \log \left( \frac{c A_c}{\xi} \right) \\
& = & \frac{\xi}{c} \log \left( \frac{e c A_c}{\xi} \right) + O(\xi^2) \; .\end{aligned}$$ This completes the proof of the lemma.
Proof of Lemma \[lemma:O2\_term\] {#proof-of-lemmalemmao2_term .unnumbered}
=================================
Consider any ${{\mbox{\boldmath $\eta$}}}^{(t)}$ which satisfies the optimization constraints (\[eq:eta\_sum\_constraint\])-(\[eq:nonnegative\_eta\_constraint\]). Since $\alpha = \sum_{t \in I_v} \alpha_t$, $\alpha$ small implies that $\alpha_t$ is small for every $t \in I_v$. From constraint (\[eq:sum\_xi\_constraint\]) we conclude that $\eta_{i,j}^{(t)}$ is small for every $t \in I_v$, $(i,j) \in S_t^{-}$, and so $\eta_{0,0}^{(t)}$ is close to $1$ for all $t \in I_v$. Formally, for any $t \in I_v$ the term in the sum over ${{\mbox{\boldmath $\eta$}}}^{(t)}$ in (\[eq:X\_function\]) corresponding to $(i,j) = (0,0)$ may be written as (here we use (\[eq:eta\_sum\_constraint\]), and the Taylor series of $\log\left(1-x\right)$ around $x=0$) $$\begin{gathered}
\eta^{(t)}_{0,0} \log \left( \frac{1}{\eta^{(t)}_{0,0}} \right) = \Big( \sum_{(i,j) \in S_t^{-}} \eta^{(t)}_{i,j} - 1 \Big) \log \Big( 1 - \sum_{(i,j) \in S_t^{-}} \eta^{(t)}_{i,j} \Big) \\
= \Big( \sum_{(i,j) \in S_t^{-}} \eta^{(t)}_{i,j} - 1 \Big) \Bigg(- \sum_{(i,j) \in S_t^{-}} \eta^{(t)}_{i,j} + O \Big( \Big( \sum_{(i,j) \in S_t^{-}} \eta^{(t)}_{i,j} \Big)^2 \Big) \Bigg) \\
= \sum_{(i,j) \in S_t^{-}} \eta^{(t)}_{i,j} + O \Big( \Big( \sum_{(i,j) \in S_t^{-}} \eta^{(t)}_{i,j} \Big)^2 \Big) \end{gathered}$$ Therefore we have $$\left| F_t({{\mbox{\boldmath $\eta$}}}^{(t)}) \right| \le k_t \Big( \sum_{(i,j) \in S_t^{-}} \eta^{(t)}_{i,j} \Big)^2
\label{eq:F_t_bound}$$ for some $k_t > 0$ independent of $\{ \eta^{(t)}_{i,j} \}_{(i,j) \in S_t^{-}}$. It follows that $$\left| \sum_{t \in I_v} \delta_t F_t({{\mbox{\boldmath $\eta$}}}^{(t)}) \right| \le \sum_{t \in I_v} \delta_t \left| F_t({{\mbox{\boldmath $\eta$}}}^{(t)}) \right| \le \sum_{t \in I_v} \delta'_t \Big( \sum_{(i,j) \in S_t^{-}} \eta^{(t)}_{i,j} \Big)^2
\label{eq:F_t_ineq_1}$$ where $\delta'_t = k_t \delta_t$ for each $t \in I_v$. Also, by (\[eq:sum\_xi\_constraint\]) we have $\sum_{(i,j) \in S_t^{-}} \eta^{(t)}_{i,j} \le \alpha_t / \delta_t$ and therefore $$\sum_{t \in I_v} \delta'_t \Big( \sum_{(i,j) \in S_t^{-}} \eta^{(t)}_{i,j} \Big)^2 \le \sum_{t \in I_v} \left( \frac{\delta'_t}{\delta_t^2} \right) \alpha_t^2
\label{eq:F_t_ineq_2}$$ Denote $\delta = \max_{t \in I_v} \{ \delta'_t / \delta_t^2 \}$; then, combining (\[eq:F\_t\_ineq\_1\]) and (\[eq:F\_t\_ineq\_2\]), $$\left| \sum_{t \in I_v} \delta_t F_t({{\mbox{\boldmath $\eta$}}}^{(t)}) \right| \le \delta \sum_{t \in I_v} \alpha_t^2 < \delta \left( \sum_{t \in I_v} \alpha_t \right) ^2 = \delta \alpha^2$$ and thus the expression $\sum_{t \in I_v} \delta_t F_t({{\mbox{\boldmath $\eta$}}}^{(t)})$ is $O(\alpha^2)$, as desired.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the EC under Seventh FP grant agreement ICT OPTIMIX n. INFSO-ICT-214625 and in part by the University of Bologna (ISA-ESRF fellowship).
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[^1]: Here we make use of the following general result [@Wilf]. Let $a_i$ be the number of ways of obtaining an outcome $i\in\mathbb{Z}$ in experiment ${{\mathcal{A}}}$, and let $b_j$ be the number of ways of obtaining an outcome $j\in\mathbb{Z}$ in experiment ${{\mathcal{B}}}$. Also let $c_k$ be the number of ways of obtaining an outcome $(i,j)$ in the combined experiment $({{\mathcal{A}}}, {{\mathcal{B}}})$ with sum $i+j=k$. Then the generating functions $A(x)=\sum_i a_i x^i$, $B(x)=\sum_j b_j x^j$ and $C(x)=\sum_k c_k x^k$ are related by $C(x) = A(x) B(x)$.
[^2]: Observe that as $m\rightarrow \infty$, $\sum_t \exp ( m Z_t ) \rightarrow \exp ( m \max_t \{Z_t\} )$
[^3]: We use the following result on bivariate generating functions [@Wilf]. Let $a_{i,j}$ be the number of ways of obtaining an outcome $(i,j)\in\mathbb{Z}^2$ in experiment ${{\mathcal{A}}}$, and let $b_{k,l}$ be the number of ways of obtaining an outcome $(k,l)\in\mathbb{Z}^2$ in experiment ${{\mathcal{B}}}$. Also let $c_{p,q}$ be the number of ways of obtaining an outcome $((i,j),(k,l))$ in the combined experiment $({{\mathcal{A}}}, {{\mathcal{B}}})$ with sums $i+k=p$ and $j+l=q$. Then the generating functions $A(x,y)=\sum_{i,j} a_{i,j} x^i y^j$, $B(x,y)=\sum_{k,l} b_{k,l} x^k y^l$ and $C(x,y)=\sum_{p,q} c_{p,q} x^p y^q$ are related by $C(x,y) = A(x,y) B(x,y)$.
[^4]: Here we use the following standard notation: the real-valued function $f(x)$ is said to be $O (g(x))$ if and only if there exist positive real numbers $k$ and $\epsilon$, both independent of $x$, such that $$\left| f(x) \right| \le k g(x) \quad \forall \; 0 \le x \le \epsilon \; .$$
|
---
author:
- 'C.S. Gerving, T.M. Hoang, B.J. Land, M. Anquez, C.D. Hamley, and M.S. Chapman'
bibliography:
- 'QPref.bib'
title: 'Non-equilibrium dynamics of an unstable quantum pendulum'
---
****
A pendulum prepared perfectly inverted and motionless is a prototype of unstable equilibria and corresponds to an unstable hyperbolic fixed point in the dynamical phase space. Unstable fixed points are central to understanding Hamiltonian chaos in classical systems [@Tabor]. In many-body quantum systems, mean-field approximations fail in the vicinity of unstable fixed points and lead to dynamics driven by quantum fluctuations [@Sachdev01; @Dziarmaga10]. Here, we measure the non-equilibrium dynamics of a many-body quantum pendulum initialized to a hyperbolic fixed point of the phase space. The experiment uses a spin-1 Bose condensate [@Ho98; @Ohmi98; @Stenger99], which exhibits Josephson dynamics in the spin populations that correspond in the mean-field limit to motion of a non-rigid mechanical pendulum [@Smerzi97; @Zhang05]. The condensate is initialized to a minimum uncertainty spin state, and quantum fluctuations lead to non-linear spin evolution along a separatrix and non-Gaussian probability distributions that are measured to be in good agreement with exact quantum calculations up to 0.25 s. At longer times, atomic loss due to the finite lifetime of the condensate leads to larger spin oscillation amplitudes compared to no loss case as orbits depart from the separatrix. This demonstrates how decoherence of a many-body system can result in more apparent coherent behaviour. This experiment provides new avenues for studying macroscopic spin systems in the quantum limit and for investigations of important topics in non-equilibrium quantum dynamics [@Polkovnikov11].
A pendulum initialized to a hyperbolic fixed point is metastable in the classical limit. Phase orbits passing close to these points have exponentially diverging periods, and the orbits passing exactly through these points form a separatrix between librational and rotational motion of the pendulum with an infinite period. If the pendulum is prepared perfectly in this orientation, the classical equations of motion predict that it will not evolve. In reality, even if perfect preparation was possible, thermal fluctuations of the pendulum would perturb the pendulum from the metastable orientation and lead to oscillation. Even at zero temperature, unavoidable quantum fluctuations would lead to evolution [@Cook86; @Leibscher09]. Although mechanical pendulums operating at the quantum limit are currently unavailable in the lab, it is possible to study quantum many-body systems that have similar dynamical behavior [@Albiez05; @Chang05; @Levy07].
The focus of this work is spin-1 atomic Bose condensates [@Ho98; @Ohmi98; @Stenger99] with ferromagnetic interactions tightly confined in optical traps such that spin domain formation is energetically suppressed. In this case, the non-trivial dynamical evolution of the system occurs only in the internal spin variables, and the mean-field dynamics of the system can be described by a non-rigid pendulum similar to the two site Bose-Hubbard model [@Smerzi97; @Zhang05]. The system is fully integrable in both the quantum [@Law98] and classical [@Pu99; @Zhang05] limits, and exhibits a rich array of non-linear phenomena including Hamiltonian monodromy [@Lamacraft11]. Furthermore, the condensate features a tunable Hamiltonian with a quantum phase transition that permits quenching of the condensate to highly-excited spin states. Together, these provide unique capabilities to explore non-equilibrium quantum dynamics that are not captured by mean-field approaches and can be solved exactly with Schrödinger’s equation.
In these experiments, we study the evolution of a quenched spin-1 condensate prepared in a metastable state corresponding to a hyperbolic fixed point in the spin-nematic phase space that ultimately evolves far beyond the perturbative limit. The quantum solution of the problem at zero magnetic field yields intricate spin-mixing dynamics that exhibit non-linear quantum revivals [@Law98] and a *quantum carpet* of highly non-Gaussian fluctutations [@Diener06]. At finite fields, the dynamics are similar [@Chen09; @Heinze10], although they occur on a time-scale favorable for experimental observation. In both cases, the evolution occurs along a separatrix of the phase space and is driven by quantum fluctuations that are absent from the mean-field theory solutions [@Pu99; @Heinze10].
The equilibrium states, domain formation and spin dynamics of spinor condensates have been studied in many experiments [@Stenger99; @Schmaljohann04; @Chang04; @Chang05; @Kronjager06; @Black07; @Liu09; @Leslie09; @Klempt10; @Bookjans11b; @Gross11; @Lucke11; @Hamley12]. In particular, observation of coherent spin oscillations have confirmed the mean-field pendulum model for small condensates [@Chang05; @Kronjager06; @Black07]. Spin evolution has been previously observed from metastable spin states in many experiments [@Stenger99; @Chang04; @Klempt10; @Bookjans11b; @Gross11; @Lucke11; @Hamley12], however, the experiments have not yet demonstrated spin dynamics in agreement with quantum calculations, except in the perturbative, low-depletion limit at very short times (where a Bogoliubov expansion around the mean field can be used) [@Leslie09; @Klempt10; @Hamley12] or for conditions where the mean-field approach suffices. Here, by using low-noise atom detection techniques and careful state preparation, we are able to observe quantum spin dynamics that agree well with quantum calculations and demonstrate a rich array of non-Gaussian fluctuations.
We begin by discussing the exact quantum model for spin-1 condensate small enough to be described by a single domain. The quantum states of the system can be described in a Fock basis, $|N_1,N_0,N_{-1}\rangle$, where $N_i$ are the number of atoms in the three spin-1 Zeeman states. The spin dynamics, including the effects of a magnetic field, are governed by the interaction Hamiltonian [@Law98; @Chen09; @Heinze10]: $$\begin{aligned}
\label{Hamilton}
\mathcal{H} &=& \lambda [(
\hat{N}_{1} - \hat{N}_{-1})^2
+ (2 \hat{N}_{0}-1) (\hat{N}_{1}+ \hat{N}_{-1}) \nonumber\\
& &~~+ 2 \hat{a}_{0}^\dag \hat{a}_{0}^\dag \hat{a}_{1} \hat{a}_{-1}
+ 2 \hat{a}_{1}^\dag \hat{a}_{-1}^\dag \hat{a}_{0} \hat{a}_{0}] \nonumber\\
& &~~+q(\hat{N}_{1}+ \hat{N}_{-1}).\end{aligned}$$
Here, $\hat{a}_{i} $ are the bosonic annihilation operators for the three spin states and $\hat{N}_{i}=\hat{a}_{i}^\dag \hat{a}_{i}$. $\lambda$ and $q (\propto B^2 )$ characterize the inter-spin and Zeeman energies, respectively. The spin-dependent binary collisions restrict the dynamical evolution to states that conserve both the total number of atoms $N=\sum_i N_i$ and the projection of angular momentum along the quantization axis $M=N_1-N_{-1}$. Starting from the initial state $|0,N,0\rangle$, consisting of all $N$ atoms in the $m_f = 0 $ state, the evolution is constrained to final states of the form $\sum_p c_p |p,N-2p,p\rangle$. Hence, the solution to the quantum many-body problem is fully enumerated by the time-dependence of the Fock state amplitudes, $c_p(t)$.
![**Phase Space** The spin state immediately following the quench is depicted on two relevant phase spaces of the spin-1 system: the $\{\rho_0,\theta_s\}$ phase space (bottom-left) and the $\{S_x,Q_{yz},Q_{zz}\}$ spin-nematic Bloch sphere (right). A zoom-in of the hyperbolic fixed point at the pole is shown (top-left) with arrows indicating the orbit directions. The $\rho_0,\theta_s$ phase space represents a Mercator projection of the $\{S_x,Q_{yz},Q_{zz}\}$ sub-space.[]{data-label="PhaseSpace"}](NonEquilPaperPhaseSpace)
The semi-classical dynamics of the system take the form of a non-rigid pendulum [@Zhang05]. Mean field states of a spin-1 condensates can be written as $\psi=(\zeta_{+1},\zeta_0,\zeta_{-1})^T$ where $\zeta_i=\sqrt{\rho_i} e^{i \theta_{i}}$, and $\rho_i =|\zeta_i|^2 =N_i/N$ are the fractional spin populations. The conservation of magnetization $m=(N_1-N_{-1})/N$ constrains the populations $\rho_{\pm1}=(1-\rho_0 \pm m)/2 $, and for the $m=0$ case that is relevant for these experiments, the spin dynamics are determined by the Hamiltonian: $$\mathcal{H} = \lambda' x^2-\lambda'(1-x^2) \text{cos} \theta_s-q x
\label{mf}$$
Here, $x=(\rho_0-1/2)/2$ and $\theta_s = \theta_{+1}+\theta_{-1}-2\theta_0$ are canonically conjugate variables and $\lambda' = 2N \lambda$. This Hamiltonian has the form of a classical non-rigid pendulum and is similar to the double-well Bose-Hubbard model that has been used to study Josephson effects in condensates. The Hamiltonian can also be written using a phase space of the spin vector $S_i$ and nematic (quadrupole) tensor $Q_{ij}$ matrix operators for the spin-1 system: $\mathcal{H} = \lambda' \sum_i S_i^2 + q Q_{zz}/2$. The phase spaces for both of these forms are shown in , where it is clear that the $\rho_0,\theta_s$ phase space corresponds to a projection of the spin-nematic phase space.
The initial state of the system following the quench, $|0,N,0\rangle$, is indicated in the different phase spaces in using quasi-probability distributions of the initial state determined from the quantum uncertainties [@Hamley12]. In the spin-nematic space, the state corresponds to a minimum uncertainty state centered at the pole. The pole is a hyperbolic fixed point lying at the intersection of the separatrix that separates the librational and rotational orbits of the system. In the projected $\rho_0,\theta_s$ phase space, the distribution in $\rho_0$ is tightly packed at the top of the phase space with random spinor phase. In the absence of quantum fluctuations, the state initialized the hyperbolic fixed point is non-evolving. However, quantum fluctuations populate a family of orbits that straddle the fixed point, and subsequent evolution leads to phase flow along the unstable manifolds of the separatrix. In the short term, this creates squeezed states with negligible change in $\rho_0$ [@Hamley12]. For longer times, the system evolves along the separatrix, which forms a closed homoclinic orbit in the spin-nematic space.
![**Time evolution of spin populations.** Probability density of the fractional population of the condensate in the $m_f=0$ state, $\rho_0$, as a function of time. The curves show the mean, $\bar{\rho}_0$ (black line) and $\pm$ the standard deviation, $\sigma$ (blue lines). **(a)** Experimental data showing the results of 50 runs at each evolution time placed into 40 bins. The mean and standard deviation curves have been smoothed using a cubic spline. **(b)** Quantum calculation using the initial atom number, magnetic field ramp, and atom loss rate measured in the experiment. The Fock state probabilities $|c_p|^2$, placed into 100 bins, are plotted. []{data-label="Raw"}](ProbDensScale2 "fig:")\
![**Time evolution of spin populations.** Probability density of the fractional population of the condensate in the $m_f=0$ state, $\rho_0$, as a function of time. The curves show the mean, $\bar{\rho}_0$ (black line) and $\pm$ the standard deviation, $\sigma$ (blue lines). **(a)** Experimental data showing the results of 50 runs at each evolution time placed into 40 bins. The mean and standard deviation curves have been smoothed using a cubic spline. **(b)** Quantum calculation using the initial atom number, magnetic field ramp, and atom loss rate measured in the experiment. The Fock state probabilities $|c_p|^2$, placed into 100 bins, are plotted. []{data-label="Raw"}](ExpDataWMeanStdDev4 "fig:")\
![**Time evolution of spin populations.** Probability density of the fractional population of the condensate in the $m_f=0$ state, $\rho_0$, as a function of time. The curves show the mean, $\bar{\rho}_0$ (black line) and $\pm$ the standard deviation, $\sigma$ (blue lines). **(a)** Experimental data showing the results of 50 runs at each evolution time placed into 40 bins. The mean and standard deviation curves have been smoothed using a cubic spline. **(b)** Quantum calculation using the initial atom number, magnetic field ramp, and atom loss rate measured in the experiment. The Fock state probabilities $|c_p|^2$, placed into 100 bins, are plotted. []{data-label="Raw"}](QMCWMeanStdDev775 "fig:")
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
**** ****
![**Full probability distributions of $\rho_0$.** Evolution on the semi-classical phase space (left column) and histograms of the measured spin population, $\rho_0$ (right) for different evolution times after the quench, 15 ms, 130 ms, 140 ms, 170 ms, and 240 ms. **a.** The simulations use the semi-classical equations of motion together with a quasi-probability distribution for the initial state. The mean value for $\rho_0$ and $\theta_s$ are indicated with a black dot. **b.** The histogram bars for each evolution time depict the measured probability density of $\rho_0$ for over 900 experimental runs, and the red line represents the simulation. The purple bar in each histogram represents the bin in which the mean of $\rho_0$ is located. []{data-label="PDF"}](econtour0ms "fig:") ![**Full probability distributions of $\rho_0$.** Evolution on the semi-classical phase space (left column) and histograms of the measured spin population, $\rho_0$ (right) for different evolution times after the quench, 15 ms, 130 ms, 140 ms, 170 ms, and 240 ms. **a.** The simulations use the semi-classical equations of motion together with a quasi-probability distribution for the initial state. The mean value for $\rho_0$ and $\theta_s$ are indicated with a black dot. **b.** The histogram bars for each evolution time depict the measured probability density of $\rho_0$ for over 900 experimental runs, and the red line represents the simulation. The purple bar in each histogram represents the bin in which the mean of $\rho_0$ is located. []{data-label="PDF"}](histo0ms "fig:")
![**Full probability distributions of $\rho_0$.** Evolution on the semi-classical phase space (left column) and histograms of the measured spin population, $\rho_0$ (right) for different evolution times after the quench, 15 ms, 130 ms, 140 ms, 170 ms, and 240 ms. **a.** The simulations use the semi-classical equations of motion together with a quasi-probability distribution for the initial state. The mean value for $\rho_0$ and $\theta_s$ are indicated with a black dot. **b.** The histogram bars for each evolution time depict the measured probability density of $\rho_0$ for over 900 experimental runs, and the red line represents the simulation. The purple bar in each histogram represents the bin in which the mean of $\rho_0$ is located. []{data-label="PDF"}](econtour130ms "fig:") ![**Full probability distributions of $\rho_0$.** Evolution on the semi-classical phase space (left column) and histograms of the measured spin population, $\rho_0$ (right) for different evolution times after the quench, 15 ms, 130 ms, 140 ms, 170 ms, and 240 ms. **a.** The simulations use the semi-classical equations of motion together with a quasi-probability distribution for the initial state. The mean value for $\rho_0$ and $\theta_s$ are indicated with a black dot. **b.** The histogram bars for each evolution time depict the measured probability density of $\rho_0$ for over 900 experimental runs, and the red line represents the simulation. The purple bar in each histogram represents the bin in which the mean of $\rho_0$ is located. []{data-label="PDF"}](histo130ms "fig:")
![**Full probability distributions of $\rho_0$.** Evolution on the semi-classical phase space (left column) and histograms of the measured spin population, $\rho_0$ (right) for different evolution times after the quench, 15 ms, 130 ms, 140 ms, 170 ms, and 240 ms. **a.** The simulations use the semi-classical equations of motion together with a quasi-probability distribution for the initial state. The mean value for $\rho_0$ and $\theta_s$ are indicated with a black dot. **b.** The histogram bars for each evolution time depict the measured probability density of $\rho_0$ for over 900 experimental runs, and the red line represents the simulation. The purple bar in each histogram represents the bin in which the mean of $\rho_0$ is located. []{data-label="PDF"}](econtour140ms "fig:") ![**Full probability distributions of $\rho_0$.** Evolution on the semi-classical phase space (left column) and histograms of the measured spin population, $\rho_0$ (right) for different evolution times after the quench, 15 ms, 130 ms, 140 ms, 170 ms, and 240 ms. **a.** The simulations use the semi-classical equations of motion together with a quasi-probability distribution for the initial state. The mean value for $\rho_0$ and $\theta_s$ are indicated with a black dot. **b.** The histogram bars for each evolution time depict the measured probability density of $\rho_0$ for over 900 experimental runs, and the red line represents the simulation. The purple bar in each histogram represents the bin in which the mean of $\rho_0$ is located. []{data-label="PDF"}](histo140ms "fig:")
![**Full probability distributions of $\rho_0$.** Evolution on the semi-classical phase space (left column) and histograms of the measured spin population, $\rho_0$ (right) for different evolution times after the quench, 15 ms, 130 ms, 140 ms, 170 ms, and 240 ms. **a.** The simulations use the semi-classical equations of motion together with a quasi-probability distribution for the initial state. The mean value for $\rho_0$ and $\theta_s$ are indicated with a black dot. **b.** The histogram bars for each evolution time depict the measured probability density of $\rho_0$ for over 900 experimental runs, and the red line represents the simulation. The purple bar in each histogram represents the bin in which the mean of $\rho_0$ is located. []{data-label="PDF"}](econtour170ms "fig:") ![**Full probability distributions of $\rho_0$.** Evolution on the semi-classical phase space (left column) and histograms of the measured spin population, $\rho_0$ (right) for different evolution times after the quench, 15 ms, 130 ms, 140 ms, 170 ms, and 240 ms. **a.** The simulations use the semi-classical equations of motion together with a quasi-probability distribution for the initial state. The mean value for $\rho_0$ and $\theta_s$ are indicated with a black dot. **b.** The histogram bars for each evolution time depict the measured probability density of $\rho_0$ for over 900 experimental runs, and the red line represents the simulation. The purple bar in each histogram represents the bin in which the mean of $\rho_0$ is located. []{data-label="PDF"}](histo170ms "fig:")
![**Full probability distributions of $\rho_0$.** Evolution on the semi-classical phase space (left column) and histograms of the measured spin population, $\rho_0$ (right) for different evolution times after the quench, 15 ms, 130 ms, 140 ms, 170 ms, and 240 ms. **a.** The simulations use the semi-classical equations of motion together with a quasi-probability distribution for the initial state. The mean value for $\rho_0$ and $\theta_s$ are indicated with a black dot. **b.** The histogram bars for each evolution time depict the measured probability density of $\rho_0$ for over 900 experimental runs, and the red line represents the simulation. The purple bar in each histogram represents the bin in which the mean of $\rho_0$ is located. []{data-label="PDF"}](econtour210ms "fig:") ![**Full probability distributions of $\rho_0$.** Evolution on the semi-classical phase space (left column) and histograms of the measured spin population, $\rho_0$ (right) for different evolution times after the quench, 15 ms, 130 ms, 140 ms, 170 ms, and 240 ms. **a.** The simulations use the semi-classical equations of motion together with a quasi-probability distribution for the initial state. The mean value for $\rho_0$ and $\theta_s$ are indicated with a black dot. **b.** The histogram bars for each evolution time depict the measured probability density of $\rho_0$ for over 900 experimental runs, and the red line represents the simulation. The purple bar in each histogram represents the bin in which the mean of $\rho_0$ is located. []{data-label="PDF"}](histo210ms "fig:")
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\
We now turn to the experimental results. The experiment begins with a rubidium-87 condensate containing $4 \times 10^4$ atoms, initialized in the $f=1,m_f=0$ hyperfine state and held in a high magnetic field. The condensate is rapidly quenched by lowering the field, and the spin populations are measured for different evolution times. The experiment is repeated many times in order to acquire sufficient statistics to determine the full probability distributions of the populations. The main results of the paper are shown in , which shows the measured probability density of $\rho_0 = N_0/N$ versus evolution time, which is effectively a determination of the probabilities $|c_p|^2$. The experimental results are compared with a quantum calculation using a spinor energy, $2\lambda N = -2 \pi \hbar \times 7.5~\mathrm{Hz}$, chosen to match the population dynamics. Both the experiment and quantum solutions exhibit population evolution that is in good overall agreement. In particular, both exhibit a long pause (80 ms) before any population evolution is apparent. After this pause, the spin population executes a regular damped oscillation. Population evolution from the metastable state is exponentially sensitive to initial population in the $m_F= \pm1$ states [@Klempt10]. At the earliest evolution time studied (15 ms), the total population in these states is measured to be $<30$ atoms which represents an upper bound limited by atom detection noise [@Hamley12]. Initial populations at this level effect the duration of the initial pause and first oscillation minimum, but not the overall character of the evolution [@Diener06] (see Supplemental Information). For evolution times beyond $>250$ ms, it is necessary to include in the theory the effects of atomic loss due to the lifetime of the condensate $\tau=1.8$ s, which is discussed in more detail below.
It is clear that the mean and standard deviation are insufficient to fully characterize the distribution of $\rho_0$ for both the experiment and theory, since for much of the evolution the mean does not pass through the highest probability density, and the asymmetry indicates a significant skew in the distribution. This point is reinforced in , which shows the full probability distributions for several evolution times, along with the theoretical predictions. The highly non-Gaussian nature of the distributions provide compelling evidence of the quantum nature of the spin dynamics. The physical origin of these non-Gaussian fluctuations is dispersion of neighboring orbits about the separatrix. Immediately following the quench, the distribution in $\rho_0$ is tightly packed at the top of the phase space with random spinor phase. This state corresponds to a minimum uncertainty state of the spin-nematic subspace shown in [@Hamley12]. As evolution proceeds, the phase, $\theta_s$, converges towards the separatrix separating the librational and rotational trajectories, and the population starts to evolve along it. The separatrix has a divergent period [@Zhang05], and so the states disperse significantly due to the different evolution rates of nearby energy contours. It is this dispersion, together with the shape of the orbit, that gives rise to the highly non-Gaussian probability distributions.
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![**Long-term evolution of $\rho_0$.** The top graph shows the quantum calculation without loss. The middle graphs shows the calculation including the effects of uncorrelated atom loss. The bottom graph shows the measured data. In each plot, the mean value is shown as a solid line, and the shaded envelopes correspond to the standard deviation[]{data-label="Moments2"}](rho0longSimConstC "fig:")\
![**Long-term evolution of $\rho_0$.** The top graph shows the quantum calculation without loss. The middle graphs shows the calculation including the effects of uncorrelated atom loss. The bottom graph shows the measured data. In each plot, the mean value is shown as a solid line, and the shaded envelopes correspond to the standard deviation[]{data-label="Moments2"}](rho0longSimVaryC-7p75 "fig:")\
![**Long-term evolution of $\rho_0$.** The top graph shows the quantum calculation without loss. The middle graphs shows the calculation including the effects of uncorrelated atom loss. The bottom graph shows the measured data. In each plot, the mean value is shown as a solid line, and the shaded envelopes correspond to the standard deviation[]{data-label="Moments2"}](rho0longdat1 "fig:")
In order to characterize the non-Gaussian distribution, we determine several central moments, $u_k = \langle (\rho - \bar{\rho})^k \rangle$ from the data. The first six central moments are shown in compared with the quantum simulation. Overall the measured moments are in good agreement with the predicted moments from the simulation. The population revival in the second oscillation predicted from the simulation is clearly seen in the first four moments, but is less obvious in higher moments.
We now turn to a discussion of the role of atomic loss in the dynamical evolution. The lifetime of the condensate $\tau=1.8$ s is only a factor of 10 larger than the spin evolution timescale ($\sim$150 ms), hence one expects that loss plays an important role in the dynamics. We explore this question in where we compare quantum calculations without loss, quantum calculations including uncorrelated loss and the experimental data. Uncorrelated atom loss is incorporated into the calculation using quantum Monte Carlo (QMC) techniques with the collapse operators $C_i = \sqrt{1/\tau}\, \hat{a}_i$. The loss causes the overall magnetization $M$ to execute a random walk with a restoring tendency towards $M=0$ and decreases the spinor dynamical rate, which scales as $\lambda \propto N^{-3/5}$. (Supplementary Information)
For the first 250 ms of evolution corresponding to the first spin oscillation, the effects of loss are not discernable between the two calculations, and the experimental data are in good agreement with both. Beyond 250 ms, there are significant differences between the two quantum calculations. The spin population of the calculation without loss nearly returns to the initial value and then experiences a long pause followed by complex multi-frequency oscillations. The calculation with loss however exhibits steady oscillations with one dominant frequency and a slowly decreasing amplitude centered on the ground state populations. In the semi-classical picture, the apparent damping of the calculation without loss derives from the dispersion about the separatrix in . The effect of loss is to eventually move the orbits away from the separatrix, which turns off this dispersion and leads to more regular oscillations.
While the inclusion of loss into the model makes a significant improvement in the agreement of long term dynamics ($>250$ ms) with the experimental results, it is clear that this simple loss model is inadequate to fully replicate the measurements at longer time scales. While the experimental data and the simulations with loss are qualitatively similar, there is clearly more dissipation in the experiment as the amplitude of the oscillations damp more quickly and the standard deviation decreases. In future work, we intend to further investigate the damping of the spin dynamics and its connection to thermalization of isolated quantum systems subject to loss. Similar investigations are on-going using 1-D condensate systems [@Kinoshita06; @Hofferberth07; @Hofferberth08; @Trotzky12], and it will be interesting to explore the similarities and differences in these completely different systems. Finally, we believe that our results point the way to a host of fascinating explorations of out-of-equilibrium quantum spin systems [@Dziarmaga10; @Polkovnikov11].
**Methods**
We prepare a condensate of $N= 38,500\pm500$ $^{87}$Rb atoms in the $|f=1,m_f=0 \rangle$ hyperfine state in a high magnetic field ($2~\mathrm{G}$). The condensate is tightly confined in an optical dipole trap with trap frequencies of $ 250~\mathrm{Hz}$. To initiate dynamical evolution, the condensate is quenched below the quantum critical point by lowering the magnetic field to a value $210~\mathrm{mG}$ and then allowed to freely evolve for a set time. The trap is then turned off and a Stern-Gerlach field is applied to separate the $m_f$ components during 22 ms time-of-flight expansion. The atoms are probed for $400~\mu\mathrm{s}$ with three pairs of orthogonal laser beams, and the resulting fluorescence signal is collected by a CCD camera with $>90\%$ quantum efficiency.
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title: 'GEMSEC: Graph Embedding with Self Clustering'
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Introduction {#sec:introduction}
============
Community detection is one of the most important problems in network analysis due to its wide applications ranging from the analysis of collaboration networks to image segmentation, the study of protein-protein interaction networks in biology, and many others [@van2012robust; @backstrom2006group; @papadopoulos2012community]. Communities are usually defined as groups of nodes that are connected to each other more densely than to the rest of the network. Classical approaches to community detection depend on properties such as graph metrics, spectral properties and density of shortest paths [@leskovec2014mining]. Random walks and randomized label propagation [@walktrap; @gregory2010finding] have also been investigated.
Embedding the nodes in a low dimensional Euclidean space enables us to apply standard machine learning techniques. This space is sometimes called the [*feature space*]{} – implying that it represents abstract structural features of the network. Embeddings have been used for machine learning tasks such as labeling nodes, regression, link prediction, and graph visualization, see [@goyal2017graph] for a survey. Graph embedding processes usually aim to preserve certain predefined differences between nodes encoded in their embedding distances. For social network embedding, a natural priority is to preserve community membership and enable community detection.
Recently, sequence-based methods have been developed as a way to convert complex, non-linear network structures into formats more compatible with vector spaces. These methods sample sequences of nodes from the graph using a randomized mechanism (e.g. random walks), with the idea that nodes that are “close” in the graph connectivity will also frequently appear close in a sampling of random walks. The methods then proceed to use this random-walk-proximity information as a basis to embed nodes such that socially close nodes are placed nearby. In this category, *Deepwalk* [@deepwalk] and *Node2Vec* [@grover2016node2vec] are two popular methods.
While these methods preserve the proximity of nodes in the graph sense, they do not have an explicit preference for preserving social communities. Thus, in this paper, we develop a machine learning approach that considers clustering when embedding the network and includes a parameter to control the closeness of nodes in the same community. Figure \[fig:zachary\](a) shows the embedding obtained by the standard *Deepwalk* method, where communities are coherent, but not clearly separated in the embedding. The method described in this paper, called *GEMSEC*, is able to produce clusters that are tightly embedded and separated from each other (Fig. \[fig:zachary\](b)).
Our Contributions
-----------------
*GEMSEC* is an algorithm that considers the two problems of embedding and community detection simultaneously, and as a result, the two solutions of embedding and clustering can inform and improve each other. Through iterations, the embedding converges toward one where nodes are placed close to their neighbors in the network, while at the same time clusters in the embedding space are well separated. The algorithm is based on the paradigm of sequence-based node embedding procedures that create $d$ dimensional feature representations of nodes in an abstract feature space. Sequence-based node embeddings embed pairs of nodes close to each other if they occur frequently within a small window of each other in a random walk. This problem can be formulated as minimizing the negative log-likelihood of observed neighborhood samples (Sec. \[sec:model\]) and is called the skip-gram optimization [@mikolov_1]. We extend this objective function to include a clustering cost. The formal description is presented in Subsection \[sec:cluster\]. The resulting optimization problem is solved with a variant of mini-batch gradient descent [@adampaper].
The detailed algorithm is presented in Subsection \[sec:algorithm\]. By enforcing clustering on the embedding, *GEMSEC* reveals the natural community structure (e.g. Figure \[fig:zachary\]).Our approach improves over existing methods of simultaneous embedding and clustering [@cavallari2017learning; @wang2017community; @ye2018deep] and shows that community sensitivity can be directly incorporated into the skip-gram style optimization to obtain greater accuracy and efficiency.
In social networks, nodes in the same community tend to have similar groups of friends, which is expressed as high neighborhood overlap. This fact can be leveraged to produce clusters that are better aligned with the underlying communities. We achieve this effect using a regularization procedure – a smoothness regularization added to the basic optimization achieves more coherent community detection. The effect can be seen in Figure \[fig:regularization\], where a somewhat uncertain community affiliation suggested by the randomized sampling is sharpened by the smoothness regularization. This technique is described in Subsection \[sec:reglularization\].
In experimental evaluation we demonstrate that *GEMSEC* outperforms – in clustering quality – the state of the art neighborhood based [@deepwalk; @grover2016node2vec], multi-scale [@tang2015line; @perozzidontwalk] and community aware embedding methods [@cavallari2017learning; @wang2017community; @ye2018deep]. We present new social datasets from the streaming service Deezer and show that the clustering can improve music recommendations. The clustering performance of *GEMSEC* is found to be robust to hyperparameter changes, and the runtime complexity of our method is linear in the size of the graphs.
To summarize, the main contributions of our work are:
1. *GEMSEC*: a sequence sampling-based learning model which learns an embedding of the nodes at the same time as it learns a clustering of the nodes.
2. Clustering in *GEMSEC* can be aligned to network neighborhoods by a smoothness regularization added to the optimization. This enhances the algorithm’s sensitivity to natural communities.
3. Two new large social network datasets are introduced – from Facebook and Deezer data.
4. Experimental results show that the embedding process runs linearly in the input size. It generally performs well in quality of embedding and in particular outperforms existing methods on cluster quality measured by modularity and subsequent recommendation tasks.
We start with reviewing related work in the area and relation to our approach in the next section. A high-performance Tensorflow reference implementation of *GEMSEC* and the datasets that we collected can be accessed online[^1].
Related Work {#sec:related_work}
============
There is a long line of research in metric embedding – for example, embedding discrete metrics into trees [@fakcharoenphol2004tight] and into vector spaces [@matouvsek2002lectures]. Optimization-based representation of networks has been used for routing and navigation in domains such as sensor networks and robotics [@yu2011spherical; @huang2014bounded]. Representations in hyperbolic spaces have emerged as a technique to preserve richer network structures [@sarkar2011low; @de2018representation; @zeng2010resilient].
Recent advances in node embedding procedures have made it possible to learn vector features for large real-world graphs [@deepwalk; @tang2015line; @grover2016node2vec]. Features extracted with these *sequence-based node embedding* procedures can be used for predicting social network users’ missing age [@goyal2017graph], the category of scientific papers in citation networks [@perozzidontwalk] and the function of proteins in protein-protein interaction networks [@grover2016node2vec]. Besides supervised learning tasks on nodes the extracted features can be used for graph visualization [@goyal2017graph], link prediction [@grover2016node2vec] and community detection [@cavallari2017learning]. Sequence based embedding commonly considers variations in the sampling strategy that is used to obtain vertex sequences – truncated random walks being the simplest strategy [@deepwalk]. More involved methods include second-order random walks [@grover2016node2vec], skips in random walks [@perozzidontwalk] and diffusion graphs [@rozemberczki2018fast]. It is worth noting that these models implicitly approximate matrix factorizations for different matrices that are expensive to factorize explicitly [@qiu2018network]. Our work extends the literature of node embedding algorithms which are community aware. Earlier works in this category did not directly extend the skip-gram embedding framework. *M-NMF* [@wang2017community] applies computationally expensive non-negative matrix factorization with a modularity constraint term. The procedure *DANMF* [@ye2018deep] uses hierarchical non-negative matrix factorization to create community-aware node embeddings. *ComE* [@cavallari2017learning] is a more scalable approach, but it assumes that in the embedding space the communities fit a gaussian structure, and aims to model them by a mixture of Gaussians. In comparison to these methods, *GEMSEC* provides greater control over community sensitivity of the embedding process, it is independent of the specific neighborhood sampling methods and is computationally efficient.
Graph Embedding with Self Clustering {#sec:model}
====================================
For a graph $G=(V,E)$, a node embedding is a mapping $f:V\to \mathbb{R}^d$ where $d$ is the dimensionality of the embedding space. For each node $v \in V$ we create a $d$ dimensional representation. Alternatively, the embedding $f$ is a $|V| \times d$ real-valued matrix. In sequence-based embedding, sequences of neighboring nodes are sampled from the graph. Within a sequence, a node $v$ occurs in the [*context*]{} of a window $\omega$ within the sequence. Given a sample $S$ of sequences, we refer to the collection of windows containing $v$ as $N_{S}(v)$. Earlier works have proposed random walks, second-order random walks or branching processes to obtain $N_S(v)$. In our experiments, we used unweighted first and second-order random walks for node sampling [@deepwalk; @grover2016node2vec].
Our goal is to minimize the negative log-likelihood of observing neighborhoods of source nodes conditional on feature vectors that describe the position of nodes in the embedding space. Formally, the optimization objective is: $${\small
\min_{f}\quad \sum_{v \in V}- \log P(N_S(v)|f(v))\label{eq:negative_loglikelihood}}$$ for a suitable probability function $P(\cdot|\cdot)$. To define this $P$, we consider two standard properties (see [@grover2016node2vec]) expected of the embedding $f$ in relation to $N_{S}$. First, it should be possible to factorize $P(N_S(v)|f(v))$ in line with *conditional independence* with respect to $f(v)$. Formally: $$\begin{aligned}
{\small
P(N_S(v)|f(v)) = \prod\limits_{n_i \in N_S(v)}P(n_i \in N_S(v)\mid f(v),f(n_i)).\label{eq:condi}}\end{aligned}$$ Second, it should satisfy *symmetry in the feature space*, meaning that source and neighboring nodes have a symmetric effect on each other in the embedding space. A softmax function on the pairwise dot products of node representations with $f(v)$ to get $P(n_i \in N_S(v)\mid f(v),f(n_i))$ express such a property:
$$\begin{aligned}
{\small
P(n_i \in N_S(v)\mid f(v),f(n_i)) = \frac{\exp (f(n_i)\cdot f(v))}{\sum\limits_{u \in V}\exp(f(u)\cdot (f(v))}.\label{eq:dotprod}}\end{aligned}$$
Substituting and into the optimization function, we get:
$$\begin{aligned}
\min_{f}&\sum\limits_{v \in V} \left [\ln \left(\smashoperator{\sum\limits_{u\in V}} \exp(f(v)\cdot f(u))\right)-\smashoperator{\sum\limits_{n_i\in N_S(v)}}f(n_i)\cdot f(v) \right].\label{eq:classic_opti}\end{aligned}$$
The partition function in Equation enforces nodes to be embedded in a low volume space around the origin, while the second term forces nodes with similar sampled neighborhoods to be embedded close to each other.
Learning to Cluster {#sec:cluster}
-------------------
Next, we extend the optimization to pay attention to the clusters it forms. We include a clustering cost similar to $k$-means, measuring the distance from nodes to their cluster centers. This augmented optimization problem is described by minimizing a loss function over the embedding $f$ and position of cluster centers $\mu$, that is, ${\displaystyle}\min_{f,\mu}\mathcal{L}$, where:
$$\begin{aligned}
\mathcal{L} = &\quad \underbrace{\sum\limits_{v \in V} \left [\ln \left(\sum\limits_{u\in V} \exp(f(v)\cdot f(u))\right)-\sum\limits_{n_i\in N_S(v)}f(n_i)\cdot f(v) \right]}_{\text{Embedding cost}}\nonumber\\
&\quad +\underbrace{\gamma \cdot \sum_{v \in V} \min_{c\in C} \left \|f(v)-\mu_c \right \|_2}_{\text{Clustering cost}}.\label{eq:proper_opti}\end{aligned}$$
In Equation we have $C$ the set of cluster centers – the $c^{th}$ cluster mean is denoted by $\mu_c$. Each of these cluster centers is a $d$-dimensional vector in the embedding space. The idea is to minimize the distance from each node to its nearest cluster center. The weight coefficient of the clustering cost is given by the hyperparameter $\gamma$. Evaluating the partition function in the proposed objective function for all of the source nodes has a $\mathcal{O}(|V|^2)$ runtime complexity. Because of this, we approximate the partition function term with negative sampling which is a form of noise contrastive estimation [@mikolov_1; @gutmann2010noise].
$$\begin{aligned}
\frac{\partial \mathcal{L}}{\partial f(v^{\ast})}=&\underbrace{\frac{\sum\limits_{u\in V} \exp(f(v^{\ast})\cdot f(u))\cdot f(u)}{\sum\limits_{u\in V} \exp(f(v^{\ast})\cdot f(u))}}_{\text{Partition function gradient}}-
\underbrace{\sum\limits_{n_i\in N_S(v^\ast)}f(n_i)}_{\text{Neighbor direction}}\nonumber \\
&+\underbrace{\gamma\cdot \frac{f(v^\ast)-\mu_c}{\left \|f(v^\ast)-\mu_c \right \|_2}}_{\text{Closest cluster direction}}\label{eq:feature_gradient}\end{aligned}$$
The gradients of the loss function in Equation \[eq:proper\_opti\] are important in solving the minimization problem. As a result we can obtain the gradients for node representations and cluster centers. Examining in more detail, the gradient of the objective function $\mathcal{L}$ with respect to the representation of node $v^{\ast} \in V$ is described by Equation if $\mu_c$ is the closest cluster center to $f(v^{\ast})$.
The gradient of the partition function pulls the representation of $v^{\ast}$ towards the origin. The second term moves the representation of $v^{\ast}$ closer to the representations of its neighbors in the embedding space while the third term moves the node closer to the closest cluster center. If we set a high $\gamma$ value the third term dominates the gradient. This will cause the node to gravitate towards the closest cluster center which might not contain the neighbors of $v^{\ast}$. An example is shown in Figure \[nodecapture\]. If the set of nodes that belong to cluster center $c$ is $V_c$, then the gradient of the objective function with respect to $\mu_c$ is described by $$\frac{\partial \mathcal{L}}{\partial \mu_c}=-\gamma\cdot \sum\limits_{v\in V_c}\frac{f(v)-\mu_c}{\left \|f(v)-\mu_c \right \|_2}.\label{eq:center_gradient}$$
In Equation \[eq:center\_gradient\] we see that the gradient moves the cluster center by the sum of coordinates of nodes in the embedding space that belong to cluster $c$. Second, if a cluster ends up empty it will not be updated as elements of the gradient would be zero. Because of this, cluster centers and embedding weights are initialized with the same uniform distribution. A wrong initialization just like the one with an empty cluster in Subfigure \[wronginit\] can affect clustering performance considerably.
GEMSEC algorithm {#sec:algorithm}
----------------
We propose an efficient learning method to create *GEMSEC* embeddings which is described with pseudo-code by Algorithm \[GEMSEC\_algo\]. The main idea behind our procedure is the following. To avoid the clustering cost overpowering the graph information (as in Fig. \[nodecapture\]), we initialize the system with a low weight $\gamma_{0}\in [0,1]$ for clustering, and through iterations anneal it to $1$.
The embedding computation proceeds as follows. The weights in the model are initialized based on the number of vertices, embedding dimensions and clusters. After this, the algorithm makes $N$ sampling repetitions in order to generate vertex sequences from every source node. Before starting a sampling epoch, it shuffles the set of vertices. We set the clustering cost coefficient $\gamma$ (line 7) according to an exponential annealing rule described by Equation . The learning rate is set to $\alpha$ (line 8) with a linear annealing rule (Equation ).
$$\begin{aligned}
\gamma&=\gamma_0 \cdot \left(10^{\frac{-t\cdot \log_{10} \gamma_0}{w \cdot l \cdot |V|\cdot N}} \right)\label{eq:exp_anneal}\\
\alpha &=\alpha_0-(\alpha_0-\alpha_F)\cdot \frac{t}{w \cdot l \cdot |V|\cdot N}\label{eq:lin_anneal}\end{aligned}$$
The sampling process reads sequences of length $l$ (line 9) and extracts features using the context window size $\omega$ (line 10). The extracted features, gradient, current learning rate and clustering cost coefficient determine the update to model weights by the optimizer (line 11). In the implementation we utilized a variant of stochastic gradient descent – the *Adam* optimizer [@adampaper]. We approximate the first cost term with noise contrastive estimation to make the gradient descent tractable, drawing $k$ noise samples for each positive sample. If the node sampling is done by first-order random walks the runtime complexity of this procedure will be $\mathcal{O}((\omega\cdot k + |C|)\cdot l\cdot d\cdot |V|\cdot N)$ while *DeepWalk* with noise contrastive estimation has a $\mathcal{O}(\omega\cdot k\cdot l\cdot d\cdot |V|\cdot N)$ runtime complexity.
Smoothness Regularization for coherent community detection {#sec:reglularization}
----------------------------------------------------------
We have seen in Subsection \[sec:cluster\] that there is a tension between what the clustering objective considers to be clusters and what the real communities are in the underlying social network. We can incorporate additional knowledge of social network communities using a machine learning technique called regularization.
We observe that social networks have natural local properties such as homophily, strong ties between members of a community, etc. Thus, we can incorporate such social network-specific properties in the form of regularization to find more natural embeddings and clusters.
This regularization effect can be achieved by adding a term $\Lambda$ to the loss function: [$$\begin{aligned}
\Lambda & =\lambda \cdot\smashoperator{\sum_{(v,u) \in E_S}} w_{(v,u)}\cdot\left \|f(v)-f(u) \right \|_2\label{eq:smoother},
\end{aligned}$$]{} where the weight function $w$ determines the [*social network cost*]{} of the embedding with respect to properties of the edges traversed in the sampling. We use the neighborhood overlap of an edge – defined as the fraction of neighbors common to two nodes of the edge relative to the union of the two neighbor sets[^2]. In experiments on real data, neighborhood overlap is known to be a strong indicator of the strength of relation between members of a social network [@onnela2007structure]. Thus, by treating neighborhood overlap as the weight $w_{v,u}$ of edge $(v,u)$, we can get effective social network clustering, which is confirmed by experiments in the next section. The coeffeicient $\lambda$ lets us tune the contribution of the social network cost in the embedding process. In experiments, the regularized version of the algorithms is found to be more robust to changes in hyperparameters.
The effect of the regularization can be understood intuitively through an example. For this exposition, let us consider matrix representations of the social network describing closeness of nodes. In fact, other skip-gram style learning processes like [@deepwalk; @grover2016node2vec] are known to approximate the factorization of a similarity matrix $M$ such as [@qiu2018network]: $$M_{u,v} = \log\left(\frac{\text{vol}(G)}{\omega}\sum\limits_{r=1}^{\omega} \frac{\sum \limits_{P\in \mathcal{P}^r_{v,u}} \prod\limits_{a\in P\setminus \left\{v\right\}} \frac{1}{\deg(a)}}{\deg(v)}\right)-\log(k)$$ where $P^r_{v,u}$ is the set of paths going from $v$ to $u$ with length $r$. Elements of the target matrix $M$ grow with number of paths of length at most $\omega$ between the corresponding nodes. Thus $M$ is intended to represent level of connectivity between nodes in terms of a raw graph feature like number of paths.
The barbell graph in Figure \[fig:barbell\_graph\] is a typical example with an obvious community structure we can use to analyze the matter. The optimization procedure used by Deepwalk [@deepwalk] aims to converge to a target matrix $M_{u,v}$ shown in Figure \[fig:barbell\_target\]. Observe that this matrix has fuzzy edges around the communities of the graph, showing a degree of uncertainty. An actual approximation by running the Deepwalk is shown in Figure \[fig:deepwalk\_reconstruct\], which naturally incorporates further uncertainty due to sampling. A much more clear output with sharp communities can be obtained by applying a regularized optimization. This can be seen in Figure \[fig:smooth\_deepwalk\_reconstruct\].
Experimental Evaluation {#sec:experiments}
=======================
In this section we evaluate the cluster quality obtained by the *GEMSEC* variants, their scalability, robustness and predictive performance on a downstream supervised task.
Results show that *GEMSEC* outperforms or is at par with existing methods in all measures.
Datasets
--------
For the evaluation of *GEMSEC* real-world social network datasets are used which we collected from public APIs specifically for this work. Table \[fig:stats\] shows these social networks have a variety of size, density, and level of clustering. We used graphs from two sources:
- *Facebook page networks:* These graphs represent mutual like networks among verified Facebook pages – the types of sites included TV shows, politicians, athletes, and artists among others.
- *Deezer user-user friendship networks:* We collected friendship networks from the music streaming site Deezer and included $3$ European countries (Croatia, Hungary, and Romania). For each user, we curated the list of genres loved based on the songs liked by the user.
Standard parameter settings
---------------------------
A fixed standard parameter setting is used our experiments, and we indicate any deviations. Models using first order random walk sampling strategy are referenced as $\textit{GEMSEC}$ and $\textit{Smooth GEMSEC}$, second order random walk variants are named as $\textit{GEMSEC}_2$ and $\textit{Smooth GEMSEC}_2$. Random walks with length $80$ are used and $5$ truncated random walks per source node were used. Second-order random walk control hyperparameters [@grover2016node2vec] *return* and *in-out* were chosen from $\left \{2^{-2},2^{-1},1,2,4\right\}$. A window size of $5$ is used for features. Each embedding has $16$ dimensions and we extract $20$ cluster centers. A parameter sweep over hyperparameters was used to obtain the highest average modularity. Initial learning rate values are chosen from $\left \{10^{-2},5\cdot 10^{-3},10^{-3}\right\}$ and the final learning rate is chosen from $\left \{10^{-3},5\cdot 10^{-4},10^{-4}\right\}$. Noise contrastive estimation uses $10$ negative examples. The initial clustering cost coefficient is chosen from $\left \{10^{-1},10^{-2},10^{-3}\right\}$. The smoothness regularization term’s hyperparameter is $0.0625$ and Jaccard’s coefficient is the penalty weight.
Cluster Quality
---------------
Using Facebook page networks we evaluate the clustering performance. Cluster quality is evaluated by modularity – we assume that a node belongs to a single community. Our results are summarized in Table \[fig:clust\_performance\] based on 10 experimental repetitions and errors in parentheses correspond to two standard deviations. The baselines use the hyperparameters from the respective papers. We used 16-dimensional embeddings throughout. The embeddings obtained with non-community-aware methods were clustered after the embedding by $k$-means clustering to extract 20 cluster centers. Specifically, comparisons are made with:
1. *Overlap Factorization* [@ahmed2013distributed]: Factorizes the neighborhood overlap matrix to create features.
2. *DeepWalk* [@deepwalk]: Approximates the sum of the adjacency matrix powers with first order random walks and implicitly factorizes it.
3. *LINE* [@tang2015line]: Implicitly factorizes the sum of the first two powers for the normalized adjacency matrix and the resulting node representation vectors are concatenated together to form a multi-scale representation.
4. *Node2vec* [@grover2016node2vec]: Factorizes a neighbourhood matrix obtained with second order random walks. The *in-out* and *return* parameters of the second-order random walks were chosen from the $\left\{2^{-2},2^{-1},1,2,4\right\}$ set to maximize modularity.
5. *Walklets* [@perozzidontwalk]: Approximates with first order random walks each adjacency matrix power individually and implicitly factorizes the target matrix. These embeddings are concatenated to form a multi-scale representation of nodes.
6. *ComE* [@cavallari2017learning]: Uses a Gaussian mixture model to learn an embedding and clustering jointly using random walk features.
7. *M-NMF* [@wang2017community]: Factorizes a matrix which is a weighted sum of the first two proximity matrices with a modularity based regularization constraint.
8. *DANMF* [@ye2018deep]: Decomposes a weighted sum of the first two proximity matrices hierarchically to obtain cluster memberships with an autoencoder-like non-negative matrix factorization model.
*Smooth GEMSEC*, $\textit{GEMSEC}_2$ and $\textit{Smooth GEMSEC}_2$ consistently outperform the neighborhood conserving node embedding methods and the competing community aware methods. The relative advantage of $\textit{Smooth GEMSEC}_2$ over the benchmarks is highest on the Athletes dataset as the clustering’s modularity is 3.44% higher than the best performing baseline. It is the worst on the Media dataset with a disadvantage of 0.35% compared to the strongest baseline. Use of smoothness regularization has sometimes non-significant, but definitely positive effect on the clustering performance of *Deepwalk*, *GEMSEC* and $\textit{GEMSEC}_2$.
Sensitivity Analysis for hyperparameters
----------------------------------------
We tested the effect of hyperparameter changes to clustering performance. The Politicians Facebook graph is embedded with the standard parameter settings while the initial and final learning rates are set to be $10^{-2}$ and $5\cdot 10^{-3}$ respectively, the clustering cost coefficient is 0.1 and we perturb certain hyperparameters. The second-order random walks used *in-out* and *return* parameters of $4$. In Figure \[fig:sensi\] each data point represents the mean modularity calculated from $10$ experiments. Based on the experimental results we make two observations. First, *GEMSEC* model variants give high-quality clusters for a wide range of parameter settings. Second, introducing smoothness regularization makes *GEMSEC* models more robust to hyperparameter changes. This is particularly apparent across varying the number of clusters. The length of truncated random walks and the number of random walks per source node above a certain threshold has only a marginal effect on the community detection performance.
Music Genre Recommendation
--------------------------
Node embeddings are often used for extracting features of nodes for downstream predictive tasks. In order to investigate this, we use social networks of Deezer users collected from European countries. We predict the genres (out of $84$) of music liked by people. Following the embedding, we used logistic regression with $\ell_2$ regularization to predict each of the labels and $90\%$ of the nodes were randomly selected for training. We evaluated the performance of the remaining users. Numbers reported in Table \[fig:pred\_perfor\] are $F_1$ scores calculated from $10$ experimental repetitions. $\textit{GEMSEC}_2$ significantly outperforms the other methods on all three countries’ datasets. The performance advantage varies between $3.03\%$ and $4.95\%$. We also see that *Smooth GEMSEC*$_2$ has lower accuracy, but it is able to outperform *DeepWalk, LINE, Node2Vec, Walklets, ComE, M-NMF* and *DANMF* on all datasets.
Scalability and computational efficiency
----------------------------------------
To create graphs of various sizes, we used the Erdos-Renyi model and with an average degree of $20$. Figure \[fig:performance\] shows the log of mean runtime against the log of the number of nodes. Most importantly, we can conclude that doubling the size of the graph doubles the time needed for optimizing *GEMSEC*, thus the growth is linear. We also observe that embedding algorithms that incorporate clustering have a higher cost, and regularization also produces a higher cost, but similar growth.
Conclusions {#sec:conclusion}
===========
We described *GEMSEC* – a novel algorithm that learns a node embedding and a clustering of nodes jointly. It extends existing embedding modes. We showed that smoothness regularization is used to incorporate social network properties and produce natural embedding and clustering. We presented new social datasets, and experimentally, our methods outperform a number of strong community aware node embedding baselines.
Acknowledgements
================
Benedek Rozemberczki and Ryan Davies were supported by the Centre for Doctoral Training in Data Science, funded by EPSRC (grant EP/L016427/1).
[^1]: https://github.com/benedekrozemberczki/GEMSEC
[^2]: Neighbor sets $N(a)$ and $N(b)$ of nodes $a$ and $b$, the neighborhood overlap of $(a,b)$ is defined as the Jaccard similarity $\frac{N(a)\cap N(b)}{N(a)\cup N(b)}$.
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abstract: 'We present a detailed near-infrared chemical abundance analysis of 10 red giant members of the Galactic open cluster NGC 752. High-resolution (R$\simeq$45000) near-infrared spectral data were gathered with the Immersion Grating Infrared Spectrograph (IGRINS), providing simultaneous coverage of the complete $H$ and $K$ bands. We derived the abundances of H-burning (C, N, O), $\alpha$ (Mg, Si, S, Ca), light odd-Z (Na, Al, P, K), Fe-group (Sc, Ti, Cr, Fe, Co, Ni) and neutron-capture (Ce, Nd, Yb) elements. We report the abundances of S, P, K, Ce, and Yb in NGC 752 for the first time. Our analysis yields solar metallicity and solar abundance ratios for almost all of the elements heavier than the CNO group in NGC 752. O and N abundances were measured from a number of OH and CN features in the $H$ band, and C abundances were determined mainly from CO molecular lines in the $K$ band. High excitation lines present in both near-infrared and optical spectra were also included in the C abundance determinations. Carbon isotopic ratios were derived from the R-branch band heads of first overtone (2$-$0) and (3$-$1) $^{12}$CO and (2$-$0) $^{13}$CO lines near 23440 Å and (3$-$1) $^{13}$CO lines at about 23730 Å. The CNO abundances and ratios are all consistent with our giants having completed “first dredge-up” envelope mixing of CN-cyle products. We independently assessed NGC 752 stellar membership from Gaia astrometry, leading to a new color-magnitude diagram for this cluster. Applications of Victoria isochrones and MESA models to these data yield an updated NGC 752 cluster age (1.52 Gyr) and evolutionary stage indications for the program stars. The photometric evidence and spectroscopic light element abundances all suggest that the most, perhaps all of the program stars are members of the helium-burning red clump in this cluster.'
author:
- |
G. Böcek Topcu$^{1}$[^1], M. Afşar$^{1,2}$, C. Sneden$^{2}$, C. A. Pilachowski$^{3}$, P. A. Denissenkov$^{4}$, D. A. VandenBerg$^{4}$, D. Wright$^{2}$, G. N. Mace$^{2}$, D. T. Jaffe$^{2}$, E. Strickland$^{2}$, H. Kim$^{5}$, and K. R. Sokal$^{2}$\
$^{1}$Department of Astronomy and Space Sciences, Ege University, 35100 Bornova, İzmir, Turkey\
$^{2}$Department of Astronomy and McDonald Observatory, The University of Texas, Austin, TX 78712\
$^{3}$Indiana University, Department of Astronomy SW319, 727 E 3rd Street, Bloomington, IN 47405 USA\
$^{4}$Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8W 2Y2, Canada\
$^{5}$Gemini Observatory, Casilla 603, La Serena, Chile
bibliography:
- 'totbib\_752.bib'
date: Accepted 2019 October 18
title: '*Chemical Abundances Of Open Clusters From High-Resolution Infrared Spectra. II. NGC 752*'
---
\[firstpage\]
stars: abundances – stars: atmospheres. Galaxy: open clusters and associations: individual: NGC 752
Introduction {#intro}
============
Open star clusters provide important snapshots of the chemistry of the Galactic disk with time because they can be photometrically tagged with ages, which are difficult to assess for individual field stars. Most open clusters (OCs) are young, $t$ $\leq$ 2 Gyr. Very few of them are truly old, $t$ $>$ 7 Gyr; only Be 17 and NGC 6791 appear to have ages approaching 10 Gyr (, @salaris04, @brogaard12). Fortunately there are many intermediate-age clusters close enough to the Sun that chemical compositions of their brighter members can be studied at high spectroscopic resolution. The recent Gaia DR2 catalog now provides opportunities for more accurate membership and evolutionary state data for OC red giant members. Several groups are conducting extensive OC abundance studies using echelle spectrographs in the optical spectral region.
Optical spectroscopy of OCs has a fundamental observational limit caused by Galactic disk dust extinction. While more than 1000 OCs have been cataloged, most are too obscured to yield detailed information at optical wavelengths. This problem is especially acute for clusters at small Galactocentric radii. High-resolution spectroscopy at more transparent infrared wavelengths ($IR$, $\lambda$ $\geq$ 1 $\mu$m) is essential for further progress in OC chemical composition studies.
We have begun a program to use $H$ and $K$ band high-resolution spectroscopy to determine reliable metallicities and elemental abundance ratios for OCs spanning a large range of Galactocentric distances. Special emphasis is put on determining accurate abundances for the CNO group and other light elements, which have many transitions in these $IR$ bands. As steps toward this goal we first are performing combined optical/infrared spectroscopic analyses of three relatively nearby and well-studied intermediate-age OCs that suffer only small amounts of interstellar dust extinction: NGC 6940, NGC 752, and M67. Our $IR$ spectra are obtained with the Immersion Grating Infrared Spectrograph (IGRINS), which offers complete $H$ and $K$ spectral coverage (1.45$-$2.5 $\mu$m), with a band gap of only 0.01 $\mu$m lost instrumentally between the two bands (the gap due to telluric absorption is $\sim$0.2 $\mu$m). IGRINS delivers high spectral resolution similar to those of most optical spectrographs used for abundance analysis.
For NGC 6940, [@bocek19], hereafter Paper 1, reported analyses of IGRINS spectra of 12 red giant members, the same stars with atmospheric parameters and detailed abundance sets derived from high-resolution optical spectra by [@bocek16] (hereafter BT16). Among their principal results were: (a) good agreement in all cases of optical and $IR$ abundances; (b) determination for the first time in NGC 6940 the abundances of S, P, K, Ce, and Yb, and much strengthened abundances of Mg, Al, Si, and Ca; (c) derivation of much more reliable abundances of the CNO group; (d) discovery of one star with evidence of high-temperature proton fusion products; and (e) improved assessment of the NGC 6940 color-magnitude diagram, with clear assignment of most of the program stars to the He-burning red clump.
In this paper we report results of a similar optical/IR study for 10 red giant (RG) members of NGC 752. This relatively nearby OC has been the subject of several abundance studies (, @pilachowski88 [@carrera11; @reddy12]). Recently [@lum19] have presented an extensive large-sample analysis of 6 giant and 23 main-sequence stars. Our optical spectroscopic investigation was published in [@bocek15] (hereafter BT15). In this study we focus especially on the CNO abundances and ratios, interpreting the results within a more complete wavelength window. This larger spectral coverage leads us to a better analysis of the evolutionary status of the RG members. In addition to the red giant abundances derived from IGRINS spectra, we have revisited the questions of the distance and age of NGC 752 with *Gaia* DR2 data, and have re-considered the evolutionary states of these stars via new stellar isochrone computations. The Gaia kinematic and photometric data leading to NGC 752 membership, distance, and age are discussed in §\[gaia\]. In §\[obs\] we summarize the IGRINS observations and reductions. The methods used to derive the chemical abundances and the temperatures of the target stars are described in §\[modopt\] and §\[ldrcomp\], respectively, while the results of the abundance determinations are given in section §\[irabs\]. The fitting of isochrones to the CMD of NGC 752, resulting in our best estimate of the cluster age, is discussed in §\[age\]. We compare stellar model predictions to the observed cluster RC stars in §\[isoch\], and summarize the main results of this investigation in §\[cocl\].
Membership Assignment Using *Gaia* DR2 {#gaia}
======================================
[@lccccccc@]{} Star & Gaia DR2& $\pi_{\rm (\it Gaia)}$ & $\mu_\alpha$ $_{\rm (\it Gaia)}$ & $\mu_\delta$ $_{\rm
(\it Gaia)}$ & RV$^{\rm a}$ & RV$_{\rm (Paper~1)}$ & RV$_{\rm (\it Gaia)}$\
& identifications & (mas yr$^{-1}$) & (mas yr$^{-1}$) & (mas yr$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$)\
MMU 1 & 342554191959774720 & 2.081 $\pm$ 0.049 & 9.780 $\pm$ 0.082 & $-12.003$ $\pm$ 0.090 & 5.34 $\pm$ 0.24 & 4.73 $\pm$ 0.20 & 5.29 $\pm$ 0.35\
MMU 3 & 342554187663431424 &2.101 $\pm$ 0.047 & 9.670 $\pm$ 0.078 & $-11.821$ $\pm$ 0.082 & 4.69 $\pm$ 0.29 & 4.11 $\pm$ 0.20 & 5.38 $\pm$ 0.20\
MMU 11& 343118619382072832 & 2.251 $\pm$ 0.066 & 9.811 $\pm$ 0.094 & $-12.244$ $\pm$ 0.099 & 5.08 $\pm$ 0.26 & 4.45 $\pm$ 0.19 & 5.63 $\pm$ 0.11\
MMU 24 & 342929297223676160 & 2.158 $\pm$ 0.049 & 9.602 $\pm$ 0.102 & $-11.940$ $\pm$ 0.094 & 4.79 $\pm$ 0.18 & 4.86 $\pm$ 0.19 & 5.70 $\pm$ 0.23\
MMU 27 & 342536702852966784 & 2.174 $\pm$ 0.049 & 9.620 $\pm$ 0.084 & $-11.701$ $\pm$ 0.097 & 4.06 $\pm$ 0.18 & 4.39 $\pm$ 0.19 & 4.93 $\pm$ 0.18\
MMU 77 & 342532923281905408 & 2.205 $\pm$ 0.050 & 9.827 $\pm$ 0.077 & $-11.905$ $\pm$ 0.080 & 4.89 $\pm$ 0.24 & 4.58 $\pm$ 0.20 & 4.83 $\pm$ 1.23\
MMU 137 & 342937195667536512 & 2.149 $\pm$ 0.042 & 9.535 $\pm$ 0.085 & $-11.883$ $\pm$ 0.082 & 5.17 $\pm$ 0.24 & 5.59 $\pm$ 0.20 & 5.60 $\pm$ 0.18\
MMU 295 & 342893803614055168 & 2.201 $\pm$ 0.046 & 9.404 $\pm$ 0.104 & $-11.593$ $\pm$ 0.087 & 5.29 $\pm$ 0.27 & 6.32 $\pm$ 0.23 &\
MMU 311 & 342890127122193280 & 2.229 $\pm$ 0.056 & 9.701 $\pm$ 0.112 & $-11.331$ $\pm$ 0.106 & 5.69 $\pm$ 0.24 & 5.19 $\pm$ 0.19 & 5.67 $\pm$ 0.12\
MMU 1367 & 342899537393760512 & 2.251 $\pm$ 0.052 & 9.688 $\pm$ 0.094 & $-11.794$ $\pm$ 0.106 & 4.69 $\pm$ 0.24 & 3.98 $\pm$ 0.19 & 5.09 $\pm$ 0.23\
RV (cluster mean) & & & & & 4.97 $\pm$ 0.24 & 4.82 $\pm$ 0.20 & 5.35 $\pm$ 0.31\
\
![Vector point diagram of our data set, where each dot shows the proper motion components of a star in right ascension ($\mu_{\alpha}$) and declination ($\mu_{\delta}$). The 95% confidence ellipse in the lower right corner encircles most of the cluster stars and represents the intrinsic cluster center and dispersion. The colors, defined in the side bar, signify NGC 752 membership probabilities for each star. The total data set consisted of more stars more widely separated in proper motion from the field and cluster groups; they have been omitted for clarity in this plot.[]{data-label="membership_vpd"}](membership_1.pdf){width="1\linewidth"}
[@lccccccccccc@]{} Star & RA$^{\rm a}$ & DEC$^{\rm a}$ & V$^{\rm b}$ & H$^{\rm c}$ & K$^{\rm c}$ & G$^{\rm a}$ & (G$_{\rm BP}-G_{\rm RP})^{\rm d}$ & $(B - V)_{0}$ & $(V - K)_{0}$ & Date & S/N\
& ($2000$) & ($2000$) & & & & & & & &(UT) & (s)\
MMU 1 & 01 55 12.62 & 37 50 14.55 & 9.50 & 7.37 & 7.23 & 9.23 & 1.13 & 0.92 & 2.17 & 02 12 2015 & 108\
MMU 3 & 01 55 15.29 & 37 50 31.30 & 9.57 & 7.32 & 7.20 & 9.28 & 1.16 & 0.96 & 2.27 & 02 12 2015 & 106\
MMU 11 & 01 55 27.67 & 37 59 55.24 & 9.29 & 7.16 & 7.04 & 9.03 & 1.12 & 0.93 & 2.15 & 02 12 2015 & 120\
MMU 24 & 01 55 39.37 & 37 52 52.51 & 8.92 & 6.67 & 6.55 & 8.65 & 1.16 & 0.98 & 2.28 & 03 12 2015 & 109\
MMU 27 & 01 55 42.39 & 37 37 54.57 & 9.16 & 6.90 & 6.80 & 8.88 & 1.17 & 0.98 & 2.27 & 03 12 2015 & 117\
MMU 77 & 01 56 21.64 & 37 36 08.43 & 9.38 & 7.05 & 6.92 & 9.09 & 1.21 & 0.99 & 2.36 & 03 12 2015 & 117\
MMU 137 & 01 57 03.11 & 38 08 02.65 & 8.93 & 6.66 & 6.54 & 8.64 & 1.18 & 0.99 & 2.29 & 04 12 2015 & 119\
MMU 295 & 01 58 29.82 & 37 51 37.57 & 9.30 & 7.17 & 7.04 & 9.05 & 1.13 & 0.93 & 2.17 & 04 12 2015 & 141\
MMU 311 & 01 58 52.90 & 37 48 57.23 & 9.06 & 6.80 & 6.64 & 8.77 & 1.20 & 1.00 & 2.33 & 04 12 2015 & 113\
MMU 1367& 01 59 14.80 & 38 00 55.29 & 9.01 & 6.77 & 6.65 & 8.72 & 1.18 & 0.98 & 2.26 & 26 11 2018 & 113\
\
\
\
\
For determination of our NGC 752 member set, we created a Gaussian mixture model. This model was built using proper motion data from the *Gaia* [@GAIA16] Data Release 2 [@GAIA18b]. All stars that had *Gaia* DR2 proper motions and resided within 75$^\prime$ of the approximate NGC 752 cluster center, $\alpha_{2000} = 1^h57^m41.0^s$ and $\delta_{2000} = +37^\circ47^\prime6^{\prime\prime}$, were considered for membership. A vector point diagram for these stars is shown in Figure \[membership\_vpd\]. Each dot represents the proper motion components of a single candidate star.
Fitting mixture models is an applied statistical method that allows for the Bayesian determination of membership probabilities for individual stars. Open cluster applications of membership models like ours date at least back to [@sanders71], whose model is fundamentally similar to ours: the sum of two normal probability densities $-$ bivariate in the case of proper motion data $-$ is fit to observed right ascension and declination proper motion components. Our probability density function is of the form $$\begin{aligned}
\label{eq:model_form}
\Phi(\mu_{x_i}, \mu_{y_i}, \epsilon_{x_i}, \epsilon_{y_i}) = \phi_c + \phi_f \textnormal{,}\end{aligned}$$
where $\mu_{x_i}$ and $\mu_{y_i}$ are the proper motion components for the $i^{th}$ star in our data set, and $\epsilon_{x_i}$ and $\epsilon_{y_i}$ are their respective errors.
$\phi_c$ and $\phi_f$ are Gaussians and represent the cluster and field star distributions, respectively. Both Gaussians are symmetrical and elliptical. It is common to make the assumption of a circular cluster distribution, but we have left it as elliptical for increased accuracy. In addition, we have adopted a method derived by [@zhao90] that takes into account not only the intrinsic dispersions of $\phi_c$ and $\phi_f$ but also the observed *Gaia* DR2 errors for each individual star. In total, there are 11 parameters needed to characterize the distributions $\phi_c$ and $\phi_f$ in our model. After we solved for these, cluster membership probabilities were calculated. Figure \[membership\_vpd\] presents the final probabilities for stars within the proper motion ranges plotted. See the appendix for more detail on the exact forms of $\phi_c$ and $\phi_f$ and the techniques used to solve for the parameters.
In a mixture model such as ours, it is inevitable that some field stars will pass the cluster membership probability cutoff simply due to random chance. To mitigate this effect, we imposed parallax bounds on the remaining stars. By examination of the parallax density of the stars remaining after the proper motion cutoff, we found that the parallax distribution for NGC 752 peaked at $\simeq$2.235 mas with a base width of $\simeq$0.8 mas. We therefore set parallax bounds of 1.735 mas to 2.735 mas. All stars that passed a 50% proper motion model cutoff and fell within these parallax bounds were considered to be physical cluster members.
Our membership study was performed independently of any other NGC 752 membership analysis. Our computations yield cluster parameters of $\mu_\alpha$ = 9.827$\pm$0.017 mas/yr, $\mu_\delta$ = $-$11.782$\pm$0.019 mas/yr, and parallax $p$= 2.229$\pm$0.009 mas. This corresponds to a distance of 448 pc and a true distance modulus $(m-M)_0 = 8.26$, which has been adopted in the fitting of isochrones to the observed CMD in §\[age\]. We caution the reader that our parallax uncertainty for NGC 752 is purely statistical; a more realistic estimate would take into account possible systematic *Gaia* uncertainties, which [@arenou18] suggest can be as large as 0.03 mas. Recently [@cantat18] have performed a membership study of NGC 752 using very different methods. They suggest cluster parameters $\mu_\alpha$ = 9.810$\pm$0.019 mas/yr, $\mu_\delta$ = $-$11.713$\pm$0.019 mas/yr, and $p$ = 2.239$\pm$0.005 mas. Although our study focused on proper motions and we do not claim to have determined precise parallax estimates, our results are essentially in agreement with their work.
In Table \[tab-motions\] we have listed the program stars with their *Gaia* DR2 identifications, parallaxes and proper motions. Table \[tab-motions\] also contains radial velocities (RVs) from *Gaia* [@GAIA18b], optical (BT15) and $IR$ spectra. The $IR$ RVs were measured applying a similar method described in Paper 1 using at least 10 spectral orders that are less affected by the atmospheric telluric lines. The mean cluster RVs from these measurements agree well within the mutual uncertainties: $\langle RV\rangle_{\rm opt}$ $=$ $4.82 \pm 0.20$ kms$^{-1}$ $(\sigma=0.71)$, $\langle RV\rangle_{\rm IR}$ $=$ $4.97 \pm 0.24$ kms$^{-1}$ $(\sigma=0.45)$, and $\langle RV\rangle_{\rm Gaia}$ $=$ $5.35 \pm 0.31$ kms$^{-1}$ $(\sigma=0.33)$ (from 9 RGs).
Observations and Data Reduction {#obs}
===============================
---------- --------------------- --------------------------------------- ---------------------------------- ------ --------------- -------------- ---------- ------- -------------- ---------- ------- -------- ---------- --------
Star [[T\_[eff]{}]{}]{} [[T\_[eff]{}]{}]{}$_{(\rm \it Gaia)}$ [[T\_[eff]{}]{}]{}$_{\rm (LDR)}$ $\xi_{t}$ \[/H\] $\sigma$ \# \[/H\] $\sigma$ \# \[/H\] $\sigma$ \#
(K) (K) (K) (km s$^{-1}$) (opt) (opt) (opt) (opt) (opt) (opt) ($IR$) ($IR$) ($IR$)
MMU 1 5005 4929 5075 2.95 1.07 $ 0.04 $ 0.07 58 $ -0.02 $ 0.03 12 0.00 0.06 17
MMU 3 4886 4953 5010 2.76 1.10 $ -0.05 $ 0.07 62 $ -0.07 $ 0.06 10 -0.03 0.05 20
MMU 11 4988 4956 5045 2.80 1.14 $ 0.03 $ 0.07 59 $ -0.01 $ 0.04 11 -0.01 0.06 21
MMU 24 4839 4914 4986 2.42 1.23 $ -0.05 $ 0.07 57 $ -0.11 $ 0.04 12 -0.05 0.05 20
MMU 27 4966 4878 4948 2.73 1.16 $ 0.08 $ 0.07 57 $ -0.06 $ 0.03 11 0.06 0.05 20
MMU 77 4874 4850 4944 2.80 1.15 $ 0.04 $ 0.08 62 $ -0.06 $ 0.06 11 0.03 0.04 19
MMU 137 4832 4850 4970 2.51 1.29 $ -0.08 $ 0.06 58 $ -0.16 $ 0.05 9 -0.09 0.04 19
MMU 295 5039 5050 5053 2.88 1.10 $ 0.07 $ 0.06 58 $ -0.01 $ 0.04 10 0.03 0.06 20
MMU 311 4874 4846 4959 2.68 1.24 $ 0.07 $ 0.09 62 $ 0.01 $ 0.06 10 0.05 0.06 21
MMU 1367 4831 4831 4985 2.42 1.22 $ -0.02 $ 0.07 59 $ -0.08 $ 0.03 12 -0.02 0.08 19
---------- --------------------- --------------------------------------- ---------------------------------- ------ --------------- -------------- ---------- ------- -------------- ---------- ------- -------- ---------- --------
We gathered IGRINS $H$- and $K$-band high resolution spectra for the 10 NGC 752 RG members studied in the optical spectral region by BT15. The stars chosen for that paper were selected from the radial velocity membership catalog of [@mermilliod08], before the release of Gaia DR2 astrometric data. The membership analysis presented here confirms that our targets belong to NGC 752. Additionally, several stars not included here appear to be RG members (this study and B. Twarog, private communication). Future spectroscopic study of these stars would be welcome. The log of the IGRINS observations is given in Table \[tab-basic\] along with the basic parameters of program stars. These stars are all red giants with similar parameters, as indicated by spectroscopic analyses ([[T\_[eff]{}]{}]{} $\sim$ 4900 K, $\sim$ 2.7; BT15) and by photometric data ($V$ $\simeq$ 9.2, $M_V$ $\simeq$ 1.0, $(B-V)_0$ $\simeq$ 0.97). Three other stars with similar photometric characteristics satisfy our NGC 752 membership criteria: BD+37 404 (MMU 2054), BD+36 328 (MMU 1533), and BD+37 422 (MMU 110, HD 11811). The derived distance for BD+37 422 is $\sim$30 pc from the cluster mean, but this star is a known spectroscopic binary, so its photometric and astrometric data should be treated with caution. None of these three stars appears to have been subjected to a comprehensive atmospheric and abundance analysis; future spectroscopic studies of them would be of some interest.
Characteristics of the IGRINS instrument have been presented by [@yuk10] and [@park14]. This spectrograph employs a silicon immersion grating (@gully12) to achieve resolving power $R$ $\equiv$ $\lambda/\Delta\lambda$ $\simeq$ 45000 for the entire $H$ and $K$ bands (1.45 – 2.5 $\micron$) in a single exposure. Almost all of the observations were made with IGRINS installed at the Cassegrain focus of the 2.7m Harlan J. Smith Telescope at McDonald Observatory in 2015 December (@mace16). One object, MMU 1367, was observed with IGRINS on Lowell Observatory’s 4.3m Discovery Channel Telescope (@mace18). Typical exposure times were 300s and used ABBA nod sequences along the spectrograph slit length. We also observed telluric standards with spectral types of B9IV to A0V. They were observed right after each science exposures at very close airmasses to the ones at which program stars were observed. The spectra used in this analysis were reduced using the IGRINS pipeline (@lee17). The pipeline performs flat-field correction, A-B frame subtractions to remove skyline emission, wavelength correction using OH emission and telluric absorption, and optimal spectral extraction. Due to their high rotational velocities ($\sim$150 ), telluric stars come with extremely broadened absorption features that can be easily distinguished from the atmospheric telluric lines. After removing the extremely broadened features from the spectra of telluric standards, we used the *telluric* task of IRAF[^2] to remove the contamination of atmospheric absorption lines from the spectra of our program stars.
Model Atmospheres and Abundances from the Optical Region {#modopt}
========================================================
Model atmospheric parameters (Table \[tab-model\]) of the 10 RG members of NGC 752 were previously presented in BT15, along with the abundances for 26 species of 23 elements present in the optical spectral region (see Table 10 in BT15). In this study, we newly report abundances of species , , and from those spectra. We present optical sulfur abundances for our targets for the first time in NGC752 using the triplet centered at 6757.17 Å. [@takeda16] reported non-local thermodynamic equilibrium (non-LTE) corrections for [S [i]{}]{}, estimated them to be $\lesssim 0.1$ dex for G-K giants for this blended S feature. We have also repeated the analyses for optical lines, adopting new values from [@lawler19]. Detailed description of the elemental abundance analysis methods in the optical region were provided in BT15, in which we also derived the solar photospheric abundances following the same procedure applied for the program stars, to obtain the differential values of stellar abundances relative to the Sun. Here we used the same method as in Paper 1, adopting the [@asplund:09] solar photospheric abundances for both regions in order to achieve consistency between optical and $IR$ data sets. These slightly revised relative optical abundances are listed in the upper part of Table \[tab-abunds\]. Mean abundances of the species and their standard deviations are given in columns 12 and 13 of this table. For the \[X/Fe\] calculations, we took star-by-star species differences using both and abundances as appropriate. We will discuss the differences between the NGC752 and NGC6940 abundance sets in §\[comp6940\].
![Left panel: comparison of optical and $H$-band LDR [[T\_[eff]{}]{}]{} values with those derived spectroscopically in Paper I. Full symbols represents NGC 752 RGs, open symbols are NGC 6940 RGs. Circle’s (red) are IR LDR and square symbols (blue) are optical LDR [[T\_[eff]{}]{}]{} values. Right panel: comparison of *Gaia* temperatures along with the spectroscopic [[T\_[eff]{}]{}]{}’s of the NGC 752 (full red circles) and NGC 6940 (open grey symbols). The dashed line represents equality of the temperatures for both panels.[]{data-label="teff"}](teff_2){width="\columnwidth"}
Temperature determination using IGRINS Data {#ldrcomp}
===========================================
Accurate effective temperatures, gravities, and microturbulent velocities are required for abundance analyses. In our OC studies we have adopted traditional line-by-line equivalent width ($EW$) analyses to derive atmospheric parameters [[T\_[eff]{}]{}]{}, , , and \[M/H\]. For NGC752 these parameters derived in BT15 are listed in Table \[tab-model\] and we have used them for all of the abundances reported in this paper. However, for heavily dust-obscured clusters optical parameter determinations will not be possible, and $IR$-based methods will be needed. Here we explore $IR$ [[T\_[eff]{}]{}]{} estimates.
Line-depth ratios (LDR) have proven to be good temperature indicators in several studies (e.g. @gray91 [@kovt06; @biazzo07a; @biazzo07b]). The LDR method is based on depth ratios of high-excitation atomic lines (relatively sensitive to [[T\_[eff]{}]{}]{}) to low-excitation lines (much less sensitive to [[T\_[eff]{}]{}]{}). This spectroscopic method has some attractive features: LDR temperatures are not affected by interstellar reddening and extinction, and they also only weakly depend on other atmospheric parameters for solar metallicity RGs. In our previous optical studies (BT15, BT16), we used the line pairs and equations of [@biazzo07a; @biazzo07b]. Recently, [@fukue15] have applied the LDR method to the $IR$ spectra, and have found nine pairs of absorption lines in the $H$-band (1.4 $-$ 1.8 $\mu$m) to be good [[T\_[eff]{}]{}]{} indicators. The application of LDR method to the $IR$ spectra brings new opportunities, such as access to the most dust obscured stars and the determination of the [[T\_[eff]{}]{}]{} without any information from the optical region.
We applied the [@fukue15] relationships to NGC6940 IGRINS data in Paper 1, and now we have calculated the LDR effective temperatures also for the NGC 752 RGs. [[T\_[eff]{}]{}]{} results from this LDR method are listed in Table \[tab-model\] along with other [[T\_[eff]{}]{}]{} values obtained from the optical region. In Figure \[teff\] (left panel), we compare the temperatures derived from both optical and $IR$ LDRs with the spectroscopic temperatures. This comparison indicates that $T_{\rm eff,spec}$ values derived from traditional line-by-line Fe and Ti $EW$ analyses and $T_{\rm eff,LDR}$ agree well for [[T\_[eff]{}]{}]{} $\geq$ 4900 K. For cooler RG stars the LDR temperatures become systematically larger. The $T_{\rm eff,LDR}$ of MMU1367, the coolest ($T_{\rm eff,spec}$ = 4831 K) member among others, is 154 K away from its spectroscopic temperature. This star and MMU311 also deviate similarly in the optical (BT15, Table 6). The $IR$-based LDR temperatures of NGC6940 shown in the left panel of Figure \[teff\] suggest a similar effect in that cluster also.
The LDR calibration issue is not of importance in our work, as most program stars are warmer than 4900 K, but it should be revisited in the future with larger sets of spectroscopic data in the $IR$. Considering the NGC752 sample as a whole, on average LDR and spectroscopic temperature are in reasonable accord: $\langle T_{\rm eff,LDR} - T_{\rm eff,spec}\rangle_{\rm IR}$ $=$ $84 \pm 18$ K, and $\langle T_{\rm eff,LDR} - T_{\rm eff,spec}\rangle_{\rm opt}$ $=$ $81 \pm 18$ K. Overall the $IR$ LDRs provide reliable temperatures in the absence of information from the optical region for giant stars with solar metallicities for the temperature range considered here.
The right panel of Figure \[teff\] shows comparisons of *Gaia* temperatures (Table \[tab-model\]) vs. $T_{\rm eff,spec}$ for the RGs of NGC752 and NGC6940. By inspection, *Gaia* and spectroscopic agree well for NGC752, and for the whole sample $\langle T_{\rm eff, Gaia} - T_{\rm eff, spec.}\rangle_{\rm NGC~752}$ $=$ $-8 \pm 17$ K. For NGC6940 the lack of agreement between spectroscopic and *Gaia* temperatures is clear in Figure \[teff\]: $\langle T_{\rm eff, Gaia} - T_{\rm eff, spec.}\rangle_{\rm NGC~6940}$ $=$ $-264 \pm 26$ K. *Gaia* temperatures are photometrically based and thus depend on interstellar extinction corrections. For NGC6940 $E(B-V)$ = 0.21, while for NGC752 the reddening is very small, $E(B-V)$ = 0.035 . This probably is related to the poor [[T\_[eff]{}]{}]{} correlation for NGC6940, but resolution of the question is beyond the scope of this paper.
[@lrrrrrrrrrrrcl@]{} Species &\
$\rm{[X/Fe]}$& 1 & 3 & 11 & 24 & 27 & 77 & 137 & 295 & 311 & 1367 & mean & $\sigma$ & \#$_{max}$\
\
&$ 0.20 $&$ 0.16 $&$ 0.21 $&$ 0.27 $&$ 0.16 $&$ 0.15 $&$ 0.30 $&$ 0.17 $&$ 0.20 $&$ 0.23 $&$ 0.20 $& 0.05 & 4\
&$ -0.06 $&$ 0.03 $&$ 0.00 $&$ 0.08 $&$ -0.02 $&$ -0.02 $&$ 0.03 $&$ -0.09 $&$ -0.07 $&$ 0.00 $&$ -0.01 $& 0.05 & 2\
&$ -0.08 $&$ -0.02 $&$ -0.08 $&$ 0.00 $&$ -0.08 $&$ -0.04 $&$ 0.02 $&$ -0.09 $&$ -0.07 $&$ -0.03 $&$ -0.04 $& 0.04 & 2\
&$ 0.18 $&$ 0.22 $&$ 0.20 $&$ 0.27 $&$ 0.17 $&$ 0.22 $&$ 0.29 $&$ 0.19 $&$ 0.22 $&$ 0.27 $&$ 0.22 $& 0.04 & 15\
$^{*}$ &$ 0.02 $&$ 0.13 $&$ 0.04 $&$ 0.04 $&$ -0.01 $&$ 0.07 $&$ 0.17 $&$ -0.04 $&$ 0.04 $&$ 0.06 $&$ 0.05 $& 0.06 & 2\
$^{*}$ &$ 0.47 $&$ 0.61 $&$ 0.49 $&$ 0.54 $&$ 0.51 $&$ 0.44 $&$ 0.47 $&$ 0.51 $&$ 0.47 $&$ 0.52 $&$ 0.50 $& 0.05 & 1\
&$ 0.13 $&$ 0.17 $&$ 0.12 $&$ 0.16 $&$ 0.09 $&$ 0.13 $&$ 0.13 $&$ 0.11 $&$ 0.12 $&$ 0.15 $&$ 0.13 $& 0.02 & 10\
$^{*}$ &$ 0.08 $&$ 0.02 $&$ 0.06 $&$ 0.06 $&$ 0.14 $&$ 0.12 $&$ 0.09 $&$ 0.08 $&$ 0.09 $&$ 0.03 $&$ 0.08 $& 0.01 & 6\
&$ -0.04 $&$ -0.05 $&$ -0.08 $&$ -0.09 $&$ -0.01 $&$ -0.01 $&$ -0.07 $&$ -0.02 $&$ -0.06 $&$ -0.11 $&$ -0.05 $& 0.03 & 11\
&$ 0.14 $&$ 0.07 $&$ 0.10 $&$ 0.03 $&$ 0.11 $&$ 0.04 $&$ 0.02 $&$ 0.13 $&$ 0.02 $&$ 0.01 $&$ 0.07 $& 0.05 & 4\
&$ -0.03 $&$ -0.04 $&$ -0.08 $&$ -0.08 $&$ -0.02 $&$ 0.01 $&$ -0.06 $&$ -0.04 $&$ -0.04 $&$ -0.11 $&$ -0.05 $& 0.04 & 12\
&$ 0.07 $&$ 0.10 $&$ 0.03 $&$ 0.07 $&$ 0.07 $&$ 0.05 $&$ 0.09 $&$ -0.01 $&$ -0.01 $&$ 0.04 $&$ 0.05 $& 0.04 & 14\
&$ 0.17 $&$ 0.17 $&$ 0.18 $&$ 0.26 $&$ 0.20 $&$ 0.23 $&$ 0.20 $&$ 0.25 $&$ 0.19 $&$ 0.18 $&$ 0.20 $& 0.03 & 3\
&$ -0.17 $&$ -0.23 $&$ -0.18 $&$ -0.15 $&$ -0.24 $&$ -0.21 $&$ -0.22 $&$ -0.30 $&$ -0.24 $&$ -0.07 $&$ -0.20 $& 0.06 & 3\
&$ -0.07 $&$ -0.07 $&$ -0.10 $&$ -0.08 $&$ -0.08 $&$ -0.05 $&$ -0.06 $&$ -0.10 $&$ -0.06 $&$ -0.09 $&$ -0.07 $& 0.02 & 5\
&$ 0.09 $&$ 0.07 $&$ 0.07 $&$ 0.09 $&$ 0.10 $&$ 0.12 $&$ 0.09 $&$ 0.12 $&$ 0.11 $&$ 0.07 $&$ 0.09 $& 0.02 & 29\
&$ -0.18 $&$ -0.31 $&$ -0.26 $&$ -0.26 $&$ -0.27 $&$ -0.17 $&$ -0.28 $&$ -0.37 $&$ -0.31 $&$ -0.30 $&$ -0.27 $& 0.06 & 1\
&$ -0.07 $&$ 0.02 $&$ -0.02 $&$ 0.12 $&$ -0.04 $&$ 0.03 $&$ 0.09 $&$ -0.05 $&$ 0.05 $&$ 0.04 $&$ 0.02 $& 0.06 & 1\
&$ -0.02 $&$ -0.04 $&$ 0.05 $&$ 0.08 $&$ 0.04 $&$ 0.10 $&$ -0.05 $&$ 0.00 $&$ -0.04 $&$ -0.06 $&$ 0.01 $& 0.06 & 4\
&$ 0.22 $&$ 0.17 $&$ 0.17 $&$ 0.21 $&$ 0.25 $&$ 0.36 $&$ 0.26 $&$ 0.20 $&$ 0.19 $&$ 0.18 $&$ 0.22 $& 0.06 & 4\
$^{*}$ &$ 0.08 $&$ 0.10 $&$ 0.07 $&$ 0.09 $&$ 0.18 $&$ 0.14 $&$ 0.14 $&$ 0.08 $&$ 0.10 $&$ 0.08 $&$ 0.11 $& 0.03 & 4\
&$ 0.20 $&$ 0.04 $&$ 0.24 $&$ 0.22 $&$ 0.33 $&$ 0.33 $&$ 0.27 $&$ 0.19 $&$ 0.20 $&$ 0.16 $&$ 0.22 $& 0.08 & 3\
&$ 0.11 $&$ 0.21 $&$ 0.05 $&$ 0.21 $&$ 0.16 $&$ 0.25 $&$ 0.21 $&$ 0.10 $&$ 0.15 $&$ 0.14 $&$ 0.16 $& 0.06 & 2\
log $\epsilon$(Li) &$ 0.15 $&$ 1.25 $&$ 1.00 $&$ <0.0 $&$ 0.95 $&$ 1.34 $&$ <0.0 $&$ <0.0 $&$ 0.78 $&$ <0.0 $&$ $& & 1\
&$ 25 $&$ 20 $&$ 25 $&$ 13 $&$ 17 $&$ 25 $&$ 15 $&$ 20 $&$ 15 $&$ 20 $&$ 19.5 $&$ 4.5 $& CN\
C &$ -0.39 $&$ -0.28 $&$ -0.27 $&$ -0.27 $&$ -0.37 $&$ -0.39 $&$ -0.21 $&$ -0.32 $&$ -0.37 $&$ -0.28 $&$ -0.31 $& 0.06 & C$_{2}$, CH,\
N &$ 0.51 $&$ 0.45 $&$ 0.47 $&$ 0.50 $&$ 0.50 $&$ 0.46 $&$ 0.48 $&$ 0.48 $&$ 0.48 $&$ 0.47 $&$ 0.48 $& 0.02 & CN\
O &$ -0.15 $&$ -0.16 $&$ -0.14 $&$ -0.14 $&$ -0.11 $&$ -0.10 $&$ -0.10 $&$ -0.11 $&$ -0.13 $&$ -0.14 $&$ -0.13 $& 0.02 & \[O I\]\
& & & & & & & & & & & & &\
\
&$ 0.11 $&$ 0.02 $&$ 0.15 $&$ 0.16 $&$ 0.10 $&$ 0.12 $&$ 0.11 $&$ 0.11 $&$ 0.07 $&$ 0.19 $&$ 0.12 $& 0.05 & 5\
&$ -0.05 $&$ -0.01 $&$ -0.02 $&$ 0.01 $&$ -0.03 $&$ 0.03 $&$ 0.02 $&$ -0.05 $&$ -0.01 $&$ 0.02 $&$ -0.01 $& 0.03 & 11\
&$ 0.02 $&$ -0.05 $&$ 0.04 $&$ 0.04 $&$ 0.01 $&$ 0.05 $&$ 0.04 $&$ 0.04 $&$ 0.00 $&$ 0.04 $&$ 0.02 $& 0.03 & 6\
&$ 0.07 $&$ 0.10 $&$ 0.13 $&$ 0.14 $&$ 0.10 $&$ 0.08 $&$ 0.15 $&$ 0.10 $&$ 0.05 $&$ 0.16 $&$ 0.11 $& 0.04 & 11\
&$ -0.08 $&$ 0.16 $&$ 0.06 $&$ 0.15 $&$ -0.11 $&$ 0.09 $&$ 0.09 $&$ 0.04 $&$ 0.07 $&$ -0.01 $&$ 0.05 $& 0.09 & 2\
&$ 0.02 $&$ 0.04 $&$ 0.05 $&$ 0.03 $&$ -0.05 $&$ -0.02 $&$ 0.10 $&$ 0.02 $&$ -0.01 $&$ -0.01 $&$ 0.02 $& 0.04 & 10\
&$ -0.03 $&$ 0.02 $&$ -0.04 $&$ -0.07 $&$ -0.11 $&$ 0.00 $&$ -0.11 $&$ -0.24 $&$ -0.08 $&$ 0.00 $&$ -0.06 $& 0.08 & 2\
&$ 0.12 $&$ 0.10 $&$ 0.10 $&$ 0.11 $&$ 0.07 $&$ 0.13 $&$ 0.10 $&$ 0.05 $&$ 0.06 $&$ 0.14 $&$ 0.10 $& 0.03 & 11\
&$ 0.01 $&$ -0.11 $&$ -0.02 $&$ -0.14 $&$ -0.10 $&$ -0.02 $&$ 0.01 $&$ -0.06 $&$ 0.04 $&$ -0.05 $&$ -0.04 $& 0.06 & 2\
&$ -0.02 $&$ -0.09 $&$ -0.05 $&$ -0.09 $&$ -0.05 $&$ 0.02 $&$ -0.06 $&$ -0.10 $&$ -0.07 $&$ -0.15 $&$ -0.06 $& 0.05 & 10\
&$ -0.12 $&$ -0.08 $&$ -0.13 $&$ -0.17 $&$ -0.16 $&$ -0.10 $&$ -0.12 $&$ -0.16 $&$ -0.12 $&$ -0.22 $&$ -0.14 $& 0.04 & 1\
&$ 0.00 $&$ -0.04 $&$ 0.00 $&$ -0.03 $&$ -0.10 $&$ -0.10 $&$ 0.00 $&$ -0.13 $&$ -0.03 $&$ -0.08 $&$ -0.05 $& 0.05 & 3\
&$ -0.01 $&$ 0.11 $&$ -0.11 $&$ -0.06 $&$ -0.03 $&$ 0.02 $&$ 0.07 $&$ 0.11 $&$ 0.03 $&$ -0.09 $&$ 0.01 $& 0.08 & 1\
&$ 0.04 $&$ -0.02 $&$ 0.03 $&$ 0.03 $&$ 0.00 $&$ 0.03 $&$ -0.01 $&$ -0.03 $&$ 0.01 $&$ -0.02 $&$ 0.01 $& 0.03 & 6\
&$ 0.13 $&$ 0.05 $&$ 0.13 $&$ 0.10 $&$ 0.06 $&$ 0.20 $&$ 0.15 $&$ 0.16 $&$ 0.13 $&$ 0.02 $&$ 0.11 $& 0.06 & 9\
&$ 0.43 $&$ 0.15 $&$ 0.34 $&$ $&$ 0.16 $&$ 0.19 $&$ 0.30 $&$ 0.35 $&$ 0.21 $&$ 0.05 $&$ 0.24 $& 0.12 & 1\
&$ 0.10 $&$ 0.04 $&$ 0.06 $&$ 0.02 $&$ -0.06 $&$ 0.07 $&$ 0.04 $&$ 0.10 $&$ 0.00 $&$ 0.00 $&$ 0.04 $& 0.05 & 1\
&$ 28 $&$ 28 $&$ 28 $&$ 20 $&$ 22 $&$ 30 $&$ 20 $&$ 25 $&$ 23 $&$ 16 $&$ 25.0 $& 3.6& CO\
C &$ -0.32 $&$ -0.41 $&$ -0.31 $&$ -0.37 $&$ -0.32 $&$ -0.30 $&$ -0.27 $&$ -0.31 $&$ -0.31 $&$ -0.36 $&$ -0.33 $& 0.04 & CO,\
N &$ 0.58 $&$ 0.48 $&$ 0.49 $&$ 0.45 $&$ 0.41 $&$ 0.37 $&$ 0.42 $&$ 0.48 $&$ 0.39 $&$ 0.31 $&$ 0.44 $& 0.08 & CN\
O &$ 0.13 $&$ 0.05 $&$ 0.01 $&$ -0.06 $&$ 0.06 $&$ 0.05 $&$ -0.04 $&$ 0.13 $&$ 0.00 $&$ -0.14 $&$ 0.02 $& 0.08 & OH\
Abundances from the Infrared Region {#irabs}
===================================
![NGC 752 cluster mean elemental abundances from optical (blue symbols) and $IR$ (red symbols) spectral region. The data for this figure are from Table \[tab-abunds\]. For elements represented by two species (Cr and Ti), the average of the species is displayed.[]{data-label="InfOpt"}](InfOpt){width="\columnwidth"}
We determined the abundances of 20 elements in NGC752 from the IGRINS $H$ and $K$ band spectra. Of these elements, 18 also have optical region abundances reported by BT15. We applied synthetic analyses to all transitions with the same atomic and molecular line lists and methods described in [@afsar18]; see Paper 1 for more detailed discussion. We derived the abundances of H-burning (C, N, O), $\alpha$ (Mg, Si, S, Ca), light odd-Z (Na, Al, P, K), Fe-group (Sc, Ti, Cr, Fe, Co, Ni), and neutron-capture () (Ce, Nd, Yb) elements, and also determined ratios. The relative $IR$ abundances for our NGC 752 RGs are listed in the second part of Table \[tab-abunds\]. In Figure \[InfOpt\], we plot the mean $IR$ abundances along with the optical ones from BT15, updated as described in §\[modopt\]. The figure shows general agreement between $IR$ and optical abundances. Defining $\Delta^{\rm IR}_{\rm opt}$\[A/B\] = \[A/B\]$_{\rm IR}$ $-$ \[A/B\]$_{\rm opt}$, we find $\langle\Delta^{\rm IR}_{\rm opt}$\[X/Fe\]$\rangle$ = 0.07 $\pm$ 0.04 ($\sigma$ = 0.15) for 18 species with both optical and $IR$ abundances. Figure \[Abd\] shows optical and $IR$ abundances of each species for all program RGs vs. effective temperature. This figure shows that in a small temperature range ($\sim$175 K) that is covered by our RG sample, abundances do not show significant changes with temperature. Both figure \[InfOpt\] and \[Abd\] indicate the optical/$IR$ agreement for most of the species with a few exceptions: O, Sc, and . The $IR$ abundances of these species deviate more than 0.15 dex from their optical counterparts. We will discuss these deviations in the subsections below.
Fe-group elements: {#fegroup}
------------------
We have investigated NGC752 Fe abundances from about 20 [Fe [i]{}]{} transitions. As noted in our previous studies, there are no known useful [Fe [ii]{}]{} transitions in the IGRINS spectral range. The [Fe [i]{}]{} transitions were adopted from [@afsar18]. In Table \[tab-model\] we list the optical and $IR$ Fe abundances for each RG. Optical [Fe [i]{}]{} and [Fe [ii]{}]{} abundances were derived by using the $EW$ method (BT15). The optical cluster means are: $\rm \langle[\ion{Fe}{i}/H]\rangle_{\rm opt}$ $= 0.01$ ($\sigma = 0.07$) and $\rm \langle[\ion{Fe}{ii}/H]\rangle_{\rm opt}$ $= -0.06$ ($\sigma = 0.04$). The 0.07 dex difference between the neutral and ionized iron abundances stays, in general, within the uncertainty limits. The cluster mean from the $IR$ [Fe [i]{}]{} lines is $\rm \langle[\ion{Fe}{i}/H]\rangle_{\rm IR}$ $= 0.00$ ($\sigma = 0.06$), clearly in agreement with the optical values within the mutual uncertainties. The metallicity of NGC 752 from the neutral- and ionized-species Fe and Ti lines, $\rm \langle[M/H]\rangle$ $= -0.07 \pm 0.04$ (BT15), also agrees well with these values and indicates a solar metallicity for NGC 752.
For other Fe-group elements, we derived abundances from species , , , , and . For Sc we used two weak $K$ band transitions, taking into account their hyperfine structures. We applied synthetic spectrum analysis to these absorption lines and the difference from the optical is \[$\rm \ion{Sc}{ii} / Fe]$$_{\rm opt}$ - \[$\rm \ion{Sc}{i} / Fe]$$_{\rm IR}$ = 0.12 dex. The difference between two spectral regions resembles the difference between neutral and ionized species of Cr in the optical and Ti both in the optical and $IR$. We calculated Ti abundances from 10 lines and the one line at 15783 Å. Although optical and $IR$ abundances are in agreement, for the difference is \[$\rm \ion{Ti}{ii} / Fe]$$_{\rm opt}$ - \[$\rm \ion{Ti}{ii} /
Fe]$$_{\rm IR}$ = 0.20 dex. This situation was also discussed in Paper 1 and [@afsar18]. For 12 RGs of NGC 6940, the difference between the optical and $IR$ abundances was 0.16 dex, and for the three RHB stars presented in [@afsar18] was also 0.16 dex. The $H$-band line comes with a CO blend but for the temperature range for our stars its contamination of feature is negligible. Further investigation of the $IR$ line is needed to better understand the discrepancy between optical and $IR$ abundances. The other Fe-group elements have agreement between optical and IR transitions. The mean \[X/Fe\] abundance from Table \[Abd\] for Fe-group elements is $\langle \rm [X/Fe]\rangle$ $_{\rm IR}= -0.05~(\sigma = 0.05)$ for six species and optical mean is $\langle \rm [X/Fe]\rangle$ $_{\rm opt}$ $= -0.01~(\sigma = 0.14)$ for 11 species including , , , and .
Alpha elements {#alphas}
--------------
We derived the abundances of Mg, Si, S and Ca in NGC752 from their neutral species using the lines in both $H$ and $K$ bands (Table \[Abd\]). For the $IR$ abundance, we made use of about ten transitions in the $H$ and $K$ bands. In the top panels of Figure \[SK\] observed and synthetic spectra of two lines in the $K$ band and the combined absorption of three closely-spaced lines in the optical domain for MMU 77 are shown. Sulfur abundances are about solar for both optical and $IR$ spectral regions; $\langle \rm [\ion{S}{i}/Fe]\rangle_{\rm opt} = 0.05~(\sigma = 0.06)$ and $\langle \rm [\ion{S}{i}/Fe]\rangle_{\rm IR} = 0.02~(\sigma = 0.04)$. Optical Ca and Si abundances from BT15 have a small line-to-line scatter about 0.03 dex, but Mg, on the other hand, obtained from the spectrum synthesis of two strong Mg lines at 5528 and 5711 Å resulted in $\sim$0.20 dex difference. Mg abundances from the $IR$ region were derived from about ten absorption lines with a mean standard deviation of about 0.08 for 10 RGs, which suggests greater reliability for $IR$-based Mg abundances. Mean abundances for $\alpha$ elements $\langle \rm [\alpha/Fe]\rangle \equiv \langle[Mg,Si,S,Ca/Fe]\rangle$, for both optical and $IR$ regions are $\langle \rm [\alpha/Fe]\rangle_{\rm opt} = 0.10~(\sigma=0.10)$ and $\langle \rm [\alpha/Fe]\rangle_{\rm IR} = 0.06~(\sigma=0.06)$, which are well in agreement and slightly above solar.
![Observed (points) and synthetic spectra (colored lines) of transitions for sulfur and potassium in $IR$ and optical wavelengths. In each panel the blue line (in the top) represents a synthesis with no contribution by the element of interest, the red line (in the middle) is for the abundance that best matches the observed spectrum, and the green line (on the bottom) represent the synthesis larger than the best fit by 0.5 dex. []{data-label="SK"}](SK){width="\columnwidth"}
Odd-Z light elements {#oddz}
--------------------
The odd-Z light elements investigated in this study are Na, Al and rarely-studied P and K. Their abundances are in Table \[Abd\]. Na abundances were derived from four neutral $K$ band transitions: 22056.4, 22083.7, 23348.4 and 23348.1 Å. To our knowledge, possible non-LTE effects on these transitions have not yet been investigated. The optical and $IR$ Na abundances are both above the solar values. The $IR$ mean for NGC 752 (Table \[Abd\]) is $\langle \rm [\ion{Na}{i}/Fe]\rangle_{\rm IR} = 0.12~(\sigma=0.05)$, while the optical mean is $\langle \rm [\ion{Na}{i}/Fe]\rangle_{\rm opt} = 0.20~(\sigma=0.05)$. Al abundances were obtained from two lines in the $H$ and four lines in the $K$ band. $H$ band abundances are always $\sim$0.1 dex lower than the $K$ band abundances and the lower $H$ band abundances are more in accord with the optical ones: $\langle \rm [\ion{Al}{i}/Fe]\rangle_{\rm IR} = 0.02~(\sigma=0.03)$ and $\langle \rm [\ion{Al}{i}/Fe]\rangle_{\rm opt} = -0.04~(\sigma=0.04)$.
Phosphorus abundances were determined from two weak $H$-band transitions at 15711.5 and 16482.9 Å. As illustrated in Figure \[P\] for MMU 77, the [P [i]{}]{} lines always have central depths $\lesssim$5% for the members studied here. The P abundance difference obtained from these two lines is 0.13 dex, but this is an extreme case; for other program stars the agreement is much better, usually $<$0.1 dex. The mean phosphorus abundance, $\langle \rm [\ion{P}{i}/Fe]\rangle_{\rm IR} = 0.04~(\sigma=0.09)$ is consistent with the solar value.
As noted in §\[modopt\] we derived NGC752 optical-region K abundances, using the very strong [K [i]{}]{} resonance line at 7698.97 Å (lower right panel of Figure \[SK\]). Our derived mean K abundance for the cluster is large, $\langle \rm [\ion{K}{i}/Fe]\rangle_{\rm opt} = 0.50~(\sigma=0.05)$, but this resonance line is subject to significant non-LTE effects. [@takeda02] and [@Mucciarelli17] have computed non-LTE corrections between 0.2 and 0.7 dex for disk/halo stars of various [[T\_[eff]{}]{}]{}$-$combinations. [@afsar18] found $\sim$0.6 dex higher abundances for the 7699 Å line in three RHB stars. Taking into account the non-LTE corrections suggested for 7698.97 Å[K [i]{}]{} line leads to a conclusion of solar K abundance for our targets. But since this is not based on our own calculations, we have chosen to keep the LTE abundance in Table \[Abd\].
![Observed and synthetic spectra of the phosphorus in the $IR$. The symbols and lines have the same meanings as they do in Figure \[SK\]. []{data-label="P"}](P){width="\columnwidth"}
Potassium abundances from the $IR$ region were derived from two lines at 15163.1 and 15168.4 Å which are affected by CN contamination. We illustrate this with observed/synthetic spectrum comparisons in the lower left panel of Figure \[SK\]. Unlike the optical resonance line, the $H$ band lines yield approximately solar abundances: $\rm \langle[\ion{K}{i}/Fe]\rangle_{IR} = -0.06~(\sigma=0.08)$. This consistency suggests that at most very small (or no) non-LTE corrections may be needed for these K lines. Non-LTE studies of all detectable lines in cool stars will be welcome.
Following the detailed description of HF analyses in [@pilachowski15], we have also inspected the unblended HF feature located at 23358.3 Å in the $K$ band region. Unfortunately no obvious absorption of fluorine is detectable in our targets.
Elements {#ncapels}
---------
In this study we have obtained abundances of three elements from their ionized species transitions: Ce and Nd (mostly due to the *s*-process in the solar-system), and Yb (mostly from the *r*-process). Ce abundances were derived from about four transitions, Nd from one weak transition at 16262.04 Å, and Yb also from one weak line at 16498.4 Å. is blended with CO but that contamination is weak enough to be neglected for the temperature/gravity range for our stars. Mean abundances of all three elements are about/above solar, $\langle \rm [\ion{Ce}{ii}/Fe]\rangle_{\rm IR} = 0.11~(\sigma=0.06)$, $\langle \rm [\ion{Nd}{ii}/Fe]\rangle_{\rm IR} = 0.24~(\sigma=0.12)$ and $\langle \rm [\ion{Yb}{ii}/Fe]\rangle_{\rm IR} = 0.04~(\sigma=0.05)$ (Table \[Abd\]). We have also analyzed optical Ce abundances from 5274.23, 5330.56, 5975.82 and 6043.37 Å transitions. The mean value for the NGC 752 RGs is $\langle \rm [\ion{Ce}{ii}/Fe]\rangle_{\rm opt} = 0.11~(\sigma=0.03)$, which is in harmony with the $IR$ abundance with smaller star-to-star scatter. In BT15 we derived the Nd abundances from two lines at 5255.5 and 5319.8 Å; the overabundance is similar what we found from the $H$ band transition, $\langle \rm [\ion{Nd}{ii}/Fe]\rangle_{\rm opt} = 0.22~(\sigma=0.08)$. In Paper 1, the RGs of NGC 6940, which were analyzed in the same manner with the NGC 752 RGs in this study, showed slightly overabundance in *r*-process and more in the *s*-process elements. We observe a similar behavior; the simple mean of the optical and $IR$ La, Ce and Nd abundances is $\langle \rm [\textit{s}-process/Fe]\rangle \simeq 0.18$, while mean of Eu and Yb is $\langle \rm [\textit{r}-process/Fe]\rangle \simeq 0.10$.
![Observed and synthetic spectra illustrating the carbon isotopic ratio of NGC 752 MMU 77 in both optical and $IR$ regions. The upper panel is centred on the triplet or [$^{\rm 13}$CN]{} red system (2-0) lines, and the bottom panel shows the [$^{\rm 13}$CO]{} (3-1) R-branch band head region. The blue, red, and green synthesis represent =100, 25, and 5 in the upper panel, and =100, 30, and 10 in the bottom panel, respectively.[]{data-label="Ciso"}](Ciso){width="\columnwidth"}
The CNO Group {#cnoiso}
-------------
The IGRINS spectral range contains many useful OH, CN and CO molecular bands that can be used to obtain CNO abundances. We have followed the same iterative scheme used in Paper 1 to obtain the CNO abundances. We have also determined carbon abundances from its neutral transitions in both optical and $IR$ spectral regions.
There are many OH molecular lines in the $H$ band but most of them are very weak for our temperature and metallicity range and also blended with other lines and/or molecular bands. We were able to use about 10 OH molecular lines located between 15200$-$17700 Å; the abundances for each star in Table \[Abd\] are simple means of the abundances derived from individual OH features. The resulting cluster mean is $\langle \rm [O/Fe]\rangle_{\rm IR} = 0.02~(\sigma=0.09)$ (Table \[Abd\]). In BT15, we were able use only the \[[O [i]{}]{}\] line at 6300.3 Åto determine optical O abundances, and noted that this feature is plagued with [Ni [i]{}]{} and CN contamination. The calculated mean for the cluster from this line is $\langle \rm[O/Fe]\rangle_{\rm opt} = -0.13~(\sigma=0.02)$. Having the advantage of obtaining O abundances from many OH features, we rely more on the O abundance we determine from the $IR$ region.
---------- ------ ------ ------- ------ ------ ------ ------
Star CH C$_2$ CO mean mean
opt opt opt $IR$ $IR$ opt $IR$
MMU 1 8.04 7.90 8.03 8.10 8.11 7.99 8.11
MMU 3 8.16 7.78 7.98 8.00 7.98 7.97 7.99
MMU 11 8.20 7.98 8.08 8.10 8.11 8.08 8.11
MMU 24 8.16 7.80 7.95 8.00 8.02 7.97 8.01
MMU 27 8.14 7.93 8.05 8.13 8.20 8.04 8.17
MMU 77 8.05 7.88 8.05 8.11 8.21 7.99 8.16
MMU 137 8.21 7.83 8.00 8.07 8.07 8.01 8.07
MMU 295 8.18 7.95 8.08 8.11 8.18 8.07 8.15
MMU 311 8.14 7.90 8.04 8.17 8.18 8.03 8.18
MMU 1367 8.18 7.83 8.03 8.06 8.03 8.01 8.05
---------- ------ ------ ------- ------ ------ ------ ------
: log$\epsilon$ Abundances of Carbon in optical and infrared regions.[]{data-label="tabc"}
Carbon abundances were derived from multiple optical and $IR$ species. The summary of the results for each star are given in Table \[tabc\]. In the $IR$, we used primarily the CO molecular features in the $K$ band: $^{12}$CO first overtone, $\Delta$v = 2, (2-0) and (3-1) bands at 23400 and 23700 Å. The scatter based on different abundance measurements for a single RG is very small, about $\sim$0.03 dex. The mean C abundance from the CO molecular lines is $\rm \langle[C/Fe]\rangle_{CO} = -0.32~(\sigma=0.06)$, a value which would be expected after first dredge-up and envelope mixing in metal-rich disk stars. There are second overtone $^{12}$CO band heads also in the $H$ band but due to relatively high temperatures of our programme stars they are too weak for detection. In BT15 we obtained the carbon abundances from the CH G band, the Swan band heads of C$_{2}$ (0-0) at 5155 Å and the (0-1) at 5635 Å (Figure 9 in BT15). These molecular bands are heavily blended with other atomic transitions and the C$_{2}$ bands are weak in strength, which makes the spectral analysis challenging in these regions. But from those features BT15 derived $\rm \langle[C/Fe]\rangle_{CH, C_{2}} = -0.41~(\sigma=0.03)$. Considering the analytical difficulties for CH and C$_{2}$, the $\sim$0.1 dex difference from the $IR$ CO result indicates reasonable accord.
We obtained the carbon abundances also from the high-excitation lines. Carbon abundances derived from the transitions agree very well with CO results, $\rm \langle[\ion{C}{i}/Fe]\rangle_{IR} = -0.34~(\sigma=0.04)$. As a further check we also determined the C abundances from synthetic spectrum analyses of three high-excitation lines located in the optical at 5052.1, 5380.3 and 8335.1 Å. The line-to-line C abundance scatter from these transitions is about 0.1 dex, and the mean abundance for the cluster is $\rm \langle[\ion{C}{i}/Fe]\rangle_{opt} = -0.22~(\sigma=0.10)$, on average only $\sim$0.14 dex higher compare to the mean C abundance obtained from other features mentioned above.
In Table \[tabc\] we have listed the individual and mean carbon abundances. The quoted carbon abundances in this table are the average of the molecular and high-excitation carbon abundances and they are in relatively good agreement; $\rm \langle[C/Fe]\rangle_{IR} = -0.33~(\sigma=0.04)$, $\rm \langle[C/Fe]\rangle_{opt} = -0.31~(\sigma=0.06)$.
We obtained nitrogen abundances from the CN molecular transitions in the H-band. We used about 18 CN features between 15000 and 15500 Å, and calculated N abundances. The mean $IR$ N abundance is $\rm \langle[N/Fe]\rangle_{IR} = 0.44~(\sigma=0.08)$. Optical nitrogen abundances were obtained from $^{12}$CN and $^{13}$CN red system lines in the 7995$-$8040 Å region in BT15, and the means are in accord with those from the $IR$, $\rm \langle[N/Fe]\rangle_{opt} = 0.48~(\sigma=0.02)$.
Finally, we measured the ratios from the first overtone $^{12}$CO ($\Delta$v = 2) (2-0) and (3-1) band lines, which are accompanied by the $^{13}$CO band heads near 23440 and 23730 Å. These are more robust features for ratio determination than the standard optical $^{13}$CN feature near 8003 Å used by BT15. The top panel of Figure \[Ciso\] shows that the $^{13}$CN triplet, the strongest feature of this band system, is barely detectable in MMU 77 (nor is it much stronger in any NGC 752 star). In contrast, the $^{13}$CO features shown in the bottom panel of this figure are much stronger. We compare the optical and $IR$ values in Table \[tab-iso\]. They are in reasonable accord, given the extreme weakness of the CN bands.
---------- ----------- ----------- -----------
Stars $^{13}$CN $^{13}$CO $^{13}$CO
(8004 Å) (23440 Å) (23730 Å)
MMU 1 25 25 30
MMU 3 25 25 30
MMU 11 25 25 30
MMU 24 13 20 20
MMU 27 17 19 25
MMU 77 25 30 30
MMU 137 15 20 20
MMU 295 20 25 25
MMU 311 15 23 22
MMU 1367 17 15 16
---------- ----------- ----------- -----------
: Carbon isotopic ratios of optical and infrared regions.[]{data-label="tab-iso"}
Abundance Uncertainties {#uncertaities}
-----------------------
Detailed investigation of the internal and external uncertainty levels of the atmospheric parameters and their effects on the elemental abundances were provided in BT15, in which we calculated an average uncertainty limit of about 150 K by comparing the spectroscopically derived [[T\_[eff]{}]{}]{} values with the literature, photometric and LDR temperatures. In Table 8 of BT15 we list the sensitivity of derived abundances to the model atmosphere changes within uncertainty limits for the star MMU 77. Additional investigation of LDR temperatures from the $IR$ data has shown that our temperature uncertainty limit has remained almost the same as determined in BT15, considering the highest LDR and spectral temperature difference of 154 K for MMU 1367 (see §\[ldrcomp\]). Therefore, in Table \[temp\], we present the sensitivity of derived abundances in the elements only newly studied in this work adopting the same atmospheric parameter uncertainties in BT15. The uncertainties were determined using the $IR$ spectra of same star, MMU 77, as applied in BT15. In general, abundance changes are mostly well within 1$\sigma$ level of the \[X/Fe\] values (Table \[Abd\]). However, the sensitivity level of abundance to the change in temperature stands out. The temperature sensitivity of some $IR$ Sc lines has been previously noticed by [@thor18], based on lines identified in $K$ band of cool M giants observed with NIRSPEC/Keck II. They reported up to 0.2 dex uncertainties in Sc abundances mostly originated from the temperature sensitivity for stars [[T\_[eff]{}]{}]{} < 3800 K. Although our stars have higher temperatures and the lines we used are different than those that @thor18 discussed, caution should be taken in interpreting the $IR$ Sc abundances for our stars until the underlying physical process for the temperature sensitivity of Sc lines are better understood.
--------- ------------------------------- ----------------- ----------------------
Species $\Delta$${{T_{\rm eff}}}$ (K) $\Delta$ $\Delta$$\xi_{t}$ ()
$-$150 / +150 $-$0.25 / +0.25 $-$0.3 / +0.3
0.01 / 0.01 0.09 / $-$0.09 0.01 / 0.00
$-$0.06 / 0.09 0.09 / $-$0.05 $-$0.03 / 0.03
0.10 / $-$0.11 0.02 / $-$0.01 $-$0.01 / 0.05
0.18 / $-$0.18 0.01 / 0.01 0.01 / 0.00
0.08 / $-$0.06 0.12 / $-$0.11 $-$0.03 / 0.03
0.05 / $-$0.05 0.10 / $-$0.10 $-$0.02 / 0.03
--------- ------------------------------- ----------------- ----------------------
: Sensitivity of elemental abundances to the model atmosphere parameter uncertainties for MMU 77.[]{data-label="temp"}
Comparison withNGC 6940 {#comp6940}
-----------------------
We have now derived metallicities and relative abundance ratios for OCs NGC6940 and NGC752 with high resolution spectra in the optical spectral region (BT15, BT16) and infrared (BT19, this study). The NGC6940 optical data were obtained with the Hobby-Eberly Telescope and its high-resolution echelle spectrometer [@tull98], and those for NGC752 with the 2.7m Smith Telescope and Tull echelle spectrometer [@tull95]; both data sets have high resolution ($R$ $\simeq$ 60,000) and high $S/N$ $\geq$ 100. The $H$ and $K$ band spectra for the two clusters were gathered with IGRINS set up as described in §\[obs\] and observed in identical fashions.
![Differences between relative abundances \[X/Fe\] in NGC752 and NGC6940 in the optical and $IR$ spectral regions. The dotted line at $\Delta_{ 752}^{6940}$\[X/Fe\] = 0.00 indicates equality between \[X/Fe\] values in the two clusters. The dashed line at $\Delta_{ 752}^{6940}$\[X/Fe\] = 0.04 represents the mean value for all abundances, excluding the aberrant values for [K [i]{}]{} in the $IR$ and [Cu [i]{}]{} in the optical spectral region. []{data-label="n752n6940"}](n752n6940){width="\columnwidth"}
Our derived metallicities for the two clusters suggest that NGC752 is slightly more metal-rich than NGC6940. Defining $\Delta_{ 752}^{6940}$X = X$_{\rm NGC6940}$ $-$ X$_{\rm NGC752}$, from optical data $\Delta_{ 752}^{6940}$\[[Fe [i]{}]{}/H\]$_{\rm opt}$ = $+$0.05 and $\Delta_{ 752}^{6940}$\[[Fe [ii]{}]{}/H\]$_{\rm opt}$ = $+$0.08, but these differences are well within the observational/analytical uncertainties. The $IR$ metallicities are essentially identical: $\Delta_{ 752}^{6940}$\[[Fe [i]{}]{}/H\]$_{\rm IR}$ = $+$0.02. We conclude, in agreement with past studies, that both NGC6940 and NGC752 have solar metallicities.
The general accord between the two clusters extends to the abundance ratios of individual elements. In Figure \[n752n6940\] we show abundance differences for all species studied in the optical and $IR$ regions. The uncertainties shown in the figure are approximate, being averages of the $\sigma$ values of the abundances in each cluster. Excluding the aberrant points for optical [Cu [i]{}]{} and $IR$ [K [i]{}]{}, we derive <$\Delta_{ 752}^{6940}$\[X/Fe\]> = $+$0.045 (+0.06 in the optical, +0.03 in the $IR$). The Cu difference is not well determined, as the NGC 752 optical spectra permitted use of only one [Cu [i]{}]{} feature. At present we lack an explanation for the 0.2 dex abundance difference between the $IR$-based [K [i]{}]{} lines in NGC 6940 and NGC 752. This issue will be considered again in our future studies of M67 and other OCs. In Table \[tab-diff\], we have listed the abundance differences between two clusters that generate Figure 8. Table \[tab-diff\] also contains the comparison with the recent optical abundances from the literature. The comparison with three studies [@carrera11; @reddy12; @lum19] shows a general accord in uncertainty limits in both regions.
Species N6940-N752 Lum - us Carrera-us Reddy-us
--------- ------------ ---------- ------------ ----------
C 0.07 0.09
N $-0.02$ $-0.20$
O 0.04 -0.02
0.07 -0.07 $-0.23$ $-0.08$
0.08 0.06 $-0.03$ 0.00
0.00 0.36 $-0.02$ 0.19
0.00 $-0.11$ $-0.28$ $-0.11$
0.11
$-0.02$ $-0.08$ $-0.15$ $-0.10$
0.03 $-0.09$ $-0.09$
$-0.01$ 0.11 $-0.02$
0.04 $-0.03$ $-0.11$
0.01 0.18 0.08
0.01 0.00 $-0.03$ $-0.08$
0.04 $-0.30$ $-0.22$
0.12 0.24 0.17 0.07
$-0.05$ 0.18 0.04
0.00 $-0.09$ $-0.12$ $-0.10$
0.28 0.16
0.07 $-0.08$ $-0.12$
0.11 0.04 0.02
$-0.01$ $-0.09$
0.00 $-0.04$ 0.02
0.04 $-0.07$ $-0.16$
$-0.03$ $-0.09$
average 0.04 0.00 $-0.09$ $-0.04$
sigma 0.07 0.15 0.14 0.10
C 0.05 0.11
N 0.08 $-0.16$
O $-0.03$ $-0.17$
0.16 0.01 $-0.15$ 0.00
0.03 0.06 $-0.03$ 0.00
0.05 0.30 $-0.08$ 0.13
0.04 0.00 $-0.17$ 0.00
0.06
0.05
0.23
0.05 $-0.05$ $-0.12$ $-0.07$
0.04 0.11
0.07 0.12 $-0.01$
0.09 0.18 0.10
0.06 0.10 0.07 0.02
$-0.01$ 0.10 $-0.04$
0.05 $-0.01$ $-0.04$ $-0.02$
0.13 $-0.04$ 0.02
0.10 $-0.09$ $-0.18$
0.03
average 0.07 0.03 $-0.07$ 0.00
sigma 0.06 0.13 0.08 0.08
: Abundance differences.[]{data-label="tab-diff"}
The Age of NGC 752 {#age}
==================
In Paper I stellar evolutionary models with a solar abundance set were fitted to the CMD of NGC$\,$6940, yielding an age of 1.15 Gyr. To well within the uncertainties, the metallicity that we have determined for NGC$\,$752 is the same as that of NGC$\,$6940. Both clusters appear to have the same helium abundance as well, given that (as discussed below) models for $Y=0.270$ provide equally good fits to the luminosities of the core He-burning red clump (RC) stars if (a) distance moduli based on *Gaia* parallaxes are adopted, and (b) the observed RGs in the clusters are mostly in the RC evolutionary stage. Inspection of the CMD for NGC 6940 (Figure 1 of Paper 1) suggests that most RGs in that cluster are not associated with the RGB evolutionary tracks, and thus truly are RC stars. For NGC 752 we discuss this issue below, but for the moment simply assume that our program stars are mostly RCs. Then to derive our best estimate of the age of NGC$\,$752, it is simply a matter of interpolating in the same model grids that were used in Paper I to identify which isochrone provides the best fit to the cluster turnoff stars.[^3]
However, this process involves the cluster reddening, for which most estimates fall in the range $0.03 \le E(B-V) \le 0.05$ [@dan94; @taylor:07; @schlafly:11; @twarog:15], and the adopted color–[[T\_[eff]{}]{}]{} relations (from @casagrande:18; hereafter CV18). Fortunately, the Sun provides a valuable constraint on both the predicted $T_{\rm eff}$ and color scales. According to CV18, their determinations of $M_{G, \odot} = 4.67$ and $(G-G_{\rm RP})_\odot = 0.49$ from reference solar spectra are accurate to within $\approx 0.01$ mag. Encouragingly, the bolometric corrections (BCs) derived from the MARCS library of theoretical stellar fluxes [@gustafsson:08], yield the same value of $M_G$ on the assumption of \[Fe/H\] $= 0.0$ and the solar values of [[T\_[eff]{}]{}]{} and , but a bluer $G-G_{\rm RP}$ color by $\approx 0.01$ mag. We have therefore applied a $+0.01$ mag zero-point correction to the synthetic colors in order that our solar model reproduces the “observed" $G-G_{\rm RP}$ color of the Sun.
![The CMD of NGC752 (black dots) and its fit with a new $1.52$ Gyr isochrone (solid black curve; dotted and dashed curves show $1.62$ Gyr and $1.42$ Gyr isochrones). The blue and red curves are the MESA evolutionary tracks computed for $M=1.82\,M_\odot$, \[Fe/H\]$=0$, $Y=0.27$, assuming that $f_\mathrm{ov} = 0.035$ in Equation 1 from Paper I. The red track has the mixing length increased by $10\%$ compared to the solar-calibrated value of $\alpha_\mathrm{MLT} = 2$ adopted for the blue track. See the text for descriptions of the solar symbol and the red dot. []{data-label="fig:ngc752Gaia"}](ngc752Gaia_cut.pdf){width="\columnwidth"}
In Figure \[fig:ngc752Gaia\] we show the best model fits to the observed $(G-G_{RP},\,G)$ color-magnitude diagram for NGC$\,$752. The upper MS and turnoff stars are fit quite well by a 1.52 Gyr isochrone for solar abundances if the the adopted reddening is $E(B-V) = 0.035$ mag. The isochrone begins to deviate slightly to the blue of the observed MS at $G \sim 14$, with the offset in color at a given magnitude rising to as much as 0.1 mag at $G \sim 18$. Inadequacies in the CV18 color–$T_{\rm eff}$ relations for cooler stars are likely responsible for this problem given that the transmission function of the $G$ filter extends well into the ultraviolet.
The tables of BCs presented by CV18 take into account the dependence of the extinction on spectral type in a fully consistent way; i.e., these transformations enable one to convert predicted luminosities and temperatures directly to absolute $G$ and $G_{RP}$ magnitudes that have been suitably corrected for an assumed reddening. In order for the resultant models to appear on the observed $(G-G_{RP},\,G)$ CMD, they must then be shifted in the vertical direction by an amount corresponding to the true distance modulus, $(m-M)_0$. In Figure \[fig:ngc752Gaia\] the solar symbol indicates where the Sun would be located if it was as distant as NGC$\,$752 and subject to the same reddening. The red filled circle represents a model at an age of 1.52 Gyr along an evolutionary track that has been calculated for a Standard Solar Model, and similarly adjusted by the adopted reddening and distance modulus. Thus, in order to satisfy the solar constraint, the reddening of NGC$\,$752 must be quite close to $E(B-V) = 0.035$ mag if it has $(m-M)_0 = 8.26$. The inferred reddening would be larger than this if the cluster is less distant, and vice versa.
Ages in the range of 1.7–2.0 Gyr were typically found for NGC$\,$752 in the mid-1990s [e.g. @dan94; @dinescu:95], but subsequent determinations have generally favored ages closer to 1.5 Gyr [e.g. @anthony-twarog:06; @twarog:15]. The earlier age determinations are especially uncertain because the stellar models used in those studies assumed little or no overshooting from convective cores during the MS phase. Because such isochrones are incapable of reproducing the observed turnoff morphology, the ages derived from them are highly questionable. In contrast, later investigations employed models that allowed for significant amounts of core overshooting, resulting in fits to the NGC$\,$752 CMD that are quite similar to that shown in Fig. \[fig:ngc752Gaia\], and they yield ages that have little ambiguity.
Recently, a similar color-magnitude diagram study by [@agueros:18] obtained an age of $1.34 \pm 0.06$ Gyr for NGC$\,$752, which is inconsistent with our determination by more than $2\,\sigma$. Although those researchers used a sophisticated Bayesian approach to determine the cluster parameters (including such observational properties as the distance, metallicity, and extinction) from fits of isochrones to the photometric data, their results will be subject to systematic errors that are very difficult to quantify. In particular, the predicted [[T\_[eff]{}]{}]{} scale is quite sensitive to, e.g., the adopted atmospheric boundary condition and the treatment of super-adiabatic convection. Errors in the adopted color transformations can further impact how well stellar models are able to reproduce an observed CMD. Consequently, one cannot rely on such isochrone predictions as the location of the giant branch relative to the turnoff to provide a useful constraint on absolute cluster ages (see, e.g., [@vandenberg:90], who show that this diagnostic may be used to obtain accurate [*relative*]{} ages of star clusters.) We suspect that the derivation by @agueros:18 of $A_V = 0.198\pm 0.0085$ mag, which is appreciably higher than most determinations, including the line-of-sight Galactic extinction [@schlafly:11], can be attributed, in part, to errors in the model [[T\_[eff]{}]{}]{} and/or color scales.
Isochrones appropriate to young and intermediate-age clusters are also very dependent on how much overshooting from convective cores during the MS phase is assumed. In fact, the MESA models [@choi:16] that were used by @agueros:18 assume a value of the overshooting parameter that is, according to our analysis (see Paper I and the next section) too low by about a factor of two. This appears to be the main reason (see below) why they obtained a significantly younger age than our determination. Unfortunately, @agueros:18 do not include a figure that compares their best-fit isochrone with the CMD of NGC$\,$752; hence it is not possible to make a visual assessment of how well the data are fitted. Our age determination should be particularly robust because we have used the Sun to calibrate the predicted [[T\_[eff]{}]{}]{} and color scales, and have adopted the *Gaia* distance and a spectroscopically derived metallicity, from which we have deduced the $E(B-V) \approx 0.035$ in order to achieve consistency with the solar constraint. Thus, nearly all of the cluster parameters are derived independently of our stellar models and the age is effectively obtained from an overlay of the isochrone that provides the best fit to the turnoff stars.
Stellar evolution modeling of NGC752 {#isoch}
====================================
In Paper I we emphasized the importance of calibrating the efficiency of convective overshooting beyond the Schwarzschild boundary of the hydrogen convective core in MS stars with $1\la M/M_\odot\la 2$. In the Victoria stellar evolution code employed here to generate isochrones, the convective overshooting is estimated using the integral equations of [@roxburgh:89] as described in [@vandenberg:06]. In particular, Figure 1 in the latter paper shows the variation of the free parameter $F_\mathrm{over}$ in Roxburgh’s equations calibrated by comparing the predicted and observed CMDs for a number of open clusters with different ages. This parameter starts to increase from $F_\mathrm{over}=0$ at $M\approx 1.14\,M_\odot$, reaches a maximum value of $F_\mathrm{over}=0.55$ at $M = 1.7\,M_\odot$ and then remains constant. In the MESA code, that we use to model the evolution of MS turn-off (MSTO) stars up to the red-clump (RC) phase, the convective overshooting is approximated by a diffusion coefficient that is exponentially decreasing outside the convective boundary on a lengthscale of $0.5f_\mathrm{ov}H_P$, where $H_P$ is a local pressure scale height. In Paper I we showed that the MESA code with $f_\mathrm{ov}=0.035$ produces a stellar evolution track for an initial mass $M=2\,M_\odot$ that is approximately equal to the MSTO mass of stars in the open cluster NGC6940, in the excellent agreement with the Victoria 1.15 Gyr isochrone generated for this cluster.
![Upper panel: the MESA evolutionary tracks of the solar-metallicity $1.82\,M_\odot$ model with the convective overshooting parameter $f_\mathrm{ov}=0.035$ (red) and of the $1.85\,M_\odot$ model with $f_\mathrm{ov}=0.016$ (green) that both fit the luminosity of the stars leaving the MS in NGC752. Lower panel: the RGB and RC evolutionary timescales of these models. []{data-label="fig10new"}](fig10new.pdf){width="\columnwidth"}
The MSTO mass for the estimated 1.52 Gyr age of NGC752 is $M\approx 1.8\,M_\odot$. This mass is high enough that the maximum value of $F_\mathrm{over} = 0.55$ should still be used according to Figure 1 in [@vandenberg:06]. Therefore we have used the same value of $f_\mathrm{ov}=0.035$ in the MESA code to model the evolution of MSTO stars in NGC752. Figure \[fig:ngc752Gaia\] demonstrates that in this case the evolutionary track for the initial mass $1.82\,M_\odot$ (blue curve) is again in an excellent agreement with a post-MSTO part of the isochrone (black curve).
We believe that most/all of our program stars are red clump (RC) members. First, their derived CNO abundances and ratios are all consistent with evolution beyond the first-ascent RGB, with completion of “first dredge-up” envelope mixing of CN-cyle products. Second, stars with the NGC 752 turnoff mass should spend little time on the upper RGB, and none are observed. Based on the comparison of the timescales of the RGB and RC evolution for our best-fit stellar model, we conclude that most of the red giants observed in NGC752 should be RC stars.
As in Paper I, the red giant branches of the track and the isochrone are $\sim0.05$ mag redder than they should be for the track to be able to almost perfectly fit the colors of the RC stars in NGC752 on the following He-core burning evolutionary phase. Possible causes of this small discrepancy are mentioned in Paper I. To show that our track can successfully reproduce colors and magnitudes of RC stars in NGC752, we have increased the solar-calibrated convective mixing length parameter $\alpha_\mathrm{MLT} = 2.0$ used for the red track by 10%. This has enhanced heat transport in the convective envelopes of our RGB and RC model stars and, as a result, changed their colors by $\sim-0.05$ (red curve in Figure \[fig:ngc752Gaia\]). We used the same remedy in Paper I. It does not solve the problem of the RGB color discrepancy, but it enables us to see how the RGB and RC evolutionary tracks will look after the true cause of this discrepancy is found and fixed.
When we reduce the convective overshooting parameter in our solar-metallicity $1.82\,M_\odot$ MESA stellar models to $f_\mathrm{ov} = 0.016$ (the value used by @choi:16 for core overshooting), and slightly increase the initial mass to $1.85\,M_\odot$ to keep the same MSTO luminosity, the morphology of their corresponding evolutionary track becomes inconsistent with the observed CMD of NGC752 (green curves in Figure \[fig10new\]). There are multiple problems: (a) the effective temperature at the end of the core H-burning phase is too high; (b) the track produced by core He-burning is too narrow in color and it does not reach the minimum luminosity of the observed RC stars (top panel); and (c) the RGB and RC evolutionary timescales are now comparable (bottom panel), which would lead us to expect comparable numbers of RGB and RC stars in NGC752, which is not observed. The last inconsistency arises because the reduced efficiency of convective H-core overshooting leads to an extended RGB evolution with the He core becoming electron degenerate and experiencing a flash at the end, while in the models with $f_\mathrm{ov}=0.035$ the He core remains non-degenerate, and He in the core is ignited quiescently.
Applying these evolutionary computations to C and N abundances, in the upper panel of Figure \[fig11new\] we compare the predicted and observed \[C/Fe\] abundance ratios for the RC stars in NGC752. The observed C abundances can be reproduced by our models only if we assume that they were already slightly reduced initially by $\simeq$ $0.1$dex, compared to the solar-scaled \[C/Fe\] ratio (the dashed black curve), because without this assumption the predicted RC abundance is \[C/Fe\]$=-0.19$. while our mean observed value is \[C/Fe\]$=-0.31\pm 0.06$. [@lum19] support a slightly subsolar initial C abundance in NGC752, deriving \[C/Fe\]= $-$0.10 ($\sigma$ =0.13, 21 stars). However, the @lum19 red giant abundance, \[C/Fe\]= $-$0.22 ($\sigma$ =0.08, 6 stars) is consistent with our predictions with solar or slightly subsolar initial C abundances.
In the lower panel of Figure \[fig11new\] we make the same kind of comparison for N. The observed optical and $IR$ values, \[N/Fe\]$=0.48\pm 0.02$ and 0.44 $\pm$ 0.08, respectively, are slightly larger than the predictions, \[N/Fe\]= 0.41 (red curve) and 0.37 (black dashed curve) The initial N abundance has been assumed to be solar. However, [@lum19] derives \[N/Fe\]= 0.12 (no stated $\sigma$), and that would raise the predicted red giant N abundance to be nearly comparable to the observed one.[^4]
Finally, for carbon isotopic ratios our model predicts = 22.2 and 20.9 for the red and black dashed tracks. They are comparable with the mean C isotopic ratios measured in the RC stars in NGC752: = 22 (optical) and 16 ($IR$). Note that our predicted C and N abundances are in a good agreement with those obtained for a non-rotating $1.8\,M_\odot$ star by [@charbonnel:10] ($=19.9$, \[C/Fe\]$=-0.18$, \[N/Fe\]$=0.37$), who did not consider any convective overshooting, but did include thermohaline mixing on the RGB. In our models, the enhanced convective overshooting significantly decreases the RGB evolution time (the lower panel in Figure \[fig10new\]), therefore if we included thermohaline mixing on the RGB its effect on the surface abundances of C and N would be even less pronounced and our assumption on the reduced initial abundance of C would still be required. According to [@charbonnel:10], rotation with a ZAMS velocity of $110\,\mathrm{km\,s}^{-1}$ only slightly changes these abundances to $=15.2$, \[C/Fe\]$=-0.19$ and \[N/Fe\]$=0.31$.
![The predicted changes of the surface C (top panel) and N (bottom panel) abundances for the solar-metallicity $1.82\,M_\odot$ stellar evolutionary tracks computed with $f_\mathrm{ov}=0.035$ and $\alpha_\mathrm{MLT}=2.2$ are compared with the C and N abundances determined for the red-clump stars in NGC752 using optical (black circles) and infrared (magenta squares) spectra. The red and black dashed curves are obtained assuming that the initial C abundance is \[C/Fe\]$=0$ and \[C/Fe\]$=-0.1$, respectively. []{data-label="fig11new"}](fig11new.pdf){width="\columnwidth"}
Summary {#cocl}
=======
This is the second of three papers that report analyses of high-resolution optical and $IR$ spectra RG members of prominent OCs. In this study, we have performed the detailed chemical abundance analysis for 10 RGs in the NGC 752 open cluster using the high-resolution near-$IR$ $H$ and $K$ band spectral data obtained with the IGRINS spectrograph. BT15 investigated the same RG members in the optical region from their high-resolution optical spectra, and here we combine data from both regions and explore the NGC 752 from a more complete wavelength window.
We revisited the CMD of NGC 752, investigating the membership assignments using *Gaia* DR2 [@GAIA18b]. We applied a Gaussian mixture model to set the parallax bounds, leading to an estimated cluster distance of 448 pc with a true distance modulus of $(m-M)_0 = 8.26$. We also remeasured the radial velocities of our targets from the $H$ and $K$ band spectra, finding a cluster mean of 4.97 $\pm$ 0.24 (Table \[tab-motions\]), which is in general agreement with both our previous optical RV (Paper 1) and *Gaia*.
We applied LDR relations reported by [@fukue15] and estimated the $IR$-LDR effective temperatures for our targets. LDR temperatures obtained from both optical and $IR$ line depth ratios are in good agreement with the spectral temperatures within $\sim$150 K, indicating that this method provides reliable temperature estimations in the cases of lack of information from the optical region. This encouraging result paves the way for dust-obscured open cluster chemical composition studies.
Adopting the model atmospheric parameters from Paper 1, we performed detailed abundance analysis for 20 elements in the $H$ and $K$ band spectral regions of our targets. The abundances for 18 of these elements were determined both in the optical and $IR$ regions. In general, we derived the abundances of H-burning, $\alpha$, light odd-Z, Fe-group, and elements, and also determined ratios from both regions. In general, they are in accord with their optical counterparts and have abundances similar to their solar-system values. $IR$ abundances of CNO and some $\alpha$ elements (such as Mg and S) were found to be more reliable compare to their optical counterparts due to more number of lines and regions used during the $IR$ spectral analysis.
In some cases, small abundance differences were seen between neutral and ionized species of the same element. In particular there is only one [Ti [ii]{}]{} line present in the $H$ band and, compare to its optical counterparts, it seems to yield sub-solar abundances for our stars in general. Further investigation is needed to better understand this issue. For Sc abundances, [Sc [ii]{}]{} and [Sc [i]{}]{} lines were used in the optical and $IR$ regions, respectively. Their mean Sc cluster abundance differs by 0.12 dex. But only two weak [Sc [i]{}]{} lines in the $K$ band with hyperfine structures were used to determine the $IR$ Sc abundances, so we do not regard this as a significant discrepancy.
To the best of our knowledge, P, S and K abundances have been derived here for the first time for our targets, and all are consistent with solar abundances in NGC 752. Potassium abundances obtained from two lines in the $H$ band indicate that these lines are likely to be less affected by non-LTE line formation problems than is the strong [K [i]{}]{} 7698.7 Å resonance line. A similar suggestion could be made for the $IR$ Na lines. They provide Na abundances $\sim$0.1 dex lower than the optical ones, which might indicate that they are also less affected by non-LTE conditions.
Five elements were identified in the spectra of NGC 752 RGs. The abundances in our stars resulted in somewhat overabundances both in the $s$-process and $r$-process elements, later being less slightly overabundant. Encouragingly, the abundances of Ce and Nd, show agreement between their optical and $IR$ values. Detection of a [Yb [ii]{}]{} line at 16498.4 Å in the $H$ band provides a unique opportunity to study this element, since the strong resonance [Yb [ii]{}]{} 3694 Å line occurs in a very crowded low-flux region of cool stars, essentially useless for abundance studies in most solar-metallicity stars.
Analyzing CNO abundances using the many available $IR$ CO, CN, and OH molecular features, we found cluster mean abundances from optical and $IR$ regions to be in reasonable agreement. We suggest that $IR$ O abundances may provide more robust O abundances than does the \[[O [i]{}]{}\] 6300.3 Å optical line. Our study multiple $^{12}$CO and $^{13}$CO first overtone band lines yields a similar endorsement: these regions provide more robust measurements of values than ones possible from the weak CN optical features near 8004 Å. Our CNO results indicate that all NGC 752 RC stars have abundances consistent with those predicted from first dredge-up predictions, , [@charbonnel:10].
We used the NGC 752 CMD to investigate the evolutionary states of 10 RG members, first concluding that they are at least mostly red clump members. The best evolutionary model for solar metallicity yielded core He-burning RC stars consistent with our stars if a helium abundance $Y$ = 0.270 is adopted. Isochrones fitted to the cluster turnoff yield an age of 1.52 Gyr for the reddening $E(B-V)$ = 0.035 mag and the turnoff mass $M=1.82\,M_\odot$ of NGC 752. Our light element abundance values, $\langle$\[C/Fe\]$\rangle$ $\simeq$ $-$0.32, $\langle$\[N/Fe\]$\rangle$ $\simeq$ $+$0.46, $\langle$\[C/Fe\]$\rangle$ $\simeq$ $-$0.05, and $\langle$$\rangle$ $\simeq$ 22, are in reasonable accord with those predicted by our MESA evolutionary models.
Acknowledgments {#acknowledgments .unnumbered}
===============
This study has been supported by the US National Science Foundation (NSF, grant AST 16-16040), and by the University of Texas Rex G. Baker, Jr. Centennial Research Endowment. This work used the Immersion Grating Infrared Spectrometer (IGRINS) that was developed under a collaboration between the University of Texas at Austin and the Korea Astronomy and Space Science Institute (KASI) with the financial support of the US National Science Foundation under grant AST-1229522, of the University of Texas at Austin, and of the Korean GMT Project of KASI. These results made use of spectra obtained at the Discovery Channel Telescope at Lowell Observatory, and the 2.7m Smith telescope at McDonald Observatory. Lowell is a private, non-profit institution dedicated to astrophysical research and public appreciation of astronomy and operates the DCT in partnership with Boston University, the University of Maryland, the University of Toledo, Northern Arizona University and Yale University. We also gathered data from the European Space Agency (ESA) mission *Gaia* ($\rm https://www.cosmos.esa.int/gaia$), processed by the *Gaia* Data Processing and Analysis Consortium (DPAC, $\rm 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 research has made use of NASA’s Astrophysics Data System Bibliographic Services; the SIMBAD database and the VizieR service, both operated at CDS, Strasbourg, France. This research has made use of the WEBDA database, operated at the Department of Theoretical Physics and Astrophysics of the Masaryk University, and the VALD database, operated at Uppsala University, the Institute of Astronomy RAS in Moscow, and the University of Vienna.
APPENDIX {#membership_model}
========
The Gaussian mixture model is of the form: $$\Phi(\mu_{x_i},\mu_{y_i}, \epsilon_{x_i},\epsilon_{y_i}) = \phi_c + \phi_f$$
where
$$\begin{aligned}
\phi_c &= \frac{1-N_f}{2\pi\sqrt{\sigma_{c,x}^2 + \epsilon_{x_i}^2}\sqrt{\sigma_{c,y}^2 + \epsilon_{y_i}^2}\sqrt{1-\rho_c^2}}exp[-\frac{\alpha}{2(1-\rho_c^2)}]\\
\\
\phi_f &= \frac{N_f}{2\pi\sqrt{\sigma_{f,x}^2 + \epsilon_{x_i}^2}\sqrt{\sigma_{f,y}^2 + \epsilon_{y_i}^2}\sqrt{1-\rho_f^2}}exp[-\frac{\beta}{2(1-\rho_f^2)}]\\\end{aligned}$$
and where $$\begin{aligned}
\alpha &= \frac{(\mu_{x_i} - \mu_{c,x})^2}{\sigma_{c,x}^2 + \epsilon_{x_i}^2} - \frac{2 \rho_c(\mu_{x_i} - \mu_{c,x})(\mu_{y_i} - \mu_{c,y})}{\sqrt{\sigma_{c,x}^2 + \epsilon_{x_i}^2}\sqrt{\sigma_{c,y}^2 + \epsilon_{y_i}^2}} + \frac{(\mu_{y_i} - \mu_{c,y})^2}{\sigma_{c,y}^2 + \epsilon_{y_i}^2}
\\
\beta &= \frac{(\mu_{x_i} - \mu_{f,x})^2}{\sigma_{f,x}^2 + \epsilon_{x_i}^2} - \frac{2 \rho_f(\mu_{x_i} - \mu_{f,x})(\mu_{y_i} - \mu_{f,y})}{\sqrt{\sigma_{f,x}^2 + \epsilon_{x_i}^2}\sqrt{\sigma_{f,y}^2 + \epsilon_{y_i}^2}} + \frac{(\mu_{y_i} - \mu_{f,y})^2}{\sigma_{f,y}^2 + \epsilon_{y_i}^2}\end{aligned}$$
The notation for the $Gaia$ DR2 proper motion data and the 11 model parameters is as follows: $$\begin{aligned}
\mu_{x_i}, \mu_{y_i} &= \text{proper motion components for $i^{th}$ star} \\
\epsilon_{x_i}, \epsilon_{y_i} &= \text{proper motion component errors for $i^{th}$ star} \\
N_f &= \text{field scaling parameter} \\
\mu_{c,x}, \mu_{c,y} &= \text{cluster center} \\
\mu_{f,x}, \mu_{f,y}& = \text{field center} \\
\sigma_{c,x}, \sigma_{c,y} &= \text{cluster std. deviations} \\
\sigma_{f,x}, \sigma_{f,y} &= \text{field std. deviations} \\
\rho_c, \rho_f &= \text{cluster and field correlation coefficients}\end{aligned}$$
While it is common to use ordinary maximum likelihood estimation to determine the parameters defining mixture models, as [@sanders71] did, we used an expectation-maximization (EM) machine-learning algorithm for finite mixtures as derived by [@dempster77]. We found that convergence of the model parameters using EM was more reliable than when applying MLE to our model. Central to the EM algorithm, the expectation of our complete-data log-likelihood function is of the form $$\begin{aligned}
\label{eq:q_function}
Q = \sum_{i=1}^{NST} T_{c_i}\log(\phi_{c_i}) + T_{f_i}\log(\phi_{f_i}) \textnormal{,}\end{aligned}$$
where $NST$ is the number of total stars in our data set, and $\phi_{c_i}$ and $\phi_{f_i}$ are simply $\phi_c$ and $\phi_f$ evaluated at the $i^{th}$ star using the current parameter guesses. $T_{c_i}$ and $T_{f_i}$ are the conditional probabilities that the $i^{th}$ star belongs to the cluster or field distribution, respectively. They are calculated with $T_{c_i} = \phi_{c_i}\mathbin{/}(\phi_{c_i} + \phi_{f_i})$ and $T_{f_i} = \phi_{f_i}\mathbin{/}(\phi_{c_i} + \phi_{f_i})$. In our EM algorithm, \[eq:q\_function\] was maximized with respect to each of the 11 parameters numerically and the probabilities, which feed into it, were in turn updated. This process was iterated until convergence of the parameters. While the conditional probabilities $T_{c_i}$ and $T_{f_i}$ changed during the process of running the EM algorithm, the final $T_{c_i}$ probabilities after parameter convergence were the probabilities that we used for cluster membership determination.
\[lastpage\]
[^1]: Contact e-mail: [email protected] (GBT); [email protected] (MA); [email protected] (CS); [email protected] (CAP); [email protected] (PAD); [email protected] (DAV); [email protected] (GNM); [email protected] (HK); [email protected] (KS); [email protected] (DTJ)
[^2]: <http://iraf.noao.edu/>
[^3]: See Paper I for a fairly detailed description of the evolutionary codes and stellar models that are used in the present series of papers — including, in particular, a discussion of the treatment of convective core overshooting during the main-sequence (MS) phase.
[^4]: The [@lum19] red giant abundance mean is \[N/Fe\]= 0.28 ($\sigma$ = 0.07), somewhat lower than our predicted and observed N values.
|
---
abstract: 'We systematically study the pump-wavelength dependence of terahertz pulse generation in thin-film spintronic THz emitters composed of a ferromagnetic Fe layer between adjacent nonmagnetic W and Pt layers. We find that the efficiency of THz generation is essentially flat for excitation by 150 fs pulses with center wavelengths ranging from 900 to 1500 nm, demonstrating that the spin current does not depend strongly on the pump photon energy. We show that the inclusion of dielectric overlayers of TiO~2~ and SiO~2~, designed for a particular excitation wavelength, can enhance the terahertz emission by a factor of of up to two in field.'
author:
- 'R.I. Herapath'
- 'S.M. Hornett'
- 'T.S. Seifert'
- 'G. Jakob'
- 'M. Kläui'
- 'J. Bertolotti'
- 'T. Kampfrath'
- 'E. Hendry'
bibliography:
- 'MyColl.bib'
nocite: '[@*]'
title: 'Impact of pump wavelength on terahertz emission of a cavity-enhanced spintronic trilayer'
---
![\[fig:schematic\]Schematic of a spintronic trilayer with added dielectric cavity, grown on 0.5 mm of sapphire (Al~2~O~3~). The near-infrared pump pulse, incident through the substrate, is partially absorbed in the metallic layers, launching a spin current from the ferromagnetic (FM) layer into the nonmagnetic (NM) layers. The inverse spin Hall effect converts this ultrashort out-of-plane spin current into an in-plane charge current resulting in the emission of THz radiation into the optical far-field. A weak in plane magnetic field (B) determines the magnetization direction, and the linear polarization of the emitted THz field.](schematic3.eps)
Terahertz (THz) radiation is non-ionizing and, therefore, safe for many applications, ranging from micro- and macroscopic imaging and spectroscopy to wireless communication[@Pawar2013; @Federici2005; @Sizov2010; @Dhillon2017]. However, the spectroscopically interesting THz frequency band near 1 THz is not easily accessible. Electronic sources such as oscillators can only provide high (milliwatt) output levels up to a few 100 GHz [@Zhang2010], while optical sources such as quantum cascade lasers are typically limited to frequencies >2 THz at room temperature [@Lewis2014a; @Dhillon2017]. To fill this “gap", considerable effort has been garnered towards sources capable of frequency mixing and optical rectification [@Burford2017], typically driven by femtosecond lasers.
To date, most THz emitting materials have been found to be insulators or semiconductors [@Lee2009]. Recently, THz emitters based on magnetic, metallic thin films have been demonstrated which emit THz radiation under illumination by femtosecond pulses[@Seifert2016a; @Huisman2017; @Yang2016Heterostructure; @Papaioannou; @Feng2018; @Torosyan2017; @Wu2017]. We here focus on trilayer thin-film emitters formed from a ferromagnetic (FM) layer between two non-ferromagnetic (NM) layers. A two-step process is thought to generate THz radiation:[@Kampfreth2013TerahertzHeterostructures] Upon excitation by the femtosecond pump pulse, an ultrashort out-of-plane spin current polarized along the FM magnetization is injected from the FM into the NM layers. Thereafter, the inverse spin Hall effect converts the laser-induced spin current into a transverse in-plane charge current within the NM layer which leads to the emission of a terahertz pulse into the optical far-field [@Kampfreth2013TerahertzHeterostructures; @Battiato2010; @Saitoh2006].
One of the most efficient films of this type[@Seifert2016a] comprises W, CoFeB and Pt layers. Importantly, Pt and W feature a spin Hall angle of opposite sign, resulting in a constructive superposition of the two charge currents in these layers. The result is an ultrabroadband THz emitter, capable of delivering pulses spanning 0.1 to 30 THz.[@Seifert2016a] With an active region only a few nanometers thick in total, these emitters can generate as much THz radiation as a phase-matched electrooptic crystal of millimeter thickness [@Seifert2016a]. Such highly efficient, but thin, THz emitters show much promise, particularly for near-field measurement or applications that benefit from the absence of phase matching.
Most studies of these THz emitters have been carried out using Ti:sapphire laser sources with wavelengths around 800 nm. However, many thin metal films show a rather wavelength-independent absorptance in the visible and near infrared, such that the THz-generation efficiency may naively be expected to be largely independent of the pump wavelength [@Seifert2016a]. Such wavelength-independent emission would be a great advantage of these types of emitters, allowing users free choice in excitation laser source.
In this paper, we investigate the pump-wavelength dependence of THz emission of W$|$CoFeB$|$Pt trilayer emitters using a continuously tuneable femtosecond laser source. We find that the efficiency of THz generation is surprisingly flat for excitation by 150 fs pulses with central wavelengths ranging from 900 to 1500 nm. This observation reveals that the photon energy has little effect on the number of electrons contributing to transport of spin polarization and that the key parameter is the total amount of energy deposited by the pump pulse. We demonstrate that the inclusion of dielectric overlayers (TiO~2~ and SiO~2~), forming a cavity with the substrate, can enhance emission by up to a factor of four in intensity in the frequency window from 0 to 2 THz.
[0.42]{} \[fig:mononrm\]
[0.43]{} \[fig:monodiele\]
\
[0.42]{} \[fig:tracenrm\]
[0.42]{} \[fig:tracediele\]
\
[0.41]{} \[fig:wavenrm\]
[0.42]{} \[fig:wavediele\]
\
A schematic of the samples is shown in Fig. \[fig:schematic\]. The emitters have each 10 mm by 7 mm in area, with layers of W, CoFeB and Pt deposited on a sapphire substrate by Ar sputtering using a Singulus Rotaris^©^ deposition system.[@Seifert2016a] The layers of W, CoFeB and Pt have a nominal thickness of 2 nm, 1.8 nm and 2 nm, respectively, giving a total thickness of 5.8 nm for the metallic layers. On six samples, an optical cavity made from alternating dielectric overlayers of TiO~2~ (thickness 113 nm, near-infrared index of refraction 2.265)[@Artimis] and SiO~2~ (185 nm, 1.455), with a total overlayer thickness of 1.495 $\mu$m, was deposited by plasma-assisted electron beam evaporation using a Leybold 1104 coating platform. Note that the overlayer thicknesses were optimized numerically to give a flat response at the target wavelength. The near-infrared transmission spectrum, normalized by the optical transmission of the sapphire substrate, was recorded for each sample using a monochromator (Oriel) and Quartz Tungsten-Halogen lamp (Thorlabs) (see Figs. \[fig:mononrm\] and \[fig:monodiele\]).
THz emission was investigated using wavelength-tunable near-infrared pump pulses (duration $\sim 150$ fs, spot diameter $\sim 4$ mm, fluence $\sim 0.1$ J m^-2^) incident normal to the sample surface. They were generated by an optical parametric amplifier (Light Conversion TOPAS) driven by an 800 nm Ti:Sapphire amplified laser ($\sim 100$ fs, repetition rate of 1 kHz). When varying the pump wavelength, we paid particular attention to keeping the other pump-pulse parameters, in particular energy, duration and focus diameter, constant. The resulting THz pulse, emitted in the forward direction, is detected through electrooptic sampling with 800 nm pulses from our amplified laser system using a 1 mm thick, (110)-oriented ZnTe crystal. For all samples, a neodymium magnet attached to the emitter holder approximately 7 mm from the edge of the film gives an in-plane magnetic field of $\sim 13$ mT, as depicted in Fig. \[fig:schematic\]. This determines the linear polarization of the emitted THz field which is always oriented perpendicularly to the magnetic field.
Figure \[fig:tracenrm\] shows a typical electrooptic signal waveform of a THz pulse emitted from the bare spintronic trilayer after excitation by a pump pulse with a center wavelength of 1040 nm. The signal amplitude grows approximately linearly for incident pump fluences at least up to 0.4 J m^-2^ (see Supplementary Fig. \[fig:powdep\]).
In Fig. \[fig:wavenrm\], we plot the integrated THz field emitted by a W$|$CoFeB$|$Pt trilayer sample for various pump wavelengths in the range from 900 to 1500 nm. Remarkably, over this range, the efficiency of THz generation is wavelength-independent. Importantly, since they originate from the same source, the focus diameter, energy and duration of the pump pulses are fairly independent of the wavelength in the range 1000 to 1300 nm. For example, the pulse duration varies by less than 12 fs across this entire range (see Supplementary Fig. \[fig:autocorr\] and Supplementary Section \[sec:length\]). An unchanged THz signal amplitude has also been reported[@Papaioannou] for the two significantly shorter wavelengths of 800 nm and 400 nm. Our results and those of Ref. [@Papaioannou] indicate that the spin current arises from the hot electrons induced by the pump pulse. Details of the involved optical transitions are insignificant. Therefore, the key parameter is the amount of energy that is deposited by the pump pulse in the electronic system.
Since the THz emission from the spintronic emitters is largely independent of pump wavelength, one can in principle enhance the emission for any particular wavelength by designing a suitable adjacent cavity. A straightforward implementation are dielectric overlayers similar to a Bragg mirror, forming a broadband dielectric cavity with the substrate. By placing the lower index material, in this case SiO~2~, next to the trilayer, an intensity maximum is formed within the active THz generation layers. Doing so, for a particular target pump wavelength, pump absorption in the metallic trilayer will be maximized, and so will the emitted THz radiation. For convenience, we choose a target pump wavelength of 1040 nm aimed at common fibre lasers, specified by the thickness and type of the dielectric layers. To achieve a stop-band with less than 5% transmission, a total of 10 dielectric layers are required. For these coated samples, the total thickness (neglecting the substrate) is 1.495 $\mu$m, corresponding to an optical thickness for THz wavelengths of approximately 9 $\mu$m, which is still very subwavelength for THz radiation.
To model the behavior of the emitter with dielectric cavity, a standard modal matching approach [@Tomas1995] is used which can also treat a source within the active region (see Supplementary Section \[sec:modal\]). In Fig. \[fig:monodiele\], we plot the calculated near-infrared transmission spectrum for the samples with dielectric overlayers (green dot-dash line). We expect a transmission minimum (Fig. \[fig:monodiele\]) and an absorptance maximum, (see Supplementary Section \[sec:modal\]) at the target wavelength of 1040 nm, with a bandwidth of around 300 nm.
In Fig. \[fig:monodiele\], the blue-shaded region represents the distribution of transmittance spectra measured for the six samples with overlayers. While the transmittance (Fig. \[fig:monodiele\]) and absorptance (Supplementary Fig. \[fig:transabs\]) of the sample without the dielectric overlayers are fairly flat over the measured range, all samples with dielectric overlayers show clear band-stop behavior. The minimum transmittance in the measurement, $\sim 3$%, is consistent with our modeling. The slight shift in wavelength of the band-stop region between the modeling and experiment likely arises from variation in the refractive indices and/or thicknesses of the dielectric layers. For example, the optical thickness needs to change by only 3.8% with respect to specified values to account for this discrepancy. The band-stop in the region of the target wavelength designates a region where we expect to see enhancement of THz emission.
In Fig. \[fig:wavediele\], we can observe the enhancement in THz emission generated by the dielectric overlayers. For an identical pump fluence, in the band-stop window (900 to 1200 nm), we find a THz field that is a factor of $\sim 2$ (intensity of a factor $\sim 4$) larger compared to the trilayer without overlayers. This is slightly larger than the 70% enhancement in field recently reported for a cavity formed from spintronic layers separated by dielectric spacers. [@Feng2018] We also observe reduction in THz emission near the near-infrared transmittance maxima at 1300 nm. In these bandpass regions, the dielectric overlayers act to reduce the near-infrared intensity in the active magnetic layer. For the band-stop region, the factor of two enhancement in THz field emission is observed for the entire fluence range investigated here (Supplementary Fig. \[fig:powdep\]).
In Fig. \[fig:THzDep\], we plot the spectral dependence of this observed THz emission enhancement for excitation by a 1040 nm pump pulse (blue dots). The highest enhancement is observed for the lowest THz frequencies. Above 1 THz, we begin to see a slight drop in the field enhancement factor (limited to 2 THz by the bandwidth of our detector). Also shown is the prediction from modeling (red line): this is calculated from the modal modeling method using the THz field emitted with and without the dielectric cavity, multiplied by the predicted pump intensity in the active magnetic layer. Our modeling predicts a similar behavior, but with an enhancement factor which decreases more quickly with frequency, an effect which arises due to the frequency dependent absorption of TiO~2~. In our model, we also observe high sensitivity to the precise index of the Pt layer, a material which shows high variation dependent on morphology. [@Kovalenko2001ThicknessSemiconductors] It is important to note that alternative dielectrics to TiO~2~ may well exhibit lower absorption losses, and could increase the operational bandwidth of cavity enhanced emitters.
To conclude, we systematically study the pump-wavelength dependence of THz emission of spintronic W$|$CoFeB$|$Pt trilayers. We find that the efficiency of THz generation is essentially flat for excitation by 150 fs pulses with central wavelengths ranging from 900 to 1500 nm, indicating that the spin current is largely independent of the pump-photon energy. We demonstrate that the inclusion of dielectric overlayers of TiO~2~ and SiO~2~, designed for a particular excitation wavelength, can enhance emission by up to a factor of two in field amplitude. The four-fold enhancement in emitted THz intensity could be further improved using cavities with higher quality factors, matched to the bandwidth of the pump pulse.
![\[fig:THzDep\] Measured THz emission enhancement for different spectral components, plotting the THz field amplitude emitted from a spintronic trilayer with the dielectric cavity divided by field emitted without cavity, when excited by a pump pulse of 1040 nm. The solid line is a calculation from modal matching theory.](THzDepv8)
The authors like to acknowledge support via the EPSRC Centre for Doctoral Training in Metamaterials (Grant No. EP/L015331/1). EH acknowledges support from EPSRC fellowship (EP/K041215/1). TK, TSS, MK and GJ acknowledge the German Research Foundation for funding through the collaborative research centers SFB TRR 227 Ultrafast spin dynamics (project B02) and SFB TRR 173 Spin+X as well as the Graduate School of Excellence Materials Science in Mainz (MAINZ, GSC 266). TK also acknowledges funding through the ERC H2020 CoG project TERAMAG/Grant No. 681917.
Supplementary Material
======================
![\[fig:powdep\] THz emission as function of pump fluence, measured with a pump wavelength of 1000 nm. Both films exhibit linear dependences over this range of fluence, with the dielectric cavity enhancing the THz field emitted. Solid lines are linear fits.](PowDepv3)
[0.4]{} \[fig:detresp1\]
[0.4]{} \[fig:detresp2\]
\
[0.42]{} \[fig:mononrmsup\]
[0.434]{} \[fig:monodielesup\]
\
[0.42]{} \[fig:absnrm\]
[0.42]{} \[fig:absdiele\]
\
\[sec:powdep\]Power dependence of THz emission
----------------------------------------------
We examine the power dependence of THz emission from the spintronic trilayer with and without the dielectric cavity attached. As seen in Fig. \[fig:powdep\], the THz signal amplitude grows linearly with respect to the pump fluence for both samples (with and without cavity) over the entire range investigated here. Non-reversible saturation was observed for pump fluences $\gg 1$ J cm^-2^, possibly due to heating of the sample above the Curie temperature of the FM layer and ablation of the metal films .[@Seifert2016a; @Yang2016Heterostructure]
\[sec:length\]Pump-pulse duration vs wavelength
-----------------------------------------------
We determine the length of the pulses generated by our optical parametric amplifier as a function of wavelength, 1000 to 1300 nm, using an autocorrelator (FR-103MC from Femtochrome). Autocorrelation curves are fitted with a Gaussian, yielding the pulse duration (full width at half intensity maximum). In Fig. \[fig:autocorr\], we see that the variation in pulse lengths across our measureable wavelength range is negligible and smaller than the noise in the measurement. Error bars represent variations in measurement, most likely resulting from environmental factors, including humidity and laser fluctuations.
\[sec:detresp\]Detector response function
-----------------------------------------
In Fig.\[fig:detresp\], we plot the detector response function $h(t)$ of our THz electrooptic detection system (1 mm thick, (110)-ZnTe crystal and 100 fs, 800 nm sampling pulses) calculated using Kampfrath et al.[@Kampfrath2007SamplingCrystals] It connects the THz electric field incident $E_{\textnormal{inc}}(t)$ onto the electrooptic crystal with the electrooptic signal $S(t)$ by the convolution $S=h*E_{\textnormal{inc}}$. Since $h(t)$ is much wider than the pump pulses ($\sim 150$ fs), we expect that the small variations of the pump-pulse duration present in Fig. \[fig:autocorr\] are invisible to the electrooptic detection.
\[sec:modal\]Modal matching calculations
----------------------------------------
The layers are considered homogeneous in the $x$-$y$ plane, where the radiation propagates along the $z$-axis. We define the in-plane component of the electric field in the semi-infinite vacuum region as a sum of forward and backward propagating plane waves of frequency [$\omega$]{} given by
$$\label{eqn:appa}
A_i \textnormal{e}^{\textnormal{i}n_i\omega z/c} + B_i \textnormal{e}^{-\textnormal{i}n_i\omega z/c},$$
where $A_i$ and $B_i$ are the amplitudes of the forward- and backward-propagating fields in the $i$-th layer, with refractive index $n_i$, and $c$ being the vacuum speed of light. Note that the forward- and backward-propagating amplitudes in the reflection and transmission regions are zero, respectively. Using Maxwell’s equations, one can then find a similar expression for the magnetic field: $$\label{eqn:appa2}
\frac{\omega}{c}A_in_i \textnormal{e}^{\textnormal{i}{n_i\omega z}/{c}} - \frac{\omega}{c}B_in_i \textnormal{e}^{-\textnormal{i}{n_i\omega z}/{c}}.$$ Then, by applying field continuity boundary condition at the interfaces between all the layers, one obtains a set of simultaneous equations that can be solved for the sets of unknowns $A_i$ and $B_i$, and, therefore, for the field amplitudes in each of the regions. We use this model to calculate 1) the near-infrared transmission of our multilayer stack (by including a source electric field of unit amplitude in the incident region), and 2) the normalized THz emission of the multilayer stack (by including a unitary source field in the magnetic layer of the stack), both normalized by the field within the sapphire substrate. Literature values for the metallic[@Seifert2016a] and dielectric[@Grischkowsky1990; @Dang2014ElectricalDeposition] THz refractive indices are used. The optical frequency parameters for TiO~2~ and SiO~2~ were provided by our commercial fabrication partners [@Artimis].
In Fig. \[fig:transabs\], we plot the calculation absorption spectra for our samples without (bold red line) and with (bold blue line) dielectric cavity. Without cavity, the absorption is fairly wavelength-independent. With the cavity, absorption is increased by around a factor of two in the band-stop region of the spectrum. In this region, we also see a commensurate increase in the the THz field emitted from samples with the cavity, as seen in Fig. \[fig:wavediele\].
|
---
abstract: 'We explore the quantum scattering of systems classically described by binary and other low order Smale horseshoes, in a stage of development where the stable island associated with the inner periodic orbit is large, but chaos around this island is well developed. For short incoming pulses we find periodic echoes modulating an exponential decay over many periods. The period is directly related to the development stage of the horseshoe. We exemplify our studies with a one-dimensional system periodically kicked in time and we mention possible experiments.'
author:
- 'C. Jung'
- 'C. Mejia-Monasterio'
- 'T.H. Seligman'
title: Quantum and classical echoes in scattering systems described by simple Smale horseshoes
---
In classical mechanics the Smale horseshoe [@smale] construction has proven to be the key point to the understanding of chaotic scattering in time independent systems with two degrees of freedom and time dependent ones with one degree of freedom [@jung-1; @rueckerl]. Though the importance of this construction in the quantum analogue of such systems has been noticed occasionally [@borondo], a study of the implications in quantum systems has not yet been undertaken. We are interested in low-order (such as binary or ternary) horseshoes, where the features encountered are comparatively simple. In the present paper we shall concentrate on situations in which the stage of development of the horseshoe is fairly low. We will discuss a binary horseshoe for which one of the fundamental periodic orbits is hyperbolic and shows homoclinic connections, while the other one is still elliptic and confined inside a large stable island. In such a situation tunneling into the island will be the most notable quantum effect. We shall show that a short pulse as incoming wave leads to periodic pulses in the outgoing wave. They survive many periods and we shall call them echoes. If we use good energy resolution instead we find narrow resonances.
We will focus our discussion on a one-dimensional kicked scattering model mainly because of the ease of calculation, but experiments with similar periodically driven models may be of interest [@raizen]. Yet the effect is not confined to such models. Two-dimensional time-independent billiards with two openings or leads can produce ternary horseshoes with a large central island, whose echoes may be seen in microwave experiments [@richter] or mesoscopic systems.
The model we use is given in terms of the Hamiltonian $$\label{hamiltonian}
H(q,p,t) = \frac{p^2}{2} + A \ V(q) \sum_{n=-\infty}^{\infty} \delta (t -
n) \ .$$ The time dependence is an infinite periodic train of delta pulses kicking the potential with period $1$. The parameter $A$ determines the strength of the potential, which is given by $$\label{potential}
V(q) = \left\{
\begin{array}{ccl}
\frac{\textstyle q^2}{\textstyle 2} + 1 &,& q < 0 \\
\\
e^{-q}(q^2 + q + 1) &,& q \geq 0 \\
\end{array}
\right..$$ Note that in the literature [@rueckerl; @gaspard] a similar but simpler potential is used, where the exponential form extends to negative values of $q$. We have changed the form of the potential to obtain better convergence of the quantum calculation. Yet the classical results for the two potentials are quite similar. We obtain a smooth development of the binary horseshoe as a function of the parameter $A$. The only difference is that for the present potential the development is not entirely monotonic [@thesis], but this does not affect our considerations.
We represent the classical dynamics by a Poincaré map which we choose as the stroboscopic map taken at times $t = n + 1/2$
$$\label {stroboscopic-map}
\begin{array}{lcl}
p_{n+1} & = & p_n - AV'(q_n+p_n/2) \\
q_{n+1} & = & q_n + p_n - A V'(q_n+p_n/2)/2 \ .
\end{array}$$
It gives us the evolution of a classical trajectory from time $n+1/2$ to time $n+3/2$.
The kicked potential has a maximum at $q=1$ and a minimum at $
q=0$. It is quite obvious that the points $p=0,\, q=0$ and $q=1,\,
p=0$ are fixed points of this map. Indeed they represent the fundamental periodic orbits, that determine the construction of the binary horseshoe, which describes the topology of the Hamiltonian flow of our system. The fixed point at $q=1$ is obviously hyperbolic, while the other one can vary according to the strength parameter $A$. For $A<4$ this point is elliptic. At this value it turns inverse hyperbolic, but many secondary islands of stability survive. The phase portrait is hyperbolic when the horseshoe becomes complete near $A=6.25$. Hyperbolic stages are also possible for smaller values of $A$ for which the horseshoe is incomplete [@rueckerl; @davis; @troll].
We shall focus our attention on the low development stages long before the original elliptic orbit bifurcates. On the other hand we wish to see a well developed chaotic region. These conditions are met for values of $A$ between 0.5 and 3. We shall use the parameter $A=0.967$. A phase portrait of the stroboscopic map for this value is shown in Fig. \[fig-1\]. This value of $A$ was chosen because somewhere between this value and $A=1$, the outermost KAM surface shown in Fig. \[fig-1\] disintegrates.
We are interested in learning about the quantum properties of such a low order horseshoe with a large stable island. For this purpose we use the unitary time evolution operator, which is rather easily obtained for kicked systems. This is the case because the stroboscopic map eq. (\[stroboscopic-map\]) can be decomposed into three transformations as follows:
$$\begin{aligned}
\label{step1}
p_{n'} & = & p_n \nonumber\\
q_{n'} & = & q_n + p_n/2 \\
% & & \nonumber\\
\label {step2}
p_{n''} & = & p_{n'} - AV'(q_{n'}) \nonumber\\
q_{n''} & = & q_{n'} \\
% & & \nonumber\\
\label {step3}
p_{n+1} & = & p_{n''} \nonumber\\
q_{n+1} & = & q_{n''} + p_{n''}/2 \ .\end{aligned}$$
The second step (\[step2\]), can be interpreted as a gauge transformation in coordinate space and the other two (\[step1\], \[step3\]), as a gauge transformations in momentum space. This implies, that the unitary operator for one time step can be written as three phases intertwined by Fourier transforms $\mathcal{F}$, which take us from coordinate to momentum space and back. Thus we obtain for the kernel of this operator in momentum-space $$\label {u-time-evolution} U (p',p) = \exp\Big[{-\frac{i}{4\hbar} \,
{p^2}}\Big] \ \mathcal{F} \ \exp \Big[\frac{i}{\hbar} \, AV(q)\Big] \
\mathcal{F}^{-1} \, \exp\Big[{-\frac{i}{4\hbar} \, {p^2}}\Big] \ .$$ This expression is simple as it involves only Fourier transforms and multiplication with phases. It is also very efficient if good fast Fourier transform (FFT) codes are used.
We shall analyze our scattering system in terms of wave packet dynamics. In all our simulations we use minimum uncertainty Gaussian wave packets given by
$$\label {wave-packet}
\Psi(q,0) = \frac{1}{\pi^{1/4}\sigma^{1/2}}
\exp \Big[{-\frac{(q-q_{\rm in})^2}{2\sigma^2} + \frac{i}{\hbar}p_{\rm in}
q}\Big] \ .$$
For a given value of the initial momentum $p_{\rm in}$, $\sigma$ will determine the duration of the pulse and the value of $\hbar$ will determine how near to the classical limit we operate. Recall that short pulses imply a poor energy resolution, while long ones can have very well defined energies. We can therefore use short pulses and consider the time evolution in configuration space or in phase space, or we can use long pulses and look at the energy dependence of some outgoing quantity. We shall start with the former and then consider the latter.
First we show in Fig. \[fig-2\] the Husimi distributions as a function of time superposed to the phase portrait shown in Fig. \[fig-1\]; the Husimi function is indicated by a colour-scale code. We choose the strength parameter $A=0.967$ and the wave packet with $\sigma = 2.5$, $\hbar = 0.01$ and $(q_{\rm in}=100,p_{\rm
in}=-1.48)$. At time $t=68$, the packet reaches the interaction region, Fig. \[fig-2\]-a. For $t=73$, Fig. \[fig-2\]-b, the packet has entered the potential well near the external fixed point; part of the packet bounces of the barrier and never enters the well. In Fig.\[fig-2\]-c, the probability that enters the potential well performs its evolution along the chaotic layer of the phase portrait at $t=78$. For $t=86$ most of the packet has left the well. The small remaining probability gets trapped in the potential well where it has tunneled through the surface to inner stable regions, Fig. \[fig-2\]-d. It is interesting to note that for $A=1$, when the KAM-torus has became a classically penetrable cantorus the quantum picture remains unchanged as diffusion is much slower than tunneling. For larger values of $A$, the Husimi distribution will move inwards to be again enclosed by the outermost KAM surface.
Next we show in Fig. \[fig-3\] the probability distribution in configuration space as a function of time in a logarithmic colour-scale code. The parameters and initial conditions are as above. We clearly distinguish the incoming pulse, the part directly scattered at the barrier and another part that entered the well, but leave directly. This direct scattering corresponds to the one shown in Fig. \[fig-2\]. At longer times we see an oscillating packet inside the well with an amplitude that corresponds to the classically forbidden region in phase space, as shown in Fig. \[fig-1\]. Each time the wave packet returns to the front side ([*i.e*]{} the largest value of $q$) of the well it sends an “echo” to infinity. The time intervals between these echoes corresponds to the oscillation times of the packet inside the stable island which in turn corresponds to the average classical winding time of the invariant surfaces in the region of phase space where the packet oscillates.
Note that the winding time near the surface of the island is a generic feature related directly to the degree of development of the horseshoe as measured by the formal parameter $\alpha$ [@thesis]. The period of rotation $T$ at the surface is given by$$T=n+3/2\ \ \ \ {\rm if} \ \ \ \ \alpha \propto 2^{-n} \ ,$$ with $n$ integer. The derivation of this result will be presented elsewhere [@mejia]. For $A=0.967$ we have $T \approx 10.5$ and thus the period of rotation we see (Fig. \[fig-3\]), is smaller than this estimate. This shortening of period is not surprising and confirms that we really see a tunneling into the stable island. As we expect, the winding time decreases as we approach the center of the island, (see Fig. \[fig-1\]).
We evaluate the total intensity inside the potential well as a function of time $I_w(t)$. In Fig. \[fig-4\] we see that the decay is oscillatory (inset), with near constant period, but with exponentially diminishing envelope.
We may suspect that the echoes we see are a pure quantum phenomenon, because of the tunneling displayed by the wave packets. Interestingly this is not entirely true. The whole sticky fractal structure of ever smaller islands rotates with the outer invariant surfaces of the main island. Therefore, a small fraction of the intensity of a packet of classically scattered particles will have a similar behaviour, though three differences are notable: First the period is equal to the one predicted at the surface and therefore, larger than the quantum period, second the wave packet spreads more rapidly and third the decay beyond the oscillations is governed by a power-law; the staying probability decays with a power of roughly $2.554 \pm 0.005$ as is expected for a mixed phase space [@karney; @chirikov]. In Fig. \[fig-5\] we plot the intensity of a packet of scattered classical particles. Note that the amplitude of the oscillation clearly indicates that it takes place in the sticky region.
Returning to quantum mechanics we can also use long pulses with high energy resolution to analyze the same phenomenon in the energy domain. In Fig. \[fig-6\] we plot the total intensity remaining in the potential well at time $t = 450$ as a function of the incoming energy for a strength parameter $A=2$. We clearly see the two periods characteristic of our problem, namely the period of the pulsed system and the one corresponding to the echoes which now is shorter as the island is smaller. We also calculated the $S$-matrix as a function of the quasienergy. It shows the same resonances we see in Fig. \[fig-6\] but, a good resolution is harder to obtain. The presence of these sharp resonances is an additional indication that tunneling between the regular and chaotic regions of the classical phase portrait occurs [@seba].
We have proposed the possibility to explore scattering systems corresponding to binary and other low order horseshoes with wave and classical scattering experiments, and find characteristic phenomena, which we call echoes, if we use short pulses as input. From a theoretical point of view it is interesting that experiments performed along these lines are sensitive to the degree of development of the corresponding horseshoe, giving us a powerful tool to explore low developed horseshoes, where it is very hard, if not impossible, to obtain the symbolic dynamics as pruning sets in at a very low level [@rueckerl].
From a practical point of view experiments with similar time dependent Hamiltonians are feasible in scattering of atoms on surfaces [@raizen] in the classical or almost classical domain, whereas the corresponding experiments with electromagnetic waves seem quite feasible with ternary horseshoes generated in appropriate cavities [@richter]. Similar experiments can be performed on mesoscopic scale.
We acknowledge useful discussions with F. Leyvraz, P. Seba and H.-J. Stöckmann. Financial support by DGAPA-UNAM, project IN109000 and by CONACyT, project 25-192-E is acknowledged. C.M. acknowledges a fellowship by DGEP-UNAM.
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---
address: 'CERN, PH Department CH-1211 Genève 23, Switzerland'
author:
- 'F. BEAUDETTE'
title: Top physics prospects at LHC
---
(80,100) (-40,-210)
Top quark pair production
=========================
A top factory
-------------
Whereas the centre-of-mass energy of the collisions at the LHC is seven times higher than at the TeVatron, the production cross section of top quark pairs is about hundred times larger. It reaches [@mangano] $\sigma_{\ttb} \approx 840$pb$(1\pm5\%_{\rm scale}\pm3\%_{\rm PDF})$ where the error terms represent the systematic uncertainties related to the choice of the renormalization and factorization scales and to the proton function structures respectively. At $\sqrt{s}=14$TeV, the top quarks are mostly produced by gluon fusion (90%). The quark annihilation, dominant at the TeVatron, amounts to only 10% of the top quark pair production.
In the Standard Model (SM), the top quark always decays into a $\rm W$ and a $\rm b$ quark. As a result, the topology of the final state is mostly driven by the decay channels of the $\rm W$’s. The events where one of the $\rm W$’s decays into a lepton [^1] have a clear signature: one isolated lepton, missing transverse energy from the undetected neutrino and at least four jets of which two b jets. At “low” luminosity, $L=10^{33}\,{\rm cm}^{-2}{\rm s}^{-1}$, there will be such so-called “lepton+jet” event every 4s while one top quark pair will be produced every second. The LHC will thus be a “top factory”.
Top observation and cross section measurement
---------------------------------------------
The production cross section is so large that the top signal will be visible after the equivalent of one week of data taking at low luminosity [@mangianotti] in the lepton+jet channel. By requiring one isolated lepton with a transverse momentum , and exactly four jets with high transverse energy , the top signal is clearly visible above the $\rm W+4$ jets background in the invariant mass distribution of the most energetic three jets (Fig. \[pallin\]).
![Invariant mass of the three most energetic jets in the selected sample with 150pb$^{-1}$ of integrated luminosity, as obtained from a fast simulation of the ATLAS detector. The dashed blue line shows the W+4 jets background and the dots with error bars represent the signal plus background expectations.[]{data-label="pallin"}](top_initial.eps){width="8cm"}
The cross section can thus be measured. With the large number of events collected, the statistical error will soon be negligible. After “one month” at low luminosity, it will be at the level of 0.4%. The overall error will be dominated by the systematic uncertainty related to the luminosity measurement. A 5% uncertainty is achievable [@yellowbook]. Because of its strong dependence on the top mass [@mangano], a measurement of the production cross section together with a precise measurement of the top mass will provide a test of QCD. Alternatively, within the SM, the cross section measurement provides a mass estimate, with a potential accuracy of 3GeV/$c^{2}$ precision can be reached. A direct measurement can, however, be done with a better precision.
The top mass measurement
========================
Why measuring (precisely) the top mass ?
----------------------------------------
Because of its mass, the top plays a particular rôle in the electroweak sector. In the SM, the W and Z boson masses are connected through the relation [@yellowbook] $m^2_{\rm W}(1-\frac{m^2_{\rm W}}{m^2_{\rm Z}}) = \frac{\pi\alpha}{\sqrt{2}G_\mu}\frac{1}{1-\Delta r}$, where $G_\mu$ is the Fermi constant and $\Delta r$ contains the one-loop corrections. The top mass arises in $\Delta r$ via the loops in the W and Z boson propagators [@willenbrock] and gives rise to terms proportional to $m^2_{\rm t}/m^2_{\rm Z}$. Similarly, the Higgs boson loops give terms proportional to $\log{m_{\rm H}/m_{\rm Z}}$. The relationship thus obtained between the Higgs boson and top quark masses is currently used as an indirect prediction of the Higgs boson mass [@lepeww]: for . The allowed region in the plane for different Higgs boson masses is displayed in Fig. \[ew\] as well as the direct and direct measurements of $m_{\rm W}$ and $m_{\rm t}$.
![Direct and indirect measurements of the $\rm W$ boson and top quark masses with lines of constant Higgs boson masses in the Standard Model.[]{data-label="ew"}](w05_mt_mw_contours.epsi){width="7.5cm"}
At LHC, a direct measurement of the Higgs boson mass will be carried out towards a consistency test of the SM by checking the relation between $m_{\rm t}$, $m_{\rm W}$ and $m_{\rm H}$. To ensure a similar accuracy in the combination, the precision on $m_{\rm t}$ and $m_{\rm W}$ must fulfil [@willenbrock2] $\Delta m_{\rm t} \approx 0.7\times 10^{-2} \Delta m_{\rm W}$ corresponding to the slope of the constant Higgs boson mass lines in Fig. \[ew\]. As can be seen in Table \[table1\], a 2GeV/$c^2$ precision on $m_{\rm t}$ will allow a consistency check of the SM with similar relevance as a 15MeV$/c^{2}$ accuracy on $m_{\rm W}$, reachable at LHC. The LHC, however, can even do better and achieve a 1GeV/$c^2$ on $m_{\rm t}$ as described in the following. A higher precision might be needed in case of new physics discovery. Such an accuracy would be obtained with an $\rm e^+ e^-$ linear collider [@lincol].
-------------------- -------------------- ------------------------------------- --------------------
Expected precision $\Delta m_{\rm W}$ $\Delta m_{\rm W}/0.7\times10^{-2}$ $\Delta m_{\rm t}$
(MeV/$c^2$) (GeV/$c^2$) (GeV$/c^2$)
TeVatron 25 4 3
LHC 15 2 1
LC 6 1 0.1
-------------------- -------------------- ------------------------------------- --------------------
: Expected precisions on the W boson and top quark masses at present and future colliders.
\[table1\]
Measurement in the lepton+jet channel
-------------------------------------
The lepton+jet channel is the golden channel for the top mass measurement. Indeed, the leptonic-decaying W boson allows the top events to be efficiently triggered and selected. After the selection of the events with an energetic isolated lepton ($p_T>20$GeV$/c$) and a missing transverse energy in excess of 20GeV, the characteristics of the $\ttb$ events are then used to improve the purity of the sample. The events must contain at least four energetic jets ($E_T>20$GeV) of which two identified b jets. The $\bbb$+jets, W+jets and Z+jets backgrounds are highly suppressed by this event selection [@atlassummary].
The top mass is reconstructed from the two light jets from the W decay and the b jet from the top decay. As a result, the jet energy scale and angular resolutions are crucial. The non b-jet pair minimizing the $(M_{\rm jj}-m_{\rm W})^2$ difference, where $M_{\rm jj}$ is the invariant mass of the two jets, is assumed to originate from the hadronically decaying W. A difference smaller than 20GeV$/c^2$ is required. It is finally combined with the b jet giving the highest reconstructed top transverse momentum.
The cone algorithm used to reconstruct the jets tends to underestimate the opening angle between the two jets from the W [@roy]. An in-situ calibration can however be applied to correct the jet energies and directions.
The distribution of the three-jet invariant mass is displayed in Fig. \[finaltopmass\]. The reconstructed top quark mass is deduced from the fit value of the peak. The combinatorial background is dominant. With an integrated luminosity of , the statistical uncertainty on the top mass is at the level of .
![Distribution of the jjb invariant mass of the selected events as obtained from a fast simulation of the ATLAS detector for a 10fb$^{-1}$ integrated luminosity. The shaded area represents the combinatorial background.[]{data-label="finaltopmass"}](finalMjjb.epsi){width="7cm"}
The systematic uncertainties are summarized in Ref. [@atlassummary]. The main two sources of systematic uncertainty are the final state radiation (FSR) and the b-jet energy scale. The FSR systematic error is conservatively evaluated as 20% of the shift in the fit top mass when disabling the FSR and amounts to 1GeV$/c^2$. At LHC, the light and b-jet energy scales are expected to be determined with a 1% precision [@atlastdr]. In this analysis, the b-jet energy scale systematic uncertainty is 0.7GeV$/c^2$ whereas the light-jet energy scale uncertainty is mostly canceled by the in-situ calibration and amounts to 0.2GeV$/c^2$. Altogether a 1.3GeV$/c^2$ error on the top mass is achievable. The effect of the FSR can be lowered down to 0.5GeV$/c^2$ if a kinematic fit is implemented. Indeed, the events with large FSR tend to have a high $\chi^2$ and can be removed from the analysis. The systematic uncertainty thus becomes 0.9GeV$/c^2$, dominated by the b-jet energy scale determination. As explained in the next section, it is possible to get rid of the heavy-flavour-jet-energy-scale related uncertainty.
Measurement in leptonic final state with $\jp$
----------------------------------------------
A determination of the top mass quark mass can be carried out in the lepton+jet events where a $\jp$ arises from the b quark associated to the leptonic decaying W (Fig. \[jpsichannel\]). The top quark is partially reconstructed from the isolated lepton coming from the W and corresponding b quark [@cmsjpsi].
[>m[7.5cm]{} >m[5.5cm]{} ]{} ![Diagram of the top decay to leptonic final state with $\jp$(left). Example of lepton-$\jp$ invariant mass in the four-lepton final state as obtained from a fast simulation of the CMS detector after four years at high LHC luminosity (right).[]{data-label="jpsichannel"}](graphe2.eps "fig:"){width="7.3cm"} & ![Diagram of the top decay to leptonic final state with $\jp$(left). Example of lepton-$\jp$ invariant mass in the four-lepton final state as obtained from a fast simulation of the CMS detector after four years at high LHC luminosity (right).[]{data-label="jpsichannel"}](sbac.eps "fig:"){width="5.3cm"}
To solve the twofold ambiguities on the b quark origin, a flavour identification, requiring a muon of the same electric charge as the isolated lepton, is applied. The $\jp$ can be precisely identified and reconstructed when it decays into a muon pair. As a result, one isolated lepton and three non isolated muons are required, two of them being consistent with the $\jp$. This configuration is very seldom: one thousand events per year will be collected at high luminosity ($L=10^{34}\,{\rm cm}^{-2}{\rm s}^{-1}$). The isolated lepton-$\jp$ invariant mass is determined (Fig. \[jpsichannel\]) and the fit value of the peak turns out to depend linearly on the generated top mass [@cmsjpsi] (Fig. \[jpsichannel2\]).
The background is essentially combinatorial, and its shape can be extracted from the data. The main systematic uncertainty comes from the b-quark fragmentation and is the combination of the uncertainty on the b hadron spectrum in top decays and that of the $\jp$ spectrum in b hadron decays. The B factories can help in the determination of the latter. An overall error on the top mass of the order of 1GeV$/c^2$ can be achieved.
![Correlation between the fit value of the peak of the lepton-$\jp$ invariant mass distribution ($\rm M^{\rm max}$) and the generated top mass as obtained from a fast simulation of the CMS detector.[]{data-label="jpsichannel2"}](fitl2.eps){width="7cm"}
Search for single top
=====================
The top quark can also be produced by electroweak interaction. In this case, one single top is produced at a time. The total production cross section reaches is 310pb. The production diagrams at tree level are displayed in Fig. \[singletop2\]. The dominant process is the W-gluon fusion with a cross section of about 250pb. In the $t$ and $s$ channels, the production rate of top quarks is about 50% higher than anti-tops [@lukas] , while at the TeVatron, they are identical. The associate production cross section is about 50pb and is one of the dominant backgrounds to the search for the Higgs boson in the $\rm H\rightarrow WW^* \rightarrow$$\ell\nu \ell\nu$ channel [@atlastdr].
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Electroweak top production diagrams in the $t$ and $s$ channels (left) and associate production (right) []{data-label="singletop2"}](singletopfig1.eps "fig:"){width="6cm"} ![Electroweak top production diagrams in the $t$ and $s$ channels (left) and associate production (right) []{data-label="singletop2"}](singletopfig3.eps "fig:"){width="6cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Spectrum of the Wb invariant mass for the selected events obtained with a CMS full simulation. The open histogram represents the signal plus background expectations, the shaded histograms shows the background considered: $\ttb$, W+2 jets and W+3 jets.[]{data-label="finalsingletop"}](singletopfig19.eps){width="7cm"}
The event preselection, as in the lepton+jet channels, requires a leptonically decaying W. The different event topologies need dedicated final selections. The $t$ channel is taken here as an example. The full analysis is described in Ref [@singletop]. The b jet from the initial gluon splitting is lost in the beam pipe. The events with one forward non b-tagged jet and one central b-tagged jet (coming from the top) are selected. The Wb invariant mass is then computed, the twofold ambiguity on the neutrino longitudinal momentum is solved by choosing the smallest one. This is true in only 55% of the cases. The result is shown in Fig. \[finalsingletop\]. The overall efficiency (including the W to lepton branching ratio) is 0.3%. More than 6000 events are expected in 10fb$^{-1}$ of integrated luminosity. The main backgrounds are $\ttb$ and W+$\geq$2 jets. A signal-to-background ratio of 3.5 is obtained.
The single top production cross section can be measured with a 10% precision, which is equivalent to a 5% precision on the measurement of the $\rm V_{tb}$(=1) element of the CKM matrix. The single-top polarization can also be measured in this channel with a 1.6% statistical precision [@yellowbook] with 10fb$^{-1}$.
Associate Higgs boson production
================================
For small Higgs boson masses ($\lesssim$ 130GeV$/c^2$), the $\rm H\rightarrow \bbb$ decay channel is dominant. Unfortunately, it is impossible to efficiently trigger the acquisition of these events due to the huge di-jet $\bbb$ background present at LHC. To observe the $\bbb$ decay of the Higgs boson, an associate production mode (with W, Z bosons or with a $\ttb$ pair) has to be considered. The $\ttb\rm H$ production diagrams are presented in Fig. \[ttbh1\]. These channels allow the top Yukawa coupling to be measured. The cross section is small: $\sigma(m_{\rm H}=120$GeV$/c^2)=0.8$pb, while the $\ttb\bbb$ background has a 3pb cross section.
![Associate $\ttb\rm H$ production diagrams[]{data-label="ttbh1"}](tthfig1.eps){width="8cm"}
The lepton+jet events are first selected. The final state is intricate, since in addition of the “usual” lepton+jet event, two additional b jets from the Higgs boson are present. As a result, the event selection requires at least six jets in the final state of which exactly four b jets.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Signal plus background expectation for the Higgs boson reconstructed mass in the $\ttb\rm H$ channel with $\rm H\rightarrow \bbb$ in the CMS and ATLAS detectors for $m_H=$115GeV$/c^2$ and $m_H=$120GeV$/c^2$ with 30fb$^{-1}$ and 100fb$^{-1}$ respectively. In both cases, fast simulations of the detectors have been used.[]{data-label="ttbh"}](ttbHCMS.epsi "fig:"){width="7cm"} ![Signal plus background expectation for the Higgs boson reconstructed mass in the $\ttb\rm H$ channel with $\rm H\rightarrow \bbb$ in the CMS and ATLAS detectors for $m_H=$115GeV$/c^2$ and $m_H=$120GeV$/c^2$ with 30fb$^{-1}$ and 100fb$^{-1}$ respectively. In both cases, fast simulations of the detectors have been used.[]{data-label="ttbh"}](ttbarh1.eps "fig:"){width="6.6cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Both W’s are fully reconstructed. The two b’s from the top decays have to be identified and the pair giving the “best” reconstructed top quarks pair is chosen. The remaining two b’s are combined to reconstruct the Higgs boson. The resulting invariant mass distribution is shown in Fig. \[ttbh\], showing a nice agreement between the ATLAS and CMS analyses and a peak due to the presence of the Higgs boson.
The shape of the background can be extracted from $\ttb\rm jj$ data. With 30fb$^{-1}$, 40 signal events are expected [@yellowbook], with a significance of 3.6$\sigma$. A 16% precision on the Yukawa coupling should be reached. The combination of the low and high luminosity runs giving a integrated luminosity of 100fb$^{-1}$ will allow a 4.8$\sigma$ significance and a 12% precision on the Yukawa coupling to be reached. All these numbers are relative to a Higgs boson mass of 120GeV$/c^2$.
Conclusion {#conclusion .unnumbered}
==========
The physics of the top quark will be one of the LHC main topics. Many exciting analyses will be carried out. Only a few of them have been summarized in this paper. Most of the analyses can be done with the the first 10fb$^{-1}$. Due to the large production cross section, the statistical uncertainty will be, in most of the cases, quickly negligible.
The top mass measurement will be a key issue. A precision can be reached provided that an excellent understanding of the detectors to control the systematic uncertainties. The study of the top quark sector highlights several theoretical challenges like the high order QCD calculations and the b fragmentation. Finally, most of the analyses presented in this report make use of fast simulation of the ATLAS and CMS detectors. As a result, the systematic studies are ahead of us.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank the conference organizing committee for their hospitality and financial support.
References {#references .unnumbered}
==========
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[^1]: Hereafter, “lepton” means electron or muon
|
ø 6.25in 22.0cm -0.5in -0.5in
0.35cm [**Keshav N. Shrivastava**]{} 0.25cm [*School of Physics, University of Hyderabad,\
Hyderabad 500046, India*]{}
The usual cyclotron resonance occurs at $\omega_c=eB/m^*c$. The new resonances occur at $\omega_{c\pm}={1\over 2}g_{\pm}eB/m^*c$ where ${1\over 2}g_{\pm}=({\it l}+{1\over 2}\pm s)/(2{\it l}
+1)$. The energy in the centre of two resonance frequencies varies as the square root of the two-dimensional density of the electrons due to spin access in the Gaussian model. The frequencies $\omega_{c\pm}$ are linearly proportional to the magnetic field except near crossing point where the linear combination of wave functions must be made, i.e., ${1\over \sqrt 2}(|n,{\it l},
\uparrow>\pm|n, {\it l}, \downarrow>)$. 1.0cm [*PACS numbers:*]{}76.40.+b, 73.20.Mf 0.10cm Corresponding author: [email protected]\
Fax: +91-40-2301 0145.Phone: 2301 0811. 1.0cm
[**1. Introduction**]{}
When an external magnetic field is applied to the electrons they go into cyclotron orbits. When energy is swept, there is a resonance at $\omega_c=eB/m^*c$. Since $e$ and $c$ are already known and $B$ can be measured accurately, a measurement of $\omega_c$ leads to a measurement of the effective mass of the electron. Here $\omega_c$ is called the cyclotron frequency, $e$ the charge of the electron, $c$ the velocity of light, $m^*$ the effective mass of the electron and $B$ the external magnetic field\[1\].
In this paper, we propose that there must exist new resonances at $\omega_{c+}={1\over 2}g_+eB/m^*c$ and at $\omega_{c-}={1\over 2}
g_-eB/m^*c$. The $g_{\pm}$ are related by Kramers time reversed states and the energy at the centre of these states is proportional to the square root of the two-dimensional number density of electrons. We have found\[2-7\] the factor before the charge during our study of the quantum Hall effect where it is used to describe the effective fractional charge of the quasiparticles. It has recently been noted that the fractional charge which were not understood in the begining are due to electron clusters where spin $1/2$ is not sufficient. In these clusters the spin may be larger than $1/2$ such as $3/2$, $5/2$, $7/2$, etc. The repulsive Coulomb interactions align the electron spins ferromagnetically so that the spin of a cluster depends on the number of electrons\[5\]. Since usually a large magnetic field is present, the spins align parallel to the field although some may also be directed opposite to the magnetic field. So the electrons align even though there is no ferromagnetism. Our report of the new resonances makes use of the experimental measurements carried out by Syed et al\[6\].
[**2. Theory**]{}.\
The cyclotron resonance consists of a single resonance line at, \_c=[eBm\^\*c]{}. We predict new resonance lines at, \_[c+]{}=[12]{}g\_+[eBm\^\*c]{} and at, \_[c-]{}=[12]{}g\_-[eBm\^\*c]{} where g\_ =[[*l*]{}+[12]{}s 2[*l*]{}+1]{} as described in ref.2. The spin is not restricted to s=1/2 only. When there are electron clusters, it may be larger value also. The s=+1/2 is the Kramers time reversed state of s=-1/2. Therefore, ${1\over 2}g_{\pm}$ is having two values. When we reverse one spin from the $N_{\uparrow}$ state and put it in the $N_{\downarrow}$ state, the spin of the system changes by 2s. This is a text book problem which shows that, = s proportional to N\^[1/2]{} as given by Kittel and Kroemer\[7\] for ordinary Gaussian
distribution. Since, the energy in the centre of two Kramers conjugate states will be proportional to $N_{\uparrow}-N_{\downarrow}=2s$, we expect that it varies as the square root of 2-dimensional electron density. Thus we have two new resonances at $\omega_{c+}$ and $\omega_{c-}$ with Kramers symmetry and Gaussian distribution for the central energy.
[**2. Analysis of data**]{}
We will show that all of the above discussed properties can be extracted from the experimental work of Syed et al\[6\] and the new resonances at $\omega_{c\pm}$ can be identified from the data. The far infrared transmission data of a two-dimensional electron gas (2DEG) of density $1.14\times 10^{12}cm^{-2}$ in AlGaN/GaN at 12.5 T shows a strong resonance at 6.9 meV and a weaker one to 5.1 meV. We assign 6.9 meV to $g_+\mu_B12.5\times 10^4$ and 5.1 meV to $g_-\mu_B 12.5\times 10^4$. The value of $g_+$ is obtained as follows. g\_+9.27410\^[-21]{}12.5 10\^4=6.910\^[-3]{} 1.60210\^[-12]{} where the value of the Bohr magneton is $\mu_B=9.274\times10^{-21}$ erg/Gauss and the magnetic field is $12.5\times
10^4$ Gauss.The resonance occurs at $6.9\times
10^{-3}$ eV and we multiply it by $1.602\times
10^{-12}$ to obtain erg units. This gives, g\_+=9.5353 Similarly using the resonance at 5.1 meV, we obtain, g\_-=7.047 From the above two values we obtain =0.5749 and = 0.4250. The sum of the above two numbers is 0.9999. According to one of our theorems $\nu_++\nu_-=1$. Therefore 0.9999 is just what we expected. Therefore the interpretation of resonances at 6.9 meV and at 5.1 meV in terms of $g_+$ and $g_-$ is correct. Thus the new radiation at $\omega_{c+}$ and at $\omega_{c-}$ is discovered. It shows that the usual cyclotron resonance occuring at $\omega_c$ is flanked by two new resonances at $\omega_{c\pm}$. Some times, the prefactors may be zero or one, in which case $\omega_{c\pm}$ will occur in such a way that $\omega_c$ will not occur.
The spin of $g_+$ is + and the spin of $g_-$ is -, so when one spin is removed from $N_{\uparrow}$ and put in $N_{\downarrow}$, the spin excess is 2s. For Gaussian distribution, the centre of two new resonances varies as the square root of the number density of two-dimensional electrons. Indeed, the variation of this point is already plotted in ref.6 and the measured value agrees with the predicted square root of the number density.
The resonance condition varies linearly with magnetic field. However, there is a crossing point or the centre of the two lines at $\omega_{c\pm}$. The energy levels at ${1\over2}g_-\omega_c
(n+{1\over 2})$ are narrowly spaced. When the magnetic field is increased, these narrowly spaced levels separate out untill their separation can become equal to those of ${1\over 2}g_+\omega_c
(n+{1\over 2})$. Thus there is a crossing point. The states are characterized by ${\it l}$ and $s$ and the Landau level number $n$. Thus the states are of the form $|n,{\it l},
\uparrow>$ and $|n,{\it l},\downarrow>$. Near the crossing point, the states get mixed so that the proper way of writing the wave function becomes, (|n, [*l*]{},> |n’, [*l*]{}’,>)
This is the reason why energy as a function of magnetic field bends near the crossing point. The resonances above 6 meV are $g_+$ type and below 6 meV are $g_-$ type. When energy is plotted as a function of magnetic field, bending occurs just as predicted. The prediction of new radiation at $\omega_{c\pm}$ is thus confirmed by the experiments.
[**3. Conclusions**]{}.
We predict new resonances at $\omega_{c\pm}$. Their frequency locks the spin as given in the expressions. The centre of these lines varies as the square root of the number density of two-dimensional electron gas. The energy bends near the central point due to mixing of states. This represents fundamentally different resonance than the cyclotron resonance, known since 1953, because the value of the spin enters in the frequency whereas the frequency of the cyclotron resonance is independent of the same. The cyclotron resonance is only one frequency, whereas ${\it many\,\,\, different
\,\,\, values}$ can be detected in the new resonances because of the many different values of ${\it l}$ and $s$. Some of the details of the new resonance frequencies can be derived from the results given in ref.2.
1.25cm
[**4. References**]{}
1. G. Dresselhaus, A. F. Kip and C. Kittel, Phys. Rev. [**92**]{}, 827 (1953).
2. K.N. Shrivastava, Introduction to quantum Hall effect,\
Nova Science Pub. Inc., N. Y. (2002).
3. K. N. Shrivastava, cond-mat/0212552.
4. K. N. Shrivastava, cond-mat/0303309, cond-mat/0303621.
5. K. N. Shrivastava, cond-mat/0302610.
6. S. Syed, M. J. Manfra, Y. J. Wang, H. L. Stormer and R. J. Molner, cond-mat/0305358.
7. C. Kittel and H. Kroemer, Thermal Physics, W. H. Freeman and Co., San Francisco, 1980, Second Edition, p.22.
0.1cm
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abstract: 'We present a measurement of the inclusive jet cross section in $p{\bar p}$ interactions at $\sqrt{s}=1.96$ TeV using 385 ${\rm pb}^{-1}$ of data collected with the CDF II detector at the Fermilab Tevatron. The results are obtained using an improved cone-based jet algorithm (Midpoint). The data cover the jet transverse momentum range from 61 to 620 GeV/[*c*]{}, extending the reach by almost 150 GeV/[*c*]{} compared with previous measurements at the Tevatron. The results are in good agreement with next-to-leading order perturbative QCD predictions using the CTEQ6.1M parton distribution functions.'
title: 'Measurement of the Inclusive Jet Cross Section in [$p\bar p$]{} Interactions at [$\sqrt{s}=1.96$]{} TeV Using a Cone-based Jet Algorithm'
---
The differential jet production cross section at the Tevatron probes the world’s highest momentum transfers in particle collisions, is potentially sensitive to a wide variety of new physics, such as quark compositeness [@b:compositness], and tests perturbative QCD (pQCD) over more than eight orders of magnitude. There was great interest when the inclusive jet cross section measured by the CDF collaboration at the center of mass energy $\sqrt{s}=1.8$ TeV [@b:CDFRunIPRL; @b:CDFRunIPRD] exhibited an excess in the high transverse energy $E_T$ region when compared to next-to-leading order (NLO) QCD predictions obtained using then-current parton distribution functions (PDFs) [@b:cteq4m]. Global PDF analysis by CTEQ [@b:cteq4hj; @b:cteq5hj] demonstrated that the excess could be explained by an enhanced gluon distribution at high momentum fraction $x$ ($x>0.3$). Recent global PDF fits (CTEQ6, CTEQ6.1, MRST2004) [@b:cteq6m; @b:cteq6_jet; @b:mrst2004], which include the Run I Tevatron jet data [@b:CDFRunIPRL; @b:d0RunI], find an increased gluon density at high $x$ and provide a good description of the Run I Tevatron data. The gluon distribution is still poorly constrained at high $x$ (see e.g. Ref.[@b:cteq6_jet]) and contributes significantly to the theoretical uncertainty for many interesting processes at the Tevatron and the LHC. The increase in $\sqrt{s}$ from 1.8 to 1.96 TeV, together with higher luminosity in Run II, allows more precise jet production measurements with a significantly extended kinematic range.
Jet algorithms cluster together objects such as partons or particles, or energies measured in calorimeter cells. The clustering algorithms rely on the association of these objects based on transverse momenta (the $k_T$ algorithm), or angles (the cone algorithm), relative to a jet axis. A measurement using the $k_T$ algorithm is reported in Ref. [@b:kT_prl]. In this letter, we report the results of an inclusive jet measurement using a cone algorithm for the rapidity region $0.1<|y|<0.7$ [@b:coordinate]. Cone jet algorithms, rather than $k_T$ algorithms, have been used dominantly at hadron collider experiments mainly due to the simplicity in constructing corrections for the underlying event and for multiple interactions in the same bunch crossing [@b:RunII_workshop]. It is worth noting that, previously, results from a cone algorithm [@b:d0RunI] and $k_T$ algorithm [@b:d0RunIKt] by the DØ collaboration showed only marginal agreement at low $p_T$ where corrections for multiple interactions and underlying event become important. We use the Midpoint algorithm, an improved iterative cone clustering algorithm [@b:RunII_workshop]. It is difficult to use previous iterative cone algorithms [@b:CDFRunIPRD; @b:d0RunI] with higher order pQCD calculations due to the presence of infrared singularities. The Midpoint algorithm places additional seeds between stable cones having a separation of less than twice the size of the clustering cones; the use of these additional seeds reduces the problem with infrared singularities. The CDF II detector is a magnetic spectrometer which is described in detail elsewhere [@b:CDFIITDR]. Here we describe briefly those components that are crucial to this measurement. The central detector consists of a silicon vertex detector inside a cylindrical drift chamber. Surrounding the tracking detectors is a superconducting solenoid which provides a 1.4 T magnetic field. Outside the solenoid is the central calorimeter, covering a pseudorapidity ($\eta$) [@b:coordinate] range up to 1.1. The central calorimeter consists of 48 modules, segmented into towers of granularity $\Delta \eta \times \Delta\phi \approx 0.1 \times 0.26$ and divided into electromagnetic (CEM) and hadronic (CHA) sections. The CEM is a lead-scintillator calorimeter; the CHA is an iron-scintillator calorimeter with a depth of approximately 4.7 interaction lengths. The energy resolution of the CEM for electrons is $\sigma(E_T) / E_T = 13.5\%/\sqrt{E_T ({\rm GeV})} \oplus 2\%$ while the average energy resolution of the CHA for charged pions is $\sigma(E_T) / E_T = 50\%/\sqrt{E_T ({\rm GeV})} \oplus 3\%$. The forward region, $1.1 < |\eta| < 3.6$, is covered by the “Plug Calorimeters” consisting of lead-scintillator for the electromagnetic section and iron-scintillator for the hadronic section. The region between the central and forward calorimeters, $0.7 < |\eta| <1.3$, is covered by an iron-scintillator hadron calorimeter with similar segmentation to the central calorimeter.
This measurement uses a data sample corresponding to an integrated luminosity of 385 ${\rm pb}^{-1}$ collected between February 2002 and August 2004. The data were collected using four trigger paths. The Level 1 trigger requires a calorimeter trigger tower, consisting of two calorimeter towers adjacent in $\eta$, to have either $E_T>5$ GeV or $E_T>10$ GeV. At Level 2, the calorimeter towers are clustered using a nearest neighbor algorithm. Four trigger paths with cluster $E_T >$ 15, 40, 60, and 90 GeV are used. Events in these paths are required to pass jet $E_T >$ 20 (J20), 50 (J50), 70 (J70), and 100 (J100) GeV thresholds at Level 3, where the clustering is performed using a cone algorithm with a cone radius $R_{cone}=0.7$.
Cosmic ray events are rejected by a cut on the missing transverse energy (${\not\!\!E_T}$) significance [@b:missingEt]. For J20, J50, J70, and J100, we remove events having a ${\not\!\!E_T}$ significance greater than 4, 5, 5, and 6 GeV$^{1/2}$, respectively. In addition, all events containing jets with $p_T>360$ GeV/[*c*]{} and passing the analysis cuts were visually scanned, and no cosmic ray events were found. The efficiency of the ${\not\!\!E_T}$ significance cut is 100% for jets at low $p_T$ (65 GeV/[*c*]{}) and decreases to 92 % for jets at high $p_T$ (550 GeV/[*c*]{}). We reconstruct $z$-vertices by fits to tracks in the event and a beamline constraint, and select the vertex with the highest total $p_T$ of the associated tracks as the event vertex. In order to ensure that the jet energy is well measured, the event vertex is required to be within 60 cm of the center of the detector along the beamline. The efficiency for this cut is determined to be 95% from the beam profile measured using a minimum bias sample. Jets are required to have a rapidity $|y|$ between 0.1 and 0.7 to reduce the effects of the gap between calorimeter modules and at the transition region between the central and plug calorimeters.
There are two essential stages for any jet algorithm. First, the objects (partons, particles, or calorimeter towers) belonging to a cluster are identified. With the Midpoint algorithm the cluster is a cone of radius 0.7 in $(y,\phi)$ space. For reasons dealing with problems of unclustered energy endemic to iterative cone algorithms [@b:building], the clustering radius is at first set to $R_{cone}/2 (=0.35)$, and then later expanded to its full size as discussed below. Second, the kinematic variables defining the jet are calculated from the objects comprising a cluster. The Midpoint algorithm makes use of four-vectors throughout the clustering process. The four-vector for each tower is computed as a sum of vectors for the electromagnetic and hadronic compartments of the tower; the vector for each compartment is defined by assigning a mass-less vector with magnitude equal to the deposited energy and with direction from the event vertex to the center of each compartment [@b:RunII_workshop]. The detector towers are sorted in order of descending $p_T$. Only towers passing a seed cut, $p_T^{tower}>p_T^{seed}$, are used as starting points for the initial jet cones. The seed threshold is chosen to be 1 GeV/[*c*]{}; its value has a negligible effect on jets in the kinematic region used in this measurement. A tower passing the threshold of 100 MeV/[*c*]{} is clustered into a cone and eventually into a jet if the separation from the axis of the cone in $(y,\phi)$ is smaller than $R_{cone}/2$. There is no threshold for particle and parton clustering. After each iteration the jet centroid position is updated. The jet clustering is repeated until all of the jet cones are stable. A cone is stable when the tower list is unchanged from the previous iteration. After all stable cones have been determined, the clustering radius is expanded to the full size ($R_{cone}$). The use of the smaller initial cone results in an expected cross section approximately 5% larger due to the inclusion of jet energy that would have remained unclustered in the default Midpoint algorithm [@b:RunII_workshop]. At this point, an additional seed is defined at the midpoint between any two cones separated by less than $2R_{cone}$ and the iteration process is repeated. Two overlapping cones, if present, are merged into a single jet if the shared energy is larger than $75\%$ ($f_{merge} = 0.75$) of the energy of the jet with lower $p_T$; otherwise the shared towers are assigned to the nearest jet. This splitting/merging procedure is iterated until the tower assignments to jets are stable. The jet kinematic properties are defined using a four-vector recombination scheme [@b:RunII_workshop]. The inclusive differential jet cross section is defined as: $$\frac{ d^2\sigma}{dp_T dy }
= \frac{ 1}{\Delta y}\frac{ 1}{\int L dt}\frac{N_{jet}/\epsilon }{\Delta p_T },$$ where $N_{jet}$ is the number of jets in the $p_T$ range $\Delta p_T$, $\epsilon$ is the trigger, ${\not\!\!E_T}$ significance cut and $z$-vertex cut efficiency, $\int L dt$ is the effective integrated luminosity, and $\Delta y = 1.2$ is the rapidity interval used in the analysis. A trigger efficiency greater than $99.5\%$ is required to include the jets collected by a given trigger threshold. The measured calorimeter level jet cross section must be corrected for detector effects and for energy from additional $p\bar p$ interactions in the same bunch crossing (pile-up). For this sample, the average number of additional $p\bar p$ interactions is about 0.9. The pile-up corrections subtract $0.93(\pm0.14)$ GeV/[*c*]{} for each additional $z$-vertex from the measured jet $p_T$ [@b:JetCorNIM].
The detector response corrections are determined from a detector simulation and a jet fragmentation model. The detector response is determined using a [geant]{}-based detector simulation [@b:cdfsim] in which a parametrized shower simulation ([gflash]{} [@b:gflash]) is used for the calorimeter simulation. The [gflash]{} parameters are tuned to test-beam data for electrons and high-$p_T$ charged pions and to the collision data for low-$p_T$ charged hadrons [@b:JetCorNIM]. [pythia]{} 6.216 [@b:pythia], with Tune A [@b:TuneA; @b:jetshape], is used for the production and fragmentation of the jets. Tune A refers to the values of the parameters describing multiple-parton interactions and initial state radiation which have been adjusted to reproduce the energy observed in the region transverse to the leading jet in the data from Run I. It has also been shown to provide a reasonable description of the measured energy distribution inside a jet [@b:jetshape].
The measured $p_T$ spectrum must be corrected for detector effects before it can be compared to theoretical predictions. We cluster the final state stable particles [@b:cluster] in [pythia]{} using the same algorithm as the one used to cluster calorimeter towers. The resulting jets contain all the particles from the $p\overline{p}$ collision, including those from the hard scatter, multiple parton-parton interactions and beam remnants. The correction, done in two correlated steps, is determined from a large sample of [pythia]{} events, passed through the CDF detector simulation. First, a $p_T$-dependent correction is determined by matching the particle jet to the corresponding calorimeter jet and is applied to each measured jet. A binned spectrum is formed from the corrected $p_T$ of each jet. The bin widths are chosen commensurate with jet energy resolution and statistics. The $p_T$ correction ranges from 1.17 at low $p_T$ (65 ${\rm GeV}/{\it c}$) to 1.04 at high $p_T$ (550 ${\rm GeV}/{\it c}$). The spectrum must be further corrected for bin-to-bin jet migration due to the finite energy resolution of the calorimeter. This unfolding correction depends on the detector energy resolution and the true spectrum as well as the $p_T$-dependent correction that was applied in the first step. The [pythia]{} events are reweighted to match the experimental spectrum before the correction factors are calculated. A bin-by-bin unfolding correction is then determined by taking the ratio of the binned hadron level cross section and calorimeter level cross section corrected by the $p_T$-dependent correction. The size of the unfolding correction varies from 1.30 at low $p_T$ to 2.31 at high $p_T$. The applied corrections remove the detector effects from the raw cross section and the corrected hadron level cross section can now be compared to theoretical predictions.
The main systematic uncertainties on the measured inclusive jet cross section arise from four sources: the jet energy scale, the jet energy resolution, the unfolding of the measured cross section to the hadron level cross section, and the luminosity. The dominant source of uncertainty is from the jet energy scale. The energy scale is known to better than 3% over the entire transverse momentum range, leading to an uncertainty on the jet cross section varying from 10% at low $p_T$ to ${}^{+58}_{-39}$% at high $p_T$, comparable to the uncertainty achieved by CDF in Run I. The uncertainty due to the jet $p_T$ resolution is determined by the $p_T$ resolution difference between the data and the [pythia]{} Monte Carlo. The uncertainty on the cross section varies from about 6% at low $p_T$ to about 10% at high $p_T$. The uncertainty associated with the unfolding correction is determined by correcting a [herwig]{} 6.5 [@b:Herwig] dijet sample using the corrections derived from the [pythia]{} sample. This uncertainty is determined to be less than 5% at high $p_T$ and lesss than 10% at low $p_T$. The luminosity uncertainty is 6%, independent of $p_T$ and is not included in the quoted systematic error. Other effects considered were determined to have a negligible effect on the cross section. Adding all of these contributions in quadrature yields a total experimental systematic uncertainty on the inclusive jet cross section varying from approximately 15% at low $p_T$ to approximately ${}^{+60}_{-40}$ % at high $p_T$. To compare the data with predictions for jets of partons as obtained from NLO calculations, the data must be further corrected for underlying event and hadronization effects. It is also possible to correct the NLO predictions for the same effects; the two corrections are simply the inverse of each other. For the former, we correct for the energy in the jet cone not associated with the hard scatter, i.e., from the collisions of other partons in the proton and antiproton. The latter corrects for particles outside the jet cone originating from partons whose trajectories lie inside the jet cone. It does not correct for the effects of hard gluon emission outside the jet cone, which are already accounted for in the NLO prediction. The bin-by-bin hadron-to-parton corrections are obtained by applying the Midpoint clustering algorithm to the hadron level and to the parton level outputs of the [pythia]{} Tune A dijet Monte Carlo samples, generated with and without an underlying event. The sample without the underlying event was generated by turning off multiple parton interactions. The underlying event correction results in a decrease of the cross section varying from 22% at low $p_T$ to 4% at high $p_T$; the hadronization correction increases the cross section by 13% at low $p_T$, and by 3.5% at high $p_T$. [herwig]{} provides consistent results on the hadronization corrections, but predicts smaller underlying event energy; the difference in the underlying event correction is taken as the underlying event correction uncertainty. In previous measurements at the Tevatron [@b:CDFRunIPRL; @b:d0RunI], the hadronization corrections were not applied to the data. The inclusive jet cross section is shown in Fig. \[f:R2Log\], and Table \[t:XSec1\] lists the cross sections with the statistical and systematic uncertainties at the hadron and parton levels. Also included in Table \[t:XSec1\] are the explicit factors applied to the hadron level cross section to obtain the parton level cross section. The experimental and theoretical jet cross sections are obtained by averaging over the transverse momentum bins. Current NLO theoretical predictions for inclusive jet production exist only at the parton level, for which the final state consists only of 2 or 3 partons [@b:eks; @b:jetrad; @b:jet++]. For our comparisons with theory we use the calculation of EKS [@b:eks]. The ratio of the inclusive jet cross section, corrected to the parton level, to the NLO QCD predictions using the CTEQ6.1M PDFs is shown in Fig. \[f:R2Lin1\]. The Midpoint jet algorithm has been applied to the 2 or 3 partons in the final state of the EKS calculation. In order to mimic the properties of the splitting/merging step, present at the experimental level but not at the NLO parton level, a parameter $R_{sep}$ with a value of 1.3, has been introduced [@b:building]. Two partons are clustered within the same jet if (1) they are within $R_{cone}$ (0.7 for this measurement) of the jet centroid and (2) within $R_{sep} \times R_{cone}$ of each other. The use of $R_{sep}=1.3$ results in a reduction of the theoretical cross section prediction by approximately 5%, roughly independent of jet transverse momentum, as compared to the prediction obtained when $R_{sep}$ is not used in the calculation. In the EKS predictions, the renormalization and factorization scales ($\mu_R$ and $\mu_F$) have both been set to $p_T^{jet}/2$. Using a scale of $p_T^{jet}$ ($2p_T^{jet}$) rather than $p_T^{jet}/2$ leads to a theoretical prediction for the jet cross section lower by approximately 10% (20%) over the entire $p_T$ range and a larger $\chi^2$ in the global PDF fits [@b:cteq6m]. The gluon distribution has been determined in the global fits, primarily by the Tevatron Run I jet data, using a renormalization and factorization scale of $p_T^{jet}/2$. Thus, for self-consistency, this scale should be used in the NLO comparisons.
We show in Fig. \[f:R2Lin1\] the experimental uncertainties for the inclusive jet cross section and the theoretical uncertainties estimated from the 40 CTEQ6.1M error PDFs [@b:cteq6m]. The PDF uncertainty is the dominant theoretical uncertainty for most of the transverse momentum range. The correction for underlying event and hadronization is model dependent. The error associated with this correction is added in quadrature to the total experimental error and shown in Fig. \[f:R2Lin1\] as the outer shaded band. The data are in good agreement with the NLO QCD predictions, which is consistent with what is reported in Ref. [@b:kT_prl].
It is important to emphasize that the CTEQ6.1M gluon density is already “enhanced” at high $x$ and so automatically leads to a larger prediction for the jet cross section than older PDFs such as CTEQ5M. Also shown in Fig. \[f:R2Lin1\] is the prediction using the latest PDF set from the MRST group [@b:mrst2004]. The MRST2004 PDFs also contain an enhanced higher $x$ gluon, leading to reasonable agreement with the CDF jet measurement.
In conclusion, we have measured the inclusive jet cross section in the range $61 < p_T < 620$ GeV/[*c*]{} using an improved iterative cone clustering algorithm, Midpoint. The new measurement extends the jet transverse momentum range over previous measurements at the Tevatron by about 150 GeV/[*c*]{}. The data are well described by NLO QCD predictions using CTEQ6.1M PDFs, within the theoretical (PDF) and experimental uncertainties. No new physics is indicated in the high $p_T$ region. Inclusion of these data in future global PDF fits will provide further constraints on the gluon distribution at large $x$.
We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Culture, Sports, Science and Technology of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; the Swiss National Science Foundation; the A.P. Sloan Foundation; the Bundesministerium für Bildung und Forschung, Germany; the Korean Science and Engineering Foundation and the Korean Research Foundation; the Particle Physics and Astronomy Research Council and the Royal Society, UK; the Russian Foundation for Basic Research; the Comisión Interministerial de Ciencia y Tecnología, Spain; in part by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00292; and the Academy of Finland.
------------------- ------------------------------------------------------------------ ---------------------------------- -------------------------------------------------------------------
$p_T$ $\frac{d^2 \sigma^{hadron}}{d p_T d y}$ $C^{h\to p}$ $\frac{d^2 \sigma^{parton}}{d p_T d y}$
(GeV/[*c*]{}) (nb/(GeV/[*c*]{})) (nb/(GeV/[*c*]{}))
61-67 $( 9.03 \pm 0.09 \stackrel{+ 1.26}{_{- 1.20} } )\times 10^{ 0}$ $ 0.889 \pm 0.008 \pm 0.116 $ $( 8.02 \pm 0.11 \stackrel{+ 1.53}{_{- 1.49} } )\times 10^{ 0}$
67-74 $( 5.17 \pm 0.05 \stackrel{+ 0.70}{_{- 0.65} } )\times 10^{ 0}$ $ 0.903 \pm 0.008 \pm 0.104 $ $( 4.67 \pm 0.06 \stackrel{+ 0.83}{_{- 0.80} } )\times 10^{ 0}$
74-81 $( 2.92 \pm 0.03 \stackrel{+ 0.39}{_{- 0.35} } )\times 10^{ 0}$ $ 0.916 \pm 0.009 \pm 0.092 $ $( 2.67 \pm 0.04 \stackrel{+ 0.45}{_{- 0.42} } )\times 10^{ 0}$
81-89 $( 1.70 \pm 0.02 \stackrel{+ 0.23}{_{- 0.20} } )\times 10^{ 0}$ $ 0.927 \pm 0.009 \pm 0.082 $ $( 1.57 \pm 0.02 \stackrel{+ 0.26}{_{- 0.23} } )\times 10^{ 0}$
89-97 $( 1.02 \pm 0.01 \stackrel{+ 0.14}{_{- 0.12} } )\times 10^{ 0}$ $ 0.936 \pm 0.007 \pm 0.073 $ $( 0.95 \pm 0.01 \stackrel{+ 0.15}{_{- 0.13} } )\times 10^{ 0}$
97-106 $( 5.90 \pm 0.04 \stackrel{+ 0.83}{_{- 0.69} } )\times 10^{-1}$ $ 0.945 \pm 0.007 \pm 0.064 $ $( 5.57 \pm 0.05 \stackrel{+ 0.87}{_{- 0.75} } )\times 10^{-1}$
106-115 $( 3.53 \pm 0.02 \stackrel{+ 0.51}{_{- 0.42} } )\times 10^{-1}$ $ 0.952 \pm 0.007 \pm 0.057 $ $( 3.36 \pm 0.03 \stackrel{+ 0.53}{_{- 0.44} } )\times 10^{-1}$
115-125 $( 2.07 \pm 0.01 \stackrel{+ 0.31}{_{- 0.25} } )\times 10^{-1}$ $ 0.958 \pm 0.007 \pm 0.050 $ $( 1.98 \pm 0.02 \stackrel{+ 0.31}{_{- 0.26} } )\times 10^{-1}$
125-136 $( 1.23 \pm 0.01 \stackrel{+ 0.19}{_{- 0.15} } )\times 10^{-1}$ $ 0.963 \pm 0.007 \pm 0.044 $ $( 1.18 \pm 0.01 \stackrel{+ 0.19}{_{- 0.16} } )\times 10^{-1}$
136-158 $( 5.84 \pm 0.03 \stackrel{+ 0.94}{_{- 0.76} } )\times 10^{-2}$ $ 0.970 \pm 0.007 \pm 0.035 $ $( 5.67 \pm 0.05 \stackrel{+ 0.94}{_{- 0.77} } )\times 10^{-2}$
158-184 $( 2.10 \pm 0.01 \stackrel{+ 0.36}{_{- 0.30} } )\times 10^{-2}$ $ 0.977 \pm 0.007 \pm 0.026 $ $( 2.05 \pm 0.02 \stackrel{+ 0.36}{_{- 0.30} } )\times 10^{-2}$
184-212 $( 7.47 \pm 0.05 \stackrel{+ 1.36}{_{- 1.16} } )\times 10^{-3}$ $ 0.983 \pm 0.007 \pm 0.019 $ $( 7.34 \pm 0.07 \stackrel{+ 1.35}{_{- 1.15} } )\times 10^{-3}$
212-244 $( 2.67 \pm 0.02 \stackrel{+ 0.52}{_{- 0.46} } )\times 10^{-3}$ $ 0.987 \pm 0.006 \pm 0.014 $ $( 2.63 \pm 0.02 \stackrel{+ 0.52}{_{- 0.45} } )\times 10^{-3}$
244-280 $( 8.88 \pm 0.10 \stackrel{+ 1.89}{_{- 1.69} } )\times 10^{-4}$ $ 0.990 \pm 0.006 \pm 0.009 $ $( 8.79 \pm 0.11 \stackrel{+ 1.87}{_{- 1.67} } )\times 10^{-4}$
280-318 $( 3.03 \pm 0.05 \stackrel{+ 0.72}{_{- 0.64} } )\times 10^{-4}$ $ 0.992 \pm 0.007 \pm 0.006 $ $( 3.01 \pm 0.06 \stackrel{+ 0.71}{_{- 0.63} } )\times 10^{-4}$
318-360 $( 9.53 \pm 0.27 \stackrel{+ 2.57}{_{- 2.21} } )\times 10^{-5}$ $ 0.993 \pm 0.006 \pm 0.004 $ $( 9.46 \pm 0.27 \stackrel{+ 2.55}{_{- 2.20} } )\times 10^{-5}$
360-404 $( 2.53 \pm 0.14 \stackrel{+ 0.79}{_{- 0.65} } )\times 10^{-5}$ $ 0.994 \pm 0.008 \pm 0.003 $ $( 2.51 \pm 0.14 \stackrel{+ 0.79}{_{- 0.64} } )\times 10^{-5}$
404-464 $( 6.34 \pm 0.61 \stackrel{+ 2.42}{_{- 1.81} } )\times 10^{-6}$ $ 0.994 \pm 0.010 \pm 0.002 $ $( 6.31 \pm 0.61 \stackrel{+ 2.40}{_{- 1.80} } )\times 10^{-6}$
464-530 $( 1.36 \pm 0.29 \stackrel{+ 0.65}{_{- 0.45} } )\times 10^{-6}$ $ 0.994 \pm 0.013 \pm 0.002 $ $( 1.36 \pm 0.29 \stackrel{+ 0.64}{_{- 0.44} } )\times 10^{-6}$
530-620 $( 2.78 \pm 1.24 \stackrel{+ 1.64}{_{- 1.11} } )\times 10^{-7}$ $ 0.994 \pm 0.008 \pm 0.003 $ $( 2.76 \pm 1.24 \stackrel{+ 1.63}{_{- 1.10} } )\times 10^{-7}$
------------------- ------------------------------------------------------------------ ---------------------------------- -------------------------------------------------------------------
![\[f:R2Log\] The measured inclusive jet differential cross section corrected to the parton level compared to the NLO pQCD prediction of the EKS calculation using CTEQ6.1M. ](DataOverNLO_Parton_Log_new){width="1.00\linewidth"}
![\[f:R2Lin1\] The ratio of the data corrected to the parton level over the NLO pQCD prediction of the EKS calculation using CTEQ6.1M. Also shown are the experimental systematic errors and the theoretical errors from the PDF uncertainty. The ratio of MRST2004/CTEQ6.1M is shown as the dashed line. An additional 6% uncertainty on the determination of the luminosity is not shown. ](DataOverNLO_Full_new){width="1.00\linewidth"}
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abstract: 'Quasars, as the most luminous persistent sources in the Universe, have broad applications for cosmological studies. In particular, they can be employed to directly measure the expansion history of the Universe, similarly to SNe Ia. The advantage of quasars is that they are numerous, cover a broad range of redshifts, up to $z = 7$, and do not show significant evolution of metallicity with redshift. The idea is based on the relation between the time delay of an emission line and the continuum, and the absolute monochromatic luminosity of a quasar. For intermediate redshift quasars, the suitable line is Mg II. Between December 2012 and March 2014, we performed five spectroscopic observations of the QSO CTS C30.10 ($z = 0.900$) using the South African Large Telesope (SALT), supplemented with photometric monitoring, with the aim of determining the variability of the line shape, changes in the total line intensity and in the continuum. We show that the method is very promising.'
---
![Time evolution of the V-band flux and the MgII line intensity as measured (upper panel), and after a shift by 280 days corresponding to a plausible time delay (lower panel).[]{data-label="fig2"}](obs1_temp13.eps){width="2.6in"}
![Time evolution of the V-band flux and the MgII line intensity as measured (upper panel), and after a shift by 280 days corresponding to a plausible time delay (lower panel).[]{data-label="fig2"}](linia_v2.eps){width="2.6in"}
Introduction
============
Quasars represent the high luminosity tail of active galactic nuclei (AGN). Among their multiple applications are probing the intergalactic medium (Borodoi et al. 2014) or providing information on massive black hole growth (Kelly et al. 2011), but they have been also proposed as promising tracers of the expansion of the Universe. The latter two aspects mostly rely on the presence of the Broad Emission Lines in quasar spectra. The BLR (Broad Line Region) is unresolved but the spectral variability allows to measure the size of the BLR from the time delay between the lines and of the continuum. For sources at redshifts $0.4 < z < 1.5$ the suitable line for such study is Mg II, monitored in the optical range. This delay is then used to determine the absolute quasar luminosity (see Czerny et al. 2013) employing the idea of dust origin of the BLR (Czerny & Hryniewicz 2011).
Results
=======
Using the South African Large Telescope (SALT), we obtained five spectra of the QSO CTS C30.10, taken in a period of 15 months. All the spectra were analysed separately, in a relatively narrow spectral range of 2700 – 2900 Å in the rest frame. We used 16 different FeII pseudo-continuum templates and we fit the spectra with the continuum and the Mg II line at the same time. The Mg II line in this source had to be modelled by two separate kinematic components, meaning that CTS C30.10 is a type B source. We considered two components in emission with a double Lorentzian line shape, which provided the best fit. Using photometry from the Optical Gravitational Lensing Experiment (OGLE), we were able to calibrate the spectra properly and to obtain the calibrated line and continuum luminosity. The time dependence of the SALT Mg II flux, and OGLE and CATALINA Survey continuum luminosity are shown in Fig. 2 (Modzelewska et al. 2014). The monitoring of the distant quasar for 15 months has not allowed yet for any firm conclusion on the time delay between the continuum and the Mg II line; however, we can try to make some preliminary estimates based on the fact that the continuum had a clear maximum just at the beginning of our monitoring campaign.
Summary
=======
Reverberation studies of quasars can be used as new cosmology probes of the expansion of the Universe. The understanding of the formation of the BLR in AGN, and in particular of the properties of the Mg II line, is also important in a much broader context. The measurement important for cosmological applications is the time delay, and this can be determined well in type B sources. Our monitoring has been too short so far to allow for a detection, but the variability pattern of the line and the continuum seems to suggest a delay of about 300 days.
2014, *ApJ*, 784, 108 2008, *ApJ*, 675, 83 2011, *A&A*, 525, L8 2013, *A&A*, 556, A97 2010,*ApJ*, 719, 1315 2014, *A&A*, in press, (arXiv1408.1520)
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abstract: |
In this paper, we study the Starobinsky model of inflation in the context of gravity’s rainbow theory. We propose that gravity rainbow functions can be written in the power-law form of the Hubble parameter. We present a detailed derivation of the spectral index of curvature perturbation and the tensor-to-scalar ratio and compare the predictions of our models with Planck 2015 data. We discover that in order to be consistent with Planck data up to $2\sigma$C.L., the viable values of $N_{k}$[*e*]{}-folds would satisfy $42\lesssim N_{k}\lesssim 87$ and the rainbow parameter $\lambda$ is nicely constrained to be $\lambda \lesssim6.0$.\
[PACS numbers: 98.80.-k, 04.50.Kd, 95.30.Sf]{}
author:
- 'Auttakit Chatrabhuti$^{1}$ and Vicharit Yingcharoenrat$^{2}$'
- 'Phongpichit Channuie$^{3}$'
title: Starobinsky Model in Rainbow Gravity
---
Introduction
============
In the past three decades, the cosmological inflation model has been well established as the leading paradigm for the very early universe both theoretically and experimentally. It provides solutions to important problems of standard big bang cosmology such as the flatness and horizon problem and gives a mechanism for seeds of the formation of the large scale structure of the universe. Although the inflation model agrees remarkably well with the observational data, it still suffers some fundamental issues, such as the fine-tuning slow-roll potential, the initial conditions, and the trans-Planckian problem. In addition, the standard mechanism of inflation requires a scalar field (an inflaton) to drive the exponential expansion of the very early universe but physical nature and fundamental origin of an inflaton are still an open question.
Since a period of inflation may occur in the very early universe, one can, in principle, expect that inflation comes from a semi-classical theory of quantum gravity with some quantum correction of Einstein-Hilbert action at high energy. The simplest example is $f(R)$-gravity theories, where the modification of the action of Einstein-Hilbert action is intended to generalize the Ricci scalar to be some function of the Ricci scalar itself. (for a review see [@De_Felice_10] and references therein). In the 4-dimensional action in $f(R)$ gravity, we write the action $S=\frac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g}f(R)$ where $\kappa^{2} = 8\pi G$, $g$ is the determinant of the metric $g_{\mu\nu}$. The Ricci scalar R is defined by $R = g^{\mu\nu}R_{\mu\nu}$. One can linearize the action by introducing an auxiliary field $\phi$ such that $S=\frac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g}[F(\phi)(R-\phi) + F(\phi)]$ where we have defined $F(\phi)=dF(\phi)/d\phi$. It is obvious that by making the variation of the action with respect to $\phi$, we simply get $\phi = R$. Thus, a scalar field $\phi$, which will be the inflaton in the scalar field representation, will propagate the freedom degree from modified gravity. In summary, in the case of $f(R)$-gravity for inflation, we can pass to its scalar-field representation to achieve some useful simplification whose mechanism has been well studied in literature, e.g. [@Barrow:1988xh; @Maeda:1987xf]. The well-known example is the “Starobinsky model” for inflation [@Starobinsky_80; @Mukhanov_81; @Starobinsky_83] with a correction quadratic in the Ricci scalar in the modified gravity framework, and of an exponential potential in the scalar field framework. Starobinsky model offers the clear origin of the inflaton field stemming from gravitational theory by identifying it with the spin-0 part of spacetime metric.
Adding quantum corrections to the action is not the only possible modification to Einstein’s gravity. Almost all approaches to quantum gravity suggest that the standard energy momentum dispersion relation is deformed near the Planck scale. This feature was predicted from string field theory [@Kostelecky_89], loop quantum gravity [@Gambini_98], non-commutative geometry [@Carroll_01]. The modification of the dispersion relation generally takes the form $$\varepsilon^2 \tilde{f}^2(\varepsilon) - p^2\tilde{g}^2(\varepsilon) = m^2.
\label{dispersion}$$ The functions $\tilde{f}(\varepsilon)$ and $\tilde{g}(\varepsilon)$ are commonly known as the rainbow functions assuming $\tilde{f}(\varepsilon) \rightarrow1$ and $ \tilde{g}(\varepsilon) \rightarrow1$ in the limit $\varepsilon/M \rightarrow 0$ where $M$ is the energy scale that quantum effects of gravity become important.
One of the interesting approaches that naturally produce modified dispersion relations is called doubly special relativity[@Amelino-Camelia_01; @Amelino-Camelia_02; @Procaccini_05]. In addition to the invariance of the speed of light, it extends special relativity by including an invariant energy scale, usually assumed to be the Planck energy. Magueijo and Smolin [@Magueijo_04] generalized this idea to include curvature i.e. doubly general relativity. In their approach, the spacetime metric felt by a free particle depends on the energy or momentum of the probe particle. Thus, spacetime is represented by a one parameter family of metrics parametrized by energy of the probe $\varepsilon$ , forming a rainbow of metrics, and hence this approach is called gravity’s rainbow. The rainbow modified metric can be written as $$g(\varepsilon) = \eta^{ab} \ \tilde{e}_a(\varepsilon) \otimes \tilde{e}_b(\varepsilon) .$$ The energy dependence of the frame field $\tilde{e}_a(\varepsilon)$ can be written in terms of the energy independence frame field $e_a$ as $\tilde{e}_0(\varepsilon) = \frac{1}{\tilde{f}(\varepsilon)} e_0$ and $\tilde{e}_i(\varepsilon) = \frac{1}{\tilde{g}(\varepsilon)} e_i$ for $i = 1,2,3$. In the study of cosmology, the conventional Friedmann-Robertson-Walker (FRW) metric for the homogeneous and isotropic universe is replaced by a rainbow metric of the form $$ds^2 = -\frac{1}{\tilde{f}^2(\varepsilon)} dt^2 + a^2(t)\delta_{ij}dx^idx^j.
\label{FRW}$$ For simplicity, we chose $\tilde{g}(\varepsilon) = 1$ and only considered the spatially flat case with $K = 0$. In general, the rainbow function $\tilde{f}(\varepsilon)$ does not depend on space-time coordinates since for any specific operation of measurement the probe’s energy $\varepsilon$ can be treated as a constant. However, instead of considering any specific measurement, the author in [@Ling07] suggested that this formalism can be generalized to study semi-classical effects of relativistic particles on the background metric during a longtime process. For the very early universe, we can choose massless particles which dominate the universe at that period such as gravitons and inflatons as our probes. In this case, we need to consider the evolution of the probe’s energy with cosmic time, denoted as $\varepsilon(t)$. As a result the rainbow functions $\tilde{f}(\varepsilon)$ depends on time only implicitly through the energy of particles.
The rainbow universe formalism was studied in the case of $f(R) = R$ [@Ling07; @Ling08] by using modified Friedmann equations and the early universe is driven by thermally fluid substance. The cosmological linear perturbations of this model was studied in [@Wang14]. In this paper, we extend this formalism to more general $f(R)$ gravity, in particular the Starobinsky model. Qualitatively, Eq. (\[FRW\]) implies that time and space are scaled with different scaling functions i.e. the rainbow function and the scale factor respectively. By choosing suitable form of $\tilde{f}(\varepsilon)$, one can solve the horizon problem without any need of inflationary expansion [@Moffat_93; @Albrecht_99; @Barrow_99]. In our case, however, the quadratic term in the Ricci scalar will lead to an inflationary solution for $a(t)$. We will show how the observational data put the constraint on the scaling function $\tilde{f}(\varepsilon)$.
We organize the paper as follows. In section \[equation\], we derive the equations of motion for $f(R)$ theory in the framework of rainbow universe where cosmological evolution of probes is taken into account and written down as an inflationary solution to these equations for the case of Starobinsky model. Then we turn to investigate cosmological perturbation in the rainbow universe in section \[perturbation\] where the spectral index of scalar perturbation and the tensor-to-scalar ratio of the model will be calculated. In section \[data\], we use the Planck 2015 data set to constraint the rainbow parameter of our model. Finally, we present our conclusion in section \[con\].
$f(R)$ theory with Rainbow gravity’s effect {#equation}
===========================================
In addition to the modified FRW metric (\[FRW\]), modifications to General Relativity are expected to be possible in the very early universe where some corrections to Einstein’s gravity may emerge at high curvature. The simplest class of modified gravity theory is $f(R)$ gravity, where the Einstein-Hilbert term in the action is replaced by a function of the Ricci scalar. Now taking this into account we derive the field equations by following the formalism in $f(R)$ theory described in [@De_Felice_10]. We start by considering the 4-dimensional action in $f(R)$ gravity: $$S = \frac{1}{2\kappa^2} \int d^4x\sqrt{-g} f(R) + \int d^4x \sqrt{-g}\mathcal{L}_M,
\label{Action}$$ where $\kappa^2 = 8\pi G$ and $\mathcal{L}_M$ is the matter Lagrangian. By varying the action (\[Action\]) with respect to $g_{\mu\nu}$, we obtain the field equations:
$$\begin{aligned}
\Sigma_{\mu\nu} \equiv F(R)R_{\mu\nu}(g) - \frac{1}{2}f(R)g_{\mu\nu}-\nabla_\mu\nabla_\nu F(R)+g_{\mu\nu} \square F(R) = \kappa^2 T_{\mu\nu}^{(M)},
\label{eom}\end{aligned}$$
where $F(R) \equiv \partial f(R)/\partial R$ and $\square F = (1/\sqrt{-g})\partial_\mu (\sqrt{-g}g^{\mu\nu}\partial_\nu F)$. The energy-momentum tensor of matter fields is defined by $T_{\mu\nu}^{(M)} \equiv - \frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g} \mathcal{L}_M)}{\delta g^{\mu\nu}}$. $\Sigma_{\mu\nu}$ and $ T^{(M)}_{\mu\nu}$ both satisfy the continuity equation i.e. $\nabla^\mu \Sigma_{\mu\nu} = 0$ and $\nabla^\mu T^{(M)}_{\mu\nu} =0 $ respectively. In cosmology, the energy-momentum tensor of matter is given in the perfect fluid form as $T^{\mu (M)}_{\nu} = \text{diag}(-\rho_M,P_M,P_M,P_M)$, where $\rho_M$ is the energy density and $P_M$ is the pressure. In the following, we will derive cosmological solutions to field equations (\[eom\]). In so doing, let us assume the modified FRW background (\[FRW\]), we obtain $$\begin{aligned}
3(F H^2 + H \dot{F}) + \dot{F}\frac{\dot{\tilde{f}}}{\tilde{f}} = \frac{FR-f(R)}{2\tilde{f}^2} + \frac{\kappa^2 \rho_M}{\tilde{f}^2},
\label{fR_eom_1}\end{aligned}$$ $$\begin{aligned}
\ddot{F} - H\dot{F} + 2 F \dot{H} + 2 FH\frac{\dot{\tilde{f}}}{\tilde{f}} = -\frac{\kappa^2}{\tilde{f}^2}(\rho_M+P_M),
\label{fR_eom_2}\end{aligned}$$ where the Hubble parameter $H = \dot{a}/a$ and $\dot{a} = da/dt$.
Inflation with the effect of Rainbow Gravity {#Inflation}
--------------------------------------------
In the following let us focus on the Starobinsky’s model [@Starobinsky_80] in which $f(R)$ takes the following form $$f(R) = R + R^2/(6M^2),$$ where $M$ has a dimension of mass and is chosen to be the same as the mass scale for quantum gravity in Eq. (\[dispersion\]) for simplicity. This is the first inflation model related to the conformal anomaly in quantum gravity. The $R^2$ term is responsible for the de Sitter expansion. Because of the linear term in $R$, the inflationary expansion ends when the term $R^2/(6M^2)$ becomes smaller than the linear term $R$. This gives the graceful exit to inflation.
In contrast to previous studies on rainbow universe [@Ling07; @Ling08; @Wang14], the $R^2$-term plays a leading role in the very early universe instead of relativistic matter. We will set $\rho_M = 0$ and $P_M =0$ from now on. One can also consider this assumption in a scalar field framework of $f(R)$ theory. For the case of Starobinsky model, its classical dynamics would be equivalent to the system of one scalar (an inflaton) with an exponential potential. So, it is reasonable to assume that inflatons dominated the very early universe and neglect contributions from matter and radiation. Then the combination of equation (\[fR\_eom\_1\]) and (\[fR\_eom\_2\]) gives us
$$\begin{aligned}
\ddot{H}-\frac{\dot{H}^2}{2H} + \frac{1}{2\tilde{f}^2}M^2H + \frac{11}{6} H\left( \frac{\dot{\tilde{f}}}{\tilde{f}}\right)^2 + \frac{1}{3}\left(\frac{\dot{\tilde{f}}}{ \tilde{f}}\right)^3 +\frac{10}{3}\dot{H}\frac{\dot{\tilde{f}}}{\tilde{f}}+\frac{\dot{H}}{H}\left( \frac{\dot{\tilde{f}}}{\tilde{f}}\right)^2+H\frac{\ddot{\tilde{f}}}{\tilde{f}} + \frac{1}{3}\frac{\dot{\tilde{f}}\ddot{\tilde{f}}}{\tilde{f}^2}+\frac{1}{3}\frac{\ddot{H}}{H}\frac{\dot{\tilde{f}}}{\tilde{f}} = -3H^2\left(\frac{\dot{\tilde{f}}}{\tilde{f}} +\frac{\dot{H}}{H}\right)\,.\label{R2_01}\end{aligned}$$
and $$\ddot{R} + 3H\dot{R}+\frac{4\dot{\tilde{f}}\dot{R}}{3\tilde{f}} + \frac{M^2R}{\tilde{f}^2} = 0.
\label{R2_02}$$ Now, we consider the evolution of modified FRW universe. For explicitness, we need to choose the form of $\tilde{f}(\varepsilon)$ and this can be explained in the “Einstein frame," where the scalar degree of freedom takes a canonical form with a potential. Under the conformal transformation $(g_E)_{\mu\nu} = \Omega^2 g_{\mu\nu}$ with $\Omega^2 = F$, the action in Einstein frame can be written as $$S_E = \int d^4x\sqrt{-g_E} \left[\frac{1}{2\kappa^2} R_E -\frac{1}{2}g_E^{\mu\nu} \partial_\mu \phi_E \partial_\nu \phi_E - V(\phi_E) \right],$$ where the subscribe $E$ denote quantities in Einstein frame. We define the scalar field (or the inflaton) by $\kappa\phi_E \equiv \sqrt{3/2}\ln F$ and its potential is determined by $V(\phi_E) = (FR-f)/(2\kappa^2F^2)$. Note that we neglect contribution from other matters.
In this framework it is natural to choose an inflaton as our probe during the inflation. We assume that probe’s energy $\varepsilon_E(t)$ is proportional to the total energy density of the inflaton, $\rho_{\phi_E}$. During the slow-roll inflation, we can assume further that $\rho_{\phi_E} \approx V(\phi_E)$ and it can be treated as a function of the Ricci Scalar in the Jordan frame, $R$. In the case of FRW metric ($\tilde{f} = 1$), for the Starobinsky model, it is well known that $R \approx 12H^2 - M^2$ during inflation [@De_Felice_10]. Hence, we can now assume that the probe energy $\varepsilon_E(t)$ can be expressed in terms of a geometric expansion rate $H$ and this conclusion should also be applied in the Jordan frame. Based on these assumptions, we choose the rainbow function to be explicitly parametrized in the power-law form of the Hubble parameter i.e. $\tilde{f}^2 = 1 + (H/M)^{2\lambda}$ with the rainbow parameter $\lambda > 0$ and $\tilde{f} =1$ for late time. For the inflationary period, we are interested in the limit $H^2 \gg M^2$. Then, the rainbow function can be approximated as $\tilde{f} \approx (H/M)^\lambda$. Equation (\[R2\_01\]) can be written as
$$\begin{aligned}
\frac{1}{2(1+\lambda)}\frac{M^2H}{\tilde{f}^2} + 3H\dot{H} + \frac{1}{6}(17\lambda-3)\frac{\dot{H}^2}{H} + \frac{2\lambda^2\dot{H}^3}{3H^3} + (1+\frac{\lambda}{3}\frac{\dot{H}}{H^2})\ddot{H} =0.
\label{R2_03}\end{aligned}$$
During inflation $H$ is very slowly varying. We can keep the first two terms in Equation (\[R2\_03\]) and neglect others. This gives us $$\dot{H} \simeq - \frac{M^{2\lambda+2}}{6(1+\lambda)}H^{-2\lambda}.$$ We then obtain the solution $$\begin{aligned}
&H \simeq H_i - \frac{M^2}{6(1+\lambda)}\left(\frac{M}{H_i}\right)^{2\lambda}(t-t_i),\label{H_evolution}\\&
a \simeq a_i \exp\left\{H_i(t-t_i) - \frac{M^2}{12(1+\lambda)}\left(\frac{M}{H_i} \right)^{2\lambda}(t-t_i)^2 \right\},\\&
R \simeq 12H^2 \left(\frac{H}{M}\right)^{2\lambda},\end{aligned}$$ where $H_i$ and $a_i$ are Hubble parameter and scale factor at the onset of inflation ($t = t_i$), respectively. The slow-roll parameter is defined by $$\epsilon_1 = -\frac{\dot{H}}{H^2} = \frac{1}{6(1+\lambda)}\left(\frac{M}{H} \right)^{2+2\lambda},$$ which is less than unity in the limit $H^2 \gg M^2$. One can also check that the approximate relation $3H\dot{R} +\frac{M^2R}{\tilde{f}^2} \simeq 0$ holds in Eq. (\[R2\_02\]) by using $R \simeq 12H^2 \left(\frac{H}{M}\right)^{2\lambda}$. The end of inflation (at time $t = t_f$) can be determined by the condition $\epsilon_1 \simeq 1$, i.e. $H_f \simeq M/[6(1+\lambda)]^{1/(2+2\lambda)}$. The number of $e$-foldings from $t = t_i$ to $t = t_f$ is defined by $$\begin{aligned}
N &\equiv \int_{t_i}^{t_f} Hdt \nonumber\\&\simeq H_i(t_f-t_i) - \frac{M^2}{12(1+\lambda)}\left(\frac{M}{H_i} \right)^{2\lambda}(t_f-t_i)^2.\end{aligned}$$ Since inflation ends at $t_f \simeq t_i + 6(1+\lambda) H_i^{1+2\lambda}/M^{2+2\lambda}$, we can show that $$N \simeq 3(1+\lambda)\left(\frac{H_i}{M} \right)^{2+2\lambda} \simeq \frac{1}{2\epsilon_1(t_i)}. \label{N_fold}$$ Note that for $\lambda = 0$ the model reverts to the Starobinsky model.
Cosmological perturbation in Rainbow gravity {#perturbation}
============================================
In this section, we focus on cosmological linear perturbation generated during inflation by following the formalism presented in [@De_Felice_10]. From the point of view of a probe particle with energy $\varepsilon$, a general perturbed metric about the flat FRW universe would take the form
$$\begin{aligned}
ds^2 = -\frac{1+2\alpha}{\tilde{f}^2(\varepsilon)}dt^2 - \frac{2a(t)(\partial_i\beta-S_i)}{\tilde{f}(\varepsilon)}dtdx^i + a^2(t)(\delta_{ij}+2\psi\delta_{ij}+2\partial_i\partial_j\gamma+2\partial_jF_i+h_{ij})dx^idx^j\,, \label{B1}\end{aligned}$$
where $\alpha$, $\beta$, $\psi$, $\gamma$ are scalar perturbations, $S_i$, $F_i$ are vector perturbations, and $h_{ij}$ are tensor perturbations. The rainbow function $\tilde{f}(\varepsilon)$ can be viewed as a smooth background function. As usual, we will focus on scalar and tensor perturbations by setting vector perturbations $S_i=F_i=0$. Before discussing the evolution of cosmological perturbations, we construct a number of gauge-invariant quantities required to eliminate unphysical modes. Let us consider the gauge transformation $t \rightarrow \hat{t} = t+\delta t$ and $x^i \rightarrow \hat{x}^i = x^i + \delta^{ij}\partial_j \delta x$. Here $\delta t$ and $\delta x$ denote the infinitesimal scalar functions of the spacetime coordinates. Then we can show that the scalar metric perturbations transform as: $$\begin{aligned}
\alpha &\rightarrow& \hat{\alpha} = \alpha +\frac{\dot{\tilde{f}}}{\tilde{f}}\delta t - \dot{\delta t}, \\
\beta &\rightarrow& \hat{\beta} = \beta - \frac{\delta t}{a\tilde{f}} + a \tilde{f} \dot{\delta x},\\
\psi &\rightarrow& \hat{\psi} = \psi - H\delta t,\\
\gamma &\rightarrow& \hat{\gamma} = \gamma - \delta x.\end{aligned}$$ Note that the tensor perturbations $h_{ij}$ are invariant under the gauge transformation. We can define gauge invariant quantities under the above gauge transformations as $$\begin{aligned}
\Phi &=& \alpha - \tilde{f}\frac{d}{dt}\left[ a^2\tilde{f}\left(\dot{\gamma} + \frac{\beta}{a\tilde{f}} \right)\right],\\
\Psi &=& -\psi + a^2\tilde{f}^2H\left(\dot{\gamma}+\frac{\beta}{a\tilde{f}} \right),\\
\mathcal{R} &=& \psi - \frac{H\delta F}{\dot{F}}.\end{aligned}$$ We can choose the longitudinal gauge $\beta = 0$ and $\gamma=0$. In this gauge $\Phi = \alpha$ and $\Psi = -\psi$. Thus, without tensor perturbations, the line element takes the form $$\begin{aligned}
ds^2 = -\frac{1+2\Phi}{\tilde{f}^2(t)}dt^2 + a^2(t)(1-2\Psi)\delta_{ij}dx^idx^j .\label{B2}\end{aligned}$$ For later convenience, we define the perturbed quantity $$\begin{aligned}
A \equiv 3(H\Phi+\dot{\Psi}) . \label{B3}\end{aligned}$$ By using the perturbed metric (\[B2\]) and Eq. (\[eom\]), we come up with the following equations
$$\begin{aligned}
\frac{\nabla^2\Psi}{a^2}+\tilde{f}^2HA = -\frac{1}{2F}\left[3\tilde{f}^2\left(H^2 + \dot{H} + \frac{\dot{\tilde{f}}}{\tilde{f}}\right)\delta F + \frac{\nabla^2\delta F}{a^2}-3\tilde{f}^2H\delta\dot{F} + 3\tilde{f}^2H\dot{F}\Phi + \tilde{f}^2\dot{F}A+\kappa^2\delta\rho_M\right]\ , \label{B4}\end{aligned}$$
$$\begin{aligned}
H\Phi+\dot{\Psi}=-\frac{1}{2F}(H\delta F+\dot{F}\Phi-\delta\dot{F}) \ , \label{B5} \end{aligned}$$
and
$$\begin{aligned}
\dot{A} + \left(2H+\frac{\dot{\tilde{f}}}{\tilde{f}}\right)A+3\dot{H}\Phi + \frac{\nabla^2\Phi}{a^2\tilde{f}^2}+\frac{3H\Phi\dot{\tilde{f}}}{\tilde{f}}\nonumber\\= \frac{1}{2F}\left[3\delta\ddot{F}+3\left(H+\frac{\dot{\tilde{f}}}{\tilde{f}}\right)\delta\dot{F}-6H^2\delta F -\frac{\nabla^2\delta F}{a^2\tilde{f}^2} - 3\dot{F}\dot{\Phi}-\dot{F}A - 3\left(H+\frac{\dot{\tilde{f}}}{\tilde{f}}\right)\dot{F}\Phi-6\ddot{F}\Phi+\frac{\kappa^2}{\tilde{f}^2}(3\delta P_M+\delta\rho_M) \right] .
\label{B6}\end{aligned}$$
These equations describe evolutions of the scalar perturbations. We shall solve the above equations in the context of the inflationary universe.
Curvature perturbations
-----------------------
Let us consider scalar perturbations generated during inflation without taking into account the perfect fluid i.e. $\delta\rho_M =0$ and $\delta P_M=0$. We can choose the gauge condition $\delta F =0$, so that $\mathcal{R} = \psi = -\Psi$. Since the spatial curvature $^{(3)}\mathcal{R}$ on the constant-time hypersurface is related to $\psi$ via the relation $^{(3)}\mathcal{R} = - 4\nabla^2\psi/a^2$, the quantity $\mathcal{R}$ is often called the curvature perturbation on the uniform-field hypersurface. By setting $\delta F$ to be zero, Eq. (\[B5\]) gives us $$\begin{aligned}
\Phi = \frac{\dot{\mathcal{R}}}{H+\dot{F}/2F} \ , \label{B8}\end{aligned}$$ and from the equation (\[B5\]), we get $$\begin{aligned}
A = -\frac{1}{H+\dot{F}/2F}\left[\frac{\nabla^2\mathcal{R}}{a^2\tilde{f}^2}+\frac{3H\dot{F}\dot{\mathcal{R}}}{2F(H+\dot{F}/2F)}\right] \ . \label{B9}\end{aligned}$$ By using the background equation (\[fR\_eom\_2\]), Eq. (\[B4\]) gives
$$\begin{aligned}
\dot{A}+\left(2H+\frac{\dot{F}}{2F}\right)A+\frac{\dot{\tilde{f}}A}{\tilde{f}}+\frac{3\dot{F}\dot{\Phi}}{2F}+\left[\frac{3\ddot{F}+6H\dot{F}}{2F}+\frac{\nabla^2}{a^2\tilde{f}^2}\right]\Phi+\frac{3\dot{F}}{2F}\frac{\Phi\dot{\tilde{f}}}{\tilde{f}} = 0. \label{B10}\end{aligned}$$
Substituting Eq. (\[B8\]) and (\[B9\]) into Eq. (\[B10\]), we can show that the curvature perturbation satisfies the following equation in Fourier space $$\begin{aligned}
\ddot{\mathcal{R}} + \frac{1}{a^3Q_s}\frac{d}{dt}(a^3Q_s)\dot{\mathcal{R}} + \frac{\dot{\tilde{f}}}{\tilde{f}}\dot{\mathcal{R}} + \frac{k^2}{a^2\tilde{f}^2} \mathcal{R}= 0 \ , \label{B13}\end{aligned}$$ where $k$ is a comoving wave number and $Q_s$ is defined by $$\begin{aligned}
Q_s \equiv \frac{3\dot{F}^2}{2\kappa^2F(H+\dot{F}/2F)^2} \ . \label{B12}\end{aligned}$$ To simplify Eq. (\[B13\]) any further we will introduce the new variables $z_s = a\sqrt{Q_s}$ and $u = z_s\mathcal{R}$. The above equation can be expressed as $$\begin{aligned}
u'' + \left(k^2-\frac{z_s''}{z_s}\right)u = 0 \ , \label{B14}\end{aligned}$$ where a prime denotes a derivative with respect to the new time coordinates $\eta = \int (a\tilde{f})^{-1} dt$. To calculate the spectrum of curvature perturbations, let us further introduce the Hubble flow parameters (also known as slow-roll parameters) as $$\begin{aligned}
\epsilon_1 \equiv -\frac{\dot{H}}{H^2}, \ \ \epsilon_3 \equiv \frac{\dot{F}}{2HF}, \ \ \epsilon_4 \equiv \frac{\dot{E}}{2HE}\ ,\end{aligned}$$ where $E \equiv 3\dot{F}^2/2\kappa^2$. Using these definitions, $Q_s$ can be rewritten as $$\begin{aligned}
Q_s = \frac{E}{FH^2(1+\epsilon_3)^2} \ . \label{B16}\end{aligned}$$ Assuming that parameters $\epsilon_i$ are nearly constants during the inflation for $i=1,3,4$ and $\tilde{f} \simeq (H/M)^{\lambda}$, we have $\eta = -1/[(1-(1+\lambda)\epsilon_1)\tilde{f}aH]$. If $\dot{\epsilon_i}\simeq 0$, it follows that the term $z_s''/z_s$ is given by $$\begin{aligned}
\frac{z_s''}{z_s} = \frac{\nu^2_{\mathcal{R}} - 1/4}{\eta^2} \ , \label{B18}\end{aligned}$$ with $$\begin{aligned}
\nu_{\mathcal{R}}^2 = \frac{1}{4} + \frac{(1+\epsilon_1 - \epsilon_3+\epsilon_4)(2-\lambda\epsilon_1 -\epsilon_3+\epsilon_4)}{(1-(\lambda+1)\epsilon_1)^2} \ .\label{B19}\end{aligned}$$ With these assumptions, the explicit solution for Eq. (\[B14\]) can be found as $$\begin{aligned}
u = \frac{\sqrt{\pi|\eta|}}{2}\textmd{e}^{i(1+2\nu_{\mathcal{R}})\pi/4}\left[c_1\textmd{H}_{\nu_{\mathcal{R}}}^{(1)}(k|\eta|)+c_2\textmd{H}_{\nu_{\mathcal{R}}}^{(2)}(k|\eta|)\right] \ , \label{B20}\end{aligned}$$ where $c_1$, $c_2$ are integration constants and $\textmd{H}_{\nu_{\mathcal{R}}}^{(1)}(k|\eta|)$, $\textmd{H}_{\nu_{\mathcal{R}}}^{(2)}(k|\eta|)$ are the Hankel functions of the first kind and the second kind respectively. In the asymptotic past $k\eta \rightarrow -\infty$, the solution to Eq. (\[B14\]) is $u \rightarrow \textmd{e}^{-ik\eta}/\sqrt{2k}$, this implies $c_1=1$ and $c_2=0$. Thus we have $$\begin{aligned}
u = \frac{\sqrt{\pi|\eta|}}{2}\textmd{e}^{i(1+2\nu_{\mathcal{R}})\pi/4}\textmd{H}_{\nu_{\mathcal{R}}}^{(1)}(k|\eta|) \ . \label{B21}\end{aligned}$$ The power spectrum of curvature perturbations is defined by $$\begin{aligned}
\mathcal{P}_{\mathcal{R}} \equiv \frac{4\pi k^3}{(2\pi)^3}|\mathcal{R}|^2 \ . \label{B22}\end{aligned}$$ By using Eq. (\[B21\]) and the relation $u = z_s\mathcal{R}$, we will get $$\begin{aligned}
\mathcal{P}_{\mathcal{R}} = \frac{1}{Q_s}\left[(1-(1+\lambda)\epsilon_1)\frac{\Gamma(\nu_{\mathcal{R}})H}{2\pi\Gamma(3/2)}\left(\frac{H}{M}\right)^\lambda\right]^2\left(\frac{k|\eta|}{2}\right)^{3-2\nu_{\mathcal{R}}} \ . \label{B23}\end{aligned}$$ Note that for $k|\eta| \rightarrow 0$ we have $\textmd{H}_{\nu_{\mathcal{R}}}^{(1)}(k|\eta|) \rightarrow -(i/\pi)\Gamma(\nu_{\mathcal{R}})(k|\eta|/2)^{-\nu_{\mathcal{R}}}$ and $P_{\mathcal{R}}$ should be evaluated at $k=aH$ because $\mathcal{R}$ is fixed after the Hubble radius crossing. Now, we can define the spectral index $n_{\mathcal{R}}$ as $$\begin{aligned}
n_{\mathcal{R}} - 1 = \left.\frac{d\textmd{ln}\mathcal{P}_{\mathcal{R}}}{d\textmd{ln}k}\right|_{k=aH} = 3 - 2\nu_{\mathcal{R}} \ . \label{B25}\end{aligned}$$ During the inflationary epoch, we assume that $|\epsilon_i | \ll 1$ for all $i$, then the spectral index is reduced to $$\begin{aligned}
n_{\mathcal{R}} - 1 \simeq -2(\lambda+2)\epsilon_1+2\epsilon_3-2\epsilon_4 \ . \label{B27}\end{aligned}$$ Giving that $|\epsilon_i|$ are much smaller than one, $n_{\mathcal{R}} \simeq 1$, the spectrum is nearly scale-invariant. Subsequently, the power spectrum of curvature perturbation is $$\begin{aligned}
\mathcal{P}_{\mathcal{R}} \approx \frac{1}{Q_s}\left(\frac{H}{2\pi}\right)^2\left(\frac{H}{M}\right)^{2\lambda} \ . \label{B28}\end{aligned}$$
Tensor perturbations
--------------------
In this section we derive appropriate expressions of the power spectrum and the spectral index of tensor perturbations. In general, the tensor perturbation $h_{ij}$ can be generally written as $$\begin{aligned}
h_{ij} = h_{+}e^+_{ij} + h_{\times}e^\times_{ij} \ , \label{C2}\end{aligned}$$ where $e^+_{ij}$ and $e^\times_{ij}$ are the polarization tensors corresponding to the two polarization states of $h_{ij}$. Let $\vec{k}$ be in the direction along the z-axis, then the non-vanishing components of polarization tensors are $e^+_{xx} = -e^+_{yy} = 1$ and $e^\times_{xy} = e^\times_{yx} = 1$. By neglecting the scalar and vector perturbation, the perturbed FRW metric in (\[B1\]) can be written as
$$\begin{aligned}
ds^2 = -\frac{dt^2}{\tilde{f}(\varepsilon)^2} + a^2(t)h_{\times}dxdy + a^2(t)\left[(1+h_{+})dx^2+(1-h_{+})dy^2+dz^2\right] .\label{C1}\end{aligned}$$
By using the equation of motion (\[eom\]), we can show that the Fourier components $h_\chi$ satisfy the equation $$\begin{aligned}
\ddot{h}_\chi + \frac{(a^3F)^\cdot}{a^3F}\dot{h}_\chi + \frac{\dot{\tilde{f}}}{\tilde{f}}\dot{h}_\chi + \frac{k^2}{a^2\tilde{f}^2}h_\chi = 0 \ , \label{C5}\end{aligned}$$ where $\chi$ denotes $+$ and $\times$. We now follow the similar procedure to the one given in the case of curvature perturbation. Let us introduce the new variables $z_t = a\sqrt{F}$ and $u_\chi = z_t h_\chi /\sqrt{16\pi G}$, then Eq. (\[C5\]) can be written as $$\begin{aligned}
u''_\chi + \left(k^2-\frac{z_t''}{z_t}\right)u_\chi = 0 \ . \label{C6}\end{aligned}$$ Note that the massless scalar field $u_\chi$ has dimension of mass. By choosing $\dot{\epsilon}_i=0$, we get $$\begin{aligned}
\frac{z_t''}{z_t} = \frac{\nu^2_t-1/4}{\eta^2} \ , \label{C7}\end{aligned}$$ with $$\begin{aligned}
\nu^2_t = \frac{1}{4} + \frac{(1+\epsilon_3)(2-(1+\lambda)\epsilon_1+\epsilon_3)}{(1-(1+\lambda)\epsilon_1)^2} \ . \label{C8}\end{aligned}$$ Hence, the solution to the equation (\[C6\]) can be expressed in terms of the Hankel functions as in Eq. (\[B21\]). Taking into account polarization states, the power spectrum of tensor perturbations $P_T$ after the Hubble radius crossing is $$\begin{aligned}
\mathcal{P}_T &= 4\times\frac{16\pi G}{a^2F}\frac{4\pi k^3}{(2\pi)^3}|u_\chi|^2 \nonumber\\&= \frac{16}{\pi}\left(\frac{H}{m_{pl}}\right)^2\frac{1}{F}\left[(1-(1+\lambda)\epsilon_1)\frac{\Gamma(\nu_t)}{\Gamma(3/2)}\left(\frac{H}{M}\right)^\lambda\right]^2\times\nonumber\\&\,\,\,\,\,\times\left(\frac{k|\eta|}{2} \right)^{3-2\nu_t}.
\label{C12}\end{aligned}$$ Note that we have used $\tilde{f} \simeq (H/M)^{\lambda}$. Since all of the slow-roll parameters are very small during the inflation ($|\epsilon_i| \ll 1$), $\nu_t$ can be estimated as $$\begin{aligned}
\nu_t \simeq \frac{3}{2} + (1+\lambda)\epsilon_1 + \epsilon_3 \ . \label{C13}\end{aligned}$$ In addition, the spectral index of tensor perturbations is determined by $$\begin{aligned}
n_T = \left.\frac{d \textmd{ln}\mathcal{P}_T}{d\textmd{ln}k}\right|_{k=aH} = 3-2\nu_t \simeq -2(1+\lambda)\epsilon_1 - 2\epsilon_3 \ , \label{C16}\end{aligned}$$ the power spectrum $\mathcal{P}_T$ can also be rewritten as $$\begin{aligned}
\mathcal{P}_T \simeq \frac{16}{\pi}\left(\frac{H}{m_{pl}}\right)^2\frac{1}{F}\left(\frac{H}{M}\right)^{2\lambda} \ . \label{C17}\end{aligned}$$ The tensor-to-scalar ratio $r$ can be obtained as $$\begin{aligned}
r \equiv \frac{\mathcal{P}_T}{\mathcal{P}_R} \simeq \frac{64\pi}{m_{pl}^2}\frac{Q_s}{F} \ . \label{C20}\end{aligned}$$ By substituting $Q_s$ from Eq. (\[B16\]), we therefore get $$\begin{aligned}
r = 48\epsilon_3^2 \ . \label{C21}\end{aligned}$$
The Spectra of Perturbations based on Starobinsky’s model in Gravity’s rainbow theory
-------------------------------------------------------------------------------------
Let us start by finding relations among the Hubble flow parameters. In the absence of matter field with an assumption that $ | \epsilon_i | \ll 1$ during the inflation, Eq. (\[fR\_eom\_2\]) gives us $$\begin{aligned}
\epsilon_3 \simeq -(1+\lambda)\epsilon_1 \ . \label{C23}\end{aligned}$$ For Starobinsky model, $ f(R) = R + R^2/6M^2$, inflation occurred in the limit $R \gg M^2$ and $|\dot{H}| \ll H^2$. We can approximate $F(R) \simeq \frac{4H^2\tilde{f}^2}{M^2}$. By assuming that $|\epsilon_i| \ll 1$ during the inflation, this leads to $$\begin{aligned}
\epsilon_4 \simeq -(1+2\lambda)\epsilon_1 \ . \label{C26}\end{aligned}$$ The spectral index of scalar perturbations and the tensor-to-scalar ratio can be rewritten in terms of $\epsilon_1$ as $$\begin{aligned}
n_\mathcal{R} - 1 \simeq -4\epsilon_1\ \ \text{and } \ r \simeq 48(\lambda+1)^2\epsilon_1^2 . \label{C28} \end{aligned}$$ Let $t_k$ be the time at the Hubble radius crossing ($k=aH$). From Eq. (\[H\_evolution\]), as long as the condition $\frac{M^2(t_k-t_i)}{6(1+\lambda)}(M/H_i)^{2\lambda} \ll H_i$ is satisfied, we can approximate $H(t_k) \simeq H_i$. The number of $e$-fold from $t=t_k$ to the end of the inflation can be estimated as $N_k \simeq 1/2\epsilon_1(t_k)$.
{width="0.42\linewidth"} {width="0.42\linewidth"} {width="0.42\linewidth"} {width="0.42\linewidth"} {width="0.42\linewidth"}
In the following, we will express the power spectrum, the spectral index, and the tensor-to-scalar ratio in terms of $e$-folding number $N_k$. By substituting the expression for $Q_s$ in Eq. (\[B16\]) into Eq. (\[B28\]) and using the relation (\[C23\]), we then obtain $$\begin{aligned}
\mathcal{P_R} &= \frac{1}{3\pi F}\left(\frac{H}{m_{pl}}\right)^2\left(\frac{H}{M}\right)^{2\lambda}\frac{1}{(1+\lambda)^2\epsilon_1^2} \nonumber\\&\simeq \frac{1}{12\pi}\left(\frac{M}{m_{pl}}\right)^2\frac{1}{(1+\lambda)^2\epsilon_1^2} \ . \label{C31}\end{aligned}$$ According to the equation (\[N\_fold\]), the quantities $\mathcal{P_R}$, $n_\mathcal{R}$, and $r$ can be written in terms of $N_k$ as $$\begin{aligned}
\mathcal{P_R} = \frac{1}{3\pi}\left(\frac{M}{m_{pl}}\right)^2\frac{N_k^2}{(1+\lambda)^2} \ , \label{C32}\end{aligned}$$ $$\begin{aligned}
n_\mathcal{R} - 1 = -\frac{2}{N_k} \ , \label{C33}\end{aligned}$$ and $$\begin{aligned}
r = \frac{12(1+\lambda)^2}{N_k^2} \ . \label{C34}\end{aligned}$$ Note that the spectral index of scalar perturbations $n_\mathcal{R}$ does not depend on the rainbow parameter.
Fitting 2015 Planck data {#data}
========================
In this examination, we confront the results predicted by our model with Planck 2015 data. In order to be consistent with Planck data, we find that the value of $\lambda$ cannot be arbitrary. For concreteness, Fig.(\[fig1\]) and Fig.(\[fig2\])show that to be within $1\sigma$C.L.of Planck’15 contours value of $\lambda$ cannot be larger than $3.6$ with the sizable number of [*e*]{}-folds. We find for $\lambda = 3.6$ and $N_{k}=60-65$ [*e*]{}-folds that the predictions lie well on the boundary of $1\sigma$region of the Planck data. Moreover, we discover that the predictions are consistent with the Planck data up to $2\sigma$C.L. if the value of $\lambda$ satisfies $\lambda\lesssim 6.0$. Given $\lambda = 6.0$ and $N_{k}=66-72$ [*e*]{}-folds, the predictions lie on the boundary of $2\sigma$region of the Planck data.
{width="0.42\linewidth"} {width="0.42\linewidth"} {width="0.42\linewidth"} {width="0.42\linewidth"} {width="0.42\linewidth"}
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We also consider the situation in which the values of $\lambda$ are arbitrary but keep $N_{k}$ fixed. In this case, we obtain from Fig.(\[fig3\]) the window of viable values of $N_{k}$[*e*]{}-folds providing our predictions consistent with the Planck contours. Surprisingly, we discover that in order to be consistent with Planck data up to $2\sigma$C.L., the values of $N_{k}$[*e*]{}-folds would satisfy $42\lesssim N_{k}\lesssim 87$ with a proper choice of $\lambda$. However, the values of $\lambda$ cannot be larger than $6.0$.
In some realistic scenario, one can apply $N_{k}=70$ [@De_Felice_10]. In this specific case, we find that our predictions nicely satisfy the Planck data at $1\sigma$C.L. when we opt $\lambda\lesssim 3.0$. Note an extreme (and highly implausible) situation where the number of [*e*]{}-folds can be even higher, possibly up to 100 [@Liddle:2003as].
Moreover, using parameters of the base $\Lambda$CDM cosmology reported by Planck 2015 [@Ade:2015lrj] for $\mathcal{P_R}$ at the scale $k=0.05\,{\rm Mpc^{-1}}$ and taking the value $N_{k}=70$, we find from Eq.(\[C32\]) that the mass $M$ is constrained to be $$\begin{aligned}
M &\simeq 2\times 10^{-6}(1+\lambda)\,m_{pl}\nonumber\\&\sim \left(0.34 - 1.70\right)\times 10^{14}\,\,{\rm GeV}\,, \label{C35}\end{aligned}$$ with the lower value obtained for the reduced Planck mass of $2.44\times 10^{18}\,\,{\rm GeV}$ and the higher one for the standard one $1.22\times 10^{19}\,\,{\rm GeV}$.
Conclusion {#con}
==========
In this paper, we have studied the Starobinsky model in the framework of rainbow gravity and obtained the equations of motion where the energy of probe particles varies with the cosmological time. By assuming the rainbow function in the power-law form of the Hubble constant, we have derived the inflationary solution to the equations of motion and calculate the spectral index of curvature perturbation and the tensor-to-scalar ratio.
By comparing the results predicted by our model with Planck 2015 data, we found that the model is in agreement with the data up to $2\sigma$ if the rainbow parameter satisfies $\lambda\lesssim 6.0$ with the sizable number of $N_{k}$ [*e*]{}-folds associated with $42\lesssim N_{k}\lesssim 87$. We have also used the scalar power spectrum amplitude reported by Planck 2015 to determine the mass $M$ and found, for $\lambda\lesssim 3.0$ assuming $N_{k}=70$, that $M \sim \left(0.34 - 1.70\right)\times 10^{14}\,\,{\rm GeV}$. We are grateful to Ahpisit Ungkitchanukit and Khamphee Karwan for discussions and helpful suggestions. P.C. thanks Peeravit Koad for technical assistance. The work of A.C. and V.Y. is supported by “CUniverse” research promotion project by Chulalongkorn University (grant reference CUAASC) and P.C. is financially supported by the Thailand Research Fund (TRF) under the project of the TRF Grant for New Researcher with Grant No. TRG5780143. This work is supported in part by the Special Task Force for Activating Research (STAR) Project, Ratchadaphiseksomphot Endowment Fund, Chulalongkorn University.
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---
abstract: |
It is shown that the Hecke-Kiselman algebra associated to a finite directed graph is an automaton algebra in the sense of Ufnarovskii. Consequently, its Gelfand-Kirillov dimension is an integer if it is finite. This answers a question stated in [@mecel_okninski]. As a consequence, it is proved that the Hecke-Kiselman algebra associated to an oriented cycle admits a finite Gröbner basis.\
**2010 Mathematics Subject Classification**: 16S15, 16S36, 20M05, 20M35.\
**Key words**: Hecke-Kiselman algebra, monoid, simple graph, automaton algebra, Gröbner basis.
author:
- Arkadiusz Mȩcel and Jan Okniński
title: 'Gröbner basis and the automaton property of Hecke–Kiselman algebras'
---
Introduction
============
In the paper [@maz] of Ganyushkin and Mazorchuk a finitely generated monoid $\operatorname{HK}_{\Theta}$ was defined for an arbitrary finite simple digraph $\Theta$ with $n$ vertices $\{1, \ldots, n\}$ by specifying generators and relations.
- $\operatorname{HK}_{\Theta}$ is generated by idempotents $ x_i^2 = x_i$, where $1 \leq i \leq n$,
- if the vertices $i$, $j$ are not connected in $\Theta$, then $x_ix_j = x_jx_i$,
- if $i$, $j$ are connected by an arrow $i \to j$ in $\Theta$, then $x_ix_jx_i = x_jx_ix_j = x_ix_j$,
- if $i$, $j$ are connected by an (unoriented) edge in $\Theta$, then $x_ix_jx_i = x_jx_ix_j$.
If the graph $\Theta$ is unoriented (has no arrows), the monoid $\operatorname{HK}_{\Theta}$ is isomorphic to the so-called $0$-Hecke monoid $\operatorname{H}_0(W)$, where $W$ is the Coxeter group of the graph $\Theta$, see [@den]. The latter monoid plays an important role in representation theory. In the case $\Theta$ is oriented (all edges are arrows) and acyclic, the monoid $\operatorname{HK}_{\Theta}$ is finite and it is a homomorphic image of the so-called Kiselman monoid $K_n$, see [@maz], [@kun]. It is worth mentioning that a characterization of general finite digraphs $\Theta$ such that the monoid $\operatorname{HK}_{\Theta}$ is finite remains an open problem, see [@aragona].\
The aim of this paper is to continue the study of the semigroup algebra $A = k[\operatorname{HK}_{\Theta}]$ over a field $k$, in the case when $\Theta$ is an oriented graph, that was started in [@mecel_okninski], where it was shown that the growth of $A$ is either polynomial or the monoid $\operatorname{HK}_{\Theta}$ contains a noncommutative free submonoid. The main result of the present paper states that the algebra $A$ is automaton in the sense of Ufnarovskii [@ufnar], which means that the set of normal words of $A$ forms a regular language. In other words, the set of normal words of $A$ is determined by a finite automaton.
\[main\] Assume that $\Theta$ is a finite simple oriented graph. Then $A = k[\operatorname{HK}_{\Theta}]$ is an automaton algebra, with respect to any deg-lex order on the underlying free monoid of rank $n$. Consequently, the Gelfand-Kirillov dimension $\operatorname{GKdim}(A)$ of $A$ is an integer if it is finite.
In the case when the digraph $\Theta$ is unoriented, the corresponding monoid algebra is known to be automaton. Indeed, as mentioned above: in this case $\operatorname{HK}_{\Theta} = \operatorname{H}_0(W)$, where $W$ is the Coxeter group of the graph $\Theta$. In fact, one can prove that the reduced words for $W$ and $\operatorname{H}_0(W)$ are the same, and two words represent the same element of the Coxeter group if and only if they represent the same element of the Coxeter monoid, see [@tsa]. However, the set of normal forms of elements of a Coxeter group is known to be regular, see [@bri].\
We note that it was proved in [@mecel_okninski] that the following conditions are equivalent: 1) $k[\operatorname{HK}_{\Theta}]$ is a PI-algebra, 2) $\operatorname{HK}_{\Theta}$ does not contain a noncommutative free submonoid, 3) $\operatorname{GKdim}( k[\operatorname{HK}_{\Theta}])$ is finite, 4) $\Theta$ does not contain two different oriented cycles connected by an oriented path. Theorem \[main\] answers a question raised in [@mecel_okninski].\
The key method used to obtain this result is the description of a Gröbner basis of Hecke-Kiselman algebras. It is known that if the leading terms of the elements of this basis form a regular subset of the corresponding free monoid, then the algebra is automaton, see [@ufnar], Theorem 2 on page 97. Consequently, our methods involve the monoid $\operatorname{HK}_{\Theta}$ only, rather than certain ring theoretical aspects of the algebra $k[\operatorname{HK}_{\Theta}]$. The obtained Gröbner basis is crucial for the approach to the structure of such algebras, which will be pursued in a forthcoming paper.\
The class of automaton algebras was introduced by Ufnarovskii in [@ufn]. The main motivation was to study a class of finitely generated algebras that generalizes the class of algebras that admit a finite Gröbner basis with respect to some choice of generators and an ordering on monomials. The difficulty here lies in the fact that there are infinitely many generating sets as well as infinitely many admissible orderings on monomials to deal with. There are examples of algebras with finite Gröbner bases with respect to one ordering, and infinite bases with respect to the other. Up until recently it was not known whether for any of known examples of automaton algebras with infinite Gröbner bases with respect to certain orderings one could find a better ordering that would yield a finite Gröbner basis. First counterexamples were found by Iyudu and Shkarin in [@iyu].\
There are many results indicating that the class of automaton algebras not only has better computational properties but also several structural properties that are better than in the class of arbitrary finitely generated algebras. For example, in this context one can refer to results on the Gelfand-Kirillov dimension, results on the radical in the case of monomial automaton algebras [@ufn], results on prime algebras of this type [@bell], and also structural results concerned with the special case of finitely presented monomial algebras [@okn]. In particular, finitely generated algebras of the following types are automaton: commutative algebras, algebras defined by not more than two quadratic relations, algebras for which all the defining relations have the form $[x_ix_j] = 0$, for some pairs of generators, see [@ufnar]. Moreover, algebras that are finite modules over commutative finitely generated subalgebras are also of this type [@cedo_okninski]. Several aspects of automaton algebras have been recently studied also in [@iyu], [@piontkovski1], [@piontkovski2].\
In Section 2 we introduce the necessary definitions and auxiliary results. Next, in Section 3, we determine a Gröbner basis of $k[\operatorname{HK}_{\Theta}]$, from which the main result follows. Finally, in Section 4, we prove that in the case when the graph $\Theta$ is a cycle, $k[\operatorname{HK}_{\Theta}]$ has a finite Gröbner basis. An example is given to show that this is not true for arbitrary Hecke-Kiselman algebras of oriented graphs, even in the case when the algebra satisfies a polynomial identity.
Definitions and the necessary background {#dwa}
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Let $F$ denote the free monoid on the set $X$ of $n\ge
3$ free generators $x_1,\dotsc,x_n$. Let $k$ be a field and let $k[F] =k\langle x_1,\dotsc,x_n\rangle$ denote the corresponding free algebra over $k$. Assume that a well order $<$ is fixed on $X$ and consider the induced degree-lexicographical order on $F$ (also denoted by $<$). Let $A$ be a finitely generated algebra over $k$ with a set of generators $r_1, \ldots,
r_n$ and let $\pi: k[F] \to A$ be the natural homomorphism of $k$-algebras with $\pi(x_i) = r_i$. We will assume that $\ker
(\pi)$ is spanned by elements of the form $w-v$, where $w,v\in F$ (in other words, $A$ is a semigroup algebra). Let $I$ be the ideal of $F$ consisting of all leading monomials of $\ker(\pi)$. The set of normal words corresponding to the chosen presentation for $A$ and to the chosen order on $F$ is defined by $N(A) = F \setminus I$. One says that $A$ is an automaton algebra if $N(A)$ is a regular language. That means that this set is obtained from a finite subset of $F$ by applying a finite sequence of operations of union, multiplication and operation $*$ defined by $T^* = \bigcup_{i \geq 1}T^i$, for $T \subseteq F$. If $T = \{w\}$ for some $w \in F$, then we write $T^* = w^*$.\
For every $x \in X$ and $w \in F$ by $|w|_x$ we mean the number of occurrences of $x$ in $w$. By $|w|$ we denote the length of the word $w$. The support of the word $w$, denoted by $\operatorname{supp}(w)$, stands for the set of all $x \in X$ such that $|w|_x >
0$. We say that the word $w = x_1\cdots x_r \in F$ is a subword of the word $v \in F$, where $x_i \in X$, if $ v = v_1x_1\cdots v_r
x_r v_{r+1}$, for some $v_1, \ldots, v_{r+1} \in F$. If $v_2,
\ldots, v_r$ are trivial words, then we say that $w$ is a factor of $v$.\
Describing the normal words of a finitely generated algebra $A$ is related to finding a Gröbner basis of the ideal $J=\ker(\pi)$. Recall that a subset $G$ of $J$ is called a Gröbner basis of $J$ (or of $A$) if $0\notin G$, $J$ is generated by $G$ as an ideal and for every nonzero $f\in J$ there exists $g\in G$ such that the leading monomial $\overline{g}\in F$ of $g$ is a factor of the leading monomial $\overline{f}$ of $f$. If $G$ is a Gröbner basis of $A$, then a word $w\in F$ is normal if and only if $w$ has no factors that are leading monomials in $g\in G$.\
The so-called diamond lemma is often used in this context. We will follow the approach and terminology of [@ber]. By a reduction in $k[F]$ determined by a pair $(w,w')\in F^2$, where $w' < w$ (the deg-lex order of $F$), we mean any operation of replacing a factor $w$ in a word $f \in F$ by the factor $w'$. For a set $T \subseteq F^2$ of such pairs (these pairs will be called reductions as well) we say that the word $f \in F$ is $T$-reduced if no factor of $f$ is the leading term $w$ of a reduction $(w, w')$ from the set $T$. The deg-lex order on $F$ satisfies the descending chain condition, which means there is no infinite decreasing chain of elements in $F$. This means that a $T$-reduced form of a word $w \in F$ can always be obtained in a finite series of steps. The linear space spanned by $T$-reduced monomials in $k[F]$ is denoted by $R(T)$.\
The diamond lemma gives necessary and sufficient conditions for the set $N(A)$ of normal words to coincide with the set of $T$-reduced words in $F$. The key tool is the notion of ambiguity. Let $\sigma = (w_{\sigma}, v_{\sigma})$, $\tau = (w_{\tau}, v_{\tau})$ be reductions in $T$. By an overlap ambiguity we mean a quintuple $(\sigma, \tau, l, w, r)$, where $1
\neq l, w, r \in F$ are such that $w_{\sigma} = wr$ and $w_{\tau}
= lw$. A quintuple $(\sigma, \tau, l, w, r)$ is called an inclusive ambiguity if $w_{\sigma} = w$ and $w_{\tau} =
lwr$. For brevity we will denote these ambiguities as $l(wr)
= (lw)r$ and $l(w)r = (lwr)$, respectively. We will also say that they are of type $\sigma$-$\tau$. We say that the overlap (inclusive, respectively) ambiguity is resolvable if $v_{\tau}r$ and $lv_{\sigma}$ ($v_{\tau}$ and $lv_{\sigma}r$, respectively) have equal $T$-reduced forms. We use the following simplified version of Bergman’s diamond lemma.
\[diamond\] Let $T$ be a reduction set in the free algebra $k[F]$ over a field $k$, with a fixed deg-lex order in the free monoid $F$ over $X$. Then the following conditions are equivalent:
- all ambiguities on $T$ are resolvable,
- each monomial $f \in F$ can be uniquely $T$-reduced,
- if $I(T)$ denotes the ideal of $k[F]$ generated by $\{ w - v: (w, v) \in T \}$ then $k[F]=I(T) \oplus R(T)$ as vector spaces.
Moreover if the conditions above are satisfied then the $k$-algebra $A = k[F]/I(T)$ can be identified with $R(T)$ equipped with a $k$-algebra structure with $f \cdot g$ defined as the $T$-reduced form of $fg$, for $f,g\in R(T)$. In this case, $\{ w - v: (w, v) \in
T \}$ is a Gröbner basis of $A$.
Gröbner basis in the oriented graphs case
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In this section we will prove that for any oriented graph $\Theta = (V(\Theta), E(\Theta))$, the language of normal words of the Hecke-Kiselman algebra $k[\operatorname{HK}_{\Theta}]$ is regular, and thus that the algebra is always automaton.\
For $t \in V(\Theta)$ and $w \in F = \langle V(\Theta)\rangle$ we write $w
\nrightarrow t$ if $|w|_t = 0$ and there are no $x \in \operatorname{supp}(w)$ such that $x \rightarrow t$ in $\Theta$. Similarly, we define $t
\nrightarrow w$: again we assume that $|w|_t = 0$ and there is no arrow $t \to y$, where $y \in \operatorname{supp}(w)$. In the case when $t \nrightarrow w$ and $w \nrightarrow t$, we write $t \nleftrightarrow w$.
\[basis\] Let $\Theta$ be a finite simple oriented graph with vertices $V(\Theta) = \{x_1, x_2, \ldots, x_n\}$. Extend the natural ordering $x_1 < x_2 < \cdots < x_n$ on the set $V(\Theta) $ to the deg-lex order on the free monoid $F = \langle V(\Theta) \rangle$. Consider the following set $T$ of reductions on the algebra $k[F]$:
- $(twt, tw)$, for any $t \in V(\Theta)$ and $w \in F$ such that $w \nrightarrow t$,
- $(twt, wt)$, for any $t \in V(\Theta)$ and $w \in F$ such that $t \nrightarrow w$,
- $(t_1wt_2, t_2t_1w)$, for any $t_1, t_2 \in V(\Theta)$ and $w \in F$ such that $t_1 > t_2$ and $t_2 \nleftrightarrow t_1w$.
Then the set $\{w - v, \text{ where } (w, v) \in T\}$ forms a Gröbner basis of the algebra $k[\operatorname{HK}_{\Theta}]$.
Clearly, $w>v$ for every pair $(w,v)\in T$. Moreover, it is easy to see that $w$ and $v$ represent the same element of $\operatorname{HK}_{\Theta}$. It remains to use the diamond lemma. We will prove that all overlap and inclusive ambiguities of the reduction system $T$ are resolvable. We begin with a simple observation.
\[tosamo\] Assume that $t \in V(\Theta)$ and $w \in F$ are such that $t \nleftrightarrow w$. Then the words $tw$ and $wt$ have equal $T$-reduced forms.
We argue by induction on the length $|w|$ of $w$. If $w = 1$, the assertion is clear. If $w \in V(\Theta)$, then we either have $tw {\xrightarrow}{(iii)} wt$ or $wt {\xrightarrow}{(iii)} tw$, since $t \nleftrightarrow w$. We proceed with the induction step. Assume that $w =
y_1 \cdots y_k$, where $y_i \in V(\Theta)$, for $i = 1, \ldots,
k$. If $y_1 > t$, then we apply (iii) to $wt$ and we are done. If there exists $i > 1$ such that $y_i > t$, then $y_1\cdots y_kt
{\xrightarrow}{(iii)} y_1\cdots y_{i-1}ty_{i}\cdots y_k$. Now we apply the induction hypothesis to the words $ty_1\cdots y_{i-1}$ and $y_1\cdots y_{i-1}t$ and $T$-reduce them to some $w' \in F$. Thus we get that $tw$ and $wt$ can be both $T$-reduced to $w'y_{i}\cdots y_k$. Finally, if $y_i < t$, for all $i$, then by using reduction (iii) $k$ times we get: $$ty_1\cdots y_k
{\xrightarrow}{(iii)} y_1ty_2 \cdots y_k {\xrightarrow}{(iii)} y_1y_2ty_3\cdots y_k
{\xrightarrow}{(iii)} \cdots {\xrightarrow}{(iii)} y_1\cdots y_kt.$$
We will now list overlap and inclusive ambiguities of all possible types (x)-(y) of pairs of reductions in $T$, where (x), (y) $\in \{$(i), (ii), (iii)$\}$.\
There are two overlap and one inclusive ambiguity of type (i)-(i):
1. $tw_1(tw_2t) = (tw_1t)w_2t$, for $t \in V(\Theta)$ and $w_1, w_2 \in F$ such that $w_1w_2 \nrightarrow t$,
2. $ t_1w_1(t_2w_2t_1w_3t_2) = (t_1w_1t_2w_2t_1)w_3t_2$, for $t_1, t_2 \in V(\Theta)$, $w_1, w_2, w_3 \in F$ such that $w_1t_2w_2 \nrightarrow t_1$ and $w_2t_1w_3 \nrightarrow t_2$,
3. $(t_1w_1t_2w_2t_2w_3t_1) = t_1w_1(t_2w_2t_2)w_3t_1$, for $t_1, t_2 \in V(\Theta)$, $w_1, w_2, w_3 \in F$ such that $w_1t_2w_2t_2w_3 \nrightarrow t_1$ and $w_2 \nrightarrow t_2$.
There are two overlap and one inclusive ambiguity of type (i)-(ii):
4. $ tw_1(tw_2t) = (tw_1t)w_2t$, for $t \in V(\Theta)$, $w_1, w_2 \in F$ such that $t \nrightarrow w_1$, $w_2 \nrightarrow t$,
5. $ t_1w_1(t_2w_2t_1w_3t_2) = (t_1w_1t_2w_2t_1)w_3t_2$, for $t_1, t_2 \in V(\Theta)$ and $w_1, w_2, w_3 \in F$ such that $w_2t_1w_3 \nrightarrow t_2$, $t_1 \nrightarrow w_1t_2w_2$,
6. $(t_1w_1t_2w_2t_2w_3t_1) = t_1w_1(t_2w_2t_2)w_3t_1$, for $t_1, t_2 \in V(\Theta)$ and $w_1, w_2, w_3 \in F$ such that $w_1t_2w_2t_2w_3 \nrightarrow t_1$ and $t_2 \nrightarrow w_2$.
There are two overlap and three inclusive ambiguities of type (i)-(iii):
7. $ t_1w_1(t_2w_2t_2) = (t_1w_1t_2)w_2t_2$, for $t_1, t_2 \in V(\Theta)$, $w_1, w_2 \in F$ such that $t_1 > t_2$, $w_2 \nrightarrow t_2$ and $t_2 \nleftrightarrow t_1w_1$,
8. $t_1w_1(t_2w_2t_3w_3t_2) = (t_1w_1t_2w_2t_3)w_3t_2$, for $t_1, t_2, t_3 \in V(\Theta)$, $w_1, w_2, w_3 \in F$ such that $t_1 > t_3,$ $w_2t_3w_3 \nrightarrow t_2$ and $t_3 \nleftrightarrow t_1w_1t_2w_2$,
9. $(t_1w_1t_2w_2t_1) = (t_1w_1t_2)w_2t_1$, for $t_1, t_2 \in V(\Theta)$, $w_1, w_2 \in F$ such that $t_1 > t_2$, $w_1t_2w_2 \nrightarrow t_1$ and $t_2 \nleftrightarrow t_1w_1$,
10. $(t_1w_1t_2w_2t_3w_3t_1) = t_1w_1(t_2w_2t_3)w_3t_1$, for $t_1, t_2, t_3 \in V(\Theta)$, $w_1, w_2, w_3 \in F$ such that $t_2 > t_3,$ $w_1t_2w_2t_3w_3 \nrightarrow t_1$ and $t_3 \nleftrightarrow t_2w_2$,
11. $(t_1w_1t_2w_2t_1) = t_1w_1(t_2w_2t_1)$, for $t_1, t_2 \in V(\Theta)$, $w_1, w_2 \in F$ such that $t_2 > t_1$, $w_1t_2w_2 \nrightarrow t_1$ and $t_1 \nleftrightarrow t_2w_2$.
There are two overlap and one inclusive ambiguity of type (ii)-(i):
12. $ tw_1(tw_2t) = (tw_1t)w_2t$, for $t \in V(\Theta)$, $w_1, w_2 \in F$ such that $w_1 \not\to t$, $t \nrightarrow w_2$,
13. $ t_1w_1(t_2w_2t_1w_3t_2) = (t_1w_1t_2w_2t_1)w_3t_2$, for $t_1, t_2 \in V(\Theta)$ and $w_1, w_2, w_3 \in F$ such that $t_2 \nrightarrow w_2t_1w_3$, $w_1t_2w_2 \nrightarrow t_1$,
14. $(t_1w_1t_2w_2t_2w_3t_1) = t_1w_1(t_2w_2t_2)w_3t_1$, for $t_1, t_2 \in V(\Theta)$ and $w_1, w_2, w_3 \in F$ such that $t_1 \nrightarrow w_1t_2w_2t_2w_3$ and $w_2 \nrightarrow t_2$.
There are two overlap and one inclusive ambiguity of type (ii)-(ii):
15. $tw_1(tw_2t) = (tw_1t)w_2t$, for $t \in V(\Theta)$ and $w_1, w_2 \in F$ such that $t \nrightarrow w_1w_2$,
16. $ t_1w_1(t_2w_2t_1w_3t_2) = (t_1w_1t_2w_2t_1)w_3t_2$, for $t_1, t_2 \in V(\Theta)$, $w_1, w_2, w_3 \in F$ such that $t_1 \nrightarrow w_1t_2w_2$ and $t_2 \nrightarrow w_2t_1w_3 $,
17. $(t_1w_1t_2w_2t_2w_3t_1) = t_1w_1(t_2w_2t_2)w_3t_1$, for $t_1, t_2 \in V(\Theta)$, $w_1, w_2, w_3 \in F$ such that $t_1 \nrightarrow w_1t_2w_2t_2w_3$ and $t_2 \nrightarrow w_2$.
There are two overlap and three inclusive ambiguities of type (ii)-(iii):
18. $ t_1w_1(t_2w_2t_2) = (t_1w_1t_2)w_2t_2$, for $t_1, t_2 \in V(\Theta)$, $w_1, w_2 \in F$ such that $t_1 > t_2,$ $t_2 \nrightarrow w_2$ and $t_2 \nleftrightarrow t_1w_1$,
19. $t_1w_1(t_2w_2t_3w_3t_2) = (t_1w_1t_2w_2t_3)w_3t_2$, for $t_1, t_2, t_3 \in V(\Theta)$, $w_1, w_2, w_3 \in F$ such that $t_1 > t_3$, $t_2 \nrightarrow w_2t_2w_3$ and $t_3 \nleftrightarrow t_1w_1t_2w_2$,
20. $(t_1w_1t_2w_2t_1) = (t_1w_1t_2)w_2t_1$, for $t_1, t_2 \in V(\Theta)$, $w_1, w_2 \in F$ such that $t_1 > t_2$, $t_1 \nrightarrow w_1t_2w_2$ and $t_2 \nleftrightarrow t_1w_1$,
21. $(t_1w_1t_2w_2t_3w_3t_1) = t_1w_1(t_2w_2t_3)w_3t_1$, for $t_1, t_2, t_3 \in V(\Theta)$, $w_1, w_2, w_3 \in F$ such that $t_2 > t_3,$ $t_1 \nrightarrow w_1t_2w_2t_3w_3$ and $t_3 \nleftrightarrow t_2w_2$,
22. $(t_1w_1t_2w_2t_1) = t_1w_1(t_2w_2t_1)$, for $t_1, t_2 \in V(\Theta)$, $w_1, w_2 \in F$ such that $t_2 > t_1$, $t_1 \nrightarrow w_1t_2w_2$ and $t_1 \nleftrightarrow t_2w_2$.
There are two overlap and two inclusive ambiguities of type (iii)-(i):
23. $t_1w_1(t_1w_2t_2) = (t_1w_1t_1)w_2t_2$, for $t_1, t_2 \in V(\Theta)$, $w_1, w_2 \in F$ such that $t_1 > t_2,$ $w_1 \nrightarrow t_1$ and $t_2 \nleftrightarrow t_1w_2$,
24. $t_1w_1(t_2w_2t_1w_3t_3) = (t_1w_1t_2w_2t_1)w_3t_3$, for $t_1, t_2, t_3 \in V(\Theta)$, $w_1, w_2, w_3 \in F$ such that $t_2 > t_3,$ $w_1t_2w_2 \nrightarrow t_1$ and $t_3 \nleftrightarrow t_2w_2t_1w_3$,
25. $(t_1w_1t_1w_2t_2) = (t_1w_1t_1)w_2t_2,$ for $t_1, t_2 \in V(\Theta)$, $w_1, w_2 \in F$ such that $t_1 > t_2$, $w_1 \nrightarrow t_1$ and $t_2 \nleftrightarrow t_1w_1t_1w_2$,
26. $(t_1w_1t_2w_2t_2w_3t_3) = t_1w_1(t_2w_2t_2)w_3t_3$, for $t_1, t_2, t_3 \in V(\Theta)$, $w_1, w_2, w_3 \in F$ such that $t_1 > t_3$, $w_2 \nrightarrow t_2$ and $t_3 \nleftrightarrow t_1w_1t_2w_2t_2w_3$.
There are two overlap and two inclusive ambiguities of type (iii)-(ii):
27. $t_1w_1(t_1w_2t_2) = (t_1w_1t_1)w_2t_2$, for $t_1, t_2 \in V(\Theta)$, $w_1, w_2 \in F$ such that $t_1 > t_2,$ $t_1 \nrightarrow w_1$, $t_2 \nleftrightarrow t_1w_2$,
28. $t_1w_1(t_2w_2t_1w_3t_3) = (t_1w_1t_2w_2t_1)w_3t_3$, for $t_1, t_2, t_3 \in V(\Theta)$, $w_1, w_2, w_3 \in F$ such that $t_2 > t_3$, $t_1 \nrightarrow w_1t_2w_2$ and $t_3 \nleftrightarrow t_2w_2t_1w_3$,
29. $(t_1w_1t_1w_2t_2) = (t_1w_1t_1)w_2t_2,$ for $t_1, t_2 \in V(\Theta)$, $w_1, w_2 \in F$ such that $t_1 > t_2$, $t_1 \nrightarrow w_1$ and $t_2 \nleftrightarrow t_1w_1t_1w_2$,
30. $(t_1w_1t_2w_2t_2w_3t_3) = t_1w_1(t_2w_2t_2)w_3t_3$, for $t_1, t_2, t_3 \in V(\Theta)$, $w_1, w_2, w_3 \in F$ such that $t_1 > t_3$, $t_2 \nrightarrow w_2$ and $t_3 \nleftrightarrow t_1w_1t_2w_2t_2w_3$.
There are two overlap and three inclusive ambiguities of type (iii)-(iii):
31. $ t_1w_1(t_2w_2t_3) = (t_1w_1t_2)w_2t_3$, for $t_1, t_2, t_3 \in V(\Theta)$, $w_1, w_2 \in F$ such that $t_1 > t_2$, $t_2 > t_3$, $t_3 \nleftrightarrow t_2w_2$ and $t_2 \nleftrightarrow t_1w_1$,
32. $t_1w_1(t_2w_2t_3w_3t_4) = (t_1w_1t_2w_2t_3)w_3t_4$, for $t_1, t_2, t_3, t_4 \in V(\Theta)$, $w_1, w_2, w_3 \in F$ such that $t_1 > t_3,$ $t_2 > t_4$, $t_4 \nleftrightarrow t_2w_2t_3w_3$ and $t_3 \nleftrightarrow t_1w_1t_2w_2$,
33. $(t_1w_1t_2w_2t_3) = (t_1w_1t_2)w_2t_3$, for $t_1, t_2, t_3 \in V(\Theta)$, $w_1, w_2 \in F$ such that $t_1 > t_2$, $t_1 > t_3$ and $t_3 \nleftrightarrow t_1w_1t_2w_2$, $t_2 \nleftrightarrow t_1w_1$,
34. $(t_1w_1t_2w_2t_3w_3t_4) = t_1w_1(t_2w_2t_3)w_3t_4$, for $t_1, t_2, t_3, t_4 \in V(\Theta)$, $w_1, w_2, w_3 \in F$ such that $t_1 > t_4$, $t_2 > t_3$, $t_4\nleftrightarrow t_1w_1t_2w_2t_3w_3$ and $t_3 \nleftrightarrow t_2w_2$,
35. $(t_1w_1t_2w_2t_3) = t_1w_1(t_2w_2t_3)$, for $t_1, t_2, t_3 \in V(\Theta)$. $w_1, w_2 \in F$ such that $t_1 > t_3, t_2 > t_3$, $t_3 \nleftrightarrow t_1w_1t_2w_2$.
We will now solve these ambiguities.
1. $tw_1(tw_2t) {\xrightarrow}{(i)} tw_1tw_2 {\xrightarrow}{(i)} tw_1w_2$,\
$(tw_1t)w_2t {\xrightarrow}{(i)} tw_1w_2t {\xrightarrow}{(i)} tw_1w_2$.
2. $t_1w_1(t_2w_2t_1w_3t_2) {\xrightarrow}{(i)} t_1w_1t_2w_2t_1w_3 {\xrightarrow}{(i)} t_1w_1t_2w_2w_3,$\
$(t_1w_1t_2w_2t_1)w_3t_2 {\xrightarrow}{(i)} t_1w_1t_2w_2w_3t_2 {\xrightarrow}{(i)} t_1w_1t_2w_2w_3,$ since $w_2w_3 \nrightarrow t_2$.
3. $(t_1w_1t_2w_2t_2w_3t_1) {\xrightarrow}{(i)} t_1w_1t_2w_2t_2w_3 {\xrightarrow}{(i)} t_1w_1t_2w_2w_3$,\
$t_1w_1(t_2w_2t_2)w_3t_1 {\xrightarrow}{(i)} t_1w_1t_2w_2w_3t_1 {\xrightarrow}{(i)} t_1w_1t_2w_2w_3,$ since $w_1t_2w_2w_3 \nrightarrow t_1$.
4. $tw_1(tw_2t) {\xrightarrow}{(i)} tw_1tw_2 {\xrightarrow}{(ii)} w_1tw_2,$\
$(tw_1t)w_2t {\xrightarrow}{(ii)} w_1tw_2t {\xrightarrow}{(i)} w_1tw_2$.
5. $t_1w_1(t_2w_2t_1w_3t_2) {\xrightarrow}{(i)} t_1w_1t_2w_2t_1w_3 {\xrightarrow}{(ii)} w_1t_2w_2t_1w_3,$\
$(t_1w_1t_2w_2t_1)w_3t_2 {\xrightarrow}{(ii)}w_1t_2w_2t_1w_3t_2 {\xrightarrow}{(i)} w_1t_2w_2t_1w_3.$
6. $(t_1w_1t_2w_2t_2w_3t_1) {\xrightarrow}{(i)} t_1w_1t_2w_2t_2w_3 {\xrightarrow}{(ii)} t_1w_1w_2t_2w_3,$\
$t_1w_1(t_2w_2t_2)w_3t_1 {\xrightarrow}{(ii)}t_1w_1w_2t_2w_3t_1 {\xrightarrow}{(i)} t_1w_1w_2t_2w_3,$ since $w_1w_2t_2w_3 \nrightarrow t_1$.
7. $t_1w_1(t_2w_2t_2) {\xrightarrow}{(i)} t_1w_1t_2w_2 {\xrightarrow}{(iii)} t_2t_1w_1w_2$,\
$(t_1w_1t_2)w_2t_2 {\xrightarrow}{(iii)} t_2t_1w_1w_2t_2$. Since $t_2 \nleftrightarrow t_1w_1$, then $t_1w_1w_2 \nrightarrow t_2$ and we have:\
$t_2t_1w_1w_2t_2 {\xrightarrow}{(i)} t_2t_1w_1w_2.$
8. $t_1w_1(t_2w_2t_3w_3t_2) {\xrightarrow}{(i)} t_1w_1t_2w_2t_3w_3 {\xrightarrow}{(iii)} t_3t_1w_1t_2w_2w_3,$\
$(t_1w_1t_2w_2t_3)w_3t_2 {\xrightarrow}{(iii)} t_3t_1w_1t_2w_2w_3t_2 {\xrightarrow}{(i)} t_3t_1w_1t_2w_2w_3$, since $w_2w_3 \nrightarrow t_2$.
9. $(t_1w_1t_2w_2t_1) {\xrightarrow}{(i)} t_1w_1t_2w_2 {\xrightarrow}{(iii)} t_2t_1w_1w_2$,\
$(t_1w_1t_2)w_2t_1 {\xrightarrow}{(iii)} t_2t_1w_1w_2t_1 {\xrightarrow}{(i)} t_2t_1w_1w_2$, since $w_1w_2 \nrightarrow t_1$.
10. $(t_1w_1t_2w_2t_3w_3t_1) {\xrightarrow}{(i)} t_1w_1t_2w_2t_3w_3 {\xrightarrow}{(iii)} t_1w_1t_3t_2w_2w_3$,\
$t_1w_1(t_2w_2t_3)w_3t_1 {\xrightarrow}{(iii)} t_1w_1t_3t_2w_2w_3t_1 {\xrightarrow}{(i)} t_1w_1t_3t_2w_2w_3$, since $w_1t_3t_2w_2w_3 \nrightarrow t_1$.
11. $(t_1w_1t_2w_2t_1) {\xrightarrow}{(i)} t_1w_1t_2w_2$,\
$t_1w_1(t_2w_2t_1) {\xrightarrow}{(iii)} t_1w_1t_1t_2w_2 {\xrightarrow}{(i)} t_1w_1t_2w_2$, since $w_1 \nrightarrow t_1$.
12. $tw_1(tw_2t) {\xrightarrow}{(ii)} tw_1w_2t$,\
$(tw_1t)w_2t {\xrightarrow}{(i)} tw_1w_2t$.
13. $t_1w_1(t_2w_2t_1w_3t_2) {\xrightarrow}{(ii)} t_1w_1w_2t_1w_3t_2 {\xrightarrow}{(i)} t_1w_1w_2w_3t_2$, since $w_1w_2 \nrightarrow t_1$,\
$(t_1w_1t_2w_2t_1)w_3t_2 {\xrightarrow}{(i)} t_1w_1t_2w_2w_3t_2 {\xrightarrow}{(ii)} t_1w_1w_2w_3t_2$, since $t_2 \nrightarrow w_2w_3.$
14. $(t_1w_1t_2w_2t_2w_3t_1) {\xrightarrow}{(ii)} w_1t_2w_2t_2w_3t_1 {\xrightarrow}{(i)} w_1t_2w_2w_3t_1$,\
$t_1w_1(t_2w_2t_2)w_3t_1 {\xrightarrow}{(i)} t_1w_1t_2w_2w_3t_1 {\xrightarrow}{(ii)} w_1t_2w_2w_3t_1$, since $t_1 \nrightarrow w_1t_2w_2w_3$.
15. $tw_1(tw_2t) {\xrightarrow}{(ii)} tw_1w_2t {\xrightarrow}{(ii)} w_1w_2t$,\
$(tw_1t)w_2t {\xrightarrow}{(ii)} w_1tw_2t {\xrightarrow}{(ii)} w_1w_2t$.
16. $t_1w_1(t_2w_2t_1w_3t_2) {\xrightarrow}{(ii)} t_1w_1w_2t_1w_3t_2 {\xrightarrow}{(ii)} w_1w_2t_1w_3t_2$, since $t_1 \nrightarrow w_1w_2$,\
$(t_1w_1t_2w_2t_1)w_3t_2 {\xrightarrow}{(ii)} w_1t_2w_2t_1w_3t_2 {\xrightarrow}{(ii)} w_1w_2t_1w_3t_2$.
17. $(t_1w_1t_2w_2t_2w_3t_1) {\xrightarrow}{(ii)} w_1t_2w_2t_2w_3t_1 {\xrightarrow}{(ii)} w_1w_2t_2w_3t_1$,\
$t_1w_1(t_2w_2t_2)w_3t_1 {\xrightarrow}{(ii)} t_1w_1w_2t_2w_3t_1 {\xrightarrow}{(ii)} w_1w_2t_2w_3t_1$, since $t_1 \nrightarrow w_1w_2t_2w_3$.
18. $t_1w_1(t_2w_2t_2) {\xrightarrow}{(ii)} t_1w_1w_2t_2$,\
$(t_1w_1t_2)w_2t_2 {\xrightarrow}{(iii)} t_2t_1w_1w_2t_2$. Since $t_2 \nleftrightarrow t_1w_1$, we have $t_2 \nrightarrow t_1w_1w_2$ and thus:\
$t_2t_1w_1w_2t_2 {\xrightarrow}{(ii)} t_1w_1w_2t_2.$
19. $t_1w_1(t_2w_2t_3w_3t_2) {\xrightarrow}{(ii)} t_1w_1w_2t_3w_3t_2 {\xrightarrow}{(iii)} t_3t_1w_1w_2w_3t_2$, since $t_1 > t_3$ and $t_3 \nleftrightarrow t_1w_1w_2$,\
$(t_1w_1t_2w_2t_3)w_3t_2 {\xrightarrow}{(iii)} t_3t_1w_1t_2w_2w_3t_2 {\xrightarrow}{(ii)} t_3t_1w_1w_2w_3t_2$, since $t_2 \nrightarrow w_2w_3$.
20. $(t_1w_1t_2w_2t_1) {\xrightarrow}{(ii)} w_1t_2w_2t_1$,\
$(t_1w_1t_2)w_2t_1 {\xrightarrow}{(iii)} t_2t_1w_1w_2t_1 {\xrightarrow}{(ii)} t_2w_1w_2t_1$.\
Since $t_2 \nleftrightarrow w_1$, we can use Observation \[tosamo\] to reduce $w_1t_2$ and $t_2w_1$ to the same form.
21. $(t_1w_1t_2w_2t_3w_3t_1) {\xrightarrow}{(ii)} w_1t_2w_2t_3w_3t_1 {\xrightarrow}{(iii)} w_1t_3t_2w_2w_3t_1$,\
$t_1w_1(t_2w_2t_3)w_3t_1 {\xrightarrow}{(iii)} t_1w_1t_3t_2w_2w_3t_1 {\xrightarrow}{(ii)} w_1t_3t_2w_2w_3t_1$.
22. $(t_1w_1t_2w_2t_1) {\xrightarrow}{(ii)} w_1t_2w_2t_1 {\xrightarrow}{(iii)} w_1t_1t_2w_2$,\
$t_1w_1(t_2w_2t_1) {\xrightarrow}{(iii)} t_1w_1t_1t_2w_2 {\xrightarrow}{(ii)} w_1t_1t_2w_2,$ since $t_1 \nrightarrow w_1$.
23. $t_1w_1(t_1w_2t_2) {\xrightarrow}{(iii)} t_1w_1t_2t_1w_2$. Since $t_2 \nleftrightarrow t_1w_2$, we have $w_1t_2 \nrightarrow t_1$ and:\
$t_1w_1t_2t_1w_2 {\xrightarrow}{(i)} t_1w_1t_2w_2$.\
$(t_1w_1t_1)w_2t_2 {\xrightarrow}{(i)} t_1w_1w_2t_2$. Since $t_2 \nleftrightarrow w_2$, then by Observation \[tosamo\] we can reduce $t_2w_2$ and $w_2t_2$ to the same form.
24. $t_1w_1(t_2w_2t_1w_3t_3) {\xrightarrow}{(iii)} t_1w_1t_3t_2w_2t_1w_3$. Since $t_3 \nleftrightarrow t_2w_2t_1w_3$, then $w_1t_3t_2w_2 \nrightarrow t_1$ and:\
$t_1w_1t_3t_2w_2t_1w_3 {\xrightarrow}{(i)} t_1w_1t_3t_2w_2w_3$.\
$(t_1w_1t_2w_2t_1)w_3t_3 {\xrightarrow}{(i)} t_1w_1t_2w_2w_3t_3 {\xrightarrow}{(iii)} t_1w_1t_3t_2w_2w_3$, since $t_3 \nleftrightarrow t_2w_2w_3$ and $t_2 > t_3$.
25. $(t_1w_1t_1w_2t_2) {\xrightarrow}{(iii)} t_2t_1w_1t_1w_2 {\xrightarrow}{(i)} t_2t_1w_1w_2$,\
$(t_1w_1t_1)w_2t_2 {\xrightarrow}{(i)} t_1w_1w_2t_2 {\xrightarrow}{(iii)} t_2t_1w_1w_2$, since $t_1 > t_2$ and $t_2 \nleftrightarrow t_1w_1w_2$.
26. $(t_1w_1t_2w_2t_2w_3t_3) {\xrightarrow}{(iii)} t_3t_1w_1t_2w_2t_2w_3 {\xrightarrow}{(i)} t_3t_1w_1t_2w_2w_3$,\
$t_1w_1(t_2w_2t_2)w_3t_3 {\xrightarrow}{(i)} t_1w_1t_2w_2w_3t_3 {\xrightarrow}{(iii)} t_3t_1w_1t_2w_2w_3$, since $t_3 \nleftrightarrow t_1w_1t_2w_2w_3$ and $t_1 > t_3$.
27. $t_1w_1(t_1w_2t_2) {\xrightarrow}{(iii)} t_1w_1t_2t_1w_2$. Since $t_2 \nleftrightarrow w_2$, we have $t_1 \nrightarrow w_1t_2$, and thus:\
$t_1w_1t_2t_1w_2 {\xrightarrow}{(ii)} w_1t_2t_1w_2$.\
$(t_1w_1t_1)w_2t_2 {\xrightarrow}{(ii)} w_1t_1w_2t_2 {\xrightarrow}{(iii)} w_1t_2t_1w_2$.
28. $t_1w_1(t_2w_2t_1w_3t_3) {\xrightarrow}{(iii)} t_1w_1t_3t_2w_2t_1w_3$. Since $t_3 \nleftrightarrow t_2w_2t_1w_3$, we have $t_1 \nrightarrow w_1t_3t_2w_2$ and:\
$t_1w_1t_3t_2w_2t_1w_3 {\xrightarrow}{(ii)} w_1t_3t_2w_2t_1w_3.$\
$(t_1w_1t_2w_2t_1)w_3t_3 {\xrightarrow}{(ii)} w_1t_2w_2t_1w_3t_3 {\xrightarrow}{(iii)} w_1t_3t_2w_2t_1w_3$.
29. $ (t_1w_1t_1w_2t_2) {\xrightarrow}{(iii)} t_2t_1w_1t_1w_2 {\xrightarrow}{(ii)} t_2w_1t_1w_2$,\
$(t_1w_1t_1)w_2t_2 {\xrightarrow}{(ii)} w_1t_1w_2t_2 {\xrightarrow}{(iii)} w_1t_2t_1w_2$, since $t_1 > t_2$ and $t_2 \nleftrightarrow t_1w_2$. Here, again we can see that $w_1 \nleftrightarrow t_2$ and thus $t_2w_1$ and $w_1t_2$ can be reduced to the same word, by Observation \[tosamo\].
30. $(t_1w_1t_2w_2t_2w_3t_3) {\xrightarrow}{(iii)} t_3t_1w_1t_2w_2t_2w_3 {\xrightarrow}{(ii)} t_3t_1w_1w_2t_2w_3$,\
$t_1w_1(t_2w_2t_2)w_3t_3 {\xrightarrow}{(ii)} t_1w_1w_2t_2w_3t_3 {\xrightarrow}{(iii)} t_3t_1w_1w_2t_2w_3$, since $t_3 \nleftrightarrow t_1w_1w_2t_2w_3$ and $t_1 > t_3$.
31. $t_1w_1(t_2w_2t_3) {\xrightarrow}{(iii)} t_1w_1t_3t_2w_2 {\xrightarrow}{(iii)} t_2t_1w_1t_3w_2$, since $t_1 > t_2$ and $t_2 \nleftrightarrow t_1w_1t_3$,\
$(t_1w_1t_2)w_2t_3, {\xrightarrow}{(iii)} t_2t_1w_1w_2t_3$.\
Since $t_3 \nleftrightarrow w_2$ then by Observation \[tosamo\] $t_3w_2$ and $w_2t_3$ can be reduced to the same form.
32. $t_1w_1(t_2w_2t_3w_3t_4) {\xrightarrow}{(iii)} t_1w_1t_4t_2w_2t_3w_3 {\xrightarrow}{(iii)} t_3t_1w_1t_4t_2w_2w_3$, since $t_3 \nleftrightarrow t_1w_1t_4t_2w_2$ and $t_1 > t_3$,\
$(t_1w_1t_2w_2t_3)w_3t_4 {\xrightarrow}{(iii)} t_3t_1w_1t_2w_2w_3t_4 {\xrightarrow}{(iii)} t_3t_1w_1t_4t_2w_2w_3$, since $t_4 \nleftrightarrow t_2w_2w_3$ and $t_2 > t_4$.
33. $ (t_1w_1t_2w_2t_3) {\xrightarrow}{(iii)} t_3t_1w_1t_2w_2 {\xrightarrow}{(iii)} t_3t_2t_1w_1w_2$, since $t_1 > t_2$ and $t_2 \nleftrightarrow t_1w_1$,\
$ (t_1w_1t_2)w_2t_3 {\xrightarrow}{(iii)} t_2t_1w_1w_2t_3 {\xrightarrow}{(iii)} t_2t_3t_1w_1w_2$, since $t_1 > t_3$ and $t_3 \nleftrightarrow t_1w_1w_2$.\
Since $t_2 \nleftrightarrow t_3$, we either have $t_2t_3 {\xrightarrow}{(iii)} t_3t_2$, or $t_3t_2 {\xrightarrow}{(iii)} t_2t_3$.
34. $(t_1w_1t_2w_2t_3w_3t_4) {\xrightarrow}{(iii)} t_4t_1w_1t_2w_2t_3w_3 {\xrightarrow}{(iii)} t_4t_1w_1t_3t_2w_2w_3$,\
$t_1w_1(t_2w_2t_3)w_3t_4 {\xrightarrow}{(iii)} t_1w_1t_3t_2w_2w_3t_4 {\xrightarrow}{(iii)} t_4t_1w_1t_3t_2w_2w_3$, since $t_1 > t_4$ and $t_4 \nleftrightarrow t_1w_1t_3t_2w_2w_3$.
35. $(t_1w_1t_2w_2t_3) {\xrightarrow}{(iii)} t_3t_1w_1t_2w_2,$\
$t_1w_1(t_2w_2t_3) {\xrightarrow}{(iii)} t_1w_1t_3t_2w_2 {\xrightarrow}{(iii)} t_3t_1w_1t_2w_2$, since $t_1 > t_3$ and $t_3 \nleftrightarrow t_1w_1$.
We have checked that all ambiguities of the reduction system $T$ are resolvable. Thus the diamond lemma can be applied and the result follows.
We are ready to prove our first main result.\
We have $$N_{\Theta} = N_{(i)} \cup N_{(ii)} \cup N_{(iii)},$$ where $N_{\Theta}$ stands for the set of leading terms in pairs from the set $T$ considered in Theorem \[basis\], and $N_{(i)}, N_{(ii)}, N_{(iii)}$ are the sets of leading terms from the three families (i), (ii), (iii) of reductions in $T$, respectively. We only need to show that the sets $N_{(i)}, N_{(ii)}, N_{(iii)}$ are regular.\
Indeed, observe that $N_{(i)} = \{tvt\, | \, t \in V(\Theta), v \in F, v \nrightarrow t\} =
\bigcup\limits_{t \in V(\Theta)}\{tvt \mid v \nrightarrow t\}$ which is a finite union of sets of the form $t\langle Y_t \rangle t$, where $Y_t$ is the subset of $V(\Theta)$ consisting of all generators $z$ such that $z \nrightarrow t$. All these summands are clearly regular. Thus $N_{(i)}$ is regular. A similar argument works for $N_{(ii)}$.\
Finally, $$N_{(iii)} = \bigcup\limits_{x,z \in V(\Theta), x<z, x\nleftrightarrow z} z\langle X_x\rangle x,$$ where $X_x \subseteq V(\Theta)$ is the subset consisting of all generators $y \in V(\Theta)$ such that $y \nleftrightarrow x$. Again, these summands are clearly regular. Therefore, the set $N_{(iii)}$ is regular as a union of regular sets.\
As a result, the entire set $N_{\Theta}$ is regular and it is well known that this implies that the algebra $k[\operatorname{HK}_{\Theta}]$ is automaton, see [@ufnar], p. 97. The fact that $\operatorname{GKdim}(k[\operatorname{HK}_{\Theta}])$ is an integer, if it is finite, follows, see [@ufnar], Theorem 3 on page 97 and Theorem 1 on page 90.
Gröbner basis of a cycle monoid {#cycle}
===============================
Let $C_n$ denote the Hecke–Kiselman monoid associated to the cycle consisting of $n\ge 3$ vertices. The aim of this section is to prove that in the case of $k[C_n]$ one can find a finite subset of the Gröbner basis obtained in the previous section such that it itself forms a Gröbner basis of $k[C_n]$. Our interest in this special case comes from the fact that the structure of the algebra $k[C_{n}]$ is crucial for the study of an arbitrary algebra $k[\operatorname{HK}_{\Theta}]$.\
Recall that the monoid $C_n$ is defined by generators $x_1,\dotsc,x_n$ subject to the following relations: $$\begin{gathered}
x_i^2=x_i,\\
x_ix_{i+1}x_i=x_{i+1}x_ix_{i+1}=x_ix_{i+1},
\end{gathered}$$ for all $i=1,\dotsc,n$ (with the convention that indices are taken modulo $n$) and $$x_ix_j=x_jx_i$$ for all $i,j=1,\dotsc,n$ satisfying $1<i-j<n-1$ (note that for $n=3$ there are no relations of this type).\
The natural order $x_1 < x_2 < \cdots < x_n$ is considered on the set of generators and the corresponding deg-lex order on the free monoid $F$. We also adopt the following notation in this section. When we write a word of the form: $x_i \cdots
x_j$, we mean that consecutive generators from $x_i$ up to $x_j$ if $ i<j$ (or down to $x_{j}$, if $i
> j$) appear in this word. For instance, $x_2 \cdots x_5$ denotes $x_2x_3x_4x_5$ and $x_6\cdots x_3$ stands for the word $x_6x_5x_4x_3$.\
Consider two sets $S$ and $S'$ of reductions on $k[F]$. The first one is a subset of the system $T$ considered in the previous section that consists of all pairs of the form:
1. $(x_ix_i,x_i)$ for all $i\in\{1,\dotsc,n\}$,
2. $(x_jx_i,x_ix_j)$ for all $i,j\in\{1,\dotsc,n\}$ such that $1<j-i<n-1$,
3. $(x_n(x_1\dotsm x_i)x_j,x_jx_n(x_1\dotsm x_i))$ for all $i,j\in\{1,\dotsc,n\}$ such that $i+1<j<n-1$,
4. $(x_iux_i,x_iu)$ for all $i\in\{1,\dotsc,n\}$ and $1\ne u\in F$ such that $u \nrightarrow x_i$. Here, $i-1 = n$, for $i = 1$ (we say, for the sake of simplicity, that the word $x_iux_i$ is of type $(4x_i)$),
5. $(x_ivx_i,vx_i)$ for all $i\in\{1,\dotsc,n\}$ and $1\ne v\in F$ such that $x_i \nrightarrow u$. Here $i+1 = 1$, for $i =n$ (similarly, we say that the word $x_ivx_i$ is of type $(5x_i)$).
The second set of reductions is a subset $S'$ of $S$ consisting of:
- all pairs of type (1)-(3),
- all pairs $(x_iux_i,x_iu)$ of type (4) such that $|u|_{x_{j}}\le 1$, for $j\in\{1,\dotsc,n\}\setminus\{i,i-1\}$,
- all pairs $(x_ivx_i,vx_i)$ of type (5) such that $|v|_{x_{j}}\le 1$ for $j\in\{1,\dotsc,n\}\setminus\{i,i+1\}$,
- all pairs $(x_izx_i,zx_i)$ of type (5) such that $i < n$ and: $$x_izx_i = x_i(x_{i_1}\cdots x_{j_1})(x_{i_2}\cdots x_{j_2}) \cdots (x_{i_k} \cdots x_{j_k})x_n(x_1\cdots x_i),$$ where $i_1 < i_2 < \cdots < i_k < n$ and $j_1 < j_2 < \cdots < j_k < n$.
We will say that the word $x_iux_i$ that appears in (ii) is of type $(4x_{i}')$, the word $x_ivx_i$ that appears in (iii) is of type $(5x_{i}')$, and the word $x_izx_i$ that appears in (iv) is of type $(5x_{i}'')$. We will also say that a word $x \in F$ is of type (1), (2), or (3), respectively, if $x$ is the leading term of one of the reductions of the corresponding type.\
One can recognize reductions of type (1) and (4) as subsets of the reduction set (i) from Theorem \[main\]. Similarly, reductions of types (2), (3) are special cases of reductions of type (iii) and reductions of type (5) correspond to the subset (ii) of $T$. It is convenient to explicitly distinguish five families of reductions of the system $S$, as they will be repeatedly used in the process of reducing the size of the Gröbner basis obtained in the previous section.\
We will prove two facts concerning the reduction sets $S$ and $S'$.
\[rq\] Let $T$ be a reduction set on $k[C_n]$ obtained in Section 2. If $w \in F$ is $T$-reduced, then it is also $S$-reduced.
\[sprim\] Every $S'$-reduced word in $F$ is $S$-reduced.
The first lemma is a simple observation that is an intermediate step towards the main result of this section.\
Assume, to the contrary, that some word $w \in F$ is $S$-reduced, but not $T$-reduced. Clearly, it is enough to consider the case where $w$ is of the form (iii) from the definition of $T$, namely $v = x_kwx_i$, where $k > i$ and $x_i \nleftrightarrow x_kw$. We will use inductive argument to show that $v$ is not $S$-reduced, which leads to a contradiction.\
Of course, if $|w| = 0$ then $x_kx_i {\xrightarrow}{(2)} x_ix_k$, so $x_kx_i$ is $S$-reducible. We proceed with the inductive step. Let $|w| > 0$ and let $w = x_{i_1}\cdots x_{i_r}$, for some $x_{i_s} \in \{x_1,\ldots, x_n \} $ such that $x_{i_s} \nleftrightarrow x_i$, for $1 \leq s \leq r$. If for any $s$ we have $i_s > i$ then the factor $x_{i_s}\cdots x_{i_r}x_i$ is of the form (iii) and thus it is not $S$-reduced, by the induction hypothesis. So we only need to consider the case where $i_s \leq i < k$, for all $s$. In particular, we have $i_1 \leq i < k$. We consider two cases.\
Case 1. $k = n$. Here we must have $i_1 = 1$. Otherwise, an $S$-reducible factor $x_nx_{i_1}$ appears in $v$ and the induction step follows. If an $S$-reducible factor of the form (3) appears in $v$, then we are done, so we may only consider the case where $i_2 = 2, i_3 = 3, \ldots, x_r = r$. However, it follows that $r < i$, since $i_s < i$, for all $s$. Since $x_i \nleftrightarrow w = x_nx_1\cdots x_r$, we must have $i > r+1$, which means that $v$ is of the form (3) and $w$ is thus $S$-reducible. The induction step follows again.\
Case 2. $k < n$. In this case we either have $i_1 < k-1$ and an $S$-reducible factor $x_kx_{i_1}$ appears in $v$, which yields the induction step, or $i_1 = k - 1$. In the latter case we have $ v = x_kx_{k-1}x_{i_2} \cdots x_rx_i$. However now we can repeat the argument for $i_1$ to obtain that the only relevant case is $i_2 = k-2.$ Indeed, we have $i_2 \neq k-1$, $i_2 \neq k$ and $i_2 \leq i < k$. If we were to assume that $i_2 < k-2$, then the $S$-reducible factor $x_{k-1}x_{i_2}$ would appear in $v$, which would immediately yield the inductive step. After repeating this process we are left with the case when $v x_i= x_kx_{k-1}x_{k-2} \cdots x_m \cdot x_i$. However, since $k > i$ and $x_i \nleftrightarrow x_kw$, we have $k > m-1$, so we get and an $S$-reducible factor $x_mx_i$. Thus, the induction step follows again.\
We have shown that the word $v$ of the form (iii) is $S$-reducible, which yields a contradiction. The assertion follows.
Before proving Lemma \[sprim\], we will prove the following fact concerning certain special family of words.
\[pe\] Assume that $1 \neq p \in F$ is such that $|p|_{x_{n}} = 0$ and $p$ does not contain factors of the forms $(1)$-$(3)$, $(4x_i')$, $(5{x_i}')$, where $1 \leq i \leq n$. Then there exists $k \in \mathbb{N}$ such that $p$ is of the form: $$(x_{i_1}\cdots x_{j_1})(x_{i_2}\cdots x_{j_2}) \cdots (x_{i_k} \cdots x_{j_k}), \label{formp}$$ where $i_1 < i_2 < \cdots < i_k$ and $j_1 < j_2 < \cdots < j_k$, if $k > 1$.
We need some additional notation. We will say that a factor $v$ of a word $w
\in F$ is a block if $v$ is of the form $x_{i}\cdots x_{j}$, for some $1 \leq i,j < n$, but there is no factor $v'$ of $w$ such that $v$ is a factor of $v'$, the latter is also of the form $x_{i'}\cdots x_{j'}$, for some $1 \leq i',j' < n$, and $v \neq
v'$. The length of a block $v$ is defined as the number $|j-i+1|$. The block is called increasing if $i \leq j$ and decreasing if $i \geq j$ (note that $|p|_{x_{n}} = 0$).\
Take $p \neq 1$ such that $|p|_{x_{n}} = 0$. Since $p$ cannot have subwords of the form $x_j x_{j+1}x_{j}$ or $x_{j}x_{j-1}x_{j}$ (conditions $(4x_i')$, $(5{x_i}')$, respectively), it follows that $p$ is (in a unique way) a product of blocks and, by definition, the product of two consecutive blocks is not a block. If $p$ is a product of an exactly one block then there is nothing to prove – $p$ is of the form . Assume that $p$ is a product of at least two blocks and take two consecutive blocks of the form $(x_{i_s}\cdots
x_{j_s})(x_{i_{s+1}}\cdots x_{j_{s+1}})$. Observe first, that we cannot have $i_{s+1} \leq j_s +1$. Indeed, if $i_{s+1} <j_s - 1$, then a factor of type (2) would appear in $p$, a contradiction. If we had $i_{s+1} = j_s \pm 1$, then either the product of the two blocks $(x_{i_s}\cdots x_{j_s})(x_{i_{s+1}}\cdots x_{j_{s+1}})$ is a block itself, or a factor of one of the forms $x_{j_s}x_{j_{s}-1}x_{j_s}$, $x_{j_s}x_{j_{s}+1}x_{j_s}$ appears in $p$, again a contradiction. Of course, we cannot have $i_{s+1}
= j_s$, as this yields a factor of type (1) in $p$.
We will prove that $i_s < i_{s+1}$. Note that we cannot have $i_s = i_{s+1}$ since this immediately gives a factor $x_{i_s}\cdots x_{j_s}x_{i_{s}}$ of type $(4x_{i_s}')$ or $(5x_{i_s}')$ in $p$, a contradiction. Assume, to the contrary, that $i_{s+1} < i_{s}$. We already know that must have $i_{s+1} > j_s +1$, so $j_s +1 < i_{s+1} < i_{s}$ and thus the first block is decreasing of length $> 1$ and the factor of the form $x_{i_{s+1}} \cdots x_{j_s} x_{i_{s+1}}$ of type $(5x_{i_{s+1}}')$ appears in $p$, a contradiction. So $i_s <
i_{s+1}$. The inequality $j_s < j_{s+1}$ is proved in a completely analogous way.
Assume, to the contrary, that some word $w \in F$ is $S'$-reduced, but not $S$-reduced. We may choose $w$ to be minimal with respect to the deg-lex order on $F$. It is clear that $w$ may only be of the form $(4x_i)$ or $(5x_i)$.\
We will first consider the case $(4x_i)$; in other words $w = x_iux_i$, for some $u \neq 1$, $|u|_{x_{i-1}} = |u|_{x_{i}} = 0$ (if $i = 1$, then $i-1 = n$).\
First, observe that $i \neq n$. Indeed, if $i = n$, then as $w$ is S’-reduced and $|u|_{x_{n}} = 0$, $u$ is of the form and $$w = x_n(x_{i_1}\cdots x_{j_1})(x_{i_2}\cdots x_{j_2}) \cdots (x_{i_k} \cdots x_{j_k})x_n, \label{forman}$$ for some $k$ and $i_1 < i_2 < \cdots < i_k$ and $j_1 < j_2 <
\cdots < j_k$, if $k > 1$. As $|w|_{x_{n-1}} = 0$ and $x_nx_{i_1}$ cannot be of the form (2), we have $i_1 = 1$ and the first block of $u$ is increasing. If $k > 1$, however, then $i_2 > j_1 + 1$, since otherwise a factor $x_{i_2}\cdots x_{j_1}x_{i_2}$ of the form ($4x_{i_2}'$) appears in $w$, which is impossible. But if $i_2
\neq n-1$, then $x_nx_{i_1}\cdots x_{j_1}x_{i_2}$ is a factor of type (3) in $w$, a contradiction. Thus $k = 1$. In this case, however, $w = x_n(x_{1}\cdots x_{j_1})x_n$ is of the form $(4x_{n}')$, again a contradiction. Therefore $i \neq n$.\
Let $t = \max\{l: |u|_{x_{l}} \neq 0\}$. Of course, $t > 1$ as otherwise $w$ is $S'$-reducible. Moreover, $t > i$ since otherwise $w$ has a prefix $x_ix_m$ with $m<i-1$, which is a word of the form (2), a contradiction. We consider two cases: $1 < t
<n$ and $t=n$.
- Case 1. $1 < t < n$. Since $w$ is $S'$-reduced and $|w|_{x_{n}} = 0$, then by Observation \[pe\] $w$ is of the form , for some $k$ and $i_1 < \cdots < i_k$, $j_1 < \cdots < j_k$, if $k > 1$. But since $w$ is of the form $(4x_i)$, the first block of $w$ must begin with $x_i$, and the last block must end with $x_i$. If the length of the first block $x_i \cdots x_{j_1}$ was greater than 1, then this block must have been increasing, since $|w|_{x_{i-1}} = 0$. However, in this case $i = i_1 < j_1 \leq j_k = i$, which is impossible. Thus the first block of $w$ consists just of $x_i$. If $k = 1$, then $w = x_i$, a contradiction. If $k > 1$, then $j_k > j_1 $, which is impossible, as $i_1 = j_1 < j_k = i$. Again, a contradiction.
- Case 2. $t = n$. Then $i\neq 1$ because we are in the case $(4x_i)$. Consider the last appearance of $x_n$ in $w$, namely let $w = x_ipx_nqx_i$, where $p, q \in X$ and $|q|_{x_{n}} =
0$. First, assume that $q = 1$. Then $i$ must be equal to $n-1$ since otherwise we would have a factor of type (2) in $w$. Hence $w = x_{n-1}px_nx_{n-1}$. If $p = 1$, then $w$ is of type $(4x_{n-1}')$, which is impossible as $w$ is $S'$-reduced. Thus $p = x_np'$, since otherwise $w$ contains a factor $x_{n-1}x_s$ of type (2). Thus $w$ has a proper factor $x_np''x_n$ of type $(4x_n)$, contradicting the minimality of the word $w$. Thus we may assume that $q \neq 1$.
Since $w$ is $S'$-reduced, also $qx_i$ is $S'$-reduced and since $|qx_i|_{x_{n}} = 0$, as $i < n$, we can apply Observation \[pe\] and assume that it is of the form , for some $k$ and $i_1
< \cdots < i_k$ and $j_1 < \cdots < j_k$, if $k > 1$. However, since $x_nx_{i_1}$ is a factor of $w$ we must have $i_1 = n-1$ or $i_1 = 1$, as otherwise $w$ has a factor of type (2). We consider these subcases now:
- If $i_1 = n-1$, then there is only one block in the decomposition of $qx_i$, otherwise another block of $qx_i$ would have to begin with $x_{i_2}$, where $i_2 > i_1$ and also $n >i_2$. This is impossible. Therefore $w = x_ipx_nx_{n-1} \cdots x_i$. If $p = 1$ then $w$ is the form $(4x_{i}')$, a contradiction. Assume that $p \neq 1$. Then $x_ipx_n\cdots x_{i+1}$ cannot contain two occurrences of $x_{i+1}$ as that would yield a factor of the form $(4x_{i+1})$ in $w$, which contradicts its minimality. Thus $|x_ipx_n\cdots x_{i+2}|_{x_{i+1}} = 0$ and we can see that $x_ipx_n\cdots x_{i+2}$ cannot contain two occurrences of $x_{i+2}$. Continuing this way, we can see that $|p|_{x_{l}} = 0$, for $n \geq l > i-1$. Thus $p = x_mp'$, for some $p'$, for some $m
< i-1$ and thus we have a factor $x_ix_m$ of type (2) in $w$, a contradiction.
- If $i_1 = 1$, then $qx_i$ is of the form $(x_{1}\cdots x_{j_1}) \cdots (x_{i_k}
\cdots x_i)$. We cannot have $k = 1$, since in that case, we would have a factor of the form $x_1\cdots x_i$ in $w$. Its length would be greater than 1, since $i \neq 1$. Therefore, $w$ would contain $x_{i-1}$, a contradiction. If $k > 1$ then as in the case of words of the form we have $i_2 = n-1$. This easily implies that $k = 2$ and $w = x_ipx_n(x_{1}\cdots x_{j_1})(x_{n-1}
\cdots x_i)$. Next, if $p = 1$ then, since $1 \leq j_1 < i - 1$ the word $w$ is of the form $(4x_{i}')$, whence $w$ is $S'$-reducible, a contradiction. Let $p \neq 1$. As in the previous subcase, we can easily see that $|p|_{x_{l}}=0$ for $l=i+1, \ldots , n-1$. Again, if $|p|_{x_{n}} \neq 0$, the minimality of $w$ is violated, and thus $|p|_{x_{n}} = 0$. Therefore $p =
x_mp'$, for some $m < i-1$. As in the previous case, a factor $x_ix_m$ of type (2) appears in $w$, a contradiction.
We have proved that if $w$ is $S'$-reduced of the form $(4x_i)$, then it is also $S$-reduced. Assume now that $w$ is a minimal $S'$-reduced word of the form $(5x_i)$ with respect to the deg-lex order on $F$. Namely, $w = x_iux_i$, for some $u \neq 1$, $|u|_{x_{i+1}} = |u|_{x_{i}} = 0$. Formally, we need to note that $i+1 = 1$, if $i = n$, but we will begin with showing that in fact $i \neq n$.\
Assume the contrary, that $w = x_nux_n$ is of the form $(5x_n)$. Thus $|u|_{x_{n}} = 0$ and by Observation \[pe\] $u$ must be of the form , for some $k$ and $i_1, \ldots, i_k$, $j_1,
\ldots, j_k$. Since $x_nx_{i_1}$ cannot be a factor of the form (2), and $i_1 \neq 1$, we must have $i_1 = n-1$. This implies, that $u$ is a product of only one block $x_{n-1}\cdots x_{j_1}$. Since $j_1 > 1$ we can see that $w$ is of the form $(5x_n')$, a contradiction. Thus $i < n$.\
Our approach will be similar to that from the first part of the proof. Again, consider $t = \max\{l: |u|_{x_{l}} \neq 0\}$. Clearly, $t > 1$, as otherwise $w$ is $S'$-reducible. The proof breaks into two cases:
- Case 1. $t < n$. Since $w$ is $S'$-reduced and $|w|_{x_{n}} = 0$, it satisfies the conditions of Observation \[pe\]. Thus it must be of the form , namely $w = (x_{i_1}\cdots
x_{j_1})(x_{i_2}\cdots x_{j_2}) \cdots (x_{i_k} \cdots x_{j_k}), $ where $i_1 < i_2 < \cdots < i_k$ and $j_1 < j_2 < \cdots < j_k$, if $k > 1$. Of course, $w$ cannot consist of only one block $x_{i_1}\cdots x_{j_1}$, since otherwise we have $i_1 = j_1 = i$ and thus $w=x_{i}$, a contradiction. Hence $k >
1$. We claim that $j_l \geq i$, for all $l > 1$. Indeed, if we had $j_l < i$, for some $1 < l \leq k$, then the block $x_{i_l}\cdots x_{j_l}$ would have to be decreasing, as $i_l > i_1 = i$, and thus it would contain $x_{i+1}$, a contradiction with the fact that $w$ is of the form $(5x_i)$. Thus $j_l \geq i$, for all $l > 1$. Consider the second block $x_{i_2}\cdots x_{j_2}$ of $w$. Of course $i_2 > i_1 = i$. If we had $j_2 = i$, then the entire second block of $w$ would be decreasing and it would contain $x_{i+1}$, a contradiction. So $j_2 > i$. It follows that $i = j_k \geq j_2 > i$, and we arrive at a contradiction, again.
- Case 2. $ t= n$. Notice that $i\neq n-1$ in this case. We assume, again, that $w = x_ipx_nqx_i$, where $|q|_{x_{n}} = 0$. To avoid the appearance of a factor of type (2) in $w$, we must restrict ourselves to one of the following subcases: (a) $q = 1$, (b) $q = x_1q'$, or (c) $q = x_{n-1}q'$, where $q' \in F$.
- Subcase (a). If $q = 1$, then $w = x_ipx_nx_i$. Therefore, $i = n-1$ or $i=1$, otherwise we have a factor of the form (2) in $w$. The first case was excluded in the beginning of Case 2. So $w = x_1px_nx_1$. Thus $p \neq 1$, as otherwise $w$ is of type $(5x_1')$. Also, observe that $|p|_{x_{n}} = 0$, since otherwise we would have a proper factor $x_np'x_n$ of $w$ such that $|p'|_{x_{n}} = |p'|_{x_{l}} = 0$ and thus this factor would be of the form $(5x_n)$. This violates the minimality of $w$ as a minimal $S'$-reduced and $S$-reducible word with respect to the deg-lex order in $F$. This means that $x_1p$ satisfies the conditions of Observation \[pe\] and is of the form , so that $w$ is of the form $(5x_1'')$. This contradicts the fact that it is $S'$-reduced.
- Subcase (b). $q = x_1q'$. Again, $qx_i$ is of the form and as $i < n-1$ it follows, using the same arguments as in the case of words of the form , that $qx_i$ must be a single block and thus $w =
x_ipx_nx_1\cdots x_i$. Now, by an argument used in the subcase (b) of Case 2 in the first part of the proof, when we considered words $w$ of type $(4x_i)$, we can assume that $|p|_{x_{j}} =0$ for $j=i-1,i-2, \ldots ,1, n$. So $ |p|_{x_{n}} = 0$ allows us to apply Observation \[pe\] to prove that $p$ is of the form . This yields a contradiction, as $w$ is again proved to be of the form $(5x_i'')$.
- Subcase (c). $q = x_{n-1}q'$. [Once again, $qx_i$ is of the form . As the first block of $qx_i$ begins with $x_{n-1}$ we can see, as before, that this is in fact the only block of this word. Otherwise another block of $qx_i$ would have to begin with $x_{i_2}$, where $i_2 > i_1 = n-1$ and also $n >i_2$. This is impossible. Thus $qx_i = x_{n-1}q'x_i$ is a a single decreasing block of length greater than $1$ which is impossible, as $|u|_{x_{i+1}} = 0$.]{}
The subcases (a)-(c) have been proved to lead to a contradiction. Therefore, also in the case when $t = n$ we can see that no $w$ can be $S'$-reduced but $S$-reducible.
So, every $S'$-reduced word is $S$-reduced. Thus, Lemma \[sprim\] is proved.
\
It now follows easily from Lemma \[rq\] that the reduction system $S$ satisfies the diamond lemma, because the reduction system $T$ satisfies this. And similarly, Lemma \[sprim\] implies then that the reduction system $S'$ satisfies the diamond lemma. Consequently, we have proved the following theorem.
\[fingr\] $G'=\{w-w':(w,w')\in S'\}{\subseteq}k[F]$ forms a finite Gröbner basis and $G=\{w-w':(w,w')\in S\}{\subseteq}k[F]$ forms a Gröbner basis of the algebra $k[\operatorname{HK}_{C_{n}}]$. Consequently, all $S'$-reduced words form a basis of $k[\operatorname{HK}_{C_{n}}]$.
As mentioned before, the fact that in the particular case of a cycle graph, even a finite Gröbner basis can be obtained, strengthens the assertion of Theorem \[main\] in view of [@iyu].\
We conclude with an example showing that the above result cannot be extended to arbitrary Hecke-Kiselman algebras of oriented graphs, even in the case of PI-algebras.
Let $\Theta$ be the graph obtained by adjoining an outgoing arrow to the cycle $C_3$:
(6,3.5)(0,0.5)
(1,1) (0.9,0.7)[b]{} (2,1) (2.1,0.7)[c]{} (1.5,2) (1.7,2.1)[a]{} (1.5,3) (1.7,3.1)[d]{} (1.1,1)(1.9,1) (1,1.1)(1.4,1.9) (1.95,1.1)(1.55,1.95) (1.47,2.9)(1.47,2.1)
then $V(\Theta )=\{a,b,c,d\}$ and we consider the deg-lex order on the free monoid $F=\langle a,b,c,d\rangle$ defined by $a<b<c<d$. Then the algebra $k[\operatorname{HK}_{\Theta}]$ does not have a finite Gröbner basis and it is a PI-algebra of Gelfand-Kirillov dimension $2$.
It is easy to see that the set $N_{\Theta}$ used in the proof of Theorem \[main\] is the union of the following subsets of $F$:
- $N_{(i)}= a\langle b,d\rangle a \cup b \langle c,d\rangle b
\cup c\langle a,d\rangle c \cup d\langle b,c\rangle d $,
- $N_{(ii)}=a\langle
c\rangle a \cup b\langle a,d\rangle b \cup c\langle b,d\rangle c
\cup d\langle a,b,c\rangle d$,
- $N_{(iii)}= d\langle d \rangle b \cup d\langle d \rangle c$.
In particular, all words $d(abc)^kd, k\geq 1$, are in $N_{(ii)}$, but they do not have factors that are another words of $N_{\Theta}$. It follows that the algebra $k[\operatorname{HK}_{\Theta}]$ does not have a finite Gröbner basis (with respect to the indicated presentation and deg-lex order). By [@mecel_okninski], this is a PI-algebra. Moreover, $k[C_{3}]$ is of linear growth with reduced words being factors of two infinite words $(abc)^{\infty}$ and $(acb)^{\infty}$ and reduced words in $F$ are in the set $F'\cup F'dF'$, where $F'=\langle a,b,c\rangle \subseteq F$. So, $\operatorname{GKdim}(k[\operatorname{HK}_{\Theta}]) \leq 2$. On the other hand, words of the form $(abc)^kd(abc)^m$, $k,m\geq 1$, do not have factors in $N_{\Theta}$, so they are reduced. Therefore $\operatorname{GKdim}(k[\operatorname{HK}_{\Theta}]) = 2$.
[**Acknowledgment.**]{} The second author was partially supported by the National Science Centre grant 2016/23/B/ST1/01045 (Poland).
[99]{} =-2pt Aragona R., Andrea A.D., Hecke-Kiselman monoids of small cardinality, Semigroup Forum 86 (2013), 32–40. Bell J., Colak P., Primitivity of finitely presented monomial algebras, J. Pure Appl. Algebra 213 (2009), no. 7, 1299–1305. Bergman G.M., The diamond lemma for ring theory, Advances in Mathematics 29 (1978), 178–218. Brink B., Howlett R.B., A finiteness property and an automatic structure for Coxeter groups, Math. Ann. 296 (1993), 179–190. Cedó F., Okniński J., On a class of automaton algbras, Proc. Edinb. Math. Soc. 60 (2017), 31–38. Denton T., Hivert F., Schilling A., Thiery N.M., On the representation theory of finite $J$-trivial monoids, Seminaire Lotharingien de Combinatoire 64 (2011), Art. B64d. Ganyushkin O., Mazorchuk V., On Kiselman quotients of $0$-Hecke monoids, Int. Electron. J. Algebra 10(2) (2011), 174–191. Iyudu N., Shkarin S., Quadratic automaton algebras and intermediate growth, J. Comb. Algebra 2 (2018), 147–167. Kudryavtseva G., Mazorchuk V., On Kiselman’s semigroup, Yokohama Math. J., 55(1) (2009), 21–46. Mȩcel A., Okniński J., Growth alternative for Hecke–Kiselman monoids, Publicacions Matemàtiques, to appear (2019). Okniński J., Semigroup Algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 138, Marcel Dekker Inc., New York, 1991. Piontkovski D., Algebras of linear growth and the dynamical Mordell–Lang conjecture, preprint, arXiv:1706.06470. Piontkovski D., Homogeneous finitely presented monoids of linear growth, preprint, arXiv:1712.06022. Tsaranov S.V., Representation and classification of Coxeter monoids, Eur. J. Comb 11 (1990), 189-204. Ufnarovskii V.A., On the use of graphs for calculating the basis, growth and Hilbert series of associative algebras, Math. Sb. 180 (1989), 1548–1560. Ufnarovskii V.A., Combinatorial and Asymptotic Methods in Algebra, in: Encyclopedia of Mathematical Sciences vol. 57, pp.1–196, Springer, 1995.
-------------------------- -- -------------------------
Arkadiusz Mȩcel Jan Okniński
`[email protected]` `[email protected]`
Institute of Mathematics
University of Warsaw
Banacha 2
02-097 Warsaw, Poland
-------------------------- -- -------------------------
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abstract: 'The use of multiple antennas in a transmit and receive antenna array for MIMO wireless communication allows the spatial degrees of freedom in rich scattering environments to be exploited. However, for *line-of-sight* (LOS) MIMO channels with *uniform linear arrays* (ULAs) at the transmitter and receiver, the antenna separations at the transmit and receive array need to be optimized to maximize the spatial degrees of freedom and the channel capacity. In this paper, we first revisit the derivation of the optimum antenna separation at the transmit and receive ULAs in a LOS MIMO system, and provide the *general* expression for the optimum antenna separation product, which consists of *multiple* solutions. Although only the solution corresponding to the smallest antenna separation product is usually considered in the literature, we exploit the multiple solutions for a LOS MIMO design over a *range* of distances between the transmitter and receiver. In particular, we consider the LOS MIMO design in a *vehicle-to-vehicle* (V2V) communication scenario, over a range of distances between the transmit and receive vehicle.'
author:
-
title: LOS MIMO Design based on Multiple Optimum Antenna Separations
---
at (current page.south) ;
at (current page.north) ;
Introduction
============
The spatial degrees of freedom offered by a MIMO system with a transmit and receive antenna array can be exploited in the presence of a rich scattering environment. However, in LOS MIMO channels with little or no scattering, the channel responses can become highly correlated, leading to a MIMO channel of rank 1. Nevertheless, with a proper placement of the antennas in the arrays [@Driessen99; @Gesbert02; @Haustein03], the channel capacity and rank of the LOS MIMO channel can be maximized.
With ULAs at the transmitter and receiver, the best antenna placement is obtained by optimizing the separation between the antennas in the transmit and receive arrays. Although for ULAs there are multiple solutions [@Bohagen05; @Sarris07] for the optimum antenna separation product, i.e. the product of the antenna separation at the transmit and receive array, the one corresponding to the *smallest* antenna separation product is usually considered, as this leads to the smallest arrays [@Bohagen05].
The previous cited works consider a fixed distance between the transmit and receive array. However, for many applications, a LOS MIMO channel needs to be designed for a *range* of distances between the transmitter and receiver. Since the optimum antenna separation depends on the distance between the transmitter and receiver, there is a performance degradation when the distance is varied for a given antenna placement of the transmit and receive arrays. To reduce the sensitivity to distance variations between the transmitter and receiver, non-uniform linear arrays have been proposed [@Torkildson09; @Zhou13]. The optimum antenna placement in such cases was found using an exhaustive search, with the aim of maximizing the range where a minimum condition number or capacity can be guaranteed.
In this paper, we first revisit the derivation of the optimum antenna separation for LOS MIMO systems with ULAs at the transmitter and receiver. In contrast to prior work, we provide the *general* expression for the optimum antenna separation product, which consists of multiple solutions. In addition, we propose to use the multiple solutions for the LOS MIMO design over a range of distances. In particular, we consider the LOS MIMO design for a V2V communication scenario over a range of distances between the transmit and receive vehicle. Although the optimum antenna placement can not be met at all distances, we exploit the fact that some antenna separations are optimum at *several* distances. We show that *larger* antenna separations can be beneficial in certain cases. This paper is organized as follows. Section \[Sec:ChMod\] introduces the LOS MIMO channel model. The optimum antenna separation is derived in Section \[Sec:OptAntSep\]. The V2V scenario is described in Section \[Sec:V2V\], where numerical results for the LOS MIMO design are presented. We conclude the paper with Section \[Sec:Con\].
LOS MIMO Channel Model {#Sec:ChMod}
======================
In this paper, we use lower case and capital boldface letters to denote vectors and matrices, respectively. In addition, $(\bullet)^{{\operatorname{T}}}$ and $(\bullet)^{{\operatorname{H}}}$ denote the transpose and conjugate transpose, respectively. The cardinality of the set $\mathcal{P}$ is denoted by $|\mathcal{P}|$.
We consider a MIMO channel with a pure LOS between a transmitter and a receiver consisting of a ULA with $N > 1$ and $M > 1$ antennas, respectively. The antenna separation at the *transmit* (Tx) and *receive* (Rx) ULA is $d_{\text{\tiny Tx}}$ and $d_{\text{\tiny Rx}}$, respectively. The distance between the first antenna of the Tx ULA, placed at the origin, and the first antenna of the Rx ULA is given by $R$ as shown in Fig. \[fig:LOSMIMO\]. With the Tx array placed on the $xz$-plane, we assume an arbitrary orientation of the arrays given by the angles $\theta_{\text{\tiny Tx}}$, $\theta_{\text{\tiny Rx}}$ and $\phi_{\text{\tiny Rx}}$ as shown in Fig. \[fig:LOSMIMO\], where $0 \le \theta_{\text{\tiny Tx}} \le \frac{\pi}{2}$ and $0 \le \theta_{\text{\tiny Rx}} \le \frac{\pi}{2}$. The carrier frequency and wavelength of the signal are given by $f_{\text{c}}$ and $\lambda$, respectively.
(xyz cs:x=0) – (xyz cs:x=12) node\[above\] [$x$]{}; (xyz cs:y=0) – (xyz cs:y=5) node\[right\] [$z$]{}; (xyz cs:z=0) – (xyz cs:z=-3.5) node\[above\] [$y$]{}; at (0,0) (Tx1) ; at (-0.5,1) (Tx2) ; at (-1.5,3) (TxN) ; (Tx1) node\[below\] – (Tx2) node\[below\] [ $d_{\text{\tiny Tx}}$]{}; (Tx2) – (TxN) ; (0,2) arc (90:115:2) node\[above\] [ $\theta_{\text{\tiny Tx}}$]{}; at (-0.2,4) () [Tx ULA with $N$ antennas]{}; (xyz cs:x=8,y=0) – (xyz cs:x=8,y=5) node\[right\] [$z^{\prime}$]{}; (xyz cs:x=8,z=0) – (xyz cs:x=8,z=-3.5) node\[above\] [$y^{\prime}$]{}; at (8,0) (Rx1) ; at (8.6,0.8) (Rx2) ; at (9.8,2.4) (RxN) ; (Rx1) – (Rx2) node\[left\] [ $d_{\text{\tiny Rx}}$]{}; (Rx2) – (RxN) ; at (9.1,3.4) () [Rx ULA with $M$ antennas ]{}; (RxN) – (xyz cs:x=9.8,y=-1.4) ; (Rx1) node\[below\] – (xyz cs:x=9.8,y=-1.4) ; (xyz cs:x=8,z=-2.3) – (xyz cs:x=9.8,y=-1.4) ; (xyz cs:x=11) – (xyz cs:x=9.8,y=-1.4) ; (8,2) arc (90:53:2) node\[midway, above\] [$\theta_{\text{\tiny Rx}}$]{}; (9,0) arc (0:-37:1) node\[midway,right\] [ $\phi_{\text{\tiny Rx}}$]{};
The *normalized* channel matrix for the LOS MIMO system is denoted as $$\begin{aligned}
{\mathbf{H}} = \left[ \begin{array}{cccc}
{\mathbf{h}}_{1} & {\mathbf{h}}_{2} & \cdots & {\mathbf{h}}_{N}
\end{array} \right] \in \mathbb{C}^{M \times N},
\label{def_H}\end{aligned}$$ where the $n$-th column of ${\mathbf{H}}$, i.e. ${\mathbf{h}}_{n}$, corresponds to the channel vector from the $n$-th antenna at the Tx array to the $M$ antennas at the Rx antenna array. With the path loss included in the receive SNR, the *normalized* channel vector ${\mathbf{h}}_{n} \in \mathbb{C}^{M}$ is determined with ray tracing, i.e. with the spherical wave model instead of the planar wave assumption, and is given as: $$\begin{aligned}
{\mathbf{h}}_{n} = \left[ \begin{array}{ccc}
\text{exp}\left( {\operatorname{j}}2\pi \frac{r_{1,n}}{\lambda}\right), & \cdots, & \text{exp}\left( {\operatorname{j}}2\pi \frac{r_{M,n}}{\lambda}\right)
\end{array} \right]^{{\operatorname{T}}}.
\label{def_hn}\end{aligned}$$ where $r_{m,n}$ corresponds to the path length between the $n$-th Tx antenna and the $m$-th Rx antenna, for $n=1,\ldots,N$ and $m=1,\ldots,M$, respectively. The path length $r_{m,n}$ can be obtained from the coordinates $(x^{\text{\tiny Tx}}_n,y^{\text{\tiny Tx}}_n,z^{\text{\tiny Tx}}_n)$ of the $n$-th Tx antenna and the coordinates $(x^{\text{\tiny Rx}}_m,y^{\text{\tiny Rx}}_m,z^{\text{\tiny Rx}}_m)$ of the $m$-th Rx antenna, which from Fig. \[fig:LOSMIMO\] are given by $$\begin{aligned}
\begin{array}{ll}
\text{$n$-th Tx ant.}\!: & x^{\text{\tiny Tx}}_n =-(n-1) d_{\text{\tiny Tx}} \sin \theta_{\text{\tiny Tx}}, \\
& y^{\text{\tiny Tx}}_n = 0, \quad \quad z^{\text{\tiny Rx}}_m = (n-1) d_{\text{\tiny Tx}} \cos \theta_{\text{\tiny Tx}} \\
\text{$m$-th Rx ant.}\!: & x^{\text{\tiny Rx}}_m = R \!+\!(m\!-\!1) d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \cos \phi_{\text{\tiny Rx}}, \\
&\hspace{-12ex} y^{\text{\tiny Rx}}_m \!=\! (m\!-\!1) d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \sin \phi_{\text{\tiny Rx}},
\quad\, z^{\text{\tiny Rx}}_m \!=\!(m\!-\!1) d_{\text{\tiny Rx}} \cos \theta_{\text{\tiny Rx}}.
\end{array}\end{aligned}$$ With the above coordinates, $r_{m,n}$ can be determined as follows $$\begin{aligned}
r_{m,n} &\boldsymbol{=} \left(\left(x^{\text{\tiny Rx}}_m-x^{\text{\tiny Tx}}_n\right)^{2} + \left(y^{\text{\tiny Rx}}_m-y^{\text{\tiny Tx}}_n\right)^{2} + \left(z^{\text{\tiny Rx}}_m-z^{\text{\tiny Tx}}_n\right)^{2} \right)^{\frac{1}{2}} \nonumber \\
&\hspace{-5ex} \boldsymbol{=}\! \! \left(\!\left(R \!+\!(m\!-\!1) d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \! \cos \phi_{\text{\tiny Rx}}\!\! +\! (n\!-\!1) d_{\text{\tiny Tx}} \sin \theta_{\text{\tiny Tx}} \!\right)^{2} \!\!\! +\! \big(\!(m\!-\!1) \times \right. \nonumber \\
& \hspace{-5ex} \left. d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \sin \phi_{\text{\tiny Rx}}\big)^{2} \!\! + \!
\left((m\!-\!1) d_{\text{\tiny Rx}} \! \cos \theta_{\text{\tiny Rx}} \! - \!(n\!-\!1) d_{\text{\tiny Tx}} \cos \theta_{\text{\tiny Tx}} \right)^{2} \right)^{\!\frac{1}{2}} \nonumber \\
&\hspace{-5ex} \boldsymbol{=} \Big(R \!+\!(m\!-\!1) d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \cos \phi_{\text{\tiny Rx}} + (n-1) d_{\text{\tiny Tx}} \sin \theta_{\text{\tiny Tx}} \Big) \Big( 1 + \nonumber \\
& \hspace{-6ex} \frac{\left( \! (m\!-\!1) d_{\text{\tiny Rx}} \sin \! \theta_{\text{\tiny Rx}} \sin \!\phi_{\text{\tiny Rx}}\right)^{2} \!\!\! + \!
\left(\!(m\!-\!1) d_{\text{\tiny Rx}} \cos\! \theta_{\text{\tiny Rx}}\! \! - \!(n\!-\!1) d_{\text{\tiny Tx}} \!\cos \! \theta_{\text{\tiny Tx}}\!\right)^{\!2}}{\left(R \!+\!(m\!-\!1) d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \cos \phi_{\text{\tiny Rx}} + (n-1) d_{\text{\tiny Tx}} \sin \theta_{\text{\tiny Tx}}\right)^{2}} \!\bigg)^{\!\!\!\frac{1}{2}} \nonumber \\
& \hspace{-5ex}\boldsymbol{\approx} R \!+\!(m\!-\!1) d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \cos \phi_{\text{\tiny Rx}} + (n-1) d_{\text{\tiny Tx}} \sin \theta_{\text{\tiny Tx}} + \nonumber \\
& \hspace{-6ex} \frac{\left(\! (m\!-\!1) d_{\text{\tiny Rx}}\! \sin \theta_{\text{\tiny Rx}} \! \sin \phi_{\text{\tiny Rx}}\right)^{2} \!\!\! + \!
\left(\!(m\!-\!1) d_{\text{\tiny Rx}} \!\cos \theta_{\text{\tiny Rx}} \! - \!(n\!-\!1) d_{\text{\tiny Tx}}\! \cos \theta_{\text{\tiny Tx}}\right)^{2}}{2 R}, \label{def_rmn}\end{aligned}$$ where the last step results from the first order approximation of the Taylor series of $\sqrt{1+a}$ with $a \ll 1$, i.e. $\sqrt{1+a}\approx 1 + \frac{a}{2}$, and from $R \approx R+ (m\!-\!1) d_{\text{\tiny Rx}} \sin \theta_{\text{\tiny Rx}} \cos \phi_{\text{\tiny Rx}} + (n-1) d_{\text{\tiny Tx}} \sin \theta_{\text{\tiny Tx}}$ in the denominator of the argument of the square root, where both approximations hold if the distance $R$ between the transmitter and receiver is much larger than Tx and Rx array dimensions.
Optimum Antenna Separation {#Sec:OptAntSep}
==========================
Consider the case when $N \le M$, such that[^1] $\text{rank}\left({\mathbf{H}}\right) \le N$. As discussed in [@Bohagen05], the capacity of the LOS MIMO system at high SNR is maximized if ${\mathbf{H}}^{\text{H}}{\mathbf{H}} = M {\mathbf{1}}_{N}$, i.e. if the columns of ${\mathbf{H}}$ are orthogonal. For this case, ${\mathbf{H}}$ achieves the maximum rank of $N$ and the $N$ eigenvalues of ${\mathbf{H}}^{\text{H}}{\mathbf{H}}$ are all equal to $M$, as $\text{tr}\left({\mathbf{H}}^{\text{H}}{\mathbf{H}}\right)=MN$.
Solution of the Orthogonality Condition
---------------------------------------
In order to design the channel matrix ${\mathbf{H}}$ of the LOS MIMO system to have orthogonal columns, from (\[def\_H\]) we need to have $$\begin{aligned}
{\mathbf{h}}_{k}^{{\operatorname{H}}} {\mathbf{h}}_{l} = 0, \quad \quad \text{for} \quad k \ne l; \quad k,l=1,\cdots,N.
\label{orth_cond}\end{aligned}$$ Using (\[def\_hn\]), we can write $$\begin{aligned}
{\mathbf{h}}_{k}^{{\operatorname{H}}} {\mathbf{h}}_{l}
&= \sum_{m=1}^{M} \text{exp}\left({\operatorname{j}}2\pi \frac{r_{m,l}-r_{m,k}}{\lambda} \right) \nonumber \\
&\!\!\!\!\overset{\text{(a)}}{\approx} \! \sum_{m=1}^{M} \! \text{exp}\left(\! {\operatorname{j}}2\pi \! \left(\!\frac{\gamma}{\lambda}\! -\!\frac{d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} \cos{\theta_{\text{\tiny Tx}}} \cos{\theta_{\text{\tiny Rx}}}}{\lambda R} (l\!-\!k) (m\!-\!1) \!\right) \!\right) \nonumber \\
&\!\!\!\!\overset{\text{(b)}}{=} \Gamma \cdot \sum_{m^{\prime}=0}^{M-1} \!\! \text{exp}\left({\operatorname{j}}2\pi \frac{d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} \cos{\theta_{\text{\tiny Tx}}} \cos{\theta_{\text{\tiny Rx}}}}{\lambda R} (k-l) m^{\prime} \right) \nonumber \\
&\!\!\!\!\overset{\text{(c)}}{=} \Gamma \cdot
\frac{1-\text{exp}\left({\operatorname{j}}2\pi \frac{d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} \cos{\theta_{\text{\tiny Tx}}} \cos{\theta_{\text{\tiny Rx}}}}{\lambda R} M (k-l) \right)}{1-\text{exp}\left({\operatorname{j}}2\pi \frac{d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} \cos{\theta_{\text{\tiny Tx}}} \cos{\theta_{\text{\tiny Rx}}}}{\lambda R} (k-l) \right)},
\label{inner_prod}\end{aligned}$$ where step (a) results from $$\begin{aligned}
r_{m,l}-r_{m,k} \approx &
\, \gamma - \frac{d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} \cos \theta_{\text{\tiny Tx}} \cos \theta_{\text{\tiny Rx}}}{R} (l\!-\!k) (m\!-\!1),\end{aligned}$$ which follows from using the approximation (\[def\_rmn\]) for $r_{m,n}$, and where $\gamma \!=\! (l\!-\!k) d_{\text{\tiny Tx}} \sin \theta_{\text{\tiny Tx}} - \frac{ \left((l-1)^{2}\! - (k-1)^{2} \right)d^{2}_{\text{\tiny Tx}} \cos^{2} \theta_{\text{\tiny Tx}}}{2 R}$. For step (b), we use the substitutions $m^{\prime}=m\!-\!1$ and $\Gamma\! \!= \!\text{exp}\left({\operatorname{j}}2\pi \! \frac{\gamma}{\lambda}\right)$, with $\Gamma$ being independent of $m^{\prime}$. For step (c), we employ the expression for the finite sum of a geometric series for $w\ne 1$: $$\begin{aligned}
\sum_{m^{\prime}=0}^{M-1} w^{m^{\prime}} = \frac{1 - w^M}{1-w} ,\end{aligned}$$ with $w = \text{exp}\left({\operatorname{j}}2\pi \frac{d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} \cos{\theta_{\text{\tiny Tx}}} \cos{\theta_{\text{\tiny Rx}}}}{\lambda R} (k-l) \right) $.
Given that ${\mathbf{h}}_{k}^{{\operatorname{H}}} {\mathbf{h}}_{l}$ depends on $(k-l)$, as observed from (\[inner\_prod\]), and that $|{\mathbf{h}}_{k}^{{\operatorname{H}}} {\mathbf{h}}_{l}|=|{\mathbf{h}}_{l}^{{\operatorname{H}}} {\mathbf{h}}_{k}|$, the conditions given in (\[orth\_cond\]) required to have orthogonal columns of ${\mathbf{H}}$ are equivalent to $$\begin{aligned}
{\mathbf{h}}_{k}^{{\operatorname{H}}} {\mathbf{h}}_{l} = 0, \quad \quad \text{for} \quad (k -l)=1,\cdots,N-1.
\label{orth_cond_0}\end{aligned}$$ From (\[inner\_prod\]) and as $\Gamma \ne 0$, the [equivalent]{} orthogonality conditions in are fulfilled[^2] if $$\begin{aligned}
\frac{1-\text{e}^{{\operatorname{j}}2\pi \delta M q}}{1-\text{e}^{{\operatorname{j}}2\pi \delta q }} =0, \quad \forall \, q\in\{1, 2,\! \cdots\!, N\!-\!1\},
\label{ortho_cond_1}\end{aligned}$$ where we introduce $q=k-l$ and define $$\begin{aligned}
\delta \overset{\Delta}{=}
\frac{d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} \cos{\theta_{\text{\tiny Tx}}} \cos{\theta_{\text{\tiny Rx}}}}{\lambda R}.
\label{delta}\end{aligned}$$
Solving with respect to $\delta$, allows us to determine the optimum antenna separations $d_{\text{\tiny Tx}}$ and $d_{\text{\tiny Rx}}$ of the Tx and Rx ULAs, which lead to a channel matrix ${\mathbf{H}}$ that maximizes the capacity of the LOS MIMO system. To satisfy (\[ortho\_cond\_1\]), the numerator of the expression in (\[ortho\_cond\_1\]) needs to be zero while the denominator is non-zero, i.e. $$\begin{aligned}
\text{e}^{{\operatorname{j}}2\pi \delta M q}=1, \quad \forall \, q \in \{1, 2,\! \cdots\!, N\!-\!1\},
\label{num_zero_0}\end{aligned}$$ while $$\begin{aligned}
\text{e}^{{\operatorname{j}}2\pi \delta q} \ne 1, \quad \forall \, q \in\{1, 2,\! \cdots\!, N\!-\!1\}.
\label{den_zero_0}\end{aligned}$$
As the solution of (\[num\_zero\_0\]) for $q=1$ is also a solution of (\[num\_zero\_0\]) for $q=2,\cdots,N-1$, the solution of (\[num\_zero\_0\]) for all $q$ results from $\text{e}^{{\operatorname{j}}2\pi \delta M }=1$, i.e. the solution of (\[num\_zero\_0\]) is $$\begin{aligned}
\delta = \frac{p}{M}, \quad \forall \,\, p \in \mathbb{Z}_{+}, \label{num_zero}
$$ where $\mathbb{Z}_{+}$ represents the set of positive integers. The set of negative integers is excluded from the solution since all the terms in $\delta$ are positive, as can be seen in .
On the other hand, to avoid the denominator of the expression in being zero for any value of $q$, from we get $$\begin{aligned}
&\delta \ne \frac{p_1}{q}, \quad \, \, \forall \, p_1 \in \mathbb{Z}_{+}, \,\, q\in\{1, 2,\! \cdots\!, N\!-\!1\}.
\label{den_zero} \end{aligned}$$
Thus, given and , we have that (\[ortho\_cond\_1\]) is fulfilled if $\delta = \frac{p}{M}$ for $p \in \mathbb{Z}_{+}$ but excluding the integers $p$ for which $\frac{p}{M} = \frac{p_1}{q}$ for $q= 1, 2,\cdots, N\!-\!1$, i.e. when $$\begin{aligned}
\delta = \frac{p}{M}, \quad \forall \, p \in \mathbb{Z}_{+} \setminus \left\{p^{\prime}: p^{\prime} \!=\! \frac{p_1 M}{q}, \!\!\! \begin{array}{c} p_1 \in \mathbb{Z}_{+} , p^{\prime} \in \mathbb{Z}_{+}, \\
q\in\{1, 2,\! \cdots\!, N\!-\!1\}
\end{array}
\!\!\!\! \right\} \!\! . \label{ortho_sol_0}
$$ Writing $q$ as the product of *any* two (positive integer) factors, i.e. $q=q_1 q_2$, $p^{\prime}=\frac{p_1 M}{q}$ is an integer if $\frac{p_1}{q_1}$ and $\frac{M}{q_2}$ are both integers. As there is always a $p_1\in \mathbb{Z}_{+}$ such that $\frac{p_1}{q_1} \in \mathbb{Z}_{+}$, we only need to consider when $\frac{M}{q_2}$ is an integer for any factor $q_2$ of $q$. Given that $q_2 \le q \le N-1$, $\forall q$, we have that $p^{\prime}=\frac{p_1 M}{q}$ is an integer if $\frac{M}{q}$ is an integer for $q=1, \ldots, N-1$. The possible values, in ascending order, of $\frac{M}{q}$ for $q=1, \ldots, N-1$, are $\frac{M}{N-1}$, $\frac{M}{N-2}$, …$\frac{M}{2}$, $M$, out of which those that are integers (recall that $N \le M$), correspond to the *divisors* of $M$ which are larger than or equal to $\frac{M}{N-1}$. Let us denote the set of divisors of $M$ which satisfy this condition as $\mathcal{D}_M(N)$, i.e. $$\begin{aligned}
\mathcal{D}_{M}(N)= \left\{\nu: \nu \,\, | \, \, M, \,\,\nu \ge \frac{M}{N-1} \right\} ,
\label{div_set}\end{aligned}$$ where $a \,\, | \, \, b$ means that $a$ is a divisor of $b$. Given , we can rewrite the solution for the orthogonality conditions as $$\begin{aligned}
\delta = \frac{p}{M}, \quad \forall \, p \in \mathbb{Z}_{+} \setminus \, \left\{
p^{\prime} \, \nu, \, \, p^{\prime} \in \mathbb{Z}_{+}, \, \nu \in \mathcal{D}_M(N) \right\},
\label{ortho_sol}\end{aligned}$$ i.e. is fulfilled if $\delta = \frac{p}{M}$ for the set of positive integers $p$ *excluding* the multiples of divisors of $M$ which are larger than or equal to $\frac{M}{N-1}$.
Prior solutions of (\[ortho\_cond\_1\]) provided in the literature, e.g. as in [@Sarris07], include only a subset of the possible integers $p$ given in (\[ortho\_sol\]). In addition, in contrast to prior work, our derived expression (\[ortho\_sol\]) shows the dependency on $N$, which corresponds to the number of Tx antennas and the $\text{rank}({\mathbf{H}})$. We discuss this dependency with two examples: $N\!=\!2$ and $N\!=\!M$. For $N\!=\!2$, $\frac{M}{N-1}=M$ such that from , $\mathcal{D}_M(2)=\{M\}$. On the other hand, for $N\!=\!M$, $\frac{M}{N-1}=1+\frac{1}{M-1}$ such that $\mathcal{D}_M(M) = \left\{\nu: \nu \,\, | \, \, M, \,\,\nu > 1 \right\}$, i.e. $\mathcal{D}_M(M)$ consists of all the divisors[^3] of $M$ *except* $1$. As $|\mathcal{D}_M(M)| \ge |\mathcal{D}_M(2)|$, we see that in general a larger set of positive integers $p$ are excluded in (\[ortho\_sol\]) when $N=M>2$ compared to when $N=2$. This is a consequence of the fact that the orthogonality conditions in (\[ortho\_cond\_1\]) becomes more stringent with increasing $N$: for $N=2$, only two channel vectors need to be orthogonal, whereas for $N=M$, $M$ orthogonal channel vectors need to be designed.
Design of LOS MIMO Systems
--------------------------
Using , we rewrite (\[ortho\_sol\]) in terms of the *antenna separation product* (ASP) [@Bohagen05], i.e. in terms of the product of the antenna separation at the transmitter and receiver $$\begin{aligned}
&\quad d_{\text{\tiny Tx}} d_{\text{\tiny Rx}} = p \cdot \frac{\lambda R}{M \cos{\theta_{\text{\tiny Tx}}} \cos{\theta_{\text{\tiny Rx}}}}, \label{ortho_sol_asp}
\\ &\forall \, p \in \mathbb{Z}_{+} \setminus \, \left\{
p^{\prime} \, \nu, \, \, p^{\prime} \in \mathbb{Z}_{+}, \, \nu \in \mathcal{D}_M(N) \right\},
\nonumber\end{aligned}$$ for $N \le M$. For $N > M$, the optimum solution for the ASP results from exchanging $N$ with $M$ in the expression above.
By setting the antenna separations $d_{\text{\tiny Tx}}$ and $d_{\text{\tiny Rx}}$ of the Tx and Rx ULAs according to (\[ortho\_sol\_asp\]), the channel matrix ${\mathbf{H}}$ of the LOS MIMO system can be designed to have orthogonal columns, for a given distance $R$ between the arrays and a given orientation of the arrays. Although multiple solutions for the ASP exist[^4], only the first solution of , i.e. $p=1$, is usually considered in the literature as this leads to the smallest antenna separations and hence, to the smallest arrays [@Bohagen05; @Sarris07].
However, for certain applications, *other* solutions for the ASP, i.e. $p>1$, might be of interest. Take for instance the LOS MIMO design over a *range* of distances between the transmitter and receiver, which is relevant for many applications. As observed in (\[ortho\_sol\_asp\]), the optimum antenna separations at the Tx and Rx arrays depends on the *fixed* distance $R$ between the arrays. Thus, varying the distance between the transmitter and receiver with a given optimum antenna separation, leads to a capacity reduction, i.e. reduced $\text{rank}({\mathbf{H}})$ or non-equal eigenvalues of ${\mathbf{H}}^{\text{H}}{\mathbf{H}}$. To reduce the sensitivity to distance variations, non-uniform linear arrays have been proposed [@Torkildson09; @Zhou13], where the optimum antenna placement is found via an exhaustive search, in order to maximize the range for which a certain metric can be guaranteed. In this paper, we propose the use of ULAs for the LOS MIMO design over a set of distances between the transmitter and receiver, by exploiting the multiple solutions for the ASP given in (\[ortho\_sol\_asp\]). In particular, we consider the LOS MIMO design for a V2V link as discussed next.
LOS MIMO Design for V2V {#Sec:V2V}
=======================
Due to the importance of V2V communication in future wireless networks, e.g. 5G, we consider the LOS MIMO design for a V2V link between two vehicles located in the same lane, where the front car (Tx car) is communicating with a rear car (Rx car) separated by a longitudinal distance $D$ as shown in Fig. \[fig:V2V\]. The Tx car is equipped in the *rear* bumper with a Tx ULA consisting of $N$ antennas, whereas the Rx car is equipped in the *front* bumper with a Rx ULA consisting of $M$ antennas. The *maximum* length of the Tx and the Rx ULA is assumed to be $L_{\text{\tiny Tx}}$ and $L_{\text{\tiny Rx}}$, respectively. From Fig. \[fig:V2V\], we can see that the Tx ULA and the Rx ULA are always parallel and hence, the orientation of both arrays are the same, i.e. $\theta_{\text{\tiny Tx}}=\theta_{\text{\tiny Rx}}$ (c.f. Fig. \[fig:LOSMIMO\]). We assume a pure LOS channel between the Tx ULA and the Rx ULA, as well as the same speed for the Tx and Rx car. We assume a carrier frequency of $f_{\text{c}}=28$ GHz ($\lambda \approx 10.7$ mm) and a normalized LOS channel as discussed in Section \[Sec:ChMod\], with a fixed receive $\text{SNR}=$ 13 dB for the considered distances $D$, i.e. with perfect sidelink power control.
(0,-2.2) – (0,20.2) ; (5,-2.5) – (5,20.5) ; (-5,-2.5) – (-5,20.5) ; (0,-2.7) – (5,-2.7) node\[align=center, midway,below=5pt\] [[ Lane Width]{}\
[ 3.5 m]{}]{}; (-1.1,4.4) – (7.5,4.4) ; (-1.1,13.6) – (6,13.6) ; (-0.6,4.4) – (-0.6,13.6) node\[align=center,midway,left\] [\
[between cars ]{}]{}; (RxCar) at (2.9,1.1)
(-1.35,1.4) to\[out=-160,in=20\] (-1.58,1.3) to\[out=-170,in=130\] (-1.65,1.05) to\[out=20,in=-160\] (-1.25,1.15) ;
(1.35,1.4) to\[out=-20,in=160\] (1.58,1.3) to\[out=-10,in=50\] (1.65,1.05) to\[out=160,in=-20\] (1.25,1.15) ;
(-1.2,-2.5) to\[out=115,in=-90\] (-1.45,-1.3) to\[out=90,in=-95\] (-1.35,0) to\[out=90,in=-90\] (-1.35,2.2) to\[out=105,in=-90\] (-1.45,3.2) to\[out=85,in=-180\] (-0.2,4.3) to\[out=0,in=180\] (0.2,4.3) to\[out=0,in=95\] (1.45,3.2) to\[out=-90,in=75\] (1.35,2.2) to\[out=-90,in=90\] (1.35,0) to\[out=-90,in=95\] (1.45,-1.3) to\[out=-90,in=65\] (1.2,-2.5) to\[out=-170,in=-10\] (-1.2,-2.5) ;
(-1.15,3.85) to\[out=10,in=-135\] (-0.7,4) to\[out=60,in=-95\] (-0.6,4.23) to\[out=-150,in=70\] (-1.15,3.85);
(1.15,3.85) to\[out=-170,in=-45\] (0.7,4) to\[out=120,in=-85\] (0.6,4.23) to\[out=-30,in=110\] (1.15,3.85);
(-1.05,2.55) to \[out=30,in=150\] (1.05,2.55);
(-1.05,2.9) to \[out=20,in=-100\] (-0.35,4);
(1.05,2.9) to \[out=160,in=-80\] (0.35,4);
(0,1.2) to\[out=180,in=0\] (-0.75,1.2) to\[out=140,in=-80\] (-1.05,2) to\[out=80,in=100\] (1.05,2) to\[out=-100,in=40\] (0.75,1.2) to\[out=180,in=0\] (0,1.2) ;
(0,-1) to\[out=180,in=0\] (-0.7,-1) to\[out=-150,in=100\] (-0.65,-1.7) to\[out=-60,in=-120\] (0.65,-1.7) to\[out=80,in=-30\] (0.7,-1) to\[out=180,in=0\] (0,-1) ;
(-1.12,1.4) to\[out=-90,in=88\] (-1.15,-0.35) to\[out=-93,in=120\] (-0.95,-0.85) to\[out=85,in=-90\] (-0.85,0.1) to\[out=92,in=-70\] (-1.12,1.4) ;
(1.12,1.4) to\[out=-90,in=92\] (1.15,-0.35) to\[out=-87,in=60\] (0.95,-0.85) to\[out=95,in=-90\] (0.85,0.1) to\[out=88,in=-110\] (1.12,1.4) ;
at (0,4.1) (ula) ;
at (-.9,4.1) (Rx1) ; at (-.4,4.1) (Rx1) ; at (.05,4.1) (Rx1) ; at (.25,4.1) (Rx1) ; at (.45,4.1) (Rx1) ; at (.9,4.1) (Rx1) ;
; (TxCar) at (2,16.9)
(-1.35,1.4) to\[out=-160,in=20\] (-1.58,1.3) to\[out=-170,in=130\] (-1.65,1.05) to\[out=20,in=-160\] (-1.25,1.15) ;
(1.35,1.4) to\[out=-20,in=160\] (1.58,1.3) to\[out=-10,in=50\] (1.65,1.05) to\[out=160,in=-20\] (1.25,1.15) ;
(-1.2,-2.5) to\[out=115,in=-90\] (-1.45,-1.3) to\[out=90,in=-95\] (-1.35,0) to\[out=90,in=-90\] (-1.35,2.2) to\[out=105,in=-90\] (-1.45,3.2) to\[out=85,in=-180\] (-0.2,4.3) to\[out=0,in=180\] (0.2,4.3) to\[out=0,in=95\] (1.45,3.2) to\[out=-90,in=75\] (1.35,2.2) to\[out=-90,in=90\] (1.35,0) to\[out=-90,in=95\] (1.45,-1.3) to\[out=-90,in=65\] (1.2,-2.5) to\[out=-170,in=-10\] (-1.2,-2.5) ;
(-1.15,3.85) to\[out=10,in=-135\] (-0.7,4) to\[out=60,in=-95\] (-0.6,4.23) to\[out=-150,in=70\] (-1.15,3.85);
(1.15,3.85) to\[out=-170,in=-45\] (0.7,4) to\[out=120,in=-85\] (0.6,4.23) to\[out=-30,in=110\] (1.15,3.85);
(-1.05,2.55) to \[out=30,in=150\] (1.05,2.55);
(-1.05,2.9) to \[out=20,in=-100\] (-0.35,4);
(1.05,2.9) to \[out=160,in=-80\] (0.35,4);
(0,1.2) to\[out=180,in=0\] (-0.75,1.2) to\[out=140,in=-80\] (-1.05,2) to\[out=80,in=100\] (1.05,2) to\[out=-100,in=40\] (0.75,1.2) to\[out=180,in=0\] (0,1.2) ;
(0,-1) to\[out=180,in=0\] (-0.7,-1) to\[out=-150,in=100\] (-0.65,-1.7) to\[out=-60,in=-120\] (0.65,-1.7) to\[out=80,in=-30\] (0.7,-1) to\[out=180,in=0\] (0,-1) ;
(-1.12,1.4) to\[out=-90,in=88\] (-1.15,-0.35) to\[out=-93,in=120\] (-0.95,-0.85) to\[out=85,in=-90\] (-0.85,0.1) to\[out=92,in=-70\] (-1.12,1.4) ;
(1.12,1.4) to\[out=-90,in=92\] (1.15,-0.35) to\[out=-87,in=60\] (0.95,-0.85) to\[out=95,in=-90\] (0.85,0.1) to\[out=88,in=-110\] (1.12,1.4) ;
at (0,-2.5) (ula) ;
at (-.9,-2.5) (Rx1) ; at (-.4,-2.5) (Rx1) ; at (.05,-2.5) (Rx1) ; at (.25,-2.5) (Rx1) ; at (.45,-2.5) (Rx1) ; at (.9,-2.5) (Rx1) ;
; (2.225,2) – (0.875,16) ; (2,4.4) – (1.1,13.6) node\[midway,right\] [$R$]{}; (0,13.5) – (6,14.0784) ; (1,4.3) – (7.5,4.8784) ; (7,4.4) arc (0:5:5) node\[midway,xshift=9pt\] [ $\theta_{\text{\tiny Rx}}$]{}; (5.5,13.6) arc (0:5:5) node\[midway,xshift=9pt\] [ $\theta_{\text{\tiny Tx}}$]{}; (-6.5,1.5) node\[left,align=center\] [ [Rx Car with]{}\
[Rx ULA in]{}\
[front bumper]{} ]{} – (2.9,0.5); (-6.5,17) node\[left,align=center\] [ [Tx Car with]{}\
[Tx ULA in]{}\
[rear bumper]{} ]{} – (2,16) ; (1.85,4.5) – node\[above=10pt\] [ $\le L_{\text{\tiny Rx}}$]{} (3.9,4.5); (.95,13.5) – node\[below=10pt\] [ $\le L_{\text{\tiny Tx}}$]{} (3.05,13.5); (6,11) node\[right,align=center\] [ [Tx ULA with]{}\
[$N$ antennas]{} ]{} – (3.05,13.5); (6,7) node\[right,align=center\] [ [Rx ULA with]{}\
[$M$ antennas]{} ]{} – (3.9,4.5);
We consider the LOS MIMO design over a range of distances $D$ between the two cars with $10 \le D \le 100$. Due to lack of space, we do not consider the horizontal displacement of the two cars within the lane (of width equal to $3.5$ m), which leads to slightly different orientation angles of the arrays. We assume the cars are facing each other, such that $\theta_{\text{\tiny Tx}}=\theta_{\text{\tiny Rx}}=0$ and $R=D$. Furthermore, we assume the same number of antennas in the Tx and Rx array and set it to $3$, i.e. $N=M=3$, as well as the same antenna separation $d$ at both the Tx and Rx array, i.e. $d=d_{\text{\tiny Tx}}^{}=d_{\text{\tiny Rx}}^{}$. The maximum length of the arrays is assumed to be equal and set to $1.8$ m, in order to fit in the bumpers of a standard car, i.e. $L_{\text{\tiny Tx}}= L_{\text{\tiny Rx}} = 1.8$ m. Note that for the considered distances, $D \gg L_{\text{\tiny Tx}}= L_{\text{\tiny Rx}} = 1.8$.
From (\[ortho\_sol\_asp\]) with $d_{\text{\tiny Tx}}^{}=d_{\text{\tiny Rx}}^{}=d$, $R=D$, and $N=M=3$, the optimum antenna separation for both arrays is given by $$\begin{aligned}
&d_{\text{}}^{} = \sqrt{ p \cdot \frac{\lambda D}{M}} \quad \quad \text{for} \quad p \in \left\{1,2,4,5,7,8,\cdots \right\}, \label{ortho_sol_ex} \end{aligned}$$ where only the multiples of $M=3$ are excluded from the set of positive integers for the possible values of $p$ in . To observe the multiple solutions for the optimum antenna separation $d$ which maximize the capacity, i.e. which result in an orthogonal LOS MIMO channel with $3$ equally strong eigenmodes, we plot $d$ given in (\[ortho\_sol\_ex\]) as a function of the longitudinal distance $D$ between the cars for the first eight values of $p$. As mentioned before, only the solution corresponding to $p=1$ is usually considered in the literature, as this corresponds to the smallest optimum antenna separation which then results in the shortest Tx and Rx arrays.
, ///in [1/1/myblue/o,2/2/myorange/square,3/4/myyellow/+,4/5/mypurple/diamond,5/7/mygreen/triangle,6/8/myred/asterisk,7/10/mylightblue/pentagon,8/11/magenta/x]{} [ ; ]{}
(axis cs: 0,0.5976) – (axis cs: 100,0.5976);
at (axis cs:70,0.6) \[anchor=south west\] [$d=0.5976$]{};
However, the curves for $p>1$ result in larger antenna separations which also maximize the channel capacity. For a given distance $D$ between the arrays, the optimum antenna separation increases with $\sqrt{p}$ as can be shown in (\[ortho\_sol\_ex\]). This results in an increasing length of the arrays with $p$, given by $(M-1) \sqrt{ p \cdot \frac{\lambda D}{M}}$. Due to the maximum length of the Tx and Rx arrays (car bumpers) in our V2V scenario given by $L_{\text{\tiny Tx}}= L_{\text{\tiny Rx}} = 1.8$ m, we consider only those solutions for $d$ which are less than or equal to $\frac{1.8}{M-1}$, i.e. with $M=3$, we consider only the optimum antenna separations which fulfill $$\begin{aligned}
d \le 0.9 .\end{aligned}$$ With this constraint, we observe from Fig. \[fig:res1\] there are at least two possible antenna separations which guarantee a $3 \times 3$ orthogonal LOS MIMO channel for each distance $D$ in the considered range of distances up to $100$ m.
More interestingly we observe in Fig. \[fig:res1\] that some antenna separations are optimum at *several* distances! For example, $d=0.5976$ is an optimum antenna separation at $D~=~10,\, 12.5,\, 14.2857,\, 20,\, 25,\, 50$ and $100$ m, which can be obtained from (\[ortho\_sol\_ex\]). At these distances, the LOS MIMO channel matrix with $d=0.5976$ is orthogonal with three equally strong eigenmodes as shown in Fig. \[fig:res2\], where the eigenvalues of ${\mathbf{H}}(D){\mathbf{H}}^{\text{H}}(D)$ are depicted for the considered range of distances between the cars. As $\text{tr}\left({\mathbf{H}}(D){\mathbf{H}}^{\text{H}}(D)\right)=MN=9$, the capacity with ${\mathbf{H}}(D)$ is maximized when the three eigenvalues of ${\mathbf{H}}(D){\mathbf{H}}^{\text{H}}(D)$ are equal to $3$. The channel matrix ${\mathbf{H}}(D) \in \mathbb{C}^{3 \times 3}$ corresponds to a LOS MIMO system given by (\[def\_H\]), (\[def\_hn\]) and (\[def\_rmn\]) with a Tx and Rx ULA consisting of $3$ antennas with an antenna separation of $0.5976$ and a distance $D$ between the Tx and Rx arrays. ${\mathbf{H}}(D)$ is given as a function of $D$ to highlight its dependency on the distance $R=D$ between the arrays via (\[def\_rmn\]). From Fig. \[fig:res2\], we also see that at certain distances some eigenvalues go to zero and hence, the LOS channel ${\mathbf{H}}(D)$ becomes rank deficient, e.g. at $D=34$ and $D=68$ the channel rank is $1$ and $2$, respectively.
,
table\[x index=0, y index=1\] [Eigs\_vs\_Distance\_R\_M3sol\_2\_D50.txt]{}; table\[x index=0, y index=2\] [Eigs\_vs\_Distance\_R\_M3sol\_2\_D50.txt]{}; table\[x index=0, y index=3\] [Eigs\_vs\_Distance\_R\_M3sol\_2\_D50.txt]{};
at (axis cs:50,7) \[anchor=south west,align=left\] [\
]{};
To elaborate further on the performance over the considered range of distances, we depict in Fig. \[fig:res3\] the capacity of the LOS MIMO channel for the described V2V link for three different antenna separations $d=0.5,\,0.5976,\,0.7$ for the Tx and Rx ULAs. In this case, the maximum capacity with an SNR of $13$ dB is $13.18$ bps/Hz, whereas the capacity is $10.72$ and $7.50$ bps/Hz when one or two eigenmodes go to zero, respectively. As can be seen in Fig. \[fig:res3\], the maximum capacity with $d=0.5976$ is achieved for the set of distances mentioned previously. For $d=0.5$ and $d=0.7$, the maximum capacity is achieved at other sets of distances. In fact, we observe a *stretching* and *shift* to the right of the capacity curve as the antenna separation $d$ increases, which can be explained as follows. Given that (\[ortho\_sol\_ex\]) can be rewritten as $\frac{d^{2}}{D}=p\frac{\lambda}{M}$, we can find other pairs of antenna separation $d^{\prime}$ and distance $D^{\prime}$ which achieve the same value $p \cdot\frac{\lambda}{M}$, i.e. $$\begin{aligned}
\frac{d^{\prime,2}}{D^{\prime}} = \frac{d^{2}}{D}, \quad \quad \text{such that} \quad \quad
D^{\prime} = D \, \frac{d^{\prime,2}}{d^{2}}.
\label{Ddexpression}\end{aligned}$$ For instance, from Fig. \[fig:res1\] the optimum antenna separation for $p=2$ at $D=50$ is $d=0.5976$. From (\[Ddexpression\]), the distance $D^{\prime}$ which achieves the same value $p \cdot\frac{\lambda}{M}$ as the previous setting but with $d^{\prime}=0.7$ is given by $D^{\prime}=50\cdot\frac{0.7^2}{0.5976^2}=68.8$ m. Thus, in Fig. \[fig:res3\] the point on the capacity curve for $d=0.5976$ at $D=50$ is shifted to the right by a factor of $\frac{0.7^2}{0.5976^2} \approx 1.37$ when the antenna separation $d=0.7$ is employed. The stretching of the capacity curve can also be explained in a similar manner.
,
table\[x index=0, y index=2\] [Cap\_vs\_Distance\_R\_M3d05.txt]{};
table\[x index=0, y index=1\] [Cap\_vs\_Distance\_R\_M3d05.txt]{};
table\[x index=0, y index=1\] [Cap\_vs\_Distance\_R\_M3sol\_2\_D50.txt]{};
table\[x index=0, y index=1\] [Cap\_vs\_Distance\_R\_M3d07.txt]{};
(axis cs: 35,6) node\[right,,align=center\] [[$2$ eigenvalues]{}\
[go to zero]{}]{} – (axis cs: 34,7.39) ;
(axis cs: 70,9.1) node\[right,align=center\] [[$1$ eigenvalue]{}\
[goes to zero]{}]{} – (axis cs: 68,10.6) ;
at (axis cs:50,13.18) () ; at (axis cs:68.8,13.18) () ;
(axis cs: 50,13.8) – (axis cs: 68.8,13.8) node\[above,midway,black\] [ ]{};
Conclusion {#Sec:Con}
==========
We have derived the general expression for the optimum antenna separation product for maximizing the capacity of a LOS MIMO channel with a Tx and Rx ULA. The expression leads to multiple solutions of the optimum antenna separation product which depend on the number of Tx and Rx antennas. We have proposed to exploit the multiple solutions for the LOS MIMO design over a range of distances between the transmitter and receiver, such as for V2V. We have shown that larger antenna separations can be beneficial and that some antenna separations are optimum at several distances. The provided results can serve as guidelines for the LOS MIMO design for V2V. Future work includes considering non-uniform linear arrays as well as the ground reflection in the V2V link.
Acknowledgment {#acknowledgment .unnumbered}
==============
[The authors would like to acknowledge support of this work under the 5GPPP European Project 5GCAR (grant agreement number 761510).]{}
[1]{}
P. F. Driessen and G. Foschini, “On the Capacity Formula for Multiple Input-Multiple Output Wireless Channels: A Geometric Interpretation,” in *IEEE Trans. Commun.*, vol. 47, no. 2, pp. 173-176, Feb. 1999.
D. Gesbert, H. Bolcskei, D. A. Gore and A. J. Paulraj, “Outdoor MIMO Wireless Channels: Models and Performance Prediction,” in *IEEE Trans. Commun*, Vol. 50, No. 12, pp. 1926-1934, Dec. 2002.
T. Haustein and U. Kruger, “Smart Geometrical Antenna Design exploiting the LOS Component to enhance a MIMO System based on Rayleigh-Fading in Indoor Scenarios,” in *IEEE PIMRC*, pp. 1144-1148, Sep. 2003.
F. Bohagen, P. Orten and G. E. Oien, “Construction and Capacity Analysis of High-Rank Line-of-Sight MIMO Channels,” in *IEEE WCNC*, pp. 432-437, Mar. 2005.
I. Sarris, A. R. Nix, “Design and Performance Assessment of High Capacity MIMO Architectures in the Presence of a Line-of-Sight Component,” in *IEEE Trans. Veh. Technol.*, vol. 56, no. 4, pp. 2194-2202, Jul. 2007.
E. Torkildson, C. Sheldon, U. Madhow and M. Rodwell, “Nonuniform Array Design for Robust Millimeter Wave MIMO Links,” in *IEEE Globecom*, Dec. 2009
L. Zhou and Y. Ohashi, “Design of Non-uniform Antenna Arrays for Robust mmWave LoS MIMO Communications,” in *IEEE PIMRC*, pp. 1397-1401, Sep. 2013.
[^1]: The case $N > M$ can be derived in a similar manner by simply interchanging the tranmsitter and the receiver.
[^2]: Due to the approximation (\[def\_rmn\]) for $r_{m,n}$, (\[orth\_cond\_0\]) can only be fulfilled approximately with (\[ortho\_cond\_1\]). As the error introduced with (\[def\_rmn\]) is negligible for practical systems [@Sarris07], we assume in the following that (\[ortho\_cond\_1\]) can be met with equality.
[^3]: If $M$ is prime, $\mathcal{D}_M(M)=\{M\}$ and hence, (\[ortho\_sol\]) is independent of $N$.
[^4]: Despite infinite solutions, not all solutions fulfill (\[orth\_cond\_0\]) in practice. As $p\! \rightarrow \!\infty$, the length of the arrays increase such that the assumption that the distance between the arrays is much larger than the array dimensions becomes invalid.
|
---
author:
- |
Mo Chen[^1], Qie Hu, Jaime F. Fisac, Kene Akametalu, Casey Mackin[^2], Claire J. Tomlin[^3]\
*University of California, Berkeley*
bibliography:
- 'references.bib'
title: 'Reachability-Based Safety and Goal Satisfaction of Unmanned Aerial Platoons on Air Highways'
---
Nomenclature {#nomenclature .unnumbered}
============
---------------------- --- --------------------------------------------------------------
${c}$ = Cost map
${\mathbb{P}}$ = A path between two points
${C}$ = Cumulative cost of a path
${V}$ = Value function of partial differential equations
${\mathbb{H}}$ = Air highway
${\hat{d}}$ = Direction of travel of air highway
${\mathbb{S}}$ = A sequence of air highways
${\mathcal{W}}$ = Waypoint
$x$ = System state (of a vehicle)
${p}=({p}_x, {p}_y)$ = Horizontal position
${v}=({v}_x, {v}_y)$ = Horizontal velocity
${\bar{p}}$ = Target position
${\bar{v}}$ = Target velocity
${d_\text{sep}}$ = Separation distance of vehicles within a platoon
${t_\text{faulty}}$ = Time limit for descent during potential conflict
${Q_{i}}$ = $i$th vehicle
${\mathcal{Q}_{i}}$ = Set of vehicles for vehicle ${Q_{i}}$ to consider for safety
---------------------- --- --------------------------------------------------------------
\
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported in part by NSF under CPS:ActionWebs (CNS-0931843) and CPS:FORCES (CNS1239166), by NASA under grants NNX12AR18A and UCSCMCA-14-022 (UARC), by ONR under grants N00014-12-1-0609, N000141310341 (Embedded Humans MURI), and MIT\_5710002646 (SMARTS MURI), and by AFOSR under grants UPenn-FA9550-10-1-0567 (CHASE MURI) and the SURE project.
[^1]: PhD Candidate, Department of Electrical Engineering and Computer Sciences
[^2]: PhD Student, Department of Electrical Engineering and Computer Sciences
[^3]: Professor, Department of Electrical Engineering and Computer Sciences, Member AIAA
|
---
abstract: 'Vector spin glasses are known to show two different kinds of phase transitions in presence of an external field: the so-called de Almeida–Thouless and Gabay–Toulouse lines. While the former has been studied to some extent on several topologies (fully connected, random graphs, finite-dimensional lattices, chains with long-range interactions), the latter has been studied only in fully connected models, which however are known to show some unphysical behaviors (e.g. the divergence of these critical lines in the zero-temperature limit). Here we compute analytically both these critical lines for XY spin glasses on random regular graphs. We discuss the different nature of these phase transitions and the dependence of the critical behavior on the field distribution. We also study the crossover between the two different critical behaviors, by suitably tuning the field distribution.'
author:
- Cosimo Lupo
- 'Federico Ricci-Tersenghi'
bibliography:
- 'myBiblio.bib'
title: 'Comparison of Gabay–Toulouse and de Almeida–Thouless instabilities for the spin glass XY model in a field on sparse random graphs'
---
Introduction
============
Vector spin glass models [@BinderYoung1986; @Book_MezardEtAl1987; @Book_FischerHertz1991] go beyond the much more studied discrete spin glass models (e.g. Ising and Potts models) by taking into account also small fluctuations in spin variables. A direct consequence of this is the presence of many more soft modes even at very low temperatures, which may change the critical behavior of the model.
Compared to Ising spin glasses, analytic studies on vector spin glasses are scarce and mostly related to fully connected models [@KirkpatrickSherrington1978; @deAlmeidaEtAl1978; @GabayToulouse1981; @ToulouseGabay1981; @CraggEtAl1982; @CraggSherrington1982; @GabayEtAl1982; @ElderfieldSherrington1982a; @ElderfieldSherrington1982b; @NobreEtAl1989; @SharmaYoung2010] (as usual finite-dimensional vector models can be studied approximately via a perturbative renormalization group at first order in $\epsilon=6-d$ [@HarrisEtAl1976; @ChenLubensky1977; @MooreBray1982], but the outcomes from this approach are still very much debated even for the simplest Ising models [@CharbonneauYaida2017]). Unfortunately, fully connected spin glass models have some undesirable features: e.g. the coupling strength must be scaled as $1/\sqrt{N}$ — with $N$ being the system size — in order to have a good thermodynamic limit, and the critical line in the temperature versus field plane diverges in the zero-temperature limit. These unrealistic features strongly ask for the solution of the *diluted* mean-field version of vector spin glass models, where coupling strength does not need to be scaled with the system size. However, previous works on the diluted version are even scarcer [@SkantzosEtAl2005; @CoolenEtAl2005; @BraunAspelmeier2006; @MarruzzoLeuzzi2015; @MarruzzoLeuzzi2016; @LupoRicciTersenghi2017a], and none of these works discusses the physics of vector spin glass models in presence of an external field.
It is worth reminding that in $m$-component vector models with $m\ge 2$ the effect of the external field may be drastically different from what happens in Ising ($m=1$) models. For example, when the external field has the same direction on each spin variable, the longitudinal and the transverse responses may be very different (and the divergence of the latter defines the Gabay–Toulouse critical line); an effect impossible to observe in spin glass models with Ising variables.
Our main aim is to understand the nature of the phase transitions taking place in presence of an external field in vector spin glass models defined on sparse random graphs (i.e. having a finite coordination number). To this aim, we focus on the simplest vector spin model, namely the XY model ($m=2$), and we study the phase diagrams and the critical behavior in presence of a uniform external field and eventually of a random field extracted according to different probability distributions.
It is worth reminding that sparse random graphs do not have short loops (their density scales as $1/N$) and so chiral ordering does not play any role on these topologies. Nevertheless our results may help elucidating the importance of the chiral ordering in finite-dimensional regular lattices, since we are going to show which kind of long-range order can actually take place without the need for a nonzero chiral order parameter.
The structure of the manuscript is the following. In Section \[sec:fully\_conn\] we summarize the main results about vector spin glasses in a field on fully connected graphs, showing the existence of two different kinds of phase transitions: the de Almeida–Thouless (dAT) one and the Gabay–Toulouse (GT) one. Then, in Section \[sec:XYmodel\_sparse\_graphs\] we define the XY model on sparse random graphs and show how to solve it via the belief-propagation algorithm. In Section \[sec:GT\_dAT\_lines\] we compute the critical lines by studying the stability of the replica symmetric solution under different types of external field, eventually recognizing them as GT or dAT critical lines. The different kinds of symmetry breaking taking place on GT and dAT critical lines are analyzed in Section \[sec:symBreak\]. Then, in Section \[sec:intermediate\] we study the crossover between GT and dAT critical behaviors. Our concluding remarks are reported in Section \[sec:concl\]. Finally, in Appendix \[app:SK\_limit\] we explain with full details how to recover the replica results cited in Section \[sec:fully\_conn\] via an alternative and simpler derivation, based on the dense limit of the belief-propagation equations, also proving the equivalence of the two approaches.
The fully connected case {#sec:fully_conn}
========================
The most generic Hamiltonian of vector spin glasses in a field reads $$\mathcal{H}[\{\boldsymbol{\sigma}_i\}]=-\sum_{(i,j)}J_{ij}\,\boldsymbol{\sigma}_i\cdot\boldsymbol{\sigma}_j -\sum_i \boldsymbol{H}_i\cdot\boldsymbol{\sigma}_i
\label{eq:H_vector_pm_J}$$ with spins $\{\boldsymbol{\sigma}_i\}$ being $m$-dimensional vectors of unit norm. The field $\boldsymbol{H}_i$ is represented by a $m$-dimensional vector as well, while couplings $J_{ij}$’s are as usual drawn from a suitable probability distribution $\mathbb{P}_J$ with support also on negative values.
Our work focuses on the sparse topology, that turns out to provide results that are closer to the finite-dimensional case. However, we first provide a brief summary of the results already obtained in the fully connected case — referring to Appendix \[app:SK\_limit\] for more details —, since they justify some choices we will make in the following.
In the scalar case ($m=1$, i.e. Ising spins) — where $J_{ij}$’s are Gaussian distributed with zero mean and variance $1/N$, while field $H$ is homogeneous — the system exhibits a paramagnetic phase for large enough values of $H$ and $T$, correctly described within a replica symmetric (RS) ansatz [@SherringtonKirkpatrick1975]. However, such solution turns out to be unstable when crossing a well defined line in the $H$ vs $T$ plane, named de Almeida–Thouless line [@deAlmeidaThouless1978]. A distinctive feature of the dAT line $H_\text{dAT}(T)$ is the $3/2$ exponent of its expansion at small fields, $H_\text{dAT} \simeq \tau^{3/2}$ with $\tau \equiv T_c-T$. Moreover, in fully connected models, the dAT line $H_\text{dAT}(T)$ diverges in the $T\to 0$ limit (a rather unphysical feature). Below this line, the assumption of symmetry between replicas is wrong and hence a scheme of replica symmetry breaking (RSB) has to be taken into account, eventually leading to the Parisi solution [@Parisi1980b], that actually represents the correct solution, at least for models on fully connected graphs. Notice that the case of a random quenched Gaussian-distributed field does not qualitatively change the above picture [@Bray1982], since a suitable gauge transformation maps the model back to the one with strictly positive fields [@Book_Nishimori2001].
Moving to the vector case ($m \geqslant 2$), again the RS paramagnetic solution is stable for large enough $H$ or $T$. However, the stability of this solution now depends on the distribution of the external field, and in particular on its direction. Indeed, Gabay and Toulouse showed in Ref. [@GabayToulouse1981] that the paramagnetic solution in presence of a *uniform* field becomes unstable towards RSB along a critical line $H_\text{GT}(T)$ very different from the dAT line: e.g., at small fields it behaves as $H_\text{GT} \simeq \tau^{1/2}$. At the Gabay–Toulouse critical line, the degrees of freedom transverse to the field direction show spontaneous symmetry breaking, highlighted by a nonzero value of the transverse overlap $q_{\perp}$. The freezing of longitudinal degrees of freedom seems to occur at lower temperatures, along a line with features reminiscent of the dAT line (however, this computation would require the use of the full RSB ansatz below the GT line, not taken into account in Ref. [@GabayToulouse1981]).
Later works [@CraggEtAl1982; @GabayEtAl1982] then showed that RSB actually involves both transverse and longitudinal degrees of freedom along the same line — the GT one — though in a different manner: $q_{\perp}$ suddenly shows a *strong* RSB as soon as the GT line is crossed, with a strong dependence on the Parisi parameter $x$. Instead $q_{\parallel}$, i.e. the longitudinal overlap with respect to the direction of the field, *weakly* depends on $x$ until the dAT line is crossed, when a strong RSB occurs along the field direction as well. Hence, the dAT line in vector spin glasses with a uniform field has been recognized as a crossover between a weak and a strong RSB along the longitudinal direction, rather than a sharp phase transition from a RS to a RSB region, which at variance occurs at the GT line.
The situation changes when considering a random field, where randomness can affect the field strength, its direction or both. It has been pointed out by Sharma and Young [@SharmaYoung2010] that the key ingredient to avoid the GT line and hence recover the dAT line as a sharp RS-RSB phase transition also for vector spin glasses is the randomness in the direction of the external field, while the randomness in its strength is not essential. Indeed, the crucial observation is that the GT line is also linked to a breaking in the spin symmetry (the inversion symmetry with respect to the direction given by the external field), while dAT line is not linked to any change in spin symmetry. Moreover, the resulting line of RS instability turns out to have the same $3/2$ exponent and the same features of the dAT line in the Ising case.
The XY model on sparse graphs {#sec:XYmodel_sparse_graphs}
=============================
Let us now move to the diluted case. Without any loss of generality, we choose to study the $m=2$ case, that is the so-called XY model [^1]. This is a particularly simple vector model, since each spin can be described by a single *continuous* degree of freedom $\theta_i\in[0,2\pi)$, that we assume to represent the direction of the vector spin $\boldsymbol{\sigma}_i$. Analogously, also the field on the $i$-th site can be described by its modulus $H_i$ and its direction $\phi_i\in[0,2\pi)$. Moreover, keeping in mind the key observation by Sharma and Young, we fix $H_i=H$ on each site and let only directions $\{\phi_i\}$ to vary according to a suitable probability distribution $\mathbb{P}_{\phi}$. The corresponding Hamiltonian reads $$\mathcal{H}[\{\theta_i\}]=-\sum_{(ij)\in\mathcal{E}}J_{ij}\,\cos{(\theta_i-\theta_j)} -H\sum_i \cos{(\theta_i-\phi_i)}$$ where $\mathcal{E}$ is the edge set of the interacting graph $G$. The couplings $J_{ij}$’s are random quenched variables distributed according to the symmetric bimodal distribution $$\mathbb{P}_J(J_{ij}) = \frac{1}{2}\delta(J_{ij}-J)+\frac{1}{2}\delta(J_{ij}+J)\,.$$
Our main task is to characterize the instability of the XY model in an external field when the underlying graph is no longer a fully connected graph, but a sparse random graph [@Book_Bollobas2001]. Indeed, it is well known that many results of the mean-field approach provided by fully connected topologies are not representative of what actually happens in the finite-dimensional case: among all, the lack of strong spatial heterogeneities and the impossibility of defining and studying correlation functions. Contrarily, on sparse random graphs one can naturally define distances between spins, long-range correlations and local heterogeneities.
In particular, we focus on the ensemble of Random Regular Graphs (RRG) of fixed connectivity $C=3$, namely each vertex has exactly $C=3$ neighbors. These graphs have the crucial property of being *locally tree-like*, i.e. each neighborhood of a given site contains no loops with high probability, eventually tending to one in the thermodynamic limit. This feature allows us to invoke the Bethe approximation [@Bethe1935] and hence to exploit the Belief-Propagation (BP) algorithm [@Book_Pearl1988; @YedidiaEtAl2003; @Book_MezardMontanari2009] to solve the model.
Notice that this approach is equivalent to the RS cavity method [@MezardParisi1987] and it turns out to be always correct for models defined on trees and on large enough random graphs, given the correlations between spins decay fast enough [@MezardParisi2001; @Book_MezardMontanari2009]. When the RS solution becomes unstable towards RSB, one can then use the ansatz based on the 1-step replica symmetry breaking (1RSB) scheme [@MezardParisi2001; @MezardParisi2003] (the full-RSB scheme has not been developed yet within the cavity approach [@Parisi2017]).
Since our interest is in identifying critical lines between RS and RSB phases, we are going to use a RS formalism, i.e. the BP algorithm, focusing specifically on the stability of the BP fixed point.
In the Bethe approximation [@Bethe1935], each physical observable can be computed starting from just the one-point $\eta_i(\theta_i)$ and the two-point $\eta_{ij}(\theta_i,\theta_j)$ marginals. In turn, their computation is based on the knowledge of the cavity marginals $\{\eta_{i\to j}(\theta_i)\}$ through the following relations
$$\begin{split}
\eta_i(\theta_i) &= \frac{1}{\mathcal{Z}_i}\,e^{\,\beta H\cos{(\theta_i-\phi_i)}}\\
&\qquad\times\prod_{k\in\partial i}\int d\theta_k\,e^{\,\beta J_{ik}\cos{(\theta_i-\theta_k)}}\,\eta_{k\to i}(\theta_k)
\end{split}
\label{eq:full}$$
$$\eta_{ij}(\theta_i,\theta_j) = \frac{1}{\mathcal{Z}_{ij}}\,e^{\,\beta J_{ij}\cos{(\theta_i-\theta_j)}}\,\eta_{i\to j}(\theta_i)\,\eta_{j\to i}(\theta_j)$$
where $\partial i$ is the set of neighbors of the $i$-th spin, while $\mathcal{Z}_i$ and $\mathcal{Z}_{ij}$ are normalizing constants.
Cavity marginals satisfy the set of self-consistency equations going under the name of BP equations [@YedidiaEtAl2003; @Book_MezardMontanari2009]: $$\begin{gathered}
\eta_{i\to j}(\theta_i) = \mathcal{F}[\{\eta_{k\to i}\},\{J_{ik}\},\phi_i]\equiv \frac{1}{\mathcal{Z}_{i\to j}}\,e^{\,\beta H\cos{(\theta_i-\phi_i)}}\\
\times\prod_{k\in\partial i\setminus j}\int d\theta_k\,e^{\,\beta J_{ik}\cos{(\theta_i-\theta_k)}}\,\eta_{k\to i}(\theta_k)
\label{eq:BP_XY_eqs}\end{gathered}$$ with $\mathcal{Z}_{i\to j}$ ensuring the correct normalization. The physical meaning of $\eta_{i\to j}(\theta_i)$ is that of the probability distribution of the variable $\theta_i$ in a modified graph where edge $(i,j)$ has been removed.
When there is no external field ($H=0$), the BP equations (\[eq:BP\_XY\_eqs\]) are solved by the simple paramagnetic solution $\eta_{i\to j}(\theta_i)=1/(2\pi)$ for each directed edge, which turns out to be stable only above a certain critical temperature $T_c$. Slightly below $T_c$, an approximated solution can still be analytically obtained, based on a Fourier expansion [@LupoRicciTersenghi2017a]. Instead, when $T \ll T_c$ or when a field is present, the BP equations (\[eq:BP\_XY\_eqs\]) need to be solved numerically.
Since we are not interested in a given realization of the quenched disorder, but rather in the average over the disorder distribution, we solve the BP equations (\[eq:BP\_XY\_eqs\]) in distribution sense. In practice we look for the probability distribution of cavity marginals $P[\eta_{i\to j}]$ solving the following equation $$\begin{split}
P[\eta_{i\to j}] &= \mathbb{E}_{G,J,\phi} \int\prod_{k=1}^{C-1}\mathcal{D}\eta_{k\to i}\,P[\eta_{k\to i}]\\
&\qquad\times\delta\Bigl[\eta_{i\to j}-\mathcal{F}[\{\eta_{k\to i}\},\{J_{ik}\},\phi_i]\Bigr]
\label{eq:BP_XY_eqs_distr}
\end{split}$$ with $\mathbb{E}_{G,J,\phi}$ indicating the average over the ensemble of RRGs with $C=3$ and over the coupling and field probability distributions. The fixed point $\{\eta^*_{i\to j}\}$ of BP self-consistency equations (\[eq:BP\_XY\_eqs\]) so becomes a fixed point for their probability distribution, $P^*[\eta]$. The advantage brought by this approach is that the set of distributional equations (\[eq:BP\_XY\_eqs\_distr\]) can be efficiently solved via the Population Dynamics Algorithm (PDA), firstly introduced in Ref. [@AbouChacraEtAl1973] and then revisited and refined in Refs. [@MezardParisi2001; @MezardParisi2003].
A crucial issue arising when numerically solving BP equations — both on a given instance of the quenched disorder or in the PDA approach — regards the discretization of continuous variables. Indeed the marginals $\eta(\theta)$ are functions over the $[0,2\pi)$ interval and would in principle require an infinite number of parameters to be described. The most effective approach [@LupoRicciTersenghi2017a] is to discretize such an interval in $Q$ bins of width $2\pi/Q$ each. The resulting model is no longer endowed with the $\mathrm{O}(2)$ continuous symmetry, but with the discrete $Z_Q$ symmetry, and it is known as the *Q-state clock model* [@NobreSherrington1986; @IlkerBerker2013; @IlkerBerker2014; @MarruzzoLeuzzi2015; @LupoRicciTersenghi2017a; @CaglarBerker2017a; @CaglarBerker2017b].
In a previous work [@LupoRicciTersenghi2017a] we showed that the $Q$-state clock model provides an efficient and reliable approximation of the XY model, in both the weak and the strong disorder regimes, with deviations in physical observables decreasing exponentially fast in $Q$. This result allows us to safely use $Q=64$ in numerical simulations. Notice that BP equations for the $Q$-state clock model can be numerically solved with a computational effort that scales as $O(Q^2 N)$, with $N$ being the size of the graph (or equivalently the population size $\mathcal{N}$ in the PDA approach). Hence the exponential convergence in $Q$ actually provides a huge enhancement in numerical simulations.
Computing critical lines in sparse models {#sec:GT_dAT_lines}
=========================================
The linear stability of the fixed point $P^*[\eta]$ of (\[eq:BP\_XY\_eqs\_distr\]) provides the stability of the RS ansatz. We look at the global growth rate of perturbations $\{\delta\eta_{i\to j}(\theta_i)\}$ to fixed-point cavity marginals. Such perturbations evolve according to the following equations [@LupoRicciTersenghi2017a] $$\delta\eta_{i\to j} = \sum_{k\in\partial i\setminus j}\Biggl{|}\frac{\delta \mathcal{F}[\{\eta_{k\to i}\},\{J_{ik}\},\phi_i]}{\delta \eta_{k\to i}}\Biggr{|}_{\{\eta^*_{k\to i}\}}\delta\eta_{k\to i}
\label{eq:pert}$$ which are nothing but the linearized version of (\[eq:BP\_XY\_eqs\]). We solve these equations via PDA, evolving a population of $\mathcal{N}$ pairs $(\eta_{i\to j},\delta\eta_{i\to j})$, actually pairs of vectors of length $Q$. We measure the global growth rate $\lambda_{\text{BP}}$ of perturbations as follows $$\lambda_{\text{BP}} \equiv \lim_{t\to\infty}\frac{1}{t\,\mathcal{N}}\sum_{(i\to j)}\ln\int |\delta\eta_{i\to j}(\theta)| d\theta$$ where the integral of the absolute value of the perturbation is actually performed summing over the $Q$ discrete values. So when $\lambda_{\text{BP}}$ is positive the RS fixed point is unstable, while it is stable if $\lambda_{\text{BP}}<0$. This approach is known as *Susceptibility Propagation* (SuscP). Notice that, as usual in sparse models, a strong heterogeneity characterizes the population of cavity messages, with the corresponding perturbations spanning several orders of magnitude. Hence, we chose to average the logarithm of the norm of the perturbations over the population, and this in turn make the estimate of $\lambda_{\text{BP}}$ more robust and reliable.
However, the precise determination of the critical point requires to use some precautions, because the BP equations have multiple solutions and some of these solutions (e.g. the paramagnetic one) change their stability at the critical point. Thus at the critical point the iterative solution of BP equations may take a large time to converge to the right solution. In order to avoid such a critical slowing down, we solve the BP equations at a given temperature using as initial condition the fixed point reached at a nearby temperature: we call ‘cooling’ and ‘heating’ these two protocols to solve the BP equations, depending on whether the temperature is decreased or increased in successive rounds. Although the critical slowing down is much reduced, these two protocols have the problem that may get stuck in a solution, even when this solution becomes unstable. This is well illustrated by the cooling data at $\Delta=0$ in Fig. \[fig:roundTrip\]. We try to solve this problem by perturbing a little bit the initial condition before starting the iterative search for the solution to the BP equations: we add to each component of the $\eta$ marginals independent random numbers $\Delta |z|$ with $z$ being a Gaussian random variable of zero mean and unitary variance. The resulting stability parameter $\lambda_{\text{BP}}$ averaged over iterations in the time range $t\in[151,300]$ is shown in Fig. \[fig:roundTrip\]. We clearly see that when increasing $\Delta$, the population dynamics algorithm leaves sooner the unstable fixed point (e.g. the paramagnetic fixed point in the low-temperature region).
![Stability parameter $\lambda_{\text{BP}}$ for the spin glass XY model on a $C=3$ RRG at zero field. Data are collected during cooling and heating numerical experiments with 300 iterations for temperature, and averaged over the last 150 iterations. The black dot marks the exact value for the critical temperature.[]{data-label="fig:roundTrip"}](lambdaBP_zeroField_roundTrip){width="\columnwidth"}
For $H=0$, a second-order phase transition occurs between the high-temperature RS-stable phase and the low-temperature RS-unstable phase, with a critical temperature $T_c=1/\beta_c$ given by [@SkantzosEtAl2005; @CoolenEtAl2005; @LupoRicciTersenghi2017a]: $$\left[\frac{I_1(\beta_c J)}{I_0(\beta_c J)}\right]^2=\frac{1}{C-1}
\label{eq:Tc}$$ where $C$ is the degree of the ensemble of RRG considered, while $I_0(\cdot)$ and $I_1(\cdot)$ are the modified Bessel functions of the first kind respectively of order zero and one [@Book_AbramowitzStegun1964]. Critical temperatures for some values of $C$ are reported in Table \[tab:Tc\_vs\_C\]. The strength of the coupling constants $J=1/\sqrt{C-1}$ has been chosen in order to approach the critical temperature $T_c=1/2$ in the fully connected limit (indeed, when normalizing $m$-dimensional spin vectors to unity, $T_c$ is equal to $1/m$ in the fully connected limit).
$C$ $T_c/J$ $T_c$
----- --------- --------
3 0.4859 0.3436
4 0.7012 0.4048
6 0.9977 0.4462
8 1.2234 0.4624
12 1.5805 0.4765
16 1.8704 0.4829
20 2.1211 0.4866
: Critical temperatures $T_c$ for the XY model on random $C$-regular graphs with no external field and unbiased random couplings $J_{ij}\in\{+J,-J\}$. The coupling strength $J=1/\sqrt{C-1}$ has been chosen such that $\lim_{C\to\infty} T_c = 1/2$.
\[tab:Tc\_vs\_C\]
The exact critical temperature at $H=0$ is reported in Fig. \[fig:roundTrip\] by a black dot. It is clear that the best way to estimate such a critical temperature from the stability parameter $\lambda_{\text{BP}}$ is to check when the data gathered during the cooling experiment cross the axis. Such a crossing point is almost independent on the value of $\Delta$ and can be very well computed either by interpolating the data in a temperature range that includes $T_c$ or by linearly extrapolating the data collected at $T>T_c$.
On the contrary, we notice that the data in the heating experiment are of no help in identifying precisely $T_c$ for two reasons. Firstly, the stability parameter $\lambda_{\text{BP}}$ is very close to zero in a broad temperature range below $T_c$, thus inducing a very large statistical error on the estimate of $T_c$. Secondly, there are systematic effects that make $\lambda_{\text{BP}}$ slightly negative close to $T_c$, thus producing a biased estimate of $T_c$. A further data inspection reveals that these systematic effects are due to a very slow convergence of the population dynamics to the paramagnetic fixed point, even in presence of the $\Delta$ perturbation. In summary, a random perturbation is good for leaving the trivial fixed point, but is not as good to reach it again from a random configuration.
![Stability parameter $\lambda_{\text{BP}}$ for the spin glass XY model on a $C=3$ RRG at zero field. All the points reported have been measured in the stationary regime. The full green line refers to the analytic evaluation of $\lambda_{\text{BP}}$ on the paramagnetic solution. The inset shows the power-law behavior below the critical point, $\lambda_{\text{BP}} \propto \tau^\alpha$, with $\alpha=1.6(1)$.[]{data-label="fig:H0"}](lambdaBP_zeroField_Cooling_Delta1e-2){width="\columnwidth"}
Having discussed the possible problems arising in the numerical determination of the critical temperature, we show in Fig. \[fig:H0\] only the data that have been collected in the stationary regime at the stable fixed point. Some points are missing for temperatures slightly below $T_c$, but they are not really necessary in the determination of $T_c$, which is achieved by using only data with $T\ge T_c$. Being at $H=0$, we can also plot with a full line the analytic expression for $\lambda_{\text{BP}}$ that holds at the trivial paramagnetic fixed point. Instead, the behavior of the stability parameter below the critical temperature is well fitted by the power law $\lambda_{\text{BP}} \propto \tau^\alpha$ with $\alpha=1.6(1)$.
At this point, once understood how to effectively locate the transition from the RS-stable region to the RS-unstable one, we can switch on the external field. We will focus on two diametrically opposite field distributions, trying to recover also in the sparse case the well-known GT and dAT transition lines studied on fully connected graphs: firstly a uniform field and then a randomly oriented field with a flat distribution of the local field direction.
The uniform field case
----------------------
In order to check if the GT line also appears in the sparse case, we fix the field direction to be the same on each site, e.g. the $\hat{x}$ direction with no loss of generality: $\mathbb{P}_{\phi}(\phi_i) = \delta(\phi_i)$.
In Fig. \[fig:unif\_many\_H\] we show the stability parameter $\lambda_{\text{BP}}$ versus $T$ with a uniform field of several intensities. We are plotting all the data collected during a cooling protocol, but from the discussion above we know that points slightly below the critical temperature should be discarded. We notice that the main effect of the field is to shift the data leftwards in the plot, that is the same instability parameter is achieved at a lower temperature.
![Stability parameter $\lambda_{\text{BP}}$ for the spin glass XY model on a $C=3$ RRG with a uniform external field of intensity $H$. The two panels show data with different ranges of fields. The lower one makes evident the leftward shift of the curves when increasing the field strength $H$.[]{data-label="fig:unif_many_H"}](lambdaBP_UF_severalH_Cooling_Delta1e-2){width="\columnwidth"}
From data in Fig. \[fig:unif\_many\_H\] we estimate the critical temperature for each value of $H$ from a fit in the $T>T_c$ region. We repeat the measurements for several connectivities $C$ and we summarize in Fig. \[fig:GT\_lines\] the results. We draw the corresponding critical lines in the $(T,H)$ plane and we observe that all they seem to have the same behavior at small fields, namely the scaling $H_c(T) \propto \tau^{1/2}$ that holds for the GT line in the fully connected model. An evidence of this is shown in the inset of Fig. \[fig:GT\_lines\], where we draw the critical lines in the $(T,H^2)$ plane: zooming on the interesting region of small fields, we observe a clear linear behavior in $\tau$ (such a linear behavior is soon lost due to the fact the $H_c(T)$ curves change concavity at moderately small field values). Notice that no error bars have been reported in the main plot of Fig. \[fig:GT\_lines\], because they would have not been appreciable, since critical points have been estimated with a statistical error of order $O(10^{-4})$.
Together with the critical curves for the diluted case with different connectivities, we also report the GT line for the fully connected graph (i.e. in the SK limit), computed as explained in Appendix \[app:SK\_limit\]. It is evident the collapse of the former ones onto the latter one in the large-$C$ limit, with the most important dependence in $C$ being in the location of the zero-field critical point, while the functional form of the instability line seems to have already converged to the dense limit. So we can safely identify the critical lines reported for different $C$ values as the corresponding GT transition lines.
![Critical lines in a uniform field for a spin glass XY model on a random $C$-regular graph. The corresponding line in the fully connected model (SK limit) is given by the black curve. The inset shows evidence for the $H_c(T) \propto \tau^{1/2}$ behavior, typical of the GT transition.[]{data-label="fig:GT_lines"}](GT_line_severalC_Jrescaled_H2_vs_T){width="\columnwidth"}
The random field case
---------------------
In order to study the onset of the dAT instability in the disordered XY model, and following the suggestion of Ref. [@SharmaYoung2010], we now consider the model where the external field is constant in intensity, but has random directions $\{\phi_i\}$ uniformly drawn in $[0,2\pi)$.
Since the field has a different (random) direction on each site, it is no longer possible to define global order parameters respectively parallel and perpendicular to the field direction; in other words, the overlaps $q_{\parallel}$ and $q_{\perp}$, used in the replica calculation to define the GT instability (see Appendix \[app:SK\_limit\]), are now useless. Eventually it will be possible to define the instabilities parallel and perpendicular to the field direction only locally, as it will be discussed in the next section. For the moment, we study the global growth rate of perturbations to the BP fixed point, averaged over the population, that is the SuscP algorithm.
![Stability parameter $\lambda_{\text{BP}}$ for the spin glass XY model on a $C=3$ RRG with a randomly oriented external field of fixed intensity $H$. At variance with the uniform-field case, the curve $\lambda_{\text{BP}}(T)$ mainly moves downwards when increasing $H$, while smoothing away the zero-field singularity.[]{data-label="fig:dAT_many_H"}](lambdaBP_RF_severalH_Cooling_Delta1e-2){width="\columnwidth"}
![Critical lines in a field of random direction for a spin glass XY model a random $C$-regular graph. The corresponding line in the fully connected model (SK limit) is given by the black curve. The inset shows evidence for the $H_c(T) \propto \tau^{3/2}$ behavior, typical of the dAT transition.[]{data-label="fig:dAT_lines"}](dAT_line_severalC_Jrescaled_H2_3_vs_T){width="\columnwidth"}
In Fig. \[fig:dAT\_many\_H\] we show the instability parameter $\lambda_{\text{BP}}$ versus the temperature for several values of the field intensity $H$. At variance with the uniform-field case, now the curve moves mostly downwards with $H$ in the entire low-temperature region. The most dramatic effect, with respect to the uniform-field case, is that the stability parameter $\lambda_{\text{BP}}$ changes a lot even for very small fields, smoothing away the zero-field singularity (compare Fig. \[fig:dAT\_many\_H\] with the lower panel in Fig. \[fig:unif\_many\_H\]).
In Fig. \[fig:dAT\_lines\] we plot the corresponding critical lines in the $(T,H)$ plane for different connectivities $C$. Close to the respective zero-field critical points, the behavior is clearly $H_c(T) \propto \tau^{3/2}$, typical of the dAT line. Again, a fast convergence towards the SK limit (the black line, computed via equations in Appendix \[app:SK\_limit\]) can be detected, with the most important dependence in $C$ given by the location of $T_c(H=0)$. The evidence for the dAT-like behavior of these $H_c(T)$ lines is shown in the inset of Fig. \[fig:dAT\_lines\], where critical lines are plotted in the $(T,H^{2/3})$ plane, following the expected linear trend.
GT vs dAT: different ways of breaking the spin symmetries {#sec:symBreak}
=========================================================
![GT and dAT critical lines computed in the XY model with $J_{ij}=\pm1$ on a $C=3$ random regular graph.[]{data-label="fig:GT_vs_dAT"}](dAT_plus_GT_line_C3){width="\columnwidth"}
In Fig. \[fig:GT\_vs\_dAT\] we show together the GT and the dAT critical lines for the XY model on a $C=3$ RRG. As explained in the previous Section, the GT line has been computed by applying a uniform field with constant direction, while the dAT line has been obtained applying a uniform field of random directions. The overall shape of the two critical lines, including the exponent relating $H$ to $\tau$ in the vicinity of the zero-field critical point, is very similar to the fully connected case. The main difference with respect to the fully connected case is the lack of a divergence of the critical fields in the $T\to 0$ limit, as expected for the diluted case. An estimate of them, say $H_{\text{GT}}$ and $H_{\text{dAT}}$ respectively, can be obtained via an extrapolation from the finite-temperature datasets, though quite noisy due to the diverging slope of the two critical curves close to the $T=0$ axis. A more precise and reliable location of $H_{\text{GT}}$ and $H_{\text{dAT}}$ can be achieved directly in the zero-temperature setting [@LupoRicciTersenghi2017a]; however, the zero-temperature BP approach requires some further precautions about the way perturbations are iteratively computed, both in the PDA [@LupoRicciTersenghi2017a] as well as on a given instance of the model .
We are now interested in understanding which symmetries get broken along these two different critical lines. In fully connected models, the relation between the GT transition line and the freezing of the transverse degrees of freedom of spins with respect to the direction of the field is already known since the original work of Gabay and Toulouse [@GabayToulouse1981]. Indeed, it is a transition from the solution $q_{\perp}=0$ to the one $q_{\perp}\neq 0$. At the same time, the dAT line — later interpreted as a crossover — has been naturally linked to the freezing of the longitudinal degrees of freedom. However, the strong connection between these instabilities and the distribution of the *direction* of the field has been pointed out only recently by Sharma and Young [@SharmaYoung2010].
Here we want to reach a deeper understanding of the kind of instabilities becoming critical on the GT and dAT lines. To this aim, we perform a local analysis by computing, for each spin, the direction along which the most probable fluctuation may take place. We are interested in understanding whether this local fluctuation is parallel or perpendicular to the external field on the same spin (remind that in the random case the field direction changes from spin to spin and so the projection according to any global direction would be useless).
In the PDA we store $\mathcal{N}$ pairs $(\eta_{i\to j},\delta\eta_{i\to j})$ of cavity marginals and corresponding (linear) perturbations. Once the BP fixed point $\mathbb{P}^*[\eta]$ for the cavity marginals has been reached, the perturbations provide the direction along which such fixed point gets most easily destabilized. Then our analysis proceeds spin by spin. For each spin $i$, we extract randomly $C$ pairs from the fixed-point population, we compute the full marginal $\eta_i$ by using Eq. (\[eq:full\]) and the corresponding perturbation $\delta\eta_i$ by using Eq. (\[eq:pert\]) with the sum running over the same $C$ randomly chosen elements. The following local vectors
$$\boldsymbol{m}_i \equiv \int \, d\theta_i \, \eta_i(\theta_i) \, \Bigl(\cos{\theta_i},\sin{\theta_i}\Bigr)$$
$$\delta\boldsymbol{m}_i \equiv \int \, d\theta_i \, \delta\eta_i(\theta_i) \, \Bigl(\cos{\theta_i},\sin{\theta_i}\Bigr)$$
provide the required information: $\boldsymbol{m}_i$ is the local magnetization, while $\delta\boldsymbol{m}_i$ points along the direction of the most probable local fluctuation. The scalar product between $\delta\boldsymbol{m}_i$ and the field $\boldsymbol{H}_i$ on the same spin makes explicit the kind of perturbation to the BP fixed point: indeed a transverse perturbation would yield a scalar product close to zero, while a longitudinal perturbation would correspond to a scalar product close to one (in absolute value). In order to be more quantitative, let us define the following local parameter $$\cos{\vartheta_i} \equiv \frac{\delta\boldsymbol{m}_i\cdot\boldsymbol{H}_i}{\norm{\delta\boldsymbol{m}_i}\norm{\boldsymbol{H}_i}} = \frac{\delta\boldsymbol{m}_i\cdot\boldsymbol{H}_i}{\delta m_i\,H}
\label{eq:def_cosVarTheta}$$ and let us compute its distribution by using the SuscP algorithm. Its distribution for several points along the dAT and the GT lines is depicted in Fig. \[fig:histo\_1d\_ScalProd\] for a $C=3$ RRG.
{width="\textwidth"}
{width="\textwidth"}
The interpretation of the GT line as an instability in the transverse direction and that of the dAT line as an instability in the longitudinal direction — with respect to the direction of the local field $\boldsymbol{H}_i$ — is quite well confirmed by the two histograms of $\cos{\vartheta_i}$. Notice that the occurrence of transverse excitations also on the dAT line — even though with a smaller probability with respect to longitudinal excitations — is due to the fact that the field strength $H$ is not so large along such line, hence the energy cost of a transverse perturbation is surely larger than the cost of a longitudinal perturbation, but not enough to suppress them. On the other hand, on the GT line the higher the field strength $H$, the stronger the transverse behavior of perturbations.
{width="\textwidth"}
The two different behaviors can be better appreciated if discriminated according to the strength of the local *effective* field, given by the sum of the local field $\boldsymbol{H}_i$ and of the messages coming from the nearest-neighbor spins. A simple estimate of this strength is given by the polarization of the site marginal, namely by the modulus of the site magnetization $\boldsymbol{m}_i$. Indeed, a value of $m_i$ close to zero is representative of a weak local effective field, hence of a spin that can be easily excited along different directions with almost the same energetic cost. Instead, a strongly polarized spin is identified by a local magnetization $m_i$ close to one, hence the most likely perturbation is of course the most energetically favorable one.
In Fig. \[fig:histo\_2d\_ScalProd\] we report the joint probability distribution of $(m_i,\cos{\vartheta}_i)$ for the same points of Fig. \[fig:histo\_1d\_ScalProd\] along both instability lines. Again the difference between the basic behaviors of GT and dAT lines is quite clear, with a preference for $\cos{\vartheta_i}=0$ in the former case and for $\cos{\vartheta_i}=\pm 1$ in the latter case. In addition to this, also the dependence on the specific point of the line is evident. Indeed, when temperature is large, the local effective field is typically weak and hence the energetic cost of the two kinds of excitations is similar. So on the GT line we can also observe a nonnegligible fraction of longitudinal perturbations, conversely on the dAT line. Instead, when lowering the temperature and hence getting closer to the $T=0$ axis, the site marginals strongly polarize ($m_i\to 1$) and hence likely perturbations become more and more energetically favorable with respect to the unlikely ones. This results in well defined peaks for both lines, with the probability of an unlikely perturbation going to zero with $T$.
So the correspondence between the two transitions in field and the breaking of spin symmetries is well established, as well as the simultaneous breaking of replica symmetry in both cases.
Intermediate behaviors {#sec:intermediate}
======================
The two cases analyzed so far — a constant field for the GT line and a random field with a flat distribution of the field local direction for the dAT line — represent the two extremal cases in the distribution of the field direction (always keeping in mind that the field strength can be safely set equal to $H$ for all the sites without any loss of generality). Now we want to discuss some intermediate cases, in order to check which instability, between the GT-like and the dAT-like, is the dominant one in a more general case.
Since we actually solve the $Q$-state clock model, we prefer to work with probability distributions of the field direction $\phi$ taking values in the discrete set of $Q$ elements $\mathcal{S}=\{0,2\pi/Q,\ldots,2\pi(Q-1)/Q\}$. There are still infinitely many distributions that interpolate between a delta function in $\phi=0$ and a uniform distribution over $\mathcal{S}$. For convenience, let us make a change of variables, taking $\phi = 2\pi\kappa/Q$ with $\kappa$ being an integer number in the range $0\le\kappa<Q$. We choose to work with the following two classes of distributions parametrized by a single number:
- $0 \le \kappa < Q'$ uniformly with probability $1/Q'$;
- $\kappa = 0$ with probability $1-w(Q-1)/Q$ and\
$0<\kappa<Q$ uniformly with probability $w/Q$.
The ranges for the two parameters are $1\le Q' \le Q$ in the first class, with $Q'$ integer, and $0 \le w \le 1$ in the second one, with $w$ real-valued. It is easy to check that the extremal values for these parameters recover the field distributions used in the previous sections to study GT and dAT critical lines, respectively.
{width="\textwidth"}
In Fig. \[fig:firstClass\] we plot the critical lines obtained for $C=3$ using the first class of field distributions with different values of the parameter $Q'$. Remind that $Q'=1$ and $Q'=Q=64$ correspond respectively to GT and dAT lines. In the left panel we see that even with the smallest nontrivial value $Q'=2$ the critical line moves sensibly: so the loss of the perfect alignment among the local directions of the external field seems to have a visible effect on the critical properties of the model. In the right panel we study in more detail the behavior of the critical lines close to the zero-field critical point: while the extremal case $Q'=1$ follows a power law with the GT-like exponent $1/2$, for $Q'>1$ the data seem to follow the dAT-like exponent $3/2$ (dashed lines have slopes $1/2$ and $3/2$, respectively). So, to the best of our numerical evidences, the GT-like critical behavior seems to be relegated to the singular $Q'=1$ case, where all the external fields are perfectly aligned.
Given that in the first class of distributions there is a minimal perturbation $O(1/Q)$ to the GT-like distribution, we study now the second class of field distributions, where the intensity of the perturbation with respect to the $\delta(\phi)$ distribution is given by the continuous parameter $w$. In Fig. \[fig:secondClass\] we show the results obtained with the second class of interpolating functions. Also in this case we notice that even the smallest $w=0.01$ perturbation produces a sensible effect on the critical line, that changes from a GT-like shape to a dAT-like shape (see left panel). Moreover, the analysis in the vicinity of the zero-field critical point shown in the right panel strongly suggests that for any $w>0$ the critical lines have the exponent $3/2$ corresponding to the dAT line. If any GT-like behavior is eventually present it would show up only in a region of extremely small values of $\tau$ and $H$ which is not easily accessible numerically.
These observations are coherent with the claim that a GT-like transition is possible if and only if the model admits the solution $q_{\perp}=0$, whose loss of stability just defines the GT line. Since this is possible only in the case of a homogeneous field over the whole system, hence our claim is that any infinitesimal perturbation to the homogeneous distribution of the field would make the GT transition disappear in favour of the dAT transition, so greatly enhancing the stability of the paramagnetic solution. GT transition is then a singular case, while the most generic and robust mechanism of RSB for a vector spin glass in a field is hence represented by the dAT transition.
Conclusions {#sec:concl}
===========
We have shown how to compute critical lines in the $(T,H)$ plane for a XY spin glass model on a random regular graph. We have used different distributions of the field direction in order to probe different critical behaviors. We have identified GT-like and dAT-like critical behaviors. The corresponding critical lines in the $(T,H)$ plane are similar to the fully connected case in the vicinity of the zero-field critical point, $H_{\text{GT}}\propto \tau^{1/2}$ and $H_{\text{dAT}} \propto \tau^{3/2}$, but differ sensibly at low temperatures (as in the Ising case [@ParisiEtAl2014]).
We have then shown how different are the local fluctuations that become critical in the two cases: they are strongly orthogonal to the local field in the GT case, while they are mostly longitudinal in the dAT case.
Finally, we have analyzed intermediate cases, where the fields are neither fully aligned nor completely random in direction. These cases have never been studied before, to the best of our knowledge. The comparison of the results obtained with two classes of field direction distributions interpolating between the delta function in $\phi=0$ and the flat distribution $\phi\in[0,2\pi)$ seems to suggest that the GT-like critical behavior is very unstable with respect to any small perturbation. In practice we only observe the dAT-like critical behavior for any field distribution that deviates (even by a tiny amount of order $10^{-2}$) from the situation with all the external fields perfectly aligned.
The overall picture resulting from our analysis is that the GT-like critical behavior can take place only if all the fields are perfectly aligned, while the dAT-like behavior is much more robust and generic, representing the mechanism through which replica symmetry typically breaks for vector spin glass models in a field.
The authors thank Giorgio Parisi for useful discussions. This research has been supported by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No \[694925\]).
GT and dAT lines in the large connectivity limit {#app:SK_limit}
================================================
In the main text we refer to the computation of the GT and dAT lines in an external field (respectively homogeneous over the whole system or randomly oriented on each site) on fully connected graphs, i.e. in the SK model, that has been already accomplished via the standard replica approach in the literature. In this appendix we want to pursue a twofold goal: first of all, we recall the replica results, explicitly writing them for the $m=2$ case, i.e. the XY model; then, we obtain the saddle-point equations for the fully connected XY model in a more straightforward and simpler way, via the large-connectivity limit of the belief-propagation equations; finally, we prove the equivalence of the two approaches, so providing a more direct physical interpretation of the quantities appearing in the replica computations.
Replica results
---------------
### The uniform-field case {#the-uniform-field-case-1 .unnumbered}
On the fully connected geometry, the replica trick [@Book_MezardEtAl1987] allows to succesfully solve the spin glass vector model in an external magnetic field, leading to the detection of the GT line or the dAT line depending on the distribution of local directions of the field. In particular, in the homogeneous case, the RS computation has been carried out for generic values of the number $m$ of spin components by Gabay and Toulouse [@GabayToulouse1981] and later by Cragg, Sherrington and Gabay [@CraggEtAl1982]. For the XY model, the saddle-point equations describing the paramagnetic solution ($q_{\perp}=0$) read:
$$q_{\parallel} = \int_{-\infty}^{\infty}\frac{dz}{\sqrt{2\pi}}e^{-z^2/2}\biggl(\frac{P_{01}}{P_{00}}\biggr)^2$$
$$x = -1+\int_{-\infty}^{\infty}\frac{dz}{\sqrt{2\pi}}e^{-z^2/2}\biggl(2-\frac{P_{20}}{P_{00}}\biggr)$$
with $x$ known as the *quadrupolar parameter*. Notice that, as usual in the replica computations, the spins are not taken with unit norm, rather $\sum_{\mu=1}^m \sigma^2_{\mu}=m$ (i.e. $2$ for the XY model). Functions $P_{\mu\nu}$’s appearing inside the Gaussian averages are then defined for the $m=2$ case as follows: $$\begin{aligned}
P_{\mu\nu}=\int_{-\sqrt{2}}^{\sqrt{2}}dS\,&e^{\,\beta(z\sqrt{q_{\parallel}}+H)S+(\beta^2/2)(2x-q_{\parallel}) S^2}\\
&\times(2-S^2)^{(\mu-1)/2}S^{\nu}
\label{eq:Bessel_like_functions_replica}
\end{aligned}$$ Such solution is stable until the following condition is satisfied: $$\beta^2\int_{-\infty}^{\infty}\frac{dz}{\sqrt{2\pi}}e^{-z^2/2}\biggl(\frac{P_{20}}{P_{00}}\biggr)^2=1
\label{eq:stab_cond_GT_replica}$$ Then, below the corresponding critical line, the stable solution is characterized by a nonvanishing transverse overlap $q_{\perp}$, together with a breaking of the replica symmetry. However, here we restrict ourselves to the RS analysis, being enough for our purposes.
The small-field expansion of the condition in Eq. (\[eq:stab\_cond\_GT\_replica\]) yields the well-known $1/2$ exponent of the GT line: $$H_c \propto \tau^{1/2}$$ while in the opposite limit we have an exponential divergence of the inverse critical temperature: $$\beta_c \propto e^{\,H^2/4}$$
Since these equations are obtained by using spins with norm $m=2$, it is useful to rewrite them for spins with unit norm, accordingly to all the computations of the main text: $$S \rightarrow \tilde{S} \equiv S/\sqrt{2}$$ Coherently with this choice, a dimensional analysis in the Hamiltonian leads to the corresponding rescaling of temperature and field: $$\beta \rightarrow \tilde{\beta} \equiv 2\beta \quad , \quad H \rightarrow \tilde{H} \equiv H/\sqrt{2}$$ Bessel-like functions (\[eq:Bessel\_like\_functions\_replica\]) then become: $$\begin{aligned}
P_{\mu\nu}&=2^{(\mu+\nu)/2}\int_{-1}^{1}d\tilde{S}\,e^{\,\tilde{\beta}(z\sqrt{\tilde{q}_{\parallel}}+\tilde{H})\tilde{S}+(\tilde{\beta}^2/2)(\tilde{x}-\tilde{q}_{\parallel})\tilde{S}^2}\\
&\qquad\qquad\qquad\times(1-\tilde{S}^2)^{(\mu-1)/2}\tilde{S}^{\nu}\\
&\equiv 2^{(\mu+\nu)/2}\tilde{P}_{\mu\nu}
\end{aligned}$$ so that we finally get also the proper rescaling of the longitudinal overlap $q_{\parallel}$ and of the quadrupolar parameter $x$ moving between the two normalizations: $$\tilde{x} \equiv x \quad , \quad \tilde{q}_{\parallel} \equiv q_{\parallel}/2$$
Looking at the definition of the Bessel-like functions (\[eq:Bessel\_like\_functions\_replica\]), it is easy to recognize $\tilde{S}$ as the projection of the unit spin $\tilde{\boldsymbol{S}}$ onto the $\hat{x}$ axis, namely $\tilde{S}=\cos{\theta}$. Moving to the angular variable $\theta$, then, we get: $$\begin{aligned}
\tilde{P}_{\mu\nu}&=\int_{0}^{2\pi}d\theta\,e^{\,\tilde{\beta}(z\sqrt{\tilde{q}_{\parallel}}+\tilde{H})\cos{\theta}+(\tilde{\beta}^2/2)(\tilde{x}-\tilde{q}_{\parallel})\cos^2{\theta}}\\
&\qquad\qquad\times\sin^{\mu}{\theta}\cos^{\nu}{\theta}
\end{aligned}$$ namely we get a sort of average of the quantity $\sin^{\mu}{\theta}\cos^{\nu}{\theta}$ over $\theta\in[0,2\pi]$ via the exponential measure $\exp{[\tilde{\beta}(z\sqrt{\tilde{q}_{\parallel}}+\tilde{H})\cos{\theta}+(\tilde{\beta}^2/2)(\tilde{x}-\tilde{q}_{\parallel})\cos^2{\theta}]}$. More concretely, we can introduce the following short-hand notation for such (normalized) angular averages: $$\braket{\sin^{\mu}(\theta)\cos^{\mu}(\theta)} \equiv \frac{\tilde{P}_{\mu\nu}}{\tilde{P}_{00}}$$
In this way, one can easily recognize the physical meaning of the longitudinal overlap $q_{\parallel}$: it represents the Gaussian average of the square average magnetization along the field direction $$\begin{aligned}
\tilde{q}_{\parallel}&=\int_{-\infty}^{\infty}\frac{dz}{\sqrt{2\pi}}e^{-z^2/2}\biggl(\frac{\tilde{P}_{01}}{\tilde{P}_{00}}\biggr)^2\\
&\equiv\mathbb{E}_z[\braket{\cos{\theta}}^2]
\end{aligned}$$ where $\mathbb{E}_z[\cdot]$ is indeed the expectation value over the Gaussian variable $z$. In the same manner, the quadrupolar parameter $x$ can be easily expressed in terms of angular averages: $$\begin{aligned}
\tilde{x} &= 2\int_{-\infty}^{\infty}\frac{dz}{\sqrt{2\pi}}e^{-z^2/2}\biggl(1-\frac{\tilde{P}_{20}}{\tilde{P}_{00}}\biggr)-1\\
&\equiv 2\mathbb{E}_z[\braket{\cos^2{\theta}}]-1\\
&=\mathbb{E}_z[\braket{\cos{2\theta}}]
\end{aligned}$$ as well as the transverse overlap $\tilde{q}_{\perp}$, representing the quadratic fluctuations in the direction transverse to the field: $$\tilde{q}_{\perp}=\mathbb{E}_z[\braket{\sin{\theta}}^2]$$ hence vanishing in the paramagnetic phase.
Under this light, the replica saddle-point equations in the RS ansatz acquire a clear physical meaning: as long as the solution is paramagnetic, all the marginals are polarized in the direction of the field, with no freezing in the transverse direction. In the cold phase, instead, the marginals acquire incoherent transverse components, that result in a $\tilde{q}_{\perp}$ different from zero. Consequently, in this latter case, a further term proportional to $\sqrt{\tilde{q}_{\perp}}\sin{\theta}$ should be added in the exponential measure appearing in the definition of $\tilde{P}_{\mu\nu}$’s. In addition, notice that the three parameters $\tilde{q}_{\parallel}$, $\tilde{q}_{\perp}$ and $\tilde{x}$ are enough to describe both the phases — still in the RS ansatz —, since the candidate for a fourth parameter, $\mathbb{E}_z[\braket{\sin{2\theta}}]$, can be expressed in terms of the other ones due to the constraint on the spin normalization.
Finally, the stability condition (\[eq:stab\_cond\_GT\_replica\]) becomes in the unit-norm frame: $$\tilde{\beta}^2\int_{-\infty}^{\infty}\frac{dz}{\sqrt{2\pi}}e^{-z^2/2}\biggl(\frac{\tilde{P}_{20}}{\tilde{P}_{00}}\biggr)^2=1$$ namely, in terms of the angular variable $\theta$: $$\tilde{\beta}^2\,\mathbb{E}_z[\braket{\sin^2{\theta}}^2]=1$$ which is nothing but the marginality condition for the growth rate of $\tilde{q}_{\perp}$, as it can be shown by expanding around the vanishing solution $\tilde{q}_{\perp}=0$. Such marginality condition will be even clearer when analyzing the large-$C$ limit of the cavity equations.
![The GT line computed via the replica approach in the unit-norm frame.[]{data-label="fig:GT_line_replica"}](GT_line_FC){width="\columnwidth"}
The corresponding critical line is reported in Fig. \[fig:GT\_line\_replica\], with the axes rescaled according to the unit-norm choice for the spins. One could easily recognize the square-root singularity close to the zero-field axis and the exponential divergence close to the zero-temperature axis.
### The Gaussian-field case {#the-gaussian-field-case .unnumbered}
At variance, the diametrically opposite case is represented by a randomly oriented field with a flat distribution over the local directions of the field. In particular, since in replica computations one usually deals with Gaussian-distributed couplings, it is comfortable to introduce a Gaussian-distributed field as well, so that Gaussian integrals can be straightforwardly performed. Following Sharma and Young [@SharmaYoung2010], we consider each component of the field $\boldsymbol{H}$ as independently distributed according to a Gaussian of zero mean and variance $\sigma^2_H$: $$H_{\mu} \sim \mathcal{N}(0,\sigma^2_H)
\label{eq:Gauss_field}$$ Hence, the rotational invariance $\mathrm{O}(m)$ is restored, corresponding to a unique order parameter $q$ in the RS frame, self-consistently given — for the XY model — by the following equation: $$q = \int_0^{\infty} d\rho \, \rho \, e^{-\rho^2/2}\biggl[\frac{I_1(\Delta_{\mathcal{G}})}{I_0(\Delta_{\mathcal{G}})}\biggr]^2$$ with $\Delta_{\mathcal{G}}$ containing $q$ itself, $\beta$ and the variance of the external field: $$\Delta_{\mathcal{G}} \equiv \sqrt{2}\beta\sqrt{q+\sigma^2_H}\,\rho
\label{eq:def_Delta_Gauss}$$
Another consequence of the rotational invariance is the absence of the quadrupolar parameter $x$, indeed being related to the breaking of the $\mathrm{O}(2)$ symmetry.
Finally, the stability of the paramagnetic RS solution can be studied via the usual techniques from the Hessian in the replica space [@Book_MezardEtAl1987; @SharmaYoung2010], obtaining the following marginality condition: $$\beta^2\chi_0 = 1
\label{eq:margCond_dAT_GaussField}$$ with $\chi_0$ given by: $$\begin{aligned}
\chi_0 = 2\int_0^{\infty} d\rho \, \rho \, e^{-\rho^2/2} \biggl[ 2\frac{I^2_1(\Delta_{\mathcal{G}})}{\Delta^2_{\mathcal{G}}\,I^2_0(\Delta_{\mathcal{G}})} + 2\frac{I_1(\Delta_{\mathcal{G}})\,I_2(\Delta_{\mathcal{G}})}{\Delta_{\mathcal{G}}\,I^2_0(\Delta_{\mathcal{G}})}&&\\
+\frac{I^2_2(\Delta_{\mathcal{G}})}{I^2_0(\Delta_{\mathcal{G}})} - 2\frac{I^3_1(\Delta_{\mathcal{G}})}{\Delta_{\mathcal{G}}\,I^3_0(\Delta_{\mathcal{G}})}
- 2\frac{I^2_1(\Delta_{\mathcal{G}})\,I_2(\Delta_{\mathcal{G}})}{I^3_0(\Delta_{\mathcal{G}})} + \frac{I^4_1(\Delta_{\mathcal{G}})}{I^4_0(\Delta_{\mathcal{G}})} \biggr]&&\end{aligned}$$
As usual, it is easy to map these equations into the corresponding ones for the unit spins. Indeed, we already know the rescaling of $\beta$ ($\tilde{\beta}=2\beta$) and $q$ ($\tilde{q}=q/2$); then, $\sigma_H$ should rescale exactly as $H$: $$\tilde{\sigma}_H = \sigma_H / \sqrt{2}$$ and finally we get the proper rescaling also for $\Delta_{\mathcal{G}}$: $$\tilde{\Delta}_{\mathcal{G}} \equiv \tilde{\beta}\sqrt{\tilde{q}+\tilde{\sigma}^2_H}\rho$$ i.e. $\tilde{\Delta}_{\mathcal{G}}=\Delta_{\mathcal{G}}$. The equation for $\tilde{q}$, then, reads: $$\tilde{q} = \frac{1}{2}\int_0^{\infty} d\rho \, \rho \, e^{-\rho^2/2}\biggl[\frac{I_1(\tilde{\Delta}_{\mathcal{G}})}{I_0(\tilde{\Delta}_{\mathcal{G}})}\biggr]^2$$ and finally the marginality condition (\[eq:margCond\_dAT\_GaussField\]) becomes: $$\tilde{\beta^2}\tilde{\chi}_0=1$$ where we have defined $\tilde{\chi}_0 \equiv \chi_0/4$.
### The random-field case with constant intensity $H$ {#the-random-field-case-with-constant-intensity-h .unnumbered}
Since in the main text we have not used a Gaussian-distributed field, rather a randomly oriented field with a constant intensity $H$, we would like here to obtain the corresponding dAT line in the fully connected limit, since in principle it could be different from the one recalled above. To this aim, it is enough to look at the definition of the quantity $\Delta_{\mathcal{G}}$ in Eq. (\[eq:def\_Delta\_Gauss\]): $q+\sigma^2_H$ indeed represents the *total* variance of the Gaussian field acting on each site, composed by an “intrinsic” variance $q$ (due to the contributions from the neighbours) and an “external” contribution $\sigma^2_H$ (due to the proper magnetic field $\boldsymbol{H}$).
Hence, in the case of a randomly oriented field with constant intensity, we just get rid of $\sigma_H$. However, a counterpart should put into the first moment of the external field: in more detail, a bias $H\cos{\phi}$ should be considered along the $\hat{x}$ direction and $H\sin{\phi}$ along the $\hat{y}$ direction, forcing us to move from polar coordinates $(\rho,\vartheta)$ to Cartesian coordinates $(z_x,z_y)$. Finally, we must average over $\phi$ via the flat distribution $1/2\pi$. The argument of Bessel functions consequently changes from $\Delta_{\mathcal{G}}$ to $\Delta_{\mathcal{R}}$ so defined: $$\Delta_{\mathcal{R}} \equiv \sqrt{2}\beta\sqrt{(H\cos{\phi}+z_x\sqrt{q})^2+(H\sin{\phi}+z_y\sqrt{q})^2}$$ and finally, via a gauge transformation over the local direction $\phi$ of the external field — since the sum of all the messages coming from the neighbours is $\mathrm{O}(2)$ symmetric as well — we can get rid of the average over $\phi$, getting the following definition for $\Delta_{\mathcal{R}}$: $$\Delta_{\mathcal{R}} \equiv \sqrt{2}\beta\sqrt{(H+z_x\sqrt{q})^2+(z_y\sqrt{q})^2}$$ and the following self-consistency equation for $q$: $$\begin{aligned}
q &= \int_{-\infty}^{\infty}dz_x\,dz_y\,\frac{e^{-(z^2_x+z^2_y)/2}}{2\pi}\biggl[\frac{I_1(\Delta_{\mathcal{R}})}{I_0(\Delta_{\mathcal{R}})}\biggr]^2\\
&\equiv \mathbb{E}_{\boldsymbol{z}}\biggl[\frac{I^2_1(\Delta_{\mathcal{R}})}{I^2_0(\Delta_{\mathcal{R}})}\biggr]
\label{eq:q_dAT_GaussField}
\end{aligned}$$ with $\mathbb{E}_{\boldsymbol{z}}[\cdot]$ being a short-hand notation for the Gaussian average over $\boldsymbol{z}=(z_x,z_y)$.
Finally, also the marginality condition (\[eq:margCond\_dAT\_GaussField\]) reads formally the same, i.e. $\beta^2\chi_0=1$, once coherently moved from $\Delta_{\mathcal{G}}$ to $\Delta_{\mathcal{R}}$: $$\begin{aligned}
\chi_0 = 2\,\mathbb{E}_{\boldsymbol{z}}\biggl[ 2\frac{I^2_1(\Delta_{\mathcal{R}})}{\Delta^2_{\mathcal{R}}\,I^2_0(\Delta_{\mathcal{R}})} + 2\frac{I_1(\Delta_{\mathcal{R}})\,I_2(\Delta_{\mathcal{R}})}{\Delta_{\mathcal{R}}\,I^2_0(\Delta_{\mathcal{R}})} + \frac{I^2_2(\Delta_{\mathcal{R}})}{I^2_0(\Delta_{\mathcal{R}})}&&\\
- 2\frac{I^3_1(\Delta_{\mathcal{R}})}{\Delta_{\mathcal{R}}\,I^3_0(\Delta_{\mathcal{R}})}- 2\frac{I^2_1(\Delta_{\mathcal{R}})\,I_2(\Delta_{\mathcal{R}})}{I^3_0(\Delta_{\mathcal{R}})} + \frac{I^4_1(\Delta_{\mathcal{R}})}{I^4_0(\Delta_{\mathcal{R}})} \biggr]&&\end{aligned}$$
Also in this case, the mapping to the unit-norm frame is quite straightforward, being: $$\tilde{\beta} = 2\beta \quad , \quad \tilde{q} = q/2 \quad , \quad \tilde{H} = H/\sqrt{2}$$ and from them the definition of $\tilde{\Delta}_{\mathcal{R}}$: $$\tilde{\Delta}_{\mathcal{R}} \equiv \tilde{\beta}\sqrt{(\tilde{H}+\sqrt{\tilde{q}}\,z_x)^2+(\sqrt{\tilde{q}}\,z_y)^2}$$ so that $\tilde{\Delta}_{\mathcal{R}}=\Delta_{\mathcal{R}}$. The equation (\[eq:q\_dAT\_GaussField\]) for $q$ then becomes: $$\tilde{q} = \frac{1}{2}\mathbb{E}_{\boldsymbol{z}}\biggl[\frac{I^2_1(\tilde{\Delta}_{\mathcal{R}})}{I^2_0(\tilde{\Delta}_{\mathcal{R}})}\biggr]$$ and finally we get again that the marginality condition reads $\tilde{\beta}^2\tilde{\chi}_0=1$ with $\tilde{\chi}_0\equiv\chi_0/4$.
At this point, we can compare the two choices for the local distribution of the external field. As anticipated, they yield different shapes of the dAT line in the $T$ vs $H$ plane, as it can be appreciated in Fig. \[fig:dAT\_line\_replica\]. First of all, they have the same behaviour in the small-field limit, namely $H \propto \tau^{3/2}$, but a different coefficient in front of such term. This is due to the fact that in the Gaussian-field case the stability of the paramagnetic phase is enhanced by the rare presence of some exceptionally intense field, while this phenomenon is not possible in the case of the random field with fixed modulus $H$.
![The dAT line computed via the replica approach in the unit-norm frame, obtained when using respectively a Gaussian distribution for the field components (purple curve) and a randomly oriented field with constant intensity (green curve).[]{data-label="fig:dAT_line_replica"}](dAT_line_FC){width="\columnwidth"}
Secondly, and most importantly, a rather different behaviour when approaching the zero-temperature limit. Indeed, in the $\beta\to\infty$ limit, both $\Delta_{\mathcal{G}}$ and $\Delta_{\mathcal{R}}$ diverge linearly with $\beta$, so that $\chi_0$ can be expanded in power series of $1/\Delta$. The first nonvanishing contribution of $\chi_0$ is given by the term $2/\Delta^2 \propto \beta^{-2}$, as expected. So when substituting into the marginality condition $\beta^2\chi_0=1$ we get (in the $m=2$-norm setting, so to match with the literature results): $$\frac{2\beta^2}{\Delta^2_{\mathcal{G},\mathcal{R}}}=1
\label{eq:margCond_dAT_zeroTemp}$$ where $q$ within $\Delta_{\mathcal{G},\mathcal{R}}$ can be already set equal to $1$, so neglecting higher-order corrections.
In the Gaussian case, such condition explicitly becomes: $$\begin{aligned}
&\int_{0}^{\infty} d\rho \, \rho \, \frac{e^{-\rho^2/2}}{(1+\sigma^2_H)\rho^2} = 1\\
&\qquad\qquad \Rightarrow \quad \frac{1}{1+\sigma^2_H}\int_{0}^{\infty} d\rho \, \frac{e^{-\rho^2/2}}{\rho} = 1
\label{eq:margCond_dAT_zeroTemp_GaussField}
\end{aligned}$$ which can not be satisfied for any finite value of $\sigma_H$, being the integral in $\rho$ divergent. Hence, according to the prediction by Sharma and Young [@SharmaYoung2010], the dAT line approaches the zero-temperature axis only asymptotically when considering a Gaussian distribution for the field components in the $m=2$ case, while it touches the $T=0$ axis at a finite value of $\sigma_H$ for $m \geqslant 3$.
Analogously, in the case of a randomly oriented field with constant intensity $H$, the marginality condition at $T=0$ reads: $$\int_{-\infty}^{\infty} dz_x \, dz_y \, \frac{e^{-(z^2_x+z^2_y)/2}}{2\pi}\,\frac{1}{(H+z_x)^2+z_y^2} = 1
\label{eq:margCond_dAT_zeroTemp_GaugeGlassField}$$ from which, via some manipulations, we get again a divergent integral on the left hand side of the marginality condition $$\int_{1}^{\infty} d\rho \, \frac{e^{-H^2(\rho-1)/2\rho}}{2\rho} = \infty \quad \forall\,H$$ implying a divergent value of the critical field $H$ in the $T\to 0$ limit.
The divergence of the integrals in the two cases can be then exploited in order to check the rate at which the critical variance $\sigma_H$ and the critical field $H$, respectively, diverge in the $\beta\to\infty$ limit. Indeed, from the inspection of Fig. \[fig:dAT\_line\_replica\] it is clear that they approach the $T=0$ axis in a rather different manner, with the curve $H(T)$ converging faster than the curve $\sigma_H(T)$. To this aim, let us define the function $f(\Delta)$ such that its Gaussian average over $\boldsymbol{z}$ gives $\chi_0$: $$f(\Delta): \quad \chi_0 \equiv \mathbb{E}_{\boldsymbol{z}}[f(\Delta)]$$ Moreover, we already know its behaviour in the two opposite regimes of small- and large-argument limits, valid in both cases of a Gaussian field and a randomly oriented field with constant intensity: $$f(\Delta=0)=1 \quad , \quad f(\Delta\gg 1) \simeq \frac{2}{\Delta^2}$$
So let us now analyze the condition $\beta^2\chi_0=1$ for large but finite values of $\beta$ in the Gaussian case. We have: $$\beta^2\int_0^{\infty} d\rho \, \rho \, e^{-\rho^2/2} f(\Delta_{\mathcal{G}}) = 1$$ The argument $\Delta_{\mathcal{G}}$ becomes $\sqrt{2}\beta\sigma_H\rho$, being $q$ negligible with respect to the critical value of $\sigma^2_H$ in the low-temperature limit. Then, the divergence of the integral in the $\beta\to\infty$ limit can be controlled by dividing the integration domain in two regions, respectively $\mathcal{A} \equiv [0,\epsilon]$ and $\mathcal{B} \equiv [\epsilon,\infty)$. In the region $\mathcal{A}$, we get that the Gaussian weight can be neglected; then, we perform a change of variables, $\sqrt{2}\beta\sigma_H\rho \equiv x$: $$\begin{aligned}
&\beta^2\int_{\mathcal{A}} d\rho \, \rho \, e^{-\rho^2/2} f(\sqrt{2}\beta\sigma_H\rho)\\
&\qquad\simeq \beta^2\int_{\mathcal{A}} d\rho \, \rho \, f(\sqrt{2}\beta\sigma_H\rho)\\
&\qquad\simeq \frac{1}{2\sigma^2_H}\int_{0}^{\sqrt{2}\beta\sigma_H\epsilon} dx \, x \, f(x)\\
&\qquad\simeq \frac{1}{2}\beta^2\epsilon^2
\end{aligned}$$ having exploited the limit $\lim_{x\to 0}f(x)=1$. Since the integral on the region $\mathcal{A}$ has to be finite in the $\beta\to\infty$ limit, then $\epsilon$ should scale as the inverse power of it: $$\epsilon \sim \frac{1}{\beta}$$ Let us now move to the integration over the $\mathcal{B}$ region. In this region, $f$ can be approximated with the first term of its expansion for large arguments, giving: $$\begin{aligned}
&\beta^2\int_{\mathcal{B}} d\rho \, \rho \, e^{-\rho^2/2} f(\sqrt{2}\beta\sigma_H\rho)\\
&\qquad\simeq \beta^2\int_{\mathcal{B}} d\rho \, \rho \, e^{-\rho^2/2} \, \frac{2}{2\beta^2\sigma^2_H\rho^2}\\
&\qquad\simeq \frac{1}{\sigma^2_H}\int_{\epsilon}^{\infty} d\rho \, \frac{e^{-\rho^2/2}}{\rho}\\
&\qquad\simeq -\frac{1}{\sigma^2_H}\ln{\epsilon}
\end{aligned}$$ Finally, when taking $\epsilon \sim 1/\beta$ for both the contributions, we get that the marginality condition reads: $$\frac{1}{2}+\frac{1}{\sigma^2_H}\ln{\beta} = 1$$ from which the scaling of the inverse critical temperature $\beta_c$ with $\sigma_H$: $$\beta_c \propto e^{\,\sigma^2_H/2}
\label{eq:dAT_stab_largeSigmaH_GaussField}$$ that can be also appreciated in the upper panel of Fig. \[fig:dAT\_line\_FC\_TempCloseToZero\].
![Convergence to zero of the critical temperature along the dAT line when increasing the field variance for the Gaussian case (upper panel) or the field strength for the randomly oriented case with fixed $H$ (lower panel). The linear trend for large values of $\sigma_H$ and $H$ confirms the analytic results (\[eq:dAT\_stab\_largeSigmaH\_GaussField\]) and (\[eq:dAT\_stab\_largeH\_GaugeGlassField\]), respectively. Axes scale refer to the $m=2$-norm choice for the spins, while error bars are due to the numeric precision used in the computation.[]{data-label="fig:dAT_line_FC_TempCloseToZero"}](dAT_line_FC_TempCloseToZero){width="\columnwidth"}
An analogous reasoning leads to the prediction of the growth of $\beta$ with $H$ along the dAT line in the random-field case with fixed $H$. Indeed, we have that the integral in the marginality condition $$\beta^2\int_{-\infty}^{\infty} dz_x \, dz_y \, \frac{e^{-(z^2_x+z^2_y)/2}}{2\pi}\,f(\Delta_{\mathcal{R}}) = 1$$ can be again divided in two regions, $\mathcal{A}$ and $\mathcal{B}$, where $\mathcal{A}$ is the disk of radius $\epsilon$ centered around the point $(-H,0)$ and $\mathcal{B}$ is the remaining portion of the $(z_x,z_y)$ plane. As before, in the region $\mathcal{A}$ the Gaussian weight can be considered constant; then, we move to polar coordinates and perform the change of coordinates $x \equiv \sqrt{2}\beta\rho$ $$\begin{aligned}
&\beta^2\int_{\mathcal{A}} dz_x \, dz_y \, \frac{e^{-(z^2_x+z^2_y)/2}}{2\pi}\,f(\Delta_{\mathcal{R}})\\
&\qquad\simeq \beta^2\int_{\mathcal{A}} dz_x \, dz_y \, \frac{e^{-H^2/2}}{2\pi}\,f(\Delta_{\mathcal{R}})\\
&\qquad\simeq \beta^2 \, e^{-H^2/2}\int_0^{\epsilon} d\rho \, \rho \, f(\sqrt{2}\beta\rho)\\
&\qquad\simeq \frac{1}{2}e^{-H^2/2}\int_0^{\sqrt{2}\beta\epsilon} dx \, x \, f(x)\\
&\qquad\simeq \frac{1}{2}\beta^2\epsilon^2 \, e^{-H^2/2}
\end{aligned}$$ where again the proper rescaling of the radius $\epsilon$ of the region $\mathcal{A}$ should be as $1/\beta$ when increasing $\beta$. Then, considering the integral over the region $\mathcal{B}$, we can substitute $f$ by its large-argument expansion, and then move again to polar coordinates: $$\begin{aligned}
&\beta^2\int_{\mathcal{B}} dz_x \, dz_y \, \frac{e^{-(z^2_x+z^2_y)/2}}{2\pi}\,f(\Delta_{\mathcal{R}})\\
&\qquad\simeq \beta^2\int_{\mathcal{B}} dz_x \, dz_y \, \frac{e^{-(z^2_x+z^2_y)/2}}{2\pi}\,\frac{2}{2\beta^2[(H+z_x)^2+z^2_y]}\\
&\qquad\simeq e^{-H^2/2}\int_{\epsilon}^{\infty} d\rho \, \frac{e^{-\rho^2/2}}{\rho}\int_0^{2\pi}d\vartheta\,\frac{e^{H\rho\cos{\vartheta}}}{2\pi}\\
&\qquad\simeq e^{-H^2/2}\int_{\epsilon}^{\infty} d\rho \, \frac{e^{-\rho^2/2}}{\rho}I_0(H\rho)\\
&\qquad\simeq -e^{-H^2/2}\ln{\epsilon}
\end{aligned}$$ So, taking again $\epsilon \sim 1/\beta$, we have: $$\frac{1}{2}e^{-H^2/2}+e^{-H^2/2}\ln{\beta} = 1$$ from which the scaling of $\beta$ with $H$ along the dAT line in the large-field region: $$\beta_c \propto \exp{\left\{e^{H^2/2}-\frac{1}{2}\right\}}
\label{eq:dAT_stab_largeH_GaugeGlassField}$$ numerically confirmed by the lower panel of Fig. \[fig:dAT\_line\_FC\_TempCloseToZero\].
These computations just confirm the feeling given by Fig. \[fig:dAT\_line\_replica\] that the dAT line approaches the $T=0$ axis much more rapidly in the random-field case with respect to the Gaussian-field case, gaining an exponential factor. An analogous “exponential speedup” can be observed in the Ising model, where the dAT line in the case of a Gaussian field goes as $\beta \propto \sigma_H$ [@Bray1982], while in the case of a field with constant intensity it goes as $\beta \propto \exp{\{H^2/2\}}$ [@deAlmeidaThouless1978]. The reason lies in the observation that in the case of Gaussian-distributed field, with finite probability we may observe small enough fields that make the system more unstable with respect to the case of a field with constant intensity at the same temperature $T$.
The SK limit from the BP equations
----------------------------------
Even though providing a formal tool through which solve spin glass models on fully connected graphs, the replica method is often quite involved, so that the physical interpretation of what is actually happening at the critical point remains hidden. At variance, the belief-propagation method bases on a very intuitive idea, symmetries are always exploited in a clear manner and the phase transitions can be typically detected via a standard analysis of the linear stability of fixed points.
In this spirit, we would like to recover the replica results via a suitable large-$C$ expansion of the BP equations, that at variance have been numerically solved in the main text in the case $C=O(1)$. To this aim, it is more convenient to use the factor-graph notation [@Book_MezardMontanari2009] with both $\eta$’s and $\hat{\eta}$’s cavity marginals — though still considering just pairwise interactions —, then rewriting them as large-deviation functions in $\beta$, as done in the zero-temperature limit [@LupoRicciTersenghi2017a; @Thesis_Lupo2017]: $$\eta \equiv \exp{(\beta h)} \quad , \quad \hat{\eta} \equiv \exp{(\beta u)}$$ Moreover, in order to lighten the notation and also to generalize the result to the case $m>2$, in this section we denote each spin with the unit vector $\boldsymbol{\sigma}_i$ rather with its angular variables. We will go back to the XY case when making explicit the distribution of the local direction of the external field.
Along a given directed edge $k\to i$, we have both the variable-to-check cavity message $\eta_{k\to i}(\boldsymbol{\sigma}_k) \equiv \exp{[\beta h_{k\to i}(\boldsymbol{\sigma}_k)]}$ and the check-to-variable cavity message $\hat{\eta}_{k\to i}(\boldsymbol{\sigma}_i) \equiv \exp{[\beta u_{k\to i}(\boldsymbol{\sigma}_i)]}$, that transform into each other when encountering the interaction node: $$\hat{\eta}_{k\to i}(\boldsymbol{\sigma}_i) \cong \int{d\boldsymbol{\sigma}_k} \, e^{\,\beta J_{ik}\boldsymbol{\sigma}_i\cdot\boldsymbol{\sigma}_k}\eta_{k\to i}(\boldsymbol{\sigma}_k)
\label{eq:BP_XY_eq_etahat}$$ apart from a normalizing multiplicative constant, or exploiting the large-deviation formalism: $$e^{\,\beta u_{k\to i}(\boldsymbol{\sigma}_i)} \cong \int{d\boldsymbol{\sigma}_k} \, e^{\,\beta[J_{ik}\boldsymbol{\sigma}_i\cdot\boldsymbol{\sigma}_k+h_{k\to i}(\boldsymbol{\sigma}_k)]}
\label{eq:BP_XY_eq_u}$$ Eventually, once arrived into the node $i$, the external field acting on it (about which we do not make any assumption for the moment) and the bias given by the other neighbours allow to write the expression for the variable-to-check cavity message $\eta_{i\to j}(\boldsymbol{\sigma}_i)$ along the directed edge $i\to j$: $$\eta_{i\to j}(\boldsymbol{\sigma}_i) \cong e^{\,\beta \boldsymbol{H}_i\cdot\boldsymbol{\sigma}_i}\prod_{k\in\partial i\setminus{j}}\hat{\eta}_{k\to i}(\boldsymbol{\sigma}_i)
\label{eq:BP_XY_eq_eta}$$ again up to a multiplicative constant, namely: $$h_{i\to j}(\boldsymbol{\sigma}_i) \simeq \boldsymbol{H}_i\cdot\boldsymbol{\sigma}_i+\sum_{k\in\partial i\setminus{j}} u_{k\to i}(\boldsymbol{\sigma}_i)
\label{eq:BP_XY_eq_h}$$ up to an additive constant. If Eqs. (\[eq:BP\_XY\_eq\_etahat\]) and (\[eq:BP\_XY\_eq\_eta\]) are put together, one gets back the pairwise BP equations seen in the main text.
In the large-$C$ limit — when $C$ becomes of order $N$ — exchange couplings $J_{ij}$’s have to be taken of order $1/\sqrt{C-1} \sim 1/\sqrt{N}$; then, the compatibility function can be expanded up to the second order in $J_{ij}$: $$e^{\,\beta J_{ik}\boldsymbol{\sigma}_i\cdot\boldsymbol{\sigma}_k} \simeq 1+\beta J_{ik}\boldsymbol{\sigma}_i\cdot\boldsymbol{\sigma}_k+\frac{\beta^2}{2}J^2_{ik}(\boldsymbol{\sigma}_i\cdot\boldsymbol{\sigma}_k)^2$$ from which, when integrating over the spin $\boldsymbol{\sigma}_k$ as in (\[eq:BP\_XY\_eq\_u\]): $$\begin{aligned}
&e^{\,\beta u_{k\to i}(\boldsymbol{\sigma}_i)}\\
&\quad\simeq 1 + \beta J_{ik}\braket{\boldsymbol{\sigma}_i\cdot\boldsymbol{\sigma}_k}_k+\frac{\beta^2}{2}J^2_{ik}\braket{(\boldsymbol{\sigma}_i\cdot\boldsymbol{\sigma}_k)^2}_k\\
&\quad\simeq e^{\,\beta J_{ik}\braket{\boldsymbol{\sigma}_i\cdot\boldsymbol{\sigma}_k}_k+(\beta^2/2)J^2_{ik}\bigl[\braket{(\boldsymbol{\sigma}_i\cdot\boldsymbol{\sigma}_k)^2}_k-\braket{\boldsymbol{\sigma}_i\cdot\boldsymbol{\sigma}_k}^2_k\bigr]}\\
&\quad\simeq e^{\,\beta J_{ik}\boldsymbol{\sigma}_i\cdot\braket{\boldsymbol{\sigma}_k}_k+(\beta^2/2)J^2_{ik}\boldsymbol{\sigma}_i\cdot\bigl(\braket{\boldsymbol{\sigma}_k\boldsymbol{\sigma}^{\intercal}_k}_k-\braket{\boldsymbol{\sigma}_k}_k\braket{\boldsymbol{\sigma}^{\intercal}_k}_k\bigr)\cdot\boldsymbol{\sigma}_i}
\end{aligned}$$ where $\boldsymbol{\sigma}^{\intercal}$ is the transpose vector of $\boldsymbol{\sigma}$, and where $$\braket{(\cdot)}_k \equiv \frac{\int d\boldsymbol{\sigma}_k (\cdot) \exp{[\beta h_{k\to i}(\boldsymbol{\sigma}_k)]}}{\int d\boldsymbol{\sigma}_k \exp{[\beta h_{k\to i}(\boldsymbol{\sigma}_k)]}}$$ so to take into account also the proper normalization constant.
At this point, we exploit the second BP equation, (\[eq:BP\_XY\_eq\_h\]), to compute the cavity field $h_{i\to j}(\theta_i)$: $$\begin{gathered}
\beta h_{i\to j}(\boldsymbol{\sigma}_i) \simeq \beta\boldsymbol{H}_i\cdot\boldsymbol{\sigma}_i+\beta\sum_{k\in\partial i\setminus j}J_{ik}\,\boldsymbol{\sigma}_i\cdot\braket{\boldsymbol{\sigma}_k}_k\\
\qquad\times (\beta^2/2)\sum_{k\in\partial i\setminus j}J^2_{ik}\,\boldsymbol{\sigma}_i\cdot\bigl(\braket{\boldsymbol{\sigma}_k\boldsymbol{\sigma}^{\intercal}_k}_k-\braket{\boldsymbol{\sigma}_k}_k\braket{\boldsymbol{\sigma}^{\intercal}_k}_k\bigr)\cdot\boldsymbol{\sigma}_i\end{gathered}$$ Since the r.h.s. also contains the $h$’s cavity fields — hidden into the expectation values $\braket{\cdot}_k$’s —, such set of equations can be closed by using the following ansatz, presented in Ref. [@JavanmardEtAl2016]: $$\beta h(\boldsymbol{\sigma}) \equiv \beta \boldsymbol{\xi}\cdot\boldsymbol{\sigma} + \frac{\beta^2}{2}\boldsymbol{\sigma}\cdot\mathbb{C}\cdot\boldsymbol{\sigma}
\label{eq:largeC_matrixAnsatz_generic}$$ where $\boldsymbol{\xi}$ is a $m$-component vector and $\mathbb{C}$ is a $m \times m$ symmetric matrix. So we get a set of cavity equations for these $\boldsymbol{\xi}$’s and $\mathbb{C}$’s:
$$\boldsymbol{\xi}_{i\to j} = \boldsymbol{H}_i + \sum_{k\in\partial i\setminus j}J_{ik}\braket{\boldsymbol{\sigma}_k}_k
\label{eq:largeC_matrixAnsatz_xi}$$
$$\mathbb{C}_{i\to j} = \sum_{k\in\partial i\setminus j}J^2_{ik}\,\bigl[\braket{\boldsymbol{\sigma}_k\boldsymbol{\sigma}^{\intercal}_k}_k-\braket{\boldsymbol{\sigma}_k}_k\braket{\boldsymbol{\sigma}^\intercal_k}_k\bigr]
\label{eq:largeC_matrixAnsatz_C}$$
\[eq:largeC\_matrixAnsatz\]
Finally, since we are summing over $C=O(N)$ neighbours with the couplings that are randomly distributed with zero mean and $O(1/N)$ variance, we get for the central-limit theorem that all the sites and the directed edges behave the same. Getting rid of the edge indexes, we get that $\boldsymbol{\xi}$ is a Gaussian-distributed vector with mean $\boldsymbol{M}$ and covariance matrix $\mathbb{Q}$: $$\boldsymbol{\xi} \sim \mathcal{N}(\boldsymbol{M},\mathbb{Q})$$ while $\mathbb{C}$ becomes a deterministic quantity, due to the system-wide average ($J^2\approx 1/N$) on the r.h.s. of (\[eq:largeC\_matrixAnsatz\_C\]).
Since the Hamiltonian of vector spin glass models is generally $\mathrm{O}(m)$-invariant in absence of the external field, we expect such a symmetry to be eventually broken to $\mathrm{O}(m-1)$, either spontaneously or due to the presence of the external field. Hence, there exists a suitable rotation that makes the $\mathbb{Q}$ and $\mathbb{C}$ matrices diagonal.
The exponential measure $\exp{[\beta h(\boldsymbol{\sigma})]}$ appearing in the average $\braket{\cdot}$ can be then rewritten in terms of few parameters $$\beta h(\boldsymbol{\sigma}) = \beta \sum_{\mu=1}^m \bigl(M_{\mu}+z_{\mu}\sqrt{\mathbb{Q}_{\mu\mu}}\bigr)\sigma_{\mu} + \frac{\beta^2}{2}\sum_{\mu=1}^m\mathbb{C}_{\mu\mu}\sigma^2_{\mu}
\label{eq:largeC_exponential_ansatz}$$ with $z_{\mu} \sim \mathcal{N}(0,1)$, leading to a set of self-consistency equations for them:
$$M_{\mu} \equiv \mathbb{E}_{\boldsymbol{z}}[\xi_{\mu}] = H_{\mu}
\label{eq:largeC_matrixAnsatz_parameter_M}$$
$$\mathbb{Q}_{\mu\mu} \equiv \mathbb{V}_{\boldsymbol{z}}[\xi_{\mu}] = \mathbb{E}_{\boldsymbol{z}}\bigl[\braket{\sigma_{\mu}}^2\bigr]
\label{eq:largeC_matrixAnsatz_parameter_Q}$$
$$\mathbb{C}_{\mu\mu} = \mathbb{E}_{\boldsymbol{z}}\bigl[\braket{\sigma^2_{\mu}}-\braket{\sigma_{\mu}}^2\bigr]
\label{eq:largeC_matrixAnsatz_parameter_C}$$
\[eq:largeC\_matrixAnsatz\_parameters\]
where $\mathbb{E}_{\boldsymbol{z}}[\cdot]$ refers to the expectation value with respect to the Gaussian variables $z_{\mu}$’s, while $\mathbb{V}_{\boldsymbol{z}}[\cdot]$ is the corresponding variance. Eventually, $H_{\mu}$ is the expectation value of the field along the direction $\mu$.
In the end, we further exploit the breaking of the $\mathrm{O}(m)$ rotational symmetry to — at most — $\mathrm{O}(m-1)$ and the normalization constraint for the spins. Consequently, assuming as $\mu=1$ — i.e. the $\hat{x}$ axis — the direction along which the symmetry is eventually broken, we can redefine the matrix $\mathbb{C}$ up to a diagonal shift, $\mathbb{C}' \equiv \mathbb{C}-\mathbb{C}_{\mu\mu}\mathbb{I}$, getting the following saddle-point equations:
$$M_{\mu} =
\left\{
\begin{aligned}
&H_x \quad &&\text{for } \mu=1\\
&0 \quad &&\text{for } \mu=\{2,3,\dots,m\}
\end{aligned}
\right.
\label{eq:largeC_matrixAnsatz_parameter_M_new}$$
$$\mathbb{Q}_{\mu\mu} =
\left\{
\begin{aligned}
&\mathbb{E}_{\boldsymbol{z}}\bigl[\braket{\sigma_x}^2\bigr] \quad &&\text{for } \mu=1\\
&\mathbb{E}_{\boldsymbol{z}}\bigl[\braket{\sigma_{\mu}}^2\bigr] \quad &&\text{for } \mu=\{2,3,\dots,m\}
\end{aligned}
\right.
\label{eq:largeC_matrixAnsatz_parameter_Q_new}$$
$$\mathbb{C}'_{\mu\mu} =
\left\{
\begin{aligned}
&\mathbb{E}_{\boldsymbol{z}}\Bigl[\braket{\sigma^2_x}-\braket{\sigma_x}^2
-\bigl(\braket{\sigma^2_{\mu}}-\braket{\sigma_{\mu}}^2\bigr)\Bigr] \quad\text{for } \mu=1&&\\
&0 \qquad\qquad\qquad\qquad\qquad\;\;\text{for } \mu=\{2,3,\dots,m\}&&
\end{aligned}
\right.
\label{eq:largeC_matrixAnsatz_parameter_C_new}$$
\[eq:largeC\_matrixAnsatz\_parameters\_new\]
where, again, $H_x$ is the expectation value of the external field along the $\hat{x}$ axis, while it has zero mean along the other directions. Finally, these equations can be completely solved only once made explicit the distribution of the local direction of the field.
### The uniform field case {#the-uniform-field-case-2 .unnumbered}
Let us now go back to the XY model. In the uniform-field case, assuming that the symmetry is broken along the $\hat{x}$ axis, self-consistency equation (\[eq:largeC\_matrixAnsatz\_parameter\_M\_new\]) becomes: $$M_x = H \quad , \quad M_y = 0$$ and hence we can directly get rid of $\boldsymbol{M}$, by plugging $H$ into the other equations. Then, in Eq. (\[eq:largeC\_matrixAnsatz\_parameter\_Q\_new\]), $\mathbb{Q}_{xx}$ is surely larger than zero at any temperature — due to the presence of the external field —, while $\mathbb{Q}_{yy}$ is either positive or zero depending on whether the transverse symmetry is broken or not, respectively. Finally, in the quadratic term in the exponential measure $\exp{[\beta h(\boldsymbol{\sigma})]}$ — over which perform the average $\braket{\cdot}$ —, we are left with the only term $\mathbb{C}'_{xx}$, as explained before. The large-$C$ ansatz for $h(\theta)$ hence reads: $$\begin{aligned}
\beta h(\theta) &=\beta\bigl(H+z_x\sqrt{\mathbb{Q}_{xx}}\bigr)\cos{\theta}+\beta z_y\sqrt{\mathbb{Q}_{yy}}\sin{\theta}\\
&\qquad + \frac{\beta^2}{2}\mathbb{C}'_{xx}\cos^2{\theta}
\label{eq:h_ansatz_GT}
\end{aligned}$$ In terms of the angular variable $\theta$, the self-consistency equations for $\mathbb{Q}_{xx}$, $\mathbb{Q}_{yy}$ and $\mathbb{C}'_{xx}$ read:
$$\mathbb{Q}_{xx} = \mathbb{E}_{\boldsymbol{z}}\bigl[\braket{\cos{\theta}}^2\bigr]
\label{eq:largeC_matrixAnsatz_GT_parameter_Qxx}$$
$$\mathbb{Q}_{yy} = \mathbb{E}_{\boldsymbol{z}}\bigl[\braket{\sin{\theta}}^2\bigr]
\label{eq:largeC_matrixAnsatz_GT_parameter_Qyy}$$
$$\begin{aligned}
\mathbb{C}'_{xx} &= \mathbb{E}_{\boldsymbol{z}}\bigl[\braket{\cos^2{\theta}}-\braket{\cos{\theta}}^2-\bigl(\braket{\sin^2{\theta}}-\braket{\sin{\theta}}^2\bigr)\bigr]\\
&=\mathbb{E}_{\boldsymbol{z}}\bigl[2\braket{\cos^2{\theta}}-1-\braket{\cos{\theta}}^2+\braket{\sin{\theta}}^2\bigr]
\label{eq:largeC_matrixAnsatz_GT_parameter_Cxx}
\end{aligned}$$
\[eq:largeC\_matrixAnsatz\_GT\_parameters\]
The paramagnetic solution is the one with no breaking of the transverse symmetry, namely $\mathbb{Q}_{yy}=0$. The corresponding values of $\mathbb{Q}_{xx}$ and $\mathbb{C}'_{xx}$ have then to be determined according to Eqs. (\[eq:largeC\_matrixAnsatz\_GT\_parameter\_Qxx\]) and (\[eq:largeC\_matrixAnsatz\_GT\_parameter\_Cxx\]), with the Gaussian average meant to be over the sole $z_x$ variable. Eventually, it is straightforward to obtain the stability condition for such solution, by looking at Eq. (\[eq:largeC\_matrixAnsatz\_GT\_parameter\_Qyy\]) and expanding the r.h.s. at the first order in $\mathbb{Q}_{yy}$: $$\begin{aligned}
\mathbb{Q}_{yy} &= \mathbb{E}_{\boldsymbol{z}}\bigl[\braket{\sin{\theta}}^2\bigr]\\
&\simeq \mathbb{E}_{\boldsymbol{z}}\bigl[(\beta z_y \sqrt{\mathbb{Q}_{yy}}\braket{\sin^2{\theta}}_{\mathbb{Q}_{yy}=0})^2\bigr]\\
&= \beta^2\,\mathbb{E}_{z_x}\bigl[\braket{\sin^2{\theta}}^2_{\mathbb{Q}_{yy}=0}\bigr]\,\mathbb{Q}_{yy}
\end{aligned}$$ so that the paramagnetic solution is stable as long as $\beta^2\,\mathbb{E}_{z_x}\bigl[\braket{\sin^2{\theta}}^2_{\mathbb{Q}_{yy}=0}\bigr]<1$, while the critical line is identified by the condition: $$\beta^2\,\mathbb{E}_{z_x}\bigl[\braket{\sin^2{\theta}}^2_{\mathbb{Q}_{yy}=0}\bigr]=1
\label{eq:largeC_GT_stability}$$
At this point, we would like to prove the equivalence between this approach and the replica one. To this aim, it is enough to compare the ansatz over the cavity field $h(\boldsymbol{\sigma})$ that we exploited here, Eq. (\[eq:largeC\_exponential\_ansatz\]), with the exponent in the definition of $\tilde{P}_{\mu\nu}$ functions in the replica computation, Eq. (\[eq:Bessel\_like\_functions\_replica\]). Indeed, when taking also into account the proper sine factor in the replica computations — absent in the aforementioned equations, related to the $\tilde{q}_{\perp}=0$ solution —, it is easy to map into each other the various quantities appearing in both the approaches: $$\begin{gathered}
\tilde{q}_{\parallel} \, \Leftrightarrow \, \mathbb{Q}_{xx} \quad , \quad \tilde{q}_{\perp} \, \Leftrightarrow \, \mathbb{Q}_{yy}\\
\tilde{x}-\tilde{q}_{\parallel}+\tilde{q}_{\perp} \, \Leftrightarrow \, \mathbb{C}'_{xx}
\end{gathered}$$ Consistently with these identifications, all the saddle-point equations can be mapped exactly one into each other, as well as the marginality condition corresponding to the location of the GT line.
Notice that the large-$C$ limit of the cavity equations allows not only to recover the replica results in a simpler way, but in addition it provides a clearer physical picture of the symmetry breaking related to the GT transition. Indeed, the longitudinal and the transverse overlaps are directly identified with the quadratic fluctuations of the magnetization components along the field or perpendicular to it, respectively, with the GT instability given by the appearance of these latter ones.
### The random field case with constant intensity $H$ {#the-random-field-case-with-constant-intensity-h-1 .unnumbered}
In the randomly oriented field case, instead, the $\mathrm{O}(m)$ symmetry is not explicitly broken by the field, since its local direction is uniformly distributed over the $m$-dimensional unit sphere. This has three important consequences: *i)* the vector $\boldsymbol{M}$ identically vanishes; *ii)* the matrix $\mathbb{Q}$ becomes a multiple of the identity, $\mathbb{Q}=q\mathbb{I}$; *iii)* also the matrix $\mathbb{C}$ becomes a multiple of the identity, and by the norm constraint of the spins it can be finally set equal to zero.
Hence, we have that the generic ansatz (\[eq:largeC\_matrixAnsatz\_generic\]) for $h(\theta)$ reduces just to the first term, namely a scalar product, that for the XY model reads: $$\begin{aligned}
\beta h(\theta) &= \beta\xi\cos{(\vartheta-\theta)}\\
&=\beta(\xi_x\cos{\theta}+\xi_y\sin{\theta})
\label{eq:h_ansatz_dAT_1}
\end{aligned}$$ where $\xi$ is the modulus of $\boldsymbol{\xi}$ and $\vartheta$ gives its direction. Component-wise, in the case of a randomly oriented field with constant intensity $H$, $\boldsymbol{\xi}$ is then given by: $$\begin{aligned}
\xi_x &= H\cos{\phi}+z_x\sqrt{q}\\
\xi_y &= H\sin{\phi}+z_y\sqrt{q}
\end{aligned}$$ where $\phi$ is the local direction of the external field, over which we should average.
Since both $\phi$ and the local direction of $\boldsymbol{z}$ are uniformly distributed over the unit circle, by a gauge transformation we can set the former to zero — as already seen in the replica computations —, so getting rid of the average over it. We are then left with the only Gaussian average over $\boldsymbol{z}$. Consequently, Eq. (\[eq:h\_ansatz\_dAT\_1\]) becomes: $$\beta h(\theta) = \beta\bigl(H+z_x\sqrt{q}\bigr)\cos{\theta}+\beta z_y\sqrt{q}\sin{\theta}
\label{eq:h_ansatz_dAT_2}$$ A direct consequence of the vectorial shape of $h$ is that the angular average $\braket{\cdot}$ can now be analytically computed in terms of Bessel functions: $$\braket{\cos{\theta}} = \frac{I_1(\beta \xi)}{I_0(\beta \xi)}\cos{\vartheta} \quad , \quad \braket{\sin{\theta}} = \frac{I_1(\beta \xi)}{I_0(\beta \xi)}\sin{\vartheta}$$
Eqs. (\[eq:largeC\_matrixAnsatz\_parameter\_Q\_new\]) reduce to a unique one for $q$, which is indeed the unique parameter to be self-consistently determined: $$q = \frac{1}{2}\mathbb{E}_{\boldsymbol{z}}\bigl[\braket{\cos{\theta}}^2+\braket{\sin{\theta}}^2\bigr] = \frac{1}{2}\mathbb{E}_{\boldsymbol{z}}\biggl[\frac{I^2_1(\beta \xi)}{I^2_0(\beta \xi)}\biggr]
\label{eq:largeC_matrixAnsatz_dAT_parameter_q}$$ with $\xi$ given by: $$\xi = \sqrt{\bigl(H+z_x\sqrt{q}\bigr)^2+\bigl(z_y\sqrt{q}\bigr)^2}$$
Despite the resulting saddle-point equation is by far simpler than the one obtained in the uniform-field case, the stability of the paramagnetic phase can not be analyzed as simply. Indeed, $q$ is larger than zero both in the paramagnetic and in the ordered phase, so that it is not possible to expand around a vanishing solution. However, we can still rely on the linear-stability analysis, but now looking at the growth rate of a perturbation $\delta h(\theta)$ — i.e. $\delta\boldsymbol{\xi}$ — under BP iterations.
In more detail, let us come back to the edge-dependent notation, namely before exploiting the central-limit theorem. We have that, being $h(\theta)=\boldsymbol{\xi}\cdot\boldsymbol{\sigma}$ a scalar product, the same happens to $u(\theta')=\boldsymbol{u}\cdot\boldsymbol{\sigma}'$. So we get that each interaction node acts as: $$\boldsymbol{u}_{k\to i} = J_{ik}\braket{\boldsymbol{\sigma}_k}_k = J_{ik}\frac{I_1(\beta\xi_{k\to i})}{I_0(\beta\xi_{k\to i})}\frac{\boldsymbol{\xi}_{k\to i}}{\xi_{k\to i}}
\label{eq:largeC_matrixAnsatz_dAT_evolutionMatrix}$$ Hence, a small perturbation $\delta\boldsymbol{\xi}$ propagates as: $$\delta\boldsymbol{u}_{k\to i} = \mathbb{A}_{k\to i} \, \delta\boldsymbol{\xi}_{k\to i}$$ with $\mathbb{A}_{k\to i}$ being the symmetric $2 \times 2$ matrix that comes from the linearization of Eq. (\[eq:largeC\_matrixAnsatz\_dAT\_evolutionMatrix\]), i.e. (getting rid of the edge indexes): $$\mathbb{A} \equiv
\begin{pmatrix}
\frac{\partial u_x}{\partial \xi_x} && \frac{\partial u_x}{\partial \xi_y}\\
\frac{\partial u_y}{\partial \xi_x} && \frac{\partial u_y}{\partial \xi_y}
\end{pmatrix}$$
The matrix $\mathbb{A}_{k\to i}$ affects the “incoming” perturbation $\delta\boldsymbol{\xi}_{k\to i}$ in two different ways: a rescaling of its norm and a change in its direction. Then, once reached the node $i$, in order to get the outgoing $\delta\boldsymbol{\xi}_{i\to j}$, we have to sum all the incoming perturbations $\delta\boldsymbol{u}_{k\to i}$’s, whose directions are incoherent, being the $\mathrm{O}(2)$ symmetry preserved. Hence, what we should look at is the growth rate of the norm of these perturbations: $$\norm{\delta\boldsymbol{\xi}_{i\to j}}^2 = \sum_{k\in\partial i\setminus j}\norm{\delta\boldsymbol{u}_{k\to i}}^2 = \sum_{k\in\partial i\setminus j}\norm{\mathbb{A}_{k\to i}\delta\boldsymbol{\xi}_{k\to i}}^2$$ Finally, in the large-$C$ limit, we can as usual exploit the central-limit theorem, getting: $$\norm{\delta\boldsymbol{\xi}}^2 = \mathbb{E}_{\boldsymbol{z}}\biggl[\frac{\lambda^2_1+\lambda^2_2}{2}\biggr]\norm{\delta\boldsymbol{\xi}}^2\;,$$ where $\lambda_{1,2}$ are the eigenvalues of a generic $\mathbb{A}$ matrix and the factor $1/2$ comes from the mean value of the projection of $\boldsymbol{\xi}$ over the eigenvectors of $\mathbb{A}$.
The marginality condition is then obtained by considering a unitary growth rate for the norm of the perturbations: $$\mathbb{E}_{\boldsymbol{z}}\biggl[\frac{\lambda^2_1+\lambda^2_2}{2}\biggr]=1$$ Explicitly computing $\lambda_1$ and $\lambda_2$, finally, we get the marginality condition which refers to the dAT line for the randomly oriented field with constant intensity $H$: $$\begin{gathered}
\frac{\beta^2}{2}\mathbb{E}_{\boldsymbol{z}} \biggl[ \frac{I^2_1(\beta\xi)}{(\beta\xi)^2\,I^2_0(\beta\xi)} + \frac{1}{4} + \frac{I^4_1(\beta\xi)}{I^4_0(\beta\xi)} + \frac{I^2_2(\beta\xi)}{4 I^2_0(\beta\xi)}\\
- \frac{I^2_1(\beta\xi)}{I^2_0(\beta\xi)} + \frac{I_2(\beta\xi)}{2 I_0(\beta\xi)} - \frac{I^2_1(\beta\xi)\,I_2(\beta\xi)}{I^3_0(\beta\xi)} \biggr] = 1
\label{eq:largeC_matrixAnsatz_dAT_marginal_stability}\end{gathered}$$
Also in this case, the cavity approach is completely equivalent to the replica computations. Indeed, noticing that the rescaled argument $\tilde{\Delta}_{\mathcal{R}}$ of Bessel functions in the replica approach is exactly equal to $\beta\xi$ in the present computation, we suddenly recognize that the saddle point equation for $q$ is the same. Moreover, also the marginality condition $\tilde{\beta}^2\tilde{\chi}_0=1$ of the replica computation is perfectly equivalent with the Eq. (\[eq:largeC\_matrixAnsatz\_dAT\_marginal\_stability\]) derived via the cavity computation. Although it is not easy to match analytically the expressions entering the Gaussian integrations in the two methods, we have numerically checked their identity.
### The Gaussian field case {#the-gaussian-field-case-1 .unnumbered}
The self-consistency equations for a Gaussian distributed field can be easily derived from the ones obtained for the randomly oriented field with constant intensity. The ansatz for the components of the vector $\boldsymbol{\xi}$ has to be properly modified as $$\begin{aligned}
\xi_x &= z_x\sqrt{q+\sigma^2_H}\\
\xi_y &= z_y\sqrt{q+\sigma^2_H}
\end{aligned}$$ then we have that the saddle-point equation for $q$ reads $$q = \frac{1}{2}\mathbb{E}_{\boldsymbol{z}}\biggl[\frac{I^2_1(\beta \xi)}{I^2_0(\beta \xi)}\biggr] = \frac{1}{2}\int_0^{\infty} d\rho \, \rho \, e^{-\rho^2/2} \biggl[\frac{I_1(\beta\xi)}{I_0(\beta\xi)}\biggr]^2$$ with $\xi = (q+\sigma^2_H)\sqrt{z^2_x+z^2_y} = (q+\sigma^2_H)\rho$, in polar coordinates. The argument to get the marginality condition for the paramagnetic solution follows exactly the same steps as in the previous case, leading to an expression analogous to Eq. (\[eq:largeC\_matrixAnsatz\_dAT\_marginal\_stability\]): $$\begin{gathered}
\frac{\beta^2}{2}\int_0^{\infty} d\rho \, \rho \, e^{-\rho^2/2} \biggl[ \frac{I^2_1(\beta\xi)}{(\beta\xi)^2\,I^2_0(\beta\xi)} + \frac{1}{4} + \frac{I^4_1(\beta\xi)}{I^4_0(\beta\xi)}\\
+ \frac{I^2_2(\beta\xi)}{4 I^2_0(\beta\xi)} - \frac{I^2_1(\beta\xi)}{I^2_0(\beta\xi)} + \frac{I_2(\beta\xi)}{2 I_0(\beta\xi)} - \frac{I^2_1(\beta\xi)\,I_2(\beta\xi)}{I^3_0(\beta\xi)} \biggr] = 1\end{gathered}$$ again written in polar coordinates.
Finally, $\beta\xi$ has exactly the same expression of $\tilde{\Delta}_{\mathcal{G}}$ in the replica computations; once recognized this, the saddle point equation for $q$ and the marginality condition can be recognized as equivalent between the two approaches.
### The generic case {#the-generic-case .unnumbered}
By exploiting the cavity formalism for large connectivities developed in this appendix, we can also solve the model in case of a generic distribution of the external field, namely neither uniform nor perfectly $\mathrm{O}(2)$-symmetric.
The general reasoning for obtaining the saddle-point equations should follow the same steps of the uniform case, since for the most generic distribution of the external field we have that the matrix $\mathbb{C}$ does not vanish. The saddle-point equations for the parameters $\mathbb{Q}_{xx}$, $\mathbb{Q}_{yy}$ and $\mathbb{C}'_{xx}$ can be then straightforwardly obtained starting from the generic expression (\[eq:largeC\_matrixAnsatz\_parameters\_new\]).
More caution has then to be payed to the computation of the stability condition of the paramagnetic solution. Indeed, it is a generalization of the reasoning followed for the $\mathrm{O}(2)$-symmetric field, though taking also into account that incoming fields $\boldsymbol{\xi}_{k\to i}$ may have a directional bias given by the external field. So when exploiting the central-limit theorem, we get both a condition for the growth of the first moment of $\boldsymbol{\xi}$ and one for the growth of its fluctuations, each one giving a well-defined critical line in the $(T,H)$ plane; the paramagnetic solution actually becomes marginally stable in correspondence of the highest among these two critical lines.
[^1]: As long as $m\ge 2$, we have both longitudinal and transverse degrees of freedom, and all the possible scenarios may take place. For $m\to\infty$ the RSB phase shrinks as $1/m$, and one would need to rescale temperatures in order to get a sensible result.
|
---
abstract: 'In this work, we investigate the chiral corrections to the magnetic moments of the spin-$3\over 2$ doubly charmed baryons systematically up to next-to-next-to-leading order with the heavy baryon chiral perturbation theory. The numerical results are given up to next-to-leading order: $\mu_{\Xi^{*++}_{cc}}=1.72\mu_{N}$, $\mu_{\Xi^{*+}_{cc}}=-0.09\mu_{N}$, $\mu_{\Omega^{*+}_{cc}}=0.99\mu_{N}$. As a by-product, we have also calculated the magnetic moments of the spin-$3\over 2$ doubly bottom baryons and charmed bottom baryons: $\mu_{\Xi^{*0}_{bb}}=0.63\mu_{N}$, $\mu_{\Xi^{*-}_{bb}}=-0.79\mu_{N}$, $\mu_{\Omega^{*-}_{bb}}=0.12\mu_{N}$, $\mu_{\Xi^{*+}_{bc}}=1.12\mu_{N}$, $\mu_{\Xi^{*0}_{bc}}=-0.40\mu_{N}$, $\mu_{\Omega^{*0}_{bc}}=0.56\mu_{N}$.'
author:
- 'Lu Meng, Hao-Song Li, Zhan-Wei Liu, Shi-Lin Zhu'
date: 'Received: date / Revised version: date'
title: 'Magnetic moments of the spin-${3\over 2}$ doubly heavy baryons'
---
Introduction {#Sec1}
============
A doubly charmed baryon was first reported by the SELEX Collaboration in the decay model $\Xi^{+}_{cc}\rightarrow\Lambda_{c}^{+}K^{-}\pi^{+}$ with the mass $M_{\Xi^{+}_{cc}}=3519\pm1\rm{MeV}$ [@Mattson:2002vu]. However, no other collaborations confirmed the observation [@Ratti:2003ez; @Aubert:2006qw; @Chistov:2006zj]. Recently, the spin-$1 \over 2$ doubly charmed baryon $\Xi_{cc}^{++}$ was reported by the LHCb Collaboration in the $\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+}$ mass spectrum with the mass $M_{\Xi^{++}_{cc}}$=3621.40$\pm$0.72 (stat.) $\pm$0.27(syst.)$\pm$0.14($\Lambda^{+}_{c}$)MeV [@Aaij:2017ueg].
The production of the doubly heavy baryons have been discussed in Refs. [@Chang:2006eu; @Chang:2006xp; @Chang:2007xa; @Chang:2007pp; @Zheng:2015ixa; @Chen:2014frw; @Chen:2014hqa; @Huan-Yu:2017emk]. Apart from the spin-$1 \over 2$ doubly charmed baryons, there exist the spin-$3 \over 2$ doubly charmed baryons as degenerate states of the spin-$1\over 2$ ones in the heavy quark limit. In the past decades, the masses and other properties of spin-$1\over 2$ and spin-$3\over 2$ doubly charmed baryons have been investigated extensively in literature [@Bagan:1992za; @Roncaglia:1995az; @SilvestreBrac:1996bg; @Ebert:1996ec; @Tong:1999qs; @Itoh:2000um; @Gershtein:2000nx; @Kiselev:2001fw; @Kiselev:2002iy; @Narodetskii:2001bq; @Lewis:2001iz; @Ebert:2002ig; @Mathur:2002ce; @Flynn:2003vz; @Vijande:2004at; @Chiu:2005zc; @Migura:2006ep; @Albertus:2006ya; @Martynenko:2007je; @Tang:2011fv; @Liu:2007fg; @Roberts:2007ni; @Patel:2007gx; @Valcarce:2008dr; @Liu:2009jc; @Namekawa:2012mp; @Alexandrou:2012xk; @Aliev:2012ru; @Namekawa:2013vu; @Karliner:2014gca; @Sun:2014aya; @Chen:2015kpa; @Sun:2016wzh; @Shah:2016vmd; @Chen:2016spr; @Kiselev:2017eic; @Chen:2017sbg; @Hu:2005gf; @Yu:2017zst; @Li:2017ndo; @Meng:2017udf; @Guo:2017vcf; @Wang:2017mqp; @Wang:2017azm; @Lu:2017meb; @Xiao:2017udy], where the mass splittings between the spin-$1 \over 2$ and spin-$3\over 2$ doubly charmed baryons are from several tens MeV to one hundred MeV. Since the real or virtual photons are usually used as probes to explore the inner structures of baryons, the electromagnetic form factors of the doubly charmed baryons are very important. Especially the magnetic moments encode crucial information of their underlying structure. In Refs. [@SilvestreBrac:1996bg; @Patel:2007gx; @Lichtenberg:1976fi; @JuliaDiaz:2004vh; @Faessler:2006ft; @Dhir:2009ax; @Branz:2010pq; @Sharma:2010vv; @Bose:1980vy; @Bernotas:2012nz; @Jena:1986xs; @Oh:1991ws; @Can:2013zpa; @Li:2017cfz], the magnetic moments of the spin-$1 \over 2$ and spin-$3 \over 2$ doubly charmed baryons have been investigated.
The magnetic moments of the spin-$3\over 2$ doubly charmed baryons were first investigated in the naive quark model by Lichtenberg [@Lichtenberg:1976fi]. After that, many authors have employed the quark model to calculate the spin-$3\over 2$ doubly charmed baryon magnetic moments [@Albertus:2006ya; @Patel:2007gx; @Sharma:2010vv]. Apart from the quark models, the magnetic moments of the spin-$3 \over 2$ doubly charmed baryons have been predicted employing the effective quark mass and screened charge scheme in Ref. [@Dhir:2009ax]. The magnetic moments of the spin-$3 \over 2$ doubly charmed baryons have been also calculated in the MIT bag model [@Bose:1980vy; @Bernotas:2012nz] and Skymion model [@Oh:1991ws].
Compared with the quark model, the chiral perturbation theory (ChPT) [@Weinberg:1978kz; @Scherer:2002tk] provides a systematic framework to calculate the electromagnetic form factors of the baryons order by order. However, the baryon mass does not vanish in the chiral limit, which introduces another energy scale besides the $\Lambda_{\chi}=4\pi f_\pi$. In order to employ ChPT in the baryon sector, the heavy baryon chiral perturbation theory (HBChPT) was proposed [@Jenkins:1990jv; @Jenkins:1992pi; @Bernard:1992qa; @Bernard:1995dp]. The magnetic moments of the octet, decuplet and spin-$1 \over
2$ doubly charmed baryons have been calculated in the framework of HBChPT [@Jenkins:1992pi; @Meissner:1997hn; @Li:2016ezv; @Li:2017cfz]. The decuplet to octet and spin-$3 \over 2$ to spin-$1 \over 2$ doubly charmed baryons transition magnetic moments were also investigated [@Jenkins:1990jv; @Hemmert:1996xg; @Gellas:1998wx; @Gail:2005gz; @Li:2017vmq; @Li:2017pxa].
In this paper, we investigate the magnetic moments of the spin-$3\over 2$ doubly heavy baryons within the framework of HBChPT. We use the quark model to estimate the low energy constants (LECs), since there does not exist any experiment data. The numerical results are given to the next-to-leading order while the analytical results are presented to the next-to-next-to-leading order.
We first discuss the electromagnetic form factors of the spin-$3\over 2$ baryons in Sec. \[SecEM\]. The chiral Lagrangians are constructed in Sec. \[SecLag\]. In Sec. \[SecAnaly\], we calculate the magnetic moments analytically order by order. In Sec. \[SecNO\], with the help of quark model, we estimate the LECs and give the numerical results of the magnetic moments to the next-to-leading order. A short summary is given in Sec. \[SecConcld\]. All the coefficients of the loop corrections are collected in the \[AppCG\].
Electromagnetic form factors of the spin-$\frac{3}{2}$ doubly charmed baryons {#SecEM}
==============================================================================
For the spin-$\frac{3}{2}$ doubly charmed baryons, one can parameterize the general electromagnetic current matrix elements [@Nozawa:1990gt], which satisfy the gauge invariance, parity conservation and time-reversal invariance: $$\langle
T(p^{\prime})|J_{\mu}|T(p)\rangle=\bar{u}^{\rho}(p^{\prime})O_{\rho\mu\sigma}(p^{\prime},p)u^{\sigma}(p),$$ with $$\begin{split}
O_{\rho\mu\sigma}(p^{\prime},p)=&-g_{\rho\sigma}(A_{1}\gamma_{\mu}+\frac{A_{2}}{2M_{T}}P_{\mu})\\
&-\frac{q_{\rho}q_{\sigma}}{(2M_{T})^{2}}(C_{1}\gamma_{\mu}+\frac{C_{2}}{2M_{T}}P_{\mu}),
\end{split}$$ where $p$ and $p'$ are the momenta of the spin-$3\over 2$ doubly charmed baryons. $P=p+p'$, $q=p'-p$. $M_T$ is the doubly charmed baryon mass and $u_{\sigma}$ is the Rarita-Schwinger spinor [@Rarita:1941mf].
In the heavy baryon limit, the baryon field can be decomposed into the large component $\mathcal{T}$ and small component $\mathcal{H}$, $$T=e^{-iM_Tv\cdot x}(\mathcal{T}+\mathcal{H})$$ $$\mathcal{T}=e^{iM_Tv\cdot x}{1+\slashed{v}\over 2}T$$ $$\mathcal{H}=e^{iM_Tv\cdot x}{1-\slashed{v}\over 2}T$$ where $v_\mu$ is the velocity of the baryon. In the heavy baryon limit, the matrix elements of the electromagnetic current $J_{\mu}$ can be re-parametrized as [@Li:2016ezv] $$\langle
\mathcal{T}(p^{\prime})|J_{\mu}|\mathcal{T}(p)\rangle=\bar{u}^{\rho}(p^{\prime})\mathcal{O}_{\rho\mu\sigma}(p^{\prime},p)u^{\sigma}(p),$$ with $$\begin{split}
\mathcal{O}_{\rho\mu\sigma}(p^{\prime},p)&=-g_{\rho\sigma}\left[v_{\mu}F_{1}(q^{2})+\frac{[S_{\mu},S_{\alpha}]}{M_{T}}q^{\alpha}F_{2}(q^{2})\right]\\
&-\frac{q^{\rho}q^{\sigma}}{(2M_{T})^{2}}\left[v_{\mu}F_{3}(q^{2})+\frac{[S_{\mu},S_{\alpha}]}{M_{T}}q^{\alpha}F_{4}(q^{2})\right]
\end{split}$$ where $S_{\mu}={i\over 2}\gamma_5\sigma_{\mu\nu}v^\nu$ is the covariant spin-operator. The charge (E0), electro-quadrupole (E2), magnetic-dipole (M1), and magnetic octupole (M3) form factors read $$\begin{aligned}
G_{E0}(q^{2})&=&(1+\frac{2}{3}\tau)[F_{1}+\tau(F_{1}-F_{2})]\nonumber\\
&&-\frac{1}{3}\tau(1+\tau)[F_{3}+\tau(F_{3}-F_{4})],\\
G_{E2}(q^{2})&=&[F_{1}+\tau(F_{1}-F_{2})]\nonumber \\
&&-\frac{1}{2}(1+\tau)[F_{3}+\tau(F_{3}-F_{4})],\\
G_{M1}(q^{2})&=&(1+\frac{4}{5}\tau)F_{2}-\frac{2}{5}\tau(1+\tau)F_{4},\\
G_{M3}(q^{2})&=&F_{2}-\frac{1}{2}(1+\tau)F_{4}.
\end{aligned}$$ where $\tau=-\frac{q^{2}}{(2M_{T})^{2}}$. When $q^2=0$, we obtain the magnetic moment $\mu_{T}=G_{M1}(0){e\over 2M_T}$.
Chiral Lagrangians {#SecLag}
==================
The leading order chiral Lagrangians
------------------------------------
To calculate the chiral corrections to the magnetic moments, we construct the relevant chiral Lagrangians. The doubly charmed baryon fields read $$B=\left(\begin{array}{c}
\Xi_{cc}^{++}\\
\Xi_{cc}^{+}\\
\Omega_{cc}^{+}
\end{array}\right), T^{\mu}=\left(\begin{array}{c}
\Xi_{cc}^{*++}\\
\Xi_{cc}^{*+}\\
\Omega_{cc}^{*+}
\end{array}\right)^\mu,\Rightarrow \left(\begin{array}{c}
ccu\\
ccd\\
ccs
\end{array}\right).$$ where the $B$ and $T^{\mu}$ are spin-${1\over 2}$ and spin-${3\over
2}$ doubly chamed baryon fields respectively. We follow the notations in Refs. [@Li:2016ezv; @Scherer:2002tk; @Bernard:1995dp] to define the basic chiral effective Lagrangians of the pseudoscalar mesons. The pseudoscalar meson fields are introduced as follows, $$\phi=\left(\begin{array}{ccc}
\pi^{0}+\frac{1}{\sqrt{3}}\eta & \sqrt{2}\pi^{+} & \sqrt{2}K^{+}\\
\sqrt{2}\pi^{-} & -\pi^{0}+\frac{1}{\sqrt{3}}\eta & \sqrt{2}K^{0}\\
\sqrt{2}K^{-} & \sqrt{2}\bar{K}^{0} & -\frac{2}{\sqrt{3}}\eta
\end{array}\right).$$ The chiral connection and axial vector field are defined as [@Scherer:2002tk; @Bernard:1995dp], $$\Gamma_{\mu}=\frac{1}{2}\left[u^{\dagger}(\partial_{\mu}-ir_{\mu})u+u(\partial_{\mu}-il_{\mu})u^{\dagger}\right],$$ $$u_{\mu}=
i\left[u^{\dagger}(\partial_{\mu}-ir_{\mu})u-u(\partial_{\mu}-il_{\mu})u^{\dagger}\right],$$ where $$\begin{aligned}
&&u^{2}=\mathit{U}=\exp(i\phi/f_{0})\\
&&r_{\mu}=l_{\mu}=-eQA_{\mu},\end{aligned}$$ For the Lagrangians with the baryon fields, $Q=Q_B=\rm{diag}(2,1,1)$ and for the pure meson Lagrangians $Q=Q_M=\rm{diag}(2/3,-1/3,-1/3)$. $f_0$ is the decay constant of the pseudoscalar meson in the chiral limit. The experimental value of the pion decay constant $f_{\pi}\approx$ 92.4 MeV while $f_{K}\approx$ 113 MeV, $f_{\eta}\approx$ 116 MeV.
The leading order ($\mathcal{O}(p^{2})$) pure meson Lagrangian is $$\begin{aligned}
&&\mathcal{L}_{\pi\pi}^{(2)}=\frac{f_{0}^{2}}{4}{\rm
Tr}[\nabla_{\mu}U(\nabla^{\mu}U)^{\dagger}] \label{Lag:meson1},\\
&&\nabla_{\mu}U=\partial_{\mu}U-ir_{\mu}U+iUl_{\mu},\end{aligned}$$ where the superscript denotes the chiral order. The leading order doubly charmed baryon Lagrangians and meson-baryon interaction Lagrangians read $$\begin{aligned}
{\cal L}_{TT}^{(1)}&=&\bar{T}^{\mu}[-g_{\mu\nu}(i\slashed{D}-M_{T})+i(\gamma_{\mu}D_{\nu}+\gamma_{\nu}D_{\mu})\nonumber \\
&-&\gamma_{\mu}(i\slashed{D}+M_{T})\gamma_{\nu}]T^{\nu}
+\frac{H}{2}\left(\bar{T}^{\mu}g_{\mu\nu}\slashed{u}\gamma_{5}T^{\nu}\right), \label{Lag:op1tt} \\
\mathcal{L}_{BB}^{(1)}&=&\bar{B}(i\slashed{D}-M_{B}+\frac{\tilde{g}_{A}}{2}\gamma^{\mu}\gamma_{5}u_{\mu})B ,\label{Lag:op1bb}\\
\mathcal
{L}_{BT}^{(1)}&=&\frac{C}{2}\left(\bar{T}^{\mu}u_{\mu}B+\bar{B}u_{\mu}T^{\mu}\right).
\label{Lag:op1bt}\end{aligned}$$ We use the subscript to denote the two particles involved in the Lagrangians. $M_B$ is the spin-${1\over 2 }$ doubly charmed baryon mass. $\tilde{g}_A$, $C$ and $H$ are the coupling constants. The covariant derivative is defined as $D_{\mu}\equiv
\partial_{\mu}+\Gamma_{\mu}$. Both $\mathcal{L}_{TT}$ and $\mathcal{L}_{BB}$ contain the free and interaction terms.
In the framework of HBChPT, the leading order nonrelativistic Lagrangians read $$\begin{aligned}
{\cal L}_{TT}^{(1)}&=&\bar{{\cal T}}^{\mu}\left[-iv\cdot Dg_{\mu\nu}\right]{\cal T}^{\nu}+H\left(\bar{{\cal T}}^{\mu}g_{\mu\nu}u\cdot S{\cal T}^{\nu}\right), \label{Lag:op1rdtt} \\
{\cal L}_{BT}^{(1)}&=&\frac{C}{2}\left(\bar{{\cal T}}^{\mu}u_{\mu}{\cal B}+\bar{{\cal B}}u_{\mu}{\cal T}^{\mu}\right),\label{Lag:op1rdbt} \\
{\cal L}_{BB}^{(1)}&=&{\cal \bar{B}}i(v\cdot D-\delta){\cal B}+{\cal
\bar{B}}\tilde{g}_{A}S\cdot u{\cal B}, \label{Lag:op1rdbb}\end{aligned}$$ where $\delta\equiv M_B-M_T$. Since the spin-$3\over 2$ doubly charmed baryons have not been observed in the experiments, we take two values the mass splitting, $\delta=$-100 MeV and $\delta=$-70 MeV, in our work. We do not consider the mass difference among the doubly charmed baryon triplets. The coupling constants are estimated with the help of quark model in Refs. [@Li:2017cfz; @Li:2017pxa], $\tilde{g}_{A}=-{1\over
5}g_N=-0.25$, $C=-\frac{2\sqrt{3}}{5}g_{N}=-0.88 $ and $H=-\frac{3}{5}g_{N}=-0.76 $, where $g_N=1.267$ is the nucleon axial charge. For the pseudoscalar meson masses, we use $m_{\pi}=0.140$ GeV, $m_{K}=0.494$ GeV, and $m_{\eta}=0.550$ GeV. We use the nucleon mass $M_N=0.938\rm{GeV}$.
The next-to-leading order chiral Lagrangians
--------------------------------------------
The $\mathcal{O}(p^{2})$ Lagrangians contribute to the magnetic moments of the spin-${3\over 2}$ doubly charmed baryons $$\mathcal{L}_{TT}^{(2)}=\frac{-ib_{1}^{tt}}{2M_{T}}\bar{T}^{\mu}\hat{F}_{\mu\nu}^{+}T^{\nu}+\frac{-ib_{2}^{tt}}{2M_{T}}\bar{T}^{\mu}\langle
F_{\mu\nu}^{+}\mathcal{\rangle}T^{\nu}, \label{Lag:op2tt}$$ $${\cal
L}_{BB}^{(2)}=\frac{b_{1}^{bb}}{8M_{T}}\bar{B}\sigma^{\mu\nu}\hat{F}_{\mu\nu}^{+}B+\frac{b_{2}^{bb}}{8M_{T}}\bar{B}\sigma^{\mu\nu}\langle
F_{\mu\nu}^{+}\rangle B, \label{Lag:op2bb}$$ $$\begin{aligned}
&&{\cal L}_{BT}^{(2)}=b_{1}^{bt}\frac{i}{4M_{T}}\bar{B}\hat{F}_{\mu\nu}^{+}\gamma^{\nu}\gamma_{5}T^{\mu}+b_{2}^{bt}\frac{i}{4M_{T}}\bar{B}\langle F_{\mu\nu}^{+}\rangle\gamma^{\nu}\gamma_{5}T^{\mu} \nonumber \\
&&-b_{1}^{bt}\frac{i}{4M_{T}}\bar{T}^{\mu}\hat{F}_{\mu\nu}^{+}\gamma^{\nu}\gamma_{5}B-b_{2}^{bt}\frac{i}{4M_{T}}\bar{T^{\mu}}\langle
F_{\mu\nu}^{+}\rangle\gamma^{\nu}\gamma_{5}B, \label{Lag:op2bt}\end{aligned}$$ where the coefficients $b_{1,2}^{tt,bb,bt}$ are the LECs which contribute to the magnetic moments at tree level. The chiral covariant QED field strength tensors $F_{\mu\nu}^{\pm}$ are defined as $$\begin{aligned}
F_{\mu\nu}^{\pm} & = & u^{\dagger}F_{\mu\nu}^{R}u\pm
uF_{\mu\nu}^{L}u^{\dagger},\\
F_{\mu\nu}^{R} & = &
\partial_{\mu}r_{\nu}-\partial_{\nu}r_{\mu}-i[r_{\mu},r_{\nu}],\\
F_{\mu\nu}^{L} & = &
\partial_{\mu}l_{\nu}-\partial_{\nu}l_{\mu}-i[l_{\mu},l_{\nu}].\end{aligned}$$ Since the $Q_B$ is not traceless, the operator ${F}_{\mu\nu}^{+}$ can be divided into two parts, $\hat{F}_{\mu\nu}^{+}$ and $\langle{F}_{\mu\nu}^{+}\rangle$. $\langle F^+_{\mu\nu}\rangle
\equiv \text{Tr}(F^+_{\mu\nu})$. The operator $\hat{F}_{\mu\nu}^{+}=F_{\mu\nu}^{+}-\frac{1}{3}\rm \langle
F_{\mu\nu}^{+}\rangle$ is traceless and transforms as the adjoint representation under the chiral transformation. Recall that the direct product of the representation of SU(3) group $3\otimes\bar{3}
= 1\oplus8$. Therefore, there are two independent interaction terms in the $\mathcal{O}(p^{2})$ Lagrangians for the magnetic moments of the doubly charmed baryons. The nonrelativistic Lagrangians corresponding to Eqs. (\[Lag:op2tt\]-\[Lag:op2bt\]) are: $$\begin{aligned}
&\mathcal{L}_{TT}^{(2)}=\frac{-ib_{1}^{tt}}{2M_{T}}\bar{\mathcal{T}}^{\mu}\hat{F}_{\mu\nu}^{+}\mathcal{T}^{\nu}+\frac{-ib_{2}^{tt}}{2M_{T}}\bar{\mathcal{T}}^{\mu}\langle F_{\mu\nu}^{+}\mathcal{\rangle T}^{\nu}, \label{Lag:op2rdtt} \\
&{\cal L}_{BT}^{(2)}=b_{1}^{bt}\frac{i}{2M_{T}}\bar{{\cal B}}\hat{F}_{\mu\nu}^{+}S^{\nu}{\cal T}^{\mu}+b_{2}^{bt}\frac{i}{2M_{T}}\bar{{\cal B}}\langle F_{\mu\nu}^{+}\rangle S^{\nu}{\cal T}^{\mu} \nonumber \\
&-b_{1}^{bt}\frac{i}{2M_{T}}\bar{{\cal T}}^{\mu}\hat{F}_{\mu\nu}^{+}S^{\nu}{\cal B}-b_{2}^{bt}\frac{i}{2M_{T}}\bar{{\cal T}^{\mu}}\langle F_{\mu\nu}^{+}\rangle S^{\nu}{\cal B}, \label{Lag:op2rdbt}\\
&{\cal L}_{BB}^{(2)}=-\frac{ib_{1}^{bb}}{4M_{T}}\bar{{\cal
B}}[S^{\mu},S^{\nu}]\hat{F}_{\mu\nu}^{+}{\cal B}-{\cal
\bar{B}}\frac{ib_{2}^{bb}}{4M_{T}}[S^{\mu},S^{\nu}]\langle
F_{\mu\nu}^{+}\rangle{\cal B}, \nonumber \\\label{Lag:op2rdbb}\end{aligned}$$
We also need the second order pseudoscalar meson and doubly charmed baryon interaction Lagrangians $$\begin{aligned}
{\cal
L}_{TT}^{(2)}=&&\frac{ig_{1}^{tt}}{4M_{T}}\bar{T}^{\mu}\{u_{\rho},u_{\sigma}\}\sigma^{\rho\sigma}g_{\mu\nu}T^{\nu}\nonumber\\
&&+\frac{ig_{2}^{tt}}{4M_{T}}\bar{T}^{\mu}[u_{\rho},u_{\sigma}]\sigma^{\rho\sigma}g_{\mu\nu}T^{\nu},\label{Lag:op2g}\end{aligned}$$ where $g_{1,2}^{tt}$ are the coupling constants. Recall that for SU(3) group representations, $$\begin{aligned}
3\otimes\bar{3} & = & 1\oplus8\label{Eq:flavor1},\\
8\otimes8 & = &
1\oplus8_{1}\oplus8_{2}\oplus10\oplus\bar{10}\oplus27.\label{Eq:flavor2}\end{aligned}$$ Both $u_{\mu}$ and $u_{\nu}$ transform as the adjoint representation. The two terms in Eq. (\[Lag:op2g\]) correspond to the product of $u_{\mu}$ and $u_{\nu}$ belonging to the $8_1$ and $8_2$ flavor representations, respectively. The $g_{1}^{tt}$ term vanishes because of the anti-symmetric Lorentz structure. Thus, there is only one linearly independent LEC $g_{2}^{tt}$, which contributes to the spin-$3\over 2$ doubly charmed baryon magnetic moments up to $\mathcal{O}(p^3)$. The second order pseudoscalar meson and baryon nonrelativistic Lagrangian reads $$\begin{aligned}
{\cal L}_{TT}^{(2)}=\frac{g_{2}^{tt}}{2M_{T}}\bar{{\cal
T}}^{\mu}[S^{\rho},S^{\sigma}][u_{\rho},u_{\sigma}]g_{\mu\nu}{\cal
T}^{\nu} \label{Lag:op2grd}\end{aligned}$$ The above Lagrangian contributes to the doubly charmed baryon magnetic moments in the diagram (j) of the Fig. \[fig:allloop\].
The higher order chiral Lagrangians
------------------------------------
To calculate the $\mathcal{O}(p^{3})$ magnetic moments at the tree level, we also need the $\mathcal{O}(p^{4})$ electromagnetic chiral Lagrangians. The possible flavor structures are listed in Table \[Table:Flavor structure\], where $\chi^{+}$=diag(0,0,1) at the leading order. Recalling the flavor representation in Eqs. (\[Eq:flavor1\]), (\[Eq:flavor2\]), the leading order term of the operator $[\hat{F}_{\mu\nu}^{+},\hat{\chi}_{+}]$ vanishes after expansion since both $F^{+}_{\mu\nu}$ and $\chi^+$ are diagonal. Meanwhile, the $\langle
F_{\mu\nu}^{+}\rangle\langle\chi_{+}\rangle$ and $\langle
F_{\mu\nu}^{+}\rangle\hat{\chi}_{+}$ terms can be absorbed into Eq. (\[Lag:op2tt\]) by renormalizing the LECs $b_1^{tt}$ and $b_2^{tt}$. Thus, the independent $\mathcal{O}(p^4)$ Lagrangians read: $$\begin{split}
{\cal {\cal L}}^{(4)}=&\frac{-ia_{1}}{2M_{T}}\bar{T}^{\mu}\langle
F_{\mu\nu}^{+}\rangle\hat{\chi}_{+}T^{\nu}
+\frac{-ia_{2}}{2M_{T}}\bar{T}^{\mu}\langle\hat{F}_{\mu\nu}^{+}\hat{\chi}_{+}\rangle
T^{\nu}\\
&+\frac{-ia_{3}}{2M_{T}}\bar{T}^{\mu}\{\hat{F}_{\mu\nu}^{+},\hat{\chi}_{+}\}\hat{\chi}_{+}T^{\nu}.
\label{op4r}
\end{split}$$
[c|c|c|c|c|c|c]{} Group representation & $1\otimes1\rightarrow1$ & $1\otimes8\rightarrow8$ & $8\otimes1\rightarrow8$ & $8\otimes8\rightarrow1$ & $8\otimes8\rightarrow8_{1}$ & $8\otimes8\rightarrow8_{2}$\
Flavor structure &$\langle F_{\mu\nu}^{+}\rangle \langle\chi_{+}\rangle$ & $\langle
F_{\mu\nu}^{+}\rangle \hat{\chi}_{+}$ & $\hat{F}_{\mu\nu}^{+}\langle\chi_{+}\rangle$ & $\langle\hat{F}_{\mu\nu}^{+}\hat{\chi}_{+}\rangle$ & $[\hat{F}_{\mu\nu}^{+},\hat{\chi}_{+}]$ & $\{\hat{F}_{\mu\nu}^{+},\hat{\chi}_{+}\}$\
Formalism up to one-loop level {#SecAnaly}
==============================
We adopt the standard power counting scheme as in Refs. [@Ecker:1994gg; @Meissner:1997ws]. The chiral order $D_{\chi}$ of a Feynman diagram is $$D_{\chi}=2L+1+\sum_d (d-2)N^M_d+\sum_d (d-1)N^{MB}_d, \label{pwct}$$ with $L$ the number of loops and $N^M_d$, $N^{MB}_d$ the number of the $d$ dimension vertices from the meson and meson-baryon Lagrangians, respectively. The chiral order of the magnetic moment $\mu_T$ is $(D_{\chi}-1)$.
The leading order ($\mathcal{O}(p^1)$) magnetic moments come from the tree diagram in Fig. \[fig:tree\] with the $\mathcal{O}(p^2)$ vertex. The magnetic moment is $$\mu^{(1)}=2\alpha\frac{M_{N}}{M_{T}}\mu_{N}\label{mu1}$$ The $\mu_N$ is the nucleon magneton. $\alpha$ are the Clebsch-Gordan coefficients, which are collected in Table \[Table:cgloop\]. Up to the leading order, there are two unknown LECs, $b_1^{tt}$ and $b_2^{tt}$.
There are four diagrams (a)-(d) which contribute to the next-to-leading order magnetic moments of the spin-$3 \over 2$ doubly charmed baryons, as shown in Fig. \[fig:allloop\]. All the vertices in (a)-(d) come from Eq. (\[Lag:meson1\]) and Eqs. (\[Lag:op1rdtt\])(\[Lag:op1rdbt\]). The diagrams (c) and (d) vanish in the heavy baryon mass limit. In particular, the amplitudes of the diagrams (c) and (d) are denoted as $\mathcal{M}_c$ and $\mathcal{M}_d$. We have $$\begin{aligned}
\mathcal{M}_c & \propto & \int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{l^{2}-m^{2}_\phi+i\epsilon}\frac{(S\cdot l)}{f_{0}}
\frac{-iP_{\rho\sigma}^{3/2}}{v\cdot l+i\epsilon}S_\mu\protect\\
\nonumber
& \propto & S\cdot v=0,\\
\mathcal{M}_d & \propto &
\int\frac{d^{d}l}{(2\pi)^{d}}\frac{i}{l^{2}-m^{2}_\phi+i\epsilon}\frac{
l_\sigma}{f_{0}}
\frac{i}{v\cdot l- \omega +i\epsilon}g_{\mu\rho}\protect\\ \nonumber
& \propto & g_{\mu\rho}v_{\sigma},
\end{aligned}$$ where $P_{\rho\sigma}^{3/2}$ is the non-relativistic spin-$\frac32$ projector. $\mathcal{M}_d$ vanishes since $v_{\sigma}u^{\sigma}=0$. In other words, diagrams (c) and (d) do not contribute to the magnetic moments in the leading order of the heavy baryon expansion. The $\mathcal{O}(p^2)$ magnetic moment is $$\mu^{(2)}=\sum_{\phi=\pi,K}({\cal A}^{\phi}+{\cal
B}^{\phi})\beta_{a}^{\phi},$$ $${\cal
A}^{\phi}=-\frac{H^{2}M_{N}m_{\phi}}{96f_{\phi}^{2}}\mu_{N},$$ $$\begin{split}
{\cal
B}^{\phi}=-\frac{C^{2}M_{N}\mu_{N}}{64\pi^{2}f_{\phi}^{2}}[&2\sqrt{m_{\phi}^{2}-\delta^{2}}\arccos\left(\frac{\delta}{m_{\phi}}\right)\\
&-\delta\left(\ln\frac{m_{\phi}^{2}}{\lambda^{2}}-1\right)]
\end{split}$$ where $\beta^\phi_a$ are the Clebsch-Gordan coefficients, which are collected in Fig. \[Table:cgloop\]. Up to $\mathcal{O}(p^2)$, there are five unknown LECs, $b_{1,2}^{tt}$, $C$, $H$ and $\tilde{g}_A$. $C$, $H$ and $\tilde{g}_A$ can be estimated with the quark model.
There are eight loop diagrams (e)-(l) in Fig. \[fig:allloop\], which contribute at $\mathcal{O}(p^3)$. For the (e)-(h) diagrams, the photon-baryon vertices are from the $\mathcal{O}(p^2)$ interaction terms in Eqs. (\[Lag:op2rdtt\]-\[Lag:op2rdbb\]), while the meson-baryon vertices are from Eqs. (\[Lag:op1rdtt\])(\[Lag:op1rdbt\]). The vertex in diagram (i) is from Eq. (\[Lag:op2rdtt\]). The meson-baryon vertex in diagram (j) is from the Lagrangian in Eq. (\[Lag:op2g\]) while the meson-photon vertex is from the Lagrangian in Eq. (\[Lag:meson1\]). The loops (k) and (l) represent the wave function renormalization. The photon-baryon vertices are from Lagrangians in Eq. (\[Lag:op2rdtt\]), while the meson-baryon vertices are from the interaction in Eqs. (\[Lag:op1rdtt\])(\[Lag:op1rdbt\]). The contributions to the magnetic moments from the eight loop diagrams read: $$\begin{split}
\mu^{(3)}_{\text {loop}}=&\sum_{\phi=\eta,\pi,K}({\cal
E}^{\phi}\gamma_{e}^{\phi}+{\cal F}^{\phi}\gamma_{f}^{\phi}+{\cal
G}^{\phi}\gamma_{g}^{\phi}+{\cal H}^{\phi}\gamma_{h}^{\phi})\\
&+\sum_{\phi=\pi,K}({\cal I}^{\phi}\delta^{\phi}+{\cal
J}^{\phi}\eta^{\phi}) +\sum_{\phi=\pi,\eta,K}\left({\cal
K}^{\phi}+{\cal L}^{\phi}\right)\xi^{\phi}\mu^{(1)}
\end{split}$$ where
$$\begin{aligned}
&&{\cal E}^{\phi}=\frac{H^{2}m_{\phi}^{2}}{864\pi^{2}f_{\phi}^{2}}\frac{M_{N}}{M_{T}}\mu_{N}\left(33\ln\frac{m_{\phi}^{2}}{\lambda^{2}}+70\right)\\
&&{\cal F}^{\phi}=\frac{C^{2}}{64\pi^{2}f_{\phi}^{2}}\frac{M_{N}}{M_{T}}\mu_{N}\left[2\delta^{2}+\left(m_{\phi}^{2}-2\delta^{2}\right)\ln\frac{m_{\phi}^{2}}{\lambda^{2}}+4\delta\sqrt{m_{\phi}^{2}-\delta^{2}}\arccos\left(\frac{\delta}{m_{\phi}}\right)\right]\\
&&{\cal G}^{\phi}={\cal H}^{\phi}=\frac{CH}{864\pi^{2}f_{\phi}^{2}\delta}\frac{M_{N}}{M_{T}}\mu_{N}\left[-2\left(\delta^{3}+3\pi m_{\phi}^{3}\right)+\left(6\delta^{3}-9\delta m_{\phi}^{2}\right)\ln\left(\frac{m_{\phi}^{2}}{\lambda^{2}}\right)+12\left(m_{\phi}^{2}-\delta^{2}\right)^{3/2}\arccos\left(\frac{\delta}{m_{\phi}}\right)\right]\\
&&{\cal I}^{\phi}={\cal J}^{\phi}=\frac{m_{\phi}^{2}}{64\pi^{2}f_{\phi}^{2}}\frac{M_{N}}{M_{T}}\text{ln}\frac{m_{\phi}^{2}}{\lambda^{2}}\mu_{N}\\
&&{\cal K}^{\phi}=-\frac{H^{2}m_{\phi}^{2}}{576\pi^{2}f_{\phi}^{2}}\left(15\ln\frac{m_{\phi}^{2}}{\lambda^{2}}+26\right)\\
&&{\cal
L}^{\phi}=\frac{-C^{2}}{64\pi^{2}f_{\phi}^{2}}\left[2\delta^{2}+\left(m_{\phi}^{2}-2\delta^{2}\right)\ln\frac{m_{\phi}^{2}}{\lambda^{2}}+4\delta\sqrt{m_{\phi}^{2}-\delta^{2}}\arccos\left(\frac{\delta}{m_{\phi}}\right)\right]\end{aligned}$$
The $\gamma^{\phi}_{e-h}$,$\delta^{\phi}$, $\eta^{\phi}$ and $\xi^{\phi}$ are loop coefficients, which are listed in Table \[Table:cgloop\].
Apart from the loop diagrams, there is a tree diagram which contributes to the $\mathcal{O}(p^3)$ magnetic moments. The Lagrangian is given in Eq. \[op4r\]. The contribution reads $$\mu_{\text{\text{tree}}}^{(3)}=2\phi\frac{M_{N}}{M_{T}}\mu_{N}$$ where the coefficients $\phi$ are given in Table \[Table:cgtree\]. Up to $\mathcal{O}(p^3)$ magnetic moments, there are thirteen LECs, $\tilde{g}_A$, $C$, $H$, $b_{1,2}^{tt,bb,bt}$, $g_2^{tt}$, and $a_{1,2,3}$.
![The $\mathcal{O}(p^{2})$ and $\mathcal{O}(p^{4})$ tree level diagrams where the spin-$3\over 2$ doubly charmed baryon is denoted by the double solid line. The solid dot and black square represent the second- and fourth-order couplings respectively.[]{data-label="fig:tree"}](treeccq){width="0.99\hsize"}
{width="0.9\hsize"}
Numberical results and discussions {#SecNO}
==================================
Since the spin-$3 \over 2$ doubly charmed baryons have not been observed in the experiments, we do not have any experimental inputs to fit the LECs. However, we can employ quark model to determine some of the LECs. The numerical results are given in Table \[Table:NO\].
There are two unknown LECs $b_{1,2}^{tt}$ up to the leading order ($\mathcal{O}(p^1)$) magnetic moments. Unlike the light baryons, the charge matrix $Q_{B}$ of the doubly charmed baryons is not traceless. The heavy quarks contribute to the trace part of the charge matrix. Thus, in the second column of Table \[Table:NO\], the $b_{1}^{tt}$ terms come from the light quark contribution. The $b_2^{tt}$ terms are the same for the three doubly charmed baryons and arise solely from the two charm quarks.
At the quark level, the flavor and spin wave function of the $\Xi_{cc}^{*++}$ reads: $$\begin{aligned}
|\Xi_{cc}^{*++};S_3={3\over 2}\rangle =|c\uparrow c\uparrow
u\uparrow\rangle, \label{xiwavefunc}\end{aligned}$$ where the arrow denote the third component of the spin. The magnetic moments of the baryons in the quark model are the matrix elements of the following operator $\mu$ sandwiched between the wave functions, $$\mu=\sum_i\mu_i\sigma_3^i, \label{magmomen}$$ where $\mu_i$ is the magnetic moment of the quark. $$\mu_i={e_i\over 2m_i},\quad i=u,d,s,c,b.$$ We adopt the constituent quark masses from Ref.[@Lichtenberg:1976fi] as the set A with $m_u=m_d=336$ MeV, $m_s=540$ MeV, $m_c=1660$ MeV and $m_b=4700$ MeV. We adopt the constituent quark masses from Ref. [@Karliner:2014gca] as the set B with $m_u=m_d=363$ MeV, $m_s=538$ MeV, $m_c=1711$ MeV and $m_b=5044$ MeV. The magnetic moments from the naive quark model estimation with set A and B are given in Table \[quark model\]. The two sets lead to the similar magnetic moments. We choose the results of set A in the following calculation.
[l|c|cc]{} Baryons & Magnetic moments & Set A & Set B\
$\Xi_{cc}^{*++}$ & $2\mu_{c}+\mu_{u}$ & 2.61 & 2.45\
$\Xi_{cc}^{*+}$ & $2\mu_{c}+\mu_{d}$ & -0.18 & -0.13\
$\Omega_{cc}^{*+}$ & $2\mu_{c}+\mu_{s}$ & 0.17 & 0.15\
$\Xi_{bb}^{*0}$ & $2\mu_{b}+\mu_{u}$ & 1.73 & 1.60\
$\Xi_{bb}^{*-}$ & $2\mu_{b}+\mu_{d}$ & -1.06 & -0.99\
$\Omega_{bb}^{*-}$ & $2\mu_{b}+\mu_{s}$ & -0.71 & -0.71\
$\Xi_{bc}^{*+}$ & $\mu_{b}+\mu_{c}+\mu_{u}$ & 2.17 & 2.03\
$\Xi_{bc}^{*0}$ & $\mu_{b}+\mu_{c}+\mu_{d}$ & -0.62 & -0.56\
$\Omega_{bc}^{*0}$ & $\mu_{b}+\mu_{c}+\mu_{s}$ & -0.27 & -0.28\
The $\mathcal{O}(p^1)$ magnetic moments of the spin-$3\over 2$ doubly charmed baryons are given in the second column in Table \[Table:NO\]. The numerical results from the quark model are given in the third column. In the quark model, the light quark parts contribute to the $b_{1}^{tt}$ terms, which are proportional to the light quark charge. The heavy quark parts contribute to the $b_{2}^{tt}$ terms, which are the same for the three doubly charmed baryons.
Up to $\mathcal{O}(p^{2})$, we must take the loop corrections into consideration. At this order, there exist three new LECs $\tilde{g}_A$, $C$ and $H$, which are estimated in the quark model [@Li:2017cfz; @Li:2017pxa]. The numerical results are given in the third column of Table \[Table:NO\].
In Table \[Table:NO\], the magnetic moments of the $\Xi^{*++}_{cc}$ and $\Xi^{*+}_{cc}$ are dominated by the leading order term while the magnetic moment of $\Omega_{cc}^{*+}$ is dominated by the chiral corrections. At the leading order, since three quarks in $\Xi_{cc}^{*++}$ all have the positive charge, their contributions to the magnetic moments are constructive. For the $\Xi_{cc}^{*+}$ and $\Omega_{cc}^{*+}$, the contribution of the heavy quarks and light quark cancel out to a large extent, which leads to small magnetic moments at the leading order.
At the next-to-leading order, both $\pi^+$ and $K^+$ mesons contribute to the chiral correction of $\mu_{\Xi^{*++}_{cc}}$, while only $\pi^+$ ($K^+$) contributes to $\mu_{\Xi^{*+}_{cc}}$ ($\mu_{\Omega^{*+}_{cc}}$). The chiral corrections in the loops (a) and (b) are proportional to the pseudoscalar meson mass, i.e., $\sim{
m_\phi \over M_T}$. Therefore, the chiral corrections for $\mu_{\Xi^{*++}_{cc}}$ and $\mu_{\Omega_{cc}^{*+}}$ are much larger than that for $\mu_{\Xi_{cc}^{*+}}$. It is interesting to note that the chiral correction from the $K^+$ loop for the $\mu_{\Omega_{cc}^{*+}}$ is much lager than the leading order contribution. Such a unique feature might be exposed by future lattice QCD simulation.
Up to $\mathcal{O}(p^{3})$, eight new LECs, $b_{1,2}^{bb,bt}$, $g_2^{tt}$ and $a_{1,2,3}$ are introduced. Since it is impossible to fix all these LECs due to lack of experimental data, we do not present the numerical results up to this order.
[l|cc|ccc|ccc]{} Baryons & ${\cal O}(p^{1})$ & quark model & $\delta$/MeV & ${\cal O}(p^{2})$ I & Total I & $\delta$/MeV & ${\cal O}(p^{2})$ II & Total II\
$\Xi_{cc}^{*++}$ & $-\frac{4}{3}\frac{M_{N}}{M_T}b_{1}^{tt}-\frac{8}{3}\frac{M_{N}}{M_T}b_{2}^{tt}$ & $2\mu_{c}+\mu_{u}=2.61$ & -100 & -0.90 & 1.72 & -70 & -1.02 & 1.59\
$\Xi_{cc}^{*+}$ & $\frac{2}{3}\frac{M_{N}}{M_T}b_{1}^{tt}-\frac{8}{3}\frac{M_{N}}{M_T}b_{2}^{tt}$ & $2\mu_{c}+\mu_{d}=-0.18$ & -100 & 0.09 & -0.09 & -70 & 0.19 & 0.02\
$\Omega_{cc}^{*+}$ & $\frac{2}{3}\frac{M_{N}}{M_T}b_{1}^{tt}-\frac{8}{3}\frac{M_{N}}{M_T}b_{2}^{tt}$ & $2\mu_{c}+\mu_{s}=0.17$ & -100 & 0.81 & 0.99 & -70 & 0.82 & 1.00\
With the same formalism, we have also calculated the magnetic moments of the doubly bottom baryons and charmed bottom baryons. Since the $b$ quark is much heavier than the $c$ quark, we adopt the mass splitting $\delta=-40$ MeV and $\delta=-20$ MeV for the doubly bottom baryons and $\delta=-60$ MeV and $\delta=-40$ MeV for the charmed bottom baryons. The above mass difference in our work is consistent with that in Refs. [@Martynenko:2007je; @Patel:2007gx; @Karliner:2014gca; @Shah:2016vmd]. The leading order magnetic moments and the LECs in the next-to-leading order magnetic moments are estimated by the quark model. We present the numerical results of the doubly bottom baryon and charmed bottom baryons magnetic moments to next-to-leading order in Table \[Table:mubbq\].
[l|c|ccc|ccc]{} Baryons & ${\cal O}(p^{1})$ & $\delta$/MeV & ${\cal O}(p^{2})$ I & Total I & $\delta$/MeV & ${\cal O}(p^{2})$ II & Total II\
$\Xi_{bb}^{*0}$ & $2\mu_{b}+\mu_{u}=1.73$ & -40 & -1.10 & 0.63 & -20 & -1.15 & 0.58\
$\Xi_{bb}^{*-}$ & $2\mu_{b}+\mu_{d}=-1.06$ & -40 & 0.27 & -0.79 & -20 & 0.31 & -0.75\
$\Omega_{bb}^{*-}$ & $2\mu_{b}+\mu_{s}=-0.71$ & -40 & 0.83 &0.12 & -20 & 0.83 & 0.12\
$\Xi_{bc}^{*+}$ & $\mu_{b}+\mu_{c}+\mu_{u}=2.17$ & -60 & -1.05 & 1.12 & -40 & -1.10 & 1.07\
$\Xi_{bc}^{*0}$ & $\mu_{b}+\mu_{c}+\mu_{d}=-0.62$ & -60 & 0.22 & -0.40 & -40 & 0.27 & -0.35\
$\Omega_{bc}^{*0}$ & $\mu_{b}+\mu_{c}+\mu_{s}=-0.27$ & -60 & 0.82 & 0.56 & -40 & 0.83 & 0.56\
Conclusions {#SecConcld}
===========
The doubly heavy baryons are particularly interesting because their chiral dynamics is solely dominated by the single light quark. In this work, we have employed the heavy baryon chiral perturbation theory to calculate the magnetic moments of the spin-$3 \over 2$ doubly charmed baryons, which reveal the information of their inner structure. Due to the large mass of the doubly heavy baryons, the recoil corrections are expected to be very small. We have derived our analytical expressions up to the next-to-next-to-leading order, which may be useful to the possible chiral extrapolation of the lattice simulations of the doubly charmed baryon electromagnetic properties.
With the help of quark model, we have estimated the LECs and presented the numerical results up to next-to-leading order: $\mu_{\Xi^{*++}_{cc}}=1.72\mu_{N}$, $\mu_{\Xi^{*+}_{cc}}=-0.09\mu_{N}$, $\mu_{\Omega^{*+}_{cc}}=0.99\mu_{N}$ for $\delta=-100$ MeV and $\mu_{\Xi^{*++}_{cc}}=1.59\mu_{N}$, $\mu_{\Xi^{*+}_{cc}}=0.02\mu_{N}$, $\mu_{\Omega^{*+}_{cc}}=1.00\mu_{N}$ for $\delta=-70$ MeV. As by-products, we have also calculated the magnetic moments of the spin-$3\over 2$ doubly bottom baryons and charmed bottom baryons.
For comparison, we have listed the spin-$3\over 2$ doubly charmed baryon magnetic moments from some other model calculations in Table \[Table:Comp\] including quark model [@Lichtenberg:1976fi], MIT bag model [@Bose:1980vy; @Bernotas:2012nz], Skymion model [@Oh:1991ws], nonrelativistic quark model (NRQM) [@Albertus:2006ya], hyper central quark model (HCQM) [@Patel:2007gx], effective mass and screened charge scheme [@Dhir:2009ax] and chiral constituent quark model ($\chi$CQM) [@Sharma:2010vv]. We define the relative changes $\rho$ of these models with respect to the naive estimates as $\mu=(1+\rho)\mu_{\text{QM}}$, where $\mu_{\text{QM}}$ is the magnetic moments estimated with the naive quark model. For the $\Xi_{cc}^{*++}$ baryon, one notices that the $|\rho|<0.1$ for most models, and $|\rho|<0.4$ for all models. Thus, all these approaches lead to more or less similar numerical results for the magnetic moments of $\Xi_{cc}^{*++}$. For the $\Xi_{cc}^{*+}$ baryon, the $|\rho|<2.5$ for most models including our work while the $|\rho|>4$ from the Skymion models. For the $\Omega_{cc}^{*+}$ baryon, the $|\rho|< 2$ in all the models except our work and one Skymion calculation. Thus, various models lead to quite different predictions for the magnetic moments of the $\Xi_{cc}^{*+}$ and $\Omega_{cc}^{*+}$, which may be used to distinguish these models.
In the numerical analysis, we have truncated the chiral expansions at $\mathcal{O}(p^{2})$ and omitted all the $\mathcal{O}(p^{3})$ higher order chiral corrections because of too many unknown LECs at this oder. When more experimental measurements become available in the future, the numerical analysis in the present work can be further improved. In principle, all the low energy constants shall be extracted through fitting to the experimental data instead of using the estimation from the quark model. The $\mathcal{O}(p^{3})$ chiral corrections may turn out to be non-negligible and should be included.
There is good hope that the spin-$3\over 2$ doubly charmed baryons will be observed through its radiative or weak decays in the coming future. We hope our numerical calculation may be useful for future experimental measurements of their magnetic moments. There are several LECs in our analytical results to be determined by the future progresses in the experiment and theory, which will help check the chiral expansion convergence of the three doubly charmed baryons.
[l|ccc]{} & $\Xi_{cc}^{*++}$ & $\Xi_{cc}^{*+}$ & $\Omega_{cc}^{*+}$\
Quark model [@Lichtenberg:1976fi] & 2.60 & -0.19 & 0.17\
Bag model l [@Bose:1980vy] & 2.54 & 0.20 & 0.39\
Bag model 2 [@Bernotas:2012nz] & 2.00 & 0.16 & 0.33\
Skymion 1 [@Oh:1991ws] & 3.16 & -0.98 & -0.20\
Skymion 2 [@Oh:1991ws] & 3.18 & -1.17 & 0.03\
NRQM [@Albertus:2006ya] & 2.67 & -0.31 & 0.14\
HCQM [@Patel:2007gx] & 2.75 & -0.17 & 0.12\
Effective mass [@Dhir:2009ax] & 2.41 & -0.11 & 0.16\
Screened charge [@Dhir:2009ax] & 2.52 & 0.04 & 0.21\
$\chi\text{CQM}$ [@Sharma:2010vv] & 2.66& -0.47 & 0.14\
This work I & 1.72 & -0.09 & 0.99\
This work II & 1.59 & 0.02 & 1.00\
L. Meng is very grateful to X. L. Chen and W. Z. Deng for very helpful discussions. This project is supported by the National Natural Science Foundation of China under Grants 11575008, 11621131001 and 973 program. This work is also supported by the Fundamental Research Funds for the Central Universities of Lanzhou University under Grants 223000–862637.
COEFFICIENTS OF THE LOOP CORRECTIONS {#AppCG}
====================================
In this appendix, we collect the explicit formulae for the chiral expansion of the doubly charmed baryon magnetic moments in Tables \[Table:cgtree\] and \[Table:cgloop\].
[lcc|ccc|cc|cc|ccc]{} & $\beta_{a}^{\pi}$ & $\beta_{a}^{K}$ & $\gamma_{e-h}^{\pi}$ & $\gamma_{e-h}^{K}$ & $\gamma_{e-h}^{\eta}$ & $\delta^{\pi}$ & $\delta^{K}$ & $\eta^{\pi}$ & $\eta^{K}$ & $\xi^{\pi}$ & $\xi^{K}$ & $\xi^{\eta}$\
$\Xi_{cc}^{*++}$ & $4$ & $4$ & $4b_{2}$ & $\frac{2}{3}\left(-b_{1}+4b_{2}\right)$ & $\frac{2}{9}\left(b_{1}+2b_{2}\right)$ & $-4b_{1}^{tt}$ & $-4b_{1}^{tt}$ & $-8g_{2}^{tt}$ & $-8g_{2}^{tt}$ & $3$ & $2$ & $\frac{1}{3}$\
$\Xi_{cc}^{*+}$ & $-4$ & & $b_{1}+4b_{2}$ & $\frac{2}{3}\left(-b_{1}+4b_{2}\right)$ & $\frac{1}{9}\left(-b_{1}+4b_{2}\right)$ & $4b_{1}^{tt}$ & & $8g_{2}^{tt}$ & & $3$ & $2$ & $\frac{1}{3}$\
$\Omega_{cc}^{*+}$ & & $-4$ & & $\frac{2}{3}\left(b_{1}+8b_{2}\right)$ & $\frac{4}{9}\left(-b_{1}+4b_{2}\right)$ & & $4b_{1}^{tt}$ & & $8g_{2}^{tt}$ & & $4$ & $\frac{4}{3}$\
[lc|c]{} & $\alpha$ & $\phi$\
$\Xi_{cc}^{*++}$ & $\frac{2}{3}b_{1}^{tt}+\frac{4}{3}b_{2}^{tt}$ & $-\frac{4}{9}a_{1}-\frac{1}{9}a_{2}-\frac{4}{9}a_{3}$\
$\Xi_{cc}^{*+}$ & $-\frac{1}{3}b_{1}^{tt}+\frac{4}{3}b_{2}^{tt}$ & $-\frac{4}{9}a_{1}-\frac{1}{9}a_{2}+\frac{2}{9}a_{3}$\
$\Omega_{cc}^{*+}$ & $-\frac{1}{3}b_{1}^{tt}+\frac{4}{3}b_{2}^{tt}$ & $\frac{8}{9}a_{1}-\frac{1}{9}a_{2}-\frac{4}{9}a_{3}$\
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---
abstract: 'Coded caching and delivery is studied taking into account the correlations among the contents in the library. Correlations are modeled as common parts shared by multiple contents; that is, each file in the database is composed of a group of subfiles, where each subfile is shared by a different subset of files. The number of files that include a certain subfile is defined as the *level of commonness* of this subfile. First, a correlation-aware *uncoded* caching scheme is proposed, and it is shown that the optimal placement for this scheme gives priority to the subfiles with the highest levels of commonness. Then a correlation-aware *coded* caching scheme is presented, and the cache capacity allocated to subfiles with different levels of commonness is optimized in order to minimize the delivery rate. The proposed correlation-aware coded caching scheme is shown to remarkably outperform state-of-the-art correlation-ignorant solutions, indicating the benefits of exploiting content correlations in coded caching and delivery in networks.'
author:
- 'Qianqian Yang and Deniz Gündüz\'
bibliography:
- 'report.bib'
title: Centralized Coded Caching of Correlated Contents
---
Introduction {#intro}
============
In *proactive caching*, popular contents are stored in user devices during off-peak traffic periods even before they are requested by the users [@GregoryDtoD; @Bastug:CM:14; @samuel2017]. Proactive caching is considered as a promising solution for the recent explosive growth of wireless data traffic, and can alleviate both the network congestion and the latency during peak traffic periods (see [@GregoryDtoD; @Bastug:CM:14; @MaddahAliCentralized; @samuel2017], and references therein).
Proactive caching typically takes place in two phases: the first phase takes place during off-peak traffic periods, when users’ caches are filled as a function of the whole library of files, referred to as the *placement phase*; while the second, *delivery phase*, takes place during the peak traffic period when the users’ demands are revealed and satisfied simultaneously. In contrast to traditional uncoded caching schemes, which simply employ orthogonal unicast transmissions during the *delivery phase*, recently proposed *coded caching*[@MaddahAliCentralized] creates coded multicasting opportunities, significantly reducing the amount of data that needs to be delivered to the users to satisfy their demands, even when these demands are distinct. Coded caching benefits from the aggregate cache capacity across the network, rather than local cache capacities as in conventional uncoded caching [@MaddahAliCentralized]. This significant improvement has motivated intense research interest on coded caching in recent years [@MaddahAliDecentralized; @MohammadQianDenizITW; @MohammadDenizTCom; @VilardeboCodedCaching; @JiArXivNonuniform; @NiesenNonuniform; @PedarsaniOnlineCaching; @yang2016coded]. Some works follow the simplified model proposed in [@MaddahAliCentralized], and aim to improve the fundamental limits of caching [@MohammadDenizTCom; @MohammadQianDenizITW; @yu2017exact]; others consider more realistic settings, such as decentralized caching [@MaddahAliDecentralized], nonuniform popularities across files [@NiesenNonuniform; @JiArXivNonuniform], audience retention rate aware caching [@yang2017audience], or heterogeneous quality of service requirements[@yang2016coded].
An important feature of video contents, which is the main source of the recent explosive traffic growth, is that, there may be significant overlaps among different files, e.g., the recordings of the same event from different angles and different cameras, or even the frames of the same scene in the same video. In [@hassanzadeh2016correlation], Hassanzadeh et al. propose a correlation-aware caching scheme, which groups the contents in the library into two sets according to their correlations as well as popularity, where each file in second set is compressed with respect to a file in the first set. This scheme is shown to outperform correlation-ignorant caching schemes. A more information theoretic formulation for caching of correlated sources is considered in [@timo2016rate], focusing on a special scenario with two receivers and one cache. A similar information-theoretic analysis is carried out in [@hassanzadeh:Arxiv:17] for two files and two receivers, each with its own cache.
In this paper, we consider a server with a library of $N$ correlated files, serving $K$ users equipped with local caches. Different from [@hassanzadeh2016correlation], which only exploits a fixed level of common information among the files, we consider a more general model in order to fully exploit the potential correlations among different subsets of files. We model each file in the server to be composed of a group of subfiles, such that each subfile is shared by a different subset of files. The number of files to which each subfile belongs is defined as its *level of commonness*. Equivalently, in our model, any subset of files $\mathcal{S}$ share a common part that is independent of the rest of the library, and shared exclusively by the files in $\mathcal{S}$.
We first propose a correlation-aware *uncoded* caching scheme, and show that the optimal placement for this scheme is achieved by giving priority to the subfiles with the highest levels of commonness in the placement phase. We then propose a correlation-aware *coded* caching scheme, and derive a closed-form expression of the achievable delivery rate, based on which the cache capacity is optimally allocated to the subfiles according to their level of commonness. Then, we compare the performance of the proposed correlation-aware schemes with those that ignore the correlation, and the cut-set bound; and show that, exploiting file correlations in coded caching can significantly reduce the delivery rate.
*Notations:* The set of integers $\left\{ i, ..., j \right\}$, where $i \le j$, is denoted by $\left[ i:j \right]$, particularly, $\left\{1, ..., j \right\}$ is denoted by $\left[j \right]$. For sets $\mathcal{A}$ and $\mathcal{B}$, we define $\mathcal{A} \backslash \mathcal{B}\triangleq\{x: x \in \mathcal{A}, x\notin \mathcal{B}\}$, and $\left| \mathcal{A} \right|$ denotes the cardinality of $\mathcal{A}$. $\binom{j}{i}$ represents the binomial coefficient if $j\geq i$; otherwise, $\binom{j}{i}=0$. For event $E$, $\mathbbm{1}\{E\}=1$ if $E$ is true; and $\mathbbm{1}\{E\}=0$, otherwise.
System Model {#sys}
============
We consider a server with a database of $N$ correlated files, $W_1, ..., W_N$, where each file consists of $2^{N-1}$ independent subfiles, e.g., $W_i=\bigcup\limits_{\substack{\mathcal{S}\subset [N]\\ i \in \mathcal{S}}}\overline{W}_{\mathcal{S}}$, $\forall i\in [N]$. Here, $\overline{W}_{\mathcal{S}}$ denotes the subfile shared exclusively by the subset of files $\{W_i: i \in \mathcal{S}\}$. For simplicity, we assume that for $\mathcal{S}\subset [N]$, $|\overline{W}_{\mathcal{S}}|=F_l$, if $|\mathcal{S}|=l$, i.e., the common subfiles shared exclusively by $l$ files are of the same size of $F_l$ bits. Let $\mathbf{F} \triangleq (F_1, \ldots, F_N)$. As a result, each file in the library is also of the same size of $F$ bits, given by $$F=\sum\limits_{l=1}^N \binom{N-1}{l-1}F_l.$$
For $\mathcal{S}\subset [N]$, $|\mathcal{S}|=l$, we say that the subfiles $\overline{W}_{\mathcal{S}}$ have a commonness level of $l$. For example, $\overline{W}_{\{1, 2, 3\}}$ and $\overline{W}_{\{3, 4, 5\}}$ both have level $3$ commonness. For brevity, we refer to all the subfiles with level $l$ commonness as $l$-subfiles, $l=1, ..., N$. We consider $K$ users connected to the server through a shared, error-free link, each equipped with a cache of size $MF$ bits.
We consider centralized caching; that is, the server has the knowledge of the active users during the placement phase, though not the knowledge of their demands. Centralized caching allows the server to fill the user caches in a coordinated manner. After the placement phase, each user requests a single file from the library, where $d_k \in [N]$ denotes user $k$’s request, $k\in [K]$. All the requests are satisfied simultaneously over the error-free shared link.
An $(\mathbf{F}, M, R)$ caching code for this system consists of:
- **$K$ caching functions** $f_{k}$, $k \in [K]$, $$f_{k}: \underbrace{[2^F] \times \cdots \times [2^F]}\limits_{N~\text{files}}\rightarrow [2^{MF}],$$ such that the contents of user $k$’s cache at the end of the placement phase, denoted by $Z_k$, is given by $Z_k=f_{k}(\{W_i\}_{i=1}^N)$;
- **a delivery function** $g$, $$g: \underbrace{[2^F] \times \cdots \times [2^F]}\limits_{N~\text{files}} \times \mathbf{D} \rightarrow [ 2^{RF}],$$ where $\mathbf{D}\triangleq (d_1, ..., d_K)$, such that a single message of $RF$ bits, $X_{\mathbf{D}}=g((W_1, ..., W_N), \mathbf{D})$, is sent by the server over the shared link according to users’ demands;
- **$K$ decoding functions** $h_k$, $k \in [K]$, $$h_k: \mathbf{D} \times [2^{MF}]\times [2^{RF}] \rightarrow [2^{F}],$$ where $\hat{W}_{d_k}=h_k (\mathbf{D}, Z_k, X_{\mathbf{D}})$, is the reconstruction of $W_{d_k}$ at user $k$.
A user cache capacity-delivery rate pair $(M, R)$ is *achievable* for a system described above, if there exists a sequence of $(\mathbf{F}, M, R)$ codes such that for any demand realization $\mathbf{D} \subset [N]^K$, $$\lim_{F_1, \ldots, F_N \rightarrow \infty} \Pr \left\{\bigcup\limits_{k\in [K]} \Big\{\hat{W}_{d_k}\neq W_{d_k}\Big\}\right\}=0.$$
For a system with $N$ files and $K$ users, our goal is to characterize the minimum achievable rate $R$ as a function of the user cache capacity $M$, i.e., $R^*(M)\triangleq \inf\{R: (M, R) \mbox{ is achievable}\}$.
Correlation-aware Uncoded Caching and Delivery (CAUC) Scheme {#s:Uncoded}
============================================================
We first present an uncoded caching and delivery scheme exploiting the correlation among files, referred to as CAUC.
### Placement phase
Each user caches the same $p_lF_l$ bits from each $l$-subfile, where $0 \leq p_l\leq 1$, $l\in [N]$, such that $$\label{cachecapacity}
MF=\sum\limits_{l=1}^N \binom{N}{l}p_lF_l,$$ which meets the limitation of the cache capacities. We refer to $\mathbf{P} \triangleq (p_{1}, ..., p_N)$ as the *cache allocation vector*, which will be specified in the sequel.
### Delivery phase
The server delivers the remaining bits of each requested subfile that have not been cached by the users, i.e., $\overline{W}_{\mathcal{S}}$ for which $\sum\limits_{k=1}^{K}\mathbbm{1}\{d_k \in \mathcal{S}\} \geq 1$.
In the worst case, when the demand combination is the most distinct, i.e., users request distinct files for the case $N\geq K$, or each file is requested by at least one user for the case $N <K$, the delivery rate is given by $$\label{uncodedrate}
R_{CAUC}(\mathbf{P})=\sum\limits_{l=1}^{N}(1-p_l)F_l{\small\left(\binom{N}{l}-\binom{\min\{N-K, 0\}}{l}\right)}.$$ The optimal $\mathbf{P}^*$ can be derived by solving the following optimization problem $$\begin{aligned}
&\min~~~R_{CAUC}(\mathbf{P})\\
&\text{such that}~~~\sum\limits_{l=1}^N \binom{N}{l}p_lF_l\leq MF,
\end{aligned}$$ which, straightforwardly, leads to: $p^*_l=1$, if $C(l) \leq MF$; $p^*_l=\frac{MF-C(l+1)}{\binom{N}{l}F_l}$, if $C(l+1) < MF< C(l)$; and $p^*_l=0$, otherwise; where we have defined $C(l) \triangleq \sum\limits_{i=l}^N \binom{N}{i}F_i$, for $l\in [N]$. We remark that the optimal cache allocation gives priority to the subfiles with the highest level of commonness.
Correlation-aware Coded Caching and Delivery (CACC) Scheme {#s:Coded}
==========================================================
In this section, we present a correlation-aware coded caching and delivery scheme, referred to as CACC. Similarly to the CAUC scheme, we allocate different cache capacities to subfiles of different levels of commonness, again specified by the cache allocation vector, $\mathbf{P}=(p_{1}, ..., p_N)$, which satisfies the constraint in , such that each user caches $p_lF_l$ bits from each $l$-subfile, $l\in [N]$. In the following, we first present how coded caching and delivery of the subfiles with the same level of commonness is carried out, and then specify the allocation of cache capacity.
Coded Caching and Delivery of $l$-subfiles
------------------------------------------
Here, for a given cache allocation vector $\mathbf{P}$, we present the coded caching and delivery of $l$-subfiles, $l\in[N]$. We define $t_l\triangleq Kp_l$, $0\leq t_l \leq K$. If $t_l=0$, users do not cache the $l$-subfiles at all, while if $t_l=K$, each user stores all the $l$-subfiles in its cache. In the following, we focus on the cases where $t_l \in [K-1]$.
### Placement Phase
We employ the prefetching scheme proposed by [@MaddahAliCentralized] for the subfiles rather than the files themselves: each $l$-subfile is partitioned into $\binom{K}{t_l}$ disjoint parts, each with approximately the same size of $F_l/\binom{K}{t_l}$ bits. We label these $\binom{K}{t_l}$ disjoint parts of each $l$-subfile $\overline{W}_{\mathcal{S}}$ by $\overline{W}_{\mathcal{S}}^{\mathcal{A}}$, where $|\mathcal{A}|=t_l,~ \mathcal{A}\subset [K]$; that is, we have $\overline{W}_{\mathcal{S}}= \bigcup_{\mathcal{A}: |\mathcal{A}|=t_l,~ \mathcal{A}\subset [K]} \overline{W}_{\mathcal{S}}^{\mathcal{A}}$. Each of these parts, $\overline{W}_{\mathcal{S}}^{\mathcal{A}}$, is placed into the cache of user $k$ if $k \in \mathcal{A}$. Thus, each user caches a total of $\binom{K-1}{t_l-1}$ disjoint parts of each $l$-subfile with a total size of $t_lF_l/K$ bits, which sums up to $p_lF_l$ bits.
### Delivery Phase
We first focus on the case when $N\leq K$. There are a total of $\binom{N}{l}$ $l$-subfiles. We denote the set of these $l$-subfiles by $\mathcal{W}^l=\left\{\overline{W}_{\mathcal{S}}: \mathcal{S} \subset [N], |\mathcal{S}|=l\right\}$. Each user requires a total of $\binom{N-1}{l-1}$ $l$-subfiles, i.e., user $k$ needs to recover subfiles in $\left\{\overline{W}_{\mathcal{S}}: \mathcal{S} \subset [N], |\mathcal{S}|=l, d_k\in \mathcal{S}\right\}$, $\forall k\in [K]$. For each user, we can regard these $\binom{N-1}{l-1}$ $l$-subfiles as $\binom{N-1}{l-1}$ distinct demands. Our delivery scheme for the $l$-subfiles operates in $\binom{N-1}{l-1}$ steps, and satisfies one demand of each user at each step.
We define $\mathbf{C}_j\triangleq(c_{1j}, ..., c_{Nj})$, where $c_{ij} \in \{\mathcal{S}: \mathcal{S} \subset [N], |\mathcal{S}|=l, i\in \mathcal{S}\}$, $\forall i\in [N]$, $j\in [\binom{N-1}{l-1}]$, which specifies which subfile should be delivered in the $j$th step of the delivery phase. $\mathbf{C}_j$ is generated by Algorithm \[groupingscheme\] by setting $\mathcal{R}=[N]$ and $\overline{\mathcal{R}}=\emptyset$. Note that these vectors are generated independently of the number of users or their demands.
Consider $N=5$ and $l=2$. From Algorithm \[groupingscheme\] we obtain: $$\begin{aligned}
&\mathbf{C_1}=(\{1,2\}, \{1,2\}, \{3,4\}, \{3,4\}, \{1,5\});\\ &\mathbf{C}_2=(\{1,5\}, \{2,3\},\{2,3\}, \{4,5\},\{4, 5\});\\
&\mathbf{C}_3=(\{1,3\}, \{2,5\}, \{1,3\}, \{2,4\}, \{2,5\});\\
&\mathbf{C}_4=(\{1,4\}, \{2,4\}, \{3,5\}, \{1,4\}, \{3, 5\}).\end{aligned}$$ This means, for example, that, in the first step, subfiles $W_{12}, W_{34},$ and $W_{15}$ will be delivered (if there is a user requesting them).
We denote by $d^j_k\triangleq c_{d_kj}$ the demand of user $k$ to be satisfied in the $j$th step, i.e., user $k$ recovers $\overline{W}_{d^j_k}$ after the $j$th step. We emphasize that $\bigcup\limits_{j=1}^{\binom{N-1}{l-1}}\overline{W}_{c_{ij}}=\{\overline{W}_{\mathcal{S}}: \mathcal{S} \in [N], |\mathcal{S}|=l, i\in \mathcal{S}\}$, $\forall i \in [N]$; that is all the required $l$-subfiles will be recovered by each user after step $\binom{N-1}{l-1}$ for any demand combination.
\[ex:2\] Consider $N=K=5$ and $l=2$ as in Example 1. Consider distinct demands, i.e., $\mathbf{D} = \{1, 2, 3, 4, 5\}$. Thus, based on $\mathbf{C}_1$, we have $d^1_1=d^1_2=\{1, 2\}$, $d^1_3=d^1_4=\{3, 4\}$, and $d^1_5=\{1, 5\}$; that is, at the end of the first step, users 1 and 2 should recover $W_{12}$, users 3 and 4 should recover $W_{34}$, while user 5 should recover $W_{15}$.
\[ex:3\] With the same setting as in Example 2, consider now a non-distinct demand combination $\mathbf{D}=\{1, 1, 1, 3, 4\}$. Based on $\mathbf{C}_1$, we have $d^1_1=d^1_2$ $=d^1_3$ $=\{1, 2\}$, and $d^1_4$ $=d^1_5=\{3, 4\}$; that is, at the end of the first step users 1, 2 and 3 should recover $W_{12}$, while users 4 and 5 should recover $W_{34}$.
Based on the delivery scheme proposed in [@yu2017exact], we present our coded transmission scheme in Algorithm \[deliveryscheme\] according to $\mathbf{C}_j$, $j\in [\binom{N-1}{l-1}]$, where $\overline{\mathcal{R}}=\emptyset$, $\mathcal{R}=[N]$. In Algorithm \[deliveryscheme\], we define $A_j$ as the number of distinct $d^j_k$ for each $j \in [\binom{N-1}{l-1}]$. We note that, among the CODED DELIVERY and RANDOM DELIVERY procedures of Algorithm. \[deliveryscheme\], the one that requires a smaller delivery rate is performed.
**Example \[ex:2\] - continued.** In Example \[ex:2\], assume that $t_l=1$, i.e., each $l$-subfile is divided into $K$ disjoint parts of equal size, and each disjoint part is cached exactly by one user. Based on $\mathbf{D}$, we have $A_1=3$. Assume that $\mathcal{U}_1=\{1, 3, 5\}$. Then, the server sends $\overline{W}_{\{1, 2\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 2\}}^{\{2\}}$, $\overline{W}_{\{3, 4\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 2\}}^{\{3\}}$, $\overline{W}_{\{3, 4\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 2\}}^{\{4\}}$, $\overline{W}_{\{1, 5\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 2\}}^{\{5\}}$, $\overline{W}_{\{3, 4\}}^{\{2\}}\bigoplus\overline{W}_{\{1, 2\}}^{\{3\}}$, $\overline{W}_{\{3, 4\}}^{\{3\}}\bigoplus\overline{W}_{\{3, 4\}}^{\{4\}}$, $\overline{W}_{\{3, 4\}}^{\{5\}}\bigoplus\overline{W}_{\{1, 5\}}^{\{3\}}$, $\overline{W}_{\{1, 5\}}^{\{2\}}\bigoplus\overline{W}_{\{1, 2\}}^{\{5\}}$, $\overline{W}_{\{1, 5\}}^{\{4\}}\bigoplus\overline{W}_{\{3, 4\}}^{\{5\}}$. By receiving these coded bits, users $1$ and $2$ can recover $\overline{W}_{\{1, 2\}}$ together with the contents of their own caches. Similarly, users $3$ and $4$ can recover $\overline{W}_{\{3, 4\}}$, while user $5$ recovers $\overline{W}_{\{1, 5\}}$. In the same manner, by coded transmission based on $\mathbf{C}_2$, user $1$ can recover $\overline{W}_{\{1, 5\}}$, users $2$ and $3$ recover $\overline{W}_{\{2, 3\}}$, and users $4$ and $5$ recover $\overline{W}_{\{4, 5\}}$ in the second step. After four delivery steps based on $\mathbf{C}_1$, …, $\mathbf{C}_4$ each user decodes all the $2$-subfiles of their requests. The total number of bits delivered in these four steps is $36F_2/5$.
**Example \[ex:3\] - continued.** Assume again that $t=1$. Based on $\mathbf{D}$, we have $A_1=2$, and let $\mathcal{U}_1=\{1, 4\}$. Then, the server sends $\overline{W}_{\{1, 2\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 2\}}^{\{2\}}$, $\overline{W}_{\{1, 2\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 2\}}^{\{3\}}$, $\overline{W}_{\{3, 4\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 2\}}^{\{4\}}$, $\overline{W}_{\{3, 4\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 2\}}^{\{5\}}$, $\overline{W}_{\{3, 4\}}^{\{2\}}\bigoplus\overline{W}_{\{1, 2\}}^{\{4\}}$, $\overline{W}_{\{3, 4\}}^{\{3\}}\bigoplus\overline{W}_{\{1, 2\}}^{\{4\}}$, $\overline{W}_{\{3, 4\}}^{\{5\}}\bigoplus\overline{W}_{\{3, 4\}}^{\{4\}}$, such that users $1, 2$ and $3$ can recover $\overline{W}_{\{1, 2\}}$, while users $4$ and $5$ can recover $\overline{W}_{\{3, 4\}}$. Based on $\mathcal{C}_2$, the server sends $\overline{W}_{\{1, 5\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 5\}}^{\{2\}}$, $\overline{W}_{\{1, 5\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 5\}}^{\{3\}}$, $\overline{W}_{\{2, 3\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 5\}}^{\{4\}}$, $\overline{W}_{\{4, 5\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 5\}}^{\{5\}}$, $\overline{W}_{\{2, 3\}}^{\{2\}}\bigoplus\overline{W}_{\{1, 5\}}^{\{4\}}$, $\overline{W}_{\{2, 5\}}^{\{3\}}\bigoplus\overline{W}_{\{1, 5\}}^{\{4\}}$, $\overline{W}_{\{2, 3\}}^{\{5\}}\bigoplus\overline{W}_{\{4, 5\}}^{\{4\}}$, $\overline{W}_{\{4, 5\}}^{\{2\}}\bigoplus\overline{W}_{\{1, 5\}}^{\{5\}}$, $\overline{W}_{\{4, 5\}}^{\{3\}}\bigoplus\overline{W}_{\{1, 5\}}^{\{5\}}$, such that users $1, 2$ and $3$ can recover $\overline{W}_{\{1, 5\}}$, while user $4$ and user $5$ can recover $\overline{W}_{\{2, 3\}}$ and $\overline{W}_{\{4, 5\}}$, respectively. Based on $\mathbf{C}_3$, the server sends $\overline{W}_{\{1, 3\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 3\}}^{\{2\}}$, $\overline{W}_{\{1, 3\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 3\}}^{\{3\}}$, $\overline{W}_{\{1, 3\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 3\}}^{\{4\}}$, $\overline{W}_{\{2, 4\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 3\}}^{\{5\}}$, $\overline{W}_{\{2, 4\}}^{\{2\}}\bigoplus\overline{W}_{\{1, 3\}}^{\{5\}}$, $\overline{W}_{\{2, 4\}}^{\{3\}}\bigoplus\overline{W}_{\{1, 3\}}^{\{5\}}$, $\overline{W}_{\{2, 4\}}^{\{4\}}\bigoplus\overline{W}_{\{1, 3\}}^{\{5\}}$, based on which users $1$,$2$,$3$ and $4$ can recover $\overline{W}_{\{1, 3\}}$, while user $5$ recovers $\overline{W}_{\{2, 4\}}$. Finally, based on $\mathbf{C}_4$, the server sends $\overline{W}_{\{1, 4\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 4\}}^{\{2\}}$, $\overline{W}_{\{1, 4\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 4\}}^{\{3\}}$, $\overline{W}_{\{1, 4\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 4\}}^{\{5\}}$, $\overline{W}_{\{3, 5\}}^{\{1\}}\bigoplus\overline{W}_{\{1, 4\}}^{\{4\}}$, $\overline{W}_{\{3, 5\}}^{\{2\}}\bigoplus\overline{W}_{\{1, 4\}}^{\{4\}}$, $\overline{W}_{\{3, 5\}}^{\{3\}}\bigoplus\overline{W}_{\{1, 4\}}^{\{4\}}$, $\overline{W}_{\{3, 5\}}^{\{5\}}\bigoplus\overline{W}_{\{1, 4\}}^{\{4\}}$, such that users $1, 2, 3$ and $4$ are able to recover $\overline{W}_{\{1, 4\}}$, while user $4$ recovers $\overline{W}_{\{3, 5\}}$. Thus, all the users are able to decode the $2$-subfiles they requested. The total number of bits delivered in this case is $30F_2/5$.
Next, we consider the case $N>K$. We first select a subset of $K$ files $\mathcal{R}$, $\mathcal{R} \subset [N]$, such that $|\mathcal{R}|=K$, and $d_k \in \mathcal{R}$ for $k=1, ..., K$. For any subset $\overline{\mathcal{R}} \in [N]\setminus \mathcal{R}$, $|\overline{\mathcal{R}}|=s$, $s \in [\max\{l-K, 0\}: \min\{l-1, N-K\}\}]$, Algorithm \[groupingscheme\] is applied to a subset of $l$-subfiles $\mathcal{W}^l=\{\overline{W}_{\mathcal{S}\cup\overline{\mathcal{R}}}: \mathcal{S} \subset \mathcal{R}, |\mathcal{S}|=l-|\overline{\mathcal{R}}|\}$ to derive $\mathbf{C}_j$, $j\in[\binom{|\mathcal{R}|-1}{l-\mathcal{\overline{R}}-1}]$, based on which Algorithm \[deliveryscheme\] is applied to enable each user to decode its demanded $l$-subfiles in $\mathcal{W}^l$. Therefore, each user can decode all the $l$-subfiles it is demanding.
Achievable rate
---------------
The following theorem presents the delivery rate achieved by the proposed coded caching and delivery scheme for any demand combination, for a given cache allocation vector $\mathbf{P}$.
\[theorem:rate\] For the caching system described in Section \[sys\], given a cache allocation vector $\mathbf{P}$, the following delivery rate is achievable $$\label{averagerate}
R_{CACC}(\mathbf{P})=\sum\limits_{l=1}^N R_l(t_l),$$ where $t_l=p_lK$, and for $t_l \in [0: K]$, $$\label{Rleq}
R_l(t_l)= \min\{\alpha_l(t_l), m_l(t_l)\},$$ and $$\label{ratecodeddelivery}
\begin{aligned}
&\alpha_l(t_l) \triangleq \sum\limits_{s=\max\{l-K, 0\}}^{\max\{\min\{l-1, N-K\}, 0\}}\binom{N-K}{s}\binom{\min\{N, K\}-1}{l-s-1}\cdot\\
&\left[\binom{K}{t_l+1}-\binom{\max\{K-\lceil \frac{\min\{N, K\}}{l-s} \rceil -1, 0\}}{t_l+1}\right]\frac{F_l}{F\binom{K}{t_l}},
\end{aligned}$$ $$\begin{aligned}\label{varphi}
m_l(t_l) \triangleq \left(\binom{N}{l}-\binom{\min\{N-K, 0\}}{l}\right) (F_l-t_lF_l/K)/F.
\end{aligned}$$ For $t_l \notin [0: K]$, $R_l(t_l)$ is given by the lower convex envelop of the above achievable points.
We show that $R_l(t_l)$ given above is achievable $t_l \in [0:K]$. The lower convex envelop of these integer points can then be achieved by memory sharing. For the case $N\leq K$, recall that the requested $l$-subfiles are delivered in $\binom{N-1}{l-1}$ steps based on $\mathbf{C}_j$, $j=[\binom{N-1}{l-1}]$, derived by Algorithm \[groupingscheme\]. In each step, the server sends at most $\lceil N/l \rceil+1$ $l$-subfiles. Therefore, for any demand combination, the number of distinct $d^j_k$ based on $\mathbf{C}_j$, i.e., $A_j\leq \lceil N/l \rceil+1$, $n=1, ..., \binom{N-1}{l-1}$. Similar to the delivery scheme proposed in [@yu2017exact], by the CODED DELIVERY procedure of Algorithm \[deliveryscheme\], the server broadcasts binary sums that help at least one user in $\mathcal{U}_j$ based on $\mathbf{C}_j$. The total number of such subsets of $t_1+1$ users that contain at least one user in $\mathcal{U}_j$ is given by $\binom{K}{t_l+1}-\binom{\max\{K-\lceil N/l \rceil -1, 0\}}{t_l+1}$. Hence, given $t_l \in \{1, ..., K-1\}$, the total number of bits sent by CODED DELIVERY procedure of Algorithm \[deliveryscheme\] for the delivery of the $l$-subfiles is bounded by (normalized by $F$): $$\label{rateneqkcodeddelivery}
R_l(t_l)\leq\binom{N-1}{l-1}\left[\binom{K}{t_l+1}-\binom{\max\{K-\lceil N/l \rceil -1, 0\}}{t_l+1}\right]\frac{F_l}{F\binom{K}{t_l}}$$ The right hand side (RHS) of \[rateneqkcodeddelivery\] is equal to for $N\leq K$. Since each user caches $t_lF_l/K$ bits of each $l$-subfile, according to [@MaddahAliDecentralized Appendix A], the number of bits sent by the RANDOM DELIVERY procedure is bounded by $$\label{rateneqkrandomdelivery}
R_l(t_l) \leq \binom{N}{l} (F_l-t_lF_l/K)/F.$$ The RHS equals to for $N\leq K$. Hence, for $t_l \in \{1, ..., K-1\}$, we have proven $R_l(t_l)$ given in is achievable for $N\leq K$.
We then focus on the case where $N> K$. For any $ \max\{l-
K, 0\} \leq s\leq \min\{l-1, N-K\}$, there are a total of $\binom{N-K}{s}$ subsets $\overline{\mathcal{R}}$ such that $\overline{\mathcal{R}} \in [N]\setminus \mathcal{R}$, $|\overline{\mathcal{R}}|=s$. Following the similar analysis for the case where $N \leq K$, given $\mathcal{R}$ containing all the demanded files such that $|\mathcal{R}|=K$, and any $\overline{\mathcal{R}}$ such that $\overline{\mathcal{R}} \in [N]\setminus \mathcal{R}$, $|\overline{\mathcal{R}}|=s$, requested $l$-subfiles in $\mathcal{W}^l=\{\overline{W}_{\mathcal{S}\cup\overline{\mathcal{R}}}: \mathcal{S} \subset \mathcal{R}, |\mathcal{S}|=l-|\overline{\mathcal{R}}|\}$ are sent in $\binom{K-1}{l-s-1}$ step. At each step, there are at most $\lceil \frac{N}{l-s} \rceil+1$ $l$-subfiles to be sent. Therefore, with similar arguments, the total number of bits sent by CODED DELIVERY procedure of Algorithm \[deliveryscheme\] in each step is bounded by $\binom{K-1}{l-s-1}\left(\binom{K}{t_l+1}-\binom{\max\{K-\lceil \frac{N}{l-s} \rceil -1, 0\}}{t_l+1}\right)\frac{F_l}{F\binom{K}{t_l}}$, while the number of bits send by the RANDOM DELIVERY procedure is bounded by $\binom{K}{l-s}(F_l-t_lF_l/K)/F$. By summing over all $\binom{N-K}{s}$ subsets $\overline{\mathcal{R}}$ for each $s \in [\max\{l-K, 0\}: \min\{l-1, N-K\}]$, we have $$\label{rateneqkcodeddelivery1}
\begin{aligned}
&R_l(t_l)\leq\sum\limits_{s=\max\{l-K, 0\}}^{\min\{l-1, N-K\}}\binom{N-K}{s}\binom{ K-1}{l-s-1}\cdot\\
&\left(\binom{K}{t_l+1}-\binom{\max\{K-\lceil \frac{N}{l-s} \rceil -1, 0\}}{t_l+1}\right)\frac{F_l}{F\binom{K}{t_l}},
\end{aligned}$$ and, $$\label{rateneqkrandomdelivery1}
R_l(t_l) \leq \sum\limits_{s=\max\{l-K, 0\}}^{\min\{l-1, N-K\}}\binom{N-K}{s}\binom{K}{l-s} (F_l-t_lF_l/K)/F,$$ by which, we have proven the correctness of for the case $N>K$.
At each step of sending $l$-subfiles in $\mathcal{W}^l=\{\overline{W}_{\mathcal{S}\cup\overline{\mathcal{R}}}: \mathcal{S} \subset \mathcal{R}, |\mathcal{S}|=l-|\overline{\mathcal{R}}|\}$, there are sometimes $\lceil \frac{N}{l-s} \rceil+1$ and sometimes $\lceil \frac{N}{l-s} \rceil$ distinct demands, while when $N$ is a multiple of $l-s$, there are always $\frac{N}{l-s}$ distinct demands ($\mathcal{R}=[N]$, $\overline{\mathcal{R}}=\emptyset$, $s=0$, for the case $N\leq K$). To obtain a closed-form expression for the achievable delivery rate, we simply assume $\lceil \frac{N}{l-s} \rceil+1$ distinct demands at each step. Note that, the more the number of distinct demands at each step, the larger the delivery rate. Therefore $R_{CACC}(\mathbf{P})$ in is an upper bound on the actual achievable delivery rate of CACC.
Allocation of Cache Capacity
----------------------------
We can further optimize the cache content distribution $\mathbf{P}$ by solving:
\[optimization\] $$\begin{aligned}
\min~ & R_{CACC}(\mathbf{P})\label{object1}\\
\mathrm{such~that~} & \sum\limits_{l=1}^N \binom{N}{l}p_lF_l\leq MF,\label{constrain}\end{aligned}$$
where the objective is to minimize the achievable delivery rate under the cache capacity constraint. The problem in can be solved numerically.
\[fig:1\] {width="1.1\linewidth"}
Lower Bound {#s:lower_bound}
===========
In this section, we present a lower bound derived using cut-set arguments.
\[cutset\](Cut-set Bound) For the caching problem described in Section \[sys\], the optimal achievable delivery rate is lower bounded by
$$\begin{aligned}
R^*(M)\geq&\operatorname*{max}\limits_{p\in [1: \min\{N, K\}]} \sum\limits_{s=0}^{N-p\lfloor N/p\rfloor}\sum\limits_{l=1}^{p\lfloor N/p\rfloor} \binom{N-p\lfloor N/p\rfloor}{s}\cdot \nonumber \\
&~~~~~~~ \binom{p\lfloor N/p\rfloor}{l}\frac{F_{l+s}}{\lfloor N/p\rfloor}-\frac{pM}{\lfloor N/p\rfloor}.\end{aligned}$$
The proof will be provided in a longer version of the paper.
Numerical results {#s:numerical}
=================
In this section, we numerically compare the delivery rates of the proposed correlation-aware caching schemes CAUC and CACC with the lower bound and the state-of-the-art coded caching scheme from [@yu2017exact], which does not take the content correlations into account. We refer to the later scheme as the correlation-ignorant coded caching scheme (CICC).
We consider $N=10$ files and $k=10$ users. Each user is equipped with a cache of size $F$ bits, i.e., $M=1$. We denote by $r_l$ the ratio of $l$-subfiles among each file, i.e., $r_l\triangleq\binom{N-1}{l-1}F_l/F$. Note that, we have $\sum_{l=1}^K r_l =1$. In Fig. \[fig:1\], we assume that the files have only pairwise correlations, that is, $r_3=\cdots=r_{10}=0$, and we plot the delivery rate as a function of $r_2$. Meanwhile, in Fig. \[fig:2\], we assume that each file consists of a private part, i.e., $1$-subfile, and a common subfile that is shared by all the files in the library, i.e., $10$-subfile, i.e., $r_2=\cdots=r_{9}=0$. We plot the delivery rate as a function of $r_{10}$.
We observe in both figures that the delivery rate achieved by the correlation-ignorant scheme, CICC, remains the same no matter how high the ratio of common subfiles, while the delivery rates of the correlation-aware schemes, CAUC and CACC, decrease as the ratio of the common subfiles increases. Obviously, CACC achieves a lower delivery rate than both CAUC and CICC, since it benefits both from incorporating the correlations among the files as well as coded multicasting. When the ratio of the common subfiles is sufficiently large, even without coded multicasting CAUC achieves a lower delivery rate than CICC. It can also be observed that the delivery rates of correlation-aware schemes decrease faster with the percentage of common subfiles in Fig. \[fig:2\] than in Fig. \[fig:1\]. That is because the gain from exploiting correlation is more pronounced as the common parts are shared among more files. While there is a gap between the cut-set lower bound and the achievable delivery rate, we note that the gap is smaller in Fig \[fig:2\], where the level of commonness is higher.
\[fig:2\] {width="1.1\linewidth"}
Conclusions
===========
We have studied coded caching taking into account the available correlations among the files in the library. To capture arbitrary correlations, we assume that each file consists of a number of subfiles, each of which is shared by a different subset of files in the library, and the number of files that share a certain subfile is defined as its level of commonness. We proposed both a correlation-aware uncoded caching scheme, the optimal placement of which is proven to be caching the subfiles with the highest levels of commonness, and a correlation-aware coded caching scheme (CACC), the placement of which is optimized in terms of the achievable delivery rate. The proposed CACC scheme, or even the uncoded caching scheme when the correlation among files is strong enough, is shown to significantly outperform the best known achievable delivery rate by correlation-unaware solution in the literature.
|
---
author:
- |
Samuel Herrmann and Pierre Vallois\
Institut de Mathématiques Elie Cartan - UMR 7502\
Nancy-Université, CNRS, INRIA\
B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex, France\
{herrmann,vallois}@iecn.u-nancy.fr
bibliography:
- 'biblio.bib'
title:
- From persistent random walk to the telegraph noise
- From persistent random walks to the telegraph noise
---
The setting of persistent random walks. {#section_not}
=======================================
1\) The simplest way to present and define a persistent random walk with value in $\Zset$ is to introduce the process of its increments $(Y_t, \ t\in\Nset)$. In the classical symmetric random walk case, this process is just a sequence of independent random variables satisfying $\P(Y_t=1)=\P(Y_t=-1)=\frac{1}{2}$ for any $t\ge 0$. Here we shall introduce some short range memory in these increments in order to create the persistence phenomenon. Namely $(Y_t)$ is a $\{-1,1\}$-valued Markov chain: the law of $Y_{t+1}$ given $\mathcal{F}_t=\sigma(Y_0, Y_1,\ldots, Y_t)$ depends only on the value of $Y_t$. This dependence is represented by the transition probability $\pi(x,y)=\P(Y_{t+1}=y\vert Y_{t}=x)$ with $(x,y)\in\{-1,1\}^2$: $$\begin{aligned}
\pi=\left(\begin{array}{cc} 1-\alpha & \alpha\\
\beta & 1-\beta\end{array}\right)\quad\quad 0<\alpha<1,\quad
0<\beta<1.\end{aligned}$$ The persistent random walk is the corresponding process of partial sums: $$\label{25*1} X_t=\sum_{i=0}^t Y_i\quad\mbox{with }\quad X_0=Y_0=1\
\mbox{or }\ -1.$$ Let us discuss two particular cases:
- If $\alpha+\beta=1$, then increments are independent and therefore the short range memory disappears. $(X_t,\ t\in\Nset)$ is a classical Bernoulli random walk.
- The symmetric case $\alpha=\beta$ was historically suggested by Fürth [@furthbook20] and precisely defined by Taylor [@taylor]. Goldstein [@goldstein50] developed the calculation of the random walk law and clarified the link between this process and the so-called telegraph equation. Some nice presentation of these results can be found in Weiss’ book [@weissbook94] and [@weiss02]. This particular short memory process is often called either *persistent or correlated random walk* or *Kac walks* (see, for instance, [@eckstein00]). An interesting presentation of different limiting distributions for this correlated random walk has been given by Renshaw and Henderson [@renshaw81].
2\) Recently, Vallois and Tapiero [@vallois07] studied the influence of the persistence phenomenon on the first and second moments of a counting process whose increments takes their values in $\{0,1\}$ instead of $\{-1,1\}$. They obtained some nearly linear behaviour for the expectation. Using the transformation $y\to 2y-1$, it is easy to deduce that, in our setting, we have: $$\label{11b}
\E_{-1}[X_t]:=\E[X_t\vert
X_0=Y_0=-1]=\frac{\alpha-\beta}{1-\rho}\,(t+1)-\frac{2\alpha}{(1-\rho)^2}\,
(1-\rho^{t+1}).$$ $$\label{11c}
\E_{+1}[X_t]:=\E[X_t\vert
X_0=Y_0=+1]=\frac{\alpha-\beta}{1-\rho}\,(t+1)-\frac{2\beta}{(1-\rho)^2}\,
(1-\rho^{t+1}).$$ An application to insurance has been given in [@vallois08].\
It is actually possible to determine the moment generating function (see Proposition \[a+a-1\] in Section \[section preuve2\]). $$\Phi(\lambda,t)=\E[\lambda^{X_t}],\quad(\lambda\in\Rset_+^*).$$ However it seems difficult to invert this transformation; i.e. to give the law of $X_t$.\
3) This leads us to investigate limit distributions. It is well-known that the correctly normalized symmetric random walk converges towards the Brownian motion. Let us define the time and space normalizations. Let $\alpha_0$ and $\beta_0$ denote two real numbers satisfying: $$\label{*1**}
0\le\alpha_0\le 1,\quad 0\le\beta_0\le 1.$$ Let $\Delta_x$ be a positive small parameter so that: $$\label{*2**}
0\le \alpha_0+c_0\Delta_x\le 1,\quad 0\le \beta_0+c_1\Delta_x\le 1,$$ where $c_0$ and $c_1$ belong to $\Rset$ (see in subsection \[\*\*\*\] the allowed range of parameters).\
Let $(Y_t,\, t\in\Nset)$ be a Markov chain whose transition probabilities are given by the matrix: $$\begin{aligned}
\label{1-4B}
\pi^\Delta=\left(\begin{array}{cc}
1-\alpha_0-c_0\Delta_x & \alpha_0+c_0\Delta_x\\
\beta_0+c_1\Delta_x & 1-\beta_0-c_1\Delta_x
\end{array}\right).\end{aligned}$$ Let $(X_t,\ t\in\Nset)$ be the random walk associated with $(Y_t)$ (cf. ). Define the normalized random walk $(Z^\Delta_s, \, s\in\Delta_t\Nset)$ by the relation: $$\label{scal}
Z^\Delta_s=\Delta_x X_{s/\Delta_t}, \quad (\Delta_t>0, \ \Delta_x>0).$$ Set $(\tilde{Z}^\Delta_s,\, s\ge 0)$ the continuous time process obtained by linear interpolation of $(Z_s^\Delta)$.\
We introduce two essential parameters: $$\label{double1}
\rho_0=1-\alpha_0-\beta_0\quad\mbox{(the asymmetry coefficient),}$$ $$\label{double2}
\eta_0=\beta_0-\alpha_0.$$ In this paper, we will aim at showing the existence of a normalization (i.e. to express $\Delta_t$ in terms of $\Delta_x$) which depends on $\alpha_0$, $\beta_0$, so that $(\tilde{Z}^\Delta_s)$ converges in distribution, as $\Delta_x\to 0$.\
Our main results and the organization of the paper will be given in Section \[results\].
The main results {#results}
================
Case : $\rho_0=1$ {#section2.1}
-----------------
Obviously $\rho_0=1$ implies that $\alpha_0=\beta_0=0$, and the transition probabilities matrix is given as $$\begin{aligned}
\pi^\Delta=\left(\begin{array}{cc}
1-c_0\Delta_x & c_0\Delta_x\\
c_1\Delta_x & 1-c_1\Delta_x
\end{array}\right)\quad (c_0,c_1>0).\end{aligned}$$ In order to describe the limiting process, we introduce a sequence of independent identically exponentially distributed random variables $(e_n,n\ge 1)$ with $\E[e_n]=1$. We construct the following counting process: $$\label{**1.1}
N_t^{c_0,c_1}=\sum_{k\ge 1}1_{\{\lambda_1 e_1+\lambda_2 e_2+\ldots+\lambda_k e_k\le t\}},$$ where $$\begin{aligned}
\label{.*.}
\lambda_k=\left\{\begin{array}{l}
1/c_0\quad \mbox{if}\ k\ \mbox{is odd}\\
1/c_1\quad \mbox{otherwise.}
\end{array}\right.\end{aligned}$$ Finally we define $$\label{def_1.2}
{Z^{c_0,c_1}}_t=\int_0^t (-1)^{{N^{c_0,c_1}}_u}du.$$ For simplicity of notations, in the symmetric case (i.e. $c_0=c_1$), $N^{c_0}_t$ (resp. $Z^{c_0}_t$) will stand for $N^{c_0,c_0}_t$ (resp. $Z^{c_0,c_0}_t$). The process $(Z_t^{c_0})$ has been introduced by Stroock (in [@stroock] p. 37). It is possible to show that if we rescale $(Z_t^{c_0})$, this process converges in distribution to the standard Brownian motion. This property has been widely generalized. For instance Bardina and Jolis [@bardina00] have given weak approximation of the Brownian sheet from a Poisson process in the plane.
\[cas1\] Let $\Delta_x=\Delta_t$ and $Y_0=X_0=-1$. Then the interpolated persistent random walk $(\tilde{Z}^\Delta_s,\ s\ge
0)$ converges in distribution, as $\Delta_x\to 0$, to the process $(-{Z^{c_0,c_1}}_s,\ s\ge 0)$.\
In particular if $c_0=c_1$, then $(N_u^{c_0})$ is the Poisson process with parameter $c_0$.\
If $Y_0=X_0=1$ then the interpolated persistent random walk $(\tilde{Z}^\Delta_s,\ s\ge
0)$ converges in distribution, as $\Delta_x\to 0$, to the process $({Z^{c_1,c_0}}_s,\ s\ge 0)$.
See Section \[sectionpreuve1\].
Next, in Section \[properties\_\*\], we investigate the process $({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_t;\ t\ge 0)$. In particular we prove that it is Markov, we determine its semigroup and the law of $({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_t)$, $t$ being fixed. This permits to prove, when $c_0=c_1$, the well-known relation (cf. [@weiss02], [@eckstein00], [@goldstein50], [@griego71]) between the solutions of the wave equation and the telegraph equation. For this reason the process $({Z^{c_0,c_1}}_t)$ will be called the integrated telegraph noise (ITN for short).\
We emphasize that our approach based on stochastic processes gives a better understanding of analytical properties.\
We will give in Section \[extensions\] below two extensions of Theorem \[cas1\] to the cases where $(Y_t)$ is\
1) a Markov chain which takes its values in $\{y_1,\ldots,y_k\}$,\
2) a Markov chain with order $2$ and valued in $\{-1,1\}$.
Case : $\rho_0\neq1$ {#section2.2}
--------------------
In this case, the limit process is Markov. We shall prove two kind of convergence results. The first one corresponds to the law of large numbers and the second one looks like functional central limit theorem.\
Recall that $(\tilde{Z}^\Delta_t,\, t\ge 0)$ is the linear interpolation of $(Z_t^\Delta)$ and $\rho_0$ (resp. $\eta_0$) has been defined by (resp. ).
\[conv1\] 1) Suppose that $r\Delta_t=\Delta_x$ with $r>0$. Then $\tilde{Z}^\Delta_t$ converges to the deterministic limit $-\frac{rt\eta_0}{1-\rho_0}$ when $\Delta_x\to 0$.\
2) Suppose that $r\Delta_t=\Delta_x^2$ with $r>0$, then the process $(\xi^\Delta_t,\ t\ge 0)$ defined by $$\xi^{\Delta}_t=\tilde{Z}^\Delta_t+\frac{t\sqrt{r}\eta_0}{(1-\rho_0)\sqrt{\Delta_t}}$$ converges in distribution to the process $(\xi^0_t,\ t\ge 0)$, as $\Delta_x\to 0$, where $$\label{juil3}
\xi^0_t=2r\Big(\frac{-\overline{\tau}}{1-\rho_0}+\frac{\eta_0
\tau}{(1-\rho_0)^2}\Big)t+\sqrt{\frac{r(1+\rho_0)}{1-\rho_0}\Big(1-
\frac{\eta_0^2}{(1-\rho_0)^2}\Big)}W_t,$$ ($W_t$, $t\ge 0$) is a one-dimensional Brownian motion, $
\tau=(c_0+c_1)/2$ and $\overline{\tau}=(c_1-c_0)/2$.
See Section \[section preuve2\].
Gruber and Schweizer have proved in [@gruber06] a weak convergence result for a large class of generalized correlated random walks. However these results and ours can be only compared in the case $\alpha_0=\beta_0$.\
Note that $$1-\frac{\eta_0^2}{(1-\rho_0)^2}=0\Longleftrightarrow \alpha_0=0\quad\mbox{or}\quad\beta_0=0.$$ Suppose for instance that $\alpha_0=0$. Then $\beta_0,c_0>0$ and $$\xi^{\Delta}_t=\tilde{Z}^{\Delta}_t+\frac{t\sqrt{r}}{\sqrt{\Delta_t}}\quad\mbox{and}\quad \xi^0_t=\frac{2rc_0}{\beta_0}\, t.$$ Obviously, the diffusion coefficient of $(\xi^0_t)$ can also cancel when $\rho_0=-1$.\
Since $\rho_0=-1\Longleftrightarrow\alpha_0=\beta_0=1$, then $c_0,c_1<0$ and $$\xi^{\Delta}_t=\tilde{Z}^{\Delta}_t\quad\mbox{and}\ \xi^0_t=-r\overline{\tau}t.$$ This shows that, in the symmetric case (i.e. $c_0=c_1$), we have $\xi^0_t=0$. This means that the normalization is not the right one since the limit is null. Changing the rescaling we can obtain a non-trivial limit.
\[cas\_limit\] Suppose $\alpha_0=\beta_0=1$, $c_0=c_1<0$ and $r\Delta_t=\Delta_x^3$ with $r>0$.\
The interpolated persistent walk $(\tilde{Z}^\Delta_t,\ t\ge 0)$ converges in law, as $\Delta_x\to 0$, to $(\sqrt{-r c_0}W_t,\, t\ge 0)$ where $(W_t)$ is a standard Brownian motion.
See subsection \[scas\_limit\]
Organization of the paper
-------------------------
The third section presents few properties of the process $({Z^{c_0,c_1}}_t,\, t\ge 0)$ which has been defined by . Theorem \[cas1\] will be proven in Section \[sectionpreuve1\]. Section \[extensions\] will be devoted to two extensions of Theorem \[cas1\]. In subsection \[mgf\] we determine the generating function of $X_t$ (recall that $X_t$ has been defined by ). This is the main tool which permits to prove Theorem \[conv1\] and Proposition \[cas\_limit\] (see subsections \[\*\*\*\] and \[scas\_limit\]).
Properties of the integrated telegraph noise {#properties_*}
============================================
The aim of this section is to study the two dimensional process $({Z^{c_0,c_1}}_t,\
{N^{c_0,c_1}}_t;\ t\ge 0)$ introduced in and . In the particular symmetric case $c_0=c_1$, the study is simpler since the process $(N^{c_0}_t,\ t\ge0)$ is a Poisson process with rate $c_0$ ($\E(N^{c_0}_t)=c_0 t$) and $N_0^{c_0}=0$. However we shall study the general case.\
First, we determine in Proposition \[loi\_\_1\] the conditional density of ${Z^{c_0,c_1}}_t$ given ${N^{c_0,c_1}}_t=n$. As a by product we obtain the distribution of ${Z^{c_0,c_1}}_t$ (see Proposition \[bessel\_\_\*\]). Second, we prove in Proposition \[markk\] that $({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_t,\, t\ge 0)$ is Markov and we determine its semi-group. We conclude this section by showing that the solution of the telegraph equation can be expressed in terms of the associated wave equation and $(Z_t^{c_0,c_0})_{t\ge 0}$. For this reason, $({Z^{c_0,c_1}}_t)_{t\ge 0}$ will be called the integrated telegraph noise (ITN for short). Recall that: $$\label{constantes_c}
\tau=\frac{c_0+c_1}{2},\quad\quad \overline{\tau}=\frac{c_1-c_0}{2}.$$
\[loi\_\_1\] 1) $\P({N^{c_0,c_1}}_t=0)=e^{-tc_0}$ and given ${N^{c_0,c_1}}_t=0$, we have ${Z^{c_0,c_1}}_t=t$.\
2) The counting process takes even values with probability: $$\label{pair__*}
\P({N^{c_0,c_1}}_t=2k)=\frac{(c_0c_1)^k\alpha_k(t)}{2^{2k}k!(k-1)!}\, e^{-\tau t}\quad\mbox{with}\
\alpha_k(t)=\int_{-t}^t (t-z)^{k-1}(t+z)^{k} e^{\overline{\tau}z}dz,$$ and the conditional distribution of ${Z^{c_0,c_1}}_t$ is given by $$\label{cond_pair}
\P({Z^{c_0,c_1}}_t\in dz\vert {N^{c_0,c_1}}_t=2k)=\frac{1}{\alpha_k(t)}(t-z)^{k-1}(t+z)^k e^{\overline{\tau}z}1_{[-t,t]}(z)\quad (k\ge 1).$$ 3) The counting process takes odd values with probability: $$\label{impair__*}
\P({N^{c_0,c_1}}_t=2k+1)=\frac{c_0^{k+1}c_1^k\tilde{\alpha}_k(t)}{2^{2k+1}(k!)^2}\, e^{-\tau t}\quad\mbox{with}\
\tilde{\alpha}_k(t)=\int_{-t}^t (t-z)^{k}(t+z)^{k} e^{\overline{\tau}z}dz,$$ and the conditional distribution of ${Z^{c_0,c_1}}_t$ is given by $$\label{cond_impair}
\P({Z^{c_0,c_1}}_t\in dz\vert {N^{c_0,c_1}}_t=2k+1)=\frac{1}{\tilde{\alpha}_k(t)}(t-z)^{k}(t+z)^k e^{\overline{\tau}z}1_{[-t,t]}(z)\quad (k\ge 0).$$
In the particular symmetric case $c_0=c_1$, the conditional density function of $Z^{c_0}_t$ given $N^{c_0}_t=n$ is the centered beta density, i.e. $$\label{eq:loi cond1_*}
\mbox{for}\ n=2k,\ k\in\mathbb{N^*}:\quad
f_n(t,z)=\chi_{2k}\frac{(t+z)^k(t-z)^{k-1}}{t^{2k}}1_{[-t,t
]}(z),$$ $$\label{eq:loi cond2_*}
\mbox{for}\ n=2k+1,\ k\in\mathbb{N}:\quad
f_n(t,z)=\chi_{2k+1}\frac{(t+z)^k(t-z)^{k}}{t^{2k+1}}1_{[-t, t
]}(z),$$ with $$\begin{aligned}
\chi_{2k+1}=\chi_{2k+2}=\frac{1}{2^{2k+1}B(k+1,k+1)}=\frac{(2k+1)!}{2^{2k+1}(k!)^2}\quad
(k\ge 0),\end{aligned}$$ ($B$ is the beta function (first Euler function): $B(r,s)=\frac{\Gamma(r)\Gamma(s)}{\Gamma(r+s)}$).
[*Proof of Proposition \[loi\_\_1\]*]{}. Associated with $n\ge 0$ and a bounded continuous function $f$, we define $$\Delta_n(f)=\E\Big[f({Z^{c_0,c_1}}_t)1_{\{{N^{c_0,c_1}}_t=n\}}\Big].$$ [**a)**]{} When $n=0$, we obtain $$\Delta_0(f)=\E\Big[ f({Z^{c_0,c_1}}_t)1_{\{t<\lambda_1 e_1\}} \Big].$$ If $t<\lambda_1 e_1$, then ${Z^{c_0,c_1}}_t=t$ and $$\Delta_0(f)=f(t)\P(t<\lambda_1 e_1)=f(t)e^{-tc_0}.$$ [**b)**]{} When $n\ge 1$, using we obtain $$\Delta_n(f)=\E\Big[f({Z^{c_0,c_1}}_t)1_{\{\lambda_1 e_1+\ldots+\lambda_n e_n \le t< \lambda_1 e_1+\ldots +\lambda_{n+1}e_{n+1}\}}\Big].$$ If $\lambda_1 e_1+\ldots+\lambda_n e_n \le t< \lambda_1 e_1+\ldots +\lambda_{n+1}e_{n+1}$ then $$\begin{aligned}
{Z^{c_0,c_1}}_t&=&\int_0^{\lambda_1 e_1}(-1)^0du+\int_{\lambda_1 e_1}^{\lambda_1 e_1+\lambda_2 e_2}(-1) du+\ldots+\int_{\lambda_1 e_1+\ldots+\lambda_{n-1}e_{n-1}}^{\lambda_1 e_1+\ldots+\lambda_{n}e_{n}}(-1)^{n-1}du\\
&+&\int_{\lambda_1 e_1+\ldots+\lambda_{n}e_{n}}^t (-1)^n du.\end{aligned}$$ Hence $$\label{resum__*}
{Z^{c_0,c_1}}_t=\lambda_1 e_1-\lambda_2e_2+\lambda_3e_3+\ldots +(-1)^{n-1}\lambda_n e_n +(-1)^n(t-\lambda_1 e_1-\ldots-\lambda_n e_n).$$ [**c)**]{} Evaluation of $\Delta_{2k}(f)$, $k\ge 1$.\
We introduce two sequences of random variables associated with $(e_n)$: $$\label{pair-impair}
\xi_k^e=e_2+\ldots+e_{2k},\quad \xi^o_k=e_1+\ldots+e_{2k-1},\quad (k\ge 1).$$ By , and we obtain the simpler expression $$\Delta_{2k}(f)=\E\Big[ f(t-2\xi_k^e/c_1)1_{\{ \xi^o_k/c_0+\xi_k^e/c_1\le t< \xi^o_k/c_0+\xi_k^e/c_1+ e_{2k+1}/c_0\}} \Big].$$ Note that from our assumptions, $\xi^e_k$, $\xi^o_k$ and $e_{2k+1}$ are independent r.v.’s, $\xi^o_k$ and $\xi^e_k$ are both gamma distributed with parameter $k$. Consequently: $$\begin{aligned}
\Delta_{2k}(f)&=&\frac{1}{((k-1)!)^2}\int_{D_t }\exp\{-c_0(t-y/c_0-x/c_1)\}
f(t-2 x/c_1)e^{-x-y}x^{k-1}y^{k-1}dx\,dy\\
&=&\frac{c_0^ke^{-tc_0}}{k!(k-1)!}\int_{0}^{tc_1}f(t-2 x/c_1) x^{k-1}(t-x/c_1)^k
\exp\Big\{ \Big( \frac{c_0}{c_1}-1 \Big)x \Big\}dx,\end{aligned}$$ where $D_t=\Rset_+^2\cap\{y/c_0+ x/c_1\le t \}$. Using the change of variable $z=t-2x/c_1$, we obtain $x=c_1\frac{t-z}{2}$, $t-x/c_1=\frac{t+z}{2}$ and $$\label{fin_pair__*}
\Delta_{2k}(f)=\frac{(c_0c_1)^k}{2}\frac{e^{-(c_0+c_1)t/2}}{k!(k-1)!}
\int_{-t}^tf(z)\Big(\ \frac{t-z}{2} \Big)^{k-1}\Big( \frac{t+z}{2} \Big)^k
\exp\{ ( c_1-c_0 )z/2\}dz.$$ Finally and imply and .\
[**d)**]{} Evaluation of $\Delta_{2k+1}(f)$ for $k\ge 0$. The arguments are similar to those presented in part c). On the event $\xi^o_{k+1}/c_0+\xi_k^e/c_1\le t<\xi^o_{k+1}/c_0+\xi_k^e/c_1+ e_{2k+2}/c_1$, we have: ${Z^{c_0,c_1}}_t=2\xi^o_{k+1}/c_0-t$; this implies $$\Delta_{2k+1}(f)=\E\Big[ 1_{\{ \xi_{k+1}^o/c_0+\xi_k^e/c_1\le t \}}
\exp\Big( -c_1(t-\xi_{k+1}^o/c_0-\xi_k^e/c_1) \Big) f(2\xi_{k+1}^o/c_0-t) \Big].$$ Since $\xi_{k+1}^o$ and $\xi_k^e$ are independent and gamma distributed with parameter $k+1$ (resp. $k$), we get $$\label{fin_impair__*}
\Delta_{2k+1}(f)=\frac{c_0^{k+1}c_1^k}{2(k!)^2}e^{-(c_0+c_1)t/2}\int_{-t}^tf(z)
\Big(\ \frac{t-z}{2} \Big)^{k}\Big( \frac{t+z}{2} \Big)^k\exp\Big\{ ( c_1-c_0 )z/2 \Big\}dz.$$ This leads directly to and .\
Let us recall the definition of the modified Bessel functions: $$I_\nu(\xi)=\sum_{m\ge0}\frac{(\xi/2)^{\nu+2m}}{m!\Gamma(\nu+m+1)}.$$
\[bessel\_\_\*\] The distribution of ${Z^{c_0,c_1}}_t$ is given by $$\label{distrib__*}
\P({Z^{c_0,c_1}}_t\in dx)=e^{-c_0t}\delta_t(dx)+e^{-\tau t} f(t,x) 1_{[-t,t]}(x),$$ where $$\label{express__*}
f(t,x)=\frac{1}{2}\Big[ \sqrt{\frac{c_0c_1(t+x)}{t-x}}
I_1\Big( \sqrt{c_0c_1(t^2-x^2)} \Big) +c_0I_0\Big(\sqrt{c_0c_1(t^2-x^2)} \Big)
\Big]e^{\overline{\tau}x}.$$
Let us focus our attention to the symmetric case $c_0=c_1$. We can introduce some randomization of the initial condition as follows: let $\epsilon$ be a $\{-1,1\}$-valued random variable, independent from the Poisson process $N^{c_0}_t$, with $p:=\P(\epsilon=1)=1-\P(\epsilon=-1)$. It is easy to deduce from that we have $$\label{der}
\P(\epsilon Z_t^{c_0}/t\in dx)=\Big(p\delta_1(dx)+(1-p)\delta_{-1}(dx)+g(t,x)dx\Big)e^{-c_0 t},$$ with $$g(t,x)=\frac{c_0 t}{2}\Big\{I_0\Big(c_0
t\sqrt{1-x^2}\Big)+\frac{1+(2p-1)x}{\sqrt{1-x^2}}I_1\Big(c_0
t\sqrt{1-x^2}\Big)\Big\}1_{[-1,1]}(x)$$ and $\delta_1(dx)$ (resp. $\delta_{-1}(dx)$) is the Dirac measure at $1$ (resp. $-1$).\
In the particular case $p=1/2$, $x\to g(t,x)$ is an even function. G.H. Weiss ([@weiss02] p.393) proved using an analytic method based on Fourier-Laplace transform.
[*Proof of Proposition \[bessel\_\_\*\].*]{} The proof is a direct consequence of the expression of Proposition \[loi\_\_1\]. Indeed, for each bounded continuous function $\varphi$ we denote $$\Delta=\E[\varphi({Z^{c_0,c_1}}_t)]=\varphi(t)e^{-c_0 t}+\sum_{k\ge 1}\Delta_{2k}(\varphi)
+\sum_{k\ge 0}\Delta_{2k+1}(\varphi)=\varphi(t)e^{-c_0 t}+\Delta_e+\Delta_o,$$ where $\Delta_n(\varphi)=\E[\varphi({Z^{c_0,c_1}}_t)1_{\{ {N^{c_0,c_1}}_t=n \}}]$. Using and we get $$\Delta_e=e^{-\tau t}\int_{-t}^t \varphi(z)S_e(z)e^{\overline{\tau}z}dz,$$ with $$\begin{aligned}
S_e(z)&=&\frac{1}{2}\sum_{k\ge 1}\frac{(c_0c_1)^k}{k!(k-1)!}
\Big( \frac{t-z}{2} \Big)^{k-1}\Big( \frac{t+z}{2} \Big)^{k}\\
&=&\frac{1}{2}\sqrt{c_0c_1}\sqrt{\frac{t+z}{t-z}}\sum_{k\ge 0}
\frac{1}{k!(k+1)!}\Big( \frac{\sqrt{c_0c_1(t^2-z^2)}}{2} \Big)^{2k+1}\\
&=&\frac{1}{2}\sqrt{c_0c_1}\sqrt{\frac{t+z}{t-z}}I_1\Big( \sqrt{c_0c_1(t^2-z^2)} \Big).\end{aligned}$$ For the odd indexes, by and we get $$\Delta_o=e^{-\tau t}\int_{-t}^t \varphi(z)S_o(z)e^{\overline{\tau}z}dz,$$ with $$S_o(z)=\frac{1}{2}\sum_{k\ge 0}\frac{c_0^{k+1}c_1^k}{(k!)^2}\Big( \frac{t^2-z^2}{4} \Big)^{k}
=\frac{c_0}{2}I_0\Big( \sqrt{c_0c_1(t^2-z^2)} \Big).$$
\[markk\] 1) $({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_t;\ t\ge 0)$ is a $\Rset\times\Nset$-valued Markov process.\
2) Let $s\ge 0$ and $n\ge 0$. Conditionally on ${Z^{c_0,c_1}}_s=x$ and ${N^{c_0,c_1}}_s=n$, $\Big(({Z^{c_0,c_1}}_{t+s},{N^{c_0,c_1}}_{t+s}), \, t\ge 0\Big)$ is distributed as $$\begin{aligned}
\left\{\begin{array}{ll}
\Big(\Big( x+\int_0^{t}(-1)^{{N^{c_0,c_1}}_u}du, n+{N^{c_0,c_1}}_{t} \Big),\ t\ge 0\Big)&\mbox{when $n$ is even},\\[8pt]
\Big(\Big( x-\int_0^{t}(-1)^{{N^{c_1,c_0}}_u}du, n+{N^{c_1,c_0}}_{t} \Big),\ t\ge 0\Big)&\mbox{otherwise}.
\end{array}\right.\end{aligned}$$
Note that Propositions \[markk\] and \[loi\_\_1\] permit to determine the semigroup of $\Big(({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_t),\, t\ge 0\Big)$ i.e. $\P({Z^{c_0,c_1}}_t\in dx,\, {N^{c_0,c_1}}_t=n\vert {Z^{c_0,c_1}}_s=y,\,{N^{c_0,c_1}}_s=m)$ where $t>s$, $n\ge m$ and $y\in[-s,s]$.
[*Proof of Proposition \[markk\]*]{}. Let $t>s\ge 0$. Using we get $${Z^{c_0,c_1}}_t={Z^{c_0,c_1}}_s+(-1)^{{N^{c_0,c_1}}_s}\int_0^{t-s}(-1)^{\tilde{N}^s_u}du,$$ where $\tilde{N}^s_u={N^{c_0,c_1}}_{s+u}-{N^{c_0,c_1}}_s$, $u\ge 0$.\
Note that $(\tilde{N}_u^{s};\ u\ge 0)\overset{(d)}{=}({N^{c_0,c_1}}_u;\ u\ge 0)$ if ${N^{c_0,c_1}}_s\in 2\Nset$ and $(\tilde{N}_u^{s};\ u\ge 0)\overset{(d)}{=}({N^{c_1,c_0}}_u;\ u\ge 0)$ if ${N^{c_0,c_1}}_s\in 2\Nset+1$. This shows Proposition \[markk\].\
Next, we determine (in Proposition \[trlp\] below) the Laplace transform of the r.v. ${Z^{c_0,c_1}}_t$. It is possible to use the distribution of ${Z^{c_0,c_1}}_t$ (cf Proposition \[bessel\_\_\*\]), but this method has the disadvantage of leading to heavy calculations. We develop here a method which uses the fact that $({Z^{c_0,c_1}}_s;\ s\ge 0)$ is a stochastic process given by . The key tool is Lemma \[essai\_mark\] below. Roughly speaking Lemma \[essai\_mark\] gives the generator of the Markov process $({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_t)$. Lemma \[essai\_mark\] is an important ingredient in the proof of Proposition \[telegraph\*\*\] besides.
\[essai\_mark\] Let $F:\Rset\times\Nset\to\Rset$ denote a bounded and continuous function such that $z\to F(z,n)$ is of class $\mathcal{C}^1$ for all $n$. Then $$\begin{aligned}
\label{eq:essai_mark} \frac{d}{dt}\E[ F({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_t) ]&=\E\Big[&
\frac{\partial F}{\partial
z}({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_t)(-1)^{{N^{c_0,c_1}}_t}\Big]\nonumber \\
&+\E\Big[&\Big( F({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_t+1)-F({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_t) \Big)\nonumber\\
&&\times\Big( c_11_{\{ {N^{c_0,c_1}}_t\in 2\Nset+1 \}}+c_01_{\{ {N^{c_0,c_1}}_t\in
2\Nset \}} \Big) \Big].\end{aligned}$$
Let us denote by $\Delta(t)=\E[F({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_t)]$. In order to compute the $t$-derivative we shall decompose the increment of $t\to\Delta(t)$ in a sum of two terms: $$\frac{\Delta(t+h)-\Delta(t)}{h}=B_h+C_h,$$ with $$B_h=\frac{1}{h}\Big\{
\E[F({Z^{c_0,c_1}}_{t+h},{N^{c_0,c_1}}_{t+h})]-\E[F({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_{t+h})] \Big\},$$ $$C_h=\frac{1}{h}\Big\{ \E[F({Z^{c_0,c_1}}_{t},{N^{c_0,c_1}}_{t+h})]-\E[F({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_{t})]
\Big\}.$$ Since $F(\cdot,n)$ is continuously differentiable with respect to the variable $z$ and $t\to{Z^{c_0,c_1}}_t$ is differentiable (cf ), using the change of variable formula we obtain $$\frac{1}{h}\Big\{ F({Z^{c_0,c_1}}_{t+h},{N^{c_0,c_1}}_{t+h})-F({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_{t+h})
\Big\}=\frac{1}{h}\int_t^{t+h}\frac{\partial F}{\partial
z}({Z^{c_0,c_1}}_u,{N^{c_0,c_1}}_{t+h})(-1)^{{N^{c_0,c_1}}_u}du.$$ Therefore $$\label{..1} \lim_{h\to 0}B_h=\E\Big[ \frac{\partial F}{\partial
z}({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_t)(-1)^{{N^{c_0,c_1}}_t} \Big].$$ In order to study the limit of $C_h$, we consider two cases: ${N^{c_0,c_1}}_t\in 2\Nset$ and ${N^{c_0,c_1}}_t\in
2\Nset +1$: $$\begin{aligned}
C_h&=&\frac{1}{h}
\E\Big[\Big(F({Z^{c_0,c_1}}_{t},{N^{c_0,c_1}}_{t}+\tilde{N}_h^{c_1,c_0})-F({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_{t})\Big)1_{\{
{N^{c_0,c_1}}_t\in 2\Nset +1
\}}\Big]\\
&+&\frac{1}{h}
\E\Big[\Big(F({Z^{c_0,c_1}}_{t},{N^{c_0,c_1}}_{t}+\tilde{N}_h^{c_0,c_1})-F({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_{t})\Big)1_{\{
{N^{c_0,c_1}}_t\in 2\Nset \}}\Big],\end{aligned}$$ where $\tilde{N}_h={N^{c_0,c_1}}_{t+h}-{N^{c_0,c_1}}_t$.\
According to Proposition \[markk\], conditionally on ${Z^{c_0,c_1}}_t$ and ${N^{c_0,c_1}}_t\in 2\Nset$ (resp. ${N^{c_0,c_1}}_t\in 2\Nset+1$), $\tilde{N}_h$ is distributed as ${N^{c_0,c_1}}_h$ (resp. ${N^{c_1,c_0}}_h$). Note that Proposition \[loi\_\_1\] implies that $\P({N^{c_0,c_1}}_h\ge 2)=o(h)$ and $$\P(N_h^{c_0,c_1}=1)=\frac{c_0}{2}\Big(\frac{e^{\overline{\tau} h}-e^{-\overline{\tau} h}}{\overline{\tau}}\Big)e^{-\tau h}=c_0 h+o(h).$$ Consequently $$\begin{aligned}
\label{..2} \lim_{h\to 0}
C_h&=&c_1\E\Big[\Big(F({Z^{c_0,c_1}}_{t},{N^{c_0,c_1}}_{t}+1)-F({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_{t})\Big)1_{\{ {N^{c_0,c_1}}_t\in
2\Nset +1 \}}\Big]\nonumber\\
&+&c_0\E\Big[\Big(F({Z^{c_0,c_1}}_{t},{N^{c_0,c_1}}_{t}+1)-F({Z^{c_0,c_1}}_t,{N^{c_0,c_1}}_{t})\Big)1_{\{ {N^{c_0,c_1}}_t\in 2\Nset
\}}\Big].\end{aligned}$$ Then, and clearly imply Lemma \[essai\_mark\].
Let us introduce the two quantities: $$\label{lapp}
L_e(t)=\E\Big[ e^{-\mu {Z^{c_0,c_1}}_t}1_{\{ {N^{c_0,c_1}}_t\in 2\Nset \} }
\Big]\ \mbox{and}\ L_o(t)=\E\Big[ e^{-\mu {Z^{c_0,c_1}}_t}1_{\{ {N^{c_0,c_1}}_t\in
2\Nset+1 \} } \Big],\ (t\ge 0,\mu\in\Rset).$$ Since $\vert {Z^{c_0,c_1}}_t\vert\le t$, then $L_e(t)$ and $L_o(t)$ are well defined for any $\mu\in\Rset$. Note that $\mu\to L_e(t)$ (resp. $\mu\to L_o(t)$) is a Laplace transform. We have mentioned the $t$-dependency only because it will play an important role in our proof of Proposition \[trlp\] below.
\[trlp\] Let $L_e(t)$ and $L_o(t)$ be defined by . Then $$\label{nn*} L_e(t)=\frac{1}{\sqrt{\mathcal{E}}}\left(
(-\mu+\overline{\tau})\sinh(t\sqrt{\mathcal{E}})+\sqrt{\mathcal{E}}\cosh(t\sqrt{\mathcal{E}})
\right)e^{-\tau t},$$ $$\label{nn**}
L_o(t)=\frac{c_0}{\sqrt{\mathcal{E}}}\sinh(t\sqrt{\mathcal{E}})e^{-\tau
t},$$ $$\label{resum**} \E[e^{-\mu{Z^{c_0,c_1}}_t}]=\frac{1}{\sqrt{\mathcal{E}}}\Big[
(-\mu+\tau)\sinh(t\sqrt{\mathcal{E}})+\sqrt{\mathcal{E}}\cosh(t\sqrt{\mathcal{E}})
\Big]e^{-\tau t},$$ where $\mathcal{E}=\mu^2-2\overline{\tau}\mu+\tau^2$.
Applying Lemma \[essai\_mark\] with the particular function $F(z,n)=e^{-\mu z}1_{\{n\in 2\Nset \}}$, we have: $$\begin{aligned}
\frac{d}{dt}L_e(t)&=&-\mu\E\Big[
e^{-\mu{Z^{c_0,c_1}}_t}(-1)^{{N^{c_0,c_1}}_t}1_{\{{N^{c_0,c_1}}_t\in
2\Nset\}} \Big]\\
&+&E\Big[ e^{-\mu{Z^{c_0,c_1}}_t}\Big( 1_{\{ {N^{c_0,c_1}}_t\in 2\Nset+1 \}}-1_{\{
{N^{c_0,c_1}}_t\in 2\Nset \}} \Big)\\&&\times \Big(c_1 1_{\{ {N^{c_0,c_1}}_t\in 2\Nset+1
\}}+c_01_{\{ {N^{c_0,c_1}}_t\in 2\Nset \}} \Big)\Big]\end{aligned}$$ We deduce $$\frac{d}{dt}L_e(t)=-(\mu+c_0) L_e(t)+c_1L_o(t).$$ Similarly $$\begin{aligned}
\frac{d}{dt}L_o(t)&=&-\mu\E\Big[
e^{-\mu{Z^{c_0,c_1}}_t}(-1)^{{N^{c_0,c_1}}_t}1_{\{{N^{c_0,c_1}}_t\in
2\Nset+1\}} \Big]\\
&+&E\Big[ e^{-\mu{Z^{c_0,c_1}}_t}\Big( 1_{\{ {N^{c_0,c_1}}_t\in 2\Nset \}}-1_{\{ {N^{c_0,c_1}}_t\in
2\Nset+1 \}} \Big)\\&&\times \Big(c_1 1_{\{ {N^{c_0,c_1}}_t\in 2\Nset+1
\}}+c_01_{\{ {N^{c_0,c_1}}_t\in 2\Nset \}} \Big)\Big].\end{aligned}$$ We get therefore $$\frac{d}{dt}L_o(t)=(\mu-c_1) L_o(t)+c_0L_e(t).$$ To sum up $$\begin{aligned}
\frac{d}{dt}\left(\begin{array}{l}L_e(t)\\
L_o(t)\end{array}\right)=\left(\begin{array}{cc}-\mu-c_0 & c_1\\
c_0 & \mu-c_1\end{array}\right)\left(\begin{array}{l}L_e(t)\\
L_o(t)\end{array}\right).
\end{aligned}$$ We deduce the expressions of $L_e(t)$ and $L_o(t)$: $$\label{eta1}
L_e(t)=a_+e^{\lambda_+ t}+a_-e^{\lambda_-t}\quad\quad L_o(t)=b_+e^{\lambda_+
t}+b_-e^{\lambda_-t},$$ where $\lambda_\pm=-\tau\pm\sqrt{\mu^2-2\overline{\tau}\mu+\tau^2}=
-\tau\pm\sqrt{\mathcal{E}}$.\
The constants $a_\pm$ and $b_\pm$ are evaluated with the initial conditions: $$L_e(0)=\P({N^{c_0,c_1}}_0\in 2\Nset)=1,\quad\quad L_o(0)=\P({N^{c_0,c_1}}_0\in
2\Nset+1)=0,$$ $$\frac{dL_e}{dt}(0)=-(\mu+c_0)L_e(0)+c_1L_o(0)=-\mu-c_0,$$ $$\frac{dL_o}{dt}(0)=(\mu-c_1)L_o(0)+c_0L_e(0)=c_0.$$ We obtain $$\label{eta2}
a_+=\frac{1}{2\sqrt{\mathcal{E}}}(-\mu+\overline{\tau}+\sqrt{\mathcal{E}})\quad\mbox{and}\quad
a_-=\frac{1}{2\sqrt{\mathcal{E}}}(\mu-\overline{\tau}+\sqrt{\mathcal{E}}),$$ $$\label{eta3}
b_+=\frac{c_0}{2\sqrt{\mathcal{E}}}\quad\mbox{and}\quad b_-=-\frac{c_0}{2\sqrt{\mathcal{E}}}$$ Using , and , Proposition \[trlp\] follows.
It is easy to deduce two direct consequences of Proposition \[trlp\]. First, taking $\mu=0$ we obtain $\P({N^{c_0,c_1}}_t\in 2\Nset)$ and $\P({N^{c_0,c_1}}_t\in 2\Nset+1)$. Second, taking the expectation in we get the mean of ${Z^{c_0,c_1}}_t$.
We have: $$\P({N^{c_0,c_1}}_t\in 2\Nset)=\frac{1}{\tau}\Big[ \overline{\tau}\sinh(\tau
t)+\tau\cosh(\tau t) \Big]e^{-\tau t},$$ $$\P({N^{c_0,c_1}}_t\in 2\Nset+1)=\frac{c_0}{\tau} \sinh(\tau t)e^{-\tau t},$$ and $$\E[{Z^{c_0,c_1}}_t]=\frac{\overline{\tau}}{\tau}t+\frac{c_0}{2\tau^2}(1-e^{-2\tau
t}).$$
\[remark1\] The Laplace transform with respect to the time variable can also be explicitly computed. We define $F(\mu,s)=\int_0^\infty
e^{-st}\E[e^{-\mu {Z^{c_0,c_1}}_t}]dt$. Integrating with respect to $dt$ we get $$\begin{aligned}
F(\mu,s)&=&\frac{1}{2\sqrt{\mathcal{E}}}\Big(\sqrt{\mathcal{E}}+(-\mu+\tau)
\Big)\frac{1}{s-\sqrt{\mathcal{E}}+\tau}+\frac{1}{2\sqrt{\mathcal{E}}}
\Big(\sqrt{\mathcal{E}}-(-\mu+\tau)\Big)\frac{1}{s+\sqrt{\mathcal{E}}+\tau}\\
&=&\frac{(\sqrt{\mathcal{E}}-\mu+\tau)(s+\sqrt{\mathcal{E}}+\tau)+
(\sqrt{\mathcal{E}}+\mu-\tau)(s-\sqrt{\mathcal{E}}+\tau)}{2\sqrt{\mathcal{E}}
((s+\tau)^2-\mathcal{E})}\\
&=&\frac{2s\sqrt{\mathcal{E}}+4\tau\sqrt{\mathcal{E}}-2\mu
\sqrt{\mathcal{E}}}{2\sqrt{\mathcal{E}}((s+\tau)^2-\mathcal{E})}=
\frac{ s+ 2\tau-\mu}{(s+\tau)^2-\mathcal{E}}\end{aligned}$$ In the symmetric case, $\mathcal{E}$ equals $\mu^2+c_0^2$, then $$\label{...}
F(\mu,s)=\frac{s+ 2c_0-\mu }{s^2+2sc_0-\mu^2}.$$ Let $(Z_t)$ be the symmetrization of $(Z^{c_0}_t)$ which is defined by an initial randomization: $$Z_t=\epsilon Z_t^{c_0},\quad t\ge 0,$$ where $\epsilon$ is independent of $Z_t^{c_0}$ and $\P(\epsilon=\pm 1)=1/2$.\
Relation implies $$\int_0^\infty e^{-st}\E[e^{-\mu Z_t}]dt=\frac{s+ 2c_0}{s^2+2sc_0-\mu^2}.$$ This identity has been obtained by Weiss in [@weiss02].
Let us now present a link between the ITN process and the telegraph equation in the particular symmetric case $c_0=c_1=c>0$. Recall that $(N_t^c)$ is a Poisson process with parameter $c$.\
Let $f:\Rset\to\Rset$ be a function of class $\mathcal{C}^2$ whose first and second derivatives are bounded. We define $$u(x,t)=\frac{1}{2}\Big\{f(x+at)+f(x-at)\Big\},\quad x\in\Rset,\ t\ge 0.$$ Then (cf [@eckstein00]) $u$ is the unique solution of the wave equation $$\begin{aligned}
\left\{\begin{array}{l}
\displaystyle\frac{\partial^2 u}{\partial t^2}=a^2\frac{\partial^2 u}{\partial x^2},\\
\displaystyle u(x,0)=f(x),\quad \frac{\partial u}{\partial
t}(x,0)=0.
\end{array}\right.\end{aligned}$$
\[telegraph\*\*\] The function $$w(x,t)=\E\Big[ u\Big(x,\int_0^t (-1)^{N^c_s}ds\Big)\Big],\quad
(x\in\Rset, t\ge 0)$$ is the solution of the telegraph equation (TE) $$\begin{aligned}
\left\{\begin{array}{l}
\displaystyle\frac{\partial^2 w}{\partial t^2}+2c\frac{\partial w}{\partial t}=
a^2\frac{\partial^2 w}{\partial x^2},\\[8pt]
\displaystyle w(x,0)=f(x),\quad \frac{\partial w}{\partial
t}(x,0)=0.
\end{array}\right.\end{aligned}$$
This result can be proved using asymptotic analysis applied to difference equation associated with the persistent random walk [@goldstein50] or using Fourier transforms [@weiss02]. Here we shall present a new proof.\
[*Proof of Proposition \[telegraph\*\*\].*]{} Applying twice Lemma \[essai\_mark\] to $(z,n)\to u(x,z)$ and $(z,n)\to \frac{\partial u}{\partial t}(x,z)(-1)^n$ we obtain: $$\frac{\partial w}{\partial t}(x,t)=\E\Big[ \frac{\partial u}{\partial t}
\Big(x,\int_0^t (-1)^{N^c_s}ds\Big) (-1)^{N^c_t} \Big].$$ and $$\frac{\partial^2 w}{\partial t^2}(x,t)=\E\Big[ \frac{\partial^2 u}{\partial t^2}
\Big(x,\int_0^t (-1)^{N^c_s}ds\Big)\Big]-2c\E\Big[ \frac{\partial u}{\partial t}
\Big(x,\int_0^t (-1)^{N^c_s}ds\Big) (-1)^{N^c_t} \Big].$$ Since $u$ solves the wave equation we have $$\frac{\partial^2 w}{\partial t^2}(x,t)=a^2\frac{\partial^2 w}{\partial x^2}(x,t)
-2c\frac{\partial w}{\partial t}(x,t).$$ The function $w$ is actually the solution of the telegraph equation. It is easy to prove that $w$ satisfies the boundary conditions.\
Let us note that Proposition \[telegraph\*\*\] can be extended to the asymmetric case $c_0\neq c_1$. In this general case the telegraph equation is replaced by a linear system of partial differential equations.
\[telegraph\_gen\] 1) In [@eckstein00], [@griego71], an extension of Proposition \[telegraph\*\*\] has been proved. Let $A$ be the generator of a strongly continuous group of bounded linear operators on a Banach space. If $w$ is the unique solution of this abstract “wave equation”: $$\frac{\partial^2 w}{\partial t^2}=A^2 w;\ w(\cdot,0)=f,\
\frac{\partial w}{\partial t}(\cdot,0)=Ag\quad (f,g\in\mathcal{D}(A))$$ then $u(x,t)=\E\Big[ w\Big(x,\int_0^t (-1)^{N^c_s}ds\Big) \Big]$ solves the abstract “telegraph equation”: $$\frac{\partial^2 u}{\partial t^2}=A^2 u-2c\frac{\partial u}{\partial t},\
\ u(\cdot, 0)=f,\ \ \frac{\partial u}{\partial t}(\cdot, 0)=Ag.$$ 2) In the same vein as [@griego71], Enriquez [@enriquez07] has introduced processes with jumps to represent solutions of some linear differential equations and biharmonic equations in the presence of a potential term. Moreover useful references are given in [@enriquez07].\
3) It is easy to deduce from Lemma \[essai\_mark\] that the functions $$w_e(x,t)=\E\Big[ u\Big(x,\int_0^t (-1)^{N^{c_0,c_1}_s}ds\Big)
1_{\{N^{c_0,c_1}_t\in 2\Nset \}}\Big],\quad
(x\in\Rset, t\ge 0)$$ $$w_o(x,t)=\E\Big[ u\Big(x,\int_0^t (-1)^{N^{c_0,c_1}_s}ds\Big)
1_{\{N^{c_0,c_1}_t\in 2\Nset+1 \}}\Big],\quad
(x\in\Rset, t\ge 0)$$ are solutions of the general telegraph system (TS) $$\begin{aligned}
\left\{\begin{array}{l}
\displaystyle\frac{\partial^2 w_e}{\partial t^2}
=(c_0c_1-c_0^2)w_e+(c_0c_1-c_1^2)w_o-2c_0\frac{\partial w_e}{\partial t}
+a^2\frac{\partial^2 w_e}{\partial x^2},\quad \\[8pt]
\displaystyle\frac{\partial^2 w_o}{\partial t^2}
=(c_0c_1-c_0^2)w_e+(c_0c_1-c_1^2)w_o-2c_1\frac{\partial w_o}{\partial t}
+a^2\frac{\partial^2 w_o}{\partial x^2},\quad \\[8pt]
\displaystyle w_e(x,0)=f(x),\quad w_o(x,0)=0\quad \frac{\partial w_e}{\partial
t}(x,0)=-c_0 f(x)\quad \frac{\partial w_o}{\partial
t}(x,0)=c_0f(x).
\end{array}\right.\end{aligned}$$
[ ]{} [ ]{}
Convergence of the persistent walk to the ITN {#sectionpreuve1}
=============================================
Suppose $\rho_0=1$. The aim of this section is to prove the convergence of the interpolated persistent random walk towards the generalized integrated telegraph noise (ITN) i.e. Theorem \[cas1\]. Let us start with preliminary results.\
First, let us recall that $(X_n, n\in\Nset)$ is the persistent random walk starting in $0$ defined by the increments process $(Y_n,\,n\in\Nset)$ (see Section \[section\_not\]) with transition probabilities $$\begin{aligned}
\pi^\Delta=\left(\begin{array}{cc}
1-c_0\Delta_t & c_0\Delta_t\\
c_1\Delta_t & 1-c_1\Delta_t
\end{array}\right).\end{aligned}$$ Let $(T_k;\ k\ge 1)$ be the sign changes sequence of times : $$\begin{aligned}
\label{*1} \left\{\begin{array}{l}
T_1=\inf\{t\ge 1 \ : Y_t\neq Y_0\}\\[8pt]
T_{k+1}=\inf\{t> T_k\ : Y_t\neq Y_{T_{k}}\};\quad k\ge 1.
\end{array}\right.\end{aligned}$$ We put $T_0=0$ and $$\label{*2}
A_k=T_k-T_{k-1}\quad k\ge 1$$ Let $N_t$ be the number of times over $[0,t]$ so that the sign of $(Y_n)$ changes: $$\label{*3} N_t=\sum_{j\ge 1}1_{\{T_j\le t\}}$$ The definition of $N_t$ implies that: $$N_t=k\Longleftrightarrow T_k\le t<T_{k+1}$$ We suppose in this subsection that $Y_0=-1$.\
We deduce from the identities above: $$\label{*4} X_t=\sum_{j=1}^k (-1)^j
A_j+(-1)^{k+1}(t-T_k+1)\quad\mbox{where}\ k=N_t.$$ By we obtain: $$\label{*5} T_k=A_1+\ldots+A_k\quad k\ge 1.$$ Hence the equations , and permit to emphasize the bijective correspondence between $(X_n;\
n\in\Nset)$ and $(A_k;\ k\in\Nset)$.\
We introduce the normalization of $(X_n;\ n\in\Nset)$ given by with $\Delta_x=\Delta_t$: $$\label{4.4b}
Z^\Delta_s=\Delta_t X_{s/\Delta_t}\quad (s/\Delta_t\in\Nset).$$ Let us define: $$\label{*6} N^\Delta_s=\sum_{j\ge 1}1_{\{\sum_{i=1}^j\Delta_t
A_j\le s\}}\quad s\ge 0.$$ Let us note that $$N^\Delta_s=N_{s/\Delta_t}\quad\mbox{if}\ s/\Delta_t\in\Nset.$$ That permits to extend the definition of $Z^\Delta_s$ to any $s\ge
0$ by setting $$\label{*7} \tilde{Z}_s^\Delta=\sum_{j=1}^k (-1)^j(\Delta_t
A_j)+(-1)^{k+1}(s-\Delta_t T_k+\Delta_t)\quad k=N^\Delta_s.$$ Obviously $\tilde{Z}_s^\Delta=Z^\Delta_{s}$ if $s/\Delta_t\in\Nset$.\
In order to study the asymptotic behaviour of $(\tilde{Z}_s^\Delta)$ as $\Delta_t\to 0$, we shall first prove the convergence in distribution of $(\Delta_t A_j)_{j\ge 1}$ and $(N^\Delta_s)_{s\ge 0}$.\
We recall that some random variable $\xi$ is exponentially distributed with parameter $\lambda>0$ if its density is given by $\frac{1}{\lambda}\, e^{-x/\lambda}1_{\{x\ge 0\}}$.
\[convloi\] The random variables $(A_k)$ are independent and $\Delta_t A_{2k}$ (resp. $\Delta_t A_{2k+1}$) converges in distribution, as $\Delta_t\to 0$, to the exponential law with parameter $\frac{1}{c_1}$ (resp. $\frac{1}{c_0}$).
Since $(Y_n)$ is a Markov chain, then the $(A_k)$ are independent. First let us study the convergence in distribution of the sequence $\Delta_t A_{2k}$. We use the Laplace transform of $\Delta_t A_{2k}$: $\varphi(\mu)=\E[e^{-\mu\Delta_tA_{2k}}]$, $\mu\ge 0$. Since $A_{2k}$ is geometrically distributed with parameter $c_1\Delta_t$, we obtain $$\begin{aligned}
\label{25**2}
\varphi(\mu)&=&\sum_{j=1}^\infty e^{-\mu\Delta_t j}(1-c_1\Delta_t)^{j-1}c_1\Delta_t\nonumber\\
&=&
\frac{c_1\Delta_t}{e^{\mu\Delta_t}-(1-c_1\Delta_t)}=\frac{c_1\Delta_t}{(\mu+c_1)\Delta_t
+o(\Delta_t)}=\frac{c_1}{\mu+c_1}+o(\Delta_t)\end{aligned}$$ The function $\varphi(\mu)$ converges for any $\mu\ge 0$ to the Laplace transform of some exponential law with parameter $c_1^{-1}$. This proves the convergence in distribution of $\Delta_t A_{2k}$. Concerning $A_{2k-1}$ the arguments are similar.
Let us recall that the counting process $({N^{c_0,c_1}}_t, \ t\ge 0)$ has been defined through the sequence of jumps $(e_n; n\ge 1)$ via , and $(e_n;\ n\ge 1)$ are i.i.d. and exponentially distributed.
\[lem:convmarg\] Let $s>0$, $k\ge 1$ and $\Phi_k:\Rset^{k}\to\Rset$ be a bounded continuous function. Then\
1) $\displaystyle\lim_{\Delta_t\to 0}\P(N^\Delta_s=0)=\P({N^{c_0,c_1}}_s=0)$\
2) $\displaystyle\lim_{\Delta_t\to
0}\E[\Phi_k(\Delta_tA_1,\Delta_tA_2,\ldots,\Delta_tA_k)1_{\{N^{\Delta}_s=k\}}]=
\E[\Phi_k(\lambda_1e_1,\lambda_2e_2,\ldots,\lambda_ke_k)1_{\{{N^{c_0,c_1}}_s=k\}}],$ where $\lambda_k$ has been defined by .
1\) Statement 1) follows from: $$\begin{aligned}
\P(N^\Delta_s=0)&=&\P(N_{\lfloor s/\Delta_t\rfloor}=0)=\P(T_1\ge \lfloor s/\Delta_t\rfloor)\\
&=&\P(A_1\ge \lfloor s/\Delta_t\rfloor)=\P(\Delta_tA_1\ge
\Delta_t\lfloor s/\Delta_t\rfloor)\end{aligned}$$ where $\lfloor a \rfloor$ denotes the integer part of $a$.\
2) Set $k\ge 1$. The event $\{N^{\Delta}_s=k\}$ can be decomposed as follows: $$\{N^{\Delta}_s=k\}=\left\{\Delta_t \sum_{j=1}^k A_j\le
s\right\}\cap\left\{\Delta_t \sum_{j=1}^{k+1} A_j> s\right\}.$$ This identity imply existence of a bounded Borel function $\psi_k$:$\Rset^{k+1}\to\Rset$ so that $$\begin{aligned}
&&\Phi_k(\Delta_tA_1,\ldots,\Delta_tA_k)1_{\{N^{\Delta}_s=k\}}\\
&&=\Phi_k(\Delta_tA_1,\ldots,\Delta_tA_k)1_{\{\Delta_t \sum_{j=1}^k A_j\le s\}}
1_{\{ \Delta_t \sum_{j=1}^{k+1} A_j> s\}}\\
&&=\psi_k(\Delta_tA_1,\Delta_tA_2,\ldots,\Delta_tA_{k+1}).\end{aligned}$$ Since $\Phi_k$ is continuous, the discontinuity points of $\psi_k$ are included in: $$\mathbb{U}=\Big\{x\in\Rset^{k+1}:\ \sum_{j=1}^k x_j=s
\Big\}\cup\Big\{x\in\Rset^{k+1}:\ \sum_{j=1}^{k+1} x_j=s \Big\}.$$ By Lemma \[convloi\], $(\Delta_t A_1,\ldots,\Delta_t A_{k+1})$ converges in distribution towards $(\lambda_1 e_1,\ldots,\lambda_{k+1}e_{k+1})$ as $\Delta_t\to 0$. Since the Lebesgue measure of $\mathbb{U}$ is null, the limit law does not charge $\mathbb{U}$. We can conclude evoking for instance Theorem 14 p.247 in [@Brancovan06]).
Let us formulate a straightforward generalization of Lemma \[lem:convmarg\].
\[lem:convmarggen\] Let $n\in \Nset$, $(k_1,\ldots,k_n)\in\Nset^n$ such that $k_1\le k_2\le\ldots\le
k_n$ and $(s_1,\ldots,s_n)\in\Rset_+^n$ with $s_1\le
s_2\le\ldots\le s_n $. Let $\Phi:\Rset^{k_n}\to\Rset$ be a bounded and continuous function. Then $$\begin{aligned}
\label{conv_marg_gen} &&\lim_{\Delta_t\to
0}\E[\Phi(\Delta_tA_1,...,\Delta_tA_{k_n})1_{\{N^{\Delta}_{s_1}=k_1,...,N^{\Delta}_{s_n}=k_n
\}}]\nonumber\\
&&\quad =\E[\Phi(\lambda_1e_1,...,\lambda_{k_n}e_{k_n})1_{\{{N^{c_0,c_1}}_{s_1}=k_1,...,{N^{c_0,c_1}}_{s_n}=k_n\}}]\end{aligned}$$
\[convmarg\] The random variable $\tilde{Z}^\Delta_s$ converges in distribution towards $-{Z^{c_0,c_1}}_s$, for any $s>0$, as $\Delta_t\to 0$.
Let $f\ :\ \Rset \to\Rset$ be a continuous function which is bounded by $M$. Identities and imply that $
\E[f(\tilde{Z}^\Delta_s)]=\sum_{k=0}^\infty E_\Delta(k)$, with $$E_\Delta(k)=\E\Big[f\Big(\sum_{j=1}^k
(-1)^j\Delta_tA_j+(-1)^{k+1}\Big(s-\Delta_t\sum_{j=1}^k A_j+\Delta_t\Big)
\Big)1_{\{N^\Delta_s=k\}} \Big]$$ Applying Lemma \[lem:convmarg\] and , we obtain for any $k\ge 0$, $$\begin{aligned}
\lim_{\Delta_t\to 0}E_\Delta(k)&=&E\Big[f\Big(\sum_{j=1}^k
(-1)^j\lambda_j e_j+(-1)^{k+1}\Big(s-\sum_{j=1}^k \lambda_j e_j\Big)
\Big)1_{\{{N^{c_0,c_1}}_s=k\}} \Big]\\
&=&\E[f(-{Z^{c_0,c_1}}_s)1_{\{{N^{c_0,c_1}}_s=k \}}].\end{aligned}$$ Moreover since $f$ is bounded by $M$, we get $$\vert E_\Delta(k)\vert\le M\P(N^\Delta_s=k).$$ Suppose that $k\ge 1$. Then, using the Markov inequality and the independence property of the random sequence $(A_n,\ n\ge 0)$, we obtain $$\begin{aligned}
\P(N^\Delta_s=k)&=&\P\Big(\Delta_t\sum_{j=1}^k A_j\le
s<\Delta_t\sum_{j=1}^{k+1}A_j
\Big)\\
&\le&\P\Big(\Delta_t\sum_{j=1}^k A_j\le
s\Big)=\P\Big(\exp\Big\{-\Delta_t\sum_{j=1}^k A_j\Big\}\ge
e^{-s}\Big)\\
&\le&e^s\E\Big[\exp-\Delta_t\sum_{j=1}^k A_j \Big]=e^
s\prod_{j=1}^k\varphi_j(1)\end{aligned}$$ where $\varphi_j(\mu)=\E[e^{-\mu\Delta_tA_j}]$. Since $(Y_n)$ is a Markov chain starting at $Y_0=-1$, for any $j\ge 1$, $A_{2j-1}$ (resp. $A_{2j}$) is geometrically distributed with parameter $c_0\Delta_t$ (resp. $c_1\Delta_t$). According to we get $$\varphi_{2j}(1)=\frac{c_1\Delta_t}{e^{\Delta_t}-1+c_1\Delta_t}\le
\frac{c_1\Delta_t}{\Delta_t+c_1\Delta_t}=\frac{c_1}{1+c_1}<1.$$ By the same way, we have: $$\varphi_{2j-1}(1)\le\frac{c_0}{1+c_0}<1.$$ As a result, there exists $0<r<1$ so that $$\label{lebesgue} \P(N^\Delta_s=k)\le e^s r^k.$$ We are now allowed to apply the dominated convergence theorem: $$\lim_{\Delta_t\to 0}\E[f(\tilde{Z}^\Delta_s)]=\sum_{k\ge
0}\lim_{\Delta_t\to 0}E_\Delta(k)=\sum_{k\ge
0}\E[f(-{Z^{c_0,c_1}}_s)1_{\{{N^{c_0,c_1}}_s=k\}}]=\E[f(-{Z^{c_0,c_1}}_s)].$$
\[convmarg\_gen\] For any $(s_1,\ldots,s_n)\in\Rset_+^n$ such that $s_1\le s_2\le\ldots\le s_n $, the random vector $
(\tilde{Z}^\Delta_{s_1},\ldots,\tilde{Z}^\Delta_{s_n})$ converges in distribution to $(-{Z^{c_0,c_1}}_{s_1},\ldots,-{Z^{c_0,c_1}}_{s_n})$, as $\Delta_t$ tends to $0$.
We follow the approach developed in the proof of Proposition \[convmarg\]. Let $f:\Rset^n\to\Rset$ be a bounded and continuous function. We have: $$\E\Big[f(\tilde{Z}^\Delta_{s_1},\ldots,\tilde{Z}^\Delta_{s_n})\Big]=\sum_{k_1,\ldots,k_n}
E_\Delta(k_1,\dots,k_n),$$ where the sum is extended to $(k_1,\ldots,k_n)\in\Nset^n$ so that $k_1\le k_2\le\ldots\le k_n$ and $$E_\Delta(k_1,\dots,k_n)=\E\Big[f(\tilde{Z}^\Delta_{s_1},\ldots,\tilde{Z}^\Delta_{s_n})1_{\{
N^{\Delta}_{s_1}=k_1,\ldots,N^{\Delta}_{s_n}=k_n\}}\Big].$$ Identity implies the existence of a bounded continuous function $\psi_n:\Rset^{k_n}\to\Rset$ so that $$E_\Delta(k_1,\dots,k_n)=\E\Big[\psi_n(\Delta_tA_1,\ldots,\Delta_tA_{k_n})1_{\{
N^{\Delta}_{s_1}=k_1,\ldots,N^{\Delta}_{s_n}=k_n\}}\Big].$$ Applying Lemma \[lem:convmarggen\], we get $$\lim_{\Delta_t\to
0}E_\Delta(k_1,\dots,k_n)=\E[\psi_{n}(\lambda_1
e_1,\ldots,\lambda_{k_n}e_{k_n})1_{\{{N^{c_0,c_1}}_{s_1}=k_1,\ldots,{N^{c_0,c_1}}_{s_n}=k_n\}}].$$ According to the definition of the process ${Z^{c_0,c_1}}_s$, we may deduce : $$\lim_{\Delta_t\to 0}E_\Delta(k_1,\dots,k_n)=\E[f(-{Z^{c_0,c_1}}_{s_1},\ldots,-{Z^{c_0,c_1}}_{s_n})
1_{\{{N^{c_0,c_1}}_{s_1}=k_1,\ldots,{N^{c_0,c_1}}_{s_n}=k_n\}}].$$ In order to obtain that $$\lim_{\Delta_t\to
0}\E[f(\tilde{Z}^\Delta_{s_1},\ldots,\tilde{Z}^\Delta_{s_n})]=\E[f(-{Z^{c_0,c_1}}_{s_1},\ldots,-{Z^{c_0,c_1}}_{s_n}
)],$$ it suffices (cf the proof of Proposition \[convmarg\]) to prove that $$\sum_{k_1,\ldots, k_{n-1}}\sup_{\Delta_t}\vert
E_\Delta(k_1,\dots,k_n)\vert<\infty.$$ Since $f$ is bounded, $$\vert E_\Delta(k_1,\dots,k_n)\vert\le M\P(N_{s_n}^\Delta=k_n)$$ Using moreover we get $$\sum_{k_1,\ldots, k_{n}}\vert E_\Delta(k_1,\dots,k_n)\vert\le M
e^{s_n}\sum_{k_n}(k_n)^{n-1}r^{k_n}<\infty$$ since $r<1$.
We are now able to complete the proof of Theorem \[cas1\]. Since $(\tilde{Z}^\Delta_s)$ and $({Z^{c_0,c_1}}_s)$ are both continuous processes, the convergence of the process $(\tilde{Z}^\Delta_s)$ to the process $(-{Z^{c_0,c_1}}_s)$ will be proved as soon as the following measure tension criterium (cf Theorem 8.3 p.56 in [@Billingsley99]) holds : for all $\eps>0$ and $\eta_0$, there exists some constants $\delta\in]0,1[$ and $\mu>0$ such that $$\label{tension}
\frac{1}{\delta} \P\Big(\sup_{s\le u\le s+\delta}\vert
\tilde{Z}^\Delta_{u}-\tilde{Z}^\Delta_s\vert\ge\eps
\Big)\le\eta_0,\quad\mbox{for any}\ \Delta_t\le\mu.$$ Since $(\tilde{Z}^\Delta_s,\ s\ge 0)$ is the interpolated persistent random walk, its slope is always equal to $1$ or $-1$. Hence we obtain for any $(u,s)\in\Rset_+^2$, $$\vert \tilde{Z}^\Delta_u-\tilde{Z}^\Delta_s\vert \le\vert
u-s\vert.$$ Consequently $$\sup_{s\le u\le s+\delta}\vert
\tilde{Z}^\Delta_u-\tilde{Z}^\Delta_s\vert \le \delta.$$ By choosing $\delta=\eps/2$ we get the tension criterium and so the convergence of the process $(\tilde{Z}^\Delta_s)$ to the process $(-{Z^{c_0,c_1}}_s)$.
Two extensions of Theorem \[cas1\] {#extensions}
==================================
First of all, the extensions presented in this section concerns the regime $\Delta_x=\Delta_t$.
The case when $(Y_t)$ takes $k$ values.
---------------------------------------
Let us introduce our parameters. Let $k\ge 2$, $y_1,\ldots,y_k$ denote $k$ real numbers, and $(c(i,j);\ 1\le i,j\le k)$ a matrix so that $$\label{cadre}
c(i,j)\ge 0\ \mbox{for any}\ i\neq j,\ c(i,i)=0,\ \sum_{l=1}^k c(i,l)>0\ \forall i.$$ We directly consider the asymptotic regime. Let $(Y_t)$ be a $\{y_1,\ldots, y_k\}$-valued Markov chain, with transition probability matrix: $$\begin{aligned}
\label{gen:matrix}
\pi^\Delta(y_i,y_j)=\left\{\begin{array}{ll}
c(i,j)\Delta_t & i\neq j\\[5pt]
1-\Big( \sum_{l=1}^k c(i,l) \Big)\Delta_t & i=j,
\end{array}\right.\end{aligned}$$ where $\Delta_t>0$ is supposed to be small so that $$c(i,j)\Delta_t\le 1,\quad \Big( \sum_{l=1}^k c(i,l) \Big)\Delta_t<1.$$ Similarly to the case $k=2$ and $y_1=-1$, $y_2=1$, we are interested in the linear interpolation $(\tilde{Z}_s^\Delta;\ s\ge 0)$ of the process $(Z^\Delta_s;\ s\ge 0)$ defined by .
\[the:ext1\] Suppose $Y_0=y_i$. Then $(\tilde{Z}^\Delta_s;\ s\ge 0)$ converges in distribution, as $\Delta_t\to 0$, to the process $\Big( \int_0^t R_sds;\ t\ge 0\Big)$ where $(R_s)$ is a $\{y_1,\ldots, y_k\}$-valued continuous-time Markov chain starting at level $y_i$, whose dynamic is the following: $(R_t)$ stays on level $y_i$ an exponential time with parameter $1/\Big( \sum_{l=1}^k c(j,l) \Big)$ and jumps to $y_{j'}$ ($j'\neq j$) with probability $c(j,j')/\Big( \sum_{l=1}^k c(j,l) \Big)$.
In the case $k=2$, $y_1=-1$ and $y_2=1$, then $((-1)^{{N^{c_0,c_1}}_t};\ t\ge 0)$ (cf ) may be chosen as a realization of $(R_t)$ when it starts at $R_0=-1$.
[*Proof of Theorem \[the:ext1\].*]{} We proceed as in the proof of Theorem \[cas1\] developed in Section \[sectionpreuve1\].\
Let $(T_n)_{n\ge 1}$ be the sequence of stopping times defined by . Then: $$\begin{aligned}
X_t=\left\{\begin{array}{ll}
y_i(t+1) & 0\le t<T_1\\[5pt]
y_i T_1+Y_{T_1}(t-T_1+1) & T_1\le t <T_2.
\end{array}\right.\end{aligned}$$ Recall that $(Z_s^\Delta;\ s/\Delta_t\in\Nset)$ has been defined by . From the relations above, it is easy to deduce: $$\begin{aligned}
Z_s^\Delta=\left\{\begin{array}{l}
y_i(s+\Delta_t)\quad 0\le s\le \Delta_t T_1\\[5pt]
y_i(\Delta_t T_1)+Y_{T_1}(s-\Delta_t T_1+\Delta_t)\quad \Delta_t T_1\le s<\Delta_t T_2.
\end{array}\right.\end{aligned}$$ Let us determine the limit distribution of $(\Delta_t T_1, Y_{T_1})$ as $\Delta_t\to 0$. Set $$V^\Delta(\lambda, j)=\E\Big[ e^{-\lambda \Delta_t T_1}1_{\{ Y_{T_1}=y_{j} \}} \Big],
\quad \lambda>0,\ j\neq i.$$ Proceeding as in the proof of Lemma \[convloi\], we obtain: $$V^\Delta(\lambda, j)=\frac{e^{-\lambda\Delta_t} c(i,j)\Delta_t}{1-\Big[1-\Big( \sum_{l=1}^k c(i,l)\Big)\Delta_t\Big]e^{-\lambda\Delta_t}}.$$ Using standard analysis, we deduce that $(\Delta_t T_1, Y_{T_1})$ converges in distribution as $\Delta_t\to 0$ to $(e'_1, U_1)$ where: $$\E\Big[ e^{-\lambda e'_1}1_{\{ U_1=j\}} \Big]=\frac{c(i,j)}{\lambda+\sum_{l=1}^k c(i,l)}.$$ As a result, $e'_1$ and $U_1$ are independent, $e'_1$ is exponentially distributed with parameter $1/\sum_{l=1}^k c(i,l)$ and $$\P(U_1=j)=\frac{c(i,j)}{\sum_{l=1}^k c(i,l)}.$$ Using the approach developed in Section \[sectionpreuve1\], we can prove Theorem \[the:ext1\]. The details are left to the reader.
The case when $(Y_t)$ is a Markov chain of order $2$. {#ext2}
-----------------------------------------------------
Let $(Y_t)$ be a Markov chain with order $2$. For simplicity we suppose that it takes its values in $\{-1,1\}$. Obviously $(Y_t,Y_{t+1})_{t\ge 0}$ is a Markov chain with state space $$E=\{(-1,-1),(-1,1),(1,-1),(1,1)\}.$$ Let $\pi^\Delta$ be the transition probability matrix: $$\begin{aligned}
\label{matrixx}
\pi^\Delta=\left(\begin{array}{cccc}
1-c_0\Delta_t & c_0\Delta_t & 0 & 0\\
0 & 0 & 1-p_0 & p_0\\
p_1 & 1-p_1 & 0 & 0\\
0 & 0 & c_1 \Delta_t & 1-c_1\Delta_t
\end{array}\right)\end{aligned}$$ where $\Delta_t,c_0,c_1,p_0,p_1>0$ and $c_0\Delta_t, c_1\Delta_t, p_0, p_1<1$.\
Let us introduce: $$\label{defv}
v_i=\frac{p_i}{1-(1-p_0)(1-p_1)},\quad c'_i=c_iv_i,\ i=0,1.$$ Recall that $(Z^\Delta_t)$ and $(\tilde{Z}^\Delta_t)$ have been defined by , resp. , $({N^{c_0,c_1}}_t)$ is the counting process defined by , and $${Z^{c_0,c_1}}_t=\int_0^t (-1)^{{N^{c_0,c_1}}_u}du,\ t\ge 0.$$
\[the:ext2\] 1) Suppose that $Y_0=Y_1=-1$ (resp. $Y_0=Y_1=1$) then $(\tilde{Z}^\Delta_s;\ s\ge 0)$ converges in distribution, as $\Delta_t\to 0$, to $(-Z^{c'_0,c'_1}_s;\ s\ge 0)$ (resp. $(Z^{c'_1,c'_0}_s;\ s\ge 0)$).\
2) Suppose $Y_0=1$ and $Y_1=-1$ (resp. $Y_0=-1$, $Y_1=1$) then $(\tilde{Z}^\Delta_s;\ s\ge 0)$ converges in distribution, as $\Delta_t\to 0$, to $$\Big((\epsilon-1)\int_0^s(-1)^{{N^{c'_0,c'_1}}_u}du+\epsilon\int_0^s(-1)^{{N^{c'_1,c'_0}}_u}du;\ s\ge 0 \Big)$$ where $\epsilon$ is independent from $({N^{c'_0,c'_1}}_u)$, $({N^{c'_1,c'_0}}_u)$ and $$\P(\epsilon=0)=1-\P(\epsilon=1)=v_1\quad (\mbox{resp.}\ \P(\epsilon=1)=1-\P(\epsilon=0)=v_0).$$
1\) Note that $(Y_t)_{t\in\Nset}$ is a Markov chain if and only if $1-c_0\Delta_t=p_1$ and $1-c_1\Delta_t=p_0$. If we replace formally $p_0$ (resp. $p_1$) by $1-c_1\Delta_t$ (resp. $1-c_0\Delta_t$) in and take the limit $\Delta_t\to 0$, we obtain $v_i=p_i$ and $c'_i=c_i$. We recover Theorem \[cas1\].\
2) The fact that $(Y_t)$ is a Markov chain with order $2$ does not modify drastically the limit. The limit process can be expressed in terms of processes of the type $(Z_s^{\alpha,\beta};\ s\ge 0)$.
[*Proof of Theorem \[the:ext2\].*]{} [**1)**]{} We only consider the case $Y_0=Y_1=1$. Let us define $T_1$, $T_2$ and $T_3$ as follows: $$T_1=\inf\{ t\ge 1,\ Y_t=-1 \},\quad T_2=\inf\{ t\ge T_1+1,\ Y_t=Y_{t-1} \},
\quad T_3=\inf\{ t\ge T_2+1,\ Y_t\neq Y_{T_2} \}.$$ Using the definition (cf ) of $(X_t)$ we easely obtain: $$\begin{aligned}
X_t=\left\{\begin{array}{l}
t+1\quad 0\le t<T_1\\
T_1+\hat{X}_t\quad T_1\le t<T_2
\end{array}\right.\end{aligned}$$ where $\hat{X}_t$ equals either $-1$ or $0$.\
Moreover, when $T_2\le t<T_3$, we have: $$\begin{aligned}
X_t=\left\{\begin{array}{l}
T_1-2-(t-T_2)\quad \mbox{if}\ T_2-T_1\ \mbox{is odd}\\
T_1+1+(t-T_2)\quad \mbox{otherwise.}
\end{array}\right.\end{aligned}$$ According to , we can deduce: $$\begin{aligned}
Z_s^\Delta=\left\{\begin{array}{ll}
s+\Delta_t & 0\le s\le \Delta_t T_1\\[5pt]
\Delta_t T_1+\Delta_t\hat{X}_{s/\Delta_t}& \Delta_t T_1\le s<\Delta_t T_2\\[5pt]
\Delta_t T_1-2\Delta_t-(s-\Delta_t T_2) & \Delta_t T_2\le s<\Delta_t T_3,\ Y_{T_2}=-1\\[5pt]
\Delta_t T_1+\Delta_t+s-\Delta_t T_2 & \Delta_t T_2\le s<\Delta_t T_3,\ Y_{T_2}=1
\end{array}\right.\end{aligned}$$ (note that $T_2-T_1$ is odd if and only if $Y_{T_2}=-1$).\
[**2) a)**]{} Proceeding as in the proof of Theorem \[cas1\], we can prove that $\Delta_t T_1$ converges in distribution, as $\Delta_t\to 0$, to $e'_1$, where $e'_1$ is exponentially distributed with parameter $1/c_1$. Then $(\tilde{Z}_s^\Delta; \ 0\le s\le \Delta_t T_1)\stackrel{(d)}{\longrightarrow}
(s;\ s\le e'_1)$, as $\Delta_t\to 0$.\
[**b)**]{} The distribution of $T_2-T_1$ does not depend on $\Delta_t$. Moreover $\vert\hat{X}_{\cdot}\vert\le 1$, then the limit of the length of the interval $[\Delta_t T_1,\Delta_t T_2]$ is null. We have $$\P( Y_{T_2}=-1)=\sum_{l\ge 0} \Big( (1-p_1)(1-p_0) \Big)^l p_1=v_1.$$ [**c)**]{} Using the strong Markov property, we easely show that $(\tilde{Z}_{s+\Delta_t T_2}^\Delta; \ 0\le s\le \Delta_t (T_3-T_1))\stackrel{(d)}{\longrightarrow}
(e'_1+Y_{T_1}s;\ 0\le s\le e'_2)$, as $\Delta_t\to 0$, where $(e'_1, Y_{T_1})$ (resp. $(e'_1,e'_2)$) are independent r.v.’s and conditionally on $Y_{T_2}=1$ (resp. $Y_{T_2}=-1$) $e'_2$ is exponentially distributed with parameter $1/c_1$ (resp. $1/c_0$).\
[**d)**]{} Let us summarize the former analysis. We have proved that $(\tilde{Z}_s^\Delta;\ s\ge 0)\stackrel{(d)}{\longrightarrow}(\int_0^s \hat{R}_udu,\ s\ge 0)$, where $(\hat{R}_u)$ is a continuous-time Markov chain which takes its values in $\{-1,1\}$ and $\hat{R}_0=1$. Moreover the dynamic of $(\hat{R}_u)$ is the following: $(\hat{R}_u)$ stays in $1$ (resp. $-1$) an exponential time with parameter $1/c_1$ (resp. $1/c_0$) and moves to $-1$ (resp. $1$) with probability $v_1$ (resp. $v_0$). Note that $(\hat{R}_u)$ is allowed to stay in the same site. It is classical (cf [@ross03]) to prove that $(\hat{R}_u)_{u\ge 0}\stackrel{(d)}{=}({Z^{c'_1,c'_0}}_u)_{u\ge 0}$ where $c'_0$ and $c'_1$ are defined by .
Convergence of the persistent random walk towards the Brownian motion with drift {#section preuve2}
================================================================================
In subsection \[mgf\] below we determine the generating function of $X_t$, where $X_t$ is the persistent random walk defined by . This allows to prove Theorem \[conv1\] and Proposition \[cas\_limit\] in subsections \[\*\*\*\], \[scas\_limit\].
The moment generating function of $X_t$ {#mgf}
---------------------------------------
Let us recall that the increments process $(Y_t,\, t\in\mathbb{N})$ is a Markov chain valued in the state space $E=\{-1,1\}$. Its transition probability is given by $$\begin{aligned}
\pi=\left(\begin{array}{cc} 1-\alpha & \alpha\\
\beta & 1-\beta\end{array}\right)\quad\quad 0<\alpha<1,\quad
0<\beta<1.\end{aligned}$$ The persistent random walk $(X_t,\, t\in\mathbb{N})$ is defined by the partial sum: $$X_t=\sum_{i=0}^t Y_i\quad\mbox{with}\quad X_0=Y_0=1\ \mbox{or}\
-1.$$
\[\*\*\*1\] Let us define the functions $a_t$ and $b_t$: $$\label{eq:***1}
a_t(j)=\P(X_t=j, Y_t=-1)\quad\mbox{and}\quad b_t(j)=\P(X_t=j,
Y_t=1).$$ Then, $$\label{aj} a_{t+1}(j)=(1-\alpha)a_t(j+1)+\beta b_t(j+1)$$ $$\label{bj} b_{t+1}(j)=\alpha a_t(j-1)+(1-\beta)b_t(j-1).$$
Using the Markov property of $(Y_t)$ we have: $$\begin{aligned}
a_{t+1}(j)&=&\P(X_{t+1}=j, Y_{t+1}=-1,
Y_t=-1)+\P(X_{t+1}=j, Y_{t+1}=-1, Y_t=1)\\
&=&\P(X_{t}=j+1, Y_{t+1}=-1, Y_t=-1)+\P(X_{t}=j+1, Y_{t+1}=-1,
Y_t=1)\\
&=& (1-\alpha)a_t(j+1)+\beta b_t(j+1).\end{aligned}$$ The second recursive formula involving $(b_t(j))$ can be obtained similarly.
Let us define the moment generating function $\Phi(\lambda,t)=\E[\lambda^{X_t}],\quad(\lambda>0)$. We decompose $\Phi(\lambda,t)$ as $$\label{phi}
\Phi(\lambda,t)=\Phi_-(\lambda,t)+\Phi_+(\lambda,t),$$ with $$\label{phi_decomp}
\Phi_-(\lambda,t)=\E[\lambda^{X_t}1_{\{Y_t=-1\}}],\quad\quad
\Phi_+(\lambda,t)=\E[\lambda^{X_t}1_{\{Y_t=1\}}].$$
1\) $\Phi_-(\lambda,0)=\frac{1}{\lambda}\P(Y_0=-1)$ and $\Phi_+(\lambda,0)=\lambda\P(Y_0=1)$.\
2) The moment generating function verifies the following induction equations: $$\begin{aligned}
\label{rec1} \Phi_-(\lambda,t+1)=\frac{1-\alpha}{\lambda}\
\Phi_-(\lambda,t)+\frac{\beta}{\lambda}\ \Phi_+(\lambda,t)\\
\label{rec2}\Phi_+(\lambda,t+1)=\alpha\lambda
\Phi_-(\lambda,t)+(1-\beta)\lambda \Phi_+(\lambda,t)\end{aligned}$$
Definition implies that $$\Phi_-(\lambda,t)=\sum_{j\in\mathbb{Z}}\lambda^j
a_t(j)=\sum_{j=-t-1}^{t+1}\lambda^j a_t(j).$$ Hence, $$\begin{aligned}
\Phi_-(\lambda,t+1)&=&\sum_j \lambda^j
a_{t+1}(j)=(1-\alpha)\sum_j\lambda^j a_t(j+1)+\beta\sum_j
\lambda^j
b_t(j+1)\\
&=&(1-\alpha)\frac{1}{\lambda}\sum_j
\lambda^{j+1}a_t(j+1)+\frac{\beta}{\lambda}\sum_j
\lambda^{j+1}b_t(j+1)\\
&=&\frac{1-\alpha}{\lambda}\Phi_-(\lambda,t)+\frac{\beta}{\lambda}\Phi_+(\lambda,t).\end{aligned}$$ The proof of is similar.
\[\*\*\*3\] Let $f(\lambda,t)$ be equal to either $\Phi_-(\lambda,t)$ or $\Phi_+(\lambda,t)$, then $$\label{seconddegr}
f(\lambda,t+2)-\Big(\frac{1-\alpha}{\lambda}+(1-\beta)\lambda\Big)f(\lambda,t+1)
+(1-\alpha-\beta)f(\lambda,t)=0.$$
By , we get $$\label{poly}
\Phi_+(\lambda,t)=\Big\{\Phi_-(\lambda,t+1)-\frac{1-\alpha}{\lambda}\Phi_-(\lambda,t)\Big\}
\frac{\lambda}{\beta}.$$ Replacing $t$ by $t+1$ in , we obtain $$\label{poly_2}
\Phi_+(\lambda,t+1)=\Big\{\Phi_-(\lambda,t+2)-\frac{1-\alpha}{\lambda}
\Phi_-(\lambda,t+1)\Big\}\frac{\lambda}{\beta}.$$ Using successively , and , we have: $$\begin{aligned}
\alpha\lambda\Phi_-(\lambda,t)&=&\Phi_+(\lambda,t+1)-(1-\beta)\lambda\Phi_+(\lambda,t)\\
&=&\frac{\lambda}{\beta}\left\{\Phi_-(\lambda,t+2)-\frac{1-\alpha}{\lambda}\Phi_-(\lambda,t+1)
\right\}\\
&&-\frac{1-\beta}{\beta}\lambda^2\left\{\Phi_-(\lambda,t+1)-\frac{1-\alpha}{\lambda}
\Phi_-(\lambda,t)\right\}.\end{aligned}$$ Finally $$\Phi_-(\lambda,t+2)-\left(\frac{1-\alpha}{\lambda}+(1-\beta)\lambda
\right)\Phi_-(\lambda,t+1)+\Big((1-\alpha)(1-\beta)-\alpha\beta\Big)\Phi_-(\lambda,t)=0.$$ The proof concerning $f(\lambda,t)=\Phi_-(\lambda,t)$ is similar and is left to the reader.
In order to obtain the explicit form of $\Phi_-(\lambda,t)$ and $\Phi_+(\lambda,t)$ in terms of $\lambda$ and $t$, it suffices to compute the roots $\theta_-$ and $\theta_+$ of the following polynomial $$\label{polybis}
\theta^2-\left(\frac{1-\alpha}{\lambda}+(1-\beta)\lambda\right)\theta+1-\alpha-\beta=0$$ Its discriminant equals $$\label{5A}
\mathcal{D}=\Big(\frac{1-\alpha}{\lambda}+(1-\beta)\lambda
\Big)^2-4(1-\alpha-\beta).$$ It is clear that $$\begin{aligned}
\label{5AA}
\mathcal{D}&=&\Big(\frac{1-\alpha}{\lambda}+(1-\beta)\lambda+2\sqrt{\rho}\Big)
\Big(\frac{1-\alpha}{\lambda}+(1-\beta)\lambda-2\sqrt{\rho}\Big)\nonumber\\
&=&\frac{1}{\lambda}\Big(\frac{1-\alpha}{\lambda}+(1-\beta)\lambda+2\sqrt{\rho}\Big)
\Big((1-\beta)\lambda^2-2\sqrt{\rho}\lambda+1-\alpha\Big).\end{aligned}$$ Since the discriminant of $\lambda\to (1-\beta)\lambda^2-2\sqrt{\rho}\lambda+1-\alpha$ is equal to $-4\alpha\beta$ then $\mathcal{D}>0$ for any $\lambda>0$.\
Consequently the roots of are: $$\label{B1}
\theta_\pm=\frac{1}{2}\Big(\frac{1-\alpha}{\lambda}+(1-\beta)\lambda
\pm\sqrt{\mathcal{D}}\Big).$$ We deduce the following result.
\[a+a-1\] 1) The moment generating function $\Phi(\lambda,t)$ satisfies $$\label{puiss} \Phi(\lambda,t)=a_+\theta_+^t+a_-\theta_-^t$$ with $$a_+=\frac{1-\alpha+\lambda(\lambda\alpha-\theta_-)}{\lambda^2\sqrt{\mathcal{D}}}
\quad\quad\mbox{and}\quad
a_-=\frac{1}{\lambda}-a_+\quad\mbox{if}\ X_0=Y_0=-1$$ and $$a_+=\frac{(1-\beta)\lambda^2+\beta-\lambda\theta_-}{\sqrt{\mathcal{D}}}\quad
\quad\mbox{and}\quad
a_-=\lambda-a_+\quad\mbox{if}\ X_0=Y_0=1.$$
Suppose that $X_0=Y_0=-1$. Let us first determine the values of the generating function at time $t=0$ and $t=1$: $$\Phi(\lambda,0)=\Phi_+(\lambda,0)+\Phi_-(\lambda,0)=\frac{1}{\lambda}\P(Y_0=-1)
+\lambda\P(Y_0=1)=\frac{1}{\lambda}=a_++a_-$$ Moreover, using and with $t=0$, we get $$\Phi(\lambda,1)=\Phi_+(\lambda,1)+\Phi_-(\lambda,1)=\Big(\frac{1-\alpha}{\lambda}
+\alpha\lambda
\Big)\Phi_-(\lambda,0)=\frac{1-\alpha}{\lambda^2}+\alpha=
a_+\theta_++ a_-\theta_-.$$ It is clear that Lemma \[\*\*\*3\] and $\Phi(\lambda,t)=\Phi_+(\lambda,t)+\Phi_-(\lambda,t)$ implies that $\Phi(\lambda,t)$ satisfies . Then follows by standard arguments. The second case $X_0=Y_0=1$ can be proved in a similar way.
Proof of Theorem \[conv1\] {#***}
--------------------------
We keep the notations given in Section \[section\_not\]. Let $\alpha_0$ and $\beta_0$ be two real numbers in $[0,1]$. Let $\Delta_x$ be a small space parameter so that: $$0\le \alpha_0+c_0\Delta_x\le 1,\quad 0\le \beta_0+c_1\Delta_x\le 1,$$ where $c_0$ and $c_1$ belong to $\Rset$.\
Note that $\alpha_0>0$ (resp. $\beta_0>0$) implies that $\alpha_0+c_0\Delta_x> 0$ (resp. $\beta_0+c_1\Delta_x> 0$) when $\Delta_x$ is small enough. If $\alpha_0<1$ (resp. $\beta_0<1$), similarly $\alpha_0+c_0\Delta_x< 1$ (resp. $\beta_0+c_1\Delta_x< 1$) as soon as $\Delta_x$ is small. In the case $\alpha_0=1$ (resp. $\beta_0=1$) $c_0$ (resp. $c_1$) has to be chosen in $]-\infty,0]$.\
We assume that the coefficients of the transition probability matrix $\pi^\Delta$ of the Markov chain $(Y_t)$ satisfy: $$\label{*3**}
\alpha=\alpha_0+c_0\Delta_x,\quad \beta=\beta_0+c_1\Delta_x$$ i.e. $\pi^\Delta$ is given by . $(X_t)$ is defined by and $(Z_s^\Delta)$ is the normalized persistent random walk: $$Z^\Delta_s=\Delta_x X_{s/\Delta_t}, \quad (\Delta_t>0, \ \Delta_x>0, \ s\in\Delta_t\Nset).$$ $(\tilde{Z}_t^\Delta;\ t\ge 0)$ denotes the linear interpolation of $(Z_t^\Delta)$.\
Recall that $\rho_0=1-\alpha_0-\beta_0$ and $\eta_0=\beta_0-\alpha_0$. Note that $\rho_0\neq 1\Longleftrightarrow \alpha_0+\beta_0\neq 0$
\[conv11\] Let $\rho_0\neq1$,\
1) if $r\Delta_t=\Delta_x$ with $r>0$ then $\tilde{Z}^\Delta_t$ converges towards the deterministic limit $-\frac{rt\eta_0}{1-\rho_0}$ as $\Delta_x$ tends to $0$.\
2) if $r\Delta_t=\Delta_x^2$ with $r>0$, $\tilde{Z}^\Delta_t+\frac{t\sqrt{r}\eta_0}{(1-\rho_0)\sqrt{\Delta_t}}$ converges in distribution to the Gaussian law with mean $$\label{moyenne} m=rt\Big(\frac{-\overline{c}}{1-\rho_0}+\frac{\eta_0
c}{(1-\rho_0)^2} \Big)$$ and variance $$\label{variance}
\sigma^2=\frac{r(1+\rho_0)}{1-\rho_0}\Big(1-\frac{\eta_0^2}{(1-\rho_0)^2}\Big)t,$$ where $$\label{encorcoeff} c=c_0+c_1\quad \mbox{and}\quad
\overline{c}=c_1-c_0.$$
We shall prove the statement under the condition $X_0=Y_0=-1$. If $X_0=Y_0=+1$, the limit is obtained by changing the sign and replacing $c_0$ (resp. $c_1$) by $c_1$ (resp. $c_0$).\
[**1)**]{} Let $\Phi(\lambda,t)$ be the generating function associated with $X_t$. In order to determine the limit distribution of $Z_t^\Delta$, let us introduce: $$\label{lapla} \phi(\mu,t)=\mathbb{E}_{-1}[e^{-\mu \tilde{Z}_t^\Delta}],$$ where $\E_{-1}$ denotes the expectation when $Y_0=-1$. Observe that $$\label{5.22b} \phi(\mu,t)=\Phi(e^{-\mu\Delta_x},\frac{t}{\Delta_t})=
\mathbb{E}_{-1}[e^{-\mu \Delta_x X(t/\Delta_t)}],$$ when $t/\Delta_t\in\Nset$.\
According to Proposition \[a+a-1\], when $t/\Delta_t\in\Nset$, $\phi(\mu,t)$ can be expressed in terms of $a_+$, $a_-$ and $\sqrt{\mathcal{D}}$.\
First let us study the asymptotic expansion of the discriminant $\mathcal{D}$ as $\Delta_x\to 0$. It is convenient to set: $$\label{star}
\bar{\delta}=c_0\Delta_x\quad\mbox{and}\quad \hat{\delta}=c_1\Delta_x.$$ Applying with $\alpha=\alpha_0+\bar{\delta}$ and $\beta=\beta_0
+\hat{\delta}$ we have: $$\mathcal{D}=\left((1-\alpha_0-\bar{\delta})e^{\mu
\Delta_x}+(1-\beta_0-\hat{\delta})e^{-\mu
\Delta_x}\right)^2-4(1-\alpha_0-\beta_0-\bar{\delta}-\hat{\delta}).$$ By we get $$\begin{aligned}
\label{delta}
\mathcal{D}&=\Big(&(2-\alpha_0-\beta_0)+\Delta_x\Big(\mu(\beta_0-\alpha_0)-c_0-c_1\Big)\\
&&+\Delta_x^2\left(\frac{\mu^2}{2}\,(2-\alpha_0-\beta_0)+\mu(c_1-c_0)\right)
+o(\Delta_x^2)\Big)^2\nonumber\\
&-&4\Big(1-\alpha_0-\beta_0-\Delta_x(c_0+c_1)\Big)\end{aligned}$$ It is clear that $\mathcal{D}$ admits the following asymptotic expansion, as $\Delta_x\to 0$: $$\mathcal{D}=A_0+A_1\Delta_x+A_2\Delta_x^2+o(\Delta_x^2)$$ It is usefull to note that $\alpha_0$ and $\beta_0$ can be expressed in terms of $\eta_0$ and $\rho_0$: $$\alpha_0=\frac{1-\eta_0-\rho_0}{2}\quad\mbox{and}\quad
\beta_0=\frac{1+\eta_0-\rho_0}{2}.$$ Let us compute $A_0$, $A_1$ and $A_2$ using standard analysis: $$A_0=(2-\alpha_0-\beta_0)^2-4(1-\alpha_0-\beta_0)=\alpha_0^2+\beta_0^2
+2\alpha_0\beta_0=(\alpha_0+\beta_0)^2=(1-\rho_0)^2.$$ $$\begin{aligned}
\label{5B}
A_1&=&2(2-\alpha_0-\beta_0)\Big(\mu(\beta_0-\alpha_0)-(c_0+c_1)\Big)+4(c_0+c_1)\nonumber\\
&=&2\mu(2-\alpha_0-\beta_0)(\beta_0-\alpha_0)-4(c_0+c_1)+2(\alpha_0+\beta_0)(c_0+c_1)
+4(c_0+c_1)\nonumber\\
&=&2\Big\{\mu(2-\alpha_0-\beta_0)(\beta_0-\alpha_0)+(\alpha_0+\beta_0)(c_0+c_1)\Big\}\nonumber\\
&=&2\Big(\mu\eta_0(1+\rho_0)+c(1-\rho_0)\Big).\end{aligned}$$ $$\begin{aligned}
\label{5C}
A_2&=&2(2-\alpha_0-\beta_0)\Big(\frac{\mu^2}{2}(2-\alpha_0-\beta_0)
+\mu(c_1-c_0)\Big)+\Big(\mu(\beta_0-\alpha_0)-(c_0+c_1)\Big)^2\nonumber\\
&=&\mu^2\Big((2-\alpha_0-\beta_0)^2+(\beta_0-\alpha_0)^2 \Big)\nonumber\\
&&+2\mu\Big((2-\alpha_0-\beta_0)(c_1-c_0)-(\beta_0-\alpha_0)(c_0+c_1) \Big) +(c_0+c_1)^2\nonumber\\
&=&2\mu^2\Big((\alpha_0-1)^2+(\beta_0-1)^2\Big)+4\mu\Big((1-\beta_0)c_1-(1-\alpha_0)c_0 \Big)
+(c_0+c_1)^2\nonumber\\
&=&\mu^2(\eta_0^2+(1+\rho_0)^2)+2\mu\Big((1+\rho_0)\overline{c}-\eta_0
c\Big)+c^2.\end{aligned}$$ Under the condition $\rho_0\neq 1$, we have $$\sqrt{\mathcal{D}}=(1-\rho_0)\sqrt{1+\frac{A_1}{(1-\rho_0)^2}\Delta_x
+\frac{A_2}{(1-\rho_0)^2}\Delta_x^2+o(\Delta_x^2)}$$ Hence $$\sqrt{\mathcal{D}}=B_0+B_1\Delta_x+B_2\Delta_x^2+o(\Delta_x^2)$$ with $$B_0=1-\rho_0,$$ $$\begin{aligned}
B_1&=&\frac{1}{2}\frac{A_1}{1-\rho_0}=\frac{1}{1-\rho_0}
\Big\{\mu(2-\alpha_0-\beta_0)(\beta_0-\alpha_0)+(\alpha_0+\beta_0)(c_0+c_1)\Big\}\\
&=&\mu \,\frac{\eta_0(1+\rho_0)}{1-\rho_0}+c\end{aligned}$$ $$\begin{aligned}
\label{5D}
B_2&=&\frac{1}{2}\frac{A_2}{1-\rho_0}-\frac{1}{8}\frac{A_1^2}{(1-\rho_0)^3}.\end{aligned}$$ As a result, $B_2$ is a second order polynomial function with respect to the $\mu$-variable: $$B_2=\mu^2 B_{22}+\mu B_{21}+B_{20}.$$ Identities , and imply: $$B_{20}=\frac{c^2}{2(1-\rho_0)}-\frac{\Big(2c(1-\rho_0)\Big)^2}{8(1-\rho_0)^3}=0$$ $$B_{21}=\frac{1}{2}\frac{2\Big((1+\rho_0)\overline{c}-\eta_0
c\Big)}{1-\rho_0}-\frac{1}{8}\frac{8\eta_0
c(1-\rho_0)(1+\rho_0)}{(1-\rho_0)^3}=\overline{c}\frac{1+\rho_0}{1-\rho_0}-\frac{2\eta_0
c}{(1-\rho_0)^2}$$ $$\begin{aligned}
B_{22}&=&\frac{1}{2}\frac{\eta_0^2+(1+\rho_0)^2}{1-\rho_0}
-\frac{1}{8}\frac{4\eta_0^2(1+\rho_0)^2}{(1-\rho_0)^3}=
\frac{1}{2}\frac{\Big(\eta_0^2+(1+\rho_0)^2\Big)(1-\rho_0)^2
-\eta_0^2(1+\rho_0)^2}{(1-\rho_0)^3}\\
&=&\frac{(1+\rho_0)^2}{2(1-\rho_0)}-\frac{2\eta_0^2\rho_0}{(1-\rho_0)^3}.\end{aligned}$$ Consequently $$\begin{aligned}
\label{B3}
\sqrt{\mathcal{D}}&=&1-\rho_0+\Big(\mu\,\frac{\eta_0(1+\rho_0)}{1-\rho_0}
+c\Big)\Delta_x+\Big\{\mu^2\Big(\frac{(1+\rho_0)^2}{2(1-\rho_0)}
-\frac{2\eta_0^2\rho_0}{(1-\rho_0)^3}\Big)\nonumber\\
&&+\mu\Big(\overline{c}\,\frac{1+\rho_0}{1-\rho_0}-\frac{2\eta_0
c}{(1-\rho_0)^2}\Big) \Big\}\Delta_x^2+o(\Delta_x^2).\end{aligned}$$ [**2)**]{} The first order development suffices to determine the limit of $\phi(\mu,t)$ as $\Delta_x\to 0$. Indeed $$\begin{aligned}
\label{B2}
\sqrt{\mathcal{D}}&=&1-\rho_0+\frac{\Delta_x}{1-\rho_0}\Big\{\mu\eta_0(1+\rho_0)
+c(1-\rho_0)\Big\}+o(\Delta_x).\end{aligned}$$ From and we can easely deduce $$\begin{aligned}
\theta_\pm&=&\frac{1}{2}(1+\rho_0)+\frac{\Delta_x}{2}(\mu\eta_0-c)
\pm\frac{1}{2}\Big\{1-\rho_0+\Delta_x\Big(\frac{\mu\eta_0(1+\rho_0)}{1-\rho_0}+c\Big)\Big\}
+o(\Delta_x).\end{aligned}$$ Then $$\label{R4} \theta_+=1+\Delta_x\frac{\mu\eta_0}{1-\rho_0}+o(\Delta_x)
\quad\mbox{and}\quad\theta_-=\rho_0-\Delta_x\Big(\frac{\mu\eta_0\rho_0}{1-\rho_0}
+c\Big)+o(\Delta_x).$$ Let $t'=\lfloor\frac{t}{\Delta_t} \rfloor \Delta_t$. Since $\tilde{Z}^\Delta_t=\tilde{Z}^\Delta_{t'}+(t-t')\Delta_x Y_{\lfloor t/\Delta_t\rfloor+1}$ and $\vert Y_n\vert\le 1$, then $$\label{5.31b}
\vert \phi(\mu,t)-\phi(\mu,t')\vert\le C\Delta_x\Delta_t,$$ where $C$ is a constant which only depends on $\mu$.\
Recall that identity and Proposition \[a+a-1\] lead to $$\label{5.27b}
\phi(\mu,t')=a_+\theta_+^{t'/\Delta_t}+a_-\theta_-^{t'/\Delta_t}$$ where $$\label{R3}
a_+=\frac{(1-\alpha)e^{2\mu\Delta_x}+\alpha-\theta_-e^{\mu\Delta_x}}{\sqrt{\mathcal{D}}}
\quad\quad\mbox{and}\quad a_-=e^{\mu\Delta_x}-a_+.$$ It is obvious that and imply: $\lim_{\Delta_x\to 0}a_+=1$ and $\lim_{\Delta_x\to 0}a_-=0$.\
Since $\lim_{\Delta_t\to
0}\theta_-=\rho_0$ and $-1<\rho_0<1$ then $$\label{reflim} \lim_{\Delta_x\, \Delta_t\to 0}a_-\theta_-^{t'/\Delta_t}=0.$$ Consequently, the second term in tends to $0$. It is important to note that the initial condition $X_0=Y_0=-1$ disappears. Let us study the first term in the right hand side of . Note that $\lim_{\Delta_x\to 0}\theta_+=1$, then if $\Delta_x$ is small enough, we can take the logarithm of $\theta_+$. From a straightforward calculation gives $$\log\theta_+=\Delta_x\frac{\mu\eta_0}{1-\rho_0}+o(\Delta_x)$$ Choosing $r\Delta_t=\Delta_x$ and using , and , we obtain the following limit: $$\lim_{\Delta_x\to 0}\phi(\mu,t)=\exp
\{\frac{r\mu\eta_0 t}{1-\rho_0}\}.$$ Since the convergence holds for any $\mu\in\Rset$, we can conclude (cf Theorem 3 in [@curtiss42]) that $$\lim_{\Delta_x\to 0}\E_{-1}[\exp (iu \tilde{Z}^\Delta_t)]=\exp
\Big\{-\frac{iu r\eta_0 t}{1-\rho_0} \Big\},\quad\mbox{for any}\ u\in\Rset.$$ Thus $\tilde{Z}^\Delta_t$ converges in distribution, as $\Delta_x\to 0$, to the Dirac measure at $-\frac{r\eta_0 t}{1-\rho_0}$.\
[**3)**]{} Next, we consider the convergence of the process $$\xi^\Delta_t=\tilde{Z}^\Delta_t+\frac{t\eta_0\sqrt{r}}{(1-\rho_0)\sqrt{\Delta_t}}.$$ Hence we define $$\psi(\mu,t)=\E_{-1}[e^{-\mu \xi^\Delta_t}]=e^{-\frac{\mu
t\eta_0\sqrt{r}}{(1-\rho_0)\sqrt{\Delta_t}}}\phi(\mu,t).$$ To determine the limit of $\psi(\mu,t)$ as $\Delta_t,\Delta_x\to 0$, from and we may deduce that it suffices to compute the second order development of the root $\theta_+$. Using and we get: $$\begin{aligned}
\theta_+&=&\frac{1}{2}(1+\rho_0)+\frac{\Delta_x}{2}(\mu\eta_0-c)
+\frac{\Delta_x^2}{2}\Big(\frac{\mu^2(1+\rho_0)}{2}+\mu\overline{c} \Big)\\
&+&\frac{1-\rho_0}{2}+\frac{\Delta_x}{2}\Big(\frac{\mu\eta_0(1+\rho_0)}{1-\rho_0}+c \Big)\\
&&+\frac{\Delta_x^2}{2}\Big(\mu^2\Big(\frac{(1+\rho_0)^2}{2(1-\rho_0)}
-\frac{2\eta_0^2\rho_0}{(1-\rho_0)^3}\Big)+\mu\Big(\overline{c}\frac{1+\rho_0}{1-\rho_0}
-\frac{2\eta_0 c}{(1-\rho_0)^2}\Big) \Big)+o(\Delta_x^2).\end{aligned}$$ As a result $$\label{***5D}
\theta_+=1+\Delta_x\frac{\mu\eta_0}{1-\rho_0}+\Delta_x^2\Big(\frac{\mu^2}{2}
\Big(\frac{1+\rho_0}{1-\rho_0}-\frac{2\eta_0^2\rho_0}{(1-\rho_0)^3}\Big)
+\mu\Big(\frac{\overline{c}}{1-\rho_0}-\frac{\eta_0
c}{(1-\rho_0)^2} \Big) \Big)+o(\Delta_x^2).$$ We take $r\Delta_t=\Delta_x^2$. Then
$$\begin{aligned}
\lim_{\Delta_x\to 0}\psi(\mu,t)&=&\lim_{\Delta_x\to 0}
\Big(a_+\theta_+^{rt/\Delta_x^2}\exp\Big\{-\frac{\mu r\eta_0 t}{1-\rho_0}
\frac{1}{\Delta_x}\Big\}\Big)\\
&=&\lim_{\Delta_x\to 0}\exp\Big\{-\frac{\mu r\eta_0 t}{1-\rho_0}
\frac{1}{\Delta_x}+\frac{rt}{\Delta_x^2}\log\theta_+\Big\}.\end{aligned}$$
It is straightforward to deduce $$\label{5K}
\lim_{\Delta_x\to 0}\psi(\mu,t)=\exp\Big\{-m\mu+\frac{\sigma^2\mu^2}{2}\Big\}$$ with $$\label{juil1}
m=r\Big(\frac{-\bar{c}}{1-\rho_0}+\frac{\eta_0 c}{(1-\rho_0)^2}\Big)t$$ $$\label{juil2}
\sigma^2=\frac{r(1+\rho_0)}{1-\rho_0}\Big(1-\frac{\eta_0^2}{(1-\rho_0)^2}\Big)t.$$ [**4)**]{} Since holds for any $\mu\in\Rset$, this implies that $\xi^\Delta_t$ converges in distribution, as $\Delta_x\to 0$, to the Gaussian distribution with mean $m$ and variance $\sigma^2$. (see Theorem 3 in [@curtiss42])
\[multimarginales\] Assume that $\rho_0\neq 1$ and $r\Delta_t=(\Delta_x)^2$. Let us denote $\xi^\Delta$ the process defined by $$\xi^{\Delta}_t=\tilde{Z}^\Delta_t+\frac{t\sqrt{r}\eta_0}{(1-\rho_0)\sqrt{\Delta_t}}.$$ Then $(\xi^{\Delta}_{t_1},\xi_{t_2}^{\Delta},\ldots,\xi^{\Delta}_{t_n})$ converges in distribution, as $\Delta_x\to 0$, towards $(\xi^0_{t_1},\xi^0_{t_2},\ldots,\xi^0_{t_n})$ where $\xi^0$ is given by $$\xi^0_t=r\Big(\frac{-\overline{c}}{1-\rho_0}+\frac{\eta_0 c}{(1-\rho_0)^2}\Big)t
+\sqrt{\frac{r(1+\rho_0)}{1-\rho_0}\Big(1-\frac{\eta_0^2}{(1-\rho_0)^2}\Big)}W_t.$$ ($W_t$, $t\ge 0$) is the one-dimensional Brownian motion starting at $0$.
The proof is only presented in the case $n=2$. For simplicity let $s=t_1<t_2=t$. We are interested in the limit of the random vector $(\xi^\Delta_{s},\xi^\Delta_{t})$. Let us then compute the two dimensional Fourier transform $$\Psi^\Delta(\mu,\lambda)=\E_{-1}\Big[e^{i\mu(\xi^\Delta_t-\xi^\Delta_s)}
e^{i\lambda\xi^\Delta_s}\Big],\quad (\lambda,\mu\in\Rset).$$ Since the process $(X_t, Y_t)$ is Markovian, we obtain $$\begin{aligned}
\Psi^\Delta(\mu,\lambda)&=&\E_{-1}\Big[e^{i\mu\xi^\Delta_{t-s}}\Big]
\E_{-1}\Big[1_{\{Y(s/\Delta_t)=-1\}}e^{i\lambda\xi^\Delta_s}\Big]\\
&+&\E_{+1}\Big[e^{i\mu\xi^\Delta_{t-s}}\Big]\E_{-1}\Big[1_{\{Y(s/\Delta_t)=+1\}}
e^{i\lambda\xi^\Delta_s}\Big],\end{aligned}$$ when $s/\Delta_t$ and $t/\Delta_t$ belongs to $\Nset$.\
Note that $\vert \xi^\Delta_u-\xi^\Delta_{u'}\vert\le\Delta_x\Delta_t$ when $u'=\Big\lfloor\frac{u}{\Delta_t}\Big\rfloor \Delta_t$. Consequently $$\begin{aligned}
&&\Psi^\Delta(\mu,\lambda)\underset{\Delta_x\to 0}{\sim}\E_{-1}\Big[e^{i\mu\xi^\Delta_{t'-s'}}\Big]
\E_{-1}\Big[1_{\{Y(s'/\Delta_t)=-1\}}
e^{i\lambda\xi^\Delta_{s'}}\Big]\\
&&+\E_{+1}\Big[e^{i\mu\xi^\Delta_{t'-s'}}\Big]\E_{-1}\Big[1_{\{Y(s'/\Delta_t)=+1\}}
e^{i\lambda\xi^\Delta_{s'}}\Big], \quad(s'=\lfloor s/\Delta_t\rfloor\Delta_t,\
t'=\lfloor t/\Delta_t\rfloor \Delta_t).\end{aligned}$$ According to Proposition \[conv1\], $$\lim_{\Delta_x\to
0}E_{-1}\Big[e^{i\mu\xi^\Delta_{t'-s'}}\Big]=\lim_{\Delta_x\to
0}E_{+1}\Big[e^{i\mu\xi^\Delta_{t'-s'}}\Big]=e^{(i\mu m-\frac{\sigma^2}{2}\,\mu^2)(t-s)}$$ where $m$ and $\sigma^2$ are defined by , resp. . Then we can deduce: $$\begin{aligned}
\lim_{\Delta_x\to
0}\Psi^\Delta(\mu,\lambda)&=&e^{(i\mu m-\frac{\sigma^2}{2}\,\mu^2)(t-s)}
\lim_{\Delta_x\to 0}\E_{-1}\Big[e^{i\lambda\xi^\Delta_{s'}}\Big]\\
&=&e^{(i\mu m-\frac{\sigma^2}{2}\,\mu^2)(t-s)}
\lim_{\Delta_x\to 0}\E_{-1}\Big[e^{i\lambda\xi^\Delta_{s}}\Big]\\
&=&e^{(i\mu m-\frac{\sigma^2}{2}\,\mu^2)(t-s)}e^{(i\lambda m-\frac{\sigma^2}{2}\,\lambda^2)s}\\
&=&\E\Big[\exp\{i\mu(\xi_t^0-\xi_s^0)+i\lambda\xi_s^0\} \Big]\end{aligned}$$
We are now able to end the proof of Theorem \[conv1\] (item 2). We may apply, without any change, the measure tension criterium used in the proof of convergence of $(Z^\Delta_t)$ in the case $\alpha_0=\beta_0=1$ (see the end of Section \[sectionpreuve1\]). This, and Proposition \[multimarginales\] show that $(\xi_t^\Delta)_{t\ge 0}$ converges in distribution as $\Delta_x\to 0$ to the Brownian motion with drift $(\xi_t^0)_{t\ge 0}$.
Proof of Proposition \[cas\_limit\] {#scas_limit}
-----------------------------------
We suppose $\alpha_0=\beta_0=1$, $c_1=c_0<0$ and $r\Delta_t=\Delta_x^3$ where $r>0$.\
We briefly sketch the proof of Proposition \[cas\_limit\]. The approach is similar to the one developed in the case 2) of Theorem \[conv1\]. We only prove that $\tilde{Z}^\Delta_t$ converges to the Gaussian distribution with $0$-mean and variance equals $-rc_0 t$. Using Theorem 3 in [@curtiss42], it is equivalent to show $$\lim_{\Delta_x\to 0}\E_{-1}\Big[e^{-\mu \tilde{Z}_t^\Delta} \Big]=
e^{\frac{-rc_0 t \mu^2}{2}},\quad \forall \mu\in\Rset.$$ We have already observed that we may reduce to the case $t/\Delta_t\in\Nset$; in this case we have $\tilde{Z}_t^\Delta=Z_t^\Delta$ and $$\E_{-1}\Big[ e^{-\mu Z_t^\Delta} \Big]=\Phi\Big(e^{-\mu\Delta_x},\frac{t}{\Delta_t}\Big)$$ where $\Phi(\lambda,t)$ is the moment generating function associated with $(X_t)$ (see the beginning of subsection \[mgf\]). Recall that $\Phi(\lambda,t)$ is given by identity .\
Note that: $$\alpha=\alpha_0+c_0\Delta_x=1+c_0\Delta_x,\quad \beta=\beta_0+c_0\Delta_x=1+c_0\Delta_x.$$ Since $\alpha$ and $\beta$ have to belong to $[0,1]$, this implies that $c_0<0$. Recall that $\mathcal{D}$, $\theta_+$ and $\theta_-$ are the real numbers which have been defined by resp. (with $\lambda=e^{-\mu\Delta_x}$). We have: $$\mathcal{D}=4c_0^2\Delta_x^{2}\cosh^2(\mu\Delta_x)+4(1+2c_0\Delta_x),$$ $$\theta_\pm=-c_0\Delta_x\cosh(\mu\Delta_x)\pm\sqrt{c_0^2\Delta_x^{2}\cosh^2(\mu\Delta_x)
+1+2c_0\Delta_x.
}$$ Using classical analysis we get: $$\begin{aligned}
\sqrt{\mathcal{D}}/2&=&\sqrt{1+2c_0\Delta_x+c_0^2\Delta_x^2
+o(\Delta_x^3)}=1+c_0\Delta_x+o(\Delta_x^3),\end{aligned}$$ $$\begin{aligned}
\theta_+
=1-\frac{c_0\mu^2}{2}\Delta_x^3+o(\Delta_x^3),\quad \theta_-=-1-2c_0\Delta_x+o(\Delta_x).\end{aligned}$$ $$\lim_{\Delta_x\to 0}a_+=1,\quad\lim_{\Delta_x\to 0}\theta_+^{t/\Delta_t}=
\lim_{\Delta_x\to 0}\exp\Big\{-\frac{t}{\Delta_t}\,
\frac{c_0\mu^2}{2}\Delta_x^3\Big\}=\exp\Big\{-c_0r\frac{\mu^2}{2}\,t\Big\},$$ $$\lim_{\Delta_x\to 0}a_-=0,\quad
\lim_{\Delta_x\to 0}\vert\theta_-\vert^{t/\Delta_t}=\lim_{\Delta_x\to 0}
\exp\Big\{\frac{t}{\Delta_t}\, 2c_0\Delta_x\Big\}=\lim_{\Delta_x\to 0}
\exp\Big\{\frac{2c_0 rt}{\Delta_x^2}\Big\}=0\quad (c_0<0).$$
Relation implies that the variable $Z^\Delta_t$ is asymptotically normal distributed with variance $-rc_0t$. [ ]{}
|
---
abstract: 'This paper presents a new scheme to treat escaping stars in the orbit-averaged Fokker-Planck models of globular star clusters in a galactic tidal field. The existence of a large number of potential escapers, which have energies above the escape energy but are still within the tidal radius, is taken into account in the models. The models allow potential escapers to experience gravitational scatterings before they leave clusters and thus some of them may lose enough energy to be bound again. It is shown that the mass evolution of the Fokker-Planck models are in good agreement with that of $N$-body models including the full tidal-force field. The mass-loss time does not simply scale with the relaxation time due to the existence of potential escapers; it increases with the number of stars more slowly than the relaxation time, though it tends to be proportional to the relaxation time in the limit of a weak tidal field. The Fokker-Planck models include two parameters, the coefficient $\gamma$ in the Coulomb logarithm $\ln (\gamma N)$ and the coefficient $\nu_{\rm e}$ controlling the efficiency of the mass loss. The values of these parameters are determined by comparing the Fokker-Planck models with the $N$-body models. It is found that the parameter set $(\gamma, \nu_{\rm e})=(0.11, 7)$ works well for both single-mass and multi-mass clusters, but that the parameter set $(\gamma, \nu_{\rm e})=(0.02, 40)$ is another possible choice for multi-mass clusters.'
author:
- |
K. Takahashi$^{1}$[^1] and H. Baumgardt$^{2}$\
$^{1}$Department of Informational Society Studies, Faculty of Human and Social Studies, Saitama Institute of Technology,\
1690 Fusaiji, Fukaya, Saitama 369-0293, Japan\
$^{2}$School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia
date: 'Accepted 201x xx xx. Received 2011 xx xx; in original form 2011 xx xx'
title: 'Tidal mass loss in star clusters and treatment of escapers in Fokker-Planck models'
---
\[firstpage\]
stellar dynamics – globular clusters: general – galaxies: star clusters: general – methods: numerical
Introduction {#sec:intro}
============
The numerical integration scheme of the orbit-averaged Fokker-Planck (FP) equation developed by @c79 has been one of the most useful tools for simulating the dynamical evolution of globular star clusters. In addition to two-body relaxation, many physical processes have been incorporated into FP models to achieve realistic modelling of the globular cluster evolution; these processes include tidal cutoff, binary heating, disc and bulge shocks, mass loss via stellar evolution, etc. (see @skt08 for a recent example of detailed FP modelling).
In this paper we consider the dynamical evolution of globular clusters in a steady galactic tidal field. Our main purpose is to investigate what boundary condition can give a better description of escape of stars from clusters in the tidal field. This study has been motivated by the studies by @fh00 and @b01.
@fh00 found that a large fraction of stars with energies above the escape energy (i.e. potential escapers) take much longer escape time than the dynamical time. Until their study it had been generally thought that the escape time-scale is of the order of the dynamical time and that the mass-loss times of the clusters essentially scale with the relaxation time, which is much longer than the dynamical time. The findings of @fh00 indicate that this simple scaling may be spoiled by potential escapers with long escape times.
In fact @b01 performed $N$-body simulations and showed that the mass-loss times (lifetimes) of clusters do not scale with the relaxation time $t_{\rm rh}$ but scale with $t_{\rm rh}^{3/4}$. He concluded that the reason is that some of potential escapers are scattered back to lower energies before they leave the cluster. More recently @tf05 showed that the dependence on the relaxation time changes with the strength of the tidal filed. These two studies have revealed that the behavior of potential escapers greatly influences the rate of mass loss from clusters in the tidal field.
The effects of long escape times and re-scattering of potential escapers have never been considered in previous FP models in the literature, but it was assumed that escapers leave a cluster on the dynamical time-scale, as is described in detail in Section \[sec:FP\]. Since the effect of the galactic tidal field is essentially important to the cluster evolution, it is necessary to find a way to include the effect into FP models as precisely as possible.
We should mention that Takahashi & Portegies Zwart (1998, 2000) compared FP and $N$-body models of star clusters in the tidal field and found good agreement between these two theoretical models over a wide range of initial conditions. They showed that the use of anisotropic FP models with the apocentre escape criterion [@tli97] and the dynamical-time removal of escapers [@lo87] is necessary to obtain such good agreement. However, note that in their $N$-body models the tidal force field is not included but the tidal cutoff is applied. @tpz00 confirmed that the difference between tidal cutoff and self-consistent tidal field $N$-body models is small for a particular set of initial conditions, but did not do systematic investigations on this problem.
In this study we have devised a new scheme to treat escapers in FP models. The scheme defines a region of potential escapers in phase space and allows them to be scattered again. Comparing the results of FP models calculated with the new scheme with the results of $N$-body models, we examine the accuracy of the FP models.
Fokker–Planck models of star clusters in a steady tidal field {#sec:FP}
=============================================================
Basic assumptions
-----------------
The orbit-averaged FP equation is derived under the assumption of spherical symmetry of star clusters [@c79]. Therefore the tidal field, which is not spherically symmetric, cannot be directly incorporated into orbit-averaged FP models. In FP models the effect of the tidal field is taken into account by imposing a tidal cutoff radius $r_{\rm t}$ on the cluster, which is treated as an isolated system in other respects. Under these assumptions the distribution function $f$ of stars at time $t$ depends only on the energy of a star per unit mass, $E$, and the angular momentum per unit mass, $J$.
Classical treatments of escapers
--------------------------------
First we summarise classical treatments of escapers used in FP models of previous studies.
### Escape criteria in phase space {#sec:esc_crit}
In previous studies, two kinds of criteria were adopted to define an escape region in $(E,J)$-space:
1. Energy criterion $$E > E_{\rm t} \equiv -\frac{GM}{r_{\rm t}} \label{eq:eng-crit},$$
2. Apocentre criterion $$r_{\rm a}(E,J) > r_{\rm t} \label{eq:apc-crit},$$
where $M$ is the cluster mass and $r_{\rm a}(E,J)$ is the apocentre radius of a star having energy $E$ and angular momentum $J$. It is assumed that a star is destined to escape once it enters into the escape region.
The apocentre criterion [@tli97] is considered to be more realistic, at least as long as the tidal field is modelled as a radial cut-off, and in fact gives better agreement between FP and $N$-body models [@tpz98; @tpz00]. For isotropic FP models, where the distribution function does not depend on $J$, only the energy criterion can be applied (e.g. @lo87).
### Removal of escapers
In previous studies stars in the escape region are assumed to leave the cluster inevitably, as mentioned above. It is also assumed that the time required for this travel is of the order of the dynamical time at the tidal radius. Considering this travel time, @lo87 applied the following equation to the distribution function $f$ in the escape region: $$\frac{\partial f}{\partial t}
= - \nu_{\rm e} f \left[ 1- \left( \frac{E}{E_{\rm t}} \right)^3 \right]^{1/2}
/t_{\rm tid},
\label{eq:lo}$$ where $\nu_{\rm e}$ is a dimensionless constant determining the efficiency of escape (see also @lfr91). The time-scale $t_{\rm tid}$ is an orbital time-scale at the tidal radius defined by $$t_{\rm tid}=\frac{2\pi}{\sqrt{(4\pi/3){G \rho_{\rm t}}}},$$ where $\rho_{\rm t}$ is the mean mass density within the tidal radius.
Since the dynamical time is generally much smaller than the relaxation time in globular clusters, we may assume that escapers leave the cluster immediately after they enter into the escape region, when we are interested only in the evolution on the relaxation time-scale. This assumption leads to the boundary condition $$f=0$$ on the tidal boundary (e.g. @cw90).
A new treatment of escapers {#sec:nt}
---------------------------
The boundary condition of equation (\[eq:lo\]) takes account of the fact that stars satisfying the escape criterion, i.e. potential escapers, need time to actually leave the cluster. However the effect of re-scattering of potential escapers is not considered there. Here we propose a new scheme in which the re-scattering effect is taken into account.
First we summarise basic assumptions and equations. Suppose that the cluster is on a circular orbit, with radius $R_{\rm G}$ and angular velocity $\omega$, round the centre of a spherical galaxy. We consider the motion of a star in the rotating coordinate system moving with the cluster; the origin is at the cluster centre, the $x$-axis points to the galactic centre, and the $y$-axis is in the cluster orbital plane. If the cluster and the galaxy are treated as point masses $M$ and $M_{\rm G}$ ($\gg M$) and the size of the cluster is much smaller than $R_{\rm G}$, there exists a conserved quantity known as the Jacobi integral given by $$E_{\rm J}=\frac{v^2}{2}-\frac{GM}{r}-\frac12 \omega^2 (3x^2-z^2) \label{eq:ej}$$ (cf. @s87, Chapt. 5). Here $v$ is the velocity of the star measured in the rotating frame, $r$ is the distance from the star to the cluster centre, and the angular velocity $\omega$ is given by $$\omega=\sqrt{\frac{GM_{\rm G}}{R_{\rm G}^3}} .$$ The third term on the right-side in equation (\[eq:ej\]) is a combination of the centrifugal and tidal potentials.
The effective potential is defined as $$\phi_{\rm eff}(x,y,z)=-\frac{GM}{r}-\frac12 \omega^2 (3x^2-z^2).$$ A contour plot of $\phi_{\rm eff}$ is shown, e.g., in Fig. 5.1 of @s87. The effective potential has the saddle points at $(\pm x_{\rm e},0,0)$, where $$x_{\rm e}=\left( \frac{M}{3M_{\rm G}}\right)^{1/3} R_{\rm G}$$ and $$\phi_{\rm eff}(\pm x_{\rm e},0,0) =-\frac32 \frac{GM}{x_{\rm e}}. \label{eq:pxe}$$ The equipotential surface passing through these saddle points intersects with the $y$-axis at $y=\pm y_{\rm e}$, where $$y_{\rm e}=\frac23 x_{\rm e}. \label{eq:ye}$$ The necessary condition for escape of a star from the cluster is given by $$E_{\rm J} > E_{\rm J,crit} \equiv -\frac32 \frac{GM}{x_{\rm e}}. \label{eq:ejcrit}$$ Note that equations (\[eq:pxe\]), (\[eq:ye\]), and (\[eq:ejcrit\]) are valid for any spherical galactic potential. @fh00 found that the time-scale for escape of stars with $E_{\rm J}>E_{\rm J,crit}$ varies as $$t_{\rm e} \propto ( E_{\rm J}-E_{\rm J,crit})^{-2}. \label{eq:fhte}$$
With this relation in mind we have devised a new scheme to follow the evolution of potential escapers. In this scheme the evolution of the distribution function $f$ for potential escapers is described by $$\frac{\partial f}{\partial t} =
\left( \frac{\partial f}{\partial t} \right)_{\rm coll}
-\frac{f}{t_{\rm e}(E)}, \label{eq:fp_pe}$$ where the first term on the right-side is the FP collision term and the second term represents mass loss due to escape. Here the escape time-scale $t_{\rm e}$ is given by $$\frac{1}{t_{\rm e}(E)} =
\frac{\nu_{\rm e}}{t_{\rm tid}} \left( 1-\frac{E}{E_{\rm crit}} \right)^2, \label{eq:te}$$ where $\nu_{\rm e}$ is a dimensionless numerical constant. It should be noted that energy $E$, not the Jacobi integral $E_{\rm J}$, is used in equations (\[eq:fp\_pe\]) and (\[eq:te\]). Energy $E$ does not include the centrifugal and tidal potentials. Despite this difference, we use the same critical value of energy $$E_{\rm crit} = -\frac32 \frac{GM}{r_{\rm t}}, \label{eq:ecrit}$$ where the tidal radius $r_{\rm t}$ is identified with $x_{\rm e}$. One might think that using equation (\[eq:te\]) with equation (\[eq:ecrit\]) is too crude an approximation, but it brings good agreement between FP and $N$-body models as is shown in Section \[sec:results\].
The most important difference between equations (\[eq:lo\]) and (\[eq:fp\_pe\]) is that the latter includes the collision term. Thus equation (\[eq:fp\_pe\]) allows potential escapers to be scattered back to lower energies. The effect of mass loss is included in both equations in a similar way, though the functional forms of the escape time-scale $t_{\rm e}$ are different.
In this new treatment of the tidal field, the escape criteria described in Section \[sec:esc\_crit\] are modified as follows:
1. Energy criterion $$E > E_{\rm crit} = -\frac32 \frac{GM}{r_{\rm t}} \label{eq:eng-crit_pe},$$
2. Apocentre criterion $$r_{\rm a}(E,J) > \frac23 r_{\rm t} \label{eq:apc-crit_pe}.$$
Note that $\phi_{\rm eff}(0,\pm 2r_{\rm t}/3,0) =\phi(0,\pm 2r_{\rm t}/3,0)=-3GM/2r_{\rm t}$. Equation (\[eq:fp\_pe\]) is applied only in the region where an adopted criterion is satisfied.
The Fokker-Planck code
----------------------
The FP code used in the present study is essentially the same as that used by @tpz00, but adopts the new scheme for treating escapers described above. The code calculates the evolution of the distribution function $f(E,J,t)$. Unlike @tpz00, stellar evolution is not considered in the models presented in this paper. Instead the effect of heating by three-body binaries is considered in the manner described in @t97.
For all the models presented in the present paper, 201 energy mesh points, 51 angular-momentum mesh points, and 101 radial mesh points are used. The meshes are constructed as described in @t95. When calculating the evolution of multi-mass clusters, 10 discrete mass-components are used to represent a continuous mass function.
Our FP models have two free parameters: one is $\nu_{\rm e}$ in equation (\[eq:te\]) and the other is $\gamma$ in the Coulomb logarithm $\ln (\gamma N)$ appearing in the FP collision term. How the value of $\nu_{\rm e}$ is determined is described in Section \[sec:results\]. We set $\gamma=0.11$ [@gh94a] in most of our runs and $\gamma=0.02$ [@gh96] in a part of runs for multi-mass clusters.
Results {#sec:results}
=======
Comparison with $N$-body models: single-mass clusters
-----------------------------------------------------
First we compare FP models with the full tidal field models of @b01 and additional $N$-body runs performed for this comparison. All the model clusters are composed of equal-mass stars and move on circular orbits round a point-mass galaxy. The initial distribution of stars is given by King models [@k66].
Results are presented in $N$-body units, where the initial total mass and energy of a cluster are equal to 1 and $-0.25$, respectively, and the gravitational constant $G=1$. The same units are used throughout this paper.
Here we will refer to FP models with the boundary condition of equation (\[eq:fp\_pe\]) as “FPf” models, which aim to model clusters in a self-consistent [*full tidal field*]{}. FP models with equation (\[eq:lo\]) will be called “FPd” models, where stars beyond the tidal cutoff radius are removed on the [*dynamical time-scale*]{}.
![Evolution of the cluster mass. The solid lines represent FPf models, and the dashed lines represent $N$-body models. The initial models are $W_0=3$ King models with the number of stars $N=1024$, 4096, 16384 and 65536. []{data-label="fig:massW3"}](f1.eps){width="84mm"}
Fig. \[fig:massW3\] compares FPf and $N$-body models concerning the evolution of the total mass of bound stars. The initial models are $W_0=3$ King models with the number of stars $N=1024$, 4096, 16384 and 65536. The new treatment of escapers described by equation (\[eq:fp\_pe\]) with the apocentre criterion of equation (\[eq:apc-crit\_pe\]) is employed in the FPf models. The agreement between the FPf and $N$-body models is good in all the cases. In fact the value of the parameter $\nu_{\rm e}$ in equation (\[eq:fp\_pe\]) has been determined so that good agreement is obtained by performing test runs with different values of $\nu_{\rm e}$ as was done by @tpz00. We have finally chosen the value of $\nu_{\rm e}=7$. All the FPf models shown in Fig. \[fig:massW3\] are calculated with this value.
![Evolution of the ratio of the mass of potential escapers $M_{\rm pe}$ to the total cluster mass $M$. The ratio is plotted as a function of the cluster mass at each instance. []{data-label="fig:mpeW3"}](f2.eps){width="84mm"}
Fig. \[fig:mpeW3\] shows the evolution of the ratio of the mass of potential escapers $M_{\rm pe}$ to the total cluster mass $M$ for the runs shown in Fig. \[fig:massW3\]. The agreement between the FPf and $N$-body models is fairly good also in this comparison. Note that here $M_{\rm pe}$ for the FPf models is defined as the mass of stars with $E>E_{\rm crit}$, although the apocentre criterion is used in the simulations. The mass of stars satisfying the apocentre criterion is smaller than that of stars with $E>E_{\rm crit}$, but shows a similar trend in time variation.
![Half-mass time $T_{\rm half}$ as a function of the initial half-mass relaxation time $t_{\rm rh,i}$ for the initial conditions of $W_0=3$ King models. Two types of FP models, FPf and FPd models (see text), are shown by the circles and crosses, respectively, and $N$-body models are shown by the triangles. The dotted lines represent scalings proportional to $t_{\rm rh,i}$ and $t_{\rm rh,i}^{3/4}$ (they are arbitrarily shifted in a vertical direction). []{data-label="fig:thalfW3"}](f3.eps){width="84mm"}
--------- -------------------- ------------------ ------------------ ------------------
$N$ $t_{\rm rh,i}$ $T_{\rm half}$ $T_{\rm half}$ $T_{\rm half}$
($N$-body) (FPf) (FPd)
128 $5.13\times 10^0$ $8.94\times10^1$ $8.37\times10^1$ $6.76\times10^1$
256 $8.13\times 10^0$ $1.27\times10^2$ $1.21\times10^2$ $9.93\times10^1$
512 $1.35\times 10^1$ $1.83\times10^2$ $1.71\times10^2$ $1.50\times10^2$
1024 $2.30\times 10^1$ $2.59\times10^2$ $2.51\times10^2$ $2.41\times10^2$
2048 $4.01\times 10^1$ $3.73\times10^2$ $3.73\times10^2$ $4.00\times10^2$
4096 $7.11\times 10^1$ $5.58\times10^2$ $5.56\times10^2$ $6.86\times10^2$
8192 $1.28\times 10^2$ $8.41\times10^2$ $8.33\times10^2$ $1.20\times10^3$
16384 $2.32\times 10^2$ $1.18\times10^3$ $1.26\times10^3$ $2.16\times10^3$
32768 $4.24\times 10^2$ $1.96\times10^3$ $1.92\times10^3$ $3.92\times10^3$
65536 $7.82\times 10^2$ $3.05\times10^3$ $2.96\times10^3$ $7.22\times10^3$
131072 $1.45 \times 10^3$ — $4.63\times10^3$ $1.34\times10^4$
262144 $2.71 \times 10^3$ — $7.36\times10^3$ $2.50\times10^4$
524288 $5.07 \times 10^3$ — $1.19\times10^4$ $4.68\times10^4$
1048576 $9.54 \times 10^3$ — $1.97\times10^4$ $8.82\times10^4$
2097152 $1.80 \times 10^4$ — $3.33\times10^4$ $1.67\times10^5$
--------- -------------------- ------------------ ------------------ ------------------
: Half-mass times $T_{\rm half}$ given by $N$-body, FPf, and FPd models for the initial conditions of King models with $W_0=3$.[]{data-label="tab:W3"}
Fig. \[fig:thalfW3\] shows the half-mass time $T_{\rm half}$, which is the time required for a cluster to lose a half of its initial mass, as a function of the initial half-mass relaxation time $t_{\rm rh,i}$. Here the half-mass relaxation time is defined by $$t_{\rm rh} = 0.138\frac{N^{1/2}r_{\rm h}^{3/2}}
{G^{1/2} m^{1/2} \ln (\gamma N)} \label{eq:trh}$$ (@s87, Chapt. 2) with $\gamma=0.11$ [@gh94a]. The results are summarised also in Table \[tab:W3\]. The FPf and $N$-body models show good agreement over the whole range of $N$ where the comparison is made. The scaling $T_{\rm half} \propto t_{\rm rh}^{3/4}$ gives a reasonable fit to the results of these models as @b01 found.
The results of FPd models are also shown in Fig. \[fig:thalfW3\]. In these models the parameter $\nu_{\rm e}=2.5$ is used for equation (\[eq:lo\]) [@tpz00]. The FPd models show clearly a different scaling from the other models; $T_{\rm half} \propto t_{\rm rh}$ expect for models with very short $t_{\rm rh}$ (i.e. small $N$ ).
![Same as Fig. \[fig:thalfW3\], but FPf models with the energy criterion are compared with those with the apocentre criterion and the $N$-body models. []{data-label="fig:thalfW3e"}](f4.eps){width="84mm"}
In Fig. \[fig:thalfW3e\] FPf models with the energy criterion are compared with those with the apocentre criterion as well as the $N$-body models. We have set $\nu_{\rm e}=5$ in the energy-criterion models so that their mass evolution reasonably agrees with that of the $N$-body models for small $N$. There is no significant difference between the energy-criterion models and the other models for $t_{\rm rh,i} \la 100$, but the energy-criterion models tend to lose mass much faster as $t_{\rm rh,i}$ increases. This indicates that the apocentre criterion is a better escape criterion for FPf models.
![Same as Fig. \[fig:thalfW3\], but FPf models with $N$ up to $2^{30}$ are shown. The steeper dotted line represents the relation $T_{\rm half} = t_{\rm rh,i}$. []{data-label="fig:thalfW3ex"}](f5.eps){width="84mm"}
![Logarithmic slope $\alpha = d\log T_{\rm half}/d\log t_{\rm rh,i}$ as a function of the initial half-mass relaxation time $t_{\rm rh,i}$ for the models shown in Fig. \[fig:thalfW3ex\].[]{data-label="fig:thalfW3expw"}](f6.eps){width="84mm"}
$N$ $t_{\rm rh,i}$ $T_{\rm half}$ $T_{\rm half}/t_{\rm rh,i}$
-------------------------------------- -------------------- -------------------- -----------------------------
$2^{22}\ (\approx 4.19 \times 10^6)$ $3.41 \times 10^4$ $5.77 \times 10^4$ 1.69
$2^{23}\ (\approx 8.39 \times 10^6)$ $6.47 \times 10^4$ $1.02 \times 10^5$ 1.57
$2^{24}\ (\approx 1.68 \times 10^7)$ $1.23 \times 10^5$ $1.82 \times 10^5$ 1.48
$2^{25}\ (\approx 3.36 \times 10^7)$ $2.35 \times 10^5$ $3.31 \times 10^5$ 1.41
$2^{26}\ (\approx 6.71 \times 10^7)$ $4.50 \times 10^5$ $6.09 \times 10^5$ 1.35
$2^{27}\ (\approx 1.34 \times 10^8)$ $8.62 \times 10^5$ $1.14 \times 10^6$ 1.32
$2^{28}\ (\approx 2.68 \times 10^8)$ $1.65 \times 10^6$ $2.14 \times 10^6$ 1.30
$2^{29}\ (\approx 5.37 \times 10^8)$ $3.18 \times 10^6$ $4.07 \times 10^6$ 1.28
$2^{30}\ (\approx 1.07 \times 10^9)$ $6.12 \times 10^6$ $7.77 \times 10^6$ 1.27
: Half-mass times $T_{\rm half}$ given by FPf models for the initial conditions of $W_0=3$ King models with very large $N$.[]{data-label="tab:W3ex"}
As stated above, the results of the FPf models shown in Fig. \[fig:thalfW3\] are reasonably well described by the scaling law $T_{\rm half} \propto t_{\rm rh,i}^{3/4}$. However we should not expect this scaling continues to hold in the limit of large $N$. If this scaling continues, the half-mass time measured in the units of the half-mass relaxation time, $T_{\rm half}/t_{\rm rh,i}$, would go to zero as $N \to \infty$. This must be impossible because the mass loss is driven by two-body relaxation. In order to see the scaling of $T_{\rm half}$ in the limit of large $N$, we have calculated FPf models with very large $N$, $N=2^{22}\approx 4.19\times 10^6$ to $2^{30}\approx 1.07\times 10^9$, which are much larger than typical numbers of stars in globular clusters. The results of these models are shown in Table \[tab:W3ex\] and Fig. \[fig:thalfW3ex\]. In Fig. \[fig:thalfW3ex\] we see that $T_{\rm half}$ is nearly proportional to $t_{\rm rh,i}$ for very large $N$ clusters, say, for $t_{\rm rh,i} \ga 10^5$ or $N \ga 10^7$. This trend is more qualitatively shown in Fig. \[fig:thalfW3expw\], where the change in the logarithmic slope, $$\alpha = \frac{d \log T_{\rm half}}{d \log t_{\rm rh,i}}, \label{eq:alpha}$$ is plotted. The slope $\alpha$ approaches one as $N$ increases. The ratio $T_{\rm half}/t_{\rm rh,i} \approx 1.3$ for our largest-$N$ models.
![Same as Fig. \[fig:thalfW3\], but for the initial conditions of $W_0=5$ King models. FPf models with the apocentre criterion and $N$-body models are shown. []{data-label="fig:thalfW5"}](f7.eps){width="84mm"}
--------- -------------------- ------------------ ------------------
$N$ $t_{\rm rh,i}$ $T_{\rm half}$ $T_{\rm half}$
($N$-body) (FPf)
1024 $2.19 \times 10^1$ $3.89\times10^2$ $3.92\times10^2$
2048 $3.82 \times 10^1$ $5.78\times10^2$ $6.07\times10^2$
4096 $6.77 \times 10^1$ $9.51\times10^2$ $9.77\times10^2$
8192 $1.22 \times 10^2$ $1.51\times10^3$ $1.61\times10^3$
16384 $2.21 \times 10^2$ $2.54\times10^3$ $2.67\times10^3$
32768 $4.04 \times 10^2$ $4.14\times10^3$ $4.49\times10^3$
65536 $7.45 \times 10^2$ — $7.62\times10^3$
131072 $1.38 \times 10^3$ — $1.31\times10^4$
262144 $2.58 \times 10^3$ — $2.28\times10^4$
524288 $4.83 \times 10^3$ — $4.03\times10^4$
1048576 $9.09 \times 10^3$ — $7.20\times10^4$
2097152 $1.72 \times 10^4$ — $1.30\times10^5$
--------- -------------------- ------------------ ------------------
: Half-mass times $T_{\rm half}$ given by $N$-body and FPf models for the initial conditions of King models with $W_0=5$.[]{data-label="tab:W5"}
We have performed simulations also for the initial conditions of $W_0=5$ King models. The half-mass times of $N$-body and FPf models for $W_0=5$ are summarised in Table \[tab:W5\] and are plotted in Fig. \[fig:thalfW5\]. Here we find good agreement again. The same parameter $\nu_{\rm e}=7$ is used for both the $W_0=3$ and $W_0=5$ clusters. In Fig. \[fig:thalfW5\] the slope of the $\log t_{\rm rh,i}$–$\log T_{\rm half}$ relation seems to be in between $3/4$ and 1. This point is further examined in subsection \[ssec:field\_strength\].
Dependence on the escape-time function {#ssec:escape_time}
--------------------------------------
@b01 argued that the scaling $T_{\rm half} \propto t_{\rm rh}^{3/4}$ can be explained by a steady state solution of a simple model for the evolution of potential escapers (see equation (12) of his paper). His model adopts the escape time-scale $t_{\rm e}$ of equation (\[eq:fhte\]). If a different function is assumed for $t_{\rm e}$, his model predicts a different scaling law. It is shown that the scaling $$T_{\rm half} \propto t_{\rm rh}^\frac{\beta+1}{\beta+2} \label{eq:thbeta}$$ is obtained for $t_{\rm e} \propto (E-E_{\rm crit})^{-\beta}$ (see Appendix A). It is interesting to see if this prediction is confirmed by the results of our FPf models.
![Same as Fig. \[fig:thalfW3\], but FPf models with different functional forms of $t_{\rm e}(E) \propto (E-E_{\rm crit})^{-\beta} $ ($\beta=1, 2, 3$) are compared. The dotted lines represent scalings $t_{\rm rh}^{2/3}$, $t_{\rm rh}^{3/4}$ and $t_{\rm rh}^{4/5}$, which are predicted by the simple steady-solution model for $\beta=1$, 2 and 3, respectively (see text). []{data-label="fig:thalfW3p"}](f8.eps){width="84mm"}
We have performed FP runs using a generalized form of equation (\[eq:te\]), $$\frac{1}{t_{\rm e}(E)} =
\frac{\nu_{\rm e}}{t_{\rm tid}} \left( 1-\frac{E}{E_{\rm crit}} \right)^\beta, \label{eq:tebeta}$$ with $\beta=1$ and 3. Fig. \[fig:thalfW3p\] plots the half-mass time against the initial half-mass relaxation time for these runs as well as for the standard runs, where King models with $W_0=3$ are used as initial conditions. The value of $\nu_{\rm e}$ has been adjusted so that the non-standard models should have roughly the same half-mass times with those of the standard ones for lower $N$; $\nu_{\rm e}=7/3$ and $7\times 3$ for $\beta=1$ and 3, respectively.
The results of the FPf models actually depend on $\beta$, but the degree of the dependence is weaker than predicted by equation (\[eq:thbeta\]). While this equation predicts the slopes 2/3, 3/4 and 4/5 for $\beta=1$, 2 and 3, respectively, linear least-squares fitting of the data in Fig. \[fig:thalfW3p\] gives the slopes 0.69, 0.72 and 0.75. When the fitting is done only for $N \ge 16384$, the slopes are 0.75, 0.75 and 0.77. Thus the scaling law $T_{\rm half} \propto t_{\rm rh}^{3/4}$ is not a bad approximation in all the cases investigated here. This is not consistent with equation (\[eq:thbeta\]).
Dependence on the strength of the tidal field {#ssec:field_strength}
---------------------------------------------
@tf05 found that the dependence of $T_{\rm half}$ on $t_{\rm rh,i}$ is affected by the strength of the tidal field and that the logarithmic slope $\alpha$, defined by equation (\[eq:alpha\]), approaches unity as the strength of the tidal field decreases. In order to confirm their findings, we have calculated FPf models for the initial conditions where the initial tidal radius $r_{\rm t,i}$ is greater than the King cutoff radius $r_{\rm K}$ (i.e. the radius at which the density drops to zero) for each value of $W_0$. On the other hand, all the models presented above are calculated for the initial conditions with $r_{\rm t,i}=r_{\rm K}$.
Table \[tab:W3rk\] lists the half-mass times for $W_0=3$ King models with $r_{\rm t,i}/r_{\rm K}=1.4$, 2, 4 and 6, and Fig. \[fig:thalfW3rk\] illustrates these results. In this figure the results for $W_0=3$ and $W_0=5$ King models with $r_{\rm t,i}/r_{\rm K}=1$ are also plotted. Note that the ratio $r_{\rm K}(W_0=5)/r_{\rm K}(W_0=3) \approx 1.4$. Fig. \[fig:thalfpw\] shows the variation of $\alpha$ with $t_{\rm rh,i}$.
The results shown in Figs. \[fig:thalfW3rk\] and \[fig:thalfpw\] confirm the findings of @tf05. The dependence of $T_{\rm half}$ on $t_{\rm rh,i}$ does depend on the strength of the tidal field. In the limit of $r_{\rm t,i}/r_{\rm K} \to \infty$ and $N \to \infty$, it is expected that $\alpha \to 1$.
Note that the curve for $W_0=3$ King models with $r_{\rm t,i}/r_{\rm K}=1.4$ lies very close to that for $W_0=5$ King models with $r_{\rm t,i}/r_{\rm K}=1$ in each of Figs. \[fig:thalfW3rk\] and \[fig:thalfpw\]. This indicates that the mass-loss time-scale does not depend very much on the initial concentration of the cluster but is mainly determined by the strength of the tidal field, as was found by @tf05.
--------- ------------------------------- ----------------------------- ----------------------------- ----------------------------- --
$N$ $T_{\rm half}$ $T_{\rm half}$ $T_{\rm half}$ $T_{\rm half}$
($r_{\rm t,i}/r_{\rm K}=1.4$) ($r_{\rm t,i}/r_{\rm K}=2$) ($r_{\rm t,i}/r_{\rm K}=4$) ($r_{\rm t,i}/r_{\rm K}=6$)
128 $1.62\times10^2$ $2.71\times10^2$ $6.86\times10^2$ $1.11\times10^3$
256 $2.22\times10^2$ $3.48\times10^2$ $7.61\times10^2$ $1.12\times10^3$
512 $3.02\times10^2$ $4.52\times10^2$ $8.95\times10^2$ $1.25\times10^3$
1024 $4.44\times10^2$ $6.48\times10^2$ $1.24\times10^3$ $1.73\times10^3$
2048 $6.93\times10^2$ $1.01\times10^3$ $1.97\times10^3$ $2.78\times10^3$
4096 $1.12\times10^3$ $1.68\times10^3$ $3.33\times10^3$ $4.83\times10^3$
8192 $1.87\times10^3$ $2.84\times10^3$ $5.82\times10^3$ $8.64\times10^3$
16384 $3.14\times10^3$ $4.89\times10^3$ $1.02\times10^4$ $1.54\times10^4$
32768 $5.33\times10^3$ $8.49\times10^3$ $1.80\times10^4$ $2.73\times10^4$
65536 $9.16\times10^3$ $1.49\times10^4$ $3.16\times10^4$ $4.77\times10^4$
131072 $1.59\times10^4$ $2.64\times10^4$ $5.56\times10^4$ $8.34\times10^4$
262144 $2.80\times10^4$ $4.72\times10^4$ $9.89\times10^4$ $1.48\times10^5$
524288 $4.99\times10^4$ $8.54\times10^4$ $1.78\times10^5$ $2.64\times10^5$
1048576 $8.98\times10^4$ $1.56\times10^5$ $3.24 \times 10^5$ $4.79 \times 10^5$
2097152 $1.63\times10^5$ $2.87 \times 10^5$ $6.02 \times 10^5$ $8.99 \times 10^5$
--------- ------------------------------- ----------------------------- ----------------------------- ----------------------------- --
![Same as Fig. \[fig:thalfW3\], but FPf models for the initial conditions of King models with $r_{\rm t,i} > r_{\rm K}$ are compared with the cases of $r_{\rm t,i} = r_{\rm K}$. []{data-label="fig:thalfW3rk"}](f9.eps){width="84mm"}
![Logarithmic slope $\alpha = d\log T_{\rm half}/d\log t_{\rm rh,i}$ as a function of the initial half-mass relaxation time $t_{\rm rh,i}$. The models are the same as those shown in Fig. \[fig:thalfW3rk\]. []{data-label="fig:thalfpw"}](f10.eps){width="84mm"}
Comparison with $N$-body models: multi-mass clusters {#ssec:multi-mass}
----------------------------------------------------
So far we have concentrated on single-mass clusters. Here we consider the evolution of multi-mass clusters comparing our FP models with the $N$-body models of @gb08. They performed $N$-body simulations of clusters on circular orbits around a point-mass galaxy. In their simulations the initial mass function (IMF) is given by $dN/dm \propto m^{-2.35}$ with the ratio $m_{\rm max}/m_{\rm min}=30$. Stellar evolution is not considered in their simulations. The clusters initially have the density distribution of King models with $W_0=5$. The ratio of the initial tidal radius to the King radius $r_{\rm t,i}/r_{\rm K}$ is varied from 1 to 8. The results of the simulations of @gb08 are summarised in their Table 1. Note that they use different notations from ours: $r_{\rm J}$ is for the tidal (Jacobi) radius and $r_{\rm t}$ is for the King radius.
------- -------------------- -------------------- -------------------- --------------------
$N$ $T_{\rm half}$ $T_{\rm half}$ $T_{\rm half}$ $T_{\rm half}$
($N$-body) (FPf) (FPf) (FPf)
(0.11, 7) (0.02, 7) (0.02, 40)
1024 $1.14 \times 10^2$ $1.20 \times 10^2$ $1.69 \times 10^2$ $1.17 \times 10^2$
2048 $1.74 \times 10^2$ $1.87 \times 10^2$ $2.54 \times 10^2$ $1.80 \times 10^2$
4096 $2.69 \times 10^2$ $2.86 \times 10^2$ $3.75 \times 10^2$ $2.72 \times 10^2$
8192 $4.35 \times 10^2$ $4.39 \times 10^2$ $5.59 \times 10^2$ $4.18 \times 10^2$
16384 $6.70 \times 10^2$ $6.90 \times 10^2$ $8.57 \times 10^2$ $6.59 \times 10^2$
32768 $1.06 \times 10^3$ $1.12 \times 10^3$ $1.36 \times 10^3$ $1.08 \times 10^3$
------- -------------------- -------------------- -------------------- --------------------
: Half-mass times $T_{\rm half}$ given by $N$-body [@gb08] and FPf models for the initial conditions of multi-mass King models with $W_0=5$ and $r_{\rm t,i}/r_{\rm K}=1$. Three sets of the parameters $(\gamma, \nu_{\rm e})$ are used for the FPf models.[]{data-label="tab:mm_gamma"}
![ Half-mass time $T_{\rm half}$ as a function of the initial number of stars $N$ for $W_0=5$ King models with the IMF $dN/dm \propto m^{-2.35}$ ($m_{\rm max}/m_{\rm min}=30$). FPf models with three different sets of the parameters $\gamma$ and $\nu_{\rm e}$ are compared with the $N$-body models of @gb08. []{data-label="fig:mm_gamma"}](f11.eps){width="84mm"}
FPf models are calculated for the same initial conditions as those of @gb08. The results for $r_{\rm t,i}/r_{\rm K}=1$ are summarised in Table \[tab:mm\_gamma\] and Fig. \[fig:mm\_gamma\]. There the results of the FPf models with three different sets of parameters $\gamma$ and $\nu_{\rm e}$ are reported. @gh94a estimated the best value of $\gamma=0.11$ for single-mass clusters by comparing $N$-body models with FP and gas models. Similarly @gh96 obtained $\gamma=0.02$ for multi-mass with the IMF $dN/dm \propto m^{-2.5} (m_{\rm max}/m_{\rm min}=37.5)$. We have calculated FPf models for multi-mass clusters using these two values of $\gamma$.
Fig. \[fig:mm\_gamma\] shows that the parameter set $(\gamma, \nu_{\rm e}) = (0.11,7)$ adopted for single-mass clusters gives good fit to the $N$-body models also for multi-mass clusters. On the other hand the parameter set $(\gamma, \nu_{\rm e}) = (0.02,7)$ results in a clear deviation from the $N$-body models. If we stick to $\gamma=0.02$, the value of $\nu_{\rm e}$ needs to be increased to about 40 in order to obtain good agreement with the $N$-body models. We will discuss in more detail what values of the parameters we should choose in the next section.
------- ----------------------------- ----------------------------- -----------------------------
$N$ $T_{\rm half}$ $T_{\rm half}$ $T_{\rm half}$
($r_{\rm t,i}/r_{\rm K}=2$) ($r_{\rm t,i}/r_{\rm K}=4$) ($r_{\rm t,i}/r_{\rm K}=8$)
1024 $3.34 \times 10^2$ $8.03 \times 10^2$ $1.81 \times 10^3$
2048 $5.12 \times 10^2$ $1.17 \times 10^3$ $2.41 \times 10^3$
4096 $7.79 \times 10^2$ $1.72 \times 10^3$ $3.35 \times 10^3$
8192 $1.21 \times 10^3$ $2.65 \times 10^3$ $5.08 \times 10^3$
16384 $1.95 \times 10^3$ $4.30 \times 10^3$ $8.33 \times 10^3$
32768 $3.26 \times 10^3$ $7.32 \times 10^3$ $1.45 \times 10^4$
------- ----------------------------- ----------------------------- -----------------------------
: Half-mass times $T_{\rm half}$ given by FPf models for the initial conditions of multi-mass King models with $W_0=5$ and $r_{\rm t,i}/r_{\rm K}=2$, 4, 8. The adopted parameter set is $(\gamma, \nu_{\rm e})=(0.11, 7)$.[]{data-label="tab:mm_rt"}
------- ----------------------------- ----------------------------- -----------------------------
$N$ $T_{\rm half}$ $T_{\rm half}$ $T_{\rm half}$
($r_{\rm t,i}/r_{\rm K}=2$) ($r_{\rm t,i}/r_{\rm K}=4$) ($r_{\rm t,i}/r_{\rm K}=8$)
1024 $3.68 \times 10^2$ $9.28 \times 10^2$ $2.20 \times 10^3$
2048 $5.53 \times 10^2$ $1.32 \times 10^3$ $2.87 \times 10^3$
4096 $8.25 \times 10^2$ $1.89 \times 10^3$ $3.86 \times 10^3$
8192 $1.26 \times 10^3$ $2.84 \times 10^3$ $5.63 \times 10^3$
16384 $2.01 \times 10^3$ $4.53 \times 10^3$ $8.92 \times 10^3$
32768 $3.55 \times 10^3$ $7.62 \times 10^3$ $1.52 \times 10^4$
------- ----------------------------- ----------------------------- -----------------------------
: Same as Table \[tab:mm\_rt\], but the results of FPf models with the parameter set $(\gamma, \nu_{\rm e})=(0.02, 40)$ are listed.[]{data-label="tab:mm_rt2"}
![ Same as Fig. \[fig:mm\_gamma\], but FPf models are compared with the $N$-body models of @gb08 for the clusters with $r_{\rm t,i} > r_{\rm K}$. The adopted parameter sets for the FPf models are $(\gamma, \nu_{\rm e})=(0.11, 7)$ and $(0.02, 40)$. []{data-label="fig:mm_rt"}](f12.eps){width="84mm"}
The results for the initial conditions with $r_{\rm t,i}>r_{\rm K}$ are shown in Tables \[tab:mm\_rt\] and \[tab:mm\_rt2\] and Fig. \[fig:mm\_rt\]. The results of @gb08 are not shown in these tables (see their Table 1). Fig. \[fig:mm\_rt\] shows that the FPf models with $(\gamma, \nu_{\rm e})=(0.11, 7)$ are in good agreement with the $N$-body models for $r_{\rm t,i}/r_{\rm K}=2$ and 4. The FPf models with $(\gamma, \nu_{\rm e})=(0.02, 40)$ are a little farther to the $N$-body models but still follow them rather well. However, for $r_{\rm t,i}/r_{\rm K}=8$, a noticeable difference is observed between the FPf and $N$-body models; in Fig. \[fig:mm\_rt\] the curve for the $N$-body models is approximately linear but the slopes of the curves for the FPf models apparently change with $N$. Neither parameter set reproduces the results of the $N$-body models as well as in the cases of $r_{\rm t,i}/r_{\rm K}<8$. The reason for this discrepancy is not clear at present, but there is a possibility that very early core-collapse in the models with $r_{\rm t,i}/r_{\rm K}=8$ is, at least partially, responsible for it. The FPf model with $(\gamma, \nu_{\rm e})=(0.11, 7)$ and $r_{\rm t,i}/r_{\rm K}=8$ experiences core collapse (bounce) at $t=0.006 T_{\rm half}$ for $N=1024$, and at $t=0.03 T_{\rm half}$ for $N=32768$. The Coulomb logarithm may take different values for pre-collapse and post-collapse stages (see the next section), which affects the time-scale of the evolution of FP models.
Discussion
==========
We have shown that FP models can well follow the mass evolution of star clusters in a tidal field if a new scheme for treating potential escapers is implemented. This is the first time the effect of re-scattering of potential escapers has been taken into account in FP models. Although Takahashi & Portegies Zwart (1998, 2000) showed that anisotropic FP models are in good agreement with $N$-body models for the mass evolution of star clusters in a galaxy, the tidal field is treated as a tidal cutoff rather than an actual force field. In the present study we have found that our new FP models are in good agreement with $N$-body models calculated with the inclusion of the tidal force field. Thus the new scheme has improved the accuracy of FP models.
@b01 argued that some potential escapers are scattered back to lower energies before they leave the cluster and that this complicates the scaling of the mass-loss time. The success of our models is consistent with his argument. Actually our equation for potential escapers, equation (\[eq:fp\_pe\]), can be regarded as a generalization of the equation of his toy model, his equation (12), used for explaining the scaling $T_{\rm half} \propto t_{\rm rh}^{3/4}$.
The toy model of @b01 is useful for giving us insight into the effect of potential escapers on the cluster evolution. On the other hand, the results presented in subsection \[ssec:escape\_time\] have revealed the limitation of the model. When the energy dependence of the escape time is artificially changed from the true one, the toy model does not correctly explain the results of our FP models. This failure of the toy model is not a big surprise, because it is only a simplified model based on many assumptions, some of which are not very realistic. For example, our simulations show that an exact steady state is never established, but the toy model assumes a steady state. In addition, the scaling of the cluster lifetime depends on the strength of the tidal field, as found by @tf05 and confirmed by the present study, but the toy model does not take account of the strength of the tidal field.
Our FP models show good agreement with $N$-body models not only for single-mass clusters but also for multi-mass clusters. However, we have encountered a difficulty in determining proper values of the two parameters, $\gamma$ and $\nu_{\rm e}$, in the FP models. As shown in subsection \[ssec:multi-mass\], the parameter set $(\gamma, \nu_{\rm e})=(0.11, 7)$ brings good agreement for both single-mass and multi-mass clusters. Since the escape time-scale $t_{\rm e}$ given by equation (\[eq:te\]) is expected to be independent of stellar mass, it is natural that the same value of the parameter $\nu_{\rm e}$ is applicable to both single-mass and multi-mass clusters.
On the other hand, the value of $\gamma$ is expected to depend on the stellar mass function. @h75 argued theoretically that the value of $\gamma$ is generally smaller in multi-mass clusters than in single-mass clusters. Based on the results of $N$-body simulations, @gh94a obtained a value of $\gamma=0.11$ for isolated single-mass clusters, and @gh96 obtained a much smaller value, $\gamma=0.02$, for isolated multi-mass clusters having an IMF similar to the IMF used in our simulations.
When we adopt the value of $\gamma=0.02$ for multi-mass clusters, we have to use a much larger value of $\nu_{\rm e}$, $\nu_{\rm e}=40$, than the best value of $\nu_{\rm e}=7$ for single-mass clusters, in order to obtain good agreement with $N$-body models. Thus we have not found a parameter set satisfying both the independence of $\nu_{\rm e}$ on the mass function and the dependence of $\gamma$ on it. It needs further investigation to solve this incompatibility, but even the determination of $\gamma$ itself is not a simple task. For example, @gh94b obtained the best value of $\gamma=0.035$ by examining the post-collapse evolution of $N$-body models of isolated single-mass clusters. This value is much smaller than the value of $\gamma=0.11$ obtained for pre-collapse single-mass clusters. These results suggest that the value of $\gamma$ changes along with the evolution of clusters. It may also change with radius within a cluster [@gh94a].
@fh00 theoretically estimated not only the energy dependence of the escape time-scale $t_{\rm e}$ but also its numerical coefficient, which is given in their equation (9). If we ignore the difference between energy $E$ and the Jacobi integral $E_{\rm J}$, their estimate for a $W_0=3$ King model leads to a value of $\nu_{\rm e}=29$. This is about four times larger than our best value of $\nu_{\rm e}=7$ for single-mass clusters. However, @fh00 also did numerical experiments and found that their theoretical estimate of $t_{\rm e}$ is too small; escape time-scales obtained from the numerical experiments are more than a few times larger than the theoretical one. Therefore our value $\nu_{\rm e}=7$ is not inconsistent with the result of @fh00. On the other hand, our value of $\nu_{\rm e}=40$ for multi-mass clusters with $\gamma=0.02$ is a little larger than their theoretical estimate.
Another issue not addressed in the present paper is how the mass profile of the parent galaxy affects the results. In all the simulations presented here we assume that the parent galaxy is represented by a point mass. On the other hand, @tf10 showed that the mass-loss time-scale depends on the mass profile of the parent galaxy; the time-scale increases as the mass profile gets shallower. Therefore we expect that the parameter $\nu_{\rm e}$ depends on the mass profile of the parent galaxy. This issue will be examined in a future study.
Conclusion
==========
In this paper we have developed new FP models of globular clusters in a steady galactic tidal field. Our FP models are novel in the method of treating escapers: potential escapers are allowed to experience gravitational scattering with other stars before they really leave clusters. The new method has been devised in order to construct more realistic models of star clusters in a tidal field compared to simple tidal-cutoff models as in previous studies. The mass evolution of clusters in a tidal field does not simply scale with the relaxation time, and our FP models are in good agreement with $N$-body models in this respect.
Our FP models include two parameters $\gamma$ and $\nu_{\rm e}$; $\gamma$ is the numerical factor in the Coulomb logarithm $\ln (\gamma N)$ and $\nu_{\rm e}$ adjusts the speed of the tidal mass loss. We have determined the best values of $\nu_{\rm e}$ for given values of $\gamma$ by comparing FP results with $N$-body results. For single-mass clusters the best parameter set is $(\gamma, \nu_{\rm e})=(0.11, 7)$. This parameter set is applicable to multi-mass clusters as well, but another set $(\gamma, \nu_{\rm e})=(0.02, 40)$ does work equally well as long as multi-mass clusters are concerned. The parameter $\nu_{\rm e}$ is expected to depend on the mass profile of the parent galaxy, though a point-mass galaxy is assumed in all the simulations of the present paper. Further investigation is required for the determination of the best values of the parameters $\gamma$ and $\nu_{\rm e}$ under various conditions.
While FP models are generally thought to be less faithful models of globular clusters than $N$-body models, the present study has significantly improved the accuracy of FP models. An advantage of FP models is that they can be calculated much faster than $N$-body models. Therefore FP models are particularly useful when we need to calculate a huge number of models. For example, when we try to specify the initial conditions of individual clusters, we have to perform simulations for many sets of the initial conditions, because the parameter space to be searched is very large. We believe that our FP models is quite useful for such searching.
Acknowledgments {#acknowledgments .unnumbered}
===============
Part of the work was done while the authors visited the Center for Planetary Science (CPS) in Kobe, Japan, during a visit that was funded by the HPCI Strategic Program of MEXT. We are grateful for their hospitality. HB acknowledges support by the Australian Research Council (ARC) through Future Fellowship Grant FT0991052. The numerical calculations of the Fokker-Planck models were carried out on Altix3700 and SR16000 at YITP in Kyoto University.
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Estimation of the scaling of the cluster lifetime {#sec:ap1}
=================================================
We follow the arguments given by @b01 and @h01 in order to derive the scaling law of equation (\[eq:thbeta\]).
Let $\hat{E}=(E-E_{\rm crit})/|E_{\rm crit}|$ and assume that the escape time-scale $t_{\rm e}$ has energy-dependence such as $$t_{\rm e}(\hat{E}) = t_{\rm esc} \hat{E}^{-\beta} \quad(\beta >0) .$$ Then Baumgardt’s toy model is modified as $$\frac{\partial n}{\partial t}
= \frac{k_1}{t_{\rm rh}} \frac{\partial^2 n}{\partial \hat{E}^2} - \hat{E}^\beta \frac{n}{t_{\rm esc}},
\label{eq:toy}$$ where $n(\hat{E},t)d\hat{E}$ is the number of stars with energies in the range $(\hat{E}, \hat{E}+d\hat{E})$ and $k_1$ is a constant. If we assume that the distribution of escapers is nearly in equilibrium, equation (\[eq:toy\]) shows that the width of the distribution is approximately given by $$\Delta \hat{E} \sim \left( \frac{t_{\rm esc}}{t_{\rm rh}} \right)^{\frac{1}{\beta+2}} ,$$ and the number of escapers $N_{\rm esc} \sim N \Delta \hat{E}$. The escape rate $\dot{N}_{\rm esc}$ is estimated to be $$\dot{N}_{\rm esc} \sim \frac{N_{\rm esc}}{t_{\rm e}(\Delta \hat{E})}
\sim \frac{N}{t_{\rm esc}} \left( \frac{t_{\rm esc}}{t_{\rm rh}} \right)^{\frac{\beta+1}{\beta+2}} .$$ Therefore the scaling of the half-mass time is given by $$T_{\rm half} \sim \frac{N}{\dot{N}_{\rm esc}} \sim t_{\rm rh}^\frac{\beta+1}{\beta+2} t_{\rm esc}^\frac{1}{\beta+2} .$$
\[lastpage\]
[^1]: E-mail: [email protected] (KT)
|
---
abstract: 'We provide tight finite-time convergence bounds for gradient descent and stochastic gradient descent on quadratic functions, when the gradients are delayed and reflect iterates from $\tau$ rounds ago. First, we show that without stochastic noise, delays strongly affect the attainable optimization error: In fact, the error can be as bad as non-delayed gradient descent ran on only $1/\tau$ of the gradients. In sharp contrast, we quantify how stochastic noise makes the effect of delays negligible, improving on previous work which only showed this phenomenon asymptotically or for much smaller delays. Also, in the context of distributed optimization, the results indicate that the performance of gradient descent with delays is competitive with synchronous approaches such as mini-batching. Our results are based on a novel technique for analyzing convergence of optimization algorithms using generating functions.'
author:
- |
Yossi Arjevani Ohad Shamir\
Weizmann Institute of Science\
Rehovot 7610001, Israel\
`{yossi.arjevani,ohad.shamir}@weizmann.ac.il`\
-
- |
Nathan Srebro\
TTI Chicago\
Chicago, IL 60637\
`[email protected]`\
bibliography:
- 'bib.bib'
title: A Tight Convergence Analysis for Stochastic Gradient Descent with Delayed Updates
---
Introduction
============
Gradient-based optimization methods are widely used in machine learning and other large-scale applications, due to their simplicity and scalability. However, in their standard formulation, they are also strongly synchronous and iterative in nature: In each iteration, the update step is based on the gradient at the current iterate, and we need to wait for this computation to finish before moving to the next iterate. For example, to minimize some function $F$, plain stochastic gradient descent initializes at some point ${\mathbf{w}}_0$, and computes iterates of the form $$\label{eq:sgd}
{\mathbf{w}}_{k+1} = {\mathbf{w}}_{k}-\eta (\nabla F({\mathbf{w}}_k)+{\boldsymbol{\xi}}_k)~,$$ where $\nabla F({\mathbf{w}}_k)$ is the gradient of $F$ at ${\mathbf{w}}_k$, $\eta$ is the step size and ${\boldsymbol{\xi}}_1,{\boldsymbol{\xi}}_2,\ldots$ are independent zero-mean noise terms. Unfortunately, in several important applications, a direct implementation of this is too costly. For example, consider a setting where we wish to optimize a function $F$ using a distributed platform, consisting of several machines with shared memory. We can certainly implement gradient descent, by letting one of the machines compute the gradient at each iteration, but this is clearly wasteful, since just one machine is non-idle at any given time. Thus, it is highly desirable to use methods which parallelize the computation. One approach is to employ *mini-batch gradient* methods, which parallelize the computation of the stochastic gradient, and their analysis is relatively well understood (e.g. [@dekel2012optimal; @cotter2011better; @shamir2014distributed; @takac2013mini]). However, these methods are still generally iterative and synchronous in nature, and hence can suffer from problems such as having to wait for the slowest machine at each iteration.
A second and popular approach is to utilize *asynchronous* gradient methods. With these methods, each update step is not necessarily based just on the gradient of the current iterate, but possibly on the gradients of earlier iterates (often called *stale updates*). For example, when optimizing a function using several machines, each machine might read the current iterate from a shared parameter server, compute the gradient at that iterate, and then update the parameters, even though other machines might have performed other updates to the parameters in the meantime. Although such asynchronous methods often work well in practice, analyzing them is much trickier than synchronous methods.
In our work, we focus on arguably the simplest possible variant of these methods, where we perform plain stochastic gradient descent on a convex function $F$ on ${\mathbb{R}}^d$, with a fixed delay of $\tau>0$ in the gradient computation: $$\label{eq:dsgd}
{\mathbf{w}}_{k+1} = {\mathbf{w}}_{k}-\eta (\nabla F({\mathbf{w}}_{k-\tau})+{\boldsymbol{\xi}}_k)~,$$ where we assume that ${\mathbf{w}}_0={\mathbf{w}}_1=\ldots={\mathbf{w}}_{\tau}$. Compared to [Eq. (\[eq:sgd\])]{}, we see that the gradient is computed with respect to ${\mathbf{w}}_{k-\tau}$ rather than ${\mathbf{w}}_k$. Already in this simple formulation, the precise effect of the delay on the convergence rate is not completely clear. For example, for a given number of iterations $k$, how large can $\tau$ be before we might expect a significant deterioration in the accuracy? And under what conditions? Although there exist some prior results in this direction (which we survey in the related work section below), these questions have remained largely open.
In this paper, we aim at providing a tight, finite-time convergence analysis for stochastic gradient descent with delays, focusing on the simple case where $F$ is a convex quadratic function. Although a quadratic assumption is non-trivial, it arises naturally in problems such as least squares, and is an important case study since all smooth and convex function are locally quadratic close to their minimum (hence, our results should still hold in a local sense). In future work, we hope to show that our results are also applicable more generally.
First, we consider the case of *deterministic* delayed gradient descent (DGD, defined in [Eq. (\[eq:dsgd\])]{} with ${\boldsymbol{\xi}}_k=\mathbf{0}$). Assuming the step size $\eta$ is chosen appropriately, we prove that $$\begin{aligned}
F({\mathbf{w}}_k)-F({\mathbf{w}}^*)~\leq~ 5\mu\|{\mathbf{w}}_0 - {\mathbf{w}}^*\|^2\exp\left(-\frac{\lambda
(k+1)}{10\mu(\tau+1)}\right)\end{aligned}$$ after $k$ iterations, over the class of $\lambda$-strongly convex $\mu$-smooth quadratic functions with a minimum at ${\mathbf{w}}^*$, and $$\begin{aligned}
F({\mathbf{w}}_k)-F({\mathbf{w}}^*)~\leq~\frac{17\mu\|{\mathbf{w}}_0 - {\mathbf{w}}^*\|^2(\tau+1)}{k+1}\end{aligned}$$ over the class of $\mu$-smooth convex quadratic functions with minimum at ${\mathbf{w}}^*$. In terms of iteration complexity, the number of iterations $k$ required to achieve a fixed optimization error of at most $\epsilon$ in the strongly convex and the convex cases is therefore $${\mathcal{O}}\left(\tau\cdot\kappa\ln\left(\frac{\mu{\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2}{\epsilon}\right)\right)~~~~
\text{and}~~~~
{\mathcal{O}}\left(\tau\cdot \frac{\mu{\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2}{\epsilon}\right)
\label{eq:itercomp}$$ respectively, where $\kappa\coloneqq \mu/\lambda$ is the so-called condition number[^1]. When $\tau$ is a bounded constant, these bounds match the known iteration complexity of standard gradient descent without delays [@nesterov2004introductory]. However, as $\tau$ increases, both bounds deteriorate linearly with $\tau$. Notably, in our setting of delayed gradients, this implies that DGD is no better than a trivial algorithm, which performs a single gradient step, and then waits for $\tau$ rounds till the delayed gradient is received, before performing the next step (thus, the algorithm is equivalent to non-delayed gradient descent with $k/\tau$ gradient steps, resulting in the same linear deterioration of the iteration complexity with $\tau$).
Despite these seemingly weak guarantees, we show that they are in fact tight in terms of $\tau$, by proving that this linear dependence on $\tau$ is unavoidable with standard gradient-based methods (including gradient descent). The dependence on the other problem parameters in our lower bounds is a bit weaker than our upper bounds, but can be matched by an *accelerated* gradient descent procedure (see [Sec. \[section:deterministic\_delayed\]]{} for more details).
In the second part of our paper, we consider the case of *stochastic* delayed gradient descent (SDGD, defined in (\[eq:dsgd\])). Assuming ${\boldsymbol{\xi}}_k$ satisfies ${\mathbb{E}}[{\|{\boldsymbol{\xi}}_k\|}^2]\leq \sigma^2$ and that the step size $\eta$ is appropriately tuned, we prove that $${\mathbb{E}}\left[F({\mathbf{w}}_k)-F({\mathbf{w}}^*)\right]~\leq~\tilde{{\mathcal{O}}}\left(\frac{\sigma^2}{\lambda
k}+\mu {\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2\exp\left(-\frac{\lambda k
}{10\mu\tau}\right)\right)~.
\label{eq:sdgdstrong}$$ for $\lambda$-strongly convex, $\mu$-smooth quadratic functions with minimum at ${\mathbf{w}}^*$, and $${\mathbb{E}}\left[F({\mathbf{w}}_k)-F({\mathbf{w}}^*)\right]~\leq~ \tilde{{\mathcal{O}}}\left(\frac{
{\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}\sigma}{\sqrt{k}}
+ \frac{{\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2 \mu\tau }{k}\right).
\label{eq:sdgdconvex}$$ for $\mu$-smooth convex quadratic functions. In terms of iteration complexity, these correspond to $$\tilde{{\mathcal{O}}}\left(\frac{\sigma^2}{\lambda
\epsilon}+\tau\cdot
\kappa\ln\left(\frac{\mu{\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2}{\epsilon}\right)\right)
~~~\text{and}~~~
\tilde{{\mathcal{O}}}\left(\frac{{\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2\sigma^2}{\epsilon^2}+\tau\cdot
\frac{{\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2\mu}{\epsilon}
\right)~,
\label{eq:itercompstoch}$$ in the strongly convex and convex cases respectively, where again $\kappa:=\mu/\lambda$. As in the deterministic case, when $\tau$ is a bounded constant, these bounds match the known iteration complexity bounds for standard gradient descent without delays [@bubeck2015convex; @shamir2013stochastic]. Moreover, these bounds match the bounds for the deterministic case in [Eq. (\[eq:itercomp\])]{} when $\sigma^2=0$ (i.e. zero noise), as they should. However, in sharp contrast to the deterministic case, the dependence on $\tau$ in [Eq. (\[eq:itercompstoch\])]{} is quite different: The delay $\tau$ only appears in second-order terms (as $\epsilon\rightarrow 0$), and its influence becomes negligible when $\epsilon$ is small enough. The same effect can be seen in [Eq. (\[eq:sdgdstrong\])]{} and [Eq. (\[eq:sdgdconvex\])]{}: Once the number of iterations $k$ is large enough, the first term in both bounds dominates, and $\tau$ no longer plays a role. More specifically:
- In the strongly convex case, the effect of the delay becomes negligible once the target accuracy $\epsilon$ is sufficiently smaller than $\tilde{{\mathcal{O}}}(\sigma^2/(\mu\tau))$, or when the number of iterations $k$ is sufficiently larger than $\tilde{\Omega}(\tau\mu/\lambda)$. In other words, assuming the condition number $\mu/\lambda$ is bounded, we can have the delay $\tau$ nearly as large as the total number of iterations $k$ (up to log-factors), without significant deterioration in the convergence rate. Note that this is a mild requirement, since if $\tau\geq k$, the algorithm receives no gradients and makes no updates.
- In the convex case, the effect of the delay becomes negligible once the target accuracy $\epsilon$ is sufficiently smaller than $\tilde{{\mathcal{O}}}(\sigma^2/(\mu\tau))$, or when the number of iterations $k$ is sufficiently larger than $\tilde{\Omega}(({\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}\mu\tau/\sigma)^2)$. Compared to the strongly convex case, here the regime is the same in terms of $\epsilon$, but the regime in terms of $k$ is more restrictive: We need $k$ to scale quadratically (rather than linearly) with $\tau$. Thus, the maximal delay $\tau$ with no performance deterioration is order of $\sqrt{k}$.
Finally, it is interesting to compare our bounds to those of *mini-batch* stochastic gradient descent (SGD), which can be seen as a synchronous gradient-based method to cope with delays, especially in distributed optimization and learning problems [@dekel2012optimal; @cotter2011better; @agarwal2011distributed]. In mini-batch SGD, each update step is performed only after accumulating and averaging a mini-batch of $b$ stochastic gradients, all with respect to the same point: $$\forall k\in \{0,b,2b,\ldots\},~~{\mathbf{w}}_{k+b} = {\mathbf{w}}_k-\eta\cdot
\frac{1}{b}\sum_{i=0}^{b-1}\left(\nabla
F({\mathbf{w}}_k)+\xi_{k+i}\right)~,$$ Although the algorithm makes an update only every $b$ stochastic gradient computations, the averaging reduces the stochastic noise, and helps speed up convergence. Moreover, this can be seen as a particular type of algorithm with delayed updates (with the delay correspond to $b$), as we use $\nabla F({\mathbf{w}}_k)$ to compute iterate ${\mathbf{w}}_{k+b}$. The important difference is that it is an inherently synchronous method, that waits for all $b$ stochastic gradients to be computed before performing an update step. Remarkably, the bounds we proved above for delayed SGD are essentially identical to those known for mini-batch SGD, with the delay $\tau$ replaced by the mini-batch size $b$ (at least in the convex case where mini-batch SGD has been more thoroughly analyzed). This indicates that an asynchronous method like delayed SGD can potentially match the performance of synchronous methods like mini-batch SGD, even without requiring synchronization – an important practical advantage.
Analyzing gradient descent with delays is notoriously tricky, due to the dependence of the updates on iterates produced many iterations ago. The technique we introduce for deriving our upper bounds is primarily based on *generating functions*, and might be useful for studying other optimization algorithms. We discuss this approach more thoroughly in Section \[section:generating\_functions\_and\_framework\]. The rest of the paper is devoted mostly to presenting the formal theorems and an explanation of how they are derived (with technical details relegated to the supplementary material).
Related Work {#related-work .unnumbered}
------------
There is a huge literature on asynchronous versions of gradient-based methods (see for example the seminal book [@bertsekas1989parallel]), including treating the effect of delay. However, most of these do not consider the setting we study here. For example, there has been much recent interest in asynchronous algorithms, in a model where there is a delay in updating individual *coordinates* in a shared parameter vector (e.g., the Hogwild! algorithm of [@recht2011hogwild], or more recently [@mania2015perturbed; @leblond18improved]). Of course, this is a different model than ours, where the updates use a full gradient vector. Other works (such as [@sirb2016decentralized]) focus on a setting where different agents in a network can perform local communication, which is again a different model than ours. Yet other works focus on sharp but asymptotic results, and do not provide guarantees after a fixed number $k$ of iterations (e.g., [@chaturapruek2015asynchronous]).
Moving closer to our setting, [@nedic2001distributed] showed convergence for delayed gradient descent, with the result implying an $\sqrt{\tau/k}$ convergence rate for convex functions. A similar bound on average regret has been shown in an adversarial online learning setting, for general convex functions, and this bound is known to be optimal [@joulani2013online]. These results differ from our setting, in that they consider possibly non-smooth functions, in which the dependence on $k$ is no better than $1/\sqrt{k}$ even without delays and no noise, and where the delay $\tau$ always plays a significant role. In contrast, we focus here on smooth functions, where rates better than $1/\sqrt{k}$ are possible, and where the effect of $\tau$ is more subtle. In [@feyzmahdavian2014delayed], the authors study a setting very similar to ours in the deterministic case, and manage to prove a linear convergence rate, but for a less standard algorithm, different than the one we study here (with iterates of the form ${\mathbf{w}}_{t+1} = {\mathbf{w}}_{t-\tau}-\nabla
F({\mathbf{w}}_{t-\tau})$).
Perhaps the works closest to ours are [@agarwal2011distributed; @feyzmahdavian2016asynchronous], which study stochastic gradient descent with delayed gradients. Moreover, they consider a setting more general than ours, where the delay at each iteration is any integer up to $\tau$ (rather than fixed $\tau$), and the functions are not necessarily quadratic. On the flip side, their bounds are significantly weaker. For example, for smooth convex functions and an appropriate step size, [@agarwal2011distributed Corollary 1] show a bound of $${\mathcal{O}}\left(\frac{\sigma}{\sqrt{k}}+\frac{\tau^2+1}{\sigma^2
k}\right).$$ in terms of $k,\tau,\sigma$. Note that this bound is vacuous in the deterministic or near-deterministic case (where $\sigma^2\approx 0$), and is weaker than our bounds. With a different choice of the step size, it is possible to get a non-vacuous bound even if $\sigma^2\rightarrow 0$, but the dependence on $\tau$ becomes even stronger. [@feyzmahdavian2016asynchronous] improve the bound to $${\mathcal{O}}\left(\frac{\sigma}{\sqrt{k}}+\frac{\tau^2+1}{k}\right)~~~\text{and}~~~
{\mathcal{O}}\left(\frac{\sigma^2}{k}+\frac{\tau^4+1}{k^2}\right).$$ in the convex and strongly convex case respectively. Even if $\sigma^2=0$, the iteration complexity is ${\mathcal{O}}(\tau^2/\epsilon)$ and ${\mathcal{O}}(\tau^2/\sqrt{\epsilon})$, and implies a quadratic dependence on $\tau$ (whereas in our bounds the scaling is linear). When $\sigma^2$ is positive, the effect of delay on the bound is negligible only up to $\tau={\mathcal{O}}(\sqrt[4](k))$ (in contrast to $\tilde{{\mathcal{O}}}(\sqrt{k})$ or even $\tilde{{\mathcal{O}}}(k)$ in our bounds). We note that there are several other works which study a similar setting (such as [@sra2015adadelay]), but do not result in bounds which improve on the above. Finally, we note that [@langford2009slow] attempt to show that for stochastic gradient descent with delayed updates, the dependence on the delay $\tau$ is negligible after sufficiently many iterations. Unfortunately, as pointed out in [@agarwal2011distributed], the analysis contains a bug which make the results invalid.
Framework and the Generating Functions Approach {#section:generating_functions_and_framework}
===============================================
Throughout, we will assume that $F$ is a convex quadratic function specified by $$\begin{aligned}
\label{quadratic_problem}
F({\mathbf{w}}) \coloneqq \frac{1}{2}{\mathbf{w}}^\top A{\mathbf{w}}+ {\mathbf{b}}^\top{\mathbf{w}}+c,\end{aligned}$$ where $A\in{\mathbb{R}}^{d\times d}$ is a positive semi-definite matrix whose eigenvalues $a_1,\dots,a_d$ are in $[0,\mu]$ (where $\mu$ is the smoothness parameter), ${\mathbf{b}}\in {\mathbb{R}}^d$ and $c\in {\mathbb{R}}$. To make the optimization problem meaningful, we further assume that $F$ is bounded from below, which implies that it has some minimizer ${\mathbf{w}}^*\in{\mathbb{R}}^d$ at which the gradient vanishes (for completeness, we provide a proof in [Lemma \[lem:bounded\_quadratic\]]{} in the supplementary material). Letting ${\mathbf{e}}_k = {\mathbf{w}}_k-{\mathbf{w}}^*$, it is easily verified that $$\begin{aligned}
\label{eq:convex_value_error}
F({\mathbf{w}}_k)-F({\mathbf{w}}^*) = \frac{1}{2}\left\| \sqrt{A} ({\mathbf{w}}-{\mathbf{w}}^*)\right\|^2 =
\frac{1}{2}\left\| \sqrt{A} {\mathbf{e}}_k\right\|^2,\end{aligned}$$ so our goal will be to analyze the dynamics of ${\mathbf{e}}_k$.
To explain our technique, consider the iterates of DGD on the function $F$, which can be written as ${\mathbf{w}}_{k+1}={\mathbf{w}}_k-\eta \nabla F({\mathbf{w}}_{k-\tau}) ={\mathbf{w}}_k-\eta (A{\mathbf{w}}_{k-\tau} +{\mathbf{b}})$. Since $\nabla F({\mathbf{w}}^*)=0$, we have ${\mathbf{w}}^*={\mathbf{w}}^*-\eta (A{\mathbf{w}}^* +{\mathbf{b}})$, by which it follows that the error term ${\mathbf{e}}_k = {\mathbf{w}}_k-{\mathbf{w}}^*$, satisfies the recursion ${\mathbf{e}}_{k+1} = {\mathbf{e}}_k -\eta A {\mathbf{e}}_{k-\tau}$, and (by definition of the algorithm) ${\mathbf{e}}_0={\mathbf{e}}_1=\ldots={\mathbf{e}}_\tau$. By some simple arguments, our analysis then boils down to bounding the elements of the scalar-valued version of this sequence, namely $$\begin{aligned}
\label{basic_dynamics}
\begin{aligned}
b_0&=\dots=b_\tau\in{\mathbb{R}},\\
b_{k+1} &= b_k - \alpha b_{k-\tau},~k\ge \tau,
\end{aligned}\end{aligned}$$ for some integer $\tau\ge0$ and non-negative real number $\alpha\ge0$. To analyze this sequence, we rely on tools from the area of generating functions, which have proven very effective in studying growth rates of sequences in many areas of mathematics. We now turn to briefly describe these functions and our approach (for general surveys on generating functions, see [@wilf2005generatingfunctionology; @flajolet2009analytic; @stanley1986enumerative], to name a few).
Generally speaking, generating functions are formal power series associated with infinite sequences of numbers . Concretely, given a sequence $(b_k)$ of numbers in a ring $R$, we define the corresponding generating function as a formal power series in $z$, defined as $
f(z) = \sum_{k=0}^\infty b_k z^k
$. The set of all formal power series in $z$ over $R$ is denoted by $R[[z]]$. Moreover, given two power series defined by sequences $(a_k)$ and $(c_k)$, we can define their addition as the power series corresponding to $(a_k+c_k)$, and their multiplication as the coefficients of the Cauchy product of the power series, namely $(\sum_k a_k z^k)(\sum_k c_k z^k) = \sum_k
(\sum_{l=0}^{k}a_l c_{k-l}) z^k$. In particular, over the reals, ${\mathbb{R}}[[z]]$ endowed with addition and multiplication is a commutative ring, and the set of matrices with elements in ${\mathbb{R}}[[z]]$ (with the standard addition and multiplication operations) forms a matrix algebra, denoted by $\mathcal{M}({\mathbb{R}}[[z]])$. We will often use the fact that any matrix, whose entries are power series with scalar coefficients, can also be written as a power series with matrix-valued coefficients: More formally, $\mathcal{M}({\mathbb{R}}[[z]])$ is naturally identified with the ring of formal power series with real matrix coefficients $\mathcal{M}({\mathbb{R}})[[z]]$. To extract the coefficients of a given $M(z)\in \mathcal{M}(R[[z]])$, we shall use the conventional bracket notation $[z^k]M(z)$, defined to be a matrix whose entries are the $k$’th coefficients of the respective formal power series.
Returning to [Eq. (\[basic\_dynamics\])]{}, we write $(b_k)$ as a formal power series denoted by $f(z)$, and proceed as follows, $$\begin{aligned}
f(z) &=
\sum_{k=0}^{\tau} b_k z^k + \sum_{k=\tau+1}^\infty (b_{k-1} - \alpha b_{k-\tau-1}) z^k
= \sum_{k=0}^{\tau} b_k z^k + \sum_{k=\tau+1}^\infty b_{k-1} z^k -\alpha
\sum_{k=\tau+1}^\infty b_{k-\tau-1} z^k\notag\\
&= \sum_{k=0}^{\tau} b_k z^k +z\left( f(z) - \sum_{k=0}^{\tau-1} b_{k}z^k\right) -\alpha z^{\tau+1}f(z)
= b_0 +(z-\alpha z^{\tau+1})f(z)~.\label{eq:generating_function_derivation}\end{aligned}$$ Denoting $$\pi_\alpha(z) ~\coloneqq~ 1-z+\alpha z^{\tau+1}$$ and rearranging terms gives $$\begin{aligned}
\label{eq:coefficients_of_b}
f(z) = \frac{ b_0 }{\pi_\alpha(z)}\quad \implies\quad
b_k = [z^k] f(z) = [z^{k}] \frac{b_0}{\pi_\alpha(z)}~\end{aligned}$$ (by a well-known fact, $\pi_\alpha(z)$ is invertible in ${\mathbb{R}}[[z]]$, as its constant term 1 is trivially invertible in ${\mathbb{R}}$ – see surveys mentioned above). We now see that the problem of bounding the coefficients $(b_k)$ is reduced to that of estimating the coefficients of the rational function ${1}/{\pi_\alpha(z)}$, written as a power series. Note that for the analogous problem where the elements of the sequence are vectors $({\mathbf{b}}_k)_{k=0}^\infty$ and the factor $\alpha$ is replaced by $\alpha
A$ for some square matrix $A$, the same derivation as above yields $\sum_{k=0}^\infty {\mathbf{b}}_k z^k = (I-z+\alpha A z)^{-1}{\mathbf{b}}_0$ (likewise, $I-z+ A z$ is invertible in $\mathcal{M}({\mathbb{R}})[[z]]$ as its constant term $I$ is invertible in $\mathcal{M}({\mathbb{R}})$).
To estimate the coefficients of ${1}/{\pi_\alpha(z)}$, we form its corresponding partial fraction decomposition. First, we note that as a polynomial of degree $\tau+1$, $\pi_{\alpha}(z)$ has $\tau+1$ roots $\zeta_1,\ldots,\zeta_{\tau+1}$ (possibly complex-valued, and all non-zero since $\pi_\alpha(0)=1$ for any $\alpha\in{\mathbb{R}}$). Assuming $\alpha$ is chosen so that all the roots are distinct (equivalently, $\pi'_\alpha(\zeta_i)\neq0,$ for $i\in[\tau+1]$), we have by a standard derivation $$\begin{aligned}
\frac{1}{\pi_\alpha(z)} = \sum_{i=1}^{\tau+1} \frac{1}{\pi_\alpha'(\zeta_i) (z-\zeta_i) }
= \sum_{i=1}^{\tau+1} \frac{-1}{\pi_\alpha'(\zeta_i) \zeta_i }\cdot \frac{1}{1- \frac{z}{\zeta_i} }
= \sum_{i=1}^{\tau+1} \frac{-1}{\pi_\alpha'(\zeta_i) \zeta_i} \sum_{k=0}^\infty \left(\frac{z}{\zeta_i}\right)\!^k.\end{aligned}$$ Thus, $$\begin{aligned}
\label{z_coefficient}
[z^k] \left(\frac{1}{\pi_\alpha(z)}\right) = \sum_{i=1}^{\tau+1}
\frac{-1}{\pi_\alpha'(\zeta_i) \zeta_i^{k+1}}~.\end{aligned}$$ To bound the magnitude of $1/\zeta_i$ and $\pi'_\alpha(\zeta_i)$, we invoke the following lemma, whose proof (in the supplementary material) relies on some tools from complex analysis:
\[lem:roots\_bound\] Let $\alpha\in\left(0,{1}/{20 (\tau+1)}\right]$, and assume $|\zeta_1|\le|\zeta_2|\le\dots\le|\zeta_{\tau+1}|$, then
1. $\zeta_1$ is a real scalar satisfying $1/\zeta_1 \le 1- \alpha$, and for $i>1$, $|1/\zeta_i|\le
1-\frac{3}{2(\tau+1)}$.
2. $|\pi'_\alpha(\zeta_i)|>1/2$, for any $i\in[\tau+1]$.
With this lemma at hand, we have $$\begin{aligned}
\begin{aligned}
\left|[z^k] \left(\frac{1}{\pi_\alpha(z)}\right)\right|
&\le
2(1-\alpha)^{k+1} + 2\tau\left(1-\frac{3}{2(\tau+1)}\right)^{k+1}
\le 2(1-\alpha)^{k+1} \left(1+\tau\exp\left(-\frac{k+1}{\tau+1}\right)\right)~,
\end{aligned}\end{aligned}$$ where the last inequality is due to [Lemma \[lem:k\_regime\]]{} (provided in the supplementary material). Moreover, one can use elementary arguments to show that $|[z^k]1/\pi_\alpha(z)|\le 1$ for any $k\ge0$, as long as $\alpha\in [0,1/\tau]$ (see [Lemma \[lem:bound\_1\]]{} in the supplementary material). Overall, for any $\tau\ge0$, we have $$\begin{aligned}
\label{ineq:coefficients_bound}
\begin{cases}
\left|[z^k] \left(\frac{1}{\pi_\alpha(z)}\right)\right|
\le 1 & 0\le k \le (\tau+1)\ln(2(\tau+1))-1,\\
\left|[z^k] \left(\frac{1}{\pi_\alpha(z)}\right)\right|
\le 3(1-\alpha)^{k+1}& k \ge (\tau+1)\ln(2(\tau+1)),
\end{cases}\end{aligned}$$ which, using [Eq. (\[eq:coefficients\_of\_b\])]{}, gives the desired bounds on the elements $(b_k)$ defined in [Eq. (\[basic\_dynamics\])]{}.
Deterministic Delayed Gradient Descent {#section:deterministic_delayed}
======================================
We start by analyzing the convergence of DGD for $\lambda$-strongly convex and $\mu$-smooth quadratic functions, where the eigenvalues of $A$ are assumed to lie in $[\lambda,\mu]$ for some $\mu\geq \lambda>0$.
Following the same line of the derivation as in [Eq. (\[eq:generating\_function\_derivation\])]{}, we obtain $\mathbf{e}(z) = (I-Iz+\eta A z^{\tau+1})^{-1} {\mathbf{e}}_0$. Letting $[d]:=\{1,2,\ldots,d\}$, it follows that for any $k\ge
(\tau+1)\ln(2(\tau+1))$, $$\begin{aligned}
\label{ineq:generating_bound}
\begin{aligned}
\|{\mathbf{e}}_k\|&=\|[z^k]\left((I-Iz+\eta A z^{\tau+1})^{-1}{\mathbf{e}}_0\right)\|
\stackrel{(a)}{=}\|[z^k]\left((I-Iz+\eta A z^{\tau+1})^{-1}\right){\mathbf{e}}_0\|\\
&\stackrel{(b)}{\le} \max_{i\in[d]}\left|[z^k] \frac{1}{\pi_{\eta a_i}(z)}
\right|\|{\mathbf{e}}_0\| \stackrel{(c)}{\le} 3 \max_{i\in[d]} (1-\eta
a_i)^{k+1}\|{\mathbf{e}}_0\| \stackrel{(d)}{\le} 3(1-\eta \lambda)^{k+1}{\|{\mathbf{e}}_0\|}~,
\end{aligned}\end{aligned}$$ where $(a)$ follows by the linearity, $(b)$ by eigendecomposition of $A$ (that reveals that the spectral norm of a matrix polynomial equals the absolute value of the same polynomial in one of its eigenvalues), $(c)$ by Ineq. \[ineq:coefficients\_bound\] for $\eta
\mu\in\left(0,{1}/{(20(\tau+1))}\right]$, and $(d)$ by the fact that $a_i\geq \lambda$ for all $i$. Moreover, by [Eq. (\[eq:convex\_value\_error\])]{} and the fact that all eigenvalues of $A$ are at most $\mu$, we arrive at the following bound:
\[thm:convergence\] For any delay $\tau\ge0$ and $k\ge (\tau+1)\ln(2(\tau+1))$, running DGD with step size $\eta\in\left(0,{1}/{(20\mu(\tau+1))}\right]$ on a $\mu$-smooth, $\lambda$-strongly convex quadratic function yields $$\begin{aligned}
F({\mathbf{w}}_k)-F({\mathbf{w}}^*)
&\le 5\mu\left(1-\eta\lambda\right)^{2(k+1)}\|{\mathbf{w}}_0 - {\mathbf{w}}^*\|^2.
\end{aligned}$$ In particular, setting $\eta=\Omega(1/\mu\tau)$, we get that $$F({\mathbf{w}}_k)-F({\mathbf{w}}^*) ~\leq~
5\mu{\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2\exp\left(-\Omega\left(\frac{k\lambda}{\mu\tau}\right)\right)~.$$
Note that the assumption that $k\geq
(\tau+1)\ln(2(\tau+1))$ is very mild, since if $k\leq \tau$ then the algorithm trivially makes no updates after $k$ rounds.
We now turn to analyze the case of $\mu$-smooth convex quadratic functions, where the eigenvalues of the matrix $A$ are assumed to lie in $[0,\mu]$. Following the same derivation as in Ineq. \[ineq:generating\_bound\] and using Ineq. \[eq:convex\_value\_error\], we have for any $k\ge (\tau+1)\ln(2(\tau+1))$ and $\eta\in\left(0,{1}/{(20\mu(\tau+1))}\right]$, $$\begin{aligned}
\begin{aligned}
F({\mathbf{w}}_k)-F({\mathbf{w}}^*) &= \frac{1}{2}\|\sqrt{A}{\mathbf{e}}_k\|^2 =
\frac{1}{2}\|\sqrt{A}[z^k]\left((I-Iz+\eta Az^{\tau+1})^{-1}\right){\mathbf{e}}_0\|^2\\
&\stackrel{(a)}\le \frac{1}{2}\left( 3\max_{i\in[d]} \sqrt{a_i}(1-\eta
a_i)^{k+1}\right)^2\|{\mathbf{e}}_0\|^2
\stackrel{(b)}\le
\frac{9}{4e\eta (k+1)}\|{\mathbf{e}}_0\|^2~,
\end{aligned}\end{aligned}$$ where $e= 2.718...$ is Euler’s number, $(a)$ is by the fact that the spectral norm of a matrix polynomial equals the absolute value of the same polynomial in one of its eigenvalues, and $(b)$ is by the fact that $\sqrt{a_i}(1-\eta a_i)^{k+1}\le {1}/{\sqrt{2e \eta (k+1)}}$ for any $i\in[d]$ (see [Lemma \[lem:convex\_power\_bound\]]{} in the supplementary material).We have thus arrived at the following bound for the convex case:
\[thm:convex\_convergence\] For any delay $\tau\ge0$ and $k\ge (\tau+1)\ln(2(\tau+1))$, running DGD with step size $\eta\in\left(0,{1}/{(20\mu(\tau+1))}\right]$ on a $\mu$-smooth convex quadratic function yields $$\begin{aligned}
F({\mathbf{w}}_k)-F({\mathbf{w}}^*) &\le
\frac{9}{4e\eta(k+1)}{\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2~.
\end{aligned}$$ In particular, if we set $\eta=\Omega(1/\mu\tau)$, we get that $$F({\mathbf{w}}_k)-F({\mathbf{w}}^*)\leq {\mathcal{O}}\left(\frac{\mu\tau {\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2}{k}\right)~.$$
As discussed in the introduction, the theorems above imply that a delay of $\tau$ increases the iteration complexity by a factor of $\tau$. We now show lower bounds which imply that this linear dependence on $\tau$ is unavoidable, for a large family of gradient-based algorithms (of which gradient descent is just a special case). Specifically, we will consider any iterative algorithm producing iterates ${\mathbf{w}}_0,{\mathbf{w}}_1,\ldots$ which satisfies the following: $$\label{eq:assumplowbound}
{\mathbf{w}}_0=\ldots={\mathbf{w}}_{\tau}=\mathbf{0}~~~\text{and}~~~\forall k\geq t,~
{\mathbf{w}}_{k+1} \in \text{span}\{ \nabla F({\mathbf{w}}_{0}), \nabla F({\mathbf{w}}_{1}),\dots, \nabla
F({\mathbf{w}}_{k-\tau}) \}~.$$ This is a standard assumption in proving optimization lower bounds (see [@nesterov2004introductory]), and is satisfied by most standard gradient-based methods, and in particular our DGD algorithm. We also note that this algorithmic assumption can be relaxed at the cost of a more involved proof, similar to [@nemirovskyproblem; @woodworth2016tight] in the non-delayed case.
\[thm:lower\_bound\] Consider any algorithm satisfying [Eq. (\[eq:assumplowbound\])]{}. Then the following holds for any $k\ge \tau+1$ and sufficiently large dimensionality $d$:
- There exists a $\mu$-smooth, $\lambda$-strongly convex function $F$ over ${\mathbb{R}}^d$, such that $$\begin{aligned}
F({\mathbf{w}}_k) - F({\mathbf{w}}^*) ~\ge~ \frac\lambda4\exp\left(
-\frac{5k}{\left(\sqrt{\mu/\lambda}-1\right)(\tau+1)}\right){\|{\mathbf{w}}_0-{\mathbf{w}}^*\|^2}~.
\end{aligned}$$
- There exists a $\mu$-smooth, convex quadratic function $F$ over ${\mathbb{R}}^d$, such that $$\begin{aligned}
F({\mathbf{w}}_k) - F({\mathbf{w}}^*)~\ge~
\frac{\mu(\tau+1)^2{\|{\mathbf{w}}_0 - {\mathbf{w}}^*\|}^2}{45k^2}~.
\end{aligned}$$
The proof of the theorem is very similar to standard optimization lower bounds for gradient-based methods without delays (e.g. [@nesterov2004introductory; @lan2015optimal]), and is presented in the supplementary material. In fact, our main contribution is to recognize that the proof technique easily extends to incorporate delays.
In terms of iteration complexity, these bounds correspond to $\Omega\left(\tau\cdot \sqrt{\mu/\lambda}\cdot
\ln\left(\lambda{\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2/\epsilon\right)
\right)$ in the strongly convex case, and $\Omega\left(\tau\cdot
\sqrt{\mu{\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2/\epsilon^2}\right)$ in the convex case, which show that the linear dependence on $\tau$ is inevitable. The dependence on the other problem parameters is somewhat better than in our upper bounds, but this is not just an artifact of the analysis: In our delayed setting, the lower bounds can be matched by running *accelerated* gradient descent (AGD) [@nesterov2004introductory], where each time we perform an accelerated gradient descent step, and then stay idle for $\tau$ iterations till we get the gradient of the current point. Overall, we perform $k/\tau$ accelerated gradient steps, and can apply the standard analysis of AGD to get an iteration complexity which is $\tau$ times the iteration complexity of AGD without delays. These match the lower bounds above up to constants. We believe it is possible to prove a similar upper bound for AGD performing an update with a delayed gradient at every iteration (like our DGD procedure), but the analysis is more challenging than for plain gradient descent, and we leave it to future work.
Stochastic Delayed Gradient Descent {#section:stochstic_delayed}
===================================
In this section, we study the case of noisy (stochastic) gradient updates, and the SDGD algorithm, in which the influence of the delay is quite different than in the noiseless case. Instantiating SDGD for quadratic $F({\mathbf{w}})$ (defined in (\[quadratic\_problem\])) results in the following update rule $$\begin{aligned}
\label{stochastic_dynamics}
{\mathbf{w}}_{k+1}={\mathbf{w}}_k-\eta \nabla F({\mathbf{w}}_{k-\tau} + \epsilon_k) ={\mathbf{w}}_k-\eta
(A{\mathbf{w}}_{k-\tau} +{\mathbf{b}}+ {\boldsymbol{\xi}}_k)~,\end{aligned}$$ where ${\boldsymbol{\xi}}_{k},~k\ge0$ are independent zero-mean noise terms satisfying ${\mathbb{E}}[\|{\boldsymbol{\xi}}_t\|^2]\le\sigma^2$. As before, in terms of the error term ${\mathbf{e}}_k={\mathbf{w}}_k-{\mathbf{w}}^*$, [Eq. (\[stochastic\_dynamics\])]{} reads as ${\mathbf{e}}_{k+1}
={\mathbf{e}}_k-\eta A{\mathbf{e}}_{k-\tau} -\eta {\boldsymbol{\xi}}_k$. Given a realization of $({\boldsymbol{\xi}}_{k})$, we denote its associated formal power series by $g(z)\coloneqq
\sum_{k=\tau}^{\infty} {\boldsymbol{\xi}}_k z^k$. By an analysis similar to before, we get that the formal power series of the error terms $({\mathbf{e}}_k)$ satisfies $$\begin{aligned}
{\mathbf{e}}(z) = (I-Iz + \eta A z^{\tau+1})^{-1}({\mathbf{e}}_0 -\eta g(z))~.\end{aligned}$$ We can now bound the error terms by extracting the corresponding coefficients of ${\mathbf{e}}(z)$. Letting $D\coloneqq (I-Iz + \eta A z^{\tau+1})^{-1}$, we have for any $k\ge(\tau+1)\ln(2(\tau+1))$ $$\begin{aligned}
2\cdot{\mathbb{E}}[F({\mathbf{w}}_k)-F({\mathbf{w}}^*)]&={\mathbb{E}}\left[\left\|\sqrt{A}{\mathbf{e}}_k\right\|^2\right]
= {\mathbb{E}}\left[\left\|\sqrt{A}~[z^k] \left(D({\mathbf{e}}_0 -\eta g(z))
\right)\right\|^2\right]\notag\\
&\stackrel{(a)}{=}
\left\|\sqrt{A}~[z^k] D{\mathbf{e}}_0\right\|^2 + \eta^2
{\mathbb{E}}\left[\left\|\sqrt{A}~[z^k]\left(D g(z)
\right)\right\|^2\right]\notag\\
&\stackrel{(b)}{=}
\|\sqrt{A}~[z^k] D{\mathbf{e}}_0\|^2 + \eta^2{\mathbb{E}}\left[\|\sqrt{A}\sum_{i=0}^k
\left([z^i]D
\right){\boldsymbol{\xi}}_{k-i}\|^2\right]\notag\\
&\stackrel{(c)}{\le}
\|\sqrt{A}~[z^k] D\|^2\|{\mathbf{e}}_0\|^2 + \eta^2\sigma^2\sum_{i=0}^k
\left\|\sqrt{A}~[z^i]D \right\|^{2},
\label{ineq:stochastic}\end{aligned}$$ where $(a)$ follows by the linearity of the bracket operation $[z^k]$ and the assumption that ${\mathbb{E}}[{\boldsymbol{\xi}}_k]=0$ for all $k$ (hence ${\mathbb{E}}[g(z)]=0$), $(b)$ follows by the Cauchy product for formal power series, and $(c)$ by the hypothesis that ${\boldsymbol{\xi}}_k$ are independent and satisfy ${\mathbb{E}}[\|{\boldsymbol{\xi}}_k\|^2]\le\sigma^2$ for all $k$. We then upper bound both terms, building on Ineq. \[ineq:coefficients\_bound\] (see the supplementary material for a full derivation), resulting in the following theorem:
\[thm:stochastic\_rate\] Assuming the step $\eta$ satisfies $\eta\in
(0,\frac{1}{20\mu(\tau+1)}]$, and $k\geq (\tau+1)\ln(2(\tau+1))$, the following holds for SDGD:
- For $\lambda$-strongly convex, $\mu$-smooth quadratic convex functions, ${\mathbb{E}}[F({\mathbf{w}}_k)-F({\mathbf{w}}^*)]$ is at most $$5 \mu \exp(-2\eta \lambda(k+1)){\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2
~+~\frac{\eta^2\sigma^2}{2}\left(\mu(\tau+1)\ln(2(\tau+1))
+\frac{1+e+\ln(\frac{1}{\eta\lambda})}{e\eta}\right)~.$$ In particular, by tuning $\eta$ appropriately, $${\mathbb{E}}\left(F({\mathbf{w}}_k)-F({\mathbf{w}}^*)\right)
~\leq~\tilde{{\mathcal{O}}}\left(\frac{\sigma^2}{\lambda
k}+\mu{\|{\mathbf{e}}_0\|}^2\exp\left(-\frac{\lambda k
}{10\mu\tau}\right)\right)~.$$
- For $\mu$-smooth quadratic convex functions, ${\mathbb{E}}[F({\mathbf{w}}_k)-F({\mathbf{w}}^*)]$ is at most $$ \frac{9{\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2}{4e\eta(k+1)}
~+~
\eta^2\sigma^2\left(\mu(\tau+1)\ln(2(\tau+1))
+\frac{9}{2e\eta }(1+\ln(k+1))\right)~.$$ In particular, by tuning $\eta$ appropriately, $${\mathbb{E}}\left(F({\mathbf{w}}_k)-F({\mathbf{w}}^*)\right)
~\leq~
\tilde{{\mathcal{O}}}\left(\frac{{\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}\sigma}{\sqrt{k}}
+\frac{{\|{\mathbf{w}}_0-{\mathbf{w}}^*\|}^2\mu\tau}{
k}\right)~.$$
As discussed in the introduction in detail, the theorem implies that the effect of $\tau$ is negligible once $k$ is sufficiently large.
Proof of [Lemma \[lem:roots\_bound\]]{} {#app:section:roots}
=======================================
Recall that $\pi_\alpha(z) = 1-z+\alpha z^{\tau+1}$, and its roots, denoted by $\zeta_i$, are ordered such that $|\zeta_1|\le|\zeta_2|\le\dots\le|\zeta_{\tau+1}|$. In order to bound from above the magnitude of $1/\zeta_i$, we analyze a related polynomial $p_\alpha(z) = z^{\tau+1}\pi_a(1/z)$ which takes the following explicit form $$p_\alpha(z) ~=~ z^{\tau+1}-z^{\tau}+\alpha~=~ (z-1)z^{\tau}+\alpha.$$ The roots of $p_\alpha$ are precisely $1/\zeta_i$ (note that, $\pi_a(0) =
1\neq 0 $, hence $\zeta_i\neq0,~i\in\{1,\ldots,\tau+1\}$). Thus, bounding from above (below) the magnitude of the roots of $p_\alpha(z)$ gives an upper (lower) bound for $|1/\zeta_i|$.
We first establish that for any $\alpha \in \left(0,\frac{1}{20(\tau+1)}\right]$, $p_\alpha$ has a real-valued root in $\left(1-\frac{1}{2(\tau+1)},1-\alpha\right]$. Indeed, for any such $\alpha$, we have on the one hand, $$p_{\alpha}(1-\alpha) ~=~ -\alpha(1-\alpha)^{\tau}+\alpha~=~ \alpha\left(1-(1-\alpha)^{\tau}\right)~\ge~0,$$ and on the other hand (using the fact that $(1-1/2x)^x \ge 1/2$ for all $x\ge1$), $$\begin{aligned}
p_\alpha\left(1-\frac{1}{2(\tau+1)}\right) &=
-\frac{1}{2(\tau+1)}\left(1-\frac{1}{2(\tau+1)}\right)^\tau + \alpha
\notag\\
&=
-\frac{1}{2(\tau+1)}\left(\left(1-\frac{1}{2(\tau+1)}\right)^{\tau+1}\right)^{\frac{\tau}{\tau+1}}
+ \alpha
\notag\\&\le
-\frac{1}{2(\tau+1)}\left(\frac{1}{2}\right)^{\frac{\tau}{\tau+1}} + \alpha
< -\frac{1}{20(\tau+1)}+\alpha \le0,
\label{ineq:lower_bound_dominant}
\end{aligned}$$ so by continuity of $p_z$, we get that a real-valued root exists in $\left(1-\frac{1}{2(\tau+1)},1-\alpha\right]$.
Next, we show that $\tau$ non-dominant roots of $p_\alpha$ are of absolute value smaller than $R= 1-\frac{3}{2(\tau+1)}$. To this end, we invoke Rouché’s theorem, which states that for any two holomorphic functions $f,g$ in some region $K\subseteq
\mathbb{C}$ with closed contour $\partial K$, if $|g(z)|<|f(z)|$ for any $z\in
\partial K$, then $f$ and $f+g$ have the same number of zeros (counted with multiplicity) inside $K$. In particular, choosing $f(z)=-z^\tau$, $g(z)=z^{\tau+1}+\alpha$ and $K=\{z:|z|\le R\}$, it follows that if $|z^{\tau+1}+\alpha|<|-z^\tau|$ for all $z$ such that $|z|=R$, then $f+g$ (which equals our polynomial $p_\alpha$) has the same number of zeros as $f=-z^\tau$ inside $K$ (namely, exactly $\tau$). However, since $p_\alpha$ is a degree $\tau+1$ polynomial, it has exactly $\tau+1$ roots, so the only root of absolute value larger than $R$ is the real-valued one we found earlier. It remains to verify the condition $|z^{\tau+1}+\alpha|<|-z^\tau|$ for all $z$ such that $|z|=R$. For that, it is sufficient to show that $|z^{\tau+1}|+\alpha < |z^{\tau}|$ for all such $z$, or equivalently, $R^\tau> \alpha+R^{\tau+1}$. $$\begin{aligned}
R^{\tau}&~=~\left(1-\frac{3}{2(\tau+1)}\right)^{\tau} ~=~
\left(1-\frac{3}{2(\tau+1)}\right)^{\tau}
-\left(1-\frac{3}{2(\tau+1)}\right)^{\tau+1}+\left(1-\frac{3}{2(\tau+1)}\right)^{\tau+1}\\
&~=~ \frac{3}{2(\tau+1)}\left(1-\frac{3}{2(\tau+1)}\right)^{\tau}+R^{\tau+1}.
\end{aligned}$$ By the inequality $1-1/(x+1)\ge\exp(-1/x)$ (see [Lemma \[lem:tech1\]]{} below), we have $$\begin{aligned}
1-\frac{3}{2(\tau+1)}
\ge
\exp\left(\frac{-1}{2/3\tau-1/3}\right) \quad\implies\quad
\left(1-\frac{3}{2(\tau+1)}\right)^{\tau}
\ge
\exp\left(\frac{-\tau}{2/3\tau-1/3}\right)
\end{aligned}$$ It is straightforward to verify that $$\begin{aligned}
\frac{-\tau}{2/3\tau-1/3}\ge -3,~\tau\ge1,
\end{aligned}$$ implying that $$\begin{aligned}
R^{\tau}&~=~ \frac{3}{2(\tau+1)}\left(1-\frac{3}{2(\tau+1)}\right)^{\tau}+R^{\tau+1} \\
&~\ge~ \frac{3}{2e^3(\tau+1)}+R^{\tau+1}\\
&~>~ \frac{1}{20(\tau+1)}+R^{\tau+1}~\ge~\alpha+R^{\tau+1}~,
\end{aligned}$$ where in the last inequality we used the assumption that $\alpha\in
\left(0,\frac{1}{20(\tau+1)}\right]$. As mentioned earlier, the roots of $p_\alpha$ are exactly the reciprocals of the roots of $\pi_\alpha$, therefore we conclude $$\begin{aligned}
\label{ineq:bound_on_non_dominant_roots}
\left|\frac{1}{\zeta_i}\right|\le 1-\frac{3}{2(\tau+1)},~ i\in[\tau].
\end{aligned}$$ We now turn to bound $|\pi'_\alpha(\zeta_i)|$ from above. By definition, any root of $\pi_a$ satisfies $\alpha\zeta_i^{\tau+1}-\zeta_i + 1 =0$. Thus, $\alpha\zeta_i^{\tau} =\frac{\zeta_i-1}{\zeta_i}$ (note that as mentioned in the first part of the proof, $\zeta_i\neq0$). This, in turn, gives $$\begin{aligned}
\label{eq:roots_in_derivarive}
\pi'_\alpha(\zeta_i) &= \alpha(\tau+1)\zeta_i^\tau - 1 = \frac{(\tau+1)(\zeta_i-1)}{\zeta_i} - 1 \nonumber
\\&= \frac{(\tau+1)(\zeta_i-1) - \zeta_i}{\zeta_i} \nonumber
\\&= \frac{(\tau+1)\zeta_i-(\tau+1) - \zeta_i}{\zeta_i} \nonumber
\\&= \frac{\tau\zeta_i-(\tau+1)}{\zeta_i}
= \tau - \frac{(\tau+1)}{\zeta_i}
= (\tau+1)\left( \frac{\tau}{\tau+1} - \frac{1}{\zeta_i} \right).
\end{aligned}$$ In the previous parts of the proof, we showed that the distance from any root of $p_\alpha$ to the contour $\{z~|~|z|=1-1/(\tau+1)\}$ is bounded from below by $\frac{1}{2(\tau+1)}$ (Ineq. \[ineq:lower\_bound\_dominant\] and Ineq. \[ineq:bound\_on\_non\_dominant\_roots\]), therefore $$\begin{aligned}
|\pi'_\alpha(\zeta_i)| = (\tau+1)\left| 1- \frac{1}{\tau+1} -
\frac{1}{\zeta_i} \right| \ge
\frac{\tau+1}{2(\tau+1)}=\frac{1}{2},~i=1,\dots,\tau+1~,
\end{aligned}$$ thus concluding the proof.
Technical Lemmas {#app:section:technical_lemmas}
=================
\[lem:bound\_1\] For any $\alpha\in[0,1/\tau]$ and $k\ge0$, it holds that $|[z^k]1/\pi_\alpha(z)|\le1$.
Recall that by [Eq. (\[eq:coefficients\_of\_b\])]{}, $b_k = [z^{k}] \frac{b_0}{\pi_\alpha(z)}$. Therefore, suffices it to prove that $(b_k)$ (defined in \[basic\_dynamics\]) with $b_0=1$ and $\alpha\in[0,1/\tau]$, satisfies $|b_k|\le1$ for any $k\ge0$.
For the sake of simplicity, we slightly extend $(b_k)$ to the negative indices by defining $b_{-\tau}=b_{-\tau+1}=\dots =b_{-1}=1$. We proceed by full induction. The base case holds trivially by the definition of the initial conditions of $b_k$. For the induction step, suppose that $|b_0|,\dots,|b_k|\le1$. We have $b_{k+1}=b_{k}-\alpha b_{k-\tau}$, and therefore $$\begin{aligned}
b_{k+1}=(1-\alpha)b_k+\alpha(b_k-b_{k-\tau}) = (1-\alpha)b_k+\alpha\sum_{i=k-\tau}^{k-1} (b_{i+1}-b_{i}).
\end{aligned}$$ Using the recurrence relation again, this equals $$\begin{aligned}
(1-\alpha)b_k+\alpha\sum_{i=k-\tau}^{k-1} (-\alpha b_{i-\tau}) = (1-\alpha)b_k-\alpha\left(\alpha\sum_{i=k-2\tau}^{k-\tau-1} b_i\right).
\end{aligned}$$ By the induction hypothesis, this equals $(1-\alpha)b_k+\alpha r_k$, where $|r_k|\leq \alpha\tau\leq 1$. Thus, $b_{k+1}$ is a weighted average of $b_k$ and $r_k$ which are both in $[-1,+1]$ by the induction hypothesis and the above, implying that we must have $b_{k+1} \in [-1,+1]$ as well. Thus, proving the induction step.
\[lem:bounded\_quadratic\] Let $F({\mathbf{w}}) \coloneqq \frac{1}{2}{\mathbf{w}}^\top A {\mathbf{w}}+ {\mathbf{b}}^\top {\mathbf{w}},~A\in{\mathbb{R}}^{d\times d}, {\mathbf{b}}\in{\mathbb{R}}^d$ be a convex quadratic function defined over ${\mathbb{R}}^d$. If $F$ is bounded from below, then $F$ has a minimizer at which the gradient vanishes.
Since $F$ is convex and twice differentiable, $A$ is positive semidefinite. In particular, we have ${\mathbb{R}}^d = \ker(A) \oplus \text{im}(A)$ (namely, the direct sum of the null space and the image space of $A$). Thus, ${\mathbf{b}}$ can be expressed as a sum of two orthogonal vectors ${\mathbf{b}}= {\mathbf{b}}^\perp + \bar{{\mathbf{b}}}$, where ${\mathbf{b}}^\perp\in\ker(A)$ and $\bar{{\mathbf{b}}}\in \text{im}(A)$. For any $\alpha\in{\mathbb{R}}$, we have $$\begin{aligned}
F(\alpha {\mathbf{b}}^\perp) = \frac{1}{2}(({\mathbf{b}}^\perp)^\top A {\mathbf{b}}^\perp)\alpha^2 + \alpha {\mathbf{b}}^\top{\mathbf{b}}^\perp
= \alpha\|{\mathbf{b}}^\perp\|^2.\end{aligned}$$ By the hypothesis, $F$ is bounded from below, hence ${\mathbf{b}}^\perp$ must vanish (otherwise we can take $\alpha\rightarrow -\infty$ and make $F$ as negative as we wish). In particular, ${\mathbf{b}}=\bar{{\mathbf{b}}}\in\text{im}(A)$. Let ${\mathbf{y}}\in{\mathbb{R}}^d$ be such that $A{\mathbf{y}}={\mathbf{b}}$, then $\nabla F(-{\mathbf{y}})=A(-{\mathbf{y}})+{\mathbf{b}}=0$. Lastly, $F$ is convex, therefore $-{\mathbf{y}}$ must be a (global) minimizer, thus concluding the proof.
\[lem:tech1\] For any $x>0$, it holds that $1-1/(x+1)\ge\exp(-1/x)$.
Since $(\ln(1+x))'=1/(1+x)>0$ for any $x>-1$, it follows by the mean-value theorem that for any $x>0$ $$\begin{aligned}
\ln(1+x) = \ln(1+x) -\ln(1) =\frac{1}{1+\xi} x,\end{aligned}$$ for some $\xi\in (0, x)$, hence $\ln(1+x) \le x$ for any $x>0$. In particular, for any $x>0$ we have $$\begin{aligned}
\ln\left(1+\frac{1}{x}\right) \le \frac{1}{x}\implies \ln\left(\frac{x}{x+1}\right) \ge \frac{-1}{x}.\end{aligned}$$ Taking the exponent of both sides yields the desired lower bound.
\[lem:k\_regime\] Let $\tau\ge0$. If $\alpha\in(0,1/(20(\tau+1)]$ then $$\begin{aligned}
\left(\frac{1-\frac{3}{2(\tau+1)}}{1-\alpha}\right)^{k+1} &
\le \exp\left(-\frac{k+1}{\tau+1}\right).
\end{aligned}$$ In particular, for $k\ge(\tau+1)\ln(2(\tau+1))-1$, we have $$\begin{aligned}
\label{k_regime_ineq}
1+\tau\left(\frac{1-\frac{3}{2(\tau+1)}}{1-\alpha}\right)^{k+1} \le 3/2.
\end{aligned}$$
$$\begin{aligned}
\left(\frac{1-\frac{3}{2(\tau+1)}}{1-\alpha}\right)^{k+1} &=
\left(\frac{1-\alpha+\alpha - \frac{3}{2(\tau+1)}}{1-\alpha}\right)^{k+1}
= \left(1 + \frac{\alpha - \frac{3}{2(\tau+1)}}{1-\alpha}\right)^{k+1}
\le \left(1 + \alpha - \frac{3}{2(\tau+1)}\right)^{k+1},
\end{aligned}$$
where the latter inequality follows from that fact that $\alpha < \frac{1}{20(\tau+1)}<\frac{3}{2(\tau+1)}$. Now, $$\begin{aligned}
\left(1 + \alpha - \frac{3}{2(\tau+1)}\right)^{k+1}&\le
\exp\left((k+1)(\alpha - \frac{3}{2(\tau+1)})\right)
\le \exp\left((k+1)\left(\frac{1}{20(\tau+1)}- \frac{3}{2(\tau+1)}\right)\right)\\
&= \exp\left(-\frac{k+1}{\tau+1}\left(\frac{3}{2}-\frac{1}{20}\right)\right)
\le \exp\left(-\frac{k+1}{\tau+1}\right).
\end{aligned}$$ Lastly, to derive Ineq. \[k\_regime\_ineq\], we have $$\begin{aligned}
1+\tau\left(\frac{1-\frac{3}{2(\tau+1)}}{1-\alpha}\right)^{k+1}
&\le 1+(\tau+1)\exp\left(-\frac{k+1}{\tau+1}\right)
= 1+\exp\left(\ln(\tau+1)-\frac{k+1}{\tau+1}\right)
\le 1+1/2,
\end{aligned}$$ where the last inequality by the assumption $k\ge(\tau+1)\ln(2(\tau+1))-1$.
Proof of [Thm. \[thm:lower\_bound\]]{} {#section:lower_bounds_deterministic}
======================================
The proof technique is based on a construction, first presented in [@nesterov2004introductory Section 2.1.2], which has been proven effective in various settings of optimization since then.
First, we address the strongly convex case. Given $\mu>\lambda>0$, we consider the following function (devised by [@lan2015optimal]): $$\begin{aligned}
\label{def:strongly_convex_construction}
F({\mathbf{w}}) \coloneqq \frac{\mu(\kappa-1)}{4}\left(\frac12{\langleA{\mathbf{w}}, {\mathbf{w}}\rangle}-
{\langle{\boldsymbol{\epsilon}}_1,{\mathbf{w}}\rangle}\right) + \frac{\lambda}{2}{\|{\mathbf{w}}\|}^2,\end{aligned}$$ where $\kappa=\mu/\lambda$ as before, ${\boldsymbol{\epsilon}}_1$ denotes the first unit vector, and $A$ is a $d\times d$ matrix defined as follows $$\begin{aligned}
\label{def:lower_bound_hessian}
A = \left(\begin{matrix}
2& -1 & 0 & 0& \cdots & 0 & 0&0\\
-1& 2 & -1 & 0&\cdots & 0 & 0&0\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\
0& 0& 0 &0& \cdots & -1 & 2&-1\\
0& 0& 0 &0& \cdots & 0 & -1 &\frac{\sqrt{\kappa}+1}{\sqrt{\kappa}+3}
\end{matrix}\right).\end{aligned}$$ It can be easily verified that $F$ is $\mu$-smooth and $\lambda$-strongly convex function. Moreover, by [@lan2015optimal Lemma 8], it follows that the minimizer of $f$ is ${\mathbf{w}}^*=(q,q^2,\dots,q^d)$ where $q=(\sqrt{\kappa}-1)/(\sqrt{\kappa}+1)$. In particular, if ${\mathbf{w}}\in{\mathbb{R}}^d$ is a vector whose all non-zero entries are located in the first $m$ coordinates, where $m$ is such that $d\ge m/2 +{\log(1/2)}/{\log(q^2)}$, then $$\begin{aligned}
\label{ineq:lower_bound_rate}
\frac{\|{\mathbf{w}}\|^2}{\|{\mathbf{w}}^*\|^2} \ge
\frac{ \sum_{i=m+1}^d q^{2i}}{\sum_{i=1}^d q^{2i} }
=q^{2(m+1)}\frac{ 1- q^{2(d-m-1)}}{1- q^{2d} }
\ge \frac12 q^{2(m+1)}\ge \frac12
\exp\left(-\frac{4(m+1)}{\sqrt{\kappa}-1}\right),\end{aligned}$$ where the last two inequalities follow from [@lan2015optimal Lemma 9.b] and [Lemma \[lem:tech1\]]{}, respectively. Therefore, by bookkeeping which entries of the iterates are non-zero, we can bound from below the distance to the minimizer. To this end, we will need the following lemma which, based on the tridiagonal structure of the Hessian of $F$, determines the non-zero entries:
\[proof:lem:non\_zero\_entries\] Let $F:{\mathbb{R}}^d\to{\mathbb{R}}$ be a convex quadratic function specified as follows $F({\mathbf{w}})\coloneqq \frac{c}{2} {\mathbf{w}}^\top A {\mathbf{w}}+ d {\boldsymbol{\epsilon}}_1^\top {\mathbf{w}}$, where $A$ is a tridiagonal matrix and $c,d$ are real scalars. Assuming that the iterates produced by a given optimization algorithm satisfy ${\mathbf{w}}_0=\dots={\mathbf{w}}_{\tau}=0$ and $$\begin{aligned}
\forall k\ge\tau,~{\mathbf{w}}_{k+1} \in \text{span}\{ \nabla F({\mathbf{w}}_{0}), \nabla F({\mathbf{w}}_{1}),\dots, \nabla F({\mathbf{w}}_{k-\tau}) \},
\end{aligned}$$ then $ {\mathbf{w}}_k \in \text{span}\{{\boldsymbol{\epsilon}}_0,{\boldsymbol{\epsilon}}_1,\dots,{\boldsymbol{\epsilon}}_{\lfloor
k/(\tau+1)\rfloor } \}$ for all $k\ge0$ (where ${\boldsymbol{\epsilon}}_0$ denotes the vector of all zeros, and ${\boldsymbol{\epsilon}}_i$ denote the $i$’th standard unit vector).
First, note that, given a vector ${\mathbf{w}}\in{\mathbb{R}}^d$, such that ${\mathbf{w}}\in\text{span}\{
{\boldsymbol{\epsilon}}_0, {\boldsymbol{\epsilon}}_1,\dots,{\boldsymbol{\epsilon}}_m \}$ for some $m\ge0$, we have $$\begin{aligned}
\nabla F({\mathbf{w}}) = cA{\mathbf{w}}+ d {\boldsymbol{\epsilon}}_1.
\end{aligned}$$ Since the entries of ${\mathbf{w}}$ are all zero start from the $m+1$ coordinate, $cA{\mathbf{w}}$ is a linear combination of the first $m$ columns of $A$. Being a $A$ tridiagonal matrix, it follows that all the entries of $cA{\mathbf{w}}$ are zero, except for its first $m+1$ coordinates, that is, $cA{\mathbf{w}}\in\text{span}\{{\boldsymbol{\epsilon}}_0,{\boldsymbol{\epsilon}}_1\dots, {\boldsymbol{\epsilon}}_{m+1}\}$. Together, $\nabla F({\mathbf{w}}) = cA{\mathbf{w}}+ d {\boldsymbol{\epsilon}}_1 \in \text{span}\{{\boldsymbol{\epsilon}}_1\dots, {\boldsymbol{\epsilon}}_{m+1}\}$.
We proceed by full induction. For $k=0,\dots,\tau$, the claim holds trivially. Now, assume the claim holds for all $i\le k$, where $k\ge\tau$, we show that the claim holds for $k+1$. By the induction hypothesis, $ {\mathbf{w}}_i \in \text{span}\{{\boldsymbol{\epsilon}}_0,{\boldsymbol{\epsilon}}_1,\dots,{\boldsymbol{\epsilon}}_{\lfloor i/(\tau+1)\rfloor } \}$ for all $i\le k$. Therefore, by the first part of the proof, we have, $ \nabla F ({\mathbf{w}}_i) \in \text{span}\{{\boldsymbol{\epsilon}}_1,{\boldsymbol{\epsilon}}_2,\dots,{\boldsymbol{\epsilon}}_{\lfloor i/(\tau+1)\rfloor +1 } \}$ for all $i\le k$, by which we conclude that $ \text{span}\{\nabla F ({\mathbf{w}}_0),\nabla F ({\mathbf{w}}_1),\dots,\nabla F ({\mathbf{w}}_{k-\tau})\} \subseteq \text{span}\{{\boldsymbol{\epsilon}}_1,{\boldsymbol{\epsilon}}_2,\dots,{\boldsymbol{\epsilon}}_{\lfloor (k-\tau)/(\tau+1)\rfloor +1 } \}$. Thus, by the linear span assumption, it follows that $$\begin{aligned}
{\mathbf{w}}_{k+1} \in \text{span}\{{\boldsymbol{\epsilon}}_1,{\boldsymbol{\epsilon}}_2,\dots,{\boldsymbol{\epsilon}}_{\lfloor (k-\tau)/(\tau+1)\rfloor +1 }\}.
\end{aligned}$$ Observing that, $$\begin{aligned}
\lfloor (k-\tau)/(\tau+1)\rfloor +1 = \lfloor (k-\tau)/(\tau+1)+1 \rfloor = \lfloor (k+1)/(\tau+1) \rfloor,
\end{aligned}$$ concludes the proof.
Overall, by [Lemma \[proof:lem:non\_zero\_entries\]]{}, the $k$’th iterate $w_k$, has all its entries zero, expect for (possibly) the first $\lfloor k/(\tau+1) \rfloor$ first coordinates. By Ineq. \[ineq:lower\_bound\_rate\], for any $\tau+1\le k \le 2\left(d- \frac{\log(1/2)}{2\log(q)}\right)$, we then have $$\begin{aligned}
\frac{\|{\mathbf{w}}_k\|^2}{\|{\mathbf{w}}^*\|^2} &\ge
\frac12 \exp\left(-\frac{4( \lfloor k/(\tau+1) \rfloor +1)}{\sqrt{\kappa}-1}\right)
\ge
\frac12 \exp\left(-\frac{4( k/(\tau+1) +1)}{\sqrt{\kappa}-1}\right)\\
&\ge
\frac12 \exp\left(-\frac{5 k }{(\sqrt{\kappa}-1)(\tau+1)}\right).\end{aligned}$$
For the convex case, we use a construction (devised by [@nesterov2004introductory]) similar to that of the strongly convex case. Let $\mu>0$ be fixed and consider the following function $$\begin{aligned}
F_k({\mathbf{w}}) \coloneqq \frac{\mu}{4}\left(\frac12{\langleA_k{\mathbf{w}}, {\mathbf{w}}\rangle}-
{\langle{\boldsymbol{\epsilon}}_1,{\mathbf{w}}\rangle}\right),\end{aligned}$$ where $A_k$ is a $d\times d$ matrix defined as follows $$\begin{aligned}
A = \left(\begin{matrix}
2& -1 & 0 & 0& \cdots & 0 & 0&0\\
-1& 2 & -1 & 0&\cdots & 0 & 0&0\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots& 0_{k,d-k}\\
0& 0& 0 &0& \cdots & -1 & 2&-1\\
0& 0& 0 &0& \cdots & 0 & -1 &2\\
& & & & 0_{d-k,k}& & & & 0_{d-k,d-k}
\end{matrix}\right),\end{aligned}$$ where $0_{m,n}$ is an $m\times n$ zero matrix. Given an iteration number $k$ such that $\tau+1\le k\le \frac12(d-1)(\tau+1)$, we take our function $F$ to be $F_{2\lfloor \frac{k}{\tau+1}\rfloor+1}({\mathbf{w}})$. Using [Lemma \[proof:lem:non\_zero\_entries\]]{}, the only (possibly) non-zero entries of the $k$’th iterate ${\mathbf{w}}_k$ are the first $\lfloor {k}/{(\tau+1)} \rfloor $ coordinates. Thus, following the same lines of proof as in [@nesterov2004introductory Theorem 2.1.6] yields $$\begin{aligned}
\frac{F({\mathbf{w}}_k) - F({\mathbf{w}}^*)}{{\|{\mathbf{w}}^*\|}^2}\ge
\frac{3L}{32(k/(\tau+1)+1)^2}\ge
\frac{L(\tau+1)^2}{45k^2}~.\end{aligned}$$
Proof of [Thm. \[thm:stochastic\_rate\]]{} {#proof:thm:stochastic_rate}
==========================================
We will first state and prove the following auxiliary lemma:
\[lem:convex\_power\_bound\] The following holds for any $\eta >0$:
- For any $k\ge1$, $$\begin{aligned}
\max_{\{a~:~0 <a<1/\eta\} }a(1-\eta a)^k \le \frac{1}{e\eta k},
\end{aligned}$$ where $e=2.718...$ is Euler’s number. In particular, $\sum_{i=0}^k
\max_{\{a~:~0<a<1/\eta\} }a(1-\eta
a)^{2(i+1)} \le \frac{1}{2e\eta }H_k \le \frac{1}{2e\eta
}(1+\ln(k+1))$, where $H_k$ denotes the $k$’th harmonic number.
- If, in addition, we assume that $a>\lambda$ for some constant $\lambda>0$, then $$\begin{aligned}
\sum_{i=0}^k \max_{\{a~:~\lambda< a<1/\eta\} }a(1-\eta a)^{2(i+1)} \le
\frac{1+e+\ln( \frac{1}{\eta\lambda}) }{e\eta}.
\end{aligned}$$
By the well-known inequality $1+x\le \exp(x),~x\in{\mathbb{R}}$, and since for the domain over which we optimize it holds that $1-\eta a >0$, we have for any $k\ge 1$ $$\begin{aligned}
a(1-\eta a)^k\le a\exp(-\eta a k).
\end{aligned}$$ Let us denote the latter by $\psi(a) \coloneqq a\exp(-\eta ak)$, and derive for it the desired upper bound.
Taking the derivative of $\psi$ and setting to zero, gives $$\begin{aligned}
(1-a \eta k)\exp(-\eta a k) =0.
\end{aligned}$$ Therefore, the only stationary point of $\psi$ is $a^*=\frac{1}{\eta k }$. Since $\psi'$ is positive for $a<a^*$ and negative for $a>a^*$, it follows that $a^*$ is a global maximum, at which the value of $\psi$ is $\frac{1}{e\eta k}$, concluding the first part of the proof.
Now, let $\lambda>0$. Since, the only maximizer of $\psi$ is at $a=\frac{1}{\eta
k}$, if $\lambda \ge\frac{1}{2\eta (i+1)}$, or equivalently $i\ge\frac{1}{2\eta\lambda}-1$, then $\max_{\{a~:~\lambda< a<1/\eta\}
}a(1-\eta a)^{2(i+1)}\le \lambda(1-\eta \lambda)^{2(i+1)}$. Therefore, $$\begin{aligned}
\sum_{i=1}^k \max_{\{a~:~\lambda< a<1/\eta\} }a(1-\eta a)^{2(i+1)} &\le
\sum_{i=1}^{\lfloor \frac{1}{2\eta\lambda}-1\rfloor } \max_{\{a~:~\lambda<
a<1/\eta\} }a(1-\eta a)^{2(i+1)}\\
&+
\sum_{i=\lceil \frac{1}{2\eta\lambda}-1\rceil }^{k } \max_{\{a~:~\lambda<
a<1/\eta\} }a(1-\eta a)^{2(i+1)}\\
&\le
\sum_{i=1}^{\lfloor \frac{1}{2\eta\lambda}-1\rfloor } \frac{1}{e\eta k}
+
\sum_{i=\lceil \frac{1}{2\eta\lambda}-1\rceil }^{k } \lambda(1-\eta \lambda)^{2(i+1)}
\\
&\le \frac{1}{e\eta }(1+\ln( \frac{1}{2\eta\lambda})) + \frac{1}{\eta}\\
&\le \frac{1+e+\ln( \frac{1}{\eta\lambda}) }{e\eta}
\end{aligned}$$
We now turn to prove [Thm. \[thm:stochastic\_rate\]]{} itself. By Ineq. \[ineq:stochastic\] we have $$\begin{aligned}
\label{ineq:stochastic_rate1}
2{\mathbb{E}}[F({\mathbf{w}}_k)-F({\mathbf{w}}^*)]&\le \|\sqrt{A}[z^k] D\|^2\|{\mathbf{e}}_0\|^2 +
\eta^2\sigma^2\sum_{i=0}^k \left\|\sqrt{A}[z^i]D \right\|^{2}.\end{aligned}$$ We will bound each of the terms above separately. Assuming $\eta
\in\left(0,\frac{1}{20\mu(\tau+1)}\right]$ we have by Ineq. \[ineq:generating\_bound\] and Ineq. \[ineq:coefficients\_bound\], $$\begin{aligned}
\label{ineq:stochastic_rate0}
\begin{aligned}
\|\sqrt{A}[z^k] D\|^2 &=
\|\sqrt{A}[z^k]\left((I-Iz+\eta A z^{\tau+1})^{-1}\right)\|^2\\
&\le \max_{i\in[d]}\left|\sqrt{a_i}[z^k] \frac{1}{\pi_{\eta a_i}(z)} \right|^2\\
&\le
\begin{cases}
\max_{i\in[d]} a_i & 0\le k \le (\tau+1)\ln(2(\tau+1))-1,\\
9\max_{i\in[d]} a_i (1-\alpha)^{2(k+1)}& k \ge (\tau+1)\ln(2(\tau+1)),
\end{cases}
\end{aligned}\end{aligned}$$ Thus, for the first term, assuming $k \ge (\tau+1)\ln(2(\tau+1))$, we have $$\begin{aligned}
\label{ineq:stochastic_rate2}
\begin{aligned}
\|\sqrt{A}[z^k] D\|^2 &\le 9 \max_{i\in[d]} a_i (1-\eta a_i)^{2(k+1)}
\le 9 \mu \max_{i\in[d]} (1-\eta a_i)^{2(k+1)}\\
&\le 9 \mu \exp(-2\eta \lambda(k+1)).
\end{aligned}\end{aligned}$$ Bounding the second term in Ineq. \[ineq:stochastic\_rate1\] is somewhat more involved and requires separating into the two regimes stated in Ineq. \[ineq:stochastic\_rate0\]: $$\begin{aligned}
\label{ineq:stochastic_rate3}
\begin{aligned}
\sum_{i=0}^k \left\|\sqrt{A}[z^i]D \right\|^{2}
&\le
\sum_{i=0}^{\lceil(\tau+1)\ln(2(\tau+1))\rceil-1} \left\|\sqrt{A}[z^i]D \right\|^{2}
+\sum_{i=\lceil(\tau+1)\ln(2(\tau+1))\rceil}^{k} \left\|\sqrt{A}[z^i]D \right\|^{2}\\
&\le \mu(\tau+1)\ln(2(\tau+1))
+9\sum_{i=0}^{k} \max_{i\in[d]} a_i(1-\eta a_i)^{2(i+1)}
\end{aligned}\end{aligned}$$ We proceed by considering the strongly convex case and the convex case separately. For the strongly convex case we have by [Lemma \[lem:convex\_power\_bound\]]{} $$\begin{aligned}
\sum_{i=0}^k \left\|\sqrt{A}[z^i]D \right\|^{2}
&\le \mu(\tau+1)\ln(2(\tau+1))
+9\sum_{i=0}^{k} \max_{i\in[d]} a_i(1-\eta a_i)^{2(i+1)}\\
&\le
\mu(\tau+1)\ln(2(\tau+1))
+\frac{1+e+\ln(\frac{1}{\eta\lambda})}{\eta}.\end{aligned}$$ Together with Ineq. \[ineq:stochastic\_rate1\] and Ineq. \[ineq:stochastic\_rate2\], this implies that for $k \ge (\tau+1)\ln(2(\tau+1))$, $$\begin{aligned}
2{\mathbb{E}}[&F({\mathbf{w}}_k)-F({\mathbf{w}}^*)]\le \|\sqrt{A}[z^k] D\|^2\|{\mathbf{e}}_0\|^2 +
\eta^2\sigma^2\sum_{i=0}^k \left\|\sqrt{A}[z^i]D \right\|^{2}\\
&\le 9 \mu \exp(-2\eta \lambda(k+1))\|{\mathbf{e}}_0\|^2 + \eta^2\sigma^2\left(\mu(\tau+1)\ln(2(\tau+1))
+\frac{1+e+\ln(\frac{1}{\eta\lambda})}{e\eta}\right)~,\end{aligned}$$ resulting in the first bound stated in the theorem. To get the second bound, we show how to optimally tune the step size $\eta$ (up to log factors). Ignoring the log factors, the bound above is $${\mathbb{E}}\left(F({\mathbf{w}}_k)-F({\mathbf{w}}^*)\right)~\leq~
\tilde{{\mathcal{O}}}\left(\mu{\|{\mathbf{e}}_0\|}^2\exp(-2\eta\lambda
k)+\eta^2\sigma^2\left(\mu\tau+\frac{1}{\eta}\right)\right)~.$$ Moreover, since we assume that $\eta\leq {\mathcal{O}}(1/\mu\tau)$, we get that $\mu\tau$ is dominated (up to constants) by $1/\eta$, so we can simplify the above to $$\label{eq:strconvtilde}
{\mathbb{E}}\left(F({\mathbf{w}}_k)-F({\mathbf{w}}^*)\right)~\leq~
\tilde{{\mathcal{O}}}\left(\mu{\|{\mathbf{e}}_0\|}^2\exp(-2\eta\lambda
k)+\eta\sigma^2\right)~.$$ We now consider three cases:
- If $0\leq \frac{\ln(\lambda\mu{\|{\mathbf{e}}_0\|}^2 k/\sigma^2)}{2\lambda
k}\leq
\frac{1}{20(\mu\tau)}$, we can pick $\eta = \frac{\ln(\lambda
\mu{\|{\mathbf{e}}_0\|}^2
k/\sigma^2)}{2\lambda k}$, and get that [Eq. (\[eq:strconvtilde\])]{} is $$\begin{aligned}
\tilde{{\mathcal{O}}}\left(\frac{\sigma^2}{\lambda k}\right)~=~
\tilde{{\mathcal{O}}}\left(\mu{\|{\mathbf{e}}_0\|}^2\exp\left(-\frac{\lambda k
}{10\mu\tau}\right)+\frac{\sigma^2}{\lambda k}\right)
\end{aligned}$$
- If $ \frac{\ln(\lambda\mu{\|{\mathbf{e}}_0\|}^2 k/\sigma^2)}{2\lambda
k} <0$, it follows that $\mu{\|{\mathbf{e}}_0\|}^2\leq
\frac{\sigma^2}{\lambda k}$. In that case, we pick $\eta=0$, and get that [Eq. (\[eq:strconvtilde\])]{} is $$\tilde{{\mathcal{O}}}\left(\mu{\|{\mathbf{e}}_0\|}^2\right)~\leq~\tilde{{\mathcal{O}}}\left(\frac{\sigma^2}{\lambda
k}\right)~=~
\tilde{{\mathcal{O}}}\left(\mu{\|{\mathbf{e}}_0\|}^2\exp\left(-\frac{\lambda k
}{10\mu\tau}\right)+\frac{\sigma^2}{\lambda k}\right)~.$$
- If $\frac{\ln(\lambda\mu{\|{\mathbf{e}}_0\|}^2 k/\sigma^2)}{2\lambda
k}>
\frac{1}{20(\mu\tau)}$, we pick $\eta=\frac{1}{20(\mu\tau)}$, and get that [Eq. (\[eq:strconvtilde\])]{} is $$\tilde{{\mathcal{O}}}\left(\mu{\|{\mathbf{e}}_0\|}^2\exp\left(-\frac{\lambda
k}{10\mu\tau}\right)+\frac{\sigma^2}{\mu\tau}\right)~\leq~
\tilde{{\mathcal{O}}}\left(\mu{\|{\mathbf{e}}_0\|}^2\exp\left(-\frac{\lambda k
}{10\mu\tau}\right)+\frac{\sigma^2}{\lambda k}\right)~.$$
Collecting the three cases above, we get a bound of $$\tilde{{\mathcal{O}}}\left(\mu{\|{\mathbf{e}}_0\|}^2\exp\left(-\frac{\lambda k
}{10\mu\tau}\right)+\frac{\sigma^2}{\lambda k}\right)$$ as required.
For the convex case, we have by Ineq. \[ineq:stochastic\_rate1\], Ineq. \[ineq:stochastic\_rate0\] and [Lemma \[lem:convex\_power\_bound\]]{}, that for $k \ge (\tau+1)\ln(2(\tau+1))$ $$\begin{aligned}
{\mathbb{E}}[&F({\mathbf{w}}_k)-F({\mathbf{w}}^*)]\\
&\le
\frac{9}{4e\eta(k+1)}\|{\mathbf{e}}_0\|^2 + \frac{\eta^2\sigma^2}{2}\left(\mu(\tau+1)\ln(2(\tau+1))
+\frac{9}{2e\eta }(1+\ln(k+1))\right)~,\end{aligned}$$ resulting in the third bound in the theorem. To get the fourth bound, we now show how to optimally tune the step size $\eta$ (up to log factors). Ignoring the log factors, the bound above is $$\tilde{{\mathcal{O}}}\left(\frac{{\|{\mathbf{e}}_0\|}^2}{\eta
k}+\eta^2\sigma^2\left(\mu\tau+\frac{1}{\eta}\right)\right)~.$$ As in the strongly convex case, since we assume $\eta\leq {\mathcal{O}}(1/(\mu\tau)$, we can simplify the above to $$\tilde{{\mathcal{O}}}\left(\frac{{\|{\mathbf{e}}_0\|}^2}{\eta
k}+\eta\sigma^2\right)~.$$ We now consider two cases:
- If $\frac{{\|{\mathbf{e}}_0\|}}{\sigma\sqrt{k}}\leq \frac{1}{20(\mu\tau)}$, we choose $\eta=\frac{{\|{\mathbf{e}}_0\|}}{\sigma\sqrt{k}}$, and get $$\tilde{{\mathcal{O}}}\left(\frac{{\|{\mathbf{e}}_0\|}\sigma}{\sqrt{k}}\right)~=~
\tilde{{\mathcal{O}}}\left(\frac{{\|{\mathbf{e}}_0\|}^2\mu\tau}{
k}+\frac{{\|{\mathbf{e}}_0\|}\sigma}{\sqrt{k}}\right)~.$$
- If $\frac{{\|{\mathbf{e}}_0\|}}{\sigma\sqrt{k}}> \frac{1}{20(\mu\tau)}$, we choose $\eta=\frac{1}{20(\mu\tau)}$, and get $$\tilde{{\mathcal{O}}}\left(\frac{{\|{\mathbf{e}}_0\|}^2\mu\tau}{
k}+\frac{\sigma^2}{\mu\tau}\right)~\leq~
\tilde{{\mathcal{O}}}\left(\frac{{\|{\mathbf{e}}_0\|}^2\mu\tau}{
k}+\frac{{\|{\mathbf{e}}_0\|}\sigma}{\sqrt{k}}\right)~.$$
Collecting the two cases above, we get a bound of $$\tilde{{\mathcal{O}}}\left(\frac{{\|{\mathbf{e}}_0\|}^2\mu\tau}{
k}+\frac{{\|{\mathbf{e}}_0\|}\sigma}{\sqrt{k}}\right)$$ as required.
[^1]: Following standard convention, we use here the ${\mathcal{O}}$-notation to hide constants, and tilde $\tilde{{\mathcal{O}}}$-notation to hide constants and factors polylogarithmic in the problem parameters.
|
---
abstract: 'We remark that the power diagrams from computer science are the spines of amoebas in algebraic geometry, or the hypersurfaces in tropical geometry. Our concept of a Morse poset generalizes to power diagrams. We show that there exists a discrete Morse function on the coherent triangulation, dual to the power diagram, such that its critical set equals the Morse poset of the power diagram. In the final section we use Maslov dequantization to compute the medial axis.'
address:
- 'Mathematisch Instituut, Universiteit Utrecht, PO Box 80010, 3508 TA Utrecht The Netherlands.'
- 'Department of Mathematics, Hokkaido University Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan'
author:
- 'D. Siersma'
- 'M. van Manen'
title: Power diagrams and their applications
---
Introduction
============
Power diagrams were introduced by Aurenhammer in [@MR873251], as a generalization of the more commonly known Voronoi diagrams. They recently surfaced in algebraic geometry, as the spines of amoebas, see [@MR2040284]. Another appearance of the power diagram is what other authors call a tropical hypersurface, see [@AG0306366]. They also appear when string theorists use toric geometry, see for instance [@MR1969655].
The treatment they receive in [@MR2040284] is different from the one in [@MR873251]. The essential tool that Passare and Rullg[å]{}rd use is the Legendre transform, a tool that first seems to have been introduced in convex analysis in [@MR0000846]. The essential tool that Aurenhammer uses is that the power diagram of some points in ${{\mathbb R}}^n$ is a projection of a convex hull in ${{\mathbb R}}^{n+1}$. We spell out the relation between the classical theory of Voronoi diagrams and that of the tropical hypersurface. Note that what we call here a Delaunay triangulation is called a coherent triangulation in chapter 7 of [@MR1264417]. Such triangulations are called regular in [@MR1311028].
After that we will explore the relationship between Voronoi diagrams and power diagrams. In particular given a power diagram $\mathcal{T}$ of $N$ points $\colP1N$ in ${{\mathbb R}}^n$ we will construct a Voronoi diagram $\Upsilon$ of $N$ points in ${{\mathbb R}}^{n+1}$ a hyperplane section of which is the diagram $\mathcal{T}$. Thus power diagrams are affine Voronoi diagrams in a very precise sense.
Then we focus on higher order structures. With a Voronoi diagram there come the $k$-th order Voronoi diagrams and the Delaunay triangulation. The $k$-th order power diagram is again a power diagram, see [@MR1939695], and to it there is associated the $k$-th order coherent triangulation. The union of all the power diagrams is a hyperplane arrangement. We conclude that not every picture that looks like a power diagram is actually a power diagram.
The main new point of view we present in this article is the relation between tropical hypersurfaces and discrete Morse theory introduced by Forman in [@MR1612391]. This seems to have gone unnoticed before. The triangulations are polyhedral complexes, and the affine linear functions that define the power diagram give rise to discrete Morse function on the dual simplicial complex. We apply this third way of looking at power diagrams to a notion we introduced before, the Morse poset. Morse posets are seen to be determining a discrete vectorfield.
We apply our newly found knowledge to the problem of enumerating different Morse posets.
In the final section we show another application of tropical geometry to computational geometry. We use Maslov dequantization to determine the medial axis.
Many of our results can in some explicit form or another be found in the literature. We have gone to great lengths to include appropriate references, but in all likelihood we are not complete. It has been remarked by other authors also that many results in tropical geometry - as the field has come to be known - are just well-known theorems from other fields, but now in a different guise. This article just reinforces that opinion.
Power diagrams and tropical hypersurfaces.
==========================================
In this section we show the equivalence of two definitions of the power diagram. First we discuss the original notion of Aurenhammer, then we show that these are equivalent to the definitions of Passare *et. al.*.
The definition of Aurenhammer
-----------------------------
The original definition of a power diagram is by means of distance function with weights. Take a point set $\colP1N\subset{{\mathbb R}}^n$. Throughout this article we assume that $\dim(\operatorname{CH}(\colP1N))=n$. Assign a weight $w_i$ to each point $P_i$. Then write down the functions: $$\label{eq:20}
g_i \colon {{\mathbb R}}^n \rightarrow{{\mathbb R}}\quad g_i(x)=\frac12\lVert x -P_i \rVert - \frac12w_i\quad g(x) = \min_{1\leq i \leq N}g_i(x)$$ Here we have used the following notation: $$\lVert x\rVert = \sum_{i=1}^nx_i^2$$
For a subset $\alpha\subset\colP1N$ the set $\operatorname{Pow}(\alpha)$ is the closure of $$\{ x \in{{\mathbb R}}^n\,\mid\,
g_i(x) = g(x) \,\,\, P_i\in\alpha\text{ and }
g_j(x) > g(x) \,\,\, P_j\not\in\alpha
\}$$ The sets $\operatorname{Pow}(\{ P_i \} )$ are sometimes called cells.
The *separator* $\operatorname{Sep}(\{P_i,P_j\})$ between $P_i$ and $P_j$ is the hyperplane. $$\{ x\in{{\mathbb R}}^n \,\mid\, g_i(x)=g_j(x) \}$$
It is no restriction to assume that $w_i>0$. We may add some number to all of the functions $g_i$: the cells will not change. In fact we can add any function $h\colon{{\mathbb R}}^n\rightarrow {{\mathbb R}}$ to the $g_i$. We will get the same power diagram. Power diagrams are polyhedral subdivisions. We repeat definition 1 in [@MR2040284].
A *polyhedral subdivision* $\mathcal{T}$ of a polyhedron $K\subset{{\mathbb R}}^n$ is a subdivision of $K$ in polyhedra $K_i$, such that
- The union of all sets $K_i\in\mathcal{T}$ is $K$.
- If $K_i$ and $K_j$ are both in $\mathcal{T}$ then so is their intersection.
- Every compact subset $L$ of $K$ intersects only a finite number of the $K_i$.
The sets $\operatorname{Pow}(\alpha)$ for $\alpha\subset\colP1N$ are a polyhedral subdivision of ${{\mathbb R}}^n$. This polyhedral subdivision is called the *power diagram*.
The cells cover ${{\mathbb R}}^n$. They are the intersection of a finite number of halfspaces of the form: $\{ x\in{{\mathbb R}}^n \,\mid\, g_i(x)\leq g_j(x) \}$. So they are polyhedra.
Besides many similarities , there are a few crucial differences between power diagrams and Voronoi diagrams. In a power diagram the cell of a point may well be empty. It might happen that $g_i(x) > g_j(x)$, for all $i\neq j$ and all $x\in{{\mathbb R}}^n$.
Also, whereas in Voronoi diagrams, $P_i$ is always contained in its own cell, in a power diagram the cell of $P_i$ might not be empty and still $P_i$ does not lie in its own cell!
The power diagram can be constructed much like the Voronoi diagram. Recall that the Voronoi diagram is the upper convex hull to the tangent planes to a parabola $x_0=a(\sum_{i=1}^n x_i^2 )$. For the case of the power diagrams we do much the same thing. Look at figure \[fig:12\]. We take cylinders with radius $\sqrt{w_i}$ around the lines $x=P_i$ in ${{\mathbb R}}^{n+1}$. The intersection of the cylinders with the parabola is a torsion zero space curve. We draw the planes in which they lie ( left figure ) and we consider the upper convex hull of these planes (right figure). The projection of its singular sides is the power diagram.
![Power lifting to the parabola[]{data-label="fig:12"}](newblaat1.eps "fig:"){width="40.00000%"} ![Power lifting to the parabola[]{data-label="fig:12"}](newblaat2.eps "fig:"){width="40.00000%"}
The above construction is also described in [@MR873251], section 4.1. When the $r_i=0$ the construction reduces to the well-known construction of Voronoi diagrams. In that case, of course, one has to take the tangent planes to the paraboloid.
Tropical hypersurface.
----------------------
The power diagram can be defined using only affine functions. With amoebas in mind, this was done in [@MR2040284]. With tropical hypersurfaces in mind, this was done in [@AG0306366]. The separator between $P_i$ and $P_j$ is a hyperplane. It is given by $$g_i(x)=g_j(x) \,\Leftrightarrow\,
\langle x , P_i \rangle
- \frac{\lVert P_i \rVert -w_i}2 =
\langle x , P_j \rangle
- \frac{\lVert P_j \rVert -w_j}2$$ We can define the power diagram solely using the affine functions $f_i(x)=\langle x , P_i \rangle + c_i$, where the coefficient $c_i$ is $$\label{eq:15}
c_i =
- \frac{\lVert P_i \rVert -w_i}2$$ Consider the function $$f(x)= \max_{i=1, \cdots , N} f_i(x)$$ The following proposition shows that we can define the power diagram solely using affine functions.
For a subset $\alpha\subset\colP1N$ the set $\operatorname{Pow}(\alpha)$ is the closure of $$\{ x \in{{\mathbb R}}^n\,\mid\,
f_i(x) = f(x) \,\,\, P_i\in\alpha\text{ and }
f_j(x) < f(x) \,\,\, P_j\not\in\alpha
\}$$
In [@AG0306366], $f$ is defined as the minimum of the functions $f_i$. Taking the minimum or taking the maximum is just a convention though.
The affine definition of power diagrams is more elegant. We lose the immediate appeal of the construction in figure \[fig:12\], but two statements are now immediately clear.
The intersection of a hyperplane with a power diagram is again a power diagram. The image of a power diagram under an affine map ${{\mathbb R}}^n\rightarrow{{\mathbb R}}^n$ is again a power diagram.
So a power diagram can be sliced, after which we obtain a new power diagram. In fact, every power diagram in ${{\mathbb R}}^n$ is a slice of a Voronoi diagram in ${{\mathbb R}}^{n+1}$.
Let $\mathcal{T}$ be a power diagram in ${{\mathbb R}}^n$. There is a Voronoi diagram $\Upsilon$ in ${{\mathbb R}}^{n+1}$ and a hyperplane $H$ such that $H\cap\Upsilon=\mathcal{T}$.
We will explicitly construct such an $\Upsilon$. Let $\colP1N$ be the set of points in ${{\mathbb R}}^n$ that determine $\mathcal{T}$. Write $$Q_i=(P_i,P_i^\prime ) \quad \bar{x}=(x,x^\prime)$$ We have $$\lVert \bar{x}-Q_i \rVert = \lVert x - P_i \rVert + (x^\prime - P_i^\prime)^2$$ So when $x^\prime=0$ we get $$\lVert \bar{x}-Q_i \rVert = \lVert x - P_i \rVert + P_i^{\prime 2}$$ Now let the power diagram be given by functions $g_i$ as in equation . It is no restriction to assume that $w_i<0$, because only the differences $w_i-w_j$ matter for the power diagram. Thus we can choose $P_i^\prime=\sqrt{-w_i}=r_i$.
The Legendre transform
----------------------
Dual to the Voronoi diagram is the Delaunay triangulation. In a similar way there exists a dual of the power diagram. This dual cell complex is not always a triangulation, similar to the case with the Delaunay triangulation. If we use the tropical point of view ( with affine functions ) we can best explain the dual object using the Legendre transform, see [@MR1301332], which we explain below. This dual object is also considered in [@MR1264417], where it has the name coherent triangulation.
We start with some definitions. Let $P$ be a polyhedron in ${{\mathbb R}}^{n+1}$. Let $v$ be a vector in ${{\mathbb R}}^n$. The *lower faces* of $P$ with respect to $v$ are those faces $F$ of $P$ such that $$\forall x \in F\,\, \forall \lambda \in {{\mathbb R}}_{>0} \colon x - \lambda v \notin P$$ A polyhedral subdivision of a polytope in ${{\mathbb R}}^n$ is called *regular* if it is the projection of the lower faces of a polytope in ${{\mathbb R}}^{n+1}$. Not every polyhedral subdivision is regular, see chapter 5 in [@MR1311028], or chapter 7 in [@MR1264417].
The Legendre transform of a convex function $f$, with domain $D\subset {{\mathbb R}}^n$ is $$\hat{f}(\xi)=\sup_{x\in D}\left(\langle \xi, x \rangle - f(x)\right)$$ When the supremum does not exists, we put $\hat{f}(\xi)=\infty$. The domain $\operatorname{Dom}(\hat{f})$ of $\hat{f}$ are those $\xi$ for which $\hat{f}(\xi) < \infty$.
The Legendre transform of $f(x)=\frac12\lVert x \rVert$ is the function itself. The Legendre transform of a linear function $f_i=\langle x , P_i \rangle +c_i$ is $ < \infty$ only when $\xi =P_i$. Theorem 2.2.7 of [@MR1301332] reads:
\[sec:legendre-transform\] Let $f = \sup_{\alpha\in A}f_\alpha(x)$ be the maximum of a number of lower semi-continuous convex function. Then $f$ is also a lower semi-continuous convex function. Furthermore $\hat{f}$ is the infimum over all finite sums: $$\hat{f}(\xi)=\inf_{\sum\lambda_\alpha\xi_\alpha=\xi,\lambda_\alpha>0,\sum_\alpha\lambda_\alpha=1}
\sum_\alpha\lambda_\alpha\hat{f_\alpha}(\xi_\alpha)$$
With theorem \[sec:legendre-transform\] we calculate the Legendre transform of the function that determines the power diagram. We have $$f=\max_{1 \leq i \leq N} \langle x , P_i \rangle + c_i$$ The domain where $\hat{f} < \infty$ is the convex hull of the points $\colP1N$. We have $$\hat{f_i}(P_i)=-c_i \Rightarrow \hat{f}(\xi)=\inf \left( -\sum_i\lambda_i c_i \right)$$ where the infimum is taken over all $\lambda_i$ such that $$\sum_{1\leq i \leq N}\lambda_iP_i = \xi\text{ and }
\forall i\colon \lambda_i \geq 0
\text{ and } \sum_{i=1}^N \lambda_i = 1$$ It is no restriction to assume that only $\leq n+1$ of the $\lambda_i$ are non-zero because any point in the convex hull of the $P_i$ can be expressed as a sum of $\dim\operatorname{CH}(\colP1N)$ of the $P_i$. The infima are best thought of in a geometric way. Take the convex hull $\operatorname{CH}( \{ ( P_i, -c_i) \}_{i=1}^N )$. The lower faces form the graph of $\hat{f}$ over the convex hull $\operatorname{CH}(\colP1N)$. We summarize our discussion in a theorem.
\[thm:legendre-transform\] Let $\mathcal{T}$ be a power diagram. Let $\Upsilon$ be the lower convex hull of the lifted points $(P_i, -c_i)$ wrt. to the vector $( 0 , \cdots , 0 , 1 )$. The polyhedral complex $\Upsilon$ is the graph of the Legendre transform $\hat{f}$ of $f$. The domain $\operatorname{Dom}(\hat{f})$ of $\hat{f}$ is the convex hull $\operatorname{CH}(\colP1N)$.
The geometrical construction is directly related to the construction of figure \[fig:12\]. If we put the upper convex hull of theorem \[thm:legendre-transform\] in figure \[fig:12\] we get figure \[fig:11\].
![The power lift and the Legendre transform[]{data-label="fig:11"}](newblaat3.eps "fig:"){width="40.00000%"} ![The power lift and the Legendre transform[]{data-label="fig:11"}](newblaat4.eps "fig:"){width="40.00000%"}
What exactly happens here can best understood by writing out the equations. In figure \[fig:12\] the graph of $x_0=\frac12\lVert x \rVert$ is drawn. Then the cylinders in the picture are $\lVert x - P_i \rVert = r_i^2$. Or $\lVert x -P_i \rVert = w_i$. Hence the three planes of which we determine the upper hull are $$\label{eq:21}
x_0=\langle x , P_j \rangle +\frac{r_j^2-\lVert P_j \rVert}{2}$$ The triangle in figure \[fig:11\] is the plane $x_0=-\sum \lambda_i c_i$ where the $\lambda_i$ are defined by $x=\sum\lambda_iP_i$. In the point $P_i$ we have that the graph of $f(x)=\max_if_(x)$ lies $\frac{w_i}2$ above the paraboloid $x_0=\frac12\lVert x \rVert$, whilst the graph of $\hat{f}(\xi)$ lies $\frac{w_i}2$ below the paraboloid there.
Theorem \[thm:legendre-transform\] gives a dual triangulation of the power diagram.
Given a power diagram $\mathcal{T}$ specified by $N$ affine functions, the coherent triangulation $\sigma$ is the set of points $\xi\in\operatorname{CH}(\colP1N)$ where $\hat{f}$ is not differentiable.
Since the coherent triangulation is the projection of the lower convex hull, it has also the structure of a polyhedral complex. In the case when all the weights $w_i$ are zero this coherent triangulation is of course the Delaunay triangulation.
A number of points in $\colP1N$ in ${{\mathbb R}}^n$ is affinely in general position when the convex hull of each subset of length $n+1$ of $\colP1N$ is full dimensional. For Voronoi diagrams this definition is not sufficient. It might still be that $n+2$ lie on a sphere. In that case the Delaunay triangulation is not a simplicial complex. For the Delaunay triangulation to really be a triangulation we need that the points $\colQ1N$ are affinely general position in ${{\mathbb R}}^{n+1}$ where $$Q_i = ( P_i , \frac12\lVert P_i \rVert )$$
A power diagram $f=\max f_i$ is *in general position* when the points $Q_i$ are affinely in general position in ${{\mathbb R}}^{n+1}$. Here $$Q_i=(P_i, -c_i)\text{, } 1 \leq i \leq N$$
The following theorem is left to the reader.
The coherent triangulation is a simplicial complex when the power diagram is in general position.
Consider the function $$(\xi , x) \mapsto F(x,\xi )= f(x) + \hat{f}(\xi) - \langle x , \xi \rangle$$ This function hides the *Gateau differential*. Namely, let $h\colon{{\mathbb R}}^n\rightarrow{{\mathbb R}}$ be a function. If the limit $$h^\prime(x ; \xi ) = \lim_{t \downarrow 0} \frac{h(x+t \xi ) - h(x)}{t}$$ exists, it is called the Gateau differential. See [@MR1301332], theorem 2.1.22. For a convex set $K\subset{{\mathbb R}}^n$ the *supporting function* of $K$ is the function $$\xi \mapsto \sup_{x\in K} \langle x , \xi \rangle$$ Theorem 2.2.11. in [@MR1301332] says that the Gateau differential $\xi \rightarrow f^\prime( x ; \xi )$ is the supporting function of $$\left\{ \xi \,\mid\, F(x,\xi) = 0 \right\}$$ Let us see an example of what that means. Take $\xi$ to be one of the points of $\colP1N$. Then $\hat{f}(P_i)= - c_i$. So the set of $x$ where $F(x,\xi)=0$ are those for which $f(x)=\langle x, P_i \rangle + c_i$.
The Gateau differential is called Clarke’s generalized derivative in [@MR1481622]. The set of which it is the support function is called $\partial f(x)$. It is stated in that article that $\partial f(x)$ is the convex hull of the gradients of the functions $f_i$ for which $f_i(x)=f(x)$.
That last statement and the theorem 2.1.22 are all equivalent to what is neatly formulated in proposition 1 of [@MR2040284]:
There is a subdivision of the convex hull $\operatorname{CH}(\colP1N)$, dual to the power diagram $\mathcal{T}$. The cell $\operatorname{Del}(\alpha)$ dual to $\operatorname{Pow}(\alpha)\in\mathcal{T}$ is $$\operatorname{Del}(\alpha) = \{ \xi \mid F(x,\xi) = 0 \quad\forall x \in\operatorname{Pow}(\alpha)\}$$ Reversely $$\operatorname{Pow}(\alpha) = \{ x \mid F(x,\xi) = 0 \quad\forall\xi\in\operatorname{Del}(\alpha)\}$$
Similar statements are in the review by Pennaneac’h, see [@HenPen]. However Pennaneac’h does not discuss the relation between the figures \[fig:12\] and \[fig:11\]. Also she does not make clear what the relationship between this theorem and the well-known theory of Voronoi diagrams/ Delaunay tesselations is. Her review however is heartily recommended.
Disappearing faces
==================
We have noted before that faces in the power diagram might disappear. This will happen for some $P_j\in\colP1N$ iff.$$f_j(x) < f(x)\quad\forall x\in {{\mathbb R}}^n$$ Cells in the power diagram correspond to vertices in the coherent triangulation. In that case it is very easy to see which vertices disappear. They are the ones lying in the epigraph of the Legendre transform. Remember that the epigraph of a function $h\colon D\rightarrow{{\mathbb R}}$ is defined as: $$\{ ( x, t )\,\mid\, h(x) < t\quad x\in D \}$$ Let us sum up some consequences:
If $P_j$ lies outside the convex hull of $\operatorname{CH}(\colP1N\setminus\{P_j\})$ then the cell $\operatorname{Pow}(\{P_j\})$ is not empty. Proof: the lifted point $(P_j, -c_j)$ forcibly is part of the lower hull of the points $(P_i, -c_i)$.
The transition on the power diagrams where a cell disappears is exemplified in figure \[fig:9\]. In the pictures the convex hull of the points $(P_i,-c_i)$ is drawn. The graph of the Legendre transform $\hat{f}$ is formed by the lower faces of that polytope. In the ground plane we see the projection of those lower faces: the coherent triangulation dual to the power diagram. In the ground plane also the circles with radius $\sqrt{w_i}$ are drawn.
On the left hand side of figure \[fig:9\] we see a point with a small circle around it. This point disappears when we make one of the circles in the pictures ( the one on the upper right ) bigger. Such has been done on the left hand side of figure \[eq:9\]. The upper right circle is much bigger there and the lifted point corresponding to the small circle lies inside the epigraph of the Legendre transform $\hat{f}$.
![Transition on power diagrams[]{data-label="fig:9"}](appeared.eps "fig:"){height="40.00000%"} ![Transition on power diagrams[]{data-label="fig:9"}](disappeared.eps "fig:"){height="40.00000%"}
We thus see that there are three sorts of events on planar coherent triangulations:
- Convex hull events and edge flips
- Disappearing vertices.
Convex hull events and edge flips already occur in families of Voronoi diagrams. A detailed study of these phenomena can be found in [@MR1925090]. However in a dual triangulation of a Voronoi diagram, none of the cells $\operatorname{Pow}(\{P_i\})$ can be empty.
The $k$-th order power diagram and hyperplane arrangements
==========================================================
Given a power diagram we define a $k$th order power diagram. Consider all subsets $I$ of length $k$ in $\{ 1 , \cdots , N \}$. Denote $$P_I=\frac1{\lvert I \rvert}\sum_{i\in I} P_i$$ and $$f_I=\frac1{\lvert I \rvert}\sum_{i\in I} f_i = \langle x , P_I \rangle
+ \frac1{\lvert I \rvert}\sum_{i\in I}c_i$$ Denote also $$f^{(k)}= \max_{I\subset\{1, \cdots , N\}\,\lvert I \rvert =k} f_I$$ So $f^{(1)}=f$.
The $k$-th order power diagram $\operatorname{Pow}^{(k)}$ is the power diagram of the $f_I$. Denote $\operatorname{Del}^{(k)}$ its dual.
Our definition is admittedly rather arbitrary. One can take any weighted sum of the points.
The $k$-th order power diagram is a polyhedral complex. The point $x$ lies in the cell $I$ of the $k$-th power diagram iff. $f_i(x)\geq f_j(x)$ $\forall i\in I$ and $\forall j\in \{ 1 , \cdots , N\}\setminus I$.
The first part of the lemma is obvious. The second part of the lemma boils down to the following obvious statement. Let $\{a_i\}_{i=1}^N$ be a set of real numbers. If $I$ is the index set of the $k$ biggest of the $\{a_i\}_{i=1}^N$ then $a_I$ is the biggest of the numbers $$\{ \frac1k \sum_{j\in J} a_j\,\mid\, \lvert J\rvert=k\}$$
There are two sources of cells in $\operatorname{Pow}^{(k+1)}$.
- Let $\operatorname{Pow}^{(k)}(I_1)$ and $\operatorname{Pow}^{(k)}(I_2)$ be two power cells in the $k$-th order power diagram. Assume that the two cells are adjacent, that is there is an edge between the two vertices in the dual coherent triangulation $\operatorname{Del}^{(k)}$. This means that $I_1\cap I_2$ has $k+1$ distinct elements. Thus the cell of $I_1\cap I_2$ in $\operatorname{Pow}^{(k+1)}$ is not empty.
- Let the cell of $I\subset \colP1N$ be an empty cell. That is: $$f_I(x) \leq f_J(x) \quad \forall x \in {{\mathbb R}}^n \quad\forall J\subset\colP1N
\quad
\lvert J \rvert =k$$ but that at some point $x_0\in{{\mathbb R}}^n$ there is only one $j\in\colP1N\setminus I$ for which $f_j(x_0)$ is bigger than all $f_i(x_0)$ for $i\in I$.
The union of the power diagrams of order $k$ is a hyperplane arrangement. The hyperplanes are the separators. Each such separator is equipped with a polyhedral subdivision, that is: the separator between $P_i$ and $P_j$ is a hyperplane $V$ divided in at most $N$ closed convex subsets $K_l$. If $x$ lies in the interior of $K_l$ then $x$ lies on the $l$-th order power diagram.
That there is a polyhedral subdivision on the separator is rather obvious. The points of the $k$-th diagram is the intersection of two closed convex subsets, so its intersection is also convex.
The aforementioned hyperplane arrangement is not a generic arrangement. Generically, at most $n$ hyperplanes intersect. In this case, $n+1$ hyperplanes intersect when they correspond to some cell in a higher order diagram. A hyperplane arrangement does not determine in general a power diagram.
Drawings that look like a power diagram need not be one, just as not all triangulations are coherent. For examples of the latter phenomenon see for instance section 5.1 in [@MR1311028], where coherent triangulations are called *regular*. As an example of the former, see figure \[fig:34\]. Not all directions will lead to an intersection of the three arrows inside the cell.
Figure \[fig:34\]:Four regions, marked by $P_1$, $P_2$ , $P_3$ and $P_4$. $Q_1=(1,-2)$, $Q_2=(2,2)$ and $Q_3=(-2,0)$. If the halflines from the $Q_i$ are allowed to move in the specified directions we always have a polyhedral subdivision, but only a power diagram when the back ends of the arrows intersect inside the triangle.
\[fig:34\]
The Morse poset
===============
In [@SieVanM] we introduced the Morse poset for Voronoi diagrams. Here we generalize the Morse poset, so that it is defined for power diagrams.
Consider the function $$g=\min_{ 1 \leq i \leq N} g_i(x)$$ This function is a topological Morse function. When does it have a critical point?
The function $g$ does not have critical points outside the convex hull of $\colP1N$.
Let $x$ be a point outside the convex hull of $\colP1N$. Hyperplanes in $T_x{{\mathbb R}}^n$ can be identified with affine hyperplanes in ${{\mathbb R}}^n$ that pass through $x$. Choose a hyperplane $H$ in $x$ that is parallel to some hyperplane that separates $x$ from $\operatorname{CH}(\colP1N)$. It follows that all gradients $\frac{\partial g_i}{\partial x} = x- P_i$ point outward from $x$ at one side of $H$. Thus the function $\min_i g_i$ does not have a critical point at $x$.
Consider the coherent triangulation dual to a power diagram. In the special case of a Delaunay triangulation we have defined a poset of active simplices in the triangulation. They were defined using the critical points of $\min g_i$. We repeat the procedure for power diagrams.
First define the separator of a cell $\alpha\in\operatorname{Del}(\colP1N)$. The separator $\operatorname{Sep}(\alpha)$ of $\alpha$ is the intersection of all the separators of $P_i$ and $P_j$ such that $\{ P_i , P_j \} \subset \alpha$. If $\alpha$ is a vertex then the separator is defined to be just the whole of ${{\mathbb R}}^n$. Also, for a polyhedron $P$ in ${{\mathbb R}}^{n+1}$ the relative interior is the interior of the polyhedron as a subset of the affine span $\operatorname{Aff}(P)$ of $P$. If $\dim\alpha> 0$ then put $$\label{eq:24}
c(\alpha)=\operatorname{Sep}(\alpha)\cap \operatorname{Aff}( \alpha )$$ For a vertex put $c(\{ i \}) = P_i$. We call the simplex in the coherent triangulation *active* if
- $c(\alpha)$ is contained in the relative interior $\operatorname{Interior}( \operatorname{CH}( \alpha ) )$ of $\operatorname{CH}(\alpha)$, and
- $g( c(\alpha) ) = g_i(c(\alpha))$ for some $P_i \in \alpha$
With this definition it might well happen that a vertex is not active. Here is a typical counter intuitive example. Take three points, say $P_1=(-1,0)$, $P_2=(1,1)$ and $P_3=(1,-1)$. Then place a large circle around $P_1$: $r_1=5$. Place two smaller circles around $P_2$ and $P_3$: $r_2=\sqrt\frac34$ and $r_3=\sqrt\frac56$. If we look from below at the graph of $\min g_i$ we see a picture alike the one in figure \[fig:13\].
![Typical counterintuitive example[]{data-label="fig:13"}](typicalcounterintuitive.eps){width="7cm"}
The *Morse poset* of $\{ ( P_i, w_i ) \}_{i=1 , \cdots , N}$ consists of the active subsets of the coherent triangulation. We write $$g_\alpha(x) = \min_{i\in\alpha}g_i(x)\text{ and }
f_\alpha(x) = \max_{i\in\alpha}f_i(x)$$
There is a one to one correspondence between critical points of $\min_i g_i$ and active cells in the coherent triangulation. An active cell of dimension $d$ corresponds to a critical point of index $d$.
The function $g(x)=\min_i g_i(x)$ can only have a critical point at $x_0$ when at least two of the $g_i(x_0)$ are equal to $g(x)$. The index set for which $g_i(x_0)=g(x_0)$ is the face $\alpha$. The function can also only have a critical point when $x_0\in\operatorname{CH}(\alpha)$. We can thus assume $x_0=c(\alpha)$.
The index of the critical point can be calculated with theorem 2.3 in [@MR1481622]. First note that for all $x\in {{\mathbb R}}^n$ the Hessian of $g_i(x)$ is always the unit matrix $\mathbf{I}$.
Consider the subspace $$T(x)=\cap_{i\in\alpha} \ker \frac{\partial g_i}{\partial x}(x_0)
=\cap_{i\in\alpha}\ker(x-P_i)$$ Clearly $T(c(\alpha))$ has the same dimension as $\operatorname{Sep}(\alpha)$. The dimension of $T$ is $n-\dim\alpha$.
The index of the critical point is $n$ minus the dimension of $T(x)$, it is thus the dimension of $\alpha$.
Conversely if $\alpha$ is active then $g_\alpha$ restricted to $\operatorname{CH}(\alpha)$ has a supremum at $c(\alpha)$. $$g(x)\leq g_\alpha(x)\leq g_\alpha(c(\alpha))\quad \forall x\in\operatorname{CH}(\alpha)$$ Also there is a neighborhood $U$ of $c(\alpha)$ where $g(x)=g_\alpha(c(\alpha))$ and so $g$ has a critical point of index $\dim(\alpha)$.
Note that the reasoning is independent of genericity conditions.
Finally we remark that though power diagrams are affinely defined, activity is not retained under affine volume preserving linear transformations. Through such transformations the power diagram and its dual coherent triangulation do not change. The Morse poset though, can change drastically after an affine transformation.
Discrete Morse theory
=====================
We recall the main definitions of the discrete Morse theory developed by Forman. A very good introduction to the subject is [@MR900443]. We show that, though the coherent triangulation is not in general a simplicial complex, it has another nice property: for every two faces $\alpha\subset\gamma$ with $2+\dim(\alpha)=\dim(\gamma)$ all there are two faces $\beta$ of the coherent triangulation in between $\alpha$ and $\gamma$. Hence, discrete Morse functions on the coherent triangulation have the same nice properties as discrete Morse functions on a simplicial complex.
Let $\mathcal{T}$ be a polyhedral subdivision. By orientating all the faces $\mathcal{T}$ becomes a complex.
A function $h\colon\mathcal{T}\rightarrow{{\mathbb R}}$ is called a *discrete Morse function* if for all $\beta\in\mathcal{T}$ $$\label{eq:22}
\# \{ \alpha\in\mathcal{T}\,\mid\, 1 + \dim(\alpha)=\dim(\beta)\,\,\,
\alpha\subset\beta
\,\,\, h(\alpha)\geq h(\beta) \} \leq 1$$ and $$\label{eq:23}
\# \{ \alpha\in\mathcal{T}\,\mid\, \dim(\alpha)=1+ \dim(\beta)\,\,\,
\beta\subset\alpha
\,\,\, h(\alpha)\leq h(\beta) \} \leq 1$$ In case both numbers are zero for some $\beta\in\mathcal{T}$, $\beta$ is called *critical*.
Analogous to the case of a power diagram and its dual coherent triangulation, we can define a Morse poset of a discrete Morse function $h$ as the set of critical faces of $h$.
If the power diagram is in general position $\mathcal{T}$ is a simplicial complex. Discrete Morse function on a simplicial complex have a very nice property, see lemma 2.6 in [@MR1612391]. If $\alpha$ is a face of $\beta$ and $\beta$ is a face of $\gamma$. Label the vertices: $$\alpha=\colP1{k}\quad\beta=\colP1{k+1}\quad\gamma=\colP1{k+2}$$ Because we are dealing with a simplicial complex there is another face between $\alpha$ and $\gamma$: $\delta=\{ P_1, \cdots ,P_k, P_{k+2} \}$: $$\UseTips
\xymatrix @C=1em @R=1em {
& \beta=\colP1{k+1} \ar@{^{(}->}[dr] & \\
\alpha=\colP1k \ar@{^{(}->}[dr] \ar@{^{(}->}[ur] & & \gamma=\colP1{k+2} \\
& \delta = \{ P_1, \cdots ,P_k, P_{k+2} \} \ar@{^{(}->}[ur] &
}$$ Let $f$ be a discrete Morse function on the simplicial complex. Suppose that $f(\alpha) \geq f(\beta) \geq f(\gamma)$, that is both and hold. We also have $f(\alpha) < f(\delta) < f(\gamma)$, or $$f(\gamma) > f(\delta) > f(\alpha) \geq f(\beta) \geq f(\gamma)$$ This is a contradiction, so we see that on simplicial complexes at most one and can hold.
However this argument is not restricted to simplicial complexes. All that is needed is that if $\alpha$ and $\gamma$ are two cells with $\alpha\subset\gamma$ and $2+\dim\alpha=\dim\gamma$ then there are at least two cells in between $\alpha$ and $\gamma$.
\[thm:simplicialproperty\] For a coherent triangulation $\sigma$ as in the above between any two cells $\alpha$ and $\gamma$ with $2+\dim\alpha=\dim\gamma$ and $\alpha\subset\gamma$ there are $\beta_1$ and $\beta_2$, both cells in the coherent triangulation, both having dimension $1+\dim\alpha$ such that $\alpha\subset\beta_i\subset\gamma$.
Consider the cone $\operatorname{Cone}(\gamma,\alpha)$ of $\gamma$ over $\alpha$: $$\operatorname{Cone}(\gamma,\alpha)\overset{\textrm{def}}{=}\{
\beta \,\mid\, \alpha\subseteq \beta\subseteq\gamma\}$$ Intersect that cone with the $n-k+1$ dimensional plane through zero and orthogonal to the $k-1$-dimensional plane affinely spanned by $\alpha$. The intersection is a cone in the two dimensional plane. It has two extremal vectors $\xi_1$ and $\xi_2$. These correspond to the faces $\beta_1$ and $\beta_2$ we were looking for.
Hence even if the coherent triangulation we have is not a triangulation then we still have that not both and can hold.
If we mod out $\operatorname{Cone}(\gamma,\alpha)$ by the directions $\operatorname{Aff}(\alpha)$ along $\alpha$ we get a vertex of a polyhedron and the faces between $\alpha$ and $\gamma$ are the edges of a connected fan. So if we have that there can be any number of $\beta_{i1}$ and $\beta_{i2}$ between $\alpha$ and $\gamma$. $$\alpha\rightarrow
\left\{
\begin{matrix}
\beta_{11} \\
\vdots \\
\beta_{l1} \\
\end{matrix}
\right.
\begin{matrix}
\subset \\ \vdots \\ \subset
\end{matrix}
\left.
\begin{matrix}
\beta_{11} \\
\vdots \\
\beta_{l1} \\
\end{matrix}
\right\}
\rightarrow
\gamma$$ On any coherent triangulation there exists a discrete Morse function: $h\colon\alpha\mapsto\dim\alpha$. So let us have a discrete Morse function $h$ on the coherent triangulation. The gradient of $h$ consists of arrows $\alpha\rightarrow\beta$, drawn when $\alpha\subset\beta$ and $h(\alpha)\geq h(\beta)$. In general such a collection of arrows is called a *discrete vectorfield*. A $V$-path is a sequence of arrows such that the simplex pointed to contains a second simplex that points to another one, of higher dimension. A *closed $V$-path* is a circular $V$-path as in equation . Theorem 3.5 in [@MR900443] states that a discrete vectorfield is the gradient of a discrete Morse function if and only if it contains no closed $V$-paths. $$\label{eq:10}
\alpha_1 \rightarrow \beta_1 \supset
\alpha_2 \rightarrow \beta_2 \supset \cdots \supset
\alpha_k \rightarrow \beta_k \supset \alpha_1$$ The dual to the coherent triangulation, the power diagram, is also a polyhedral complex. Given a discrete Morse function $\mathfrak{g}$ on the coherent triangulation, clearly $-\mathfrak{g}$ is a discrete Morse function on the power diagram.
![A cone of simplices filled up with a discrete vectorfield[]{data-label="fig:2"}](./conenew.eps){height="3cm"}
From Morse poset to a discrete vectorfield
==========================================
In this section we will prove the main new theorem of this article.
\[sec:from-morse-poset\] On $\operatorname{Del}(\colP1N)$ there exists a discrete Morse function $h$ such that the Morse poset of $h$ equals the Morse poset of $g(x)=\min_{1\leq i \leq N}g_i(x)$.
In fact we will not construct $h$ directly. Instead we will construct a discrete vectorfield without closed $V$-paths. In the reasoning below we disregard the disappearing faces.
A face $\alpha$ can be non-active for one of two reasons
- The separator $\operatorname{Sep}(\alpha)$ can lie outside $\operatorname{Interior}(\operatorname{CH}(\alpha))$, or if that is not the case:
- Some point $P_i\in\colP1N\setminus\alpha$ can lie closer to $c(\alpha)$ then the points in $\alpha$, i.e.$$\label{eq:5}
g_i(c(\alpha))\leq g_j(c(\alpha))\quad \forall j\in \alpha$$
Suppose that $\alpha$ is not active for the “Down” reason. Then in the case of Voronoi diagrams $\alpha$ cannot be a vertex or an edge. In general with power diagrams $\alpha$ cannot be a vertex.
Suppose that $\alpha$ is not active for the “Up” reason and that $\dim\operatorname{Del}(\alpha)=n$. Then we have $$\{c(\alpha)\}= \cap_{j\in\alpha}\operatorname{Pow}(\{P_j\})\textrm{:}$$ we cannot have . We conclude that no $\alpha$, not active for the “Up”-reason has $\dim\operatorname{Del}(\alpha)=n$.
We start by showing that for every face, not active for the “Down” reason, there is a unique lower dimensional simplex not active for the “Up” reason. Then we construct for each face not active for the “Up” reason a part of the discrete vectorfield.
\[thm:downreason\] Let $\beta\in\operatorname{Del}(\colP1N)$ be a face not active for the “Down”-reason. Then the closest point to $\operatorname{CH}(\beta)$ from $c(\beta)$ is $c(\alpha)$ for some proper face $\alpha$ of $\beta$. At $c(\alpha)$ we have $$\label{eq:6}
g_j(c(\alpha)) \leq g_\alpha(c(\alpha)) \quad \forall j\in\beta\setminus\alpha$$ Hence $\alpha$ is not active for the “Up”-reason. Conversely if holds for $\beta\supset\alpha$ then $\beta$ is not active for the “Down”-reason.
We start with “$\Rightarrow$”.
A special case we first have to handle is when $c(\beta)$ lies on the relative boundary of $\operatorname{CH}(\beta)$. This means that $c(\beta)\in\operatorname{CH}(\alpha)$ for some proper face $\alpha$ of $\beta$. Because $\operatorname{Sep}(\beta)\subset
\operatorname{Sep}(\alpha)$ it follows that $c(\beta)=c(\alpha)$. Equation follows automatically.
Now we can safely assume that the distance from $c(\beta)$ to $\operatorname{CH}(\beta)$ is $>0$. Denote $y$ the closest point on $\operatorname{CH}(\beta)$ from $c(\beta)$. If $\alpha$ is a vertex we are done, so suppose that $\alpha$ is not a vertex. The line from $y$ to $c(\beta)$ is orthogonal to $\operatorname{Aff}(\alpha)$ and hence parallel to $\operatorname{Sep}(\alpha)$. Clearly $c(\beta)$ lies in $\operatorname{Sep}(\alpha)$. So $y$ also lies in $\operatorname{Sep}(\alpha)$. But $y$ also lies in $\operatorname{Aff}(\alpha)$. So, by the definition of $c(\alpha)$ in equation we have that $y=c(\alpha)$.
Because $\operatorname{CH}(\beta)$ is a convex set the hyperplane with normal $c(\beta)-c(\alpha)$ passing through $c(\alpha)$ separates $\operatorname{Interior}(CH(\beta))$ from $c(\beta)$. Thus the angle $P_k$ to $c(\alpha)$ to $c(\beta)$ must be obtuse. Consequently: $$\label{eq:7}
\lVert c(\beta) - P_k \rVert \geq \lVert c(\beta) - c(\alpha) \rVert
+ \lVert c(\alpha) - P_k \rVert$$ if $P_k$ is a vertex in $\beta\supset\alpha$. By Pythagoras, equality holds if $k\in\alpha$: $$\label{eq:9}
\lVert c(\beta) - P_k \rVert = \lVert c(\beta) - c(\alpha) \rVert
+ \lVert c(\alpha) - P_k \rVert$$ Then equation becomes, taking the factor $\frac12$ we used in the definition of $g$ in equation into account: $$\label{eq:8}
g_k(c(\beta)) \geq \frac12 \lVert c(\beta) - c(\alpha) \rVert + g_k(c(\alpha))
\quad k \in \beta\setminus\alpha$$ Again, equality holds when $k\in\alpha$. Putting this together we get , and we see that $\alpha$ is not active for the “Up”-reason.
Next do the “$\Leftarrow$” part. We assume and $\alpha\subset\beta$. From we get for $j\in\beta\setminus\alpha$: $$\label{eq:1}
g_j(c(\alpha)) +\frac12 \lVert c(\beta) - c(\alpha) \rVert \leq
g_\alpha(c(\alpha))+\frac12 \lVert c(\beta) - c(\alpha) \rVert = g_\alpha(c(\beta))=
g_j(c(\beta))$$ The special case to handle first is where equality holds in . Then $c(\beta)=c(\alpha)$ and so $c(\beta)$ is not an element of the relative interior $\operatorname{Interior}(\operatorname{CH}(\beta))$. So $\beta$ is not active for the “Down”-reason.
Now we can safely assume strict inequality in . The simplest case is a $\beta$ for which $\beta = \alpha \cup \{ j \}$. The three points $P_j$, $c(\beta)$ and $c(\alpha)$ all lie in $\operatorname{Aff}(\beta)$. From we get with strict inequality. The line segment from $c(\alpha)$ to $c(\beta)$ is thus an outward normal to $\operatorname{CH}(\beta)$ in $\operatorname{Aff}(\beta)$. Consequently $\beta$ is not active for the “Down”-reason. The general case is similar.
Let again $\alpha$ be not active for the “Up”-reason. Denote $K$ the set of indices for which holds. Denote $\operatorname{Up}(\alpha)$ the set of simplices: $$\label{eq:4}
\operatorname{Up}(\alpha)=\{ \beta\in\operatorname{Del}(\colP1N) \,\mid\,
\alpha\subset\beta\subset\alpha\cup K \}$$ Lemma \[thm:downreason\] characterizes the elements of $\operatorname{Up}(\alpha)$ as those $\beta\supset\alpha$ for which $c(\alpha)$ is the closest point on $\operatorname{CH}(\beta)$ from $c(\beta)$. The complement of the Morse poset is divided into different subsets $\operatorname{Up}(\alpha)$, one for each face not active for the “Up”-reason.
In $\operatorname{Up}(\alpha)$ identify those faces $\delta$ such that there are no $\gamma$ in $\operatorname{Up}(\alpha)$ with $\delta$ a proper face of $\gamma$. Then we see that $\operatorname{Up}(\alpha)$ contains different cones $\operatorname{Cone}(\delta,\alpha)$. These cones can be filled up by a discrete vectorfields, as is shown in figure \[fig:2\]. It is possible to take a number of simplicial cones such that no discrete vectorfield exists that fills up these cones completely. For instance, take two triangles: $\P12$, $\PP123$ and $\PP124$. Obviously there is no way to fill these cones up with a discrete vectorfield. Therefore, we need to prove that $\operatorname{Up}(\alpha)$ can be filled up with a discrete vectorfield.
We will establish another criterion for which a $\beta\in\operatorname{Del}(\colP1N)$ with $\alpha\subset\beta$ is an element of $\operatorname{Up}(\alpha)$. Take the point $c(\alpha)$ outside the polytope $\operatorname{Pow}(\alpha)$ in $\operatorname{Sep}(\alpha)$ and consider the faces of $\operatorname{Pow}(\alpha)$ for which the outward pointing normal makes an angle smaller than 90 degrees with $x-c(\alpha)$. Then we get a number of faces $\beta_1, \cdots , \beta_l$ in the Delaunay triangulation that we see as faces $\operatorname{Pow}(\beta_1), \cdots ,
\operatorname{Pow}(\beta_l)$ on the relative boundary $\partial(\operatorname{Pow}(\alpha))$ of $\operatorname{Pow}(\alpha)$. This is illustrated in figure \[fig:6\].
![View from $c(\alpha)$ to $\operatorname{Pow}(\alpha)$[]{data-label="fig:6"}](ladiedie.eps){height="2cm"}
A precise formulation is contained in the following lemma.
\[thm:up-polyhedron\] Let $\beta\supset\alpha$ be an element of $\operatorname{Del}(\colP1N)$. Let $x$ be a point in the interior of $\operatorname{Pow}(\beta)$. If, either $x=c(\alpha)$, or
1. the line through $x$ and $c(\alpha)$ intersects $\operatorname{Pow}(\alpha)$ in a segment, and
2. the segment $[x,c(\alpha)]$ contains no point of the interior of $\operatorname{Pow}(\alpha)$
then $\beta\in\operatorname{Up}(\alpha)$. Conversely, if 1 and 2 hold for some $\beta\supset\alpha$, with $\alpha$ not active for the “Up”-reason, then $\beta\in\operatorname{Up}(\alpha)$.
We first need to treat the case $x=c(\alpha)$. In that case $c(\alpha)$ lies on the relative boundary of $\operatorname{Pow}(\alpha)$. So $x\in \operatorname{Pow}(\beta)$, $\beta=\alpha\cup I$. And $f_j(c(\alpha)) = f_\alpha(c(\alpha))$, for all $j\in I$. Obviously $\operatorname{Up}(\alpha)$ consists exactly of all those $\gamma\in\operatorname{Del}(\colP1N)$ with $$\alpha\subseteq \gamma \subseteq \beta=\alpha\cup I$$
Now we can assume $x\not=c(\alpha)$. Again $x$ lies on the relative boundary of $\operatorname{Pow}(\beta)$ and $\beta=\alpha\cup I$ for some maximal set $I$. We put $x_t=tx+(1-t)c(\alpha)$. Hence $x_1=x$ and $x_0=c(\alpha)$. It also follows for $j\in I$ $$f_j(x_t)= t f_j(x)+(1-t)f_j(c(\alpha))=
t f_\alpha(x) +(1-t)f_j(c(\alpha))$$ and thus $$f_j(x_t)-f_\alpha(x_t)=(1-t)(f_j(c(\alpha))-f_\alpha(c(\alpha)))$$ Now we have $$f_j(c(\alpha))\geq f_\alpha(c(\alpha))\,\Leftrightarrow\,
\exists t>1\,\colon\, f_j(x_t)\leq f_\alpha(x_t)$$ So that $$\exists t > 1\,\, x_t\in\operatorname{Pow}(\alpha)\,
\Rightarrow\,f_j(c(\alpha))\geq f_\alpha(c(\alpha))$$ And this is exactly what we needed to prove.
Now suppose that $\beta\supset\alpha$, and $\alpha$ not active for the “Up”-reason, but that 1 and 2 do not hold.
The only point of $\operatorname{Pow}(\alpha)$ on the line through $x$ and $c(\alpha)$ is $x$. It follows that $\beta\setminus\alpha$ contains more than one index, i.e. $\beta=\alpha\cup\{j_1 , \cdots , j_r\}$ with $r\geq 2$. In a sufficiently small neighborhood of $x$ the halfspaces $H_i=\{f_{j_i}\leq f_\alpha\}$ define $\operatorname{Pow}(\alpha)$ as a polyhedron in $\operatorname{Sep}(\alpha)$. Because $\{x_t\,\mid\,t\in{{\mathbb R}}\}$ touches $\operatorname{Pow}(\alpha)$ there exist an index $k$, say $k=j_1$, and a $t<1$ such that $f_j(x_t)-f_\alpha(x_t)$ and hence $f_k(c(\alpha))<f_\alpha(c(\alpha))$. And so $\beta\supset\alpha$ is not an element of $\operatorname{Up}(\alpha)$.
From lemma \[thm:up-polyhedron\] we see that we can fill up $\operatorname{Up}(\alpha)$ with arrows of a discrete vectorfield. Indeed, with its dual on $\operatorname{Pow}(\alpha)$ we can obviously do this because it is contractible, so we can also do it with $\operatorname{Up}(\alpha)$.
We conclude that the complement of the Morse poset can be divided into separate parts $\operatorname{Up}(\alpha)$, for all $\alpha$ not active for the “Up”-reason. Each of these parts cones represents what Matveev calls a polyhedral collapse in [@MR1997069]. To show that we indeed get a discrete vectorfield we need to prove one more thing: there should be no closed $V$-path.
Suppose that we have a closed $V$-path. Inside one $\operatorname{Up}(\alpha)$ there is no such closed $V$-path, so at some simplex the $V$-path should jump from one $\operatorname{Up}(\gamma_1)$ to another $\operatorname{Up}(\gamma_2)$, as in figure \[fig:jump\].
![A $V$-path, with a jump from $\operatorname{Up}(\gamma_1)$ to $\operatorname{Up}(\gamma_2)$.[]{data-label="fig:jump"}](vpath.eps){height="2cm"}
\[sec:from-morse-poset-4\] Suppose that we have a $V$-path that jumps from $\operatorname{Up}(\gamma_1)$ to $\operatorname{Up}(\gamma_2)$ as in equation . $$\label{eq:3}
\begin{matrix}
\alpha_i & \rightarrow & \beta_i & & & \subset & \operatorname{Cone}(\gamma_1\cup I_1,\gamma_1) \\
& & \cup & & & & \\
& &\alpha_{i+1} & \rightarrow & \beta_{i+1} & \subset & \operatorname{Cone}(\gamma_2\cup I_2,\gamma_2)
\end{matrix}$$ Then $g_{\gamma_1}(c(\gamma_1))>g_{\gamma_2}(c(\gamma_2))$.
Because $\gamma_1$ is a proper subset of $\gamma_1\cup\gamma_2$ we see that: $c(\gamma_1\cup\gamma_2)\notin\operatorname{Interior}(\operatorname{CH}(\gamma_1\cup\gamma_2))$. The closest point to $\operatorname{CH}(\gamma_1\cup\gamma_2)$ from $c(\gamma_1\cup\gamma_2)$ is $c(\gamma_1)$, as we have shown in lemma \[thm:downreason\]. In particular $$\label{eq:11}
\lVert c(\gamma_1 \cup \gamma_2) - c(\gamma_1) \rVert <
\lVert c(\gamma_1 \cup \gamma_2) - c(\gamma_2) \rVert$$ Here, the inequality is strict. The closest point is unique, because $\operatorname{CH}(\gamma_1\cup\gamma_2)$ is convex.
For the reasons explained in the proof of lemma \[thm:downreason\] we have $$\label{eq:12}
g_{\gamma_1\cup\gamma_2}(c(\gamma_1\cup\gamma_2))=
g_{\gamma_1}(c(\gamma_1\cup\gamma_2))=
\frac12 \lVert c(\gamma_1\cup\gamma_2) - c(\gamma_1) \rVert + g_{\gamma_1}(c(\gamma_1))$$ The same identity holds with $\gamma_1$ and $\gamma_2$ exchanged, thus: $$\label{eq:13}
\frac12 \lVert c(\gamma_1\cup\gamma_2) - c(\gamma_1) \rVert +
g_{\gamma_1}(c(\gamma_1)) =
\frac12 \lVert c(\gamma_1\cup\gamma_2) - c(\gamma_2) \rVert +
g_{\gamma_2}(c(\gamma_2))$$ Putting and together we get the desired result.
Lemma \[sec:from-morse-poset-4\] says that to each $\operatorname{Up}(\alpha)$ appearing in the $V$-path we can associate a number, and passing from one $\operatorname{Up}(\alpha)$ to another that number strictly decreases. Hence we can not have a closed $V$-path. The proof of theorem \[sec:from-morse-poset\] is complete.
We are not entirely satisfied with the proof of our main theorem. What is lost in the proof is a direct relation between $g$ and the discrete Morse function that gives the discrete vectorfield we construct. We tried using $\alpha\rightarrow\sup_{x\in\operatorname{CH}(\alpha)}g(x)$ as a function on the coherent triangulation but this approach did not give the desired results.
We were neither entirely satisfied with discrete Morse theory, because it seems to be ill equipped for dealing with polyhedral collapses: our vectorfield is not unique. The non-uniqueness only lies in the choice of the arrows possible in $\operatorname{Up}(\alpha)$. One could also try to develop a generalization of discrete Morse theory in order to include polyhedral collapses.
Examples
========
Let us return to the counterintuitive example of figure \[fig:13\]. There the Morse poset is as in figure \[fig:14\]. In the picture we see three cones, consisting each of two faces in the simplest triangulation one can think of.
![Morse poset and discrete vectorfield of figure \[fig:13\][]{data-label="fig:14"}](newcounterposet.eps){height="3cm"}
Another example is the one called $4300I$ in our previous article [@SieVanM]. Let the active subsets be $\{\P12,\P23,\P34 \}$. Then if $\alpha=\PPP1234$ the face $\P14$ is not active for the “Up”-reason. In that case both $\PP134$ and $\PP124$ and $\PPP1234$ are not active for the “Down” reason. There is one cone consisting of the polyhedral collapse: $$\P14 \rightarrow \PP124 \quad \PP134 \rightarrow \PPP1234$$ The decomposition of that cone into arrows is not unique. We might just as well write $$\P14 \rightarrow \PP134 \quad \PP124 \rightarrow \PPP1234$$ Another application is the classification of all Morse posets of Voronoi diagrams for $4$ points in the plane in [@VoronoiDist]. Unfortunately there were two cases missing, which are now included in figure \[fig:8\].
![$3$ and $4$ points in ${{\mathbb R}}^2$: all Morse posets from Voronoi diagrams with the accompanying discrete vectorfields[]{data-label="fig:8"}](fourtwo.eps){width="\textwidth"}
The medial axis and Maslov dequantization
=========================================
It is well known that the medial axis of a compact embedded submanifold $M$ in ${{\mathbb R}}^n$ can be calculated as an approximation of a Voronoi diagram, see for instance [@Bern280947]. We use tropical geometry to give a new interpretation of that fact. Recall that the *corner locus* of a convex function $f$ is the set of points where $f$ is not differentiable.
We have seen in the above that the Voronoi diagram can also be calculated using the affine functions $f_i$, instead of the distance function. The corner locus of the function $$f(x)=\max_i \langle x , P_i \rangle -\frac12\lVert P_i \rVert$$ is the Voronoi diagram. It is natural to try the same procedure with the embedding $\gamma\colon M\rightarrow {{\mathbb R}}^n$. We will show next that the medial axis is a corner locus as well.
Let $M$ be a compact smooth manifold, embedded by $\gamma\colon M\rightarrow{{\mathbb R}}^n$ in ${{\mathbb R}}^n$. Write for $s\in M$: $$f_s(x)=\langle x , \gamma(s) \rangle - \frac12\lVert \gamma(s)\rVert$$ The function $f$ defined by $$\label{eq:17}
f(x) \overset{\mathrm{def}}{=} \sup_{s\in M}f_s(x) =
\lim_{h\rightarrow \infty}\frac{\log\left(\int_M h^{f_s(x)} ds \right)}{\log(h)}$$ is a convex function, and satisfies the stated equality. Its corner locus is the medial axis of $M$.
The function $f(x)$ is convex, because it is the supremum of a number of convex functions. Because $M$ is compact the supremum is attained for one or more points $s=s(x)$ in $M$. We see that $x\in{{\mathbb R}}^n$ and $s\in M$ , for which $f(x)=f_s(x)$, are related by $$\label{eq:16}
\langle x - \gamma(s), \dot{\gamma}(s) \rangle = 0$$ I.e. $x$ lies on the normal of one or more $s\in M$. Of those that satisfy we choose one that has the highest value of $f_{s_i}(x)$, and thus the lowest value of $$\frac12\lVert x \rVert - f_{s_i}(x) = \frac12\lVert x - \gamma(s_i)\rVert$$ The $s$ we get from $\sup_{s\in M}f_s$ corresponds to the point closest to $x$ on the submanifold $M$. And the points where $f(x)$ is not differentiable form exactly the medial axis of $M$. It remains to prove the formula for $\sup_{s\in M}f_s$ in .
For this we use the so-called Maslov dequantization. Maslov dequantization is a fancy name for the identity: $$\label{eq:14}
\lim_{h\rightarrow\infty} \log_h(h^a + h^b) = \max(a,b)$$ that holds for any two real numbers $a$ and $b$.
For practical purposes this procedure works well. The integrals can in most cases not be calculated explicitly, but if one uses numerical integration one can plot $f$, as is illustrated in figure \[fig:33\].
The Maslov dequantization as in equation is strongly related to a construction in toric geometry, described in [@MR1234037], section 4.2. Namely, let again $$f_i(x) = \langle x , P_i \rangle +c_i \qquad i=1 , \cdots , N$$ be affine functions defining a power diagram. Define the following function $$H\colon{{\mathbb R}}^n\rightarrow{{\mathbb R}}^n\quad
H(x) = \frac{\sum_{i=1}^N P_i e^{f_i(x)}}{\sum_{i=1}^N e^{f_i(x)}}$$ It is proved in [@MR1234037] that $H$ is a real analytic diffeomorphism from ${{\mathbb R}}^n$ to the interior of $\operatorname{CH}(\colP1N)$, when that interior is not empty. One recognizes $H$ as a gradient: $$H(x)=\frac{\partial}{\partial x}\left(\log\left( \sum_{i=1}^N e^{f_i(x)}\right)\right)$$ Now consider the gradient: $$H_h(x)=\frac{\partial}{\partial x}\left(\log_h\left( \sum_{i=1}^N h^{f_i(x)}\right)\right)
=\frac{\sum_{i=1}^N P_i h^{f_i(x)}}{\sum_{i=1}^N h^{f_i(x)}}$$
Suppose that the interior of $\operatorname{CH}(\colP1N)$ is not empty. The map $H_h$ is a real analytic diffeomorphism from ${{\mathbb R}}^n$ to the interior of $\operatorname{CH}(\colP1N)$.
The $f_i$ are of the form $\langle x, P_i \rangle+c_i$. Put $\tilde{f}_i(u)=\langle u, P_i \rangle +\log(h)c_i$. We already know that $$u \longrightarrow \tilde{H}(u)=\frac{\sum_{i=1}^N P_i e^{\tilde{f}_i(u)}}{
\sum_{i=1}^N e^{\tilde{f}_i(u)}}$$ is a real analytic diffeomorphism from ${{\mathbb R}}^n$ to the interior of $\operatorname{CH}(\colP1N)$. Because we have $H_h(x)=\tilde{H}(x \log(h))$ the proof is complete.
Taking infinitely many points and a Riemann sum we see that the gradient of the function $\log_h\left(\int_M h^{f_s(x)} ds \right)$ is a diffeomorphism from ${{\mathbb R}}^n$ to the region in ${{\mathbb R}}^n$ bounded by the compact embedded manifold $M\subset{{\mathbb R}}^n$. Now in the limit $h\rightarrow\infty$, the image of $M$ is the medial axis. This is another way of saying that the dual of the medial axis in the sense of the theorem of Passare and Rullg[å]{}rd is the manifold $M$ itself. \[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
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abstract: |
In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$ as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in
C(M,\mathbb{R}^1)$ as initial condition converges to a backward weak KAM solution faster than the Lax-Oleinik semigroup in the time-independent case.
address:
- '$^{1}$ School of Mathematical Sciences and Key Lab of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, China'
- '$^{2}$ College of Mathematics, Jilin University, Changchun 130012, China'
author:
- 'Kaizhi Wang$^{1,\,2}$ and Jun Yan$^{1}$'
date: December 2010
title: 'A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems'
---
Introduction
============
Let $M$ be a compact and connected smooth manifold. Denote by $TM$ its tangent bundle and $T^*M$ the cotangent one. Consider a $C^\infty$ Lagrangian $L: TM\times\mathbb{R}^1\to\mathbb{R}^1$, $(x,v,t)\mapsto L(x,v,t)$. We suppose that $L$ satisfies the following conditions introduced by Mather [@Mat91]:
0.2cm
- **Periodicity**. $L$ is 1-periodic in the $\mathbb{R}^1$ factor, i.e., $L(x,v,t)=L(x,v,t+1)$ for all $(x,v,t)\in TM\times\mathbb{R}^1$.
- **Positive Definiteness**. For each $x\in M$ and each $t\in\mathbb{R}^1$, the restriction of $L$ to $T_xM\times
t$ is strictly convex in the sense that its Hessian second derivative is everywhere positive definite.
- **Superlinear Growth**. $\lim_{\|v\|_x\to+\infty}\frac{L(x,v,t)}{\|v\|_x}=+\infty$ uniformly on $x\in M$, $t\in\mathbb{R}^1$, where $\|\cdot\|_x$ denotes the norm induced by a Riemannian metric on $T_xM$. By the compactness of $M$, this condition is independent of the choice of the Riemannian metric.
- **Completeness of the Euler-Lagrange Flow**. The maximal solutions of the Euler-Lagrange equation, which in local coordinates is: $$\frac{d}{dt}\frac{\partial L}{\partial
v}(x,\dot{x},t)=\frac{\partial L}{\partial
x}(x,\dot{x},t),$$ are defined on all of $\mathbb{R}^1$.
0.2cm
The Euler-Lagrange equation is a second order periodic differential equation on $M$ and generates a flow of diffeomorphisms $\phi^L_t:TM\times\mathbb{S}^1\to
TM\times\mathbb{S}^1$, $t\in\mathbb{R}^1$, where $\mathbb{S}^1$ denotes the circle $\mathbb{R}^1/\mathbb{Z}$, defined by
$$\phi^L_t(x_0,v_0,t_0)=(x(t+t_0),\dot{x}(t+t_0),(t+t_0)\
\mathrm{mod}\ 1),$$ where $x:\mathbb{R}^1\to M$ is the maximal solution of the Euler-Lagrange equation with initial conditions $x(t_0)=x_0$, $\dot{x}(t_0)=v_0$. The completeness and periodicity conditions grant that this correctly defines a flow on $TM\times\mathbb{S}^1$.
We can associate with $L$ a Hamiltonian, as a function on $T^*M\times\mathbb{R}^1$: $H(x,p,t)=\sup_{v\in T_xM}\{\langle
p,v\rangle_x-L(x,v,t)\}$, where $\langle \cdot,\cdot\rangle_x$ represents the canonical pairing between the tangent and cotangent space. The corresponding Hamilton-Jacobi equation is
$$\begin{aligned}
\label{1-1}
u_t+H(x,u_x,t)=c(L),\end{aligned}$$
where $c(L)$ is the Ma$\mathrm{\tilde{n}}\mathrm{\acute{e}}$ critical value [@Man97] of the Lagrangian $L$. In terms of Mather’s $\alpha$ function $c(L)=\alpha(0)$.
In this paper we also consider time-independent Lagrangians on $M$. Let $L_a:TM\to\mathbb{R}^1$, $(x,v)\mapsto L_a(x,v)$ be a $C^2$ Lagrangian satisfying the following two conditions:
0.2cm
- **Positive Definiteness**. For each $(x,v)\in TM$, the Hessian second derivative $\frac{\partial^2L_a}{\partial v^2}(x,v)$ is positive definite.
- **Superlinear Growth**. $\lim_{\|v\|_x\to+\infty}\frac{L_a(x,v)}{\|v\|_x}=+\infty$ uniformly on $x\in
M$.
0.2cm
Since $M$ is compact, the Euler-Lagrange flow $\phi^{L_a}_t$ is complete under the assumptions (H2’) and (H3’).
For $x\in M$, $p\in T_x^*M$, the conjugated Hamiltonian $H_a$ of $L_a$ is defined by: $H_a(x,p)=\sup_{v\in T_xM}\{\langle
p,v\rangle_x-L(x,v)\}$. The corresponding Hamilton-Jacobi equation is
$$\begin{aligned}
\label{1-2}
H_a(x,u_x)=c(L_a).\end{aligned}$$
The Lax-Oleinik semigroup (hereinafter referred to as L-O semigroup) [@Hop; @Lax; @Ole] is well known in several domains, such as PDE, Optimization and Control Theory, Calculus of Variations and Dynamical Systems. In particular, it plays an essential role in the weak KAM theory (see [@Fat1; @Fat97b; @Fat98a; @Fat4] or [@Fat-b]).
Let us first recall the definitions of the L-O semigroups associated with $L_a$ (time-independent case) and $L$ (time-periodic case), respectively. For each $u\in
C(M,\mathbb{R}^1)$ and each $t\geq0$, let
$$\begin{aligned}
\label{1-3}
T^a_tu(x)=\inf_\gamma\Big\{u(\gamma(0))+\int_0^tL_a(\gamma(s),\dot{\gamma}(s))ds\Big\}\end{aligned}$$
for all $x\in M$, and
$$\begin{aligned}
\label{1-4}
T_tu(x)=\inf_\gamma\Big\{u(\gamma(0))+\int_0^tL(\gamma(s),\dot{\gamma}(s),s)ds\Big\}\end{aligned}$$
for all $x\in M$, where the infimums are taken among the continuous and piecewise $C^1$ paths $\gamma:[0,t]\to M$ with $\gamma(t)=x$. In view of (\[1-3\]) and (\[1-4\]), for each $t\geq 0$, $T_t^a$ and $T_t$ are operators from $C(M,\mathbb{R}^1)$ to itself. It is not difficult to check that $\{T^a_t\}_{t\geq 0}$ and $\{T_n\}_{n\in\mathbb{N}}$ are one-parameter semigroups of operators, which means $T^a_0=I$ (unit operator), $T^a_{t+s}=T^a_t\circ T^a_s$, $\forall t,\ s\geq 0$, and $T_0=I$, $T_{n+m}=T_n\circ T_m$, $\forall n,\ m\in\mathbb{N}$, where $\mathbb{N}=\{0,1,2,\cdots\}$. $\{T^a_t\}_{t\geq 0}$ and $\{T_n\}_{n\in\mathbb{N}}$ are called the L-O semigroup associated with $L_a$ and $L$, respectively.
The L-O semigroup is used to obtain backward weak KAM solutions (viscosity solutions) first by Lions, Papanicolaou and Varadhan [@LPV] on the $n$-torus $\mathbb{T}^n$ and later by Fathi [@Fat1] for arbitrary compact manifolds. More precisely, for the time-independent case, Fathi [@Fat1] proves that there exists a unique $c_0\in \mathbb{R}^1$ ($c_0=c(L_a)$), such that the semigroup $\hat{T}^a_t:u\to T^a_tu+c_0t$, $t\geq 0$ has a fixed point $u^*\in C(M,\mathbb{R}^1)$ and that any fixed point is a backward weak KAM solution of (\[1-2\]). In the particular case $M=\mathbb{T}^n$, the backward weak KAM solution obtained by Fathi is just the viscosity solution obtained earlier by Lions, Papanicolaou and Varadhan. Moreover, Fathi points out that the above results for the time-independent case are still correct for the time-periodic dependent case [@Fat-b]. Furthermore, for the time-independent case, he shows in [@Fat4] that for every $u\in C(M,\mathbb{R}^1)$, the uniform limit $\lim_{t\to+\infty}\hat{T}^a_tu=\bar{u}$ exists and is a fixed point of $\{\hat{T}^a_t\}_{t\geq 0}$, i.e., $\bar{u}$ is a backward weak KAM solution of (\[1-2\]). In the same paper Fathi raises the question as to whether the analogous result holds in the time-periodic case. This would be the convergence of $T_nu+nc(L)$, $\forall u\in C(M,\mathbb{R}^1)$, as $n\to+\infty$, $n\in\mathbb{N}$. In view of the relation between $T_n$ and the Peierls barrier $h$ (see [@Mat93] or [@Fat5; @Ber; @Con]), if the liminf in the definition of the Peierls barrier is not a limit, then the L-O semigroup in the time-periodic case does not converge. Fathi and Mather [@Fat5] construct examples where the liminf in the definition of the Peierls barrier is not a limit, thus answering the above question negatively.
The main aim of the present paper is to introduce a new kind of Lax-Oleinik type operator with parameters (hereinafter referred to as new L-O operator) associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. The new L-O operator associated with the time-independent Lagrangian is a special case of the one associated with the time-periodic Lagrangian. We show that
- for the time-periodic Lagrangian $L$, the new family of L-O operators with an arbitrary continuous function on $M$ as initial condition converges to a backward weak KAM solution of (\[1-1\]).
- for the time-independent Lagrangian $L_a$, the new family of L-O operators is a one-parameter semigroup of operators, and the new L-O semigroup with an arbitrary continuous function on $M$ as initial condition converges to a backward weak KAM solution of (\[1-2\]) faster than the L-O semigroup.
Without loss of generality, we will from now on always assume $c(L_a)=c(L)=0$. We view the unit circle $\mathbb{S}^1$ as the fundmental domain in $\mathbb{R}^1: [0,1]$ with two endpoints identified.
We are now in a position to introduce the new L-O operators mentioned above associated with $L$ and $L_a$, respectively.
Time-periodic case
------------------
For each $n\in\mathbb{N}$ and each $u\in C(M,\mathbb{R}^1)$, let
$$\tilde{T}_nu(x)=\inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}\inf_{\gamma}\Big\{u(\gamma(0))+\int_{0}^k
L(\gamma(s),\dot{\gamma}(s),s)ds\Big\}$$ for all $x\in M$, where the second infimum is taken among the continuous and piecewise $C^1$ paths $\gamma:[0,k]\rightarrow M$ with $\gamma(k)=x$. One can easily check that for each $n\in\mathbb{N}$, $\tilde{T}_n$ is an operator from $C(M,\mathbb{R}^1)$ to itself, and that $\{\tilde{T}_n\}_{n\in\mathbb{N}}$ is a semigroup of operators.
\[def1\] For each $\tau\in[0,1]$ and each $n\in \mathbb{N}$, let $\tilde{T}_n^\tau=T_\tau\circ\tilde{T}_n$. Then for each $u\in
C(M,\mathbb{R}^1)$,
$$\tilde{T}_n^\tau u(x)=(T_\tau\circ\tilde{T}_nu)(x)=
\inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}\inf_{\gamma}\Big\{u(\gamma(0))+\int_{0}^{\tau+k}
L(\gamma(s),\dot{\gamma}(s),s)ds\Big\}$$ for all $x\in M$, where the second infimum is taken among the continuous and piecewise $C^1$ paths $\gamma:[0,\tau+k]\rightarrow
M$ with $\gamma(\tau+k)=x$.
For each $\tau\in[0,1]$ and each $n\in \mathbb{N}$, since $\tilde{T}_n^\tau=T_\tau\circ\tilde{T}_n$ and $T_\tau$, $\tilde{T}_n$ are both operators from $C(M,\mathbb{R}^1)$ to itself, then $\tilde{T}_n^\tau$ is an operator from $C(M,\mathbb{R}^1)$ to itself too. We also provide an alternative direct proof of the continuity of $\tilde{T}_n^\tau u$ for each $u\in C(M,\mathbb{R}^1)$ in Section 3. We call $\tilde{T}_n^\tau$ [*the new L-O operator associated with $L$*]{}. Note that for $\tau\in[0,1]\setminus\{0\}$, $\{\tilde{T}_n^\tau\}_{n\in\mathbb{N}}$ is not a semigroup of operators, while in the particular case $\tau=0$, $\{\tilde{T}_n^0\}_{n\in\mathbb{N}}=\{\tilde{T}_n\}_{n\in\mathbb{N}}$ is a semigroup of operators as mentioned above. For each $n\in\mathbb{N}$ and each $u\in C(M,\mathbb{R}^1)$, let $U^u_n(x,\tau)=\tilde{T}_n^\tau u(x)$ for all $(x,\tau)\in
M\times[0,1]$. Then $U^u_n$ is a continuous function on $M\times[0,1]$.
Now we come to the main result:
\[th1\] For each $u\in C(M,\mathbb{R}^1)$, the uniform limit $\bar{u}=\lim_{n\to+\infty}U^u_n$ exists and
$$\bar{u}(x,\tau)=\inf_{y\in M}\big(u(y)+h_{0,\tau}(y,x)\big)$$ for all $(x,\tau)\in M\times\mathbb{S}^1$. Furthermore, $\bar{u}$ is a backward weak KAM solution of the Hamilton-Jacobi equation
$$\begin{aligned}
\label{1-5}
u_\tau+H(x,u_x,\tau)=0.\end{aligned}$$
For the definition of the (extended) Peierls barrier $h$, see [@Mat93] or [@Fat5; @Ber; @Con]. For completeness’ sake, we recall the definition in Section 3.
In addition, we discuss the relation among uniform limits $\lim_{n\to+\infty}U^u_n$, backward weak KAM solutions and viscosity solutions of (\[1-5\]). Let $\bar{u}\in
C(M\times\mathbb{S}^1,\mathbb{R}^1)$. Then the following three statements are equivalent.
- There exists $u\in C(M,\mathbb{R}^1)$ such that the uniform limit $\lim_{n\to+\infty}U^u_n=\bar{u}$.
- $\bar{u}$ is a backward weak KAM solution of (\[1-5\]).
- $\bar{u}$ is a viscosity solution of (\[1-5\]).
See Propositions \[pr3-5\], \[pr3-6\] for details.
0.2cm
Time-independent case
---------------------
Just like the time-periodic case, for each $n\in\mathbb{N}$ and each $u\in C(M,\mathbb{R}^1)$, let
$$\tilde{T}_n^au(x)=\inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}\inf_{\gamma}\Big\{u(\gamma(0))+\int_{0}^k
L_a(\gamma(s),\dot{\gamma}(s))ds\Big\}$$ for all $x\in M$, where the second infimum is taken among the continuous and piecewise $C^1$ paths $\gamma:[0,k]\rightarrow M$ with $\gamma(k)=x$. For each $n\in\mathbb{N}$, $\tilde{T}_n^a$ is an operator from $C(M,\mathbb{R}^1)$ to itself, and $\{\tilde{T}_n^a\}_{n\in\mathbb{N}}$ is a semigroup of operators.
For each $\tau\in[0,1]$ and each $n\in \mathbb{N}$, let $\tilde{T}_n^{a,\tau}=T_\tau^a\circ\tilde{T}_n^a$. Then for each $u\in C(M,\mathbb{R}^1)$,
$$\tilde{T}_n^{a,\tau} u(x)=(T_\tau^a\circ\tilde{T}_n^au)(x)=
\inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}\inf_{\gamma}\Big\{u(\gamma(0))+\int_{0}^{\tau+k}
L_a(\gamma(s),\dot{\gamma}(s))ds\Big\}$$ for all $x\in M$, where the second infimum is taken among the continuous and piecewise $C^1$ paths $\gamma:[0,\tau+k]\rightarrow
M$ with $\gamma(\tau+k)=x$. For each $\tau\in[0,1]$ and each $n\in
\mathbb{N}$, $\tilde{T}_n^{a,\tau}$ is an operator from $C(M,\mathbb{R}^1)$ to itself. Furthermore, it is not difficult to show that for each $\tau\in[0,1]$ and each $u\in
C(M,\mathbb{R}^1)$, the uniform limit $\lim_{n\to+\infty}\tilde{T}_n^{a,\tau}u$ exists and $\lim_{n\to+\infty}\tilde{T}_n^{a,\tau}u=\lim_{n\to+\infty}T_n^au=\bar{u}$, which is a backward weak KAM solution of (\[1-2\]), see Remark \[re4-1\]. It means that the parameter $\tau$ does not effect the convergence of $\{\tilde{T}_n^{a,\tau}u\}_{n\in\mathbb{N}}$. Therefore, without any loss of generality, we take $\tau=0$ and thus consider the operator $\tilde{T}_n^{a,0}=\tilde{T}_n^a$. In order to compare the new family of L-O operators to the [*full*]{} L-O semigroup $\{T^a_t\}_{t\geq 0}$, it is convenient to define [*the new L-O operator associated with $L_a$*]{} as follows.
\[def2\] For each $u\in C(M,\mathbb{R}^1)$ and each $t\geq0$, let $$\tilde{T}^a_tu(x)=\inf_{t\leq \sigma\leq
2t}\inf_{\gamma}\Big\{u(\gamma(0))+\int_0^\sigma
L_a(\gamma(s),\dot{\gamma}(s))ds\Big\}$$ for all $x\in M$, where the second infimum is taken among the continuous and piecewise $C^1$ paths $\gamma:[0,\sigma]\rightarrow
M$ with $\gamma(\sigma)=x$.
It is easy to check that $\{\tilde{T}^a_t\}_{t\geq
0}:C(M,\mathbb{R}^1)\to C(M,\mathbb{R}^1)$ is a one-parameter semigroup of operators. We call it [*the new L-O semigroup associated with $L_a$*]{}. We show that $u\in C(M,\mathbb{R}^1)$ is a fixed point of $\{\tilde{T}^a_t\}_{t\geq 0}$ if and only if it is a fixed point of $\{T^a_t\}_{t\geq 0}$, and that for each $u\in
C(M,\mathbb{R}^1)$, the uniform limit $\lim_{t\to+\infty}\tilde{T}^a_tu=\lim_{t\to+\infty}T^a_tu=\bar{u}$. For more properties of $\tilde{T}^a_t$ we refer to Section 4.
How fast does the L-O semigroup converge? It is an interesting question which is well worth discussing. We believe that there is a deep relation between dynamical properties of Mather sets (or Aubry sets) and the rate of convergence of the L-O semigroup. To the best of our knowledge there are now two relative results: In [@Itu], Iturriaga and S$\mathrm{\acute{a}}$nchez-Morgado prove that if the Aubry set consists in a finite number of hyperbolic periodic orbits or hyperbolic fixed points, the L-O semigroup converges exponentially. Recently, in [@Wan] the authors deal with the rate of convergence problem when the Mather set consists of degenerate fixed points. More precisely, consider the standard Lagrangian in classical mechanics $L^0_a(x,v)=\frac{1}{2}v^2+U(x)$, $x\in \mathbb{S}^1$, $v\in
\mathbb{R}^1$, where $U$ is a real analytic function on $\mathbb{S}^1$ and has a unique global minimum point $x_0$. Without loss of generality, one may assume $x_0=0$, $U(0)=0$. Then $c(L^0_a)=0$ and $\tilde{\mathcal{M}}_0=\{(0,0)\}$, where $\tilde{\mathcal{M}}_0$ is the Mather set with cohomology class 0 [@Mat91]. An upper bound estimate of the rate of convergence of the L-O semigroup is provided in [@Wan] under the assumption that $\{(0,0)\}$ is a degenerate fixed point: for every $u\in C(\mathbb{S}^1,\mathbb{R}^1)$, there exists a constant $C>0$ such that
$$\|T^a_tu-\bar{u}\|_\infty\leq\frac{C}{\sqrt[k-1]{t}}, \quad
\forall t>0,$$ where $k\in \mathbb{N}$, $k\geq 2$ depends only on the degree of degeneracy of the minimum point of the potential function $U$.
Naturally, we also care the problem of the rate of convergence of the new L-O semigroup. We compare the rate of convergence of the new L-O semigroup to the rate for the L-O semigroup as follows. First, we show that for each $u\in C(M,\mathbb{R}^1)$, $\|\tilde{T}^a_tu-\bar{u}\|_\infty\leq\|T^a_tu-\bar{u}\|_\infty$, $\forall t\geq 0$. It means that the new L-O semigroup converges faster than the L-O semigroup.
Then, in particular, we consider a class of $C^2$ positive definite and superlinear Lagrangians on $\mathbb{T}^n$
$$\begin{aligned}
\label{1-6}
L^1_a(x,v)=\frac{1}{2}\langle
A(x)(v-\omega),(v-\omega)\rangle+f(x,v-\omega), \quad x\in
\mathbb{T}^n,\ v\in\mathbb{R}^n,\end{aligned}$$
where $A(x)$ is an $n\times n$ matrix, $\omega\in\mathbb{S}^{n-1}$ is a given vector, and $f(x,v-\omega)=O(\|v-\omega\|^3)$ as $v-\omega\rightarrow 0$. It is clear that $c(L^1_a)=0$ and $\tilde{\mathcal{M}}_0=\tilde{\mathcal{A}}_0=\tilde{\mathcal{N}}_0=\cup_{x\in\mathbb{T}^n}(x,\omega)$, which is a quasi-periodic invariant torus with frequency vector $\omega$ of the Euler-Lagrange flow associated to $L^1_a$, where $\tilde{\mathcal{A}}_0$ and $\tilde{\mathcal{N}}_0$ are the Aubry set and the Ma$\mathrm{\tilde{n}}\mathrm{\acute{e}}$ set with cohomology class 0 [@Mat93], respectively. For the Lagrangian system (\[1-6\]), we obtain the following two results on the rates of convergence of the L-O semigroup and the new L-O semigroup, respectively.
\[th2\] For each $u\in C(\mathbb{T}^n,\mathbb{R}^1)$, there is a constant $K>0$ such that
$$\|T^a_tu-\bar{u}\|_\infty\leq\frac{K}{t}, \quad \forall t>0,$$ where $K$ depends only on $n$ and $u$.
We recall the notations for Diophantine vectors: for $\rho>n-1$ and $\alpha>0$, let
$$\mathcal{D}(\rho,\alpha)=\Big\{\beta\in \mathbb{S}^{n-1}\ |\
|\langle\beta,k\rangle|\geq\frac{\alpha}{|k|^\rho},\ \forall
k\in\mathbb{Z}^n\backslash\{0\}\Big\},$$ where $|k|=\sum_{i=1}^n|k_i|$.
\[th3\] Given any frequency vector $\omega\in\mathcal{D}(\rho,\alpha)$, for each $u\in C(\mathbb{T}^n,\mathbb{R}^1)$, there is a constant $\tilde{K}>0$ such that
$$\|\tilde{T}^a_tu-\bar{u}\|_\infty\leq
\tilde{K}t^{-(1+\frac{4}{2\rho+n})}, \quad \forall t>0,$$ where $\tilde{K}$ depends only on $n$, $\rho$, $\alpha$ and $u$.
Finally, we construct an example (Example \[ex1\]) to show that the result of Theorem \[th2\] is sharp in the sense of order. Therefore, in view of Theorems \[th2\], \[th3\] and Example \[ex1\], we conclude that the new L-O semigroup converges faster than the L-O semigroup in the sense of order when the Aubry set $\tilde{\mathcal{A}}_0$ of the Lagrangian system (\[1-6\]) is a quasi-periodic invariant torus with Diophantine frequency vector $\omega\in\mathcal{D}(\rho,\alpha)$.
We hope that the new L-O operator introduced in the present paper will contribute to the development of the Mather theory and the weak KAM theory. At the end of this section, we refer the reader to some good introductory books (lecture notes), survey articles and most recent research articles on the Mather theory and the weak KAM theory: [@Mat-b; @Fat-b; @Con-b; @Sor-b; @Man92; @Man96; @Eva04; @Eva08; @Kal; @Arn11; @Arn; @Ber08; @Ber082; @Eva09; @Gom08; @Gom10].
The rest of the paper is organized as follows. In Section 2 we introduce the basic language and notation used in the sequel. In Section 3 we first study the basic properties of the new L-O operator associated with $L$ and then prove Theorem \[th1\]. The last part of the section is devoted to the discussion of the relation among uniform limits $\lim_{n\to+\infty}U^u_n$, backward weak KAM solutions and viscosity solutions of (\[1-5\]). In Section 4 we first study the basic properties of the new L-O semigroup associated with $L_a$ and then give the proofs of Theorems \[th2\] and \[th3\]. At last, we construct the example mentioned above (Example \[ex1\]).
Notation and terminology
========================
Consider the flat $n$-torus $\mathbb{T}^n$, whose universal cover is the Euclidean space $\mathbb{R}^n$. We view the torus as a fundamental domain in $\mathbb{R}^n$
$$\overline{A}=\underbrace{[0,1]\times\dots\times[0,1]}_{n\
\mathrm{times}}$$ with opposite faces identified. The unique coordinates $x=(x_1,\dots,x_n)$ of a point in $\mathbb{T}^n$ will belong to the half-open cube
$$A=\underbrace{[0,1)\times\dots\times[0,1)}_{n\ \mathrm{times}}.$$ In these coordinates the standard universal covering projection $\pi:\mathbb{R}^n\rightarrow\mathbb{T}^n$ takes the form
$$\pi(\tilde{x})=([\tilde{x}_1],\dots,[\tilde{x}_n]),$$ where $[\tilde{x}_i]=\tilde{x}_i$ mod 1, denotes the fractional part of $\tilde{x}_i$ ($\tilde{x}_i=\{\tilde{x}_i\}+[\tilde{x}_i]$, where $\{\tilde{x}_i\}$ is the greatest integer not greater than $\tilde{x}_i$). We can now define operations on $\mathbb{T}^n$ using the covering projection: each operation is simply the projection of the usual operation with coordinates in $\mathbb{R}^n$. Thus the flat metric $d_{\mathbb{T}^n}$ may be defined for any pair of points $x$, $y\in\mathbb{T}^n$ by $d_{\mathbb{T}^n}(x,y)=\|x-y\|$, where $\|\cdot\|$ is the usual Euclidean norm on $\mathbb{R}^n$. And the distance between points on the torus is at most $\frac{\sqrt{n}}{2}$. For $x\in\mathbb{T}^n$ and $R>0$, $B_R(x)=\{y\in\mathbb{T}^n|\
d_{\mathbb{T}^n}(x,y)<R\}$ denotes the open ball of the radius $R$ centered on $x$ in $\mathbb{T}^n$.
We choose, once and for all, a $C^\infty$ Riemannian metric on $M$. It is classical that there is a canonical way to associate to it a Riemannian metric on $TM$. We use the same symbol “$d$" to denote the distance function defined by the Riemannian metric on $M$ and the distance function defined by the Riemannian metric on $TM$. Denote by $\|\cdot\|_x$ the norm induced by the Riemannian metric on the fiber $T_xM$ for $x\in M$, and by $\langle
\cdot,\cdot\rangle_x$ the canonical pairing between $T_xM$ and $T_x^*M$. In particular, for $M=\mathbb{T}^n$, we denote $\langle
\cdot,\cdot\rangle_x$ by $\langle \cdot,\cdot\rangle$ for brevity. We use the same notation $\langle \cdot,\cdot\rangle$ for the standard inner product on $\mathbb{R}^n$. However, this should not create any ambiguity.
We equip $C(M,\mathbb{R}^1)$ and $C(M\times\mathbb{S}^1,\mathbb{R}^1)$ with the usual uniform topology (the compact-open topology, or the $C^0$-topology) defined by the supremum norm $\|\cdot\|_\infty$. We use $u\equiv
const.$ to denote a constant function whose values do not vary.
The new L-O operator: time-periodic case
========================================
In this section we first discuss some basic properties of the new L-O operator associated with $L$, i.e., $\{\tilde{T}_n^\tau\}$, and then study the uniform convergence of $U_n^u$, $\forall u\in
C(M,\mathbb{R}^1)$, as $n\to+\infty$. At last, we discuss the relation among uniform limits $\lim_{n\to+\infty}U^u_n$, backward weak KAM solutions and viscosity solutions of (\[1-5\]).
Basic properties of the new L-O operator
----------------------------------------
Recall the definition of the new L-O operator associated with $L$. For each $\tau\in[0,1]$, each $n\in \mathbb{N}$ and each $u\in
C(M,\mathbb{R}^1)$,
$$\tilde{T}_n^\tau u(x)=\inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}\inf_{\gamma}\Big\{u(\gamma(0))+\int_{0}^{\tau+k}
L(\gamma(s),\dot{\gamma}(s),s)ds\Big\}$$ for all $x\in M$, where the second infimum is taken among the continuous and piecewise $C^1$ paths $\gamma:[0,\tau+k]\rightarrow
M$ with $\gamma(\tau+k)=x$.
First of all, we show that for each $\tau\in[0,1]$ and each $n\in\mathbb{N}$, $\tilde{T}_n^\tau$ is an operator from $C(M,\mathbb{R}^1)$ to itself. For this, noticing that $\tilde{T}_n^\tau u(x)\in\mathbb{R}^1$ for all $x\in M$, we only need to prove the following result.
\[pr3-1\] For each $\tau\in[0,1]$, each $n\in \mathbb{N}$ and each $u\in
C(M,\mathbb{R}^1)$, $\tilde{T}_n^\tau u$ is a continuous function on $M$.
Following Mather ([@Mat93], also see [@Ber]), it is convenient to introduce, for $t'\geq t$ and $x$, $y\in M$, the following quantity:
$$F_{t,t'}(x,y)=\inf_\gamma\int_t^{t'}L(\gamma(s),\dot{\gamma}(s),s)ds,$$ where the infimum is taken over the continuous and piecewise $C^1$ paths $\gamma:[t,t']\to M$ such that $\gamma(t)=x$ and $\gamma(t')=y$.
By the definition of $\tilde{T}_n^\tau$, for each $u\in
C(M,\mathbb{R}^1)$ and each $x\in M$, we have
$$\begin{aligned}
\tilde{T}_n^\tau u(x)=\inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}\inf_{y\in M}\big(u(y)+F_{0,\tau+k}(y,x)\big).\end{aligned}$$
Since the function $(y,x)\mapsto F_{0,\tau+k}(y,x)$ is continuous for each $n\leq k\leq 2n$, $k\in\mathbb{N}$ (see [@Ber]), then from the compactness of $M$ the function $x\mapsto \inf_{y\in
M}\big(u(y)+F_{0,\tau+k}(y,x)\big)$ is also continuous for each $n\leq k\leq 2n$, $k\in\mathbb{N}$. Therefore, the function $x\mapsto\tilde{T}_n^\tau u(x)$ is continuous on $M$.
\[pr3-2\] For given $\tau\in[0,1]$, $n\in\mathbb{N}$, $u\in
C(M,\mathbb{R}^1)$ and $x\in M$, there exist $n\leq k_0\leq2n$, $k_0\in\mathbb{N}$ and a minimizing extremal curve $\gamma:[0,\tau+k_0]\to M$ such that $\gamma(\tau+k_0)=x$ and
$$\tilde{T}_n^\tau u(x)=u(\gamma(0))+\int_{0}^{\tau+k_0}
L(\gamma(s),\dot{\gamma}(s),s)ds.$$
Recall that
$$\begin{aligned}
\tilde{T}_n^\tau u(x)=\inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}\inf_{y\in M}\Big(u(y)+F_{0,\tau+k}(y,x)\Big).\end{aligned}$$
For each $k$, the function $y\mapsto u(y)+F_{0,\tau+k}(y,x)$ is continuous on $M$. Thus, from the compactness of $M$ there exist $y^k\in M$ such that
$$\begin{aligned}
\tilde{T}_n^\tau u(x)=\inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}\Big(u(y^k)+F_{0,\tau+k}(y^k,x)\Big).\end{aligned}$$
Then it is clear that there is $n\leq k_0\leq2n$, $k_0\in\mathbb{N}$ such that
$$\tilde{T}_n^\tau u(x)=u(y^{k_0})+F_{0,\tau+k_0}(y^{k_0},x).$$ It follows from Tonelli’s theorem (see, for example, [@Mat91]) that there exists a minimizing extremal curve $\gamma:[0,\tau+k_0]\to M$ such that $\gamma(0)=y^{k_0}$, $\gamma(\tau+k_0)=x$ and
$$F_{0,\tau+k_0}(y^{k_0},x)=\int_{0}^{\tau+k_0}
L(\gamma(s),\dot{\gamma}(s),s)ds.$$ Hence,
$$\tilde{T}_n^\tau u(x)=u(\gamma(0))+\int_{0}^{\tau+k_0}
L(\gamma(s),\dot{\gamma}(s),s)ds.$$
\[pr3-3\]
- For $u$, $v\in C(M,\mathbb{R}^1)$, if $u\leq
v$, then $\tilde{T}_n^\tau u\leq\tilde{T}_n^\tau v$, $\forall
\tau\in[0,1]$, $\forall n\in\mathbb{N}$.
- If $c$ is a constant and $u\in C(M,\mathbb{R}^1)$, then $\tilde{T}_n^\tau(u+c)=\tilde{T}_n^\tau u+c$, $\forall
\tau\in[0,1]$, $\forall n\in\mathbb{N}$.
- For each $u$, $v\in C(M,\mathbb{R}^1)$, $\|\tilde{T}_n^\tau u-\tilde{T}_n^\tau
v\|_\infty\leq\|u-v\|_\infty$, $\forall
\tau\in[0,1]$, $\forall n\in\mathbb{N}$.
For each $\tau\in[0,1]$, each $n\in\mathbb{N}$ and each $x\in M$,
$$\begin{aligned}
\tilde{T}_n^\tau u(x) & =\inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}\inf_{y\in
M}\big(u(y)+F_{0,\tau+k}(y,x)\big)\\
& \leq \inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}\inf_{y\in
M}\big(v(y)+F_{0,\tau+k}(y,x)\big)\\
&= \tilde{T}_n^\tau v(x),\end{aligned}$$
which proves (1). (2) results from the definition of $\tilde{T}_n^\tau$ directly. To prove (3), we notice that for each $x\in M$,
$$-\|u-v\|_\infty+v(x)\leq u(x)\leq\|u-v\|_\infty+v(x).$$ From (1) and (2), for each $x\in M$ we have
$$\tilde{T}_n^\tau v(x)-\|u-v\|_\infty\leq \tilde{T}_n^\tau u(x)\leq
\tilde{T}_n^\tau v(x)+\|u-v\|_\infty, \quad \forall \tau\in[0,1],
\ \forall n\in\mathbb{N}.$$ Hence, $\|\tilde{T}_n^\tau u-\tilde{T}_n^\tau
v\|_\infty\leq\|u-v\|_\infty$, $\forall \tau\in[0,1]$, $\forall
n\in\mathbb{N}$.
Uniform convergence of $U^u_n$
------------------------------
Here we deal with the uniform convergence of $U^u_n$, $\forall
u\in C(M,\mathbb{R}^1)$, as $n\to+\infty$. We show that for each $u\in C(M,\mathbb{R}^1)$ the uniform limit $\bar{u}=\lim_{n\to+\infty}U^u_n$ exists and
$$\bar{u}(x,\tau)=\inf_{y\in M}\big(u(y)+h_{0,\tau}(y,x)\big)$$ for all $(x,\tau)\in M\times\mathbb{S}^1$. This is an immediate consequence of Proposition \[pr3-4\] below.
Following Ma$\mathrm{\tilde{n}}\mathrm{\acute{e}}$ [@Man97] and Mather [@Mat93], define the action potential and the extended Peierls barrier as follows.
0.1cm
[*Action Potential*]{}: for each $(\tau,\tau')\in\mathbb{S}^1\times\mathbb{S}^1$, let
$$\Phi_{\tau,\tau'}(x,x')=\inf F_{t,t'}(x,x')$$ for all $(x,x')\in M\times M$, where the infimum is taken on the set of $(t,t')\in\mathbb{R}^2$ such that $\tau=[t]$, $\tau'=[t']$ and $t'\geq t+1$.
0.1cm
[*Extended Peierls Barrier*]{}: for each $(\tau,\tau')\in\mathbb{S}^1\times\mathbb{S}^1$, let
$$\begin{aligned}
\label{3-1}
h_{\tau,\tau'}(x,x')=\liminf_{t'-t\to+\infty}F_{t,t'}(x,x')\end{aligned}$$
for all $(x,x')\in M\times M$, where the liminf is restricted to the set of $(t,t')\in\mathbb{R}^2$ such that $\tau=[t]$, $\tau'=[t']$.
0.1cm
From the above definitions, it is not hard to see that
$$\begin{aligned}
\label{3-2}
\Phi_{\tau,\tau'}(x,x')\leq h_{\tau,\tau'}(x,x'), \quad \forall
(x,\tau),\ (x',\tau')\in M\times\mathbb{S}^1\end{aligned}$$
and
$$\begin{aligned}
\label{3-3}
h_{\tau,t}(x,y)\leq h_{\tau,s}(x,z)+\Phi_{s,t}(z,y), \quad \forall
(x,\tau),\ (y,t),\ (z,s)\in M\times\mathbb{S}^1.\end{aligned}$$
It can be shown that the extended Peierls barrier $h_{\tau,\tau'}$ is Lipschitz and that, the liminf in (\[3-1\]) can not always be replaced with a limit, which leads to the non-convergence of the L-O semigroup associated with $L$ [@Fat5]. See [@Sor-b] for more details about the action potential and the extended Peierls barrier. Before stating Proposition \[pr3-4\], we introduce the following lemma.
\[le3-1\] If $t>0$ is fixed, there exists a compact subset $\mathcal{C}_t\subset TM\times\mathbb{S}^1$ such that for each minimizing extremal curve $\gamma:[a,b]\to M$ with $b-a\geq t$, we have
$$(\gamma(s),\dot{\gamma}(s),[s])\in \mathcal{C}_t, \quad \forall
s\in[a,b].$$
The lemma may be proved by small modifications of the proof found in [@Fat-b Corollary 4.3.2].
\[pr3-4\] $$\lim_{n\to+\infty}\inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}F_{\tau,\tau'+k}(x,x')=h_{\tau,\tau'}(x,x')$$ uniformly on $(\tau,\tau',x,x')\in
\mathbb{S}^1\times\mathbb{S}^1\times M\times M$.
Throughout this proof we use $C$ to denote a generic positive constant not necessarily the same in any two places. Since the proof is rather long, it is convenient to divide it into two steps.
Step 1. In the first step, we show that
$$\begin{aligned}
\label{3-4}
\lim_{n\to+\infty}\inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}F_{\tau,\tau'+k}(x,x')=h_{\tau,\tau'}(x,x'), \quad \forall
(\tau,\tau',x,x')\in \mathbb{S}^1\times\mathbb{S}^1\times M\times
M.\end{aligned}$$
For each $\tau,\tau'\in\mathbb{S}^1$ and each $x$, $x'\in M$, by the definition of $h_{\tau,\tau'}$, we have $\liminf_{k\to+\infty}F_{\tau,\tau'+k}(x,x')=h_{\tau,\tau'}(x,x')$. Then there exist $\{k_i\}_{i=1}^{+\infty}$ such that $k_i\to+\infty$ and $F_{\tau,\tau'+k_i}(x,x')\to
h_{\tau,\tau'}(x,x')$ as $i\to+\infty$. Tonelli’s theorem guarantees the existence of the minimizing extremal curves $\gamma_{k_i}:[\tau,\tau'+k_i]\to M$ with $\gamma_{k_i}(\tau)=x$, $\gamma_{k_i}(\tau'+k_i)=x'$ and $A(\gamma_{k_i})=F_{\tau,\tau'+k_i}(x,x')$, where
$$A(\gamma_{k_i})=\int_{\tau}^{\tau'+k_i}
L(\gamma_{k_i},\dot{\gamma}_{k_i},s)ds.$$ Thus, we have $A(\gamma_{k_i})\to h_{\tau,\tau'}(x,x')$ as $i\to+\infty$. Then for every $\varepsilon>0$, there exists $I\in\mathbb{N}$ such that
$$|A(\gamma_{k_i})-h_{\tau,\tau'}(x,x')|<\varepsilon$$ if $i\geq I$, $i\in \mathbb{N}$. And it is clear that for each $k_i$, $(\gamma_{k_i}(s),\dot{\gamma}_{k_i}(s),[s]):[\tau,\tau'+k_i]\to
TM\times\mathbb{S}^1$ is a trajectory of the Euler-Lagrange flow.
To prove (\[3-4\]), it suffices to show that for $n\in\mathbb{N}$ large enough, we can find a curve $\tilde{\gamma}:[\tau,\tau'+k_0]\to M$ with $\tilde{\gamma}(\tau)=x$, $\tilde{\gamma}(\tau'+k_0)=x'$, where $n\leq k_0\leq2n$, $k_0\in\mathbb{N}$, such that
$$|A(\tilde{\gamma})-A(\gamma_{k_I})|\leq C\varepsilon$$ for some constant $C>0$. In fact, if such a curve exists, then
$$\inf_{k\in\mathbb{N} \atop n\leq
k}F_{\tau,\tau'+k}(x,x')\leq\inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}F_{\tau,\tau'+k}(x,x')\leq A(\tilde{\gamma})\leq
A(\gamma_{k_I})+C\varepsilon\leq
h_{\tau,\tau'}(x,x')+C\varepsilon.$$ By letting $n\to+\infty$, from the arbitrariness of $\varepsilon>0$, we have
$$\begin{aligned}
h_{\tau,\tau'}(x,x')& = \liminf_{k\to+\infty}F_{\tau,\tau'+k}(x,x')\\
& = \lim_{n\to+\infty}\inf_{k\in\mathbb{N} \atop n\leq
k}F_{\tau,\tau'+k}(x,x')\\
& \leq \lim_{n\to+\infty}\inf_{k\in\mathbb{N}
\atop n\leq k\leq 2n}F_{\tau,\tau'+k}(x,x')\\
& \leq h_{\tau,\tau'}(x,x'),\end{aligned}$$
which implies that
$$\lim_{n\to+\infty}\inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}F_{\tau,\tau'+k}(x,x')=h_{\tau,\tau'}(x,x').$$
Our task is now to construct the curve mentioned above. Note that for the above $\varepsilon>0$, there exists $I'\in\mathbb{N}$ such that there exists
$$(z_{k_i},v_{z_{k_i}},t_{z_{k_i}})\in
O_i:=\{(\gamma_{k_i}(s),\dot{\gamma}_{k_i}(s),[s])\ |\ \tau\leq
s\leq\tau'+k_i\}\subset TM\times\mathbb{S}^1$$ such that
$$d((z_{k_i},v_{z_{k_i}},t_{z_{k_i}}),\tilde{\mathcal{M}}_0)<\varepsilon,$$ if $i\geq I'$, $i\in\mathbb{N}$, where $\tilde{\mathcal{M}}_0$ is the Mather set of cohomology class 0. As usual, distance is measured with respect to smooth Riemannian metrics. Since $\tilde{\mathcal{M}}_0$ is compact and by the a priori compactness given by Lemma \[le3-1\], $O_i$ is contained in the compact subset $\mathcal{C}_{k_{I'}-1}$ of $TM\times\mathbb{S}^1$ for each $i\geq I'$, then it doesn’t matter which Riemannian metrics we choose to measure distance.
Let $I=\max\{I,I'\}$. Then $|A(\gamma_{k_I})-h_{\tau,\tau'}(x,x')|<\varepsilon$ and there exists $(z_0,v_{z_0},t_{z_0})\in
O_I=\{(\gamma_{k_I}(s),\dot{\gamma}_{k_I}(s),[s])\ |\ \tau\leq
s\leq\tau'+k_I\}$ such that
$$\begin{aligned}
\label{3-5}
d((z_0,v_{z_0},t_{z_0}),\tilde{\mathcal{M}}_0)<\varepsilon.\end{aligned}$$
In view of (\[3-5\]), there exists an ergodic minimal measure $\mu_e$ on $TM\times\mathbb{S}^1$ [@Mat91] such that $\mu_e(\mathrm{supp}\mu_e\cap
B_{2\varepsilon}(z_0,v_{z_0},t_{z_0}))=\Delta>0$, where $B_{2\varepsilon}(z_0,v_{z_0},t_{z_0})$ denotes the open ball of radius $2\varepsilon$ centered on $(z_0,v_{z_0},t_{z_0})$ in $TM\times\mathbb{S}^1$. Set $A_{2\varepsilon}=\mathrm{supp}\mu_e\cap
B_{2\varepsilon}(z_0,v_{z_0},t_{z_0})$. Since $\mu_e$ is an ergodic measure, then
$$\mu_e(\bigcup_{t=1}^{+\infty}\phi^L_{-t}(A_{2\varepsilon}))=1.$$ Thus, for any $0<\Delta'<\Delta$, there exists $T>0$ such that
$$\mu_e(\bigcup_{t=1}^{T'}\phi^L_{-t}(A_{2\varepsilon}))\geq
1-\Delta',$$ if $T'\geq T$. From this, we may deduce that for each $n\in\mathbb{N}$,
$$\begin{aligned}
\label{3-6}
\Big(\bigcup_{t=1}^T\phi^L_{-t}(A_{2\varepsilon})\Big)\cap\phi^L_n(A_{2\varepsilon})
\neq\emptyset.\end{aligned}$$
For, otherwise, there would be $n_0\in\mathbb{N}$ such that
$$\begin{aligned}
\mu_e\Big(\big(\bigcup_{t=1}^T\phi^L_{-t}(A_{2\varepsilon})\big)\cup\phi^L_{n_0}(A_{2\varepsilon})\Big) & =\mu_e\Big(\bigcup_{t=1}^T\phi^L_{-t}(A_{2\varepsilon})\Big)+\mu_e(\phi^L_{n_0}(A_{2\varepsilon}))\\
& \geq 1-\Delta'+\Delta>1,\end{aligned}$$
which contradicts that $\mu_e$ is a probability measure.
For a given $n\in\mathbb{N}$ large enough with $\max\{k_I,T+1\}\leq\{\frac{n}{2}\}$, from (\[3-6\]) there exist $(e_0,v_{e_0},t_{e_0})$, $(\bar{e}_0,v_{\bar{e}_0},t_{\bar{e}_0})\in A_{2\varepsilon}$ and $1\leq t\leq T$ such that
$$\begin{aligned}
\label{3-7}
\phi^L_{-t}(e_0,v_{e_0},t_{e_0})=(e,v_e,t_e)=\phi^L_n(\bar{e}_0,v_{\bar{e}_0},t_{\bar{e}_0})\end{aligned}$$
for some $(e,v_e,t_e)\in \tilde{\mathcal{M}}_0$. Since $(e_0,v_{e_0},t_{e_0})\in A_{2\varepsilon}$, then
$$\begin{aligned}
\label{3-8}
d((e_0,v_{e_0},t_{e_0}),(z_0,v_{z_0},t_{z_0}))<2\varepsilon.\end{aligned}$$
Set $(z_1,v_{z_1},t_{z_1})=\phi^L_{t_{e_0}-t_{z_0}}(z_0,v_{z_0},t_{z_0})$. Then $t_{z_1}=t_{e_0}$ and from (\[3-8\]) we have
$$\begin{aligned}
\label{3-9}
d((e_0,v_{e_0},t_{e_0}),(z_1,v_{z_1},t_{e_0}))<C\varepsilon\end{aligned}$$
for some constant $C>0$. Set $(z_2,v_{z_2},\tau)=\phi^L_{\tau-t_{e_0}}(z_1,v_{z_1},t_{e_0})$ and $(e_1,v_{e_1},\tau)=\phi^L_{\tau-t_{e_0}}(e_0,v_{e_0},t_{e_0})$. Then by the differentiability of the solutions of the Euler-Lagrange equation with respect to initial values, we have
$$\begin{aligned}
\label{3-10}
d((e_1,v_{e_1},\tau),(z_2,v_{z_2},\tau))<C\varepsilon\end{aligned}$$
for some constant $C>0$.
Since $(e_0,v_{e_0},t_{e_0})$, $(\bar{e}_0,v_{\bar{e}_0},t_{\bar{e}_0})\in A_{2\varepsilon}$, then
$$\begin{aligned}
\label{3-11}
d((e_0,v_{e_0},t_{e_0}),
(\bar{e}_0,v_{\bar{e}_0},t_{\bar{e}_0}))<4\varepsilon.\end{aligned}$$
Set $(\bar{e}_1,v_{\bar{e}_1},t_{e_0})=\phi^L_{t_{e_0}-t_{\bar{e}_0}}(\bar{e}_0,v_{\bar{e}_0},t_{\bar{e}_0})$. Then from (\[3-11\]) we have
$$\begin{aligned}
\label{3-12}
d((e_0,v_{e_0},t_{e_0}),
(\bar{e}_1,v_{\bar{e}_1},t_{e_0}))<C\varepsilon\end{aligned}$$
for some constant $C>0$. Set $(\bar{e}_2,v_{\bar{e}_2},\tau)=\phi^L_{\tau-t_{e_0}}(\bar{e}_1,v_{\bar{e}_1},t_{e_0})$. Recall that $(e_1,v_{e_1},\tau)=\phi^L_{\tau-t_{e_0}}(e_0,v_{e_0},t_{e_0})$. Then from the differentiability of the solutions of the Euler-Lagrange equation with respect to initial values, we have
$$\begin{aligned}
\label{3-13}
d((e_1,v_{e_1},\tau),(\bar{e}_2,v_{\bar{e}_2},\tau)<C\varepsilon\end{aligned}$$
for some constant $C>0$.
Note that since $(z_0,v_{z_0},t_{z_0})\in
O_I=\{(\gamma_{k_I}(s),\dot{\gamma}_{k_I}(s),[s])\ |\ \tau\leq
s\leq\tau'+k_I\}$, where $O_I$ is an orbit of the Euler-Lagrange flow, then $(z_2,v_{z_2},\tau)\in O_I$. And thus, there exists $k_{I_1}$, $k_{I_2}\in\mathbb{N}$ with $k_{I_1}+k_{I_2}=k_I$ such that
$$(z_2,v_{z_2},\tau)=(\gamma_{k_I}(\tau+k_{I_1}),\dot{\gamma}_{k_I}(\tau+k_{I_1}),\tau).$$
We are now in a position to construct the curve we need. We treat the case $k_{I_1}\neq 0$, $k_{I_2}\neq 0$ and the remaining cases can be treated similarly. Let $\alpha_1:[\tau,\tau+k_{I_1}]\to M$ with $\alpha_1(\tau)=x$ and $\alpha_1(\tau+k_{I_1})=\bar{e}_2$ be a Tonelli minimizer such that $A(\alpha_1)=F_{\tau,\tau+k_{I_1}}(x,\bar{e}_2)$. Since $\gamma_{k_I}:[\tau,\tau'+k_I]\to M$ is a minimizing extremal curve, then $\gamma_{k_I}|_{[\tau,\tau+k_{I_1}]}$ is also a minimizing extremal curve and thus $A(\gamma_{k_I}|_{[\tau,\tau+k_{I_1}]})=F_{\tau,\tau+k_{I_1}}(x,z_2)$. Therefore, by the Lipschtiz property of the function $F_{\tau,\tau+k_{I_1}}$ (see, for example, [@Ber]), (\[3-10\]) and (\[3-13\]) we have
$$\begin{aligned}
\label{3-14}
|A(\alpha_1)-A(\gamma_{k_I}|_{[\tau,\tau+k_{I_1}]})|=|F_{\tau,\tau+k_{I_1}}(x,\bar{e}_2)-F_{\tau,\tau+k_{I_1}}(x,z_2)|\leq
Dd(\bar{e}_2,z_2)\leq C\varepsilon\end{aligned}$$
for some constant $C>0$, where $D>0$ is a Lipschitz constant of $F_{t_1,t_2}$ which is independent of $t_1$, $t_2$ with $t_1+1\leq
t_2$.
Let $\beta(s)=p\phi^L_{s-(\tau+k_{I_1})}(\bar{e}_2,v_{\bar{e}_2},\tau)$, $s\in\mathbb{R}^1$, where $p:TM\times\mathbb{S}^1\to M$ denotes the projection. Then $(\beta(s),\dot{\beta}(s),[s])=\phi^L_{s-(\tau+k_{I_1})}(\bar{e}_2,v_{\bar{e}_2},\tau)$, $s\in\mathbb{R}^1$, and $(\beta(\tau+k_{I_1}),\dot{\beta}(\tau+k_{I_1}))=(\bar{e}_2,v_{\bar{e}_2})$. Hence , from (\[3-7\]) we have
$$(e,v_e,t_e)=(\beta(l),\dot{\beta}(l),[l]),$$ where $l=\tau+k_{I_1}+(t_{e_0}-\tau)+(t_{\bar{e}_0}-t_{e_0})+n$, and
$$\begin{aligned}
(e_1,v_{e_1},\tau) = (\beta(l'),\dot{\beta}(l'),[l']),\end{aligned}$$
where $l'=l+t+(\tau-t_{e_0})=\tau+k_{I_1}+n+t+t_{\bar{e}_0}-t_{e_0}$. Then $[l']=[\tau+k_{I_1}+n+t+t_{\bar{e}_0}-t_{e_0}]=[\tau+t+t_{\bar{e}_0}-t_{e_0}]=\tau$, which means that $t+t_{\bar{e}_0}-t_{e_0}\in\mathbb{Z}$. Notice that $0\leq t+t_{\bar{e}_0}-t_{e_0}\leq
T+t_{\bar{e}_0}-t_{e_0}\leq\{\frac{n}{2}\}$. Thus,
$$\begin{aligned}
\label{3-15}
n\leq k_I+n+t+t_{\bar{e}_0}-t_{e_0}\leq k_I+n+\{\frac{n}{2}\}\leq
2n.\end{aligned}$$
Let $m=n+t+t_{\bar{e}_0}-t_{e_0}\in\mathbb{Z}$ and $\alpha_2=\beta|_{[\tau+k_{I_1},\tau+k_{I_1}+m]}$. Then $\alpha_2(\tau+k_{I_1})=\beta(\tau+k_{I_1})=\bar{e}_2$ and $\alpha_2(\tau+k_{I_1}+m)=\beta(\tau+k_{I_1}+m)=e_1$. In view of $(\bar{e}_0,v_{\bar{e}_0},t_{\bar{e}_0})\in
A_{2\varepsilon}\subset\tilde{\mathcal{M}}_0$ and the definitions of $\beta$ and $\alpha_2$, $(\alpha_2(s),\dot{\alpha}_2(s),[s])$ is a trajectory of the Euler-Lagrange flow in $\tilde{\mathcal{M}}_0$. According to [@Mat91 Proposition 3] and the definition of $h_{\tau,\tau}$, we have
$$A(\alpha_2)=F_{\tau+k_{I_1},\tau+k_{I_1}+m}(\bar{e}_2,e_1)=h_{\tau,\tau}(\bar{e}_2,e_1).$$ Hence, on account of the Lipschitz property of $h_{\tau,\tau}$ and (\[3-13\]),
$$|A(\alpha_2)-h_{\tau,\tau}(e_1,e_1)|=
|h_{\tau,\tau}(\bar{e}_2,e_1)-h_{\tau,\tau}(e_1,e_1)|\leq\bar{D}d(\bar{e}_2,e_1)\leq
C\varepsilon$$ for some constant $C>0$, where $\bar{D}$ is a Lipschitz constant of $h_{\tau,\tau}$. Since $(e_1,\tau)\in\mathcal{M}_0$, where $\mathcal{M}_0\subset M\times\mathbb{S}^1$ is the projected Mather set, then $h_{\tau,\tau}(e_1,e_1)=0$, and thus
$$\begin{aligned}
\label{3-16}
|A(\alpha_2)|\leq C\varepsilon.\end{aligned}$$
Let $\alpha_3:[\tau+k_{I_1}+m,\tau'+k_I+m]\to M$ with $\alpha_3(\tau+k_{I_1}+m)=e_1$ and $\alpha_3(\tau'+k_I+m)=x'$ be a Tonelli minimizer such that
$$A(\alpha_3)=F_{\tau+k_{I_1}+m,\tau'+k_I+m}(e_1,x')=F_{\tau+k_{I_1},\tau'+k_I}(e_1,x').$$ Since $\gamma_{k_I}:[\tau,\tau'+k_I]\to M$ is a minimizing extremal curve, then $\gamma_{k_I}|_{[\tau+k_{I_1},\tau'+k_I]}$ is also a minimizing extremal curve and thus
$$A(\gamma_{k_I}|_{[\tau+k_{I_1},\tau'+k_I]})=F_{\tau+k_{I_1},\tau'+k_I}(z_2,x').$$ Therefore, from the Lipschitz property of $F_{\tau+k_{I_1},\tau'+k_I}$ and (\[3-10\]), we have
$$\begin{aligned}
\label{3-17}
\begin{split}
|A(\alpha_3)-A(\gamma_{k_I}|_{[\tau+k_{I_1},\tau'+k_I]})|
&=|F_{\tau+k_{I_1},\tau'+k_I}(e_1,x')-F_{\tau+k_{I_1},\tau'+k_I}(z_2,x')|\\
&\leq Dd(e_1,z_2)\\
&\leq C\varepsilon
\end{split}\end{aligned}$$
for some constant $C>0$.
Consider the curve $\tilde{\gamma}:[\tau,\tau'+k_I+m]\to M$ connecting $x$ and $x'$ defined by
$$\tilde{\gamma}(s)= \left\{\begin{array}{ll}
\alpha_1(s),\quad & s\in[\tau,\tau+k_{I_1}],\\[2mm]
\alpha_2(s),\quad & s\in[\tau+k_{I_1},\tau+k_{I_1}+m],\\[2mm]
\alpha_3(s),\quad & s\in[\tau+k_{I_1}+m,\tau'+k_I+m].
\end{array}\right.$$ By (\[3-15\]), $n\leq k_0:=k_I+m\leq 2n$. From (\[3-14\]), (\[3-16\]) and (\[3-17\]), we have
$$|A(\tilde{\gamma})-A(\gamma_{k_I})|\leq C\varepsilon$$ for some constant $C>0$. It is clear that $\tilde{\gamma}$ is just the curve we need, and we have proved (\[3-4\]).
Step 2. For each $n\in\mathbb{N}$ and each $(\tau,\tau',x,x')\in[0,1]\times[0,1]\times M\times M$, let
$$\mathcal{F}_n(\tau,\tau',x,x')=\inf_{k\in\mathbb{N} \atop n\leq
k\leq 2n}F_{\tau,\tau'+k}(x,x').$$ Then, to complete the proof of Proposition \[pr3-4\], it suffices to show that $\{\mathcal{F}_n\}_{n=2}^{+\infty}$ are equicontinuous. Notice that $(\tau,\tau',x,x')\mapsto
F_{\tau,\tau'+k}(x,x')$ is a Lipschitz function on $[0,1]\times[0,1]\times M\times M$ for every $k\geq 2$, $k\in\mathbb{N}$, and that the Lipschitz constant $\tilde{D}$ is independent of $k$, see [@Ber 3.3 LEMMA]. Hence, for each $n\geq 2$, $n\in\mathbb{N}$ the function $(\tau,\tau',x,x')\mapsto
\mathcal{F}_n(\tau,\tau',x,x')$ is also Lipschitz with the same Lipschitz constant $\tilde{D}$, and thus $\{\mathcal{F}_n\}_{n=2}^{+\infty}$ are equicontinuous. The proof is now complete.
Recall that for each $n\in\mathbb{N}$ and each $u\in
C(M,\mathbb{R}^1)$,
$$U^u_n(x,\tau)=\tilde{T}_n^\tau u(x)=\inf_{k\in\mathbb{N} \atop
n\leq k\leq 2n}\inf_{y\in
M}\big(u(y)+F_{0,\tau+k}(y,x)\big)=\inf_{y\in
M}\big(u(y)+\mathcal{F}_n(0,\tau,y,x)\big)$$ for all $(x,\tau)\in M\times[0,1]$. Since
$$\begin{aligned}
\big|U^u_n(x,\tau)-\inf_{y\in
M}\big(u(y)+h_{0,\tau}(y,x)\big)\big|&= \big|\inf_{y\in
M}\big(u(y)+\mathcal{F}_n(0,\tau,y,x)\big)-\inf_{y\in
M}\big(u(y)+h_{0,\tau}(y,x)\big)\big|\\
& \leq \sup_{y\in M}|\mathcal{F}_n(0,\tau,y,x)-h_{0,\tau}(y,x)|,\end{aligned}$$
then from Proposition \[pr3-4\], we conclude that the uniform limit $\bar{u}=\lim_{n\to+\infty}U^u_n$ exists, and
$$\begin{aligned}
\label{3-18}
\bar{u}(x,\tau)=\inf_{y\in M}\big(u(y)+h_{0,\tau}(y,x)\big)\end{aligned}$$
for all $(x,\tau)\in M\times\mathbb{S}^1$, thus proving the first assertion of Theorem \[th1\].
$\lim_{n\to+\infty}U_n^u$, backward weak KAM solutions and viscosity solutions
------------------------------------------------------------------------------
Here we discuss the relation among uniform limits $\lim_{n\to+\infty}U^u_n$, backward weak KAM solutions and viscosity solutions of (\[1-5\]). Following Fathi [@Fat1], as done by Contreras et al. in [@Con], we give the definition of the backward weak KAM solution as follows.
\[def3\] A backward weak KAM solution of the Hamilton-Jacobi equation (\[1-5\]) is a function $u:M\times\mathbb{S}^1\to\mathbb{R}^1$ such that
- u is dominated by $L$, i.e., $$u(x,\tau)-u(y,s)\leq\Phi_{s,\tau}(y,x), \quad
\forall (x,\tau),\ (y,s)\in M\times\mathbb{S}^1.$$ We use the notation $u\prec L$.
- For every $(x,\tau)\in M\times\mathbb{S}^1$ there exists a curve $\gamma:(-\infty,\tilde{\tau}]\to M$ with $\gamma(\tilde{\tau})=x$ and $[\tilde{\tau}]=\tau$ such that $$u(x,\tau)-u(\gamma(t),[t])=\int_t^{\tilde{\tau}}L(\gamma(s),\dot{\gamma}(s),s)ds,\quad
\forall t\in(-\infty,\tilde{\tau}].$$
We denote by $\mathcal{S}_-$ the set of backward weak KAM solutions. Let us recall two known results [@Con] on backward weak KAM solutions, which will be used later in the paper.
\[le3-2\] Given a fixed $(y,s)\in M\times\mathbb{S}^1$, the function
$$(x,\tau)\mapsto h_{s,\tau}(y,x),\quad (x,\tau)\in
M\times\mathbb{S}^1$$ is a backward weak KAM solution.
\[le3-3\] If $\mathcal{U}\subset \mathcal{S}_-$, let $\underline{u}(x,\tau):=\inf_{u\in\mathcal{U}}u(x,\tau)$ then either $\underline{u}\equiv-\infty$ or $\underline{u}\in\mathcal{S}_-$.
We define the projected Aubry set $\mathcal{A}_0$ as follows:
$$\mathcal{A}_0:=\{(x,\tau)\in M\times\mathbb{S}^1\ |\
h_{\tau,\tau}(x,x)=0\}.$$ Note that $\mathcal{A}_0=\Pi\tilde{\mathcal{A}_0}$, where $\Pi:TM\times\mathbb{S}^1\to M\times\mathbb{S}^1$ denotes the projection and $\tilde{\mathcal{A}_0}$ denotes the Aubry set in $TM\times\mathbb{S}^1$, i.e., the union of global static orbits. See for instance [@Ber] for the definition of static orbits and more details on $\tilde{\mathcal{A}_0}$.
From the definition of $\mathcal{A}_0$, (\[3-2\]) and (\[3-3\]), it is straightforward to show that if $(x,\tau)\in\mathcal{A}_0$, then
$$\begin{aligned}
\label{3-19}
h_{\tau,s}(x,y)=\Phi_{\tau,s}(x,y)\end{aligned}$$
for all $(y,s)\in M\times\mathbb{S}^1$. Define an equivalence relation on $\mathcal{A}_0$ by saying that $(x,\tau)$ and $(y,s)$ are equivalent if and only if
$$\begin{aligned}
\label{3-20}
\Phi_{\tau,s}(x,y)+\Phi_{s,\tau}(y,x)=0.\end{aligned}$$
By (\[3-19\]), it is simple to see that (\[3-20\]) is equivalent to
$$h_{\tau,s}(x,y)+h_{s,\tau}(y,x)=0.$$ The equivalent classes of this relation are called static classes. Let $\mathrm A$ be the set of static classes. For each static class $\Gamma\in \mathrm A$ choose a point $(x,0)\in\Gamma$ and let $\mathbb{A}_0$ be the set of such points.
Contreras et al. [@Con] characterize backward weak KAM solutions of the Hamilton-Jacobi equation (\[1-5\]) in terms of their values at each static class and the extended Peierls barrier. See [@Con01] for similar results in the time-independent case.
\[th4\] The map $\{f:\mathbb{A}_0\to\mathbb{R}^1\ |\ f\prec
L\}\to\mathcal{S}_-$
$$f\mapsto
u_f(x,\tau)=\min_{(p,0)\in\mathbb{A}_0}(f(p,0)+h_{0,\tau}(p,x))$$ is a bijection.
\[pr3-5\] $$\{\bar{u}\in C(M\times\mathbb{S}^1,\mathbb{R}^1)\ |\ \exists\ u\in
C(M,\mathbb{R}^1),\
\bar{u}=\lim_{n\to+\infty}U_n^u\}=\mathcal{S}_-.$$
Proposition \[pr3-5\] tells us two things: (i) For each $u\in
C(M,\mathbb{R}^1)$, $\bar{u}=\lim_{n\to+\infty}U_n^u$ is a backward weak KAM solution of (\[1-5\]), which proves the second assertion of Theorem \[th1\]. (ii) For each $w\in\mathcal{S}_-$ there exists $w_0\in C(M,\mathbb{R}^1)$ such that $w=\lim_{n\to+\infty}U_n^{w_0}$. Moreover, we know from the proof of Proposition \[pr3-5\] that $w_0(x)=w(x,0)$ for all $x\in M$.
First we show that for each $u\in C(M,\mathbb{R}^1)$, $\bar{u}=\lim_{n\to+\infty}U^u_n$ is a backward weak KAM solution of (\[1-5\]). By (\[3-18\]) we have
$$\bar{u}(x,\tau)=\inf_{y\in M}\big(u(y)+h_{0,\tau}(y,x)\big)$$ for all $(x,\tau)\in M\times\mathbb{S}^1$. Combining Lemmas \[le3-2\] and \[le3-3\] we get that $\bar{u}\in\mathcal{S}_-$.
Then we prove that for each $w\in\mathcal{S}_-$, there exists $w_0\in C(M,\mathbb{R}^1)$ such that $w=\lim_{n\to+\infty}U^{w_0}_n$. From Theorem \[th4\] there exists $f:\mathbb{A}_0\to\mathbb{R}^1$ with $f\prec L$ such that for each $(x,\tau)\in M\times\mathbb{S}^1$,
$$\begin{aligned}
w(x,\tau) & =\min_{(p,0)\in\mathbb{A}_0}\big(f(p,0)+h_{0,\tau}(p,x)\big)\\
& =\min_{(p,0)\in\mathbb{A}_0}\Big(f(p,0)+\min_{y\in
M}\big(h_{0,0}(p,y)+h_{0,\tau}(y,x)\big)\Big)\\
& =\min_{y\in
M}\Big(\min_{(p,0)\in\mathbb{A}_0}\big(f(p,0)+h_{0,0}(p,y)\big)+h_{0,\tau}(y,x)\Big)\\
& =\min_{y\in M}\big(w(y,0)+h_{0,\tau}(y,x)\big).\end{aligned}$$
Let $w_0(x)=w(x,0)$ for all $x\in M$. Then by Proposition \[pr3-4\] and (\[3-18\]), the uniform limit $\bar{w}_0=\lim_{n\to+\infty}U^{w_0}_n$ exists and
$$\bar{w}_0(x,\tau)= \min_{y\in
M}\big(w_0(y)+h_{0,\tau}(y,x)\big)=\min_{y\in
M}\big(w(y,0)+h_{0,\tau}(y,x)\big)$$ for all $(x,\tau)\in M\times\mathbb{S}^1$. Therefore, $w=\bar{w}_0=\lim_{n\to+\infty}U^{w_0}_n$.
\[pr3-6\] Let $u\in C(M\times\mathbb{S}^1,\mathbb{R}^1)$. Then $u$ is a backward weak KAM solution of (\[1-5\]) if and only if it is a viscosity solution of (\[1-5\]).
Let $u\in C(M\times\mathbb{S}^1,\mathbb{R}^1)$ and $u_0(x)=u(x,0)$ for all $x\in M$. If $u$ is a backward weak KAM solution of (\[1-5\]), then from Proposition \[pr3-5\] we have $u=\lim_{n\to+\infty}U^{u_0}_n$. Recall that
$$U^{u_0}_n(x,\tau)=\tilde{T}^\tau_nu_0(x)=(T_\tau\circ\tilde{T}_nu_0)(x).$$ It is a standard result that for each $n\in\mathbb{N}$, $U^{u_0}_n(x,\tau)=(T_\tau\circ\tilde{T}_nu_0)(x)$ is a viscosity solution of (\[1-5\]), see [@Fat5] for instance. Since $u$ is the uniform limit of $\{U^{u_0}_n\}_{n=1}^{+\infty}$, then from the stability of viscosity solution of (\[1-5\]) [@Fat-b], $u$ is also a viscosity solution of (\[1-5\]).
Suppose now that $u$ is a viscosity solution of (\[1-5\]). Let $U^{u_0}(x,t)=T_tu_0(x)$ for all $(x,t)\in M\times[0,+\infty)$. Then $U^{u_0}$ is a viscosity solution of (\[1-5\]) with $U^{u_0}(x,0)=T_0u_0(x)=u_0(x)$. Since $u$ can be considered as a 1-periodic in time viscosity solution on $M\times[0,+\infty)$ and the Cauchy Problem
$$\left\{
\begin{array}{ll}
v_t+H(x,v_x,t)=0, & \mathrm{on}\ M\times(0,+\infty),\\
v(x,0)=u_0(x), & \mathrm{on}\ M
\end{array}
\right.$$ is well posed in the viscosity sense (see, for example, [@Lio82] or [@Ber04]), then $u(x,t)=U^{u_0}(x,t)=T_tu_0(x)$ for all $(x,t)\in
M\times[0,+\infty)$. Since $u$ is 1-periodic in time, for each $(x,\tau)\in M\times[0,1]$ we have
$$u(x,\tau)=u(x,\tau+k)=\inf_\gamma\{u_0(\gamma(0))+\int_0^{\tau+k}L(\gamma,\dot{\gamma},s)ds\}, \quad \forall k\in\mathbb{N},$$ where the infimum is taken among the continuous and piecewise $C^1$ paths $\gamma:[0,\tau+k]\to M$ with $\gamma(\tau+k)=x$. Hence,
$$u(x,\tau)=\inf_{k\in\mathbb{N} \atop n\leq k\leq
2n}\inf_{y\in M}\big(u_0(y)+F_{0,\tau+k}(y,x)\big)=U^{u_0}_n(x,\tau),\quad \forall n\in\mathbb{N}.$$ Then by letting $n\to+\infty$, from Proposition \[pr3-5\] we have $u=\lim_{n\to+\infty}U^{u_0}_n\in\mathcal{S}_-$.
The new L-O operator: time-independent case
===========================================
As mentioned in the Introduction, in this section we first discuss the main properties of the new L-O semigroup associated with $L_a$ and then give the proofs of Theorems \[th2\] and \[th3\]. Finally, we construct an example to show that the new L-O semigroup converges faster than the L-O semigroup in the sense of order when the Aubry set $\tilde{\mathcal{A}}_0$ of the Lagrangian system (\[1-6\]) is a quasi-periodic invariant torus with Diophantine frequency vector $\omega\in\mathcal{D}(\rho,\alpha)$.
Main properties of the new L-O semigroup
----------------------------------------
Let us recall the definition of the new L-O operator $\tilde{T}^a_t$ associated with $L_a$. For each $t\geq0$ and each $u\in C(M,\mathbb{R}^1)$, $$\tilde{T}^a_tu(x)=\inf_{t\leq \sigma\leq
2t}\inf_{\gamma}\big\{u(\gamma(0))+\int_0^\sigma L_a(\gamma(s),\dot{\gamma}(s))ds\big\}$$ for all $x\in M$, where the second infimum is taken among the continuous and piecewise $C^1$ paths $\gamma:[0,\sigma]\rightarrow M$ with $\gamma(\sigma)=x$.
Obviously, $\tilde{T}^a_tu(x)=\inf_{t\leq \sigma\leq 2t}T^a_\sigma u(x)$. Moreover, it is straightforward to check that for each $t\geq 0$, $\tilde{T}^a_t$ is an operator from $C(M,\mathbb{R}^1)$ to itself, and that $\{\tilde{T}^a_t\}_{t\geq 0}$ is a semigroup of operators.
\[pr4-1\] For given $t>0$, $u\in C(M,\mathbb{R}^1)$ and $x\in M$, there exist $\sigma\in[t,2t]$ and a minimizing extremal curve $\gamma:[0,\sigma]\rightarrow
M$ such that $\gamma(\sigma)=x$ and
$$\tilde{T}^a_tu(x)=u(\gamma(0))+\int_0^\sigma L_a(\gamma,\dot{\gamma})ds.$$
Since $\sigma\mapsto T^a_\sigma u(x)$ is continuous on $[t,2t]$ and $\tilde{T}^a_tu(x)=\inf_{t\leq \sigma\leq 2t}T^a_\sigma u(x)$, then there is $\sigma_0\in[t,2t]$ such that $\tilde{T}^a_tu(x)=T^a_{\sigma_0}u(x)$. From the property of the operator $T^a_{\sigma_0}$ (see [@Fat-b Lemma 4.4.1]), there exists a minimizing extremal curve $\gamma:[0,\sigma_0]\rightarrow M$ such that $\gamma(\sigma_0)=x$ and
$$\tilde{T}^a_tu(x)=T^a_{\sigma_0}u(x)=u(\gamma(0))+\int_0^{\sigma_0}L_a(\gamma,\dot{\gamma})ds.$$
Some fundamental properties of $\tilde{T}^a_t$ are discussed in the following proposition.
\[pr4-2\]
- For $u$, $v\in C(M,\mathbb{R}^1)$, if $u\leq
v$, then $\tilde{T}^a_tu\leq\tilde{T}^a_tv$, $\forall t\geq
0$.
- If $c$ is a constant and $u\in C(M,\mathbb{R}^1)$, then $\tilde{T}^a_t(u+c)=\tilde{T}^a_tu+c$, $\forall t\geq
0$.
- For each $u$, $v\in C(M,\mathbb{R}^1)$ and each $t\geq 0$, $\|\tilde{T}^a_tu-\tilde{T}^a_tv\|_\infty\leq\|u-v\|_\infty$.
- For each $u\in C(M,\mathbb{R}^1)$, $\lim_{t\rightarrow
0^+}\tilde{T}^a_tu=u$.
- For each $u\in C(M,\mathbb{R}^1)$, $(t,x)\mapsto\tilde{T}^a_tu(x)$ is continuous on $[0,+\infty)\times
M$.
The property (3) means that the semigroup $\{\tilde{T}^a_t\}_{t\geq 0}$ is continuous at the origin or of class $C_0$ [@Kel].
Since $T^a_t$ has the monotonicity property (see [@Fat-b Corollary 4.4.4]), then
$$\tilde{T}^a_tu(x)=\inf_{t\leq \sigma\leq 2t}T^a_\sigma u(x)\leq\inf_{t\leq
\sigma\leq 2t}T^a_\sigma v(x)=\tilde{T}^a_tv(x), \quad \forall t>0, \ \forall
x\in M,$$ i.e., (1) holds. (2) results from the definition of $\tilde{T}^a_t$ directly. Note that for any $x\in M$,
$$-\|u-v\|_\infty+v(x)\leq u(x)\leq\|u-v\|_\infty+v(x).$$ By the properties of $T^a_\sigma$ (see [@Fat-b Corollary 4.4.4]), for each $t\geq 0$ we have
$$T^a_\sigma v(x)-\|u-v\|_\infty\leq T^a_\sigma u(x)\leq
T^a_\sigma v(x)+\|u-v\|_\infty, \quad \forall \sigma\in[t,2t].$$ Taking the infimum on $\sigma$ over $[t,2t]$ yields
$$\inf_{t\leq \sigma\leq 2t}T^a_\sigma v(x)-\|u-v\|_\infty\leq\inf_{t\leq \sigma\leq
2t}T^a_\sigma u(x)\leq\inf_{t\leq \sigma\leq 2t}T^a_\sigma v(x)+\|u-v\|_\infty,
\quad \forall x\in M,$$ and thus (3) holds.
Next we prove (4). For each $u\in C(M,\mathbb{R}^1)$ and each $\varepsilon>0$, there is $w\in C^1(M,\mathbb{R}^1)$ such that $\|u-w\|_\infty<\varepsilon$ since $C^1(M,\mathbb{R}^1)$ is a dense subset of $C(M,\mathbb{R}^1)$ in the topology of uniform convergence. Thus, we have
$$\begin{aligned}
\label{4-1}
\begin{split}
\|\tilde{T}^a_tu-u\|_\infty &
\leq\|\tilde{T}^a_tu-\tilde{T}^a_tw\|_\infty
+\|\tilde{T}^a_tw-w\|_\infty+\|w-u\|_\infty\\
&\leq 2\|w-u\|_\infty+\|\tilde{T}^a_tw-w\|_\infty\\
&\leq 2\varepsilon+\|\tilde{T}^a_tw-w\|_\infty, \quad \forall
t\geq 0,
\end{split}\end{aligned}$$
where we have used (3). Since $M$ is compact, then $w$ is Lipschitz. Denote the Lipschitz constant of $w$ by $K_w$, and by the superlinearity of $L_a$ there exists $C_{K_w}\in\mathbb{R}^1$ such that
$$L_a(x,v)\geq K_w\|v\|_x+C_{K_w}, \quad \forall(x,v)\in TM.$$
For each $x\in M$, each $t\geq 0$ and each continuous and piecewise $C^1$ path $\gamma:[0,\sigma]\to M$ with $\gamma(\sigma)=x$ and $t\leq \sigma\leq
2t$, since
$$d(\gamma(0),\gamma(\sigma))\leq\int_0^\sigma\|\dot{\gamma}(s)\|_{\gamma(s)}ds,$$ then
$$\int_0^\sigma L_a(\gamma,\dot{\gamma})ds\geq
K_wd(\gamma(0),\gamma(\sigma))+C_{K_w}\sigma\geq
w(\gamma(\sigma))-w(\gamma(0))+C_{K_w}\sigma.$$ Thus, by the definition of $T^a_\sigma$ we have
$$T^a_\sigma w(x)\geq w(x)+C_{K_w}\sigma.$$ Taking the infimum on $\sigma$ over $[t,2t]$ on both sides of this last inequality yields
$$\begin{aligned}
\label{4-2}
\tilde{T}^a_tw(x)\geq w(x)+O(t), \quad \mathrm{as}\ t\rightarrow
0^+,\end{aligned}$$
where $O(t)$ is independent of $x$. Using the constant curve $\gamma_x:[0,\sigma]\rightarrow M$, $s\mapsto x$, we have
$$T^a_\sigma w(x)\leq w(x)+L_a(x,0)\sigma.$$ Taking the infimum on $\sigma$ over $[t,2t]$, we obtain
$$\begin{aligned}
\label{4-3}
\tilde{T}^a_tw(x)\leq w(x)+O(t), \quad \mathrm{as}\ t\rightarrow
0^+,\end{aligned}$$
where $O(t)$ is independent of $x$. Combining (\[4-1\]), (\[4-2\]) and (\[4-3\]), we have
$$\lim_{t\rightarrow 0^+}\|\tilde{T}^a_tu-u\|_\infty=0,$$ i.e., (4) holds.
Finally, we prove (5). For any $(t_0,x_0)\in[0,+\infty)\times M$, from the semigroup property and (3) we have
$$\begin{aligned}
\label{4-4}
\begin{split}
|\tilde{T}^a_tu(x)-\tilde{T}^a_{t_0}u(x_0)|&
\leq|\tilde{T}^a_tu(x)-\tilde{T}^a_tu(x_0)|+|\tilde{T}^a_tu(x_0)-\tilde{T}^a_{t_0}u(x_0)|\\
&\leq
|\tilde{T}^a_tu(x)-\tilde{T}^a_tu(x_0)|+\|\tilde{T}^a_tu-\tilde{T}^a_{t_0}u\|_\infty\\
&\leq|\tilde{T}^a_tu(x)-\tilde{T}^a_tu(x_0)|+\|\tilde{T}^a_{|t-t_0|}u-u\|_\infty.
\end{split}\end{aligned}$$
From (\[4-4\]), $\tilde{T}^a_tu\in C(M,\mathbb{R}^1)$ and (4), we conclude that (5) holds.
The proposition below establishs a relationship between $\tilde{T}^a_t$ and $T^a_t$.
\[pr4-3\]
- For each $u\in C(M,\mathbb{R}^1)$, the uniform limit $\lim_{t\rightarrow+\infty}\tilde{T}^a_tu$ exists and $$\lim_{t\rightarrow+\infty}\tilde{T}^a_tu=\lim_{t\rightarrow+\infty}T^a_tu=\bar{u}.$$
- For each $t\geq 0$ and each $u\in C(M,\mathbb{R}^1)$, $\|\tilde{T}^a_tu-\bar{u}\|_\infty\leq\|T^a_tu-\bar{u}\|_\infty.$
- $u\in C(M,\mathbb{R}^1)$ is a fixed point of $\{\tilde{T}^a_t\}_{t\geq
0}$ if and only if it is a fixed point of $\{T^a_t\}_{t\geq0}.$
From (1) $\lim_{t\rightarrow+\infty}\tilde{T}^a_tu$ exists and is a backward weak KAM solution of the Hamilton-Jacobi equation $H_a(x,u_x)=0$. (2) essentially says that the new L-O semigroup converges faster than the L-O semigroup. (3) implies that $u\in
C(M,\mathbb{R}^1)$ is a backward weak KAM solution if and only if it is a fixed point of $\{\tilde{T}^a_t\}_{t\geq 0}$.
\[re4-1\] Just as we mentioned earlier, for each $\tau\in[0,1]$ and each $u\in
C(M,\mathbb{R}^1)$, the uniform limit $\lim_{n\to+\infty}\tilde{T}_n^{a,\tau}u$ exists and
$$\lim_{n\to+\infty}\tilde{T}_n^{a,\tau}u=\lim_{n\to+\infty}T_n^au=\bar{u}.$$ It can be proved by slight modifications of the proof of (1) in Proposition \[pr4-3\].
First we prove (1). Assume by contradiction that there exist $\varepsilon_0>0$, $t_n\rightarrow+\infty$ and $x_n\in M$ such that
$$|\tilde{T}^a_{t_n}u(x_n)-\bar{u}(x_n)|\geq\varepsilon_0.$$ From the compactness of $M$, without loss of generality we assume that $x_n\rightarrow x_0$, $n\rightarrow+\infty$. In view of the definition of $\tilde{T}^a_t$, there exist $\sigma_n\in[t_n,2t_n]$ such that
$$|T^a_{\sigma_n}u(x_n)-\bar{u}(x_n)|\geq\varepsilon_0.$$ Let $n\rightarrow+\infty$. Since $(\sigma,x)\mapsto T^a_\sigma u(x)$ is continuous, then we have
$$\lim_{\sigma\rightarrow+\infty}T^a_\sigma u(x_0)\neq\bar{u}(x_0),$$ which contradicts $\lim_{\sigma\rightarrow+\infty}T^a_\sigma u=\bar{u}$.
Next we show (2). For each $t\geq 0$ and each $x\in M$, there exists $t\leq \sigma_x\leq 2t$ such that
$$|\tilde{T}^a_tu(x)-\bar{u}(x)|=|T^a_{\sigma_x}u(x)-\bar{u}(x)|.$$ Since $\bar{u}$ is a fixed point of $\{T^a_t\}_{t\geq 0}$, then we have $|T^a_{\sigma_x}u(x)-\bar{u}(x)|=|T^a_{\sigma_x}u(x)-T^a_{\sigma_x}\bar{u}(x)|\leq
\|T^a_{\sigma_x}u-T^a_{\sigma_x}\bar{u}\|_\infty=\|T^a_{\sigma_x-t}\circ
T^a_tu-T^a_{\sigma_x-t}\circ T^a_t\bar{u}\|_\infty\leq
\|T^a_tu-T^a_t\bar{u}\|_\infty=\|T^a_tu-\bar{u}\|_\infty$, where we have used the non-expansiveness property of $T^a_{\sigma_x-t}$ (see [@Fat-b Corollary 4.4.4]). Hence (2) holds.
At last, we show (3). Suppose that $u$ is a fixed point of $\{T^a_t\}_{t\geq 0}$, i.e., $T^a_tu=u$, $\forall t\geq 0$. Then $\lim_{t\rightarrow+\infty}T^a_tu=u$. From (2) we have
$$\|\tilde{T}^a_tu-u\|_\infty\leq\|T^a_tu-u\|_\infty=0, \quad
\forall t\geq 0,$$ which implies that $u$ is a fixed point of $\{\tilde{T}^a_t\}_{t\geq 0}$. Suppose conversely that $u$ is a fixed point of $\{\tilde{T}^a_t\}_{t\geq 0}$. Then from (1) $\lim_{t\rightarrow+\infty}\tilde{T}^a_tu=u=\lim_{t\rightarrow+\infty}T^a_tu$. Hence $u$ is a backward weak KAM solution of $H_a(x,u_x)=0$ and a fixed point of $\{T^a_t\}_{t\geq 0}$.
Rates of convergence of the L-O semigroup and the new L-O semigroup
-------------------------------------------------------------------
Recall the $C^2$ positive definite and superlinear Lagrangian (\[1-6\])
$$\begin{aligned}
L^1_a(x,v)=\frac{1}{2}\langle
A(x)(v-\omega),(v-\omega)\rangle+f(x,v-\omega), \quad x\in
\mathbb{T}^n,\ v\in\mathbb{R}^n.\end{aligned}$$
The conjugated Hamiltonian $H^1_a:\mathbb{T}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^1$ of $L_a^1$ has the following form
$$H^1_a(x,p)=\langle\omega,p\rangle+\frac{1}{2}\langle
A^{-1}(x)p,p\rangle+g(x,p),$$ where $g(x,p)=O(\|p\|^3)$ as $p\rightarrow 0$. It is clear that $H^1_a(x,0)=0$ and thus $w\equiv const.$ is a smooth viscosity solution of the corresponding Hamilton-Jacobi equation $H^1_a(x,u_x)=0$. In view of the Legendre transform,
$$L^1_a(x,v)=L^1_a(x,v)-\langle
w_x,v\rangle\geq-H^1_a(x,w_x)=-H^1_a(x,0)=0, \quad \forall
(x,v)\in\mathbb{T}^n\times\mathbb{R}^n.$$ Furthermore, if $(x,v)\in\tilde{\mathcal{M}}_0=\cup_{x\in\mathbb{T}^n}(x,\omega)$, then $w_x=\frac{\partial L}{\partial v}(x,v)$ (see [@Fat-b Theorem 4.8.3]), from which we have
$$L^1_a(x,v)=L^1_a(x,v)-\langle
w_x,v\rangle=-H^1_a(x,w_x)=-H^1_a(x,0)=0.$$ Hence
$$L^1_a\geq 0, \quad \forall (x,v)\in\mathbb{T}^n\times\mathbb{R}^n$$ and in particular,
$$L^1_a|_{\cup_{x\in\mathbb{T}^n}(x,\omega)}=0.$$
For each $u\in C(\mathbb{T}^n,\mathbb{R}^1)$, because of $c(L^1_a)=0$ we have $\lim_{t\rightarrow+\infty}T_t^au=\bar{u}$. Note that both $w\equiv const.$ and $\bar{u}$ are viscosity solutions of $H^1_a(x,u_x)=0$. Hence $\bar{u}\equiv const.$ since the viscosity solution of $H^1_a(x,u_x)=0$ is unique up to constants when $\mathcal{A}_0=\mathbb{T}^n$ [@Lia], where $\mathcal{A}_0$ is the projected Aubry set.
### Rate of convergence of the L-O semigroup
We present here the proof of Theorem \[th2\]. For this, the following lemma is needed.
\[le4-1\] For each $u\in C(\mathbb{T}^n,\mathbb{R}^1)$, $\bar{u}\equiv\min_{x\in\mathbb{T}^n}u(x)$.
For any $x\in\mathbb{T}^n$, from the definition of $T^a_t$ we have
$$\bar{u}(x)=\lim_{t\rightarrow+\infty}T^a_tu(x)
=\lim_{t\rightarrow+\infty}\inf_{z\in\mathbb{T}^n}
\{u(z)+\int_0^tL^1_a(\gamma_z,\dot{\gamma}_z)ds\},$$ where $\gamma_z:[0,t]\rightarrow\mathbb{T}^n$ is a Tonelli minimizer with $\gamma_z(0)=z$, $\gamma_z(t)=x$. Since $L^1_a\geq
0$, then $\bar{u}(x)\geq\min_{z\in\mathbb{T}^n}u(z)$ and therefore it suffices to show that $\bar{u}(x)\leq\min_{z\in\mathbb{T}^n}u(z)$.
Take $y\in\mathbb{T}^n$ with $u(y)=\min_{z\in\mathbb{T}^n}u(z)$. Consider the following two curves
$$\gamma_\omega:[0,t]\rightarrow\mathbb{T}^n,\ s\mapsto \omega s+y$$ and
$$\gamma_{\omega'}:[0,t]\rightarrow\mathbb{T}^n,\ s\mapsto \omega'
s+y$$ with $\gamma_{\omega'}(t)=x$, where $\omega'\in\mathbb{S}^{n-1}$ and $t>0$. It is clear that $\gamma_{\omega'}$ is a curve in $\mathbb{T}^n$ connecting $y$ and $x$. Let $\Delta=\gamma_{\omega'}(t)-\gamma_\omega(t)=x-(\omega t+y)$. Then $\|\Delta\|\leq\frac{\sqrt{n}}{2}$ and $\dot{\gamma}_{\omega'}\equiv\omega'=\frac{\Delta}{t}+\omega$. Therefore, we have
$$\begin{aligned}
T^a_tu(x) & \leq
u(\gamma_{\omega'}(0))+\int_0^tL^1_a(\gamma_{\omega'},\dot{\gamma}_{\omega'})ds\\
& = u(y)+\int_0^t\Big(\frac{1}{2}\langle
A(\gamma_{\omega'})(\omega'-\omega),(\omega'-\omega)\rangle+f(\gamma_{\omega'},\omega'-\omega)\Big)ds\\
& = u(y)+\int_0^t\Big(\frac{1}{2}\Big\langle
A(\gamma_{\omega'})\frac{\Delta}{t},\frac{\Delta}{t}\Big\rangle+f(\gamma_{\omega'},\frac{\Delta}{t})\Big)ds\\
& \leq u(y)+\frac{C}{t}+O(\frac{1}{t^2}),\end{aligned}$$
where $C$ is a constant, which depends only on $n$.
From the arguments above we know that for any $\varepsilon>0$, there exists $T>0$ such that for any $t>T$ there exists $\gamma_{\omega'}:[0,t]\rightarrow\mathbb{T}^n$ with $\gamma_{\omega'}(t)=x$, and
$$T^a_tu(x)\leq
u(\gamma_{\omega'}(0))+\int_0^tL^1_a(\gamma_{\omega'},\dot{\gamma}_{\omega'})ds\leq
\min_{z\in\mathbb{T}^n}u(z)+\varepsilon.$$ Hence $\bar{u}(x)=\lim_{t\rightarrow+\infty}T^a_tu(x)\leq\min_{z\in\mathbb{T}^n}u(z)$.
*Proof of Theorem \[th2\].* In order to prove our result, it is sufficient to show that for each $u\in
C(\mathbb{T}^n,\mathbb{R}^1)$, there exists a constant $K>0$ such that the following two inequalities hold.
$$T^a_tu(x)-\bar{u}(x)\leq \frac{K}{t}, \quad \forall t>0,\ \forall
x\in\mathbb{T}^n; \eqno (\mathrm{I1})$$
$$\bar{u}(x)-T^a_tu(x)\leq \frac{K}{t}, \quad \forall t>0,\ \forall
x\in\mathbb{T}^n. \eqno (\mathrm{I2})$$ Obviously, (I2) holds. In fact, for each $t>0$ and each $x\in\mathbb{T}^n$, from the definition of $T^a_t$ we have
$$T^a_tu(x) =\inf_{z\in\mathbb{T}^n}
\{u(z)+\int_0^tL^1_a(\gamma_z,\dot{\gamma}_z)ds\},$$ where $\gamma_z:[0,t]\rightarrow\mathbb{T}^n$ is a Tonelli minimizer with $\gamma_z(0)=z$, $\gamma_z(t)=x$. In view of $L^1_a\geq 0$ and Lemma \[le4-1\], we have
$$T^a_tu(x) =\inf_{z\in\mathbb{T}^n}
\{u(z)+\int_0^tL^1_a(\gamma_z,\dot{\gamma}_z)ds\}\geq\min_{z\in\mathbb{T}^n}u(z)=\bar{u}(x).$$ Thus $\bar{u}(x)-T^a_tu(x)\leq 0$, $\forall t>0,\ \forall
x\in\mathbb{T}^n$ and (I2) holds.
Next we prove (I1). It suffices to show that there exists a constant $C>0$ such that for sufficiently large $t>0$,
$$\begin{aligned}
\label{4-5}
T^a_tu(x)-\bar{u}(x)\leq \frac{C}{t}, \quad \forall
x\in\mathbb{T}^n,\end{aligned}$$
where $C$ depends only on $n$. In deed, since $(s,z)\mapsto
T_su(z)$ is continuous on $[0,\infty)\times\mathbb{T}^n$, if (\[4-5\]) holds, then there exists a constant $K>0$ such that
$$T^a_tu(x)-\bar{u}(x)\leq\frac{K}{t}, \quad \forall t>0,\ \forall
x\in\mathbb{T}^n,$$ where $K$ depends only on $n$ and $u$.
Take $y\in\mathbb{T}^n$ with $u(y)=\min_{z\in\mathbb{T}^n}u(z)$. Let us consider the following curve in $\mathbb{T}^n$
$$\gamma_\omega:[0,t]\rightarrow\mathbb{T}^n,\ s\mapsto \omega s+y,$$ where $t>0$. Then for each $x\in\mathbb{T}^n$, let
$$\gamma_{\omega'}:[0,t]\rightarrow\mathbb{T}^n,\ s\mapsto \omega'
s+y$$ be a curve in $\mathbb{T}^n$ connecting $y$ and $x$, where $\omega'\in\mathbb{S}^{n-1}$. Let $\Delta=\gamma_{\omega'}(t)-\gamma_\omega(t)=x-(\omega t+y)$. Then $\|\Delta\|\leq\frac{\sqrt{n}}{2}$ and $\dot{\gamma}_{\omega'}\equiv\omega'=\frac{\Delta}{t}+\omega$. Hence,
$$\begin{aligned}
T^a_tu(x) & \leq
u(\gamma_{\omega'}(0))+\int_0^tL^1_a(\gamma_{\omega'},\dot{\gamma}_{\omega'})ds\\
& = u(y)+\int_0^t\Big(\frac{1}{2}\langle
A(\gamma_{\omega'})(\omega'-\omega),(\omega'-\omega)\rangle+f(\gamma_{\omega'},\omega'-\omega)\Big)ds\\
& = u(y)+\int_0^t\Big(\frac{1}{2}\Big\langle
A(\gamma_{\omega'})\frac{\Delta}{t},\frac{\Delta}{t}\Big\rangle+f(\gamma_{\omega'},\frac{\Delta}{t})\Big)ds\\
& \leq u(y)+\frac{C_1}{t}+O(\frac{1}{t^2}),\end{aligned}$$
where $C_1$ is a constant which depends only on $n$. From Lemma \[le4-1\], we have $T^a_tu(x)-\bar{u}(x)\leq\frac{C}{t}$ for $t>0$ large enough, where $C$ is a constant which still depends only on $n$, i.e., (\[4-5\]) holds. $\Box$
### Rate of convergence of the new L-O semigroup
To complete the proof of Theorem \[th3\], we review preliminaries on the ergodization rate for linear flows on the torus $\mathbb{T}^n$, i.e., the rate at which the image of a point fills the torus when subjected to linear flows. There is a direct relationship between the rate of convergence of the new L-O semigroup and the ergodization rate for linear flows on the torus $\mathbb{T}^n$. Let us recall the following result of Dumas’ [@Dum] concerning the estimate of ergodization time.
For each $t\in\mathbb{R}^1$ and each $\omega\in\mathbb{S}^{n-1}$, consider the one-parameter family of translation maps $\omega_t:\mathbb{T}^n\rightarrow\mathbb{T}^n$, $x\mapsto x+\omega
t$. A rectilinear orbit of $\mathbb{T}^n$ with direction vector $\omega$ and initial condition $x$ is defined as the image of $x$ under the linear flow $\omega_t$ over some closed interval $[t_0,t_1]\subset\mathbb{R}^1$, i.e.,
$$\bigcup_{t_0\leq t\leq t_1}\omega_t(x).$$
Given $R>0$, the direction vector $\omega\in\mathbb{S}^{n-1}$ is said to ergodize $\mathbb{T}^n$ to within $R$ after time $T$ if
$$\begin{aligned}
\label{4-6}
\bigcup_{0\leq t\leq T}\omega_t(B_R(x))=\mathbb{T}^n\end{aligned}$$
for all $x\in\mathbb{T}^n$.
As defined in the Introduction, for $\rho>n-1$ and $\alpha>0$,
$$\mathcal{D}(\rho,\alpha)=\Big\{\beta\in \mathbb{S}^{n-1}|\
|\langle\beta,k\rangle|>\frac{\alpha}{|k|^\rho},\ \forall
k\in\mathbb{Z}^n\backslash\{0\}\Big\},$$ whose elements can not be approximated by rationals too rapidly.
\[th5\] Let $0<R\leq 1$. Given any highly nonresonant direction vector $\omega\in\mathcal{D}(\rho,\alpha)$, rectilinear orbits of $\mathbb{T}^n$ with direction vector $\omega$ will ergodize $\mathbb{T}^n$ to within $R$ after time T, where
$$T=\frac{2\|V_*\|_\triangle}{\alpha\pi R^{\rho+n/2}}$$ is independent of $\omega$.
The constant $\|V_*\|_\triangle$ is a Sobolev norm of a certain “smoothest test function" and it depends only on $n$ and $\rho$. See [@Dum] for complete details.
We are now in a position to give the proof of Theorem \[th3\].
*Proof of Theorem \[th3\].* Our purpose is to show that for each $u\in C(\mathbb{T}^n,\mathbb{R}^1)$, there exists a constant $\tilde{K}>0$ such that the following two inequalities hold.
$$\tilde{T}^a_tu(x)-\bar{u}(x)\leq
\tilde{K}t^{-(1+\frac{4}{2\rho+n})}, \quad \forall t>0,\ \forall
x\in\mathbb{T}^n; \eqno (\mathrm{I3})$$
$$\bar{u}(x)-\tilde{T}^a_tu(x)\leq
\tilde{K}t^{-(1+\frac{4}{2\rho+n})}, \quad \forall t>0,\ \forall
x\in\mathbb{T}^n. \eqno (\mathrm{I4})$$
First we show (I4). For each $t>0$ and each $x\in\mathbb{T}^n$, by the definition of $\tilde{T}^a_t$ we have
$$\tilde{T}^a_tu(x) =\inf_{t\leq \sigma\leq
2t}\inf_{z\in\mathbb{T}^n} \{u(z)+\int_0^\sigma
L^1_a(\gamma_z,\dot{\gamma}_z)ds\},$$ where $\gamma_z:[0,\sigma]\rightarrow\mathbb{T}^n$ is a Tonelli minimizer with $\gamma_z(0)=z$, $\gamma_z(\sigma)=x$. In view of $L^1_a\geq 0$ and Lemma \[le4-1\], we have
$$\tilde{T}^a_tu(x) =\inf_{t\leq \sigma\leq
2t}\inf_{z\in\mathbb{T}^n} \{u(z)+\int_0^\sigma
L^1_a(\gamma_z,\dot{\gamma}_z)ds\}\geq\min_{z\in\mathbb{T}^n}u(z)=\bar{u}(x).$$ Thus $\bar{u}(x)-\tilde{T}^a_tu(x)\leq 0$, $\forall t>0,\ \forall
x\in\mathbb{T}^n$, i.e., (I4) holds.
Then it remains to show (I3). When $R=1$, according to Theorem \[th5\] the ergodization time $T=\frac{2\|V_*\|_\triangle}{\alpha\pi}$. For any $t\geq T$, let $R_t=\sqrt[\rho+n/2]{\frac{2\|V_*\|_\triangle}{\alpha\pi t}}$. Then $0<R_t\leq 1$.
Take $y\in\mathbb{T}^n$ with $u(y)=\min_{z\in\mathbb{T}^n}u(z)$. Let $y_t=\omega_t(y)=\omega t+y$. For $R_t$ defined above, since $\omega\in\mathcal{D}(\rho,\alpha)$, then from Theorem \[th5\] and (\[4-6\]) we have
$$\bigcup_{0\leq\varsigma\leq
t}\omega_\varsigma(B_{R_t}(y_t))=\mathbb{T}^n.$$
Therefore, for each $x\in\mathbb{T}^n$, there exists $0\leq\varsigma'\leq t$ such that $d_{\mathbb{T}^n}(\omega_{\varsigma'}(y_t),x)\leq R_t$, i.e., $d_{\mathbb{T}^n}(\omega(t+\varsigma')+y,x)\leq R_t$. Equivalently this means that there exists $t\leq \sigma'\leq 2t$ such that
$$d_{\mathbb{T}^n}(\omega \sigma'+y,x)\leq R_t,$$ where $\sigma'=t+\varsigma'$. Consider the following curve in $\mathbb{T}^n$
$$\gamma_{\omega'}:[0,\sigma']\rightarrow\mathbb{T}^n,\ s\mapsto
\omega's+y$$ with $\gamma_{\omega'}(\sigma')=x$, where $\omega'\in\mathbb{S}^{n-1}$. It is clear that $\gamma_{\omega'}$ connects $y$ and $x$. Let $\Delta=\gamma_{\omega'}(\sigma')-\omega_{\sigma'}(y)=x-(\omega
\sigma'+y)$. Then $\|\Delta\|=d_{\mathbb{T}^n}(x,\omega
\sigma'+y)\leq R_t$ and $\dot{\gamma}_{\omega'}\equiv\omega'=\frac{\Delta}{\sigma'}+\omega$. Hence we have
$$\begin{aligned}
\tilde{T}^a_tu(x)-\bar{u}(x) & \leq u(\gamma_{\omega'}(0))+\int_0^{\sigma'}L^1_a(\gamma_{\omega'},\dot{\gamma}_{\omega'})ds-\bar{u}(x)\\
& =\int_0^{\sigma'}\Big(\frac{1}{2}\langle
A(\gamma_{\omega'})(\omega'-\omega),(\omega'-\omega)\rangle+f(\gamma_{\omega'},\omega'-\omega)\Big)ds\\
&\leq \frac{CR_t^2}{t}\end{aligned}$$
for sufficiently large $t>0$ and some constant $C>0$. Since $R_t^2=(\frac{2\|V_*\|_\triangle}{\alpha\pi
t})^{\frac{2}{\rho+n/2}}$, then for $t>0$ large enough we have
$$\tilde{T}^a_tu(x)-\bar{u}(x)\leq C_1t^{-(1+\frac{4}{2\rho+n})},
\quad \forall x\in\mathbb{T}^n,$$ where $C_1$ is a constant which depends only on $n$, $\rho$ and $\alpha$. From (5) of Proposition \[pr4-2\], $(s,z)\mapsto
\tilde{T}^a_s u(z)$ is continuous on $[0,\infty)\times\mathbb{T}^n$. Hence there exists a constant $\tilde{K}>0$ such that
$$\tilde{T}^a_tu(x)-\bar{u}(x)\leq
\tilde{K}t^{-(1+\frac{4}{2\rho+n})}, \quad \forall t>0,\ \forall
x\in\mathbb{T}^n,$$ where $\tilde{K}$ depends only on $n$, $\rho$, $\alpha$ and $u$, i.e., (I3) holds. $\Box$
### An example
\[ex1\] Consider the following integrable $C^2$ Lagrangian
$$\bar{L}^1_a(x,v)=\frac{1}{2}\langle v-\omega,v-\omega\rangle,
\quad x\in\mathbb{T}^n,\ v\in\mathbb{R}^n,\
\omega\in\mathbb{S}^{n-1}.$$
It is easy to see that $\bar{L}^1_a$ is a special case of $L^1_a$. For $\bar{L}^1_a$, we show that there exist $u\in
C(\mathbb{T}^n,\mathbb{R}^1)$, $x^0\in\mathbb{T}^n$ and $t_m\to+\infty$ as $m\to+\infty$ such that
$$|T^a_{t_m}u(x^0)-\bar{u}(x^0)|=O(\frac{1}{t_m}), \quad
m\to+\infty,$$ which implies that the result of Theorem \[th2\] is sharp in the sense of order.
Recall the universal covering projection $\pi:\mathbb{R}^n\to\mathbb{T}^n$. Let $x^0\in\mathbb{T}^n$ such that each point $\tilde{x}^0\in\mathbb{R}^n$ in the fiber over $x^0$ ($\pi\tilde{x}^0=x^0$) is the center of each fundamental domain in $\mathbb{R}^n$. Define a continuous function on $\mathbb{R}^n$ as follows: for $\tilde{x}\in\mathbb{R}^n$
$$\tilde{u}(\tilde{x})=\left\{
\begin{array}{ll}
\delta-\|\tilde{x}-\tilde{x}^0\|, &
\|\tilde{x}-\tilde{x}^0\|\leq\delta,\\
0, & \mathrm{otherwise},
\end{array}
\right.$$ where $0<\delta<\frac{1}{2}$. We then define a continuous function on $\mathbb{T}^n$ as $u(x)=\tilde{u}(\tilde{x})$ for all $x\in\mathbb{T}^n$, where $\tilde{x}$ is an arbitrary point in the fiber over $x$. Thus, from Lemma \[le4-1\], $\bar{u}\equiv\min_{x\in\mathbb{T}^n}u(x)=0$.
Now fix a point $\tilde{x}^0_0$ in the fiber over $x^0$. Then there exist $\{\tilde{x}^0_m\}_{m=1}^{+\infty}$ in the fiber over $x^0$ and $t_m\to+\infty$ as $m\to+\infty$ such that $\|(\tilde{x}^0_m-\omega
t_m)-\tilde{x}^0_0\|\leq\frac{\delta}{2}$. Let $\tilde{z}_m=\tilde{x}^0_m-\omega t_m$. Then $\|\tilde{z}_m-\tilde{x}^0_0\|\leq\frac{\delta}{2}$. For each $t_m$ there exists $y_m\in\mathbb{T}^n$ such that
$$T^a_{t_m}u(x^0)=u(y_m)+\int_0^{t_m}\bar{L}^1_a(\gamma_{y_m},\dot{\gamma}_{y_m})ds,$$ where $\gamma_{y_m}:[0,t_m]\to\mathbb{T}^n$ is a Tonelli minimizer with $\gamma_{y_m}(0)=y_m$, $\gamma_{y_m}(t_m)=x^0$. In view of the lifting property of the covering projection, there is a unique curve $\tilde{\gamma}_{y_m}:[0,t_m]\to\mathbb{R}^n$ with $\pi\tilde{\gamma}_{y_m}=\gamma_{y_m}$ and $\tilde{\gamma}_{y_m}(t_m)=\tilde{x}^0_m$. Set $\tilde{y}_m=\tilde{\gamma}_{y_m}(0)$. Then $\pi\tilde{y}_m=y_m$. Moreover, $\tilde{\gamma}_{y_m}$ has the following form
$$\tilde{\gamma}_{y_m}(s)=\omega's+\tilde{y}_m, \quad s\in[0,t_m],$$ where $\omega'\in\mathbb{S}^{n-1}$. It is clear that $\tilde{\gamma}_{y_m}(0)=\tilde{y}_m$ and $\tilde{y}_m=\tilde{x}^0_m-\omega't_m$.
If $\|\tilde{y}_m-\tilde{z}_m\|\leq\frac{\delta}{4}$, then from $\|\tilde{z}_m-\tilde{x}^0_0\|\leq\frac{\delta}{2}$ we have $\|\tilde{y}_m-\tilde{x}^0_0\|\leq\frac{3\delta}{4}$. Hence,
$$\begin{aligned}
\label{4-7}
\begin{split}
T^a_{t_m}u(x^0)&=u(y_m)+\int_0^{t_m}\bar{L}^1_a(\gamma_{y_m},\dot{\gamma}_{y_m})ds\\
&\geq\tilde{u}(\tilde{y}_m)\geq\delta-\frac{3\delta}{4}=\frac{\delta}{4}.
\end{split}\end{aligned}$$
From (\[4-7\]), we may deduce that there can only be a finite number of $\tilde{y}_m$’s such that $\|\tilde{y}_m-\tilde{z}_m\|\leq\frac{\delta}{4}$. For, otherwise, there would be $\{t_{m_i}\}_{i=1}^{+\infty}$ and $\{\tilde{y}_{m_i}\}_{i=1}^{+\infty}$ such that
$$T^a_{t_{m_i}}u(x^0)\geq\frac{\delta}{4}, \quad i=1,2,\cdots,$$ which contradicts $\lim_{i\to+\infty}T^a_{t_{m_i}}u(x^0)=\bar{u}(x^0)=0$.
For $\tilde{y}_m$ with $\|\tilde{y}_m-\tilde{z}_m\|>\frac{\delta}{4}$, we have
$$\frac{\delta}{4}<\|\tilde{y}_m-\tilde{z}_m\|=\|\tilde{x}_m^0-\omega't_m-(\tilde{x}_m^0-\omega
t_m)\|=\|\omega-\omega'\|t_m.$$ Thus,
$$\begin{aligned}
\label{4-8}
\begin{split}
T^a_{t_m}u(x^0)&=u(y_m)+\int_0^{t_m}\bar{L}^1_a(\gamma_{y_m},\dot{\gamma}_{y_m})ds\\
&\geq\frac{1}{2}t_m\|\omega-\omega'\|^2=\frac{1}{2}\frac{t^2_m\|\omega-\omega'\|^2}{t_m}\geq\frac{\delta^2}{32t_m}.
\end{split}\end{aligned}$$
Therefore, from (\[4-8\]) and Theorem \[th2\] we have
$$|T^a_{t_m}u(x^0)-\bar{u}(x^0)|=|T^a_{t_m}u(x^0)|=O(\frac{1}{t_m}),
\quad m\to+\infty.$$
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---
abstract: 'The mixing set with a knapsack constraint arises as a substructure in mixed-integer programming reformulations of chance-constrained programs with stochastic right-hand-sides over a finite discrete distribution. Recently, Luedtke et al. (2010) and Küçükyavuz (2012) studied valid inequalities for such sets. However, most of their results were focused on the equal probabilities case (equivalently when the knapsack reduces to a cardinality constraint), with only minor results in the general case. In this paper, we focus on the general probabilities case (general knapsack constraint). We characterize the valid inequalities that do not come from the knapsack polytope and use this characterization to generalize the inequalities previously derived for the equal probabilities case. We also show that one can separate over a large class of inequalities in polynomial time.'
author:
- |
Ahmad Abdi and Ricardo Fukasawa\
Department of Combinatorics and Optimization†\
University of Waterloo\
`{a3abdi,rfukasaw}@math.uwaterloo.ca`
title: On the Mixing Set with a Knapsack Constraint
---
Introduction {#intro}
============
Many optimization problems in real world applications allow to some extent a number of violated constraints, which results in a decrease in the quality of service (QoS) and also a decrease in the cost of production. These optimization problems have been a main motive to study probabilistic (in particular, chance-constrained) programming. A difficulty when dealing with these optimization problems is that the feasible region is not necessarily convex. In this paper, we consider mixed-integer programming (MIP) reformulations of chance-constrained programs with joint probabilistic constraints in which the right-hand-side vector is random with a finite discrete distribution. This model was first proposed in Sen [@sen1992], studied in Ruszczyński [@ruszczynski2002], and extended by Luedtke et. al [@luedtke2010] and Küçükyavuz [@kucukyavuz2012]. This reformulation gives rise to a mixing-type set [@gunluk2001] subject to an additional knapsack constraint, which is the focus of this paper.
Formally, consider the following chanced-constrained programming problem $$\begin{array}{ccl}$(PLP)$ &\min & c^T x \\
&{\rm s.t.} & {\bf P}(f(x) \geq \xi) \geq 1-\epsilon \\ & & x \in X, \end{array}$$ where $X\subset \mathbb{R}^n$ is a polyhedron, $f:X\to \mathbb{R}^d_+$ is a linear function, $\xi$ is a random variable in $\mathbb{R}^d$ with finite discrete distribution, $\epsilon \in (0,1)$, and $c\in \mathbb{R}^n$. Suppose that $\xi$ takes values from $\xi_1,\ldots,\xi_n$ with probabilities $\pi_1,\ldots,\pi_n$, respectively. We may assume that $\xi_j\geq 0$ for all $j\in [n]:=\{1,\ldots,n\}$. (Otherwise, replace $\xi_j$ by $\xi_j-\xi'$ and reset $f(x):=f(x)-\xi'$, where $\xi'$ is chosen so that $\xi_j\geq \xi'$ for all $j\in [n]$.) Also, by definition, $\pi_j>0$ for each $j\in [n]$ and $\sum_{j=1}^{n}{\pi_j}=1$. We can reformulate the chance constraint in (PLP) using linear inequalities and auxiliary binary variables as follows: let $z\in \{0,1\}^n$ where $z_j=0$ guarantees that $f(x)\geq \xi_j$. Then (PLP) is equivalent to $$\begin{array}{ccl}$(PLP)$ &\min & c^T x \\
&{\rm s.t.} & y=f(x) \\ & & y+z_j\xi_j\geq \xi_j ~~~~ \forall j\in [n]\\ & & \sum_{j=1}^n\pi_jz_j\leq \epsilon \\ & & z\in \{0,1\}^n \\ & & x \in X. \end{array}$$ Observe that we may assume that $\pi_j\leq \epsilon$ for all $j\in [n]$, for if $\pi_j>\epsilon$ for some $j\in [n]$, then we must have $z_j=0$ for all feasible solutions $(x,y,z)$ to the above system, and so may as well drop the index $j$. Now let $$\mathcal{D}:=\left\{(y,z)\in \mathbb{R}^d_+ \times \{0,1\}^n:\sum_{j=1}^n{\pi_jz_j}\leq \epsilon, ~y+\xi_jz_j\geq \xi_j ~ \forall j\in [n]\right\}.$$ Then (PLP) can be rewritten as $$\begin{array}{ccl}$(PLP)$ &\min & c^T x \\
&{\rm s.t.} & f(x)\in \text{proj}_y \mathcal{D}\\ & & x \in X. \end{array}$$ This motivates us to study the set $\mathcal{D}$. For each $k\in [d]$, let $$\mathcal{D}_k:=\left\{(y_k,z)\in \mathbb{R}_+ \times \{0,1\}^n:\sum_{j=1}^n{\pi_jz_j}\leq \epsilon, ~y_k+\xi_{jk}z_j\geq \xi_{jk} ~ \forall j\in [n]\right\}.$$ Then observe that $$\mathcal{D}=\bigcap_{k\in [d]}\left\{(y,z)\in \mathbb{R}^d_+ \times \{0,1\}^n: (y_k,z)\in \mathcal{D}_k\right\}.$$ Therefore, in order to study the set $\mathcal{D}$, a first step is to study the lower dimensional sets $\mathcal{D}_k$.
Fix $k\in [d]$ and for notational convenience, let $h_j:=\xi_{jk}$ for each $j\in [n]$. Let $\sum_{j\in [n]}{a_j z_j}\leq p$ be a valid inequality for $\mathcal{D}_k$ where $a\in \mathbb{R}_+^{n}$, $p\in \mathbb{R}_+$, $a_j\leq p$ for all $j\in [n]$, and $\sum_{j\in [n]}a_j>p$. Observe that this inequality may be the knapsack constraint $\sum_{j=1}^n{\pi_jz_j}\leq \epsilon$. Now let $$\mathcal{Q}:=\left\{(y,z)\in \mathbb{R}_+ \times \{0,1\}^n:\sum_{j\in [n]}{a_jz_j}\leq p, ~y+h_jz_j\geq h_j ~ \forall j\in [n]\right\}.$$ Note that the assumption that $a_j\leq p$ for all $j\in [n]$ implies that $\mathcal{Q}$ is a full-dimensional set. (The points $(h_1+1,0), (h_1,e_1),\ldots,(h_1,e_n)$ are in $\mathcal{Q}$, where $e_j$ is the $j$-th $n$-dimensional unit vector.) Also, note that the assumption that $\sum_{j\in [n]}a_j>p$ implies that $y\geq h_n\geq 0$ for all $y\in \mathcal{Q}$. Observe that the set $\mathcal{Q}$ contains as a substructure the intersection of a mixing set, first introduced by Günlük and Pochet [@gunluk2001], and a knapsack constraint. Various structural properties of conv$(\mathcal{Q})$ were studied in [@luedtke2010] and [@kucukyavuz2012] when the knapsack constraint $\sum_{j\in [n]}a_jz_j\leq p$ is just a cardinality constraint. In [@luedtke2010], a characterization of all valid inequalities of conv$(\mathcal{Q})$ was given, and in both [@luedtke2010] and [@kucukyavuz2012], explicit classes of facet-defining inequalities were introduced.\
Outline of Our Work
In this paper, we do not make any assumptions on the knapsack constraint. In Sect$.$ \[char\], we characterize the set of all valid inequalities for conv$(\mathcal{Q})$, and give a general cutting plane generating algorithm. In Sect$.$ \[fdisection\], we give necessary conditions for facet-defining inequalities of conv$(\mathcal{Q})$. In Sect$.$ \[explicitfdisub\], we introduce an explicit class of facet-defining inequalities that subsumes the facet-defining inequalities found in [@luedtke2010] and [@kucukyavuz2012]. Finally, in Sect$.$ \[heuristicsep\], using our ideas regarding characterization of all valid inequalities, we introduce a polynomial time heuristic separation algorithm for conv$(\mathcal{Q})$.
The Coefficient Polyhedron $\mathcal{G}$ {#char}
========================================
Let $$\mathcal{P}:=\left\{z\in \{0,1\}^{{n}}:\sum_{j\in [n]}a_jz_j\leq p \right\}.$$ Observe that $\mathcal{P}=\text{proj}_z\mathcal{Q}$ and $\text{conv}(\mathcal{P})=\text{proj}_z (\text{conv}(\mathcal{Q}))$. The focus of this paper is to study the class of valid inequalities for $\text{conv}(\mathcal{Q})$ that do not arise from $\text{conv}(\mathcal{P})$. We will first show that any such inequality has a particular form.
\[genform\] Suppose that $$\begin{aligned}
\gamma y+\sum_{j\in [n]}\alpha_jz_j\geq \beta\label{gvdi} \end{aligned}$$ is a valid inequality for $\emph{conv}(\mathcal{Q})$ for some $\alpha\in \mathbb{R}^n, \gamma,\beta\in \mathbb{R}$. Then $\gamma\geq 0$. Moreover, if $\gamma=0$ then $(\ref{gvdi})$ is a valid inequality for $\emph{conv}(\mathcal{P})$.
Observe that $(1,0)\in \text{cone}(\text{conv}(\mathcal{Q}))$. This implies that $\gamma\geq 0$. Moreover, since $\text{conv}(\mathcal{P})=\text{proj}_z (\text{conv}(\mathcal{Q}))$, it follows that if $\gamma=0$ then $(\ref{gvdi})$ is a valid inequality for $\text{conv}(\mathcal{P})$.
As a side note, the problem of finding a characterization for the class of all valid inequalities of conv$(\mathcal{P})$ is a very difficult problem and it has been extensively studied; the seminal works may be found in [@balas1975; @balas1978; @hammer1975; @wosley1975]. Recall that only those valid inequalities for $\text{conv}(\mathcal{Q})$ are of interest that do not come from $\text{conv}(\mathcal{P})$. As a result, by rescaling the coefficients, if necessary, we may assume that (\[gvdi\]) has the following form: $$\begin{aligned}
y+\sum_{j\in [n]}\alpha_jz_j\geq \beta.\label{gvdi2} \end{aligned}$$
It turns out that it is possible to characterize the inequalities of type (\[gvdi2\]) by considering the set of all vectors $(\alpha,\beta)$ that give valid inequalities of type (\[gvdi2\]). In this section, we will explicitly find this set, which happens to be a polyhedron. This polyhedron and its formulation will help us throughout the paper with various results on the structure of conv$(\mathcal{Q})$.
Let $\nu:=\max\{k:\sum_{j\leq k}a_j\leq p\}$. Notice that if $(y^*,z^*)\in \mathcal{Q}$ then $y^*\geq h_{\nu+1}$. Define, for each $0\leq k\leq \nu$, the knapsack set $$\mathcal{P}_k:=\left\{z\in \{0,1\}^{[n]}:\sum_{j> k}a_jz_j\leq p-\sum_{j\leq k}a_j \right\}.$$ Observe that $\mathcal{P}=\mathcal{P}_0 \supset \mathcal{P}_1\supset \cdots \supset \mathcal{P}_{\nu}$. Define the polyhedron $$\mathcal{G}:=\left\{(\alpha,\beta)\in \mathbb{R}^n\times \mathbb{R}: (\ref{Gconstp})\right\}$$ where $$\begin{aligned}
\sum_{j\leq k}\alpha_j+\sum_{j> k}\alpha_jz^*_j+h_{k+1} \geq \beta ~~~~\forall~ z^*\in \mathcal{P}_k, \forall ~0\leq k\leq \nu.\label{Gconstp} \end{aligned}$$ The following theorem proves that $\mathcal{G}$ is the desired set, and is one of the main results of this section.
\[validthm\] Choose $(\alpha,\beta)\in \mathbb{R}^n\times \mathbb{R}$. Then $(\ref{gvdi2})$ is a valid inequality for *conv*$(\mathcal{Q})$ if and only if $(\alpha,\beta)\in \mathcal{G}$.
We refer to $\mathcal{G}$ as the [*coefficient polyhedron*]{} of $\mathcal{Q}$. Before proving the above lemma, we would like to point out that $\mathcal{G}$ has an alternate formulation with $O(n)$ non-linear inequalities. For $\alpha\in \mathbb{R}^n$ and $0\leq k\leq \nu$, define the [*minimizer*]{} function $$\begin{aligned}
f_k(\alpha):=\min \left\{\sum_{j> k}\alpha_jz_j:z\in \mathcal{P}_k \right\}. \label{f}\end{aligned}$$ Then it is easy to see that $$\mathcal{G}=\left\{(\alpha,\beta)\in \mathbb{R}^n\times \mathbb{R}:\sum_{j\leq k}\alpha_j+ f_k(\alpha) +h_{k+1}\geq \beta ~\forall ~0\leq k\leq \nu\right\}.$$ As one may expect, the minimizers $f_k$ will play a central role when looking for structural properties of $\mathcal{G}$. For instance, as long as the minimizers $f_k$ can be efficiently solved to optimality, the polyhedron $\mathcal{G}$ can be efficiently described by a system of $O(n)$ non-linear inequalities. Notice that this is not too surprising since if $f_0$ can be efficiently solved for all $\alpha\in \mathbb{R}$, then the reverse polar of conv$(\mathcal{P})$ can be efficiently described, and this yields an efficient method to obtain valid inequalities for conv$(\mathcal{Q})$, which correspond to points in $\mathcal{G}$. For more details on this approach, see Sen [@sen1992]. However, note that our purpose here is [*not*]{} necessarily finding fast and practical algorithms to obtain valid inequalities but rather obtaining structural results about conv$(\mathcal{Q})$. The coefficient polyhedron $\mathcal{G}$ and the minimizers $f_k$ will be useful to better understand the structure of the polyhedron conv$(\mathcal{Q})$.
It is now time to prove Theorem \[validthm\].
Suppose that $$\begin{aligned}
y+\sum_{j\in [n]}\alpha_jz_j\geq \beta \label{genvdi}\end{aligned}$$ is a valid inequality of conv$(\mathcal{Q})$ for some $(\alpha,\beta)\in \mathbb{R}^n\times \mathbb{R}$. Take $0\leq k\leq \nu$. Let $z^*\in \{0,1\}^n$ be an optimal solution to (\[f\]) with $z^*_j=1$ for all $1\leq j\leq k$. Note that such an optimal solution exists due to how $\mathcal{P}_k$ and $f_k$ are defined. Let $y^*=h_{k+1}$. Observe that $(y^*,z^*)\in \mathcal{Q}$. Hence, since (\[genvdi\]) is valid for $\mathcal{Q}$, it follows that $$\begin{aligned}
\beta&\leq y^*+\sum_{j\in [n]}\alpha_jz^*_j\\
&= h_{k+1}+\sum_{j\leq k}\alpha_j+\sum_{j> k}\alpha_jz^*_j\\
&= h_{k+1}+\sum_{j\leq k}\alpha_j+f_k(\alpha). \end{aligned}$$ Since this holds for all $0\leq k\leq \nu$, it follows that $(\alpha,\beta)\in \mathcal{G}$.
Conversely, suppose that $(\alpha,\beta)\in \mathcal{G}$. Let $(y^*,z^*)\in \mathcal{Q}$. Observe that $y^*\geq h_{\nu+1}$ by definition. Temporarily let $h_0:=+\infty$. Suppose that $h_k> y^*\geq h_{k+1}$ for some $0\leq k\leq \nu$. Note that $z^*_j=1$ for all $j\leq k$ and so $z^*\in \mathcal{P}_k$. Then $$\begin{aligned}
y^*+\sum_{j\in [n]}\alpha_jz^*_j&\geq h_{k+1}+\sum_{j\leq k}\alpha_j+ \sum_{j>k}\alpha_jz^*_j\\
&\geq h_{k+1}+\sum_{j\leq k}\alpha_j+ f_k(\alpha)\\
&\geq \beta~~\text{ since } (\alpha,\beta)\in \mathcal{G} \end{aligned}$$ and so (\[genvdi\]) is valid for $(y^*,z^*)$. Therefore, (\[genvdi\]) is a valid inequality for $\mathcal{Q}$ and hence for $\text{conv}(\mathcal{Q})$, as claimed.
The above characterization theorem can be viewed as a generalization of a similar characterization given in Luedtke et. al [@luedtke2010], which is only applicable to the case when the knapsack constraint $\sum_{j\in [n]}a_jz_j\leq p$ is a cardinality constraint. For the sake of completeness, here we will state the result, which is a slightly modified version of Theorem 3 in [@luedtke2010].
\[jimcharQ\] Suppose that $a_j=1$ for all $j\in [n]$. Then any valid inequality for $\emph{conv}(\mathcal{Q})$ with nonzero coefficient on $y$ can be written in the form $$\begin{aligned}
y+\sum_{j\in [n]}\alpha_jz_j\geq \beta. \label{jimchar} \end{aligned}$$ Furthermore, $(\ref{jimchar})$ is valid for $\emph{conv}(\mathcal{Q})$ if and only if $$\begin{aligned}
\sum_{j\leq k-1}\alpha_j+\min_{S\in \mathcal{S}_k} \sum_{j\in S}\alpha_j+h_k\geq \beta~~\forall~ 1\leq k\leq p+1,\label{constraints}\end{aligned}$$ where $\mathcal{S}_k:=\{S\subset \{k,\ldots,n\}:|S|\leq p-k+1\}$.
Observe that in the case when $a_j=1$ for all $j\in [n]$, we have that $\nu=p$ and $$f_{k-1}(\alpha)=\min_{S\in \mathcal{S}_k} \sum_{j\in S}\alpha_j.$$ As a result, the inequalities (\[constraints\]) are equivalent to $$\sum_{j\leq k}\alpha_j+f_k(\alpha)+h_{k+1}\geq \beta~~\forall ~0\leq k\leq \nu,$$ which is equivalent to $(\alpha,\beta)\in \mathcal{G}$. Hence, Theorem \[jimcharQ\] follows from Theorem \[validthm\] and Lemma \[genform\].
Using Theorem \[validthm\], we can find cutting planes valid for conv$(\mathcal{Q})$ by solving a linear program.
\[septhm\] Suppose $(y^*,z^*)\in \mathbb{R}_+\times \mathbb{R}^n_+$ satisfies $z^*\in \emph{conv}(\mathcal{P})$. Let $$\begin{aligned}
LP^*:=\min\left\{\sum_{j\in [n]}\alpha_jz^*_j-\beta: (\alpha,\beta)\in \mathcal{G}\right\}.\label{sepopt}\end{aligned}$$ Then $(y^*,z^*)\in \emph{conv}(\mathcal{Q})$ if and only if $y^*+LP^*\geq 0$. Also, if $y^*+LP^*<0$ and $(\alpha^*,\beta^*)$ is an optimal solution to *(\[sepopt\])*, then $y+\sum_{j\in [n]}\alpha^*_jz_j\geq \beta^*$ is a valid inequality for $\emph{conv}(\mathcal{Q})$ which is violated by $(y^*,z^*)$.
If $(y^*,z^*)\in \text{conv}(\mathcal{Q})$, then $y^*+LP^*\geq 0$ by Theorem \[validthm\]. Conversely, if $y^*+LP^*\geq 0$ then $$y^*+\sum_{j\in [n]}\alpha_jz^*_j\geq \beta$$ for all $(\alpha,\beta)\in \mathcal{G}$. Hence, by Theorem \[validthm\], $(y^*,z^*)$ satisfies all inequalities of the type (\[gvdi\]) with $\gamma>0$. Moreover, since $z^*\in \text{conv}(\mathcal{Q})$, $(y^*,z^*)$ also satisfies all inequalities of the type (\[gvdi\]) with $\gamma=0$. Hence, there is no valid inequality for $\text{conv}(\mathcal{Q})$ that separates $(y^*,z^*)$ from $\text{conv}(\mathcal{Q})$, so $(y^*,z^*)\in \text{conv}(\mathcal{Q})$.
Observe that the running time of solving the linear program (\[sepopt\]) is polynomial in $n$ and $T_{\mathcal{P}}$, where $T_{\mathcal{P}}$ is the running time of optimizing $f_k$ over $\mathcal{P}_k$, for all $0\leq k \leq \nu$. Hence, solving (\[sepopt\]) is efficient if minimizing over $\mathcal{P}_k$ can be accomplished efficiently, for all $0\leq k\leq \nu$.
Facet-Defining Inequalities of conv($\mathcal{Q}$) {#fdisection}
==================================================
The following theorem describes some interesting properties for facet-defining inequalities of $\text{conv}(\mathcal{Q})$. This helps us understand better the structure of the polyhedron conv($\mathcal{Q}$).
\[genfdithm\] Suppose that the inequality $$\begin{aligned}
y+\sum_{j\in [n]}\alpha_jz_j\geq \beta\label{genfdi}\end{aligned}$$ is facet-defining for *conv*$(\mathcal{Q})$ for some $(\alpha, \beta)\in \mathbb{R}^n\times \mathbb{R}$. Then
1. $(\alpha,\beta)$ is an extreme point of $\mathcal{G}$,
2. $\beta=h_1+f_0(\alpha)$, and
3. if $\alpha_k<0$ for some $1\leq k\leq n$, then $a_k>0$.
\(i) This is true since otherwise (\[genfdi\]) would be the convex combination of two distinct valid inequalities of conv($\mathcal{Q}$), which cannot be the case since (\[genfdi\]) is a facet-defining inequality for conv($\mathcal{Q}$).
\(ii) Let $z^*\in \mathcal{P}_0$ be a solution that attains the minimum of $f_0$. It is then by definition clear that $(h_1, z^*)\in \mathcal{Q}$. Thus by (\[genfdi\]) we get that $$\beta\leq h_1+\sum_{j\in [n]}\alpha_jz^*_j=h_1+f_0(\alpha)$$ and so $\beta\leq h_1+f_0(\alpha)$. Since conv$(\mathcal{Q})$ is full-dimensional and (\[genfdi\]) is a facet-defining inequality different from $z_1\leq 1$, it follows that there is a point $(y',z')\in \mathcal{Q}$ on the facet defined by (\[genfdi\]) such that $z'_1=0$. Note that this implies that $y'\geq h_1$. Note also that $z'\in \mathcal{P}_0$. We thus have $$\beta=y'+\sum_{j\in [n]}\alpha_jz'_j\geq h_1+f_0(\alpha)$$ which implies that $\beta\geq h_1+f_0(\alpha)$. Hence, $\beta=h_1+f_0(\alpha)$, as claimed.
\(iii) Suppose not, and assume that $a_k=0$ for some $1\leq k\leq n$ with $\alpha_k<0$. Since $a_k=0$ it follows that $$\text{proj}_{\mathbb{R}\times \mathbb{R}^{[n]\setminus \{k\}}}\mathcal{Q}=\left\{(y,z)\in \mathbb{R}_+\times \{0,1\}^{[n]\setminus \{k\}}:\sum_{j\in [n]\setminus \{k\}}a_jz_j\leq p,~y+h_iz_i\geq h_i~\forall i\in [n]\setminus \{k\}\right\}.$$ We claim that $$\begin{aligned}
y+\sum_{j\in [n]\setminus \{k\}}\alpha_jz_j\geq \beta-\alpha_k\label{projvalid}\end{aligned}$$ is a valid inequality for $\text{proj}_{\mathbb{R}\times \mathbb{R}^{[n]\setminus \{k\}}}\mathcal{Q}$. Let $(y^*,z^k)\in \text{proj}_{\mathbb{R}\times \mathbb{R}^{[n]\setminus \{k\}}}\mathcal{Q}$. Define $z^*\in \mathbb{R}^n$ as follows: $z^*_j=z^k_j$ if $j\neq k$ and $z^*_k=1$. Since $a_k=0$ it follows that $(y^*,z^*)\in \mathcal{Q}$. Thus, since (\[genfdi\]) is valid for $\mathcal{Q}$, it follows that $$\beta-\alpha_k\leq y^*+\sum_{j\in [n]}\alpha_jz^*_j-\alpha_k= y^*+\sum_{j\in [n]\setminus \{k\}}\alpha_jz^*_j,$$ and so (\[projvalid\]) is valid for $\text{proj}_{\mathbb{R}\times \mathbb{R}^{[n]\setminus \{k\}}}\mathcal{Q}$, and so it is also a valid inequality for $\mathcal{Q}$. However, the facet-defining inequality (\[genfdi\]) is the sum of (\[projvalid\]) and the inequality $-\alpha_k(1-z_k)\geq 0$, which is valid for $\mathcal{Q}$ since $\alpha_k<0$. But then (\[genfdi\]) is dominated by (\[projvalid\]), a contradiction. Thus, $a_k> 0$.
In the next section, we give an explicit class of facet-defining inequalities of $\text{conv}(\mathcal{Q})$, which may be useful for generating cutting planes in a branch-and-cut algorithm.
An Explicit Class of Facet-Defining Inequalities for $\text{conv}(\mathcal{Q})$ {#explicitfdisub}
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In this section, we will first give an overview of the known explicit class of facet-defining inequalities for conv($\mathcal{Q}$). We will then introduce a new explicit class of facet-defining inequalities that subsumes all the previously known classes.
Chronologically speaking, the first class of facet-defining inequalities for conv$(\mathcal{Q})$ are called the [*strengthened star inequalities*]{} (see [@luedtke2010; @atamturk2000]).
\[starineqthm\] The strengthened star inequalities $$\begin{aligned}
y+\sum_{j=1}^{a}(h_{t_j}-h_{t_{j+1}})z_{t_j}\geq h_{t_1}~~~~\forall~T=\{t_1,\ldots,t_a\}\subset \{1,\ldots,\nu\} \label{starineq}\end{aligned}$$ with $t_1<\cdots<t_a$ and $h_{t_{a+1}}:=h_{\nu+1}$ are valid for $\emph{conv}(\mathcal{Q})$. Moreover, $(\ref{starineq})$ is facet-defining for $\emph{conv}(\mathcal{Q})$ if and only if $h_{t_1}=h_1$.
As shown in [@gunluk2001; @atamturk2000; @guan2007], the (strengthened) star inequalities can be separated in polynomial time and are sufficient to describe the convex hull of $$\mathcal{R}:=\left\{(y,z)\in \mathbb{R}_+ \times \{0,1\}^n:y+h_iz_i\geq h_i ~ \forall i\in [n]\right\}.$$ However, as it turns out, when a knapsack constraint is enforced in $\mathcal{R}$ to obtain $\mathcal{Q}$, the convex hull becomes much more complex.
Later, Luedtke et. al [@luedtke2010] found a larger and subsuming class of facet-defining inequalities for conv$(\mathcal{Q})$ in the case when the knapsack constraint $\sum_{j\in [n]}a_jz_j\leq p$ is just a cardinality constraint, i.e. $a_j=1$ for all $j\in [n]$. Subsequently, Küçükyavuz [@kucukyavuz2012] introduced an even larger and subsuming class facet-defining inequalities for conv$(\mathcal{Q})$, called the $(T,\Pi_L)$ [*inequalities*]{}, which again only applies to the case when the knapsack constraint $\sum_{j\in [n]}a_jz_j\leq p$ is just a cardinality constraint. Here, we only state the latter class.
Suppose that $a_j=1$ for all $j\in [n]$. Take a positive integer $m\leq \nu=p$. Suppose that
1. $T:=\{t_1,\ldots,t_a\}\subset \{1,\ldots,m\}$, where $t_1<\ldots <t_a$; and
2. $L\subset \{m+2,\ldots,n\}$ and take a permutation of the elements in $L$, $\Pi_L:=\{\ell_1,\ldots,\ell_{p-m}\}$ such that $\ell_j> m+j$ for all $1\leq j\leq p-m$.
Set $t_{a+1}:=m+1$. Let $\Delta_1:=h_{m+1}-h_{m+2}$, and for $2\leq j\leq p-m$ define $$\Delta_j:= \max\left\{\Delta_{j-1}, h_{m+1}- h_{m+1+j}-\sum{\left(\Delta_i:\ell_i> m+j,i<j\right)}\right\}.$$ Then the $(T,\Pi_L)$ inequality $$\begin{aligned}
y+\sum_{j=1}^{a}(h_{t_j}-h_{t_{j+1}}) z_{t_j}+\sum_{j=1}^{p-m}{\Delta_j(1-z_{q_j})} \geq h_{t_1} \label{TPiineq}\end{aligned}$$ is valid for $\emph{conv}(\mathcal{Q})$. Furthermore, $(\ref{TPiineq})$ is facet-defining inequality for $\emph{conv}(\mathcal{Q})$ if and only if $h_{t_1}=h_1$.
Observe that the $(T,\emptyset)$ inequalities are simply the strengthened star inequalities.
We now introduce a larger and subsuming class of facet-defining inequalities for conv$(\mathcal{G})$ in the general setting that coincides with the $(T,\Pi_L)$ inequalities in the case when $a_j=1$ for all $j\in [n]$. For all $1\leq m\leq n$, let $$s_m:=\sum_{j=1}^ma_j,$$ and let $s_0:=0$.
\[explicitfdi\] Take an integer $0\leq m\leq \nu$ such that $p-s_m$ is an integer. For each $1\leq j\leq p-s_m$, let $k(j):=\max\{k:j\geq s_k-s_m\}$. Let
1. $T:=\{t_1,\ldots,t_a\}\subset \{1,\ldots,m\}$ where $t_1<\ldots <t_a$;
2. $S:=\{q_1,\ldots,q_s\}\subset \{m+2,\ldots,n\}$ where $s=p-s_m$ and $q_j> k(j)$ for all $1\leq j \leq p-s_m$; and
3. $S$ is chosen so that $a_j=1$ for all $j\in S$, and $a_j\leq s_m$ for all $j\notin S$.
Set $t_{a+1}=m+1$. Let $\alpha_{q_1}:=h_{k(1)+1}-h_{m+1}$, and for $2\leq j\leq p-s_m$, define $$\begin{aligned}
\alpha_{q_j}:= \min\left\{\alpha_{q_{j-1}}, h_{k(j)+1}-h_{m+1}-\sum{\left(\alpha_{q_i}:q_i> k(j),i<j\right)}\right\}.\label{coeff}\end{aligned}$$ Then $$\begin{aligned}
y+\sum_{j=1}^{a}(h_{t_j}-h_{t_{j+1}}) z_{t_j}+\sum_{i\in S}\alpha_iz_i \geq h_{t_1}+\sum_{i\in S}\alpha_i \label{fdieg1} \end{aligned}$$ is a valid inequality for *conv*$(\mathcal{Q})$. Furthermore, $(\ref{fdieg1})$ is a facet-defining inequality for *conv*$(\mathcal{Q})$ if and only if $h_{t_1}=h_1$.
We would like to explain how this theorem implies that the $(T,\Pi_L)$ inequalities (\[TPiineq\]) are facet-defining for conv$(\mathcal{Q})$ when $a_j=1$ for all $j\in [n]$. Let $S:=L$ and $q_i:=\ell_i$ for all $1\leq i\leq p-m$. Observe that $s_m=m$ is an integer, $|S|=p-m=p-s_m$, and $k(j)=m+j$ for all $1\leq j\leq p-s_m$. Note that $q_j=\ell_j>m+j=k(j)$ for all $1\leq j\leq p-s_m$. Also, note that $a_k=1\leq s_m$ for all $k\in [n]\setminus (T\cup S)$ since $m\geq 1$. Lastly, observe that $\alpha_j=-\Delta_j$ for all $j\in S$. Hence, by Theorem \[explicitfdi\], the $(T,\Pi_L)$ inequalities (\[TPiineq\]) are facet-defining for conv$(\mathcal{Q})$ when $a_j=1$ for all $j\in [n]$.
Observe that Theorem \[explicitfdi\] can also be applied to any scalar multiple of the knapsack constraint, and this will potentially give us more facet-defining inequalities. That is, one can apply Theorem \[explicitfdi\] to $\sum_{j\in [n]}da_jz_j\leq dp$, for any arbitrary positive real number $d$.
In Sect. \[sep\], we give a polynomial time separation algorithm to separate over a subset of (\[fdieg1\]). This separation algorithm is analogous to and an appropriate generalization of the one given in Küçükyavuz [@kucukyavuz2012]. Finally in Sect. \[prooffdi\], we give a proof of Theorem \[explicitfdi\].
Separation of the new class of FDIs (\[fdieg1\]) {#sep}
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In this section, we give a polynomial time separation algorithm over a subset of the inequalities (\[fdieg1\]). This algorithm is analogous to the separation algorithm given in [@kucukyavuz2012].
\[sepoverfdis\] Take $0\leq m\leq \nu$ and $0\leq r\leq p-s_m$. Let $A_m:=\{m+2\leq j\leq n: a_j=1\}$. Suppose that $a_j\leq s_m$ for all $j\in [n]$, $k(1)<k(2)<\cdots<k(r)<k(r+1)$ and $F:=\{k(1)+1,\ldots,k(r)+1\}\subset A_m$. Then we can find the most violated inequality $(\ref{fdieg1})$ with $m$ as above and $S=F\cup G$ with $G\subset \{k(p-s_m)+1,\ldots,n\}$ in $O(p^3)$.
Suppose that $m$ and $r$ are given as above, and that $S=F\cup G$ with $G\subset \{k(p-s_m)+1,\ldots,n\}$. With this choice of $S$, we must have that $q_j=k(j)+1$ for all $1\leq j\leq r$ (note (ii)). As a result, $\alpha_{q_j}$ in (\[coeff\]) simplifies to $\alpha_{q_j}=\min\{\alpha_{q_{j-1}}, h_{k(j)+1}-h_{m+1}\}$ for $2\leq j\leq r$. Moreover, for $q_i\in G\subset \{k(p-s_m)+1,\ldots,n\}$, we have $q_i> k(p-s_m)\geq k(j)$ for all $r+1\leq j\leq p-s_m$. Hence, $\alpha_{q_j}=\min\{\alpha_{q_{j-1}}, h_{k(j)+1}-h_{m+1}-\sum_{i=r+1}^{j-1}\alpha_{q_i}\}$ for $r+1\leq j\leq p-s_m$. Observe that, assuming $S=F\cup G$, the coefficients $\alpha_{q_j}$ do not depend on a particular choice of $G$, but depend only on $\alpha_r$.
Let $(y^*,z^*)\in \mathbb{R}_+\times \mathbb{R}^n_+$. We now give an algorithm to to identify the most violated inequality (\[fdieg1\]) with $S=F\cup G$ and $G\subset \{k(p-s_m)+1,\ldots,n\}$. Note that the problem of finding the best $T$ in inequalities (\[fdieg1\]) can be solved as a shortest path problem on a directed acyclic graph with $O(p^2)$ arcs. For details, see [@kucukyavuz2012]. Note that we always include a $t_i$ in $T$ for which $h_{t_i}=h_1$.
To find the set $G$ that gives the most violated inequality (\[fdieg1\]) in the desired form, we keep an ordered list of the elements in $\{k(p-s_m)+1,\ldots,n\}$, denoted by $Z$, in decreasing order of $z^*_j$ for $k(p-s_m)+1\leq j\leq n$ and we choose the first $p-s_m-r$ elements in $Z$ to be in the set $G$.
As a result, the most violated inequality $(\ref{fdieg1})$ with $m$ as above and $S=F\cup G$ with $G\subset \{k(p-s_m)+1,\ldots,n\}$ in $O(p^3)$.
\[sepoverfdisgen\] Take $0\leq m\leq \nu$. Let $A_m:=\{m+2\leq j\leq n: a_j=1\}$. Suppose that $a_j\leq s_m$ for all $j\in [n]$, and $k(1)<k(2)<\cdots<k(p-s_m)$. Then we can find the most violated inequality $(\ref{fdieg1})$ with $m$ as above and $S=\{k(1)+1,\ldots,k(r)+1\}\cup G$ with $G\subset \{k(p-s_m)+1,\ldots,n\}$ over all $0\leq r\leq p-s_m$ in $O(p^4)$.
Proof of Theorem \[explicitfdi\] {#prooffdi}
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As we will see, the function $f_k$ plays a central role in the proof of Theorem \[explicitfdi\]. In the following lemma, which is needed for the proof, we compute $f_k$ for each $k$ under the nice assumptions on $S$ given in Theorem \[explicitfdi\]. Recall that $s_k=\sum_{j=1}^ka_j$ for all $1\leq k\leq n$, and $s_0=0$.
Let $R\cup S$ be a partition of $[n]$, $\alpha\in \mathbb{R}^n$ and $0\leq m\leq \nu$ such that the following are satisfied:
1. $p-s_m$ is an integer, $|S|=p-s_m$ and $a_j=1$ for all $j\in S$;
2. if $i\in R$ then $\alpha_i\geq 0$, and if $i\in S$ then $\alpha_i\leq 0$;
3. $S\subset \{m+2,\ldots, n\}$ and $\alpha_{q_1}\geq \cdots\geq \alpha_{q_{|S|}}$ for some permutation $(q_1,\ldots,q_{|S|})$ on $S$;
4. $i> s_k-s_m$ implies that $q_i>k$ for all $i\in \{1,\ldots,p-s_m\}$ and $k\in \{m+1,\ldots,\nu\}$.
Then $$f_k(\alpha)=\left\{\begin{array}{ll} \sum_{j\in S}\alpha_j ~~~~~~~~~~~~~~~~~~ \text{ if } 0\leq k\leq m; \\ \sum{(\alpha_{q_i}:i> s_k-s_m)} ~~~\text{ if } m+1\leq k\leq \nu.\end{array} \right.$$
If $0\leq k\leq m$, then $z^*\in \{0,1\}^{[n]}$ defined as $$z^*_j:=\left\{\begin{array}{ll} 1 ~~~ \text{ if } j\in S; \\ 0 ~~~ \text{ otherwise.} \end{array} \right.$$ is a feasible point for $\mathcal{P}_k$ since $$\sum_{j>k}a_jz^*_j=\sum_{j\in S,j>k}z^*_j=|S|= p-s_m\leq p-\sum_{j\leq k}a_j.$$ Hence, $$\sum_{j\in S}\alpha_j\leq f_k(\alpha)\leq \sum_{j\in S}\alpha_j$$ and so $f_k(\alpha)=\sum_{j\in S}\alpha_j$. Now choose $m+1\leq k\leq \nu$. We will first find a lower bound for $f_k(\alpha)$. Let $z\in \mathcal{P}_k$. Then $$\begin{aligned}
|\{j\in S:j> k,z_j=1\}|=\sum_{j\in S,j>k}z_j=\sum_{j\in S,j>k}a_jz_j \leq p-\sum_{j\leq k}a_j &=p-s_k\\
&=|S|-s_k+s_m\\
&= |\{q_i\in S:i>s_k-s_m\}|.\end{aligned}$$ Observe that $i\geq s_k-s_m$ implies that $q_i>k$. As a result, since $\alpha_{q_1}\geq \cdots\geq \alpha_{q_{|S|}}$, we get that $$\sum_{j\in S,j>k}\alpha_jz_j\geq \sum{\left(\alpha_{q_i}:i> s_k-s_m\right)}.$$ Since this is true for all $z\in \mathcal{P}_k$, it follows that $$f_k(\alpha)\geq \sum{\left(\alpha_{q_i}:i> s_k-s_m\right)}.$$ Furthermore, we claim that equality holds above. Define $z^*\in \{0,1\}^{[n]}$ as follows: for $q_i\in S$ let $$z^*_{q_i}:=\left\{\begin{array}{ll} 1 ~~~ \text{ if } i> s_k-s_m \text{ or } q_i\leq k; \\ 0 ~~~ \text{ otherwise,} \end{array} \right.$$ and for $i\in R$ let $z^*_i:=0$. We have $$\begin{aligned}
\sum_{i>k}a_iz^*_i=\sum_{q_i>k}z^*_{q_i}
&=\sum (z^*_{q_i}:q_i>k, i>s_k-s_m)\\
&=\sum (z^*_{q_i}: i>s_k-s_m)~\text{ since } i>s_k-s_m \text{ implies } q_i>k\\
&= |S|-(s_k-s_m)\\
&=p-s_k=p-\sum_{j\leq k}a_j.\end{aligned}$$ Hence, $z^*\in \mathcal{P}_k$. However, $$f_k(\alpha)\leq \sum_{j\in S,j>k}\alpha_jz^*_j=\sum{\left(\alpha_{q_i}:i> s_k-s_m\right)}\leq f_k(\alpha)$$ and so $$f_k(\alpha)= \sum{\left(\alpha_{q_i}:i> s_k-s_m\right)}.$$ Hence, we are done.
Now we are ready to prove Theorem \[explicitfdi\]. We restate Theorem \[explicitfdi\] for convenience.
[**Restatement of Theorem \[explicitfdi\].**]{} Take an integer $0\leq m\leq \nu$ such that $p-s_m$ is an integer. For each $1\leq j\leq p-s_m$, let $k(j):=\max\{k:j\geq s_k-s_m\}$. Let
1. $T:=\{t_1,\ldots,t_a\}\subset \{1,\ldots,m\}$ where $t_1<\ldots <t_a$;
2. $S:=\{q_1,\ldots,q_s\}\subset \{m+2,\ldots,n\}$ where $s=p-s_m$ and $q_j> k(j)$ for all $1\leq j \leq p-s_m$; and
3. $S$ is chosen so that $a_j=1$ for all $j\in S$, and $a_j\leq s_m$ for all $j\notin S$.
Set $t_{a+1}=m+1$. Let $\alpha_{q_1}:=h_{k(1)+1}-h_{m+1}$, and for $2\leq j\leq p-s_m$, define $$\alpha_{q_j}:= \min\left\{\alpha_{q_{j-1}}, h_{k(j)+1}-h_{m+1}-\sum{\left(\alpha_{q_i}:q_i> k(j),i<j\right)}\right\}.$$ Then $$\begin{aligned}
y+\sum_{j=1}^{a}(h_{t_j}-h_{t_{j+1}}) z_{t_j}+\sum_{i\in S}\alpha_iz_i \geq h_{t_1}+\sum_{i\in S}\alpha_i \label{fdieg} \end{aligned}$$ is a valid inequality for conv$(\mathcal{Q})$. Furthermore, $(\ref{fdieg})$ is a facet-defining inequality for conv$(\mathcal{Q})$ if and only if $h_{t_1}=h_1$.
Let $R=[n]\setminus S$ and define, for $i\in R$, $$\alpha_i:=\left\{\begin{array}{ll} h_{t_j}-h_{t_{j+1}} ~~~ \text{ if } i=t_j \text{ for some } 1\leq j\leq m; \\ 0 ~~~~~~~~~~~~~~~ \text{ otherwise.} \end{array} \right.$$ We first show that $(\alpha, h_{t_1}+\sum_{i\in S}\Delta_i)\in \mathcal{G}$. Observe that if $i\in R$ then $\alpha_i\geq 0$, and if $i\in S$ then $\alpha_i\leq 0$. Also, note that $\alpha_{q_1}\geq \cdots\geq \alpha_{q_{|S|}}$. Let $0\leq k\leq \nu$. If $0\leq k\leq m$, then by the previous lemma, $f_k(\alpha)=\sum_{i\in S}\alpha_i$. Suppose that $t_j< k+1\leq t_{j+1}$ for some $j\in \{0,1,\ldots,a\}$ where $t_0:=0$. Then $$\sum_{i\leq k}\alpha_i+f_k(\alpha)-h_{t_1}-\sum_{i\in S}\alpha_i= \sum_{i=1}^{j}(h_{t_i}-h_{t_{i+1}})-h_{t_1}=h_{t_1}-h_{t_{j+1}}-h_{t_1}\geq -h_{k+1}.$$ Otherwise, assume that $m+1\leq k\leq \nu$. Note that in this case we have $$\sum_{i\in R,i\leq k}\alpha_i=\sum_{i\in T}\alpha_i=h_{t_1}-h_{m+1}.$$ Also, by the previous lemma, we have $$f_k(\alpha)= \sum{\left(\alpha_{q_i}:i> s_k-s_m\right)}.$$ Then $$\begin{aligned}
&\sum_{i\in R,i\leq k}\alpha_i+\sum_{i\in S,i\leq k}\alpha_i+f_k(\alpha)-h_1-\sum_{i\in S}\alpha_i\\
&=h_{t_1}-h_{m+1}+\sum{\left(\alpha_{q_i}:q_i\leq k\right)} +\sum{\left(\alpha_{q_i}:i> s_k-s_m\right)}-h_{t_1}-\sum_{i\in S}\alpha_i\\
&=-h_{m+1}-\alpha_{q_j}- \sum{\left(\alpha_{q_i}:q_i>k,i<j\right)} ~~\text{ for } j=s_k-s_m\\
&\geq -h_{k+1}.\end{aligned}$$ As a result, $(\alpha, h_{t_1}+\sum_{i\in S}\alpha_i)\in \mathcal{G}$ and so by Theorem \[validthm\], we get that (\[fdieg\]) is a valid inequality for conv$(\mathcal{Q})$.
Observe that if (\[fdieg\]) is facet-defining for conv$(\mathcal{Q})$, then by Theorem \[genfdithm\] (ii), we must have that $h_{t_1}+\sum_{i\in S}\alpha_i=h_1+f_0(\alpha)$. However, by the previous lemma, we know that $f _0(\alpha)=\sum_{i\in S}\alpha_i$. Hence, $h_{t_1}=h_1$ is a necessary condition for (\[fdieg\]) to be facet-defining. Conversely, assume that $h_{t_1}=h_1$. We will find $n+1$ affinely independent points in $\mathcal{Q}$ that satisfy (\[fdieg\]) at equality.
For each $k:=t_j\in T$, let $y^k=h_{t_j}$ and define $z^k\in \{0,1\}^{[n]}$ as follows: $$z^k_i:=\left\{\begin{array}{ll} 1 ~~~ \text{ if } i< k \text{ or } i\in S; \\ 0 ~~~ \text{ otherwise.} \end{array} \right.$$ Note that $z^k_i=1$ for all $i<k$. Also, we have $$\sum_{i=1}^{n}a_iz^k_i= \sum_{i<k}a_iz^k_i+ \sum_{i\in S}z^k_i=s_{k-1}+|S|=s_{k-1}+p-s_m\leq p.$$ Hence, $(y^k,z^k)\in \mathcal{Q}$. Moreover, $$\begin{aligned}
y^k+\sum_{i=1}^{a}(h_{t_i}-h_{t_{i+1}}) z^k_{t_i}+\sum_{i\in S}\alpha_iz^k_i=h_{t_j}+\sum_{t_i<t_j}(h_{t_i}-h_{t_{i+1}})+\sum_{i\in S}\alpha_i&=h_{t_j}+h_1-h_{t_j}+\sum_{i\in S}\alpha_i\\
&=h_1+\sum_{i\in S}\alpha_i.\end{aligned}$$
For each $k:=q_j\in S$, define $$\ell(j):=\max\left\{1\leq \ell\leq j:\alpha_{q_j}=h_{k(\ell)+1}-h_{m+1}-\sum{\left(\alpha_{q_i}:q_i> k(\ell),i<\ell\right)} \right\}.$$ Now let $y^k:=h_{k(\ell(j))+1}$ and define $z^k\in \{0,1\}^{[n]}$ as follows: $$z^k_t:=\left\{\begin{array}{ll} 0 ~~~ \text{ if } t=q_i \text{ and } q_i>k(\ell(j)) \text{ and } i<\ell(j), \text{ or } t=q_j, \text{ or } t\in R \text{ and } t>k(\ell(j));\\ 1 ~~~ \text{ otherwise.} \end{array} \right.$$ Note that $z^k_i=1$ for all $i\leq k(\ell(j))$. Also, we have $$\begin{aligned}
\sum_{i=1}^{n}a_iz^k_i&= \sum_{i\leq k(\ell(j))}a_i+
|\{q_i\in S:q_i>k(\ell(j)), i\geq \ell(j), i\neq j \}|\\
&=s_{k(\ell(j))}+|\{q_i\in S:q_i>k(\ell(j)), i\geq \ell(j) \}|-1\\
&=s_{k(\ell(j))}+|\{q_i\in S:i\geq \ell(j) \}|-1\\
&=s_{k(\ell(j))}+|S|-\ell(j)\\
&=s_{k(\ell(j))}+p-s_m-\ell(j)\\
&\leq p ~~\text{ by definition of } k(\cdot).\end{aligned}$$ Hence, $(y^k,z^k)\in \mathcal{Q}$. Moreover, $$\begin{aligned}
& y^k+\sum_{i=1}^{a}(h_{t_i}-h_{t_{i+1}}) z^k_{t_i}+\sum_{i=1}^{s}\alpha_{q_i}z^k_{q_i}\\
&=h_{k(\ell(j))+1}+ h_1-h_{m+1}+\sum_{i\in S}\alpha_i-\Delta_{q_j}-\sum {\left(\alpha_{q_i}:q_i> k(\ell(j)),i<\ell(j)\right)} \\
&= h_{k(\ell(j))+1}+ h_1-h_{m+1}+\sum_{i\in S}\alpha_i+h_{m+1}-h_{k(\ell(j))+1}\\
&=h_1+\sum_{i\in S}\alpha_i.\end{aligned}$$
For all $k\in R\setminus T$, let $y^k:=h_1$ and define $z^k\in \{0,1\}^{[n]}$ as follows: $$z^k_i:=\left\{\begin{array}{ll} 0 ~~~ \text{ if } i\in R\setminus \{k\};\\ 1 ~~~ \text{ otherwise.} \end{array} \right.$$ We have $$\sum_{i=1}^{n}a_iz^k_i= a_k+|S|=a_k+p-s_m\leq p ~~~ \text{ by (iii).}$$ Hence, $(y^k,z^k)\in \mathcal{Q}$. Moreover, $$y^k+\sum_{i=1}^{a}(h_{t_i}-h_{t_{i+1}}) z^k_{t_i}+\sum_{i\in S}\alpha_iz^k_i=h_1+\sum_{i\in S}\alpha_i.$$
Lastly, let $y^0=h_{m+1}$ and define $z^0\in \{0,1\}^{[n]}$ as follows: $$z^0_i:=\left\{\begin{array}{ll} 1 ~~~ \text{ if } i< m+1 \text{ or } i\in S; \\ 0 ~~~ \text{ otherwise.} \end{array} \right.$$ Note that $z^0_i=1$ for all $i<m+1$. Also, we have $$\sum_{i=1}^{n}a_iz^k_i= \sum_{i<m+1}a_iz^k_i+ \sum_{i\in S}z^k_i=s_{m}+|S|=s_m+p-s_m= p.$$ Hence, $(y^0,z^0)\in \mathcal{Q}$. Moreover, $$\begin{aligned}
y^0+\sum_{i=1}^{a}(h_{t_i}-h_{t_{i+1}}) z^0_{t_i}+\sum_{i\in S}\alpha_iz^0_i&=h_{m+1}+\sum_{i=1}^{a}(h_{t_i}-h_{t_{i+1}})+\sum_{i\in S}\alpha_i\\
&=h_{m+1}+h_1-h_{m+1}+\sum_{i\in S}\alpha_i\\
&=h_1+\sum_{i\in S}\alpha_i.\end{aligned}$$
Hence, the face defined by (\[fdieg\]) contains $z^0,z^1,\ldots,z^{n}$, which are $n+1$ affinely independent points in $\mathcal{Q}$. As a result, (\[fdieg\]) is a facet-defining inequality for conv($\mathcal{Q}$).
Heuristic Separation over $\text{conv}(\mathcal{Q})$ {#heuristicsep}
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In this section, we give a polynomial time algorithm that separates over a subset of inequalities of the type $$y+\sum_{j\in [n]}\alpha_jz_j\geq \beta~~~~\forall~\alpha\in \mathbb{R}^n:\alpha_j\geq 0 \text{ if and only if } j\in R,$$ for a fixed subset $R$ of $[n]$. When $a_i=a_j$ for all $i,j\notin R$, this separation is exact.
Let $R\cup S$ be a partition of $[n]$. Define the polyhedron $$\mathcal{G}(R):=\left\{(\delta,\Delta,h)\in \mathbb{R}^R_+\times \mathbb{R}^S_+\times \mathbb{R}: (\ref{Gconst})\right\}$$ where $$\begin{aligned}
\sum_{j\in R,j\leq k}\delta_j+ \sum_{j\in S,j> k}\Delta_j(1-z^*_j) +h_{k+1}&\geq h ~~~~\forall z^*\in \mathcal{P}_k, \forall ~0\leq k\leq \nu.\label{Gconst} \end{aligned}$$ The following theorem explains the importance of the polyhedron $\mathcal{G}(R)$.
\[RSvalidthm\] Let $R\cup S$ be a partition of $[n]$. Choose $(\delta,\Delta,h)\in \mathbb{R}_+^R\times \mathbb{R}_+^S\times \mathbb{R}$. Then $$y+\sum_{j\in R}\delta_jz_j+\sum_{j\in S}\Delta_j(1-z_j)\geq h$$ is a valid inequality for $\emph{conv}(\mathcal{Q})$ if and only if $(\delta,\Delta,h)\in \mathcal{G}(R)$.
The proof is very similar to that of Theorem \[validthm\] and is therefore omitted.
Again, we would like to point out that $\mathcal{G}(R)$ in general has exponentially many constraints. However, $\mathcal{G}(R)$ has an alternate formulation with $O(n)$ non-linear constraints: for $\Delta\in \mathbb{R}_+^S$ and $k\in \mathbb{Z}$, let $$g_k(\Delta):=\min \left\{\sum_{j\in S,j> k}\Delta_j(1-z_j):z\in \mathcal{P}_k \right\}.$$ Then we have $$\mathcal{G}(R)=\left\{(\delta,\Delta,h)\in \mathbb{R}_+^R\times \mathbb{R}_+^S\times \mathbb{R}: \sum_{j\in R,j\leq k}\delta_j+g_k(\Delta)+h_{k+1} \geq h ~\forall ~0\leq k\leq \nu \right\}.$$ We will now find a polyhedron that is contained in $\mathcal{G}(R)$ and is equal to $\mathcal{G}(R)$ when $a_i=a_j$ for all $i,j\in S$, and can be described efficiently.
Let $\Delta=(\Delta_j:j\in S)\subset \mathbb{N}$ where $S$ is some index set, and let $k,l\in \mathbb{N}$. Define $\Delta [k,l]$ to be the sum of the smallest $l-|\{j\in S:j\leq k\}|$ elements in $\{\Delta_j: j> k\}$.
Let $R\cup S$ be a partition of $[n]$, and suppose that $m_S:=\min\{a_j:j\in S\}>0$.
\[approxobs\] Given $\Delta\in \mathbb{R}^S_+$ and $0\leq k\leq \nu$, we have $$g_k(\Delta)\geq \Delta\left[k,|S|-\left \lfloor\frac{p-s_k}{m_S}\right\rfloor\right].$$ Moreover, the inequality is tight when $a_j=m_S$ for all $j\in S$.
Choose $z\in \mathcal{P}_k$. We have that $$|\{j\in S:j>k, z_j=1\}|= \sum_{j\in S, j>k}z_j\leq \frac{1}{m_S}\sum_{j\in S, j>k}a_jz_j\leq \frac{1}{m_S}\left(p-\sum_{j\leq k}a_j\right)=\frac{p-s_k}{m_S}.$$ Therefore, $$\begin{aligned}
|\{j\in S:j> k, z_j=0\}|&=|\{j\in S:j>k\}|-|\{j\in S:j>k, z_j=1\}|\\
&\geq |S|-|\{j\in S:j\leq k\}|-\left\lfloor\frac{p-s_k}{m_k}\right\rfloor.\end{aligned}$$ As a result, we obtain that $\sum_{j\in S, j>k}\Delta_j(1-z_j)\geq \Delta\left[k,|S|-\left \lfloor\frac{p-s_k}{m_k}\right\rfloor\right].$ Since this is true for all $z\in \mathcal{P}_k$, it follows that $$g_k(\Delta)\geq \Delta\left[k,|S|-\left \lfloor\frac{p-s_k}{m_k}\right\rfloor\right],$$ as claimed. In the case when $a_j=m_S$ for all $j\in S$, consider the point $z^*\in \{0,1\}^n$ defined as follows: $z^*_j=1$ if $j\leq k$ or $\Delta_j$ corresponds to one of the largest $\left\lfloor \frac{p-s_k}{m_S}\right \rfloor$ elements in $\{\Delta_j:j\in S:j>k\}$, and $z^*_j=0$ otherwise. Then $z^*\in \mathcal{P}_k$ and $\sum_{j\in S, j>k}\Delta_j(1-z^*_j)=\Delta\left[k,|S|-\left \lfloor\frac{p-s_k}{m_k}\right\rfloor\right].$ Hence, when $a_j=m_S$ for all $j\in S$, we have that $g_k(\Delta)= \Delta\left[k,|S|-\left \lfloor\frac{p-s_k}{m_S}\right\rfloor\right].$
Define the polyhedron $$\mathcal{A}(R):=\left\{(\delta,\Delta,h)\in \mathbb{R}_+^R\times \mathbb{R}_+^S\times \mathbb{R}: (\ref{ARSconst})\right\}$$ where $$\begin{aligned}
\sum_{j\in R, j\leq k}\delta_j+ \Delta\left[k,|S|-\left \lfloor\frac{p-s_k}{m_S}\right\rfloor\right] +h_{k+1}\geq h ~~~~ \forall ~0\leq k\leq \nu.\label{ARSconst} \end{aligned}$$ Note that Observation \[approxobs\] implies that $\mathcal{A}(R)\subset \mathcal{G}(R)$, and that $\mathcal{A}(R)= \mathcal{G}(R)$ when $a_j=m_S$ for all $j\in S$.
For each $0\leq k\leq \nu$, let $\beta^k:=|S|-|\{j\in S:j\leq k\}|-\left\lfloor\frac{p-s_k}{m_S}\right\rfloor$. We have $$\begin{array}{ccl} \Delta\left[k,|S|-\left\lfloor\frac{p-s_k}{m_S}\right\rfloor\right] = &\min & \sum_{j\in S, j>k} \Delta_jx_j\\
&{\rm s.t.} & \sum_{j\in S,j>k}x_j=\beta^k, \\ & & x_j\in [0,1] ~~\forall ~j\in S,j>k.\end{array}$$ (Note that the constraint matrix of the above linear program is totally unimodular.) Hence, by LP duality, we obtain that $$\begin{array}{ccl} \Delta\left[k,|S|-\left\lfloor\frac{p-s_k}{m_S}\right\rfloor\right] = &\max & \beta^k\gamma^k+\sum_{j\in S,j>k} \rho^k_j\\
&{\rm s.t.} & \gamma^k+\rho^k_j\leq \Delta_j~~~~~~~\forall ~j\in S,j>k, \\ & & \rho^k_j\leq 0 ~~~~~~~~~~~~~~~~\forall ~j\in S,j>k.\end{array}$$ Now let $\rho^k_j=0$ for all $j\in S$ with $j\leq k$. So $\rho^k\in \mathbb{R}^S_-$ for all $0\leq k\leq \nu$. With this, define the polyhedron $$\Theta(R):=\left\{\left(\delta,\Delta,h, \left(\gamma^k\right),\left(\rho^k\right)\right)\in \mathbb{R}_+^R\times \mathbb{R}_+^S\times \mathbb{R} \times \mathbb{R}^{\nu+1}\times \mathbb{R}_-^{S\times {\nu+1}}: (\ref{thetaconst1})-(\ref{thetaconst3})\right\}$$ where $$\begin{aligned}
\sum_{j\in R, j\leq k}\delta_j+\beta^k\gamma^k+\sum_{j\in S} \rho^k_j+h_{k+1} &\geq h ~~~~~~~\forall ~0\leq k\leq \nu, \label{thetaconst1}\\
\gamma^k+\rho^k_j&\leq \Delta_j~~~~~~\forall ~j\in S,j>k, \forall ~0\leq k\leq \nu,\label{thetaconst2}\\
\rho^k_j&=0~~~~~~~~\forall~j\in S,j\leq k,\forall ~0\leq k\leq \nu.\label{thetaconst3} \end{aligned}$$
Hence, by weak and strong LP duality we get the following.
\[RSprojlemma\] $\mathcal{A}(R)=\emph{proj}_{\mathbb{R}^R\times \mathbb{R}^S\times \mathbb{R}}\Theta(R)$.
This lemma implies the following theorem.
Let $R\cup S$ be a partition of $[n]$ such that $m_S>0$. Let $(y^*,z^*)\in \mathbb{R}\times \mathbb{R}^n_+$ and $$\begin{aligned}
LP^*:=\min\left\{\sum_{j\in R}\delta_jz^*_j+\sum_{j\in S}\Delta_j(1-z^*_j)- h:(\delta,\Delta,h)\in \mathcal{A}(R)\right\}.\label{RSsepopt}\end{aligned}$$ If $y^*+LP^*<0$ and $(\delta^*,\Delta^*,h^*)$ is an optimal solution to $(\ref{RSsepopt})$, then $(y^*,z^*)\notin \emph{conv}(\mathcal{Q})$ and $y+\sum_{j\in R}\delta^*_jz_j+\sum_{j\in S}\Delta^*_j(1-z_j)\geq h^*$ is a valid inequality for $\emph{conv}(\mathcal{Q})$ which is violated by $(y^*,z^*)$. Furthermore, when $a_j=m_S$ for all $j\in S$, separation over all inequalities of the type $$\begin{aligned}
y+\sum_{j\in R}\delta_jz_j+\sum_{j\in S}\Delta_j(1-z_j)\geq h~~~(\delta,\Delta,h)\in \mathbb{R}^R_+\times \mathbb{R}^S_+\times \mathbb{R},\label{RStype}\end{aligned}$$ can be accomplished in polynomial time.
Suppose that $y^*+LP^*< 0$ and that $(\delta^*,\Delta^*,h^*)$ is an optimal solution to $(\ref{RSsepopt})$. Then, by Lemma \[RSprojlemma\], we know that $(\delta^*,\Delta^*,h^*)\in \mathcal{A}(R)\subset \mathcal{G}(R)$, and so by Corollary \[RSvalidthm\], $y+\sum_{j\in R}\delta^*_jz_j+\sum_{j\in S}\Delta^*_j(1-z_j)\geq h^*$ is a valid inequality for $\text{conv}(\mathcal{Q})$. Since $y^*+LP^*<0$, it follows that $(y^*,z^*)$ violates this inequality and so $(y^*,z^*)\notin \text{conv}(\mathcal{Q})$. When $a_j=m_S$ for all $j\in S$, we have that $\mathcal{G}(R)=\mathcal{A}(R)= \text{proj}_{\mathbb{R}^R\times \mathbb{R}^S\times \mathbb{R}}\Theta(R)$, and so by solving the linear program (\[RSsepopt\]) (in polynomial time), one can separate over all inequalities of the type (\[RStype\]).
Observe that the above algorithm yields a heuristic separation algorithm over all inequalities of the type $$y+\sum_{j\in [n]}\alpha_jz_j\geq \beta~~~~\forall~\alpha\in \mathbb{R}^n:\alpha_j\geq 0 \text{ if and only if } j\in R,$$ for a fixed subset $R$ of $[n]$. Furthermore, this separation algorithm is exact when $a_i=m_S>0$ for all $i\in S$.
Conclusion {#concl}
==========
In this paper, our main purpose is to recognize some structural properties of the convex hull of the mixing set subject to a knapsack constraint arising in chanced-constrained programming. We start off by characterizing the set of all the valid inequalities for this polyhedron. This characterization helps us in two ways. Firstly, it helps us find a new class of explicit facet-defining inequalities that subsumes the class of strengthened star-inequalities, which were the [*only*]{} known explicit class previously known for the general knapsack constraint. Secondly, it helps us in finding a polynomial time heuristic separation algorithm for the polyhedron. We also give necessary conditions for the facet-defining inequalities of the polyhedron.
A complete characterization of the facet-defining inequalities of the convex hull of the mixing set subject to a knapsack constraint arising in chanced-constrained programming remains an open problem. We also intend to perform computational experiments with the proposed inequalities to measure their effectiveness.
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---
abstract: 'In cosmological first-order phase transitions, gravitational waves are generated by the collisions of bubble walls and by the bulk motions caused in the fluid. A sizeable signal may result from fast-moving walls. In this work we study the hydrodynamics associated to the fastest propagation modes, namely, ultra-relativistic detonations and runaway solutions. We compute the energy injected by the phase transition into the fluid and the energy which accumulates in the bubble walls. We provide analytic approximations and fits as functions of the net force acting on the wall, which can be readily evaluated for specific models. We also study the back-reaction of hydrodynamics on the wall motion, and we discuss the extrapolation of the friction force away from the ultra-relativistic limit. We use these results to estimate the gravitational wave signal from detonations and runaway walls.'
author:
- |
Leonardo Leitao[^1] and Ariel Mégevand[^2]\
*IFIMAR (CONICET-UNMdP)\
*Departamento de Física, Facultad de Ciencias Exactas y Naturales,\
*UNMdP, Deán Funes 3350, (7600) Mar del Plata, Argentina***
title: 'Hydrodynamics of ultra-relativistic bubble walls'
---
Introduction
============
A first-order phase transition of the universe proceeds by nucleation and expansion of bubbles, and may have different cosmological consequences, depending on the velocity of bubble growth. For instance, the generation of the baryon asymmetry of the universe in the electroweak phase transition is most efficient for non-relativistic bubble walls, and is suppressed as the bubble wall velocity approaches the speed of sound in the plasma [@baryo]. In contrast, the formation of gravitational waves may be sizeable if the wall velocity is supersonic [@gw]. These cosmological consequences generally depend not only on the wall velocity but also on the bulk motions of the plasma caused by the wall. For instance, gravitational waves are generated by bubble collisions [@gw; @gwcol; @hk08] as well as by turbulence [@gwturb; @cds09; @cds10] and sound waves [@gwsound].
The propagation of the phase transition fronts (bubble walls) is affected by hydrodynamics in a non-trivial manner (see, e.g., [@hidrowall; @ikkl94; @ms09; @kn11; @inst]). The wall motion is driven essentially by the difference of pressure between the two phases. This force grows with the amount of supercooling, i.e., the further down the temperature descends below the critical temperature, the larger the pressure difference between phases. As a consequence, the driving force is very sensitive to the (inhomogeneous) reheating which occurs due to the release of latent heat.
Besides, the microscopic interactions of the particles of the plasma with the wall cause a friction force on the latter (see, e.g., [@mp95]). Computing the friction force is a difficult task, and for many years only the non-relativistic (NR) case was studied [@nrfric]. In this approximation, a wall velocity $v_{w}\ll 1$ is assumed, and the friction force scales as $v_{w}$. Beyond the NR regime, a dependence $v_w\gamma_w$ was usually assumed, where $\gamma_w=1/\sqrt{1-v_{w}^{2}}$. As a consequence of this scaling, the wall would always reach a terminal velocity. More recently, the *total* force acting on the wall was calculated in the ultra-relativistic (UR) limit, $\gamma _{w}\gg 1$ [@bm09]. The result does not allow to discriminate the friction or the hydrodynamic effects. Nevertheless, the net force $F_{\mathrm{net}}$ is independent of $v_{w}$, which means that the friction saturates as a function of $v_w\gamma _{w}$. As a consequence, the wall may run away. For intermediate velocities, microscopic calculations of the friction were hardly attempted [@hs13; @knr14]. To compute the wall velocity, phenomenological interpolations between the NR and the UR limits have been considered in Refs. [@ekns10; @ariel13].
Leaving aside the determination of the wall velocity, the perturbations caused in the plasma by the moving wall have been extensively studied for the case of a stationary solution [@kurki85; @hidro; @lm11; @lm15]. Different hydrodynamic regimes can be established, depending on the wall velocity. For a subsonic wall the hydrodynamic solution is a weak deflagration, in which the wall is preceded by a shock wave. For a supersonic wall, we have a Jouguet deflagration if the wall velocity is smaller than the Jouguet velocity. In this case, the fluid is disturbed both in front and behind the wall. For higher wall velocities, the solution is a weak detonation. For the detonation, the velocity is so high that the fluid in front of the wall is unaffected. In this case, the wall is followed by a rarefaction wave.
The steady-state hydrodynamics can be investigated as a function of thermodynamic parameters (such as the latent heat) and of the wall velocity (i.e., considering $v_w$ as a free parameter). Thus, in particular, the kinetic energy in bulk motions of the plasma, which is relevant for the generation of gravitational waves, was computed in Refs. [@ekns10; @lm11] for the whole velocity range $0<v_w<1$. These results are useful for applications, as they do not depend on a particular calculation of the wall velocity for a specific model.
For the runaway case, the hydrodynamics was considered in Ref. [@ekns10]. However, the results rely on the decomposition of the total force into driving and friction forces, and are sensitive to approximations. The decomposition of the UR force was discussed also in Ref. [@ariel13]. Since the net force is known [@bm09], it is actually not necessary, in the UR limit, to determine the friction component in order to study the wall motion. However, identifying the forces acting on the wall is useful, in the first place, to understand the hydrodynamics, and, in the second place, to construct a phenomenological model for the friction, which allows to compute the wall velocity away from the UR limit.
In this paper we consider ultra-relativistic walls and we study, on the one hand, the hydrodynamics as a function of the wall acceleration, and, on the other hand, the role of hydrodynamics and friction in the determination of the net force. In particular, we obtain the energy in bulk fluid motions as a function of the net force $F_{\mathrm{net}}$, in the whole range of runaway solutions. We also discuss the effect of reheating on the force, and we compare with approximations used in previous approaches. We apply these results to the estimation of the gravitational wave signal from phase transitions.
The paper is organized as follows. In Sec. \[review\] we review the dynamics of the Higgs-fluid system. In Sec. \[hidro\] we consider the hydrodynamics of detonations and runaway walls for given values of the wall velocity and acceleration, while in Sec. \[micro\] we consider the wall equation of motion and we analyze the dependence of the energy distribution on thermodynamic and friction parameters. In Sec. \[gw\] we estimate the amplitude of the gravitational waves as a function of all these quantities. We summarize our conclusions in Sec. \[conclu\]. In appendix \[apend\] we find analytic results for the efficiency factor for the case of planar walls, and we provide fits for the case of spherical walls.
The Higgs-fluid system \[review\]
=================================
To describe the phase transition, we shall consider a system consisting of an order-parameter field $\phi (x)$ (the Higgs field) and a relativistic fluid (the hot plasma). The latter is characterized by a four-velocity field $ u^{\mu
}(x)$ and the temperature $T(x)$. The phase transition dynamics is mostly determined by the free-energy density, also called finite-temperature effective potential. For a given model, it is given by $$\mathcal{F}(\phi,T)=V(\phi) +V_T(\phi), \label{ftot}$$where $V(\phi)$ is the zero-temperature effective potential and $V_T(\phi)$ the finite-temperature correction. To one-loop order, the latter is given by [@quiros] $$V_T(\phi) =\sum_{i}(\pm {g_{i}})T
\int\frac{d^3p}{(2\pi)^3}\log \left( 1\mp e^{-E_i/T} \right),
\label{f1loop}$$ where the sum runs over particle species, $g_{i}$ is the number of degrees of freedom of species $i$, the upper sign stands for bosons, the lower sign stands for fermions, and $E_i=\sqrt{p^2+m_i^2(\phi)}$, where $m_i$ are the Higgs-dependent masses.
We may have a phase transition if the free-energy density has two minima $\phi
_{\pm }(T)$, corresponding to the two phases of the system. At high temperatures the absolute minimum is $\phi _{+}$, while at low temperatures the absolute minimum is $\phi _{-}$. Hence, the system is initially in a state characterized by $\phi (x)\equiv \phi _{+}$, which we shall refer to as “the $+$ phase”. Similarly, at late times the universe is in “the $-$ phase”, characterized by $\phi (x)\equiv \phi _{-}$. In the case of a first-order phase transition, there is a temperature range in which these minima coexist in the free energy, separated by a barrier. The critical temperature $T_{c}$ is given by the condition $ \mathcal{F}(\phi _{+},T_{c})=\mathcal{F}(\phi _{-},T_{c})$. Below the critical temperature, bubbles of the $-$ phase appear, inside which we have $\phi =\phi _{-}$.
The growth of a bubble can be studied by considering the equations for the variables $\phi,u^\mu,T$. The dynamics of the fluid variables can be obtained from the conservation of the energy-momentum tensor. For the Higgs-fluid system we have (see, e.g. [@ikkl94]) $$T_{\mu \nu }=\partial _{\mu }\phi \partial _{\nu }\phi -g_{\mu \nu }\left[
\frac{1}{2}\partial _{\alpha }\phi \partial ^{\alpha }\phi -\mathcal{F}(\phi
,T)\right] -u_{\mu }u_{\nu }T\frac{\partial \mathcal{F}}{\partial T}(\phi
,T),$$ with $g^{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$. Conservation of $T^{\mu \nu
}$ gives $$\partial _{\mu }\left[ T\frac{\partial \mathcal{F}}{\partial T}u^{\mu
}u^{\nu }-\mathcal{F}g^{\mu \nu }\right] =\square \phi \,\partial ^{\nu
}\phi . \label{constmn}$$ These equations govern the fluid dynamics and also contain the interaction of the fluid with the scalar field $\phi$. The evolution of $\phi$ is governed by a finite-temperature equation of motion of the form [@mp95] $$\square \phi +\frac{\partial \mathcal{F}}{\partial \phi }+
\sum_i g_i\frac{dm^2_i}{d\phi}\int\frac{d^3p}{(2\pi)^32E_i}
\delta f_i=0.
\label{fieldeq}$$ Here, the derivative of the finite-temperature effective potential takes into account quantum and thermal corrections to the tree-level field equation, where the thermal corrections are calculated from the equilibrium distribution functions $f_i^{\mathrm{eq}}(p)=1/(e^{E_i/T}\mp 1)$. On the other hand, $\delta
f_i$ are the deviations from the equilibrium distributions. The last term constitutes a damping due to the presence of the plasma. Computing $\delta f_i$ generally involves solving a system of Boltzmann equations which take into account all the particles interactions.
In the bubble configuration, the bubble wall separates the two phases, i.e., the regions with $\phi=\phi_+$ and $\phi=\phi_-$. Thus, by definition, the field varies only inside the bubble wall. As a consequence, away from the wall, Eqs. (\[constmn\]) give equations for the fluid alone, $$\partial _{\mu
}T_{\mathrm{fl}}^{\mu \nu }=0,\quad \mbox{with}\quad T_{\mathrm{fl}}^{\mu \nu }=u^{\mu }
u^{\nu }w-g^{\mu \nu }p, \label{tmnp}$$ where $p$ is the pressure and $w$ is the enthalpy density. In each phase, these quantities are given by the free-energy density $\mathcal{F}_\pm(T)=\mathcal{F}(\phi_\pm,T)$ through the well-known thermodynamic relations $p=-\mathcal{F}, w=Tdp/dT=e+p$, where $e$ is the energy density. The energy involved in the wall and fluid motions and in the reheating comes from the difference of energy density between the two phases. This energy is released at the phase transition fronts. The latent heat is defined as $L=e_+(T_c)-e_-(T_c)$. For the treatment of hydrodynamics, the wall can be assumed to be infinitely thin. Therefore, we shall simplify the system by considering the fluid equations (\[tmnp\]) together with an equation of motion for the wall (rather than for the Higgs field).
An equation for the wall can be obtained from Eq. (\[fieldeq\]) by multiplying by $\partial_\mu \phi$ and integrating in the direction perpendicular to the wall (see Sec. \[micro\]). In this way, from the first term in (\[fieldeq\]) we obtain a term which is proportional to the wall acceleration. If we ignore hydrodynamics (i.e., temperature gradients), the second term gives the difference between the pressures on each side of the wall, $p_--p_+$. This is a positive force acting on the wall. In contrast, the deviations from equilibrium $\delta f_i$ in the last term turn out to oppose the wall motion. It is well known that, for a small wall velocity $v_w$, the last term in (\[fieldeq\]) gives a term proportional to $-v_w$ in the wall equation, i.e., a friction force. As a consequence of the friction, the wall may reach a steady-state regime of constant velocity. However, it is known that such a steady state does not always exist, either due to instabilities which make the wall motion turbulent [@inst], or just because the friction is not high enough to prevent the wall to run away [@bm09]. In the latter case, the wall quickly reaches velocities $v_{w}\simeq 1$, with increasingly high values of the gamma factor.
Interestingly, the ultra-relativistic case turns out to be much simpler than the non-relativistic one. This is because particles which cross the UR wall do not have time to interact, and Boltzmann equations need not be considered. In this case, it is simpler to compute the complete occupancies $f_i$ rather than the deviations $\delta f_i$, i.e., to consider the second and third terms of Eq. (\[fieldeq\]) simultaneously. Macroscopically, this amounts to calculating the [total]{} force acting on the wall. The result (for particle masses which vanish in the $+$ phase) is a net force given by [@bm09] $$F_{\mathrm{net}}=V(\phi_+)-V(\phi_-)-\sum_i g_i c_i\frac{T_+^2m_i^2(\phi_-)}{24},
\label{Fnet}$$ where $c_i=1$ ($1/2$) for bosons (fermions), and $T_+$ is the temperature of the unperturbed fluid in front of the wall. Notice that this force does not depend on the wall velocity. As a consequence, if the wall reaches the UR regime with a positive $F_{\mathrm{net}}$, then it will run away.
In order to determine the actual value of $v_{w}$, the force acting on the wall in the whole range $0<v_{w}<1$ is needed. The friction force seems to be generally a growing function of $v_{w}$, although the usual NR approximations break-down around the speed of sound [@knr14]. The fact that $F_{\mathrm{net}}$ becomes independent of $v_{w}$ in the ultra-relativistic limit implies that the friction force saturates as a function of $v_{w}\gamma
_{w}$. A phenomenological model for the friction force, which interpolates between the NR and UR regimes, was introduced in Ref. [@ariel13]. It consists in replacing the last term in the field equation (\[fieldeq\]) with a simpler damping term, $$\mathcal{K}=\frac{f(\phi )\,u^{\mu }\partial _{\mu }\phi }{\sqrt{1+[g(\phi
)\,u^{\mu }\partial _{\mu }\phi ]^{2}}}, \label{Kfeno}$$ where $f$ and $g$ are scalar functions which can be chosen suitably to give the correct $\phi$ dependence of the friction. Considering Eq. (\[Kfeno\]) in the wall frame, it was shown in [@ariel13] that this term gives a friction force which has the correct velocity dependence in the NR and UR limits. In Sec. \[micro\] we shall repeat the derivation in the plasma frame.
Hydrodynamics {#hidro}
=============
Let us consider a wall moving with velocity $v_w$. We will assume that the wall is infinitely thin, and that the symmetry of the problem is such that the velocity of the fluid is perpendicular to the wall (e.g., spherical or planar symmetry). Thus, the fluid is characterized by two variables, namely, the temperature $T$ and a single component of the velocity, $v$. We are interested in supersonic walls, i.e., with $v_w>c_+$, where $c_+=\sqrt{dp_+/de_+}$ is the speed of sound in the plasma in the $+$ phase. Concretely, we shall only consider wall velocities which are so high that the fluid in front of the wall is unperturbed. Therefore, in the $+$ phase the fluid velocity $v_+$ vanishes and the temperature $T_{+}$ is set by the nucleation temperature. We will also assume that the fluid behind the wall is in local equilibrium, so that the variables $T$ and $v$ are well defined everywhere. Inside the bubble, the fluid variables are given by Eqs. (\[tmnp\]). Their values $v_-,T_-$ next to the wall can be obtained by integrating the equations $\partial _{\mu }T^{\mu \nu
}=0$ across the interface. Besides, we have the boundary condition that the fluid velocity vanishes at the bubble center.
Detonations
-----------
In the steady-state case, it is usual to consider the reference frame of the wall, so that time derivatives vanish. We shall consider instead the rest frame of the unperturbed fluid in front of the wall (the plasma frame), where it is easier to take the limit $v_{w}\rightarrow 1$. We thus have $v_+=0$. We can obtain $v_{-}$, $T_{-}$ as functions of $v_{w},T_{+}$ from the continuity equations for energy and momentum.
Consider a piece of wall of surface area $A$ which is small enough that it can be regarded as planar. Locally, we place the coordinate system so that the wall moves in the positive $z$-direction with velocity $v_{w}$. Thus, we only need to consider time and $z$ components of $T^{\mu \nu }$. In a small time $\Delta
t$ the wall moves a distance $\Delta z=v_{w}\Delta t$. During this time, we have an incoming energy flux from the left of the wall, given by $T_{-}^{0z}$, and an outgoing flux to the right, given by $T_{+}^{0z}$. Therefore, a net energy $(T_{-}^{0z}-T_{+}^{0z})A\Delta t$ will accumulate in the interface unless it is transferred to the plasma. The change of energy in the plasma in the volume $A\Delta z$ is given by $(T_{-}^{00}-T_{+}^{00})Av_w\Delta t$. For a steady-state wall the energy balance gives $$T_{-}^{0z}-T_{+}^{0z}=(T_{-}^{00}-T_{+}^{00})v_{w}. \label{enbal}$$ Similarly, considering the momentum density $T^{z0}$ and momentum flux $
T^{zz}$, we obtain $$T_{-}^{zz}-T_{+}^{zz}=(T_{-}^{z0}-T_{+}^{z0})v_{w}. \label{mombal}$$
On each side of the wall, we have $T^{\mu \nu }=T_{\mathrm{fl} }^{\mu \nu }$ given by Eq. (\[tmnp\]). In our reference frame we have $u^{\mu
}=(\gamma,0,0,\gamma v)$, with $\gamma=1/\sqrt{1-v^2}$. Since $v_+=0$ and $v_->0$, we have $$T_{+}^{00}=e_{+},\quad T_{+}^{0z}=T_{+}^{z0}=0,\quad T_{+}^{zz}=p_{+},
\label{tmas}$$ and $$T_{-}^{00}=w_{-}\gamma _{-}^{2}-p_{-},\quad
T_{-}^{0z}=T_{-}^{z0}=w_{-}\gamma _{-}^{2}v_{-},\quad
T_{-}^{zz}=w_{-}\gamma _{-}^{2}v_{-}^{2}+p_{-}. \label{tmenos}$$ Inserting Eqs. (\[tmas\]-\[tmenos\]) in (\[enbal\]-\[mombal\]), we obtain the system of equations $$\begin{aligned}
w_{-}(v_{w}-v_{-}) &=&(p_{-}+e_{+})v_{w}(1-v_{-}^{2}), \label{junc1} \\
w_{-}(v_{w}-v_{-})v_{-} &=&(p_{-}-p_{+})(1-v_{-}^{2}), \label{junc2}\end{aligned}$$ from which we readily obtain $$v_{-}v_{w}=\frac{p_{-}-p_{+}}{e_{+}+p_{-}},\quad \frac{v_{-}}{v_{w}}=\frac{
e_{-}-e_{+}}{e_{-}+p_{+}}. \label{velplasma}$$ These expressions are similar, but different, to the usual expressions for $v_{+}$ and $v_{-}$ in the wall frame.
Since the variables $w,p,e,T$ are related by the equation of state (EOS), Eqs. (\[velplasma\]) can be solved for, say, $w_-$ and $v_-$ as functions of $w_+$ and $v_w$. It is not difficult to see that the derivatives $\partial
w_-/\partial v_w|_{T_+}$ and $\partial v_-/\partial v_w|_{T_+}$ diverge for $v_w$ such that[^3] $$\frac{v_w-v_-}{1-v_wv_-}= c_-, \label{Jouguet}$$ where $c_-=\sqrt{dp_-/de_-}$ is the speed of sound in the $-$ phase. The left-hand side of Eq. (\[Jouguet\]) gives the value of the fluid velocity in the reference frame of the wall. Therefore, the mentioned divergence occurs when the outgoing flow velocity in the wall frame reaches the speed of sound. This indicates that the hydrodynamics becomes too strong at this point. For a detonation, the incoming flow velocity is given by $v_w$, which is supersonic. Detonations are divided into weak detonations, for which the outgoing flow is supersonic too, strong detonations, for which the outgoing flow is subsonic, and Jouguet detonations, which are characterized by the condition (\[Jouguet\]). This means that in the plasma frame, weak detonations correspond to smaller values of $v_-$ (i.e., to solutions which do not perturb the fluid strongly) while strong detonations correspond to higher values of $v_-$.
The behavior of $v_-$ as a function of $v_w$ is shown in Fig. \[figvmedet\] for a simple equation of state (the bag EOS) considered below. The upper branch corresponds to strong detonations, the lower branch corresponds to weak detonations, and the red dots indicate the Jouguet point.
=7.5cm
The abovementioned divergence of $\partial v_-/\partial v_w|_{T_+}$ can be observed in the figure. It causes $v_w$ to be a minimum at the Jouguet point. As a consequence, for detonations the wall velocity is in the range $v_{J}\leq
v_{w}<1$, where $v_{J}(T_+)$ is velocity of the Jouguet detonation. It is well known that strong detonations are not compatible with the solutions for the fluid profile behind the wall. Therefore, the upper curves in Fig. \[figvmedet\] do not correspond to physical solutions. Weak detonations become weaker (i.e., $v_-$ decreases) for higher wall velocities. In the limit $v_w\to 1$, Eqs. (\[velplasma\]) become $$e_{-}-p_{-}=e_{+}-p_{+},\quad v_{-}=\frac{p_{-}-p_{+}}{e_{+}+p_{-}}=\frac{
e_{-}-e_{+}}{e_{+}+p_{-}}. \label{hydreldet}$$ The first of these equations gives the temperature $T_{-}$ as a function of $
T_{+}$ for an ultra-relativistic stationary solution. The second one gives the fluid velocity behind the interface.
Runaway walls
-------------
If the wall is accelerated, we have to take into account the fact that a part of the energy accumulates in the wall [@bm09]. In the time $\Delta t$, an amount of energy $A\Delta \sigma $ is accumulated in a surface area $A$ of the interface, where $\sigma $ is the surface energy density. Hence, the energy balance now gives $$T_{-}^{0z}-T_{+}^{0z}=(T_{-}^{00}-T_{+}^{00})v_{w}+\frac{d\sigma }{dt}.
\label{juncr1}$$ Similarly, since the momentum of a piece of wall is given by $A\sigma v_{w} $, we have $$T_{-}^{zz}-T_{+}^{zz}=(T_{-}^{z0}-T_{+}^{z0})v_{w}+\frac{d(v_{w}\sigma )}{dt}.
\label{juncr2}$$
After a certain (generally short) period of time, the accelerated wall will either reach a terminal velocity or accelerate to ultra-relativistic velocities. The ultra-relativistic accelerated regime is similar to the steady-state case in the sense that the wall velocity is essentially a constant, $v_{w}\simeq 1$ (although $\gamma_w$ and $\sigma$ vary). In this limit we have $$\frac{d\sigma }{dt}=\frac{d(v_{w}\sigma )}{dt}=F_{\mathrm{net}},
\label{dsigf}$$ where $F_{\mathrm{net}}$ is the net force per unit area acting on the wall. Inserting Eqs. (\[tmas\]-\[tmenos\]) in (\[juncr1\]-\[juncr2\]) we obtain $$e_{-}-p_{-}=e_{+}-p_{+}-2F_{\mathrm{net}},\quad v_{-}=\frac{w_{-}-w_{+}}{
w_{-}+w_{+}}. \label{eqsrun}$$ For $F_{\mathrm{net}}=0$, Eqs. (\[eqsrun\]) match the ultra-relativistic detonation case, Eq. (\[hydreldet\]). From Eqs. (\[eqsrun\]) we may obtain the temperature $T_{-}$ and the velocity $v_{-}$ as functions of $F_{\mathrm{net}}$ and $T_{+}$. We see that, for a constant net force, $v_-$ and $T_-$ are constant, like in the stationary case.
Fluid profiles
--------------
The profiles of $v$ and $T$ behind the wall are a solution of Eqs. (\[tmnp\]) with boundary conditions $v=v_{-}$, $T=T_{-}$ at the wall. For a system with spherical, cylindrical or planar symmetry, the problem is 1+1 dimensional, since the fluid profile depends only on time and on the distance $r$ from the center, axis or plane of symmetry [@kurki85]. Besides, since there is no distance scale in the fluid equations, it is customary to assume the similarity condition, namely, that the solutions depend only on the variable $\xi =r/t$. With this assumption, one obtains the equation for the wall velocity [@lm11] $$\gamma ^{2}(1-v\xi )\left[ \frac{1}{c_-^{2}}\left( \frac{\xi -v}{1-\xi v}
\right) ^{2}-1\right] v^{\prime }=j\frac{v}{\xi }, \label{fluideqv}$$ where a prime indicates a derivative with respect to $\xi$, and $j=2$, $1$, or $0$ for spherical, cylindrical, or planar walls, respectively. The enthalpy profile is given by the equation $$\frac{w^{\prime }}{w}=\left( \frac{1}{c_-^{2}}+1\right) \frac{\xi -v}{
1-\xi v}\gamma ^{2}v^{\prime }. \label{fluideqw}$$
It is important to note that the similarity condition is compatible with a wall which is placed at a fixed value of $\xi$, namely, $\xi _{w}=v_{w}$. For an accelerated wall, this condition will not be compatible, in general, with the boundary conditions at the interface. Nevertheless, in the ultra-relativistic limit, the wall position corresponds essentially to the constant value $\xi
_{w}=1$. Indeed, as we have seen, the values of $ T_{-}$ and $v_{-}$ are constant in this limit for a constant $F_{\mathrm{net}}$. Therefore, the fluid profiles for the runaway solution can be obtained from Eqs. (\[fluideqv\]-\[fluideqw\]), like in the detonation case.
The bag EOS
-----------
In order to solve the hydrodynamic equations we need to consider a particular equation of state. We shall consider the bag EOS, in which the two phases consist of radiation and vacuum energy. This approximation has been widely used for simplicity, and also in order to obtain model-independent results which depend on a few physical quantities. Setting the vacuum energy in the low-temperature phase to zero, the model depends on three physical parameters, which we may choose to be the critical temperature $T_{c}$, the latent heat $L$, and the radiation constant of the high-temperature phase, $a$. Thus, we write $$p_{+}(T)=\frac{1}{3}aT^{4}-\frac{L}{4},\quad p_{-}(T)=\frac{1}{3}\left( a-\frac{
3L}{4T_{c}^{4}}\right) T^{4}.$$ The energy density of the high-temperature phase is of the form $
e_{+}(T)=aT^{4}+\varepsilon $, where the false-vacuum energy density is given by $\varepsilon =L/4$. In the low-temperature phase, the energy density is of the form $e_{-}(T)=a_{-}T^{4}$, with a radiation constant given by $a_{-}=a(1-
3\alpha_{c})$, where $$\alpha_{c}=\varepsilon/(aT_{c}^{4}).$$ We define the usual bag variable $$\alpha\equiv\varepsilon/(aT_+^4)=L/(3{w_{+}}). \label{Lbar}$$ The enthalpy density is given by $w_{\pm}=(4/3)a_\pm T_{\pm}^{4}$ (with $a_+\equiv a$). For the bag EOS the speed of sound is the same in both phases, $c_{\pm}=1/\sqrt{3}$.
From Eqs. (\[velplasma\]) we obtain the fluid variables behind a weak detonation wall, $$\begin{aligned}
v_{-}&=&\frac{3\alpha-1+3(1+\alpha)v_{w}^{2}-\sqrt{\left[ 3\alpha-1+3(1+\alpha
)v_{w}^{2}\right] ^{2}-12\alpha(2+3\alpha)v_{w}^{2}}}{2(2+3\alpha)v_{w}},
\label{vme} \\
\frac{w_{-}}{w_{+}}&=&\frac{\gamma _{w}^{2}}{3}\left[ 1-3\alpha+3(1+\alpha
)v_{w}^{2}-2\sqrt{\left( 1-3\alpha+3(1+\alpha)v_{w}^{2}\right)
^{2}-12v_{w}^{2}}\right] \label{wmewma}\end{aligned}$$ (there is also a solution with a $+$ sign in front of the square roots, corresponding to strong detonations). Notice that the fluid velocity and the enthalpy ratio depend only on the variable $\alpha$ and the wall velocity. On the other hand, the temperature is given by $$\frac{T_{-}^{4}}{T_{+}^{4}}=\frac{1}{1-3\alpha_{c}}\,\frac{w_{-}}{w_{+}}.
\label{tmetmabag}$$ The Jouguet velocity is obtained by considering the condition (\[Jouguet\]) together with Eqs. (\[velplasma\]). For the bag EOS we obtain $$v_{J}=\frac{\sqrt{2\alpha+3\alpha^{2}}+1}{\sqrt{3}(1+\alpha)}.$$ The ultra-relativistic limit can be obtained either by taking the limit $v_{w}\rightarrow 1$ in Eqs. (\[vme\]-\[wmewma\]), or directly from (\[hydreldet\]). We have $$\frac{w_{-}}{w_{+}}=1+3\alpha,\quad v_{-}=\frac{3\alpha}{2+3\alpha}\quad
\mbox{(UR detonation).} \label{tmevmeurdeto}$$
For a runaway wall we obtain, from Eqs. (\[eqsrun\]), $$\frac{w_{-}}{w_{+}}=1+3(\alpha-\bar{F}),\quad v_{-}=\frac{3(\alpha-\bar{F})}{
2+3(\alpha-\bar{F})}\quad \mbox{(runaway)}, \label{tmevmerun}$$ where $$\bar{F}\equiv \frac{F_{\mathrm{net}}}{aT_+^4}=\frac{4}{3}\frac{F_{\mathrm{net}}}{w_+}.
\label{Fbar}$$ Notice that Eqs. (\[tmevmerun\]) match the detonation case for $F_{\mathrm{net}}=0$. On the other hand, as $F_{\mathrm{net}}$ increases, $T_{-}$ and $v_{-}$ decrease. Hence, the hydrodynamics becomes weaker for larger acceleration. This behavior is similar to the detonation case, in which the higher the wall velocity, the weaker the hydrodynamics (see Fig. \[figvmetme\]). This is related to the fact that the hydrodynamics obstructs the wall motion [@ms09; @kn11]. Moreover, we see that for $\bar
F=\alpha$ we have $w_{-}=w_{+}$ and $v_{-}=0$, i.e., the fluid remains unperturbed after the passage of the wall.
The condition $\bar F=\alpha$ sets a maximum value for the net force, which is given by the false vacuum energy density, $F_{\mathrm{\max }}=\varepsilon$. To understand this physically, notice that the force which drives the wall motion is essentially given by the pressure difference between the two phases. This force vanishes at the critical temperature and reaches its maximum at zero temperature. At $T=0$ the pressure difference is just given by the zero-temperature effective potential, and coincides with the false-vacuum energy density. In the bag model, this is given by the parameter $\varepsilon $ (at finite temperature there is also a friction force due to the plasma, but at zero temperature the friction force vanishes). Therefore, $F_{\mathrm{net}}$ can reach the maximum value $\varepsilon$ if the phase transition occurs at $T_{+}=0$. However, such an extreme supercooling is not likely in concrete physical models.
For the bag EOS it is relatively simple to obtain the fluid profiles, since $c_-$ is a constant. However, except for the planar case, the fluid equations (\[fluideqv\]-\[fluideqw\]) must be solved numerically. Behind the wall, the solutions which fulfil the boundary condition of a vanishing fluid velocity at $\xi=0$ are rarefaction waves, in which $v(\xi)$ actually vanishes for $0<\xi <c_{-}$ and grows for $\xi>c_-$ up to the boundary value $v_{-}$ at $\xi=\xi_w$ (see e.g. [@ekns10; @lm11]). The temperature and pressure also decrease away from the wall. These can be computed from the enthalpy profile, which is readily obtained by integrating Eq. (\[fluideqw\]), $$\frac{w}{w_-}=\exp\left[\left(\frac{1}{c_-^2}+1\right)\int_{v_-}^{v}
\frac{\xi-v}{1-\xi v}\gamma^2\,dv\right]. \label{entprof}$$ In Fig. \[figprof\] we show some profiles for the case of spherically-symmetric bubbles.
Efficiency factors
------------------
For the bag EOS, the efficiency factor is defined as the fraction of the released vacuum energy $\varepsilon $ which goes into bulk motions of the fluid[^4], $$\kappa_{\mathrm{fl}} \equiv\frac{E_{\mathrm{kin}}}{\varepsilon V_{b}},$$ where $V_b$ is the volume of the bubble. We have, for the different wall symmetries [@lm11], $$\kappa_{\mathrm{fl}}=\frac{j+1}{\varepsilon
v_{w}^{j+1}}\int_{0}^{\infty }\!d\xi \,\xi ^{j}\,w\,\gamma ^{2}v^{2}.$$ As can be seen from Eq. (\[entprof\]), for detonations or runaway walls the profile of $w/w_-$ does not depend on $w_-$ but only on the profile of $v$, which only depends on the boundary values $v_w,v_{-}$. As a consequence, we can write $$\kappa _{\mathrm{fl}}=
\frac{j+1}{v_{w}^{j+1}}\,\frac{4}{3\alpha}\frac{w_{-}}{w_{+}}
\,I(v_{w},v_{-}), \label{kappa}$$ with $$I(v_{w},v_{-})=
\int_{c_{-}}^{v_{w}}\!d\xi \,\xi ^{j}\,\frac{w}{w_{-}}\,\gamma ^{2}v^{2}.
\label{integ}$$ For detonations, $v_-$ and $w_-/w_+$ are functions of $\alpha$ and $v_w$, and therefore the efficiency factor depends only on these quantities, $\kappa_{\mathrm{fl}} =\kappa_{\mathrm{fl}} ^{\mathrm{det}}(\alpha,v_{w})$. On the other hand, for runaway walls we have $v_w=1$, while $w_-/w_+$ and $v_{-}$ depend on $\alpha$ and $\bar F$. From Eqs. (\[kappa\]) and (\[tmevmerun\]), in this case $\kappa_{\mathrm{fl}}$ is of the form $$\kappa _{\mathrm{fl}}^{\mathrm{run}}(\alpha,\bar{F})=\frac{j+1}{3}\,\frac{4(1+3\alpha-3\bar{F})}{\alpha
}\,I_{1}(v_{-}), \label{kapparun}$$ where $I_{1}(v_{-})=I(1,v_{-})$.
For the planar case the integral (\[integ\]) can be done analytically [@lm11], while for spherical or cylindrical walls it must be calculated numerically. Notice that $w_-$ and $v_-$ are the same for all these cases, but the rarefaction profiles differ. In Fig. \[figplesf\] we show the value of the efficiency factor for spherical and planar walls. The cylindrical case lies between the other two. For steady-state walls we plotted $\kappa
_{\mathrm{fl}}$ as a function of the wall velocity (left panel), while for runaway walls we plotted it as a function of the net force (right panel).
For the range of values shown in the figure, the difference between the two wall symmetries is always less than a 10%, while this relative difference is exceeded only for $F_{\mathrm{net}}$ very close to $F_{\max }=\varepsilon$, where $\kappa _{\mathrm{fl}}\rightarrow 0$. As already discussed, it is not likely that such values of $F_{\mathrm{net}}$ will be reached in a physical model. Such a small difference is interesting, since for planar walls we obtain analytic results (see the appendix). In the appendix we also give fits for spherical detonations and runaway walls, where the relative error is smaller than a 3% in the whole detonation range and in most of the runaway range.
For comparison, we show in Fig. \[figkesf\] the value of $\kappa_{\mathrm{fl}}$ in the different regimes, for some values of the bag parameter $\alpha $ (for the calculation in the cases of weak and Jouguet deflagrations, see [@lm11]). The different types of stationary solutions are divided by the points $v_w=c_-$ and $v_w=v_J(\alpha)$.
As already discussed, the hydrodynamics of weak detonations becomes weaker as the wall velocity increases, and it becomes even weaker for runaway walls. This is reflected in the efficiency factor, which is monotonically decreasing in these regimes. On the other hand, for small wall velocities (i.e., for weak deflagrations) the efficiency factor increases with the wall velocity, and is maximal for supersonic (Jouguet) deflagrations.
In the runaway case, a portion of the released vacuum energy goes into kinetic energy of the wall. This will increase the efficiency for gravitational wave generation through direct bubble collisions. This fraction is given by $$\kappa_{\mathrm{wall}}\equiv\frac{\Delta E_{\mathrm{wall}}}{\varepsilon \Delta V_{b}}.$$ Considering a small piece of the thin interface, the energy which goes to the corresponding volume $\Delta V_b=Av_w\Delta t$ is given by $\Delta
E_{\mathrm{wall}}=\Delta \sigma A$. Therefore, we have, from Eq. (\[dsigf\]) and the definitions (\[Lbar\],\[Fbar\]), $$\kappa_{\mathrm{wall}}=\frac{d\sigma /dt}{
\varepsilon v_{w}}
=\frac{F_{\mathrm{net}}}{\varepsilon}=\frac{\bar F}{\alpha}. \label{enwall}$$
The wall equation of motion \[micro\]
=====================================
We shall now consider the equation of motion for the wall. At a given point of the thin interface we may place the $z$ axis perpendicular to the wall. Then the field equation (\[fieldeq\]) with the damping (\[Kfeno\]) becomes $$(\partial _{0}^{2}-\partial _{z}^{2})\phi +\partial \mathcal{F}/\partial
\phi +\mathcal{K}=0 . \label{fieldeq0z}$$ If we describe the wall at rest by a certain field profile $\phi _{0}(z)$ which varies between the values $\phi _{-}$ and $\phi _{+}$, then, in the plasma frame, we have $$\phi(z,t) =\phi _{0}(\gamma _{w}\left(z-z_{w}\right)), \label{fieldprof}$$ where $z_w$ depends on time and we have $v_{w}=\dot{z}_{w}$, $ \gamma
_{w}=1/\sqrt{1-v_{w}^{2}}$. The wall corresponds to the range of $z$ where $\phi $ varies, and we define the wall position $z_{w}(t)$ by the condition $\int (\partial_z\phi )^{2}(z-z_{w})\,dz\,=0$. Multiplying Eq. (\[fieldeq0z\]) by $\partial_z\phi $ and integrating across the wall, we obtain the equation $$\sigma _{0}\gamma _{w}^{3}\dot{v}_{w}=\int_{-}^{+}\frac{\partial \mathcal{F}
}{\partial \phi }\,\frac{\partial \phi }{\partial z}\,dz+\int_{-}^{+}\mathcal{K}
\,\frac{\partial \phi }{\partial z}\,dz, \label{eqwall}$$ where $\int_{-}^{+}dz$ means integration between points on each side of the wall (where $\partial_z\phi $ vanishes) and $$\sigma _{0}=\int_{-}^{+}\left[ \phi _{0}^{\prime }(z)\right] ^{2}dz.$$ This integral gives the surface energy density of the wall at rest. We see that it appears in the left-hand side of Eq. (\[eqwall\]) multiplying the proper acceleration $\gamma _{w}^{3}\dot{v}_{w}$. Hence, the terms in the right-hand side are forces (per unit area) acting on the wall. Since the force which drives the wall motion is finite, the factor $\gamma _{w}^{3}$ implies that, in the plasma frame, $\dot v_w$ decreases as the wall approaches the speed of light.
The force terms in Eq. (\[eqwall\]) depend not only on the Higgs profile but also on the fluid profiles. The first of them is very sensitive to hydrodynamics. For a constant temperature, this term gives the pressure difference $p_{-}-p_{+}$. Since it is always positive, we shall refer to it as the driving force $F_{\mathrm{dr}}$. The term containing $\mathcal{K}$ represents the microscopic departures from equilibrium caused by the moving wall. It is always negative and velocity dependent. We shall refer to this term as the friction force $F_{\mathrm{fr}}$. Thus, Eq. (\[eqwall\]) can be written as $$F_{\mathrm{net}}=F_{\mathrm{dr}}+F_{\mathrm{fr}} . \label{eqforces}$$
Driving force and hydrodynamic obstruction
------------------------------------------
The driving force can be written in the form [@mm14deto] $$F_{\mathrm{dr}}=p_{-}-p_{+}-\int_{-}^{+}\frac{\partial \mathcal{F}}{\partial
T^2}\,{dT^2}, \label{Fdr}$$ which we shall approximate for definite calculations by $$F_{\mathrm{dr} }\simeq p_{-}-p_{+}-\left\langle \frac{\partial \mathcal{F}}{\partial
T^{2}}\right\rangle (T_{+}^{2}-T_{-}^{2}), \label{Fdrapp}$$ where we have approximated the value of ${\partial \mathcal{F}}/{\partial
T^{2}}$ inside the wall by $$\left\langle \frac{\partial \mathcal{F}}{\partial
T^{2}}\right\rangle\equiv \frac{1}{2}\left(\frac{\partial \mathcal{F_+}}{\partial
T_+^{2}}+\frac{\partial \mathcal{F_-}}{\partial
T_-^{2}}\right).$$ For the bag EOS, Eq. (\[Fdrapp\]) takes the simple form $$F_{\mathrm{dr}}=\frac{L}{4}\left( 1-\frac{T_{-}^{2}T_{+}^{2}}{T_{c}^{4}}
\right) = \varepsilon-\varepsilon \frac{T_{-}^{2}T_{+}^{2}}{T_{c}^{4}}.
\label{Fdrbag}$$ In this approximation, the first term is the zero-temperature part of the force. Indeed, as already mentioned, the false vacuum energy density is given by the zero-temperature effective potential. Thus, for a physical model the bag constant $\varepsilon$ would be given by $V(\phi_+)-V(\phi_-)$. The second term in Eq. (\[Fdrbag\]) is the temperature-dependent part of the driving force, and contains the effect of hydrodynamics. For weak detonations, this effect is to increase $T_-$ with respect to the outside temperature $T_+$, and hence to decrease the driving force. In this decomposition, the term $F_{\mathrm{hyd}}
=-(L/4) {T_{-}^{2}T_{+}^{2}}/{T_{c}^{4}}$ can be seen as a force which opposes the wall motion and depends indirectly on the wall velocity (through the dependence of the temperature on $v_{w}$). However, this hydrodynamic obstruction does not behave as a fluid friction, since it *decreases* with the wall velocity. Indeed, the reheating behind the wall is highest for $v_w$ close to the Jouguet point and lowest for $v_{w}\rightarrow 1$. It is worth remarking that the approximation (\[Fdrbag\]) preserves this important effect of reheating, while simpler approximations such as setting $T_{-}=T_{+}$ in Eq. (\[Fdr\]) (see e.g. [@ekns10]) *overestimate* the driving force.
Friction force
--------------
As already discussed, the friction must be obtained from microphysics considerations which are much more involved than the calculation of $F_{\mathrm{dr}}$. Here we shall use instead the phenomenological damping (\[Kfeno\]). Hence, we have $$F_{\mathrm{fr}}=\int_{-}^{+}\frac{f(\phi )\,u^{\mu }\partial _{\mu }\phi }{\sqrt{1+[g(\phi
)\,u^{\mu }\partial _{\mu }\phi ]^{2}}}\,\partial_z\phi \,dz . \label{Ffrdef}$$ For the field profile (\[fieldprof\]), we have $$u^\mu\partial_\mu\phi=\gamma(\partial_0\phi+v\partial_z\phi) =
\phi_0'\gamma[\gamma_w(v-v_w) + \dot\gamma_w(z-z_w)]. \label{udfi}$$ The term $\dot\gamma_w(z-z_w)$ vanishes in the stationary case. In the runaway regime we can also neglect it, since $\phi_0'$ vanishes out of the thin interface[^5]. We thus have $$F_{\mathrm{fr}}=\int_{-}^{+}\frac{\gamma_w\gamma(v-v_w)
f(\phi_0)(\phi_0')^2}{\sqrt{1+\gamma_w^2\gamma^2(v-v_w)^2 g^2(\phi_0) (\phi_0')^2}} \,dz ,
\label{Ffrfenpl}$$ where we have used the change of variable of integration $\gamma_w
(z-z_w)\to z$. It is easy to see that in the limit $\gamma_w\to \infty$ we have $F_{\mathrm{fr}}\sim$ constant, while for $v_w\ll 1$ (which implies $v\ll 1$ as well) we have $F_{\mathrm{fr}}\sim v_w$. The result (\[Ffrfenpl\]) is equivalent to that of Ref. [@ariel13], as can be seen from the transformation $\gamma v\to \gamma\gamma_w(v-v_w)$ from the wall frame to the plasma frame.
The integral in Eq. (\[Ffrfenpl\]) is of the form $\int
[\phi_0'(z)]^2F(z)\,dz$, where the function $[\phi_0'(z)]^2$ vanishes outside the wall and peaks at the center of the latter. Therefore, we can write this integral as $F(\bar z)\int [\phi_0'(z)]^2dz=F(\bar z)\sigma_0$, where $\bar z$ is some point near the wall center. We thus obtain $$F_{\mathrm{fr}}=
\frac{\eta\gamma_w\bar\gamma(\bar v-v_w)}{\sqrt{1+
\lambda^2\gamma_w^2\bar\gamma^2(\bar v-v_w)^2}}, \label{Ffrfeno}$$ where $\bar\gamma,\bar v$ are the values of $\gamma,v$ at the center of the wall, $\eta=\sigma_0f(\phi_0(\bar z))$, and $\lambda$ is similarly given by the function $g$ and details of the wall profile. We shall regard $\eta$ and $\lambda$ as free parameters which can be chosen appropriately to give the correct numerical values of the friction in the NR and UR limits. On the other hand, we shall approximate the value of $\bar v$ by the average[^6] $$\bar v=({v_-+v_+})/{2}=v_-/2, \label{vmedia}$$ and $\bar\gamma=1/\sqrt{1-\bar v^2}$.
For non-relativistic velocities, Eq. (\[Ffrfeno\]) gives a friction force which is proportional to the relative velocity, $F_{\mathrm{fr} }=-\eta
\,(v_{w}-\bar v)$, as expected. For a specific model, the value of $\eta $ can be obtained by comparison with the result of a non-relativistic microphysics calculation. Therefore, we use the notation $\eta= \eta _{NR}$. It is out of the scope of the present work to compute the friction for specific models. General approximations for $\eta _{NR}$ as a function of the parameters for a variety of models can be found in Ref. [@ms10]. The friction coefficient depends on temperature. In particular, it decreases as $T$ decreases, since the friction depends on the particle populations. Nevertheless, in contrast to $F_{\mathrm{dr}}$, the friction is not sensitive to the temperature difference $T_{c}-T$. Therefore, it is not very sensitive to hydrodynamics. For specific calculations, in this work we shall assume that, roughly, $\eta_{NR}\sim
T_{+}^{4}$.
In the ultra-relativistic case $\gamma_w\gg 1$, Eq. (\[Ffrfeno\]) gives $F_{\mathrm{fr}}= -\eta/\lambda$. Therefore, we define the UR friction coefficient $\eta _{UR}=-\eta/\lambda$, so that $F_{\mathrm{fr}}= -\eta
_{UR}v_w$. The value of this parameter for a specific model can be obtained from the microphysics result (\[Fnet\]). This result, however, gives the total UR force $F_{\mathrm{net}}$ rather than the friction. Notice that the last term in Eq. (\[Fnet\]) includes the finite-temperature part of the driving force, $F_{\mathrm{hyd}}$, as well as the friction. We may obtain the UR friction force as $F_{\mathrm{fr}}=F_{\mathrm{net}}-F_{ \mathrm{dr}}$, taking into account the UR limit of the driving force. The latter is given by Eqs. (\[Fdrbag\]), (\[tmetmabag\]), and (\[tmevmerun\]). We obtain $$\frac{\eta_{UR}}{aT_+^4}=
\alpha-\bar{F}-\alpha_{c}
\sqrt{\frac{1+3(\alpha-\bar{F})}{1-3\alpha_{c}}} . \label{etaur}$$ For a given model, the quantities $\alpha=L/(3w_+)$ and $\bar
F=4F_{\mathrm{net}}/(3w_+)$ can be obtained as functions of the nucleation temperature $T_+$. We remark that, although decomposing the total force into driving and friction forces is not relevant for the runaway regime, determining the UR value of the friction component is relevant for a correct use of Eq. (\[Ffrfeno\]) as an interpolation between the NR and UR cases. In terms of the friction coefficients $\eta_{NR},\eta_{UR}$, we have $$F_{\mathrm{fr}}=-\frac{\eta_{NR}\eta_{UR}\,
\gamma_w\bar\gamma(v_w-\bar v)}{\sqrt{\eta_{UR}^2+
\eta_{NR}^2\,\gamma_w^2\bar\gamma^2(v_w-\bar v)^2}}\label{Ffrfeneta}.$$
It is worth commenting on previous approaches. A similar, but simpler, phenomenological model for the friction was considered in Ref. [@ekns10]. The approximation involves a single free parameter and is equivalent to setting $\lambda=1$ in Eq. (\[Ffrfeno\]). Although this model gives a friction which saturates at high $\gamma_w$, it is numerically incorrect as it corresponds to the case $\eta _{UR}=\eta _{NR}$ (besides, the approximation $T_-=T_+$ was used in [@ekns10] for the driving force; we shall discuss on this approximation below). The phenomenological model (\[Kfeno\]) was already considered in Ref. [@ariel13]. The resulting friction is equivalent to Eq. (\[Ffrfeno\]). However, in [@ariel13] the hydrodynamics was neglected for runaway walls (but not for stationary solutions). That is, the relation $T_{-}=T_{+}$ was assumed for the runaway case. This results in a different value of $\eta_{UR}$, as the right-hand side of Eq. (\[etaur\]) becomes $\alpha-\bar F-\alpha_c$. Since, on the other hand, the hydrodynamics was taken into account for detonations, the stationary solutions did not match continuously the accelerated ones. This is not correct since, as we have seen, the hydrodynamics of a runaway wall is similar to that of the detonation, and matches the latter for $F_{\mathrm{net}}=0$.
The wall velocity
-----------------
From Eqs. (\[eqforces\]), (\[Fdrbag\]) and (\[Ffrfeneta\]), we have, for detonations or runaway walls, $$\frac{F_{\mathrm{net}}}{aT_+^4}=
\alpha-\alpha_c\frac{T_{-}^{2}}{T_{+}^{2}}
-\frac{\bar\eta_{NR}\bar\eta_{UR}\,(v_w-\bar v_-)}{\sqrt{\bar\eta_{UR}^2(1-v_w^2)
(1-\bar v_-^2)+
\bar\eta_{NR}^2\,(v_w-\bar v_-)^2}} \label{eqwallbag}$$ where $\bar v=v_-/2$, and we use the notation $\bar\eta=\eta/(aT_+^4)$ for the two friction coefficients. In the UR limit, Eq. (\[eqwallbag\]) becomes $$\bar F=\alpha-{\alpha_c}{T_{-}^{2}}/{T_{+}^{2}}
-\bar\eta_{UR}. \label{eqwallbagur}$$ We remark again that the latter equation is just a decomposition of the UR net force, which actually defines the value of the friction coefficient $\eta_{UR}$, while the former gives an equation of motion for the wall away from that limit. In particular, if the microphysics computation of the net force, Eq. (\[Fnet\]), gives $F_{\mathrm{net}}<0$, it means that, in fact, the wall will never reach the UR regime. Nevertheless, the UR calculation is still useful and Eq. (\[eqwallbagur\]) makes sense. The interpretation is that the UR friction is so high that the driving force cannot compensate it. In this case we just obtain $\bar\eta_{UR}$ from Eq. (\[etaur\]), and then compute the steady-state wall velocity by setting $F_{\mathrm{net}}=0$ in Eq. (\[eqwallbag\]).
To solve for $v_w$, we must use Eqs. (\[vme\]-\[tmetmabag\]) for $v_-$ and $T_-$. It is worth mentioning that, for detonations, the result does not depend on the wall being spherical or planar, since all the quantities appearing in Eq. (\[eqwallbag\]) are the same in the two cases. This is because the relations between $v_-,T_-$ and $v_+,T_+$ are the same for spherical or planar walls. Besides, for detonations the conditions in front of the wall (i.e., $v_+,T_+$) are also the same (in contrast, for deflagrations, the fluid in front of the wall is perturbed differently for planar or spherical walls).
We show the result in Fig. \[figfric\] (solid line) for fixed values of the friction parameters and varying the bag quantity $\alpha$. For concreteness, and in order to compare with previous results, we consider the case $\eta
_{UR}=\eta _{NR}$ (for other cases and different parameter variations, see [@ariel13]). The vertical dashed lines delimit the weak detonation solutions. Increasing $\alpha$ generally increases the driving force and, consequently, the wall velocity. The figure does not show the deflagration cases, for which $v_w\lesssim c_-$ (there is a discontinuity between deflagrations and detonations).
The dotted line in Fig. \[figfric\] is obtained by neglecting the reheating in the calculation of the driving force, i.e., setting $T_{-}=T_{+}$, for which the driving-force term in Eq. (\[eqwallbag\]) becomes $\alpha-\alpha_c$. This was used as an approximation in Ref. [@ekns10]. We consider it here in order to appreciate the role of hydrodynamics. Quantitatively, we see that this approximation overestimates the driving force, as we obtain higher values of the velocity. Besides, we observe a significant qualitative difference between the two results at the lower end of the detonation curve. This end corresponds to the Jouguet point. Since the hydrodynamics becomes very strong near this point, Eq. (\[eqwallbag\]) gives two solutions for $v_{w}$, while the approximation $ T_{-}=T_{+}$ completely misses this effect. In Ref. [@mm14deto] it was shown that weak detonations corresponding to the lower branch of solutions are unstable.
The value of $\alpha$ for which the detonation reaches the ultra-relativistic regime in Fig. \[figfric\] can be obtained from Eq. (\[eqwallbagur\]) which, for $\bar F=0$, gives $\alpha
={\alpha_c}{T_{-}^{2}}/{T_{+}^{2}}+{\bar\eta_{UR}}$. Notice that $T_-$ actually depends on $\alpha$ through Eqs. (\[tmetmabag\],\[tmevmeurdeto\]), ${T_{-}^{4}}/{T_{+}^{4}}=({1+3\alpha})/({1-3\alpha_{c}})$. Thus, for given values of $\alpha_c$ and $\bar\eta_{UR}$ we have a quadratic equation for $\alpha$ (or for $T_-$), which yields $$\alpha= \alpha_{c}\,{T_{0}^{2}}/{T_{+}^{2}}+{\bar\eta _{UR}}\equiv\alpha_0,
\label{L0deeta}$$ where $T_0$ is the corresponding value of $T_-$, given by $$\frac{T_{0}^{4}}{T_{+}^{4}}=\frac{3\alpha_{c}}{2(1-3\alpha_{c})}
\left[1+\sqrt{ 1+\frac{4(1+3\bar\eta _{UR})(1-3\alpha_c)}{
3\alpha_c^2}}\right]. \label{tmedeeta}$$ For $\alpha>\alpha_0$, the steady-state equation gives $v_w>1$, which actually indicates that the friction force cannot compensate the driving force and we have $F_{\mathrm{net}}>0$, i.e., a runaway wall.
In the runaway regime, we have a proper acceleration which is proportional to the net force. It is interesting to calculate the value of $F_{\mathrm{net}}$ corresponding to Fig. \[figfric\], which can be obtained from Eq. (\[eqwallbagur\]) \[although for a given model one would rather compute $F_{\mathrm{net}}$ directly from Eq. (\[Fnet\]), and then determine $\eta_{UR}$\]. We must take into account the dependence of $T_{-}$ on $F_{\mathrm{net}}$, which is given by Eqs. (\[tmetmabag\],\[tmevmerun\]), $$T_-^4/T_+^4=(1+3\alpha-3\bar F)/(1-3\alpha_c). \label{tmeur}$$ From Eqs. (\[tmeur\]) and (\[eqwallbagur\]) we obtain quadratic equations for $\bar F$ and $T_-/T_+$ as functions of $\alpha$, $\alpha_c$, and $\bar\eta_{UR}$. Nevertheless, the dependence on $\alpha$ cancels in the equation for $T_-$, and we obtain $$T_-=T_0,\quad
\bar{F}=\alpha-\alpha_cT_0^2/T_+^2-\bar\eta_{UR}=\alpha-\alpha_{0},
\label{Fdeeta}$$ with $\alpha_0$ and $T_0$ given by Eqs. (\[L0deeta\]-\[tmedeeta\]). On the other hand, if we use the approximation $T_-=T_+$, we just neglect Eq. (\[tmeur\]), while Eq. (\[eqwallbagur\]) gives $\bar F=\alpha-\alpha_c
-\bar\eta_{UR}$. This result can also be written in the form $\bar
F=\alpha-\alpha_{0}$, but the value of $\alpha_0$ is different, namely, $\alpha_0=\alpha_c+\bar\eta_{UR}$. In Fig. \[figfricF\] we plot the net force, normalized to its maximum value $\varepsilon$, as a function of $\alpha$ for the parameters of Fig. \[figfric\]. We see that neglecting the hydrodynamics gives a higher wall acceleration.
Comparing the solid lines of Figs. \[figfric\] and \[figfricF\], we see that the detonation solution matches the runaway solution at $\alpha=\alpha_0$. In Ref. [@ariel13] this matching does not occur, due to the assumption of different hydrodynamics. As a consequence, the two kinds of solutions were found to coexist in a small parameter range. We do not find such a coexistence of detonation and runaway solutions here, since the hydrodynamics is continuous with $v_w$ and $\bar F$. In fact, coexistence of solutions could arise also due to strong hydrodynamics, even if the hydrodynamics varies continuously with the parameters. For instance, we have multiple weak-detonation solutions near the Jouguet point, even though $v_-,T_-$ are continuous functions of $\alpha,v_w$. This does not happen in the UR limit, since the perturbations of the fluid vary continuously with $\alpha,v_w$, and $\bar F$ and, besides, the hydrodynamics is weaker.
Microphysics and released energy
--------------------------------
In Sec. \[hidro\] we computed the fractions of the released vacuum energy which go into bulk motions of the fluid and into kinetic energy of the bubble wall, as functions of the quantities $v_w$, $\bar F=(4/3)F_{\mathrm{net}}/w_+$, and $\alpha=L/(3w_+)$. For a given model, the nucleation temperature and the thermodynamical quantities $L,w_+$ can be calculated from the finite-temperature effective potential (\[ftot\]), and the net force can be readily computed from Eq. (\[Fnet\]). This gives the values of $\alpha$ and $\bar F$. The steady-state wall velocity can be obtained from Eq. (\[eqwallbag\]), after determining the friction coefficients $\eta_{UR}$ and $\eta_{NR}$. The value of $\eta_{UR}$ can be obtained from $\bar F$ using Eq. (\[etaur\]), while $\eta_{NR}$ must be obtained from a microphysics calculation. Such a computation is beyond the scope of this paper. Here, we shall only consider the energy distribution among the fluid and the wall for the parameter variation of Figs. \[figfric\] and \[figfricF\]. As already discussed, this parameter variation becomes rather artificial in the runaway regime. It is useful, though, for a comparison with previous results.
The fraction of energy accumulated in the interface, $\kappa_{\mathrm{wall}}$, is just given by $F_{\mathrm{net}}/\varepsilon$, which is plotted in Fig. \[figfricF\]. In Fig. \[figbudget\] we consider a wider range of runaway solutions, and we plot separately (in the right panel) the result obtained by using the approximation $T_{-}=T_{+}$ in the calculation of the driving force. The value of $\kappa_{\mathrm{wall}}$ is represented by the height of the light shade. Thus, the curves delimiting this region in the left and right panels of Fig. \[figbudget\] correspond, respectively, to the solid and dotted lines of Fig. \[figfricF\]. Following Ref. [@ekns10], we plot the value of $\kappa _{\mathrm{fl}}$ (for spherical bubbles) added to that of $\kappa_{\mathrm{wall}}$. This gives the upper curves delimiting the dark shade regions. Hence, the fraction of $\varepsilon$ which goes into bulk fluid motions is represented by the dark shade. Accordingly, the white region indicates the portion of $\varepsilon $ which goes into reheating[^7]. The vertical line separates the detonation and runaway regimes. We include the complete detonation range (which is different in the two panels).
The right panel of Fig. \[figbudget\] agrees with the results of Ref. [@ekns10]. The values of the parameters, namely, $\eta _{NR}=\eta
_{UR}=0.2$, $a_{-}/a_{+}=1-3\alpha_{c}=0.85$, correspond to one of the cases considered in that work (cf. the left panel of Fig. 12 in [@ekns10]). We observe that the two panels of Fig. \[figbudget\] are qualitatively similar, particularly for the runaway regime, where the hydrodynamics is weaker. The difference is more apparent for detonations, where the hydrodynamics is strongest. In fact, the Jouguet point is never reached in the left panel. The quantitative difference between the two calculations is better appreciated in Figs. \[figfric\] and \[figfricF\]. For a given $\alpha$, neglecting the hydrodynamics gives higher wall velocities and accelerations and, hence, a larger $\kappa_{\mathrm{wall}}$ and a smaller $\kappa_{\mathrm{fl}}$. It is worth emphasizing that in both panels we have used the results from Sec. \[hidro\] in terms of $\alpha$, $v_w$ and $\bar F$, and the discrepancy originates in the computation of $v_w$ and $\bar F$.
For the runaway case we may obtain simple semi-analytical expressions for the efficiency factors as functions of the quantities $\alpha$, $\alpha_c$, and $\bar\eta_{UR}$. From Eqs. (\[enwall\]) and (\[Fdeeta\]), we have $$\kappa_{\mathrm{wall}}=1-{\alpha_{0}}/{\alpha}. \label{ewallfric}$$ This is valid for the two plots of Fig. \[figbudget\], with different values of $\alpha_0(\alpha_c,\bar\eta_{UR})$. Since $\kappa_{\mathrm{wall}}$ gives the fraction of $\varepsilon$ which goes to kinetic energy of the wall, Eq. (\[ewallfric\]) indicates that a fraction $\alpha_{0}/\alpha$ goes to the fluid (either to bulk motions or reheating). Moreover, using Eq. (\[Fdeeta\]) in Eqs. (\[kapparun\]) and (\[tmevmerun\]), we have $$\kappa _{\mathrm{fl}}^{\mathrm{run}}=
\frac{4(1+3\alpha_{0})I_{1}(v_-)}{\alpha}, \quad \mbox{with}\quad
v_-=\frac{3\alpha_{0}}{2+3\alpha_{0}}.$$ Hence, the runaway efficiency factor can be written as $$\kappa _{\mathrm{fl}}^{\mathrm{run}}=\kappa _{UR}^{\mathrm{det }}\,{\alpha_{0}}/{\alpha},
\label{krunfric}$$ where $\kappa _{UR}^{\mathrm{det}}\equiv\kappa
_{\mathrm{fl}}^{\mathrm{det}}(\alpha_0,v_w=1)$ is the efficiency factor of the UR detonation. The functional dependence of Eqs. (\[ewallfric\]) and (\[krunfric\]) on $\alpha$ agrees with the results of Ref. [@ekns10]. The quantitative difference, which is illustrated by the two panels of Fig. \[figbudget\], is due to different values of $\alpha_0$ and $\kappa
_{UR}^{\mathrm{det }}$ (in the notation of [@ekns10], $ \alpha_0=\alpha
_{\infty}$ and $\kappa _{UR}^{\mathrm{det }}=\kappa _{\infty }$). As already discussed, this discrepancy is due essentially to a different treatment of hydrodynamics.
In Ref. [@ekns10] an expression for the quantity $\alpha_0$ is provided in terms of microphysics parameters. We may obtain a similar expression as follows. If we identify the difference $V(\phi_+)-V(\phi_-)$ in Eq. (\[Fnet\]) with the bag constant $\varepsilon$ (notice, though, that the minima, particularly $\phi_-$, are temperature dependent), then the UR net force vanishes for $$\varepsilon_0=\sum_i g_i c_i{T_+^2m_i^2(\phi_-)}/{24}. \label{eps0}$$ For a given $T_+$, this is the value of $\varepsilon$ corresponding to $\alpha_0$. Thus, we have $\alpha_0=4\varepsilon_0/(3w_+)$, which can be used in Eqs. (\[ewallfric\]-\[krunfric\]). We remark that this approach involves more approximations than those used in Sec. \[hidro\], where we obtained $\kappa_{\mathrm{fl}} ^{\mathrm{run}}$ directly as a function of $\bar F$.
Gravitational waves {#gw}
===================
The efficiency factors $\kappa_{\mathrm{fl}}$ and $\kappa_{\mathrm{wall}}$ give the fractions of the released energy which go into fluid motions and into the wall, respectively. The values of these factors are the key quantities in the different mechanisms of gravitational wave (GW) generation in a first-order phase transition. Three mechanisms of GW generation have been considered in the literature, namely, bubble collisions, turbulence, and sound waves. The bubble collisions mechanism assumes that the energy-momentum tensor which sources the GWs is concentrated in thin spherical shells [@gwcol; @hk08]. For detonations, this is not a bad approximation during the phase transition, since a portion $\kappa_{\mathrm{fl}}\varepsilon\Delta V_b$ of the released vacuum energy $\varepsilon\Delta V_b$ is concentrated as kinetic energy of the fluid in a region which follows the bubble wall supersonically. This is also a good approximation for runaway walls, for which another portion $\kappa_{\mathrm{wall}}\varepsilon\Delta V_b$ of the vacuum energy is accumulated in the infinitely thin interface. Hence, the total energy involved in this mechanism is proportional to $\kappa_{\mathrm{tot}}=\kappa_{\mathrm{wall}}+ \kappa_{\mathrm{fl}}$ (with $\kappa_{\mathrm{wall}}=0$ in the detonation case).
On the other hand, fluid motions can remain long after the completion of the phase transition and continue producing GWs. This may happen by two mechanisms, namely, magnetohydrodynamic (mhd) turbulence [@gwturb] or sound waves [@gwsound]. Since these are long-lasting sources, they are generally more efficient than bubble collisions. However, the energy involved in these “fluid motions” mechanisms is proportional to $\kappa_{\mathrm{fl}}$ alone. In the runaway regime the hydrodynamics becomes weaker and the energy in the fluid is suppressed. As a consequence, it is not clear a priori whether these mechanisms will still dominate over bubble collisions.
Depending on the generation mechanism, the peak frequency of the GW spectrum is determined by a characteristic time or a characteristic length of the source. For a first-order phase transition, the time scale is given by its duration $\Delta t$, while the length scale is given by the average bubble radius $R\sim
v_w\Delta t$. For detonations or runaway walls, we have $R\sim\Delta t$. Therefore, the characteristic frequency at the formation of GWs is given by $f_{p*}\sim 1/\Delta t$. The corresponding frequency today (after redshifting) would be given by $$f_p\sim 10^{-5}\,\mathrm{Hz}\left( \frac{g_{\ast }
}{100}\right) ^{1/6}\left( \frac{T}{100\,\mathrm{GeV}}\right) \frac{1 }{H\Delta t},$$ where $H$ is the Hubble rate during the phase transition, $g_*$ is the number of relativistic degrees of freedom, and $T$ is the temperature at which the phase transition occurred, namely, $T\approx T_+\lesssim T_c$.
It is interesting to consider the electroweak phase transition, for which we have $T_c\simeq 100$GeV and $g_*\simeq 100$. The duration of the phase transition may vary from $\Delta t\sim 10^{-5}H^{-1}$ for very weak phase transitions to $ \Delta t\sim H^{-1}$ for very strong phase transitions. For the sake of concreteness, we shall consider $H\Delta t =10^{-1}$, corresponding to strong phase transitions, which is consistent with having detonations or runaway walls. This gives $f_p\sim 0.1\mathrm{mHz}$, which is close to the peak sensitivity of the planned space-based observatory eLISA [@elisa]. It is customary to express the energy density of gravitational radiation in terms of the quantity $$h^2\Omega _{GW}\left( f\right) =\frac{h^2}{\rho _{c}}\frac{d\rho
_{GW}}{d\log f},$$ where $\rho _{GW}$ is the energy density of the GWs, $f$ is the frequency, and $\rho _{c} $ is the critical energy density today, $\rho _{c}=3H_{0}^{2}/8\pi
G$, with $H_0=100\, h\, \mathrm{ km \, s}^{-1} \mathrm{Mpc}^{-1}$, and $h=0.72$. The peak sensitivity of eLISA may be in the range $\Omega_{GW}\sim
10^{-14}-10^{-10}$, depending on its final configuration.
An approximation for the spectrum of GWs from bubble collisions was given in Ref. [@hk08]. For the peak amplitude that would be observed today we have $$h^{2}\Omega _{\mathrm{\mathrm{coll}}}=1.67\times 10^{-5}\left(
\frac{\kappa_{\mathrm{tot}} \alpha}{1+\alpha}
\right) ^{2}\left( \frac{100}{g_{\ast
}}\right)^{1/3} \left(\frac{0.11v_{w}^{3}}{
0.42+v_{w}^{2}}\right) \left(\Delta t H\right)^{2}. \label{omcol}$$ The spectrum from mhd turbulence was calculated using analytic approximations in Ref. [@cds09]. For the peak amplitude we have [@cds09; @cds10] $$h^{2}\Omega _{\mathrm{turb}}=2.6\times 10^{-5}
\left( \frac{\kappa_{\mathrm{fl}} \alpha}{1+\alpha}\right) ^{3/2}
\left( \frac{100}{g_{\ast }}\right) ^{1/3}\frac{{v_w\Delta t H}
}{1+4\pi 3.5/(v_w\Delta tH) }.
\label{omturb}$$ Regarding the GW spectrum from sound waves, a fit to the numerical results of Ref. [@gwsound] was given in [@elisasci]. For the peak intensity we have $$h^{2}\Omega _{sw}=2.65\times 10^{-6}\left(\frac{\kappa_{\mathrm{fl}}
\alpha}{1+\alpha}\right) ^{2}\left( \frac{100}{g_{\ast
}}\right) ^{1/3}{v_w\Delta t H}. \label{omsw}$$ The quantity $\alpha=\varepsilon/(aT_+^4)$ gives the ratio of the vacuum energy density to the radiation energy density. For the electroweak phase transition we generally have $\alpha<1$, i.e., radiation dominates. Hence, for $g_*\sim
100$, $v_w\sim 1$, and $\Delta t H\sim 10^{-1}$ we have $h^{2}\Omega_\mathrm{coll}\sim 10^{-8}(\kappa_{\mathrm{tot}}\alpha)^2$, $h^{2}\Omega_\mathrm{turb}\sim 10^{-8}(\kappa_{\mathrm{fl}}\alpha)^{3/2}$, and $h^{2}\Omega_\mathrm{sw}\sim 10^{-7}(\kappa_{\mathrm{fl}}\alpha)^{2}$. We see that the numerical values are similar, and the precise results will depend on details of the phase transition dynamics. In particular, the values of $\alpha$ and the efficiency factors will determine which of these sources is dominant.
The efficiency factors depend on $\alpha$. Besides, they depend on the wall velocity (in the detonation case) and on the net force (in the runaway case). In Fig. \[figgw\] we plot the GW intensities (\[omcol\]-\[omsw\]) as functions of the wall velocity and the net force, for some values of $\alpha$.
We see that the different sources dominate in different parameter regions. However, in the case of stationary walls the GW signal from bubble collisions is generally smaller, as expected. Besides, in the detonation case all the curves behave similarly as functions of $v_w$. This is due to the dependence of the GW amplitudes on $\kappa_\mathrm{fl}$, which decreases with $v_w$ (cf. Figs. \[figplesf\] and \[figkesf\]). In contrast, for runaway walls, the GW signal from bubble collisions grows with the net force, while the other signals decrease. This is because the former depends on $\kappa_\mathrm{wall}=F_{\mathrm{net}}/\varepsilon$, as already discussed. Also, as expected, all the signals grow with $\alpha$ for a given value of $v_w$ or $F_{\mathrm{net}}/\varepsilon$.
The quantities $v_w,F_{\mathrm{net}}/\varepsilon$ and $\alpha$ are not actually independent. As we have seen, for fixed values of the friction parameters, $v_w$ and $F_{\mathrm{net}}/\varepsilon$ are increasing functions of $\alpha$. In such a case, $\kappa_\mathrm{fl}$ actually decreases with $\alpha$, as shown in Fig. \[figbudget\], while $\kappa_\mathrm{wall}$ increases. In Fig. \[figgwfric\] we plot the GW amplitudes corresponding to that parameter variation.
We see that the decrease of $\kappa_\mathrm{fl}(\alpha)$ is reflected in the GW amplitudes, even though the latter depend on the product $\kappa_\mathrm{fl}(\alpha)\, \alpha$. Indeed, the signals from fluid motions generally decrease with $\alpha$, while the signal from bubble wall collisions grows in the runaway regime due to the increase of $\kappa_{\mathrm{wall}}$.
It is important to notice that this behavior of the GW signals with the quantity $\alpha$ was obtained by fixing several parameters, such as the friction coefficients $\eta_{NR},\eta_{UR}$, the bag parameter $\alpha_c=\varepsilon/(aT_c^4)$ (which is equivalent to fixing $a_-/a_+$), as well as the time scale $\Delta t$. In a concrete model, all these quantities will vary together with $\alpha$, as all of them depend on the model parameters. We shall consider concrete models elsewhere. In any case, values of $\alpha$ in the range $0.3\lesssim \alpha\lesssim 1$ are possible in a very strong electroweak phase transition. Hence, Fig. \[figgwfric\] shows that the GWs generated by any of the mechanisms at this phase transition may be observable by eLISA.
Conclusions {#conclu}
===========
Several hydrodynamic modes are possible for the growth of a bubble in a cosmological first-order phase transition. These include steady-state walls propagating as deflagrations or detonations [@hidro], accelerated (runaway) walls [@bm09], or even turbulent motion associated with wall corrugation [@inst]. Which of these propagation modes will a phase transition front take, depends on several factors, such as the amount of supercooling and the friction of the bubble wall with the plasma. In this work we have studied the fastest of these modes, namely, ultra-relativistic detonations and runaway solutions, which may give an important gravitational wave signal from the phase transition.
The generation of gravitational waves depends on the released energy, which is usually measured by the ratio $\alpha$ of the vacuum energy density to the radiation energy density. It is also important how this energy is distributed, as the wall moves, among the bubble wall and the fluid. For a steady-state wall, all the released energy goes to the fluid, either to reheating or to bulk motions. The fraction $\kappa_{\mathrm{fl}}$ of the released vacuum energy which goes into bulk motions is relevant for the formation of gravitational waves through turbulence or sound waves. For runaway walls, there is also a fraction $\kappa_{\mathrm{wall}}$ of the vacuum energy which goes into kinetic energy of the wall. This is relevant for gravitational wave generation from direct bubble collisions. Thus, $\kappa_{\mathrm{fl}}$ is an efficiency coefficient for the injection of kinetic energy in the fluid, while $\kappa_{\mathrm{wall}}$ is as an efficiency coefficient for accelerating the wall.
We have studied, on the one hand, the hydrodynamics of a phase transition front, for given values of the wall velocity and acceleration, i.e., considering these variables as free parameters. Thus, we obtained results for $\kappa_{\mathrm{wall}}$ and $\kappa_{\mathrm{fl}}$ as functions of the velocity $v_w$ and the net force $F_{\mathrm{net}}$ acting on the wall. In this way, the results do not depend on the very involved computation of the wall dynamics, for which several approximations are generally needed. On the other hand, we have also studied the wall dynamics, taking into account the back-reaction of hydrodynamics on the wall motion, and we have discussed on the calculation of the wall velocity and acceleration as functions of thermodynamic and friction parameters.
We have computed the efficiency factor $\kappa_{\mathrm{fl}}$ for different wall symmetries, namely, spherical, cylindrical and planar walls. This complements the work of Ref. [@lm11], where we performed a similar analysis for stationary solutions. Here we considered the runaway case. For planar walls we obtained analytic results. The result for spherical bubbles is quite similar to the planar case, which can thus be used as an analytic approximation for the former. Besides, we provide fits for the spherical case as functions of $v_w$, $F_{\mathrm{net}}$, and thermodynamic parameters.
For the analysis of the wall dynamics, we considered a phenomenological model for the friction, which was introduced in Ref. [@ariel13] and depends on two free parameters. Thus, the model can reproduce the correct value of the friction force in the NR limit, $F_{\mathrm{fr}}\sim \eta_{NR}\, v_w$ as well as in the UR limit, $F_{\mathrm{fr}}\sim \eta_{UR}$. In the UR case it is actually more straightforward, for a given model, to compute the net force $F_{\mathrm{net}}$. The determination of the friction component of the UR force is relevant for the use of this phenomenological interpolation, which allows to calculate the wall velocity away from the UR limit. We have clarified the decomposition of $F_{\mathrm{net}}$ into driving and friction forces, taking hydrodynamics effects correctly into account.
Some of the issues discussed in the present paper were previously considered in Ref. [@ekns10] (for the case of spherical walls). However, a simpler phenomenological friction was used in that work, which depends on a single friction coefficient, corresponding to the particular case $\eta_{NR}=\eta_{UR}$. Moreover, some results, particularly those for the efficiency factor in the runaway regime, are given in terms of this friction coefficient (cf. Figs. 10 and 12 in [@ekns10]). Therefore, those results depend on the wall dynamics. As we have seen, the effect of reheating on the driving force was neglected in Ref. [@ekns10], which leads to a different distribution of the released energy. Concrete expressions for the runaway efficiency factors are given in [@ekns10] as functions of $\alpha$ and the UR detonation limits $ \alpha_0,\kappa^{\mathrm{det}}_{UR}$. In contrast, we obtained these quantities directly as functions of $\alpha$ and $F_{\mathrm{net}}$. Thus, we provide clean results for $\kappa_{\mathrm{wall}}$ and $\kappa_{\mathrm{fl}}$, which can be used to compute the production of gravitational waves in a phase transition.
In physical models, strongly first-order phase transitions (i.e., those with a relatively high value of the order parameter, $\phi>T$), generally have large amounts of released energy and considerable supercooling. This gives large values of $\alpha$, as well as high wall velocities or even runaway walls. We have explored the efficiency factors and the generation of gravitational waves for such parameter variations.
For detonations and runaway walls, the efficiency factor $\kappa_{\mathrm{fl}}$ decreases with the wall velocity and acceleration, while it increases with the quantity $\alpha$. However, for given values of the friction parameters, $v_w$ and $F_{\mathrm{net}}$ are increasing functions of $\alpha$, and it turns out that $\kappa_{\mathrm{fl}}$ decreases with $\alpha$. In contrast, $\kappa_{\mathrm{wall}}$ increases with $F_{\mathrm{net}}$ and $\alpha$. As a consequence, the gravitational wave production through fluid motions generally decreases with $\alpha$, while the production through bubble collisions generally increases. This does not mean, though, that stronger phase transitions will be less efficient in producing gravitational waves through fluid motions. In our parameter variations we have fixed some quantities which, for a concrete model, will change as $\alpha$ changes. In Ref. [@lms12] the electroweak phase transition was considered for several extensions of the Standard Model (all of which gave steady-state walls). In those cases, stronger phase transitions gave stronger GW signals from fluid motions. We shall consider models with even stronger phase transitions in a forthcoming paper [@lm15b]. As we have seen, for parameters which are characteristic of such strongly first-order electroweak phase transitions, the gravitational waves may be observed in the planned observatory eLISA.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by Universidad Nacional de Mar del Plata, Argentina, grant EXA699/14, and by FONCyT grant PICT 2013 No. 2786. The work of L.L. was supported by a CONICET fellowship.
Approximations for the fluid efficiency factor {#apend}
==============================================
In Ref. [@lm11] the hydrodynamics was studied for the stationary case, for spherical, cylindrical and planar walls. Although the fluid profiles are different in the three cases (corresponding to the spreading of the released energy in 3, 2, and 1 dimensions, respectively), the total energy in fluid motions is quite similar, particularly for detonations. Therefore, a relatively good approximation for the efficiency factor is to consider a planar wall, for which one obtains analytic results. One expects the same to hold for runaway walls.
For planar symmetry, the shape of the rarefaction wave behind the wall is quite simple. We have constant fluid velocity and enthalpy between the wall and a point which follows the wall with velocity $v
_{0}=(v_{-}+c_{-})/(1+v_{-}c_{-})$. Hence, the rarefaction actually begins behind that point. In the variable $\xi=z/t$, this point lies at a fixed position $\xi =v_{0}$, while the wall is at $\xi=v_w$. Between $\xi =c_{-}$ and $\xi =v _{0}$ we have $$v=\frac{\xi -c_{-}}{1-c_{-}\xi },\quad w=w_{-}\left( \frac{1-v_{-}}{1+v_{-}}
\frac{1+v}{1-v}\right) ^{\frac{1}{2}(c_{-}+\frac{1}{c_{-}})}.$$ For the bag EOS, we have $c_-=1/\sqrt{3}$, and the values of $v_-$ and $w_-$ are given by Eqs. (\[vme\]-\[wmewma\]) for detonations and by Eqs. (\[tmevmerun\]) for runaway walls. For this profile, the integral $I(v_w,v_-)$ in Eqs. (\[kappa\]-\[kapparun\]) is given by [@lm11] $$I=\gamma _{-}^{2}v_{-}^{2}\left( v_{w}-v _{0}\right) +\frac{3\left( 2-
\sqrt{3}\right) ^{\frac{2}{\sqrt{3}}}}{4}\left[ \frac{1-v_{-}}{1+v_{-}}
\right] ^{\frac{2}{\sqrt{3}}}\left[ f\left( v _{0}\right) -f\left(
c_{-}\right) \right] , \label{kappapl}$$ where $f\left( \xi \right) =\left( \frac{1+\xi }{1-\xi }\right) ^{\frac{2}{
\sqrt{3}}}\left\{ \frac{2}{\sqrt{3}}-1+\left( 1-\xi \right) \left[
2-\,_{2}F_{1}(1,1,\frac{2}{\sqrt{3}}+1,\frac{1+\xi }{2})\right] \right\} $, and $_{2}F_{1}$ is the hypergeometric function [@grads]. The efficiency factors $\kappa^{\mathrm{det}}_{\mathrm{pl}},\kappa^{\mathrm{run}}_{\mathrm{pl}}$ for the planar case are obtained by inserting Eq. (\[kappapl\]) in Eq. (\[kappa\]) (with $j=0$) or directly in Eq. (\[kapparun\]) for the runaway case. The result is shown in Fig. \[figplesf\], together with that for spherical walls.
Since the planar and spherical results are so similar, we may construct a fit for the spherical case by just approximating the difference between the two results. Indeed, correcting the planar results with a factor $1.03 + 0.1
\sqrt{v_w - v_J(\alpha)}$ is a good approximation. We thus have $$\begin{aligned}
\kappa_{\mathrm{fl}}^{\mathrm{det}}&=&\kappa^{\mathrm{det}}_{\mathrm{pl}}(\alpha,v_w)
\left(1.03 + 0.1 \sqrt{v_w - v_J(\alpha)}\right),
\label{fitpl1}
\\
\kappa_{\mathrm{fl}}^{\mathrm{run}}&=&\kappa^{\mathrm{run}}_{\mathrm{pl}}(\alpha,\bar F)
\left(1.03 + 0.1 \sqrt{1 - v_J(\alpha)}\right).
\label{fitpl2}\end{aligned}$$ In Fig. \[figfit\] we compare these fits with the numerical result. In the whole detonation range, and for $F_{\mathrm{net}}/\varepsilon<0.9$ in the runaway case, the relative error is smaller than a 3%.
In the runaway regime it is easy to find a simple fit (which does not rely on the analytic formulas of the planar case), since the integral $I_{1}(v_{-})= I(1,v_-)$ depends on the single parameter $v_-$. In the whole range $0<v_{-}<1$, this function is well approximated by the polynomial $I_{1}(v_{-})\simeq 0.15v_{-}^{2}-0.132v_{-}^{3}+0.065v_{-}^{4}$, with a relative error which is smaller than a 3% for $v_->10^{-3}$. Inserting in Eq. (\[kapparun\]), we have $$\kappa _{\mathrm{fl}}^{\mathrm{run}}\simeq (4/\alpha)(1+3\alpha-3\bar{F}
)(0.15v_{-}^{2}-0.132v_{-}^{3}+0.065v_{-}^{4})\,, \label{fit}$$ with $v_-(\alpha,\bar{F})$ given by Eq. (\[tmevmerun\]). The result is shown in the right panel of Fig. \[figfit\].
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[^1]: E-mail address: [email protected]
[^2]: Member of CONICET, Argentina. E-mail address: [email protected]
[^3]: This can be seen by differentiating Eqs. (\[junc1\]-\[junc2\]) for fixed $T_+$ and using the relation $dp_-=dw_-/(1+c_-^2)$ (see also [@mm14deto]).
[^4]: The total released energy density at finite temperature, $\Delta
e(T_{+})$, is actually higher than $\varepsilon $. At $T=T_{c}$ we have $\Delta
e=L=4\varepsilon $, while $\Delta e=\varepsilon $ occurs only at $T=0$. See [@lm15] for an alternate definition of an efficiency factor.
[^5]: More precisely, $\dot\gamma_w$ is proportional to the proper acceleration $\gamma_w^3\dot v_w$ which, according to Eq. (\[eqwall\]), is bounded by $\sim F_{\mathrm{dr}}/\sigma_0$, while $z-z_w$ is bounded by $l_0/\gamma_w$, where $l_0$ is the wall width at rest.
[^6]: In Ref. [@ariel13], a different approximation was used, in which the whole function of $\bar v$ in Eq. (\[Ffrfeno\]) was replaced by its average value. We have checked that there is no significant numerical difference.
[^7]: In fact, thermal energy is released as well as vacuum energy (whence $L>\varepsilon$). As a consequence, the white regions in Fig. \[figbudget\] only represent a part of the total energy which goes into reheating of the plasma. For a more detailed discussion, see [@lm15].
|
---
abstract: 'Outdoor acoustic event detection is an exciting research field but challenged by the need for complex algorithms and deep learning techniques, typically requiring many computational, memory, and energy resources. These challenges discourage IoT implementations, where an efficient use of resources is required. However, current embedded technologies and microcontrollers have increased their capabilities without penalizing energy efficiency. This paper addresses the application of sound event detection at the very edge, by optimizing deep learning techniques on resource-constrained embedded platforms for the IoT. The contribution is two-fold: firstly, a two-stage student-teacher approach is presented to make state-of-the-art neural networks for sound event detection fit on current microcontrollers; secondly, we test our approach on an ARM Cortex M4, particularly focusing on issues related to 8-bits quantization. Our embedded implementation can achieve 68% accuracy in recognition on Urbansound8k, not far from state-of-the-art performance, with an inference time of 125 ms for each second of the audio stream, and power consumption of 5.5 mW in just 34.3 kB of RAM.'
author:
- '\'
bibliography:
- 'mybib.bib'
title: |
Compact recurrent neural networks for\
acoustic event detection on low-energy low-complexity platforms
---
|
---
abstract: 'A method is proposed for studying wave and particle transport in disordered waveguide systems of dimension higher than unity by means of exact one-dimensionalization of the dynamic equations in the mode representation. As a particular case, the $T=0$ conductance of a two-dimensional quantum wire is calculated, which exhibits ohmic behaviour, with length-dependent conductivity, at any conductor length exceeding the electron quasi-classical mean free path. The unconventional diffusive regime of charge transport is found in the range of conductor lengths where the electrons are commonly considered as localized. In quantum wires with more than one conducting channel, each being identified with the extended waveguide mode, the inter-mode scattering is proven to serve as a phase-breaking mechanism that prevents interference localization without real inelasticity of interaction.'
address: |
Institute for Radiophysics & Electronics, National Academy of Sciences of Ukraine,\
12 Acad. Proskura St., Kharkov 61085, Ukraine
author:
- 'Yu. V. Tarasov'
title: |
Elastic Scattering as a Cause of Quantum Dephasing:\
The Conductance of Two-Dimensional Imperfect Conductors
---
Introduction {#intro}
============
Since the original formulation of the localization problem by Anderson [@anders58], the question of whether electronic states in disordered systems are localized at any strength of disorder or a mobility edge can be defined by a critical level of disorder has become a central issue. In the former case this yields an insulating-type behaviour of large samples, while in the latter the metallic-type transport of conducting electrons is allowed due to the existence of extended states.
It was subsequently ascertained that the answer to this question depends substantially on the dimensionality of the disordered system. For the case of one-dimensional (1D) conductors in the limit of vanishing temperature one can prove in a mathematically rigorous way (see Ref. and references therein) that the spectrum of the electrons, subject to an arbitrarily small but finite random potential in infinitely long samples, is purely discrete, i.e. all electron states are necessarily localized irrespective of their energy. As a result, the DC conductivity vanishes for such systems, whereas in finite 1D samples the conductance falls exponentially as the length of the conductor grows.
In contrast to the 1D case, no mathematically rigorous theory of localization exists for two- (2D) and tree-dimensional (3D) random systems. Formulation of the problem in terms of the renormalization group[@wegner76] (RG) led to the one-parameter scaling hypothesis of localization[@abrah79] and appeared to provide a considerable progress in studying systems of dimensionality greater than unity. The main conviction gained from this development was that all the electron states in both 1D and 2D disordered systems are localized at arbitrary small disorder. Hence, the metal-insulator transition (MIT) is usually believed to occur only in 3D systems, while (sufficiently large) 1D and 2D systems are always Anderson insulators.[@myremark_1]
This opinion, established quite long ago, has recently been challenged after unexpected experimental detection of MIT in dilute 2D electron and hole systems.[@kravch9596] Originally obtained on Si-MOSFETs, the results have entailed numerous experimental findings of MIT in other dilute 2D systems, [@popov97; @coler97; @simmon98; @papad98; @hanein98] leading to the well-founded belief that the effect is of a rather general nature. To elucidate the unconventional experimental facts, different theoretical approaches were put forward including quite disputable ones (see the discussion in Ref. ). Among the variety of options the greatest anticipations in explaining the experiments are mostly associated with accounting for the Coulomb interaction of carriers.[@shepelyan94; @weinman97] Yet the transport theories for correlated particles in the presence of disorder still cannot claim for general acceptance because of the substantial controversy in the interpretation of the role of interaction within different parameter domains which correspond to diffusive[@berkovits96; @schmitteckert98] and localized[@efros95; @talamantes96; @vojta98] regimes.
Traditional understanding of electron localization in 2D disordered systems and numerous experimental facts which indicate the metallic nature of charge transport at low temperatures have brought about active progress in the line of study that is identified with [*quantum dephasing*]{}. The main efforts in this direction are focused on the detection of various [*phase-breaking mechanisms*]{} responsible for the delocalization of quasi-particles primarily localized by disorder. However, in spite of a large body of versions on offer the problem still remains open.
Our intention in this contribution is to re-examine the conventional one-particle approach without appealing to scattering mechanisms other than those characteristic of quenched disorder. Evidence will be given that even within this elementary model the experimental findings of Refs. , at first sight curious, are as a matter of fact quite natural. Numerous attempts to interpret the results of Ref. within the framework of a single-particle approach were made, in particular, by improving the scaling approach.[@dobrosavl97] In this study, however, a fundamentally different strategy is chosen which is an alternative to the RG analysis.[@ma76] We prefer to obtain the observables directly, while conclusions (though indirect) about the localization of electron states are made on the basis of the results.
It is instructive to recall that working out, even without a profound spectral analysis, practical asymptotic methods for calculating the disorder-averaged many-particle characteristics (conductivity, density-density correlator, etc.) turned out to be more helpful for the establishment of a highly advanced theory of 1D random systems than the development of rigorous mathematical foundations.[@berezinski73; @abrikryzh78; @kanercheb87; @kanertar88] The usefulness of such an approach can be attributed to the fact that in the context of the above-mentioned essentially perturbative methods one has managed to trace with the required accuracy the effect of mutual interference of quantum waves corresponding to multiply backscattered current carriers. In such a way, physical results entirely consistent with the anticipations based on mathematical predictions were eventually obtained. The present research was primarily induced by long-standing discontent associated with the lack of arguments of a comparable standard, either in favour of localization or against it, as applied to 2D systems of degenerate electrons subject to a static random potential.
Commonly, the presence of inelastic scattering mechanisms is held responsible as a main cause of preventing quantum interference and, thus, Anderson localization.[@bu_im_lan83] Among these are the electron-phonon and electron-electron interactions and other conceivable methods of energy interchange between the electron bath and the environment.[@mello99] These can lead to the loss of phase (meaning energy) memory or, in other words, phase coherence of electronic states. Note in this connection that in the 1D case the demand of energy coherence admits a large transfer of [*momentum*]{} for onefold scattering of degenerate electrons in the backward direction. This leads inevitably to considerable local breaking of [*spatial*]{} (instead of [*temporal*]{}) phases of the wavefunctions. It was recognized that such a large violation of spatial phases is quite helpful when deriving a constructive theory of 1D quantum transport. On the basis of such arguments, the selection of efficient subsets of terms in perturbative expansion which are responsible for the effective interference of electronic waves scattered iteratively in a [*backward*]{} direction was suggested.[@berezinski73] This interference finally results in the formation of true localized states, even when one starts their perturbative construction from plain-wave-like trial functions which belong to a mathematical class different from that of localized functions.
An analogous scenario of the perturbative formation of localized states in 2D systems cannot give the same result, since for sufficiently isotropic scattering the spatial coherence of the wavefunctions is already violated at distances of the order of the quasi-classical mean free path even for weak energy scattering. The coherence is maintained efficiently only within a small phase volume, thus resulting in the relative smallness of the interference corrections, usually known as weak localization corrections[@altaronkhmel82; @altarkhmellark82] for non-one-dimensional systems.
Nevertheless, the analysis of the problem of multidimensional localization with the use of a one-dimensional scenario turns out to be quite productive. As we shall prove below the problem of electron transport in 2D open system of waveguide type can be reduced without any approximations to the set of one-dimensional (though non-Hermitian) problems for the quantum waves propagating in individual conducting channels. The channels will be identified with extended waveguide modes. For the corresponding dynamic equations that are one-dimensional, an opportunity arises for making substantial use of [*spatial*]{} phases of the wavefunctions instead of their [*temporal*]{} parts. This turns out to be preferable from the technical point of view for solving stationary problems of the electron, as well as classical wave, transport. The reason for the usefulness of such a procedure lies in the fact that in one-dimensional problems spatial averaging has been shown to be highly advantageous, leading to the reduction of the perturbative expansions of the physical observables to a summable series.[@berezinski73; @abrikryzh78]
Being exactly quantum in nature, the waveguide approach used here is, to a certain degree, less obvious than semi-classical ones usually applied in most localization theories. Thus, in its context the clarity of the path-based interpretation is substantially lost. At the same time, the benefit of our method is that one-dimensional ‘channel’ equations enable one to distinguish unambiguously the coherent intra-mode scattering, which is easily interpreted from the standpoint of 1D localization theory, and the inter-mode scattering which corresponds, although not quite directly, to isotropic scattering of semi-classical electrons. The quantum states in different channels are specified by different longitudinal momenta. This difference essentially suppresses interference of primary and scattered electronic waves, if they belong to different channels. As a result, the inter-mode scattering turns out to be an intrinsic origin of the inability for 2D electrons to be localized by weak static disorder.
The ‘one-dimensionalization’ procedure suggested in this paper gives an opportunity to highlight the role of spatial coherence in the interference of electronic waves even in a single-particle approximation.[@feynman65] The results obtained here by conventional perturbative methods enable us to conclude about the unrealizability of the strong (Anderson) localization in systems whose spatial dimensionality is greater than unity, irrespective of their size. It should be particularly emphasized that decoherence appearing due to inter-mode scattering is unconnected to genuine inelasticity in the interaction of electrons with disorder. However, the difference in [*longitudinal energy*]{} between the conducting channels could be treated as a source of ‘hidden inelasticity’, if one is accustomed to such an interpretation.
The paper is organized as follows. In the next section, the problem is formulated using linear response theory. In section \[Uni\_proc\] we develop a method of exact one-dimensionalization which is a central point of the paper. Then, in section \[trial\_green\], the trial Green functions supremely important for the developed technique are analyzed with the aid of a two-scale perturbation method. A spectral analysis of the electron system is given in section \[Spectrum\]. In the final two sections we present asymptotic expressions for the conductance and discuss the results. A pair of tedious but important calculations is presented in two appendices.
Statement of the problem {#statement}
========================
A common approach used in studies of random systems of various dimensionality is to take a hyper-cube of a certain linear size and vary the size while searching for the conductance scaling. Such an approach seems to be natural when studying spectral properties of closed systems. At the same time, it is not quite appropriate for solving transport problems as applied to structures of waveguide type, in particular, quantum conductors of arbitrary length.
In this paper, we examine the case of a 2D imperfect rectangular sample of length $L$ in the $x$-direction and width $D$ in the lateral direction $y$. Degenerate non-interacting spinless electrons are assumed to be confined between the hard-wall side boundaries $y=\pm D/2$, whereas in the direction of current ($x$) we suppose the system to be open at the strip ends $x=\pm L/2$. The dimensionless conductance $g(L)$ (in units $e^2/\pi\hbar$) is calculated directly from linear response theory,[@kubo57] whence at zero temperature we have
$$g(L)=-\frac{4}{L^2}\int\!\!\!\!\int d\bbox{r}\,d\bbox{r}'
\frac{\partial G(\bbox{r},\bbox{r}')}{\partial x}\cdot
\frac{\partial G^*(\bbox{r},\bbox{r}')}{\partial x'} \ .
\label{Kubo_cond}$$
Here the integration with respect to $\bbox{r}=(x,y)$ is performed over the area occupied by the conductor
$$x\in(-L/2,L/2)\ ,\qquad y\in(-D/2,D/2) \ .
\label{cond_area}$$
$G(\bbox{r},\bbox{r}')$ is the retarded Green function of the conducting electrons. Within the isotropic Fermi liquid model this function is governed by the equation
$$\left[ \Delta+k_F^2+i0-V(\bbox{r}) \right]G(\bbox{r},\bbox{r}')=
\delta(\bbox{r}-\bbox{r}') \ .
\label{StartEq}$$
We adopt the system of units with $\hbar=2m=1$ ($m$ is the electron effective mass), so that $\Delta$ is a two-dimensional Laplace operator, $k_F$ is the Fermi wavenumber of the electrons, $V(\bbox{r})$ is the ‘bulk’ static random potential.
The potential $V(\bbox{r})$ in equation (\[StartEq\]) will be regarded as a short-range one and not necessarily isotropic. The term ‘short-range’ implies the characteristic spatial interval over which the potential is substantially varied to be small compared with the ‘macroscopic’ lengths of the problem, namely the electron mean free path and the conductor length. Being considered as a statistical variable, the potential $V(\bbox{r})$ will be specified by a zero mean value and the binary correlation function
$$\left< V(\bbox{r})V(\bbox{r}') \right>={\cal Q}W(\bbox{r}-\bbox{r}') \ .
\label{VrVr}$$
Here the angular brackets denote ensemble averaging and $W(\bbox{r})$ is some function normalized to unity. The explicit form of this function is not so important. In many cases $W(\bbox{r})$ is approximated by the delta-function, $W(\bbox{r})=\delta(\bbox{r})$. However, the method developed in this contribution permits one to consider not only isotropic and not necessarily local scattering events. In addition, the choice of $W(\bbox{r})$ in the form of a delta-function, apart from restricting the physical applicability of the model, is not quite convenient from the technical viewpoint. Thus, for example, when calculating the corrections to the mode spectrum, equations (\[T-renorm\]), a formal problem can arise of the divergence of the evanescent mode contribution. It is certainly absent provided the potential $W(\bbox{r})$ is not exactly point-supported — this problem is familiar in quantum mechanics.[@BazZeldPerel; @LandauLif] To get rid of the divergences in the problems of dimension more than one it is merely sufficient to choose the function $W(\bbox{r})$ to be less singular than $\delta(\bbox{r})$.[@LifGredPas] Therefore, in order to focus on the main questions, in place of the correlation equality (\[VrVr\]), we shall use the expression below
$$\left< V(\bbox{r})V(\bbox{r}') \right>={\cal Q}W(x-x')\delta(y-y') \ .
\label{VrVr_2}$$
Additionally, this form of equation (\[VrVr\]) allows us to consider anisotropic scattering. Similar to $W(\bbox{r})$ from (\[VrVr\]), the function $W(x)$ in (\[VrVr\_2\]) is normalized to unity.
The one-dimensionalization procedure {#Uni_proc}
====================================
The general scheme {#UNI-GEN}
------------------
It seems intuitively natural for an open system with the prescribed direction of quasi-particle transport to be considered as to some extent a one-dimensional object. However, the realizability of ‘one-dimensionalization’ at the level of dynamic equations, which would be quite important from a mathematical perspective, is not [*a priori*]{} apparent. Here the term ‘one-dimensionalization’ means reduction of the two-dimensional differential equation (\[StartEq\]) to a set of one-dimensional equations. Although a system of waveguide type can often be regarded as a collection of one-dimensional quantum channels, the latter are not independent in general. Normally, they are strongly coupled with each other through static or dynamic inhomogeneities present in the problem.
Nevertheless, in what follows we intend to show that just the waveguide nature of a system under consideration enables the mathematical description of the transport problem for a 2D region (\[cond\_area\]) to be reduced to a set of independent strictly one-dimensional, although non-Hermitian, problems posed on the interval $x\in(-L/2,L/2)$, regardless of the strength of the disorder. To perform the reduction one should merely pass to the mode representation, i.e. Fourier transform in the transverse coordinate $y$, using some complete set {$\phi(y)$} of eigenfunctions of the transverse free-electron Hamiltonian, namely the Laplace operator in Eq. (\[StartEq\]). The conductance (\[Kubo\_cond\]) acquires the form
$$g(L)=
- \frac{4}{L^2} \int\!\!\!\!\int_L dx dx'\sum_{n,n'=1}^{N_c}
\frac{\partial G_{nn'}(x,x')}{\partial x}
\frac{\partial G_{nn'}^*(x,x')}{\partial x'} \ ,
\label{Cond-mode}$$
where $N_c=[k_FD/\pi]$ is the number of conducting channels or, in other words, extended waveguide modes. Equation (\[StartEq\]) is then transformed into a set of coupled ordinary differential equations for the Fourier components $G_{nn'}(x,x')$,
$$\bigg[ \frac{\partial^2}{\partial x^2}+k_n^2+i0
- V_n(x)\bigg]G_{nn'}(x,x')
-\sum_{m=1\atop(m\neq n)}^{\infty} U_{nm}(x)G_{mn'}(x,x')
=\delta_{nn'}\delta(x-x') \ .
\label{ModeEq}$$
Here $k_n^2=k_F^2-(n\pi/D)^2$ is the longitudinal energy of the $n$th mode, $U_{nm}(x)$ is the inter-mode matrix element of the potential $V(\bbox{r})$,
$$U_{nm}(x)=\int_D dy\, \varphi_n(y)V(\bbox{r})\varphi_m(y) \ .
\label{Unm}$$
Note the difference between the summation limits in Eqs. (\[Cond-mode\]) and (\[ModeEq\]). Restriction of the summation in (\[Cond-mode\]) by the number of conducting channels implies, strictly speaking, weakness of the electron scattering, to be specified in Sec. \[WEAK-APPROX\]. Under the same assumption formula (\[Kubo\_cond\]) itself is valid, where the products of the Green functions of the same kind (both retarded and advanced) have already been omitted. In equation (\[ModeEq\]), the summation is naturally performed over a complete set of waveguide modes.
Particular attention should be drawn to the fact that the term containing the diagonal (intra-mode) matrix element $U_{nn}(x)\equiv V_n(x)$ is initially detached from the sum of Eq. (\[ModeEq\]), so that the matrix $\|U_{nm}\|$ hereafter is held off-diagonal in the discrete mode variable. This little technical trick enables one to reduce the problem of finding the overall of the functions $G_{nn'}(x,x')$ to the solution of a subset of purely one-dimensional closed equations for the mode-diagonal functions $G_{nn}(x,x')$ only. To this end we first introduce the auxiliary trial Green functions $G_n^{(V)}(x,x')$ ($n\in\aleph$), each obeying the equation
$$\left[\frac{\partial^2}{\partial x^2}+k_n^2+i0- V_n(x)\right]
G_n^{(V)}(x,x')=\delta(x-x') \ ,
\label{trial_Gn}$$
and Sommerfeld’s radiation conditions[@BassFuks79; @Vladimirov67] at the strip ends $x=\pm
L/2$. These conditions seem to be natural to impose on an [*open*]{} system. For the case of the 1D equation (\[trial\_Gn\]) they acquire the form[@Klyatskin86]
$$\left.\left(\frac{\partial}{\partial x}\mp ik_n\right)
G_n^{(V)}(x,x')\right|_{x=\pm L/2}=0 \ .
\label{1Dradcond}$$
It implies that the field of the $n$th mode radiated by the point source placed at $x'\in(-L/2,L/2)$ reaches the corresponding ($\pm$) end of the interval and then propagates unscattered with the conserved momentum $k_n$ beyond the ends of the conductor. Possible scattering in the leads attached to the strip from the left and right should be taken into account separately.
Although a solution of the stochastic problem (\[trial\_Gn\]) and (\[1Dradcond\]) cannot be obtained in quadratures, in the case of weak scattering specified below by inequalities (\[weakscat\]) the main features of the solution can be extracted by due consideration with any desired accuracy in the [*intra-mode*]{} potential $V_n(x)$. The necessary analysis is presented in the next section. As for now, we merely consider all the functions $G_n^{(V)}(x,x')$ as [*a priori*]{} known ones whose properties are specified solely by the elastic intra-mode scattering. With these functions chosen as an initial approximation for the exact mode functions $G_{nn}$, only the inter-mode scattering associated with the off-diagonal matrix $U_{nm}(x)$ will then be taken as a perturbation. To implement this intent, we turn from equation (\[ModeEq\]) to the consequent integral equation,
$$G_{nn'}(x,x')=G_n^{(V)}(x,x')\delta_{nn'}
+\sum_{m=1}^{\infty}\int_L dx_1\,
{\sf R}_{nm}(x,x_1)G_{mn'}(x_1,x') \ .
\label{Mode_inteq}$$
Here the kernel $${\sf R}_{nm}(x,x_1)=G_n^{(V)}(x,x_1)U_{nm}(x_1)
\label{kern_ R}$$ already contains only those harmonics of the potential $V(\bbox{r})$ which are responsible for the inter-mode scattering. A thorough study of system (\[Mode\_inteq\]) leads to a conjecture that all non-diagonal mode elements $G_{mn}$ ($m\neq
n$) can be expressed only via the diagonal element $G_{nn}$ by means of some linear operator $\hat{\sf K}$,
$$G_{mn}(x,x')=\int_Ldx_1\,{\sf K}_{mn}(x,x_1)G_{nn}(x_1,x') \ .
\label{G_mn-sol}$$
To specify this operator, one should separate the term with the diagonal (in the mode variable) Green function on the right-hand side of equation (\[Mode\_inteq\]) and substitute all non-diagonal Green functions in the form of equation (\[G\_mn-sol\]). Then, after renaming the mode variables, we arrive at the following equation for the matrix elements of the operator $\hat{\sf K}$:
$${\sf K}_{mn}(x,x')={\sf R}_{mn}(x,x')
+\!\!\sum_{k=1\atop (k\neq n)}^{\infty}
\int_Ldx_1\,{\sf R}_{mk}(x,x_1){\sf K}_{kn}(x_1,x') \ .
\label{K_mn}$$
Equation (\[K\_mn\]) belongs to a class of multi-channel Lippmann-Schwinger equations that are known to be extremely singular, in contrast to their single-channel counterparts.[@Taylor72] Nevertheless, by choosing the trial Green function $G_n^{(V)}$ as a zero approximation and perturbing it only by the inter-mode potentials, one manages to avoid the above mentioned singularity. Note that mode indices $m$ and $k$ in Eq. (\[K\_mn\]) take all the positive integer values except for the value $n$. This urges one to interpret the functions appearing in (\[K\_mn\]) as matrix elements of operators acting in two-dimensional mixed coordinate-mode space ($x,m$) which does not include the mode $n$ (the notation ${\sf{\overline M}_n}$ will be used for that space). The presence in Eq. (\[K\_mn\]) of the right-hand index $n$, which does not belong to ${\sf{\overline
M}_n}$, can be ensured by introducing the projection operator ${\bf P}_n$ that will make the mode index of any operator standing next to it (both from the left or right) equal to $n$. With this convention accepted, it follows from equation (\[K\_mn\]) that the operator $\hat{\sf K}$ implementing relation (\[G\_mn-sol\]) has the form
$$\hat{\sf K}=\left(\openone-\hat{\sf R}\right)^{-1}\hat{\sf R}{\bf P}_n \ .
\label{HAT_K}$$
Here $\hat{\sf R}$ is a 2D operator acting on ${\sf{\overline
M}_n}$ and is specified by the matrix elements (\[kern\_ R\]).
As for the mathematical correctness of the operator representation (\[HAT\_K\]), it depends on the existence of the inverse operator $\left(\openone-\hat{\sf R}\right)^{-1}$. This point is discussed in Appendix \[K\_exist\], where we provide a proof that detachment of the intra-mode potential $V_n(x)$ in Eq. (\[ModeEq\]) prevents the possible singularity.
Thus, equations (\[G\_mn-sol\]) and (\[HAT\_K\]) reduce the problem of finding the complete Green function $G(\bbox{r},\bbox{r}')$ within the 2D region (\[cond\_area\]) to calculation of its diagonal mode components only. In order to do that it is necessary to put $n'=n$ in Eq. (\[ModeEq\]) and substitute all non-diagonal components $G_{mn}$ from equation (\[G\_mn-sol\]). As a result, the closed equation for the diagonal component $G_{nn}$ is deduced,
$$\left[\frac{\partial^2}{\partial x^2}+k_n^2
+i0-V_n(x)-\hat{\cal T}_n\right]
G_{nn}(x,x')=\delta(x-x') \ .
\label{1Deq}$$
In equation (\[1Deq\]), in addition to the [*local*]{} intra-mode potential $V_n(x)$, the [*operator*]{} potential ${\hat{\cal T}}_n$ has arisen,
$${\hat{\cal T}}_n={\bf P}_n\hat{\cal U}\left(\openone-
\hat{\sf R}\right)^{-1}\hat{\sf R}{\bf P}_n =
{\bf P}_n \hat{\cal U}\left(\openone-
\hat{\sf R}\right)^{-1}{\bf P}_n \ .
\label{Tn}$$
Here $\hat{\cal U}$ is the inter-mode operator specified on ${\sf{\overline M}_n}$ by the matrix elements
$$\biglb<x,l\bigrb|\hat{\cal U}\biglb|x',m\bigrb>=U_{lm}(x)\delta(x-x') \ .
\label{calU}$$
The potential ${\hat{\cal T}}_n$ has quite an interesting interpretation. In the right-hand side of equation (\[Tn\]) there is a conventional $T$-matrix[@Taylor72; @Newton68] enveloped by the projective operators, one of which removes the excitation from mode $n$ and the other restores it back to the same mode after the appropriate scattering events within the subspace ${\sf{\overline M}_n}$. Hence, the potential ${\hat{\cal
T}}_n$, although corresponding to effectively intra-mode scattering, actually includes all inter-mode scattering events undergone by the excitation while passing over ‘closed paths’ in the mode space. The intra- and inter-mode scattering mechanisms turn out to be attributed to different potentials in equation (\[1Deq\]), facilitating significantly the subsequent interpretation of the results. In what follows the potentials $V_n(x)$ and ${\hat{\cal T}}_n$ will be referred to as those responsible for [*direct*]{} intra-mode and inter-mode scattering, respectively.
Finally, in this subsection we express the conductance (\[Cond-mode\]) through the diagonal elements of the mode Green matrix. After rearranging the terms in Eq. (\[Cond-mode\]) and using relation (\[G\_mn-sol\]) we obtain
$$\begin{aligned}
g(L)=-\frac{4}{L^2}\sum_{n=1}^{N_c}
&&\int\!\!\!\!\int_L dx\,dx'\Bigg[
\frac{\partial G_{nn}(x,x')}{\partial x}
\cdot\frac{\partial G_{nn}^*(x,x')}{\partial x'}
\nonumber \\
+\sum_{m=1\atop(m\neq n)}^{N_c}
&&\int\!\!\!\!\int_L dx_1dx_2\,
\frac{\partial {\sf K}_{mn}(x,x_1)}{\partial x}G_{nn}(x_1,x')
{\sf K}_{mn}^*(x,x_2)\frac{\partial G_{nn}^*(x_2,x')}{\partial x'}\Bigg]\ .
\label{Cond_2}\end{aligned}$$
Expression (\[Cond\_2\]) jointly with equation (\[1Deq\]) completes, in principle, the ‘one-dimensionalization’ procedure introduced at the beginning of this section. In this form the problem under consideration is convenient for a numerical treatment at any disorder strength, because the solution of the 2D problem governed by equation (\[StartEq\]) is reduced to a [*finite set*]{} of purely 1D problems (\[trial\_Gn\]) and (\[1Deq\]). At the same time, assuming weak electron scattering (in the semi-classical sense), we manage to proceed with our analytical consideration and obtain the results.
The weak scattering approximation {#WEAK-APPROX}
---------------------------------
It is natural to specify the intensity of electron scattering in terms of characteristic spatial scales inherent to the problem. Henceforth we recognize the scattering as weak provided the following inequalities hold:
$$k_F^{-1},r_c\ll\ell \ .
\label{weakscat}$$
Here $r_c$ is the correlation radius of the potential $V(\bbox{r})$, $\ell=2k_F/{\cal Q}$ is the [*semiclassical*]{} mean free path of electrons evaluated within the model of a $\delta$-correlated 2D random potential, i.e. $W(\bbox{r})=\delta(\bbox{r})$ in Eq. (\[VrVr\]).
Estimation of the norm of the operator $\hat{\sf R}$ specified on ${\sf{\overline M}_n}$ by the matrix elements (\[kern\_ R\]) results in
$$\|\hat{\sf R}\|^2\sim \frac{D/L}{k_F\ell} \ .
\label{normR}$$
Under conditions (\[weakscat\]), this enables us to find an expansion to lowest order in the impurity potential of the operator $\hat{\sf K}$, Eq. (\[HAT\_K\]), so that it becomes approximately equal to $\hat{\sf R}$ almost regardless of the conductor aspect ratio. The exact operator ${\hat{\cal T}}_n$ from (\[Tn\]) can, in turn, be replaced by the approximate value
$${\hat{\cal T}}_n\approx {\bf P}_n\hat{\cal U}{\hat
G}^{(V)}\hat{\cal U}{\bf P}_n
\label{T_approx}$$
where the operator ${\hat G}^{(V)}$ is specified on ${\sf{\overline M}_n}$ by the matrix elements $$\left<x,k\right|{\hat G}^{(V)}\left|x',m\right>=
\delta_{km}G_m^{(V)}(x,x') \ .$$
Besides the reduction of the $T$-operator (\[Tn\]) to truncated form (\[T\_approx\]), a substitution of the approximate matrix elements of the operator $\hat{\sf K}$ brings the second term of conductance (\[Cond\_2\]) to the form
$$\int\!\!\!\!\int_L dx_1dx_2\, U_{mn}(x_1)U_{mn}(x_2)
\frac{\partial G_m^{(V)}(x,x_1)}{\partial x}{G_m^{(V)}}^*(x,x_2)
G_{nn}(x_1,x')\frac{\partial G_{nn}^*(x_2,x')}{\partial x'} \ ,
\label{aux_g2}$$
which is convenient for performing the ensemble averaging. It will be shown below that all Green functions in (\[aux\_g2\]), and not only the trial ones, may be thought of as independent of the inter-mode potentials $U_{mn}$. Yet correlation of those potentials with the intra-mode one, $V_n(x)$, governing the trial Green functions, can be disregarded in view of Eq. (\[inter-intra\]). As a result, after averaging conductance (\[Cond\_2\]) with the use of (\[aux\_g2\]) and (\[VrVr\_2\]), the expression, which will be subject to a further analysis, takes the form
$$\begin{aligned}
\big<g(L)\big>=-&&\frac{4}{L^2}\!\sum_{n=1}^{N_c}
\!\int\!\!\!\!\int_L\!\!dx\,dx'\!\left[
\Big<\frac{\partial G_{nn}(x,x')}{\partial x}
\frac{\partial G_{nn}^*(x,x')}{\partial x'}\Big> \right.
\nonumber \\
+&&
\frac{{\cal Q}}{D}\sum_{m=1\atop(m\neq n)}^{N_c}
\int\!\!\!\!\int_Ldx_1\,dx_2 \left.W(x_1-x_2)
\Big<{G_m^{(V)}}^*(x,x_2)\frac{\partial}{\partial x}G_m^{(V)}(x,x_1)\Big>
\Big<G_{nn}(x_1,x')\frac{\partial}{\partial x'}
{G_{nn}}^*(x_2,x')\Big>\right] \ .
\label{G(L)_AV}\end{aligned}$$
Analysis of the trial Green functions {#trial_green}
=====================================
The trial Green functions $G_m^{(V)}(x,x')$ enter the potential ${\hat{\cal T}}_n$, Eq. (\[T\_approx\]), thus determining the exact mode functions $G_{nn}(x,x')$, and the second term of the conductance (\[G(L)\_AV\]). Although these functions appear as subsidiary mathematical objects, they are of great concern for the problem and therefore deserve particular consideration. The study of the trial functions is also instructive, since it may provide useful insights into the analysis of some misinterpretation regarding 2D localization which existed until recently in the literature.
Note first that the perturbative solution of equation (\[trial\_Gn\]) depends substantially on whether the corresponding unperturbed waveguide mode is either extended or evanescent. The Green functions of evanescent modes with $n>N_c$ are localized even without any perturbation. To find them in the limit of weak scattering, it is sufficient to restrict oneself to zero-order perturbation in the potential $V_n(x)$,
$$G_n^{(V)}(x,x')= -\frac{1}{2|k_n|}\exp\biglb(-|k_n||x-x'|\bigrb) \ ,
\hspace{1.5cm} n>N_c \ .
\label{G_evan}$$
The problem is much more involved for the extended modes, $n<
N_c$. Inasmuch as the function $G_n^{(V)}(x,x')$ is defined as a solution of the strictly one-dimensional problem (\[trial\_Gn\]) and (\[1Dradcond\]), to find it correctly in the context of localization theory the plain-wave-based zero approximation is not quite appropriate. This stems from the fact that in such an approximation it is rather difficult to account for the interference of multiply backscattered waves. Instead we apply the two-scale perturbation method analogous to that used in the theory of non-linear oscillations[@BogolMitr74]. This method showed itself well for solving the problem of charge transport in extremely narrow, namely single-mode, surface-corrugated conducting strips.[@MakTar98] Below an outline of the method is given together with some essential results. The details of their derivation are deferred to Appendix \[MOMENTS-GN\].
The method used in Ref. is based on the representation of the 1D Green function, which is the solution of the boundary-value problem (\[trial\_Gn\]) and (\[1Dradcond\]), via the solutions of the appropriate Cauchy problems,
$$G_n^{(V)}(x,x')=\bbox{\cal W}_n^{-1}
\big[ \psi_+(x|n)\psi_-(x'|n)\Theta(x-x') +
\psi_+(x'|n)\psi_-(x|n)\Theta(x'-x) \big] \ ,
\label{Green-Cochi}$$
Here the functions $\psi_{\pm}(x|n)$ are linearly independent solutions of the homogeneous equation (\[trial\_Gn\]) with the radiation conditions analogous to (\[1Dradcond\]) satisfied at the strip ends $x=\pm L/2$, according to the ‘sign’ index. The Wronskian of those functions is $\bbox{\cal W}_n$, $\Theta(x)$ is Heaviside’s step function. This reformulation of a boundary-value problem in terms of an initial-value problem will allow one to perform averaging over the disorder later on.[@Klyatskin86]
It is advantageous to represent the functions $\psi_{\pm}(x|n)$ as superpositions of modulated harmonic waves propagating in opposite directions along the $x$-axis,
$$\psi_{\pm}(x|n) = \pi_{\pm}(x|n)\exp(\pm ik_n x) -
i\gamma_{\pm}(x|n)\exp(\mp ik_n x) \ .
\label{psi-pm}$$
Within the framework of the weak scattering approximation (\[weakscat\]), the amplitudes $\pi_{\pm}(x|n)$ and $\gamma_{\pm}(x|n)$ vary slowly as compared with the ‘fast’ exponentials $\exp(\pm ik_n x)$, so that the radiation conditions for $\psi_{\pm}(x|n)$ are reformulated as the ‘initial’ conditions for the smooth amplitudes as follows:
$$\pi_{\pm}(\pm L/2|n) = 1 \ , \qquad \qquad \gamma_{\pm}(\pm L/2|n) = 0 \ .
\label{In_cond}$$
In view of the smoothness of $\pi_{\pm}$ and $\gamma_{\pm}$, differential equations for them can be derived by means of averaging the equations for $\psi_{\pm}(x|n)$ over an arbitrary-valued spatial interval intermediate between the ‘microscopic’ lengths $k_n^{-1}$ and $r_c$ on the one hand, and the ‘macroscopic’ lengths on the other hand. Among the latter lengths are the scattering length, to be specified in the course of the solution, and the sample length $L$. For weak scattering, the equations for $\pi_{\pm}$ and $\gamma_{\pm}$ are reduced to the coupled first-order ones,
$$\begin{array}{ccl}
&&\pi'_{\pm}(x|n)\pm i\eta_n(x)\pi_{\pm}(x|n)
\pm\zeta_{n\pm}^*(x)\gamma_{\pm}(x|n) =0\ , \\[6pt]
&&\gamma'_{\pm}(x|n)\mp i\eta_n(x)\gamma_{\pm}(x|n)
\pm\zeta_{n\pm}(x)\pi_{\pm}(x|n) =0\ .
\end{array}
\label{PiGamma_eq}$$
Here the variable coefficients $\eta_n(x)$ and $\zeta_{n\pm}(x)$ are random fields associated with the intra-mode potential $V_n(x)$ in the following way:
$$\eta_n(x)=\frac{1}{2k_n}\int\limits_{x-l}^{x+l}\frac{dt}{2l}V_n(t) \ ,
\hspace{1.5cm}
\zeta_{n\pm}(x)=\frac{1}{2k_n}\int\limits_{x-l}^{x+l}\frac{dt}{2l}
{\rm e}^{\mp 2ik_nt}V_n(t) \ .
\label{EtaZeta}$$
For the intermediate property of the averaging interval $2l$, the fields $\eta_n(x)$ and $\zeta_{n\pm}(x)$ are, in fact, nothing but the ‘narrow’ sets of spatial harmonics of the potential $V_n(x)$ with the momenta close to zero and $\pm 2k_n$, respectively. The real field $\eta_n(x)$ is responsible for the ‘forward’ scattering, while the complex one $\zeta_{n\pm}(x)$ for the ‘backward’ scattering of the $n$th waveguide mode.
Under the assumption of weak scattering, only binary correlators of the random potentials govern the majority of statistical characteristics of physical quantities. It was shown in Ref. that only the two correlation functions, $\left<\eta_n(x)\eta_n(x')\right>$ and $\left<\zeta_{n\pm}(x)\zeta_{n\pm}^*(x')\right>$, of modified random fields (\[EtaZeta\]) may be thought of as non-vanishing. Calculation with the use of model (\[VrVr\_2\]) readily gives
\[Eta\_Zeta\] $$\begin{aligned}
\left<\eta_n(x)\eta_n(x')\right>&=&
\frac{1}{L_f^{(V)}(n)}F_l(x-x') \ ,
\label{EtaEta}\\
\left<\zeta_{n\pm}(x)\zeta_{n\pm}^*(x')\right>&=&
\frac{1}{L_b^{(V)}(n)}F_l(x-x') \ .
\label{ZetaZeta}\end{aligned}$$
Here
$$L_f^{(V)}(n)=\frac{2D}{3{\cal Q}}(2k_n)^{2}
\qquad\text{and}\qquad
L_b^{(V)}(n)=\frac{2D}{3{\cal Q}}\frac{(2k_n)^{2}}{\widetilde{W}(2k_n)}
\label{Lf(n)Lb(n)}$$
are the forward and backward mode scattering lengths, respectively; $\widetilde{W}(q)$ is the Fourier transform of $W(x)$ from Eq. (\[VrVr\_2\]). The function
$$F_l(x)=\int\limits_{-\infty}^{\infty}\frac{dq}{2\pi}{\text e}^{iqx}
\frac{\sin^2(ql)}{(ql)^2}=
\frac{1}{2l}\left(1-\frac{|x|}{2l}\right)\Theta(2l-|x|)
\label{F_l}$$
in Eqs. (\[Eta\_Zeta\]) is sharp at the scale of ‘macroscopic’ lengths, so it can be regarded as the $\delta$-function in the ‘distributional’ sense, $F_l(x)\to\delta(x)$.
Equations (\[PiGamma\_eq\]) and correlation relations (\[Eta\_Zeta\]) enable one to obtain the entire statistical information concerning the trial Green function for any extended mode $n$. In Appendix \[MOMENTS-GN\] derivation of all plain moments of that function is presented with the following result:
$$\begin{aligned}
&&\Big<\left[G_n^{(V)}(x,x')\right]^\mu\Big>=
\left(\frac{-i}{2k_n}\right)^\mu
\exp\left[i\mu k_n|x-x'|-\frac{\mu}{2}
\left(\frac{\mu}{L_f^{(V)}(n)}+
\frac{1}{L_b^{(V)}(n)}\right)|x-x'|\right] \ ,
\nonumber \\
&&\mu\in\aleph \ .
\label{moments}\end{aligned}$$
It is noteworthy that from Eqs. (\[Lf(n)Lb(n)\]) we have the following estimate:
$$L_{f,b}^{(V)}(n)\sim N_c\ell\cos^2\vartheta_n \ ,
\label{Lfb_estim}$$
where $\vartheta_n$ is a ‘sliding angle’ of the $n$th mode with respect to the $x$-axis, $|\sin\vartheta_n|=n\pi/k_FD$. The value of (\[Lfb\_estim\]) is coincident, in order of magnitude, with the [*localization length*]{} widely believed to be characteristic for multi-mode quasi-one-dimensional (Q1D) waveguide systems (see, e.g., Refs.). However, in the present theory lengths (\[Lf(n)Lb(n)\]) are nothing but the extinction lengths of the [*auxiliary*]{} excitations that are not subjected to inter-mode scattering.
Besides plain moments (\[moments\]), important characteristics of the random function $G_n^{(V)}(x,x')$ are the mixed moments $\Big<\left[G_n^{(V)}(x,x')\right]^\mu
\left[{G_n^{(V)}}^*(x,x')\right]^\nu\Big>$. At $\mu=\nu$ all of them are smooth (not oscillatory) functions of the argument $|x-x'|$ whose spatial decrease is determined by the [*one-dimensional localization length*]{} $\xi_n=4L_b^{(V)}(n)$.[@LifGredPas] The second term in (\[G(L)\_AV\]) contains one of the simplest correlators of this type, the ‘density-current’ correlator $\big<G^*\nabla G\big>$. It was already studied in applications to the problem of classical wave transport.[@FreylTaras91] We do not present here the exact expression for this correlator since only the fact of its exponential decrease at the localization length $\xi_n$ is of significance for our analysis,
$$\Big<{G_m^{(V)}}^*(x,x')\frac{\partial}{\partial x}G_m^{(V)}(x,x')\Big>
\propto\exp\left(-\frac{|x-x'|}{\xi_n}\right) \ .
\label{corr_dens-curr}$$
The mode states spectrum: Perturbative treatment {#Spectrum}
================================================
For calculation of the diagonal Green function $G_{nn}(x,x')$ in equation (\[1Deq\]) by means of the perturbation technique, it would make a good sense to reconstruct the operator potential ${\hat{\cal T}}_n$ so that the mean value equals zero. To that end one has to define the result of the action of the operator $\langle {\hat{\cal T}}_n\rangle$ on the function $G_{nn}$. For $n>N_c$ (evanescent modes) the function $G_{nn}$ can be left in its unperturbed form (\[G\_evan\]) due to the weakness of scattering. For $n< N_c$, with the operator nature of the potential $\langle {\hat{\cal T}}_n\rangle$ and reduced form (\[T\_approx\]) of the $T$-operator taken into account, the following integral has to be calculated:
$$\langle {\hat{\cal T}}_n\rangle G_{nn}(x,x')=
\int_L dx_1 {\sf T}_n(x,x_1)G_{nn}(x_1,x') \ .
\label{hat_T-Gnn}$$
Here the kernel ${\sf T}_n(x,x_1)$ is given by
$${\sf T}_n(x,x_1)=\sum_{m=1\atop(m\neq n)}^{\infty}
\left<U_{nm}(x)G_m^{(V)}(x,x_1)U_{mn}(x_1)\right> \ .
\label{krn_<T>}$$
On performing the ensemble averaging in Eq. (\[krn\_<T>\]), it is justifiable to neglect the correlation between the inter-mode potentials $U_{nm}(x)$ and the intra-mode ones, $V_m(x)$. Within the model of point-like scatterers such correlation is entirely absent. Yet even in the case of disorder specified by correlation function (\[VrVr\_2\]) it is not difficult, using the definition (\[Unm\]) and the hard-wall model of side boundaries, to ascertain the equality
$$\left<U_{nm}(x)V_m(x')\right>=0 \ .
\label{inter-intra}$$
This allows us to couple the inter-mode potentials in Eq. (\[krn\_<T>\]) only with each other, not affecting the trial functions $G_m^{(V)}(x,x_1)$. The averaging procedure thus makes the operator $\langle {\hat{\cal
T}}_n\rangle$ effectively local.
Since equation (\[1Deq\]) is one-dimensional in the space variable, the exact Green function $G_{nn}(x,x')$ can be represented in a form similar to that used for the trial function $G_n^{(V)}(x,x')$, Eq. (\[Green-Cochi\]). Specifically, under weak scattering conditions the function $G_{nn}(x,x')$, being considered as a function of the first argument ($x$) only, is composed of two slightly modulated exponential summands of the appearance $\phi_{\pm}(x)=t_{\pm}(x)\exp(\pm ik_nx)$. By applying the operator $\langle {\hat{\cal T}}_n\rangle$ to the functions $\phi_{\pm}(x)$ one can factor the smooth amplitudes $t_{\pm}(x)$ out of the integral thus arriving at the result
$$\langle {\hat{\cal T}}_n\rangle \phi_{\pm}(x)=
\Sigma(k_n)\phi_{\pm}(x) \ ,
\label{<T>->Sigma}$$
where
$$\Sigma(k_n)=\frac{\cal Q}{D}\int_L dx_1 W(x-x_1)
\exp\left[\mp ik_n(x-x_1)\right]\sum_{m=1\atop(m\neq n)}^{\infty}
\left<G_m^{(V)}(x,x_1)\right> \ .
\label{Sigma(k_n)}$$
For deriving equation (\[Sigma(k\_n)\]), the correlation equality was used
$$\left<U_{nm}(x)U_{kn}(x_1)\right>=\frac{\cal Q}{D}W(x-x_1)\delta_{mk}
\label{<UnmUmn>}$$
which results from definition (\[Unm\]) and the correlation model (\[VrVr\_2\]). The sharp function $W(x-x_1)$ present in Eq. (\[Sigma(k\_n)\]) allows us to replace the trial functions $G_m^{(V)}(x,x_1)$ by the unperturbed free mode Green functions $G_m^{(0)}(x,x_1)$. Then, in the case of even $W(x)$, the factor $\Sigma(k_n)$ acquires the simple form
$$\Sigma(k_n)=\frac{{\cal Q}}{D}\sum_{m=1\atop (m\neq n)}^{\infty}
\int\limits_{-\infty}^{\infty}\frac{dq}{2\pi}\widetilde{W}(q+k_n)
\widetilde{G}_m^{(0)}(q) \ .
\label{Sigma-int_q}$$
Here $\widetilde{G}_m^{(0)}(q)$ is the Fourier transform of the function $G_m^{(0)}(x)$, and it is independent of the sign of the momentum $k_n$.
From the above analysis it follows that the action of the operator $\langle
{\hat{\cal T}}_n\rangle$ on the function $G_{nn}(x,x')$ is reduced to multiplying by, in general, the complex-valued quantity $\Sigma(k_n)$. Using the explicit form of $\widetilde{G}_m^{(0)}(q)$,
$$\widetilde{G}_m^{(0)}(q)=\frac{1}{(k_m+i0)^2-q^2} \ ,
\label{G(q)}$$
for both real and imaginary parts of the mode ‘self-energy’ $\Sigma(k_n)=\Delta k_n^2-i/\tau_n^{(\varphi)}$ the expressions are obtained as follows:
\[T-renorm\] $$\begin{aligned}
\Delta k_n^2&=&\frac{{\cal Q}}{D}\sum_{m=1\atop (m\neq n)}^{\infty}
{\cal P}\!\!\!\int_{-\infty}^{\infty}\frac{dq}{2\pi}
\frac{\widetilde{W}(q+k_n)}{k_m^2-q^2} \ ,
\label{Corr_k2}
\\
\frac{1}{\tau_n^{(\varphi)}}&=&\frac{{\cal Q}}{4D}
\sum_{m=1\atop(m\neq n)}^{Nc}\frac{1}{k_m}
\left[\widetilde{W}(k_n-k_m)+\widetilde{W}(k_n+k_m)\right] \ .
\label{Atten_n}\end{aligned}$$
The symbol ${\cal P}$ in (\[Corr\_k2\]) denotes the principal value. Under the conditions of weak scattering the real part $\Delta k_n^2$ is always small, $|\Delta k_n^2|\ll k_n^2$, so it can be disregarded without serious consequences. At the same time, ‘dissipative’ term (\[Atten\_n\]) plays a crucial role for the further analysis and cannot be omitted. As a result, equation (\[1Deq\]) takes the form
$$\left[\frac{\partial^2}{\partial x^2}+\kappa_n^2
+i0-V_n(x)-\Delta{\hat{\cal T}}_n\right]
G_{nn}(x,x')=\delta(x-x') \ ,
\label{1Deq_fin}$$
where $\kappa_n^2=k_n^2+i/\tau_n^{(\varphi)}$ and $\Delta{\hat{\cal
T}}_n={\hat{\cal T}}_n-\langle{\hat{\cal T}}_n\rangle$. Thus, for the analysis of equation (\[1Deq\_fin\]) we shall regard the set of renormalized energies $\kappa_n^2$ ($n=1,2,\ldots$) as representing the new ‘unperturbed’ spectrum of the system, instead of the original spectrum {$k_n^2$}. The perturbation theory can now be developed making use of the appropriate zero-mean potentials $V_n(x)$ and $\Delta{\hat{\cal T}}_n$.
Note the difference in summation regions for Eqs. (\[Corr\_k2\]) and (\[Atten\_n\]). The summation in (\[Atten\_n\]) turns out to be restricted by the number of [*conducting channels*]{} (extended waveguide modes), because only for $n\leq N_c$ the disorder-averaged trial Green functions in Eq. (\[Sigma(k\_n)\]) are essentially complex-valued (see Eq. (\[moments\]) at $\mu=1$). The level broadening $1/\tau_n^{(\varphi)}$ implies obligatory presence in the conductor of other extended modes besides the $n$th mode itself. In the case of an extremely narrow strip with $N_c=1$ the sum (\[Atten\_n\]) contains no terms, and thus the system should exhibit true one-dimensional properties. Specifically, the electrons in such systems can be transferred within two regimes only, [*ballistic*]{} and [*localized*]{}, and the conductance should go down [*exponentially*]{} with the length $L$ exceeding the localization length $\xi_1$.[@MakTar98]
On increasing the conductor width, as soon as the wire ceases to be single-mode ($N_c\geq 2$), the situation changes drastically. The $n$th-mode spectrum acquires the level broadening (\[Atten\_n\]) and is subjected to both the potentials $V_n(x)$ and $\Delta{\hat{\cal T}}_n$. The physical reason for spectrum ‘complexification’ with the availability of more than one extended mode in the conductor is actually the randomization of the [*spatial*]{} phase of the electron wavefunction in going between the states with different mode energies and, consequently, mode momenta. It is just the uncertainty of those momenta that destroys the spatial coherence of one-dimensional quantum waves governed by equation (\[1Deq\_fin\]). This destruction prevents interferential localization of the true mode states of conducting electrons contrary to their trial states.
To evaluate the ‘phase-breaking’ effect of the term (\[Atten\_n\]) note that at any $N_c>1$ the estimate $1/\tau_n^{(\varphi)}\sim{\cal Q}$ is valid. In particular, in the extreme case of a multimode conductor ($N_c\gg 1$) with point-like scatterers, replacing the summation in Eq. (\[Atten\_n\]) by an integration we obtain
$$1/\tau_n^{(\varphi)}\approx{\cal Q}/4 \ .
\label{Ph-Br_Nc>>1}$$
It is noteworthy that for a large number of conducting sub-bands each level width becomes independent of the mode number $n$ and can thus be thought of as a universal dephasing rate inherent to the 2D conductor in general. Although it is commonly believed that the phase breaking stems from inelastic scattering, the level broadening (\[Atten\_n\]) has nothing to do with such a scattering mechanism. The only important reason for the existence of an imaginary part of the mode energy is that the conductor has to possess more than one extended mode.
Besides the phase breaking contribution $1/\tau_n^{(\varphi)}$, the role of varying potentials $V_n(x)$ and $\Delta{\hat{\cal T}}_n$ should also be studied. These zero-mean potentials can be assessed through evaluation of the corresponding Born scattering rates $1/\tau_n^{(V)}$ and $1/\tau_n^{(\cal T)}$. Estimation of the operator norms $\|\hat V_n\|^2$ and $\|\Delta{\hat{\cal T}}_n\|^2$ yields
\[Born\_rates\] $$\begin{aligned}
\frac{\tau_n^{(V)}}{\tau_n^{(\cal T)}}&\sim&
N_c\text{min}\left(1,\frac{L}{N_c\ell}\right) \ ,
\label{estim_V/T} \\
\frac{\tau_n^{(\varphi)}}{\tau_n^{(\cal T)}}&\sim&
\frac{1}{\cos^2\vartheta_n}\text{min}\left(1,\frac{L}{N_c\ell}\right) \ .
\label{estim_phi/T}\end{aligned}$$
It hence follows that scattering caused by the non-local potential $\Delta{\hat{\cal T}}_n$ is more efficient than that attributed to the local intra-mode potential $V_n(x)$.
On the other hand, the inter-mode scattering caused by the operator potential ${\hat{\cal T}}_n$ in Eq. (\[1Deq\]) is already taken largely into account through the imaginary renormalization of the mode energy in (\[1Deq\_fin\]). From (\[estim\_phi/T\]) it can be seen that the scattering rate $1/\tau_n^{(\cal T)}$ is always small compared to the level width (\[Atten\_n\]) provided that the wire is not extremely long in the $x$-direction, i.e. if the length $L$ does not fall into the interval $L\gg N_c\ell$. Yet even within this interval the quantity $1/\tau_n^{(\cal T)}$ cannot exceed the level broadening. If that is the case, the search for strong Anderson localization at any length of the multimode ($N_c\geq 2$) conducting strip has no sense even without any inelastic scattering mechanisms. Probably, it is necessary to invent some extra conditions to make the interferential localization possible in such conductors, e.g., magnetic field,[@Malin97] surface roughness,[@nieto98] etc.
Calculation of the conductance {#Eval_Cond}
==============================
Equations (\[1Deq\_fin\]) and (\[trial\_Gn\]) provide a way to perform a direct calculation of the disorder-averaged conductance (\[G(L)\_AV\]). In this section, to avoid cumbersome expressions the results will be given for the case of a large number of conducting channels, $N_c\gg 1$. Nevertheless, all the estimates, as well as the final formulae, are valid for an arbitrary finite number of modes, $N_c\gtrsim 1$. In what follows, two summands in equation (\[G(L)\_AV\]) will be considered separately inasmuch as the calculation techniques and the contribution of these terms in the total conductance differ significantly. The first summand, $\left<g^{(1)}(L)\right>$, will be conventionally referred to as the diagonal conductance, since it does not clearly contain any quantities characteristic of the inter-mode scattering. The second term in (\[G(L)\_AV\]), $\left<g^{(2)}(L)\right>$, will be conventionally referred to as the non-diagonal conductance.
Note that the effect of phase breaking, which manifests itself strongly through complexification of the excitation spectrum at $N_c>1$, enables one to obtain the solution of equation (\[1Deq\_fin\]) perturbatively in the potentials $V_n$ and $\Delta{\hat{\cal T}}_n$, i.e. to neglect, in the leading approximation, the interference of quantum waves multiply scattered by those potentials. It follows from estimates (\[Born\_rates\]) that such an approach is absolutely justified for conductors of length $L\ll N_c\ell$. Yet even in the case of a Q1D wire of length $L\gg N_c\ell$, when the potential $\Delta{\hat{\cal T}}_n$ in Eq. (\[1Deq\_fin\]) cannot be disregarded as follows from estimation (\[estim\_phi/T\]), the effect of this potential will be shown to be negligibly small.
The diagonal conductance {#diag_cond}
------------------------
The presence of the local potential $V_n(x)$ in Eq. (\[1Deq\_fin\]) does not substantially complicate the calculation of the conductance. By applying the method of Ref. , this potential can be taken into account with just the same accuracy as was done diagrammatically in Ref. . Accounting for this local potential leads to purely interferential corrections, which we are not concerned with in this paper.
As regards the operator potential $\Delta{\hat{\cal T}}_n$, to treat it perturbatively keeping in mind the estimate (\[Born\_rates\]) it is advantageous to go over from the differential equation (\[1Deq\_fin\]) to the consequent integral one,
$$G_{nn}(x,x')=G_{nn}^{(0)}(x,x')+\Big(\hat G_{nn}^{(0)}
\Delta{\hat{\cal T}}_n\hat G_{nn}\Big)(x,x') \ ,
\label{Gnn_Dyson}$$
Here $G_{nn}^{(0)}(x,x')$ obeys equation (\[1Deq\_fin\]) with $\Delta{\hat{\cal T}}_n=0$. In the leading approximation in the parameter $L/N_c\ell\ll 1$, the function $G_{nn}^{(0)}$ has the simple ‘unperturbed’ form
$$G_{nn}^{(0)}(x,x')=\frac{1}{2ik_n}
\exp\Big\{\big[ik_n-1/l_n^{(\varphi)}\big]|x-x'|\Big\}
\label{Gnn^0}$$
which nonetheless includes most of the inter-mode-scattering effects. In Eq. (\[Gnn\^0\]) the mode coherence length $l_n^{(\varphi)}$ is associated with the $n$th level broadening (\[Atten\_n\]), namely $l_n^{(\varphi)}=2k_n\tau_n^{(\varphi)}$. In the limit $N_c\gg 1$ its value equals
$$l_n^{(\varphi)}=4\ell\cos\vartheta_n \ .
\label{l_n-approx}$$
Substitution of the Green function (\[Gnn\^0\]) into the first summand of Eq. (\[G(L)\_AV\]) readily gives the diagonal part of the conductance
$$\big<g^{(1)}(L)\big>=\sum_{n=1}^{N_c}\frac{l_n^{(\varphi)}}{L}
\left[1-\frac{l_n^{(\varphi)}}{L}\exp\left(-\frac{L}{l_n^{(\varphi)}}\right)
\sinh\frac{L}{l_n^{(\varphi)}}\right] \ .
\label{g(1)}$$
In the limit $N_c\gg 1$, by replacing the sum in Eq. (\[g(1)\]) with the integral and substituting the coherence length in the form (\[l\_n-approx\]), we arrive at the following asymptotic expressions for the conductance (\[g(1)\]):
\[g(1)\_asymp\] $$\begin{aligned}
L\ll\ell&\qquad\text{---}\qquad&
\big<g^{(1)}(L)\big>\approx
N_c\left(1-\frac{\pi}{12}\frac{L}{\ell}\right) \ ,
\label{g(1)_ball} \\
L\gg\ell&\qquad\text{---}\qquad&
\big<g^{(1)}(L)\big>\approx \pi N_c\ell/L \ .
\label{g(1)_diff}\end{aligned}$$
In the limit of $L\gg N_c\ell$, it follows from (\[estim\_phi/T\]) that the zero-order approximation in $\Delta{\hat{\cal T}}_n$ is, strictly speaking, insufficient for calculating the function $G_{nn}(x,x')$. Nevertheless, since the scattering rate $1/\tau_n^{(\cal T)}$ can only be less than or of the order of the level width $1/\tau_n^{(\varphi)}$, the appropriate correction to the conductance can be reasonably estimated by substituting into $g^{(1)}(L)$ the function $G_{nn}(x,x')$ obtained from (\[Gnn\_Dyson\]) to the first order in $\Delta{\hat{\cal T}}_n$. A simple but tedious calculation brings about the following estimation of the corresponding correction to the conductance:
$$\big<\Delta g^{(1)}(L)\big>\sim \left(\frac{\ell}{L}\right)^2 \ .
\label{Delta_g(1)}$$
This quantity is apparently small compared with (\[g(1)\_diff\]). The result in (\[Delta\_g(1)\]) indicates that one may account for the inter-mode scattering in the problem under consideration by means of the phase breaking factor which shows itself only in smearing of the mode energy levels.
The non-diagonal term of the conductance {#nondiag_cond}
----------------------------------------
When calculating the second term $\big<g^{(2)}(L)\big>$ in Eq. (\[G(L)\_AV\]) two essentially different correlators have to be evaluated, the first containing the trial mode Green functions and the second composed of the exact ones. As regards the correlator of the trial functions $G_m^{(V)}$, from result (\[moments\]) it is clear that at $L\ll N_c\ell$ those functions can be replaced by the free ones, independently of the mode. Using the function $G_{nn}$ in the form (\[Gnn\^0\]) we obtain
$$\big<g^{(2)}(L)\big>=-\frac{{\cal Q}}{4L^2D}
\sum_{n=1}^{N_c}\frac{\left(l_n^{(\varphi)}\right)^3}{k_n}
\exp\left(-\frac{L}{l_n^{(\varphi)}}\right)
\left[\frac{L}{l_n^{(\varphi)}}\cosh\frac{L}{l_n^{(\varphi)}}-
\sinh\frac{L}{l_n^{(\varphi)}}\right]
\sum_{m=1\atop(m\neq n)}^{N_c}\frac{1}{k_m} \ .
\label{g(2)}$$
At $N_c\gg 1$ this expression is substantially simplified giving the asymptotics as follows:
\[g(2)\_assymp\] $$\begin{aligned}
L\ll\ell \hspace{.5cm} &\qquad\text{---}\qquad&
\big<g^{(2)}(L)\big>\approx -\frac{\pi}{24}N_c\frac{L}{\ell} \ ,
\label{g(2)_ball} \\
\ell\ll L\ll N_c\ell &\qquad\text{---}\qquad&
\big<g^{(2)}(L)\big>\approx -\pi N_c\ell/2L \ .
\label{g(2)_diff}\end{aligned}$$
In the case of extremely long conductors with $L\gg N_c\ell$, the trial Green functions in (\[G(L)\_AV\]) cannot be replaced by unperturbed ones so that the effect of 1D localization should be taken into account properly. The ‘density-current’ correlator standing in (\[G(L)\_AV\]) was studied in Ref. . The result obtained there closely corresponds to analogous results for the ‘density-density’ and ‘current-current’ correlators. The main feature of the correlator is that it decays exponentially at the localization length $\xi_m=4L_b^{(V)}(m)$. Keeping this in mind, it is not difficult to estimate the non-diagonal term in (\[G(L)\_AV\]) as
$$\big<g^{(2)}(L)\big>\sim
\left(\frac{N_c\ell}{L}\right)^2 \ .
\label{g(2)_loc}$$
Results and discussion {#DISCUSSION}
======================
While comparing results (\[g(2)\_assymp\]) and (\[g(2)\_loc\]) with those given by equations (\[g(1)\_asymp\]) it can be seen that the non-diagonal part of the conductance is not small relative to the diagonal one only if the conductor length falls within the interval $\ell\ll L\ll N_c\ell$. In this case the non-diagonal term is half the size of its diagonal counterpart and has the opposite sign. Hence in the whole range of the conductor length (with the width being kept constant) the conductance is described by the following asymptotic expressions:
$$\begin{array}{rll}
\text{(i)} & L<\ell\ : & \big<g(L)\big>\approx N_c \\[6pt]
\text{(ii)} & \ell\ll L\ll N_c\ell\ :\qquad & \big<g(L)\big>\approx
(\pi/2) N_c\ell/L\gg 1 \\[6pt]
\text{(iii)} & N_c\ell\ll L\ : & \big<g(L)\big>\approx
\pi N_c\ell/L\ll 1 \ .
\end{array}
\label{g(L)_fin}$$
Strictly speaking, result (\[g(L)\_fin\]) is valid exactly when the number of channels is large, $N_c\gg 1$. Nevertheless, even in the case of $N_c\gtrsim 1$ only an insignificant difference occurs, produced by the dependence of the coherence length $l_n^{(\varphi)}$ on the mode number $n$. This dependence differs somewhat in the non-semi-classical limit from that given by equation (\[l\_n-approx\]), but the difference reduces to a numerical factor of the order of unity only.
The result given by Eq. (\[g(L)\_fin\]) allows one to distinguish three regimes of charge transport in a quantum conductor, depending on its aspect ratio. Regime (i) corresponds to entirely ballistic transport, both from the semi-classical and quantum points of view. The result obtained exhibits a natural stepwise dependence of the conductance on the transverse size of the wire.
In regimes (ii) and (iii) the semi-classical motion should be regarded as diffusive. This opinion is consistent with the conventional view of classical diffusion since the mean free path $\ell$ is small compared with the sample length $L$, although it can be in arbitrary relation to the conductor width. Furthermore, such an interpretation is supported by the ohmic, i.e. inversely proportional to $L$, dependence of the conductance in both of the indicated regimes. At the same time, it should be particularly emphasized that only in regime (ii), commonly called the [*weak localization*]{} regime, is the result given by the classical kinetic theory reproduced exactly. The expression for the diffusion coefficient in regime (iii) differs from that pertinent to regime (ii) by a factor of two.
Regime (iii) is often called [*localized*]{}, because it is usually supposed that in wires of such a length (commonly referred to as Q1D systems) the Anderson localization should manifest itself to a considerable extent, thus leading to an exponential fall in the conductance. However, from the above analysis it follows that conductors with more than one quantum channel interconnected through scattering mechanisms, even elastic ones, should not exhibit an exponential dependence of kinetic coefficients on the sample length. Such a behaviour is characteristic for the single-mode wires only, which is entirely consistent with theoretical predictions for one-dimensional disordered systems.
To associate findings of this paper with convictions that have prevailed hitherto, it is helpful to examine the electron transport in regimes (ii) and (iii) starting with the trial waveguide states governed by the homogeneous equation (\[trial\_Gn\]). Those states are certainly fictitious, they would exist provided the inter-mode scattering were disregarded. If so, the system would indeed represent a set of $N_c$ independent one-dimensional conducting channels where the true interferential localization should take place as a result of direct intra-mode scattering from the potentials $V_n$. For all of the channels, a hierarchy of localization lengths would exist similar to that representative of equation (\[Lfb\_estim\]). The length region in (ii) corresponds to the condition when the majority of the trial states are extended. In contrast, in regime (iii) all of those states are localized, which is consistent with the expectation of an exponential fall of the conductance with the growth of the sample length.
In reality one certainly cannot disregard the inter-mode scattering in the case of arbitrary quenched disorder. That scattering results in quite strong coupling of the channels or, rather, the trial mode states. The coupling has shown itself through the complexification of true mode spectrum in Eq. (\[1Deq\_fin\]). Such a complexification suggests that for any extended mode in a multi-mode strip all other extended modes can be thought of as a phase-breaking reservoir destroying quantum interference and hence strong (exponential) localization. Only weak localization corrections due to the [*local*]{} intra-mode potentials can be detected in the both of the diffusive regimes (\[g(L)\_fin\]). A comprehensive analysis of the matter is beyond the scope of this paper and will be presented elsewhere.
The existence of different diffusion regimes (ii) and (iii) can be interpreted as the dependence of the diffusion coefficient on the conductor aspect ratio. This dependence can hardly be extracted in the framework of the semi-classical approach. It results from the fact that under a gradual transition from regime (ii) to (iii), with the growth of the conductor length, the [*trial waveguide states*]{} undergo sequential localization. This should reduce the probability for the corresponding excitations to leave the conductor through the current terminals, whereas the probability of their scattering into other extended modes should increase. When all the trial states become finally localized, the diffusion coefficient stabilizes at the value corresponding to regime (iii). The dimensionless conductance of such a long wire is less than unity as a consequence of the conventional Ohm’s law. It seems that this smallness has previously been the reason to suppose all genuine states in Q1D conductors to be localized.
The author is grateful to N. M. Makarov for stimulating discussions, A. V. Moroz for help in the interpretation of the results and K. Ilyenko for reading the manuscript.
On regularity of the operator $\protect\hat{\sf K}$, E. (\[HAT\_K\]) {#K_exist}
====================================================================
To ascertain that unity is not among the characteristic numbers of the operator $\hat{\sf R}$, let us take advantage of the operator identity
$$\log\det(\openone-\hat{\sf R})=\text{Tr}\log(\openone-\hat{\sf R})=
\text{Tr}\sum_{k=1}^{\infty}\frac{(-1)^k}{k}\hat{\sf R}^k \ .
\label{Simil}$$
The last equality in (\[Simil\]) presumes that the operator norm is limited by $\|\hat{\sf R}\|<1$. This is entirely consistent with the estimate (\[normR\]).
Consider the traces of the first two terms in sum (\[Simil\]),
$$\begin{aligned}
\text{Tr}\,\hat{\sf R}&=&\sum_{n=1}^{\infty}
\int_Ldx\,G_n^{(V)}(x,x)U_{nn}(x) \ ,
\label{First}
\\
\text{Tr}\,{\hat{\sf R}}^2&=&\sum_{n,m=1}^{\infty}
\int\!\!\!\!\int_Ldxdx'\,G_n^{(V)}(x,x')U_{nm}(x')
G_m^{(V)}(x',x)U_{mn}(x) \ .
\label{Second}\end{aligned}$$
Since at $n>N_c$ the function $G_n^{(V)}(x,x')$ in the case of weak scattering has the form (\[G\_evan\]), it is easy to conclude that the divergence of the logarithm in (\[Simil\]) can arise from the first term, given in (\[First\]), provided $U_{nn}(x)\not\equiv 0$. The divergence stems from locality, i.e. coincidence of the arguments, of the Green functions. It manifests itself not only on average but also at a given realization of the random potential entering this term. By separating the diagonal, i.e. intra-mode, potential $V_n(x)\equiv U_{nn}(x)$ and making the matrix $\|U_{nm}\|$ off-diagonal we prevent the singularity of the operator $\hat{\sf K}$ given by Eq. (\[HAT\_K\]).
Statistical moments of the trial Green functions {#MOMENTS-GN}
================================================
To perform the ensemble-averaging of the random function $\Phi_{\mu}(x,x'|n)=\big[G_n^{(V)}(x,x')\big]^\mu$, $\mu\in\aleph$, in accordance with representation (\[Green-Cochi\]) and (\[psi-pm\]), we first decompose the function $G_n^{(V)}(x,x')$ into the sum of four terms each containing narrow packets only of spatial harmonics with phases close to $\pm k_n(x\pm x')$. In doing so one should use the asymptotic expression for the Wronskian $\bbox{\cal W}_n$,
$$\bbox{\cal W}_n=2ik_n\big[\pi_{+}(x|n)\pi_{-}(x|n)+
\gamma_{+}(x|n)\gamma_{-}(x|n)\big] \ .
\label{W_appr}$$
This results from the assumption of smoothness of the amplitude functions in Eq. (\[psi-pm\]). Then, after substituting (\[psi-pm\]) and (\[W\_appr\]) into (\[Green-Cochi\]), the function $G_n^{(V)}(x,x')$ can be represented in the form of a scalar product
$$G_n^{(V)}(x,x')=\pmatrix{{\rm e}^{ik_nx}&\!\!\!\!;\ {\rm e}^{-ik_nx}}
{{\widetilde G_1\ \widetilde G_3 }\choose
{\widetilde G_4\ \widetilde G_2}}
{{{\rm e}^{-ik_nx'}} \choose {{\rm e}^{ik_nx'}} } \ .
\label{scal_prod}$$
Here $\widetilde G_i(x,x'|n)$ are the smooth amplitudes constructed from the envelopes $\pi_{\pm}(x|n)$ and $\gamma_{\pm}(x|n)$ as follows:
\[G\_matr\] $$\begin{aligned}
\widetilde G_1(x,x'|n)&=&\frac{-i}{2k_n}A_n(x)
\left[\Theta_+\frac{\pi_{-}(x'|n)}{\pi_{-}(x|n)}-
\Theta_-\frac{\gamma_{+}(x'|n)}{\pi_{+}(x|n)}\Gamma_{-}(x|n)\right] \ ,
\label{G_1} \\[6pt]
\widetilde G_2(x,x'|n)&=&\frac{-i}{2k_n}A_n(x)
\left[\Theta_-\frac{\pi_{+}(x'|n)}{\pi_{+}(x|n)}
-\Theta_+\Gamma_{+}(x|n)\frac{\gamma_{-}(x'|n)}{\pi_{-}(x|n)}\right] \ ,
\label{G_2} \\[6pt]
\widetilde G_3(x,x'|n)&=&\frac{-1}{2k_n}A_n(x)
\left[\Theta_+\frac{\gamma_{-}(x'|n)}{\pi_{-}(x|n)}+
\Theta_-\frac{\pi_{+}(x'|n)}{\pi_{+}(x|n)}\Gamma_{-}(x|n)\right] \ ,
\label{G_3} \\[6pt]
\widetilde G_4(x,x'|n)&=&\frac{-1}{2k_n}A_n(x)
\left[\Theta_-\frac{\gamma_{+}(x'|n)}{\pi_{+}(x|n)}
+\Theta_+\Gamma_{+}(x|n)\frac{\pi_{-}(x'|n)}{\pi_{-}(x|n)}\right] \ .
\label{G_4}\end{aligned}$$
The notation used in Eqs. (\[G\_matr\]) is $$\hspace{-1cm}
A_n(x)=\left[1+\Gamma_{+}(x|n)\Gamma_{-}(x|n)\right]^{-1} \ ,
\qquad \Gamma_{\pm}(x|n)=\frac{\gamma_{\pm}(x|n)}{\pi_{\pm}(x|n)}
\ , \qquad \Theta_{\pm}=\Theta[\pm(x-x')] \ .$$ Before averaging the functions (\[G\_matr\]) over the random potential, let us note some useful features of the dynamic system (\[PiGamma\_eq\]). Since $\pi_{\pm}(x|n)$ and $\gamma_{\pm}(x|n)$ are the [*causal*]{} functionals of the fields $\eta_n(x)$, $\zeta_{n\pm}(x)$ and $\zeta_{n\pm}^*(x)$, they are determined by the values of those fields on the intervals $(x,L/2]$ and $[-L/2,x)$ for the functionals labelled by the indexes ($+$) and ($-$), correspondingly. The Green function elements (\[G\_matr\]), which will be subjected to ensemble averaging, are constructed in such a fashion that supports of the random functions entering the functionals of ‘plus’ and ‘minus’ type do not meet. Due to the random fields being effectively $\delta$-correlated, see Eqs. (\[Eta\_Zeta\]) and (\[F\_l\]), averaging of the functionals with different sign indexes can be performed independently.
It also follows from equations (\[PiGamma\_eq\]) and conditions (\[In\_cond\]) that all terms of functional series for $\pi_{\pm}(x|n)$ contain an equal number of fields $\zeta_{n\pm}$ and $\zeta_{n\pm}^*$, whereas $\gamma_{\pm}(x|n)$ has an extra functional factor $\zeta_{n\pm}$. Since for weak scattering all fields $\eta_n(x)$, $\zeta_{n\pm}(x)$ and $\zeta_{n\pm}^*(x)$ are approximately [*Gaussian*]{} random processes, only the first summands in square brackets of Eqs. (\[G\_1\]) and (\[G\_2\]) remain non-zero after averaging, while the quantities $\big<\widetilde G_{3,4}\big>$ vanish. By the same arguments, the factor $A_n(x)$ in (\[G\_matr\]) can be replaced by unity.
In view of the statistical independence of the functions $\eta_n(x)$ and $\zeta_{n\pm}(x)$, it is convenient to average over the real field $\eta_n(x)$ already at the initial stage. To that end, it is advantageous to perform the following phase transformation of the amplitudes $\pi_{\pm}$ and $\gamma_{\pm}$:
$$\begin{aligned}
\pi_{\pm}(x|n)&=&\widetilde\pi_{\pm}(x|n)\exp\bigg[\pm i\int_x^{\pm L/2}
\!\!\!\!\eta_n(x_1)\,dx_1\bigg] \ , \nonumber
\\[-7pt]\label{phase_ren}\\[-7pt]
\gamma_{\pm}(x|n)&=&\widetilde\gamma_{\pm}(x|n)\exp\bigg[\mp i\int_x^{\pm
L/2} \!\!\!\!\eta_n(x_1)\,dx_1\bigg] \ .
\nonumber\end{aligned}$$
The new amplitudes $\widetilde\pi_{\pm}$ and $\widetilde\gamma_{\pm}$ obey the equations
$$\begin{array}{ccl}
&&\widetilde\pi'_{\pm}(x|n)\pm\widetilde\zeta_{n\pm}^*(x)
\widetilde\gamma_{\pm}(x|n) =0\ , \\[6pt]
&&\widetilde\gamma'_{\pm}(x|n)\pm\widetilde\zeta_{n\pm}(x)
\widetilde\pi_{\pm}(x|n) =0\ ,
\end{array}
\label{tildePiGamma_eq}$$
where the random field $\widetilde\zeta_{n\pm}(x)$ is related to $\zeta_{n\pm}(x)$ by the equality
$$\widetilde\zeta_{n\pm}(x)=\zeta_{n\pm}(x)\exp\bigg[\pm 2i\int_x^{\pm L/2}
\!\!\!\!\eta_n(x_1)\,dx_1\bigg] \ .
\label{tilde-zeta}$$
This latter condition does not modify correlation properties of the backscattering fields, Eqs. (\[Eta\_Zeta\]). Then performing a Fourier transformation of the function $\Phi_{\mu}(x,x'|n)$ over $x'$, we arrive at the expression conveniently decomposed into the sum of ‘plus’ and ‘minus’ functionals,
$$\widetilde\Phi_{\mu}(x,q|n)=\left(-
\frac{1}{2k_n}\right)^\mu{\rm e}^{iqx}
\left[\widetilde\Phi_{\mu}^{(+)}(x,q|n)+
\widetilde\Phi_{\mu}^{(-)}(x,q|n)\right] \ .
\label{Phi_N+-}$$
Here the functions $\widetilde\Phi_{\mu}^{(\pm}$ are given by
$$\widetilde\Phi_{\mu}^{(\pm)}(x,q|n)=\pm \int_x^{\pm L/2}
\!\!\!\!dx_1\left[\frac{\widetilde\pi_{\pm}(x_1|n)}
{\widetilde\pi_{\pm}(x|n)}\right]^\mu
\exp\left[-iq(x-x_1)+i\mu k_n|x-x_1| \pm i\mu\int_{x_1}^{x}
\eta_n(x_2)\,dx_2\right] \ .
\label{Phi^(pm)}$$
Averaging functions (\[Phi\^(pm)\]) over the random field $\eta_n(x)$ with the use of (\[EtaEta\]) readily yields
$$\widetilde\Phi_{\mu}^{(\pm)}(x,q|n)=\pm \int_x^{\pm L/2}
\!\!\!\!dx_1\left[\frac{\widetilde\pi_{\pm}(x_1|n)}
{\widetilde\pi_{\pm}(x|n)}\right]^\mu
\exp\left\{-iq(x-x_1)+\left[i\mu k_n -\frac{\mu^2}{L_f^{(V)}(n)}
\right]|x-x_1|\right\} \ .
\label{Phi^(pm)av-eta}$$
To then perform averaging over the fields $\zeta_{n\pm}(x)$ it is convenient to use the dynamic equations for the functions $\widetilde\Phi_{\mu}^{(\pm)}(x,q|n)$ and $\widetilde\Gamma_{\pm}(x|n)=
{\widetilde\gamma_{\pm}(x|n)}/{\widetilde\pi_{\pm}(x|n)}$. They read
$$\mp\frac{d\widetilde\Phi_{\mu}^{(\pm)}(x,q|n)}{dx}=
1-\left[\frac{\mu^2}{2L_f^{(V)}(n)}-i\mu k_n\mp iq\right]
\widetilde\Phi_{\mu}^{(\pm)}(x,q|n)-
\mu\widetilde\zeta_{n\pm}^*(x)\widetilde\Gamma_{\pm}(x|n)
\widetilde\Phi_{\mu}^{(\pm)}(x,q|n) \ ,
\label{Phi^pm_eq}$$
$$\pm \frac{d\widetilde\Gamma_{\pm}(x|n)}{dx}=
-\widetilde\zeta_{n\pm}(x)+
\widetilde\zeta_{n\pm}^*(x)\widetilde\Gamma_{\pm}^2(x|n) \ .
\label{Gamma_pm-eq}$$
These equations stem from definitions (\[Phi\^(pm)\]) and system (\[tildePiGamma\_eq\]) along with the obvious ‘initial’ conditions
$$\widetilde\Phi_{\mu}^{(\pm)}(\pm L/2,q|n)=0 \ ,\hspace{1.5cm}
\widetilde\Gamma_{\pm}(\pm L/2|n)=0 \ .
\label{In-cond}$$
Averaging of (\[Phi\^pm\_eq\]) with the use of Furutsu-Novikov formula[@Klyatskin86] gives the equation
$$\frac{d\big<\widetilde\Phi_{\mu}^{(\pm)}(x,q|n)\big>}{dx}=
1-\left[\frac{\mu}{2}\left(\frac{\mu}{L_f^{(V)}(n)}+
\frac{1}{L_b^{(V)}(n)}\right)-i\mu k_n\mp iq\right]
\big<\widetilde\Phi_{\mu}^{(\pm)}(x,q|n)\big> \ ,
\label{Phi_fineq}$$
from which the result (\[moments\]) arises immediately.
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|
---
author:
- 'Thomas Bitoun[^1]'
title: 'Lagrangianité de cycles associés à un $\mathcal{D}$-module holonôme'
---
Introduction {#introduction .unnumbered}
============
[frenchb]{}
Cette thèse traite d’algèbre et de géométrie algébrique, plus précisément de théorie algébrique des $\mathcal{D}$-modules, avec une forte présence de la caractéristique positive. On sait le rôle joué par la variété caractéristique en théorie des $\mathcal{D}$-modules, nous en étudions ici un analogue lié aux $p$-courbures. Commençons par quelques rappels sur l’anneau des opérateurs différentiels.
#### L’anneau des opérateurs différentiels.
Pour un morphisme lisse de schémas $X\to S,$ on entend, dans ce travail, par anneau des opérateurs différentiels, le faisceau d’anneaux $D_{X/S}$ sur $X$ engendré par l’anneau des fonctions $\mathcal{O}_X$ et les champs de vecteurs relatifs $\mathcal{T}_{X/S},$ imposant que le commutateur d’un champ de vecteurs et d’une fonction soit la dérivée de celle-ci suivant celui-là et que le commutateur de deux champs de vecteurs soit leur crochet de Lie. Notons que cet anneau est noethérien et que si $S$ est le spectre d’un corps de caractéristique nulle, on retrouve l’anneau des opérateurs différentiels algébriques usuels. Nos considérations ont pour point de départ les propriétés remarquables de $D_{X/S}$ lorsque $S$ est le spectre d’un corps de caractéristique positive $p.$ En effet, d’après [@BMR], $D_{X/S}$ est alors fini sur son centre $Z(D_{X/S})$ et le spectre de ce dernier s’identifie au fibré cotangent $T^*(X/S)'$ de $X/S$ (tordu par le Frobenius, noté ’ ici et plus bas). De plus cette identification se fait via la $p$-courbure $$\psi_{\nabla}:\partial\mapsto(\nabla(\partial))^p-\nabla(\partial^{[p]}),$$ notion typique à la caractéristique positive, rendant compte du défaut de compatibilité d’une connexion intégrable $\nabla$ à la puissance $p$-ième $\partial^{[p]}$ des champs de vecteurs $\partial.$ Enfin, il est également observé dans [@BMR], et c’est une condition technique essentielle pour la suite, que $D_{X/S}$ est une algèbre d’Azumaya sur le cotangent $T^*(X/S)'.$
Explicitons le cas de l’espace affine $\mathbb{A}^n_S,$ $S$ affine d’anneau $R.$ L’anneau des opérateurs différentiels est alors la faisceau associé à l’algèbre de Weyl d’indice $n,$ $A_n(R),$ c’est-à-dire le produit tensoriel sur $R$ $$\otimes_{i=1}^{i=n}(R<x_i,\partial_i>/(\partial_ix_i-x_i\partial_i=1)).$$ On constate bien que si $R$ est un corps de caractéristique positive $p,$ $A_n(R)$ est finie sur son centre, qui s’identifie à l’algèbre de polynômes $\otimes_{i=1}^{i=n}R[x_i^p,\partial_i^p].$ L’opérateur de $p$-courbure est ici celui qui au champ de vecteur $\partial_i$ correspondant à la $i$-ième coordonnée, associe l’élément central $\partial_i^p-\partial_i^{[p]}=\partial_i^p$ de $A_n(R).$ (La puissance $p$-ième $\partial_i^{[p]}$ du champ de vecteur particulier $\partial_i$ s’annule.)
#### Les $p$-supports.
Supposons que $S$ soit un schéma affine d’anneau $R,$ une algèbre de type fini sur $\mathbb{Z},$ intègre et de corps de fractions de caractéristique nulle. Donc telle que pour chaque idéal maximal $\mathfrak{m}$ de $R,$ le corps $k(\mathfrak{m}):=R/\mathfrak{m}$ soit de caractéristique positive. Pour un $S$-schéma lisse $X,$ donnons-nous un module $M$ de type fini sur $D_{X/S},$ noté aussi $D_{X/R}.$ Sa réduction $k(\mathfrak{m})\otimes_RM,$ pour un idéal maximal $\mathfrak{m}$ de $R,$ est alors un module de type fini sur $D_{X_{\mathfrak{m}}/k(\mathfrak{m})}$ et en particulier, par les rappels ci-dessus, un module de type fini sur le centre $Z(D_{X_{\mathfrak{m}}/k(\mathfrak{m})})$ de $D_{X_{\mathfrak{m}}/k(\mathfrak{m})},$ ce dernier s’identifiant à l’anneau des fonctions sur le fibré cotangent $T^*(X_{\mathfrak{m}}/k(\mathfrak{m}))'.$ On définit le $p$-support de $M$ en $\mathfrak{m}$ comme la sous-variété de $T^*(X_{\mathfrak{m}}/k(\mathfrak{m}))',$ support de $k(\mathfrak{m})\otimes_RM$ vu comme module sur $Z(D_{X_{\mathfrak{m}}/k(\mathfrak{m})}).$ L’objet central de cette thèse, ainsi que l’analogue promis de la variété caractéristique, est la collection des $p$-supports de $M,$ pour l’ensemble des idéaux maximaux de $R.$ On se permet en fait d’identifier les collections provenant de $M$ et de ses localisés $M[1/r]$ par des éléments non-nuls $r$ de $R,$ ce qu’on exprime parfois en disant “la collection des $p$-supports pour $p$ suffisamment grand”.
Remarquons enfin qu’on construit des modules comme ci-dessus en épaississant (ou déformant) les $\mathcal{D}$-modules de type fini, où $\mathcal{D}$ est l’anneau des opérateurs différentiels sur une variété lisse sur $k,$ un corps de caractéristique nulle. En effet, l’anneau $\mathcal{D}$ étant noethérien, tout module de type fini $M_0$ est de présentation finie et peut donc s’épaissir en un module $M$ de type fini sur $D_{X/R}$ tel que $M_0\cong k\otimes_RM,$ pour $X$ lisse sur $R$ comme ci-dessus, $R$ étant par exemple le sous-anneau de $k$ engendré par les coefficients des relations d’une présentation finie de $M_0.$
#### Résultats.
Rappelons que le fibré cotangent d’une variété équidimensionnelle lisse $X$ sur un corps est canoniquement muni d’une forme symplectique et que ses sous-variétés lagrangiennes sont celles qui sont de même dimension que $X$ en chacun de leurs points et sur un ouvert dense desquelles la forme symplectique s’annulle. Rappelons aussi que l’holonômie est une condition de finitude fondamentale sur les $\mathcal{D}$-modules, satisfaite par exemple par tous les modules provenant des fibrés à connexion intégrable. Notre résultat principal est le
#### Théorème 1
(cf. le Théorème \[thm:main thm\])
*Soit $X$ un schéma lisse purement de dimension relative $n$ sur $S$ affine d’anneau $R$ comme ci-dessus et soit $M$ un module à gauche de type fini sur $D_{X/R}.$ Supposons que $k\otimes_RM$ soit un $\mathcal{D}$-module holonôme, pour $k$ le corps des fractions de $R$ ($k$ est donc de caractéristique nulle). Alors les $p$-supports de $M$ sont des sous-variétés lagrangiennes, “pour $p$ suffisamment grand” (i.e. il existe un élément non-nul $r$ de $R$ tel que les $p$-supports de $M[1/r]$ soient des sous-variétés lagrangiennes).*\
Pour l’obtenir, on démontre également l’assertion qui suit. Rappelons cette fois-ci qu’une algèbre d’Azumaya sur une variété est scindée si elle est isomorphe à l’algèbre des endomorphismes d’un fibré vectoriel.\
#### Théorème 2
(cf. le Théorème \[thm:splitting\])
*Dans la situation du théorème 1 et sous les mêmes hypothèses, l’algèbre d’Azumaya des opérateurs différentiels se scinde sur le lieu régulier du $p$-support, “pour $p$ suffisamment grand” (i.e. il existe un élément non-nul $r$ de $R$ tel que pour tout idéal maximal $\mathfrak{m}$ de $R[1/r],$ l’algèbre d’Azumaya $D_{X_{\mathfrak{m}}/k(\mathfrak{m})}$ se scinde sur le lieu régulier du $p$-support de $k(\mathfrak{m})\otimes_RM$).*\
L’énoncé du théorème 1 s’inspire du théorème classique d’intégrabilité de la variété caractéristique [@Gabber]. Il ne semble toutefois pas se déduire aisément de son modèle. Qui plus est, les objets sur lesquels ces deux assertions portent ont des différences notables, comme il se voit sur les exemples. Mettons en évidence que les $p$-supports peuvent dépendre non-trivialement de $p$ et qu’ils ne sont en général pas des sous-variétés coniques du cotangent, c’est-à-dire que contrairement à la variété caractéristique, ils ne sont pas nécessairement préservés par l’action du groupe multiplicatif dans les fibres.
#### Exemples.
- Soient $R=\mathbb{Z},$ $X=\mathbb{A}^n_{\mathbb{Z}}$ et soit $M$ le $D_{X/R}$-module à gauche de type fini correspondant à la $R$-connexion intégrable $$\nabla=d+dg$$ sur $\mathcal{O}_X,$ où $g$ est une section globale de $\mathcal{O}_X.$ De l’identité $(\partial_i+\partial g/\partial x_i)^p=(\partial_i)^p+(\partial g/\partial x_i)^p$ dans $A_n(\mathbb{Z}/p\mathbb{Z})$ [@Katz 5.2.4], on déduit que le $p$-support de $\mathbb{Z}/p\mathbb{Z}\otimes_{\mathbb{Z}}M$ $\subset T^*\mathbb{A}^{n'}_{\mathbb{Z}/p\mathbb{Z}}=T^*\mathbb{A}^n_{\mathbb{Z}/p\mathbb{Z}}$ est la réduction modulo $p$ du graphe de $dg$ $\subset T^*\mathbb{A}^n_{\mathbb{Z}}.$ Les $p$-supports ne sont donc pas nécessairement coniques.
- Soit $R=\mathbb{Z}[\lambda]$ le sous-anneau de $\mathbb{C}~$ engendré par $\lambda\in\mathbb{C}~$ et soit $X=specR[x,x^{-1}]=\mathbb{A}^1_R-\{0\}\subset \mathbb{A}^1_R=specR[x].$ Considérons le $D_{X/R}$-module à gauche de type fini $M$ correspondant à la $R$-connexion intégrable $$\nabla=d+\lambda~dx/x$$ sur $\mathcal{O}_X.$ On déduit de l’identité $(x\partial)^p=x^p\partial^p+x\partial$ dans $A_1(\mathbb{Z}/p\mathbb{Z})$ [@Hochschild lemma 1] que pour tout $\mathfrak{m}$ idéal maximal de $R$ tel que $k(\mathfrak{m})$ soit de caractéristique positive $p,$ le $p$-support de $k(\mathfrak{m})\otimes_RM$ $\subset T^*X'_{\mathfrak{m}}\subset T^*\mathbb{A}^{1'}_{k(\mathfrak{m})}=T^*\mathbb{A}^1_{k(\mathfrak{m})}$ est décrit par l’équation $xy=\lambda^p-\lambda$ (mod $p$), où $y$ est la section globale de $\mathcal{O}_{T^*\mathbb{A}^1_{k(\mathfrak{m})}}$ correspondant à $dx.$ Si $\lambda$ n’est pas rationnel, on obtient ainsi des exemples de dépendance non-triviale en $p$ des $p$-supports.
- Signalons enfin que d’après [@Katz], les connexions de Gauss-Manin ont des $p$-courbures nilpotentes. Leurs $p$-supports sont donc réduits à la section nulle du cotangent, tout comme leur variété caractéristique.
Avant de commenter le contenu des différentes sections, attirons l’attention sur l’article [@Kontsevich] (voir aussi [@AutWeyl]), qui est consacré à des conjectures basées sur les $p$-supports. On s’attend notamment à ce que leur considération clarifie la théorie des $\mathcal{D}$-modules holonômes irréguliers.
#### Organisation du texte.
Dans la section \[sec:preliminaries\], nous fixons les notations et rappelons les faits nécessaires sur l’algèbre des opérateurs différentiels et ses modules, la géométrie symplectique du cotangent et le calcul différentiel en caractéristique positive.
Dans la section \[sec:statement\] figurent l’énoncé du résultat principal ainsi qu’un plan de sa démonstration (auquel on renvoie aussi pour une description du contenu de la thèse).
La section \[sec:dimension of the p-supports\] traite de l’équidimension des $p$-supports. On y applique des techniques développées pour la variété caractéristique [@GabberLevasseur] (exposées dans [@Bjork2 A:IV]), cf. aussi [@Kash 2.4].
La section \[sec:reduction\] contient la réduction de la démonstration du théorème principal au cas de l’espace affine. On y utilise l’image directe des $D_{X/S}$-modules.
Dans la section \[sec:bound\], on majore les rangs génériques des restrictions d’un module à ses $p$-supports ainsi que les degrés de ces derniers [@Kontsevich conjecture 1], dans le cas de l’espace affine. Ces majorations sont essentielles aux démonstrations des théorèmes 1 et 2 énoncés ci-dessus.
La première partie de la section \[sec:Brauer group and forms\] est consacrée à la démonstration du théorème 2 sur le scindage de l’algèbre d’Azumaya des opérateurs différentiels. Dans la deuxième partie, on expose le lien entre la classe de cette algèbre dans le groupe de Brauer et la forme canonique sur le cotangent.
La section \[sec:lagrangianity\] contient les derniers résultats nécessaires à la démonstration du théorème principal, ainsi que la démonstration proprement dite. On s’y base sur la majoration de la section \[sec:bound\] pour uniformément compactifier un ouvert dense des $p$-supports et on étudie, suivant une suggestion de M.Kontsevich, l’action de l’opérateur de $p$-courbure sur l’ordre des pôles des formes différentielles.
Signalons aussi que plusieurs parties débutent par des descriptions détaillées.
Remerciements {#remerciements .unnumbered}
-------------
Cette thèse a été effectuée sous la direction de Maxim Kontsevich. Je le remercie chaleureusement pour sa générosité, sa disponibilité et le beau sujet qu’il m’a donné. La sûreté de son intuition est un abri précieux.
J’ai aussi eu la chance de passer plusieurs semestres auprès de Joseph Bernstein. Ils ont été importants pour moi.
Je remercie mes rapporteurs Pierre Berthelot et Roman Bezrukavnikov pour leurs commentaires ainsi que Luc Illusie, Gérard Laumon et Michel Van den Bergh d’avoir accepté de faire partie de mon jury.
Une première version de ce manuscrit a été relue par Simon Riche, Olivier Schiffmann et Sasha Yomdin.
J’ai eu d’utiles discussions avec Ahmed Abbes, Ofer Gabber, Luc Illusie, Laurent Lafforgue, Gérard Laumon, Kay Rülling, Claude Sabbah et Michel Van den Bergh.
Enfin je souhaite saluer le support de Dimitri Ara, Hussein Mourtada, du MEN, de l’IHES et surtout de la Fondation Borgers, à Bruxelles et Paris. Merci!
Preliminaries {#sec:preliminaries}
=============
Conventions {#subsec:general assumptions}
-----------
Schemes are assumed to be noetherian, positive characteristics to be non zero and morphisms of algebras to send $1\mapsto 1$. Local coordinates of a smooth scheme $X/S$ mean local étale relative coordinates in the neighborhood of a point of $X$. As a rule we define notions and state results for left modules, we often omit to mention that they easily adapt to right modules.
$D_{X/S}$ {#subsec:$D_{X/S}$}
---------
Let $S$ be a scheme and let $X$ be a smooth $S$-scheme of relative dimension $n.$ Let the tangent sheaf $T_{X/S}$ be the $\mathcal{O}_X$-module dual to the locally free sheaf of relative differentials $\Omega^1_{X/S}$, it is endowed with a Lie bracket. The ring of PD-differential operators of $X/S$ [@BO §4], noted $D_{X/S}$, is a sheaf of noncommutative rings on $X.$ It is the enveloping algebra of the Lie algebroid $T_{X/S}$ [@BMR 1.2]. Thus $D_{X/S}$ is generated by the structure sheaf $\mathcal{O}_X$ and the tangent sheaf $T_{X/S}$, subject to relations $f.\partial=f\partial,\ \partial.f-f.\partial=\partial(f)$ and $\partial.\partial'-\partial'.\partial=[\partial,\partial']$ for $f$ and $\partial, \partial'$ local sections of $\mathcal{O}_X$ and $T_{X/S}$ respectively. Note that it is compatible with base change. In terms of local étale relative coordinates $\{x_1,...,x_n\}$, for the sections $\{\partial_1,...,\partial_n\}$ of $T_{X/S}$, dual to $\{dx_1,...,dx_n\}$, one has $D_{X/S}=\bigoplus_I \mathcal{O}_X.\partial^{I}$, summing over non negative multi-indices. By definition of an enveloping algebra, endowing an $\mathcal{O}_X$-module with a compatible left $D_{X/S}$-module structure or with an integrable connection are equivalent. Left multiplication by $\mathcal{O}_X$ makes $D_{X/S}$ into a quasi-coherent $\mathcal{O}_X$-module. Moreover,
\[prop:filtration D\]
The sheaf of rings $D_{X/S}$ has a natural positive filtration $D_{X/S}= \bigcup_{m\geq 0} D_{X/S,\leq m}$, defined by $D_{X/S,\leq 0}:=\mathcal{O}_X$ and $D_{X/S,\leq m+1}:=T_{X/S}.D_{X/S,\leq m}+D_{X/S,\leq m}$, whose associated graded sheaf of rings $grD_{X/S}$ is canonically isomorphic to $\mathcal{O}_{T^*(X/S)}$, the structure sheaf of the cotangent bundle of $X/S$.
where the cotangent bundle of $X/S$ is $T^*(X/S)/X:=Spec_X(Sym_{\mathcal{O}_X}T_{X/S})$, also denoted $V(T_{X/S})$ [@EGAII 1.7.8]. Therefore $D_{X/S}$ is a sheaf of coherent noetherian rings [@BerthI 2.2.5 and 3.1.2]. One has the familiar finiteness condition for modules,
\[prop:coherentD\]
A left $D_{X/S}$-module is coherent [@EGAI 0.5.3] if and only if it is quasi-coherent as an $\mathcal{O}_X$-module and its module of sections over any open of an affine covering is a finitely generated left module over the ring of sections of $D_{X/S}$ [@BerthI 3.1.3 (ii)].
Furthermore on X affine the functor of global sections is an equivalence from the category of coherent left $D_{X/S}$-modules to the category of finitely generated left modules over the global sections of $D_{X/S}$ [@BerthI 3.1.3 (iii)]. Note also that coherence is preserved under base change.
Similar results hold for right $D_{X/S}$-modules.
There is also a notion of good filtration on a coherent left $D_{X/S}$-module [@BerthIntro 5.2.3.]. Namely recall that by \[prop:filtration D\], $D_{X/S}$ is a positively filtered sheaf of rings, then
A filtration on a coherent left $D_{X/S}$-module, that is a filtration by coherent sub-$\mathcal{O}_X$-modules compatible with the filtration on $D_{X/S}$ is said to be good if it is bounded below and if the associated graded module over $grD_{X/S} \cong \mathcal{O}_{T^*(X/S)}$ is coherent.
Note that coherent left $D_{X/S}$-modules admit good filtrations [@BerthIntro 5.2.3 (iv)] and that for a good filtration $\Gamma$ on a module $M$, the support of the graded coherent $grD_{X/S} \cong \mathcal{O}_{T^*(X/S)}$-module $gr^{\Gamma}M$ in $T^*(X/S)$ is independent of $\Gamma$, [@Borel V 2.2, lemma]. Hence the following is well-defined,
\[defi:singular support\]
The singular support of a coherent left $D_{X/S}$-module $M$ is the support of the associated graded module to a good filtration, it is a well-defined closed subset $SS(M)$ of $T^*(X/S)$.
The singular support behaves well under exact sequences, indeed, here are some easy consequences of coherence,
\[prop:induced good filtrations\]
Let $$0 \rightarrow M' \xrightarrow{\phi'} M \xrightarrow{\phi''} M''\rightarrow 0$$ be a short exact sequence of coherent left $D_{X/S}$-modules and let $\Gamma$ be a good filtration on M. Then $gr^{\Gamma}M=0$ if and only if $M=0$. Moreover, for the induced filtrations $\Gamma'_.:=\phi'^{-1}(\Gamma_.)$ on $M'$ and $\Gamma''_.:=\phi''(\Gamma_.)$ on $M''$, the natural short sequence $$0 \rightarrow gr^{\Gamma'}M' \xrightarrow{gr\phi'} gr^{\Gamma}M \xrightarrow{gr\phi''} gr^{\Gamma''}M''\rightarrow 0$$ is exact. In particular, $\Gamma'$ and $\Gamma''$ are good and $SS(M)=SS(M') \cup SS(M'')$.
Assume that $S$ is the spectrum of a field of characteristic zero, then $D_{X/S}$ is the usual algebra of differential operators $D_X$, also noted $\mathcal{D}$. Moreover the natural filtration is the usual filtration by the order of differential operators and the notions of coherent module, good filtration and singular support specialize to their $\mathcal{D}$-module counterparts [@Borel VI §1]. Recall that in characteristic zero the dimension of the singular support satisfies the following fundamental inequality, [@Borel VI 1.10(iii)],
Let $M$ be a non zero coherent left $\mathcal{D}$-module. Then $dimSS(M)\geq n = dimX$.
The modules for which this lower bound is reached are called holonomic, indeed,
\[defi:holonomic\]
A coherent left $\mathcal{D}$-module is said to be holonomic either if its singular support is of dimension $n = dimX$ or if it is zero.
There’s the following homological characterization of holonomicity, [@Borel VI 1.12],
\[thm:homological characterization of holonomicity\]
A coherent left $\mathcal{D}$-module M is holonomic if and only if $\mathcal{E}\mathnormal{xt}^i_{\mathcal{D}}(M,\mathcal{D})=0$ for all $i\neq n = dimX$.
$D_{X/S}$ in positive characteristic {#subsec:D char p}
------------------------------------
If $S$ is a scheme of positive characteristic $p$, then so is $X$ and let $X^{(p/S)}$, or simply $X'$, be the base change of $X/S$ by the Frobenius endomorphism of $S$, raising the local sections to their $p$-th power. There is a $S$-morphism $F_{X/S}:X \to X^{(p/S)}$ associated to the Frobenius endomorphism of $X$ and called the relative Frobenius of $X/S$ [@SGA5 §1]. Moreover, as the $p$-th iterate of a derivation is again a derivation, one may associate to a local section $\partial$ of $T_{X/S}$ the local section $\partial^{[p]}$ of $T_{X/S}$ corresponding to its $p$-th iterate. Comparing it with the $p$-th power of the element corresponding to $\partial$ in $D_{X/S}$, one gets a $p$-linear map $c:\partial \mapsto \partial^p-\partial^{[p]}$ from $T_{X/S}$ to $D_{X/S}$, which actually lands in the center $Z(D_{X/S})$ of $D_{X/S}$ [@BMR 1.3.1]. By adjunction one deduces from $c$ an $\mathcal{O}_{X'}$-linear morphism $c':T_{X'/S} \to {F_{X/S}}_*Z(D_{X/S})$.
\[prop:centerD\] ([@BMR 1.3.2])
The $\mathcal{O}_{X'}$-linear morphism $c':T_{X'/S} \to {F_{X/S}}_*Z(D_{X/S})$ extends to an $\mathcal{O}_{X'}$-linear isomorphism $\mathcal{O}_{T^*(X'/S)} \to {F_{X/S}}_*Z(D_{X/S}).$
It turns $\mathcal{D}_{X/S}:={F_{X/S}}_*D_{X/S}$ into a central $\mathcal{O}_{T^*(X'/S)}$-algebra.
Note that in local étale coordinates as above one has ${F_{X/S}}_*Z(D_{X/S})=\bigoplus_I \mathcal{O}_{X'}.\partial^{pI}$ hence $\mathcal{D}_{X/S}$ is a locally free $\mathcal{O}_{T^*(X'/S)}$-module of rank $p^{2n}.$ Moreover,
\[thm:Azumaya\] ([@BMR 2.2.3])
$\mathcal{D}_{X/S}$ is an Azumaya algebra of rank $p^n$ on $T^*(X'/S).$
Recall that an Azumaya algebra is a relative central simple algebra. The notion has many equivalent characterizations [@BrauerI 5.1] one of which is that an Azumaya algebra of rank $r$ on a scheme $Y$ is a sheaf of $\mathcal{O}_Y$-algebras, coherent as an $\mathcal{O}_Y$-module and isomorphic to a rank $r$ matrix algebra $M_r(\mathcal{O}_Y)$ on a flat covering.
In the case of $\mathcal{D}_{X/S},$ let $\mathcal{A}_{X/S}$ be the centralizer of $\mathcal{O}_X$ in $\mathcal{D}_{X/S}$ and let’s denote $(\mathcal{D}_{X/S})_{\mathcal{A}_{X/S}}$ the rank $p^n$ locally free $\mathcal{A}_{X/S}$-module $\mathcal{D}_{X/S},$ $\mathcal{A}_{X/S}$ acting by right multiplication. Then $\mathcal{A}_{X/S}$ is a faithfully flat ${F_{X/S}}_*Z(D_{X/S})$-algebra and the morphism $\mathcal{D}_{X/S} \otimes_{{F_{X/S}}_*Z(D_{X/S})} \mathcal{A}_{X/S} \to \mathcal{E}nd_{\mathcal{A}_{X/S}}((\mathcal{D}_{X/S})_{\mathcal{A}_{X/S}})$ given by left multiplication by $\mathcal{D}_{X/S}$ and right multiplication by $\mathcal{A}_{X/S}$ is an isomorphism [@BMR 2.2.2] thus realizing $\mathcal{D}_{X/S}$ as an Azumaya algebra of rank $p^n$ on $T^*(X'/S).$
Symplectic geometry of the cotangent bundle {#subsec:symplectic geometry}
-------------------------------------------
Let $S$ be a scheme and let $Y$ be a smooth $S$-scheme of relative dimension $n$. Recall that the cotangent bundle of $Y/S$ is the $Y$-scheme $T^*(Y/S) \xrightarrow{p_Y} Y:= V((\Omega^1_{Y/S})^*)=Spec_Y(Sym_{\mathcal{O}_Y}(\Omega^1_{Y/S})^*)$ and hence that the sheaf of germs of $Y$-sections of $T^*(Y/S)/Y$ is canonically identified with $\Omega^1_{Y/S}$ [@EGAII 1.7.9]. Moreover $T^*(Y/S)$ is a smooth $Y$-scheme of relative dimension $n$ [@EGAIV 17.3.8], smooth of relative dimension $2n$ as an $S$-scheme. For $f:X\to Y$ a $S$-morphism of smooth $S$-schemes, the pullback of differentials $\Omega^1_{X/S}\xleftarrow{f^*} f^*\Omega^1_{Y/S}$ [@EGAIV 16.4.3.6] gives rise to the $X$-morphism $T^*(X/S) \xleftarrow{f_d} X\times_Y T^*(Y/S)$ called the cotangent map. It is part of the cotangent diagram of $f$,
$$\xymatrix{
T^*(X/S) & X\times_Y T^*(Y/S) \ar[l]_-{f_d} \ar[d]^{f_\pi}\\
& T^*(Y/S) }$$ where $f_\pi$ is the canonical projection. Let $U\subset Y$ be an open subset, one sees right-away on the definitions that if $s_{\alpha}$ is the section of $T^*(Y/S)/U$ corresponding to $\alpha \in \Gamma(U,\Omega^1_{Y/S})$ then $f_d \circ X\times_Ys_{\alpha}$ corresponds to $(f^*)^{ad}\alpha\in \Gamma(f^{-1}U,\Omega^1_{X/S}).$
Note the
\[lmm:cotangent of an immersion\] If $f:X\to Y$ is an immersion (resp. a closed immersion) then $f_d$ is smooth and surjective and $f_{\pi}$ is an immersion (resp. a closed immersion). Moreover $f_d$ admits a section locally on $X.$
**Proof:** The morphism $f_d$ is smooth and surjective by [@EGAIV 17.2.5], [@EGAII 1.7.11(iii)], [@EGAIV 17.3.8] and stability under base change of surjective smooth morphisms, $f_{\pi}$ is an immersion (resp. a closed immersion) by [@EGAI 4.3.1(i)] and local sections of the locally split “conormal” short exact sequence of [@EGAIV 17.2.5] induce [@EGAII 1.7.11(i)] local sections of $f_d.$\
The cotangent bundle of $Y/S$ carries a canonical global $S$-relative $1$-form $\theta_{Y/S}$ corresponding to the section $T^*(Y/S) \xrightarrow{\Delta_{T^*(Y/S)/Y}} T^*(Y/S)\times_Y T^*(Y/S) \xrightarrow{(p_Y)_d} T^*(T^*(Y/S)/S)$ of the cotangent bundle $T^*(T^*(Y/S)/S) \xrightarrow{p_{T^*(Y/S)}} T^*(Y/S)$, where $\Delta_{T^*(Y/S)/Y}$ is the diagonal of $T^*(Y/S) \xrightarrow{p_Y} Y$. Let $\{y_1,...,y_n\}$ be local étale coordinates on $Y$, then in terms of the associated local étale coordinates $\{y_1,...,y_n;\xi_1,...,\xi_n\}$ on $T^*(Y/S)$, where $\{\xi_1,...,\xi_n\}$ are dual to $\{dy_1,...,dy_n\}$, $\theta_{Y/S}=\sum_{i=1}^{i=n}\xi_idy_i$. Note that the canonical form is compatible with base change and with the cotangent diagram, the latter in the sense that ${f_\pi}^*\theta_{Y/S}={f_d}^*\theta_{X/S}$.
Suppose that $S$ is the spectrum of a field $k$ and let’s drop the reference to the base $S$ from the notations. So $Y$ is a smooth $k$-scheme of pure dimension $n$. The nondegenerate global exact $2$-form $\omega_Y:=d\theta_Y$ on $T^*Y$ is called the symplectic form.
A subscheme $X \overset{i}\hookrightarrow T^*Y$ is said to be a lagrangian subscheme of $(T^*Y,\omega_Y)$ if it contains a dense open $U\subset X$ on which the symplectic form vanishes, $(i^*\omega_Y)|U=0$ and if at each of its points $x$ it is of dimension $n=dim_xY$.
We’ll use the
\[lmm:isotropy\] Let $f:X\to Y$ be an immersion of smooth $k$-schemes and let $Z_Y \overset{i}\hookrightarrow T^*Y$ and $Z_X \overset{j}\hookrightarrow T^*X$ be reduced subschemes. Suppose that $f_{\pi}^{-1}Z_Y=f_d^{-1}Z_X$ and that $f_{\pi}^{-1}Z_Y \xrightarrow{f_{\pi}}Z_Y$ is surjective. Then $\omega_Y$ vanishes on a dense open subset of $Z_Y$ if and only if $\omega_X$ vanishes on a dense open subset of $Z_X$.
**Proof:** Note that by lemma \[lmm:cotangent of an immersion\], $f_d|Z_X$ is smooth and surjective and $f_{\pi}|Z_Y$ is an immersion. Since by hypothesis $f_{\pi}|Z_Y$ is surjective, it is a nilimmersion [@EGAI 4.5.16] hence, $Z_Y$ being reduced, an isomorphism. Moreover by [@EGAIV 17.2.3(ii)], [@BourbakiAIII §7 n$^{\rm o}$2 prop.4] and flatness of smooth morphisms, the pullback of forms $f_d^*:\Omega^2_{Z_X,z} \to \Omega^2_{f_d^{-1}Z_X,\tilde{z}}$ is injective for all $z=f_d(\tilde{z})$. Since $f_{\pi}^*\omega_Y=f_d^*\omega_X$ as $f_{\pi}^*\theta_Y=f_d^*\theta_X$ and $(f_d|Z_X)\circ(f_{\pi}|Z_Y)^{-1}$ and $(f_{\pi}|Z_Y)\circ(f_d|Z_X)^{-1}$ preserve open dense subsets, the lemma follows.\
Differential calculus in positive characteristic {#subsec:calculus in char p}
------------------------------------------------
Let $Y$ be a smooth equidimensional scheme over a field $k$ of positive characteristic $p,$ let $Y\xrightarrow{F_{/k}} Y'$ be the relative Frobenius of $Y/k$ and let $W:Y' \to Y$ be the canonical projection. The differential $d=d_{Y/k}:\mathcal{O}_Y \to \Omega^1_{Y/k}$ is $F_{/k}^{-1}\mathcal{O}_{Y'}$-linear hence the differential of the complex ${F_{/k}}_*\Omega^{\bullet}_{Y/k}$ is $\mathcal{O}_{Y'}$-linear, where $\Omega^{\bullet}_{Y/k}=(\Omega^{\bullet}_{Y/k}, d_{Y/k}= d)$ is the de Rham complex of $Y/k$. Let $Z^i({F_{/k}}_*\Omega^{\bullet}_{Y/k}):= ker ({F_{/k}}_*d: {F_{/k}}_*\Omega^{i}_{Y/k} \to {F_{/k}}_*\Omega^{i+1}_{Y/k}),$ $B^i({F_{/k}}_*\Omega^{\bullet}_{Y/k}):= im ({F_{/k}}_*d: {F_{/k}}_*\Omega^{i-1}_{Y/k} \to {F_{/k}}_*\Omega^{i}_{Y/k})$ and $\mathcal{H}^{i}({F_{/k}}_*\Omega^{\bullet}_{Y/k}):=
Z^i({F_{/k}}_*\Omega^{\bullet}_{Y/k})/B^i({F_{/k}}_*\Omega^{\bullet}_{Y/k}),$ the exterior product of differential forms endowes $\bigoplus_{i\in \mathbb{Z}} \mathcal{H}^{i}({F_{/k}}_*\Omega^{\bullet}_{Y/k})$ and $\bigoplus_{i\in \mathbb{Z}} \Omega^{i}_{Y'/k}$ with structures of graded $\mathcal{O}_{Y'}$-algebras. They are canonically isomorphic [@Katz 7.2],
\[thm:Cartier isomorphism\]
There is a unique morphism of graded $\mathcal{O}_{Y'}$-algebras $$C_Y^{-1}: \bigoplus_{i\in \mathbb{Z}} \Omega^{i}_{Y'/k} \to \bigoplus_{i\in \mathbb{Z}} \mathcal{H}^{i}({F_{/k}}_*\Omega^{\bullet}_{Y/k})$$ such that for all local sections $y$ of $\mathcal{O}_Y, C_Y^{-1}(d(W^*y))=\text{ the class of }y^{p-1}dy \in \mathcal{H}^{1}({F_{/k}}_*\Omega^{\bullet}_{Y/k}).$ It is an isomorphism, compatible with étale localization on $Y.$
The composed morphism $$\bigoplus_{i\in \mathbb{Z}} Z^i({F_{/k}}_*\Omega^{\bullet}_{Y/k}) \to \bigoplus_{i\in \mathbb{Z}} \mathcal{H}^{i}({F_{/k}}_*\Omega^{\bullet}_{Y/k}) \xrightarrow{\text{inverse of }C_Y^{-1}} \bigoplus_{i\in \mathbb{Z}} \Omega^{i}_{Y'/k}$$ where $\bigoplus_{i\in \mathbb{Z}} Z^i({F_{/k}}_*\Omega^{\bullet}_{Y/k}) \to \bigoplus_{i\in \mathbb{Z}} \mathcal{H}^{i}({F_{/k}}_*\Omega^{\bullet}_{Y/k})$ is the quotient, is denoted $C_Y$ and called the Cartier operator.
Recall ([@Illusie; @dR-W 2.1.18]) that there is an exact sequence of abelian sheaves on $Y'$ $$0\to \mathcal{O}_{Y'}^* \xrightarrow{{F_{/k}}^*} {F_{/k}}_*\mathcal{O}_Y^* \xrightarrow{{F_{/k}}_*dlog} {F_{/k}}_*Z^1(\Omega^{\bullet}_{Y/k})\cong Z^1({F_{/k}}_*\Omega^{\bullet}_{Y/k}) \xrightarrow{W^{\star}-C_Y} \Omega^{1}_{Y'/k}$$ where $dlog(y):=dy/y$ for local sections $y$ of $\mathcal{O}_{Y}^*$ and $W^{\star}$ is induced by the morphism ${F_{/k}}_*\Omega^{1}_{Y/k} \xrightarrow{{F_{/k}}_*((W^*)^{ad})} {F_{/k}}_*W_*\Omega^{1}_{Y'/k}$ coming from the pullback of forms $W^*,$ note that since $F_{/k}\circ W$ is the Frobenius endomorphism of $Y'$ its underlying map is the identity and hence ${F_{/k}}_*W_*\Omega^{1}_{Y'/k} = (F_{/k}\circ W)_*\Omega^{1}_{Y'/k}$ and $\Omega^{1}_{Y'/k}$ are identified as abelian sheaves on $Y'$. Moreover since $W^{\star}-C_Y$ is étale locally surjective [@Illusie; @dR-W 2.1.18], the above sequence induces an exact sequence of étale sheaves on $Y'$ $$0\to \mathbb{G}_{\,m/Y'} \xrightarrow{{F_{/k}}^*} {F_{/k}}_*\mathbb{G}_{\,m/Y} \xrightarrow{{F_{/k}}_*dlog} {F_{/k}}_*Z^1(\Omega^{\bullet}_{Y/k}) \xrightarrow{W^{\star}-C_Y} \Omega^{1}_{Y'/k} \to 0$$ where for $U' \xrightarrow{f'} Y'$ étale and $U \xrightarrow{f} Y$ its base change by $F_{/k},$ ${F_{/k}}_*Z^1(\Omega^{\bullet}_{Y/k})(U' \xrightarrow{f'} Y'):= f'^*{F_{/k}}_*Z^1(\Omega^{\bullet}_{Y/k}) \cong {F_{/k}}_*Z^1(\Omega^{\bullet}_{U/k})$ and $\Omega^{1}_{Y'/k}(U' \xrightarrow{f'} Y'):=f'^*\Omega^{1}_{Y'/k} \cong \Omega^{1}_{U'/k}$ are the étale sheaves associated to the coherent $\mathcal{O}_{Y'}$-modules ${F_{/k}}_*Z^1(\Omega^{\bullet}_{Y/k})$ and $\Omega^{1}_{Y'/k}$[@Milne II 1.6]. Let’s call it the $p$-curvature exact sequence and $W^{\star}-C_Y$ the $p$-curvature operator.
Statement of the result and first reductions {#sec:statement}
============================================
The $p$-support {#subsec:p-support}
---------------
Let $X$ be a smooth scheme over a field $k$ of positive characteristic $p$. Recall from \[subsec:D char p\], often dropping the base from the notations, that there is an $\mathcal{O}_{X'}$-linear isomorphism $\mathcal{O}_{T^*(X')} \to {F_{X/k}}_*Z(D_X)$ identifying $\mathcal{O}_{T^*(X')}$ with the center ${F_{X/k}}_*Z(D_X)$ of $\mathcal{D}_X:={F_{X/k}}_*D_{X}$ (\[prop:centerD\]). Let $M$ be a coherent left $D_X$-module (\[prop:coherentD\]), then $\mathcal{M}:={F_{X/k}}_*M$ is coherent as an $\mathcal{O}_{T^*(X')}$-module.
The $p$-support[^2]of $M$ is the support of the coherent $\mathcal{O}_{T^*(X')}$-module $\mathcal{M}:={F_{X/k}}_*M$. It is a closed subset $p$-supp($M$) of $T^*(X')$, endowed with its reduced subscheme structure.
Note using [@SGA5 prop.1.a), prop.2.c)2)] and [@EGAI 9.3.2(i)] that the $p$-support is compatible with étale localization on $X.$
Further recall that $T^*(X')$ carries a nondegenerate exact 2-form $\omega_{X'}$ which is called the symplectic form.
([@AutWeyl 5.2])
The symplectic form $\omega_{X'}$ or rather the corresponding Poisson bracket has a natural deformation theoretic interpretation related to the lifting of $X$ modulo $p^2.$
The statement {#subsec:The statement}
-------------
Let $S$ be a scheme of finite type over $\mathbb{Z}$. Recall that the closed points of $S$ have finite residue fields and that there are lots of them, indeed every nonempty locally closed subset of $S$ contains a closed point [@EGAIV 10.4.6 and 10.4.7].
\[thm:main thm\]
Let $S$ be an integral scheme dominant and of finite type over $\mathbb{Z}$, let $X$ be a smooth $S$-scheme of relative dimension $n$ and let $M$ be a coherent left $D_{X/S}$-module. Suppose that the fiber of $M$ at the generic point of $S$ is a holonomic left $\mathcal{D}$-module (\[defi:holonomic\]). Then there is a dense open subset $U$ of $S$ such that the $p$-support of the fiber of $M$ at each closed point $u$ of $U$ is a lagrangian subscheme of $(T^*(X'_u),\omega_{X'_u})$.
About the proof
---------------
A complete proof is given in \[subsec:conclusion\] and rests on most of the results of the paper. Let us roughly outline the argument, which has two parts, in accordance with the definition of a lagrangian subvariety.
A first part bears on the dimension and equidimension of the $p$-support, notions which are of cohomological nature (vanishing of some double $\mathcal{E}xt$’s), and occupies section \[sec:dimension of the p-supports\].
A second part handles the vanishing of the symplectic form on the regular locus of the $p$-support.
Its starting point is twofold, namely, there is a natural map from 1-forms to the Brauer group which sends the canonical form to the class of the Azumaya algebra of differential operators (subsection \[subsec:Brauer group via the p-curvature sequence\]), and the latter splits on the regular locus of the $p$-support (theorem \[thm:splitting\]).
Thus the restriction of the canonical form to the regular locus of the $p$-support is in the kernel of the above map, which may be described in terms of the $p$-curvature operator (proposition \[prop:kernel of phi\]). Further considering the action of the $p$-curvature operator on the order of poles along the boundary of a compactification (proposition \[prop:log poles\]), one shows that the restriction of the symplectic form, that is the exterior derivative of the canonical form, has logarithmic poles. The result then follows from the vanishing of globally exact forms with logarithmic poles, in characteristic zero.
Let us mention that crucial to the argument is an estimate of some degrees and ranks of modules, in the case of the affine space (theorem \[thm:bound\]).
Note finally that in the above sketch, we should have written “for $p$ large enough” several times.
First reductions
----------------
Here we carry out some standard reductions (\[rmk:reduction to affine\]) and deal with an easy case of theorem \[thm:main thm\] (\[rmk:generic fiber zero\]). It is organized into two remarks.
\[rmk:reduction to affine\] The conclusion of \[thm:main thm\] depends on $S$ only up to restricting to a dense open subset and so do its hypotheses. Moreover \[thm:main thm\] is compatible with Zariski (even étale) localization on $X.$ Indeed lagrangianity and the $p$-support are compatible with localization while the hypotheses are stable by restriction to open coverings. Hence in the proof of \[thm:main thm\] one may further assume that $S$ is affine, regular (by lemma \[lmm:reg locus\] below) and that $X$ is regular [@EGAIV 17.5.8 (iii)], affine and integral [@EGAI 4.5.7].
\[lmm:reg locus\]
Let S be an integral scheme of finite type over $\mathbb{Z}$. Then the regular locus of S is non empty and open in S.
**Proof:** The regular locus is open by [@EGAIV 6.12.6]. It is non empty since the generic stalk is a field and hence is regular.
\[rmk:generic fiber zero\]
If the fiber of $M$ at the generic point of $S$ is zero, theorem \[thm:main thm\] is easy as there is a dense open subset $U$ of $S$ such that $M|_{U}=0.$ More generally if $S$ is integral and $M$ is a coherent left (resp. right) $D_{X/S}$-module such that the fiber of $M$ at the generic point of $S$ is zero, then there is a dense open subset $U$ of $S$ such that $M|_{U}=0.$
Indeed one may assume that $X$ and $S$ are affine and thus consider a left (resp. right) module over the ring of global sections of $D_{X/S}.$ By the hypotheses, this module has a finite generating family $\{m_1,...,m_l\}$ and each $m_i$ is annihilated by a non zero global section $r_i$ of $\mathcal{O}_S.$ Since $\mathcal{O}_S$ acts through the center of $D_{X/S},$ the open subset of $S$ determined by the product of these global sections fulfills the statement.
Thus we may assume that the fiber of $M$ at the generic point of $S$ is non zero.
Dimension of the $p$-supports {#sec:dimension of the p-supports}
=============================
Theorem \[thm:main thm\] boils down to assertions about the dimensions of some schemes and the vanishing of a certain 2-form. Let us start by the dimensions,
Statement
---------
\[thm:purity\]
Let $S$ be an integral scheme dominant and of finite type over $\mathbb{Z}$, let $X$ be a smooth $S$-scheme of relative dimension $n$ and let $M$ be a coherent left $D_{X/S}$-module. Suppose that the fiber of $M$ at the generic point of $S$ is a non zero holonomic left $\mathcal{D}$-module (\[defi:holonomic\]). Then there is a dense open subset $U$ of $S$ such that the $p$-support of the fiber of $M$ at each closed point $u$ of $U$ is equidimensional of dimension $n = dimX.$
The proof is deferred till \[subsec:equidim of the p-supports\]. It is based on the notion of pure coherent sheaf (\[subsec:pure sheaves\]) and its characterization in terms of duality (\[thm:ext and coherent purity\]). This is relevant as purity implies equidimensionality (\[prop:pure equidimensional\]) and holonomic $\mathcal{D}$-modules satisfy strong duality properties (\[thm:homological characterization of holonomicity\]). One concludes using the Azumayaness of the algebra of differential operators (\[prop:pure Azumaya\]).
In view of \[rmk:reduction to affine\], we may and shall assume that $S$ and $X$ are regular, integral and affine.
Pure coherent sheaves {#subsec:pure sheaves}
---------------------
Recall that the (co)dimension of a coherent sheaf is the (co)dimension of its support and let us call a coherent sheaf equidimensional if its support is equidimensional. There is a strengthening of equidimensionality which has a very convenient interpretation in terms of duality theory. Indeed, let us fix an affine scheme $Y,$
A coherent sheaf on $Y$ is pure if all its non zero coherent subsheaves are of the same dimension.
It is easily seen to imply equidimensionality,
\[prop:pure equidimensional\]
A coherent sheaf on $Y$ is pure if and only if all its associated points [@EGAIV 3.1.1, 3.1.2] are of the same dimension. In particular a pure coherent sheaf on $Y$ is equidimensional.
Here’s the interpretation in terms of duality theory,
\[thm:ext and coherent purity\]
Suppose that $Y$ is regular and equidimensional. A coherent sheaf $\mathcal{F}$ on $Y$ is pure if and only if there’s a non negative integer $c$ such that $$\mathcal{E}\mathnormal{xt}^l(\mathcal{E}\mathnormal{xt}^l(\mathcal{F},\mathcal{O}_Y),\mathcal{O}_Y)=0$$ for all $l \neq c$. If $\mathcal{F}$ is not zero then $c$ is its codimension.
**Proof:** The proof is in the literature. Indeed, [@Bjork2 A:IV 2.6] applies by [@Bjork2 A:IV 3.4] and gives the result, using [@Borel V 2.2.3] to see that purities here and there coincide and to get the above index.\
Equidimensionality of the $p$-supports {#subsec:equidim of the p-supports}
--------------------------------------
Recall the notations and hypotheses of \[thm:purity\] and use \[rmk:reduction to affine\]. In particular $X/S$ is smooth of relative dimension $n$ with $S$ and $X$ regular, affine and integral.
Here are two lemmas and two propositions, preliminary to the proof of theorem \[thm:purity\], which follows.
\[lmm:graded flatness\]
Let $M$ be a left module over a ring $R$ and let $\{M_i\}_{i\in \mathbb{Z}}$ be an exhaustive increasing filtration of M by left sub-R-modules. Suppose that there is $i_0$ such that $M_{i_0}=0$ and for all $i > i_0$ the left R-modules $M_i/M_{i-1}$ are flat, then $M$ is flat. Suppose further that for all $i$ the $M_i/M_{i-1}$ are free, then $M$ is free.
**Proof:** By hypothesis $M_{i_0+1} \cong M_{i_0+1}/(M_{i_0}=0)$ is flat. Since the $M_i/M_{i-1}$ are flat for all $i \geq i_0+1$ and extensions of flat modules are flat [@BourbakiACI §2 n$^{\rm o}$5 prop.5], $M_i$ is flat for all $i \geq i_0+1$. So $M$ is a union of flat submodules, hence is flat by [@BourbakiACI §2 n$^{\rm o}$3 prop.2(ii)]. If the $M_i/M_{i-1}$ are free, then the union for all $i \geq i_0+1$ of an arbitrary lift of a basis of $M_i/M_{i-1}$ is a basis of $M,$ thus $M$ is free.\
\[lmm:ext fibers\]
Let $M$ be a coherent left $D_{X/S}$-module. Then there is a dense open subset $U$ of $S$ such that for all $l$ and for all $s \in U$, the canonical map $$(\mathcal{E}\mathnormal{xt}^l_{D_{X/S}}(M,D_{X/S}))_s \to \mathcal{E}\mathnormal{xt}^l_{D_{X_s}}(M_s,D_{X_s})$$ is an isomorphism, where the subscript $s$ denotes restriction to the fiber.
**Proof:** Since coherent left $D_{X/S}$-modules form an abelian category, the proof of [@EGAIV 9.4.3] goes through here. Indeed provided the above abelianity, the proof of [@EGAIV 9.4.2] carries reducing to associated graded to good filtrations and using [@Kash A.17] and lemma \[lmm:graded flatness\] to conclude. There are only finitely many $l$’s to consider since by [@Bjork2 A:IV 4.5], both target and domain of the above morphism are zero for $l > dimT^*(X/S) \geq dimT^*X_s$, $T^*(X/S)$ and $T^*X_s$ being the respective spectra of the regular rings $grD_{X/S}$ and $grD_{X_s}$.\
\[prop:holonomic fibers\]
Let $M$ be a coherent left $D_{X/S}$-module. Suppose that the fiber of $M$ at the generic point of $S$ is a holonomic left $\mathcal{D}$-module. Then there is a dense open subset $U$ of $S$ such that for all $l \neq n$ and all $s \in U$, $$\mathcal{E}\mathnormal{xt}^l_{D_{X_s}}(M_s,D_{X_s})=0.$$
**Proof:** By lemma \[lmm:ext fibers\] and theorem \[thm:homological characterization of holonomicity\], the fiber of $\mathcal{E}\mathnormal{xt}^l_{D_{X/S}}(M,D_{X/S})$ at the generic point of $S$ vanishes for all $l \neq n.$ Hence by remark \[rmk:generic fiber zero\] and lemma \[lmm:ext fibers\], for each $l \neq n$ there is a dense open subset $U_l$ of $S$ such that for all $s \in U_l,$ $\mathcal{E}\mathnormal{xt}^l_{D_{X_s}}(M_s,D_{X_s})=0.$ Since by the proof of \[lmm:ext fibers\] there are at most finitely many $l$’s to consider, this proves the proposition.\
\[prop:pure Azumaya\]
Recall \[subsec:p-support\], let $Y$ be a smooth equidimensional scheme over a field $k$ of positive characteristic $p$ and let $M$ be a coherent left $D_{Y/k}$-module. Then $$\mathcal{E}\mathnormal{xt}^l_{D_{Y/k}}(M,D_{Y/k})=0 \text{ if and only if } \mathcal{E}\mathnormal{xt}^l_{\mathcal{O}_{T^*(Y')}}(\mathcal{M},\mathcal{O}_{T^*(Y')})=0,$$ where $\mathcal{M}:={F_{Y/k}}_*M$ and $l$ is an integer.
**Proof:** Since $F_{Y/k}$ is affine, $\mathcal{E}\mathnormal{xt}^l_{D_{Y/k}}(M,D_{Y/k})=0$ if and only if $$0={F_{Y/k}}_*\mathcal{E}\mathnormal{xt}^l_{D_{Y/k}}(M,D_{Y/k}) \cong \mathcal{E}\mathnormal{xt}^l_{{F_{Y/k}}_*D_{Y/k}}({F_{Y/k}}_*M,{F_{Y/k}}_*D_{Y/k}).$$ With the notations of \[subsec:p-support\], $\mathcal{E}\mathnormal{xt}^l_{{F_{Y/k}}_*D_{Y/k}}({F_{Y/k}}_*M,{F_{Y/k}}_*D_{Y/k})$ is $\mathcal{E}\mathnormal{xt}^l_{\mathcal{D}_Y}(\mathcal{M},\mathcal{D}_Y).$ Let us show that $$\mathcal{E}\mathnormal{xt}^l_{\mathcal{D}_Y}(\mathcal{M},\mathcal{D}_Y)=0 \text{ if and only if } \mathcal{E}\mathnormal{xt}^l_{\mathcal{O}_{T^*(Y')}}(\mathcal{M},\mathcal{O}_{T^*(Y')})=0.$$ Both of them are coherent sheaves on $T^*(Y')$ (\[thm:Azumaya\]) hence their respective vanishings may be checked on a flat covering $\mathcal{U} \xrightarrow{\pi} T^*(Y')$ of $T^*(Y')$, which since $\mathcal{D}_Y$ is an Azumaya algebra over $\mathcal{O}_{T^*(Y')}$ (\[thm:Azumaya\]), may be chosen to split $\mathcal{D}_Y$, that is $(\mathcal{D}_Y)_\mathcal{U}:=\pi^*\mathcal{D}_Y \simeq M_r(\mathcal{O}_{\mathcal{U}})$, see \[subsec:D char p\]. Moreover tensoring, $\mathcal{O}^r_{\mathcal{U}} \otimes_{\mathcal{O}_{\mathcal{U}}}-$, with the $(M_r(\mathcal{O}_{\mathcal{U}}),\mathcal{O}_{\mathcal{U}})$-bimodule $\mathcal{O}^r_{\mathcal{U}}$ induces an equivalence between the categories of coherent $\mathcal{O}_{\mathcal{U}}$-modules and coherent left $M_r(\mathcal{O}_{\mathcal{U}})$-modules. Note that the coherent sheaf $(\mathcal{O}^r_{\mathcal{U}})^*$ is sent to $\mathcal{O}^r_{\mathcal{U}} \otimes_{\mathcal{O}_{\mathcal{U}}}(\mathcal{O}^r_{\mathcal{U}})^*\cong M_r(\mathcal{O}_{\mathcal{U}})$ and let $\mathcal{F}$ be a coherent sheaf sent to the coherent left $(\mathcal{D}_Y)_\mathcal{U} \simeq M_r(\mathcal{O}_{\mathcal{U}})$-module $\mathcal{M}_\mathcal{U}:=\pi^*\mathcal{M}\simeq \mathcal{O}^r_{\mathcal{U}} \otimes_{\mathcal{O}_{\mathcal{U}}}\mathcal{F}$. Then $$\pi^*\mathcal{E}\mathnormal{xt}^l_{\mathcal{D}_Y}(\mathcal{M},\mathcal{D}_Y)\simeq \mathcal{E}\mathnormal{xt}^l_{(\mathcal{D}_Y)_\mathcal{U}}(\mathcal{M}_\mathcal{U},(\mathcal{D}_Y)_\mathcal{U})$$ $$\simeq \mathcal{E}\mathnormal{xt}^l_{M_r(\mathcal{O}_{\mathcal{U}})}(\mathcal{O}^r_{\mathcal{U}} \otimes_{\mathcal{O}_{\mathcal{U}}}\mathcal{F},\mathcal{O}^r_{\mathcal{U}} \otimes_{\mathcal{O}_{\mathcal{U}}}(\mathcal{O}^r_{\mathcal{U}})^*)\simeq_{\mathcal{O}_{\mathcal{U}}-mod} \mathcal{E}\mathnormal{xt}^l_{\mathcal{O}_{\mathcal{U}}}(\mathcal{F},(\mathcal{O}^r_{\mathcal{U}})^*)\text{ vanishes}$$ if and only if $\mathcal{E}\mathnormal{xt}^l_{\mathcal{O}_{\mathcal{U}}}(\mathcal{F},\mathcal{O}_{\mathcal{U}})\text{ vanishes}$ if and only if $$\mathcal{E}\mathnormal{xt}^l_{\mathcal{O}_{\mathcal{U}}}(\mathcal{O}^r_{\mathcal{U}} \otimes_{\mathcal{O}_{\mathcal{U}}}\mathcal{F},\mathcal{O}_{\mathcal{U}})\simeq \mathcal{E}\mathnormal{xt}^l_{\mathcal{O}_{\mathcal{U}}}(\mathcal{M}_\mathcal{U},\mathcal{O}_{\mathcal{U}})
\simeq \pi^*\mathcal{E}\mathnormal{xt}^l_{\mathcal{O}_{T^*(Y')}}(\mathcal{M},\mathcal{O}_{T^*(Y')}) \text{ vanishes.}$$\
**Proof of theorem \[thm:purity\]:** Note that if the fiber of $M$ at the generic point of $S$ is non zero then $M$ is non zero. Therefore by generic freeness [@EisenbudCA theorem 14.4] applied to the associated graded to a good filtration on $M$ and lemma \[lmm:graded flatness\], there is a dense open subset $W$ of $S$ on which $M$ is faithfully flat, hence $M_s\neq 0$ and $(F_{X_s/k(s)})_*M_s\neq 0$ for all $s\in W$. By proposition \[prop:holonomic fibers\], there is a dense open subset $U$ of $W$ such that $\mathcal{E}\mathnormal{xt}^l_{D_{X_s}}(M_s,D_{X_s})=0$ for all $l\neq n$ and all $s\in U$ which by proposition \[prop:pure Azumaya\] is equivalent to $\mathcal{E}\mathnormal{xt}^l_{\mathcal{O}_{T^*(X'_s)}}((F_{X_s/k(s)})_*M_s,\mathcal{O}_{T^*(X'_s)})=0$ for all $l\neq n$ and all $s\in U$. In particular $\mathcal{E}\mathnormal{xt}^l_{\mathcal{O}_{T^*(X'_s)}}(\mathcal{E}\mathnormal{xt}^l_{\mathcal{O}_{T^*(X'_s)}}((F_{X_s/k(s)})_*M_s,\mathcal{O}_{T^*(X'_s)}),\mathcal{O}_{T^*(X'_s)})=0$ for all $l\neq n$ and all $s\in U,$ implying by theorem \[thm:ext and coherent purity\] that for all $s\in U,$ $(F_{X_s/k(s)})_*M_s$ is a pure non zero coherent $\mathcal{O}_{T^*(X'_s)}$-module of dimension $n,$ hence equidimensional of dimension $n$ by \[prop:pure equidimensional\].\
The purity of the coherent $\mathcal{O}_{T^*(X'_s)}$-module $(F_{X_s/k(s)})_*M_s$ guarantees that it has no embedded associated points.
Reduction to $\mathbb{A}^n$ {#sec:reduction}
===========================
It is convenient (see section \[sec:bound\]) to further reduce the proof of theorem \[thm:main thm\] to modules on $\mathbb{A}^n_S$. In order to do so we shall use the direct image of $D_{X/S}$-modules.
Direct image of $D_{X/S}$-modules for a closed immersion {#subsec:closed immersion}
--------------------------------------------------------
Let $X/S$ be smooth of relative dimension $n$, then the invertible $\mathcal{O}_X$-module $\Omega_{X/S} := \wedge^n\Omega^1_{X/S}$ is endowed with a right $D_{X/S}$-module structure, defined via the Lie derivative [@Kash 1.4(a)] (see also [@BerthII 1.2.1]). Moreover for $M$ a left $D_{X/S}$-module and for $N$, $N'$ right $D_{X/S}$-modules, $N\otimes_{\mathcal{O}_X} M$ (resp. $\mathcal{H}om_{\mathcal{O}_X}(N,N')$) is naturally a right (resp. left) $D_{X/S}$-module, [@Borel VI 3.4]. In particular, denote by $M_r$ the right $D_{X/S}$-module $\Omega_{X/S}\otimes_{\mathcal{O}_X} M$ and by $N_l$ the left $D_{X/S}$-module $\mathcal{H}om_{\mathcal{O}_X}(\Omega_{X/S},N)$. In local étale relative coordinates $\{x_1,...,x_n\}$, trivializing $\Omega_{X/S}$, exchanging left and right, that is going from $M$ to $M_r$ and from $N$ to $N_l$, is expressed by making a differential operator $P=\sum_I P_I\partial^{I} \in\bigoplus_I \mathcal{O}_X.\partial^{I}\simeq D_{X/S}$ act through its adjoint $P^t:=\sum_I (-1)^{|I|}\partial^{I}P_I$, where $|I|$ is the length $I_1+...+I_n$ of the multi-index $I$, [@BerthII 1.2.7].
Let $Y/S$ be a smooth morphism of relative dimension $m$ and let $X \xrightarrow{f} Y$ be a $S$-morphism. Then for a left $D_{Y/S}$-module $M$, $f^*M:=\mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y}f^{-1}M$ is naturally endowed with a left $D_{X/S}$-module structure, via the morphism $T_{X/S} \to f^*T_{Y/S}$ dual to the pullback of differentials $\Omega^1_{X/S}\xleftarrow{f^*} f^*\Omega^1_{Y/S}$. In particular the pullback $f^*D_{Y/S}$ of the $(D_{Y/S},D_{Y/S})$-bimodule $D_{Y/S}$ is naturally a $(D_{X/S},f^{-1}D_{Y/S})$-bimodule, noted $D_{X \to Y}$.
Further suppose (see \[rmk:reduction to affine\]) that $X$, $Y$ and $S$ are affine. Then to a left $D_{X/S}$-module $M$ is associated the left $D_{Y/S}$-module $f_0(M):= (f_*(M_r\otimes_{\mathcal{D}_X}D_{X \to Y}))_l$. Note that $f_0$ is compatible with base change.
If $X \xrightarrow{f} Y$ is a closed immersion, then $f_0(M)$ is called the direct image of $M$ by $f$ and has the following description [@Borel VI §7]. In local coordinates $\{y_1,...,y_n,y_{n+1},...,y_m\}$ of $Y$ around a point of $X$ in which $X$ is described by $\{y_{n+1}=...=y_m=0\},$ $f_0(M)$ is $(M_r[\partial_{n+1},...,\partial_m])_l$ where we omit $f_*$ and where $D_{Y/S} \simeq\bigoplus_I \mathcal{O}_Y.\partial^{I}$ acts on $M_r[\partial_{n+1},...,\partial_m]:=M_r \otimes_{\mathcal{O}_S}\mathcal{O}_S[\partial_{n+1},...,\partial_m]$ on the right using the commutation rules of $D_{Y/S}$ and the restriction $\mathcal{O}_Y \to f_*\mathcal{O}_X$ and one goes from left to right and vice-versa via the adjoint.
Hence $f_0$ preserves coherence and if $S$ is of positive characteristic the above description shows that the $p$-support of $f_0(M)$ is $f'_{\pi}({f'_d}^{-1}(p\text{-}supp(M)))$ where $M$ is a coherent left $D_{X/S}$-module, $X' \xrightarrow{f'} Y'$ is the closed immersion induced by $f$ on the relative Frobeniuses of $X$ and $Y$ and we used notations of \[subsec:symplectic geometry\], in particular $f'_{\pi}$ is a closed immersion by \[lmm:cotangent of an immersion\].
Reduction {#subsec:reduction to affine space}
---------
Recall \[rmk:reduction to affine\], in particular $X/S$ is smooth of relative dimension $n$ and one may assume that $X$ and $S$ are affine. Hence for some $m$, there is a closed immersion $X \overset{f}\hookrightarrow \mathbb{A}^m_S$ over $S$. Let $M$ be a left $D_{X/S}$-module as in the statement of \[thm:main thm\], then $f_0(M)$ is a coherent left $D_{\mathbb{A}^m_S/S}$-module (\[subsec:closed immersion\]) and by compatibility of direct image with base change (\[subsec:closed immersion\]) and [@Borel VI 7.8(iii)], it is holonomic at the generic fiber of $S$. Hence in view of the compatibility of $f_0$ with base change and the description of $p\text{-}supp((f_s)_0(M_s))$ for a closed point $s$ of $S$ given in \[subsec:closed immersion\], theorem \[thm:purity\] and lemma \[lmm:isotropy\] further reduce the proof of theorem \[thm:main thm\] to the case $X/S=\mathbb{A}^n_S$.
A bound on degrees and ranks {#sec:bound}
============================
Thanks to \[subsec:reduction to affine space\], in the proof of theorem \[thm:main thm\], one may restrict one’s attention to modules $M$ over $D_{\mathbb{A}^n_S/S}.$
In addition to the natural filtration (\[prop:filtration D\]), $D_{\mathbb{A}^n_S/S}$ has a filtration whose associated graded pieces are finite over $S,$ the Bernstein filtration. We use it to refine part of the comparison (\[thm:purity\]) between fibers of $M$ at closed points and at the generic point of $S.$
More precisely, we get an estimate (theorem \[thm:bound\]), bounding the degrees of the $p$-supports (for a suitable projective embedding) as well as the generic ranks, of the fibers at “almost all” closed points of a module $M$ as above. These bounds are crucial in the proofs of theorems \[thm:splitting in affine space case\] and \[thm:main thm\].
Bernstein filtration
--------------------
Let $S=spec(R)$ (\[rmk:reduction to affine\]) and $\mathbb{A}^n_S=spec(R[x_1,...,x_n])$, then the ring of global sections of $D_{\mathbb{A}^n_S/S}$ is $R[x_1,...,x_n]\langle\partial_1,...,\partial_n\rangle/ \langle[\partial_i,\partial_j]=0, [\partial_i,x_j]=\delta_{i,j}\rangle$, the $n$-th Weyl algebra with coefficients in $R$, $A_n(R)$. The filtration on $A_n(R)=\bigoplus_{\alpha,\beta} R x^{\alpha}\partial^{\beta}$, $\alpha$, $\beta$ multi-indices, by the total order in $x$ and $\partial$, $\mathcal{B}_lA_n(R):=\bigoplus_{|\alpha|+|\beta|\leq l} R x^{\alpha}\partial^{\beta}$ is called the *Bernstein filtration*. Note that the associated graded ring $gr^{\mathcal{B}}A_n(R)$ is the $R$-algebra of polynomials in the classes $x_1,...,x_n,y_1,...,y_n$ of $x_1,...,x_n,\partial_1,...,\partial_n$, respectively, graded by the order of polynomials. In particular the $\mathcal{B}_lA_n(R)/\mathcal{B}_{l-1}A_n(R)$ are finite free modules over $R$.
A good filtration $\Gamma$ on a left $A_n(R)$-module $M$ is an increasing exhaustive filtration on $M,$ compatible with $\mathcal{B},$ which is bounded below and such that the associated graded module $gr^{\Gamma}M$ is finite over the algebra of polynomials $gr^{\mathcal{B}}A_n(R)$. In particular the $\Gamma_lM/\Gamma_{l-1}M$ and hence the $\Gamma_lM$ are finite R-modules. Note that good filtrations exist on finitely generated left $A_n(R)$-modules, [@Bjork1 Ch.1 2.7].
Suppose that $R$ is a field $K$. Let $M$ be a finitely generated left $A_n(K)$-module and let $\Gamma$ be a good filtration on $M$. Then for $l$ large enough, the function $l\mapsto dim_K\Gamma_lM$ coincides with a polynomial $\mathcal{H}_{M,\Gamma} \in \mathbb{Q}[t]$, [@Bjork1 Ch.1 3.3]. Moreover, let $d$ (resp. $a_d$) be the degree (resp. the leading coefficient) of $\mathcal{H}_{M,\Gamma}$, then $d!a_d$ is a non negative integer and $d(M):=d$ and $e(M):=d!a_d$ are independent of $\Gamma$ and called the dimension and multiplicity of $M$, respectively [@Bjork1 p.8].
\[lmm:fibers dimension and multiplicity\]
Suppose that $R$ is a domain and let $M$ be a finitely generated left $A_n(R)$-module. Then there is a dense open subset $U$ of $S:=spec(R)$ such that the functions $s\mapsto d(M_s)$ and $s\mapsto e(M_s)$ are constant on $U$.
**Proof:** Let $\Gamma$ be a good filtration on $M$. Then by generic freeness [@EisenbudCA theorem 14.4], there is a dense open subset $U$ of $S$ such that for all $l$, $(\Gamma_lM/\Gamma_{l-1}M)|_U$ is free over $U$. In particular the $(\Gamma_lM/\Gamma_{l-1}M)|_U$ are flat over $U$, hence for all $l$ and all $s\in U$, $(\Gamma_lM/\Gamma_{l-1}M)_s\cong(\Gamma_lM)_s/(\Gamma_{l-1}M)_s$ and $(\Gamma)_s$ is a good filtration on $M_s$. The lemma follows since for all $s\in U$ and all $l$, $dim_{k(s)}(\Gamma_lM)_s=\sum_{i=-\infty}^{i=l} dim_{k(s)}(\Gamma_iM)_s/(\Gamma_{i-1}M)_s$ and $$dim_{k(s)}(\Gamma_lM)_s/(\Gamma_{l-1}M)_s=dim_{k(s)}(\Gamma_lM/\Gamma_{l-1}M)_s$$ is the rank of the free module $\Gamma_lM/\Gamma_{l-1}M|_U$ over $U$.\
Induced filtration over the center {#subsec:induced filtration over the center}
----------------------------------
Let $K$ be a field of positive characteristic $p$. Then the center $ZA_n(K)$ of $A_n(K)=K[x_1,...,x_n]\langle\partial_1,...,\partial_n\rangle/ \langle[\partial_i,\partial_j]=0, [\partial_i,x_j]=\delta_{i,j}\rangle$ is the algebra of polynomials $K[x_1^p,...,x_n^p,\partial_1^p,...,\partial_n^p]$. It is graded by the degree of polynomials, where $degree(x_i^p)=degree(\partial_j^p)=1$ and the associated increasing filtration is denoted $\mathcal{C}$. The Rees ring $R_n(\mathcal{C})$ of the filtered ring $(ZA_n(K),\mathcal{C})$ is the naturally graded ring $\bigoplus_{i=0}^{i=\infty} \mathcal{C}_iZA_n(K)$. Note that the graded algebra morphism $K[t_0,x_1^p,...,x_n^p,\partial_1^p,...,\partial_n^p] \to R_n(\mathcal{C}):=\bigoplus_{i=0}^{i=\infty} \mathcal{C}_iK[x_1^p,...,x_n^p,\partial_1^p,...,\partial_n^p]$ sending $t_0\mapsto 1 \in \mathcal{C}_1K[x_1^p,...,x_n^p,\partial_1^p,...,\partial_n^p]$ and $x_i^p (\text{resp. }\partial_j^p) \mapsto x_i^p (\text{resp. }\partial_j^p)\in \mathcal{C}_1K[x_1^p,...,x_n^p,\partial_1^p,...,\partial_n^p]$ is an isomorphism. Keeping the same notation for $t_0$ and its image under this isomorphism, the natural map $R_n(\mathcal{C})/t_0R_n(\mathcal{C}) \to gr^{\mathcal{C}}ZA_n(K)$ is an isomorphism of graded algebras. Note also that summing components induces an isomorphism $R_n(\mathcal{C})_{(t_0)}\widetilde{\to} ZA_n(K)$ where $R_n(\mathcal{C})_{(t_0)}$ is the subring of degree $0$ elements of the graded ring $R_n(\mathcal{C})_{t_0}$.
An increasing $\mathcal{C}$-compatible filtration $G$ on a $ZA_n(K)$-module $M$ is said to be good if the associated Rees module $\mathcal{R}(M,G):=\bigoplus_{i=-\infty}^{i=\infty} G_iM$ over the Rees ring $\bigoplus_{i=0}^{i=\infty} \mathcal{C}_iZA_n(K)$ is finitely generated. This implies in particular that $G$ is bounded below. Moreover one easily sees that a filtration $G$ on $M$ is good if and only if $G$ is bounded below and the associated graded module $gr^GM$ is finitely generated over $gr^{\mathcal{C}}ZA_n(K)$, [@Bjork2 A:III 1.29].
Let $\Gamma$ be a filtration on the left $A_n(K)$-module $M$, then $p\Gamma$, $(p\Gamma)_lM:=\Gamma_{pl}M$ endows $M$ with the structure of a filtered module over the center $(ZA_n(K),\mathcal{C})$.
\[lmm:pGamma is good\] Let $\Gamma$ be a good filtration on the left $A_n(K)$-module $M$, then $p\Gamma$ is a good filtration on $M$ seen as a $(ZA_n(K),\mathcal{C})$-module.
**Proof:** Since $\Gamma$ is bounded below then so is $p\Gamma$. Let’s show that the $gr^{\mathcal{C}}ZA_n(K)$-module $gr^{p\Gamma}M$ is finitely generated. For this let $F$ be the filtration on $ZA_n(K)$ induced by $\mathcal{B}$, in particular $degree_F(x_i^p)=degree_F(\partial_j^p)=p$ and let $F(\Gamma)$ be the $F$-compatible filtration on $M$ defined by $F(\Gamma)_lM:=\Gamma_{pm}M$, where $pm$ is the greatest integer multiple of $p$ such that $pm\leq l$. Note that $\text{class of }x_i^p (\text{resp. }\partial_j^p)\mapsto \text{class of }x_i^p (\text{resp. }\partial_j^p)$ induces an isomorphism of $K$-algebras $gr^{\mathcal{C}}ZA_n(K)\to gr^FZA_n(K)$ with which the $K$-module isomorphism $gr^{p\Gamma}M\to gr^{F(\Gamma)}M$ defined by $(p\Gamma)_lM/(p\Gamma)_{l-1}M= \Gamma_{pl}M/\Gamma_{p(l-1)}M= F(\Gamma)_{pl}M/F(\Gamma)_{pl-1}M$ is compatible. Hence $gr^{p\Gamma}M$ is finitely generated over $gr^{\mathcal{C}}ZA_n(K)$ if and only if $gr^{F(\Gamma)}M$ is finitely generated over $gr^FZA_n(K)$. Consider the finite exhaustive filtration of $gr^{F(\Gamma)}M$ by graded sub-$gr^FZA_n(K)$-modules, $0 =(gr^{F(\Gamma)}M)_0 \subset (gr^{F(\Gamma)}M)_1 \subset...\subset (gr^{F(\Gamma)}M)_p= gr^{F(\Gamma)}M$ such that $(gr^{F(\Gamma)}M)_i \cap F(\Gamma)_lM/F(\Gamma)_{l-1}M$ is the image of the natural map $\Gamma_{p(m-1)+i}M/(F(\Gamma)_{l-1}M\cap\Gamma_{p(m-1)+i}M) \to F(\Gamma)_lM(:=\Gamma_{pm}M)/F(\Gamma)_{l-1}M$ with $pm$, as above, the greatest integer multiple of $p$ such that $pm\leq l$. Denote by $gr(gr^{F(\Gamma)}M)$ the graded $gr^FZA_n(K)$-module $\bigoplus_{i=1}^{i=p}(gr^{F(\Gamma)}M)_i/(gr^{F(\Gamma)}M)_{i-1}$. Note also that $gr^{\Gamma}M$ seen as a module over $gr^FZA_n(K) \hookrightarrow gr^{\mathcal{B}}A_n(K)$ decomposes as a direct sum of graded sub-$gr^FZA_n(K)$-modules $\bigoplus_{i=1}^{i=p}(gr^{\Gamma}M)_i$ where $(gr^{\Gamma}M)_i:=\bigoplus_{l\in\mathbb{Z}}gr_{pl+i}^{\Gamma}M$ and let $F_*gr^{\Gamma}M$ be the graded $gr^FZA_n(K)$-module $\bigoplus_{i=1}^{i=p}(gr^{\Gamma}M)_i[i-p]$. Then for all $i,$ the morphism of graded $gr^FZA_n(K)$-modules $(gr^{F(\Gamma)}M)_i/(gr^{F(\Gamma)}M)_{i-1}\to (gr^{\Gamma}M)_i[i-p]$ induced by $\Gamma_{p(m-1)+i}M \to (gr^{\Gamma}M)_i[i-p]\text{ in degree }pm=(gr^{\Gamma}M)_i\text{ in degree }p(m-1)+i=gr_{p(m-1)+i}^{\Gamma}M$ is an isomorphism. These induce an isomorphism of graded $gr^FZA_n(K)$-modules $$gr(gr^{F(\Gamma)}M) \simeq F_*gr^{\Gamma}M.$$ Since $gr^{\Gamma}M$ is finitely generated over $gr^{\mathcal{B}}A_n(K)$ by hypothesis and $gr^{\mathcal{B}}A_n(K)\cong K[x_1,...,x_n,y_1,...,y_n]$ is finite over $gr^FZA_n(K)\cong K[x_1^p,...,x_n^p,y_1^p,...,y_n^p]$, $F_*gr^{\Gamma}M$ is finitely generated over $gr^FZA_n(K)$. Then $gr^{F(\Gamma)}M$ has an exhaustive finite filtration whose subquotients are finitely generated over $gr^FZA_n(K)$, hence it is finitely generated, giving the lemma.\
Let $M$ be a left $A_n(K)$-module and let $\Gamma$ be a good filtration on $M.$ Since $p\Gamma$ is a good filtration on the $(ZA_n(K),\mathcal{C})$-module $M$ (\[lmm:pGamma is good\]) the Rees module of $(M,p\Gamma),$ which is a finitely generated graded module over the Rees ring $R_n(\mathcal{C})\simeq K[t_0,x_1^p,...,x_n^p,\partial_1^p,...,\partial_n^p],$ has a Hilbert polynomial $\mathcal{H}_{\mathcal{R}(M,p\Gamma)}$. The latter can be computed in terms of the Hilbert polynomial of $(M,\Gamma)$ and in particular one gets the degree and leading coefficient of $\mathcal{H}_{\mathcal{R}(M,p\Gamma)}$ in terms of the dimension and multiplicity of $M.$ Indeed,
\[prop:Hilbert polynomial of pGamma\]
Let $M$ be a left $A_n(K)$-module and let $\Gamma$ be a good filtration on $M$. Then the Hilbert polynomial $\mathcal{H}_{\mathcal{R}(M,p\Gamma)}(t)$ of the Rees module of $(M,p\Gamma)$ is $\mathcal{H}_{M,\Gamma}(pt)$. In particular, the degree of $\mathcal{H}_{\mathcal{R}(M,p\Gamma)}$ is $d(M)$ and its leading coefficient times $d(M)!$ is $e(M)p^{d(M)}$.
**Proof:** For $l$ large enough, the Hilbert polynomial $\mathcal{H}_{M,\Gamma}(l)$ coincides with $l\mapsto dim_K\Gamma_lM$ while $\mathcal{H}_{\mathcal{R}(M,p\Gamma)}(l)$ does so with $l\mapsto dim_K(p\Gamma)_lM=dim_K\Gamma_{pl}M$. The proposition follows.\
The bound {#subsec:bound}
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There is a geometric picture of the Rees construction in which the affine scheme $spec(ZA_n(K))\cong spec(R_n(\mathcal{C})_{(t_0)})$ is identified with the affine open $D_+(t_0)$ associated to $t_0$ in the homogeneous prime spectrum $Proj(R_n(\mathcal{C}))$ of $R_n(\mathcal{C})$ [@EGAII 2.4.1] and in which its complement, the reduced closed subscheme $V_+(t_0)$ is identified with the closed subscheme $Proj(gr^{\mathcal{C}}ZA_n(K))\cong Proj(R_n(\mathcal{C})/t_0R_n(\mathcal{C}))$ of $Proj(R_n(\mathcal{C}))$, [@EGAII 2.9.2 (i)]. Making these identifications, let $G$ be a good filtration on a finitely generated $(ZA_n(K),\mathcal{C})$-module $M$ then the coherent sheaf $\widetilde{\mathcal{R}(M,G)}$ on $Proj(R_n(\mathcal{C}))$ extends $\widetilde{M}$ and its restriction to the complement $Proj(gr^{\mathcal{C}}ZA_n(K))$ of $spec(ZA_n(K))$ is isomorphic to $\widetilde{gr^GM}$. Moreover it’s easy to see that the support of $\widetilde{\mathcal{R}(M,G)}$ is the closure of $supp(\widetilde{M})$ in $Proj(R_n(\mathcal{C}))$. Note that here $$Proj(R_n(\mathcal{C})) \simeq Proj(K[t_0,x_1^p,...,x_n^p,\partial_1^p,...,\partial_n^p])\simeq\mathbb{P}^{2n}_K,$$ $$spec(ZA_n(K)) \simeq spec(K[x_1^p,...,x_n^p,\partial_1^p,...,\partial_n^p])\simeq \mathbb{A}^{2n}_K \text{ and}$$ $$Proj(gr^{\mathcal{C}}ZA_n(K))\simeq Proj(K[x_1^p,...,x_n^p,\partial_1^p,...,\partial_n^p])\simeq\mathbb{P}^{2n-1}_K.$$
The leading coefficient of the Hilbert polynomial of $\widetilde{\mathcal{R}(M,G)}$ is related to the top-dimensional irreducible components of its support through the following,
\[prop:mu, ranks and degrees\]
Let $Y \overset{i}\hookrightarrow \mathbb{P}^{m}_K$ be a closed subscheme and let $\mathcal{F}$ be a coherent sheaf of dimension $d$ on $Y$. Set $\mu(\mathcal{F}):=d!a_d$ where $a_d$ is the leading coefficient of the Hilbert polynomial of $\mathcal{F}$ with respect to $i$. Then $$\Sigma_z rk_z(\mathcal{F})deg(\overline{\{z\}}) \leq\mu(\mathcal{F})$$ where the sum is over the generic points of the $d$-dimensional irreducible components of $supp(\mathcal{F})$, $rk_z(\mathcal{F}):=dim_{k(z)}(\mathcal{F}_z\otimes k(z))$ and $deg(\overline{\{z\}})$ is the degree of $\overline{\{z\}}^{red}$ with respect to $i$.
**Proof:** By [@Kleiman lemma B.4] and [@EGAIV 5.3.1], $\mu(\mathcal{F})=\Sigma_z length_{\mathcal{O}_{Y,z}}(\mathcal{F}_z)\mu(\mathcal{O}_{\overline{\{z\}}^{red}})$ summing over the generic points of the $d$-dimensional irreducible components of $supp(\mathcal{F}).$ Let $z$ be as above, then by additivity of the length under short exact sequences $length_{\mathcal{O}_{Y,z}}(\mathcal{F}_z)\geq length_{k(z)}(\mathcal{F}_z\otimes k(z))=dim_{k(z)}(\mathcal{F}_z\otimes k(z))=:rk_z(\mathcal{F}).$ This gives the proposition since $deg(\overline{\{z\}}):=\mu(\mathcal{O}_{\overline{\{z\}}^{red}}).$\
\[thm:bound\]
Let $S$ be an integral scheme dominant and of finite type over $\mathbb{Z}$ and let $M$ be a coherent left $D_{\mathbb{A}^n_S/S}$-module. Suppose that the fiber of $M$ at the generic point of $S$ is a non zero holonomic left $\mathcal{D}$-module. Then there is a dense open subset $U$ of $S$ such that for each closed point $u\in U$ and each $z$ generic point of an irreducible component of $p$-supp($M_u$) $$deg(\overline{\{z\}})\leq e(M) \text{ and }rk_z(M_u) \leq e(M)p^n$$ where $e(M)$ is the multiplicity for the Bernstein filtration of the fiber of $M$ at the generic point of $S,$ $deg(\overline{\{z\}})$ is the degree of the reduced closure of the image of $z$ in $\mathbb{P}^{2n}_{k(u)}$ by the open immersion of the Rees construction and $rk_z(M_u):=dim_{k(z)}(({F_{\mathbb{A}^n/k(u)}}_*M_u)_z\otimes k(z)).$
**Proof:** The proof reduces to the case $S$ is integral and affine $=spec(R)$. Hence it is equivalent (\[prop:coherentD\]) to consider a finitely generated left $A_n(R)$-module $M$. By \[lmm:fibers dimension and multiplicity\] there is a dense open subset $U_e$ of $S$ such that for each closed point $u\in U_e$, $d(M_u)=n$ and $e(M_u)=e(M).$ Then for such a $u$ set $p:=char(k(u))$ and let $\Gamma$ be a good filtration on the left $A_n(k(u))$-module $M_u$. By \[prop:Hilbert polynomial of pGamma\] $\widetilde{\mathcal{R}(M_u,p\Gamma)}$ is of dimension $n$ and $\mu(\widetilde{\mathcal{R}(M_u,p\Gamma)})=e(M)p^n$. Hence since $supp(\widetilde{\mathcal{R}(M_u,p\Gamma)})=\overline{p \text{-}supp(M_u)}$ in which $p$-$supp(M_u)=\overline{p \text{-}supp(M_u)}\cap spec(ZA_n(K))$ is open, \[prop:mu, ranks and degrees\] gives $\Sigma_z rk_z(M_u)deg(\overline{\{z\}})\leq e(M)p^n$ where the sum is over the generic points of the $n$-dimensional irreducible components of $p$-supp($M_u$), which are all its irreducible components if $u\in U \subset U_e$ where $U \subset U_e$ is a dense open subset provided by theorem \[thm:purity\]. In particular for each $z$ generic point of an irreducible component of $p$-supp($M_u$), $rk_z(M_u)deg(\overline{\{z\}})\leq e(M)p^n$.
Moreover ${F_{\mathbb{A}^n/k(u)}}_*M_u$ being a left module over an Azumaya algebra of rank $p^n$ (\[thm:Azumaya\]), $({F_{\mathbb{A}^n/k(u)}}_*M_u)_z\otimes \overline{k(z)}$ is by [@BrauerI 5.1 (i)] a left module over $M_{p^n}(\overline{k(z)}),$ where $\overline{k(z)}$ is an algebraic closure of $k(z).$ Hence there is a finite dimensional $\overline{k(z)}$-vector space $V$ such that $({F_{\mathbb{A}^n/k(u)}}_*M_u)_z\otimes \overline{k(z)} \simeq\overline{k(z)}^{p^n} \otimes_{\overline{k(z)}}V$ where $\overline{k(z)}^{p^n}$ is the standard left $M_{p^n}(\overline{k(z)})$-module. In particular $$rk_z(M_u):=dim_{k(z)}(({F_{\mathbb{A}^n/k(u)}}_*M_u)_z\otimes k(z))=dim_{\overline{k(z)}}(({F_{\mathbb{A}^n/k(u)}}_*M_u))_z\otimes \overline{k(z)})$$ is divisible by $p^n$, thus proving the theorem.\
The Brauer group and differential forms {#sec:Brauer group and forms}
=======================================
Here we prove, in a first part, that “the Azumaya algebra of differential operators splits on the regular locus of the $p$-support of a holonomic $\mathcal{D}$-module, for $p$ large enough” (theorem \[thm:splitting\]).
In a second part, we consider a map arising from the $p$-curvature exact sequence (\[subsec:calculus in char p\]), which sends 1-forms to the Brauer group. It maps the canonical form to the class of the Azumaya algebra of differential operators (prop. \[prop:canonical form maps to algebra of differential operators\]) and we describe its kernel (prop. \[prop:kernel of phi\]).
Splittings of Azumaya algebras on the support of modules
--------------------------------------------------------
Let $Y$ be a scheme and let $\mathcal{A}$ be an Azumaya algebra on $Y$. Since $\mathcal{A}$ is a coherent $\mathcal{O}_Y$-module, it is a coherent noetherian ring and a left $\mathcal{A}$-module is coherent if and only if it is coherent as an $\mathcal{O}_Y$-module. The Azumaya algebra $\mathcal{A}$ is said to split on $Y$ if its class $[\mathcal{A}]$ in the Brauer group $Br(Y)$ of $Y$ [@BrauerI §2] is trivial.
Let $M$ be a coherent left $\mathcal{A}$-module and let $z$ be the generic point of an irreducible component of the support of the coherent $\mathcal{O}_Y$-module $M.$ The next proposition relates $rk_z(M)$ (\[prop:mu, ranks and degrees\]) to the order of $[\mathcal{A}|_{(\overline{\{z\}}^{red})^{reg}}]$ in $Br((\overline{\{z\}}^{red})^{reg}).$
Let us first prove a lemma,
\[lmm:Azumaya acting on a vector bundle\]
Let $Y$ be a scheme and let $\mathcal{A}$ be an Azumaya algebra of rank $r$ on $Y$. Suppose that $\mathcal{A}$ acts on the left on a locally free sheaf $\mathcal{V}$ of rank $v$. Then $r$ divides $v=lr$ and $l[\mathcal{A}]=0$ in $Br(Y).$
**Proof:** By hypothesis there is a morphism of $\mathcal{O}_Y$-algebras $\mathcal{A} \to \mathcal{E}nd_{\mathcal{O}_Y}(\mathcal{V}), 1 \mapsto 1.$ It is injective by [@EGAI 0.5.5.4] since the fiber of $\mathcal{A}$ at each point of $Y$ is a simple algebra [@BrauerI 5.1 (i)]. Therefore one may view $\mathcal{A}$ as a subalgebra of $\mathcal{E}nd_{\mathcal{O}_Y}(\mathcal{V})$ and in particular consider $\mathcal{C}_{\mathcal{E}nd_{\mathcal{O}_Y}(\mathcal{V})}(\mathcal{A})$ the commutant of $\mathcal{A}$ in $\mathcal{E}nd_{\mathcal{O}_Y}(\mathcal{V}),$ which is a coherent subalgebra of $\mathcal{E}nd_{\mathcal{O}_Y}(\mathcal{V}).$ By [@AusGold theorem 3.3], the natural morphism of $\mathcal{O}_Y$-algebras $\mathcal{A}\otimes_{\mathcal{O}_Y}\mathcal{C}_{\mathcal{E}nd_{\mathcal{O}_Y}(\mathcal{V})}(\mathcal{A})\to \mathcal{E}nd_{\mathcal{O}_Y}(\mathcal{V})$ is an isomorphism and $\mathcal{C}_{\mathcal{E}nd_{\mathcal{O}_Y}(\mathcal{V})}(\mathcal{A})$ is an Azumaya algebra on $Y$. Hence by the behaviour of ranks under tensor products, $\mathcal{C}_{\mathcal{E}nd_{\mathcal{O}_Y}(\mathcal{V})}(\mathcal{A})$ is of constant rank $l$, $v=lr$. By definition of the Brauer group $0=[\mathcal{E}nd_{\mathcal{O}_Y}(\mathcal{V})]=[\mathcal{A}]+[\mathcal{C}_{\mathcal{E}nd_{\mathcal{O}_Y}(\mathcal{V})}(\mathcal{A})]$ in $Br(Y).$ The lemma follows since for each Azumaya algebra $\mathcal{B}$ of rank $n$ on $Y$, $n[\mathcal{B}]=0$ in $Br(Y)$ [@BrauerI §2] giving $0=l[\mathcal{A}]+l[\mathcal{C}_{\mathcal{E}nd_{\mathcal{O}_Y}(\mathcal{V})}(\mathcal{A})]=l[\mathcal{A}]$ in $Br(Y).$\
\[prop:order on the support\]
Suppose that $Y$ is of finite type over a field $K$. Let $\mathcal{A}$ be an Azumaya algebra of rank $r$ on $Y$, let $M$ be a coherent left $\mathcal{A}$-module and let $z$ be the generic point of an irreducible component of $supp(M)$. Then $r$ divides $rk_z(M)=l_z(M)r$ and $$l_z(M)[\mathcal{A}|_{(\overline{\{z\}}^{red})^{reg}}]=0$$ in $Br((\overline{\{z\}}^{red})^{reg}).$
**Proof:** Since the vector space $M_z\otimes k(z)$ is of dimension $rk_z(M)$ and acted upon on the left by the rank $r$ Azumaya algebra $\mathcal{A}_z\otimes k(z),$ lemma \[lmm:Azumaya acting on a vector bundle\] implies that $rk_z(M)=l_z(M)r$ and $l_z(M)[\mathcal{A}_z\otimes k(z)]=0$ in $Br(k(z)).$ Moreover since $Y$ is of finite type over a field, so is $\overline{\{z\}}^{red}$ and $(\overline{\{z\}}^{red})^{reg} \hookrightarrow \overline{\{z\}}^{red}$ is a non empty open subscheme by [@EGAIV 6.12.5]. Hence $\mathcal{A}_z\otimes k(z)\cong (\mathcal{A}|_{\overline{\{z\}}^{red}})_z\otimes k(z) \cong (\mathcal{A}|_{(\overline{\{z\}}^{red})^{reg}})_z\otimes k(z)$ and the proposition follows from the canonical embedding $Br((\overline{\{z\}}^{red})^{reg}) \hookrightarrow Br(k(z))$ [@Milne IV 2.6].\
The above proposition combined with the second estimate of theorem \[thm:bound\] leads to the
\[thm:splitting in affine space case\]
Let S be an integral scheme dominant and of finite type over $\mathbb{Z}$ and let $M$ be a coherent left $D_{\mathbb{A}^n_S/S}$-module. Suppose that the fiber of M at the generic point of S is a non zero holonomic left $\mathcal{D}$-module. Then there is a dense open subset $U$ of $S$ such that for each closed point $u\in U$ and each $z$ generic point of an irreducible component of $p$-supp($M_u$) the Azumaya algebra ${F_{\mathbb{A}^n/k(u)}}_*D_{\mathbb{A}^n_{k(u)}}$ on $T^*(\mathbb{A}^{n'}_{k(u)})$ splits on $(\overline{\{z\}}^{red})^{reg}.$
**Proof:** By theorem \[thm:bound\] and using its notations, there is a dense open subset $U_b$ of $S$ such that for each closed point $u\in U_b$ and each $z$ generic point of an irreducible component of $p$-supp($M_u$), $rk_z(M_u) \leq e(M)p^n$ where $p:=chark(u).$ Moreover by proposition \[prop:order on the support\], $rk_z(M_u)=l_z(M_u)p^n\leq e(M)p^n,$ thus $l_z(M_u)\leq e(M)$ and $l_z(M_u)[{F_{\mathbb{A}^n/k(u)}}_*D_{\mathbb{A}^n_{k(u)}}|_{(\overline{\{z\}}^{red})^{reg}}]=0$ in $Br((\overline{\{z\}}^{red})^{reg}).$ Note that by definition $l_z(M_u)\neq 0$ and hence for $u \in U$ the open dense subset of $U_b$ defined by inverting all the primes $\leq e(M)$, $l_z(M_u)$ and $p^n$ are coprime, that is there are integers $a$ and $b$ such that $1=a.l_z(M_u)+b.p^n.$ Since ${F_{\mathbb{A}^n/k(u)}}_*D_{\mathbb{A}^n_{k(u)}}$ is of rank $p^n$, $p^n[{F_{\mathbb{A}^n/k(u)}}_*D_{\mathbb{A}^n_{k(u)}}|_{(\overline{\{z\}}^{red})^{reg}}]=0$ by [@BrauerI §2] and the theorem follows from $[{F_{\mathbb{A}^n/k(u)}}_*D_{\mathbb{A}^n_{k(u)}}|_{(\overline{\{z\}}^{red})^{reg}}]
=1[{F_{\mathbb{A}^n/k(u)}}_*D_{\mathbb{A}^n_{k(u)}}|_{(\overline{\{z\}}^{red})^{reg}}]
$
$=(a.l_z(M_u)+b.p^n)[{F_{\mathbb{A}^n/k(u)}}_*D_{\mathbb{A}^n_{k(u)}}|_{(\overline{\{z\}}^{red})^{reg}}]
$
$=a.l_z(M_u)[{F_{\mathbb{A}^n/k(u)}}_*D_{\mathbb{A}^n_{k(u)}}|_{(\overline{\{z\}}^{red})^{reg}}]+b.p^n.[{F_{\mathbb{A}^n/k(u)}}_*D_{\mathbb{A}^n_{k(u)}}|_{(\overline{\{z\}}^{red})^{reg}}]$
$=0$ in $Br((\overline{\{z\}}^{red})^{reg}).$\
As in \[subsec:reduction to affine space\], theorem \[thm:splitting in affine space case\] implies the apparently more general
\[thm:splitting\] Let $S$ be an integral scheme dominant and of finite type over $\mathbb{Z}$, let $X$ be a smooth $S$-scheme of relative dimension $n$ and let $M$ be a coherent left $D_{X/S}$-module. Suppose that the fiber of $M$ at the generic point of $S$ is a holonomic left $\mathcal{D}$-module. Then there is a dense open subset $U$ of $S$ such that for each closed point $u\in U$ the Azumaya algebra ${F_{X_u/k(u)}}_*D_{X_u}$ on $T^*(X'_u)$ splits on $(p\text{-}supp(M_u))^{reg}.$
**Proof:** By [@BrauerI 2.1] applied to the Zariski site and [@EGAIV 21.11.1] the case $i=2$ of lemma \[lmm:vanishing of Zariski cohomology\] below implies that on a regular (noetherian) scheme for an Azumaya algebra to be split is a Zariski local condition. Therefore by remark \[rmk:reduction to affine\] one may further assume that $S$ and $X$ are regular integral and affine and in particular that there is a closed immersion $X \overset{f}\hookrightarrow \mathbb{A}^m_S$ over $S.$ Specializing to a closed point $u$ of positive characteristic $p$ of $S$ it follows from the description of $p\text{-}supp(f_0(M_u))$ in \[subsec:closed immersion\] and from [@EGAIV 17.5.8(iii)] $f'_d$ being smooth that $p\text{-}supp(f_0(M_u))^{reg}=f'_{\pi}({f'_d}^{-1}((p\text{-}supp(M_u))^{reg})).$ Moreover by [@BezrBr 3.7] ${f'_d}^*({F_{X_u/k(u)}}_*D_{X_u})$ splits on ${f'_d}^{-1}(p\text{-}supp(M_u)^{reg})$ if ${F_{\mathbb{A}^m/k(u)}}_*D_{\mathbb{A}^m_{k(u)}}$ splits on $p\text{-}supp(f_0(M_u))^{reg}.$ Since by lemma \[lmm:cotangent of an immersion\] $f'_d$ Zariski locally admits a section there is a Zariski open covering of $X'$ above which the pullback of Brauer classes ${f'_d}^*$ is injective and therefore to split being Zariski local on a regular noetherian scheme, ${f'_d}^*$ induces an injective morphism $Br(p\text{-}supp(M_u)^{reg}) \to Br({f'_d}^{-1}(p\text{-}supp(M_u)^{reg})).$ So ${F_{X_u/k(u)}}_*D_{X_u}$ splits on $p\text{-}supp(M_u)^{reg}$ if ${F_{\mathbb{A}^m/k(u)}}_*D_{\mathbb{A}^m_{k(u)}}$ splits on $p\text{-}supp(f_0(M_u))^{reg}.$ Note that if the fiber of $M$ at the generic point of $S$ is zero the theorem holds by remark \[rmk:generic fiber zero\] and that a regular noetherian scheme is the sum of its irreducible components by [@EGAI 2.1.9(iii)], in particular an Azumaya algebra splits on a regular noetherian scheme if and only if it splits on its irreducible components. Thus since by \[subsec:closed immersion\] the fiber of $f_0(M)$ at the generic point of $S$ is non zero if so is that of $M$ and since $f_0(M)$ satisfies the other hypotheses of \[thm:splitting in affine space case\] by \[subsec:reduction to affine space\], \[thm:splitting\] reduces to theorem \[thm:splitting in affine space case\].\
\[lmm:vanishing of Zariski cohomology\]
Let $Y$ be a noetherian scheme. If $Y$ is locally factorial then the Zariski cohomology $H^i(Y,\mathcal{O}_Y^*)=0$ for all $i\geq 2.$
**Proof:** By definition of the sheaf $\mathcal{D}iv_Y$ of Cartier divisors there’s an exact sequence of abelian sheaves $0\to\mathcal{O}_Y^*\to\mathcal{K}_Y^*\to\mathcal{D}iv_Y\to 0$ on $Y$ where $\mathcal{K}_Y$ is the sheaf of meromorphic functions and $\mathcal{O}_Y^*\to\mathcal{K}_Y^*$ is the natural injection. If $Y$ is locally factorial then it is the sum of its (finitely many) irreducible components, each of which is integral [@EGAI 4.5.5]. Hence if $Y_i \overset{f_i}\hookrightarrow Y$ is the open immersion of the $i$-th irreducible component then $\mathcal{K}_Y^* \cong \Pi_i {f_i}_*\mathcal{K}_{Y_i}^*$ where $\mathcal{K}_{Y_i}^*$ is isomorphic to the constant sheaf associated to $k(y_i)^*$ for $y_i$ the generic point of $Y_i.$ In particular $\mathcal{K}_Y^*$ is flasque. Since $\mathcal{D}iv_Y$ is flasque by [@EGAIV 21.6.11], $\mathcal{K}_Y^*\to\mathcal{D}iv_Y$ is a flasque right resolution of $\mathcal{O}_Y^*.$ This gives the result as sheaf cohomology may be computed using flasque resolutions.\
The Brauer group via the $p$-curvature sequence {#subsec:Brauer group via the p-curvature sequence}
-----------------------------------------------
Let $Y$ be a smooth equidimensional scheme over a perfect field $K$ of positive characteristic $p$. Composing the coboundary morphisms of the étale cohomology long exact sequences of the two short exact sequences of étale sheaves on $Y',$ $0\to \mathbb{G}_{\,m/Y'} \xrightarrow{{F_{/K}}^*} {F_{/K}}_*\mathbb{G}_{\,m/Y} \xrightarrow{{F_{/K}}_*dlog} Im{F_{/K}}_*dlog\to 0$ and $0\to coker{F_{/K}}^*\xrightarrow{{F_{/K}}_*dlog} {F_{/K}}_*Z^1(\Omega^{\bullet}_{Y/K}) \xrightarrow{W^{\star}-C_{Y}} \Omega^{1}_{Y'/K} \to 0$ deduced from the $p$-curvature exact sequence (\[subsec:calculus in char p\]) $$0\to \mathbb{G}_{\,m/Y'} \xrightarrow{{F_{/K}}^*} {F_{/K}}_*\mathbb{G}_{\,m/Y} \xrightarrow{{F_{/K}}_*dlog} {F_{/K}}_*Z^1(\Omega^{\bullet}_{Y/K}) \xrightarrow{W^{\star}-C_{Y}} \Omega^{1}_{Y'/K} \to 0,$$ one gets a morphism $H^0(Y',\Omega^{1}_{Y'/K}) \to H^1(Y',coker{F_{/K}}^*\cong Im{F_{/K}}_*dlog) \to H^2(Y',\mathbb{G}_{\,m/Y'})$ which by construction factors through $$H^0(Y',\Omega^{1}_{Y'/K}) \twoheadrightarrow coker H^0(W^{\star}-C_{Y}) \to kerH^2({F_{/K}}^*)\hookrightarrow H^2(Y',\mathbb{G}_{\,m/Y'}),$$ where $H^0(Y',\Omega^{1}_{Y'/K}) \twoheadrightarrow coker H^0(W^{\star}-C_{Y})$ and $kerH^2({F_{/K}}^*)\hookrightarrow H^2(Y',\mathbb{G}_{\,m/Y'})$ are the canonical coker and ker morphisms. Since by [@Hoobler 2.1] the canonical embedding $Br(Y')\hookrightarrow H^2(Y',\mathbb{G}_{\,m/Y'})$ [@BrauerI 2.1] is an isomorphism on the $p$-torsion (:= the kernel of multiplication by $p$) $Br(Y')_p\widetilde{\hookrightarrow} H^2(Y',\mathbb{G}_{\,m/Y'})_p=kerH^2({F_{/K}}^*),$ this leads to a morphism $$\phi_Y:
H^0(Y',\Omega^{1}_{Y'/K}) \twoheadrightarrow coker H^0(W^{\star}-C_{Y}) \xrightarrow{\overline{\phi_Y}} Br(Y')_p \subset Br(Y').$$
\[rmk:other description of phi\]
Here’s another description of $\phi_Y$[@Ogus-Vologodsky Rem. 4.3]. Let $\alpha \in H^0(Y',\Omega^{1}_{Y'/K}),$ then $\phi_Y(\alpha)=[s_{\alpha}^*({F_{Y/K}}_*D_{Y})] \in Br(Y')$ where $Y' \overset{s_{\alpha}}\to T^*(Y'/K)$ is the section of $T^*(Y'/K)/Y'$ corresponding to $\alpha$ (\[subsec:symplectic geometry\]).
It depends functorially on $Y,$ namely
\[lmm:functoriality of phi\]
Let $Z\xrightarrow{f} Y$ be a $K$-morphism of smooth $K$-schemes and let $\alpha \in H^0(Y',\Omega^1_{Y'/K}),$ then $f'^*\phi_Y(\alpha)=\phi_Z((f'^*)^{ad}\alpha)$ where $f'^*$ on the left (resp. on the right) is the pullback of classes in the Brauer group (resp. pullback of forms) by $f'$.
**Proof:** By \[rmk:other description of phi\], $\phi_Z((f'^*)^{ad}\alpha)=[s_{(f'^*)^{ad}\alpha}^*({F_{Z/K}}_*D_{Z})]$ and $[s_{(f'^*)^{ad}\alpha}^*({F_{Z/K}}_*D_{Z})]=[(f'_d \circ Z\times_Ys_{\alpha})^*({F_{Z/K}}_*D_{Z})]$ since $s_{(f'^*)^{ad}\alpha}=f'_d \circ Z\times_Ys_{\alpha}$ (\[subsec:symplectic geometry\]). Moreover by [@BezrBr 3.7], $[{f'_d}^*({F_{Z/K}}_*D_{Z})]=[{f'_{\pi}}^*({F_{Y/K}}_*D_{Y})]$ hence $[(f'_d \circ Z\times_Ys_{\alpha})^*({F_{Z/K}}_*D_{Z})]=[(Z\times_Ys_{\alpha})^*{f'_d}^*({F_{Z/K}}_*D_{Z})]
=[(Z\times_Ys_{\alpha})^*{f'_{\pi}}^*({F_{Y/K}}_*D_{Y})]
=[(f'_{\pi} \circ Z\times_Ys_{\alpha})^*({F_{Y/K}}_*D_{Y})]=[(s_{\alpha}\circ f')^*({F_{Y/K}}_*D_{Y})]=[{f'}^*{s_{\alpha}}^*({F_{Y/K}}_*D_{Y})]
={f'}^*[{s_{\alpha}}^*({F_{Y/K}}_*D_{Y})]=f'^*\phi_Y(\alpha)$ by \[rmk:other description of phi\] and using the equality $f'_{\pi} \circ Z\times_Ys_{\alpha}=s_{\alpha}\circ f'.$\
Further diagram chasing through the cohomology long exact sequences gives control over the kernel of $\phi_Y,$ say Zariski locally,
\[prop:kernel of phi\]
Suppose further that $Y$ is affine. Then there is an exact sequence (compatible with restriction to affine open subsets) $$Pic(Y)\to coker H^0(W^{\star}-C_{Y}) \xrightarrow{\overline{\phi_Y}} Br(Y')_p \to 0.$$
**Proof:** It is a special case of [@Hoobler 1.7].\
For $Y=T^*X$ where X is a smooth equidimensional $K$-scheme, $\phi_{T^*X}$ relates the canonical $1$-form $\theta_{X'}$ on $T^*(X')$ to the class of the Azumaya algebra ${F_{X/K}}_*D_X$ (\[subsec:D char p\]) in $Br(T^*(X')).$ Indeed, note that the pullback of forms $W^*\Omega^1_{X/K} \xrightarrow{W^*}\Omega^1_{X'/K}$ (\[subsec:calculus in char p\]) is an isomorphism [@EGAIV 16.4.5] hence induces an isomorphism $(T^*X)'\to T^*(X'),$ use it to identify $(T^*X)'$ and $T^*(X')$ and denote the resulting $K$-scheme $T^*X',$ then by [@Ogus-Vologodsky prop. 4.4 and 4.2] we have the
\[prop:canonical form maps to algebra of differential operators\]
$\phi_{T^*X}(\theta_{X'})=[{F_{X/K}}_*D_X] \in Br(T^*X').$
Lagrangianity {#sec:lagrangianity}
=============
In this section, we complete the proof of the main theorem \[thm:main thm\].
Namely, the first bound of theorem \[thm:bound\] allows one to construct a smooth compactification of an open dense subset of the $p$-support (prop. \[prop:compactification\]), “uniformly in $p$”, by reducing the problem to characteristic zero. From the description of the canonical form restricted to the $p$-support in terms of the $p$-curvature operator, given in section \[sec:Brauer group and forms\], and the analysis of the latter’s action on the order of poles of differential forms (prop. \[prop:log poles\]), we get that the symplectic form has logarithmic poles along the boundary of the above compactification. We then conclude using Hodge theory in characteristic zero (\[subsec:conclusion\]).
Poles and logarithmic poles {#subsec:log poles}
---------------------------
Let $S$ be a scheme, let $\overline{Y}$ be a smooth $S$-scheme and let $D$ be a closed subscheme of $\overline{Y}$. The closed subscheme $D$ is said to be a divisor with normal crossings relative to $S$ if there is an étale covering $\mathcal{U} \xrightarrow{\pi} \overline{Y}$ and at each point $\upsilon\in\mathcal{U}$ local étale coordinates $\{u_1,...,u_{n_{\upsilon}}\}: V_{\upsilon}\to \mathbb{A}^{n_{\upsilon}}_S$ in which the closed subscheme $\pi^{-1}D:= \mathcal{U}\times_{\overline{Y}}D$ is described by the equation $u_1...u_{r_\upsilon}=0$ for some $r_\upsilon \leq n_\upsilon.$ The notion of divisor with normal crossings relative to the base is stable under étale localization on $\overline{Y}$ and base change. Note that the ideal sheaf $\mathcal{I}\subset \mathcal{O}_{\overline{Y}}$ defining $D$ is invertible and set for an $\mathcal{O}_{\overline{Y}}$-module $\mathcal{F}$ and $n\in \mathbb{Z},$ $\mathcal{F}(nD):=\mathcal{F}\otimes_{\mathcal{O}_{\overline{Y}}}\mathcal{I}^{\otimes_{\mathcal{O}_{\overline{Y}}}(-n)}.$ Then the inclusion $Y \overset{j}\hookrightarrow \overline{Y}$ of the open subscheme $\overline{Y}-D$ is affine and if $m$ is a nonnegative integer there is a canonical embedding ${\Omega}^m_{\overline{Y}/S}(nD) \hookrightarrow j_*{\Omega}^m_{Y/S}$ sending $\eta\otimes t^{\otimes(-n)} \mapsto \eta_{|Y}/t^n$ where $\eta$ and $t$ are respectively a local section of ${\Omega}^m_{\overline{Y}/S}$ and a local equation of $D.$ We use this embedding to view ${\Omega}^m_{\overline{Y}/S}(nD)$ as a subsheaf of $j_*{\Omega}^m_{Y/S}.$ Note that by noetherianity and [@EGAI 1.4.1 c1)], for every local section $\eta$ of $j_*{\Omega}^m_{Y/S}$ there is an $n$ such that $\eta \in{\Omega}^m_{\overline{Y}/S}(nD).$
A local section of $j_*{\Omega}^m_{Y/S}$ which is in ${\Omega}^m_{\overline{Y}/S}(nD)$ is said to have poles of order at most $n$ along $D.$ One defines [@DeligneLNM II §3] a subcomplex $(\Omega^{\bullet}_{\overline{Y}/S}(logD), d)$ of $j_*(\Omega^{\bullet}_{Y/S}, d_{Y/S})$ by the condition that a local section $\eta$ of $j_*\Omega^m_{Y/S}$ belongs to $\Omega^m_{\overline{Y}/S}(logD)$ if and only if $\eta$ and $(j_*d_{Y/S})\eta$ have poles of order at most $1$ along $D.$ It is called the logarithmic de Rham complex of $D \subset\overline{Y}/S$ and a local section of $j_*\Omega^m_{Y/S}$ which is in $\Omega^m_{\overline{Y}/S}(logD)$ is said to have logarithmic poles along $D.$ Moreover in local étale coordinates $\{u_1,...,u_{n_{\upsilon}}\}$ in the neighborhood of a point $\upsilon\in \mathcal{U} \xrightarrow{\pi} \overline{Y}$ where as above $\pi$ is an étale covering and $\pi^{-1}D:= \mathcal{U}\times_{\overline{Y}}D$ is described by the equation $u_1...u_{r_\upsilon}=0$ for some $r_\upsilon \leq n_\upsilon,$ $\Omega^1_{\overline{Y}/S}(logD)$ is free of basis $\{du_1/u_1,...,du_{r_\upsilon}/u_{r_\upsilon},du_{r_\upsilon+1},...,du_{n_{\upsilon}}\},$ hence for all $m,$ $\Omega^m_{\overline{Y}/S}(logD) \cong \Lambda^m_{\mathcal{O}_{\overline{Y}}}\Omega^1_{\overline{Y}/S}(logD)$ and the $\mathcal{O}_{\overline{Y}}$-module $\Omega^m_{\overline{Y}/S}(logD)$ is locally free of finite type.
Compactification of the $p$-supports {#subsec:compactification}
------------------------------------
Let us fix coordinates on $\mathbb{A}^n_{\mathbb{Z}}=spec(\mathbb{Z}[x_1,...,x_n]).$ For any scheme $S$ they induce, compatibly with base change, coordinates on $\mathbb{A}^n_S$ hence on $T^*(\mathbb{A}^n_S),$ as well as an open immersion $T^*(\mathbb{A}^n_S)\overset{r}\hookrightarrow\mathbb{P}^{2n}_S$ (by the Rees construction associated to the increasing filtration by the order of polynomials, see \[subsec:induced filtration over the center\] and \[subsec:bound\]). Moreover if $S$ is of positive characteristic, the choice of coordinates induces base change compatible identifications $\mathbb{A}^{n'}_S = \mathbb{A}^n_S$ and $T^*(\mathbb{A}^{n'}_S)=T^*(\mathbb{A}^n_S).$ Note also that if $S$ is the spectrum of a field of positive characteristic, the open immersion $T^*(\mathbb{A}^{n'}_S)=T^*(\mathbb{A}^n_S)\overset{r}\hookrightarrow\mathbb{P}^{2n}_S$ matches that of \[subsec:bound\].
Provided the above open immersions and identifications, here are some consequences of the first estimate of theorem \[thm:bound\].
Let $S$ be an integral scheme dominant and of finite type over $\mathbb{Z}$ and let $M$ be a coherent left $D_{\mathbb{A}^n_S/S}$-module. Suppose that the fiber of $M$ at the generic point of $S$ is a non zero holonomic left $\mathcal{D}$-module. Then by theorem \[thm:bound\] there is a dense open subset $U$ of $S$ such that for each closed point $u\in U$ and each $z$ generic point of an irreducible component of $p$-supp($M_u$), the degree of the reduced closure $\overline{\{r(z)\}}^{red}$ of the image of $z \in T^*(\mathbb{A}^{n'}_{k(u)})$ by the open immersion $T^*(\mathbb{A}^{n'}_{k(u)})=T^*(\mathbb{A}^n_{k(u)})\overset{r}\hookrightarrow\mathbb{P}^{2n}_{k(u)}$ is bounded above independently of $u$ (by $e(M)$ the $\mathcal{D}$-module multiplicity for the Bernstein filtration of the fiber of $M$ at the generic point of $S$). Hence since the $k(u)$ are perfect, by [@EGAIV 4.6.1, 4.2.8] and the invariance of the Hilbert polynomial under fields base change, [@FGA; @Hilb 2.4 and 2.1(b)] imply that the Hilbert polynomial of $\overline{\{r(z)\}}^{red}$ belongs to a finite set $\Phi$, independent of $u.$
Let $\mathcal{H}_{e(M)}:= \amalg_{P\in\Phi} \mathcal{H}ilb^P_{\mathbb{P}^{2n}_{\mathbb{Z}}}$ where $\mathcal{H}ilb^P_{\mathbb{P}^{2n}_{\mathbb{Z}}}$ is the Hilbert scheme of $\mathbb{P}^{2n}_{\mathbb{Z}}$ of index $P$ [@FGA; @Hilb p.17] and let $\mathcal{Z}_{e(M)}\subset\mathbb{P}^{2n}_{\mathbb{Z}}\times_{spec(\mathbb{Z})}\mathcal{H}_{e(M)}$ be the associated universal flat closed subscheme. Then by [@FGA; @Hilb 3.2] $\mathcal{H}_{e(M)}$ is projective over $spec(\mathbb{Z}),$ in particular it is noetherian.
The first estimate of theorem \[thm:bound\] is used in the proof of the main theorem (\[thm:main thm\]) through the following
\[prop:compactification\]
Let $T$ be a noetherian scheme and let $\mathcal{H}_{e(M),T}:=\mathcal{H}_{e(M)}\times_{spec{\mathbb{Z}}}T.$ Then there are a strictly positive integer $N$ and a finite partition of $\mathcal{H}_{e(M),T}[1/N]$ into locally closed irreducible subsets $\mathcal{S}_i$ such that if the $\mathcal{S}_i$ are endowed with their reduced subschemes structure and $\mathcal{Z}_i:=\mathcal{Z}_{e(M)}\times_{\mathcal{H}_{e(M)}}\mathcal{S}_i,$ then for each $i,$ the generic point of $\mathcal{S}_i$ is of characteristic zero and there are a smooth open subset $\mathcal{U}_i\subset\mathcal{Z}_i^{red}$ surjecting onto the base $\mathcal{S}_i$ and an open $\mathcal{S}_i$-immersion $\mathcal{U}_i\cap T^*(\mathbb{A}^n_{\mathcal{S}_i})=:\mathcal{Y}_i \overset{j}\hookrightarrow \overline{\mathcal{Y}_i}$ into a smooth projective $\mathcal{S}_i$-scheme which is the complement of a divisor $\mathcal{D}_i$ with normal crossings relative to $\mathcal{S}_i.$
Moreover, for each i, let $\theta_i$ be the restriction of the canonical form $\theta_{\mathbb{A}^n_{\mathcal{S}_i}/\mathcal{S}_i}$ on $ T^*(\mathbb{A}^n_{\mathcal{S}_i})$ to $\mathcal{Y}_i.$ Then the $\mathcal{S}_i$ can also be chosen such that $d\theta_i$ has logarithmic poles along $\mathcal{D}_i$ as soon as there is a fiber on which it has logarithmic poles and such that if $d\theta_i$ vanishes on the generic fiber of $\mathcal{Y}_i$ then it vanishes on the whole of $\mathcal{Y}_i.$
Note that by noetherianity, there is a nonnegative integer $m$ such that for each $i$ the restriction $\theta_i$ of the canonical form $\theta_{\mathbb{A}^n_{\mathcal{S}_i}/\mathcal{S}_i}$ on $ T^*(\mathbb{A}^n_{\mathcal{S}_i})$ to $\mathcal{Y}_i$ has poles of order at most $m$ along $\mathcal{D}_i.$
**Proof:** Let us consider the analogous statement, with $\mathcal{H}_{e(M),T}$ replaced by one of its subschemes and note that its fulfillment for $(\mathcal{H}_{e(M),T})^{red}$ implies the proposition. Hence it is enough to prove the statement for all reduced closed subschemes of $\mathcal{H}_{e(M),T}.$ Let $\mathcal{S}$ be such a scheme. By noetherian induction on the set of reduced closed subschemes of $\mathcal{H}_{e(M),T}$ satisfying its conclusion, one may assume that the statement holds for all reduced closed subschemes of $\mathcal{S}$ whose underlying space is $\neq\mathcal{S}.$ There is a strictly positive integer $N$ such that the generic points of the irreducible components of $\mathcal{S}[1/N]$ are of characteristic zero. If $\mathcal{S}[1/N]$ is empty then the statement holds. Suppose thus that $\mathcal{S}[1/N]$ is not empty. Let $z\in\mathcal{S}[1/N]$ be the generic point of an irreducible component and let $\mathcal{Z}_z$ be the fiber of $(\mathcal{Z}_{e(M)}\times_{\mathcal{H}_{e(M)}}\mathcal{S}[1/N])^{red}$ at $z.$ Then $\mathcal{Z}_z\subset\mathbb{P}^{2n}_{k(z)}$ is a scheme of finite type over $k(z),$ a field of characteristic zero, it is reduced by [@EGAIV 8.7.2 a)] and hence contains an open dense $k(z)$-smooth subset $U\subset\mathcal{Z}_z$ by [@EGAIV 17.15.12]. By the resolution of singularities in characteristic zero, there is an open immersion $Y \overset{j}\hookrightarrow \overline{Y}$ of the quasi-projective variety $Y:=U\cap T^*(\mathbb{A}^n_{k(z)})\subset\mathbb{P}^{2n}_{k(z)}$ into a smooth projective scheme $\overline{Y}$ over $k(z)$ which is the complement of a divisor $D$ with normal crossings relative to $k(z).$ Hence by [@EGAIV 8.10.5 and 17.7.8] there is an open affine neighborhood $\mathcal{T}$ of $z,$ which can be chosen, integral by [@EGAI 2.1.9(ii)] since $\mathcal{S}$ is reduced and such that there is a non-empty smooth open subset $\mathcal{U}\subset(\mathcal{Z}_{e(M)}\times_{\mathcal{H}_{e(M)}}\mathcal{T})^{red}$ surjecting onto $\mathcal{T}$ as smooth morphisms are open and an open $\mathcal{T}$-immersion $\mathcal{U}\cap T^*(\mathbb{A}^n_\mathcal{T})=:\mathcal{Y} \overset{j}\hookrightarrow \overline{\mathcal{Y}}$ into a smooth projective $\mathcal{T}$-scheme which is the complement of a divisor $\mathcal{D}$ with normal crossings relative to $\mathcal{T}.$
Let $\theta$ be the restriction of the canonical form $\theta_{\mathbb{A}^n_{\mathcal{T}}/\mathcal{T}}$ on $T^*(\mathbb{A}^n_{\mathcal{T}})$ to $\mathcal{Y}.$ If $d\theta$ vanishes on the generic fiber of $\mathcal{Y}$ then there is a dense open subset $V\subset \mathcal{T}$ on which $d\theta$ vanishes, $d\theta|_{\mathcal{Y}|_V}=0.$ If $d\theta$ does not vanish on the generic fiber of $\mathcal{Y},$ then set $V:=\mathcal{T}.$ There is also a dense open subset $W\subset V$ such that $d\theta|_{\mathcal{Y}|_W}$ has logarithmic poles along $\mathcal{D}|_W$ as soon as there is a fiber on which it has logarithmic poles. Indeed, since having logarithmic poles is an étale local condition, one may assume that $(\overline{\mathcal{Y}}|_V,\mathcal{D}|_V)=(\mathbb{A}^n_V=spec(V[y_1,...,y_n]),\{y_1...y_r=0\})$ for some $0\leq r\leq n$ and one concludes by considering the vanishing loci in $V$ of the remainders of divisions by the $y_l$’s.
By noetherian induction the statement holds for the reduced closed subscheme of $\mathcal{S}$ whose underlying space is $\mathcal{S}-W.$ Hence combining with the above on $W,$ the statement holds for $\mathcal{S}.$ This proves the proposition.\
Action of the $p$-curvature operator on the order of poles
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Let $\overline{Y}$ be a smooth scheme over a field $k$ of positive characteristic $p,$ let $D$ be a divisor with normal crossings relative to $k$ (\[subsec:log poles\]) and let $Y \overset{j}\hookrightarrow \overline{Y}$ be the inclusion of the open subscheme $\overline{Y}-D.$ Base changing by the Frobenius endomorphism of $k,$ one sees that the closed subscheme $D'\subset \overline{Y}'$ is a divisor with normal crossings relative to $k$ and that $Y' \overset{j'}\hookrightarrow \overline{Y}'$ is the open subscheme $\overline{Y}'-D'.$ Suppose that $Y$ is equidimensional and let $\mathcal{I}m(W^{\star}-C_Y)$ be the image of the morphism $Z^1({F_{/k}}_*\Omega^{\bullet}_{Y/k}) \xrightarrow{W^{\star}-C_Y} \Omega^{1}_{Y'/k}$ of abelian sheaves on $Y'$(\[subsec:calculus in char p\]).
Then there’s the following inclusion of abelian subsheaves of $j'_*\Omega^{2}_{Y'/k}$
\[prop:log poles\]
$$d({\Omega}^1_{\overline{Y}'/k}((p-1)D') \cap j'_*\mathcal{I}m(W^{\star}-C_Y)) \subset \Omega^2_{\overline{Y}'/k}(logD')$$
**Proof:** Let $\eta$ be a local section of ${\Omega}^1_{\overline{Y}'/k}((p-1)D') \cap j'_*\mathcal{I}m(W^{\star}-C_Y) \subset j'_*\Omega^{1}_{Y'/k}$ and let $\mathcal{U} \xrightarrow{\pi} \overline{Y}$ be an étale covering as in the definition of a divisor with normal crossings (\[subsec:log poles\]). Since the pullback $\pi'^*\eta$ is a local section of $${\Omega}^1_{\mathcal{U}'/k}((p-1)\pi'^{-1}D') \cap {j'_{\mathcal{U}}}_*\mathcal{I}m(W^{\star}-C_\mathcal{U})$$ where $j_{\mathcal{U}}$ is the open immersion $\mathcal{U}-\pi^{-1}D \hookrightarrow \mathcal{U}$ and since being a section of $\Omega^2_{\overline{Y}'/k}(logD')$ is an étale local condition, one may assume that at each point $y\in\overline{Y}$ there are local étale coordinates $\{y_1,...,y_n\}: V_y\to \mathbb{A}^n_k$ in which the closed subscheme $D$ is described by the equation $y_1...y_r=0$ for some $r \leq n,$ $n$ and $r$ depending on $y.$
Let us prove that $${\Omega}^1_{\overline{Y}'/k}((p-1)D') \cap j'_*\mathcal{I}m(W^{\star}-C_Y) \subset j'_*B^1\Omega^\bullet_{Y'/k} + \Omega^1_{\overline{Y}'/k}(logD')$$ where $B^1\Omega^{\bullet}_{Y'/k}:= im (\mathcal{O}_{Y'} \xrightarrow{d} \Omega^{1}_{Y'/k})$ are the exact 1-forms. It implies the proposition. Let $\eta$ be a local section of $j'_*\mathcal{I}m(W^{\star}-C_Y)$, by lemma \[lmm:j and Im commute\] below the canonical inclusion $\mathcal{I}mj'_*(W^{\star}-C_Y) \hookrightarrow j'_*\mathcal{I}m(W^{\star}-C_Y)$ is an isomorphism hence locally there is a section $\zeta$ of $j'_*Z^1({F_{/k}}_*\Omega^{\bullet}_{Y/k})$ such that $\eta=j'_*(W^{\star}-C_Y)\zeta.$ Moreover if a local section $\zeta$ of $j'_*Z^1({F_{/k}}_*\Omega^{\bullet}_{Y/k}) \subset j'_*{F_{/k}}_*\Omega^1_{Y/k}={F_{/k}}_*j_*\Omega^1_{Y/k}$ does not belong to ${F_{/k}}_*{\Omega}^1_{\overline{Y}/k}((p-1)D)$ then as $C_Y$ is $p^{-1}$-linear and sends $Z^1({F_{/k}}_*\Omega^{\bullet}_{\overline{Y}/k})$ to $\Omega^1_{\overline{Y}'/k},$ $\eta=j'_*(W^{\star}-C_Y)\zeta$ does not belong to ${\Omega}^1_{\overline{Y}'/k}((p-1)D')$ [@Illusie; @dR-W 0.2.5.4]. Note also that the étale coordinates $\{y_1,...,y_n\}: V_y\to \mathbb{A}^n_k$ above determine a splitting on $Y'\cap V_y'$ of the canonical short exact sequence $$0\to B^1({F_{/k}}_*\Omega^{\bullet}_{Y/k})\to Z^1({F_{/k}}_*\Omega^{\bullet}_{Y/k})\xrightarrow{C_Y}{\Omega}^1_{Y'/k} \to 0$$ given in terms of local sections by ${\Omega}^1_{Y'/k} \to Z^1({F_{/k}}_*\Omega^{\bullet}_{Y/k}): \Sigma_{i=1}^{i=n} a_idy'_i\mapsto \Sigma_{i=1}^{i=n} {F_{/k}}^*(a_i){y_i}^{p-1}dy_i.$ It induces a splitting on $V_y'$ of the direct image $j'_*$ of the above short exact sequence and hence a local section $\zeta$ of $j'_*Z^1({F_{/k}}_*\Omega^{\bullet}_{Y/k})|_{V_y'}$ may uniquely be written as a sum $\zeta=\beta+ \Sigma_{i=1}^{i=n} {F_{/k}}^*(a_i){y_i}^{p-1}dy_i$ where $\beta$ is a local section of $j'_*B^1({F_{/k}}_*\Omega^{\bullet}_{Y/k})$ and the $a_i$’s are local sections of $j'_*\mathcal{O}_{Y'}.$ Note finally that if $\zeta$ is a section of ${F_{/k}}_*{\Omega}^1_{\overline{Y}/k}((p-1)D)$ then so is $\Sigma_{i=1}^{i=n} {F_{/k}}^*(a_i){y_i}^{p-1}dy_i$ and that if $\Sigma_{i=1}^{i=n} {F_{/k}}^*(a_i){y_i}^{p-1}dy_i$ is not a section of ${F_{/k}}_*{\Omega}^1_{\overline{Y}/k}(1D)$ then it is not a section of ${F_{/k}}_*{\Omega}^1_{\overline{Y}/k}((p-1)D)$ either. Indeed the proofs of both assertions reduce to $(\overline{Y},D)=(\mathbb{A}^n_k=spec(k[y_1,...,y_n]),\{y_1...y_r=0\})$ with étale coordinates $\{y_1,...,y_n\}$ since the pullback by $\{y_1',...,y_n'\}: V_y'\to \mathbb{A}^{n'}_k$ preserves the splitting and the order of poles. There the second assertion is a direct consequence of the factoriality of rings of polynomials with coefficients in a field while the first can be proved as follows.
Let $\zeta=dg + \Sigma_{i=1}^{i=n} {F_{/k}}^*(a_i){y_i}^{p-1}dy_i$ be the above decomposition of a closed form on an affine open $\{f\neq 0\},$ for a rational function $g$ and a polynomial $f.$ Suppose that $\zeta\in {F_{/k}}_*{\Omega}^1_{\overline{Y}/k}((p-1)D).$ Then $\Sigma_{i=1}^{i=n} {F_{/k}}^*(a_i){y_i}^{p-1}dy_i\in {F_{/k}}_*{\Omega}^1_{\overline{Y}/k}((p-1)D).$ Indeed, multiplying by a high enough power of $f^p,$ one may assume that $\zeta$ is a global section. Moreover by uniqueness of the decomposition and the corresponding splitting for forms without poles, one may also assume that $dg=\Sigma_{i=1}^{i=n}\partial_i(g)dy_i$ and $\Sigma_{i=1}^{i=n} {F_{/k}}^*(a_i){y_i}^{p-1}dy_i$ are sections of ${F_{/k}}_*{\Omega}^1_{\overline{Y}/k}(pD).$ Further using uniqueness of the decomposition and the splitting for forms without poles, and multiplying by $(y_1...y_r)^p,$ if $\Sigma_{i=1}^{i=n} {F_{/k}}^*(a_i){y_i}^{p-1}dy_i$ was not a section of $ {F_{/k}}_*{\Omega}^1_{\overline{Y}/k}((p-1)D)$ then there would be an $i$ such that $y_1...y_r$ divides $\partial_i(g)+{F_{/k}}^*(a_i){y_i}^{p-1}$ but not ${F_{/k}}^*(a_i){y_i}^{p-1},$ where $g$ and $a_i$ are polynomials. In particular there should be $1\leq j\leq r$ such that $j\neq i$ and $y_j$ divides $\partial_i(g)+{F_{/k}}^*(a_i){y_i}^{p-1}$ but not ${F_{/k}}^*(a_i){y_i}^{p-1}.$ Expressing as polynomials in $y_j$ and considering the degree zero terms would provide an equality $\partial_i(g_0)+{F_{/k}}^*((a_i)_0){y_i}^{p-1}=0$ with $(a_i)_0\neq 0.$ Since this cannot happen in characteristic $p,$ the assertion holds.
Hence if $\zeta=\beta+ \Sigma_{i=1}^{i=n} {F_{/k}}^*(a_i){y_i}^{p-1}dy_i$ belongs to ${F_{/k}}_*{\Omega}^1_{\overline{Y}/k}((p-1)D)$ then $\Sigma_{i=1}^{i=n} {F_{/k}}^*(a_i){y_i}^{p-1}dy_i$ is a section of ${F_{/k}}_*{\Omega}^1_{\overline{Y}/k}(1D)\cap j'_*Z^1({F_{/k}}_*\Omega^{\bullet}_{Y/k}) \subset Z^1({F_{/k}}_*\Omega^{\bullet}_{\overline{Y}/k}(log D)).$ Thus $$\eta=j'_*(W^{\star}-C_Y)\zeta=j'_*(W^{\star}-C_Y)(\beta+ \Sigma_{i=1}^{i=n} {F_{/k}}^*(a_i){y_i}^{p-1}dy_i)$$ $$=j'_*(W^{\star})\beta+ j'_*(W^{\star}-C_Y)(\Sigma_{i=1}^{i=n} {F_{/k}}^*(a_i){y_i}^{p-1}dy_i)$$ is a section of $j'_*B^1\Omega^\bullet_{Y'/k} + \Omega^1_{\overline{Y}'/k}(logD'),$ as $C_Y$ preserves forms with logarithmic poles [@Katz 7.2]. This concludes the proof of the proposition.\
\[lmm:j and Im commute\]
The canonical inclusion $\mathcal{I}m(j'_*(W^{\star}-C_Y)) \hookrightarrow j'_*\mathcal{I}m(W^{\star}-C_Y)$ is an isomorphism.
**Proof:** The exact sequence of abelian sheaves on $Y'$ (\[subsec:calculus in char p\]) $$0\to \mathcal{O}_{Y'}^* \xrightarrow{{F_{/k}}^*} {F_{/k}}_*\mathcal{O}_Y^* \xrightarrow{{F_{/k}}_*dlog} Z^1({F_{/k}}_*\Omega^{\bullet}_{Y/k}) \xrightarrow{W^{\star}-C_Y} \Omega^{1}_{Y'/k}$$ provides two short exact sequences $$0\to coker{F_{/k}}^* \xrightarrow{{F_{/k}}_*dlog} Z^1({F_{/k}}_*\Omega^{\bullet}_{Y/k}) \xrightarrow{W^{\star}-C_Y} \mathcal{I}m(W^{\star}-C_Y) \to 0$$ and $$0\to \mathcal{O}_{Y'}^* \xrightarrow{{F_{/k}}^*} {F_{/k}}_*\mathcal{O}_Y^* \to coker{F_{/k}}^* \to 0.$$ The associated long exact sequence to the first one $$\begin{gathered}
0\to j'_*coker{F_{/k}}^* \xrightarrow{j'_*{F_{/k}}_*dlog} j'_*Z^1({F_{/k}}_*\Omega^{\bullet}_{Y/k}) \xrightarrow{j'_*(W^{\star}-C_Y)} j'_*\mathcal{I}m(W^{\star}-C_Y) \to R^1j'_*coker{F_{/k}}^*\to...\end{gathered}$$ shows that the lemma follows from the vanishing of $R^1j'_*coker{F_{/k}}^*$ which in turn by the long exact sequence associated to the second short exact sequence $$...\to R^1j'_*\mathcal{O}_{Y'}^* \to R^1j'_*({F_{/k}}_*\mathcal{O}_Y^*) \to R^1j'_*coker{F_{/k}}^* \to R^2j'_*\mathcal{O}_{Y'}^*\to...$$ would follow from the vanishings of $R^1j'_*({F_{/k}}_*\mathcal{O}_Y^*)$ and $R^2j'_*\mathcal{O}_{Y'}^*.$ Since the direct image ${F_{/k}}_*$ preserves flasque sheaves and is exact as $F_{/k}$ is a homeomorphism, $R^q{F_{/k}}_*(G)=0$ for all abelian sheaves $G$ and all $q>0$ by [@HartshorneAG III 8.3] and $R^1j'_*({F_{/k}}_*\mathcal{O}_Y^*) \cong R^1(j'_*\circ {F_{/k}}_*)(\mathcal{O}_Y^*)=R^1({F_{/k}}_* \circ j_*)(\mathcal{O}_Y^*) \cong {F_{/k}}_*R^1j_*(\mathcal{O}_Y^*).$ Hence by [@EGAIV 17.15.2 and 21.11.1] both $R^1j'_*({F_{/k}}_*\mathcal{O}_Y^*) \cong {F_{/k}}_*R^1j_*(\mathcal{O}_Y^*)$ and $R^2j'_*\mathcal{O}_{Y'}^*$ vanish by \[lmm:vanishing of higher direct images\] below, thus proving the lemma.\
\[lmm:vanishing of higher direct images\]
Let $U \overset{j}\hookrightarrow Y$ be an open immersion. Suppose that $Y$ is a locally factorial noetherian scheme. Then $R^qj_*(\mathcal{O}_U^*)=0$ for all $q>0.$
**Proof:** By [@HartshorneAG III 8.1] $R^qj_*(\mathcal{O}_U^*)$ is the abelian sheaf associated to the presheaf $V \mapsto H^q(U\cap V,\mathcal{O}_{U\cap V}^*),$ $V$ open in $Y.$ Since by \[lmm:vanishing of Zariski cohomology\] $H^q(U\cap V,\mathcal{O}_{U\cap V}^*)=0$ for all $q\geq 2,$ $R^qj_*(\mathcal{O}_U^*)=0$ for all $q\geq 2.$
For $q=1,$ $R^1j_*(\mathcal{O}_U^*)$ is the abelian sheaf associated to the presheaf $V \mapsto H^1(U\cap V,\mathcal{O}_{U\cap V}^*)\cong Pic(U\cap V),$ $V$ open in $Y.$ Let $\mathcal{L}_{U\cap V} \in Pic(U\cap V).$ By [@EGAIV 21.6.11] it extends to an invertible sheaf $\mathcal{L}_V \in Pic(V)$ hence $\mathcal{L}_{U\cap V}$ is trivial on the trace of an open covering of $V$ trivializing $\mathcal{L}_V$ and in particular the corresponding section of the associated sheaf $R^1j_*(\mathcal{O}_U^*)$ is $0.$ Thus $R^1j_*(\mathcal{O}_U^*)=0,$ finishing the proof.\
Conclusion {#subsec:conclusion}
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Recall the statement \[thm:main thm\]
Let $S$ be an integral scheme dominant and of finite type over $\mathbb{Z}$, let $X$ be a smooth $S$-scheme of relative dimension $n$ and let $M$ be a coherent left $D_{X/S}$-module. Suppose that the fiber of $M$ at the generic point of $S$ is a holonomic left $\mathcal{D}$-module. Then there is a dense open subset $U$ of $S$ such that the $p$-support of the fiber of $M$ at each closed point $u$ of $U$ is a lagrangian subscheme of $(T^*(X'_u),\omega_{X'_u})$.
**Proof:** By the remark \[rmk:generic fiber zero\], we may assume that the fiber of $M$ at the generic point of $S$ is non zero. Hence by theorem \[thm:purity\], there is a dense open subset $U_a$ of $S$ such that the $p$-support of the fiber of $M$ at each closed point $u$ of $U_a$ is equidimensional of dimension $n.$ Since by \[subsec:reduction to affine space\] one may further suppose that $X/S=\mathbb{A}^n_S/S,$ theorem \[thm:splitting in affine space case\] implies that there is a dense open subset $U_b$ of $U_a$ such that for each closed point $u\in U_b$ and each $z$ generic point of an irreducible component of $p$-supp($M_u$) the Azumaya algebra ${F_{\mathbb{A}^n/k(u)}}_*D_{\mathbb{A}^n_{k(u)}}$ on $T^*(\mathbb{A}^{n'}_{k(u)})$ splits on $(\overline{\{z\}}^{red})^{reg}.$ Therefore, $k(u)$ being perfect (\[subsec:The statement\]), by \[prop:canonical form maps to algebra of differential operators\], \[lmm:functoriality of phi\] and \[prop:kernel of phi\] the restriction of the canonical form $\theta_{\mathbb{A}^{n'}_{k(u)}}$ to each of the $(\overline{\{z\}}^{red})^{reg}$ is a section of $\mathcal{I}m(W^{\star}-C_{(\overline{\{z\}}^{red})^{reg}}),$ where we identified $(\overline{\{z\}}^{red})^{reg}$ and $(\overline{\{z\}}^{red})^{reg'}$ by perfection of $k(u).$
Moreover, by \[subsec:compactification\] and proposition \[prop:compactification\], there is a dense open subset $U_c$ of $U_b$ such that for each closed point $u\in U_c$ and each $z$ generic point of an irreducible component of $p$-supp($M_u$), there are a smooth open dense subscheme $Y_z$ of $(\overline{\{z\}}^{red})^{reg}$ and an open immersion $Y_z \overset{j}\hookrightarrow \overline{Y_z}$ into a smooth projective scheme, which is the complement of a divisor $D_z$ with normal crossings relative to $k(u).$ In addition there is a nonnegative integer $m,$ independent of $u,$ such that the restriction of the canonical form to $Y_z$ has poles of order at most $m$ along $D_z.$ Hence inverting all primes $\leq m,$ one may suppose that $m\leq char(k(u))-1$ and thus by proposition \[prop:log poles\], $\omega_{\mathbb{A}^{n'}_{k(u)}}|_{Y_z}:=d\theta_{\mathbb{A}^{n'}_{k(u)}}|_{Y_z}$ has logarithmic poles along $D_z.$
Actually by proposition \[prop:compactification\], there is a finite set $\Xi$ such that for each $i\in\Xi$ there are an integral scheme $\mathcal{S}_i$ whose generic point is of characteristic zero, a smooth $\mathcal{S}_i$-scheme $\mathcal{Y}_i,$ an open immersion $\mathcal{Y}_i \overset{j}\hookrightarrow \overline{\mathcal{Y}_i}$ into a smooth projective $\mathcal{S}_i$-scheme which is the complement of a divisor $\mathcal{D}_i$ with normal crossings relative to $\mathcal{S}_i$ and a relative 1-form $\theta_i \in {\Omega}^1_{\overline{\mathcal{Y}_i}/\mathcal{S}_i}(m\mathcal{D}_i)$ such that for each closed point $u\in U_c$ and each $z$ generic point of an irreducible component of $p$-supp($M_u$), there is $i(z)\in\Xi$ such that $Y_z \overset{j}\hookrightarrow \overline{Y_z}$ and $\theta_{\mathbb{A}^{n'}_{k(u)}}|_{Y_z}$ are deduced from $\mathcal{Y}_{i(z)} \overset{j}\hookrightarrow \overline{\mathcal{Y}_{i(z)}}$ and $\theta_{i(z)},$ base changing by a $k(u)$-point of $\mathcal{S}_{i(z)}.$ Moreover by construction of the $\mathcal{S}_i$’s (\[prop:compactification\]), if $d\theta_{\mathbb{A}^{n'}_{k(u)}}|_{Y_z}\in \Omega^2_{\overline{Y_z}/k}(logD_z)$ then $d\theta_{i(z)}\in {\Omega}^2_{\overline{\mathcal{Y}_i}/\mathcal{S}_i}(log\mathcal{D}_i),$ hence in particular $d\theta_0\in {\Omega}^2_{\overline{\mathcal{Y}}_0}(logD_0),$ where here and below, the subscript 0 denotes restriction to the generic fiber. As the generic fiber is over a field of characteristic zero, the canonical inclusion ${\Omega}^\bullet_{\overline{\mathcal{Y}}_0}(logD_0)\subset (j_0)_*{\Omega}^\bullet_{\mathcal{Y}_0}$ is a quasi-isomorphism [@Hodge; @II 3.1.8], implying that the class of $d\theta_0$ in the hypercohomology of the logarithmic de Rham complex is zero. Hence $d\theta_0$ vanishes by the degeneracy at $E_1$ of the logarithmic Hodge to de Rham spectral sequence [@Hodge; @II 3.2.13 (ii) and 3.2.14] and so, by construction of the $\mathcal{S}_i$’s (\[prop:compactification\]), the symplectic form $\omega_{\mathbb{A}^{n'}_{k(u)}}:=d\theta_{\mathbb{A}^{n'}_{k(u)}}$ vanishes on the open dense subset $Y_z$ of $(\overline{\{z\}}^{red})^{reg}.$ Thus by the above there is a dense open subset $U$ of $U_c\subset U_a$ such that for each closed point $u$ of $U,$ the symplectic form vanishes on a dense open subset of $p$-supp($M_u$). Since by definition of $U_a$, $p$-supp($M_u$) is equidimensional of dimension $n$ for all closed points $u\in U_a,$ this concludes the proof of the theorem.\
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Exposé XV in SGA 5, LNM 589.
[^1]: [email protected]
[^2]: The $p$ of $p$-support is the p of positive characteristic, prime characteristic and $p$-curvature.
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[ **The design, construction and performance of the\
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abstract: 'The commutator $[a,b] = ab - ba$ in a free Zinbiel algebra (dual Leibniz algebra) is an anticommutative operation which satisfies no new relations in arity 3. Dzhumadildaev discovered a relation $T(a,b,c,d)$ which he called the tortkara identity and showed that it implies every relation satisfied by the Zinbiel commutator in arity 4. Kolesnikov constructed examples of anticommutative algebras satisfying $T(a,b,c,d)$ which cannot be embedded into the commutator algebra of a Zinbiel algebra. We consider the tortkara triple product $[a,b,c] = [[a,b],c]$ in a free Zinbiel algebra and use computer algebra to construct a relation $TT(a,b,c,d,e)$ which implies every relation satisfied by $[a,b,c]$ in arity 5. Thus, although tortkara algebras are defined by a cubic binary operad (with no Koszul dual), the corresponding triple systems are defined by a quadratic ternary operad (with a Koszul dual). We use computer algebra to construct a relation in arity 7 satisfied by $[a,b,c]$ which does not follow from the relations of lower arity. It remains an open problem to determine whether there are further new identities in arity $\ge 9$.'
address: 'Department of Mathematics and Statistics, University of Saskatchewan, Canada'
author:
- Murray Bremner
title: On tortkara triple systems
---
[^1]
Introduction
============
Leibniz algebras were first studied by Blokh [@Blokh1965] under the name D-algebras and were given their present name by Loday [@Loday1993]. A vector space $L$ over a field $\mathbb{F}$ with a bilinear multiplication $L \times L \to L$ denoted $(a,b) \mapsto [a,b]$ is called a (left) Leibniz algebra if it satisfies the (left) derivation identity, $$[a,[b,c]] \equiv [[a,b],c] + [b,[a,c]],$$ where the symbol $\equiv$ indicates that the equation holds for all values of the arguments. The corresponding symmetric operad is quadratic and hence possesses a Koszul dual which was first studied by Loday [@Loday1995]. Algebras over the Zinbiel (dual Leibniz operad) are defined by the (right) Zinbiel identity, $$(ab)c \equiv a(bc) + a(cb).$$ This identity first appeared in work of Schützenberger [@Schutzenberger1959 equation (S2), p. 18] on combinatorics of free Lie algebras. In every Zinbiel algebra the anticommutator $a \circ b = ab + ba$ is associative (and commutative): $$\begin{aligned}
&
( a \circ b ) \circ c - a \circ ( b \circ c )
\\[-1pt]
&=
(ab)c + (ba)c + c(ab) + c(ba) - a(bc) - a(cb) - (bc)a - (cb)a
\\[-1pt]
&=
a(bc) + a(cb) + b(ac) + b(ca) + \cdots - b(ca) - b(ac) - c(ba) - c(ab)
= 0.\end{aligned}$$ Zinbiel algebras have therefore been called precommutative algebras; see Aguiar [@Aguiar2000]. Zinbiel algebras are commutative dendriform (preassociative) algebras. Kawski [@Kawski2009] discusses applications of Zinbiel algebras in control theory. Livernet [@Livernet1998] studies a generalization of rational homotopy based on Leibniz and Zinbiel algebras.
An important feature shared by both the Leibniz and Zinbiel operads is that they are regular in the sense that in every arity $n$ the homogeneous component is isomorphic to the regular $S_n$-module $\mathbb{F} S_n$. Indeed, both the Leibniz and Zinbiel identities can be interpreted as directed rewrite rules, $$[[a,b],c] \longmapsto [a,[b,c]] - [b,[a,c]],
\qquad\quad
(ab)c \longmapsto a(bc) + a(cb).$$ These rules are the inductive step in the proof that every monomial in either operad can be rewritten as a linear combination of right-normed monomials (formed by using only left multiplications). Thus in each arity we require only one association type, and so any (multilinear) monomial in any association type can be identified with a linear combination of permutations of its arguments. Regular parameterized one-relation operads have recently been classified by the author and Dotsenko [@BD2017].
In both the Leibniz and the Zinbiel operads, the generating binary operation has no symmetry (it is neither commutative nor anticommutative), so it is of interest to consider the same operads with a different set of generators, namely the commutator and anticommutator. This process is called polarization and has been studied in detail by Markl & Remm [@MR2006]. Basic results on the polarizations of the Leibniz and Zinbiel operads are found in two papers by Dzhumadildaev [@Askar2007; @Askar2008]. We make no further comment on the Leibniz case, since our focus in this paper is on the Zinbiel operad, and in particular its binary and ternary suboperads generated by the Zinbiel commutator $[a,b] = ab-ba$ (the tortkara product) and the iterated Zinbiel commutator (the tortkara triple product): $$\label{ttp}
\begin{array}{l@{\,}l}
[a,b,c] = [[a,b],c]
&
= (ab)c - (ba)c - c(ab) + c(ba)
\\[1mm]
&
= a(bc) + a(cb) - b(ac) - b(ca) - c(ab) + c(ba).
\end{array}$$ The normal form of $[a,b,c]$ includes all permutations of $a, b, c$ in the association type $\ast(\ast\ast)$; the sign pattern reflects the fact that both sides alternate in $a, b$.
Dzhumadildaev [@Askar2007] showed that the Zinbiel commutator does not satisfy any new (quadratic) relation in arity 3, but does satisfy new (cubic) relations of arity 4, all of which are consequences of what is known as the tortkara identity: $$\label{tortkaraidentity}
(ab)(cd) + (ad)(cb) \equiv J(a,b,c)d + J(a,d,c)b,$$ where $J(a,b,c) = (ab)c + (bc)a + (ca)b$. For further results on identical relations for Zinbiel algebras, and the speciality problem for tortkara algebras, see Dzhumadildaev & Tulenbaev [@DT2005], Naurazbekova & Umirbaev [@NU2010], and Kolesnikov [@Kolesnikov2016]. Since the suboperad of the Zinbiel operad generated by the commutator is cubic, it is natural ask whether the operad of the corresponding triple systems is quadratic.
A similar case is that of Jordan algebras, where the original binary operad is defined by a commutative product $a \circ b$ satisfying the cubic Jordan identity, which (if the characteristic of $\mathbb{F}$ is not 2) is equivalent to the multilinear identity, $$\begin{array}{l}
((a \circ b) \circ c) \circ d + ((b \circ d) \circ c) \circ a + ((d \circ a) \circ c) \circ b
\equiv {}
\\[1pt]
(a \circ b) \circ (c \circ d) + (b \circ d) \circ (c \circ a) + (d \circ a) \circ (c \circ b).
\end{array}$$ On the other hand, the Jordan triple product is defined by the following expression, which reduces to $abc + cba$ in any special Jordan algebra: $$\{a,b,c\} = \tfrac12 \big( ( a \circ b ) \circ c + a \circ ( b \circ c ) - b \circ ( a \circ c ) \big).$$ This product satisfies a twisted form of the ternary derivation identity: $$\{a,b,\{c,d,e\}\} \equiv \{\{a,b,c\},d,e\} - \{c,\{b,a,d\},e\} + \{c,d,\{a,b,e\}\}.$$ For further information on Jordan algebras and triple systems, we refer to McCrimmon [@McCrimmon2004] and Meyberg [@Meyberg1972]; see also Loos & McCrimmon [@LM1977].
In this paper we explain how computer algebra may be used to verify that the tortkara triple product satisfies a (quadratic) relation of arity 5 which is not a consequence of the skewsymmetry in arity 3. For the $S_5$-submodule of all such relations we determine an explicit generator (Theorem \[theorem5\]) and the decomposition into irreducible representations (Corollary \[corollary5\]). We then extend these computations to arity 7, and verify that there exist relations in seven variables for the tortkara triple product which are new in the sense that they do not follow from the known relations of lower arity. We determine an explicit new relation, which however does not generate all the new relations, and the decomposition into irreducible representations of the $S_7$-submodule of all new relations (Theorem \[theorem7\]). It is an open question whether there exist further new relations in arity $\ge 9$.
Although this paper is written from the point of view of algebraic operads, we require very little background in that topic; the standard reference for the theoretical aspects is Loday & Vallette [@LV2012], and for algorithmic methods the reader may refer to the author & Dotsenko [@BD2016].
Relations of arity 5
====================
We write $\mathbf{Zinb}$ for the symmetric operad generated by one binary operation denoted $ab$ with no symmetry satisfying the Zinbiel relation: $$\label{zinbiel}
(ab)c \equiv a(bc) + a(cb).$$ We write $\mathbf{TTS}$ (tortkara triple system) for the suboperad of $\mathbf{Zinb}$ generated by the tortkara triple product . The operad $\mathbf{TTS}$ is a quotient of the free symmetric operad $\mathbf{SkewTS}$ generated by one ternary operation $[a,b,c]$ satisfying $$[a,b,c] + [b,a,c] \equiv 0.$$ We use the same symbol $[a,b,c]$ for the generators of both $\mathbf{TTS}$ and $\mathbf{SkewTS}$.
\[skewbasis5\] A basis for the homogeneous space $\mathbf{SkewTS}(5)$ of arity 5 consists of the following 90 multilinear monomials where $\sigma \in S_5$ acts on $\{ a,b,c,d,e \}$:
- 60 monomials $[[a^\sigma\!,b^\sigma\!,c^\sigma],d^\sigma\!,e^\sigma]$ where $a^\sigma \prec b^\sigma$ in lex order, and
- 30 monomials $[a^\sigma\!,b^\sigma\!,[c^\sigma\!,d^\sigma\!,e^\sigma]]$ where $a^\sigma \prec b^\sigma$ and $c^\sigma \prec d^\sigma$ in lex order.
Immediate.
\[zinbielbasis5\] A basis for the homogeneous space $\mathbf{Zinb}(5)$ of arity 5 consists of the 120 multilinear monomials $a^\sigma ( b^\sigma ( c^\sigma ( d^\sigma e^\sigma ) ) )$ where $\sigma \in S_5$ acts on $\{ a,b,c,d,e \}$; hence we may identify $\mathbf{Zinb}(5)$ with the regular $S_5$-module $\mathbb{F}S_5$.
Repeated application of the Zinbiel relation as the rewrite rule $$(xy)z \,\longmapsto\, x(yz) + x(zy),$$ allows us to express any nonassociative monomial as a linear combination of right-normed monomials. There are 14 binary association types in arity 5: $$\begin{array}{lllll}
(((\ast\ast)\ast)\ast)\ast, &\quad
((\ast(\ast\ast))\ast)\ast, &\quad
((\ast\ast)(\ast\ast))\ast, &\quad
(\ast((\ast\ast)\ast))\ast, &\quad
(\ast(\ast(\ast\ast)))\ast,
\\
((\ast\ast)\ast)(\ast\ast), &\quad
(\ast(\ast\ast))(\ast\ast), &\quad
(\ast\ast)((\ast\ast)\ast), &\quad
(\ast\ast)(\ast(\ast\ast)),
\\
\ast(((\ast\ast)\ast)\ast), &\quad
\ast((\ast(\ast\ast))\ast), &\quad
\ast((\ast\ast)(\ast\ast)), &\quad
\ast(\ast((\ast\ast)\ast)), &\quad
\ast(\ast(\ast(\ast\ast))).
\end{array}$$ We present explicit formulas for the normal forms of these association types with the identity permutation of the arguments $a, b, c, d, e$; in the first nine cases, the sum is over all permutations $\sigma \in S_4$ acting on $\{ a, b, c, d \}$; in the next three cases, the sum is over all permutations $\tau \in S_3$ acting on $\{ c, d, e \}$; the notation $x \prec y$ indicates that $x$ must appear to the left of $y$: $$\begin{aligned}
{2}
&
(((ab)c)d)e = a \sum b^\sigma ( c^\sigma ( d^\sigma e^\sigma ) ),
&\qquad
&
((a(bc))d)e = a \sum_{b \prec c} b^\sigma ( c^\sigma ( d^\sigma e^\sigma ) ),
\\
&
((ab)(cd))e = a \sum_{c \prec d} b^\sigma ( c^\sigma ( d^\sigma e^\sigma ) ),
&\qquad
&
(a((bc)d))e = a \sum_{b \prec c, \, b \prec d} b^\sigma ( c^\sigma ( d^\sigma e^\sigma ) ),
\\
&
(a(b(cd)))e = a \sum_{b \prec c \prec d} b^\sigma ( c^\sigma ( d^\sigma e^\sigma ) ),
&\qquad
&
((ab)c)(de) = a \sum_{d \prec e} b^\sigma ( c^\sigma ( d^\sigma e^\sigma ) ),
\\
&
(a(bc))(de) = a \sum_{b \prec c, \, d \prec e} b^\sigma ( c^\sigma ( d^\sigma e^\sigma ) ),
&\qquad
&
(ab)((cd)e) = a \sum_{c \prec d, \, c \prec e} b^\sigma ( c^\sigma ( d^\sigma e^\sigma ) ),
\\
&
(ab)(c(de)) = a \sum_{c \prec d \prec e} b^\sigma ( c^\sigma ( d^\sigma e^\sigma ) ),
&\qquad
&
a(((bc)d)e) = a \big( b \sum_{\tau \in S_3} ( c^\tau ( d^\tau e^\tau ) ) \big),
\\
&
a((b(cd))e) = a \big( b \sum_{c \prec d} ( c^\tau ( d^\tau e^\tau ) ) \big),
&\qquad
&
a((bc)(de)) = a \big( b \sum_{d \prec e} ( c^\tau ( d^\tau e^\tau ) ) \big),
\\
&
a(b((cd)e)) = a(b(c(de))) + a(b(c(de))),
&\qquad
&
\text{$a(b(c(de)))$ is already in normal form}.\end{aligned}$$ Figure \[zinbielnormalform\] presents a recursive algorithm for computing the Zinbiel normal form of a nonassociative monomial.
\[expansionlemma\] The expansions of the ternary monomials $[[a,b,c],d,e]$ and $[a,b,[c,d,e]]$ have the following Zinbiel normal forms: $$\begin{aligned}
&
[[a,b,c],d,e] =
\left[
\begin{array}{c}
++++++++++++++++++++++++
\\[-2pt]
------------------------
\\[-2pt]
------++++++--++-+--++-+
\\[-2pt]
------++++++++--+---+++-
\\[-2pt]
------++++++++--+-++---+
\end{array}
\right]
\\
&
[a,b,[c,d,e]] =
\left[
\begin{array}{c}
++---+++++++-------+--++
\\[-2pt]
--+++-------+++++++-++--
\\[-2pt]
------++++++--++-+--++-+
\\[-2pt]
++++++------++--+-++--+-
\\[-2pt]
+-++---+--++++--+---++-+
\end{array}
\right]\end{aligned}$$ The notation is as follows: each normal form consists of a linear combination of the 120 right-normed monomials consisting of all permutations of the arguments $a,b,c,d,e$; every coefficient is $\pm 1$. The $5 \times 24$ sign matrices displayed above contain the sequence of coefficients in which the standard row-column order of matrix entries corresponds to the lex order of permutations.
For each ternary monomial, we first use equation twice to compute the expansion, obtaining a linear combination of 36 nonassociative monomials with coefficients $\pm 1$, and we then use the algorithm of Figure \[zinbielnormalform\] to compute the Zinbiel normal form of each nonassociative monomial; the result is a linear combination with coefficients $\pm 1$ of all 120 right-normed monomials.
\[E5lemma\] With respect to the ordered basis of Lemma \[skewbasis5\], the coefficient vectors of the relations of arity 5 satisfied by the tortkara triple product may be identified with the nonzero vectors in the nullspace of the $120 \times 90$ matrix $E_5$ defined as follows: the $(i,j)$ entry of $E_5$ is the coefficient of the $i$-th basis monomial for $\mathbf{Zinb}(5)$ in lex order (Lemma \[zinbielbasis5\]) in the expansion of the $j$-th basis monomial for $\mathbf{SkewTS}(5)$ in lex order (Lemma \[skewbasis5\]), where the expansions are obtained by applying the appropriate permutation $\sigma \in S_5$ to the arguments of the two basic expansions (Lemma \[expansionlemma\]).
The matrix $E_5$ represents the expansion map $\mathbf{SkewTS}(5) \longrightarrow \mathbf{Zinb}(5)$ with respect to the given ordered bases of the homogeneous components.
\[theorem5\] The nullspace of the expansion matrix $E_5$ of Lemma \[E5lemma\] has dimension 30 and is generated as an $S_5$-module by the following relation: $$\begin{aligned}
&
TT(a,b,c,d,e) =
[[a,b,c],d,e]
- [[a,c,b],d,e]
+ [[b,c,d],e,a]
+ [[b,d,c],a,e]
\\[-1pt]
& {}
- [[b,e,d],c,a]
- [[c,d,b],a,e]
+ [[c,d,b],e,a]
+ [[d,e,b],c,a]
+ [a,c,[b,d,e]]
\\[-1pt]
& {}
- [a,e,[c,d,b]]
- [c,d,[a,b,e]]
+ [c,d,[a,e,b]]
- [c,d,[b,e,a]]
- [d,e,[b,c,a]].\end{aligned}$$
Over a field $\mathbb{F}$, a basis for the nullspace of an $m \times n$ matrix $E$ of rank $r$ may be computed by the standard algorithm: first compute the row canonical form (RCF) of $E$, identify the $n{-}r$ free columns (those which do not contain one of the $r$ leading 1s), set the free variables equal to the standard basis vectors in $\mathbb{F}^{n-r}$, and solve for the leading variables. However, for a matrix $E$ with integer entries, there is a fraction-free method: since $\mathbb{Z}$ is a Euclidean domain (and hence a PID), we may combine Gaussian elimination and the Euclidean algorithm for GCDs to compute the Hermite normal form (HNF) of $E$. More precisely, we compute the HNF, say $H$, of the transpose $E^t$, and keep track of the row operations to produce an invertible integer matrix $U$ for which $UE^t = H$. Since the bottom $n{-}r$ rows of $H$ are zero, it follows that the bottom $n{-}r$ rows of $U$ belong to (in fact form a basis of) the right nullspace of $E^t$ (which is the left nullspace of $E$); we call this $(n{-}r) \times n$ matrix $N$. Furthermore, we may remain in the category of free $\mathbb{Z}$-modules (instead of vector spaces over $\mathbb{F}$), and apply the LLL algorithm for lattice basis reduction to $N$ to produce another matrix $N'$ whose rows generate the same free $\mathbb{Z}$-module as the rows of $N$, but whose Euclidean lengths are much smaller. As a measure of the size of the basis, we use the base 10 logarithm of the product of the squared lengths of the basis vectors. For the original basis $N$ this measure is $\approx 40.847$; after applying the LLL algorithm with the standard value $3/4$ of the reduction parameter, we obtain a basis with measure $39.851$; using the higher reduction parameter $999/1000$, we obtain a basis $N'$ with measure $\approx 35.656$. The square lengths of the rows of the original basis $N$ are as follows (multiplicities are written as exponents): $14^3$, $16^3$, $20^2$, $22$, $24^{13}$, $26$, $28^2$, $30$, $32^2$, $36^2$; for the reduced basis $N'$ the square lengths are $14^{13}$, $16^{14}$, $18$, $20$, $22$. The shortest basis vector has not changed, but the total length of the basis has dropped by more than $10^5$; the shortest vector corresponds to the relation $TT(a,b,c,d,e)$. We then apply all 120 permutations of $a, b, c, d, e$ to $TT(a,b,c,d,e)$, store the coefficient vectors in a $120 \times 90$ matrix, compute its RCF over $\mathbb{Q}$, and verify that this RCF coincides with the RCF of $N$. For more about the application of LLL to identical relations for nonassociative structures, see [@BP2009].
\[corollary5\] Let $[\lambda]$ denote the irreducible representation of $S_5$ corresponding to the partition $\lambda$ of 5. The nullspace of the expansion matrix $E_5$ of Lemma \[E5lemma\] has the following $S_5$-module structure where the exponents indicate multiplicities: $$[221] \oplus [311] \oplus [32]^2 \oplus [41]^2 \oplus [5].$$
Using the reduced basis $N'$ from the proof of Theorem \[theorem5\], we compute the $30 \times 30$ matrices representing the action of a set of conjugacy class representatives on the nullspace of $E_5$, and obtain the character $[30, -6, 2, 0, 0, 0, 0]$. Comparing this with the character table for $S_5$, we obtain the indicated decomposition.
Relations of arity 7
====================
The (somewhat redundant) consequences in arity 7 of the relation $TT(a,b,c,d,e)$ with respect to operadic partial composition are these eight relations: $$\label{partial}
\left\{ \;
\begin{array}{l@{\;\;\;}l@{\;\;\;}l}
TT([a,f,g],b,c,d,e), &
TT(a,[b,f,g],c,d,e), &
TT(a,b,[c,f,g],d,e),
\\[2pt]
TT(a,b,c,[d,f,g],e), &
TT(a,b,c,d,[e,f,g]),
\\[2pt] {}
[TT(a,b,c,d,e),f,g], &
[f,TT(a,b,c,d,e),g], &
[f,g,TT(a,b,c,d,e)].
\end{array}
\right.$$
\[theorem7\] The multilinear relation in arity 7 displayed in Figure \[newarity7\] has 60 terms and coefficients $\pm 1$, $\pm 2$; it is satisfied by the tortkara triple product in the Zinbiel operad, but it is not a consequence of the skewsymmetry in arity 3 or the 14-term relation in arity 5 displayed in Theorem \[theorem5\].
Let $\mathrm{Con}(7) \subset \mathbf{SkewTS}(7)$ be the $S_7$-submodule generated by the consequences with respect to operadic partial composition of the relation $TT(a,b,c,d,e)$ displayed in Theorem \[theorem5\], and let $\mathrm{ConNew}(7) \subset \mathbf{SkewTS}(7)$ be the $S_7$-submodule generated by those consequences together with the 60-term relation in Figure \[newarity7\]. The quotient module $\mathrm{ConNew}(7)/\mathrm{Con}(7)$ has dimension 106 and the following multiplicity-free decomposition into the direct sum of irreducible representations: $$\label{connew7}
\mathrm{ConNew}(7)/\mathrm{Con}(7) \cong [421] \oplus [322] \oplus [3211] \oplus [31111].$$ The dimensions of the irreducible summands are respectively 35, 21, 35, 15.
Let $\mathrm{All}(7) \subset \mathbf{SkewTS}(7)$ be the $S_7$-module consisting of all relations satisfied by the tortkara triple product in arity 7. The quotient module $\mathrm{All}(7)/\mathrm{Con}(7)$ has dimension 246; it decomposes as follows: $$\label{isotypic7}
\mathrm{All}(7)/\mathrm{Con}(7)
\cong
[511] \oplus [421]^2 \oplus [4111]^2 \oplus [322] \oplus [3211]^2 \oplus [31111]^2.$$ The dimensions of the isotypic components are respectively 15, 70, 40, 21, 70, 30.
$$\boxed{
\begin{array}{l}
2 [dg[[efb]ca]] % type 3 count 1
-2 [dg[[efc]ba]] % type 3 count 2
+2 [ef[[dgb]ca]] % type 3 count 3
-2 [ef[[dgc]ba]] % type 3 count 4
+2 [dg[ef[abc]]] % type 4 count 5
\\[2pt] {}
-2 [dg[ef[acb]]] % type 4 count 6
+2 [ef[dg[abc]]] % type 4 count 7
-2 [ef[dg[acb]]] % type 4 count 8
- [[abd]g[efc]] % type 6 count 9
- [[abe]f[dgc]] % type 6 count 10
\\[2pt] {}
+ [[abf]e[dgc]] % type 6 count 11
+ [[abg]d[efc]] % type 6 count 12
+ [[acd]g[efb]] % type 6 count 13
+ [[ace]f[dgb]] % type 6 count 14
- [[acf]e[dgb]] % type 6 count 15
\\[2pt] {}
- [[acg]d[efb]] % type 6 count 16
+ [[adb]g[efc]] % type 6 count 17
- [[adc]g[efb]] % type 6 count 18
+ [[adg]b[efc]] % type 6 count 19
- [[adg]c[efb]] % type 6 count 20
\\[2pt] {}
+ [[aeb]f[dgc]] % type 6 count 21
- [[aec]f[dgb]] % type 6 count 22
+ [[aef]b[dgc]] % type 6 count 23
- [[aef]c[dgb]] % type 6 count 24
- [[afb]e[dgc]] % type 6 count 25
\\[2pt] {}
+ [[afc]e[dgb]] % type 6 count 26
- [[afe]b[dgc]] % type 6 count 27
+ [[afe]c[dgb]] % type 6 count 28
- [[agb]d[efc]] % type 6 count 29
+ [[agc]d[efb]] % type 6 count 30
\\[2pt] {}
- [[agd]b[efc]] % type 6 count 31
+ [[agd]c[efb]] % type 6 count 32
- [[bda]g[efc]] % type 6 count 33
- [[bdg]a[efc]] % type 6 count 34
- [[bea]f[dgc]] % type 6 count 35
\\[2pt] {}
- [[bef]a[dgc]] % type 6 count 36
+ [[bfa]e[dgc]] % type 6 count 37
+ [[bfe]a[dgc]] % type 6 count 38
+ [[bga]d[efc]] % type 6 count 39
+ [[bgd]a[efc]] % type 6 count 40
\\[2pt] {}
+ [[cda]g[efb]] % type 6 count 41
+ [[cdg]a[efb]] % type 6 count 42
+ [[cea]f[dgb]] % type 6 count 43
+ [[cef]a[dgb]] % type 6 count 44
- [[cfa]e[dgb]] % type 6 count 45
\\[2pt] {}
- [[cfe]a[dgb]] % type 6 count 46
- [[cga]d[efb]] % type 6 count 47
- [[cgd]a[efb]] % type 6 count 48
- [[dga]b[efc]] % type 6 count 49
+ [[dga]c[efb]] % type 6 count 50
\\[2pt] {}
+ [[dgb]a[efc]] % type 6 count 51
-2 [[dgb]c[efa]] % type 6 count 52
- [[dgc]a[efb]] % type 6 count 53
+2 [[dgc]b[efa]] % type 6 count 54
- [[efa]b[dgc]] % type 6 count 55
\\[2pt] {}
+ [[efa]c[dgb]] % type 6 count 56
+ [[efb]a[dgc]] % type 6 count 57
-2 [[efb]c[dga]] % type 6 count 58
- [[efc]a[dgb]] % type 6 count 59
+2 [[efc]b[dga]] % type 6 count 60
\end{array}
}$$
For arity 7, the sizes of the matrices involved are too large to use rational arithmetic, so we use arithmetic modulo $p = 101$ to keep the time and space requirements within reasonable bounds.
*Step 1:* We first construct the consequences of the new relation of arity 5; the monomials in these consequences require straightening (using anticommutativity in the first two arguments) in order to ensure that they involve only the six standard association types in arity 7: $$\label{types7}
\left\{ \quad
\begin{array}{l@{\qquad}l@{\qquad}l}
[[[\ast,\ast,\ast],\ast,\ast],\ast,\ast], & % 1
[[\ast,\ast,[\ast,\ast,\ast]],\ast,\ast], & % 2
[\ast,\ast,[[\ast,\ast,\ast],\ast,\ast]], % 3
\\[2pt] {}
[\ast,\ast,[\ast,\ast,[\ast,\ast,\ast]]], & % 4
[[\ast,\ast,\ast],[\ast,\ast,\ast],\ast], & % 5
[[\ast,\ast,\ast],\ast,[\ast,\ast,\ast]]. % 6
\end{array}
\right.$$ These association types have respectively 1, 2, 2, 3, 3, 2 skew-symmetries: $$\label{symmetries7}
\left\{ \quad
\begin{array}{r@{\qquad}r}
[[[abc]de]fg] + [[[bac]de]fg] \equiv 0,
\\[1pt] {}
[[ab[cde]]fg] + [[ba[cde]]fg] \equiv 0,
&
[[ab[cde]]fg] + [[ab[dce]]fg] \equiv 0,
\\[1pt] {}
[ab[[cde]fg]] + [ba[[cde]fg]] \equiv 0,
&
[ab[[cde]fg]] + [ab[[dce]fg]] \equiv 0,
\\[1pt] {}
[ab[cd[efg]]] + [ba[cd[efg]]] \equiv 0,
&
[ab[cd[efg]]] + [ab[dc[efg]]] \equiv 0,
\\[1pt] {}
&
[ab[cd[efg]]] + [ab[cd[feg]]] \equiv 0,
\\[1pt] {}
[[abc][def]g] + [[bac][def]g] \equiv 0,
&
[[abc][def]g] + [[abc][edf]g] \equiv 0,
\\[1pt] {}
&
[[abc][def]g] + [[def][abc]g] \equiv 0,
\\[1pt] {}
[[abc]d[efg]] + [[bac]d[efg]] \equiv 0,
&
[[abc]d[efg]] + [[abc]d[feg]] \equiv 0.
\end{array}
\right.$$ The total number of multilinear monomials is therefore $$\dim \mathbf{SkewTS}(7)
=
\left( \tfrac12 + \tfrac14 + \tfrac14 + \tfrac18 + \tfrac18 + \tfrac14 \right) 7!
=
7560.$$
*Step 2:* We initialize to zero a block matrix of size $( 7560 + 5040 ) \times 7560$. For each of the eight consequences we apply all permutations of the arguments, straighten using skewsymmetry, store the resulting coefficient vectors in the lower block ($5040 \times 7560$), and compute the row canonical form (RCF); nonzero entries in the upper block are retained for subsequent iterations. The cumulative ranks obtained are 1785, 2730, 3150, 3150, 3150, 4410, 4410, 4794 which shows that consequences 4, 5, 7 are not required as generators of the $S_7$-module $\mathrm{Con}(7)$ of all consequences of $TT(a,b,c,d,e)$ in arity 7 and that $\dim \mathrm{Con}(7) = 4794$.
*Step 3:* We insert the identity permutation of the arguments $a, \dots, g$ into the association types , expand them using the tortkara triple product , obtaining six sums each with 216 terms and coefficients $\pm 1$, normalize the terms using the algorithm of Figure \[zinbielnormalform\], and sort the resulting permutations in lex order, obtaining six sequences of $\pm$ signs of length 5040 which are the analogues in arity 7 of the two sequences of length 120 in Lemma \[expansionlemma\]. For each association type $t$, we apply the permutations corresponding to the multilinear monomials of type $t$ to the expansion of type $t$, and store the resulting sequences of $\pm$ signs in the columns of the expansion matrix of size $5040 \times 7560$. We compute the RCF and find that the rank is 2520, so the nullity is 5040. The nullspace of this matrix is the $S_7$-module $\mathrm{All}(7)$ containing all the (multilinear) identical relations satisfied by the tortkara triple product in arity 7. Comparing this with the result of Step 1, we see that the quotient module of new relations, $\mathrm{New}(7) = \mathrm{All}(7) / \mathrm{Con}(7)$, has dimension $5040 - 4794 = 246$.
*Step 4:* From the full rank matrix of size $2520 \times 7560$ produced by Step 2, we extract a $5040 \times 7560$ matrix whose row space is the nullspace of the expansion matrix, and compute its RCF. We then sort the rows of the RCF by increasing number of nonzero entries, obtaining a minimum of 17 and a maximum of 1397. We then process these sorted rows as in Step 1 to determine which of the nullspace basis vectors do not belong to the $S_7$-module generated by the consequences of $TT(a,b,c,d,e)$ and the previously processed nullspace basis vectors. A relation with 60 terms increases the rank from $\dim \mathrm{Con}(7) = 4794$ to $\dim \mathrm{ConNew}(7) = 4900$ (this is the relation displayed in Figure \[newarity7\]); a relation with 798 terms then increases the rank from 4900 to 4970; and finally a relation with 985 terms increases the rank from 4900 to $\dim \mathrm{All}(7) = 5040$. The coefficients modulo $p = 101$ of the relation with 60 terms are $1, 50, 51, 100$; multiplying by 2 and using symmetric representatives gives the coeffients $2, -1, 1, -2$.
*Step 5:* We confirm the results of Steps 1–3 using the representation theory of the symmetric group. These computational methods have been described in detail elsewhere [@BMP2016], so we omit most of the details and provide only a brief outline. The basic idea is to break the computation down into smaller subproblems corresponding to the irreducible representations of $S_7$. Let $\lambda$ be a partition of 7, with corresponding simple $S_7$-module $[\lambda]$ of dimension $d_\lambda$. For each $s \in \mathbb{F} S_7$, let $R_\lambda(s)$ be the $d_\lambda \times d_\lambda$ matrix for $s$ in the natural representation which may be computed according to Clifton’s algorithm [@BMP2016 Figure 3]. For each $\lambda$, we construct matrices $[\mathrm{Sym}(\lambda)]$, $[\mathrm{Con}(\lambda)]$, $[\mathrm{New}(\lambda)]$ of sizes $13d_\lambda \times 6d_\lambda$, $8d_\lambda \times 6d_\lambda$, $d_\lambda \times 6d_\lambda$ respectively, which contain the blocks $R_\lambda(s)$ representing the terms in each association type for the symmetries , the consequences , and the new 60-term relation (Figure \[newarity7\]). We stack these matrices vertically and compute the RCFs; the ranks give the multiplicities of the irreducible representation $[\lambda]$ in the corresponding $S_7$-modules (Figure \[mults7\]). Comparing the last two rows we obtain the decomposition .
$$\begin{array}{cc@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c}
& 7 & 61 & 52 & 51^2 & 43 & 421 & 41^3 & 3^21 & 32^2 & 321^2 & 31^4 & 2^31 & 2^21^3 & 21^5 & 1^7
\\ \midrule
\begin{bmatrix}
\mathrm{Sym}(\lambda)
\end{bmatrix}
& 6 & 35 & 77 & 81 & 71 & 172 & 95 & 95 & 92 & 145 & 57 & 50 & 44 & 14 & 0
\\[1mm]
\begin{bmatrix}
\mathrm{Sym}(\lambda) \\
\mathrm{Con}(\lambda)
\end{bmatrix}
& 6 & 35 & 80 & 84 & 79 & 193 & 108 & 116 & 114 & 188 & 78 & 75 & 74 & 31 & 5
\\[3mm]
\begin{bmatrix}
\mathrm{Sym}(\lambda) \\
\mathrm{Con}(\lambda) \\
\mathrm{New}(\lambda) \\
\end{bmatrix}
& 6 & 35 & 80 & 84 & 79 & 194 & 108 & 116 & 115 & 189 & 79 & 75 & 74 & 31 & 5
\\ \midrule
\end{array}$$
$$\begin{array}{cc@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c@{\;\;}c}
& 7 & 61 & 52 & 51^2 & 43 & 421 & 41^3 & 3^21 & 32^2 & 321^2 & 31^4 & 2^31 & 2^21^3 & 21^5 & 1^7
\\ \midrule
\begin{bmatrix}
\mathrm{Exp}(\lambda)
\end{bmatrix}
& 0 & 1 & 4 & 5 & 5 & 15 & 10 & 10 & 11 & 20 & 10 & 9 & 10 & 5 & 1
\\[1mm]
\begin{bmatrix}
\mathrm{Nul}(\lambda)
\end{bmatrix}
& 6 & 35 & 80 & 85 & 79 & 195 & 116 & 115 & 114 & 190 & 80 & 75 & 74 & 31 & 5
\\ \midrule
\end{array}$$
*Step 6:* For each $\lambda$, we construct the matrix $[\mathrm{Exp}(\lambda)]$ of size $d_\lambda \times 6d_\lambda$ in which the $k$-th block $R_\lambda(s)^t$ is the transpose of the representation matrix for the normalized Zinbiel expansion of the skew-ternary monomial with the identity permutation of $a, \dots, g$ in association type $k$. (For an explanation of the transpose, see [@BM2013 §5].) From this we compute the matrix $[\mathrm{Nul}(\lambda)]$ in RCF whose row space is the isotypic component of type $[\lambda]$ in the $S_7$-module $\mathrm{All}(7)$. Comparing the last row of Figure \[expnul7\] with the second-last row of Figure \[mults7\] gives the decomposition .
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[^1]: The research of the author was supported by a Discovery Grant from NSERC, the Natural Sciences and Engineering Research Council of Canada.
|
---
abstract: '$5d$ iridium oxides are of huge interest due to the potential for new quantum states driven by strong spin-orbit coupling. The strontium iridate Sr$_2$IrO$_4$ is particularly in the spotlight because of the novel $j_\text{eff}=1/2$ state consisting of a quantum superposition of the three $t_{2g}$ orbitals with nearly equal population, which stabilizes an unconventional Mott insulating state. Here, we report an anisotropic and aspherical magnetization density distribution measured by polarized neutron diffraction in a magnetic field up to 5 T at 4 K, which strongly deviates from a local [$j_{\mathrm{eff}}=1/2\ $]{}picture. Once reconstructed by the maximum entropy method and multipole expansion model refinement, the magnetization density shows cross-shaped positive four lobes along the crystallographic tetragonal axes with a large spatial extent, showing that the $xy$ orbital contribution is dominant. Theoretical considerations based on a momentum-dependent composition of the [$j_{\mathrm{eff}}=1/2\ $]{}orbital and an estimation of the different contributions to the magnetization density casts the applicability of an effective one-orbital [$j_{\mathrm{eff}}=1/2\ $]{}Hubbard model into doubt. The analogy to the superconducting copper oxide systems might thus be weaker than commonly thought.'
author:
- 'Jaehong Jeong$^1$, Benjamin Lenz$^2$, Arsen Gukasov$^1$, Xavier Fabreges$^1$, Andrew Sazonov$^3$, Vladimir Hutanu$^3$, Alex Louat$^4$, Cyril Martins$^5$, Silke Biermann$^2$, Véronique Brouet$^4$, Yvan Sidis$^1$ & Philippe Bourges$^1$'
title: 'Magnetization density distribution of Sr$_2$IrO$_4$: Deviation from a local [$j_{\mathrm{eff}}=1/2\ $]{}picture'
---
Laboratoire Léon Brillouin, CEA-CNRS, Université Paris-Saclay, CEA Saclay, 91191 Gif-sur-Yvette, France
Centre de Physique Théorique (CPHT), Ecole Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau Cedex, France
Institute of Crystallography, RWTH Aachen University and Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), 85747 Garching, Germany
Laboratoire de Physique des Solides, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
Laboratoire de Chimie et Physique Quantiques, UMR 5626, Université Paul Sabatier, 118 route de Narbonne, 31400 Toulouse, France
Sr$_2$IrO$_4$ possesses a tetragonal structure with $I4_1/acd$ space group, in which the IrO$_6$ octahedra are rotated by $\approx$11$^\circ$ around the $c$-axis with an opposite phase for the neighboring Ir ions [@huang1994; @ye2013] and it orders antiferromagnetically below $T_\text{N}\approx230$ K [@kim2008]. Strong spin-orbit coupling (SOC) stabilizes an unconventional Mott insulating ground state, which is commonly described by a spin-orbital product state within a so-called $j_\text{eff}=1/2$ model [@kim2008; @kim2009; @jackeli2009; @wang2011]. In this model, $5d$ electrons at the Ir$^{4+}$ ($5d^5$) ions occupy the $t_{2g}$ states with an effective angular momentum $l_\text{eff}=1$, which are split by the relatively large SOC into a $j_\text{eff}=1/2$ doublet and a $j_\text{eff}=3/2$ quartet. The Coulomb repulsion induces a gap in the narrow half-filled $j_\text{eff}=1/2$ band, and stabilizes the Mott insulating state with the pseudospin $j_\text{eff}=1/2$ [@kim2008; @jackeli2009], which consists of three equally populated orbital components in the $t_{2g}$ band (Fig. \[fig:jeff\]a): $$\left|j_\text{eff}=\frac{1}{2}, \pm\frac{1}{2}\right> = \frac{1}{\sqrt{3}} \left( \left|xy,\pm\sigma\right> \pm
\left|yz,\mp\sigma\right> + i\left|xz,\mp\sigma\right> \right).
\label{Eq:1}$$ While resonant and inelastic X-ray scattering [@kim2009; @kim2012] gave credit to a description in terms of $J_\text{eff}=1/2$ states, this simple description has been questionned owing to the tetragonal distortion that is not negligible [@chapon2011; @haskel2012; @morettisala2014]. Strictly speaking, the $j_\text{eff}=1/2$ model is realized only for a perfect cubic symmetry. In the presence of a tetragonal distortion accompanied by a compression or elongation of the octahedra, the $t_{2g}$ band is split into three Kramers doublet states, which consist of the mixing between $j_\text{eff}=1/2$ and $j_\text{eff}=3/2$ states [@abragam2012; @jackeli2009; @chapon2011; @perkins2014; @morettisala2014].
In addition, a strong hybridization between Ir $5d$ and O $2p$ orbitals, which seems to be natural for a large spatial extent of $5d$ orbitals, has been proposed to account for a large reduction of the ordered magnetic moment [@kim2008] as well as for AFM exchange interactions between the nearest-neighboring Ir ions and for the canted magnetic moments following the octahedral rotations [@jackeli2009; @perkins2014]. The strong hybridization of the $d$-orbitals with the $p$-orbitals of the ligand oxygen is reminiscent of K$_2$IrCl$_6$ [@lynn1976] and the isostructural ruthenate Ca$_{1.5}$Sr$_{0.5}$RuO$_4$ [@gukasov2002], where similar covalency effects have been reported. In Sr$_2$IrO$_4$, recent muon spin relaxation measurements have suggested the formation of oxygen moments [@miyazaki2015], and charge redistribution between adjacent IrO$_2$ and SrO layers has been revealed using electron spin resonance measurement [@bogdanov2015]. Further, unusual magnetic multipoles have been proposed to be observed by neutron diffraction [@lovesey2014] and recently a hidden magnetic order having the same symmetry as a loop-current state has been observed by polarized neutron diffraction [@jeong2017].
{width="12"}
The magnetic moments of Ir ions are confined in the $ab$-plane and track the staggered octahedral rotation in an $-++-$ sequence along the $c$-axis in the unit cell [@ye2013]. Owing to this canted AFM structure, each IrO$_2$ layer has a weak ferromagnetic (WFM) moment along the principal crystallographic axis in the $ab$-plane. At zero magnetic field, this WFM is compensated due to the $-++-$ stacking sequence whereas, in a magnetic field higher than $H_c\approx0.3$ T applied in the ab plane [@kim2008; @ye2013], a net homogeneous WFM moment appears in the plane (inset of Fig. \[fig:mag\]b). Remarkadly, this WFM moment follows the direction of applied magnetic field in the $ab$-plane [@fruchter2016; @nauman2017] and attains a saturation value of $\approx$0.08$\mu_\text{B}$/Ir in the field of 1 T [@fruchter2016]. Therefore, the WFM moment does not interlock with the rotation of IrO$_6$ octahedra [@porras2019] in contrast with the AFM staggered moment.
The existence of this WFM allows us to probe the magnetization density distribution in crystals by polarized neutron diffraction (PND). This technique is unique because it provides direct information about the 3-dimensional distribution of the magnetization throughout the unit cell, which in turn allows for a determination of the symmetry of occupied orbitals. This method is well established for paramagnetic and ferromagnetic (FM) systems. It gave access, e.g., to the $3d$-orbital population in ferromagnetic insulator YTiO$_3$ [@akimitsu2001]. It has been also successfully used in the study of FM ruthenate Ca$_{1.5}$Sr$_{0.5}$RuO$_4$, isostructural to Sr$_2$IrO$_4$, where an anomalously high spin density at the oxygen site and the $xy$ character of the Ru $d$-orbitals have been reported [@gukasov2002]. While in conventional spin density studies either a positive or a negative spin component can be present at a given ion, in the iridates thanks to the spin-orbit coupling both positive and negative densities can coexist at the same Ir site (Fig. \[fig:jeff\]a) [@morettisala2014]. To access this intra-atomic variation of magnetization density high resolution polarized hot neutron diffraction data are needed. Here we have performed PND (i) to establish the symmetry of the Ir $5d$ orbitals occupied by unpaired electrons and (ii) to check the presence of unpaired electron density on the oxygen ligand.
{width="\columnwidth"}
The typical experimental setup for PND, shown in Fig. \[fig:setup\]a, consists of a neutron polarizer, a flipping device that reverses the incident neutron polarization, a magnet and a detector. The sample is magnetized by a magnetic field applied along the vertical axis and scattering intensities of Bragg reflections for the two opposite states (spin-up and spin-down) of the incident polarization are measured. They are used to calculate the so-called flipping ratio, allowing access to the Fourier components of the magnetization density, as $$R_\text{PND}=\frac{I_\uparrow}{I_\downarrow}=\frac{F_N^2+2p \sin\alpha F_N F_M + \sin^2\alpha {F_M}^2} {F_N^2 - 2 p e \sin\alpha F_N F_M + \sin^2\alpha {F_M}^2},
\label{eq:FR}$$ where $F_N$ is the nuclear structure factor and $F_M$ is the projection of the magnetic structure factor along the vertically applied magnetic field. $p$ and $e$ are the polarization efficiency of the polarizer and flipper, respectively, and $\alpha$ is the angle between the scattering and the magnetic interaction vector (see Supplementary section 2).
The flipping ratios $R_\text{PND}$ of more than 280 $(hkl)$ reflections were measured in the weakly ferromagnetic state at 2 K for two magnetic field orientations, $H\|[010]$ and $H\|[\bar{1}10]$ (well above the critical field $H_c\approx0.3$ T [@kim2009; @porras2019]). The measured intensities for two orientations were averaged (see Supplementary section 3). As shown in Fig. \[fig:formfactor\]b, the magnetic structure factors $F_M$ were directly obtained from the measured flipping ratios by using Eq. (\[eq:FR\]) and known nuclear structure factors $F_N$. For convenience, the amplitudes are given in Bohr magnetons, normalized by the number of Ir atoms (8) in the unit cell, and taken in absolute values to remove alternating signs of the phase factor. The amplitude, $F_M(0)$, is imposed in agreement with the saturation moment (0.08$\mu_\text{B}$/Ir) given by the uniform magnetization measurement [@fruchter2016].
In the *dipole approximation*, $F_M(Q)$ is usually described by a smooth decreasing function of $Q$, the magnetic form factor, corresponding to a linear combination of radial integrals calculated from the electronic radial wave function. Instead in Fig. \[fig:formfactor\]b, one observes a large distribution of the measured structure factor indicating unusually large anisotropy. That large anisotropy is explained by a predominance of $xy$-orbital as shown below using the reconstruction of the magnetization density in real space. The theoretical radial integrals $\left<j_n\right>$ for an isolated Ir$^{4+}$ ion [@kobayashi2011] are also shown in Fig. \[fig:formfactor\]b for comparison. We recall that $\left<j_0\right>$ describes a spherical form factor of the magnetic moment, while $\left<j_2\right>$,$\left<j_4\right>$ and higher-order integrals are needed to describe the departures from spherical symmetry. As seen from Fig. \[fig:formfactor\] except for the $(0,0,l)$ reflections, decreasing gradually with increasing $Q$, the majority of reflections strongly deviate from any expected smooth curve. Moreover, while the $(0,0,4n)$, $(2,0,4n)$ and $(2,2,4n)$ reflections are close to the $\left<j_0\right>$ curve in a small $Q$ region, the $(1,1,4n+2)$ and $(2,1,2n+1)$ reflections deviate from it quite strongly. This indicates an aspherical magnetization density, which is typical of ions with one or two unpaired electrons in the $d$-orbitals [@lynn1976; @shamoto1993; @zaliznyak2004]. In addition, one can see that high-$Q$ reflections like $(4,0,0)$, $(4,2,0)$ and $(4,4,0)$ ones show anomalously large values. That suggests large $\left<j_2\right>$ and $\left<j_4\right>$ terms, indicative of admixtures of $J$-manifolds and the inadequacy of the $j_\text{eff}=1/2$ model.
Next, a real space visualization has been performed by a reconstruction of the magnetization density, using two different very well-established and widely used approaches; a model-free maximum entropy method (MEM) [@papoular1990] and a quantitative refinement using the multipole expansion of the density function [@coppens1997]. Both techniques have advantages and limits and should be employed where they are the most efficient. Typically, no assumption is made for the initial magnetization distribution in MEM whereas the $d$-orbitals shape is constrained in the multipole expansion.
Since the crystal structure is centrosymmetric, the magnetization density can be directly reconstructed from the measured magnetic structure factors by MEM [@papoular1990]. Fig. \[fig:magden\]a-d, shows the 3-dimensional magnetization density reconstructed by using a conventional flat density prior. There are three key features to be noted in the figure. First, the magnetization density at Ir sites has four positive density lobes directed along the $a$, $b$ axes, corresponding to a dominant positive magnetization density of $d_{xy}$ orbital symmetry (Fig. \[fig:magden\]b). The two other components of the effective $j_\text{eff}=1/2$ state model, $d_{yz}$ and $d_{xz}$, which would form an axially symmetric ring-shaped density with negative lobes above and below the $xy$ plane (see Fig. \[fig:jeff\]a), does not appear as seen in Fig. \[fig:magden\]c,d. Thus the WFM density originates predominantly from the $xy$ orbital, at variance with a naive $j_\text{eff}=1/2$ picture. Second, positive density lobes are very strongly elongated, in such a way that some magnetization density is delocalized well beyond of the IrO$_6$ octahedra. It is supported by very large spatial extent of the $t_{2g}$ orbitals (reaching the nearest neighbouring Ir atoms) found by core-to-core RIXS and *ab initio* calculation [@agrestini2017]. It also could give a support to a direct Ir-Ir exchange mechanism, via electron hopping between the neighboring ions. Third, contrary to the expectation of strong iridium oxygen ligand hybridization, no visible magnetization density appears at the oxygen sites. Actually, no significative polarization dependence has been found in any of dozens measured $(2,1,2n+1)$ reflections where oxygen atoms contribute. This is in contrast with the isostructural $4d$ compound Ca$_{1.5}$Sr$_{0.5}$RuO$_4$, where $\sim$20% of the magnetic moment is transferred to the in-plane O sites [@gukasov2002]. However, one can notice the presence of a negative magnetic density mostly along the Ir-O direction existing between the large positive lobes. In fact, a significant negative magnetic density as large as half of the net moment is essential for a better description in the MEM analysis (see Supplementary section 4).
{width="\columnwidth"}
To confirm the symmetry found by MEM, multipole expansion model was perfomed for an alternative refinement of the WFM density. It is composed of radial and angular parts: Slater-type radial wave functions and real spherical harmonic density functions (see Supplementary section 5). In Fig. \[fig:magden\]e-h, the magnetization density distribution with the best refinement is shown. The main positive magnetization density lobes located between the local $x$- and $y$-axis appear clearly, which corresponds to the $d_{xy}$ symmetry. Therefore, the multipole expansion model fully confirms the $d_{xy}$ symmetry fround by MEM. Further, between the positive lobes, negative density lobes occurs as well which are more pronounced in the multipole refinement than in the MEM results. They are about 60% of positive ones and surprisingly have $d_{x^2-y^2}$ symmetry, requiring an admixing of the $e_g$ orbital to the ground state. A benefit of the multipole method is to determine the contribution of each $d$-orbital to the magnetization. Using the orbital-multipole relations [@coppens1997], the magnetic moments on each orbitals were obtained as: $+0.48$, $-0.051$, $-0.035$ and $-0.314\mu_\text{B}$/Ir for $d_{xy}$, $d_{yz/xz}$, $d_{z^2}$ and $d_{x^2-y^2}$, respectively. Thus a positive $d_{xy}$ and to a lesser extent a negative $d_{x^2-y^2}$ orbital are dominant in the refinement, while the $d_{yz/xz}$ orbitals are barely populated.
It is obvious that the refinement of multipoles with a single radial exponent cannot fit the widely delocalized density. Therefore, we introduce in the refinement a second radial exponent to describe the delocalized Ir density. Such a model shows a considerably better agreement factor ($R_w\sim0.09$) compared to the model with a single radial exponent ($R_w\sim0.20$) (see Supplementary section 5). It confirms the anomalously large spatial extent of the magnetization density of Ir found by the MEM analysis. Further, we have also examined a model with magnetic density at the O1 and O2 sites, but contrary to expectations, no evidence for the existence of the oxygen moment was found.
This result of predominant $d_{xy}$-orbital can be understood from a modelization of Sr${}_2$IrO${}_4$ based on a spin-orbit generalization of the multi-orbital Heisenberg model, (see supplemental section 6). Key to our proposed effective low-energy model is the observation that the hole in the $t_{2g}$-manifold resides in a $\mathbf{k}$-dependent effective $\alpha=1/2$ Wannier state $\varphi_{\mathbf{k},\alpha}$. We thereby account explicitly for the strong $\mathbf{k}$-dependence of $t_{2g}$ components in the $j_{\mathrm{eff}}$ states revealed both by ab initio calculations [@martins2010] and photoemission experiments [@louat2018]. In terms of Fourier-transformed spin operators of such a hole, $\mathbf{s}_{i,\alpha}$, the model is given by a Hamiltonian of the form $$\label{Eq:Ham}
\mathcal{H} = \sum_{i,j,\alpha,\beta}\mathbf{s}_{i,\alpha}\mathbf{J}_{i\alpha,j\beta}\mathbf{s}_{j,\beta},$$ where $\mathbf{J}_{i\alpha,j\beta}$ denotes the tensor of spin interactions in real-space. Away from half-filling, a similar $t-J$ model in orbital space has been derived for iron pnictides [@si2008].\
Solving such a model is beyond the scope of this work. However, to get a qualitative idea of the underlying physics, we proceed by making a few further assumptions. First, if we suppose that we only have to retain the diagonal terms of $\mathbf{J}$, Eq.(\[Eq:Ham\]) decomposes into a sum of three Heisenberg models, one for each of the three $t_{2g}$ components. Let us furthermore assume that we can consider each component separately. In this case, the spin exchange of the $xz-$ and $yz-$ components of $\mathbf{J}$ is essentially described by quasi-1D Heisenberg chains in $x-$ and $y-$direction, suggesting an antiferromagnetic alignment at low temperature. In contrast, the $xy-$component is characterized by longer-ranged exchange of nearest neighbor ($J_1$) and next-nearest neighbor exchange ($J_2$). It is well described by the $J_1-J_2$ Heisenberg model on the square lattice, which has a quantum disordered singlet ground state at zero temperature for $0.4\lesssim J_2/J_1\lesssim 0.6$ [@mambrini2006]. Here, the ratio is $J_2/J_1\approx 1/3$ in close proximity to the disordered ground state phase, such that even small thermal fluctuations can destabilize an antiferromagnetic alignment and render the $xy-$component disordered.
As a consequence, in this picture the $xy$ magnetic component aligns much easier along an external magnetic field than the antiferromagnetically ordered $xz-$ and $yz-$components, which are less susceptible to such a perturbation. Projecting the magnetization density onto t2g components hence reveals a predominant xy-character, which follows the field direction in accordance with the measurements. In summary, using PND we have evidenced a magnetization density distribution in Sr$_2$IrO$_4$ that is inconsistent with the naive local [$j_{\mathrm{eff}}=1/2\ $]{}picture. The measured magnetic structure factor shows a strong axial anisotropy and anomalous values at large $Q$, which indicate an aspherical magnetization density distribution with a significant orbital contribution. Real space visualization exhibits a dominant $d_{xy}$ orbital character with highly elongated lobes of Ir magnetization densities towards the next Ir atoms. Although a strong $d$-$p$ hybridization is expected in Sr$_2$IrO$_4$, the magnetization density at the ligand oxygen sites is barely present. Our results elucidate that the ground state of Sr$_2$IrO$_4$ substantially deviates from the commonly accepted local [$j_{\mathrm{eff}}=1/2\ $]{}state with equally populated $t_{2g}$ orbitals. Rather, the hole resides in an orbital that results from a strongly non-local (that is, $\mathbf{k}$-dependent) superposition of Wannier functions of $t_{2g}$ character. These considerations give an additional twist to the exotic properties of Sr${}_2$IrO${}_4$ and the possibilities of modeling them as well as to the relationship to superconducting copper oxides.
Methods {#methods .unnumbered}
=======
PND measurements were performed on three different polarized neutron diffractometers, *6T2* and *5C1* at the *ORPHÉE* reactor, LLB CEA Saclay and *POLI* at the *FRM-II*, Garching. To increase spatial resolution by using high $hkl$ indices, thermal and hot polarized neutron diffractometers were used with two different neutron wavelengths: $\lambda$=1.4 and 0.84 Å. . Overall, four sets on *6T2* up to $s<0.6$ Å$^{-1}$ under a magnetic field up of 5 T at 4 K. Three additional sets of flipping ratios were measured on *5C1* and two sets at *POLI* up to $s<0.8$ Å$^{-1}$. The magnetic field was applied for two different sample orientations with $[010]$- and $[\bar{1}10]$-axes oriented parallel to the vertical magnetic field. A detailed description is available in the Supplementary material.
The data that support the findings of this study are available from the corresponding authors upon request.
We acknowledge supports from the projects NirvAna (contract ANR-14-OHRI-0010) and SOCRATE (ANR-15-CE30-0009-01) of the French Agence Nationale de la Recherche (ANR), by the Investissement d’Avenir LabEx PALM (GrantNo. ANR-10-LABX-0039-PALM) and by the European Research Council under grant agreement CorrelMat-617196 for financial support. J.J. was supported by an Incoming CEA fellowship from the CEA-Enhanced Eurotalents program, co-funded by FP7 Marie-Sklodowska-Curie COFUND program (Grant Agreement 600382). We acknowledge computing time at IDRIS/GENCI Orsay (Project No. t2017091393). We are grateful to the CPHT computer support team. We thank Stephen Backes and Hong Jiang for useful discussions and for sharing with us their cRPA data prior to publication \[S. Backes, H. Jiang et al., unpublished.\]. Instrument POLI at Maier-Leibnitz Zentrum (MLZ) Garching is operated in cooperation between RWTH Aachen University and Forschungszentrum Jülich GmbH (Jülich-Aachen Research Alliance JARA).
J.J., X.F., A.G., A.S. and V.H. performed the polarized neutron experiments at LLB Saclay and FRM-II. J.J., A.G and A.S. analyzed the neutron data. A.L. and V.B. grew the single crystals and performed their characterization. J.J. co-aligned single crystals. B.L. and S.B. developed the theoretical model, and B.L. and C.M. performed the calculations. J.J., A.G., B.L., Y.S. and P.B. wrote the manuscript with contributions from all authors. P.B. supervised the project.
The authors declare that they have no competing interests.
Correspondence and requests for materials should be addressed to Jaehong Jeong (email: [email protected]) or Philippe Bourges (email: [email protected]).
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Backscattering can be described as a phenomenon which causes the radiation of a massless field to disperse outside those null cones defined by the initial impulse. It is known in older mathematical literature as the ‘breakdown of Huyghens principle’ [@Hadamard] and it is interesting for a number of reasons. First, as a non-Minkowski spacetime effect, it offers a new way of testing general relativity. Second, it can be important in astrophysics, both in order to explain the radiation coming to us from regions adjoining a black hole and to infer information about the sources of the radiation. Third, a strict upper bound on the magnitude of the backscattering effect, such as we derive here, offers numerical relativists an independent test of the correctness of their numerical codes.
Our model is a spherically symmetric massless scalar field emitted from a region near to, but outside, a non-rotating black hole. These simplifying assumptions should not seriously restrict the validity of our conclusions. For example, the propagation of electromagnetic fields, we believe, should obey inequalities similar to the ones derived below. These estimates, while they break down close to the horizon, allow us to distinguish the region in which the backscattering may play a significant role from that in which it is of no importance. They offer, to our knowledge, the first quantitative measure of this strong field effect. Others who work on backscattering adopt quite different approaches [@NS].
We consider a foliation of the spacetime by using the polar gauge slicing condition, $tr K = K_r^r$; that is, with a diagonal line element $$ds^2 = - \beta (R, t)\gamma (R, t)dt^2 + {\beta (R, t)\over \gamma (R, t)} dR^2 +
R^2(r, t) d\Omega^2~,
\label{4.1}$$ where $t$ is the time, $R$ is a radial coordinate which coincides with the areal radius and $d\Omega^2 = d\theta^2 + \sin^2\theta d\phi^2$ is the line element on the unit sphere, $0\le \phi < 2\pi $ and $0\le \theta \le \pi $; $\beta$ and $\gamma$ go to $+1$ at infinity.
For a massless scalar field the stress-energy is $T_{\mu \nu }= \nabla_{\mu }\phi \nabla_{\nu }\phi -g_{\mu \nu }\nabla_{\alpha }\phi
\nabla^{\alpha }\phi/2 $. The matter energy density is $\rho =-T_0^0 $ and the matter current density is $J = -T_{0R}/\beta$. $(\partial_0 \pm \gamma\partial_R )$ are the outgoing and ingoing null directions. We define radiation amplitudes $$\begin{aligned}
h_+(R,t) &= h(R, t) &={1\over 2\gamma} (-\partial_0 +\gamma\partial_R ) (R\phi )\\
h_-(R,t) &= h(-R, t)&={1\over 2\gamma} (\partial_0 +\gamma\partial_R ) (R\phi ).
\label{4.2}\end{aligned}$$ One can show [@Malec1997] that $$\beta (R) = e^{-8\pi \int_R^{\infty } {dr\over r} \left[( h_+ -\hat h)^2 +
(h_- -\hat h)^2\right]}~,
\label {4.3}$$ and $$\gamma(R) = 1 - {2m_0 \over R} + {1 \over R} \int_R^{\infty}[1 - \beta (r)]dr~,
\label{5.2}$$ where $\hat h= -{1\over 2R}\int_{R}^{\infty}dr[h_+(r) + h_-(r)]={1\over 2}\phi~.$ $\gamma $ can be expressed in the following useful form [@Malec1997] $$\gamma (R) = \left( 1- {2m_0\over R} +{ 2 m_{ext}(R)\over R}\right) \beta (R)
\label {4.8}$$ where $m_0$ is the asymptotic (total) mass and $m_{ext}$ is a contribution to the asymptotic mass coming from the exterior of a sphere of a radius $R$, $$m_{ext}(R) = 4\pi\int_{R}^{\infty }{\gamma \over \beta } \left( [h_+(r)-\hat h ]^2+
[h_-(r)-\hat h ]^2\right) dr~. \label {4.9}$$ The scalar field equation $\nabla_{\mu }\partial^{\mu } \phi =0 $ can be written as a single first order equation on a ‘symmetrized’ domain $-\infty \le R\le \infty $ [@Malec1997] by writing $h(R) = h_+(R)$ and $h(-R) = h_-(R)$ as $$(\partial_0 +\gamma \partial_R)h = (h -\hat h ) {\gamma -\beta \over R}~.
\label {4.6}$$ Eq. (\[4.6\]), together with the definitions of $h, \hat h , \beta $, and $ \gamma $, is equivalent to the Einstein equations coupled to the scalar field. The external mass changes along an outgoing null cone according to $$(\partial_0 + \gamma\partial_R)m_{ext}(R) = -8\pi\gamma^2 (h_- - \hat h)^2 ~.
\label{4.10}$$ The polar gauge allows us to express the metric directly in terms of the matter, Eqs. (\[4.3\] - \[5.2\]), where all the integrals are in the exterior region. The local and global Cauchy problems for the above system are solvable in an external region bounded from the interior by a null cone ([@Malec1996b], [@Malec1997]).
Following Eq.(\[4.8\]), $(\gamma -\beta)/R= -2m(R)R\beta/|R|^3$ where $m(R) = m_0 - m_{ext}(R)$ is the Hawking mass at a radius $R$. Eq.(\[4.6\]) gives a ‘red-shift’ due to the $h$ term on the right-hand-side (determined by the mass function, $m(R)$, rather than the Schwarzschild mass, $m_0$) and a ‘backscattering’ due to the $\hat h$ term.
Let us define $$\ln\left[1 - {2\tilde{m}(R) \over R}\right] = -\int_R^{\infty}{2m(r) dr
\over r^2(1 - 2m(r)/r)}~,
\label{4.13}$$ where the integral is taken along an outgoing null ray. This allows us to rewrite Eq.(\[4.6\]) as $$(\partial_0 + \gamma \partial_R)(1 - {2\tilde{m}(R) \over R})h =
\hat h (1 - {2\tilde{m}(R) \over R}){2m(R)R\beta \over |R|^3}~.
\label{4.15}$$ Thus $(1 - 2\tilde{m}(R)/R)$ is the redshift factor and the right-hand-side of Eq.(\[4.15\]) determines the backscattering.
It is natural to write Eq.(\[4.3\]) as $$\beta (R) = e^{-8\pi \left(\int_R^{\infty }+\int_{-R}^{-\infty }\right)
{\beta \over \gamma r} {\gamma \over \beta}( h -\hat h)^2 dr }~,
\label {5.4}$$ and by using Eq.(\[4.8\]) we get that $${\beta \over \gamma r} = {1 \over r - 2m(r)} \le {1 \over r - 2m_0}~.
\label{5.5}$$ The factor $\beta/\gamma r$ can be taken out of the integral in Eq.(\[5.4\]) and replaced with its value at $r = R$. The remainder is then essentially right hand side of Eq.(\[4.9\]). Let us choose an $\epsilon$ and an $R_A$ such that $m_{ext}(R_A)/m_0 < \epsilon$ and $R_A > 2m_0(1+\epsilon)$. For any $R \ge R_A$ we obtain $$1 \ge \beta(R) \ge \beta(R_A) \ge e^{-2{m_{ext}(R_A) \over R_A - 2m_0}}
\simeq 1 - O(m_{ext}/\epsilon m_0)~.
\label{5.6}$$ In the same vein, using Eq.(\[4.8\]) one gets $\gamma (R) \simeq 1 - 2m_0/R +
O(m_{ext}/\epsilon m_0)$. Thus the effect of the matter on the geometry can be controlled. Notice that in regions sufficiently close to the horizon, even if $m_{ext}/m_0 \ll 1$, a small cloud of matter can still strongly influence the geometry. In what follows, however, we will restrict our attention to the region outside $R=3m_0$, since only in that region of spacetime can we get sensible analytic estimates. This is equivalent to choosing $\epsilon \ge 1/2$.
For $R>3m_0$ the scalar wave equation Eq.(\[4.15\]) can be approximated as $$(\partial_0 +\gamma \partial_R)(1 - {2m_0 \over R})h = \hat h ( 1 - {2m_0 \over R})
{2m_0R \over |R|^3}~,
\label {5.8}$$ with the error terms of order $m_{ext}/m_0$. In the limit of $m_{ext}/m_0 \ll 1$ the equation describes a scalar field propagating on a fixed Schwarzschild background.
One can show, improving a coefficient in an inequality of [@Malec1997], that $$|\hat h|\le {\sqrt{ m_{ext}}\over R^{1/2} \sqrt{8\pi (1-{2m_0\over R})}}.
\label {5.9}$$ (A similar estimate with $m_0$ rather than $m_{ext}$, appears in [@Demetrios].) Let us introduce the Regge-Wheeler coordinate [@MTW] $ r^* = R+2m_0 \ln (R/2m_0 - 1) $ so that $\gamma \partial_R = \partial_{r^*}$. The solution of Eq.(\[5.8\]) can be estimated above and below by solutions of $$(\partial_0 + \partial_{r^*}) \Bigl( (1-{2m_0\over R})h \Bigr) =^+_- { 2m_0 (1-{2m_0\over R})
\sqrt{ m_{ext}}\over R^{5/2} \sqrt{8\pi (1-{2m_0\over R})}}.
\label {5.10}$$ Eqns.(\[5.10\]) are solved by $$\begin{aligned}
&\left(1-{2m_0\over R(r)}\right)h(r^* , t) = h_0(r^* - t)\nonumber\\& \pm \sqrt{{m_{ext}\over 2\pi
}}m_0
\int_{(r - t, 0)}^{(r, t)}{\left(1 - {2m_0 \over R}\right) \over R^2 \sqrt{ R-2m_0}}dv~,
\label {5.12}\end{aligned}$$ here the $dv$ in Eq.(\[5.12\]) is $dr^*$, with $dR/dr^* = 1 -2m_0/R$. If we ignore the second term then the standard redshift is obtained, $$h(r^*(0) + \tau, \tau) ={\left(1 - {2m_0 \over R(r^*(0))}\right)\over
\left(1 - {2m_0 \over R(r^*(0) + \tau)}\right)} h(r^*(0), 0)~,
\label{3.10}$$ while the integral in Eq.(\[5.12\]) can be solved to give $$\begin{aligned}
&\sqrt{{m_{ext} \over 2\pi}}m_0
\int {dR\over R^2 \sqrt{( R-2m_0}} =\nonumber\\ & \sqrt{{m_{ext} \over 16 \pi m_0}}
\left[\sqrt{{2 m_0 \over R}\left(1 - {2m_0 \over R}\right)} +
\arctan \sqrt{{R - 2 m_0 \over 2 m_0}}\right] ~.
\label{5.15}\end{aligned}$$
Thus the total backscattering is bounded by a term of order $\sqrt{m_{ext}/m_0}$. In the limit where $m_{ext}/m_0$ is small, we recover the usual gravitational redshift along an outgoing null ray with an error of order $\sqrt{m_{ext}/m_0}$. This holds even in a strong gravitational field and no ‘quasi-static’ assumption need be made. We need not assume $R \gg m_0$, but only $R \ge 3 m_0$.
Let us assume that initial data at $t = 0$ represents a pure outgoing wave. In other words, $h_-(t = 0) \equiv 0$. We select an $R_A \ge 3m_0$ on the initial slice such that $m_{ext}(R_A)/m_0 \ll 1$. Finally we assume, in addition to $\phi \rightarrow 0$ at infinity, that $\phi = 0$ for $R < R_A$. This guarantees that the radiation is bounded away from the black hole. Consider the future outgoing lightray from $(R_A, 0)$. This is well outside any event horizon, which would be at approximately $R = 2m_0$, so this lightray is really outgoing all the way to null infinity. Note that $m_{ext}(R_A, 0)$ is the maximum value of $m_{ext}$ over the whole wedge bounded by the initial slice and the outgoing null ray. We will estimate $h_-$ and $\hat h$ along this null ray. The integration of Eq.(\[4.10\]) along this ray yields an estimate of the total energy flux across this surface in the inward direction. This will be the total energy loss from the outgoing wave due to backscatter.
Choose a point on the null ray from $R_A$ and label it by $(R_1, T_1)$. To calculate $h_-$ at this point, consider the ingoing future null ray which passes through this point and integrate Eq.(\[5.10\]) along this ingoing ray. This will start from the initial hypersurface at some point $(R_2, 0)$ with $R_2
> R_A$. Along this null ray $R$ monotonically decreases while $m_{ext}$ monotonically increases.
To get an explicit estimate, the integral in Eq.(\[5.12\]) can be further approximated; since $R$ monotonically decreases along the ingoing lightray, we can replace the $\sqrt{1 - 2m_0/R}$ by $\sqrt{1 - 2m_0/R_1}$. This yields
$$\begin{aligned}
&
\int_{R_2}^{R_1} {dR\over R^2 \sqrt{ R-2m_0}} \le \nonumber \\ & \sqrt{{R_1
\over R_1 - 2m_0}} \int {dR \over R^{5/2}}
\le{2 \over 3 R_1^{3/2}}\sqrt{{R_1 \over (R_1 - 2m_0)}} ~.
\label{6.4} \end{aligned}$$
Thus we arrive at $$|\gamma h_-(R)| \le {2\over 3} \sqrt{{m_{ext} \over 2\pi m_0}}
\left[{m_0^{3/2} \over R\sqrt{R - 2m_0}}\right]~.
\label{6.6}$$
We can write, using Eq.(\[4.2\]) and (\[6.6\]), $$(\partial_0 + \partial_{r^*})(R\hat{h}) = \gamma h_- \le {2 \over 3}
\sqrt{{m_{ext} \over 2\pi m_0}}\left[{m_0^{3/2} \over R(R - 2m_0)^{1/2}}\right]
\label{6.7}$$ and a similar inequality with a minus sign to give a lower bound of $\hat h$.
These equations are integrated along the outgoing null ray from $R_A$ to give $$|R\hat{h}(R,t)| \le |R\hat{h}(R_A,0)| +
{2 \over 3} \sqrt{{m_{ext} \over 2\pi }}m_0 \int{dR \over (R - 2m_0)^{3/2}}~,
\label{6.8}$$ Since we demand that $\phi(R_A, 0) = 0$, the first term in Eq.(\[6.8\]) vanishes. Thus we can bound $\hat{h}$ by $$\begin{aligned}
&|\hat{h}(R,t)| \le
{4 \over 3}\sqrt{{m_{ext} \over 2\pi m_0}}{m_0^{3/2} \over R(R_A - 2m_0)^{1/2}} -\nonumber \\ &
{4 \over 3} \sqrt{{m_{ext} \over 2\pi m_0}}{m_0^{3/2} \over R(R - 2m_0)^{1/2}}~.
\label{6.8a}\end{aligned}$$ The last term in Eq.(\[6.8a\]) is strictly larger than $|h_-|$ as given by Eq.(\[6.6\]) if $R > R_A > 4m_0$ (i. e., when [*all*]{} radiation is placed outside $4m_0$). Thus $$|\hat{h} - h_-| \le |\hat{h}| + |h_-| \le {4\over 3} \sqrt{{m_{ext}
\over 2 \pi m_0}}{m_0^{3/2}R_A \over R(R_A - 2m_0)^{3/2}} ~.
\label{6.9}$$
Therefore, from Eq.(\[4.10\]) the total change in $m_{ext}$ satisfies $$\Delta m_{ext} \le
m_{ext} {16\over 9}
\left( {2m_0 \over R_A} \right)^2
\left({1 - m_0/R_A \over 1 - 2m_0/R_A}\right)~.\label{6.10}$$
From this expression we can see how sensitive the amount of backscattering is to the location of the innermost null cone. This estimate becomes meaningless if $R_2 \approx
3.5m_0$ because $\Delta m_{ext}>m_{ext}$. However if $R_A = 6m_0$ we get that less than 25% of the total energy in the exterior field is backscattered. In the case of a neutron star, where $2m_0/R \le 0.1$ (on the surface of the star), we have an upper bound for the backscattered energy of 2% of $m_{ext}$.
In deriving the bound Eq.(\[6.10\]), a number of truncations and approximations were used. While most of them are sharp in the sense that configurations exist that turn the inequality into an equality, we do not believe that all of them can be sharp simultaneously. Therefore Eq.(\[6.10\]) is clearly an overestimate. We present it here, not because it is the best that can be done with this technique, but because it is simple to derive, it is in analytic form, and is easy to interpret physically.
For example, in Eq.(\[6.4\]) we ignored the term depending on $R_2$. We can include this term if the initially outgoing pulses are far enough from the apparent horizon [@EMNOMTCH], so as to sharpen the estimate to $m_{ext}$
$$\Delta m_{ext} \le
{ 16\alpha^2 \over 9} \Bigl( {2m_0\over R_2}\Bigr)^2 {(1 - m_0/R_2) \over 1-2m_0/ R_2}m_{ext}~,
\label{6.20}$$
where $\alpha^2 \simeq 0.4$. Applying this to the case of neutron stars, when the conditions assumed above hold true, we find that the maximal amount of backscattered radiation cannot exceed 1 percent.
Eq.(\[5.12\]) can be integrated in closed form, (Eq.(\[5.15\])), but in Eq.(\[6.4\]) we approximated it to get a simpler expression because we needed to integrate the result again. Numerical integration would obviate the need for this approximation entirely, but the result would be much less transparent.
The tail term can also be bounded using this method. The initially outgoing field, $h_+$, generates a weaker ingoing field, $h_-$, which enters the ‘no-radiation’ zone behind the wavefront. This, in turn, scatters again off the gravitational field to generate a new outgoing field, which turns up at null infinity at a later time, after the first burst of outgoing radiation has gone. This is the so-called ‘tail’ term. Eq.(\[6.8a\]) gives us an estimate for $\hat{h}$ in the ‘no-radiation’ zone. Substituting this back into the scalar wave equation (\[5.8\]), an estimate of the second order $h_+$ along the outgoing null ray is $$\begin{aligned}
&|\gamma h_+(R,t)| \le \nonumber \\ &
{4 \over 3}\sqrt{{m_{ext} \over 2\pi m_0}}{m_0^{5/2} \over \sqrt{R_A - 2m_0}}
\int_{R_A}^R \left[ {1 \over R^3} - {\sqrt{R_A - 2m_0}\over R^3\sqrt{R - 2m_0}}\right]dR~.
\label{6.12}\end{aligned}$$ In the limit as $R \rightarrow \infty$ along the null cone we get $$|h_+(\infty,\infty)| \le {4 \over 9}
\sqrt{{m_{ext} \over 2\pi m_0}}{(m_0/R_2)^{5/2} \over \sqrt{1 - 2m_0/R_2}}~. \label{6.14}$$ This should be compared with the leading term in (\[5.12\]), $h_0$. Using the definition of the external mass $m_{ext}$, we get that the tail term is smaller than the leading term by a factor $(m_0/R_0)^2$.
We expect that photons will behave in a way similar to the massless spin zero field that we have discussed here. Gravitational physics predicts two phenomena for a radiating plasma surrounding a black hole or a neutron star: Redshift diminishes the intensity and frequency of the outgoing radiation while the total energy in the radiation (as measured by the mass function) remains unchanged. Backscattering, the other effect, weakens the overall outgoing radiation, so that the total energy that reaches infinity is reduced. In addition, backscattering changes the shape of extended signals - the leading part weakens while the rest of the impulse gains in intensity. This conclusion follows from a careful analysis of (\[4.6\]) and is supported by the numerical results of [@EMNOMTCH], which show that up to $10\%$ of the radiation emitted at $R = 3m_0$ is ‘shifted’ inside the main impulse; even if it reaches infinity, it does so with a significant delay.
There are two astrophysical situations where those two different aspects of backscattering will play a role. The net efficiency of the black hole - matter system is reduced below what is expected when only the redshift factors are taken into account. This may be of significance in modelling ‘too faint’ galactic nuclei fuelled by black holes, such as the nuclei of M87 [@Rey96]. Second, backscattering may leave imprints in X ray bursts [@Lewin] resulting from energetic processes on the surface of a neutron star or in collisions of compact bodies, such as neutron star - neutron star, neutron star - black hole, or black hole - planetoid mergers. Backscattering will deform the peak and contribute a tail term to the radiation emitted from such short-lived sources. Gamma ray bursts (GRB) are believed by some [@p95] to arise from such collisions and should reveal traces of backscattering. The absence of these effects would give great support to the fireball scenario of GRB’s [@p97].
Another more immediate application is in numerical relativity. Much work has been done in constructing codes to analyse the Einstein - massless scalar field model [@choptuik]. Bounds on the magnitude of the backscattering effect, such as derived here, would offer numerical relativists a reliable test of the correctness and long-time stability of their codes.
J. Hadamard, [*Lectures on Cauchy’s problem in linear partial differential equations*]{}, Yale University Press, Yale, New Haven, 1923. N. Sanchez, [*The black hole: scatterer, absorber and emitter of particles*]{}, hep-th/9711068 and references within; A. Bachelot, [*J. Math. Pures Appl.*]{} [**76**]{}, 155(1997); B.C. Nolan, [*Class. Quantum Grav.*]{} [**14**]{}, 1295(1997). C. Misner, K. Thorne, J. A. Wheeler, [*Gravitation*]{}, Freeman, San Francisco,1973. E. Malec [*Classical and Quantum Gravity*]{} [**13**]{}, 1849(1996). E. Malec, [*Journal of Mathematical Physics*]{} [**38**]{}, 3650(1997). D. Christodoulou, [*Commun. Math. Phys.*]{} [**106**]{}, 487(1987). E. Malec, N. Ó Murchadha, T. Chmaj, [*Spherical scalar waves and gravity - redshift and backscattering*]{}, in preparation. C. S. Reynolds et al., [*The “quiescent” black hole in M87*]{}, astro-ph/9610097. W. H. G. Lewin, J. von Paradijs and R. Tamm, in: X-Ray Binaries (Eds:W. G. Lewis, J. von Paadijs and E. van den Heuwel), Cambridge University Press (1995). for a review see B. Paczyński, [*Publ. Astron. Soc. Pac.*]{} [**167**]{}(718), 1167(1995). B. Paczyński, [*Gamma-ray bursts as hypernovae*]{}, astro-ph/9706232. M. Choptuik, [*Phys. Rev. Lett.*]{} [**70**]{}, 9(1993).
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author:
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Simon Van Eyndhoven, Tom Francart, and Alexander Bertrand, .\
[^1]
bibliography:
- 'bibliografie\_morerecent.bib'
title: 'EEG-informed attended speaker extraction from recorded speech mixtures with application in neuro-steered hearing prostheses'
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EEG signal processing, speech enhancement, auditory attention detection, brain-computer interface, auditory prostheses, blind source separation, multi-channel Wiener filter
[^1]: This work was carried out at the ESAT Laboratory of KU Leuven, in the frame of KU Leuven Research Council BOF/STG-14-005, CoE PFV/10/002 (OPTEC), Research Projects FWO nr. G.0931.14 ‘Design of distributed signal processing algorithms and scalable hardware platforms for energy-vs-performance adaptive wireless acoustic sensor networks’, and . The project HANDiCAMS acknowledges the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open grant number: 323944. The scientific responsibility is assumed by its authors.
S. Van Eyndhoven and A. Bertrand are with KU Leuven, Department of Electrical Engineering (ESAT), Stadius Center for Dynamical Systems, Signal Processing and Data Analytics, Kasteelpark Arenberg 10, box 2446, 3001 Leuven, Belgium (e-mail: , ). T. Francart is with KU Leuven, Department of Neurosciences, Research Group Experimental Oto-rhino-laryngology (e-mail: [email protected]).
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abstract: 'We propose a method for quantum enhanced phase estimation based on continuous variable (CV) quantum teleportation. The phase shift probed by a coherent state can be enhanced by repeatedly teleporting the state back to interact with the phase shift again using a supply of two-mode squeezed vacuum states. In this way, both super resolution and super sensitivity can be obtained due to the coherent addition of the phase shift. The protocol enables Heisenberg limited sensitivity and super-resolution given sufficiently strong squeezing. The proposed method could be implemented with current or near-term technology of CV teleportation.'
author:
- 'J. Borregaard'
- 'T. Gehring'
- 'J. S. Neergaard-Nielsen'
- 'U. L. Andersen'
title: Super sensitivity and super resolution with quantum teleportation
---
Quantum correlations can be used in a number of ways to enhance metrological performance [@giovannetti2004quantum; @giovannetti2011advances; @Escher2011; @Deganreview2017]. Highly entangled states such as NOON and GHZ states can enable Heisenberg limited sensitivity yielding a square root improvement with the number of photons over the standard quantum limit (SQL) [@pan2012; @Rafal2015; @Bouchard2017]. This kind of improvement is particularly useful for probing fragile systems where photon damage limits the allowed number of probe photons. This can be the case in, e.g. imaging of biological systems such as live cells [@Frigault2009], molecules [@Celebrano2011], and proteins [@Piliarik2014]. While this effect has been demonstrated in experiments for small probe sizes [@Mitchell2004; @Afek2010; @Wolfgramm2012; @Facon2016], scaling up the size of the entangled states remains a technological barrier due to their fragility to loss and noise. Other strategies based on the more experimentally accessible squeezed vacuum states have also shown to beat the SQL in various settings [@Caves1981; @Ligo2011; @Ligo2013; @Taylor2013; @Berni2015; @Schafermeier2016]. An alternative strategy is to perform multi-pass protocols with a single probe. This enables both Heisenberg limited sensitivity and super-resolution [@note1] for phase estimation without entangled resources by applying the phase shift to the same probe multiple times [@giovannetti2006quantum; @vanDam2007; @Demkowicz2010]. Its experimental demonstration was realized by surrounding the phase shift system with mirrors to measure a transversally distributed phase shift [@Higgins2007] or an image [@Juffmann2016] with Heisenberg-limited sensitivity. While these approaches have demonstrated the effect of sub-shot noise scaling without entanglement, the former demonstration could only measure a transversally distributed phase shift while both demonstrations were based on post-selection, rendering the efficiency very low.
Here we propose a fundamentally different method based on quantum teleportation for realizing quantum enhanced phase measurements. The essence of our proposal is to repeatedly teleport back the probe to coherently apply the phase shift multiple times (see Fig. \[fig:figure1\]). This circumvents the need for physically redirecting a probe state to the same phase shift multiple times and allows to keep the entangled resources separate from the potentially lossy phase shifting system. We describe how this protocol can be implemented with current technology of continuous variable teleportation using two-mode squeezed states and an initial coherent state as a probe.
In the general setup, we consider some initial probe in a state $\ket{\psi_0}$ which is subject to an unknown phase shift described by a unitary $U(\phi)=e^{i\phi\hat{n}}$, where $\hat{n}$ is the number operator. The goal is to estimate the phase $\phi$. After the interaction, an entangled state is used to teleport the output state $U(\phi)\ket{\psi_0}$ back to interact with the phase shift again. This process is then iterated $m$ times. If the teleportation is perfect, this would correspond to the transformation $\ket{\psi_0}\to \left(U(\phi)\right)^{(m+1)}\ket{\psi_0}=U((m+1)\phi)\ket{\psi_0}$ of the input state where $m$ is the number of teleportations. By coherently applying the phase $(m+1)$ times, the signal can have both super resolution and super sensitivity since it will now depend on $(m+1)\phi$ instead of just $\phi$ [@giovannetti2006quantum; @vanDam2007].
\[t\] ![Sketch of the setup considered. Consecutive pulses of two-mode squeezed vacuum states (illustrated by connected dots) are being supplied by interferring the outputs of two single mode squeezers on a balanced beam splitter. The second mode is delayed by some time $T$ such that it coincides in time with the first mode of the next pair. The second mode is subject to feedback (quadrature displacement) based on previous measurements before a phase shift $U(\phi)=e^{i\phi\hat{n}}$ acts on it. Here, $\hat{n}$ is the photon number operator. Initially, the second mode contains a state $\ket{\psi_0}$ []{data-label="fig:figure1"}](figure1.pdf "fig:"){width="49.00000%"}
As a physical realization of this protocol, we consider the setup illustrated in Fig. \[fig:figure1\] where consecutive two-mode squeezed vacuum states are supplied by interferring the output of two single mode squeezed vacuum sources on a balanced beam splitter [@Furusawa1998]. One mode is delayed by some time $T$ and will subsequently be subject to an unknown phase shift described by the unitary $U(\phi)$. Feedback based on previous measurements is applied before the phase shift. The delay $T$ is chosen such that the phase shifted mode can be interfered with the first mode of the next two-mode squeezed vacuum state on a balanced beam splitter before measurement. This setup is inspired by Ref. [@Yokoyama2013] where the continuous generation of continuous variable cluster states is demonstrated. We choose the measurements and the feedback such that the CV teleportation protocol of Ref. [@Braunstein1998], is realized. In this teleportation protocol, the momentum quadrature of one of the output modes and the position quadrature of the other is measured, which can be achieved with homodyne detection. The feedback then consists of a displacement of the momentum and position quadratures based on the measurement outcomes. For perfect teleportation, infinitely many photons are, in principle, needed in the two-mode squeezed vacuum states. The number of photons actually obtaining the phase shift will nonetheless only depend on the initial input state. For situations where the phase shift is obtained by interaction with some fragile system, the effective number of probe photons actually interacting with the system will be $\sim (m+1) n_{0}$ where $n_{0}$ is the number of photons in the initial state. We will show that Heisenberg limited sensitivity in terms of probe photons can be reached with a simple coherent state as input state and two-mode squeezed states containing on average $\sim m$ photons. Furthermore, the phase resolution can be enhanced by a factor of $m+1$.
We consider a coherent state $|\alpha\rangle$ with ${\langle \hat{p} \rangle}=\alpha$ as the initial probe state $\ket{\psi_0}$. In the setup in Fig. \[fig:figure1\], we can think of displacing the initial vacuum mode of the lower arm before the first measurements. After the interaction of $U(\phi)$, the state will be $\ket{\alpha e^{i\phi}}$. This state is now teleported back to the second mode of the first two-mode squeezed vacuum state following the CV protocol of Ref. [@Braunstein1998]. The two-mode squeezed vacuum state has squeezing parameter $r$ such that ${\langle \left(\hat{x}_2-\hat{x}_3\right)^2 \rangle}=e^{-2r}/2$ , where $\hat{x}_2$, $\hat{x}_3$ are the position quadratures for the two modes. The first mode of the two-mode squeezed vacuum state is mixed with the probe state on a balanced beamsplitter. The output modes of the beamsplitter have position quadratures $\hat{x}'_{1}=(\hat{x}_1+\hat{x}_2)/\sqrt{2}$ and $\hat{x}'_{2}=(\hat{x}_1-\hat{x}_2)/\sqrt{2}$ with similar expressions for the momentum quadratures. Here $\hat{x}_1$ is the position quadrature of the probe state. The quadratures $\hat{p}'_1$ and $\hat{x}'_2$ are now measured giving measurement outcomes $\{p'_1,x'_2\}$. Finally, a feedback implements the displacements $\hat{x}_3\to\hat{x}'_3=\hat{x}_3+g_x\sqrt{2}x'_2$ and $\hat{p}_3\to\hat{p}'_3=\hat{p}_3+g_p\sqrt{2}p'_1$, which concludes the teleportation protocol of Ref. [@Braunstein1998].
\[t\]\
The feedback displaces the quadratures such that the teleported state, $\ket{\psi_{1}}$ will be close to $\ket{\alpha e^{i\phi}}$. The quality of the teleportation will depend on the amount of squeezing contained in the two-mode squeezed vacuum state and the feedback strength quantified by the gains $g_x$ and $g_p$. In the limit of high squeezing, perfect teleportation is obtained for $g_x=g_p=1$. The protocol now repeats itself $m$ times corresponding to $m$ teleportations being performed. Finally, the position quadrature ($\hat{x}_m$) of the final state, $\ket{\psi_{\text{m}}}$ is measured. Assuming gains of $g_x=g_p=1$, the mean and variance of $\hat{x}_m$ is $$\begin{aligned}
{\langle \hat{x}_m \rangle}&=&\bra{\psi_{\text{m}}}\hat{x}_m\ket{\psi_{\text{m}}}=\alpha\sin((m+1)\phi) \label{eq:mean} \\
\text{Var}(\hat{x}_m)&=&\bra{\psi_{\text{m}}}\left(\hat{x}_m^2-{\langle \hat{x}_m \rangle}^2\right)\ket{\psi_{\text{m}}}=\frac{1+2me^{-2r}}{4}. \label{eq:var}\end{aligned}$$ It is clear from Eq. (\[eq:mean\]) that the signal exhibits super-resolution in $\phi$ by a factor of $(m+1)$. The sensitivity of the measurement can be quantified as [@Rafal2015] $$\label{eq:sensitivity}
\sigma_{m}=\frac{\sqrt{\text{Var}(\hat{x}_m)}}{|\delta{\langle \hat{x}_m \rangle}/\delta\phi|}=\frac{\sqrt{1+2me^{-2r}}}{2(m+1)\alpha|\cos((m+1)\phi)|}.$$ Note that the sensitivity exhibits a linear decrease in the number of teleportations $m$ as long as $|\cos((m+1)\phi)|\approx1$ and the squeezing is sufficiently strong such that $2me^{-2r}\ll1$. If $m$ consecutive coherent states $\ket{\alpha}$ had been employed, the sensitivity would have a scaling of $\propto1/(\sqrt{m}\alpha)$. The average number of probe photons, $\bar{n}_{m}$ contained in the state $\ket{\psi_{m}}$ is $$\bar{n}_{m}=\alpha^2+me^{-2r},$$ thus the total average number of probe photons that have interacted with the phase shift operator will be $$\label{eq:photonnumber}
\bar{n}_{\text{total}}=\sum_{i=0}^{m}\bar{n}_{i}=(m+1)\alpha^2+\frac{1}{2}m(m+1)e^{-2r}.$$ If the coherent state contains one photon ($\alpha=1$) on average, we have that $\bar{n}_{\text{total}}=(m+1)(1+\frac{1}{2}me^{-2r})$ and the sensitivity is $$\sigma_{m}=\frac{\sqrt{\left(1+\frac{1}{2}me^{-2r}\right)^2\left(1+2me^{-2r}\right)}}{2\bar{n}_{\text{total}}|\cos((m+1)\phi)|}.$$ Thus if $me^{-2r}\ll1$, the sensitivity exhibits Heisenberg scaling in the number of photons for $|\cos((m+1)\phi)|\approx1$. This sensitivity is similar to what could be obtained using NOON states of $(m+1)$ photons and single photon detection and expresses the ultimate scaling allowed by quantum mechanics [@giovannetti2004quantum].
One of the dominant experimental limitation of the proposed protocol will arguably be the amount of squeezing in the two-mode squeezed vacuum states. This will limit how many teleportations can be performed before the extra noise from the imperfect teleportations will dominate the signal. We therefore consider what the optimum strategy is given a constraint on the amount of squeezing. We consider both a limitation on the amount of squeezing and on the total average number of photons that can interact with the phase shift system. We then optimize over the number of teleportations $m$ and the size of the coherent probe state $\alpha$, to find the strategy that provides the maximum sensitivity for these limitations. Furthermore, we also allow for arbitrary gains $g_x$ and $g_p$. The result of the optimization is shown in Fig. \[fig:figure2\] where we illustrate the performance relative to a standard coherent state protocol with matched average photon number. For such an approach, the sensitivity is simply $\sigma_{\text{coh}}=1/(2\sqrt{\bar{n}_{\text{total}}}|\cos(\phi)|)$ where $\bar{n}_{\text{total}}$ is the average number of probe photons. For $|\cos(\phi)|\approx1$, the coherent state approach exhibits sensitivity at the SQL. Fig. \[fig:figure2\] shows the two effects of the imperfect teleportation; noise is added in the $\hat{x}$-quadrature (see Eq. (\[eq:var\])) and more photons are added to the probe state (see Eq. (\[eq:photonnumber\])). In the minimization, the error from the extra photons added by an imperfect teleportation has smaller weight for higher $n_{\text{total}}$. In the limit where $n_{\text{total}}\gg e^{2r}$, the enhancement is $\sim e^{r}/\sqrt{2}$ and equal gains of $g_x=g_p=1$ are optimal. This is the limit where the extra photons added to the probe state does not have any significant effect on the optimum performance. We note that a similar enhancement in sensitivity could be obtained by using a squeezed coherent state as probe [@Caves1981; @Bondurant1984]. For such protocols, the squeezed photons, however, interact with the phase shift system, which is not the case here. Consequently, this protocol also works in the limit $n_{\text{total}}\ll e^{2r}$ where an enhancement of $\sim\!\!\left(n_{\text{total}}e^{2r}/2\right)^{\frac{1}{4}}$ can be obtained for $g_x=g_p=1$. Note that our numerical optimization shows that larger enhancement can also be obtained for optimized gains in this limit (see Fig. \[fig:figure2\]).
One of the technological challenges of using highly entangled quantum states for enhanced phase measurements is that they are very fragile to losses. Multi-pass protocols share this fragility since losses grow exponentially with the number of passes through the sample [@Demkowicz2010]. This means that if the losses are too high, the sensitivity enhancement of the multi-pass protocol proposed here will vanish. Note, however that while approaches based on NOON states rely on single photon detection this protocol is based on homodyne detection, which in practice is much more efficient. Since imperfect photon detection will add to the overall loss this means that the effective loss may be substantially reduced with this protocol.
\[t\]\
We investigate the performance of the proposed protocol in the presence of both losses acting on the probe state corresponding to a lossy phase shift system and losses acting on the two-mode squeezed vacuum states. We model the losses with fictitious beamsplitters where the unused output port is traced out. To model the lossy phase shift system a fictitious beamsplitter of transmission $\eta_1$ is inserted after the phase shift $U(\phi)$ (see Fig. \[fig:figure1\]). For the loss in the two-mode squeezed vacuum state, fictitious beamsplitters both with transmission $\eta_2$ are inserted for each of the modes. For simplicity, we have assumed equal losses for both modes. Assuming equal gains of $g_x=g_p=1$, the signal and sensitivity after $m$ teleportations for $\{\eta_1,\eta_2\}<1$ is $$\begin{aligned}
{\langle \hat{x} \rangle}_m&=&\alpha\eta_1^{\frac{m+1}{2}}\sin((m+1)\phi) \label{eq:losssignal} \\
\sigma_m&=&\frac{\sqrt{1+2\eta_1\frac{1-\eta_1^m}{1-\eta_1}\left(\eta_2e^{-2r}+1-\eta_2\right)}}{2(m+1)\alpha\eta_1^{\frac{m+1}{2}}|\cos((m+1)\phi)|}. \label{eq:sensitivityloss}\end{aligned}$$ As expected, the loss on the probe state ($\eta_1$) enters in the expression for the sensitivity exponentially in $m$, while loss on the two-mode squeezed vacuum states ($\eta_2$) only has a linear effect in $m$. The effect of $\eta_2<1$ on the sensitivity is equivalent to having a limited squeezing of $r_{\text{lim}}=-\frac{1}{2}\ln\left(\eta_2e^{-2r}+1-\eta_2\right)$. This also holds when considering the average number of total probe photons incident on the phase shift system. For $m$ teleportations and gains of $g_x=g_p=1$, we have that $$\begin{aligned}
\bar{n}_{\text{total}}&=&\frac{1-\eta_1^{m+1}}{1-\eta_1}\alpha^2\nonumber \\
&&+\frac{m(1-\eta_1)-\eta_1(1-\eta_1^m)}{(1-\eta_1)^2}\left(\eta_2e^{-2r}+1-\eta_2\right). \qquad \label{eq:photonloss}\end{aligned}$$ Note that by taking the limit $\eta_1\to1$ for $\eta_2=1$, Eqs. (\[eq:losssignal\])-(\[eq:photonloss\]) reduces to Eqs. (\[eq:mean\]), (\[eq:sensitivity\]), and (\[eq:photonnumber\]). If excess noise on the squeezed states is included by mixing in thermal states of average photon number $\bar{n}$ instead of vacuum in the fictitious beam splitters ($\eta_2$), one would have that $r_{\text{lim}}=-\frac{1}{2}\ln\left(\eta_2e^{-2r}+(1+2\bar{n})(1-\eta_2\right))$. We will assume that $\bar{n}\ll1$ such that excess noise can be neglected. To see the effect of finite losses, we again compare the protocol to the simple coherent strategy for which $\sigma_{\text{coh}}=1/(2\sqrt{\eta_1 n_{\text{total}}}|\cos(\phi)|)$ in the presence of loss. The result of the optimization is shown in Fig. \[fig:figure3\]. While a small improvement was found by optimizing the gains, near optimal performance is reached for $g_x=g_p=1$. The error from losses in the two-mode squeezed vacuum state limits the gain in the same way as finite squeezing does for the lossless case. Consequently, when these losses dominate the error, the enhancement is $\sim1/\sqrt{2(1-\eta_2)}$ and no enhancement is possible for $\eta_2\sim1/2$. When losses in the probe state limit the enhancement, the optimum performance is effectively found as a tradeoff between the $\sqrt{m+1}$ enhancement due to the teleportation and the exponential reduction due to the loss. As a result, we find that the enhancement is $~\sim\sqrt{2/(3(1-\eta_1))}$ and no enhancement is possible for $\eta_1\sim1/3$. We note that while losses quickly reduce the enhancement, the scheme still exhibits enhanced sensitivity compared to the standard coherent state probe even for substantial losses.
Our method can be easily extended to a multi-mode scheme to demonstrate Heisenberg-limited imaging. This can be realized by replacing the single-mode teleportation scheme with a multi-mode scheme in which multiple higher-order spatial modes are simultaneously teleported [@Sokolov2001]. Using such a multi-mode approach, sub-shot noise and eventually Heisenberg-limited microscopy can be realized
In conclusion, we have shown how both super-sensitivity and super-resolution can be obtained for an optical phase measurement using continuous variable quantum teleportation based on two-mode squeezed vacuum states. For negligible losses, the protocol can exhibit Heisenberg limited sensitivity ($\sim1/N$) for squeezing $2Ne^{-2r}\ll1$ and increase the resolution by a factor of $N$, where $N$ is the number of probes. While this is equivalent to the enhancement possible with $N$-photon NOON states and single photon detection [@Rafal2015], the protocol proposed here relies on homodyne detection, which generally is more efficient than single photon detection. As a consequence of the super-resolution, the phase to be estimated should, in principle, be localized within a window of $1/N$ to reach the Heisenberg limit as for a NOON or GHZ state approach [@giovannetti2004quantum; @pan2012]. However, methods developed to estimate arbitrary phases [@Berni2015; @Xiang2010], in particular, for NOON [@mitchell2005] and GHZ states [@kessler2014] might also be employed in a straightforward way to this scheme. The latter method uses GHZ states of different sizes in order to estimate the digits of the phase allowing for arbitrary phase estimation [@kessler2014]. The same technique could be employed here by operating with different number of teleportations before readout, which effectively corresponds to sending entangled states of varying sizes. We have also studied the effect of photon loss on the scheme both for loss in the two-mode squeezed vacuum states used for teleportation (limits the effective squeezing) and for loss on the probes corresponding to a lossy phase shift system. While loss quickly reduces the performance, the protocol may still provide super-sensitivity for loss on the order of several percent. For an effective squeezing of 13 dB ($r=1.5$), a 6 dB enhancement of the sensitivity ($\sigma^2$) may be obtained with $100$ probe photons and 10% loss in the phase shift system.
While the specific protocol studied here employed CV teleportation of a coherent state with two-mode squeezed vacuum states, the generic setup of teleporting back a probe state to interact with the phase shift system multiple times may be extended to other scenarios. In particular, non-Gaussian states such as photon subtracted two-mode squeezed states [@Kaushik2015] may be considered for enhanced teleportation performance or different probe states providing better single-shot estimation [@Bouchard2017]. A discrete variable variant of the protocol could also be envisioned using 1D-cluster states emitted by single quantum emitters [@Lindner2009]. Here every second qubit could probe the phase shift while the remaining qubits are used to teleport the phase information on from probe to probe.
We would like to thank Matthias Christandl for valuable feedback on the manuscript and helpful discussions. We also acknowledge funding from Center for Macroscopic Quantum States (bigQ DNRF142) and Qubiz - Quantum Innovation Center. JB acknowledges financial support from the European Research Council (ERC Grant Agreements no 337603), the Danish Council for Independent Research (Sapere Aude), VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059).
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---
abstract: 'Let ${\mathcal C}$ be a pocategory, $FI({\mathcal C})$ the finitary incidence ring of ${\mathcal C}$ and ${\varphi}$ a Jordan isomorphism of $FI({\mathcal C})$ onto an associative ring $A$. We study the problem of decomposition of ${\varphi}$ into the (near-)sum of a homomorphism and an anti-homomorphism. In particular, we obtain generalizations of the main results of [@Akkurts-Barker; @BFK].'
address:
- 'Departamento de Matemática, Universidade Estadual de Maringá, Maringá — PR, CEP: 87020–900, Brazil'
- 'Departamento de Matemática, Universidade Estadual de Maringá, Maringá — PR, CEP: 87020–900, Brazil'
- 'Departamento de Matemática, Universidade Federal de Santa Catarina, Campus Reitor João David Ferreira Lima, Florianópolis — SC, CEP: 88040–900, Brazil'
author:
- Rosali Brusamarello
- 'Érica Z. Fornaroli'
- Mykola Khrypchenko
bibliography:
- 'bibl.bib'
title: |
Jordan Isomorphisms\
of the Finitary Incidence Ring\
of a Partially Ordered Category
---
Introduction {#intro .unnumbered}
============
The study of Jordan maps on the ring of upper triangular matrices was iniciated by L. Molnár and P. Šemrl [@Molnar-Semrl98]. They proved in [@Molnar-Semrl98 Corollary 4] that each Jordan automorphism on the ring $T_n(R)$ of upper triangular matrices is either an automorphism or an anti-automorphism, where $ R$ is a field with at least $3$ elements. K. I. Beidar, M. Brešar and M. A. Chebotar generalized this result in [@BeBreChe], where they considered the case when $R$ is a $2$-torsionfree unital commutative ring without non-trivial idempotents and $n\ge 2$ and showed that each Jordan isomorphism of $T_n( R)$ onto a $R$-algebra is either an isomorphism or an anti-isomorphism. D. Benkovič, using the new notion of near-sum, described in [@Benkovic05 Theorem 4.1] all Jordan homomorphisms $T_n(R)\to A$, where $R$ is an arbitrary $2$-torsionfree commutative ring and $A$ is an $R$-algebra. E. Akkurt, M. Akkurt and G. P. Barker extended in [@Akkurts-Barker Theorem 2.1] Benkovič’s result to structural matrix algebras $T_n(R,\rho)$, where $\rho$ is either a partial order or a quasi-order each of whose equivalence classes contains at least $2$ elements.
It should be noted that the structural matrix algebra $T_n( R,\rho)$ is isomorphic to the incidence algebra $I(P,R)$ of the ordered set $P=(\{1, \ldots, n\},\rho)$ over $R$ as defined in [@SpDo]. A generalization of incidence algebras to the case of non-locally finite posets appeared in [@Khripchenko-Novikov09], where the authors defined the so-called finitary incidence algebras $FI(P,R)$. R. Brusamarello, E. Z. Fornaroli and M. Khrypchenko [@BFK] extended one of the main results of [@Akkurts-Barker Theorem 2.1] to the case of $FI(P,R)$, namely, they showed that each $R$-linear Jordan isomorphism of $FI(P,R)$ onto an $R$-algebra $A$ is the near-sum of a homomorphism and an anti-homomorphism, where $P$ is an arbitrary partially ordered set and $R$ is a commutative $2$-torsionfree unital ring.
Our initial goal was to generalize [@BFK Theorem 3.13] to the case of a quasiordered set $P$. This case is technically more complicated, and to deal with it, we used the notion of the finitary incidence ring $FI({\mathcal C})$ of a pocategory, introduced by M. Khrypchenko in [@Khr10-quasi]. This permitted to us to obtain generalizations of both [@Akkurts-Barker Theorem 2.1] and [@BFK Theorem 3.13] at the same time. Moreover, the methods elaborated in our paper work for non-necessarily $R$-linear Jordan isomorphisms of incidence rings.
The structure of the article is as follows.
In \[sec-prelim\] we recall the definitions and the main properties of Jordan homomorphisms and finitary incidence algebras. We recall, for instance, that as abelian group, $FI({\mathcal C})=D({\mathcal C})\oplus FZ({\mathcal C})$, where $D({\mathcal C})$ is the subring of $FI({\mathcal C})$ of all diagonal elements and $FZ({\mathcal C})$ is the ideal of $FI({\mathcal C})$ consisting of ${\alpha}\in FI({\mathcal C})$ with ${\alpha}_{xy}=0_{xy}$ for $x=y$.
In \[sec-jiso-FI(C)\] we study the decomposition of a Jordan isomorphism ${\varphi}:FI({\mathcal C}) \to A$ into a near-sum. First we show that ${\varphi}|_{FZ({\mathcal C})}$ decomposes as the sum of two additive maps $\psi, {\theta}:FI({\mathcal C})\to A$. Then, using several technical lemmas, we prove in \[psi-and-0-hom-and-anti-hom\] that the maps $\psi$ and ${\theta}$ are a homomorphism and an anti-homomorphism, respectively. The main result of \[sec-jiso-FI(C)\] is \[vf-near-sum-of-tl-psi-and-tl-theta\], which says that ${\varphi}:FI({\mathcal C}) \to A$ the near-sum of two additive maps ${\tilde}\psi,{\tilde}{\theta}:FI({\mathcal C})\to A$ with respect to $D({\mathcal C})$ and $FZ({\mathcal C})$. Moreover, we give necessary and sufficient conditions under which ${\tilde}\psi$ and ${\tilde}{\theta}$ are a homomorphism and an anti-homomorphism, respectively.
\[jord-iso-FI(P\_R)\] is devoted to the decomposition of a Jordan isomorphism ${\varphi}:FI({\mathcal C}) \to A$ as the sum of a homomorphism and an anti-homomorphism. In \[vf|\_D(C)-is-the-sum-of-psi-and-theta\] we give necessary and sufficient conditions under which ${\varphi}|_{D(C)}$ admits such a decomposition. Finally, we restrict ourselves to the case ${\mathcal C}={\mathcal C}(P,R)$, where $P$ is a quasiordered set such that $1<|\bar{x}|<\infty$ for all $\bar{x}\in \bar{P}$, $R$ is a commutative ring and $A$ is an $R$-algebra. We prove in \[vf=psi+0\] that each $R$-linear Jordan isomorphism ${\varphi}:FI(P,R) \to A$ is the sum of a homomorphism and an anti-homomorphism.
Preliminaries {#sec-prelim}
=============
Jordan homomorphisms {#jordan-homo}
--------------------
Let $R$ and $S$ be associative rings. An additive map $\varphi:R\to S$ is called a *Jordan homomorphism*, if it satisfies $$\begin{aligned}
{\varphi}(r^2)&={\varphi}(r)^2,\label{vf(r^2)}\\
{\varphi}(rsr)&={\varphi}(r){\varphi}(s){\varphi}(r),\label{vf(rsr)}\end{aligned}$$ for all $r,s\in R$. A bijective Jordan homomorphism is called a *Jordan isomorphism*.
Each homomorphism, as well as an anti-homomorphism, is a Jordan homomorphism. The sum of a homomorphism $\psi:R\to S$ and an anti-homomorphism ${\theta}:R\to
S$ is a Jordan homomorphism, if $\psi(r){\theta}(s)={\theta}(s)\psi(r)=0$ for all $r,s\in R$. A more general construction was introduced by D. Benkovič in [@Benkovic05]. Suppose that $R$ can be represented as the direct sum of additive subgroups $R_0\oplus R_1$, where $R_0$ is a subring of $R$ and $R_1$ is an ideal of $R$. Let $\psi,{\theta}:R\to S$ be additive maps, such that $\psi|_{R_0}={\theta}|_{R_0}$ and $\psi(r){\theta}(s)={\theta}(s)\psi(r)=0$ for all $r,s\in
R_1$. Then the *near-sum* of $\psi$ and ${\theta}$ (with respect to $R_0$ and $R_1$) is the additive map ${\varphi}:R\to S$, which satisfies ${\varphi}|_{R_0}=\psi|_{R_0}={\theta}|_{R_0}$ and ${\varphi}|_{R_1}=\psi|_{R_1}+{\theta}|_{R_1}$. If $\psi$ is a homomorphism and ${\theta}$ is an anti-homomorphism, then one can show that ${\varphi}$ is a Jordan homomorphism in this case.
Let us now mention some basic facts on Jordan homomorphisms. Applying ${\varphi}$ to $(r+s)^2$ and using \[vf(r\^2)\], we get $$\begin{aligned}
\label{vf(rs+sr)}
{\varphi}(rs+sr)={\varphi}(r){\varphi}(s)+{\varphi}(s){\varphi}(r).\end{aligned}$$ The substitution of $r$ by $r+t$ in \[vf(rsr)\] gives $$\begin{aligned}
{\varphi}(rst+tsr)={\varphi}(r){\varphi}(s){\varphi}(t)+{\varphi}(t){\varphi}(s){\varphi}(r).\end{aligned}$$ We shall also use the following fact (see Corollary 2 of [@Jacobson-Rickart50 Theorem 1]). If $e$ is an idempotent, such that $er=re$, then $$\begin{aligned}
\label{vf(e)vf(r)=vf(r)vf(e)}
{\varphi}(r){\varphi}(e)={\varphi}(e){\varphi}(r)={\varphi}(re).\end{aligned}$$ In particular, if $R$ has identity $1$, then ${\varphi}(1)$ is the identity of ${\varphi}(R)$. Another particular case of \[vf(e)vf(r)=vf(r)vf(e)\]: if $er=re=0$, then $$\begin{aligned}
\label{vf(e)vf(r)=0}
{\varphi}(e){\varphi}(r)={\varphi}(r){\varphi}(e)=0.\end{aligned}$$
From now on, all rings will be associative and with $1$.
Finitary incidence rings {#fin-inc-ring}
------------------------
Recall from [@Khr16] (see also [@Khr10-quasi] for a slightly stronger definition) that a *pocategory* is a preadditive small category ${\mathcal C}$ with a partial order $\le$ on the set ${\operatorname{\mathrm{Ob}}{{\mathcal C}}}$ of its objects. Denote by $I({\mathcal C})$ the set of the formal sums $$\begin{aligned}
\label{formal-sum-in-I(C)}
\alpha=\sum_{x\le y}\alpha_{xy}e_{xy},\end{aligned}$$ where $x,y\in{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$, $\alpha_{xy}\in{\operatorname{\mathrm{Mor}}(x,y)}$ and $e_{xy}$ is a symbol. It is an abelian group under the addition coming from the addition of morphisms in ${\mathcal C}$. We shall also consider the series $\alpha$ of the form \[formal-sum-in-I(C)\], whose indices run through a subset $X$ of the ordered pairs $(x,y)$, $x,y\in{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$, $x\le y$, in which case $\alpha_{xy}$ will be meant to be the zero $0_{xy}$ of ${\operatorname{\mathrm{Mor}}(x,y)}$ for $(x,y)\not\in X$.
The sum \[formal-sum-in-I(C)\] is called a [*finitary series*]{}, whenever for any pair of $x,y\in{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$ with $x<y$ there exists only a finite number of $u,v\in{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$, such that $x\le u<v\le y$ and $\alpha_{uv}\ne 0_{uv}$. The set of finitary series, denoted by $FI({\mathcal C})$, is an additive subgroup of $I({\mathcal C})$, and it is closed under the convolution $$\begin{aligned}
\alpha\beta=\sum_{x\le y}\left(\sum_{x\le z\le y}\alpha_{xz}\beta_{zy}\right)e_{xy},\end{aligned}$$ where $\alpha,\beta\in FI({\mathcal C})$. Thus, $FI({\mathcal C})$ is a ring, called the [*finitary incidence ring of ${\mathcal C}$*]{}. The identity element $\delta$ of $FI({\mathcal C})$ is the series $\delta=\sum_{x\in{\operatorname{\mathrm{Ob}}{{\mathcal C}}}}{\mathrm{id}}_xe_{xx}$, where ${\mathrm{id}}_x$ is the identity morphism from ${\operatorname{\mathrm{End}}(x)}$.
An element ${\alpha}\in FI({\mathcal C})$ will be said to be [*diagonal*]{}, if ${\alpha}_{xy}=0_{xy}$ for $x\ne y$. The subring of $FI({\mathcal C})$ consisting of the diagonal elements will be denoted by $D({\mathcal C})$. Clearly, as an abelian group, $FI({\mathcal C})=D({\mathcal C})\oplus FZ({\mathcal C})$, where $FZ({\mathcal C})$ is the ideal of $FI({\mathcal C})$ consisting of ${\alpha}\in FI({\mathcal C})$ with ${\alpha}_{xy}=0_{xy}$ for $x=y$. Thus, each ${\alpha}\in FI({\mathcal C})$ can be uniquely decomposed as ${\alpha}={\alpha}_D+{\alpha}_Z$, where ${\alpha}_D\in
D({\mathcal C})$ and ${\alpha}_Z\in FZ({\mathcal C})$.
Observe that $$\begin{aligned}
\label{af_xye_xy-cdot-bt_uve_uv}
{\alpha}_{xy}e_{xy}\cdot {\beta}_{uv}e_{uv}=
\begin{cases}
{\alpha}_{xy}{\beta}_{uv}e_{xv}, & \mbox{if $y=u$},\\
0, & \mbox{otherwise}.
\end{cases}\end{aligned}$$ In particular, the elements $e_x:={\mathrm{id}}_x e_{xx}$, $x\in {\operatorname{\mathrm{Ob}}{(}}{\mathcal C})$, are pairwise orthogonal idempotents of $FI({\mathcal C})$, and for any ${\alpha}\in FI({\mathcal C})$ $$\begin{aligned}
\label{e_x-alpha-e_y}
e_x{\alpha}e_y=\begin{cases}
{\alpha}_{xy}e_{xy}, & \mbox{ if }x\le y,\\
0, & \mbox{ otherwise}.
\end{cases}\end{aligned}$$ Consequently, for all ${\alpha},{\beta}\in FI({\mathcal C})$ $$\begin{aligned}
{\alpha}={\beta}&{\Leftrightarrow}\forall x\le y:\ e_x{\alpha}e_y=e_x{\beta}e_y\notag\\
&{\Leftrightarrow}\begin{cases}
\forall x<y:& e_x{\alpha}e_y+e_y{\alpha}e_x=e_x{\beta}e_y+e_y{\beta}e_x,\\
\forall x:& e_x{\alpha}e_x=e_x{\beta}e_x.
\end{cases}\end{aligned}$$
Given $X\subseteq {\operatorname{\mathrm{Ob}}{{\mathcal C}}}$, we shall use the notation $e_X$ for the diagonal idempotent $\sum_{x\in X}{\mathrm{id}}_x e_{xx}$. In particular, $e_x=e_{\{x\}}$. Note that $e_Xe_Y=e_{X\cap Y}$, so $e_xe_X=e_x$ for $x\in X$, and $e_xe_X=0$ otherwise.
Let $(P,\preceq)$ be a preordered set and $R$ a ring. We recall the construction of the pocategory ${\mathcal C}(P,R)$, introduced in [@Khr16].
Denote by $\sim$ the natural equivalence relation on $P$, namely, $x\sim y\Leftrightarrow x\preceq y\preceq x$, and by $\bar P$ the quotient set $P/\sim$. Define ${\operatorname{\mathrm{Ob}}{{\mathcal C}}}(P,R)$ to be $\bar P$ with the induced partial order $\leq$. For any pair $\bar x,\bar y \in {\operatorname{\mathrm{Ob}}{{\mathcal C}}}(P,R)$, let ${\operatorname{\mathrm{Mor}}(\bar{x},\bar{y})}=RFM_{\bar x\times \bar y}(R)$, where $RFM_{\bar I\times \bar J}(R)$ denotes the additive group of row-finite matrices over $R$, whose rows are indexed by the elements of $I$ and columns by the elements of $J$. The composition of morphisms in ${\mathcal C}(P,R)$ is the matrix multiplication, which is defined by the row-finiteness condition.
The *finitary incidence ring of $P$ over $R$*, denoted by $FI(P,R)$, is by definition $FI({\mathcal C}(P,R))$. Furthermore, $D(P,R):=D({\mathcal C}(P,R))$ and $FZ(P,R):=FZ({\mathcal C}(P,R))$.
Jordan isomorphisms of $FI({\mathcal C})$ {#sec-jiso-FI(C)}
=========================================
Let $A$ be a ring, ${\mathcal C}$ an arbitrary pocategory and ${\varphi}:FI({\mathcal C})\to A$ a Jordan isomorphism. We first adapt the ideas from [@Akkurts-Barker; @BFK] to decompose ${\varphi}|_{FZ({\mathcal C})}$ as the sum of two additive maps $\psi,{\theta}:FZ({\mathcal C})\to A$.
Decomposition of ${\varphi}|_{FZ({\mathcal C})}$
------------------------------------------------
The following two lemmas are straightforward generalizations of [@BFK Lemmas 3.1–3.2], so we omit their proofs.
Let ${\varphi}:FI({\mathcal C})\to A$ be a Jordan homomorphism. Then for any ${\alpha}\in FI({\mathcal C})$ one has $$\begin{aligned}
\forall x<y:\
{\varphi}({\alpha}_{xy}e_{xy})&={\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_y)+{\varphi}(e_y){\varphi}({\alpha}){\varphi}(e_x),\label{vf(af_xye_xy)=vf(e_x)vf(af)vf(e_y)+vf(e_y)vf(af)vf(e_x)}\\
\forall x:\ {\varphi}({\alpha}_{xx}e_{xx})&={\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_x).\label{vf(af_xxe_x)=vf(e_x)vf(af)vf(e_x)}
\end{aligned}$$
\[a=b-in-A\] Let ${\varphi}:FI({\mathcal C})\to A$ be a Jordan isomorphism. Then for all $a,b\in A$ $$a=b{\Leftrightarrow}\begin{cases}
\forall x<y:& {\varphi}(e_x)a{\varphi}(e_y)+{\varphi}(e_y)a{\varphi}(e_x)={\varphi}(e_x)b{\varphi}(e_y)+{\varphi}(e_y)b{\varphi}(e_x),\\
\forall x:& {\varphi}(e_x)a{\varphi}(e_x)={\varphi}(e_x)b{\varphi}(e_x).
\end{cases}$$
Observe also from \[vf(af\_xye\_xy)=vf(e\_x)vf(af)vf(e\_y)+vf(e\_y)vf(af)vf(e\_x),vf(e)vf(r)=0\] that $$\begin{aligned}
{\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_y)&={\varphi}(e_x){\varphi}({\alpha}_{xy}e_{xy}){\varphi}(e_y),\label{vf(e_x)vf(af)vf(e_y)=vf(e_x)vf(af_xye_xy)vf(e_y)}\\
{\varphi}(e_y){\varphi}({\alpha}){\varphi}(e_x)&={\varphi}(e_y){\varphi}({\alpha}_{xy}e_{xy}){\varphi}(e_x),\label{vf(e_y)vf(af)vf(e_x)=vf(e_y)vf(af_xye_xy)vf(e_x)}\end{aligned}$$ for all $x<y$.
The next two lemmas will lead us to the definition of $\psi$ and ${\theta}$.
\[terms-of-psi-and-0\] Given a Jordan isomorphism ${\varphi}:FI({\mathcal C})\to A$, ${\alpha}\in FI({\mathcal C})$ and $x<y$, there exists a (unique) pair of ${\alpha}'_{xy},{\alpha}''_{xy}\in{\operatorname{\mathrm{Mor}}(x,y)}$, such that $$\begin{aligned}
{\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_y)={\varphi}({\alpha}'_{xy}e_{xy}),\label{defn-of-af'_xy}\\
{\varphi}(e_y){\varphi}({\alpha}){\varphi}(e_x)={\varphi}({\alpha}''_{xy}e_{xy}).\label{defn-of-af''_xy}
\end{aligned}$$
We construct ${\alpha}'_{xy}$, the construction of ${\alpha}''_{xy}$ is similar. Since ${\varphi}$ is bijective, there exists a unique ${\beta}\in FI({\mathcal C})$, such that ${\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_y)={\varphi}({\beta})$. Apply \[vf(af\_xye\_xy)=vf(e\_x)vf(af)vf(e\_y)+vf(e\_y)vf(af)vf(e\_x)\] to ${\beta}$ and use \[vf(e)vf(r)=0,vf(r\^2)\] to obtain $$\begin{aligned}
{\varphi}({\beta}_{xy}e_{xy})&={\varphi}(e_x){\varphi}({\beta}){\varphi}(e_y)+{\varphi}(e_y){\varphi}({\beta}){\varphi}(e_x)\\
&={\varphi}(e_x){\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_y){\varphi}(e_y)+{\varphi}(e_y){\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_y){\varphi}(e_x)\\
&={\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_y)\\
&={\varphi}({\beta}).
\end{aligned}$$ Now take ${\alpha}'_{xy}={\beta}_{xy}$.
\[af’-and-af”-are-finitary-series\] Let ${\varphi}:FI({\mathcal C})\to A$ be a Jordan isomorphism, ${\alpha}\in FZ({\mathcal C})$ and $\{{\alpha}'_{xy}\}_{x<y},\{{\alpha}''_{xy}\}_{x<y}\subseteq{\operatorname{\mathrm{Mor}}({\mathcal C})}$ defined by \[defn-of-af’\_xy,defn-of-af”\_xy\]. Then $$\begin{aligned}
{\alpha}'&=\sum_{x<y}{\alpha}'_{xy}e_{xy},\label{defn-of-af'}\\
{\alpha}''&=\sum_{x<y}{\alpha}''_{xy}e_{xy}\label{defn-of-af''}
\end{aligned}$$ are finitary series, such that $$\begin{aligned}
\label{af=af'+af''}
{\alpha}={\alpha}'+{\alpha}''.\end{aligned}$$ Moreover, the maps ${\alpha}\mapsto{\alpha}'$ and ${\alpha}\mapsto{\alpha}''$ are additive.
We first prove \[af=af’+af”\] and thus reduce all the assertions about ${\alpha}''$ to the corresponding assertions about ${\alpha}'$. Indeed, \[af=af’+af”\] easily follows from \[vf(af\_xye\_xy)=vf(e\_x)vf(af)vf(e\_y)+vf(e\_y)vf(af)vf(e\_x),defn-of-af’\_xy,defn-of-af”\_xy\] and bijectivity of ${\varphi}$, since $$\begin{aligned}
{\varphi}({\alpha}_{xy}e_{xy})&={\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_y)+{\varphi}(e_y){\varphi}({\alpha}){\varphi}(e_x)\\
&={\varphi}({\alpha}'_{xy}e_{xy})+{\varphi}({\alpha}''_{xy}e_{xy})\\
&={\varphi}(({\alpha}'_{xy}+{\alpha}''_{xy})e_{xy}).
\end{aligned}$$
Now suppose that ${\alpha}'_{uv}\ne 0_{uv}$ for an infinite number of ordered pairs $x\le u<v\le y$. Then ${\varphi}(e_u){\varphi}({\alpha}){\varphi}(e_v)\ne 0$ by \[defn-of-af’\_xy\]. It follows from \[vf(e\_x)vf(af)vf(e\_y)=vf(e\_x)vf(af\_xye\_xy)vf(e\_y)\] that ${\varphi}({\alpha}_{uv}e_{uv})\ne 0$. Hence, ${\alpha}_{uv}\ne
0_{uv}$ for $x\le u<v\le y$ contradicting the fact that ${\alpha}\in FI({\mathcal C})$. Thus, ${\alpha}'\in FI({\mathcal C})$, and additivity of ${\alpha}\mapsto{\alpha}'$ is explained by additivity of ${\varphi}$ and distributivity of multiplication in $A$.
Thus, with any Jordan isomorphism ${\varphi}:FI({\mathcal C})\to A$ we may associate $\psi,{\theta}:FZ({\mathcal C})\to A$ given by $$\begin{aligned}
\psi({\alpha})&={\varphi}({\alpha}'),\label{defn-of-psi}\\
{\theta}({\alpha})&={\varphi}({\alpha}''),\label{defn-of-0}\end{aligned}$$ where ${\alpha}\in FZ({\mathcal C})$ and ${\alpha}',{\alpha}''$ are defined by means of \[defn-of-af’\_xy,defn-of-af”\_xy,defn-of-af’,defn-of-af”\]. By \[af’-and-af”-are-finitary-series\] the maps $\psi$ and ${\theta}$ are well defined and additive.
\[vf|\_FZ-is-sum\] Let ${\varphi}:FI({\mathcal C})\to A$ be a Jordan isomorphism. Then $$\begin{aligned}
{\varphi}|_{FZ({\mathcal C})}=\psi+{\theta}.
\end{aligned}$$
If ${\alpha}\in FZ({\mathcal C})$, then $\psi({\alpha})={\varphi}({\alpha}')$ and ${\theta}({\alpha})={\varphi}({\alpha}'')$ by \[defn-of-psi,defn-of-0\], so ${\varphi}({\alpha})=\psi({\alpha})+{\theta}({\alpha})$ thanks to \[af=af’+af”\].
Properties of $\psi$ and ${\theta}$
-----------------------------------
In this subsection we prove that the maps $\psi$ and ${\theta}$ are in fact a homomorphism and an anti-homomorphism. We first show that they satisfy the properties analogous to the ones given in [@BFK Propositions 3.5, 3.12], as the next lemma shows.
\[vf(e\_x)psi(af)vf(e\_y)-etc\] Let ${\varphi}:FI({\mathcal C})\to A$ be a Jordan isomorphism, $\psi,{\theta}:FZ({\mathcal C})\to A$ the associated maps given by \[defn-of-psi,defn-of-0\] and ${\alpha}\in
FZ({\mathcal C})$. Then for all $x<y$ $$\begin{aligned}
{\varphi}(e_x)\psi({\alpha}){\varphi}(e_y)&={\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_y),\label{vf(e_x)psi(af)vf(e_y)=vf(e_x)vf(af)vf(e_y)}\\
{\varphi}(e_y){\theta}({\alpha}){\varphi}(e_x)&={\varphi}(e_y){\varphi}({\alpha}){\varphi}(e_x)\notag \end{aligned}$$ and $$\begin{aligned}
\label{vf(e_y)psi(af)vf(e_x)=vf(e_x)0(af)vf(e_y)=vf(e_x)psi(af)vf(e_x)=vf(e_x)0(af)vf(e_x)=0}
{\varphi}(e_y)\psi({\alpha}){\varphi}(e_x)={\varphi}(e_x)\psi({\alpha}){\varphi}(e_x)={\varphi}(e_x){\theta}({\alpha}){\varphi}(e_y)={\varphi}(e_x){\theta}({\alpha}){\varphi}(e_x)=0.
\end{aligned}$$
By \[vf(r\^2),vf(e\_x)vf(af)vf(e\_y)=vf(e\_x)vf(af\_xye\_xy)vf(e\_y),defn-of-af’\_xy\] $$\begin{aligned}
{\varphi}(e_x)\psi({\alpha}){\varphi}(e_y)&={\varphi}(e_x){\varphi}({\alpha}'){\varphi}(e_y)={\varphi}(e_x){\varphi}({\alpha}'_{xy}e_{xy}){\varphi}(e_y)\\
&={\varphi}(e_x){\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_y){\varphi}(e_y)={\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_y).
\end{aligned}$$ Similarly, by \[vf(e)vf(r)=0,vf(e\_y)vf(af)vf(e\_x)=vf(e\_y)vf(af\_xye\_xy)vf(e\_x),defn-of-af’\_xy\] $$\begin{aligned}
{\varphi}(e_y)\psi({\alpha}){\varphi}(e_x)&={\varphi}(e_y){\varphi}({\alpha}'){\varphi}(e_x)={\varphi}(e_y){\varphi}({\alpha}'_{xy}e_{xy}){\varphi}(e_x)\\
&={\varphi}(e_y){\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_y){\varphi}(e_x)=0.
\end{aligned}$$ Finally, by \[vf(af\_xxe\_x)=vf(e\_x)vf(af)vf(e\_x)\] $$\begin{aligned}
{\varphi}(e_x)\psi({\alpha}){\varphi}(e_x)={\varphi}(e_x){\varphi}({\alpha}'){\varphi}(e_x)={\varphi}({\alpha}'_{xx}e_{xx})=0,
\end{aligned}$$ the latter equality being explained by the fact that ${\alpha}'\in FZ({\mathcal C})$.
The identities involving ${\theta}$ are proved in an analogous way using \[vf(af\_xxe\_x)=vf(e\_x)vf(af)vf(e\_x),vf(e\_y)vf(af)vf(e\_x)=vf(e\_y)vf(af\_xye\_xy)vf(e\_x),vf(e\_x)vf(af)vf(e\_y)=vf(e\_x)vf(af\_xye\_xy)vf(e\_y),defn-of-af”\_xy\].
The following lemma completes the previous one.
\[psi(af\_x-e\_xy)\] Under the conditions of \[vf(e\_x)psi(af)vf(e\_y)-etc\] one has $$\begin{aligned}
\psi({\alpha}_{xy}e_{xy})&={\varphi}(e_x)\psi({\alpha}){\varphi}(e_y),\label{psi(af_xe-e_xy)=vf(e_x)psi(af)vf(e_y)}\\
{\theta}({\alpha}_{xy}e_{xy})&={\varphi}(e_y){\theta}({\alpha}){\varphi}(e_x)\label{0(af_xe-e_xy)=vf(e_y)0(af)vf(e_x)}
\end{aligned}$$ for all $x<y$.
Given arbitrary $u<v$ such that $(u,v)\ne(x,y)$, we see by \[vf(e\_x)psi(af)vf(e\_y)=vf(e\_x)vf(af)vf(e\_y),vf(e\_x)vf(af)vf(e\_y)=vf(e\_x)vf(af\_xye\_xy)vf(e\_y)\] that $$\begin{aligned}
{\varphi}(e_u)\psi({\alpha}_{xy}e_{xy}){\varphi}(e_v)&= {\varphi}(e_u){\varphi}({\alpha}_{xy}e_{xy}){\varphi}(e_v)={\varphi}(e_u){\varphi}(({\alpha}_{xy}e_{xy})_{uv}e_{uv}){\varphi}(e_v),
\end{aligned}$$ and ${\varphi}(e_v)\psi({\alpha}_{xy}e_{xy}){\varphi}(e_u)=0$ thanks to \[vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0\]. Evidently, $$\begin{aligned}
{\varphi}(e_u){\varphi}(e_x)\psi({\alpha}){\varphi}(e_y){\varphi}(e_v)= 0 = {\varphi}(e_v){\varphi}(e_x)\psi({\alpha}){\varphi}(e_y){\varphi}(e_u)
\end{aligned}$$ in view of \[vf(e)vf(r)=0\]. Moreover, $$\begin{aligned}
{\varphi}(e_x)\psi({\alpha}_{xy}e_{xy}){\varphi}(e_y)&={\varphi}(e_x){\varphi}({\alpha}_{xy}e_{xy}){\varphi}(e_y)={\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_y)\\
&={\varphi}(e_x)\psi({\alpha}){\varphi}(e_y)={\varphi}(e_x)^2\psi({\alpha}){\varphi}(e_y)^2
\end{aligned}$$ by \[vf(e\_x)psi(af)vf(e\_y)=vf(e\_x)vf(af)vf(e\_y),vf(e\_x)vf(af)vf(e\_y)=vf(e\_x)vf(af\_xye\_xy)vf(e\_y),vf(r\^2)\], and $$\begin{aligned}
{\varphi}(e_y)\psi({\alpha}_{xy}e_{xy}){\varphi}(e_x)= 0 = {\varphi}(e_y){\varphi}(e_x)\psi({\alpha}){\varphi}(e_y){\varphi}(e_x)
\end{aligned}$$ by \[vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0,vf(e)vf(r)=0\]. Finally, $$\begin{aligned}
{\varphi}(e_u)\psi({\alpha}_{xy}e_{xy}){\varphi}(e_u)=0={\varphi}(e_u){\varphi}(e_x)\psi({\alpha}){\varphi}(e_y){\varphi}(e_u)
\end{aligned}$$ because of \[vf(e)vf(r)=0,vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0\]. Thus, \[psi(af\_xe-e\_xy)=vf(e\_x)psi(af)vf(e\_y)\] holds by \[a=b-in-A\]. The proof of \[0(af\_xe-e\_xy)=vf(e\_y)0(af)vf(e\_x)\] is analogous.
We shall also need a technical result, which deals with some kind of a restriction of a finitary series to a subset of ordered pairs.
\[defn-of-af|\_X\^Y\] Given ${\alpha}\in FI({\mathcal C})$ and $X,Y\subseteq{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$, define $$\begin{aligned}
\label{af|_X^Y=sum_x-in-X-y-in-Y-af_xye_xy}
{\alpha}|_X^Y=\sum_{x\in X, y\in Y, x\le y}{\alpha}_{xy}e_{xy}.
\end{aligned}$$
We shall write ${\alpha}|_x^Y$ for ${\alpha}|_{\{x\}}^Y$, ${\alpha}|_X^y$ for ${\alpha}|_X^{\{y\}}$ and ${\alpha}|_x^y$ for ${\alpha}|_{\{x\}}^{\{y\}}$. We shall also use the following shorter notations: $$\begin{aligned}
{\alpha}|_X:={\alpha}|_X^{{\operatorname{\mathrm{Ob}}{{\mathcal C}}}},\ {\alpha}|^X:={\alpha}|_{{\operatorname{\mathrm{Ob}}{{\mathcal C}}}}^X.\end{aligned}$$
Let us first consider some basic properties of this operation on finitary series.
\[properties-of-af|\_X\^Y\] For all ${\alpha},{\beta}\in FI({\mathcal C})$ and $X,Y,U,V\subseteq{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$ one has
1. $({\alpha}|_X^Y)|_U^V={\alpha}|_{X\cap U}^{Y\cap V}$;\[(af|\_X\^Y)\_U\^V\]
2. $({\alpha}+{\beta})|_X^Y={\alpha}|_X^Y+{\beta}|_X^Y$;\[(af+bt)|\_X\^Y=af|\_X\^Y+bt|\_X\^Y\]
3. $({\alpha}{\beta})|_X^Y={\alpha}|_X\cdot{\beta}|^Y$.\[(af-bt)|\_X\^Y=af|\_X-bt|\^Y\]
Items \[(af|\_X\^Y)\_U\^V,(af+bt)|\_X\^Y=af|\_X\^Y+bt|\_X\^Y\] are obvious. For \[(af-bt)|\_X\^Y=af|\_X-bt|\^Y\] consider first a pair $x\le y$ with $x\in
X$ and $y\in Y$. Then $$\begin{aligned}
(({\alpha}{\beta})|_X^Y)_{xy}=({\alpha}{\beta})_{xy}=\sum_{x\le z\le y}{\alpha}_{xz}{\beta}_{zy}=\sum_{x\le z\le
y}\left({\alpha}|_X\right)_{xz}\left({\beta}|^Y\right)_{zy}=\left({\alpha}|_X\cdot{\beta}|^Y\right)_{xy}.
\end{aligned}$$ Now let $u\le v$, such that $u\not\in X$. Then $$\begin{aligned}
\label{((af-bt)|_X^Y)_uv=(af|_X-bt|^Y)_uv=0}
(({\alpha}{\beta})|_X^Y)_{uv}= 0 = ({\alpha}|_X\cdot{\beta}|^Y)_{uv},
\end{aligned}$$ as $({\alpha}|_X)_{uw}=0_{uw}$ for all $w\in{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$. Similarly \[((af-bt)|\_X\^Y)\_uv=(af|\_X-bt|\^Y)\_uv=0\] holds, when $v\not\in Y$.
\[vf(e\_X)vf(af)vf(e\_Y)-in-terms-of-af|\_X\^Y\] Let ${\varphi}:FI({\mathcal C})\to A$ be a Jordan isomorphism and $\psi,{\theta}:FZ({\mathcal C})\to A$ the associated maps given by \[defn-of-psi,defn-of-0\]. Then for all ${\alpha}\in FZ({\mathcal C})$ and $X,Y\subseteq{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$ one has $$\begin{aligned}
{\varphi}(e_X)\psi({\alpha}){\varphi}(e_Y)&={\varphi}(e_X)\psi\left({\alpha}|_X^Y\right){\varphi}(e_Y),\label{vf(e_X)psi(af)vf(e_Y)=vf(e_X)psi(af|_X^Y)vf(e_Y)}\\
{\varphi}(e_X){\theta}({\alpha}){\varphi}(e_Y)&={\varphi}(e_X){\theta}\left({\alpha}|_Y^X\right){\varphi}(e_Y).\label{vf(e_X)0(af)vf(e_Y)=vf(e_X)0(af|_Y^X)vf(e_Y)}
\end{aligned}$$
Observe that $e_X=e_{X\setminus Y}+e_{X\cap Y}$, $e_Y=e_{Y\setminus X}+e_{X\cap Y}$ and $$\begin{aligned}
{\alpha}|_X^Y={\alpha}|_{X\setminus Y}^{Y\setminus X}+{\alpha}|_{X\setminus Y}^{X\cap Y}+{\alpha}|_{X\cap Y}^{Y\setminus X}+{\alpha}|_{X\cap Y}^{X\cap Y},
\end{aligned}$$ so in view of distributivity of multiplication it suffices to prove \[vf(e\_X)psi(af)vf(e\_Y)=vf(e\_X)psi(af|\_X\^Y)vf(e\_Y),vf(e\_X)0(af)vf(e\_Y)=vf(e\_X)0(af|\_Y\^X)vf(e\_Y)\] in the following two cases:
1. $X\cap Y=\emptyset$;\[case-X-and-Y-disjoint\]
2. $X=Y$.\[case-X=Y\]
*Case \[case-X-and-Y-disjoint\].* Assume that $X$ and $Y$ are disjoint. To show that \[vf(e\_X)psi(af)vf(e\_Y)=vf(e\_X)psi(af|\_X\^Y)vf(e\_Y)\] holds, we apply \[a=b-in-A\]. Notice that multiplying any side of \[vf(e\_X)psi(af)vf(e\_Y)=vf(e\_X)psi(af|\_X\^Y)vf(e\_Y)\] by ${\varphi}(e_u)$ on the left and on the right, we get zero, as $e_u$ is orthogonal either to $e_X$ or to $e_Y$, or to both of them. Now take $u<v$ and consider $$\begin{aligned}
\label{vf(e_u)vf(e_X)vf(af)vf(e_Y)vf(e_v)+vf(e_v)vf(e_X)vf(af)vf(e_Y)vf(e_u)}
{\varphi}(e_u){\varphi}(e_X)\psi({\alpha}){\varphi}(e_Y){\varphi}(e_v)+{\varphi}(e_v){\varphi}(e_X)\psi({\alpha}){\varphi}(e_Y){\varphi}(e_u).
\end{aligned}$$ If $u\not\in X\sqcup Y$ or $v\not\in X\sqcup Y$, then \[vf(e\_u)vf(e\_X)vf(af)vf(e\_Y)vf(e\_v)+vf(e\_v)vf(e\_X)vf(af)vf(e\_Y)vf(e\_u)\] is zero, since $e_u$ and $e_v$ are orthogonal both to $e_X$ and to $e_Y$. If $u,v\in X$, then $u,v\not\in Y$, i.e. $e_u$ and $e_v$ are orthogonal to $e_Y$, so \[vf(e\_u)vf(e\_X)vf(af)vf(e\_Y)vf(e\_v)+vf(e\_v)vf(e\_X)vf(af)vf(e\_Y)vf(e\_u)\] is again zero. By symmetry the same holds, when $u,v\in Y$. If $u\in Y$ and $v\in X$, then $u\not\in X$ and $v\not\in Y$, hence by \[vf(e)vf(r)=vf(r)vf(e)\] equality \[vf(e\_u)vf(e\_X)vf(af)vf(e\_Y)vf(e\_v)+vf(e\_v)vf(e\_X)vf(af)vf(e\_Y)vf(e\_u)\] becomes ${\varphi}(e_v)\psi({\alpha}){\varphi}(e_u)$, which is zero thanks to \[vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0\]. Notice also that all these four subcases do not depend on ${\alpha}$, so everything remains valid with ${\alpha}$ replaced by ${\alpha}|_X^Y$. Finally, let $u\in X$ and $v\in Y$. Then $u\not\in Y$, $v\not\in X$, and thus \[vf(e\_u)vf(e\_X)vf(af)vf(e\_Y)vf(e\_v)+vf(e\_v)vf(e\_X)vf(af)vf(e\_Y)vf(e\_u)\] equals ${\varphi}(e_u)\psi({\alpha}){\varphi}(e_v)$. The latter is $$\begin{aligned}
{\varphi}(e_u){\varphi}({\alpha}){\varphi}(e_v)&={\varphi}(e_u){\varphi}({\alpha}_{uv}e_{uv}){\varphi}(e_v)={\varphi}(e_u){\varphi}(({\alpha}|_X^Y)_{uv}e_{uv}){\varphi}(e_v)\notag\\
&={\varphi}(e_u){\varphi}({\alpha}|_X^Y){\varphi}(e_v)={\varphi}(e_u)\psi({\alpha}|_X^Y){\varphi}(e_v)\label{vf(e_u)vf(af)vf(e_v)=vf(e_u)psi(af_X^Y)vf(e_v)}
\end{aligned}$$ according to \[vf(e\_x)psi(af)vf(e\_y)=vf(e\_x)vf(af)vf(e\_y),vf(e\_x)vf(af)vf(e\_y)=vf(e\_x)vf(af\_xye\_xy)vf(e\_y),af|\_X\^Y=sum\_x-in-X-y-in-Y-af\_xye\_xy\]. If, maintaining the assumptions on $u$ and $v$, we substitute ${\alpha}|_X^Y$ for ${\alpha}$ in \[vf(e\_u)vf(e\_X)vf(af)vf(e\_Y)vf(e\_v)+vf(e\_v)vf(e\_X)vf(af)vf(e\_Y)vf(e\_u)\], then we get ${\varphi}(e_u)\psi(({\alpha}|_X^Y)|_X^Y){\varphi}(e_v)$. But this is the same as ${\varphi}(e_u)\psi({\alpha}|_X^Y){\varphi}(e_v)$ by \[(af|\_X\^Y)\_U\^V\] of \[properties-of-af|\_X\^Y\], which in view of \[vf(e\_u)vf(af)vf(e\_v)=vf(e\_u)psi(af\_X\^Y)vf(e\_v)\] completes the proof of \[vf(e\_X)psi(af)vf(e\_Y)=vf(e\_X)psi(af|\_X\^Y)vf(e\_Y)\]. The same technique is used to prove \[vf(e\_X)0(af)vf(e\_Y)=vf(e\_X)0(af|\_Y\^X)vf(e\_Y)\], and in this situation the only non-trivial subcase will be $u\in Y$ and $v\in X$, which explains why $X$ and $Y$ are “switched” in the right-hand side of \[vf(e\_X)0(af)vf(e\_Y)=vf(e\_X)0(af|\_Y\^X)vf(e\_Y)\].
*Case \[case-X=Y\].* Let $X=Y$. For \[vf(e\_X)psi(af)vf(e\_Y)=vf(e\_X)psi(af|\_X\^Y)vf(e\_Y)\] we shall again use \[a=b-in-A\] skipping some details, as the structure of the proof will be similar to the one in Case \[case-X-and-Y-disjoint\]. For any $u\in{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$ the multiplication of any side of \[vf(e\_X)psi(af)vf(e\_Y)=vf(e\_X)psi(af|\_X\^Y)vf(e\_Y)\] by $e_u$ on the left and on the right gives zero either by \[vf(e)vf(r)=0\], when $u\not\in X$, or by \[vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0\], when $u\in X$. Now, given $u<v$, we see that both of the summands of \[vf(e\_u)vf(e\_X)vf(af)vf(e\_Y)vf(e\_v)+vf(e\_v)vf(e\_X)vf(af)vf(e\_Y)vf(e\_u)\] are zero for any ${\alpha}$, when $\{u,v\}\not\subseteq X$. If $u,v\in X$, then \[vf(e\_u)vf(e\_X)vf(af)vf(e\_Y)vf(e\_v)+vf(e\_v)vf(e\_X)vf(af)vf(e\_Y)vf(e\_u)\] reduces to ${\varphi}(e_u)\psi({\alpha}){\varphi}(e_v)+{\varphi}(e_v)\psi({\alpha}){\varphi}(e_u)$, whose second summand is zero by \[vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0\], and the first one is ${\varphi}(e_u)\psi({\alpha}|_X^Y){\varphi}(e_v)$ as in \[vf(e\_u)vf(af)vf(e\_v)=vf(e\_u)psi(af\_X\^Y)vf(e\_v)\]. Since ${\varphi}(e_v)\psi({\alpha}|_X^Y){\varphi}(e_u)$ is also zero by \[vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0\], then $$\begin{aligned}
{\varphi}(e_u)\psi({\alpha}){\varphi}(e_v)+{\varphi}(e_v)\psi({\alpha}){\varphi}(e_u)={\varphi}(e_u)\psi({\alpha}|_X^Y){\varphi}(e_v)+{\varphi}(e_v)\psi({\alpha}|_X^Y){\varphi}(e_u),
\end{aligned}$$ and we are done as in Case \[case-X-and-Y-disjoint\]. The proof of \[vf(e\_X)0(af)vf(e\_Y)=vf(e\_X)0(af|\_Y\^X)vf(e\_Y)\] is totally symmetric to the proof of \[vf(e\_X)psi(af)vf(e\_Y)=vf(e\_X)psi(af|\_X\^Y)vf(e\_Y)\].
In particular, taking $X={\operatorname{\mathrm{Ob}}{{\mathcal C}}}$ or $Y={\operatorname{\mathrm{Ob}}{{\mathcal C}}}$ in \[vf(e\_X)vf(af)vf(e\_Y)-in-terms-of-af|\_X\^Y\] and observing that $e_{{\operatorname{\mathrm{Ob}}{{\mathcal C}}}}$ is the identity element which is preserved by ${\varphi}$, we get the following formulas.
\[vf(e\_X)psi(af)-etc\] Let ${\varphi}:FI({\mathcal C})\to A$ be a Jordan isomorphism and $\psi,{\theta}:FZ({\mathcal C})\to A$ the associated maps given by \[defn-of-psi,defn-of-0\]. Then for all ${\alpha}\in FZ({\mathcal C})$ and $X\subseteq{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$ $$\begin{aligned}
{\varphi}(e_X)\psi({\alpha})&={\varphi}(e_X)\psi\left({\alpha}|_X\right),\label{vf(e_X)psi(af)=vf(e_X)psi(af|_X)}\\
\psi({\alpha}){\varphi}(e_X)&=\psi\left({\alpha}|^X\right){\varphi}(e_X),\label{psi(af)vf(e_X)=psi(af|^X)vf(e_X)}\\
{\varphi}(e_X){\theta}({\alpha})&={\varphi}(e_X){\theta}\left({\alpha}|^X\right),\label{vf(e_X)0(af)=vf(e_X)0(af|^X)}\\
{\theta}({\alpha}){\varphi}(e_X)&={\theta}\left({\alpha}|_X\right){\varphi}(e_X).\label{0(af)vf(e_X)=0(af|_X)vf(e_X)}\end{aligned}$$
The next lemma shows that the maps ${\alpha}\mapsto{\alpha}'$ and ${\alpha}\mapsto{\alpha}''$ are compatible with the operation defined by \[af|\_X\^Y=sum\_x-in-X-y-in-Y-af\_xye\_xy\].
\[’-of-restriction\] For any ${\alpha}\in FZ({\mathcal C})$ and $U,V{\subseteq}{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$ one has $$\begin{aligned}
\left({\alpha}|_U^V\right)'&={\alpha}'|_U^V,\label{(af|_U^V)'=af'|_U^V}\\
\left({\alpha}|_U^V\right)''&={\alpha}''|_U^V.\label{(af|_U^V)''=af''|_U^V}
\end{aligned}$$
We first observe that for all $x<y$ $$\begin{aligned}
{\varphi}(({\alpha}_{xy}e_{xy})')=\psi({\alpha}_{xy}e_{xy})={\varphi}(e_x)\psi({\alpha}){\varphi}(e_y)={\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_y)={\varphi}({\alpha}'_{xy}e_{xy})
\end{aligned}$$ by \[psi(af\_xe-e\_xy)=vf(e\_x)psi(af)vf(e\_y),vf(e\_x)psi(af)vf(e\_y)=vf(e\_x)vf(af)vf(e\_y),defn-of-af’\_xy,defn-of-psi\]. Since ${\varphi}$ is bijective, it follows that $$\begin{aligned}
\label{(af_xye_xy)'=af'_xye_xy}
({\alpha}_{xy}e_{xy})'={\alpha}'_{xy}e_{xy}.
\end{aligned}$$ Now let $u\in U$ and $v\in V$. Then using \[(af\_xye\_xy)’=af’\_xye\_xy\] we have $$\begin{aligned}
\left({\alpha}|_U^V\right)'_{uv}=\left(\left({\alpha}|_U^V\right)_{uv}e_{uv}\right)'_{uv}=\left({\alpha}_{uv}e_{uv}\right)'_{uv}={\alpha}'_{uv}=\left({\alpha}'|_U^V\right)_{uv}.
\end{aligned}$$ And if $u\not\in U$ or $v\not\in V$, then $$\begin{aligned}
\left({\alpha}|_U^V\right)'_{uv}=\left(\left({\alpha}|_U^V\right)_{uv}e_{uv}\right)'_{uv}=0_{uv}=\left({\alpha}'|_U^V\right)_{uv}.
\end{aligned}$$ Here we applied \[(af\_xye\_xy)’=af’\_xye\_xy\] and used that $\left({\alpha}|_U^V\right)_{uv}=0_{uv}= \left({\alpha}'|_U^V\right)_{uv}$ . This proves \[(af|\_U\^V)’=af’|\_U\^V\]. Equality \[(af|\_U\^V)”=af”|\_U\^V\] follows by additivity, as ${\alpha}={\alpha}'+{\alpha}''$.
We proceed by showing that the maps ${\alpha}\mapsto{\alpha}'$ and ${\alpha}\mapsto{\alpha}''$ are also compatible with the multiplication in $FI({\mathcal C})$.
\[af’-hom-and-af”-anti-hom\] The map ${\alpha}\mapsto{\alpha}'$ (resp. ${\alpha}\mapsto{\alpha}''$) is a homomorphism (resp. anti-homomorphism) of (non-unital) rings $FZ({\mathcal C})\to FZ({\mathcal C})$.
It was proved in \[af’-and-af”-are-finitary-series\] that ${\alpha}\mapsto{\alpha}'$ is additive, so it remains to show that $$\begin{aligned}
\label{af'bt'=(af-bt)'}
{\alpha}'{\beta}'=({\alpha}{\beta})'
\end{aligned}$$ for all ${\alpha},{\beta}\in FZ({\mathcal C})$. Clearly, both sides of \[af’bt’=(af-bt)’\] belong to $FZ({\mathcal C})$. Now, given $x<y$, we have $$\begin{aligned}
({\alpha}'{\beta}')_{xy}e_{xy}&=\sum_{x<z<y}{\alpha}'_{xz}e_{xz}\cdot{\beta}'_{zy}e_{zy},\\
({\alpha}{\beta})'_{xy}e_{xy}&=(({\alpha}{\beta})_{xy}e_{xy})'=\sum_{x<z<y}({\alpha}_{xz}e_{xz}\cdot{\beta}_{zy}e_{zy})',
\end{aligned}$$ where the latter equality is explained by \[(af\_xye\_xy)’=af’\_xye\_xy\] and the additivity of the map ${\alpha}\mapsto{\alpha}'$. Thus, it suffices to prove that $$\begin{aligned}
\label{af'_xze_xz-bt'_zye_zy=(af_xze_xz-bt_zye_zy)'}
{\alpha}'_{xz}e_{xz}\cdot{\beta}'_{zy}e_{zy}=({\alpha}_{xz}e_{xz}\cdot{\beta}_{zy}e_{zy})'
\end{aligned}$$ for all $x<z<y$. Since ${\varphi}$ is bijective, \[af’\_xze\_xz-bt’\_zye\_zy=(af\_xze\_xz-bt\_zye\_zy)’\] is equivalent to $$\begin{aligned}
\label{vf(af'bt')=vf((af-bt)')}
{\varphi}({\alpha}'_{xz}e_{xz}\cdot{\beta}'_{zy}e_{zy})={\varphi}(({\alpha}_{xz}e_{xz}\cdot{\beta}_{zy}e_{zy})').
\end{aligned}$$ By \[vf(rs+sr),af\_xye\_xy-cdot-bt\_uve\_uv,defn-of-af’\_xy,vf(r\^2),vf(e)vf(r)=0\] we have $$\begin{aligned}
{\varphi}({\alpha}'_{xz}e_{xz}\cdot{\beta}'_{zy}e_{zy})&={\varphi}({\alpha}'_{xz}e_{xz}){\varphi}({\beta}'_{zy}e_{zy})+{\varphi}({\beta}'_{zy}e_{zy}){\varphi}({\alpha}'_{xz}e_{xz})\notag\\
&\quad-{\varphi}({\beta}'_{zy}e_{zy}\cdot{\alpha}'_{xz}e_{xz})\notag\\
&={\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_z)^2{\varphi}({\beta}){\varphi}(e_y)+{\varphi}(e_z){\varphi}({\beta}){\varphi}(e_y){\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_z)\notag\\
&={\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_z){\varphi}({\beta}){\varphi}(e_y).\label{vf(af'_xze_xz-bt'_zye_zy)=vf(e_x)vf(af)vf(e_z)vf(bt)vf(e_y)}
\end{aligned}$$ Now, by \[defn-of-psi,vf(e\_x)psi(af)vf(e\_y)=vf(e\_x)vf(af)vf(e\_y),psi(af\_xe-e\_xy)=vf(e\_x)psi(af)vf(e\_y),vf(rs+sr),vf(af\_xye\_xy)=vf(e\_x)vf(af)vf(e\_y)+vf(e\_y)vf(af)vf(e\_x)\] we get $$\begin{aligned}
{\varphi}(({\alpha}_{xz}e_{xz}\cdot{\beta}_{zy}e_{zy})')&=\psi({\alpha}_{xz}e_{xz}\cdot{\beta}_{zy}e_{zy})\notag\\
&=\psi(({\alpha}_{xz}{\beta}_{zy})e_{xy})\notag\\
&={\varphi}(e_x){\varphi}(({\alpha}_{xz}{\beta}_{zy})e_{xy}){\varphi}(e_y)\notag\\
&={\varphi}(e_x){\varphi}({\alpha}_{xz}e_{xz}\cdot{\beta}_{zy}e_{zy}){\varphi}(e_y)\notag\\
&={\varphi}(e_x)({\varphi}({\alpha}_{xz}e_{xz}){\varphi}({\beta}_{zy}e_{zy})+{\varphi}({\beta}_{zy}e_{zy}){\varphi}({\alpha}_{xz}e_{xz})){\varphi}(e_y)\notag\\
&\quad-{\varphi}(e_x){\varphi}({\beta}_{zy}e_{zy}\cdot{\alpha}_{xz}e_{xz}){\varphi}(e_y)\notag\\
&={\varphi}(e_x){\varphi}({\alpha}){\varphi}(e_z){\varphi}({\beta}){\varphi}(e_y).\label{vf((af_xze_xz-bt_zye_zy)')=vf(e_x)vf(af)vf(e_z)vf(bt)vf(e_y)}
\end{aligned}$$ Comparing \[vf(af’\_xze\_xz-bt’\_zye\_zy)=vf(e\_x)vf(af)vf(e\_z)vf(bt)vf(e\_y),vf((af\_xze\_xz-bt\_zye\_zy)’)=vf(e\_x)vf(af)vf(e\_z)vf(bt)vf(e\_y)\], we get the desired equality \[vf(af’bt’)=vf((af-bt)’)\].
The proof of the statement about ${\alpha}\mapsto{\alpha}''$ is analogous.
\[psi-and-0-hom-and-anti-hom\] Let ${\varphi}:FI({\mathcal C})\to A$ be a Jordan isomorphism and $\psi,{\theta}:FZ({\mathcal C})\to A$ as defined in \[defn-of-psi,defn-of-0\]. Then $\psi$ and ${\theta}$ are a homomorphism and an anti-homomorphism $FZ({\mathcal C})\to A$, respectively.
Let ${\alpha},{\beta}\in FZ({\mathcal C})$ and $x<y$ in ${\operatorname{\mathrm{Ob}}{{\mathcal C}}}$. By \[vf(e\_X)psi(af)=vf(e\_X)psi(af|\_X),psi(af)vf(e\_X)=psi(af|\^X)vf(e\_X),vf(rs+sr),defn-of-psi,(af|\_U\^V)’=af’|\_U\^V\] $$\begin{aligned}
{\varphi}(e_x)\psi({\alpha})\psi({\beta}){\varphi}(e_y)&={\varphi}(e_x)\psi({\alpha}|_x)\psi({\beta}|^y){\varphi}(e_y)\\
&={\varphi}(e_x){\varphi}(({\alpha}|_x)'){\varphi}(({\beta}|^y)'){\varphi}(e_y)\\
&={\varphi}(e_x){\varphi}({\alpha}'|_x){\varphi}({\beta}'|^y){\varphi}(e_y)\\
&={\varphi}(e_x){\varphi}({\alpha}'|_x{\beta}'|^y+{\beta}'|^y{\alpha}'|_x){\varphi}(e_y)\\
&\quad-{\varphi}(e_x){\varphi}({\beta}'|^y){\varphi}({\alpha}'|_x){\varphi}(e_y)\\
&={\varphi}(e_x){\varphi}({\alpha}'|_x{\beta}'|^y){\varphi}(e_y),
\end{aligned}$$ because ${\beta}'|^y{\alpha}'|_x=0$ and $$\begin{aligned}
{\varphi}(e_x){\varphi}({\beta}'|^y){\varphi}({\alpha}'|_x){\varphi}(e_y)&={\varphi}(e_x){\varphi}(({\beta}|^y)'){\varphi}(({\alpha}|_x)'){\varphi}(e_y)\\
&={\varphi}(e_x)\psi({\beta}|^y)\psi({\alpha}|_x){\varphi}(e_y)\\
&={\varphi}(e_x)\psi({\beta}|_x^y)\psi({\alpha}|_x^y){\varphi}(e_y)\\
&={\varphi}(e_x)\psi({\beta}_{xy}e_{xy})\psi({\alpha}_{xy}e_{xy}){\varphi}(e_y)\\
&={\varphi}(e_x)\psi({\beta}){\varphi}(e_y){\varphi}(e_x)\psi({\alpha}){\varphi}(e_y)\\
&=0\end{aligned}$$ by \[vf(e)vf(r)=0,psi(af\_xe-e\_xy)=vf(e\_x)psi(af)vf(e\_y)\]. Now using \[properties-of-af|\_X\^Y,defn-of-af’\_xy,vf(e\_x)psi(af)vf(e\_y)=vf(e\_x)vf(af)vf(e\_y),af’-hom-and-af”-anti-hom\] we get $$\begin{aligned}
{\varphi}(e_x){\varphi}({\alpha}'|_x{\beta}'|^y){\varphi}(e_y)&={\varphi}(e_x){\varphi}(({\alpha}'{\beta}')|_x^y){\varphi}(e_y)\\
&={\varphi}(e_x){\varphi}(({\alpha}{\beta})'|_x^y){\varphi}(e_y)\\
&={\varphi}(e_x){\varphi}(({\alpha}{\beta})'_{xy}e_{xy}){\varphi}(e_y)\\
&={\varphi}(e_x){\varphi}({\alpha}{\beta}){\varphi}(e_y)\\
&={\varphi}(e_x)\psi({\alpha}{\beta}){\varphi}(e_y),\end{aligned}$$ whence $$\begin{aligned}
{\varphi}(e_x)\psi({\alpha})\psi({\beta}){\varphi}(e_y)={\varphi}(e_x)\psi({\alpha}{\beta}){\varphi}(e_y).\end{aligned}$$ Now, by \[vf(e\_X)psi(af)=vf(e\_X)psi(af|\_X),psi(af)vf(e\_X)=psi(af|\^X)vf(e\_X),vf(rs+sr),defn-of-psi,(af|\_U\^V)’=af’|\_U\^V\] $$\begin{aligned}
{\varphi}(e_y)\psi({\alpha})\psi({\beta}){\varphi}(e_x)&={\varphi}(e_y)\psi({\alpha}|_y)\psi({\beta}|^x){\varphi}(e_x)\\
&={\varphi}(e_y){\varphi}(({\alpha}|_y)'){\varphi}(({\beta}|^x)'){\varphi}(e_x)\\
&={\varphi}(e_y){\varphi}({\alpha}'|_y){\varphi}({\beta}'|^x){\varphi}(e_x)\\
&={\varphi}(e_y){\varphi}({\alpha}'|_y{\beta}'|^x+{\beta}'|^x{\alpha}'|_y){\varphi}(e_x)\\
&\quad-{\varphi}(e_y){\varphi}({\beta}'|^x){\varphi}({\alpha}'|_y){\varphi}(e_x)\\
&=-{\varphi}(e_y){\varphi}({\beta}'|^x){\varphi}({\alpha}'|_y){\varphi}(e_x),
\end{aligned}$$ because ${\alpha}'|_y{\beta}'|^x= {\beta}'|^x{\alpha}'|_y=0$, by \[properties-of-af|\_X\^Y\]. But $$\begin{aligned}
{\varphi}(e_y){\varphi}({\beta}'|^x){\varphi}({\alpha}'|_y){\varphi}(e_x)&={\varphi}(e_y){\varphi}(({\beta}|^x)'){\varphi}(({\alpha}|_y)'){\varphi}(e_x)\\
&={\varphi}(e_y)\psi({\beta}|^x)\psi({\alpha}|_y){\varphi}(e_x)\\
&={\varphi}(e_y)\psi({\beta}|_y^x)\psi({\alpha}|_y^x){\varphi}(e_x)\\
&=0,\end{aligned}$$ because ${\beta}|_y^x={\alpha}|_y^x=0$. On the other hand, ${\varphi}(e_y)\psi({\alpha}{\beta}){\varphi}(e_x)=0$, by \[vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0\]. Hence $$\begin{aligned}
{\varphi}(e_x)\psi({\alpha})\psi({\beta}){\varphi}(e_y)+{\varphi}(e_y)\psi({\alpha})\psi({\beta}){\varphi}(e_x)= {\varphi}(e_x)\psi({\alpha}{\beta}){\varphi}(e_y)+{\varphi}(e_y)\psi({\alpha}{\beta}){\varphi}(e_x).\end{aligned}$$
Finally, consider $x=y$. By \[vf(e\_X)psi(af)=vf(e\_X)psi(af|\_X),psi(af)vf(e\_X)=psi(af|\^X)vf(e\_X),vf(rs+sr),defn-of-psi,properties-of-af|\_X\^Y,(af|\_U\^V)’=af’|\_U\^V\] $$\begin{aligned}
{\varphi}(e_x)\psi({\alpha})\psi({\beta}){\varphi}(e_x)&={\varphi}(e_x)\psi({\alpha}|_x)\psi({\beta}|^x){\varphi}(e_x)\\
&={\varphi}(e_x){\varphi}(({\alpha}|_x)'){\varphi}(({\beta}|^x)'){\varphi}(e_x)\\
&={\varphi}(e_x){\varphi}({\alpha}'|_x){\varphi}({\beta}'|^x){\varphi}(e_x)\\
&={\varphi}(e_x){\varphi}({\alpha}'|_x{\beta}'|^x+{\beta}'|^x{\alpha}'|_x){\varphi}(e_x)\\
&\quad-{\varphi}(e_x){\varphi}({\beta}'|^x){\varphi}({\alpha}'|_x){\varphi}(e_x)\\
&= {\varphi}(e_x){\varphi}({\beta}'|^x{\alpha}'|_x){\varphi}(e_x)\\
&\quad-{\varphi}(e_x){\varphi}({\beta}'|^x){\varphi}({\alpha}'|_x){\varphi}(e_x).
\end{aligned}$$ By \[vf(af\_xxe\_x)=vf(e\_x)vf(af)vf(e\_x)\], ${\varphi}(e_x){\varphi}({\beta}'|^x{\alpha}'|_x){\varphi}(e_x)= {\varphi}(({\beta}'|^x{\alpha}'|_x)_{xx}e_x)=0$, because ${\beta}'|^x{\alpha}'|_x\in
FZ({\mathcal C})$ and using \[vf(e\_X)psi(af)=vf(e\_X)psi(af|\_X),psi(af)vf(e\_X)=psi(af|\^X)vf(e\_X),defn-of-psi,(af|\_U\^V)’=af’|\_U\^V\] $$\begin{aligned}
{\varphi}(e_x){\varphi}({\beta}'|^x){\varphi}({\alpha}'|_x){\varphi}(e_x)&={\varphi}(e_x){\varphi}(({\beta}|^x)'){\varphi}(({\alpha}|_x)'){\varphi}(e_x)\\
&={\varphi}(e_x)\psi({\beta}|^x)\psi({\alpha}|_x){\varphi}(e_x)\\
&={\varphi}(e_x)\psi({\beta}|_x^x)\psi({\alpha}|_x^x){\varphi}(e_x)\\
&=0,\end{aligned}$$ because ${\beta}|_x^x={\alpha}|_x^x = 0$. We also have ${\varphi}(e_x)\psi({\alpha}{\beta}){\varphi}(e_x)=0$, by \[vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0\]. It follows that $$\begin{aligned}
{\varphi}(e_x)\psi({\alpha})\psi({\beta}){\varphi}(e_x)={\varphi}(e_x)\psi({\alpha}{\beta}){\varphi}(e_x).\end{aligned}$$ Therefore, by \[a=b-in-A\], we conclude that $\psi({\alpha}{\beta})= \psi({\alpha})\psi({\beta})$.
In an analogous way one proves that ${\theta}$ is an anti-homomorphism $FZ({\mathcal C})\to A$.
Decomposition of ${\varphi}$ into a near-sum
--------------------------------------------
The idea now is to extend $\psi$ and $\theta$ to the whole $FI({\mathcal C})$ in order to obtain a decomposition of $\varphi$ into a near-sum.
\[a-in-D(cC)-bt-in-FZ(cC)\] Let ${\varphi}:FI({\mathcal C})\to A$ be a Jordan isomorphism and $\psi:FZ({\mathcal C})\to A$ as defined at \[defn-of-psi\]. If $\alpha\in D({\mathcal C})$ and $\beta \in
FZ({\mathcal C})$, then $$\begin{aligned}
\label{psi(af.bt)=vf(af)psi(bt)}
\psi(\alpha\beta)={\varphi}(\alpha)\psi(\beta).
\end{aligned}$$
We will use \[a=b-in-A\]. Let $x<y$. Since $\alpha\beta\in FZ({\mathcal C})$, we have $$\begin{aligned}
{\varphi}(e_x)\psi({\alpha}{\beta}){\varphi}(e_y)+{\varphi}(e_y)\psi({\alpha}{\beta}){\varphi}(e_x) & =
{\varphi}(e_x){\varphi}({\alpha}{\beta}){\varphi}(e_y)\label{vf(e_x)psi(af-bt)vf(e_y)+vf(e_y)psi(af-bt)vf(e_x)=vf(e_x)vf(af-bt)vf(e_y)} \\
&={\varphi}(e_x){\varphi}(({\alpha}{\beta})_{xy}e_{xy}){\varphi}(e_y)\label{vf(e_x)vf(af-bt)vf(e_y)=vf(e_x)vf((af-bt)_xye_xy)vf(e_y)} \\
&={\varphi}(e_x){\varphi}({\alpha}_{xx}{\beta}_{xy}e_{xy}){\varphi}(e_y), \label{vf(e_x)psi(af bt)vf(e_y)+vf(e_y)psi(afbt)vf(e_x)}
\end{aligned}$$ where equality \[vf(e\_x)psi(af-bt)vf(e\_y)+vf(e\_y)psi(af-bt)vf(e\_x)=vf(e\_x)vf(af-bt)vf(e\_y)\] follows from \[vf(e\_x)psi(af)vf(e\_y)=vf(e\_x)vf(af)vf(e\_y),vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0\], equality \[vf(e\_x)vf(af-bt)vf(e\_y)=vf(e\_x)vf((af-bt)\_xye\_xy)vf(e\_y)\] follows from \[vf(e\_x)vf(af)vf(e\_y)=vf(e\_x)vf(af\_xye\_xy)vf(e\_y)\], and \[vf(e\_x)psi(af bt)vf(e\_y)+vf(e\_y)psi(afbt)vf(e\_x)\] uses the fact that ${\alpha}\in D({\mathcal C})$.
On the other hand, since $e_y{\alpha}={\alpha}e_y$, by \[vf(e)vf(r)=vf(r)vf(e)\] we have $${\varphi}(e_y){\varphi}({\alpha})\psi({\beta}){\varphi}(e_x)={\varphi}({\alpha}){\varphi}(e_y)\psi({\beta}){\varphi}(e_x)=0, \label{vf(e_y)vf(af)psi(bt)vf(e_x)=vf(af)vf(e_y)psi(bt)vf(e_x)=0}$$ where the last equality follows from \[vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0\]. Furthermore, by \[vf(r\^2),vf(e)vf(r)=vf(r)vf(e),vf(af\_xxe\_x)=vf(e\_x)vf(af)vf(e\_x),vf(e\_x)psi(af)vf(e\_y)=vf(e\_x)vf(af)vf(e\_y)\] $$\begin{aligned}
{\varphi}(e_x){\varphi}({\alpha})\psi({\beta}){\varphi}(e_y) &= {\varphi}(e_x) {\varphi}({\alpha}) {\varphi}(e_x) \psi({\beta}){\varphi}(e_y)={\varphi}(e_x){\varphi}({\alpha}_{xx}e_{xx})
{\varphi}({\beta}){\varphi}(e_y).\label{vf(e_x)psi(af)psi(bt)vf(e_y)=vf(af_xx.e_xx)vf(bt)vf(e_y)}
\end{aligned}$$ Now by \[vf(rs+sr)\] $$\begin{aligned}
{\varphi}({\alpha}_{xx}e_{xx}){\varphi}({\beta})={\varphi}({\alpha}_{xx}e_{xx}{\beta}+{\beta}{\alpha}_{xx}e_{xx})-{\varphi}({\beta}){\varphi}({\alpha}_{xx}e_{xx}).
\end{aligned}$$ Since ${\varphi}({\alpha}_{xx}e_{xx}){\varphi}(e_y)=0$ in view of \[vf(e)vf(r)=0\], we obtain from \[vf(e\_x)psi(af)psi(bt)vf(e\_y)=vf(af\_xx.e\_xx)vf(bt)vf(e\_y),vf(e\_x)psi(af)vf(e\_y)=vf(e\_x)vf(af)vf(e\_y),vf(e\_X)psi(af)vf(e\_Y)=vf(e\_X)psi(af|\_X\^Y)vf(e\_Y)\] $$\begin{aligned}
{\varphi}(e_x){\varphi}({\alpha})\psi({\beta}){\varphi}(e_y)&={\varphi}(e_x){\varphi}({\alpha}_{xx}e_{xx}{\beta}+{\beta}{\alpha}_{xx}e_{xx}){\varphi}(e_y)\\
&={\varphi}(e_x)\psi({\alpha}_{xx}e_{xx}{\beta}+{\beta}{\alpha}_{xx}e_{xx}){\varphi}(e_y)\\
&={\varphi}(e_x)\psi(({\alpha}_{xx}e_{xx}{\beta}+{\beta}{\alpha}_{xx}e_{xx})|_x^y){\varphi}(e_y)\\
&={\varphi}(e_x)\psi({\alpha}_{xx}{\beta}_{xy}e_{xy}){\varphi}(e_y)\\
&={\varphi}(e_x){\varphi}({\alpha}_{xx}{\beta}_{xy}e_{xy}){\varphi}(e_y).
\end{aligned}$$ Combining this with \[vf(e\_x)psi(af bt)vf(e\_y)+vf(e\_y)psi(afbt)vf(e\_x),vf(e\_y)vf(af)psi(bt)vf(e\_x)=vf(af)vf(e\_y)psi(bt)vf(e\_x)=0\] we have $${\varphi}(e_x)\psi({\alpha}{\beta}){\varphi}(e_y)+{\varphi}(e_y)\psi({\alpha}{\beta}){\varphi}(e_x)= {\varphi}(e_x){\varphi}({\alpha})\psi({\beta}){\varphi}(e_y) + {\varphi}(e_y){\varphi}({\alpha})\psi({\beta}){\varphi}(e_x).$$ Moreover, for any $x\in {\operatorname{\mathrm{Ob}}{{\mathcal C}}}$ $${\varphi}(e_x){\varphi}({\alpha})\psi({\beta}){\varphi}(e_x) = {\varphi}({\alpha}){\varphi}(e_x)\psi({\beta}){\varphi}(e_x)=0.$$ The last equality follows from \[vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0\], as well as the equality ${\varphi}(e_x)\psi({\alpha}{\beta}){\varphi}(e_x)=0$. Hence \[psi(af.bt)=vf(af)psi(bt)\] holds by \[a=b-in-A\].
\[af-in-FZ(cC)-bt-in-D(cC)\] By a similar computation one proves that if $\alpha\in FZ({\mathcal C})$ and $\beta \in D({\mathcal C})$, then $\psi(\alpha\beta)=\psi(\alpha){\varphi}(\beta)$.
Let us now extend $\psi$ and ${\theta}$ to the additive maps ${\tilde}\psi$ and ${\tilde}{\theta}$ defined on the whole ring $FI({\mathcal C})$ by means of $$\begin{aligned}
{\tilde}\psi({\alpha})&={\varphi}({\alpha}_D)+\psi({\alpha}_Z),\label{tl-psi(af)=vf(af_D)+psi(af_Z)}\\
{\tilde}{\theta}({\alpha})&={\varphi}({\alpha}_D)+{\theta}({\alpha}_Z).\label{tl-0(af)=vf(af_D)+0(af_Z)}\end{aligned}$$
The following theorem is the main result of \[sec-jiso-FI(C)\].
\[vf-near-sum-of-tl-psi-and-tl-theta\] Let ${\varphi}:FI({\mathcal C})\to A$ be a Jordan isomorphism and ${\tilde}\psi,{\tilde}\theta:FI({\mathcal C})\to A$ as defined in \[tl-psi(af)=vf(af\_D)+psi(af\_Z),tl-0(af)=vf(af\_D)+0(af\_Z)\]. Then ${\varphi}$ is the near sum of ${\tilde}\psi$ and ${\tilde}{\theta}$ with respect to $D({\mathcal C})$ and $FZ({\mathcal C})$. Moreover,
1. ${\tilde}\psi$ is a homomorphism if and only if ${\varphi}|_{D({\mathcal C})}$ is a homomorphism;\[tl-psi-homo<=>vf|\_D-homo\]
2. ${\tilde}\theta$ is an anti-homomorphism if and only if ${\varphi}|_{D({\mathcal C})}$ is an anti-homomorphism.\[tl-0-homo<=>vf|\_D-anti-homo\]
In particular, if both \[tl-psi-homo<=>vf|\_D-homo,tl-0-homo<=>vf|\_D-anti-homo\] are true, then $D({\mathcal C})$ is a commutative ring.
Clearly ${\tilde}\psi$ and ${\tilde}{\theta}$ are additive maps such that ${\varphi}|_{D({\mathcal C})}= {\tilde}\psi|_{D({\mathcal C})}={\tilde}{\theta}|_{D({\mathcal C})}$ and, by \[vf|\_FZ-is-sum\], ${\varphi}|_{FZ({\mathcal C})}={\tilde}\psi|_{FZ({\mathcal C})}+{\tilde}{\theta}|_{FZ({\mathcal C})}$.
It remains to prove that, given ${\alpha},{\beta}\in FZ({\mathcal C})$, the products ${\tilde}\psi({\alpha}){\tilde}{\theta}({\beta})$ and ${\tilde}{\theta}({\beta}){\tilde}\psi({\alpha})$ are zero. We shall show that ${\tilde}\psi({\alpha}){\tilde}{\theta}({\beta})=0$, leaving the proof of ${\tilde}{\theta}({\beta}){\tilde}\psi({\alpha})=0$, which is analogous, to the reader.
Let ${\alpha},{\beta}\in FZ({\mathcal C})$. Then ${\tilde}\psi({\alpha})=\psi({\alpha})$ and ${\tilde}{\theta}({\beta})={\theta}({\beta})$. So we need to prove that $\psi({\alpha}){\theta}({\beta})=0$. Our main tool will be \[a=b-in-A\]. Taking an arbitrary $x\in{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$, we have by \[vf(e\_X)psi(af)=vf(e\_X)psi(af|\_X),0(af)vf(e\_X)=0(af|\_X)vf(e\_X),defn-of-psi,defn-of-0\] $$\begin{aligned}
\label{vf(e_x)psi(af)0(bt)vf(e_x)=vf(e_x)psi(af|_x)0(bt|_x)vf(e_x)}
{\varphi}(e_x)\psi({\alpha}){\theta}({\beta}){\varphi}(e_x)={\varphi}(e_x)\psi({\alpha}|_x){\theta}({\beta}|_x){\varphi}(e_x)={\varphi}(e_x){\varphi}(({\alpha}|_x)'){\varphi}(({\beta}|_x)''){\varphi}(e_x).\end{aligned}$$ But $$\begin{aligned}
{\varphi}(({\alpha}|_x)'){\varphi}(({\beta}|_x)'')={\varphi}(({\alpha}|_x)'({\beta}|_x)''+({\beta}|_x)''({\alpha}|_x)')-{\varphi}(({\beta}|_x)''){\varphi}(({\alpha}|_x)')\end{aligned}$$ thanks to \[vf(rs+sr)\], and $$\begin{aligned}
{\varphi}(e_x){\varphi}(({\alpha}|_x)'({\beta}|_x)''){\varphi}(e_x)= 0 = {\varphi}(e_x){\varphi}(({\beta}|_x)''({\alpha}|_x)'){\varphi}(e_x)\end{aligned}$$ in view of \[vf(af\_xxe\_x)=vf(e\_x)vf(af)vf(e\_x)\] and the fact that $({\alpha}|_x)',({\beta}|_x)''\in FZ({\mathcal C})$. Therefore, \[vf(e\_x)psi(af)0(bt)vf(e\_x)=vf(e\_x)psi(af|\_x)0(bt|\_x)vf(e\_x)\] equals $$\begin{aligned}
-{\varphi}(e_x){\varphi}(({\beta}|_x)''){\varphi}(({\alpha}|_x)'){\varphi}(e_x)&=-{\varphi}(e_x){\theta}({\beta}|_x)\psi({\alpha}|_x){\varphi}(e_x)\\
&=-{\varphi}(e_x){\theta}({\beta}|_x^x)\psi({\alpha}|_x^x){\varphi}(e_x)\\
&=0.\end{aligned}$$ Here we have used \[defn-of-psi,defn-of-0,vf(e\_X)0(af)=vf(e\_X)0(af|\^X),psi(af)vf(e\_X)=psi(af|\^X)vf(e\_X)\] and the easy observations that ${\alpha}|_x^x={\alpha}_{xx}e_{xx}=0$ and similarly ${\beta}|_x^x=0$. Now take $x<y$ and notice from \[vf(rs+sr),vf(e\_X)psi(af)=vf(e\_X)psi(af|\_X),0(af)vf(e\_X)=0(af|\_X)vf(e\_X)\] that $$\begin{aligned}
{\varphi}(e_x)\psi({\alpha}){\theta}({\beta}){\varphi}(e_y)&={\varphi}(e_x)\psi({\alpha}|_x){\theta}({\beta}|_y){\varphi}(e_y)\notag\\
&={\varphi}(e_x){\varphi}(({\alpha}|_x)'){\varphi}(({\beta}|_y)''){\varphi}(e_y)\notag\\
&={\varphi}(e_x){\varphi}(({\alpha}|_x)'({\beta}|_y)''+({\beta}|_y)''({\alpha}|_x)'){\varphi}(e_y)\label{vf(e_x)vf((af|_x)'(bt|_y)''+(bt|_y)''(af|_x)')vf(e_y)}\\
&\quad-{\varphi}(e_x){\varphi}(({\beta}|_y)''){\varphi}(({\alpha}|_x)'){\varphi}(e_y).\label{-vf(e_x)vf((bt|_y)'')vf((af|_x)')vf(e_y)}\end{aligned}$$ The summand \[-vf(e\_x)vf((bt|\_y)”)vf((af|\_x)’)vf(e\_y)\] is $$\begin{aligned}
-{\varphi}(e_x){\theta}({\beta}|_y)\psi({\alpha}|_x){\varphi}(e_y)=-{\varphi}(e_x){\theta}({\beta}|_y^x)\psi({\alpha}|_x^y){\varphi}(e_y)=0\end{aligned}$$ in view of \[defn-of-psi,defn-of-0,vf(e\_X)0(af)=vf(e\_X)0(af|\^X),psi(af)vf(e\_X)=psi(af|\^X)vf(e\_X)\] and the fact that ${\beta}|_y^x=0$. Furthermore, use \[(af-bt)|\_X\^Y=af|\_X-bt|\^Y,(af+bt)|\_X\^Y=af|\_X\^Y+bt|\_X\^Y,(af|\_X\^Y)\_U\^V\] of \[properties-of-af|\_X\^Y\], \[af=af’+af”,vf(e\_x)psi(af)vf(e\_y)=vf(e\_x)vf(af)vf(e\_y),vf(e\_X)psi(af)vf(e\_Y)=vf(e\_X)psi(af|\_X\^Y)vf(e\_Y)\] to transform \[vf(e\_x)vf((af|\_x)’(bt|\_y)”+(bt|\_y)”(af|\_x)’)vf(e\_y)\] into $$\begin{aligned}
&{\varphi}(e_x)\psi(({\alpha}|_x)'({\beta}|_y)''+({\beta}|_y)''({\alpha}|_x)'){\varphi}(e_y)\\
&\quad={\varphi}(e_x)\psi((({\alpha}|_x)'({\beta}|_y)''+({\beta}|_y)''({\alpha}|_x)')|_x^y){\varphi}(e_y)\\
&\quad={\varphi}(e_x)\psi(({\alpha}|_x)'|_x({\beta}|_y)''|^y+({\beta}|_y)''|_x({\alpha}|_x)'|^y){\varphi}(e_y)\\
&\quad={\varphi}(e_x)\psi(({\alpha}|_x)'|_x({\beta}|_y-({\beta}|_y)')|^y+({\beta}|_y-({\beta}|_y)')|_x({\alpha}|_x)'|^y){\varphi}(e_y)\\
&\quad=-{\varphi}(e_x)\psi(({\alpha}|_x)'|_x({\beta}|_y)'|^y+({\beta}|_y)'|_x({\alpha}|_x)'|^y){\varphi}(e_y)\\
&\quad=-{\varphi}(e_x)\psi((({\alpha}|_x)'({\beta}|_y)'+({\beta}|_y)'({\alpha}|_x)')|_x^y){\varphi}(e_y)\\
&\quad=-{\varphi}(e_x)\psi(({\alpha}|_x)'({\beta}|_y)'+({\beta}|_y)'({\alpha}|_x)')){\varphi}(e_y)\\
&\quad=-{\varphi}(e_x){\varphi}(({\alpha}|_x)'({\beta}|_y)'+({\beta}|_y)'({\alpha}|_x)'){\varphi}(e_y).\end{aligned}$$ In view of \[vf(rs+sr),defn-of-psi,defn-of-0,vf(e\_X)psi(af)=vf(e\_X)psi(af|\_X),psi(af)vf(e\_X)=psi(af|\^X)vf(e\_X)\] the latter is $$\begin{aligned}
&-{\varphi}(e_x){\varphi}(({\alpha}|_x)'){\varphi}(({\beta}|_y)'){\varphi}(e_y)-{\varphi}(e_x){\varphi}(({\beta}|_y)'){\varphi}(({\alpha}|_x)'){\varphi}(e_y)\\
&\quad=-{\varphi}(e_x)\psi({\alpha}|_x)\psi({\beta}|_y){\varphi}(e_y)-{\varphi}(e_x)\psi({\beta}|_y)\psi({\alpha}|_x){\varphi}(e_y)\\
&\quad=-{\varphi}(e_x)\psi({\alpha}|_x)\psi({\beta}|_y^y){\varphi}(e_y)-{\varphi}(e_x)\psi(({\beta}|_y)|_x)\psi({\alpha}|_x){\varphi}(e_y)\\
&\quad=0,\end{aligned}$$ as ${\beta}|_y^y = 0 =({\beta}|_y)|_x$. Consequently, ${\varphi}(e_x)\psi({\alpha}){\theta}({\beta}){\varphi}(e_y)=0$. The argument that proves ${\varphi}(e_y)\psi({\alpha}){\theta}({\beta}){\varphi}(e_x)=0$ is totally symmetric (just switch $x$ and $y$ in the proof above). Thus, $$\begin{aligned}
{\varphi}(e_x)\psi({\alpha}){\theta}({\beta}){\varphi}(e_y)+{\varphi}(e_y)\psi({\alpha}){\theta}({\beta}){\varphi}(e_x)=0\end{aligned}$$ for all $x<y$. By \[a=b-in-A\] the product $\psi({\alpha}){\theta}({\beta})$ is zero.
For the second statement of the theorem take ${\alpha},{\beta}\in FI({\mathcal C})$. Write ${\alpha}={\alpha}_D+{\alpha}_Z$ and ${\beta}={\beta}_D+{\beta}_Z$. Then $${\alpha}{\beta}={\alpha}_D{\beta}_D + {\alpha}_D{\beta}_Z + {\alpha}_Z{\beta}_D + {\alpha}_Z{\beta}_Z.$$ Since $D({\mathcal C})$ is a subring of $FI({\mathcal C})$ and $FZ({\mathcal C})$ is an ideal of $FI({\mathcal C})$, we have that ${\alpha}_D{\beta}_D\in D({\mathcal C})$ and ${\alpha}_D{\beta}_Z,{\alpha}_Z{\beta}_D,{\alpha}_Z{\beta}_Z\in FZ({\mathcal C})$. If ${\tilde}\psi$ is a homomorphism, then, obviously, ${\tilde}\psi|_{D({\mathcal C})}={\varphi}|_{D({\mathcal C})}$ is a homomorphism. Conversely, combining the fact that ${\tilde}\psi|_{D({\mathcal C})}={\varphi}|_{D({\mathcal C})}$ is a homomorphism with \[psi-and-0-hom-and-anti-hom,a-in-D(cC)-bt-in-FZ(cC),af-in-FZ(cC)-bt-in-D(cC)\], one can show that ${\tilde}\psi({\alpha}{\beta})={\tilde}\psi({\alpha}){\tilde}\psi({\beta})$, that is, ${\tilde}\psi$ is a homomorphism. This proves \[tl-psi-homo<=>vf|\_D-homo\]. The proof of \[tl-0-homo<=>vf|\_D-anti-homo\] is similar.
As a consequence, we obtain [@BFK Theorem 3.13] without using the results of [@Akkurts-Barker] and thus without the restriction that $R$ is $2$-torsionfree.
\[vf-near-sum-for-FI(P.R)\] Let $P$ be a poset, $R$ a commutative ring and $A$ an $R$-algebra. Then each $R$-linear Jordan isomorphism ${\varphi}:FI(P,R)\to A$ is the near sum of a homomorphism and an anti-homomorphism with respect to $D(P,R)$ and $FZ(P,R)$.
Indeed, it was proved in [@BFK Proposition 3.3] that ${\varphi}|_{D(P,R)}$ is a homomorphism and an anti-homomorphism at the same time.
Jordan isomorphisms of $FI(P,R)$ {#jord-iso-FI(P_R)}
================================
Observe that the condition that ${\varphi}|_{D({\mathcal C})}$ is a homomorphism or an anti-homomorphism from \[vf-near-sum-of-tl-psi-and-tl-theta\] may fail for ${\mathcal C}={\mathcal C}(P,R)$, where $P$ is a quasiordered set, which is not a poset and $R$ is a commutative ring. Indeed, suppose that $1<|P|<\infty$ and $x\le y$ for all $x,y\in P$, so that $P=\bar x=\{y\in P\mid y\sim x\}$ for an arbitrary fixed $x\in P$. In this case $FI({\mathcal C})$ coincides with $D({\mathcal C})$ and is isomorphic to the full matrix ring $M_n(R)$, where $n=|P|$. If $R$ has a non-trivial idempotent $e$, then the map $J(A)=eA+(1-e)A^T$, where $A^T$ is the transpose of $A$, is a Jordan automorphism of $M_n(R)$, which is neither a homomorphism, nor an anti-homomorphism. This is a particular case of the example given in the introduction of [@BeBreChe]. Notice also that in this case $J$ is the sum of a homomorphism and an anti-homomorphism, which is true for an arbitrary Jordan homomorphism of $M_n(R)$ by [@Jacobson-Rickart50 Theorem 7].
The above example shows that it would be natural to find some sufficient conditions under which a Jordan isomorphism ${\varphi}:FI({\mathcal C})\to A$ could be decomposed as the sum of a homomorphism and an anti-homomorphism. Our final aim will be to prove the existence of such a decomposition in the case ${\mathcal C}={\mathcal C}(P,R)$, but we start with the results which hold in the general situation.
Decomposition of ${\varphi}|_{D({\mathcal C})}$
-----------------------------------------------
Since we already know by \[psi-and-0-hom-and-anti-hom,vf|\_FZ-is-sum\] that ${\varphi}|_{FZ({\mathcal C})}=\psi+{\theta}$, where $\psi$ is a homomorphism and ${\theta}$ is an anti-homomorphism, our first goal will be to find (under certain conditions) a similar decomposition for ${\varphi}|_{D({\mathcal C})}$.
For each $x\in{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$ we introduce the following notations $$\begin{aligned}
D({\mathcal C})_x&=\{{\alpha}_{xx}e_{xx}\mid {\alpha}_{xx}\in{\operatorname{\mathrm{Mor}}(x,x)}\}\subseteq D({\mathcal C}),\\
{\varphi}_x&={\varphi}|_{D({\mathcal C})_x}:D({\mathcal C})_x\to {\varphi}(D({\mathcal C})_x).\end{aligned}$$ Observe that $D({\mathcal C})_x$ is a ring with identity $e_x$, $D({\mathcal C})_x\cong{\operatorname{\mathrm{Mor}}(x,x)}$ and $D({\mathcal C})\cong\prod_{x\in{\operatorname{\mathrm{Ob}}{{\mathcal C}}}}D({\mathcal C})_x$.
\[vf(eRe)-subring\] Let ${\varphi}:R\to S$ be a Jordan isomorphism of associative rings. Then for any idempotent $e\in R$, the set ${\varphi}(eRe)$ is a ring under the operations of $S$.
Clearly, ${\varphi}(eRe)$ is a subgroup of the additive group of $S$. Let $r,s\in eRe$. Since ${\varphi}$ is surjective, there exists $t\in R$ such that $$\begin{aligned}
\label{vf(r)vf(s)=vf(t)}
{\varphi}(r){\varphi}(s)={\varphi}(t).
\end{aligned}$$ The idempotent $e$ is the identity of $eRe$, so ${\varphi}(e){\varphi}(r)={\varphi}(r)$ and ${\varphi}(s){\varphi}(e)={\varphi}(s)$ by \[vf(e)vf(r)=vf(r)vf(e)\]. Therefore, in view of \[vf(r)vf(s)=vf(t),vf(rsr)\], $$\begin{aligned}
{\varphi}(ete)={\varphi}(e){\varphi}(t){\varphi}(e)={\varphi}(e){\varphi}(r){\varphi}(s){\varphi}(e)={\varphi}(r){\varphi}(s)={\varphi}(t),
\end{aligned}$$ whence $t=ete$ thanks to the injectivity of ${\varphi}$. Thus, ${\varphi}(eRe)$ is closed under the multiplication in $S$, so it is a ring with identity ${\varphi}(e)$.
\[vf(D(C)\_x)-subring\] Let ${\varphi}:FI({\mathcal C})\to A$ be a Jordan isomorphism. Then for each $x\in{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$ the set ${\varphi}(D({\mathcal C})_x)$ is a ring under the operations of $A$.
Indeed, $D({\mathcal C})_x=e_xFI({\mathcal C})e_x$.
\[vf|\_D(C)-is-the-sum-of-psi-and-theta\] Let ${\varphi}: FI({\mathcal C})\to A$ be a Jordan isomorphism. Then ${\varphi}|_{D({\mathcal C})}:D({\mathcal C})\to A$ is the sum of a homomorphism and an anti-homomorphism if and only if for each $x\in{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$ the restriction ${\varphi}_x:D({\mathcal C})_x\to {\varphi}(D({\mathcal C})_x)$ is the sum of a homomorphism and an anti-homomorphism.
The “only if” part is obvious, so we only need to prove the “if” part. Let $$\begin{aligned}
\label{vf_x=psi_x+0_x}
{\varphi}_x=\psi_x + {\theta}_x\end{aligned}$$ be the decomposition of ${\varphi}_x$, where $\psi_x,{\theta}_x:D({\mathcal C})_x\to {\varphi}(D({\mathcal C})_x)$ are a homomorphism and an anti-homomorphism, respectively. For ${\alpha}=\sum_{x\in {\operatorname{\mathrm{Ob}}{{\mathcal C}}}}{\alpha}_{xx}e_{xx}\in D({\mathcal C})$ we have $\psi_{x}({\alpha}_{xx}e_{xx})\in {\varphi}(D({\mathcal C})_x)$ for each $x\in {\operatorname{\mathrm{Ob}}{{\mathcal C}}}$, so there exists ${\tilde}{\alpha}_{xx}e_{xx}\in
D({\mathcal C})_x$ such that $$\begin{aligned}
\label{psi_x(af_xx.e_xx)=vf(tl-af_xx.e_xx)}
\psi_{x}({\alpha}_{xx}e_{xx})={\varphi}({\tilde}{\alpha}_{xx}e_{xx}).\end{aligned}$$ Similarly, for each $x \in {\operatorname{\mathrm{Ob}}{{\mathcal C}}}$, there exists ${\tilde}{{\tilde}{{\alpha}}}_{xx}e_{xx}\in D({\mathcal C})_x$ such that $$\begin{aligned}
\label{0_x(af_xx.e_xx)=vf(tl-tl-af_xx.e_xx)}
{\theta}_{x}({\alpha}_{xx}e_{xx})={\varphi}({\tilde}{{\tilde}{{\alpha}}}_{xx}e_{xx}).\end{aligned}$$ Let $$\begin{aligned}
{\tilde}{\alpha}= \sum_{x\in {\operatorname{\mathrm{Ob}}{{\mathcal C}}}}{\tilde}{\alpha}_{xx}e_{xx} \ \ \ \ \text{and} \ \ \ \ {\tilde}{{\tilde}{{\alpha}}}= \sum_{x\in {\operatorname{\mathrm{Ob}}{{\mathcal C}}}}{\tilde}{{\tilde}{{\alpha}}}_{xx}e_{xx}\end{aligned}$$ and define $\psi, {\theta}:D({\mathcal C})\to A$ by $$\begin{aligned}
\label{psi(af)=vf(tl-af)-and-0(af)=vf(tl-tl-af)}
\psi({\alpha})={\varphi}({\tilde}{\alpha}) \ \ \ \ \text{and} \ \ \ \ {\theta}({\alpha})={\varphi}({\tilde}{{\tilde}{{\alpha}}}).\end{aligned}$$ For each $x\in {\operatorname{\mathrm{Ob}}{{\mathcal C}}}$, by \[vf\_x=psi\_x+0\_x,psi\_x(af\_xx.e\_xx)=vf(tl-af\_xx.e\_xx),0\_x(af\_xx.e\_xx)=vf(tl-tl-af\_xx.e\_xx)\] $$\begin{aligned}
{\varphi}({\alpha}_{xx}e_{xx}) & ={\varphi}_{x}({\alpha}_{xx}e_{xx}) =\psi_{x}({\alpha}_{xx}e_{xx})+{\theta}_{x}({\alpha}_{xx}e_{xx})\\
& = {\varphi}({\tilde}{\alpha}_{xx}e_{xx})+{\varphi}({\tilde}{{\tilde}{{\alpha}}}_{xx}e_{xx})\\
& = {\varphi}(({\tilde}{\alpha}_{xx} + {\tilde}{{\tilde}{{\alpha}}}_{xx})e_{xx}).\end{aligned}$$ Since ${\varphi}$ is injective, ${\alpha}_{xx}e_{xx}=({\tilde}{\alpha}_{xx}+{\tilde}{{\tilde}{{\alpha}}}_{xx})e_{xx}$, for each $x\in {\operatorname{\mathrm{Ob}}{{\mathcal C}}}$. Hence, $$\begin{aligned}
{\alpha}={\tilde}{\alpha}+{\tilde}{{\tilde}{{\alpha}}},\end{aligned}$$ and consequently $${\varphi}({\alpha})={\varphi}({\tilde}{\alpha}+ {\tilde}{{\tilde}{{\alpha}}})=\psi({\alpha})+{\theta}({\alpha})=(\psi+{\theta})({\alpha})$$ in view of \[psi(af)=vf(tl-af)-and-0(af)=vf(tl-tl-af)\]. Thus ${\varphi}|_{D({\mathcal C})}=\psi+{\theta}$.
Now, we show that $\psi$ is a homomorphism. Let us first prove that $\psi$ is additive. Take $${\alpha}=\sum_{x\in {\operatorname{\mathrm{Ob}}{{\mathcal C}}}}{\alpha}_{xx}e_{xx}, \ \ {\beta}=\sum_{x\in {\operatorname{\mathrm{Ob}}{{\mathcal C}}}}{\beta}_{xx}e_{xx} \in D({\mathcal C}).$$ For each $x\in {\operatorname{\mathrm{Ob}}{{\mathcal C}}}$, using \[psi\_x(af\_xx.e\_xx)=vf(tl-af\_xx.e\_xx)\], we have $$\begin{aligned}
{\varphi}(({\tilde}{\alpha}_{xx}+{\tilde}{\beta}_{xx})e_{xx}) & ={\varphi}({\tilde}{\alpha}_{xx}e_{xx})+ {\varphi}({\tilde}{\beta}_{xx}e_{xx})=\psi_{x}({\alpha}_{xx}e_{xx})+ \psi_{x}({\beta}_{xx}e_{xx})\\
& = \psi_{x}(({\alpha}+{\beta})_{xx}e_{xx})={\varphi}(\widetilde{({\alpha}+{\beta})}_{xx}e_{xx}),\end{aligned}$$ so ${\tilde}{\alpha}+{\tilde}{\beta}=\widetilde{{\alpha}+{\beta}}$. It follows by \[psi(af)=vf(tl-af)-and-0(af)=vf(tl-tl-af)\] that $$\begin{aligned}
\psi({\alpha}+{\beta}) & ={\varphi}(\widetilde{{\alpha}+{\beta}}) = {\varphi}({\tilde}{\alpha}+{\tilde}{\beta})={\varphi}({\tilde}{\alpha})+{\varphi}({\tilde}{\beta})=\psi({\alpha})+\psi({\beta}).\end{aligned}$$ In order to show that $\psi({\alpha}{\beta})=\psi({\alpha})\psi({\beta})$, we will use \[a=b-in-A\]. Let $u\in {\operatorname{\mathrm{Ob}}{{\mathcal C}}}$. By \[vf(rsr),e\_x-alpha-e\_y,psi(af)=vf(tl-af)-and-0(af)=vf(tl-tl-af)\] we have $$\begin{aligned}
{\varphi}(e_u)\psi({\alpha}{\beta}){\varphi}(e_u) & = {\varphi}(e_u){\varphi}(\widetilde{{\alpha}{\beta}}){\varphi}(e_u)= {\varphi}(e_u\widetilde{{\alpha}{\beta}}e_u) = {\varphi}((\widetilde{{\alpha}{\beta}})_{uu} e_{uu}) \\
& = \psi_u(({\alpha}{\beta})_{uu}e_{uu})= \psi_u({\alpha}_{uu}e_{uu})\psi_u({\beta}_{uu}e_{uu}) = {\varphi}({\tilde}{\alpha}_{uu}e_{uu}){\varphi}({\tilde}{\beta}_{uu}e_{uu})\\
& = {\varphi}(e_u{\tilde}{\alpha}e_u){\varphi}(e_u{\tilde}{\beta}e_u) = {\varphi}(e_u){\varphi}({\tilde}{\alpha}){\varphi}(e_u){\varphi}(e_u){\varphi}({\tilde}{\beta}){\varphi}(e_u).\end{aligned}$$ In view of the fact that $e_u$ is a central idempotent of $D({\mathcal C})$ and \[vf(e)vf(r)=vf(r)vf(e)\], the last product equals $$\begin{aligned}
{\varphi}(e_u)^2{\varphi}({\tilde}{\alpha}){\varphi}({\tilde}{\beta}){\varphi}(e_u)^2 = {\varphi}(e_u){\varphi}({\tilde}{\alpha}){\varphi}({\tilde}{\beta}){\varphi}(e_u) = {\varphi}(e_u)\psi({\alpha})\psi({\beta}){\varphi}(e_u).
\end{aligned}$$ Now, consider $u,v\in {\operatorname{\mathrm{Ob}}{{\mathcal C}}}$, $u<v$. Since the central idempotents $e_u$ and $e_v$ are orthogonal, equalities \[vf(e)vf(r)=0,psi(af)=vf(tl-af)-and-0(af)=vf(tl-tl-af)\] imply $$\begin{aligned}
{\varphi}(e_u)\psi({\alpha}{\beta}){\varphi}(e_v)+ {\varphi}(e_v)\psi({\alpha}{\beta}){\varphi}(e_u) & = {\varphi}(e_u){\varphi}(\widetilde{{\alpha}{\beta}}){\varphi}(e_v)+{\varphi}(e_v){\varphi}(\widetilde{{\alpha}{\beta}}){\varphi}(e_u)\\
& = {\varphi}(e_u){\varphi}(e_v){\varphi}(\widetilde{{\alpha}{\beta}})+{\varphi}(e_v){\varphi}(e_u){\varphi}(\widetilde{{\alpha}{\beta}}) = 0.
\end{aligned}$$ Similarly, $$\begin{aligned}
& {\varphi}(e_u)\psi({\alpha}) \psi({\beta}){\varphi}(e_v)+ {\varphi}(e_v)\psi({\alpha})\psi({\beta}){\varphi}(e_u) \\
& = {\varphi}(e_u){\varphi}({\tilde}{\alpha}){\varphi}({\tilde}{\beta}){\varphi}(e_v)+{\varphi}(e_v){\varphi}({\tilde}{\alpha}){\varphi}({\tilde}{\beta}){\varphi}(e_u)=0.\end{aligned}$$ It follows that $${\varphi}(e_u)\psi({\alpha}{\beta}){\varphi}(e_v)+ {\varphi}(e_v)\psi({\alpha}{\beta}){\varphi}(e_u) = {\varphi}(e_u)\psi({\alpha}) \psi({\beta}){\varphi}(e_v)+ {\varphi}(e_v)\psi({\alpha})\psi({\beta}){\varphi}(e_u).$$ By \[a=b-in-A\], $\psi({\alpha}{\beta})=\psi({\alpha})\psi({\beta})$, and therefore $\psi$ is a homomorphism.
The proof that $\theta$ is an anti-homomorphism is analogous.
Decomposition of ${\varphi}$ into a sum
---------------------------------------
Let ${\varphi}: FI({\mathcal C})\to A$ be a Jordan isomorphism and write ${\varphi}|_{FZ({\mathcal C})}=\psi_Z+{\theta}_Z$, where $\psi_Z$ and ${\theta}_Z$ are given by \[defn-of-psi,defn-of-0\]. Assume also that for all $x\in{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$ the map ${\varphi}_x$ is the sum \[vf\_x=psi\_x+0\_x\] of a homomorphism $\psi_x$ and an anti-homomorphism ${\theta}_x$, and let ${\varphi}|_{D({\mathcal C})}=\psi_D+{\theta}_D$ be the corresponding decomposition of ${\varphi}|_{D({\mathcal C})}$ constructed in \[vf|\_D(C)-is-the-sum-of-psi-and-theta\]. Define $\psi, {\theta}:FI({\mathcal C})\to A$ by $$\begin{aligned}
\psi({\alpha}) &= \psi_D({\alpha}_D)+\psi_Z({\alpha}_Z), \label{psi(af)=psi_D+psi_Z}\\
{\theta}({\alpha}) &= {\theta}_D({\alpha}_D)+{\theta}_Z({\alpha}_Z).\label{0(af)=0_D+0_Z}\end{aligned}$$ We shall show that the properties of $\psi$ and ${\theta}$ are determined by the local behavior of these maps. More precisely, given $x,y\in{\operatorname{\mathrm{Ob}}{{\mathcal C}}}$, we denote by $\{x,y\}$ the full subcategory of $C$ whose objects are $x$ and $y$. Identifying $FI(\{x,y\})$ with $e_{\{x,y\}}FI({\mathcal C})e_{\{x,y\}}{\subseteq}FI({\mathcal C})$, we have the following result.
\[psi-and-0-on-FI(xy)\] The map $\psi$ defined by \[psi(af)=psi\_D+psi\_Z\] is a homomorphism if and only if for all $x<y$ and ${\alpha},{\beta}\in FI(\{x,y\})$ $$\begin{aligned}
\psi_x({\alpha}_{xx}e_{xx})\psi_Z({\beta}_{xy}e_{xy})&=\psi_Z({\alpha}_{xx}{\beta}_{xy}e_{xy}),\label{psi_D(af_xx.e_xx)psi_Z(bt_xy.e_xy)=psi_Z(af_xx.bt_xy.e_xy)}\\
\psi_Z({\beta}_{xy}e_{xy})\psi_y({\alpha}_{yy}e_{yy})&=\psi_Z({\beta}_{xy}{\alpha}_{yy}e_{xy}).\label{psi_Z(bt_xy.e_xy)psi_D(af_yy.e_yy)=psi_Z(bt_xy.af_yy.e_xy)}
\end{aligned}$$ Similarly, ${\theta}$ given by \[0(af)=0\_D+0\_Z\] is an anti-homomorphism if and only if for all $x<y$ and ${\alpha},{\beta}\in FI(\{x,y\})$ $$\begin{aligned}
{\theta}_Z({\beta}_{xy}e_{xy}){\theta}_x({\alpha}_{xx}e_{xx})&={\theta}_Z({\alpha}_{xx}{\beta}_{xy}e_{xy}),\label{0_Z(bt_xy.e_xy)0_D(af_xx.e_xx)=0_Z(af_xx.bt_xy.e_xy)}\\
{\theta}_y({\alpha}_{yy}e_{yy}){\theta}_Z({\beta}_{xy}e_{xy})&={\theta}_Z({\beta}_{xy}{\alpha}_{yy}e_{xy}).\label{0_D(af_yy.e_yy)0_Z(bt_xy.e_xy)=0_Z(bt_xy.e_xy.af_xx)}
\end{aligned}$$
Since $\psi_Z$ and $\psi_D$ are homomorphisms, it is clear from \[psi(af)=psi\_D+psi\_Z\] that $\psi$ is a homomorphism if and only if for all ${\alpha}\in D({\mathcal C})$ and ${\beta}\in FZ({\mathcal C})$ $$\begin{aligned}
\psi_D({\alpha})\psi_Z({\beta})&=\psi_Z({\alpha}{\beta}),\label{psi_D(af)psi_Z(bt)=psi_Z(af.bt)}\\
\psi_Z({\beta})\psi_D({\alpha})&=\psi_Z({\beta}{\alpha}).\label{psi_Z(bt)psi_D(af)=psi_Z(bt.af)}
\end{aligned}$$ Given arbitrary $x<y$, by \[psi(af\_xe-e\_xy)=vf(e\_x)psi(af)vf(e\_y),vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0\] we have $$\begin{aligned}
{\varphi}(e_x)\psi_Z({\alpha}{\beta}){\varphi}(e_y)+{\varphi}(e_y)\psi_Z({\alpha}{\beta}){\varphi}(e_x)=\psi_Z({\alpha}_{xx}{\beta}_{xy}e_{xy}).
\end{aligned}$$ Now, since $e_x$ is a central idempotent of $D({\mathcal C})$, we obtain by \[psi\_x(af\_xx.e\_xx)=vf(tl-af\_xx.e\_xx),psi(af)=vf(tl-af)-and-0(af)=vf(tl-tl-af),psi(af\_xe-e\_xy)=vf(e\_x)psi(af)vf(e\_y),vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0,vf(e)vf(r)=vf(r)vf(e),vf(af\_xxe\_x)=vf(e\_x)vf(af)vf(e\_x)\] $$\begin{aligned}
{\varphi}(e_x)\psi_D({\alpha})\psi_Z({\beta}){\varphi}(e_y)&={\varphi}(e_x){\varphi}({\tilde}{\alpha})\psi_Z({\beta}){\varphi}(e_y)\\
&={\varphi}(e_x){\varphi}({\tilde}{\alpha}){\varphi}(e_x)\psi_Z({\beta}_{xy}e_{xy})\\
&={\varphi}({\tilde}{\alpha}_{xx}e_{xx})\psi_Z({\beta}_{xy}e_{xy})\\
&=\psi_x({\alpha}_{xx}e_{xx})\psi_Z({\beta}_{xy}e_{xy})
\end{aligned}$$ and $$\begin{aligned}
{\varphi}(e_y)\psi_D({\alpha})\psi_Z({\beta}){\varphi}(e_x)&={\varphi}(e_y){\varphi}({\tilde}{\alpha})\psi_Z({\beta}){\varphi}(e_x)\\
&={\varphi}({\tilde}{\alpha}){\varphi}(e_y)\psi_Z({\beta}){\varphi}(e_x)=0.
\end{aligned}$$ Since also $$\begin{aligned}
{\varphi}(e_x)\psi_Z({\alpha}{\beta}){\varphi}(e_x)=0={\varphi}(e_x)\psi_D({\alpha})\psi_Z({\beta}){\varphi}(e_x)
\end{aligned}$$ thanks to \[psi(af)=vf(tl-af)-and-0(af)=vf(tl-tl-af),vf(e\_y)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_y)=vf(e\_x)psi(af)vf(e\_x)=vf(e\_x)0(af)vf(e\_x)=0,vf(e)vf(r)=vf(r)vf(e)\], we see that \[psi\_D(af)psi\_Z(bt)=psi\_Z(af.bt)\] is equivalent to \[psi\_D(af\_xx.e\_xx)psi\_Z(bt\_xy.e\_xy)=psi\_Z(af\_xx.bt\_xy.e\_xy)\] in view of \[a=b-in-A\]. Similarly, \[psi\_Z(bt)psi\_D(af)=psi\_Z(bt.af)\] is equivalent to \[psi\_Z(bt\_xy.e\_xy)psi\_D(af\_yy.e\_yy)=psi\_Z(bt\_xy.af\_yy.e\_xy)\].
The proof of the statement for ${\theta}$ is analogous.
\[vf|\_FI(xy)\] Let ${\mathcal C}={\mathcal C}(P,R)$, where $R$ is a commutative ring and $P$ is a quasiordered set such that $1<|{\bar{x}}|<\infty$ for every class ${\bar{x}}\in \bar{P}$. Let $A$ be an $R$-algebra. Then for each $R$-linear Jordan isomorphism ${\varphi}:FI({\mathcal C})\to A$ and for every ${\bar{x}}\in \bar{P}$ there exists a decomposition of ${\varphi}_{{\bar{x}}}$ into the sum $\psi_{{\bar{x}}}+{\theta}_{{\bar{x}}}$ of a homomorphism $\psi_{{\bar{x}}}$ and an anti-homomorphism ${\theta}_{{\bar{x}}}$, such that \[psi\_D(af\_xx.e\_xx)psi\_Z(bt\_xy.e\_xy)=psi\_Z(af\_xx.bt\_xy.e\_xy),psi\_Z(bt\_xy.e\_xy)psi\_D(af\_yy.e\_yy)=psi\_Z(bt\_xy.af\_yy.e\_xy),0\_Z(bt\_xy.e\_xy)0\_D(af\_xx.e\_xx)=0\_Z(af\_xx.bt\_xy.e\_xy),0\_D(af\_yy.e\_yy)0\_Z(bt\_xy.e\_xy)=0\_Z(bt\_xy.e\_xy.af\_xx)\] hold.
Let $\bar x<\bar y$ and consider $Q{\subseteq}P$, such that $\bar Q=\{\bar x,\bar y\}$. Observe that $Q$ is a finite quasiordered set, whose classes contain at least 2 elements. Moreover, $$\begin{aligned}
FI(Q,R)\cong FI(\{\bar x,\bar y\}),
\end{aligned}$$ so, the restriction ${\varphi}_{\bar x,\bar y}$ of ${\varphi}$ to $FI(\{\bar x,\bar y\})$ can be identified with a Jordan isomorphism $FI(Q,R)\to B$, where $B={\varphi}(FI(\{\bar x,\bar y\}))$ is an $R$-algebra by \[vf(eRe)-subring\] and $R$-linearity of ${\varphi}$. It follows from Case 2 of [@Akkurts-Barker] that $$\begin{aligned}
\label{vf_bar_x_bar_y=vf_1+vf_2}
{\varphi}_{\bar x,\bar y}={\varphi}_1+{\varphi}_2,
\end{aligned}$$ where ${\varphi}_1,{\varphi}_2:FI(\{\bar x,\bar y\})\to B$ are a homomorphism and an anti-homomorphism, respectively. Moreover, $$\begin{aligned}
\label{vf_1-and-vf_2-on-FI(xy)}
{\varphi}_1({\alpha})={\varphi}_{\bar x,\bar y}({\alpha})f,\ {\varphi}_2({\alpha})={\varphi}_{\bar x,\bar y}({\alpha})g
\end{aligned}$$ for some pair of central orthogonal idempotents $f,g\in B$ whose sum is the identity of $B$.
By the construction of $f$ in [@Akkurts-Barker] we see that $f=f_{\bar x}+f_{\bar y}$, where $f_{\bar x}$ and $f_{\bar y}$ are orthogonal idempotents, $f_{\bar x}$ is a polynomial of the values of $({\varphi}_{\bar x,\bar y})_{\bar x}={\varphi}_{\bar x}$, and $f_{\bar y}$ is a polynomial of the values of ${\varphi}_{\bar y}$. The idempotent $g$ also has a similar decomposition $g=g_{\bar x}+g_{\bar y}$. Therefore, for all ${\alpha}\in D({\mathcal C})_{\bar x}$ using \[vf(e)vf(r)=vf(r)vf(e),vf(e)vf(r)=0,vf\_1-and-vf\_2-on-FI(xy)\] we have $$\begin{aligned}
\label{vf_1-and-vf_2-on-D(C)}
{\varphi}_1({\alpha})={\varphi}_{\bar x,\bar y}({\alpha})f={\varphi}_{\bar x}({\alpha})f={\varphi}_{\bar x}({\alpha}){\varphi}(e_{\bar x})(f_{\bar x}+f_{\bar y})={\varphi}_{\bar x}({\alpha})f_{\bar x},
\end{aligned}$$ which shows that $({\varphi}_1)_{\bar x}$ depends only on $\bar x$ and does not depend on $\bar y$ with $\bar x<\bar y$. By the similar reason $({\varphi}_2)_{\bar x}$ depends only on $\bar x$. Thus, we may define $$\begin{aligned}
\label{psi_D-and-0_D-from-vf_1-and-vf_2-on-D(C)}
\psi_{\bar x}=({\varphi}_1)_{\bar x},\ \ {\theta}_{\bar x}=({\varphi}_2)_{\bar x}.
\end{aligned}$$ It follows from \[vf\_bar\_x\_bar\_y=vf\_1+vf\_2\] that ${\varphi}_{{\bar{x}}}=\psi_{\bar x}+{\theta}_{\bar x}$.
Now take $u\in\bar x$, $v\in\bar y$ and denote by ${\epsilon}_{uv}\in{\operatorname{\mathrm{Mor}}(\bar x,\bar y)}$, ${\epsilon}_{uu}\in{\operatorname{\mathrm{Mor}}(\bar x,\bar x)}$, ${\epsilon}_{vv}\in{\operatorname{\mathrm{Mor}}(\bar y,\bar y)}$ the corresponding matrix units. Analyzing the proof of Case 2 of [@Akkurts-Barker], one has for all $r\in R$ $$\begin{aligned}
\label{vf(r.e_uv.e_bar_x_bary)f=f_uv}
{\varphi}(r{\epsilon}_{uv}e_{\bar x\bar y})f={\varphi}({\epsilon}_{uu}e_{\bar x\bar x}){\varphi}(r{\epsilon}_{uv}e_{\bar x\bar y}){\varphi}({\epsilon}_{vv}e_{\bar y\bar y}).
\end{aligned}$$ But $e_{\bar x}={\epsilon}_{uu}e_{\bar x\bar x}+(e_{\bar x}-{\epsilon}_{uu}e_{\bar x\bar x})$, where $$\begin{aligned}
(e_{\bar x}-{\epsilon}_{uu}e_{\bar x\bar x})\cdot r{\epsilon}_{uv}e_{\bar x\bar y}=r{\epsilon}_{uv}e_{\bar x\bar y}\cdot (e_{\bar x}-{\epsilon}_{uu}e_{\bar x\bar x})=0.
\end{aligned}$$ Hence, ${\varphi}({\epsilon}_{uu}e_{\bar x\bar x}){\varphi}(r{\epsilon}_{uv}e_{\bar x\bar y})={\varphi}(e_{\bar x}){\varphi}(r{\epsilon}_{uv}e_{\bar x\bar y})$ by \[vf(e)vf(r)=0\]. Analogously, we obtain ${\varphi}(r{\epsilon}_{uv}e_{\bar x\bar y}){\varphi}({\epsilon}_{vv}e_{\bar y\bar y})={\varphi}(r{\epsilon}_{uv}e_{\bar x\bar y}){\varphi}(e_{\bar y})$. It follows from \[vf(r.e\_uv.e\_bar\_x\_bary)f=f\_uv,vf\_1-and-vf\_2-on-FI(xy),vf(e\_x)psi(af)vf(e\_y)=vf(e\_x)vf(af)vf(e\_y),psi(af\_xe-e\_xy)=vf(e\_x)psi(af)vf(e\_y)\] that $$\begin{aligned}
{\varphi}_1(r{\epsilon}_{uv}e_{\bar x\bar y})={\varphi}(e_{\bar x}){\varphi}(r{\epsilon}_{uv}e_{\bar x\bar y}){\varphi}(e_{\bar y})=\psi_Z(r{\epsilon}_{uv}e_{\bar x\bar y}).
\end{aligned}$$ Consequently, $$\begin{aligned}
\label{vf_1-coincides-with-psi}
{\varphi}_1({\alpha}_{\bar x\bar y}e_{\bar x\bar y})=\psi_Z({\alpha}_{\bar x\bar y}e_{\bar x\bar y})
\end{aligned}$$ for arbitrary ${\alpha}_{\bar x\bar y}\in{\operatorname{\mathrm{Mor}}(\bar x,\bar y)}$. Similarly $$\begin{aligned}
{\varphi}_2({\alpha}_{\bar x\bar y}e_{\bar x\bar y})={\theta}_Z({\alpha}_{\bar x\bar y}e_{\bar x\bar y}).
\end{aligned}$$ Since ${\varphi}_1$ is a homomorphism, we have by \[psi\_D-and-0\_D-from-vf\_1-and-vf\_2-on-D(C),vf\_1-coincides-with-psi\] $$\begin{aligned}
\psi_{\bar x}({\alpha}_{\bar x\bar x}e_{\bar x\bar x})\psi_Z({\beta}_{\bar x\bar y}e_{\bar x\bar y})&={\varphi}_1({\alpha}_{\bar x\bar x}e_{\bar x\bar x}){\varphi}_1({\beta}_{\bar x\bar y}e_{\bar x\bar y})={\varphi}_1({\alpha}_{\bar x\bar x}{\beta}_{\bar x\bar y}e_{\bar x\bar y})\\
&=\psi_Z({\alpha}_{\bar x\bar x}{\beta}_{\bar x\bar y}e_{\bar x\bar y}).
\end{aligned}$$ Thus, $\psi_{\bar x}$ satisfies \[psi\_D(af\_xx.e\_xx)psi\_Z(bt\_xy.e\_xy)=psi\_Z(af\_xx.bt\_xy.e\_xy)\]. The proof that it also satisfies \[psi\_Z(bt\_xy.e\_xy)psi\_D(af\_yy.e\_yy)=psi\_Z(bt\_xy.af\_yy.e\_xy)\] is similar. Analogously one proves \[0\_Z(bt\_xy.e\_xy)0\_D(af\_xx.e\_xx)=0\_Z(af\_xx.bt\_xy.e\_xy),0\_D(af\_yy.e\_yy)0\_Z(bt\_xy.e\_xy)=0\_Z(bt\_xy.e\_xy.af\_xx)\].
We are ready to prove the main result of \[jord-iso-FI(P\_R)\].
\[vf=psi+0\] Let $R$ be a commutative ring and $P$ a quasiordered set such that $1< |{\bar{x}}| < \infty$ for all ${\bar{x}}\in \bar{P}$. Let $A$ be an $R$-algebra. Then each $R$-linear Jordan isomorphism ${\varphi}:FI(P,R)\to A$ is the sum of a homomorphism and an anti-homomorphism.
By \[vf|\_FI(xy)\], for every $\bar x \in \bar P$ there is a decomposition ${\varphi}_{\bar x}=\psi_{\bar x}+{\theta}_{\bar x}$ of ${\varphi}_{\bar x}$ where $\psi_{\bar x}$ and ${\theta}_{\bar x}$ are a homomorphism and an anti-homomorphism, respectively, for which \[psi\_D(af\_xx.e\_xx)psi\_Z(bt\_xy.e\_xy)=psi\_Z(af\_xx.bt\_xy.e\_xy),psi\_Z(bt\_xy.e\_xy)psi\_D(af\_yy.e\_yy)=psi\_Z(bt\_xy.af\_yy.e\_xy),0\_Z(bt\_xy.e\_xy)0\_D(af\_xx.e\_xx)=0\_Z(af\_xx.bt\_xy.e\_xy),0\_D(af\_yy.e\_yy)0\_Z(bt\_xy.e\_xy)=0\_Z(bt\_xy.e\_xy.af\_xx)\] hold. By \[vf|\_D(C)-is-the-sum-of-psi-and-theta,psi-and-0-on-FI(xy)\] this leads to a decomposition ${\varphi}|_{D({\mathcal C})}=\psi_D+{\theta}_D$, such that the maps $\psi$ and ${\theta}$ given by \[psi(af)=psi\_D+psi\_Z,0(af)=0\_D+0\_Z\] are a homomorphism and an anti-homomorphism, respectively. Obviously, ${\varphi}=\psi+{\theta}$.
|
---
abstract: 'We study, with numerical methods, the fractal properties of the domain walls found in slow quenches of the kinetic Ising model to its critical temperature. We show that the equilibrium interfaces in the disordered phase have critical percolation fractal dimension over a wide range of length scales. We confirm that the system falls out of equilibrium at a temperature that depends on the cooling rate as predicted by the Kibble – Zurek argument and we prove that the dynamic growing length once the cooling reaches the critical point satisfies the same scaling. We determine the dynamic scaling properties of the interface winding angle variance and we show that the crossover between critical Ising and critical percolation properties is determined by the growing length reached when the system fell out of equilibrium.'
author:
- 'Hugo <span style="font-variant:small-caps;">Ricateau</span>'
- 'Leticia F. <span style="font-variant:small-caps;">Cugliandolo</span>'
- 'Marco <span style="font-variant:small-caps;">Picco</span>'
bibliography:
- 'coarsening.bib'
date:
-
-
title: '**Critical percolation in the slow cooling of the bi-dimensional ferromagnetic Ising model**'
---
Introduction {#introduction .unnumbered}
============
In recent years, the interplay between percolation and coarsening [@CorberiPoliti; @Bray94; @HenkelPleimling10] in bi-dimensional spin models was studied in quite some detail. A series of papers proved that the critical and sub-critical instantaneous quench dynamics of the $2d$ ferromagnetic Ising model rather quickly approach a critical percolation state (in a time-scale that scales typically, as a small power of the system size) and later undergo the expected coarsening phenomenon that progressively makes the short length scales acquire the properties of the equilibrium target state. More precisely, in the quenches performed the evolution starts from a totally random initial configuration mimicking equilibrium at infinite temperature and later evolve with different microscopic stochastic spin updates. This feature was demonstrated with extensive numerical simulations of the Glauber –Ising model for ferromagnetism [@ArBrCuSi07; @SiArBrCu07; @BlCoCuPi14; @BlCuPiTa17] and the Kawasaki model for phase separation [@SiSaArBrCu09; @TaCuPi16] quenched into their symmetry broken phases. The effects of weak disorder were considered in [@SiArBrCu08; @InCoCuPi16]. The voter model dynamics was investigated in [@TaCuPi15] and, especially relevant for the present study, quenches to the critical point of the $2d$ ferromagnetic Ising model were considered in [@BlCuPi12]. The early approach to critical percolation also explained why zero temperature quenches of the $2d$ Ising model often get blocked in metastable states with infinitely long-lived flat interfaces [@SpKrRe01; @SpKrRe02; @BaKrRe09; @OlKrRe12; @OlKrRe11a; @OlKrRe11b; @OlKrRe13]. Metastable states in quenches from the critical point to zero temperature were considered in [@BlPi13].
In statistical physics studies, quenches are taken to be instantaneous. Indeed, the relevant time scales in experimental realisations are such that the cooling time is much shorter than all other time scales. Instead, in field theoretical models of cosmology, there was interest in determining the cooling rate dependencies induced by a very slow quench across a second order phase transition. The original Kibble arguments for the existence of spatial regions that are not causally connected long after going through the phase transition [@Kibble76] were complemented by a scaling proposal by Zurek [@Zurek85; @Zurek96]. This argument allows one to estimate the correlation length reached when the system falls out of equilibrium approaching a critical point from the symmetric phase with a weak finite speed. The interest in counting the number of topological defects left over after crossing the phase transition triggered by cosmology [@Kibble07], prompted condensed-matter experimental physicists to try these measurements in the lab. This kind of experiments were first performed in Helium 3 [@Bauerle-etal] and liquid crystals [@Chuang-etal] more than twenty years ago. The subject was recently revived by the realisation of cold atom experiments in which the samples are taken across the critical region with a finite speed [@Lamporesi-etal; @Chomaz-etal; @Navon-etal; @Donatello-etal]. New studies in ion crystals [@delCampo-etal; @Ulm-etal13; @Silvi-etal16] and $2d$ colloidal suspensions [@Maret] have also been recently performed. Two recent reviews give a more complete summary of the status of this field [@delCampo10; @Beugnon17].
Studies of cooling rate dependencies in statistical physics models were performed in a number of papers. The $2d$ Ising model with non-conserved order parameter dynamics was considered in [@Biroli10] and the $2d$ xy (planar spins) model in [@Jelic11] (the latter is relevant to discuss the recent experimental activity in Bose – Einstein condensates and colloidal suspensions). In the former model the phase transition is a conventional second order one, from a symmetric to a symmetry broken phase, while in the latter case the transition is of Berezinskii – Kosterlitz – Thouless (BKT) kind and the target phase is a critical one. The aim of these papers was to show that, contrary to what was usually claimed in the KZ literature, the dynamics are not frozen after the system falls out of equilibrium close to the critical point, be it second order or BKT. The critical or subcritical dynamics, at continuously changing control parameters, let the dynamic correlation length go on growing in time. Scaling arguments were used in these papers to derive the dependence of the growing correlation length, and hence the number of topological defects, as a function of time and cooling rate and they were favourably compared to the outcome of numerical simulations. Exact results for the one dimensional Ising chain and a variety of cooling procedures were derived in [@Krapivsky10]. The spherical ferromagnetic model with exponentially fast cooling was treated, also analytically, in [@Picone03]. A one-dimensional non-equilibrium lattice gas model with a phase transition was treated in [@Priyanka]. Extensive numerical simulations of models for two dimensional atomic gases were very recently presented in [@Comaron17a; @Comaron17b]. The evolution of the order parameter in the finite dimensional Ising model slowly cooled to the critical point were studied with different microscopic stochastic rules in [@Liu-etal14].
In this paper we revisit the slow cooling of the $2d$ Ising model [@Biroli10; @Liu-etal14] paying now special attention to the geometric properties of the domain structures formed when approaching the critical point. The paper is organised as follows. In \[sec:model\] we present the model and the observables. summarises some features of the equilibrium state that are useful for the dynamic study. We then present a short account of instantaneous quenches in \[sec:quenches\] and we finally enter the heart of the results on the cooling rate measurements in \[sec.cooling\]. The last section sums up our results.
The model and the observables {#sec:model}
=============================
The bi-dimensional ferromagnetic Ising model
--------------------------------------------
We focus on the emblematic ferromagnetic Ising model $$H\lr{(\lr{\{\sigma_i\}})}=-J\sum_{\lr{<i,j>}}\sigma_i\,\sigma_j\eqpc$$ with $J>0$, and spin variables taking only two values, $\sigma_i=\pm1$. In particular, we study its bi-dimensional $d=2$ realisation on the square lattice, so that the symbol $\lr{<i,j>}$ represents a sum over nearest neighbours only. The total number of spins in the system is $L\times L$ with $L$ the linear length of the lattice measured in units of the lattice spacing $a$. The canonical equilibrium properties as a function of the parameter $K=\beta\,J$, with $\beta$ the inverse temperature, are described by the partition function $$Z\lr{(K)}=\sum_{\mathclap{\lr{\{\sigma_i=\pm1\}}}}\exp{K\sum_{\lr{<i,j>}}\sigma_i\,\sigma_j}\eqpd$$ Hereafter we work with units such that $J=1$.
The model is endowed with microscopic Monte Carlo stochastic dynamics for the individual spins. The microscopic update rule is defined as follows: we randomly chose a spin $i\in\lr{\llbracket0,L^2\llbracket}$ in the system. The spin is flipped ($\sigma_i=-\sigma_i$) with a probability $$p=\min\lr{(1,e^{-\beta\,\delta H})}\eqpc$$ where $\delta H$ is the energy change due by the potential flip of the selected spin. $\beta\,\delta H$ can only takes five different values: $-8K$, $-4K$, $0$, $4K$, or $8K$. The process is controlled by the parameter $K$ given by the external inverse temperature of the bath $\beta$ times the exchange parameter $J$. Repeating this process $L^2$ times constitutes one unit of time in the kinetic Ising model. Hereafter, the time appearing in dynamical studies is always in this unit. We use a square lattice with linear size $L=1024$ and periodic boundary conditions.
Percolation
-----------
Site percolation [@Stauffer94; @Christensen02; @Saberi15; @Delfino15] is a purely geometric problem in which particles are placed at the sites of a lattice with probability $p$. This model undergoes a phase transition at $p_{\text{c}}$, a critical value of $p$ that depends on the geometry and dimension of the lattice. In $d=2$, and for a square lattice, $p_{\text{c}}\sim0.59$. The two phases correspond to one with no cluster spanning the system from one end to the other in any spatial directions ($p<p_{\text{c}}$) and one in which there is one cluster percolating across the lattice ($p>p_{\text{c}}$). At the critical point the behaviour is similar to the one at a thermodynamic second order critical point with universal critical exponents characterising various geometric quantities that one can define.
The Ising model can be thought of as a percolation problem after performing a one-to-one mapping between spins and occupation numbers. For example, an infinite temperature configuration in which the spins take $\pm$ values with probability $\nicefrac{\displaystyle{1}}{\displaystyle{2}}$ is a random percolation configuration with $p=\nicefrac{\displaystyle{1}}{\displaystyle{2}}$. It is, therefore, below the threshold for percolation of a cluster of occupied sites on the square lattice.
Critical behaviour: fractal domains
-----------------------------------
We will investigate the properties of geometric domains in the kinetic Ising model, that is to say, ensembles of connected spins pointing in the same direction (surrounded by a domain of the opposite orientation when in the bulk, or reaching the boundaries of the system if open boundary conditions are used).
At the critical point of the $2d$ ferromagnetic Ising model the geometric domains are fractal objects. Their typical area and typical interface length are $$A_{\text{c}}=\ell^{D_A}\eqpc\qquad\qquad\ell_{\text{c}}=\ell^{D_\ell}\eqpc$$ with $\ell$ a typical length. The Hausdorff dimensions are given by $$D_A=1+\frac{3\,\kappa}{32}+\frac{2}{\kappa}\eqpc\qquad\qquad D_\ell=1+\frac{\kappa}{8}\eqpc$$ with $\kappa$ a universal parameter that characterises the critical point. At the thermodynamic critical point of the Ising model in dimension two, $\kappa=3$. Instead, at the percolation threshold $\kappa=6$. In this study we only analyse interface properties so we will only use $D_\ell$ in the rest of the paper.
Observables
-----------
We used a small number of observables that are enough to characterise the growing length and geometric properties of the interfaces. We define them in this section.
### Space time correlation function and correlation length
In equilibrium, the correlation of the spin fluctuations $$C_{\text{c}}\lr{(r=\lr{|i-j|})}=\lr{<\sigma_i\,\sigma_j>}-\lr{<\sigma_i>}\,\lr{<\sigma_j>}\eqpc
\label{eq:def-corr}$$ where $r\in\lr{\llbracket0,\nicefrac{\displaystyle{L}}{\displaystyle{2}}\rrbracket}$, allows one to extract the equilibrium correlation length $\xi_{\text{eq}}$ with different studies of its decaying properties over distance. For example, one can use a fit to the expected form close to the critical point, $$C_{\text{c}}\lr{(r)}=\frac{e^{\nicefrac{\scriptstyle{-r}}{\scriptstyle{\xi}_{\text{eq}}}}}{r^{a}}\eqpc$$ or extract it from the weighted integral $$\xi_{\text{eq}}=\frac{\displaystyle{\Gamma\lr{(\zeta-a)}\,\int_0^\Lambda r^\zeta\,C_{\text{c}}\lr{(r)}\,\d{r}}}{\displaystyle{\Gamma\lr{(\zeta-a+1)}\,\int_0^\Lambda r^{\zeta-1}\,C_{\text{c}}\lr{(r)}\,\d{r}}}
\label{eq:corr-length}$$ with a convenient choice of the power $\zeta$ and the cut-off length $\Lambda$. In particular, we will use $\zeta=2$ and $\Lambda$ is chosen the largest possible such that $C_{\text{c}}\lr{(r)}$ remains larger than its statistical fluctuations.
In dynamical studies, the space-time correlation is defined just as in eq. (\[eq:def-corr\]) where the spins are time-dependent variables. The average is taken over different histories (random noises) of the dynamics. After a quench, while the system is far from equilibrium, the spin average vanishes and the connected and plain correlations simply coincide. The procedure in the right-hand-side of eq. (\[eq:corr-length\]) can then be applied to extract the dynamic growing length $\xi(t)$ that characterises the growth of equilibrium structures close, at, and below $T_{\text{c}}$.
### Variance of the interfaces winding angle
The winding angle, $\theta\lr{(\ell)}$, is measured on any bi-dimensional curve as a function of the curvilinear abscissa, $\ell$, as follows. We first choose an origin point for the curvilinear abscissa. Then, we measure a reference angle, $\theta_0$, between a chosen fixed direction and the tangent to the curve at the origin of the curvilinear lengths. Now, for each point on the curve, we define $\eta\lr{(\ell)}$, the local angle between the same chosen fixed direction as earlier and the tangent to the curve at $\ell$. Finally, the winding angle is obtained by integrating the variation of the local angle along the curve: $$\theta\lr{(\ell)}=\theta_0+\int_0^\ell\d{\eta}
\label{eq:wav-equil}$$ (note that on a square lattice $\eta$ can only take four values). For closed curves, after one turn (*ie* returning to the origin), we have $\Delta\theta=2n\pi$, where $n\in\mathbb{Z}$ is the number of loops. In particular, since the curve is an interface, it cannot cross itself and $\Delta\theta=0$ or $\Delta\theta=\pm2\pi$ (where the sign changes whether the curve rotates clockwise or anticlockwise). The former means that the interface encloses a finite area, and the latter means that the interface spans the system from one border to another one.
The moments of these angles can then be computed by taking their desired power and performing the equilibrium or dynamic statistical averages.
For a fractal curve the average of the angle vanishes and its variance satisfies [@SaDu87] $$\lr{<\theta^2\lr{(\ell)}>}=C+\frac{4\,\kappa}{8+\kappa}\,\log{\ell}\eqpc$$ where $\ell$ is the curvilinear distance along the curve, $C$ is a non-universal constant, and $\kappa$ takes a universal value depending on the kind of criticality.
In the dynamic model, this definition can be applied to study the evolution of the geometric properties of the interfaces in the system.
Equilibrium behaviour {#sec:equilibrium}
=====================
In this section we review some properties of the equilibrium behaviour of the $2d$ Ising model at high temperature and at the critical point that are relevant to our study.
![ Equilibrium behaviour above the Curie point ($T\geq T_{\text{c}}$). Panel (**a**) shows the winding angle variance as a function of the curvilinear length on the interfaces, at different temperatures. The two straight lines, $\kappa=3$ and $\kappa=6$, are the expected slopes for the Ising and percolation universality classes, respectively. Panel (**b**) displays, as a function of $T$, the value of $\kappa$ extracted from the slope of $\left<\theta^2\left(\log{\ell}\right)\right>$ at short length $\ell$. The horizontal axis, the same as on the graphic below, is a logarithmic scale where we added the two extreme points, $0$ and $\infty$. The values of $\kappa$ corresponding to the two universality classes (Ising and percolation) are labeled on the graph. The lower row shows, in (**c**), typical snapshots of the equilibrium state of the system ($L=128$), at different temperatures above $T_{\text{c}}$, and in (**d**) the average occupancy rates of the first largest clusters when approaching the critical temperature. The $n^{\text{th}}$ average occupancy is the fraction of the system occupied, on average, by the $n^{\text{th}}$ largest cluster. Except for the snapshots, all the results presented in this figure were obtained using $L=1024$. []{data-label="equilibrium.plots"}](./equilibrium.pdf)
We start by recalling a number of thermal features above the critical temperature. With this, we want to establish a reference equilibrium behaviour for the relevant observables.
Away from the critical point correlations span a finite distance. The equilibrium correlation length diverges at the Curie critical temperature, and in the close vicinity of the critical point, it does as a power law, $$\xi_{\text{eq}}\lr{(\tau)}\sim\tau^{-\nu}\qquad\text{where}\qquad\tau=\frac{T-T_{\text{c}}}{T_{\text{c}}}>0
\label{equilibrium.xi.taunu}$$ is the distance to the critical point[^1], and $\nu=1$ is the universal critical exponent of the Ising universality class associated to the correlation length. is only valid in a close vicinity of the critical temperature ($\tau\ll1$); far from it, there are extra corrections to add, but we do not need them here. Another limitation of \[equilibrium.xi.taunu\] is that it is only valid for an infinite system; if the system size ($L$) is finite, it limits the growth of the correlation length to a saturation threshold that scales with the system size as $\xi_{\text{eq}}\lr{(\tau=0)}=\bar{\xi}_{\text{eq}}\sim L$.
Let us now discuss the equilibrium behaviour of the variance of the winding angle (<span style="font-variant:small-caps;">wav</span>), *ie* the nature of the interfaces between domains, see \[equilibrium.plots\] (**a**). We observe that the <span style="font-variant:small-caps;">wav</span> increases logarithmically on short curvilinear length scales; the value of $\kappa$ extracted from the slope of $\lr{<\theta^2\lr{(\log\ell)}>}$ is close to $6$ at high temperature and close to $3$ at $T_{\text{c}}$. This means that, on short length scales, the interfaces of the domains are subject to a conformal invariance (with the criticality of percolation at high temperature and the one of Ising at $T_{\text{c}}$). There is nothing surprising here. Firstly, at the Ising critical point the domains obviously have the criticality of the corresponding universality class. Secondly, at high temperature, the Ising model is a percolation problem (correlations are so short that one could argue that the spins are randomly chosen to point up or down with half probability, $p=\nicefrac{\displaystyle{1}}{\displaystyle{2}}$). A typical configuration is, therefore, one of a site percolation problem away from its critical point (recall that, on a square lattice, the critical percolation threshold is at $p_{\text{c}}\approx0.593>\nicefrac{\displaystyle{1}}{\displaystyle{2}}=p$). In consequence, on average, there are no percolating clusters in these configurations. This means that the conformal invariance disappears at sufficiently long length scales: $\ell\sim{\lr{|p-p_{\text{c}}|}}^{-\nu_{\text{p}}D_{\ell}}\sim10^2$, where $\nu_{\text{p}}=\nicefrac{\displaystyle{4}}{\displaystyle{3}}$ is the percolation correlation length critical exponent and $D_{\ell}=\nicefrac{\displaystyle{7}}{\displaystyle{4}}$ is the fractal dimension of the interface of a percolation cluster. This leads us to our second remark: at high temperature and long length scales, the <span style="font-variant:small-caps;">wav</span> does not grow logarithmically anymore; it increases much faster. This is, in fact, due to the finite size of the domains. Indeed, since we are far from the critical percolation threshold, the domains remain small, and the overall curvature necessary to close their interface is responsible for a faster growth of the <span style="font-variant:small-caps;">wav</span>. When the temperature decreases the domains swell (like the correlation length), and the <span style="font-variant:small-caps;">wav</span> stops its logarithmic growth at a longer and longer length scale. Obviously, when reaching $T_{\text{c}}$, there is a true conformal invariance, and the <span style="font-variant:small-caps;">wav</span> increases logarithmically on all length scales. Considering only the short length scales, as the temperature decreases, the criticality smoothly evolves from the percolation universality class to the Ising one. This is most clearly shown in panel (**b**) in \[equilibrium.plots\] where $\kappa$ is plotted as a function of $T$. The slope is extracted from the <span style="font-variant:small-caps;">wav</span> by linear interpolation on short length scales; the longer length scales, where criticality disappears, are excluded from the interpolation set. The Ising criticality is only reached in a close vicinity of the critical point ($T<1.1\,T_{\text{c}}$).
The fact that we observe critical percolation properties in the disordered phase is related to the presence of a critical curve in the temperature-field phase diagram of the $2d$ Ising model. It separates a phase with an infinite cluster of parallel spins (at sufficiently large external field) from one without (weak field). This critical curve joins the Ising critical point (Curie temperature and zero field) with the infinite temperature limit at non-vanishing value of the external field, while remaining close to the zero field axis [@Delfino15]. The vicinity of this line at our working temperatures justifies the fact that we see (finite size) critical percolation geometric properties on the spins clusters.
The last quantity we want to discuss is the average occupancy rate of the largest clusters shown in \[equilibrium.plots\] (**c**). Firstly, at high temperature, all the clusters are more or less of the same size. Then, when temperature decreases, the bigger clusters start to grow by absorbing the smaller ones, up to a point ($T\approx1.1\,T_{\text{c}}$) where only the two biggest prevail over all the others. Having two coexisting big clusters is a feature of percolation of Ising clusters[^2]. These two clusters will coexist up to a very close vicinity of the Curie temperature ($T\lesssim1.01\,T_{\text{c}}$). In contrast, at the Ising critical point there is only one large cluster (much larger than all the others).
To summarise, at $T_{\text{c}}$, or in its very close vicinity, the system is occupied by only one large geometric cluster having the Ising criticality ($\kappa\approx3$) at all length scales. See the snapshot at $T=T_{\text{c}}$ in \[equilibrium.plots\] (**c**). At high temperature, the domains are much smaller. However, on short length scales, they have the geometric properties of critical percolation ($\kappa\approx5.5$, which is only $5\%$ different from the slope expected with $\kappa=6$). Finally, in between, the criticality smoothly changes from the percolation one to the Ising one in the range $\lr{[T_{\text{c}},1.1\,T_{\text{c}}]}$; the coexistence of the two biggest clusters ends much closer to the critical point ($T\lesssim1.01\,T_{\text{c}}$).
Instantaneous quenches {#sec:quenches}
======================
In this section we recall some features of the dynamics after instantaneous quenches to zero temperature and the critical point, as interpreted from the geometric point of view that we adopt in this paper.
Quench to T=0
-------------
![ Out-of-equilibrium evolution in post-quench dynamics (from $T=2\,T_{\text{c}}$ to $T=0$). Panel (**a**) represents the winding angle variance (<span style="font-variant:small-caps;">wav</span>) at different times following the quench. These times are reported in panel (**b**), and are chosen such that the constraint $a\ll\xi\left(t\right)\ll L$ is fulfilled. Panel (**c**) still represents the <span style="font-variant:small-caps;">wav</span>, but with a different scaling: the horizontal axis is rescaled following \[instaquench.0.wav.scaling\], and $\ell_{\text{d}}\left(t\right)$ is evaluated through its theoretical expression ($\sim t^{\nicefrac{\scriptstyle{1}}{\scriptstyle{z_{\text{d}}}}}$). Panel (**b**) shows the evolution over time of the correlation length $\xi\left(t\right)$ extracted from the space-time two point correlation function. Its theoretical time-dependence is shown with a dashed line; the range of validity of this prediction is highlighted by the grey shading ($a\ll\xi\left(t\right)\ll L$). Panel (**d**) represents, as a function of time, the average occupancy rates of the first largest clusters (see \[equilibrium.plots\]). All the results presented in this figure were obtained using a system size $L=1024$. []{data-label="instaquench.0.plots"}](./instaquench0.pdf)
The second situation of interest is the one of an instantaneous quench to zero temperature. We consider the following procedure: starting from an equilibrium state at $T=2\,T_{\text{c}}$, at $t=0$ we suddenly change the temperature of the bath to zero, *ie* $$T\lr{(t)}=\lr{\{\begin{array}{ll}
2\,T_{\text{c}}&t\leq0\\
0&t>0
\end{array}.}$$ and we observe the further evolution of the system.
In such a procedure, the growing length is known to increase as a power law, $$\xi\lr{(t)}\sim t^{\nicefrac{\scriptstyle{1}}{\scriptstyle{z_{\text{d}}}}}\eqpc$$ where $z_{\text{d}}=2$ is the dynamical exponent [@AlCa79; @Bray94]. Of course, this result holds only for $t$ such that $a\ll\xi\lr{(t)}\ll L$: $\xi\lr{(t)}$ cannot be smaller than the lattice spacing, and it is bounded by the finite system size. The growing length extracted from the correlation function after such a quench is compared to the theoretical expectation in \[instaquench.0.plots\] (**b**).
Let us now discuss how the behaviour of the <span style="font-variant:small-caps;">wav</span> evolves in time, as displayed in \[instaquench.0.plots\] (**a**). At the initial time, the system is in equilibrium, and the <span style="font-variant:small-caps;">wav</span> behaves as described in eq. (\[eq:wav-equil\]), see \[equilibrium.plots\] (**a**). Then, the zero temperature dynamics start to smooth the interfaces: first, on short length scales, then, on longer length scales. This is the first part of the curve and the <span style="font-variant:small-caps;">wav</span> does not increase since a smooth interface has no criticality. In the meantime, the clusters swell, and since the system has not yet realised, at long length scales, that it is at zero temperature (and should have smooth interfaces), it develops the criticality of percolation. This is the second part of the curve; the <span style="font-variant:small-caps;">wav</span> restarts to grow logarithmically. The typical (curvilinear abscissa) length scale that separates these two behaviours is denoted $\ell_{\text{d}}\lr{(t)}$, and is related to the typical size of the domains: $$\ell_{\text{d}}\lr{(t)}\sim {\xi\lr{(t)}}^D\sim t^{\nicefrac{\scriptstyle{D}}{\scriptstyle{z_{\text{d}}}}}=\sqrt{t}\eqpc$$ since $D=1$ is the (fractal) dimension of the smooth interfaces on short length scales. The <span style="font-variant:small-caps;">wav</span> has a universal behaviour in time that is highlighted by the rescaling $$ \ell\to\frac{\ell}{\ell_{\text{d}}\lr{(t)}}\eqpc
\label{instaquench.0.wav.scaling}$$ once again, while $\xi(t)$ is in the range $a\ll\xi\lr{(t)}\ll L$. See \[instaquench.0.plots\] (**c**).
(**d**) shows the evolution of the average size of the largest clusters. Starting from a high-temperature equilibrium state, all the clusters are almost of the same size. Next, in the early dynamics, they all grow in the same way. As soon as the correlation length starts to grow, the larger clusters progressively swallow the smaller ones. Indeed, the smaller the clusters, the faster they disappear. This is the so-called coarsening dynamics. In particular, the second largest cluster lengthly coexists with the largest one. As already mentioned, this long coexistence of two large clusters having almost the same size (the third cluster is far smaller) is a typical feature of percolation of Ising clusters (see \[footnote.equilibrium.percoisingclusters\] page ).
In the course of this process, the quench protocol went through the Ising critical point, and there is no track of it. Now the question is: what happens if, like in real experiments, we cannot do the quench instantaneously? What is the influence of the time spent near the Curie temperature? will address these questions. However, let us first explore the dynamics after an instantaneous quench to the Curie temperature.
Quench to T=Tc {#sec.quench.atTc}
--------------
![ Out-of-equilibrium evolution in critical post-quench dynamics (temperature is instantaneously taken from $T=2\,T_{\text{c}}$ to $T=T_{\text{c}}$). The graphics are organised in the same manner as in \[instaquench.0.plots\]. However, panel (**c**) is now scaled following \[instaquench.Tc.wav.scaling\], where $\ell_{\text{c}}\left(t\right)$ is evaluated through its theoretical expression ($\sim t^{\nicefrac{\scriptstyle{D_{\text{c}}}}{\scriptstyle{z_{\text{c}}}}}$). Moreover, in the lower right figure, we added with dashed lines the equilibrium values of the average occupancy rates of the first largest clusters at $T_{\text{c}}$. []{data-label="instaquench.Tc.plots"}](./instaquenchTc.pdf)
The process is the same as above, except that the temperature immediately after $t=0$ is now the Curie temperature: $$T\lr{(t)}=\lr{\{\begin{array}{ll}
2\,T_{\text{c}}&t\leq0\\
T_{\text{c}}&t>0
\end{array}.}$$
In this situation, the correlation length still grows as a power law, $$\xi\lr{(t)}\sim t^{\nicefrac{\scriptstyle{1}}{\scriptstyle{z_{\text{c}}}}}\eqpc$$ with $z_{\text{c}}\approx2.17$ the critical dynamical exponent [@Hohenberg-Halperin; @BlCuPi12; @NightingaleBlote00; @Duclut]. The growth of the correlation length is slightly slower than in the previous situation since $\nicefrac{\displaystyle{1}}{\displaystyle{z_{\text{c}}}}\approx0.461<\nicefrac{\displaystyle{1}}{\displaystyle{2}}$. See panel (**b**) in \[instaquench.Tc.plots\]. Again, this result is only true for $\xi(t)$ in the range $a\ll\xi\lr{(t)}\ll L$.
Concerning the <span style="font-variant:small-caps;">wav</span>, it behaves exactly as in the zero temperature quench except that, instead of the smooth zero temperature thermal state, it is the Ising criticality that develops over short length-scales, see \[instaquench.Tc.plots\] (**a**). The typical (curvilinear abscissa) length scale that separates the Ising criticality from the percolation one, $\ell_{\text{c}}\lr{(t)}$, scales differently with the correlation length: $$\ell_{\text{c}}\lr{(t)}\sim {\xi\lr{(t)}}^{D_{\text{c}}}\sim t^{\nicefrac{\scriptstyle{D_{\text{c}}}}{\scriptstyle{z_{\text{c}}}}}\eqpd$$ Since the interfaces on short length scales are not smooth anymore, their fractal dimension is given by $$D_{\text{c}}=1+\frac{\kappa_{\text{c}}}{8}=1.375\eqpc$$ where $\kappa_{\text{c}}=3$ is the same universal parameter as in the pre-factor in front of the logarithmic growth of the <span style="font-variant:small-caps;">wav</span>. Note that $\nicefrac{\displaystyle{D_{\text{c}}}}{\displaystyle{z_{\text{c}}}}\approx0.634>\nicefrac{\displaystyle{1}}{\displaystyle{2}}$. The <span style="font-variant:small-caps;">wav</span> still has a universal behaviour, now highlighted by the rescaling $$\lr{<\theta^2\lr{(\ell,t)}>}\to\lr{<\theta^2\lr{(\ell,t)}>}-\frac{4\,\kappa_{\text{c}}}{8+\kappa_{\text{c}}}\,\log{\ell}\qquad\text{and}\qquad\ell\to\frac{\ell}{\ell_{\text{c}}\lr{(t)}}\eqpc
\label{instaquench.Tc.wav.scaling}$$ where $\nicefrac{\displaystyle{4\,\kappa_{\text{c}}}}{\displaystyle{\lr{(8+\kappa_{\text{c}})}}}\approx1.09$, and still while $\xi(t)$ is in the range $a\ll\xi\lr{(t)}\ll L$, see panel (**c**) in \[instaquench.Tc.plots\].
Finally, the average sizes of the largest clusters evolve in a very similar way to the one found in the $T=0$ quenches: the only perceptible differences are that the smallest clusters do not disappear (thanks to the thermal fluctuations), and the dynamics are slightly slower (since $z_{\text{c}}>z_{\text{d}}$). See panel (**d**) in \[instaquench.0.plots\].
Effects of a finite cooling rate {#sec.cooling}
================================
![ (**a**) Description of the cooling process. Temperature is linearly decreases from $T=2\,T_{\text{c}}$ at $t=0$ to $T=T_{\text{c}}$ at $t=\tau_{\text{Q}}$. $\tau_{\text{Q}}$ controls the cooling rate, and the larger the values it takes, the slower the cooling. The right column (**b**) shows typical snapshots of the system ($L=128$) in the course of the cooling process, and for different values of the cooling rate. Panel (**c**) displays the evolution of the correlation length extracted from the space-time correlation function in the course of cooling in a system with $L=1024$ (note that the maximum value of $\xi$ is close to 10, much shorter than the system size). We have also represented the equilibrium correlation length at the corresponding temperatures. []{data-label="cooling.intro.plots"}](./coolingintro.pdf)
In the present section, we will discuss how the time spent in the vicinity of the Ising critical point affects the dynamics.
Let us first describe the process considered in the remainder of this paper. The system is initially placed in an equilibrium state at $T=2\,T_{\text{c}}$ (*ie* $T\lr{(t)}=2\,T_{\text{c}}$ for all $t\leq0$). Next, at $t=0$ the temperature of the bath is linearly cooled following $$\frac{T\lr{(t)}}{T_{\text{c}}}=2-\frac{t}{\tau_{\text{Q}}}\eqpc$$ where $\tau_{\text{Q}}$ is the cooling time up to the Curie temperature (see \[cooling.intro.plots\]). In the present study, we only consider the dynamics above $T_{\text{c}}$ (*ie* $t\in\lr{[0,\tau_{\text{Q}}]}$). Studies of the cooling rate effects on the coarsening dynamics that is at work close and below the critical point, even after annealing, have been presented in [@Biroli10] for the $2d$ Ising model, in [@Jelic11] for the $2d$ xy model, in [@Priyanka] for a one-dimensional non-equilibrium lattice gas model with a phase transition between a fluid phase with homogeneously distributed particles and a jammed phase with a macroscopic hole cluster, and in [@Comaron17a; @Comaron17b] for time-dependent dissipative and stochastic Gross – Pitaievskii models relevant to describe micro-cavity polaritons and cold boson gases.
The Kibble - Zurek mechanism
----------------------------
Starting from a thermal state, the system will follow the equilibrium conditions dictated by the changing environment as long as it can: *ie* up to a time, called $\hat{t}$, when the time needed to thermalise becomes too long with respect to the relative rate of variation of temperature. Next, the system falls out-of-equilibrium and its further evolution will be discussed later. Obviously, the slower the cooling, the later the system will fall out-of-equilibrium. For an infinite system size, the time required to thermalise at the Ising critical point is infinite; it actually scales as $L^{z_{\text{c}}}$, and, unless cooling rates are scaled with the system size in a convenient way, the system will necessarily fall out-of-equilibrium at a certain point. Conversely, for finite-size systems, there exists a sufficiently slow cooling rate such that the system never goes out-of-equilibrium; we will discuss this point in \[sec.cooling.atTc\]. We suppose the cooling to be sufficiently slow so that the system falls out-of-equilibrium only in a close vicinity of the critical point. On the one hand, in equilibrium, the correlation length depends on the distance from the critical point as $\tau^{-\nu}$. On the other hand, close to $T_{\text{c}}$, the dynamic correlation length grows in time as $\xi\lr{(t)}\sim{\lr{(t+\sharp\,{\xi_0}^{z_{\text{c}}})}}^{\nicefrac{\scriptstyle{1}}{\scriptstyle{z_{\text{c}}}}}$, where $\xi_0$ is the initial correlation length and $\sharp$ some constant factor. After an instantaneous quench from a state with correlation length $\xi(t=0)=\xi_0$ to a temperature at a distance $\tau>0$ from the critical point, the thermal state is reached after a time $t^{\text{th}}\lr{(\tau)}$ such that $\xi\lr{(t=t^{\text{th}}\lr{(\tau)})}\sim\xi_{\text{eq}}\lr{(\tau)}\sim\tau^{-\nu}$. Assuming that the instantaneous quench is performed from $2\,T_{\text{c}}$ and that the correlation length vanishes at this temperature, $\xi_0=0$, we have $t^{\text{th}}\lr{(\tau)}\sim\tau^{-\nu z_{\text{c}}}$. This is good estimate for the time needed to equilibrate at a distance $\tau$ from criticality.
Now, following the argument proposed by Zurek [@Zurek85], the system falls out-of-equilibrium at a time $\hat t$, when the time needed to reach $T_{\text{c}}$, $\tau_{\text{Q}}-\hat{t}$ in the linear cooling procedure, becomes smaller than the time needed to thermalise at the current temperature $\hat T$ (the standard notation is such that the temperature and time at which the system falls out of equilibrium are noted by hats). Hence, we have $$\tau_{\text{Q}}-\hat{t}\sim t^{\text{th}}\lr{(\hat{\tau})}\sim{\hat{\tau}}^{-\nu z_{\text{c}}}\sim{\lr{(1-\frac{\hat{t}}{\tau_{\text{Q}}})}}^{-\nu z_{\text{c}}}\sim{\lr{(\frac{\tau_{\text{Q}}-\hat{t}}{\tau_{\text{Q}}})}}^{-\nu z_{\text{c}}}\eqpc$$ where $\hat{\tau}$ is the distance from the critical temperature at $\hat{t}$. Therefore, the system falls out-of-equilibrium at $$\hat{t}=\tau_{\text{Q}}-\sharp\,{\tau_{\text{Q}}}^{\nicefrac{\scriptstyle{\nu z_{\text{c}}}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}}\eqpc$$ where $\nicefrac{\displaystyle{\nu z_{\text{c}}}}{\displaystyle{\lr{(1+\nu z_{\text{c}})}}}\approx0.685$ and $\sharp$ another constant factor.
In many papers dealing with the slow cooling of atomic systems the assumption is that, after $\hat t$, the system remains frozen and correlations do not grow beyond the correlation length present at this time, $$\hat{\xi}=\xi\lr{(\hat{t})}\sim\hat{\tau}^{-\nu}\sim{\lr{(\frac{{\tau_{\text{Q}}}^{\nicefrac{\scriptstyle{\nu z_{\text{c}}}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}}}{\tau_{\text{Q}}})}}^{-\nu}\sim{\tau_{\text{Q}}}^{\nicefrac{\scriptstyle{\nu}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}}\eqpc$$ where $\nicefrac{\displaystyle{\nu}}{\displaystyle{\lr{(1+\nu z_{\text{c}})}}}\approx0.315$. This, however, is not correct in coarsening systems as already discussed in [@Biroli10; @Jelic11; @Priyanka; @Comaron17a], for example.
The out-of-equilibrium dynamics {#sec.cooling.atTc}
-------------------------------
At early times such that $t<\hat{t}$, the system evolves in equilibrium and the correlation length grows as the equilibrium one at the temperature reached at the measuring time: $$\xi_{{}_<}\lr{(t)}\sim\xi_{\text{eq}}\lr{(\tau)}\sim{\lr{(1-\frac{t}{\tau_{\text{Q}}})}}^{-\nu}\eqpd$$ When $t$ exceeds $\hat{t}$, the correlation length does not grow fast enough, and the system falls out of equilibrium. Now, since the equilibrium correlation length soon becomes much longer than the dynamic growing length, we can make the simplifying assumption that the system behaves as if it were in contact with a bath right at the Curie temperature. This proposal amounts to treating the problem as after an instantaneous quench at $t=\hat{t}$ from $\tau=\hat{\tau}$ to $\tau=0$. Hence, the correlation length continues to grow, but now as $$\xi_{{}_>}\lr{(t)}\sim{\lr{(t-\hat{t}+\sharp\,{\hat{\xi}}^{z_{\text{c}}})}}^{\nicefrac{\scriptstyle{1}}{\scriptstyle{z_{\text{c}}}}}\eqpc$$ where the term in ${\hat{\xi}}^{z_{\text{c}}}$ takes into account the non-vanishing correlation length at $t=\hat{t}$.
Then, imposing the consistency of the correlation length before and after $\hat{t}$, $$\xi_{{}_>}\lr{(\hat{t})}=\xi_{{}_<}\lr{(\hat{t})}=\hat{\xi}\sim{\tau_{\text{Q}}}^{\nicefrac{\scriptstyle{\nu}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}}\eqpc
\label{eq.xidynamics.afterthat}$$ we have $$\xi\lr{(t)}\sim\lr{\{\begin{array}{ll}
{\lr{(1-\nicefrac{\displaystyle{t}}{\displaystyle{\tau_{\text{Q}}}})}}^{-\nu}&t\leq\hat{t}\\
{\lr{(t-\tau_{\text{Q}}+\sharp\,{\tau_{\text{Q}}}^{\nicefrac{\scriptstyle{\nu z_{\text{c}}}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}})}}^{\nicefrac{\scriptstyle{1}}{\scriptstyle{z_{\text{c}}}}}&t\geq\hat{t}
\end{array}.}\eqpd$$ where the second line is obtained from \[eq.xidynamics.afterthat\] where we have replaced $\hat{t}=\tau_{\text{Q}}-\sharp\,{\tau_{\text{Q}}}^{\nicefrac{\scriptstyle{\nu z_{\text{c}}}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}}$ and $\hat{\xi}\sim{\tau_{\text{Q}}}^{\nicefrac{\scriptstyle{\nu}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}}$. In particular, when reaching the critical point, $$\xi\lr{(t=\tau_{\text{Q}})}=\bar{\xi}\sim{\tau_{\text{Q}}}^{\nicefrac{\scriptstyle{\nu}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}}\sim\hat{\xi}\eqpd
\label{cooling.Tc.xi.scaling}$$ While Zurek assumes that the system is frozen immediately after falling out-of-equilibrium, here we claim that the dynamic growing length, $\xi\lr{(t)}$, continues to grow after $\hat{t}$. However, the growth between $\hat t$ and $\tau_{\text{Q}}$, when the cooling reaches $T_{\text{c}}$, only the pre-factor and not the scaling with $\tau_{\text{Q}}$ that is not modified. Therefore, if the interest is set upon the scaling properties of the system at the critical point (and not far below it) one can assume that the dynamic correlation length takes the form it had at $\hat{t}$.
The next section will be devoted to putting eq. (\[cooling.Tc.xi.scaling\]) to the test.
![ Dependence on the cooling rate after a linear cooling to the critical point. Panel (**a**) represents the <span style="font-variant:small-caps;">wav</span> for different cooling times and panel (**c**) shows the same quantity, after the rescaling proposed in \[cooling.Tc.wav.scaling\]; $\kappa_{\text{c}}=3$ and $D_{\text{c}}=1.375$ are the same as in \[sec.quench.atTc\]. The graphics in (**b**) shows, as a function of the cooling rate, the measured correlation length when reaching the critical point; the dashed line is its predicted evolution (see \[cooling.Tc.xi.scaling\]). The last figure, (**d**), represents the average occupancy rates of the largest clusters, and the dashed lines highlight the expected values for an infinitely slow annealing (*ie* the values in equilibrium at $T_{\text{c}}$). All the results presented in these graphics were obtained with a system size of $L=1024$. []{data-label="cooling.Tc.plots"}](./coolingTc.pdf)
We are now going to describe the state of the system when reaching the critical point, and how it depends on the cooling rate. First of all, we can easily check that $\xi\lr{(\tau_{\text{Q}})}=\bar{\xi}\sim\hat\xi$ is quite an accurate prediction, see panel (**b**) in \[cooling.Tc.plots\].
Next, let us analyse how the <span style="font-variant:small-caps;">wav</span> behaves: as exposed in \[sec.quench.atTc\], the interfaces present two critical properties: the Ising one on short length scales, and the percolation one otherwise. These features are proven in panel (**a**) in \[cooling.Tc.plots\]. The length scale that separates the two behaviours scales with the effective correlation length when reaching the critical point. Thus, the rescaling $$\lr{<\theta^2\lr{(\ell,t)}>}\to\lr{<\theta^2\lr{(\ell,t)}>}-\frac{4\,\kappa_{\text{c}}}{8+\kappa_{\text{c}}}\,\log{\ell}\quad\text{and}\quad\ell\to\frac{\ell}{{\bar{\xi}}^{D_{\text{c}}}}\to\ell\,{\tau_{\text{Q}}}^{\nicefrac{\scriptstyle{-\nu D_{\text{c}}}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}}\eqpc
\label{cooling.Tc.wav.scaling}$$ where $\nicefrac{\displaystyle{-\nu D_{\text{c}}}}{\displaystyle{\lr{(1+\nu z_{\text{c}})}}}\approx-0.434$, used in panel (**c**) in \[cooling.Tc.plots\], highlights the universal behaviour of the <span style="font-variant:small-caps;">wav</span>. The quality of this scaling provides a second proof of the accuracy of the prediction (\[cooling.Tc.xi.scaling\]).
Consequently, when reaching the critical point, the system is (at least) thermalised up to a scale $s$, as soon as the cooling is slower than $\tau_{\text{Q}}^{{\text{th}}_s}$ which is such that $$s\sim\bar{\xi}\sim{\tau_{\text{Q}}^{{\text{th}}_s}}^{\nicefrac{\scriptstyle{\nu}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}}\quad\Rightarrow\quad\tau_{\text{Q}}^{{\text{th}}_s}\sim s^{z_{\text{c}}+\nicefrac{\scriptstyle{1}}{\scriptstyle{\nu}}}\eqpd$$ We recall that for an infinitely fast quench to $T=T_{\text{c}}$, the scale $s$ is thermalised after a time $$t^{{\text{th}}_s}\sim s^{z_{\text{c}}}\eqpd$$ Since $z_{\text{c}}+\nicefrac{\displaystyle{1}}{\displaystyle{\nu}}\approx3.17>2.17\approx z_{\text{c}}$, an instantaneous quench is more efficient than a linear cooling to create the structures of the Ising critical point; the time spent far from $T_{\text{c}}$ is not useful to develop the Ising criticality, the system develops, instead, the percolation one.
This feature can also be observed by looking at the average sizes of the largest clusters by comparing panels (**d**) in \[instaquench.Tc.plots,cooling.Tc.plots\]. Indeed, on \[instaquench.Tc.plots\], at $t=10^5\sim\nicefrac{\displaystyle{t^{{\text{th}}_{L=1024}}}}{\displaystyle{10}}$, the second largest cluster has already started to be swallowed by the first one, and the third and fourth have almost reached their equilibrium average sizes. In contrast, on \[cooling.Tc.plots\], at $\tau_{\text{Q}}=10^5$, all the largest clusters are still far from their equilibrium average sizes. Moreover, the first and second are still of the same order of magnitude.
These results confirm that the dynamics are affected by the Ising critical point only in its close vicinity, and the time spent far from it is not helpful to get closer to equilibrium at $T_{\text{c}}$.
Dynamics before reaching the critical point {#sec.cooling.dynamics}
-------------------------------------------
![ Approach to the critical point; dependency on the cooling rate. Panel (**a**) shows the increase of the correlation length during cooling for different cooling rates; the equilibrium correlation length is also shown. Panel (**c**) represents the same quantities, but with a different scaling (following \[cooling.dyna.xi.scaling\]). In panel (**b**), we show the evolution of the slopes of the <span style="font-variant:small-caps;">wav</span> when approaching the critical point together with the equilibrium one. $\kappa_{\text{S}}$ is extracted from the slope of the <span style="font-variant:small-caps;">wav</span> at short curvilinear length scales and is expected to have the Ising criticality when reaching the Curie temperature. $\kappa_{\text{L}}$ is extracted from the slope of the <span style="font-variant:small-caps;">wav</span> at long curvilinear length scales and corresponds to the percolation criticality. All the results presented in these graphics were obtained with a system size of $L=1024$. []{data-label="cooling.time.plots"}](./coolingdyna.pdf)
In the previous section, we have shown that the behaviour when reaching the critical point does not really rely on the exact out-of-equilibrium dynamics in the range $t\in\lr{[\hat{t},\tau_{\text{Q}}]}$; whether the system remains frozen or evolves like in a post-quench dynamics. In this last section we try to clarify the situation.
Let us consider that the system’s typical length continues to grow as after an instantaneous quench after $\hat{t}$; the correlation length should then grow as $$\begin{aligned}
\xi\lr{(t)}&\sim{\lr{(t-\tau_{\text{Q}}+\sharp\,{\tau_{\text{Q}}}^{\nicefrac{\scriptstyle{\nu z_{\text{c}}}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}})}}^{\nicefrac{\scriptstyle{1}}{\scriptstyle{z_{\text{c}}}}}\\
&\sim{\tau_{\text{Q}}}^{\nicefrac{\scriptstyle{1}}{\scriptstyle{z_{\text{c}}}}}\,{\lr{(\sharp\,{\tau_{\text{Q}}}^{\nicefrac{\scriptstyle{-1}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}}-\tau)}}^{\nicefrac{\scriptstyle{1}}{\scriptstyle{z_{\text{c}}}}}\eqpc
\end{aligned}$$ and forgetting the dependence in the cooling rate, as $$\xi\lr{(t)}\sim{\lr{(\sharp-\tau)}}^{\nicefrac{\scriptstyle{1}}{\scriptstyle{z_{\text{c}}}}}\eqpc
\label{cooling.time.xi.tau}$$ where the $\sharp$ factor has changed but is still positive. Thus, for $\sharp\gg\tau$ (or $\tau$ small enough), the correlation length is almost constant, and the system seems to be frozen. Moreover, the shape described by \[cooling.time.xi.tau\] is in a quite good agreement with the numerical results presented in \[cooling.time.plots\](**a**).
Let us now recall that the correlation length at the time or temperature at which the system falls out-of-equilibrium scales as $$\hat{\xi}\sim{\tau_{\text{Q}}}^{\nicefrac{\scriptstyle{\nu}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}}\eqpd$$ This is only valid while $\tau_{\text{Q}}$ is such that $\hat{\xi}\leq\xi_{\text{eq}}\lr{(\tau=0)}$. Beyond this point, the correlation length saturates to $\hat{\xi}=\xi_{\text{eq}}\lr{(\tau=0)}$; especially for an infinitely slow cooling (equilibrium). Doing the rescaling $$\xi\lr{(t)}\to\frac{\xi\lr{(t)}}{\hat{\xi}}\quad\text{and}\quad\tau\to\tau\,{\hat{\xi}}^\nu
\label{cooling.dyna.xi.scaling}$$ (where $\hat{\xi}$ is taken as its saturation value for the equilibrium curve), the panel (**c**) in \[cooling.time.plots\] shows that the correlation length has a universal behaviour. This is in agreement with a power law growth of the correlation length, as assumed in \[cooling.time.xi.tau\]. Nonetheless, universality disappears far from the critical point since the equilibrium correlation length is subject to non-algebraic corrections in this region.
Let us finally discuss the “change in criticality”, as measured by the evolution of the parameter $\kappa$ in the course of the cooling process and compare it to the equilibrium ($\kappa_{\text{eq}}$). As done before, $\kappa$ is extracted from the <span style="font-variant:small-caps;">wav</span>. (**b**) represents the evolution of the criticality on short ($\kappa_{\text{S}}$) and long ($\kappa_{\text{L}}$) curvilinear abscissa length scales. The long length scales have almost always the criticality of percolation ($\kappa=6$) except for very slow cooling rates and in the vicinity of the Curie temperature where the criticality starts to be affected by the Ising critical point. On short length scales, the system is able to achieve the equilibration process, and the criticality corresponds to the equilibrium one discussed in \[sec:equilibrium\] and represented by $\kappa_{\text{eq}}$. However, for the fastest cooling rates, *eg* $\tau_{\text{Q}}=10^1$, even the short scales cannot follow the equilibrium.
Conclusions {#conclusions .unnumbered}
===========
The purpose of this work was to study the influence of the cooling rate on the dynamics close to a second order critical point (between a symmetric and symmetry broken phase; here, for Ising models, the $\mathbb{Z}_2$ symmetry). More precisely, we analysed the evolution of the geometric and scaling properties of the interfaces between domain walls close and at the critical point.
In order to set the stage, we first studied the fractal properties of the interfaces in equilibrium at various temperatures in the disordered phase. The analysis of the <span style="font-variant:small-caps;">wav</span> allowed us to reach our first conclusion:
- In equilibrium at $T>T_{\text{c}}$ the long-scale properties of the interfaces are the ones of critical percolation until a crossover length-scale that decreases with increasing temperature. A temperature dependent crossover towards critical Ising fractality at short-length scales arises close to the critical point, visible only below, say, $T=1.1\,T_{\text{c}}$.
So far, the influence of critical percolation on the dynamics of the $2d$ Ising – Glauber model after *instantaneous* quenches from infinite temperature to the critical point [@BlCuPi12] and below it [@ArBrCuSi07; @SiArBrCu07; @BlCoCuPi14; @BlCuPiTa17] was studied. The equilibrium result just mentioned indicates that this critical percolation geometry is present in high temperature equilibrium configurations.
Next, we recalled some basic features of the coarsening dynamics following an instantaneous quench from equilibrium at $T= 2\,T_{\text{c}}$ both to zero and the Curie temperatures. On the one hand, we confirmed that correlation functions scale with a growing length that increases algebraically with time. On the other hand, we highlighted the non-trivial evolution of the geometry of the domains of parallel spins. The critical percolation geometry of the interfaces present in the initial state is progressively transformed, starting from the short scales, towards the one of the target temperature: smooth at zero temperature and critical Ising at $T_{\text{c}}$.
We then explained the Kibble – Zurek mechanism [@Kibble76; @Zurek85] allowing one to estimate when the system falls out-of-equilibrium while approaching a critical point from the symmetric phase with a finite speed. While Zurek assumes that the system freezes after falling out-of-equilibrium, following [@Biroli10; @Jelic11; @Comaron17b] we argued that the correlation length continues to grow in this regime as if the system were instantaneously quenched to the critical point. Our argument does not affect the scaling in the cooling rate predicted by Zurek, but offers a more accurate description of the growing of the correlation length after the system has fallen out-of-equilibrium. We examined this scaling numerically and we found that
- after a slow linear cooling with rate $\tau_{\text{Q}}$ to $T_{\text{c}}$, the dynamic growing length extracted from the analysis of the space-time correlation function scales as $\xi(\tau_{\text{Q}})\sim\tau_{\text{Q}}^{\nicefrac{\scriptstyle{\nu}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}}$.
During the slow cooling process, while far from the critical point, the interfaces keep the fractal properties of critical percolation over a wide range of length scales, up to a temperature dependent crossover length. However, when approaching temperatures close enough to the critical point, we observe that
- the winding angle variance satisfies a scaling with respect to $\xi(\tau_{\text{Q}})$, and besides, the interfaces with critical Ising properties span the length scales that are shorter than ${\tau_{\text{Q}}}^{\nicefrac{\scriptstyle{\nu D_{\text{c}}}}{\scriptstyle{\lr{(1+\nu z_{\text{c}})}}}}$.
Finally, our results prove that the Ising critical point influences the dynamics only in its close vicinity. Therefore,
- an instantaneous quench procedure is much more efficient to create the structures of the Curie temperature than a slow annealing.
As a matter of fact, the time spent far from the Ising critical point does not contribute to the thermalisation of the system; instead, the dynamics of the system is governed by critical percolation.
This study is also a complement to works that try to elucidate the role played by the initial conditions on the post-quench dynamics of the Ginzburg – Landau scalar field theory [@BrayHumayunNewman] and, more recently, the kinetic Ising model [@ChakrabortyDas15; @ChakrabortyDas16; @Corberi16] as well as the influence of a non-vanishing cooling rate on the scaling properties of discrete models close to their phase transition [@Liu-etal14]. In the latter paper the emphasis was set on the scaling properties of the order parameter and how these depend, or not, on the microscopic stochastic updates. We focus instead on the geometrical and scaling properties of the structures when slowly approaching the critical point.
#### Acknowledgements.
L. F. C. is a member of Institut Universitaire de France. We thank A. Tartaglia for very useful discussions and M. Henkel for his helpful comments on the manuscript.
[^1]: here we are only interested in the behaviour above the Curie temperature (*ie* $T>T_{\text{c}}$).
[^2]: \[footnote.equilibrium.percoisingclusters\] in site percolation, at $p=p_{\text{c}}$, the largest cluster (the percolating one) is much larger than the second one (of approximately one order of magnitude). In Ising models two percolating clusters are in competition: the up spins one and the down spins one; generally, in a $q$-state Potts model, the $q$ largest clusters are of the same order of magnitude while the $q+1$-th will be much smaller.
|
---
abstract: 'Dwarf spheroidal galaxies (dSphs) are the most compact dark matter-dominated objects observed so far. The Pauli exclusion principle limits the number of fermionic dark matter particles that can compose a dSph halo. This results in a well-known lower bound on their particle mass. So far, such bounds were obtained from the analysis of individual dSphs. In this paper, we model dark matter halo density profiles via the semi-analytical approach and analyse the data from eight ‘classical’ dSphs assuming the same mass of dark matter fermion in each object. First, we find out that modelling of Carina dSph results in a much worse fitting quality compared to the other seven objects. From the combined analysis of the kinematic data of the remaining seven ‘classical’ dSphs, we obtain a new $2\sigma$ lower bound of $m\gtrsim 190$eV on the dark matter fermion mass. In addition, by combining a sub-sample of four dSphs – Draco, Fornax, Leo I and Sculptor – we conclude that 220eV fermionic dark matter appears to be preferred over the standard CDM at about 2$\sigma$ level. However, this result becomes insignificant if all seven objects are included in the analysis. Future improvement of the obtained bound requires more detailed data, both from ‘classical’ and ultra-faint dSphs.'
author:
- |
D. Savchenko$^{1}$[^1] and A. Rudakovskyi$^{1}$\
$^{1}$Bogolyubov Institute of Theoretical Physics, Metrolohichna Str. 14-b, 03143, Kyiv, Ukraine
bibliography:
- 'refs.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: New mass bound on fermionic dark matter from a combined analysis of classical dSphs
---
\[firstpage\]
dark matter – galaxies: haloes – galaxies: dwarf – galaxies: kinematics and dynamics – methods: statistical
Introduction
============
The nature of dark matter (DM) is one of the major questions in modern physics. The mass of DM particle candidates, which exist in numerous extensions of the Standard Model, varies in very wide range – from $\sim10^{-22}$eV for ultra-light DM [e.g., @Hu:00; @Hui:2016ltb; @Lee:18] up to TeVs for WIMPs [see, e.g., @Roszkowski:2017nbc; @Arcadi:2017kky and references therein] or up to $\sim10^{13}$GeV for WIMPZILLAs [e.g. @Chung:1998zb].
The Pauli principle forbids packing too many fermions into a gravitationally bound object. Therefore, the average phase-space density of such an object with mass $M$ enclosed within a region of radius $R$, $\bar{F}\sim \frac{M}{R^3\sigma^3}$, cannot exceed some maximum $f_\text{max}(m)$, where $m$ is the mass of fermion, and $\sigma$ is the particle velocity dispersion. This allows one to obtain the lower bound $m\simeq0.5$keV [@Bode:00; @Dalcanton:00; @Boyarsky:08a; @Horiuchi:13], based on the extended Tremaine–Gunn [@Tremaine:79] approach [see also @Gorbunov:08; @Shao:12; @Wang:2017] from the analysis of compact DM dominated objects – dwarf spheroidal satellites (dSphs). This approach requires an estimator of the dynamical mass $M$ within a sphere of some radius $R$ [@Wolf:09; @Walker:11; @Campbell:16], see also @Kowalczyk:12 for a detailed study of the mass estimator uncertainties, and @Boyarsky:08a for the estimate of the phase-space volume occupied by the DM particles.
Another method for constraining the mass of the fermionic DM particle uses direct comparison between the detailed prediction of the kinematics of dSph and the observational data [see, e.g., @Domcke:2014kla; @Randall:16; @DiPaolo:2017geq]. It does not require an estimate of the averaged phase-space density over a spatial region. Direct modelling of kinematics also allows one to incorporate the anisotropy of the velocity dispersion into analysis. Moreover, unlike the Tremaine–Gunn approach, this method allows one to combine the data on several objects to produce better limits on the particle mass. In return, it requires a (semi-)analytical model of the DM density profile and stellar density profile. Many analytical models of fermionic DM halo density profiles have been developed so far; see, e.g., @Ruffini:83 [@Bilic:97; @Angus:09; @deVega:13; @deVega:14; @Merafina:14; @Domcke:2014kla; @Ruffini:14; @Chavanis:14b; @Arguelles:16; @Randall:16; @Rudakovskyi:18rs; @Giraud:2018gxl; @Barranco:2018gjg].
In this paper, we present a new lower bound on the mass of fermionic DM particle, based on the observed kinematics [@Bonnivard:2015xpq] and photometry [@McConnachie:12] data of ‘classical’ dSphs, and assuming the DM density model of @Rudakovskyi:18rs. In comparison to @Domcke:2014kla, @DiPaolo:2017geq, this approach allows us not only to analyse individual dSphs, but also to perform *combined* statistical analysis based on the total $\chi^2$ goodness-of-fit statistics assuming the same dark matter particle mass in all of them. Thereby, we aim at utilising fully the statistical power of the approach.
This paper is organised as follows: in Sec. \[sec:methods\] our methods are described (a short description of our model of fermionic DM halo is also included), the obtained results are summarised in Sec. \[sec:results\] and discussed in Sec. \[sec:discussion\]. We use the recent *Planck* [@Planck-2018] cosmological parameters for our calculations.
Methods {#sec:methods}
=======
We use the semi-analytical method proposed in [@Rudakovskyi:18rs] to obtain the density profile of a dark matter halo. It predicts a cored halo for the general case of warm fermionic dark matter without any extra assumptions about the particle model. Here we briefly summarise this method.
For a fermionic dark matter model with particle mass $m_\mathrm{DM}$ and $g$ initial degrees of freedom (hereafter, we assume $g = 2$) the phase space density cannot exceed [@Boyarsky:08a] $$f_{\text{max}} = \frac{g m_\text{DM}^4}{2(2\pi\hbar)^3}.$$ For a steady-state isotropic spherically symmetrical dark matter halo (see [@Rudakovskyi:18rs] for a discussion on the applicability of this assumption) the phase space density $f$ is obtained by using the Eddington transformation [@Eddington:1916; @Binney-Tremaine:08book] $$f(\mathcal{E}) = \frac{1}{\pi^2\sqrt{8}}\frac{d}{d\mathcal{E}}\int_{\mathcal{E}}^{0} \frac{d\rho}{d\Phi}\frac{d\Phi}{\sqrt{\mathcal{E}-\Phi}}\,,$$ where $\Phi$ is the local gravitational potential. We perform the iterative procedure starting from the NFW profile and truncating the phase space density so that it does not exceed the limiting value: $$f_\text{tNFW}(\mathcal{E})\, {=}\, \left \{
\begin{array}{ll}
\!\!f(\mathcal{E}) , & f(\mathcal{E}) < f_\text{max} \, ,\\
\!\! f_\text{max}, & f(\mathcal{E}) \geq f_\text{max} \, .
\end{array} \right.$$ After this, we reconstruct the mass density [@Binney-Tremaine:08book] $$\rho_\text{tNFW}(r) = 4\pi\int_{\Phi(r)}^0 f_\text{tNFW}(\mathcal{E})\sqrt{2\left(\mathcal{E} - \Phi(r)\right)} d\mathcal{E}$$ for the subsequent step. [@Rudakovskyi:18rs] shows good convergence of this procedure after several iterations. We call the obtained profile tNFW (stands for truncated Navarro–Frenk–White) hereafter. The density profiles obtained in this model are in a good agreement with numerical $N$-body simulations [@Shao:12; @Maccio:12b; @Maccio:12err], see more in [@Rudakovskyi:18rs].
Given the density distribution of a dark matter halo, we follow the logic of [@Domcke:2014kla] and [@DiPaolo:2017geq] to obtain the velocity dispersion along the line of sight. Specifically, we solve the spherical Jeans equation for the radial velocity dispersion $\sigma_\mathrm{r}$, $$\label{eq:sigmar}
\left(\frac{\partial}{\partial r} + \frac{2\beta}{r}\right) (n_\star \sigma_\mathrm{r}^2) = -n_\star \frac{G M(r)}{r^2}\, ,$$ with the stellar velocity dispersion anisotropy $\beta=1-\sigma^2_\perp/\sigma^2_\mathrm{r}$. In the above, $M(r)$ is the dark matter mass distribution, and $n_\star$ is the stellar number density, which we represent by the Plummer profile [@Plummer:1911] $$n_\star(r)=n_0\left(1+r^2/r_\mathrm{h}^2\right)^{-5/2}.$$ The half-light radii $r_h$ for the objects of interest were taken from [@McConnachie:12] and are given in Table \[tab:objdata\]. We then calculate the velocity dispersion along the line of sight: $$\label{eq:sigmalos}
\sigma_\mathrm{los}^2(R)=\frac{1}{\Sigma_\star}\int_{R^2}^{\infty}\text{d}r^2\frac{n_\star}{\sqrt{r^2-R^2}}\sigma_\mathrm{r}^2\left[1-\beta \frac{R^2}{r^2}\right],$$ where $\Sigma_\star(R)=\int_{R^2}^{\infty}\text{d}r^2n_\star(r)/\sqrt{r^2-R^2}$ [@Binney-Tremaine:08book; @DiPaolo:2017geq].
We model the binned data on the velocity dispersion for eight classical dSphs taken from [@Bonnivard:2015xpq]. For every mass of the dark matter particle in the 100eV – 900eV range with logarithmic split we use brute-force grid optimisation over the tNFW profile parameters $c_{200}$, $M_{200}$, and velocity dispersion anisotropy $\beta$ to minimise the objective $\chi^2$ statistics $$\chi^2 = \sum_{i} \frac{\left(\sigma_\mathrm{los,obs}(r_i)-\sigma_\mathrm{los,th}(r_i)\right)^2}{\delta^2(r_i)}\, ,$$ where $\sigma_\mathrm{los,obs}(r_i)$ denotes the $i$’th observational point, $\delta^2(r_i)$ is its $1\sigma$ error, and the $\sigma_\mathrm{los,th}(r_i)$ is the predicted value at this point; the summation is performed over the observational points.
Results {#sec:results}
=======
The dependence of the best-fitting $\chi^2$ statistics on the particle mass for every individual object is plotted in Fig. \[fig:tnfw\_byobj\], and the best-fitting model parameters are summarised in Table \[tab:objdata\].
Object $r_\mathrm{h}$, kpc $\chi^2/N_\mathrm{df}$ $m_\text{DM}$, eV $M_{200},\,10^{8}M_\odot$ $c_{200}$ $\beta$ $r_\mathrm{c}$, kpc
------------ --------------------- ------------------------ ------------------- --------------------------- ----------- --------- ---------------------
Carina 0.25 37.5/21 561 111.7 5 0.21 0.25
Draco 0.221 4.1/7 255 177.8 10 0.34 0.66
Fornax 0.71 28.7/46 171 9.57 53 -0.05 0.93
Leo1 0.251 10.4/13 310 155.7 8 0.44 0.54
Leo2 0.176 5.5/8 650 127.6 9 0.61 0.17
Sculptor 0.283 43.2/33 220 6.01 59 0.10 0.59
Sextans 0.695 16.1/13 650 875.6 2 -0.38 0.22
Ursa Minor 0.181 11.8/14 561 4.92 36 -1.32 0.15
{width="97.00000%"}
The goodness-of-fit is acceptable for every object except Carina dSph, which is the only dSph from our selection that has best-fitting $\chi^2$ higher than two standard deviations ($2\sqrt{2N_\mathrm{df}}$) above the mean value $\chi^2_\mathrm{mean}=N_\mathrm{df}$ of the chi-squared distribution. Therefore, we exclude Carina dSph from the subsequent combined analysis.
Apart from the individual fits, we are interested in the combined goodness-of-fit. We consider the overall $\chi^2$ to be the sum of chi-squared statistics of the individual fits for every dark matter particle mass. The overall best fit is obtained for the particle mass of 342 with $\chi^2=124.7$ for 134 degrees of freedom. This value of mass, however, cannot be statistically distinguished from the higher values, as the differences between the corresponding chi-squares are negligible. For comparison, we fitted the data using the Navarro–Frenk–White profile [@Navarro:95; @Navarro:96], typical to the standard CDM dark matter model. The best-fitting $\chi^2$ statistics is 125.1 for 134 degrees of freedom, so none of this models is preferred by our analysis.
![Overall best-fitting $\chi^2$ statistics as a function of the dark matter particle mass. In the limit of high mass the curve approaches the value obtained in the fit with the Navarro–Frenk–White profile, as the tNFW halo model approaches that of the NFW in this limit. The dashed line shows the $2\sigma$ confidence bound on the particle mass.[]{data-label="fig:tnfw_chi2"}](fig2.pdf){width="\columnwidth"}
![Overall best-fitting $\chi^2$ as a function of dark matter particle mass for the combined analysis of only four selected objects: Draco, Fornax, Leo1 and Sculptor. The minimum at 220eV indicating the preferred particle mass is clearly visible. The depth of the dip corresponds to $2\sigma$ significance ($\Delta\chi^2=4$). However, it becomes negligibly small ($\Delta\chi^2=0.4$) when the rest of objects are included into analysis, see Fig. \[fig:tnfw\_chi2\].[]{data-label="fig:tnfw_4obj"}](fig3.pdf){width="\columnwidth"}
Using the dependence of the overall best-fitting statistics on the particle mass, we can build the confidence range for the mass via the standard approach, described in Sec. 15.6 of @Press:07book. The lower bound on the particle mass is the value for which $\chi^2=\chi^2_\text{best-fit}+\Delta\chi^2$, where for $2\sigma$ confidence level $\Delta\chi^2 = 4$. The resulting mass bound of $m_{2\sigma}\simeq190$eV is shown in Fig. \[fig:tnfw\_chi2\].
In Fig. \[fig:tnfw\_sigma\] we show the effect of particle mass on the velocity dispersion profile in all objects. It is clearly seen that small particle masses strongly modify this profile.
{width="89.00000%"}
We also combine four objects that show notable local minimum on $\chi^2$ vs mass dependence, namely, Draco, Fornax, Leo1 and Sculptor. The combined fitting statistics in this case is plotted in Fig. \[fig:tnfw\_4obj\]. The minimum $\chi^2_\mathrm{min}=87.4$ for 99 degrees of freedom is obtained for particle mass $m=220$eV, whereas the Navarro–Frenk–White profile fits the data with $\chi^2=91.4$. Thus one can conclude that 220eV fermionic dark matter is preferred over CDM with $\Delta\chi^2=4$. However, this observation should not be treated as a strict result, because inclusion of the rest of objects into analysis reduces the local minimum to a statistically insignificant depth of $\Delta\chi^2=0.4$.
Conclusions & Discussion {#sec:discussion}
========================
In this paper, we derive a new maximally model-independent bound on the mass of fermionic dark matter particle. We use the halo model of @Rudakovskyi:18rs and the Jeans equation for modelling the line-of-sight velocity dispersion. We obtain the conservative $2\sigma$ lower bound $m\gtrsim 190$eV on the mass of fermionic dark matter particle. Fermionic DM with higher particle mass cannot be distinguished from the CDM. This result is based on the analysis of seven ‘classical’ dSphs. Our model fails to fit the kinematics of the Carina dSph. However, this galaxy shows the strongest signs of a tidal disruption among the other ‘classical’ dSphs [@Munoz:06; @Munoz:08; @Battaglia:12; @Battaglia:2013wqa; @Fabrizio:16], see also @McMonigal:14. It appears that Carina was transformed from a disky galaxy to a spheroidal via strong tidal interaction with Milky Way [@Fabrizio:16], and different sub-populations have different kinematic patterns [@Fabrizio:16; @Hayashi:2018uop].
To check the robustness of the obtained result with respect to possible uncertainties in the values of half-light radii, we repeat the analysis using upper and lower confidence bounds on $r_\text{h}$ reported by @McConnachie:12. We found that the obtained lower bound on the particle mass changes by less than 10%, being lower for the higher $r_h$ used in the model and vice-versa.
Using only the data on four selected objects, namely, Draco, Fornax, Leo1 and Sculptor, we obtain that fermionic DM with $m=220$eV particle mass is preferred over CDM on $2\sigma$ level. The significance decreases to a negligible value when the rest of objects are included into the analysis.
Conceptually, the halo built from the low-mass fermions has an extended core with low central density compared to the cases of more massive DM particles. The best-fitting halos in case of fermions with the mass $m=100$eV show $\sim2$ cores for all the objects in the analysis. This is much larger than the radial spans of the outermost points of the observable kinematics.
The behaviour of $\sigma_\mathrm{los}$ is determined by the behaviour of $\sigma_\mathrm{r}$, which is smoothed on the characteristic scale $r_h$ via integral transformation according to Eq.\[eq:sigmalos\], see Fig. \[fig:mass-mr-mlos-1\], \[fig:mass-mr-mlos-2\]. Therefore, in the following discussion we will focus on the behaviour of the radial velocity dispersion.
Eq.\[eq:sigmar\] could be rewritten in the form analogous to Eq.14 of [@DiPaolo:2017geq]: $$\label{eq:logder}
\frac{\partial\mathrm{ln}\sigma_\mathrm{r}^2}{\partial \mathrm{ln}r}=-\frac{1}{\sigma_\mathrm{r}^2}\frac{G M(r)}{r}-\frac{\partial\mathrm{ln}n_\star}{\partial\mathrm{ln}r}-2\beta\, .$$ According to this equation, the logarithmic slope of $\sigma_\mathrm{r}$ depends on three different terms. The first negative term dominates on large scales. On the scales $\gtrsim r_\mathrm{h}$, the influence of the second positive term $-\frac{\partial\mathrm{ln}n_\star}{\partial\mathrm{ln}r}=5 r^2 / (r^2+r_h^2)$ is also significant. A density profile with a few-kpc core is similar to a constant-density profile for $r\ll r_\mathrm{c}$, and such halo has much lower mass enclosed into radii $\lesssim 1$ kpc compared to a more cusped one (see Fig. \[fig:mass-mr-mlos-1\], \[fig:mass-mr-mlos-2\]). In this case the first term in Eq. \[eq:logder\] is larger on the scales $\lesssim 1$ kpc compared to the case of more dense halos. Therefore, the logarithmic slope of $\sigma_\mathrm{r}$ is larger for halos built from low-mass fermions on the scales $r_\mathrm{h} \lesssim r \lesssim r_\mathrm{c}$. Such slope is not compatible with the data, and could be partially corrected by the third term of the Eq. \[eq:logder\] with positive $\beta$. However, large positive $\beta$ leads to fast decreasing profile of $\sigma_\mathrm{r}$ for such halos in the low-$r$ region[^2]. The dip produced in this case is reflected in the profile of $\sigma_\text{los}$, which also limits the ability to choose very large $\beta$. Generally speaking, the discussed behaviour of the radial and line-of-sight velocity dispersions is reflected in the decrease of the best-fitting $\beta$ values with an increase of $r_\mathrm{c}/r_\mathrm{h}$, see Table \[tab:corerh\].
Despite the large spread of the neighbouring points, the observational data can be regarded as “flat”, i.e. preferring $\sigma_\text{los}$ profiles without large dips or high slope. Taking into account the discussion above, one can conclude that the halos built with low-mass particles contradict such “flatness”.
In this context we must mention that Sextans has the largest scatter between the nearby observational points among other dSphs. The values in the neighbouring points often have more than 1-sigma differences. Also, the value of $r_\mathrm{h}$ in Sextans is about 0.7 kpc, or 1.5 times larger than the maximal radial span ($r_\mathrm{max}$) of the available kinematic data. As we mentioned above, $\sigma_\mathrm{los}$ is smoothed against $\sigma_\mathrm{r}$ with the characteristic radius $r_\mathrm{h}$. Because the ratio of $r_\mathrm{max}/r_\mathrm{h}$ for Sextans is the smallest among the classical dSphs, the corresponding level of $sigma_\mathrm{los}$ “smoothness” is the largest. This leads to the fact that all best-fits have close values of goodness-of-fit statistics and similar shape. In contrast, objects with sufficiently large $r_\mathrm{max}/r_\mathrm{h}$ ratios (such as Sculptor, Ursa Minor, Fornax) demonstrate the largest variations among profiles with different values of the dark matter masses.
Also note that our best-fitting parameters for Fornax $m=171$eV, $M_{200}=9.57\cdot10^8$$M_{\odot}$, $C_{200}=53$ correspond to a profile with $r_\text{c}=0.93$, which is in a good agreement with @Amorisco:2012rd[^3]. While, in general, our analysis shows no significant preference for the cored dark matter profiles over the cusped ones (obtained in the $\Lambda CDM$ model), Draco, Fornax, Leo1 and Sculptor may give a possible hint on such preference.
The main advantage of our analysis is the combined study of several objects: we *simultaneously* fit the data for seven classical dSphs. While the fits of the data of individual objects show different preferred particle masses (see Fig. \[fig:tnfw\_byobj\]) and lead to different bounds, the combined analysis ensures robustness of the results. Moreover, when modelling several object, we are able to produce stronger bound. For example, the strongest limit of 100eV in @DiPaolo:2017geq is obtained by analysing the smallest dwarfs, whereas the analysis of the classical dwarfs only leads to the mass limit of few tens of electron-volts.
In general, the dark matter halo profile of @DiPaolo:2017geq systematically prefers lower particle masses due to its fully degenerate nature, which produces sharp cut-off in the density profile. Unlike in @DiPaolo:2017geq, in our model the DM halo has two regions: a fully degenerate core and non-degenerate dispersed outskirts. Fig. \[fig:profiles\] shows the fast clipping of this profile and the smaller core size, compared with more “blured” tNFW profile.
![Dark matter density profile in the tNFW model with particle mass $m=$380eV, $M_{200}=1.5\times10^{10}\,M_\odot$, $c_{200}=10$ compared with the fully degenerate COS3 profile of @DiPaolo:2017geq with the same central density and particle mass. The NFW profile here has the corresponding asymptotic behaviour. Also plotted is the Burkert cored profile @Burkert:95 with $r_0=0.38\,\unit{kpc}$ (twice the fully degenerate COS3 profile core size).[]{data-label="fig:profiles"}](fig5.pdf){width="\columnwidth"}
Recent direct measurements of 3D stellar kinematics in Sculptor [@Massari:2017] and kinematics data modelling via the Schwarzschild method [@Kowalczyk:18] revealed that the stellar velocity dispersions in the dwarf spheroidal galaxies are likely to be non-isotropic, but the uncertainties in the value of $\beta$ are very large. Therefore, we assume, for simplicity, that this quantity is constant on all radii. Inclusion of non-zero stellar velocity anisotropy into the analysis leads to a lower DM mass bound compared to the previous findings [e.g. @Boyarsky:08a]. In the case of non-zero $\beta$, we found that DM particle masses in wide range are statistically indistinguishable. This agrees qualitatively with the results of @DiPaolo:2017geq and @Randall:16 for models of non-fully degenerate fermionic halos. This $\beta$-degeneracy could be overcome by assuming multiple stellar sub-populations [@Battaglia:2008jz; @Walker:11; @Agnello:2012uc; @Amorisco:2012rd] or by using the Virial equations instead of the Jeans equations [@Richardson:2014mra]. However, the existing data, which does not include proper 3D stellar kinematics with possible asphericity of stellar populations, is not enough to completely break this degeneracy [@Kowalczyk:12; @Genina:17; @Hayashi:2018uop].
The effects of supernova feedback [@Navarro:96b; @Pontzen:11; @Oh:11; @Teyssier:12; @Zolotov:12], other stellar feedback mechanisms [@Chan:15; @Onorbe:15], and dynamical friction [@ElZant:04; @Sanchez:06; @Romano:08; @DelPopolo:15] could cause additional flattening of the dark matter profile and reduction of the central phase-space density. These mechanisms are thus degenerate with the dark-matter-induced core generation. Inclusion of these effects could increase the lower mass bound.
In the future, progress in the exploration of DM microphysics may be achieved via studying the ultra-faint dwarfs (UFDs), which are the most DM dominated galaxies that we know [see, e.g., @Bullock:17; @Simon:19]. Their compactness also gives an opportunity to test the dark matter distribution on the smallest scales, e.g., dozens of parsecs. Also, the star-formation processes in UFDs should not be powerful enough to change substantially their internal density structure [@Onorbe:15]. However, even the most recent studies [e.g., @Fritz:18; @Simon:18a] allow one to obtain spectra only for only dozens of stars in the ultra-faint Milky Way satellites (unlike ‘classical’ dwarfs, where spectra of hundreds or thousands of stars are measured). These data are not enough to obtain any detailed line-of-sight velocity dispersion profile. Lengthy observations on $\sim10$ m or planned extremely large telescopes may obtain the spectra of many more stars [@Strigari:18; @Weisz:2019bkv; @Drlica-Wagner:19; @Simon:19b].
[lcccccc]{} Object & m, eV & $m_{200}$ & $c_{200}$ & $\beta$ & $r_\mathrm{c}$, kpc & $r_\mathrm{h}$, kpc\
Carina & 100 & 35.98 & 120 & 0.74 & 2.11 & 0.25\
Carina & 220 & 119.4 & 6 & 0.52 & 0.99 & 0.25\
Carina & 650 & 111.7 & 5 & 0.19 & 0.20 & 0.25\
\
Draco & 100 & 69.47 & 120 & 0.67 & 1.80 & 0.221\
Draco & 220 & 119.4 & 12 & 0.40 & 0.78 & 0.221\
Draco & 650 & 717.4 & 6 & 0.10 & 0.17 & 0.221\
\
Fornax & 100 & 33.92 & 200 & 0.41 & 2.10 & 0.710\
Fornax & 220 & 10.93 & 26 & -0.24 & 0.67 & 0.710\
Fornax & 650 & 13.34 & 19 & -0.50 & 0.15 & 0.710\
\
Leo1 & 100 & 52.27 & 200 & 0.83 & 1.87 & 0.251\
Leo1 & 220 & 145.7 & 9 & 0.58 & 0.86 & 0.251\
Leo1 & 650 & 127.6 & 8 & 0.27 & 0.18 & 0.251\
\
Leo2 & 100 & 76.32 & 85 & 1.00 & 1.82 & 0.176\
Leo2 & 220 & 111.7 & 12 & 0.85 & 0.78 & 0.176\
Leo2 & 650 & 127.6 & 9 & 0.61 & 0.17 & 0.176\
\
Sculptor & 100 & 52.27 & 200 & 0.64 & 1.87 & 0.283\
Sculptor & 220 & 6.010 & 59 & 0.10 & 0.59 & 0.283\
Sculptor & 650 & 11.68 & 19 & -0.38 & 0.15 & 0.283\
\
Sextans & 100 & 25.91 & 77 & -0.05 & 2.34 & 0.695\
Sextans & 220 & 145.7 & 4 & -0.20 & 1.11 & 0.695\
Sextans & 650 & 875.6 & 2 & -0.38 & 0.22 & 0.695\
\
UMi & 100 & 80.55 & 160 & 0.69 & 1.70 & 0.181\
UMi & 220 & 6.012 & 160 & 0.06 & 0.52 & 0.181\
UMi & 650 & 4.311 & 38 & -1.64 & 0.12 & 0.181\
{width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"}
{width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"}
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to D. Iakubovskyi and Yu. Shtanov for collaboration and valuable comments. We thank the anonymous Referee for the comments that significantly improved the quality of the paper. This work was supported by the grant for young scientist’s research laboratories of the National Academy of Sciences of Ukraine. The work of A.R. was also partially supported by the ICTP through AF-06.
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: The asymptotic behaviour of the radial velocity dispersion is $\sigma_\mathrm{r}\sim r^{-2\beta} + C r^2$ in the region of small $r$ for cored halos, where $C$ is some constant.
[^3]: Despite that [@Amorisco:2012rd] used the Burkert profile, the core radius $r_0$ is defined as $\rho(r_0)=\frac{\rho_0}{4}$, which is similar to our definition
|
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abstract: 'We study the gauge dependence of the one-loop effective action for the abelian $6D$, ${\cal N}=(1,0)$ supersymmetric gauge theory formulated in harmonic superspace. We introduce the superfield $\xi$-gauge, construct the corresponding gauge superfield propagator, and calculate the one-loop two-and three-point Green functions with two external hypermultiplet legs. We demonstrate that in the general $\xi$-gauge the two-point Green function of the hypermultiplet is divergent, as opposed to the Feynman gauge $\xi =1$. The three-point Green function with two external hypermultiplet legs and one leg of the gauge superfield is also divergent. We verified that the Green functions considered satisfy the Ward identity formulated in ${\cal N}=(1,0)$ harmonic superspace and that their gauge dependence vanishes on shell. Using the result for the two- and three-point Green functions and arguments based on the gauge invariance, we present the complete divergent part of the one-loop effective action in the general $\xi$-gauge.'
author:
- |
I.L. Buchbinder[^1]\
\
\
,\
,\
\
E.A. Ivanov[^2]\
,\
\
B.S. Merzlikin[^3]\
,\
,\
,\
,\
,\
\
K.V. Stepanyantz[^4]\
,\
,\
,\
title: 'Gauge dependence of the one-loop divergences in $6D$, ${\cal N} = (1,0)$ abelian theory'
---
Introduction
============
Gauge theories with extended supersymmetries in higher dimensions attract a considerable attention for a long time [@Howe:1983jm; @Howe:2002ui; @Bossard:2009sy; @Bossard:2009mn; @Fradkin:1982kf; @Marcus:1983bd; @Smilga:2016dpe; @Bork:2015zaa]. On the one hand, such theories are non-renormalizable due to the dimensionful coupling constant (see, e.g., [@Gates:1983nr; @Buchbinder:1998qv]). On the other hand, one can expect an improvement of the ultraviolet behavior due to the extended supersymmetry. It is very interesting to check this conjecture on the explicit examples of higher-dimensional supersymmetric theories. To be more realistic, one can expect that the full canceling of divergences is presumably possible only in the lowest loops even in the maximally extended theories (see, e.g., [@Marcus:1984ei]). The problem reveals clear analogies with the most interesting case of gravity. However, the analysis in supersymmetric gauge theories is much simpler.
In order to fully display the underlying properties of theories with some symmetries it is highly desirable to be aware of the regularization and quantization schemes which do not break these symmetries. In the case of extended supersymmetries these purposes can be achieved within the harmonic superspace approach [@Galperin:1984av; @Galperin:1985bj; @Galperin:1985va; @Galperin:2001uw; @Buchbinder:2001wy; @Buchbinder:2016wng]. For $6D$ supersymmetric gauge theories (which will be the subject of the present paper) this formalism [@Howe:1983fr; @Howe:1985ar; @Zupnik:1986da; @Ivanov:2005qf; @Ivanov:2005kz; @Buchbinder:2014sna] ensures manifest ${\cal N}=(1,0)$ supersymmetry. With the use of the background field method in harmonic superspace [@Buchbinder:2001wy; @Buchbinder:1997ya], gauge symmetry can also be made manifest. For these reasons the harmonic superspace formalism seems to be most suitable for quantum calculations in $6D$ supersymmetric theories (note that $6D\,,$ ${\cal N}=(1,0)$ theories are in general anomalous, see, e.g., [@Townsend:1983ana; @Smilga:2006ax; @Kuzenko:2015xiz; @Kuzenko:2017xgh]).
Recently, some explicit calculations based on the harmonic superspace method were done for ${\cal N}=(1,0)$ and ${\cal N}=(1,1)$ gauge theories [@Buchbinder:2016gmc; @Buchbinder:2016url; @Buchbinder:2017ozh; @Buchbinder:2017gbs; @Buchbinder:2017xjb], following the general pattern of Ref. [@Bossard:2009mn]. These calculations were basically performed in the Feynman gauge $\xi=1$, which ensures the simplest form of the propagator of the gauge superfield. This considerably simplifies the calculation of quantum corrections. However, the gauge dependence of the results obtained by the harmonic superspace technique has not been yet analyzed. Meanwhile, the calculations in the non-minimal gauges are frequently rather useful as compared to those in the Feynman gauge, because they are capable to make manifest divergences in the lower loops. For example, for ${\cal N}=1$ supersymmetric gauge theories in the one-loop approximation ghosts are not renormalized in the Feynman gauge, while divergences appear for $\xi\ne 1$ [@Aleshin:2016yvj]. For calculations in higher orders, the knowledge of gauge dependence in the lower-order approximations is also essential, see, e.g., [@Kazantsev:2018kjx]. These are the reasons why a vast literature is devoted to calculations in non-minimal gauges. As a characteristic example, let us mention a recent paper [@Chetyrkin:2017mwp].
In the present paper we consider the simplest $6D\,,$ ${\cal N}=(1,0)$ supersymmetric gauge theory, namely, ${\cal N}=(1,0)$ supersymmetric electrodynamics, and investigate the structure of the gauge-dependent contributions to the effective action by the harmonic superspace technique. In particular, we demonstrate that (unlike the case of the Feynman gauge considered, e.g., in [@Buchbinder:2016gmc]) the two-point Green function of hypermultiplets is divergent already at the one-loop level. The gauge-dependent divergences are also present in the gauge multiplet - hypermultiplet Green functions. In this paper we explicitly calculate the one-loop three-point Green function and find its divergent part. Moreover, we derive the Ward identity in the harmonic superspace and verify that the Green functions obtained by calculating harmonic supergraphs satisfy this identity, as expected. This result is a non-trivial verification of the correctness of our calculations. One more test, which has also been done in this paper, is the demonstration of the property that the gauge dependence of the effective action vanishes on shell (this is a consequence of the general theorem, see Refs. [@DeWitt:1965jb; @Boulware:1980av; @Voronov:1981rd; @Voronov:1982ph; @Voronov:1982ur; @Lavrov:1986hr]). Using the results for the two- and three-point Green functions, we also restore the complete result for the one-loop divergences, based on the gauge invariance of the theory under consideration.
The paper is organized as follows: In Sect. \[Section\_Electrodynamics\] we recall some basic points of the formulation of $6D\,,$ ${\cal N}=(1,0)$ supersymmetric electrodynamics in harmonic superspace. We present the superfield action for this theory, write down the Ward identity, and formulate the harmonic superspace Feynman rules. In particular, we construct the propagator of the gauge superfield in the non-minimal gauges which are analogs of the $\xi$-gauges in the usual electrodynamics. In Sect. \[Section\_Divergences\], using these Feynman rules, we investigate the gauge dependence of the one-loop two-point Green functions of the gauge superfield and the hypermultiplet. We also calculate the one-loop three-point gauge superfield - hypermultiplet Green function. Checking the Ward identities for these Green functions is the subject of Sect. \[Section\_Ward\]. The vanishing of the gauge dependence on shell in the approximation we are considering is demonstrated in Sect. \[Section\_Shell\]. The total divergent part of the one-loop effective action (which is an infinite series in $V^{++}$) is constructed in Sect. \[Section\_Total\_Divergences\], by invoking the arguments based on the gauge invariance. Also we verify that the gauge dependence of the expression obtained vanishes on shell. Some technical details are collected in two Appendices.
Harmonic superspace formulation of $6D\,,$ ${\cal N}=(1,0)$ electrodynamics {#Section_Electrodynamics}
===========================================================================
The harmonic superspace action
------------------------------
The harmonic superspace is very convenient for formulating $6D$, ${\cal N}=(1,0)$ supersymmetric theories, because it ensures manifest supersymmetry at all steps of quantum calculations. It is parametrized by the coordinate set $(x^M,\theta^{ai}, u_i^\pm)$ which will be referred to as the central basis. Here $x^M$ with $M=0,\ldots 5$ are the usual coordinates of the six-dimensional Minkowski space. The Grassmann anticommuting coordinates $\theta^{ai}$ with $a=1,\ldots 4$ and $i=1,2$ form a left-handed $6D$ spinor. The harmonic variables $u_i^\pm$ satisfy the condition $u^{+i} u_i^- = 1$, with $u_i^- = (u^{+i})^*$. The analytic basis of the harmonic superspace is parametrized by the coordinates
$$x^M_A = x^M + \frac{i}{2}\theta^{-}\gamma^M \theta^+;\qquad \theta^{\pm a} = u^\pm_i \theta^{ai}, \quad u^{\pm}_i\,,$$
where $\gamma^M$ are $6D$ $\gamma$-matrices. The coordinate subset $(x^M_A, \theta^{+ a}, u^\pm_i)$ parametrizes the analytic harmonic subspace which is closed on its own under $6D\,, {\cal N}=(1,0)$ supersymmetry transformations.
It is convenient to define the spinor covariant derivatives
$$D^+_a = u^{+}_i D_{a}^i;\qquad D^-_a = u^{-}_i D_{a}^i,$$
such that $\{D^+_a, D^{-}_b\} = i(\gamma^M)_{ab}\partial_M$, and to introduce the notation
$$(D^+)^4 = -\frac{1}{24}\varepsilon^{abcd} D_a^+ D_b^+ D_c^+ D_d^+.$$
Also we will need the harmonics derivatives in the central basis
$$D^{++} = u^{+i} \frac{\partial}{\partial u^{-i}};\qquad D^{--} = u^{-i} \frac{\partial}{\partial u^{+i}};\qquad
D^0 = u^{+i} \frac{\partial}{\partial u^{+i}} - u^{-i} \frac{\partial}{\partial u^{-i}}.$$
They satisfy the commutation relations of the $SU(2)$ algebra. The analytic basis form of these derivatives can be easily found and is given, e.g., in [@Bossard:2015dva].
For constructing the ${\cal N}=(1,0)$ invariants we need the invariant superspace integration measures:
$$\begin{aligned}
\label{Measure_Total}
&& \int d^{14}z = \int d^6x\,d^8\theta;\qquad \int d\zeta^{(-4)} = \int d^6x\, d^4\theta^+;\\
\label{Measure_Analytic}
&& \int d^6x\,d^8\theta = \int d^6x\,d^4\theta^{+} (D^+)^4.\end{aligned}$$
In this paper we consider ${\cal N}=(1,0)$ supersymmetric electrodynamics, which is a particular abelian case of ${\cal N}=(1,0)$ supersymmetric Yang–Mills theory with hypermultiplets. The harmonic superspace form of the action of $6D$, ${\cal N}=(1,0)$ supersymmetric Yang–Mills theory was pioneered in Ref. [@Zupnik:1986da]. As opposed to the analogous $4D$, ${\cal N}=2$ construction, the gauge theory coupling constant $f_0$ in $6D$ has the dimension $m^{-1}\,$. In the harmonic superspace approach the gauge superfield $V^{++}(z,u)$ satisfies the analyticity condition
$$D^+_a V^{++} = 0$$
and is real with respect to the special conjugation denoted by $\widetilde{}$ , i.e. $\widetilde{V^{++}} = V^{++}$. The hypermultiplets are described by the analytic superfield $q^+$ and its $\widetilde{}$ -conjugate $\widetilde q^+$.
Like in the non-supersymmetric case, the action of ${\cal N}=(1,0)$ electrodynamics is quadratic in the gauge superfield. It can be written as
$$\label{Electodynamics_Action}
S = \frac{1}{4f_0^2} \int d^{14}z\,\frac{du_1 du_2}{(u_1^+ u_2^+)^2} V^{++}(z,u_1) V^{++}(z,u_2) - \int d\zeta^{(-4)} du\,\widetilde q^+ \nabla^{++} q^+.$$
where
$$\label{Gauge_Covariant_Derivative}
\nabla^{++} = D^{++} + i V^{++}$$
and $D^{++}$ is taken in the analytic basis. The gauge transformations has the form
$$\label{Gauge_Transformations}
V^{++} \to V^{++} - D^{++} \lambda; \qquad q^+ \to e^{i\lambda} q^+; \qquad \widetilde q^+ \to e^{-i\lambda} \widetilde q^+,$$
where $\lambda$ is an analytic superfield parameter which is real with respect to the $\widetilde{}$ -conjugation.
It is useful to introduce the non-analytic superfield
$$\label{V--}
V^{--}(z,u) = \int du_1\,\frac{V^{++}(z,u_1)}{(u^+ u_1^+)^2}.$$
It satisfies the conditions $D^{++} V^{--} = D^{--} V^{++}$ and transforms as
$$V^{--} \to V^{--} - D^{--}\lambda$$
under the gauge transformations. Starting from this superfield, it is possible to construct the analytic superfield $F^{++} = (D^+)^4 V^{--}$, which is gauge invariant in the abelian case.
For further use, we also define the non-analytic superfield $q^-$ as a solution of the equation
$$\label{Q-_Definition}
q^+ = \nabla^{++} q^- = (D^{++} + i V^{++}) q^-.$$
From this definition one can derive that the gauge transformations act on $q^-$ as
$$q^- \to e^{i\lambda} q^-.$$
In the explicit form the solution of Eq. (\[Q-\_Definition\]) can be expressed as the series
$$\begin{aligned}
\label{Q-_Expansion}
&&\hspace*{-8mm} q^- = \int \frac{du_1}{(u^+ u_1^+)} q_1^+ -i \int \frac{du_1\,du_2}{(u^+ u_1^+)(u_1^+ u_2^+)} V^{++}_1 q_2^+ - \int \frac{du_1\,du_2\,du_3}{(u^+ u_1^+)(u_1^+ u_2^+)(u_2^+ u_3^+)} V^{++}_1 V^{++}_2 q_3^+ + \ldots\nonumber\\
&&\hspace*{-8mm} =(-i)^{n-1} \sum\limits_{n=1}^\infty \int du_1 \ldots du_n\, \frac{V^{++}_1 \ldots V^{++}_{n-1}}{(u^+ u_1^+) \ldots (u_{n-1}^+ u_n^+)} q_n^+,\end{aligned}$$
where subscripts numerate the harmonic “points”.
For quantizing the theory (\[Electodynamics\_Action\]) it is necessary to fix the gauge. This can be done by adding the gauge-fixing term to the action,
$$\label{Gauge_Fixing_Term}
S_{\mbox{\scriptsize gf}} = - \frac{1}{4f_0^2\xi_0} \int d^{14}z\, du_1 du_2 \frac{(u_1^- u_2^-)}{(u_1^+ u_2^+)^3} D_1^{++} V^{++}(z,u_1) D_2^{++} V^{++}(z,u_2),$$
where $\xi_0$ is the bare gauge-fixing parameter. This term corresponds to the $\xi$-gauge in the usual electrodynamics. In particular, the Feynman gauge amounts to the choice $\xi_0=1$. In the abelian case we are considering it is not necessary to introduce the ghosts superfields. Therefore, the generating functional for our theory can be written as
$$\label{Generating_Functional}
Z = \int DV^{++}\,D\widetilde q^+\, Dq^+\, \exp\Big\{i(S+S_{\mbox{\scriptsize gf}} + S_{\mbox{\scriptsize sources}})\Big\},$$
where $S_{\mbox{\scriptsize sources}}$ is a sum of the source terms,
$$\label{Sources}
\int d\zeta^{(-4)}\,du\, \Big[V^{++} J^{(+2)} + j^{(+3)} q^+ + \widetilde j^{(+3)} \widetilde q^+\Big].$$
Here $J^{(+2)}$ is the analytic source for the gauge superfield, while $j^{(+3)}$ and $\widetilde j^{(+3)}$ denote sources for the hypermultiplet superfields. The effective action is constructed from the generating functional for the connected Green functions $W = -i\ln Z$ by making the Legendre transformation,
$$\label{Gamma_Definition}
\Gamma = W - S_{\mbox{\scriptsize sources}},$$
where it is necessary to express the sources in terms of the fields with the help of the equations
$$V^{++} = \frac{\delta W}{\delta J^{(+2)}};\qquad q^+ = \frac{\delta W}{\delta j^{(+3)}};\qquad \widetilde q^+ = \frac{\delta W}{\delta \widetilde j^{(+3)}}.$$
Ward identity
-------------
In the abelian gauge theory at the quantum level the gauge invariance is encoded in the Ward identity [@Ward:1950xp], which is a particular case of the Slavnov–Taylor identities [@Taylor:1971ff; @Slavnov:1972fg]. The harmonic superspace analog of this identity can be formulated, using the standard technique. For this purpose we make the transformation (\[Gauge\_Transformations\]) in the generating functional (\[Generating\_Functional\]) which evidently remains invariant. Taking into account that the classical action is gauge invariant, in the lowest order in $\lambda$ we obtain
$$\begin{aligned}
\label{Original_Ward}
0 = \Big\langle \int d\zeta^{(-4)}\,du\, \Big[ - \frac{\delta S_{\mbox{\scriptsize gf}}}{\delta V^{++}} D^{++}\lambda
- J^{(+2)} D^{++}\lambda + i j^{(+3)} \lambda q^+ - i \widetilde j^{(+3)} \lambda \widetilde q^+\Big]\Big\rangle,\end{aligned}$$
where we used the notation
$$\Big\langle A(V^{++}, q^+, \widetilde q^+)\Big\rangle = \frac{1}{Z} \int DV^{++}\,D\widetilde q^+\, Dq^+\, A(V^{++}, q^+, \widetilde q^+) \exp\Big\{i(S+S_{\mbox{\scriptsize gf}} + S_{\mbox{\scriptsize sources}})\Big\}.$$
Integrating in Eq. (\[Original\_Ward\]) by parts with respect to the derivatives $D^{++}$, using an arbitrariness of $\lambda$, and expressing the result in terms of superfields, we obtain
$$\begin{aligned}
0 = D^{++} \frac{\delta S_{\mbox{\scriptsize gf}}}{\delta V^{++}}
-D^{++}\frac{\delta\Gamma}{\delta V^{++}} - i q^+ \frac{\delta\Gamma}{\delta q^+} + i \widetilde q^+ \frac{\delta\Gamma}{\delta\widetilde q^+},\end{aligned}$$
where $\Gamma$ is the effective action defined by Eq. (\[Gamma\_Definition\]), and we also took into account that the gauge-fixing term is quadratic in the gauge superfield. Introducing
$$\Delta\Gamma = \Gamma - S_{\mbox{\scriptsize gf}},$$
the Ward identity can be written in a more compact form,
$$\begin{aligned}
\label{Generating_Ward_Identity}
D^{++}\frac{\delta\Delta\Gamma}{\delta V^{++}} = - i q^+ \frac{\delta\Delta\Gamma}{\delta q^+} + i \widetilde q^+ \frac{\delta\Delta\Gamma}{\delta\widetilde q^+}.\end{aligned}$$
It is important that this equation is valid for arbitrary non-zero values of the involved superfields. Differentiating Eq. (\[Generating\_Ward\_Identity\]) with respect to various superfields we derive an infinite set of identities relating the longitudinal part of the $(n+1)$-point Green functions to the $n$-point Green functions. For example, differentiating with respect to $V^{++}_2$ and setting all fields equal to zero at the end, we obtain that quantum corrections to the two-point Green function of the gauge superfield are transversal,
$$\label{Transversality}
D^{++}_1 \frac{\delta^2\Delta\Gamma}{\delta V^{++}_1 \delta V^{++}_2} = 0.$$
Differentiating Eq. (\[Generating\_Ward\_Identity\]) with respect to $q^+_2$ and $\widetilde q^+_3$ and setting the fields equal to zero at the end give an analog of the usual Ward identity relating three- and two-point Green functions:
$$\begin{aligned}
\label{Ward_Identity}
&& D^{++}_1 \frac{\delta^3\Delta\Gamma}{\delta V^{++}_1 \delta q^+_2 \delta\widetilde q^+_3} = - i (D_1^+)^4 \delta^{14}(z_1-z_2) \delta^{(-3,3)}(u_1,u_2) \frac{\delta^2\Delta\Gamma}{\delta q^+_1 \delta\widetilde q^+_3}\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad + i (D^+_1)^4\delta^{14}(z_1-z_3)\delta^{(-3,3)}(u_1,u_3) \frac{\delta^2\Delta\Gamma}{\delta q^+_2 \delta\widetilde q^+_1}.\qquad\end{aligned}$$
When deriving this equation, we have taken into account the property implied by the Grassmann analyticity
$$\label{Analytic_Derivative}
\frac{\delta q^+_1}{\delta q^+_2} = (D^+_1)^4 \delta^{14}(z_1-z_2) \delta^{(-3,3)}(u_1,u_2)\,,$$
where
$$\delta^{14}(z_1-z_2) = \delta^6(x_1-x_2) \delta^8(\theta_1-\theta_2).$$
It is convenient to multiply the identity (\[Ward\_Identity\]) with the analytic superfields $\lambda_1$, $q^+_2$, and $\widetilde q^+_3$, and integrate the expression obtained over both analytic arguments,
$$\begin{aligned}
\label{Ward_Identity_Integrated}
&& \int d\mu\, \widetilde q^+_3 D^{++}\lambda_1 q^+_2\, \frac{\delta^3\Delta\Gamma}{\delta V^{++}_1 \delta q^+_2 \delta\widetilde q^+_3} = i \int d \zeta^{(-4)}_1 du_1\, d\zeta^{(-4)}_3\, du_3\, \widetilde q^+_3 \lambda_1 q^+_1\, \frac{\delta^2\Delta\Gamma}{\delta q^+_1 \delta\widetilde q^+_3}\qquad\nonumber\\
&& - i \int d \zeta^{(-4)}_1 du_1\, d\zeta^{(-4)}_2\, du_2\, \widetilde q^+_1 \lambda_1 q^+_2\, \frac{\delta^2\Delta\Gamma}{\delta q^+_2 \delta\widetilde q^+_1},\qquad\end{aligned}$$
where
$$\int d\mu = \int d\zeta^{(-4)}_1\, du_1\, d\zeta^{(-4)}_2\, du_2\, d\zeta^{(-4)}_3\, du_3.$$
This form of the Ward identity is most convenient, when checking it for one or another particular class of diagrams.
The Feynman rules
-----------------
For the explicit calculation of quantum correction it is necessary to formulate the relevant Feynman rules. This can be accomplished quite similarly to the $4D$, ${\cal N}=2$ case considered in detail in Refs. [@Galperin:1985bj; @Galperin:1985va]. To find the propagator of the gauge superfield in the $\xi$-gauge, we consider the sum of the gauge superfield action and the gauge-fixing term
$$\begin{aligned}
\label{Gauge_Part_Of_Total_Action}
&& S_{\mbox{\scriptsize gauge}} + S_{\mbox{\scriptsize gf}} = \frac{1}{4f_0^2}\Big(1-\frac{1}{\xi_0}\Big) \int d^{14}z\, du_1 du_2 \frac{1}{(u_1^+ u_2^+)^2} V^{++}(z,u_1) V^{++}(z,u_2) \nonumber\\
&& + \frac{1}{4f_0^2\xi_0} \int d\zeta^{(-4)}\, du\, V^{++}(z,u) \partial^2 V^{++}(z,u), \qquad \label{Quadr}\end{aligned}$$
where we made use of the identity
$$\label{U_Identity}
D_1^{++} \frac{1}{(u_1^+ u_2^+)^3} = \frac{1}{2} (D_1^{--})^2 \delta^{(3,-3)}(u_1,u_2)$$
and took into account that, when acting on the analytic superfields,
$$\frac{1}{2} (D^+)^4 (D^{--})^2 \Rightarrow \partial^2.$$
Following Ref. [@Buchbinder:2017ozh], we consider the free theory and solve the equation of motion for the superfield $V^{++}$ in the presence of the source term,
$$\frac{1}{2\xi_0 f_0^2} \partial^2 V^{++}(z,u_1) + \frac{1}{2f_0^2}\Big(1-\frac{1}{\xi_0}\Big) \int du_2 \frac{1}{(u_1^+ u_2^+)^2} (D_1^+)^4 V^{++}(z,u_2) + J^{(+2)}(z,u_1) = 0.$$
The solution can be presented as
$$V^{++}(z,u_1) = -\frac{2\xi_0 f_0^2}{\partial^2} J^{(+2)}(z,u_1) + \frac{2f_0^2(\xi_0-1)}{\partial^4} \int du_2 \frac{1}{(u_1^+ u_2^+)^2} (D_1^+)^4 J^{(+2)}(z,u_2),$$
whence one extracts the $\xi$-gauge form of the propagator of the gauge superfield
$$\begin{aligned}
\label{Gauge_Propagator}
&& G_V^{(2,2)}(z_1,u_1;z_2,u_2) = - 2 f_0^2 \Big(\frac{\xi_0}{\partial^2} (D_1^+)^4 \delta^{(2,-2)}(u_2,u_1)\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad - \frac{\xi_0-1}{\partial^4} (D_1^+)^4 (D_2^+)^4 \frac{1}{(u_1^+ u_2^+)^2}\Big) \delta^{14}(z_1-z_2).\qquad\end{aligned}$$
The second term vanishes in the Feynman gauge $\xi_0=1$. Such a choice considerably simplifies calculation of quantum corrections. However, the purpose of the present paper is to investigate the $\xi_0$-dependence of various Green functions for the generic choice of $\xi_0$.
In left part of Fig. \[Figure\_Propagators\], the propagator (\[Gauge\_Propagator\]) is depicted by the wavy line ending on the points 1 and 2.
For completeness, we also present the expression for the hypermultiplet propagator,
$$\label{Hypermultiplet_Propagator}
G_q^{(1,1)}(z_1,u_1;z_2,u_2) = (D_1^+)^4 (D_2^+)^4 \frac{1}{\partial^2} \delta^{14}(z_1-z_2) \frac{1}{(u_1^+ u_2^+)^3},$$
which is denoted by the solid line on the right.
(0,2.7) (5,1.2)[![The propagators of the gauge superfield $V^{++}$ and of the hypermultiplets.[]{data-label="Figure_Propagators"}](propagator_v.eps "fig:")]{} (4.4,0.1)[(gauge multiplet)]{} (9,1.2)[![The propagators of the gauge superfield $V^{++}$ and of the hypermultiplets.[]{data-label="Figure_Propagators"}](propagator_q.eps "fig:")]{} (8.5,0.1)[(hypermultiplet)]{}
The only vertex of the theory (\[Electodynamics\_Action\]) is presented in Fig. \[Figure\_Vertex\] and stands for the interaction of the hypermultiplet with the gauge superfield
$$\label{Vertex}
S_I = - i \int d\zeta^{(-4)}\, du\, \widetilde q^{+} V^{++} q^+.$$
(0,2) (7,0)[![The only vertex comes from the interaction of the hypermultiplet with the gauge superfield.[]{data-label="Figure_Vertex"}](vertex_quantum.eps "fig:")]{}
The superficial degree of divergence in the theory under consideration has been calculated in Ref. [@Buchbinder:2016gmc]:
$$\label{Divergence_Degree}
\omega = 2L - N_q - \frac{1}{2} N_D.$$
Here $L$ is a number of loops, $N_q$ is a number of external hypermultiplet legs, and $N_D$ is a number of spinor supersymmetric covariant derivatives acting on external legs. This formula implies that in the one-loop approximation only diagrams without external hypermultiplet legs or with two such legs can be divergent.
Gauge dependence of the one-loop divergences {#Section_Divergences}
============================================
Two-point function of the gauge superfield
------------------------------------------
In the one-loop approximation the two-point function of the gauge superfield $V^{++}$ is divergent. In the abelian case this divergence comes only from the diagram pictured in Fig. \[Figure\_Gauge\_Diagram\]. However, this diagram does not contain propagators of the gauge superfield and is therefore gauge-independent.
(0,2) (6.5,0)[![The diagram representing the one-loop two-point Green function in the abelian case.[]{data-label="Figure_Gauge_Diagram"}](1loop_dg9.eps "fig:")]{}
Thus, in the one-loop approximation this Green function in the $\xi$-gauge is the same as in the Feynman gauge. It is given by the expression [@Buchbinder:2016gmc]
$$\label{Gauge_Contribution}
\int \frac{d^6p}{(2\pi)^6} \int d^8\theta\, du_1\, du_2\, V^{++}(p,\theta,u_1) V^{++}(-p,\theta,u_2) \frac{1}{(u_1^+ u_2^+)^2} \Big[\frac{1}{4f_0^2} - \frac{i}{2} \int \frac{d^6k}{(2\pi)^6} \frac{1}{k^2 (k+p)^2}\Big].\qquad$$
The corresponding divergent part of the effective action is gauge-independent and in the dimensional reduction scheme[^5] can be written as
$$\label{Gauge_Divergence}
-\frac{1}{6\varepsilon (4\pi)^3}\int d\zeta^{(-4)}\, du\, (F^{++})^2,$$
where $\varepsilon = 6-D$.
Two-point hypermultiplet Green function
---------------------------------------
In the one-loop approximation the two-point Green function of the hypermultiplet is contributed to by the single logarithmically divergent diagram presented in Fig. \[Figure\_Hypermultiplet\].
(0,2) (6.5,0)[![The two-point Green function of the hypermultiplet in the one-loop approximation[]{data-label="Figure_Hypermultiplet"}](dga1.eps "fig:")]{}
In the Feynman gauge this superdiagram vanishes. However, it includes the propagator of the gauge superfield, for which reason we can expect that the result for it is in fact gauge-dependent. Using the Feynman rules defined above, the expression for this diagram in the generic $\xi$-gauge can be written as
$$\begin{aligned}
\label{Hypermultiplet_Diagram}
&& -2if_0^2 \int d\zeta^{(-4)}_1\, du_1\, d\zeta^{(-4)}_2\, du_2\, \widetilde q^+(z_1,u_1) q^+(z_2,u_2) \frac{1}{(u_1^+ u_2^+)^3} \frac{(D_1^+)^4 (D_2^+)^4}{\partial^2} \delta^{14}(z_1-z_2)\qquad\nonumber\\
&& \times \Big(\frac{\xi_0}{\partial^2} (D_1^+)^4 \delta^{(2,-2)}(u_2,u_1) - \frac{\xi_0-1}{\partial^4} (D_1^+)^4 (D_2^+)^4 \frac{1}{(u_1^+ u_2^+)^2}\Big) \delta^{14}(z_1-z_2).\end{aligned}$$
The derivatives $(D_1^+)^4 (D_2^+)^4$ in the hypermultiplet propagator can be used to convert the integrations over $d\zeta^{(-4)}$ into those over $d^{14}z$,
$$\begin{aligned}
\label{Hypermultiplet_Green}
&& -2if_0^2 \int d^{14}z_1\, du_1\, d^{14}z_2\, du_2\, \widetilde q^+(z_1,u_1) q^+(z_2,u_2) \frac{1}{(u_1^+ u_2^+)^3}\,
\frac{1}{\partial^2} \delta^{14}(z_1-z_2)\qquad\nonumber\\
&& \times \Big(\frac{\xi_0}{\partial^2} (D_1^+)^4 \delta^{(2,-2)}(u_2,u_1) - \frac{\xi_0-1}{\partial^4} (D_1^+)^4 (D_2^+)^4 \frac{1}{(u_1^+ u_2^+)^2}\Big) \delta^{14}(z_1-z_2).\label{Exp22}\end{aligned}$$
Taking into account the identities
$$\begin{aligned}
\label{First_Identity}
&& \delta^{8}(\theta_1-\theta_2)\, (D^{+}_1)^4 \delta^{8}(\theta_1-\theta_2) =0,\vphantom{\Big(}\\
\label{Second_Identity}
&& \delta^{8}(\theta_1-\theta_2)\, (D^{+}_1)^4 (D^{+}_2)^4 \delta^{8}(\theta_1-\theta_2) = (u_1^+ u_2^+)^4\, \delta^{8}(\theta_1-\theta_2)\vphantom{\Big(}\,,\qquad\end{aligned}$$
we find that the first term in this expression vanishes, reducing (\[Hypermultiplet\_Green\]) to the form
$$2if_0^2 \int d^{6}x_1\, d^{6}x_2\,d^8\theta\, du_1\, du_2\, \widetilde q^+(x_1,\theta, u_1) q^+(x_2,\theta,u_2)
\frac{(\xi_0-1)}{(u_1^+ u_2^+)}\, \frac{1}{\partial^2} \delta^{6}(x_1-x_2)\, \frac{1}{\partial^4} \delta^{6}(x_1-x_2).$$
This expression can be rewritten in the momentum representation as
$$\label{Hypermultiplet_Green_Function}
- 2if_0^2 \int \frac{d^6p}{(2\pi)^6}\, \frac{d^6k}{(2\pi)^6} \frac{1}{k^4 (k+p)^2} \int d^8\theta\, du_1\, du_2\, \frac{(\xi_0-1)}{(u_1^+ u_2^+)} \widetilde q^+(p,\theta, u_1) q^+(-p,\theta\,u_2).$$
We observe that this expression is logarithmically divergent and does not vanish, unless the Feynman gauge is chosen. If the theory is regularized by dimensional reduction, the corresponding contribution to the divergent part takes the form
$$\label{Two_Point_Function_Divergence}
-\frac{2f_0^2}{\varepsilon (4\pi)^3} \int d^{14}z\, du_1\, du_2\, \frac{(\xi_0-1)}{(u_1^+ u_2^+)} \widetilde q^+(z, u_1) q^+(z, u_2).$$
Three-point gauge-hypermultiplet Green function
-----------------------------------------------
According to Eq. (\[Divergence\_Degree\]), all diagrams containing two external hypermultiplet legs are logarithmically divergent, irrespective of the number of the external gauge legs. That is why in calculating the one-loop divergences it is necessary to take into account such Green functions. The simplest of them is the three-point gauge superfield - hypermultiplet. In the one-loop approximation, it is contributed to by the single supergraph depicted in Fig. \[Figure\_One-Loop\_Vertex\].
(0,2.5) (6.5,0.)[![The diagram representing the three-point gauge-hypermultiplet function in the one-loop approximation.[]{data-label="Figure_One-Loop_Vertex"}](vertex_matter.eps "fig:")]{}
Calculating this diagram by Feynman rules in the general $\xi$-gauge, we obtain
$$\begin{aligned}
&& - 2f_0^2 \int d\zeta^{(-4)}_1\,du_1\,d\zeta^{(-4)}_2\,du_2\,d\zeta^{(-4)}_3\,du_3\,\widetilde q^+(z_1,u_1)
q^+(z_3,u_3) V^{++}(z_2,u_2) \Big(\frac{\xi_0}{\partial^2} (D_1^+)^4 \qquad\nonumber\\
&&\times \delta^{(2,-2)}(u_3,u_1) - \frac{(\xi_0-1)}{\partial^4} (D_1^+)^4 (D_3^+)^4 \frac{1}{(u_1^+ u_3^+)^2}\Big) \delta^{14}(z_1-z_3)\, \frac{1}{(u_1^+ u_2^+)^3}\frac{(D_1^+)^4 (D_2^+)^4}{\partial^2}\qquad\nonumber\\
&&\times \delta^{14}(z_1-z_2)\, \frac{1}{(u_2^+ u_3^+)^3}\frac{(D_2^+)^4 (D_3^+)^4}{\partial^2} \delta^{14}(z_2-z_3).\label{Expr33}\end{aligned}$$
To work out this expression, we, first, convert the integrals over $d\zeta^{(-4)}$ in it into integrals over $d^{14}z$ using Eq. (\[Measure\_Analytic\]):
$$\begin{aligned}
&& - 2f_0^2 \int d^{14}z_1\,d^{14}z_2\,d^{14}z_3\,du_1\,du_2\,du_3\,\widetilde q^+(z_1,u_1) q^+(z_3,u_3) V^{++}(z_2,u_2)
\Big(\frac{\xi_0}{\partial^2} (D_1^+)^4 \qquad\nonumber\\
&& \times \delta^{(2,-2)}(u_3,u_1) - \frac{(\xi_0-1)}{\partial^4} (D_1^+)^4 (D_3^+)^4 \frac{1}{(u_1^+ u_3^+)^2}\Big)
\delta^{14}(z_1-z_3)\, \frac{1}{(u_1^+ u_2^+)^3 (u_2^+ u_3^+)^3}
\qquad \nonumber\\
&& \times \frac{(D_2^+)^4}{\partial^2} \delta^{14}(z_1-z_2)\, \frac{1}{\partial^2} \delta^{14}(z_2-z_3).\vphantom{\frac{1}{2}}\end{aligned}$$
Next, we integrate by parts with respect to $(D^+_2)^4$ (assuming that $D_2^+$ acts on $z_1$), taking into account that
$$\delta^8(\theta_1-\theta_2)\prod\limits_{n=1}^N D^+_{i_n a_n} \delta^8(\theta_1-\theta_2) = 0 \qquad \mbox{for arbitrary odd $N$}.$$
In the term containing the harmonic $\delta$-function we further integrate over $du_3$. Integrating also over $\theta_2$, we finally obtain for (\[Expr33\]):
$$\begin{aligned}
&& 2f_0^2 \int d^{6}x_1\,d^{6}x_2\,d^{6}x_3\,d^8\theta_1\,d^8\theta_3\,\delta^8(\theta_1-\theta_3)\, \Bigg\{
\int du_1\, du_2\, \widetilde q^+(x_1,\theta_1,u_1) q^+(x_3,\theta_3,u_1) \nonumber\\
&& \times V^{++}(x_2,\theta_1, u_2) \frac{\xi_0}{(u_1^+ u_2^+)^6}\, \frac{(D_1^+)^4 (D_2^+)^4}{\partial^2} \delta^{14}(z_1-z_3)\,\frac{1}{\partial^2} \delta^{6}(x_1-x_2)\,
\,\frac{1}{\partial^2} \delta^{6}(x_2-x_3)
\nonumber\\
&& + \int du_1\,du_2\,du_3\, V^{++}(x_2,\theta_1, u_2) q^+(x_3,\theta_3,u_3) \frac{(\xi_0-1)}{(u_1^+ u_3^+)^2 (u_1^+ u_2^+)^3 (u_2^+ u_3^+)^3}\frac{1}{\partial^2} \delta^{6}(x_1-x_2) \qquad\nonumber\\
&&\times \frac{1}{\partial^2} \delta^{6}(x_2-x_3)\, \Bigg[(D_2^+)^4 \widetilde q^+(x_1,\theta_1,u_1)\, \frac{(D_1^+)^4 (D_3^+)^4}{\partial^4} \delta^{14}(z_1-z_3)
+ \widetilde q^+(x_1,\theta_1,u_1)\nonumber\\
&&\times \frac{(D_2^+)^4 (D_1^+)^4 (D_3^+)^4}{\partial^4} \delta^{14}(z_1-z_3) - \frac{1}{4}\varepsilon^{abcd} D_{2a}^+ D_{2b}^+\, \widetilde q^+(x_1,\theta_1,u_1)\, \frac{D_{2c}^+ D_{2d}^+ (D_1^+)^4 (D_3^+)^4}{\partial^4} \nonumber\\
&&\times \delta^{14}(z_1-z_3)\Bigg] \Bigg\}.\end{aligned}$$
As the further step, we use the identities (\[First\_Identity\]), (\[Second\_Identity\]) together with
$$\begin{aligned}
\label{Third_Identity}
&& \delta^{8}(\theta_1-\theta_2)\, D^{+}_{2a} D^{+}_{2b} (D^{+}_1)^4 (D^{+}_3)^4\delta^{8}(\theta_1-\theta_2)\vphantom{\Big(}\nonumber\\
&&\qquad\qquad\qquad\qquad = -i(\gamma^M)_{ab} (u_2^+ u_1^+)\, (u_2^+ u_3^+)\, (u_1^+ u_3^+)^3 \,\delta^{8}(\theta_1-\theta_2) \partial_M;\vphantom{\Big(}\qquad\\
\label{Fifth_Identity}
&& \delta^{8}(\theta_1-\theta_2)\, (D^{+}_2)^4 (D^{+}_1)^4 (D^{+}_3)^4\delta^{8}(\theta_1-\theta_2)\vphantom{\Big(}\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad = (u_1^+ u_2^+)^2\, (u_1^+ u_3^+)^2\, (u_2^+ u_3^+)^2\,\delta^{8}(\theta_1-\theta_2) \partial^2 \vphantom{\Big(}\qquad\end{aligned}$$
in order to do the integrals over the Grassmann coordinate $\theta_2$. After renaming $\theta_1 \to \theta$, the expression for the diagram in question in the momentum representation is written as
$$\begin{aligned}
\label{Three-Point_Function}
&& 2f_0^2 \int \frac{d^{6}p}{(2\pi)^6}\,\frac{d^{6}q}{(2\pi)^6}\,\frac{d^{6}k}{(2\pi)^6}\,d^8\theta\,\Bigg\{-\int du_1\,du_2\,\widetilde q^+(q+p,\theta,u_1) V^{++}(-p,\theta,u_2) q^+(-q,\theta,u_1)
\nonumber\\
&& \times \frac{\xi_0}{k^2 (q+k)^2 (q+k+p)^2} \frac{1}{(u_1^+ u_2^+)^2}
+ \int du_1\,du_2\,du_3\,\Bigg[ (D^+_{2})^4\,\widetilde q^+(q+p,\theta,u_1)\,V^{++}(-p,\theta, u_2)\nonumber\\
&&\times q^+(-q,\theta,u_3)\,
\frac{(\xi_0-1)}{k^4 (q+k)^2 (q+k+p)^2}\, \frac{(u_1^+ u_3^+)^2}{(u_1^+ u_2^+)^3 (u_2^+ u_3^+)^3}
- \widetilde q^+(q+p,\theta,u_1)\, V^{++}(-p,\theta, u_2)\nonumber\\
&& \times q^+(-q,\theta,u_3)
\frac{(\xi_0-1)}{k^2 (q+k)^2 (q+k+p)^2} \frac{1}{(u_1^+ u_2^+) (u_2^+ u_3^+)}
- D^+_{2a} D^+_{2b}\,\widetilde q^+(q+p,\theta,u_1)\nonumber\\
&&\times V^{++}(-p,\theta, u_2)\, q^+(-q,\theta,u_3)\,
\frac{(\xi_0-1)(\widetilde\gamma^M)^{ab} k_M}{2 k^4 (q+k)^2 (q+k+p)^2}\, \frac{(u_1^+ u_3^+)}{(u_1^+ u_2^+)^2 (u_2^+ u_3^+)^2}
\Bigg]\Bigg\},\end{aligned}$$
where $(\widetilde\gamma^M)^{ab} = \varepsilon^{abcd} (\gamma^M)_{cd}/2$. The divergent part of this expression can now be found after the Wick rotation. There remains only one divergent integral
$$\int \frac{d^6k}{(2\pi)^6}\frac{1}{k^2 (k+q)^2 (k+q+p)^2}\,,$$
which, after regularizing it by dimensional reduction, is reduced to
$$-i\int \frac{d^DK}{(2\pi)^6}\frac{1}{K^2 (K+Q)^2 (K+Q+P)^2} = -\frac{i}{\varepsilon (4\pi)^3} + \mbox{finite terms},$$
where the capital letters denote Euclidean momentums. Thus, the divergent part of the diagram in Fig. 5 can be presented as
$$\label{Three_Point_Function_Divergence}
\frac{2if_0^2}{\varepsilon (4\pi)^3} \int d^{14}z \Bigg\{\int du_1\,du_2\,\widetilde q^+_1 V^{++}_2 q^+_1
\frac{\xi_0}{(u_1^+ u_2^+)^2}
+ \int du_1\,du_2\,du_3\, \widetilde q^+_1\, V^{++}_2 q^+_3 \frac{(\xi_0-1)}{(u_1^+ u_2^+) (u_2^+ u_3^+)}
\Bigg\},$$
where the subscripts on the superfields refer to the relevant harmonic arguments.
Verification of the Ward identities
===================================
\[Section\_Ward\]
To be convinced of the correctness of the results obtained in the previous sections, let us check that the two- and three-point Green functions derived above satisfy the Ward identities.
First, for completeness, we verify the Ward identity (\[Transversality\]). The two-point Green function of the gauge superfield is obtained by differentiating Eq. (\[Gauge\_Contribution\]) with respect to $V^{++}$, using Eq. (\[Analytic\_Derivative\]). This gives
$$\frac{\delta^2\Delta\Gamma}{\delta V^{++}_1 \delta V^{++}_2} = G_V(i\partial_M) \frac{1}{(u_1^+ u_2^+)^2} (D_1^{+})^4 (D_2^+)^4 \delta^{14}(z_1-z_2),$$
where
$$G_V(p_M) = \frac{1}{2f_0^2} - i \int \frac{d^6k}{(2\pi)^6}\frac{1}{k^2 (k+p)^2} + \ldots$$
Therefore,
$$\begin{aligned}
&&\hspace*{-6mm} D_1^{++}\frac{\delta^2\Delta\Gamma}{\delta V^{++}_1 \delta V^{++}_2} = G_V(i\partial_M) D_1^{--} \delta^{(2,-2)}(u_1,u_2)\cdot (D_1^{+})^4 (D_2^+)^4 \delta^{14}(z_1-z_2) = G_V(i\partial_M)\nonumber\\
&&\hspace*{-6mm} \times \Big[D_1^{--} \Big(\delta^{(2,-2)}(u_1,u_2) (D_1^{+})^4 (D_2^+)^4 \Big) - \delta^{(2,-2)}(u_1,u_2) \Big(D_1^{--} (D_1^{+})^4\Big) (D_2^+)^4\Big]\delta^{14}(z_1-z_2) = 0.\nonumber\\\end{aligned}$$
Thus, we have verified that the Ward identity (\[Transversality\]) is indeed satisfied.
The two-point Green function of the hypermultiplet is obtained by differentiating Eq. (\[Hypermultiplet\_Green\_Function\]) with respect to $q^+$ and $\widetilde q^+$. These derivatives are calculated with the help of Eq. (\[Analytic\_Derivative\]). We obtain
$$\frac{\delta^2\Gamma}{\delta q_2^+\,\delta \widetilde q_1^+} = G_q(i\partial_M) \frac{1}{(u_1^+ u_2^+)} (D^+_1)^4 (D^+_2)^4 \delta^{14}(z_1-z_2),$$
where
$$G_q(p_M) = - 2if_0^2 \int \frac{d^6k}{(2\pi)^6} \frac{(\xi_0-1)}{k^4 (k+p)^2} +\ldots$$
The three-point gauge superfield - hypermultiplet Green function can be constructed quite similarly, starting from Eq. (\[Three-Point\_Function\]), but we prefer not to present the expression for it explicitly. Instead, we will check for it the Ward identity in the form (\[Ward\_Identity\_Integrated\]). From Eq. (\[Three-Point\_Function\]) we obtain
$$\begin{aligned}
\label{Three-Point_With_Lambda}
&& \int d\zeta^{(-4)}_1\,du_1\, d\zeta^{(-4)}_2\,du_2\, d\zeta^{(-4)}_3\,du_3\, \widetilde q^+_3 D^{++}\lambda_1 q^+_2\, \frac{\delta^3\Delta\Gamma}{\delta V^{++}_1 \delta q^+_2 \delta\widetilde q^+_3} \nonumber\\
&& = 2f_0^2 \int \frac{d^{6}p}{(2\pi)^6}\,\frac{d^{6}q}{(2\pi)^6}\,\frac{d^{6}k}{(2\pi)^6}\,d^8\theta\,\Bigg\{-\int du_1\,du_2\,\widetilde q^+(q+p,\theta,u_2) D^{++}_1\lambda(-p,\theta,u_1) q^+(-q,\theta,u_2)
\nonumber\\
&& \times \frac{\xi_0}{k^2 (q+k)^2 (q+k+p)^2} \frac{1}{(u_1^+ u_2^+)^2}
+ \int du_1\,du_2\,du_3\,\Bigg[ (D^+_{1})^4\,\widetilde q^+(q+p,\theta,u_3)\,D^{++}_1\lambda(-p,\theta, u_1)\nonumber\\
&&\times q^+(-q,\theta,u_2)\,
\frac{(\xi_0-1)}{k^4 (q+k)^2 (q+k+p)^2}\, \frac{(u_3^+ u_2^+)^2}{(u_3^+ u_1^+)^3 (u_1^+ u_2^+)^3}
- \widetilde q^+(q+p,\theta,u_3)\, D^{++}_1\lambda(-p,\theta, u_1)\nonumber\\
&& \times q^+(-q,\theta,u_2)
\frac{(\xi_0-1)}{k^2 (q+k)^2 (q+k+p)^2} \frac{1}{(u_3^+ u_1^+) (u_1^+ u_2^+)}
- D^+_{1a} D^+_{1b}\,\widetilde q^+(q+p,\theta,u_3)\nonumber\\
&&\times D^{++}_1\lambda(-p,\theta, u_1)\, q^+(-q,\theta,u_2)\,
\frac{(\xi_0-1)(\widetilde\gamma^M)^{ab} k_M}{2 k^4 (q+k)^2 (q+k+p)^2}\, \frac{(u_3^+ u_2^+)}{(u_3^+ u_1^+)^2 (u_1^+ u_2^+)^2}
\Bigg]\Bigg\}.\end{aligned}$$
Next, we integrate by parts with respect to the harmonic derivatives $D^{++}_1$, taking into account the identity
$$\label{Harmonic_Identity}
D^{++}_1 \frac{1}{(u_1^+ u_2^+)^n} = \frac{1}{(n-1)!} (D_1^{--})^{n-1}\delta^{(n,-n)}(u_1,u_2) = \frac{(-1)^{n-1}}{(n-1)!} (D_2^{--})^{n-1}\delta^{(2-n,n-2)}(u_1,u_2) .$$
After some algebra (described in Appendix \[Appendix\_Ward\]), this gives
$$\begin{aligned}
\label{Three-Point_Ward}
&&\hspace*{-6mm} \int d\mu\, \widetilde q^+_3 D^{++}\lambda_1 q^+_2\, \frac{\delta^3\Delta\Gamma}{\delta V^{++}_1 \delta q^+_2 \delta\widetilde q^+_3} = - 2f_0^2 \int \frac{d^6p}{(2\pi)^6}\, \frac{d^6q}{(2\pi)^6}\, \frac{d^6k}{(2\pi)^6} \frac{1}{k^4 (k+q+p)^2} \int d^8\theta\, du_1\, du_3\,\qquad\nonumber\\
&&\hspace*{-6mm} \times \frac{(\xi_0-1)}{(u_1^+ u_3^+)} \widetilde q^+(q+p,\theta, u_3) \lambda(-p,\theta,u_1) q^+(-q,\theta\,u_1) - 2f_0^2 \int \frac{d^6p}{(2\pi)^6}\, \frac{d^6q}{(2\pi)^6}\, \frac{d^6k}{(2\pi)^6} \frac{1}{k^4 (k+q)^2} \nonumber\\
&&\hspace*{-6mm} \times \int d^8\theta\, du_1\, du_2\, \frac{(\xi_0-1)}{(u_1^+ u_2^+)} \widetilde q^+(q+p,\theta, u_1) \lambda(-p,\theta,u_1) q^+(-q,\theta\,u_2).\end{aligned}$$
The right-hand side of this equation can be rewritten as
$$\begin{aligned}
&& i \int d \zeta^{(-4)}_1 du_1\, d\zeta^{(-4)}_3\, du_3\, \widetilde q^+_3 \lambda_1 q^+_1\, \frac{\delta^2\Gamma}{\delta q^+_1 \delta\widetilde q^+_3}
\nonumber\\
&& - i \int d \zeta^{(-4)}_1 du_1\, d\zeta^{(-4)}_2\, du_2\, \widetilde q^+_1 \lambda_1 q^+_2\, \frac{\delta^2\Gamma}{\delta q^+_2 \delta\widetilde q^+_1},\qquad\end{aligned}$$
thus demonstrating that the Green functions (\[Hypermultiplet\_Green\_Function\]) and (\[Three-Point\_Function\]) satisfy the Ward identity (\[Ward\_Identity\_Integrated\]), as it should be. Obviously, they also satisfy the Ward identity in the original form (\[Ward\_Identity\]). This completes checking the correctness of our calculation.
The vanishing of the gauge dependence on shell
==============================================
\[Section\_Shell\]
According to the general theorem of Refs. [@DeWitt:1965jb; @Boulware:1980av; @Voronov:1981rd; @Voronov:1982ph; @Voronov:1982ur; @Lavrov:1986hr], the gauge-dependent terms should disappear on shell. Let us verify that our results are in agreement with this statement.
It is convenient to represent the effective action in the form
$$\Gamma = \Gamma_{\xi_0=1} + \widetilde\Gamma,$$
where
$$\begin{aligned}
&& \Gamma_{\xi_0=1} = S + S_{\mbox{\scriptsize gf}} -\frac{i}{2} \int \frac{d^6p}{(2\pi)^6} \int d^8\theta\, du_1\, du_2\, V^{++}(p,\theta,u_1) V^{++}(-p,\theta,u_2) \frac{1}{(u_1^+ u_2^+)^2} \nonumber\\
&&\times \int \frac{d^6k}{(2\pi)^6}\frac{1}{k^2 (k+p)^2}
- \int \frac{d^{6}p}{(2\pi)^6}\,\frac{d^{6}q}{(2\pi)^6}\,d^8\theta\, du_1\,du_2\,\widetilde q^+(q+p,\theta,u_1) V^{++}(-p,\theta,u_2) \qquad
\nonumber\\
&& \times q^+(-q,\theta,u_1) \frac{1}{(u_1^+ u_2^+)^2}\int\frac{d^{6}k}{(2\pi)^6}\, \frac{2f_0^2}{k^2 (q+k)^2 (q+k+p)^2} + \ldots\end{aligned}$$
is the effective action in the Feynman gauge and
$$\begin{aligned}
\label{Gamma_Gauge_Dependent}
&& \widetilde\Gamma =
- 2if_0^2 \int \frac{d^6p}{(2\pi)^6}\, \frac{d^6k}{(2\pi)^6} \frac{1}{k^4 (k+p)^2} \int d^8\theta\, du_1\, du_2\, \frac{(\xi_0-1)}{(u_1^+ u_2^+)} \widetilde q^+(p,\theta, u_1)\, q^+(-p,\theta\,u_2)\nonumber\\
&& + 2f_0^2 \int \frac{d^{6}p}{(2\pi)^6}\,\frac{d^{6}q}{(2\pi)^6}\,\frac{d^{6}k}{(2\pi)^6}\,d^8\theta\,\Bigg\{-\int du_1\,du_2\,\widetilde q^+(q+p,\theta,u_1) V^{++}(-p,\theta,u_2) q^+(-q,\theta,u_1)
\nonumber\\
&& \times \frac{(\xi_0-1)}{k^2 (q+k)^2 (q+k+p)^2} \frac{1}{(u_1^+ u_2^+)^2}
+ \int du_1\,du_2\,du_3\,\Bigg[ (D^+_{2})^4\,\widetilde q^+(q+p,\theta,u_1)\,V^{++}(-p,\theta, u_2)\nonumber\\
&&\times q^+(-q,\theta,u_3)\,
\frac{(\xi_0-1)}{k^4 (q+k)^2 (q+k+p)^2}\, \frac{(u_1^+ u_3^+)^2}{(u_1^+ u_2^+)^3 (u_2^+ u_3^+)^3}
- \widetilde q^+(q+p,\theta,u_1)\, V^{++}(-p,\theta, u_2)\nonumber\\
&& \times q^+(-q,\theta,u_3)
\frac{(\xi_0-1)}{k^2 (q+k)^2 (q+k+p)^2} \frac{1}{(u_1^+ u_2^+) (u_2^+ u_3^+)}
- D^+_{2a} D^+_{2b}\,\widetilde q^+(q+p,\theta,u_1)\nonumber\\
&&\times V^{++}(-p,\theta, u_2)\, q^+(-q,\theta,u_3)\,
\frac{(\xi_0-1)(\widetilde\gamma^M)^{ab} k_M}{2k^4 (q+k)^2 (q+k+p)^2}\, \frac{(u_1^+ u_3^+)}{(u_1^+ u_2^+)^2 (u_2^+ u_3^+)^2}
\Bigg]\Bigg\}
+ \ldots\end{aligned}$$
stands for the gauge-dependent remainder of the effective action.
The purpose of this section is to demonstrate, by an explicit calculation, that in the approximation considered, $\widetilde\Gamma$ indeed vanishes on shell. To this end, we use the equations of motion for the hypermultiplets following from the action (\[Electodynamics\_Action\]),
$$0 = \nabla^{++} q^+ = D^{++} q^+ + iV^{++} q^+;\qquad 0 = \nabla^{++} \widetilde q^+ = D^{++} \widetilde q^+ - iV^{++} \widetilde q^+.$$
In Appendix \[Appendix\_Shell\] (after some lengthy calculations) we demonstrate that, with these equations taken into account, the gauge-dependent part of the one-loop effective action can be cast in the form
$$\begin{aligned}
\label{On_Shell_Result}
&& \widetilde\Gamma = 2f_0^2 \int \frac{d^{6}p}{(2\pi)^6}\,\frac{d^{6}q}{(2\pi)^6}\,d^4\theta^+\,du\, \widetilde q^+(q+p,\theta,u) V^{++}(-p,\theta,u) q^+(-q,\theta,u)
\nonumber\\
&& \times \Big((q+p)^2+q^2\Big) \int \frac{d^{6}k}{(2\pi)^6}\, \frac{(\xi_0-1)}{k^2 (q+k)^2 (q+k+p)^2} + O\left((V^{++})^2\right).\qquad\end{aligned}$$
On shell, where $q^2=0$ and $(q+p)^2=0$,[^6] this expression vanishes. Thereby we have proved that the gauge dependence is vanishing on shell.
Note that, while deriving this result, we ignored all terms proportional to $(V^{++})^k$ for $k\ge 2$, because in this paper we limit our attention only to the diagrams without external gauge superfield legs at all, and to those having a single gauge superfield leg. In this approximation, terms of higher orders in $V^{++}$ are irrelevant.
The total divergent part of the one-loop effective action
=========================================================
\[Section\_Total\_Divergences\]
So far we investigated gauge dependence of the two- and three-point Green functions only. In particular, we demonstrated that the corresponding one-loop divergences are gauge-dependent. However, according to Eq. (\[Divergence\_Degree\]), the Green functions with an arbitrary number of external gauge legs (and two external hypermultiplet legs) are also divergent. Nevertheless, the total divergent part of the one-loop effective action can be found using the reasoning based on the gauge invariance. Actually, the one-loop divergences corresponding to the two- and three-point Green functions (see Eqs. (\[Gauge\_Divergence\]), (\[Two\_Point\_Function\_Divergence\]), and (\[Three\_Point\_Function\_Divergence\])) have the form
$$\begin{aligned}
\label{Lowest_Divergences}
&& \Gamma^{(1)}_\infty = -\frac{1}{6\varepsilon (4\pi)^3}\int d\zeta^{(-4)}\, du\, (F^{++})^2
-\frac{2f_0^2}{\varepsilon (4\pi)^3} \int d^{14}z\, du_1\, du_2\, \frac{(\xi_0-1)}{(u_1^+ u_2^+)} \widetilde q^+_1 q^+_2
+ \frac{2if_0^2}{\varepsilon (4\pi)^3}
\nonumber\\
&& \times \int d^{14}z \Bigg\{\int du_1\,du_2\,\widetilde q^+_1 V^{++}_2 q^+_1
\frac{\xi_0}{(u_1^+ u_2^+)^2}
+ \int du_1\,du_2\,du_3\, \widetilde q^+_1\, V^{++}_2 q^+_3 \frac{(\xi_0-1)}{(u_1^+ u_2^+) (u_2^+ u_3^+)}
\Bigg\}\qquad\nonumber\\
&& + O\Big(\widetilde q^+ (V^{++})^2 q^+\Big).\vphantom{\frac{1}{2}}\end{aligned}$$
The first term in this equation is gauge invariant. The expression corresponding to the first term in the curly brackets can also be rewritten in the explicitly gauge invariant form,
$$\begin{aligned}
&& \frac{2if_0^2}{\varepsilon (4\pi)^3} \int d^{14}z\, du_1\,du_2\,\widetilde q^+_1 V^{++}_2 q^+_1
\frac{\xi_0}{(u_1^+ u_2^+)^2} = \xi_0\, \frac{2if_0^2}{\varepsilon (4\pi)^3} \int d^{14}z\, du\, \widetilde q^+ V^{--} q^+\nonumber\\
&& = \xi_0\,\frac{2if_0^2}{\varepsilon (4\pi)^3} \int d\zeta^{(-4)}\, du\, \widetilde q^+ F^{++} q^+.\end{aligned}$$
According to Eq. (\[Q-\_Expansion\]), the remaining two terms in Eq. (\[Lowest\_Divergences\]) are the lowest terms in the series expansion of the gauge invariant expression
$$- \frac{2 f_0^2 (\xi_0-1)}{\varepsilon (4\pi)^3} \int d^{14}z\, du\, \widetilde q^+\, q^-$$
in powers of $V^{++}$. Thus, the divergent part of the one-loop effective action can be written as
$$\begin{aligned}
\label{One-Loop_Divergence}
&& \Gamma^{(1)}_\infty = -\frac{1}{6\varepsilon (4\pi)^3}\int d\zeta^{(-4)}\, du\, (F^{++})^2 + \frac{2if_0^2 \xi_0}{\varepsilon (4\pi)^3} \int d\zeta^{(-4)}\, du\, \widetilde q^+ F^{++} q^+\nonumber\\
&& - \frac{2 f_0^2 (\xi_0-1)}{\varepsilon (4\pi)^3} \int d^{14}z\, du\, \widetilde q^+\, q^-.\end{aligned}$$
Note that this expression does not include $O\Big(\widetilde q^+ (V^{++})^2 q^+\Big)$, because for obtaining the gauge invariant expression such terms should contain $F^{++}$ in which the number $N_D=4$ of spinor derivatives acts on $V^{--}$. However, according to Eq. (\[Divergence\_Degree\]) these terms are finite and do not contribute to the divergent part of the one-loop effective action. Therefore, Eq. (\[One-Loop\_Divergence\]) provides the exact result for the divergent part of the effective action of the theory in question.
Note that on shell the gauge dependence of Eq. (\[One-Loop\_Divergence\]) vanishes. Actually, on shell, as the consequence of the equation of motion $\nabla^{++}q^+ = 0\,$, we have the chain of relations
$$(\nabla^{++})^2 q^- = 0 \;\Rightarrow\; (\nabla^{++})^2 \nabla^{--}q^- = 0 \; \Rightarrow\; \nabla^{++} \nabla^{--} q^- = 0 \; \Rightarrow\; \nabla^{--} q^- = 0\,.$$
Acting on the latter equation by $\nabla^{++}$ it is easy to find $$q^- = \nabla^{--} q^+. \label{q-q+2}$$
In deriving these relations, we made use of the well known properties $D^{++}\omega^{-n} = 0 \rightarrow \omega^{-n} = 0,\; D^{--}\omega^{+ m} = 0 \rightarrow \omega^{+ m} = 0$ for $n\geq 1, m\geq 1$.
As a consequence of (\[q-q+2\]), we obtain that on shell
$$\begin{aligned}
&& \int d^{14}z\, du\, \widetilde q^+\, q^- = \int d\zeta^{(-4)}\, du\, (D^+)^4\Big(\widetilde q^+\, \nabla^{--} q^+\Big)\nonumber\\
&&\qquad\qquad = \int d\zeta^{(-4)}\, du\, \widetilde q^+\, (D^+)^4\Big( (D^{--} +iV^{--}) q^+\Big) = i \int d\zeta^{(-4)}\, du\, \widetilde q^+\, F^{++} q^+.\qquad\end{aligned}$$
Thus, on shell, the one-loop divergence (\[One-Loop\_Divergence\]) takes the form
$$\Gamma^{(1)}_\infty = -\frac{1}{6\varepsilon (4\pi)^3}\int d\zeta^{(-4)}\, du\, (F^{++})^2 + \frac{2if_0^2}{\varepsilon (4\pi)^3} \int d\zeta^{(-4)}\, du\, \widetilde q^+ F^{++} q^+.$$
We see that this expression does not depend on the parameter $\xi$ and, hence, on the gauge choice.
Summary
=======
\[Section\_Summary\]
In this paper, using the $6D\,,$ ${\cal N}=(1,0)$ harmonic superspace formalism, we studied the gauge dependence of the one-loop effective action for ${\cal N}=(1,0)$ supersymmetric quantum electrodynamics. As compared to the case of the Feynman gauge, in the general $\xi$-gauge some new divergences appear. In particular, we demonstrated that in the general case the hypermultiplet Green function is divergent already in the one-loop approximation, as opposed to the case of the Feynman gauge, in which this divergence vanishes. Moreover, we calculated the three-point gauge - hypermultiplet Green function in the general $\xi$-gauge. To check the correctness of the calculation, we have verified the relevant Ward identity. Also it was checked that the gauge dependence vanishes on shell. Taking into account the gauge invariance, we also restored the divergent part of the one-loop effective action with terms of higher orders in the gauge superfield $V^{++}$. It is given by Eq. (\[One-Loop\_Divergence\]) and contains a new term which is absent in the Feynman gauge. We demonstrated that the gauge dependence of this general expression also vanishes on shell.
It would be interesting to investigate the gauge dependence in the non-abelian case. In particular, from the results of this paper we can expect that in the general $\xi$-gauge the $6D\,,$ ${\cal N}=(1,1)$ sypersymmetric Yang–Mills theory is not finite even in the one-loop approximation, while the divergent terms are vanishing on shell.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the grant of Russian Science Foundation, project No. 16-12-10306.
Ward identity in harmonic superspace
====================================
\[Appendix\_Ward\]
Let us show how to pass from Eq. (\[Three-Point\_With\_Lambda\]) to its equivalent form (\[Three-Point\_Ward\]). After integrating by parts with respect to the derivatives $D^{++}_1$ and using the identity (\[Harmonic\_Identity\]), we obtain
$$\begin{aligned}
&& \int d\mu\, \widetilde q^+_3 D^{++}\lambda_1 q^+_2\, \frac{\delta^3\Delta\Gamma}{\delta V^{++}_1 \delta q^+_2 \delta\widetilde q^+_3}\nonumber\\
&& = 2f_0^2 \int \frac{d^{6}p}{(2\pi)^6}\,\frac{d^{6}q}{(2\pi)^6}\,\frac{d^{6}k}{(2\pi)^6}\,d^8\theta\,\Bigg\{-\int du_1\,du_2\,\widetilde q^+(q+p,\theta,u_2) \lambda(-p,\theta,u_1) q^+(-q,\theta,u_2)
\nonumber\\
&& \times \frac{\xi_0}{k^2 (q+k)^2 (q+k+p)^2} D_2^{--} \delta^{(0,0)}(u_1,u_2)
+ \int du_1\,du_2\,du_3\,\Bigg[ (D^+_{1})^4\,\widetilde q^+(q+p,\theta,u_3)\nonumber\\
&&\times \lambda(-p,\theta, u_1) q^+(-q,\theta,u_2)\,
\frac{(\xi_0-1)(u_3^+ u_2^+)^2}{k^4 (q+k)^2 (q+k+p)^2}\, \Bigg( \frac{1}{2 (u_1^+ u_2^+)^3} (D^{--}_3)^2 \delta^{(-1,1)}(u_1,u_3)\nonumber\\
&& + \frac{1}{2 (u_1^+ u_3^+)^3} (D^{--}_2)^2 \delta^{(-1,1)}(u_1,u_2) \Bigg)
- \widetilde q^+(q+p,\theta,u_3)\, \lambda(-p,\theta, u_1) q^+(-q,\theta,u_2) \nonumber\\
&&\times \frac{(\xi_0-1)}{k^2 (q+k)^2 (q+k+p)^2}\Bigg(\frac{1}{(u_1^+ u_3^+)} \delta^{(1,-1)}(u_1,u_2) + \delta^{(1,-1)}(u_1,u_3)\frac{1}{(u_1^+ u_2^+)}\Bigg)
\nonumber\\
&& - D^+_{1a} D^+_{1b}\,\widetilde q^+(q+p,\theta,u_3) \lambda(-p,\theta, u_1)\, q^+(-q,\theta,u_2)\,
\frac{(\xi_0-1)(\widetilde\gamma^M)^{ab} k_M}{2 k^4 (q+k)^2 (q+k+p)^2}\,(u_3^+ u_2^+)\nonumber\\
&&\times \Bigg(\frac{1}{(u_1^+ u_2^+)^2} D^{--}_3 \delta^{(0,0)}(u_1,u_3) + \frac{1}{(u_1^+ u_3^+)^2} D^{--}_2 \delta^{(0,0)}(u_1,u_2)\Bigg)
\Bigg]\Bigg\}. \label{Expr10}\end{aligned}$$
Then we integrate by parts with respect to the derivatives $D^{--}$ and take off one harmonic integral with the help of the delta functions. Taking into account that the first term vanishes as a consequence of the analyticity of the superfields $\lambda$, $\widetilde q^+$, and $q^+$, the expression (\[Expr10\]) can be further rewritten as
$$\begin{aligned}
&&\hspace*{-8mm} 2f_0^2 \int \frac{d^{6}p}{(2\pi)^6}\,\frac{d^{6}q}{(2\pi)^6}\,\frac{d^{6}k}{(2\pi)^6}\, \frac{(\xi_0-1)}{k^4 (q+k)^2 (q+k+p)^2} \int d^8\theta\, du_1\,\lambda(-p,\theta, u_1) \Bigg\{\int du_2\, \frac{1}{(u_1^+ u_2^+)}\nonumber\\
&&\hspace*{-8mm}\times q^+(-q,\theta,u_2)\, \Bigg[ \frac{1}{2}(D^+_{1})^4\, (D_1^{--})^2 \widetilde q^+(q+p,\theta,u_1) - k^2 \widetilde q^+(q+p,\theta,u_1)
+ \frac{1}{2} (\widetilde\gamma^M)^{ab} k_M\, D^+_{1a} \nonumber\\
&&\hspace*{-8mm}\times\, D^+_{1b} D_1^{--}\,\widetilde q^+(q+p,\theta,u_1) \Bigg]
+ \int du_3\, \frac{1}{(u_1^+ u_3^+)} \Bigg[ \frac{1}{2} (D^+_{1})^4\,\widetilde q^+(q+p,\theta,u_3)\, (D_1^{--})^2 q^+(-q,\theta,u_1) \nonumber\\
&&\hspace*{-8mm} - k^2 \widetilde q^+(q+p,\theta,u_3)\, q^+(-q,\theta_1,u_1)
- \frac{1}{2} (\widetilde\gamma^M)^{ab} k_M\, D^+_{1a} D^+_{1b}\,\widetilde q^+(q+p,\theta,u_3)\, D_1^{--} q^+(-q,\theta,u_1) \Bigg] \Bigg\}.\nonumber\\\end{aligned}$$
Once again, integrating by parts and taking into account that
$$\frac{1}{2} (D^{+})^4 (D^{--})^2 = \partial^2;\qquad (\widetilde\gamma^M)^{ab} D^+_{1a} D^+_{1b} D_1^{--} = - 4i\partial^M$$
on the analytic superfields, this expression can be cast in the form
$$\begin{aligned}
&&\hspace*{-6mm} - 2f_0^2 \int \frac{d^6p}{(2\pi)^6}\, \frac{d^6q}{(2\pi)^6}\, \frac{d^6k}{(2\pi)^6} \frac{(\xi_0-1)}{k^4 (k+q)^2} \int d^8\theta\, du_1\, du_2\, \frac{1}{(u_1^+ u_2^+)}
\widetilde q^+(q+p,\theta, u_1) \lambda(-p,\theta,u_1) \qquad\nonumber\\
&&\hspace*{-6mm} \times q^+(-q,\theta\,u_2) - 2f_0^2 \int \frac{d^6p}{(2\pi)^6}\, \frac{d^6q}{(2\pi)^6}\, \frac{d^6k}{(2\pi)^6} \frac{(\xi_0-1)}{k^4 (k+q+p)^2} \int d^8\theta\, du_1\, du_3\, \frac{1}{(u_1^+ u_3^+)} \widetilde q^+(q+p,\theta, u_3) \nonumber\\
&&\hspace*{-6mm} \times \lambda(-p,\theta,u_1) q^+(-q,\theta\,u_1),\vphantom{\frac{1}{2}}\end{aligned}$$
where we have also used the relations
$$\begin{aligned}
(q+p)^2 + k^2 + 2 k_M (q+p)^M = (q+k+p)^2\,, \qquad q^2 + k^2 + 2 k_M q^M = (q+k)^2.\end{aligned}$$
Gauge-dependent part of the effective action and the hypermultiplet equations of motion
=======================================================================================
\[Appendix\_Shell\]
In this appendix we verify that the gauge-dependent part of the effective action vanishes on shell. This is an important non-trivial check of the correctness of our calculations.
First, we consider the two-point Green function of the hypermultiplet given by Eq. (\[Hypermultiplet\_Green\_Function\]). Using the identity
$$\label{Identity1}
\qquad\frac{1}{(u_1^+ u_2^+)} = D^{++}_1 \frac{(u_1^- u_2^+)}{(u_1^+ u_2^+)^2} + D^{--}_1 \delta^{(1,-1)}(u_1,u_2) = D^{++}_1 D^{++}_2 \frac{(u_1^- u_2^-)}{(u_1^+ u_2^+)^2} + D^{--}_1 \delta^{(1,-1)}(u_1,u_2),\qquad$$
we rewrite it as
$$\begin{aligned}
&& \widetilde\Gamma^{(2)} = - 2if_0^2 \int \frac{d^6p}{(2\pi)^6}\,\frac{d^6k}{(2\pi)^6} \frac{(\xi_0-1)}{k^4 (k+p)^2} \int d^8\theta\, du_1\, du_2\, \Big( D^{++}_1 D^{++}_2\frac{(u_1^- u_2^-)}{(u_1^+ u_2^+)^2} \qquad\nonumber\\
&& + D^{--}_1 \delta^{(1,-1)}(u_1,u_2) \Big) \widetilde q^+(p,\theta, u_1) q^+(-p,\theta\,u_2).\qquad\end{aligned}$$
The second term in this expression vanishes due to the analyticity of the hypermultiplet superfield,
$$\int d^8\theta\, du D^{--} \widetilde q^+(p,\theta,u) q^+(-p,\theta,u) = \int d^4\theta^+\, du\, (D^+)^4 \Big(D^{--} \widetilde q^+(p,\theta,u) q^+(-p,\theta,u)\Big) = 0.$$
After integrating by parts with respect to the harmonic derivatives, the considered contribution to the effective action can be represented as
$$\label{Hypwermultiplet_Gauge_Depandence}
- 2if_0^2 \int \frac{d^6p}{(2\pi)^6}\, \frac{d^6k}{(2\pi)^6} \frac{(\xi_0-1)}{k^4 (k+p)^2}
\int d^8\theta\, du_1\, du_2\, \frac{(u_1^- u_2^-)}{(u_1^+ u_2^+)^2} D^{++} \widetilde q^+(p,\theta, u_1)\, D^{++} q^+(-p,\theta\,u_2).$$
Using the equations of motion for the hypermultiplets
$$0 = \nabla^{++} q^+ = \big(D^{++} + i V^{++}\big)q^+;\qquad 0 = \nabla^{++} \widetilde q^+ = \big(D^{++} - i V^{++}\big)\widetilde q^+,$$
we see that on shell the expression (\[Hypwermultiplet\_Gauge\_Depandence\]) is proportional to $\widetilde q^+ (V^{++})^2 q^+$. However, in this paper we do not consider terms quadratic in the gauge superfield $V^{++}$. This implies that, within the accuracy of our approximation, the part of the one-loop effective action corresponding to the hypermultiplet two-point function vanishes on shell.
Next, we consider the gauge dependent part of the three-point gauge superfield - hypermultiplet Green function. It corresponds to the terms proportional to $\widetilde q^+ V^{++} q^+$ in the expression (\[Gamma\_Gauge\_Dependent\]). We will demonstrate that $\widetilde \Gamma^{(3)}$ vanishes on shell (in the approximation when all terms with more than one $V^{++}$ are omitted).
Using the identity
$$\label{Identity2}
\frac{1}{(u_1^+ u_2^+)^2} = D_2^{++} \frac{(u_2^- u_1^+)}{(u_2^+ u_1^+)^3} + \frac{1}{2} (D_2^{--})^2 \delta^{(2,-2)}(u_2,u_1)$$
and discarding terms quadratic in $V^{++}$ (coming from $D^{++} q^+$ and $D^{++}\widetilde q^+$ after using the equations of motion), we obtain
$$\begin{aligned}
\label{Term1}
&&\int d^8\theta\, du_1\, du_2\, \widetilde q^+(q+p,\theta,u_1) V^{++}(-p,\theta,u_2) q^+(-q,\theta,u_1) \frac{1}{(u_1^+ u_2^+)^2}\nonumber\\
&& \longrightarrow \frac{1}{2} \int d^8\theta\, du\, \widetilde q^+(q+p,\theta,u) (D^{--})^2 V^{++}(-p,\theta,u) q^+(-q,\theta,u) \nonumber\\
&& = \frac{1}{2} \int d^4\theta^+\, du\, \widetilde q^+(q+p,\theta,u) (D^+)^4 (D^{--})^2 V^{++}(-p,\theta,u) q^+(-q,\theta,u) \qquad\nonumber\\
&& = -p^2 \int d^4\theta^+\, du\, \widetilde q^+(q+p,\theta,u) V^{++}(-p,\theta,u) q^+(-q,\theta,u),\end{aligned}$$
where the arrow indicates that we omitted some terms vanishing on shell, as well as $O((V^{++})^2)$ terms.
Using Eq. (\[Identity1\]) twice, we have
$$\begin{aligned}
&& \int d^8\theta\, du_1\, du_2\, du_3\,\widetilde q^+(q+p,\theta,u_1) V^{++}(-p,\theta, u_2)\, q^+(-q,\theta,u_3) \frac{1}{(u_1^+ u_2^+) (u_2^+ u_3^+)}\qquad\nonumber\\
&& \longrightarrow - \int d^8\theta\, du\, D^{--}\widetilde q^+(q+p,\theta,u) V^{++}(-p,\theta,u) D^{--} q^+(-q,\theta,u) \nonumber\\
&& = - 2 q^M (q+p)_M \int d^4\theta^+\, du\, \widetilde q^+(q+p,\theta,u) V^{++}(-p,\theta,u) q^+(-q,\theta,u).\end{aligned}$$
The remaining terms vanish. Indeed, let us consider the expression
$$\int du_1\, du_2\, du_3\, D^+_{2a} D^+_{2b}\,\widetilde q^+(q+p,\theta,u_1) V^{++}(-p,\theta, u_2)\, q^+(-q,\theta,u_3)\,
\frac{(u_1^+ u_3^+)}{(u_1^+ u_2^+)^2 (u_2^+ u_3^+)^2} \nonumber\\$$
and make use of the relation $(u_1^+ u_3^+) = D^{++}_1 D^{++}_3 (u_1^- u_3^-)$. Then, after integrating by parts with respect to the harmonic derivatives $D^{++}_1$ and $D^{++}_3$, up to the terms quadratic in $V^{++}$, we observe that on shell the resulting expression is proportional to $(u_1^- u_1^-) = 0$,
$$\begin{aligned}
&& (92) \longrightarrow \int du_1\, du_2\, du_3\, D^+_{2a} D^+_{2b}\,\widetilde q^+(q+p,\theta,u_1) V^{++}(-p,\theta, u_2)\, q^+(-q,\theta,u_3)\,(u_1^- u_3^-)\nonumber\\
&&\times D^{--}_1 \delta^{(2,-2)}(u_1,u_2) D^{--}_3 \delta^{(2,-2)}(u_3,u_2) = 0.\vphantom{\frac{1}{2}}\end{aligned}$$
Similarly, using the identity $(u_1^+ u_3^+)^2 = D^{++}_1 D^{++}_3\Big((u_1^- u_3^-)(u_1^+ u_3^+)\Big)$, we obtain
$$\begin{aligned}
&& \int du_1\, du_2\, du_3\,(D^+_{2})^4\,\widetilde q^+(q+p,\theta,u_1) V^{++}(-p,\theta, u_2)\, q^+(-q,\theta,u_3)\,
\frac{(u_1^+ u_3^+)^2}{(u_1^+ u_2^+)^3 (u_2^+ u_3^+)^3} \nonumber\\
&& \longrightarrow \frac{1}{4} \int du_1\, du_2\, du_3\,(D^+_{2})^4\,\widetilde q^+(q+p,\theta,u_1) V^{++}(-p,\theta, u_2)\, q^+(-q,\theta,u_3)\, (u_1^- u_3^-)(u_1^+ u_3^+)\qquad\nonumber\\
&& \times (D^{--}_1)^2 \delta^{(2,-2)}(u_1,u_2) (D^{--}_3)^2 \delta^{(2,-2)}(u_3,u_2) = 0.\vphantom{\frac{1}{2}}\end{aligned}$$
Finally, collecting all terms, we conclude that the exploiting of the hypermultiplet equations of motion allows us to rewrite the part of $\widetilde\Gamma$ corresponding to the three-point gauge superfield - hypermultiplet Green function in the form
$$\begin{aligned}
&& \widetilde\Gamma^{(3)} = 2f_0^2 \int \frac{d^{6}p}{(2\pi)^6}\,\frac{d^{6}q}{(2\pi)^6}\,\frac{d^{6}k}{(2\pi)^6}\,
\frac{(\xi_0-1)}{k^2 (q+k)^2 (q+k+p)^2} \Big((q+p)^2 + q^2\Big)\nonumber\\
&&\times \int d^4\theta^+\, du\, \widetilde q^+(q+p,\theta,u) V^{++}(-p,\theta,u) q^+(-q,\theta,u).\end{aligned}$$
For the on-shell hypermultiplets the relations $q^2 =0$ and $(q+p)^2=0$ are valid, so this expression vanishes. The conclusion is that the gauge-dependent contributions to the effective action are indeed canceled on shell in the approximation we stick to.
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[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: [email protected]
[^5]: Here we use the regularization by dimensional reduction [@Siegel:1979wq]. However, for calculating power divergences one should use another regularization, e.g., some modifications of the higher covariant derivative regularization [@Slavnov:1971aw; @Slavnov:1972sq]. At least for $4D$, ${\cal N}=2$ supersymmetric theories such a regularization can be formulated in the harmonic superspace [@Buchbinder:2015eva].
[^6]: These equations can be derived directly from the hypermultiplet free equation of motion, see Ref. [@Bossard:2015dva] for details.
|
---
author:
- 'M. Santander-García'
- 'P. Rodríguez-Gil'
- 'O. Hernandez'
- 'R. L. M. Corradi'
- 'D. Jones'
- 'C. Giammanco'
- 'J. E. Beckman'
- 'C. Carignan'
- 'K. Fathi'
- 'M. M. Rubio-Díez'
- 'F. Jiménez-Luján'
- 'C. R. Benn'
bibliography:
- 'msantander.bib'
date: 'A&A: Received April 21, 2010; accepted June 10, 2010'
title: 'The kinematics of the quadrupolar nebula M 1–75 and the identification of its central star[^1]'
---
[The link between the shaping of bipolar planetary nebulae and their central stars is still poorly understood.]{} [The kinematics and shaping of the multipolar nebula M 1–75 are hereby investigated, and the location and nature of its central star are briefly discussed.]{} [Fabry-Perot data from GH$\alpha$FAS on the WHT sampling the Doppler shift of the \[N[ii]{}\] 658.3 nm line are used to study the dynamics of the nebula, by means of a detailed 3-D spatio-kinematical model. Multi-wavelength images and spectra from the WFC and IDS on the INT, and from ACAM on the WHT, allowed us to constrain the parameters of the central star.]{} [The two pairs of lobes, angularly separated by $\sim$22$^\circ$, were ejected simultaneously approx. $\sim$3500-5000 years ago, at the adopted distance range from 3.5 to 5.0 kpc. The larger lobes show evidence of a slight degree of point symmetry. The shaping of the nebula could be explained by wind interaction in a system consisting of a post-AGB star surrounded by a disc warped by radiative instabilities. This requires the system to be a close binary or a single star which engulfed a planet as it died. On the other hand, we present broad- and narrow-band images and a low S/N optical spectrum of the highly-reddened, previously unnoticed star which is likely the nebular progenitor. Its estimated $V-I$ colour allows us to derive a rough estimate of the parameters and nature of the central star.]{}
Introduction
============
Planetary nebulae (PNe) represent the terminal breath of 90$\%$ of the stars in the Universe. However, their shaping mechanism is still poorly understood.
Bipolar PNe are undoubtedly the most challenging case. Several attempts have been made to explain their shaping (see the review by [@balick02]), breaking spherical symmetry by invoking elements which fall in two distinct categories: [*a)*]{} rapid stellar rotation and/or magnetic fields [e.g. @garciasegura99; @blackman01a], and [*b)*]{} a close interacting companion to the star (e.g. @nordhaus06, for a review see @demarco09). This latter hypothesis seems to be gaining some ground as close binary systems are progressively being found (e.g. [@miszalski09a]; [@miszalski10]) at the cores of bipolar PNe.
Spatio-kinematical modelling of PNe constitutes an excellent tool to test theoretical models. It provides us with important parameters to be matched by the different models of formation, such as the 3-D morphologies and velocity fields of the outflows, their kinematical age (once disentangled from the distance to the nebula) and their orientation to the line of sight.
M 1–75 (PN G068.8-00.0, $\alpha$ = 20 04 44.086 $\delta$ = +31 27 24.42 J2000) is a good example of a complex nebula. It displays a seemingly irregular horseshoe-like central region, out of which two systems of faint lobes emerge. It was first classified as quadrupolar by @manchado96b, and a tentative attempt to recover its kinematic parameters was done by @dobrincic08.
In this paper we present Fabry-Perot interferometry of , from which we derive a detailed spatio-kinematical model (section 3). We also report the first imaging and spectroscopic detection of its central star (section 4). We then discuss both results and their implications in the shaping of the nebula in section 5.
Observations and Data reduction
===============================
Fabry-Perot interferometric data
--------------------------------
The \[N[ii]{}\] 658.3 nm emission of M 1-75 was scanned with GH$\alpha$FAS (Galaxy H$\alpha$ Fabry-Perot System) on the 4.2 m WHT (William Herschel Telescope) on July 6, 2007, as part of its commissioning programme. The nebula was observed in high-resolution mode with the OM4 etalon (resolving power R$\sim$18000, effective finesse $\Im_\mathrm{e}$=24) and a plate scale of 0$''$.2 pixel$^{-1}$. The free spectral range was 8.62 $\AA$ or 392 km s$^{-1}$ split into 48 channels, thus leading to a velocity step of 8.16 km s$^{-1}$ per channel. The total exposure time of the scanning was 1.9 hr, and the seeing 0$''$.8. The instrumental response function (IRF) was measured by fitting a Lorentzian to the profile of a Neon lamp line and resulted in an instrumental width (FWHM) of 18.6 km s$^{-1}$.
The data were reduced following the standard procedure for GH$\alpha$FAS data, which are described in @hernandez08. Several artifacts persisted through the data reduction process. These include slight contamination by H$\alpha$ emission from adjacent orders (specially in the first and last channels of the datacube), a ghost of the inner region of the nebula, and an arc-shaped artifact which runs across several channels, at different locations (see Fig. \[F1\]).
Broad and narrow-band imaging
-----------------------------
Several images of M 1–75 in the light of different filters (U, B, V, I, H$\alpha$ and Strömgren Y) were taken with ACAM (Auxiliary-port Camera) on the WHT and with the WFC (Wide Field Camera) on the 2.5 m INT (Isaac Newton Telescope). The log of the observations can be found in Table 1.
All these data were reduced following standard [IRAF]{}[^2] procedures.
[lccccc]{} Date & Telescope/ & filter & Band & Exp. time & seeing\
& Instrument & ref. & & (s) &\
2009 Jun 11 & WHT/ACAM & \#17 & I & 3$\times$120 & 1$''$.4\
2009 Jun 12 & INT/WFC & \#201 & Str. Y & 2$\times$600 & 1$''$.3\
2009 Jun 12 & INT/WFC & \#197 & H$\alpha$ & 120 & 1$''$.3\
2009 Sep 10 & INT/WFC & \#204 & U & 1200 & 1$''$.6\
2009 Sep 10 & INT/WFC & \#204 & B & 1200 & 1$''$.6\
2009 Sep 10 & INT/WFC & \#204 & V & 1800 & 1$''$.6\
2009 Sep 10 & INT/WFC & \#204 & I & 600 & 1$''$.6\
2009 Sep 10 & INT/WFC & \#204 & Str. Y & 600 & 1$''$.6\
\[T1\]
Long-slit spectroscopy
----------------------
An 3600 s spectrum with the slit at parallactic angle (P.A.=284$^\circ$), crossing the centre of the inner nebula, was taken with IDS (Intermediate Dispersion Spectrograph) on the INT on March 9, 2009. The R300V grating was used, centered at 540 nm and effectively covering from 430 to 810 nm at a resolving power R$\sim$1500. The slit width was 1$''$, while the seeing was 1$''$.8. HD 192281 was chosen as the standard star to account for flux and sensitivity calibration.
A low-resolution (resolving power R ranging from 290 and 570), 40 min spectrum of M1-75 was taken with ACAM on the WHT on June 11, 2009, with the 400 lines mm$^{-1}$ transmission VPH (Volume Phase Holographic) disperser, covering the wavelength range between 350 and 950 nm. The 1$''$ wide slit was positioned at P.A.=0$^\circ$ in order to get the light from the two central star candidates (see section 4). The seeing was 2$''$.8, and the standard star was HD 338808.
The spectra were de-biased, flat-fielded, distortion-corrected and wavelength calibrated (from copper-argon and copper-neon arc lamps) using standard [IRAF]{} routines. After extraction of the selected nebular and central star features from the orthogonal 2-D spectra, the 1-D spectra were telluric and sensitivity corrected using the spectrum of the spectrophotometric standard star.
An improved spatio-kinematical model
====================================
The GH$\alpha$FAS \[N[ii]{}\] 658.3 nm integrated image of M 1–75 is shown in Fig. \[F2\]. While no central star (CSPN) is visible in this image, the nebula clearly shows two pairs of lobes with different orientations. They are nested in a central, brighter rim resembling a horseshoe. Both systems of lobes appear distorted and fragmented, and their faint outer edges are difficult to track near the poles.
The lobes of M 1–75 were the subject of a spatio-kinematical model by @dobrincic08. From two slit spectra, approximately along each pair of lobes, a \[N[ii]{}\] image from @manchado96a, and simple assumptions such as ballistic expansion, they determined the larger and smaller lobes to lie at inclinations of 87$^{\circ}$ and 65$^{\circ}$, respectively, and to be likely co-eval, with kinematical ages of 2700 and 2400 years per kpc of distance to the nebula, respectively.
Fabry-Perot interferometry (and GH$\alpha$FAS, in particular) represents a significant step forward in spatio-kinematical modelling of planetary nebulae. Not only it allows for a resolution in wavelength comparable to high-resolution echelle spectrographs, but the series of “Doppler-map” images it produces span the whole nebula, instead of being limited by a narrow slit whose orientation has to be decided *a priori* based on previous images. From the spatio-kinematical point of view, a single GH$\alpha$FAS data cube encompasses everything that is needed (i.e. information of the emission both in the plane of the sky and along the line of sight), its quality being only limited by seeing.
In particular, it is noteworthy to remark the faint, high velocity emission regions offset from the axis of the larger lobes (see Fig. \[F3\]), near the polar caps (especially in the southwest region). Those certainly would have remained unnoticed, had we been constrained by a narrow slit oriented along the lobes’ “expected” axis.
Solf-Ulrich model
-----------------
The data cube, with Doppler-shift images spread across 48 channels, allowed us to build a spatio-kinematical model of both systems of lobes (see @santander04b for a detailed description of the method). Our first approach for each pair of lobes consisted of fitting a Solf-Ulrich [@solf85] surface to the data. This analytical model is described, in spherical coordinates, by:
$$r = tD^{-1} [v_\mathrm{equator} + (v_\mathrm{polar} - v_\mathrm{equator}) \sin{|\theta|}^\gamma]$$
where $r$ is the angular distance to the centre of the nebula (i.e. the adopted central star, see section 4), $tD^{-1}$ the kinematical age of the outflow per unit distance to the nebula, $v_\mathrm{polar}$ and $v_\mathrm{equator}$ the velocities of the model at the pole and equator respectively, $\theta$ the latitude angle of the model, and $\gamma$ a dimensionless shaping factor. This assumes that each gas particle travels in the radial direction, with a velocity proportional to its distance to the central source (i.e. in a “Hubble-like” flow).
The two-dimensional generatrix is rotated around the symmetry axis and homogeneously populated with particles to produce a three-dimensional model, and then inclined to the plane of the sky. The resulting geometrical shape and velocity field —once offset by a certain systemic velocity— are then used to generate a simplified image of the nebula and a series (one per GH$\alpha$FAS channel) of Fabry-Perot synthetic interferometric images for direct comparison with the \[N[ii]{}\] integrated image and GH$\alpha$FAS channels data. The irregular surface brightness distribution is beyond the scope of this paper and therefore has not been fitted.
In order to find the best representation of the nebular geometry and expansion, we allow the parameters to vary over a large range of values and visually compare each resulting model to the integrated image and each of the 48 GH$\alpha$FAS channels, until we obtain the best fit. The range of uncertainty is also derived by eye, by individually changing each parameter away from the best fit, until the resulting model is no longer a fair fit to the data. Note that, although the inclination of each pair of lobes cannot be directly determined from the horseshoe —a clearly non-elliptical waist—, this fact does not prevent us from finding its value with certain degree of accuracy, given that the aforementioned parameters are disentangled from one another in the results they produce (i.e. there are no degeneracies in the resulting model).
A fair overall fit to the data was obtained for the large and small lobes (Figs. \[F3\] and \[F4\]), although the model of the former does not account for the offset emission near the poles. The systemic velocity of both system of lobes was found to be $v_\mathrm{sys_{LSR}}\sim$13$\pm$4 km s$^{-1}$. They were also found to share a similar kinematical age of $\sim$1000 yr kpc$^{-1}$, within uncertainties. The orientations on the sky are instead different —the larger lobes lie inclined 58$^{\circ}$ to the line of sight, while the inclination of the smaller pair is 79$^{\circ}$. Note that these results (ages, velocities and inclinations) are essentially different from the fit by @dobrincic08, resulting from two slit positions. We were unable to fit a model with their parameters to the GH$\alpha$FAS data. As their model is based just on two slits rather than the 2-D full kinematical information present in the GH$\alpha$FAS data this is possibly to be expected. The parameters corresponding to our best results for the smaller and larger lobes, together with the uncertainties, are shown in Tables 2 and 3 respectively.
However, no standard Solf-Ulrich model can reproduce the high-velocity emitting region offset from the larger lobes axis. Instead, a modified, point-symmetric Solf-Ulrich model can account for these structures while still fairly fitting the inner regions.
Point-symmetry, modified Solf-Ulrich model
------------------------------------------
In order to find a better fit to the GH$\alpha$FAS data, we introduced the following modified Solf-Ulrich model:
$$r = tD^{-1} [v_\mathrm{equator} + (v_\mathrm{polar} - v_\mathrm{equator}) \sin{|\theta|}^{\gamma(\theta)}]$$
where $\gamma(\theta)$ is described by
$$\gamma(\theta) = \gamma_\mathrm{equator} + (\gamma_\mathrm{polar}-\gamma_\mathrm{equator}) \ {(\frac{2 \theta}{\pi})}^\epsilon$$
where $\gamma_\mathrm{equator}$ and $\gamma_\mathrm{polar}$ are the values of the shape factor $\gamma$ at the equator and poles respectively, while $\epsilon$ is the power of the dependance (i.e. 1 linear, 2 quadratic, etc.). This dependance of $\gamma$ with the latitude, although increasing the number of free parameters, allows us to better sample the degree of collimation of the nebula at different latitudes.
The next step was adding point-symmetry to the model. We achieved this in a simple way by defining the nebular axes x, y, z (x along the line of sight towards the viewer, y towards the right, and z upwards along the nebula main axis), and then horizontally projecting the model’s z axis on to curves given by
$$x' = k_x \ z^p$$
and
$$y' = k_y \ z^p$$
where $k_x$ and $k_y$ are constants, and $p$ is an odd integer (so that it produces a point-symmetric structure). The modified model allows two independent degrees of point symmetry, along the x and y axes, respectively (in a corkscrew fashion). We finally rotated the model by a $\phi$ angle around the z axis before inclining it to the line of sight and produced the synthetic image and GH$\alpha$FAS channel data as described in section 3.1.
Only the large lobes were fit with this model. The best fit values along with their uncertainties —fully consistent with the standard Solf-Ulrich model except for the curvature— are listed in the lower part of Table 3.
Almost all the emission from the large lobes, including the aforementioned offset region, was found to be faithfully accounted for by the latter model (see Fig. \[F5\]), which added a slight corkscrew-like curvature.
------------------------------------ ------- ------------
Parameter Value Range
Small lobes
$tD^{-1}$ (yr kpc$^{-1}$) 925 (800-1000)
$v_\mathrm{equator}$ (km s$^{-1}$) 15-20 (:)
$v_\mathrm{polar}$ (km s$^{-1}$) 105 (90-125)
$\gamma$ 4.5 (4-5)
$P.A.$ ($^\circ$) 359 (355-1)
$i \ (^\circ)$ 79 (76-82)
$v_\mathrm{sys_{LSR}}$ 13 (11-15)
------------------------------------ ------- ------------
: Best-fitting parameters for the small lobes. “:” means uncertain.
\[T2\]
[c c c]{} Parameter & Value & Range\
Large lobes\
Solf-Ulrich model\
$tD^{-1}$ (yr kpc$^{-1}$) & 1000 & (900-1150)\
$v_\mathrm{equator}$ (km s$^{-1}$) & 25 & (23-31)\
$v_\mathrm{polar}$ (km s$^{-1}$) & 180 & (160-200)\
$\gamma$ & 7 & (6.5-7.5)\
$P.A.$ ($^\circ$) & 337 & (336-339)\
$i \ (^\circ)$ & 58 & (54-63)\
$v_\mathrm{sys_{LSR}}$ & 13 & (11-15)\
Point-symmetric model\
$tD^{-1}$ (yr kpc$^{-1}$) & 1000 & (900-1100)\
$v_\mathrm{equator}$ (km s$^{-1}$) & 25 & (23-31)\
$v_\mathrm{polar}$ (km s$^{-1}$) & 190 & (170-210)\
$\gamma_\mathrm{equator}$ & 1 & (1-3)\
$\gamma_\mathrm{polar}$ & 14 & (13-15)\
$\epsilon$ & 0.6 & (0.55-0.7)\
$P.A.$ ($^\circ$) & 338 & (337-339)\
$i \ (^\circ)$ & 58 & (55-62)\
$k_x$ & 2$\times$10$^{-5}$ & (1-3)$\times$10$^{-5}$\
$k_y$ & 8$\times$10$^{-5}$ & (6-9)$\times$10$^{-5}$\
$p$ & 3 & -\
$\phi$ ($^\circ$) & 166 & (150-185)\
$v_\mathrm{sys_{LSR}}$ & 13 & (11-15)\
\[T3\]
The central star
================
The WFC Strömgren Y image (see Fig. \[F6\] top right), where the nebular emission is practically absent, shows two faint stars inside the horseshoe region of the nebula. The star labelled as A is offset $\sim$5$''$ with respect to star B, which lies approximately at the centre of the nebular emission. In order to gain some insight on the possibility of either star being the CSPN, we took multi-colour (U, B, V and I) WFC images of the nebula (see Fig. \[F6\]) and an ACAM 40 min low-resolution spectrum of both star candidates.
Unfortunately, the spectrum of each star only shows the nebular emission lines together with a continuum whose signal to noise is too low ($\sim$10-15) to allow us to detect any photospheric spectral signatures of a white dwarf. Instead, once the nebular emission has been accounted for, we can estimate the visual magnitudes of both stars. This results in $m_\mathrm{v}\sim19.3$ for star A and $m_\mathrm{v}\sim21.4$ for star B.
On the other hand, star A is barely visible in the images in the light of the U and B bands, and clearly visible in the V and I bands, while star B is only visible in the latter bands. Although those bands are highly contaminated by strong emission from the nebula, we were able to roughly estimate the $V-I$ colour of star B. In order to do this, we added 7 rows ($\sim$1.5$\times$FWHM) centred on the star and with P.A.=138$^\circ$, where the nebular contamination is minimum. For each image, a gaussian was fitted to the star profile, taking a linear fit between the base of the wings of the profile as background. Once the quantum efficiency of the detector (EEV 4280), the filter transmissions and the atmospheric extinction have been taken into account, we found an observed $V-I\sim$2.0 for star B.
The dereddening of this value is not straightforward, due to the extinction variation across the nebula. The extinction values found in the literature range from c$_\mathrm{b}$=2.29 to c$_\mathrm{b}$=2.9 [@hua88; @bohigas01] from the Balmer decrement for different regions of the nebula. From our own ACAM and IDS spectra, we computed an extinction value c$_\mathrm{b}$=1.9$\pm$0.1 at the location of star B, close to the values found by other authors. If we assume the star to lie within the nebula, we can deredden its $V-I$ colour using the @fitzpatrick04 extinction law, assuming R=3.1, to obtain an intrinsic $(V-I)_\mathrm{0}\sim$0 for star B.
Discussion
==========
The shaping of the nebula
-------------------------
Our spatio-kinematical modelling confirms the presence of two pairs of nested lobes in the nebula of M 1–75, although with essentially different velocities and ages from those found by @dobrincic08. For reasons outlined in section 3, we consider our results more reliable. The outer lobes show some evidence of departure from axial symmetry in the polar regions, which we have modelled by applying a slight degree of point-symmetry to an axisymmetric flow. The inner lobes show a different orientation; the spatial angle between their symmetry axes is $\sim$22$^\circ$. The expansion pattern of both systems of lobes is adequately described by a simple Hubble-flow law. In other words, each lobe is the result of a brief, organised shaping process, followed by ballistic expansion. Both systems of lobes share the same age, within uncertainties.
As the lobes expand, their inner regions would interact and lose their integrity in the process, as shocks progressively convert their kinetic energy to heat. This could explain the broken and essentially irregular structure of the central horseshoe. However, the presence of shocks in the horseshoe is controversial: @guerrero95 and @riera90 found some indications that shock excitation in the horseshoe does not play a significant role in the shaping, while @bohigas01 found proof of shock wave excitation of the H$_\mathrm{2}$ emitting region, which is tightly correlated to that emitting in \[N [ii]{}\]. On the other hand, the slight twist at the polar tips of the outer lobes appear to follow the same ballistic expansion pattern as the rest of the structure, thus not arising late in the evolution of the nebula. This could be a clue to the stellar ejection process, perhaps happening in a rapidly rotating frame.
Given all the aforementioned, it is not trivial to depict a formation scenario for M 1–75. The classic Generalised Interacting Stellar Winds (GISW, @balick87b, a refined version of the original ISW by @kwok78) model. In this model, the isotropic, fast and tenuous wind from a post asymptotic giant branch (AGB) star interacts with the anisotropic, slow and dense winds previously deployed by the star during the AGB stage and shapes a bipolar nebula, is not sufficient to explain either the presence of a multipolar structure, or the slight degree of point-symmetry of the larger lobes. Instead, one has to invoke a mechanism such as the warped-disc proposed by @icke03: if the post-AGB is surrounded by a disc, warped by radiative instabilities, the wind interaction could result in a multipolar nebula with some degree of point-symmetry in the external regions (e.g. NGC 6537). The origin of the disc itself (i.e. the necessary equatorial density enhancement), however, would still require either the CSPN to be actually a close binary, or to have engulfed one of its planets as it died. A more complex approach is that of @blackman01a, in which a low-mass companion originates a disc blowing its own wind. A misalignment between the stellar and disc magnetic and rotational axes could give rise to a quadrupolar structure, while the point-symmetry observed in the tips of the larger lobes would require precession of the magnetic axes. We consider the model proposed by @manchado96b, in which a fast precessing disc is responsible for the different orientation of each structure, as a less likely scenario, for it would require both structures to have been ejected in extremely rapid succession (in $\sim$300-500 yr to still fit our spatio-kinematical model uncertainties, considering the distance range adopted below and the spatial angular separation of the inner and outer lobes), and it would not explain the point-symmetryic structure.
It is noteworthy to remark that all these models require the CSPN to be/have been a close-binary system (or at least a single star with a close massive hot Jupiter) for the disc to have formed. There is, to our knowledge, no plausible model in the extensive literature able to produce a quadrupolar (not to mention point-symmetric) PN out of a single star.
The central star
----------------
Although it constitutes the cornerstone for many parameters of PNe and their central stars, such as the kinematical age of the nebula or their total luminosity, the distance to these objects is poorly known in most cases. M 1–75 is no exception. In the literature one can find several distances based on different methods, such as the statistical distance range, 2.6-3.7 kpc by @cahn92, or the Galactic rotation curve distance of 5.3 kpc by @burton74. The more recent extinction-distance method of @sale09, easily applicable to the INT Photometric H$\alpha$ Survey (IPHAS) data sample and reliable in most PNe [@giammanco10], does not help in the case of nebulae with a significant amount of internal extinction. In the case of M 1–75, the extinction value lies far above the plateau of the field stars in the extinction-distance graph, confirming a significant internal extinction in this nebula (another possibility would be that stellar H$\alpha$, possibly from a cooler companion, scatters from dust in or near the inner horseshoe, thereby increasing the H$\alpha$/H$\beta$ ratio; this is ruled out, however, by this ratio in the core being lower than in the horseshoe). Given the lack of evidence favouring a particular distance estimate, rather than adopting a specific distance we will consider a more conservative, intermediate distance range between 3.5 and 5.0 kpc.
Probably due to its internal extinction, so far there has been no clear evidence of the CSPN of M 1–75, other than a slight enhancement of the isophotes in an \[O [iii]{}\] image [@hua88]. Our images in the light of Strömgren Y, followed by low resolution spectra, have detected two candidates to CSPN (Fig. \[F6\]), stars A and B (at the position pointed out by Hua), of apparent magnitudes m$_v\sim$19.3 and 21.4 respectively. Unfortunately, the low signal to noise ratio in a 40 min spectrum with ACAM ($\sim$15 in the continuum of the brightest star around 700 nm) prevents us from detecting and analysing any photospheric features, leading us to think that any future research on these stars will require an 8-m class telescope.
Although several nebulae have offset CSPN, it is unlikely that star A is the central star of M 1–75. Even in the most extreme case known (MyCn 18; @sahai99a), the star is nowhere in contact with the equatorial waist of its nebula. In fact, to explain such an offset ($\sim$5$''$), one would need to invoke a high proper motion central star travelling $\sim$10-20 km s$^{-1}$ faster than its own nebula since the ejection, 3500-5000 yr ago at the adopted distance range. The nebula would have to have been heavily braked by interaction with the ISM, being distorted in the process. However, making a simple extrapolation from the PN-ISM interaction models for a round nebula by @wareing07, every symmetry in the system would have been long lost at such a late stage of interaction.
Therefore we can safely rule out star A and assume star B, at the centre of the nebula, as the CSPN of M 1–75. The visual magnitude we have derived for this star is consistent with the estimate by @hua88 who, assuming a distance of 2.8 kpc [@acker78], suggested a hot (log T$_\mathrm{eff}$=$5.3$) core with a mass of about 0.57-0.6 M$_{\odot}$ with a luminosity of log L/L$_\odot$=2.36. The kinematical age of the nebula found in this work is coherent with the luminosity and T$_\mathrm{eff}$ of the fading evolutionary track of a hydrogen-burning high-mass ($\sim$0.6-0.8 $M_\odot$) core [@schonberner93; @mendez97].
Based on the extremely high N/O=2.85 and He/H= 0.18 of the nebula, @guerrero95 hinted towards the possibility of the CSPN actually being a post-common envelope close binary with a total initial mass between 4 and 6 M$_\odot$. In fact, the $(V-I)_0\sim$0 colour estimated in this work is considerably redder than the value of $V-I$= -0.9 one would expect from a single blackbody of log T$_{eff}$=5.3. This might suggest the presence of a fainter (L$_\mathrm{bol}<$ 10$^{-3}\times$L$_{\mathrm{bol}_\mathrm{WD}}$), much colder (T$_\mathrm{eff}\lesssim$10000 K) companion star, as together they would produce $V-I$ and luminosities coherent with the observations. This would be consistent with the increasing number of confirmed binary cores hosting bipolar PNe ([@miszalski09a]; [@miszalski10]), but would need to be proved via a direct method (e.g. photometric monitoring).
Summary and Conclusions
=======================
A spatio-kinematical model of the M 1–75 nebula has been presented. Two pairs of lobes emerge from the core, their expansion patterns well described by a Hubble-like flow, their kinematical ages ($\sim$1000 yr kpc$^{-1}$) being similar within uncertainties, while their orientations differ by $\sim$22$^\circ$. The outer lobes have been found to be slightly point-symmetric. The implications of these results on the different shaping theories have been briefly discussed, and a model invoking a close companion star (or a hot Jupiter planet) has been favoured.
On the other hand, the $V-I$ colour and brightness of the CSPN —first identified in this work— are compatible with the presence of a close companion provided its T$_\mathrm{eff}$ is less than 10000 K and its luminosity less than 10$^{-3}$ times that of the white dwarf.
MSG would like to thank Mariano Santander for his help with the point-symmetric model, and Guillermo García-Segura for his insight on single stars and quadrupolar nebulae.
[^1]: Based on observations made with the 4.2 m William Herschel Telescope and the 2.5 m Isaac Newton Telescope, both operated on the island of la Palma by the Isaac Newton Group of Telescopes in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.
[^2]: [IRAF]{} is distributed by National Optical Astronomy Observatories.
|
---
abstract: 'Stressing the role of dual coalgebras, we modify the definition of affine schemes over the ’field with one element’. This clarifies the appearance of Habiro-type rings in the commutative case, and, allows a natural noncommutative generalization, the study of representations of discrete groups and their profinite completions being our main motivation.'
address: |
Department of Mathematics, University of Antwerp\
Middelheimlaan 1, B-2020 Antwerp (Belgium)\
[[email protected]]{}
author:
- Lieven Le Bruyn
title: '(non)commutative F-un geometry'
---
Commutative F-un geometry
=========================
In this section we will recall the definition of affine schemes over the mythical field $\mathbb{F}_1$ with one element, originally due to Christophe Soulé [@Soule] and refined later by Alain Connes and Katia Consani [@CC]. This approach is based on functors from abelian groups to sets satisfying a universal property with respect to an integral- and a complex affine scheme. We will modify this definition slightly by replacing these affine schemes by integral- resp. complex dual coalgebras. This amounts to restricting to étale local data of the affine schemes and has the additional advantage that the definition can be extended verbatim to the noncommutative world as we will outline in the next section. Another advantage of the coalgebra approach is that it inevitably leads to the introduction of the Habiro ring [@Habiro] in the easiest example, that of the multiplicative group. This might be compared to recent work by Yuri I. Manin [@Manin] and Matilde Marcolli [@Marcolli].
For a commutative ring ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}$ we will denote with ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k-calg}}}$, resp. ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k-alg}}}$, the category of all commutative ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}$-algebras, resp. the category of all ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}$-algebras. and with morphisms all ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}$-algebra morphisms. For two objects $A,B$ in ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k-alg}}}$ we will denote the set of all ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}$-algebra morphisms from $A$ to $B$ by $(A,B)_{{{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}}$.
Grothendieck introduced the category ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k-caff}}}$ of all affine schemes living over a commutative ring ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}$ to be the category dual to the category ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k-calg}}}$ of all commutative ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}$-algebras, that is, ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k-caff}}} = ({{\text{\em \usefont{OT1}{cmtt}{m}{n} k-calg}}})^o$. One way to realize this duality is to associate to a commutative ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}$-algebra $A$ a covariant functor, [*the functor of points*]{} ${{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}_A$, $${{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}_A~:~{{\text{\em \usefont{OT1}{cmtt}{m}{n} k-calg}}} \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} sets}}} \qquad B \mapsto (A,B)_{{{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}}$$ Alternatively, one can associate to $A$ a more classical geometric object, [*the affine scheme*]{} ${{\text{\em \usefont{OT1}{cmtt}{m}{n} spec}}}(A)$. This consists of a topological space $spec(A)$, the set of all prime ideals of $A$ equipped with the Zariski topology, together with a sheaf of rings ${\mathcal{O}}_A$ on it, called the structure sheaf of $A$. The ring $A$ is recovered as the ring of global sections. Whereas both approaches are equivalent, it should be clear that the functorial point of view lends itself more easily to generalizations.
F-un or $\mathbb{F}_1$, the field with one element, is a virtual object which might be thought of as a ’ring’ living under ${\mathbb{Z}}$. $\mathbb{F}_1$-believers base their f-unny intuition on the following two mantras :
- [$\mathbb{F}_1$ forgets about additive data and retains only multiplicative data.]{}
- [$\mathbb{F}_1$-objects only acquire flesh when extended to ${\mathbb{Z}}$ (or ${\mathbb{C}}$).]{}
As an example, an $\mathbb{F}_1$-vectorspace is merely a set $V$ as there is no addition of vectors and just one element to use for scalar multiplication. Hence, the dimension of $V$ equals the cardinality of $V$ as a set. Next one should specify the classical objects one obtains after ’extending’ $V$ to the integers or to the complex numbers. The correct integral version of a vectorspace is a lattice, so one defines $V \otimes_{\mathbb{F}_1} {\mathbb{Z}}$ to be the free ${\mathbb{Z}}$-lattice ${\mathbb{Z}}V$ on $V$. Analogously, one defines the extension of $V$ to the complex numbers, $V \otimes_{\mathbb{F}_1} {\mathbb{C}}$ to be the complex vectorspace ${\mathbb{C}}V$ with basis the set $V$.
But then, linear maps between $\mathbb{F}_1$-vectorspaces will be just set-maps and invertible maps are bijections, whence the group $GL_n(\mathbb{F}_1)$ is the symmetric group $S_n$. For a group $G$, an $n$-dimensional representation over $\mathbb{F}_1$ will then be a groupmorphism $\rho : G \rTo S_n$, that is, a permutation representation of $G$. Irreducible $G$-representations over $\mathbb{F}_1$ are then transitive permutation representations, and so on.
{#commutative}
In analogy with the finite field case, one expects there to be a unique $n$-dimensional field extension of $\mathbb{F}_1$ which we will denote by $\mathbb{F}_{1^n}$. This has to be a set with $n$ elements allowing a multiplication, whence the proposal to take $\mathbb{F}_{1^n} = C_n$ the cyclic group of order $n$. Extending $\mathbb{F}_{1^n}$ to the integers or complex numbers we should obtain a commutative algebra of rank resp. dimension $n$. Christophe Soulé [@Soule] proposed to take the integral- and complex group-algebras $$\mathbb{F}_{1^n} \otimes_{\mathbb{F}_1} {\mathbb{Z}}\simeq {\mathbb{Z}}C_n \quad \text{and} \quad \mathbb{F}_{1^n} \otimes_{\mathbb{F}_1} {\mathbb{C}}\simeq {\mathbb{C}}G$$ More generally, he proposed to take as the category of all commutative $\mathbb{F}_1$-algebras the category of all finite (!) abelian groups, that is, $\mathbb{F}_1-{{\text{\em \usefont{OT1}{cmtt}{m}{n} calg}}} = {{\text{\em \usefont{OT1}{cmtt}{m}{n} abelian}}}$. For any abelian group $G$ we then have to make sense of the extended algebras which we take again to be the group-algebras $$G \otimes_{\mathbb{F}_1} {\mathbb{Z}}\simeq {\mathbb{Z}}G \quad \text{and} \quad G \otimes_{\mathbb{F}_1} {\mathbb{C}}\simeq {\mathbb{C}}G$$ Having a notion for commutative $\mathbb{F}_1$-algebras, Soulé takes Grothendieck functor of points approach to define [*affine $\mathbb{F}_1$-schemes*]{}. This should be a covariant functor $$X~:~{{\text{\em \usefont{OT1}{cmtt}{m}{n} abelian}}} \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} sets}}}$$ connecting nicely to the functor of points of an affine integral- and complex-scheme. More precisely, Soulé [@Soule] and later Connes and Consani [@CC] require the following data
- [a complex affine commutative algebra $A \in {\mathbb{C}}-{{\text{\em \usefont{OT1}{cmtt}{m}{n} calg}}}$]{}
- [an integral algebra $B \in {\mathbb{Z}}-{{\text{\em \usefont{OT1}{cmtt}{m}{n} calg}}}$ such that $B \otimes_{{\mathbb{Z}}} {\mathbb{C}}\rInto A$]{}
- [ a natural transformation $ev : X \rTo h_A$, called the ’evaluation’ map]{}
- [ an inclusion of functors $i : X \rInto h_B$]{}
satisfying the following universal property : given any integral algebra $C \in {\mathbb{Z}}-{{\text{\em \usefont{OT1}{cmtt}{m}{n} calg}}}$, any natural transformation $f : X \rTo h_C$ and any natural transformation $g : h_A \rTo h_{C \otimes_Z {\mathbb{C}}}$ making the upper square commute $$\xymatrix{X \ar[rr]^{ev} \ar[rd]^f \ar[dd]^i & & h_A \ar[rd]^{g} & \\
& h_C \ar[rr]^{- \otimes {\mathbb{C}}} & & h_{C \otimes {\mathbb{C}}} \\
h_B \ar@{.>}[ru]^{\exists} \ar[rr]^{- \otimes {\mathbb{C}}} & & h_{B \otimes {\mathbb{C}}} \ar@{.>}[ru] &}$$ there ought to be a natural transformation $h_B \rTo h_C$ making the entire diagram commute. This means that ${{\text{\em \usefont{OT1}{cmtt}{m}{n} spec}}}(B)$ is the best affine integral scheme approximating the functor $X$. Note that by Yoneda’s lemma this means that one can reconstruct from the ${\mathbb{C}}$-algebra morphism $\psi : C \otimes {\mathbb{C}}\rTo A$ determining the natural transformation $g=- \circ \psi$ a ${\mathbb{Z}}$-algebra morphism $\phi : C \rTo B$ compatible with the inclusion $B \otimes {\mathbb{C}}\rInto A$. This means that for every abelian group $G$ we have a commuting diagram $$\xymatrix{X(G) \ar[rr]^{ev} \ar[rd]^f \ar[dd]^i & & (A,{\mathbb{C}}G)_{{\mathbb{C}}} \ar[rd]^{- \circ \psi} & \\
& (C,{\mathbb{Z}}G)_{{\mathbb{Z}}} \ar[rr]^{- \otimes {\mathbb{C}}} & & (C \otimes {\mathbb{C}},{\mathbb{C}}G)_{{\mathbb{C}}} \\
(B,{\mathbb{Z}}G)_{{\mathbb{Z}}} \ar[ru]^{- \circ \phi} \ar[rr]^{- \otimes {\mathbb{C}}} & & (B \otimes {\mathbb{C}},{\mathbb{C}}G)_{{\mathbb{C}}} \ar[ru] &}$$
{#example}
The archetypical example being the multiplicative group. Consider the forgetful functor $$\mathbb{G}_m~:~{{\text{\em \usefont{OT1}{cmtt}{m}{n} abelian}}} \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} sets}}} \qquad G \mapsto G$$ Take $A = {\mathbb{C}}[q^{\pm}]$ and $B = {\mathbb{Z}}[q^{\pm}]$, then their functors of points are exactly the multiplicative group scheme, that is give the groups of units $$h_A(D) = D^* \quad \text{and} \quad h_B(C)=C^*$$ for all $D \in {\mathbb{C}}-{{\text{\em \usefont{OT1}{cmtt}{m}{n} calg}}}$ and $C \in {\mathbb{Z}}-{{\text{\em \usefont{OT1}{cmtt}{m}{n} calg}}}$. We can then take both $i$ and $ev$ the natural transformation taking $F(G)=G$ to the subgroup of units $G \subset ({\mathbb{Z}}G)^* \subset ({\mathbb{C}}G)^*$.
Remains only to prove the universal property. Let the natural transformation $g : h_{{\mathbb{C}}[q^{\pm}]} \rTo h_{C \otimes {\mathbb{C}}}$ be determined by the ${\mathbb{C}}$-algebra morphism $\psi : C \otimes {\mathbb{C}}\rTo {\mathbb{C}}[q^{\pm}]$ and let $N$ be a natural number larger than the degree of all $\psi(c)$ where $c$ is one of the ${\mathbb{Z}}$-algebra generators of $C$. Consider the finite cyclic group $C_N = \langle g \rangle$, then tracing the element $g$ around the above diagram gives the commutative diagram $$\begin{diagram}
C & \rInto & C \otimes {\mathbb{C}}& \rTo^{\psi} & {\mathbb{C}}[q^{\pm}] \\
\dTo^{\phi} & & & & \dOnto^{\pi} \\
{\mathbb{Z}}C_N & & \rInto & & {\mathbb{C}}C_N = \frac{{\mathbb{C}}[q^{\pm}]}{(q^N-1)} \end{diagram}$$ where $\phi = f(g)$. Repeating this argument, $\pi(\psi(c)) = \psi(c) = \phi(c)$ for all ${\mathbb{Z}}$-generators of $C$, whence we have that $\psi(C) \subset {\mathbb{Z}}[q^{\pm}]$ giving the required natural transformation $h_{{\mathbb{Z}}[q^{\pm}]} \rTo h_C$.
{#section-3}
Observe that Soulé uses only finite abelian groups and hence we do not require the full functor of points, but rather the restricted functors $${{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}'_A~:~{{\text{\em \usefont{OT1}{cmtt}{m}{n} k-fd.calg}}} \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} sets}}} \qquad B \mapsto (A,B)_{{{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}}$$ where ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k-fd.calg}}}$ is the category of all [*finite dimensional*]{} commutative ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}$-algebras. On the ’geometric’ level we might still use the affine scheme ${{\text{\em \usefont{OT1}{cmtt}{m}{n} spec}}}(A)$ as this object contains more information than ${{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}'_A$, but we’d rather use a slimmer geometric object having the same amount of information as the restricted functor of points. It will turn out that the object we propose can be extended verbatim to the noncommutative world, whereas trying to extend affine schemes is known to lead to major difficulties.
{#section-4}
Let us consider the complex case first. For $A \in {\mathbb{C}}-{{\text{\em \usefont{OT1}{cmtt}{m}{n} calg}}}$, we define the (finite) dual coalgebra $A^o$ to be the collection of all ${\mathbb{C}}$-linear maps $\lambda : A \rTo {\mathbb{C}}$ whose kernel contains a cofinite ideal $I \triangleleft A$. The dual maps to the multiplication and unit map of $A$ then define a coalgebra structure on $A^o$, see for example Sweedler’s monograph [@Sweedler]. For $B$ a finite dimensional ${\mathbb{C}}$-algebra, any ${\mathbb{C}}$-algebra morphism $A \rTo B$ dualizes to a ${\mathbb{C}}$-coalgebra map $B^* \rTo A^o$ and as a coalgebra is the limit of its finite dimensional sub-coalgebras we see that the dual coalgebra $A^o$ contains the same information as the restricted functor of points ${{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}'_A$. We will now turn $A^o$ into our desired ’geometric’ object.
As $A$ is commutative, any finite dimensional quotient $A/I \simeq L_{\mathfrak{m}_1} \oplus \hdots \oplus L_{\mathfrak{m}_k}$ splits into a direct sum of locals and hence the dual subcoalgebra $(A/I)^*$ is the direct sum of pointed coalgebras $(L_{\mathfrak{m}})^*$ which are subcoalgebras of the enveloping algebra of the abelian Lie-algebra of tangent-vectors $(\mathfrak{m}/\mathfrak{m}^2)^*$. Taking limits we have that $$A^o = \bigoplus_{\mathfrak{m} \in {{\text{\em \usefont{OT1}{cmtt}{m}{n} max}}}(A)} P_{\mathfrak{m}}$$ with $P_{\mathfrak{m}} \subset U((\mathfrak{m}/\mathfrak{m}^2)^*)$. In particular, we obtain the maximal ideals ${{\text{\em \usefont{OT1}{cmtt}{m}{n} max}}}(A)$ as the group-like elements of $A^o$, or equivalently, as the direct factors of the coradical $corad(A^o)$. Elements of $A$ naturally evaluate on $A^o$ (and hence on the coradical) and induce the usual Zariski topology on ${{\text{\em \usefont{OT1}{cmtt}{m}{n} max}}}(A)$.
We thus recover from the dual coalgebra $A^o$ the maximal ideal spectrum of $A$. But, $A^o$ contains a lot more local information. This is best seen by taking the full dual algebra $A^{o*}$ of $A^o$ giving rise to a Taylor-embedding (sending a function to its Taylor series expansions in all points) $$A \rInto A^{o*} = \prod_{\mathfrak{m} \in {{\text{\em \usefont{OT1}{cmtt}{m}{n} max}}}(A)} \hat{{\mathcal{O}}}_{A,\mathfrak{m}}$$ where $\hat{{\mathcal{O}}}_{A,\mathfrak{m}}$ is the $\mathfrak{m}$-adic completion of $A$ (that is the stalk of the structure sheaf in the étale topology).
Concluding, the restricted functor of points ${{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}'_A$, or equivalently the dual coalgebra $A^o$, contains enough information to recover the analytic (or étale) local information in all the closed points of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} spec}}}(A)$.
{#section-5}
An affine F-un scheme $X : {{\text{\em \usefont{OT1}{cmtt}{m}{n} abelian}}} \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} sets}}}$ connects to the complex picture via the evaluation natural transformation $ev : X \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}'_A$. The discussion above leads to the introduction of an analytic ring of functions $\mathbb{F}_1[X]^{an}$ of which we now have a complex interpretation $$\mathbb{F}_1[X]^{an} \otimes_{\mathbb{F}_1} {\mathbb{C}}= \bigcap_{\mathfrak{m} \in Im(ev)} \hat{{\mathcal{O}}}_{A,\mathfrak{m}}$$ With $Im(ev)$ we denote the images of all maps ${{\text{\em \usefont{OT1}{cmtt}{m}{n} max}}}({\mathbb{C}}G) \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} max}}}(A)$ coming from the algebra maps $A \rTo {\mathbb{C}}G$ contained in $ev(F(G)) \subset {{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}'_A({\mathbb{C}}G)$.
For the example \[example\] of the forgetful functor, we have $A = {\mathbb{C}}[q^{\pm}]$ and hence ${{\text{\em \usefont{OT1}{cmtt}{m}{n} max}}}(A) = {\mathbb{C}}^*$ and $${\mathbb{C}}[q^{\pm}]^{o*} = \prod_{\alpha \in {\mathbb{C}}^*} {\mathbb{C}}[[q-\alpha]]$$ For any finite abelian group $G$, ${{\text{\em \usefont{OT1}{cmtt}{m}{n} max}}}({\mathbb{C}}G)$ is the set of characters of $G$ and under the evaluation map an element $g \in F(G)=G$ maps a character $\chi$ to its value $\chi(g)$, which are of course all roots of unity. Hence, if we vary over all finite abelian groups we obtain $$\mathbb{F}_1[q^{\pm}]^{an} \otimes_{\mathbb{F}_1} {\mathbb{C}}= \bigcap_{\lambda \in \mu_{\infty}} {\mathbb{C}}[[q-\lambda]]$$ Observe that $\mu_{\infty}$, the set of all roots of unity, is a Zariski dense set in ${{\text{\em \usefont{OT1}{cmtt}{m}{n} max}}}({\mathbb{C}}[q^{\pm}]) = {\mathbb{C}}^*$.
{#section-6}
Whereas the new complex picture based on the dual coalgebra is still pretty close to the usual affine scheme, this changes drastically in the integral picture. For a ${\mathbb{Z}}$-algebra $B$ we have to consider the restricted functor of points $${{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}'_B~:~{\mathbb{Z}}-{{\text{\em \usefont{OT1}{cmtt}{m}{n} fp.calg}}} \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} sets}}} \qquad C \mapsto (B,C)_{{\mathbb{Z}}}$$ where ${\mathbb{Z}}-{{\text{\em \usefont{OT1}{cmtt}{m}{n} fp.calg}}}$ is the category of all commutative ${\mathbb{Z}}$-algebras which are finite projective ${\mathbb{Z}}$-modules. Again, this restricted functor contains the same information as the dual ${\mathbb{Z}}$-coalgebra $$B^o = \underset{\rightarrow}{lim}~Hom_{{\mathbb{Z}}}(B/I,{\mathbb{Z}})$$ where the limit is taken over all ideals $I \triangleleft B$ such that $B/I$ is a projective ${\mathbb{Z}}$-module of finite rank. If we try to mimic the complex description of the dual coalgebra we are led to consider a certain subset of all coheight one prime ideals of $B$ $${{\text{\em \usefont{OT1}{cmtt}{m}{n} submax}}}(B) = \{ P \in spec(B)~|~\text{$B/P$ is a free ${\mathbb{Z}}$-module of finite rank} \}$$ Note that closed points in ${{\text{\em \usefont{OT1}{cmtt}{m}{n} spec}}}(B)$ are [*not*]{} contained in ${{\text{\em \usefont{OT1}{cmtt}{m}{n} submax}}}(B)$. Therefore we face the problem that different elements $P,P' \in {{\text{\em \usefont{OT1}{cmtt}{m}{n} submax}}}(B)$ are usually not comaximal and hence that we no longer have a direct sum decomposition of $B^o$ over this set (as was the case for the complex dual coalgebra).
As we will recall in the next section, we are familiar with such situations in noncommutative algebra, where even maximal ideals can belong to the same ’clique’, that is, that the corresponding simple representations have nontrivial extensions. Using this noncommutative intuition, we therefore impose a clique-relation on the elements of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} submax}}}(B)$ $$P \leftrightarrow P' \qquad \text{iff} \qquad P+P' \not= B$$ This relation should be thought of as a ’nearness’ condition. Observe that any $P \in {{\text{\em \usefont{OT1}{cmtt}{m}{n} submax}}}(B)$ determines a finite collection of points in ${{\text{\em \usefont{OT1}{cmtt}{m}{n} max}}}(B \otimes_{{\mathbb{Z}}} {\mathbb{C}})$ and hence we can extend this nearness relation on the points of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} max}}}(B)$. Observe that this relation is clearly invariant under the action of the absolute Galois group $Gal(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$.
The different cliques determine the direct sum decomposition of the ${\mathbb{Z}}$-coalgebra $B^o$ and hence also of the Taylor-like ring of functions $B^{o*}$. Fully describing the dual ${\mathbb{Z}}$-coalgebra $B^o$ usually is a very difficult task and therefore, as in the complex case, when we are studying F-un geometry we restrict to that part determined by the elements in $Im(i)$ where $i~:~F \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}'_B$ is the inclusion of functors determined by the affine F-un scheme $F~:~{{\text{\em \usefont{OT1}{cmtt}{m}{n} abelian}}} \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} sets}}}$.
{#section-7}
Let us consider again the example of the multiplicative group and indicate how the ${\mathbb{Z}}$-coalgebra approach leads to the introduction of the Habiro ring.
The ideals $I \triangleleft B= {\mathbb{Z}}[q^{\pm}]$ such that $B/I$ is a free ${\mathbb{Z}}$-module of finite rank are precisely the principal ideals $I = (f(q))$ where $f(q)$ is a monic polynomial. Hence, $${{\text{\em \usefont{OT1}{cmtt}{m}{n} submax}}}({\mathbb{Z}}[q^{\pm}]) = \{ (p(q))~:~\text{$p(q)$ is monic and irreducible} \}$$ Because ${\mathbb{Z}}[q^{\pm}]$ is a unique factorization domain we can decompose any monic polynomial uniquely into irreducible factors $$f(q) = p_1(q)^{n_1} \hdots p_k(q)^{n_k}$$ and we would like to use this fact, as in the complex case, to decompose the (linear duals) finite rank ${\mathbb{Z}}$-algebra quotients over ${{\text{\em \usefont{OT1}{cmtt}{m}{n} submax}}}({\mathbb{Z}}[q^{\pm}])$. However, $$\frac{{\mathbb{Z}}[q^{\pm}]}{(f(q))} \not= \frac{{\mathbb{Z}}[q^{\pm}]}{(p_1(q))^{n_1}} \oplus \hdots \oplus \frac{{\mathbb{Z}}[q^{\pm}]}{(p_k(q))^{n_k}}$$ as the different primes $(p_i(q))$ and $(p_j(q))$ do not have to be comaximal. This problem makes it impossible to split the description of the dual coalgebra over the ’points’ as in the complex case. Hence, we have no other option but to describe it as a direct limit $${\mathbb{Z}}[q^{\pm}]^o = \underset{\rightarrow}{lim}~(\frac{{\mathbb{Z}}[q^{\pm}]}{(f(q))})^*$$ where the limit is considered with respect to divisibility of polynomials as there are natural inclusions of ${\mathbb{Z}}$-coalgebras $$(\frac{{\mathbb{Z}}[q^{\pm}]}{(f(q))})^* \rInto (\frac{{\mathbb{Z}}[q^{\pm}]}{(g(q))})^* \qquad \text{whenever} \qquad f(q) | g(q)$$ As in the complex case we are then interested in the dual algebra of ${\mathbb{Z}}[q^{\pm}]^o$ and the natural algebra map $${\mathbb{Z}}[q^{\pm}]^o \rInto ({\mathbb{Z}}[q^{\pm}]^o)^* = \underset{\leftarrow}{lim}~\frac{{\mathbb{Z}}[q^{\pm}]}{(f(q))}$$ and it is clear that in the description of the algebra on the right-hand side completions at principal ideals will constitute a main ingredient.
While we can do all these calculations to some extend, we are primarily interested in that part of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} submax}}}({\mathbb{Z}}[q^{\pm}])$ in the image of the inclusion functor, that is $$Im(i) = {\mathbb{N}}= \{ (\Phi_1(q)), (\Phi_2(q)), \hdots , (\Phi_n(q)), \hdots \} \subset {{\text{\em \usefont{OT1}{cmtt}{m}{n} submax}}}({\mathbb{Z}}[q^{\pm}])$$ We will confuse the natural number $n$ with the corresponding cyclotomic polynomial $\Phi_n(q)$ or with the height one prime generated by it. With this identification ${\mathbb{N}}$ is the integral analog of the set of all roots of unity $\boldsymbol{\mu}_{\infty}$ in the complex case.
In the case of cyclotomic polynomials we have complete information about possible co-maximality
- [If $\frac{m}{n} \not= p^k$ for some prime number $p$, then $(\Phi_m(q),\Phi_n(q))=1$ that is the cyclotomic prime ideals are comaximal.]{}
- [If $\frac{m}{n}=p^k$ for some prime number $p$, then $\Phi_m(q) \equiv \Phi_n(q)^d~{{\text{\em \usefont{OT1}{cmtt}{m}{n} mod}}}~(p)$ for some integer $d$, hence the cyclotomic primes are not comaximal.]{}
Therefore, the relevant clique-relation is $$n \leftrightarrow m \qquad \text{if and only if} \qquad \frac{m}{n}=p^{\pm k}$$ inducing on the complex level the $Gal(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$-invariant nearness condition on roots of unity $\lambda, \mu \in \mu_{\infty}$ $$\lambda \leftrightarrow \mu \qquad \text{iff} \qquad \frac{\lambda}{\mu}~\text{is of order $p^k$}$$ for some prime number $p$.
Yuri I. Manin argues in [@Manin] that we should take the analogy between the integral affine scheme ${{\text{\em \usefont{OT1}{cmtt}{m}{n} spec}}}({\mathbb{Z}}[q^{\pm}])$ and the (complex) affine plane more seriously and that, besides the arithmetic axis, one should also consider a projection to the ’geometric axis’ (which should then be viewed as the affine $\mathbb{F}_1$-scheme corresponding to $\mathbb{F}_1[q^{\pm}]$. He proposed that the zero sets of the cyclotomic polynomials $\Phi_n(q)$ for all integers $n$ should be considered as the union of the fibers in this second projection. That is, we should have the following picture :
$$\rotatebox{90}{
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\tlabel[cc](20,-10){(2)}
\tlabel[cc](30,-10){(3)}
\tlabel[cc](50,-10){(5)}
\tlabel[cc](70,-10){(7)}
\tlabel[cc](130,-10){(p)}
\tlabel[cc](130,210){${{\text{\em \usefont{OT1}{cmtt}{m}{n} spec}}}(\mathbb{F}_p[q^{\pm}])$}
\tlabel[cl](210,37){${{\text{\em \usefont{OT1}{cmtt}{m}{n} spec}}}({\mathbb{Z}}[q^{\pm}])$}
\tlabel[cc](220,20){$\begin{diagram} \\ \dOnto \end{diagram}$}
\tlabel[cl](210,0){${{\text{\em \usefont{OT1}{cmtt}{m}{n} spec}}}({\mathbb{Z}})$}
\tlabel[cc](110,-30){\fbox{\textcolor{green}{ARITHMETIC AXIS}}}
\tlabel[cc](-50,115){\rotatebox{270}{\fbox{\textcolor{blue}{GEOMETRIC AXIS}}}}
\tlabel[cc](-30,60){\rotatebox{270}{$1$}}
\tlabel[cc](-30,70){\rotatebox{270}{$2$}}
\tlabel[cc](-30,80){\rotatebox{270}{$3$}}
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\tlabel[cc](230,140){\rotatebox{270}{${{\text{\em \usefont{OT1}{cmtt}{m}{n} spec}}}({\mathbb{Z}}[\zeta_n])$}}
\tlabel[cc](10,230){\rotatebox{270}{$\begin{diagram} {{\text{\em \usefont{OT1}{cmtt}{m}{n} spec}}}({\mathbb{Z}}[q^{\pm}]) \\ \dOnto \\ {{\text{\em \usefont{OT1}{cmtt}{m}{n} spec}}}(\mathbb{F}_1[q^{\pm}]) \end{diagram}$}}
\end{mfpic}}$$
Note that this is an over-simplification. Whereas the different green fibers for the projection to the arithmetic axis are clearly comaximal, the blue fibers are not. For example, the zero sets ${\mathbb{V}}(\Phi_2(q))$ and ${\mathbb{V}}(\Phi_1(q))$ share the maximal ideal $(2,q-1)$. The clique-relation encodes how the blue fibers intersect each other.
The clique-relation is important to relate different completions occurring in the F-un determined part of the algebra $({\mathbb{Z}}[q^{\pm}]^o)^*$ as was proved by Kazuo Habiro [@Habiro]. Let us define for any subset $S \subset {\mathbb{N}}$ the completion $${\mathbb{Z}}[q^{\pm}]^S = \underset{\underset{p \in \Phi_S^*}{\leftarrow}}{lim}~\frac{{\mathbb{Z}}[q^{\pm}]}{(p)}$$ where $\Phi_S^*$ is the set of monic polynomials generated by all $\Phi_n(q)$ for $n \in S$. Among the many precise results proved in [@Habiro] we mention these two
1. [If $S' \subset S$ and if every clique-component of $S$ contains an element from $S'$, then the natural map is an inclusion $$\rho^S_{S'}~:~{\mathbb{Z}}[q^{\pm}]^S \rInto {\mathbb{Z}}[q^{\pm}]^{S'}$$]{}
2. [If $S$ is a saturated subset of ${\mathbb{N}}$ meaning that for every $n \in S$ also its divisor-set $\langle n \rangle = \{ m | n \}$ is contained in $S$, then $${\mathbb{Z}}[q^{\pm}]^S = \bigcap_{n \in S} {\mathbb{Z}}[q^{\pm}]^{\langle n \rangle} = \bigcap_{n \in S} \widehat{{\mathbb{Z}}[q^{\pm}]}_{(q^n-1)}$$ where the terms on the right-hand side are the $I$-adic completions where $I=(q^n-1)$.]{}
Using these properties it is then natural to define the integral version of the ring of analytic functions on the multiplicative group scheme over ${\mathbb{F}}_1$ to be $${\mathbb{F}}_1[q^{\pm}]^{an} \otimes_{{\mathbb{F}}_1} {\mathbb{Z}}\simeq \bigcap_{n \in {\mathbb{N}}} \widehat{{\mathbb{Z}}[q^{\pm}]}_{(q^n-1)} = {\mathbb{Z}}[q^{\pm}]^{{\mathbb{N}}}$$ This ring has a description very similar to that of the profinite integers replacing factorials by q-factorials $${\mathbb{Z}}[q^{\pm}]^{{\mathbb{N}}} = \underset{\underset{n}{\leftarrow}}{lim}~\frac{{\mathbb{Z}}[q^{\pm}]}{((q^n-1)(q^{n-1}-1) \hdots (q-1))}$$ and as such its elements have a unique description as formal Laurent polynomials over ${\mathbb{Z}}$ of the form $$\sum_{n=0}^{\infty} a_n(q) (q^n-1)(q^{n-1}-1)\hdots (q-1) \in {\mathbb{Z}}[[q^{\pm}]] \qquad \text{with} \qquad deg(a_n(q)) < n$$ We observe that any such formal power series can be evaluated at a root of unity. Some elements of ${\mathbb{Z}}[q^{\pm}]^{{\mathbb{N}}}$ have been discovered before. For example, Maxim Kontsevich observed in his investigations on Feynman integrals that the formal power series $$\sum_{n=0}^{\infty} (1-q)(1-q^2) \hdots (1-q^n)$$ has a properly defined value in every root of unity. Subsequently, Don Zagier [@Zagier] proved the strange equality $$\sum_{n=0}^{\infty} (1-q)(1-q^2) \hdots (1-q^n) = - \frac{1}{2} \sum_{n=1}^{\infty} n \chi(n) q^{(n^2-1)/24}$$ where $\chi$ is the quadratic character of conductor $12$. The strange fact about this equality is that the two sides never make sense simultaneously. The left hand side diverges for all points within the unit circle and outside the unit circle and can be evaluated at roots of unity whereas the right hand side converges only within the unit circle and diverges everywhere else. What Zagier meant by this equality is that for all $\alpha \in \boldsymbol{\mu}_{\infty}$ the evaluation of the left hand side coincides with the radial limit of the function on the right hand side. Don Zagier says that the function on the right ’leak through roots of unity’.
Noncommutative F-un geometry
============================
In this section we will extend Soulé’s definition of an affine $\mathbb{F}_1$-scheme to the noncommutative case. Our main motivation is the study of finite dimensional representations of discrete groups, such as the braid groups or the modular group. We have seen that irreducible finite dimensional $\mathbb{F}_1$-representations of a group $\Gamma$ are exactly the finite transitive permutation representations $\Gamma/\Lambda$ where $\Lambda$ is of finite index in $\Gamma$. That is, all finite dimensional $\mathbb{F}_1$-representation theory of $\Gamma$ comes from its profinite completion $\hat{\Gamma} = \underset{\leftarrow}{lim}~\Gamma/\Lambda$, the limit taken over all finite index normal subgroups.
In the previous section we have worked out the special case when $\Gamma = {\mathbb{Z}}$. Here, the simple representations of $\hat{{\mathbb{Z}}}$ are the roots of unity $\mu_{\infty}$ and they are Zariski closed in all simples ${\mathbb{C}}^* = {{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}({\mathbb{Z}})$. The clique-relation on $\mu_{\infty}$ was compatible with the action of the absolute Galois group and the Habiro ring ’feels’ the inclusion $\mu_{\infty} \subset {\mathbb{C}}^*$, that is it contains the tangent information in a Galois-compatible way.
Here we extend some of these results to the case of a non-Abelian discrete group $\Gamma$ satisfying the property $\bullet$ : for every finite collection of elements $\{ g_1,\hdots,g_k \} \subset \Gamma$ there is a finite index subgroup $\Lambda \subset \Gamma$ such that the natural projection map gives an embedding $\{ g_1,\hdots,g_k \} \rInto \Gamma/\Lambda$. We will prove that such groups determine a noncommutative affine $\mathbb{F}_1$-scheme, the F-un information being given by the finite dimensional permutation representations, or equivalently, the representation theory of the profinite completion $\hat{\Gamma}$. We will show that ${{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}(\hat{\Gamma})$ is Zariski dense in ${{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}(\Gamma)$ and compute the tangent information of this embedding. That is, to a finite dimensional permutation representation $P=\Gamma/\Lambda$ we will associate a noncommutative gadget (a quiver, relations and a dimension vector) encoding all possible deformations of $P$ which are still $\Gamma$-representations. In relevant situations, including the case when $\Gamma$ is the modular group ${\operatorname{PSL}}_2({\mathbb{Z}})$ (in which case the permutation representations are Grothendieck’s ’dessins d’enfants’) some subsidiary noncommutative gadgets can be derived from this tangent information, such as the necklace Lie algebra [@LBBocklandt] and the singularity type [@LBBocklandtSymens]. It is to be expected that most of these noncommutative gadgets associated to dessins are in fact Galois invariants.
{#section-8}
If we take commutative $\mathbb{F}_1$-algebras to be abelian groups, it make sense to identify the category of all $\mathbb{F}_1$-algebras with ${{\text{\em \usefont{OT1}{cmtt}{m}{n} groups}}}$ the category of all finite groups. Likewise, we have to extend Grothendieck’s functor of points to all, that is including also noncommutative, algebras. With these modifications we can extend Soulé’s definition to the noncommutative world.
Define an affine noncommutative $\mathbb{F}_1$-scheme to be a covariant functor $$X~:~{{\text{\em \usefont{OT1}{cmtt}{m}{n} groups}}} \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} sets}}}$$ from the category ${{\text{\em \usefont{OT1}{cmtt}{m}{n} groups}}}$ of all finite groups to ${{\text{\em \usefont{OT1}{cmtt}{m}{n} sets}}}$. We require that there is an affine ${\mathbb{C}}$-algebra $A$ and an evaluation natural transformation $ev : X \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}_A = (A,-)_{{\mathbb{C}}}$, giving for every finite group $G$ an evaluation map $X(G) \rTo (A,{\mathbb{C}}G)_{{\mathbb{C}}}$. Moreover, there should be a ’best’ integral affine algebra $B$ with an inclusion of functors $X \rInto {{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}_B = (B , -)_{{\mathbb{Z}}}$.
That is, for every finite group $G$ we have an inclusion $X(G) \rInto (B,{\mathbb{Z}}G)_{{\mathbb{Z}}}$. Here, ’best’ means that for every ${\mathbb{Z}}$-algebra $C$ and every natural transformation $X \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}_C = (C,-)_{{\mathbb{Z}}}$ and every ${\mathbb{C}}$-algebra morphism $\psi : {\mathbb{C}}\otimes C \rTo A$ making the upper square in the diagram below commute for every finite group $G$ $$\xymatrix{X(G) \ar[rr]^{ev} \ar[rd]^f \ar[dd]^i & & (A,{\mathbb{C}}G)_{{\mathbb{C}}} \ar[rd]^{g=- \circ \psi} & \\
& (C,{\mathbb{Z}}G)_{{\mathbb{Z}}} \ar[rr]^{- \otimes {\mathbb{C}}} & & (C \otimes {\mathbb{C}}, {\mathbb{C}}G)_{{\mathbb{C}}} \\
(B,{\mathbb{Z}}G)_{{\mathbb{Z}}} \ar@{.>}[ru]^{\exists - \circ \phi} \ar[rr]^{- \otimes {\mathbb{C}}} & & (B \otimes {\mathbb{C}},{\mathbb{C}}G)_{{\mathbb{C}}} \ar@{.>}[ru] &}$$ there exists a ${\mathbb{Z}}$-algebra morphism $\phi : C \rTo B$ making the entire diagram commute.
{#dessins}
Our first example of a noncommutative F-un scheme is Grothendieck’s theory of ’dessins d’enfants’. Let $X_{{\mathbb{C}}}$ be a Riemann surface (projective algebraic curve) defined over $\overline{{\mathbb{Q}}}$, then Belyi proved that there is a degree $d$ map $\pi : C \rOnto \mathbb{P}^1_{{\mathbb{C}}}$ ramified only in the points $\{ 0,1,\infty \}$. The open interval $] 0,1 [$ lifts to $d$ intervals on $C$. The endpoints of different lifts can be identified on $X$ indicating how the different sheets should be glued together in a neighborhood of the ramification point. The resulting graph with $d$ edges on $C$ is then called the [*dessin*]{} of $C$ and as the absolute Galois group $Gal(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ acts on the collection of all such curves, it also acts on the dessins. Writing out this action allows one to gain insight in the absolute Galois group. Hence it is a very important problem to find new Galois invariants of dessins.
We will be particularly interested in [*modular*]{} dessins, that is such that the preimages of $0$ all have valency 1 or 2 and the preimages of 1 all have valency 1 or 3 in the graph. Alternatively, this means that the curve can be viewed as the compactification of a quotient $C = \mathbb{H}/\Lambda$ of the upper-halfplane under the action of a subgroup $\Lambda$ of finite index in the modular group $\Gamma = PSL_2({\mathbb{Z}})$. That is, modular dessins are equivalent to finite dimensional permutation representations of the modular group. Therefore, one is interested in the functor $$X~:~{{\text{\em \usefont{OT1}{cmtt}{m}{n} groups}}} \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} sets}}} \qquad G \mapsto G_{(2)} \times G_{(3)}$$ sending a group to the set of all permutation representations of $\Gamma$ determined by elements of $G$. As $\Gamma \simeq C_2 \ast C_3$ is the free product of a cyclic group of order 2 with a cyclic group of order 3, this functor sends a finite group $G$ to the set product of its elements of order 2 with the elements of order 3 : $G_{(2)} \times G_{(3)}$. This functor determines a noncommutative affine $\mathbb{F}_1$-scheme as we can take as the complex- and integral group-algebras $$A = {\mathbb{C}}\Gamma \quad \text{and} \quad B = {\mathbb{Z}}\Gamma$$ of the modular group. As any ${\mathbb{C}}$-algebra morphism $A = {\mathbb{C}}\Gamma \rTo {\mathbb{C}}G$ is determined by the images of the order two (resp. three) generators $x$ and $y$ we can take as the evaluation and inclusion maps $$ev~:~G_{(2)} \times G_{(3)} \rTo ({\mathbb{C}}\Gamma, {\mathbb{C}}G)_{{\mathbb{C}}} \qquad (g_2,g_3) \mapsto \begin{cases} x \mapsto g_2 \\ y \mapsto g_3 \end{cases}$$ $$i~:~G_{(2)} \times G_{(3)} \rInto ({\mathbb{Z}}\Gamma, {\mathbb{Z}}G)_{{\mathbb{Z}}} \qquad (g_2,g_3) \mapsto \begin{cases} x \mapsto g_2 \\ y \mapsto g_3 \end{cases}$$ We can repeat the argument of \[example\] verbatim to prove that these data indeed define a noncommutative $\mathbb{F}_1$-scheme using the fact that the modular group $\Gamma$ satisfies condition $\bullet$.
{#TQFT}
The second example is motivated by 2-dimensional TQFT. To a Riemann surface $C$ of genus $g$ and any finite group $G$ one associates as topological invariant $Z_G(C)$ the number of fields on $C$ with gauge group $G$, or equivalently, the number of $G$-covers on $C$. By Frobenius-Schur this number is equal to $$Z_G(C) = \sum_{\chi} (\frac{| G |}{dim~\chi})^{2g-2}$$ where the sum runs over all irreducible representations $\chi$ of the finite group $G$. As the number of $G$-covers is equal to the number of group-morphisms $\pi_1(C) \rTo G$ from the fundamental group $\pi_1(C) = \langle x_1,\hdots,x_g,y_1,\hdots,y_g \rangle / (\prod x_iy_ix_i^{-1} y_i^{-1} )$, this motivates the functor $$X~:~{{\text{\em \usefont{OT1}{cmtt}{m}{n} groups}}} \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} sets}}} \quad G \mapsto \{ (a_1,\hdots,a_g,b_1,\hdots,b_g) \in G^{2g}~:~\prod a_ib_ia_i^{-1}b_i^{-1} = 1 \}$$ This functor is again an affine noncommutative $\mathbb{F}_1$-scheme as we can take the integral- and complex group-algebras $A = {\mathbb{C}}\pi_1(C)$ and $B= {\mathbb{Z}}\pi_1(C)$ and the natural evaluation and inclusion maps. Once again, the defining “bestness” property is verified using the fact that $\pi_1(C)$ satisfies condition $\bullet$.
Also in this example, the $\mathbb{F}_1$-info is given by all finite permutation representations of the fundamental group $\pi_1(C)$. That is, the F-un information is contained in the profinite completion $\widehat{\pi_1(C)}$.
{#section-9}
These two examples illustrate that any discrete group $\Gamma$ satisfying condition $\bullet$ determines a noncommutative affine $\mathbb{F}_1$-scheme. The corresponding functor assigns to a group $G$ the set of all groupmorphisms $\Gamma \rTo G$ and takes as the complex- and integral algebras the complex and integral group-algebra of $\Gamma$.
As in the commutative case we do not require the full strength of the functor of points ${{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}_A : {{\text{\em \usefont{OT1}{cmtt}{m}{n} k-alg}}} \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} sets}}}$ for a given (not necessarily commutative) ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}$-algebra $A$, but it suffices, for applications to F-un geometry, to restrict to finite dimensional ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}$-algebras $${{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}'_A~:~{{\text{\em \usefont{OT1}{cmtt}{m}{n} k-fd.alg}}} \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} sets}}} \qquad C \mapsto (A,C)_{{{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}}$$ If ${{\text{\em \usefont{OT1}{cmtt}{m}{n} k}}}$ is a field, the information contained in this restricted functor of points is equivalent to that contained in the dual coalgebra $A^o$. For this reason we want to associate noncommutative geometric data (say, a topological space and function) to the dual ${\mathbb{C}}$-coalgebra $A^o$ where $A$ is the complex algebra determining the evaluation natural transformation $ev : X \rTo {{\text{\em \usefont{OT1}{cmtt}{m}{n} h}}}'_A$.
Observe that in [@LBdualcoalgebra] we initiated the description of the dual coalgebra of any affine ${\mathbb{C}}$-algebra $A$ in terms of the $A_{\infty}$-structure on the Yoneda space of all finite dimensional simple $A$-representations. For the applications we have in mind here, that is, virtually free groups $G$ (such as the modular group $\Gamma = PSL_2({\mathbb{Z}})$), for which the group algebras ${\mathbb{C}}G$ is formally smooth by [@LBqurves], or 2-Calabi-Yau algebras such as ${\mathbb{C}}\pi_1(C)$, we do not require the full power of $A_{\infty}$-theory and can give, at least in principle, an explicit description of the dual coalgebra.
The geometric space associated to an affine ${\mathbb{C}}$-algebra $A$ will be the set of isomorphism classes of finite dimensional $A$-representations, which as in the commutative case, is the set of direct summands of the coradical of the dual coalgebra $${{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}(A) = corad(A^o)$$ In [@LBdualcoalgebra] we introduced a Zariski topology on ${{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}(A)$ in terms of the measuring $A^o \otimes A \rTo {\mathbb{C}}$. Here we will follow a slightly different approach based on noncommutative functions.
For a ${\mathbb{C}}$-algebra $A$ we define the [*noncommutative functions*]{} to be the ${\mathbb{C}}$-vectorspace quotients $${{\text{\em \usefont{OT1}{cmtt}{m}{n} functions}}}(A) = \mathfrak{g}_A = \frac{A}{[A,A]_{vect}}$$ where $[A,A]_{vect}$ is the subvectorspace (and [*not*]{} the ideal) spanned by all commutators in $A$. Note that in the classical case where $A = {\mathbb{C}}[X]$ is the commutative coordinate ring of an affine variety $X$, there is nothing to divide out and hence in this case we recover the coordinate ring $\mathfrak{g}_A = {\mathbb{C}}[X]$. If $A = {\mathbb{C}}G$ the group-algebra of a finite group $G$, then $\mathfrak{g}_A$ is the space dual to the space of character-functions of $G$. Hence, in both cases the linear functionals $\mathfrak{g}^*$ suffice to separate the [*points*]{} of $A$, that is ${{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}(A)$. We will show that for a general affine ${\mathbb{C}}$-algebra $A$ we do indeed have an embedding $${{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}(A) \rInto \mathfrak{g}^*$$ Consider the (commutative) affine scheme ${{\text{\em \usefont{OT1}{cmtt}{m}{n} rep}}}_n A$ of all $n$-dimensional representations. A quick and dirty way to describe its coordinate ring ${\mathbb{C}}[{{\text{\em \usefont{OT1}{cmtt}{m}{n} rep}}}_n A]$ is to take a finite set of algebra generators $\{ a_1,\hdots,a_m \}$ of $A$, consider a set of $mn^2$ commuting variables $\{ x_{ij}(k) : 1 \leq i,j \leq n, 1 \leq k \leq m \}$ and consider the ideal $I_n(A)$ of the polynomial algebra ${\mathbb{C}}[x_{ij}(k)~:~i,j,k]$ generated by all entries of all $n \times n$ matrices $f(X_1,\hdots,X_m)$ where $f(a_1,\hdots,a_m)$ runs over all relations holding in $A$ and where $X_k$ is the generic $n \times n$ matrix $(x_{ij}(k))_{i,j}$. Then, $${\mathbb{C}}[{{\text{\em \usefont{OT1}{cmtt}{m}{n} rep}}}_n A] = \frac{{\mathbb{C}}[ x_{ij}(k)~:~i,j,k ]}{I_n(A)}$$ On the affine scheme ${{\text{\em \usefont{OT1}{cmtt}{m}{n} rep}}}_n A$ there is a natural action of $GL_n$, the orbits of which correspond exactly to the isomorphism classes of $n$-dimensional $A$-representations. Basic GIT-stuff tells us that one can classify the [*closed*]{} orbits by points of the quotient-scheme ${{\text{\em \usefont{OT1}{cmtt}{m}{n} iss}}}_n A = {{\text{\em \usefont{OT1}{cmtt}{m}{n} rep}}}_n A / GL_n$ corresponding to the affine ring of invariants $${\mathbb{C}}[{{\text{\em \usefont{OT1}{cmtt}{m}{n} iss}}}_n A] = {\mathbb{C}}[{{\text{\em \usefont{OT1}{cmtt}{m}{n} rep}}}_n A]^{GL_n}$$ and Artin proved that the closed orbits are precisely the isoclasses of [*semi-simple*]{} representations.
Let us bring in our quotient $\mathfrak{g} _A= \frac{A}{[A,A]_{vect}}$. We can evaluate its elements on all points of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} rep}}}_n A$ by [*taking traces*]{}. That is, each $g \in \mathfrak{g}$ defines a function $${{\text{\em \usefont{OT1}{cmtt}{m}{n} rep}}}_n A \rTo {\mathbb{C}}\qquad M \mapsto tr(g)(M)$$ That is, lift $g$ to an element $a \in A$, write $a=f(a_1,\hdots,a_m)$ in terms of its generators, then if $(m_1,\hdots,m_k)$ are the matrices describing the $n$-dimensional representation $M$, then we define $$tr(g)(M) = Tr(f(m_1,\hdots,m_k))$$ where $Tr$ is the standard trace map on $M_n({\mathbb{C}})$. Observe that this does not depend on the chosen lift $a$ as all traces of elements from $[A,A]_{vect}$ vanish. Observe that via this trace-trick we can view elements of $\mathfrak{g}^*$ indeed as [*generalized characters*]{} as each representation defines a linear functional $$\chi_M~:~\mathfrak{g} \rTo {\mathbb{C}}\qquad g \mapsto tr(g)(M)$$
It is a classical result that the ring of invariants ${\mathbb{C}}[{{\text{\em \usefont{OT1}{cmtt}{m}{n} rep}}}_n A]^{GL_n}$ is generated by the invariant functions $tr(g)$ when $g$ runs over $\mathfrak{g}$. So, indeed, linear functionals on $\mathfrak{g}$ do separate $n$-dimensional semi-simple representations (whence a fortiori also simples). Actually, we only showed separation of simples for a fixed $n$, but clearly one recovers the dimension from $tr(1)$. That is, we have proved that for any affine ${\mathbb{C}}$-algebra $A$, the generalized character values give an embedding $${{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}(A) \rInto \mathfrak{g}^*_A$$ We will make the set ${{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}(A)$ into a topological space by taking as the basic opens $$\mathbb{X}(g,\lambda) = \{ S \in {{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}} A~|~\chi_S(g) \not= \lambda \}$$ for all $g \in \mathfrak{g}_A$ and all $\lambda \in {\mathbb{C}}$. For example, all simples of dimension $n$ form a closed subset. The obtained topology we will call the [*Zariski topology*]{} on ${{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}(A)$.
Our use of this topology is to prove a denseness result similar to the fact that roots of unity $\mu_{\infty}$ are Zariski dense in ${\mathbb{C}}^*$. Let $G$ be a discrete group, as every finite dimensional $\hat{G}$ representation factors over a finite group quotient of $G$ (and hence is semi-simple) we deduce that the dual coalgebra $({\mathbb{C}}G)^o$ is co-semi-simple and hence $${{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}({\mathbb{C}}\hat{G}) = ({\mathbb{C}}G)^o = corad(({\mathbb{C}}G)^o)$$ We claim that when $G$ is a discrete group satisfying condition $\bullet$, then $$\overline{{{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}({\mathbb{C}}\hat{G})} = {{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}({\mathbb{C}}G)$$ That is, the subset of simple representations of the profinite completion is Zariski dense in the noncommutative space ${{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}({\mathbb{C}}G)$. Observe that in the two examples given before, ${{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}({\mathbb{C}}\hat{G})$ is the image of the evaluation map determined by the F-un geometry, hence this result is a direct generalization of the commutative situation for the multiplicative group.
To prove this claim observe that the space of noncommutative functions $\mathfrak{g} = \mathfrak{g}_{{\mathbb{C}}G}$ has as ${\mathbb{C}}$-basis the conjugacy classes of elements of $G$. Hence, any linear functional $\chi \in \mathfrak{g}^*$ is a linear combination $$\chi = \lambda_1 \chi_1 + \hdots + \lambda_k \chi_k$$ where the $\chi_i$ are character functions corresponding to distinct conjugacy classes of $G$. Vanishing of $\chi$ on the whole of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} simp}}}({\mathbb{C}}\hat{G})$ would imply that the characters $\lambda_1,\hdots,\lambda_k$ are linearly dependent on every finite quotient $G/H$, which is impossible by the assumption on $G$.
{#section-10}
Let us recall briefly the main result of [@LBdualcoalgebra] describing the dual coalgebra $A^o$ of a general affine ${\mathbb{C}}$-algebra $A$ and indicate the geometric information contained in it. Let $Q$ be a possibly infinite quiver and ${\mathbb{C}}Q$ the vectorspace spanned on all paths in $Q$ of positive length. Then ${\mathbb{C}}Q$ is given a coalgebra structure (the [*path coalgebra*]{}) $$\Delta(p) = \sum_{p=p_1.p_2} p_1 \otimes p_2 \qquad \epsilon(p) = \delta_{p,vertex}$$ where $p_1.p_2$ is the concatenation of paths and the counit maps non-vertex paths to zero.
Starting from $A$ we will construct a huge quiver $Q_A$ having as its vertices the isoclasses of finite dimensional simple representations and with the number of arrows between them $$\#(S \rTo S' ) = dim_{{\mathbb{C}}}~Ext^1_A(S,S')$$ We will now describe a certain subcoalgebra of the path coalgebra ${\mathbb{C}}Q_A$ and as any coalgebra is the direct limit of its finite dimensional subcoalgebras we may restrict attention to a finite collection of simples and consider the semi-simple representation $M = S_1 \oplus \hdots \oplus S_k$ with restricted path-coalgebra ${\mathbb{C}}Q_A | M$. There is a natural $A_{\infty}$-algebra structure on the Yoneda Ext-algebra $Ext^{\bullet}_A(M,M)$, in particular there are higher multiplication maps $$m_i~:~\underbrace{Ext^1_A(M,M) \otimes \hdots \otimes Ext^1_A(M,M)}_i \rTo Ext^2_A(M,M)$$ defining a linear map, called the homotopy Maurer-Cartan map $$HMC_M = \oplus_i m_i~:~{\mathbb{C}}Q_A | M \rTo Ext^2_A(M,M)$$ The main result of [@LBdualcoalgebra] asserts that the dual coalgebra $A^o$ is Morita-Takeuchi equivalent to the largest subcoalgebra of ${\mathbb{C}}Q_A$ contained in the kernel of $HMC_M$ for all semi-simple representations $M$.
We will now describe the geometric content of the dual coalgebra. Recall that in the commutative case we had that the full linear dual of the dual coalgebra $({\mathbb{C}}[X]^o)^* = \prod_x \hat{{\mathcal{O}}}_{X,x}$ gave us back all the completed local rings at points of $X$. In the general case, assume as above that $M = S_1 \oplus \hdots \oplus S_k$ is a semi-simple representation with all simple factors distinct.The action of $A$ on $M$ gives rise to an epimorphism $$A \rOnto^{\pi_M} B_M=M_{n_1}({\mathbb{C}}) \oplus \hdots \oplus M_{n_k}({\mathbb{C}})$$ and let us denote $\mathfrak{m} = Ker(\pi_M)$. If $C_M$ is the maximal subcoalgebra of ${\mathbb{C}}Q_A | M$ contained in the kernel of the $HMC_M$, then we can generalize the commutative situation as follows. The $\mathfrak{m}$-adic completion of $A$ is Morita equivalent to the full linear dual of $C_M$ $$\hat{A}_{\mathfrak{m}} \sim_M (C_M)^*$$ This means that all $\mathfrak{m}$-adic completion of $A$ can be computed from the dual coalgebra $A^o$ and that each of them is a ring Morita equivalent to (the completion of) a path algebra of the quiver $(Q_A | M)^*$ modulo certain relations coming from the $A_{\infty}$-structure.
{#section-11}
Recall that a ${\mathbb{C}}$-algebra $A$ is said to be [*smooth*]{} if and only if the kernel of the multiplication map $$\Omega^1_A = Ker( A \otimes A \rTo^m A)$$ is a projective $A$-bimodule. Because $Ext^2_A(M,N)=0$ for all finite dimensional $A$-representations when $A$ is smooth, we have from the above general result that the $\mathfrak{m}$-adic completion $\hat{A}_{\mathfrak{m}}$ is Morita-equivalent to the completion of the path algebra ${\mathbb{C}}(Q_A | M)^*$ where we recall that this quiver depends only on the dimensions of the ext-groups $Ext^1_A(S_i,S_j)$.
In fact, in this case we do not have to use the full strength of the general result and deduce this fact from the formal neighborhood theorem for smooth algebras due to Cuntz and Quillen [@CuntzQuillen §6]. Note that $Ker(\pi_M)= \mathfrak{m}$ has a natural $B=B_M$-bimodule structure. In analogy with the Zariski tangent space in the commutative case, we define $$T_M = \left( \frac{\mathfrak{m}}{\mathfrak{m}^2} \right)^{*}$$ Because $B$ is a semi-simple algebra the simple $B$-bimodules are either of the form $M_{n_i}({\mathbb{C}})$ (with trivial action of the other components of $B$) or $M_{n_i \times n_j}({\mathbb{C}})$ with the component $M_{n_i}({\mathbb{C}})$ (resp. $M_{n_j}({\mathbb{C}})$) acting by left (resp. right) multiplication and all other actions being trivial. That is, there is a natural one-to-one correspondence between $${{\text{\em \usefont{OT1}{cmtt}{m}{n} bimod}}}~B \leftrightarrow {{\text{\em \usefont{OT1}{cmtt}{m}{n} quiver}}}_n$$ isoclasses of $B$-bimodules and quivers $n$ vertices (the number of simple components). Under this correspondence, $B$-bimodule duals corresponds to taking the opposite quiver. Hence, the tangent space $T_M$ can be identified with a quiver on the vertices $\{ S_1,\hdots,S_n \}$ which we will now show is the opposite quiver of $Q_A | M$.
By the formal tubular neighborhood theorem of Cuntz and Quillen [@CuntzQuillen §6] (using the fact that semi-simple algebras are formally smooth) we have an isomorphism of completed algebras between the $\mathfrak{m}$-adic completion of $A$ $$\hat{A}_{\mathfrak{m}} = \underset{\leftarrow}{lim}~A/\mathfrak{m}^n$$ where $\mathfrak{m} = Ker(\pi)$ as above, and, the completion (with respect to the natural gradation) of the tensor-algebra $T_B(\mathfrak{m}/\mathfrak{m}^2)$. That is, when we view $T_M$ as a quiver, then there is a Morita-equivalence $$\hat{A}_{\mathfrak{m}} \underset{M}{\sim} \widehat{{\mathbb{C}}T_M^{\vee}}$$ between the completion $\hat{A}_{\mathfrak{m}}$ and the completion (with respect to the gradation giving all arrows degree one) of the path-algebra ${\mathbb{C}}T_M^{\vee}$ of the opposite quiver $T_M^{\vee}$.
Under this Morita-equivalence the semi-simple $\hat{A}_{\mathfrak{m}}$-representation $M = S_1 \oplus \hdots \oplus S_n$ corresponds to the sum of the vertex-simples ${\mathbb{C}}e_1 \oplus \hdots \oplus {\mathbb{C}}e_n$, with the simple $S_i$ corresponding to the vertex-simple ${\mathbb{C}}e_i$ (the $e_i$ are the vertex-idempotents in the path algebra). Hence, also by Morita-equivalence we have an isomorphism $$Ext^1_{\hat{A}_{\mathfrak{m}}}(S_i,S_j) \simeq Ext^1_{\widehat{{\mathbb{C}}T_M^{\vee}}}({\mathbb{C}}e_i,{\mathbb{C}}e_j)$$ Finally, because all ext-information is preserved under completions, and, because we know from representation-theory that the dimension of the ext-space between two vertex-simples for any quiver ${Q}$, $dim_{{\mathbb{C}}}~Ext^1_{{\mathbb{C}}Q}({\mathbb{C}}e_i,{\mathbb{C}}e_j)$ is equal to the number of arrows starting in vertex $v_i$ and ending in vertex $v_j$, we are done!
Clearly, computing all $Ext^1_A(S,S')$ can still be a laborious task. However, it was proved in [@LBqurves] that all these dimensions follow often from a finite set of calculations when $A$ is a smooth algebra. The component semigroup ${{\text{\em \usefont{OT1}{cmtt}{m}{n} comp}}}(A)$ is the set of all connected components of the schemes ${{\text{\em \usefont{OT1}{cmtt}{m}{n} rep}}}_n~A$, for all $n \in {\mathbb{N}}$, with addition induced by the direct sum of finite dimensional representations.
The [*one quiver*]{} of $A$, ${{\text{\em \usefont{OT1}{cmtt}{m}{n} one}}}(A)$ is a full subquiver of $Q_A$ with one simple representant for every component which is a generator of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} comp}}}(A)$ (note that such generators are determined by the fact that the component consists entirely of simples). Now, if $S$ and $T$ are two finite dimensional $A$-representations belonging to the connected components $\alpha$ and $\beta$ in ${{\text{\em \usefont{OT1}{cmtt}{m}{n} comp}}}(A)$ then we can write for certain $a_i,b_i \in {\mathbb{N}}$ $$\alpha = \sum a_i g_i \quad \text{and} \quad \beta = \sum b_i g_i$$ with the $g_i$ the generator components. Then, $\epsilon=(a_i)_i$ and $\eta = (b_i)_i$ are dimension vectors for the one quiver. The main result of [@LBqurves] asserts now that $$dim_{{\mathbb{C}}}~Ext^1_A(S,T) = - \chi_{{{\text{\em \usefont{OT1}{cmtt}{m}{n} one}}}(A)}(\epsilon,\eta)$$ so that all ext-dimensions, and hence all $\mathfrak{m}$-adic completions of $A$ can be deduced from knowledge of the one quiver.
{#complexmodular}
We will now make all these calculations explicit in the case of prime interest to us, which is the modular group $\Gamma = PSL_2({\mathbb{Z}})$, that is, we will describe the dual coalgebra $({\mathbb{C}}\Gamma)^o$, at least in principle. Because $\Gamma \simeq C_2 \ast C_3$ we have that the group-algebra is the free algebra product of two semi-simple group algebras $${\mathbb{C}}\Gamma \simeq {\mathbb{C}}C_2 \ast {\mathbb{C}}C_3$$ and as such is a smooth algebra. In fact, a far more general result holds : whenever $G$ is a [*virtually free group*]{} (that is $G$ contains a free subgroup of finite index), then the group algebra ${\mathbb{C}}G$ is smooth by [@LBqurves].
If $V$ is an $n$-dimensional $\Gamma$ representation, we can decompose it into eigenspaces for the action of $C_2 = \langle u \rangle$ and $C_3 = \langle v \rangle$ (let $\rho$ denote a primitive third root of unity) : $$V_+ \oplus V_- = V_1 \oplus V_2 = V \downarrow_{C_2} = V = V \downarrow_{C_3} = W_1 \oplus W_2 \oplus W_3 = W_1 \oplus W_{\rho} \oplus W_{\rho^2}$$ If the dimension of $V_i$ is $a_i$ and that of $W_j$ is $b_j$, we say that $V$ is a $\Gamma$-representation of [*dimension vector*]{} $\alpha = (a_1,a_2;b_1,b_2,b_3)$. Choosing a basis $B_1$ of $V$ wrt. the decomposition $V_1 \oplus V_2$ and a basis $B_2$ wrt. $W_1 \oplus W_2 \oplus W_3$, we can view the basechange matrix $B_1 \rTo B_2$ as an $\alpha$-dimensional representation $V_Q$ of the quiver ${Q} = {Q}_{\Gamma}$ $${Q}_{\Gamma} = \qquad \xymatrix@=.6cm{
& & & & {*+[o][F-]{\scriptscriptstyle }} \\
{*+[o][F-]{\scriptscriptstyle }} \ar[rrrru] \ar[rrrrd] \ar[rrrrddd] & & & & \\
& & & & {*+[o][F-]{\scriptscriptstyle }} \\
{*+[o][F-]{\scriptscriptstyle }} \ar[rrrru] \ar[rrrruuu] \ar[rrrrd] & & & & \\
& & & & {*+[o][F-]{\scriptscriptstyle }} }$$ For a general quiver ${Q}$ on $k$ vertices, a weight $\theta \in {\mathbb{Z}}^k$ acts on the dimension vectors via the usual (Euclidian) scalar inproduct. A ${Q}$-representation of dimension vector $\alpha \in {\mathbb{N}}^k$ is said to be $\theta$-[*stable*]{} if and only if $\theta.\alpha=0$ and for every proper non-zero subrepresentation $W \subset V$ of dimension vector $\beta < \alpha$ we have that $\theta.\beta > 0$.
Bruce Westbury [@Westbury] has shown that $V$ is an irreducible $\Gamma$-representation if and only if $V_Q$ is a [*$\theta$-stable*]{} $Q$-representation where $\theta = (-1,-1;1,1,1)$ and that the two notions of isomorphism coincide. The [*Euler-form*]{} $\chi_{{Q}}$ of the quiver $Q$ is the bilinear form on $\mathbb{Z}^{\oplus 5}$ determined by the matrix $$\chi_{{Q}} = \begin{bmatrix} 1 & 0 & -1 & -1 & -1 \\ 0 & 1 & -1 & -1 & -1 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix}$$ Westbury also showed that if there exists a $\theta$-stable $\alpha$-dimensional $Q$-representation, then there is an $1 - \chi_{{Q}}(\alpha,\alpha)$ dimensional family of isomorphism classes of such representations (and a Zariski open subset of them will correspond to isomorphism classes of irreducible $\Gamma$-representations).
We will describe the one quiver ${{\text{\em \usefont{OT1}{cmtt}{m}{n} one}}}({\mathbb{C}}\Gamma)$. By the above it follows that both the component semigroup ${{\text{\em \usefont{OT1}{cmtt}{m}{n} comp}}}~{\mathbb{C}}\Gamma$ and the semigroup of ${\mathbb{Z}}^5$ generated by all $\theta$-stable ${Q}$-representations are generated by the following six connected components, belonging to the dimension vectors $$g_{ij} = (\delta_{1i},\delta_{2i},\delta_{3i};\delta_{1j},\delta_{2j})$$ and if we order and relabel these generators as $$a=g_{11}, b = g_{22}, c=g_{31}, d=g_{12}, e=g_{21}, f=g_{32}$$ we can compute from the Euler-form of ${Q}$ that the one-quiver of the modular group algebra is the following hexagonal graph $${{\text{\em \usefont{OT1}{cmtt}{m}{n} one}}}({\mathbb{C}}\Gamma) = \qquad
\xymatrix@=.6cm{
& {*+[o][F-]{\scriptscriptstyle a}} \ar@/^/[rd] \ar@/^/[ld] & \\
{*+[o][F-]{\scriptscriptstyle f}} \ar@/^/[ru] \ar@/^/[dd] & & {*+[o][F-]{\scriptscriptstyle b}} \ar@/^/[lu] \ar@/^/[dd] \\
& & \\
{*+[o][F-]{\scriptscriptstyle e}} \ar@/^/[uu] \ar@/^/[rd] & & {*+[o][F-]{\scriptscriptstyle c}} \ar@/^/[uu] \ar@/^/[ld] \\
& {*+[o][F-]{\scriptscriptstyle d}} \ar@/^/[lu] \ar@/^/[ru] &}$$ which is the origin of a lot of [*hexagonal moonshine*]{} in the representation theory of the modular group. In particular it follows from symmetry of the one quiver that the quiver $Q_{{\mathbb{C}}\Gamma}$ is also symmetric!
{#section-12}
Recall that an affine ${\mathbb{C}}$-algebra $A$ is said to be 2-Calabi-Yau if $gldim(A)=2$ and for any pair $S,T$ of finite dimensional $A$-representations, there exists a natural duality $$Ext^i_A(S,T) \simeq (Ext^{2-i}_A(T,S))^*$$ satisfying an additional sign condition. Raf Bocklandt [@Bocklandt] succeeded in extending the results on smooth algebras recalled before to the setting of 2-Calabi-Yau algebras. From the duality condition it is immediate that the quiver $Q_A$ is symmetric, that is, for every arrow $S \rTo^a T$ there is a paired arrow in the other direction $T \rTo^{a^*} S$. Bocklandt’s result asserts that the $\mathfrak{m}$-adic completion $\hat{A}_{\mathfrak{m}}$ with $\mathfrak{m} = Ker(\pi_M)$ is Morita equivalent to the completion of the path algebra of the (dual) quiver $Q_A | M$ modulo the preprojective relation $$\sum_a [a,a^*] = 0$$ Further, he extends the idea of the one quiver to the 2-Calabi-Yau setting, allowing to compute the quiver $Q_A$ often from a finite set of calculations, using earlier results due to Crawley-Boevey [@CB].
The group algebra ${\mathbb{C}}\pi_1(C)$ of the fundamental group of a genus $g$ Riemann surface is 2-Calabi-Yau by a result of Maxim Kontsevich. In [@Bocklandt §7.1] it is shown that the one-quiver of ${\mathbb{C}}\pi_1(C)$ consists of one vertex, corresponding to any one-dimensional simple representation, and $2g$ loops. From this and the results by Crawley-Boevey it follows that when $M=S_1^{\oplus e_1} \oplus \hdots \oplus S_k^{\oplus e_k}$ is a semi-simple ${\mathbb{C}}\pi_1(C)$-representation wit the simple factor $S_i$ having dimension $n_i$, then ${\mathbb{C}}Q_{{\mathbb{C}}\pi_1(C)} | M$ consists of $k$ vertices (corresponding to the distinct simple components $S_i$), such that the $i$-th vertex has exactly $2(g-1)n_i^2+2$ loops and there are exactly $2n_in_j(g-1)$ directed arrows from vertex $i$ to vertex $j$.
This information allows us then to compute all $\mathfrak{m}$-adic completions of ${\mathbb{C}}\pi_1(C)$ as Morita equivalent to the completion of the path algebra of this quiver modulo the preprojective relation.
{#section-13}
In \[complexmodular\] we described the structure of the path-coalgebra ${\mathbb{C}}Q_{{\mathbb{C}}\Gamma}$ which is Morita-Takeuchi equivalent to the dual complex coalgebra $({\mathbb{C}}\Gamma)^o$. Describing the integral dual coalgebra $({\mathbb{Z}}\Gamma)^o$ is a lot more complicated and will involve a good deal of knowledge of the integral (and modular) representation theory of the modular group.
Observe that the calculations in \[complexmodular\] are valid for every algebraically closed field, so we might as well describe the coalgebra $(\overline{{\mathbb{Q}}} \Gamma)^o$ and study the action of the absolute Galois group $Gal(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ on it, giving us an handle on the rational dual coalgebra $$({\mathbb{Q}}\Gamma)^o = ((\overline{{\mathbb{Q}}} \Gamma)^o)^{Gal(\overline{{\mathbb{Q}}}/{\mathbb{Q}})}$$ which brings us closer to $({\mathbb{Z}}\Gamma)^o$. But, as in the case of the multiplicative group in the previous section, we do not require the full structure of this dual coalgebra but rather the image of the F-un data in it. As observed before, the $\mathbb{F}_1$-representation theory of $\Gamma$ is equivalent to the study of all finite dimensional transitive permutation representations of $\Gamma$ and hence to conjugacy classes of finite index subgroups of $\Gamma$.
We will recall the combinatorial description of those, due to R. Kulkarni in [@Kulkarni] in terms of generalized Farey symbols. Starting from this symbol we then describe how to associate a dessin, its monodromy group an finally to derive from it the modular content, that is the noncommutative gadget describing all $\Gamma$-representations deforming to the given permutation representation. In the next subsection we will give some interesting examples.
A [*generalized Farey sequence*]{} is an expression of the form $$\{ \infty=x_{-1},x_0,x_1,\hdots,x_n,x_{n+1}=\infty \}$$ where $x_0$ and $x_n$ are integers and some $x_i=0$. Moreover, all $x_i= \frac{a_i}{b_i}$ are rational numbers in reduced form and ordered such that $$| a_i b_{i+1} - b_i a_{i+1} | = 1 \qquad \text{for all $1 \leq i < n$}$$ The terminology is motivated by the fact that the classical [*Farey sequence*]{} $F(n)$, that is the ordered sequence of all rational numbers $0 \leq \frac{a}{b} \leq 1$ in reduced form with $b \leq n$, has this remarkable property.
A [*Farey symbol*]{} is a generalized Farey sequence $\{ \infty=x_{-1},x_0,x_1,\hdots,x_n,x_{n+1}=\infty \}$ such that for all $-1 \leq i \leq n$ we add one of the following symbols to two consecutive terms $$\xymatrix{x_i \ar@{-}[r]_{\bullet} & x_{i+1}} \quad \text{or} \quad \xymatrix{x_i \ar@{-}[r]_{\circ} & x_{i+1}} \quad \text{or} \quad \xymatrix{x_i \ar@{-}[r]_{k} & x_{i+1}}$$ where each of the occurring integers $k$ occur in pairs.
To connect Farey symbols with cofinite subgroups of the modular group $\Gamma$ we need to recall the [*Dedekind tessellation*]{} of the upper-half plane $\mathbb{H}$. Recall that the [*extended modular group*]{} $\Gamma^* = PGL_2(\mathbb{Z})$ acts on $\mathbb{H}$ via the natural action of $\Gamma$ on it together with the extra symmetry $z \mapsto - \overline{z}$. The Dedekind tessellation is the tessellation by fundamental domains for the action of $\Gamma^*$ on $\mathbb{H}$. It splits every fundamental domain for $\Gamma$ in two hyperbolic triangles, usually depicted as a black and a white one. Here is a depiction of the upper part of the Dedekind tessellation
[-1]{}[2]{}[0]{}[1.5]{}
Here, every red edge is a $\Gamma$-translate of the edge $[i,\infty]$, a blue edge a $\Gamma$-translate of $[\rho,\infty]$ where $\rho$ is a primitive sixth root of unity and every black edge is a $\Gamma$-translate of the circular arc $[i,\rho]$. Observe that every hyperbolic triangle of this tessellation has one edge of all three colors. Moving counterclockwise along the border of a triangle we either have the ordering red-blue-black (in which case we call this triangle a [*white*]{} triangle) or blue-red-black (and then we call it a [*black*]{} triangle). Any pair of a white and black triangle make a fundamental domain for the action of $\Gamma$.
Observe that any hyperbolic geodesic connecting two consecutive terms of a generalized Farey sequence consists of two red edges (connected at an intersection with black edges. We call these intersection points [*even points*]{} (later in the theory of dessins they will be denoted by a $\bullet$). A point where three blue edges come together with three black edges will be called an [*odd point*]{} (later denoted by $\xymatrix{{*+[o][F-]{\scriptscriptstyle }}}$).
A generalized Farey sequence therefore determines a hyperbolic polygonal region of $\mathbb{H}$ bounded by the (red) full geodesics connecting consecutive terms. The extra information contained in a Farey symbol tell us how to identify sides of this polygon (as well as how to extend it slightly in case of $\bullet$-connections) as follows :
- [For $\xymatrix{x_i \ar@{-}[r]_{\circ} & x_{i+1}}$ the two red edges making up the geodesic connecting $x_i$ with $x_{i+1}$ are identified.]{}
- [For $\xymatrix{x_i \ar@{-}[r]_{k} & x_{i+1}}$ (with paired $\xymatrix{x_j \ar@{-}[r]_{k} & x_{j+1}}$) these two full geodesics (each consisting of two red edges) are identified.]{}
- [For $\xymatrix{x_i \ar@{-}[r]_{\bullet} & x_{i+1}}$ we extend the boundary of the polygon by adding the two triangles just outside the full geodesic and identify the two blue edges forming the adjusted boundary.]{}
In this way, we associate to a Farey symbol a compact surface. Next, we will construct a [*cuboid tree diagram*]{} out of it, that is, a tree embedded in $\mathbb{H}$ such that all internal vertices are $3$-valent. Take as the vertices all odd-points lying in the interior of the polygonal region together with together with all even (red) and odd (blue) points on the boundary. We connect these vertices with the black lines in the interior of the polygonal region and add an involution on the red leaf-vertices determined by the side-pairing information contained in the Farey-symbol.
Finally, we will also associate to it a [*bipartite cuboid graph*]{} (aka a ’dessin d’enfants’). Start with the cuboid tree diagram and divide all edges in two (that is, add also the even internal points connecting the two black edges making up an edge in the tree diagram) and connect two red leaf-vertices when they correspond to each other under the involution.
For example, consider the Farey symbol $$\xymatrix{\infty \ar@{-}[r]_{1} & 0 \ar@{-}[r]_{\bullet} & \frac{1}{3} \ar@{-}[r]_{\bullet} & \frac{1}{2} \ar@{-}[r]_{\bullet} & 1 \ar@{-}[r]_{1} & \infty}$$ The boundary of the polygonal region determined by the symbol is indicated by the slightly thicker red and blue edges. The vertices of the cuboid tree are the red, blue and black points and the edges are the slightly thicker black edges.
[-1]{}[2]{}[0]{}[1.5]{}
(.5,-.1)[$\frac{1}{2}$]{} (.333,-.1)[$\frac{1}{3}$]{} (.666,-.1)[$\frac{2}{3}$]{} (0,-.1)[$0$]{} (1,-.1)[$1$]{}
Because the two red leaf-vertices correspond to each other under the involution, the corresponding bipartite cuboid diagram (or modular dessin) is $$\xymatrix@=1cm{
& {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 1}}|{\bullet}|(.75){{\bf 2}}}@(ur,ul) { \ar@{-}|(.25){{\bf 3}}|{\bullet}|(.75){{\bf 4}}}[d] & \\
& {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 5}}|{\bullet}|(.75){{\bf 6}}}[r] { \ar@{-}|(.25){{\bf 7}}|{\bullet}|(.75){{\bf 8}}}[d] & {*+[o][F-]{\scriptscriptstyle }} \\
& {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 9}}|{\bullet}|(.75){{\bf 10}}}[ld] { \ar@{-}|(.25){{\bf 11}}|{\bullet}|(.75){{\bf 12}}}[rd] & \\
{*+[o][F-]{\scriptscriptstyle }} & & {*+[o][F-]{\scriptscriptstyle }} }$$
Such a dessin encodes the data of a Belyi covering $C \rOnto \mathbb{P}^1_{{\mathbb{C}}}$ ramified only in the points $\{ 0,1,\infty \}$. The inverse images of $0$ will be represented by a $\xymatrix{{*+[o][F-]{\scriptscriptstyle }}}$-vertex, those of $1$ by a $\bullet$-vertex. Of relevance for us are dessins which are [*modular quilts*]{} meaning that every $\bullet$-vertex is $2$-valent and every $\xymatrix{{*+[o][F-]{\scriptscriptstyle }}}$-vertex is $1$- or $3$-valent.
Given a modular dessin, denote each of the edges by a different number between $1$ and $d$ (the degree of $\pi$), then the [*monodromy group*]{} $G_{\pi}$ of $\pi$ is the subgroup of $S_d$ generated by the order three element $\sigma_0$ obtained by cycling round every $\xymatrix{{*+[o][F-]{\scriptscriptstyle }}}$ -vertex counterclockwise and the order two element $\sigma_1$ obtained by recording the two edges ending at every $\bullet$-vertex. This defines an exact sequence of groups $$1 \rTo G \rTo \Gamma \rTo G_{\pi} \rTo 1$$ and the projective curve $C$ corresponding to the modular dessin can be identified with a compactification of $\mathbb{H} / G$ where $\mathbb{H}$ is the upper half-plane on which $G \subset \Gamma$ acts via Möbius transformations.
The $d$-dimensional permutation representation $M = \Gamma/G$ decomposes into irreducible representations for the monodromy group $G_{\pi}$, say $$M = X_1^{\oplus e_1} \oplus \hdots \oplus X_k^{\oplus e_k}$$ with every $X_i$ an irreducible $G_{\pi}$ and hence also irreducible $\Gamma$-representation. The [*modular content*]{} of the dessin, or of the permutation representation, is the quiver on $k$ vertices $Q_{\pi} = Q_{{\mathbb{C}}\Gamma} | M$ together with the dimension vector $\alpha_{\pi} = (e_1,\hdots,e_k)$ determined by the multiplicities of the simples in the permutation representation.
Roughly speaking, the modular content $(Q_{\pi},\alpha_{\pi})$ encodes how much the curve $C$, the dessin or the permutation representation ’sees’ of the modular group. That is, the quotient variety ${{\text{\em \usefont{OT1}{cmtt}{m}{n} iss}}}_{\alpha_{\pi}} Q_{\pi} = {{\text{\em \usefont{OT1}{cmtt}{m}{n} rep}}}_{\alpha_{\pi}} Q_{\pi} / GL(\alpha_{\pi})$ classifies all semi-simple $d$-dimensional $\Gamma$-representations deforming to the permutation representation $M$. As such, it is a new noncommutative gadget associated to a classical object, the curve $C$. It would be interesting to know whether the modular content is a Galois invariant of the dessin, or more generally, what subsidiary information derived from it is a Galois invariant.
We now give an algorithm to compute the modular content, using the group-theory program GAP, starting from the modular quilt $D$.
1. [Determine the permutations $\sigma_0,\sigma_1 \in S_d$ described above, that is obtained by walking around the $\bullet$-vertices (for $\sigma_1$) and the $\xymatrix{{*+[o][F-]{\scriptscriptstyle }}}$-vertices (for $\sigma_0$) in $D$ and feed them to GAP as `s0,s1`.]{}
2. [Calculate the monodromy group $G_{\pi}$ via `G:=Group(s0,s1)` and determine its character table via `chars:=CharacterTable(G);)`]{}
3. [Determine the $G_{\pi}$-character of the permutation representation by calling `ConjugacyClasses(G)`. This returns a list of $S_d$-permutations representing the conjugacy classes of $G_{\pi}$. To determine the character-value we only need to count the numbers missing in the cycle decomposition of the permutation. Let $\chi$ be the obtained character which is the list `chi`.]{}
4. [Determine the irreducible components of $\chi$ and their multiplicities via `MatScalarProducts(chars,Irr(chars),[chi]);`. The non-zero entries form the dimension vector $\alpha_{\pi}$ and they determine the simple factors $X_1,\hdots,X_k$.]{}
5. [Determine the conjugacy classes of $\sigma_0$ and $\sigma_1$. For example, the number of the conjugacy class in the character table is found by `FusionConjugacyClasses(Group(s0),G);`. Alternatively, one can use `IsConjugate(G,s0,s);` for a suitable element representant obtained via `ConjugacyClasses(G);`. Assume $\sigma_0$ (resp. $\sigma_1$) belongs to the $a$-th (resp. $b$-th) conjugacy class.]{}
6. [From the character values of $X_i$ in the $a$-th and $b$-th column of `Display(chars);` one deduces the dimension vector $\alpha_i=(a_1(i),a_2(i);b_1(i),b_2(i),b_3(i))$ of the ${Q}_{\Gamma}$-representation corresponding to $X_i$.]{}
7. [Finally, the number of arrows (and loops) in the quiver ${Q}_{\pi}$ between the vertices corresponding to $X_i$ and $X_j$ is given by $\delta_{ij}-\chi_{{Q}_{\Gamma}}(\alpha_i,\alpha_j)$.]{}
{#section-14}
As the modular content encodes all possible $\Gamma$-representation deformations of the permutation representation, it is often a huge object which makes it difficult to extract interesting deformations from it. Sometimes though, a true gem reveals itself. In the previous subsection we used the generalized Farey-symbol $$\xymatrix{\infty \ar@{-}[r]_{1} & 0 \ar@{-}[r]_{\bullet} & \frac{1}{3} \ar@{-}[r]_{\bullet} & \frac{1}{2} \ar@{-}[r]_{\bullet} & 1 \ar@{-}[r]_{1} & \infty}$$ Note that it consists of half of the Farey-sequence $F(3)$ (those $\leq \frac{1}{2}$). Generalizing this construction for all classical Farey sequences leads to an intriguing class of examples. The $n$-th [*Iguanodon Farey-symbol*]{} is the Farey symbol $$\xymatrix{\infty \ar@{-}[r]_{1} & 0 \ar@{-}[r]_{\bullet} & \frac{1}{n} \ar@{-}[r]_{\bullet} & \hdots & \frac{1}{2} \ar@{-}[l]^{\bullet} \ar@{-}[r]_{\bullet} & 1 \ar@{-}[r]_{1} & \infty}$$ where the rational numbers occurring are precisely those Farey numbers in $F(n)$ smaller or equal to $\frac{1}{2}$.
The terminology is explained by depicting the first few bipartite cuboid diagrams associated to Farey sequences [$$\xymatrix{
& & & & & & & & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 3}}|{\bullet}|(.75){{\bf 4}}}[d] { \ar@{-}|(.25){{\bf 1}}|{\bullet}|(.75){{\bf 2}}}@(ul,r) \\
{*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 50}}|{\bullet}|(.75){{\bf 49}}}[r] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 51}}|{\bullet}|(.75){{\bf 52}}}[d] { \ar@{-}|(.25){{\bf 42}}|{\bullet}|(.75){{\bf 41}}}[r] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 43}}|{\bullet}|(.75){{\bf 44}}}[d] { \ar@{-}|(.25){{\bf 30}}|{\bullet}|(.75){{\bf 29}}}[r] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 26}}|{\bullet}|(.75){{\bf 25}}}[r] { \ar@{-}|(.25){{\bf 31}}|{\bullet}|(.75){{\bf 32}}}[d] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 18}}|{\bullet}|(.75){{\bf 17}}}[r] { \ar@{-}|(.25){{\bf 27}}|{\bullet}|(.75){{\bf 28}}}[d] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 16}}|{\bullet}|(.75){{\bf 15}}}[r] { \ar@{-}|(.25){{\bf 19}}|{\bullet}|(.75){{\bf 20}}}[d] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 12}}|{\bullet}|(.75){{\bf 11}}}[r] { \ar@{-}|(.25){{\bf 13}}|{\bullet}|(.75){{\bf 14}}}[d] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 8}}|{\bullet}|(.75){{\bf 7}}}[r] { \ar@{-}|(.25){{\bf 9}}|{\bullet}|(.75){{\bf 10}}}[d] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 5}}|{\bullet}|(.75){{\bf 6}}}[d] \\
& {*+[o][F-]{\scriptscriptstyle }} & {*+[o][F-]{\scriptscriptstyle }} & {*+[o][F-]{\scriptscriptstyle }} & {*+[o][F-]{\scriptscriptstyle }} & {*+[o][F-]{\scriptscriptstyle }} & {*+[o][F-]{\scriptscriptstyle }} & {*+[o][F-]{\scriptscriptstyle }} & {*+[o][F-]{\scriptscriptstyle }} \\
& & & & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 54}}|{\bullet}|(.75){{\bf 53}}}[ru] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 34}}|{\bullet}|(.75){{\bf 33}}}[ru] & & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 22}}|{\bullet}|(.75){{\bf 21}}}[u] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 24}}|{\bullet}|(.75){{\bf 23}}}[lu] \\
& & & & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 56}}|{\bullet}|(.75){{\bf 55}}}[ruu] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 36}}|{\bullet}|(.75){{\bf 35}}}[ruu] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 46}}|{\bullet}|(.75){{\bf 45}}}[ru] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 38}}|{\bullet}|(.75){{\bf 37}}}[ru] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 40}}|{\bullet}|(.75){{\bf 39}}}[u] \\
& & & & & & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 48}}|{\bullet}|(.75){{\bf 47}}}[ruu] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 58}}|{\bullet}|(.75){{\bf 57}}}[ru] & {*+[o][F-]{\scriptscriptstyle }} { \ar@{-}|(.25){{\bf 60}}|{\bullet}|(.75){{\bf 59}}}[u]}$$ ]{} Here, the diagram corresponding to Farey sequence $F(n)$ is the full subfigure on the first $m(n)$ (half)edges $$\begin{array}{c|cccccccc}
n & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
m(n) & 8 & 12 & 16 & 24 & 28 & 40 & 48 & 60
\end{array}$$ The monodromy groups corresponding to the $n$-th Iguanodon symbol are $$\begin{array}{c|cccccccc}
n & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
& L_2(7) & M_{12} & A_{16} & M_{24} & A_{28} & A_{40} & A_{48} & A_{60} \\
\hline \\ \hline
n & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\
\hline
& A_{68} & A_{88}& A_{96} & A_{120} & A_{132} & A_{148} & A_{164} & A_{196}
\end{array}$$ This can be verified by hand (and GAP) using the above picture for $n \leq 9$ and by using the SAGE-package `kfarey.sage` for higher $n$. It is plausible that the monodromy groups of the Iguanodon symbols are all simple groups and it is quite remarkable that the Mathieu groups $M_{12}$ and $M_{24}$ appear in this sequence of alternating groups.
Now, let us compute the modular content of these permutation representations. The action of the monodromy group is clearly 2-transitive implying that as a ${\mathbb{C}}G_{\pi}$-representation, the permutation representation splits into two irreducibles, one of which being clearly the trivial representation. Note also that the character of the generator of order $2$ is equal to zero as there are no $\bullet$-end points. Further, $\circ$-endpoints appear in pairs and add another 4 half-edges, that is 4 dimensions, to the permutation space. By induction we see that the dimension of the permutation representation is always of the form $4n$ with $\chi(\sigma_1)=0$ and $\chi(\sigma_0)=n$.
By the argument recalled in \[complexmodular\] it follows that the dimension vector of the $Q_{\Gamma}$-quiver representation corresponding to the permutation representation is $$\alpha_{4n}= \qquad \xymatrix@=.5cm{
& & & & {*+[o][F-]{\scriptscriptstyle 2n}} \\
{*+[o][F-]{\scriptscriptstyle 2n}} \ar[rrrru] \ar[rrrrd] \ar[rrrrddd] & & & & \\
& & & & {*+[o][F-]{\scriptscriptstyle n}} \\
{*+[o][F-]{\scriptscriptstyle 2n}} \ar[rrrru] \ar[rrrruuu] \ar[rrrrd] & & & & \\
& & & & {*+[o][F-]{\scriptscriptstyle n}} }$$ By 2-transitivity the dimension vectors of the two simple components $S$ and $T$ are $$\alpha_T=\xymatrix@=.4cm{
& & & & {*+[o][F-]{\scriptscriptstyle 1}} \\
{*+[o][F-]{\scriptscriptstyle 1}} \ar[rrrru] \ar[rrrrd] \ar[rrrrddd] & & & & \\
& & & & {*+[o][F-]{\scriptscriptstyle 0}} \\
{*+[o][F-]{\scriptscriptstyle 0}} \ar[rrrru] \ar[rrrruuu] \ar[rrrrd] & & & & \\
& & & & {*+[o][F-]{\scriptscriptstyle 0}} } \qquad \text{and} \qquad
\alpha_S= \xymatrix@=.4cm{
& & & & {*+[o][F-]{\scriptscriptstyle \overset{2n}{-1}}} \\
{*+[o][F-]{\scriptscriptstyle \overset{2n}{-1}}} \ar[rrrru] \ar[rrrrd] \ar[rrrrddd] & & & & \\
& & & & {*+[o][F-]{\scriptscriptstyle n}} \\
{*+[o][F-]{\scriptscriptstyle 2n}} \ar[rrrru] \ar[rrrruuu] \ar[rrrrd] & & & & \\
& & & & {*+[o][F-]{\scriptscriptstyle n}} }$$ But then, by the algorithm we have that the modular content $(Q_{\pi},\alpha_{\pi})$ of the permutation representation can be depicted as
$$\xymatrix{{*+[o][F-]{\scriptscriptstyle 1}} \ar@/^/[rrr] & & & {*+[o][F-]{\scriptscriptstyle 1}} \ar@/^/[lll] \ar@{=>}@(ur,dr)^{n^2}}$$
The $n^2$ loops in the vertex corresponding to the simple factor $S$ indicate that the moduli space of semi-stable $Q_{\Gamma}$-representations $M_{\theta}^{ss}(Q_{\Gamma},\alpha_S)$ is $n^2$-dimensional and as $S$ is a smooth point in it, there is an $n^2$-dimensional family of simple $\Gamma$-representations in the neighborhood of $S$. More interesting is the fact that there is just one arrow in each direction between the two vertices.
This implies that the permutation representation is a smooth point in the moduli space of semi-simple $\Gamma$-representations, a rare fact for higher dimensional decomposable representations (see the paper [@LBBocklandtSymens] for more details on singularities of quiver-representations). Further, this implies that there is a unique (!) curve of simple $4n$-dimensional $\Gamma$-representations degenerating to the given permutation representation! Certainly in the case of the sporadic Mathieu groups it would be interesting to study these curves (and their closures in the moduli space $M^{ss}_{\theta}(Q_{\Gamma},\alpha_{4n})$) in more detail.
[10]{}
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|
---
abstract: |
We prove a central limit theorem under diffusive scaling for the displacement of a random walk on $\Z^d$ in stationary and ergodic doubly stochastic random environment, under the ${\cH_{-1}}$-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result [@komorowski_landim_olla_12], where it is assumed that the stream tensor is in $\cL^{\max\{2+\delta, d\}}$, with $\delta>0$. Our proof relies on an extension of the *relaxed sector condition* of [@horvath_toth_veto_12], and is technically rather simpler than existing earlier proofs of similar results by Oelschläger [@oelschlager_88] and Komorowski, Landim and Olla [@komorowski_landim_olla_12].
[MSC2010: 60F05, 60G99, 60K37]{}
[Key words and phrases:]{} random walk in random environment, central limit theorem, Kipnis-Varadhan theory, sector condition.
author:
- |
[Gady Kozma$^{1}$]{}\
[$^1$ Weizmann Institute, Rehovot, IL]{}\
[$^2$ School of Mathematics, University of Bristol, UK]{}\
[$^3$ Rényi Institute, Budapest, HU]{}
title: 'Central Limit Theorem for Random Walks in Doubly Stochastic Random Environment: ${\cH_{-1}}$ Suffices'
---
by 3pt
Introduction: setup and main result {#s:Introduction: setup and main result}
===================================
Since its appearance in the probability and physics literature in the mid-seventies the general topics of *random walks/diffusions in random environment* became the most complex and robust area of research. For a general overview of the subject and its historical development we refer the reader to the surveys Kozlov [@kozlov_85], Zeitouni [@zeitouni_01], Biskup [@biskup_11] or Kumagai [@kumagai_14], written at various stages of this rich story. The main problem considered in our paper is that of diffusive limit in the doubly stochastic (and hence, a priori stationary) case.
The random walk and the H -1-condition {#ss:The random walk}
--------------------------------------
Let $(\Omega, \cF, \pi, \tau_z:z\in\Z^d)$ be a probability space with an ergodic $\Z^d$-action. Denote by $\cE_{+}:=\{e_1,\dots,e_d:
e_i\in\Z^{d}, \ \ e_i\cdot e_j=\delta_{i,j}\}$ the standard generating basis in $\Z^d$ and let $\cE:=\{\pm e_j: e_j\in \cE_{+}\} =
\{k\in\Z^d: |k|=1\}$ be the set of possible steps of a nearest-neighbour walk on $\Z^d$. Assume that bounded measurable functions $p_k:\Omega\to[0,s^*]$, $k\in\cE$ are given ($s^*$ denotes the common bound), and assume the $p_k$ satisfy *bistochasticity*, by which we mean the following property $$\begin{aligned}
\label{bistoch}
\sum_{k\in\cE}p_k(\omega)
=
\sum_{k\in\cE}p_{-k}(\tau_k\omega).\end{aligned}$$ Lift these functions to the lattice $\Z^d$ by defining $$\begin{aligned}
\label{jump_probab_field}
P_k(x)=P_k(\omega,x):=p_k(\tau_x\omega).\end{aligned}$$ Given these, define the continuous time nearest neighbour random walk $X(t)$ as a continuous time Markov chain on $\Z^d$, with $X(0)=0$ and conditional jump rates $$\begin{aligned}
\label{the walk}
{\ensuremath{\mathbf{P}_{\omega}\left(X(t+dt)= x+k\bigm|X(t)=x\right)}} = P_k(\omega, x) dt + \ordo(dt),\end{aligned}$$ where the subscript $\omega$ denotes that the random walk $X(t)$ is a continuous time Markov chain on $\Z^d$ *conditionally*, with $\omega\in\Omega$ sampled according to $\pi$. Note that is equivalent to $$\begin{aligned}
\notag
\sum_{k\in\cE}P_k(\omega,x)= \sum_{k\in\cE}P_{-k}(\omega,x+k),\end{aligned}$$ which is exactly bistochasticity of the random walk defined in above. Since the $p_k$ are bounded, so will be the total jump rate of the walk $$\begin{aligned}
\notag
p(\omega):=\sum_{k\in\cE}p_k(\omega)\le 2d s^*.\end{aligned}$$ Thus, there is no difference between the long time asymptotics of this walk and the discrete time (possibly lazy) walk $n\mapsto X_n\in\Z^d$ with jump probabilities $$\begin{aligned}
\notag
{\ensuremath{\mathbf{P}_{\omega}\left(X_{n+1}= y\bigm|X_n=x\right)}} =
\begin{cases}
(2d s^*)^{-1} P_k(\omega, x)
&\text{if }y-x=k\in\cE,
\\[2pt]
1-(2d s^*)^{-1} \sum_{l\in\cE}P_l(\omega, x)
&\text{if }y-x=0,
\\[2pt]
0
&\text{if }y-x\not\in \cE \cup\{0\}.
\end{cases}\end{aligned}$$ We speak about continuous time walk only for reasons of convenience, in order to easily quote facts and results form Kipnis-Varadhan theory of CLT for additive functionals of Markov processes, without tedious reformulations.
We formulate our problem and prove our main result in the context of nearest neighbour walks. This is only for convenience reason. The main result of this paper holds true for finite range bistochastic RWREs under the appropriate conditions. For more details on this see the remark after Theorem 1, further down in the paper.
We will use the notation ${\ensuremath{\mathbf{P}_{\omega}\left(\cdot\right)}}$, ${\ensuremath{\mathbf{E}_{\omega}\left(\cdot\right)}}$ and ${\ensuremath{\mathbf{Var}_{\omega}\left(\cdot\right)}}$ for *quenched* probability, expectation and variance. That is: probability, expectation, and variance with respect to the distribution of the random walk $X(t)$, *conditionally, with given fixed environment $\omega$*. The notation ${\ensuremath{\mathbf{P}\left(\cdot\right)}}:=\int_\Omega{\ensuremath{\mathbf{P}_{\omega}\left(\cdot\right)}} {{\mathrm d}}\pi(\omega)$, ${\ensuremath{\mathbf{E}\left(\cdot\right)}}:=\int_\Omega{\ensuremath{\mathbf{E}_{\omega}\left(\cdot\right)}} {{\mathrm d}}\pi(\omega)$ and ${\ensuremath{\mathbf{Var}\left(\cdot\right)}}:=\int_\Omega{\ensuremath{\mathbf{Var}_{\omega}\left(\cdot\right)}} {{\mathrm d}}\pi(\omega) + \int_\Omega{\ensuremath{\mathbf{E}_{\omega}\left(\cdot\right)}}^2 {{\mathrm d}}\pi(\omega) - {\ensuremath{\mathbf{E}\left(\cdot\right)}}^2$ will be reserved for *annealed* probability, expectation and variance. That is: probability, expectation and variance with respect to the random walk trajectory $X(t)$ *and* the environment $\omega$, sampled according to the distribution $\pi$.
It is well known (and easy to check, see e.g. [@kozlov_85]) that due to double stochasticity the annealed set-up is stationary and ergodic in time: the process of *the environment as seen from the position of the random walker* (to be formally defined soon) is stationary and ergodic in time under the probability measure $\pi$ and consequently the random walk $t\mapsto X(t)$ will have stationary and ergodic annealed increments.
Next we define, for $k\in\cE$, $v_k:\Omega\to[-s^*,s^*]$, $s_k:\Omega\to[0,s^*]$, and $\psi,\varphi:\Omega\to\R^d$, $$\begin{aligned}
\label{v and phi}
&
v_k(\omega):=\frac{p_k(\omega)-p_{-k}(\tau_k\omega)}{2},
&&
\varphi(\omega):= \sum_{k\in\cE} k v_k(\omega),
\\
\label{s and psi}
&
s_k(\omega):=\frac{p_k(\omega)+p_{-k}(\tau_k\omega)}{2},
&&
\psi(\omega):= \sum_{k\in\cE} k s_k(\omega).\end{aligned}$$ Their corresponding lifting to $\Z^d$ are $$\begin{aligned}
\label{vlifting}
&
V_k(x)=V_k(\omega,x):=v_k(\tau_x\omega),
&&
\Phi(x)=\Phi(\omega,x):=\varphi(\tau_x\omega),
\\[8pt]
\notag
&
S_k(x)=S_k(\omega,x):=s_k(\tau_x\omega),
&&
\Psi(x)=\Psi(\omega,x):=\psi(\tau_x\omega).\end{aligned}$$ Note that $$\begin{aligned}
\notag
-s^*\le v_k(\omega)\le s^*,
\qquad
0\le s_k(\omega)\le s^*,
\qquad
{\ensuremath\left|{\varphi(\omega)}\right|}\le 2\sqrt{d} s^*,
\qquad
{\ensuremath\left|{\psi(\omega)}\right|}\le \sqrt{d} s^*,
\qquad
\text{a.s.}\end{aligned}$$ The local *quenched* drift of the random walk is $$\begin{aligned}
\notag
{\ensuremath{\mathbf{E}_{\omega}\left(dX(t)\bigm|X(s): 0\le s\le t\right)}} = \left(\Psi(\omega,X(t))+\Phi(\omega,X(t))\right) dt +\ordo(dt).\end{aligned}$$ Note that from and the definitions , it follows that for $\pi$-almost all $\omega\in\Omega$ $$\begin{aligned}
\label{divfree}
v_k(\omega)&=-v_{-k}(\tau_k\omega),&
\sum_{k\in\cE}v_k(\omega)
&=0,
\\
\label{ssymm}
s_k(\omega)&=s_{-k}(\tau_k\omega),&
\sum_{k\in\cE}s_k(\omega)&=:s(\omega).\end{aligned}$$ Equation means that $V_k:\Z^d\to [-s^*,s^*]$ is $\pi$-almost surely a bounded and *sourceless flow* on $\Z^d$, or, equivalently, $\Phi:\Z^d\to\R^d$ is a bounded *divergence-free* vector field on $\Z^d$. On the other hand, implies that $$\begin{aligned}
\label{psiisgrad}
\psi_i(\omega)
=
s_{e_i}(\omega)-s_{e_i}(\tau_{-e_i}\omega),
\qquad
\Psi_i(\omega,x)
=
S_{e_i}(\omega,x)-S_{e_i}(\omega,x-e_i).\end{aligned}$$ That is, the vector field $\Psi:\Z^d\to\R^d$ is component-wise a directional derivative. It follows in particular that $$\begin{aligned}
\label{nodrift1}
{\ensuremath{\mathbf{E}\left(\Psi\right)}}=0.\end{aligned}$$ We assume that a similar condition holds for the drift field $\Phi$, too: $$\begin{aligned}
\label{nodrift2}
{\ensuremath{\mathbf{E}\left(\Phi\right)}}=\sum_{k\in\cE} k \int_\Omega v_k(\omega) \,{{\mathrm d}}\pi(\omega)=0,\end{aligned}$$ which due to , in the nearest neighbour set-up, is obviously the same as assuming that for $k\in\cE$ $$\begin{aligned}
\label{nodrift}
\int_\Omega v_k(\omega) \,{{\mathrm d}}\pi(\omega)
=0.\end{aligned}$$ From and it follows that in the *annealed* mean drift of the walk is nil: $$\begin{aligned}
\notag
{\ensuremath{\mathbf{E}\left(X(t)\right)}}
=
\int_\Omega {\ensuremath{\mathbf{E}_{\omega}\left(X(t)\right)}} d\pi(\omega)
=
0.\end{aligned}$$ Under these conditions the law of large numbers $$\begin{aligned}
\label{lln}
\lim_{t\to\infty} t^{-1} X(t) =0,
\qquad
\mathrm{a.s.}\end{aligned}$$ follows directly from the ergodic theorem.
Our next assumption is an ellipticity condition for the symmetric part of the jump rates: there exists another constant $s_*\in(0,s^*]$ such that for $\pi$-almost all $\omega\in\Omega$ and all $k\in\cE$ $$\begin{aligned}
\label{ellipt}
s_k(\omega)\ge s_*,
\quad
\pi\text{-a.s.}\end{aligned}$$ Note, that no ellipticity condition is imposed on the jump probabilities $\left(p_k\right)_{k\in\cE}$: it may happen that $p_k=0$ with positive $\pi$-probability. Using a time change we may assume $s_*=1$, and we will occasionally make this assumption for simplicity.
Regarding fluctuations around the law of large numbers , we will soon prove that under the ellipticity condition a *diffusive lower bound* holds: for any fixed vector $v\in\R^d$ $$\begin{aligned}
\label{diffusive lower bound}
\varliminf_{t\to \infty} t^{-1} {\ensuremath{\mathbf{E}\left((v\cdot X(t))^2\right)}} >0.\end{aligned}$$ Explicit lower bound will be provided in Proposition \[prop:diffusive\_bounds\] below.
A *diffusive upper bound* also holds under a subtle condition on the covariances of the drift field $\Phi: \Z^d \to \R^d$. Denote $$\begin{aligned}
\label{covariance matrix of Phi}
&
C_{ij}(x)
:=
{\ensuremath{\mathbf{Cov}\left(\Phi_i(0),\Phi_j(x)\right)}}
=
\int_\Omega \varphi_i(\omega)\varphi_j(\tau_x\omega)d\pi(\omega),
&&
x\in\Z^d,
\\
\notag
&
D_{ij}(x)
:=
{\ensuremath{\mathbf{Cov}\left(\Psi_i(0),\Psi_j(x)\right)}}
=
\int_\Omega \psi_i(\omega)\psi_j(\tau_x\omega)d\pi(\omega),
&&
x\in\Z^d,
\\
\label{Fourier transform of covariance matrix of Phi}
&
\wh C_{ij}(p)
:=
\sum_{x\in\Z^d} e^{\sqrt{-1}x\cdot p} C_{ij}(x),
&&
p\in [-\pi,\pi)^d,
\\
\notag
&
\wh D_{ij}(p)
:=
\sum_{x\in\Z^d} e^{\sqrt{-1}x\cdot p} D_{ij}(x),
&&
p\in [-\pi,\pi)^d.\end{aligned}$$ The Fourier transform is meant as a distribution on $[-\pi,\pi)^d$. More precisely, by Herglotz’s theorem, $\wh C$ and $\wh D$ are positive definite $d\times d$ matrix-valued *measures* on $[-\pi,\pi)^d$. Hence is equivalent to $\wh C_{ij}(\{0\})=0$, for all $i,j=1,\dots,d$.
The fact that $\Psi$ is a spatial derivative of an $L^2$ function implies that $$\begin{aligned}
\label{apriorihcond}
\int_{[-\pi,\pi)^d}
\left(
\sum_{j=1}^d (1-\cos p_j)
\right)^{-1}
\sum_{i=1}^d \wh D_{ii}(p) \, {{\mathrm d}}p <\infty.\end{aligned}$$ A similar *infrared bound* imposed on the covariances of the field $x\mapsto\Phi(x)$ is the notorious *$\cH_{-1}$-condition* referred to in the title of this paper.
#### ${\cH_{-1}}$-condition
(first formulation): We assume $$\begin{aligned}
\label{hcond1}
\int_{[-\pi,\pi)^d}
\left(\sum_{j=1}^d(1-\cos p_j) \right)^{-1} \sum_{i=1}^d \wh C_{ii}(p) \, {{\mathrm d}}p <\infty.\end{aligned}$$
For later use we define the positive definite and bounded $d\times d$ matrices $$\begin{aligned}
\label{ctilde}
&
\wt C_{ij}:=
\int_{[-\pi,\pi)^d}
\left(\sum_{j=1}^d (1-\cos p_j) \right)^{-1}
\wh C_{ij}(p)
\, {{\mathrm d}}p <\infty,
\\
\label{dtilde}
&
\wt D_{ij}:=
\int_{[-\pi,\pi)^d}
\left(\sum_{j=1}^d (1-\cos p_j) \right)^{-1}
\wh D_{ij}(p)
\, {{\mathrm d}}p <\infty.\end{aligned}$$
The probabilistic content of the infrared bounds and is the following. Let $t\mapsto S(t)$ be a continuous time simple symmetric random walk on $\Z^d$ with jump rate $1$, fully independent of the random fields $x\mapsto(\Phi(x), \Psi(x))$. Then and are in turn equivalent to $$\begin{aligned}
\notag
\lim_{T\to\infty} T^{-1}
{\ensuremath{\mathbf{E}\left({\ensuremath\left|{\int_{0}^{T} \Psi(S(t)){{\mathrm d}}t}\right|}^2\right)}}
<\infty,\end{aligned}$$ and
#### ${\cH_{-1}}$-condition
(second formulation): $$\begin{aligned}
\label{hcond2}
&
\lim_{T\to\infty} T^{-1}
{\ensuremath{\mathbf{E}\left({\ensuremath\left|{\int_{0}^{T} \Phi(S(t)){{\mathrm d}}t}\right|}^2\right)}}
<\infty.\end{aligned}$$
The expectations in the last two expressions are taken over the random walk $t\mapsto S(t)$ *and* the random scenery $x\mapsto(\Phi(x), \Psi(x))$. We omit the straightforward proof of these equivalences. Two more equivalent formulations of the $\cH_{-1}$-condition / will appear later in the paper.
The infrared bounds and imply a diffusive upper bound: for any fixed vector $v\in\R^d$ $$\begin{aligned}
\label{diffusive upper bound}
\varlimsup_{t\to\infty} t^{-1} {\ensuremath{\mathbf{E}\left((v\cdot X(t))^2\right)}} <\infty.\end{aligned}$$ An explicit upper bound will be provided in Proposition \[prop:diffusive\_bounds\] below.
Now, and jointly *suggest* that the central limit theorem $$\begin{aligned}
\label{clt}
t^{-1/2} X(t) \Rightarrow \cN(0, \sigma^2)\end{aligned}$$ should hold with some non-degenerate $d\times d$ covariance matrix $\sigma^2$. Attempts to prove the CLT under the minimal conditions of bistochasticity , ellipticity , no drift and $\cH_{-1}$ have a notorious history. In Kozlov [@kozlov_85] a similar CLT is announced under the somewhat restrictive condition that the random field $x\mapsto P(x)$ in be finitely dependent. However, as pointed out in Komorowski and Olla [@komorowski_olla_03a] the proof in [@kozlov_85] is incomplete. In the same paper [@komorowski_olla_03a] the CLT is stated, but as pointed out in [@komorowski_landim_olla_12] this proof is yet again defective. Finally, in [@komorowski_landim_olla_12] a complete proof is given, however, with more restrictive conditions: instead of the $\cH_{-1}$-condition a rather stronger integrability condition on the field $x\mapsto \Phi(x)$ is assumed. See the comments in section \[app:Historical remarks\]. More detailed historical comments on this story can be found in the notes after chapter 3 of [@komorowski_landim_olla_12]. Our main result in the present paper is a complete proof of the CLT , under the conditions listed above.
Central limit theorem for the random walk {#ss:Central limit theorem for the random walk}
-----------------------------------------
We define the *environment process*, as seen from the random walker: $$\begin{aligned}
\notag
\eta(t):=\tau_{X(t)}\omega\end{aligned}$$ This is a pure jump process on $\Omega$ with bounded total jump rates. So, its construction does not pose any technical difficulty. As already mentioned, it is well known (and easy to check, see e.g. Kozlov [@kozlov_85]) that due to condition the probability measure $\pi$ is stationary and ergodic for the Markov process $t\mapsto\eta(t)$. We will denote by $(\cF_t)_{t\ge0}$ the filtration generated by this process: $$\begin{aligned}
\notag
\cF_t:= \sigma(\eta(s): 0\le s\le t).\end{aligned}$$
It is most natural to decompose $X(t)$ as $$\begin{aligned}
\notag
X(t)
=
&
\left\{
X(t)-
\int_{0}^{t} \left(\psi(\eta(s))+\varphi(\eta(s))\right){{\mathrm d}}s
\right\}
+
\int_{0}^{t} \left(\psi(\eta(s))+\varphi(\eta(s))\right){{\mathrm d}}s.
\\[8pt]
\label{martingale decomposition}
=:
&
M(t)+I(t).\end{aligned}$$ In this decomposition the first term is clearly a square integrable $(\cF_t)$-martingale with stationary and ergodic increments and conditional covariances (or, quadratic variation) $$\begin{aligned}
\label{qvar}
{\ensuremath{\mathbf{E}\left(dM_i(t)dM_j(t)\bigm|\cF_t\right)}}=
\delta_{i,j}\left(p_{e_i}(\eta(t))+p_{-e_i}(\eta(t))\right) dt.\end{aligned}$$ Thus, due to the martingale CLT (see e.g. [@helland_82]) $$\begin{aligned}
\notag
t^{-1/2}M(t)\Rightarrow \cN(0,\sigma_M^2),\end{aligned}$$ where $$\begin{aligned}
\label{martvar}
\left(\sigma_M^2\right)_{ij}
=
2 \delta_{i,j} \int_{\Omega}s_{e_i}(\omega) d\pi(\omega).\end{aligned}$$ The difficulty is caused by the compensator integral term $I(t)$.
The following proposition quantifies assertions and .
\[prop:diffusive\_bounds\] Let $t\mapsto X(t)$ be a random walk in doubly stochastic random environment with no drift . Then the ellipticity and $\cH_{-1}$ conditions imply the following diffusive lower and upper bounds: For any vector $v\in\R^d$ $$\begin{aligned}
\label{diffusive bounds}
2s_* {\ensuremath\left|{v}\right|}^2
\le
\operatorname*{\overline{\underline{\lim}}}_{t\to\infty} t^{-1} {\ensuremath{\mathbf{E}\left((v\cdot X(t))^2\right)}}
\le
6s^* {\ensuremath\left|{v}\right|}^2
+
\frac{24}{s_*}\sum_{i,j=1}^d\left(\wt C_{ij}+ \wt D_{ij}\right)v_iv_j,\end{aligned}$$ where $\wt C_{ij}$ and $\wt D_{ij}$ are the matrices defined in and .
The proof of Proposition \[prop:diffusive\_bounds\] is postponed to the next section. Note that the ellipticity condition is relevant in both (lower and upper) bounds, while the $\cH_{-1}$-condition is relevant for the upper bound only.
Let us formally state the main result of the present paper.
\[thm:main\] Let $t\mapsto X(t)$ be a nearest neighbour random walk in random environment, which is bistochastic , has no drift and is elliptic . If in addition the *${\cH_{-1}}$-condition* holds then
\(i) The asymptotic covariance matrix $$\begin{aligned}
\notag
(\sigma^2)_{ij}
:=
\lim_{t\to\infty} t^{-1}{\ensuremath{\mathbf{E}\left(X_i(t)X_j(t)\right)}}\end{aligned}$$ exists, and it is finite and non-degenerate $$\begin{aligned}
\label{bounds on sigmasquared}
2 s_* I
\le
\sigma^2
\le
6 s^* I_d
+
24 s_*^{-1} \left(\wt C+\wt D\right),\end{aligned}$$ where $I$ is the $d\times d$ unit matrix and $\wt C$, $\wt D$ are the matrices defined in , .
\(ii) Moreover, for any $m\in\N$, $t_1,\dots,t_m\in\R_+$ and any continuous and bounded test function $F:\R^{md}\to\R$ $$\begin{aligned}
\notag
\lim_{T\to\infty}
\int_\Omega
{\ensuremath\left|{
{\ensuremath{\mathbf{E}_{\omega}\left(F\left(\frac{X(Tt_1)}{\sqrt{T}}, \dots, \frac{X(Tt_m)}{\sqrt{T}}\right)\right)}}
-
{\ensuremath{\mathbf{E}\left(F(W(t_1), \dots, W(t_m))\right)}}
}\right|} {{\mathrm d}}\pi(\omega)=0,\end{aligned}$$ where $t\mapsto W(t)\in\R^d$ is a Brownian motion with $$\begin{aligned}
\notag
&
{\ensuremath{\mathbf{E}\left(W_i(t)\right)}}=0,
&&
{\ensuremath{\mathbf{E}\left(W_i(s)W_j(t)\right)}}=\min\{s,t\}(\sigma^2)_{ij}\end{aligned}$$
[**Remark**]{} on the jump range of the walk. Throughout the paper we speak about nearest neighbour random walk with jump range $\cE$. However, we could consider a more general setup, with jump range $\cU\subset\Z^d$, with the assumptions that (i) ${\ensuremath\left|{\cU}\right|}<\infty$; (ii) the jump rates are bounded: $p_k(\omega)\le s^*$ almost surely for $k\in\cU$; (iii) the ellipticity condition holds for a subset $\cU^{\prime}\subset\cU$ which generates $\Z^d$. Under these more general assumptions Theorem 1 remains still valid. The proof remains essentially the same apart of notational changes.
It is worth noting here that (unlike in the self-adjoint/reversible cases) the $\cH_{-1}$-condition is certainly stronger than assuming just finiteness of the asymptotic variance of the walk, . So $\cH_{-1}$ seems to be a sufficient but by no means necessary condition for the CLT to hold. The following question arises very naturally.
Let $X$ be a stationary, ergodic random walk in a bistochastic random environment, and assume ${\ensuremath{\mathbf{E}\left(|X(t)|^2\right)}}\le Ct$. Does it follow that $X$ satisfies a central limit theorem?
#### Structure of the paper. {#structure-of-the-paper. .unnumbered}
The proof of this theorem is the content of sections \[s:In the Hilbert space\]-\[s:The operator B and proof of Theorem 1\]. Section \[s:In the Hilbert space\] contains Hilbert space generalities and most of the notation. Section \[s:Relaxed sector condition\] describes and slightly extends the relaxed sector condition of [@horvath_toth_veto_12] on which we rely. Proofs of the extensions are given in an Appendix (the proofs are similar to those of [@horvath_toth_veto_12], but the statements are stronger). In section \[s:The operator B and proof of Theorem 1\] we check the conditions of the relaxed sector condition for the concrete case. Remarks, comments (historical and other) and concrete examples are postponed to sections 5-7. Let us remark that assuming that $s_k$ is constant for all $k\in\cE$, in other words that the walk is *divergence-free*, removes a number of technical difficulties in the proof. Readers who prefer to see the easier version can see it in the first arxiv version of this paper [@circular]
In the Hilbert space {#s:In the Hilbert space}
=====================
Spaces and operators {#ss:Spaces and operators}
--------------------
It is most natural to put ourselves into the Hilbert space over $\C$ $$\begin{aligned}
\notag
\cH:=\Big\{f\in\cL^2(\Omega,\pi):\int_\Omega f {{\mathrm d}}\pi=0\Big\}.\end{aligned}$$ We denote by $T_x$, $x\in\Z^d$, the spatial shift operators $$\begin{aligned}
\notag
T_x f(\omega)
:=
f(\tau_x\omega),\end{aligned}$$ and note that they are unitary: $$\begin{aligned}
\label{shiftops}
T_x^*=T_{-x}=T_x^{-1}.\end{aligned}$$ The $\cL^2$-gradients $\nabla_k$, $k\in\cE$, respectively, $\cL^2$-Laplacian $\Delta$, are: $$\begin{aligned}
\notag
&
\nabla_k
:=
T_k-I,
&&
\nabla_k^*
=
\nabla_{-k},
&&
{\ensuremath\left\|{\nabla_k}\right\|}\le 2,
\\[8pt]
\label{Deltaisselfadjoint}
&
\Delta
:=
\sum_{l\in\cE}\nabla_l
=
-
\frac{1}{2}
\sum_{l\in\cE}\nabla_l\nabla_{-l},
&&
\Delta^*
=
\Delta\le0,
&&
{\ensuremath\left\|{\Delta}\right\|}\le 4d.\end{aligned}$$ We remark that the norm inequalities above are in fact equalities in any non-degenerate case, but we will not need this fact. Due to ergodicity of the $\Z^d$-action $(\Omega, \cF, \pi, \tau_z:z\in\Z^d)$, $$\begin{aligned}
\label{kerDelta}
\Ker(\Delta)=\{0\}.\end{aligned}$$ Indeed, $\Delta f=0$ implies that $0={\ensuremath\left\langle{f,\Delta f}\right\rangle} = -\frac12\sum_{k\in\cE} {\ensuremath\left\langle{\nabla_k f,\nabla_k f}\right\rangle}$ and since all terms are non-negative, they must all be 0 and $f$ must be invariant to translations. Ergodicity to $\Z^d$ actions means that $f$ is constant, and since our Hilbert space is that of functions averaging to zero, $f$ must be zero.
We define the bounded positive operator $\sqd$ in terms of the spectral theorem (applied to the bounded positive operator ${\ensuremath\left|{\Delta}\right|}:=-\Delta$). Note that due to $\Ran {\ensuremath\left|{\Delta}\right|}$ is dense in $\cH$, and hence so is $\Ran{\ensuremath\left|{\Delta}\right|}^{1/2}$ which is a superset of it. Hence it follows that $\nsqd :=\left(\sqd\right)^{-1}$ is an (unbounded) positive self-adjoint operator with $\Dom \nsqd = \Ran \sqd$ and $\Ran \nsqd = \Dom \sqd = \cH$. Note that the dense subspace $\Dom \nsqd = \Ran \sqd$ is invariant under, and the operators $\sqd$ and $\nsqd$ commute with the translations $T_x$, $x\in\Z^d$.
We define the *Riesz operators*: for all $k\in\cE$ $$\begin{aligned}
\label{Gamma_k}
\Gamma_k:\Dom \nsqd \to \cH,
\qquad
\Gamma_k=|\Delta|^{-1/2}\nabla_k=\nabla_k|\Delta|^{-1/2},\end{aligned}$$ and note that for any $f\in\Dom \nsqd $ $$\begin{aligned}
\notag
{\ensuremath\left\|{\Gamma_k f}\right\|}^2
=
{\ensuremath\left\langle{\nsqd f,\nabla_{-k}\nabla_k \nsqd f}\right\rangle}
\le
{\ensuremath\left\langle{\nsqd f,{\ensuremath\left|{\Delta}\right|} \nsqd f}\right\rangle}
=
{\ensuremath\left\|{f}\right\|}^2.\end{aligned}$$ Thus, the operators $\Gamma_k$, $k\in\cE$, extend as bounded operators to the whole space $\cH$. The following properties are easy to check: $$\begin{aligned}
\label{Gammaadj}
&
\Gamma_k^*=\Gamma_{-k},
&&
{\ensuremath\left\|{\Gamma_k}\right\|}\le 1,
&&
\frac12
\sum_{l\in\cE}\Gamma_l\Gamma_l^*=I.\end{aligned}$$ As before, in fact ${\ensuremath\left\|{\Gamma_k}\right\|}=1$ in any non-degenerate case, but we will not need this fact.
A third equivalent formulation of the ${\cH_{-1}}$-condition / is the following:
#### ${\cH_{-1}}$-condition
(third formulation): $$\begin{aligned}
\label{hcond3}
\varphi_i\in\Dom \nsqd = \Ran \sqd,
\qquad\qquad
i=1,\dots,d. \end{aligned}$$ In the case of nearest neighbour walks this is further equivalent to $$\begin{aligned}
\label{hcond3bis}
v_k\in\Dom \nsqd = \Ran \sqd,
\qquad\qquad
k\in\cE.\end{aligned}$$
\(i) Conditions and are equivalent.
\(ii) Furthermore, in the case of nearest neighbour walks conditions and are also equivalent.
\(i) Recall that is formulated in terms of continuous time simple random walk $S$. In operator theory language $$\begin{aligned}
\label{eq:SDelta}
{\ensuremath{\mathbf{E}_{\omega}\left(\Phi_i(S(t))\right)}}
=
e^{t\Delta}\varphi_i(\omega).\end{aligned}$$ Hence $$\begin{aligned}
\frac{1}{t}{\ensuremath{\mathbf{E}\left( {\ensuremath\left|{\int_{0}^{t} \Phi(S(s)){\,{\mathrm d}}s}\right|}^2\right)}}
&\stackrel{(*)}{=}
\;\sum_{i=1}^d\int_{0}^{t}
\frac{t-s}{t}
{\ensuremath{\mathbf{E}\left(
2\Phi_i(0)
\Phi_i(S(s))
\right)}}
{\,{\mathrm d}}s
\\
&\stackrel{\clap{$\scriptstyle\textrm{(\ref{eq:SDelta})}$}}
{=}
\;\sum_{i=1}^d 2
\int_{0}^{t}
\frac{t-s}{t}
\langle\varphi_i,e^{s\Delta}\varphi_i\rangle
{\,{\mathrm d}}s\end{aligned}$$ where $(*)$ follows from space stationarity of $\Phi$ (recall that $S$ is independent of $\Phi$, so $\Phi_i(S)$ is just some average of some fixed translations of $\Phi_i$). An application of the spectral theorem for $|\Delta|$ shows that this is bounded in $t$ if and only if all $\varphi_i\in\Dom|\Delta|^{-1/2}$, $i=1,\dots,d$.
\(ii) To conclude from $\varphi\in\Dom|\Delta|^{-1/2}$ that $v\in\Dom|\Delta|^{-1/2}$ we recall that $\varphi_i = (I+T_{-e_i})v_{e_i}=(2I+\nabla_{-e_i})v_{e_i}$. Since $\Gamma_{-e_i}=|\Delta|^{-1/2}\nabla_{-e_i}$ is bounded, we get that $\nabla_{-e_i} v_{e_i} \in \Dom(|\Delta|^{-1/2})$. Rearranging gives $$\begin{aligned}
\notag
\varphi_i-2v_{e_i} \in \Dom(|\Delta|^{-1/2})\end{aligned}$$ which shows that $\varphi_i\in\Dom(|\Delta|^{-1/2})$ if and only if so is $v_{e_i}$.
[**Remark.**]{} Note that equivalence of and holds *only* in the case of nearest neighbour jumps. If a larger jump range $\cU$ is allowed (see the remark after the formulation of Theorem 1) then is stronger than . However, the formulation will not be used in the proof of our main result. It will have a role only in the complementary section \[app:The stream tensor field\], which is not part of the proof. That part could also be reformulated in the context of finite jump rate, relying only on but as the main result does not rely on it we will not bother to do that.
Finally, we also define the multiplication operators $M_k, N_k$, $k\in\cE$, $$\begin{aligned}
\label{vmultipl}
&
M_k f(\omega):= v_k(\omega) f(\omega),
&&
M_k^*=M_k,
&&
{\ensuremath\left\|{M_k}\right\|}\le s^*,
\\[8pt]
\label{smultipl}
&
N_k f(\omega):= (s_k(\omega)-s_*) f(\omega),
&&
N_k^*=N_k\ge 0,
&&
{\ensuremath\left\|{N_k}\right\|}\le s^*\end{aligned}$$ (recall that $s^*$ is the overall *upper bound* on $p$ and $s_*$ is the *lower bound* on the symmetric parts $s$ in the ellipticity condition (\[ellipt\])). It is easy to check that the following commutation relations hold due to and $$\begin{aligned}
\label{Mnablacommute}
&
\sum_{l\in\cE} M_l \nabla_l
=
-
\sum_{l\in\cE} \nabla_{-l} M_{l},
\\
\notag
&
\sum_{l\in\cE} N_l \nabla_l
=
\sum_{l\in\cE} \nabla_{-l} N_{l}
=
-
\frac12
\sum_{l\in\cE} \nabla_{-l} N_{l} \nabla_{l}.\end{aligned}$$
The *infinitesimal generator* of the stationary environment process $t\mapsto\eta(t)$, acting on the Hilbert space $\cL^2(\Omega,\pi)$ is: $$\begin{aligned}
\notag
Lf(\omega)
= p_k(\omega)(f(\tau_k\omega)-f(\omega)),\end{aligned}$$ which in terms of the operators introduced above is written as $$\begin{aligned}
\label{infgen2}
L=
-D
-T
+A,\end{aligned}$$ with $$\begin{aligned}
\label{opD}
D
&
:=
-s_*\Delta,
\\
\notag
T
&
:=
-
\sum_{l\in\cE} N_l \nabla_l
=
\frac12
\sum_{l\in\cE} \nabla_{-l} N_{l} \nabla_{l},
\\
\label{opA}
A
&
:=
\sum_{l\in\cE} M_l \nabla_l
=
-
\sum_{l\in\cE} \nabla_{-l} M_{l}.\end{aligned}$$ Note that $D=D^*$, $T=T^*$, $A=-A^*$ and $$\begin{aligned}
\label{DboundsT}
0
\le
T
\le
d s^*s_*^{-1}
D.\end{aligned}$$ The inequalities are meant in operator sense. The last one follows from $$\begin{aligned}
\notag
D^{-1/2}TD^{-1/2}
=
\frac{1}{2s_*}\sum_{l\in\cE}\Gamma_{-l}N_l\Gamma_{l},\end{aligned}$$ and hence, due to and $$\begin{aligned}
\notag
{\ensuremath\left\|{D^{-1/2}TD^{-1/2}}\right\|}\le \frac{d s^*}{s_*}\end{aligned}$$ follows, which implies the upper bound in .
Proof of Proposition \[prop:diffusive\_bounds\] {#ss:Proof of Proposition 1}
-----------------------------------------------
We decompose the displacement process $t\to X(t)$ in such a way that the forward-and-backward martingale part will be uncorrelated with the rest. The variance of this forward-and-backward martingale will serve as lower bound for the variance of the displacement. Let $$\begin{aligned}
\notag
&
u_k(\omega)
:=
\sgn( v_k(\omega) ) \min\{ {\ensuremath\left|{v_k(\omega)}\right|} , s_* \},
&&
w_k(\omega)
:=
\sgn( v_k(\omega) ) {\left( {\ensuremath\left|{v_k(\omega)}\right|} - s_* \right) }_+,
\\[8pt]
\notag
&
q_k(\omega)
:=
s_*+u_k(\omega),
&&
r_k(\omega)
:=
\left( s_k(\omega) - s_* \right) + w_k(\omega).\end{aligned}$$ Note that the skew symmetry of $v_k(\omega)$ is inherited by $u_k(\omega)$ and $w_k(\omega)$: $$\begin{aligned}
\label{vecvec}
u_k(\omega)+ u_{-k}(\tau_k\omega)=0,
\qquad
w_k(\omega)+ w_{-k}(\tau_k\omega)=0.\end{aligned}$$ Further on, $$\begin{aligned}
\notag
&
u_k(\omega)+ w_k(\omega)=v_k(\omega),
&&
q_k(\omega)+ r_k(\omega)=p_k(\omega),
&&
q_k(\omega)\ge0,
&&
r_k(\omega)\ge0.\end{aligned}$$ We further define $$\begin{aligned}
\notag
&
q(\omega):=\sum_{l\in\cE}q_l(\omega)\ge0,
&&
\wt\varphi(\omega):=\sum_{l\in\cE}lq_l(\omega)\in\R^d,
\\
\notag
&
r(\omega):=\sum_{l\in\cE}r_l(\omega)\ge0,
&&
\wt\psi(\omega):=\sum_{l\in\cE}lr_l(\omega)\in\R^d,\end{aligned}$$ and note that $$\begin{aligned}
\notag
&
q(\omega)+r(\omega)=p(\omega),
&&
\wt\varphi(\omega)+\wt\psi(\omega)
=
\varphi(\omega)+\psi(\omega).\end{aligned}$$ Now let $0=\theta_0<\theta_1<\theta_2<\dots$ be the successive jump times of the environment process $t\mapsto \eta(t)$ (or, what is the same, of the random walk $t\mapsto X(t)$): $$\begin{aligned}
\notag
\theta_0:=0,
\qquad
\theta_{n+1}:=\inf\{t>\theta_n: \eta(t)\not=\eta(\theta_n)\},\end{aligned}$$ and define *extra* random variables $\alpha_n\in\{0,1\}$, $n=0, 1,2,\dots$ with the following conditional law, given the trajectory $t\mapsto\eta(t)$: for $N\in\N$ and $a_n\in\{0,1\}$, $n=0,1,\dots,N$, $$\begin{aligned}
\notag
{\ensuremath{\mathbf{P}\left(\alpha_n=a_n, \ \ n=0,1,\dots,N\bigm|\eta(t)_{t\ge0}\right)}}
=
\prod_{n=0}^N \left(\frac{q(\eta(\theta_{n}))}{p(\eta(\theta_{n}))}\right)^{a_n} \left(\frac{r(\eta(\theta_{n}))}{p(\eta(\theta_{n}))}\right)^{1-a_n}.\end{aligned}$$ In plain words, conditionally on the trajectory $t\mapsto\eta(t)$, the random variables $\alpha_n$, $n=0,1,2,\dots$, are independent biased coin tosses, with probability of head or tail (1 or 0 respectively) equal to the value of $\frac{q(\eta(t))}{p(\eta(t))}$, respectively, $\frac{r(\eta(t))}{p(\eta(t))}$, in the interval $t\in[\theta_n,\theta_{n+1})$. Now, extend piecewise continuously $$\begin{aligned}
\notag
\alpha(t):=\sum_{n=0}^\infty \alpha_n {\ensuremath{\mathbbm{1}_{\{t\in(\theta_n,\theta_{n+1}]\}}}}.\end{aligned}$$ Mind, that $t\mapsto\alpha(t)$ is defined as a *caglad*, not a cadlag process. We decompose the displacement $t\mapsto X(t)$ as follows: $$\begin{aligned}
\notag
X(t)
=
K(t)+L(t)+J(t),\end{aligned}$$ where $$\begin{aligned}
\notag
&
K(t)
:=
\int_0^t \alpha(s) dX(s) - \int_0^t \wt\varphi(s)ds,
\\
\notag
&
L(t)
:=
\int_0^t (1-\alpha(s)) dX(s) - \int_0^t \wt\psi(s)ds,
\\
&
\notag
J(t)
:=\int_0^t \left(\wt\varphi(s) + \wt\psi(s) \right)ds.\end{aligned}$$ Note the following three facts.
\(1) $t\mapsto K(t)$ and $t\mapsto L(t)$, being driven by conditionally independent Poisson flows, are *uncorrelated martingales*, with respect to their own joint filtration.
\(2) $t\mapsto K(t)$ is forward-and-backward martingale with respect to its own past, respectively, future filtration. This is due to and to the fact that the symmetric part of its jump rates is constant, $s_*$. Indeed, $$\begin{aligned}
{\ensuremath{\mathbf{E}\left(K(t+dt)-K(t)\bigm|\eta_t=\omega\right)}}
& =
\sum_{l\in\cE} l q_l(\omega) - \wt\varphi(\omega) dt
=
\sum_{l\in\cE} l u_l(\omega) - \wt\varphi(\omega) dt
=
0 dt.
\\
{\ensuremath{\mathbf{E}\left(K(t)-K(t-dt)\bigm|\eta_t=\omega\right)}}
& =
-\!
\sum_{l\in\cE} l q_{-l}(\tau_l\omega) - \wt\varphi(\omega)dt
=
-\!
\sum_{l\in\cE} l u_{-l}(\tau_l\omega) - \wt\varphi(\omega)dt\\
&=
\sum_{l\in\cE} l u_l(\omega) -\wt\varphi(\omega) dt
=
0 dt,\end{aligned}$$ and hence the claim.
\(3) $t\mapsto J(t)$, being an integral, is forward-and-backward predictable with respect to the same filtrations.
From these three facts it follows that the process $t\mapsto K(t)$ is *uncorrelated* with $t\mapsto L(t)+J(t)$. Hence, for any vector $v\in\R$ $${\ensuremath{\mathbf{E}\left((v\cdot X(t))^2\right)}} =
{\ensuremath{\mathbf{E}\left((v\cdot K(t))^2\right)}}+{\ensuremath{\mathbf{E}\left((v\cdot (L(t)+J(t)))^2\right)}}
\ge
{\ensuremath{\mathbf{E}\left((v\cdot K(t))^2\right)}}
=
2 s_* {\ensuremath\left|{v}\right|}^2.
\qedhere$$
We provide upper bounds on the variance of the various terms on the right hand side of the decomposition $X=M+I$ .
As shown in - the variance of the martingale term $M(t)$ on the right hand side of is computed explicitly: for $v\in\R^d$, $$\begin{aligned}
\label{martupp}
\frac{1}{t}{\ensuremath{\mathbf{E}\left((v\cdot M(t))^2\right)}}
=
\sum_{i=1}^d v_i^2 \int_{\Omega}(p_{e_i}(\omega)+p_{-e_i}(\omega)) d\pi(\omega)
\le
2s^*{\ensuremath\left|{v}\right|}^2.\end{aligned}$$
In order to bound the variance of the integral term $I(t)$ on the right hand side of we quote Proposition 2.1.1 in Olla [@olla_01] (alternatively, Lemma 2.4 in [@komorowski_landim_olla_12] contains the same result with a different constant).
\[lemma:hbound\] Let $t\mapsto\eta(t)$ be a stationary and ergodic Markov process on the probability space $(\Omega,\pi)$, whose infinitesimal generator acting on $\cL^2(\Omega,\pi)$ is $L$. Let $g\in\cL^2(\Omega,\pi)$ such that $\int_\Omega g d\pi=0$. Then $$\begin{aligned}
\notag
\varlimsup_{t\to\infty} \frac{1}{t}
{\ensuremath{\mathbf{E}\left(\max_{0\le s \le t}{\ensuremath\left|{\int_0^s g(\eta(u)) {{\mathrm d}}u}\right|}^2\right)}}
\le
16 \lim_{\lambda\to0}(g, (\lambda I -L-L^*)^{-1}g).\end{aligned}$$
(Olla denotes the right-hand side by $||g||_{-1}$ — his definition of $||g||_{-1}$, (2.1.2) ibid., is different but it is easy to see that it is equivalent to the above, up to a factor of 2).
The decomposition of the infinitesimal generator gives that $-L-L^*\ge 2s_*|\Delta|$, and hence by Löwner’s theorem (see [@carlen_10 Theorem 2.6] or [@lowner_34]) $(-L-L^*)^{-1}\le
1/(2s_*)|\Delta|^{-1}$. It then follows that for any vector $v\in\R^d$ $$\begin{aligned}
\label{phiupp}
&
\varlimsup_{t\to\infty}
t^{-1} {\ensuremath{\mathbf{E}\left(\left(\int_0^t v\cdot\varphi(\eta(s))ds\right)^2\right)}}
\le
\frac{8}{s_*} ((v\cdot\varphi),|\Delta|^{-1}(v\cdot\varphi))
=
\frac{8}{s_*} \sum_{i,j=1}^d v_i \wt C_{ij} v_j,
\\
\label{psiupp}
&
\varlimsup_{t\to\infty}
t^{-1} {\ensuremath{\mathbf{E}\left(\left(\int_0^t v\cdot\psi(\eta(s))ds\right)^2\right)}}
\le
\frac{8}{s_*} ((v\cdot\psi),|\Delta|^{-1}(v\cdot\psi))
=
\frac{8}{s_*} \sum_{i,j=1}^d v_i \wt D_{ij} v_j.\end{aligned}$$ From , by applying the Cauchy-Schwarz inequality we readily obtain $${\ensuremath{\mathbf{E}\left(\!\left(v\cdot X(t)\right)^2\!\right)}}
\le
3
{\ensuremath{\mathbf{E}\left(\!\left(v\cdot M(t)\right)^2\!\right)}}
+
3
{\ensuremath{\mathbf{E}\left(\!\left(\int_0^tv\cdot \varphi(\eta(s) ds\right)^{\!2}\right)}}
+
3
{\ensuremath{\mathbf{E}\left(\!\left(\int_0^tv\cdot \psi(\eta(s) ds\right)^{\!2}\right)}}.$$ Finally, the upper bound in follows from here, due to , and .
Relaxed sector condition {#s:Relaxed sector condition}
========================
In this section we recall and slightly extend the *relaxed sector condition* from [@horvath_toth_veto_12]. This is a functional analytic condition on the operators $D$, $T$ and $A$ from which ensures that the efficient martingale approximation à la Kipnis-Varadhan of integrals of the type of $I(t)$ in exists.
A clarification is due here. The relaxed sector condition (Theorem RSC1 below), is essentially equivalent to the condition that the range $L\cH_{-1}$ of the infinitesimal generator $L$ be *dense in the $\cH_{-1}$-topology of $\cL^2(\Omega, \pi)$.* (defined by the symmetric part $S:=(L+L^*)/2$ of the infinitesimal generator). This latter one appears in earlier work (see e.g. Olla [@olla_01]). But, to the best of our knowledge it has never been exploited *directly, without stronger sufficient assumptions*. The *strong* and *graded sector conditions* of Varadhan [@varadhan_95], respectively of Sethuraman, Varadhan and Yau [@sethuraman_varadhan_yau_00], are stronger sufficient conditions for this to hold, and applicable in various circumstances. Nevertheless, the equivalent formulation in [@horvath_toth_veto_12] proved to be a very useful one, applicable in conditions where the graded sector condition does not work. In particular, in the context of the present paper. Let us also stress that the graded sector condition itself gets a very transparent and handy proof through the relaxed sector condition. For more details see [@horvath_toth_veto_12].
Since in the present case the infinitesimal generator $L=-D-T+A$ and all operators in the decomposition are *bounded* we recall the result of [@horvath_toth_veto_12] in a slightly restricted form: we do not have to worry now about domains and cores of the various operators $D$, $T$ or $A$. This section will be fairly abstract.
Kipnis-Varadhan theory {#ss:Kipnis-Varadhan theory}
----------------------
Let $(\Omega, \cF, \pi)$ be a probability space: the state space of a *stationary and ergodic* pure jump Markov process $t\mapsto\eta(t)$ with bounded jump rates. We put ourselves in the complex Hilbert space $\cL^2(\Omega, \pi)$. Denote the *infinitesimal generator* of the semigroup of the process by $L$. Since the process $\eta(t)$ has bounded jump rates the infinitesimal generator $L$ is a bounded operator. We denote the *self-adjoint* and *skew-self-adjoint* parts of the generator $L$ by $$\begin{aligned}
\notag
S:=-\frac12(L+L^*)\ge0
\qquad
A:=\frac12(L-L^*).\end{aligned}$$ We assume that $S$ is itself ergodic i.e. $$\begin{aligned}
\notag
\Ker(S)=\{c\mathbbm{1} : c\in\C\},\end{aligned}$$ and restrict ourselves to the subspace of codimension 1, orthogonal to the constant functions: $$\begin{aligned}
\notag
\cH:=\{f\in\cL^2(\Omega,\pi): {\ensuremath\left\langle{\mathbbm{1},f}\right\rangle}=0\}.\end{aligned}$$ In the sequel the operators $(\lambda I+ S)^{\pm1/2}$, $\lambda\ge0$, will play an important rôle. These are defined in terms of the spectral theorem applied to the self-adjoint and positive operator $S$. The *unbounded* operator $S^{-1/2}$ is self-adjoint on its domain $$\begin{aligned}
\notag
\Dom(S^{-1/2})
=
\Ran(S^{1/2})
=
\{f\in\cH:
{\ensuremath\left\|{S^{-1/2}f}\right\|}^2
:=
\lim_{\lambda\to0}{\ensuremath\left\|{(\lambda I + S)^{-1/2}f}\right\|}^2
<
\infty
\}.\end{aligned}$$ Let $f\in\cH$. We ask about CLT/invariance principle for the rescaled process $$\begin{aligned}
\label{rescaledintegral}
Y_N(t):=\frac{1}{\sqrt{N}}\int_0^{Nt} f(\eta(s)) {{\mathrm d}} s\end{aligned}$$ as $N\to\infty$.
We denote by $R_\lambda$ the *resolvent* of the semigroup $s\mapsto e^{sL}$: $$\begin{aligned}
\label{resolvent_def}
R_\lambda
:=
\int_0^\infty e^{-\lambda s} e^{sL} {{\mathrm d}} s
=
\big(\lambda I-L\big)^{-1}, \qquad \lambda>0,\end{aligned}$$ and given $f\in\cH$, we will use the notation $$\begin{aligned}
\notag
u_\lambda:=R_\lambda f.\end{aligned}$$
The following theorem is a direct extension to general non-reversible setup of the Kipnis-Varadhan Theorem [@kipnis_varadhan_86]. It yields the *efficient martingale approximation* of the additive functional . See Tóth [@toth_86], or the surveys [@olla_01] and [@komorowski_landim_olla_12].
\[thm:kv\] With the notation and assumptions as before, if the following two limits hold in (the norm topology of) $\cH$: $$\begin{aligned}
\label{conditionA}
\lim_{\lambda\to0}
\lambda^{1/2} u_\lambda=0,
\hskip3cm
\lim_{\lambda\to0} S^{1/2} u_\lambda=v \in\cH,\end{aligned}$$ then $$\begin{aligned}
\notag
\sigma^2
:=
2\lim_{\lambda\to0}{\ensuremath\left\langle{u_\lambda,f}\right\rangle}
=
2{\ensuremath\left\|{v}\right\|}^2
\in
[0,\infty),\end{aligned}$$ exists, and there also exists a zero mean, $\cL^2$-martingale $M(t)$, adapted to the filtration of the Markov process $\eta(t)$, with stationary and ergodic increments and variance $$\begin{aligned}
\notag
{\ensuremath{\mathbf{E}\left(M(t)^2\right)}}=\sigma^2t,\end{aligned}$$ such that for $t\in(0,\infty)$ $$\begin{aligned}
\notag
\lim_{N\to\infty}
{\ensuremath{\mathbf{E}\left({\ensuremath\left|{Y_N(t) - \frac{M(Nt)}{\sqrt{N}}}\right|}^2\right)}} =0.\end{aligned}$$
With the same setup and notation, for any $m\in\N$, $t_1,\dots,t_m\in\R_+$ and $F:\R^{m}\to\R$ continuous and bounded $$\begin{aligned}
\notag
\lim_{N\to\infty}
\int_\Omega
{\ensuremath\left|{
{\ensuremath{\mathbf{E}_{\omega}\left(F(Y_N(t_1), \dots, Y_N(t_m))\right)}}
-
{\ensuremath{\mathbf{E}\left(F(W(t_1), \dots, W(t_m))\right)}}
}\right|}
{{\mathrm d}}\pi(\omega)=0,\end{aligned}$$ where $t\mapsto W(t)\in\R$ is a 1-dimensional Brownian motion with variance ${\ensuremath{\mathbf{E}\left(W(t)^2\right)}}=\sigma^2 t$.
Relaxed sector condition {#ss:Relaxed sector condition}
------------------------
Let, for $\lambda>0$, $$\begin{aligned}
\label{Clambda_def}
C_\lambda:=(\lambda I + S)^{-1/2} A (\lambda I + S)^{-1/2}.\end{aligned}$$ These are bounded and skew-self-adjoint.
\[thm:rsc1\] Assume that there exist a dense subspace $\cC\subseteq \cH$ and an operator $C:\cC\to \cH$ which is essentially skew-self-adjoint on the core $\cC$ and such that for any vector $\psi\in \cC$ there exists a sequence $\psi_\lambda\in\cH$ such that $$\begin{aligned}
\label{Clambdalimit}
\lim_{\lambda\to 0}{\ensuremath\left\|{\psi_\lambda-\psi}\right\|}=0.
\qquad
\text{and}
\qquad
\lim_{\lambda\to 0}{\ensuremath\left\|{C_\lambda\psi_\lambda-C\psi}\right\|}=0.\end{aligned}$$ Then for any $f\in\Dom(S^{-1/2})$ the limits hold and thus the martingale approximation and CLT of Theorem KV follow.
#### Remarks
1\. The conditions of Theorem RSC1 can be shown to be equivalent to that the sequence of bounded skew-self-adjoint operators $C_\lambda$ converges in the *strong graph limit* sense to the unbounded skew-self-adjoint operator $C$, see Lemma \[lem:strrescvg\] (ii) below. For various notions of graph limits of operators over Hilbert or Banach spaces see chapter VIII of [@reed_simon_vol1_vol2_75], especially Theorem VIII.26 ibid.\
2. Theorem RSC1 is a slightly stronger reformulation of Theorem 1 from [@horvath_toth_veto_12] where the condition was slightly stronger. There it was assumed that for any $\varphi\in\cC$, $\lim_{\lambda\to 0}{\ensuremath\left\|{C_\lambda\varphi-C\varphi}\right\|}=0$. It turns out that the weaker and more natural condition suffices and this has some importance in our next extension, Theorem RSC2. For sake of completeness we give the proof of this theorem in the Appendix.\
2. The operator $C$ is heuristically some version of $S^{-1/2} A S^{-1/2}$. However, it is not sufficient that a naturally densely defined version of $S^{-1/2} A S^{-1/2}$ is skew-Hermitian. One must show that its closure is actually skew-self-adjoint. The conditions of Theorem RSC1 require to be careful with domains and with point-wise convergence as $\lambda\to 0$, as above.
RSC refers to *relaxed sector condition*: indeed, as shown in [@horvath_toth_veto_12] this theorem contains the *strong sector condition* of [@varadhan_95] and the *graded sector condition* of [@sethuraman_varadhan_yau_00] as special cases. See the comments at the beginning of Section \[s:Relaxed sector condition\] for the precise relation of RSC to other sector conditions. For comments on history, content and variants of Theorem KV we refer the reader to the monograph [@komorowski_landim_olla_12]. For some direct consequences of Theorem RSC1 see [@horvath_toth_veto_12].
Now, we slightly extend the validity of Theorem RSC1. Assume that the symmetric part of the infinitesimal generator decomposes as $$\begin{aligned}
\notag
S=D+T,\end{aligned}$$ where $D=D^*$, $T=T^*$ and the “diagonal” part $D$ dominates $T$ in the following sense: there exists $c<\infty$ so that $$\begin{aligned}
\label{Ddominates}
0\le T \le cD.\end{aligned}$$ Further, denote $$\begin{aligned}
\label{Blambda_def}
B_\lambda:=(\lambda I + D)^{-1/2} A (\lambda I + D)^{-1/2}.\end{aligned}$$ The following statement is actually a straightforward consequence of Theorem RSC1.
\[thm:rsc2\] Assume that there exist a dense subspace $\cB\subseteq \cH$ and an operator $B:\cB\to \cH$ which is essentially skew-self-adjoint on the core $\cB$ and such that for any vector $\varphi\in \cB$ there exists a sequence $\varphi_\lambda\in\cH$ such that $$\begin{aligned}
\label{Blambdalimit}
\lim_{\lambda\to 0}{\ensuremath\left\|{\varphi_\lambda-\varphi}\right\|}=0.
\qquad
\text{and}
\qquad
\lim_{\lambda\to 0}{\ensuremath\left\|{B_\lambda\varphi_\lambda-B\varphi}\right\|}=0.\end{aligned}$$ Then for any $f\in\Dom(D^{-1/2})$ the limits hold and thus the martingale approximation and CLT of Theorem KV follow.
The proof of Theorem RSC2 is also postponed to the Appendix.
The operator B and proof of Theorem \[thm:main\] {#s:The operator B and proof of Theorem 1}
================================================
We apply Theorem RSC2 to our concrete setup, with the operators $D$ and $A$ defined using and (\[opA\]) respectively. Recall that without loss of generality we have fixed $s_*=1$ (see the remark after the ellipticity condition ). Let $$\begin{aligned}
\notag
\cB
:=
\Dom\nsqd
=
\Ran\sqd,\end{aligned}$$ and recall from and the definition of the operators $\Gamma_l$ and $M_l$, $l\in\cE$. Define the *unbounded* operator $B:\cB\to\cH$ $$\begin{aligned}
\notag
B
:=
-
\sum_{l\in\cE}\Gamma_{-l} M_l \nsqd.\end{aligned}$$ (The definition of $B$ uses our assumption that $s_*=1$, otherwise with our definitions of $D$ and $A$ we would have needed a factor of $1/s_*$ before it). First we verify , i.e. that $B_\lambda\to B$ *pointwise* on the core $\cB$, where the bounded operator $B_\lambda$ is expressed by inserting the explicit form of $D$ and $A$, , respectively, , into the definition of $B_\lambda$: $$\begin{aligned}
\notag
B_\lambda = -\sum_{l\in\cE}(\lambda I - \Delta)^{-1/2}\nabla_{-l}M_l(\lambda I - \Delta)^{-1/2}.\end{aligned}$$ From the spectral theorem for the commutative $C^*$-algebra generated by the shift operators $T_{e_i}$, $i=1,\dots,d$, (see e.g. Theorem 1.1.1 on page 2 of [@arveson_76]) we obtain that $\|(\lambda I-\Delta)^{-1/2}\nabla_l\|\le 1$, $\|(\lambda I-\Delta)^{-1/2}{\ensuremath\left|{\Delta}\right|}^{1/2}\|\le 1$, and, moreover, for any $\varphi\in\cH$ $$\begin{aligned}
\notag
&
(\lambda I - \Delta)^{-1/2} \nabla_l \varphi
\to
\Gamma_l \varphi
&&
(\lambda I - \Delta)^{-1/2} {\ensuremath\left|{\Delta}\right|}^{1/2} \varphi
\to
\varphi,
&&
\text{as }\lambda\searrow0.\end{aligned}$$ When $\varphi\in\cB$ we get $(\lambda I-\Delta)^{-1/2}\varphi\to|\Delta|^{-1/2}\varphi$ which allows to write $$\begin{aligned}
B_\lambda\varphi&=-\sum_{l\in\cE d}(\lambda
I-\Delta)^{-1/2}\nabla_{-l}M_l(\lambda I-\Delta)^{-1/2}\varphi\\
&=-\sum_{l\in\cE d}(\lambda
I-\Delta)^{-1/2}\nabla_{-l}M_l|\Delta|^{-1/2}\varphi+O(\Vert(\lambda I-\Delta)^{-1/2}\varphi
- |\Delta|^{-1/2}\varphi\Vert).\end{aligned}$$ Hence follows readily for any $\varphi\in\cB$.
With established, we need to show that $B$ is essentially skew-self-adjoint on $\cB$. We start with a light lemma.
\[lem:B\^\*\] (i) $B:\cB\to\cH$ is skew-Hermitian, i.e. $\langle\varphi,B\psi\rangle=-\langle B\varphi,\psi\rangle$ for all $\varphi,\psi\in\cB$.\
(ii) The full domain of $B^*$ is $$\begin{aligned}
\label{domain of B^*}
{\cB}^*
=
\{f\in\cH: \sum_{l\in\cE} M_l \Gamma_l f \in\cB\}\end{aligned}$$ and $B^*$ acts on $\cB^*$ by $$\begin{aligned}
\label{B^*}
B^*
:=
-
\nsqd \sum_{l\in\cE} M_l \Gamma_l.\end{aligned}$$
[**Remark:**]{} It is of crucial importance here that $\cB^*$ in is the *full* domain of the adjoint operator $B^*$, i.e. the subspace of all $f$ such that the linear functional $g\mapsto {\ensuremath\left\langle{f,Bg}\right\rangle}$ is bounded on $\cB$. It will not be enough for our purposes just to show that $\cB^*$ is some core of definition.
\(i) Let $f,g\in\cB$. Then, due to $$\begin{aligned}
\notag
{\ensuremath\left\langle{f,Bg}\right\rangle}
&
=
-
\sum_{l\in\cE} {\ensuremath\left\langle{\nsqd f,\nabla_{-l} M_l \nsqd g}\right\rangle}
\\
\notag
&
\stackrel{\clap{$\scriptstyle\textrm{(\ref{Mnablacommute})}$}}{=}
\phantom{-}
\sum_{l\in\cE} {\ensuremath\left\langle{\nabla_{-l} M_l \nsqd f,\nsqd g}\right\rangle}
=
-{\ensuremath\left\langle{Bf,g}\right\rangle},\end{aligned}$$ (ii) Next, $$\begin{aligned}
\notag
\Dom(B^*)
&=
\Big\{
f\in\cH:
(\exists c(f)<\infty)
(\forall g\in\cB):
\Big|\Big\langle f, \sum_{l\in\cE} \Gamma_{-l}M_l \nsqd g\Big\rangle\Big|
\le
c(f){\ensuremath\left\|{g}\right\|}
\Big\}
\\
\notag
&=
\Big\{
f\in\cH:
(\exists c(f)<\infty)
(\forall g\in\cB):
\Big|\Big\langle\sum_{l\in\cE} M_l \Gamma_{l} f,\nsqd g\Big\rangle\Big|
\le
c(f){\ensuremath\left\|{g}\right\|}
\Big\}
\\
\notag
&
=
\Big\{
f\in\cH:
\sum_{l\in\cE} M_l \Gamma_{l} f \in \cB
\Big\},\end{aligned}$$ as claimed. In the last step we used the fact that $\cB$ is the *full domain* of the self-adjoint operator $\nsqd$. The action of $B^*$ follows from straightforward manipulations.\[page:other conditions of RSC2\]
Note that Lemma \[lem:B\^\*\] in particular implies that $\cB\subseteq\cB^*$, that $B^*:\cB^*\to\cH$ is in principle an extension of $-\overline{B}$ and hence the operator $B:\cB\to\cH$ is closable as the adjoint of any operator is automatically closed. We actually ought to prove that $$\begin{aligned}
\notag
B^*=-\overline{B}.\end{aligned}$$ We apply von Neumann’s criterion (see e.g. Theorem VIII.3 of Reed and Simon [@reed_simon_vol1_vol2_75]): If for *some* $\alpha>0$, $$\begin{aligned}
\label{vonNeumann's criterion}
\Ker(B^* \pm \alpha I)=\{0\}\end{aligned}$$ then $B^*=-\overline{B}$. For reasons which will become clear very soon we will choose $\alpha=s^*$. (Actually any $\alpha\ge s^*$ would work equally well.) Thus (\[vonNeumann’s criterion\]) is equivalent to showing that the equations $$\begin{aligned}
\label{vonNeumannplus}
\sum_{l\in\cE}M_l \Gamma_l \mu + s^* \sqd \mu = 0
\\
\label{vonNeumannminus}
\sum_{l\in\cE}M_l \Gamma_l \mu - s^* \sqd \mu = 0\end{aligned}$$ admit only the trivial solution $\mu=0$. We will prove this for . The other case is done very similarly.
Note that assuming $\mu\in\cB$ the problem becomes fully trivial. Indeed, inserting $\mu={\ensuremath\left|{\Delta}\right|}^{1/2}\chi$ in and taking inner product with $\chi$ we get $$\begin{aligned}
\notag
\sum_{l\in\cE} \langle M_l \nabla_l \chi,\chi\rangle - s^* \langle\Delta \chi,\chi\rangle =0.\end{aligned}$$ The first term is pure imaginary while the second term is real , giving that $\langle \Delta\chi,\chi\rangle=0$ which, due to , admits only the trivial solution $\chi=0$. The point is that $\mu$ is not necessarily in $\cB$ so $|\Delta|^{-1/2}\mu$ is not necessarily well defined as an element of $\cH$. Nevertheless, we are able to define a scalar random field $\Psi:\Omega\times\Z^d \to \R$ of stationary increments (rather than stationary) which can be thought of as the lifting of $|\Delta|^{-1/2}\mu$ to the lattice $\Z^d$.
Let, therefore $\mu$ be a putative solution for and define, for each $k\in\cE$, $$\begin{aligned}
\label{udef}
u_k:=\Gamma_k \mu.\end{aligned}$$ These are vector components and they also satisfy the *gradient condition*: $\forall \ k,l\in\cE$ $$\begin{aligned}
\label{ugradvec}
&
u_k+T_ku_{-k}=0,
&&
u_k+T_ku_l=u_l+T_lu_k.\end{aligned}$$ Note also that $$\begin{aligned}
\notag
\sum_{l\in\cE} u_l = \sqd \mu.\end{aligned}$$ The eigenvalue equation becomes $$\begin{aligned}
\label{vonNeumann3}
\sum_{l\in\cE} v_l u_l + s^* \sum_{l\in\cE}u_l=0.\end{aligned}$$ We lift this equation to $\Z^d$. By defining the lattice vector fields $V, U:\Omega\times\Z^d\to\R^d$ as $$\begin{aligned}
\notag
&
V_k(\omega,x):=v_k(\tau_x\omega),
&&
U_k(\omega,x):=u_k(\tau_x\omega),\end{aligned}$$ we obtain the following lifted version of equation $$\begin{aligned}
\label{vonNeumann3_lifted}
\sum_{l\in\cE} V_l(\omega,x) U_l(\omega,x) + s^* \sum_{l\in\cE}U_l(\omega,x)=0.\end{aligned}$$ Note that $U$ is the $\Z^d$-gradient of a scalar field $\Psi:\Omega\times\Z^d\to\R$, determined uniquely by $$\begin{aligned}
\label{Psidef}
&
\Psi(\omega,0)=0,
&&
\Psi(\omega, x+k) -\Psi(\omega, x) = U_k(\omega, x).\end{aligned}$$ As promised, the scalar field $\Psi$ has stationary increments (or, in the language of ergodic theory: it is a cocycle), i.e. $$\begin{aligned}
\label{Psistatincr}
\Psi(\omega, y) - \Psi(\omega, x) = \Psi(\tau_x\omega, y-x) - \Psi(\tau_x\omega, 0).\end{aligned}$$ The equation gets the form $$\begin{aligned}
\label{harmonic1}
s^*
\sum_{l\in\cE}(\Psi(\omega,x+l)-\Psi(\omega,x))
+
\sum_{l\in\cE}V_l(\omega,x)(\Psi(\omega,x+l)-\Psi(\omega,x))
=0,\end{aligned}$$ Denote the first term by $\operatorname{lap}\Psi$ and the second by $\operatorname{grad}\Psi$ (these are the usual $\Z^d$ Laplacian and gradient, respectively), so the equation becomes $$\begin{aligned}
\label{harmonic2}
s^* \operatorname{lap}\Psi + V\cdot \operatorname{grad}\Psi = 0.\end{aligned}$$ We prove that equation / admits $\Psi\equiv 0$ as the only solution satisfying ${\ensuremath{\mathbf{E}\left(\Psi(x)\right)}}=0$ for all $x\in\Z^d$. This will be done using *an auxiliary* random walk in random environment which will be denoted by $Y$. We remark that in the specific case where $X$ is *divergence free* i.e. $s\equiv 1$, or in general when $s$ is constant, we get that $Y$ is the same as $X$, but in general they differ. We define the environment for $Y$ on the same probability space $\Omega$ as $X$. The transfer rates $p^Y_k$, $k\in\cE$ are given by $$\begin{aligned}
\notag
p^Y_k(\omega)= s^* + v_k(\omega).\end{aligned}$$ In other words, we take from $X$ the anti-symmetric part $v_k=(p_k-p_{-k})/2$ but replace the symmetric part with the constant $s^*$. The walk $Y$ is also bistochastic, so all results proved so far (in particular, stationarity and ergodicity of $Y$’s environment process $t\mapsto \eta^Y(t):=\tau_{Y(t)}\omega$, and the diffusive lower and upper bounds for $t\mapsto Y(t)$) are in force.
Note that equation / means exactly that for a given $\omega\in\Omega$ fixed (that is: in the quenched setup) the field $\Psi(\omega, \cdot):\Z^d\to\R$ is *harmonic* for the random walk $Y(t)$. Thus the process $$\begin{aligned}
\label{marti}
t\mapsto R(t):=\Psi(Y(t))\end{aligned}$$ is a martingale (with $R(0)=0$) in the quenched filtration $\sigma\left( \omega,Y(s)_{0\le s\le t} \right)_{t\ge0}$. Hence, $t\mapsto R(t)$ is a martingale in its own filtration $\sigma\left( R(s)_{0\le s\le t} \right)_{t\ge0}$, too. We will soon show that ${\ensuremath{\mathbf{E}\left(R(t)^2\right)}}<\infty$. From stationarity and ergodicity of the environment process $t\mapsto \eta_t^Y$ and it follows that the process $t\mapsto R(t)$ has stationary and ergodic increments with respect to the annealed measure ${\ensuremath{\mathbf{P}\left(\cdot\right)}}:= \int_\Omega {\ensuremath{\mathbf{P}_{\omega}\left(\cdot\right)}} d\pi(\omega)$. Indeed, let $F(R(\cdot))$ be an arbitrary bounded and measurable functional of the process $t\mapsto R(t)$, $t\ge0$. Using , a straightforward computation shows that $$\begin{aligned}
\notag
{\ensuremath{\mathbf{E}_{\omega}\left(F(R(t_0+\cdot)-R(t_0))\right)}}
=
{\ensuremath{\mathbf{E}_{\omega}\left(\mathbf{E}_{\eta(t_0)}\left(F(R(\cdot)\right)\right)}},\end{aligned}$$ Hence, by stationarity and ergodicity of the environment process $t\mapsto\eta(t)$, the claim follows.
Thus, the process $t\mapsto R(t)$ is a martingale (with $R(0)=0$) with stationary and ergodic increments, in its own filtration $\sigma\left(R(s)_{0\le s\le t} \right)_{t\ge0}$, with respect to the annealed measure ${\ensuremath{\mathbf{P}\left(\cdot\right)}}$.
\[lem:variance of R\] Let $\mu$ be a solution of equation , $\Psi$ the harmonic field constructed in and $R(t)$ the martingale defined in . Then $$\begin{aligned}
\label{martingale variance}
{\ensuremath{\mathbf{E}\left(R(t)^2\right)}} = 2s^*{\ensuremath\left\|{\mu}\right\|}^2 t.\end{aligned}$$
Since $t\mapsto R(t)$ is a martingale with stationary increments (with respect to the annealed measure ${\ensuremath{\mathbf{P}\left(\cdot\right)}}$), we automatically have ${\ensuremath{\mathbf{E}\left(R(t)^2\right)}}=\varrho^2 t$ with some $\varrho\ge0$. We now compute $\varrho$. $$\begin{aligned}
\notag
\varrho^2
&
:=
\lim_{t\to0}
\frac{{\ensuremath{\mathbf{E}\left(R(t)^2\right)}}}{t}
\stackrel{1}{=}
\lim_{t\to0} \int_\Omega
\frac{{\ensuremath{\mathbf{E}_{\omega}\left(\Psi(\omega, Y(t))^2\right)}}}{t}
{\mathrm d}\pi(\omega)
\\
\notag
&
\stackrel{2}{=}
\int_\Omega \lim_{t\to0}
\frac{{\ensuremath{\mathbf{E}_{\omega}\left(\Psi(\omega, Y(t))^2\right)}}}{t}
{\mathrm d}\pi(\omega)
\stackrel{3}{=}
\sum_{l\in\cE}\int_{\Omega}
\left(s^* + v_l(\omega)\right) {\ensuremath\left|{u_l(\omega)}\right|}^2
{{\mathrm d}}\pi(\omega)
\\
\notag
&
\stackrel{4}{=}
s^*
\sum_{l\in\cE}\int_{\Omega}
{\ensuremath\left|{u_l(\omega)}\right|}^2
{{\mathrm d}}\pi(\omega)
\stackrel{5}{=}
s^*
\sum_{l\in\cE}
{\ensuremath\left\|{\Gamma_l\mu}\right\|}^2
\stackrel{6}{=}
2 s^* {\ensuremath\left\|{\mu}\right\|}^2.\end{aligned}$$ Step 1 is annealed averaging. Step 2 is easily justified by dominated convergence. Step 3 drops out from explicit computation of the conditional variance of one jump. In step 4 we used that due to and $v_{-l}(\omega){\ensuremath\left|{u_{-l}(\omega)}\right|}^2 = -
v_{l}(\tau_{-l}\omega){\ensuremath\left|{u_{l}(\tau_{-l}\omega)}\right|}^2$ and translation invariance of the measure $\pi$ on $\Omega$. In step 5 we use the definition of $u_l$. Finally, in the last step 6 we used the third identity of .
\[prop:Neumann\] The unique solution of / is $\mu=0$, and consequently the operator $B$ is essentially skew-self-adjoint on the core $\cB$.
Let $\mu$ be a solution of the equation , $\Psi$ the harmonic field constructed in and $R(t)$ the martingale defined in . From the martingale central limit theorem (see e.g. [@helland_82]) and it follows that $$\begin{aligned}
\label{martingale CLT}
\frac{R(t)}{\sqrt{t}}
\Rightarrow \cN(0,2s^*{\ensuremath\left\|{\mu}\right\|}^2),
\qquad
\text{ as } t\to\infty.\end{aligned}$$ On the other hand we are going to prove that $$\begin{aligned}
\label{toprob0}
\frac{R(t)}{\sqrt{t}}
\toprob0,
\qquad
\text{ as }t\to\infty.\end{aligned}$$ Jointly, and clearly imply $\mu=0$, as claimed in the proposition.
The proof of will combine\
(A) the (sub)diffusive behaviour of the displacement $$\begin{aligned}
\notag
\varlimsup_{T\to\infty}T^{-1}{\ensuremath{\mathbf{E}\left(Y(T)^2\right)}}<\infty,\end{aligned}$$ which follows from the ${\cH_{-1}}$-condition, see ; and\
(B) the fact that the scalar field $x\mapsto \Psi(x)$ having zero mean and stationary increments, cf. , increases *sublinearly* with ${\ensuremath\left|{x}\right|}$. The sublinearity is the issue here. Since $\Psi$ has stationary, mean zero increments, due to the individual (pointwise) ergodic theorem, it follows that *in any fixed direction* $\Psi$ increases sublinearly almost surely. However, this does not warrant that $\Psi$ increases sublinearly uniformly in $\Z^d$, $d\ge 2$, which is the difficulty we will now tackle.
Let $\delta>0$ and $K<\infty$. Then $$\begin{aligned}
\label{R large}
{\ensuremath{\mathbf{P}\left({\ensuremath\left|{R(t)}\right|}>\delta\sqrt{t}\right)}}
\le
{\ensuremath{\mathbf{P}\left(\{{\ensuremath\left|{R(t)}\right|}>\delta\sqrt{t}\} \cap \{{\ensuremath\left|{Y(t)}\right|}\le K\sqrt{t}\}\right)}}
+
{\ensuremath{\mathbf{P}\left({\ensuremath\left|{Y(t)}\right|}> K\sqrt{t}\right)}}.\end{aligned}$$ From (sub)diffusivity and Chebyshev’s inequality it follows directly that $$\begin{aligned}
\label{X large}
\lim_{K\to\infty}
\varlimsup_{t\to\infty}
{\ensuremath{\mathbf{P}\left({\ensuremath\left|{Y(t)}\right|}> K\sqrt{t}\right)}}
=0.\end{aligned}$$ We present two proofs of $$\begin{aligned}
\label{R large and X small}
\lim_{t\to\infty}
{\ensuremath{\mathbf{P}\left(\{{\ensuremath\left|{R(t)}\right|}>\delta\sqrt{t}\} \cap \{{\ensuremath\left|{Y(t)}\right|}\le K\sqrt{t}\}\right)}}
=0,\end{aligned}$$ with $\delta>0$ and $K<\infty$ fixed. One with bare hands, valid in $d=2$ only, and another one valid in any dimension which relies on a heat kernel (upper) bound from Morris and Peres [@morris_peres_05].
\[Proof of in $d=2$, with bare hands\] We follow here the approach of [@berger_biskup_07] where the argument was applied in a different context. In order to keep it short (as another full proof valid in all dimensions follows) we assume separate ergodicity i.e. that $(\Omega, \cF, \pi, \tau_{e_i})$ is ergodic for both $i=1,2$.
First note that $$\begin{aligned}
\label{bound R by Psi}
{\ensuremath{\mathbf{P}\left(\{{\ensuremath\left|{R(t)}\right|}>\delta\sqrt{t}\} \cap \{{\ensuremath\left|{Y(t)}\right|}\le K\sqrt{t}\}\right)}}
\le
{\ensuremath{\mathbf{P}\left( \max_{{\ensuremath\left|{x}\right|}<K\sqrt{t}} {\ensuremath\left|{\Psi(x)}\right|}> \delta\sqrt{t}\right)}}.\end{aligned}$$ Next, since $\Psi$ is harmonic with respect to the random walk $Y(t)$, it obeys the *maximum principle* (this is true for any random walk, no special property of $Y$ is used here). Thus $$\begin{aligned}
\label{maximum principle}
\max_{{\ensuremath\left|{x}\right|}_\infty\le L}{\ensuremath\left|{\Psi(x)}\right|}
=
\max_{{\ensuremath\left|{x}\right|}_\infty = L}{\ensuremath\left|{\Psi(x)}\right|},\end{aligned}$$ where ${\ensuremath\left|{x}\right|}_\infty:=\max\{{\ensuremath\left|{x_1}\right|},{\ensuremath\left|{x_2}\right|}\}$. By spatial stationarity $$\begin{aligned}
\label{shiftit}
\begin{gathered}
\max_{{\ensuremath\left|{x_1}\right|}\le L} {\ensuremath\left|{\Psi(x_1,-L)-\Psi(0,-L)}\right|}
\sim
\max_{{\ensuremath\left|{x_1}\right|}\le L} {\ensuremath\left|{\Psi(x_1,0)}\right|}
\sim
\max_{{\ensuremath\left|{x_1}\right|}\le L} {\ensuremath\left|{\Psi(x_1,+L)-\Psi(0,+L)}\right|},
\\
\max_{{\ensuremath\left|{x_2}\right|}\le L} {\ensuremath\left|{\Psi(-L,x_2)-\Psi(-L,0)}\right|}
\sim
\max_{{\ensuremath\left|{x_2}\right|}\le L} {\ensuremath\left|{\Psi(0,x_2)}\right|}
\sim
\max_{{\ensuremath\left|{x_2}\right|}\le L} {\ensuremath\left|{\Psi(+L,x_2)-\Psi(+L,0)}\right|},
\end{gathered}\end{aligned}$$ where $\sim$ stands for equality in distribution. Now, note that $\Psi(x_1,0)$ and $\Psi(0, x_2)$ are Birkhoff sums: $$\begin{aligned}
\notag
\Psi(x_1,0)
=
\sum_{j=0}^{x_1-1} u_{e_1}(\tau_{je_1}\omega),
\qquad
\Psi(0, x_2)
=
\sum_{j=0}^{x_2-1} u_{e_2}(\tau_{je_2}\omega),\end{aligned}$$ where $u_{e_1}(\omega)$ and $u_{e_2}(\omega)$ are zero mean and square integrable (recall the definition of $u$, ). Hence, by the ergodic theorem $$\begin{aligned}
\label{ergthm}
L^{-1}
\max\big\{
\max_{{\ensuremath\left|{x_1}\right|}\le L} {\ensuremath\left|{\Psi(x_1,0)}\right|},
\max_{{\ensuremath\left|{x_2}\right|}\le L} {\ensuremath\left|{\Psi(0,x_2)}\right|}
\big\}
\to0,
\qquad
\text{a.s., as }
L\to\infty.\end{aligned}$$ Putting together , and we readily obtain, for any $\vareps>0$, $$\begin{aligned}
\label{Psi is sublinear}
\lim_{L\to\infty}
{\ensuremath{\mathbf{P}\left(\max_{{\ensuremath\left|{x}\right|}_\infty\le L}{\ensuremath\left|{\Psi(x)}\right|}\ge\vareps L\right)}}
=0.\end{aligned}$$ Finally, follows by applying to the right hand side of .
\[Proof of in all $d\ge 2$\] We start with the following uniform upper bound on the (quenched) heat kernel of the walk $Y(t)$.
\[prop:heat kernel bound\] There exists a constant $C=C(d, s^*)$ (depending only on the dimension $d$ and the upper bound $s^*$ on the jump rates) such that for $\pi$-almost all $\omega\in\Omega$ and all $t>0$ $$\begin{aligned}
\label{heat kernel bound}
&
\sup_{x\in\Z^d} {\ensuremath{\mathbf{P}_{\omega}\left(Y(t)=x\right)}}\le C t^{-d/2},
&&
\pi\text{-a.s.}\end{aligned}$$
This bound follows from Theorem 2 of Morris and Peres [@morris_peres_05] through Lemma \[lem:Morris-Peres\], below, which states essentially the same bound for discrete-time lazy random walks on $\Z^d$ (recall that a random walk is called lazy if there is a lower bound on the probability of the walker staying put at any given point).
\[lem:Morris-Peres\] Let $V:\Z^d\to[-1,1]^{\cE}$ be a (deterministically given) field such that for all $k\in\cE$ and $x\in\Z^d$ $$\begin{aligned}
\label{lifted conditions}
V_k(x)+V_{-k}(x+k)=0,
\qquad
\sum_{l\in\cE}V_l(x)=0.\end{aligned}$$ Define the discrete-time nearest-neighbour, lazy random walk $n\mapsto Y_n$ on $\Z^d$ with transition probabilities $$\begin{aligned}
\label{lazywalk}
{\ensuremath{\mathbf{P}\left(Y_{n+1}=y\bigm|Y_n=x\right)}}
=
p_{x,y}
:=
\begin{cases}
\frac12
&\text{ if }
y=x,
\\
\frac{1}{4d}(1+V_k(x))
&\text{ if }
y=x+k, \ \ k\in\cE,
\\
0
&\text{ if }
{\ensuremath\left|{y-x}\right|}>1.
\end{cases}\end{aligned}$$ Then there exists a constant $C=C(d)$ depending only on dimension such that for any $x,y\in\Z^d$ $$\begin{aligned}
\label{discrete heat kernel bound}
{\ensuremath{\mathbf{P}\left(Y_n=y\bigm|Y_0=x\right)}}\le C n^{-d/2}.\end{aligned}$$
For $A,B\subset\Z^d$, such that $A\cap B=\emptyset$ let $$\begin{aligned}
\notag
Q(A,B)
:=
\sum_{x\in A, y\in B}
p_{x,y}.\end{aligned}$$ For notational reasons we extend the definition of $V_k(x)$, $k\in\cE$, $x\in\Z^d$, as follows $$\begin{aligned}
\notag
V_z(x):=
\begin{cases}
V_k(x)& \text{if } z=k\in\cE,
\\
0 & \text{if } z\not\in\cE.
\end{cases}\end{aligned}$$ For $S\subset\Z^d$, ${\ensuremath\left|{S}\right|}<\infty$ let $\partial S:=\{(x,y): x\in S, y\in \Z^d\setminus S, \Vert x-y\Vert=1\}$ and note that by the isoperimetric inequality for $\Z^d$ $$\begin{aligned}
\label{isoperi}
{\ensuremath\left|{\partial S}\right|}
\ge C {\ensuremath\left|{S}\right|}^{(d-1)/d}, \end{aligned}$$ with some dimension-dependent constant $C$. (This discrete isoperimteric inequality is a simple corollary of the classic isoperimetric inequality in $\R^d$. See also Theorem V3.1 in [@chavel_01] for a general discretisation result for isoperimteric inequalities.)
We have $$\begin{aligned}
\notag
Q(S,S^c)
&
=
\sum_{x\in S, y\in S^c} \frac{1}{4d} (1+V_{y-x}(x))
\\
\notag
&
=
\frac{1}{4d} {\ensuremath\left|{\partial S}\right|}
+
\frac{1}{4d}
\left(
\sum_{x\in S, y\in\Z^d}
V_{y-x}(x)
-
\sum_{x\in S, y\in S}
V_{y-x}(x)
\right)
\\
\label{cut}
&
=
\frac{1}{4d} {\ensuremath\left|{\partial S}\right|},\end{aligned}$$ where the last equality follows from $$\begin{aligned}
\notag
\sum_{x\in S, y\in\Z^d}
V_{y-x}(x)
&
=
\sum_{x\in S} \sum_{l\in\cE} V_l(x) =0,
\\
\notag
\sum_{x\in S, y\in S}
V_{y-x}(x)
&
=
\frac12
\sum_{x\in S, y\in S}
\left(
V_{y-x}(x)
+
V_{x-y}(y)
\right)
=0,\end{aligned}$$ both of which are consequences of . Yet another consequence of is that the uniform counting measure on $\Z^d$ is stationary to our walk. Hence the isoperimetric profile $\Phi(r)$ (in the sense of Morris and Peres [@morris_peres_05]) is given by $$\begin{aligned}
\notag
\Phi(r)
:=
\inf_{0<{\ensuremath\left|{S}\right|}\le r}\frac{Q(S,S^c)}{{\ensuremath\left|{S}\right|}}.\end{aligned}$$ Theorem 2 of [@morris_peres_05] (specified to our setup) states that for any $0<\vareps\le 1$, if $$\begin{aligned}
\label{morris-peres thm2}
n> 1 + 4 \int_{4}^{4/\vareps}\frac{{{\mathrm d}} u}{u\Phi^2(u)}\end{aligned}$$ then, for any $x, y \in \Z_d$ $$\begin{aligned}
\notag
{\ensuremath{\mathbf{P}\left(X_n=y\bigm|X_0=x\right)}} \le \vareps.\end{aligned}$$ From and the isoperimetric inequality we have $$\begin{aligned}
\label{isope}
C_1 r^{-1/d} \le \Phi(r) \le C_2 r^{-1/d},\end{aligned}$$ with the constants $0<C_1<C_2<\infty$ depending only on the dimension. Finally, from and we readily get .
In order to obtain from , note that $Y(t)=Y_{\nu(t)}$ where $Y_n$ is a discrete time lazy random walk defined in and , with $V_k(x)=v_k(\tau_x\omega)/s^*$ and $t\mapsto\nu(t)$ is a Poisson birth process with intensity $s^*t$ independent of the discrete time walk $Y_n$. Thus $$\begin{aligned}
\notag
{\ensuremath{\mathbf{P}_{\omega}\left(Y(t)=x\right)}}
&
=
e^{-s^*t/2}\sum_{n=0}^\infty \frac{(s^*t/2)^n}{n!} {\ensuremath{\mathbf{P}_{\omega}\left(Y_n=x\right)}}
\\
\notag
&
\le
e^{-s^*t/2}\left(1+\sum_{n=1}^\infty \frac{(s^*t/2)^n}{n!} Cn^{-d/2}\right)
\\
\notag
&
\le
C(d,s^*) t^{-d/2}\end{aligned}$$ This completes the proof of Proposition \[prop:heat kernel bound\].
### Remarks. {#remarks. .unnumbered}
1. The point in Proposition \[prop:heat kernel bound\] is that it provides uniform upper bound in any (deterministic) environment which satisfies conditions , and thus allows decoupling of the expectation with respect to the walk and with respect to the environment.
2. In Lemma \[lem:Morris-Peres\] the “amount of laziness” could be any $\delta\in(0,1)$, with appropriate minor changes in the formulation and proof.
3. Alternative proofs of Proposition \[prop:heat kernel bound\] are also valid, using either Nash-Sobolev or Faber-Krahn inequalities, see e.g. Kumagai [@kumagai_14]. These alternative proofs – which we do not present here – are more analytic in flavour. Their advantage is robustness: these proofs are also valid in continuous-space setting (see section \[app:Historical remarks\] below).
We now return to the proof of . By Chebyshev’s inequality $$\begin{aligned}
\label{cheb}
{\ensuremath{\mathbf{P}\left(\{{\ensuremath\left|{R(t)}\right|}>\delta\sqrt{t}\} \cap \{{\ensuremath\left|{Y(t)}\right|}\le K\sqrt{t}\}\right)}}
\le
\delta^{-2}t^{-1}{\ensuremath{\mathbf{E}\left({\ensuremath\left|{R(t)}\right|}^2{\ensuremath{\mathbbm{1}_{\{{\ensuremath\left|{Y(t)}\right|}\le K\sqrt{t}\}}}}\right)}}\end{aligned}$$ Since the scalar field $\Psi$ has stationary increments, cf. , and zero mean, we get from the $\cL^2$ ergodic theorem that for $k\in\cE$ $$\begin{aligned}
\notag
\lim_{n\to\infty}n^{-2}{\ensuremath{\mathbf{E}\left({\ensuremath\left|{\Psi(n k)}\right|}^2\right)}} =0,\end{aligned}$$ and, consequently, $$\begin{aligned}
\label{Psi sublinear}
\lim_{|x|\to\infty}{\ensuremath\left|{x}\right|}^{-2}{\ensuremath{\mathbf{E}\left({\ensuremath\left|{\Psi(x)}\right|}^2\right)}} =0.\end{aligned}$$ Applying in turn the heat kernel bound of Proposition \[prop:heat kernel bound\] and the limit on the right hand side of we obtain $$\begin{aligned}
\notag
t^{-1}{\ensuremath{\mathbf{E}\left({\ensuremath\left|{R(t)}\right|}^2 {\ensuremath{\mathbbm{1}_{\{{\ensuremath\left|{Y(t)}\right|}\le K\sqrt{t}\}}}}\right)}}
\le
Ct^{-d/2-1}\sum_{|x|\le K\sqrt{t}}{\ensuremath{\mathbf{E}\left(|\Psi(x)|^2\right)}}
\to
0,
\qquad
\textrm{ as }t\to\infty.\end{aligned}$$ Here the first expectation is both on the random walk $Y(t)$ and on the field $\omega$, while the second is just on the field $\omega$. The point is that with the help of the uniform heat kernel bound of Proposition \[prop:heat kernel bound\] we can *decouple* the two expectations.
This concludes the proof of in arbitrary dimension.
We conclude the proof of the Proposition \[prop:Neumann\] by noting that from , and we readily get which, together with implies indeed that $\mu=0$. So holds with $\alpha=s^*$. We showed that $\Ker(B^*+ s^* I)=\{0\}$, the proof that $\Ker(B^*-s^* I)=\{0\}$ is done in the same way with $Y$ defined using $-V$ instead of $V$. Thus the operator $B:\cB\to\cH$ is indeed essentially skew-self-adjoint.
Proposition \[prop:Neumann\] verifies that the operator $B$ is essentially skew-self-adjoint. The other conditions of Theorem RSC2 are verified . Thus Theorem RSC2 may be applied and we get that for any $f\in\Dom(|\Delta|^{-1/2})$, the time average $\int_0^Nf(\eta(t))$ may be approximated by a Kipnis-Varadhan martingale. The third formulation of the ${\cH_{-1}}$ condition gives that $v_k\in\Dom(|\Delta|^{-1/2})$ while it is always true that $s_k\in\Dom(|\Delta|^{-1/2})$, . Applying Theorem RSC2 with $f=v_k+s_k$ for each $k\in\{1,\dotsc,d\}$ gives that the compensator $I$ from the decomposition $X=M+I$ can be approximated with a Kipnis-Varadhan martingale which, we recall, is a stationary martingale $M'$ which is adapted to the filtration of the environment process $\eta$. Hence $M+M'$ is also a stationary martingale and has a CLT. Proposition \[prop:diffusive\_bounds\] gives the bounds .
The stream tensor field {#app:The stream tensor field}
=======================
The content of this section is not a part of the proof of our main result, but it is an important part of the story and sheds light on the role and limitations of the ${\cH_{-1}}$-condition in this context. We formulate this section in the context of nearest neighbour jumps and part (ii) of Proposition \[prop:helmholtz\] (below) as presented here relies on the equivalence of and which is valid only in the nearest neighbour case. However, we remark that this statement, too, can be easily reformulated for general finite jump rates, but in this case some modifications in the definition of the lattice stream tensor are due and the formulation becomes less transparent. We omit these not particularly instructive details, noting that it is doable with minimum effort.
The following proposition establishes the existence of the stream tensor field and is essentially Helmholtz’s theorem. It is the $\Z^d$ lattice counterpart of Proposition 11.1 from [@komorowski_landim_olla_12]. Recall the definition of the field $V:\Omega\times\Z^d\to[-s^*,s^*]^{\cE}$ from .
\[prop:helmholtz\] (i) There exists an antisymmetric tensor field ${H}:\Omega\times\Z^d\to\R^{\cE\times\cE}$ such that for all $x\in\Z^d$ we have ${H}_{k,l}(\cdot,x)\in\cH$ and $$\begin{aligned}
\label{Thetasymms}
{H}_{l,k}(\omega,x)
=
{H}_{-k,l}(\omega,x+k)
=
{H}_{k,-l}(\omega,x+l)
=
-{H}_{k,l}(\omega,x),\end{aligned}$$ with stationary increments $$\begin{aligned}
\notag
{H}(\omega,y)-{H}(\omega,x)={H}(\tau_x\omega, y-x)-{H}(\tau_x\omega, 0),\end{aligned}$$ such that $$\begin{aligned}
\label{V=curlTheta}
V_k(\omega,x)=\sum_{l\in\cE} {H}_{k,l}(\omega,x).\end{aligned}$$ The realization of the tensor field ${H}$ is uniquely determined by the “pinning down” condition below.
\(ii) The ${\cH_{-1}}$-condition holds if and only if there exist ${h}_{k,l}\in\cH$, $k,l\in\cE$, such that $$\begin{aligned}
\label{thetatensor}
{h}_{l,k}
=
T_k {h}_{-k,l}
=
T_l {h}_{k,-l}
=
-
{h}_{k,l}\end{aligned}$$ and $$\begin{aligned}
\label{v=curltheta}
v_k(\omega)=\sum_{l\in\cE}{h}_{k,l}(\omega).\end{aligned}$$ In this case the tensor field ${H}$ can be realized as the stationary lifting of ${h}$: $$\begin{aligned}
\label{lifting of theta}
{H}_{k,l}(\omega,x)={h}_{k,l}(\tau_x\omega).\end{aligned}$$
\(i) For $k,l,m\in\cE$ define $$\begin{aligned}
\notag
g_{m;k,l}
:=
\Gamma_m\big(\Gamma_l v_k - \Gamma_k v_l\big),\end{aligned}$$ where $\Gamma_l=|\Delta|^{-1/2}\nabla_l$ are the Riesz operators defined in , and note that for all $k,l,m,n\in\cE$ $$\begin{aligned}
\label{gistensor}
&
g_{m;l,k}
=
T_kg_{m;-k,l}
=
T_lg_{m;k,-l}
=
-g_{m;k,l},
\\[8pt]
\label{gisgrad}
&
g_{m;l,k}+T_mg_{n;l,k}
=
g_{n;l,k}+T_ng_{m;l,k},
\\[8pt]
\label{curlg=gradv}
&
\sum_{l\in\cE}g_{m;k,l} = \nabla_m v_k.\end{aligned}$$ means that that keeping the subscript $m\in\cE$ fixed, $g_{m;k,l}$ has exactly the symmetries of a $\cL^2$-tensor variable indexed by $k,l\in\cE$. means that, on the other hand, keeping $k,l\in\cE$ fixed, $g_{m;k,l}$ is a $\cL^2$-gradient in the subscript $m\in\cE$. Finally, means that the $\cL^2$-divergence of tensor $g_{m;\cdot,\cdot}$ is actually the $\cL^2$-gradient of the vector $v_\cdot$.
Let $G_{m;k,l}:\Omega\times\Z^d\to\R$ be the lifting $G_{m;k,l}(\omega,x):=g_{m;k,l}(\tau_x\omega)$. By , for any $k,l\in\cE$ fixed $\left(G_{m;k,l}(\omega,x)\right))_{m\in\cE}$ is a lattice gradient. The increments of ${H}_{k,l}$ are defined by $$\begin{aligned}
\label{gradTheta=G}
{H}_{k,l}(\omega,x+m)-{H}_{k,l}(\omega,x)
=
G_{m;k,l}(\omega,x),
\ \ \ m\in\cE.\end{aligned}$$ This is consistent, due to .
Next, in order to uniquely determine the tensor field $H$, we “pin down” its values at $x=0$. For $e_i,e_j\in\cE_{+}$ choose $$\begin{aligned}
\label{Theta0}
\begin{aligned}
&
{H}_{e_i,e_j}(\omega,0)
=
0,
&&
{H}_{-e_i,e_j}(\omega,0)
=
-g_{-e_i;e_i,e_j}(\omega),
\\[8pt]
&
{H}_{e_i,-e_j}(\omega,0)
=
g_{-e_j;e_i,e_j}(\omega),
&&
{H}_{-e_i,-e_j}(\omega,0)
=
-g_{-e_i;e_i,e_j}(\omega)+g_{-e_j;e_i,e_j}(\tau_{-e_i}\omega).
\end{aligned}\end{aligned}$$ The tensor field ${H}$ is fully determined by and . Due to and , , respectively, will hold, indeed.
\(ii) We show equivalence with $v_k\in\Dom(|\Delta|^{-1/2})$ . First we prove the *only* if part. Assume and let $$\begin{aligned}
\notag
{h}_{k,l}
=
\Gamma_l {\ensuremath\left|{\Delta}\right|}^{-1/2} v_k - \Gamma_k {\ensuremath\left|{\Delta}\right|}^{-1/2} v_l
=
{\ensuremath\left|{\Delta}\right|}^{-1/2} \big( \Gamma_l v_k - \Gamma_k v_l\big).\end{aligned}$$ Hence and are readily obtained. Next we prove the if part. Assume that there exist $h_{k,l}\in\cH$ with the symmetries and $v_k$ is realized as in . Then we have $$\begin{aligned}
\notag
v_k
=
\sum_{l\in\cE} h_{k,l}
=
\frac12 \sum_{l\in\cE} \left(h_{k,l} + h_{k,-l}\right)
=
-\frac12 \sum_{l\in\cE} \nabla_l h_{k,-l}
=
-\frac12 {\ensuremath\left|{\Delta}\right|}^{1/2} \sum_{l\in\cE} \Gamma_l h_{k,-l},\end{aligned}$$ which shows indeed .
#### ${\cH_{-1}}$-condition
(fourth formulation): The drift vector field $V$ is realized as the curl of a *stationary and square integrable*, zero mean tensor field ${H}$, as shown in .
[**Remark.**]{} If the $\cH_{-1}$-condition does not hold it may still be possible that there exists a *non-square integrable* tensor variable ${h}:\Omega\to\R^{\cE\times\cE}$ which has the symmetries and with $v:\Omega\to\R^{\cE}$ realized as in . Then let ${H}:\Omega\times\Z^d\to\R^{\cE\times\cE}$ be the stationary lifting and we still get with a *stationary but not square integrable* tensor field. Note that this is not decidable in terms of the covariance matrix or its Fourier transform . The question of diffusive (or super-diffusive) asymptotic behaviour of the walk $t\mapsto X(t)$ in these cases is fully open.
In the next proposition – which essentially follows an argument from Kozlov [@kozlov_85] – we give a sufficient condition for the ${\cH_{-1}}$-condition to hold.
\[prop:suff cond for H-1\] If $p\mapsto \wh C(p)$ is twice continuously differentiable function in a neighbourhood of $p=0$ then the ${\cH_{-1}}$-condition holds.
For the duration of this proof we introduce the notation $$\begin{aligned}
\notag
B_{k,l}(x):={\ensuremath{\mathbf{E}\left(V_k(0)V_l(x)\right)}},
\quad
\wh B_{k,l}(p) := \sum_{x\in \Z^d} e^{\sqrt{-1}x\cdot p} B_{k,l} (x),\end{aligned}$$ with $k,l\in\cE, x\in\Z^d, p\in[-\pi,\pi]^d$. Hence for $i,j\in\{1,\dots,d\}$ $$\begin{aligned}
\notag
\wh C_{ij}(p)
=
\wh B_{e_i,e_j}(p) - \wh B_{-e_i,e_j}(p) - \wh B_{e_i,-e_j}(p) + \wh B_{-e_i,-e_j}(p).\end{aligned}$$ (The identity is meant in the sense of distributions.)
Note that due to the first clause in $$\begin{aligned}
\label{Chatvector}
\wh B_{k,l}(p)
=
-e^{\sqrt{-1}p\cdot k}\wh B_{-k,l}(p)
=
-e^{-\sqrt{-1}p\cdot l}\wh B_{k,-l}(p)
=
e^{\sqrt{-1}p\cdot (k-l)}\wh B_{-k,-l}(p).\end{aligned}$$ Using in the above expression of $C(p)$ in terms of $B(p)$, direct computations yield $$\begin{aligned}
\notag
\wh C_{ij} = \left(1+e^{-\sqrt{-1}p\cdot e_i}\right) \left(1+e^{\sqrt{-1}p\cdot e_j}\right) \wh B_{e_i,e_j}(p).\end{aligned}$$ Thus, the regularity condition imposed on $p\mapsto C(p)$ is equivalent to assuming the same regularity about $p\mapsto \wh B(p)$.
Next, due to the second clause of $$\begin{aligned}
\label{Chatdivfree}
\sum_{k\in\cE}\wh B_{k,l}(p)
=
\sum_{l\in\cE}\wh B_{k,l}(p)
=
0,\end{aligned}$$ and, from and again by direct computations we obtain $$\begin{aligned}
\label{equiv0}
\sum_{k,l\in\cE}
(1-e^{-\sqrt{-1}p\cdot k})(1-e^{\sqrt{-1}p\cdot l})\wh B_{k,l}(p)\equiv0.\end{aligned}$$ At $p=0$ we apply $\partial^2/\partial p_i\partial p_j$ to and get $$\begin{aligned}
\label{C(0)=0}
\wh C_{ij}(0)
=
\sum_{k,l\in\cE}
k_il_j \wh B_{k,l}(0)=0,
\qquad
i,j=1,\dots,d.\end{aligned}$$ Since $\wh C_{j,i}(p) = \wh C_{ij}(-p) = \overline{\wh C_{ij}(p)} $ and $p\mapsto \wh C(p)$ is assumed to be twice continuously differentiable at $p=0$, from it follows that $$\begin{aligned}
\notag
\wh C(p)=\Ordo({\ensuremath\left|{p}\right|}^2),
\qquad
\text{ as }
{\ensuremath\left|{p}\right|}\to0,\end{aligned}$$ which implies .
In particular it follows that sufficiently fast decay of correlations of the divergence-free drift field $V(x)$ implies the ${\cH_{-1}}$-condition . Note that the divergence-free condition is crucial in this argument.
Historical remarks {#app:Historical remarks}
==================
There exist a fair number of important earlier results to which we should compare Theorem \[thm:main\].
1. In Kozlov [@kozlov_85], Theorem II.3.3 claims the same result under the supplementary restrictive condition that the random field of jump probabilities $x\mapsto P(x)$ in be *finitely dependent*. However, as pointed out by Komorowski and Olla [@komorowski_olla_02], the proof is incomplete there. Also, the condition of finite dependence of the field of jump probabilities is a very serious restriction.
2. In Komorowski and Olla [@komorowski_olla_03a], Theorem 2.2, essentially the same result is announced as above. However, as noted in section 3.6 of [@komorowski_landim_olla_12] this proof is yet again incomplete.
3. To our knowledge the best fully proved result is Theorem 3.6 of [@komorowski_landim_olla_12] where the same result is proved under the condition that the stream tensor field $x\mapsto H(x)$ of Proposition \[prop:helmholtz\] be stationary and in $\cL^{\max\{2+\delta, d\}}$, $\delta>0$, rather than $\cL^2$. Note that the conditions of our theorem only request that the tensor field $x\mapsto H$ be square integrable. The proof of Theorem 3.6 in [@komorowski_landim_olla_12] is very technical, see sections 3.4 and 3.5 of the monograph.
4. The special case when the tensor field ${H}$ is actually in $\cL^{\infty}$ is fundamentally simpler. In this case the so-called *strong sector condition* of Varadhan [@varadhan_95] applies directly. This was noticed in [@komorowski_olla_03a]. See also section 3.3 of [@komorowski_landim_olla_12] and section \[app:Examples\] below.
5. Examine the following *diffusion* problem is as follows. Let $t\mapsto X(t)\in \R^d$ be the strong solution of the SDE $$\begin{aligned}
\label{sde}
{{\mathrm d}}X(t)
=
{{\mathrm d}}B(t) + \Phi(X(t)){{\mathrm d}}t,\end{aligned}$$ where $B(t)$ is standard $d$-dimensional Brownian motion and $\Phi:\R^d\to\R^d$ is a stationary and ergodic (under space-shifts) vector field on $\R^d$ which has zero mean $$\begin{aligned}
\notag
{\ensuremath{\mathbf{E}\left(\Phi(x)\right)}}=0,\end{aligned}$$ and is almost surely *divergence-free*: $$\begin{aligned}
\label{divfree in continuous space}
\operatorname{div}\Phi \equiv 0,
\ \ \
\mathrm{a.s.}\end{aligned}$$ It is analogous to the discrete-space problem studied in this paper in the case that $s_k$ is constant for all $k\in\cE$. In this case the $\cH_{-1}$-condition is $$\begin{aligned}
\label{hcond in continuous space}
\sum_{i=1}^d\int_{\R^d} {\ensuremath\left|{p}\right|}^{-2} \wh C_{ii}(p) {{\mathrm d}}p <\infty,\end{aligned}$$ where $$\begin{aligned}
\notag
\wh C_{ij}(p)
:=
\int_{\R^d} {\ensuremath{\mathbf{E}\left(\Phi_i(0)\Phi_j(x)\right)}} e^{\sqrt{-1}p\cdot x} {{\mathrm d}}x,
\qquad
p\in\R^d.\end{aligned}$$ It is a fact that, similarly to the $\Z^d$ lattice case, under minimally restrictive regularity conditions, a stationary and square integrable divergence-free drift field $x\mapsto \Phi(x)$ on $\R^d$ can be written as the curl of an antisymmetric stream tensor field with stationary increments $H:\R^d\to\R^{d\times d}$: $$\begin{aligned}
\notag
\Phi_i(x) = \sum_{j=1}^d \frac{\partial H_{ji}}{\partial x_j}(x).\end{aligned}$$ This is essentially Helmholtz’s theorem. See Proposition 11.1 of [@komorowski_landim_olla_12], which is the continuous-space analogue of Proposition \[prop:helmholtz\] of section \[app:The stream tensor field\] above. As shown in [@komorowski_landim_olla_12], the ${\cH_{-1}}$-condition is equivalent with the fact that the stream tensor ${H}$ is *stationary* (not just of stationary increments) *and square integrable*. The case of bounded ${H}$ was first considered in Papanicolaou and Varadhan [@papanicolaou_varadhan_81]. This paper is historically the first instant where the problem of diffusion in stationary divergence-free drift field was considered with mathematical rigour. Homogenization and central limit theorem for the diffusion , in *bounded* stream field, ${H}\in\cL^{\infty}$, was first proven in Osada [@osada_83]. Today the strongest result in the continuous space-time setup is due to Oelschläger [@oelschlager_88] where homogenization and CLT for the displacement is proved for square-integrable stationary stream tensor field, ${H}\in\cL^2$. Oelschläger’s proof consists in truncating the stream tensor and bounding the error. If the stream tensor field is stationary *Gaussian* then – as noted by Komorowski and Olla [@komorowski_olla_03b] – the *graded sector condition* of [@sethuraman_varadhan_yau_00] can be applied. See also chapters 10 and 11 of [@komorowski_landim_olla_12] for all existing results on the diffusion model , .
6. Attempts to apply Oelschläger’s method in the discrete ($\Z^d$ rather than $\R^d$) setting run into enormous technical difficulties, see chapter 3 of [@komorowski_landim_olla_12] and seemingly this approach can’t be fully accomplished beyond the overly restrictive condition ${H}\in\cL^{\max\{2+\delta, d\}}$. The main result of this paper, Theorem \[thm:main\] fills this gap between the restrictive condition ${H}\in\cL^{\max\{2+\delta, d\}}$ of Theorem 3.6 in [@komorowski_landim_olla_12] and the minimal restriction ${H}\in\cL^{2}$. The content of our Theorem \[thm:main\] is the discrete $\Z^d$-counterpart of Theorem 1 in Oelschläger [@oelschlager_88]. We also stress that our proof is conceptually and technically much simpler that of Theorem 3.6 in [@komorowski_landim_olla_12] or Theorem 1 in [@oelschlager_88]. The continuous space-time diffusion model — under the same regularity conditions as those of Oelschläger [@oelschlager_88] can be treated in a very similar way reproducing this way Theorem 1 of [@oelschlager_88] in a conceptually and technically simpler way. In order to keep this paper relatively short and transparent, those details will be presented elsewhere.
7. There exist results on *super-diffusive* behaviour of the random walk in doubly stochastic random environment , or diffusion in divergence-free random drift field , , when the ${\cH_{-1}}$-condition fails to hold. In Komorowski and Olla [@komorowski_olla_02] and Tóth and Valkó [@toth_valko_12] the diffusion model , is considered when the drift field $\Phi$ is Gaussian and the stream tensor field ${H}$ is *genuinely* delocalized: of stationary increment but not stationary. Super-diffusive bounds are proved.
Examples {#app:Examples}
========
Before formulating concrete examples let us spend a few words about the physical motivation and phenomenology of the problem considered. The continuous case discussed in the previous section, diffusion in divergence-free drift field, cf. - may model the drifting of a suspended particle in stationary turbulent incompressible flow. Very similarly, the lattice counterpart with jump rates satisfying describe a random walk whose local drift is driven by a stationary source- and sink-free flow. The interest in the asymptotic description of this kind of displacement dates back to the discovery of turbulence. However, divergence-free environments appear in many other natural contexts, too. See e.g. [@komorowski_landim_olla_12 chapter 11] or a surprising recent application to group theory by Bartholdi and Erschler [@bartholdi_erschler_11].
A phenomenological picture of these walks can be formulated in terms of randomly oriented cycles. Imagine that a translation invariant random “soup of cycles” — that is, a Poisson point process of oriented cycles — is placed on the lattice, and the walker is drifted along by these whirls. Now, local small cycles contribute to the diffusive behaviour. But occasionally very large cycles may cause on the long time scale faster-than-diffusive transport. Actually, this happens: in Komorowski and Olla [@komorowski_olla_02] and Tóth and Valkó [@toth_valko_12] anomalous *superdiffusive* behaviour is proved in particular cases when the $\cH_{-1}$-bound doesn’t hold. Our result establishes that on the other hand, the $\cH_{-1}$-bound ensures not only boundedness of the diffusivity but also normal behaviour under diffusive scaling.
And now, to some examples:
1. *Stationary and bounded stream field:* When there exists a *bounded* tensor valued variable ${h}:\Omega\to\R^{\cE\times\cE}$ with the symmetries and such that holds we define the multiplication operators $M_{k,l}$, $k,l\in\cE$, acting on $f\in\cH$: $$\begin{aligned}
\label{thetamultipl}
M_{k,l} f(\omega):= {h}_{k,l}(\omega) f(\omega).\end{aligned}$$ These are bounded selfadjoint operators and they inherit the symmetries of ${h}$ (recall the shift operators $T_k$, $k\in\cE$ from ): $$\begin{aligned}
\label{Nsymmetries}
\begin{gathered}
M_{l,k}=T_k M_{-k,l}T_{-k}=T_l M_{k,-l}T_{-l}=-M_{k,l},
\\[8pt]
\sum_{l\in\cE}M_{k,l}=M_k.
\end{gathered}\end{aligned}$$ As an alternative to , using , the skew-self-adjoint part of the infinitesimal generator is expressed as $$\begin{aligned}
\label{Aalt}
A=\sum_{k,l\in\cE} \nabla_{-k}M_{k,l}\nabla_l.\end{aligned}$$ In [@komorowski_olla_03a] and [@komorowski_landim_olla_12] this form of the operator $A$ is used. The operators $M_{k,l}$ are bounded and so is the operator $$\begin{aligned}
\label{bibabu}
B
:=
{\ensuremath\left|{\Delta}\right|}^{-1/2} A {\ensuremath\left|{\Delta}\right|}^{-1/2}
=
\sum_{k,l\in\cE} \Gamma_{-k}M_{k,l}\Gamma_l\end{aligned}$$ which plays a key rôle in our proof. Due to boundedness of $B$ the *strong sector condition* is valid in these cases and the central limit theorem for the displacement readily follows. See [@komorowski_olla_03a] and section 3.3 of [@komorowski_landim_olla_12].
Finitely dependent constructions of this type appear in Kozlov [@kozlov_85]. The so-called *cyclic walks* analysed in [@komorowski_olla_03a] and in section 3.3 of [@komorowski_landim_olla_12] are also of this nature. When the tensor variables ${h}:\Omega\to\R^{\cE\times\cE}$ in are in $\cL^2\setminus \cL^\infty$, the multiplication operators $M_{k,l}$ defined in are *unbounded*, the representation of the skew-self-adjoint part of the infinitesimal generator and the operator $B$ defined in become just *formal*. Nevertheless, Theorem 1 in Oelschläger [@oelschlager_88] and theorem 3.6 in [@komorowski_landim_olla_12] are proved by controlling approximations of $h_{k,l}$ and the unbounded operators $M_{k,l}$ by truncations at high levels.
2. *Stationary, square integrable but unbounded stream field:* We let, in arbitrary dimension $d$, $\Psi:\Z^d+(1/2,\dots, 1/2)\to\Z$ be a stationary, scalar, Lipschitz field with Lipschitz constant 1. As shown in Peled [@peled_10], such fields exist in sufficiently high dimension. Define ${H}:\Z^d\to\R^{\cE_{2}\times\cE_{2}}$ by $$\begin{aligned}
\notag
&
{H}_{e_i,e_j}(x)
:=
\frac 1{d}\Psi(x+(e_i+e_j)/2),
&&
x\in\Z^d,
\ \
1\le i<j \le d,\end{aligned}$$ and extend to $\left({H}_{k,l}(x)\right)_{k,l\in\cE}$ by the symmetries . The tensor field ${H}:\Z^d\to\R^{\cE_{2}\times\cE_{2}}$ defined this way will be stationary and $\cL^2$, but not necessary in $\cL^\infty$ — the uniform graph homomorphism of Peled [@peled_10], for example, is not bounded. Nevertheless, $V$ is bounded by 1, as it should, since $|H_{k,l}(x)+H_{-k,l}(x)|=|H_{k,l}(x)-H_{k,l}(x-k)|\le \frac 1d$ and $V$ is a sum of $d$ such terms.
3. *Randomly oriented Manhattan lattice:* Let $u_i:\Z^{d-1}\to \{-1,+1\}$, $i=1,\dots, d$, be translation invariant and ergodic, zero mean random fields, which are independent between them. Denote their covariances $$\begin{aligned}
\notag
c_i(y)
&
:=
{\ensuremath{\mathbf{E}\left(u_i(0), u_i(y)\right)}},
&&
y\in\Z^{d-1},
\\[8pt]
\notag
\hat c_i(p)
&
:=
\sum_{y\in\Z^{d-1}} e^{\sqrt{-1}p\cdot y} c_i(y),
&&
p\in[-\pi,\pi)^{d-1}.\end{aligned}$$ Define now the lattice vector field $$\begin{aligned}
\notag
V_{\pm e_i}(x)
:=
\pm u_i(x_1, \dots, x_{i-1}, \cancel{x_i}, x_{i+1}, \dots, x_d).\end{aligned}$$ Then the random vector field $V$ will satisfy all conditions in and $t\mapsto X(t)$ will actually be a random walk on the lattice $\Z^d$ whose line-paths parallel to the coordinate axes are randomly oriented in a shift-invariant and ergodic way. This oriented graph is called the *randomly oriented Manhattan lattice*. The covariances $C$ and $\wh C$ defined in , respectively, will be $$\begin{aligned}
\notag
C_{ij} (x)
&
=
\delta_{i,j} c_i(x_1, \dots, x_{i-1}, \cancel{x_i}, x_{i+1}, \dots, x_d),
\\[8pt]
\notag
\wh C_{ij}(p)
&
=
\delta_{i,j} \delta(p_i) \hat c_i(p_1, \dots, p_{i-1}, \cancel{p_i}, p_{i+1}, \dots, p_d).\end{aligned}$$ The $\cH_{-1}$-condition is in this case $$\begin{aligned}
\label{hcond for manhattan}
\sum_{i=1}^d
\int_{[-\pi,\pi]^{d-1}}
\wh D(q)^{-1}
\hat c_{i}(q) {{\mathrm d}}q <\infty.\end{aligned}$$ In the particular case when the random variables $u_i(y)$, $i\in\{1,\dots, d\}$, $y\in\Z^{d-1}$, are *independent fair coin-tosses*, $\hat c_i(q)\equiv1$. In this case, for $d=2,3$ the ${\cH_{-1}}$-condition fails to hold, the tensor field ${H}$ is *genuinely* of stationary increments. In these cases super-diffusivity of the walk $t\mapsto X(t)$ can be proved with the method of Tarrès, Tóth and Valkó [@tarres_toth_valko_12] (in the $2d$ case), respectively, of Tóth and Valkó [@toth_valko_12] (in the $3d$ case). In dimensions $d\ge4$ the ${\cH_{-1}}$-condition (and thus ) holds and the central limit theorem for the displacement follows from our Theorem \[thm:main\].
Appendix: Proof of Theorem RSC1 and Theorem RSC2
================================================
\[Proof of Theorem RSC1\]
Since the operators $C_\lambda$, $\lambda>0$, defined in are a priori and the operator $C$ is by assumption skew-self-adjoint, we can define the following bounded operators (actually contractions): $$\begin{aligned}
\label{eq:Klambda_contraction}
&
K_\lambda:=(I-C_\lambda)^{-1},
&&
{\ensuremath\left\|{K_\lambda}\right\|}\le1,
&&
\lambda>0,
\\
&
K:=(I-C)^{-1},
&&
{\ensuremath\left\|{K}\right\|}\le1.\notag\end{aligned}$$ Hence, we can write the resolvent $R_\lambda =(\lambda I-L)^{-1}$ as $$\begin{aligned}
\label{resolvent_master}
R_\lambda
=
(\lambda+S)^{-1/2}
K_\lambda
(\lambda+S)^{-1/2}.\end{aligned}$$
\[lem:Klambdastcvg\] Assume that the sequence of bounded operators $K_\lambda$ converges to $K$ in the strong operator topology: $$\begin{aligned}
\label{Klambdastcvg}
K_\lambda{\stackrel{\mathrm{st.op.top.}}{\longrightarrow}}K,
\qquad \text{as}\qquad\lambda\to0.\end{aligned}$$ Then for any $f\in\Dom(S^{-1/2})=\Ran(S^{1/2})$, the limits in hold.
\[Proof of Lemma \[lem:Klambdastcvg\]\] From the spectral theorem applied to the positive operator $S$, it is obvious that, as $\lambda\to0+$, $$\begin{aligned}
&
{\ensuremath\left\|{\lambda^{1/2}(\lambda+S)^{-1/2}}\right\|}\le1,
&&
\lambda^{1/2}(\lambda+S)^{-1/2}{\stackrel{\mathrm{st.op.top.}}{\longrightarrow}}0,
\notag\\[8pt]
\label{SlambdaS}
&
{\ensuremath\left\|{S^{1/2}(\lambda+S)^{-1/2}}\right\|}\le1,
&&
S^{1/2}(\lambda+S)^{-1/2}{\stackrel{\mathrm{st.op.top.}}{\longrightarrow}}I.\end{aligned}$$ We can write $f = S^{1/2} g$ with $g\in\cH$. Now, using , we get $$\begin{aligned}
\lambda^{1/2}u_\lambda
&=
\lambda^{1/2}(\lambda+S)^{-1/2}
K_\lambda
(\lambda+S)^{-1/2}S^{1/2} g,
\\[8pt]
S^{1/2}u_\lambda
&=
S^{1/2}(\lambda+S)^{-1/2}
K_\lambda
(\lambda+S)^{-1/2}S^{1/2} g.\end{aligned}$$ We get $$\begin{aligned}
S^{1/2}u_\lambda &=
S^{1/2}(\lambda + S)^{1/2} K_\lambda(\lambda+S)^{-1/2}S^{1/2}g
\stackrel{\textrm{$(\ref{SlambdaS})$}}{=}
S^{1/2}(\lambda+S)^{-1/2}K_\lambda(g+o(1))\\
\textrm{By (\ref{Klambdastcvg},\ref{eq:Klambda_contraction})}\qquad
&= S^{1/2}(\lambda+S)^{-1/2}(Kg+o(1))
\stackrel{\textrm{$(\ref{SlambdaS})$}}{=}
Kg+o(1)\end{aligned}$$ where the notation $o(1)$ is for convergence in norm as $\lambda\to 0$. Verifying the other condition of is similar.
In the next lemma, we formulate a sufficient condition for to hold.
\[lem:strrescvg\] Let $C_n$, $n\in\N$, and $C=C_\infty$ be densely defined closed (possibly unbounded) operators over the Hilbert space $\cH$. Let also $\cC_n$ and $\cC$ be a cores of definition of the operators $C_n$ and $C$, respectively. Assume that some (fixed) $\mu\in\C$ is in the intersection of the resolvent set of all operators $C_n$, $n\le\infty$, and $$\begin{aligned}
\label{unifbound}
\sup_{1\le n\le\infty}{\ensuremath\left\|{(\mu I - C_n)^{-1}}\right\|}<\infty,\end{aligned}$$ and for any $h\in\cC$ there exists a sequence $h_n\in\cC_n$ such that the following limits hold $$\begin{aligned}
\label{stcvgoncore}
\lim_{n\to\infty}{\ensuremath\left\|{h_n-h}\right\|}=0.
\qquad
\text{and}
\qquad
\lim_{n\to\infty}{\ensuremath\left\|{C_n h_n -C h}\right\|}=0.\end{aligned}$$ Then (i) and (ii) below hold.\
(i) $$\begin{aligned}
\label{strescvg}
(\mu I - C_n)^{-1}
{\stackrel{\mathrm{st.op.top.}}{\longrightarrow}}(\mu I - C)^{-1}.\end{aligned}$$ (ii) The sequence of operators $C_n$ converges *in the strong graph limit* sense to $C$.
\[Proof of Lemma \[lem:strrescvg\]\] (i) Since $\cC$ is a core for the densely defined closed operator $C$ and $\mu$ is in the resolvent set of $C$, the subspace $\wh{\cC}:=\{\wh h=(\mu I - C) h\,:\, h\in\cC\}$ is dense in $\cH$. For $\wh h\in\wh\cC$ let $h:= (\mu I - C)^{-1} \wh h\in\cC$ and choose a sequence $h_n\in\cC_n$ for which holds. Then $$(\mu I-C_n)^{-1} \wh h - (\mu I-C)^{-1} \wh h
=
\left( \mu (\mu I-C_n)^{-1} -I \right) (h-h_n)
+
(\mu I-C_n)^{-1} (C_n h_n -Ch),$$ and hence $$\begin{aligned}
&
{\ensuremath\left\|{(\mu I-C_n)^{-1} \wh h - (\mu I-C)^{-1} \wh h}\right\|}
\\
&\hskip3cm
\le
\left( {\ensuremath\left|{\mu}\right|}{\ensuremath\left\|{(\mu I-C_n)^{-1}}\right\|} +1\right) {\ensuremath\left\|{h-h_n}\right\|}
+
{\ensuremath\left\|{\mu I-C_n)^{-1}}\right\|} {\ensuremath\left\|{C_n h_n -Ch}\right\|}
\to 0. \end{aligned}$$ due to and . Since this is valid on the *dense* subspace $\wh\cC\subset\cH$, using again , we conclude .
\(ii) The proof of the “if” part of Theorem VIII. 26 in [@reed_simon_vol1_vol2_75] can be transposed without any essential alteration.
To finish the proof of Theorem RSC1 first apply Lemma \[lem:strrescvg\](i) to $C_\lambda$, $\lambda\to0+$, defined in , $C$ assumed (essentially) skew self-adjoint, and $\mu=1$. Note that $\mu=1$ is indeed in the resolvent set of all these operators and, indeed $\sup_{\lambda>0}{\ensuremath\left\|{(I-C_\lambda)^{-1}}\right\|}<\infty$ and ${\ensuremath\left\|{(I-C)^{-1}}\right\|}<\infty$, as required in , since the operators $C_\lambda$ are bounded and skew-self-adjoint and the operator $C$ is assumed to be essentially skew-self-adjoint. From Lemma \[lem:strrescvg\](i) it follows that that holds. Finally, quoting Lemma \[lem:Klambdastcvg\] we conclude the proof of Theorem RSC1.
From $0\le T\le cD$ it follows that $$\begin{aligned}
\label{Ddominates2}
0\le D \le S \le (1+c) D\end{aligned}$$ Let $$\begin{aligned}
\notag
&
V_\lambda:=
(\lambda I+D)^{1/2}
(\lambda I+S)^{-1/2},
&&
V=V_0:=
D^{1/2}
S^{-1/2}.\end{aligned}$$ The operator $V$ is a priori defined on $\Dom(S^{-1/2})=\Ran(S^{1/2})$, but as we see next, it extends by continuity to a bounded and invertible linear operator defined on the whole space $\cH$. Due to the following bounds hold uniformly for $\lambda\ge0$: $$\begin{aligned}
\notag
&
{\ensuremath\left\|{V_\lambda}\right\|}
=
{\ensuremath\left\|{V_\lambda^*}\right\|}
\le 1,
&&
{\ensuremath\left\|{V_\lambda^{-1}}\right\|}
=
{\ensuremath\left\|{(V_\lambda^{-1})^*}\right\|}
\le
\sqrt{1+c}.\end{aligned}$$ Let us show that bound on ${\ensuremath\left\|{V_\lambda}\right\|}$, the bound on ${\ensuremath\left\|{V_\lambda^{-1}}\right\|}$ is similar. We write $$\begin{aligned}
{\ensuremath\left\|{V_\lambda\varphi}\right\|}^2
&=\langle(\lambda I+D)^{1/2}(\lambda I+S)^{-1/2}\varphi,
(\lambda I+D)^{1/2}(\lambda I+S)^{-1/2}\varphi\rangle \\
& =
\langle(\lambda I+S)^{-1/2}\varphi,
(\lambda I+D)(\lambda I+S)^{-1/2}\varphi\rangle\\
&\le
\langle(\lambda I+S)^{-1/2}\varphi,
(\lambda I+S)(\lambda I+S)^{-1/2}\varphi\rangle
={\ensuremath\left\|{\varphi}\right\|}^2.\end{aligned}$$ From here, first of all, it follows that $$\begin{aligned}
\notag
\Dom(S^{-1/2})=\Dom(D^{-1/2}),\end{aligned}$$ and thus the $\cH_{-1}$-conditions $f\in\Dom(S^{-1/2})$, respectively, $f\in\Dom(D^{-1/2})$ in Theorem RSC1, respectively, Theorem RSC2, are actually the same. It is also easy to see that for any $\varphi\in\cH$ $$\lim_{\lambda\to0}
V_\lambda\varphi= V\varphi
\qquad
\text{and}
\qquad
\lim_{\lambda\to0}
V_\lambda^{-1}\varphi= V^{-1}\varphi.$$ That is, $V_\lambda{\stackrel{\mathrm{st.op.top.}}{\longrightarrow}}V$ and $V^{-1}_\lambda{\stackrel{\mathrm{st.op.top.}}{\longrightarrow}}V^{-1}$, as $\lambda\to0$, where ${\stackrel{\mathrm{st.op.top.}}{\longrightarrow}}$ stands for convergence in the strong operator topology.
Next write the operators $C_\lambda$ and $C$ from Theorem RSC1, as $$\begin{aligned}
\notag
&
C_\lambda
=
V^*_{\lambda} B_\lambda V_{\lambda},
&&
C
=
V^* B V.\end{aligned}$$ Now, from the fact that $V_\lambda$ and $V_\lambda^{-1}$ are all bounded, uniformly in $\lambda\ge0$, it readily follows that: (a) one can use $\cC=V^{-1}\cB$ as a core for the operator $C$; (b) $C$ is essentially skew-self-adjoint on $\cC$ if so was $B$ on $\cB$; and (c) the limit follows from by straightforward manipulations. Indeed, for $\psi\in\cC$ define $\varphi:=V\psi\in\cB$ and let $\varphi_\lambda \in\cH$ be such that the limits in hold. Define $\psi_\lambda:=V_\lambda^{-1}\varphi_\lambda$. Then the limits in clearly hold: $$\begin{aligned}
{\ensuremath\left\|{\psi_\lambda-\psi}\right\|}
&=
{\ensuremath\left\|{V_\lambda^{-1}\varphi_\lambda - V^{-1}\varphi}\right\|}
\le
{\ensuremath\left\|{V_\lambda^{-1}}\right\|}{\ensuremath\left\|{\varphi_\lambda-\varphi}\right\|}
+
{\ensuremath\left\|{V^{-1}_\lambda\varphi-V^{-1}\varphi}\right\|}
\to0,
\\[10pt]
{\ensuremath\left\|{C_\lambda\psi_\lambda-C\psi}\right\|}
&=
{\ensuremath\left\|{V^{*}_\lambda B_\lambda\varphi_\lambda - V^{*}B\varphi}\right\|}
\le
{\ensuremath\left\|{V_\lambda^{*}}\right\|}{\ensuremath\left\|{B_\lambda\varphi_\lambda-B\varphi}\right\|}
+
{\ensuremath\left\|{V^{*}_\lambda B\varphi-V^{*} B\varphi}\right\|}
\to0.\qedhere\end{aligned}$$
[**Acknowledgements:**]{} We thank an anonymous reviewer for thorough criticism, and in particular for the recommendation to formulate a more generally valid version of the main result. The research of BT is partially supported by OTKA (HU) grant K 100473 and by EPSRC (UK) Fellowship, grant no. EP/P003656/1. The research of GK is partially supported by the Israel Science Foundation and the Jesselson Foundation. Both authors acknowledge mobility support by The Leverhulme Trust (UK) through the International Network “Laplacians, Random Walks, Quantum Spin Systems”.
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---
abstract: 'Deligne and Kato proved a formula computing the dimension of the nearby cycles complex of an $\ell$-adic sheaf on a relative curve over an excellent strictly henselian trait. In this article, we reprove this formula using Abbes-Saito’s ramification theory.'
address: 'Haoyu Hu, IHES, Le Brois-Marie, 35 Rue de Chartres, 91440 Bures-sur-Yvette, France.'
author:
- Haoyu Hu
title: 'Ramification and nearby cycles for $\ell$-adic sheaves on relative curves'
---
Introduction
============
{#set intro}
Let $R$ be an excellent strictly henselian discrete valuation ring of residue characteristic $p>0$, $S=\operatorname{Spec}(R)$, $s$ (resp. $\eta$, resp. $\bar\eta$) the closed point (resp. the generic point, resp. a geometric generic point) of $S$. Let $\mathfrak{X}$ be a smooth relative curve over $S$, $x$ a closed point of the special fiber $\fX_s$, $X$ the strict henselization of $\fX$ at $x$, $U$ a non-empty open sub-scheme of $X_{\eta}$, and $u:U{\rightarrow}X_{\eta}$ the canonical injection. Let $\L$ a finite field of characteristic $\ell\neq p$, and $\sF$ a locally constant constructible étale sheaf of $\L$-module on $U$. The spaces of nearby cycles of $\sF$ $$\Psi^i_x(u_!\sF)=\mathrm{H}^i_{\text{\'et}}(X_{\bar\eta},u_!\sF)\ \ \ (i\geqslant 0)$$ vanish when $i\geqslant 2$ ([@sga7ii] XIII, [@fu] 9.2.2) and the dimension of $\Psi^0_x(u_!\sF)$ is easy to compute. The aim of this article is to reprove a Deligne-Kato’s formula that computes the dimension of $\Psi_x^1(u_!\sF)$ [@lau; @kato; @vc; @kato; @scdv] using Abbes-Saito’s ramification theory [@as; @i; @as; @ii].
{#section}
Let $\fp$ be the generic point of the special fiber $X_s$. We denote by $\k(\fp)$ the residue field of $\fp$, which is the fraction field of a strictly henselian discrete valuation ring. Assume first that $\sF$ can be extended to a locally constant constructible sheaf $\wt \sF$ on an open sub-scheme $\wt U$ of $X$ containing $\fp$. Then Deligne computes the dimension of $\Psi^1_x(u_!\sF)$. Let $\operatorname{sw}_{\fp}(\wt{\sF})$ be the Swan conductor of the pull-back of $\wt\sF$ on $\operatorname{Spec}(\k(\fp))$ and let $$\varphi(s)=\operatorname{sw}_{\fp}(\wt\sF)+\operatorname{rank}(\sF).$$ On the other hand, for any $t\in X_{\bar\eta}-U_{\bar\eta}$, let $\operatorname{sw}_{t}(\sF)$ be the Swan conductor of the pull-back of $\sF$ on $\operatorname{Spec}({\mathcal{O}}_{X_{\bar\eta},t}){\times}_XU$, and let $$\varphi(\eta)=\sum_{t\in X_{\bar\eta}-U_{\bar\eta}}(\operatorname{sw}_t(\sF)+\operatorname{rank}(\sF)).$$ Then, Deligne’s formula is ([@lau] 5.1.1) $$\label{deligne}
\dim_{\L}\Psi^0_x(u_!\sF)-\dim_{\L}\Psi^1_x(u_!\sF)=\varphi(s)-\varphi(\eta).$$
{#kato mtds}
Kato generalized Deligne’s formula for any $\sF$. His formula has the same form as . The definition of the invariant $\varphi(\eta)$ is the same as above, but $\varphi(s)$ cannot be defined by the same method. He provided two definitions of $\varphi(s)$. The first one uses a ramification theory for valuation rings of rank two, that he developed for this purpose [@kato; @vc]. The second one uses his notion of Swan conductors with differential values [@kato; @scdv]. Both methods rely on Epp’s partial semi-stable reduction theorem [@epp]. In this article, we define the invariant $\varphi(s)$ in terms of ramification theory of Abbes and Saito [@as; @i; @as; @ii]. The case when $\sF$ has rank $1$ is due to Abbes and Saito ([@as; @ft] Appendix A).
{#section-1}
Let $K$ be a complete discrete valuation field, ${\mathcal{O}_K}$ its integer ring, ${\mathfrak m}_K$ the maximal ideal of ${\mathcal{O}_K}$ and $F$ the residue field of ${\mathcal{O}_K}$. We assume that $F$ is of finite type over a perfect field $F_0$ of characteristic $p$. We denote by ${\overline}K$ a separable closure of $K$, by ${\mathcal{O}}_{{\overline}K}$ the integral closure of ${\mathcal{O}_K}$ in ${\overline}K$, by ${\overline}F$ the residue field of ${\mathcal{O}}_{{\overline}K}$, by $v$ the valuation of ${\overline}K$ normalized by $v(K^{{\times}})=\Z$ and by $G_K$ the Galois group of ${\overline}K/K$. Abbes and Saito defined a decreasing filtration $G^r_{K,\log}$ ($r\in \Q_{\geqslant 0}$) of $G_K$, called the logarithmic ramification filtration. For any rational number $r\geqslant0$, we put $G_{K,\log}^{r+}={\overline}{\bigcup_{b>r}G^b_{K,\log}}$. Then $P=G_{K,\log}^{0+}$ is the wild inertia subgroup of $G_K$ ([@as; @i] 3.15). For any rational number $r>0$, the graded piece $$\operatorname{Gr}^r_{\log}G_K=G_{K,\log}^r\big/G^{r+}_{K,\log}$$ is abelian and killed by $p$ ([@saito; @cc] 1.24, [@as; @iii] Th. 2).
For any $r\in\Q$, we denote by ${\mathfrak m}^r_{{\overline}K}$ (resp. ${\mathfrak m}^{r+}_{{\overline}K}$) the set of elements of ${\overline}K$ such that $v(x)\geqslant r$ (resp. $v(x)>r$). Let $\O^1_F(\log)$ be the $F$-vector space $$\O^1_F(\log)=(\O^1_{F/F_0}{\oplus}(F{\otimes}_{\Z} K^{{\times}}))/({\mathrm d}\bar a-\bar a{\otimes}a;\;a\in{\mathcal{O}_K}^{{\times}}),$$ where $\bar a$ is the residue class of $a$ in $F$. We have a canonical exact sequence of finite dimensional $F$-vector spaces $$0{\rightarrow}\O^1_F{\rightarrow}\O^1_F(\log){\rightarrow}F{\rightarrow}0.$$ For any rational number $r>0$, there exists a canonical injective homomorphism ([@saito; @cc] 1.24, [@as; @iii] Th. 2), called the [*refined Swan conductor*]{}, $$\operatorname{rsw}:\operatorname{Hom}_{\F_p}(\operatorname{Gr}^r_{\log}G_K,\F_p){\rightarrow}\O^1_F(\log){\otimes}_F{\mathfrak m}^{-r}_{{\overline}K}/{\mathfrak m}^{-r+}_{{\overline}K}.$$
Let $M$ be a finite dimensional ${\L}$-vector space on which $P$ acts through a finite discrete quotient, $$M={\oplus}_{r\in\Q_{\geqslant 0}} M^{(r)}$$ the slope decomposition of $M$ (cf. \[slope decom lemma\]), and for any rational number $r>0$, $$M^{(r)}={\oplus}_{\chi}M^{(r)}_{\chi}$$ the central character decomposition of $M^{(r)}$, where the sum runs over finitely many characters $\chi:\operatorname{Gr}^r_{\log}G_K{\rightarrow}{\L}_{\chi}^{{\times}}$ such that ${\L}_{\chi}$ is a finite extension of ${\L}$ (cf. \[center char decomp\]). Enlarging ${\L}$, we may assume that for all rational number $r>0$ and for all central characters $\chi$ of $M^{(r)}$, ${\L}={\L}_{\chi}$. We fix a non-trivial character $\psi_0:\F_p{\rightarrow}{\L}^{{\times}}$. Since $\operatorname{Gr}^r_{\log}G_K$ is abelian and killed by $p$, $\chi$ factors uniquely through $\operatorname{Gr}^r_{\log}G_K{\rightarrow}\F_p{\xrightarrow}{\psi_0} {\L}^{{\times}}$. We denote abusively by $\c:\operatorname{Gr}^r_{\log}G_K{\rightarrow}\F_p$ the induced character. We fix a uniformizer $\p$ of ${\mathcal{O}_K}$. We define [*Abbes-Saito’s characteristic cycle*]{} of $M$ and denote by ${\mathrm{CC}}_{\psi_0}(M)$ the following section $${\mathrm{CC}}_{\psi_0}(M)={\bigotimes}_{r\in\Q_{> 0}}{\bigotimes}_{\c\in X(r)}(\operatorname{rsw}(\c){\otimes}\p^{r})^{\dim_{\L} M^{(r)}_{\c}}\in (\O^1_F(\log){\otimes}_{F}{\overline}F)^{{\otimes}\dim_{A}M/M^{(0)}}.$$
{#cc=kcc intro}
In the following, we assume that $p$ is not a uniformizer of $K$ (i.e. either $K$ has characteristic $p$ or $K$ has characteristic zero and $p$ is not a uniformizer of ${\mathcal{O}_K}$). Let $L$ be a finite Galois extension of $K$ of Group $G$. We assume that $L/K$ has ramification index one and that the residue field extension is non-trivial, purely inseparable and monogenic ; we say that the extension $L/K$ is of type (II) (c.f. \[type\]). Let $M$ be a finite ${\L}$-vector space on which $G_K$ acts through $G$. We prove that, for any rational number $r>0$, and any central character $\chi:\operatorname{Gr}^r_{\log}G_K{\rightarrow}\F_p$ of $M^{(r)}$, we have (\[thlog factor xi\]) $$\operatorname{rsw}(\c)\in \O^1_F{\otimes}_{F}{\mathfrak m}^{-r}_{{\overline}K}/{\mathfrak m}^{-r+}_{{\overline}K}.$$ Hence, we have ${\mathrm{CC}}_{\psi_0}(M)\in (\O^1_F{\otimes}_F{\overline}F)^{{\otimes}m}$, where $m=\dim_AM/M^{(0)}$ (\[cc in O\]). On the other hand, using Kato’s theory of Swan conductors with differential values, we can define [*Kato’s characteristic cycle*]{} ${\mathrm{KCC}}_{\psi_0(1)}(M)$ . Our main result is the following equality $$\label{general equal intro}
{\mathrm{CC}}_{\psi_0}(M)={\mathrm{KCC}}_{\psi_0(1)}(M).$$ Using Kato’s theory, we deduce a Hasse-Arf type theorem (\[hasse arf cc\]) $${\mathrm{CC}}_{\psi_0}(M)\in (\O^1_F)^m{\subset}(\O^1_F{\otimes}_F{\overline}F)^m,$$ and an induction formula for Abbes-Saito’s characteristic cycle.
{#section-2}
Under the assumptions of \[set intro\], we can now give the new definition of $\varphi(s)$. Firstly, by Epp’s results [@epp], we can reduce to the case where $\sF$ is trivialized by a Galois étale connected covering $U'$ of $U$ such that the special fiber of the normalization $X'$ of $X$ in $U'$ is reduced. We denote by ${\widehat}{\mathcal{O}}_{X,\fp}$ the completion of ${\mathcal{O}}_{X,\fp}$, by $K_{\fp}$ the fraction field of ${\widehat}{\mathcal{O}}_{X,\fp}$ and by $\sF_{\fp}$ the representation of ${\mathrm{Gal}}(K^{\mathrm{sep}}_{\fp}/K_{\fp})$ corresponding to the pull-back of $\sF$ on $\operatorname{Spec}({\widehat}{\mathcal{O}}_{X,\fp}){\times}_XU$. The latter factors through the Galois group of a finite Galois extension $L_{\fp}$ of $K_{\fp}$, which is of type (II) over an unramified extension of $K_{\fp}$. We fix a uniformizer $\p$ of $R$ and a non-trivial character $\psi_0:\F_p{\rightarrow}\L^{{\times}}$. We still have ${\mathrm{CC}}_{\psi_0}(\sF_{\fp})\in (\O^1_{\k(\fp)})^{{\otimes}m}$ (cf. \[remark general equal\]). We denote by $\operatorname{ord}_{\fp}$ the valuation of $\kappa(\fp)$ normalized by $\operatorname{ord}_{\fp}(\kappa(\fp)^{{\times}})=\Z$ and abusively by $\operatorname{ord}_{\fp}:\O^1_{\kappa(\fp)}-\{0\}{\rightarrow}\Z$ the map defined by $\operatorname{ord}_{\fp}(\a{\mathrm d}\b)=\operatorname{ord}_{\fp}(\a)$, if $\a,\b\in\kappa(\fp)^{{\times}}$ and $\operatorname{ord}_{\fp}(\b)=1$. The latter can be uniquely extended to $(\O^1_{\k(\fp)})^{{\otimes}r}-\{0\}$ for any integer $r\geqslant 1$. We denote by ${\overline}{\sF}_{\fp}$ the restriction to $\operatorname{Spec}(\kappa(\fp))$ of the direct image of $\sF_{\fp}$ by the map $\operatorname{Spec}(K_{\fp}){\rightarrow}\operatorname{Spec}({\widehat}{{\mathcal{O}}}_{X,\fp})$. It corresponds to a representation of ${\mathrm{Gal}}({\overline}{\k(\fp)}/\k(\fp))$. The invariant $\varphi(s)$ is defined by $$\label{phi s}
\varphi(s)=-\operatorname{ord}_{\fp}({\mathrm{CC}}_{\psi_0}(\sF_{\fp}))+\operatorname{sw}_s({\overline}{\sF}_{\fp})+\operatorname{rank}({\overline}{\sF}_{\fp}).$$ In fact, Kato’s second definition of $\varphi(s)$ ([@kato; @scdv] 4.4) is obtained by replacing ${\mathrm{CC}}_{\psi_0}(\sF_{\fp})$ by ${\mathrm{KCC}}_{\psi_0(1)}(\sF_{\fp})$ in . Hence, from , we deduce that Deligne-Kato’s formula holds true with our definition (cf. \[deligne kato\]).
{#section-3}
Deligne-Kato’s formula has already had important applications. For instance, Deligne’s formula could be used in Laumon’s work on local Fourier transform ([@lautf] 2.4.3) and Kato’s formula was recently used in the work of Obus and Wewers on local lifting problem [@ow]. We would like to mention that Laumon’s formula of the rank of the local Fourier transform is a direct application of the formulation of Deligne-Kato’s formula using . Indeed, it was reproved in ([@as; @ft] Appendix B) by reducing to the rank 1 case by Brauer theorem.
{#section-4}
This article is organized as follows. We briefly introduce Kato’s swan conductor with differential values and Abbes-Saito’s ramification theory in $\S$3 and $\S$4, respectively. We study in $\S$5 the ramification of extensions of type (II). We recall tubular neighborhoods and normalized integral models in $\S$6. We study the isogeny associated to an extension of type (II) in $\S$7 in the equal character case and in $\S$8 in the unequal characteristic case. Using the results of these two sections, we prove the main theorem \[theorem rsw\] in $\S$9. In $\S$10, the heart of this article, we compare Kato’s characteristic cycle and Abbes-Saito’s characteristic cycle. The last section is devoted to Deligne-Kato’s formula by using Abbes-Saito’s characteristic cycle.
Acknowledgement {#acknowledgement .unnumbered}
---------------
This article is a part of the author’s thesis at Université Paris-Sud and Nankai University. The author would like to express his deepest gratitude to his supervisors Ahmed Abbes and Lei Fu for leading him to this area and for patiently guiding him in solving this problem. The author is also grateful to Fonds Chern and Fondation Mathématiques Jacques Hadamard for their support during his stay in France.
Notation
========
{#basic notes}
In this article, $K$ denotes a complete discrete valuation field, ${\mathcal{O}_K}$ its integer ring, ${\mathfrak m}_K$ the maximal ideal of ${\mathcal{O}_K}$ and $F$ the residue field of ${\mathcal{O}_K}$. We assume that the characteristic of $F$ is $p>0$. We fix a uniformizer $\p$ of ${\mathcal{O}_K}$. Let ${\overline}K$ be a separable closure of $K$, $G_K$ the Galois group of ${\overline}K$ over $K$, ${\mathcal{O}}_{{\overline}K}$ the integral closure of ${\mathcal{O}_K}$ in ${\overline}K$, ${\overline}F$ the residue field of ${\mathcal{O}}_{{\overline}K}$ and $v$ the valuation of ${\overline}K$ normalized by $v(K^{{\times}})=\Z$. We denote by ${\text{F\'E}_{/K}}$ the category of finite étale $K$-algebras. For any object $K'$ of ${\text{F\'E}_{/K}}$, we denote by ${\mathcal{O}}_{K'}$ the integer ring of $K'$ and by ${\mathfrak m}_{K'}$ the radical of ${\mathcal{O}}_{K'}$.
{#space}
For a field $k$ and one dimensional $k$-vector spaces $V_1,\dots, V_m$, we denote by $k\< V_1,\dots,V_m \>$ the $k$-algebra $$\label{plus} {\bigoplus}\limits_{(i_1,...,i_m)\in {\Z}^m} V_1^{{\otimes}i_1}{\otimes}\cdots {\otimes}V_m^{{\otimes}i_m},$$ and by $(k\<V_1,\dots,V_m\>)^{\times}$ its group of units. An element of $(k\<V_1,\dots,V_m\>)^{{\times}}$ is contained in some vector space $V_1^{{\otimes}i_1}{\otimes}\cdots {\otimes}V_m^{{\otimes}i_m}$. Such an element $x$ will be denoted by $[x]$ and we adopt the additive notation, i.e. $[x]+[y]=[x\cdot y]$ and $-[x]=[x^{-1}]$. If for each $1\le i\le m$, $e_i$ is a non-zero element of $V_i$, we have an isomorphism $$k\<V_1,\dots,V_m\>{{\xrightarrow}{\sim}}k[X_1,\dots,X_m,X_1^{-1},\dots,X_m^{-1}],\ \ \
e_i{\mapsto}X_i,$$ and hence an isomorphism $$\label{conductor space}
(k\<V_1,\dots,V_m\>)^{\times}{{\xrightarrow}{\sim}}k^{\times}{\oplus}\Z^m.$$
Kato’s Swan conductors with differential values
===============================================
{#section-5}
In this section, we fix a finite separable extension $L$ of $K$ of ramification index $e$ contained in ${\overline}K$. We denote by ${\mathcal{O}_L}$ its integer ring and by $E$ the residue field of ${\mathcal{O}_L}$.
{#rkrl}
We denote the group $(F\<{{\mathfrak m}_K/{\mathfrak m}_K^2}\>)^{{\times}}$ by $R_K$ and the group $(E\<{{\mathfrak m}_L/{\mathfrak m}_L^2}\>)^{{\times}}$ by $R_L$ (cf. \[space\]). The canonical isomorphisms $$\label{iso cs1}
E{\otimes}_F({{\mathfrak m}_K/{\mathfrak m}_K^2}){{\xrightarrow}{\sim}}{\mathfrak m}_{L}^e/{\mathfrak m}_{L}^{e+1},$$ $$\label{iso cs2}
({{\mathfrak m}_L/{\mathfrak m}_L^2})^{{\otimes}e}{{\xrightarrow}{\sim}}{\mathfrak m}_{L}^e/{\mathfrak m}_{L}^{e+1},$$ induce an injective homomorphism of $F$-algebras $$F\<{{\mathfrak m}_K/{\mathfrak m}_K^2}\>{\rightarrow}E\<{{\mathfrak m}_L/{\mathfrak m}_L^2}\>$$ and hence an injective homomorphism $R_K{\rightarrow}R_{L}$.
{#type}
Kato’s theory applies if the extension $L/K$ is of one of the following types ([@kato; @scdv], 1.5):
- $L/K$ is totally ramified (i.e. $F=E$) ;
- the ramification index of $L/K$ is $1$ and the residue field extension $E/F$ is purely inseparable and monogenic.
Observe that in both cases, ${\mathcal{O}_L}$ is monogenic over ${\mathcal{O}_K}$. These two cases do not cover all finite separable extensions.
In the remaining part of this section, we assume that $L/K$ is of type (II). We denote by $p^n$ the degree of the residue extension $E/F$. We choose an element $h\in{\mathcal{O}_L}$ such that its reduction $\bar h\in E$ is the generator of $E/F$ and a lifting $a\in {\mathcal{O}_K}$ of $\bar a=\bar h^{p^n}\in F$.
\[frob\] Let $V$ be the kernel of the canonical morphism $\O^1_F{\rightarrow}\O^1_E$. Denote by $\varrho$ the morphism $E{\rightarrow}F,\;b{\mapsto}b^{p^n}$, by $\phi$ the morphism $F{\rightarrow}F,\;b{\mapsto}b^{p^n}$, and by $\varphi$ the morphism $E{\rightarrow}E,\;b{\mapsto}b^{p^n}$.
- The $F$-vector space $V$ is of dimension $1$, generated by ${\mathrm d}\bar a$.
- The $E$-vector space $\O^1_{E/F}$ is of dimension $1$, generated by ${\mathrm d}\bar h$.
- The canonical morphism $F{\otimes}_{\varrho,E}\O^1_{E/F}{\rightarrow}\O^1_{F/\phi(F)}=\O^1_F$ associated to $F{\rightarrow}E{\xrightarrow}{\varrho} F$ is injective with image $V$.
- For any $1$-dimensional $E$ vector space $W$, the morphism $$E{\otimes}_{\varphi,E}W{\rightarrow}W^{{\otimes}p^n}, \ \ \ y{\otimes}z {\mapsto}yz^{{\otimes}{p^n}}$$ is an isomorphism.
- There exist a canonical $E$-linear isomorphism $$\label{vo}
E{\otimes}_F V{{\xrightarrow}{\sim}}(\O^1_{E/F})^{{\otimes}p^n},$$ that maps $y{\otimes}{\mathrm d}\bar a$ to $y({\mathrm d}\bar h)^{{\otimes}p^n}$.
(i), (ii), (iv) are obvious. We have two canonical exact sequences of differential modules corresponding to the extensions $\phi:F{\rightarrow}E{\xrightarrow}{\varrho} F$ and $\varphi:E{\xrightarrow}{\varrho}F{\rightarrow}E$, $$F{\otimes}_{\varrho,E}\O^1_{E/F}{\xrightarrow}{\b}\O^1_F{\rightarrow}\O^1_{F/\varrho(E)}{\rightarrow}0,$$ $$E{\otimes}_{F}\O^1_{F/\varrho(E)}{\rightarrow}\O^1_E{\rightarrow}\O^1_{E/F}{\rightarrow}0.$$ Since the canonical morphism $\O^1_F{\rightarrow}\O^1_E$ factors as $$\O^1_F{\rightarrow}\O^1_{F/\varrho(E)}{\rightarrow}E{\otimes}_{F}\O^1_{F/\varrho(E)}{\rightarrow}\O^1_E,$$ the image of $F{\otimes}_{\varrho,E}\O^1_{E/F}$ in $\O^1_E$ is $\{0\}$. Hence the image of $\b$ lies in $V$. Since the kernel of $\O^1_F{\rightarrow}\O^1_{F/\varrho(E)}$ is not zero (as it contains ${\mathrm d}\bar a$) and since $F{\otimes}_{\varrho,E}\O^1_{E/F}$ is of dimension 1, $\b$ is injective. Hence $\b$ induces an isomorphism $$\b:F{\otimes}_{\varrho,E}\O^1_{E/F}{{\xrightarrow}{\sim}}V.$$ From (ii) and (iv), we obtain an isomorphism $$\b':E{\otimes}_{\varphi,E}\O^1_{E/F}{\rightarrow}(\O^1_{E/F})^{{\otimes}p^n}, \ \ \ y{\otimes}z {\mathrm d}\bar h{\mapsto}yz^{p^n}({\mathrm d}\bar h)^{{\otimes}p^n}.$$ We take for the isomorphism $\b'\circ({\mathrm{id}}_E{\otimes}\b)^{-1}$.
{#2type}
Let $V$ be the kernel of the canonical morphism $\O^1_F{\rightarrow}\O^1_E$ (\[frob\]). We put $${S_{K,L}}=(F\<{{\mathfrak m}_K/{\mathfrak m}_K^2}, V\>)^{\times} \quad \mathrm{and} \quad{S_{L/K}}=(E\<{{\mathfrak m}_L/{\mathfrak m}_L^2}, \O_{E/F}^1\>)^{\times}.$$ From and , we obtain an injective homomorphism of $F$-algebras $$F\<{{\mathfrak m}_K/{\mathfrak m}_K^2},V\>{\hookrightarrow}E\<{{\mathfrak m}_L/{\mathfrak m}_L^2},\O^1_{E/F}\>,$$ which induces an injective homomorphism $$\label{injs} {S_{K,L}}{\hookrightarrow}{S_{L/K}}.$$
{#section-6}
Let $L'$ be a subfield of $L$ containing $K$, ${\mathcal{O}}_{L'}$ its integer ring and $E'$ its residue field. When $L'\neq L$ (resp. $L'\neq K$), the extension $L/L'$ (resp. $L'/K$) is of type (II) ; we consider $S_{L',L}$ (resp. $S_{L'/K}$) as a subgroup of $S_{L/K}$ containing ${S_{K,L}}$, by functoriality. If $K\neq L'\neq L$, the following canonical maps $$\ker(\O^1_F{\rightarrow}\O^1_{E'}){\rightarrow}\ker(\O^1_F{\rightarrow}\O^1_E),$$ $$\O^1_{E/F}{\rightarrow}\O^1_{E/E'},$$ $$\ker(\O^1_{E'}{\rightarrow}\O^1_E){\rightarrow}\O^1_{E'/F}$$ are isomorphisms by considering dimensions, which give the following relations: $${S_{K,L}}=S_{K,L'}{\subset}S_{L'/K}=S_{L',L}{\subset}S_{L/L'}=S_{L/K}.$$
{#different}
Let $i$ be the maximal integer such that $\operatorname{Tr}_{L/K} (m^i_L)=O_K$. The surjective homomorphism $\operatorname{Tr}_{L/K}:{\mathfrak m}^i_L/{\mathfrak m}^{i+1}_L{\rightarrow}O_K/{\mathfrak m}_K=F$ induces an $E$-isomorphism $${\mathfrak m}^i_L/{\mathfrak m}^{i+1}_L{{\xrightarrow}{\sim}}\operatorname{Hom}_F(E,F),\ \ \ b{\mapsto}(a{\mapsto}\operatorname{Tr}_{L/K}(ab)),$$ and hence a basis of $({{\mathfrak m}_L/{\mathfrak m}_L^2})^{{\otimes}(-i)}{\otimes}_E\operatorname{Hom}_F(E,F)$, that we call Kato’s different of $L/K$ and denote by $\di(L/K)$ ([@kato; @scdv] 2.1).
{#section-7}
Following Kato ([@kato; @scdv] 2.3), there is an $F$-linear map $\operatorname{Tr}_{E/F}:\O_E^1{\rightarrow}\O^1_F$ characterized by $$\operatorname{Tr}_{E/F}{\left}({\frac}{{\mathrm d}x}{x}{\right})={\frac}{{\mathrm d}x^{p^n}}{x^{p^n}},\quad \operatorname{Tr}_{E/F}{\left}(x^i{\frac}{{\mathrm d}x}{x}{\right})=0,$$ for any $x\in E^{{\times}}$ and $1\le i\le p^n-1$. Its image is $V$ (\[frob\]) and it induces an isomorphism $$\label{ohom}
\O^1_{E/F}{\xrightarrow}{\sim}\operatorname{Hom}_F(E,V),\quad \o{\mapsto}(a{\mapsto}\operatorname{Tr}_{E/F}(a\o)).$$ Hence we obtain a sequence of isomorphisms $$\label{r to s}
\operatorname{Hom}_F(E,F){\xrightarrow}{\eqref{ohom}}\O^1_{E/F}{\otimes}_F V^{{\otimes}(-1)}{\xrightarrow}{\eqref{vo}}\O^1_{E/F}{\otimes}_E(\O^1_{E/F})^{{\otimes}(-p^n)}=(\O^1_{E/F})^{{\otimes}(1-p^n)},$$ by which $E\<{{\mathfrak m}_L/{\mathfrak m}_L^2}\>{\otimes}_E\operatorname{Hom}_F(E,F)$ is a sub-$E\<{{\mathfrak m}_L/{\mathfrak m}_L^2}\>$-module of $E\<{{\mathfrak m}_L/{\mathfrak m}_L^2},\O^1_{E/F}\>$. Hence we may consider $\di(L/K)$ (\[different\]) as an element of $S_{L/K}$.
Let $L'$ be a subfield of $L$ containing $K$. If $L=L'$ (resp. $L'=K$), we put $\di(L/L')=[1]$ (resp. $\di(L'/K)=[1]$). Then, we have $$\label{relation diff}
\di(L/K)=\di(L/L')+\di(L'/K)\in{S_{L/K}}.$$
We consider $\di(L'/K)\in S_{L'/K}{\subseteq}{S_{L/K}}$.
{#def s}
In the rest of this section, we assume that the extension $L/K$ is Galois of group $G$. For any $\s\in G-\{1\}$, we put $$s_G(\s)=[{\mathrm d}\bar h]-[h-\s(h)]\in{S_{L/K}},$$ where the term $[{\mathrm d}\bar h]$ corresponds to the element ${\mathrm d}\bar h$ in $\O^1_{E/F}$ and the term $[h-\s(\s)]$ corresponds abusively to the class of $h-\s(h)\in ({\mathfrak m}_L/{\mathfrak m}_L^2)^{{\otimes}v(h-\s(h))}$. The definition of $s_G(\s)$ is independent of the choice of the generator $h$ ([@kato; @scdv] 1.8). We also put $$\label{s1}
s_G(1)=-\sum_{\s\in G-\{1\}}s_G(\s)\in{S_{L/K}}.$$ We have ([@kato; @scdv] (2.4)) $$\label{sg1}
s_G(1)=\di(L/K).$$
Let $H$ be a normal subgroup of $G$. Then for any element $\t\in G/H-\{1\}$, we have $$s_{G/H}(\t)=\sum_{\substack{\s\in G\\\s{\mapsto}\t}}s_G(\s).$$
{#<c,1>}
In the following of this section, let $C$ be an algebraically closed field of characteristic zero, $\x$ a primitive $p$-th root of $1$ in $C$ and $\widetilde{\Z}$ the integral closure of $\Z$ in $C$. For any finite group $H$, we denote by $R_C(H)$ the Grothendieck group of finitely generated $C[H]$-modules. For an element $\c\in R(H)$, let $\<\c, 1\>=\frac{1}{\sharp H}\sum_{\s\in H}\operatorname{tr}_{\c}(\s)$.
{#def swan}
For an element $\c\in R_C(G)$, we put $$\begin{aligned}
s_G(\c)&=&\sum_{\s\in G} s_G(\s){\otimes}\operatorname{tr}_{\c}(\s)\in{S_{L/K}}{\otimes}_{\Z}\widetilde{\Z},\nonumber\\
\ve(\x)&=&\sum_{r\in\F_p^{{\times}}{\subseteq}E^{{\times}}}[r]{\otimes}\x^r\in {S_{L/K}}{\otimes}_{\Z}\widetilde{\Z}.\nonumber\end{aligned}$$ Kato defined the [*Swan conductor with differential values*]{} of $\c$ as $$\label{swan}
\operatorname{sw}_{\x}(\c)=s_G(\c)+(\dim\c-\<\c,1\>)\ve(\x)\in{S_{L/K}}{\otimes}\widetilde{\Z}.$$ For any $r\in \F^{{\times}}_p$, we have $\operatorname{sw}_{\x^r}(\c)=\operatorname{sw}_{\x}(\c)+(\dim{\c}-\<\c,1\>)[r]$.
\[scdv quotient\] Let $H$ be a normal subgroup of $G$, $\v$ an element in $R_C(G/H)$ and $\v'$ the image of $\v$ under the canonical map $R_C(G/H){\rightarrow}R_C(G)$. Then, we have $$s_G(\v')=s_{G/H}(\v)\quad \rm{and} \quad \operatorname{sw}_{\x}(\v')=\operatorname{sw}_{\x}(\v).$$
\[ind\] Let $H$ be a subgroup of $G$. For any $\th\in R_C(H)$, we have $$s_G(\operatorname{Ind}^G_H\th)=[G:H]{\left}(s_H(\th)+\dim\th\cdot\di(L^H/K){\right})$$ $$\label{indsw}
\operatorname{sw}_{\x}(\operatorname{Ind}^G_H\th)=[G:H]{\left}(\operatorname{sw}_{\x}(\th)+(\dim\th-\<\th,1\>)\cdot\di(L^{H}/K){\right}).$$
By , and , equation can be written as $$\label{indsw good}
\operatorname{sw}_{\x}(\operatorname{Ind}^G_H\th)=[G:H]{\left}(\operatorname{sw}_{\x}(\th)-(\dim\th-\<\th,1\>){\left}(\sum_{\s\in G-H} ([{\mathrm d}\bar h]-[h-\s(h)]){\right}){\right}).$$
\[class field\] For any $\c\in R_C(G)$, we have $$\operatorname{sw}_{\x}(\c)\in{S_{K,L}}{\subset}{S_{L/K}}{\otimes}_{\Z}\widetilde{\Z}.$$
This is a generalization of Hasse-Arf’s theorem. It can be reduced to the case where $G$ is cyclic of rank $p^s$ and $\c$ is 1-dimensional by the induction formula \[ind\] and Brauer theorem. Then the proof relies on the higher dimensional class field theory of Kato ([@kato; @scdv] 3.6, 3.7).
{#kcc}
For an element $\c\in R_C(G)$, the Swan conductor with differential values $\operatorname{sw}_{\x}(\c)$ is given by $$\operatorname{sw}_{\x}(\c)=-\sharp G(\dim_C\c-\<\c,1\>)[{\mathrm d}\bar h]+\Delta,$$ where $$\Delta=\sum_{\s\in G-\{1\}}[h-\s(h)]{\otimes}(\dim_C\c-\operatorname{tr}_\c(\s))+(\dim_C\c-\<\c ,1\>)\varepsilon(\x)\in R_L{\otimes}_{\Z}\widetilde\Z.$$ From and \[class field\], we have $\sharp G[{\mathrm d}\bar h]=[{\mathrm d}\bar a]$ and $\Delta\in R_K$. Hence, we get $$\operatorname{sw}_{\x}(\c)=[\p^c]+[\D']-m[{\mathrm d}\bar a]\in {S_{K,L}},$$ where $\p$ is the uniformizer of ${\mathcal{O}_K}$ fixed in (\[basic notes\]), $c$ is an integer, $m=\dim_C\c-\<\c,1\>$ and $\D'\in F$ such that $[\p^c\D']=\D$. We define [*Kato’s characteristic cycle*]{} of $\c$ and denote by ${\mathrm{KCC}}_{\x}(\c)$ the element $$\label{kcc formula}
{\mathrm{KCC}}_{\x}(\c)=\D'({\mathrm d}\bar a)^m\in (\O^1_F)^{{\otimes}m}.$$
\[ext swan 1\] If the extension $L/K$ is not of type (II), but there exists a subfield $K'$ of $L$ containing $K$ such that $K'/K$ is an unramified extension and $L/K'$ is of type (II), we define $$\operatorname{sw}_{\x}(\c)=\operatorname{sw}_{\x}(\operatorname{Res}^G_{{\mathrm{Gal}}(L/K')}\c).$$ Denote by ${\mathcal{O}}_{K'}$ the integer ring of $K$, ${\mathfrak m}_{K'}$ the maximal ideal of ${\mathcal{O}}_{K'}$ and $F'$ the residue field of ${\mathcal{O}}_{K'}$. Observe that $\operatorname{sw}_{\x}(\c)$ is fixed by ${\mathrm{Gal}}(K'/K)$ and that the ${\mathrm{Gal}}(K'/K)$-invariant part of $F'\<{\mathfrak m}_{K'}/{\mathfrak m}^2_{K'},\ker(\O^1_{F'}{\rightarrow}\O^1_E)\>$ is $F\<{{\mathfrak m}_K/{\mathfrak m}_K^2},\ker(\O^1_F{\rightarrow}\O^1_E)\>$. Thus $\operatorname{sw}_{\x}(\c)$ is still contained in ${S_{K,L}}$.
\[ext swan 2\] Let $A$ be an algebraically closed field of characteristic $\ell\notin\{0,p\}$. We denote by $A'$ an algebraic closure of the fraction field of the ring of Witt vectors $W(A)$. Let $\c$ be an element of $R_A(G)$ and let $\hat \c$ be a pre-image of $\c$ in $R_{A'}(G)$ ([@serre; @gr] 16.1 Th. 33). We denote by $\hat {\x}$ the $p$-th root of unity in $A'$ lifting of a primitive $p$-th root of unity $\x$ in $ A$. Then we put $$\operatorname{sw}_{\x}(\c)=\operatorname{sw}_{\hat{\x}}(\hat{\c}).$$ This definition is independent of the choice of $\hat{\c}$ because of ([@serre; @gr] 18.2 Th. 42) and .
Abbes-Saito’s ramification theory
=================================
{#note}
Abbes and Saito defined two decreasing filtrations ${G_K}^r$ and ${G_{K,\log}}^r$ ($r\in\Q_{>0}$) of $G_K$ by closed normal subgroups called the ramification filtration and the logarithmic ramification filtration, respectively ([@as; @i], 3.1, 3.2).
{#section-8}
We denote by $G_K^0$ the group $G_K$. For any $r\in \Q_{\geqslant 0}$, we put $$G_K^{r+}={\overline}{\bigcup_{s\in\Q_{>r}}G_K^s} \quad\mathrm{and}\quad \operatorname{Gr}^r G_K=G_K^r/G^{r+}_K.$$ Let $L$ be a finite separable extension of $K$. For a rational number $r\geqslant 0$, we say that the ramification of $L/K$ is bounded by $r$ (resp. by $r+$) if $ G_K^r$ (resp. $G_K^{r+}$) acts trivially on $\operatorname{Hom}_K(L, {\overline}K)$ via its action on ${\overline}K$. We define the [*conductor*]{} $c$ of $L/K$ as the infimum of rational numbers $r>0$ such that the ramification of $L/K$ is bounded by $r$. Then $c$ is a rational number and $L/K$ is bounded by $c+$ ([@as; @i] 6.4). If $c>0$, the ramification of $L/K$ is not bounded by $c$.
{#section-9}
We denote by ${G_{K,\log}}^0$ the inertia subgroup of ${G_K}$. For any $r\in \Q_{\geqslant 0}$, we put $${G_{K,\log}}^{r+}={\overline}{\bigcup_{s\in\Q_{>r}}{G_{K,\log}}^s} \quad\mathrm{and}\quad {\mathrm{Gr}^r_{\log}{G_K}}={G_{K,\log}}^r\big/{G_{K,\log}}^{r+}.$$ By ([@as; @i] 3.15), $P={G_{K,\log}}^{0+}$ is the wild inertia subgroup of $G_{K}$, i.e. the $p$-Sylow subgroup of $G_{K,\log}$. Let $L$ be a finite separable extension of $K$. For a rational number $r\geqslant 0$, we say that the logarithmic ramification of $L/K$ is bounded by $r$ (resp. by $r+$) if ${G_{K,\log}}^r$ (resp. ${G_{K,\log}}^{r+}$) acts trivially on $\operatorname{Hom}_K(L, {\overline}K)$ via its action on ${\overline}K$. We define the [*logarithmic conductor*]{} $c$ of $L/K$ as the infimum of rational numbers $r>0$ such that the ramification of $L/K$ is bounded by $r$. Then $c$ is a rational number and $L/K$ is bounded by $c+$ ([@as; @i] 9.5). If $c>0$, the ramification of $L/K$ is not bounded by $c$.
\[center\] For every rational number $r>0$, the group ${\mathrm{Gr}^r_{\log}{G_K}}$ is abelian and is contained in the center of $P/{G_{K,\log}}^r$.
\[slope decom lemma\] Let $M$ be a $\Z[{\frac}{1}{p}]$-module on which $P={G_{K,\log}}^{0+}$ acts through a finite discrete quotient, say by $\r: P{\rightarrow}\operatorname{Aut}_{\Z}(M)$. Then,
- The module $M$ has a unique direct sum decomposition $$\label{slopedecom}
M={\bigoplus}_{r\in\Q_{\geqslant0}} M^{(r)}$$ into $P$-stable submodules $M^{(r)}$, such that $M^{(0)}=M^P$ and for every $r>0$, $$(M^{(r)})^{{G_{K,\log}}^r}=0\quad \mathrm{and}\quad (M^{(r)})^{{G_{K,\log}}^{r+}}=0.$$
- If $r>0$, then $M^{(r)}=0$ for all but the finitely many values of $r$ for which $\r({G_{K,\log}}^{r+})\neq\r({G_{K,\log}}^r)$.
- For any $r\geqslant 0$, the functor $M{\mapsto}M^{(r)}$ is exact.
- For $M$, $N$ as above, we have $\operatorname{Hom}_{P-mod}(M^{(r)}, N^{(r')})=0$ if $r\neq r'$.
The decomposition is called the [*slope decomposition*]{} of $M$. The values $r\geqslant0$ for which $M^{(r)}\neq 0$ are called the [*slopes*]{} of $M$. We say that $M$ is [*isoclinic*]{} if it has only one slope.
{#l L psi}
In the following of this section, we fix a prime number $\ell$ different from $p$, a local $\Z_{\ell}$-algebra $\L$ which is of finite type as a $\Z_{\ell}$-module and a non-trivial character $\psi_0:\F_p{\rightarrow}\L^{{\times}}$.
\[[@as; @rc] 6.7\]\[center char decomp\] Let $M$ be a $\L$-module on which $P$ acts $\L$-linearly through a finite discrete quotient, which is isoclinic of slope $r>0$. So the $P$ action on $M$ factors through the group $P/{G_{K,\log}}^{r+}$.
- Let $X(r)$ be the set of isomorphism classes of finite characters $\c:{\mathrm{Gr}^r_{\log}{G_K}}{\rightarrow}\L^{{\times}}_{\c}$ such that $\L_{\c}$ is a finite étale $\L$-algebra, generated by the image of $\c$, and having a connected spectrum. Then $M$ has a unique direct sum decomposition $$\label{cen char decom form}
M={\bigoplus}_{\c\in X(r)}M_{\c}.$$ Each $M_{\c}$ is a $P$ stable sub-$\L$-module such that $\L[{G_{K,\log}}^r]$ acts on $M_{\c}$ through $\L_{\c}$.
- There are finitely many characters $\c\in X(r)$ for which $M_{\c}\neq0$.
- Fix $\c\in X(r)$, for all isoclinic $M$ of slope $r$, the functor $M{\rightarrow}M_{\c}$ is exact.
- For $M$, $N$ as above, we have $\operatorname{Hom}_{\L}(M_{\c}, N_{\c'})=0$ if $\c\neq \c'$.
The decomposition is called the [*central character decomposition*]{} of $M$. The characters $\c: {\mathrm{Gr}^r_{\log}{G_K}}{\rightarrow}\L^{{\times}}_{\c}$ for which $M_{\c}\neq0$ are called the central characters of $M$ ([@as; @rc] 6.8).
Let $P_0$ be a finite discrete quotient of $P/{G_{K,\log}}^{r+}$ through which $P$ acts on $M$ and let $C_0$ be the image of ${\mathrm{Gr}^r_{\log}{G_K}}$ in $P_0$. By \[center\], we know that $C_0$ is contained in the center of $P_0$. The connected components of $\operatorname{Spec}(\L[C_0])$ correspond to the isomorphism classes of characters $\c:C_0{\rightarrow}\L^{{\times}}_{\c}$, where $\L_{\c}$ is finite étale $\L$-algebra, generated by the image of $\c$, and having a connected spectrum. If $p^nC=0$, and $\L$ contains a primitive $p^n$-th root of 1, then $\L_{\c}=\L$ for every $\c$ such that $M_{\c}\neq 0$.
\[slope center decom p to 0\] Let $A$ be a $\L$-algebra and $M$ a left $A$-module on which $P$ acts $A$-linearly through a finite discrete quotient. Then,
- In the slope decomposition $M={\bigoplus}_r M^{(r)}$, each $M^{(r)}$ is a sub-$A$-module of $M$. For any $A$-algebra $B$, the decomposition of $B{\otimes}_A M$ is given by $B{\otimes}_A M={\bigoplus}_r(B{\otimes}_A M{(r)})$.
- If $M$ is isoclinic, then in the central character decomposition $M={\bigoplus}_{\c}M_{\c}$, each $M_{\c}$ is a sub-$A$-module of $M$. For any $A$-algebra $B$, the central character decomposition of $B{\otimes}_A M$ is given by $B{\otimes}_A M={\bigoplus}_{\c}(B{\otimes}_A M_{\c})$.
{#isogeny V}
Let $V$ be a finite dimensional ${\overline}F$-vector space and we denote by $V^*$ its dual space. We consider $V$ as a smooth abelian algebraic group over ${\overline}F$, i.e. $\operatorname{Spec}(\operatorname{Sym}(V^*))$. Let $\p_1^{{\mathrm{alg}}}(V)$ be the quotient of $\p_1^{{\mathrm{ab}}}(V)$ classifying étale isogenies. Then $\p_1^{{\mathrm{alg}}}(V)$ is a profinite group killed by $p$ and the group $\operatorname{Hom}(\p_1^{{\mathrm{alg}}}(V),\F_p)$ is canonical identified with the dual space $V^*$ by pulling-back the Lang’s isogeny $\mathbb{A}^1{\rightarrow}\mathbb{A}^1:\;t{\mapsto}t^p-t$ by linear forms (cf. [@wrcb] 1.19).
{#iso}
For the rest of this section, we assume that $F$ is of finite type over a perfect subfield $F_0$. We define the $F$-vector space $\O^1_F(\log)$ by $$\O^1_F(\log)=(\O^1_{F/F_0}{\oplus}(F{\otimes}_{\Z} K^{{\times}}))/({\mathrm d}\bar a-\bar a{\otimes}a;\;a\in{\mathcal{O}_K}^{{\times}}).$$ Then we have an exact sequence of finite dimensional $F$-vector spaces $$\label{Omegalog}
0{\longrightarrow}\O^1_F{\longrightarrow}\O^1_F(\log){\xrightarrow}{\mathrm{res}}F{\longrightarrow}0,$$ where $\mathrm{res}((0,a{\otimes}b))=a\cdot v(b)$ for $a\in F$ and $b\in K^{{\times}}$. If $K$ has characteristic $p$, we put $${\widehat}\O^1_{{\mathcal{O}_K}/F_0}={\varprojlim}_n\O^1_{({\mathcal{O}_K}/{\mathfrak m}_K^n)/F_0}.$$ We have an exact sequence of $F$-vector spaces $$\label{hatO exact p}
0{\rightarrow}{\mathfrak m}_K/{\mathfrak m}_K^2{\rightarrow}{\widehat}\O^1_{{\mathcal{O}_K}/F_0}{\otimes}_{{\mathcal{O}_K}}F{\rightarrow}\O^1_F{\rightarrow}0.$$ If $K$ has characteristic zero and $p$ is not a uniformizer of ${\mathcal{O}_K}$, we denote by ${\mathcal{O}}_{K_0}$ the ring of Witt vectors $W(F_0)$ regarded as a sub-algebra of ${\mathcal{O}_K}$. Then, we put $${\widehat}\O^1_{{\mathcal{O}_K}/{\mathcal{O}}_{K_0}}={\varprojlim}_n\O^1_{({\mathcal{O}_K}/{\mathfrak m}_K^n)/{\mathcal{O}}_{K_0}}.$$ We have an exact sequence of $F$-vector spaces $$\label{hatO exact 0}
0{\rightarrow}{\mathfrak m}_K/{\mathfrak m}_K^2{\rightarrow}{\widehat}\O^1_{{\mathcal{O}_K}/{\mathcal{O}}_{K_0}}{\otimes}_{{\mathcal{O}_K}}F{\rightarrow}\O^1_F{\rightarrow}0.$$
For any rational number $r$, we put $${\mathfrak m}^r_{{\overline}K}=\{x\in {\overline}K\,|\,v(x)\geqslant r\},\ \ \ {\mathfrak m}^{r+}_{{\overline}K}=\{x\in {\overline}K\,|\,v(x)> r\},$$ $$\begin{aligned}
{\Th^{(r)}_{{\overline}F,\log}}&=&\operatorname{Hom}_F\big(\O^1_F(\log), {\mathfrak m}^r_{{\overline}K}/{\mathfrak m}^{(r+)}_{{\overline}K}\big),\nonumber\\
{\Xi^{(r)}_{{\overline}F}}&=&\operatorname{Hom}_F\big(\O^1_F, {\mathfrak m}^r_{{\overline}K}/{\mathfrak m}^{(r+)}_{{\overline}K}\big)\label{xi}.\end{aligned}$$ When $K$ has characteristic $p$ (resp. characteristic zero and $p$ is not a uniformizer of ${\mathcal{O}_K}$), for any rational number $r>0$, we denote by ${\Th^{(r)}_{{\overline}F}}$ the ${\overline}F$-vector space $$\label{thnon}
{\Th^{(r)}_{{\overline}F}}=\operatorname{Hom}_F\big({\widehat}\O^1_{{\mathcal{O}_K}/F_0}{\otimes}_{{\mathcal{O}_K}}F, {\mathfrak m}^r_{{\overline}K}/{\mathfrak m}^{(r+)}_{{\overline}K}\big)$$ $$\big(\text{resp.}\ \ \ {\Th^{(r)}_{{\overline}F}}=\operatorname{Hom}_F\big({\widehat}\O^1_{{\mathcal{O}_K}/{\mathcal{O}}_{K_0}}{\otimes}_{{\mathcal{O}_K}}F, {\mathfrak m}^r_{{\overline}K}/{\mathfrak m}^{(r+)}_{{\overline}K}\big)\ \ \big).$$ By , and , when $p$ is not a uniformizer of $K$, we have homomorphisms $${\Th^{(r)}_{{\overline}F,\log}}{\rightarrow}{\Xi^{(r)}_{{\overline}F}}{\rightarrow}{\Th^{(r)}_{{\overline}F}}.$$
By ([@as; @ii] 5.12), we have a canonical surjection $$\label{pi1 gr}
\p_1^{{\mathrm{ab}}}({\Th^{(r)}_{{\overline}F,\log}}){\rightarrow}{\mathrm{Gr}^r_{\log}{G_K}}.$$
\[isogeny\] For every rational number $r>0$, the canonical surjection factors through the quotient $\p_1^{{\mathrm{alg}}}({\Th^{(r)}_{{\overline}F,\log}})$. In particular, the abelian group ${\mathrm{Gr}^r_{\log}{G_K}}$ is killed by $p$ and the surjection induces an injective homomorphism $$\label{rsw}
\operatorname{rsw}:\operatorname{Hom}({\mathrm{Gr}^r_{\log}{G_K}}, \F_p){\rightarrow}\operatorname{Hom}_{{\overline}F}({\mathfrak m}^r_{{\overline}K}/{\mathfrak m}^{r+}_{{\overline}K}, \O^1_F(\log){\otimes}{\overline}F).$$
The morphism is called the [*refined Swan conductor*]{}.
{#cc}
Let $M$ be a free $\L$-module of finite type on which $P$ acts $\L$-linearly through a finite discrete quotient. Let $$M={\bigoplus}_{r\in\Q_{\geqslant 0}}M^{(r)}$$ be the slope decomposition of $M$ and for each rational number $r>0$, let $$M^{(r)}={\bigoplus}_{\c\in X(r)}M^{(r)}_{\c}$$ be the central character decomposition of $M^{(r)}$. We notice that each $M^{(r)}_{\c}$ is a free $\L$-module. Enlarging $\L$, we may assume that for all rational number $r>0$ and $\c\in X(r)$, $\L=\L_{\c}$ (\[center char decomp\]). Each $\c$ factors uniquely through (\[l L psi\]) $$\operatorname{Gr}^r_{\log}G_K{\rightarrow}\F_p{\xrightarrow}{\psi_0}\L^{{\times}}.$$ We denote abusively by $\c$ the induced character $\operatorname{Gr}^r_{\log}G_K{\rightarrow}\F_p$. We define the [*Abbes-Saito characteristic cycle*]{} ${\mathrm{CC}}_{\psi_0}(M)$ of $M$ by $$\label{cc formula}
{\mathrm{CC}}_{\psi_0}(M)={\bigotimes}_{r\in\Q_{> 0}}{\bigotimes}_{\c\in X(r)}(\operatorname{rsw}(\c){\otimes}\p^{r})^{\dim_{\L} M^{(r)}_{\c}}\in (\O^1_F(\log){\otimes}_{F}{\overline}F)^{{\otimes}\dim_{\L}M/M^{(0)}}.$$
Ramification of extensions of type (II)
=======================================
{#ram type ii notation}
In this section, we assume that the residue field $F$ of ${\mathcal{O}_K}$ is of finite type over a perfect field $F_0$ of characteristic $p$. Let $L$ be a finite Galois extension of $K$ of group $G$ and type (II) (\[type\]), ${\mathcal{O}_L}$ the integer ring of $L$ and $E$ the residue field of ${\mathcal{O}_L}$. We denote by $p^n$ the degree of the residue extension $E/F$. We choose an element $h\in{\mathcal{O}_L}$ such that its residue class $\bar h\in E$ is a generator of $E/F$. We have ${\mathcal{O}_L}={\mathcal{O}_K}[h]$. Let $f(T)\in{\mathcal{O}_K}[T]$ be the minimal polynomial of $h$: $$\label{f}
f(T)=T^{p^n}+a_{p^n-1}T^{p^n-1}+\cdots+a_0.$$ Notice that $\bar a_0=\bar h^{p^n}\in F$. We put $$\label{conductor}
c=\sup_{\s\in G-\{1\}}v(h-\s(h))+\sum_{\s\in G-\{1\}} v(h-\s(h)),$$ which is an integer $\geqslant p^n$.
For any rational number $r\geqslant0$, we denote by $G^r$ (resp. $G^r_{\log})$ the image of $G^r_K$ (resp. $G_{K,\log}^r$) in $G$ ([@as; @i] 3.1). Using the monogenic presentation ${\mathcal{O}_L}={\mathcal{O}_K}[T]/(f(T))$, we obtain that, for any rational number $r>1$, $G^r=G^r_{\log}$([@as; @i] 3.1, 3.2) and that the conductor of $L/K$ is $c$ ([@as; @i] 6.6). By \[isogeny\], the normal subgroup $G^c$ of $G$ is commutative and killed by $p$. In the following, we put $\sharp G^{c}=p^s$.
{#ram type ii notation2}
For any integer $j\geqslant1$, we denote by $D^j$ the $j$-dimensional closed poly-disc of radius one over $K$ and by $\mathring D^j$ the $j$-dimensional open disc of radius one over $K$. For a rational number $r\geqslant 0$, the $j$-dimensional closed poly-disc of radius $r$ is denoted by $D^{j,(r)}=\{(x_1,...,x_j)\in D^j|v(x_i)\geqslant r\}$. Let $$\tilde f: D^1{\rightarrow}D^1,\ \ \ x{\mapsto}f(x),$$ be the morphism induced by $f$. For any rational number $r\geqslant0$, it is easy to see that $\tilde f^{-1}(D^{1,(r)})$ is a disjoint union of closed discs with the same radius, i.e. there exists a rational number $\r(r)\geqslant0$ such that $$\tilde f^{-1}(D^{1,(r)})=\coprod_{1\leqslant j\leqslant i}{\left}(x_j+D^{1,(\r(r))}{\right}),$$ where the $x_j$’s are zeros of $f$. The function $\r:\Q_{\geqslant 0}{\rightarrow}\Q_{\geqslant 0}$ is called the [*Herbrand function*]{} of the extension $L/K$. By ([@as; @ii] 6.6), we have $\r(c)=\sup_{\s\in G-\{1\}}v(h-\s(h))$ and $$\label{property Gc}
G^c=\{\s\in G;v(h-\s(h))\geqslant\r(c)\}.$$
{#u}
We denote by $u$ the map $$\label{def u}
u:G{\rightarrow}E, \ \ \ \s{\mapsto}{\left}\{\begin{array}{ll}
u_{\s}={\overline}{{\left}({\frac}{h-\s(h)}{\p^{v(h-\s(h))}}{\right})},&\text{if}\ \ \s\neq 1,\\
u_{\s}=0,&\text{if}\ \ \s=1.
\end{array}{\right}.$$ The restriction $u|_{G^c}:G^c{\rightarrow}E$ of $u$ to $G^c$ is an injective homomorphism of groups. Indeed, for any $\s\in G^c-\{1\}$, we have $v(h-\s(h))=\r(c)$. Hence, for $\s_1,\s_2\in G^c$, we have $$\begin{aligned}
u_{\s_1\s_2}={\overline}{{\left}( {\frac}{h-\s_1\s_2(h)}{\p^{\r(c)}} {\right})}
={\overline}{{\left}( {\frac}{h-\s_1(h)+\s_1(h-\s_2(h))}{\p^{\r(c)}} {\right})}=u_{\s_1}+u_{\s_2}.\nonumber\end{aligned}$$
\[isogeny f\] The polynomial $f_{c}(T)=f(\p^{\r(c)}T+h)/\p^{c}\in L[T]$ has integral coefficients. Its reduction ${\overline}{f_{c}}\in E[T]$ is an additive polynomial of degree $p^s=\sharp G^{c}$ with a non-zero linear term.
We have $$f_{c}(T)=T\prod_{\s\in G-\{1\}}{\frac}{\p^{\r(c)}T+h-\s(h)}{\p^{v(h-\s(h))}}\in {\mathcal{O}_L}[T].$$ Hence $$\label{f_g}
{\overline}{f_{c}}(T)=T\prod_{\s\in G-\{1\}}{\overline}{{\left}({\frac}{\p^{\r(c)}T+h-\s(h)}{\p^{v(h-\s(h))}}{\right})}=\prod_{\s\in G-G^{c}}u_{\s}\cdot\prod_{\s\in G^{c}}{\left}(T+u_{\s}{\right}).$$ Choose an $\F_p$-basis $\t_1,...,\t_s$ of $G^{c}$, we get $$\begin{aligned}
\prod_{\s\in G^{c}}{\left}(T+u_{\s}{\right})&=&\prod_{(j_1,...,j_s)\in \F^s_p}(T+j_1u_{\t_{1}}+\cdots+j_s u_{\t_{s}}).\end{aligned}$$ We conclude by the lemma below.
\[isogeny lemma\] Let $C$ be a field of characteristic $p$. For any integer $r\geqslant0$, let $x_1,...,x_r$ be $r$ elements of $C$ such that for any $(j_1,...,j_r)\in \F^n_p-\{0\}$, $j_1x_1+\cdots+j_rx_r\neq 0$. Then we have $$\label{math ind}
\prod_{(j_1,...,j_r)\in \F^n_p}(T+j_1x_1+\cdots+j_rx_r)=T^{p^r}+\l_{r-1}T^{p^{r-1}}+\cdots+\l_1T^p+\l_0T\in C[T],$$ where $$\l_0=\prod_{(j_1,...,j_r)\in \F^r_p-\{0\}}(j_1x_1+\cdots+j_rx_r)\neq 0.$$
We proceed by induction on $r$. If $r=1$, $$\prod_{j_1\in \F_p}(T+j_1x_1)=T^p-x_1^{p-1}T,$$ which satisfies . Assume that holds for $(r-1)$-tuples where $r\geqslant 2$, let $(x_1,...,x_r)\in C^r$ be as in the lemma. We put $$g_{r-1}(T)=\prod_{(j_1,...,j_{r-1})\in \F^{r-1}_p}(T+j_1x_1+\cdots+j_{r-1}x_{r-1}).$$ Then, we have $$\begin{aligned}
\prod_{(j_1,...,j_r)\in \F^r_p}(T+j_1x_1+\cdots+j_rx_r)&=&\prod_{j_r\in\F_p}(g_{r-1}(T+j_r x_r))\\
&=&\prod_{j_r\in\F_p}(g_{r-1}(T)+j_r g_{r-1}(x_r))\\
&=&g^p_{r-1}(T)-g^{p-1}_{r-1}(x_r)g_{r-1}(T),\end{aligned}$$ which satisfies since $g_{r-1}$ does.
In the following of this section, we assume that $p$ [*is not a uniformizer of*]{} $K$.
\[p\^2\] Suppose $c>2$. Then, for any $1\leqslant i \leqslant p^n-1$, we have $v(a_i)\geqslant 2$ .
From the equation $f(T)=\prod_{\s\in G}(T-\s(h))$, for any $1\leqslant i\leqslant p^n-1$, we obtain $$\begin{aligned}
\label{decomp a}
a_i&=&(-1)^{(p^n-i)}\sum_{\{\s_1,...,\s_{p^n-i}\}{\subseteq}G} \s_{1}(h)\s_{2}(h)\cdots\s_{p^n-i}(h)\\\nonumber
&=&(-1)^{(p^n-i)}\sum_{\{\s_1,...,\s_{p^n-i}\}{\subseteq}G}(\s_{1}(h)-h+h)\cdots(\s_{p^n-i}(h)-h+h)\\\nonumber
&=&(-1)^{(p^n-i)}{\left}( {p^n\choose i} h^{p^n-i}+{p^n-1\choose i} h^{p^n-i-1}\sum_{\s\in G}(\s(h)-h)+\Delta {\right}),\end{aligned}$$ where $v(\Delta)\geqslant 2$. Since the integer ${p^n\choose i}$ is divided by $p$, $v({p^n\choose i}h^{p^n})\geqslant 2$. Hence it is sufficient to show that $$v\Bigg(\sum_{\s\in G}(\s(h)-h)\Bigg)\geqslant 2.$$
Assume first that for any $\s\in G-\{1\}$, $v(h-\s(h))=\r(c)$, i.e. $G=G^{c}$. It suffices to treat the case where $\r(c)=1$. In this case, $\sharp G=c>2$ . From \[ram type ii notation\], $G$ is an $\F_p$-vector space of dimension $n$ and we choose an $\F_p$-basis $\t_1,...,\t_n$ of $G$. By \[u\], we have $$\begin{aligned}
\sum_{\s\in G}u_{\s}&=&\sum_{(j_1,...,j_n)\in \F^n_p}(j_1u_{\t_1}+\cdots+j_nu_{\t_n})\\
&=&{\frac}{p^n(p-1)}{2}(u_{\t_1}+\cdots+u_{\t_n})=0,\end{aligned}$$ which means that $v(\sum_{\s\in G-\{1\}}(\s(h)-h))\geqslant \r(c)+1= 2$. Assume next that for $\s\in G-\{1\}$, the $v(h-\s(h))$’s are not equal. Let $c'$ be the smallest jump of the ramification filtration of $G$ and let $\sharp(G^{c'+})=p^{n'}$ for some integer $n'<n$. Let $\varsigma_1=1,\varsigma_2,...,\varsigma_{p^{n-n'}}$ be liftings of all the elements of $G/G^{c'+}$ in $G$. Observe that for any $\varsigma\in G-G^{c'+}$ and $\s\in G^{c'+}$, we have $u_{\varsigma\s}=u_{\varsigma}$. Hence $$\sum_{\varsigma\in G-G^{c'+}} u_{\varsigma}=\sum_{j=2}^{p^{n-n'}} \sum_{\s\in G^{c'+}}u_{\varsigma_j}=p^{n'}\sum_{j=2}^{p^{n-n'}}u_{\varsigma_j}=0.$$ Hence $v(\sum_{\s\in G-G^{c'+}}(\s(h)-h))\geqslant 2$. Meanwhile, $v(\sum_{\s\in G^{c'+}}(\s(h)-h))\geqslant 2$, hence we obtain the inequality $v(\sum_{\s\in G}(\s(h)-h))\geqslant 2$.
\[thlog factor xi\] The composition of the canonical homomorphisms (\[isogeny\]) $$\p^{{\mathrm{alg}}}_1({\Th^{(c)}_{{\overline}F,\log}}){\rightarrow}\operatorname{Gr}^c_{\log}G_K {\rightarrow}G^c$$ factors through $\p^{{\mathrm{alg}}}_1({\Xi^{(c)}_{{\overline}F}})$ . In particular, for any non-trivial character $\c:G^c{\rightarrow}\F_p$, we have $\operatorname{rsw}(\c)\in \O^1_F{\otimes}_{F}{\mathfrak m}^{-c}_{{\overline}K}/{\mathfrak m}^{-c+}_{{\overline}K}$.
The proof of this proposition is given in \[proof of proposition\].
{#notation for rsw}
For a non-trivial character $\c:G^{c}{\rightarrow}\F_p$, we denote by $\bar f_{c,\c}(T)$ the polynomial (\[u\]) $$\label{notation ker c}
\bar f_{c,\c}(T)=\prod_{\s\in \ker\c}(T+u_{\s})\in {\overline}F[T],$$ and by $\t\in G^{c}$ a lifting of $1\in \F_p$. Recall that $\bar f_{c,\c}$ is an additive polynomial with a non-zero linear term (\[isogeny lemma\]), and that $\bar f_{c,\c}(u_{\t})$ is independent of the choice of $\t$.
\[theorem rsw\] For any non-trivial character $\c:G^c{\rightarrow}\F_p$, the refined Swan conductor $\operatorname{rsw}(\c)$ is given by $$\operatorname{rsw}(\c)=-{\mathrm d}\bar a_0{\otimes}{\frac}{\p^{-c}}{{\left}(\prod_{\s\in G-G^{c}}u_{\s}{\right})\bar f_{c,\c}^p(u_{\t})}\in \O^1_F{\otimes}_{F}{\mathfrak m}^{-c}_{{\overline}K}/{\mathfrak m}^{-c+}_{{\overline}K}.$$
The proof of this theorem is given in \[proof of theorem\].
\[cc in O\] Let $M$ be a finite dimensional $\L$-vector space with a non-trivial linear $G$-action. Then, with the notation of $\ref{l L psi}$, we have $${\mathrm{CC}}_{\psi_0}(M)\in (\O^1_F{\otimes}_F {\overline}F)^{{\otimes}r},$$ where $r=\dim_{\L}M/M^{(0)}$ (\[slope decom lemma\]).
Tubular neighborhoods and normalized integral models
====================================================
{#section-10}
Let $R$ be an ${\mathcal{O}_K}$-algebra. Following ([@as; @ii] 1), we say that $R$ is formally of finite type over ${\mathcal{O}_K}$ if it is semi-local with radical ${\mathfrak m}_R$, ${\mathfrak m}_R$-adically complete, Noetherian and if the quotient $R/{\mathfrak m}_R$ is of finite type over $F$. We say that $R$ is topologically of finite type over ${\mathcal{O}_K}$ if it is $\p$-adically complete, Noetherian and if the quotient $R/\p R$ is of finite type over $F$.
{#dejong}
We denote by ${\mathrm{AFS}_{{\mathcal{O}}_{K}}}$ the category of affine Noetherian adic formal schemes $\fX$ over $\operatorname{Spf}({\mathcal{O}_K})$ such that the closed sub-scheme $\fX_{{\mathrm{red}}}$ defined by the largest ideal of definition of $\fX$, is a scheme of finite type over $\operatorname{Spec}(F)$. Let $A$ be a finite flat algebra over ${\mathcal{O}_K}$, and $i:\operatorname{Spf}(A){\rightarrow}\fX$ a closed immersion in ${\mathrm{AFS}_{{\mathcal{O}}_{K}}}$. For any rational number $r>0$, following ([@dejong] 7.1 and [@am] 2.1), we associate to $i$ a $K$-affinoid variety $X^r$, called the [*tubular neighborhood of $i$ of thickening $r$*]{}, as follows. Let $\fX=\operatorname{Spf}({\mathcal{A}})$, $I$ be the ideal of ${\mathcal{A}}$ which defines the immersion $i$ and $t,s>0$ be two integer such that $r=t/s$. Let ${\mathcal{A}}\<I^s/\p^t\>$ be the $\p$-adic completion of the subalgebra of ${\mathcal{A}}{\otimes}_{{\mathcal{O}_K}}K$ generated by ${\mathcal{A}}$ and $f/\p^t$ for $f\in I^s$. Then ${\mathcal{A}}\<I^s/\p^t\>{\otimes}_{{\mathcal{O}_K}} K$ is a $K$-affinoid algebra which depends only on $r$. We denote by $X^r$ the $K$-affinoid variety $\operatorname{Sp}({\mathcal{A}}\<I^s/\p^t\>{\otimes}_{{\mathcal{O}_K}} K)$. For rational numbers $r'>r>0$, there exists a canonical morphism $X^{r'}{\rightarrow}X^r$ which makes $X^{r'}$ a rational sub-domain of $X^r$. The admissible union of the affinoid spaces $X^{r}$ for $r\in\Q_{\geqslant 0}$ is a quasi-separated rigid variety over $K$.
\[fini theo\] Let ${\mathcal{R}}$ be a geometrically reduced $K$-affinoid algebra. Then, there exists a finite separable extension $K'$ of $K$ such that the supremum norm unit ball ([@bgr] 3.8.1) $$\label{model over K'}
{\mathcal{R}}_{{\mathcal{O}}_{K'}}=\{f\in {\mathcal{R}}{\otimes}_K K';|f|_{\sup}\leqslant 1\}{\subseteq}{\mathcal{R}}{\otimes}_K K'$$ has a reduced geometric closed fiber ${\mathcal{R}}_{{\mathcal{O}}_{K'}}{\otimes}_{{\mathcal{O}}_{K'}}{\overline}F$. Moreover, the formation of ${\mathcal{R}}_{{\mathcal{O}}_{K'}}$ commutes with any finite extension of $K'$.
{#def nim}
Let ${\mathcal{R}}$ be a geometrically reduced $K$-affinoid algebra. We consider the collection of ${\mathcal{O}}_{K'}$-formal scheme $\operatorname{Spf}({\mathcal{R}}_{{\mathcal{O}}_{K'}})$, where $K'$ and ${\mathcal{R}}_{{\mathcal{O}}_{K'}}$ are as in \[fini theo\], as a unique model of $\operatorname{Sp}({\mathcal{R}})$ over ${\mathcal{O}}_{{\overline}K}$. We call it the [*normalized integral model*]{} over ${\mathcal{O}}_{{\overline}K}$. We say that the the normalized integral model of $\operatorname{Sp}({\mathcal{R}})$ is defined over $K'$ if the supremum norm unit ball ${\mathcal{R}}_{{\mathcal{O}}_{K'}}$ has a reduced geometric special fiber. We call this reduced geometric special fiber over ${\overline}F$ the special fiber of the normalized integral model of $\operatorname{Sp}({\mathcal{R}})$ over ${\mathcal{O}}_{{\overline}K}$.
\[iso sp gen\] Let $X$ be a geometrically reduced affinoid variety over $K$, $\fX$ its normalized integral model over ${\mathcal{O}}_{{\overline}K}$ and ${\overline}\fX$ the special fiber of $\fX$. Then the set of geometric connected components of $X$ and ${\overline}\fX$ are isomorphic.
{#geo monodromy}
Let $X$ be a geometrically reduced affinoid variety over $K$, $\fX$ its normalized integral model over ${\mathcal{O}}_{{\overline}K}$ and ${\overline}\fX$ the special fiber of $\fX$. If $\fX$ is defined over a finite Galois extension $K'$ of $K$, we denote by $\fX_{{\mathcal{O}}_{K'}}$ the normalized integral model of $X$ over ${\mathcal{O}}_{K'}$. The natural $K'$-semi-linear action of $G_K$ on $X{\otimes}_K K'$ extends to an ${\mathcal{O}}_{K'}$-semi-linear action of $G_K$ on $\fX_{{\mathcal{O}}_{K'}}$. If $K''$ is another finite Galois extension of $K$ containing $K'$, then $\fX'_{{\mathcal{O}}_{K''}}=\fX_{{\mathcal{O}}_{K'}}{\otimes}_{{\mathcal{O}}_{K'}}{\mathcal{O}}_{K''}$ and the semi-linear action of $G_K$ on both sides are compatible. Hence, it induces an ${\overline}F$-semi-linear action of $G_K$ on the special fiber ${\overline}\fX$, called the [*geometric monodromy*]{} ([@as; @i] 4.5).
Isogenies associated to extensions of type (II): the equal characteristic case
==============================================================================
{#first pr char p}
In this section, we assume that $K$ has characteristic $p$ and that the residue field $F$ of ${\mathcal{O}_K}$ is of finite type over a perfect field $F_0$. For an object $L$ of ${\text{F\'E}_{/K}}$ and an integer $r\geqslant 1$, we denote by $({\mathcal{O}}_{L}/{\mathfrak m}^r_{L}){\widehat}{\otimes}_{F_0} {\mathcal{O}_K}$ the completion of $({\mathcal{O}}_{L}/{\mathfrak m}^r_{L}){\otimes}_{F_0} {\mathcal{O}_K}$ relatively to the kernel of the homomorphism $$\label{surj first}
({\mathcal{O}}_{L}/{\mathfrak m}^r_{L}){\otimes}_{F_0} {\mathcal{O}_K}{\rightarrow}{\mathcal{O}}_{L}/{\mathfrak m}^r_{L},\ \ \ a{\otimes}b{\rightarrow}ab,$$ and by ${\mathcal{O}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K}$ the projective limit $${\varprojlim}_r ({\mathcal{O}}_{L}/{\mathfrak m}^r_{L}){\widehat}{\otimes}_{F_0} {\mathcal{O}_K}.$$ We will always consider ${\mathcal{O}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K}$ as an ${\mathcal{O}_K}$-algebra by the homomorphism $$\label{OK alg}
{\mathcal{O}_K}{\rightarrow}{\mathcal{O}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K},\ \ \ u{\mapsto}1{\otimes}u,$$ (in the following, we always abbreviate $1{\otimes}u$ by $u$) and we will consider it as an ${\mathcal{O}}_{L}$-algebra by $${\mathcal{O}}_{L}{\rightarrow}{\mathcal{O}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K},\ \ \ v{\mapsto}v{\otimes}1.$$ There is a canonical surjective homomorphism $$\label{surj third}
{\mathcal{O}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K}{\rightarrow}{\mathcal{O}}_{L}$$ induced by the surjections . We denote by $I_{L}$ its kernel.
\[A ot O char p\] Let $L$ be an object of ${\text{F\'E}_{/K}}$.
- The ${\mathcal{O}_K}$-algebra ${\mathcal{O}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K}$ is formally of finite type and formally smooth over ${\mathcal{O}_K}$ and the morphism $({\mathcal{O}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K})/{\mathfrak m}_{{\mathcal{O}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\hat{{\otimes}}}}
\addtolength{\dhatheight}{-0.70ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K}}{\rightarrow}{\mathcal{O}}_{L}/{\mathfrak m}_{{\mathcal{O}}_{L}}$ is an isomorphism.
- Any morphism $L{\rightarrow}L'$ of ${\text{F\'E}_{/K}}$ induces an isomorphism $$\label{origin reps}
{\mathcal{O}}_{L'}{\otimes}_{{\mathcal{O}}_{L}}({\mathcal{O}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K}){{\xrightarrow}{\sim}}{\mathcal{O}}_{L'}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K}.$$
{#nim char p}
Let $L$ be an object of ${\text{F\'E}_{/K}}$. By \[A ot O char p\], $\operatorname{Spf}({\mathcal{O}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K})$ is an object of ${\mathrm{AFS}_{{\mathcal{O}}_{K}}}$ (\[dejong\]). For any rational number $r>0$ and integer numbers $s,t>0$ such that $r=t/s$, we denote by ${\mathcal{R}}^r_{L}$ the $K$-affinoid algebra $${\mathcal{R}}^r_{L}=({\mathcal{O}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K})\<I_{L}^s/\p^t\>{\otimes}_{{\mathcal{O}_K}}K,$$ by $X^r_{L}=\operatorname{Sp}({\mathcal{R}}^r_{L})$ the tubular neighborhood of thickening $r$ of the closed immersion $\operatorname{Spf}({\mathcal{O}}_{L}){\rightarrow}\operatorname{Spf}({\mathcal{O}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K})$ , (\[dejong\]), which is smooth over $K$ ([@as; @ii] 1.7). By \[fini theo\], there exists a finite separable extension $K'$ of $K$ such that the normalized integral model of $X^r_L$ over ${\mathcal{O}}_{{\overline}K}$ is defined over $K'$ (\[def nim\]). We denote by ${\mathcal{R}}^r_{L,{\mathcal{O}}_{K'}}$ the supremum norm unit ball of ${\mathcal{R}}^r_{L}{\otimes}_K K'$ (\[model over K’\]), by $\fX^r_L$ the normalized integral model of $X^r_L$ over ${\mathcal{O}}_{{\overline}K}$ and by ${\overline}\fX^r_L$ the special fiber of $\fX^r_L$ (\[def nim\]).
{#X int r}
Let $m$ be the dimension of the $F$-vector space $\O^1_F$, which is finite. By ([@as; @ii] 1.14.3), there is an isomorphism of ${\mathcal{O}_K}$-algebras $$\label{iso o ot o}
{\mathcal{O}_K}[[T_0,...,T_m]]{{\xrightarrow}{\sim}}{\mathcal{O}}_{K}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K},$$ such that the composition of it and ${\mathcal{O}_K}[[T_0,...,T_m]]{{\xrightarrow}{\sim}}{\mathcal{O}}_{K}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K}{\rightarrow}{\mathcal{O}_K}$ maps $T_i$ to $0$. Here the ${\mathcal{O}_K}$-algebra structure of ${\mathcal{O}}_{K}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K}$ is as in . If $r$ is an integer $\geqslant 1$, we have an isomorphism of $K$-affinoid algebras $$\label{iso affinoid K}
K\<T_0/\p^r,...,T_m/\p^r\>{{\xrightarrow}{\sim}}{\mathcal{R}}^r_K.$$ The normalized integral model $\fX^r_K$ is defined over ${\mathcal{O}_K}$, and we have an isomorphism $$\label{iso nim K}
{\mathcal{O}_K}\<T_0/\p^r,...,T_m/\p^r\>{{\xrightarrow}{\sim}}({\mathcal{O}}_{K}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K})\<I_K/\p^r\>={\mathcal{R}}^r_{K,{\mathcal{O}_K}}.$$ Hence the closed fiber ${\overline}\fX^r_K$ is isomorphic to the affine scheme $$\operatorname{Spec}{\overline}F[T_0/\p^r,...,T_m/\p^r].$$ In general, for any rational number $r>0$, the $K$-affinoid variety $X^r_K$ is isomorphic to $D^{m+1,(r)}$ and the rigid space $X_K=\cup_{r>0}X^r_K$ is isomorphic to $\mathring D^{m+1}$ (\[ram type ii notation2\]).
By ([@as; @ii] 1.13, 2.4), for any rational number $r>0$, there exists a canonical isomorphism ${\overline}\fX^r_K{{\xrightarrow}{\sim}}{\Th^{(r)}_{{\overline}F}}$ (\[thnon\]) which is compatible with the geometric monodromy on ${\overline}\fX^r_K$ and the natural $G_K$-action on ${\Th^{(r)}_{{\overline}F}}$ (via its action on ${\mathfrak m}^r_{{\overline}K}/{\mathfrak m}^{r+}_{{\overline}K}$). If $r$ is an integer, it is constructed as follows. Firstly, we have a natural ring isomorphism $$\label{alg C to X}
{\bigoplus}_{i=0}^{\infty}I_K^i/I_K^{i+1}{\otimes}_{{\mathcal{O}_K}}{\mathfrak m}^{-ir}_{K}/{\mathfrak m}^{-ir+1}_{K}{\rightarrow}{\mathcal{R}}^r_{K,{\mathcal{O}_K}}/{\mathfrak m}_K{\mathcal{R}}^r_{K,{\mathcal{O}_K}},\ \ \ {\overline}b{\otimes}{\overline}c{\mapsto}{\overline}{bc},$$ by and . Extending scalars, we have $$\label{X to C}
{\overline}\fX^r_K{{\xrightarrow}{\sim}}\operatorname{Spec}{\left}({\bigoplus}_{i=0}^{\infty}I_K^i/I_K^{i+1}{\otimes}_{{\mathcal{O}_K}}{\mathfrak m}^{-ir}_{{\overline}K}/{\mathfrak m}^{-ir+}_{{\overline}K}{\right}).$$ Then, from ([@as; @ii] 1.14.3, 2.4), we have an isomorphism of free ${\mathcal{O}_K}$-modules $$\label{Omega iso I}
\hO_{{\mathcal{O}_K}/F_0}{\rightarrow}I_K/I_K^2 ,\ \ \ {\mathrm d}t{\mapsto}{\overline}{1{\otimes}t-t{\otimes}1},$$ which induces the isomorphism ${\overline}\fX^r_K{\rightarrow}{\Th^{(r)}_{{\overline}F}}$.
{#p to G^r}
Let $L$ be a finite Galois extension of $K$ of group $G$ and conductor $r>1$. By ([@as; @i] 7.2), the natural action of $G$ on ${\mathcal{O}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K}$ induces an ${\mathcal{O}}_{{\overline}K}$-linear action of $G$ on $\fX^r_L$ making it an étale $G$-tosor over $\fX^r_K$. In particular, $X^r_{L}$ and ${\overline}\fX^r_{L}$ are étale $G$-tosors of $ X^r_K$ and ${\overline}\fX^r_K$, respectively. The geometric monodromy action of $G_K$ on ${\overline}\fX^r_{L}$ (\[geo monodromy\]) commutes with the action of $G$. Let ${\overline}\fX^r_{L,0}$ be a connected component of ${\overline}\fX^r_{L}$. The stabilizers of ${\overline}\fX^r_{L,0}$ via these two actions are $G^r$ and $G^r_K$, respectively. Then, we get an isomorphism $G^r{{\xrightarrow}{\sim}}\operatorname{Aut}({\overline}\fX^r_{L,0}/{\overline}\fX^r_K)$ and a surjection $G^r_K{\rightarrow}\operatorname{Aut}({\overline}\fX^r_{L,0}/{\overline}\fX^r_K)$ which implies that $G^r$ is commutative (cf. [@as; @ii] 2.15.1). Composing with ${\overline}\fX^r_K{{\xrightarrow}{\sim}}{\Th^{(r)}_{{\overline}F}}$, the étale covering ${\overline}\fX^r_{L,0}{\rightarrow}{\Th^{(r)}_{{\overline}F}}$ induces a surjective homomorphism ([@as; @ii] 2.15.1) $$\p^{{\mathrm{ab}}}_1({\Th^{(r)}_{{\overline}F}}){\rightarrow}\operatorname{Gr}^rG_K{\rightarrow}G^r.$$
{#section-11}
In the rest of this section, let $L/K$ be a finite Galois extension of type (II) and we take again the notation and assumptions of \[ram type ii notation\] and \[ram type ii notation2\]. By and the proof of ([@as; @ii] 1.6), for any rational number $r>0$, we have an isomorphism $$\label{cart1}
{\mathcal{R}}_K^r{\otimes}_{{\mathcal{O}_K}{
\settoheight{\dhatheight}{\ensuremath{\hat{{\otimes}}}}
\addtolength{\dhatheight}{-0.70ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K}}({\mathcal{O}_L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K}) {{\xrightarrow}{\sim}}{\mathcal{R}}_L^r.$$ It induces, for any rational numbers $r>r'>0$, an isomorphism $${\mathcal{R}}^{r}_K{\otimes}_{{\mathcal{R}}^{r'}_K}{\mathcal{R}}_{L}^{r'}{{\xrightarrow}{\sim}}{\mathcal{R}}^{r}_L,$$ which gives a Cartesian diagram of rigid spaces $$\label{tub morphism}
\xymatrix{\relax
X^{r}_L\ar[d]\ar[r]&X_L\ar[d]\\
X^{r}_K\ar[r]&X_K}$$ where $X_K=\bigcup_{r>0}X^r_K$ and $X_L=\bigcup_{r>0}X^r_L$.
We put $$\bmf(T)=T^{p^n}+(a_{p^n-1}{\otimes}1)T^{p^n-1}+\cdots+(a_0{\otimes}1)\in ({\mathcal{O}_K}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{F_0}{\mathcal{O}_K})[T].$$ From and , we have a surjection $$\t_L:{\mathcal{R}}^r_K\<T\>{\rightarrow}{\mathcal{R}}^r_L,\ \ \ T{\mapsto}h{\otimes}1,$$ which induces an isomorphism that we denote abusively also by $$\label{key representation}
\t_L:{\mathcal{R}}^r_K\<T\>/\bmf(T){{\xrightarrow}{\sim}}{\mathcal{R}}^r_L.$$ In other terms, we have a co-Cartesian diagram of homomorphisms of ${\mathcal{R}}^r_K$-algebras $$\label{cocart}
\xymatrix{\relax
{\mathcal{R}}^r_L&{\mathcal{R}}^r_K\<T\>\ar[l]_{\t_L}\\
{\mathcal{R}}^r_K\ar[u]&{\mathcal{R}}^r_K\<T\>\ar[u]_{\phi}\ar[l]_{\t_K}
}$$ where $\phi(T)=\bmf(T)$ and $\t_K(T)=0$. Hence, taking the union of the $K$-affinoid varieties associated to each of the $K$-affinoid algebras in for $r\in\Q_{> 0}$, we have a Cartesian diagram $$\label{tub explicit}
\xymatrix{\relax
X_L\ar[d]\ar[r]^-(0.5){i_L}&X_K{\times}D^1\ar[d]^{{\mathbf{f}}}\\
X_K\ar[r]^-(0.5){i_K}&X_K{\times}D^1}$$ where $i_L$, ${\mathbf{f}}$ and $i_K$ are the morphisms induced by $\t_L$, $\phi$ and $\t_K$.
{#ab}
In the following, for any $0\leqslant i\leqslant p^n-1$, we denote by $\a_i$ the element $ a_i-a_i{\otimes}1\in I_K$ (\[first pr char p\]). When the conductor $c>2$, for each $1\leqslant i\leqslant p^n-1$, $v(a_i)\geqslant 2$ (\[p\^2\]). Let $a'_i=\p^{-2}a_i\in{\mathcal{O}_K}$. We denote by $\a'_i$ the element $a'_i-a'_i{\otimes}1\in I_K$ and by $\b$ the element $\p-\p{\otimes}1\in I_K$. Then, we have $$\a_i=(a'_i-\a'_i)(2\p\b-\b^2)+\p^2\a'_i.$$ Since $\a'_i$, $\b\in I_K {\subset}\p^c{\mathcal{R}}^c_{K,{\mathcal{O}_K}}$, we have $\a_i\in \p^{c+1}{\mathcal{R}}^{c}_{K,{\mathcal{O}_K}}$.
When $c=2$, we have $p=2$, $\sharp G=2$, $\r(c)=1$ and $a''_1=\p^{-1} a_1\in {\mathcal{O}_K}$. We denote by $\a''_1$ the element $ a''_1-a''_1{\otimes}1\in I_K$. Then we have $$\a_1=(a''_1-\a''_1)\b+\p \a''_1.$$ Since $\a''_1,\b\in\p^c{\mathcal{R}}^c_{K,{\mathcal{O}_K}}$, we have $\a_1\in \p^c{\mathcal{R}}^{c}_{K,{\mathcal{O}_K}}$, and ${\overline}{\a_1/\p^c}={\overline}{a''_1\b/\p^c}\in {\mathcal{R}}^{c}_{K,{\mathcal{O}_K}}\big/\p{\mathcal{R}}^{c}_{K,{\mathcal{O}_K}}$.
We put $$\bmf_0(T)=\sum_{0\leqslant i\leqslant p^n-1}(\a_i/\p^c)\cdot T^i\in {\mathcal{R}}^c_{K,{\mathcal{O}_K}}[T].$$ We have $$\bmf(T)=f(T)-\sum_{0\leqslant i\leqslant p^n-1}\a_iT^i=f(T)-\p^{c} \bmf_0(T).$$
In the rest of this section, we fix an embedding $L{\rightarrow}{\overline}K$.
\[component\] The $K$-affinoid $X^{c}_L$ has $\sharp(G/G^{c})=p^{n-s}$ geometric connected components. Let $\s_1,...,\s_{p^{n-s}}$ be liftings of all the elements of $G/G^{c}$ in $G$. We have $$\label{disjoint union}
i_L(X^{c}_L){\subseteq}\coprod_{1\leqslant j\leqslant p^{n-s}} X^{c}_K{\times}(\s_j(h)+D^{1,(\r(c))}){\subseteq}X_K{\times}D^1,$$ and each disc of the disjoint union contains exact one geometric connected component of $X^{c}_L$.
By the Cartesian diagrams and , we have $$i_L(X^{c}_L)={\mathbf{f}}^{-1}(i_K(X^{c})){\subseteq}X^{c}_K{\times}D^1{\subseteq}X_K{\times}D^1.$$ Taking in account the isomorphisms and , for any point $$(t_0,...,t_m, t)\in X^{c}_K{\times}D^1-\coprod_{1\leqslant k\leqslant p^{n-s}} X^{c}_K{\times}(\s_k(h)+D^{1,(\r(c))}),$$ we have $v(f(t))<c$ and $v((\a_i/\p^{c})(t_0,...t_m)t^i)\geqslant 0$. Hence $v(f(t)-\p^{c} \bmf_0(t_0,...,t_m,t))<c$ which means ${\mathbf{f}}(t_0,..,t_m,t)=(t_1,...,t_m,\bmf(t_0,...,t_m,t))\not\in i_K(X^c_K)$. Thus holds. By the proof of ([@as; @ii] 2.15), $X^{c}_L$ has exactly $p^{n-s}$ geometric connected components. Moreover, for any $1\leqslant j\leqslant p^{n-s} $, $f(\s_j(h))-\p^{c} \bmf_0(0,...,0,\s_j(h))=0$, hence each disc $X^c_K{\times}(\s_j(h)+D^{1,(\r(c))})$ contains at least one geometric connected component of $X^{c}_L$.
In the following, we denote by ${\overline}\fX^c_{L,0}$ the connected component of ${\overline}\fX^c_L$ corresponding to the connected component $X^c_{L,0}$ of $X^c_L$ containing $(0,...,0,h)\in X^c_K{\times}D^1$ defined over $L$.
\[thnong isogeny\] There exists a canonical Cartesian diagram $$\label{cartesian diagram isogeny}
\xymatrix{\relax
{\overline}\fX^c_{L,0}\ar[d]\ar[r]^{\nu}&\A^1_{{\overline}F}\ar[d]^{{\overline}{f_c}}\\
{\Th^{(c)}_{{\overline}F}}\ar[r]^{\mu}&\A^1_{{\overline}F}}$$ where ${\overline}{f_{c}}$ is defined in , such that if $\xi$ is the canonical coordinate of $\A^1_{{\overline}F}$, we have $$\mu^*(\xi)={\left}\{\begin{array}{ll}{\mathrm d}a_0{\otimes}\p^{-c},&\text{if}\ \ c>2,\\
(a''_1h{\mathrm d}\p+{\mathrm d}a_0){\otimes}\p^{-2},&\text{if}\ \ c=2.
\end{array}{\right}.$$ Moreover, for any $\s\in G^c$, the following diagram $$\label{G action diagram}
\xymatrix{\relax
{\overline}\fX^c_{L,0}\ar[d]_{\s}\ar[r]^{\nu}&\A^1_{{\overline}F}\ar[d]^{d_{\s}}\\
{\overline}\fX^c_{L,0}\ar[r]^{\nu}&\A^1_{{\overline}F}}$$ where $d_{\s}^{*}(\x)=\x-u_{\s}$ (\[u\]), is commutative.
We consider the $K$-affinoid algebra ${\mathcal{R}}^c_K$ (resp. ${\mathcal{R}}^c_L$) as a sub-ring of the $L$-affinoid algebra ${\mathcal{R}}^c_K {\otimes}_K L$ (resp. ${\mathcal{R}}^c_L {\otimes}_K L$). By , we have $$X^c_{L,0}=i^{-1}_L(X^c_K{\times}(h+D^{1,(\r(c))}))\cap X^c_L.$$ Hence $X^c_{L,0}$ is presented by the $L$-affinoid algebra $$\label{connect comp over L}
({\mathcal{R}}^{c}_L{\otimes}_{K}L)\<T'\>/(\p^{\r(c)}T'+h-h{\otimes}1).$$ By the isomorphism , is isomorphic to $$({\mathcal{R}}^{c}_K{\otimes}_{K}L)\<T,T'\>/(\bmf(T), \p^{\r(c)}T'+h-T),$$ which, after eliminating $T$ by the relation $\p^{\r(c)}T'+h-T=0$, is $$\label{connected comp}
({\mathcal{R}}^{c}_K{\otimes}_{K}L)\<T'\>/(\bmf(\p^{\r(c)}T'+h)).$$ In both cases, by \[isogeny f\] and \[ab\], we have $$\bmf(\p^{\r(c)}T'+h)/\p^{c}\in {\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}\<T'\>,$$ $$\bmf(\p^{\r(c)}T'+h)/\p^{c+1}\notin{\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}\<T'\>.$$ Then the image of ${\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}\<T'\>$ by the canonical surjection $$({\mathcal{R}}^c_K{\otimes}_KL)\<T'\>{\rightarrow}({\mathcal{R}}^{c}_K{\otimes}_{K}L)\<T'\>/(\bmf(\p^{\r(c)}T'+h)),$$ is $$\label{oB}
{\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}\<T'\>/(\bmf(\p^{\r(c)}T'+h)/\p^{c}).$$ Extending the scalars from ${\mathcal{O}_L}$ to ${\overline}F$, we obtain the following ${\overline}F$-algebra:
- if $c>2$, $$\label{clos fiber >2}
({\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}{\otimes}_{{\mathcal{O}_L}}{\overline}F)[T']/({\overline}{ f_{c}}(T')-{\overline}{\a_0/\p^{c}})~;$$
- if $c=2$, $$\label{clos fiber =2}
({\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}{\otimes}_{{\mathcal{O}_L}}{\overline}F)[T']/({\overline}{f_2}(T')- {\overline}{(\a_0+a_1''h\b)/\p^{2}}).$$
From isomorphisms , and the canonical exact sequence , we know that when $c>2$ (resp. $c=2$), ${\overline}{\a_0/\p^c}$ (resp. ${\overline}{(\a_0+a_1''h\b)/\p^{2}}$) is a non-zero linear term in ${\overline}F{\otimes}_{{\mathcal{O}_L}}{\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}$. Hence and are all reduced. Then, by ([@as; @i] 4.1), $$\operatorname{Spf}({\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}\<T'\>/(\bmf(\p^{\r(c)}T'+h)/\p^{c}))$$ is the normalized integral model of $X^c_{K,0}$ defined over ${\mathcal{O}_L}$. Hence ${\overline}\fX^{c}_{L,0}$ is defined by the ${\overline}F$-algebra (resp. ) when $c>2$ (resp. $c=2$). We put $$\nu:{\overline}\fX^c_{L,0}{\rightarrow}\A^1_{{\overline}F}=\operatorname{Spec}({\overline}F[\x]),\ \ \ \nu^*(\x)=T'.$$ It follows form the isomorphism ${\overline}\fX^{c}_K{{\xrightarrow}{\sim}}{\Th^{(c)}_{{\overline}F}}$ (\[X int r\]) that is Cartesian.
For any $\s\in G^c$, let $y_{\s}(x)$ be a polynomial $b_rx^r +\cdots +b_0\in {\mathcal{O}_K}[x]$, where $ r\leqslant p^n-1$, such that $y_{\s}(h)=(h-\s(h))/\p^{\r(c)}\in {\mathcal{O}_L}$. We denote by $\mathbbm y_{\s}$ the polynomial $$\mathbbm y_{\s}(x)=(b_r{\otimes}1)x^r+\cdots+(b_0{\otimes}1)\in {\mathcal{R}}^c_K[x].$$ The action of $\s$ on ${\mathcal{R}}^c_K\<T\>/\bmf(T)$ (isomorphic to $ {\mathcal{R}}^c_L$ ) is given by $:T{\mapsto}T-(\p^{\r(c)}{\otimes}1)\mathbbm y(T)$. Hence the action of $\s$ on is given by $$T'{\mapsto}T'-\mathbbm y_{\s}(\p^{\r(c)}T'+h)-((\p^{\r(c)}{\otimes}1-\p^{\r(c)})/\p^{\r(c)})\mathbbm y_{\s}(\p^{\r(c)}T'+h)$$ and the induced action on is given by the same formula. Since $\p^{\r(c)}{\otimes}1-\p^{\r(c)}\in \p^{c}{\mathcal{R}}^c_{K,{\mathcal{O}_K}}$ and $c>\r(c)$, the reduction of $(\p^{\r(c)}{\otimes}1-\p^{\r(c)})/\p^{\r(c)}$ to the geometric special fiber is $0$. For any $0\leqslant j\leqslant r$, $b_j{\otimes}1- b_j\in \p^c{\mathcal{R}}^c_{K,{\mathcal{O}_K}}$. Then, the reduction of $\mathbbm y_{\s}(\p^{\r(c)}T'+h)$ to the geometric special fiber is (\[u\]) $${\overline}{\mathbbm y_{\s}(\p^{\r(c)}T'+h)}={\overline}{y_{\s}(\p^{\r(c)}T'+h)}={\overline}{y_{\s}(h)}=u_{\s}.$$ Hence, diagram is commutative.
Isogenies associated to extensions of type (II): the unequal characteristic case
================================================================================
{#section-12}
In this section, we assume that $K$ has characteristic $0$ and that the residue field $F$ of ${\mathcal{O}_K}$ is of finite type over a perfect field $F_0$. Let $K_0$ be the fraction field of the ring of Witt vectors $W(F_0)=O_{K_0}$ considered as a subfield of $K$. We denote by $m$ the dimension of the $F$-vector space $\O^1_F$, which is finite.
{#present cartier}
Let $L$ be an object of ${\text{F\'E}_{/K}}$. We call an ${\mathcal{O}}_{K_0}$-[*presentation of Cartier type*]{} of ${\mathcal{O}_L}$ a pair $({\mathcal{A}}_L, j:{\mathcal{A}}_L{\rightarrow}{\mathcal{O}_L})$, where ${\mathcal{A}}_L$ is a complete semi-local Noetherian ${\mathcal{O}}_{K_0}$-algebra formally smooth of relative dimension $m+1$ over ${\mathcal{O}}_{K_0}$ and $j$ a surjective homomorphism of ${\mathcal{O}}_{K_0}$-algebra inducing an isomorphism ${\mathcal{A}}_L/{\mathfrak m}_{{\mathcal{A}}_L}{{\xrightarrow}{\sim}}{\mathcal{O}}_L/{\mathfrak m}_L$ such that the kernel of $j$ is generated by a non-zero divisor of ${\mathcal{A}}_L$.
Let $L_1$, $L_2$ be two objects of ${\text{F\'E}_{/K}}$ and $({\mathcal{A}}_{L_1},j_1:{\mathcal{A}}_{L_1}{\rightarrow}{\mathcal{O}}_{L_1})$, $({\mathcal{A}}_{L_2},j_2:{\mathcal{A}}_{L_2}{\rightarrow}{\mathcal{O}}_{L_2})$ two ${\mathcal{O}}_{K_0}$-presentations of Cartier type. A morphism $(g,{\mathbbm{g}})$ from $({\mathcal{A}}_{L_1},j_1)$ to $({\mathcal{A}}_{L_2},j_2)$ is a pair of ${\mathcal{O}}_{K_0}$-homomorphisms $g:{\mathcal{O}}_{L_1}{\rightarrow}{\mathcal{O}}_{L_2}$ and $ {\mathbbm{g}}:{\mathcal{A}}_{L_1}{\rightarrow}{\mathcal{A}}_{L_2} $ such that the diagram $$\label{A0 ra OL}
\xymatrix{\relax
{\mathcal{A}}_{L_1}\ar[d]_{{\mathbbm{g}}}\ar[r]^{j_1}&{\mathcal{O}}_{L_1}\ar[d]^g\\
{\mathcal{A}}_{L_2}\ar[r]^{j_2}&{\mathcal{O}}_{L_2}}$$ is commutative. We say that $(g,{\mathbbm{g}})$ is [*finite and flat*]{} if ${\mathbbm{g}}$ is finite and flat and if the diagram is co-Cartesian.
\[step 0 lemma\]
- Any object of ${\text{F\'E}_{/K}}$ admits an ${\mathcal{O}}_{K_0}$-presentation of Cartier type.
- Let $g:L_1{\rightarrow}L_2$ be a morphism of ${\text{F\'E}_{/K}}$, and $({\mathcal{A}}_{L_1},j_1)$, $({\mathcal{A}}_{L_2},j_2)$ two ${\mathcal{O}}_{K_0}$-presentations of Cartier type. Then there exist a morphism ${\mathbbm{g}}:{\mathcal{A}}_{L_1}{\rightarrow}{\mathcal{A}}_{L_2}$ such that $(g,{\mathbbm{g}})$ is a morphism of ${\mathcal{O}}_{K_0}$-presentations of Cartier type.
- Let $g:L_1{\rightarrow}L_2$ be a morphism of ${\text{F\'E}_{/K}}$ and $(g,{\mathbbm{g}})$ a morphism between ${\mathcal{O}}_{K_0}$-presentations of Cartier type $({\mathcal{A}}_{L_1},j_1)$ and $({\mathcal{A}}_{L_2},j_2)$. If a uniformizer $\p_0$ of $K_0$ is not a uniformizer of any factor of ${\mathcal{O}}_{L_1}$, then $(g,{\mathbbm{g}})$ is finite and flat.
{#A ot O}
Let $L$ be an object of ${\text{F\'E}_{/K}}$, and $({\mathcal{A}}_{L},j:{\mathcal{A}}_L{\rightarrow}{\mathcal{O}_L})$ an ${\mathcal{O}}_{K_0}$-presentation of Cartier type. We denote by $({\mathcal{A}}_L/{\mathfrak m}^r_{{\mathcal{A}}_L}){\widehat}{\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}$ the formal completion of $({\mathcal{A}}_L/{\mathfrak m}^r_{{\mathcal{A}}_L}){\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}$ relatively to the kernel of the homomorphism $$\label{surj first char 0}
({\mathcal{A}}_L/{\mathfrak m}^r_{{\mathcal{A}}_L}){\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}{\rightarrow}{\mathcal{O}_L}/{\mathfrak m}^r_{{\mathcal{O}_L}},\ \ \ a{\otimes}b{\mapsto}ab,$$ and by ${\mathcal{A}}_L{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}$ the projective limit $$\label{AwdhatO}
{\mathcal{A}}_L{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}={\varprojlim}_r(({\mathcal{A}}_L/{\mathfrak m}^r_{{\mathcal{A}}_L}){\widehat}{\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}).$$ We will always consider ${\mathcal{A}}_L{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}$ as an ${\mathcal{O}_K}$-algebra by the homomorphism $${\mathcal{O}_K}{\rightarrow}{\mathcal{A}}_L{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K},\ \ \ u{\mapsto}1{\otimes}u,$$ (in the following, we always abbreviate $1{\otimes}u$ by $u$) and we will consider it as an ${\mathcal{A}}_L$-algebra by $${\mathcal{A}}_{L}{\rightarrow}{\mathcal{A}}_L{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K},\ \ \ v{\mapsto}v{\otimes}1.$$ There is a canonical surjective homomorphism $$\label{surj third char 0}
{\mathcal{A}}_L{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}{\rightarrow}{\mathcal{O}}_{L},$$ induced by the surjections . We denote by $I_{L}$ its kernel.
Let $L$ be an object of ${\text{F\'E}_{/K}}$, and $({\mathcal{A}}_{L},j:{\mathcal{A}}_{L}{\rightarrow}{\mathcal{O}}_{L})$ an ${\mathcal{O}}_{K_0}$-presentation of Cartier type. Then,
- The ${\mathcal{O}_K}$-algebra ${\mathcal{A}}_L{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}$ is formally of finite type and formally smooth over ${\mathcal{O}_K}$ and the morphism ${\mathcal{A}}_L{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}/{\mathfrak m}_{{\mathcal{A}}_L{
\settoheight{\dhatheight}{\ensuremath{\hat{{\otimes}}}}
\addtolength{\dhatheight}{-0.70ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}}{\rightarrow}{\mathcal{O}_L}/{\mathfrak m}_{{\mathcal{O}_L}}$ is an isomorphism.
- Let $L'$ be another object in ${\text{F\'E}_{/K}}$ and $({\mathcal{A}}_{L'},j':{\mathcal{A}}_{L'}{\rightarrow}{\mathcal{O}}_{L'})$ an ${\mathcal{O}}_{K_0}$-presentation of Cartier type. If a uniformizer $\p_0$ is not a uniformizer of any factor of ${\mathcal{O}_L}$, then, any morphism $({\mathcal{A}}_{L},j){\rightarrow}({\mathcal{A}}_{L'},j')$ induces an isomorphism $$\label{key iso char 0}
{\mathcal{A}}_{L'}{\otimes}_{{\mathcal{A}}_L}({\mathcal{A}}_L{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}){{\xrightarrow}{\sim}}{\mathcal{A}}_{L'}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}.$$
Part (i) is proved in ([@as; @ii] 2.9). For part (ii), we may assume $L$ and $L'$ are fields. We denote by $e$ the ramification index of the extension $L'/L$. For any integer $r\geqslant 1$, we have the following canonical commutative diagram $$\xymatrix{
{\mathcal{A}}_L\ar[r]^-(0.5){\mathrm{pr}_1}\ar[d]&({\mathcal{A}}_{L}/{\mathfrak m}_{{\mathcal{A}}_{L}}^r){\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}\ar[d]\ar[r]^-(0.5){\eqref{surj first char 0}}&{\mathcal{O}_L}/{\mathfrak m}_L^r\ar[d]\\
{\mathcal{A}}_{L'}\ar[r]^-(0.5){\mathrm{pr}'_1}&({\mathcal{A}}_{L'}/{\mathfrak m}_{{\mathcal{A}}_{L}}^r{\mathcal{A}}_{L'}){\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}\ar[r]^-(0.5){g_{L'}}&{\mathcal{O}}_{L}/{\mathfrak m}_{L'}^{er}}$$ such that each square is Cartesian. We denote by $({\mathcal{A}}_{L'}/{\mathfrak m}_{{\mathcal{A}}_{L}}^r{\mathcal{A}}_{L'}){\widehat}{\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}$ the formal completion of $({\mathcal{A}}_{L'}/{\mathfrak m}_{{\mathcal{A}}_{L}}^r{\mathcal{A}}_{L'}){\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}$ relatively to the kernel of $g_{L'}$. Since ${\mathcal{A}}_{L}$ is a Noetherian local ring, by \[step 0 lemma\](iii) and Nakayama’s lemma, ${\mathcal{A}}_{L'}$ is a finite free ${\mathcal{A}}_L$-module. Then, we have $${\mathcal{A}}_{L'}{\otimes}_{{\mathcal{A}}_{L}}({\mathcal{A}}_{L}/{\mathfrak m}_{{\mathcal{A}}_{L}}^r){\widehat}{\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}{{\xrightarrow}{\sim}}({\mathcal{A}}_{L'}/{\mathfrak m}_{{\mathcal{A}}_{L}}^r{\mathcal{A}}_{L'}){\widehat}{\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}.$$ After taking projective limit on both sides, we obtain $$\label{iso 1 char 0}
{\mathcal{A}}_{L'}{\otimes}_{{\mathcal{A}}_{L}}({\mathcal{A}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}){{\xrightarrow}{\sim}}{\varprojlim}_r(({\mathcal{A}}_{L'}/{\mathfrak m}_{{\mathcal{A}}_{L}}^r{\mathcal{A}}_{L'}){\widehat}{\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}).$$ By the proof of ([@as; @ii] 2.7.3), we obtain that ${\mathfrak m}_{{\mathcal{A}}_{L'}}^e{\subseteq}{\mathfrak m}_{{\mathcal{A}}_L}{\mathcal{A}}_{L'}{\subseteq}{\mathfrak m}_{{\mathcal{A}}_{L'}}$. Hence, for any integer $r\geqslant 1$, we have two surjections $$({\mathcal{A}}_{L'}/{\mathfrak m}^{er}_{{\mathcal{A}}_{L'}}){\widehat}{\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}{\twoheadrightarrow}({\mathcal{A}}_{L'}/{\mathfrak m}^{r}_{{\mathcal{A}}_L}{\mathcal{A}}_{L'}){\widehat}{\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}{\twoheadrightarrow}({\mathcal{A}}_{L'}/{\mathfrak m}^{r}_{{\mathcal{A}}_{L'}}){\widehat}{\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K},$$ which imply $$\label{iso 2 char 0}
{\varprojlim}_r(({\mathcal{A}}_{L'}/{\mathfrak m}_{{\mathcal{A}}_{L}}^r{\mathcal{A}}_{L'}){\widehat}{\otimes}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}){{\xrightarrow}{\sim}}{\mathcal{A}}_{L'}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}.$$ Combining and , we get (ii).
{#nim char 0}
Let $L$ be an object of ${\text{F\'E}_{/K}}$, and $({\mathcal{A}}_{L},j:{\mathcal{A}}_{L}{\rightarrow}{\mathcal{O}}_{L})$ an ${\mathcal{O}}_{K_0}$-presentation of Cartier type. We will introduce objects analogue of those defined in $\S7$, and denote them by the same notation. For any rational number $r>0$ and integer numbers $s,t>0$ such that $r=t/s$, we denote by ${\mathcal{R}}^r_{L}$ the $K$-affinoid algebra $${\mathcal{R}}^r_{L}=({\mathcal{A}}_L{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K})\<I_{K}^s/\p^t\>{\otimes}_{{\mathcal{O}_K}}K,$$ by $X^r_{L}=\operatorname{Sp}({\mathcal{R}}^r_{L})$ the tubular neighborhood of thickening $r$ of the immersion $$\operatorname{Spf}({\mathcal{O}}_{L}){\rightarrow}\operatorname{Spf}({\mathcal{A}}_L{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}),$$ which is smooth over $K$ ([@as; @ii] 1.7). By \[fini theo\], there exists a finite separable extension $K'$ of $K$ such that the normalized integral model of $X^c_L$ is defined over $K'$ (\[def nim\]). We denote by ${\mathcal{R}}^r_{L,{\mathcal{O}}_{K'}}$ the supremum norm unit ball of ${\mathcal{R}}^r_{L}{\otimes}_K K'$ (\[model over K’\]), by $\fX^r_L$ the normalized integral model of $X^r_L$ over ${\mathcal{O}}_{{\overline}K}$ and by ${\overline}\fX^r_L$ the special fiber of $\fX^r_L$.
{#section-13}
In the following of this section, we assume that $p$ is not a uniformizer of $K$. By ([@as; @ii] 1.14.3), there is an isomorphism of ${\mathcal{O}_K}$-algebras $$\label{iso o ot o char 0}
{\mathcal{O}_K}[[T_0,...,T_m]]{{\xrightarrow}{\sim}}{\mathcal{A}}_K{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K},$$ such that the composition of it and ${\mathcal{O}_K}[[T_0,...,T_m]]{\rightarrow}{\mathcal{A}}_K{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}{\rightarrow}{\mathcal{O}_K}$ maps $T_i$ to $0$. If $r$ is an integer $\geqslant 1$, we have an isomorphism of $K$-affinoid algebras $$\label{iso affinoid K char 0}
K\<T_0/\p^r,...,T_m/\p^r\>{{\xrightarrow}{\sim}}{\mathcal{R}}^r_K.$$ The normalized integral model $\fX^r_K$ is defined over ${\mathcal{O}_K}$, and we have an isomorphism $$\label{iso nim K char 0}
{\mathcal{O}_K}\<T_0/\p^r,...,T_m/\p^r\>{{\xrightarrow}{\sim}}({\mathcal{A}}_K{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K})\<I_K/\p^r\>={\mathcal{R}}^r_{K,{\mathcal{O}_K}}.$$ Hence the geometric closed fiber ${\overline}\fX^r_K$ is isomorphic to the affine scheme $$\operatorname{Spec}{\overline}F[T_0/\p^r,...,T_m/\p^r].$$ In general, for any rational number $r>0$, the $K$-affinoid variety $X^r_K$ is isomorphic to $D^{m+1,(r)}$ and the rigid space $X_K=\bigcup_{r>0}X^r_K$ is isomorphic to $\mathring D^{m+1}$ (\[ram type ii notation2\]).
By ([@as; @ii] 2.11.2), we have an isomorphism $$\label{I/I2 iso O char 0}
(I_K/I^2_K){\otimes}_{{\mathcal{O}_K}}F{\rightarrow}{\widehat}\O^1_{{\mathcal{O}_K}/{\mathcal{O}}_{K_0}}{\otimes}_{{\mathcal{O}_K}} F,$$ such that for any $x\in{\mathcal{O}_K}$ and $\wt x$ a lifting in ${\mathcal{A}}_K$, the image of $({\overline}{1{\otimes}x-\widetilde x{\otimes}1}){\otimes}1$ is ${\mathrm d}x {\otimes}1.$ From ([@as; @ii] 1.14.3, 2.11.2), for any rational number $r>0$, the inverse of gives an isomorphism ${\overline}\fX^r_K{{\xrightarrow}{\sim}}{\Th^{(r)}_{{\overline}F}}$. When $r$ is an integer, the construction of the isomorphism is similar to the equal characteristic case (\[X int r\]).
\[ker A to O\] From , we notice that for any element $\wt x\in \ker({\mathcal{A}}_K{\rightarrow}{\mathcal{O}_K})$, the class $({\overline}{\wt x{\otimes}1}){\otimes}1$ vanishes in $(I_K/I_K^2){\otimes}_{{\mathcal{O}_K}}F$. It is equivalent to say that $\wt x{\otimes}1\in I_K^2+\p I_K$.
{#th to X char 0}
Let $L$ be a finite Galois extension of $K$ of group $G$ and conductor $c>1$. Let $(g,{\mathbbm{g}})$ be a finite and flat morphism from $({\mathcal{A}}_K,j_K:{\mathcal{A}}_K{\rightarrow}{\mathcal{O}_K})$ to $({\mathcal{A}}_L,j_L:{\mathcal{A}}_L{\rightarrow}{\mathcal{O}_L})$ (\[present cartier\]). By , ${\mathbbm{g}}$ induces a finite flat morphism ${\mathbbm{g}}{\otimes}{\mathrm{id}}:{\mathcal{A}}_K{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}{\rightarrow}{\mathcal{A}}_{L}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}$. Hence, for any rational number $r>0$, it gives a morphism of smooth $K$-affinoid varieties $X^r_L{\rightarrow}X^r_K$ ([@as; @ii] 1.6) which induces morphisms $\fX^r_L{\rightarrow}\fX^r_K$ and ${\overline}\fX^r_K{\rightarrow}{\overline}\fX^r_L$. For any $\s\in G$, there is a morphism ${\mathbbm{g}}_{\s}$ making the following diagram commutative (\[step 0 lemma\] iii) $$\label{gs}
\xymatrix{\relax
{\mathcal{A}}_{L}\ar[d]_{{\mathbbm{g}}_{\s}}\ar[r]^{j_L}&{\mathcal{O}}_{L}\ar[d]^{\s}\\
{\mathcal{A}}_{L}\ar[r]^{j_L}&{\mathcal{O}}_{L}}$$ The pair $(\s,{\mathbbm{g}}_{\s})$ induces automorphisms of $X^r_L$, $\fX^r_L$ and ${\overline}\fX^r_L$. Notice that, ${\mathbbm{g}}_{\s}$ is not unique in general and may not be an ${\mathcal{A}}_K$-homomorphism. Hence the automorphisms of $X^r_L$, $\fX^r_L$ and ${\overline}\fX^r_L$ induced by all possible ${\mathbbm{g}}_{\s}$ may not be uniquely determined by $\s\in G$. Luckily, by ([@as; @ii] 2.13), the induced automorphism of ${\overline}\fX^c_L$ is ${\overline}\fX_K^c$-invariant and independent of the choice of ${\mathbbm{g}}_{\s}$. Hence ${\overline}\fX^c_L{\rightarrow}{\overline}\fX^c_K$ is a finite étale $G$-torsor ([@as; @ii] 1.16.2). The geometric monodromy action of $G_K$ on ${\overline}\fX^c_{L}$ commutes with the action of $G$. Let ${\overline}\fX^c_{L,0}$ be a connected component of ${\overline}\fX^c_{L}$. The stabilizers of ${\overline}\fX^c_{L,0}$ via these two actions are $G^c$ and $G^c_K$, respectively ([@as; @ii] 2.15.1). Then, we get an isomorphism $G^c{{\xrightarrow}{\sim}}\operatorname{Aut}({\overline}\fX^c_{L,0}/{\overline}\fX^c_K)$ and a surjection $G^c_K{\rightarrow}\operatorname{Aut}({\overline}\fX^c_{L,0}/{\overline}\fX^c_K)$ which imply that $G^c$ is commutative (cf. [@as; @ii] 2.15.1). Composing with ${\overline}\fX^r_K{{\xrightarrow}{\sim}}{\Th^{(r)}_{{\overline}F}}$, the étale covering ${\overline}\fX^c_{L,0}{\rightarrow}{\Th^{(r)}_{{\overline}F}}$ induces a surjective homomorphism ([@as; @ii] 2.15.1) $$\p^{{\mathrm{ab}}}_1({\Th^{(r)}_{{\overline}F}}){\rightarrow}\operatorname{Gr}^cG_K{\rightarrow}G^c.$$
{#section-14}
In the following of this section, we assume the finite Galois extension $L/K$ of type (II) and we take again the notation and assumptions of \[ram type ii notation\] and \[ram type ii notation2\]. Let $(g,{\mathbbm{g}})$ be a finite and flat morphism as in \[th to X char 0\]. Let $\widetilde h$ be a lifting of $h\in {\mathcal{O}_L}$ in ${\mathcal{A}}_L$. Since ${\mathcal{A}}_K$ is a Noetherian local ring, by \[step 0 lemma\] (iii) and Nakayama’s lemma, we have that ${\mathcal{A}}_L$ is a finite free ${\mathcal{A}}_K$-module of rank $\sharp G$ and that ${\mathcal{A}}_L={\mathcal{A}}_K[\wt h]$. Let $$\wt f(T)=T^{p^n}+\wt a_{p^n-1}T_{p^n-1}+\cdots+\wt a_0\in{\mathcal{A}}_K[T],$$ be a lifting of $f[T]\in{\mathcal{O}_K}[T]$ such that $\wt h$ is a zero. We have an isomorphism $$\label{AL=AKh}
{\mathcal{A}}_K[T]/(\wt f(T)){{\xrightarrow}{\sim}}{\mathcal{A}}_L,\ \ \ T{\mapsto}\wt h.$$ By and the proof of ([@as; @ii] 1.6), we have an isomorphism $$\label{cart1 char 0}
{\mathcal{R}}_K^r{\otimes}_{{\mathcal{A}}_K{
\settoheight{\dhatheight}{\ensuremath{\hat{{\otimes}}}}
\addtolength{\dhatheight}{-0.70ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}}({\mathcal{A}}_L{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K}) {{\xrightarrow}{\sim}}{\mathcal{R}}_L^r.$$ It induces, for any rational numbers $r>r'>0$, an isomorphism $${\mathcal{R}}^{r}_K{\otimes}_{{\mathcal{R}}^{r'}_K}{\mathcal{R}}_{L}^{r'}{{\xrightarrow}{\sim}}{\mathcal{R}}^{r}_L,$$ which gives a Cartesian diagram of rigid spaces $$\label{tub morphism char 0}
\xymatrix{\relax
X^{r}_L\ar[d]\ar[r]&X_L\ar[d]\\
X^{r}_K\ar[r]&X_K}$$ where $X_K=\bigcup_{r>0}X^r_K$ and $X_L=\bigcup_{r>0}X^r_L$. We put $$\wt\bmf(T)=T^{p^n}+(\wt a_{p^n-1}{\otimes}1)T^{p^n-1}+\cdots+(\wt a_0{\otimes}1)\in ({\mathcal{A}}_{K}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K})[T].$$ From and , we have a surjection $$\t_L:{\mathcal{R}}^r_K\<T\>{\rightarrow}{\mathcal{R}}^r_L,\ \ \ T{\mapsto}\wt h{\otimes}1,$$ which induces an isomorphism that we denote abusively also by $$\label{key representation char 0}
\t_L:{\mathcal{R}}^r_K\<T\>/\wt\bmf(T){{\xrightarrow}{\sim}}{\mathcal{R}}^r_L.$$ In other terms, we have a co-Cartesian diagram of homomorphisms of ${\mathcal{R}}^r_K$-algebras $$\label{cocart char 0}
\xymatrix{\relax
{\mathcal{R}}^r_L&{\mathcal{R}}^r_K\<T\>\ar[l]_{\t_L}\\
{\mathcal{R}}^r_K\ar[u]&{\mathcal{R}}^r_K\<T\>\ar[u]_{\phi}\ar[l]_{\t_K},
}$$ where $\phi(T)=\wt\bmf(T)$ and $\t_K(T)=0$. Hence, taking the union of the $K$-affinoid varieties associated to each of the $K$-affinoid algebras in for $r\in\Q_{\geqslant 0}$, we obtain a Cartesian diagram $$\label{tub explicit char 0}
\xymatrix{\relax
X_L\ar[d]\ar[r]^-(0.5){i_L}&X_K{\times}D^1\ar[d]^{\wt{\mathbf{f}}}\\
X_K\ar[r]^-(0.5){i_K}&X_K{\times}D^1,}$$ where $i_L$, $\wt{\mathbf{f}}$ and $i_K$ are the morphisms induced by $\t_L$, $\phi$ and $\t_K$.
{#ab char 0}
In the following, for any $0\leqslant i\leqslant p^n-1$, we denote by $\a_i$ the element $ a_i-\wt a_i{\otimes}1\in I_K$ and fix $\wt\p\in{\mathcal{A}}_K$ a lifting of $\p\in{\mathcal{O}_K}$. When the conductor $c>2$, for each $1\leqslant i\leqslant p^n-1$, $v(a_i)\geqslant 2$ (\[p\^2\]). Let $ a'_i= \p^{-2} a_i\in{\mathcal{O}_K}$ and $\wt a'_i\in {\mathcal{A}}_K$ a lifting of $a'_i$. Then we have $\wt a_i=\wt\p^2\wt a_i'+\wt y_i$, where $\wt y_i\in\ker({\mathcal{A}}_K{\rightarrow}{\mathcal{O}_K})$. We denote by $\a'_i$ the element $a'_i-\wt a'_i{\otimes}1\in I_K$ and by $\b$ the element $\p-\wt\p{\otimes}1\in I_K$. Then, we have $$\a_i=(a'_i-\a'_i)(2\p\b-\b^2)+\p^2\a'_i+\wt y_i{\otimes}1.$$ Since $\a'_i$, $\b\in I_K {\subset}\p^c{\mathcal{R}}^c_{K,{\mathcal{O}_K}}$ and $\wt y_i{\otimes}1\in I_K^2+\p I_K{\subset}\p^{c+1}{\mathcal{R}}^c_{K,{\mathcal{O}_K}}$ (\[ker A to O\]), we have $\a_i\in \p^{c+1}{\mathcal{R}}^{c}_{K,{\mathcal{O}_K}}$. When $c=2$, we have $p=2$, $\sharp G=2$, $\deg f=2$ and $\r(c)=1$. Let $\wt a''_1\in {\mathcal{A}}_K$ be a lifting of $a''_1=\p^{-1}a_1$. We have $\a_1=\wt\p\wt a''_1+\wt z_1$, where $z_1\in\ker({\mathcal{A}}_K{\rightarrow}{\mathcal{O}_K})$. We denote by $\a''_1$ the element $a''_1-\wt a''_1{\otimes}1\in I_K$. Then we have $$\a_1=(a''_1-\a''_1)\b+\p \a''_1+\wt z_1{\otimes}1.$$ Since $\a''_1,\b\in\p^c{\mathcal{R}}^c_{K,{\mathcal{O}_K}}$ and $\wt z_1{\otimes}1\in I_K^2+\p I_K{\subset}\p^{c+1}{\mathcal{R}}^c_{K,{\mathcal{O}_K}}$, we have $\a_1\in \p^c{\mathcal{R}}^{c}_{K,{\mathcal{O}_K}}$, and ${\overline}{\a_1/\p^c}={\overline}{a''_1\b/\p^c}\in {\mathcal{R}}^{c}_{K,{\mathcal{O}_K}}\big/\p{\mathcal{R}}^{c}_{K,{\mathcal{O}_K}}$.
Put $$\wt\bmf_0(T)=\sum_{0\leqslant i\leqslant p^n-1}(\a_i/\p^c)\cdot T^i\in {\mathcal{R}}^c_{K,{\mathcal{O}_K}}[T].$$ We have $$\wt\bmf(T)=f(T)-\sum_{0\leqslant i\leqslant p^n-1}\a_iT^i=f(T)-\p^{c} \wt\bmf_0(T).$$
In the following, we fix an embedding $L{\rightarrow}{\overline}K$.
\[component char 0\] The $K$-affinoid $X^{c}_L$ has $\sharp(G/G^{c})=p^{n-s}$ geometric connected components. Let $\s_1,...,\s_{p^{n-s}}$ be liftings of all the elements of $G/G^{c}$ in $G$. We have $$i_L(X^{c}_L){\subseteq}\coprod_{1\leqslant j\leqslant p^{n-s}} X^{c}_K{\times}(\s_j(h)+D^{1,(\r(c))}){\subseteq}X_K{\times}D^1.$$
The proof is the same as in the equal characteristic case (\[component\]).
In the following, we denote by ${\overline}\fX^c_{L,0}$ the connected component of ${\overline}\fX^c_L$ corresponding to the connected component $X^c_{L,0}$ of $X^c_L$ containing $(0,...,0,h)\in X^c_K{\times}D^1$ defined over $L$.
\[hatO isogeny char 0\] There exists a canonical Cartesian diagram $$\label{cartesian diagram isogeny char 0}
\xymatrix{\relax
{\overline}\fX^c_{L,0}\ar[d]\ar[r]^{\nu}&\A^1_{{\overline}F}\ar[d]^{{\overline}{f_c}}\\
{\Th^{(c)}_{{\overline}F}}\ar[r]^{\mu}&\A^1_{{\overline}F},}$$ where ${\overline}{f_{c}}$ is defined in and if $\xi$ is the canonical coordinate of $\A^1_{{\overline}F}$, we have $$\mu^*(\xi)={\left}\{\begin{array}{ll}{\mathrm d}a_0{\otimes}\p^{-c},&\text{if}\ \ c>2,\\
(a''_1h{\mathrm d}\p+{\mathrm d}a_0){\otimes}\p^{-2},&\text{if}\ \ c=2.
\end{array}{\right}.$$ Moreover, for any $\s\in G^c$, the following diagram $$\label{G action diagram char 0}
\xymatrix{\relax
{\overline}\fX^c_{L,0}\ar[d]_{\s}\ar[r]^{\nu}&\A^1_{{\overline}F}\ar[d]^{d_{\s}}\\
{\overline}\fX^c_{L,0}\ar[r]^{\nu}&\A^1_{{\overline}F},}$$ where $d_{\s}^{*}(\x)=\x-u_{\s}$ (\[u\]), is commutative.
We consider the $K$-affinoid algebra ${\mathcal{R}}^c_K$ (resp. ${\mathcal{R}}^c_L$) as a sub-ring of the $L$-affinoid algebra ${\mathcal{R}}^c_K {\otimes}_K L$ (resp. ${\mathcal{R}}^c_L {\otimes}_K L$). By , we have $$X^c_{L,0}=i^{-1}_L(X^c_K{\times}(h+D^{1,(\r(c))}))\cap X^c_L.$$ Hence $X^c_{L,0}$ is presented by the $L$-affinoid algebra $$\label{connect comp over L char 0}
({\mathcal{R}}^{c}_L{\otimes}_{K}L)\<T'\>/(\p^{\r(c)}T'+h-\wt h{\otimes}1).$$ By the isomorphism , is isomorphic to $$({\mathcal{R}}^{c}_K{\otimes}_{K}L)\<T,T'\>/(\wt\bmf(T), \p^{\r(c)}T'+h-T),$$ which, after eliminating $T$ by the relation $\p^{\r(c)}T'+h-T=0$, is $$\label{connected comp char 0}
({\mathcal{R}}^{c}_K{\otimes}_{K}L)\<T'\>/(\wt\bmf(\p^{\r(c)}T'+h)).$$ In both cases, by \[isogeny f\] and \[ab char 0\], we have $$\wt\bmf(\p^{\r(c)}T'+h)/\p^{c}\in {\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}\<T'\>,$$ $$\wt\bmf(\p^{\r(c)}T'+h)/\p^{c+1}\notin{\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}\<T'\>.$$ Then the image of ${\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}\<T'\>$ in through the canonical surjective map $$({\mathcal{R}}^c_K{\otimes}_KL)\<T'\>{\rightarrow}({\mathcal{R}}^{c}_K{\otimes}_{K}L)\<T'\>/(\wt\bmf(\p^{\r(c)}T'+h)),$$ is $$\label{oB char 0}
{\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}\<T'\>/(\wt\bmf(\p^{\r(c)}T'+h)/\p^{c}).$$ Extending scalars from ${\mathcal{O}_L}$ to ${\overline}F$, we obtain the following ${\overline}F$-algebra:
- if $c>2$, $$\label{clos fiber >2 char 0}
({\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}{\otimes}_{{\mathcal{O}_L}}{\overline}F)[T']/({\overline}{ f_{c}}(T')-{\overline}{\a_0/\p^{c}}).$$
- if $c=2$, $$\label{clos fiber =2 char 0}
({\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}{\otimes}_{{\mathcal{O}_L}}{\overline}F)[T']/({\overline}{f_2}(T')- {\overline}{(\a_0+a_1''h\b)/\p^{2}}).$$
From the isomorphism and the canonical exact sequence , we know that when $c>2$ (resp. $c=2$), ${\overline}{\a_0/\p^c}$ (resp. ${\overline}{(\a_0+a_1''h\b)/\p^{2}}$) is a non-zero linear term in ${\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}{\otimes}_{{\mathcal{O}_L}}{\overline}F$. Hence and are all reduced. Then, by ([@as; @i] 4.1), $$\operatorname{Spf}({\mathcal{R}}^{c}_{K,{\mathcal{O}_L}}\<T'\>/(\wt\bmf(\p^{\r(c)}T'+h)/\p^{c}))$$ is the normalized integral model of $X^c_{K,0}$ defined over ${\mathcal{O}_L}$. Hence ${\overline}\fX^{c}_{L,0}$ is defined by the ${\overline}F$-algebra (resp. ) when $c>2$ (resp. $c=2$). We put $$\nu:{\overline}\fX^c_{L,0}{\rightarrow}\A^1_{{\overline}F}=\operatorname{Spec}({\overline}F[\x]),\ \ \ \nu^*(\x)=T'.$$ It follows form the isomorphism ${\overline}\fX^{c}_K{\rightarrow}{\Th^{(c)}_{{\overline}F}}$ that is Cartesian.
For any $\s\in G^c$, let $y_{\s}(x)=b_rx^r +\cdots +b_0\in {\mathcal{O}_K}[x]$ be a polynomial, such that $y_{\s}(h)=(h-\s(h))/\p^{\r(c)}\in {\mathcal{O}_L}$. We denote by $\wt y_{\s}(x)=\wt b_rx^r +\cdots +\wt b_0$ a lifting of $y_{\s}(x)$ in ${\mathcal{A}}_K[x]$ and by $\wt{\mathbbm y}(x)$ the polynomial $$\wt {\mathbbm y}(x)=(\wt b_r{\otimes}1)x^r +\cdots +(\wt b_0{\otimes}1)\in ({\mathcal{A}}_K{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K})[x].$$ Let ${\mathbbm{g}}_{\s}:{\mathcal{A}}_L{\rightarrow}{\mathcal{A}}_L$ be a homomorphism as in $\eqref{gs}$. We denote by ${\mathbf{g}}_{\s}$ the induced morphism of ${\mathbbm{g}}_{\s}$ on . By , we have $$\ker({\mathcal{A}}_L{\rightarrow}{\mathcal{O}_L})={\bigoplus}_{i=0}^{p^n-1}\ker({\mathcal{A}}_K{\rightarrow}{\mathcal{O}_K}) \wt h^i.$$ Hence, we have ${\mathbbm{g}}_{\s}(\wt h)=\wt h- \wt \p^{\r(c)}\wt y_{\s}(\wt h)+\varepsilon(\wt h)$, where $\varepsilon$ is a polynomial with coefficients in $\ker({\mathcal{A}}_K{\rightarrow}{\mathcal{O}_K})$. Then, we have $${\mathbf{g}}_{\s}(T')=T'-\wt{\mathbbm y}_{\s}(\p^{\r(c)}T'+h)+\D(T'),$$ where $$\D(T')=-((\wt\p^{\r(c)}{\otimes}1-\p^{\r(c)})/\p^{\r(c)})\wt{\mathbbm y}_{\s}(\p^{\r(c)}T'+h)+\wt\varepsilon(\p^{\r(c)}T'+h)/\p^{\r(c)},$$ and $\wt\varepsilon$ is a polynomials with coefficients in $J=\{\wt x{\otimes}1\in{\mathcal{A}}{
\settoheight{\dhatheight}{\ensuremath{\widehat{{\otimes}}}}
\addtolength{\dhatheight}{-0.35ex}
\widehat{\vphantom{\rule{1pt}{\dhatheight}}
\smash{\widehat{{\otimes}}}}}_{{\mathcal{O}}_{K_0}}{\mathcal{O}_K};\wt x\in \ker({\mathcal{A}}_K{\rightarrow}{\mathcal{O}_K})\}$. Since $J{\subseteq}\p^{c+1}{\mathcal{R}}_{K,{\mathcal{O}_K}}$ (\[ker A to O\]), $\wt\p^{\r(c)}{\otimes}1-\p^{\r(c)}\in\p^{c}{\mathcal{R}}_{K,{\mathcal{O}_K}}$ and $c>\r(c)$, it is easy to see that the reduction of $\D(T')$ to ${\overline}\fX^c_{L,0}$ is zero. For any $0\leqslant j\leqslant r$, we have $\wt b_j{\otimes}1- b_j\in \p^c{\mathcal{R}}^c_{K,{\mathcal{O}_K}}$. Then $${\overline}{\wt{\mathbbm y}_{\s}(\p^{\r(c)}T'+h)}={\overline}{\wt y_{\s}(\p^{\r(c)}T'+h)}={\overline}{\wt y_{\s}(h)}=u_{\s}.$$ Hence, by ([@as; @ii] 2.13), the diagram is commutative.
The refined Swan conductor of an extension of type (II)
=======================================================
{#section-15}
In this section, we assume either that $K$ has characteristic $p$ or that it has characteristic $0$ and that $p$ is not a uniformizer of $K$. Let $L$ be a finitely generated extension of $K$ of type (II) and we take again the notation and assumptions of \[ram type ii notation\], \[ram type ii notation2\], \[thnong isogeny\] and \[hatO isogeny char 0\].
\[isogeny xi\] The fibre product ${\overline}\fX^{c}_{L,0}{\times}_{{\Th^{(c)}_{{\overline}F}}}{\Xi^{(c)}_{{\overline}F}}$ is a connected affine scheme.
The image of ${\mathrm d}a_0{\otimes}1$ and $(a''_1h{\mathrm d}\p+{\mathrm d}a_0){\otimes}1$ by the canonical map from ${\widehat}\O^1_{{\mathcal{O}_K}/F_0}{\otimes}_{{\mathcal{O}_K}}{\overline}F$ (resp. ${\widehat}\O^1_{{\mathcal{O}_K}/{\mathcal{O}}_{K_0}}{\otimes}_{{\mathcal{O}_K}}{\overline}F $) to $\O^1_F{\otimes}_F {\overline}F$ is ${\mathrm d}\bar a_0{\otimes}1$, which is a non-zero element. So we have a Cartesian diagram $$\label{isogeny xi1}
\xymatrix{
{\overline}\fX^c_{L,0}{\times}_{{\Th^{(c)}_{{\overline}F}}}{\Xi^{(c)}_{{\overline}F}}\ar[d]\ar[r]&\A^1_{{\overline}F}\ar[d]^{{\overline}{f_c}}\\
{\Xi^{(c)}_{{\overline}F}}\ar[r]^{\mu'}&\A^1_{{\overline}F}}$$ where $\mu'^*(\xi)={\mathrm d}\bar a_0{\otimes}\p^{-c}$. Since ${\mathrm d}\bar a_0{\otimes}\p^{-c}$ is a non-zero linear term in the affine space ${\Xi^{(c)}_{{\overline}F}}$, ${\overline}\fX^{c}_{L,0}{\times}_{{\Th^{(c)}_{{\overline}F}}}{\Xi^{(c)}_{{\overline}F}}$ is connected.
{#proof of proposition}
[*Proof of \[thlog factor xi\]*]{}. By ([@as; @ii] 5.13), both in the equal and unequal characteristic case, we have a commutative diagram $$\label{as ii diagram}
\xymatrix{\relax
\p^{{\mathrm{ab}}}_1({\Th^{(c)}_{{\overline}F,\log}})\ar[r]\ar@{->>}[d]_{\g_1}&\p^{{\mathrm{ab}}}_1({\Th^{(c)}_{{\overline}F}})\ar@{->>}[d]^{\g_2}\\
G^{c}_{\log}\ar@{=}[r]&G^{c}.}$$ The surjection $\g_1$ factors through $\p^{{\mathrm{alg}}}_1({\Th^{(c)}_{{\overline}F,\log}})$ (\[isogeny\]). By \[thnong isogeny\] and \[hatO isogeny char 0\], $\g_2$ also factors through $\p^{{\mathrm{alg}}}_1({\Th^{(c)}_{{\overline}F}})$. Combining and the following canonical commutative diagram $$\xymatrix{\relax
\p^{{\mathrm{ab}}}_1({\Th^{(c)}_{{\overline}F,\log}})\ar[r]\ar@{->>}[d]&\p^{{\mathrm{ab}}}_1({\Xi^{(c)}_{{\overline}F}})\ar@{->>}[d]\ar[r]&\p^{{\mathrm{ab}}}_1({\Th^{(c)}_{{\overline}F}})\ar@{->>}[d]\\
\p^{{\mathrm{alg}}}_1({\Th^{(c)}_{{\overline}F,\log}})\ar[r]&\p^{{\mathrm{alg}}}_1({\Xi^{(c)}_{{\overline}F}})\ar[r]&\p^{{\mathrm{alg}}}_1({\Th^{(c)}_{{\overline}F}}),}$$ we obtain that $$\label {key diagram}
\xymatrix{\relax
\p^{{\mathrm{alg}}}_1({\Th^{(c)}_{{\overline}F,\log}})\ar[r]\ar@{->>}[d]&\p^{{\mathrm{alg}}}_1({\Xi^{(c)}_{{\overline}F}})\ar[r]&\p^{{\mathrm{alg}}}_1({\Th^{(c)}_{{\overline}F}})\ar@{->>}[d]\\
G^{c}_{\log}\ar@{=}[rr]& &G^{c}}$$ is commutative. The composition of morphisms $\p^{{\mathrm{alg}}}_1({\Xi^{(c)}_{{\overline}F}}){\rightarrow}\p^{{\mathrm{alg}}}_1({\Th^{(c)}_{{\overline}F}}){\rightarrow}G^c$ corresponds the isogeny ${\overline}\fX^c_{L,0}{\times}_{{\Th^{(c)}_{{\overline}F}}}{\Xi^{(c)}_{{\overline}F}}{\rightarrow}{\Xi^{(c)}_{{\overline}F}}$ (cf. ). Hence, by , we have a commutative diagram $$\label{diagram refined swan}
\xymatrix{\relax
&\operatorname{Hom}(\p^{{\mathrm{alg}}}_1({\Xi^{(c)}_{{\overline}F}}),\F_p)\ar[r]\ar[d]&\O^1_F{\otimes}_{F}{\mathfrak m}^{-c}_{{\overline}K}/{\mathfrak m}^{-c+}_{{\overline}K}\ar[d]\\
\operatorname{Hom}(G^{c},\F_p)\ar[r]\ar[ur]&\operatorname{Hom}(\p^{{\mathrm{alg}}}_1({\Th^{(c)}_{{\overline}F,\log}}),\F_p)\ar[r]&\O^1_F(\log){\otimes}_{F}{\mathfrak m}^{-c}_{{\overline}K}/{\mathfrak m}^{-c+}_{{\overline}K},}$$ which conclude \[thlog factor xi\].
{#proof of theorem}
[*Proof of \[theorem rsw\].*]{} Since the surjection $\p^{{\mathrm{alg}}}_1({\Xi^{(c)}_{{\overline}F}}){\rightarrow}G^c$ is obtained by pulling-back the isogeny ${\overline}{f_c}:\A_{{\overline}F}^1{\rightarrow}\A_{{\overline}F}^1$ by $\mu'$ (cf. ), which is an étale $G^c$-torsor with the action of $G^c$ given by $d_{\s}$ for $\s\in G^c$ , . With notation in \[notation for rsw\], we denote by $ \tilde f_{c,\c}(\xi)$ the polynomial $$\tilde f_{c,\c}(\xi)={\left}(\prod_{\s\in G-G^{c}}u_{\s}{\right})(\xi^p-\bar f_{c,\c}^{p-1}(u_{\t})\xi)\in {\overline}F[\xi].$$ Observe that $\tilde f_{c,\c}(\bar f_{c,\c}(\xi))={\overline}{f_{c}}(\xi)$, hence the isogeny ${\overline}{f_c}$ is the composition of two isogenies $$\A^1_{{\overline}F}{\xrightarrow}{\bar f_{c,\c}}\A^1_{{\overline}F}{\xrightarrow}{\tilde f_{c,\c}}\A^1_{{\overline}F}.$$ For any $\s\in \ker \c$, $\bar f_{c,\c}^*(\xi-u_{\s})=\bar f_{c,\c}^*(\xi)$, i.e. $\bar f_{c,\c}d_{\s}=\bar f_{c,\c}$. Hence the isogeny $\tilde f_{c,\c}:\A^1_{{\overline}F}{\rightarrow}\A^1_{{\overline}F}$ is an étale $(G^c/\ker\c)$-torsor. Then, the surjection $\p^{{\mathrm{alg}}}_1({\Xi^{(c)}_{{\overline}F}}){\rightarrow}G^c{\xrightarrow}{\c}\F_p$ corresponds to the pull-back of $\tilde f_{c,\c}$ by $\mu'$ and the $\F_p$-action on this torsor is given by $1^*:\xi{\mapsto}\xi-\bar f_{c,\c}(u_{\t})$. We have the following Cartesian diagram $$\label{serre}
\xymatrix{\relax
\F_p\ar[d]_{{\mathrm{id}}}\ar[r]^{\phi} & \A^1_{{\overline}F}\ar[d]_{\l_2}\ar[r]^{\tilde f_{c,\c}} & \A^1_{{\overline}F}\ar[d]^{\l_1} \\
\F_p\ar[r] & \A^1_{{\overline}F}\ar[r]^{\mathrm{L}} & \A^1_{{\overline}F}}$$ where $\mathrm{L}$ denotes the Lang’s isogeny defined by $\mathrm{L}^*(\xi)=\xi^p-\xi$. The morphisms $\l_1$, $\l_2$ and $\phi$ are given as follows $$\l_1^*(\xi)=-\xi\bigg/\bigg(\prod\limits_{\s\in G-G^{c}}u_{\s}\bigg)\bar f_{c,\c}^p(u_{\t}),$$ $$\l_2^*(\xi)=-\xi/\bar f_{c,\c}(u_{\t}),$$ $$\phi(1)=-\bar f_{c,\c}(u_{\t}).$$ The sign is chosen in order that, for any $\s\in G^c$, the translation by $\phi(\c(\s))$ is induced by $d_{\s}$. Consequently, $\p^{{\mathrm{alg}}}_1({\Xi^{(c)}_{{\overline}F}}){\rightarrow}G^c{\xrightarrow}{\c}\F_p$ corresponds to the pull-back of $\mathrm{L}$ by $\l_1\mu'$. Hence the image of $\c\in\operatorname{Hom}(G^{c},\F_p)$ in $\O^1_F{\otimes}_{F}{\mathfrak m}^{-c}_{{\overline}K}/{\mathfrak m}^{-c+}_{{\overline}K}$ is $$-{\mathrm d}\bar a_0{\otimes}{\frac}{\p^{-c}}{{\left}(\prod_{\s\in G-G^{c}}u_{\s}{\right})\bar f_{c,\c}^p(u_{\t})}\in \O^1_F{\otimes}_{F}{\mathfrak m}^{-c}_{{\overline}K}/{\mathfrak m}^{-c+}_{{\overline}K}.$$ Then the theorem follows from .
Comparison of Kato’s and Abbes-Saito’s characteristic cycles
============================================================
{#section-16}
In this section, let $L$ be a finite Galois extension of $K$ of type (II) and we take again the notation and assumptions of \[ram type ii notation\] and \[ram type ii notation2\]. Let $C$ be an algebraically closed field of characteristic zero. We fix a non-trivial character $\psi_0:\F_p{\rightarrow}C^{{\times}}$. Any character $\c:G^c{\rightarrow}C^{{\times}}$ factors uniquely through $G^c{\rightarrow}\F_p{\xrightarrow}{\psi_0} C^{{\times}}$. We denote by $\bar\c$ the induced character $G^c{\rightarrow}\F_p$.
\[rank 1 theorem\] Let $\c: G{\rightarrow}C^{{\times}}$ be a character of $G$ such that its restriction to $G^c$ is non-trivial. Let $\t\in G^c$ be a lifting of $1\in \F_p$ in $G^c$ through $\bar\c:G^c{\rightarrow}\F_p$. Then Kato’s swan conductor with differential values $\operatorname{sw}_{\psi_0(1)}(\c)$ is given by , $$\operatorname{sw}_{\psi_0(1)}(\c)=[\p^c]+[-\bar f^p_{c,\bar\c}(u_{\t})]+\sum_{\s\in G-G^c}[u_{\s}]-[{\mathrm d}\bar a_0]\in {S_{K,L}}.$$
By definition , we have $$\begin{aligned}
\operatorname{sw}_{\psi_0(1)}(\c)&=&\sum_{\s\in G-\{1\}}([h-\s(h)]-[{\mathrm d}\bar h]){\otimes}(1-\c(\s))+\sum_{r\in \F_p^{{\times}}}[r]{\otimes}\psi_0(r)\label{sw 1 dim}\\
&=&\sum_{\s\in G^c-\{1\}}[h-\s(h)]{\otimes}(1-\c(\s))+\sum_{r\in \F_p^{{\times}}}[r]{\otimes}\psi_0(r)\label{G^c}+ \nonumber\\
& &\sum_{\s\in G-G^c}[h-\s(h)]-\sum_{\s\in G-G^c}[h-\s(h)]{\otimes}\c(\s)- \nonumber\\
& &\sum_{\s\in G-\{1\}}[{\mathrm d}\bar h]{\otimes}(1-\c(\s)). \nonumber\end{aligned}$$ Choose an $\F_p$-basis $\t_1=\t,\t_2,...,\t_s$ of $G^{c}$ such that $\bar\c(\t_1)=1\in\F_p$ and, that for any $2\leqslant j\leqslant s$, $\bar\c(\t_j)=0$. Then, by \[isogeny lemma\], we have $$\begin{aligned}
\,&\,&\sum_{\s\in G^c-\{1\}}[h-\s(h)]{\otimes}(1-\c(\s))+\sum_{r\in \F_p^{{\times}}}[r]{\otimes}\psi_0(r)\label{G^c}\\
&=&[\p^{\r(c)\sharp G^c}]+\sum_{\{j_1,...,j_s\}\in\F^s_p-\{0\}}[j_1u_{\t_1}+\cdots+j_s u_{\t_s}]{\otimes}(1-\psi_0(j_1))+\sum_{r\in \F_p^{{\times}}}[r]{\otimes}\psi_0(r)\nonumber\\
&=&[\p^{\r(c)\sharp G^c}]+\sum_{r\in\F_p^{{\times}}}[\bar f_{c,\bar\c}(r u_{\t_1})]{\otimes}(1-\psi_0(r))+\sum_{r\in \F_p^{{\times}}}[r]{\otimes}\psi_0(r)\nonumber\\
&=&[\p^{\r(c)\sharp G^c}]+\sum_{r\in\F_p^{{\times}}}([\bar f_{c,\bar\c}(u_{\t_1})]+[r]){\otimes}(1-\psi_0(r))+\sum_{r\in \F_p^{{\times}}}[r]{\otimes}\psi_0(r)\nonumber\\
&=&[\p^{\r(c)\sharp G^c}]+\sum_{r\in\F_p^{{\times}}}[\bar f_{c,\bar\c}(u_{\t_1})]{\otimes}(1-\psi_0(r))+\sum_{r\in \F_p^{{\times}}}[r]\nonumber\\
&=&[\p^{\r(c)\sharp G^c}]+[-\bar f_{c,\bar\c}^p(u_{\t_1})]\in{S_{L/K}}.\nonumber\end{aligned}$$ Let $\s_1=1,\s_2...,\s_{p^{n-s}}$ be a lifting of all the element of $G/G^c$ in $G$ and denote by $J$ the set $\{\s_2,...,\s_{p^{n-s}}\}$. Observe that for any $\varsigma\in J$ and $\s\in G^c$, we have $$[h-\varsigma\s(h)]=[h-\varsigma(h)+\varsigma(h-\s(h))]=[h-\varsigma(h)].$$ Hence $$\begin{aligned}
\sum_{\s\in G-G^c}[h-\s(h)]{\otimes}\c(\s)&=&\sum_{\varsigma\in J}\sum_{\s\in G^c}[h-\varsigma\s(h)]{\otimes}\c(\varsigma\s)\label{G-G^c}\\
&=&\sum_{\varsigma\in J}\sum_{\s\in G^c}[h-\varsigma(h)]{\otimes}\c(\varsigma)\c(\s)=0.\nonumber\end{aligned}$$ Moreover, by the isomorphism , we have $$\label{dh da}
\sum_{\s\in G^c-\{1\}}[{\mathrm d}\bar h]{\otimes}(1-\c(\s))=\sharp G[{\mathrm d}\bar h]=[{\mathrm d}\bar a_0]\in {S_{K,L}}.$$ Hence, combining , , and , we obtain that $$\begin{aligned}
\operatorname{sw}_{\psi_0(1)}(M)&=&[\p^{\r(c)\sharp G^c}]+[-\bar f_{c,\c}^p(u_{\t_1})]+\sum_{\s\in G-G^c}[h-\s(h)]{\otimes}1-\sharp G[{\mathrm d}\bar h]\nonumber\\
&=&[\p^c]+[-\bar f^p_{c,\bar\c}(u_{\t})]+\sum_{\s\in G-G^c}[u_{\s}]-[{\mathrm d}\bar a_0].\nonumber\end{aligned}$$
\[reduce to rank 1 key lemma\] Let $M$ be a finite dimensional $C$-vector space with an irreducible linear action of $G$. Then, there exists a subgroup $H$ of $G$ satisfying $G^c{\subseteq}H$ and a 1-dimensional representation $\th$ of $H$, such that $M=\operatorname{Ind}_H^G\th$.
Since $M$ is irreducible and $G$ is nilpotent (hence super-solvable), there exist a subgroup $H$ of $G$ and a 1-dimensional representation $\th$ of $H$, such that $M=\operatorname{Ind}_H^G\th$ ([@serre; @gr] 8.5 Th. 16). Let $\operatorname{Res}^G_{G^c}M={\bigoplus}_i M_i$ be the canonical decomposition of $\operatorname{Res}^G_{G^c}M$ into isotypic $G^c$-representations (cf. [@serre; @gr] 2.6). Since $G^c$ is contained in the center of $G$, any $\s\in G$ defines an automorphism of the $G^c$-representation $\operatorname{Res}^G_{G^c}M$. In particular, for any $i$, $\s$ induces an automorphism of $M_i$. On the other hand, since $M$ is irreducible, $G$ permutes transitively the $M_i$’s. Hence $\operatorname{Res}^G_{G^c}M$ is isotypic. By ([@serre; @gr] 7.3 Prop. 22), we have $$\label{ind res}
\operatorname{Res}_{G^c}^GM=\operatorname{Res}_{G^c}^G\operatorname{Ind}^G_H\th={\bigoplus}_{H\backslash G/G^c}\operatorname{Ind}^{G^c}_{H\cap G^c}\operatorname{Res}^H_{H\cap G^c}\th.$$ We notice that, if $H\cap G^c\neq G^c$, since $G^c=H\cap G^c{\oplus}G^c/H\cap G^c$, $\operatorname{Ind}^{G^c}_{H\cap G^c}\operatorname{Res}^H_{H\cap G^c}\th$ is isomorphic to the tensor of the regular representation of $G^c/H\cap G^c$ with $\operatorname{Res}^H_{H\cap G^c}\th$ which is not isotypic.
\[mainkey\] Assume that $p$ is not a uniformizer of $K$. Let $M$ be a finite dimensional $C$-vector space with a linear action of $G$. Then, $$\label{cckcc}
{\mathrm{CC}}_{\psi_0}(M)={\mathrm{KCC}}_{\psi_0(1)}(M).$$
From the definitions, we may assume that $M$ is irreducible. We denote by $c_0$ the unique slope of $M$. By definitions and \[scdv quotient\], both sides of will not change if replacing $G$ by $G/G^{c_0+}$. Hence we may assume further that the unique slope of $M$ is equal to $c$. By \[reduce to rank 1 key lemma\], $M=\operatorname{Ind}_H^G\th$ where $H$ is a subgroup of $G$ containing $G^c$ and $\th$ is a character of $H$. Since the slope of $M$ is $c$, the restriction of $\th$ to $G^c$ is non-trivial . We notice that $[G:H]=\dim_CM$. Choose an $\F_p$-basis $\t_1,...,\t_s$ of $G^{c}$ such that $\bar\th(\t_1)=1\in\F_p$ and, for any $2\leqslant j\leqslant s$, $\bar\th(\t_j)=0$. Let $c'=\r(c)+\sum_{\s\in H-\{1\}}v(h-\s(h))$. Since $L/L^H$ is still of type (II), we obtain that the conductor of $L/L^H$ is $c'$, that $H^{c'}=G^c$ and, denoting by $\r'$ the Herbrand function of $L/L^H$, that $\r'(c')=\r(c)$. Using (\[rank 1 theorem\]) for the group $H$ and the representation $\th$, we have $$\label{sw th}
\operatorname{sw}_{\psi_0(1)}(\th)=[\p^{c'}]+[-\bar f^p_{c,\bar\th}(u_{\t_1})]+\sum_{\s\in H-H^{c'}}[u_{\s}]-\sharp H[{\mathrm d}\bar h].$$ Meanwhile, we have $$\label{D-D}
-\sum_{\s\in G-H} ([{\mathrm d}\bar h]-[h-\s(h)])=(\sharp H-\sharp G)[{\mathrm d}\bar h]+[\p^{c-c'}]+\sum_{\s\in G-H}u_{\s}.$$
Hence, combining , and the induction formula for Kato’s swan conductors (\[indsw good\]), we have $$\begin{aligned}
\operatorname{sw}_{\psi_0(1)}(M)&=&[G:H]{\left}(\operatorname{sw}_{\psi_0(1)}(\th)-\sum_{\s\in G-H} ([{\mathrm d}\bar h]-[h-\s(h)]{\right})\nonumber\\
&=&[G:H]{\left}( [\p^c]+[-\bar f^p_{c,\bar\th}(u_{\t_1})]-[{\mathrm d}\bar a_0]+\sum_{\s\in G-G^c}[u_{\s}]{\right}).\nonumber\end{aligned}$$ Hence Kato’s characteristic cycle ${\mathrm{KCC}}_{\psi_0(1)}(M)$ is given by $${\mathrm{KCC}}_{\psi_0(1)}(M)={\frac}{(-{\mathrm d}\bar a_0)^{{\otimes}[G:H]}}{\big({\left}(\prod_{\s\in G-G^c}u_{\s}{\right})\bar f^p_{c,\bar\th}(u_{\t_1})\big)^{[G:H]}}\in (\O^1_F)^{{\otimes}[G:H]}.$$ On the other hand, $\operatorname{Res}^G_{G^c} M={\bigoplus}_{G/H}\operatorname{Res}^H_{G^c} \th$ . Hence the Abbes-Saito’s characteristic cycle ${\mathrm{CC}}_{\psi_0}(M)$ is given by $$\label{cc ultimate}
{\mathrm{CC}}_{\psi_0}(M)={\left}(\operatorname{rsw}(\operatorname{Res}^H_{G^c}(\th)){\otimes}\p^c{\right})^{[G:H]}={\frac}{(-{\mathrm d}\bar a_0)^{{\otimes}[G:H]}}{\big({\left}(\prod_{\s\in G-G^c}u_{\s}{\right})\bar f^p_{c,\bar\th}(u_{\t_1})\big)^{[G:H]}}\in (\O^1_F{\otimes}_{F}{\overline}F)^{{\otimes}[G:H]}.$$ So, we have ${\mathrm{CC}}_{\psi_0}(M)={\mathrm{KCC}}_{\psi_0(1)}(M)$.
\[hasse arf cc\] Assume that $p$ is not a uniformizer of $K$. Let $M$ be a finite dimensional $\L$-vector space with a linear action of $G$ and $r=\dim_{\L}M/M^{(0)}$. Then, we have $${\mathrm{CC}}_{\psi_0}(M)\in (\O^1_F)^r{\subseteq}(\O^1_F{\otimes}_F {\overline}F)^r$$
It is a Hasse-Arf type result for Abbes-Saito characteristic cycle. We should mention that T. Saito ([@wrcb] 3.10) and L. Xiao [@xiao] proved independently analogue results for smooth varieties of any dimension over perfect fields.
\[indcc\] Assume that $p$ is not a uniformizer of $K$. Let $H$ be a sub-group of $G$, and $N$ a finite dimensional $C$-linear representation of $H$. We denote by $r$ the dimension of $N$ and by $r'$ the dimension of $N^{(0)}$. Then, we have $$\label{ind for cc}
{\mathrm{CC}}_{\psi_0}(\operatorname{Ind}^G_HN)={\mathrm{CC}}_{\psi_0}(N)^{{\otimes}[G:H]}{\otimes}{\frac}{({\mathrm d}\bar a_0)^{{\otimes}([G:H]-1)}}{{\left}(\prod_{\s\in G-H}u_{\s}{\right})^{[G:H]}}\in(\O^1_F)^{{\otimes}([G:H]r-r')}.$$
Indeed, follows from the induction formula for Kato’s swan conductor with differential values and \[mainkey\].
\[remark general equal\] Assume that $p$ is not a uniformizer of $K$. Let $L'$ be a finite Galois extension of $K$ of group $G'$ which contains a sub-extension $K'$ of $K$ such that $K'/K$ is unramified and $L'/K'$ is of type (II). We denote by $P'$ the Galois group of the extension $L'/K'$ and by $F'$ the residue field of ${\mathcal{O}}_{K'}$. Let $\L$ be a finite field of characteristic $\ell\neq p$ which contains a primitive $(\sharp P')$-th root of unity and let $N$ be a $\L$-vector space of finite dimension with a linear-$G'$ action. We fix a non-trivial character $\psi:\F_p{\rightarrow}\L^{{\times}}$. By \[ext swan 1\] and \[ext swan 2\], we can still define ${\mathrm{KCC}}_{\psi(1)}(N)\in(\O^1_F)^{{\otimes}r}$, where $r=\dim_{\L}N/N^{(0)}$. On the other hand, the wild inertia subgroup $P$ of $G_K$ acts on $N$ through $P'$, we can define ${\mathrm{CC}}_{\psi}(N)$ (\[cc\]). By ([@saito; @cc] 1.22) and ([@as; @iii] 3.1), we have $$\label{1}
{\mathrm{CC}}_{\psi}(\operatorname{Res}^{G'}_{P'}N)={\mathrm{CC}}_{\psi}(N)\in(\O^1_{F}(\log){\otimes}_{F}{\overline}F)^{{\otimes}r}$$ through the canonical isomorphism $\O^1_F(\log){\otimes}_F F'{{\xrightarrow}{\sim}}\O^1_{F'}(\log)$. Moreover, let $\L'$ be the algebraic closure of the fraction field of the ring of Witt vectors $W(\L)$, $N'$ a pre-image of the class of $\operatorname{Res}^{G'}_{P'}N$ in the Grothendieck ring $R_{\L'}(P')$ ([@serre; @gr] 16.1 Th. 33) and $\psi':\F_p{\rightarrow}\L'^{{\times}}$ the unique lifting of $\psi:\F_p{\rightarrow}\L^{{\times}}$. By \[slope center decom p to 0\], we deduce that $$\label{2}
{\mathrm{CC}}_{\psi'}(N')={\mathrm{CC}}_{\psi}(\operatorname{Res}^{G'}_{P'}N).$$ From \[mainkey\], we have $$\label{3}
{\mathrm{CC}}_{\psi'}(N')={\mathrm{KCC}}_{\psi(1)}(N).$$ By , and , we conclude that $$\label{general equal}
{\mathrm{CC}}_{\psi}(N)={\mathrm{KCC}}_{\psi(1)}(N)\in(\O^1_F)^{{\otimes}r}.$$
Nearby cycles of $\ell$-sheaves on relative curves
==================================================
{#section-17}
In this section, we denote by $S=\operatorname{Spec}(R)$ an excellent strictly henselian trait. Assume that the residue field of $R$ has characteristic $p$ and that $p$ is not a uniformizer of $R$. We denote by $s$ (resp. $\eta$, resp. $\bar\eta$) the closed point (resp. generic point, a geometric generic point) of $S$. A finite covering of $(S,\eta,s)$ stands for a trait $(S',\eta',s')$ equipped with a finite morphism $S'{\rightarrow}S$. Let $\L$ be a finite field of characteristic $\ell\neq p$ and fix a non-trivial character $\psi_0:\F_p{\rightarrow}\L^{{\times}}$.
{#section-18}
We define a category $\sC_S$ as follows. An object of $\sC_S$ is a normal affine $S$-scheme $H$ for which there exist an $S$-scheme of finite type and a closed point $x$ of $X_s$, such that $X-\{x\}$ is smooth over $S$ and $H$ is $S$-isomorphic to the henselization of $X$ at $x$. A morphism between two objects of $\sC_S$ is a generically étale finite morphism of $S$-schemes. Let $(S',\eta',s')$ be a finite covering of $(S,\eta,s)$. Then for any object $H$ of $\sC_S$, $H{\times}_SS'$ is an object of $\sC_{S'}$ ([@kato; @vc] 5.4).
{#section-19}
Let $H$ be an object of $\sC_S$. We denote by $P(H)$ the set of height 1 points of $H$, by $$P_s(H)=P(H)\cap H_s,\ \ \ P_{\eta}(H)=P(H)\cap H_{\eta}.$$ We have ([@kato; @vc] 5.2, [@as; @ft] A.6):
- $H_{\eta}$ is geometrically regular over $\eta$ and for any $\fp\in P_{\eta}(H)$, the residue field $\kappa(\fp)$ of $H$ at $\fp$ is a finite extension of the fraction field $K(S)$ of $S$.
- $H_s$ is a reduced henselian noetherian local scheme over $s$ of dimension 1, hence $P_s(H)$ is a finite set.
We denote by $\widetilde{H}_s$ the normalization of $H_s$, which is a finite union of strictly henselian traits. We put $$\d(H)=\dim_k(\sO_{\widetilde{H}_s}/\sO_{H_s}).$$
{#section-20}
([@as; @ft] A.7, A.8).\[triple\] Let $H$ be an object of $\sC_S$, $U$ a non-empty open sub-scheme of $H_{\eta}$ and $\sF$ a locally constant constructible étale sheaf of ${\L}$-modules over $U$. For a triple $(H,U,\sF)$ and a finite covering $(S',\eta',s')$ of $(S,\eta,s)$, we denote by $(H,U,\sF)_{S'}$ the triple $(H',U',\sF')$ where $H'=H{\otimes}_SS'$, $U'$ is the inverse image of $U$ in $H'$ and $\sF'$ is the inverse image of $\sF$ on $U'$. We call the triple $(H,U,\sF)$ [*stable*]{} if there is an étale connected Galois covering $\widetilde U$ of $U$ such that
- The pull-back of $\sF$ to $\widetilde U$ is constant.
- The normalization $\widetilde H$ of $H$ in $\widetilde U$ belongs to $\sC_S$ and the residue field of $\widetilde H$ at all points of $\widetilde H_{\eta}-\widetilde U_{\eta}$ are finite separable extensions of $\k(\eta)$.
\[be stable\] Let $(H,U,\sF)$ be a triple as \[triple\].
- If $(H,U,\sF)$ is stable, $(H,U,\sF)_{S'}$ is stable for any finite covering $S'$ of $S$.
- For any triple $(H,U,\sF)$, there exist a finite covering $(S',\eta',s')$ of $(S,\eta,s)$ such that $(H,U,\sF)_{S'}$ is stable.
Proposition (i) follows form ([@kato; @vc] 5.4) and proposition (ii) follows form [@epp].
{#section-21}
Let $(H,U,\sF)$ be a stable triple. For $\fp\in P(H)$, we denote by ${\widehat}{{\mathcal{O}}}_{H,\fp}$ the completion of the local ring of $H$ at $\fp$ and by $\kappa(\fp)$ its residue field. For $\fp\in P_s(H)$, we denote by $\widetilde{H}_{s,\fp}$ the integral closure of $H_s$ in $\kappa(\fp)$, which is a strictly henselian trait. Let $\operatorname{ord}_{s,\fp}$ be the valuation of $\kappa(\fp)$ associated to $\widetilde{H}_{s,\fp}$ normalized by $\operatorname{ord}_{s,\fp}(\kappa(\fp)^{{\times}})=\Z$. We denote also by $\operatorname{ord}_{s,\fp}:\O^1_{\kappa(\fp)}-\{0\}{\rightarrow}\Z$ the valuation defined by $\operatorname{ord}_{s,\fp}(\a{\mathrm d}\b)=\operatorname{ord}_{s,\fp}(\a)$, if $\a,\b\in\kappa(\fp)^{{\times}}$ and $\operatorname{ord}_{s,\fp}(\b)=1$. It can be canonically extended, for any integer $r>0$, to $(\O^1_{\kappa(\fp)})^{{\otimes}r}-\{0\}$. Following ([@sga7i] XVI, [@lau] and [@kato; @vc] 6.4), we call [*total dimension*]{} of $\sF$ at a point $\fp\in P(H)$, and denote by $\operatorname{dimtot}_{\fp}(\sF)$ the integer defined as follows:
- For $\fp\in P_{\eta}(H)$, we put $$\operatorname{dimtot}_{\fp}(\sF)=[\kappa(\fp):\k(\eta)](\operatorname{sw}_{\fp}(\sF)+\operatorname{rank}(\sF)),$$ where $\operatorname{sw}_{\fp}(\sF)$ is the Swan conductor of the pull-back of $\sF$ over $\operatorname{Spec}({\widehat}{{\mathcal{O}}}_{H,\fp}){\times}_HU$.
- For $\fp\in P_s(H)$, we denote by $K_{\fp}$ the fraction field of ${\widehat}{{\mathcal{O}}}_{H,\fp}$. Since the triple $(H,U,\sF)$ is stable, there exists a finite Galois extension $L_{\fp}$ of $K_{\fp}$ of ramification index one, such that the representation $\sF_{\fp}$ of ${\mathrm{Gal}}(K^{\mathrm{sep}}_{\fp}/K_{\fp})$ defined by $\sF$ factors through the quotient ${\mathrm{Gal}}(L_{\fp}/K_{\fp})$. Notice that $L_{\fp}/K_{\fp}$ factors through a field $K'_{\fp}$ such that $K'_{\fp}/K_{\fp}$ is unramified and $L_{\fp}/K'_{\fp}$ is of type (II) (\[type\]). Fixing a uniformizer $\p$ of $R$ (also a uniformizer of $K_{\fp}$), we have ${\mathrm{CC}}_{\psi_0}(\sF_{\fp})\in (\O^1_{\k(\fp)})^m$ (cf. \[remark general equal\]). We denote by ${\overline}{\sF}_{\fp}$ the restriction to $\operatorname{Spec}(\kappa(\fp))$ of the direct image of $\sF$ under $\operatorname{Spec}(K_{\fp}){\rightarrow}\operatorname{Spec}({\widehat}{{\mathcal{O}}}_{H,\fp})$ and by $\operatorname{dimtot}_{s,{\fp}}({\overline}{\sF}_{\fp})$ the sum of $\operatorname{rank}({\overline}{\sF}_{\fp})$ and the Swan conductor of ${\overline}{\sF}_{\fp}$ over $\operatorname{Spec}(\kappa(\fp))$. We put $$\label{dimtot s}
\operatorname{dimtot}_{\fp}(\sF)=-\operatorname{ord}_{s,\fp}({\mathrm{CC}}_{\psi_0}(\sF_{\fp}))+\operatorname{dimtot}_{s,{\fp}}({\overline}{\sF}_{\fp}).$$ We notice that $\operatorname{ord}_{s,\fp}({\mathrm{CC}}_{\psi_0}(\sF))$ dose not depend on the choice of $\psi_0$ and the choice of $\p$.
We put $$\begin{aligned}
\varphi_{\eta}(H,U,\sF)&=&\sum_{\fp\in H_{\eta}-U}\operatorname{dimtot}_{\fp}(\sF),\\
\varphi_{s}(H,U,\sF)&=&\sum_{\fp\in P_s(H)}\operatorname{dimtot}_{\fp}(\sF)\label{psi s as}.\end{aligned}$$
\[invar dimtot\] Let $(H,U,\sF)$ be a stable triple (\[triple\]), $(S',\eta', s')$ a finite covering of $(S,\eta,s)$. We put $(H',U',\sF')=(H,U,\sF)_{S'}$.
- For any $\fp\in P_s(H)$ and for the unique $\fp'\in P_s(H)$ above $\fp$, we have $$\operatorname{dimtot}_{\fp}(\sF)=\operatorname{dimtot}_{\fp'}(\sF').$$
- For any $\fp\in H_{\eta}-U$, we have $$\operatorname{dimtot}_{\fp}(\sF)=\sum_{\fp'}\operatorname{dimtot}_{\fp'}(\sF'),$$ where $\fp'$ runs over the points above $\fp$.
{#section-22}
Let $(H,U,\sF)$ be a triple (\[triple\]). By \[be stable\], there exists a finite covering $(S',\eta',s')$ of $(S,\eta,s)$ such that $(H,U,\sF)_{S'}$ is stable. We put $$\begin{aligned}
\varphi_{\eta}(H,U,\sF)&=&\varphi_{\eta'}((H,U,\sF)_{S'}),\\
\varphi_{s}(H,U,\sF)&=&\varphi_{s'}((H,U,\sF)_{S'}).\end{aligned}$$ By \[invar dimtot\], they don’t depend on the choice of the covering $(S',\eta',s')$.
\[deligne kato\] Let $(H,U,\sF)$ be a triple (\[triple\]), x the closed point of $H$, $u:U{\rightarrow}H_{\eta}$ the canonical open immersion. Then we have $$\label{d k formula}
\dim_{\L}(\Psi^0_x(u_!\sF))-\dim_{\L}(\Psi^1_x(u_!\sF))=\varphi_{s}(H,U,\sF)-\varphi_{\eta}(H,U,\sF)-2\d(H)\operatorname{rank}(\sF).$$
Indeed, for a stable triple $(H,U,\sF)$ and any $\fp\in P_s(H)$, $\operatorname{dimtot}_{\fp}(\sF)$ is the same as Kato’s definition in ([@kato; @scdv] 4.4) by .
The theorem \[deligne kato\] is proved by Deligne if $\sF$ is unramified at every point of $P_s(H)$ ([@lau] 5.1.1). In the general case, Kato proved the theorem with two different definitions of the invariant $\varphi_s(H,U,\sF)$ ([@kato; @vc] 6.7, [@kato; @scdv] 4.5). T. Saito give another proof with another definition of $\varphi_s(H,U,\sF)$ ([@saito; @tf]) which corresponds to the latter definition of Kato ([@kato; @scdv] 4.5). If $\sF$ is of rank 1, Abbes and Saito gave a definition of $\varphi_s(H,U,\sF)$ ([@as; @ft] A.10) using the refined Swan conductor in their ramification theory [@as; @aml], which coincides with Kato’s latter definition ([@kato; @ch1] remark after 6.8). Here, using Abbes and Saito’s ramification theory, we give the definition of $\varphi_s(H,U,\sF)$ for any rank sheaf $\sF$ which is equal to Kato’s latter formula (\[mainkey\]).
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|
---
abstract: 'We study the production of charmed hadrons with the help of ALCOR$_c$, the algebraic coalescence model for rehadronisation of charmed quark matter. Mesonic ratios are introduced as factors connecting various antibaryon to baryon ratios. The resulting simple relations could serve as tests of quark matter formation and coalescence type rehadronization in heavy ion collisions.'
author:
- |
P. Lévai, T.S. Biró, T. Csörgő, J. Zimányi\
RMKI Research Institute for Particle and Nuclear Physics,\
P. O. Box 49, Budapest, 1525, Hungary
date: 'July 22. 2000'
title: |
Simple predictions from ALCOR$_c$\
for rehadronisation of charmed quark matter
---
Charm hadron production has gained an enhanced attention in relativistic heavy ion collisions at CERN SPS (Super Proton Synchrotron). The measured anomalous suppression of the $J/\psi$ in Pb+Pb collision [@NA50supr] is considered as one of the strongest candidates for an evidence of quark-gluon plasma (QGP) formation in Pb+Pb collision at 158 GeV/nucleon bombarding energy [@QGPjpsi]. So far only the $J/\psi$ and $\psi'$ production was measured in heavy ion collisions through dilepton decay channels. However, recent efforts to measure D-meson [@letterNA6i; @NA49D] support the theoretical investigation of charm production from a different point of view. Namely, it is interesting to search for predictions on the total numbers of charmed hadrons and their ratios. The answer to this question may become very important at the RHIC accelerator, where a large number of charmed quark-antiquark pairs will be produced and a number of different charmed hadrons could be detected.
In this paper we assume that quark matter is formed in heavy ion collisions and the charm hadrons will be produced directly from this state via quark coalescence. Quark coalescence was successfully applied to describe direct hadron production from deconfined quark matter phase (see. the ALCOR [@ALCOR1; @ALCOR3], the Transchemistry [@Transchem] and the MICOR [@MICOR] models). In these models the hadronic rescatterings are assumed to be weak and they are neglected. Thus the results of quark coalescence processes were compared directly to the experimental data - and the agreement was remarkably good.
Multicharm hadron production was already investigated in a simplified quark coalescence model and first results were obtained at RHIC and LHC energies, where an appreciable number of charm quark may appear [@multic]. Here we summarize simple predictions from a non-linear algebraic coalescence model ALCOR$_c$, the extension of the ALCOR model of algebraic coalescence of strange quark matter [@ALCOR1; @ALCOR3] for the inclusion of charmed quarks, mesons and baryons.
The description of the charmed baryons has to deal with the fact that two possible $(1/2)^+$ baryon multiplets exist containing $c$, $s$ and $u$ (or $d$) quarks, one being flavor symmetric under $s$ and $d$ (or $u$) exchange and the other being antisymmetric [@csym]. The heavier (symmetric) states decay into the lighter (antisymmetric) one by emission of a $\gamma$ or a $\pi$ meson. However, if quark clusterization is the basic hadronization process, then the effect of these decay processes will be cancelled from charmed antibaryon to baryon ratios. Neglecting the difference between the light $u$ and $d$ quarks and using the notation $q$ for them, the 10 different types of produced quark clusters can be connected to the 40 lowest lying SU(4)-flavor baryon species in the following way (see e.g. Ref. [@Charmb; @PDG] for precise quark content, hadron names and masses): $$\begin{aligned}
N(qqq) &:=& \ p, \ n, \ \Delta^{++}, \ \Delta^+, \ \Delta^0,
\ \Delta^- \ ; \nonumber \\
Y(qqs) &:=& \ \Lambda^0, \ \Sigma^+, \ \Sigma^0, \ \Sigma^-, \ \Sigma^{*+},
\ \Sigma^{*0}, \ \Sigma^{*-} \ ; \nonumber \\
\Xi(qss) &:=& \ \Xi^0, \ \Xi^-, \ \Xi^{*0}, \ \Xi^{*-} \ ; \nonumber \\
\Omega(sss) &:=& \ \Omega^- \ ; \nonumber \\
Y_c(qqc) &:=& \ \Lambda_c^+, \ \Sigma_c^{++}, \ \Sigma_c^+, \ \Sigma_c^0,
\ \Sigma_c^{*++}, \ \Sigma_c^{*+}, \ \Sigma_c^{*0} \ ;\nonumber \\
\Xi_c(qsc) &:=& \ \Xi_c^+, \ \Xi_c^0, \ {\Xi_c}'{^+}, \ {\Xi_c}'{^0},
\ \Xi_c^{*+}, \ \Xi_c^{*0} \ ; \nonumber \\
\Omega_c(ssc) &:=& \ \Omega_c^0, \ \Omega_c^{*0} \ ; \nonumber \\
\Xi_{cc}(qcc) &:=& \ \Xi_{cc}^{+}, \ \Xi_{cc}^{++},
\ \Xi_{cc}^{*+}, \ \Xi_{cc}^{*++} \ ; \nonumber \\
\Omega_{cc}(scc) &:=& \ \Omega_{cc}^+, \ \Omega_{cc}^{*+} \ ; \nonumber \\
\Omega_{ccc}(ccc) &:=& \ \Omega_{ccc}^{++} \end{aligned}$$
In ALCOR, the algebraic coalescence model of rehadronization it is assumed that the number of directly produced hadrons is given by the product of the the number of quarks (or anti-quarks) from which those hadrons are produced, multiplied by coalescence coefficients $C_h$ and by non-linear normalization coefficients $b_q$, that take into account conservation of quark numbers during quark coalescence, as will be explained subsequently. The number of various hadrons and quarks is denoted by the symbol usual for that type of particles, e.q. $q$, $s$ and $c$ denote the number of light, strange and charmed quarks, respectively, $N$ denotes the number of protons, neutrons and deltas etc.
In this way the baryons and antibaryons can be described through the following clustering relations: $$\begin{aligned}
N\,(qqq)&=&
C_N \cdot (b_q\, q)^3 \hskip 3.28 truecm
{\overline N}\, ({\overline q} {\overline q} {\overline q})
= C_{\overline N} \cdot (b_{\overline q} \, {\overline q})^3
\nonumber \\
Y\,(qqs) &= &
C_{Y} \cdot (b_q\, q)^2 \cdot (b_s\, s) \hskip 2.15 truecm
{\overline Y} \, ({\overline q} {\overline q} {\overline s}) =
C_{\overline Y} \cdot (b_{\overline q}\, {\overline q})^2
\cdot (b_{\overline s}\, {\overline s}) \nonumber \\
\Xi\,(qss) &= &
C_{\Xi} \cdot (b_q\, q) \cdot (b_s\, s)^2 \hskip 2.22 truecm
{\overline \Xi}\,({\overline q} {\overline s} {\overline s}) =
C_{\overline \Xi} \cdot (b_{\overline q}\, {\overline q})
\cdot (b_{\overline s} \, {\overline s})^2 \nonumber \\
\Omega\,(sss) &= &
C_{\Omega} \cdot (b_s\, s)^3 \hskip 3.43 truecm
{\overline \Omega} \,({\overline s} {\overline s} {\overline s}) =
C_{\overline \Omega}
\cdot (b_{\overline s}\, {\overline s})^3 \nonumber \\
Y_c\,(qqc) &= &
C_{Y}^c \cdot (b_q\, q)^2 \cdot (b_c\, c) \hskip 2.1 truecm
{\overline Y}_c \, ({\overline q} {\overline q} {\overline c}) =
C_{\overline Y}^c \cdot (b_{\overline q}\, {\overline q})^2
\cdot (b_{\overline c}\, {\overline c}) \nonumber \\
\Xi_c\,(qsc) &= &
C_{\Xi}^c \cdot (b_q\, q) \cdot (b_s\, s) \cdot (b_c\, c) \hskip 1.1 truecm
{\overline \Xi}_c \, ({\overline q} {\overline s} {\overline c}) =
C_{\overline \Xi}^c \cdot (b_{\overline q}\, {\overline q})
\cdot (b_{\overline s}\, {\overline s})
\cdot (b_{\overline c}\, {\overline c}) \nonumber \\
\Omega_c\,(ssc) &= &
C_{\Omega}^c \cdot (b_s\, s)^2 \cdot (b_c\, c) \hskip 2.1 truecm
{\overline \Omega}_c \, ({\overline s} {\overline s} {\overline c}) =
C_{\overline \Omega}^c \cdot (b_{\overline s}\, {\overline s})^2
\cdot (b_{\overline c}\, {\overline c}) \nonumber \\
\Xi_{cc}\,(qcc) &= &
C_{\Xi}^{cc} \cdot (b_q\, q) \cdot (b_c\, c)^2 \hskip 2 truecm
{\overline \Xi}_{cc}\,({\overline q} {\overline c} {\overline c}) =
C_{\overline \Xi}^{cc} \cdot (b_{\overline q}\, {\overline q})
\cdot (b_{\overline c} \, {\overline c})^2 \nonumber \\
\Omega_{cc}\,(scc) &= &
C_{\Omega}^{cc} \cdot (b_s\, s) \cdot (b_c\, c)^2 \hskip 2 truecm
{\overline \Omega}_{cc}\,({\overline s} {\overline c} {\overline c}) =
C_{\overline \Omega}^{cc} \cdot (b_{\overline s}\, {\overline s})
\cdot (b_{\overline c} \, {\overline c})^2 \nonumber \\
\Omega_{ccc} \,(ccc) &= &
C_{\Omega}^{ccc} \cdot (b_c\, c)^3 \hskip 3.1 truecm
{\overline \Omega}_{ccc} \,({\overline c} {\overline c} {\overline c}) =
C_{\overline \Omega}^{ccc}
\cdot (b_{\overline c}\, {\overline c})^3
\label{cbarion}\end{aligned}$$
Mesons in the pseudoscalar and vector SU(4)-flavor multiplets are grouped in the following way: $$\begin{aligned}
\pi(q{\overline q}) &:=& \ \pi^+, \ \pi^0, \ \pi^-, \ \eta,
\ \rho^+, \ \rho^0, \ \rho^-, \ \omega; \nonumber \\
K(q{\overline s}) &:=& \ K^+, \ K^0,
\ K^{*+}, \ K^{*0}; \nonumber \\
{\overline K}({\overline q}s) &:=& \ K^-, \ {\overline K}^0,
\ K^{*-}, \ {\overline K}^{*0}; \nonumber \\
\phi(s{\overline s}) &:=& \ {\eta}', \ \phi; \nonumber \\
D({\overline q}c) &:=& \ D^+, \ D^0,
\ D^{*+}, \ D^{*0}; \nonumber \\
{\overline D}(q{\overline c}) &:=& \ D^-, \ {\overline D}^0,
\ D^{*-}, \ {\overline D}^{*0}; \nonumber \\
D_s({\overline s}c) &:=& \ D_s^+, \ D_s^{*+}; \nonumber \\
{\overline D}_s(s{\overline c}) &:=& \ D_s^-, \ D_s^{*-}; \nonumber \\
J/\psi({\overline c}c) &:=& \ \eta_c, \ J/\psi; \end{aligned}$$
Thus the number of directly produced mesons reads as
$$\begin{aligned}
\pi \, (q\overline q) &=&
C_\pi \cdot (b_q \, q) \cdot (b_{\overline q} \, {\overline q})
\hskip 2.1 truecm
J/\psi \, (c\overline c) =
C_{J/\psi} \cdot (b_c \, c) \cdot (b_{\overline c} \, {\overline c})
\nonumber \\
K \, (q \overline s) &=&
C_K \cdot (b_q \, q) \cdot (b_{\overline s} \, {\overline s})
\hskip 2.4 truecm
D \, ( {\overline q} c) =
C_D \cdot (b_{\overline q} \, {\overline q}) \cdot (b_c \, c) \nonumber \\
{\overline K} \, (\overline q s) &=&
C_{\overline K}
\cdot (b_{\overline q} \, {\overline q}) \cdot (b_s \, s)
\hskip 2.4 truecm
{\overline D} \, (q \overline c ) =
C_{\overline D}
\cdot (b_q \, q) \cdot (b_{\overline c} \, {\overline c}) \nonumber \\
\phi \, (s \overline s) &=&
C_{\phi}
\cdot (b_s \, s) \cdot (b_{\overline s} \, {\overline s})
\hskip 2.4 truecm
D_s \, ( \overline s c) =
C_D^s \cdot (b_{\overline s} \, {\overline s}) \cdot (b_c \, c) \nonumber \\
&\ & \hskip 5.3 truecm
{\overline D}_s \, (s \overline c) =
C_{\overline D}^s
\cdot (b_s \, s) \cdot (b_{\overline c} \, {\overline c})
\label{cmeson}\end{aligned}$$
As a straightforward extension to the ALCOR model, the non-linear coalescence factors $b_q$, $b_s$, $b_c$ and the $b_{\overline q}$, $b_{\overline s}$, $b_{\overline c}$ are determined unambiguously from the requirement that the number of the constituent quarks and anti-quarks do not change during the hadronization, and that all initially available quarks and anti-quarks have to end up in the directly produced hadrons. This constraint is a basic assumption in all models of quark coalescence. The correct quark counting yields to the following equations, expressing the conservation of the number of quarks: $$\begin{aligned}
q &=& 3 \ N\,(qqq) + 2\ Y\,(qqs) + \Xi\,(qss)
+ K \, (q \overline s ) + \pi \, (q \overline q) +
\nonumber\\
&&+ 2 \ Y_{c}\,(qqc)+ \Xi_c\,(qsc) + \Xi_{cc}\,(qcc)
+ {\overline D} \, (q\overline c) \\
{\overline q} &=&
3 \ {\overline N}\,
({\overline q}{\overline q}{\overline q})
+ 2 \ {\overline Y}\,
({\overline q}{\overline q}{\overline s})
+ \overline{\Xi}\,({\overline q}{\overline s}{\overline s})
+ {\overline K} \, (\overline q s ) + \pi \, (q \overline q) +
\nonumber\\
&&+ 2 \ {\overline Y}_{c}\,({\overline q}{\overline q}{\overline c})
+ {\overline \Xi}_c\,
({\overline q}{\overline s}{\overline c})
+ {\overline \Xi}_{cc}\,
({\overline q}{\overline c}{\overline c})
+ D\, (\overline q c) \\
s &=& 3\ \Omega \,(sss) + 2\ \Xi \,(qss) + Y\,(qqs)
+ {\overline K} \, (\overline q s) + \phi \, (s \overline s) +
\nonumber\\
&&+ 2 \ \Omega_{c}\,(ssc)+ \Xi_c\,(qsc) + \Omega_{cc}\,(scc)
+ {\overline D}_s \, (s\overline c) \\
{\overline s} &=&
3 \ {\overline \Omega}\,
({\overline s}{\overline s}{\overline s})
+ 2 \ {\overline \Xi}\,
({\overline q}{\overline s}{\overline s})
+ \overline{Y}\,({\overline q}{\overline q}{\overline s})
+ K \, (q \overline s ) + \phi \, (s \overline s) +
\nonumber\\
&&+ 2 \ {\overline \Omega}_{c}\,({\overline s}{\overline s}{\overline c})
+ {\overline \Xi}_c\,
({\overline q}{\overline s}{\overline c})
+ {\overline \Omega}_{cc}\,
({\overline s}{\overline c}{\overline c})
+ D_s\, (\overline s c) \\
c &=& 3 \ \Omega_{ccc}\,(ccc) + 2\ \Xi_{cc}\,(qcc) + \Lambda_c\,(qqc)
+ D \, (\overline q c) + J/\psi \, (c \overline c) +
\nonumber\\
&&+ 2 \ \Xi_{cc}\,(scc)+ \Lambda_c\,(qsc) + \Lambda_c\,(ssc)
+ D_s \, (\overline s c) \\
{\overline c} &=&
3 \ {\overline \Omega}_{ccc}\,
({\overline c}{\overline c}{\overline c})
+ 2 \ {\overline \Xi}_{cc}\,
({\overline q}{\overline c}{\overline c})
+ \overline{\Lambda}_c\,({\overline q}{\overline q}{\overline c})
+ {\overline D} \, (q \overline c ) + J/\psi \, (c \overline c) +
\nonumber\\
&&+ 2 \ {\overline \Xi}_{cc}\,({\overline s}{\overline c}{\overline c})
+ {\overline \Lambda}_c\,
({\overline q}{\overline s}{\overline c})
+ {\overline \Lambda}_c\,
({\overline s}{\overline s}{\overline c})
+ {\overline D}_s \, (s \overline c) \end{aligned}$$
These equations for $q$, $s$, $c$ and $ ({\overline q}$, ${\overline s}$, ${\overline c}) $ determine the six $b_i$ normalization factors — which are not free parameters. These constraints, together with the prescription of the coalescence factors $C_i$, complete the description of hadron production from charmed quark matter by quark coalescence, and define the ALCOR$_c$ model.
In this paper, we will evaluate only the simplest predictions from ALCOR$_c$, by considering ratios of the number of particles to the number of anti-particles and by assuming the symmetry of the coalescence process for charge conjugation, extending the results of ref. [@ALCOR3] to the case of charmed quarks, mesons and baryons.
Assuming that the coalescence coefficients $C$ for hadrons are equal to that for the corresponding anti-particles, e.g. $C_\Lambda = C_{\overline \Lambda}$, the following relations were obtained for the ratio of light and strange antibaryons and baryons [@ALCOR3]:
$$\begin{aligned}
\frac{{\overline N}\, ({\overline q} {\overline q} {\overline q})}
{N\,(qqq)} &=&
\left[ \frac{b_{\overline q} \, {\overline q}}{b_q\, q} \right]^3
\label{rn} \\
& & \nonumber \\
\frac{{\overline Y} \, ({\overline q} {\overline q} {\overline s})}
{Y\,(qqs)} &=&
\left[ \frac{b_{\overline q} \, {\overline q}}{b_q\, q} \right]^2 \cdot
\left[ \frac{b_{\overline s} \, {\overline s}}{b_s\, s} \right]
\label{rlam}\\
& & \nonumber \\
\frac{{\overline \Xi} \, ({\overline q} {\overline s} {\overline s})}
{\Xi\,(qss)} &=&
\left[ \frac{b_{\overline q} \, {\overline q}}{b_q\, q} \right] \cdot
\left[ \frac{b_{\overline s} \, {\overline s}}{b_s\, s} \right]^2
\label{rxi} \\
& & \nonumber \\
\frac{{\overline \Omega} \, ({\overline s} {\overline s} {\overline s})}
{\Omega\,(sss)} &=&
\left[ \frac{b_{\overline s} \, {\overline s}}{b_s\, s} \right]^3
\label{romega}\end{aligned}$$
Inspecting eqs. (\[rn\])-(\[romega\]) one can recognize, that the kaon to anti-kaon ratio ${\cal S}^{qs}$ has a special role as it acts as a stepping factor that connects various antibaryon to baryon rations, $${\cal S}^{qs} \equiv
\frac{K\,(q\overline s)}{{\overline K}\, ({\overline q} s )}=
\left[ \frac{b_q\, q}{b_{\overline q} \, {\overline q}} \right] \cdot
\left[ \frac{b_{\overline s} \, {\overline s}}{b_s\, s} \right] \ \ .
\label{oper_s}$$
This factor ${\cal S}^{qs}$ substitutes a light quark with a strange quark in the antibaryon to baryon ratios. Thus it shifts the antibaryon to baryon ratios and changes their strangeness content by one unit, as the following relations display: $$\begin{aligned}
{\cal S}^{qs} \left[ \frac{{\overline N}}{N} \right] &=&
\frac{{\overline Y}} {Y} \\
{\cal S}^{qs} {\cal S}^{qs} \left[ \frac{{\overline N}}{N} \right] &=&
\frac{{\overline \Xi}} {\Xi} \\
{\cal S}^{qs} {\cal S}^{qs} {\cal S}^{qs}
\left[ \frac{{\overline N}}{N} \right] &=&
\frac{{\overline \Omega}} {\Omega} \end{aligned}$$
The inverse factor, ${\cal S}^{sq} = ({\cal S}^{qs})^{-1}$ decreases the strangeness content and increases the number of light quarks in the antibaryon to baryon ratios. Note that these relations hold between the ratios of the directly produced anti-baryons to baryons and that the number of observed anti-baryons and baryons have to be corrected to the various chains of resonance decays [@ALCOR3].
Extending the above ALCOR model to the case of charmed baryons and antibaryons, further relations are obtained:
$$\begin{aligned}
\frac{{\overline Y}_c \, ({\overline q} {\overline q} {\overline c})}
{Y_c\,(qqc)} &=&
\left[ \frac{b_{\overline q} \, {\overline q}}{b_q\, q} \right]^2 \cdot
\left[ \frac{b_{\overline c} \, {\overline c}}{b_c\, c} \right]
\hskip 1.5 truecm
\frac{{\overline \Xi}_c \, ({\overline q} {\overline s} {\overline c})}
{\Xi_c\,(qsc)} =
\left[ \frac{b_{\overline q} \, {\overline q}}{b_q\, q} \right] \cdot
\left[ \frac{b_{\overline s} \, {\overline s}}{b_s\, s} \right] \cdot
\left[ \frac{b_{\overline c} \, {\overline c}}{b_c\, c} \right]
\nonumber \\
& & \nonumber \\
\frac{{\overline \Xi}_{cc} \, ({\overline q} {\overline c} {\overline c})}
{\Xi_{cc}\,(qcc)} &=&
\left[ \frac{b_{\overline q} \, {\overline q}}{b_q\, q} \right] \cdot
\left[ \frac{b_{\overline c} \, {\overline c}}{b_c\, c} \right]^2
\hskip 1.5 truecm
\frac{{\overline \Omega}_{c} \, ({\overline s} {\overline s} {\overline c})}
{\Omega_{c}\,(ssc)} =
\left[ \frac{b_{\overline s} \, {\overline s}}{b_s\, s} \right]^2 \cdot
\left[ \frac{b_{\overline c} \, {\overline c}}{b_c\, c} \right]
\nonumber \\
& & \nonumber \\
\frac{{\overline \Omega}_{ccc} \, ({\overline c} {\overline c} {\overline c})}
{\Omega_{ccc}\,(ccc)} &=&
\left[ \frac{b_{\overline c} \, {\overline c}}{b_c\, c} \right]^3
\hskip 2.9 truecm
\frac{{\overline \Omega}_{cc} \, ({\overline s} {\overline c} {\overline c})}
{\Omega_{cc}\,(scc)} =
\left[ \frac{b_{\overline s} \, {\overline s}}{b_s\, s} \right] \cdot
\left[ \frac{b_{\overline c} \, {\overline c}}{b_c\, c} \right]^2
\label{charm}\end{aligned}$$
These ratios and the ratios from eqs. (\[rn\])-(\[romega\]) can be organized into a special structure displayed in Fig.1. We can introduce two more factors ${\cal S}^{sc}$ and ${\cal S}^{cq}$ constructed as in eq.(\[oper\_s\]) but from the ratios of charmed mesons: $$\begin{aligned}
{\cal S}^{sc} \equiv \
\frac{{\overline D}_s\,(s\overline c)}{ D_s\, ({\overline s} c )} &=&
\left[ \frac{b_s\, s}{b_{\overline s} \, {\overline s}} \right] \cdot
\left[ \frac{b_{\overline c} \, {\overline c}}{b_c\, c} \right]
\label{oper_c} \\
{\cal S}^{cq} \equiv \
\frac{ D\, ({\overline q} c )}{{\overline D}\,(q\overline c)} &=&
\left[ \frac{b_c\, c}{b_{\overline c} \, {\overline c}} \right] \cdot
\left[ \frac{b_{\overline q} \, {\overline q}}{b_q\, q} \right]
\label{oper_q}\end{aligned}$$
The factor ${\cal S}^{sc}$ substitutes a strange quark with a charm one and the factor ${\cal S}^{cq}$ changes the charm quark into a light one. These properties lead to the following identity: $${\cal S}^{qs} \cdot {\cal S}^{sc} \cdot {\cal S}^{cq} \equiv 1$$
This identity can be rewritten as an identity between the mesonic ratios: $$\frac{{\overline D_s} / D_s}{{\overline D} / D} =
{\overline K}/{K}$$
A comparison of this simple relation with experimental data could serve as test of quark matter formation and coalescence type rehadronization in heavy ion collisions.
The inverse of the step factors is defined as ${\cal S}^{ji} = ({\cal S}^{ij})^{-1}$. The structure of the antibaryon to baryon ratios in ALCOR$_c$ is visualized in a geometric manner in Fig. 1. This way, more complicated but definitely interesting relations can be obtained. Since the baryons with one charm quark (or antiquark) can be measured most easily, one may consider the following relations as candidates for an experimental test: $$\frac{{\overline \Xi}_c \, ({\overline q} {\overline s} {\overline c})}
{\Xi_c \,(qsc)}
= {\cal S}^{qs} \left[\frac{{\overline Y}_c}{Y_c} \right]
= {\cal S}^{qc} \left[\frac{{\overline Y}}{Y} \right]
= {\cal S}^{sc} \left[\frac{{\overline \Xi}}{\Xi} \right]
= {\cal S}^{sq} \left[\frac{{\overline \Omega_c}}{\Omega_c} \right].$$ These yield the following simple relation between baryonic and mesonic ratios: $$\begin{aligned}
\frac{ {\overline Y}/ Y}{{\overline Y}_c / Y_c}
& = & D_s / {\overline D_s} \ ,\\
\frac{ {\overline N}/ N}{{\overline Y}_c / Y_c}
& = & D / {\overline D } \ ,\\
\frac{ {\overline N}/ N}{{\overline Y} / Y}
& = & {\overline K} / K \ .\end{aligned}$$
[ [**Fig. 1.**]{} The application of mesonic step factors ${\cal S}^{qs}$, ${\cal S}^{sc}$ and ${\cal S}^{cq}$ on the antibaryon to baryon ratios. The arrows are indicating the three corresponding directions of shifting. ]{}
A number of similar expressions can be derived from Figure 1, picking up a given ratio and following all the paths to reach that from its neighbors.
[*In summary* ]{}, we have made simple predictions from the ALCOR$_c$ model, extending the ALCOR model of algebraic coalescence and rehadronization of quark matter to the case when charmed quarks and final state hadrons are present in a significant number. We found that the various ${\overline M}/M$ mesonic ratios connect different ${\overline B}/B$ ratios. The agreement between the obtained theoretical relations and those in the measured data could serve as proof or disproof of the formation of quark matter in heavy ion collisions followed by a fast hadronization via quark coalescence. The predictions made in this paper are independent from the detailed values of coalescence coefficients, we have assumed only their symmetry for charge conjugation. The calculations of the absolute numbers of produced particles from ALCOR$_c$ requires the specification of these coalescence coefficients from calculations of cross-sections.
[**Acknowledgment:**]{} This work was supported by the OTKA Grants No. T029158, T025579 and T024094.
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|
---
abstract: 'We have discovered a very radio-loud Narrow-Line Seyfert 1 candidate: PKS 2004-447. This Seyfert is consistent with the formal definition for NLS1s, although it does not have quite the same spectral features as some typical members of this subclass. Only ROSAT survey data is available at X-ray wavelengths, so it has not been possible to compare this source with other NLS1s at these wavelengths. A full comparison of this source with other members of the subclass will improve our physical understanding of NLS1s. In addition, using standard calculations, we estimate the central black hole to have a mass of $\sim 5 \times 10^6 M_{\odot}$. This does not agree with predictions in the literature, that radio-loud AGN host very massive black holes.'
author:
- 'A. Y. K. N. Oshlack,'
- 'R. L. Webster'
- 'and M. T. Whiting'
title: 'A Very Radio-Loud Narrow-Line Seyfert 1: PKS 2004-447'
---
Introduction
============
The radio-loud source PKS 2004-447, $z=0.24$, was identified as a candidate Narrow-Line Seyfert 1 (NLS1) from low resolution optical spectra of a subsample of quasars identified in the Parkes Half-Jansky Flat-Spectrum Sample [@dri97](hereafter PHFS). The PHFS is designed to efficiently select radio-loud AGN by looking for flat or inverted radio spectra over the range of 2.7 and 5.0 GHz. It contains 323 radio bright ($>0.5$Jy at 2.7 GHz), generally compact radio sources. Higher resolution spectroscopy of this object was obtained giving more accurate velocity widths of the emission lines indicating this may be an NLS1 but more accurate spectroscopy is still needed to be able to clearly separate the broad and narrow components of H$\beta$. In any case this object has very interesting implications to the study of AGN.
NLS1s are generally considered to be an extreme but common subclass of AGN. They are defined by their optical emission line properties, such that the H$\beta$ line is both strong (with a flux ratio \[\]/H$\beta<3$, similar to Seyfert 1 galaxies) and narrow (the H$\beta_{FWHM}<2000$km s$^{-1}$) [@ost85]. These properties are correlated with strong emission and a strong soft X-ray excess, amongst other properties, for most objects in the subclass. NLS1s do not appear to form a distinct class but are instead connected to the “standard” broad-line Seyfert population through a continuum of properties. @bor92 have shown that NLS1s cluster at one end of the region defined by the first principal component described in their study of the optical spectra of 87 low-redshift BQS quasars. The first principal component is thought to describe the major physical property in quasar structure that is responsible for spectral differences between AGN, independent of orientation. @bor92 suggest that the physical parameter driving this eigenvector is $\dot{M} / \dot{M}_{Edd}$, where $\dot{M}$ is the rate of mass accretion onto the central massive object. NLS1s are thought to be accreting at a rate closer to the Eddington limit ($\dot{M}_{Edd}$) [@bor92; @lao97]. NLS1s are generally radio-quiet objects, with only three previously identified radio-loud objects, PKS 0558-504 [@rem86], RGB J0044+193 [@sie99] and J0134.2-4258 [@gru00]. @bor92 find that radio-loud QSOs and the NLS1s (with a strong, soft X-ray excess) lie at opposite ends of the primary eigenvector.
Although the criteria for NLS1s are well defined, it is unclear whether these phenomenological attributes reflect a single underlying physical mechanism. Thus the discovery of a very radio-loud NLS1 may indicate that the observational definition of NLS1s requires refinement. Alternatively, radio-loud NLS1s may provide a more stringent test of the models of NLS1s. Three important consequences of the identification of a radio-loud NLS1 will be considered. Firstly, are the radio-loud NLS1s the same class of objects as most others in the class. Secondly, are radio-loud NLS1s consistent with any of the popular models for NLS1s and thirdly, do radio-loud quasars require large mass black holes. This object challenges that assertion as is further discussed in section 4
In section 2, we present the observational data on PKS 2004-477. In section 3, this object is compared to the three other radio-loud NLS1s which have been identified. The central black hole mass is determined using standard techniques in section 4, and possible models for NLS1s are discussed in section 5. Finally our conclusions are presented in section 6.
Observational Data
==================
Optical Spectrum
----------------
PKS 2004-447 was first identified as a NLS1 candidate from a low resolution spectrum obtained using the RGO/FORS spectrograph at the AAT in 1984 and published in @dri97. A higher resolution confirmation spectrum was taken using the double beam spectrograph (DBS) on the ANU 2.3m telescope, 1st August 2000, and is shown in Fig 1. The conditions were not photometric. The spectrum was reduced using standard procedures in the IRAF. The spectrum has a resolution of$\sim 2.2$Å.
Typically, in NLS1s, emission contaminates the spectrum making accurate measurements of H$\beta$ and \[\] difficult. In order to account for this and to get an estimate on the strength of the emission in this object we employed the subtraction method introduced by @bor92 and now commonly used. We use a template spectrum taken from the prototype strong Fe emitter I Zw 1 and scale it and shift it until the width and intensity match those seen in our object. We did this over the region 5050-5500Å using a $\chi ^2$-minimisation to get the best value for the scaling. We find that this spectral region contains very little emission with an equivalent width EW$< 10$Å for the whole complex in the region 5050-5450 Å. Compared to values measured in @bor92, this is extremely low (although we are fitting a slightly different region of the spectrum). We measured the line widths and fluxes using Lorentzian fits to the emission lines and the results are summarised in Table 1. The width of $H\beta _{FWHM}=1447$km s$^{-1}$and the flux ratio \[\]/H$\beta= 1.6$, fit the criteria for classification as a NLS1. The DBS spectrum does not extend to wavelengths of the ($\lambda$4570) lines. However the previous low resolution spectrum indicates emission in the region 4435-4700 Å, blueward of the $H\beta$ line (Fig 2). The low resolution spectrum also shows strong H$\alpha$ and evidence for H$\gamma$ emission and it also indicates that the stregth of the H$\beta$ emission, relative to \[\] has varied between the two epochs.
PKS 2004-447 is also optically variable. The magnitude of this object measured from the COSMOS/UKST Southern Sky Catalogue is $B_{J}=18.1$, while more recent photometry obtained on the ANU 2.3m telescope gave $B=19.5$ [@fra00], indicating a drop in flux by a factor of $\approx 4$ over an interval of several years. Simultaneous optical/IR photometry are shown in Fig 3. The continuum is very red and there is no evidence of a big blue bump. The absolute magnitude is between -19.0 and -21.2, for $q_{0} =0.5$ and $H_0 =100$, clearly placing this in the Seyfert luminosity class.
[|p[3.5cm]{}||p[2.5cm]{}|p[2.5cm]{}|p[2.5cm]{}|]{} & H$\beta$ (4861) & (4959) & (5007)\
FWHM(km s$^{-1}$) & 1447 & 951 & 754\
flux relative to H$\beta$ & 1.0&0.61 &1.61\
EW (Å)& 31.6&19.6 &50.6\
Offset (km s$^{-1}$)& $+28$&$-78$&0\
Radio Emission
--------------
Under the selection criteria, the PHFS objects are detected above 0.5 Jy at 2.7GHz and with a spectral slope $\alpha < 0.5$ (F$_{\nu}\propto \nu^{-\alpha}$) taken from non-simultaneous observations at 2.7GHz and 5.0 GHz. Using this method PKS 2004-447 had a 2.7 GHz flux of 0.81 Jy and a spectral index $\alpha_{r} =
0.36$. Subsequent simultaneous observations of this source were taken using the ATCA (23rd November 1995) and the radio flux had varied since the Parkes observations, demonstrating long term variability at these frequencies as well. The radio spectrum is shown in Fig 3. A powerlaw fit to the simultaneous data gives a spectral index $\alpha_{r} = 0.67$; thus PKS 2004-447 is actually a steep spectrum source. Calculating $R$, where $R$ is the ratio of radio to optical flux ($f_{4.85GHz}/f_{B}$), [@kel89], gives values in the range $1710 < R < 6320$ depending on value used for the optical magnitude. Obviously, this source is very radio-loud.
The radio image is unresolved in the ATCA observations. The visibility of PKS 2004-477 was measured using the Parkes-Tidbinbilla Interferometer on a 275 km baseline at 2.3 GHz [@dun93]. The average of two measurements gave a visibility of 0.680 corresponding to an angular size of $\sim 0.036 \arcsec$ or $\sim 85 pc$. Since this source size is derived from a 2-element interferometer, it is measured along one baseline.
There are two previous studies of the radio properties of NLS1s. The most recent and well defined is that of @mor00, who obtained high resolution observations of 24 NLS1s at 20cm and 3.6cm using the VLA. All but one of the sources were detected, with 20cm fluxes up to $10^{25} W\,
Hz^{-1}$, which is higher, on average, than classical Seyferts. Moran found most NLS1s unresolved at $\sim 1 \arcsec$, with generally quite steep ($\alpha \approx 1.1-1.2$) radio spectra, though the spectral index of PKS 2004-477 is consistent with the flattest values. In other words these could be considered as low luminosity compact steep spectrum (CSS) sources. There is also a few cases of variability at 20cm, again, as seen in PKS 2004-477. Thus PKS 2004-477 is compact, steep-spectrum and variable in the radio, which are similar properties to other NLS1s @mor00. This is further discussed in section 5.
X-ray Emission
--------------
NLS1s have unusual X-ray properties. They generally display a large soft X-ray excess, a steep hard X-ray spectrum and large amplitude X-ray variability over short and long timescales. @sie98 investigated the X-ray properties of the PHFS using the ROSAT All Sky Survey. PKS 2004-447 was detected with a total flux in the 0.1-2.4 keV range of $0.427 \pm 0.218 \times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$ corresponding to $L_X = 2.9 \times 10^{43} $erg s$^{-1}$. We find an optical–to–X-ray slope of $\alpha_{ox}=-1.2$. This is consistent with values found by @xu99 for a ROSAT selected sample of NLS1s.
Previous Detections of Radio-Loud NLS1s
=======================================
Most NLS1s are radio-quiet and only three radio-loud NLS1s are known [@sie99]. PKS 0558-504 was optically identified on the basis of X-ray positions from the High Energy Astronomical Observatory [@rem86]. It was noted as not only having very narrow hydrogen emission lines, but also strong emission. It has a radio flux of 113 mJy at 4.85 GHz, an optical magnitude $m_B = 14.97$ and a redshift $z=0.137$. This gives $R \approx 27$ [@sie99]. This source exhibits strong X-ray variability on medium (months) and short (days, hours) time scales [@gli00] and an X-ray flare indicating relativistic beaming [@rem91].
@sie99 identified RGB J0044+193 as a radio-loud NLS1. RGB J0044+193 has only a moderate radio flux, but is calculated to have a radio-to-optical flux ratio $R \approx 31$. It was identified as a radio-loud X-ray source with a redshift of $0.181$ in a cross-correlation of the ROSAT All-Sky Survey and the Green Bank 5GHz radio survey [@lau98]. It has the bluest optical spectrum of any NLS1 observed to date. @sie99 measure an optical continuum slope of $\alpha = 1.3$ redward of 5000Å and a slope of $-3.1$ blueward of 5000Å where $f_{\nu} \propto \nu^{-\alpha}$. This source was detected in the 87GB survey with $f_{4.85GHz}=24$ mJy. A high resolution follow-up using the VLA at 4.85 GHz gave a flux of only 7 mJy [@lau97], indicating that this source is extended or variable. The latter is supported by the fact that the radio source is unresolved on the VLA map and it is not detected at 1.4 GHz in the NRAO/VLA Sky Survey, which is sensitive down to 2.5 mJy. This suggests that the classification of the source as radio-loud is uncertain. In all other respects, RGB J0044+193 is indistinguishable from radio-quiet NLS1s.
RX J0134.2-4258 was discovered by @gru00 to have the steepest soft X-ray spectrum observed during the ROSAT All-Sky Survey. It has an optical magnitude of $m_V =16.2$ and $z=0.237$. It was detected in the PMN survey [@wri94] with a flux of 0.055 Jy at 4.85 GHz, and was subsequently re-observed at the VLA to have a flux of 0.025 Jy at 8.4 GHz. This gives a ratio of $R=71$, and a radio spectral index of $\alpha = 1.4$ for non-simultaneous observations. The source is highly variable in the X-ray, and there is evidence for variability at other wavelengths. The emission is strong, unlike that of PKS 2004-477.
Black Hole Mass
===============
There seems to be a growing consensus that a more massive black hole is needed to produce a radio-loud quasar [@lao00; @kas00; @mcl00; @pet00]. PKS 2004-447 appears to directly conflict with this statement. Calculations of the black hole mass rely on the assumption that the dynamics of the BLR gas are dominated by the central black hole. To calculate the black hole mass for this object we use the results of @kas00, which are derived from a reverberation study of 17 quasars. @kas00 empirically determine linear relationship between $R_{BLR}$ and the luminosity ($\lambda L_{\lambda}$) which we use to estimate the radius of the BLR-emitting gas. From equation 5 in @kas00. $$R_{BLR} = (32.9)\left[\frac{\lambda L_{\lambda}(5100
{\text \AA})}{10^{44} {\text erg~s}^{-1}}\right]^{0.700} \text{lt-days}$$ We use the luminosity taken from a linear interpolation between photometric data points [@fra00] at the rest wavelength of 5100Å. Following @kas00, the mass of the black hole is given by $M_{BH}=rv^{2}G^{-1}$. To determine $v$, the velocity, we correct $v_{FWHM}$ of the H$\beta$ emission line by a factor of $\sqrt{3}/2$ to account for velocities in three dimensions. The mass is then $$M = 1.464 \times 10^{5} \left(\frac{R_{BLR}}{\text{
lt-days}}\right)\left(\frac{v_{FWHM}}{10^{3}{\rm km~s}^{-1}}\right)^{2}
M_{\odot}$$ Using the cosmology $q_{0} =0.5$ and $H_{0}=100$km s$^{-1}$Mpc$^{-1}$, this gives us a value for the central black hole mass of $5.4 \times 10^6 M_{\odot}$. This mass is two orders of magnitude lower than those obtained by other authors for radio-loud quasars. In a study of the black hole masses of quasars from the @bor92 sample, it was found that all quasars with $M_{BH}<3 \times 10^8
M_{\odot}$ are radio-quiet [@lao00]. The data from this sample is shown in Fig 4, with the mass of PKS 2004-447 added. It can be seen that PKS 2004-447 is quite different than the rest of the sample. We note that @lao00 assumes a flatter relationship than @kas00, but estimates that this difference results in a discrepancy of $<50\%$ in the mass estimates. This corresponds to a shift of $<0.3$ in log$M_{BH}$, which does not qualitatively affect the result in figure 4. Similarly @mcl00, in their study of radio-loud and radio-quiet objects, came to the conclusion that a radio-loud quasar requires a mass $M_{BH}>6 \times 10^8 M_{\odot}$. This again,is more than two orders of magnitude greater than the mass we derive for PKS 2004-447.
We speculate that the reason our object has a black hole mass so much smaller than those seen in previous studies is not entirely due to it being an unusual object, but more likely due to the selection criteria in the previous samples. All previous studies have used optical selection to compile their sample. This selection technique preferentially selects radio-loud quasars which are blue with very broad emission lines. However samples of radio-selected quasars such as the PHFS show that the general radio-loud population can be quite different [@web95; @fra00]. In these samples it may be quite common to have low mass black holes producing large radio-jets and this issue will be addressed in a future paper.
Discussion of Physical models
=============================
Classification
--------------
Since the subclass of NLS1s are defined by phenomenological measurements, rather than a specific physical model, it is possible that more than one physical mechanism will produce the defining parameters. First we should consider whether PKS 2004-447 should be defined as an NLS1. Certainly it meets the formal definition of @ost85. In the X-ray, the flux ratios are consistent with other NLS1s, but pointed observations are required to obtain spectral information. In the optical, the emission is much weaker than other NLS1s. Apart from the obvious strength of the radio flux, the size of the radio source, and its spectral index are consistent with other radio measurements [@mor00].
The object’s radio properties fit with the definition of a Compact Steep Spectrum radio source (CSS) according to the definition from @fan95 of: $P_{tot} > 10^{26}$W Hz$^{-1}$; a size smaller or comparable to the optical galaxy scale; and the spectral index $\alpha_{r} >0.5$. The CSS sources tend to have spectra with narrow emission lines typical of radio-galaxies. The broad component of H$\beta$ seems to be extremely weak or absent [@mor97]. @gel94 find an average value \[\]/H$\beta \sim
5$ indicating that the H$\beta$ flux is too weak to be an NLS1. In the study of @gel94 all CSS sources with measurable H$\beta$ are of quasar luminosity. So PKS 2004-447 is unusual in both categories: firstly it has an optical spectrum which is unusual for CSS source; and secondly has radio power unusual for NLS1s.
The source is clearly in the Seyfert luminosity range, as are a significant fraction of the sources in PHFS. The differences in optical spectral properties, (such as the weakness of the emission) between PKS 2004-477 and NLS1s in general raise the possibility that the NLS1-defining characteristics of this source are due to a different physical mechanism from other objects in the subclass. This possibility may be resolved when an X-ray spectrum is obtained.
Physical Models
---------------
Finding unusual objects in a particular class is a useful method for distinguishing between different possible physical models. Although there have been a variety of physical models suggested for NLS1s [@tan99 see for a summary] it seems that the model with the strongest observational support is the high accretion rate, low black hole mass model. This model suggests that NLS1s are objects where the black hole is accreting at a rate closer to the Eddington limit compared to broad-line Seyferts. Since there is a higher rate of accretion, a given luminosity corresponds to a black hole with a smaller mass. If the size of the BLR is determined by the bolometric luminosity, then, at a given radius, the broad line clouds will have slower velocities and the emission lines will be narrower. This model was developed to explain the soft X-ray excess and the steep X-ray powerlaw, by analogy with the X-ray spectra of galactic black hole candidates accreting in their “high”state [@pou95]. For high accretion rates, the soft thermal emission from the disk becomes energetically dominant, producing the soft X-ray excess, and the X-ray powerlaw spectrum can be steepened as a result of Compton cooling of the electrons by the soft X-ray photons.
An alternative model for the narrow Hydrogen lines suggests we are viewing the source near the axis of the accretion disk. If the velocity width of the Hydrogen emission lines is dominated by the rotation of the accretion disk, then they will appear narrow. The radio emission in PKS is moderately steep spectrum, which weakens the argument that we observe this source near the axis. Indeed it also weakens arguments which suggest that the radio emission is strongly boosted by beaming. However the radio source is very compact, with no evidence at any frequency for double-lobed emission.
Conclusion
==========
PKS 2004-447, detected in the PHFS, is an unusual source. To summarize the major points regarding PKS 2004-447:
1. It has H$\beta$ width and strength consistent with being classified as an NLS1 as defined by @ost85.
2. Its radio properties are very unusual for a Seyfert Galaxy having the following characteristics: very strong radio flux, 0.81 Jy at 2.7 GHz; very radio-loud, $R>1700$; steep radio spectral index $\alpha_{r}=0.67$; compact radio source; some evidence of long term variability. These properties are consistent with it being classified as a Compact Steep Spectrum source.
3. It has been detected in the RASS with a flux $0.427 \times
10^{-12}$ erg cm$^{-2}$ s$^{-1}$ but no spectral information is available.
4. Following the procedure set out by @kas00, we calculate the black hole mass to be $5.4 \times 10^{6} M_{\odot}$. This mass is more than 2 orders of magnitude lower than those seen previously for radio-loud AGN and challenges previous results that a large black hole mass is needed to produce radio-loud AGN.
Additional observations of the X-ray spectrum will further constrain models and may provide evidence that the black hole is accreting at a rate closer to the Eddington limit. This would be an indication that the underlying physical mechanism in this object is similar to those in other NLS1s, and that the radio power of the object is actually connected to a different parameter.
We wish to thank Dirk Grupe for providing us with the template used in the analysis.
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abstract: 'We discuss two-sided bounds for moments and tails of quadratic forms in Gaussian random variables with values in Banach spaces. We state a natural conjecture and show that it holds up to additional logarithmic factors. Moreover in a certain class of Banach spaces (including $L_r$-spaces) these logarithmic factors may be eliminated. As a corollary we derive upper bounds for tails and moments of quadratic forms in subgaussian random variables, which extend the Hanson-Wright inequality.'
address:
- 'Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland'
- 'Institute of Mathematics, University of Warsaw, Banacha 2, 02–097 Warsaw, Poland.'
author:
- 'Rados[ł]{}aw Adamczak'
- 'Rafa[ł]{} Lata[ł]{}a'
- 'Rafa[ł]{} Meller'
date: 'preliminary version 12.09.2018'
title: 'Hanson-Wright inequality in Banach Spaces'
---
Introduction and main results
=============================
The Hanson-Wright inequality gives an upper bound for tails of real quadratic forms in independent subgaussian random variables. Recall that a random variable $X$ is called $\alpha$-*subgaussian* if for every $t > 0$, $\Pr(|X| \ge t) \le 2\exp(-t^2/2\alpha^2)$. The Hanson-Wright inequality states that for any sequence of independent mean zero $\alpha$-subgaussian random variables $X_1,\ldots,X_n$ and any symmetric matrix $A=(a_{ij})_{i,j\leq n}$ one has $$\label{eq:HW}
\Pr\left(\left|\sum_{i,j=1}^n a_{ij}(X_iX_j-\Ex (X_iX_j))\right|\geq t\right)
\leq 2\exp\left(-\frac{1}{C}\min\left\{\frac{t^2}{\alpha^4\|A\|_{\mathrm{HS}}},\frac{t}{\alpha^2\|A\|_{\mathrm{op}}}\right\}\right),$$ where in the whole article we use the letter $C$ to denote universal constants which may differ at each occurrence. Estimate was essentially established in [@HW] in the symmetric and in [@W] in the mean zero case (in fact in both papers the operator norm of $A$ was replaced by the operator norm of $(|a_{ij}|)$, which in general could be much bigger, proofs of may be found in [@BM] and [@RV]).
The Hanson-Wright inequality has found numerous applications in high-dimensional probability and statistics, as well as in random matrix theory (see e.g., [@vershynin_2018]). However in many problems one faces the need to analyze not a single quadratic form but a supremum of a collection of them or equivalently a norm of a quadratic form with coefficients in a Banach space. While in the literature there are inequalities addressing this problem (see ineq. below), they are usually expressed in terms of quantities which themselves are troublesome to analyze. The main objective of this article is to provide estimates on vector-valued quadratic forms which can be applied more easily and are of optimal form.
The main step in modern proofs of the Hanson-Wright inequality is to get a bound similar to in the Gaussian case. The extension to general subgaussian variables is then obtained with use of the by now standard tools of probability in Banach spaces, such as decoupling, symmetrization and the contraction principle. Via Chebyshev’s inequality to obtain a tail estimate it is enough to bound appropriately the moments of quadratic forms in the case when $X_i=g_i$ are standard Gaussian $\mathcal{N}(0,1)$ random variables. One may in fact show that (cf. [@LaSM; @LaAoP]) $$\left(\Ex\left|\sum_{i,j=1}^n a_{ij}(g_ig_j-\delta_{ij})\right|^p\right)^{1/p}
\sim p\|A\|_{\mathrm{op}}+\sqrt{p}\|A\|_{\mathrm{HS}}, \label{mom}$$ where $\delta_{ij}$ is the Kronecker delta, and $\sim$ stands for a comparison up to universal multiplicative constants.
Following the same line of arguments, in order to extend the Hanson-Wright bound to the Banach space setting we first estimate moments of centered vector-valued Gaussian quadratic forms, i.e. quantities $$\left\|\sum_{i,j=1}^n a_{ij}(g_ig_j-\delta_{ij})\right\|_p=\left(\Ex\left\|\sum_{i,j=1}^n a_{ij}
(g_ig_j-\delta_{ij})\right\|^p\right)^{1/p}, \quad p\geq 1,$$ where $A=(a_{ij})_{i,j\leq n}$ is a symmetric matrix with values in a normed space $(F,\|\ \|)$. We note that (as mentioned above) there exist two-sided estimates for the moments of Gaussian quadratic forms with vector-valued coefficients. To the best of our knowledge they were obtained first in [@Bo] and then they were reproved in various context by several authors (see e.g., [@AG; @Le; @LT]). They state that for $p\ge 1$, $$\begin{aligned}
\label{eq:Borel-Arcones-Gine}
\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p \sim& \Ex \left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\| + \sqrt{p}\Ex \sup_{\|x\|_2\le 1} \left \| \sum_{ij} a_{ij}x_i g_j\right\| \nonumber\\
&+ p\sup_{\|x\|_2\le 1,\|y\|_2\le 1} \left\| \sum_{ij}a_{ij} x_i y_j\right\|.\end{aligned}$$
Unfortunately the second term on the right hand side of is usually difficult to estimate. The main effort in this article will be to replace it by quantities which even if still involve expected values of Banach space valued random variables in many situations can be handled more easily. More precisely, we will obtain inequalities in which additional suprema over Euclidean spheres are placed outside the expectations, which reduces the complexity of the involved stochastic processes. As one of the consequences we will derive two-sided bounds in $L_r$ spaces involving only purely deterministic quantities.
Our first observation is a simple lower bound
\[prop:lower2d\] Let $(a_{ij})_{i,j \leq n}$ be a symmetric matrix with values in a normed space . Then for any $p\geq 1$ we have $$\begin{aligned}
\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p
&\geq
\frac{1}{C}\Bigg(\Ex\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|
+\sqrt{p}\sup_{\|x\|_2\leq 1}\Ex\left\|\sum_{i\neq j}a_{ij}x_ig_j\right\|
\\
&\phantom{aaaaa}+\sqrt{p}\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|
+p\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|
\Bigg).\end{aligned}$$
This motivates the following conjecture.
\[conj1\_2d\] Under the assumptions of Proposition \[prop:lower2d\] we have $$\begin{aligned}
\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p
&\leq
C\Bigg(\Ex\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|
+\sqrt{p}\sup_{\|x\|_2\leq 1}\Ex\left\|\sum_{i\neq j}a_{ij}x_ig_j\right\|
\\
&\phantom{aaaaa}+\sqrt{p}\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|
+p\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|
\Bigg).\end{aligned}$$
We are able to show that the conjectured estimate holds up to logarithmic factors.
\[thm:uppper2d1\] Let $(a_{ij})_{i,j \leq n}$ be a symmetric matrix with values in a normed space $(F,\| \ \cdot \|)$. Then for any $p\geq 1$ the following two estimates hold $$\begin{aligned}
\notag
\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p
\leq&
C\Bigg(\log(ep)\Ex\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|
+\sqrt{p}\sup_{\|x\|_2\leq 1}\Ex\left\|\sum_{i\neq j}a_{ij}x_ig_j\right\|
\\
\label{eq:upperest1}
+&\sqrt{p}\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|
+p\log(ep)\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|
\Bigg)\end{aligned}$$ and $$\begin{aligned}
\notag
\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p
\leq&
C\Bigg(\Ex\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|
+\sqrt{p}\sup_{\|x\|_2\leq 1}\Ex\left\|\sum_{i\neq j}a_{ij}x_ig_j\right\|
\\
\label{eq:upperest2}
+&\sqrt{p}\log(ep)\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|
+p\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|
\Bigg).\end{aligned}$$
One of the main reasons behind the appearance of additional logarithmic factors is lack of good Sudakov-type estimates for Gaussian quadratic forms. Such bounds hold for linear forms and as a result we may show the following ($(g_{i,j})_{i,j\leq n}$ below denote as usual i.i.d. ${\mathcal N}(0,1)$ random variables).
\[thm:upper2d2\] Under the assumptions of Theorem \[thm:uppper2d1\] we have $$\begin{aligned}
\notag
\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p
\leq
C\Bigg(&\Ex\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|+\Ex\left\|\sum_{i\neq j}a_{ij}g_{ij}\right\| \\
\notag
&+\sqrt{p}\sup_{\|x\|_2\leq 1}\Ex\left\|\sum_{i\neq j}a_{ij}x_ig_j\right\|
+\sqrt{p}\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\| \\
&+p\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|
\Bigg).\label{eq:upperest3}\end{aligned}$$
In particular we know that Conjecture \[conj1\_2d\] holds in Banach spaces, in which Gaussian quadratic forms dominate in mean Gaussian linear forms, i.e. in Banach spaces $(F,\|\ \|)$ for which there exists a constant $\lambda<\infty$ such for any finite symmetric matrix $(a_{ij})_{i,j\leq n}$ with values in $F$ one has $$\Ex\left\|\sum_{i\neq j}a_{ij}g_{ij}\right\|\leq \lambda \Ex\left\|\sum_{ i\neq j}a_{ij}g_ig_j\right\|. \label{war}$$ It is easy to check (see Proposition \[prop:estLr\] below) that such property holds for $L_r$-spaces with $\lambda=\lambda(r)\leq Cr$.
For non-centered Gaussian quadratic forms $S=\sum_{i,j}a_{ij}g_ig_j$ one has $\|S\|_p\sim \|\Ex S\|+\|S- \Ex S\|_p$, so Proposition \[prop:lower2d\] yields $$\begin{aligned}
\left\|\sum_{ij}a_{ij}g_ig_j\right\|_p
&\geq
\frac{1}{C}\Bigg(\Ex\left\|\sum_{ij}a_{ij}g_ig_j\right\|
+\sqrt{p}\sup_{\|x\|_2\leq 1}\Ex\left\|\sum_{i\neq j}a_{ij}x_ig_j\right\|
\\
&\phantom{aaaaa}+\sqrt{p}\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|
+p\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|
\Bigg).\end{aligned}$$ and Theorem \[thm:upper2d2\] implies $$\begin{aligned}
\left\|\sum_{ij}a_{ij}g_ig_j\right\|_p
&\leq
C\Bigg(\Ex\left\|\sum_{ij}a_{ij}g_ig_j\right\|+\Ex\left\|\sum_{i\neq j}a_{ij}g_{ij}\right\|
+\sqrt{p}\sup_{\|x\|_2\leq 1}\Ex\left\|\sum_{i\neq j}a_{ij}x_ig_j\right\|
\\
&\phantom{aaaaa}+\sqrt{p}\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|
+p\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|
\Bigg).\end{aligned}$$
Proposition \[prop:lower2d\] and Theorem \[thm:upper2d2\] may be expressed in terms of tails.
\[thm:tails2d\] Let $(a_{ij})_{i,j \leq n}$ be a symmetric matrix with values in a normed space $(F,\| \ \cdot \ \|)$. Then for any $t>0$,$$\Pr\left(\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|
\geq t+\frac{1}{C}\Ex\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|\right)
\geq \frac{1}{C}\exp\left(-C\min\left\{\frac{t^2}{U^2},\frac{t}{V}\right\}\right),$$ where $$\begin{aligned}
U&=\sup_{\|x\|_2\leq 1}\Ex\left\|\sum_{i \neq j}a_{ij}x_ig_j\right\|
+\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|, \label{u}
\\
V&=\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|. \label{v}\end{aligned}$$ Moreover, for $t>C(\Ex\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\|+\Ex\|\sum_{i\neq j}a_{ij}g_{ij}\|)$ we have $$\Pr\left(\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|\geq t\right)
\leq 2\exp\left(-\frac{1}{C}\min\left\{\frac{t^2}{U^2},\frac{t}{V}\right\}\right).$$
As a corollary we get a Hanson-Wright-type inequality for Banach space valued quadratic forms in general independent subgaussian random variables.
\[thm:hw\] Let $X_1,X_2,\ldots,X_n$ be independent mean zero $\alpha$-subgaussian random variables. Then for any symmetric matrix $(a_{ij})_{i,j\leq n}$ with values in a normed space $(F,\| \ \cdot \ \|)$ and $t>C\alpha^2(\Ex\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\|+\Ex\|\sum_{i \neq j}a_{ij}g_{ij}\|)$ we have $$\label{eq:HWnorm}
\Pr\left(\left\|\sum_{ij}a_{ij}(X_iX_j-\Ex(X_iX_j))\right\|\geq t\right)
\leq 2\exp\left(-\frac{1}{C}\min\left\{\frac{t^2}{\alpha^4 U^2},\frac{t}{\alpha^2 V}\right\}\right),$$ where $U$ and $V$ are as in Theorem \[thm:tails2d\].
It is not hard to check that in the case $F=\er$ we have $U\sim \|(a_{ij})\|_{\mathrm{HS}}$ and $V=\|(a_{ij})\|_{\mathrm{op}}$. Moreover, $$\Ex\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|+\Ex\left\|\sum_{i \neq j}a_{ij}g_{ij}\right\|
\leq 2\|(a_{ij})\|_{\mathrm{HS}},$$ so the right hand side of is at least 1 for $t<C'(\Ex\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\|+\Ex\|\sum_{i \neq j}a_{ij}g_{ij}\|)$ and sufficiently large $C$. Hence holds for any $t>0$ in the real case and is equivalent to the Hanson-Wright bound.
Proposition \[prop:diagest\] below shows that we may replace in all estimates above the term $\sup_{\|x\|_2\leq 1}\Ex\|\sum_{i \neq j}a_{ij}x_ig_j\|$ by $\sup_{\|x\|_2\leq 1}\Ex\|\sum_{ij}a_{ij}x_ig_j\|$.
We are able to derive similar estimates as discussed in this paper for Banach space valued Gaussian chaoses of arbitrary degree. Formulas are however more complicated and the proof is more technical. For these reasons we decided to include details in a separate forthcoming paper [@ALM].
The organization of the paper is as follows. In the next section we discuss a few corollaries of Theorems \[thm:upper2d2\] and \[thm:hw\]. In Section 3 we prove Proposition \[prop:lower2d\] and show that it is enough to bound separately moments of diagonal and off-diagonal parts of chaoses. In Section 4 we reduce Theorems \[thm:uppper2d1\] and \[thm:upper2d2\] to the problem of estimating means of suprema of certain Gaussian processes. In Section 5 we show how to bound expectations of such suprema – the main new ingredient are entropy bounds presented in Corollary \[cor:entrestch\] (derived via volumetric-type arguments). Unfortunately our entropy bounds are too weak to use the Dudley integral bound. Instead, we present a technical chaining argument (of similar type as in [@LaAoP]). In the last section we conclude the proofs of main Theorems.
Consequences and extensions
===========================
$L_r$-spaces
------------
We start with showing that $L_r$ spaces for $r<\infty$, satisfy with $\lambda=Cr$, so Theorem \[thm:upper2d2\] implies Conjecture \[conj1\_2d\] for $L_r$ spaces (and as a consequence the Hanson-Wright inequality). Moreover, in this case one may express all parameters without any expectations as is shown in the proposition below.
\[prop:estLr\] For any symmetric matrix $(a_{ij})_{i,j\leq n}$ with values in $L_r=L_r(X,\mu)$, $1\leq r<\infty$ and $x_1,\ldots,x_n\in \er$ we have $$\begin{aligned}
&\frac{1}{C}\left\|\sqrt{\sum_{ij}a_{ij}^2}\right\|_{L_r}
\leq \Ex\left\|\sum_{ij}a_{ij}g_{ij}\right\|_{L_r}
\leq C\sqrt{r}\left\|\sqrt{\sum_{ij}a_{ij}^2}\right\|_{L_r}, \label{loc1}
\\
&\frac{1}{C}\left\|\sqrt{\sum_{j}\left(\sum_j a_{ij}x_i\right)^2}\right\|_{L_r}
\leq \Ex\left\|\sum_{ij}a_{ij}x_{i}g_j\right\|_{L_r}
\leq C\sqrt{r}\left\|\sqrt{\sum_{j}\left(\sum_i a_{ij}x_i\right)^2}\right\|_{L_r}, \label{loc2}
\\
&\frac{1}{C\sqrt{r}}\left\|\sqrt{\sum_{ij}a_{ij}^2}\right\|_{L_r}
\leq \Ex\left\|\sum_{ij}a_{ij}(g_{i}g_j-\delta_{ij})\right\|_{L_r}
\leq Cr\left\|\sqrt{\sum_{ij}a_{ij}^2}\right\|_{L_r}. \label{loc3}\end{aligned}$$
For any $a_i$’s in $L_r$ the Gaussian concentration yields $$\begin{aligned}
\Ex \left\| \sum_i a_i g_i \right\|_{L_r} \leq \left(\Ex \left\| \sum_i a_i g_i \right\|^r_{L_r} \right)^{1/r} \leq C\sqrt{r}\Ex \left\| \sum_i a_i g_i \right\|_{L_r}. %\label{eq1}\end{aligned}$$ Since $$\begin{aligned}
\left(\Ex \left\| \sum_i a_i g_i \right\|^r_{L_r} \right)^{1/r}&=\left(\int_X \Ex \left| \sum_i a_i (x) g_i \right|^r d \mu(x) \right)^{1/r}\\
&=\left(\int_X \Ex|g_1|^r \Big(\sum_i a^2_i(x)\Big)^{r/2} d \mu(x) \right)^{1/r}
\sim \sqrt{r} \left\| \sqrt{\sum_i a^2_i}\right\|_{L_r},\end{aligned}$$ estimates , follow easily. The proof of is analogous. It is enough to observe that from [@delaPenaGine Theorem 3.2.10] $$\begin{aligned}
\Ex\left\|\sum_{ij}a_{ij}(g_{i}g_j-\delta_{ij})\right\|_{L_r}
\leq \left( \Ex\left\|\sum_{ij}a_{ij}(g_{i}g_j-\delta_{ij})\right\|_{L_r}^r\right)^{1/r}
\leq C r \Ex\left\|\sum_{ij}a_{ij}(g_{i}g_j-\delta_{ij})\right\|_{L_r}\end{aligned}$$ and imples for any $x \in X$, $$\frac{\sqrt{r}}{C} \sqrt{\sum_{ij} a^2_{ij}(x) }
\leq \left( \Ex \left|\sum_{ij}a_{ij}(x)(g_{i}g_j-\delta_{ij}) \right| ^{r} \right)^{1/r}
\leq C r \sqrt{\sum_{ij} a^2_{ij}(x)}.$$
The above proposition, together with Proposition \[prop:lower2d\] and Theorems \[thm:upper2d2\] and \[thm:hw\] immediately yield the following corollaries (in particular they imply that Conjecture \[conj1\_2d\] holds in $L_r$ spaces with $r$-dependent constants)
For any symmetric matrix $(a_{ij})_{ij}$ with values in $L_r$ and $p\geq 1$ we have $$\begin{aligned}
\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p
\sim^r&
\left\|\sqrt{\sum_{ij}a_{ij}^2}\right\|_{L_r}
+\sqrt{p} \sup_{\|x\|_2\leq 1}\left\|\sqrt{\sum_{j}\left(\sum_{ i\neq j} a_{ij}x_i\right)^2}\right\|_{L_r}
\\
&+\sqrt{p}\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|_{L_r}
+p\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|_{ L_r}.\end{aligned}$$ The implicit constants in the estimates for moments can be taken to be equal to $Cr$ in the upper bound and $r^{-1/2}/C$ in the lower bound.
Let $X_1,X_2,\ldots,X_n$ be independent mean zero $\alpha$-subgaussian random variables. Then for any symmetric finite matrix $(a_{ij})_{i,j\leq n}$ with values in $L_r=L_r(X,\mu)$, $1\leq r<\infty$ and $t>C\alpha^2r\|\sqrt{\sum_{ij}a_{ij}^2}\|_{L_r}$ we have $$\label{eq:HWnorm1}
\Pr\left(\left\|\sum_{ij}a_{ij}(X_iX_j-\Ex(X_iX_j))\right\|_{L_r}\geq t\right)
\leq 2\exp\left(-\frac{1}{C}\min\left\{\frac{t^2}{\alpha^4 r U^2},\frac{t}{\alpha^2 V }\right\}\right),$$
where $$\begin{aligned}
U&=\sup_{\|x\|_2\leq 1}\left\|\sqrt{\sum_{j}\left(\sum_{ i\neq j }a_{ij}x_i\right)^2}\right\|_{L_r}
+\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|_{L_r},
\\
V&=\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|_{L_r}.\end{aligned}$$
Spaces of type 2
----------------
Recall that a normed space $F$ is of type 2 with constant $\lambda$ if for every positive integer $n$ and $v_1,\ldots,v_n \in F$, $$\Ex\left\|\sum_{i=1}^n v_i \varepsilon_i\right\| \le \lambda\sqrt{\sum_{i=1}^n \|v_i\|^2},$$ where $\varepsilon_1,\varepsilon_2,\ldots$ is a sequence of independent Rademacher variables.
By standard symmetrization inequalities one easily obtains that if $F$ is of type two with constant $\lambda$ then for any independent random variables $X_i$, $$\Ex \left \| \sum_i a_i (X_i^2 - \Ex X_i^2)\right\| \le 2\lambda \sqrt{\sum_i \|a_i\|^2 \Ex X_i^4}$$ and if $\Ex X_i=0$, then decoupling arguments combined with symmetrization and Khintchine-Kahane inequalities give $$\Ex\left \|\sum_{i\neq j} a_{ij}X_i X_j\right\| \le C\lambda^2 \sqrt{\sum_{i\neq j} \|a_{ij}\|^2 \Ex X_i^2 \Ex X_j^2}.$$
Therefore, Theorem \[thm:hw\] gives immediately the following
Let $X_1,X_2,\ldots,X_n$ be independent mean zero $\alpha$-subgaussian random variables and let $F$ be a normed space of type two constant $\lambda$. Then for any symmetric finite matrix $(a_{ij})_{i,j\leq n}$ with values in $F$ and $t>C\lambda^2 \alpha^2 \sqrt{\sum_{ij}\|a_{ij}\|^2}$ we have $$%\label{eq:HWnorm}
\Pr\left(\left\|\sum_{ij}a_{ij}(X_iX_j-\Ex(X_iX_j))\right\|\geq t\right)
\leq 2\exp\left(-\frac{1}{C}\min\left\{\frac{t^2}{\alpha^4U^2},\frac{t}{\alpha^2V}\right\}\right),$$ where $$\begin{aligned}
U&=\lambda \sup_{\|x\|_2\leq 1}\sqrt{\sum_{j}\Big\|\sum_{i\neq j} a_{ij}x_i\Big\|^2}
+\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|,
\\
V&=\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|.\end{aligned}$$
We note that from Theorem \[thm:hw\] one can also derive similar inequalities for suprema of quadratic forms over VC-type classes of functions appearing e.g., in the analysis of randomized $U$-processes (cf. e.g., [@delaPenaGine Chapter 5.4]).
Random vectors with dependencies
--------------------------------
Let us assume that $X= (X_1,\ldots,X_n)$ is an image of a standard Gaussian vector in $\er^n$ under an $\alpha$-Lipschitz map. In particular, by the celebrated Caffarelli contraction principle [@Caffarelli_contraction], this is true if $X$ has density of the form $e^{-V}$, where $\nabla^2 V \ge \alpha^{-2} {\rm Id}$. As observed by Ledoux and Oleszkiewicz [@LO Corollary 1], by combining the well known comparison result due to Pisier [@Pisier] with a stochastic domination-type argument, one gets that for any smooth function $f\colon \er^n \to F$, and any $p\ge 1$, $$\begin{aligned}
\label{eq:Pisier}
\|f(X) - \Ex f(X)\|_p \le \frac{\pi\alpha}{2}\|\langle \nabla f(X), G\rangle\|_p,\end{aligned}$$ where here and subsequently $G_n$ is a standard Gaussian vector in $\er^n$ independent of $X$ and for $a \in F^n$, $b \in \er^n$ we denote $\langle a,b\rangle = \sum_{i=1}^n a_i b_i$. This inequality together with Theorem \[thm:upper2d2\] allow us to implement a simple argument from [@AW] and obtain inequalities for quadratic forms and more general $F$-valued functions of the random vector $X$. Below, we will denote the second partial derivatives of $f$ by $\partial_{ij} f$. For the sake of brevity, we will focus on moment estimates, clearly tail bounds follow from them by an application of the Chebyshev inequality.
\[cor:lip-image\] Let $X$ be an $\alpha$-Lipschitz image of a standard Gaussian vector in $\er^n$ and let $f\colon \er^n \to F$ be a function with bounded derivatives of order two. Assume moreover that $\Ex \nabla f(X) = 0$. Then for any $p \ge 2$, $$\begin{aligned}
\| f(X) - \Ex f(X)\|_p \le&
C\alpha^2\sup_{z\in \er^n} \Bigg(\Ex\left\|\sum_{ij}\partial_{ij}f(z)(g_ig_j-\delta_{ij})\right\|
+\Ex\left\|\sum_{ i\neq j}\partial_{ij}f(z)g_{ij}\right\| \nonumber \\
&+\sqrt{p}\sup_{\|x\|_2\leq 1}\Ex\left\|\sum_{i\neq j}\partial_{ij}f(z)x_ig_j\right\|\nonumber
\\
&+\sqrt{p}\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}\partial_{ij}f(z)x_{ij}\right\|
+p\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}\partial_{ij}f(z)x_iy_j\right\|
\Bigg).\label{eq:bounded-Hessian}\end{aligned}$$ In particular if $X$ is of mean zero, then $$\begin{aligned}
\label{eq:lip-HW}
\left\|\sum_{ij}a_{ij}(X_iX_j-\Ex(X_iX_j))\right\|_p
\le&
C\alpha^2\Bigg(\Ex\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|+\Ex\left\|\sum_{i\neq j}a_{ij}g_{ij}\right\|
\nonumber
\\
&+\sqrt{p}\sup_{\|x\|_2\leq 1}\Ex\left\|\sum_{i\neq j}a_{ij}x_ig_j\right\|
\nonumber+\sqrt{p}\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|
\\
&+p\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|
\Bigg)\end{aligned}$$ and the inequality is satisfied.
Let $G_n = (g_1,\ldots,g_n)$, $G_n'=(g_1',\ldots,g_n')$ be independent standard Gaussian vectors in $\er^n$, independent of $X$. By an iterated application of (the second time conditionally on $G_n$) we have $$\begin{aligned}
\Ex \|f(X) - \Ex f(X)\|^p
&\le C^p\alpha^p\Ex\|\langle \nabla f(X),G_n\rangle\|^p
\le C^{2p}\alpha^{2p} \Ex \left\|\sum_{ij} \partial_{ij} f(X) g_ig_j'\right\|^p
\\
& \le \tilde{C}^{2p}\alpha^{2p} \Ex \left\|\sum_{ij} \partial_{ij} f(X) (g_ig_j-\delta_{ij})\right\|^p ,\end{aligned}$$ where the last inequality follows by [@AG Theorem 2.2]. To finish the proof of it is now enough to apply Theorem \[thm:upper2d2\] conditionally on $X$ and replace the expectation in $X$ by the supremum over $z \in \er^n$.
The inequality follows by a direct application of .
Lower bounds
============
In this part we show Proposition \[prop:lower2d\] and the lower bound in Theorem \[thm:tails2d\]. We start with a simple lemma.
\[lem:lower1\] Let $W= \|\sum_{i\neq j}a_{ij}g_ig_j\|_p+\|\sum_{i}a_{ii}(g_i^2-1)\|_p$. Then for any $p\geq 1$, $$\frac{1}{3}W \leq \left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p \leq W.$$
Let $(\ve_i)_{i}$ be a sequence of i.i.d. symmetric $\pm 1$ r.v’s independent of $(g_i)_i$. We have by symmetry of $g_i$ and Jensen’s inequality, $$\begin{aligned}
\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p&=
\left\|\sum_{ij}a_{ij}(\ve_i\ve_jg_ig_j-\delta_{ij})\right\|_p
\geq \left\|\Ex_\ve\sum_{ij}a_{ij}(\ve_i\ve_jg_ig_j-\delta_{ij})\right\|_p\\
&=\left\|\sum_{i}a_{ii}(g_i^2-1)\right\|_p.\end{aligned}$$ To conclude we use the triangle inequality in $L_p$ and get $$\left\|\sum_{i\neq j}a_{ij}g_ig_j\right\|_p\leq \left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p+
\left\|\sum_{i}a_{ii}(g_i^2-1)\right\|_p\leq 2\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p.$$
Adding the inequalities above yields the first estimate of the lemma. The second one follows trivially from the triangle inequality.
Obviously $$\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p\geq \Ex\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|.$$ Moreover, denoting by $\|\cdot\|_*$ the norm in the dual of $F$, we have $$\begin{aligned}
\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p
&\geq \sup_{\|\varphi\|_*\leq 1}\left\|\sum_{ij}\varphi(a_{ij})(g_ig_j-\delta_{ij})\right\|_p
\\
&\geq \frac{1}{C}\left(\sqrt{p}\sup_{\|\varphi\|_*\leq 1}\|(\varphi(a_{ij}))_{ij}\|_{\mathrm{HS}}
+p\sup_{\|\varphi\|_*\leq 1}\|(\varphi(a_{ij}))_{ij}\|_{\mathrm{op}}\right)
\\
&=\frac{1}{C}\left(\sqrt{p}\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|
+p\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|\right),\end{aligned}$$ where in the second inequality we used .
Lemma \[lem:lower1\] and the decoupling inequality of Kwapień [@K] (see also [@dlPMS]) yield $$\label{eq:decoupling-lower-bound}
\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p
\geq\frac{1}{3}\left\|\sum_{i\neq j}a_{ij}g_ig_j\right\|_p
\geq \frac{1}{C}\left\|\sum_{i\neq j}a_{ij}g_ig_j'\right\|_p,$$ where $(g_i')_{i}$ denotes an independent copy of $(g_i)_i$.
For any finite sequence $(b_i)_i$ in $(F,\|\ \cdot \ \|)$ we have $$\begin{aligned}
\left\|\sum_{i}b_ig_i\right\|_p\geq \sup_{\|\varphi\|_*\leq 1}\left\|\sum_{i}\varphi(b_i)g_i\right\|_p
=\sup_{\|\varphi\|_*\leq 1}\|(\varphi(b_i))_i\|_2 \cdot \|g_1\|_p\geq
\frac{\sqrt{p}}{C}\sup_{\|x\|_2\leq 1}\left\|\sum_i x_ib_i\right\|. \label{ine:1}\end{aligned}$$ Thus, by and the Fubini Theorem, we get $$\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|_p\geq
\frac{ \sqrt{p}}{C}\sup_{\|x\|_2\leq 1}\left\|\sum_{i\neq j}a_{ij}x_ig_j' \right\|_p
\geq \frac{\sqrt{p}}{C}\sup_{\|x\|_2\leq 1}\Ex\left\|\sum_{i\neq j}a_{ij}x_ig_j\right\|.$$
Reduction to a bound on the supremum of a Gaussian process
==========================================================
In this section we will reduce the upper estimates of Theorems \[thm:uppper2d1\] and \[thm:upper2d2\] to an estimate on expected value of a supremum of a certain Gaussian process. The arguments in this part of the article are well-known, we present them for the sake of completeness. In particular we will demonstrate the upper bounds given in .
The first lemma shows that we may easily bound the diagonal terms.
\[lem:diag\] For $p\geq 1$ we have $$\begin{aligned}
\left\|\sum_{i}a_{ii}(g_i^2-1)\right\|_p
&\leq C\left(\Ex\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|
+\sqrt{p}\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|\right.
\\
&\left.\phantom{aaaaaa}+p\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|\right).\end{aligned}$$
Let $X_i$ be a sequence of i.i.d. standard symmetric exponential r.v’s. A simple argument (cf. proof of Lemma 9.5 in [@AL]) shows that $$\begin{aligned}
\left\|\sum_{i}a_{ii}(g_i^2-1)\right\|_p\sim \left\|\sum_{i}a_{ii}g_i g'_i\right\|_p \sim \left\|\sum_{i}a_{ii}X_i\right\|_p, \label{ine:2}\end{aligned}$$ the latter quantity was bounded in [@LaSMlc Theorem 1], thus $$\begin{aligned}
&\left\|\sum_{i}a_{ii}(g_i^2-1)\right\|_p
\sim
\left\|\sum_{i}a_{ii}(g_i^2-1)\right\|_1+\sqrt{p}\sup_{\|x\|_2\leq 1}\left\|\sum_{i}a_{ii}x_i\right\|
+p\sup_{i}\|a_{ii}\|
\\
&\leq C\left(\Ex\left\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\right\|+\sqrt{p}\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|+p\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|\right),\end{aligned}$$ where in the last inequality we used Lemma \[lem:lower1\].
The next proposition implies that in all our main results we can replace the term $\sup_{\|x\|_2\leq 1}\Ex\|\sum_{i \neq j}a_{ij}x_ig_j\|$ by $\sup_{\|x\|_2\leq 1}\Ex\|\sum_{ij}a_{ij}x_ig_j\|$.
\[prop:diagest\] Under the assumption of Proposition \[prop:lower2d\] we have for $p\geq 1$,
$$\begin{aligned}
\sqrt{p}\sup_{\|x\|_2\leq 1}\Ex\left\|\sum_{i}a_{ ii}{x_ig_i}\right\|
\leq
&C\left(\Ex\left\|\sum_{ij}a_{ij}(g_{ij}-\delta_{ij})\right\|
+\sqrt{p}\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|\right.
\\
&+\left. p\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|\right)\end{aligned}$$
Let $(g_i')_i$ be an independent copy of the sequence $(g_i)_i$. Denoting by $\Ex'$ the expectation with respect to the variables $(g_i')_i$, we may estimate $$\begin{aligned}
\sqrt{p}\sup_{\|x\|_2\leq 1}\Ex\left\|\sum_{i}a_{ii}{x_ig_i}\right\|&\leq \Ex \sqrt{p}\sup_{\|x\|_2\leq 1} \left\|\sum_{i}a_{ii}{x_ig_i}\right\|
\leq \Ex \left(\Ex'\left \|\sum_{i}a_{ii} g_ig_i'\right\|^p\right)^{1/p} \\
& \leq C \left\|\sum_i a_{ii} g_i g_i'\right\|_p \leq C \left\|\sum_i a_{ii} (g_i^2 - 1)\right\|_p,\end{aligned}$$ where the second inequality follows from applied conditionally on $(g_i)_i$, the third one from Jensen’s inequality and the last one from . The assertion of the proposition follows now by Lemma \[lem:diag\].
For the off-diagonal terms we use first the concentration approach.
\[prop:red\] For $p\geq 1$ we have $$\begin{gathered}
\left\|\sum_{i\neq j}a_{ij}g_ig_j\right\|_p\leq
C\Bigg(\Ex\left\|\sum_{i\neq j}a_{ij}g_ig_j\right\|
+\sqrt{p}\Ex\sup_{\|x\|_2\leq 1}\left\|\sum_{i\neq j}a_{ij}x_ig_j\right\|\\
+p\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|
\Bigg).\end{gathered}$$
Let $$\begin{aligned}
A:=\Bigg\{ z\in \er^n\colon\
&\left\|\sum_{i\neq j}a_{ij}z_iz_j\right\|\leq 4\Ex\left\|\sum_{i\neq j}a_{ij}g_ig_j\right\|,
\\
&\sup_{\|x\|_2\leq 1}\left\|\sum_{i\neq j}a_{ij}x_iz_j\right\|\leq
4\Ex \sup_{\|x\|_2\leq 1}\left\|\sum_{i\neq j}a_{ij}x_ig_j\right\|
\Bigg\}.\end{aligned}$$
Then $\gamma_{n}(A)\geq \frac{1}{2}$ by the Chebyshev inequality. Gaussian concentration gives $\gamma_{n}(A+tB_{2}^{n})\geq 1-e^{-t^2/2}$ for $t\geq 0$. It is easy to check that for $z\in A+tB_{2}^n$ we have $$\left\|\sum_{i\neq j}a_{ij}z_iz_j\right\|\leq 4S(t),$$ where $$S(t)=\Ex\left\|\sum_{i\neq j}a_{ij}g_ig_j\right\|+2t\Ex \sup_{\|x\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_ig_j\right\|
+t^2\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{i\neq j}a_{ij}x_iy_j\right\|.$$ So $$\Pr\left(\left\|\sum_{i\neq j}a_{ij}g_ig_j\right\|> 4S(t)\right)\leq e^{-t^2/2}\quad \mbox{for }t\geq 0.$$ Integrating by parts we get $\|\sum_{i\neq j}a_{ij}g_ig_j\|_p\leq CS(\sqrt{p})$ for $p\geq 1$, which ends the proof.
Observe that for any symmetric matrix by using the decoupling bound [@K] we obtain $\Ex\|\sum_{i\neq j}a_{ij}g_ig_j\|\sim \Ex\|\sum_{i\neq j}a_{ij}g_ig_j'\|$. Moreover introducing decoupled chaos enables us to release the assumptions of the symmetry of the matrix and zero diagonal.
Taking into account the above observations, Conjecture \[conj1\_2d\] reduces to the statement that for any $p\geq 1$ and any finite matrix $(a_{ij})$ in $(F,\|\ \cdot \ \|)$ we have $$\begin{aligned}
\Ex\sup_{\|x\|_2\leq 1}\left\|\sum_{ij}a_{ij}g_ix_j\right\|
&\leq
C\Bigg(\frac{1}{\sqrt{p}}\Ex\left\|\sum_{ij}a_{ij}g_ig'_j\right\|
+\sup_{\|x\|_2\leq 1}\Ex\left\|\sum_{ij}a_{ij}g_ix_j\right\| \nonumber
\\
&\phantom{aaaaa}+\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|
+\sqrt{p}\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|
\Bigg).\label{conj:supgauss0}\end{aligned}$$
Let us rewrite in another language. We may assume that $F=\er^m$ for some finite $m$ and $a_{ij}=(a_{ijk})_{k\leq m}$. Let $T=B_{F^*}$ be the unit ball in the dual space $F^*$. Then takes the following form.
\[conj:supgauss\]
Let $p\geq 1$. Then for any triple indexed matrix $(a_{ijk})_{i,j\leq n, k\leq m}$ and bounded nonempty set $T\subset\er^m$ we have $$\begin{aligned}
\label{eq:corollary-supremum}
\Ex\sup_{\|x\|_2\leq 1, t\in T}\left|\sum_{ijk}a_{ijk}g_ix_jt_k\right|
&\leq
C\Bigg(\frac{1}{\sqrt{p}}\Ex\sup_{t\in T}\left|\sum_{ijk}a_{ijk}g_ig'_jt_k\right|
\nonumber\\
&\phantom{aa}
+\sup_{\|x\|_2\leq 1}\Ex\sup_{t\in T}\left|\sum_{ijk}a_{ijk}g_ix_jt_k\right|
+\sup_{t\in T}\left(\sum_{ij}\left(\sum_k a_{ijk}t_k\right)^2\right)^{1/2}
\nonumber\\
&\phantom{aa}+\sqrt{p}\sup_{\|x\|_2\leq 1,t\in T}\left(\sum_i\left(\sum_{jk}a_{ij}x_jt_k\right)^2\right)^{1/2}
\Bigg).\end{aligned}$$
Obviously it is enough to show this for finite sets $T$.
Estimating suprema of Gaussian processes {#sec:supgauss}
========================================
To estimate the supremum of a centered Gaussian process $(G_{v})_{v\in V}$ one needs to study the distance on $V$ given by $d(v,v'):=(\Ex |G_v-G_{v'}|^2)^{1/2}$ (cf. [@Ta_book]). In the case of the Gaussian process from Conjecture \[conj:supgauss\] this distance is defined on $B_2^n\times T\subset \er^n\times \er^m$ by the formula $$d_A((x,t),(x',t')):=\left(\sum_{i}\left(\sum_{jk}a_{ijk}(x_jt_k-x'_jt'_k)\right)^2\right)^{1/2}
=\alpha_A(x\otimes t-x'\otimes t'),$$ where $x\otimes t=(x_jt_k)_{j,k}\in \er^{nm}$ and $\alpha_A$ is a norm on $\er^{nm}$ given by $$\alpha_A(y):=\left(\sum_{i}\left(\sum_{jk}a_{ijk}y_{jk}\right)^2\right)^{1/2},$$ (as in Conjecture \[conj:supgauss\] in this section we do not assume that the matrix $(a_{ijk})_{ijk}$ is symmetric or that it has $0$ on the generalized diagonal).
Let $$B((x,t),d_A,r)=\left\{(x',t')\in \er^n\times T\colon\ \alpha_A(x\otimes t-x'\otimes t')\leq r\right\}$$ be the closed ball in $d_A$ with center at $(x,t)$ and radius $r$.
Observe that $$\mathrm{diam}(B_2^n\times T,d_A)\sim \sup_{\|x\|_2\leq 1,t\in T}\left(\sum_i\left(\sum_{jk}a_{ijk}x_jt_k\right)^2\right)^{1/2}.$$
Now we try to estimate entropy numbers $N(B_2^n\times T,d_A,\ve)$ for $\ve>0$ (recall that $N(S,\rho,\ve)$ is the smallest number of closed balls with the diameter $\ve$ in metric $\rho$ that cover set $S$). To this end we first introduce some notation. For a nonempty bounded set $S$ in $\er^m$ let $$\beta_{A,S}(x):=\Ex\sup_{t\in S}\left|\sum_{ijk}a_{ijk}g_ix_jt_k\right|, \quad x\in\er^n.$$
Observe that $\beta_{A,S}$ is a norm on $\er^n$. Moreover, by the classical Sudakov minoration ([@Su] or [@LT Theorem 3.18]) for any $x\in \er^n$ there exists a set $S_{x,\ve}\subset S$ of cardinality at most $\exp(C\ve^{-2})$ such that $$\forall_{t\in S}\ \exists_{t'\in S_{x,\ve}}\ \alpha_A(x\otimes (t-t'))\leq \ve \beta_{A,S}(x).$$ For a finite set $S\subset \er^m$ and $\ve>0$ define a measure $\mu_{\ve,S}$ on $\er^n\times S$ in the following way $$\mu_{\ve,S}(C):=\int_{\er^n} \sum_{t\in S_{x,\ve}}\delta_{(x,t)}(C)d\gamma_{n,\ve}(x),$$ where $\gamma_{n,\ve}$ is the distribution of the vector $\ve G_n$ (recall that $G_n$ is the standard Gaussian vector in $\er^n$). Since $S$ is finite, we can choose sets $S_{x,\ve}$ in such a way that there are no problems with measurability.
To bound $N(B_2^n\times T,d_A,\ve)$ we need two lemmas.
[@LaAoP Lemma 1]\[lem1\] For any norms $\alpha_1,\alpha_2$ on $\er^n$, $y\in B_2^n$ and $\ve>0$, $$\gamma_{n,\ve}\left(x\colon\ \alpha_1(x-y)\leq 4\ve\Ex\alpha_1(G_n),\
\alpha_2(x)\leq 4\ve \Ex\alpha_2(G_n)+\alpha_2(y)\right)
\geq \frac{1}{2}\exp(-\ve^{-2}/2).$$
\[lem:meas2d\] For any finite set $S$ in $\er^m$, any $(x,t)\in B_2^n\times S$ and $\ve>0$ we have $$\mu_{\ve,S} \left(B\left((x,t),d_A,r(\ve)\right)\right) \geq \frac{1}{2}\exp(-\ve^{-2}/2),$$ where $$r(\ve)=r(A,S,x,t,\ve)
=4\ve^2\Ex\beta_{A,S}(G_n)+\ve\beta_{A,S}(x)+4\ve\Ex\alpha_A(G_n\otimes t).$$
Let $$U=\left\{x'\in \er^n\colon\ \beta_{A,S}(x')\leq 4\ve\Ex\beta_{A,S}(G_n)+\beta_{A,S}(x),
\alpha_A((x-x')\otimes t)\leq 4\ve \Ex\alpha_A(G_n\otimes t)\right\}.$$ For any $x'\in U$ there exists $t'\in S_{x',\ve}$ such that $\alpha_A(x'\otimes(t-t'))\leq \ve \beta_{A,S}(x')$. By the triangle inequality $$\alpha_A(x\otimes t-x'\otimes t')\leq \alpha_A((x-x')\otimes t)+\alpha_A(x'\otimes(t-t'))\leq r(\ve).$$ Thus, by Lemma \[lem1\], $\mu_{\ve,S} \left(B\left((x,t),d_A,r(\ve)\right)\right) \geq\gamma_{n,\ve}(U)\geq \frac{1}{2}\exp(-\ve^{-2}/2)$.
Having Lemma \[lem:meas2d\] we can estimate the entropy numbers by a version of the usual volumetric argument.
\[cor:entrestch\] For any $\ve>0$, $U\subset B_2^n$ and $S\subset \er^m$, $$\label{eq:entrestch}
N\left(U\times S,d_A,8\ve^2\Ex\beta_{A,S}(G_n)
+2\ve\sup_{x\in U}\beta_{A,S}(x)+8\ve\sup_{t\in S}\Ex\alpha_A(G_n\otimes t)\right)
\leq \exp(C\ve^{-2})$$ and for any $\delta>0$,
$$\begin{aligned}
\sqrt{\log N(U\times S,d_A,\delta)}\leq
C\bigg(&\delta^{-1}\left(\sup_{x\in U}\beta_{A,S}(x)+\sup_{t\in S}\Ex\alpha_A(G_n\otimes t)\right)\\
&+\delta^{-1/2}(\Ex\beta_{A,S}(G_n))^{1/2}\bigg).\end{aligned}$$
Let $r=4\ve^2\Ex\beta_{A,S}(G_n)+\ve\sup_{x\in U}\beta_{A,S}(x)+4\ve\sup_{t\in S}\Ex\alpha_A(G_n\otimes t)$ and $N=N(U\times S,d_A,2r)$. Then there exist points $(x_i,t_i)_{i=1}^N$ in $U\times S$ such that $d_A((x_i,t_i),(x_j,t_j))> 2r$. To show we consider two cases.
If $\ve>2$ then $$\begin{aligned}
2r
&\geq 4\sup_{x\in U}\beta_{A,S}(x)\geq 4\sup_{(x,t)\in U\times S}\Ex\left|\sum_{ijk}a_{ijk}g_it_jx_k\right|\\
&=4\sqrt{\frac{2}{\pi}}\sup_{(x,t)\in U\times S}\left(\sum_{i}\left(\sum_{jk}a_{ijk}t_jx_k\right)^2\right)^{1/2}\geq\diam(U\times S,d_A)\end{aligned}$$ so $N=1\leq \exp(C\ve^{-2})$.
If $\ve<2$, note that the balls $B((x_i,t_i),d_A,r)$ are disjoint and, by Lemma \[lem:meas2d\], each of these balls has $\mu_{\ve,S}$ measure at least $\frac{1}{2}\exp(-\ve^{-2}/2)\geq\exp(-5\ve^{-2})$. On the other hand we obviously have $\mu_{\ve,S}(\er^n\times S)\leq \exp(C\ve^{-2})$. Comparing the upper and lower bounds on $\mu_{\ve,S}(\er^n\times S)$ gives in this case.
The second estimate from the assertion is an obvious consequence of the first one.
\[Dud\] The classical Dudley’s bound on suprema of Gaussian processes (see e.g., [@delaPenaGine Corollary 5.1.6]) gives $$\Ex\sup_{\|x\|_2\leq 1, t\in T}\left|\sum_{ijk}a_{ijk}g_ix_jt_k\right|
\leq C\int_{0}^{\diam(B_2^n\times T,d_A)}\sqrt{\log N(B_2^n\times T,d_A,\delta)}d\delta.$$ Observe that $$\begin{aligned}
\int_{0}^{\diam(B_2^n\times T,d_A)}\delta^{-1/2}(\Ex\beta_{A,T}(G_n))^{1/2}d\delta
&=2\sqrt{\diam(B_2^n\times T,d_A)\Ex\beta_{A,T}(G_n)}
\\
&\leq\frac{1}{\sqrt{p}}\Ex\beta_{A,T}(G_n)+\sqrt{p}\diam(B_2^n\times T,d_A)\end{aligned}$$ appears on the right hand side of . Unfortunately the other term in the estimate of $\log^{1/2} N(B_2^n\times T,d_A,\delta)$ is not integrable. The remaining part of the proof is devoted to improve on Dudley’s bound.
We will now continue along the lines of [@LaAoP]. We will need in particular to partition the set $T$ into smaller pieces $T_i$ such that $\sup_{t,s\in T_i}\Ex\alpha_A(G_n\otimes (t-t'))$ is small on each piece. To this end we apply the following Sudakov-type estimate for chaoses, derived by Talagrand ([@Ta_ch] or [@Ta_book Section 8.2]).
\[thm:minchaos2d\] Let $\mathcal{A}$ be a subset of $n$ by $n$ real valued matrices and $d_2$, $d_{\infty}$ be distances associated to the Hilbert-Schmidt and operator norms respectively. Then $$\ve\log^{1/4} N(\mathcal{A},d_2,\ve)\leq C\Ex\sup_{a\in \mathcal{A}}\sum_{ij}a_{ij}g_ig_j'
\quad \mbox{ for } \ve>0$$ and $$\ve\log^{1/2} N(\mathcal{A},d_2,\ve)\leq C\Ex\sup_{a\in \mathcal{A}}\sum_{ij}a_{ij}g_ig_j'
\quad \mbox{ for } \ve>C\sqrt{\mathrm{diam}(\mathcal{A},d_{\infty})\Ex\sup_{a\in \mathcal{A}}\sum_{ij}a_{ij}g_ig_j'}.$$
To make the notation more compact let for $T\subset \er^m$ and $V\subset \er^n\times\er^m$, $$\begin{aligned}
s_A(T)&:=\Ex\beta_{A,T}(G_n)=\Ex\sup_{t\in T}\left|\sum_{ijk}a_{ijk}g_ig'_jt_k\right|,
%%\quad \mbox{ for } T\subset \er^m,
\\
F_A(V)&:=\Ex\sup_{(x,t)\in V}\sum_{ijk}a_{ijk}g_ix_jt_k
\\
\Delta_{A,\infty}(T)&:=\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1, t,t'\in T}
\left|\sum_{ijk}a_{ijk}x_iy_j(t_k-t'_k)\right|,
\\
\Delta_A(V)&:=\mathrm{diam}(V,d_A)=\sup_{(x,t),(x',t')\in V}\alpha_A(x\otimes t-x'\otimes t').
%%\quad \mbox{ for } V\subset \er^n\times\er^m,\end{aligned}$$
\[cor:decomp1\] Let $T$ be a subset of $\er^m$. Then for any $r>0$ there exists a decomposition $T-T=\bigcup_{i=1}^N T_i$ such that, $N\leq e^{Cr}$ and $$\sup_{t,t'\in T_i}\Ex\alpha_A(G_n\otimes (t-t'))\leq
\min\left\{r^{-1/4}s_A(T),r^{-1/2}s_A(T)+C\sqrt{s_A(T)\Delta_{A,\infty}(T)}\right\}.
%%r^{-1/4}\Ex\sup_{t\in T}\sum_{ijk}a_{ijk}g_ig'_jt_k.$$
We use Theorem \[thm:minchaos2d\] with $\mathcal{A}=\{(\sum_{k}a_{ijk}t_k)_{ij}\colon\ t\in T-T\}$. It is enough to observe that $$\Ex\sup_{b\in \mathcal{A}}\left|\sum_{ij}b_{ij}g_ig_j'\right|=s_A(T-T)\leq 2s_A(T), \quad
\mathrm{diam}(\mathcal{A},d_{\infty})= 2\Delta_{A,\infty}(T)$$ and $$\Ex \alpha_A(G_n\otimes t-t') \leq \left\|\left(\sum_k a_{ijk}(t_k-t_k')\right)_{ij}\right\|_{HS}.$$
On the other hand the dual Sudakov minoration (cf. formula (3.15) in [@LT]) yields the following
\[cor:decomp2\] Let $U$ be a subset of $B_2^n$. Then for any $r>0$ there exists a decomposition $U=\bigcup_{i=1}^N U_i$ such that $N\leq e^{Cr}$ and $$\sup_{x,x'\in U_i}\beta_{A,T}(x-x')\leq r^{-1/2}s_A(T).
%%\Ex\beta_{A,T}(G_n).$$
Putting the above two corollaries together with Corollary \[cor:entrestch\] we get the following decomposition of subsets $B_2^n\times T$.
\[cor:maindecomp\] Let $V\subset \er^n\times \er^m$ be such that $V-V\subset B_2^n\times (T-T)$. Then for $r\geq 1$ we may find a decomposition $V=\bigcup_{i=1}^N((x_i,t_i)+V_i)$ such that $N\leq e^{Cr}$ and for each $1\leq i\leq N$,\
i) $(x_i,t_i)\in V$, $V_i-V_i\subset V-V$, $V_i\subset B_2^n\times (T-T)$,\
ii) $\sup_{(x,t)\in V_i}\beta_{A,T}(x)\leq r^{-1/2}s_A(T)$,\
iii) $\sup_{(x,t)\in V_i}\Ex\alpha_A(G_n\otimes t)\leq
\min\left\{r^{-1/4}s_A(T),r^{-1/2}s_A(T)+C\sqrt{s_A(T)\Delta_{A,\infty}(T)}\right\}$,\
iv) $\Delta_A(V_i)\leq \min\left\{r^{-3/4}s_A(T),r^{-1}s_A(T)+r^{-1/2}\sqrt{s_A(T)\Delta_{A,\infty}(T)}\right\}$.
The assertion is invariant under translations of the set $V$ thus we may assume that $(0,0)\in V$ and so $V\subset V-V\subset B_2^n\times (T-T)$. By Corollaries \[cor:decomp1\] and \[cor:decomp2\] we may decompose $B_2^n=\bigcup_{i=1}^{N_1}U_i$, $T-T=\bigcup_{i=1}^{N_2} T_i$ in such a way that $N_1,N_2\leq e^{Cr}$ and $$\begin{aligned}
\sup_{x,x'\in U_i}\beta_{A,T}(x-x')&\leq r^{-1/2}s_A(T),
\\
\sup_{t,t'\in T_i}\Ex\alpha_A(G_n\otimes (t-t'))&\leq\min\left\{r^{-1/4}s_A(T),r^{-1/2}s_A(T)+C\sqrt{s_A(T)\Delta_{A,\infty}(T)}\right\}.\end{aligned}$$ Let $V_{ij}:=V\cap (U_i\times T_j)$. If $V_{ij}\neq \emptyset$ we take any point $(x_{ij},y_{ij})\in V_{ij}$ and using Corollary \[cor:entrestch\] with $\ve=r^{ -1/2}/C$ we decompose $$V_{ij}-(x_{ij},y_{ij})=\bigcup_{k=1}^{N_3}V_{ijk}$$ in such a way that $N_3\leq e^{Cr}$ and $$\begin{aligned}
&\Delta_A(V_{ijk})
\\
&\leq \frac{1}{C} \Bigg(r^{-1}s_A(T)+r^{-1/2} \sup_{x'\in U_{i}} \beta_{A,T}(x'-x_{ij})
+r^{-1/2}\sup_{y'\in T_j} \Ex \alpha_A (G_n\otimes (y'-y_{ij})) \Bigg)
%\\
%&\leq \frac{1}{C} \left(r^{-1}\Ex\beta_{A,T}(G_n)+r^{-1/2} \sup_{x,x'\in U_i}\beta_{A,T}(x-x')+r^{-1/2}\sup_{t,t'\in T_i}\Ex\alpha_A(G_n\otimes (t-t')) \right)
\\
&\leq \min\left\{r^{-3/4}{ s_A(T)},r^{-1}s_A(T)+r^{-1/2}\sqrt{s_A(T)\Delta_{A,\infty}(T)}\right\}.\end{aligned}$$ The final decomposition is obtained by relabeling of the decomposition $V = \bigcup_{ijk} ((x_{ij},y_{ij}) + V_{ijk})$.
\[rem:decomp\] We may also use a trivial bound in iii): $$\sup_{(x,t)\in V_i}\Ex\alpha_A(G_n\otimes t)\leq \sup_{t,t'\in T}\Ex\alpha_A(G_n\otimes (t-t'))
\leq 2\sup_{t\in T}\Ex\alpha_A(G_n\otimes t),$$ this will lead to the following bound in iv): $$\Delta_A(V_i)\leq r^{-1}s_A(T)+r^{-1/2}\sup_{t\in T}\Ex\alpha_A(G_n\otimes t).$$
\[sud:decomp\] By using Sudakov minoration instead of Theorem \[thm:minchaos2d\] we may decompose the set $T=\bigcup_{i=1}^N T_i$, $N\leq \exp(Cr)$ in such a way that $$\forall_{i \leq N} \sup_{t,t' \in T_i}\Ex\alpha_A(G_n\otimes (t-t'))\leq r^{-1/2}\Ex \sup_{t \in T} \sum_{ijk} a_{ijk}g_{ij}t_k.$$ This will lead to the following bounds in iii) and iv): $$\begin{aligned}
\sup_{(x,t)\in V_i}\Ex\alpha_A(G_n\otimes t) &\leq r^{-1/2} \Ex \sup_{t \in T} \sum_{ijk} a_{ijk}g_{ij}t_k \\
\Delta_A(V_i)&\leq r^{-1} \left( \Ex \sup_{t \in T} \sum_{ijk} a_{ijk}g_{ij}t_k+s_A(T) \right).\end{aligned}$$
\[lem:translate\] Let $V$ be a subset of $B_2^n\times (T-T)$. Then for any $(y,s)\in \er^n\times \er^m$ we have $$F_A(V+(y,s))\leq F_A(V)+2\beta_{A,T}(y)+C\Ex\alpha_A(G_n\otimes s).$$
We have $$F_A(V+(y,s))
\leq F_A(V)+\Ex\sup_{(x,t)\in V}\sum_{ijk}a_{ijk}g_iy_jt_k
+\Ex\sup_{(x,t)\in V}\sum_{ijk}a_{ijk}g_ix_js_k.$$ Obviously, $$\Ex\sup_{(x,t)\in V}\sum_{ijk}a_{ijk}g_iy_jt_k
\leq\Ex \sup_{t,t'\in T}\left|\sum_{ijk}a_{ijk}g_iy_j(t_k-t'_k)\right|\leq 2\beta_{A,T}(y).$$ Moreover, $$\begin{aligned}
\Ex\sup_{(x,t)\in V}\sum_{ijk}a_{ijk}g_ix_js_k
& \leq \left(\Ex\sup_{x\in B_2^n}\left|\sum_{ijk}a_{ijk}g_ix_js_k\right|^2\right)^{1/2}=\left(\sum_{ij}\left(\sum_k a_{ijk}s_k\right)^2\right)^{1/2}\\
&=(\Ex\alpha_A(G_n\otimes s)^2)^{1/2}\leq C\Ex\alpha_A(G_n\otimes s),\end{aligned}$$ where in the second inequality we used the comparison of moments of Gaussian variables [@delaPenaGine Theorem 3.2.10].
\[osz:main\] For any nonempty finite set $T$ in $\er^m$ and $p\geq 1$ we have $$\begin{gathered}
F_A(B_2^n\times T)\leq C\Bigg(\frac{\log (ep)}{\sqrt{p}}s_A(T)+\sup_{\|x\|_2\leq 1}\beta_{A,T}(x)\\
+\sup_{t\in T}\Ex\alpha_A(G_n\otimes t)+\log (ep)\sqrt{p}\Delta_A(B_2^n\times T)\Bigg), \label{wzor1}\end{gathered}$$ $$\begin{gathered}
F_A(B_2^n\times T)\leq C\Bigg(\frac{1}{\sqrt{p}}s_A(T)+\sup_{\|x\|_2\leq 1}\beta_{A,T}(x)\\
+\log(ep)\sup_{t\in T}\Ex\alpha_A(G_n\otimes t)+\sqrt{p}\Delta_A(B_2^n\times T)\Bigg), \label{wzor2}\end{gathered}$$ $$\begin{gathered}
F_A(B_2^n\times T)\leq C\Bigg(\frac{1}{\sqrt{p}}s_A(T)+\frac{1}{\sqrt{p}}\Ex \sup_{t \in T} \sum a_{ijk} g_{ij} t_k+\sup_{\|x\|_2\leq 1}\beta_{A,T}(x) \\
+\sup_{t\in T}\Ex\alpha_A(G_n\otimes t)+\sqrt{p}\Delta_A(B_2^n\times T)\Bigg). \label{wzor3}\end{gathered}$$
First we prove Let $l_0\in \ensuremath{\mathbb N}$ be such that $2^{l_0-1}\leq p< 2^{l_0}$. Define $$\Delta_0:=\Delta_A(B_2^n\times T),\quad \tilde{\Delta}_0:=\sup_{x\in B_2^n}\beta_{A,T}(x)
+\sup_{t\in T}\Ex\alpha_A(G_n\otimes t),$$ $$\Delta_l=2^{-3l/4}p^{-3/4}s_A(T),\quad \tilde{\Delta}_l=2^{-l/4}p^{-1/4}s_A(T), \quad l>l_0.$$ and for $1\leq l\leq l_0$, $$\begin{aligned}
\Delta_l&:=2^{-l}p^{-1}s_A(T)+2^{-l/2}p^{-1/2}\sqrt{s_A(T)\Delta_{A,\infty}(T)},\\
\tilde{\Delta}_l&:=2^{-l/2}p^{-1/2}s_A(T)+C\sqrt{s_A(T)\Delta_{A,\infty}(T)}.\end{aligned}$$
Let for $l=0,1,\ldots$ and $m=1,2,\ldots$ $$\begin{aligned}
c(l,m):=\sup\big\{F_A(V)\colon\
&V-V\subset B_2^n\times (T-T),\#V\leq m,
\\
&\Delta_A(V)\leq \Delta_l,
\sup_{(x,t)\in V}(\beta_{A,T}(x)+\Ex\alpha_A(G_n\otimes t))\leq 2\tilde{\Delta}_l\big\}.\end{aligned}$$
Obviously $c(l,1)=0$. We will show that for $m>1$ and $l\geq 0$ we have $$\label{eq:iteration}
c(l,m)\leq c(l+1,m-1)+C\left(2^{l/2}\sqrt{p}\Delta_l+\tilde{\Delta}_l\right).$$ To this end take any set $V$ as in the definition of $c(l,m)$ and apply to it Corollary \[cor:maindecomp\] with $r=2^{l+1}p$ to obtain decomposition $V=\bigcup_{i=1}^N ((x_i,t_i)+V_i)$. We may obviously assume that all $V_i$ have smaller cardinality than $V$. Conditions i)-iv) from Corollary \[cor:maindecomp\] easily imply that $F_A(V_i)\leq c(l+1,m-1)$.
Gaussian concentration (cf. [@LaAoP Lemma 3]) yields $$F_A(V)=F_A\left(\bigcup_{i}((x_i,t_i)+V_i)\right)\leq
C\sqrt{\log N}\Delta_A(V)+\max_{i}F_A((x_i,t_i)+V_i).$$ Estimate follows since $$\sqrt{\log N}\Delta_A(V)\leq C2^{l/2}\sqrt{p}\Delta_l$$ and for each $i$ by Lemma \[lem:translate\] we have (recall that $(x_i,t_i)\in V$) $$F_A((x_i,t_i)+V_i)\leq F_A(V)+2\beta_{A,T}(x_i)+C\Ex\alpha_A(G_n\otimes t_i)\leq
c(l+1,m-1)+C\tilde{\Delta}_l.$$
Hence $$\begin{aligned}
c(0,m)
&\leq C\left(\sum_{l=0}^{\infty} 2^{l/2}\sqrt{p}\Delta_l+\sum_{l=0}^\infty \tilde{\Delta}_l\right)
\\
&\leq C\left(\sqrt{p}\Delta_0+\tilde{\Delta}_0 +\frac{1}{\sqrt{p}}s_A(T)+l_0\sqrt{s_A(T)\Delta_{A,\infty}(T)}+
2^{-l_0/4}p^{-1/4}s_A(T)\right).\end{aligned}$$
Since $\log_2 p < l_0 \leq \log_2 p +1$ and $\sqrt{s_A(T)\Delta_{A,\infty}(T)}\leq \frac{1}{\sqrt{p}}s_A(T)+\sqrt{p}\Delta_{A,\infty}(T)$ and clearly $\Delta_{A,\infty}(T)\leq \Delta_A(B_2^n\times T)$ we get for all $m \ge 1$, $$\begin{gathered}
c(0,m)\leq C\Bigg(\frac{\log (ep)}{\sqrt{p}}s_A(T)+\sup_{\|x\|_2\leq 1}\beta_{A,T}( x)\\
+\sup_{t\in T}\Ex\alpha_A(G_n\otimes t)+\log (ep)\sqrt{p}\Delta_A(B_2^n\times T)\Bigg).\end{gathered}$$
To conclude the proof of it is enough to observe that $$F_A(B_2^n\times T)=2F_A\left(\frac{1}{2}B_2^n\times T\right)\leq 2\sup_{m\geq 1}c(0,m).$$
The proofs of and are the same as the proof of . The only difference is that for $1\leq l\leq l_0$ we change the definitions of $\Delta_l$, $\tilde{\Delta}_l$ and we use Remarks \[rem:decomp\] and \[sud:decomp\] respectively. In the first case we take $$\begin{aligned}
\Delta_l&:=2^{-l}p^{-1}s_A(T)+2^{-l/2}p^{-1/2}\sup_{t \in T} \Ex \alpha_A(G_n \otimes t) \\
\tilde{\Delta}_l&:=2^{-l/2}p^{-1/2}s_A(T)+\sup_{t \in T} \Ex \alpha_A(G_n \otimes t),\end{aligned}$$ while in the second $$\begin{aligned}
\Delta_l:=2^{-l}p^{-1}\left(s_A(T)+\Ex \sup_{t\in T}\sum_{ijk} a_{ijk} g_{ij}t_k\right)\\
\tilde{\Delta}_l=2^{-l/2}p^{-1/2}\left( s_A(T)+\Ex \sup_{t\in T}\sum_{ijk} a_{ijk} g_{ij}t_k\right).\end{aligned}$$
Proofs of main results
======================
By Lemmas \[lem:lower1\], \[lem:diag\] and Proposition \[prop:red\] we need only to establish - with $\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\|_p$ replaced by $\sqrt{p}\Ex\sup_{\|x\|_2\leq 1}\|\sum_{i\neq j}a_{ij}g_ix_j\|$. We may assume that $F=\er^m$ and $a_{ii}=0$, so taking for $T$ the unit ball in the dual space $F^*$ we have $$\left\|\sum_{i\neq j}a_{ij}g_ix_j\right\|=\sup_{t\in T}\sum_{ijk}a_{ijk}g_ix_jt_k.$$ Then, using the notation introduced in Section \[sec:supgauss\], $$\Ex\sup_{\|x\|_2\leq 1}\left\|\sum_{i\neq j}a_{ij}g_ix_j\right\|=F_A(B_2^n\times T),\quad
\Ex\left\|\sum_{i\neq j}a_{ij}g_ix_j\right\|=\beta_{A,T}(x),$$ $$\sup_{\|(x_{ij})\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_{ij}\right\|=\sup_{t\in T}(\Ex\alpha_A^2(G_n\otimes t))^{1/2}
\sim \sup_{t\in T}\Ex\alpha_A(G_n\otimes t),$$ $$\sup_{\|x\|_2\leq 1,\|y\|_2\leq 1}\left\|\sum_{ij}a_{ij}x_iy_j\right\|\sim \Delta_A(B_2^n\times T),$$ $$\Ex\left\|\sum_{i\neq j}a_{ij}g_{ij}\right\|=\Ex\sup_{t\in T}\sum_{ijk}a_{ijk}g_{ij}t_k\quad \mbox{and}\quad
\Ex\left\|\sum_{i\neq j}a_{ij}g_{i}g_j\right\|\sim s_A(T),$$ where the last estimate follows by decoupling. We conclude the proof invoking Proposition \[osz:main\].
Let $S=\|\sum_{ij}a_{ij}(g_ig_j-\delta_{ij})\|$. By the Paley-Zygmund inequality (see e.g., [@delaPenaGine Corollary 3.3.2]) and comparison of moments of Gaussian quadratic forms (see e.g. [@delaPenaGine Theorem 3.2.10]) we have for $p\geq 1$, $$\Pr\left(S\geq \frac{1}{2}(\Ex|S|^p)^{1/p}\right)=\Pr\left(|S|^p\geq \frac{1}{2^p}\Ex S^p\right)
\geq \left(1-\frac{1}{2^p}\right)^2\frac{(\Ex S^p)^2}{\Ex S^{2p}}\geq C_1^{-2p}.$$ Thus, the lower bound on tails of $S$ follows by Proposition \[prop:lower2d\] and substitution $p=1+C\min\{t^2/U^2,t/V\}$.
To derive the upper bound we use Theorem \[thm:upper2d2\], estimate $\Pr(S\geq e\|S\|_p)\leq e^{-p}$ for $p\geq 1$ and make an analogous substitution.
Recall that for $r > 0$ the $\psi_r$-norm of a random variable $Y$ is defined as $$\|Y\|_{\psi_r} = \inf\Big\{a>0\colon \Ex\exp\Big(\Big(\frac{|Y|}{a}\Big)^r\Big) \le 2\Big\}.$$ (formally for $r < 1$ this is a quasi-norm, but it is customary to use the name $\psi_r$-norm for all $r$). By [@AW Lemma 5.4] if $k$ is a positive integer and $Y_1,\ldots,Y_n$ are symmetric random variables such that $\|Y\|_{\psi_{2/k}} \le M$, then $$\begin{aligned}
\label{eq:comparison-psi-r}
\left\|\sum_{i=1}^n a_i Y_i\right\|_p \le C_k M \left\|\sum_{i=1}^n a_i g_{i1}\cdots g_{ik}\right\|_p,\end{aligned}$$ where $g_{ik}$ are i.i.d. standard Gaussian variables (we remark that the lemma in [@AW] is stated only for $F = \er$ but its proof, based on contraction principle, works in any normed space).
To prove the theorem we will again establish a moment bound and then combine it with Chebyshev’s inequality. Similarly as in the Gaussian setting we will treat the diagonal and off-diagonal part separately. Let $\varepsilon_1,\ldots,\varepsilon_n$ be a sequence of i.i.d. Rademacher variables independent of $X_i$’s. For $p\ge 1$ we have $$\left\|\sum_i a_{ii}(X_i^2-\Ex X_i^2)\right\|_p \le 2 \left\|\sum_i a_{ii}\varepsilon_i X_i^2\right\|_p \le C \alpha^2 \left\|\sum_i a_{ii} g_{i1}g_{i2}\right\|_p,$$ where in the first inequality we used symmetrization and in the second one together with the observation $\|\varepsilon_i X_i^2\|_{\psi_1} \le C\alpha^2$ (which can be easily proved by integration by parts).
Now by [@AL Lemma 9.5], $$\begin{aligned}
\left\|\sum_i a_{ii} g_{i1}g_{i2}\right\|_p \le C\left\|\sum_i a_{ii}\varepsilon_i g_i^2 \right\|_p \le 2C \left\|\sum_i a_{ii}(g_i^2-1) \right\|_p,\end{aligned}$$ and thus $$\begin{aligned}
\label{eq:diagonal-subgaussian}
\left\|\sum_i a_{ii}(X_i^2-\Ex X_i^2)\right\|_p \leq C \alpha^2 \left\|\sum_i a_{ii}(g_i^2-1) \right\|_p.\end{aligned}$$ The estimate of the off-diagonal part is analogous, the only additional ingredient is decoupling. Denoting $(X_i')_{i=1}^n$ an independent copy of the sequence $(X_i)_{i=1}^n$ and by $(\varepsilon_i)_{i=1}^n$, $(\varepsilon_i')_{i=1}^n$ (resp. $(g_i)_{i=1}^n$, $(g_i')_{i=1}^n$ ) independent sequences of Rademacher (resp. standard Gaussian) random variables, we have $$\begin{aligned}
\left\|\sum_{i\neq j} a_{ij} X_iX_j\right\|_p &\sim \left\|\sum_{i\neq j} a_{ij} X_iX_j'\right\|_p \sim \left\|\sum_{i\neq j} a_{ij} \varepsilon_iX_i\varepsilon_i'X_j'\right\|_p \nonumber \\
&\le C\alpha^2\left\|\sum_{i\neq j} a_{ij} g_ig_j'\right\|_p \sim \alpha^2 \left\|\sum_{i\neq j} a_{ij} g_ig_j\right\|_p,\label{eq:off-diagonal-subgaussian}\end{aligned}$$ where in the first and last inequality we used decoupling, the second one follows from iterated conditional application of symmetrization inequalities and the third one from iterated conditional application of (note that by integration by parts we have $\|\varepsilon_i X_i\|_{\psi_2} \le C\alpha$).
Combining inequalities and with Lemma \[lem:lower1\] we obtain $$\left\|\sum_{ij} a_{ij} (X_iX_j - \Ex X_iX_j)\right\|_p \le C\alpha^2 \left\|\sum_{ij} a_{ij}(g_ig_j - \delta_{ij})\right\|_p.$$ To finish the proof of the theorem it is now enough to invoke moment estimates of Theorem \[thm:upper2d2\] and use Chebyshev’s inequality in $L_p$.
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---
abstract: 'Domelike magnetic-flux-density distributions previously have been observed experimentally and analyzed theoretically in superconducting films with edges, such as in strips and thin plates. Such flux domes have been explained as arising from a combination of strong geometric barriers and weak bulk pinning. In this paper we predict that, even in films with bulk pinning, flux domes also occur when vortices and antivortices are produced far from the film edges underneath current-carrying wires, coils, or permanent magnets placed above the film. Vortex-antivortex pairs penetrating through the film are generated when the magnetic field parallel to the surface exceeds $H_{c1}+K_c$, where $H_{c1}$ is the lower critical field and $K_c = j_c d$ is the critical sheet-current density (the product of the bulk critical current density $j_c$ and the film thickness $d$). The vortices and antivortices move in opposite directions to locations where they join others to create separated vortex and antivortex flux domes. We consider a simple arrangement of a pair of current-carrying wires carrying current $I_0$ in opposite directions and calculate the magnetic-field and current-density distributions as a function of $I_0$ both in the bulk-pinning-free case ($K_c= 0$) and in the presence of bulk pinning, characterized by a field-independent critical sheet-current density ($K_c > 0$).'
author:
- 'John R. Clem'
- Yasunori Mawatari
title: Flux domes in superconducting films without edges
---
Introduction\[Sec\_Intro\]
==========================
The hysteretic penetration of magnetic flux into a superconductor has long been a subject of experimental and theoretical interest. One phenomenon that has often been observed in flat samples in an increasing perpendicular applied magnetic field is a domelike magnetic-flux-density distribution centered in the middle of the sample, surrounded by a superconducting flux-free zone. This phenomenon has been investigated experimentally and interpreted theoretically in both type-I superconductors[@DeSorbo64; @Baird64; @Haenssler67; @Huebener72; @Clem73; @Provost74; @Fortini76; @Fortini80; @Castro99] and weak-pinning type-II superconductors.[@Indenbom94; @Schuster94; @Zeldov94; @Maksimov95a; @Maksimov95b; @Morozov96; @Benkraouda96; @Doyle97; @Brandt99a; @Brandt99b; @Mawatari03] In these previous investigations, the flux domes have been explained as arising from an energy barrier of geometric origin at the edge of the sample. Once this barrier is overcome, flux tubes or vortices escape from the edge and are driven to the middle by Meissner screening currents, which flow on the sample’s surface.
In this paper we consider a type-II superconducting film without edges and predict that when it is subjected to local magnetic fields produced by current-carrying wires above the sample, vortex and antivortex magnetic-flux domes are produced in the film when the currents in the wires are sufficiently large. When bulk pinning is weak, we predict that the vortex and antivortex domes are separated from each other, but as the bulk pinning increases, the space between the vortex and antivortex domes shrinks to zero.
To investigate these effects in an easily calculable geometry, we consider in Sec. \[Sec\_SC-wire\] a simple model in which the local magnetic fields are produced by a pair of infinitely long straight wires. The resulting two-dimensional geometry allows us to calculate all the magnetic-field and sheet-current distributions analytically. In Sec. \[Sec\_Meissner\] we discuss the Meissner-state response of the film before any penetration of vortices into the film, and in Sec. \[Sec\_Vortices\] we discuss the distributions produced by vortices and antivortices that have penetrated through the film thickness. We then present the magnetic-field and sheet-current distributions associated with the flux domes, both in the absence of bulk pinning \[Sec. \[Sec\_Kc=0\]\] and in the presence of weak \[Sec. \[Sec\_Kc>0\]\] and strong \[Sec. \[Sec\_Kc\_strong\]\] bulk pinning. In Sec. \[Sec\_conclusion\] we summarize our results, discuss the generality of the predicted effects in more easily realizable experimental configurations, consider similar phenomena in type-I superconductors, and discuss possible extensions of this work. Calculations of screening effects in the Meissner state are presented in Appendix A, and derivations of the complex field and complex potential are given in Appendix B.
Superconducting films and linear wires\[Sec\_SC-wire\]
======================================================
We consider a simple geometry in which vortex and antivortex flux domes are produced far from the film edges. For simplicity we consider an infinite type-II superconducting film and a pair of infinitely long current-carrying wires, as shown in Fig. 1.
![Wire 1 at $(x,y)=(0,y_1)$ carries current $I_1= I_0$ in the $z$ direction and wire 2 at $(x,y)=(0,y_2)$ carries current $I_2 = -I_0$ above an infinite type-II superconducting film in the $xz$ plane.[]{data-label="fig1"}](Fig01.eps){width="8cm"}
A wire, carrying current $I_1=I_0$ parallel to the $z$ axis, is situated at $(x,y)=(0,y_1)$, where $y_1>0$. The return current $I_2 = -I_0$ is carried by a second wire at $(x,y)=(0,y_2)$, where $y_2>y_1$. The wire radius $r_w$ is assumed to be much smaller than either $y_1$ or the wire separation $y_2-y_1$. A superconducting film, infinitely extended in the $xz$ plane, is situated at $-d/2<y<d/2$, where the film thickness $d$ is larger than the London penetration depth $\lambda$ and $d \ll y_1$. As discussed in Appendix A, these conditions guarantee that when the film is in the Meissner state, the magnetic field below the film is negligibly small.
Flux pinning in the film is characterized by the critical current density $j_c$, which is assumed to be constant (independent of magnetic field), as in Bean’s critical state model, [@Bean62] and isotropic (independent of vortex direction) However, the relevant physical quantity here is the critical sheet-current density $K_c = j_c d$. Since we are interested in the case for which $d \ll y_1$, in the following we ignore the finite thickness of the film, as this simplification allows us to obtain simple analytic expressions for the magnetic-field and current-density distributions.
We introduce the complex field ${{\cal H}}(\zeta)=H_y(x,y)+iH_x(x,y)$, which is an analytic function of $\zeta=x+iy$ except for poles at $\zeta = \zeta_1 =i
y_1$ and $\zeta = \zeta_2 =
i y_2$ and a branch cut at $y = 0$. The Biot-Savart law for the complex field is given by $${{\cal H}}(\zeta)= {{\cal H}}_0(\zeta) +\frac{1}{2\pi}
\int_{-\infty}^{+\infty}du \frac{K_z(u)}{\zeta -u} ,
\label{Biot-Savart}$$ where $${{\cal H}}_0(\zeta) =\frac{I_0}{2\pi} \frac{1}{\zeta-iy_1} -\frac{I_0}{2\pi}
\frac{1}{\zeta-iy_2} ,
\label{H0}$$ is the complex field arising from the pair of wires alone (see Fig. 1) and $K_z(x)$ is the sheet current in the film. The complex potential describing the field generated by the wires alone, defined by ${\cal G}_0(\zeta)=\int_0^\zeta{{\cal H}}_0(\zeta')d\zeta'$, is $${\cal G}_0(\zeta)=\frac{I_0}{\pi}\ln\Big(\frac{1+i\zeta/y_1}{1+i\zeta/y_2}\Big),$$ and the contour lines of the real part of ${\cal G}_0(\zeta)$ correspond to the magnetic field lines of ${{\cal H}}_0(\zeta)$. At the upper ($\zeta=x+i\epsilon$) and lower ($\zeta=x-i\epsilon$) surfaces of the superconducting film, (where we take $\epsilon = d/2$ to be a positive infinitessimal, since $d \ll y_1$,) the perpendicular and parallel magnetic fields $H_y(x,0)={\mbox{Re}\,}{{\cal H}}(x\pm i\epsilon)$ and $H_x(x,\pm \epsilon)={\mbox{Im}\,}{{\cal H}}(x\pm
i\epsilon)$ are obtained from Eq. as $$\begin{aligned}
H_y(x,0) &=& H_{0y}(x,0) +\frac{\rm P}{2\pi}
\int_{-\infty}^{+\infty}du \frac{K_z(u)}{x-u} ,
\label{Hy_x+-i0}\\
H_x(x,\pm \epsilon) &=& H_{0x}(x,0) \mp K_z(x)/2 ,
\label{Hx_x+-i0}\end{aligned}$$ where $H_{0y}(x,0) = {\mbox{Re}\,}{{\cal H}}_0(x)$, $H_{0x}(x,0) = {\mbox{Im}\,}{{\cal H}}_0(x)$, and P denotes the principal value integral. The complex potential is defined by ${\cal
G}(\zeta)=\int_{i\epsilon}^\zeta {{\cal H}}(\zeta')d\zeta'$, and the contour lines of the real part of ${\cal G}(\zeta)$ correspond to the magnetic field lines of ${{\cal H}}(\zeta)$.
Meissner-state response\[Sec\_Meissner\]
----------------------------------------
We first consider the magnetic-field distribution when the film is in the (vortex-free) Meissner state and wires 1 and 2 carry dc currents $I_1=I_0 > 0$ and $I_2 = -I_0 < 0$ after monotonically increasing in magnitude from zero. As discussed in Appendix A, when the current $I_0$ is small and the thickness $d$ is larger than $\lambda$, the magnetic field is practically zero below the film, where $y={\mbox{Im}\,}\zeta <0$. The field distribution above the film can be obtained by adding to ${{\cal H}}_0(\zeta)$ and ${\cal G}_0(\zeta)$ the contributions ${{\cal H}}_I(\zeta)$ and ${\cal G}_I(\zeta)$ due to image wires at $\zeta = -\zeta_1 = -iy_1$ and $\zeta = -\zeta_2 = -iy_2$ carrying currents $-I_0$ and $+I_0$, respectively: $$\begin{aligned}
{{\cal H}}_I(\zeta)& =&-\frac{I_0}{2\pi} \frac{1}{\zeta+iy_1} +\frac{I_0}{2\pi}
\frac{1}{\zeta+iy_2} , \\
\label{HI}
{\cal
G}_I(\zeta)&=&\frac{I_0}{\pi}\ln\Big(\frac{1-i\zeta/y_2}{1-i\zeta/y_1}\Big).
\label{GI}\end{aligned}$$ The resulting complex field ${{\cal H}}_M(\zeta)={{\cal H}}_0(\zeta)+{{\cal H}}_I(\zeta)$ is $${{\cal H}}_M(\zeta)=
\begin{cases} \displaystyle
i \frac{I_0}{\pi}\Big(\frac{y_1}{\zeta^2+y_1^2} -\frac{y_2}{\zeta^2+y_2^2}\Big)
& \mbox{for } {\mbox{Im}\,}\zeta >0 , \\
0 & \mbox{for } {\mbox{Im}\,}\zeta <0 .
\end{cases}
\label{cH_linear}$$ The subscript $M$ is a reminder that this field describes the Meissner-state response to the applied field given in Eq. . Note that $$H_{Mx}(x,\epsilon)=
\frac{I_0}{\pi}\Big(\frac{y_1}{x^2+y_1^2} -\frac{y_2}{x^2+y_2^2}\Big).
\label{HMx}$$ As is usual using the method of images, $H_{Mx}(x,\epsilon) = 2 H_{0x}(x,0) =2
{\mbox{Im}\,}{{\cal H}}_0(x)$ \[see Eq. \]. The corresponding complex potential ${\cal G}_M(\zeta)=\int_{i\epsilon}^\zeta{\cal H}_M(\zeta')d\zeta'$ for ${\mbox{Im}\,}\zeta
>0$ is given by $$\begin{aligned}
{\cal G}_M(\zeta) =
i\frac{I_0}{\pi}\Big[\arctan\Big(\frac{\zeta}{y_1}\Big)
-\arctan\Big(\frac{\zeta}{y_2}\Big)\Big].
\label{cG_linear}\end{aligned}$$ We may take ${\cal G}_M(\zeta)=0$ for ${\mbox{Im}\,}\zeta <0.$
![Contour plot of the real part of the complex potential ${\cal G}_M(x+iy)$ vs $x$ and $y$ for $y_1 = 1$ and $y_2 = 2$. The contours correspond to magnetic field lines of the complex field ${{\cal H}}_M(x+iy)$ describing the Meissner-state response of the superconducting film to currents in the wires shown in Fig. 1. The contours near the wires at $\zeta = iy_1$ and $\zeta
=iy_2$ are nearly circular.[]{data-label="fig2"}](Fig02.eps){width="8cm"}
The perpendicular magnetic field and sheet-current density are thus given by $H_{My}(x,0)=0$ and $$K_{Mz}(x) = -\frac{I_0}{\pi}\Big( \frac{y_1}{x^2+y_1^2}-
\frac{y_2}{x^2+y_2^2}\Big) .
\label{KMz_linear}$$ The net current induced in the superconducting film is $\int_{-\infty}^{+\infty}K_{Mz}(x)dx =0$, as expected.
The maximum magnetic field parallel to the top surface of the film is $$H_{Mx}(0,\epsilon) = \frac{I_0}{\pi}\Big( \frac{1}{y_1}-
\frac{1}{y_2}\Big) ,
\label{HMx0eps}$$ and the maximum magnitude of the sheet-current density is $$|K_{Mz}(0)| = \frac{I_0}{\pi}\Big( \frac{1}{y_1}-
\frac{1}{y_2}\Big) .
\label{KMzmax}$$
![Parallel magnetic field $H_{Mx}(x,\epsilon)/H_{c1}$ vs $x$ (solid curve) in the Meissner state at the top surface of the superconducting film when $I_0=I_{c1}$, $H_{Mx}(0,\epsilon)=H_{c1}$, $y_1 = 1$, and $y_2 = 2$. The dashed curve shows the corresponding sheet-current density $K_{Mz}(x)/H_{c1}$ vs $x$. Note that $K_{Mz}(0) = -H_{c1}.$[]{data-label="fig3"}](Fig03.eps){width="8cm"}
We now define two critical currents as follows. Let $I_{c1}$ denote the value of $I_0$ at which $H_{Mx}(0,\epsilon)$ in Eq. reaches the lower critical field $H_{c1}$, $$I_{c1}= \frac{\pi H_{c1}y_1y_2}{y_2-y_1} ,
\label{Ic1}$$ and let $I_{c0}$ denote the value of $I_0$ at which $|K_{Mz}(0)|$ in Eq. reaches the critical sheet-current density $K_c$, $$I_{c0}= \frac{\pi K_{c}y_1y_2}{y_2-y_1}.
\label{Ic0}$$ Both of these critical currents play important roles in determining the details of how magnetic flux penetrates through the film. In the following sections we discuss in turn the cases for which $K_c$ = 0 ($I_{c0}=0$) and $K_c >
0$ ($I_{c0} > 0$).
Vortex-generated fields and currents\[Sec\_Vortices\]
-----------------------------------------------------
Consider a closely spaced row of vortices along the $z$ axis carrying magnetic flux $\Phi'$ per unit length up through the film. Ignoring spatial variation on the scale of the London penetration depth $\lambda$ or the intervortex spacing, the magnetic field in the space $y > 0$ ($y < 0$) appears as if produced by a line of positive (negative) magnetic monopoles. At a distance $r$ from the origin, the magnitude of the magnetic field is $h =
\Phi'/\mu_0 \pi r$. Expressing this result in terms of a complex magnetic field ${{\cal H}}_v(\zeta) = H_{vy}(x,y)+iH_{vx}(x,y)$ and extending it to a distribution of vortices or antivortices generating a magnetic field $H_{vy}(x,0)$ in the plane of the film, we see that the vortex-generated complex magnetic field can be expressed as $${{\cal H}}_v(\zeta)=\pm \frac{i}{\pi}\int_{-\infty}^{+\infty}du
\frac{H_{vy}(x,0)}{\zeta-u},
\label{Hv}$$ where $\zeta = x +iy$ and the upper (lower) sign in Eq. holds for $y
>0$ ($y<0$). The corresponding vortex-generated sheet-current density $K_{vz}(x) = H_{vx}(x-i\epsilon)-H_{vx}(x+i\epsilon)$ is $$K_{vz}(x)=-\frac{\rm P}{\pi}\int_{-\infty}^{+\infty}du
\frac{2H_{vy}(x,0)}{x-u},
\label{Kvz}$$ while the Biot-Savart law yields another relation between $K_{vz}(x)$ and $H_{vy}(x,0)$: $$2H_{vy}(x,0) = \frac{\rm P}{\pi}
\int_{-\infty}^{+\infty}du \frac{K_{vz}(u)}{x-u}.$$
In the following sections, the complex field can always be regarded as a linear superposition of the Meissner-state and vortex-generated complex fields: ${{\cal H}}(\zeta) = {{\cal H}}_M(\zeta) + {{\cal H}}_v(\zeta).$ The complex potential ${\cal G}(\zeta) = \int_{i\epsilon}^\zeta {{\cal H}}(\zeta') d\zeta'$ can be written as ${\cal G}(\zeta) ={\cal
G}_M(\zeta) +{\cal
G}_v(\zeta)$. Similarly, the sheet-current density can always be expressed as $K_z(x) = K_{Mz}(x) + K_{vz}(x)$.
Flux domes in the absence of bulk pinning\[Sec\_Kc=0\]
------------------------------------------------------
In bulk-pinning-free films ($K_c= 0$), the first vortex enters the film at $x = 0$, where the maximum magnetic field at the top surface is equal to the lower critical field $H_{c1}$.[@foot1] This occurs at the current $I_0 = I_{c1}$, given in Eq. (see Fig. 3). An initially tiny vortex loop expands in radius, and a portion of the loop is driven to the bottom of the film surface, where it annihilates, resulting in a separated vortex-antivortex pair. The vortex (antivortex) carries magnetic flux $\phi_0$ in the $+y$ ($-y$) direction, where $\phi_0 = h/2e$ is the superconducting flux quantum. Responding to the Lorentz force $K_{Mz}(x)
\phi_0$, the vortex moves in the $x$ direction until it comes to rest at the point $x = x_0 =
\sqrt{y_1y_2}$, where $K_{Mz}(x)$ = 0, as can be seen from Eq. and Fig. 3. The antivortex moves in the opposite direction and comes to rest at the point $x = -x_0$.
For increasing values of $I_0$ in the range $I_{c1} < I_0 < 2 I_{c1}$, the magnetic field distribution perpendicular to the film can be characterized as having a positive vortex-generated magnetic flux dome in the region $ b < x < a$, where $0 <b < x_0 < a$, and a negative antivortex-generated flux dome in the region $-a < x < -b$. The complex magnetic field ${{\cal H}}(\zeta) = {{\cal H}}_M(\zeta) +{{\cal H}}_v(\zeta)$ is given in Eq. but with $K_c = 0$. Subtracting the Meissner-state complex field ${{\cal H}}_M(\zeta)$, we obtain the following expression for the vortex-generated complex magnetic field, $${{\cal H}}_v(\zeta)\!\!=\!\!\frac{I_0}{2\pi} \Big\{ \frac{y_1[\mp i
+\phi(\zeta)/s_1]}{\zeta^2+y_1^2}
\! -\!\frac{y_2[\mp i
+\phi(\zeta)/s_2]}{\zeta^2+y_2^2}
\Big\} ,
\label{Hv0}$$ where $ s_j= \sqrt{(a^2+y_j^2)(b^2+y_j^2)}$, and the upper (lower) sign holds when $\zeta = x+iy$ is in the upper (lower) half plane. Writing $\phi(\zeta)=\phi_1(x,y)+i\phi_2(x,y)$, we find that $\phi_1(x,y)$ is an odd function of $x$ and an even function of $y$ \[$\phi_1(x,y)=-\phi_1(-x,y)=\phi_1(x,-y)$\], while $\phi_2(x,y)$ is an even function of $x$ and an odd function of $y$ \[$\phi_2(x,y)=\phi_2(-x,y)=-\phi_2(x,-y)$\]. Thus $H_{vy}(x,y)$ is an odd function of $x$ but an even function of $y$, while $H_{vx}(x,y)$ is an odd function of $y$ but an even function of $x$. Just above (below) the real axis, $$\phi(x \pm i\epsilon)=
\begin{cases} \displaystyle
\pm i \tilde \phi(x),
& |x|<b \mbox { or } |x| > a, \\
\tilde \phi(x), & b<|x|<a, \\
\end{cases}
\label{phixpm}$$ where $$\tilde \phi(x)=
\begin{cases} \displaystyle
\sqrt{(a^2-x^2)(b^2-x^2)},
& |x|<b , \\
{\rm sgn}(x) \sqrt{(a^2-x^2)(x^2-b^2)}, & b<|x|<a, \\
- \sqrt{(x^2-a^2)(x^2-b^2)}, & |x|>a .
\end{cases}
\label{tildephi}$$ From Eq. we obtain the following values of $H_{vy}(x,0)$, $H_{vx}(x,\epsilon)=-H_{vx}(x,-\epsilon),$ and $K_{vz}(x)=H_{vx}(x,-\epsilon)-H_{vx}(x,\epsilon)$, $$\begin{aligned}
H_{vy}(x,0) =
\begin{cases} \displaystyle
0, & |x|<b \mbox { or } |x| > a,\\
\frac{I_0}{2\pi}\Big(\frac{y_1}{s_1(x^2+y_1^2)} \\-
\frac{y_2}{s_2(x^2+y_2^2)}\Big)\tilde \phi(x), &
b<|x|<a,
\end{cases}
\label{Hvyx0_zero}\end{aligned}$$ $$\begin{aligned}
H_{vx}(x, \epsilon) =
\begin{cases} \displaystyle
-\frac{I_0}{2\pi}\Big(\frac{y_1[1\!-\! \tilde \phi(x)/s_1]}{x^2+y_1^2}
- \frac{y_2[1\!-\!\tilde \phi(x)/s_2]}{x^2+y_2^2}\Big),\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\
|x|<b \mbox
{ or } |x| > a,\\
-\frac{I_0}{2\pi}\Big(\frac{y_1}{x^2+y_1^2}
- \frac{y_2}{x^2+y_2^2}\Big), b<|x|<a,
\end{cases}
\label{Hvxxeps_zero}\end{aligned}$$ $$\begin{aligned}
K_{vz}(x) =
\begin{cases} \displaystyle
\frac{I_0}{\pi}\Big(\frac{y_1[1\!-\! \tilde \phi(x)/s_1]}{x^2+y_1^2}
- \frac{y_2[1\!-\!\tilde \phi(x)/s_2]}{x^2+y_2^2}\Big),\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\
|x|<b \mbox
{ or } |x| > a,\\
\frac{I_0}{\pi}\Big(\frac{y_1}{x^2+y_1^2}
- \frac{y_2}{x^2+y_2^2}\Big),\;\;\; b<|x|<a,
\end{cases}
\label{Kzx_zer0}\end{aligned}$$
As discussed in Appendix B, the requirement that $ \int_{-\infty}^{+\infty}
K_z(x) dx=0 $ leads to the condition that $ab = x_0^2 = y_1 y_2$ when there is no bulk pinning. A second condition relating $a$ , $b$ and $I_0$ follows from the requirement that $H_x(0,\epsilon)=H_{Mx}(0,\epsilon)+H_{vx}(0,\epsilon)= H_{c1}$, which yields $$I_0=\Big(\frac{2}{1+y_1y_2(s_1+s_2)/s_1s_2}\Big)I_{c1}.
\label{I0ab}$$ Combining these two conditions and eliminating $a$, we obtain the following connection between $I_0$ and $b$, $$I_0=\Big(\frac{2}{1+b(y_1+y_2)/\sqrt{(y_1^2+b^2)(y_2^2+b^2)}}\Big)I_{c1}.$$ When $I_0 =I_{c1}$, $a = b = x_0 = \sqrt{y_1
y_2}$. As $I_0$ increases above $I_{c1}$, $b$ decreases monotonically, as shown in Fig. \[fig4\] and the lower solid curve in Fig. \[fig11\].
![Plot of the vortex dome’s left boundary $b$ vs $I_0/I_{c1}$ for the bulk-pinning-free case, where $b$ is in units of $x_0 = \sqrt{y_1y_2}$, for $I_{c1} < I_0 < 2I_{c1}$ and $y_1/y_2 =$ 0.01, 0.03, 0.1, 0.3, and 1 (bottom to top). The right boundary of the vortex dome is at $a = x_0^2/b$.[]{data-label="fig4"}](Fig04.eps){width="8cm"}
In the limit as $I_0
\to 2 I_{c1}$, we see that $b \to 0$, such that $a = y_1 y_2/b
\to \infty$. For $I_0 > 2 I_{c1}$, the magnetic-flux-filled region extends from $-\infty$ to $+\infty$, $K_z(x) = 0$, and ${{\cal H}}(\zeta) =
{{\cal H}}_0(\zeta)$ everywhere, since the superconducting film is then completely incapable of screening the magnetic field produced by the two wires.
Figures 5 and 6 show plots of $H_y(x,0) = H_{vy}(x,0)$ and $K_z(x)$ vs $x$ for several values of $I_0$ in the range $I_{c1} \le I_0 \le 2I_{c1}$.
![The perpendicular magnetic field $H_y(x,0)/H_{c1} = H_{vy}(x,0)/H_{c1}$ vs $x$ exhibits vortex and antivortex flux domes at $b < |x| <a$ in the absence of bulk pinning. With $y_1 =1$, $y_2 = 2$, and $x_0 = \sqrt{y_1y_2} = 1.414$, when $I_0 = 1.001 I_{c1}$, tiny domes are centered at $\pm x_0$. As $I_0$ increases, the domes become taller and wider, as shown for $I_0 = 1.026 I_{c1}$ when $b = 1$ (dotted), $I_0 = 1.212 I_{c1}$ when $b = 0.5$ (short dash), and $I_0 = 2 I_{c1}$ when $b = 0$ and $a = \infty$ (long dash). For $I_0 \ge 2I_{c1}$, the superconducting film is no longer capable of screening, $K_z(x) = 0$ everywhere, and $H_y(x,0) =
H_{0y}(x,0)$ \[Eq. \].[]{data-label="fig5"}](Fig05.eps){width="8cm"}
![Sheet-current-density $K_z(x)/H_{c1}$ vs $x$ in the absence of bulk pinning for $y_1 =1$, $y_2 = 2$, and $x_0 = \sqrt{y_1y_2} = 1.414$. When $I_0 = I_{c1}$, $K_z(x)$ (solid) is equal to $K_{Mz}(x)$, shown in Fig. 3. With increasing $I_0$, regions of $K_z(x) = 0$ develop underneath the vortex and antivortex domes at $b <
|x| <a$, beginning at $x = \pm x_0$. Shown are results for $I_0 = 1.026 I_{c1}$ when $b = 1$ (dotted) and $I_0 = 1.212 I_{c1}$ when $b = 0.5$ (dashed). For $I_0 \ge 2I_{c1}$, the superconducting film is no longer capable of screening, and $K_z(x) = 0$ everywhere.[]{data-label="fig6"}](Fig06.eps){width="8cm"}
Figure 7 shows plots of $H_x(0,\pm\epsilon)$ = $H_{Mx}(0,\pm\epsilon)+H_{vx}(0,\pm\epsilon)$ vs $I_0$.
![Parallel magnetic fields at $x=0$ at the top and bottom surfaces $H_x(0,\epsilon)$ and $H_x(0,-\epsilon)$ vs $I_0$ in the absence of bulk pinning. When $0 \le I_0 \le I_{c1}$, $H_x(0,\epsilon) = H_{Mx}(0,\epsilon)$ \[Eq. \] and $H_x(0,-\epsilon) = H_{Mx}(0,-\epsilon)=0$. When $ I_{c1} \le I_0 \le
2 I_{c1}$, $H_x(0,\epsilon) =H_{Mx}(0,\epsilon) +H_{vx}(0,\epsilon)= H_{c1}$, and $H_x(0,-\epsilon) = H_{vx}(0,-\epsilon) = -H_{vx}(0,\epsilon)$. When $ I_0 \ge
2 I_{c1}$, $H_x(0,\pm\epsilon) = H_{0x}(0,0) =H_{c1}I_0/2I_{c1}$ \[Eq. \].[]{data-label="fig7"}](Fig07.eps){width="8cm"}
![The magnitude of the sheet-current-density at the origin $-K_z(0) =
H_x(0,\epsilon)-H_x(0,-\epsilon)$ vs $I_0$ in the absence of bulk pinning (see Fig. 7). []{data-label="fig8"}](Fig08.eps){width="8cm"}
For $0 \le I_0 \le I_{c1}$, we have $H_x(0,\epsilon) = H_{Mx}(0,\epsilon) =
H_{c1}I_0/I_{c1}$ and $H_x(0,-\epsilon) = H_{Mx}(0,-\epsilon) =
0.$ For $I_{c1} \le I_0 \le 2 I_{c1}$, we have $H_x(0,\epsilon) =
H_{Mx}(0,\epsilon) + H_{vx}(0,\epsilon) = H_{c1}$ and $H_x(0,-\epsilon) =
H_{vx}(0,-\epsilon) = H_{c1}(I_0/I_{c1}-1)$, where $H_{Mx}(0,\epsilon) = H_{c1}I_0/I_{c1}$ and $H_{vx}(0,\pm \epsilon) = \pm
H_{c1}(1-I_0/I_{c1}).$ When $I_0 = 2 I_{c1}$, $H_x(0,\pm
\epsilon) = H_{c1}$. For $I_0 > I_{c1}$, $H_x(0,\pm \epsilon) = H_{0x}(0,0) =
H_{c1}I_0/2I_{c1}$, and in addition $H_x(x,\pm \epsilon) = H_{Mx}(x, \pm
\epsilon) + H_{vx}(x, \pm \epsilon) = H_{0x}(x,0)$ for all $x$. The macroscopic magnetic-field distribution is then the same as it would be if the superconducting film were absent.
Figure 8 shows a plot of $-K_z(0)$ vs $I_0$, where $K_z(0) =
H_x(0,-\epsilon)-H_x(0,\epsilon)$. For $0 \le I_0 \le I_{c1}$, $-K_z(0) = H_{c1}I_0/I_{c1}$, and for $I_{c1} \le I_0 \le 2 I_{c1}$, $-K_z(0) = H_{c1}(2-I_0/I_{c1})$. For $I_0 > 2I_{c1}$, $K_z(x) = 0$ for all $x$, i.e., everywhere in the film.
The vortex-generated complex potential ${\cal G}_v(\zeta)={\cal
G}(\zeta)-{\cal G}_M(\zeta)$ \[see Eqs. and \] in the absence of bulk pinning is $$\begin{aligned}
{\cal G}_v(\zeta)\!\! &=\!\!& \frac{I_0}{2\pi}
\left[ g_v(\zeta,y_1) -g_v(\zeta,y_2) \right] ,
\label{cG_v}\end{aligned}$$ where $$\begin{aligned}
g_v(\zeta,y)
&=& \mp i\arctan (\zeta/y) \nonumber\\
&& {}\mp\frac{i}{asy} [ a^2y^2\bm{E}(\theta,k)
+y^2(b^2+y^2)\bm{F}(\theta,k)\nonumber\\
&& -(a^2+y^2)(b^2+y^2)\bm{\Pi}
(\theta,-b^2/y^2,k) ],
\label{g_v} \\
s &=& \sqrt{(a^2+y^2)(b^2+y^2)}, \\
\theta &=& \arcsin(\zeta/b), \mbox{ and } \\
k &=& b/a , \end{aligned}$$ where $\bm E$, $\bm K$, and $\bm \Pi$ are incomplete elliptic integrals and the upper (lower) signs hold in the upper (lower) half $\zeta$ plane.
![Contour plot of the real part of the vortex-generated complex potential in the absence of bulk pinning \[Eq. \] ${\cal G}_v(x+iy)$ vs $x$ and $y$ for $y_1 = 1$, $y_2 = 2$, $a = 2$, and $b = 1$, as for the dotted curves in Figs. 5 and 6. The contours correspond to magnetic field lines of the vortex-generated complex field ${{\cal H}}_v(x+iy)$ given in Eq. . []{data-label="fig9"}](Fig09.eps){width="8cm"}
Shown in Fig. 9 is a contour plot of the real part of ${\cal G}_v(x+iy)$. These contours correspond to the magnetic field lines of the vortex-generated magnetic field. The magnetic field flows in a generally counterclockwise direction, carried by vortices in the region $b < x <
a$ up through the film and by antivortices in the region $-a < x < -b$ back down through the film. A contour plot of the real part of ${\cal G}(x+iy)=
{\cal G}_M(x+iy)+{\cal G}_v(x+iy)$ would also show the magnetic field lines generated by the two wires, as in Fig. 2.
Flux domes in the presence of weak bulk pinning\[Sec\_Kc>0\]
---------------------------------------------------------------
In superconducting films in which bulk pinning is present and is characterized by a field-independent critical sheet-current density $K_c = j_c d >0$, the first vortex enters the film at $x = 0$, when the maximum parallel magnetic field at the top surface $H_{Mx}(0,\epsilon)$ \[Eq. \] is equal to the lower critical field $H_{c1}$.[@foot1] As in Sec. \[Sec\_Kc=0\], this again occurs at the current $I_0 = I_{c1}$, given in Eq. . According to critical-state theory,[@Campbell72; @Clem79] this vortex advances toward the bottom surface at the leading edge of a curving flux front of nearly parallel vortices of thickness $d_p = [H_{Mx}(0,\epsilon)-H_{c1}]/j_c$, and it reaches the bottom surface when $d_p = d$, i.e., when $H_{Mx}(0,\epsilon) = H_{c1} + K_c$ or $I_0 = I_{c1} + I_{c0}$ \[see Eqs. , , and \]. The positive end of the first vortex is then at $x = x_p$ and its negative end is at $x = -x_p$, where $x_p$ is the solution of $H_{Mx}(x_p,\epsilon) = H_{c1}$ \[see Eq. \]. It can be shown that $x_p \gg d$ when $d \ll y_1$ except when $K_c \ll H_{c1}$.
Once the first vortex reaches the bottom surface, the portion at $x \approx 0$ annihilates with its image, and the vortex divides into two halves. The half in the region $x > 0$, which we call a vortex, carries magnetic flux $\phi_0$ [*up*]{} from the bottom to the top surface, and the half in the region $x < 0$, which we call an antivortex, carries magnetic flux $\phi_0$ [*down*]{} from the top to the bottom surface. Since $H_{Mx}(0,-\epsilon)=0$ and $H_{Mx}(0,\epsilon) = H_{c1} + K_c$ at $I_0=I_{c1}
+ I_{c0}$, the sheet-current density is initially $K_{Mz}(0) =
-H_{c1} - K_c$. Because $|K_{Mz}(0)| > K_c$, the vortex separates from the rest of the flux front and is driven in the $x$ direction by the Lorentz force $K_{Mz}(x)\phi_0$ until it comes to rest at $x = x_c$, where $|K_{Mz}(x_c)|=K_c$ \[see Eq. \] and the Lorentz force is balanced by the pinning force. Similarly, the antivortex moves in the opposite direction and comes to rest at the point $x = -x_c$. As we will show below, $0 < x_c < x_0 =
\sqrt{y_1y_2}$ when $ K_c>0$ \[see Fig. 10\].
For increasing values of $I_0$ in the range $I_{c1}+I_{c0} < I_0 <
2I_{c1}+ I_{c0}$, the magnetic field distribution perpendicular to the film can be characterized as having a positive vortex-generated magnetic flux dome in the region $ b < x < a$, where $0 <b < x_c < a$, and a negative antivortex-generated flux dome in the region $-a < x < -b$. The complex magnetic field ${{\cal H}}(\zeta) = {{\cal H}}_M(\zeta) +{{\cal H}}_v(\zeta)$ is given by Eq. . Subtracting the Meissner-state complex field ${{\cal H}}_M(\zeta)$, we obtain the following expression for the vortex-generated complex magnetic field, $$\begin{aligned}
{{\cal H}}_v(\zeta)\!\!&=\!\!&\frac{I_0}{2\pi} \Big\{ \frac{y_1[\mp i
+\phi(\zeta)/s_1]}{\zeta^2+y_1^2}
\! -\!\frac{y_2[\mp i
+\phi(\zeta)/s_2]}{\zeta^2+y_2^2}
\Big\}\nonumber \\
&&\pm i K_c/2,
\label{Hvweak} \end{aligned}$$ where $ s_j= \sqrt{(a^2+y_j^2)(b^2+y_j^2)}$, and the upper (lower) sign holds when $\zeta = x+iy$ is in the upper (lower) half plane. Following a procedure similar to that used in Sec. \[Sec\_Kc=0\], we obtain the following values of $H_{vy}(x,0)$, $H_{vx}(x,\epsilon)=-H_{vx}(x,-\epsilon),$ and $K_{vz}(x)=H_{vx}(x,-\epsilon)-H_{vx}(x,\epsilon)$, $$\begin{aligned}
H_{vy}(x,0) =
\begin{cases} \displaystyle
0, & |x|<b \mbox { or } |x| > a,\\
\frac{I_0}{2\pi}\Big(\frac{y_1}{s_1(x^2+y_1^2)} \\-
\frac{y_2}{s_2(x^2+y_2^2)}\Big)\tilde \phi(x), &
b<|x|<a,
\end{cases}
\label{Hvyx0_small}\end{aligned}$$ $$\begin{aligned}
H_{vx}(x, \epsilon) =
\begin{cases} \displaystyle
-\frac{I_0}{2\pi}\Big(\frac{y_1[1\!-\! \tilde \phi(x)/s_1]}{x^2+y_1^2}
- \frac{y_2[1\!-\!\tilde \phi(x)/s_2]}{x^2+y_2^2}\Big)\\
\;\; +K_c/2, \;\;\;\;\;\;\;
|x|<b \mbox
{ or } |x| > a,\\
-\frac{I_0}{2\pi}\Big(\frac{y_1}{x^2+y_1^2}
- \frac{y_2}{x^2+y_2^2}\Big) +K_c/2,\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; b<|x|<a,
\end{cases}
\label{Hvxxeps_small}\end{aligned}$$ $$\begin{aligned}
K_{vz}(x) =
\begin{cases} \displaystyle
\frac{I_0}{\pi}\Big(\frac{y_1[1\!-\! \tilde \phi(x)/s_1]}{x^2+y_1^2}
- \frac{y_2[1\!-\!\tilde \phi(x)/s_2]}{x^2+y_2^2}\Big)\\\;\;-K_c,
\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;
|x|<b \mbox
{ or } |x| > a,\\
\frac{I_0}{\pi}\Big(\frac{y_1}{x^2+y_1^2}
- \frac{y_2}{x^2+y_2^2}\Big)-K_c,\; b<|x|<a,
\end{cases}
\label{Kzx_small}\end{aligned}$$
![Plot of $\tilde x_c = x_c/x_0$ vs $K_c/H_{c1}$ in the presence of bulk pinning, where $x_c$ is the position of the first entering vortex when $I_0$ just exceeds $I_{c1}+I_{c0}$ and $x_0 =
\sqrt{y_1y_2}$. Curves are shown for $y_1/y_2 =$ 0.01, 0.03, 0.1, 0.3, and 1 (bottom to top). []{data-label="fig10"}](Fig10.eps){width="8cm"}
For $I_{c1}+I_{c0} < I_0 < 2I_{c1}+I_{c0}$, the requirement that $\int_{-\infty}^{+\infty}K_z(x) dx = 0$ leads to Eq. . A second condition on $a$, $b$, and $I_0$ follows from the requirement that $H_x(0,\epsilon) = H_{c1}+K_c$, which from Eq. yields: $$\begin{aligned}
{{\cal H}}_x(0,\epsilon)&=&\frac{I_0}{2\pi} \Big[
\frac{(1+ab/s_1)}{y_1}
\! -\!\frac{(1+ab/s_2)}{y_2}
\Big]+\frac{K_c}{2}\nonumber \\
&=& H_{c1}+K_c.
\label{Hweakcondition} \end{aligned}$$ Elimination of $I_0$ between Eqs. and yields the equation $$\frac{y_1y_2(y_1s_2-y_2s_1)}{(y_2-y_1)s_1s_2+ab(y_2s_2-y_1s_1)}=
\frac{K_c/2H_{c1}}{1+K_c/2H_{c1}}.
\label{abKcweak}$$ Numerical solutions of Eqs. and yield $a$ and $b$ as a function of $I_0$ for any given value of $K_c$. As discussed above, when $I_0$ just exceeds $I_{c1}+I_{c0}$, the first vortex (antivortex) comes to rest at $x_c$ (-$x_c$). The equation determining the value of $\tilde x_c = x_c /x_0 = x_c
/\sqrt{y_1y_2}$ can be obtained from Eq. by setting $a = b = x_c$ and making use of Eqs. and : $$\frac{(1-\tilde x_c^2)}{(\tilde x_c^2+y_1/y_2)(\tilde x_c^2+y_2/y_1)}
=\frac{K_c/H_{c1}}{1+K_c/H_{c1}}.
\label{xc}$$ Figure 10 shows plots of $\tilde x_c$ vs $K_c/H_{c1}$ for various values of $y_1/y_2$, obtained by numerically solving Eq. . Note that for each case $\tilde x_c = 1$ when $K_c$ = 0, and $\tilde x_c \to 0$ when $K_c/H_{c1} \to \infty$. For $K_c/H_{c1}\gg 1$, $$\tilde x_c \approx 1/\sqrt{(1+y_1/y_2+y_2/y_1)(K_c/H_{c1})}.
\label{tildexclimit}$$
Numerical solutions for $a$ and $b$ when $I_{c1}+I_{c0} < I_0 < 2I_{c1}+I_{c0} $ are shown by the dotted curves in Fig. 11 for the case that $y_1 = 1$, $y_2 = 2$, $x_0 = \sqrt{2}$, and $K_c = H_{c1}/2$, such that $I_{c0} = I_{c1}/2$. Note that $a = b = x_c = 0.585 x_0 = 0.827$ at $I_0 = I_{c1}+I_{c0}$ when vortex penetration first occurs. From Eqs. , , and , we see that $b = 0$ when $I_0 = 2I_{c1}+I_{c0}.$ When $I_0 > 2I_{c1}+I_{c0}$, $b$ remains equal to zero, and the value of $a$ must be obtained from Eq. . Figure 11 also shows similar plots of $a$ and $b$ for $K_c = H_{c1}$ and $K_c = 2 H_{c1}$
![Plots of $a$ (upper curves), the right boundary of the vortex dome, and $b$ (lower curves), the left boundary, vs $I_0/I_{c1}$ for $y_1 = 1$ and $y_2 = 2$: $K_c=0$ (solid curves separated by the dot at $ a= b=x_c
= 1.414$), $K_c = H_{c1}/2$ (dotted curves and dot at $x_c = 0.827$), $K_c = H_{c1}$ (short-dashed curves and dot at $x_c =
0.651$), and $K_c = 2H_{c1}$ (long-dashed curves and dot at $x_c =
0.493$). Note that $b=0$ when $I_0/I_{c1} \ge 2 +
K_c/H_{c1}$.[]{data-label="fig11"}](Fig11.eps){width="8cm"}
Figures 12 and 13 show plots of $H_y(x,0) = H_{vy}(x,0)$ and $K_z(x)$ vs $x$ for several values of $I_0$ in the range $I_{c1} +I_{c0} < I_0 \le 2I_{c1} +I_{c0}$.
![When $I_0 > I_{c1}+I_{c0}$, the perpendicular magnetic field $H_y(x,0)/H_{c1} = H_{vy}(x,0)/H_{c1}$ vs $x$ initially exhibits separated vortex and antivortex flux domes at $b < |x|
<a$ even in the presence of bulk pinning, as shown here for $K_c=H_{c1}/2$, $y_1 =1$, $y_2 = 2$, and $x_c = 0.827$. When $I_0$ is just above $I_{c1}+I_{c0} = 1.5
I_{c1}$, tiny domes are centered at $\pm x_c$. As $I_0$ increases, the domes become taller and wider, as shown for $I_0 = 1.545 I_{c1}$ when $b = 0.600$ and $a = 1.077$ (dotted curves), $I_0 = 1.683 I_{c1}$ when $b = 0.400$ and $a = 1.337$ (short-dashed curves), $I_0 = 1.966 I_{c1}$ when $b = 0.200$ and $a = 1.630$ (medium-dashed curves), and $I_0 = 2 I_{c1} + I_{c0} = 2.5 I_{c1}$ when $b = 0$ and $a = 1.933$ (long-dashed curves).[]{data-label="fig12"}](Fig12.eps){width="8cm"}
![Sheet-current density $K_z(x,0)/H_{c1}$ vs $x$ in the presence of bulk pinning, shown here for $K_c=H_{c1}/2$, $y_1 =1$, $y_2 = 2$, and $x_c = 0.827$. When $I_0 = I_{c1}+I_{c0}$, $K_z(x)$ (solid) is equal to $K_{Mz}(x)$, shown in Fig. 3. With increasing $I_0$, regions of $K_z(x) = -K_c$ develop underneath the vortex and antivortex domes at $b <
|x| <a$, beginning at $x = \pm x_c$, while $|K_z(x)| > K_c$ in the vortex-free region $-b < x < b$. Shown are results for $I_0 = 1.545 I_{c1}$ when $b = 0.600$ and $a = 1.077$ (dotted curves), $I_0 = 1.683 I_{c1}$ when $b = 0.400$ and $a = 1.337$ (short-dashed curves), $I_0 = 1.966 I_{c1}$ when $b = 0.200$ and $a = 1.630$ (medium-dashed curves), and $I_0 = 2 I_{c1} + I_{c0} = 2.500 I_{c1}$ when $b = 0$ and $a = 1.933$ (long-dashed curves). For $I_0 \ge 2I_{c1}+I_{c0}$, $K_z(x) =
-K_c$ in the region $-a < x < a$ and $|K_z(x)| < K_c$ outside this region.[]{data-label="fig13"}](Fig13.eps){width="8cm"}
Figure 14 shows plots of $H_x(0,\pm\epsilon)$ = $H_{Mx}(0,\pm\epsilon)+H_{vx}(0,\pm\epsilon)$ vs $I_0$.
![Parallel magnetic fields at $x=0$ at the top and bottom surfaces $H_x(0,\epsilon)$ and $H_x(0,-\epsilon)$ vs $I_0$ in the presence of bulk pinning, shown here for $K_c=H_{c1}/2$, such that $I_{c1}+I_{c0}= 1.5
I_{c1}$ and $2I_{c1}+I_{c0}= 2.5 I_{c1}$. When $0
\le I_0
\le I_{c1}+I_{c0}$, $H_x(0,\epsilon) = H_{Mx}(0,\epsilon)$ \[Eq. \] and $H_x(0,-\epsilon) = H_{Mx}(0,-\epsilon)=0$. When $ I_{c1}+I_{c0} \le I_0 \le
2I_{c1}+I_{c0}$, $H_x(0,\epsilon) =H_{Mx}(0,\epsilon) +H_{vx}(0,\epsilon)=
H_{c1}+K_c$, and $H_x(0,-\epsilon) = H_{vx}(0,-\epsilon) = -H_{vx}(0,\epsilon)$. When $ I_0 \ge
2I_{c1}+I_{c0}$, $H_x(0,\pm\epsilon) = H_{0x}(0,0)\pm K_c/2$.[]{data-label="fig14"}](Fig14.eps){width="8cm"}
![Magnitude of the sheet-current density at the origin $-K_z(0) =
H_x(0,\epsilon)-H_x(0,-\epsilon)$ vs $I_0$ in the presence of weak bulk pinning, shown here for $K_c=H_{c1}/2$, such that $I_{c2} =2I_{c1}-I_{c0}
= 1.5 I_{c1}$ (see Fig. 14). []{data-label="fig15"}](Fig15.eps){width="8cm"}
For $0 \le I_0 \le I_{c1}+I_{c0}$, we have $H_x(0,\epsilon) = H_{Mx}(0,\epsilon)
= H_{c1}I_0/I_{c1}$ and $H_x(0,-\epsilon) = H_{Mx}(0,-\epsilon) =
0.$ For $I_{c1}+I_{c0} \le I_0 \le 2I_{c1}+I_{c0}$, we have $H_x(0,\epsilon) =
H_{Mx}(0,\epsilon) + H_{vx}(0,\epsilon) = H_{c1}+K_c$ and $H_x(0,-\epsilon) =
H_{vx}(0,-\epsilon)$, where $H_{Mx}(0,\epsilon) = H_{c1}I_0/I_{c1}$ and $H_{vx}(0,\pm \epsilon) = \mp
H_{c1}(I_0-I_{c1}-I_{c0})/I_{c1}.$ When $I_0 = 2I_{c1}+I_{c0}$, $H_x(0,\epsilon) = H_{c1}+K_c$ and $H_x(0,-\epsilon) = H_{c1}$. For $I_0 \ge 2I_{c1}+I_{c0}$, $H_x(0,\pm \epsilon) = H_{0x}(0,0) \pm K_c/2 =
H_{c1}I_0/2I_{c1} \pm K_c/2$.
Figure 15 shows a plot of $-K_z(0)$ vs $I_0$, where $K_z(0) =
H_x(0,-\epsilon)-H_x(0,\epsilon)$. For $0 \le I_0 \le I_{c1}+I_{c0}$, $-K_z(0) = (H_{c1}+K_c)I_0/(I_{c1}+I_{c0})$, and for $I_{c1}+I_{c0} \le I_0 \le
2I_{c1}+I_{c0}$, $-K_z(0) = K_c +H_{c1}(2I_{c1}+I_{c0}-I_{0})/I_{c1}$. For $I_0 \ge
2I_{c1}+I_{c0}$, $-K_z(0) = K_c$.
The vortex-generated complex potential ${\cal G}_v(\zeta)={\cal
G}(\zeta)-{\cal G}_M(\zeta)$ \[see Eqs. and \] in the presence of weak bulk pinning ($0 < K_c < H_{c1}$) is, when $I_{c1} < I_0 < I_{c2}$, such that $0 < b < a$, $$\begin{aligned}
{\cal G}_v(\zeta)\!\! &=\!\!& \frac{I_0}{2\pi}
\left[ g_v(\zeta,y_1) -g_v(\zeta,y_2) \right]
\pm i\frac{K_c}{2}\zeta ,
\label{cG_vweak}\end{aligned}$$ where $g_v(\zeta,y)$ is given in Eq. .
![Contour plot of the real part of the vortex-generated complex potential in the presence of weak bulk pinning \[Eq. \] ${\cal G}_v(x+iy)$ vs $x$ and $y$ for $y_1 = 1$, $y_2 = 2$, $K_c = H_{c1}/2,$ $I_0 = 1.683 I_{c1}$, $a = 1.337$, and $b = 0.4$, as for the short-dashed curves in Figs. 12 and 13. The contours correspond to magnetic field lines of the vortex-generated complex field ${{\cal H}}_v(x+iy)$ given in Eq. . []{data-label="fig16"}](Fig16.eps){width="8cm"}
Shown in Fig. 16 is a contour plot of the real part of ${\cal G}_v(x+iy)$. These contours correspond to the magnetic field lines of the vortex-generated magnetic field. The magnetic field flows in a generally counterclockwise direction, carried by vortices in the region $b < x <
a$ up through the film and by antivortices in the region $-a < x < -b$ back down through the film. A contour plot of the real part of ${\cal G}(x+iy)=
{\cal G}_M(x+iy)+{\cal G}_v(x+iy)$ would also show the magnetic field lines generated by the two wires, as in Fig. 2.
When $I_0 > 2I_{c1} + I_{c0}$, the gap between the vortex dome and the antivortex dome is closed ($b = 0$), and the magnetic-field distribution thus can be characterized as a dome of vortices in the region $0 < x < a$ carrying magnetic flux up through the film and an adjacent dome of antivortices in the region $-a < x < 0$ carrying an equal amount of magnetic flux back down through the film. The outer boundaries of these domes ($\pm a$) depend upon the values of $I_0$ and $K_c$, and the magnetic-field and sheet-current-density distributions can be calculated as follows. The complex magnetic field ${{\cal H}}(\zeta) = {{\cal H}}_M(\zeta) +{{\cal H}}_v(\zeta)$ is as given in Eq. , except that $b=0$ and $\phi(\zeta)$, $s_1$, and $s_2$ are now given by Eqs. , , and . Similarly, the vortex-generated complex magnetic field ${{\cal H}}_v(\zeta)$ is as given in Eq. , and $H_{vy}(x,0)$, $H_{vx}(x,\epsilon)=-H_{vx}(x,-\epsilon),$ and $K_{vz}(x)=H_{vx}(x,-\epsilon)-H_{vx}(x,\epsilon)$ are as given in Eqs. , , and , except that now just above or below the real axis, $$\phi(x \pm i\epsilon)=
\begin{cases} \displaystyle
\tilde \phi(x), & 0<|x|<a, \\
\pm i \tilde \phi(x),
& |x| > a,
\end{cases}
\label{phixpm_Kcbig}$$ where $$\tilde \phi(x)=
\begin{cases} \displaystyle
x \sqrt{a^2-x^2}, & 0<|x|<a, \\
- |x|\sqrt{x^2-a^2}, & |x|>a .
\end{cases}
\label{tildephi_Kcbig}$$ For given values of $I_0$ and $K_c$ the value of $a$ in the above equations is determined by Eq. , which follows from the requirement that $\int_{-\infty}^{+\infty}K_z(x)dx = 0$, and by Eq. (but with $b=0$), which follows from the requirement that $H_x(0,\epsilon) = H_{c1}+K_c$. Numerical solutions for $a$ obtained in this way are shown in Fig. \[fig11\] vs $I_0/I_{c1}$ for $I_0 > 2I_{c1} + I_{c0}$ (where $b=0$) for the case of $y_1 = 1$ and $y_2 = 2$. Figures \[fig17\] and \[fig18\] show plots of $H_y(x,0) = H_{vy}(x,0)$ and $K_z(x)$ vs $x$ for several values of $I_0 \ge 2I_{c1} + I_{c0}$ when $K_c = H_{c1}/2$.
![When $I_0 \ge 2I_{c1} +I_{c0}$ in the presence of bulk pinning, the perpendicular magnetic field $H_y(x,0)/H_{c1}
= H_{vy}(x,0)/H_{c1}$ vs $x$ exhibits adjacent vortex and antivortex flux domes at $0 < |x| <a$, as shown here for $K_c=H_{c1}/2$, $I_{c0} = 0.5 I_{c1}$, $y_1 =1$, and $y_2 = 2$. As $I_0$ increases, the domes become taller and wider, as shown for $I_0 = 2.5 I_{c1}$ when $a = 1.933$ (solid curves), $I_0 = 4.25 I_{c1}$ when $a = 2.504$ (short-dashed curves), and $I_0 = 6 I_{c1}$ when $a = 2.913$ (long-dashed curves).[]{data-label="fig17"}](Fig17.eps){width="8cm"}
![Sheet-current density $K_z(x,0)/H_{c1}$ vs $x$ when $I_0 \ge 2I_{c1} +I_{c0}$ in the presence of bulk pinning, shown here for $K_c=H_{c1}/2$, $I_{c0} = 0.5 I_{c1}$, $y_1 =1$ and $y_2 = 2$. Shown are results for $I_0 = 2.5 I_{c1}$ when $a = 1.933$ (solid curve), $I_0 = 4.25 I_{c1}$ when $a = 2.504$ (short-dashed curve), and $I_0 = 6 I_{c1}$ when $a = 2.913$ (long-dashed curve), corresponding to the cases shown in Fig. \[fig17\].[]{data-label="fig18"}](Fig18.eps){width="8cm"}
When $I_0 > 2I_{c1}+I_{c0}$, such that $a > 0$ and $b=0$, the vortex-generated complex potential ${\cal G}_v(\zeta)={\cal G}(\zeta)-{\cal G}_M(\zeta)$ \[see Eqs. , , and \] in the presence of bulk pinning is $$\begin{aligned}
{\cal G}_v(\zeta)\!\! &=\!\!& \frac{I_0}{2\pi}
\left[ g_v(\zeta,y_1) -g_v(\zeta,y_2) \right]
\pm i\frac{K_c}{2}\zeta ,
\label{cG_vstrong}\end{aligned}$$ where $$\begin{aligned}
g_v(\zeta,y)
& = \mp i\arctan (\zeta/y)+
\sqrt{\frac{a^2-\zeta^2}{a^2+y^2}}-\frac{a}{\sqrt{a^2+y^2}} \nonumber \\
&
-\mbox{arctanh} \sqrt{\frac{a^2-\zeta^2}{a^2+y^2}}
+\mbox{arctanh} \frac{a}{\sqrt{a^2+y^2}},
\label{gvstrong}\end{aligned}$$ and the upper (lower) signs hold in the upper (lower) half $\zeta$ plane.
![Contour plot of the real part of the vortex-generated complex potential in the presence of bulk pinning \[Eq. \] ${\cal G}_v(x+iy)$ vs $x$ and $y$ for $y_1 = 1$, $y_2 = 2$, $K_c = H_{c1}/2,$ $I_0 = 2 I_{c1}+I_{c0} =
2.5 I_{c1}$, $a = 1.933$, and $b=0$, as for the solid curves in Figs. \[fig17\] and \[fig18\]. The contours correspond to magnetic field lines of the vortex-generated complex field ${{\cal H}}_v(x+iy)$ given by Eq. . []{data-label="fig19"}](Fig19.eps){width="8cm"}
Shown in Fig. \[fig19\] is a contour plot of the real part of ${\cal G}_v(x+iy)$. These contours correspond to the magnetic field lines of the vortex-generated magnetic field. The magnetic field flows in a generally counterclockwise direction, carried by vortices in the region $0 < x <
a$ up through the film and by antivortices in the region $-a < x < 0$ back down through the film. A contour plot of the real part of ${\cal G}(x+iy)=
{\cal G}_M(x+iy)+{\cal G}_v(x+iy)$ would also show the magnetic field lines generated by the two wires, as in Fig. 2.
Flux domes in the presence of strong bulk pinning\[Sec\_Kc\_strong\]
--------------------------------------------------------------------
We consider here briefly the case of superconducting films in which bulk pinning, characterized by a field-independent critical sheet-current density $K_c = j_c d >0$, is so strong that $K_c \gg H_{c1}$. In principle, the process of vortex entry is qualitatively the same as discussed in Sec. \[Sec\_Kc>0\]. However, in the limit that $H_{c1}/K_c \to 0$, the width of the current region $I_{c1}+I_{c0}$ to $2I_{c1}+I_{c0}$ (in which there is a gap of width $2b$ between the vortex and antivortex domes) shrinks to zero, and $I_{c0}$ becomes the only critical current of practical interest. Essentially, as soon as $I_0$ exceeds $I_{c0}$, the vortices penetrating from the top surface divide in such a way as to produce adjacent vortex and antivortex flux domes in the regions $0 < x < a$ and $-a < x < 0$, and the sheet-current density is $K_z(x) = - K_c$ in these regions. The perpendicular magnetic field $H_y(x,0)$, sheet-current density $K_z(x)$ and vortex-generated complex potential ${\cal G}_v(x+iy)$ can be calculated as discussed in Sec. \[Sec\_Kc>0\] for the case that $b = 0$, and plots of all these quantities look very similar to those in Figs. \[fig17\], \[fig18\], and \[fig19\].
Figure 20 shows plots of $H_x(0,\pm\epsilon)$ = $H_{Mx}(0,\pm\epsilon)+H_{vx}(0,\pm\epsilon)$ vs $I_0$ in the limit $H_{c1}/K_c \to 0$.
![Parallel magnetic fields at $x=0$ at the top and bottom surfaces $H_x(0,\epsilon)$ and $H_x(0,-\epsilon)$ vs $I_0$ for strong bulk pinning in the limit as $H_{c1}/K_c \to 0.$ When $0
\le I_0
\le I_{c0}$, $H_x(0,\epsilon) = H_{Mx}(0,\epsilon)$ \[Eq. \] and $H_x(0,-\epsilon) = H_{Mx}(0,-\epsilon)=0$. When $ I_0 \ge
I_{c0}$, $H_x(0,\pm\epsilon) = H_{0x}(0,0)\pm K_c/2$ \[see Eq. \].[]{data-label="fig20"}](Fig20.eps){width="8cm"}
![Magnitude of the sheet-current density at the origin $-K_z(0) =
H_x(0,\epsilon)-H_x(0,-\epsilon)$ vs $I_0$ for strong bulk pinning in the limit as $H_{c1}/K_c \to 0$ (see Fig. 20). []{data-label="fig21"}](Fig21.eps){width="8cm"}
For $0 \le I_0 \le I_{c0}$, we have $H_x(0,\epsilon) = H_{Mx}(0,\epsilon) =
K_{c}I_0/I_{c0}$ and $H_x(0,-\epsilon) = H_{Mx}(0,-\epsilon) =
0.$ For $I_0 \ge I_{c0}$, $H_x(0,\pm \epsilon) = H_{0x}(0,0) \pm K_c/2 =
K_{c}I_0/2I_{c0} \pm K_c/2$. Figure 21 shows a plot of $-K_z(0)$ vs $I_0$ in the same limit, where $K_z(0) =
H_x(0,-\epsilon)-H_x(0,\epsilon)$. For $0 \le I_0 \le I_{c0}$, $-K_z(0) = K_{c}I_0/I_{c0}$, and for $I_0 \ge I_{c0}$, $-K_z(0)
= K_c$.
Summary and Discussion\[Sec\_conclusion\]
=========================================
In this paper we have predicted that separated vortex and antivortex flux domes can be produced in weak-pinning type-II superconducting films subjected to local magnetic fields generated by current-carrying wires above the film’s surface and far from the edges. To calculate these effects analytically, we have chosen an idealized geometry of two parallel infinitely long wires above an infinite superconducting film. However, the basic phenomenon of the creation of separated vortex and antivortex flux domes without nucleation at the film’s edges is far more general than in the geometry we have considered here.
For example, consider the case of a bulk-pinning-free superconducting film of finite size in the $xz$ plane and a small coil at $(x,y,z) = (0,y_0,0)$ a short distance above the film. Suppose the coil produces a magnetic dipole moment ${\bm m} = -\hat x m$. When $m$ is small, the film remains in the Meissner state, and if the magnetic field below the film is very small, the coil-generated dipole magnetic field and its image produce a magnetic field at the film’s top surface ${\bm H}_M =
\hat x H_{Mx} +
\hat z H_{Mz}$, where $H_{Mx}(x,\epsilon,z) = 2m(y_0^2-2x^2+z^2)/R_0^5$, $H_{Mz}(x,\epsilon,z) = -6mxz/R_0^5$, and $R_0 = \sqrt{x^2+y_0^2+z^2}$. The induced sheet-current density is ${\bm K}_M = \hat x K_{Mx} +\hat z K_{Mz}$, where $K_{Mx}(x,z) = H_{Mz}(x,\epsilon,z)$ and $K_{Mz}(x,z) =
-H_{Mx}(x,\epsilon,z)$, such that $\nabla \cdot {\bm K}_M = 0$. In the plane $z = 0$ the magnetic-field and sheet-current distributions resemble those shown in Figs. 2 and 3. The maximum magnetic field parallel to the top surface occurs at the origin, where ${\bm H}_M(0,\epsilon,0) = (2m/y_0^3)\hat x $, and the first flux penetration through the film occurs when $m$ increases to the value $m_c =
H_{c1}y_0^3/2$. The penetrating vortex splits into a vortex and an antivortex, and the vortex is driven by the Meissner screening currents via the corresponding Lorentz force ${\bm F}(x,z) = \phi_0 {\bm H}_M(x,\epsilon,z)$ to the point $(x,y,z) = (y_0/\sqrt{2},0,0)$, while the antivortex is similarly driven to $(x,y,z) = (-y_0/\sqrt{2},0,0)$. A further increase in the coil’s current ($m >
m_c$) will result in more nucleating vortices and antivortices and cause the development of vortex and antivortex flux domes centered at $(x,y,z) =
(\pm y_0/\sqrt{2},0,0)$. Vortex and antivortex flux domes also could be produced by bringing a permanent magnet with magnetic dipole moment ${\bm m} = -\hat x m$ close to the film.
Similar effects should occur in superconducting films with bulk pinning. In particular, when the maximum magnetic field parallel to the surface, accounting for image fields, exceeds $H_{c1} + K_c$, vortex and antivortex flux domes should be produced with properties similar to those described in Secs. \[Sec\_Kc>0\] and \[Sec\_Kc\_strong\]. In the above geometry with a small coil or a permanent magnet producing a magnetic moment ${\bm m} = -\hat x m$, we expect that the main effect of bulk pinning will be to reduce the separation between the vortex and antivortex flux domes.
In this paper we have confined our attention to the case in which an initially flux-free film is subjected to locally applied magnetic fields increasing in magnitude. We expect that interesting hysteretic effects, similar to those in a finite-width film with a geometrical barrier,[@Benkraouda96] will occur when ac magnetic fields are applied. For example, consider a bulk-pinning-free superconducting film in the geometry studied in Sec. \[Sec\_SC-wire\] but with $I_0$ being cycled. As $I_0$ is increased from zero, we expect vortex and antivortex domes to develop as predicted in Sec. \[Sec\_Kc=0\], where the dome boundaries $a$ and $b$ are determined in part by the condition that $H_x(0,\epsilon) =
H_{c1}$ \[Eq. \]. Suppose that these values are $a_{max}$ and $b_{max}$ when $I_0$ increases to some maximum value $I_{max}$, where $I_{c1} < I_{max} < 2
I_{c1}$, $a_{max} = x_0^2/b_{max} > x_0,$ and $b_{max} < x_0 = \sqrt{y_1 y_2}$. If $I_0$ is now decreased, the condition that $H_x(0,\epsilon) = H_{c1}$ is replaced by the condition that the magnetic flux under each flux dome remains constant; i.e., ${\mbox{Re}\,}[{\cal G}(a)-{\cal
G}(b)]$ = const. As a result, as $I_0$ decreases, the width of each dome increases while its height decreases; i.e., $b$ decreases ($b < b_{max}$) and $a$ increases ($a > a_{max}$). The resulting values of $b$ and $a$ can be calculated as functions of $I_0$ using the relations $a = x_0^2/b$ and $$I_0 \Delta g(a,b) = I_{max} \Delta g(a_{max},b_{max}),$$ where \[see Eqs. and \] $$\begin{aligned}
\Delta g(a,b)& =& {\rm Re}\{[g_0(a,y_1)-g_0(a,y_2)] \nonumber \\
&&-[g_0(b,y_1)-g_0(b,y_2)]\}.\end{aligned}$$ So long as $b >0$, vortex-antivortex annihilation cannot occur because the sheet current flowing in the region $|x| < b$ still keeps vortices and antivortices apart. In fact, a subsequent increase of $I_0$ would produce reversible changes in the width and height of the vortex and antivortex domes, provided $I_0 <
I_{max}$. However, the vortex-free gap of width $2b$ closes when $I_0$ is decreased to the value $I_{ex}$ at which $b \to 0$ and $a
\to
\infty$, where $$I_{ex}=I_{max} \Delta g(a_{max},b_{max})/\Delta g(\infty,0)$$ and $\Delta g(\infty,0) = \ln (y_2/y_1)$. When $I_0 = I_{ex}$, the sheet-current density becomes everywhere zero, the film appears as if it were completely incapable of screening, and the magnetic field distribution (averaged over a length of the order of the intervortex separation) is essentially the same as it would be in the absence of the film. On the other hand, viewing the field distribution as a linear superposition of the Meissner-state response and a vortex-antivortex distribution, we see that when the gap of width $2b$ closes, vortex-antivortex annihilation begins to occur at $x = 0$, and magnetic flux begins to exit from the vortex and antivortex domes. As $I_0$ decreases from $I_{ex}$ to zero, the magnitude of the magnetic flux under each dome decreases to zero. When $I_0 = 0$, the film is again flux-free, and as $I_0$ further decreases to $-I_{max}$, the behavior is very similar to that for increasing $I_0$, except that the roles of vortices and antivortices are interchanged.
We expect that similar but somewhat more complicated hysteretic effects will occur in the presence of bulk pinning. In the case of strong bulk pinning ($K_c \gg
H_{c1}$), the role of $H_{c1}$ can be neglected, and the hysteretic properties can be calculated analytically as in Ref. , which treats the response of a superconducting film to currents in linear wires in arrangements similar to that discussed in Sec. \[Sec\_SC-wire\]. Such calculations illuminate the fundamental physics underlying the ac technique introduced by Claassen [*et al.*]{}[@Claassen91] to determine the critical current density $j_c$ in superconducting films. This technique employs a small coil, placed just above the film, carrying a sinusoidal current. When the current amplitude exceeds the value at which the maximum induced sheet-current density reaches $K_c=j_c d$, a third-harmonic voltage appears in the coil.[@Claassen91; @Poulin93; @Mawatari02; @Yamasaki03] A similar technique was introduced by Hochmuth and Lorenz.[@Hochmuth94]
The effects discussed in this paper and applied to type-II superconducting films are quite general and also should be observed in type-I superconductors. Magnetic flux domes consisting of intermediate-state regions containing multiply quantized flux tubes have been observed in type-I strips in which the geometric barrier plays a dominant role.[@Castro99] It is therefore likely that separated domes of positive and negative magnetic flux produced in response to nearby current-carrying wires, coils, or permanent magnets also will be observed in weak-pinning type-I superconducting films, foils, or plates when the net parallel field at the surface exceeds the bulk thermodynamic critical field $H_c$ (or, when bulk pinning is present, $H_c +K_c$). In type-I superconductors, however, we expect that the magnetic flux will enter in the form of the intermediate state. The analog of a vortex dome will be an intermediate-state region consisting either of an array of multiply quantized flux tubes or a meandering normal-superconducting domain structure carrying magnetic flux up through the film, while the analog of an antivortex dome will be a similar intermediate-state region carrying magnetic flux down through the film. Such magnetic structures should be observable by magneto-optics or related means by placing the magnetic-field source on one side of the sample and the magnetic-field detector on the opposite side.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
We thank V. G. Kogan for stimulating discussions. This manuscript has been authored in part by Iowa State University of Science and Technology under Contract No. W-7405-ENG-82 with the U.S. Department of Energy.
\[A\]Screening Effects
======================
The primary purpose of this paper is to describe in detail how magnetic flux penetrates through a superconducting film in the form of vortices and antivortices, using the assumption that the magnetic field below the film is initially negligibly small. In the following we present results confirming that this is an excellent approximation when $d > \lambda$ and $d \ll y_1$.
Consider the experimental configuration shown in Fig. 1 but allow for the finite thickness $d$ of the superconducting film in the region $|z| < d/2$. When the film, characterized by the London penetration depth $\lambda$, is in the Meissner state, the vector potential and the magnetic field throughout all space, as well as the supercurrent density in the film, can be calculated as described in Ref. . The results for the $x$ components of the magnetic fields $H_x(0,\mp d/2)$ at the bottom and top surfaces of the film, expressed in units of $H_{0x}(0,0)$ \[Eq. \], are
$$\begin{aligned}
\frac{H_x(0,-d/2)}{H_{0x}(0,0)}&=&\frac{2 y_1 y_2}{(y_2 - y_1)}
\int_0^\infty \Big[ \frac{Qq}
{2qQ \cosh(Qd) +(q^2 + Q^2) \sinh(Qd)}\Big](e^{-qy_1}-e^{-qy_2})e^{qd/2} dq,
\label{Hbot} \\
\frac{H_x(0,+d/2)}{H_{0x}(0,0)}&=&\frac{2 y_1 y_2}{(y_2 - y_1)}
\int_0^\infty \Big[ \frac{Q[q \cosh(Qd)+Q \sinh(Qd)]}
{2qQ \cosh(Qd) +(q^2 + Q^2) \sinh(Qd)}\Big] (e^{-qy_1}-e^{-qy_2})e^{qd/2} dq,
\label{Htop}\end{aligned}$$
where $Q = \sqrt{q^2+\lambda^{-2}}$. These quantities are plotted as the solid curves in Figs. \[fig22\] and \[fig23\] as functions of $\lambda/d$ for the case that $d = y_1/1000$ and $y_2 =2 y_1$.
![Magnetic field just below the film $H_x(0,-d/2)/H_{x0}(0,0)$ vs $\lambda/d$, shown here for $d = y_1/1000$ and $y_2 = 2y_1$ as calculated from Eq. (solid curve), Eq. (long-dashed curve), and Eq. (short-dashed curve). []{data-label="fig22"}](Fig22.eps){width="8cm"}
![Magnetic field just above the film $H_x(0,+d/2)/H_{x0}(0,0)$ vs $\lambda/d$, shown here for $d = y_1/1000$ and $y_2 = 2y_1$ as calculated from Eq. (solid curve) and Eq. (dashed curve). A plot of the same quantity using Eq. would be indistinguishable from the solid curve.[]{data-label="fig23"}](Fig23.eps){width="8cm"}
When $\lambda \to 0$, the film screens perfectly, such that $H_x(0,-d/2)=0$ and $H_x(0,+d/2)=2H_{x0}(0,d/2)$. In the opposite limit, when $\lambda \to \infty$, $H_x(0,-d/2)=H_{x0}(0,-d/2)$ and $H_x(0,d/2)=H_{x0}(0,d/2)$, where $H_{x0}(0,y) = I_0(y_2-y_1)/2\pi( y_1-y)(y_2-y)$ is the magnetic field produced by the wires in the plane $x=0$ for $y<y_1$ in the film’s absence.
Equations and reduce to simpler expressions when $d \ll y_1$ and either $\lambda < d$ or, if $\lambda > d$, the Pearl length[@Pearl64] $\Lambda = \lambda^2/d$ obeys $\Lambda \ll y_1$. Then to good approximation $q$ can be set equal to zero inside the brackets and $Q$ can be replaced by $1/\lambda$. The resulting integrals yield $$\begin{aligned}
\frac{H_x(0,-d/2)}{H_{0x}(0,0)}&=&\frac{2 (y_1 + y_2) \lambda}{y_1 y_2
\sinh{(d/\lambda)}}
\label{Hbotsinh} \\
\frac{H_x(0,+d/2)}{H_{0x}(0,0)}&=&2-\frac{2 (y_1 + y_2) \lambda}{y_1 y_2
\tanh{(d/\lambda)}}.
\label{Htoptanh}\end{aligned}$$ As shown by the long-dashed curves in Figs. \[fig22\] and \[fig23\], these expressions are excellent approximations, indistinguishable from the solid curves, when $\lambda/d < 1$ or $\Lambda \ll y_1$ when $\lambda/d > 1$. However, the long-dashed curves deviate significantly from the solid curves for $\lambda/d >
10$, which is to be expected, since for the parameters used for the figures, $\Lambda \approx y_1$ when $\lambda/d \approx 30$.
To evaluate Eqs. and for all values of the Pearl length[@Pearl64] $\Lambda =
\lambda^2/d$ when $d
\ll y_1$ and $\lambda \gg d$, it is a good approximation to ignore $q^2$ relative to $1/\lambda^2$ inside the brackets and to replace $Q$ by $1/\lambda$ but to retain the terms proportional to $q$ in the denominators. This approximation is equivalent to the assertion that when $d \ll \lambda$, the only length that determines the screening properties of the film is $\Lambda$. This procedure yields the following approximate results: $$\begin{aligned}
H_x(0,-d/2)/H_{0x}(0,0)&=&1-I,
\label{HbotLambda} \\
H_x(0,+d/2)/H_{0x}(0,0)&=&1+I,
\label{HtopLambda} \end{aligned}$$ where $$\begin{aligned}
I&=&\frac{y_1 y_2}{(y_2 - y_1)}
\int_0^\infty\frac{1}
{1+2q\Lambda}(e^{-qy_1}-e^{-qy_2}) dq,
\label{F1} \\
&=&\frac{y_1 y_2}{2(y_2 - y_1)\Lambda}[e^{y_2/2\Lambda}{\rm{Ei}}(-y_2/2\Lambda)
\nonumber \\
&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
-e^{y_1/2\Lambda}{\rm{Ei}}(-y_1/2\Lambda)],
\label{F2}\end{aligned}$$ and Ei($x$) is the exponential integral. On the scale of Fig. \[fig22\], the approximation for $H_x(0,-d/2)$ given in Eq. , shown as the short-dashed curve, is indistinguishable from the solid curve for values of $\lambda/d > 3$. Surprisingly, on the scale of Fig. \[fig23\], the approximation for $H_x(0,d/2)$ given in Eq. is indistinguishable from the solid curve for all values of $\lambda/d$.
\[B\]Complex field and complex potential
========================================
In the following we derive the complex field ${{\cal H}}(\zeta)$ and complex potential $\cal G(\zeta)$ satisfying the boundary conditions that the perpendicular ($y$) component of the magnetic field in the plane of the film obeys $H_y(x,0)=0$ for $|x| < b$ or $|x| > a$ and that the sheet-current density in the $z$ direction obeys $K_z(x)
= - K_c$ for $b < |x| < a$. We begin by considering the function $$F(\zeta)=[{{\cal H}}(\zeta)-{{\cal H}}_{\|}]/\phi(\zeta),
\label{AF}$$ where ${{\cal H}}(\zeta)$ is defined in Eq. , $${{\cal H}}_{\|}(\zeta)=i \frac{I_0}{2\pi}\Big(\frac{y_1}{\zeta^2+y_1^2}
-\frac{y_2}{\zeta^2+y_2^2}\Big),
\label{Hpara}$$ and $$\phi(\zeta)=\sqrt{(a^2-\zeta^2)(\zeta^2-b^2)}.
\label{Aphi}$$ For $|\zeta| \to \infty$, $\phi(\zeta) \to \mp i\zeta^2$, where the upper (lower) sign holds in the upper (lower) half plane. Just above (below) the real axis, $$\phi(x \pm i\epsilon)=
\begin{cases} \displaystyle
\pm i \tilde \phi(x),
& |x|<b \mbox { or } |x| > a, \\
\tilde \phi(x), & b<|x|<a, \\
\end{cases}
\label{Aphixpm}$$ where $$\tilde \phi(x)=
\begin{cases} \displaystyle
\sqrt{(a^2-x^2)(b^2-x^2)},
& |x|<b , \\
{\rm sgn}(x) \sqrt{(a^2-x^2)(x^2-b^2)}, & b<|x|<a, \\
- \sqrt{(x^2-a^2)(x^2-b^2)}, & |x|>a .
\end{cases}
\label{Atildephi}$$ As can be seen from Eqs. and , $F(\zeta)$ is an analytic function of $\zeta = x + iy$ except for poles at $\zeta = \pm i y_1$ and $\zeta = \pm i y_2$, and a branch cut along the real axis.
Next consider the following integral around the closed contour $C$ consisting of a line just above the real axis at $\zeta' = u + i \epsilon$ from $u=-\infty$ to $u=+\infty$, the infinite circle at $\zeta' = R e^{i\theta}$ from $\theta = 0$ to $\theta = 2 \pi$ with $R\to
\infty$, and a line just below the real axis at $\zeta' = u - i \epsilon$ from $u=+\infty$ to $u=-\infty$, $$\oint_C d\zeta' \frac{F(\zeta')}{\zeta'-\zeta}
=\int_{-\infty}^{+\infty}du\frac{F(u+i\epsilon)-F(u-i\epsilon)}{u-\zeta},
\label{Fintegral}$$ where the integral around the infinite circle vanishes because $|F(\zeta)|
\propto |H(\zeta)/\zeta^2|\to 0$ as $|\zeta| \to \infty$. Using the residue theorem, accounting for the poles of the integrand on the left-hand side of Eq. at $\zeta' = \zeta, i y_1, i y_2, -i y_1,$ and $-iy_2$, we obtain $$F(\zeta)-F_0(\zeta)
=\frac{1}{2 \pi i}
\int_{-\infty}^{+\infty}du\frac{F(u+i\epsilon)-F(u-i\epsilon)}{u-\zeta},
\label{F-F0}$$ where $$\begin{aligned}
F_0(\zeta)&=&\frac{I_0}{2\pi}\Big(\frac{y_1}{s_1(\zeta^2+y_1^2)}
-\frac{y_2}{s_2(\zeta^2+y_2^2)}\Big),\\
s_1&=&\sqrt{(a^2+y_1^2)(b^2+y_1^2)}, \\
s_2&=&\sqrt{(a^2+y_2^2)(b^2+y_2^2)}.
\label{F0s1s2}\end{aligned}$$ From Eqs. , , and , we find that $${{\cal H}}(x\pm i\epsilon)-{{\cal H}}_{\|}(x\pm i\epsilon)=H_y(x,0)\mp i K_z(x)/2,$$ such that Eqs. and yield $$F(u+i\epsilon)-F(u-i\epsilon)=
\begin{cases} \displaystyle
-2iH_y(x,0)/\tilde \phi(x),
& |x|<b, \\
-iK_z(x)/\tilde \phi(x), & b<|x|<a ,\\
-2iH_y(x,0)/\tilde \phi(x), & |x|>a .\\
\end{cases}
\label{Fdiff}$$ However, $H_y(x,0) = 0$ for $|x|<b$ or $|x|>a$, and $K_z(x) = -K_c$ for $b <
|x| < a$. Using Eqs. and and evaluating the integrals, we obtain the following expressions for the complex field ${{\cal H}}(\zeta)$ and complex potential ${\cal G}(\zeta) =\int_{i\epsilon}^\zeta {{\cal H}}(\zeta') d \zeta'$: $$\begin{aligned}
{{\cal H}}(\zeta)\!\!&=\!\!&\frac{I_0}{2\pi} \Big\{ \frac{y_1[i
+\phi(\zeta)/s_1]}{\zeta^2+y_1^2}
\! -\!\frac{y_2[i
+\phi(\zeta)/s_2]}{\zeta^2+y_2^2}
\Big\}\nonumber \\
&&\pm i K_c/2 ,
\label{cH_zero}\\
{\cal G}(\zeta)\!\! &=\!\!& \frac{I_0}{2\pi}
\left[ g_0(\zeta,y_1) -g_0(\zeta,y_2) \right]
\pm i\frac{K_c}{2}\zeta ,
\label{cG_zero}\end{aligned}$$ where $$\begin{aligned}
g_0(\zeta,y) &=& \int_{\pm i\epsilon}^\zeta d\zeta'
\frac{y[i+\phi(\zeta')/s]}{\zeta'^2+y^2}
\nonumber\\
&=& i\arctan (\zeta/y) \nonumber\\
&& {}\mp\frac{i}{asy} [ a^2y^2\bm{E}(\theta,k)
+y^2(b^2+y^2)\bm{F}(\theta,k)\nonumber\\
&& -(a^2+y^2)(b^2+y^2)\bm{\Pi}
(\theta,-b^2/y^2,k) ],
\label{g0} \\
s &=& \sqrt{(a^2+y^2)(b^2+y^2)}, \\
\theta &=& \arcsin(\zeta/b), \mbox{ and } \\
k &=& b/a , \end{aligned}$$ where $\bm E$, $\bm K$, and $\bm \Pi$ are incomplete elliptic integrals.
When $b=0$, the following replacements can be made in the above expressions, $$\begin{aligned}
\phi(\zeta)&=&\zeta \sqrt{a^2-\zeta^2}, \label{phi0}\\
s_1&=&y_1\sqrt{a^2+y_1^2}, \label{s10}\\
s_2&=&y_2\sqrt{a^2+y_2^2}, \label{s20}\end{aligned}$$ and $\bm E$, $\bm K$, and $\bm \Pi$ can be evaluated to obtain $$\begin{aligned}
g_0(\zeta,y)
& = i\arctan (\zeta/y)+
\sqrt{\frac{a^2-\zeta^2}{a^2+y^2}}-\frac{a}{\sqrt{a^2+y^2}} \nonumber \\
&
-\mbox{arctanh} \sqrt{\frac{a^2-\zeta^2}{a^2+y^2}}
+\mbox{arctanh} \frac{a}{\sqrt{a^2+y^2}}.
\label{g0strong}\end{aligned}$$
It follows from Eq. that the integral $\int {{\cal H}}(\zeta)
d\zeta$ around the great circle at $|\zeta| \to \infty$ yields $i \int K_z(x)
dx$ along the real axis. The requirement that the film carries no net current is thus equivalent to the requirement that $\int {{\cal H}}(\zeta)
d\zeta = 0.$ Using Eq. and the property that for $|\zeta| \to
\infty$, $\phi(\zeta) \to \mp i\zeta^2$, where the upper (lower) sign holds in the upper (lower) half plane, we obtain the requirement that when $K_c = 0$, $y_1/s_1 =y_2/s_2$. Solving the latter equation, we obtain the following condition relating $a$ and $b$, $$ab = x_0^2=y_1 y_2.
\label{condition1}$$ Similarly, when $ K_c>0$ and $b >0$, the requirement that the film carries no net current leads to the condition that $$\frac{I_0}{\pi}\Big(\frac{y_1}{\sqrt{(a^2+y_1^2)(b^2+y_1^2)}}
-\frac{y_2}{\sqrt{(a^2+y_2^2)(b^2+y_2^2)}}\Big)=K_c.
\label{condition2}$$ When $ K_c>0$ but $b =0$, the same requirement leads to the condition that $$\frac{I_0}{\pi}\Big(\frac{1}{\sqrt{a^2+y_1^2}}-\frac{1}{\sqrt{a^2+y_2^2}}\Big)=K_c.
\label{condition3}$$
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|
**QUANTUM GROUPS AND NON-COMMUTATIVE COMPLEX ANALYSIS**
S. Sinel’shchikov and L. Vaksman
The problem of uniform approximation by analytic polynomials on a compact $K\subset\mathbb{C}$ made an essential impact to the function theory and the theory of commutative Banach algebras. This problem was solved by S. Mergelyan. Much later a theory of uniform algebras was developed and an abstract proof of Mergelyan’s theorem was obtained.
The work by W. Arveson [@Arv1] starts an investigation of non-commutative analogs for uniform algebras. In particular, a notion of the Shilov boundary for a subalgebra of a $C^*$-algebra have been introduced therein. So, the initial results of non-commutative complex analysis were obtained. We assume the basic concepts of that work known to the reader.
In mid’90-s an investigation of quantum analogs for bounded symmetric domains has been started within the framework of the quantum group theory [@Drinf1]. The simplest of those is a unit ball in $\mathbb{C}^n$. Our goal is to explain that the quantum sphere is the Shilov boundary for this quantum domain. The subsequent results of the authors on non-commutative function theory and quantum groups are available at www.arxiv.org . Specifically, we obtained some results on weighted Bergman spaces, the Berezin transform, and the Cauchy-Szegö kernels for quantum bounded symmetric domains introduced in [@SV1].
In what follows the complex numbers are assumed as a ground field and all the algebras are assumed to be unital. In what follows $q\in(0,1)$.
To introduce a quantum unit ball, consider a $*$-algebra $\operatorname{Pol}(\mathbb{C}^n)_q$ given by the generators $z_1,z_2,\ldots,z_n$ and the defining relations $z_jz_k=qz_kz_j$ for $j<k$, $$\begin{aligned}
z_j^*z_k=qz_kz_j^*,\quad j\ne k,\qquad
z_j^*z_j=q^2z_jz_j^*+(1-q^2)\left(1-\sum_{k>j}z_kz_k^*\right).\end{aligned}$$ This $*$-algebra has been introduced by W. Pusz and S. Woronowicz [@PWor] where one can find a description (up to unitary equivalence) of its irreducible $*$-representations $T$. One can demonstrate that $0<\|f\|\overset{\mathrm{def}}{=}\sup\limits_T\|T(f)\|<\infty$ for all non-zero $f\in\operatorname{Pol}(\mathbb{C}^n)_q$ and that its $C^*$-enveloping algebra $C(\mathbb{B})_q$ is a q-analogue for the $C^*$-algebra of continuous functions in the closed unit ball in $\mathbb{C}^n$. A plausible description of this $C^*$-algebra has been obtained by D. Proskurin and Yu. Samoilenko [@PrSam1].
To introduce a quantum unit sphere, consider a closed two-sided ideal $J$ of the $C^*$-algebra $C(\mathbb{B})_q$ generated by $1-\sum\limits_{j=1}^nz_jz_j^*$. Obviously, the $C^*$-algebra $C(\partial\mathbb{B})_q\overset{\mathrm{def}}{=}C(\mathbb{B})_q/J$ is a q-analogue for the algebra we need. Thus the canonical onto morphism $$j_q:C(\mathbb{B})_q\to C(\partial\mathbb{B})_q$$ is a $q$-analogue for the restriction operator of a continuous function onto the boundary of the ball.
The closed subalgebra $A(\mathbb{B})_q\subset C(\partial\mathbb{B})_q$ generated by $z_1,z_2,\ldots,z_n$ is a $q$-analogue for the algebra of continuous functions in the closed ball which are holomorphic in its interior.
Let $j_{A(\mathbb{B})_q}$ be the restriction of the homomorphism $j_q$ onto the subalgebra $A(\mathbb{B})_q$.
The homomorphism $j_{A(\mathbb{B})_q}$ is completely isometric.
This result is a q-analogue of the well known maximum principle for holomorphic functions. By an Arveson’s definition [@Arv1] this means that $J$ is a boundary ideal for the subalgebra $A(\mathbb{B})_q$. A proof of this theorem elaborates the methods of quantum group theory and theory of unitary dilations [@SeNaFo].
One can use the deep result of M. Hamana [@Ham1] (on existence of the Shilov boundary) to prove the following simple
$J$ is the largest boundary ideal for $A(\mathbb{B})_q$.
Thus the quantum sphere is the Shilov boundary of the quantum ball. A more detailed exposition of this talk is available in [@Vaks].
The second named author would like to express his gratitude to M. Livšitz and V. Drinfeld who taught him the theory of non-selfadjoint linear operators and quantum groups, respectively.
[1]{}
W. B. Arveson. Subalgebras of ${C}^*$-algebras. , 123:141–122, 1969.
V. Drinfeld. Quantum groups. In A. M. Gleason, editor, [*Proceedings of the International Congress of Mathematicians (Berkeley, 1986)*]{}, pages 798–820. Amer. Math. Soc., Providence, RI, 1987.
M. Hamana. Injective envelopes of operator systems. , 15:773 – 785, 1979.
$\!\!$Nagy and C. Foiaş. . Masson, Académiai Kiado, 1967.
D. Proskurin and Yu. Samoilenko. Stability of the [$C^*$]{}-algebra associated with the twisted [CCR]{}. , 5(4):456–460, 2002.
W. Pusz and S. Woronowich. Twisted second quantization. , 27:231 – 257, 1989.
S. Sinel’shchikov and L. Vaksman. On q-analogues of bounded symmetric domains and [D]{}olbeault complexes. , 1:75–100, 1998.
L. Vaksman. Maximum principle for holomorphic functions in the quantum ball. , 10(1):12 – 28, 2003.
|
**Ill-posedness for the Navier-Stokes equations in critical Besov spaces $\dot B^{-1}_{\infty,q}$**
Baoxiang Wang[^1] \
**Abstract. We study the Cauchy problem for the incompressible Navier-Stokes equations in two and higher spatial dimensions $$\begin{aligned}
u_t -\Delta u+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= \delta u_0. \label{NS}\end{aligned}$$ For arbitrarily small $\delta>0$, we show that the solution map $\delta u_0 \to u$ in critical Besov spaces $\dot B^{-1}_{\infty,q}$ ($\forall \ q\in [1,2]$) is discontinuous at origin. It is known that the Navier-Stokes equation is globally well-posed for small data in $BMO^{-1}$ ([@KoTa01]). Taking notice of the embedding $\dot B^{-1}_{\infty,q} \subset BMO^{-1}$ ($q\le 2$), we see that for sufficiently small $\delta>0$, $u_0\in \dot B^{-1}_{\infty,q} $ ($q\le 2$) can guarantee that has a unique global solution in $BMO^{-1}$, however, this solution is instable in $ \dot B^{-1}_{\infty,q} $ and the solution can have an inflation in $\dot B^{-1}_{\infty,q} $ for certain initial data. So, our result indicates that two different topological structures in the same space may determine the well and ill posedness, respectively.\
**Key words and phrases. Navier-Stokes equations; Besov spaces, Ill-posedness.\
[**2000 Mathematics Subject Classifications.**]{} 35Q30, 35K55.\
****
Introduction {#sect1}
============
We study the ill-posedness for the Cauchy problem of the incompressible Navier-Stokes equations (NS): $$\begin{aligned}
u_t -\Delta u+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= \delta u_0. \label{1.1}\end{aligned}$$ where $t\in \mathbb{R}^{+}=[0,\infty)$, $x\in \mathbb{R}^{n} \ (n\geq 2)$, $u=(u_1,...,u_n)$ denotes the flow velocity vector and $p(t,x)$ describes the scalar pressure. $0<\delta \ll 1$ denotes a small parameter, $\nabla=(\partial_1,...,\partial_n)$ is the gradient operator, $\Delta=\partial^2_1+...+\partial^2_n$ is the Laplacian, $u_0(x)=(u^0_1,...,u^0_n)$ is a given velocity with $\mathrm{div}
u_0=\partial_1 u^0_1+...+\partial_n u^0_n =0$. It is easy to see that can be rewritten as the following equivalent form: $$\begin{aligned}
u_t -\Delta u+\mathbb{P}\ \textrm{div}(u\otimes u)=0, \ \
u(0, x)= \delta u_0,
\label{2.1}\end{aligned}$$ where $\mathbb{P}=I-\nabla \Delta^{-1}{\rm div}$ is the projection operator onto the divergence free vector fields.
It is known that is essentially equivalent to the following integral equation: $$\begin{aligned}
\label{1.3}
u(t) = e^{ t \Delta } u_0 + \int^t_0 e^{ (t-\tau) \Delta } \mathbb{P}\ \textrm{div}(u\otimes u)(\tau) d\tau.\end{aligned}$$
Note that is scaling invariant in the following sense: if $u$ solves , so does $u_{\lambda}(t, x)=\lambda u(\lambda^{2 }t, \lambda x)$ and $p_\lambda (t, x)=\lambda^2 p(\lambda^{2 }t, \lambda x)$ with initial data $\lambda u_0(\lambda x)$. A function space $X$ defined in $\mathbb{R}^n$ is said to be a *critical space* for if the norms of $u_{\lambda}(0,x)$ in $X$ are equivalent for all $\lambda>0$ (i.e., $\|u_\lambda(0,\cdot)\|_X\sim \|u_0\|_X$). It is easy to see that the following spaces are critical spaces for NS: $$\begin{aligned}
& \mbox{Besov type spaces:} \ \dot{B}^{n/p-1}_{p, q} \subset \dot{B}^{-1}_{\infty, r} \subset \dot{B}^{-1}_{\infty, \infty}\ (p< \infty, \ q\le r), \nonumber\\
& \mbox{Triebel type spaces:} \ L^n = \dot{F}^{0}_{n, 2} \subset \dot{F}^{n/p-1}_{p, q} \subset \dot{F}^{-1}_{\infty, r} \ (n< p< \infty). \nonumber\end{aligned}$$
NS has been extensively studied in the past twenty years; cf. [@BaBiTa12; @Can97; @Ch99; @ChGaPa11; @EsSeSv03; @FoTe89; @Iw10; @Ka84; @KeKo11; @KoTa01; @Pl96] and references therein. For the initial data in critical Besov type spaces, Cannone [@Can97], Planchon [@Pl96] and Chemin [@Ch99] obtained global solutions in 3D for small data in critical Besov spaces $\dot{B}^{3/p-1}_{p, q}$ for all $p<\infty, \ q\le \infty$. In the case $p=\infty$, Bae, Biswas and Tadmor [@BaBiTa12] can show the global well posedness of the solutions in 3D for the sufficiently small initial data in $\dot{B}^{-1}_{\infty, q} \cap \dot{B}^{0}_{3, \infty}$ with $1\le q<\infty$. Bourgain and Pavlovic [@BoPa08] showed the *ill-posedness* of NS in $\dot{B}^{-1}_{\infty,\infty}$, i.e., the solution map is discontinuous in $\dot{B}^{-1}_{\infty,\infty}$, Germain [@Ge08] proved that the solution map of NS is not $C^2$ in $\dot{B}^{-1}_{\infty, q} $ for any $q>2$, Yoneda [@Yo10] can show that the solution map is discontinuous in $\dot{B}^{-1}_{\infty, q} $ for any $q>2$ and in fact, he constructed a logarithmic type Besov space $V$ very near to $\dot{B}^{-1}_{\infty, 2}$ so that NS is ill-posed in $V$. Up to now, the largest Besov-type space on initial data for which NS is well-posed is still unknown.
For the initial data in critical Triebel type spaces, Koch and Tataru [@KoTa01] obtained the global well-posedness for small initial data in $BMO^{-1}= \dot{F}^{-1}_{\infty, 2}$, Yoneda [@Yo10] pointed out that his argument implies that NS is ill-posed in $\dot{F}^{-1}_{\infty, q}$ for all $q>2$ (cf. also [@DeYa13]). Now let us recall Koch and Tataru’s result:
\[NSWellposed\] [([@KoTa01], Global well-posedness for small data in $BMO^{-1}$)]{} Let $u_0 \in {BMO}^{-1}$ with $\|u_0\|_{BMO^{-1}} \le 1$. Then there exists a $\delta_0>0$ such that for any $0<\delta \le \delta_0$, NS has a unique global solution $u(\delta,t) \in X$ with $\|u\|_{X} \le C\delta$, where $$\|u\|_X := \sup_{t>0} t^{1/2}\|u(t,\cdot)\|_{L^\infty(\mathbb{R}^n)} + \sup_{x\in \mathbb{R}^n, R>0} |B(x, R)|^{-1/2} \|u(t,y)\|_{L^2_{t,y}([0,R^2]\times B(x,R))}.$$
Moreover, Auscher, Dubois and Tchamitchian [@AuDuTc06] obtained that Koch and Tataru’s solution is stable and belongs to $L^\infty (0,\infty; BMO^{-1})$. Recalling that the inclusions $ \dot{B}^{s}_{p, \min\{p,q\}} \subset \dot{F}^{s}_{p, q} \subset \dot{B}^{s}_{p, \max\{p,q\}}$ ($q>1$), one sees that $\dot{B}^{-1}_{\infty, 2}\subset \dot{F}^{-1}_{\infty, 2}$. According to their results, $u_0\in \dot{B}^{-1}_{\infty, 2}$ and $0<\delta\ll 1$ imply that NS has a unique global solution in $BMO^{-1}$. So, it seems natural to conjecture that NS is also globally well-posed in $\dot{B}^{-1}_{\infty, 2}$ for sufficiently small $\delta>0$. However, in this paper we will show the following negative result:
\[NSIll\] [(Ill-posedness of the solution)]{} Let $n\ge 2$, $1\le q\le 2$, $0<\delta \ll 1$. Let $u(\delta, t)$ be Koch and Tataru’s solution of . Then the solution map $\delta u_0 \to u(\delta, t)$ in $\dot{B}^{-1}_{\infty, q}(\mathbb{R}^n)$ is discontinuous at $\delta=0$. More precisely, for any $N\gg 1$, there exists $u_0 \in \dot{B}^{-1}_{\infty, q}(\mathbb{R}^n)$ with $\|u_0\|_{\dot{B}^{-1}_{\infty, q}} \lesssim 1$, $t \le 1/N$ such that $$\left\| u (\delta, t )\right\|_{\dot{B}^{-1}_{\infty, q}(\mathbb{R}^n)} \ge (\log N)^{1/2 q}.$$
We remark that our result also holds for all $2\le q<\infty$. In the proof of Theorem \[NSIll\] we have no any conditions on $q\ge 1$. The techniques used in this paper are quite different from the arguments as in [@BoPa08; @Ge08; @Yo10].
Throughout this paper, $C\ge 1, \ c\le 1$ will denote constants which can be different at different places, we will use $A\lesssim B$ to denote $A\leqslant CB$. We denote by $L^p=L^p(\mathbb{R}^n)$ the Lebesgue space on which the norm is written as $\|\cdot\|_p$. Now let us recall the definition of Besov type spaces; cf. [@BL; @Tr]. Let $\psi: \mathbb{R}^n\rightarrow
[0, 1]$ be a smooth cut-off function which equals $1$ on the ball $B(0,5/4):=\{\xi\in \mathbb{R}^n: |\xi|\le 5/4\}$ and equals $0$ outside the ball $B(0,3/2)$. Write $$\begin{aligned}
\label{phi}
\varphi(\xi):=\psi(\xi)-\psi(2\xi), \ \ \varphi_j(\xi)=\varphi(2^{-j}\xi),\end{aligned}$$ $\triangle_j:=\mathscr{F}^{-1}\varphi_j \mathscr{F}, \ j\in \mathbb{Z}$ are said to be the dyadic decomposition operators[^2]. One easily sees that ${\rm supp} \varphi_j \subset B(0, 2^{j+1})\setminus B(0, 2^{j-1})$ and $$\begin{aligned}
\label{phij}
\varphi_j(\xi)=1, \ \ \mbox{if } \xi \in B(0,5 \cdot 2^{j-2})\setminus B(0,3\cdot 2^{j-2}).\end{aligned}$$ Let $s\in \mathbb{R}$, $1\le p,q\le \infty$. The norms in homogeneous Besov and Triebel spaces are defined as follows: $$\begin{aligned}
\label{Besov} \|f\|_{\dot{B}^s_{p, q}}=
\left(\sum_{j=-\infty}^{+\infty}2^{jsq}\|\triangle_j f\|^q_{p}\right)^{1/q}, \ \
\|f\|_{ \dot{F}^s_{p, q} } = \left\|
\left(\sum_{j=-\infty}^{+\infty} |2^{js } \triangle_j
f |^q \right)^{1/q} \right\|_{p}\end{aligned}$$ with the usual modification for $q=\infty$, where we further assume $p\neq \infty$ in $\dot{F}^s_{p, q}$. Since the definition of $\dot{F}^s_{\infty, q}$ is slightly different from , we leave it into Section \[other\]. In this paper, we will frequently us the following Bernstein’s multiplier estimate (see [@BL; @WaHuHaGu11]):
\[Bern\] [(Bernstein’s multiplier estimate)]{} Let $L\in \mathbb{N}, \ L>n/2, \ \theta=n/2L$. We have $$\begin{aligned}
\label{1.7}
\|\mathscr{F}^{-1} \rho\|_{1} \lesssim \|\rho\|^{1-\theta}_{2} \left(\sum^n_{i=1} \|\partial^L_i \rho\|_2^{\theta} \right).\end{aligned}$$
Let us consider Taylor’s expansion of Koch and Tataru’s solution $u(\delta,t )$ of NS, one can find a small $\delta_0>0$ such that for any $0<\delta \le \delta_0$, $$\begin{aligned}
\label{1.8}
u(\delta, t) = \sum^\infty_{r=0} \frac{\delta^r}{r!} \frac{\partial^r u}{\partial \delta^r}(0,t) \ \ \ \mbox{in} \ \ X \cap L^\infty(0,\infty; BMO^{-1}).\end{aligned}$$ Recall that Koch and Tataru applied the contraction mapping method to show the global well-posedness of NS for small data in $BMO^{-1}$, we see that can be obtained by iterations. If NS is globally well-posed in $\dot{B}^{-1}_{\infty, q}$ ($q\in [1,2]$), the solution must coincide with the solution $u(\delta,t)$ in $ X$. So, if we can find some initial data $u_0 \in \dot{B}^{-1}_{\infty, q}$ such that the second iteration $ \delta^2 \partial_\delta^2 u (0,t)/2$ has an inflation in $\dot{B}^{-1}_{\infty, q}$ and the norm of the other terms in are much less than that of the second iteration, then our result is shown. More precisely, for some subset $\mathbb{A} \subset \mathbb{N}$, $$\begin{aligned}
\| u(\delta, t)\|_{\dot{B}^{-1}_{\infty, q}} \ge & \left( \sum_{j\in \mathbb{A}} 2^{-jq} \|\triangle_j u(\delta, t) \|^q_\infty \right)^{1/q} \nonumber\\
\ge & \frac{\delta^2}{2} \left( \sum_{j\in \mathbb{A}} 2^{-jq} \|\triangle_j \partial_\delta^2 u (0,t) \|^q_\infty \right)^{1/q} \label{1.10} \\
& - \left( \sum_{j\in \mathbb{A}} 2^{-jq} \left\|\triangle_j \sum_{r\ge 1, \ r\neq 2} \frac{\delta^r}{r!} \partial_\delta^r u (0,t) \right\|^q_\infty \right)^{1/q}. \label{1.11}\end{aligned}$$ We will show that contributes the main part and it will be quite large after any short time and is much less than .
The paper is organized as follows. In Section \[sect2\] we prove Theorem \[NSIll\] for higher dimensions $n\ge 3$ and in Section \[sect3\] we continue its proof for 2D case. In Section \[other\] we compare the solutions in different critical spaces.
Proof of Theorem \[NSIll\]: $n\ge 3$ {#sect2}
====================================
Estimates on second iteration
------------------------------
Let $u(\delta,t)$ be the solution of with $0<\delta \ll 1$, we see that $$\begin{aligned}
& \label{seconditerat} u(\delta, t)|_{\delta=0} =0, \ \ \frac{\partial u}{\partial\delta}\Big|_{\delta=0} = e^{t\Delta } u_0, \nonumber\\
& \frac{\partial^2 u}{\partial\delta^2}\Big|_{\delta=0} = 2 \int^t_0 e^{(t-\tau)\Delta } \mathbb{P} (e^{\tau\Delta } u_0 \cdot \nabla ) e^{\tau\Delta } u_0 d\tau.\end{aligned}$$ Our aim is to choose a suitable $u_0 \in \dot B^{-1}_{\infty, q}$ which is localized in one dyadic frequency $|\xi| \sim 2^k$ and to find a time $t\sim 2^{-2k}$ verifying $ \| \frac{\partial^2 u}{\partial\delta^2}(0,t) \|_{\dot{B}^{-1}_{\infty, q}(\mathbb{R}^n)} \gtrsim k^{1/q}.$ Let $k\in 16\mathbb{N}=\{16, 32, 48,...\}$, $k\gg 1$, $0<\varepsilon \ll 1$[^3]. Denote $$\begin{aligned}
a_l = 2^l (\varepsilon, 2\varepsilon, \sqrt{1-5\varepsilon^2},0,...,0), \ \
b_l =a_l/2, \ \ c_k = \frac{2^k}{\sqrt{n}} (1,1,...,1). \label{ck}\end{aligned}$$ We write $$\begin{aligned}
\mathbb{N}_k = \{l \in 4\mathbb{N}: \ k/4\le l \le k/2\}. \label{ck}\end{aligned}$$ Let $\psi$ be as in and $\varrho(\xi) = \psi(4 \xi)$ and $$\label{Phi}
\begin{cases}
\widehat{\Phi^+_l}(\xi) = e^{{\rm i}\xi a_l} (\varrho (\xi-c_k-b_l) + \varrho (\xi-c_k+b_l)),\\
\widehat{\Phi^-_l}(\xi) = e^{{\rm i}\xi a_l} (\varrho (\xi+c_k-b_l) + \varrho (\xi+c_k+b_l)).\\
\end{cases}$$ We now introduce the initial data $u_0=(u^0_1,...,u^0_n)$: $$\label{u0}
\begin{cases}
\widehat{u^0_1} (\xi) & = 2^k \sum_{l\in \mathbb{N}_k} ( \widehat{\Phi^+_l}(\xi) + \widehat{\Phi^-_l}(\xi)),\\
\widehat{u^0_2} (\xi) & = -\frac{\xi_1}{\xi_2} \widehat{u^0_1} (\xi) = - 2^k \sum_{l\in \mathbb{N}_k} \frac{\xi_1}{\xi_2} ( \widehat{\Phi^+_l}(\xi) + \widehat{\Phi^-_l}(\xi))
\end{cases}$$ and $u^0_3(x) = ... = u^0_n (x) =0.$ One can rewrite $ u^0_1 $ and $ u^0_2 $ as $$\begin{aligned}
u^0_1 (x) & = 2^k \sum_{l\in \mathbb{N}_k} ( \cos{ (x + a_l)(c_k+b_l)} + \cos{ (x + a_l)( b_l - c_k)}) \check{\varrho} (x+a_l)
\label{u01r}\\
u^0_2 (x) & = -u^0_1(x) + 2^k \sum_{l\in \mathbb{N}_k} \mathscr{F}^{-1} \frac{\xi_2-\xi_1}{\xi_2} ( \widehat{\Phi^+_l}(\xi) + \widehat{\Phi^-_l}(\xi)).
\label{u02r}
\end{aligned}$$ Now we give some explanations to the initial data. We can assume that $\varrho(\xi) $ is radial so that $\check{\varrho}$ is a real function. From $\widehat{u^0_2}= - (\xi_1/\xi_2) \widehat{u^0_1}$ it follows that ${\rm div} u_0=0$. Introducing $c_k$ is to guarantee that $\widehat{u^0_1}$ and $\widehat{u^0_2}$ are supported in the dyadic $\{\xi: 2^{k-1} \le |\xi| \le 2^{k+1}\}$, it follows that the frequency of $u_0$ is very high if $k$ is very large. The nonlinear interaction will pullback $\widehat{u^0_1}*\widehat{u^0_1}$ into much lower frequencies. In order to $\widehat{u^0_1}*\widehat{u^0_1}$ concentrated in different dyadic regions as many as possible, different $b_l$ is introduced. $a_l$ is used for controlling the superpositions of all $\check{\varrho}(\cdot+a_l)$ in .
\[NSlem1\] Let $1\leq q\le \infty$, $k\gg -\log \varepsilon$. Then $$\begin{aligned}
\label{boundu012}
\|u^0_i\|_{ \dot{B}^{-1}_{\infty,
q} }\lesssim 1, \ \ i=1,2.\end{aligned}$$
[*Proof.*]{} Since $\check{\varrho}$ is a Schwartz function, we have $$\begin{aligned}
|\check{\varrho} (x)| \lesssim (1+|x|)^{-N}, \ N \gg 1.
\label{rapiddecay}
\end{aligned}$$ It follows from and that $$\begin{aligned}
\| u^0_1 \|_\infty \lesssim 2^k \left\| \sum_{l\in \mathbb{N}_k} ( 1+|x+a_l|)^{-N} \right\|_\infty \lesssim 2^k.
\label{2.11}
\end{aligned}$$ Hence, noticing that ${\rm supp} \widehat{u^0_1}$ is included in $\{\xi: |\xi| \sim 2^k\}$, we have $$\begin{aligned}
\| u^0_1 \|_{\dot B^{-1}_{\infty, q}} \lesssim 1.
\label{boundu01}
\end{aligned}$$ Applying Young’s inequality and Bernstein’s multiplier estimate, we have for any $l\in \mathbb{N}_k$ ($k\gg -\log \varepsilon$), $$\begin{aligned}
\left\| \mathscr{F}^{-1} \frac{\xi_2-\xi_1}{\xi_2} e^{{\rm i}\xi a_l} \varrho(\xi+c_k - b_l) \right\|_\infty & \le \left\| \mathscr{F}^{-1} \frac{\xi_2-\xi_1}{\xi_2} \psi(\xi+c_k - b_l)\right\|_1 \|\check{\varrho}\|_\infty \nonumber\\
& \lesssim \left\| \mathscr{F}^{-1} \frac{\xi_2-\xi_1+ \varepsilon 2^{l-1}}{\xi_2- 2^k/\sqrt{n} +2\varepsilon 2^{l-1}} \psi(\xi)\right\|_1 \nonumber\\
& \lesssim \varepsilon 2^{-k/2}. \label{2.13}
\end{aligned}$$ Similarly, one can estimate the other terms in $\Phi^{\pm}_l$ and we have $$\begin{aligned}
\left\| \mathscr{F}^{-1} \frac{\xi_2-\xi_1}{\xi_2} \widehat{\Phi^{\lambda}_l}(\xi)\right\|_\infty \lesssim \varepsilon 2^{-k/2}, \quad \lambda=+,-. \label{2.14}
\end{aligned}$$ Collecting , –, $$\begin{aligned}
\| u^0_2 \|_{\dot B^{-1}_{\infty, q}} \lesssim 1 + \sum_{l\in \mathbb{N}_k} \left\| \mathscr{F}^{-1} \frac{\xi_2-\xi_1}{\xi_2} (\widehat{\Phi^+_l}(\xi)+ \widehat{\Phi^-_l}(\xi)) \right\|_\infty \lesssim 1+ \varepsilon k 2^{-k/2} \lesssim 1.
\end{aligned}$$ So, we have the desired bounds of $u^0_i, \ i=1,2.$ $\hfill \Box$
Recall that for the solution $u=(u_1(\delta,t),...,u_n(\delta,t))$ with initial data $\delta u_0$, $$\begin{aligned}
\frac{\partial^2 u_1}{\partial\delta^2}\Big|_{\delta=0} & = \int^t_0 e^{(t-\tau)\Delta } \left[ \sum_{i=1}^2 \partial_i (e^{\tau\Delta } u^0_i e^{\tau\Delta } u^0_1) - \partial_1 \sum_{i,j =1}^2 \frac{\partial_i \partial_j}{\Delta} (e^{\tau\Delta } u^0_i e^{\tau\Delta } u^0_j) \right]d\tau \nonumber\\
& := F_1(t,x) -F_2(t,x). \label{secondder}\end{aligned}$$ It follows that $$\begin{aligned}
& \left(\sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \frac{\partial^2 u_1}{\partial\delta^2}\Big|_{\delta=0} \right\|^q_\infty \right)^{1/q} \nonumber\\
& \quad \ge \left(\sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j F_1 \right\|^q_\infty \right)^{1/q} - \left(\sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j F_2 \right\|^q_\infty \right)^{1/q} \nonumber\\
& \quad : = A_1-A_2. \label{secondder}\end{aligned}$$ In the following, our aim is to show that for $t=\varepsilon^2 2^{-2k}$, $A_1 \gtrsim \varepsilon^3 k^{1/q}$ and $A_2 \ll \varepsilon^3 k^{1/q}$. Since $k$ can be arbitrarily large, one immediately has $\left\| \partial^2 u_1/\partial\delta^2 |_{\delta=0} \right\|_{\dot B^{-1}_{\infty, q}} \to \infty$ as $k\to \infty.$ For convenience, we denote $$\begin{aligned}
R_i(f,g) & = \int^t_0 e^{(t-\tau)\Delta } \partial_i (e^{\tau\Delta } f \ e^{\tau\Delta } g) d\tau; \label{2.18}\\
\widetilde{R}_i(f,g) & = \int^t_0 e^{(t-\tau)\Delta } \partial_i (e^{\tau\Delta } f \ e^{\tau\Delta } g - fg) d\tau. \label{2.19}\end{aligned}$$ It follows that $$\begin{aligned}
R_i(f,g) = \Delta^{-1} ( e^{t\Delta }-1) \partial_i (fg) + \widetilde{R}_i(f,g).\end{aligned}$$ Hence, we have $$\begin{aligned}
A_1 \ge & \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\|\triangle_j \Delta^{-1} ( e^{t\Delta }-1) [\partial_1 (u^0_1 u^0_1) + \partial_2 (u^0_1 u^0_2)] \|^q_\infty \right)^{1/q} \nonumber\\
& - \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\|\triangle_j [ \widetilde{R}_1(u^0_1, u^0_1) + \widetilde{R}_2 (u^0_1, u^0_2)] \|^q_\infty \right)^{1/q} := A_{11} -A_{12}. \label{2.21}\end{aligned}$$ In view of Taylor’s expansion $e^x=\sum_{r\ge 0} x^r/r! $, we have $$\begin{aligned}
A_{11} \ge & t \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\|\triangle_j [\partial_1 (u^0_1 u^0_1) + \partial_2 (u^0_1 u^0_2)] \|^q_\infty \right)^{1/q} \nonumber\\
& - t \sum_{r\ge 2} \frac{1}{r!} \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\| \triangle_j (t\Delta)^{r-1} [\partial_1 (u^0_1 u^0_1) + \partial_2 (u^0_1 u^0_2)] \|^q_\infty \right)^{1/q} \nonumber\\
\ge & t \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\|\triangle_j (\partial_1- \partial_2)(u^0_1 u^0_1) \|^q_\infty \right)^{1/q} \label{2.22} \\
& - t \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\|\triangle_j \partial_2 (u^0_1+ u^0_2) u^0_1 \|^q_\infty \right)^{1/q} \label{2.23}\\
& - t \sum_{r\ge 2} \frac{1}{r!} \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\|\triangle_j (t\Delta)^{r-1} [\partial_1 (u^0_1 u^0_1) + \partial_2 (u^0_1 u^0_2)] \|^q_\infty \right)^{1/q}. \label{2.24}\end{aligned}$$ We will show that $A_{12}$, $\eqref{2.23}$ and $\eqref{2.24}$ are much less than $\eqref{2.22}$. First, we have
\[NSlem2\] Let $1\leq q\le \infty$, $t=\eta 2^{-2k}$. Then $$\begin{aligned}
t \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\|\triangle_j (\partial_1- \partial_2)(u^0_1 u^0_1) \|^q_\infty \right)^{1/q} \gtrsim \eta \varepsilon k^{1/q}. \label{2.22bound}\end{aligned}$$
[**Proof.**]{} For the sake of convenience, we denote $$\begin{aligned}
& \widehat{\Phi^{++}_l} = e^{{\rm i}\xi a_l} \varrho (\xi- c_k- b_l) , \quad \widehat{\Phi^{+-}_l} = e^{{\rm i}\xi a_l} \varrho (\xi- c_k+ b_l) \label{NSlem2-2} \\
& \widehat{\Phi^{-+}_l} = e^{{\rm i}\xi a_l} \varrho (\xi+ c_k- b_l) , \quad \widehat{\Phi^{--}_l} = e^{{\rm i}\xi a_l} \varrho (\xi + c_k+ b_l). \label{NSlem2-3}\end{aligned}$$ Let us observe that $$\begin{aligned}
\widehat{u^0_1} * \widehat{u^0_1} = & 2^{2k} \sum_{l,m \in \mathbb{N}_k} (\widehat{\Phi^{++}_l} * \widehat{\Phi^{++}_m} + \widehat{\Phi^{+-}_l} * \widehat{\Phi^{+-}_m}) \nonumber\\
& + 2^{2k} \sum_{l,m \in \mathbb{N}_k} (\widehat{\Phi^{-+}_l} * \widehat{\Phi^{-+}_m} + \widehat{\Phi^{--}_l} * \widehat{\Phi^{--}_m}) \nonumber\\
& + 2^{2k+1} \sum_{l,m \in \mathbb{N}_k} (\widehat{\Phi^{++}_l} * \widehat{\Phi^{+-}_m} + \widehat{\Phi^{-+}_l} * \widehat{\Phi^{--}_m}) \nonumber\\
& + 2^{2k+1} \sum_{l,m \in \mathbb{N}_k} (\widehat{\Phi^{++}_l} * \widehat{\Phi^{--}_m} + \widehat{\Phi^{+-}_l} * \widehat{\Phi^{-+}_m}) \nonumber\\
& + 2^{2k+1} \sum_{l,m \in \mathbb{N}_k} (\widehat{\Phi^{++}_l} * \widehat{\Phi^{-+}_m} + \widehat{\Phi^{--}_l} * \widehat{\Phi^{+-}_m}) \nonumber\\
:= & \widehat{U}_1+...+\widehat{U}_5. \nonumber\\\label{NSlem2-1}\end{aligned}$$ Since ${\rm supp} \varrho (\cdot -a) * \varrho (\cdot -b) \subset B(a+b, 1)$, we see that $$\begin{aligned}
& {\rm supp} \widehat{U_1} \cup
{\rm supp} \widehat{U_2} \cup
{\rm supp} \widehat{U_3} \subset B(2 c_k, \ 2^{2+k/2}) \cup B(-2 c_k, \ 2^{2+k/2}). \label{NSlem2-5}\end{aligned}$$ It follows that $
\triangle_j (U_1+U_2+U_3 )=0, \ j\in \mathbb{N}_k.
$ Moreover, noticing that ${\rm supp} \widehat{\Phi^{++}_l} * \widehat{\Phi^{--}_m} \subset B(b_l -b_m, \ 1) $, we see that ${\rm supp} \widehat{\Phi^{++}_l} * \widehat{\Phi^{--}_l} \subset B(0,1) $ and ${\rm supp} \widehat{\Phi^{++}_l} * \widehat{\Phi^{--}_m} \subset \{\xi: \ 2^{l-2} \le |\xi| < 2^{l-1}\} $ if $m<l$. Hence, we have $
\triangle_j U_4=0, \ j\in \mathbb{N}_k.
$ It follows that for any $j\in \mathbb{N}_k$, $$\begin{aligned}
\triangle_j (u^0_1 u^0_1) = \triangle_j U_5 . \label{NSlem2.29}\end{aligned}$$ Let us rewrite $U_5$ as $$\begin{aligned}
\widehat{ U_5}
= & 2^{2k+1} \sum_{l \in \mathbb{N}_k} (\widehat{\Phi^{++}_l} * \widehat{\Phi^{-+}_l} + \widehat{\Phi^{+-}_l} * \widehat{\Phi^{--}_l}) \nonumber\\
& + 2^{2k+1} \sum_{l,m \in \mathbb{N}_k, l\neq m} (\widehat{\Phi^{++}_l} * \widehat{\Phi^{-+}_m} + \widehat{\Phi^{+-}_l} * \widehat{\Phi^{--}_m}):= \widehat{U_{51}} +\widehat{U_{52}}. \label{NSlem2.31}\end{aligned}$$ We easily see that $\triangle_j( \Phi^{++}_l \Phi^{-+}_l)=0$ if $l \neq j$. Moreover, noticing that $\varphi_j(\xi) =1$ for $|\xi|\in [3\cdot 2^{j-2}, 5\cdot 2^{j-2}]$, we have $\triangle_j ( \Phi^{++}_j \Phi^{-+}_j )= \Phi^{++}_j \Phi^{-+}_j$. So, $$\begin{aligned}
\triangle_j U_5 = 2^{2k+1} (\Phi^{++}_j \Phi^{-+}_j + \Phi^{+-}_j \Phi^{--}_j) + \triangle_j (U_{52}) . \label{2.30}\end{aligned}$$ In view of and , $$\begin{aligned}
& t\left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\|\triangle_j (\partial_1- \partial_2)(u^0_1 u^0_1) \|^q_\infty \right)^{1/q} \nonumber\\
& \ \ \geqslant \eta \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|(\partial_1- \partial_2) (\Phi^{++}_j \Phi^{-+}_j + \Phi^{+-}_j \Phi^{--}_j) \right \|^q_\infty \right)^{1/q} \nonumber\\
& \ \ \ \ - \eta \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|(\partial_1- \partial_2) \triangle_j U_{52} \right \|^q_\infty \right)^{1/q}.
\label{NSlem2-6}\end{aligned}$$ Noticing that $$\begin{aligned}
\Phi^{+\pm}_l = e^{{\rm i} (x + a_l)(c_k \pm b_l)} \check{\varrho} (x+a_l), \quad \Phi^{- \pm}_l = e^{{\rm i} (x + a_l)(-c_k \pm b_l)} \check{\varrho} (x+a_l),
\end{aligned}$$ one sees that $$\begin{aligned}
\Phi^{++}_j \Phi^{-+}_j + \Phi^{+-}_j \Phi^{--}_j = \cos {[ a_j (x + a_j)] } \ ( \check{\varrho} (x+a_j))^2.
\end{aligned}$$ It follows that $$\begin{aligned}
(\partial_1 -\partial_2) [\Phi^{++}_j \Phi^{-+}_j + \Phi^{+-}_j \Phi^{--}_j ] = & - \varepsilon 2^j \sin {[ a_j (x + a_j)] } \ ( \check{\varrho} (x+a_j))^2 \nonumber\\
& + \cos {[ a_j (x + a_j)] } \ (\partial_1 -\partial_2) ( \check{\varrho} (x+a_j))^2.
\end{aligned}$$ By choosing $x= -a_j + \pi/2a_j$, we have $$\begin{aligned}
\|(\partial_1 -\partial_2) (\Phi^{++}_j \Phi^{-+}_j + \Phi^{+-}_j \Phi^{--}_j ) \|_\infty \ge \varepsilon 2^j \ \check{\varrho} (\pi/2a_j)^2 \ge \varepsilon 2^{j-1} \check{\varrho} (0)^2, \label{lem2.37}
\end{aligned}$$ where we have applied the continuity of $\check{\varrho}$ and assume that $\check{\varrho} (\pi/2a_j)^2 \ge \check{\varrho} (0)^2/2$. By , $$\begin{aligned}
\eta \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|(\partial_1- \partial_2) (\Phi^{++}_j \Phi^{-+}_j + \Phi^{+-}_j \Phi^{--}_j) \right \|^q_\infty \right)^{1/q} \gtrsim \eta \varepsilon k^{1/q}.
\label{NSlem2-7}\end{aligned}$$ For any $j, l, m \in \mathbb{N}_k $ with $l<m$, if $m\neq j$, then we have $\triangle_j (\Phi^{++}_m \Phi^{-+}_l)=0$. Hence, applying the multiplier estimates, we have $$\begin{aligned}
\left\| \partial_i \triangle_j \left(\sum_{m,l\in \mathbb{N}_k; m> l} \Phi^{++}_m \Phi^{-+}_l \right) \right \|_\infty \lesssim 2^j \left\| \sum_{l<j, l\in \mathbb{N}_k} \Phi^{++}_j \Phi^{-+}_l \right \|_\infty, \ \ i=1,2.
\label{NSlem2-9}\end{aligned}$$ Using the rapid decay , we have $$\begin{aligned}
\left\| \sum_{l<j, l\in \mathbb{N}_k} \Phi^{++}_j \Phi^{-+}_l \right \|_\infty & \lesssim \left\| \sum_{l<j, l\in \mathbb{N}_k} (1+|x+a_j|)^{-N} (1+|x+a_l|)^{-N} \right \|_\infty \nonumber\\
& \lesssim \left\| \sum_{l<j, l\in \mathbb{N}_k} (1+|x|)^{-N} (1+|x+a_l-a_j|)^{-N} \right \|_\infty .
\label{NSlem2-10}\end{aligned}$$ In , by separating $\mathbb{R}^n$ into two different regions $\{x: |x|\le 3\cdot 2^{j-3}\}$ and $\{x: |x|> 3\cdot 2^{j-3}\}$, we easily see that the right hand side of can be bounded by $j 2^{-Nj}$. So, $$\begin{aligned}
\left\| \partial_i \triangle_j \left(\sum_{m,l\in \mathbb{N}_k; m> l} \Phi^{++}_m \Phi^{-+}_l \right) \right \|_\infty \lesssim 2^j j 2^{-Nj}, \ \ i=1,2 .
\label{NSlem2-11}\end{aligned}$$ By symmetry, also holds if one substitutes the summation $\sum_{m,l\in \mathbb{N}_k; m> l}$ by $\sum_{m,l\in \mathbb{N}_k; m< l}$. It follows from that $$\begin{aligned}
\left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\| \partial_i \triangle_j \left(\sum_{m,l\in \mathbb{N}_k; m\neq l} \Phi^{++}_m \Phi^{-+}_l \right) \right \|^q_\infty \right)^{1/q} \lesssim k 2^{-k}, \ \ i=1,2.
\label{NSlem2-12}\end{aligned}$$ Using the same way as in , we can estimate another term in $U_{52}$ and $$\begin{aligned}
\left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\| \partial_i \triangle_j U_{52} \right \|^q_\infty \right)^{1/q} \lesssim k 2^{-k}, \ \ i=1,2.
\label{NSlem2-13}\end{aligned}$$
Collecting the estimates as in , and , we have $$\begin{aligned}
t \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\|\triangle_j (\partial_1- \partial_2)(u^0_1 u^0_1) \|^q_\infty \right)^{1/q} \gtrsim \eta \varepsilon k^{1/q} - C \eta 2^{-k/4} - C\eta k 2^{-k}. \label{2.22boundin}\end{aligned}$$ By choosing $k\gg -\log \varepsilon$, we have the lower bound as desired in . $\hfill\Box$
\[NSlem3\] Let $1\leq q\le \infty$, $t=\eta 2^{-2k}$. Then $$\begin{aligned}
t \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\|\triangle_j \partial_2 (u^0_1+ u^0_2) u^0_1 \|^q_\infty \right)^{1/q} \lesssim \eta\varepsilon k^{1+1/q} 2^{-k/2} \label{2.39}\end{aligned}$$
[**Proof.** ]{} By , we have $$\begin{aligned}
u^0_1 + u^0_2 &= 2^k \sum_{l\in \mathbb{N}_k} \mathscr{F}^{-1} \frac{\xi_2-\xi_1}{\xi_2} (\widehat{\Phi^+_l}(\xi)+ \widehat{\Phi^-_l}(\xi)). \label{2.40}
\end{aligned}$$ By the multiplier estimate and , $$\begin{aligned}
& t \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\|\triangle_j \partial_2 (u^0_1+ u^0_2) u^0_1 \|^q_\infty \right)^{1/q} \nonumber\\
& \quad
\lesssim t k^{1/q} \| u^0_1+ u^0_2 \|_\infty \| u^0_1 \|_\infty \lesssim \eta 2^{-k} k^{1/q} \| u^0_1+ u^0_2 \|_\infty. \label{2.41}\end{aligned}$$ It follows from , and that $$\begin{aligned}
\| u^0_1+ u^0_2 \|_\infty & \lesssim 2^{k} \sum_{l\in \mathbb{N}_k} \sum_{\lambda,\mu=\pm 1} \left\| \mathscr{F}^{-1} \left(\frac{\xi_2-\xi_1}{\xi_2} \psi(\xi + \lambda c_k - \mu b_l)\right)\right\|_1 \nonumber\\
& \lesssim \varepsilon k 2^{k/2}. \label{2.42}\end{aligned}$$ In view of and , we have . $\hfill\Box$
\[NSlem4\] Let $1\leq q\le \infty$, $t=\eta 2^{-2k}$. Then $$\begin{aligned}
t \sum_{r\ge 2} \frac{1}{r!} \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\| (t\Delta)^{r-1} \triangle_j \partial_i (u^0_\alpha u^0_\beta) \|^q_\infty \right)^{1/q} \lesssim \eta^2 k^{1/q} 2^{-k}, \ i,\alpha,\beta=1,2. \label{2.43}\end{aligned}$$
[**Proof.**]{} Using Bernstein’s multiplier estimates, we have for $i=1,2$, $$\begin{aligned}
\|\partial_i \mathscr{F}^{-1} (t|\xi|^2)^{r-1} \varphi_j \mathscr{F} f \|_\infty & \lesssim 2^j (t2^{2j})^{r-1} \| \mathscr{F}^{-1} |\xi|^{2(r-1)} \varphi \|_1 \|f\|_\infty \nonumber\\
& \lesssim 2^j (t2^{2j})^{r-1} 4^r r^n \|f\|_\infty.
\label{2.43}\end{aligned}$$ Noticing that $j\in \mathbb{N}_k$ implies that $j\le k/2$, one has that for $i=1,2$, $$\begin{aligned}
& t \sum_{r\ge 2} \frac{1}{r!} \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\| \mathscr{F}^{-1} (t|\xi|^2)^{r-1} \varphi_j \mathscr{F} \partial_i (u^0_\alpha u^0_\beta) \|^q_\infty \right)^{1/q} \nonumber\\
& \quad \lesssim t \sum_{r\ge 2} \frac{ 4^r r^n }{r!} \left( \sum_{j\in \mathbb{N}_k} ((t2^{2j})^{q(r-1)} \| u^0_\alpha u^0_\beta \|^q_\infty \right)^{1/q} \nonumber\\
& \quad \lesssim \eta \sum_{r\ge 2} \frac{ 4^r r^n }{r!} k^{1/q} (t2^{k})^{ (r-1)} \lesssim \eta^2 k^{1/q} 2^{-k}.
\label{2.44}\end{aligned}$$ Collecting Lemmas \[NSlem2\]–\[NSlem4\], we immediately have
\[NSlem5\] Let $1\leq q\le \infty$, $t=\eta 2^{-2k}$, $0<\eta \le \varepsilon^2 \ll 1, \ k\gg -\log \varepsilon$. Then $$\begin{aligned}
A_{11} \gtrsim \eta \varepsilon k^{1/q}. \label{2.45}\end{aligned}$$
In the following we will show that $A_{12}$ is much less than $A_{11}$. One can rewrite $\widetilde{R}_i$ as $$\begin{aligned}
\widetilde{R}_i(f,g) = \int^t_0 e^{(t-\tau)\Delta } \partial_i [e^{\tau\Delta } f \ ( e^{\tau\Delta }-1) g + g( e^{\tau\Delta }-1)f ] d\tau. \label{2.46}\end{aligned}$$
\[NSlem6\] Let $1\leq q\le \infty$, $t=\eta 2^{-2k}$. Then $$\begin{aligned}
\left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\| \triangle_j \widetilde{R}_i(u^0_\alpha, u^0_\beta)\|^q_\infty \right)^{1/q} \lesssim \eta^2 k^{1/q}, \ \ i, \alpha, \beta=1,2. \label{2.47}\end{aligned}$$
[**Proof.**]{} Using the fact that $e^{t\Delta}: L^\infty \to L^\infty$, we have $$\begin{aligned}
& \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\| \triangle_j \widetilde{R}_1(u^0_1, u^0_1) \|^q_\infty \right)^{1/q} \nonumber\\
& \quad \lesssim t \sup_{\tau\in [0,t]} \left( \sum_{j\in \mathbb{N}_k} \| \triangle_j [e^{\tau\Delta } u^0_1 \ ( e^{\tau\Delta }-1) u^0_1 + u^0_1( e^{\tau\Delta }-1)u^0_1 ] \|^q_\infty \right)^{1/q} \nonumber\\
& \quad \lesssim t k^{1/q} \sup_{\tau\in [0,t]} \| u^0_1 \|_\infty \| ( e^{\tau\Delta }-1) u^0_1 \|_\infty \nonumber\\
& \quad \lesssim \eta k^{1/q} 2^{-k} \sup_{\tau\in [0,t]} \| ( e^{\tau\Delta }-1) u^0_1 \|_\infty .
\label{2.49}\end{aligned}$$ Using Taylor’s expansion, one has that $$\begin{aligned}
\| ( e^{\tau\Delta }-1) u^0_1 \|_\infty \leqslant \sum^\infty_{r=1} \frac{\tau^r}{r!}\|\mathscr{F}^{-1}|\xi|^{2r} \mathscr{F} u^0_1 \|_\infty. \label{2.50}\end{aligned}$$ Since ${\rm supp} \ \widehat{u^0_1} \subset \{\xi: 2^k \le |\xi| \le 2^k+ 2^{k/2}+1\}$, we see that $$\begin{aligned}
\|\mathscr{F}^{-1}|\xi|^{2r} \mathscr{F} u^0_1 \|_\infty & \leqslant \|\mathscr{F}^{-1}(|\xi|^{2r}\varphi_k)\|_1 \|u^0_1\|_\infty \nonumber\\
& \leqslant 2^{2kr}\|\mathscr{F}^{-1}(|\xi|^{2r}\varphi)\|_1 \|u^0_1\|_\infty \lesssim r^n 4^r 2^{2kr+1} \lesssim 8^r 2^{2kr+1}.
\label{2.51}\end{aligned}$$ It follows from and that $$\begin{aligned}
\| ( e^{\tau\Delta }-1) u^0_1 \|_\infty \leqslant C2^k (e^{8\tau 2^{2k}}-1) \leqslant C2^k (e^{8\eta}-1) \leqslant C\eta 2^k. \label{2.52}\end{aligned}$$ By and , we have the result, as desired. The other cases can be proven in a similar way. $\hfill \Box$
If we take $\eta=\varepsilon^2,$ then we see that $\eta^2 k^{1/q} \ll \eta \varepsilon k^{1/q}$. So, $A_{11} \gg A_{12}$. In the following we need to control $A_2$ and show that it is much less than $A_1$. For convenience, we use the same notations as in and . One can rewrite $F_2$ as $$\begin{aligned}
F_2(t,x) = \sum^2_{\alpha,\beta=1} \frac{\partial_\alpha\partial_\beta}{\Delta} R_1(u^0_\alpha, u^0_\beta). \label{2.53}\end{aligned}$$
\[NSlem7\] Let $1\leq q\le \infty$, $0<\varepsilon \ll 1,$ $t=\eta 2^{-2k}$, $\eta \le \varepsilon ^2$. Then $$\begin{aligned}
A_2 \lesssim \eta \varepsilon^2 k^{1/q}. \label{2.54}\end{aligned}$$
[Proof.]{} Recalling that $$\begin{aligned}
R_1(f,g) = (e^{t\Delta}-1) \Delta^{-1}\partial_1 (fg) + \widetilde{R}_1(f,g), \label{2.55}\end{aligned}$$ we see that $$\begin{aligned}
A_2 \le & \sum^2_{\alpha, \beta=1} \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \frac{\partial_\alpha\partial_\beta}{\Delta} (e^{t\Delta}-1) \Delta^{-1} \partial_1 (u^0_\alpha u^0_\beta) \right\|^q_\infty \right)^{1/q} \nonumber\\
& + \sum^2_{\alpha, \beta=1} \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \frac{\partial_\alpha\partial_\beta}{\Delta} \widetilde{R}_1(u^0_\alpha, u^0_\beta) \right\|^q_\infty \right)^{1/q} \nonumber\\
:= & A_{21} + A_{22}. \label{2.56}\end{aligned}$$ The estimate of $A_{22} $ is very easy. Noticing that $$\begin{aligned}
\left\|\triangle_j \frac{\partial_\alpha\partial_\beta}{\Delta} f \right\|_\infty
\lesssim \sum^{1}_{\ell =-1} \left\|\mathscr{F}^{-1} \frac{\xi_\alpha\xi_\beta}{|\xi|^2} \varphi_{j+\ell}(\xi) \right\|_1 \|\triangle_j f\|_\infty \lesssim \|\triangle_j f\|_\infty, \label{2.57}\end{aligned}$$ from Lemma \[NSlem6\] we immediately have $$\begin{aligned}
A_{22} \lesssim \sum^2_{\alpha, \beta=1} \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \widetilde{R}_1(u^0_\alpha, u^0_\beta) \right\|^q_\infty \right)^{1/q} \lesssim \eta^2 k^{1/q}. \label{2.58}\end{aligned}$$ Now we estimate $A_{21}$. Using Taylor’s expansion and , $$\begin{aligned}
\left\|\triangle_j \frac{\partial_\alpha\partial_\beta}{\Delta} (e^{t\Delta}-1) \Delta^{-1} \partial_1(u^0_\alpha u^0_\beta) \right\|_\infty
& \lesssim t \left\|\triangle_j \frac{\partial_\alpha\partial_\beta}{\Delta} \partial_1 (u^0_\alpha u^0_\beta) \right\|_\infty \nonumber\\
& \quad + t \sum^\infty_{r=2} \frac{1}{r!} \left\|\triangle_j (t\Delta)^{r-1} \partial_1 (u^0_\alpha u^0_\beta) \right\|_\infty.
\label{2.59}\end{aligned}$$ Hence, we have $$\begin{aligned}
A_{21} \le & t \sum^2_{\alpha, \beta=1} \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \frac{\partial_\alpha\partial_\beta}{\Delta} \partial_1 (u^0_\alpha u^0_\beta) \right\|^q_\infty \right)^{1/q} \nonumber\\
& + t \sum^2_{\alpha, \beta=1} \sum^\infty_{r=2} \frac{1}{r!}\left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j (t\Delta)^{r-1} \partial_1 (u^0_\alpha u^0_\beta) \right\|^q_\infty \right)^{1/q} \nonumber\\
:= & \sum_{\alpha, \beta=1,2} A^{\alpha\beta}_{211} + A_{212}. \label{2.60}\end{aligned}$$ By Lemma \[NSlem4\], $$\begin{aligned}
A_{212} \lesssim \eta^2 k^{1/q} 2^{-k}. \label{2.61}\end{aligned}$$ So, it suffices to bound $A^{\alpha\beta}_{211}$. We divide the estimates of $A^{\alpha\beta}_{211}$ into the following three cases.
[*Case*]{} 1. $\alpha=\beta =1$. From , and , it follows that for $j\in \mathbb{N}_k$, $$\begin{aligned}
\triangle_j (u^0_1 u^0_1) = 2^{2k+1} (\Phi^{++}_j \Phi^{-+}_j + \Phi^{+-}_j \Phi^{--}_j) + \triangle_j U_{52},
\label{2.62}\end{aligned}$$ we see that $$\begin{aligned}
A^{11}_{211} = & t \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \frac{\partial^3_1 }{\Delta} (u^0_1 u^0_1) \right\|^q_\infty \right)^{1/q} \nonumber\\
\le & 2\eta \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\| \frac{\partial^3_1 }{\Delta} \triangle_j \left(\Phi^{++}_j \Phi^{-+}_j + \Phi^{+-}_j \Phi^{--}_j\right) \right\|^q_\infty \right)^{1/q} \nonumber\\
& + 2\eta \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\| \frac{\partial^3_1 }{\Delta} \triangle_j U_{52} \right\|^q_\infty \right)^{1/q} \nonumber\\
:= & B_1+B_2. \label{2.63}\end{aligned}$$ Using the Bernstein’s multiplier estimate and , $$\begin{aligned}
B_2 \lesssim \eta \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\| \triangle_j \partial_1 U_{52} \right\|^q_\infty \right)^{1/q} \lesssim \eta k 2^{-k}. \label{2.64}\end{aligned}$$ For the estimate of $B_1$, noticing that ${\rm supp} \ \widehat{\Phi^{++}_j} * \widehat{\Phi^{-+}_j} \subset B(a_j,1)$, we have from the multiplier estimate, $$\begin{aligned}
\left\| \triangle_j \frac{\partial^3_1 }{\Delta} \left( \Phi^{++}_j \Phi^{-+}_j \right) \right\|_\infty & \lesssim 2^j \left\| \mathscr{F}^{-1} \frac{\xi^2_1 }{|\xi|^2} \psi(\xi-a_j) \right\|_1 \left\| \Phi^{++}_j \Phi^{-+}_j \right\|_\infty \nonumber\\
& \lesssim 2^j \left\| \mathscr{F}^{-1} \frac{(\xi_1 +\varepsilon 2^j)^2 }{|\xi+a_j|^2} \psi(\xi) \right\|_1.
\label{2.65}\end{aligned}$$ Since the third coordinate of $a_j$ is larger than $2^j/2$, we have from the Bernstein’s multiplier estimate that ($j\ge k/4\gg -\log \varepsilon$) $$\begin{aligned}
\left\| \mathscr{F}^{-1} \frac{(\xi_1 +\varepsilon 2^j)^2 }{|\xi+a_j|^2} \psi(\xi) \right\|_1 \lesssim \varepsilon^2.
\label{2.66}\end{aligned}$$ Therefore, $$\begin{aligned}
B_1 \lesssim \eta \varepsilon^2 k^{1/q}.
\label{2.67}\end{aligned}$$ Summarizing and , one has that $$\begin{aligned}
A^{11}_{211} \lesssim \eta \varepsilon^2 k^{1/q}.
\label{2.68}\end{aligned}$$
[*Case*]{} 2. $\alpha=\beta =2$. We can rewrite $u^0_2$ as $$\begin{aligned}
u^0_2 & = -u^0_1 + \widetilde{u^0_2}, \ \ \widetilde{u^0_2}= 2^k \sum_{l\in \mathbb{N}_k} \mathscr{F}^{-1} \frac{\xi_2-\xi_1}{\xi_2} ( \widehat{\Phi^+_l}(\xi) + \widehat{\Phi^-_l}(\xi)).
\label{2.69}
\end{aligned}$$ It follows that $$\begin{aligned}
u^0_2 u^0_2 & = u^0_1 u^0_1 + \widetilde{u^0_2} \widetilde{u^0_2} - 2 u^0_1\widetilde{u^0_2} \label{2.70}
\end{aligned}$$ and $$\begin{aligned}
A^{22}_{211} \le & t \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \frac{\partial_1 \partial^2_2 }{\Delta} (u^0_1 u^0_1) \right\|^q_\infty \right)^{1/q} \nonumber\\
& + 2 t \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \frac{\partial_1 \partial^2_2 }{\Delta} (u^0_1 \widetilde{u^0_2} ) \right\|^q_\infty \right)^{1/q} \nonumber\\
& + t \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \frac{\partial_1 \partial^2_2}{\Delta} (\widetilde{u^0_2} \widetilde{u^0_2} ) \right\|^q_\infty \right)^{1/q}:= I+II+III. \label{2.71}\end{aligned}$$ Applying the same way as in the estimate of $B_1$, one has that $$\begin{aligned}
I \lesssim \eta \varepsilon^2 k^{1/q}.
\label{2.72}\end{aligned}$$ By Bernstein’s multiplier estimate and Lemma \[NSlem3\], we have $$\begin{aligned}
II \lesssim t \left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \partial_1 (u^0_1 \widetilde{u^0_2} ) \right\|^q_\infty \right)^{1/q} \lesssim \eta\varepsilon k^{1+1/q} 2^{-k/2}. \label{2.73}\end{aligned}$$ The estimate of $III$ is easier than that of $II$ and we have $$\begin{aligned}
III \lesssim \eta\varepsilon k^{2+1/q} 2^{-k}. \label{2.74}\end{aligned}$$ Collecting the estimates of , and , we have $$\begin{aligned}
A^{22}_{211} \le \eta\varepsilon^2 k^{1/q} . \label{2.75}\end{aligned}$$
[*Case*]{} 3. $\alpha=1, \ \beta =2$. Similar to Case 2, we can show that $ A^{12}_{211}$ has the same bound as $ A^{22}_{211}$ and we omit the details of the proof.
Up to now, we have shown that $$\begin{aligned}
\sum_{\alpha,\beta=1,2} A^{\alpha\beta}_{211} \le \eta\varepsilon^2 k^{1/q} . \label{2.76}\end{aligned}$$ By , and , we have $$\begin{aligned}
A_{21} \le \eta\varepsilon^2 k^{1/q} . \label{2.77}\end{aligned}$$ In view of , and , we immediately have the result, as desired. $\hfill\Box$\
\[NSlem8\] Let $1\leq q\le \infty$, $t= \varepsilon^2 2^{-2k}$. Then $$\begin{aligned}
\left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \frac{\partial^2 u_1}{\partial \delta^2}(0,t) \right \|^q_\infty \right)^{1/q} \gtrsim \varepsilon^3 k^{1/q}. \label{2.78}\end{aligned}$$
[**Proof.**]{} In view of Lemma \[NSlem6\] we have $A_{12} \lesssim \varepsilon^4 k^{1/q}$. Using and Lemma \[NSlem5\], we get $A_{1} \ge (c \varepsilon^3-C \varepsilon^4) k^{1/q}$. By Lemma \[NSlem7\] and , one has that for $t=\varepsilon^2 2^{-2k}$, $$\begin{aligned}
\left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \frac{\partial^2 u_1}{\partial \delta^2}(0,t) \right \|^q_\infty \right)^{1/q} \ge (c \varepsilon^3-2 C \varepsilon^4) k^{1/q}. \label{2.78}\end{aligned}$$ We can choose $\varepsilon $ satisfying $100C\varepsilon \le c/2$, the result follows. $\hfill \Box$
Estimates on higher order terms of $\delta$ {#sect2.2}
-------------------------------------------
Using Koch and Tataru’s global wellposedness result, we can show that the higher order terms on $\delta$ in Taylor’s expansion in $\dot B^{-1}_{\infty,q}$ are much smaller than $\delta^2\varepsilon^3 k^{1/q}$. This fact, together with Lemma \[NSlem8\] imply the inflation phenomena of the solution of NS in $\dot B^{-1}_{\infty,q}$. By induction we have for the solution $u(\delta,t)$ of NS, $$\begin{aligned}
\frac{\partial^r u}{\partial \delta^r}(0,t) = \int^t_0 e^{(t-\tau)\Delta} \mathbb{P} {\rm div} \sum^{r-1}_{m=1} {r \choose m} \left( \frac{\partial^m u}{\partial \delta^m}(0,\tau) \otimes \frac{\partial^{r-m} u}{\partial \delta^{r-m}}(0,\tau)\right) d\tau . \label{2.80}\end{aligned}$$ Let $X$ be defined in Theorem \[NSWellposed\]. Let us recall that Koch and Tataru [@KoTa01] and Auscher, Dubois and Tchamitchian [@AuDuTc06], Lemma 2.8 obtained the following estimates. $$\begin{aligned}
\left\|\int^t_0 e^{(t-\tau)\Delta} \mathbb{P} {\rm div} \left( u(\tau) \otimes v(\tau)\right) d\tau \right\|_{Y} \lesssim \|u\|_X\|v\|_X, \label{Koch2.80}\end{aligned}$$ where $Y:= X\cap L^\infty(0,\infty; BMO^{-1})$ Applying the integral equation, and Theorem \[NSWellposed\], we see that $$\begin{aligned}
\|u (\delta, t)-\delta e^{t\Delta}u_0\|_Y \le \left\|\int^t_0 e^{(t-\tau)\Delta} \mathbb{P} {\rm div} \left( u(\tau) \otimes u(\tau)\right) d\tau \right\|_Y \lesssim \|u\|^2_X \lesssim\delta^2. \label{Koch2.81}\end{aligned}$$ One sees that $$\begin{aligned}
\label{Koch2.82}
u (\delta, t)- \delta e^{t\Delta}u_0 - \frac{\delta^2}{2} \frac{\partial^2 u}{\partial \delta^2}(0,t)& = \int^t_0 e^{(t-\tau)\Delta} \mathbb{P} {\rm div} \left( u(\tau) \otimes u(\tau) - \delta^2e^{\tau\Delta}u_0 \otimes e^{\tau\Delta}u_0 \right) d\tau.\end{aligned}$$ It follows from , , and Theorem \[NSWellposed\] that $$\begin{aligned}
\label{Koch2.83}
\left\|u (\delta, t)- \delta e^{t\Delta}u_0 - \frac{\delta^2}{2} \frac{\partial^2 u}{\partial \delta^2}(0,t) \right\|_Y \lesssim \|u\|_X \left\|u (\delta, t)- \delta e^{t\Delta}u_0 \right\|_X \lesssim \delta^3.\end{aligned}$$ Let us fix $t=\varepsilon^2 2^{-2k}$ and denote $$\begin{aligned}
\label{2.81}
\widetilde{u}(\delta, t) = \sum^\infty_{r=3} \frac{\delta^r}{r!} \frac{\partial^r u}{\partial \delta^r}(0,t).\end{aligned}$$ By we see that there exist $\delta_0>0$, such that for any $0<\delta\le \delta_0$, $$\begin{aligned}
\label{2.82}
u (\delta, t) = \delta e^{t\Delta}u_0 + \frac{\delta^2}{2} \frac{\partial^2 u}{\partial \delta^2}(0,t)+ \widetilde{u}(\delta, t) \ \ \ \ \mbox{in} \ \ Y.\end{aligned}$$ By , one sees that for $\delta\le \delta_0$, $$\begin{aligned}
\label{2.82a}
\|\widetilde{u}(\delta, t) \|_{Y} \lesssim \delta^3.\end{aligned}$$ This implies that for any $\delta \le \min\{\delta_0, \ \varepsilon^4\}$, $$\begin{aligned}
\label{2.86}
\left\| \widetilde{u}(\delta, t) \right\|_{\dot B^{-1}_{\infty,\infty}} \le C \delta^2 \varepsilon^4, \ \ \forall t>0.\end{aligned}$$ Applying this fact, we can prove Theorem \[NSIll\].\
**Proof of Theorem \[NSIll\]: $n\ge 3$.**
-----------------------------------------
Let $0<\varepsilon \ll 1$ be as in Lemma \[NSlem8\], $t=\varepsilon^2 2^{-2k} $ and $\delta \le \min\{\delta_0, \ \varepsilon^4\}$ be the same one as in . Using , we have $$\begin{aligned}
\left\| u_1 (\delta,t) \right \|_{\dot B^{-1}_{\infty,1}} & \ge
\left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j u_1 (\delta,t) \right \|^q_\infty \right)^{1/q} \nonumber\\
& \ge \frac{\delta^2}{2}
\left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \frac{\partial^2 u_1}{\partial \delta^2}(0,t) \right \|^q_\infty \right)^{1/q} \nonumber\\
& \quad -\delta \|e^{t\Delta}u_0\|_{\dot B^{-1}_{\infty,q}} -
\left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \widetilde{u}_1 (\delta,t) \right \|^q_\infty \right)^{1/q}. \label{2.87}\end{aligned}$$ Applying and noticing that $\mathbb{N}_k$ has at most $k$ many indices, we have $$\begin{aligned}
\left( \sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \widetilde{u}_1 (\delta,t) \right \|^q_\infty \right)^{1/q}
& \le k^{1/q} \left\| \widetilde{u}_1 (\delta,t) \right \|_{\dot B^{-1}_{\infty,\infty}}
\le C k^{1/q} \delta^2 \varepsilon^4 . \label{2.88}\end{aligned}$$ Obviously, we have $$\begin{aligned}
\|e^{t\Delta}u_0\|_{\dot B^{-1}_{\infty,q}} \le C. \label{2.89}\end{aligned}$$ By Lemma \[NSlem8\], , and , we have $$\begin{aligned}
\left\| u_1 (\delta,t) \right \|_{\dot B^{-1}_{\infty,1}} \ge c \delta^2 \varepsilon^3 k^{1/q} -C \delta - C \delta^2 \varepsilon^4 k^{1/q}.
\label{2.90}\end{aligned}$$ Recalling that $\varepsilon$ has been chosen as in , we immediately have for $k\gg \varepsilon^{-3q} \delta^{-q}$, $$\begin{aligned}
\left\| u_1 (\delta,t) \right \|_{\dot B^{-1}_{\infty,1}} \gtrsim \varepsilon^{3} \delta^2 k^{1/q}.
\label{2.99}\end{aligned}$$ This finishes the proof of Theorem \[NSIll\] in the case $n\ge 3$. $\hfill\Box$
Proof of Theorem \[NSIll\], $n=2$ {#sect3}
=================================
Noticing that in Section \[sect2\] one needs the third coordinates of $a_l, b_l$ are not zero, the proof of higher dimensional cases cannot be straightly applied to the 2D case. However, the nonlinearity in 2D case is simpler than that of higher dimensions. We now sketch the proof in 2D case. We write $$\begin{aligned}
\partial_t u_j -\Delta u_j + B_j (u,u)=0, \ \ j=1,2, \label{2d3.1}\end{aligned}$$ where $$\begin{aligned}
B_1 (u,u) & : = \sum^2_{i=1} \partial_i (u_i u_1) - \partial_1 \Delta^{-1} \sum^2_{l, m=1} \partial_l \partial_m (u_l u_m) \nonumber\\
&= \Delta^{-1} (\partial^2_1-\partial^2_2) \partial_2 (u_1u_1) + \partial_2 (u_1(u_1+u_2)) + \Delta^{-1} \partial^2_2 \partial_1 ((u_1)^2 -(u_2)^2) \nonumber\\
& \quad -2 \Delta^{-1} \partial^2_1 \partial_2 (u_1(u_1+u_2)). \label{2d3.2}\end{aligned}$$ and $B_2(u,u)$ is similar. We have from that $$\begin{aligned}
\label{2d3.3}
\frac{\partial^2 u_1}{\partial\delta^2}\Big|_{\delta=0} = \int^t_0 e^{(t-\tau)\Delta } B_1 (e^{\tau\Delta } u_0, \ e^{\tau\Delta } u_0) d\tau.\end{aligned}$$ Let $0<\varepsilon \ll 1$, $k$ and $\mathbb{N}_k$ be the same ones as in the cases $n\ge 3$. Put $$\begin{aligned}
c_k = (\sqrt{2}/2, \ \sqrt{2}/2 )2^k, \ \ a_l = (\varepsilon, \ \sqrt{1-\varepsilon^2}) 2^l, \ \ b_l=a_l/2 \label{2d3.4}\end{aligned}$$ and let $u^0_1$ and $ u^0_2$ be the same ones as . Similarly as in , we have $$\begin{aligned}
\|u^0_1+ u^0_2\|_\infty \lesssim 2^{k/2}. \label{2d3.5}\end{aligned}$$ Let us estimate $$\begin{aligned}
\left(\sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \frac{\partial^2 u_1}{\partial\delta^2}\Big|_{\delta=0} \right\|^q_\infty \right)^{1/q}. \label{2d3.6}\end{aligned}$$ Let $t=\eta 2^{-2k}$. Denote $$\begin{aligned}
\widetilde{R}(t,u_0) =& \frac{\partial^2 u_1}{\partial\delta^2}\Big|_{\delta=0} - \int^t_0 e^{(t-\tau)\Delta } B_1 ( u_0, \ u_0) d\tau \nonumber\\
=& \int^t_0 e^{(t-\tau)\Delta } ( B_1 ( e^{\tau\Delta} u_0, \ e^{\tau\Delta} u_0)- B_1 ( u_0, \ u_0)) d\tau.
\label{3.2d7}
\end{aligned}$$ Similarly as in , every term in $\widetilde{R}(t,u_0)$ contains $(e^{\tau\Delta}-1)$, using the same way as in Lemma \[NSlem6\], we have for $t=\eta 2^{-2k}$, $$\begin{aligned}
\left(\sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \widetilde{R}(t,u_0) \right\|^q_\infty \right)^{1/q} \lesssim \eta^2 k^{1/q}. \label{2d3.8}\end{aligned}$$ We can write $$\begin{aligned}
\int^t_0 e^{(t-\tau)\Delta } B_1 ( u_0, \ u_0) d\tau =& \Delta^{-1} (e^{t\Delta}-1) B_1(u_0,\ u_0) \nonumber\\
:= & t\sum_{r\ge 1} \frac{(t\Delta)^{r-1}}{r!} \Delta^{-1} (\partial^2_1- \partial^2_2)\partial_2 (u^0_1u^0_1) + \widetilde{Q}(t, u_0).
\label{2d3.9}
\end{aligned}$$ Since every term in $\widetilde{Q}(t, u_0)$ contains $u^0_1+u^0_2$, by we can repeat the procedures as in Section \[sect2\] to obtain that $$\begin{aligned}
\left(\sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \widetilde{Q}(t,u_0) \right\|^q_\infty \right)^{1/q} \lesssim \eta k^{1/q} 2^{-k/2}. \label{2d3.10}\end{aligned}$$ In the series $ \sum_{r\ge 1} \frac{(t\Delta)^{r-1}}{r!} \Delta^{-1} (\partial^2_1- \partial^2_2)\partial_2 (u^0_1u^0_1) $, the first term $ \Delta^{-1} (\partial^2_1- \partial^2_2)\partial_2 (u^0_1u^0_1)= -\partial_2 (u^0_1u^0_1) + 2 \partial^2_1\partial_2 (u^0_1u^0_1)$ contributes the main part and using the same way as Lemma \[NSlem2\], we have $$\begin{aligned}
t \left(\sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \partial_2 (u^0_1u^0_1) \right\|^q_\infty \right)^{1/q} \gtrsim \eta k^{1/q}. \label{2d3.11}\end{aligned}$$ Moreover, taking notice of the definition of $a_l$ and $b_l$, one can repeat the procedures as in Lemma \[NSlem7\], Case 1 to obtain that $$\begin{aligned}
t \left(\sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \Delta^{-1} \partial^2_1 \partial_2 (u^0_1u^0_1) \right\|^q_\infty \right)^{1/q} \lesssim \eta \varepsilon^2 k^{1/q}. \label{2d3.12}\end{aligned}$$ Similar to Lemma \[NSlem4\], we have $$\begin{aligned}
t \sum_{r\ge 2} \frac{1}{r!} \left( \sum_{j\in \mathbb{N}_k} 2^{-jq}\| (t\Delta)^{r-1} \triangle_j \partial_2 (u^0_1 u^0_1) \|^q_\infty \right)^{1/q} \lesssim \eta k^{1/q} 2^{-k}. \label{2d3.13}\end{aligned}$$ Collecting the above estimates, we obtain that $$\left(\sum_{j\in \mathbb{N}_k} 2^{-jq} \left\|\triangle_j \frac{\partial^2 u_1}{\partial\delta^2}\Big|_{\delta=0} \right\|^q_\infty \right)^{1/q} \gtrsim \eta k^{1/q}.$$ Recall that the estimates of higher order iterations in Section \[sect2.2\] also hold for 2 dimensional case. So, the left part of the proof of Theorem \[NSIll\] in 2D case proceeds in the same way as that of higher dimensional cases. The details of the proof are omitted.
Other critical spaces {#other}
======================
Even though NS is instable in the critical Besov spaces $\dot B^{-1}_{\infty,1}$, we still have some global well-posedness results in the subspaces of $\dot B^{-1}_{\infty,1}$. Recall that Iwabuchi [@Iw10] (cf. also [@HuWa13; @Wa06]) considered the well-posedness of NS in modulation spaces, particularly in $M^{-1}_{\infty, 1}$ for which the norm is defined as follows. Let $\sigma$ be a smooth cut-off function with ${\rm supp} \ \sigma \subset [-3/4, 3/4]^n$, $\sigma_k=\sigma(\cdot-k)$ and $ \sum_{ k\in\mathbb{Z}^n}\sigma_k = 1.$ Denote $\square_k = \mathscr{F}^{-1} \sigma_k \mathscr{F}, \ k \in \mathbb{Z}^n. $ Put $$\begin{aligned}
\|f\|_{M^{-1}_{\infty,1}} = \sum_{k\in \mathbb{Z}^n} (1+|k|^2)^{-1/2} \|\Box_k f\|_\infty. \label{3.1}\end{aligned}$$ Modulation spaces was introduced by Feichtinger [@Fei83] (see also Gröchenig [@Groch01], here we adopt an equivalent norm; cf. [@WaHe07]). Lei and Lin [@LeLi11] considered the the global existence and uniqueness of 3D NS in the space $\mathcal{X}^{-1}$ for which the norm is defined by $$\begin{aligned}
\|f\|_{\mathcal{X}^{-1}} = \int_{\mathbb{R}^n} |\xi|^{-1} |\widehat{f}(\xi)| d\xi. \label{3.2}\end{aligned}$$
\[embed\] We have the following inclusions $$\begin{aligned}
\mathcal{X}^{-1} \subset M^{-1}_{\infty,1} \subset B^{-1}_{\infty,1}, \label{3.3}\end{aligned}$$ where $B^{-1}_{\infty,1}= L^p+ \dot B^{-1}_{\infty,1}$.
[**Proof.**]{} For the proof of the sharp embedding $ M^{-1}_{\infty,1} \subset B^{-1}_{\infty,1}$, one can referee to [@To04], [@SuTo07], [@WaHu07]. For any $f\in \mathcal{X}^{-1}$, by Young’s inequality, $$\begin{aligned}
\|f\|_{M^{-1}_{\infty,1}} & \lesssim \sum_{k\in \mathbb{Z}^n} \langle k\rangle^{-1} \|\sigma_k \widehat{f}\|_1 \nonumber\\
& \lesssim \sum_{k\in \mathbb{Z}^n} \int_{k+[-1,1]^n} \langle \xi\rangle^{-1} |\widehat{f}(\xi)| d\xi \lesssim \int_{\mathbb{R}^n} |\xi|^{-1} |\widehat{f}(\xi)| d\xi = \|f\|_{\mathcal{X}^{-1}}. \label{3.4}\end{aligned}$$ It follows that $\mathcal{X}^{-1} \subset M^{-1}_{\infty,1}$. $\hfill \Box$
It is worth to mention that $M^{-1}_{\infty,1}$ and $\dot B^{-1}_{\infty,1}$ have no inclusions, since the lower (higher) frequency part of $\dot B^{-1}_{\infty,1}$ is smoother (rougher) than that of $M^{-1}_{\infty,1}$. However, we have $\mathcal{X}^{-1} \subset \dot B^{-1}_{\infty,1}$, which is easily seen by imitating the proof of . To some extent, $M^{-1}_{\infty,1}$ is also the critical space of NS, using similar ideas in this paper, we can show NS is ill-posed in $M^s_{\infty,1}$ if $s<-1$.
Let us observe the initial data $u^0_1$, we easily see that $\|u^0_1\|_{M^{-1}_{\infty,1}} \sim \|u^0_1\|_{\mathcal{X}^{-1}}\sim k$, $\|u^0_1\|_{\dot B^{-1}_{\infty,1}} \sim 1$. One sees that $ M^{-1}_{\infty,1} $ and $\mathcal{X}^{-1}$ are quite similar in the higher frequency part on which the norms of $u^0_1$ are far away from $B^{-1}_{\infty,1}$ or $\dot B^{-1}_{\infty,1}$ .
Finally, let us observe the initial data $u^0_1$ and the solution $u$ in critical Triebel-Lizorkin’s spaces $\dot F^{-1}_{\infty, q}$ with $1<q<\infty$. Let us recall that $\dot F^{-1}_{\infty, q}$ is defined by (cf. [@Tr]) $$\begin{aligned}
\dot F^{-1}_{\infty,q} = & \Big\{ f\in \mathscr{Z}'(\mathbb{R}^n): \exists \ \{f_j\}_{j\in \mathbb{Z}} \ such \ that \ f = \sum_{j\in \mathbb{Z}} \triangle_j f_j \nonumber\\
& \quad \quad \quad \quad \quad \quad in \ \mathscr{Z}'(\mathbb{R}^n) \ and \ \left\| \|\{2^{-j} f_j\}\|_{\ell^q}\right\|_\infty <\infty \Big\} \label{3.5}\end{aligned}$$ for which the norm is $$\begin{aligned}
\|f\|_{\dot F^{-1}_{\infty,q}} = & \inf \left\| \|\{2^{-j} f_j\}\|_{\ell^q}\right\|_\infty, \label{3.6}\end{aligned}$$ where the infimum is taken over all of the possible expressions in .
Due to the embedding $\dot F^1_{1, q'} \subset \dot B^1_{1, q'}$, we immediately have from the duality that $\dot B^{-1}_{\infty, q} \subset \dot F^{-1}_{\infty, q}$. Considering the initial data $u_0$ defined in , we have $$\begin{aligned}
\|u_0\|_{\dot F^{-1}_{\infty,q}} \lesssim \|u_0\|_{\dot B^{-1}_{\infty,q}} \lesssim 1, \ \ \forall q>1. \label{3.7}\end{aligned}$$ According to Koch and Tataru’s result, NS has a unique small global solution in $BMO^{-1}=\dot F^{-1}_{\infty,2}$. One may ask how does the solution vanish in $BMO^{-1}$ when $\delta \to 0$? Now we outline the essential difference between $BMO^{-1}$ and $\dot B^{-1}_{\infty,2}$ for NS with initial data as in . By Lemma \[NSlem2\], we see that $(\partial_1-\partial_2)(u^0_1u^0_1)$ contributes the main part in critical Besov space $\dot B^{-1}_{\infty,q}$ which grows like $O(k^{1/q})$. However, we can show it is very small in $\dot F^{-1}_{\infty,q}$ and we have $$\begin{aligned}
t\| (\partial_1-\partial_2)(u^0_1u^0_1)\|_{\dot F^{-1}_{\infty,q}} \lesssim \varepsilon, \ \ t \le 2^{-2k}. \label{3.8}\end{aligned}$$ Taking $\tilde{\triangle}_j= \triangle_{j-1}+ \triangle_{j}+ \triangle_{j+1}$, we see that $f=\sum_j \triangle_j (\tilde{\triangle}_j f)$ and $$\begin{aligned}
t\| (\partial_1-\partial_2)(u^0_1u^0_1)\|_{\dot F^{-1}_{\infty,q}} & \le \varepsilon t\left\| \sum_{j} |\tilde{\triangle}_j (u^0_1u^0_1)| \right\|_{\infty} \nonumber\\
& \quad + t \left\| \sum_{j} |2^{-j}\tilde{\triangle}_j (\partial_1-\partial_2- \varepsilon 2^j)(u^0_1u^0_1)| \right\|_{\infty}. \label{3.9}\end{aligned}$$ In view of , we see that $$\sum_j |\tilde{\triangle}_j (u^0_1u^0_1)| = 2^{2k+1} \sum_j |\Phi^+_j \Phi^-_j | +... \lesssim 2^{2k+1} \sum_j |\check{\varrho}(\cdot+a_j)|^2+... \lesssim 2^{2k},$$ which implies that the first term in is less than $\varepsilon$ as $t\le 2^{-2k}$, however, it is $O(k^{1/q})$ in critical Besov space $\dot B^{-1}_{\infty,q}$. The second term in is a remainder term which is exponentially small like $2^{- ck}$. Hence, we have . Roughly speaking, for the solution $u$ considered in this paper, the velocities localized in different frequencies, say $\triangle_j u$ and $\triangle_{m} u$ ($m\neq j$) have concentrations in different physical regions which are far away from each other, so their superpositions in physical spaces have no inflation phenomena, this is why $u$ is small in $BMO^{-1}$.\
[**Acknowledgment.**]{} The author is supported in part by the National Science Foundation of China, grants 11271023.
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[^1]: Email: [email protected]
[^2]: $\triangle$ and $\Delta$ are different notations in this paper.
[^3]: $\varepsilon>0$ will be chosen below as in , $k$ can be arbitrarily large.
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abstract: 'We consider systems of ordinary differential equations with quadratic homogeneous right hand side. We give a new simple proof of a result already obtained in \[8,10\] which gives the necessary conditions for the existence of polynomial first integrals. The necessary conditions for the existence of a polynomial symmetry field are given. It is proved that an arbitrary homogeneous first integral of a given degree is a linear combination of a fixed set of polynomials.'
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[**On the existence of polynomial first integrals of quadratic homogeneous systems of ordinary differential equations**]{}\
[**Tsygvintsev Alexei**]{}
Introduction
============
In the present paper we study the system of ordinary differential equations with quadratic homogeneous right hand side
$$\dot x_i=f_i(x_1,\ldots,x_n), \quad
f_i=\sum^n_{j,k=1}a_{ijk}x_j x_k , \quad a_{ijk} \in {{\mathbb C}},
\quad i=1,\ldots,n. \eqno (1.1)$$
Systems of such a form arise in many problems of classical mechanics: Euler–Poincaré equations on Lie algebras, the Lotka–Volterra systems, etc.
The main concern of this paper is to find the values of the parameters $a_{ijk}$ for which equations (1.1) have first integrals.
In the paper \[2\] the necessary conditions are found for the existence of polynomial first integrals of the system $$\dot
x_i=V_i(x_1,\ldots,x_n), \quad i=1,\ldots,n, \eqno (1.2)$$ where $V_i\in {{\mathbb C}}[x_1,\ldots,x_n]$ are homogeneous polynomials of weighted degree $s\in {\mathbb N}$. In the case $s=2$ we obtain equations (1.1).
The method given in \[2\] is based on ideas of Darboux \[1,6,7\] who used a special type of particular solutions of the system (1.2) $$x_i(t)=d_i\phi(t),\quad i=1,\ldots,n,$$ where $\phi(t)$ satisfies the differential equation $\dot\phi=\lambda\phi^s$, $\lambda$ is an arbitrary number and $d=(d_1,\ldots,d_n)^T\ne 0$ is a solution of the following algebraic system $$V_i(d)=\lambda d_i,\quad i=1,\ldots,n.$$
In this paper we generalize this method.
It was shown in \[8,10\] that the weighted degree of a polynomial first integral of the system (1.1) is a certain integer linear combination of Kovalevskaya exponents (see \[9\]). In Section 2 we give a new simple proof of this result. In Section 3 a similar theorem for polynomial symmetry fields is proved. As an example, we consider the well known Halphen equations. Section 4 contains our main result. We present so called [ *base functions* ]{} and prove that every homogeneous polynomial first integral of a fixed degree is a certain linear combination of the corresponding base functions. In Section 5 we give an application of previous results to planar homogeneous quadratic systems where necessary and sufficient conditions for the existence of polynomial first integrals in terms of Kovalevskaya exponents are found.
The existence of polynomial first integral. Necessary conditions
================================================================
Following the paper \[2\], we consider solution $C=(c_1,\dots,c_n)^T\ne(0,\dots,0)^T$ of algebraic equations $$f_i(c_1,\dots,c_n)+c_i=0,\quad i=1,\dots,n.\eqno (2.1)$$ Define the [*Kovalevskaya matrix*]{} $K$ \[3\] $$K_{ij}=\frac{\partial
f_i}{\partial x_j}(C)+\delta_{ij},\quad i,j=1,\dots,n.$$ where $\delta_{ij}$ is the Kronecker symbol. Let us assume that $K$ can be transformed to diagonal form $$K=\mbox{diag}(\rho_1,\dots,\rho_n).$$ The eigenvalues $\rho_1,\dots,\rho_n$ are called [*Kovalevskaya exponents*]{}.
[**(\[3\])**]{} Vector C is an eigenvector of the matrix K with eigenvalue $\rho_1=-
1$.
Consider the following linear differential operators $$\begin{array}{ll}
D_+=\sum\limits^n_{i=1}f_i\frac{\displaystyle \partial}{\displaystyle
\partial x_i},
&\quad D_0=\sum\limits^n_{i,j=1}K_{ij}x_j
\frac{\displaystyle\partial}{\displaystyle \partial x_i},\\
U=\sum\limits^n_{i=1}x_i
\frac{\displaystyle\partial}{\displaystyle \partial x_i},&\quad
D_-=\sum\limits^n_{i=1}c_i
\frac{\displaystyle\partial}{\displaystyle \partial x_i},
\end{array} \eqno (2.2)$$ which satisfy relations $$[D_-,D_+]=D_0-U,\quad [D_0,D_-]=D_-,\eqno (2.3)$$ where $[A,B]=AB-BA$. \[lemma\][Theorem]{}[**\[8,10\]**]{}
Suppose that the system (1.1) possesses a homogeneous polynomial first integral $F_M$ of degree M, and $\rho_1=-1,\rho_2,\dots,\rho_n$ are Kovalevskaya exponents. Then there exists a set of non-negative integers $k_2,\dots,k_n$ such that $$k_2\rho_2+\cdots+k_n\rho_n=M,\quad k_2+\cdots+k_n\le M.
\eqno (2.4)$$
[*Proof.*]{} By definition of a first integral $D_+F_M=0$. Considering identities $$D^l_-(D_+F_M)=0,\quad \mbox{for}\quad
l\in {\mathbb N}, \eqno (2.5)$$ we obtain the following set of polynomials $$F_M,F_{M-1},\dots,F_{\rho+1},F_\rho,$$ defined by the recursive relations $$D_-F_{i+1}=(M-i)F_i,\quad
i=\rho,\dots,M-1,$$ where the number $1\le \rho \le M$ is determined by the condition $D_- F_\rho=0$. Using (2.3), (2.5) we deduce the following chain of equations $$\begin{array}{l}
D_0F_M=MF_M-D_+F_{M-1},\\ D_0F_{M-1}=MF_{M-1}-D_+F_{M-2},\\
\dots\\ D_0F_{\rho+1}=MF_{\rho+1}-D_+F_\rho,\\ D_0F_\rho=MF_\rho.
\hspace{10.2cm} (2.6)
\end{array}$$
Let $J_1$, …, $J_n$ – linearly independent eigenvectors of the Kovalevskaya matrix $K$ corresponding to the eigenvalues $\rho_1=-1$, $\rho_2$, …, $\rho_n$. According to Lemma 1 we can always put $J_1=C$.
Consider the linear change of variables $$x_i=\sum^n_{j=1}L_{ij}y_j,\quad i=1,\dots,n, \eqno (2.7)$$ where $L=(L_{ij})$ is a nonsingular matrix defined by $$L=(C,J_2,\dots,J_n),$$ then obviously $$L^{-1}KL=\mbox{diag}(-1,\rho_2,\dots,\rho_n).$$
With help of (2.7) and Lemma 1 one finds the following expressions for the operators $D_0$, $D_-$ in the new variables $$D_0=\sum^n_{i=1}\rho_iy_i\frac{\partial}{\partial y_i}, \quad
D_-=-\frac{\displaystyle
\partial}{\displaystyle
\partial y_1},$$ and the equation (2.6) becomes $$\left(\rho_2y_2\frac{\partial}{\partial y_2}+\cdots+\rho_ny_n
\frac{\partial}{\partial y_n}\right)F_\rho=MF_\rho. \eqno (2.8)$$ We can write the polynomial $F_\rho$ as follows $$F_\rho=\sum_{|k|=\rho}A_{k_2\dots k_n}y^{k_2}_2\cdots y^{k_n}_n,
\quad |k|=k_2+\cdots+k_n,\quad k_i \in {\mathbb Z}_+. \eqno (2.9)$$ Substituting (2.9) into (2.8), one obtains the following linear system $$(k_2\rho_2+\cdots+k_n\rho_n)A_{k_2\dots k_n}=MA_{k_2
\dots k_n}, \quad \mbox{for} \quad |k|=\rho. \eqno (2.10)$$ Taking into account that $F_\rho$ is not zero identically, we conclude that there exists at least one nonzero set $k_2,\dots,k_n$, $|k| \le M $ such that $$k_2\rho_2+\cdots+k_n\rho_n=M. \eqno (2.11)$$ This relation implies (2.4). Q.E.D.
\[lemma\][Corollary]{}
[*Remark.*]{} Theorem 2 does not impose any restrictions on $\mbox{grad}(F_M)$ calculated at the point C. Thus, it generalizes the theorem of Yoshida (\[3\], p.572), who used essentially the condition $\mbox{grad}(F_M) \ne 0$.
[The Halphen equations $$\begin{array}{l}
\dot x_1=x_3x_2-x_1x_3-x_1x_2,\\
\dot x_2=x_1x_3-x_2x_1-x_2x_3,\\
\dot x_3=x_2x_1-x_3x_2-x_3x_1,
\end{array} \eqno (2.12)$$ admit no polynomial first integrals.]{}
Indeed, the system (2.12) has Kovalevskaya exponents $\rho_1=\rho_2=\rho_3=-1$. It is easy to verify that conditions (2.4) are not fulfilled for any positive integer $M$. Moreover, as proved in \[2\], the system (2.12) has no rational first integrals.
Existence of polynomial symmetry fields. Necessary conditions
=============================================================
The first integrals are the simplest tensor invariants of the system (1.1). In \[4\] Kozlov considered tensor invariants of weight-homogeneous differential equations which include the system (1.1). In particular, he found necessary conditions for the existence of symmetry fields. Below we propose a generalization of his result.
Recall that the linear operator $
W=\sum\limits^n_{i=1}w_i(x_1,\dots,x_n) \frac{\displaystyle
\partial}{\displaystyle \partial x_i},
$ is called the [*symmetry field*]{} of (1.1), if $[W,D_+]=0$,where $D_+$ is defined by (2.2). If $w_1,\dots,w_n$ are homogeneous functions of degree $M+1$ then the degree of $W$ is $M$ (\[4\]).
Suppose that the system (1.1) possesses a polynomial symmetry field of degree $M$ and\
$\rho_1=-1,\rho_2,\dots,\rho_n$ are Kovalevskaya exponents. Then there exist non-negative integers $k_2,\dots,k_n$, $|k|\le M+1$ such that at least one of the following equalities holds $$k_2\rho_2+\cdots+k_n\rho_n=M+\rho_i,\quad i=1,\dots,n. \eqno (3.1)$$
[*Proof.*]{} Let $W_M$ be a polynomial symmetry field of degree $M$. Substituting in the proof of Theorem 2 operators $D_+$, $D_0$, $D_-$ with their commutators $[D_+,{}]$, $[D_0,{}]$, $[D_-,{}]$ respectively we repeat the same arguments.
[The Halphen equations (2.12) admit no polynomial symmetry fields.]{}
Using (3.1) we obtain that $M$ may be equal to $-1$, $0$, $1$ only. It is easy to check that (2.12) does not have symmetry fields of such degrees.
Base functions
==============
After the change of variables (2.7) the system (1.1) takes the form $$\begin{array}{l}
\dot y_1=-y^2_1+\varphi_1(y_2,\dots,y_n)\\ \dot
y_i=(\rho_i-1)y_1y_i+\varphi_i(y_2,\dots,y_n), \quad i=2,\dots,n,
\end{array} \eqno (4.1)$$ where $\varphi_i$ are quadratic homogeneous polynomials in the variables $y_2,\dots,y_n$.
According to (2.2) define operators $D_+,D_0,D_-$.
A homogeneous polynomial $P_M(y_1,\dots,y_n)$ of degree $M$ satisfying the condition $$D_-(D_+P_M)=0, \eqno(4.2)$$ is called [*base function*]{} of the system (4.1). In others words, the function $D_+P_M$ does not depend on $y_1$. It is clear that base functions of degree $M$ form a linear space $L_M$ over field ${\mathbb C}$.
If the system (4.1) has a homogeneous polynomial first integral $F_M$ of degree $M$, then $F_M\in L_M$.
Indeed, by definition, we have $D_+F_M=0$, hence, in view of (4.2), $F_M\in L_M$.
Let $J(M)=\{z\in {{\mathbb Z}^{n-1}_+}\mid z_2\rho_2+\cdots+z_n\rho_n=M,
|z|\leq M\}$ be the set of integer-valued vectors $z=(z_2,\dots,z_n)^T$ for which the condition (2.11) is fulfilled. Put $m=|J(M)|$ and suppose $J(M)\ne \emptyset$.
The dimension $d$ of $L_M$ satisfies the condition $1\le d \le m$.
[*Proof.*]{} Let us assume the set J(M) contains vectors $z^{(1)},\dots,z^{(m)}$ which are ordered by the norm\
$|z|=z_2+\cdots+z_n$ $$|z^{(1)}|\le \cdots \le |z^{(m)}|.$$ Define the vector $\rho=(\rho_2,\dots,\rho_n)^T$ and put $(\rho,z)=\rho_2z_2+\cdots+\rho_nz_n$, $|z^{(i)}|=n_i$, $i=1,\dots,m$.
Following the proof of Theorem 2, for each $i=1,\dots,m$ consider the system of linear partial differential equations $$\begin{array}{l}
D_0P_{i,n_i}=MP_{i,n_i},\\
D_0P_{i,n_i+1}=MP_{i,n_i+1}-D_+P_{i,n_i},\\
\cdots\\
D_0P_{i,M-1}=MP_{i,M-1}-D_+P_{i,M-2},\\
D_0P_{i,M}=MP_{i,M}-D_+P_{i,M-1},\\
\end{array} \eqno (4.3)$$ $$D_-P_{i,l+1}=(M-l)P_{i,l},\quad l=n_i,\dots,M-1,\eqno (4.4)$$ which defines polynomials $P_{i,n_i},\dots,P_{i,M}$ recurrently.
It follows from $(z^{(i)},\rho)=M$ that the first equation in (4.3) has the particular solution $
P_{i,n_i}=y^{z_2^{(i)}}_2\cdots y^{z_n^{(i)}}_n.
$
Equations (4.3), (4.4) define certain base function $P_{i,M}$. Indeed, according to (4.4), we have $$P_{i,M-1}=D_-P_{i,M}. \eqno
(4.5)$$ Substituting (4.5) into the last equation in (4.3), and using relations (2.3) we find $$D_0P_{i,M}=MP_{i,M}-D_+D_-P_{i,M}=MP_{i,M}-(D_-D_+-D_0+U)P_{i,M}.$$ Hence $D_-(D_+P_{i,M})=0$.
Now consider the problem on the existence of a solution of (4.3), (4.4) in form of homogeneous polynomials $P_{i,n_i},\dots,P_{i,M}$. Fix certain $i=1,\dots,m$ and put $a_i=M-n_i$. Using the relations (4.4) we can write $$\begin{array}{l}
P_{i,n_i}=I_{i,n_i},\\ P_{i,n_i+p}=\sum\limits^p_{j=0}{p-j\choose
a_i-j}y^{p- j}_1I_{i,n_i+j},\quad p=1,\dots,a_i,
\end{array} \eqno (4.6)$$ where $I_{i,k}(y_2,\dots,y_n)$ are certain homogeneous polynomials of degrees $k=n_i,\dots,M$. Notice that $I_{i,k}$ does not depend on $y_1$.
Differential operators $D_+$, $D_0$ can be represented in the form $$\begin{array}{l}
D_+=(-y^2_1+\varphi_1)\frac{\displaystyle \partial}{\displaystyle
\partial y_1}+y_1(A_0-\tilde U)+A_+,\\
D_0=-y_1\frac{\displaystyle \partial}{\displaystyle \partial y_1}+A_0,
\end{array} \eqno (4.7)$$ where $$A_+=\sum^n_{k=2}\varphi_k\frac{\partial}{\partial y_k},\quad
A_0=\sum^n_{k=2}\rho_ky_k
\frac{\partial}{\partial y_k},\quad
\tilde U=\sum^n_{k=2}y_k\frac{\partial}{\partial y_k}.
\eqno (4.8)$$ Using (4.3), (4.6), (4.7) one deduces the following equations for determination of $I$ $$\begin{array}{l}
A_0I_{i,n_i}=MI_{i,n_i},\\
A_0I_{i,n_i+1}=MI_{i,n_i+1}-A_+I_{i,n_i},\\
A_0I_{i,n_i+2}=MI_{i,n_i+2}-a_i\varphi_1I_{i,n_i}-A_+I_{i,n_i+1},\\
A_0I_{i,n_i+3}=MI_{i,n_i+3}-(a_i-1)\varphi_1I_{i,n_i+1}-
A_+I_{i,n_i+2},\\
\cdots\\
A_0I_{i,M}=MI_{i,M}-2\varphi_1I_{i,M-2}-A_+I_{i,M-1}.
\end{array} \eqno (4.9)$$ We can write each equation of (4.9) as follows $$A_0X_l=MX_l+Y_l, \eqno (4.10)$$ where $X_l,Y_l$ are homogeneous polynomials of weighted degree $l=n_i,\dots,M$. Let us assume $$X_l=\sum\limits_{|i|=l}c_{i_2\dots i_n}y^{i_2}_2\cdots y^{i_n}_n,
\quad Y_l=\sum\limits_{|i|=l}d_{i_2\dots i_n}y^{i_2}_2\cdots
y^{i_n}_n,\quad |i|=i_2+\cdots+i_n, \eqno (4.11)$$ where $c_{i_2\dots i_n}$, $d_{i_2\dots i_n}$ are constant parameters. Then substituting (4.11) into (4.10), we obtain the following linear system with respect to $c_{i_2\dots i_n}$ $$(i_2\rho_2+\cdots+i_n\rho_n-M)c_{i_2\dots i_n}=d_{i_2\dots i_n},
\eqno (4.12)$$ for $i_2,\dots,i_n=0,1,\dots$, $|i|=l$.
Suppose there exists a set $k_2,\dots,k_n$ for which the following conditions are fulfilled $$(k_2,\dots,k_n)^T\in J(M),\quad
d_{k_2\dots k_n}\ne 0,\quad |k|=l, \eqno (4.13)$$ then the solution $I_{i,n_i},\dots,I_{i,M}$ does not exist. In this case we put $P_{i,M}=0$.
If the conditions (4.13) are not satisfied, we obtain the base function $$P_{i,M}=\sum^{a_i}_{j=0}y^{a_j-j}_1I_{i,n_i+j}.\eqno
(4.14)$$ It is easy to show that polynomials $\{P_{i,M}\}^{i=1}_{i=m}$ are linearly independent over the field ${\mathbb C}$.
Taking into account that $n_1\le\cdots\le n_m$ and using (4.13), we see that in case $i=m$ we always can determine the base function $P_{i,M}$. Therefore, under the assumption $J(M)\ne\emptyset$, the space $L_M$ always contains a nonzero function. Q.E.D.
[If at least one resonance condition of the form $$(z,\rho)=M,\quad |z|\le M,\quad z\in {{\mathbb Z}^{n-1}_+},$$ is fulfilled, then there exists a base function of degree $M$.]{}
Polynomial first integrals in the case of quadratic plane vector field.
========================================================================
The first classification of integral curves of two-dimensional quadratic homogeneous systems can be found in the paper of Lyagina \[5\] and later was completed by numerous authors.
In this section we apply the previous results to this problem to illustrate the method of basis functions.
Consider the system $$\begin{array}{l}
\dot x_1=a_1x^2_1+b_1x_1x_2+d_2x^2_2,\\
\dot x_2=a_2x^2_2+b_2x_1x_2+d_1x^2_1,
\end{array} \eqno (5.1)$$ where $a_i,b_i,d_i$ are constant parameters.
Let $c^{(1)}=(c^{(1)}_1,c^{(1)}_2)^T$, $c^{(2)}=(c^{(2)}_1,c^{(2)}_2)^T$ be any two linearly independent solutions of the algebraic system (2.1). The exceptional cases when the system (2.1) has only one or admit no solutions are excluded for the discussion below.
Assume that Kovalevskaya exponents corresponding to $c^{(1)}$, $c^{(2)}$ are $$\Re_1=(-1,\rho_1)^T,\quad
\Re_2=(-1,\rho_2)^T.\eqno (5.2)$$
The system (5.1) has a homogeneous polynomial first integral of degree M if and only if there exists integer $k=1,\dots,M-1$ such that $\rho_1=M/k$ and $\rho_2$ is one of the following numbers $$\frac{M}{M-k},\frac{M}{M-k-1},\dots,\frac{M}{2},M.$$
[*Proof.*]{} Consider the following change of coordinates $$\left(
\begin{array}{l}
x_1\\
x_2
\end{array}
\right)
=
\left(
\begin{array}{ll}
c^{(1)}_1&c^{(2)}_1 \\
c^{(1)}_2&c^{(2)}_2
\end{array}
\right)
\left(
\begin{array}{l}
p_1\\
p_2
\end{array}
\right)\eqno (5.3)$$ which exists because of linear independence of vectors $c^{(1)}$, $c^{(2)}$. In coordinates $(p_1,p_2)$ the system (5.1) takes a more simple form $$\begin{array}{l}
\dot p_1=-p^2_1+(\rho_2-1)p_1p_2,\\
\dot p_2=-p^2_2+(\rho_1-1)p_1p_2.
\end{array} \eqno (5.4)$$ It is easy to show that under the change (5.3), the vectors $c^{(1)}$, $c^{(2)}$ turn into $\tilde c^{(1)}=(1,0)^T$, $\tilde
c^{(2)}=(0,1)^T$ respectively. Obviously, the system (5.4) has the same Kovalevskaya exponents (5.2). The matrix $K$, calculated for $\tilde c^{(1)}$ is $$K= \left(
\begin{array}{ll}
-1&\rho_2-1\\
0&\rho_1\\
\end{array}
\right)$$
Under the assumption $\rho_1\ne -1$, we can reduce $K$ to a diagonal form using the following change of coordinates $$\left(
\begin{array}{l}
p_1\\
p_2
\end{array}
\right)=L
\left(
\begin{array}{l}
y_1\\
y_2
\end{array}\right)$$ with the constant matrix $L$ $$L=
\left(
\begin{array}{ll}
1&\rho_2-1\\
0&\rho_1-1
\end{array}
\right)$$ The case $\rho_1=-1$ will be considered below.
Finally, equations (5.4) take the form (4.1) $$\begin{array}{l}
\dot y_1=-y_1^2+\varphi_1,\\
\dot y_2=(\rho_1-1)y_1y_2+\varphi_2,
\end{array} \eqno (5.5)$$ where $$\begin{array}{l}
\varphi_1=ay^2_2, \quad
\varphi_2=by^2_2,\\
a=(\rho_2-1)(\rho_1+\rho_2),\quad b=(\rho_1-1)(\rho_2-1)-\rho_1-1.
\end{array}$$ For the operators (4.8) we get $$A_+=\varphi_2\frac{\partial}{\partial y_2},\quad
A_0=\rho_1y_2\frac{\partial}{\partial y_2},\quad
\tilde U=y_2\frac{\partial}{\partial y_2}.$$ Let $F_M$ be a polynomial first integral of (5.5) of degree $M$.
According to Theorem 2, there exists an integer $k=1,\dots,M-1$ such that $$k\rho_1=M. \eqno (5.6)$$ We exclude the case $k=M
(\rho_1=1)$, since if $\rho_1=\pm 1$, then the system (5.1) has no polynomial first integrals. This can be shown directly using equations (5.4), (5.5).
Next, calculate the base function $P_M$ corresponding to the resonance condition (5.6). Consider the equations (4.9). It is obvious, that polynomials $I_{1,k},\dots,I_{1,M}$ can be represented in the following form $$I_{1,k+i}=\alpha_iy^{k+i}_2,\quad i=0,\dots,M-k, \eqno (5.7)$$ where $\alpha_0,\dots,\alpha_{M-k}$ are constant parameters.
Substituting (5.7) into (4.9) we obtain $$\begin{array}{l}
\alpha_0=1,\\ \alpha_1=\frac{\displaystyle bk}{\displaystyle
\rho_1(k+1)-M},\\ \alpha_i=\frac{\displaystyle
a(M-k-i+2)\alpha_{i-1}+b(k+i-1) \alpha_{i-2}}{\displaystyle
\rho_1(k+i)-M},\quad i=2,\dots,M-k.
\end{array} \eqno (5.8)$$ According to (4.14), we get the following expression for the base function $P_M$ $$P_M=\sum^{M-k}_{j=0}\alpha_j y^{M-k-j}_1y_2^{k+j}. \eqno (5.9)$$ By definition of the base function it is clear that $$D_+P_M=\delta y_2^{M+1}, \eqno (5.10)$$ where $$\delta=a\alpha_{M-1}+bM\alpha_M. \eqno (5.11)$$ Thus, the linear space $L_M$ contains only one polynomial $P_M$. Hence, taking into account Lemma 6, $F_M=\mbox{const}\cdot P_M$.
Using (5.10), we conclude that $P_M$ is a first integral if and only if $\delta =0$. In view of (5.6), (5.8), (5.11) and the above condition, we arrive at Lemma 9.
The system (5.1) possesses a homogeneous polynomial first integral of degree $M$ if and only if the following conditions are fulfilled\
a) $\rho_i$, $i=1,2$ – are positive rational numbers,\
b) $\rho_1^{-1}+\rho_2^{-1}\leq 1$,\
c) $\displaystyle \frac{M}{\rho_i}\in {\mathbb N}.$
This is an obvious consequence of Lemma 9.
As an example consider the following system $$\begin{array}{l}
\dot x_1=x_1^2-9x^2_2,\\
\dot x_2=-3x^2_1-8x_1x_2+3x_2^2.
\end{array} \eqno (5.12)$$ The vectors $c^{(1)}$, $c^{(2)}$ have the form $$c^{(1)}=(1/8,-1/8)^T,\quad c^{(2)}=(1/8,1/8)^T.$$ Calculating the corresponding Kovalevskaya exponents (5.2) one obtains $$\Re_1=(-1,3)^T,\quad \Re_2=(-1,3/2)^T.$$
We have $\rho_1=3$, $\rho_2=3/2$, $\rho_1^{-1}+\rho_2^{-1}=1$. So, the conditions a), b) of Theorem 10 are fulfilled. By the condition c) one gets $M=3l$, $l\in {\mathbb N}$. Thus, the equations (5.12) possess a cubic first integral $F_3$. Using formulas (5.8), (5.9), we obtain $$F_3=x_1^3+x_1^2x_2-x_1x^2_2-x_2^3.$$
[**Acknowledgments**]{}
We thank J.-M. Strelcyn for his useful remarks, V. Kozlov, D. Treshev, H. Yoshida, K. Emelyanov, Yu. Fedorov and L. Gavrilov for the interest to the paper.
[99]{}
G. Darboux, Mémoire sur les équations différentielles algébrique du premier ordre et du premier degré (Mélanges), Bull. Sci. Math. (1878).
Andrej J. Maciejewski, Jean-Marie Strelcyn. On algebraic non- integrability of the Halphen system. Phys. Lett. A 201 (1995)
Yoshida H. Necessary condition for existence of algebraic first integrals. Celestial Mechanics. 1983.V. 31. P. 363-399.
Kozlov V.V. Tensor invariants of quasihomogenous systems of differential equations and the asymptotic method of Kovallevskaya-Lyapunov. Mat. Zametki 51 (1992) , no. 2, 46-52; English transl. in Math. Notes 51 (1992)
Lyagina L. The invariant curves of equation $dy/dx=(ax^2+bxy+cy^2)/(dx^2+exy+fy^2)$, (Russian), Usp. Mat. Nauk, 6, no 2 (42), 171–183 (1951)
Jouanolou J.-P. Equations de Pfaff algébriques, Lect. Notes in Math. 708, Springer-Verlag, Berlin (1979)
Ollagnier J. Nowicki A. Strelcyn J.-M., On the non-existence of constants of derivations: the proof of theorem of Jouanolou and its development, Bull. Sci. math. 119 (1995), 195–233
Goriely A. Integrability, partial integrability and nonintegrability for systems of ordinary differential equations. J. Math. Phys., 37:1871–1893 (1996)
Goriely A. A brief history of Kovalevskaya exponents and modern developments. Regular and Chaotic Dynamics, vol. 5, No. 1, 3–15 (2000)
Furta S.D. On non–integrability of general systems of differential equations. Z. angew. Math. Phys., 47:112–131, (1996)
[ **Section de Mathematiques,\
Université de Genève\
2-4, rue du Lievre,\
CH-1211, Case postale 240, Suisse\
Tel l.: +41 22 309 14 03\
Fax: +41 22 309 14 09\
E–mail: [email protected]**]{}
|
---
abstract: |
RF pulse design via optimal control is typically based on gradient and quasi-Newton approaches and therefore suffers from slow convergence. We present a flexible and highly efficient method that uses exact second-order information within a globally convergent trust-region CG-Newton method to yield an improved convergence rate. The approach is applied to the design of RF pulses for single- and simultaneous multi-slice (SMS) excitation and validated using phantom and in-vivo experiments on a scanner using a modified gradient echo sequence.
[Keywords]{}pulse design, optimal control, second-order methods, simultaneous multi-slice excitation
author:
- 'Christoph Stefan Aigner[^1]'
- 'Christian Clason[^2]'
- 'Armin Rund[^3]'
- Rudolf Stollberger
bibliography:
- 'rfcontrol.bib'
date: 'November 23, 2015'
title: 'Efficient high-resolution RF pulse design applied to simultaneous multi-slice excitation'
---
=1
Introduction {#sec:intro}
============
For many applications in MRI there is still demand for the optimization of selective RF excitation, e.g., for simultaneous multi-slice excitation [@Larkman01; @Ugurbil13], UTE imaging [@Bydder12], or velocity selective excitation [@Rochefort06]. To achieve a well-defined slice profile at high field strength while meeting B$_1$ peak amplitude limitations is a challenge for RF pulse design and becomes especially critical for quantitative methods. Correspondingly, many approaches for general pulse design have been proposed in the literature. RF pulses with low flip angles can be designed using the small tip angle simplification [@Pauly89], which makes use of an approximation of the Bloch equation to compute a pulse via the Fourier transform of the desired slice profile. However, this simplification breaks down for large flip angles. The resulting excitation error for large flip angle pulses can be reduced by applying the Shinnar–Le Roux (SLR) technique [@Pauly91] or optimization methods, e.g., simulated annealing, evolutionary approaches or optimal control [@Shen01; @Wu91; @Khalifa01; @Conolly86; @Xu08; @Lapert12; @Vinding12]. The SLR method is based on the hard pulse approximation and a transformation of the excitation problem, allowing to solve the excitation problem recursively by applying fast filter design algorithms such as the Parks–McClellan algorithm [@Pauly91]. Originally, this approach only covered special pulses such as and excitation or refocusing, but Lee [@Lee07] generalized this approach to arbitrary flip angles with an exact parameter relation. Despite its limitations due to neglected relaxation terms and sensitivity to B$_1$ inhomogeneities, it found widespread use (see, e.g., [@Lee07; @Balchandani10; @Grissom12; @Ma14]) and is considered to be the gold standard for large tip angle pulse design. An alternative approach is based on optimizing a suitable functional; see, e.g., [@Ulloa04; @Xu08; @Pauly91; @Gezelter90; @Buonocore93]. In particular, optimal control (OC) approaches involve the solution of the Bloch equation describing the evolution of the magnetization vector in an exterior magnetic field [@Conolly86; @Khaneja05; @Skinner12; @Bonnard14; @Lapert12; @Grissom09; @Xu08; @Vinding12]. They often lead to better excitation profiles due to a more accurate design model and are increasingly used in MRI, for instance, to perform multidimensional and multichannel RF design [@Xu08; @Grissom09], robust 2D spatial selective pulses [@Vinding12] and saturation contrast [@Lapert12]. In addition, arbitrary flip angles and target slice profiles, as well as inclusion of additional physical effects such as, e.g., relaxation can be handled. However, so far OC approaches are limited by the computational effort and require a proper modeling of the objective. In particular, standard gradient-based approaches suffer from slow convergence, imposing significant limitations on the accuracy of the obtained slice profiles. On the other hand, Newton methods show a locally quadratic convergence, but require second-order information which in general is expensive to compute [@Anand12]. Approximating the Hessian using finite differences causes loss of quadratic convergence due to the lack of exact second-order information and typically requires significantly more iterations. Superlinear convergence can be obtained using quasi-Newton methods based on exact gradients [@Fouquieres11], although their performance can be sensitive to implementation details. The purpose of this work is to demonstrate that for the OC approach to pulse design, it is in fact possible to use exact second-order information while avoiding the need of computing the full Hessian, yielding a highly efficient numerical method for the optimal control of the full time-dependent Bloch equation. In contrast to [@Anand12] (which uses black-box optimization method and symbolically calculated Hessians based on an effective-matrix approximation of the Bloch equation), we propose a matrix-free Newton–Krylov method [@Knoll2004] using first- and second-order derivatives based on the adjoint calculus [@Hinze:2001] together with a trust-region globalization [@Steihaug83]; for details we refer to \[sec:optimization\]. Recently, similar matrix-free Newton–Krylov approaches with line search globalization were presented for optimal control of quantum systems in the context of NMR pulse sequence design [@Ciaramella15_1; @Ciaramella15_2]. In comparison, the proposed trust-region framework significantly reduces the computational effort, particularly for the initial steps far away from the optimum. The effectiveness of the proposed method is demonstrated for the design of pulses for single and simultaneous multi-slice excitation (SMS).
SMS excitation is increasingly used to accelerate imaging experiments [@Larkman01; @Nunes06; @Feinberg13; @Ugurbil13]. Conventional design approaches, based on a superposition of phase-shifted sub-pulses [@Mueller88] or sinusoidal modulation [@Larkman01], typically result in a linear scaling of the B$_1$ peak amplitude, a quadratic peak power and a linear increase in the overall RF power [@Norris11; @Auerbach13]. The required maximal B$_1$ peak amplitude of conventional multi-slice pulses therefore easily exceeds the transmit voltage of the RF amplifier. In this case, clipping will occur, while rescaling will decrease and limit the maximal flip angle of such a pulse. On the other hand, restrictions of the specific absorption rate limit the total (integrated) B$_1$ power and therefore the maximal number of slices as well as the pulse duration and flip angle. The increase of B$_1$ power can be addressed by the Power Independent of Number of Slices (PINS) technique [@Norris11], which was extended to the kT-PINS method [@Sharma13] to account for B$_1$ inhomogeneities. This approach leads to a nearly slice-independent power requirement, but the periodicity of the resulting excitation restricts the slice orientation and positioning. Furthermore, the slice profile accuracy is reduced [@Feinberg13], and a limited ratio between slice thickness and slice distance may further restrict possible applications. The combination of PINS with regular multi-band pulses was shown to reduce the overall RF power by up to (MultiPINS [@Eichner14]) and was applied to refocusing pulses in a multi-band RARE sequence with 13 slices [@Gagoski15].
A different way to reduce the maximum B$_1$ amplitude is to increase the pulse length; however, this stretching increases the minimal echo and repetition times and decreases the RF bandwidth, thus reducing the slice profile accuracy [@Auerbach13]. Applying variable rate selective excitation [@Conolly88; @Setsompop12-2] avoids this problem but leads to an increased sensitivity to slice profile degradations at off-resonance frequencies. In addition, they require specific sequence alterations, e.g., variable slice gradients or gradient blips. Instead of using the same phase for all sub-slices, the peak power can be reduced by changing the uniform phase schedule to a different phase for each individual slice [@Wong12]. Alternative approaches[@Zhu14; @Sharma15] using phase-matched excitation and refocusing pairs show that a nonlinear phase pattern can be corrected by a subsequent refocusing pulse. Another way to reduce the power deposition and SAR of SMS pulses is to combine them with parallel transmission [@Katscher08]. This allows to capitalize transmit sensitivities in the pulse design and leads to a more uniform excitation with an increased power efficiency [@Wu13; @Poser14]. Recently, Guerin et al. [@Guerin14] demonstrated that it is possible to explicitly control both global and local SAR as well as the peak power using a spokes-SMS-pTx pulse design.
The focus of this work, however, is on single channel imaging, where we apply our OC-based pulse design for efficient SMS pulse optimization using a direct description of the desired magnetization pattern. Its flexible formulation allows a trade-off between the slice profile accuracy and the required pulse power and is well suited for the reduction of power and amplitude requirements of such pulses, even for a large number of slices or large flip angles or in presence of relaxation. The efficient implementation of the proposed method allows to optimize for SMS pulses with a high spatial resolution to achieve accurate excitation profiles. The RF pulses are designed to achieve a uniform effective echo time and phase for each slice and use a constant slice-selective gradient, allowing to insert the RF pulse into existing sequences and opening up a wide range of applications.
Theory {#sec:theory}
======
This section is concerned with the description of the optimal control approach to RF pulse design as well as of the proposed numerical solution approach.
Optimal control framework
-------------------------
Our OC approach is based on the full time-dependent Bloch equation, which describes the temporal evolution of the ensemble magnetization vector ${M}(t)=({M_x}(t),{M_y}(t),{M_z}(t))^T$ due to a transient external magnetic field ${B}(t)$ as the solution of the ordinary differential equation (ODE) $$\label{eq:bloch1}
\left\{\begin{aligned}
\dot{{M}}(t) &= \gamma {B}(t) \times {M}(t) + {R}({M}(t)),\qquad t>0,\\
{M}(0) &={M^0},
\end{aligned}\right.$$ where $\gamma$ is the gyromagnetic ratio, ${M^0}$ is the initial magnetization and $${R(M}(t)) = (-{M_x}(t)/T_2,-{M_y}(t)/T_2,-({M_z}(t)-M_{0})/T_1)^T$$ denotes the relaxation term with relaxation times $T_1,T_2$ and the equilibrium magnetization $M_{0}$. To encode spatial information in MR imaging, the external magnetic field ${B}$ (and thus the magnetization vector) depends on the slice direction $z$, hence the Bloch equation can be considered as a parametrized family of three-dimensional ODEs. In the on-resonance case and ignoring spatial field inhomogeneities, the Bloch equation can be expressed in the rotating frame as $$\label{eq:bloch_rot}
\left\{\begin{aligned}
{\dot{M}}(t;z) &= {A}({u}(t);z) {M}(t;z)+ {b}(z), \qquad t>0,\\
{M}(0;z) &= {M^0}(z),
\end{aligned}\right.$$ where the control ${u}(t)=({u_x}(t),{u_y}(t))$ describes the RF pulse, $$\label{eq:matrix}
{A}({u};z) = \begin{pmatrix}
-\frac1{T_2} & \gamma {G_z}(t)z & \gamma {u_y}(t) B_1 \\
- \gamma {G_z}(t)z & -\frac1{T_2} & \gamma {u_x}(t) B_1 \\
-\gamma {u_y}(t) B_1 & -\gamma {u_x}(t) B_1 & -\frac1{T_1}
\end{pmatrix}, \qquad
{b}(z) = \begin{pmatrix} 0\\0\\ \frac{{M_{0}}}{T_1} \end{pmatrix},$$ and $G_z$ is the slice-selective gradient; see, e.g., [@Nishimura96 Chapter 6.1].
The OC approach consists in computing for given initial magnetization ${M^0}(z)$ the RF pulse ${u}(t)$, $t\in[0,T_u]$, that minimizes the squared error at read-out time $T>T_u$ between the corresponding solution ${M}(T;z)$ of and a prescribed slice profile ${M_d}(z)$ for all $z\in [-a,a]$ together with a quadratic cost term modeling the SAR of the pulse, i.e., solving $$\label{eq:costfunc}
\min\limits_{({u},{M}) \text{ satisfying \eqref{eq:bloch_rot}}} \quad J({M},{u}) = \frac{1}{2} \int_{-a}^a |{M}(T;z)-{M_d}(z)|_2^2 \,dz + \frac\alpha2\int_0^{T_u} |{u}(t)|_2^2\,dt.$$ The parameter $\alpha>0$ controls the trade-off between the competing goals of slice profile attainment and SAR reduction.
Adjoint approach {#sec:optimization}
----------------
The standard gradient method for solving consists of computing for given ${u}^k$ the gradient ${g}({u}^k)$ of $j({u}):=J({M}({u}),{u})$ and setting ${u}^{k+1} = {u}^k-s^k {g}({u}^k)$ for some suitable step length $s^k$. The gradient can be calculated efficiently using the adjoint method, which in this case yields $$\label{eq:gradient}
\begin{aligned}[t]
{g}({u}^k)(t) &= \alpha {u}(t) + \gamma B_1 \begin{pmatrix}
\int_{-a}^a {M_z}(t;z){P_y}(t;z) - {M_y}(t;z){P_z}(t;z)\,dz\\[0.5ex]
\int_{-a}^a {M_z}(t;z){P_x}(t;z) - {M_x}(t;z){P_z}(t;z)\,dz
\end{pmatrix}\\
&=: \alpha {u}(t) +
\begin{pmatrix}
\int_{-a}^a {M}(t;z){A_1} {P}(t;z)\,dz\\[0.5ex]
\int_{-a}^a {M}(t;z){A_2} {P}(t;z)\,dz
\end{pmatrix},
\qquad 0\leq t \leq T_u,
\end{aligned}$$ where ${M}$ is the solution to for ${u}={u}^k$ and $0<t\leq T$, ${P}$ is the solution to the adjoint (backward in time) equation $$\label{eq:adjoint}
\left\{\begin{aligned}
-\dot{{P}}(t;z)&= {A}({u}(t);z)^T {P}(t;z),\qquad 0\leq t< T,\\
{P}(T;z)&={M}(T;z)-{M_d}(z),
\end{aligned}\right.$$ and for the sake of brevity, we have set $${A_1} := \gamma B_1 \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 &0\end{pmatrix},\qquad
{A_2} := \gamma B_1 \begin{pmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 &0\end{pmatrix}.$$
However, this method requires a line search to converge and usually suffers from slow convergence close to a minimizer. This is not the case for Newton’s method (which is a second-order method and converges locally quadratically), where one additionally computes the Hessian ${H}({u}^k)$ of $j$ at $u^k$, solves for $\delta {u}$ in $$\label{eq:Newton_step}
{H}({u}^k)\delta {u} = -{g}({u}^k),$$ and sets ${u}^{k+1} = {u}^k+\delta {u}$. While the full Hessian ${H}({u}^k)$ is very expensive to compute in practice, solving using a Krylov method such as conjugate gradients (CG) only requires computing the Hessian *action* ${H}({u}^k){h}$ for a given direction ${h}$ per iteration; see, e.g., [@Knoll2004]. The crucial observation in our approach is that the adjoint method allows computing this action exactly (e.g., without employing finite difference approximations) and without knowledge of the full Hessian. Since Krylov methods usually converge within very few iterations, this so-called “matrix-free” approach amounts to significant computational savings. To derive a procedure for computing the Hessian action ${H}({u}^k){h}$ for a given direction ${h}$ directly, we start by differentiating with respect to ${u}$ in direction $h$ and applying the product rule. This yields $$\label{eq:Hessian}
[{H}({u}^k){h}](t) = \alpha {h}(t) + \begin{pmatrix}
\int_{-a}^a \delta {M}(t;z){A_1} {P}(t;z) + {M}(t;z){A_1}\delta {P}(t;z)\,dz\\[0.5ex]
\int_{-a}^a \delta {M}(t;z){A_2} {P}(t;z) + {M}(t;z){A_2}\delta {P}(t;z)\,dz
\end{pmatrix},
\quad 0\leq t \leq T_u,$$ where $\delta {M}$ – corresponding to the directional derivative of ${M}$ with respect to ${u}$ – is given by the solution of the linearized state equation $$\label{eq:bloch_lin}
\left\{\begin{aligned}
\dot{{\delta {M}}}(t;z) &= {A}({u}^k;z) {\delta {M}}(t;z)+ {A}'({h}){M}, \qquad 0<t\leq T,\\
\delta {M}(0;z) &= (0,0,0)^T,
\end{aligned}\right.$$ with $${A}'({h})= \gamma B_1
\begin{pmatrix}
0 & 0 & {h_y}(t)\\
0 & 0 & {h_x}(t)\\
-{h_y}(t) & - {h_x}(t) & 0
\end{pmatrix},$$ and $\delta {P}$ – corresponding to the directional derivative of ${P}$ with respect to ${u}$ – is the solution of the linearized adjoint equation $$\label{eq:adjoint_lin}
\left\{\begin{aligned}
-\dot{\delta {P}}(t;z) &= {A}({u}^k;z)^T {\delta {P}}(t;z)+ {A}'({h})^T {P},\qquad 0\leq t<T,\\
\delta {P}(T;z) &= \delta {M}(T;z).
\end{aligned}\right.$$ This characterization can be derived using formal Lagrangian calculus and rigorously justified using the implicit function theorem; see, e.g., [@Hinze:2009 Chapter 1.6]. Since can be computed by solving the two ODEs and , the cost of computing a single Hessian action is comparable to that of a gradient evaluation; cf. . This has already been observed in the context of seismic imaging [@Santosa:1988], meteorology [@Wang:1992], and optimal control of partial differential equations [@Hinze:2001], but has received little attention so far in the context of optimal control of ODEs.
One difficulty is that the Bloch equation is bilinear since it involves the product of the unknowns ${u}$ and ${M}$. Hence, the optimal control problem is not convex and the Hessian ${H}({u})$ is not necessarily positive definite (or even invertible), thus precluding a direct application of the CG-Newton method. We therefore embed Newton’s method into the trust-region framework of Steihaug [@Steihaug83], where a breakdown of the CG method is handled by a trust-region step and the trust region radius is continually adapted. This allows global convergence (i.e., for any starting point) to a local minimizer as well as transition to fast quadratic convergence; see [@Steihaug83]. As an added advantage, computational time is saved since the CG method is usually not fully resolved far away from the optimum. The full algorithm is given in \[sec:algo\].
Discretization
--------------
For the numerical computation of optimal controls, both the Bloch equation and the optimal control problem in need to be discretized. Here, the time interval $[0,T]$ is replaced by a time grid $0=t_0<\dots<t_N=T$ with time steps $\Delta t_m:=t_m -t_{m-1}$, chosen such that $t_{N_u} = T_u<T$ for some $N_u<N$. The domain $[-a,a]$ is replaced by a spatial grid $-a = z_1<\dots<z_Z = a$ with grid sizes $\Delta z_m:=z_m-z_{m-1}$. We note that for each $z_i$, the corresponding ODEs can be solved independently and in parallel. The Bloch equation is discretized using a Crank–Nicolson method, where the state ${M}$ is discretized as continuous piecewise linear functions with values ${M}_m:={M}(t_m)$, and the controls ${u}$ are treated as piecewise constant functions, i.e., ${u}=\sum_{m=1}^{N_u} {u_m} \chi_{(t_{m-1},t_m]}(t)$, where $\chi_{(a,b]}$ is the characteristic function of the half-open interval $(a,b]$.
For the efficient computation of optimal controls, it is crucial that both the gradient and the Hessian action are computed in a manner consistent with the chosen discretization. This implies that the adjoint state ${P}$ has to be discretized as piecewise constant using an appropriate time-stepping scheme [@BecMeiVex_07], and that the linearized state $\delta{M}$ and the linearized adjoint state $\delta{P}$ have to be discretized in the same way as the state and adjoint state, respectively. Furthermore, the conjugate gradient method has to be implemented using the scaled inner product $\langle {u},{v}\rangle := \sum_{m=1}^{N_u}\Delta t_m{u}_m{v}_m$ and the corresponding induced norm $\|{u}\|^2:=\langle {u},{u}\rangle$. For completeness, the resulting schemes and discrete derivatives are given in \[sec:discrete\].
Methods {#sec:methods}
=======
This section describes the computational implementation of the proposed pulse design and the experimental protocol for its validation.
Pulse design
------------
The OC approach described in \[sec:theory\] is implemented in MATLAB (The MathWorks, Inc., Natick, USA) using the Parallel Toolbox for parallel solution of the (linearized) Bloch and adjoint equation for different values of $z_i$. In the spirit of reproducible research, the code used to generate the results in this paper can be downloaded from <https://github.com/chaigner/rfcontrol/releases/v1.2>.
The initial magnetization vector is set to equilibrium, i.e., ${M^0}(z) =M_0(0,0,1)^T$. The slice-selective gradient $G_z(t)$ is extracted out of a standard Cartesian GRE sequence simulation and consists of a trapezoidal shape of length that is followed by a re-phasing part of length to correct the phase dispersion using the maximal slew rate; i.e., $T_u =$ and $T=$ with a temporal resolution of $\Delta t=$ for the single-slice excitation (see dashed line in \[fig:single:1\]) and $T_u =$ and $T=$ with a temporal resolution of $\Delta t=$ for the SMS excitation (see dashed line in \[fig:sms:a2\]). This corresponds in both cases to $N=697$ uniform time steps for the time interval $[0,T]$ and $N_u=512$ time steps for the control interval $[0,T_u]$. For the spatial computational domain, $a=$ is chosen to consider typical scanner dimensions; the domain $[-a,a]$ is discretized using $Z=5001$ equidistant points to achieve a homogeneous spatial resolution of $\Delta z=$ .
For the desired magnetization vector, we consider three examples:
#### Single-slice excitation
To validate the design procedure, we compute an optimized pulse for a single slice of a given thickness $\Delta_w$ and a flip angle of , i.e., we set $${\tilde M_d}(z) = \begin{cases} \left(0,\sin(\ang{90}),\cos(\ang{90})\right)^T &\text{if } |z|<\Delta_w/2,\\
(0,0,1)^T &\text{else,}
\end{cases}$$ as visualized in \[fig:initial:3\_zoom\]. To reduce Gibbs ringing, this vector is filtered before the optimization with a Gaussian kernel with a full width at half maximum of . For comparison, an SLR pulse [@Pauly91; @Lee07] with an identical temporal resolution and pulse duration is designed to the same specification (slice width, flip angle, full width at half maximum) using the Parks–McClellan (PM) algorithm [@Pauly91] with a in- and out-of slice ripple as usual [@Pauly91] and a bandwidth of . To achieve a fully refocused magnetization, the refocusing area of the slice selective gradient for the SLR pulse is increased by compared to the OC pulse (\[fig:single:1\]).
#### SMS excitation: phantom
RF pulses for the simultaneous excitation of two, four, and six equidistant rectangular slices with a flip angle of are computed, i.e., we set $${\tilde M_d}(z) = \begin{cases} \left(0,\sin(\ang{90}),\cos(\ang{90})\right)^T &\text{if $z$ in slice},\\
(0,0,1)^T &\text{if $z$ out of slice,}
\end{cases}$$ and apply Gauss filtering; see \[fig:initial:4\_zoom\] for the case of six slices.
Since PINS pulses are not suitable for axial or axial-oblique slice preference as they generate a periodic slice pattern extending outside the field of interest [@Ugurbil13], the optimized pulses are compared with conventional SMS pulses obtained using superposed phase-shifted sinc-based excitation pulses, again for the same slice width, flip angle and full width at half maximum.
#### SMS excitation: in-vivo
Since multi-slice in-vivo imaging using slice-GRAPPA starts to suffer from g-factor problems for more than three slices, we modify the above-described SMS pulses using a CAIPIRINHA-based excitation pattern [@Breuer06], which alternates two different pulses to achieve phase-shifted magnetization vectors in order to increase the spatial distance of aliased voxels. Here, the first vector and pulse are identical to those designed for the phantom. For odd slice numbers, the second vector is modified by adding a phase term of $\pi$ to every second slice of the desired magnetization, i.e., $${\tilde M_d}(z) = \begin{cases} \left(0,\pm\sin(\ang{90}),\cos(\ang{90})\right)^T &\text{if $z$ in odd/even slice},\\
(0,0,1)^T &\text{if $z$ out of slice}
\end{cases}$$ (before filtering). For even slice numbers, the transverse pattern has to be further shifted by $\frac{\pi}2$, i.e., $${\tilde M_d}(z) = \begin{cases} \left(\pm\sin(\ang{90}),0,\cos(\ang{90})\right)^T &\text{if $z$ in even/odd slice},\\
(0,0,1)^T &\text{if $z$ out of slice,}
\end{cases}$$ see \[fig:initial:5\_zoom\] for the case of six slices. The additional phase shift is balanced before reconstruction by subtracting a phase of $\frac{\pi}{2}$ from every second phase-encoding line of the measured k-space data. Since typical relaxation times in the human brain are at least an order of magnitude bigger than the pulse duration, relaxation effects are neglected in the optimization.
The starting point for the optimization is chosen in all cases as ${u}^0=[0,\dots,0]$. The control cost parameter is fixed at $\alpha = 10^{-4}$ for both the single-slice and the multi-slice optimization. The parameters in \[alg:trcgn\] are set to $\mathrm{tol}_N = 10^{-9}$, $\mathrm{maxit}_N = 5$, $\mathrm{tol}_C = 10^{-6}$, $\mathrm{maxit}_{C} = 50$, $\rho_0 = 1$, $\rho_{\max} = 2$, $q=2$, $\sigma_1=0.03$, $\sigma_2 = 0.25$, $\sigma_3 = 0.7$. All calculations are performed on a workstation with a four-core processor with (Intel i-P) and of RAM.
Experimental validation
-----------------------
Fully sampled experimental data for a phantom and a healthy volunteer were acquired on a MR scanner (Magnetom Skyra, Siemens Healthcare, Erlangen, Germany) using the built-in body coil to transmit the RF pulse. The MR signals were received using a body coil for the phantom experiments and a -channel head coil for the in-vivo experiments. A standard Cartesian GRE sequence was modified to import and apply external RF pulses. By changing the read-out gradient from the frequency-axis to the slice direction, the excited slice can be measured and visualized. The single-slice excitation was measured using a water filled sphere with a diameter of . To acquire a high resolution in z-direction, we used a matrix size of with a FOV of and a bandwidth of . The echo time was $T_E=$ and the repetition time $T_R=$ to get fully relaxed magnetization before the next excitation. The SMS phantom experiments were performed using a homogeneous cylinder phantom with diameter of , length of , and relaxation times $T_1=$ , $T_2=$ , and $T_2^*=$ . The sequence parameters were $T_E=$ , $T_R=$ , bandwidth , matrix size , and a field of view of .
To verify the in-vivo applicability, human brain images of a healthy volunteer were acquired using the above described GRE sequence modified to include the optimized CAIPIRINHA-based pulses. The sequence parameters were set to $T_E=$ , $T_R=$ , bandwidth , matrix size and FOV . After acquisition, the k-space data of the individual slices were separated using an offline slice-GRAPPA ( coils, kernel size of ) reconstruction [@Setsompop12; @Cauley14]. The reference scans used in the slice-GRAPPA reconstruction were performed with the same sequence using an optimized single-slice pulse (not shown here). To decrease the scanning time, we acquired k-space lines ( of the full dataset) around the k-space center for each reference scan. After this separation, a conventional Cartesian reconstruction was performed individually for each slice.
Results {#sec:res}
=======
#### Single-slice excitation
shows the results of the design of an RF pulse for the excitation of a single slice of width $\Delta_w =$ ; see \[fig:initial:3\_zoom\]. The computed pulse (after Newton iterations and a total number of CG steps taking on the above-mentioned workstation is shown in \[fig:single:1\]. (To indicate the sequence timing, the slice-selective gradient $G_z$ – although not part of the optimization – is shown dashed.) It can be seen that $u_x (t)$ is similar, but not identical, to a standard sinc shape, and that $u_y (t)$ is close to zero, which is expected due to the symmetry of the prescribed slice profile. contains a detail of the corresponding transverse magnetization ${M_{xy}}(T)=({M_x}(T)^2+{M_y}(T)^2)^{1/2}$ obtained from the numerical solution of the Bloch equation, which is confirmed by experimental phantom measurements in \[fig:single:4\] and \[sub@fig:single:3\]. Both simulation and measurement show an excitation with a steep transition between the in- and out-of-slice regions and a homogeneous flip angle distribution across the target slice.
\
compares the optimized (OC) pulse with a standard SLR pulse by showing details of the corresponding simulated magnetizations (\[fig:comp:5\] for OC and \[fig:comp:6\] for SLR; in both cases the targeted ideal magnetization is shown dashed). It can be seen that the in-slice magnetization of the optimized pulse has oscillations of higher frequency but of much smaller amplitude than that of the SLR pulse. This becomes especially visible when comparing the resulting in-slice phases (\[fig:comp:3\]).
This is achieved by allowing higher ripples close to the slice while decreasing the amplitude monotonically away from the slice. (Note that only a small central segment of this region is shown in the figures.) This leads to the total root mean squared error (RMSE) and the mean absolute error (MAE) with the ideal rectangular magnetization pattern (\[fig:comp:4\]) matching the full width at half maximum of both pulses being smaller for the OC pulse ( and , respectively) compared to the SLR pulse ( and , with an equal power demand for both pulses.
\
#### SMS excitation: phantom
shows the results of the design of RF pulses for simultaneous excitation of two, four and six equidistant slices with a separation of and a thickness $\Delta_w=$ ; see \[fig:initial:4\_zoom\]. The computational effort in all cases is similar to that in the single-slice case. The corresponding computed pulses are shown in . A graphical analysis shows that instead of higher amplitudes, the optimization distributes the total RF power (which increases with the number of slices) more uniformly over the pulse length. A central section of the corresponding optimized slice profiles are given in . It can be seen that all slices have a sharp profile which does not deteriorate as the number of slices increases (although it decreases slightly farther from the center and the bandwidth is slightly reduced). These results are validated by the experimental phantom measurements using the computed pulses: show the reconstructed excitation inside the phantom, while show the measured slice profiles along a cut parallel to the $x$-axis in the center of the previous images.
A quantitative comparison of SLR and OC-based SMS pulses from one to six simultaneous slices is given in \[tab:exp2\], which shows both the power requirement of the computed pulses, both in total B$_1$ energy $$\|{B_{1,x}}\|_2^2 = \int_0^T |B_1 u_x(t)|^2\,dt$$ and in peak B$_1$ amplitude $$\|{B_{1,x}}\|_\infty = \max_{t\in[0,T]} |B_1 u_x(t)|,$$ as well as the mean absolute error (MAE) with respect to the ideal (unfiltered) slice profiles for the in-slice and the out-of-slice regions. While both methods lead to a linear increase of the total energy with the number of slices, the peak amplitude increases more slowly for the OC pulses than for the conventional pulses. Furthermore, we remark that the peak B$_1$ amplitude for four, five and six slices remain similar. Regarding the corresponding slice profiles, the OC pulses lead to a significantly lower MAE in both the in-slice and out-of-slice regions compared to the SLR pulses. Visual inspection of shows that this is due to the fact that the out-of-slice ripples are concentrated around the in-slice regions while quickly decaying away from them.
[S\[table-format=1.0\] S\[table-format=2.1\] S\[table-format=2.1\] S\[table-format=2.1\] S\[table-format=2.2\] S\[table-format=1.3\] S\[table-format=1.3\] S\[table-format=1.4\] S\[table-format=1.4\]]{} & & & &\
& & & &\
(lr)[2-3]{} (lr)[4-5]{} (lr)[6-7]{} (lr)[8-9]{}
[slices]{} & [conv]{} & [OC]{} & [conv]{} & [OC]{} & [conv]{} & [OC]{} & [conv]{} & [OC]{}\
1 & 19.5 & 19.5 & 3.5 & 3.49 & 0.062 & 0.052 & 0.0039 & 0.0014\
2 & 38.9 & 38.1 & 7.0 & 6.78 & 0.060 & 0.052 & 0.0040 & 0.0018\
3 & 58.4 & 57.2 & 10.5 & 10.02 & 0.054 & 0.053 & 0.0039 & 0.0030\
4 & 77.9 & 76.3 & 14.0 & 12.13 & 0.065 & 0.045 & 0.0086 & 0.0031\
5 & 97.3 & 95.5 & 17.5 & 11.38 & 0.059 & 0.053 & 0.0078 & 0.0051\
6 & 116.8 & 113.9 & 21.0 & 12.63 & 0.068 & 0.053 & 0.0075 & 0.0067\
Finally, we illustrate the influence of the regularization parameter $\alpha$ in \[tab:regalpha\], where the root of mean square error (RMSE), the total B$_1$ energy as well as the B$_1$ peak of the OC SMS 6 pulses is shown for different values of the control cost parameter $\alpha$. As can be seen, a bigger $\alpha$ leads to an increase in the error between desired and controlled magnetization while both the total B$_1$ power and the peak B$_1$ amplitude are reduced, although these effects amount to less than over a range of parameters spanning two orders of magnitude. This demonstrates that the results presented here are robust with respect to the choice of the control cost parameter.
[S\[table-format=1.0e+1\] S\[table-format=1.3e+1\] S\[table-format=3.1\] S\[table-format=2.2\] ]{} & [RMSE]{} & [$\|B_{1,x}\|_2^2$]{} & [$\|B_{1,x}\|_{\infty}$]{}\
[\[a.u.\]]{} & [\[a.u.\]]{} & [\[a.u.\]]{} & [\[\]]{}\
1e-5 & 2.374e-2 & 117.0 & 12.75\
5e-5 & 2.375e-2 & 115.1 & 12.71\
1e-4 & 2.377e-2 & 113.9 & 12.62\
5e-4 & 2.437e-2 & 106.7 & 12.14\
1e-3 & 2.591e-2 & 98.9 & 11.63\
#### SMS excitation: in-vivo
The CAIPIRINHA-based modifications to the SMS pulse design (see \[fig:initial:5\_zoom\]) are illustrated in \[fig:method1\] (showing the case of five slices for the sake of variation). shows the unmodified pulse, which differs in structure from the cases with an even number of slices in, e.g., \[fig:sms:a6\] due to the different symmetry of the slice profile (see \[fig:method1:2\]). On the other hand, the pulse is very similar to the modified pulse for the alternating phase shift; see \[fig:method1:3\] for the computed pulse and \[fig:method1:4\] for the resulting slice profile. For illustration, a slice-aliased reconstruction of the acquired in-vivo data using this pulse sequence is shown in \[fig:method1:5\].
z readout
![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms2_z "fig:"){width="100.00000%"}\
![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms4_z "fig:"){width="100.00000%"}\
![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms6_z "fig:"){width="100.00000%"}
sG reconstruction
![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms2_1 "fig:"){width="15.00000%"} ![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms2_2 "fig:"){width="15.00000%"}\
![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms4_1 "fig:"){width="15.00000%"} ![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms4_2 "fig:"){width="15.00000%"} ![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms4_3 "fig:"){width="15.00000%"} ![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms4_4 "fig:"){width="15.00000%"}\
![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms6_1 "fig:"){width="15.00000%"} ![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms6_2 "fig:"){width="15.00000%"} ![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms6_3 "fig:"){width="15.00000%"} ![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms6_4 "fig:"){width="15.00000%"} ![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms6_5 "fig:"){width="15.00000%"} ![Slice-GRAPPA reconstruction of in-vivo data using CAIPIRINHA-based SMS excitation pattern for two (top), four (middle) and six (bottom) slices (left: conventional reconstruction showing the collapsed data in slice-encoding direction; right: reconstruction of GRAPPA-separated slices)[]{data-label="fig:exp3"}](result5_sms6_6 "fig:"){width="15.00000%"}
shows the image reconstruction using optimized RF pulses for simultaneous excitation of two, four and six slices with the same slice separation and thickness as above. As can be seen clearly in the first column, all three pulses lead to the desired excitation pattern in-vivo as well. The remaining columns show the slice-GRAPPA reconstructions, which illustrate that the excitation is uniform across the field of view.
Discussion {#sec:discussion}
==========
Our optimization approach is related to the basic ideas presented by Conolly et al. [@Conolly86]. In the context of MRI, the implementation of this principle was also carried out by other groups using gradient [@Xu08; @Grissom09] and quasi-Newton [@Vinding12] methods. However, these methods do not make full use of second-order information and therefore achieve at best superlinear convergence. In contrast, our Newton method makes use of exact second derivatives and is therefore quadratically convergent. In particular, the main contribution of our work is the efficient computation of exact Hessian actions using the adjoint approach and its implementation in a matrix-free trust-region CG–Newton method. The use of exact derivatives speeds up convergence of the CG method, while the trust-region framework guarantees global convergence and terminates the CG method early especially at the beginning of the optimization. Both techniques save CG steps and therefore computations of Hessian actions, allowing the use of second-order information with limited computational effort and memory requirements. Since computing a Hessian action incurs the same computational cost as a gradient evaluation (i.e., the solution of two ODEs; compare with ), we were able to compute a minimizer, e.g., for the single-slice example, with a computational effort corresponding to $32$ gradient evaluations ($4$ for the right-hand side in each Newton iteration and $28$ for the Hessian action in each CG iteration). This is less than the same number of iterations of a gradient or quasi-Newton method with line search (required in this case for global convergence), demonstrating the efficiency of the proposed approach. Therefore, our method can be used to compute RF pulses with a high temporal resolution, allowing the design of pulses for a desired magnetization on a very fine spatial scale, in particular for the excitation of a sharp slice profile.
Furthermore, the proposed algorithm does not require an educated initial guess for global convergence (to a local minimizer, which might depend on the initial guess if more than one exists) and allows for pulse optimization in non-standard situations where no analytic RF pulse exists (e.g., for large flip angles). Compared to design methods using a simplification or approximation of the Bloch equation [@Pauly89; @Pauly91], our OC based approach is capable of including relaxation terms. However, for standard in-vivo imaging applications of the human head, the relevant relaxation times are very long compared to the RF pulse duration. Thus, in our examples the influence of relaxation during excitation on the designed pulses is insignificant and has been neglected in the optimization process (although the inclusion may be indicated for other applications). The presented direct design approach allows to specify the desired magnetization in x-, y- and z-direction independently for every point in the field of view. This spatial independence of each control point allows to directly apply parallel computing to speed up the optimization process. While real-time optimization was not the aim of this work, a proof-of-concept implementation of the proposed approach on a GPU system (CUDA, double precision, GeForce GTX Ti with cores and of RAM) shows an average speedup of (e.g., instead of for the single-slice example) while yielding identical results, thus making patient-specific design feasible as well as making the gap between OC and SLR pulse design nearly negligible. This allows efficient and fast generation of accurate slice profiles – important for minimal slice gaps, optimal contrast and low systematic errors in quantitative imaging – for arbitrary flip angles and even for specialized pulses such as refocusing or inversion.
In particular, our approach can be used to design pulses for the simultaneous excitation of multiple slices, which increases the temporal efficiency of advanced imaging techniques such as diffusion tensor imaging, functional imaging or dynamic scans. In these contexts, SMS excitation is successfully used to reduce the total imaging time [@Larkman01; @Nunes06; @Feinberg13; @Ugurbil13]; however, the peak B$_1$ amplitude of conventional SMS pulses is one of the main restrictions of applying SMS imaging to high-field systems [@Ugurbil13]. The performed studies show that compared to conventional SMS design, the presented procedure yields pulses with a reduced B$_1$ peak amplitude (e.g., reduction for six simultaneous slices). Depending on the desired temporal resolution, the bandwidth and the slice profiles of the outer slices are slightly changed, which results in a decreased B$_1$ peak amplitude. It could be shown that the peak B$_1$ amplitude does not increase linearly with the number of slices, while the power requirement per slice remains constant and the overall power consumption is comparable to that of conventional pulses. To further reduce the SAR it is necessary to either change the excitation velocity using a time-varying slice selective gradient [@Conolly88], or to extend the pulse design to parallel transmit [@Katscher08; @Wu13; @Poser14; @Guerin14]. Furthermore, our OC-based pulses produce sharp slice profiles with a lower mean absolute error compared to the used PM-based SLR pulse, both in- and out-of-slice, at the cost of slightly larger out-of-slice ripples close to the in-slice regions. Of course, the ripple behavior of the SLR pulse can be balanced with the transition steepness by using different digital filter design methods (i.e. PM for minimizing the maximum ripple or a least squares linear-phase FIR filter for minimizing integrated squared error). The OC ripple amplitude close to the transition band can be further controlled by using offset-dependent weights as demonstrated by Skinner et al. [@Skinner12_2]. In addition, the computational complexity of OC methods is significantly higher than for direct or linearized methods. This implies that OC-based pulse design is advantageous in situations where high in-slice contrast and low B$_1$ peak amplitude are important, while SLR pulses should be used when minimal near-slice excitation and computational effort are crucial.
The presented OC approach is able to avoid some possible disadvantages of previously proposed design methods for SMS excitation. In particular, the OC design method prescribes each slice with the same uniform echo-time and phase in comparison to time-shifted [@Auerbach13], phase relaxation [@Wong12] and nonlinear phase design techniques [@Zhu14; @Sharma15]. On the other hand, some of their features such as different echo times [@Auerbach13] or a non-uniform phase pattern [@Wong12; @Zhu14; @Sharma15] (e.g., for spin echo experiments) can be incorporated in our approach to further reduce the B$_1$ peak amplitude. It also should be possible to combine the OC design method with other techniques analogous to MultiPINS [@Eichner14; @Gagoski15] that combine PINS with conventional multiband pulses for a further reduction of SAR. Finally, the phantom and in-vivo experiments demonstrate that it is possible to simply replace standard pulses by optimized pulses in existing imaging sequences, and that the proposed method is therefore well suited for application in a wide range of imaging situations in MRI.
Conclusions {#sec:conclusions}
===========
This paper demonstrates a novel general-purpose implementation of RF pulse optimization based on the full time-dependent Bloch equation and a highly efficient second-order optimization technique assuring global convergence to a local minimizer, which allows large-scale optimization with flexible problem-specific constraints. The power and applicability of this technique was demonstrated for SMS, where a reduced B$_1$ peak amplitude allows exciting a higher number of simultaneous slices or achieving a higher flip angle. Phantom and in-vivo measurements (on a scanner) verified these findings for optimized single- and multi-slice pulses. Even for a large number of simultaneously acquired slices, the reconstructed images show good image quality and thus the applicability of the optimized RF pulses for practical imaging applications. While the computational requirements for optimal control approaches are of course significantly greater than for, e.g., SLR-based approaches, a proof-of-concept GPU implementation indicates that this gap can be sufficiently narrowed to make patient-specific design feasible.
Due to the flexibility of the optimal control formulation and the efficiency of our optimization strategy, it is possible to consider field inhomogeneities (B$_1$, B$_0$), design complex RF pulses for parallel transmit, or to extend the framework to include pointwise constraints due to hardware limits such as peak B1 amplitude and slew rate.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is funded and supported by the Austrian Science Fund (FWF) in the context of project “SFB F3209-18” (Mathematical Optimization and Applications in Biomedical Sciences). Support from BioTechMed Graz and NAWI Graz is gratefully acknowledged. We would like to thank Markus Bödenler from the Graz University of Technology for the CUDA implementation of our algorithm.
Trust-region algorithm {#sec:algo}
======================
Set ${u}^0\equiv 0, \quad k=0, \quad {g}\equiv 1,\quad \rho = \rho_0$ ()[$\|g\|>\mathrm{tol}_N$ and $k<\mathrm{maxit}_N$]{}
Compute gradient ${g}({u}^k)$ Set ${p}^0={r}^0=-{g}({u}^k)$, $\delta {u}=0$, $i = 0$ ()[$\|{r}^i\| > \mathrm{tol}_C\|{r}^0\|$ and $i<\mathrm{maxit}_C$]{}
Compute ${H}({u}^k){p}^i$ ()[$\langle{p}^i,{H}({u}^k){p}^i\rangle<\varepsilon$]{}
Compute $\max\{\tau:\|\delta {u} + \tau {p}^i\|\leq \rho\}$ Set $\delta {u} = \delta {u} + \tau {p}^i$; **break**
Compute $\alpha = \|{r}^i\|/\langle {p}^i, {H}({u}^k){p}^i\rangle$ ()[$\|\delta {u}+\alpha {p}^i\|\geq \rho$]{}[ Compute $\max\{\tau:\|\delta {u} + \tau {p}^i\|\leq \rho\}$ Set $\delta {u} = \delta {u} + \tau {p}^i$; **break** ]{} Set ${r}^{i+1} = {r}^{i} - \alpha {H}({u}^k){p}^i$ Set ${p}^{i+1} = {r}^{i+1} + \|{r}^{i+1}\|^2/\|{r}^i\|^2 {p}^{i}$ Set $\delta {u} = \delta {u} + \alpha {p}^i$,$i = i+1$
Compute $\delta J_a = J({u}^k) - J({u}^k +\delta {u})$ Compute $\delta J_m = -\tfrac12 \langle \delta {u},{H}({u}^k)\delta {u}\rangle - \langle\delta {u}, {g}({u}^k)\rangle$ ()[$\delta J_a >\varepsilon$ and $\delta J_a > \sigma_1 \delta J_m$]{}[ Set ${u}^{k+1} = {u}^k +\delta {u}$ ]{} ()[$\delta J_a>\varepsilon$ and $|\delta J_a /\delta J_m - 1| \leq 1-\sigma_3$]{}[ Set $\rho = \min\left\{q \rho,\rho_{\max}\right\}$ ]{} ()[$\delta J_a \leq \varepsilon$]{}[ Set $\rho = \rho / q$ ]{} ()[$\delta J_a < \sigma_2 \delta J_m$]{}[ Set $\rho = \rho / q$ ]{}
Discretization {#sec:discrete}
==============
#### Cost functional:
$$J({M},{u})= \frac{1}{2} \sum_{i=1}^Z \Delta z_i |{M}_{N,i}-{M_d}(z_i)|_2^2 + \frac\alpha2 \sum_{m=1}^N \Delta t_m |{u}_m|_2^2$$
#### Bloch equation
for all $i=1,\dots,Z$: $$\begin{aligned}[t]
\left[{I}-\frac{\Delta t_m}{2}{A}({u}_m;z_i)\right] {M}_{m,i} &= \left[{I}+\frac{\Delta t_m}{2}{A}({u}_m;z_i)\right] {M}_{m-1,i} +\Delta t_m {b}, \quad m=1,\dots,N\\
{M}_{0,i} &= {M^0}(z_i)
\end{aligned}$$
#### Adjoint equation
for all $i=1,\dots,Z$: $$\begin{aligned}[t]
\left[{I}-\frac{\Delta t_m}{2}{A}({u}_m;z_i)^T\right] {P}_{m,i} &= \left[{I}+\frac{\Delta t_{m+1}}{2}{A}({u}_{m+1};z_i)^T\right] {P}_{m+1,i} , \quad m=1,\dots,N-1\\
\left[{I}-\frac{\Delta t_N}{2}{A}({u}_N;z_i)^T\right]{P}_{N,i}&={M}_{N,i}-{M_d}(z_i)\\
\end{aligned}$$
#### Discrete gradient
for all $m=1,\dots,N_u$: $\bar{{M}}_{m} := \frac12({M}_m + {M}_{m-1})$, $${g}_m = \alpha {u}_m + \gamma B_1
\begin{pmatrix}
\sum_{i=1}^Z \Delta z_i \left({P}_{m,i}^T {A_1} \bar{{M}}_{m,i}\right) \\
\sum_{i=1}^Z \Delta z_i \left({P}_{m,i}^T {A_2} \bar{{M}}_{m,i}\right)
\end{pmatrix}$$
#### Linearized state equation
for all $i=1,\dots,Z$: $$\begin{aligned}[t]
\left[{I}-\frac{\Delta t_m}{2}{A}(u_m;z_i)\right] \delta {M}_{m,i} &= \left[{I}+\frac{\Delta t_m}{2}{A}(u_m;z_i)\right] \delta {M}_{m-1,i} + \Delta t_m {A'}(\delta u_m) \bar{{M}}_{m,i} , \quad m=1,\dots,N\\
\delta {M}_{0,i} &= 0
\end{aligned}$$
#### Linearized adjoint equation
for all $i=1,\dots,Z$: $$\begin{aligned}[t]
\left[{I}-\frac{\Delta t_m}{2}{A}(u_m;z_i)^T\right] \delta {P}_{m,i} &= \left[{I}+\frac{\Delta t_{m+1}}{2}{A}(u_{m+1};z_i)^T\right] \delta {P}_{m+1,i} +\frac{\Delta t_m}{2}{A'}(\delta {u}_m)^TP_{m,i}\\
\MoveEqLeft[-1]+\frac{\Delta t_{m+1}}{2}{A'}(\delta {u}_{m+1})^T {P}_{m+1,i} , \quad m=1,\dots,N-1\\
\left[{I}-\frac{\Delta t_N}{2}{A}(u_N;z_i)^T\right]\delta {P}_{N,i}&=\delta {M}_{N,i} + \frac{\Delta t_N}{2}{A'}(\delta {u}_{N})^T{P}_{N,i}
\end{aligned}$$
#### Discrete Hessian action
for all $m=1,\dots,N_u$: $\bar {\delta {M}_{m}} := \frac12(\delta {M}_m + \delta {M}_{m-1})$ $$\quad [{H}({u}){h}]_m= \alpha {h}_{m} + \gamma B_1
\begin{pmatrix}
\sum_{i=1}^Z \Delta z_i \left(\delta {P}_{m,i}^T {A_1} {\bar M}_{m,i} + {P}_{m,i}^T {A_1} {\bar{\delta M}}_{m,i}\right)\\
\sum_{i=1}^Z \Delta z_i \left(\delta {P}_{m,i}^T {A_2} {\bar M}_{m,i} + {P}_{m,i}^T {A_2} {\bar{\delta M}}_{m,i}\right)
\end{pmatrix}$$
[^1]: Institute of Medical Engineering, Graz University of Technology, Kronesgasse 5, and BioTechMed Graz, 8010 Graz, Austria ()
[^2]: Faculty of Mathematics, University of Duisburg-Essen, 45117 Essen, Germany
[^3]: Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria
|
---
abstract: 'Understanding sequential information is a fundamental task for artificial intelligence. Current neural networks attempt to learn spatial and temporal information as a whole, limited their abilities to represent large scale spatial representations over long-range sequences. Here, we introduce a new modeling strategy called Semi-Coupled Structure (SCS), which consists of deep neural networks that decouple the complex spatial and temporal concepts learning. Semi-Coupled Structure can learn to implicitly separate input information into independent parts and process these parts respectively. Experiments demonstrate that a Semi-Coupled Structure can successfully annotate the outline of an object in images sequentially and perform video action recognition. For sequence-to-sequence problems, a Semi-Coupled Structure can predict future meteorological radar echo images based on observed images. Taken together, our results demonstrate that a Semi-Coupled Structure has the capacity to improve the performance of LSTM-like models on large scale sequential tasks.'
author:
- |
Bo Pang, Kaiwen Zha, Hanwen Cao, Jiajun Tang, Minghui Yu, Cewu Lu\*\
Shanghai Jiao Tong University\
[{pangbo, Kevin\_zha, mbd\_chw, yelantingfeng, 1475265722, lucewu}@sjtu.edu.cn]{}
title: Complex Sequential Understanding through the Awareness of Spatial and Temporal Concepts
---
Complex sequential tasks involve extremely high-dimensional spatial signal over long timescales. Neural networks have made breakthroughs in sequential learning [@graves2013generating; @sutskever2014sequence], visual understanding [@krizhevsky2012imagenet; @he2016deep; @he2017mask], and robotic tasks [@levine2016end; @schulman2015trust]. Conventional neural networks treat spatial and temporal information as a whole, processing these parts together. This limits their ability to solve complex sequential tasks involving high-dimensional spatial and temporal components [@feichtenhofer2018slowfast; @kim2017residual]. A natural idea to address this limitation is to learn the two different concepts relatively independently.
Here, we introduce a structure that decouples spatial and temporal information, implicitly learning respective spatial and temporal concepts through a deep comprehensive model. We find that such concept decomposition significantly simplifies the learning and understanding process of complex sequences. Due to the differentiable property of this structure, which we call Semi Coupled Structure (SCS), we can train it end to end with gradient descent, allowing it to effectively learn to decouple and integrate information in a goal-directed manner.
{width="\linewidth"}
Awareness of Spatial and Temporal Concept {#sec:aware}
=========================================
In the brain, there are two different pathways that feed temporal information and contextual representations respectively into the hippocampus [@kitamura2015entorhinal]. This implies that spatial and temporal concepts are learnt by different cognitive mechanisms and, moreover, that they should be synchronized in order to effectively process sequential information. Taking inspiration from this mechanism in the brain, the deep neural model that is implicitly aware of the two concepts can be formulated as: $$\begin{aligned}
\mathcal{F}[h_s(\mathbf{x}|\psi_s), h_t(\mathbf{x}|\psi_t)] \label{eq:split}
\end{aligned}$$ where $\mathbf{x}$ is the input, and $\psi_s$ and $\psi_t$ are the parameters to optimize. $h_s$ aims at extracting spatial information, while $h_t$ is designed to handle temporal learning. These two kinds of information are fed into $\mathcal{F}$ which is designed to output the final processing results, just like the hippocampus.
{width="\linewidth"}
We further advance our model by considering the fact that spatial and temporal information are deeply coupled with each other, when processed by a brain [@oliveri2009spatial]. Therefore, the model can naturally be extended as a deep nested structure to model such mutual-coupling. We define the $i^{th}$ coupling unit as: $$\begin{aligned}
\mathcal{G}_i(\mathbf{x}_i) = \mathcal{F}[h_s(\mathbf{x}_i|\psi_s^i), h_t(\mathbf{x}_i|\psi_t^i)] \label{eq:G}
\end{aligned}$$ thus, the deep spatial-temporal Semi-Coupled Structure can be expressed as: $$\begin{aligned}
\mathcal{T}[\mathbf{x};\Psi_s, \Psi_t] = \mathcal{G}_1\circ \mathcal{G}_2 \circ ... \circ \mathcal{G}_n(\mathbf{x}) \label{eq: Gs}
\end{aligned}$$ where $n$ is the depth of the deep nested model, and $\Psi_s = \{\psi_s^1,...,\psi_s^n\}$ and $\Psi_t =\{\psi_t^1,...,\psi_t^n\}$ are the parameter sets.
In this structure, spatial and temporal information are intertwined deeply and collaboratively, meanwhile, $h_s(\cdot)$ and $h_t(\cdot)$ are responsible for spatial and temporal concept processing respectively. To this end, we propose two design paradigms (see Fig. \[fig:pipeline\]).
- **Structure Paradigms** $h_s(\cdot)$ and $h_t(\cdot)$ work as their roles by their different structural designs. At a certain time stamp of the sequence, the structure of $h_t(\cdot)$ should have access to the temporal information of other stamps in the sequence like the Recurrent Neural Network (RNN). While for $h_s(\cdot)$, it has no direct connection to the samples of other time stamps, so it can focus on the spatial information, which normally can be a CNN structure.
- **Task Paradigms** Because of the deep nested structure, $h_s(\cdot)$ and $h_t(\cdot)$ will disturb each other. To make $h_s(\cdot)$ and $h_t(\cdot)$ further focus on their roles, besides the main goal: $g=\mathcal{T}[\mathbf{x}; \Psi_s, \Psi_t]$, we assign two extra sub-goals: $r_s = \mathcal{T}^1[\mathbf{x};\Psi_s, \Psi_t]$ and $r_t = \mathcal{T}^2[\mathbf{x};\Psi_s, \Psi_t]$, where $\mathcal{T}^1$ and $\mathcal{T}^2$ share the same model components and parameters with $\mathcal{T}$. We also call $r_s$ and $r_t$ as the spatial and temporal indicating goals which can reflect the qualities of spatial and temporal features. The key design is to make both two indicating goals only impact on their own parameters: $\Psi_s$ or $\Psi_t$. That is, in $\mathcal{T}^1$, ${\partial r_s} / {\partial \Psi_t} = 0$ and in $\mathcal{T}^2$, ${\partial r_t} / {\partial \Psi_s} = 0$. The specific definition of the sub-goals depends on different tasks. Taking action recognition as an example, $r_s$ can be human poses in a single frame that is unrelated to temporal information but useful for action understanding, and $r_t$ can be the estimates of optical flow.
This new modeling strategy is called Semi-Coupled Structure (SCS). It is a general framework that is easy to be revised to fit various applications. If the temporal indicating label of a specific application is difficult to provide, we find that only $r_s$ is enough to encourage $h_s(\cdot)$ to focus on spatial learning, and $h_t(\cdot)$ can naturally take the responsibility of the remain (temporal) information.
#### Discussion
The proposed SCS makes each component be responsible for a specific sub-concept (spatial or temporal). This strategy widely exists in the brain using several different encephalic regions to complete a single complex task [@wolman2012tale; @diez2015novel]. During this process, our method learns to separate temporal and spatial information, even though we do not define them separately. There are previous works that also try to separate the temporal and spatial information, but they adopt the hand-craft spatial and temporal definitions, like Two-Stream model [@simonyan2014two] which uses optical flow [@lucas1986generalized] to define temporal information and SlowFast Networks [@feichtenhofer2018slowfast] that uses asymmetrical spatial and temporal sampling density to distinguish and define them. Fig. \[fig:toyexample\] illustrates that SCS can successfully decouple the temporal and spatial information in visual sequences. In a toy experiment, the visual sequences show different geometries moving in different directions (see Fig. \[fig:toyexample\] **a** for details) and we note that our method outperforms LSTM [@hochreiter1997long] on recognizing “Which direction is the geometry going?" when it encounters a specific geometry it never saw and “what is the geometry?" when the motion is disordered. Fig. \[fig:toyexample\] **b** shows the feature maps of $h_t(\mathbf{x})$ and $h_s(\mathbf{x})$, and we can primarily recognize that $h_t(\mathbf{x})$ represents temporal-related features and $h_s(\mathbf{x})$ is for the spatial one from the viewpoint of human vision. Interestingly, our model can quantitatively indicate how important the temporal information is toward the final goal by comparing the indicating goals $r_s$ and $r_t$, thanks to the awareness of the temporal and spatial concepts. We believe this quantitative indicator will largely benefit the sequential analysis.
Performance Profiling on Academic and Reality Datasets
======================================================
Video action recognition experiments
------------------------------------
To investigate the capacity of the Semi-Coupled Structure, we conduct the first experiment on video action recognition task. We choose UCF-101 [@soomro2012ucf101], HMDB-51 [@kuehne2011hmdb] and Kinetics-400 [@carreira2017quo] datasets which consist of short videos describing human actions collected from website. Correct classification is inseparable from the comprehensive abilities of extracting temporal and spatial information: for example, distinguishing “triple jump" and “long jump" requires a structure to precisely understand temporal information, while to tell “sweeping floor" and “mopping floor" apart requires great spatial information processing ability. We find that our SCS can successfully learn the temporal information over long-range sequences on limited resources and compared to the conventional sequential models, such as LSTM stacked on CNN [@donahue2015long] and ConvLSTM [@xingjian2015convolutional], our SCS achieves remarkable improvements (see Tab. \[tab:actionResult\] for details). Unlike the previous architectures, our SCS can be trained end-to-end without the support of a backbone network (such as VGG [@simonyan2014very], ResNet [@he2016deep], and Inception [@szegedy2015going]. Due to the temporal and spatial semi-coupling, the network can reduce the interference from the temporal unit to the spatial one so that we can still get high-quality spatial features.
Object outline sequentially annotation experiments
--------------------------------------------------
Although the video action recognition task takes a sequence as input, each sequence only need to be assigned one action label. Therefore, modeling it as a pattern recognition problem instead of a sequence learning is also a way to go. For example, 3D convolution model [@ji20133d; @carreira2017quo] is widely used recently. Based on this consideration, we need a typical sequential task to further validate the SCS’s performance. We, therefore, turn to the outline annotation task.
Unlike video action recognition, outline annotation task calls for a point sequence to represent the outline of the target. Each input consists of an image with a start point to declare which object is the annotation target and an end point to indicate which direction to annotate. The annotation models are trained to give out the outline’s key points of the target object one by one from the provided start point to the end point. A new key point is generated based on the already calculated key points (Fig. \[fig:experiments\] a). The generated key points form the predicted outline and we adopt the IoU between the predicted and ground-truth outline as the evaluation metric. Because it is not easy to give out the complete sequential key points in one step only with the start and end point, it is not suitable to model this task as a pattern recognition task like the video action recognition task.
We adopt CityScapes dataset [@cordts2016cityscapes] as our data source and the target objects are all from the outdoor scene. The relatively complex backgrounds require great ability to extract spatial features. Different from the action recognition task, the temporal information lies in the sequential key point positions which act as the attentions to assist the selection of the subsequent points. As a benchmark we compare our SCS based model, a modified Polygon-RNN model [@castrejon2017annotating], with the original LSTM based Polygon-RNN model. In this case, our deep SCS model reaches an average of 70.4 in terms of IoU, 15% relative improvements over the baseline. Fig. \[fig:experiments\] c illustrates the training processes of SCSs with different depths and training strategies.
Auto-driving experiments
------------------------
Next, we want to evaluate the performance of SCS on some cutting-edge applications. Still, we start from a pattern recognition like problem: the simplified auto-driving problem. We treat the problem as a visual sequence processing task so that we only focus on the driving direction and ignore the route planning, strong driving safety and other things in the real driving environments.
A driving agent, given the sequence of driver’s perspective images, needs to decide the driving direction for the last image. It is worth noting that the agent does not know the historical direction to avoid it making “lazy decision": simply repeating the recent direction. As the previous experiments, this task also requires great ability to process spatial information to figure out the road direction and obstruction condition, and ability to capture temporal information to make coherent decisions.
We evaluate the SCS on the Comma.ai dataset [@santana2016learning] and the LiVi dataset [@chen2018lidar]. The image sequences are the driving videos in real traffic including varied scenes such as highways and mountain roads, and the behaviours of the driver are recorded as the direction label. By experiments, we find that features from $h_s$ record more features of the current road and $h_t$ records more about scene changes during driving (Fig. \[fig:toyexample\] c). This indicates that they divide the works successfully and just as the design purpose, they focus on temporal and spatial features respectively so SCS can remarkably disperse the learning pressure to different components. Again, the SCS performs substantially better than conventional LSTM models (see Tab. \[tab:drivingResult\]).
Precipitation forecasting experiments
-------------------------------------
We further apply our SCS model to precipitation forecasting task in order to test its performance on sequence generation problem. Unlike the previous experiments, where the model receives the input sequence and gives out the output sequence synchronously, we apply a form of “sequence to sequence" learning [@sutskever2014sequence] in which the input sequence is encoded into a representation and then the model gives out the output sequence based on this representation.
Our dataset, which we term as REEC-2018, contains a set of meteorological Radar Echo images for Eastern China in 2018. The metric of the radar echo is composite reflectivity (CR) which can be utilized to predict the precipitation intensity. A model, given a sequence of the radar echo images sorted in time, needs to predict a sequence of the future radar echo images from the previous evolution of CR (see Fig. \[fig:experiments\] b).
Through experiments, we find that our SCS model can successfully generate the results with original evolution trends, such as diffusion and translation. Compared with the ConvLSTM, a conventional sequence model for visual, again, our SCS gains huge performance improvements.
Discussion
==========
In summary, we have built a Semi-Coupled Structure that can learn to divide the work of extracting features automatically. A major reason for utilizing such a structure is to alleviate the interference between learning temporal and spatial features. Many techniques like STSGD and LTSC (see Methods and Fig. \[fig:STSGD\]) are proposed to make the SCS easier to train. The performances of our structure are provided by the experiments, and the theme connecting these experiments is the need to synthesize high dimensional temporal and spatial features embedded in data sequences. All the experiments demonstrate that SCS is able to process visual sequential data regardless of whether the task is sensory processing or sequence learning. Moreover, we have seen that the temporal and spatial features are handled separately by different sub-structures due to the temporal-spatial semi-coupling mechanism (see Fig. \[fig:toyexample\] and Fig. \[fig:ablation\_study\]).
Related Work
============
#### Sequence models
Sequential tasks on high dimensional signal require a model to extract spatial representations as well as temporal features. A series of prior works has shed light on these tough problems: Constrained by the computational resource, inchoate methods [@karpathy2014large; @yue2015beyond; @wang2016actionness; @weinzaepfel2015learning] do not explicitly extract temporal feature, instead, acquire global features by combining spatial information, where pooling is a common method. To extract temporal information, some researchers adopt low-level features, like optical flow [@simonyan2014two; @carreira2017quo], trajectories [@wang2011action; @wang2013dense], and pose estimation [@maji2011action] to deal with temporal information. These low-level features are easy to extract but they are handcraft to some extent, therefore, the performance is limited. Then with more computational resource, Recurrent Neural Networks (RNN) [@donahue2015long; @wu2015modeling; @srivastava2015unsupervised] are widely used, where hidden states take charge of “remembering" the history and extract the temporal features. Recently, 3D convolutional networks [@ji20133d; @carreira2017quo; @feichtenhofer2018slowfast; @wu2019long; @girdhar2019video] appear, where the temporal information is treated as the same with the spatial ones. The large 3D kernel makes this method consume a large amount of computational resource.
#### Methods to split temporal and spatial information
A simple method to split temporal-spatial information is to utilize relatively pure spatial information without temporal one to extract spatial features and pure temporal input for temporal ones. For example, two-stream models [@simonyan2014two; @carreira2017quo; @feichtenhofer2016convolutional] adopt one static image as spatial input and optical flows as temporal input. One problem of this method is that the processes of extracting spatial and temporal features are completely independent, making it impossible to extract hierarchical spatial-temporal features. Another method is to adjust the density of these two types of information. In SlowFast network [@feichtenhofer2018slowfast], the input of spatial stream has higher spatial resolution and lower temporal sampling rate, while the input of temporal stream is the opposite.
Methods
=======
\[Sec:Method\] In this section, we will introduce the detailed structure of SCS, the training method with spatial-temporal switch gradient descent, the strategy to deal with the high-dimension spatial signal and super long sequences, and the designs of the experiments.
Network for SCS
---------------
At every time-stamp $t$, the network $\mathcal{T}$, consisting of $n$ semi-coupled layers, receives an input matrix $\mathbf{x}_t$ from the dataset or environment and outputs an vector $\mathbf{y}_t$ (the main goal $g$) to approximate the target (ground truth) vector $\mathbf{z}_t$.
As mentioned above, each semi-coupled layer satisfies the structure of $\mathbf{u}^{l}_t = \mathcal{F}(h_s(\mathbf{u}^{l-1}_t), h_t(\mathbf{u}^{l-1}_t))$, where $\mathbf{u}^{l}_t$ is the output of the $l^{th}$ layer at $t^{th}$ step and $\mathbf{u}^{l-1}_t$ is the input. By defining $\mathbf{u}^0_t=\mathbf{x}_t$, we get: $$\begin{aligned}
& \mathbf{s}^l_t= h_s(\mathbf{u}^{l-1}_t;\psi_s^l) = {\rm Conv}(\mathbf{u}^{l-1}_t; \psi_s^l) \label{eq:hs}\\
& \mathbf{c}^l_t= h_t(\mathbf{u}^{l-1}_t;\psi_t^l) = {\rm Conv}([\mathbf{u}^{l-1}_t, \sigma(\mathbf{c}^l_{t-1})];\psi_t^l) \label{eq:ht}
\end{aligned}$$ where $l$ is the layer index, $\sigma(x)=1/(1+exp(-x))$ is the logistic sigmoid function, $\rm Conv$ is the convolutional neural layer, $\psi_s^l$ and $\psi_t^l$ are spatial state and temporal cell state matrix, respectively, of layer $l$ at time $t$. $\mathbf{c}^l_0=\mathbf{0}$ is true for all $l$. We adopt $\rm Conv$ here for it is an excellent spatial feature extractor and of course, we can replace $\rm Conv$ by other operators like fully connection, according to different tasks. Note that Eq. \[eq:hs\] describes the structure of $h_s$ and Eq. \[eq:ht\] describes $h_t$ which is a simple naive RNN structure. It is feasible to replace $h_t$ with LSTM architecture [@xingjian2015convolutional], but the computing complexity is too high to apply on visual tasks, so we do not practice this in this paper.
The synthesizer $\mathcal{F}$ adopts a parameter-free structure: $$\begin{aligned}
& \mathbf{u}^l_t={\rm Relu}(\mathbf{s}^l_t) \circ {\rm Sigmoid}(\mathbf{c}^l_t) \label{eq:F_ts}
\end{aligned}$$
where $\circ$ denotes element-wise multiplication, ${\rm Relu}(x)=max(0,x)$ is the rectified linear unit and ${\rm Sigmoid}(x)= 1/(1 + e ^ { - x })$ is the sigmoid function. In this way, the results of $h_t$ are normalized to $(0,1)$, so $h_t$ is treated as a control gate of $h_s$ in the viewpoint of $\mathcal{F}$.
As the network is recurrent, its outputs are a function of the complete sequence $(\mathbf{x}_1,...,\mathbf{x}_t)$. We can further encapsulate the operation of the network as $$\begin{aligned}
&(\mathbf{u}^n_1,..., \mathbf{u}^n_t)=\mathcal{T}([\mathbf{x}_1,...,\mathbf{x}_t];\Psi_s, \Psi_t)
\end{aligned}$$ where $\bm{\Psi}$ is the set of trainable network weights and $\mathbf{u}^n_t$ is the output of the $n$th layer at time stamp $t$. Finally, the output vector $\mathbf{y}_t$ is defined by the assembly of $(\mathbf{u}^n_1,..., \mathbf{u}^n_t)$: $$\begin{aligned}
\label{eq:yt}
& \mathbf{y}_t=[\mathbf{u}^n_1,..., \mathbf{u}^n_t]
\end{aligned}$$
For $\mathcal{T}^1$ and $\mathcal{T}^2$, the sub-goal networks, we adopt the same $h_s(\cdot)$ and $h_t(\cdot)$ with $\mathcal{T}$, while the synthesizer $\mathcal{F}$ is different. In $\mathcal{T}^1$, $\mathcal{F}$ and sub-goal $r_s$ (or $\mathbf{y}_t^{\mathcal{T}^1}$) are defined as: $$\begin{aligned}
\hat{\mathbf{u}}^l_t&={\rm Relu}(\mathbf{s}^l_t)\\
\mathbf{y}_t^{\mathcal{T}^1}&=[\hat{\mathbf{u}}^n_1,..., \hat{\mathbf{u}}^n_t]
\end{aligned}$$ While in $\mathcal{T}^2$, $\mathcal{F}$ and sub-goal $r_t$ (or $\mathbf{y}_t^{\mathcal{T}^2}$) are defined as: $$\begin{aligned}
\hat{\mathbf{u}}^l_t&={\rm Relu}(\mathbf{c}^l_t)\\
\mathbf{y}_t^{\mathcal{T}^2}&=[\hat{\mathbf{u}}^n_1,..., \hat{\mathbf{u}}^n_t]
\end{aligned}$$
Deep Nested Semi-coupled Structure Training
-------------------------------------------
\[sec:STSGD\]
As discussed in section \[sec:aware\], on one hand, $\mathcal{G}$ computes spatial and temporal information by separate modules. On the other hand, we adopt deep nested structure of stacked $\mathcal{G}$, inspired from the spatial and temporal coupling in human brains. But this structure leads to a difficult training process, because the deep nested structure actually merges the spatial and temporal information early in the shallow layers, which aggravates the pressure of spatial and temporal decomposition in the later layers as well as reduces the hierarchy of the decoupled features. Moreover, as the depth of layers and the length of sequences increase, both the number and the length of the back-propagation chains will grow significantly, which makes the training process much more challenging as well (see Fig. \[fig:STSGD\]).
![**Expectation numbers of back-propagation chains**. The horizontal axis is the length of the back-propagation chain and the vertical axis is the exception number of the chains. Note that with the growing of the model depth and sequence length, the number and length of the chains grow significantly. Our STSGD with large $p$ can efficiently reduce the number of long sequences.[]{data-label="fig:STSGD"}](seq_len.pdf){width="\linewidth"}
To address this challenge, the Spatial-Temporal Switch Gradient Descent (STSGD) is proposed to conduct a higher level of semi-coupling, in which the optimizer updates parameters based on either spatial or temporal information with a certain probability at each training step. As the training goes on, we reduce the degree of this separation and finally the network can learn all the information. This training strategy is also a practice of the semi-coupled mechanism: decoupling first then synthesizing.
#### Spatial-Temporal Switch Gradient Descent
STSGD is also a gradient based optimization method and the gradients are propagated by the BP algorithm [@rumelhart1988learning]. It works like a switch that turns off gradients on spatial and temporal modules with a certain probability. This scheme largely reduces the interference between $h_s(\cdot)$ and $h_t(\cdot)$ induced by the deep nested structure.
Given the definition Eq. \[eq:G\]. Its forward propagation is: $$\begin{aligned}
& \mathbf{y}_t = \mathcal{G}_n \circ \ldots \circ \mathcal{G}_1
\end{aligned}$$ where $\mathcal{G}_i = \mathcal{F}[h_t(\cdot), h_s(\cdot);\psi_i]$ is the $i^{th}$ layer of the network and $\bm{\psi}_i$ is the set of the trainable parameters in the $i^{th}$ layer. The loss between $\mathbf{y}_t$ and ground truth $\mathbf{z}_t$ is defined as: $$\begin{aligned}
E = \sum_{t=1}^T{E_t} = \sum_{t=1}^T{L(\mathbf{y}_t, \mathbf{z}_t)}
\end{aligned}$$ where $L$ is the loss function.
Then during back-propagation, according to the BPTT argorithm, the gradient of $\bm{\psi}_i$ can be presented as: $$\begin{aligned}
\frac{\partial E}{\partial \bm{\psi}_i} = & \sum_{t=1}^T\frac{\partial E}{\partial \mathcal{G}^t_{i+1}}\frac{\partial\mathcal{G}^t_{i+1}}{\partial \bm{\psi_i}}
\end{aligned}$$ and in conventional stochastic gradient descent methods, this exact gradient is adopted to update the parameters and continue back propagating. In our STSGD, we need to decouple the gradients based on the information carried by these two modules. To this end, we rewrite the gradient as: $$\begin{aligned}
\frac{\partial E}{\partial \bm{\psi}_i} & = \sum_{t=1}^T\frac{\partial E}{\partial \mathcal{G}^t_{i+1}}(\frac{\partial \mathcal{G}^t_{i+1}}{\partial \mathbf{c}^{i+1}_{t}}\frac{\partial \mathbf{c}^{i+1}_{t}}{\partial \bm{\psi}_i} + \frac{\partial \mathcal{G}^t_{i+1}}{\partial \mathbf{s}^{i+1}_{t}}\frac{\partial \mathbf{s}^{i+1}_{t}}{\partial \bm{\psi}_i})
\end{aligned}$$
In the equation, the first term in the bracket is the gradient from $h_t(\cdot)$ and the second term is from $h_s(\cdot)$. As the structures of $h_t(\cdot)$ and $h_s(\cdot)$ are designed for temporal and spatial information respectively, the gradients from them carry different concepts.
To decouple the gradients, we use a switch to prevent a certain part (spatial or temporal) from propagating its gradient in back-propagation, which can be defined as: $$\begin{aligned}
\hat{\frac{\partial E}{\partial \bm{\psi}_i}} =\sum_{t=1}^T(&\gamma_t(p_t)\frac{\partial E}{\partial \mathcal{G}^t_{i+1}}\frac{\partial \mathcal{G}^t_{i+1}}{\partial \mathbf{c}^{i+1}_{t}}\frac{\partial \mathbf{c}^{i+1}_{t}}{\partial \bm{\psi}_i}\\
+&
\gamma_t(p_s)\frac{\partial E}{\partial \mathcal{G}^t_{i+1}}\frac{\partial \mathcal{G}^t_{i+1}}{\partial \mathbf{s}^{i+1}_{t}}\frac{\partial \mathbf{s}^{i+1}_{t}}{\partial \bm{\psi}_i}) \label{eq: split}
\end{aligned}$$ where $\gamma$ is a probability function defined as: $$\gamma(p) =\left\{
\begin{aligned}
0, &{\rm~with~the~probability~of~}p\\
1, &{\rm~with~the~probability~of~}(1-p)
\end{aligned}
\right.$$
#### Discussion
This scheme partly decouples the spatial and temporal learning process by initializing $p_s$ and $p_t$ to a relative high value ($p_s=p_t=0.5$) and as the training goes on, $p$ decreases to 0 to synthesize the spatial and temporal training processes. From a macro perspective, it cuts off some paths in the back propagation with a certain probability, which reduces the number of back-propagation chains significantly, to make the training process more tractable (see Fig. \[fig:STSGD\]). According to the Assumption 4.3 in [@bottou2018optimization], if we set $p_s = p_t$, we get $E(\hat{\frac{\partial E}{\partial \bm{\psi}_i}})={\frac{\partial E}{\partial \bm{\psi}_i}}$ and this will lead to the similar convergence properties with the conventional stochastic gradient descent method.
#### Advanced STSGD
Note that, in STSGD, the same value of $p$ for $h_s(\cdot)$ and $h_t(\cdot)$ is the restriction for convergence and is a sufficient condition. But, we hope the network can learn more spatial information at the beginning, since temporal information can not be captured given a very unreliable spatial representation. After getting a relatively mature spatial representation, we hope the STSGD can shift its learning focus between spatial and temporal features. To this end, we modify the Eq. \[eq: split\] with a dynamic ratio $q\in [0,1]$ to: $$\begin{aligned}
\hat{\frac{\partial E}{\partial \bm{\psi}_i}} =\sum_{t=1}^T(&\gamma_t(q)\frac{\partial E}{\partial \mathcal{G}^t_{i+1}}\frac{\partial \mathcal{G}^t_{i+1}}{\partial \mathbf{c}^{i+1}_{t}}\frac{\partial \mathbf{c}^{i+1}_{t}}{\partial \bm{\psi}_i}\\
+&
\gamma_t(1-q)\frac{\partial E}{\partial \mathcal{G}^t_{i+1}}\frac{\partial \mathcal{G}^t_{i+1}}{\partial \mathbf{s}^{i+1}_{t}}\frac{\partial \mathbf{s}^{i+1}_{t}}{\partial \bm{\psi}_i}) \label{eq:split2}
\end{aligned}$$
Although there is no theory to guarantee the convergence of the Advanced STSGD (ASTSGD), the experiment results show it works well. Moreover, to automatically control the process of decreasing $q$, we train a small network with 3 fully connection layers which takes $r_s$, $r_t$, and $g$ as input to optimize $q$. To simplify the problem, we provide an empirical formula as another option: $$\begin{aligned}
q =q_0 + (1-q_0)\frac{\max(0, L_s- {\rm thresh})}{{\rm InitL_g} - {\rm thresh}} * (\alpha(\frac{L_g}{L_s} -1)+1)
\end{aligned}$$ where $L_s$ and $L_g$ are the loss values of $r_s$ and $g$. $q_0$ is usually set as 0.5. In this equation, we monitor the decreasing process of $L_s$ to update $q$. $\rm thresh$ is a hyper-parameter that acts as a threshold for $L_s$, considering that $L_s$ is difficult to decrease to 0 and we want $q$ get the minimum value when $L_s$ decreasing to $\rm thresh$. $\rm InitL_g$ is the initial training loss value of $g$, for example, the initial loss of an $n$-class classification problem, the initial cross entropy loss, is $ln(n)$. $\alpha$ is a hyper-parameter and the multiplier $(\alpha(L_g/L_s-1)+1)$ is designed to balance the integral and spatial information. If the task is more depended on the spatial information, we can set $\alpha$ smaller, in this way the function will have a relative big value to better learn the spatial features.
Dealing long sequences with LTSC {#sec:LTSC}
--------------------------------
Making use of the different properties between time and space, $\mathcal{T}[\mathbf{x};\Psi_s, \Psi_t]$ decouples the temporal and spatial information. As the length of the sequence grows, the enormous increase in information which leads to huge computing requirements also brings up the demand of information decoupling. Briefly, *long range temporal semi-coupling* (LTSC) decouples the original long sequence $\mathbf{D}$ into short sequences $\{\mathbf{d}_i|i = 1,...,n\}$ along the temporal dimension with a partitioning principle: $$\begin{aligned}
& \bigcup_{i=1,...,n}\{\tilde{\mathbf{d}_i}\} =\tilde{\mathbf{D}} \\
& \tilde{\mathbf{d}}_{i-1} \cap \tilde{\mathbf{d}}_i \neq \varnothing \label{eq: overlap}
\end{aligned}$$ where $\tilde{\mathbf{d}}_i$ and $\tilde{\mathbf{D}}$ are the set of $\mathbf{x}_i$ contained in sequence $\mathbf{d}_i$ and $\mathbf{D}$, and ${\mathbf{d}_i}$ is sorted by index of its first element. Eq. \[eq: overlap\] requires overlaps between the adjacent sub-sequences which are the hinge to transmit the information, for it makes sure that there is no such a point in $\mathbf{D}$ that information cannot feed forward and backward across it. For example, when splitting $\mathbf{D} = (\mathbf{x}_1, \mathbf{x}_2,\mathbf{x}_3, \mathbf{x}_4)$ into $\mathbf{d}_1=(\mathbf{x}_1, \mathbf{x}_2)$ and $\mathbf{d}_2=(\mathbf{x}_3, \mathbf{x}_4)$ in which there is no overlap, information will never transfer between $\mathbf{x}_2$ and $\mathbf{x}_3$, but if splitting $\mathbf{D}$ into $\mathbf{d}_1 = (\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3)$ and $\mathbf{d}_2=(\mathbf{x}_2, \mathbf{x}_3, \mathbf{x}_4)$, there is no such issue.
As a hyper-parameter, a high overlap $\eta$ will lead to high computing complexity and a low $\eta$ means worse ability to transmit information. In this paper, we chose 25% for $\eta$.
LTSC can be adopted in any sequence task, while in this paper, LTSC is nested with SCS network. LTSC further utilizes the overlaps among $\mathbf{d}_i$ to enhance the flow of information and we adopt an error function to shorten the distance between output of the adjacent $\mathbf{d}_i$: $$\begin{aligned}
& L_{overlap}=\sum_{i=2,...,n}{\xi(h_m({\mathbf{d}}_{i-1}), h_m({\mathbf{d}}_i))}
\end{aligned}$$ where $\xi(a, b)$ is defined as the overlap MSE function which calculates the MSE value of the overlap parts of $a$ and $b$. This arrangement makes the output of $h_m$ with adjacent input $\mathbf{d}_i$ be close in the overlap part.
Note that there are other straightforward methods to decouple the long-range temporal information like TBPTT algorithm [@williams1990efficient] or simply sampling from the original $\mathbf{D}$ [@carreira2017quo; @donahue2015long; @gu2017ava; @hou2017tube], but these do not make sure that the semantic information can transmit through the whole sequence or just discard some percentage of information.
A simple example demonstrates the smooth flow of semantic information in LTSC. The input visual sequence consists of an image of star and several of rhombus, and the star can appear at any temporal position. Our model needs to learn how far the current rhombus image is from the appeared star. With LTSC, the model can correctly output the results even if the star image appears 50 frames ago when the decoupled sequence lengths are smaller than 10. This can serve as a preliminary verification on LTSC.
{width="\linewidth"}
{width="\linewidth"}
Comparison with Deep RNN and spatial-temporal attention model
-------------------------------------------------------------
The Deep RNN framework [@pang2018deep] is the predecessor to the SCS described in this work, yet they have significant differences. Firstly, in the Deep RNN framework, the splitting of two flows is designed to make the deep recurrent structure easier to train by adding spatial shortcuts over temporal flows. While in SCS, the semi-coupling mechanism aims at endowing the model with the awareness of spatial and temporal concepts. Moreover, $h_s(\cdot)$ and $h_t(\cdot)$ have the equal status to explicitly learn the two concepts. Secondly, the Deep RNN framework has no mechanism to ensure the two flows focusing on the two kinds of information. This is not an issue for the SCS which adopts two extra independent sub-goals and two stand-alone modules tailored for spatial and temporal features. Thirdly, in the training process, Deep RNN has no way to control the training degree of the two flows, thus, no way of re-focusing on the spatial or temporal information. This problem is addressed in SCS by the STSGD mechanism.
The Spatial-Temporal Attention model (STAM) [@song2017end] also introduces spatial and temporal concepts. There are several differences between STAM and SCS. Firstly, SCS aims at extracting the temporal and spatial features (concepts) separately from the input, while STAM is designed to output the spatial and temporal attention from input features which integrate spatial and temporal information. These attentions defined on skeleton key-points rely on human skeleton assumption very much and are not general features. Secondly, STAM is designed for small-scale data format (skeleton coordinates, 20D only), thus, it is not suitable for the large scale problems which are the targets of SCS. Thirdly, STAM does not have awareness of general spatial and temporal concepts. The heads of spatial and temporal attention modules are designed for the specific tasks: spatial one for skeleton keypoints and temporal one for video frames, thus, the spatial and temporal concepts are actually human-defined, not learnt by the model itself.
Action recognition task descriptions
------------------------------------
The experiments of action recognition are conducted on UCF-101, HMDB-51 and Kinetics-400 datasets, which comprise sets of 101, 51, 400 action categories respectively. For each dataset, we follow the official training and testing splits. For each video, the frames are rescaled with their shorter side into 368 and a 224 $\times$ 224 crop is randomly sampled from the rescaled frames or their horizontal flips. Colour augmentation is used, where all the random augmentation parameters are shared among the frames of each video.
For this task, we adopt two kinds of Semi-Coupled Structures: backbone-supported one and stand-alone structure without backbone. For the backbone-supported structure, there is a CNN backbone pre-trained on ImageNet [@krizhevsky2012imagenet] (we choose VGG and InceptionV1 as examples) before the 15-layer SCS network. While for the stand-alone version, the model only consists of a 17-layer SCS network. Shortcuts between layers like ResNet [@he2016deep] are adopted in SCS networks to simplify the training process. The detailed structures are summarized in Tab. \[tab:actionstructure\].
--------------------------------------------------
layer blocks output size
------------------------------------ -------------
$7\times7, 64, {\rm stride}2$ 112
\[1.5ex\] $ \left[ \begin{aligned} 56
3\times3, 64 \\
3\times3, 64
\end{aligned}
\right] \times 2$
\[4ex\] $ \left[ \begin{aligned} 28
3\times3, 128 \\
3\times3, 128
\end{aligned}
\right] \times 2$
\[4ex\] $ \left[ \begin{aligned} 14
3\times3, 256 \\
3\times3, 256
\end{aligned}
\right] \times 2$
\[4ex\] $ \left[ \begin{aligned} 7
3\times3, 512 \\
3\times3, 512
\end{aligned}
\right] \times 2$
--------------------------------------------------
: Detailed stand-alone SCS structures for action recognition task. There are residual lines between every layer blocks.
\[tab:actionstructure\]
The main goal $g$ of the network is to minimize the cross-entropy of the softmax outputs with respect to the action categories; the final output is the average of the outputs of every time-stamp frame. The spatial goal $r_s$ is the same as the main goal and the temporal goal $r_t$ is to estimate the optical flow between the current and last input frames. For each step, the network processes a new video frame and the probability distribution over action categories is predicted based on the current processed frames.
Adopting LTSC enables our network to process much longer sequences than previous works on action recognition in which sampling methods are used to shorten the video length. This places greater stress on the long-range memory capacity of the model but preserves more temporal information in the original video. In addition, due to the deep structure of SCS network, we adopt ASTSGD.
Tab. \[tab:actionResult\] lists the complete results and hyper-parameters of the experiments on action recognition for SCS, LSTM and pure CNN model. We can see that our SCS has much better performances than LSTM, ConvLSTM, and CBM [@pang2018deep] models. Compared with CBM, the new SCS decouples spatial and temporal information and adjusts its focus (on spatial or temporal information) strategically during the learning process. Detailed analysis is shown in “Ablation Study".
Architecture Kinetics UCF-101 HMDB-51
-------------------------------------------- ---------- --------- ---------
LSTM with BB (VGG) [@donahue2015long] 53.9 86.8 49.7
3D-Fused [@feichtenhofer2016convolutional] 62.3 91.5 66.5
Stand-alone CBM [@pang2018deep] 60.2 91.9 61.7
Stand-alone SCS 61.7 92.6 65.0
15-layer ConvLSTM - 68.9 34.2
BB (VGG) supported CBM [@pang2018deep] - 79.8 40.2
BB (VGG) supported SCS - 82.1 42.5
BB (Inception) supported SCS - 87.9 52.1
: Action recognition accuracy on Kinetics, and end-to-end fine-tuning on UCF-101 and HMDB-51. Note that our SCS model applies 17 layers. “BB" denotes backbone.
\[tab:actionResult\]
#### Ablation Study
Since our SCS is a universal backbone, we conduct the ablation study on this low-level feature-driven task to show the function of each component. The results are shown in Tab. \[tab:ablsty\]. We first test the design of the spatial-temporal sub-task paradigm. From the view of the performances, it leads to 1.2% accuracy boost and from Fig. \[fig:ablation\_study\] **a**, we can see that this paradigm makes $h_s$ and $h_t$ more focus on their own functions: $h_s$ for spatial features and $h_t$ for temporal ones. Then we remove the ASTSGD from the training process, leading to 1.4% accuracy drop. In Fig. \[fig:ablation\_study\] **b**, we show the change tendency of $q$, which demonstrates that the model learns spatial information first then merges temporal features into it just as we expect. The sub-tasks and ASTSGD are the main improvements on CBM [@pang2018deep], which make $h_s$ and $h_t$ focus on their jobs, the training process controllable, and the model perform better. Without LTSC, the model can only access a short clip due to the limit of computational resource and the accuracy drops 2%.
Architecture Kinetics HMDB-51
----------------------------------------- ---------- ---------
Whole Stand-alone SCS 61.7 65.0
Stand-alone SCS w/o two sub-tasks 60.5 63.7
Stand-alone SCS w/o sub-task $T^2$ only 61.3 64.2
Stand-alone SCS w/o ASTSGD 60.8 63.2
Stand-alone SCS w/o LTSC 59.7 62.9
: Ablation study results (accuracy) on action recognition task with Kinetics and UCF-101 dataset. “w/o" denotes “without".
\[tab:ablsty\]
Outline annotation task descriptions
------------------------------------
We adopt CityScapes dataset [@cordts2016cityscapes] to conduct the outline annotation task experiments. CityScapes dataset consists of the street view images and their segmentation labels. We crop out the backgroud of the images and only preserve 8 kinds of the foregrounds: Bicycle, Bus, Person, Train, Truck, Motorcycle, Car and Rider. After cropping, there are 51k training images and 10k test images. The preserved images are resized to 224 $\times$ 224.
Similar with Polygon-RNN [@castrejon2017annotating; @acuna2018efficient], a VGG model is adopted first to extract the spatial features of the original images. Then a deep SCS network with 15 layers takes the image features and the outline point positions of time-stamp $t-1$ as input for each time-stamp $t$ and generates the outline point positions sequentially. In short, we replace the RNN part in the original Polygon-RNN model with our deep SCS network and adjust the optimizing method with our LTSC and STSGD schemes.
This task is also treated as a classification task. Each position of the image is a class and the loss function is the cross-entropy of the softmax outputs with respect to the image positions (28 $\times$ 28 $+$ 1, 784 positions in total and a terminator). The spatial goal $r_s$ and temporal goal $r_t$ are the same as the main goal: predicting the positions of the outline’s key points. Though we do not adopt different targets for $\mathcal{T}^1$ and $\mathcal{T}^2$, the independent optimization processes and asymmetrical structures allow them to focus on different information. The outline position sequence makes up a polygon area and we adopt the IoU between the predicted and ground-truth polygon area as the evaluation metric.
For each foreground, the model predicts 60 outline points at most. With a 15-layer SCS network, this sequence length requires huge computing resources, so we adopt the LTSC mechanism to split the sequence into 6 short clips and utilize ASTSGD to decouple the temporal and spatial training process of the deep structure.
Detailed results and hyper-parameters of the experiments on outline annotation for SCS-Polygon-RNN, Polygon-RNN and Polygon-RNN++ are shown in Tab. \[tab:polygonResult\]. Compared with traditional LSTM or GRU module, our model can be stacked deeply and achieve better performances with less parameters. Our SCS does not achieve the state-of-the-art performance because Polygon-RNN++ adopts many advanced tricks to improve the performance (including reinforcement learning, graph neural network, and attention module). These tricks are not the focus of this paper. Compared with CBM [@pang2018deep], the new SCS with sub-tasks and ASTSGD achieves better performances.
IoU
--------------------------- ----------- ------------------ ----------
61.4
62.2
67.2
70.2
**71.4**
\# layers \# params of RNN
Polyg-LSTM 2 0.47M 61.4
Polyg-LSTM 5 2.94M 63.0
Polyg-LSTM 10 7.07M 59.3
Polyg-LSTM 15 15.71M 46.7
Polyg-GRU 5 2.20M 63.8
Polyg-GRU 15 11.78M 64.7
Polyg-CBM [@pang2018deep] 5 1.13M 63.1
Polyg-CBM [@pang2018deep] 15 5.85M 70.4
Polyg-SCS 2 0.20M 62.9
Polyg-SCS 5 1.13M 65.8
Polyg-SCS 10 2.68M 68.0
Polyg-SCS 15 5.85M **71.0**
: Performance (IoU in %) on Cityscapes validation set (used as test set in [@castrejon2017annotating]). Note that “Polyg-LSTM" denotes the original Polygon-RNN structure with ConvLSTM cell, “Poly-GRU" for Polygon-RNN with GRU cell, and “Polyg-SCS" for Polygon-RNN with our Semi-Coupled Structure.
\[tab:polygonResult\]
Auto-driving task description
-----------------------------
Auto driving is a complex task. Completely solving it requires to conduct scene sensing, route planning, security assurance and so on. Here we simplify the task into a sequential vision task: given a short video from driver’s perspective and outputting the driving direction in the form of steering wheel angles. The experiments are conducted on comma.ai [@santana2016learning] and LiVi-Set [@chen2018lidar] datasets. These sets record the real driving behaviours of human drivers and there are various road conditions including town streets, highways and mountain roads. To make the model better focus on the road, we crop out the sky and other irrelevant information from the original images. And the final input images are resized to 192 $\times$ 64.
We compare our SCS network with conventional LSTM model and CNN model. The SCS structure is the same as the stand-alone model used in action recognition and the LSTM model adopts a CNN backbone (VGG) like LRCNs [@donahue2015long]. These two models both take a short driving video as the input and extract the temporal-spatial features. While for CNN model, we adopt the ResNet [@he2016deep] structure, and it only takes the current driving image as input and utilizes spatial features to commit predicting.
This is a regression problem and the main goal $g$ of the network is set to minimize the MSE of the predicted steering angles with the ground truth. The same with the action recognition task, the spatial goal $r_s$ is the same as the main goal and the temporal goal $r_t$ is to estimate the optical flow. We adopt sigmoid function to normalize the angles because the angles before normalization are more likely distributed around 0 and this non-linear function can, to some extent, make the distribution more uniform. Accuracy is used as the metric which is defined as: $$\begin{aligned}
Acc=\frac{\sum_i^I\lfloor{\min(\frac{\lambda}{|{\rm pred}_i-{\rm label}_i| + \epsilon }, 1)}\rfloor}{I}
\end{aligned}$$ where $I$ is the number of the samples, $\lambda$ is a threshold, and $\epsilon$ is a small value to prevent the denominator from being zero. ${\rm pred}_i$ and ${\rm label}_i$ are the predicted angle value and label angle value of sample $i$. In short, if the difference between predicted angle and label angle is less than the threshold, we treat it as an accurate prediction.
To predict the current driving direction, the models need to review a short history driving video (except the CNN model). We adopt our LTSC scheme when reviewing relative long history and we adopt STSGD for the deep SCS networks. We find that, adopting LTSC to access more temporal information makes the model achieve better performances without increasing memory resources. And STSGD relatively improves the performances by 9% on average. The detailed comparison results are shown in Tab. \[tab:drivingResult\].
[c|c|c|c|c|c|c|c]{}\
& &\
& $\lambda$=6 & $\lambda$=3 & MSE & $\lambda$=6 & $\lambda$=3 & MSE\
& p=0.0 & 31.8 & 16.9 & 0.046 & 28.8 & 15.9 & 0.049\
& p=0.3 & 34.1 & 17.4 & 0.045 & 30.5 & 16.1 & 0.048\
& p=0.5 & **35.1** & **19.4** & **0.044** & **33.4** & **17.6** & **0.046**\
& p=0.0 & 45.4 & 24.5 & 0.060 & 42.5 & 22.9 & 0.05\
& p=0.3 & 48.8 & **25.5** & 0.043 & 46.9 & 23.9 & 0.044\
& p=0.5 & **49.2** & 25.0 & **0.037** & **47.4** & **24.1** & **0.041**\
\
& &\
& $\lambda$=6 & $\lambda$=3 & MSE & $\lambda$=6 & $\lambda$=3 & MSE\
& 29.2 & 15.9 & 0.052 & 27.3 & 14.5 & 0.057\
& 43.1 & 23.8 & 0.056 & 42.3 & 21.0 & 0.058\
\
& &\
& $\lambda$=6 & $\lambda$=3 & MSE &\
& 24.5 & 13.0 & 0.057 &\
& 45.2 & 25.3 & 0.056 &\
\[tab:drivingResult\]
We adopt this experiment to show that our SCS can quantitatively indicate how important the temporal information is toward the final goal. In this analysis, we set $r_s$ and $r_t$ to the same with the main goal. By calculating the accuracy of $\mathcal{T}^1$ and $\mathcal{T}^2$, we can determine the importance of temporal and spatial information in different road conditions. The results shown in Tab. \[tab:tsimp\] are consistent with our intuition: On straight roads, $\mathcal{T}^1$ and $\mathcal{T}^2$ have similar performances, which reveals that the temporal information is not so important on this condition. While on crossroads, $\mathcal{T}^2$ performs much better than $\mathcal{T}^1$, which shows that we need more temporal information to give out steering angles. On these four conditions, the performance gaps of $\mathcal{T}^2$ and $\mathcal{T}^1$ can be ordered as: straight roads <cure roads <T-junctions <crossroads. This is reasonable and proves that $\mathcal{T}^2$ and $\mathcal{T}^1$ focus on temporal and spatial information separately. Moreover, it preliminarily shows that how can we utilize this method to reveal the importance of temporal information on a specific sample.
-- --------------- ------------- ------------- ------------- ------------- ------------- -----
$\lambda$=6 $\lambda$=3 $\lambda$=6 $\lambda$=3 $\lambda$=6 $\lambda$=3
Crossroads 18.3 10.2 29.1 16.3 10.8 6.1
T-junction 23.4 12.6 32.2 17.0 8.8 4.4
Curve road 32.9 17.1 37.6 20.1 4.7 3.0
Straight road 39.1 21.0 41.7 21.4 2.6 0.4
-- --------------- ------------- ------------- ------------- ------------- ------------- -----
: Accuracy of $\mathcal{T}^1$ and $\mathcal{T}^2$ on LiVi. Comparing their performances, we can get the importance of temporal information on different road conditions.
\[tab:tsimp\]
Precipitation forecasting experiments
-------------------------------------
MSE CSI FAR POD COR
-- --------- ------------- ------------ ------------ ------------ ------------
0.01156 0.5349 0.1733 0.5986 0.6851
p=0.0 0.01030 0.5624 0.1720 **0.6372** 0.7062
p=0.3 0.01033 0.5635 0.1702 0.6371 **0.7072**
p=0.5 **0.01022** **0.5636** **0.1682** 0.6368 **0.7072**
: Performance on the REEC-2018 validation set. Note that “p" denotes the initial probability to stop the back-propagation in STSGD.
\[tab:precipitationResult\]
The composite reflectance (CR) image received by the weather radar can reflect the precipitation situation in the specific area. By predicting the morphological changes of CR in the future we can forecast the precipitation. In this task, the models take a short period of the CR images as the input and generate the future CR images. The experiments are conducted on our REEC-2018 dataset which contains a set of CR images of Eastern China in 2018 and the CR image is recorded every 6 minutes. For better prediction, we select the top 100 rainy days from the dataset and crop a 224 $\times$ 244 pixel region as our input images. For preprocessing, we normalize the intensity value $Z$ of each pixel to $Z'$ by setting $Z'=\frac{Z-\min(\{Z_i\})}{\max(\{Z_i\}) - \min(\{Z_i\})}$, where $\{Z_i\}$ is the set of intensity values of all the pixels in the input image.
In this task, we compare our SCS network with the ConvLSTM network. Both of them consist of an encoder and a decoder which have the same structure. For our model, the encoder and the decoder are 15-layer SCS networks while there are multi-stacked ConvLSTMs in the ConvLSTM version. Encoders take one frame in the CR sequence as input for every time-stamp and then generate the intermediate representation of the observed sequence. Decoders take the intermediate representation as well as the last CR image as input and generate the CR image prediction and new intermediate representation as shown in Fig. \[fig:experiments\] d.
------------------- ------------------ ------------- -------------- -------------- ------------------------- -------- ---------- --------
BB supported Stand-alone LSTM SCS CNN+LSTM SCS ConvLSTM SCS
Batch size 16 40 8 4 128 128 8 4
Learning rate 1e-4 1e-4 1e-4 2e-4 2e-4 1e-4 1e-4 1e-4
Backbone {VGG, Inception} - VGG VGG ResNet-18 [@he2016deep] - - -
Num. layers BB layers + 15 17 {2,5 10, 15} {2,5 10, 15} 18 + 1 15 15 15
Training method STSGD STSGD - STSGD - ASTSGD - ASTSGD
LTSC setting $10\times 7$ $5\times6$ - $10\times 4$ - - - -
Feature dimension 512 512 256 256 512 512 64 128
$\lambda$ - - - - {3, 6} {3,6} - -
------------------- ------------------ ------------- -------------- -------------- ------------------------- -------- ---------- --------
\[tab:hyperparameter\]
This is a regression problem and every pixel of CR image represents the reflectance intensity of a specific geographic position. The networks are trained under the MSE loss function (the main goal $g$ is the MSE loss). The spatial goal $r_s$ is the same as the main goal. The temporal goal $r_t$ is to estimate the optical flow and pixel-wise difference between frames since every pixel has its own independent meaning: the reflectance intensity of that location. The optical flow guides $\mathcal{T}^2$ to learn the variation of wind direction while the pixel-wise difference is designed for the local precipitation changes. We evaluate the models using several metrics following [@xingjian2015convolutional], namely, mean squared error (MSE), critical success index (CSI), false alarm rate (FAR), probability of detection (POD) and correlation. Since every pixel has stand-alone meaning, we evaluate the performance at pixel level. We convert the prediction and the label to a 0/1 matrix using a threshold of 0.5 and define “hit" (prediction=label=1), “miss" (prediction=0, label=1), “falsealarm" (prediction=1, label = 0). Then the metrics are defined as: $$\begin{aligned}
& {\rm SCI}=\frac{\#{\rm hit}}{{\rm \#hit} + {\rm \#miss} + {\rm \#falsealarm}} \\
& {\rm FAR}=\frac{{\rm \#falsealarm}}{{\rm \#hit}+{\rm \#falsealarm}}\\
& {\rm POD}=\frac{{\rm \#hit}}{{\rm \#hit}+{\rm \#miss}}\\
& {\rm correlation}=\frac{\sum_{i,j}{{\rm CR\_P}_{i,j}\times{\rm CR\_L}_{i,j}}}{\sqrt{(\sum_{i,j}{{\rm CR\_P}_{i,j}^2})(\sum_{i,j}{{\rm CR\_L}_{i,j}^2})+\epsilon}}
\end{aligned}$$ where ${\rm CR\_P}$ is the predicted CR image and ${\rm CR\_P}_{i,j}$ is the 0/1 value of position (i,j) in the CR image. ${\rm CR\_L}$ is the ground-truth CR image, i.e. the label.
The models take 5 CR images as input and predict 5 future images. This is not a long sequence, so we do not adopt the LTSC scheme. STSGD is utilized in the deep SCS network. With a higher initial $p$, the model achieves better performance (see details in Tab. \[tab:precipitationResult\]), which indicates the important role of STSGD for SCS. The detailed comparison results are shown in Tab. \[tab:precipitationResult\].
Optimization
------------
The hyper-parameters are selected from grid searches and are listed in Tab. \[tab:hyperparameter\]. For all the experiments, the CNN layer is initialized with the “Xavier initialization" method followed by Batch Normalization layer [@ioffe2015batch]. All networks are trained using Adam optimizer [@kingma2014adam] and the backbones are pre-trained on ImageNet. For the huge memory consumption of the long sequential vision tasks, the batch size of each training step is relative small and we accumulate the parameters’ gradients of several training steps, then update the parameters together, which can speed up the training process to some extent. In the process of back-propagation-through-time (BPTT) [@werbos1990backpropagation], the gradients of RNN parameters was clipped to the range \[-5, 5\].
Data availability
=================
The data that support the plots within this paper are available from the corresponding author upon reasonable request.
Code availability
=================
A public version of the experiment codes will be made available with this paper, linked to from our website <http://www.mvig.sjtu.edu.cn> and [Github website](https://github.com/BoPang1996/Semi-Coupled-Structure-for-visual-sequental-tasks).
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Author contributions
====================
B.P. and C.L. conceived the idea. B.P., K.Z. and C.L. designed the experiments. B.P., K.Z., H.C., J.T. and M.Y. carried out programming, adjustment, and data analysis. B.P. and C.L. wrote the manuscript. B.P., J.T., M.Y. and all other authors contributed to the results analysis and commented on the manuscript.
Competing Interests
===================
The authors declare no competing interests.
|
---
abstract: 'We develop a Floquet scattering formalism for the description of quasistationary states of microwave photons in a one-dimensional waveguide interacting with a nonlinear cavity by means of a periodically modulated coupling. This model is inspired by the recent progress in engineering of tunable coupling schemes with superconducting qubits. We argue that our model can realize the quantum analogue of an optical chopper. We find strong periodic modulations of the transmission and reflection envelopes in the scattered few-photon pulses, including photon compression and blockade, as well as dramatic changes in statistics. Our theoretical analysis allows us to explain these non-trivial phenomena as arising from non-adiabatic memory effects.'
author:
- 'Mikhail Pletyukhov$^{1}$, Kim G. L. Pedersen$^{1}$, Vladimir Gritsev$^{2}$'
title: 'Control over few photon pulses by a time-periodic modulation of the photon-emitter coupling'
---
Introduction.
=============
Periodically driven quantum systems – or Floquet quantum systems as they are often called – may behave markedly different than their equilibrium counterparts, and it has been shown time and time again that this difference in behavior serve a whole range of potential applications.
In many-body quantum physics intensive research has recognized that the periodic driving of quantum many-body system could create new, synthetic phases of matter not accessible in equilibrium systems. This intuition, motivated by the classical example of the Kapitza pendulum [@Kapitza], has been explored and confirmed in several contexts. In particular, some proposals predict the formation of topological phases [@F6; @kitagawa_11; @KBRD] and artificial gauge systems [@Gold_Dali; @F7; @F16], as well as localized non-thermal states in isolated many-body systems [@ponte_15; @bukov_16; @foster_15; @DP; @bukov_15].
In quantum information protocols proposals for dynamical decoupling schemes [@VL; @Ban; @VKL; @Z; @KLV] and their refinements [@KL; @U; @KV] use periodic sequences of fast and strong symmetrizing pulses to reduce the parts of the system-bath interaction Hamiltonian which are sources of decoherence. Additionally, Floquet systems also naturally appear in digital quantum computation schemes [@F15].
In quantum transport various Floquet-driven quantum tunneling problems [@GH] are in the heart of physics described by an effective two-level systems, quantum wells and quantum open systems.
In this paper we seek to combine the possibilities offered by periodically driven quantum systems with the experimental flexibility available in quantum photonics, as e.g. realized in quantum optics or microwave quantum electro-dynamics. At this point it is important to stress that we do not simply talk about the time-dependence of e.g. a classical laser field where the time-dependence always trivially can be gauged away; instead we refer to quantum photonics systems where the time-dependence manifest itself directly in the steady-state observables, i.e. such that they themselves become time-dependent.
Specifically we are interested in the long time behavior of the observables which can be captured by a suitably formulated version of scattering theory.
In classical optics the most common periodically driven instruments are optical choppers and shutters [@Choppers], famous, perhaps, for their application in the first non-astronomical speed of light measurements by Hippolyte Fizeau in 1849 [@Fizeau], and used today for e.g. speed or rotation measurements, light exposure control, and off-frequency noise filtering. The prototypical chopper uses a rotating wheel with holes that periodically block the incident light beam, with the added feature of being able to control the waveform of the chopped light through the hole diameter to beam width ratio [@Choppers2].
One may imagine a quantum version of this instrument, with the light beam replaced by a weak coherent state of photons in a one-dimensional channel, and the rotating wheel by a single emitter that periodically couples to the channel. A key difference to the classical optical chopper is of course that a quantum chopper could potentially maintain a unitary evolution of the photons (when disregarding any losses).
As we later discuss, such a quantum chopper could be used for single photon pulse shaping [@kolchin], dynamical routing of single photons [@Hoi2011], and altering of the photon statistics [@Mandel; @Bouwmeester].
Due to the non-linear aspect of the emitter, a quantum chopper may also be able to modulate the statistics of the photons periodically in time. One may even speculate that the resulting periodically modulated signals may be used as input for other quantum optical instruments.
The experimental realization of a quantum chopper seems within the grasp of current nano-photonic technologies that allow for tunable and controllable manipulation of the coupling between different photonic elements. Various tunable coupling schemes have already been proposed and implemented with superconducting qubits, essentially based on the tunability of the Josephson inductance [@M-group1; @M-group2; @M-group3; @Delsing]. Dynamic control has also been demonstrated using an external coupling element between two directly coupled phase and flux qubits [@Tun1; @Tun2; @Tun3; @Tun4], between a phase qubit and a lumped element resonator [@Tun5], and between a charge qubit and a coplanar waveguide cavity [@Houck]. The latter scheme uses quantum interference to provide an intrinsic method to control the coupling. Recently a qubit architecture that incorporates fast tunable coupling and high coherence has been demonstrated, with dynamical tunability at nanosecond resolution [@M-group5].
We model the proposed quantum analogue of chopper by the following Hamiltonian, $$\begin{aligned}
H (t) &= H_0 + V(t) \nonumber \\ &= \int \mathrm{d} \omega \, \hbar \omega (a_\omega^{\dagger}
a^{{\phantom{\dagger}}}_\omega + \tilde{a}_\omega^{\dagger} \tilde{a}^{{\phantom{\dagger}}}_\omega) + \hbar
\omega_c b^{\dagger} b + \frac{U}{2} b^{\dagger 2} b^2 \nonumber \\
&+ \hbar g (t)
\int \mathrm{d} \omega \, (a_\omega^{\dagger} b + b^{\dagger} a^{{\phantom{\dagger}}}_\omega ).
\label{modelHam}\end{aligned}$$ Here $a_{\omega}= (a_{r \omega}+ a_{l \omega})/\sqrt{2}$ and $\tilde{a}_{\omega} = (a_{r \omega}- a_{l \omega})/\sqrt{2}$ describe the two waveguide fields expressed in terms of right- and left-moving modes, $g(t)$ is the coupling strength, and the emitter, described by the bosons $b, b^{\dag}$, has been generalized to a non-linear cavity characterized by a resonance frequency $\omega_c$, and a non-linearity $U$. An illustration of the model is also shown in Figure \[fig:figure0\].
![Quantum chopper model consisting of a one-dimensional transmission line supporting two counter-propagating channels, and a cavity with a non-linear spectrum. The two couple through a periodic coupling $g(t)$.[]{data-label="fig:figure0"}](fig0){width="0.8\columnwidth"}
In the next sections we first show how to solve the quasi-stationary dynamics of this system through a generalization of diagrammatic scattering theory to Floquet systems. Then we apply the Floquet scattering theory for describing open Floquet quantum systems explicitly in the few-photon limit. Various results for reflection, transmission and statistics are then summarized. The method is general and can directly be applied to more intricate quantum systems.
Floquet scattering formalism
============================
An extension of the scattering formalism for time-periodic Hamiltonians was originally proposed in Ref. [@PM] for the calculation of above-threshold-ionization spectra. Remarkably, it offered an effectively time-independent description of the quasistationary limit in terms of the Floquet states. Later, similar scattering approaches have been developed for single-particle scattering [@LR; @ER], many body scattering of non-interacting [@Moskal-b; @Moskal1] and interacting [@BC] particles in driven systems.
Let us briefly review the basic ideas of scattering theory. Suppose that at time $t_0 \to - \infty$ we inject $N$ photons into the transmission line, while the cavity is empty. In second quantization, this incoming state is given by ${|p \rangle \equiv | \{ \omega_j \} \rangle | 0 \rangle_c = \left( \prod_{j=1}^N
a_{\omega_j}^{\dagger} \right) | 0 \rangle | 0 \rangle_c}$, where the vacuum state $|0\rangle $ of the transmission line is defined by $a_{\omega} |
0 \rangle = \tilde{a}^{{\phantom{\dagger}}}_\omega |0 \rangle = 0$, and $|l \rangle_c$ is the photon number state of the cavity, $b^{\dagger} b | l
\rangle_c = l | l \rangle_c$. The energy of the incoming state equals $\varepsilon_p
= \sum_{j=1}^N \omega_j$, where we have set $\hbar =1$, as we will continue to do in the rest of this paper. After scattering, at time $t \to + \infty$, the cavity is empty again. Since the Hamiltonian conserves a number of excitations, a scattering state $S | p \rangle$ also contains $N$ photons. Here $S$ is a scattering operator which emerges from a time evolution operator in the long time limit. In case of the time-independent interaction $V$ the energy $\varepsilon_p$ of the input state is conserved in the following sense: matrix elements $S_{p'p} =
\langle p' | S | p \rangle$ appear to be proportional to delta functions $\delta (\varepsilon_{p'} - \varepsilon_{p})$, where $\varepsilon_{p'} = \sum_{j=1}^N \omega'_j$ is the energy of a state $| p' \rangle$. Moreover, $S_{p'p} = \delta_{p'p} - 2 \pi i \delta (\varepsilon_{p'} -
\varepsilon_{p}) T_{p'p} (\varepsilon_{p})$, where $T_{p'p} (E)$ is the energy-dependent $T$ operator containing all the information about scattering off the cavity. A systematic way of computing $T (E)$ has been developed in [@PG] for scatterers with an arbitrary level structure and transition matrix elements.
Following the ideas of [@PM] we now elaborate on the Floquet scattering formalism, particularly adapting it to problems of multi-particle scattering of (microwave) photons in one-dimensional waveguides interacting with (artificial) atoms. Our goal is to present a systematic way of computing the scattering operator $S$ for a time-periodic interaction, $V (t) = V (t+T)= \sum_m V^{(m)} e^{-i m \Omega t}$, with a fundamental frequency $\Omega = \frac{2 \pi}{T}$, in the Floquet-extended Hilbert space, thereby generalizing the approach of Ref. [@PG] for time-independent couplings.
We start from an equation for the evolution operator in the interaction picture $$\begin{aligned}
i\frac{d U_{\mathrm{int}} (t, t_0)}{d t} = V_{\mathrm{int}} (t) U_{\mathrm{int}} (t,t_0),
\label{diff_eq}\end{aligned}$$ where $V_{\mathrm{int}} (t) = e^{i H_0 t} V (t) e^{-i H_0 t}$. Taking the limit $t_0 \to -\infty$ we transform into the integral form $$\begin{aligned}
U_{\mathrm{int}} (t) = \hat{1} - i \int_{-\infty}^t d t' e^{\eta t'} V_{\mathrm{int}} (t') U_{\mathrm{int}} (t'),
\label{int_eq1}\end{aligned}$$ where an infinitesimal factor $\eta > 0$ is additionally introduced for convergence.
Next, we define matrix elements $U_{p' p} (t) = \langle p' | U_{\mathrm{int}} (t) | p \rangle$ in the eigenbasis $\{ | p \rangle \}$ of $H_0$, and express in the matrix form $$\begin{aligned}
U_{p' p} (t) = \delta_{p' p} &- i \int_{-\infty}^t d t ' \sum_q \sum_m e^{ i (\varepsilon_{p'} - \varepsilon_q - m \Omega - i \eta) t'}
\nonumber \\
& \times V^{(m)}_{p'q} U_{q p} (t'),
\label{int_eq2}\end{aligned}$$ Being interested in a solution of this equation at times $t>0$, satisfying the condition $\eta t \ll 1$, we look for it in the form $$\begin{aligned}
U_{p' p} (t) = \delta_{p'p} - \sum_{m'} \frac{e^{i (\varepsilon_{p'} - \varepsilon_p - m' \Omega) t}}{\varepsilon_{p'} - \varepsilon_p - m' \Omega - i \eta} \Theta_{p' p}^{(m')},
\label{ansatz}\end{aligned}$$ where $\Theta_{p' p}^{(m')}$ are constant matrices. Plugging into , we obtain the equation $$\begin{aligned}
\Theta_{p' p}^{(m')}= V_{p' p}^{(m')} - \sum_q \sum_{n} \frac{V^{(m'-n)}_{p'q} \Theta_{q p}^{(n)}}{\varepsilon_q - \varepsilon_p - n \Omega - i \eta} ,\end{aligned}$$ from which we can establish the matrices $\Theta^{(m')}$.
At large times $t$ we make in the standard replacement $\frac{e^{i \omega t}}{\omega - i \eta} \to 2 \pi i \delta (\omega)$, and thus obtain the scattering matrix $$S_{p' p} = \delta_{p'p} - 2 \pi i \sum_{m'} \delta (\varepsilon_{p'} - m' \Omega - \varepsilon_p) \Theta_{p' p}^{(m')}.
\label{Spp}$$ Finally, we introduce the matrix $T_{p'p}^{(m')} (E)$ depending on the energy parameter $E$ and obeying the equation $$\begin{aligned}
T_{p' p}^{(m')} (E)= V_{p' p}^{(m')} + \sum_q \sum_{n} \frac{V^{(m'-n)}_{p'q} T_{q p}^{(n)} (E)}{ E - (\varepsilon_q - n \Omega) + i \eta} .
\label{Teq1}\end{aligned}$$ Noticing that $T_{p'p}^{(m')} (E=\varepsilon_p)$ coincides with the matrix $\Theta_{p' p}^{(m')}$, we arrive at the expression $$S_{p' p} = \delta_{p'p} - 2 \pi i \sum_{m'} \delta (\varepsilon_{p'} -m' \Omega - \varepsilon_p) T_{p' p}^{(m')} (\varepsilon_p) ,
\label{Spp1}$$ which relates the $S$ matrix to the $T$ matrix in the time-periodic case.
As follows from , the energy $\varepsilon_p$ of an incoming state is conserved modulo an integer number of the drive frequency quanta, for each of which we need to find the corresponding $T$ matrix from the equation .
Let us consider a generalized version of $$\begin{aligned}
T_{p'p}^{m'm} (E) = V_{p'p}^{m'm}+ V_{p'q'}^{m'n'} \left[ \frac{1}{E - H'_0 + i \eta} \right]_{q'q}^{n'n} T_{qp}^{nm} (E),
\label{TVT}\end{aligned}$$ where $V_{p'p}^{m'm} \equiv V_{p'p}^{(m'-m)}$, and $H'_0 = H_0 - i {\partial}_\tau$ is the free Floquet Hamiltonian. The operator $i
{\partial}_{\tau}$ is defined by $i {\partial}_{\tau} |m \rangle = m \Omega |m \rangle$ in terms of the Floquet states $|m \rangle = e^{-i m \Omega \tau}$, such that $\langle m'
| m \rangle =\int_0^{T} \frac{d \tau}{T} e^{i (m'-m) \tau}= \delta_{m'm}$. Thus, Eq. is understood as a relation between operators which act in the Floquet-Hilbert space spanned by $\{|p \rangle \otimes |m \rangle \}$. For brevity we implicitly assume summations (integrations) over repeated discrete (continuous) indices.
Writing in the operator form $T(E) = V + V (E-H'_0 + i \eta)^{-1} T (E)$, we can easily invert this equation and get $T (E) =V + V (E-H' + i \eta)^{-1} V$, where $H' = H'_0 + V$ is the full Floquet Hamiltonian. In the matrix representation, this solution reads $$T_{p'p}^{m'm} (E) = V_{p'p}^{m'm} + V_{p' q'}^{m' n'}
\left[ \frac{1}{E - H' +i \eta} \right]_{q'q}^{n' n} V_{qp}^{nm}.
\label{TVV}$$ In turn, the solution of $T_{p'p}^{(m')} (E) = T_{p'p}^{m'0} (E) $ is obtained from in the special case $m=0$.
Let us make the following important observation: the equation for the $T$ matrix in the time-periodic case has almost the same form as its time-independent counterpart, the only difference consisting in additional summations over Floquet indices. Noticing that the Hamiltonian conserves a number of incoming photons after scattering, we can decompose $T = \sum_{N=1}^{\infty} T_N$, where $T_N$ is a normal ordered $N$-photon operator, and straightforwardly generalize the diagrammatic rules of Ref. [@PG]. Thus, in the time-periodic case we obtain $$\begin{aligned}
&T^{m' m}_N (E) = \sum_{\{m'_j\}, \{m_j\}} P_{0c} \, \vdots \, V^{m' m^{\phantom{'}}_1} \tilde{G}^{m^{\phantom{'}}_1 m'_1} (E) V^{m'_1 m^{\phantom{'}}_2} \ldots \nonumber \\
&\times V^{m'_{2N-2}, m^{\phantom{'}}_{2N-1}} \tilde{G}^{m^{\phantom{'}}_{2N-1},m'_{2N-1}} (E) V^{m'_{2N-1},m } \, \vdots \, P_{0c} ,
\label{TNormalmain}\end{aligned}$$ given by the alternating product of $2N$ interaction operators, $V$, and $2 N-1$ dressed Green’s functions, ${\tilde{G} (E) = (E-H'_0 - \Sigma)^{-1}}$, of the cavity. Here ${P_0 }$ is a projector onto the dark (i.e. nonrelaxing) state of the cavity. The Floquet components of the cavity’s self-energy $\Sigma^{mm'} \equiv \Sigma^{(m-m')} = - i \pi \sum_n \langle V^{(m-n)} V^{(n-m')} \rangle_{0}$ are given by an average in the vacuum state of a waveguide. (In particular, for the model we have $P_{0c} = |0 \rangle_c \,_c\langle 0 | $ and $\Sigma^{mm'}= - i \pi b^{\dagger} b \sum_n g^{(m-n)} g^{(n-m')} $). Finally, the symbol $\vdots (\ldots ) \vdots$ denotes a modified normal ordering, which ignores commutators between field operators contained in different $V$’s, but at the same time obliges to canonically commute a field operator contained in $V$ with $\tilde{G} (E)$ which contains $H_0$.
The expression is exact and sufficient to describe scattering an initial state with arbitrary number of photons. However, because of multiple summations over Floquet indices, it is not optimal for a theoretical analysis. In order to find a more convenient expression, we transform into the local time representation $$\begin{aligned}
T_{N\varepsilon} (\tau) &\equiv \sum_{m'} T_N^{(m')} (E) e^{-i m' \Omega \tau} \nonumber \\
&= \int_0^T \frac{d \tau_1}{T} \ldots \frac{d \tau_{2N}}{T} \delta_T (\tau -\tau_1) \nonumber \\
& \times P_{0c} \left( \vdots V (\tau_1) \tilde{G}_{\varepsilon} (\tau_1, \tau_2) V (\tau_2) \ldots \right. \nonumber \\
& \left. \ldots V (\tau_{2N-1}) \tilde{G}_{\varepsilon} (\tau_{2N-1},\tau_{2N}) V (\tau_{2N}) \vdots \right) P_{0c} ,
\label{Ttau}\end{aligned}$$ where we introduced the notations $\varepsilon=H_0-E = H_0 - \varepsilon_p$ and $$\begin{aligned}
\tilde{G}_{\varepsilon} (\tau , \tau') = \sum_{m,m'} e^{-i m \Omega \tau} \tilde{G}^{m m'} (E)
e^{i m' \Omega \tau'},
\label{Gtt}\end{aligned}$$ and used the Poisson resummation formula $$\begin{aligned}
& \sum_{m'} e^{-i m' \Omega (\tau-\tau_1)} \nonumber \\
=& T \sum_{n} \delta (\tau - \tau_1 - n T) \equiv \delta_T (\tau- \tau_1).\end{aligned}$$ Then, from and we deduce that the $N$-photon operator contribution to the nontrivial part of the scattering operator equals $$\begin{aligned}
(S-1)_N &= - i \int_{-\infty}^{\infty} d \tau e^{i (\varepsilon_{p'} - \varepsilon_p) \tau} T_{N \varepsilon} (\tau) \nonumber \\
&= - i \int_{-\infty}^{\infty} d \tau e^{i H_0 \tau} T_{N \varepsilon} (\tau) e^{- i H_0 \tau} ,\end{aligned}$$ and the scattering operator itself is given by $$\begin{aligned}
S = 1 + \sum_{N=1}^{\infty} (- i) \int_{-\infty}^{\infty} d \tau e^{i H_0 \tau} T_{N \varepsilon} (\tau) e^{- i H_0 \tau} .
\label{S1T}\end{aligned}$$
Now it is necessary to establish an explicit form of $\tilde{G}_{\varepsilon} (\tau , \tau')$ defined in . From the relations $$\begin{aligned}
& \sum_{m''} \left[ (m \Omega - \varepsilon) \delta_{m m''} - \Sigma^{m m''} \right] \tilde{G}^{m'' m'} (E) = \delta_{mm'}, \\
& \sum_{m''} \tilde{G}^{m m''} (E) \left[ (m' \Omega - \varepsilon) \delta_{m'' m'} - \Sigma^{m'' m'} \right] = \delta_{mm'},\end{aligned}$$ which are equivalent to the definition of $\tilde{G}^{mm'} (E)$, we obtain the differential equations $$\begin{aligned}
& (i {\partial}_{\tau} - \varepsilon) \tilde{G}_{\varepsilon} (\tau, \tau')- \Sigma (\tau) \tilde{G}_{\varepsilon} (\tau, \tau') = \delta_T (\tau - \tau'), \\
& (-i {\partial}_{\tau'} - \varepsilon) \tilde{G}_{\varepsilon} (\tau, \tau')- \tilde{G}_{\varepsilon} (\tau, \tau') \Sigma (\tau') = \delta_T (\tau - \tau'),\end{aligned}$$ where $\Sigma (\tau) = - i \pi \langle V^2 (\tau) \rangle_{0} = \sum_m \Sigma^{(m)} e^{-i m \Omega \tau}$. Equipping them with the periodic boundary conditions in both variables, we find a solution $$\begin{aligned}
\tilde{G}_{\varepsilon} (\tau, \tau') &= - i T \sum_n \Theta (\tau - \tau' - n T) e^{-i \bar{\varepsilon} (\tau - \tau' - n T)} \nonumber \\
& \times e^{-F_{\mathrm{osc}} (\tau) + F_{\mathrm{osc}} (\tau')},\end{aligned}$$ where $\bar{\varepsilon} = \varepsilon + \Sigma^{(0)}$ and $F_{\mathrm{osc}} (\tau) = - \sum_{m \neq 0} \frac{\Sigma^{(m)}}{m \Omega} e^{- i m \Omega \tau}$. Inserting it into and extending the finite integration ranges $0 < \tau_j <T$ to the infinite ones $-\infty < t_j < \infty$, we cast the scattering operator to the form $$\begin{aligned}
S &= 1 + \sum_{N=1}^{\infty} (-i)^{2N} \int d t_1 \ldots d t_{2N} \Theta (t_1> \ldots >
t_{2N}) \nonumber \\
&\times e^{i (H_0-E) t_1} P_{0c} \left( {\vphantom{\int^0}\smash[t]{\vdots}}\, V
(t_1) e^{-F (t_1)} e^{F (t_2)} V (t_2) e^{-F (t_2)} \ldots \right. \nonumber \\
& \times \left. V (t_{2N-1}) e^{- F (t_{2 N-1})} e^{F (t_{2 N})} V
(t_{2N}) \, {\vphantom{\int^0}\smash[t]{\vdots}}\right) P_{0c},
\label{SNmain}\end{aligned}$$ where $F (t) = i (H_0 + \Sigma^{(0)} - E) t +
F_{\mathrm{osc}} (t)$, and $E$ is the energy of an input state. In the following we identify $S$ with $\,_c \langle 0 | S | 0 \rangle_c$.
Note that a $N$-photon operator from the above sum gives only nonzero contribution, if it is applied to a $M$-photon initial state such that $N \leq M$. This means that for a $M$-photon initial state the sum can be truncated after the $M$th term.
To illustrate an application of we consider in the next section examples of a single- and two-photon scattering in the model .
Few photon scattering {#sec:few}
=====================
Let us consider the model and assume that an initial state is prepared in a form of a coherent rectangular pulse of the length $L$, which is initially located far left from the cavity and starts moving towards it in the right direction with a constant velocity $v$. In the interaction picture, this initial state is expressed by $$\begin{aligned}
| \Psi_i \rangle = e^{-|\alpha |^2/2} e^{\alpha \mathcal{A}_{r,\omega_0}^{\dagger}} | 0 \rangle ,
\label{Psi}\end{aligned}$$ where $\mathcal{A}_{r, \omega_0} = \int d \omega \phi (\omega) a_{r \omega} $ is a normalized wavepacket operator centered around the mode $\omega_0$ and broadened over the width $\sim \frac{2 \pi v}{L}$. Formally it is defined by the function $$\begin{aligned}
\phi (\omega ) = \sqrt{\frac{2 v}{\pi L}} \frac{\sin \frac{L}{2 v} (\omega - \omega_0)}{\omega - \omega_0 } , \end{aligned}$$ which approaches $\sqrt{\frac{2 \pi v}{L}} \delta (\omega - \omega_0)$ for long pulses.
For weak coherence $|\alpha | \ll 1$ we approximate the state by $$\begin{aligned}
| \Psi_i \rangle \approx e^{-|\alpha |^2/2} \left[ 1+ \alpha \mathcal{A}_{r,\omega_0}^{\dagger} + \alpha^2 \frac{(\mathcal{A}_{r,\omega_0}^{\dagger})^2}{2} \right] | 0 \rangle .
\label{Psiapp}\end{aligned}$$ Both single- and two-photon states contributing to have a well-defined energy in the long pulse limit $L \to \infty$, and therefore we can apply the scattering operator to each of them, thus obtaining a final state $| \Psi_f \rangle = S | \Psi_i \rangle$ in the two-photon approximation.
We are interested in computing – to the leading order in $\alpha$ – of average transmitted and reflected fields, and their statistical properties quantified by the second order coherence function $g^{(2)}$. In particular, defining the field operators in coordinate representation $$\begin{aligned}
a_{\sigma} (x) = \frac{1}{\sqrt{2 \pi v}} \int d \omega a_{\sigma \omega} e^{i \omega x/v}, \quad \sigma =r,l,
\label{aspace}\end{aligned}$$ we wish to find $\langle \Psi_f | a_{\sigma} (x-v t) | \Psi_f \rangle$ and $$\begin{aligned}
g_{\sigma \sigma'}^{(2)} (t, \tau_d) =& \frac{G_{\sigma \sigma'}^{(2)} (t, \tau_d)}{g_{\sigma}^{(1)} (t) g_{\sigma'}^{(1)} (t+ \tau_d) } ,
\label{g2def}\end{aligned}$$ where $$\begin{aligned}
G_{\sigma \sigma'}^{(2)} (t, \tau_d) =& \langle \Psi_f | a_{\sigma}^{\dagger} (x-v t) a^{\dagger}_{\sigma'} (x-v t- v \tau_d) \nonumber \\
& \times a_{\sigma'} (x-v t - v \tau_d)a_{\sigma} (x-v t)| \Psi_f \rangle, \label{G2def} \\
g_{\sigma}^{(1)} (t) =& \langle \Psi_f | a_{\sigma}^{\dagger} (x-v t) a_{\sigma} (x-v t)| \Psi_f \rangle ,
\label{g1def}\end{aligned}$$ and $\tau_d$ is a delay time.
Because of an explicit time dependence in the Hamiltonian , there is no time translational invariance in the long time limit (a corresponding system’s state is therefore said to be [*quasistationary*]{}), and the above defined functions also depend on the evolution time $t$ (though in a periodic way, as we will see later).
Note that the definition implies that the $x$-axis for left-moving photons ($\sigma =l$) points in the left direction.
Since in the Hamiltonian only even states (${a_{\omega} = \frac{a_{r \omega} + a_{l \omega}}{\sqrt{2}}}$) are coupled to the cavity, and odd states (${\tilde{a}_{\omega} = \frac{a_{r \omega} - a_{l \omega}}{\sqrt{2}}}$) are decoupled from it, it appears convenient to express the scattering operator in the basis of even states, also representing the initial state in terms of even and odd states. A task of finding $| \Psi_f \rangle$ essentially reduces to evaluation of $S \mathcal{A}_{\omega_0}^{\dagger} | 0 \rangle$ and $S \frac12 (\mathcal{A}_{\omega_0}^{\dagger})^2 | 0 \rangle$, where $\mathcal{A}_{\omega_0}$ is an even counterpart of $\mathcal{A}_{r, \omega_0}$. We consider these cases of single- and two-photon scattering in the following subsections.
Single-photon scattering
------------------------
Let us first establish how the scattering operator acts on a single-photon plane wave even state $a^{\dagger}_{\omega} | 0 \rangle$ with energy $E=\omega$. Truncating the sum in at $N=1$, we obtain $$\begin{aligned}
& S a_{\omega}^{\dagger} | 0 \rangle = a_{\omega}^{\dagger} | 0 \rangle \nonumber \\
&- \int d t_1 d t_{2} \Theta (t_1> t_{2}) \int d \omega_1 \int d \omega_2 e^{i (\omega_1-\omega) t_1} \nonumber \\
&\times \,_c \langle 0 | \left( {\vphantom{\int^0}\smash[t]{\vdots}}\, g
(t_1) a^{\dagger}_{\omega_1} b e^{-F (t_1)} e^{F (t_2)} g (t_2) b^{\dagger} a_{\omega_2} \, {\vphantom{\int^0}\smash[t]{\vdots}}\right) | 0 \rangle_c \, a_{\omega}^{\dagger} |0 \rangle \nonumber \\
&= a_{\omega}^{\dagger} | 0 \rangle - \int d \omega_1 \int_{-\infty}^{\infty} d t_1 e^{i (\omega_1-\omega) t_1} g (t_1) e^{-f_{1\omega} (t_1)} \nonumber \\
& \qquad \times \int_{-\infty}^{t_1} d t_2 e^{f_{1\omega} (t_2)} g (t_2) a_{\omega_1}^{\dagger} |0 \rangle \nonumber \\
&\equiv \int d \omega_1 [\delta_{\omega_1 \omega} + s_{\omega_1 \omega}] a_{\omega_1}^{\dagger} |0 \rangle .
\label{S1main}\end{aligned}$$ The function $F (t)$ for the model acquires the form $$\begin{aligned}
F (t) = i (H_0 -i \Gamma^{(0)} b^{\dagger} b -E) t + f_{osc} (t) b^{\dagger} b\end{aligned}$$ where $\Gamma^{(0)} + \dot{f}_{osc} (t) = \pi g^2 (t) \equiv \Gamma (t)$, and $f_{osc} (t)$ is fixed by the condition that it does not have a zero frequency component. To single-photon scattering contributes only a single-excitation component $\langle 1 | F (t) | 1\rangle $, and its contribution is appropriately written in terms of the functions $
f_{1 \omega} (t) = i (\omega_c - i \Gamma^{(0)} - \omega) t + f_{osc} (t)$.
Folding with the wavepacket $\phi (\omega)$ and applying the field operator $a (x-v t)$ to the obtained single-photon scattering state, we find $$\begin{aligned}
& a (x-v t) S \mathcal{A}_{\omega_0}^{\dagger} | 0 \rangle = \frac{e^{-i \omega_0 t_x}}{\sqrt{L} } [1 +2 A (t_x) ] | 0 \rangle ,
\label{aSeven} \\
& A (t_x )= - \pi g (t_x) e^{-f_1 (t_x)} \int_{-\infty}^{t_x} d t' e^{f_1 (t')} g (t') ,
\label{defA}\end{aligned}$$ where $t$ is time elapsed since the beginning of interaction and $t_x = t -x/v$ is a time lag between the pulse front and the field at point $x$. In we have also introduced $$\begin{aligned}
f_1 (t) \equiv f_{1 \omega_0} (t) = - i (\delta + i \Gamma^{(0)}) t + f_{osc} (t),
\label{f1}\end{aligned}$$ with the detuning $\delta = \omega_0 - \omega_c$.
The function $A (t_x)$ is periodic in its argument, $A (t_x) = A (t_x + T)$, and therefore we can reduce the central time of pulse evolution $t_x$ (in other words, the observation time at point $x$) to a single period: $t_x \to \tau_c \in [-T/2,T/2]$.
Transforming to the basis of right and left modes, we obtain the transmitted field (labeled by $r$, the direction of the incident field) and the reflected field (labeled by $l$, the opposite direction) $$\begin{aligned}
& a_{r,l} (-v t_x) S \mathcal{A}_{r, \omega_0}^{\dagger} | 0 \rangle \nonumber \\
&= \frac{a (-v t_x)\pm \tilde{a} (-v t_x)}{\sqrt{2}} \frac{S \mathcal{A}_{\omega_0}^{\dagger} + \tilde{\mathcal{A}}_{\omega_0}^{\dagger}}{\sqrt{2}} | 0 \rangle \nonumber \\
&= \frac{e^{-i \omega_0 t_x}}{\sqrt{L} } \left[ \frac{1 \pm 1}{2} + A (t_x) \right] | 0 \rangle.\end{aligned}$$ The transmission $t (\tau_c) = 1+ A (\tau_c)$ and reflection $r (\tau_c) = A (\tau_c)$ amplitudes give envelope shapes of the corresponding fields, and they are not constant in time. Nevertheless, they obey the normalization condition $$\begin{aligned}
\frac{1}{T} \int_0^T d \tau_c \left( | t (\tau_c) |^2 + | r (\tau_c)|^2 \right) =1,
\label{normtr}\end{aligned}$$ corresponding to a conservation of the photon number (see Appendix \[NC\] for the proof). In the linear regime, one can relate the transmission and reflection amplitudes to the equal-time first order coherences by $$\begin{aligned}
g_r^{(1)} (\tau_c) = \frac{|\alpha|^2}{L} | t (\tau_c) |^2 , \quad g_l^{(1)} (\tau_c) = \frac{|\alpha|^2}{L} | r (\tau_c) |^2 .\end{aligned}$$
Periodic time dependence of an envelope of a scattered field is the main effect of a periodic time modulation of coupling seen in a single-photon scattering. In the following we study this dependence for different modulation protocols. To evaluate $A (\tau_c)$ for $ \tau_c \in [-T/2 , T/2]$ in practice, it is convenient to split the integral range $[-\infty , \tau_c]$ in into two ranges $[-\infty , -T/2]$ and $[ -T/2, \tau_c]$. The integral over the second range can be evaluated numerically, while the integral over the first range can be converted into a geometric series by using the periodicity of $g (t)$ and $f_{osc} (t)$ which results in $$\begin{aligned}
\int_{-\infty}^{-T/2} d t' e^{-i (\delta +i \Gamma^{(0)}) t'} e^{f_{osc} (t')} g(t') = \frac{C_0}{e^{-i (\delta +i \Gamma^{(0)})T} -1}.\end{aligned}$$ Here $
C_0 =\int_{-T/2}^{T/2} d t' e^{-i (\delta +i \Gamma^{(0)}) t'} e^{f_{osc} (t')} g(t') $ is also evaluated numerically.
Before choosing specific protocols $g (t)$, let us first analyze under which conditions one can expect an interesting time behavior of an envelope $A$.
The most trivial time dependence appears in case of slow driving, when $A (\tau_c)$ instantaneously follows $g (\tau_c)$. It is captured by applying the adiabatic approximation to , which is achieved by expanding the integrand close to the upper limit given by the time of observation $\tau_c$. Physically this means that a protocol’s history influences very little the present time value of $A$. We have $$\begin{aligned}
A (\tau_c )& = - \pi g (\tau_c) \int_{-\infty}^{0} d \tau e^{f_1 ( \tau_c + \tau) -f_1 (\tau_c)} g (\tau_c + \tau) \nonumber \\
& \approx - \pi g (\tau_c ) \int_{-\infty}^{0} d \tau e^{\dot{f}_1 ( \tau_c ) \tau } \nonumber \\
& \times [g (\tau_c) + \dot{g} (\tau_c) \tau + \frac12 g (t_x) \ddot{f}_1 ( \tau_c ) \tau^2].\end{aligned}$$ Noticing that $\dot{f}_1 (\tau_c) = - i (\delta + i \Gamma (\tau_c))$, we conclude $$\begin{aligned}
A (\tau_c ) \approx - \frac{i \Gamma (\tau_c)}{\delta + i \Gamma (\tau_c)} \left[ 1 - \frac{i \dot{g} (\tau_c) }{g (\tau_c)} \frac{\delta - i \Gamma (\tau_c)}{(\delta + i \Gamma (\tau_c))^2} \right].
\label{Ainstan}\end{aligned}$$ The leading term gives the instantaneous amplitude, and the second term represents the adiabatic correction. This approximation is valid as long as the adiabaticity condition $$\begin{aligned}
\bigg| \frac{ \dot{g} (t) }{g (t)} \bigg| \ll \sqrt{\delta^2 + \Gamma^2 (t)}
\label{adiab}\end{aligned}$$ is fulfilled. Interesting and unexpected behavior shows up when this condition is violated as we explore in further detail in section \[results\].
Two-photon scattering
---------------------
Applying the scattering operator to the two-photon state with energy $E= \omega+ \omega'$ we obtain $$\begin{aligned}
& S a_{\omega}^{\dagger} a_{\omega'}^{\dagger} | 0 \rangle = \frac12 a_{\omega}^{\dagger} a_{\omega'}^{\dagger} | 0 \rangle
+ \int d \omega_1 s_{\omega_1 \omega} a_{\omega_1}^{\dagger} a_{\omega'}^{\dagger} |0 \rangle \nonumber \\
& + \int d \omega_1 d \omega_2 d \omega_3 d \omega_4 \int d t_1 d t_2 d t_3 d t_4 \nonumber \\
&\quad \times \Theta (t_1> t_2 > t_3 > t_{4}) e^{i (H_0-E) t_1}\nonumber \\
&\quad \times \,_c \langle 0 | \left( {\vphantom{\int^0}\smash[t]{\vdots}}\, g
(t_1) a_{\omega_1}^{\dagger} b e^{-F (t_1)} e^{F (t_2)} g (t_2) b^{\dagger} a_{\omega_3} \right. \nonumber \\
& \quad \times e^{-i (H_0 -E) (t_2 - t_3)} g (t_{3}) a_{\omega_2}^{\dagger} b e^{- F (t_{3})} e^{F (t_{4})} g
(t_4) b^{\dagger} a_{\omega_4} \, {\vphantom{\int^0}\smash[t]{\vdots}}\nonumber \\
& \quad + {\vphantom{\int^0}\smash[t]{\vdots}}\, g
(t_1) a^{\dagger}_{\omega_1} b e^{-F (t_1)} e^{F (t_2)} g (t_2) a^{\dagger}_{\omega_2} b e^{-F (t_2)} e^{F (t_3)} \nonumber \\
& \quad \times \left. g (t_{3}) b^{\dagger} a_{\omega_3} e^{- F (t_{3})} e^{F (t_{4})} g
(t_4) b^{\dagger} a_{\omega_4} \, {\vphantom{\int^0}\smash[t]{\vdots}}\right) |0 \rangle_{c} a_{\omega}^{\dagger} a_{\omega'}^{\dagger} | 0 \rangle
\nonumber \\
& + (\omega \leftrightarrow \omega').
\label{S2form}\end{aligned}$$ The $N=2$ contribution is represented by the two terms populating the cavity with at most one photon ($\sim bb^{\dagger}bb^{\dagger}$) and with two photons ($\sim bbb^{\dagger}b^{\dagger}$). Simplifying \[see Appendix \[twophot\]\] we obtain $$\begin{aligned}
& a (-v t_x- v \tau_d ) a (-v t_x) S \frac{\mathcal{A}_{\omega_0}^{\dagger \, 2}}{2} | 0 \rangle =\frac{e^{-i \omega_0 (2 t_x + \tau_d)}}{L} \nonumber \\
& \times \left[ 1 + 2 A (t_x) + 2 A (t_x + \tau_d) + 4 \bar{B} (t_x , \tau_d) \right] | 0 \rangle,
\label{aaS}\end{aligned}$$ where $$\begin{aligned}
\bar{B} (t_x , \tau_d) &= B (t_x , \tau_d) + A (t_x) A (t_x + \tau_d), \label{bB0} \\
B (t_x , \tau_d) &= -i U g (t_x) e^{-f_1 (t_x)} g (t_x + \tau_d) e^{-f_1 (t_x + \tau_d)} \nonumber \\
& \times \int_{-\infty}^{t_x} d t' e^{i U (t'-t_x)+2 f_1 (t')} \frac{A^2 (t')}{g^2 (t')} . \label{B0}\end{aligned}$$
The function $B$ in is associated with an inelastic contribution to the two-photon scattering: it vanishes for $U=0$. It is periodic in the argument $t_x$, therefore we can again make a replacement $t_x \to \tau_c$. For the time-independent coupling we recover the expression $$\begin{aligned}
B (\tau_d) = - A^2 \frac{U}{U -2 (\delta + i \Gamma)} e^{i (\delta + i \Gamma) \tau_d} .\end{aligned}$$
In the large $U$ limit, which corresponds to the case of a two-level system, the inelastic contribution becomes equal \[see Appendix \[twophot\]\] $$\begin{aligned}
B (t_x , \tau_d) &= -\frac{g (t_x + \tau_d)}{g (t_x)} A^2 (t_x) \nonumber \\
& \times e^{f_{osc} (t_x)-f_{osc} (t_x + \tau_d)} e^{ i (\delta + i \Gamma) \tau_d} .
\label{B0Ui}\end{aligned}$$
With help of we find analogous expressions for transmitted and reflected fields $$\begin{aligned}
& a_r (-v t_x- v \tau_d ) a_r (-v t_x) S \frac{\mathcal{A}_{r, \omega_0}^{\dagger \, 2}}{2} | 0 \rangle \nonumber \\
& =\frac{e^{-i \omega_0 (2 t_x + \tau)}}{L} \left[ t (t_x) t (t_x + \tau_d) + B (t_x , \tau_d) \right] | 0 \rangle, \\
& a_l (-v t_x- v \tau_d ) a_l (-v t_x) S \frac{\mathcal{A}_{r,\omega_0}^{\dagger \, 2}}{2} | 0 \rangle \nonumber \\
& =\frac{e^{-i \omega_0 (2 t_x + \tau_d)}}{L} \left[ r (t_x) r (t_x + \tau_d) + B (t_x , \tau_d) \right] | 0 \rangle ,\end{aligned}$$ which allow us to define the corresponding second order coherence functions $$\begin{aligned}
g_{rr}^{(2)} (\tau_c, \tau_d) &= \bigg|1 + \frac{B (\tau_c, \tau_d)}{t (\tau_c) t (\tau_c + \tau_d)} \bigg|^2, \\
g_{ll}^{(2)} (\tau_c, \tau_d) &= \bigg|1 + \frac{B (\tau_c, \tau_d)}{r (\tau_c) r (\tau_c + \tau_d)} \bigg|^2.\end{aligned}$$
Results. {#results}
========
Reflection and transmission
---------------------------
Assuming a weakly coherent initial signal in the right-moving mode $a_{r,\omega_0}$, we study in this section the linear reflection $r (\tau_c)=A (\tau_c)$ and transmission $t (\tau_c) =1 + A (\tau_c)$, which are periodic functions of the reduced central time $\tau_c \in [-T/2 , T/2]$. Their absolute values give envelope shapes of average reflected and transmitted fields, periodically changing in space and time. This behavior contrasts with the case time-independent coupling featuring constant $r=-\frac{i \Gamma}{\delta + i \Gamma}$ and $t=\frac{\delta}{\delta+ i \Gamma}$.
We apply the general results of the Section \[sec:few\] to two coupling modulation protocols: 1) “on-off” $g (t) = g_0 (1+\cos \Omega t)$; and 2) “sign change” $g (t) =
g_0 \cos \Omega t$. In the “on-off” protocol the coupling strength is periodically quenched to zero \[Fig. \[fig:g1\](a)\], while in the “sign change” protocol, the sign of $g (t)$ changes after crossing zero \[Fig. \[fig:g1\](c)\]. A notable difference between the two protocols is that the former yields a $2 \pi$-periodic modulation of a field’s amplitude \[Fig. \[fig:g1\](b)\], while the latter yields a $\pi$-periodic one \[Fig. \[fig:g1\](d)\].
For a time independent interaction, a single photon on resonance ($\delta=0$) is fully reflected ($r=-1$), regardless the value of the coupling strength. Should the adiabaticity condition be fulfilled at every time $t$ for a time periodic interaction, we would expect the reflection amplitude $r (t)$ to follow $\Gamma (t)$ instantaneously \[see Eq. \], also showing (almost) full reflection in the resonant case (up to a small fraction $\sim |\dot{g} (t)/(g (t) \Gamma (t))|$ of the transmitted photon’s probability density). However, the adiabaticity condition is strongly violated for these two protocols.
For any protocol with a momentary quench of coupling this can happen even at slow driving. In these cases the nonadiabatic behavior of $A$ does depend on a protocol’s history as we shall see later.
Moreover, at certain time instants the coupling strength in both of them is quenched, implying a momentary decoupling of microwave photons from the cavity and hence full transmission at these time instants. Since we are dealing with an open quantum system, this qualitative picture becomes even more complicated due to memory effects, and the non-adiabatic behavior can be explained as a sum over histories. Each history has the photon entering the cavity at some initial time, $\tau_i$, and leaving at some later time, $\tau_f$, with an amplitude $g(\tau_i)
g(\tau_f)$, and a weight determined by the decay probability of the photonic state in the cavity, $\exp( - \int_{\tau_i}^{\tau_f} \Gamma(\tau)
\mathrm{d}\tau)$. The reflection coefficient at $\tau_f$, given by the sum over initial times $\tau_i$, is highly influenced by the evolution within a memory window set by the decay rate of the cavity.
In the “on-off” protocol the memory window is largest for final times after the $\Omega \tau_c = -\pi$ node, meaning that the photon remains longer in the cavity and is released shortly after when the coupling strength is sufficiently increased, producing a spike in the reflection coefficient that overshoots unity \[Fig. \[fig:g1\](b)\]. In the “sign change” protocol memory effects create an additional node, that is absent in $g (t)$, close to $\Omega \tau_c = -\pi/2$ for slow drive and moving towards ${\tau_c = 0}$ for faster drives \[Fig. \[fig:g1\](d)\]. For times shortly after the $-\pi/2$ node of $g (t)$ the memory window includes histories with amplitudes of opposite signs, and their competition creates this additional node. These two examples show how different protocols may not only chop the wavepacket of the incoming photon, but also significantly alter its form.
![(Color online) The envelopes of the reflected field. **(a)** The “on-off” cosine signal, and the resulting **(b)** envelope as a function of the central time $\tau_c$ for various driving speeds. Note the perfect transmission ($A=0$) when the coupling is quenched. **(c)** The “sign change” cosine signal and the resulting **(d)** envelope. The envelope repeats itself after a half period, and in addition to the two coupling quench nodes at $\Omega \tau_c = \pm \pi/2$ an extra node develops at $\Omega \tau_c \approx -\pi/2$ (at slow drive) and moves towards $\tau_c = 0$ (at fast drive).[]{data-label="fig:g1"}](fig2){width="0.95\columnwidth"}
The resulting envelopes strongly depend on the normalized frequency $\beta =
\Omega / \Gamma^{(0)}$, where $\Gamma^{(0)}$ is the zeroth harmonic of $\Gamma(t)$. For the fast drive $\beta \gg 1$, we obtain ${A (\tau_c) \approx - \frac23
(1+\cos \Omega \tau_c)}$ in the “on-off” protocol, which means that the reflected pulse follows $g (\tau_c)$, not $\Gamma (\tau_c)$; and ${A (\tau_c)
\approx - \frac{1}{\beta} \sin 2 \Omega \tau_c}$ following $\Gamma^{(0)} \tau_c
- f_1 (\tau_c)$ in the “sign change” protocol. In the second case, $A
(\tau_c)$ is negligibly small, so that we have (almost) full transmission despite the resonance – this effect is in sharp contrast to its non-driven counterpart, where the full reflection is expected. Thus, this protocol can be used for the dynamical routing of photons. For slow drive, $\beta \ll 1$, the adiabaticity condition is fulfilled at least within some range around $\tau_c =0$, and this accounts for the formation of a plateau with $A (\tau_c) \approx - 1$, resembling full reflection in the non-driven resonant case.
As we have seen above, a momentary quench of coupling leads to a formation of nodes in the reflected field. This effect can be viewed as the quantum version of optical chopping. It is a quantum effect because a scattered single photon remains in a linear superposition of its transmitted and reflected states. It is analogous to chopping because the amplitude of the transmitted signal is periodically changed from its maximal value down to zero and back again.
To make this analogy more obvious we show in Fig. \[fig:rect\] the single photon reflection amplitudes in the resonant case $\delta=0$ for rectangular driving procedures that more closely resemble the operation of a conventional chopper: on the figure at $|g| \sim g_0$ the photon passage is shut (full reflection), while at $|g| \ll g_0$ it is open (full transmission). The envelope function $A(\tau_c)$ shows qualitatively the same effects as for the cosine signal investigated above. Note that at large $\beta$ (fast drive), the signal shaping also works for the on-off procedure, as it does for the cosine signal.
![(Color online) Envelope function $A(\tau_c)$ for rectangular driving procedures, which are non-smooth versions of the “on-off” and “sign change” protocols in Fig. \[fig:g1\]. The off component of the “on-off” signal has been set at (a more realistic) small non-zero value, $g_{off} = g_0/5$. []{data-label="fig:rect"}](figs4){width="\columnwidth"}
Second order coherence
----------------------
The second order coherences manifest nonlinear effects quantified by the value of $U$.
Only fast drives, $\beta = \Omega/\Gamma^{(0)} \gg 1$, are able to affect the correlations before they decay, and we numerically calculate $g_{ll}^{(2)}$ for fast and moderate drives in the two cosine protocols.
In the “on-off” protocol, the fast drive only induces small oscillations in the correlation function around the non-driven results, as shown in Fig. \[fig:g21\].
![(Color online) The $g^{(2)}_{ll}(\tau_c = 0, \tau_d)$ correlation function for the “on-off” protocol at fast driving, $\beta=10$. The $g^{(2)}$ correlation for the corresponding non-driven system with a decay rate set to $\Gamma^{(0)}$ are shown as dashed lines, and the (uninteresting) correlations for the driven system slightly oscillate around the non-driven antibunching curves.[]{data-label="fig:g21"}](figs1){width="0.8\columnwidth"}
In contrast, the “sign change” protocol induces huge bunching effects due to the additional node in the single-photon reflection, as can be clearly seen in Fig. \[fig:g2\](a). We also find periodic oscillations between strong bunching (red areas) and anti-bunching (blue areas) away from $\Omega \tau_c = 0$ and $\Omega \tau_c = \pm \pi$. This is a dramatic change in statistical properties of the reflected light due to the time dependence of $g (t)$ as compared to the case of constant $g$, where $g_{ll}^{(2)}$ is monotonously anti-bunched. For a moderate drive, $\beta = 1$, all oscillatory effects in the “sign change” protocol die out for delay times longer than a single drive period, as shown in Fig. \[fig:g2\](b).
![(Color online) Second order coherence of the reflected pulse $g_{ll}^{(2)}$ in the “sign change” protocol as a function of central time $\tau_c$ and delay time $\tau_d$ for the Kerr nonlinearity $|U|=4
\Gamma^{(0)}$. We show results for **(a)** the fast drive $\beta = 10$, where huge periodically repeated bunching peaks are formed and interwoven with areas of moderate bunching and anti-bunching; **(b)** the intermediate drive $\beta = 1$, showing the decay of $g_{ll}^{(2)}$ to the uncorrelated value (white area). Insets: Comparison of the cuts at $\Omega \tau_c =
- \frac{\pi}{2}$ (dashed line) with the unmodulated $g_{ll}^{(2)}$ (solid line).[]{data-label="fig:g2"}](fig3){width="\columnwidth"}
The photon compression by the “on-off” driving introduces nodes in the transmission and produces, similarly to the field quench effects in the reflected light for the “sign change” protocol, strong bunching in the transmitted light captured by $g^{(2)}_{rr}$. This picture is verified by a numerical calculation of the correlation function for fast drive, $\beta=10$, and nonlinearity, $|U| = 2 \Gamma^{(0)}$ as shown in Fig. \[fig:g23\].
![(Color online) The $g^{(2)}_{rr}$ correlation of transmitted light in the “on-off” protocol at fast driving, $\beta = 10$, with a nonlinearity $|U|=2\Gamma^{(0)}$. Note the strong periodically recurring bunching due to the wavepacket compression.[]{data-label="fig:g23"}](figs3){width="0.8\columnwidth"}
Summary.
========
We have proposed a quantum analogue of an optical chopper, operating at the few-photon level and realizable by a time-periodic modulation of the photon-emitter coupling. We have developed an exact Floquet scattering approach based on diagrammatic scattering theory and applied it to quantitatively describe scattering of microwave photons from the nonlinear cavity in two driving protocols of the coupling: “on-off” and “sign change”. In both of them we have observed interesting non-adiabatic memory effects arising due to the driving. In particular, the “on-off” protocol produces periodic compressions of the photon’s wavepacket at slow drive, while at fast drive the signal is directly encoded into the shape of the single photon pulse. The “sign change” protocol in turn gives rise to the additional nodes in the envelope at which the field is completely quenched, while at fast drive it may completely change the direction of a photon. These are two examples of chopping realizable at the quantum single-photon level. In addition, in the latter protocol we find dramatic changes in statistical properties of the reflected field showing up as strong bunching peaks in the $g^{(2)}$ function that are interwoven with periodically alternating areas of antibunching and moderate bunching — features that are in sharp contrast to their non-driven counterparts. Thus, our findings can be useful for single photon pulse shaping, dynamical routing of photons, and altering of the photon statistics in real time.
Acknowledgements
================
We are grateful to A. Fedorov and M. Hafezi for useful discussions. The work of V. G. is a part of the Delta-ITP consortium, a program of the Netherlands Organization for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).
Proof of the normalization condition {#NC}
====================================
To prove the normalization condition we need to show that $$\begin{aligned}
\int_0^T d \tau_c |A (\tau_c ) |^2 = - \textup{Re} \int_0^T d \tau_c A (\tau_c).
\label{normA}\end{aligned}$$
Let us introduce the function $$\begin{aligned}
W (t) = \int_{-\infty}^t d t' e^{f_1 (t')} g (t') \equiv - \frac{A (t)}{\pi g (t)} e^{f_1 (t)}.
\label{Wfun}\end{aligned}$$ Noticing that $\frac{d}{dt} [f_1 (t) + f_1^* (t) ] =2 \Gamma (t)=2 \pi g^2 (t)$ we integrate lhs of by parts $$\begin{aligned}
& \int_0^T d \tau_c \pi \Gamma (\tau_c) e^{-[f_1 (\tau_c) + f_1^* (\tau_c) ]} |W (\tau_c ) |^2 \nonumber \\
=& - \frac{\pi}{2} e^{-[f_1 (\tau_c) + f_1^* (\tau_c) ]} |W (\tau_c ) |^2 \bigg|_0^T \nonumber \\
&+ \frac{\pi}{2} \int_0^T d \tau_c e^{-[f_1 (\tau_c) + f_1^* (\tau_c) ]} \nonumber \\
& \times [\dot{W}^* (\tau_c ) W (\tau_c ) + W^* (\tau_c ) \dot{W} (\tau_c ) ].\end{aligned}$$ The first term vanishes because of the periodicity of the function $e^{-f_1 (t)} W (t)$, while the second term amounts to $$\begin{aligned}
\frac{\pi}{2} \int_0^T d \tau_c g (\tau_c) [e^{-f_1 (\tau_c)} W (\tau_c ) + e^{-f_1^* (\tau_c)} W^* (\tau_c ) ],\end{aligned}$$ which coincides with rhs of . Thus, is fulfilled.
Evaluation of the two-photon scattering state {#twophot}
=============================================
The form of the two-photon scattering state can be reduced to $$\begin{aligned}
& S a_{\omega}^{\dagger} a_{\omega'}^{\dagger} | 0 \rangle = \frac12 a_{\omega}^{\dagger} a_{\omega'}^{\dagger} | 0 \rangle
+ \int d \omega_1 s_{\omega_1 \omega} a_{\omega_1}^{\dagger} a_{\omega'}^{\dagger} |0 \rangle \nonumber \\
& + \int d \omega_1 d \omega_2 \int d t_1 d t_2 d t_3 d t_4 \nonumber \\
&\quad \times \Theta (t_1> t_2 > t_3 > t_{4}) e^{i (\omega_1 + \omega_2 -\omega -\omega') t_1}\nonumber \\
&\quad \times \left(
g (t_1) e^{-f_{1,E-\omega_2} (t_1)} e^{f_{1,E-\omega_2} (t_2)} g (t_2) \right. \nonumber \\
& \quad \times e^{-i (\omega_2 - \omega' ) (t_2 - t_3)} g (t_{3}) e^{- f_{1 \omega'} (t_{3})} e^{f_{1 \omega'} (t_{4})} g
(t_4) \nonumber \\
& \quad + 2 g
(t_1) e^{-f_{1,E-\omega_2} (t_1)} e^{f_{1,E-\omega_2} (t_2)} g (t_2) e^{-f_2 (t_2)} e^{f_2 (t_3)} \nonumber \\
& \quad \times \left. g (t_{3}) e^{- f_{1 \omega'} (t_{3})} e^{f_{1 \omega'} (t_{4})} g
(t_4) \right) a^{\dagger}_{\omega_1}a^{\dagger}_{\omega_2} | 0 \rangle
\nonumber \\
& + (\omega \leftrightarrow \omega'),\end{aligned}$$ where $f_2 (t)= i (2 \omega_c + U - 2 i \Gamma^{(0)} -\omega -\omega') t +2 f_{osc} (t)$. Folding it with $\phi (\omega) \phi (\omega')$ and applying the field operators $a (-v t_x - v \tau_d) a (-v t_x) $ we obtain the expression with $$\begin{aligned}
& 4 \bar{B} (t_x, \tau_d) = \int d \omega_1 d \omega_2 \int d t_1 d t_2 d t_3 d t_4 \nonumber \\
&\quad \times \Theta (t_1> t_2 > t_3 > t_{4}) e^{i (\omega_1 -\omega_0 ) t_1} e^{ i (\omega_2 - \omega_0) t_2} \nonumber \\
&\quad \times \left(
g (t_1) e^{-f_{1} (t_1)} e^{f_{1} (t_2)} g (t_2) \right. \nonumber \\
& \quad \times e^{-i (\omega_2 - \omega_0 ) (t_2 - t_3)} g (t_{3}) e^{- f_{1} (t_{3})} e^{f_{1} (t_{4})} g
(t_4) \nonumber \\
& \quad + 2 e^{-i U (t_2 -t_3)} g
(t_1) e^{-f_{1} (t_1)} e^{f_{1} (t_2)} g (t_2) e^{-2 f_1 (t_2)} e^{2 f_1 (t_3)} \nonumber \\
& \quad \times \left. g (t_{3}) e^{- f_{1} (t_{3})} e^{f_{1} (t_{4})} g (t_4) \right) \nonumber \\
&\quad \times \left( e^{-i t_x (\omega_1 - \omega_0)} e^{-i (t_x + \tau_d) (\omega_2 - \omega_0)} \right. \nonumber \\
& \quad \left. + e^{-i t_x (\omega_2 - \omega_0)} e^{-i (t_x + \tau_d) (\omega_1 - \omega_0)} \right) ,
\label{S2expl}\end{aligned}$$ and $f_1 (t)$ defined in . Performing frequency integrals in simplifies it to $$\begin{aligned}
& \bar{B} (t_x, \tau_d) =\pi^2 g (t_x + \tau_d) e^{-f_1 (t_x + \tau_d)} g (t_x) e^{-f_{1} (t_x)} \nonumber \\
&\quad \times \int d t_2 d t_4 e^{f_{1} (t_{4})} g
(t_4) e^{f_{1} (t_2)} g (t_2) \nonumber \\
& \quad \times [ \Theta (t_x + \tau_d> t_2 > t_x ) \Theta (t_x > t_{4}) \nonumber \\
& \quad + 2 \Theta (t_x > t_2 > t_4) e^{-i U (t_x -t_2)} ] .
\label{BU1}\end{aligned}$$ The second integral containing the $U$-dependent phase factor can be written in terms of the function as $$\begin{aligned}
& \int_{-\infty}^{t_x} dt_2 2 \dot{W} (t_2) W (t_2) e^{- i U (t_x -t_2)} \nonumber \\
& = W^2 (t_x) - i U \int_{-\infty}^{t_x} dt_2 W^2 (t_2) e^{-i U (t_x -t_2)}.
\label{byparts}\end{aligned}$$ Representing $$\begin{aligned}
W^2 (t_x) = \int^{t_x}_{-\infty} d t_2 e^{f_1 (t_2)} g (t_2) \int^{t_x}_{-\infty} d t_4 e^{f_1 (t_4)} g (t_4) ,\end{aligned}$$ we substitute in and obtain $$\begin{aligned}
& \bar{B} (t_x, \tau) =\pi^2 g (t_x + \tau_d) e^{-f_1 (t_x + \tau_d)} g (t_x) e^{-f_{1} (t_x)} \nonumber \\
&\quad \times \left[ \int^{t_x + \tau_d}_{-\infty} d t_2 e^{f_{1} (t_2)} g (t_2) \int^{t_x}_{-\infty} d t_4 e^{f_{1} (t_{4})} g
(t_4) \right. \nonumber \\
& \quad \left. - i U \int_{-\infty}^{t_x} dt_2 W^2 (t_2) e^{-i U (t_x -t_2)} \right] ,
\label{BU2}\end{aligned}$$ which is equivalent to , .
Note that the contribution to the inelastic part of $g^{(2)}$ vanishes in the limit $|U| \to \infty$ (rapid oscillations average the integral in lhs to zero). Thus, we obtain .
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|
---
author:
- 'Klaus M. Frahm'
- 'Dima L. Shepelyansky'
title: Linear response theory for Google matrix
---
Introduction
============
Linear response theory finds a great variety of applications in statistical physics, stochastic processes, electron transport, current density correlations and dynamical systems (see e.g. [@kubo; @hanggi; @stone; @kohn; @ruelle]). In this work we apply the approach of linear response to Google matrices of directed networks with the aim to characterize nontrivial interactions between nodes.
The concept of Google matrix and the related PageRank algorithm for the World Wide Web (WWW) has been proposed by Brin and Page in 1998 [@brin]. A detailed description of the Google matrix construction and its properties is given in [@meyer]. This approach can be applied to numerous situations and various directed networks [@rmp2015].
Here we develop the LInear Response algorithm for GOogle MAtriX (LIRGOMAX) which applies to a very general model of a weakly perturbed Google matrix or the related PageRank algorithm. As a particular application we consider a model of injection and absorption at a small number of nodes of the networks and test its efficiency on examples of the English Wikipedia network of 2017 [@wiki2017]. However, the scope of LIRGOMAX algorithm is more general. Thus, for example, it can be also applied to compute efficiently and accurately the PageRank sensitivity with respect to small modifications of individual elements of the Google matrix or its reduced version [@greduced; @politwiki; @wrwu2017; @zinovyev].
From a physical viewpoint the approach of $\;\;\;\;$ injection/absorption corresponds to a small pumping probability at a certain network node (or group of nodes) and absorbing probability at another specific node (or group of nodes). In a certain sense such a procedure reminds lasing in random media where a laser pumping at a certain frequency generates a response in complex absorbing media [@cao].
More specifically we select two particular nodes, one for injection and one for absorption, for which we use the LIRGOMAX algorithm to determine a subset of most sensitive nodes involved in a pathway between these two nodes. Furthermore we apply to this subset of nodes the REduced GOogle MAtriX (REGOMAX) algorithm developed in [@greduced; @politwiki] and obtain in this way an effective Google matrix description between nodes of the found pathway.
In general the REGOMAX algorithm determines effective interactions between selected nodes of a certain relatively small subset embedded in a global huge network. Its efficiency was recently demonstrated for the Wikipedia networks of politicians [@politwiki] and world universities [@wrwu2017], SIGNOR network of protein-protein interactions [@zinovyev] and multiproduct world trade network of UN COMTRADE [@wtn2019].
In this work our aim is to provide a first illustration of the efficiency of the LIRGOMAX algorithm combined with the reduced Google matrix analysis. Due to this we restrict in this work our considerations to the analytical description of the LIRGOMAX algorithm and the illustration of its application to two cases from the English Wikipedia network of 2017.
The paper is constructed as follows: in Section 2 we provide the analytical description of the LIRGOMAX algorithm complemented by a brief description of the Google matrix construction and the REGOMAX algorithm, in Section 3 we present certain results for two examples of the Wikipedia network, the discussion of results is given in Section 4. Additional data are also available at [@ourwebpage].
Theory of a weakly perturbed Google matrix
==========================================
Google matrix construction
--------------------------
We first briefly remind the general construction of the Google matrix $G$ from a direct network of $N$ nodes. For this one first computes the adjacency matrix $A_{ij}$ with elements $1$ if node $j$ points to node $i$ and zero otherwise. The matrix elements of $G$ have the usual form $G_{ij} = \alpha S_{ij} + (1-\alpha) / N$ [@brin; @meyer; @rmp2015], where $S$ is the matrix of Markov transitions with elements $S_{ij}=A_{ij}/k_{out}(j)$ and $k_{out}(j)=\sum_{i=1}^{N}A_{ij}\neq 0$ being the out-degree of node $j$ (number of outgoing links) or $S_{ij}=1/N$ if $j$ has no outgoing links (dangling node). The parameter $0< \alpha <1$ is the damping factor with the usual value $\alpha=0.85$ [@meyer] used here. We note that for the range $0.5 \leq \alpha \leq 0.95$ the results are not sensitive to $\alpha$ [@meyer; @rmp2015]. This corresponds to a model of a random surfer who follows with probability $\alpha$ at random one of the links available from the actual node or jumps with probability $(1-\alpha)$ to an arbitrary other node in the network.
The right PageRank eigenvector of $G$ is the solution of the equation $G P = \lambda P$ for the unit eigenvalue $\lambda=1$ [@brin; @meyer]. The PageRank $P(j)$ values represent positive probabilities to find a random surfer on a node $j$ ($\sum_j P(j)=1$). All nodes can be ordered by decreasing probability $P$ numbered by the PageRank index $K=1,2,...N$ with a maximal probability at $K=1$ and minimal at $K=N$. The numerical computation of $P(j)$ is done efficiently with the PageRank iteration algorithm described in [@brin; @meyer].
It is also useful to consider the original network with inverted direction of links. After inversion the Google matrix $G^*$ is constructed via the same procedure (using the transposed adjacency matrix) and its leading eigenvector $P^*$, determined by $G^* P^*= P^*$, is called CheiRank [@cheirank] (see also [@rmp2015]). Its values $P^*(j)$ can be again ordered in decreasing order resulting in the CheiRank index $K^*$ with highest value of $P^*$ at $K^*=1$ and smallest values at $K^*=N$. On average, the high values of $P$ ($P^*$) correspond to nodes with many ingoing (outgoing) links [@meyer; @rmp2015].
Reduced Google matrix algorithm
-------------------------------
The REGOMAX method is described in detail in $\;\;\;\;$ [@greduced; @politwiki; @zinovyev; @wrwu2017]. For a given relatively small subset of $N_r\ll N$ nodes it allows to compute efficiently a “reduced Google matrix” ${{G_{\rm R}}}$ of size $N_r \times N_r$ that captures the full contributions of direct and indirect pathways appearing in the full Google matrix $G$ between the $N_r$ selected nodes of interest. The PageRank vector $P_r$ of ${{G_{\rm R}}}$ coincides with the full PageRank vector projected on the subset of nodes, up to a constant multiplicative factor due to the sum normalization. The mathematical computation of ${{G_{\rm R}}}$ provides a decomposition of ${{G_{\rm R}}}$ into matrix components that clearly distinguish direct from indirect interactions: ${{G_{\rm R}}}= {{G_{rr}}}+ {{G_{\rm pr}}}+ {{G_{\rm qr}}}$ [@politwiki]. Here ${{G_{rr}}}$ is given by the direct links between the selected $N_r$ nodes in the global $G$ matrix with $N$ nodes. ${{G_{\rm pr}}}$ is a rank one matrix whose columns are rather close (up to constant factor) to the reduced PageRank vector $P_r$. Even though the numerical weight of ${{G_{\rm pr}}}$ is typically quite large it does not give much new interesting information about the reduced effective network structure.
The most interesting role is played by ${{G_{\rm qr}}}$, which takes into account all indirect links between selected nodes happening due to multiple pathways via the global network nodes $N$ (see [@greduced; @politwiki]). The matrix ${{G_{\rm qr}}}={{G_{\rm qrd}}}+ {{{G_{\rm qr}}}}^{(nd)}$ has diagonal (${{G_{\rm qrd}}}$) and non-diagonal (${{{G_{\rm qr}}}}^{(nd)}$) parts with ${{{G_{\rm qr}}}}^{(nd)}$ describing indirect interactions between selected nodes. The exact formulas and the numerical algorithm for an efficient numerical computation of all three components of ${{G_{\rm R}}}$ are given in [@greduced; @politwiki]. It is also useful to compute the weights ${{W_{\rm R}}}$, ${{W_{\rm pr}}}$, ${{W_{\rm rr}}}$, ${{W_{\rm qr}}}$ of ${{G_{\rm R}}}$ and its 3 matrix components ${{G_{\rm pr}}}$, ${{G_{rr}}}$, ${{G_{\rm qr}}}$ given by the sum of all its elements divided by the matrix size $N_r$. Due to the column sum normalization of ${{G_{\rm R}}}$ we obviously have ${{W_{\rm R}}}={{W_{\rm rr}}}+{{W_{\rm pr}}}+{{W_{\rm qr}}}=1$.
General model of linear response
--------------------------------
We consider a Google matrix $G(\eps)$ (with non-negative matrix elements satisfying the usual column sum normalization) depending on a small parameter $\eps$ and a general stochastic process $P(t+1)=G(\eps)\,F(\eps,P(t))$ where $F(\eps,P)$ is a general function on $\eps$ and $P$ which does NOT need to be linear in $P$. (Here $P(t)$ denotes a time dependence of the vector $P$; below and for the rest of this paper we will use the notation $P(j)$ for the $j$th component of the vector $P$).
Let $E^T=(1,\ldots,1)$ be the usual vector with unit entries. Then the condition of column sum normalization of $G(\eps)$ reads $E^T G(\eps)=E^T$. The function $F(\eps,P)$ should satisfy the condition $E^T F(\eps,P)=1$ if $E^T P=1$. At $\eps=0$ we also require that $F(0,P)=P$, i.e. $F(0,P)$ is the identity operation on $P$. We denote by $G_0=G(0)$ the Google matrix at $\eps=0$ and by $P_0$ its PageRank vector such that $G_0\,P_0=P_0$ with $E^T P_0=1$. We denote by $P$ the more general, $\eps$-dependent, solution of $$\label{eq_Psol1}
P=G(\eps)\,F(\eps,P)\quad,\quad E^T\,P=1\ .$$
### Pump model
As a first example we present the [*Pump model*]{} to model an injection- and absorption scheme. For this we choose for the Google matrix simply $G(\eps)=G_0$ (i.e. no $\eps$-dependence for $G$) and $$\label{eq_Finject}
F(\eps,P)=\frac{({\bf 1}+\eps D)P}{E^T[({\bf 1}+\eps D)P]}=
\frac{({\bf 1}+\eps D)P}{1+\eps\,e(P)}$$ with $e(P)=E^T D P$ and $D$ being a diagonal matrix with entries $D_j$ which are mostly zero and with few positive values $D_j>0$ for nodes $j$ with injection and few negative values $D_j<0$ for nodes $j$ with absorption. The non-vanishing diagonal entries of $D_j$ should be comparable (a global scaling factor can be absorbed in a redefinition of the parameter $\eps$) and we have $e(P)=\sum_{D_j\neq 0} D_j P(j)$. Physically, we multiply each entry $P(j)$ by the factor $1+\eps D_j$ (which is unity for most nodes $j$) and then we sum-normalize this vector to unity before we apply the Google matrix $G_0$ to it.
### PageRank sensitivity
As a second example we consider the PageRank sensitivity. For this we fix a pair $(i,j)$ of indices and multiply the matrix element $(G_0)_{ij}$ by $(1+\eps)$ and then we sum-normalize the column $j$ to unity which provides the $\eps$-dependent Google matrix $G(\eps)$. For the function $F(\eps,P)$ we simply choose the identity operation: $F(\eps,P)=P$. In a more explicit formula we have: $$\label{eq_Gsensdef}
\forall_{k,l}\qquad
G_{kl}(\eps)=\frac{(1+\eps\,\delta_{ki}\delta_{lj})\,(G_0)_{kl}}
{1+\eps\,\delta_{lj}\,(G_0)_{ij}}$$ where $\delta_{ki}=1$ (or $0$) if $k=i$ (or $k\neq i$). Note that the denominator is either $1$ if $l\neq j$ or the modified column sum $1+\eps\,(G_0)_{ij}$ of column $j$ if $l=j$. Then the sensitivity $D_{(j\to i)}(k)$ is defined as: $$\label{eq_sensdef}
D_{(j\to i)}(k)=\frac{P(k)-P_0(k)}{\eps P_0(k)}$$ where $P$ is the $\eps$-dependent PageRank of $G(\eps)$ computed in the usual way. We expect that this quantity has a well defined limit if $\eps\to 0$ but equation (\[eq\_sensdef\]) is numerically not very precise for very small values of $\eps$ due to the effect of loss of precision. Below we present a method to compute the sensitivity in a precise way in the limit $\eps\to 0$.
Examples of the sensitivity analysis, using directly (\[eq\_sensdef\]), were considered for the reduced Google matrix of sets of Wikipedia and other networks (see e.g. [@wrwu2017; @wtn2019]).
Linear response
---------------
### General scheme
One can directly numerically determine the $\eps$-dependent solution $P(\eps)$ of (\[eq\_Psol1\]) for some small but finite value of $\eps$ (by iterating $P^{(n+1)}=G(\eps),F(\eps,P^{(n)})$ with some suitable initial vector $P^{(0)}$) and compute the quantity $$\label{eq_Pdelta1}
\Delta P(\eps)=\frac{P(\eps)-P(0)}{\eps}\ .$$ We expect that $\Delta P(\eps)$ has a finite well defined limit if $\eps\to 0$. However, its direct numerical computation by (\[eq\_Pdelta1\]) is subject to numerical loss of precision if $\eps$ is too small. In the following, we will present a different scheme to compute $\Delta P$ which is numerically more accurate and stable and that we call linear response of Google matrix. For this we expand $G(\eps)$ and $F(\eps,P)$ up to order $\eps^1$ (neglecting terms $\sim \eps^2$ or higher): $$\label{eq_expand1}
G(\eps)=G_0+\eps G_1+\ldots \quad,\quad F(\eps,P)=P+\eps F_1(P)+\ldots$$ Furthermore we write $$\label{eq_Pexpand1}
P(\eps)=P_0+\eps\,P_1+\ldots\ .$$ The usual sum-normalization conditions for the first order corrections read as : $$\label{eq_sum1}
E^T G_1=0\quad,\quad E^T F_1(P)=0\quad,\quad
E^T P_1=0$$ if $E^T P = E^T P_0= 1$. These conditions imply that $P_1$ and also $F_1(P)$ belong to the subspace “bi-orthogonal” to the PageRank, i.e. orthogonal to the left leading eigenvector of $G_0$ which is just the vector $E^T$.
Inserting (\[eq\_expand1\]) and (\[eq\_Pexpand1\]) into (\[eq\_Psol1\]) we obtain (up to order $\eps^1$): $$\label{eq_Ppert1}
P=P_0+\eps P_1=
G_0\,P_0+\eps\Bigl[G_0\,P_1+G_1\,P_0+G_0\,F_1(P_0)\Bigr]\ .$$ Comparing the terms of order $\eps^0$ one obtains the usual unperturbed PageRank equation $P_0=G_0\,P_0$. The terms of order $\eps^1$ provide an inhomogeneous PageRank equation of the type : $$\label{eq_inhom1}
P_1=G_0\,P_1+V_0\quad,\quad V_0=G_1\,P_0+G_0\,F_1(P_0)\ .$$ The solution $P_1$ of this equation is just the limit of (\[eq\_Pdelta1\]): $$\label{eq_P1lim}
P_1=\lim_{\eps\to 0} \Delta P(\eps)=
\lim_{\eps\to 0} \frac{P(\eps)-P(0)}{\eps}\ .$$ To solve numerically (\[eq\_inhom1\]) we first determine the unperturbed PageRank $P_0$ of $G_0$ in the usual way and compute $V_0$ which depends on $P_0$. Then we iterate the equation: $$\label{eq_P1iterate}
P_1^{(n+1)}=G_0\,P_1^{(n)}+V_0$$ where for the initial vector we simply choose $P_1^{(0)}=0$. This iteration converges with the same speed as the usual PageRank algorithm versus the vector $P_1$ and it is numerically more accurate than the finite difference $\Delta P(\eps)$ at some finite value of $\eps$.
We remind that $E^T V_0=\sum_j V_0(j)=0$ and also $E^T G_0=E^T$. Therefore if at a given iteration step the vector $P_1^{(n)}$ satisfies the condition $E^T P_1^{(n)}=0$ we also have $E^T P_1^{(n+1)}=E^T G_0 P_1^{(n)}+E^T V_0= E^T P_1^{(n)}=0$.
Therefore the conditions (\[eq\_sum1\]) are satisfied by the iteration equation (\[eq\_P1iterate\]) at least on a theoretical/mathematical level. However, rounding errors may produce slight errors in the conditions (\[eq\_sum1\]) and since such numerical errors contain a contribution in the direction of the unperturbed PageRank vector $P_0$, corresponding to the eigenvector of $G_0$ with maximal eigenvalue, they do not disappear during the iteration and might even (slightly) increase with $n$. Therefore, due to purely numerical reasons, it is useful to remove such contributions by a projection after each iteration step of the vector $P_1^{(n+1)}$ on the subspace bi-orthogonal to the PageRank by: $$\label{eq_projectQ}
P_1^{(n+1)}\to Q\left(P_1^{(n+1)}\right)\quad,\quad Q(X)=X-(E^T X)\,P_0$$ where $Q(X)$ is the projection operator applied on a vector $X$. It turns out that such an additional projection step indeed increases the quality and accuracy of the convergence of (\[eq\_P1iterate\]) but even without it the iteration (\[eq\_P1iterate\]) converges numerically, however with a less accurate result.
It is interesting to note that one can “formally” solve (\[eq\_P1iterate\]) by: $$\label{eq_P1solve}
P_1=\sum_{n=0}^\infty G_0^n\,V_0=\frac{\bf 1}{{\bf 1}-G_0}\,V_0$$ which can also be found directly from the first equation in (\[eq\_inhom1\]). Strictly speaking the matrix inverse $({\bf 1}-G_0)^{-1}$ does not exist since $G_0$ has always one eigenvalue $\lambda=1$. However, since $E^T\,V_0=0$, the vector $V_0$, when expanded in the basis of (generalized) eigenvectors of $G_0$, does not contain a contribution of $P_0$ which is the eigenvector for $\lambda=1$ such that the expression (\[eq\_P1solve\]) is actually well defined. From a numerical point of view a different scheme to compute $P_1$ would be to solve directly the linear system of equations $({\bf 1}-G_0)\,P_1=V_0$ where the first (or any other suitable) equation of this system is replaced by the condition $E^T P_1=0$ resulting in a linear system with a well defined unique solution. Of course, such a direct computation is limited to modest matrix dimensions $N$ such that a full matrix inversion is possible (typically $N$ being a few multiples of $10^4$) while the iterative scheme (\[eq\_P1iterate\]) is possible for rather large matrix dimensions such that the usual PageRank computation by the power method is possible. For example for the English Wikipedia edition of 2017 with $N\approx 5\times 10^6$ the iterative computation of $P_1$ using (\[eq\_P1iterate\]) takes typically $2-5$ minutes on a recent single processor core (e.g.: Intel i5-3570K CPU) without any use of parallelization once the PageRank $P_0$ is known. (The computation of $P_0$ by the usual power method takes roughly the same time.)
### Application to the pump model
For the injection- and absorption scheme we can compute $F_1(P)$ from (\[eq\_Finject\]) as: $$\begin{aligned}
\label{eq_F1P_compute}
F(\eps,P)&=&({\bf 1}-\eps\,e(P)+\ldots)(P+\eps D P)
\\
\nonumber
&=& P+\eps\,[P-(E^T D P)P]+\ldots\ .\end{aligned}$$ Here the term $\sim \eps^1$ is just the projection of $DP$ to the subspace bi-orthogonal to $P$. This projection is the manifestation in first order in $\eps$ of the renormalization used in (\[eq\_Finject\]).
Furthermore, since for the injection- and absorption scheme we also have $G_1=0$, the vector $V_0$ in (\[eq\_inhom1\]) and (\[eq\_P1iterate\]) becomes: $$\begin{aligned}
\nonumber
V_0&=&G_0\,F_1(P_0)=G_0\,Q(D\,P_0)\\
\label{eq_V0pump}
&=&G_0 D\,P_0-(E^T D P_0)\,G_0\,P_0\\
\nonumber
&=&G_0 D\,P_0-(E^T G_0\,D P_0)\,P_0=Q(G_0\,D\,P_0)\end{aligned}$$ with $Q$ being the projector given in (\[eq\_projectQ\]). Here we have used that $E^T\,G_0=E^T$ and $G_0\,P_0=P_0$. This small calculation also shows that the projection operator can be applied before or after multiplying $G_0$ to $D\,P_0$.
### Application to the sensitivity
In this case we have $F(\eps,P)=P$ such that $F_1(P)=0$ and we have to determine $G_1$ from the expansion $G(\eps)=G_0+\eps G_1+\ldots$. Expanding (\[eq\_Gsensdef\]) up to first order in $\eps$ we obtain: $$\label{eq_G1sens}
\forall_{kl}\qquad (G_1)_{kl}=\delta_{ki}\delta_{lj}\,(G_0)_{kl}-
\delta_{lj}\,(G_0)_{ij}$$ where $(i,j)$ is the pair of indices for which we want to compute the sensitivity. Using $G_1$ we compute $V_0=G_1\,P_0$ and solve the inhomogeneous PageRank equation (\[eq\_inhom1\]) iteratively as described above to obtain $P_1$. Once $P_1$ is know we can compute the sensitivity from : $$\label{eq_sensexact}
D_{(j\to i)}(k)=\frac{P_1(k)}{P_0(k)}\ .$$ We note that equation (\[eq\_sensexact\]) is numerically accurate and corresponds to the exact limit $\eps\to 0$ while (\[eq\_sensdef\]) is numerically not very precise and requires a finite small value of $\eps$.
\[table1\]
LIRGOMAX combined with REGOMAX
------------------------------
We consider the pump model described above and we take two particular nodes $i$ with injection and $j$ with absorption. For the diagonal matrix $D$ we choose $D_i=1/P_0(i)$ and $D_j=-1/P_0(j)$ where $P_0$ is the PageRank of the unperturbed network and all other values $D_k=0$. In this way we have $e(P_0)=E^T D\,P_0=D_i\,P_0(i)+D_j\,P_0(j)=0$. Due to this the renormalization denominator in (\[eq\_Finject\]) is simply unity and all excess probability provided by the injection at node $i$ will be exactly absorbed by the absorption at node $j$. We insist that this is only due to our particular choice for the matrix $D$ and concerning the numerical procedure one can also choose different values of $D_i$ or $D_j$ with $e(P_0)\neq 0$ (which would result in some global excess probability which would be artificially injected or absorbed due the normalization denominator in (\[eq\_Finject\]) being different from unity).
Using the above values of $D_i$ and $D_j$ we compute the vector $V_0=G_0\,D\,P_0=G_0\,W_0$ (since $E^T D\,P_0=0$) where $W_0=D\,P_0$ is a vector with only two non-zero components $W_0(i)=1$ and $W_0(j)=-1$. Therefore for all $k$ we have $V_0(k)=(G_0)_{ki}-(G_0)_{kj}$. According to the above theory we know that $V_0$ and $W_0$ are orthogonal to $E^T$, i.e. $E^T V_0=E^T W_0=0$ or more explicitely $\sum_k V_0(k)=\sum_k W_0(k)=0$. For $W_0$ the last equality is obvious and the first one is due to the column sum normalization of $G_0$ meaning that $\sum_k (G_0)_{ki}=\sum_k (G_0)_{kj}=1$. Using the expression $V_0(k)=(G_0)_{ki}-(G_0)_{kj}$ we determine the solution of the linear response correction to the PageRank $P_1$ by solving iteratively the inhomogeneous PageRank equation (\[eq\_inhom1\]) as described above. The vector $P_1$ has real positive and negative entries also satisfying the condition $\sum_k P_1(k)=0$. Then we determine the 20 top nodes with strongest negative values of $P_1$ and further 20 top nodes with strongest positive values of $P_1$ which constitute a subset of 40 nodes which are the most significant nodes participating in the pathway between the pumping node $i$ and absorbing node $j$.
Using this subset we then apply the REGOMAX algorithm to compute the reduced Google matrix and its components which are analyzed in a similar way as in [@politwiki]. The advantage of the application of LIRGOMAX at the initial stage is that it provides an automatic procedure to determine an interesting subset of nodes related to the pumping between nodes $i$ and $j$ instead of using an arbitrary heuristic choice for such a subset.
The question arises if the initial two nodes $i$ and $j$ belong themselves to the subset of nodes with largest $P_1$ entries (in modulus). From a physical point of view we indeed expect that this is generically the case but there is no simple mathematical argument for this. In particular for nodes with a low PageRank ranking and zero or few incoming links this is probably not the case. However, concerning the two examples which we will present in the next section both initial nodes $i$ and $j$ are indeed present in the selected subset and even with rather top positions in the ranking (provided by ordering $|P_1|$).
LIRGOMAX for Wikipedia network
==============================
As a concrete example we illustrate the application of LIRGOMAX algorithm to the English Wikipedia network of 2017 (network data available at [@wiki2017]). This network contains $N=5416537$ nodes, corresponding to article titles, and $N_i = 122232932$ directed hyperlinks between nodes. Previous applications of the REGOMAX algorithm for the Wikipedia networks of years 2013 and 2017 are described in [@politwiki; @wrwu2017].
Case of pathway Cambridge - Harvard Universities
------------------------------------------------
As a first example of the application of the combined LIRGOMAX and REGOMAX algorithms we select two articles (nodes) of the Wikipedia network with pumping at [*University of Cambridge*]{} and absorption at [*Harvard University*]{}. The global PageRank indices of these two nodes are $K=229$ (PageRank probability $P_0(229)= 0.0001078$) and $K=129$ (PageRank probability $P_0(129)= 0.0001524$). As described above we chose the diagonal matrix $D$ as $D(229)=1/P_0(229)$ and $D(129)=-1/P_0(129)$ (other diagonal entries of $D$ are chosen as zero) and determine the vector $V_0$ used for the computation of $P_1$ (see (\[eq\_P1iterate\])) by $V_0=G_0\,W_0$ where the vector $W_0=D\,P_0$ has the nonzero components $W_0(229) = 1$ and $W_0(129) = - 1$. Both $W_0$ and $V_0$ are orthogonal to the left leading eigenvector $E^T=(1,\ldots,1)$ of $G_0$ according to the theory described in the last section.
![Linear response vector $P_1$ of PageRank for the English Wikipedia 2017 with injection (or pumping) at [*University of Cambridge*]{} and absorption at [*Harvard University*]{}. Here $K_L$ is the ranking index obtained by ordering $|P_1|$ from maximal value at $K_L=1$ down to its minimal value. Top panel shows $|P_1|$ versus $K_L$ in a double logarithmic representation for all $N$ nodes. Bottom panel shows a zoom of $P_1$ versus $K_L$ for $K_L \le 10^3$ in a double logarithmic representation with sign; blue data points correspond to $P_1>0$ and red data points to $P_1<0$. []{data-label="fig1"}](fig1){width="48.00000%"}
The subset of 40 most affected nodes with 20 strongest negative and 20 strongest positive values of the linear response correction $P_1$ to the initial PageRank $P_0$ are given in Table \[table1\]. We order these 40 nodes by the index $i=1, \ldots, 20$ for the first 20 most negative $P_1$ values and then $i=21, \ldots, 40$ for the most positive $P_1$ values. The index $K_L$ is obtained by ordering $|P_1|$ for all $N\approx 5\times 10^6$ network nodes. The table also gives the PageRank index $K$ obtained by ordering $P_0$. The first 4 positions in $K_L$ are taken by [*Harvard University; Cambridge, Massachusetts; University of Cambridge; United States*]{}. Thus, even if the injection is made for [*University of Cambridge*]{} the strongest response appears for [*Harvard University; Cambridge, Massachusetts*]{} and only then for [*University of Cambridge*]{} ($K_L=1,2,3$). We attribute this to nontrivial flows existing in the global directed network. This shows that the linear response approach provides rather interesting information about the sensitivity and interactions of nodes on directed networks. We will see below for other examples that the top nodes of the linear response vector $P_1$ can have rather unexpected features.
In general the most sensitive nodes of Table \[table1\] are rather natural. They represent countries, cities and other administrative structures related to the two universities. Other type of nodes are [*Yale University, The New York Times, American Civil War*]{} for Harvard U and [*Church of England, The Guardian, Durham University*]{} for U Cambridge (in addition to many Colleges presented in the list) corresponding to closest other universities and also newspapers appearing on the pathway between the pair of selected nodes.
Of course, the linear response vector $P_1$ extends on all $N$ nodes of the global network. We show its dependence on the ordering index $K_L$ in Figure \[fig1\]. Here the top panel represents the decay of $|P_1|$ with $K_L$ and the bottom panel shows the decay of negative and and positive $P_1$ values for $K_L \leq 10^3$. We note that among top 100 values of $K_L$ there are only 4 nodes related to U Cambridge with positive $P_1$ values while all other values of $P_1$ are negative being related to Harvard U. This demonstrates a rather different structural influence between these two universities.
![Reduced Google matrix components ${{G_{\rm R}}}$, ${{G_{\rm pr}}}$, ${{G_{rr}}}$ and ${{G_{\rm qr}}}$ for the English Wikipedia 2017 network and the subgroup of nodes given in Table \[table1\] corresponding to injection at [*University of Cambridge*]{} and absorption at [*Harvard University*]{} (see text for explanations). The axis labels correspond to the index number $i$ used in Table \[table1\]. The relative weights of these components are ${{W_{\rm pr}}}=0.920$, ${{W_{\rm rr}}}=0.036$, and ${{W_{\rm qr}}}=0.044$. Note that elements of $G_{qr}$ may be negative. The values of the color bar correspond to $\operatorname{sgn}(g)(|g|/\max|g|)^{1/4}$ where $g$ is the shown matrix element value. The exponent $1/4$ amplifies small values of $g$ for a better visibility. []{data-label="fig2"}](fig2){width="48.00000%"}
After the selection of 40 most significant nodes of the pathway between both universities (see Table \[table1\]) we apply the REGOMAX algorithm which determines all matrix elements of Markov transitions between these 40 nodes including all direct and indirect pathways via the huge global Wikipedia network with 5 million nodes.
The reduced Google matrix ${{G_{\rm R}}}$ and its three components ${{G_{\rm pr}}}$, ${{G_{rr}}}$, ${{G_{\rm qr}}}$ are shown in Figure \[fig2\]. As discussed above the weight ${{W_{\rm pr}}}=0.920$ of ${{G_{\rm pr}}}$ is close to unity and its matrix structure is rather similar to the one of ${{G_{\rm R}}}$ with strong transition lines of matrix elements corresponding to [*United States*]{} at top PageRank index $K=1\ (i=3,\ K_L=4)$ and [*United Kingdom*]{} at $K=6\ (i=28,\ K_L=257)$. The weights ${{W_{\rm rr}}}=0.036 $, ${{W_{\rm qr}}}= 0.044$ of ${{G_{rr}}}$, ${{G_{\rm qr}}}$ are significantly smaller. These values are similar to those obtained in the REGOMAX analysis of politicians and universities in Wikipedia networks [@politwiki; @wrwu2017]. Even if the weights of these matrix components are not large they represent the most interesting and nontrivial direct (${{G_{rr}}}$) and indirect (${{G_{\rm qr}}}$) interactions between the selected 40 nodes. The image of ${{G_{rr}}}$ in Figure \[fig2\] shows that the direct links between the U Cambridge block of nodes (with index $21 \leq i \leq 40$ in Table \[table1\]) and the Harvard U block of nodes (with index $1 \leq i \leq 20$ in Table \[table1\]) are rather rare and relatively weak while the links within each block are multiple and relatively strong. This confirms the appropriate selection of nodes in each block provided by the LIRGOMAX algorithm.
![Same as in Fig. \[fig2\] but for the matrix $G_{rr}+G_{qr}^{(nd)}$, where $G_{qr}^{(nd)}$ is obtained from $G_{qr}$ by putting its diagonal elements at zero; the weight of these two components is $W_{rr+qrnd}=0.066$. []{data-label="fig3"}](fig3){width="48.00000%"}
The matrix elements of the sum of two components ${{G_{rr}}}+{{{G_{\rm qr}}}}^{(nd)}$ (component ${{G_{\rm qr}}}$ is taken without diagonal elements) are shown in Figure \[fig3\]. We note that some elements are negative which is not forbidden since only the sum of all three components given by ${{G_{\rm R}}}$ should have positive matrix elements. However, the negative values are rare and relatively small compared to the values of positive matrix elements. Thus the minimal value is $-0.00216$ for the transition from [*Church of England*]{} to [*United States*]{} while other typical negative values are smaller by a factor 5-10. For comparison, the maximal value of positive element is $ 0.1135$ from [*Regent House*]{} to [*University of Cambridge*]{} and there are many other positive elements of the order of $0.03$. Thus we consider that the negative elements play no significant role. A similar conclusion was also obtained for the interactions of politicians and universities in [@politwiki; @wrwu2017].
![Network of friends for the subgroup of nodes given in Table \[table1\] corresponding to injection at [*University of Cambridge*]{} and absorption at [*Harvard University*]{} constructed from the matrix $G_{rr}+G_{qr}^{(nd)}$ using 4 top (friends) links per column (see text for explanations). The numbers used as labels for the different nodes correspond to the index $i$ of Table \[table1\]. []{data-label="fig4"}](fig4){width="48.00000%"}
From Figure \[fig3\] we see that for $G_{rr}+G_{qr}^{(nd)}$ the strongest interactions are also inside each university block. However, there are still some significant links between blocks with strongest matrix elements being $0.0120$ from [*Fellow*]{} to [*United States*]{} and $0.0050$ from [*Harvard University*]{} to [*University of Cambridge*]{} (for both directions between blocks). The link [*Fellow*]{} to [*United States*]{} is also the strongest indirect link in its off diagonal sub-block (for ${{G_{\rm qr}}}$) while the strongest direct link (for ${{G_{rr}}}$) is [*Fellow*]{} to [*Harvard University*]{}. Furthermore, the link [*Harvard University*]{} to [*University of Cambridge*]{} is also the strongest indirect link in its off diagonal sub-block (for ${{G_{\rm qr}}}$) while the strongest direct link (for ${{G_{rr}}}$) is [*Harvard College*]{} to [*University of Cambridge*]{}.
Using the transition matrix elements of $G_{rr}+G_{qr}^{(nd)}$ we construct a network of effective friends shown in Figure \[fig4\]. First, we select five initial nodes which are placed on a (large) circle: the two nodes with injection/absorption ([*University of Cambridge*]{} and [*Harvard University*]{}) and three other nodes with a rather top position in the $K_L$ ranking ([*England*]{}, (Town of) [*Cambridge*]{} and [*United States*]{}). For each of these five initial nodes we determine four friends by the criterion of largest matrix elements (in modulus) in the same column, i.e. corresponding to the four strongest links from the initial node to the potential friends. The friend nodes found in this way are added to the network and drawn on circles of medium size around their initial node (if they do not already belong to the initial set of 5 top nodes). The links from the initial nodes to their friends are drawn as thick black arrows. For each of the newly added nodes (level 1 friends) we continue to determine the four strongest friends (level 2 friends) which are drawn on small circles and added to the network (if there are not already present from a previous level). The corresponding links from level 1 friends to level 2 friends are drawn as thin red arrows.
Each node is marked by the index $i$ from the first column of Table \[table1\]. The colors of the nodes are essentially red for nodes with strong negative values of $P_1$ (corresponding to the index $i=1,\ldots,20$) and blue for nodes with strong positive values of $P_1$ (for $i=21,\ldots,40$). Only for three of the initial nodes we choose different colors which are olive for [*US*]{}, green for [*England*]{} and cyan for (the town of) [*Cambridge*]{}.
The network of Figure \[fig4\] shows a quite clear separation of network nodes in two blocks associated to the two universities with a rather small number of links between the two blocks (e.g. US is a friend of England but not vice-versa).
Case of pathway Napoleon - Alexander I of Russia
------------------------------------------------
We illustrate the application of the LIRGOMAX and REGOMAX algorithms on two other nodes of the Wikipedia network with injection (pumping) at [*Napoleon*]{} and absorption at [*Alexander I of Russia*]{}. The global PageRank indices of these two nodes are $K=201$ (PageRank probability $P_0(201)= 0.0001188$) and $K= 5822$ (PageRank probability $P_0(5822)= 1.389 \times 10^{-5}$ ) respectively. In contrast to the the previous example the two PageRank probabilities are rather different. However, this difference is compensated by our choice of the diagonal matrix $D$ with $D(201)=1/P_0(201)$ and $D(5822)=-1/P_0(5822)$ (other diagonal entries of $D$ begin zero). Again we determine the vector $V_0$ used for the computation of $P_1$ (see (\[eq\_P1iterate\])) by $V_0=G_0\,W_0$ where the vector $W_0=D\,P_0$ has the nonzero components $W_0(201) = 1$ and $W_0(5822) = - 1$. Furthermore, both $W_0$ and $V_0$ are orthogonal to the left leading eigenvector $E^T=(1,\ldots,1)$ of $G_0$.
The top nodes of $P_1$, noted by index $i$, with 20 strongest negative and 20 strongest positive values are presented in Table \[table2\]. The ranking of nodes in decreasing order of $|P_1|$ given by the index $K_L$ is shown in the second column of Table \[table2\]. It is interesting to note that the injection node [*Napoleon*]{} is only at position $K_L=29$ with a significantly smaller value of $|P_1|$ compared to [*Alexander I of Russia*]{} at $K_L=3$, [*Russian Empire*]{} at $K_L=1$ and [*Saint Petersburg*]{} at $K_L=2$. But among the positive $P_1$ values [*Napoleon*]{} is still at the first position. We attribute this relatively small $|P_1|$ value of [*Napoleon*]{} compared to the nodes of the other block to significant complex directed flows in the global Wikipedia network. Also [*Napoleon*]{} has a significantly stronger PageRank probability and thus this node produces a stronger influence on [*Alexander I of Russia*]{} than vice-versa.
In contrast to the previous case of universities Table \[table2\] contains mainly countries, a few towns and islands, and historical figures related in some manner to [*Napoleon*]{} or [*Alexander I of Russia*]{}.
The dependence of the linear response vector $P_1$ on the index $K_L$ is shown in Figure \[fig5\] (analogous to Figure \[fig1\]). The decay of $|P_1|$ with $K_L$ is shown in the top panel, being similar to the top panel of Figure \[fig1\]. The values of $P_1$ with sign are shown in the bottom panel. The difference of the $|P_1|$ values for [*Napoleon*]{} and [*Alexander I of Russia*]{} is not so significant but many nodes ($28$) from the block of [*Alexander I of Russia*]{} have larger $|P_1|$ values than $|P_1|$ of [*Napoleon*]{}.
![Same as in Fig. \[fig1\] for the subgroup of Table \[table2\] corresponding to injection at [*Napoleon*]{} and absorption at [*Alexander I of Russia*]{}. []{data-label="fig5"}](fig5){width="48.00000%"}
The results for the reduced Google matrix of 40 nodes of Table \[table2\] are shown in Figure \[fig6\]. The strongest lines of transitions in ${{G_{\rm R}}}$ and ${{G_{\rm pr}}}$ correspond to nodes with top PageRank positions of the global Wikipedia network being [*France*]{} at $K=4\ (i=23,\ K_L=144)$, [*Iran*]{} at $K=7\ (i=12,\ K_L=12)$, [*Italy*]{} at $K=12\ (i=24,\ K_L=149)$ and [*Russia*]{} at $K=17\ (i=7,\ K_L=7)$. As explained above the structure of transitions appears rather similar between ${{G_{\rm R}}}$ and ${{G_{\rm pr}}}$. The weights of all three components ${{G_{\rm pr}}}$, ${{G_{rr}}}$, ${{G_{\rm qr}}}$ are similar to those of the two universities (see caption of Figure \[fig6\]).
![As Fig. \[fig2\] for the subgroup of Table \[table2\] corresponding to injection at [*Napoleon*]{} and absorption at [*Alexander I of Russia*]{}. The relative weights of the different matrix components are ${{W_{\rm pr}}}=0.900$, ${{W_{\rm rr}}}=0.042$ and ${{W_{\rm qr}}}=0.058$.[]{data-label="fig6"}](fig6){width="48.00000%"}
![Same as in Fig. \[fig3\] for the subgroup of Table \[table2\] corresponding to injection at [*Napoleon*]{} and absorption at [*Alexander I of Russia*]{}. The weight of $G_{rr}+G_{qr}^{(nd)}$ is $W_{rr+qrnd}=0.087$.[]{data-label="fig7"}](fig7){width="48.00000%"}
The components ${{G_{rr}}}$ and ${{G_{\rm qr}}}$, shown in Figure \[fig6\], are also dominated by the two diagonal blocks related to the two initial nodes [*Napoleon*]{} and [*Alexander I of Russia*]{}. There are only a few direct links between the two blocks but the number of indirect links is substantially increased. The sum of these two components $G_{rr}+G_{qr}^{(nd)}$ is shown in Figure \[fig7\], where the diagonal elements of ${{G_{\rm qr}}}$ are omitted. The strongest couplings between the two blocks in $G_{rr}+G_{qr}^{(nd)}$ are $0.009879$ for the link from [*French campaign in Egypt and Syria*]{} to [*Ottoman Empire*]{} and $0.02439$ for the link from [*Elizabeth Alexeievna (Louise of Baden)*]{} to [*Napoleon*]{} (for both directions between the diagonal blocks).
![Same as in Fig. \[fig4\] for the subgroup of Table \[table2\] corresponding to injection at [*Napoleon*]{} and absorption at [*Alexander I of Russia*]{}. []{data-label="fig8"}](fig8){width="48.00000%"}
In analogy to Figure \[fig4\] we construct the network of friends for the subset of Table \[table2\] shown in Figure \[fig8\]. As in Figure \[fig4\], we use the four strongest transition matrix elements of $G_{rr}+G_{qr}^{(nd)}$ per column to construct links from the five top nodes to level 1 friends (thick black arrows) and from level 1 to level 2 friends (thin red arrows). As the five initial top nodes we choose [*France*]{} (cyan), [*Russian Empire*]{} (olive), [*Saint Petersburg*]{} (green), [*Napoleon*]{} (blue) and [*Alexander I of Russia*]{} (red); all other nodes of the [*Napoleon*]{} block ($21 \leq i \leq 40$ in Table \[table2\]) are shown in blue, and all other nodes of the [*Alexander I of Russia*]{} block ($1 \leq i \leq 20$ in Table \[table2\])) are shown in red; numbers inside the points correspond to the index $i$ of Table \[table2\].
The network of Figure \[fig8\] also shows a clear two block structure with relatively rare links between the two blocks. The coupling between two blocks appears due to one link from [*Alexander I of Russia*]{} to [*Prussia*]{}, which even being red is more closely related to the blued block of nodes.
For both examples network figures constructed in the same way using the other matrix components ${{G_{\rm R}}}$, ${{G_{rr}}}$ or ${{G_{\rm qr}}}$ (instead of $G_{rr}+G_{qr}^{(nd)}$) or using strongest matrix elements in rows (instead of columns) to determine follower networks are available at [@ourwebpage].
This type of friend/follower effective network schemes constructed from the reduced Google matrix (or one of its components) were already presented in [@politwiki] in the context of Wikipedia networks of politicians.
Discussion
==========
We introduced here a linear response theory for a very generic model where either the Google matrix or the associated Markov process depends on a small parameter and we developed the LIRGOMAX algorithm to compute efficiently and accurately the linear response vector $P_1$ to the PageRank $P_0$ with respect to this parameter for large directed networks. As a particular application of this approach it is in particular possible to identify the most important and sensitive nodes of the pathway connecting two initial groups of nodes (or simply a pair of nodes) with injection or absorption of probability. This group of most sensitive nodes can then be analyzed with the reduced Google matrix approach by the related REGOMAX algorithm which allows to determine effective indirect network interactions for this set of nodes. We illustrated the efficiency of the combined LIRGOMAX and REGOMAX algorithm for the English Wikipedia network of 2017 with two very interesting examples. In these examples, we use two initial nodes (articles) for injection/absorption, corresponding either to two important universities or to two related historical figures. As a result we obtain associated sets for most sensitive Wikipedia articles given in Tables \[table1\] and \[table2\] with effective friend networks shown in Figures \[fig4\] and \[fig8\].
As a further independent application the LIRGOMAX algorithm allows also to compute more accurately the PageRank sensitivity with respect to variations of matrix elements of the (reduced) Google matrix as already studied in [@wrwu2017; @wtn2019].
It is known that the linear response theory finds a variety of applications in statistical and mesoscopic physics [@kubo; @stone], current density correlations [@kohn], stochastic processes and dynamical chaotic systems [@hanggi; @ruelle]. The matrix properties and their concepts, like Random Matrix Theory (RMT), find important applications for various physical systems (see e.g. [@guhr]). However, in physics one usually works with unitary or Hermitian matrices, like in RMT. In contrast the Google matrices belong to another class of matricies rarely appearing in physical systems but being very natural to the communication networks developed by modern societies (WWW, Wikipedia, Twitter ...). Thus we hope that the linear response theory for the Google matrix developed here will also find useful applications in the analysis of real directed networks.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the Programme Investissements d’Avenir ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT (project THETRACOM); it was granted access to the HPC resources of CALMIP (Toulouse) under the allocation 2019-P0110.
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abstract: 'Runge’s method is a tool to figure out integral points on algebraic curves effectively in terms of height. This method has been generalised to varieties of any dimension, unfortunately its conditions of application are often too restrictive. In this paper, we provide a further generalisation intended to be more flexible while still effective, and exemplify its applicability by giving finiteness results for integral points on some Siegel modular varieties. As a special case, we obtain an explicit finiteness result for integral points on the Siegel modular variety $A_2(2)$.'
author:
- |
Samuel Le Fourn[^1]\
ENS de Lyon
bibliography:
- 'bibliotdn.bib'
title: 'A “tubular” variant of Runge’s method in all dimensions, with applications to integral points on Siegel modular varieties'
---
Introduction {#introduction .unnumbered}
============
One of the major motivations of number theory is the description of rational or integral solutions of diophantine equations, which from a geometric perspective amounts to understanding the behaviour of rational or integral points on algebraic varieties. In dimension one, there are many techniques and results providing a good overview of the situation such as the famous Faltings’ theorem (for genus $\geq 2$ and algebraic points) or Siegel’s theorem (for integral points and a function with at least three poles). Nevertheless, in many cases the quest for effectivity (meaning a bound on the height on these points) is still open, and effective methods are quite different from these two powerful theoretical theorems.
We focus in this paper on a method for integral points on algebraic varieties called *Runge’s method*, and its generalisations and applications for Siegel modular varieties.
To keep the introduction fluid, we first explain the principles behind Runge’s method and its applicatons to Siegel modular varieties, with simplified statements and a minimum of references and details. Afterwards, we describe precisely the structure of the article, in particular where the details we omitted first are given.
On a smooth algebraic projective curve $C$ over a number field $K$, Runge’s method proceeds as follows. Let $\phi \in K(C)$ be a nonconstant rational function on $C$. For any finite extension $L/K$, we denote by $M_L$ the set of places of $L$ (and $M_L^\infty$ the archimedean ones). For $S_L$ a finite set of places of $L$ containing $M_L^\infty$, we denote the ring of $S_L$-integers of ${{\mathcal O}}_L$ by $${{\mathcal O}}_{L,S_L} = \{ x \in L \, \, |x|_v \leq 1 \, \, {\textrm{for all }} \, v \in M_L \backslash S_L \}.$$ Now, let $r_L$ be the number of orbits of poles of $\phi$ under the action of ${\operatorname{Gal}}(\overline{L}/L)$. The *Runge condition* on a pair $(L,S_L)$ is the inequality $$\label{eqRungeconditioncourbes}
|S_L|<r_L.$$ Then, Bombieri’s generalisation ([@BombieriGubler], paragraph 9.6.5 and Theorem 9.6.6) of Runge’s theorem, the latter being formulated only for $L=K={{\mathbb{Q}}}$ and $r_{{\mathbb{Q}}}\geq 2$, states that for every pair $(L,S_L)$ satisfying Runge condition and every point $P \in C(L)$ such that $\phi(P) \in {{\mathcal O}}_{L,S_L}$, there is an *absolute* bound $B$ (only depending on $C$ and $\phi$, *not* on such a pair $(L,S_L)$) such that $$h(\phi(P)) \leq B,$$ where $h$ is the Weil height. In short, as long as the point $\phi(P)$ has few non-integrality places (the exact condition being ), there is an absolute bound on the height of $\phi(P)$. There is a very natural justification (due to Bilu) for Bombieri’s theorem: let us fix a pair $(L,S_L)$ satisfying Runge condition and $P \in C(L)$ such that $\phi(P) \in {{\mathcal O}}_{L,S_L}$. For every place $v \in M_L \backslash S_L$, as $|\phi(P)|_v$ is small, it means that $P$ is $v$-adically far from all orbits of poles of $\phi$. For $v \in S_L$, $P$ can be $v$-adically close to one of the orbits but only one of them because they are pairwise disjoint. We eliminate such an orbit if it exists, and applying the process for every $v \in S_L$, Runge’s condition guarantees that there remains at the end of the process one orbit ${{\mathcal O}}$ which is $v$-far from $P$ for *all* places $v \in M_L$. This in turn implies finiteness : indeed, choosing by Riemann-Roch an auxiliary function $g_{{\mathcal O}}\in L(C)$ whose poles are the points of ${{\mathcal O}}$, this means that $h(g_{{\mathcal O}}(P))$ is small as $P$ is far from its poles at every places, hence $P$ belongs to a finite set by Northcott condition. It is a bit more technical to obtain a bound on the height $h (\phi(P))$ (and which does not depend on $(L,S_L)$) in the general case) but it is the same idea. This justification also provides a method to bound in practice the heights of such points (when one knows well enough the auxiliary functions $g_{{\mathcal O}}$), which is called *Runge’s method*. When applicable, this method has two important assets: it gives good bounds, and it is uniform in the pairs $(L,S_L)$, which for example is not true for Baker’s method.
The goal of this paper was to find ways to transpose the ideas for Runge’s method on curves to higher-dimensional varieties, where it is generally very difficult to obtain finiteness of integral or rational points, as the extent of our knowledge is much more limited. First, let us recall a previous generalisation of Bombieri’s theorem in higher dimensions obtained by Levin ([@Levin08], Theorem 4). To sum it up in a simpler case, on a projective smooth variety $X$, the analogues of poles of $\phi$ are effective divisors $D_1, \cdots, D_r$. We have to fix a smooth integral model ${{\mathcal X}}$ of $X$ on ${{\mathcal O}}_K$, and denote by ${{\mathcal D}}_1, \cdots, {{\mathcal D}}_r$ the Zariski closures of the divisors in this model, of union ${{\mathcal D}}$, so our integral points here are the points of $({{\mathcal X}}\backslash {{\mathcal D}}) ({{\mathcal O}}_{L,S_L})$. There are two major changes in higher dimension. Firstly, the divisors have to be ample (or at least big) to obtain finiteness results (this was automatic for dimension 1). Secondly, instead of the condition $|S_L| < r$ as for curves, the *higher-dimensional Runge condition* is $$\label{eqintroRungemultidim}
m |S_L|<r,$$ where $m$ is the smallest number such that any $(m+1)$ divisors amongst $D_1, \cdots, D_r$ have empty common intersection. Levin’s theorem states in particular that when the divisors are ample, $$\left( \bigcup_{\substack{(L,S_L) \\ m |S_L|< r}} \! \! \left( {{\mathcal X}}\backslash {{\mathcal D}}\right) ({{\mathcal O}}_{L,S_L}) \right) \, \, \, {\textrm{is finite}}.$$ The issue with is that the maximal number $|S_L|$ satisfying this condition is much lowered because of $m$, even more as the ample (or big) hypothesis tends to give a lower bound on this $m$. When we tried to apply Levin’s theorem to some Siegel modular varieties with chosen divisors, we found that the higher-dimensional Runge condition was too restrictive (remember that $S_L$ contains archimedean places, so $|S_L|\geq [K:{{\mathbb{Q}}}]/2$), hence the theorem was not applicable. This was the initial motivation for a generalisation of this theorem, called “tubular Runge theorem”, designed to be more flexible in terms of Runge condition. Let us explain its principle below.
Additionally to $X$ and $D_1, \cdots, D_r$, we fix a closed subvariety $Y$ of $X$ which is meant to be “a subvariety of $X$ where the divisors $D_1, \cdots, D_r$ intersect a lot more than outside it”. More precisely, let $m_Y$ the smallest number such that any $(m_Y+1)$ divisors amongst $D_1, \cdots, D_r$ have common intersection included in $Y$. In particular, $m_Y \leq m$, and the goal is to have $m_Y$ as small as possible without asking $Y$ to be too large. Now, we fix a “tubular neighbourhood” of $Y$, which is the datum of a family ${{\mathcal V}}=(V_v)_v$ where $v$ goes through the places $v$ of ${\overline{K}}$, every $V_v$ is a neighbourhood of $Y$ in $v$-adic topology, and this family is uniformly not too small in some sense. For example, if ${{\mathcal Y}}$ is the Zariski closure of $Y$ in ${{\mathcal X}}$, we can define at a finite place $v$ the neighbourhood $V_v$ to be the set of points of ${{\mathcal X}}(\overline{K_v})$ reducing in ${{\mathcal Y}}$ modulo $v$. We say that a point $P \in X({\overline{K}})$ does *not* belong to ${{\mathcal V}}$ if $P \notin V_v$ for every place $v$ of ${\overline{K}}$, and intuitively, this means that $P$ is $v$-adically far away from $Y$ for *every* place $v$ of ${\overline{K}}$. Now, assume our integral points are not in ${{\mathcal V}}$. It implies that at most $m_Y$ divisors amongst $D_1, \cdots, D_r$ can be $v$-adically close to them, hence using the same principles of proof as Levin, this gives the *tubular Runge condition* $$\label{eqintroRungetub}
m_Y |S_L| < r.$$ With this additional data, one can now give an idea of our tubular Runge theorem.
For $X,{{\mathcal X}},Y,D_1, \cdots,D_r,m_Y$ and a tubular neighbourhood ${{\mathcal V}}$ of $Y$ as in the paragraph above, let $({{\mathcal X}}\backslash {{\mathcal D}}) ({{\mathcal O}}_{L,S_L}) \backslash {{\mathcal V}}$ be the set of points of $({{\mathcal X}}\backslash {{\mathcal D}})({{\mathcal O}}_{L,S_L})$ which do not belong to ${{\mathcal V}}$. Then, if $D_1, \cdots, D_r$ are ample, for every such tubular neighbourhood, the set $$\left( \bigcup_{\substack{(L,S_L) \\ m_Y |S_L|< r}} \! \! \left( {{\mathcal X}}\backslash {{\mathcal D}}\right) ({{\mathcal O}}_{L,S_L}) \backslash {{\mathcal V}}\right) \, \, \, {\textrm{is finite}},$$ and bounded in terms of some auxiliary height.
This is a very simplified form of the theorem : one can have $D_1, \cdots, D_r$ defined on a scalar extension of $X$ and big instead of ample, and ${{\mathcal X}}$ normal for example. The general (and more precise) version is Theorem \[thmRungetubulaire\]. As the implicit bound on the height is parametered by the tubular neighbourhood ${{\mathcal V}}$, it can be seen as a *concentration result* rather as a finiteness one : essentially, it states that the points of $({{\mathcal X}}\backslash {{\mathcal D}}) ({{\mathcal O}}_{L,S_L})$ concentrate near the closed subset $Y$. As such, we have compared it to theorems of [@CorvajaLevinZannier], notably Autissier Theorem and CLZ Theorem, in section \[sectiontubularRunge\] (in particular, our version is made to be effective, whereas these results are based on Schmidt’s subspace theorem, hence theoretically ineffective).
In the second part of our paper, we applied the method for Siegel modular varieties, both as a proof of principle and because integral points on these varieties are not very well understood, apart from Shafarevich conjecture proved by Faltings. As we will see below, this is also a case where a candidate for $Y$ presents itself, thus giving tubular neighbourhoods a natural interpretation.
For $n \geq 2$, the variety denoted by $A_2(n)$ is the variety over ${{\mathbb{Q}}}(\zeta_n)$ parametrising triples $(A,\lambda,\alpha_n)$ with $(A,\lambda)$ is a principally polarised abelian variety of dimension 2 and $\alpha_n$ is a symplectic level $n$ structure on $(A,\lambda)$. It is a quasi-projective algebraic variety of dimension 3, and its Satake compactification (which is a projective algebraic variety) is denoted by $A_2(n)^S$, the boundary being $\partial A_2(n) = A_2(n)^S \backslash A_2(n)$. The extension of scalars $A_2(n)_{{\mathbb{C}}}$ is the quotient of the half-superior Siegel space ${{\mathcal H}}_2$ by the natural action of the symplectic congruence subgroup $\Gamma_2(n)$ of ${\operatorname{Sp}}_4({{\mathbb{Z}}})$ made up with the matrices congruent to the identity modulo $n$. Now, we consider some divisors ($n^4/2 +2$ of them) defined by the vanishing of some modular forms, specifically theta functions. One finds that they intersect a lot on the boundary $\partial A_2(n)$ ($m$ comparable to $n^4$), but when we fix $Y=\partial A_2(n)$, we get $m_Y \leq (n^2 - 3)$ hence giving the *tubular Runge condition* $$(n^2 - 3) |S_L| < \frac{n^4}{2} + 2.$$
Now, the application of our tubular Runge theorem gives for every even $n \geq 2$ a finiteness result for the integral points for these divisors and some tubular neighbourhoods associated to potentially bad reduction for the finite places : this is Theorem \[thmtubularRungegeneral\]. In the special case $n=2$, as a demonstration of the effectiveness of the method, we made this result completely explicit in Theorem \[thmproduitCEexplicite\]. A simplified case of this Theorem is the following result.
\[Theorem \[thmproduitCEexplicite\], simplified case\]
Let $K$ be either ${{\mathbb{Q}}}$ or a quadratic imaginary field.
Let $A$ be a principally polarised abelian surface defined over $K$ as well as all its 2-torsion and having potentially good reduction at all finite places of $K$.
Then, if the semistable reduction of $A$ is a product of elliptic curves at most at 3 finite places of $K$, we have the explicit bound $$h_{{\mathcal F}}(A) \leq 1070,$$ where $h_{{\mathcal F}}$ is the stable Faltings height. In particular, there are only finitely many such abelian surfaces.
To conclude this introduction, we explain the structure of the paper, emphasizing where the notions sketched above and proofs are given in detail.
$$\xymatrix{
\ref{sectionnotations} \ar[d] \ar[rrd] & & \\
\ref{sectionvoistub} \ar[d] & & \ref{sectionrappelsSiegel} \ar[d] \\
\ref{sectionresultatscles} \ar[d] \ar[r] & \ref{sectiontubularRunge} \ar[rd] \ar[r]& \ref{sectionapplicationsSiegel} \ar[d] \\
\ref{sectionRungecourbes} & & \ref{sectionexplicitRunge}
}$$
Section \[sectionnotations\] is devoted to the notations used throughout the paper, including heights, $M_K$-constants and bounded sets (Definition \[defMKconstante\]). We advise the reader to pay particular attention to this first section as it introduces notations which are ubiquitous in the rest of the paper. Section \[sectionvoistub\] is where the exact definition (Definition \[defvoistub\]) and basic properties of tubular neighbourhoods are given. In section \[sectionresultatscles\], we prove the key result for Runge tubular theorem (Proposition \[propcle\]), essentially relying on a well-applied Nullstellensatz. For our purposes, in Proposition \[propreductionamplegros\], we also translate scheme-theoretical integrality in terms of auxiliary functions. In section \[sectionRungecourbes\], we reprove Bombieri’s theorem for curves (written as Proposition \[propBombieri\]) with Bilu’s idea, as it is not yet published to our knowledge (although this is exactly the principle behind Runge’s method in [@BiluParent09] for example). To finish with the theoretical part, we prove and discuss our tubular Runge theorem (Theorem \[thmRungetubulaire\]) in section \[sectiontubularRunge\].
For the applications to Siegel modular varieties, section \[sectionrappelsSiegel\] gathers the necessary notations and reminders on these varieties (subsection \[subsecabvarSiegelmodvar\]), their integral models with some discussions on the difficulties on dealing with them in dimension at least 2 (subsection \[subsecfurtherpropSiegelmodvar\]) and the important notion of theta divisors on abelian varieties and their link with classical theta functions (subsection \[subsecthetadivabvar\]). The theta functions are crucial because the divisors we use in our applications of tubular Runge method are precisely the divisors of zeroes of some of these theta functions.
In section \[sectionapplicationsSiegel\], we consider the case of abelian surfaces we are interested in, especially for the behaviour of theta divisors (subsection \[subsecthetadivabsur\]) and state in subsection \[subsectubularRungethmabsur\] the applications of Runge tubular theorem for the varieties $A_2(n)^S$ and the divisors mentioned above (Theorems \[thmtubularRungeproduitCE\] and \[thmtubularRungegeneral\]).
Finally, in section \[sectionexplicitRunge\], we make explicit Theorem \[thmtubularRungeproduitCE\] by computations on the ten fourth powers of even characteristic theta constants. To do this, the places need to be split in three categories. The finite places not above 2 are treated by the theory of algebraic theta functions in subsection \[subsecalgebraicthetafunctions\], the archimedean places by estimates of Fourier expansions in subsection \[subsecarchimedeanplaces\] and the finite places above 2 (the hardest case) by the theory of Igusa invariants and with polynomials built from our ten theta constants in subsection \[subsecplacesabove2\]. The final estimates are given as Theorem \[thmproduitCEexplicite\] in subsection \[subsecfinalresultRungeCEexplicite\], both in terms of a given embedding of $A_2(2)$ and in terms of Faltings height.
The main results of this paper have been announced in the recently published note [@LeFourn4], and apart from section \[sectionexplicitRunge\] and some improvements can be found in the author’s thesis manuscript [@LeFournthese2] (both in French).
Acknowledgements {#acknowledgements .unnumbered}
================
I am very grateful to Fabien Pazuki and Qing Liu for having kindly answered my questions and given me useful bibliographic recommandations on the subject of Igusa invariants.
Notations and preliminary notions {#sectionnotations}
=================================
The following notations are classical and given below for clarity. They will be used throughout the paper.
- $K$ is a number field.
- $M_K$ (resp. $M_K^\infty$) is the set of places (resp. archimedean places). We also denote by $M_{{\overline{K}}}$ the set of places of ${\overline{K}}$.
- $|\cdot|_\infty$ is the usual absolute value on ${{\mathbb{Q}}}$, and $|\cdot|_p$ is the place associated to $p$ prime, whose absolute value is normalised by $$|x|_p = p^{-{\operatorname{ord}}_p (x)},$$ where ${\operatorname{ord}}_p (x)$ is the unique integer such that $x = p^{{\operatorname{ord}}_p(x)} a/b$ with $p \nmid ab$. By convention, $|0|_p=0$.
- $|\cdot|_v$ is the absolute value on $K$ associated to $v \in M_K$, normalised to extend $|\cdot|_{v_0}$ when $v$ is above $v_0 \in M_{{\mathbb{Q}}}$, and the local degree is $n_v = [K_v : {{\mathbb{Q}}}_{v_0}]$, so that for every $x \in K^*$, one has sthe product formula $$\prod_{v \in M_K} |x|_v^{n_v} = 1.$$ When $v$ comes from a prime ideal ${{\mathfrak{p}}}$ of ${{\mathcal O}}_K$, we indifferently write $|\cdot|_v$ and $|\cdot|_{{\mathfrak{p}}}$.
- For any place $v$ of $K$, one defines the sup norm on $K^{n+1}$ by $$\| (x_0, \cdots, x_n) \|_v = \max_{0 \leq i \leq n} |x_i|_v.$$ (this will be used for projective coordinates of points of ${\mathbb{P}}^n (K)$).
- Every set of places $S \subset M_K$ we consider is finite and contains $M_K^ {\infty}$. We then define the ring of $S$-integers as $${{\mathcal O}}_{K,S} = \{ x \in K \, \, | \, \, |x|_v \leq 1 \textrm{ for every } v \in M_K \backslash S \},$$ in particular ${{\mathcal O}}_{K,M_K^{\infty}} = {{\mathcal O}}_K$.
- For every $P \in {\mathbb{P}}^n (K)$, we denote by $$x_P=(x_{P,0}, \cdots, x_{P,n}) \in K^{n+1}$$ any possible choice of projective coordinates for $P$, this choice being of course fixed for consistency when used in a formula or a proof.
- The logarithmic Weil height of $P \in {\mathbb{P}}^n(K)$ is defined by $$\label{eqdefinitionhauteurdeWeil}
h(P) = \frac{1}{[K : {{\mathbb{Q}}}]}\sum_{v \in M_K} n_v \log \| x_P \|_v,$$ does not depend on the choice of $x_P$ nor on the number field, and satisfies Northcott property.
- For every $n \geq 1$ and every $i \in \{0, \cdots,n\}$, the $i$-th coordinate open subset $U_i$ of ${\mathbb{P}}^n$ is the affine subset defined as $$\label{eqdefUi}
U_i = \{ (x_0 : \cdots : x_n) \, \, | \, \, x_i \neq 0 \}.$$ The normalisation function $\varphi_i : U_i \rightarrow {{\mathbb{A}}}^{n+1}$ is then defined by $$\label{eqdefvarphii}
\varphi_i (x_0 : \cdots : x_n) = \left(\frac{x_0}{x_i}, \cdots, 1, \cdots \frac{x_n}{x_i} \right).$$ Equivalently, it means that to $P \in U_i$, we associate the choice of $x_P$ whose $i$-th coordinate is 1.
For most of our results, we need to formalize the notion that some families of sets indexed by the places $v \in M_K$ are “uniformly bounded”. To this end, we recall some classical definitions (see [@BombieriGubler], section 2.6).
\[defMKconstante\]
- An *$M_K$-constant* is a family ${{\mathcal C}}= (c_v)_{v \in M_K}$ of real numbers such that $c_v=0$ except for a finite number of places $v \in M_K$. The $M_K$-constants make up a cone of ${{\mathbb{R}}}^{M_K}$, stable by finite sum and maximum on each coordinate.
- Let $L/K$ be a finite extension. For an $M_K$-constant $(c_v)_{v \in M_K}$, we define (with abuse of notation) an $M_L$-constant $(c_w)_{w \in M_L}$ by $c_w : = c_v$ if $w|v$. Conversely, if $(c_w)_{w \in M_L}$ is an $M_L$-constant, we define (again with abuse of notation) $(c_v)_{v \in M_K}$ by $c_v := \max_{w|v} c_w$, and get in both cases the inequality $$\label{eqineqinductionMKconstante}
\frac{1}{[L : {{\mathbb{Q}}}]} \sum_{w \in M_L} n_w c_w \leq \frac{1}{[K: {{\mathbb{Q}}}]} \sum_{v \in M_K} n_v c_v.$$
- If $U$ is an affine variety over $K$ and $E \subset U({\overline{K}}) \times M_{{\overline{K}}}$, a regular function $f \in {\overline{K}}[U]$ is *$M_K$-bounded on $E$* if there is a $M_K$-constant ${{\mathcal C}}= (c_v)_{v \in M_K}$ such that for every $(P,w) \in E$ with $w$ above $v$ in $M_K$, $$\log |f(P)|_w \leq c_v.$$
- An *$M_K$-bounded subset of $U$* is, by abuse of definition, a subset $E$ of $U({\overline{K}}) \times M_{{\overline{K}}}$ such that every regular function $f \in {\overline{K}}[U]$ is $M_K$-bounded on $E$.
\[remdefMKconstantes\]
There are fundamental examples to keep in mind when using these definitions:
$(a)$ For every $x \in K^*$, the family $(\log |x|_v)_{v \in M_K}$ is an $M_K$-constant.
$(b)$ In the projective space ${\mathbb{P}}^n_K$, for every $i \in \{ 0 , \cdots, n\}$, consider the set $$\label{eqdefEi}
E_i = \{ (P,w) \in {\mathbb{P}}^n({\overline{K}}) \times M_{{\overline{K}}} \, \, | \, \, |x_{P,i}|_w = \|x_P\|_w \}.$$ The regular functions $x_j/x_i$ ($j \neq i$) on ${\overline{K}}[U_i]$ (notation ) are trivially $M_K$-bounded (by the zero $M_K$-constant) on $E_i$, hence $E_i$ is $M_K$-bounded in $U_i$. Notice that the $E_i$ cover ${\mathbb{P}}^n ({\overline{K}}) \times M_{{\overline{K}}}$. We will also consider this set place by place, by defining for every $w \in M_{{\overline{K}}}$ : $$\label{eqdefEiw}
E_{i,w} = \{ P \in {\mathbb{P}}^n({\overline{K}}) \, \, |\, \, |x_{P,i}|_w = \|x_P\|_w \}.$$
$(c)$ With notations , and , for a subset $E$ of $U_i({\overline{K}})$, if the coordinate functions of $U_i$ are $M_K$-bounded on $E \times M_{{\overline{K}}}$, the height $h \circ \varphi_i$ is straightforwardly bounded on $E$ in terms of the involved $M_K$-constants. This simple observation will be the basis of our finiteness arguments.
The following lemma is useful to split $M_K$-bounded sets in an affine cover.
\[lemMKbornerecouvrement\] Let $U$ be an affine variety and $E$ an $M_K$-bounded set. If $(U_j)_{j \in J}$ is a finite affine open cover of $U$, there exists a cover $(E_j)_{j \in J}$ of $E$ such that every $E_j$ is $M_K$-bounded in $U_j$.
This is Lemma 2.2.10 together with Remark 2.6.12 of [@BombieriGubler].
Let us now recall some notions about integral points on schemes and varieties.
For a finite extension $L$ of $K$, a point $P \in {\mathbb{P}}^n(L)$ and a nonzero prime ideal ${{\mathfrak{P}}}$ of ${{\mathcal O}}_L$ of residue field $k({{\mathfrak{P}}}) = {{\mathcal O}}_L/{{\mathfrak{P}}}$, the point $P$ extends to a unique morphism ${\operatorname{Spec}}{{\mathcal O}}_{L,{{\mathfrak{P}}}} \rightarrow {\mathbb{P}}^n_{{{\mathcal O}}_K}$, and the image of its special point is *the reduction of $P$ modulo ${{\mathfrak{P}}}$*, denoted by $P_{{\mathfrak{P}}}\in {\mathbb{P}}^n (k({{\mathfrak{P}}}))$. It is explicitly defined as follows : after normalisation of the coordinates $x_P$ of $P$ so that they all belong to ${{\mathcal O}}_{L,{{\mathfrak{P}}}}$ and one of them to ${{\mathcal O}}_{L,{{\mathfrak{P}}}}^*$, one has $$\label{eqreddansPn}
P_{{\mathfrak{P}}}= (x_{P,0} \! \! \mod {{\mathfrak{P}}}: \cdots : x_{P,n} \! \! \mod {{\mathfrak{P}}}) \in {\mathbb{P}}^n_{k({{\mathfrak{P}}})}.$$
The following (easy) proposition expresses scheme-theoretic reduction in terms of functions (there will be another in Proposition \[propreductionamplegros\]). We write it below as it is the inspiratoin behind the notion of tubular neighbourhood in section \[sectionvoistub\].
\[proplienreductionpointssvaluation\] Let $S$ be a finite set of places of $K$ containing $M_K^\infty$, and ${{\mathcal X}}$ be a projective scheme on ${{\mathcal O}}_{K,S}$, seen as a closed subscheme of ${\mathbb{P}}^n_{{{\mathcal O}}_{K,S}}$.
Let ${{\mathcal Y}}$ be a closed sub-${{\mathcal O}}_{K,S}$-scheme of ${{\mathcal X}}$.
Consider $g_1, \cdots, g_s \in {{\mathcal O}}_{K,S} [X_0, \cdots, X_n]$ homogeneous generators of the ideal of definition of ${{\mathcal Y}}$ in ${\mathbb{P}}^n_{{{\mathcal O}}_{K,S_0}}$. For every nonzero prime ${{\mathfrak{P}}}$ of ${{\mathcal O}}_L$ not above $S$, every point $P \in {{\mathcal X}}(L)$, the reduction $P_{{\mathfrak{P}}}$ belongs to ${{\mathcal Y}}_{{\mathfrak{p}}}(k({{\mathfrak{P}}}))$ (with ${{\mathfrak{p}}}= {{\mathfrak{P}}}\cap {{\mathcal O}}_K$) if and only if $$\label{eqecplicitereduction2}
\forall j \in \{1, \cdots, s \}, \quad
|g_j (x_P)|_{{\mathfrak{P}}}< \|x_P\|_{{\mathfrak{P}}}^{\deg g_j}.$$
For every $j \in \{1, \cdots, s\}$, by homogeneity of $g_j$, for a choice $x_P$ of coordinates for $P$ belonging to ${{\mathcal O}}_{L,{{\mathfrak{P}}}}$ with one of them in ${{\mathcal O}}_{L,{{\mathfrak{P}}}}^*$, the inequality amounts to $$g_j(x_{P,0}, \cdots, x_{P,n}) = 0 \mod {{\mathfrak{P}}}$$ . On another hand, the reduction of $P$ modulo ${{\mathfrak{P}}}$ belongs to ${{\mathcal Y}}_{{\mathfrak{p}}}(\overline{k({{\mathfrak{P}}})})$ if and only if its coordinates satisfy the equations defining ${{\mathcal Y}}_{{\mathfrak{p}}}$ in $X_{{\mathfrak{p}}}$, but these are exactly the equations $g_1, \cdots, g_s$ modulo ${{\mathfrak{p}}}$. This remark immediately gives the Proposition by .
Definition and properties of tubular neighbourhoods {#sectionvoistub}
===================================================
The explicit expression is the motivation for our definition of *tubular neighbourhood*, at the core of our results. This definition is meant to be used by exclusion : with the same notations as Proposition \[proplienreductionpointssvaluation\], we want to say that a point $P \in X(L)$ is *not* in some tubular neighbourhood of ${{\mathcal Y}}$ if it *never* reduces in ${{\mathcal Y}}$, whatever the prime ideal ${{\mathfrak{P}}}$ of ${{\mathcal O}}_L$ is.
The main interest of this notion is that it provides us with a convenient alternative to this assumption for the places in $S$ (which are the places where the reduction is not well-defined, including the archimedean places), and also allows us to loosen up this reduction hypothesis in a nice fashion. Moreover, as the definition is function-theoretic, we only need to consider the varieties over a base field, keeping in mind that Proposition \[proplienreductionpointssvaluation\] above makes the link with reduction at finite places.
\[defvoistub\]
Let $X$ be a projective variety over $K$ and $Y$ be a closed $K$-subscheme of $X$.
We choose an embedding $X \subset {\mathbb{P}}^n_K$, a set of homogeneous generators $g_1, \cdots, g_s$ in $K[X_0, \cdots, X_n]$ of the homogeneous ideal defining $Y$ in ${\mathbb{P}}^n$ and an $M_K$-constant ${{\mathcal C}}= (c_v)_{v \in M_K}$.
The *tubular neighbourhood of $Y$ in $X$ associated to ${{\mathcal C}}$ and $g_1, \cdots, g_s$* (the embedding made implicit) is the family ${{{\mathcal V}}= (V_w)_{w \in M_{{\overline{K}}}}}$ of subsets of $X({\overline{K}})$ defined as follows.
For every $w \in M_{{\overline{K}}}$ above some $v \in M_K$, $V_w$ is the set of points $P \in X({\overline{K}})$ such that $$\label{eqdefvoistub}
\forall j \in \{1, \cdots,s\}, \quad \log |g_j(x_P)|_w < \deg (g_j) \cdot \log \|x_P \|_w + c_v.$$
As we said before, this definition will be ultimately used by exclusion:
\[defhorsdunvoistub\]
Let $X$ be a projective variety over $K$ and $Y$ be a closed $K$-subscheme of $X$.
For any tubular neighbourhood ${{\mathcal V}}= (V_w)_{w \in M_{{\overline{K}}}}$ of $Y$, we say that a point $P \in X({\overline{K}})$ *does not belong to* ${{\mathcal V}}$ (and we denote it by $P \notin {{\mathcal V}}$) if $$\forall w \in M_{{\overline{K}}}, \quad P \notin V_w.$$
\[remhorsvoistub\]
$(a)$ Comparing and , it is obvious that for the $M_K$-constant ${{\mathcal C}}=0$ and with the notations of Proposition \[proplienreductionpointssvaluation\], at the finite places $w$ not above $S$, the tubular neighbourhood $V_w$ is exactly the set of points $P \in X({\overline{K}})$ reducing in ${{\mathcal Y}}$ modulo $w$. Furthermore, instead of dealing with any homogeneous coordinates, one can if desired manipulate normalised coordinates, which makes the term $\deg(g_j) \log \|x_P\|_v$ disappear. Actually, we will do it multiple times in the proofs later, as it amounts to covering ${\mathbb{P}}^n_{{\overline{K}}}$ by the bounded sets $E_i$ (notation ) and thus allows to consider affine subvarieties when needed.
$(b)$ In a topology, a set containing a neighbourhood is one as well : here, we will define everything by being out of a tubular neighbourhood, therefore allowing sets too large would be too restrictive. One can think about this definition as a family of neighbourhoods being one by one not too large but not too small, and uniformly so in the places.
$(c)$ If $Y$ is an ample divisor of $X$ and ${{\mathcal V}}$ is a tubular neighbourhood of $Y$, one easily sees that if $P \notin {{\mathcal V}}$ then $h(\psi(P))$ is bounded for some embedding $\psi$ associated to $Y$, from which we get the finiteness of the set of points $P$ of bounded degree outside of ${{\mathcal V}}$. This illustrates why such an assumption is only really relevant when $Y$ is of small dimension.
$(d)$ A tubular neighbourhood of $Y$ can also be seen as a family of open subsets defined by bounding strictly a global arithmetic distance function to $Y$ (see [@Vojtadiophapp], paragraph 2.5).
We have drawn below three different pictures of tubular neighbourhoods at the usual archimedean norm. One consider ${\mathbb{P}}^2({{\mathbb{R}}})$ with coordinates $x,y,z$, the affine open subset $U_z$ defined by $z \neq 0$, and $E_x,E_y,E_z$ the respective sets such that $|x|,|y|,|z| = \max (|x|,|y|,|z|)$. These different tubular neighbourhoods are drawn in $U_z$, and the contribution of the different parts $E_x$, $E_y$ and $E_z$ is made clear.
The boundary of the neighbourhood is made up with segments between the indicated points
The boundary is made up with arcs of hyperbola between the indicated points.
The notion of tubular neighbourhood does not seem very intrinsic, but as the proposition below shows, it actually is.
\[propcarvoistub\]
Let $X$ be a projective variety over $K$ and $Y$ a closed $K$-subscheme of $X$.
A family ${{\mathcal V}}=(V_w)_{w \in M_{{\overline{K}}}}$ is included in a tubular neighbourhood of $Y$ in $X$ if and only if for every affine open subset $U$ of $X$, every $E \subset U({\overline{K}}) \times M_{{\overline{K}}}$ which is $M_K$-bounded in $U$, and every regular function $f \in {\overline{K}}[U]$ such that $f_{|Y \cap U} = 0$, there is an $M_K$-constant ${{\mathcal C}}$ such that $$\forall (P,w) \in E, \quad P \in V_w \Rightarrow \log |f(P)|_w < c_v$$ (intuitively, this means that every function vanishing on $Y$ is “$M_K$-small” on ${{\mathcal V}}$).
One can also give a criterion for containing a tubular neighbourhood (using generators in ${\overline{K}}[U]$ of the ideal defining $Y \cap U$). Together, these imply that the tubular neighbourhoods made up by an embedding of $X$ are essentially the same. Indeed, one can prove that for two different projective embeddings of $X$, a tubular neighbourhood as defined by the first one can be an intermediary between two tubular neighbourhoods as defined by the second embedding.
First, a family ${{\mathcal V}}$ satisfying this property is included in a tubular neighbourhood. Indeed, if we choose $g_1, \cdots, g_s$ homogeneous generators of the ideal defining $Y$ for some embedding of $X$ in ${\mathbb{P}}^n_K$, for every $i \in \{0, \cdots, n\}$, consider (using notations , and ) the $M_K$-bounded set $E_i$ and the regular functions $g_j \circ \varphi_i$ on $U_i$, $1 \leq j \leq s$. By hypothesis, (taking the maximum of all the $M_K$-constants for $0 \leq i \leq n, 1 \leq j \leq s$), there is an $M_K$-constant $(c_v)_{v \in M_K}$ such that for every $w \in M_{{\overline{K}}}$, $$\forall j \in \{1, \cdots, s \}, \forall i \in \{0, \cdots, n\}, \forall P \in E_{i,w}, \textrm{ if } \, P \in V_w, \quad \log |g_j \circ \varphi_i(P)|_w < c_v$$ because $g_j \circ \varphi_i = 0$ on $Y \cap U_i$ by construction and the $\varphi_i(P)$ are normalised coordinates for $P \in E_{i,w}$. Hence, ${{\mathcal V}}$ is included in the tubular neighbourhood of $Y$ in $X$ associated to ${{\mathcal C}}$ and the generators $g_1, \cdots, g_s$.
It now remains to prove that any tubular neighbourhood of $Y$ satisfies this characterisation, and we will do so (with the same notations as Definition \[defvoistub\]) for the tubular neighbourhood defined by a given embedding $X \subset {\mathbb{P}}^n_K$, homogeneous equations $g_1, \cdots, g_s$ defining $Y$ in ${\mathbb{P}}^n_K$ and some $M_K$-constant ${{\mathcal C}}_0 = (c_{0,v})_{v \in M_K}$ (we will use multiple $M_K$-constants, hence the numbering).
Let us fix an affine open subset $U$ of $X$ and $E$ an $M_K$-bounded set on $U$. We can cover $U$ by principal affine open subsets of $X$, more precisely we can write $$U = \bigcup_{h \in {{\mathcal F}}} U_h$$ where $h$ runs through a finite family ${{\mathcal F}}$ of nonzero homogeneous polynomials of ${\overline{K}}[X_0, \cdots, X_n]$ and $$U_h = \{ P \in X \, | \, h(P) \neq 0 \}.$$ For every such $h$, the regular functions on $U_h$ are the $s/h^k$ where $s$ is homogeneous on ${\overline{K}}[X_0, \cdots, X_n]$ of degree $k \cdot \deg(h)$ (as $X$ is a closed subvariety of ${\mathbb{P}}^n$, the only subtlety is that identical regular functions on $U_h$ can come from different fractions $s/h^k$ but this will not matter in the following).
By Lemma \[lemMKbornerecouvrement\], there is a cover $E = \cup_{h \in {{\mathcal F}}} E_h$ such that every $E_h$ is $M_K$-bounded on $U_h$. This implies that for any $i \in \{0, \cdots, n\}$, the functions $x_i^{\deg (h)} / h \in {\overline{K}}[U_f]$ are $M_K$-bounded on $E_h$, therefore we have an $M_K$-constant ${{\mathcal C}}_{1}$ such that for all $(P,w) \in E_h$ with coordinate $x_0, \cdots, x_n$, $$\label{eqfoncintercoordonnees}
\log \| x_P \|_w \leq c_{1,v} + \frac{1}{\deg(h)} \log |h(x_P)|_w.$$ Now, let $f$ be a regular function on ${\overline{K}}[U]$ such that $f_{|Y \cap U} = 0$. For every $h \in {{\mathcal F}}$, we can write $f_{|U_h}=s/h^k$ for some homogeneous $s \in {\overline{K}}[X_0, \cdots, X_n]$, therefore as a homogeneous function on $X$, one has $h \cdot s = 0$ on $Y$ (it already cancels on $Y \cap U$, and outside $U$ by multiplication by $f$). Hence, we can write $$f_{|U_h} = \sum_{j=1}^s \frac{a_{j,h} g_{j}}{h^{k_j}}$$ with the $a_{j,h}$ homogeneous on ${\overline{K}}[X_0, \cdots, X_n]$ of degree $k_j \deg(h) - \deg(g_j)$. Now, bounding the coefficients of all the $a_{j,h}$ (and the number of monomials in the archimedean case), we get an $M_K$-constant ${{\mathcal C}}_2$ such that for every $P \in {\mathbb{P}}^n({\overline{K}})$, $$\log |a_{j,h} (x_P)|_w \leq c_{2,v} + \deg(a_{j,h}) \cdot \log \|x_P \|_w.$$ Combining this inequality with and , we get that for every $h \in {{\mathcal F}}$, every $(P,w) \in E_h$ and every $j \in \{1, \cdots, s\}$ : $$\textrm{ if } P \in V_w, \quad \log \left| \frac{a_{j,h} g_j}{h^{k_j}}(P) \right|_w < c_{0,v} + c_{2,v} + k_j c_{1,v}$$ which after summation on $j \in \{1, \cdots, s \}$ and choice of $h$ such that $(P,w) \in E_h$ proves the result.
Key results {#sectionresultatscles}
===========
We will now prove the key result for Runge’s method, as a consequence of the Nullstellensatz. We mainly use the projective case in the rest of the paper but the affine case is both necessary for its proof and enlightening for the method we use.
\[propcle\]
$(a)$ (Affine version)
Let $U$ be an affine variety over $K$ and $Y_1, \cdots, Y_r$ closed subsets of $U$ defined over $K$, of intersection $Y$. For every $\ell \in \{1, \cdots, r\}$, define $g_{\ell,1}, \cdots g_{\ell,s_\ell}$ generators of the ideal of definition of $Y_\ell$ in $K[U]$, and $h_1, \cdots, h_s$ generators of the ideal of definition of $Y$ in $K[U]$. For every $M_K$-bounded set $E$ of $U$ and every $M_K$-constant ${{\mathcal C}}_0$, there is an $M_K$-constant ${{\mathcal C}}$ such that for every $(P,w) \in E$ with $w$ above $v \in M_K$, one has the following dichotomy : $$\label{eqdichoaff}
\max_{\substack{1 \leq \ell \leq r \\ 1 \leq j \leq s_i }} \log |g_{\ell,j} (P)|_w \geq c_v \quad \textrm{or} \quad \max_{1 \leq j \leq s} \log |h_j (P)|_w < c_{0,v}.$$
$(b)$ (Projective version)
Let $X$ be a normal projective variety over $K$ and $\phi_1, \cdots, \phi_r \in K(X)$. Let $Y$ be the closed subset of $X$ defined as the intersection of the supports of the (Weil) divisors of poles of the $\phi_i$. For every tubular neighbourhood ${{\mathcal V}}$ of $Y$ (Definition \[defvoistub\]), there is an $M_K$-constant ${{\mathcal C}}$ depending on ${{\mathcal V}}$ such that for every $w \in M_{{\overline{K}}}$ (above $v \in M_K)$ and every $P \in X({\overline{K}})$, $$\label{eqdichoproj}
\min_{1 \leq \ell \leq r} \log |\phi_\ell (P)|_w \leq c_v \quad \textrm{or} \quad P \in V_w.$$
This result has an immediate corollary when $Y=\emptyset$: Lemma 5 of [@Levin08], restated below.
\[corpasdepolecommun\]
Let $X$ be a normal projective variety over $K$ and $\phi_1, \cdots, \phi_r \in K(X)$ having globally no common pole. Then, there is an $M_K$-constant ${{\mathcal C}}$ such that for every $w \in M_{{\overline{K}}}$ (above $v \in M_K)$ and every $P \in X({\overline{K}})$, $$\label{eqpasdepolecommun}
\min_{1 \leq \ell \leq r} \log |\phi_\ell (P|_w \leq c_v.$$
$(a)$ As will become clear in the proof, part $(b)$ is actually part $(a)$ applied to a good cover of $X$ by $M_K$-bounded subsets of affine open subsets of $X$ (inspired by the natural example of Remark \[remdefMKconstantes\] $(b)$).
$(b)$ Besides the fact that the results must be uniform in the places (hence the $M_K$-constants), the principle of $(a)$ and $(b)$ is simple. For $(a)$, we would like to say that if a point $P$ is sufficiently close to $Y_1, \cdots, Y_r$ (i.e. the first part of the dichotomy is not satisfied) it must be close to a point of intersection of the $Y_i$, hence the generators of the intersection should be small at $P$ (second part of the dichotomy). This is not true in the affine case, taking for example the hyperbola and the real axis in ${{\mathbb{A}}}^2$, infinitely close but disjoint (hence the necessity of taking a bounded set $E$ to compactify the situation), but it works in the projective case because the closed sets are then compact.
$(c)$ Corollary \[corpasdepolecommun\] is the key for Runge’s method in the case of curves in section \[sectionRungecourbes\]. Notice that Lemma 5 of [@Levin08] assumed $X$ smooth, but the proof is actually exactly the same for $X$ normal. Moreover, the argument below follows the structure of Levin’s proof.
$(d)$ If we replace $Y$ by $Y' \supset Y$ and ${{\mathcal V}}$ by a tubular neighbourhood ${{\mathcal V}}'$ of $Y'$, the result remains true with the same proof, which is not surprising because tubular neighbourhood of $Y'$ are larger than tubular neighbourhoods of $Y$.
$(a)$ By the Nullstellensatz applied on $K[U]$ to the $Y_\ell \quad (1 \leq \ell \leq r)$ and $Y$, by hypothesis, for some power $p \in {{\mathbb{N}}}_{>0}$, there are regular functions $f_{\ell,j,m} \in K[U]$ such that for every $m \in \{1, \cdots, s\}$, $$\sum_{\substack{1 \leq \ell \leq r \\ 1 \leq j \leq s_\ell}} g_{\ell,j} f_{\ell,j,m} = h_m^p.$$ As $E$ is $M_K$-bounded on $U$, all the $f_{\ell,j,m}$ are $M_K$-bounded on $E$ hence there is an auxiliary $M_K$-constant ${{\mathcal C}}_1$ such that for all $P \in E$, $$\max_{\substack{1 \leq \ell \leq r \\ 1 \leq j \leq s_\ell \\ 1 \leq m \leq s}} \log |f_{\ell,j,m} (P)|_w \leq c_{1,v},$$ therefore $$|h_m(P)^p|_w = \left| \sum_{\substack{1 \leq \ell \leq r \\ 1 \leq j \leq s_\ell}} g_{\ell,j} (P) f_{\ell,j,m} (P) \right|_w \leq N^{\delta_v} e^{c_{1,v}} \max_{\substack{1 \leq \ell \leq r \\ 1 \leq j \leq s_\ell }} |g_{\ell,j} (P)|_w$$ where $\delta_v$ is 1 if $v$ is archimedean and 0 otherwise, and $N$ the total number of generators $g_{\ell,j}$. For fixed $w$ and $P$, either $\log |h_m(P)|_w < c_{0,v}$ for all $m \in \{ 1, \cdots, s\}$ (second part of dichotomy ), or the above inequality applied to some $m \in \{1, \cdots,s\}$ gives $$p \cdot c_{0,v} \leq \delta_v \log(N) + c_{1,v} + \max_{\substack{1 \leq \ell \leq r \\ 1 \leq j \leq s_\ell }} \log |g_{\ell,j} (P)|_w,$$ which is equivalent to $$\max_{\substack{1 \leq \ell \leq r \\ 1 \leq j \leq s_\ell }} \log |g_{\ell,j} (P)|_w \geq \delta_v \log(N) + c_{1,v} - p \cdot c_{0,v},$$ and taking the $M_K$-constant defined by $c_v := c_{1,v} + \delta_v \log(N) - p \cdot c_{0,v}$ for every $v \in M_K$ gives exactly the first part of dichotomy .
$(b)$ We consider $X$ as embedded in some ${\mathbb{P}}^n_K$ so that ${{\mathcal V}}$ is exactly the tubular neighbourhood of $Y$ in $X$ associated to an $M_K$-constant ${{\mathcal C}}_0$ and generators $g_1, \cdots, g_s$ for this embedding. We will use again the notations , and . In particular we define $X_i := X \cap U_i$ for every $i \in \{0, \cdots, n\}$. The following argument is designed to make $Y$ appear as a common zero locus of regular functions built with the $\phi_\ell$.
For every $\ell \in \{1, \cdots, r\}$, let $D_\ell$ be the positive Weil divisor of zeroes of $\phi_\ell$ on $X$. For every $i \in \{0, \cdots, n\}$, let $I_{\ell,i}$ be the ideal of $K[X_i]$ made up with the regular functions $h$ on the affine variety $X_i$ such that ${\operatorname{div}}(h) \geq (D_\ell)_{|X_i}$, and we choose generators $h_{\ell,i,1}, \cdots, h_{\ell,i,j_{\ell,i}}$ of this ideal. The functions $h_{\ell,i,j}/(\phi_\ell)_{|X_i}$ are then regular on $X_i$ and $$\forall j \in \{1, \cdots, j_{\ell,i} \}, \quad {\operatorname{div}}\left( \frac{h_{\ell,i,j}}{(\phi_\ell)_{|X_i}} \right) \geq (\phi_{\ell,i})_\infty$$ (the divisor of poles of $\phi_\ell$ on $X_i$). By construction of $I_{\ell,i}$, the minimum (prime Weil divisor by prime Weil divisor) of the ${\operatorname{div}}(h_{\ell,i,j})$ is exactly $(D_\ell)_{|X_i}$ : indeed, for every finite family of distinct prime Weil divisors $D'_1, \cdots, D'_s, D''$ on $X_i$, there is a uniformizer $h$ for $D''$ of order 0 for each of the $D'_k$, otherwise the prime ideal associated to $D''$ in $X_i$ would be included in the finite union of the others. This allows to build for every prime divisor $D'$ of $X_i$ not in the support of $(D_\ell)_{|X_i}$ a function $h \in I_{\ell,i}$ of order $0$ along $D'$ (and of the good order for every $D'$ in the support of $(D_\ell)_{|X_i}$. Consequently, the minimum of the divisors of the $h_{\ell,i,j} / (\phi_{\ell})_{|X_i}$, being naturally the minimum of the divisors of the $h / (\phi_{\ell})_{|X_i} \, \, ( h \in K[X_i])$, is exactly $(\phi_{\ell,i})_\infty$.
Thus, by definition of $Y$, for fixed $i$, the set of commmon zeroes of the regular functions $h_{\ell,i,j} / (\phi_\ell)_{|X_i} \, (1 \leq \ell \leq r, 1 \leq j \leq j_{\ell,i})$ on $X_i$ is $Y \cap X_i$, so they generate a power of the ideal of definition of $Y \cap X_i$. We apply part $(a)$ of this Proposition to the $h_{\ell,i,j} / (\phi_\ell)_{|X_i} \, (1 \leq \ell \leq r, 1 \leq j \leq j_{\ell,i})$, the $g_j \circ \varphi_i \, (1 \leq j \leq s)$ and the $M_K$-constant ${{\mathcal C}}_0$, which gives us an $M_K$-constant ${{\mathcal C}}'_i$ and the following dichotomy on $X_i$ for every $(P,w) \in E_i$ : $$\max_{\substack{1 \leq \ell \leq r \\ 1 \leq j \leq s_i }} \log \left| \frac{h_{\ell,i,j}}{\phi_\ell} (P) \right|_w \geq c'_{i,v} \quad \textrm{or} \quad \max_{1 \leq j \leq s} \log |g_j \circ \varphi_i (P)|_w < c_{0,v}.$$ Now, the $h_{\ell,i,j}$ are regular on $X_i$ hence $M_K$-bounded on $E_i$, therefore there is a second $M_K$-constant ${{\mathcal C}}''_i$ such that for every $(P,w) \in E_i$ : $$\max_{\substack{1 \leq \ell \leq r \\ 1 \leq j \leq s_i }} \log \left| \frac{h_{\ell,i,j}}{\phi_\ell} (P) \right|_w \geq c'_{i,v} \Longrightarrow \min_{1 \leq \ell \leq r} \log |\phi_\ell (P)|_w \leq c''_{i,v}.$$ Taking ${{\mathcal C}}$ as the maximum of the $M_K$-constants ${{\mathcal C}}''_i, 0 \leq i \leq n$, for every $(P,w) \in X({\overline{K}}) \times M_{{\overline{K}}}$, we choose $i$ such that $(P,w) \in E_i$ and then we have the dichotomy by definition of the tubular neighbourhood $V_w$.
To finish this section, we will give the explicit link between integral points on a projective scheme (relatively to a divisor) and integral points relatively to rational functions on the scheme. In particular, this catches up with the definition of integral points of section 2 of [@Levin08].
\[propreductionamplegros\]
Let ${{\mathcal X}}$ be a normal projective scheme over ${{\mathcal O}}_{K,S}$.
$(a)$ If ${{\mathcal Y}}$ is an effective Cartier divisor on ${{\mathcal X}}$ such that ${{\mathcal Y}}_K$ is an ample (Cartier) divisor of ${{\mathcal X}}_K$, there is a projective embedding $\psi : {{\mathcal X}}_K \rightarrow {\mathbb{P}}^n_K$ and an $M_K$-constant ${{\mathcal C}}$ such that
- The pullback by $\psi$ of the hyperplane of equation $x_0=0$ in ${\mathbb{P}}^n_K$ is ${{\mathcal Y}}_K$.
- For any finite extension $L$ of $K$ and any $w \in M_L$ not above $S$,
$$\label{eqlienreducschemasfonctionsample}
\forall P \in ({{\mathcal X}}\backslash {{\mathcal Y}}) ({{\mathcal O}}_{L,w}), \quad \log \|x_{\psi(P)} \|_w \leq c_v + \log |x_{\psi(P),0}|_w .$$
This amounts to say that if the coordinates by $\psi$ of such a $P$ are normalised so that the first one is 1, all the other ones have $w$-norm bounded by $e^{c_v}$.
$(b)$ If ${{\mathcal Y}}$ is an effective Cartier divisor on ${{\mathcal X}}$ such that ${{\mathcal Y}}_K$ is a big (Cartier) divisor of ${{\mathcal X}}_K$, there is a strict Zariski closed subset $Z_K$ of ${{\mathcal X}}_K$, a morphism $\psi : {{\mathcal X}}_K \backslash {{\mathcal Y}}_K \rightarrow {\mathbb{P}}^n_K$ which induces a closed immersion of ${{\mathcal X}}_K \backslash Z_K$ and an $M_K$-constant ${{\mathcal C}}$ such that:
- The pullback by $\psi$ of the hyperplane of equation $x_0=0$ in ${\mathbb{P}}^n_K$ is contained in ${{\mathcal Y}}_K \cup Z_K$.
- For any finite extension $L$ of $K$ and any $w \in M_L$ not above $S$, formula holds.
\[rempropamplegros\]
$(a)$ This Proposition is formulated to avoid the use of local heights, but the idea is exactly that under the hypotheses above, the fact that $P \in ({{\mathcal X}}\backslash {{\mathcal Y}}) ({{\mathcal O}}_{L,w})$ implies that the local height at $w$ of $P$ for the divisor ${{\mathcal Y}}$ is bounded.
$(b)$ The hypotheses on ampleness (or “bigness”) are only necessary at the generic fiber. If we considered ${{\mathcal Y}}$ ample on ${{\mathcal X}}$, it would give us a result with the zero $M_K$-constant (using an embedding over ${{\mathcal O}}_{K,S}$ given by ${{\mathcal Y}}$), and an equivalence, but this is not crucial here. Once again, the auxiliary functions replace the need for a complete understanding of what happens at the finite places.
$(c)$ The only difference between ample and big cases is hidden in the function $\psi$ : in the big case, the formula still holds but does not say much for points belonging in $Z_K$ because the morphism $\psi$ is not an embedding there.
$(a)$ As ${{\mathcal Y}}_K$ is ample and effective, there is a projective embedding $\psi : {{\mathcal X}}_K \rightarrow {\mathbb{P}}^n$ such that the support of the divisor ${{\mathcal Y}}_K$ is exactly the inverse image of the hyperplane $x_0=0$ by $\psi$. Let us fix such an embedding and consider for every $i \in \{1, \cdots, n\}$ the coordinate functions $\phi_i := (x_i/x_0) \circ \psi$ in $ K({{\mathcal X}}_K)$, whose poles are contained in ${{\mathcal Y}}_K$ by construction. Now, we choose a tubular neighbourhood ${{\mathcal V}}$ of ${{\mathcal Y}}_K$ defined by an embedding of ${{\mathcal X}}$ in some projective ${\mathbb{P}}^m_{{{\mathcal O}}_{K,S}}$ (which can be completely unrelated to $\psi$), homogeneous generators $g_1, \cdots, g_s$ of the ideal of definition of ${{\mathcal Y}}$ in ${\mathbb{P}}^m_{{{\mathcal O}}_{K,S}}$ and the zero $M_K$-constant. By Proposition \[propcle\] $(b)$ applied to ${{\mathcal V}}$ and $\phi_j$, we obtain an $M_K$-constant ${{\mathcal C}}_j$ such that for every finite extension $L$ of $K$ and every $w \in M_L$ (with the notations and ), $$\forall P \in {{\mathcal X}}(L) , \qquad \log |\phi_j(P)|_w \leq c_{j,v} \quad {\textrm{or}} \quad P \in V_w.$$ By construction of ${{\mathcal V}}$ and Proposition \[proplienreductionpointssvaluation\], if $w$ is not above a place of $S$ and $P \in ({{\mathcal X}}\backslash {{\mathcal Y}})({{\mathcal O}}_{L,w})$, we necessarily have $\log |\phi_j(P)|_w \leq c_{j,v}$. Taking the maximum of the $M_K$-constants ${{\mathcal C}}_1, \cdots, {{\mathcal C}}_n$, we obtain the Proposition in the ample case.
$(b)$ The proof for big divisors is the same as part $(a)$, except that we can only extend our function $\psi$ to ${{\mathcal X}}_K \backslash Z_K$ for some proper Zariski closed subset $Z_K$ such that outside of this set, $\psi$ is a closed immersion. The coordinate functions $\phi_i \in K({{\mathcal X}}_K)$, similarly defined, also have poles contained in ${{\mathcal Y}}_K$. Applying the same arguments as in part $(a)$ for points $P \in {{\mathcal X}}(L)$, we obtain the same result.
The case of curves revisited {#sectionRungecourbes}
============================
In this section, we reprove the generalisation of an old Runge theorem [@Runge1887] obtained by Bombieri ([@BombieridecompWeil] p. 305, also rewritten as Theorem 9.6.6 in [@BombieriGubler]), following an idea exposed by Bilu in an unpublished note and mentioned for the case $K={{\mathbb{Q}}}$ by [@SchoofCatalan] (Chapter 5). The aim of this section is therefore to give a general understanding of this idea (quite different from the original proof of Bombieri), as well as explain how it actually gives a *method* to bound heights of integral points on curves.
It is also a good start to understand how the intuition behind this result can be generalised to higher dimension, which will be done in the next section.
\[propBombieri\] Let $C$ be a smooth projective algebraic curve defined over a number field $K$ and $\phi \in K(C)$ not constant.
For any finite extension $L/K$, let $r_L$ be the number of orbits of the natural action of ${\operatorname{Gal}}(\overline{L}/L)$ over the poles of $\phi$. For any set of places $S_L$ of $L$ containing $M_L^{\infty}$, we say that $(L,S_L)$ satisfies the **Runge condition** if $$\label{eqconditionRunge}
|S_L|<r_L.$$
Then, the reunion $$\label{eqreunionpointsRungecondition}
\bigcup_{\substack{(L,S_L)}} \left\{ P \in C(L) \, | \, \phi(P) \in {{\mathcal O}}_{L,S_L} \right\},$$ where $(L,S_L)$ runs through all the pairs satisfying Runge condition, is **finite** and can be explicitly bounded in terms of the height $h \circ \phi$.
As a concrete example, consider the modular curve $X_0(p)$ for $p$ prime and the $j$-invariant function. This curve is defined over ${{\mathbb{Q}}}$ and $j$ has two rational poles (which are the cusps of $X_0(p)$), hence $r_L=2$ for any choice of $L$, and we need to ensure $|M_L^{\infty}| \leq |S_L|< 2$. The only possibilities satisfying Runge condition are thus imaginary quadratic fields $L$ with $S_L = \{ | \cdot |_{\infty} \}$.
We thus proved in [@LeFourn1] that for any imaginary quadratic field $L$ and any $P \in X_0(p)(L)$ such that $j(P) \in {{\mathcal O}}_L$, one has $$\log |j(P)| \leq 2 \pi \sqrt{p} + 6 \log (p) + 8.$$ The method for general modular curves is carried out in [@BiluParent09] and gives explicit estimates on the height for integral points satisfying Runge condition. This article uses the theory of modular units and implicitly the same proof of Bombieri’s result as the one we expose below.
\[remRungecourbes\]
$(a)$ The claim of an explicit bound deserves a clarification : it can actually be made explicit when one knows well enough the auxiliary functions involved in the proof below (which is possible in many cases, e.g. for modular curves thanks to the modular units). Furthermore, even as the theoretical proof makes use of $M_K$-constants and results of section \[sectionresultatscles\], they are frequently implicit in pratical cases.
$(b)$ Despite the convoluted formulation of the proof below and the many auxiliary functions to obtain the full result, its principle is as descrbibed in the Introduction. It also gives the framework to apply Runge’s method to a given couple $(C,\phi)$
We fix $K'$ a finite Galois extension of $K$ on which every pole of $\phi$ is defined. For any two distinct poles $Q,Q'$ of $\phi$, we choose by Riemann-Roch theorem a function $g_{Q,Q'} \in K'(C)$ whose only pole is $Q$ and vanishing at $Q'$. For every point $P$ of $C({\overline{K}})$ which is not a pole of $\phi$, one has ${\operatorname{ord}}_P (g_{Q,Q'}) \geq 0$ thus $g_{Q,Q'}$ belongs to the intersection of the discrete valuation rings of ${\overline{K}}(C)$ containing $\phi$ and ${\overline{K}}$ ([@Hartshorne], proof of Lemma I.6.5), which is exactly the integral closure of $K[\phi]$ in ${\overline{K}}(C)$ ([@AtiyahMacDonald], Corollary 5.22). Hence, the function $g_{Q,Q'}$ is integral on $K[\phi]$ and up to multiplication by some nonzero integer, we can and will assume it is integral on ${{\mathcal O}}_K[\phi]$.
For any fixed finite extension $L$ of $K$ included in ${\overline{K}}$, we define $f_{Q,Q',L} \in L(C)$ the product of the conjugates of $g_{Q,Q'}$ by ${\operatorname{Gal}}(\overline{L}/L)$. If $Q$ and $Q'$ belong to distinct orbits of poles for ${\operatorname{Gal}}(\overline{L}/L)$, the function $f_{Q,Q',L}$ has for only poles the orbit of poles of $Q$ by ${\operatorname{Gal}}({\overline{K}}/L)$ and cancels at the poles of $\phi$ in the orbit of $Q'$ by ${\operatorname{Gal}}({\overline{K}}/L)$ . Notice that we thus built only finitely many different functions (even with $L$ running through all finite extensions of $K$) because each $g_{Q,Q'}$ only has finitely many conjugates in ${\operatorname{Gal}}(K'/K)$.
Now, let ${{\mathcal O}}_1, \cdots, {{\mathcal O}}_{r_L}$ be the orbits of poles of $\phi$ and denote for any $i \in \{1, \cdots, r_L\}$ by $f_{i,L}$ a product of $f_{Q_i, Q'_j,L}$ where $Q_i \in {{\mathcal O}}_i$ and $Q'_j$ runs through representatives of the orbits (except ${{\mathcal O}}_i$). Again, there is a finite number of possible choices, and we obtain a function $f_{i,L} \in L(C)$ having for only poles the orbit ${{\mathcal O}}_i$ and vanishing at all the other poles of $\phi$. By our construction of the $g_{Q,Q'}$ and $f_{i,L}$, we can and do choose $n \in {{\mathbb{N}}}_{\geq 1}$ such that for every $i \in \{ 1, \cdots, r_L \}$, $\phi f_{i,L}^n$ has exactly as poles the points of ${{\mathcal O}}_i$ and is integral over ${{\mathcal O}}_K[\phi]$. This implies that for any finite place $w \in M_L$, if $|\phi(P)|_w \leq 1$ then $|f_{i,L} (P)|_w \leq 1$, but we also need such a result for archimedean places. To do this, we apply Corollary \[corpasdepolecommun\] to $f_{i,L}/\phi^k$ and $f_{i,L}$ (for any $i$) for some $k$ such that $f_{i,L}/\phi^k$ does not have poles at ${{\mathcal O}}_i$, and take the maximum of the induced $M_K$-constants (Definition \[defMKconstante\]) for any $L$ and $1 \leq i \leq r_L$. This gives an $M_K$-constant ${{\mathcal C}}_0$ independant of $L$ such that $$\forall i \in \{1, \cdots, r_L\}, \forall w \in M_{{\overline{K}}}, \forall P \in C({\overline{K}}), \log \min \left( \left| \frac{f_{i,L}}{\phi^k} (P)\right|_w, |f_{i,L}(P)|_w \right) \leq c_{0,v} \quad (w|v \in M_K).$$ In particular, the result interesting us in this case is that $$\label{eqmajofilplacesarchi}
\forall i \in \{1, \cdots, r_L\}, \forall w \in M_{{\overline{K}}}, \forall P \in C({\overline{K}}), |\phi(P)|_w \leq 1 \Rightarrow \log |f_{i,L} (P)|_w \leq c_{0,v},$$ and we can assume $c_{0,v}$ is 0 for any finite place $v$ by integrality of the $f_{i,L}$ over ${{\mathcal O}}_K[\phi]$. As the sets of poles of the $f_{i,L}$ are mutually disjoint, we reapply Corollary \[corpasdepolecommun\] for every pair $(\phi f_{i,L}^n, \phi f_{j,L}^n)$ with $1 \leq i < j \leq r_L$, which again by taking the maximum of the induced $M_K$-constants for all the possible combinations (Definition \[defMKconstante\]) gives an $M_K$-constant ${{\mathcal C}}_1$ such that for every $v \in M_K$ and every $(P,w) \in C({\overline{K}}) \times M_{{\overline{K}}}$ with $w|v$, the inequality $$\label{eqineqsaufpourunefonc}
\log |(\phi \cdot f_{i,L}^n) (P)|_w \leq c_{1,v}$$ is true for all indices $i$ except at most one (depending of the choice of $P$ and $w$).
Let us now suppose that $(L,S_L)$ is a pair satisfying Runge condition and $P \in C(L)$ with $\phi(P) \in {{\mathcal O}}_{L,S_L}$. By integrality on ${{\mathcal O}}_K[\phi]$, for every $i \in \{1, \cdots, r_L \}$, $|f_{i,L}(P)|_w \leq 1$ for every place $w \in M_L \backslash S_L$. For every place $w \in S_L$, there is at most one index $i$ not satisying hence by Runge condition and pigeon-hole principle, there remains one index $i$ (depending on $P$) such that $$\label{eqmajophifiL}
\forall w \in M_L, \quad \log |\phi(P) f_{i,L}^n (P)|_w \leq c_{1,v}.$$ With and , we have obtained all the auxiliary results we need to finish the proof. By the product formula, $$\begin{aligned}
0 & = & \sum_{w \in M_L} n_w \log |f_{i,L} (P)|_w \\
& = & \sum_{\substack{w \in M_L \\ |\phi(P)|_w >1 }} n_w \log |f_{i,L} (P)|_w + \sum_{\substack{w \in M_L^{\infty} \\ |\phi(P)|_w \leq 1}} n_w \log |f_{i,L} (P)|_w + \sum_{\substack{w \in M_L \! \! \backslash M_L^{\infty} \\ |\phi(P)|_w \leq 1}} n_w \log |f_{i,L} (P)|_w.\end{aligned}$$ Here, the first sum on the right side will be linked to the height $h \circ \phi$ and the third sum is negative by integrality of the $f_{i,L}$, so we only have to bound the second sum. From and , we obtain $$\sum_{\substack{w \in M_L^{\infty} \\ |\phi(P)|_w \leq 1}} n_w \log |f_{i,L} (P)|_w \leq \sum_{\substack{w \in M_L^{\infty} \\ |\phi(P)|_w \leq 1}} n_w c_{0,v} \leq [L:K] \sum_{v \in M_K^{\infty}} n_v c_{0,v}.$$ On another side, by (and again), we have $$\begin{aligned}
n \cdot \sum_{\substack{w \in M_L \\ |\phi(P)|_w >1 }} n_w \log |f_{i,L} (P)|_w & = & \sum_{\substack{w \in M_L \\ |\phi(P)|_w >1 }} n_w \log |\phi f_{i,L}^n(P)|_w - \sum_{\substack{w \in M_L \\ |\phi(P)|_w >1 }} n_w \log |\phi(P)|_w \\
& \leq & \left([L:K] \sum_{v \in M_K} n_v c_{1,v} \right) - [L:{{\mathbb{Q}}}] h(\phi(P)). \end{aligned}$$ Hence, we obtain $$\begin{aligned}
0 & \leq & [L:K] \sum_{v \in M_K} n_v c_{1,v} - [L:{{\mathbb{Q}}}] h(\phi(P)) + [L:K] n \sum_{v \in M_K^{\infty}} n_v c_{0,v},\end{aligned}$$ which is equivalent to $$h (\phi(P)) \leq \frac{1}{[K : {{\mathbb{Q}}}]}\sum_{v \in M_K} n_v (c_{1,v} + n c_{0,v}).$$ We thus obtained a bound on $h (\phi(P))$ independent on the choice of $(L,S_L)$ satisfying the Runge condition, and together with the bound on the degree $$[L : {{\mathbb{Q}}}] \leq 2 |S_L| < 2 r_L \leq 2 r,$$ we get the finiteness.
The main result : tubular Runge theorem {#sectiontubularRunge}
=======================================
We will now present our version of Runge theorem with tubular neighbourhoods, which generalises Theorem 4 $(b)$ and $(c)$ of [@Levin08]. As its complete formulation is quite lengthy, we indicated the different hypotheses by the letter $H$ and the results by the letter $R$ to simplify the explanation of all parts afterwards. The key condition for integral points generalising Runge condition of Proposition \[propBombieri\] is indicated by the letters TRC.
We recall that the crucial notion of tubular neighbourhood is explained in Definitions \[defvoistub\] and \[defhorsdunvoistub\], and we advise the reader to look at the simplified version of this theorem stated in the Introduction to get more insight if necessary.
\[thmRungetubulaire\]
**(H0)** Let $K$ be a number field, $S_0$ a set of places of $K$ containing $M_K^{\infty}$ and ${{\mathcal O}}$ the integral closure of ${{\mathcal O}}_{K,S_0}$ in some finite Galois extension $K'$ of $K$.
**(H1)** Let ${{\mathcal X}}$ be a normal projective scheme over ${{\mathcal O}}_{K,S_0}$ and $D_1, \cdots, D_r$ be effective Cartier divisors on ${{\mathcal X}}_{{\mathcal O}}= {{\mathcal X}}\times_{{{\mathcal O}}_{K,S_0}} {{\mathcal O}}$ such that $D_{{\mathcal O}}= \bigcup_{i=1}^r D_i$ is the scalar extension to ${{\mathcal O}}$ of some Cartier divisor $D$ on ${{\mathcal X}}$, and that ${\operatorname{Gal}}(K'/K)$ permutes the generic fibers $(D_i)_{K'}$. For every extension $L/K$, we denote by $r_L$ the number of orbits of $(D_1)_{K'}, \cdots, (D_r)_{K'}$ for the action of ${\operatorname{Gal}}(K'L/L)$.
**(H2)** Let $Y$ be a closed sub-$K$-scheme of ${{\mathcal X}}_K$ and ${{\mathcal V}}$ be a tubular neighbourhood of $Y$ in ${{\mathcal X}}_K$. Let $m_Y \in {{\mathbb{N}}}$ be the minimal number such that the intersection of any $(m_Y+1)$ of the divisors $(D_i)_{K'}$ amongst the $r$ possible ones is included in $Y_{K'}$.
**(TRC)** The **tubular Runge condition** for a pair $(L,S_L)$, where $L/K$ is finite and $S_L$ contains all the places above $S_0$, is $$m_Y |S_L| < r_L.$$
Under these hypotheses and notations, the results are the following :
**(R1)** If $(D_1)_{K'}, \cdots , (D_r)_{K'}$ are ample divisors, the set $$\label{eqensfiniRungetubample}
\bigcup_{(L,S_L)} \{P \in ({{\mathcal X}}\backslash D) ({{\mathcal O}}_{L,S_L}) \, | \, P \notin {{\mathcal V}}\},$$ where $(L,S_L)$ goes through all the pairs satisfying the tubular Runge condition, is **finite**.
**(R2)** If $(D_1)_{K'}, \cdots, (D_r)_{K'}$ are big divisors, there exists a proper closed subset $Z_{K'}$ of ${{\mathcal X}}_{K'}$ such that the set $$\left( \bigcup_{(L,S_L)} \{P \in ({{\mathcal X}}\backslash D) ({{\mathcal O}}_{L,S_L}) \, | \, P \notin {{\mathcal V}}\} \right) \backslash Z_{K'} ({\overline{K}}),$$ where $(L,S_L)$ goes through all the pairs satisfying the tubular Runge condition, is **finite**.
We separated the comments about Theorem \[thmRungetubulaire\] in two remarks below : the first one explains its hypotheses and results, the second compares it with other theorems.
\[remRungetubulaire\]
$(a)$ The need for the extensions of scalars to $K'$ and ${{\mathcal O}}$ in ***(H0)*** and ***(H1)*** is the analogue of the fact that the poles of $\phi$ are not necessarily $K$-rational in the case of curves, hence the assumption that the $(D_i)_{K'}$ are all conjugates by ${\operatorname{Gal}}(K'/K)$ and the definition of $r_L$ given in ***(H1)***. It will induce technical additions of the same flavour as the auxiliary functions $f_{Q,Q',L}$ in the proof of Bombieri’s theorem (Proposition \[propBombieri\]).
$(b)$ The motivation for the tubular Runge condition is the following : imitating the principle of proof for curves (Remark \[remRungecourbes\] $(b)$), if $P \in ({{\mathcal X}}\backslash D) ({{\mathcal O}}_{L,S_L})$, we can say that at the places $w$ of $M_L \backslash S_L$, this point is “$w$-adically far” from $D$. Now, the divisors $(D_1)_{K'}, \cdots, (D_r)_{K'}$ can intersect (which does not happen for distinct points on curves), so for $w \in S_L$, this point $P$ can be “$w$-adically close” to many divisors at the same time. More precisely, it can be “$w$-adically close” to at most $m$ such divisors, where $m=m_{\emptyset}$, i.e. the largest number such that there are $m$ divisors among $D_1, \cdots, D_r$ whose set-theoretic intersection is nonempty. This number is also defined in [@Levin08] but we found that for our applications, it often makes Runge condition too strict. Therefore, we allow the use of the closed subset $Y$ in ***(H2)***, and if we assume that our point $P$ is never too close to $Y$ (i.e. $P \notin {{\mathcal V}}$), this $m$ goes down to $m_Y$ by definition. Thus, we only need to take out $m_Y$ divisors by place $w$ in $S_L$, hence the tubular Runge condition $m_Y |S_L|< r_L$. Actually, one can even mix the Runge conditions, i.e. assume that $P$ is close to $Y$ exactly at $s_1$ places, and close from one of the divisors (but not $Y$) at $s_2$ places : following along the lines of the proof below, we obtain finiteness given the Runge condition $s_1 m_{\emptyset} + s_2 m_Y < r_L$.
$(c)$ The last main difference with the case of curves is the assumption of ample or big divisors, respectively in ***(R1)*** and ***(R2)***. In both cases, such an assumption is necessary twice. First, we need it to translate by Proposition \[propreductionamplegros\] the integrality condition on schemes to an integrality expression on auxiliary functions (such as in section 2 of [@Levin08]) to use the machinery of $M_K$-constants and the key result (Proposition \[propcle\]). Then, we need it to ensure that after obtaining a bound on the heights associated to the divisors, it implies finiteness (implicit in Proposition \[propreductionamplegros\], see also Remark \[rempropamplegros\] $(a)$).
\[remcomparaisonCLZetstratification\]
$(a)$ This theorem has some resemblance to Theorem CLZ of [@CorvajaLevinZannier] (where our closed subset $Y$ would be the analogue of the ${{\mathcal Y}}$ in that article), let us point out the differences. In Theorem CLZ, there is no hypothesis of the set of places $S_L$, no additional hypothesis of integrality (appearing for us under the form of a tubular neighbourhood), and the divisors are assumed to be normal crossing divisors, which is replaced in our case by the tubular Runge condition. As for the results themselves, the finiteness formulated by CLZ depends on the set $S_L$ (that is, it is not clear how it would prove such an union of sets such as in our Theorem is finite). Finally, the techniques employed are greatly different : Theorem CLZ uses Schmidt’s subspace theorem which is noneffective, whereas our method can be made effective if one knows the involved auxiliary functions. It might be possible (and worthy of interest) to build some bridges between the two results, and the techniques involved.
$(b)$ Theorem \[thmRungetubulaire\] can be seen as a stratification of Runge-like results depending on the dimension of the intersection of the involved divisors : at one extreme, the intersection is empty, and we get back Theorem 4 $(b)$ and $(c)$ of [@Levin08]. At the other extreme, the intersection is a divisor (ample or big), and the finiteness is automatic by the hypothesis for points not belonging in the tubular neighbourhood (see Remark \[remhorsvoistub\]). Of course, this stratification is not relevant in the case of curves. In another perspective, for a fixed closed subset $Y$, Theorem \[thmRungetubulaire\] is more a concentration result of integral points than a finiteness result, as it means that even if we choose a tubular neighbourhood ${{\mathcal V}}$ of $Y$ as small as possible around $Y$, there is only a finite number of integral points in the set , i.e. these integral points (ignoring the hypothese $P \notin {{\mathcal V}}$) must concentrate around $Y$ (at least at one of the places $w \in M_L$). Specific examples will be given in section \[sectionapplicationsSiegel\] and \[sectionexplicitRunge\].
Let us now prove Theorem \[thmRungetubulaire\], following the ideas outlined in Remark \[remRungetubulaire\].
***(R1)*** Let us first build the embeddings we need. For every subextension $K''$ of $K'/K$, the action of ${\operatorname{Gal}}(K'/K'')$ on the divisors $(D_1)_{K'}, \cdots, (D_r)_{K'}$ has orbits denoted by $O_{K'',1}, \cdots, O_{K'',r_{K''}}$. Notice that any $m_Y+1$ such orbits still have their global intersection included in $Y$ : regrouping the divisors by orbits does not change this fact.
For each such orbit, the sum of its divisors is ample by hypothesis and coming from an effective Cartier divisor on ${{\mathcal X}}_{K''}$, hence one can choose by Proposition \[propreductionamplegros\] an appropriate embedding $\psi_{K'',i} : {{\mathcal X}}_{K''} \rightarrow {\mathbb{P}}^{n_i}_{K''}$, whose coordinates functions (denoted by $\phi_{K'',i,j} = (x_j/x_0) \circ \psi_{K'',i} (1 \leq j \leq n_i)$) are small on integral points of $({{\mathcal X}}_{{\mathcal O}}\backslash O_{K'',i})$. We will denote by ${{\mathcal C}}_0$ the maximum of the (induced) $M_K$-constants obtained for by the Proposition \[propreductionamplegros\] for all possible $K''/K$ and orbits $O_{K'',i} (1 \leq i \leq r_{K''})$. The important point of this is that for any extension $L/K$, any $v \in M_K \backslash S_0$, any place $w \in M_L$ above $v$ and any $P \in ({{\mathcal X}}\backslash D) ({{\mathcal O}}_{L,w})$, choosing $L'=K' \cap L$, one has $$\label{eqinterRungetousplongements}
\max_{\substack{1 \leq i \leq r_L \\ 1 \leq j \leq n_i}} \log |\phi_{L',i,j} (P)|_w \leq c_{0,v}.$$
This is the first step to obtain a bound on the height of one of the $\psi_{K'',i} (P)$. For fixed $P$, we only have to do so for one of the $i \in \{1, \cdots, r_L \}$ as long as the bound is uniform in the choice of $(L,S_L)$ (and $P$), to obtain finiteness as each $\psi_{K'',i}$ is an embedding. To this end, one only needs to bound the coordinate functions on the places $w$ of $M_L \backslash S_L$, which is what we will do now.
For a subextension $K''$ of $K'/K$ again, by hypothesis ***(H2)*** (and especially the definition of $m_Y$), taking any set ${{\mathcal I}}$ of $m_Y+1$ couples $(i,j), 1 \leq i \leq r_{K''}, j \in \{1, \cdots, n_i\}$ with $m_Y+1$ different indices $i$ and considering the rational functions $\phi_{K'',i,j}, (i,j) \in {{\mathcal I}}$, whose common poles are included in $Y$ by hypothesis, we can apply Proposition \[propcle\] to these functions and the tubular neighbourhood ${{\mathcal V}}= (V_w)_{w \in M_{{\overline{K}}}}$. Naming as ${{\mathcal C}}_1$ the maximum of the (induced) obtained $M_K$-constants (also for all the possible $K''$), we just proved that for every subextension $K''$ of $K'/K$, every place $w \in M_{{\overline{K}}}$ (above $v \in M_K$) and any $P \in {{\mathcal X}}({\overline{K}}) \backslash V_w$, the inequality $$\label{eqineqfaussepourauplusmY}
\max_{1 \leq j \leq n_i} \log |\phi_{K'',i,j} (P)|_w \leq c_{1,v}$$ is true except for at most $m_Y$ different indices $i \in \{1, \cdots, r_{K''} \}$.
Now, let us consider $(L,S_L)$ a pair satisfying tubular Runge condition $m_Y |S_L| < r_L$ and denote $L' = K' \cap L$ again. For $P \in ({{\mathcal X}}\backslash D) ({{\mathcal O}}_{L,S_L})$ not belonging to ${{\mathcal V}}$, by , and tubular Runge condition, there remains an index $i \in \{1, \cdots, r_L\}$ (dependent on $P$) such that $$\forall w \in M_L, \quad \max_{1 \leq j \leq n_i} \log |\phi_{L',i,j} (P)|_w \leq \max(c_{0,v},c_{1,v}) \quad (w | v \in M_K).$$ This gives immediately a bound on the height of $\psi_{L',i}(P)$ independent of the choice of pair $(L,S_L)$ (except the fact that $L' = K' \cap L$) and this morphism is an embedding, hence the finiteness of the set of points $$\bigcup_{(L,S_L)} \{P \in ({{\mathcal X}}\backslash D) ({{\mathcal O}}_{L,S_L}) \, | \, P \notin {{\mathcal V}}\},$$ where $(L,S_L)$ goes through all the pairs satisfying tubular Runge condition, because $[L:{{\mathbb{Q}}}]$ is also bounded by this condition.
***(R2)***
The proof is the same as for ***(R1)*** except that we have to exclude a closed subset of ${{\mathcal X}}_{K'}$ for every big divisor involved, and their reunion will be denoted by $Z_{K'}$. The arguments above hold for every point $P \notin Z_{K'} ({\overline{K}})$ (both for the expression of integrality by auxiliary functions, and for the conclusion and finiteness outside of this closed subset), using again Propositions \[propreductionamplegros\] and \[propcle\].
Reminders on Siegel modular varieties {#sectionrappelsSiegel}
=====================================
In this section, we recall the classical constructions and results for the Siegel modular varieties, parametrising principally polarised abelian varieties with a level structure. Most of those results are extracted (or easily deduced) from these general references : Chapter V of [@CornellSilvermanArithmeticGeometry] for the basic notions on abelian varieties, [@Debarre99] for the complex tori, their line bundles, theta functions and moduli spaces, Chapter II of [@MumfordTata] for the classical complex theta functions and [@MumfordTataII] for their links with theta divisors, and Chapter V of [@ChaiFaltings] for abelian schemes and their moduli spaces.
Unless specified, all the vectors of ${{\mathbb{Z}}}^g, {{\mathbb{R}}}^g$ and ${{\mathbb{C}}}^g$ are assumed to be row vectors.
Abelian varieties and Siegel modular varieties {#subsecabvarSiegelmodvar}
----------------------------------------------
\[defibaseabvar\]
- An *abelian variety* $A$ over a field $k$ is a projective algebraic group over $k$. Each abelian variety $A_{/k}$ has a dual abelian variety denoted by $\widehat{A} = {\operatorname{Pic}}^0 (A/k)$ ([@CornellSilvermanArithmeticGeometry], section V.9).
- A *principal polarisation* is an isomorphism $\lambda : A \rightarrow \widehat{A}$ such that there exists a line bundle $L$ on $A_{\overline{k}}$ with $\dim H^0(A_{\overline{k}},L)=1$ and $\lambda$ is the morphism $${\begin{array}{c|ccl} \lambda: & A_{\overline{k}} & \longrightarrow & \widehat{A_{\overline{k}}} \\
& x & \longmapsto & T_x^* L \otimes L^{-1} \end{array}}$$ ([@CornellSilvermanArithmeticGeometry], section V.13).
- Given a pair $(A,\lambda)$, for every $n \geq 1$ prime to ${\textrm{char}}(k)$, we can define the *Weil pairing* $$A[n] \times A[n] \rightarrow \mu_n (\overline{k}),$$ where $A[n]$ is the $n$-torsion of $A(\overline{k})$ and $\mu_n$ the group of $n$-th roots of unity in $\overline{k}$. It is alternate and nondegenerate ([@CornellSilvermanArithmeticGeometry], section V.16).
- Given a pair $(A,\lambda)$, for $n \geq 1$ prime to ${\textrm{char}}(k)$, a *symplectic level $n$ structure on* $A[n]$ is a basis $\alpha_n$ of $A[n]$ in which the matrix of the Weil pairing is $$J = \begin{pmatrix} 0 & I_g \\
- I_g & 0
\end{pmatrix}.$$
- Two triples $(A,\lambda,\alpha_n)$ and $(A',\lambda',\alpha'_n)$ of principally polarised abelian varieties over $K$ with level $n$-structures are *isomorphic* if there is an isomorphism of abelian varieties $\phi : A \rightarrow A'$ such that $\phi^* \lambda' = \lambda$ and $\phi^* \alpha'_n = \alpha_n$.
In the case of complex abelian varieties, the previous definitions can be made more explicit.
\[deficomplexabvar\]
Let $g \geq 1$.
- The *half-superior Siegel space of order* $g$, denoted by ${{\mathcal H}}_g$, is the set of matrices $$\label{eqdefdemiespaceSiegel}
{{\mathcal H}}_g := \{ \tau \in M_g ({{\mathbb{C}}}) \, | \, {}^t \tau = \tau \, \, \textrm{and} \, \, {\operatorname{Im}}\tau >0 \},$$ where ${\operatorname{Im}}\tau >0$ means that this symmetric matrix of $M_g ({{\mathbb{R}}})$ is positive definite. This space is an open subset of $M_g({{\mathbb{C}}})$.
- For any $\tau \in {{\mathcal H}}_g$, we define $$\Lambda_\tau := {{\mathbb{Z}}}^g + {{\mathbb{Z}}}^g \tau \quad \textrm{and} \quad A_\tau := {{\mathbb{C}}}^g / \Lambda_\tau.$$ Let $L_\tau$ be the line bundle on $A_\tau$ made up as the quotient of ${{\mathbb{C}}}^g \times {{\mathbb{C}}}$ by the action of $\Lambda_\tau$ defined by $$\label{eqdeffibresurAtau}
\forall p,q \in {{\mathbb{Z}}}^g, \quad (p \tau + q) \cdot (z,t) = \left(z+p \tau + q, e^{ - i \pi p \tau {}^tp - 2 i \pi p {}^t z} t \right).$$ Then, $L_\tau$ is an an ample line bundle on $A_\tau$ such that $\dim H^0 (A_\tau,L_\tau)=1$, hence $A_\tau$ is a complex abelian variety and $L_\tau$ induces a principal polarisation denoted by $\lambda_\tau$ on $A_\tau$ (see for example [@Debarre99], Theorem VI.1.3). We also denote by $\pi_\tau : {{\mathbb{C}}}^g \rightarrow A_\tau$ the quotient morphism.
- For every $n \geq 1$, the Weil pairing $w_{\tau,n}$ associated to $(A_\tau,\lambda_\tau)$ on $A_\tau[n]$ is defined by $${\begin{array}{c|ccl} w_{\tau,n}: & A_\tau[n] \times A_\tau[n] & \longrightarrow & \mu_n ({{\mathbb{C}}}) \\
& (\overline{x},\overline{y}) & \longmapsto & e^{ 2 i \pi n w_\tau(x,y)} \end{array}},$$ where $x,y \in {{\mathbb{C}}}^g$ have images $\overline{x}, \overline{y}$ by $\pi_\tau$, and $w_\tau$ is the ${{\mathbb{R}}}$-bilinear form on ${{\mathbb{C}}}^g \times {{\mathbb{C}}}^g$ (so that $w_\tau(\Lambda_\tau \times \Lambda_\tau) = {{\mathbb{Z}}}$) defined by $$w_\tau (x,y) := {\operatorname{Re}}(x) \cdot {\operatorname{Im}}(\tau)^{-1} \cdot {}^t {\operatorname{Im}}(y) - {\operatorname{Re}}(y) \cdot {\operatorname{Im}}(\tau)^{-1} \cdot {}^t {\operatorname{Im}}(x)$$ (also readily checked by making explicit the construction of the Weil pairing).
- Let $(e_1, \cdots, e_g)$ be the canonical basis of ${{\mathbb{C}}}^g$. The family $$\label{eqdefalphataun}
(\pi_\tau(e_1/n), \cdots, \pi_\tau(e_g/n), \pi_\tau(e_1 \cdot \tau/n), \cdots, \pi_\tau(e_g \cdot \tau/n))$$ is a symplectic level $n$ structure on $(A_\tau, \lambda_\tau)$, denoted by $\alpha_{\tau,n}$.
- Let $J = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \in M_{2g}({{\mathbb{Z}}})$. For any commutative ring $A$, the *symplectic group of order $g$ over $A$*, denoted by ${\operatorname{Sp}}_{2g} (A)$, is the subgroup of ${\operatorname{GL}}_{2g}(A)$ defined by $$\label{eqdefgroupesymplectique}
{\operatorname{Sp}}_{2g} (A) := \{ M \in {\operatorname{GL}}_{2g} (A) \, \, | \, \, {}^t M J M = J \}, \qquad J:= \begin{pmatrix} 0 & I_g \\- I_g & 0 \end{pmatrix}.$$
For every $n \geq 1$, the *symplectic principal subgroup of degree $g$ and level $n$*, denoted by $\Gamma_g(n)$, is the subgroup of ${\operatorname{Sp}}_{2g} ({{\mathbb{Z}}})$ made up by the matrices congruent to $I_{2g}$ modulo $n$. For every $\gamma = \begin{pmatrix}A & B \\ C & D \end{pmatrix} \in {\operatorname{Sp}}_{2g} ({{\mathbb{R}}})$ and every $\tau \in {{\mathcal H}}_g$, we define $$\label{eqdefjetactionsymplectique}
j_\gamma (\tau) = C \tau + D \in {\operatorname{GL}}_g ({{\mathbb{C}}}), \quad \textrm{and} \quad \gamma \cdot \tau = (A \tau + B)(C \tau + D)^{-1},$$ which defines a left action by biholomorphisms of ${\operatorname{Sp}}_{2g} ({{\mathbb{R}}})$ on ${{\mathcal H}}_g$, and $(\gamma,\tau) \mapsto j_\gamma(\tau)$ is a left cocycle for this action ([@Klingen], Proposition I.1).
- For every $g \geq 2$, $n \geq 1$ and $k \geq 1$, a *Siegel modular form of degree $g$, level $n$ and weight $k$* is an holomorphic function $f$ on ${{\mathcal H}}_g$ such that $$\label{eqdefSiegelmodularform}
\forall \gamma \in \Gamma_g(n), \quad f (\gamma \cdot z) = \det(j_\gamma(z))^k f(z).$$
The reason for this seemingly partial description of the complex abelian varieties is that the $(A_\tau,\lambda_\tau)$ described above actually make up all the principally polarised complex abelian varieties up to isomorphism. The following results can be found in Chapter VI of [@Debarre99] except the last point which is straightforward.
\[defipropuniformcomplexabvar\]
- Every principally polarised complex abelian variety of dimension $g$ with symplectic structure of level $n$ is isomorphic to some triple $(A_\tau, \lambda_\tau,\alpha_{\tau,n})$ where $\tau \in {{\mathcal H}}_g$.
- For every $n \geq 1$, two triples $(A_\tau,\lambda_\tau,\alpha_{\tau,n})$ and $(A_{\tau'},\lambda_{\tau'},\alpha_{\tau',n})$ are isomorphic if and only if there exists $\gamma \in \Gamma_g(n)$ such that $\gamma \cdot \tau = \tau'$, and then such an isomorphism is given by $${\begin{array}{ccl} A_\tau & {\longrightarrow}& A_{\tau'} \\\
z \! \mod \Lambda_\tau & {\longmapsto}& z \! \cdot j_\gamma(\tau)^{-1} \mod \Lambda_{\tau'}
\end{array}}.$$
- The *Siegel modular variety of degree $g$ and level $n$* is the quotient $A_g(n)_{{\mathbb{C}}}:= \Gamma_g(n) \backslash {{\mathcal H}}_g$. From the previous result, it is the moduli space of principally polarised complex abelian varieties of dimension $g$ with a symplectic level $n$ structure. As a quotient, it also inherits a structure of normal analytic space (with finite quotient singularities) of dimension $g(g+1)/2$, because $\Gamma_g(n)$ acts properly discontinuously on ${{\mathcal H}}_g$.
- For every positive divisor $m$ of $n$, the natural morphism $A_g(n)_{{\mathbb{C}}}\rightarrow A_g(m)_{{\mathbb{C}}}$ induced by the identity of ${{\mathcal H}}_g$ corresponds in terms of moduli to multiplying the symplectic basis $\alpha_{\tau,n}$ by $n/m$, thus obtaining $\alpha_{\tau,m}$.
- For every $g \geq 1$ and $n \geq 1$, the quotient of ${{\mathcal H}}_g \times {{\mathbb{C}}}$ by the action of $\Gamma_g(n)$ defined as $$\label{eqdeffibreL}
\gamma \cdot (\tau,t) = (\gamma \cdot \tau, t / \det( j_\gamma(z)))$$ is a variety over ${{\mathcal H}}_g$ denoted by $L$. For a large enough power of $k$ (or if $n \geq 3$), $L^{\otimes k}$ is a line bundle over $A_g(n)_{{\mathbb{C}}}$, hence $L$ is a ${{\mathbb{Q}}}$-line bundle over $A_g(n)_{{\mathbb{C}}}$ called *line bundle of modular forms of weight one* over $A_g(n)_{{\mathbb{C}}}$. By definition , for every $k \geq 1$, the global sections of $L^{\otimes k}$ are the Siegel modular forms of degree $g$, level $n$ and weight $k$.
Let us now present the compactification of $A_g(n)_{{\mathbb{C}}}$ we will use, that is the Satake compactification (for a complete description of it, see section 3 of [@Namikawa]).
\[defipropSatakecompactification\]
Let $g \geq 1$ and $n \geq 1$. The normal analytic space $A_g(n)_{{\mathbb{C}}}$ admits a compactification called *Satake compactification* and denoted by $A_g(n)^S_{{\mathbb{C}}}$, satisfying the following properties.
$(a)$ $A_g(n)^S_{{\mathbb{C}}}$ is a compact normal analytic space (of dimension $g(g+1)/2$, with finite quotient singularities) containing $A_g(n)_{{\mathbb{C}}}$ as an open subset and the boundary $\partial A_g(n)_{{\mathbb{C}}}:= A_g(n)^S_{{\mathbb{C}}}\backslash A_g(n)_{{\mathbb{C}}}$ is of codimension $g$ (see [@CartanSatake57] for details).
$(b)$ As a normal analytic space, $A_g(n)^S_{{\mathbb{C}}}$ is a projective algebraic variety. More precisely, for ${\textrm{M}}_g(n)$ the graded ring of Siegel modular forms of degree $g$ and level $n$, $A_g(n)^S_{{\mathbb{C}}}$ is canonically isomorphic to ${\operatorname{Proj}}_{{\mathbb{C}}}{\textrm{M}}_g (n)$ ([@CartanPlongements57], “théorème fondamental”).
In particular, one can obtain naturally $A_g(n)^S_{{\mathbb{C}}}$ by fixing for some large enough weight $k$ a basis of modular forms of ${\textrm{M}}_g(n)$ of weight $k$ and evaluating them all on $A_g(n)_{{\mathbb{C}}}$ to embed it in a projective space, so that $A_g(n)^S_{{\mathbb{C}}}$ is the closure of the image of the embedding in this projective space.
$(c)$ The ${{\mathbb{Q}}}$-line bundle $L$ of modular forms of weight 1 on $A_g(n)_{{\mathbb{C}}}$ extends naturally to $A_g(n)^S_{{\mathbb{C}}}$ (and is renoted $L$), to an ample ${{\mathbb{Q}}}$-line bundle (this is a direct consequence of $(b)$).
Further properties of Siegel modular varieties {#subsecfurtherpropSiegelmodvar}
----------------------------------------------
As we are interested in the reduction of abelian varieties on number fields, one needs to have a good model of $A_g(n)_{{\mathbb{C}}}$ over integer rings, as well as some knowledge of the geometry of $A_g(n)_{{\mathbb{C}}}$. The integral models below and their properties are given in Chapter V of [@ChaiFaltings].
\[defabelianscheme\]
$(a)$ An *abelian scheme* $A \rightarrow S$ is a smooth proper group scheme whose fibers are geometrically connected. It also has a natural *dual* abelian scheme $\widehat{A} = {\operatorname{Pic}}^0 (A/S)$, and it is *principally polarised* if it is endowed with an isomorphism $\lambda : A \rightarrow \widehat{A}$ such that at every geometric point $\overline{s}$ of $S$, the induced isomorphism $\lambda_{\overline{s}} : A_{\overline{s}} \rightarrow \widehat{A}_{\overline{s}}$ is a principal polarisation of $A_{\overline{s}}$.
$(b)$ A *symplectic structure of level $n \geq 1$* on a principally polarised abelian scheme $(A,\lambda)$ over a ${{\mathbb{Z}}}[\zeta_n,1/n]$-scheme $S$ is the datum of an isomorphism of group schemes $A[n] \rightarrow ({{\mathbb{Z}}}/n{{\mathbb{Z}}})^{2g}$, which is symplectic with respect to $\lambda$ and the canonical pairing on $({{\mathbb{Z}}}/n{{\mathbb{Z}}})^{2g}$ given by the matrix $J$ (as in ).
\[defipropalgmodulispaces\]
For every integers $g \geq 1$ and $n \geq 1$ :
$(a)$ The Satake compactification $A_g(n)^S_{{\mathbb{C}}}$ has an integral model ${{\mathcal A}}_g (n)^S$ on ${{\mathbb{Z}}}[\zeta_n, 1/n]$ which contains as a dense open subscheme the (coarse, if $n \leq 2$) moduli space ${{\mathcal A}}_g(n)$ on ${{\mathbb{Z}}}[\zeta_n,1/n]$ of principally polarised abelian schemes of dimension $g$ with a symplectic structure of level $n$. This scheme ${{\mathcal A}}_g(n)^S$ is normal, proper and of finite type on ${{\mathbb{Z}}}[\zeta_n,1/n]$ ([@ChaiFaltings], Theorem V.2.5).
$(b)$ For every divisor $m$ of $n$, we have canonical degeneracy morphisms ${{\mathcal A}}_g(n)^S \rightarrow {{\mathcal A}}_g(m)^S$ extending the morphisms of Definition \[defipropuniformcomplexabvar\].
Before tackling our own problem, let us give some context on the divisors on $A_g(n)^S_{{\mathbb{C}}}$ to give a taste of the difficulties to overcome.
For every normal algebraic variety $X$ on a field $K$, the *rational Picard group* of $X$ is the ${{\mathbb{Q}}}$-vector space $${\operatorname{Pic}}(X)_{{\mathbb{Q}}}:= {\operatorname{Pic}}(X) \otimes_{{\mathbb{Z}}}{{\mathbb{Q}}}.$$
\[proprationalPicardSiegel\]
Let $g \geq 2$ and $n \geq 1$.
$(a)$ Every Weil divisor on $A_g(n)_{{\mathbb{C}}}$ or $A_g(n)^S_{{\mathbb{C}}}$ is up to some multiple a Cartier divisor, hence their rational Picard group is also their Weil class divisor group tensored by ${{\mathbb{Q}}}$.
$(b)$ For $g=3$, the Picard rational groups of $A_3(n)^S_{{\mathbb{C}}}$ and $A_3(n)_{{\mathbb{C}}}$ are equal to ${{\mathbb{Q}}}\cdot L$ for every $n \geq 1$.
$(c)$ For $g=2$, one has ${\operatorname{Pic}}_{{\mathbb{Q}}}(A_2(1)^S_{{\mathbb{C}}}) = {{\mathbb{Q}}}\cdot L$.
This result has the following immediate corollary, because $L$ is ample on $A_g(n)^S_{{\mathbb{C}}}$ for every $g \geq 2$ and every $n \geq 1$ (Definition-Proposition \[defipropSatakecompactification\] $(c)$).
A ${{\mathbb{Q}}}$-divisor on $A_g(n)_{{\mathbb{C}}}$ or $A_g(n)^S_{{\mathbb{C}}}$ with $g=3$ (or $g=2$ and $n=1$) is ample if and only if it is big if and only if it is equivalent to $a \cdot L$ with $a>0$.
\[remampledifficilepourA2\] We did not mention the case of modular curves (also difficult, but treated by different methods): the point here is that the cases $g \geq 3$ are surprisingly much more uniform because then ${\operatorname{Pic}}(A_g(n)^S_{{\mathbb{C}}}) = {\operatorname{Pic}}(A_g(1)^S_{{\mathbb{C}}})$. The reason is that some rigidity appears from $g \geq 3$ (essentially by the general arguments of [@Borel81]), whereas for $g=2$, the situation seems very complex already for the small levels (see for example $n=3$ in [@HoffmanWeintraub00]).
This is why the ampleness (or bigness) is in general hard to figure out for given divisors of $A_2(n), n >1$. We consider specific divisors in the following (namely, divisors of zeroes of theta functions), whose ampleness will not be hard to prove.
$(a)$ This is true for the $A_g(n)^S_{{\mathbb{C}}}$ by [@ArtalBartolo14] as they only have finite quotient singularities, (this result actually seems to have been generally assumed a long time ago). Now, as $\partial A_g(n)^S_{{\mathbb{C}}}$ is of codimension at least 2, the two varieties $A_g(n)^S_{{\mathbb{C}}}$ and $A_g(n)_{{\mathbb{C}}}$ have the same Weil and Cartier divisors, hence the same rational Picard groups.
$(b)$ This is a consequence of general results of [@Borel81] further refined in [@Weissauer92] (it can even be generalised to every $g \geq 3$).
$(c)$ This comes from the computations of section III.9 of [@Mumford83] (for another compactification, called toroidal), from which we extract the result for $A_2(1)_{{\mathbb{C}}}$ by a classical restriction theorem ([@Hartshorne], Proposition II.6.5) because the boundary for this compactification is irreducible of codimension 1. The result for $A_2(1)^S_{{\mathbb{C}}}$ is then the same because the boundary is of codimension 2.
Theta divisors on abelian varieties and moduli spaces {#subsecthetadivabvar}
-----------------------------------------------------
We will now define the useful notions for our integral points problem.
\[defithetadivisorabvar\]
Let $k$ be an algebraically closed field and $A$ an abelian variety over $k$.
Let $L$ be an ample symmetric line bundle on $A$ inducing a principal polarisation $\lambda$ on $A$. A *theta function associated to* $(A,L)$ is a nonzero global section $\vartheta_{A,L}$ of $L$. The *theta divisor associated to* $(A,L)$, denoted by $\Theta_{A,L}$, is the divisor of zeroes of $\vartheta_{A,L}$, well-defined and independent of our choice because $\dim H^0(A,L)=\deg(\lambda)^2 = 1$.
The theta divisor is in fact determined by the polarisation $\lambda$ itself, up to a finite ambiguity we make clear below.
\[propambiguitedivthetaAL\]
Let $k$ be an algebraically closed field and $A$ an abelian variety over $k$.
Two ample symmetric line bundles $L$ and $L'$ on $A$ inducing a principal polarisation induce the same one if and only if $L' \cong T_x^* L$ for some $x \in A|2]$, and then $$\Theta_{A,L'} = \Theta_{A,L} + x.$$
For any line bundle $L$ on $A$, let us define $${\begin{array}{c|ccl} \lambda_L: & A & \longrightarrow & \widehat{A} = {\operatorname{Pic}}^0(A) \\
& x & \longmapsto & T_x^* L \otimes L^{-1} \end{array}}.$$ This is a group morphism and the application $L \mapsto \lambda_L$ is additive from ${\operatorname{Pic}}(A)$ to ${\operatorname{Hom}}(A,\widehat{A})$, with kernel ${\operatorname{Pic}}^0(A)$ ([@MumfordAbVar], Chapter II, Corollary 4 and what follows, along with section II.8). Moreover, when $L$ is ample, the morphism $\lambda_L$ is the polarisation associated to $L$, in particular surjective. Now, for every $x \in A(k)$, if $L' \cong T_x^* L$, then $L' \otimes L^{-1}$ belongs to ${\operatorname{Pic}}^0(A)$, therefore $\lambda_{L'} = \lambda_{L}$. Conversely, if $\lambda_{L'} = \lambda_L$, one has $L' \otimes L^{-1} \in {\operatorname{Pic}}^0(A)$, hence if $L$ is ample, by surjectivity, one has $x \in A(k)$ such that $L' \cong T_x^* L$. Finally, if $L$ and $L'$ are symmetric, having $[-1]^* L \cong L$ and $[-1]^* L' \cong L'$, we obtain $T_{-x}^* L \cong T_x^* L$ but as $\lambda_L$ is an isomorphism, this implies $[2] \cdot x = 0$, hence $x \in A[2]$.
Therefore, for $\vartheta_{A,L}$ a nonzero section of $L$, $T_x^* \vartheta_{A,L}$ can be identified to a nonzero section of $L$, hence $$\Theta_{A,L'} = \Theta_{A,L} - x = \Theta_{A,L} + x.$$
When ${\textrm{char}}(k) \neq 2$, adding to a principally polarised abelian variety $(A,\lambda)$ of dimension $g$ the datum $\alpha_2$ of a symplectic structure of level 2, we can determine an unique ample symmetric line bundle $L$ with the following process called *Igusa correspondence*, devised in [@Igusa67bis]. To any ample symmetric Weil divisor $D$ defining a principal polarisation, one can associate bijectively a quadratic form $q_D$ from $A[2]$ to $\{ \pm 1 \}$ called *even*, which means that the sum of its values on $A[2]$ is $2^{g}$ ([@Igusa67bis], Theorem 2 and the previous arguments). On another side, the datum $\alpha_2$ also determines an even quadratic form $q_{\alpha_2}$, by associating to a $x \in A[2]$ with coordinates $(a,b) \in ({{\mathbb{Z}}}/2{{\mathbb{Z}}})^{2g}$ in the basis $\alpha_2$ of $A[2]$ the value $$\label{eqcorrespondanceIgusa}
q_{\alpha_2}(x) = (-1)^{a {}^t b}.$$ We now only have to choose the unique ample symmetric divisor $D$ such that $q_D = q_{\alpha_2}$ and the line bundle $L$ associated to $D$.
By construction of this correspondence ([@Igusa67bis], p. 823), a point $x \in A[2]$ of coordinates $(a,b) \in ({{\mathbb{Z}}}/2{{\mathbb{Z}}})^{2g}$ in $\alpha_2$ automatically belongs to $\Theta_{A,L}$ (with $L$ associated to $(A,\lambda,\alpha_2)$) if $a{}^t b= 1 \mod 2$. A point of $A[2]$ with coordinates $(a,b)$ such that $a {}^t b = 0 \mod 2$ can also belong to $\Theta_{A,L}$ but with even multiplicity.
This allows us to get rid of the ambiguity of choice of an ample symmetric $L$ in the following, as soon as we have a symplectic level 2 structure (or finer) ( this result is a reformulation of Theorem 2 of [@Igusa67bis]).
\[defipropthetadiviseurcanonique\]
Let $n \geq 2$ even and $k$ algebraically closed such that ${\textrm{char}}(k)$ does not divide $n$.
For $(A,\lambda,\alpha_n)$ a principally polarised abelian variety of dimension $g$ with symplectic structure of level $n$ (Definition \[deficomplexabvar\]), there is up to isomorphism an unique ample symmetric line bundle $L$ inducing $\lambda$ and associated by Igusa correspondence to the symplectic basis of $A[2]$ induced by $\alpha_n$. The *theta divisor associated to* $(A,\lambda,\alpha_n)$, denoted by $\Theta_{A,\lambda,\alpha_n}$, is then the theta divisor associated to $(A,L)$, .
The Runge-type theorem we give in section \[sectionapplicationsSiegel\] (Theorem \[thmtubularRungegeneral\]) focuses on principally polarised abelian surfaces $(A,\lambda)$ on a number field $K$ whose theta divisor does not contain any $n$-torsion point of $A$ (except 2-torsion points, as we will see it is automatic). This will imply (Proposition \[propnombrepointsdivthetajacobienne\]) that $A$ is not a product of elliptic curves, but this is not a sufficient condition, as pointed out for example in [@BoxallGrant].
We will once again start with the complex case to figure out how such a condition can be formulated on the moduli spaces, using complex theta functions ([@MumfordTata], Chapter II).
\[defipropcomplexthetafunctions\]
Let $g \geq 1$.
The holomorphic function $\Theta$ on ${{\mathbb{C}}}^g \times {{\mathcal H}}_g$ is defined by the series (convergent on any compact subset) $$\label{eqdefserietheta}
\Theta(z,\tau) = \sum_{n \in {{\mathbb{Z}}}^g} e^{ i \pi n \tau {}^t n + 2 i \pi n {}^t z}.$$
For any $a,b \in {{\mathbb{R}}}^g$, we also define the holomorphic function $\Theta_{a,b}$ by $$\label{eqdefseriethetaab}
\Theta_{a,b}(z,\tau) = \sum_{n \in {{\mathbb{Z}}}^g} e^{ i \pi (n+a) \tau {}^t (n+a) + 2 i \pi (n+a) {}^t (z+b)}.$$
For a fixed $\tau \in {{\mathcal H}}_g$, one defines $\Theta_\tau : z \mapsto \Theta(z,\tau)$ and similarly for $\Theta_{a,b,\tau}$. These functions have the following properties.
$(a)$ For every $a,b \in {{\mathbb{Z}}}^g$, $$\label{eqthetaabenfonctiontheta}
\Theta_{a,b,\tau} (z) = e^{i \pi a \tau {}^t a + 2 i \pi a {}^t (z+b)} \Theta_\tau(z + a \tau + b).$$
$(b)$ For every $p,q \in {{\mathbb{Z}}}^g$, $$\label{eqfoncthetaptranslation}
\Theta_{a,b,\tau}(z+p\tau + q) = e^{- i \pi p \tau {}^t p - 2 i \pi p{}^t z + 2 i \pi (a{}^t q - b {}^t p) } \Theta_{a,b,\tau} (z).$$
$(c)$ Let us denote by $\vartheta$ and $\vartheta_{a,b}$ the *normalised theta-constants*, which are the holomorphic functions on ${{\mathcal H}}_g$ defined by $$\label{eqdefthetaconstantes}
\vartheta(\tau) : = \Theta(0,\tau) \quad {\textrm{and}} \quad \vartheta_{a,b} (\tau) := e^{ - i \pi a {}^t b} \Theta_{a,b} (0,\tau).$$
These theta functions satisfy the following modularity property : with the notations of Definition \[deficomplexabvar\], $$\label{eqmodularitethetaconstantes}
\forall \gamma \in \Gamma_g(2), \quad \vartheta_{a,b} (\gamma \cdot \tau) = \zeta_8(\gamma) e^{ i \pi (a,b)^t V_\gamma} \sqrt{j_\gamma(\tau)} \vartheta_{(a,b)\gamma} (\tau),$$ where $\zeta_8(\gamma)$ (a $8$-th root of unity) and $V_\gamma \in {{\mathbb{Z}}}^g$ only depend on $\gamma$ and the determination of the square root of $j_\gamma(\tau)$.
In particular, for every even $n \geq 2$, if $(na,nb) \in {{\mathbb{Z}}}^{2g}$, the function $\vartheta_{a,b}^{8n}$ is a Siegel modular form of degree $g$, level $n$ and weight $4n$, which only depends on $(a,b) \! \mod {{\mathbb{Z}}}^{2g}$.
The convergence of these series as well as their functional equations and are classical and can be found in section II.1 of [@MumfordTata].
The modularity property (also classical) is a particular case of the computations of section II.5 of [@MumfordTata] (we do not need here the general formula for $\gamma \in {\operatorname{Sp}}_{2g} ({{\mathbb{Z}}})$).
Finally, by natural computations of the series defining $\Theta_{a,b}$, one readily obtains that $$\vartheta_{a+p,b+q} = e^{2 i \pi (a{}^t q - b{}^t p)} \vartheta_{a,b}.$$ Therefore, if $(na,nb) \in {{\mathbb{Z}}}^{2g}$, the function $\vartheta_{a,b}^n$ only depends on $(a,b) \! \mod {{\mathbb{Z}}}^{2g}$. Now, putting the modularity formula to the power $8n$, one eliminates the eight root of unity and if $\gamma \in \Gamma_g(n)$, one has $(a,b) \gamma = (a,b) \mod {{\mathbb{Z}}}^g$ hence $\vartheta_{a,b}^{8n}$ is a Siegel modular form of weight $4n$ for $\Gamma_g(n)$.
There is of course an explicit link between the theta functions and the notion of theta divisor, which we explain now with the notations of Definition \[deficomplexabvar\].
\[propliendivthetafonctiontheta\]
Let $\tau \in {{\mathcal H}}_g$.
The line bundle $L_\tau$ is ample and symmetric on $A_\tau$, and defines a principal polarisation on $A_\tau$. It is also the line bundle canonically associated to the 2-structure $\alpha_{\tau,2}$ and its polarisation by Igusa correspondence (Definition-Proposition \[defipropthetadiviseurcanonique\]).
Furthermore, the global sections of $L_\tau$ canonically identify to the multiples of $\Theta_{\tau}$, hence the theta divisor associated to $(A_\tau, \lambda_\tau,\alpha_{\tau,2})$ is exactly the divisor of zeroes of $\Theta_{\tau}$ modulo $\Lambda_\tau$.
Thus, for every $a,b \in {{\mathbb{R}}}^g$, the projection of $\pi_\tau(a \tau + b)$ belongs to $\Theta_{A_\tau, \lambda_\tau, \alpha_{\tau,2}}$ if and only if $\vartheta_{a,b} (\tau)=0$.
The proof below that the $L_\tau$ is the line bundle associated to $(A_\tau, \lambda_\tau, \alpha_{\tau,2})$ is a bit technical, but one has to suspect that Igusa normalised its correspondence by exactly to make it work.
One can easily see that $L_\tau$ is symmetric by writing $[-1]^* L_\tau$ as a quotient of ${{\mathbb{C}}}^g \times {{\mathbb{C}}}$ by an action of $\Lambda_\tau$, then figuring out it is the same as . Then, by simple connexity, the global sections of $L_\tau$ lift by the quotient morphism ${{\mathbb{C}}}^g \times {{\mathbb{C}}}\rightarrow L_\tau$ into functions $z \mapsto (z,f(z))$, and the holomorphic functions $f$ thus obtained are exactly the functions satisfying functional equation for $a=b=0$ because of , hence the same functional equation as $\Theta_{\tau}$. This identification is also compatible with the associated divisors, hence $\Theta_{A_\tau,L_\tau}$ is the divisor of zeroes of $\Theta_{\tau}$ modulo $\Lambda_\tau$. For more details on the theta functions and line bundles, see ([@Debarre99], Chapters IV,V and section VI.2).
We now have to check that Igusa correspondence indeed associates $L_\tau$ to $(A_\tau,\lambda_\tau,\alpha_{\tau,2})$. With the notations of the construction of this correspondence ([@Igusa67bis], pp.822, 823 and 833), one sees that the meromorphic function $\psi_x$ on $A_\tau$ (depending on $L_\tau$) associated to $x \in A_\tau[2]$ has divisor $[2]^* T_x^* \Theta_{A_\tau,L_\tau} - [2]^* \Theta_{A_\tau,L_\tau}$, hence it is (up to a constant) the meromorphic function induced on $A_\tau$ by $$f_x(z) = \frac{\Theta_{a,b,\tau} (2z)}{\Theta_\tau(2z)} \quad {\textrm{where}} \, \, x= a \tau + b \mod \Lambda_\tau.$$ Now, the quadratic form $q$ associated to $L_\tau$ is defined by the identity $$f_x(-z) = q(x) f_x(z)$$ for every $z \in {{\mathbb{C}}}^g$, but $\Theta_\tau$ is even hence $$f_x(-z) = e^{4 i \pi a^t b} f_x(z)$$ by formula . Now, the coordinates of $x$ in $\alpha_{\tau,2}$ are exactly $(2b,2a) \mod {{\mathbb{Z}}}^{2g}$ by definition, hence $q=q_{\alpha_{\tau,2}}$.
Let us finally make the explicit link between zeroes of theta-constants and theta divisors : using the argument above, the divisor of zeroes of $\Theta_\tau$ modulo $\Lambda_\tau$ is exactly $\Theta_{A_\tau,L_\tau}$, hence $\Theta_{A_\tau,\lambda_\tau,\alpha_{\tau,2}}$ by what we just proved for the Igusa correspondence. This implies that for every $z \in {{\mathbb{C}}}^g$, $\Theta_\tau(z)=0$ if and only if $\pi_\tau(z)$ belongs to $\Theta_{A_\tau,\lambda_\tau, \alpha_{\tau,2}}$, and as $\vartheta_{a,b} (\tau)$ is a nonzero multiple of $\Theta(a \tau + b,\tau)$, we finally have that $\vartheta_{a,b}(\tau)=0$ if and only if $\pi_\tau(a\tau+b)$ belongs to $\Theta_{A_\tau,\lambda_\tau, \alpha_{\tau,2}}$.
Applications of the main result on a family of Siegel modular varieties {#sectionapplicationsSiegel}
=======================================================================
We now have almost enough definitions to state the problem which we will consider for our Runge-type result (Theorem \[thmtubularRungegeneral\]). We consider theta divisors on abelian surfaces, and their torsion points.
The specific situation for theta divisors on abelian surfaces {#subsecthetadivabsur}
-------------------------------------------------------------
As an introduction and a preliminary result, let us treat first the case of theta divisors on elliptic curves.
\[lemdivthetaCE\]
Let $E$ be an elliptic curve on an algebraically closed field $k$ with ${\textrm{char}}(k) \neq 2$ and $L$ an ample symmetric line bundle defining the principal polarisation on $E$.
The effective divisor $\Theta_{E,L}$ is a 2-torsion point of $E$ with multiplicity one. More precisely, if $(e_1,e_2)$ is the basis of $E[2]$ associated by Igusa correspondence to $L$ (Definition-Proposition \[defipropthetadiviseurcanonique\]), $$\label{eqdivthetaCEexplicite}
\Theta_{E,L} = [e_1 + e_2].$$
In the complex case, this can simply be obtained by proving that $\Theta_{1/2,1/2,\tau}$ is odd for every $\tau \in {{\mathcal H}}_1$ hence cancels at 0, and has no other zeroes (by a residue theorem for example), then using Proposition \[propliendivthetafonctiontheta\].
By Riemann-Roch theorem on $E$, the divisor $\Theta_{E,L}$ is of degree 1 because $h^0(E,L)=1$ (and effective). Now, as explained before when discussing Igusa correspondence, for $a,b \in {{\mathbb{Z}}}$, $a e_1 + b e_2$ automatically belongs to $\Theta_{E,L}$ if $a b = 1 \mod 2 {{\mathbb{Z}}}$, hence $\Theta_{E,L} = [e_1+ e_2]$.
This allows to use to describe the theta divisor of a product of two elliptic curves.
\[propdivthetaproduitCE\]
Let $k$ be an algebraically closed field with ${\textrm{char}}(k) \neq 2$.
Let $(A,L)$ with $A=E_1 \times E_2$ a product of elliptic curves on $k$ and $L$ an ample symmetric line bundle on $A$ inducing the product principal polarisation on $A$. The divisor $\Theta_{A,L}$ is then of the shape $$\label{eqdivthetaproduitCE}
\Theta_{A,L} = \{x_1\} \times E_2 + E_1 \times \{x_2\},$$ with $x_i \in E_i[2]$ for $i=1,2$. In particular, this divisor has a (unique) singular point of multiplicity two at $(x_1,x_2)$, and :
$(a)$ There are exactly seven 2-torsion points of $A$ belonging to $\Theta_{A,L}$: the six points given by the coordinates $(a,b) \in ({{\mathbb{Z}}}/2{{\mathbb{Z}}})^4$ such that $a{}^t b= 1$ in a basis giving $\Theta_{A,L}$ by Igusa correspondence, and the seventh point $(x_1,x_2)$.
$(b)$ For every even $n \geq 2$ which is nonzero in $k$, the number of $n$-torsion (but not $2$-torsion) points of $A$ belonging to $\Theta_{A,L}$ is exactly $2(n^2-4)$.
By construction of $(A,L)$, a global section of $(A,L)$ corresponds to a tensor product of global sections of $E_1$ and $E_2$ (with their principal polarisations), hence the shape of $\Theta_{A,L}$ is a consequence of Lemma \[lemdivthetaCE\].
We readily deduce $(a)$ and $(b)$ from this shape, using that the intersection of the two components of $\Theta_{A,L}$ is a 2-torsion point of even multiplicity for the quadratic form hence different from the six other ones.
To explain the result for abelian surfaces which are not products of elliptic curves, we recall below a fundamental result.
\[propsurfabnonproduitCEetdivtheta\]
Let $k$ be any field.
A principally polarised abelian surface $(A,\lambda)$ on $k$ is, after a finite extension of scalars, either the product of two elliptic curves (with its natural product polarisation), or the jacobian $J$ of an hyperelliptic curve $C$ of genus 2 (with its canonical principal polarisation). In the second case, for the Albanese embedding $\phi_x : C \rightarrow J$ with base-point $x$ and an ample symmetric line bundle $L$ on $K$ inducing $\lambda$, the divisor $\Theta_{J,L}$ is irreducible, and it is actually a translation of $\phi_x(C)$ by some point of $J(\overline{k})$.
This proposition (together with the dimension 3 case, for the curious reader) is the main topic of [@OortUeno] (remarkably, its proof starts with the complex case and geometric arguments before using scheme and descent techniques to extend it to all fields).
Let us now fix an algebraically closed field $k$ with ${\textrm{char}}(k) \neq 2$.
Let $C$ be an hyperelliptic curve of genus 2, and $\iota$ its hyperelliptic involution. This curve has exactly six Weierstrass points (the fixed points of $\iota$, by definition), and we fix one of them, denoted by $\infty$. For the Albanese morphism $\phi_\infty$, the divisor $\phi_\infty(C)$ is stable by $[-1]$ because the divisor $[x] + [\iota(x)] - 2 [\infty]$ is principal for every $x \in C$. As $\Theta_{J,L}$ is also symmetric and a translation of $\phi_\infty(C)$, we know that $\Theta_{J,L} = T_x^* (\phi_\infty(C))$ for some $x \in J[2]$.
This tells us that understanding the points of $\Theta_{J,L}$ amounts to understanding how the curve $C$ behaves when embedded in its jacobian (in particular, how its points add). It is a difficult problem to know which torsion points of $J$ belong to the theta divisor (see [@BoxallGrant] for example), but we will only need to bound their quantity here, with the following result.
\[propnombrepointsdivthetajacobienne\]
Let $k$ an algebraically closed field with ${\textrm{char}}(k) \neq 2$.
Let $C$ be an hyperelliptic curve of genus 2 on $k$ with jacobian $J$, and $\infty$ a fixed Weierstrass point of $C$. We denote by $\widetilde{C}$ the image of $C$ in $J$ by the associated embedding $\phi_\infty : x \mapsto \overline{[x] - [\infty]}$.
$(a)$ The set $\widetilde{C}$ is stable by $[-1]$, and the application $${\begin{array}{ccl} \operatorname{Sym}^2(\widetilde{C}) & {\longrightarrow}& J \\\
\{P,Q\} & {\longmapsto}& P+Q
\end{array}}$$ is injective outside the fiber above 0.
$(b)$ There are exactly six 2-torsion points of $J$ belonging to $\widetilde{C}$, and they are equivalently the images of the Weierstrass points and the points of coordinates $(a,b) \in (({{\mathbb{Z}}}/2{{\mathbb{Z}}})^2)^2$ such that $a{}^t b= 1$ in a basis giving $\widetilde{C}$ by Igusa correspondence.
$(c)$ For any even $n \geq 2$ which is nonzero in $k$, the number of $n$-torsion points of $J$ belonging to $\widetilde{C}$ is bounded by $\sqrt{2} n^2 + \frac{1}{2}$.
This proposition is not exactly a new result, and its principle can be found (with slightly different formulations) in Theorem 1.3 of [@BoxallGrant] or in Lemma 5.1 of [@Pazuki12]. For the latter, it is presented as a consequence on Abel-Jacobi theorem on ${{\mathbb{C}}}$, and we will here give a more detailed proof, which is also readily valid on any field. The problem of counting (or bounding) torsion points on the theta divisor has interested many people, e.g. [@BoxallGrant] and very recently [@Torsionthetadivisors] in general dimension. Notice that the results above give the expected bound in the case $g=2$, but we do not know how much we can lower the bound $\sqrt{2} n^2$ in the case of jacobians.
As $[\infty]$ is a Weierstrass point, the divisor $2 [\infty]$ is canonical. Conversely, if a degree two divisor $D$ satisfies $\ell(D) := \dim H^0 (C,{{\mathcal O}}_C(D)) \geq 2$, then it is canonical. Indeed, by Riemann-Roch theorem, this implies that $\ell(2 [\infty] - D) \geq 1$ but this divisor is of degree 0, hence it is principal and $D$ is canonical. Now, let $x,y,z,t$ be four points of $C$ such that $\phi_\infty(x) + \phi_\infty(y) = \phi_\infty(z) + \phi_\infty(t) $ in $J$. This implies that $[x] + [y] - [z] - [t]$ is the divisor of some function $f$, and then either $f$ is constant (i.e. $\{x,y\} = \{z,t\}$), either $\ell([z] + [t]) \geq 2$ hence $[z] + [t]$ is canonical by the argument above, and in this case the points $P =\phi_\infty(z)$ and $Q=\phi_\infty(t)=0$ of $\widetilde{C}$ satisfy $P+Q=0$ in $J$, which proves $(a)$.
Now, for $n \geq 2$ even, let us denote $\widetilde{C} [n] := \widetilde{C} \cap J[n]$. The summing map from $\widetilde{C} [n]^2$ to $J[n]$ has a fiber of cardinal $|\widetilde{C}[n]|$ above 0 and at most 2 above any other point of $J[n]$ by $(a)$, hence the inequality of degree two $$|\widetilde{C}[n]|^2 \leq |\widetilde{C}[n]| + 2 (n^4 - 1),$$ from which we directly obtain $(c)$. In the case $n=2$, it is enough to see that $ 2 \phi_\infty(x) = 0$ if and only if $2 [x]$ is canonical if and only if $x$ is a Weierstrass divisor, which gives $(b)$.
We can now define the divisors we will consider for our Runge-type theorem, with the following notation.
**Convention**
Until the end of this article, the expression “a couple $(a,b) \in ({{\mathbb{Z}}}/n {{\mathbb{Z}}})^4$ (resp. ${{\mathbb{Z}}}^4, {{\mathbb{Q}}}^4$ )” is a shorthand to designate the row vector with four coefficients where $a \in ({{\mathbb{Z}}}/n{{\mathbb{Z}}})^2$ (resp. ${{\mathbb{Z}}}^2$, ${{\mathbb{Q}}}^2$ ) make up the first two coefficients and $b$ the last two coefficients.
\[defipropdivthetaA2ncomplexes\]
Let $n \in {{\mathbb{N}}}_{\geq 2}$ even.
$(a)$ A couple $(a,b) \in ({{\mathbb{Z}}}/n{{\mathbb{Z}}})^4$ is called *regular* if it is *not* of the shape $((n/2)a',(n/2)b')$ with $(a',b') \in (({{\mathbb{Z}}}/2{{\mathbb{Z}}})^2)^2$ such that $a' {}^t b' = 1 \mod 2$. There are exactly 6 couples $(a,b)$ not satisfying this condition, which we call *singular*.
$(b)$ If $(a,b) \in ({{\mathbb{Z}}}/n{{\mathbb{Z}}})^4$ is regular, for every lift $(\widetilde{a}, \widetilde{b}) \in {{\mathbb{Z}}}^4$ of $(a,b)$, the function $\vartheta_{\widetilde{a}/n, \widetilde{b}/n}^{8n}$ is a *nonzero* Siegel modular form of degree 2, weight $4n$ and level $n$, independent of the choice of lifts. The *theta divisor associated to $(a,b)$*, denoted by $(D_{n,a,b})_{{\mathbb{C}}}$, is the Weil divisor of zeroes of this Siegel modular form on $A_2(n)^S_{{\mathbb{C}}}$.
$(c)$ For $(a,b)$ and $(a',b')$ regular couples in $({{\mathbb{Z}}}/n{{\mathbb{Z}}})^4$, the Weil divisors $(D_{n,a,b})_{{\mathbb{C}}}$ and $(D_{n,a',b'})_{{\mathbb{C}}}$ are equal if and only if $(a,b) = \pm (a',b')$. Hence, the set of regular couples defines exactly $n^4/2 + 2$ pairwise distinct Weil divisors.
The singular couples correspond to what are called *odd characteristics* by Igusa. The proof below uses Fourier expansions to figure out which theta functions are nontrivial or proportional, but we conjecture the stronger result that $(D_{n,a,b})_{{\mathbb{C}}}$ and $(D_{n,a',b'})_{{\mathbb{C}}}$ are set-theoretically distinct (i.e. even without counting the multiplicities) unless $(a,b) = \pm (a',b')$. Such a result seems natural as the image of a curve into its jacobian should generically not have any other symmetry than $[-1]$, but we could not obtain it by looking at the simpler case (in $A_2(n)^S_{{\mathbb{C}}}$) of the products of elliptic curves: if $(a,b)$ and $(a',b')$ are both multiples of a primitive vector $v \in (1/n){{\mathbb{Z}}}^4$, it is tedious but straighforward to see that the theta constants $\vartheta_{a,b}$ and $\vartheta_{a',b'}$ vanish on the same products of elliptic curves. Hence, to prove that the reduced divisors of $(D_{n,a,b})_{{\mathbb{C}}}$ and $(D_{n,a',b'})_{{\mathbb{C}}}$ are distinct unless $(a,b) = \pm (a',b')$, one needs to exhibit a curve $C$ whose jacobian isomorphic to $A_\tau$ contains $\pi_\tau(a \tau + b)$ but not $\pi_\tau(a' \tau + b')$ in its theta divisor.
Notice that this will not be a problem for us later because all our arguments for Runge are set-theoretic, and Proposition \[propdivthetaproduitCE\] and \[propnombrepointsdivthetajacobienne\] are not modified if some of the divisors taken into account are equal.
$(a)$ By construction, for any even $n \geq 2$, the number of singular couples $(a,b) \in ({{\mathbb{Z}}}/n{{\mathbb{Z}}})^4$ is the number of couples $(a',b') \in ({{\mathbb{Z}}}/2{{\mathbb{Z}}})^4$ such that $a' {}^t b' = 1 \mod 2$, and we readily see there are exactly six of them, namely $$(0101), (1010), (1101), (1110), (1011) \textrm{ and }(0111).$$ For $(b)$ and $(c)$, the modularity of the function comes from Definition-Proposition \[defipropcomplexthetafunctions\] $(c)$ hence we only have to prove that it is nonzero when $(a,b)$ is regular. To do this, we will use the Fourier expansion of this modular form (for more details on Fourier expansions of Siegel modular forms, see chapter 4 of [@Klingen]), and simply prove that it has nonzero coefficients. This is also how we will prove the $\vartheta_{a,b}$ are distinct.
To shorten the notations, given an initial couple $(a,b) \in ({{\mathbb{Z}}}/n{{\mathbb{Z}}})^4$, we consider instead $(\tilde{a}/n, \tilde{b}/n) \in {{\mathbb{Q}}}^4$ for some lift $(\tilde{a}, \tilde{b})$ of $(a,b)$ in ${{\mathbb{Z}}}^4$) and by abuse of notation we renote it $(a,b)$ for simplicity. Regularity of the couple translates into the fact that $(a,b)$ is different from six possibles values modulo ${{\mathbb{Z}}}^4$, namely $$\left(0 \frac{1}{2}0\frac{1}{2}\right),\left(\frac{1}{2}0\frac{1}{2}0 \right),\left(\frac{1}{2}\frac{1}{2}0\frac{1}{2}\right), \left(\frac{1}{2}\frac{1}{2}\frac{1}{2}0\right), \left(\frac{1}{2}0\frac{1}{2}\frac{1}{2}\right)\left(0\frac{1}{2}\frac{1}{2}\frac{1}{2}\right)$$ by $(a)$, which we will assume now. We also fix $n \in {{\mathbb{N}}}$ even such that $(na,nb) \in {{\mathbb{Z}}}^4$.
Recall that $$\label{eqformulationsimplevarthetapourdevFourier}
\vartheta_{a,b} (\tau) = e^{i \pi a^t b} \sum_{k \in {{\mathbb{Z}}}^2} e^{ i \pi (k+a) \tau{}^t(k+a) + 2 i \pi k^t b}$$ by and . Therefore, for any symmetric matrix $S \in M_2({{\mathbb{Z}}})$ such that $S/(2n^2)$ is half-integral (i.e. with integer coefficients on the diagonal, and half-integers otherwise), we have $$\forall \tau \in {{\mathcal H}}_2, \quad \vartheta_{a,b} (\tau + S) = \vartheta_{a,b} (\tau),$$ because for every $k \in {{\mathbb{Z}}}^2$, $$(k+a)S^t (k+a) \in 2 {{\mathbb{Z}}}.$$ Hence, the function $\vartheta_{a,b}$ admits a Fourier expansion of the form $$\vartheta_{a,b} (\tau) = \sum_{T} a_T e^{ 2 i \pi {\operatorname{Tr}}(T\tau)},$$ where $T$ runs through all the matrices of $S_2({{\mathbb{Q}}})$ such that $(2n^2) T$ is half-integral. This Fourier expansion is unique, because for any $\tau \in {{\mathcal H}}_2$ and any $T$, we have $$(2 n^2) a_T = \int_{[0,1]^4} \vartheta_{a,b} (\tau +x) e^{- 2 i \pi {\operatorname{Tr}}(T(\tau+x))} dx.$$ In particular, the function $\vartheta_{a,b}$ is zero if and only if all its Fourier coefficients $a_T$ are zero, hence we will directly compute those, which are almost directly given by . For $a=(a_1,a_2) \in {{\mathbb{Q}}}^2$ and $k=(k_1,k_2) \in {{\mathbb{Z}}}^2$, let us define $$T_{a,k} = \begin{pmatrix}
(k_1+a_1)^2 & (k_1+a_1)(k_2 + a_2) \\ (k_1+a_1)(k_2+a_2) & (k_2+a_2)^2
\end{pmatrix},$$ so that $$\label{eqpresquedevFouriervartheta}
\vartheta_{a,b} (\tau) = e^{ i \pi a{}^t b} \sum_{k \in {{\mathbb{Z}}}^2} e^{ 2 i \pi k {}^t b} e^{ i \pi {\operatorname{Tr}}(T_{a,k} \tau)}$$ by construction. It is not yet exactly the Fourier expansion, because we have to gather the $T_{a,k}$ giving the same matrix $T$ (and this is where we will use regularity). Clearly, $$T_{a,k} = T_{a',k'} {\Longleftrightarrow}(k+a) = \pm (k' +a').$$ If $2a \notin {{\mathbb{Z}}}^2$, the function $k \mapsto T_{a,k}$ is injective, so is the Fourier expansion of $\vartheta_{a,b}$, with clearly nonzero coefficients, hence $\vartheta_{a,b}$ is nonzero.
If $2a = A \in {{\mathbb{Z}}}^2$, for every $k,k'\in {{\mathbb{Z}}}^2$, we have $(k+a) = \pm (k'+a)$ if and only if $k=k'$ or $k+k' = A$, so the Fourier expansion of $\vartheta_{a,b}$ is $$\label{eqpdevFouriervarthetamauvaiscas}
\vartheta_{a,b} (\tau) = \frac{e^{i \pi a^t b}}{2} \sum_{T} \sum_{\substack{k,k' \in {{\mathbb{Z}}}^2 \\ T_{k,a} = T_{k',a} = T}} (e^{ 2 i \pi k^t b} + e^{ 2 i \pi (-A-k)^t b}) e^{ i \pi {\operatorname{Tr}}(T \tau)}.$$ Therefore, the coefficients of this Fourier expansion are all zero if and only if, for every $k \in {{\mathbb{Z}}}^2$, $$e^{ 2 i \pi (2 k + A)^t b} = -1,$$ i.e. if and only if $b \in (1/2) {{\mathbb{Z}}}$ and $(-1)^{4 a^t b} = -1$, and this is exactly singularity of the couple $(a,b)$ which proves $(b)$.
Now, let $(a,b)$ and $(a',b')$ in $(1/n) {{\mathbb{Z}}}^4$ regular couples (translated in ${{\mathbb{Q}}}^4$ as above), such that $(na,nb)$ and $(na',nb')$ modulo ${{\mathbb{Z}}}^4$ have the same associated theta divisor on $A_2(n)^S_{{\mathbb{C}}}$. Then, the function $$\frac{\vartheta_{a,b}^{8n}}{\vartheta_{a',b'}^{8n}}$$ induces a meromorphic function on $A_2(n)^S_{{\mathbb{C}}}$ whose divisor is $0$ hence a constant function, which implies that $\vartheta_{a,b} = \lambda \vartheta_{a',b'}$ for some $
\lambda \in {{\mathbb{C}}}^*$. As these functions depend (up to a constant) only on $(a,b)$ and $(a',b') \! \mod {{\mathbb{Z}}}^4$, one can assume that all the coefficients of $(a,b)$ and $(a',b')$ belong to $[-1/2,1/2[$, and we assume first that $a,a' \notin (1/2){{\mathbb{Z}}}^2$. Looking at the Fourier expansions gives that for every $k \in {{\mathbb{Z}}}^2$, $$e^{i \pi a^t b + 2 i \pi k^t b} = \lambda e^{i \pi a'^t b' + 2 i \pi k^t b'}.$$ Hence, we have $b= b' \mod {{\mathbb{Z}}}^2$ which in turns give $a= a' \mod {{\mathbb{Z}}}^2$ The same argument when $a$ or $a'$ belongs to $(1/2) {{\mathbb{Z}}}^2$ gives by the possibilities $b=-b'$ and $a=-a' \mod {{\mathbb{Z}}}^4$.
Hence, we proved that if $\vartheta_{a,b}$ and $\vartheta_{a',b'}$ are proportional, then $(a,b)= \pm (a',b') \mod {{\mathbb{Z}}}^4$,and the converse is straightforward.
These divisors have the following properties.
\[propproprietesDnabcomplexes\]
Let $n \in {{\mathbb{N}}}_{\geq 2}$ even.
$(a)$ For every regular $(a,b) \in ({{\mathbb{Z}}}/n{{\mathbb{Z}}})^4$, the divisor $(D_{n,a,b})_{{\mathbb{C}}}$ is ample.
$(b)$ For $n=2$, the ten divisors $(D_{2,a,b})_{{\mathbb{C}}}$ are set-theoretically pairwise disjoint outside the boundary $\partial A_2(2)_{{\mathbb{C}}}:= A_2(2)^S_{{\mathbb{C}}}\backslash A_2(2)_{{\mathbb{C}}}$, and their union is exactly the set of moduli of products of elliptic curves (with any symplectic basis of the 2-torsion).
$(c)$ For $(A,\lambda,\alpha_n)$ a principally polarised complex abelian surface with symplectic structure of level $n$ :
- If $(A,\lambda)$ is a product of elliptic curves, the moduli of $(A,\lambda,\alpha_n)$ belongs to exactly $n^2 - 3$ divisors $(D_{n,a,b})_{{\mathbb{C}}}$.
- Otherwise, the point $(A,\lambda,\alpha_n)$ belongs to at most $(\sqrt{2}/2)n^2 + 1/4$ divisors $(D_{n,a,b})_{{\mathbb{C}}}$.
$(a)$ The divisor $(D_{n,a,b})_{{\mathbb{C}}}$ is by definition the Weil divisor of zeroes of a Siegel modular form of order 2, weight $4n$ and level $n$, hence of a section of $L^{\otimes 4n}$ on $A_2(n)_{{\mathbb{C}}}^S$. As $L$ is ample on $A_2(n)^S_{{\mathbb{C}}}$ (Definition-Proposition \[defipropSatakecompactification\] $(c)$), the divisor $(D_{n,a,b})_{{\mathbb{C}}}$ is ample.
Now, we know that every complex pair $(A,\lambda)$ is isomorphic to some $(A_\tau,\lambda_\tau)$ with $\tau \in {{\mathcal H}}_2$ (Definition-Proposition \[defipropuniformcomplexabvar\]). If $(A,\lambda)$ is a product of elliptic curves, the theta divisor of $(A,\lambda,\alpha_2)$ contains exactly seven 2-torsion points (Proposition \[propdivthetaproduitCE\]), only one of comes from a regular pair, i.e. $(A,\lambda,\alpha_2)$ is contained in exactly one of the ten divisors. If $(A,\lambda)$ is not a product of elliptic curves, it is a jacobian (Proposition \[propsurfabnonproduitCEetdivtheta\]) and the theta divisor of $(A,\lambda,\alpha_2)$ only contains the six points coming from singular pairs (Proposition \[propnombrepointsdivthetajacobienne\]) i.e. $(A,\lambda,\alpha_2)$ does not belong to any of the ten divisors, which proves $(b)$.
To prove $(c)$, we use the same propositions for general $n$, keeping in mind that we only count as one the divisors coming from opposite values of $(a,b)$ : for products of elliptic curves, this gives $(2n^2 - 16)/2 + 7$ divisors (the 7 coming from the 2-torsion), and for jacobians, this gives $(\sqrt{2}/2)n^2 + 1/4$ (there are no nontrivial 2-torsion points to consider here).
We will now give the natural divisors extending $(D_{n,a,b})_{{\mathbb{C}}}$ on the integral models ${{\mathcal A}}_2(n)$ (Definition-Proposition \[defipropalgmodulispaces\]).
\[defidivthetasurespacemoduleentier\]
Let $n \in {{\mathbb{N}}}_{\geq 2}$ even.
For every regular $(a,b) \in ({{\mathbb{Z}}}/n{{\mathbb{Z}}})^4$, the divisor $(D_{n,a,b})_{{\mathbb{C}}}$ is the geometric fiber at ${{\mathbb{C}}}$ of an effective Weil divisor $D_{n,a,b}$ on ${{\mathcal A}}_2(n)$, such that the moduli of a triple $(A,\lambda,\alpha_n)$ (on a field $k$ of characteristic prime to $n$) belongs to $D_{n,a,b} (k)$ if and only if the point of $A[n](\overline{k})$ of coordinates $(a,b)$ for $\alpha_n$ belongs to the theta divisor $\Theta_{A,\lambda,\alpha_n}$ (Definition-Proposition \[defipropthetadiviseurcanonique\]).
This amounts to giving an algebraic construction of the $D_{n,a,b}$ satisfying the wanted properties. The following arguments are extracted from Remark I.5.2 of [@ChaiFaltings]. Let $\pi : A \rightarrow S$ an abelian scheme and ${{\mathcal L}}$ a symmetric invertible sheaf on $A$, relatively ample on $S$ and inducing a principal polarisation on $A$. If $s : S \rightarrow A$ is a section of $A$ on $S$, the evaluation at $s$ induces an ${{\mathcal O}}_S$-module isomorphism between $\pi_*{{\mathcal L}}$ and $s^*{{\mathcal L}}$. Now, if $s$ is of $n$-torsion in $A$, for $e : S \rightarrow A$ the zero section, the sheaf $(s^* {{\mathcal L}})^{\otimes 2n}$ is isomorphic to $(e^* {{\mathcal L}})^{\otimes 2n}$, i.e. trivial. We denote by $\omega_{A/S}$ the invertible sheaf on $S$ obtained as the determinant of the sheaf of invariant differential forms on $A$, and the computations of Theorem I.5.1 and Remark I.5.2 of [@ChaiFaltings] give $8 \pi_* {{\mathcal L}}= - 4 \omega_{A/S}$ in ${\operatorname{Pic}}(A/S)$. Consequenltly, the evaluation at $s$ defines (after a choice of trivialisation of $(e^* {{\mathcal L}})^{\otimes 2n}$ and putting to the power $8n$) a section of $\omega_{A/S}^ {\otimes 4n}$. Applying this result on the universal abelian scheme (stack if $n \leq 2$) ${{\mathcal X}}_2(n)$ on ${{\mathcal A}}_2(n)$ , for every $(a,b) \in ({{\mathbb{Z}}}/n{{\mathbb{Z}}})^4$, the section defined by the point of coordinate $(a,b)$ for the $n$-structure on ${{\mathcal X}}_2(n)$ induces a global section $s_{a,b}$ of $\omega_{{{\mathcal X}}_2(n)/{{\mathcal A}}_2(n)}^{\otimes 4n}$, and we define $D_{n,a,b}$ as the Weil divisor of zeroes of this section. It remains to check that it satisfies the good properties.
Let $(A,\lambda,\alpha_n)$ be a triple over a field $k$ of characteristic prime to $n$, and $L$ the ample line bundle associated to it by Definition-Proposition \[defipropthetadiviseurcanonique\]. By construction, its moduli belongs to $D_{n,a,b}$ if and only if the unique (up to constant) nonzero section vanishes at the point of $A[n]$ of coordinates $(a,b)$ in $\alpha_n$, hence if and only if this point belongs to $\Theta_{A,\lambda,\alpha_n}$.
Finally, we see that the process described above applied to the universal abelian variety ${{\mathcal X}}_2(n)_{{\mathbb{C}}}$ of ${{\mathcal A}}_2(n)_{{\mathbb{C}}}$ (by means of explicit description of the line bundles as quotients) gives (up to invertible holomorphic functions) the functions $\vartheta_{\widetilde{a}/n,\widetilde{b}/n}^{8n}$, which proves that $(D_{n,a,b})_{{\mathbb{C}}}$ is indeed the geometric fiber of $D_{n,a,b}$ (it is easier to see that their complex points are the same, by Proposition \[propproprietesDnabcomplexes\] $(c)$ and the above characterisation applied to the field ${{\mathbb{C}}}$).
If one does not want to use stacks for $n=2$, one can consider for $(a,b) \in ({{\mathbb{Z}}}/2{{\mathbb{Z}}})^4$ the divisor $D_{4,2a,2b}$ which is the pullback of $D_{2,a,b}$ by the degeneracy morphism $A_2(4) \rightarrow A_2(2)$.
Tubular Runge theorems for abelian surfaces and their theta divisors {#subsectubularRungethmabsur}
--------------------------------------------------------------------
We can now prove a family of tubular Runge theorems for to the theta divisors $D_{n,a,b}$ (for even $n \geq 2$).
We will state the case $n=2$ first because its moduli interpretation is easier but the proofs are the same, as we explain below.
In the following results, the *boundary* of $A_2(n)^S_{{\mathbb{C}}}$ is defined as $\partial A_2(n)^S_{{\mathbb{C}}}:= A_2(n)^S_{{\mathbb{C}}}\backslash A_2(n)_{{\mathbb{C}}}$.
\[thmtubularRungeproduitCE\]
Let $U$ be an open neighbourhood of $\partial A_2(2)^S_{{\mathbb{C}}}$ in $A_2(2)^S_{{\mathbb{C}}}$ for the natural complex topology.
For any such $U$, we define ${{\mathcal E}}(U)$ the set of moduli $P$ of triples $(A,\lambda,\alpha_2)$ in ${{\mathcal A}}_2(2) ({\overline{{{\mathbb{Q}}}}})$ such that (choosing $L$ a number field of definition of the moduli) :
- The abelian surface $A$ has potentially good reduction at every finite place $w \in M_L$ (**tubular condition for finite places**).
- For any embedding $\sigma : L \rightarrow {{\mathbb{C}}}$, the image $P_\sigma$ of $P$ in ${{\mathcal A}}_2(2)_{{\mathbb{C}}}$ is outside of $U$ (**tubular condition for archimedean places**).
- The number $s_L$ of non-integrality places of $P$, i.e. places $w \in M_L$ such that
- either $w$ is above $M_L^{\infty}$ or $2$,
- or the semistable reduction modulo $w$ of $(A,\lambda)$ is a product of elliptic curves
satisfies the **tubular Runge condition** $$s_L < 10.$$
Then, for every choice of $U$, the set ${{\mathcal E}}(U)$ is **finite**.
\[thmtubularRungegeneral\]
Let $n \geq 4$ even.
Let $U$ be an open neighbourhood of $\partial A_2(n)^S_{{\mathbb{C}}}$ in $A_2(n)^S_{{\mathbb{C}}}$ for the natural complex topology.
For any such $U$, we define ${{\mathcal E}}(U)$ the set of moduli $P$ of triples $(A,\lambda,\alpha_2)$ in ${{\mathcal A}}_2(n) ({\overline{{{\mathbb{Q}}}}})$ such that (choosing $L \supset {{\mathbb{Q}}}(\zeta_n)$ a number field of definition of the triple) :
- The abelian surface $A$ has potentially good reduction at every place $w \in M_L^{\infty}$ (**tubular condition for finite places**).
- For any embedding $\sigma : L \rightarrow {{\mathbb{C}}}$, the image $P_\sigma$ of $P$ in ${{\mathcal A}}_2(n)_{{\mathbb{C}}}$ is outside of $U$ (**tubular condition for archimedean places**).
- The number $s_P$ of non-integrality places of $P$, i.e. places $w \in M_L$ such that
- either $w$ is above $M_L^{\infty}$ or a prime factor of $n$,
- or the theta divisor of the semistable reduction modulo $w$ of $(A,\lambda,\alpha_n)$ contains an $n$-torsion point which is not one of the six points coming from odd characteristics,
satisfies the **tubular Runge condition** $$(n^2 - 3) s_P < \frac{n^4}{2} + 2.$$
Then, for every choice of $U$, the set of points ${{\mathcal E}}(U)$ is **finite**.
We put an emphasis on the conditions given in the theorem to make it easier to identify how it is an application of our main result, Theorem \[thmRungetubulaire\]. The tubular conditions (archimedean and finite) mean that our points $P$ do not belong to some tubular neighbourhood ${{\mathcal V}}$ of the boundary. We of course chose the boundary as our closed subset to exclude because of its modular interpretation for finite places. The places above $M_L^{\infty}$ or a prime factor of $n$ are automatically of non-integrality for our divisors because the model ${{\mathcal A}}_2(n)$ is not defined at these places. Finally, the second possibility to be a place of non-integrality straightforwardly comes from the moduli interpretation of the divisors $D_{n,a,b}$ (Definition \[defidivthetasurespacemoduleentier\]). All this is detailed in the proof below.
To give an example of how we can obtain an explicit result in practice, we prove in section \[sectionexplicitRunge\] an explicit (and even theoretically better) version of Theorem \[thmtubularRungeproduitCE\].
It would be more satisfying (and easier to express) to give a tubular Runge theorem for which the divisors considered are exactly the irreducible components parametrising the products of elliptic curves. Unfortunately, except for $n=2$, there is a serious obstruction because those divisors are not ample, and there are even reasons to suspect they are not big. We have explained in Remark \[remampledifficilepourA2\] why proving the ampleness for general divisors on $A_2(n)^S_{{\mathbb{C}}}$ is difficult.
It would also be morally satisfying to give a better interpretation of the moduli of $D_{n,a,b}$ for $n >2$, i.e. not in terms of the theta divisor, but maybe of the structure of the abelian surface if possible (nontrivial endomorphisms ? isogenous to products of elliptic curves ?). As far as the author knows, the understanding of abelian surfaces admitting some nontrivial torsion points on their theta divisor is still very limited.
Finally, to give an idea of the margin the tubular Runge condition gives for $n>2$ (in terms of the number of places which are not “taken” by the automatic bad places), we can easily see that the number of places of ${{\mathbb{Q}}}(\zeta_n)$ which are archimedean or above a prime factor of $n$ is less than $n/2$. Hence, we can find examples of extensions $L$ of ${{\mathbb{Q}}}(\zeta_n)$ of degree $n$ such that some points defined on it still can satisfy tubular Runge condition. This is also where using the full strength of tubular Runge theorem is crucial: for $n=2$, one can compute that some points of the boundary are contained in 6 different divisors $D_{2,a,b}$, and for general even $n$, a similar analysis gives that the intersection number $m_{\emptyset}$ is quartic in $n$, which leaves a lot less margin for the places of non-integrality (or even none at all).
As announced, this result is an application of the tubular Runge theorem (Theorem \[thmRungetubulaire\]) to ${{\mathcal A}}_2(n)^S_{{{\mathbb{Q}}}(\zeta_n)}$ (Definition-Proposition \[defipropalgmodulispaces\]) and the divisors $D_{n,a,b}$ (Definition \[defidivthetasurespacemoduleentier\]), whose properties will be used without specific mention. We reuse the notations of the hypotheses of Theorem \[thmRungetubulaire\] to explain carefully how it is applied.
***(H0)*** The field of definition of $A_2(n)^S_{{\mathbb{C}}}$ is ${{\mathbb{Q}}}(\zeta_n)$, and the ring over which our model ${{\mathcal A}}_2(n)^S$ is built is ${{\mathbb{Z}}}[ \zeta_n, 1/n]$, hence $S_0$ is made up with all the archimedean places and the places above prime factors of $n$. There is no need for a finite extension here as all the $D_{n,a,b}$ are divisors on ${{\mathcal A}}_2(n)^S$.
***(H1)*** The model ${{\mathcal A}}_2(n)^S_{{\mathbb{C}}}$ is indeed normal projective, and we know that the $D_{n,a,b}$ are effective Weil divisors hence Cartier divisors up to multiplication by some constant by Proposition \[proprationalPicardSiegel\]. For any finite extension $L$ of ${{\mathbb{Q}}}(\zeta_n)$, the number of orbits $r_L$ is the number of divisors $D_{n,a,b}$ (as they are divisors on the base model), i.e. $n^4/2 + 2$ (Proposition \[propproprietesDnabcomplexes\] $(c)$).
***(H2)*** The chosen closed subset $Y$ of ${{\mathcal A}}_2(n)^S_{{\mathbb{Q}}}(\zeta_n)$ is the boundary, namely $$\partial {{\mathcal A}}_2(n)^S_{{{\mathbb{Q}}}(\zeta_n)} = {{\mathcal A}}_2(n)^S_{{{\mathbb{Q}}}(\zeta_n)} \backslash {{\mathcal A}}_2(n)_{{{\mathbb{Q}}}(\zeta_n)}.$$ We have to prove that the tubular conditions given above correspond to a tubular neighbourhood. To do this, let ${{\mathcal Y}}$ be the boundary ${{\mathcal A}}_2(n)^S \backslash {{\mathcal A}}_2(n)$ and $g_1, \cdots, g_s$ homogeneous generators of the ideal of definition of ${{\mathcal Y}}$ after having fixed a projective embedding of ${{\mathcal A}}_2(n)$. Let us find an $M_{{{\mathbb{Q}}}(\zeta_n)}$-constant such that ${{\mathcal E}}(U)$ is included in the tubular neighbourhood of $\partial {{\mathcal A}}_2(n)^S_{{\mathbb{Q}}}(\zeta_n)$ in $A_2(n)^S_{{{\mathbb{Q}}}(\zeta_n)}$ associated to ${{\mathcal C}}$ and $g_1, \cdots, g_k$. For the places $w$ not above $M_L^\infty$ or a prime factor of $n$, the fact that $P = (A,\lambda,\alpha_n)$ does not reduce in $Y$ modulo $w$ is exactly equivalent to $A$ having potentially good reduction at $w$ hence we can choose $c_v= 0$ for the places $v$ of ${{\mathbb{Q}}}(\zeta_n)$ not archimedean and not dividing $n$. For archimedean places, belonging to $U$ for an embedding $\sigma : L \rightarrow {{\mathbb{C}}}$ implies that $g_1, \cdots, g_n$ are small, and we just have to choose $c_v$ stricly larger than the maximum of the norms of the $g_i(U \cap V_j)$ (in the natural affine covering $(V_j)_j$ of the projective space), independant of the choice of $v \in M_{{{\mathbb{Q}}}(\zeta_n)}^\infty$. Finally, we have to consider the case of places above a prime factor of $n$. To do this, we only have to recall that having potentially good reduction can be given by integrality of some quotients of the Igusa invariants at finite places, and these invariants are modular forms on $\Gamma_2(1)$. We can add those who vanish on the boundary to the homogeneous generators $g_1, \cdots, g_n$ and consider $c_v=0$ for these places as well. This is explicitly done in part \[subsecplacesabove2\] for $A_2(2)$.
***(TRC)*** As said before, there are $n^4/2 + 2$ divisors considered, and their generic fibers are ample by Proposition \[propproprietesDnabcomplexes\]. Furthermore, by Propositions \[propdivthetaproduitCE\] and \[propnombrepointsdivthetajacobienne\], outside the boundary, at most $(n^2 - 3)$ can have nonempty common intersection, and this exact number is attained only for products of elliptic curves, (as $n^2 - 3 = 2(n^2 - 4)/2 + 1$, separating the regular 2-torsion pairs and regular non-2-torsion pairs up to $\pm 1$).
This gives the tubular Runge condition $$(n^2 - 3) s_L < n^4/2 + 2,$$ which concludes the proof.
For $n=2$, the union of the ten $D_{2,a,b}$ is made up with the moduli of products of elliptic curves, and they are pairwise disjoint outside $\partial A_2(2)$ (Proposition \[propproprietesDnabcomplexes\] $(b)$), hence the simply-expressed condition $s_L<10$ in this case.
The explicit Runge result for level two {#sectionexplicitRunge}
=======================================
To finish this paper, we improve and make explicit the finiteness result of Theorem \[thmtubularRungeproduitCE\], as a proof of principle of the method.
Before stating Theorem \[thmproduitCEexplicite\], we need some notations. In level two, the auxiliary functions are deduced from the ten even theta constants of characteristic two, namely the functions $\Theta_{m/2} (\tau)$ (notation ), with the quadruples $m$ going through $$\label{eqevenchartheta}
E = \{(0000),(0001),(0010),(0011),(0100),(0110),(1000),(1001),(1100),(1111) \}$$ (see subsections \[subsecthetadivabvar\] and \[subsecthetadivabsur\] for details). We recall ([@vdG82], Theorem 5.2) that these functions define an embedding $$\label{eqdefplongementpsi}
{\begin{array}{c|ccl} \psi: & A_2(2) & \longrightarrow & {\mathbb{P}}^9 \\
& \overline{\tau} & \longmapsto & (\Theta_{m/2}^4 (\tau))_{m \in E} \end{array}}$$ which induces an isomorphism between $A_2(2)^S_{{\mathbb{C}}}$ and the subvariety of ${\mathbb{P}}^9$ (with coordinates indexed by $m \in E$) defined by the linear equations $$\begin{aligned}
x_{1000} - x_{1100} + x_{1111} - x_{1001} & = & 0 \\
x_{0000} - x_{0001} - x_{0110} - x_{1100} & = & 0 \\
x_{0110} - x_{0010} + x_{1111} + x_{0011} & = & 0 \\
x_{0100} - x_{0000} + x_{1001} + x_{0011} & = & 0 \\
x_{0100} - x_{1000} + x_{0001} - x_{0010} & = & 0 \end{aligned}$$ (which makes it a subvariety of ${\mathbb{P}}^4$) together with the quartic equation $$\left( \sum_{m \in E} x_m^2 \right)^2 - 4 \sum_{m \in E} x_m^4 = 0.$$
For the attentive reader, the first linear equation has sign $(+1)$ in $x_{1111}$ whereas it is $(-1)$ in [@vdG82], as there seems to be a typographic mistake there : we have realised it during our computations on Sage in part \[subsecplacesabove2\] and found the right sign back from Igusa’s relations ([@Igusa64bis], Lemma 1 combined with the proof of Theorem 1).
There is a natural definition for a tubular neighbourhood of $Y = \partial A_2(2)$: for a finite place $v$, as in Theorem \[thmtubularRungeproduitCE\], we choose $V_v$ as the set of triples $P = \overline{(A,\lambda,\alpha_2)}$ where $A$ has potentially bad reduction modulo $v$. To complete it with archimedean places, we use the classical fundamental domain for the action of ${\operatorname{Sp}}_4({{\mathbb{Z}}})$ on ${{\mathcal H}}_2$ denoted by ${{\mathcal F}}_2$ (see [@Klingen], section I.2 for details). Given some parameter $t \geq \sqrt{3}/2$, the neighbourhood $V(t)$ of $\partial A_2(2)_{{\mathbb{C}}}^S$ in $A_2(2)^S_{{\mathbb{C}}}$ is made up with the points $P$ whose lift $\tau$ in ${{\mathcal F}}_2$ (for the usual quotient morphism ${{\mathcal H}}_2 \rightarrow A_2(1)_{{\mathbb{C}}}$) satisfies ${\operatorname{Im}}(\tau_4) \geq t$, where $\tau_4$ is the lower-right coefficient of $\tau$. We choose $V(t)$ as the archimedean component of the tubular neighbourhood for every archimedean place. The reader knowledgeable with the construction of Satake compactification will have already seen such neighbourhoods of the boundary.
Notice that for a point $P=\overline{(A,\lambda,\alpha_2)} \in A_2(2)(K)$, the abelian surface $A$ is only defined over a finite extension $L$ of $K$, but for prime ideals ${{\mathfrak{P}}}_1$ and ${{\mathfrak{P}}}_2$ of ${{\mathcal O}}_L$ above the same prime ideal ${{\mathfrak{P}}}$ of ${{\mathcal O}}_K$, the reductions of $A$ modulo ${{\mathfrak{P}}}_1$ and ${{\mathfrak{P}}}_2$ are of the same type because $P \in A_2(2) (K)$. This justifies what we mean by “semistable reduction of $A$ modulo ${{\mathfrak{P}}}$” below.
\[thmproduitCEexplicite\] Let $K$ be a number field and $P=\overline{(A,\lambda,\alpha_2)} \in A_2(2)(K)$ where $A$ has potentially good reduction at every finite place.
Let $s_P$ be the number of prime ideals ${{\mathfrak{P}}}$ of ${{\mathcal O}}_K$ such that the semistable reduction of $A$ modulo ${{\mathfrak{P}}}$ is a product of elliptic curves. We denote by $h_{{\mathcal F}}$ the stable Faltings height of $A$.
$(a)$ If $K={{\mathbb{Q}}}$ or an imaginary quadratic field and $$|s_P|<4$$ then $$h(\psi(P)) \leq 10.75, \quad h_{{\mathcal F}}(A) \leq 1070.$$
$(b)$ Let $t \geq \sqrt{3}/2$ be a real number. If for any embedding $\sigma : K \rightarrow {{\mathbb{C}}}$, the point $P_\sigma \in A_2(2)_{{\mathbb{C}}}$ does not belong to $V(t)$, and $$|s_P| + |M_K^{\infty}| < 10$$ then $$h(\psi(P)) \leq 4 \pi t + 6.14, \quad h_{{\mathcal F}}(A) \leq 2 \pi t + 535 \log(2 \pi t + 9)$$
The Runge condition for $(b)$ is a straightforward application of our tubular Runge theorem. For $(a)$, we did not assume anything on the point $P$ at the (unique) archimedean place, which eliminates six divisors when applying Runge’s method here, hence the different Runge condition here (see Remark \[remRungetubulaire\] $(b)$).
The principle of proof is very simple: we apply Runge’s method to bound the height of $\psi(P)$ when $P$ satisfies the conditions of Theorem \[thmtubularRungeproduitCE\], and using the link between this height and Faltings height given in ([@Pazuki12b], Corollary 1.3), we know we will obtain a bound of the shape $$h_{{\mathcal F}}(P) \leq f(t)$$ where $f$ is an explicit function of $t$, for every point $P$ satisfying the conditions of Theorem \[thmtubularRungeproduitCE\].
At the places of good reduction not dividing 2, the contribution to the height is easy to compute thanks to the theory of algebraic theta functions devised in [@Mumford66] and [@Mumford67]. The theory will be sketched in part \[subsecalgebraicthetafunctions\], resulting in Proposition \[propalgthetafoncetreduchorsde2\].
For the archimedean places, preexisting estimates due to Streng for Fourier expansions on each of the ten theta functions allow to make explicit how only one of them can be too small compared to the others, when we are out of $V(t)$. This is the topic of part \[subsecarchimedeanplaces\].
For the places above 2, the theory of algebraic theta functions cannot be applied. To bypass the problem, we use Igusa invariants (which behave in a well-known fashion for reduction in any characteristic) and prove that the theta functions are algebraic and “almost integral” on the ring of these Igusa invariants, with explicit coefficients. Combining these two facts in part \[subsecplacesabove2\], we will obtain Proposition \[propbornesfoncthetaaudessus2\], a less-sharp avatar of Proposition \[propalgthetafoncetreduchorsde2\], but explicit nonetheless.
Finally, we put together these estimates in part \[subsecfinalresultRungeCEexplicite\] and obtain the stated bounds on $h \circ \psi$ and the Faltings height.
Algebraic theta functions and the places of potentially good reduction outside of 2 {#subsecalgebraicthetafunctions}
-----------------------------------------------------------------------------------
The goal of this part is the following result.
\[propalgthetafoncetreduchorsde2\] Let $K$ be a number field and ${{\mathfrak{P}}}$ a maximal ideal of ${{\mathcal O}}_K$, of residue field $k({{\mathfrak{P}}})$ with characteristic different from 2. Let $P = \overline{(A, \lambda,\alpha_2)} \in A_2(2)(K)$. Then, $\psi(P) \in {\mathbb{P}}^9(K)$ and :
$(a)$ If the semistable reduction of $A$ modulo ${{\mathfrak{P}}}$ is a product of elliptic curves, the reduction of $\psi(P)$ modulo ${{\mathfrak{P}}}$ has exactly one zero coordinate, in other words every coordinate of $\psi(P)$ has the same ${{\mathfrak{P}}}$-adic norm except one which is strictly smaller.
$(b)$ If the semistable reduction of $A$ modulo ${{\mathfrak{P}}}$ is a jacobian of hyperelliptic curve, the reduction of $\psi(P)$ modulo ${{\mathfrak{P}}}$ has no zero coordinate, in other words every coordinate of $\psi(P)$ has the same ${{\mathfrak{P}}}$-adic norm.
To link $\psi(P)$ with the intrinsic behaviour of $A$, we use the theory of algebraic theta functions, devised in [@Mumford66] and [@Mumford67] (see also [@DavidPhilippon] and [@Pazuki12b]). As it is not very useful nor enlightening to go into detail or repeat known results, we only mention them briefly here. In the following, $A$ is an abelian variety of dimension $g$ over a field $k$ and $L$ an ample symmetric line bundle on $A$ inducing a principal polarisation $\lambda$. We also fix $n \geq 2$ even, assuming that all the points of $2n$-torsion of $A$ are defined over $k$ and ${\textrm{char}}(k)$ does not divide $n$ (in particular, we always assume ${\textrm{char}}(k) \neq 2$). Let us denote formally the Heisenberg group ${{\mathcal G}}(\underline{n})$ as the set $${{\mathcal G}}(\underline{n}) := k^* \times ({{\mathbb{Z}}}/n{{\mathbb{Z}}})^g \times ({{\mathbb{Z}}}/n{{\mathbb{Z}}})^g$$ equipped with the group law $$(\alpha,a,b) \cdot (\alpha',a',b') := (\alpha \alpha' e^{\frac{2 i \pi}{n} a{}^t b'},a+a',b+b')$$ (contrary to the convention of [@Mumford66], p.294, we identified the dual of $({{\mathbb{Z}}}/n{{\mathbb{Z}}})^g$ with itself). Recall that $A[n]$ is exactly the group of elements of $A(\overline{k})$ such that $T_x^* (L^{\otimes n}) \cong L ^{\otimes n}$ : indeed, it is the kernel of the morphism $\lambda_{L^{\otimes n}} = n \lambda$ from $A$ to $\widehat{A}$ (see proof of Proposition \[propambiguitedivthetaAL\]).
Given the datum of a *theta structure* on $L^{\otimes n}$, i.e. an isomorphism $\beta: {{\mathcal G}}(L^{\otimes n}) \cong {{\mathcal G}}(\underline{n})$ which is the identity on $k^*$ (see [@Mumford66], p. 289 for the definition of ${{\mathcal G}}(L^{\otimes n})$), one has a natural action of ${{\mathcal G}}(\underline{n})$ on $\Gamma(A,L^{\otimes n})$ (consequence of Proposition 3 and Theorem 2 of [@Mumford66]), hence for $n \geq 4$ the following projective embedding of $A$ : $$\label{eqplongementA}
{\begin{array}{c|ccl} \psi_\beta: & A & \longrightarrow & {\mathbb{P}}^{n^{2g} - 1}_k \\
& x & \longmapsto & \left( ((1,a,b)\cdot ( s_0^{\otimes n})) (x) \right)_{a,b \in ({{\mathbb{Z}}}/n{{\mathbb{Z}}})^g} \end{array}},$$ where $s_0$ is a nonzero section of $\Gamma(A,L)$, hence unique up to multiplicative scalar (therefore $\psi_\beta$ only depends on $\beta$). This embedding is not exactly the same as the one defined in ([@Mumford66], p. 298) (it has more coordinates), but the principle does not change at all. One calls *Mumford coordinates of $(A,L)$ associated to $\beta$* the projective point $\psi_\beta(0) \in {\mathbb{P}}^{n^{2g-1}}(k)$.
Now, one has the following commutative diagram whose rows are canonical exact sequences ([@Mumford66], Corollary of Theorem 1) $$\xymatrix{
0 \ar[r] & k^* \ar[d]^{=} \ar[r] & {{\mathcal G}}(L^{\otimes n}) \ar[r] \ar[d]^{\beta}& A[n] \ar[d]^{\alpha_n} \ar[r] & 0 \\
0 \ar[r] & k^* \ar[r] & {{\mathcal G}}(\underline{n}) \ar[r] & ({{\mathbb{Z}}}/n{{\mathbb{Z}}})^{2g} \ar[r] & 0,
}$$ where $\alpha_n$ is a symplectic level $n$ structure on $A[n]$ (Definition \[defibaseabvar\]), called *the symplectic level $n$ structure induced by $\beta$*. Moreover, for every $x \in A(k)$, the coordinates of $\psi_\beta (x)$ are (up to constant values for each coordinate, only depending on $\beta$) the $\vartheta_{A,L} ([n] x +\alpha_{n}^{-1} (a,b))$ (see Definition \[defithetadivisorabvar\]). In particular, for any $a,b \in ({{\mathbb{Z}}}/n {{\mathbb{Z}}})^g$, $$\label{eqliencoordonneesMumfordetdivtheta}
\psi_\beta(0)_{a,b} = 0 \Leftrightarrow \alpha_n^{-1} (a,b) \in \Theta_{A,L}.$$ Furthermore, for two theta structures $\beta,\beta'$ on $[n]^* L$ inducing $\alpha_n$, one sees that $\beta' \circ \beta^{-1}$ is of the shape $(\alpha,a,b) \mapsto (\alpha \cdot f(a,b),a,b)$, where $f$ has values in $n$-th roots of unity, hence $\psi_\beta$ and $\psi_{\beta'}$ only differ multiplicatively by $n$-th roots of unity.
Conversely, given the datum of a symplectic structure $\alpha_{2n}$ on $A[2n]$, there exists an unique *symmetric theta structure* on $[n]^* L$ which is *compatible* with some symmetric theta structure on $[2n]^* L$ inducing $\alpha_{2n}$ ([@Mumford66], p.317 and Remark 3 p.319). We call it the *theta structure on $[n]^* L$ induced by $\alpha_{2n}$*. Thus, we just proved that the datum of a symmetric theta structure on $[n]^*L$ is intermediary between a level $2n$ symplectic structure and a level $n$ symplectic structure (the exact congruence group is easily identified as $\Gamma_g(n,2n)$ with the notations of [@Igusa66]).
Now, for a triple $(A,L,\alpha_{2n})$ (notations of subsection \[subsecabvarSiegelmodvar\]), when $A$ is a complex abelian variety, there exists $\tau \in {{\mathcal H}}_g$ such that this triple is isomorphic to $(A_\tau,L_\tau,\alpha_{\tau,2n})$ (Definition-Proposition \[defipropuniformcomplexabvar\]). By definition of $L_\tau$ as a quotient , the sections of $L_\tau ^{\otimes n}$ canonically identify to holomorphic functions $\vartheta$ on ${{\mathbb{C}}}^g$ such that $$\label{eqsectionsLtaupuissancen}
\forall p,q \in {{\mathbb{Z}}}^g, \forall z \in {{\mathbb{C}}}^g, \quad \vartheta(z + p \tau + q) = e^{ - i \pi n \tau ^t n - 2 i \pi n ^t z} \vartheta(z),$$ and through this identification one sees (after some tedious computations) that the symmetric theta structure $\beta_\tau$ on $L_\tau^{\otimes n}$ induced by $\alpha_{\tau,2n}$ acts by $$((\alpha,a,b) \cdot \vartheta) (z) = \alpha \exp\left( \frac{i \pi}{n} \widetilde{a} \tau \widetilde{a} + \frac{2 i \pi}{n} \widetilde{a}{}^t (z+\widetilde{b}) \right) \vartheta \left( z+\frac{\widetilde{a}}{n} \tau + \frac{\widetilde{b}}{n} \right),$$ where $\widetilde{a},\widetilde{b}$ are lifts of $a,b$ in ${{\mathbb{Z}}}^g$ (the result does not depend on this choice by ). Therefore, by $\psi_\beta$ and the theta functions with characteristic (formula ), the Mumford coordinates of $(A,L,\alpha_{2n})$ (with the induced theta structure $\beta$ on $L^{\otimes n})$ are *exactly* the projective coordinates $$\left( \Theta_{\widetilde{a}/n,\widetilde{b}/n (\tau)}^n(\tau) \right)_{a,b \in \frac{1}{n} {{\mathbb{Z}}}^{2g} / {{\mathbb{Z}}}^{2g}} \in {\mathbb{P}}^{n^{2g-1}} ({{\mathbb{C}}}),$$ where the choices of lifts $\widetilde{a}$ and $\widetilde{b}$ for $a$ and $b$ still do not matter.
In particular, for every $\tau \in {{\mathcal H}}_2$, the point $\psi(\tau)$ can be intrinsically given as the squares of Mumford coordinates for $\beta_\tau$, where the six odd characteristics (whose coordinates vanish everywhere) are taken out. The result only depends on the isomorphism class of $(A_\tau,L_\tau,\alpha_{\tau,2})$, as expected.
Finally, as demonstrated in the paragraph 6 of [@Mumford67] (especially the Theorem p. 83), the theory of theta structures (and the associated Mumford coordinates) can be extended to abelian schemes (Definition \[defabelianscheme\]) (still outside characteristics dividing $2n$), and the Mumford coordinates in this context lead to an embedding of the associated moduli space in a projective space as long as the *type* of the sheaf is a multiple of 8 (which for us amounts to $8|n$). Here, fixing a principally polarised abelian variety $A$ over a number field $K$ and ${{\mathfrak{P}}}$ a prime ideal of ${{\mathcal O}}_K$ not above 2, this theory means thats given a symmetric theta structure on $(A,L)$ for $L^{\otimes n}$ where $8|n$, if $A$ has good reduction modulo ${{\mathfrak{P}}}$, this theta structure has a natural reduction to a theta structure on the reduction $(A_{{{\mathfrak{P}}}}, L_{{{\mathfrak{P}}}})$ for $L_{{\mathfrak{P}}}^{\otimes n}$, and this reduction is compatible with the reduction of Mumford coordinates modulo ${{\mathfrak{P}}}$. To link this with the reduction of coordinates of $\psi$, one just has to extend the number field $K$ of definition of $A$ so that all 8-torsion points of $A$ are defined over $K$ (in particular, the reduction of $A$ modulo ${{\mathfrak{P}}}$ is semistable), and consider a symmetric theta structure on $L^{\otimes 8}$. The associated Mumford coordinates then reduce modulo ${{\mathfrak{P}}}$, but their vanishing is linked to the belonging of $8$-th torsion points to $\Theta_{A_{{\mathfrak{P}}},L_{{\mathfrak{P}}}}$ by . The number of vanishing coordinates is then entirely determined in Propositions \[propdivthetaproduitCE\] and \[propnombrepointsdivthetajacobienne\], which proves Proposition \[propalgthetafoncetreduchorsde2\] (not forgetting the six ever-implicit odd characteristics).
Evaluating the theta functions at archimedean places {#subsecarchimedeanplaces}
----------------------------------------------------
We denote by ${{\mathcal H}}_2$ the Siegel half-space of degree 2, and by ${{\mathcal F}}_2$ the usual fundamental domain of this half-space for the action of ${\operatorname{Sp}}_4({{\mathbb{Z}}})$ (see [@Klingen], section I.2 for details). For $\tau \in {{\mathcal H}}_2$, we denote by $y_4$ the imaginary part of the lower-right coefficient of $\tau$.
\[proparchimedeanbound\] For every $\tau \in {{\mathcal H}}_2$ and a fixed real parameter $t \geq \sqrt{3}/2$, one has :
$(a)$ Amongst the ten even characteristics $m$ of $E$, at most six of them can satisfy $$|\Theta_{m/2} (\tau)| < 0.42 \max_{m' \in E} |\Theta_{m'/2} (\tau)|.$$
$(b)$ If the representative of the orbit of $\tau$ in the fundamental domain ${{\mathcal F}}_2$ satisfies $y_4 \leq t$, at most one of the ten even characteristics $m$ of $E$ can satisfy $$|\Theta_{m/2} (\tau)| < 1.22 e^{- \pi t} \max_{m' \in E} |\Theta_{m'/2} (\tau)|.$$
First, we can assume that $\tau \in {{\mathcal F}}_2$ as the inequalities $(a)$ and $(b)$ are invariant by the action of ${\operatorname{Sp}}_4({{\mathbb{Z}}})$, given the complete transformation formula of these theta functions ([@MumfordTata], section II.5). Now, using the Fourier expansions of the ten theta constants (mentioned in the proof of Definition-Proposition \[defipropdivthetaA2ncomplexes\]) and isolating their respective dominant terms (such as in [@Klingen], proof of Proposition IV.2), we obtain explicit estimates. More precisely, Proposition 7.7 of [@Strengthesis] states that, for every $\tau = \begin{pmatrix} \tau_1 & \tau_2 \\ \tau_2 & \tau_4 \end{pmatrix} \in {{\mathcal B}}_2$ (which is a domain containing ${{\mathcal F}}_2$), one has $$\begin{aligned}
\left| \Theta_{m/2}(\tau) - 1 \right| & < & 0.405, \quad {\scriptstyle m \in \{(0000)(0001),(0010),(0011) \}}. \\
\left| \frac{ \Theta_{m/2}(\tau)}{2 e^{ i \pi \tau_1/2}} - 1 \right| & < & 0.348, \quad {\scriptstyle m \in \{(0100),(0110) \}}. \\
\left| \frac{ \Theta_{m/2}(\tau)}{2 e^{ i \pi \tau_4/2}} - 1 \right| & < & 0.348, \quad {\scriptstyle m \in \{(1000),(1001) \}}. \\
\left| \frac{ \Theta_{m/2}(\tau)}{(\varepsilon_m + e^{2 i \pi \tau_2})e^{ i \pi (\tau_1 + \tau_4 - 2 \tau_3)/2}} - 1 \right| & < & 0.438,\quad {\scriptstyle m \in \{(1100),(1111) \}},
\end{aligned}$$ with $\varepsilon_m = 1$ if $m = (1100)$ and $-1$ if $m=(1111)$.
Under the assumption that $y_4 \leq t$ (which induces the same bound for ${\operatorname{Im}}\tau_1$ and $2 {\operatorname{Im}}\tau_2$), we obtain $$\begin{array}{rcccl}
0.595 & < & \left| \Theta_{m/2}(\tau) \right| & < & 1.405, \quad {\scriptstyle m \in \{(0000)(0001),(0010),(0011) \}}. \\
1.304 e^{ - \pi t/2} & < & \left| \Theta_{m/2}(\tau) \right| & < & 0.692, \quad {\scriptstyle m \in \{(0100),(0110),(1000),(1001)\}}. \\
1.05 e^{ - \pi t} & < & \left| \Theta_{m/2}(\tau) \right| & < & 0.855, \quad {\scriptstyle m = (1100)}. \\
& & \left| \Theta_{m/2}(\tau) \right| & < & 0.855, \quad {\scriptstyle m = (1111)}
\end{array}$$ Thus, we get $(a)$ with $0.595/1.405 > 0.42$, and $(b)$ with $1.05 e^{- \pi t}/ 0.855> 1.22 e^{- \pi t}$.
Computations with Igusa invariants for the case places above 2 {#subsecplacesabove2}
--------------------------------------------------------------
In this case, as emphasized before, it is not possible to use Proposition \[propalgthetafoncetreduchorsde2\], as the algebraic theory of theta functions does not work.
We have substituted it in the following way.
For every $i \in \{1, \cdots, 10\}$, let $\Sigma_i$ be the $i$-th symmetric polynomial in the ten modular forms $\Theta_{m/2}^8$, $m \in E$ (notation ). This is a modular form of level $4i$ for the whole modular group ${\operatorname{Sp}}_4({{\mathbb{Z}}})$.
Indeed, each $\Theta_{m/2}^8$ is a modular form for the congruence subgroup $\Gamma_2(2)$ of weight 4, and they are permuted by the modular action of $\Gamma_2(1)$ ([@MumfordTata], section II.5). The important point is that the $\Sigma_i$ are then polynomials in the four Igusa modular forms $\psi_4,\psi_6,\chi_{10}$ and $\chi_{12}$ ([@Igusa67bis], p.848 and 849). We can now explain the principle of this paragraph : these four modular forms are linked explicitly with the Igusa invariants (for a given jacobian of an hyperelliptic curve $C$ over a number field $K$), and the semi-stable reduction of the jacobian at some place $v|2$ is determined by the integrality (or not) of some quotients of these invariants, hence rational fractions of the modular forms. Now, with the explicit expressions of the $\Sigma_i$ in terms of $\psi_4,\psi_6,\chi_{10}$ and $\chi_{12}$, we can bound these $\Sigma_i$ by one of the Igusa invariants, and as every $\Theta_{m/2}^8$ is a root of the polynomial $$P(X) = X^{10} - \Sigma_1 X^9 + \Sigma_2 X^8 - \Sigma_3 X^7 + \Sigma_4 X^6 - \Sigma_5 X^5 + \Sigma_6 X^4 - \Sigma_7 X^4 + \Sigma_8 X^2 - \Sigma_9 X + \Sigma_{10},$$ we can infer an explicit bound above on the $\Theta_{m/2}^8/\lambda$, with a well-chosen normalising factor $\lambda$ such that these quotients belong to $K$. Actually, we will even give an approximative shape of the Newton polygon of the polynomial $\lambda^{10} P(X/\lambda)$, implying that its slopes (except maybe the first one) are bounded above and below, thus giving us a minoration of each of the $|\Theta_{m/2}|_v/\max_{m' \in E} |\Theta_{m'/2}|_v$, except maybe for one $m$. The explicit result is the following.
\[propbornesfoncthetaaudessus2\] Let $K$ be a number field, $(A,L)$ a principally polarised jacobian of dimension 2 over $K$ and $\tau \in {{\mathcal H}}_2$ such that $(A_\tau,L_\tau) \cong (A,L)$.
Let ${{\mathfrak{P}}}$ be a prime ideal of $K$ above $2$ such that $A$ has potentially good reduction at ${{\mathfrak{P}}}$, and the reduced (principally polarised abelian surface) is denoted by $(A_{{\mathfrak{P}}},L_{{\mathfrak{P}}})$. By abuse of notation, we forget the normalising factor ensuring that the coordinates $\Theta_{m/2} (\tau)^8$ belong to $K$.
$(a)$ If $(A_{{\mathfrak{P}}},L_{{\mathfrak{P}}})$ is the jacobian of a smooth hyperlliptic curve, all the $m \in E$ satisfy $$\frac{\left| \Theta_{m/2} (\tau) ^8\right|_{{\mathfrak{P}}}}{\max_{m' \in E} \left| \Theta_{m'/2} (\tau)^8 \right|_{{\mathfrak{P}}}} \geq |2|_{{\mathfrak{P}}}^{12}.$$
$(b)$ If $(A_{{\mathfrak{P}}},L_{{\mathfrak{P}}})$ is a product of elliptic curves, all the $m \in E$ except at most one satisfy $$\frac{\left| \Theta_{m/2} (\tau)^8 \right|_{{\mathfrak{P}}}}{\max_{m' \in E} \left| \Theta_{m'/2} (\tau)^8\right|_{{\mathfrak{P}}}} \geq |2|_{{\mathfrak{P}}}^{21}.$$
The most technical part is computing the $\Sigma_i$ as polynomials in the four Igusa modular forms. To do this, we worked with Sage in the formal algebra generated by some sums of $\Theta_{m/2}^4$ with explicit relations (namely, $y_0, \cdots, y_4$ in the notations of [@Igusa64bis], p.396 and 397). Taking away some timeouts probably due to the computer’s hibernate mode, the total computation time on a portable PC has been about twelve-hours-long (including verification of the results). The detail of algorithms and construction is available on a Sage worksheet [^2] (in Jupyter format). An approach based on Fourier expansions might be more efficient, but as there is no clear closed formula for the involved modular forms, we privileged computations in this formal algebra. For easier reading, we slightly modified the Igusa modular forms into $h_4,h_6,h_{10},h_{12}$ defined as $$\label{eqdefmodifiedIgusamodularforms}
\left\{
\begin{array}{rcccl}
h_4 & = & 2 \cdot \psi_4 & = & {\displaystyle \frac{1}{2} \sum_{m \in E} \Theta_{m/2}^8}\\
h_6 & = & 2^2 \cdot \psi_6 & = & {\displaystyle \sum_{\scriptscriptstyle \substack{\{m_1,m_2,m_3\} \subset E \\ \textrm{syzygous}}} \pm (\Theta_{m_1/2} \Theta_{m_2/2} \Theta_{m_3/2})^4} \\
h_{10} & = & 2^{15} \cdot \chi_{10} & = & {\displaystyle 2 \prod_{m \in E} \Theta_{m/2}^2} \\
h_{12} & = & 2^{16} \cdot 3 \cdot \chi_{12} & = & {\displaystyle \frac{1}{2} \sum_{\scriptscriptstyle \substack{C \subset E\\ C \textrm{ Göpel}}} \prod_{m \in E \backslash C} \Theta_{m/2}^4 }
\end{array}
\right.$$ ([@Igusa67bis], p.848 for details on these definitions, notably syzygous triples and Göpel quadruples). The third expression is not explicitly a polynomial in $y_0, \cdots, y_4$, but there is such an expression, given p.397 of [@Igusa64bis]. We also used to great benefit (both for understanding and computations) the section I.7.1 of [@Strengthesis].
Now, the computations on Sage gave us the following formulas (the first and last one being trivial given , they were not computed by the algorithm)
$$\begin{aligned}
\Sigma_1 & = 2 h_4 \label{eqSigma1} \\
\Sigma_2 & = \frac{3}{2} h_4^2 \label{eqSigma2} \\
\Sigma_3 & = \frac{29}{2 \cdot 3^3} h_4^3 - \frac{1}{2 \cdot 3^3} h_6^2 + \frac{1}{2 \cdot 3} h_{12} \label{eqSigma3}\\
\Sigma_4 & = \frac{43}{2^4 \cdot 3^3} h_4^4 - \frac{1}{2 \cdot 3^3} h_4 h_6^2 + \frac{23}{2 \cdot 3} h_4 h_{12} + \frac{2}{3} h_6 h_{10} \label{eqSigma4} \\
\Sigma_5 & = \frac{1}{2^2 \cdot 3^3} h_4^5 - \frac{1}{2^3 \cdot 3^3} h_4^2 h_6^2 + \frac{25}{2^3 \cdot 3} h_4^2 h_{12} - \frac{1}{2 \cdot 3} h_4 h_6 h_{10} + \frac{123}{2^2} h_{10}^2 \label{eqSigma5}\\
\Sigma_6 & = \frac{1}{2^2 \cdot 3^6} h_4^6 - \frac{1}{2^2 \cdot 3^6} h_4^3 h_6^2 + \frac{7}{2 \cdot 3^3} h_4^3 h_{12} - \frac{1}{2^2 \cdot 3} h_4^2 h_6 h_{10} \label{eqSigma6} \\
& + \frac{47}{2 \cdot 3} h_4 h_{10}^2 + \frac{1}{2^4 \cdot 3^6} h_6^4 - \frac{5}{2^3 \cdot 3^3} h_6^2 h_{12} + \frac{43}{2^4 \cdot 3} h_{12}^2 \nonumber \\
\vspace{1cm} \\
\Sigma_7 & = \frac{1}{2 \cdot 3^4} h_4^2 h_{12} - \frac{1}{2 \cdot 3^4} h_4^3 h_6 h_{10} + \frac{41}{2^3 3^2} h_4^2 h_{10}^2 - \frac{1}{2^2 \cdot 3^4} h_4 h_6^2 h_{12} \label{eqSigma7} \\ & + \frac{11}{2^2 \cdot 3^2} h_4 h_{12}^2 + \frac{1}{2^2 \cdot 3^4} h_6^3 h_{10} - \frac{19}{2^2 \cdot 3^2} h_6 h_{10} h_{12} \nonumber\\
\Sigma_8 & = \frac{1}{2^2 \cdot 3^3} h_4^3 h_{10}^2 + \frac{1}{2^2 \cdot 3^2} h_4^2 h_{12}^2 - \frac{1}{2 \cdot 3^2} h_4 h_6 h_{10} h_{12} + \frac{5}{2^3 \cdot 3^3} h_6^2 h_{10}^2 - \frac{11}{2^3} h_{10}^2 h_{12} \label{eqSigma8} \\
\Sigma_9 & = \frac{-5}{2^2 \cdot 3^2} h_4 h_{10}^2 h_{12} + \frac{7}{2^2 \cdot 3^3} h_6 h_{10}^3 + \frac{1}{3^3} h_{12}^3 \label{eqSigma9} \\
\Sigma_{10} & = \frac{1}{2^4} h_{10}^4. \label{eqSigma10}\end{aligned}$$
The denominators are always products of powers of 2 and 3. This was predicted by [@Ichikawa09], as all Fourier expansions of $\Theta_{m/2}$ (therefore of the $\Sigma_i$) have integral coefficients. Surprisingly, the result of [@Ichikawa09] would actually be false for a ${{\mathbb{Z}}}[1/3]$-algebra instead of a ${{\mathbb{Z}}}[1/6]$-algebra, as the expression of $\Sigma_3$ (converted as a polynomial in $\psi_4,\psi_6, \chi_{12}$) shows, but this does not provide a counterexample for a ${{\mathbb{Z}}}[1/2]$-algebra.
Now, let $C$ be an hyperelliptic curve of genus 2 on a number field $K$ and ${{\mathfrak{P}}}$ a prime ideal of ${{\mathcal O}}_K$ above 2. We will denote by $|\cdot|$ the norm associated to ${{\mathfrak{P}}}$ to lighten the notation. Let $A$ be the jacobian of $C$ and $J_2,J_4,J_6,J_8,J_{10}$ the homogeneous Igusa invariants of the curve $C$, defined as in ([@Igusa60], pp. 621-622) up to a choice of hyperelliptic equation for $C$. We fix $\tau \in {{\mathcal H}}_2$ such that $A_\tau$ is isomorphic to $A$, which will be implicit in the following (i.e. $h_4$ denotes $h_4(\tau)$ for example). By ([@Igusa67bis], p.848) applied with our normalisation, there is an hyperelliptic equation for $C$ (and we fix it) such that $$\begin{aligned}
J_2 & = \frac{1}{2} \frac{h_{12}}{h_{10}} \\
J_4 & = \frac{1}{2^5 \cdot 3} \left( \frac{h_{12}^2}{h_{10}^2} - 2 h_4 \right) \\
J_6 & = \frac{1}{2^7 \cdot 3^3} \left( \frac{h_{12}^3}{h_{10}^3} - 6 \frac{h_4 h_{12}}{h_{10}} + 4 h_6 \right) \\
J_8 & = \frac{1}{2^{12} \cdot 3^3} \left( \frac{h_{12}^4}{h_{10}^4} - 12 \frac{h_4 h_{12}^2}{h_{10}^2} + 16 \frac{h_6 h_{12}}{h_{10}} - 12 h_4^2 \right) \\
J_{10} & = \frac{1}{2^{13}} h_{10}.\end{aligned}$$ Let us now figure out the Newton polygons allowing us to bound our theta constants.
$(a)$ If $A$ has potentially good reduction at ${{\mathfrak{P}}}$, and this reduction is also a jacobian, by Proposition 3 of [@Igusa60], the quotients $J_2^5/J_{10}, J_4^5 /J_{10}^2, J_6^5 / J_{10}^3$ and $J_{8}^5 / J_{10}^4$ are all integral at ${{\mathfrak{P}}}$. Translating it into quotients of modular forms, this gives $$\begin{aligned}
\left| \frac{J_2^5}{J_{10}} \right| & = & |2|^8 \left| \frac{h_{12}^5}{h_{10}^6} \right| \leq 1 \\
\left|\frac{J_4^5}{J_{10}^2} \right| & = & |2|^3 \left| \frac{h_{12}^2}{h_{10}^{12/5}} - 2 \frac{h_4}{h_{10}^{2/5}}\right|^{5} \leq 1 \\
\left| \frac{J_6^5}{J_{10}^3} \right| & = & |2|^4 \left| \frac{h_{12}^3} {h_{10}^{18/5}} - 6 \frac{h_4 h_{12}}{h_{10}^{8/5}} + 4 \frac{h_6}{h_{10}^{3/5}} \right|^5 \leq 1 \\
\left| \frac{J_8^5}{J_{10}^4} \right| & = & |2|^{-8} \left| \frac{h_{12}^4}{h_{10}^{24/5}} - 12 \frac{h_4 h_{12}^2}{h_{10}^{14/5}} + 16 \frac{h_6 h_{12}}{h_{10}^{9/5}} - 12 \frac{h_4^2}{h_{10}^{4/5}} \right|^5 \leq 1.\end{aligned}$$
By successive bounds on the three first lines, we obtain $$\left| \frac{h_4}{h_{10}^{2/5}} \right| \leq |2|^{-21/5}, \quad \left| \frac{h_6}{h_{10}^{3/5}} \right| \leq |2|^{-34/5}, \quad
\left| \frac{h_{12}}{h_{10}^{6/5}} \right| \leq |2|^{-8/5}.$$ Using the expressions of the $\Sigma_i$ ( to ), we compute that for every $i \in \{1, \cdots, 10\}$, one has $\left| \Sigma_i / h_{10}^{2 i /5} \right| \leq |2|^{\lambda_i}$ with the following values of $\lambda_i$ : $$\begin{array}{c|cccccccccc}
\hline
i & 10 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \\
\lambda_i & - \frac{20}{5} & - \frac{44}{5}& - \frac{83}{5}& - \frac{112}{5}& - \frac{156}{5}& - \frac{125}{5}& - \frac{104}{5}& - \frac{73}{5}& - \frac{47}{5}& - \frac{16}{5} \\
\hline
\end{array}$$ and for $i=10$, it is an equality. Therefore, the highest slope of the Newton polygon is at most $26/5 \cdot v_{{\mathfrak{P}}}(2)$, whereas the lowest one is at least $-34/5 \cdot v_{{\mathfrak{P}}}(2)$, which gives part $(a)$ of Proposition \[propbornesfoncthetaaudessus2\] by the theory of Newton polygons.
$(b)$ If $A$ has potentially good reduction at ${{\mathfrak{P}}}$ and the semistable reduction is a product of elliptic curves, defining $$\begin{aligned}
I_4 & = & J_2^3 - 25 J_4 = \frac{h_4}{2} \label{eqdefI4} \\
I_{12} & = & - 8 J_4^3 + 9 J_2 J_4J_6 - 27 J_6^2 - J_2^2 J_8 = \frac{1}{2^{10} \cdot 3^3} (2 h_4^3 - h_6^2), \label{eqdefI12} \\
P_{48} & = & 2^{12} \cdot 3^3 h_{10}^4 J_8 = h_{12}^4 - 12 h_4 h_{12}^2 h_{10}^2 + 16 h_6 h_{12} h_{10}^3 - 12 h_4^2 h_{10}^4 \label{eqdefP48}\end{aligned}$$ (which as modular forms are of respective weights $4,12$ and $48$), by Theorem 1 (parts $(V_*)$ and $(V)$) of [@Liu93], we obtain in the same fashion that $$\label{eqbornesenfoncP481}
\left| \frac{h_4}{P_{48}^{1/12}} \right| \leq |2|^{-13/3}, \left| \frac{h_6}{P_{48}^{1/8}}\right| \leq |2|^{-3}, \quad \left| \frac{h_{10}}{P_{48}^{5/24}}\right| \leq |2|^{-4/3}.$$ Using the Newton polygon for the polynomial of defining $P_{48}$, one deduces quickly that $$\label{eqbornesenfoncP482}
\left| \frac{h_{12}}{P_{48}^{1/4}} \right| \leq |2|^{-7/2}.$$ As before, with the explicit expression of the $\Sigma_i$, one obtains that the $|\Sigma_i / P_{48}^{i/12}|$ are bounded by $|2|^{\lambda_i}$ with the following values of $\lambda$ : $$\label{eqvalSigmairedproduitCE1}
\begin{array}{c|cccccccccc}
\hline
i & 10 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \\
\lambda_i & -\frac{28}{3} & -\frac{71}{6} & \frac{-53}{3} & \frac{-55}{3} & \frac{-84}{3} & \frac{-71}{3} & \frac{-64}{3} & -14 & \frac{-29}{3} & \frac{-10}{3} \\
\hline
\end{array}$$ This implies directly that the highest slope of the Newton polygon is at most $16/3 \cdot v_{{\mathfrak{P}}}(2)$. Now, for the lowest slope, there is no immediate bound and it was expected : in this situation, $\Sigma_{10} = 2^{-4} h_{10}^4$ can be relatively very small compared to $P_{48}^{5/6}$.
As $P_{48}$ is in the ideal generated by $h_{10},h_{12}$ (in other words, is cuspidal) and dominates all modular forms $h_4,h_6,h_{10},h_{12}$, one of $h_{10}$ and $h_{12}$ has to be relatively large enough compared to $P_{48}$ . In practice, we get (with , and ) $$\left| \frac{h_{12}}{P_{48}^{1/4}} \right| \geq 1 \quad \textrm{or} \quad \left| \frac{h_{10}}{P_{48}^{5/24}} \right| \geq |2|^{13/6}.$$ Now, if $h_{10}$ is relatively very small (for example, $\left|h_{10}/P_{48}^{5/24} \right| \leq |2|^{19/6} \left|h_{12}/P_{48}^{1/4} \right|$), we immediately get $\left| h_{12}/P_{48}^{1/4} \right| = 1$ and $\left| \Sigma_9 / P_{48}^{3/4} \right| = 1$. Computing again with these estimates for $h_{10}$ and $h_{12}$, we obtain that the $\left|\Sigma_i / P_{48}^{i/12} \right|$ are bounded by $|2|^{\lambda_i}$ with the following slightly improved values of $\lambda$, $$\begin{array}{c|ccccccccc}
\hline
i & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \\
\lambda_i & 0 & -\frac{32}{3} & -\frac{51}{3} & \frac{-84}{3} & \frac{-71}{3} & \frac{-64}{3} & -14 & \frac{-29}{3} & \frac{-10}{3} \\
\hline
\end{array}$$ The value at $i=9$ is exact, hence the second lowest slope is then at least $-\frac{32}{3} \cdot v_{{\mathfrak{P}}}(2)$.
If it is not so small, we have a bound on $v_{{\mathfrak{P}}}(\Sigma_{10}/P_{48}^{6/5})$, hence the Newton polygon itself is bounded (and looks like in the first situation). In practice, one finds that the lowest slope is at least $-47/3 \cdot v_{{\mathfrak{P}}}(2)$, hence all others slopes are at least this value, and this concludes the proof of Proposition \[propbornesfoncthetaaudessus2\] $(b)$.
In characteristics $\neq 2,3$, Theorem 1 of [@Liu93] and its precise computations pp. 4 and 5 give the following exact shapes of Newton polygons (notice the different normalisation factors).
(-1,0) – (12,0); (-1/2,3) node [$v_{{\mathfrak{P}}}$]{}; (12,-1/2) node [$\Sigma_{10-i}/h_{10}^{2(10-i)/5}$]{}; (0,-1) – (0,4); (0,0) node [$\bullet$]{}; (0,0) node\[above right\] [$(0,0)$]{}; (10,0) node [$\bullet$]{}; (10,0) node\[above right\] [$(10,0)$]{}; (0,0) – (10,0);
(-1,0) – (11,0); (11,-1/2) node [$\Sigma_{10-i}/h_{12}^{(10-i)/3}$]{}; (0,-1) – (0,5); (-1/2,4) node [$v_{{\mathfrak{P}}}$]{}; (0,3) node [$\bullet$]{}; (1,0) node [$\bullet$]{}; (10,0) node [$\bullet$]{}; (0,3) – (1,0); (1,0) – (10,0);
In particular, when $A$ reduces to a jacobian, the theta coordinates all have the same ${{\mathfrak{P}}}$-adic norm and when $A$ reduces to a product of elliptic curves, exactly one of them has smaller norm : in other words, we reproved Proposition \[propalgthetafoncetreduchorsde2\], and the Newton polygons have a very characteristic shape.
The idea behind the computations above is that in cases $(a)$ and $(b)$ (with other normalisation factors), the Newton polygons have a shape close to these ones, therefore estimates can be made. It would be interesting to see what the exact shape of the Newton polygons is, to maybe obtain sharper results.
Wrapping up the estimates and end of the proof {#subsecfinalresultRungeCEexplicite}
----------------------------------------------
We can now prove the explicit refined version of Theorem \[thmtubularRungeproduitCE\], namely Theorem \[thmproduitCEexplicite\].
In case $(a)$, one can avoid the tubular assumption for the (unique) archimedean place of $K$: indeed, amongst the ten theta coordinates, there remain 4 which are large enough with no further assumption. As $|s_P|<4$, there remains one theta coordinates which is never too small (at any place). In practice, normalising the projective point $\psi(P)$ by this coordinate, one obtains with Propositions \[proparchimedeanbound\] $(a)$ (archimedean place), \[propalgthetafoncetreduchorsde2\] (finite places not above 2) and \[propbornesfoncthetaaudessus2\] (finite places above 2) $$h(\psi(P)) \leq - 4 \log(0.42) + \frac{1}{[K:{{\mathbb{Q}}}]} \sum_{v|2} n_v |2|^{21/2} \leq 10.75$$ after approximation.
In case $(b)$, one has to use the tubular neighbourhood implicitly given by the parameter $t$, namely Proposition \[proparchimedeanbound\] $(b)$ for archimedean places, again with Propositions \[propalgthetafoncetreduchorsde2\] and \[propbornesfoncthetaaudessus2\] for the finite places, hence we get $$h(\psi(P)) \leq 4 \log(e^{\pi t}/1.33) + \frac{1}{[K:{{\mathbb{Q}}}]} \sum_{v|2} n_v |2|^{21/2} \leq 4 \pi t + 6.14$$ after approximation.
Finally, we deduce from there the bounds on the stable Faltings height by Corollary 2.2 of [@Pazuki12b] (with its notations, $h_\Theta(A,L) = h(\psi(P))/4$).
It would be interesting to give an analogous result for Theorem \[thmtubularRungegeneral\], and the estimates for archimedean and finite places not above 2 should not give any particular problem. For finite places above 2, the method outlined above can only be applied if, taking the symmetric polynomials $\Sigma_1, \cdots, \Sigma_{f(n)}$ in well-chosen powers $\Theta_{\widetilde{a}/n,\widetilde{b/n}} (\tau)$ for $\widetilde{a},\widetilde{b} \in {{\mathbb{Z}}}^g$, we can figure out by other arguments the largest rank $k_0$ for which $\Sigma_{k_0}$ is cuspidal but not in the ideal generated by $h_{10}$. Doing so, we could roughly get back the pictured shape of the Newton polygon when $h_{10}$ is relatively very small (because then $\Sigma_k$ is relatively very small for $k>k_0$ by construction). Notice that for this process, one needs some way to theoretically bound the denominators appearing in the expressions of the $\Sigma_i$ in $h_4,h_6,h_{10},h_{12}$, but if this works, the method can again be applied.
[^1]: Email : [samuel.le\[email protected]]([email protected])
[^2]: This worksheet can be downloaded at <http://perso.ens-lyon.fr/samuel.le_fourn/contenu/fichiers_publis/Igusainvariants.ipynb>
|
---
author:
- |
Maro Cvitan$^a$, Predrag Dominis Prester$^{a,b}$, Silvio Pallua$^a$ and Ivica Smolić$^a$\
$^a$Theoretical Physics Department, Faculty of Science, University of Zagreb\
p.p. 331, HR-10002 Zagreb, Croatia\
$^b$ Physics Department, Faculty of Arts and Sciences, University of Rijeka\
Omladinska 14, HR-51000 Rijeka, Croatia\
E-mail: , , ,
title: 'Extremal black holes in $D=5$: SUSY vs. Gauss-Bonnet corrections'
---
Introduction {#sec:intro}
============
In recent years a lot of attention was directed towards higher curvature corrections in effective SUGRA field theories appearing in compactifications of string theories. Particularly interesting question is how these corrections are affecting black hole solutions, and in particular their entropies. One of the main successes so far of string theory is that it offers statistical explanation of black hole entropy by direct counting of microstates. In some cases it was possible to obtain not only lowest order Bekenstein-Hawking area law, but also higher corrections in string tension $\alpha'$, and even $\alpha'$ exact expressions for the entropy. These calculations are typically performed in the limit of small string coupling constant $g_s$ in the realm of perturbative string theory, where space-time is almost flat and black holes are actually not present. It is expected that these objects become black holes when one turns on $g_s$ enough so that their size becomes smaller than their corresponding Schwarzschild radius. Unfortunately, it is not known how to make direct calculations in string theory in this regime. However, when one goes in the opposite extreme where the Schwarzschild radius becomes much larger than the string length $\ell_s = \sqrt{\alpha'}$, then one can use low energy effective action where black holes appear as classical solutions.
The situation is especially interesting for BPS black holes. In this case on the perturbative string side one is counting number of states in short multiplets, which is expected to not depend on $g_s$, at least generically (this property can be violated in special circumstances like, e.g., short multiplet crossings). This means that one can directly compare statistical (or microscopic) entropy from perturbative string and macroscopic entropy from classical supergravity. By comparing the results from the both limits we have not only succeeded to do sophisticated perturbative consistency checks on the theory, but also improved our understanding both of string theory and supergravity. Developments include attractor mechanism and relation to topological strings [@topstr]. Especially fruitful and rich are results obtained for black holes in $D=4$ (for reviews see [@D4bh]).
In $D=4$ especially nice examples are provided by heterotic string compactified on $K3\times S^1\times S^1$ or $T^4\times S^1\times S^1$ [@Sen:2005iz]. The simplest BPS states correspond to large spherically symmetric black holes having 4 charges (2 electric and 2 magnetic), for which statistical entropy was found [@LopesCardoso:2004xf; @Jatkar:2005bh; @David:2006ji; @David:2006yn]. The macroscopic black hole entropy was calculated using two types of actions with higher order $R^2$ terms – supersymmetric and Gauss-Bonnet. In the regime where $g_s$ is small near the horizon (limit where electric charges are much larger than magnetic) all results are [*exactly*]{} equal (i.e., in all orders in $\alpha'$). This is surprising because in both of these effective actions one has neglected an infinite number of terms in low energy effective action and one would at best expect agreement in first order in $\alpha'$. There is an argumentation [@KraLar; @David:2007ak], based on $AdS_3$ arguments, which explains why corrections of higher order than $R^2$ are irrelevant for calculation of black hole entropy, but it still does not explain why these two particular types of corrections are working for BPS black holes.
These matches are even more surprising when one takes magnetic charges to be zero. One gets 2-charge [*small*]{} black holes which in the lowest order have null-singular horizon with vanishing area, which is made regular by inclusion of higher curvature corrections [@smallBH; @Sen:2004]. As curvature is of order $1/\alpha'$, all terms in the effective action give a priori contribution to the entropy which is of the same order in $\alpha'$. This is a consequence of the fact that here we are naively outside of the regime where effective action should be applicable.
In view of these results, it would be interesting to consider what happens in higher dimensions $D>4$. 2-charge BPS states and corresponding small extremal black holes generalize to all $D\leq9$. In [@Sen:2005kj] it was shown that simple Gauss-Bonnet correction gives correct result for the entropy of such black holes also in $D=5$, but not for $D>5$. Afterwards, in [@Prester:2005qs] it was shown that there is an effective action where higher order terms are given by linear combination of all generalized Gauss-Bonnet densities (with uniquely fixed coefficients) which gives the correct entropy for all dimensions. For large black holes things do not generalize directly. In $D=5$ simplest are 3-charge BPS black holes, but even for them statistical entropy is known only in lowest order in $\alpha'$ [@Strominger:1996]. Let us mention that the argumentation based on $AdS_3$ geometry has not been generalized to $D>4$.
Motivated by all this, in this paper we analyse near-horizon solutions and calculate macroscopic entropy for a class of five-dimensional black holes in the $N=2$ supergravities for which higher-derivative $R^2$ actions were recently obtained in [@Hanaki:2006pj]. In Sec. 2 we present $D=5$ supersymmetric action [@Hanaki:2006pj]. In Sec. 3 we review Sen’s entropy function formalism [@Sen:2005wa]. In Sec. 4 we present maximally supersymmetric $AdS_2\times S^3$ solution which describes near-horizon geometry of purely electrically charged 1/2 BPS black holes. In Sec. 5 for the case of simple $STU$ prepotential we find non-BPS solutions for all values of charges, except for some small black holes with one charge equal to 0 or $\pm1$. In Sec. 6 we show how and when solutions from Sec. 5 can be generalized. In Sec. 7 we present near horizon solutions for 3-charge black holes in heterotic string theory compactified on $K3\times S^1$ when the $R^2$ correction is given by Gauss-Bonnet density. and compare them with the results from SUSY action. We show that for small black holes Gauss-Bonnet correction keeps producing results in agreement with microscopic analyses. In Appendix A we present generalisation of Sec. 5 to general correction coefficients $c_I$, and in Appendix B derivations of results presented in Sec. 7.
While our work was in the late stages references [@Castro:2007hc; @Alishahiha:2007nn] appeared which have some overlap with our paper. In these papers near-horizon solutions and the entropy for BPS black holes for supersymmetric corrections were given, which is a subject of our Sec. \[sec:bps\]. Our results are in agreement with those in [@Castro:2007hc; @Alishahiha:2007nn]. However, we emphasize that our near-horizon solutions in Secs. \[sec:stu\] and \[sec:genpp\] for non-BPS black holes are completely new. Also, in [@Alishahiha:2007nn] there is a statement on matching of the entropy of BPS black hole for supersymmetric and Gauss-Bonnet correction. We explicitly show in Sec. \[sec:GB\] that this is valid just for first $\alpha'$ correction.
Higher derivative $N=2$ SUGRA in $D=5$ {#sec:sugra}
======================================
Bosonic part of the Lagrangian for the $N=2$ supergravity action in five dimensions is given by $$\begin{aligned}
\label{l0susy}
4\pi^2\mathcal{L}_0 &=& 2 \partial^a \mathcal{A}^\alpha_i \partial_a
\mathcal{A}_\alpha^i + \mathcal{A}^2
\left(\frac{D}{4}-\frac{3}{8}R-\frac{v^2}{2}\right)
+ \mathcal{N} \left(\frac{D}{2}+\frac{R}{4}+3v^2\right)
+ 2 \mathcal{N}_I v^{ab} F_{ab}^I \nonumber \\
&& + \mathcal{N}_{IJ} \left( \frac{1}{4} F_{ab}^I F^{Jab}
+ \frac{1}{2} \partial_a M^I \partial^a M^J \right)
+ \frac{e^{-1}}{24} c_{IJK} A_a^I F_{bc}^J F_{de}^K \epsilon^{abcde}\end{aligned}$$ where $\mathcal{A}^2 = \mathcal{A}^\alpha_{i\,ab}
\mathcal{A}_\alpha^{i\,ab}$ and $v^2 = v_{ab}v^{ab}$. Also, $$\mathcal{N} = \frac{1}{6} c_{IJK} M^I M^J M^K , \quad
\mathcal{N}_I = \partial_I \mathcal{N} = \frac{1}{2} c_{IJK} M^J M^K
, \quad
\mathcal{N}_{IJ} = \partial_I \partial_J \mathcal{N} = c_{IJK} M^K$$
A bosonic field content of the theory is the following. We have Weyl multiplet which contains the fünfbein $e_\mu^a$, the two-form auxiliary field $v_{ab}$, and the scalar auxiliary field $D$. There are $n_V$ vector multiplets enumerated by $I=1,\ldots,n_V$, each containing the one-form gauge field $A^I$ (with the two-form field strength $F^I=dA^I$) and the scalar $M^I$. Scalar fields $\mathcal{A}_\alpha^i$, which are belonging to the hypermultiplet, can be gauge fixed and the convenient choice is given by $$\label{hgfix}
\mathcal{A}^2 = -2 \;, \qquad \partial_a \mathcal{A}^\alpha_i = 0$$
One can use equations of motion for auxiliary fields to get rid of them completely and obtain the Lagrangian in a standard form: $$\label{l0noaux}
4\pi^2\mathcal{L}_0 = R - G_{IJ} \partial_a M^I \partial^a M^J
- \frac{1}{2} G_{IJ} F_{ab}^I F^{Jab}
+ \frac{e^{-1}}{24} c_{IJK} A_a^I F_{bc}^J F_{de}^K \epsilon^{abcde}$$ with $$\label{gIJ}
G_{IJ} = - \frac{1}{2} \partial_I\partial_J(\ln \mathcal{N}) =
\frac{1}{2} \left( \mathcal{N}_I \mathcal{N}_J - \mathcal{N}_{IJ}
\right)$$ and where $\mathcal{N}=1$ is implicitly understood (but only after taking derivatives in (\[gIJ\])). We shall later use this form of Lagrangian to make connection with heterotic string effective actions.
Lagrangian (\[l0noaux\]) can be obtained from 11-dimensional SUGRA by compactifying on six-dimensional Calabi-Yau spaces ($CY_3$). Then $M^I$ have interpretation as moduli (volumes of $(1,1)$-cycles), and $c_{IJK}$ as intersection numbers. Condition $\mathcal{N}=1$ is a condition of real special geometry. For a recent review and further references see [@Larsen:2006].
Action (\[l0susy\]) is invariant under SUSY variations, which when acting on the purely bosonic configurations (and after using (\[hgfix\])) are given with $$\begin{aligned}
\label{svar}
\delta\psi_\mu^i &=& \mathcal{D}_\mu\varepsilon^i + \frac{1}{2}v^{ab}
\gamma_{\mu ab}\varepsilon^i - \gamma_\mu\eta^i \nonumber \\
\delta\xi^i &=& D\varepsilon^i
- 2\gamma^c\gamma^{ab}\varepsilon^i\mathcal{D}_a v_{bc}
- 2\gamma^a\varepsilon^i\epsilon_{abcde}v^{bc}v^{de}
+ 4\gamma\cdot v\eta^i \nonumber \\
\delta\Omega^{Ii} &=& - \frac{1}{4}\gamma\cdot F^{I}\varepsilon^i
- \frac{1}{2}\gamma^a\partial_a M^{I}\varepsilon^i - M^{I}\eta^i
\nonumber \\
\delta\zeta^{\alpha} &=&
\left(3\eta^j-\gamma\cdot v\varepsilon^j\right)\mathcal{A}_j^\alpha\end{aligned}$$ where $\psi_\mu^i$ is gravitino, $\xi^i$ auxiliary Majorana spinor (Weyl multiplet), $\delta\Omega^{Ii}$ gaugino (vector multiplets), and $\zeta^{\alpha}$ is a fermion field from hypermultiplet.
In [@Hanaki:2006pj] four derivative part of the action was constructed by supersymmetric completion of the mixed gauge-gravitational Chern-Simons term $A \land \textrm{tr} (R \land R)$. The bosonic part of the action relevant for our purposes was shown to be $$\begin{aligned}
\label{l1susy}
4\pi^2\mathcal{L}_1 &=& \frac{c_{I}}{24} \left\{ \frac{e^{-1}}{16}
\epsilon_{abcde} A^{Ia} C^{bcfg} C^{de}_{\;\;\;\,fg}
+ M^I \left[ \frac{1}{8} C^{abcd} C_{abcd} + \frac{1}{12} D^2
- \frac{1}{3} C_{abcd} v^{ab} v^{cd}
\right. \right. \nonumber \\ &&
+ 4 v_{ab}v^{bc} v_{cd} v^{da} - (v_{ab}v^{ab})^2
+ \frac{8}{3} v_{ab} \hat{\mathcal{D}}^b \hat{\mathcal{D}}_c v^{ac}
+ \frac{4}{3} \hat{\mathcal{D}}^a v^{bc} \hat{\mathcal{D}}_a v_{bc}
+ \frac{4}{3} \hat{\mathcal{D}}^a v^{bc} \hat{\mathcal{D}}_b v_{ca}
\nonumber \\ && \left.
- \frac{2}{3} e^{-1} \epsilon_{abcde} v^{ab} v^{cd}
\hat{\mathcal{D}}_f v^{ef} \right]
+ F^{Iab} \left[ \frac{1}{6} v_{ab} D - \frac{1}{2} C_{abcd} v^{cd}
+ \frac{2}{3} e^{-1} \epsilon_{abcde} v^{cd}
\hat{\mathcal{D}}_f v^{ef}
\right. \nonumber \\ && \left. \left.
+ e^{-1} \epsilon_{abcde} v^{c}_{\;f} \hat{\mathcal{D}}^d v^{ef}
- \frac{4}{3} v_{ac}v^{cd} v_{db} - \frac{1}{3} v_{ab} v^2 \right]
\right\}\end{aligned}$$ where $c_I$ are some constant coefficients[^1], $C_{abcd}$ is the Weyl tensor which in five dimensions is $$C^{ab}_{\;\;\;\,cd} = R^{ab}_{\;\;\;\,cd} - \frac{1}{3}
\left( g^a_c R^b_d - g^a_d R^b_c - g^b_c R^a_d + g^b_d R^a_c \right)
+ \frac{1}{12} R \left( g^a_c g^b_d - g^a_d g^b_c \right)$$ and $\hat{\mathcal{D}}_a$ is the conformal covariant derivative, which when appearing linearly in (\[l1susy\]) can be substituted with ordinary covariant derivative $\mathcal{D}_a$, but when taken twice produces additional curvature contributions [@Castro:2007sd]: $$v_{ab} \hat{\mathcal{D}}^b \hat{\mathcal{D}}_c v^{ac} =
v_{ab} \mathcal{D}^b \mathcal{D}_c v^{ac}
+ \frac{2}{3} v^{ac} v_{cb} R_a^b + \frac{1}{12} v^2 R$$
We are going to analyse extremal black hole solutions of the action obtained by combining (\[l0susy\]) and (\[l1susy\]):[^2] $$\label{lsusy}
\mathcal{A} = \int dx^5 \sqrt{-g} \mathcal{L}
= \int dx^5 \sqrt{-g} (\mathcal{L}_0 + \mathcal{L}_1)$$ As (\[l1susy\]) is a complicated function of auxiliary fields (including derivatives) it is now impossible to integrate them out in the closed form and obtain an action which includes just the physical fields.
Near horizon geometry and entropy function formalism {#sec:ent-func}
====================================================
The action (\[lsusy\]) is quartic in derivatives and generally probably too complicated for finding complete analytical black hole solutions even in the simplest spherically symmetric case. But, if one is more modest and interested just in a near-horizon behavior (which is enough to find the entropy) of [*extremal*]{} black holes, there is a smart way to do the job - Sen’s entropy function formalism [@Sen:2005wa].[^3]
For five-dimensional spherically symmetric extremal black holes near-horizon geometry is expected to be $AdS_2\times S^3$, which has $SO(2,1)\times SO(4)$ symmetry [@Kunduri:2007vf]. If the Lagrangian can be written in a manifestly diffeomorphism covariant and gauge invariant way (as a function of metric, Riemann tensor, covariant derivative, and gauge invariant fields, but without connections) it is expected that near the horizon the complete background should respect this symmetry. Then it follows that the only fields which can acquire non-vanishing values near the horizon are scalars $\phi_s$, (purely electric) two-forms fields $F^{I}$, and (purely magnetic) three-form fields $H_{m}$. Explicitly written: $$\begin{aligned}
\label{efgen}
&& ds^2 = v_1 \left( -x^2 dt^2 + \frac{dx^2}{x^2} \right)
+ v_2\,d\Omega_{3}^2 \nonumber \\
&& \phi_s = u_s \;, \qquad s=1,\ldots,n_s \nonumber \\
&& F^{I} = - e^I \mathbf{\epsilon}_{A} \;, \qquad I=1,\ldots,n_F
\nonumber \\
&& H_{m} = 2q_m \mathbf{\epsilon}_{S} \;, \qquad m=1,\ldots,n_H\end{aligned}$$ where $v_{1,2}$, $u_s$, $e^I$ and $q_m$ are constants, $\mathbf{\epsilon}_{A}$ and $\mathbf{\epsilon}_{S}$ are induced volume-forms on $AdS_2$ and $S^3$, respectively. In case where $F^{I}$ ($H_{m}$) are gauge field strengths, $e^I$ ($q_m$) are electric field strengths (magnetic charges).
It is important to notice that all covariant derivatives in this background are vanishing.
To obtain near-horizon solutions one defines $$\label{fdef}
f(\vec{v},\vec{u},\vec{e}) = \int_{S^3} \sqrt{-g} \, \mathcal{L}$$ extremization of which over $\vec{v}$ and $\vec{u}$ gives equations of motion (EOM’s) $$\label{eomf}
\frac{\partial f}{\partial v_i}=0 \;, \qquad
\frac{\partial f}{\partial u_s}=0 \;,$$ and derivatives over $\vec{e}$ are giving (properly normalized) electric charges: $$\label{chgdef}
q_I = \frac{\partial f}{\partial e^I}$$ Finally, the entropy (equal to the Wald formula [@Wald]) is given with $$\label{entropy}
S_{BH} = 2\pi \left( q_I \, e^I - f \right)$$
Equivalently, one can define entropy function $F$ as a Legendre transform of the function $f$ with respect to the electric fields and charges $$F(\vec{v},\vec{u},\vec{e},\vec{q}) = 2\pi \left( q_I \, e^I
- f(\vec{v},\vec{u},\vec{e}) \right)$$ Now equations of motion are obtained by extremizing entropy function $$\label{eomF}
0 = \frac{\partial F}{\partial v_i} \;, \qquad
0 = \frac{\partial F}{\partial u_s} \;, \qquad
0 = \frac{\partial F}{\partial e^I}$$ and the value at the extremum gives the black hole entropy $$S_{BH} = F(\vec{v},\vec{u},\vec{e},\vec{q}) \qquad \mbox{when }
\vec{v},\vec{u},\vec{e} \mbox{ satisfy (\ref{eomF})}$$
We want next to apply entropy function formalism to the $N=2$ SUGRA from Sec. \[sec:sugra\]. In this case for the near-horizon geometry (\[efgen\]) we explicitly have $$\begin{aligned}
\label{efhere}
&& ds^2 = v_1 \left( -x^2 dt^2 + \frac{dx^2}{x^2} \right)
+ v_2\,d\Omega_{3}^2 \nonumber \\
&& F^{I}_{tr}(x) = -e^I \;,\qquad v_{tr}(x) = V \nonumber \\
&& M^I(x) = M^I \;, \qquad D(x) = D\end{aligned}$$ where $v_i$, $e^I$, $M^I$, $V$, and $D$ are constants.
Putting (\[efhere\]) into (\[l0susy\]) and (\[l1susy\]) one gets $$\begin{aligned}
\label{f0susy}
f_0 &=&
\frac{1}{4} \sqrt{v_2}
\left[\left(\mathcal{N}+3\right) \left(3v_1-v_2\right)-
4 V^2 \left(3\mathcal{N}+1\right) \frac{v_2}{v_1} \right.
\nonumber \\ && \left.
\qquad\quad + 8 V \mathcal{N}_i e^i \frac{ v_2}{v_1}
- \mathcal{N}_{ij} e^i e^j \frac{v_2}{v_1}
+ D (\mathcal{N}-1)v_1 v_2 \right] \end{aligned}$$ and $$\begin{aligned}
\label{f1susy}
f_1 &=& v_1 v_2^{3/2}
\left\{
\frac{c_I e^I}{48}
\left[-\frac{4 V^3}{3 v_1^4}+\frac{D V}{3 v_1^2}+
\frac{V}{v_1^2}\left(\frac{1}{v_1}-\frac{1}{v_2}\right)\right]
\phantom{\left(\frac{1}{v_1}\right)^2} \right.
\nonumber \\
&& + \left.
\frac{c_I M^I}{48}
\left[\frac{D^2}{12}+\frac{4 V^4}{v_1^4}+
\frac{1}{4} \left(\frac{1}{v_1}-\frac{1}{v_2}\right)^2-
\frac{2 V^2}{3v_1^2}\left(\frac{5}{v_1}+\frac{3}{v_2}\right)\right]
\right\} \;,\end{aligned}$$ correspondingly. Notice that for the background (\[efhere\]) all terms containing $\varepsilon_{abcde}$ tensor vanish. Complete function $f$ is a sum $$\label{fsusy}
f = f_0 + f_1$$ and EOM’s near the horizon are equivalent to $$\label{seom}
0 = \frac{\partial f}{\partial v_1} \;, \qquad
0 = \frac{\partial f}{\partial v_2} \;, \qquad
0 = \frac{\partial f}{\partial M^I} \;, \qquad
0 = \frac{\partial f}{\partial V} \;, \qquad
0 = \frac{\partial f}{\partial D} \;.$$
Notice that both $f_0$ and $f_1$ (and so $f$) are invariant on the transformation $e^I\to-e^I$, $V\to-V$, with other variables remaining the same. This symmetry follows from CPT invariance. We shall use it to obtain new solutions with $q_I\to-q_I$.
Solutions with maximal supersymmetry {#sec:bps}
====================================
We want to find near horizon solutions using entropy function formalism described in Sec. \[sec:ent-func\]. The procedure is to fix the set of electric charges $q_I$ and then solve the system of equations (\[seom\]), (\[chgdef\]) with the function $f$ given by (\[f0susy\]), (\[f1susy\]), (\[fsusy\]). It is immediately obvious that though the system is algebraic, it is in generic case too complicated to be solved in direct manner, and that one should try to find some additional information.
Such additional information can be obtained from supersymmetry. It is known that there should be 1/2 BPS black hole solutions, for which it was shown in [@Chamseddine:1996pi] that near the horizon supersymmetry is enhanced fully. This means that in this case we can put all variations in (\[svar\]) to zero, which for $AdS_2\times S^3$ background become $$\begin{aligned}
\label{svarBH}
0 &=& \mathcal{D}_\mu\varepsilon^i + \frac{1}{2}v^{ab}
\gamma_{\mu ab}\varepsilon^i - \gamma_\mu\eta^i \nonumber \\
0 &=& D\varepsilon^i + 4\,\gamma\cdot v\,\eta^i \nonumber \\
0 &=& - \frac{1}{4}\gamma\cdot F^{I}\varepsilon^i - M^{I}\eta^i
\nonumber \\
0 &=& \left(3\eta^j-\gamma\cdot v\,\varepsilon^j\right)
\mathcal{A}_j^\alpha\end{aligned}$$ Last equation fixes the spinor parameter $\eta$ to be $$\label{etacond}
\eta^j = \frac{1}{3}(\gamma\cdot v)\varepsilon^j$$ Using this, and the condition that $\varepsilon^i$ is (geometrical) Killing spinor, in the remaining equations one gets[^4] the following conditions $$\label{bpscond}
v_2 = 4v_1 \;, \qquad M^I = \frac{e^I}{\sqrt{v_1}} \;, \qquad
D = -\frac{3}{v_1} \;, \qquad V = \frac{3}{4}\sqrt{v_1}$$ We see that conditions for full supersymmetry are so constraining that they fix everything except one unknown, which we took above to be $v_1$. To fix it, we just need one equation from (\[seom\]). In our case the simplest is to take equation for $D$, which gives $$\label{v1bps}
v_1^{3/2} = (e)^3 - \frac{c_{I}e^I}{48}$$ where we used a notation $$(e)^3 \equiv \frac{1}{6} c_{IJK} e^I e^J e^K$$ We note that higher derivative corrections violate real special geometry condition, i.e., we have now $\mathcal{N}\neq1$.[^5]
Using (\[bpscond\]) and (\[v1bps\]) in the expression for the entropy (\[entropy\]) one obtains $$\label{sente}
S_{BH} = 16\pi (e)^3$$
Typically one is interested in expressing the results in terms of charges, not field strengths, and this is achieved by using (\[chgdef\]). As shown in [@Castro:2007hc], the results can be put in compact form in the following way. We first define scaled moduli $$\bar{M}^I \equiv \sqrt{v_1} M^I \;.$$ Solution for them is implicitly given with $$\label{mbareq}
8\,c_{IJK} \bar{M}^J \bar{M}^K = q_I + \frac{c_{I}}{8}$$ and the entropy (\[sente\]) becomes $$\label{sentm}
S_{BH} = \frac{8\pi}{3} c_{IJK}\bar{M}^I \bar{M}^J \bar{M}^K$$ A virtue of this presentation is that if one is interested only in entropy, then it is enough to consider just (\[mbareq\]) and (\[sentm\]). It was shown in [@Castro:2007ci] that (\[sentm\]) agrees with the OSV conjecture [@topstr; @Guica:2005ig], after proper treatment of uplift from $D=4$ to $D=5$ is made.
We shall be especially interested in the case when prepotential is of the form $$\mathcal{N} = \frac{1}{2}M^1c_{ij}M^iM^j \;, \qquad i,j>1$$ where $c_{ij}$ is a regular matrix with an inverse $c^{ij}$. In this case, which corresponds to $K3\times T^2$ 11-dimensional compactifications, it is easy to show that the entropy of BPS black holes is given with $$\label{seK3}
S_{BH} = 2\pi\sqrt{\frac{1}{2}|\hat{q}_1|c^{ij}\hat{q}_i\hat{q}_j} \;,
\qquad \hat{q_I}=q_I+\frac{c_I}{8}$$
$\mathcal{N} = M^1 M^2 M^3$ model – heterotic string on $T^4\times S^1$ {#sec:stu}
=======================================================================
BH solutions without corrections {#ssec:sol0}
--------------------------------
To analyse non-BPS solutions we take a simple model with $I=1,2,3$ and prepotential $$\label{stupp}
\mathcal{N} = M^{1} M^{2} M^{3} \;,$$ which is obtained when one compactifies 11-dimensional SUGRA on six-dimensional torus $T^6$. It is known [@s11-het5; @Ferrara:1996hh] that with this choice one obtains tree level effective action of heterotic string compactified on $T^4\times S^1$ which is wounded around $S^1$.
The simplest way to see this is to do the following steps. Start with the Lagrangian in the on-shell form (\[l0noaux\]), use (\[stupp\]) with the condition $\mathcal{N}=1$, introduce two independent moduli $S$ and $T$ such that $$\label{hetmod}
M^1 = S^{2/3} \;,\qquad M^2 = S^{-1/3}T^{-1} \;,\qquad M^3 = S^{-1/3}T$$ Finally, make Poincaré duality transformation on the two-form gauge field $F^1$: introduce additional 2-form $B$ with the corresponding strength $H=dB$ and add to the action a term $$\label{poidual}
\mathcal{A}_B = \frac{1}{4\pi^2} \int F^1 \land H
= - \frac{1}{8\pi^2} \int dx^5 \sqrt{-g} F^1_{ab} (*H)^{ab}$$ where $*$ is Hodge star. If one first solves for the $B$ field, the above term just forces two-form $F^1$ to satisfy Bianchi identity, so the new action is classically equivalent to the starting one. But if one solves for the $F^1$ and puts the solution back into the action, after the dust settles one obtains that Lagrangian density takes the form $$\label{lhetE}
4\pi^2\mathcal{L}_0 = R - \frac{1}{3}\frac{(\partial S)^2}{S^2}
- \frac{(\partial T)^2}{T^2}
- \frac{S^{4/3}}{12}\left(H'_{abc}\right)^2
- \frac{1}{4}S^{2/3}T^2\left(F^2_{ab}\right)^2
- \frac{S^{2/3}}{4\,T^2}\left(F^3_{ab}\right)^2$$ where 3-form field $H'$ is defined with $$\label{lwh}
H'_{abc} = \partial_a B_{bc} - \frac{1}{2}
\left( A^2_a F^3_{bc} + A^3_a F^2_{bc} \right)
+ (\mbox{cyclic permutations of } a,b,c)$$ To get the action in an even more familiar form one performs a Weyl rescaling of the metric $$g_{ab} \to S^{2/3} g_{ab}$$ where in the new metric Lagrangian (\[lhetE\]) takes the form $$\label{lhetS}
4\pi^2\mathcal{L}_0 = S \left[ R + \frac{(\partial S)^2}{S^2}
- \frac{(\partial T)^2}{T^2} - \frac{1}{12}\left(H'_{abc}\right)^2
- \frac{T^2}{4}\left(F^2_{ab}\right)^{\!2}
- \frac{1}{4\,T^2}\left(F^3_{ab}\right)^{\!2} \right]$$
One can now check[^6] that (\[lhetE\]) and (\[lhetS\]) are indeed lowest order (in $\alpha'$ and $g_s$) effective Lagrangians in Einstein and string frame, respectively, of the heterotic string compactified on $T^4\times S^1$ with the only “charges” coming from winding and momentum on $S^1$. Field $T$ plays the role of a radius of $S^1$, and field $S$ is a function of a dilaton field such that $S\sim1/g_s^2$. This interpretation immediately forces all $M^I$ to be positive.
We are interested in finding 3-charge near-horizon solutions for BH’s when the prepotential is (\[stupp\]). Applying entropy function formalism on (\[f0susy\]) one easily gets: $$\begin{aligned}
v_1 &=& \frac{1}{4} \left|q_1q_2q_3\right|^{1/3} \label{v10sol} \\
e^I &=& \frac{4 v_1^{3/2}}{q_I}
= \frac{1}{2q_I} \left|q_1q_2q_3\right|^{1/2} \label{eI0sol} \\
M^I &=& \frac{|e^I|}{\sqrt{v_1}}
= \left|\frac{q_1q_2q_3}{q_I^2}\right|^{1/3} \label{merat} \\
v_2 &=& 4 v_1 \label{ansv21} \\
D &=& - \frac{1}{v_1} \left|\textrm{sign}(q_1)+\textrm{sign}(q_2)
+\textrm{sign}(q_3)\right| \label{ansD} \\
V &=& \frac{\sqrt{v_1}}{4} \left(\textrm{sign}(q_1)
+\textrm{sign}(q_2)+\textrm{sign}(q_3)\right) \label{ansV}\end{aligned}$$ and the entropy is given with $$\label{stu0ent}
S = 2\pi \left|q_1q_2q_3\right|^{1/2}$$ In fact in this case full solutions (not only near-horizon but in the whole space) were explicitly constructed in [@Peet:1995pe].
If any of charges $q_I$ vanishes, one gets singular solutions with vanishing horizon area. Such solutions correspond to small black holes. One expects that higher order (string) corrections “blow up” the horizon and make solutions regular.
Inclusion of SUSY corrections {#ssec:stusorr}
-----------------------------
We would now like to find near horizon solutions for extremal black holes when the action is extended with the supersymmetric higher derivative correction (\[l1susy\]). We already saw in Sec. \[sec:bps\] how this can be done for the special case of 1/2 BPS solutions, i.e., in case of non-negative charges $q_I\geq0$. The question is could the same be done for general sets of charges.
Again, even for the simple prepotential (\[stupp\]) any attempt of direct solving of EOM’s is futile. In the BPS case we used vanishing of all supersymmetry variations which gave the conditions (\[merat\])-(\[stu0ent\]), which are not affected by higher derivative correction, and that enabled us to find a complete solution. Now, for non-BPS case, we cannot use the same argument, and naive guess that (\[merat\]-\[stu0ent\]) is preserved after inclusion of correction is inconsistent with EOM’s.
Intriguingly, there is something which is shared by (BPS and non-BPS) solutions (\[v10sol\])-(\[ansV\]) – the following two relations: $$\begin{aligned}
\label{anscond1}
0 &=& Dv_1 + 3 - 9\frac{v_1}{v_2} + 4\frac{V^2}{v_1} \label{asusy} \\
0 &=& \frac{(Dv_1)^2}{12} + 4\left(\frac{V}{\sqrt{v_1}}\right)^{\!4}
+ \frac{1}{4}\left(1-\frac{v_1}{v_2}\right)^{\!2}
- \frac{2}{3}\left(\frac{V}{\sqrt{v_1}}\right)^{\!2}
\left(5+3\frac{v_1}{v_2}\right)
\label{anscond2}\end{aligned}$$ The above conditions are connected with supersymmetry. The first one, when plugged in the $\mathcal{L}_0$ (\[l0susy\]), makes the first bracket (multiplying $\mathcal{A}^2$) to vanish. The second condition, when plugged in the $\mathcal{L}_1$ (\[l1susy\]), makes the term multiplying $c_{I}M^I$ to vanish. We shall return to this point in Sec. \[sec:genpp\].
What is important is that for (\[anscond1\]) and (\[anscond2\]) we needed just Eqs. (\[ansv21\])-(\[ansV\]) (and, in particular, [*not*]{} Eq. (\[merat\])). Our idea is to take (\[ansv21\])-(\[ansV\]) as an [*ansatz*]{}, plug this into all EOM’s and find out is it working also in the non-BPS case. Using the CPT symmetry it is obvious that there are just two independent cases. We can choose $$\label{ans3}
v_2 = 4 v_1 \;, \qquad D = -\frac{3}{v_1} \;, \qquad
V = \frac{3}{4}\sqrt{v_1} \;,$$ which corresponds to BPS case (see (\[bpscond\])), and $$\label{ans1}
v_2 = 4 v_1 \;, \qquad D = -\frac{1}{v_1} \;, \qquad
V = \frac{1}{4}\sqrt{v_1}$$ Though in the lowest order (\[ans3\]) appears when all charges are positive, and (\[ans1\]) when just one charge is negative (see (\[ansv21\])-(\[ansV\])), we shall [*not*]{} suppose a priori any condition on the charges.
For the start, let us restrict coefficients $c_{I}$ such that $$\label{c2het}
c_1 \equiv 24\zeta > 0 \;, \qquad c_2 = c_3 = 0 \;.$$ This choice appears when one considers heterotic string effective action on the tree level in string coupling $g_s$, but taking into account (part of) corrections in $\alpha'$.[^7] In this case we have $\zeta=1$. For completeness, we present results for general coefficients $c_{I}$ in Appendix \[sec:app1\].
Let us now start with the ansatz (\[ans1\]). The EOM’s can now be written in the following form: $$\begin{aligned}
&& b^2 b^3 e^2 e^3 = 0
\nonumber \\
&& b^1 b^3 e^1 e^3 = 0
\nonumber \\
&& b^1 b^2 e^1 e^2 = 0
\nonumber \\
&& 4 \left(b^2 b^3 - 1\right) e^2 e^3 = q_1 - \frac{\zeta}{3}
\nonumber \\
&& 4 \left(b^1 b^3 - 1\right) e^1 e^3 = q_2
\nonumber \\
&& 4 \left(b^1 b^2 - 1\right) e^1 e^2 = q_3
\nonumber \\
&& 42 v_1^{3/2} + \left(\zeta \left(6 b^1 - 1\right) +
6 \left(4 b^1 b^2 b^3 - 3(b^1+1)(b^2+1)(b^3+1)+4\right)
e^2 e^3\right) e^1 = 0
\nonumber \\
&& 18 v_1^{3/2} + \left(
6 \left(4 b^1 b^2 b^3 + (b^1+1)(b^2+1)(b^3+1)+4\right)
e^2 e^3 - \zeta \left(2 b^1 + 5\right)\right) e^1 = 0
\nonumber \\
&& 6 v_1^{3/2} + \left(\zeta(2 b^1 + 1) -
6 \left(b^1 + 1\right) \left(b^2 + 1\right) \left(b^3 + 1\right)
e^2 e^3\right) e^1 = 0
\nonumber \\
&& 6 v_1^{3/2} + \left(\zeta \left(10 b^1 + 9\right)
+ 6 \left(3b^1b^2b^3 - b^1b^2- b^2b^3- b^1b^3 - 5(b^1+b^2+b^3)
- 9\right) e^2 e^3\right) e^1 = 0
\nonumber\end{aligned}$$ where $b^I$ are defined with $$\bar{M}^I \equiv (1+b^I)e^I$$ Now there are more equations than unknowns, so the system is naively overdetermined. However, not all equations are independent and the system is solvable. First notice that first three equations imply that two of $b^I$’s should vanish, which enormously simplifies solving.
Let us summarize our results. We have found that there are six branches of solutions satisfying[^8] $M^I>0$, depending on the value of the charges $q_I$.\
\
Solutions are given with: $$\begin{aligned}
&& v_1 = \frac{1}{4}
\left|\frac{q_2q_3(q_1+\zeta/3)^2}{q_1-\zeta/3}\right|^{1/3} \\
&& \frac{e^1}{\sqrt{v_1^3}}\left(q_1-\frac{\zeta}{3}\right)
= \frac{e^2q_2}{\sqrt{v_1^3}} = \frac{e^3q_3}{\sqrt{v_1^3}}
= 4\frac{q_1-\zeta/3}{q_1+\zeta/3}
\\
&& \frac{M^3\sqrt{v_1}}{e^3} = - \frac{q_1+\zeta}{q_1-\zeta/3}
\;, \qquad \frac{M^1\sqrt{v_1}}{e^1}
= \frac{M^2\sqrt{v_1}}{e^2} = 1\end{aligned}$$ together with (\[ans1\]). The entropy is given with $$\label{sent12}
S_{BH} = 2\pi\left|q_2q_3\left(q_1-\frac{\zeta}{3}\right)\right|^{1/2}$$
For heterotic string one has $\zeta=1$ and $q_I$ are integer, so the condition can be written also as $q_1>0$.\
\
As the theory is symmetric on the exchange $(2)\leftrightarrow(3)$, the only difference from the previous case is that now we have $$\frac{M^2\sqrt{v_1}}{e^2} = - \frac{q_1+\zeta}{q_1-\zeta/3}
\;, \qquad \frac{M^1\sqrt{v_1}}{e^1}
= \frac{M^3\sqrt{v_1}}{e^3} = 1$$ and everything else is the same.\
\
Here the only difference from solutions in previous two cases is: $$\frac{M^1\sqrt{v_1}}{e^1} = - \frac{q_1-\zeta/3}{q_1+\zeta}
\;, \qquad \frac{M^2\sqrt{v_1}}{e^2}
= \frac{M^3\sqrt{v_1}}{e^3} = 1$$ For heterotic string $\zeta=1$ the bound for $q_1$ is $q_1<-1$.
Beside these three “normal” branches, there are additional three “strange” branches which appear for $|q_1|<\zeta/3$:\
\
For every of the three branches discussed above, there is an additional, mathematically connected, branch, for which the difference is that now in all branches we have $|q_1|<\zeta/3$, $q_2<0$, $q_3<0$. All formulas are the same, except that the entropy is negative $$S_{BH} = -2\pi\left|q_2q_3\left(q_1-\frac{\zeta}{3}\right)\right|^{1/2}$$ Additional reason why we call these solutions “strange” is the fact that electric fields and charges have opposite sign. It is questionable that there are asymptotically flat BH solutions with such near-horizon behaviour, and for the rest of the paper we shall ignore them.
Now we take the “BPS" ansatz (\[ans3\]). There is only one branch of solutions, valid for $q_{2,3}>0$, $q_1>-\zeta$:\
\
Solution now takes the form $$\begin{aligned}
&& v_1 = \frac{1}{4}
\left|\frac{q_2q_3(q_1+\zeta)^2}{q_1+3\zeta}\right|^{1/3} \label{hetbpsv} \\
&& \frac{e^1}{\sqrt{v_1^3}}\left(q_1+3\zeta\right)
= \frac{e^2q_2}{\sqrt{v_1^3}} = \frac{e^3q_3}{\sqrt{v_1^3}}
= 4\frac{q_1+3\zeta}{q_1+\zeta}
\label{hetbpse} \\
&& \frac{M^1\sqrt{v_1}}{e^1}
= \frac{M^2\sqrt{v_1}}{e^2}
= \frac{M^3\sqrt{v_1}}{e^3} = 1
\label{hetbpsm}\end{aligned}$$ together with (\[ans3\]). The entropy is given with $$\label{sent34}
S_{BH} = 2\pi\left|q_2q_3\left(q_1+3\zeta\right)\right|^{1/2}$$ One can check that this is equal to the BPS solution from Sec. \[sec:bps\] with the prepotential and $c_I$ given by (\[stupp\]) and (\[c2het\]).
Solutions for the cases when two or all three charges are negative are simply obtained by applying the CPT transformations $e^I\to-e^I$, $q^I\to-q^I$, $V\to-V$ on the solutions above.
Some remarks on the solutions {#ssec:srem}
-----------------------------
Let us summarize the results of Sec. 5.2. For the prepotential (\[stupp\]) and (\[c2het\]) we have found nonsingular extremal near-horizon solutions with $AdS_2\times S^3$ geometry for all values of charges $(q_1,q_2,q_3)$ except for some special cases. For black hole entropy we have obtained that supersymmetric higher order ($R^2$) correction just introduces a shift $q_1\to\hat{q}_1=q_1+a$, $$S_{BH} = 2\pi\sqrt{\left| \hat{q}_1 q_2 q_3 \right|}$$ where $a=\pm3,\pm1/3$.
For the action connected with compactified heterotic string, i.e., when $\zeta=1$ and charges are integer valued, exceptions are:
(i)
: $q_2q_3=0$
(ii)
: $q_1=0\;,\;\; q_2q_3<0$
(iii)
: $q_1=-1\;,\;\; q_2,q_3>0$ (and also with reversed signs)
It is easy to show that in order to have small effective string coupling near the horizon we need $q_2q_3\gg1$ which precludes case (i) (string loop corrections make $c_{2,3}\neq0$ which regulate case (i), see Append. \[sec:app1\]). For the cases (ii) and (iii) one possibility is that regular solutions exist, but they are not given by our Ansätze. But, our efforts to find numerical solutions also failed, so it is also possible that such solutions do not exist. This would not be that strange for cases (i) and (ii), as they correspond to black hole solutions which were already singular (small) with vanishing entropy before inclusion of supersymmetric $R^2$ corrections. But for the case (iii) it would be somewhat bizarre, because it would mean that higher order corrections turn nonsingular solution into singular.
Let us make a comment on a consequence of the violation of the real special geometry condition by supersymmetric higher-derivative corrections. We have seen that the example analysed in this section can be viewed as the tree-level effective action of heterotic string compactified on $T^4\times S^1$ supplied with part of $\alpha'$ corrections. In Sec. \[ssec:sol0\] we saw that in the lowest order a radius $T$ of $S^1$ was identified with $T^2=M^3/M^2$. From (\[hetbpsv\])-(\[hetbpsm\]) follows that in the BPS solution we have $$T^2 = \frac{q_2}{q_3}$$ which is expected from T-duality $q_2\leftrightarrow q_3$, $T\to T^{-1}$.
But, in the lowest order we also have $T^2=M^1(M^3)^2$, which gives $$T^2 = \frac{q_2}{q_3} \frac{q_1+3}{q_1+1}$$ which does not satisfy T-duality. It means that relation $T^2=M^1(M^3)^2$ receives higher-derivative corrections.[^9] That at least one of relations for $T$ is violated by corrections was of course expected from $\mathcal{N}\neq1$.[^10]
Generalisation to other prepotentials {#sec:genpp}
=====================================
A natural question would be to ask in what extend one can generalize construction from the previous section. In mathematical terms, the question is of validity of ansatz (\[ans1\]) $$\label{gans1}
v_2 = 4 v_1 \;, \qquad D = -\frac{1}{v_1} \;, \qquad
V = \frac{1}{4}\sqrt{v_1}$$ which we call Ansatz 1, and (\[ans3\]) $$\label{gans3}
v_2 = 4 v_1 \;, \qquad D = -\frac{3}{v_1} \;, \qquad
V = \frac{3}{4}\sqrt{v_1} \;,$$ which we call Ansatz 3 (Ansatz 2 and 4 are obtained by applying CPT transformation, i.e, $V\to-V$).
We have seen in Sec. \[sec:bps\] that for BPS states supersymmetry directly dictates validity of Ansatz 3 (and by symmetry also 4). The remaining question is how general is Ansatz 1.
Putting (\[gans1\]) in EOM’s one gets $$\begin{aligned}
\label{sgeneom}
&& c_{IJK}e^{J}e^{K}+2 \bar{\mathcal{N}}_{I}=2
\bar{\mathcal{N}}_{IJ}e^{J}
\nonumber \\
&& 6 \left({c_I \bar{M}^I}+168 v_1^{3/2}+24
\bar{\mathcal{N}}+48 \bar{\mathcal{N}}_{IJ}e^{I}e^{J}\right)=7 {c_I
e^I}+576 \bar{\mathcal{N}}_{I}e^{I}
\nonumber \\
&& 144 \left(3 v_1^{3/2}+5 \bar{\mathcal{N}}+2
\bar{\mathcal{N}}_{IJ}e^{I}e^{J}\right)=3 {c_I e^I}+2 {c_I
\bar{M}^I}+576 \bar{\mathcal{N}}_{I}e^{I}
\nonumber \\
&& {c_I e^I}+144 \bar{\mathcal{N}} = 2({c_I\bar{M}^I}+72 v_1^{3/2})
\nonumber \\
&& {c_I e^I}+576 \bar{\mathcal{N}}_{I}e^{I}=10 {c_I \bar{M}^I}+144
v_1^{3/2}+432 \bar{\mathcal{N}}
\nonumber \\
&& q_I - \frac{c_I}{72} = 4\bar{\mathcal{N}}_{I}
- 4\bar{\mathcal{N}}_{IJ}e^{J}\end{aligned}$$ and for the black hole entropy $$S_{BH} = 4\pi
\left( 2\bar{\mathcal{N}}-\bar{\mathcal{N}}_{IJ}e^{I}e^{J} \right)
= \frac{4\pi}{3}\hat{q}_Ie^I$$ It can be shown that two equations in (\[sgeneom\]) are not independent. In fact, by further manipulation the system can be put in the simpler form $$\begin{aligned}
&& 0 = c_{IJK}\left(\bar{M}^J-e^J\right)\left(\bar{M}^K-e^K\right)
\label{a1eom1} \\
&& \frac{c_I\bar{M}^I}{12} = c_{IJK} \left(\bar{M}^I+e^I\right)
\bar{M}^J e^K
\label{a1eom3} \\
&& v_1^{3/2} = \frac{c_Ie^I}{144} - (e)^3
\label{a1eom4} \\
&& q_I - \frac{c_I}{72} = -2\,c_{IJK}e^{J}e^{K}
\label{a1eom5} \end{aligned}$$ Still the above system is generically overdetermined as there is one equation more than the number of unknowns. More precisely, Eqs. (\[a1eom1\]) and (\[a1eom3\]) should be compatible, and this is not happening for generic choice of parameters. One can check this, e.g., by numerically solving simultaneously (\[a1eom1\]) and (\[a1eom3\]) for random choices of $c_{IJK}$, $c_I$ and $e^I$. This means that for generic prepotentials the Ansatz 1 (\[gans1\]) is not working.
However, there are cases in which the system is regular and there are physical solutions. This happens, e.g., for prepotentials of the type $$\mathcal{N} = \frac{1}{2}M^1c_{ij}M^iM^j \;, \qquad i,j>1$$ where $c_{ij}$ is a regular matrix. In this case (\[a1eom1\]) gives conditions $$0 = \left(\bar{M}^1-e^1\right)\left(\bar{M}^i-e^i\right) \;, \qquad
0 = \left(\bar{M}^i-e^i\right)c_{ij}\left(\bar{M}^j-e^j\right)$$ which has one obvious solution when $\bar{M}^i=e^i$ for all $i$. Now $\bar{M}^1$ is left undetermined, and one uses “the extra equation” (\[a1eom3\]) to get it. Black hole entropy becomes $$S_{BH} = 2\pi\sqrt{\frac{1}{2}|\hat{q}_1|c^{ij}\hat{q}_i\hat{q}_j} \;,
\qquad \hat{q_I}=q_I-\frac{c_I}{72}$$ where $c^{ij}$ is matrix inverse of $c_{ij}$. Again, the influence of higher order supersymmetric correction is just to shift electric charges $q_I\to\hat{q}_I$, but with the different value for the shift constant than for BPS black holes.
We have noted in Sec. \[ssec:stusorr\] that Ansatz 1 (\[ans1\]), which gives nonsupersymmetric solutions, has some interesting relations with supersymmetry. Another way to see this is to analyse supersymmetry variations (\[svar\]). Let us take that spinor parameters $\eta$ and $\varepsilon$ are now connected with $$\eta^i = (\gamma\cdot v)\varepsilon^i$$ The variations (\[svar\]) now become $$\begin{aligned}
\label{svarn1}
\delta\psi_\mu^i &=& \left(\mathcal{D}_\mu
+ \frac{1}{2}v^{ab}\gamma_{\mu ab}
- \gamma_\mu(\gamma\cdot v)\right)\varepsilon^i \\
\delta\xi^i &=& \left(D + 4(\gamma\cdot v)^2\right)\varepsilon^i
\label{svarn2} \\
\delta\Omega^{Ii} &=& - \left(\frac{1}{4}\gamma\cdot F^{I}
+ M^{I}\gamma\cdot v \right)\varepsilon^i
\label{svarn3} \\
\delta\zeta^{\alpha} &=&
2(\gamma\cdot v)\varepsilon^j\mathcal{A}_j^\alpha
\label{svarn4}\end{aligned}$$ One can take a gauge in which $\mathcal{A}_j^\alpha=\delta_j^\alpha$, which means that last (hypermultiplet) variation (\[svarn4\]) is now nonvanishing. But, it is easy to see that for Ansatz 1 (and when $\epsilon^i$ is Killing spinor) variations (\[svarn1\]) and (\[svarn2\]) are vanishing. Also, we have seen that solutions we have been explicitly constructed have the property that for all values of the index $I$ [*except one*]{} (which we denote $J$) we had $$\bar{M}^I=e^I \; \qquad I\neq J$$ From this follows that all variations (\[svarn3\]) except the one for $I=J$ are also vanishing. One possible explanation for such partial vanishing of variations could be that our non-BPS states of $N=2$ SUGRA are connected with BPS states of some theory with higher (e.g., $N=4$) supersymmetry.
Gauss-Bonnet correction {#sec:GB}
=======================
It is known that in some cases of black holes in $D=4$ Gauss-Bonnet term somehow effectively takes into account all $\alpha'$ string corrections. Let us now investigate what is happening in $D=5$. This means that we now add as $R^2$ correction to the 0$^{th}$ order Lagrangian (\[l0susy\]) instead of (\[l1susy\]) just the term proportional to the Gauss-Bonnet density: $$\label{l1gb}
\mathcal{L}_{GB} = \frac{1}{4\pi^2}\frac{1}{8}\frac{c_{I}M^I}{24}
\left(R_{abcd}R^{abcd}-4R_{ab}R^{ab}+R^2\right)$$ To apply entropy function formalism we start with $$\label{fGB}
f = f_0 + f_{GB}$$ where $f_0$ is again given in (\[f0susy\]) and $f_{GB}$ is $$\label{f1GB}
f_{GB} = -\frac{3}{2}\sqrt{v_2}\frac{c_{I}M^I}{24}$$
Strictly speaking, we have taken just (part of) first order correction in $\alpha'$, so normally we would expect the above action to give us at best just the first order correction in entropy. This we obtain by putting $0^{th}$-order solution in the expression $$\label{delS}
\Delta S_{BH} = -2\pi \Delta f$$ where $\Delta f$ is 1$^{st}$-order correction in $f$. It is easy to show that for the BPS $0^{th}$-order solution (\[bpscond\]) one obtains the same result for supersymmetric (\[f1susy\]) and Gauss-Bonnet (\[f1GB\]) corrections, which can be written in a form: $$\label{ealpha}
\Delta S_{BH} = 6\pi\frac{c_{I}e^I}{24}$$ It was noted in [@Castro:2007hc] that for compactifications on elliptically fibred Calabi-Yau (\[ealpha\]) agrees with the correction of microscopic entropy proposed earlier by Vafa [@Vafa:1997gr]. We note that for non-BPS black holes already first $\alpha'$ correction to entropy is different for SUSY and Gauss-Bonnet case.
From experience in $D=4$ one could be tempted to suppose that SUSY and Gauss-Bonnet solutions are exactly (not just perturbatively) equal. However, this is not true anymore in $D=5$. The simplest way to see this is to analyse opposite extreme where one of the charges is zero (small black holes). To explicitly show the difference let us analyse models of the type (obtained from $K3\times T^2$ compactifications of $D=11$ SUGRA) $$\mathcal{N}=\frac{1}{2}M^1c_{ij}M^iM^j \;, \qquad c_i=0 \;, \qquad
i,j>1$$ in the case where $q_1=0$. For the Gauss-Bonnet correction, application of entropy function formalism of Sec. \[sec:ent-func\] on (\[fGB\]) gives for the entropy (see Appendix \[sec:app2\]) $$\label{eGBK3}
S_{GB} = 4\pi\sqrt{\frac{1}{2}\frac{c_1}{24}q_ic^{ij}q_j}$$ where $c^{ij}$ is the matrix inverse of $c_{ij}$. On the other hand, from (\[seK3\]) follows that for the supersymmetric correction in the BPS case one gets $$\label{esK3}
S_{SUSY} = 2\pi\sqrt{\frac{3}{2}\frac{c_1}{24}q_ic^{ij}q_j}$$ which is differing from (\[eGBK3\]) by a factor of $2/\sqrt{3}$.
In [@Huang:2007sb] some of the models of this type were analysed from microscopic point of view and the obtained entropy of small black holes agrees with the Gauss-Bonnet result (\[eGBK3\]).
Now, the fact that simple Gauss-Bonnet correction is giving the correct results for BPS black hole entropy in both extremes, $q_1=0$ and $q_1>>1$, is enough to wonder could it be that it gives the correct microscopic entropy for all $q_1\geq0$ (as it gives for 4 and 8-charge black holes in $D=4$). Analytical results, with details of calculation, for the generic matrix $c_{ij}$ and charge $q_3$ are presented in Appendix \[sec:app2\].
Here we shall present results for the specific case, already mentioned in Sec. \[sec:stu\], of the heterotic string compactified on $T^4\times S^1$. Tree-level (in $g_s$) effective action is defined with $$\mathcal{N} = M^1 M^2 M^3 \;,\qquad c_1 = 24 \;,\qquad c_2 = c_3 =0 \;.$$ Matrix $c_{ij}$ is obviously here given with $$c_{12} = c_{21} = 1 \;, \qquad c_{11} = c_{22} = 0$$ As the simple Gauss-Bonnet correction (\[l1gb\]) does not contain auxiliary fields, we can integrate them out in the same way as it was done in the lowest-order case in Sec. \[ssec:sol0\]. For independent moduli we again use $$S \equiv (M^1)^{3/2} \;\qquad T \equiv \tilde{M}^2 = S^{1/3} M^2$$ It appears that it is easier to work in string frame, where the 0$^{th}$ order action is given in (\[lhetS\]), and the correction (\[l1gb\]) is now $$\label{lgbS}
\mathcal{L}_{GB} = \frac{1}{4\pi^2}\frac{S}{8}
\left(R_{abcd}R^{abcd}-4R_{ab}R^{ab}+R^2\right) +
\mbox{ (terms containing } \partial_a S \mbox{)}$$ We are going to be interested in near-horizon region where all covariant derivatives, including $\partial_a S$, vanish, so we can again just keep Gauss-Bonnet density term.
Application of (\[efgen\]) here gives that solution near the horizon has the form $$\begin{aligned}
\label{efhgb}
&& ds^2 = v_1 \left( -x^2 dt^2 + \frac{dx^2}{x^2} \right)
+ v_2\,d\Omega_{3}^2 \nonumber \\
&& S(x) = S \;, \qquad T(x) = T \nonumber \\
&& F^{(i)}_{tr}(x) = -e_i \;,\qquad i=2,3 \nonumber \\
&& H_{mnr} = 2 q_1 \sqrt{h_S}\, \varepsilon_{mnr}\end{aligned}$$ where $\varepsilon_{mnr}$ is totally antisymmetric tensor with $\varepsilon_{234}=1$. Observe that $q_1$ is now a magnetic charge. Using this in (\[lhetS\]) and (\[lgbS\]) gives $$f = \frac{1}{2} v_1 v_2^{3/2} S \left( -\frac{2}{v_1} +
\frac{6}{v_2} + \frac{T^2 e_2^2}{2 v_1^2} + \frac{e_3^2}{2 T^2 v_1^2}
- \frac{2 q_1^2}{v_2^3} - \frac{3}{v_1 v_2} \right)$$ Following the entropy function formalism we need to solve the system of equations $$\label{gbheteom}
0 = \frac{\partial f}{\partial v_1} \;, \qquad
0 = \frac{\partial f}{\partial v_2} \;, \qquad
0 = \frac{\partial f}{\partial S} \;, \qquad
0 = \frac{\partial f}{\partial T} \;, \qquad
q_2 = \frac{\partial f}{\partial e_2} \;, \qquad
q_3 = \frac{\partial f}{\partial e_3}$$ After some straightforward algebra we obtain $$T^2 = \left|\frac{q_2}{q_3}\right|$$ which is the same as without the correction and respecting T-duality. Also $$v_1 = \frac{v_2}{4}+\frac{1}{8} \;, \qquad
S = \frac{1}{v_2}\sqrt{\frac{2v_2+1}{2v_2+3}}\sqrt{|q_2q_3|} \;.$$ Here $v_2$ is the real root of a cubic equation $$0 = x^3 - \frac{3}{2} x^2 - q_1^2 x - \frac{q_1^2}{2}$$ which, explicitly written, is $$\begin{aligned}
v_2 &=& \frac{1}{2}+\frac{(1+i\sqrt{3}) (4q_1^2+3)}{4\,3^{1/3}
\left(-9-36q_1^2+2\sqrt{3}\sqrt{27 q_1^2+72 q_1^4-16 q_1^6}
\right)^{1/3}}
\nonumber \\
&& +\frac{(1-i\sqrt{3})\left(-9-36q_1^2+2\sqrt{3}
\sqrt{27q_1^2+72 q_1^4-16q_1^6}\right)^{1/3}}{4\,3^{2/3}}\end{aligned}$$ For the macroscopic black hole entropy we obtain $$\label{GBhet}
S_{BH} = 4\pi\sqrt{|q_2q_3|}\sqrt{v_1+\frac{3}{2}\frac{v_1}{v_2}}$$ It would be interesting to compare this result with the statistical entropy of BPS states (correspondingly charged) in heterotic string theory. Unfortunately, this result is still not known.
For small 2-charge black holes $q_1=0$, and the solution further simplifies to $$v_1 = \frac{v_2}{3} = \frac{1}{2}$$ which gives for the entropy of small black holes $$S_{BH} = 4\pi\sqrt{|q_2q_3|}$$ This solution was already obtained in [@Sen:2005kj] by starting at the beginning with $q_1=0$.[^11]
Conclusion and outlook {#sec:concl}
======================
We have shown that for some prepotentials, including important family obtained with $K3\times T^2$ compactifications of 11-dimensional SUGRA, one can find non-BPS spherically symmetric extremal black hole near horizon solutions. In particular, for the simple example of so called $STU$ theory we have explicitly constructed solutions for all values of charges with the exception of some small black holes where one of the charges is equal to 0 or $\pm1$.
One of the ideas was to compare results with the ones obtained by taking $R^2$ correction to be just given with Gauss-Bonnet density, and especially to analyse cases when the actions are connected with string compactifications, like e.g., heterotic string on $K3\times S^1$, where for some instances one can find statistical entropies. Though for Gauss-Bonnet correction (which manifestly breaks SUSY) it was not possible to calculate entropy in a closed form for generic prepotentials, on some examples we have explicitly shown that in $D=5$, contrary to $D=4$ examples, black hole entropy is different from the one obtained using supersymmetric correction (BPS or non-BPS case). Interestingly, first order corrections to entropy of BPS black holes are the same for all prepotentials, and are in agreement with the result for statistical entropy for elliptically fibred Calabi-Yau compactification [@Vafa:1997gr].
For the $K3\times T^2$ compactifications of $D=11$ SUGRA (which includes $K3\times S^1$ compactification of heterotic string) we have found explicit formula for the black hole entropy in the case of Gauss-Bonnet correction. Unfortunately, expression for statistical entropy for generic values of charges is still not known, but there are examples for which statistical entropy of BPS states corresponding to [*small*]{} black holes is known [@Huang:2007sb]. We have obtained that Gauss-Bonnet correction leads to the macroscopic entropy equal to statistical, contrary to supersymmetric correction which leads to different result. This result favors Gauss-Bonnet correction. On the other hand, for [*large*]{} black holes, it is the supersymmetric result (\[seK3\]) which agrees with OSV conjecture properly uplifted to $D=5$ [@Castro:2007ci]. We propose to resolve this issue perturbatively by calculating $\alpha'^2$ correction for 3-charge black holes in heterotic string theory compactified on $K3\times S^1$ using methods of [@Sahoo:2006pm]. Calculation is underway and results will be presented elsewhere [@Cvitan:2007hu].
It is known that theories in which higher curvature correction are given by (extended) Gauss-Bonnet densities have special properties, some of which are unique. Beside familiar ones (equations of motion are “normal” second order, in flat space and some other backgrounds they are free of ghosts and other spurious states, have well defined boundary terms and variational problem, first and second order formalisms are classically equivalent, extended Gauss-Bonnet densities have topological origin and are related to anomalies, etc), they also appear special in the approaches where black hole horizon is treated as a boundary and entropy is a consequence of the broken diffeomorphisms by the boundary condition [@CPP]. It would be interesting to understand in which way this is connected with the observed fact that these terms effectively encode a lot of near-horizon properties for a class of BPS black holes in string theory.
We would like to thank L. Bonora for stimulating discussions. This work was supported by the Croatian Ministry of Science, Education and Sport under the contract no. 119-0982930-1016. P.D.P. was also supported by Alexander von Humboldt Foundation.
Solutions for $\mathcal{N}=M^1M^2M^3$ but general $c_I$ {#sec:app1}
=======================================================
In this appendix we consider actions with $$\mathcal{N}=M^1M^2M^3$$ but arbitrary coefficients $c_I$. Let us define $$\zeta_I \equiv \frac{c_I}{24}$$ and for simplicity restrict to $\zeta_I > 0$. We shall concentrate on non-BPS solutions and Ansatz 1. For this case we can specialize the general expression for relation between electric charges and field strengths (\[a1eom5\]) as follows $$\hat{q}_{1} = -4 e^{2} e^{3} \ , \ \hat{q}_{2} = -4 e^{3} e^{1} \ ,
\ \hat{q}_{3} = -4 e^{1} e^{2}\\$$ where we introduced shifted charges $$\hat{q}_{I} \equiv q_I - \frac{\zeta_I}{3}$$ From here follow also simple relations $$\frac{e^{1} \hat{q}_{1}}{4} = \frac{e^{2} \hat{q}_{2}}{4}
= \frac{e^{3} \hat{q}_{3}}{4} = -(e)^{3}$$ We introduce definition $$\label{amev}
A_{i} = \frac{M^{i}}{e^{i}} \sqrt{v_{1}} \ , \quad i=1,2,3$$ The corresponding system of equations then follows from equation (\[sgeneom\]) $$\begin{aligned}
&& (-1 + A_{2})(-1 + A_{3})e^{2}e^{3} = 0\\
&& (-1 + A_{1})(-1 + A_{3})e^{1}e^{3} = 0\\
&& (-1 + A_{1})(-1 + A_{2})e^{1}e^{2} = 0\end{aligned}$$ $$\begin{aligned}
&& 6(7v_{1}^{3/2} + A_{1}e^{1}((4 - 4A_{2} + (-4 + A_{2})A_{3})e^{2}e^{3} + \zeta_{1}) + \nonumber\\
&& + A_{2}e^{2}(-4(-1 + A_{3})e^{1}e^{3} + \zeta_{2}) + A_{3}e^{3}(4e^{1}e^{2} + \zeta_{3})) = 7(e^{1}\zeta_{1} + e^{2}\zeta_{2} + e^{3}\zeta_{3})\\
&& 18v_{1}^{3/2} + 2e^{1}(3(4A_{3} + A_{1}(4 - 4A_{3} + A_{2}(-4 + 5A_{3})))e^{2}e^{3} - A_{1}\zeta_{1}) = \nonumber\\
&& = 3e^{1}\zeta_{1} + 3e^{2}\zeta_{2} + 2A_{2}e^{2}(12(-1 + A_{3})e^{1}e^{3} + \zeta_{2}) + (3 + 2A_{3})e^{3}\zeta_{3}\\
&& 6v_{1}^{3/2} + 2A_{1}e^{1}(-3A_{2}A_{3}e^{2}e^{3} + \zeta_{1}) + (-1 + A_{2})e^{2}\zeta_{2} + (-1 + A_{3})e^{3}\zeta_{3} = e^{1}\zeta_{1}\\
&& 6v_{1}^{3/2} + 2e^{1}(3(-4A_{2}A_{3} + A_{1}(-4A_{3} + A_{2}(-4 + 3A_{3})))e^{2}e^{3} + 5A_{1}\zeta_{1}) + \nonumber\\
&& + 10A_{2}e^{2}\zeta_{2} + (-1 + 10A_{3})e^{3}\zeta_{3} = e^{1}\zeta_{1} + e^{2}\zeta_{2}\end{aligned}$$
We shall again find solutions with one negative and two positive shifted charges, and “strange” solutions with all shifted charges negative.\
\
We first describe solutions with one charge negative, e. g., $q_1$. Then $$\label{v1qz}
\sqrt{v_{1}} = \frac{1}{2} \,
\frac{(-Q_{(3)}^{2})^{1/6}}{\hat{q}_{1}^{1/6}\hat{q}_{2}^{1/6}
\hat{q}_{3}^{1/6}}$$ $$Q_{(3)} \equiv \hat{q}_{1}\hat{q}_{2}\hat{q}_{3} + \frac{2}{3} \,
\left(\zeta_{1}\hat{q}_{2}\hat{q}_{3}
+ \zeta_{2}\hat{q}_{1}\hat{q}_{3}
+ \zeta_{3}\hat{q}_{1}\hat{q}_{2}\right)$$ $$\label{aqz}
A_{1} = - \frac{(4\zeta_{3}\hat{q}_{2} + (4\zeta_{2} + 3q_{2})
\hat{q}_{3})\hat{q}_{1}}{(4\zeta_{1}
+ 3\hat{q}_{1})\hat{q}_{2}\hat{q}_{3}} \;,\qquad A_{2} = 1
\;,\qquad A_{3} = 1$$ $$M^{1} = \frac{1}{48v_1^{2}} \, \frac{Q_{(3)}}{\hat{q}_{2}\hat{q}_{3}}
\, \frac{4\zeta_{3}\hat{q}_{2} + (4\zeta_{2} + 3\hat{q}_{2})
\hat{q}_{3}}{\hat{q}_{1} + \frac{4}{3}\zeta_{1}}$$ $$\label{m23qz}
M^{2} = 4v_1\frac{\hat{q}_{1}\hat{q}_{3}}{Q_{(3)}} \;,\qquad
M^{3} = 4v_1\frac{\hat{q}_{1}\hat{q}_{2}}{Q_{(3)}}$$ Here we are able to impose positivity restriction, $$M^{i} > 0 \ , \ i=1,2,3$$ $$M^{2} > 0 \Rightarrow \frac{\hat{q}_{3}}{Q_{(3)}} < 0 \Rightarrow Q_{(3)} < 0$$ $M^{3} > 0$ is automatically satisfied.\
Consider now $M^{1} > 0$. Note first that $Q_{3}/\hat{q}_{2}\hat{q}_{3} < 0$. Thus we obtain that $$\frac{4\zeta_{3}\hat{q}_{2} + (4\zeta_{2} + 3\hat{q}_{2})\hat{q}_{3}}{\hat{q}_{1} + \frac{4}{3}\zeta_{1}} < 0$$ But numerator is positive so we get $$\hat{q}_{1} + \frac{4}{3}\zeta_{1} < 0$$ Note that under mentioned restrictions the property $Q_{(3)} < 0$ is indeed satisfied. In fact $$\begin{aligned}
Q_{(3)} &=& \hat{q}_{1}\hat{q}_{2}\hat{q}_{3}
+ \frac{2}{3}(\zeta_{1}\hat{q}_{2}\hat{q}_{3}
+ \zeta_{2}\hat{q}_{1}\hat{q}_{3} + \zeta_{3}\hat{q}_{1}\hat{q}_{2})
\nonumber\\
&<& - \frac{4}{3} \, \zeta_{1}\hat{q}_{2}\hat{q}_{3}
+ \frac{2}{3} \, (\zeta_{1}\hat{q}_{2}\hat{q}_{3}
+ \zeta_{2}\hat{q}_{1}\hat{q}_{3} + \zeta_{3}\hat{q}_{1}\hat{q}_{2})
\nonumber\\
&=& - \frac{2}{3} \, \zeta_{1}\hat{q}_{2}\hat{q}_{3}
+ \frac{2}{3} \, (\zeta_{2}\hat{q}_{1}\hat{q}_{3}
+ \zeta_{3}\hat{q}_{1}\hat{q}_{2}) < 0\end{aligned}$$ Let us find entropy. We shall use (\[entropy\]), (\[fsusy\]) and (\[amev\]) to obtain: $$F = 8\pi(e)^{3}\left\{ A_{1}A_{2}A_{3} - (A_{1} + A_{2} + A_{3}) \right\}$$ But $A_{2} = A_{3} = 1$ so $$S_{BH} = -16\pi(e)^{3}$$ From the explicit form of the solution (\[v1qz\])-(\[m23qz\]) we have $$S_{BH} = 2\pi \,
\textrm{sign}(\frac{\hat{q}_{1}\hat{q}_{2}\hat{q}_{3}}{Q_{(3)}})
\sqrt{-\hat{q}_{1}\hat{q}_{2}\hat{q}_{3}}$$ But $\textrm{sign}(\hat{q}_{1}\hat{q}_{2}\hat{q}_{3}/Q_{(3)}) = +1$ so $$S_{BH} = 2\pi \sqrt{|\hat{q}_{1}\hat{q}_{2}\hat{q}_{3}|}$$ We finally conclude that presented solution is valid for $$\hat{q}_{3} > 0 \ , \ \hat{q}_{1} < -\frac{4}{3} \zeta_{1} \ , \ \hat{q}_{2} > 0$$ completely analogous to the first case in Sec. \[ssec:stusorr\].\
\
In this case all relations can be obtained from previous case with exchange $(1) \leftrightarrow (2)$.\
\
Analogously this case can be obtained from the first case with interchange $(1) \leftrightarrow (3)$.\
In addition to described 3 “normal” branches there are also 3 “strange” branches which give negative entropy: $$S_{BH} = -2\pi \sqrt{\left|\hat{q}_{1}\hat{q}_{2}\hat{q}_{3}\right|}$$ Such a solution may occur only if all $\hat{q}_{I}$’s are negative.
Gauss-Bonnet correction in $K3\times T^2$ compactifications {#sec:app2}
===========================================================
In this appendix we give the proof of the Eqs. (\[eGBK3\]) and (\[GBhet\]). We consider the actions with $$\mathcal{N}=\frac{1}{2}M^1c_{ij}M^iM^j \;, \qquad c_1 \equiv 24\zeta
\;,\qquad c_i=0 \;, \qquad
i,j>1$$ and when the higher order correction is proportional to the Gauss-Bonnet density, i.e., it is given with (\[l1gb\]). For such corrections one can integrate out auxiliary fields in the same manner as when there is no correction, and pass to the on-shell form of the action which is now given with (\[l0noaux\]) and (\[l1gb\]), with the condition for real special geometry $\mathcal{N}=1$ (which is here [*not*]{} violated by higher order Gauss-Bonnet corrections) implicitly understood.
Now, before going to a hard work, it is convenient to do following transformations (which is a generalisation of what we did in Sec. \[ssec:sol0\]). First we introduce scaled moduli $\bar{M}^i$ and the dilaton $S$ $$S = (M^1)^{3/2} \;\qquad \tilde{M}^i = S^{1/3} M^i$$ for which the real special geometry condition now reads $$\label{cmm1}
\frac{1}{2} c_{ij} \tilde{M}^i \tilde{M}^j = 1$$ This condition fixes one of $\tilde{M}^i$[']{}s. Then we make Poincaré duality transformation (\[poidual\]) which replaces two-form gauge field strength $F^1$ with its dual 3-form strength $H$. Finally, we pass to the string frame metric by Weyl rescaling $$g_{ab} \to S^{2/3} g_{ab} \;.$$ Again, we are interested in $AdS_2\times S^3$ backgrounds which in the present case requires $$\begin{aligned}
&& ds^2 = v_1 \left( -x^2 dt^2 + \frac{dx^2}{x^2} \right)
+ v_2\,d\Omega_{3}^2 \nonumber \\
&& F^{i}_{tr}(x) = -e^i \;, \qquad
H_{mnr} = 2 q_1 \sqrt{h_S}\,\varepsilon_{mnr} \nonumber \\
&& \tilde{M}^i(x) = \tilde{M}^i \;, \qquad S(x) = S\end{aligned}$$ where $\varepsilon_{mnr}$ is totally antisymmetric tensor satisfying $\varepsilon_{234}=1$. Observe that $q_1$ now plays the role of [*magnetic*]{} charge. We apply entropy function formalism of Sec. \[sec:ent-func\]. Function $f$ is now $$\label{fK3gb}
f = \frac{1}{2}v_1v_2^{3/2}S \left( -\frac{2}{v_1} + \frac{6}{v_2}
+ \frac{1}{v_1^2}\tilde{G}_{ij}e^ie^j - \frac{2q_1^2}{v_2^3}
- \frac{3\zeta}{v_1v_2} \right)$$ where $\tilde{G}_{ij}$ is given with $$\label{gijK3}
\tilde{G}_{ij} = \frac{1}{2}\left( c_{ik}\tilde{M}^kc_{jl}\tilde{M}^l
- c_{ij} \right)$$ To obtain solutions we need to solve extremization equations $$\label{gbeom}
0 = \frac{\partial f}{\partial v_1} \;, \qquad
0 = \frac{\partial f}{\partial v_2} \;, \qquad
0 = \frac{\partial f}{\partial S} \;, \qquad
0 = \frac{\partial f}{\partial \tilde{M}^i} \;, \qquad
q_i = \frac{\partial f}{\partial e^i}$$ From the third equation (for $S$) one obtains that $f$ is vanishing. This allows us to solve immediately first two equations (for $v_1$ and $v_2$) and obtain $$\label{gbv1}
v_1 = \frac{v_2}{4}+\frac{\zeta}{8}$$ where $v_2$ is the real positive root of a cubic equation $$0 = x^3 - \frac{3}{2}\zeta x^2 - q_1^2 x
- \frac{\zeta}{2}q_1^2$$ which is, explicitly written, $$\begin{aligned}
\label{gbv2}
v_2 &=& \frac{\zeta}{2}
+ \frac{(1+i\sqrt{3})(4q_1^2+3\zeta^2)}{4\,3^{1/3}
\left(-9\zeta^3 - 36\zeta q_1^2 + 2\sqrt{3}
\sqrt{27\zeta^4q_1^2 + 72\zeta^2q_1^4 - 16q_1^6}\right)^{1/3}}
\nonumber \\
&& +\frac{(1-i\sqrt{3})\left(-9\zeta^3 - 36\zeta q_1^2 +2\sqrt{3}
\sqrt{27\zeta^4q_1^2 + 72\zeta^2q_1^4 - 16q_1^6}
\right)^{1/3}}{4\,3^{2/3}}\end{aligned}$$ Next one can solve equations for $\tilde{M}^i$. Note that one of them is not independent because of (\[cmm1\]), but this can be easily treated, e.g., by using Lagrange multiplier method. One obtains $$0 = c_{ij}e^i\tilde{M}^j \left[ c_{nk}\tilde{M}^k
\left(c_{ij}e^i\tilde{M}^j\right) - 2 c_{nk}e^k \right]$$ from which follow conditions $$\label{gbmeq}
0 = c_{ij}e^i\tilde{M}^j \qquad \mbox{or} \qquad
(c_{nk}\tilde{M}^k)(c_{ij}e^i\tilde{M}^j) = 2 c_{nk}e^k$$ From the third equation in (\[gbeom\]) (for $S$) we obtain $$\label{gbgee}
\tilde{G}_{ij}e^ie^j = v_1 +\frac{3}{2}\zeta\frac{v_1}{v_2}$$ Last equation in (\[gbeom\]), which defines electric charges, gives $$\label{gbqeom}
q_i = S\frac{v_2^{3/2}}{v_1}\tilde{G}_{ij}e^j$$ which, together with (\[gijK3\]) and (\[gbmeq\]) gives $$\label{gbqce}
q_i = \mp S\frac{v_2^{3/2}}{v_1}c_{ij}e^j$$ where the upper (lower) sign is when first (second) condition in (\[gbmeq\]) applies. For the entropy we need $q_ie^i$, which from (\[gbqeom\]) is $$\label{gbqiei}
q_ie^i = \frac{v_2^{3/2}}{v_1}S\tilde{G}_{ij}e^ie^j$$ We need a solution for the dilaton $S$ which is obtained by contracting (\[gbqce\]) with $q_kc^{ki}$. The result is $$S = \frac{v_1}{v_2^{3/2}}\left|
\frac{2q_ic^{ij}q_j}{\tilde{G}_{ij}e^ie^j} \right|^{1/2}$$ Using this in (\[gbqiei\]) we finally get for the black hole entropy $$\label{gbentK3}
S_{BH} = 2\pi q_ie^i = 4\pi \sqrt{\tilde{G}_{ij}e^ie^j}
\sqrt{\frac{1}{2}\left|q_ic^{ij}q_j\right|}
= 4\pi \sqrt{v_1 +\frac{3}{2}\zeta\frac{v_1}{v_2}}
\sqrt{\frac{1}{2}\left|q_ic^{ij}q_j\right|}$$ where $v_1$ and $v_2$ are functions of $q_1$ and $\zeta$ given in (\[gbv1\]) and (\[gbv2\]). Observe that here entropy is nontrivial function of charge $q_1$ (obtained by solving cubic equation), contrary to the case of SUSY corrections which just introduce a constant shifts.
For small black holes, i.e., when $q_1=0$, Eqs. (\[gbv1\]), (\[gbv2\]) and (\[gbgee\]) simplify to $$v_1 = \frac{v_2}{3} = \frac{\zeta}{2} \;,\qquad
\tilde{G}_{ij}e^ie^j = \zeta$$ Plugging this in (\[gbentK3\]) gives for the entropy $$S_{BH} = 4\pi \sqrt{\frac{\zeta}{2}\left|q_ic^{ij}q_j\right|}
\qquad \mbox{for } \; q_1=0$$ which is exactly (\[eGBK3\]).
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[^1]: From the viewpoint of compactification of $D=11$ SUGRA they are topological numbers connected to second Chern class, see [@Ferrara:1996hh].
[^2]: Our conventions for Newton coupling is $G_5=\pi^2/4$ and for the string tension $\alpha'=1$.
[^3]: This formalism was used recently in near-horizon analyses of a broad classes of black holes and higher dimensional objects [@entfappl]. For generalisation to rotating black holes see [@Astefanesei:2006dd]. For comparison with SUSY entropy functions see [@Cardoso:2006xz].
[^4]: As the detailed derivation was already presented in [@Castro:2007hc] (solutions in the whole space) and in [@Alishahiha:2007nn] (near horizon solutions), we shall just state the results here.
[^5]: We emphasize that one should be cautious in geometric interpretation of this result. Higher order corrections generally change relations between fields in the effective action and geometric moduli, and one needs field redefinitions to restore the relations. Then correctly defined moduli may still satisfy condition for real special geometry.
[^6]: For example by comparing with Eqs. (2.2), (2.8) and (2.3) in [@Sen:2005kj]. Observe that, beside simple change in indices $1\to2$ and $2\to3$, one needs to divide gauge fields by a factor of two to get results in Sen’s conventions. There is also a difference in a convention for $\alpha'$, which makes normalization of charges different.
[^7]: To consider corrections in $g_s$ it would be necessary also to make corrections in the prepotential (i.e., to $c_{IJK}$).
[^8]: We note that, as was shown in $D=4$ [@Sen:2004], that corrections can change relations between fields in the action and moduli of the compactification manifold, so one should be careful when demanding physicality conditions.
[^9]: Similar observation in $D=4$ dimensions was given in [@Sen:2004].
[^10]: Notice that for some non-BPS solutions both relations are violated.
[^11]: Notice that we are using $\alpha'=1$ convention, and in [@Sen:2005kj] it is $\alpha'=16$. One can use the results from [@Prester:2005qs] to make connection between conventions.
|
---
abstract: 'Computation of document image quality metrics often depends upon the availability of a ground truth image corresponding to the document. This limits the applicability of quality metrics in applications such as hyperparameter optimization of image processing algorithms that operate on-the-fly on unseen documents. This work proposes the use of surrogate models to learn the behavior of a given document quality metric on existing datasets where ground truth images are available. The trained surrogate model can later be used to predict the metric value on previously unseen document images without requiring access to ground truth images. The surrogate model is empirically evaluated on the Document Image Binarization Competition (DIBCO) and the Handwritten Document Image Binarization Competition (H-DIBCO) datasets.'
author:
-
bibliography:
- 'refs.bib'
title: |
Learning Surrogate Models of Document Image Quality Metrics for\
Automated Document Image Processing
---
surrogate models; document image quality metrics; hyperparameter optimization
Introduction {#sec:intro}
============
Document image quality metrics are objective measures that enable assessment and quantification of characteristics of a given document image. Such metrics are crucial for enabling automatic document processing applications, such as fully-automatic document image binarization. Specifically, document image processing algorithms involve hyperparameters that must be optimized to achieve the best possible resulting image. Hyperparameter optimization techniques such as Bayesian optimization [@snoek2012practical] require formulation of an *objective function* to be maximized. Document image quality metrics are natural candidates as objective functions.
In general, document image quality is calculated by comparing the image in question to the noise-free replica of the document image, known as the ground truth reference image. There exist several popular image quality metrics in literature [@ye2013document]. A vast majority of the methods considered Optical Character Recognition (OCR) results as document quality metrics [@hale2007human; @kang2014deep; @nayef2015metric]. Simple techniques to measure the image quality, such as Mean Squared Error (MSE) do not suffice due to the complex and degraded nature of images. There is a need for more sophisticated methods to assess image quality. Popular document quality evaluation measures [@gatos2009icdar; @pratikakis2016icfhr2016] include the F-Measure, the Peak-Signal-to-Noise Ratio (PSNR), the Distance Reciprocal Distortion metric (DRD) [@lu2004distance], and the Negative Rate Metric (NRM). Computation of such metrics requires a corresponding distortion-free ground truth reference image for any given document image.
In addition to ground truth images, human opinion scores have been used as ground truth in [@lu2004distance; @obafemi2012character; @kumar2012sharpness; @alaei2015document] to automatically compute the document image quality metrics. A full reference document image quality assessment technique based on texture similarity index was introduced in [@alaei2016document] with promising results for OCR text images. There have been recent efforts to formulate image quality metrics that are not dependent on availability of ground truths. Xu et al. [@xu2016no] presented a no-reference image quality metric for document image quality assessment.
However, no-reference image quality metrics, such as [@xu2016no], are typically designed for document images with OCR text, and focus on specific aspects of degradations that are mostly character level distortion (e.g., noise around a character, partial or overlapping characters), and are not suitable to quantify high levels of degradations in historical handwritten texts. Machine-printed documents have simple layouts and fonts, unlike handwritten documents that have complex layouts and variability in writing style. Handwritten documents suffer from degradations such as paper stains, ink bleed-through, missing or faded data, poor contrast, warping effects, etc. that hamper document readability and pose challenges for document image processing algorithms [@giotis2017survey].
Such variability and severity of degradations is better captured using ground truth based document image quality metrics such as F-Measure, PSNR and DRD. Ground truth images offer a reference point, relative to which candidate images can be ranked. This immensely helps image processing algorithms in automatically evaluating the quality of processed images.
However, the reliance on the availability of ground truth images is also severely limiting. In fact, the target domain of automated document image processing consists of ground truth generation as one of the applications. Therefore, it is impractical to have access to ground truths corresponding to previously unseen document images to be processed on-the-fly. It is possible however, to have access to a *training set* of document images and corresponding ground truth images.
This work explores a novel methodology wherein document quality metric scores computed using ground truth images as reference are used to train a model that learns the relationship between the difference in image quality represented by two images, and the corresponding metric score. Given two document images - an initial image and a processed image for which the quality metric is to be computed, the trained model can be used as a *surrogate* that predicts the value of the metric. Training the surrogate model is a one-time investment, and requires access to input images with corresponding ground truth images. Post training, evaluation of the surrogate model is near-instant and does not require access to ground truth image for any given test image.
This paper is organized as follows. Section \[sec:prob\] describes the concrete problem statement. Section \[sec:doc\] discusses various document quality metrics available in literature. Section \[surrogate\] explains the proposed surrogate modeling approach in detail. Section \[sec:experiments\] demonstrated the efficacy of the proposed approach on the DIBCO and H-DIBCO datasets. Section \[sec:discussion\] discusses an alternative deep learning formulation for surrogate model training. Section \[sec:conclusion\] concludes the paper.
Problem Statement {#sec:prob}
=================
The performance of document image processing tasks such as document binarization, filtering, enhancement, text or line segmentation, and high level applications such as word spotting in a document, significantly depends on the associated hyperparameter values. In general, an automated document image processing algorithm involves automatic selection of control parameters on-the-fly. This work uses document binarization as a running example throughout the text.
Although there exist several automated document image processing methods in literature [@vats2017automatic; @howe2013document], a ground truth reference image is required to tune the associated hyperparameters. For example, an automatic document image binarization method is proposed in [@vats2017automatic], where Bayesian optimization is used to infer the hyperparameters on-the-fly. The value of hyperparameters is chosen such that the quality metrics corresponding to the binarized image, (such as F-Measure, PSNR etc.) are maximized, or error is minimized. However, the optimization of quality metrics such as the F-Measure, PSNR, DRD and NRM is dependent upon the availability of a ground truth reference image. This limits the applicability of such methods in real world document image processing applications.
This work explores the use of surrogate models to approximate any given document image quality metric. Let $X=\{{\bf x}_i\}_{i=1}^n$ be a set of document images comprising of $n$ images. Let $G=\{{\bf g}_i\}_{i=1}^n$ be the corresponding $n$ ground truth images. Let $P=\{{\bf p}_i\}_{i=1}^n$ be the set of processed images corresponding to $X$, obtained after processing using algorithm $A$, for example, a binarization algorithm. It is possible to compute and assign various quality metrics to ${\bf p}_i$ using ${\bf g}_i$ as a reference. Let $Y=\{y_i\}_{i=1}^n$ be a vector of values computed for any such quality metric $q$ corresponding to $P$.
Let ${\bf x}'$ be a previously unseen test document image, with ${\bf p}'$ being the processed image obtained using a given algorithm $A$. The goal is to learn a surrogate model that can predict the value $y'$ of the metric $q$ for a given pair $({\bf x}', {\bf p}')$. Such a model will enable instant on-the-fly performance feedback for the algorithm $A$ without the availability of corresponding $g$. The model $\hat{y}$ is in effect, a *surrogate* of the quality metric $q$. The following section explores popular document image quality metrics.
Quality Metrics {#sec:doc}
===============
The most popularly used document image quality metrics include F-Measure, PSNR, DRD and NRM. These evaluation measures compute the image quality by comparing the document image with the corresponding ground truth reference image [@gatos2009icdar; @pratikakis2016icfhr2016].
F-Measure
---------
F-Measure captures accuracy, defined as the weighted harmonic mean of Precision and Recall, $$F-Measure = \frac{ 2 \times Recall \times Precision}{Recall + Precision },
\label{eq_fm}$$ where $Recall=\frac{TP}{TP + FN}$ and $Precision=\frac{TP}{TP + FP}$. TP, FN and FP denote True Positives, False Negatives and False Positives, respectively.
Peak Signal-to-Noise Ratio (PSNR)
---------------------------------
PSNR is a popularly used metric to measure how close an image is to another image. The higher the value of PSNR, the higher the similarity between two images. PSNR is defined via the mean squared error (MSE). Given a noise-free $M$$\times$$N$ image $I$ and its noisy approximation $K$, MSE is defined as, $$MSE = \frac{\sum_{i=1}^{M}\sum_{i=1}^{N}(I(i,j) - I'(i,j))^2}{MN},
\label{eq_mse}$$ and PSNR is defined as, $$PSNR = 10 log \Big( \frac{C^2}{MSE} \Big),
\label{eq_psnr}$$ where $C$ is the difference between foreground and background image.
Distance Reciprocal Distortion metric (DRD)
-------------------------------------------
DRD is used to measure the visual distortion for all the $S$ flipped pixels in binary document images [@lu2004distance], and is defined as, $$DRD = \frac{\sum\limits_{k=1}^{S}DRD_k}{NUBN},
\label{eq_drd}$$ where $DRD_k$ is the distortion of the $k$-th flipped pixel, calculated using a $5 \times 5$ normalized weight matrix $W_{Nm}$ as, $$DRD_k = {\sum\limits_{i=-2}^{2} \sum\limits_{j=-2}^{2} |GT_k(i,j) - B_k(x,y)| \times W_{Nm}(i,j)}.
\label{eq_drdk}$$
$DRD_k$ denotes the weighted sum of the pixels in the $5 \times 5$ block of the ground truth that differ from the centered $k$-th flipped pixel at $(x,y)$ in the binarized image. NUBN is the number of non-uniform (not all black/white pixels) $8 \times 8$ blocks in the ground truth image.
\
Negative Rate Metric (NRM)
--------------------------
NRM measures the pixel-wise mismatch rate between the ground truth image and the resultant binarized image. NRM is defined as, $$NRM = \frac{NR_{FN} + NR_{FP}}{2},
\label{eq_nrm}$$ where $NR_{FN}=\frac{N_{FN}}{N_{FN} + N_{TP}}$, $NR_{FP}=\frac{N_{FP}}{N_{FP} + N_{TN}}$.
$NR_{FN}$ denotes the false negative rate, $NR_{FP}$ denotes the false positive rate, $N_{TP}$ is the number of true positives, $N_{FP}$ is the number of false positives, $N_{TN}$ is the number of true negatives and $N_{FN}$ is the number of false negatives. The lower the value of NRM, the better is the binarized image quality.
Surrogate Models for Learning Document Quality Metrics {#surrogate}
======================================================
Surrogate modeling [@gorissen2010surrogate] has emerged as a popular methodology to obtain a fast-to-evaluate approximation of a computationally expensive or data-scarce function. Since the surrogate model allows fast evaluation, it can be used in applications such as optimization, parameter space exploration and sensitivity analysis where a large number of repeated calls to the target function are required.
For example, complex simulation codes are often used during the design process of electronic devices such as antennae, microwave filters, etc. In order to study and test the effect of varying design parameters, repeated calls to simulation codes are made. Each of these calls may take several minutes to evaluate, and this hampers the design space exploration process. Globally accurate surrogate models offer near-instant evaluation and can be used in place of such simulation codes. Obtaining such a surrogate involves preparing training data by evaluating the simulation code on a carefully selected set of parameter combinations or points, which is chosen according to a statistical design or a sampling algorithm [@gorissen2010surrogate].
Automated document image processing algorithms that make use of ground truth-based image quality metrics are an excellent use-case for surrogate models. Since ground truth images are scarce, therefore it makes sense to train an accurate surrogate model of a specified image metric using the limited quantity of available ground truth images. The surrogate model can then be used to estimate the value of image quality metric on-the-fly for any input test image, and corresponding processed image.
Surrogate Model Types
---------------------
Numerous surrogate model types exist in literature with Artificial Neural Networks (ANN), Gaussian Processes (GP) and Support Vector Machines (SVM) being popular [@singh2016shape]. ANNs [@haykin2009neural] have shown excellent results in recent years, especially in applications involving visual data, and problems involving large training sets. GPs [@rasmussen2006gaussian] are very popular in design optimization applications and global surrogate modeling owing to their capability of providing the variance of prediction, in addition to the prediction itself. This aids adaptive sampling algorithms in quickly searching for optima within a mathematically principled framework. SVR models [@cortes1995support] formulate the learning problem into an optimization problem that can be solved in a straightforward manner. SVR models have proven to be robust and stable in a variety of problems, and can deal with both small and large datasets. Consequently, SVR models are a reliable choice for general use in global modeling problems. This work uses ANN, GP and SVM regression models as surrogates for the purpose of experiments. However, the framework and methodology proposed herein is independent of any particular model type. A detailed discussion on the model types is out of scope in this work, and the reader is referred to [@smola2004tutorial; @basak2007support] for SVR (support vector regression), [@rasmussen2006gaussian] for GPs and [@haykin2009neural] for ANNs.
Model Training {#approachB}
--------------
Let each document image ${\bf x}_i$ and processed image ${\bf p}_i$ be represented as a $k_1 \times k_2$ matrix. The surrogate model learns the mapping $inputs \rightarrow target$. The target is the value of the document image quality metric. The metric may also be user-defined scores. Intuitively, the inputs must represent the quantity of change or transformation the image processing algorithm $A$ has brought about in the original image ${\bf x}_i \in X$ to obtain ${\bf p}_i \in P$. The surrogate must be able to learn the value of a given image quality metric associated with the difference and nature of transformation from ${\bf x}_i$ to ${\bf p}_i$. This transformation can also be represented as a vector of metrics that represent the differences between ${\bf x}_i$ and ${\bf p}_i$. Possible candidates to measure such transformation include the metrics explained in Section \[sec:doc\], e.g., F-Measure, PSNR, DRD, etc. Let $M=\{q_j\}_{j=1}^T$ be $T$ metrics. For any given document image ${\bf x}_i$ and corresponding processed image ${\bf p}_i$, the $1 \times T$ vector $I_i$ represents the values of $T$ metrics as, $$I_i = [q_1({\bf x}_i, {\bf p}_i),\; q_2({\bf x}_i, {\bf p}_i),\; \cdots , \; q_T({\bf x}_i, {\bf p}_i)].$$ The complete $n \times T$ matrix $I$ represents the input variables to be learned by the surrogate model. The use of quality metrics as input variables immensely simplifies the learning problem as compared to the case of using raw images as input. The target vector $Y$ simply represents the values of a specific document image quality metric $q$ computed as, $$Y_i = q({\bf p}_i, {\bf g}_i).$$ The training set for the surrogate model is then $\mathcal{T} = (I,Y)$. The framework of the proposed approach is pictorially described in Fig. \[fig\_model\].
Experiments {#sec:experiments}
===========
Dataset
-------
The proposed surrogate-based approach is empirically evaluated on the images from seven well-known competition datasets: DIBCO 2009 [@gatos2009icdar], H-DIBCO 2010 [@pratikakis2010h], DIBCO 2011 [@gatos2011icdar2], H-DIBCO 2012 [@pratikakis2012icfhr], DIBCO 2013 [@pratikakis2013icdar], H-DIBCO 2014 [@ntirogiannis2014icfhr2014] and H-DIBCO 2016 [@pratikakis2016icfhr2016]. These datasets contain machine-printed and handwritten historical document images suffering from various kinds of degradations including stained paper, faded ink or ink bleed through, wrinkles and unknown graphical symbols. In total there are 86 document images, out of which 63 randomly chosen images are used for training and 23 images for testing. As an example, the framework is applied to perform automatic image binarization using Bayesian optimization as proposed in [@vats2017automatic]. The document image quality metrics used as inputs for the surrogate models include PSNR, DRD and NRM. The target image quality metric to be approximated using surrogates is the F-Measure.
Experimental Results
--------------------
The $\varepsilon$-SVR variant [@basak2007support] with the Sequential Minimal Optimization (SMO) [@platt1998sequential] solver is used for the following experiments. The hyperparameters of the SVR model are optimized using Bayesian optimization [@snoek2012practical]. The GP model uses a Gaussian kernel with hyperparameters being optimized using Maximum-Likelihood Estimation (MLE) [@rasmussen2006gaussian]. The variant of ANN used is a feed-forward back propagation neural network [@haykin2009neural] trained using the Levenberg-Marquardt algorithm [@demuth2014neural].
The error metrics used to test the accuracy of the surrogate models are Root Relative Square Error (RRSE), Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) [@graczyk2009comparative].
Model Type RRSE MAE RMSE
------------ ---------------- ---------------- ----------------
ANN 0.8781 2.9980 4.1733
SVR [**0.8053**]{} [**2.7107**]{} [**3.8272**]{}
GP 1.0633 3.3506 25.5363
Ensemble 0.8979 2.9477 5.0533
: Error estimates for the surrogate models.\[tab:resB\]
Model Type Training Time (s) Prediction time (s)
------------ ------------------- ---------------------
ANN $0.3960$ $0.0717$
SVR $37.2203$ ${\bf 0.0039}$
GP ${\bf 0.1035}$ $0.0075$
: Model training and prediction times for a test dataset of $23$ document images distinct from the training set.\[tab:times\]
Table \[tab:resB\] lists error estimates corresponding to the proposed model training approach described in Section \[approachB\]. All three error measures indicate that ANN and SVR outperform GP surrogate for the given dataset. The table also contains error estimates corresponding to an ensemble model that simply averages the predictions of ANN, SVR and GP models. It can be observed from Table \[tab:resB\] that SVR emerges as the single best performing model type.
Table \[tab:times\] reports the time in seconds taken to train the surrogate models and the total time taken by the models to predict F-Measure values of $23$ unseen test document images. It can be seen that once the model is trained, predictions are made almost instantly. This makes the surrogate model assisted approach ideal for use on-the-fly in image processing algorithms. The time taken for preprocessing and model training is a one-time investment. A relatively high value of training time for SVR is due to the time taken to optimize hyperparameters using Bayesian optimization. This was to ensure that the hyperparameters are as close to optimal, given a relatively small training set.
![Predicted value of F-Measure by each surrogate model type for test images. Surrogate models are accurate in general except for test instances 2, 10, 18 and 22.[]{data-label="fig_surrPred"}](surrogatePredictions.png){width="3in"}
Figure \[fig\_surrPred\] depicts the values of F-Measure predicted by different model types following the proposed model training approach. It can be seen that there are relatively large errors made by all model types for test instances numbers 2, 10 and 18. However, all models have been able to capture the *general trend* of the test images, except for test instances 2 and 22. Even though the error is large for test instances 10 and 18, the models have been able to learn the ’downward’ leaning behavior of F-Measure therein.
![Test images having high prediction errors.[]{data-label="fig_testImages"}](Fig_testImages.png){width="3in"}
Figure \[fig\_testImages\] shows test instances 2, 10 and 18 on which all surrogate models struggled. It can be seen that test image 2 has high variation in image contrast and intensity. Test image 10 is suffering from paper wrinkles and fold marks, in addition to pen strokes of varying intensities. Test image 18 also contains variation in pen stroke intensities. Test image 22 (not shown) includes text written with multiple inks. These characteristics are not well-represented in the training set, leading to large errors in prediction of corresponding F-Measure. Having a larger training set that captures a wide variation of paper degradations, writing styles, pen stroke intensities, etc. will improve the performance of surrogate models.
Figure \[fig\_binRes\] shows a sample test document image binarized using the method [@vats2017automatic]. The hyperparameters of the binarization algorithm are optimized using Bayesian optimization [@snoek2012practical] as described in [@vats2017automatic]. The objective function to be maximized using Bayesian optimization is the F-Measure (as predicted by the SVR surrogate model trained above). The resultant image in Figure \[fig\_bin\] is clean and validates the accurate modeling of F-Measure by the SVR surrogate.
Discussion: Raw Images as Input {#sec:discussion}
===============================
The approach discussed herein represents the inputs as image quality metrics measuring difference between $X$ and $P$. This is done to simplify the learning problem to remain within a handful of input parameters, and allows highly efficient learning and inference. It is also possible to consider the input and processed images themselves as input, without any post-processing to calculate quality metrics. The surrogate model will then learn the mapping $({\bf x}_i, {\bf p}_i) \rightarrow target$, where each ${\bf x}_i$ and ${\bf p}_i$ is a $k_1 \times k_2$ matrix. The representation of inputs $I: ({\bf x}_i, {\bf p}_i)$ as images is an ideal use-case of deep learning inspired surrogate models such as convolutional neural networks (CNNs) [@krizhevsky2012imagenet]. The caveat herein is that the training set must be sufficiently large to allow meaningful learning to proceed.
\
Conclusion {#sec:conclusion}
==========
A novel approach is presented in this paper that uses surrogate models to learn a given document image quality metric. The surrogate model is trained on a dataset comprising of inputs that quantify differences in image quality between raw input images and corresponding processed images obtained using an image processing algorithm. The target to be approximated by the surrogate model is the value of a given document image quality metric that is computed for the training set by comparing the processed candidate images to corresponding ground truth images. Post training, the surrogate can be used to quickly predict the value of the document image quality metric for any given test pair of raw and processed document images, without any need for corresponding ground truth images. The methodology is tested on well-known publicly available document image datasets. Experimental evaluation indicates that the surrogate model is able to accurately learn the relationship between differing image quality and corresponding variation in document image quality metric value. Future work includes obtaining and experimenting with larger training sets, and exploring regression convolutional neural networks as surrogate models.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was funded by the Göran Gustafsson foundation, the eSSENCE strategic collaboration on eScience, and the Riksbankens Jubileumsfond (Dnr NHS14-2068:1).
|
---
abstract: 'In this work a new asymptotically flat solution of the coupled Einstein-Born-Infeld equations for a static spherically symmetric space-time is obtained. When the intrinsic mass is zero the resulting spacetime is regular everywhere, in the sense given by B. Hoffmann and L. Infeld in 1937, and the Einstein-Born-Infeld theory leads to the identification of the gravitational with the electromagnetic mass.'
author:
- |
[**D.J. Cirilo Lombardo**]{}\
[*Bogoliubov Laboratory of Theoretical Physics Joint Institute of Nuclear Research,*]{}\
[*Dubna, Moscow Region, 141980, Russia [^1]* ]{}
title: '****'
---
Introduction and results:
=========================
The four dimensional solutions with spherical symmetry of the Einstein equations coupled to Born-Infeld fields have been well studied in the literature$^{1-4}$. In particular, the electromagnetic field of the Born Infeld monopole, in contrast to Maxwell counterpart, contributes to the ADMmass of the system (it is, the four momentum of asymptotic flat manifolds). B. Hoffmann was the first who studied such static solutions in the context of the general relativity with the idea of to obtain a consistent particle-like model$^{2}$. Unfortunately, these static Einstein-Born-Infeld (EBI) models generate conical singularities at the origin$^{2-3}$ that cannot be removed as in global monopoles or other non-localized defects of the spacetime$^{5-6}$. With the existence of this type of singularities in the space-time of the monopole we can not identificate the gravitational with the electromagnetic mass. In this work a *new* static spherically symmetric solution with Born-Infeld charge is obtained. The new metric, when the intrinsic mass of the system is zero, is *regular* everywhere in the sense that was given by B. Hoffmann and L. Infeld$^{3}$ in 1937 and the EBI theory leads to identification of the gravitational with the electromagnetic mass. This means that the metric, the electromagnetic field and their derivatives have not singularities and discontinuities in all the manifold. The fundamental feature of this solution is the lack of conical singularities at the origin. A distant observer will associate with this solution an electromagnetic mass that is a twice of the mass of the electromagnetic geon founded by M. Demianski$^{4}$ in 1986 . The energy-momentum tensor and the electric field are both regular with zero value at the origin and new parameters appear, given to the new metric surprising behaviours. The used convention$^{7-8}$ is the *spatial* of Landau and Lifshitz (1962), with signatures of the metric, Riemann and Einstein tensors all positives (+++) .
The plan of this paper is as follows: in Section 2 we give a short introduction to the Born-Infeld theory: propierties and principal features. In Section 3 the regularity condition as was given by B. Hoffmann and L. Infeld$^{3}$ in 1937 . Sections 4, 5, 6 and 7 are devoted to found the new solution and to analyze its propierties. Finally, the conclusion and comments of the results are presented in section 8.
The Born-Infeld theory:
=======================
The most significative non-linear theory of electrodynamics is, by excellence, the Born-Infeld theory$^{1,9}$. Among its many special properties is an exact SO(2) electric-magnetic duality invariance. The Lagrangian density describing Born-Infeld theory (in arbitrary spacetime dimensions) is $$\mathcal{L}_{BI}=\sqrt{-g}L_{BI}=\frac{b^{2}}{4\pi }\left\{ \sqrt{-g}-\sqrt{%
\left| \det (g_{\mu \nu }+b^{-1}F_{\mu \nu })\right| }\right\},$$ where $b$ is a fundamental parameter of the theory with field dimensions. In open superstring theory$^{10}$, for example, loop calculations lead to this Lagrangian with $b^{-1}=2\pi \alpha ^{\prime }$ ($\alpha ^{\prime }\equiv \ $inverse of the string tension) . In four spacetime dimensions the determinant in (1) may be expanded out to give $$L_{BI}=\frac{b^{2}}{4\pi }\left\{ 1-\sqrt{1+\frac{1}{2}b^{-2}F_{\mu \nu
}F^{\mu \nu }-\frac{1}{16}b^{-4}\left( F_{\mu \nu }\widetilde{F}^{\mu \nu
}\right) ^{2}}\right\},$$ which coincides with the usual Maxwell Lagrangian in the weak field limit.
It is useful to define the second rank tensor $P^{\mu \nu }$ by $$P^{\mu \nu }=-\frac{1}{2}\frac{\partial L_{BI}}{\partial F_{\mu \nu }}=\frac{%
F^{\mu \nu }-\frac{1}{4}b^{-2}\left( F_{\rho \sigma }\widetilde{F}^{\rho
\sigma }\right) \,\widetilde{F}^{\mu \nu }}{\sqrt{1+\frac{1}{2}b^{-2}F_{\rho
\sigma }F^{\rho \sigma }-\frac{1}{16}b^{-4}\left( F_{\rho \sigma }\widetilde{%
F}^{\rho \sigma }\right) ^{2}}}$$ (so that $P^{\mu \nu }\approx F^{\mu \nu }$ for weak fields) satisfying the electromagnetic equations of motion $$\nabla _{\mu }P^{\mu \nu }=0$$ which are highly non linear in $F_{\mu \nu }$. The energy-momentum tensor may be written as $$T_{\mu \nu }=\frac{1}{4\pi }\left\{ \frac{F_{\mu }^{\,\ \ \lambda }F_{\nu
\lambda }+b^{2}\left[ \Bbb{R}-1-\frac{1}{2}b^{-2}F_{\rho \sigma }F^{\rho
\sigma }\right] g_{\mu \nu }}{\Bbb{R}}\right\}$$ $$\Bbb{R}\equiv \sqrt{1+\frac{1}{2}b^{-2}F_{\rho \sigma }F^{\rho \sigma }-%
\frac{1}{16}b^{-4}\left( F_{\rho \sigma }\widetilde{F}^{\rho \sigma }\right).
^{2}}$$ Although it is by no means obvious, it may verified that equations (3)-(5) are invariant under electric-magnetic rotations of duality $%
F\longleftrightarrow *G$. We can show that the SO$\left( 2\right) $ structure of the Born-Infeld theory is more easily seen in quaternionic form$%
^{11-12}$ $$\frac{1}{R}\left( \sigma _{0}+i\sigma _{2}\overline{\Bbb{P}}\right) L=\Bbb{L}$$ $$\frac{\Bbb{R}}{\left( 1+\overline{\Bbb{P}}^{2}\right) }\left( \sigma
_{0}-i\sigma _{2}\overline{\Bbb{P}}\right) \Bbb{L}=L$$ $$\overline{\Bbb{P}}\equiv \frac{\Bbb{P}}{b} ,$$ where we defined $$L=F-i\sigma _{2}\widetilde{F}$$ $$\Bbb{L}=P-i\sigma _{2}\widetilde{P}$$ the pseudoescalar of the electromagnetic tensor $F^{\mu \nu }$
$$\Bbb{P}=-\frac{1}{4}F_{\mu \nu }\widetilde{F}^{\mu \nu }$$ and $\sigma _{0}\,$, $\sigma _{2}$ the well know Pauli matrix.
In flat space, and for purely electric configurations, the Lagrangian (2) reduces to $$L_{BI}=\frac{4\pi }{b^{2}}\left\{ 1-\sqrt{1-b^{-2}\overrightarrow{E^{2}}}%
\right\}$$ so there is an upper bound on the electric field strength $\overrightarrow{E}
$ $$\left| \overrightarrow{E}\right| \leq b \ .$$
The regularity condition
========================
The new field theory initiated in 1934 by M. Born$^{9}$ introduces in the classical equations of the electromagnetic field a characteristic length $%
r_{0}$ representing the radius of the elementary particle through the relation $$r_{0}=\sqrt{\frac{e}{b}} \ ,$$ where $e$ is the elementary charge and $b$ the fundamental field strength entering in a non-linear Lagrangian function. It was originally thought that the Lagrangian (1) was the simplest choice which would lead to a finite energy for an electric particle. This is, however, not the case. It is possible to find an infinite number of quite different action functions, each giving simple algebraic relations between the fields and each leading to a finite energy for an electric particle.
In 1937 B. Hoffmann and L. Infeld$^{3}$ introduce a regularity condition on the new field theory of M. Born$^{9}$with the main idea of to solve the lack of uniqueness of the function action. They have already seen that the condition of regularity of the field gives the restriction in the spherically symmetric electrostatic case $E_{r}=0$ for $r=0$.
In the general theory they applyed the regularity condition not only to the $%
F_{\mu \nu }$ field but also to the $g_{\mu \nu }$ field. The regularity condition for the general theory was that:
*Only those solutions of the fields equations may have physical meaning for which space-time is everywhere regular and for which the* $%
F_{\mu \nu }$* and the* $g_{\mu \nu }$* fields and those of their derivatives which enter in the field equations and the conservation laws exist everywhere.*
In the general theory of the relativity the spherically symmetric solution of the purely gravitational field equations is given by the Schwarzschild line element $$ds^{2}=-Adt^{2}+A^{-1}dr^{2}+r^{2}\left( d\theta ^{2}+\sin ^{2}\theta
\,d\varphi ^{2}\right)$$ $$A\equiv 1-\frac{2M}{r} \ ,$$ where $(-2M)$ is a constant of integration $\ M$ have having the significance of the gravitational mass of the body source of the field (we take the gravitational constant $G=1$). This line element has an essential singularity at $r=0$ and does not satisfy the regularity condition.
In the general relativity form of the original new field theory the requeriment that there be no infinities in the $g_{\mu \nu }$ forces the identification of gravitational with electromagnetic mass. In$^{3}$ B. Hoffmann and L. Infeld have used for such identification the line element of the well known monopole solution studied by B. Hoffmann$^{2}$ in 1935 $$A\equiv 1-\frac{8\pi }{r}\int_{0}^{r}\left[ (r^{4}+1)^{1/2}-r^{2}\right] dr \ ,$$ that is originated by an Einstein-Born-Infeld action as in equation (1). This line element approximates the Schwarzschild form for $r$ greater than the electronic radius but avoid the infinities of that line element for $r=0$ However is still a singularity of conical type at the pole. When $%
r\rightarrow 0$ the above expression for $A$, gives $$A\rightarrow (1-8\pi )\equiv \beta$$ so $ds^{2}$ becomes $$ds^{2}=-\beta dt^{2}+\beta ^{-1}dr^{2}+r^{2}\left( d\theta ^{2}+\sin
^{2}\theta \,d\varphi ^{2}\right) \ .$$ Thus the radio of the circumference to the radius of a small circle having its centre at the pole is, in the limit, $2\pi \beta $ and not $2\pi $. Therefore the origin (it is, at $r=0$) is a conical point and not regular. Note that, because the conical point, no coordinate can be introduced which will be non singular at $r=0$ and derivatives are actually undefined at this point.
This problem with the conical singularities at $r=0$, that destroy the regularity condition, makes that in the reference$^{3}$ B. Hoffmann and L. Infeld change the action of the Born-Infeld form as in equation (1) for other non-linear Lagrangian of logarithmic type. The new logarithmic action does not presented such difficulties at $r=0$, and makes that time ago many people changes the very nice form of the Einstein-Born-Infeld action (1) for others non-linear Lagrangians that solved the problem of the self-energy of the electron and the regularity condition given above.
In this work we presented a new *exact* spherically symmetric solution of the Einstein-Born-Infeld equations. The metric, when the intrinsic mass of the system is zero, is *regular* everywhere in the sense that was given by B. Hoffmann and L. Infeld$^{3}$ in 1937, and the EBI theory leads to identification of the gravitational with the electromagnetic mass. In this manner we also show that more strong conditions are needed for to solve the problem of the lack of uniqueness of the function action.
Statement of the problem:
=========================
We propose the following line element for the static Born-Infeld monopole
$$ds^{2}=-e^{2\Lambda }dt^{2}+e^{2\Phi }dr^{2}+e^{2F\left( r\right) }d\theta
^{2}+e^{2G\left( r\right) }\sin ^{2}\theta \,d\varphi ^{2}\ ,$$
where the components of the metric tensor are $$\begin{array}{cccc}
g_{tt}=-e^{2\Lambda } & & & g^{tt}=-e^{-2\Lambda } \\
g_{rr}=e^{2\Phi } & & & g^{rr}=e^{-2\Phi } \\
g_{\theta \theta }=e^{2F} & & & g^{\theta \theta }=e^{-2F} \\
g_{\varphi \varphi }=\sin ^{2}\theta \,e^{2G} & & & g^{\varphi \varphi }=%
\frac{e^{-2G}}{\sin ^{2}\theta }\ .
\end{array}$$ For the obtention of the Einstein-Born-Infeld equations system we use the Cartan’s structure equations method$^{13}$, that is most powerful and direct where we work with differential forms and in a orthonormal frame (tetrad). The line element (7) in the 1-forms basis takes the following form $$ds^{2}=-\left( \omega ^{0}\right) ^{2}+\left( \omega ^{1}\right) ^{2}+\left(
\omega ^{2}\right) ^{2}+\left( \omega ^{3}\right) ^{2}\ ,$$ were the forms are $$\begin{array}{cccc}
\omega ^{0}=e^{\Lambda }dt & & \Rightarrow & dt=e^{-\Lambda }\omega ^{0} \\
\omega ^{1}=e^{\Phi }dr & & \Rightarrow & dr=e^{-\Phi }\omega ^{1} \\
\omega ^{2}=e^{F\left( r\right) }d\theta & & \Rightarrow & d\theta
=e^{-F\left( r\right) }\omega ^{2} \\
\omega ^{3}=e^{G\left( r\right) }\sin \theta \,d\varphi & & \Rightarrow &
d\varphi =e^{-G\left( r\right) }\left( \sin \theta \right) ^{-1}\omega ^{3}\
.
\end{array}$$ Now, following the standard procedure of the structure equations (Appendix) for to obtain easily the components of the Riemann tensor, we can construct the Einstein equations $$G^{1}\,_{2}=-e^{-\left( F+G\right) }\frac{\cos \theta }{\sin \theta }%
\partial _{r}\left( G-F\right)$$
$$G^{0}\,_{0}=e^{-2\Phi }\Psi -e^{-2F}$$
$$\Psi \equiv \left[ \partial _{r}\partial _{r}\left( F+G\right) -\partial
_{r}\Phi \,\partial _{r}\left( F+G\right) +\left( \partial _{r}F\right)
^{2}+\left( \partial _{r}G\right) ^{2}+\partial _{r}F\,\partial _{r}G\right]$$
$$G^{1}\,_{1}=e^{-2\Phi }\left[ \partial _{r}\Lambda \,\partial _{r}\left(
F+G\right) +\partial _{r}F\,\partial _{r}G\right] -e^{-2F}$$
$$G^{2}\,_{2}=e^{-2\Phi }\left[ \partial _{r}\partial _{r}\left( \Lambda
+G\right) -\partial _{r}\Phi \,\partial _{r}\left( \Lambda +G\right) +\left(
\partial _{r}\Lambda \right) ^{2}+\left( \partial _{r}G\right) ^{2}+\partial
_{r}\Lambda \,\partial _{r}G\right]$$
$$G^{3}\,_{3}=e^{-2\Phi }\left[ \partial _{r}\partial _{r}\left( F+\Lambda
\right) -\partial _{r}\Phi \,\partial _{r}\left( F+\Lambda \right) +\left(
\partial _{r}\Lambda \right) ^{2}+\left( \partial _{r}F\right) ^{2}+\partial
_{r}F\,\partial _{r}\Lambda \right]$$
$$G^{1}\,_{3}=G^{2}\,_{3}=G^{0}\,_{3}=G^{0}\,_{2}=G^{0}\,_{1}=0\ .$$
In the tetrad defined by (10), the energy-momentum tensor of Born-Infeld takes a diagonal form, being its components the following $$-T_{00}=T_{11}=\frac{b^{2}}{4\pi }\left( \frac{\Bbb{R}-1}{\Bbb{R}}\right)$$ $$T_{22}=T_{33}=\frac{b^{2}}{4\pi }\left( 1-\Bbb{R}\right)\ ,$$ where $$\Bbb{R}\equiv \sqrt{1-\left( \frac{F_{01}}{b}\right) ^{2}}$$ of this manner, one can see from the Einstein equation (18) the characteristic property of the spherically symmetric space-times$^{14}$$$G^{1}\,_{2}=-e^{-\left( F+G\right) }\frac{\cos \theta }{\sin \theta }%
\partial _{r}\left( G-F\right) =0\,\ \ \ \ \Rightarrow \,\ \ \ G=F \ .$$ Notice for that the interval be a spherically symmetric one, the functions $%
F\left( r\right) $ and $G\left( r\right) $ must be equal. As we saw in the precedent paragraph the components of the energy-momentum tensor of BI assures this condition in a natural form. Also it is interesting to see from eqs. (17) and (18) that the energy-momentum tensor of Born-Infeld has the same form as the energy-momentum tensor of an anisotropic fluid.
Equations for the electromagnetic fields of Born-Infeld in the tetrad
=====================================================================
The equations that describe the dynamic of the electromagnetic fields of Born-Infeld in a curved spacetime are $$\nabla _{a}\Bbb{F}^{ab}=\nabla _{a}\left[ \frac{F^{ab}}{\Bbb{R}}+\frac{P}{%
b^{2}\Bbb{R}}\widetilde{F}^{ab}\right] =0\,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \left( field\,equations\,\right)\ ,$$ $$\nabla _{a}\,\,\widetilde{F}^{ab}=0\,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ (\ Bianchi^{\prime }s\,\ identity)\ , \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$$ where $$P\equiv -\frac{1}{4}F_{\alpha \beta }\widetilde{F}^{\alpha \beta }$$ $$S\equiv -\frac{1}{4}F_{\alpha \beta }F^{\alpha \beta }$$ $$\Bbb{R}\equiv \sqrt{1-\frac{2S}{b^{2}}-\left( \frac{P}{b^{2}}\right) ^{2}}\
.$$ The above equations can be solved explicitly giving the follow result $$F_{01}=A\left( r\right)$$
$$\Bbb{F}_{01}=f\ e^{-2G} \ ,$$
where $f$ is a constant. We can see from equation (19) and (21) that $$\Bbb{F}_{01}=\frac{F_{01}}{\sqrt{1-\left( \overline{F}_{01}\right) ^{2}}} \ ,$$ where we obtain the following form for the electric field of the self-gravitating B-I monopole $$F_{01}=\frac{b}{\sqrt{\left( \frac{b}{f}e^{2G}\right) ^{2}+1}}$$ we can to associate$^{1}$$$f=br_{0}^{2}\equiv Q\,\ \ \ \ \Rightarrow \,\ \ \ F_{01}=\frac{b}{\sqrt{%
\left( \frac{e^{G}}{r_{0}}\right) ^{4}+1}} \ .$$ Where $r_{0}$ is a constant with units of longitude that in reference$^{1}$ was associated to the radius of the electron. Finally the components of the energy-momentum tensor of BI takes its explicit form reemplacing the $F_{01}$ that we was found in equation (29) in expressions (17) and (18) $$-T_{00}=T_{11}=\frac{b^{2}}{4\pi }\left( 1-\sqrt{\left( \frac{r_{0}}{e^{G}}%
\right) ^{4}+1}\right)$$ $$T_{22}=T_{33}=\frac{b^{2}}{4\pi }\left( 1-\frac{1}{\sqrt{\left( \frac{r_{0}}{%
e^{G}}\right) ^{4}+1}}\right)\ .$$
Expressions (11)–(16) together with (30)–(31) and (20) are the full set of Einstein equations in explicit form.
Reduction and solutions of the system of Einstein-Born-Infeld equations
=======================================================================
Of the above expressions, we can see that $G^{0}\,_{0}=G^{1}\,_{1},$then $$\partial _{r}\partial _{r}G+\left( \partial _{r}G\right) ^{2}-\partial
_{r}G\partial _{r}\left( \Phi +\Lambda \right) =0 \ .$$ In order to reduce the eq.(32) we will proceed as follow. First we make $$\partial _{r}G\equiv \xi$$ with this change of variables, in the equation (32) we have first derivatives only $$\partial _{r}\xi +\xi ^{2}-\xi \partial _{r}\left( \Phi +\Lambda \right) =0$$ dividing the above expression (34) by $\xi $ and making the substitution $$\chi \equiv \ln \xi$$ we have been obtained the following inhomogeneous equation $$\partial _{r}\chi +e^{\chi }=\partial _{r}\left( \Phi +\Lambda \right)$$ the homogeneus part of the last equation is easy to integrate $$\chi _{h}=-\ln r \ .$$ Now, of as usual, we make in eq. (36) the following substitution $$\chi =\chi _{h}+\chi _{p}=-\ln r+\ln \mu =-\ln r+\ln \left( 1+\eta \right)\
,$$ then $$\begin{aligned}
\partial _{r}\ln \left( 1+\eta \right) +\frac{\eta }{r} &=&\partial
_{r}\left( \Phi +\Lambda \right) \Rightarrow \\
\partial _{r}\left[ \ln \left( 1+\eta \right) +\mathcal{F}\left( r\right)
-\left( \Phi +\Lambda \right) \right] &=&0 \nonumber \\
\ln \left( 1+\eta \right) +\mathcal{F}\left( r\right) -\left( \Phi +\Lambda
\right) &=&cte=0 \ , \nonumber\end{aligned}$$ where$\ \ \frac{d\mathcal{F}\left( r\right) }{dr}\equiv \frac{\eta (r)}{r}$. The constant must be put equal to zero for to obtain the correct limit. Finally the form of the exponent $G$ is $$G=\ln r+\mathcal{F}\left( r\right)\ .$$ The next step is to put $\Phi $ in function of $\Lambda $ and $G$ in the expression (13). After of tedious but straighforward computations and integrations, we obtain $$e^{2\Lambda }=1+a_{0}\,e^{-G}+e^{2G}\frac{2b^{2}}{3}-2b^{2}e^{-G}\int^{Y%
\left( r\right) }\sqrt{Y^{4}+\left( r_{0}\right) ^{4}}dY \ .$$ Where, we defined $$Y(r)=e^{G}$$ and $a(0)$ is an integration constant.
Hitherto, we know that $\mathcal{F}$ is an arbitrary function of the radial coordinate $r$ , but for to be sure of it, we must to introduce the fuction $\Lambda $ given for above equation, in the Einstein equations (14-15) and to verify that $G_{22}=G_{33}$. Successfully, this equality is verified and the functions $\Lambda ,\Phi $ and $G$ remains matematically determinate. In this manner the line element of our problem (7) takes the following form $$ds^{2}=-e^{2\Lambda }dt^{2}+e^{2\mathcal{F}\left( r\right) }\left[
e^{-2\Lambda }\left( 1+r\,\ \partial _{r}\mathcal{F}\left( r\right)
\,\right) ^{2}dr^{2}+r^{2}\left( d\theta ^{2}+\sin ^{2}\theta \,d\varphi
^{2}\right) \right]\ .$$
Analysis of the function $\mathcal{F}\left( r\right) $ from the physical point of view
--------------------------------------------------------------------------------------
The function $\mathcal{F}\left( r\right) $ must to have the behaviour in the form that the electric field of the configuration obey the following requirements for gives a regular solution in the sense that was given by B. Hoffmann and L. Infeld$^{3}$ $$\left. F_{01}\right| _{r=r_{o}}<b$$ $$\left. F_{01}\right| _{r=0}=0$$ $$\,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left. F_{01}\right|
_{r\rightarrow \infty }=0\,\ \ \ \ \ \ \ \ \ \ \ \ \text{assymptotically \
Coulomb}$$ the simplest function $\mathcal{F}\left( r\right) $ that obey the above conditions, is of the type $$e^{2\mathcal{F}\left( r\right) }=\left[ 1-\left( \frac{r_{0}}{a\left|
r\right| }\right) ^{n}\right] ^{2m} \ ,$$ where $a$ is an arbitrary constant, and the exponents $n$ and $m$ will obey the following relation $$mn>1\,\ \ \ \ \left( m,n\in \Bbb{N}\right)$$ with $$0<a<1\ \text{or}\ -1<a<0\ \ \$$ depending on $m\left( n\right) $ is even or odd and $$a\neq 0 \ ,$$ that put in sure a consistent regularization condition not only for the electric (magnetic) field but for the energy-momentum tensor (30) and (31) and the line element (42).
The analysis of the Riemann tensor indicate us that it is regular everywhere and its components goes faster than $\frac{1}{r^{3}}$ when $r\rightarrow
\infty $. With all this considerations, the metric solution to the problem is $$ds^{2}=-e^{2\Lambda }dt^{2}+\left[ 1-\left( \frac{r_{0}}{a\left| r\right| }%
\right) ^{n}\right] ^{2m}\left\{ e^{-2\Lambda }dr^{2}\left[ \frac{1-\left(
\frac{r_{0}}{a\mid r\mid }\right) ^{n}\left( mn-1\right) }{\left[ 1-\left(
\frac{r_{0}}{a\mid r\mid }\right) ^{n}\right] }\right] ^{2}+\right.$$ $$\left. +r^{2}\left( d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right)
\right\}$$ and the electric field takes the form $$F_{01}=\frac{b}{\sqrt{1+\left[ 1-\left( \frac{r_{0}}{a\left| r\right| }%
\right) ^{n}\right] ^{4m}\left( \frac{r}{r_{0}}\right) ^{4}}} \ .$$ It is interesting to note that if we violating the condition (43) taken $a=1$ and $\left. F_{01}\right| _{r=r_{o}}=b$ (limit value for the electric field in BI theory) the energy momentum diverges automatically at $r=r_{0}$. Strictely, the regularity conditions for the energy-momentum tensor (without divergences and discontinuities in the neighborhood of $r_{0},$ physical radius of the spherical source of the non-linear electromagnetic field) are $$\left. T_{ab}\right| _{r=r_{o}}=finite\ \ \ \ \ \ \Rightarrow \ \ \ \ \ \
-1<a<0\ \ or\ \ 0<a<1\ \ \ \$$ depending on parity of $m,$ $n$; and $$\left. T_{ab}\right| _{r=0}\rightarrow 0\ \ \ \ \ \ \ \Rightarrow \ \ \ \ \
\ \ \Bbb{R}\rightarrow 1\ . \ \ \ \ \$$ For the magnetic monopole case the line element is as expression (48) with the following obvious definition for the magnetic charge $$br_{0}^{2}\equiv Q_{m} \ .$$ The magnetic field takes the following form $$\begin{aligned}
F_{23} &=&\frac{b}{\left[ 1-\left( \frac{r_{0}}{a\left| r\right| }\right)
^{n}\right] ^{2m}\left( \frac{r}{r_{0}}\right) ^{2}} \\
&=&\frac{Q_{m}}{\left[ 1-\left( \frac{r_{0}}{a\left| r\right| }\right)
^{n}\right] ^{2m}r^{2}}\end{aligned}$$ and the considerations about the regularity conditions on the energy momentum tensor is as the electric monopole case.
Interesting cases for particular values of $n$ and $m$
------------------------------------------------------
Because $$\exp 2\mathcal{F}\left( r\right) =\left[ 1-\left( \frac{r_{0}}{a\left|
r\right| }\right) ^{n}\right] ^{2m}$$ is easy to see that for $m=0$$$e^{G}=r$$ and we obtains the spherically symmetric line element of Hoffmann$^{2}$ and the electric field $F_{01}$and the energy-momentum tensor $\ T_{ab\text{ }}$take the form of the well know EBI solution for the electromagnetic geon of Demiánski$^{4}$ .
By other hand, in the *limit* when: $a\rightarrow 1$, $n\rightarrow 4$ and $m\rightarrow \frac{1}{4}$ we have $$F_{01}\rightarrow \frac{b}{\sqrt{1+\left[ 1-\left( \frac{r_{0}}{\left|
r\right| }\right) ^{4}\right] \left( \frac{r}{r_{0}}\right) ^{4}}}=\frac{Q}{%
r^{2}} \ ,$$ where (as is usually taked) $br_{0}^{2}\equiv Q$ . How we see, we obtain as solution in the *limit* the Maxwellian linear field. Note that the values of $a$ and the exponents $m$ and $n$ are restricted by conditions (47).
Analysis of the metric
======================
We have the metric (42) $$ds^{2}=-e^{2\Lambda }dt^{2}+e^{2\mathcal{F}\left( r\right) }\left[
e^{-2\Lambda }\left( 1+r\,\ \partial _{r}\mathcal{F}\left( r\right)
\,\right) ^{2}dr^{2}+r^{2}\left( d\theta ^{2}+\sin ^{2}\theta \,d\varphi
^{2}\right) \right] \ ,$$ if we make the substitution $$Y\equiv r\,e^{\mathcal{F}\left( r\right) }$$ and differentiating it $$dY\equiv \,e^{\mathcal{F}\left( r\right) }\left( 1+r\,\ \partial _{r}%
\mathcal{F}\left( r\right) \,\right) dr$$ the interval (7) takes the form $$ds^{2}=-e^{2\Lambda }dt^{2}+e^{-2\Lambda }dY^{2}+Y^{2}\left( d\theta
^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right) \ ,$$ we can see that the metric (in particular the $g_{tt}$ coefficient), in the new coordinate $Y(r)$, takes the similar form like a Demianski solution for the Born-Infeld monopole spacetime : $$e^{2\Lambda }=1-\frac{2M}{Y}-\frac{2b^{2}r_{o}^{4}}{3\left( \sqrt{%
Y^{4}+r_{o}^{4}}+Y^{2}\right) }-\frac{4}{3}b^{2}r_{o}^{2}\,_{2}F_{1}\left[
1/4,1/2,5/4;-\left( \frac{Y}{r_{0}}\right) ^{4}\right] \ ,$$ here $M$ is an integration constant, which can be interpreted as an intrinsic mass, and $_{2}F_{1}$ is the Gauss hypergeometric function$^{14}$. We have pass $$g_{rr}\rightarrow g_{YY},\,\ \ \ \ \ \ \ \ \ \ \ \ \ \ g_{tt}\left( r\right)
\rightarrow g_{tt}\left( Y\right) \ .$$ Specifically, for the form of the $\mathcal{F}\left( r\right) $ given by (46), $Y$ is $$Y^{2}\equiv \left[ 1-\left( \frac{r_{o}}{a\left| r\right| }\right)
^{n}\right] ^{2m}r^{2} \ .$$ Now, with the metric coefficients fixed to a asymptotically Minkowskian form, one can study the asymptotic behaviour of our solution. A regular, asymptotically flat solution with the electric field and energy-momentum tensor both regular, in the sense of B. Hoffmann and L. Infeld is when the exponent numbers of $Y(r)$ take the following particular values: $$n=3\ \ \ and\ \ \ \ \ m=1 \ .$$ In this case, and for $r>>\frac{r_{0}}{a}$ $,$ we have the following asymptotic behaviour for $Y\left( r\right) $ and $-g_{tt}\,$, that does not depend on the $a$ parameter$^{{}}$$$Y\left( r\right) \rightarrow r\ \text{ \ \ \ \ \ }\left( r>>\frac{r_{0}}{a}%
\right)$$ $$e^{2\Lambda }\simeq 1-\frac{2M}{r}-\frac{8b^{2}r_{o}^{4}K\left( 1/2\right) }{%
3r_{o}r}+2\frac{b^{2}r_{o}^{4}}{r^{2}}+... \ .$$ A distant observer will associate with this solution a total mass $$M_{eff}=M+\frac{4b^{2}r_{o}^{4}K\left( 1/2\right) }{3r_{o}}$$ and total charge $$Q^{2}=2b^{2}r_{o}^{2} \ .$$ Notice that when the intrinsec mass $M$ is zero the line element is regular everywhere, the Riemann tensor is also regular everywhere and hence the space-time is singularity free. The electromagnetic mass $$M_{el}=\frac{4b^{2}r_{o}^{4}K\left( 1/2\right) }{3r_{o}}$$ and the charge $Q$ are the *twice* that the electromagnetic charge and mass of the Demianski solution$^{4}$ for the static electromagnetic geon. Notice that the $M_{el}$ is necessarily positive, which was not the case in the Schwarzschild line element. The other important reason for to take the constant $M=0$ is that we must regard the quantity (let us to restore by one moment the gravitational constant $G$) $$4\pi G\int_{Y(r=0)}^{Y(r)}T_{0}^{0}\left( Y\right) Y^{2}dY$$ as the *gravitational mass* causing the field at coordinate distance r from the pole. In our case $T_{0}^{0}$ is given by expression (30). This quantity is precisely (in gravitational units) $M_{el}$ given by (50), the *total electromagnetic mass* within the sphere having its center at $%
r=0$ and coordinate $r.$ We will take $M=0$ in the rest of the analysis.
On the other hand, the function $Y\left( r\right) $ for the values of the $m$ and $n$ parameters given above has the following behaviour near of the origin
$$\text{for\ }a<0\text{ \ \ \ \ \ \ \ \ when\ }r\rightarrow 0,\ Y\left(
r\right) \rightarrow \infty \ ,$$ $$\text{for\ }a>0\text{ \ \ \ \ \ \ \ \ \ when \ }r\rightarrow 0,\ Y\left(
r\right) \rightarrow -\infty \ .$$ Notice that the case $a>0$ will be excluded because in any value $%
r_{0}\rightarrow $ $Y\left( r_{0}\right) =0$ , the electric field takes the limit value $b$ and the condition (43) is violated. For $M=0$ and $a<0,$expanding the hypergeometric function, we can see that the $-g_{tt}$ coefficient has the following behaviour near the origin $$e^{2\Lambda }\simeq 1-\frac{8b^{2}r_{o}^{4}K\left( 1/2\right) }{3r_{o}}%
r^{2}\left( \frac{\left| a\right| }{r_{0}}\right) ^{3}+2b^{2}r_{0}^{4}\
r^{4}\left( \frac{\left| a\right| }{r_{0}}\right) ^{6}+...$$ The metric (see figures) and the energy-momentum tensor remains *both* regulars at the origin (it is: $g_{tt}\rightarrow -1,T_{\mu \nu
}\rightarrow 0$ $\ $ for $r\rightarrow 0$). It is not very difficult to check that (for $m=1$ and $n=3$) the maximum of the electric field (see figures) is not in $r=0$ , but in the *physical* *border* of the spherical configuration source of the electromagnetic fields (this point is located around $r_{B}=2^{1/3}\frac{r_{0}}{\left| a\right| }$). It means that $Y\left( r\right) $ maps correctely the internal structure of the source in the same form that the quasiglobal coordinate of the reference$%
^{16}$ for the global monopole in general relativity. The lack of the conical singularities at the origin is because the very well description of the manifold in the neighbourhood of $r=0$ given by $Y\left( r\right) $
Because the metric is regular ($g_{tt}=-1,$ at $r=0$ and at $r=\infty $), its derivative must change sign. In the usual gravitational theory of general relativity the derivative of $g_{tt}$ is proportional to the gravitational force which would act on a test particle in the Newtonian approximation. In Einstein-Born-Infeld theory with this new static solution, it is interesting to note that although this force is attractive for distances of the order $r_{0}<<r$ , it is actually a repulsion for very small $r$. For $r$ greater than $r_{0},$ the line element closely approximates to the Schwarzschild form. Thus the regularity condition shows that the electromagnetic and gravitational mass are the same and, as in the Newtonian theory, we now have the result that the attraction is zero in the center of the spherical configuration source of the electromagnetic field.
Conclusions
===========
In this report a *new* exact solution of the Einstein-Born-Infeld equations for a static spherically symmetric monopole is presented. The general behaviour of the geometry, is strongly modified according to the value that takes $r_{0}\ $(Born-Infeld radius$^{1, \ 9}$) and three new parameters: $a$, $m$ and $n$.
The fundamental feature of this solution is the lack of conical singularities at the origin when: $-1<a<0$ or $0<a<1$ (depends on parity of $m$ and $n$) and $mn>1$. In particular, for $m=1$ and $n=3$, with the parameter $a$ in the range given above and the intrinsic mass of the system $%
M$ is zero, the strong regularity conditions given by B. Hoffmann and L. Infeld in reference$^{3}$, holds in all the spacetime. For the set of values for the parameters given above, the solution is asymptotically flat, free of singularities in the electric field, metric, energy-momentum tensor and their derivatives (with derivative values zero for $r\rightarrow 0$); and the electromagnetic mass (ADM) of the system is a twice that the electromagnetic mass of other well known$^{2, \ 4}$ solutions for the Einstein-Born-Infeld monopole. The electromagnetic mass $M_{el}$ asymptotically is necessarily positive, which was not the case in the Schwarzschild line element.
This solution have a surprising similitude with the metric for the global monopole in general relativity given in reference$^{16-17}$ in the sense that the physic of the problem have a correct description only by means of a new radial function $Y\left( r\right) $.
Because the metric is regular ($g_{tt}=-1,$ at $r=0$ and at $r=\infty $), its derivative (that is proportional to the the force in Newtonian approximation) must change sign. In Einstein-Born-Infeld theory with this new static solution, it is interesting to note that although this force is attractive for distances of the order $r_{0}<<r$ , it is actually a repulsive for very small $r$.
With this new regular solution, we also show that more strong conditions are needed for to solve the problem of the lack of uniqueness of the function action in non-linear electrodynamics.
Acknowledgements: {#acknowledgements .unnumbered}
-----------------
I am very grateful to organizers of this Conference for their kind hospitality.
Appendix: {#appendix .unnumbered}
=========
Connections and curvature forms from the geometrical Cartan’s formulation {#connections-and-curvature-forms-from-the-geometrical-cartans-formulation .unnumbered}
-------------------------------------------------------------------------
The standard procedure of E. Cartan has its startpoint in the following equations $$d\omega ^{\alpha }=-\omega ^{\alpha }\,_{\beta }\wedge \omega ^{\beta }$$ $$\mathcal{R}^{\alpha }\,_{\beta }=d\,\omega ^{\alpha }\,_{\beta }+\omega
^{\alpha }\,_{\lambda }\wedge \omega ^{\lambda }\,_{\beta } \ ,$$ these are denominated *the structure equations*. The procedure for to obtan the Einstein equations is by mean the following steps:
$i.$ Making the exterior derivatives of $\omega ^{\alpha }$ we computing the connection 1-forms $\omega ^{\alpha }\,_{\beta }$: $$\omega ^{0}\,_{1}=\omega ^{1}\,_{0}=e^{-\Phi }\,\ \partial _{r}\Lambda \,\
\omega ^{0}\,$$ $$\omega ^{2}\,_{1}=-\omega ^{1}\,_{2}=e^{-\Phi }\,\ \partial _{r}F\left(
r\right) \,\ \omega ^{2}\,$$ $$\omega ^{3}\,_{1}=-\omega ^{1}\,_{3}=e^{-\Phi }\,\ \partial _{r}G\left(
r\right) \,\ \omega ^{3}\,$$ $$\omega ^{3}\,_{2}=-\omega ^{2}\,_{3}=\frac{\cos \theta }{sen\theta }%
e^{-F\left( r\right) }\omega ^{3}\ .$$
$ii.$ Making the exterior derivatives of $\omega ^{\alpha }\,_{\beta }$ we computing the curvature 2-forms $\mathcal{R}^{\alpha }\,_{\beta }$: $$\mathcal{R}^{0}\,_{1}=e^{-2\Phi }\left( \partial _{r}\partial _{r}\Lambda
-\partial _{r}\Phi \,\partial _{r}\Lambda +\left( \partial _{r}\Lambda
\right) ^{2}\right) \,\omega ^{1}\wedge \omega ^{0}$$ $$\mathcal{R}^{2}\,_{1}=e^{-2\Phi }\left( \partial _{r}\partial _{r}F-\partial
_{r}\Phi \,\partial _{r}F+\left( \partial _{r}F\right) ^{2}\right) \,\omega
^{1}\wedge \omega ^{2}$$ $$\mathcal{R}^{3}\,_{2}=e^{-\left( F+\Phi \right) }\,\partial _{r}\left(
G-F\right) \,\frac{\cos \theta }{sen\theta }\,\omega ^{1}\wedge \omega
^{3}+\left( e^{-2\Phi }\partial _{r}G\,\,\partial _{r}F-e^{-2F}\right)
\,\omega ^{2}\wedge \omega ^{3}$$ $$\begin{aligned}
\mathcal{R}^{3}\,_{1} &=&e^{-2\Phi }\left( \partial _{r}\partial
_{r}G-\partial _{r}\Phi \,\partial _{r}G+\left( \partial _{r}G\right)
^{2}\right) \,\omega ^{1}\wedge \omega ^{3}+ \\
&&+e^{-\left( F+\Phi \right) }\,\partial _{r}\left( G-F\right) \,\frac{\cos
\theta }{sen\theta }\,\omega ^{2}\wedge \omega ^{3}\end{aligned}$$ $$\mathcal{R}^{0}\,_{2}=-e^{-2\Phi }\,\partial _{r}\Lambda \,\,\partial
_{r}F\,\,\omega ^{0}\wedge \omega ^{2}$$ $$\mathcal{R}^{0}\,_{3}=-e^{-2\Phi }\,\partial _{r}\Lambda \,\,\partial
_{r}G\,\,\omega ^{0}\wedge \omega ^{3} \ .$$
$iii.$ The components of the Riemann tensor are easily obtained from the well know geometrical relation of Cartan: $$\mathcal{R}^{\alpha }\,_{\beta }=R^{\alpha }\,_{\beta \rho \sigma
}\,\,\omega ^{\rho }\wedge \omega ^{\sigma } \ ,$$ where we obtain explicitly
$$R^{0}\,_{110}=e^{-2\Phi }\left( \partial _{r}\partial _{r}\Lambda -\partial
_{r}\Phi \,\partial _{r}\Lambda +\left( \partial _{r}\Lambda \right)
^{2}\right)$$
$$R^{2}\,_{112}=e^{-2\Phi }\left( \partial _{r}\partial _{r}F-\partial
_{r}\Phi \,\partial _{r}F+\left( \partial _{r}F\right) ^{2 \,}\right)$$
$$R^{3}\,_{113}=e^{-2\Phi }\left( \partial _{r}\partial _{r}G-\partial
_{r}\Phi \,\partial _{r}G+\left( \partial _{r}G\right) ^{2}\right)$$
$$R^{3}\,_{213}\,\,=e^{-\left( F+\Phi \right) }\,\partial _{r}\left(
G-F\right) \,\frac{\cos \theta }{\sin \theta }$$
$$R^{3}\,_{123}\,\,=e^{-\left( F+\Phi \right) }\,\partial _{r}\left(
G-F\right) \,\frac{\cos \theta }{\sin \theta }$$ $$R^{3}\,_{223}\,\,=e^{-2\Phi }\partial _{r}G\,\,\partial _{r}F-e^{-2F}$$ $$R^{0}\,_{330}=e^{-2\Phi }\,\partial _{r}\Lambda \,\,\partial _{r}G
,$$ $$R^{0}\,_{220}=e^{-2\Phi }\,\partial _{r}\Lambda \,\,\partial _{r}F\$$ from which we can construct the Einstein equations of the usual manner.
R e f e r e n c e s {#r-e-f-e-r-e-n-c-e-s .unnumbered}
===================
$^{1}$ M. Born and L. Infeld, Proc. Roy. Soc.(London) **144**, 425 (1934).
$^{2}$ B. Hoffmann, Phys. Rev. **47**, 887 (1935).
$^{3}$ B. Hoffmann and L. Infeld, Phys. Rev. **51**, 765 (1937).
$^{4}$ M. Demianski, Found. Phys. Vol. 16 , No. 2, 187 (1986).
$^{5}$ D. Harari and C. Lousto, Phys. Rev. D **42**, 2626 (1990).
$^{6}$ M. Barriola and A. Vilenkin, Phys. Rev. Lett. **63**, 341 (1989).
$^{7}$ L. D. Landau and E. M. Lifshitz, *Teoria Clasica de los Campos*, (Reverte, Buenos Aires, 1974), p. 574.
$^{8}$ C. Misner, K. Thorne and J. A. Wheeler, *Gravitation*, (Freeman, San Francisco, 1973), p. 474.
$^{9}$ M. Born , Proc. Roy. Soc. (London) **143**, 411 (1934).
$^{10}$ R. Metsaev and A. Tseytlin, Nucl. Phys.**B 293**, 385 (1987).
$^{11}$ Yu. Stepanovsky, Electro-Magnitie Iavlenia **3**, Tom 1, 427 (1988).
$^{12}$ D. J. Cirilo Lombardo, Master Thesis, Universidad de Buenos Aires, Argentina, 2001.
$^{13}$ S. Chandrasekhar, *The Mathematical Theory of Black Holes*, (Oxford University Press, New York, 1992).
$^{14}$ D. Kramer et al., *Exact Solutions of Einstein’s Field Equations,* (Cambridge University Press, Cambridge,1980).
$^{15}$ A. S. Prudnikov, Yu. Brychov and O. Marichev, *Integrals and Series* (Gordon and Breach, New York, 1986).
$^{16}$ K. A. Bronnikov, B. E. Meierovich and E. R. Podolyak, JETP **95**, 392 (2002).
$^{17}$ D. J. Cirilo Lombardo, Preprint JINR-E2-2003-221.
[^1]: e-mails:[email protected] [email protected].
|
---
abstract: 'Deep neural network based methods have achieved promising results for CT metal artifact reduction (MAR), most of which use many synthesized paired images for training. As synthesized metal artifacts in CT images may not accurately reflect the clinical counterparts, an artifact disentanglement network (ADN) was proposed with unpaired clinical images directly, producing promising results on clinical datasets. However, without sufficient supervision, it is difficult for ADN to recover structural details of artifact-affected CT images based on adversarial losses only. To overcome these problems, here we propose a low-dimensional manifold (LDM) constrained disentanglement network (DN), leveraging the image characteristics that the patch manifold is generally low-dimensional. Specifically, we design an LDM-DN learning algorithm to empower the disentanglement network through optimizing the synergistic network loss functions while constraining the recovered images to be on a low-dimensional patch manifold. Moreover, learning from both paired and unpaired data, an efficient hybrid optimization scheme is proposed to further improve the MAR performance on clinical datasets. Extensive experiments demonstrate that the proposed LDM-DN approach can consistently improve the MAR performance in paired and/or unpaired learning settings, outperforming competing methods on synthesized and clinical datasets.'
author:
- 'Chuang Niu, Wenxiang Cong, Fenglei Fan, Hongming Shan, Mengzhou Li, Jimin Liang, Ge Wang'
bibliography:
- 'references.bib'
title: 'Low-dimensional Manifold Constrained Disentanglement Network for Metal Artifact Reduction'
---
Introduction
============
Metal objects in a patient, such as dental fillings, artificial hips, spine implants, and surgical clips, will significantly degrade the quality of computed tomography (CT) images. The main reason for such metal artifacts is that the metal objects in the field of view strongly attenuate x-rays or even completely block them so that reconstructed images from the compromised/incomplete data are corrupted in various ways, which are usually observed as bright or dark streaks. As a result, the metal artifacts significantly affect medical image analysis and subsequent clinical treatment. Particularly, the metal artifacts degrade the counters of the tumor and organs at risk, raising great challenges in determining a radio-therapeutic plan [@Gian2017; @Maerz].
Over the past decades, extensive research efforts [@Gjesteby2016] have been devoted to CT metal artifact reduction (MAR), leading to various of MAR methods. Traditionally, the projection domain methods [@Kalender; @nmar] focus on projection data inside a metal trace, and replace them with estimated data. Then, the artifact-reduced image can be reconstructed from the refined projection data using a reconstruction algorithm, such as filtered backprojection (FBP). However, the projection domain methods tend to produce secondary artifacts as it is difficult for the estimated projection values to perfectly match the ground truth. In practice, the original projection data and the corresponding reconstruction algorithm are not publicly accessible. To apply the projection based methods in the absence of original sinogram data, researchers such as reported in [@Bal] proposed a post-processing scheme that generates the sinogram through forward projection of a CT image first and then applies the projection based method on the reprojected sinogram. However, the second-round projection and reconstruction may introduce extra errors.
To overcome the limitations of the projection domain methods, researchers worked extensively to reduce metal artifacts directly in the image domain [@Hamid; @karimi]. With deep learning techniques [@deepimaging], data-driven MAR methods were recently developed based on deep neural networks [@Huang2018MetalAR; @Wang2018; @Gjesteby_2019; @zhang2018; @dudonet; @adn], demonstrating superiority over the traditional algorithms for metal artifact reduction. However, most existing deep learning based methods are fully-supervised, requiring a large number of paired training images, i.e., the artifact-affected image and the co-registered artifact-free image. In clinical scenarios, it is infeasible to acquire a large number of such paired images. Therefore, the prerequisite of these method is to simulate artifact-affected images by inserting metal objects into artifact-free images to obtain paired data. However, simulated images cannot reflect all real conditions due to the complex physical mechanism of metal artifacts and many technical factors of the imaging system, degrading the performance of the fully-supervised models. To avoid synthesized data, the recently proposed ADN [@adn] only uses clinical unpaired metal-affected and metal-free CT images to train a disentanglement network on adversarial losses, giving promising results on clinical datasets, outperforming the fully-supervised methods trained on the synthesized data. However, without accurate supervision, the proposed ADN method is far from being perfect, and cannot preserve structural details in many challenging cases.
In this study, we improve the MAR performance on clinical datasets from two aspects. First, we formulate the MAR as the artifact disentanglement while at the same time leveraging the low-dimensional patch manifold of image patches to help recover structural details. Specifically, we train a disentanglement network with ADN losses and simultaneously constrain a reconstructed artifact-free image to have a low-dimensional patch manifold. The idea is inspired by the low-dimensional manifold model (LDMM) [@LDMM] for image processing and CT image reconstruction [@cong2019]. However, how to apply the iterative LDMM algorithm to train the disentanglement network is not trivial. To this end, we carefully design an LDM-DN algorithm for simultaneously optimizing objective functions of the disentanglement network and the LDM constraint. Second, we improve the MAR performance of the disentanglement network by integrating both unpaired and paired supervision. Specifically, the unpaired supervision is the same as that used in ADN, where unpaired images come from artifact-free and artifact-affected groups. The paired supervision relies on synthesized paired images to train the model in a pixel-to-pixel manner. Although the synthesized data cannot perfectly simulate the clinical scenarios, they still provide helpful information for recovering artifact-free images from artifact-affected ones. Finally, we design a hybrid training scheme to combine both the paired and paired supervision for further improving MAR performance on clinical datasets.
The rest of this paper is organized as follows. In the next section, we review the related work. In section \[sec\_method\], we describe the proposed method, including the problem formulation, the construction of a patch set, the dimension of a patch manifold, the corresponding optimization algorithm, and the hybrid training scheme. In section \[sec\_experiment\], we evaluate the proposed LDM-DN algorithm for MAR on synthesized and clinical datasets. Extensive experiments show that our proposed method consistently outperforms the state-of-the-art ADN model and other competing methods. Finally, we conclude the paper in section \[sec\_conclusion\].
Related work
============
Metal Artifact Reduction
------------------------
CT metal artifact reduction methods can be classified into three categories, including projection domain methods, image domain methods and dual domain methods.
Projection based methods aim to correct projections for MAR. Some methods of this type [@park2016; @jiang2000; @meyer2010] directly corrects corrupted data by modeling the underlying physical process, such as beam hardening and scattering. However, the results of these methods are not satisfactory when high-atom number metals are presented. Thus, a more sophisticated way is to treat metal-affected data within the metal traces as unreliable and replaced them with the surrogates estimated by reliable ones. Linear interpolation (LI) [@Kalender] is a basic and simple method for estimating metal corrupted data. The LI method is prone to generate new artifacts and distort structures due to mismatched values and geometry between the linearly interpolated data and the unaffected data. To address this problem, the prior information is employed to generate a more accurate surrogate in various ways [@Bal; @nmar; @wj2013]. Among these methods, the state-of-the-art normalized MAR (NMAR) method [@nmar] is widely used due to its simplicity and accuracy. NAMR introduces a prior image of tissue classification for normalizing the projection data before the LI is operated. With the normalized projection data, the data mismatch caused by LI can be effectively reduced for better results than that of the generic LI method. However, the performance of NMAR largely depends on the accurate tissue classification. In practice, the tissue classification is not always accurate so that NMAR also tends to produce secondary artifacts. Recently, deep neural networks [@Bernhard2017; @liao2019; @Ghani2020] were applied for projection correction and achieved promising results. However, such learning based methods require a large number of paired projection data.
The image domain based methods directly reduce metal artifacts based on CT image post-processing. Conventionally, some methods [@Hamid; @karimi] leverage image processing techniques to estimate and remove the streak artifacts from original artifact-affected images, but these hand-crafted methods have a limited performance. From a data-driven perspective, deep learning methods for MAR have superiority over traditional approaches. For example, RL-ARCNN [@Huang2018MetalAR] is a convolutional neural network with residual learning for MAR, achieving better results than the plain CNN [@vgg]. cGANMAR [@Wang2018] regards MAR as an image-to-image transformation, and adapts the Pix2pix [@Isola_2017_CVPR] model to improve the MAR performance.
To benefit from both projection and image domains, various dual domain based methods were also proposed. DestreakNet [@Gjesteby_2019] takes a corrected image by the state-of-the-art NMAR method and a detail mapping derived from the original image as the inputs to a dual-stream network, giving better results than what achievable in a single domain. In the CNN-MAR method [@zhang2018], the CNN first takes the original image and corrected images by BHC [@Verburg2012CTMA] and LI [@Kalender] as the inputs and then produces a CNN output image, which is used to generate a prior image. Then, the projection data of the prior image is used to correct the original projection data, and the final image is reconstructed with FBP. DuDoNet [@dudonet] introduces an end-to-end dual domain network to simultaneously correct sinogram data and CT images.
All above deep learning methods for MAR require a large number of synthesized paired project datasets and/or CT images for training. A recent study [@adn] has shown that the models [@zhang2018; @Wang2018] trained on the synthesized data cannot generalize the well on the clinical datasets. Then, ADN was designed and tested on clinical unpaired data, achieving promising results. However, without a strong supervision, ADN can hardly recover structural details in challenging cases.
In this study, we introduce a novel image prior, i.e., low-dimensional manifold, and different levels of supervision to train the disentangle network for improving the MAR performance on clinical datasets. Our proposed LDM prior guided disentanglement framework and synergistic supervision scheme have the potential to empower image domain and dual domain based methods as further detailed below.
Low-dimensional manifold
------------------------
The patch set of natural images has been proved coming from a low-dimensional manifold [@Lee2003; @carlsson2008; @peyre2008; @peyre2009]. Based on this low dimensionality of the patch manifold, LDMM first computes the dimension of the patch manifold based on the differential geometry and then uses the dimension to regularize an image recovery problem, including image impainting, super-resolution, and denoising. Based on LDMM, LDMNet [@ldmnet] proposes to regularize the combination of input data and output features within a low-dimensional manifold in the context of the classification task, showing a competitive performance over popular regularizers such as low-rank and DropOut. Recently, Cong et al [@cong2019] proposed to use LDMM in regularizing the CT image reconstruction, demonstrating that the LDMM has a strong ability to recover detailed structures in CT images. Inspired by the recent results with LDMM, here we propose an LDM constrained disentanglement network with both paired and unpaired supervision for improving the MAR performance on clinical datasets.
Method {#sec_method}
======
Problem formulation {#sec_problem}
-------------------
Before the problem formulation, let us introduce the general neural network based method in the image domain. In the supervised learning mode, we have the paired data $\{x_i, x_i^{gt}\}_{i=1}^N$ available, where each artifact-affected image $x_i$ , has a corresponding artifact-free image $x_i^{gt}$, and $N$ is the number of paired images. Then, the deep neural network based model can be trained on this dataset with the loss function: $$\label{eq_sup}
\mathcal{L}_{sup}(\theta) = \frac{1}{N} \sum_{i=1}^N \ell(f_{\theta}(x_i), x_i^{gt})$$ where $\ell$ is a loss function, such as the $L1$-distance function, and $f_{\theta}(x_i)$ represents the predicted image of $x_i$ by the neural network with a parameter vector $\theta$. In practice, a large number of paired data are synthesized for training the model, as the clinical datasets only contain unpaired images.
To improve the MAR performance on clinical datasets, ADN adapts the generative adversarial learning based disentanglement network for MAR, only requiring an unpaired dataset, $\{(x_i, y_j)\}, i=1, \cdots, N_1, j=1, \cdots, N_2$, where $y_i$ represents an artifact-free image. The ADN consists of several encoders and decoders, which are trained with several loss functions, including two adversarial losses, a reconstruction loss, a cycle-consistent loss and an artifact-consistent loss. For simplicity, we denote the ADN loss functions as: $$\label{eq_adn}
\mathcal{L}_{adn}(\theta) = \frac{1}{N_1 N_2} \sum_{i=1}^{N_1} \sum_{j=1}^{N_2} \ell_{adn}(f_{\theta}(x_i), y_j)$$ where $l_{adn}$ represents the combination of all loss functions of ADN.
In this work, we introduce a general image property of known as LDM to improve the MAR performance on clinical datasets. Specifically, we assume that a patch set of artifact-free images samples a low-dimensional manifold. Therefore, we formulate the MAR problem as follows: $$\label{eq_problem}
\min_{\theta, \mathcal{M}} \mathcal{L}(\theta) + dim(\mathcal{M(P_{\theta})})$$ where $P_{\theta}$ denotes the patch set of artifact-free or artifact-corrected images, $\mathcal{M}$ is a smooth manifold isometrically embedded in the patch space, and $\mathcal{L}(\theta)$ can be any network loss functions, such as $\mathcal{L}_{sup}$ for paired learning or $\mathcal{L}_{adn}$ for unpaired learning.
To solve the above optimization problem, we need to specify the reconstruction of a patch set, the computation of a patch manifold, and the learning algorithm for simultaneously optimizing the network functions and the dimensionality of a patch manifold. In the following sections, we will describe each of them.
![Diagram of patch set construction.[]{data-label="fig:patch"}](figures/patch.png){width="60.00000%"}
Construction of a patch set {#sec_patch}
---------------------------
In this work, we adapt the state-of-the-art disentangle network in our proposed LDMM-based optimization framework under different levels of supervision. For such a disentanglement network, we leverage its two branches to construct a patch set. As shown in Fig. \[fig:patch\], one is the artifact-corrected branch that maps artifact-affected images to artifact-corrected images, and the other is the artifact-free branch that maps artifact-free images to themselves. Considering the spatial correspondence between the input/output image and its convolutional feature maps, we concatenate each feature vector along the spatial axes and its corresponding image patch to represent a patch. For the artifact-corrected branch, we take patches from the artifact-corrected images, denoted as $\{P_i(\hat{x})\}$. For the artifact-free branch, we take patches from the original images, denoted as $\{P_j(y)\}$. As we assume that the patch set of the images without artifacts samples a low-dimensional manifold, the final patch set is the concatenation of these two patch sets, denoted by $P_{\theta} = P(\hat{x}, z_x^t) \cup P(y, z_y^t)$.
In our implementation, the input image size is $H\times W$ and the step size for down-sampling the encoder features is $s$, then the patch size is $s \times s$, the shape of $z_x^t$ or $z_y^t$ is $s^2 \times \frac{H}{s} \times \frac{W}{s}$, and each patch $P_{\theta}^i \in R^{2s^2}$.
Dimension of patch manifold {#sec_dimension}
---------------------------
In this work, we adopt the definition introduced in LDMM [@LDMM] for computing the dimension of a patch manifold. Specifically, we have the following theorem.
Let $M$ be a smooth submanifold isometrically embedded in $R^d$. For any patch $\bm{p}=(p_i)_{i=1}^d \in \mathcal{M}$, $$dim(\mathcal{M})=\sum_{j=1}^d||\nabla_M\alpha_j(\bm{p})||^2$$
where $\alpha_i(\bm{p})=p_i$ is the coordinate function, $\nabla_M\alpha_j(\bm{p})$ denotes the gradient of the function $\alpha_i$ on $\mathcal{M}$. More details on the definition of $\alpha_i$ on $\mathcal{M}$ can be found in [@LDMM]. In our implementation, $\bm{p} = P_{\theta}^i \in \bm{R}_d, d=2s^2$, where the patch is parameterized by the neural network parameter vector $\theta$, as introduced in Section \[sec\_patch\].
Optimization {#sec:opti}
------------
According to the the construction of a patch set and the definition of a patch manifold dimension, we can reformulate Eq. (\[eq\_problem\]) as:
$$\label{eq_opt}
\begin{split}
&\min_{\theta, \mathcal{M}} \mathcal{L}(\theta) + \sum_{i=1}^d ||\nabla_{\mathcal{M}} \alpha_i||_{L^2(\mathcal{M})}^2,\\
& s.t. \ P_{\theta} \subset \mathcal{M}.
\end{split}$$
where $$||\nabla_{\mathcal{M}}\alpha_i||_{L^2(\mathcal{M})} = \left( \int_M ||\nabla_{\mathcal{M}}\alpha_i(\bm{p})||^2d\bm{p} \right)^{1/2}$$ To solve this problem, we design an iterative algorithm, named LDM-DN, for optimizing the LDM constrained disentanglement network based on the algorithm for image processing introduced in [@LDMM]. Specifically, given $(\theta^k, M^k)$ at step $k$ satisfying $P_{\theta^k}^k \subset \mathcal{M}^k$, step $k+1$ consists of the following sub-steps:
- Update $\theta^{k+1}$ and the perturbed coordinate functions $\alpha^{k+1}=(\alpha^{k+1}_1, \cdots, \alpha^{k+1}_d)$ as the minimizers of (\[eq\_up1\]) with the fixed manifold $\mathcal{M}^{k}$: $$\label{eq_up1}
\begin{aligned}
&\min_{\theta, \alpha} \mathcal{L}(\theta) + \sum_{i=1}^d ||\nabla_M \alpha_i||_{L^2(M)}^2,\\
&s.t. \quad \alpha(P_{\theta^k}) = P_{\theta}
\end{aligned}$$
- Update $\mathcal{M}^{k+1}$: $$\label{eq:ignore}
\mathcal{M}^{k+1} = \{(\alpha_1^{k+1}(\bm{p}), \cdots, \alpha_d^{k+1}(\bm{p})): \bm{p}\in \mathcal{M}^k\}$$
- Repeat above two sub-steps until convergence.
It is noted that if the iteration converges to a fixed point, $\alpha^{k+1}$ will be very close to the coordinate functions, and $\mathcal{M}^{k+1}$ and $\mathcal{M}^k$ will be very close to each other. Eq. (\[eq\_up1\]) is a constrained linear optimization problem. We can use the alternating direction method of multipliers to simplify the above algorithm as:
- Update $\alpha_i^{k+1}, i=1,\cdots,d$, with a fixed $P_{\theta^k}$, $$\label{eq:difficult}
(\alpha_1^{k+1}, \cdots, \alpha_d^{k+1}) = \arg \min_{\alpha_1,\cdots,\alpha_c \in H^1(\mathcal{M}^n)} \sum_{i=1}^d ||\nabla \alpha_i||^2_{L^2(\mathcal{M}^k)} + \mu ||\alpha(P_{\theta^k}) - P_{\theta^k} + d^k||^2_F.$$ where
- Update $\theta^{k+1}$, $$\theta^{k+1} = \arg \min_{\theta} \mathcal{L}(\theta) + \mu ||\alpha^{k+1}(P_{\theta^k}) - P_{\theta} + d^k||_F^2$$
- Update $d^{k+1}$, $$d^{k+1} = d^k + \alpha^{k+1}(P_{\theta^k}) - P_{\theta^{k+1}}$$
Using a standard variational approach, the solutions of the objective function (\[eq:difficult\]) can be obtained by solving the following PDE $$\label{eq:pde}
\begin{aligned}
-\Delta_{\mathcal{M}} u(\bm{p}) + \mu \sum_{y \in \Omega} \delta (\bm{p}-\bm{q})(u(\bm{q})-v(\bm{q})) &= 0, \quad \bm{p}\in \mathcal{M}\\
\frac{\partial u}{\partial n}(\bm{p}) &= 0, \quad \bm{p} \in \partial \mathcal{M}.
\end{aligned}$$ where $\partial \mathcal{M}$ is the boundary of $\mathcal{M}$, and $n$ is the out normal of $\partial \mathcal{M}$.
Eq. (\[eq:pde\]) can be solved with the point integral method. For the Laplace-Beltrami equation, the key observation is the following integral approximation: $$\int_{\mathcal{M}} \Delta_{\mathcal{M}} u(\bm{q})\bar{R}_t(\bm{p},\bm{q})d\bm{q} \approx -\frac{1}{t} \int_{\mathcal{M}} (u(\bm{p})-u(\bm{q}))R_t(\bm{p},\bm{q})d\bm{q} + 2\int_{\partial \mathcal{M}} \frac{\partial u(\bm{q})}{\partial n} \bar{R}_t(\bm{p},\bm{q})d\tau_{\bm{q}},$$ where $t>0$ is a hyper parameter and $$R_t(\bm{p},\bm{q}) = C_tR\left(\frac{|\bm{p}-\bm{q}|^2}{4t}\right).$$ $R:R^+ \rightarrow R^+$ is a positive $C^2$ function which is integrable over $[0, +\infty)$, and $C_T$ is the normalizing factor $$\bar{R}(r) = \int_r^{+\infty} R(s)ds, and \quad \bar{R}_t (\bm{p},\bm{q}) = C_t \bar{R} \left(\frac{|\bm{p}-\bm{q}|^2}{4t} \right)$$ We usually set $R(r)=e^{-r}$, then $\bar{R}_t(\bm{p},\bm{q}) = R_t(\bm{p},\bm{q}) = C_t exp\left(\frac{|\bm{p}-\bm{q}|^2}{4t} \right)$ is Gaussian.
Based on the above integral approximation, we approximate the original Laplace-Beltrami equation as: $$\int_M(u(\bm{p} - u(\bm{q})) R_t(\bm{p},\bm{q})d\bm{q} + \mu t \sum_{\bm{q}\in \omega} \bar{R}_t(\bm{p},\bm{q})(u(\bm{q})-v(\bm{q})) = 0$$ This integral equation is easy to discretize over the point cloud.
To simplify the notation, we denote the patch set $P_{\theta^k} = \{\bm{p}_i\}_{i=1}^m$, where $m$ is the number of patches, in the $k$-th iteration. We assume that the patch set samples the submanifold $\mathcal{M}$ and it is uniformly distributed. Then, the integral equation can be discretized as $$\label{eq:dis}
\frac{|\mathcal{M}|}{m} \sum_{j=1}^m R_t(\bm{p}_i, \bm{p}_j)(u_i-u_j) + \mu t \sum_{j=1}^m \bar{R}_t(\bm{p}_i,\bm{p}_j)(u_j-v_j) = 0$$ where $v_j = v(\bm{p}_j)$, and $|\mathcal{M}|$ is the volume of the manifold $\mathcal{M}$.
We rewrite Eq. (\[eq:dis\]) in the matrix form: $$(L + \bar{\mu} W)u = \bar{\mu} W v.$$ where $v=(v_1,\cdots,v_m)$, $\bar{\mu}=\frac{\mu t m}{|\mathcal{M}|}$, and $L$ is a $m\times m$ matrix, $$\label{eq_l}
L=D-W$$ $W=(w_{ij}), i,j=1,\cdots,m$ is the weight matrix, $D=diag(d_i)$ with $d_i=\sum_{j=1}^m w_{ij}$, and $$\label{eq_w}
w_{ij} = R_t(\bm{p}_i,\bm{p}_j), \quad \bm{p}_i, \bm{p}_j \in P_{\theta^k}, \quad i,j = 1,\cdots,m.$$
\
Compute the outputs of disentanglement network given a batch of data $B=\{x^a_i, y_i\}_{i=1}^{bs}$, and construct the patch set, $P_{\theta^k}$, as described in Section \[sec\_patch\]. Compute the weight matrix $W=(w_{ij})$ and $L$ with $P_{\theta^k}$, as in Eqs. (\[eq\_w\]) and (\[eq\_l\]). Solve the following linear systems to obtain $U$\
$(L+\mu W)U = \mu WV$, where $V=P_{\theta^k}-d^k$.
Update $\theta^{k+1}$ using Adam with the following loss function:\
$\mathcal{J}(\theta) = \mathcal{L}(\theta) + \lambda ||U-P_{\theta^k} + d^k||_F^2$. Construct the patch set $P_{\theta^{k+1}}$ with $\theta^{k+1}$ and update $d^{k+1}$ as follows:\
$\hat{d}^k= d^k + U - P_{\theta^{k+1}}$,\
$d^{k+1}=(\hat{d}^k-\min(\hat{d}^k))/(\max(\hat{d}^k)-\min(\hat{d}^k))$.\
$k \leftarrow k+1$.\
$\theta^* \leftarrow \theta^{(k)}$.
\[alg:opt\]
The final LDM-DN learning algorithm is described in Algorithm \[alg:opt\], where we assume that the patch set of all images samples a low-dimensional manifold. However, it is impractical to optimize the LDM problem when the number of patches is very large. To this end, we randomly select a batch of images to construct the patch set, and then estimate the coordinate functions $U$, update the network parameters $\theta$ and dual variables $d$ in each iteration. Thus, in our implementation the number of iterations in training the network is the same as that in the LDM optimization. While practically updating the dual variables $\bm{d}$ in the original LDMM algorithm [@LDMM], the values of $\bm{d}$ usually increase as the number of iterations increases. As the number of iterations is usually very large, the value of the LDM term in step 6 of Algorithm \[alg:opt\] will become increasingly large, leading to a bad solution. To overcome this problem, the dual variables are normalized in step 7 of Algorithm \[alg:opt\].
Combination of paired and unpaired learning {#sec_train}
-------------------------------------------
![Combination of paired and unpaired learning.[]{data-label="fig:train"}](figures/training.png){width="60.00000%"}
ADN only requires unpaired clinical images for training so that performance degradation of the model can be avoided when it is first trained on the synthesized dataset and then transferred to a clinical application. However, the GAN loss based unpaired supervision is not strong enough for recovering full image contents details. On the other hand, although the synthesized data may not perfectly simulate real scenarios, it does provide helpful information via accurate supervision. To benefit from both the paired learning and unpaired learning, here we design a hybrid training scheme. Specifically, during training, both unpaired clinical images and paired synthetic images are selected to construct a mini-batch, which are then fed to the corresponding branches to optimize the objective functions simultaneously, as shown in Fig. \[fig:train\].
Experimental design and results {#sec_experiment}
===============================
Datasets
--------
In our experiments we evaluated the proposed method on one synthesized dataset from DeepLesion [@yan2018] and one clinical dataset from Spineweb [^1], which are the same as those used in ADN [@adn].
For the synthesized dataset, 4,118 artifact-free CT images were randomly selected from DeepLesion. Then, the paired images with and without metal artifacts were synthesized using the method introduced in CNNMAR [@zhang2018]. Finally, 3,918 pairs of images were used for training and 200 pairs for testing. For a fair comparison, the images used for training and testing, and all pre-processing processes are the same as those used for ADN [@adn].
For the clinical dataset, 6,170 images with metal artifacts and 21,190 images without metal artifacts are selected for training, and additional 100 images with metal artifacts were selected for evaluation. The criteria for selecting these images are the same as that in the ADN study. Specifically, if an image contains pixels with HU values greater than 2,500 and the number of these pixels is larger than 400, then the image is grouped into the artifact-affected class. The images with the largest HU values less than 2,000 are grouped into the artifact-free class. Furthermore, to study the effectiveness of combining both paired and unpaired supervision, we randomly selected 6,170 images from the artifact-free group. Then, we extracted 6,170 metal objects from the images in the artifact-affected group, and used CatSim [@catsim] to simulate the paired images by inserting each extracted metal shape into a selected artifact-free image. Finally, 6,170 synthesized paired images were obtained.
Implementation details
----------------------
![Network architectures of different learning paradigms. $E_{I^a}^c, E_{I^a}^a$ denote the encoders that extract content and artifact features from artifact-affected images. $E_I$ is the encoder that extracts content features from the artifact-free images. $G_I$ and $G_{I^a}$ represent the decoders that output the artifact-free/artifact-corrected and artifact-affected images respectively. The combinations of $E_{I^a}^c \rightarrow G_I$, $E_{I^a}^a \rightarrow G_{I^a}$, and $E_I \rightarrow G_I$ construct artifact-corrected, artifact-affected, and artifact-free branches respectively. $Conv$ denotes a convolutional layer.[]{data-label="fig:arcs"}](figures/arcs.png){width="70.00000%"}
We implemented different network architecture variants for different learning paradigms, as shown in Fig. \[fig:arcs\]. For unpaired learning, we use the same architecture as ADN [@adn] as shown in Fig. \[fig:arcs\] (a), and the architectures of all other learning paradigms are the variants of ADN. In Fig. \[fig:arcs\] (b), to construct the patch set for the LDM constraint, we add two convolutional layers on the top of the encoders in the artifact-corrected and artifact-free branches respectively, as described in Section \[sec\_patch\]. For paired learning only, we simply use the encoder-decoder in the artifact-corrected branch as shown in Fig. \[fig:arcs\] (c). In combination of paired learning and the LDM constraint, we keep two encoder-decoder branches as shown in Fig. \[fig:arcs\] (d).
In Fig. \[fig:patch\] and Fig. \[fig:arcs\] (b) and (d), the extra convolutional layers are used to compress the channels of the latent code. Specifically, the input image size is $1 \times 256 \times 256$, the down-sampling rate is 8, the matrix of $Z_x$ is of $512 \times 64 \times 64$, the matrix of $Z_x^t$ is of $64 \times 64 \times 64$, the patch size is $8 \times 8$, and the dimension of the point in the patch set is 128.
We implemented the proposed method in PyTorch [^2]. To be fair, we keep all hyper parameters the same as those in DAN [@adn]. In Algorithm \[alg:opt\], we empirically set the batch size $bs=1$, $\mu=0.6$.
Results on synthesized dataset {#sec_syn}
------------------------------
### Reimplementation of ADN {#sec_ratio}
ADN-0.85 ADN-0.50 ADN-0.15
------ ---------- ---------- ----------
PSNR 34.1 34.0 34.0
SSIM 92.8 92.8 92.9
: MAR results of the models trained with different ratios of numbers of images in the artifact-free and artifact-affected groups.
\[tab:ratio\]
To simulate the unpaired learning, the synthesized paired images were divided into two groups, and then the artifact-affected images were selected from one group and the artifact-free images from the other group. In [@adn], the ratio of numbers of images in these two groups was simply set to 1:1. However, in clinical scenarios, the number of artifact-affected images is much smaller than the number of artifact-free images. Therefore, we evaluated the effectiveness of the ratio to the MAR performance in the unpaired learning setting. In Table \[tab:ratio\], *ADN-0.85*, *ADN-0.50* and *ADN-0.15* signify various ratios of artifact-affected images to all images. Table \[tab:ratio\] shows that there is little difference between the models trained with different ratios between artifact-affected and artifact-free images. In addition, we found that the metrics for the MAR performance would not strictly converge in the unpaired learning setting. Therefore, we selected the best one as the final results, which are better than the reported results in [@adn]. In practice, it is also reasonable to select the best performance model on the synthesized images first and then apply the selected model to clinical images. As a representative ratio between the artifact-affected and artifact-free images in the clinical conditions, the *ADN-0.15* serves as the baseline in all following experiments.
------ -------- ------ --------- ------ ------------- ---------- ------ ------ ------ ------ --------- ------------
CNNMAE UNet cGANMAR Sup **LDM-Sup** CycleGAN DIP MUNT DRIT ADN ADN$^*$ **LDM-DN**
PSNR 32.5 34.8 34.1 37.6 **38.0** 30.8 26.4 14.9 25.6 33.6 34.0 **35.0**
SSIM 91.4 93.1 93.4 96.1 **96.3** 72.9 75.9 7.5 79.7 92.4 92.9 **94.2**
------ -------- ------ --------- ------ ------------- ---------- ------ ------ ------ ------ --------- ------------
: Comparison results on DeepLesion.
\[tab:compare\]
![Visual comparison. The metal objects are inserted back by thresholding.[]{data-label="fig:syn_results"}](figures/syn_results.png){width="99.00000%"}
### Comparative results
On the synthesized dataset, we evaluated the quantitative and qualitative performance of our proposed method as well as the compared methods. For quantitative results, we used the peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) metrics. Table \[tab:compare\] gives the comparison results of the proposed method with the competing methods in the paired and unpaired learning settings. In Table \[tab:compare\], *ADN*$^*$ is our improved ADN, see Section \[sec\_ratio\] for details. The results show that the proposed *LDM-DN* method outperformed ADN in terms of both PSNR and SSIM metrics in the unpaired learning setting. For the paired learning part in Table \[tab:compare\], *Sup* corresponds to the network architecture in Fig. \[fig:arcs\] (c), which was trained with the paired data. It is noted that this encoder-decoder architecture contains the skip connections between the encoder and the decoder, which is the same as ADN. *LDM-Sup* adds the LDM constraints to the *Sup* during the paired training, corresponding to the architecture in Fig. \[fig:arcs\] (d). Although the paired images have accurate pixel-to-pixel supervision, the LDM based learning algorithm can further improve the performance. The above results strongly demonstrate that our proposed LDM-DN algorithm can consistently improve the existing models in the paired and unpaired learning settings.
We also visually compared the results as shown in Fig \[fig:syn\_results\]. The visual impressions are consistent with the numerical results. In the unpaired learning setting, although ADN can remove a majority of metal artifacts, the local details were not well preserved. By comparing the results of *ADN\** and *LDM-DN*, an evident improvement was made on these details. Compared with the unpaired learning (*Sup* vs.*ADN\**), the ideal paired learning gave better results on the synthesized test dataset. In this case, our proposed LDM-DN learning algorithm obtained further improvements, where the boundaries of structures are sharper visually (*LDM-Sup* vs. *Sup*). These results strongly show the effectiveness of the proposed LDM-DN algorithm.
Results on clinical dataset
---------------------------
![Clinical results from different variants.[]{data-label="fig:clinical_results"}](figures/clinical_results.png){width="99.00000%"}
In this subsection, we evaluated the proposed networks on the clinical dataset. As there are not ground truth images for evaluating the performance of the models, here we can only show the visual results of two examples in Fig. \[fig:clinical\_results\]. On the clinical dataset, we see the superiority of the proposed LDM-DN algorithm in preserving (the green boxes for *LDM-DN* vs. *ADN\** in the first example) and recovering (the blue boxes for *LDM-DN* vs. *ADN\** in the second example) local details. We also evaluated the performance of the model trained with synthesized images on the same dataset in a supervised learning manner. As shown in Section \[sec\_syn\], the results of *ADN\** is better than that in the unpaired learning on synthesized dataset. However, the results in Fig. \[fig:clinical\_results\] shows that the performance of *Sup* is definitely worse than that of the unpaired learning model trained on the clinical dataset, as the synthesized data does not reflect the real conditions. This is consistent with the observation in [@adn]. However, although the performance of *Sup* is degraded, it still shows some merits over the unpaired learning methods, such as some structures are sharper (green boxes of *Sup* vs. *ADN\** and *LDM-DN* for both examples) and some regions are better (blue boxes of *Sup* vs. *ADN\** and *LDM-DN* in the second example). Combining *ADN* and *Sup* leads to an over-correction as shown in Fig. \[fig:clinical\_results\], where some structures are over sharper (green boxes of *ADN-Sup* vs. others for both examples) and some regions are over dark (blue boxes of *ADN-Sup* vs. others for both examples). We attribute these results to that *ADN* help reduce the under-correction of *Sup* in some regions (red boxes of *ADN-Sup* vs. *Sup* for the second example) but simultaneously resulting in over-correcting some other regions (blue boxes of *ADN-Sup* vs. *Sup* for the second example). Therefore, we propose to combine all merits of unpaired learning, paired learning and LDM through a hybrid learning scheme. As the results of *LDM-DN-Sup* shown in Fig. \[fig:clinical\_results\], it can inherit all good points as analyzed above. Particularly, compared with *ADN-Sup*, LDM in *LDM-DN-Sup* constrains that the structurally similar patches, especially the adjacent patches, to be coherent without dramatic changes to be too dark or too bright. The above results on the clinical dataset strongly demonstrate the effectiveness of LDM and the superiority of the hybrid training scheme.
Conclusion {#sec_conclusion}
==========
We have proposed an LDM constrained disentanglement network for MAR. Specifically, we have designed a LDM-DN learning algorithm to simultaneously optimize the objective functions of deep neural networks and constrain the recovered images to have a low-dimensional patch manifold representation. The LDM-DN algorithm can effectively help preserve and recover structural details in CT images. Moreover, we have investigated both paired and unpaired learning based models for MAR, showing their relative advantages. Finally, we have designed a hybrid optimization scheme to combine paired learning, unpaired learning and LDM-DN learning algorithm for integrating their advantages. The experimental results on synthesized and clinical datasets have strongly demonstrated the superiority of the proposed method. We believe that the proposed LDM-DN algorithm has a great potential to solve various CT MAR problems.
[^1]: spineweb.digitalimaginggroup.ca
[^2]: https://pytorch.org/
|
---
abstract: 'Mathematical operators whose transformation rules constitute the building blocks of a multi-linear algebra are widely used in physics and engineering applications where they are very often represented as tensors. In the last century, thanks to the advances in tensor calculus, it was possible to uncover new research fields and make remarkable progress in the existing ones, from electromagnetism to the dynamics of fluids and from the mechanics of rigid bodies to quantum mechanics of many atoms. By now, the formal mathematical and geometrical properties of tensors are well defined and understood; conversely, in the context of scientific and high-performance computing, many tensor-related problems are still open. In this paper, we address the problem of efficiently computing contractions among two tensors of arbitrary dimension by using kernels from the highly optimized BLAS library. In particular, we establish precise conditions to determine if and when , the kernel for matrix products, can be used. Such conditions take into consideration both the nature of the operation and the storage scheme of the tensors, and induce a classification of the contractions into three groups. For each group, we provide a recipe to guide the users towards the most effective use of BLAS.'
author:
- 'Edoardo Di Napoli [^1]'
- 'Diego Fabregat-Traver [^2]'
- 'Gregorio Quintana-Ortí [^3]'
- 'Paolo Bientinesi$^\dagger$'
title: Towards an Efficient Use of the BLAS Library for Multilinear Tensor Contractions
---
Acknowledgements {#acknowledgements .unnumbered}
================
Financial support from the Deutsche Forschungsgemeinschaft (German Research Association) through grant GSC 111 is gratefully acknowledged. This research was in part supported by the VolkswagenStiftung through the fellowship “Computational Sciences”.
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[^1]: Jülich Supercomputing Centre, Institute for Advanced Simulation, Forschungszentrum Jülich, Wilhelm-Johnen strasse, 52425–Jülich, Germany. [[email protected]]{}.
[^2]: AICES, RWTH-Aachen University, 52056–Aachen, Germany. [{fabregat,pauldj}@aices.rwth-aachen.de]{}.
[^3]: Depto. de Ingeniería y Ciencia de Computadores, Universidad Jaume I, 12.071–Castellón, Spain. [ [email protected]]{}.
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---
abstract: |
The aim of this paper is to establish various functional inequalities for the convolution of a compactly supported measure and a standard Gaussian distribution on $\dR^d$. We especially focus on getting good dependence of the constants on the dimension. We prove that the Poincaré inequality holds with a dimension-free bound. For the logarithmic Sobolev inequality, we improve the best known results (Zimmermann, JFA 2013) by getting a bound that grows linearly with the dimension. We also establish transport-entropy inequalities for various transport costs.
**Keywords:** logarithmic Sobolev inequality, transport-entropy inequality, Poincaré inequality
**MSC2010:** 60E15; 39B62; 26D10
author:
- 'Jean-Baptiste <span style="font-variant:small-caps;">Bardet</span>, Nathaël <span style="font-variant:small-caps;">Gozlan</span>, Florent <span style="font-variant:small-caps;">Malrieu</span> and Pierre-André <span style="font-variant:small-caps;">Zitt</span>'
bibliography:
- 'convolution.bib'
title: 'Functional inequalities for Gaussian convolutions of compactly supported measures: explicit bounds and dimension dependence'
---
Introduction
============
Poincaré or logarithmic Sobolev inequalities have been extensively studied in the past decades to quantify long time behavior of Markov processes or investigate the concentration of measure property, which plays a key role for example in the topic of large random matrices.
We refer to [@Log-Sob; @Ledoux-book; @Royer; @BGL14] for a comprehensive introduction to this subject. Let us briefly recall some well-known facts about these functional inequalities to motivate the present study.
A probability measure $\nu$ on $\dR^d$ satisfies a Poincaré inequality with constant $C$ if, for any smooth function $f$ from $\dR^d$ to $\dR$, $$\int_{\dR^d} f^2d\nu - {{{\left(\int_{\dR^d} fd\nu\right)}}}^2 \leq
C\int_{\dR^d}{{{\left\vert\nabla f\right\vert}}}^2d\nu.$$ We denote by $C_P(\nu)$ the smallest constant such that this inequality holds.
Similarly, $\nu$ satisfies a logarithmic Sobolev inequality with constant $C$ if, for any smooth function $f$ from $\dR^d$ to $\dR$, $$\int_{\dR^d} f^2 \log (f^2)d\nu - {{{\left(\int_{\dR^d} f^2d\nu\right)}}}
\log {{{\left(\int_{\dR^d} f^2d\nu\right)}}} \leq C\int_{\dR^d}{{{\left\vert\nabla
f\right\vert}}}^2d\nu,$$ and we denote by $C_{LS}(\nu)$ the smallest constant such that this inequality holds.
If $\nu$ is the Gaussian distribution $\cN_d(x,\Gamma)$ on $\dR^d$ with mean $x$ and covariance matrix $\Gamma$ then the values of these optimal constants are known: $$C_P(\nu)=\frac{1}{2}C_{LS}(\mu)=\max \mathrm{Spec} (\Gamma).$$ The Bakry-Émery criterion ensures that if $\nu$ has the density $e^{-V}$ on $\dR^d$ and $\mathrm{Hess}(V)\geq \rho I_d$ then $$C_P(\nu)\leq \frac{1}{\rho} \quad \text{and}\quad C_{LS}(\mu)\leq
\frac{2}{\rho}.$$ More generally, the inequality $2 C_P(\nu)\leq C_{LS}(\nu)$ always holds. These two functional inequalities do not hold if the support of $\nu$ is not connected — one can find a non constant function whose gradient is zero $\nu$-almost surely.
The present paper focuses on the case when the probability measure $\nu$ on $\dR^d$ is given by the convolution $\mu\star \cN_d(0,\delta^2 I_d)$ where the support of $\mu$ is included in the centered ball of $\dR^d$ with radius $R$. This question has been investigated recently in [@Zim; @ZimJFA; @WW13]; we present here several improvements and related questions.
Let us fix some notation first.
- $X$ and $Z$ are two independent random variables with respective distribution $\mu$ and $\cN(0,I_d)$;
- $\gamma_\delta$ is the density of the Gaussian measure $\cN_d(0, \delta^2 I_d)$;
- $p$ is the density of the law $\mu\star\gamma_\delta$ of the random variable $S=X+\delta Z$;
- $C_d(\delta,R)$ is the supremum over all probability measures $\mu$ supported in the closed Euclidean ball $B_d(0,R)$ of the optimal constants in the logarithmic Sobolev inequality for $\mu\star\gamma_\delta$.
This notation is mainly consistent with [@Zim], except that our $\delta$ is the standard deviation of the Gaussian rather than its variance, and we denote the dimension by $d$.
Zimmermann’s results [@Zim; @ZimJFA] may be summed up as follows.
\[Bounds on logarithmic Sobolev inequality constants,[@Zim]\] \[thm:zimm\] The convolution of a compactly supported measure and a Gaussian measure satisfies a logarithmic Sobolev inequality. Moreover, there exist universal constants ${(K_i)}_{1\leq i\leq4}$ such that:
- In dimension $1$, $$C_1(\delta,R) \leq K_1 \frac{\delta^3 R}{4R^2+\delta^2}
\exp{{{\left(2\frac{R^2}{\delta^2}\right)}}} + K_2(\delta+2R)^2.$$ In particular in the low variance case $\delta \leq R$, $$C_1(\delta,R) \leq K_3 \frac{\delta^3}{R}
\exp{{{\left(2\frac{R^2}{\delta^2}\right)}}}.$$
- In dimension $d$, $C_d(\delta,R)$ is finite. In the low variance case $\delta\leq R$, it satisfies: $$C_d(\delta,R) \leq K_4 R^2 \exp{{{\left( 20d + 5\frac{R^2}{\delta^2}\right)}}}.$$
The proofs in [@Zim] rely on two main ideas. The one-dimensional case is treated by explicit computations on Hardy-like criteria taken from [@BG]. In higher dimension the author applies the Lyapunov function approach of [@CGW10]. The constants $K_i$ are explicit but quite large (for example $K_4$ may be taken equal to $289$). Let us also mention the alternate approach of [@zimTransport] in dimension $1$ by measure transportation, that unfortunately yields even worse constants. In a related note, [@WW13] answer various related questions on functional inequalities for convolutions, and give many qualitative results under relaxed assumptions, both on the support of $X$ and on the distribution of the mollifier $Z$, but without exhibiting explicit constants.
We follow here the focus of [@Zim] on quantitative estimates on the constants and their dependence on the dimension $d$. Our first result concerns the Poincaré inequality.
\[thm:Poincare\] If $\mu$ is supported in the closed Euclidean ball $B_d(0,R)$ then $\mu\star \gamma_\delta$ satisfies a Poincaré inequality and $$C_P(\mu\star\gamma_\delta) \leq \delta^2 \exp{{{\left(4\frac{R^2}{\delta^2}\right)}}}.$$
The next result is an improvement on the bounds of Theorem \[thm:zimm\].
\[thm:logSob\]
- In the large variance case $\delta>R$, the logarithmic Sobolev constants are bounded uniformly in the dimension: $$C_d(\delta,R) \leq \frac{\delta^4}{\delta^2- R^2}.$$
- In dimension $1$, for any $\delta$, $R$, $$C_1(\delta,R) \leq 4\delta^2 \exp{{{\left( \frac{8}{\pi}
\frac{R^2}{\delta^2}\right)}}}.$$
- In the small variance case $\delta\leq R$, the logarithmic Sobolev constant admits the following dimension-dependent bound: $$\label{eq:logSobDimDependant}
C_d(\delta,R) \leq {{{\left(K_1 d + K_2 \frac{R^2}{\delta^2}\right)}}} R^2
\exp{{{\left(4\frac{R^2}{\delta^2}\right)}}}$$ where $K_1$, $K_2$ are universal constants.
The stronger bound in dimension $1$ is obtained as a corollary of a bound that holds in any dimension (with a strong dependence on $d$). Its proof uses a trick by Miclo to apply the classical Holley-Stroock perturbation argument, and is much less technical than the ones in [@Zim; @zimTransport].
For the logarithmic Sobolev constant, the dependence in the dimension drops from exponential to linear: this enhancement would translate into weaker dependence assumptions in the applications to random matrices considered in [@Zim].
In view of these results, it seems natural to conjecture as in [@Zim] that $C_d(\delta,R)$ may admit a dimension free bound. Let us give some partial results in this direction.
The first is a dimension free bound for a transport-entropy inequality. We recall that if $k:\dR^d\times \dR^d \to \dR^+$ is a cost function, then the optimal transport cost related to this $k$, is defined, for all probability measures $\nu_1$ and $\nu_2$, by $$\cT_k(\nu_1,\nu_2) = \inf_{\pi} \int k(x,y)\,d\pi(x,y),$$ where the infimum is taken over the set of all couplings $\pi$ between $\nu_1$ and $\nu_2$. Let $\cT_{2,4}$ and $\cT_{2}$ denote the transportation costs associated to $(x,y)\mapsto \| x-y\|_4^2$ and $(x,y)\mapsto |x-y|^2$ (here and in the whole paper, $|\cdot|$ denotes the Euclidean norm).
\[thm:transport\] Let $\mu$ be a probability measure on $\dR^d$ supported in $B_d(0,R)$. The probability $\mu \star \gamma_\delta$ satisfies the following transport-entropy inequalities: for any probability measure $\nu$ on $\dR^d$, $$\begin{aligned}
\cT_{2,4}(\nu,\mu\star\gamma_\delta)
&\leq C(R,\delta) H(\nu |\mu\star \gamma_\delta), \\
\cT_{2}(\nu,\mu\star\gamma_\delta)
&\leq \sqrt{d} C(R,\delta) H(\nu |\mu\star \gamma_\delta),
\end{aligned}$$ where $C(R,\delta) = c'\delta^2 {{{\left(1+
\frac{R^2}{\delta^2}\right)}}}\exp{{{\left(\frac{4R^2}{\delta^2}\right)}}}$ for some universal constant $c'$.
Let us remark that the factor $\sqrt{d}$ in this last result is better than the linear factor $d$ that follows by deducing $\cT_{2}$ from the logarithmic Sobolev inequality by Otto-Villani’s theorem (see [@OV00; @BGL01]).
Finally, we are able to get bounds on the logarithmic Sobolev constant in several restricted cases.
\[Partial results\] \[thm:partial\]
- The quantity $C_d(\delta,R)$ may be bounded only in terms of $\delta$ and $R$ in the region $\delta > R/\sqrt{2}$.
- If $\mu$ is radially symmetric, then $$C_{LS}(\mu \star \gamma_\delta) \leq
4\delta^2\exp{{{\left(\frac{8}{\pi} \frac{R^2}{\delta^2}\right)}}}.$$
- If $\mu$ is a uniform discrete probability measure on $N\geq 3$ points, $$C_{LS}(\mu\star\gamma_\delta) \leq \delta^2 + 3 \log(N)
\delta^2\exp{{{\left(4\frac{R^2}{\delta^2}\right)}}}\,.$$
- The logarithmic Sobolev inequality restricted to log-convex functions holds with a constant that does not depend on the dimension.
To prove or disprove the conjecture, one is tempted to guess the measure $\mu$ that leads to the worst logarithmic Sobolev constant. A natural candidate, proposed in [@Zim Example21], is the two-point measure $1/2 (\delta_{Re_1} + \delta_{-Re_1})$ (where $e_1$ denotes the first basis vector). Note that this candidate is easily seen to satisfy a logarithmic Sobolev inequality with a bounded constant, either by the bound on discrete measures or by a simple tensorization argument of a one-dimensional convolution with a $(d-1)$-dimensional Gaussian law. To build a counterexample one would have to consider measures with a number of points that grows with the dimension.
#### Outline of the paper.
The paper is organized as follows. In Section \[sec:perturbation\] we use the perturbation idea of Holley-Stroock, by rewriting the potential of $\mu\star\gamma_\delta$ as a sum of a convex function and a bounded perturbation, proving the first two items of Theorem \[thm:logSob\]. In Section \[sec:mixing\], viewing $\mu\star\gamma_\delta$ as a mixture of Gaussian measures we prove the Poincaré and transportation inequalities (Theorems \[thm:Poincare\] and \[thm:transport\]) and establish the bound for discrete measures (third item of Theorem \[thm:partial\]). Theorem \[thm:Poincare\] yields the final bound on logarithmic Sobolev constants (the third item in Theorem \[thm:logSob\]) as an easy corollary. The various remaining results in Theorem \[thm:partial\] are proved in Section \[sec:partial\].
Perturbation arguments {#sec:perturbation}
======================
Large variance
--------------
The density $p$ of $\mu\star\gamma_\delta$ is given explicitly by : $$p(z)=\int_{\dR^d} \frac{1}{(2\pi
\delta^2)^{d/2}}\exp{{{\left(-\frac{{{{\left\vertz-x\right\vert}}}^2}{2\delta^2}\right)}}} \mu(dx)
=\frac{1}{(2\pi\delta^2)^{d/2}} \exp{{{\left( - {{{\left(
\frac{{{{\left\vertz\right\vert}}}^2}{2\delta^2} + W_\delta(z)\right)}}} \right)}}}$$ where $$\begin{aligned}
W_\delta(z)
&= - \log\int_{\dR^d} \exp{{{\left(\frac{z\cdot x}{\delta^2}
-\frac{{{{\left\vertx\right\vert}}}^2}{2\delta^2}\right)}}}\,\mu(dx) \\
&= - \log\int_{\dR^d} \exp{{{\left( \frac{z\cdot x}{\delta^2}\right)}}}\,\nu(dx)
-\log C_\nu\end{aligned}$$ for $C_\nu = \int_{\dR^d} \exp(-{{{\left\vertx\right\vert}}}^2/(2\delta^2)) \mu(dx)$ and $\nu(dx) = C_\nu^{-1} \exp(-{{{\left\vertx\right\vert}}}^2/(2\delta^2)) \mu(dx)$. Let us compute the Hessian of $(-W_\delta)$: $$\partial_{z_i}(- W_\delta)(z) = \frac{1}{\delta^2} \dE(\tilde X_i)
\quad\text{and}\quad
\partial^2_{z_i z_j}(- W_\delta)(z) = \frac{1}{\delta^4} \operatorname{Cov}(\tilde
X_i, \tilde X_j)$$ where the distribution of $\tilde X$ is proportional to $\exp(z\cdot x) d\nu$. Therefore, for any unit vector $v$, $$0\leq \operatorname{Hess}(-W_\delta) v \cdot v \leq \frac{1}{\delta^4} \operatorname{Var}( v\cdot
\tilde X).$$ Since $v\cdot \tilde X$ lives in $[-R,R]$, its variance is bounded by $R^2$, so $$\operatorname{Hess}(- \log(p)) \geq {{{\left(\frac{1}{\delta^2} - \frac{R^2}{\delta^4}\right)}}}
I_d.$$
This bound is slightly better than the one given in [@Zim] where the variance of $v\cdot\tilde X$ is bounded by $2R^2$.
In particular, if $\delta>R$, $p$ is log-concave and the Bakry-Émery criterion yields: $$C_{LS}(\mu\star\gamma_\delta) \leq \frac{\delta^4}{\delta^2- R^2}.$$ This proves the first item in Theorem \[thm:logSob\].
A perturbation argument {#sec:Miclo}
-----------------------
It turns out we can get a (dimension dependent) bound on the logarithmic Sobolev constant with a very short proof, using the following trick to decompose the logarithm of the density $p$ as a sum of a convex function and a bounded perturbation.
Let $a_d = {\dE_{}\left[{{{\left\vertZ\right\vert}}}\right]}$ be the expected value of the norm of a standard Gaussian random variable $Z$ in dimension $d$. Note that $a_d$ has an explicit expression (we will use below that $a_1=\sqrt{2/\pi}$) and is in any case smaller than $\sqrt{d}$.
\[lem:Royer\] Suppose the function $W:\dR^d\to\dR$ may be written as $W=W_c+W_l$ where $\operatorname{Hess}(W_c)\geq \rho I_d$, and $W_l$ is $l$-Lipschitz with respect to the Euclidean distance.
Then for any $\sigma > 0$, one can write $W$ as a sum $U_c + U_b$ where $\operatorname{Hess}(U_c) \geq {{{\left(\rho - \frac{la_1}{\sigma}\right)}}}I_d$ and $U_b$ is bounded by $l\sigma a_d$.
In particular the measure $Z_W^{-1} \exp(-W)$ satisfies a logarithmic Sobolev inequality and $$C_{LS}{{{\left(Z_W^{-1}\exp(-W)\right)}}} \leq \frac{4}{\rho} \exp{{{\left(
\frac{4}{\rho} l^2 a_1a_d\right)}}}\,.$$
By way of comparison, it is known (see [@AidShi94; @Aid98]) that if $\mu_0 = \exp(-V_0)dx$ satisfies a logarithmic Sobolev inequality, then $\nu = \exp(-V)dx $ satisfies a defective logarithmic Sobolev inequality, as soon as the gradient $\nabla(V-V_0)$ satisfies some exponential integrability condition. This defective inequality can be used together with the Poincaré inequality to obtain the logarithmic Sobolev inequality. This strategy is used in [@WW13] (see in particular [@WW13 Lemma 2.3] for a precise statement of the perturbation result). It is more general, since it only supposes a logarithmic Sobolev inequality for the unperturbed measure, and replaces a boundedness assumption by an integrability condition. The trade-off is that the constants are not explicit.
Since the statement of Lemma \[lem:Royer\] in [@Ledoux-revisited; @Royer] contains a typo in the convexity bound, let us provide a detailed proof.
Let $\sigma>0$ and $ U_\sigma$ be the following regularized version of $W_l$: $U_\sigma(x) = {\dE_{}\left[W_l(x+\sigma Z)\right]}$, where $Z$ is a standard $d$-dimensional Gaussian random variable. Let $U_c = W_c + U_\sigma$ and $U_b= W_l - U_\sigma$. Since $W_l$ is $l$-Lipschitz, $${{{\left\vertU_b(x)\right\vert}}} = {{{\left\vert {\dE_{}\left[W_l(x) - W_l(x+\sigma Z)\right]}\right\vert}}} \leq l\sigma
{\dE_{}\left[{{{\left\vertZ\right\vert}}}\right]} \leq l\sigma a_d\,.$$ Therefore $U_b$ is bounded.
We now turn to the convexity bound. It is enough to prove that, for any unit vector $v$ in $\dR^d$, $\operatorname{Hess}U_\sigma v \cdot v \leq \frac{lc_1}{\sigma}$. First we compute the derivatives of $U_\sigma$: $$\begin{aligned}
\partial_i U_\sigma(x)
&= (2\pi \sigma^2)^{-d/2}\frac{1}{\sigma^2}
\int_{\dR^d} W_l(y)(y_i-x_i)
\exp{{{\left( - \frac{{{{\left\vertx-y\right\vert}}}^2}{2\sigma^2}\right)}}} dy \\
&= (2\pi \sigma^2)^{-d/2}\frac{1}{\sigma^2}
\int_{\dR^d} W_l(x+z) z_i
\exp{{{\left( - \frac{{{{\left\vertz\right\vert}}}^2}{2\sigma^2}\right)}}} dz \\
\partial_{ij} U_\sigma(x)
&= (2\pi \sigma^2)^{-d/2}\frac{1}{\sigma^2}
\int_{\dR^d} \partial_j W_l(x+z) z_i
\exp{{{\left( - \frac{{{{\left\vertz\right\vert}}}^2}{2\sigma^2}\right)}}} dz.
\end{aligned}$$ Now, $$\operatorname{Hess}U_\sigma v \cdot v= (2\pi \sigma^2)^{-d/2}\frac{1}{\sigma^2}
\int_{\dR^d} (v\cdot\nabla W_l(x+z)) (v\cdot z)\exp{{{\left( -
\frac{{{{\left\vertz\right\vert}}}^2}{2\sigma^2}\right)}}} dz\,.$$ Since $W_l$ is $l$-Lipschitz, $$\begin{aligned}
{{{\left\vert\operatorname{Hess}U_\sigma v \cdot v\right\vert}}} &\leq
(2\pi \sigma^2)^{-d/2}\frac{l}{\sigma^2}
\int_{\dR^d} {{{\left\vertv\cdot z\right\vert}}}
\exp{{{\left( - \frac{{{{\left\vertz\right\vert}}}^2}{2\sigma^2}\right)}}} dz.
\end{aligned}$$ By rotation invariance of the standard Gaussian distribution, we get $${{{\left\vert\operatorname{Hess}U_\sigma v \cdot v\right\vert}}} \leq \frac{l}{\sigma^2} {\dE_{}\left[ \sigma
{{{\left\vertZ_1\right\vert}}}\right]} \leq \frac{l a_1}{\sigma}\,.$$ This implies that $\operatorname{Hess}(W_c + U_\sigma) \geq (\rho-\frac{la_1}{\sigma})I_d$, as claimed.
The final claim is a direct consequence of the obtained decomposition with $\sigma=2la_1/\rho$, the Holley–Stroock perturbation Lemma and the Bakry–Émery criterion (see [@Royer]).
Let us now use this lemma to prove the one-dimensional bound in Theorem \[thm:logSob\]. Write $-\log(p)$ as $$- \log (p(z)) = {{{\left(\frac{{{{\left\vertz\right\vert}}}^2}{2\delta^2} + \frac{d}{2}
\log(2\pi\delta^2)\right)}}} + W_\delta(z).$$ The first term is $\delta^{-2}$-convex. Since $$\nabla W_\delta(z)= -\frac{1}{\delta^2} \frac{\int_{\dR^d}\! x
\exp{{{\left(\frac{z\cdot x}{\delta^2}\right)}}} \,\nu(dx)} {\int_{\dR^d}\!
\exp{{{\left(\frac{z\cdot x}{\delta^2}\right)}}} \,\nu(dx)},$$ and $\nu(B_d(0,R))=1$, $W_\delta$ is $R/\delta^2$-Lipschitz on $\dR^d$. Lemma \[lem:Royer\] then yields $$C_{LS}(\mu\star\gamma_\delta) \leq 4\delta^2 \exp{{{\left( 4a_1a_d
R^2\delta^{-2}\right)}}}\,.$$ This gives a first dimension dependent bound that is not comparable to the one from Theorem \[thm:zimm\]. In dimension $1$, since $a_1 = \sqrt{2/\pi}$, we get the bound claimed in the second item of Theorem \[thm:logSob\].
Mixture arguments {#sec:mixing}
=================
Poincaré inequality {#sec:Poincare}
-------------------
In this section we denote by $\gamma_{x,\delta}$ the distribution $\cN_d(x,\delta^2I_d)$. Recall that $\mu \star \gamma_\delta = \int_{\dR^d} \gamma_{x,\delta} d\mu(x)$. The variance of a function $f$ under the mixture $\mu \star \gamma_\delta$ can be classically decomposed as $$\begin{aligned}
\operatorname{Var}_{\mu \star \gamma_\delta}(f)
&= \int_{\dR^d} \operatorname{Var}_{\gamma_{x,\delta}} (f) d\mu(x)
+ \operatorname{Var}_{\mu} {{{\left(x\mapsto \int f d\gamma_{x,\delta}\right)}}} \\
&= A + B.\end{aligned}$$ Since $\gamma_{x,\delta}$ satisfies the Poincaré inequality with constant $\delta^2$, the first term $A$ is bounded by $$\delta^2 \int_{\dR^d} \int_{\dR^d} {{{\left\vert\nabla f\right\vert}}}^2 d\gamma_{x,\delta}
d\mu(x) = \delta^2 \int_{\dR^d} {{{\left\vert\nabla f\right\vert}}}^2
d(\mu\star\gamma_\delta)\,.$$ For the second term $B$ let $g:x\mapsto \int_{\dR^d} f d\gamma_{x,\delta}$. Duplicating variables yields $$B = \frac{1}{2} \iint_{\dR^d\times\dR^d} (g(x) - g(y))^2 d\mu(x)
d\mu(y).$$ Now $$\begin{aligned}
(g(x) - g(y))^2 &= {{{\left( \int_{\dR^d} f d\gamma_{x,\delta}
- \int_{\dR^d} f d\gamma_{y,\delta} \right)}}}^2
= {{{\left( \int_{\dR^d} f{{{\left(1
- \frac{d\gamma_{y,\delta}}{d\gamma_{x,\delta}}\right)}}}
d\gamma_{x,\delta} \right)}}}^2 \\
&= {{{\left( \operatorname{Cov}_{\gamma_{x,\delta}}
{{{\left(f,{{{\left(1 - \frac{d\gamma_{y,\delta}}
{d\gamma_{x,\delta}}\right)}}}\right)}}}\right)}}}^2 \\
&\leq \operatorname{Var}_{\gamma_{x,\delta}}(f) \operatorname{Var}_{\gamma_{x,\delta}}
{{{\left( 1 - \frac{d\gamma_{y,\delta}}{d\gamma_{x,\delta}}\right)}}}\end{aligned}$$ by Cauchy-Schwarz inequality. For the first factor we reapply the Poincaré inequality for the Gaussian measure $\gamma_{x,\delta}$. The second factor is the $\chi^2$ divergence between the Gaussian distributions $\gamma_{x,\delta}$ and $\gamma_{y,\delta}$. An easy computation shows that this divergence is ${{{\left(\exp{{{\left({{{\left\vertx-y\right\vert}}}^2/\delta^2\right)}}} - 1\right)}}}$; since ${{{\left\vertx-y\right\vert}}}$ is bounded by $2R$, we get $$(g(x) - g(y))^2 \leq \delta^2{{{\left(\exp{{{\left(4R^2/\delta^2\right)}}}-1\right)}}}
\int_{\dR^d}{{{\left\vert\nabla f\right\vert}}}^2 d\gamma_{x,\delta}.$$ Reintegrating with respect to $\mu$ yields $$B \leq \delta^2 {{{\left(\exp(4R^2/\delta^2)-1\right)}}} \int_{\dR^d} {{{\left\vert\nabla
f\right\vert}}}^2 d(\mu\star\gamma_\delta),$$ so that the measure $\mu\star\gamma_\delta$ satisfies a Poincaré inequality with a constant $$C_P(\mu\star\gamma_\delta)\leq \delta^2 \exp(4R^2/\delta^2)\,.$$
A mild dependence on d for logarithmic Sobolev constants via Lyapunov functions
-------------------------------------------------------------------------------
The proof of the logarithmic Sobolev inequality in dimension greater than $1$ in [@Zim] is based on a criterion from [@CGW10]. This criterion uses a Lyapunov function approach to prove a so-called defective logarithmic Sobolev inequality, which can then be strengthened using the Poincaré inequality. In [@Zim], this Poincaré inequality is itself obtained by Lyapunov criteria, with constants depending exponentially on the dimension. Simply plugging our dimension-free Poincaré inequality in the argument of [@CGW10] gives a much better bound.
Let us first recall the criterion, in the form used in [@Zim], where the constants are explicitly written.
Suppose that $V$ satisfies $$\begin{aligned}
\operatorname{Hess}(V) \geq -K I_d
\end{aligned}$$ with $K \geq 0$, and there exists a “Lyapunov function”, that is, a function $W\geq 1$ such that $$\label{eq:lyap}
\Delta W - \langle \nabla V, \nabla W \rangle \leq (b-c{{{\left\vertx\right\vert}}}^2) W$$ for some positive constants $b$, $c$.
Suppose that $\nu=Z_V^{-1}\exp(-V)dx$ satisfies a Poincaré inequality with constant $C_P(\nu)$. Let $A$ and $B$ be defined by $$\begin{aligned}
A &= \frac{2}{c} {{{\left({\varepsilon}^{-1} + K/2\right)}}} + {\varepsilon}, \\
B &= \frac{2}{c} {{{\left({\varepsilon}^{-1} + K/2\right)}}}{{{\left(b+c
\int_{\dR^d} {{{\left\vertx\right\vert}}}^2 d\nu(x)\right)}}}.
\end{aligned}$$
Then $\nu$ satisfies a logarithmic Sobolev inequality and $C_{LS}(\nu)\leq A+ (B+2)C_P(\nu)$.
Zimmermann proves in [@Zim] that holds with $b = d/(8\delta^2) + R^2/(32\delta^4)$ and $c = \frac{1}{64\delta^4}$ for the function $W(x)=\exp{{{\left(\frac{1}{64\delta^4}\right)}}}$. Using the bound $K\leq R^2/\delta^4$ and choosing ${\varepsilon}=2/K$, this proves that, for $\delta\leq R$, thanks to the bound on the Poincaré constant, $$C_{LS}(\mu \star \gamma_\delta) \leq {{{\left(K_1 d + K_2
\frac{R^2}{\delta^2}\right)}}} R^2 \exp{{{\left(4\frac{R^2}{\delta^2}\right)}}}$$ for some universal constants $K_1$, $K_2$, which is the general bound announced in Theorem \[thm:logSob\].
A bound for uniform discrete measures
-------------------------------------
Suppose in this section that $\mu$ is a uniform probability measure on $N$ points in $B_d(0,R)$: $$\mu = \frac{1}{N} \sum_{i=1}^N \delta_{x_i}.$$ The distribution of $S = X+Z_\delta$ is a mixture of $N$ Gaussian laws with respective means $x_i$ and common covariance matrix $\delta^2I_d$. Poincaré and logarithmic Sobolev inequalities for mixtures of two measures have been studied by Chafaï and Malrieu in [@CM10]; Schlichting and Menz [@Sch; @MS14] have used and generalized their results to prove Eyring-Kramers formulæ. The decomposition of the variance used in Section \[sec:Poincare\] has the following analogue for entropies: $$\label{eq:dec_entropy}
\operatorname{Ent}_{\mu\star\gamma_\delta}{{{\left(f^2\right)}}} = \int_{\dR^d} \operatorname{Ent}_{\gamma_{x,\delta}}{{{\left(f^2\right)}}}
d\mu(x) + \operatorname{Ent}_\mu {{{\left(x\mapsto \int_{\dR^d} f^2 d\gamma_{x,\delta}\right)}}}.$$ To bound the second term, we use the following result, that is essentially a consequence of the discrete logarithmic Sobolev inequality for the complete graph proved by Diaconis and Saloff-Coste in [@DSC96].
\[Upper bound for the entropy when $\mu$ is discrete, [@Sch]\] \[th:Sch\] Let $\mu = \sum_{i=1}^N Z_i \mu_i$ be a finite mixture of measures. Let $Z^\star = \min_{1\leq i\leq N} (Z_i)$. Then for any $f$, $$\operatorname{Ent}_\mu{{{\left(i \mapsto \int_{\dR^d} f^2d\mu_i\right)}}} \leq
\frac{1}{\Lambda(Z_\star, 1 - Z_\star)}
{{{\left( \sum_{i=1}^N Z_i \operatorname{Var}_{\mu_i}(f) + \operatorname{Var}_\mu{{{\left(i\mapsto
\int_{\dR^d} f d\mu_i\right)}}}\right)}}},$$ where $\Lambda(p,q) = (p-q)/(\log p - \log q)$.
This follows from [@Sch Corollary 2.18], using the result from [@DSC96] instead of the alternate [@Sch Lemma 2.13].
Coming back to the decomposition , we can use the Gaussian logarithmic Sobolev inequality on the first term and Theorem \[th:Sch\] on the second term to get:
$$\operatorname{Ent}_{\mu\star\gamma_\delta}(f^2) \leq 2\delta^2 \int_{\dR^d}
{{{\left\vert\nabla f\right\vert}}}^2 d\mu(x) + \frac{1}{\Lambda(1/N, (N-1)/N)}
{{{\left( \frac{1}{N} \sum_{i=1}^N \operatorname{Var}_{\gamma_{\delta,x_i}}(f) +
\operatorname{Var}_\mu{{{\left(i\mapsto \int_{\dR^d} f d\gamma_{\delta,x_i}\right)}}}\right)}}}.$$ The last bracket is the variance $\operatorname{Var}_{\mu\star\gamma_\delta}(f)$, which is bounded thanks to the Poincaré inequality. Since $ \frac{1}{\Lambda(p,1-p)} \leq \frac{ \log(1/p)}{1-2p} $, we finally get $$C_{LS}(\mu\star\gamma_\delta)\leq 2\delta^2 + 3 \log(N)
\delta^2\exp(4R^2/\delta^2).$$
Dimension free transport-entropy inequality for the l4 norm
-----------------------------------------------------------
We now adapt the arguments of Section \[sec:Poincare\] to prove that the measure $\mu\star\gamma_\delta$ satisfies a transport-entropy inequality with a constant depending only on $R$ and $\delta$. It is more convenient in this section to state and prove all intermediate results for $\delta=1$. In the final result we come back to the general case by an immediate scaling argument.
The first step is to establish a weighted version of the Poincaré inequality.
\[lem:wP-Gauss\] For all $x \in \dR^d$, the Gaussian measure $\gamma_{x,1}$ satisfies the following weighted Poincaré inequality: for all $\mathcal{C}^1$ function $f$, $$\operatorname{Var}_{\gamma_{x,1}}(f) \leq c (1+|x|^2) \int_{\dR^d}\sum_{i=1}^d
\frac{1}{1+u_i^2} (\partial_if(u))^2\,d\gamma_{x,1}(u)\,,$$ where $c$ is a positive universal constant.
Let us first establish the result for the standard Gaussian distribution $\gamma= \cN(0,1)$ in dimension $d=1$. According to the well known Muckenhoupt criterion for Hardy type inequalities (see *e.g.* [@Log-Sob Theorem 6.2.1]), the inequality $$\int_0^\infty {{{\left(f(u)-f(0)\right)}}}^2\,d\gamma(u) \leq c \int_0^\infty
\frac{1}{1+u^2} f'(u)^2\, d\gamma(u)$$ holds for all $\mathcal{C}^1$ function $f: [0,\infty) \to \dR$, with the constant $$c = \sup_{y\geq 0} \int_y^\infty e^{-u^2/2} \,du\int_0^y
(1+u^2)e^{u^2/2}\,du <\infty.$$ Similarly, for any $\mathcal{C}^1$ function $f$ on $(-\infty,0]$, it holds $$\int_{-\infty}^0 {{{\left(f(u)-f(0)\right)}}}^2\,d\gamma(u) \leq c
\int_{-\infty}^0 \frac{1}{1+u^2} f'(u)^2\, d\gamma(u)\,.$$ Therefore, if $f$ is now $\mathcal{C}^1$ function on $\dR$, one has $$\operatorname{Var}_{\gamma}(f) \leq \int_{\dR} (f(u)-f(0))^2\,d\gamma(u) \leq c
\int_{\dR} \frac{1}{1+u^2} f'(u)^2\, d\gamma(u)\,.$$ Applying this inequality to $f(u) = g(x + u)$, $u \in \dR$, yields $$\operatorname{Var}_{\gamma_{x,1}}(g) \leq c \int_{\dR} \frac{1}{1+(v-x)^2}
g'(v)^2\, d\gamma_{x,1}(v)\,.$$ Since $1+v^2 \leq 1+2(v-x)^2 + 2x^2 \leq 2(1+x^2) (1 + (x-v)^2)$, the claim holds for the Gaussian measure $\gamma_{x,1}$ in dimension $1$.
To prove the general case, just remark that, for any $x\in \dR^d$, $\gamma_{x,1}$ is the product of the (one dimensional) measures $\gamma_{1,x_i}$. The classical tensorization property for Poincaré–type inequalities yields $$\operatorname{Var}_{\gamma_{x,1}}(g) \leq 2c \max_{i}(1+x_i^2) \int_{\dR}
\sum_{i=1}^d \frac{1}{1+v_i^2} (\partial_i g(v))^2\,
d\gamma_{x,1}(v)\,,$$ which completes the proof.
This result extends to mixture of Gaussian measures.
Let $\mu$ be a probability measure on $\dR^d$ supported in $B_d(0,R)$. The probability $\mu \star \gamma_\delta$ satisfies the following weighted Poincaré inequality: for all $\mathcal{C}^1$ function $f$ on $\dR^d$, $$\label{eq:wPoinc}
\operatorname{Var}_{\mu\star \gamma_1} (f)
\leq C(R) \int_{\dR^d} \sum_{i=1}^d \frac{1}{1+u_i^2}
(\partial_i f(u))^2\,d(\mu\star\gamma_1)(u)\,,$$ with $C(R) =c (1+R^2)e^{4R^2}$ for some universal constant $c$.
According to Lemma \[lem:wP-Gauss\], for all $x \in \dR^d$ such that $|x|\leq R$, it holds $$\operatorname{Var}_{\gamma_{x,1}}(f) \leq c (1+R^2)\int_{\dR} \sum_{i=1}^d
\frac{1}{1+u_i^2} (\partial_i f(u))^2\, d\gamma_{x,1}(u)$$ for all $\mathcal{C}^1$ function $f$ on $\dR^d$. Inserting these weighted Poincaré inequalities into the proof given in Section \[sec:Poincare\] immediately yields the desired bound.
We now arrive at a first transportation-entropy inequality.
\[thm:complicatedTransport\] Let $\mu$ be a probability measure on $\dR^d$ having its support in $B_d(0,R)$. The probability $\mu \star \gamma_1$ satisfies the following transport-entropy inequality: for any probability measure $\nu$ on $\dR^d$, $$\cT_k(\nu,\mu\star\gamma_1) \leq c'(1+R^2)\exp(4R^2) H(\nu |\mu\star
\gamma_1)\,,$$ where $c'$ is a universal constant and $\cT_k$ is the optimal transport cost related to the cost function $$k(x,y) = \min{{{\left(|x-y|^2 ; |x-y|\right)}}} + \min{{{\left(\|x-y\|_4^4,
\|x-y\|_4^2\right)}}},\qquad \forall x,y \in \dR^d\,.$$
Before proving this result, let us show how to deduce Theorem \[thm:transport\] as a corollary. The Euclidean and $\ell^4$ norms on $\dR^d$ satisfy: $$\forall z\in \dR^d, \quad \|z\|_4 \leq {{{\left\vertz\right\vert}}} \leq d^{1/4} \|z\|_4\,.$$ This gives the following lower bound on the cost $k$: $$\begin{aligned}
k(x,y) &= \min{{{\left(|x-y|^2 ; |x-y|\right)}}} + \min{{{\left(\|x-y\|_4^4, \|x-y\|_4^2\right)}}} \\
&\geq \min{{{\left(\|x-y\|_4^2 ; \|x-y\|_4\right)}}} + \min{{{\left(\|x-y\|_4^4, \|x-y\|_4^2\right)}}} \\
&\geq \|x-y\|_4^2.\end{aligned}$$ By Theorem \[thm:complicatedTransport\] we get $$\begin{aligned}
\cT_{2,4}(\nu,\mu\star\gamma_1)
\leq \cT_k(\nu,\mu\star\gamma_1)
\leq c'{{{\left(1+R^2\right)}}}\exp{{{\left(4R^2\right)}}} H(\nu | \mu\star\gamma_1); \end{aligned}$$ where we recall that $\cT_{2,4}$ is the transportation cost associated to $(x,y)\mapsto \|x-y\|^4$. The inequality for a general $\delta$ follows by a simple scaling argument. The inequality for the Euclidean cost $\cT_2$ is proved in the same way, by bounding $k(x,y)$ from below by $d^{-1/2} {{{\left\vertx-y\right\vert}}}^2$. This concludes the proof of Theorem \[thm:transport\].
We proceed in two steps.
#### 1. A transport-entropy inequality with an intricate cost.
Let us define three functions $\alpha$, $\omega$ and $T$ by $$\begin{aligned}
\forall u \in \dR, \quad
\omega(u) &= \mathrm{sign} (u) \left({{{\left\vertu\right\vert}}}+\frac{u^2}{2}\right)\,; \\
\forall u \in \dR, \quad
\alpha(u) &= \min(u^2 ; |u|)\,; \\
\forall x\in \dR^d, \quad
T(x) &= (\omega(x_1),\ldots,\omega(x_d))\,. \end{aligned}$$
According to [@Goz10 Theorem 4.6], the weighted Poincaré inequality implies (and is actually equivalent to) the following transport cost inequality: for all probability measure $\nu$ on $\dR^d$, $$\cT_{\tilde{k}}(\nu,\mu\star \gamma_1) \leq H(\nu|\mu\star
\gamma_\delta),$$ where the cost function $\tilde{k}$ is defined by $$\label{eq:cost}
\tilde{k}(x,y) = \alpha \left(\frac{1}{D} {{{\left\vertT(x) - T(y)\right\vert}}} \right)\,,
\qquad \forall x,y \in \dR^d$$ and where $D = c'' \sqrt{C(R)}$ for some universal constant $c''$.
For the sake of completeness, let us give the short proof of the implication we need. Let us begin by showing that the measure $\tilde{\mu} := T_\# (\mu \star \gamma_{1})$ satisfies the usual Poincaré inequality with the constant $2C(R).$ Indeed, if $f$ is a $\mathcal{C}^1$ function, applying the weighted Poincaré inequality to $g=f\circ T$ and using the elementary bound $(\omega'(v))^2 \leq 2(1+ v^2)$ yields: $$\begin{aligned}
\operatorname{Var}_{\tilde{\mu}}(f)
&\leq C(R) \int
\sum_{i=1}^d \frac{1}{1+v_i^2} \omega'(v_i)^2 (\partial_i f)^2(T(v))\, d(\mu\star \gamma_1)(v) \\
&\leq 2C(R) \int {{{\left\vert\nabla f\right\vert}}}^2(u) d\tilde{\mu}(u). \end{aligned}$$ According to a well known result by Bobkov, Gentil and Ledoux [@BGL01 Corollary 5.1] showing the equivalence between the Poincaré inequality and a transport inequality involving a quadratic-linear cost, the probability $\tilde{\mu}$ satisfies the following: for any probability measure $\nu$ on $\dR^d$, $$\cT_{\rho}(\nu,\tilde{\mu}) \leq H(\nu | \tilde{\mu}),$$ where the cost function $\rho : \dR^d\times \dR^d \to \dR^+$ is defined by $$\rho(x,y) = \alpha\left( \frac{1}{D} {{{\left\vertx-y\right\vert}}}\right),\qquad x,y \in
\dR^d\,,$$ where $D=c''\sqrt{C(R)}$ for some universal constant $c''$. Let $\nu$ be a probability measure on $\dR^d$ and let $(\tilde{X},\tilde{Y})$ be an optimal coupling between $\tilde{\nu}:= T_\# \nu$ and $\tilde{\mu}$ (for the transport cost $\cT_\rho$) and denote by $X = T^{-1}(\tilde{X})$ and $Y = T^{-1}(\tilde{Y})$. Then $(X,Y)$ is a coupling between $\nu$ and $\mu\star \gamma_1$ and it holds $${\dE_{}\left[\tilde{k}(X,Y)\right]}
= {\dE_{}\left[\rho (T(X),T(Y))\right]}
= {\dE_{}\left[\rho (\tilde{X} ,\tilde{Y})\right]}
= \mathcal{T}_\rho(\tilde{\nu},\tilde{\mu}) \leq H(\tilde{\nu} | \tilde{\mu})
= H(\nu | \mu\star \gamma_1),$$ where the last equality comes from the fact that if $\nu \ll \mu\star \gamma_1$, then $\tilde{\nu} \ll \tilde{\mu}$ with $$\frac{d\tilde{\nu}}{d\tilde{\mu}}(u)
= \frac{d\nu}{d(\mu\star \gamma_1)} {{{\left(T^{-1}(u)\right)}}},
\qquad \forall u\in \dR^d.$$ This concludes the first step.
#### A lower bound on the cost function $\tilde{k}$.
We now bound $\tilde{k}(x,y)$ from below by the more convenient cost function $k(x,y)$. According to [@Goz10 Lemma 2.6], $|\omega(u)-\omega(v)| \geq \omega (|u-v|/2)$, for all $u,v \in
\dR$. Therefore, for all $x$, $y$ in $\dR^d$: $$\begin{aligned}
{{{\left\vertT(x) - T(y)\right\vert}}}^2
&= \sum_i {{{\left\vert\omega(x_i) - \omega(y_i)\right\vert}}}^2 \\
&\geq \sum_i \omega{{{\left( \frac{{{{\left\vertx_i - y_i\right\vert}}}}{2}\right)}}}^2 \\
&= \sum_i {{{\left( \frac{1}{2} {{{\left\vertx_i - y_i\right\vert}}} + \frac{1}{8} {{{\left\vertx_i - y_i\right\vert}}}^2\right)}}}^2 \\
&\geq \frac{1}{4} \sum_i {{{\left\vertx_i - y_i\right\vert}}}^2 + \frac{1}{64} \sum_i {{{\left\vertx_i - y_i\right\vert}}}^4 \\
&\geq \frac{1}{32}{{{\left( \frac{1}{2} {{{\left\vertx-y\right\vert}}}^2 + \frac{1}{2} \|x-y\|_4^4\right)}}}.\end{aligned}$$
Using the inequality $\alpha(au) \geq \alpha(a)\alpha(u)$ for all $a,u \in \dR$ ([@Goz10 Lemma 2.6]) and the concavity of the function $u\mapsto \alpha(\sqrt{u})$, $u\in \dR^+$, this leads to the following bound on the cost function $\tilde{k}$: $$\begin{aligned}
\tilde{k}(x,y)
&\geq \alpha{{{\left( \frac{1}{D\sqrt{32}}
{{{\left( \frac{1}{2} {{{\left\vertx-y\right\vert}}}^2 + \frac{1}{2} \|x-y\|_4^4\right)}}}^{1/2}\right)}}} \\
&\geq \frac{1}{2}\alpha{{{\left(\frac{1}{D\sqrt{32}}\right)}}} {{{\left(\alpha( |x-y|)
+ \alpha(\|x-y\|_4^2)\right)}}}\,,\end{aligned}$$ Finally, it is easy to check that $\alpha{{{\left(\frac{1}{D\sqrt{32}}\right)}}} \geq \frac{c'''}{C(R)} $ for some universal constant $c'''$, which completes the proof.
If one could improve the conclusion in the result by Bobkov, Gentil, Ledoux and conclude that $\tilde{\mu}$ satisfies the transport inequality with the cost function $$(x,y) \mapsto \sum_{i=1}^d \alpha\left( \frac{1}{D}
{{{\left\vertx_i-y_i\right\vert}}}\right)$$ instead of $\rho$, then one would conclude that $\mu$ satisfies Talagrand’s inequality, with respect to the Euclidean norm, with a dimension free constant.
Special cases and extensions {#sec:partial}
============================
Spherically symmetric measures
------------------------------
We prove in this section the following claim of Theorem \[thm:partial\]:
\[thm:symmetry\] If $\mu$ is a spherically symmetric measure with support in $B_d(0,R)$, then $\mu\star\gamma_\delta$ satisfies a logarithmic Sobolev inequality and $$C_{LS}(\mu \star \gamma_\delta) \leq 4\delta^2\exp{{{\left(\frac{8}{\pi}
\frac{R^2}{\delta^2}\right)}}}.$$
Let us recall that $\mu\star\gamma_\delta$ is the law of the random variable $S=X+\delta Z$. By assumption, the law $\mu$ of $X$ is spherically symmetric, that is, invariant by any vectorial rotation of $\dR^d$. Since $Z$ has the same invariance, this implies that the density $p(z)$ of $S$ only depends on the norm of $z$, thus we can write: $$p(z)=p(|z|e_1) =\int_{\dR^d}\! \frac{1}{(2\pi
\delta^2)^{d/2}}\exp{{{\left(-\frac{1}{2\delta^2}
{{{\left({{{\left(|z|-x_1\right)}}}^2+\sum_{i=2}^dx_i^2\right)}}}\right)}}} d\mu(x_1,x_2,\ldots,x_d)\,.$$ Denoting, for all $r\in\dR$, $$\hat{p}_\delta(r)=\int_{\dR}\! \frac{1}{(2\pi
\delta^2)^{1/2}}\exp{{{\left(-\frac{{{{\left(|z|-x_1\right)}}}^2}{2\delta^2}\right)}}}
d\hat\mu_1(x_1)$$ the density of the convolution of $\gamma_\delta$ with the first marginal $\hat\mu_1$ of the measure $$\frac{1}{(2\pi
\delta^2)^{(d-1)/2}}\exp{{{\left(-\frac{1}{2\delta^2}\sum_{i=2}^dx_i^2\right)}}}
d\mu(x_1,x_2,\ldots,x_d)\,,$$ one has $p(z)=\hat{p}_\delta(|z|)$.
Since the one-dimensional measure $\hat{\mu}_1$ is supported in the interval $[-R,R]$, the method from Section \[sec:Miclo\] apply. Using Lemma \[lem:Royer\], with $\sigma=2Ra_1$, we obtain a decomposition $$-\log(\hat{p}_\delta(r))=w_\sigma(r)+w_b(r)\,,$$ where $w_\sigma\,:\,\dR\rightarrow\dR$ is $1/(2\delta^2)$-convex and $w_b\,:\,\dR\rightarrow\dR$ is bounded by $2(Ra_1/\delta)^2$.
Since the measure $\hat{\mu}_1$ is symmetric, the function $\hat{p}_\delta$ is even, so that $w_\sigma$ and $w_b$ constructed in the proof of Lemma \[lem:Royer\] are even too.
This entails a decomposition of $p$ on $\dR^d$ as a sum $$-\log(p(z))=W_\sigma(z)+W_b(r)$$ by taking $W_\sigma(z)=w_\sigma(|z|)$ and $W_b(z)=w_b(|z|)$. The function $W_b$ is of course bounded by $2(Ra_1/\delta)^2$. We prove in Lemma \[lem:convexite\] below that $W_c$ is convex. The conclusion follows by the same reasoning as in Section \[sec:Miclo\].
\[lem:convexite\] Let $w:\,\dR\rightarrow\dR$ be a $\cC^2$, even, and $\rho$-convex function. Then $W:\,\dR^d\rightarrow\dR$ defined by $W(z)=w(|z|)$ for all $z\in\dR^d$ is also $\cC^2$ and $\rho$-convex.
Let us denote $N(z)=|z|$. For any $z\neq0$, one computes $$\begin{aligned}
\nabla N(z)&=\frac1{|z|}z\\
\operatorname{Hess}N(z)&=\frac1{|z|}{{{\left(I_d-\frac1{|z|^2}zz^T\right)}}}
\end{aligned}$$ By composition with $w$, one deduces, for any $z\neq 0$, $$\begin{aligned}
\nabla W(z)&=\frac{w'(|z|)}{|z|}z\\
\operatorname{Hess}W(z)&=\frac{w''(|z|)}{|z|^2}zz^T+\frac{w'(|z|)}{|z|}
{{{\left(I_d-\frac1{|z|^2}zz^T\right)}}}\,.
\end{aligned}$$ These two quantities converge respectively to $0$ and $w''(0)I_d$ when $z\to 0$. By a classical continuation lemma, this implies that $W$ is $\cC^2$ with $\nabla W(0) = 0$ and $\operatorname{Hess}W(0) = w''(0)I_d$.
By assumption, $w''(|z|)\geq \rho$ for any $z\in\dR^d$. Furthermore, for any $z\neq0$, $\frac{w'(|z|)}{|z|}\geq \rho$ (since the assumptions imply that $0$ is a minimum of $w$). Finally, noting that $zz^T$ and ${{{\left(I_d-\frac1{|z|^2}zz^T\right)}}}$ are the orthogonal projections on $\operatorname{Vect}(z)$ and $z^\perp$, one gets that $\operatorname{Hess}W(z)\geq \rho I_d$ for any $z\in\dR^d$.
Dimension free log-Sobolev for R/sqrt(2)<delta<R
------------------------------------------------------
The first item of Theorem \[thm:logSob\] states that for $\delta>R$, the probability measure $\mu\star \gamma_\delta$ satisfies a logarithmic Sobolev inequality with an explicit, dimension free, constant. In this section, we improve on this result by proving the first point of Theorem \[thm:partial\].
The proof of the following result relies on the connections between functional inequalities and concentration of measure inequalities. The well known Herbst argument shows that the logarithmic Sobolev inequality implies a Gaussian concentration of measure phenomenon. More precisely, if $\mu$ is a probability measure on $\dR^d$ satisfying the logarithmic Sobolev inequality with a constant $C_{LS}$, then for any $1$-Lipschitz function $f:\dR^d \to \dR$, it holds $$\mu \left( f \geq m + t\right) \leq e^{-t^2/{C_{LS}}},\qquad \forall t
\geq0,$$ where $m = \int f\,d\mu$ (see *e.g.* Theorem 5.3 of [@Ledoux-book]). On the other hand, a recent result by E. Milman [@Mil10] shows that conversely under some curvature assumptions a sufficiently strong Gaussian concentration of measure inequality implies back the logarithmic Sobolev inequality. It appears that in the range of parameters $R /\sqrt{2}<\delta<R$ the measure $\mu\star \gamma_\delta$ is sufficiently concentrated to apply Milman’s result.
\[thm:deltaMoyen\] Suppose that $R /\sqrt{2}<\delta<R$, then $\mu\star \gamma_\delta$ satisfies a logarithmic Sobolev inequality with a constant depending only on $R$ and $\delta$ and not on $d$.
Let us examine the concentration properties of $X+\delta Z$ where $X$ and $Z$ are independent random variables with respective laws $\mu$ and $\cN_d(0,I_d)$. If $f : \dR^d \to \dR$ is a $1$-Lipschitz function, then denoting by $m = {\dE_{}\left[f(X+\delta Z)\right]}$, it holds for any $t \geq0$ $$\begin{aligned}
{\dP_{}\left[ f(X + \delta Z) \geq m + t\right]}
& = {\dE_{X}\left[ {\dP_{}\left[ f(X + \delta Z) \geq m + t \,|\, X\right]}\right]} \\
& \leq {\dE_{X}\left[ \exp\left(-\frac{1}{2\delta^2}
\left[t+m-{\dE_{Z}\left[f(X+\delta Z)\right]}\right]_+^2\right)\right]}
\end{aligned}$$ where the second inequality follows from the concentration inequality satisfied by $\delta Z$ (which is for instance a consequence of the fact that $\gamma_\delta$ satisfies the logarithmic Sobolev inequality with the constant $2\delta^2$). Now, for any $x\in B_d(0,R)$, $${{{\left\vertm - {\dE_{Z}\left[f(x+\delta Z)\right]}\right\vert}}} = \left| {\dE_{X}\left[ {\dE_{Z}\left[ f(X+\delta
Z) - f(x+ \delta Z) \right]}\right]} \right| \leq {\dE_{X}\left[ {{{\left\vertX-x\right\vert}}}\right]} \leq
2R.$$ Therefore ${\dE_{Z}\left[f(X+\delta Z)\right]} \leq 2R + m$ almost surely, hence $${\dP_{}\left[ f(X + \delta Z) \geq m + t\right]} \leq
\exp\left(-\frac{1}{2\delta^2} \left[t-2R\right]_+^2\right).$$ In particular, for any $0<{\varepsilon}<1$, it holds $${\dP_{}\left[f(X + \delta Z) \geq m + t\right]} \leq
\exp\left(-\frac{{\varepsilon}}{2\delta^2}t^2 \right), \qquad \forall t >
\frac{2R}{1-\sqrt{{\varepsilon}}} := t_{\varepsilon}.$$
On the other hand, the density of the law of $X+\delta Z$ is of the form $e^{-V_\delta}$, with a function $V_\delta$ such that $\mathrm{Hess}\, V_{\delta} \geq \frac{1}{\delta^2} -
\frac{R^2}{\delta^4} = - \kappa_\delta$. In this range of parameters, $\kappa_\delta >0$. According to Theorem 1.2 of [@Mil10], as soon as $\frac{{\varepsilon}}{2\delta^2} \geq \frac{1}{2} \kappa_\delta$ (which means that $R/\delta <\sqrt{1+{\varepsilon}}$), the probability measure $\mu$ satisfies a Gaussian isoperimetric inequality, which in turn implies the logarithmic Sobolev inequality with a constant depending only on the parameters ${\varepsilon}, R, \delta$.
Dimension free log-Sobolev for log-convex functions {#sec:Maurey}
---------------------------------------------------
Recall the following results by Maurey.
Let $X$ be a bounded random variable such that ${{{\left\vertX\right\vert}}} \leq R$ a.s. Then $X$ satisfies the so called convex $\tau$-property : $${\dE_{}\left[e^{Q_{4R^2}f(X)}\right]} {\dE_{}\left[e^{-f(X)}\right]} \leq 1,$$ for any convex function $f : \dR^d \to \dR$, where $Q_sf(x) = \inf_{y\in \dR^d}{{{\left\{ f(y) + \frac{{{{\left\vertx-y\right\vert}}}^2}{4s} \right\}}}}$, $s>0.$
On the other hand, the Gaussian random variable $\delta Z$ with law $\cN_d(0,\delta I_d)$ satisfies the following $\tau$-property $${\dE_{}\left[e^{Q_{\delta^2} f (\delta Z)}\right]} {\dE_{}\left[e^{-f(\delta Z)}\right]} \leq 1,$$ for any function $f: \dR^d \to \dR$ ([@Ma91 Theorem 2]).
By the tensorization property of the convex $\tau$-property ([@Ma91]), one concludes that $(X,\delta Z)$ satisfies the following $\tau$-property $${\dE_{}\left[e^{\tilde{Q}f (X,\delta Z)}\right]} {\dE_{}\left[e^{-f(X,\delta Z)}\right]} \leq 1,$$ for any convex function $f : \dR^d\times \dR^d \to \dR$, where $$\tilde{Q}f(x_1,x_2) = \inf_{(y_1,y_2) \in \dR^d \times \dR^d}
\left\{f(y_1,y_2) + \frac{1}{16R^2} {{{\left\vertx_1-y_1\right\vert}}}^2 +
\frac{1}{4\delta^2} {{{\left\vertx_2-y_2\right\vert}}}^2\right\}.$$ In particular, applying the inequality above to $f(x_1,x_2) = g(x_1+x_2)$, and using the fact that $$\inf_{y_1+y_2 =y} \left\{ \frac{1}{16R^2} {{{\left\vertx_1-y_1\right\vert}}}^2 +
\frac{1}{4\delta^2} {{{\left\vertx_2-y_2\right\vert}}}^2 \right\} =
\frac{1}{4C(\delta,R)}{{{\left\vertx_1+x_2 - y\right\vert}}}^2,$$ with $C(\delta,R) = \delta^2 + 4R^2$, one concludes that $X+ \delta Z$ satisfies $${\dE_{}\left[e^{Q_C g (X+\delta Z)}\right]}{\dE_{}\left[e^{-g(X+\delta Z)}\right]} \leq 1,$$ for any convex function $g: \dR^d \to \dR.$
According to [@GRST14], this inequality is equivalent to the following transport type inequality $$\overline{\mathcal{T}}_2(\nu_1,\nu_2) \leq C(\delta,R)
\left(H(\nu_1|\mu\star\gamma_\delta) +
H(\nu_2|\mu\star\gamma_\delta)\right),$$ for all probability measures $\nu_1,\nu_2$ on $\dR^d$, where $H(\,\cdot\,|\mu\star\gamma_\delta)$ denotes the relative entropy functional and $$\overline{\mathcal{T}}_2(\nu_1,\nu_2) = \inf_{X_1 \sim \nu_1,\ X_2
\sim \nu_2} {\dE_{}\left[\ |X_1 -{\dE_{}\left[X_2 |X_1\right]}|^2 \ \right]}.$$ It is also shown in [@GRST14] that this transport inequality implies the following logarithmic Sobolev inequality $$\mathrm{Ent}_{\mu\star\gamma_\delta} (e^f) \leq 8(\delta^2 + 4 R^2)
\int |\nabla f |^2 e^f \,d\mu\star\gamma_\delta,$$ for any *convex* function $f: \dR^d \to\dR$. This proves the fourth item of Theorem \[thm:partial\].
Jean-Baptiste <span style="font-variant:small-caps;">Bardet</span>, e-mail: `jean-baptiste.bardet(AT)univ-rouen.fr`
<span style="font-variant:small-caps;">LMRS, Université de Rouen, Avenue de l’Université, BP 12, Technopôle du Madrillet, 76801 Saint-Étienne-du-Rouvray, France.</span>
Nathaël <span style="font-variant:small-caps;">Gozlan</span>, e-mail: `natael.gozlan(AT)u-pem.fr`
<span style="font-variant:small-caps;">LAMA UMR 8050, CNRS-Université-Paris-Est-Marne-La-Vallée, 5, boulevard Descartes, Cité Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France.</span>
Florent <span style="font-variant:small-caps;">Malrieu</span>, e-mail: `florent.malrieu(AT)univ-tours.fr`
<span style="font-variant:small-caps;">Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964), Université François-Rabelais, Parc de Grandmont, 37200 Tours, France.</span>
Pierre-André <span style="font-variant:small-caps;">Zitt</span>, e-mail: `pierre-andre.zitt(AT)u-pem.fr`
<span style="font-variant:small-caps;">LAMA UMR 8050, CNRS-Université-Paris-Est-Marne-La-Vallée, 5, boulevard Descartes, Cité Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France.</span>
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**On the Bimodal Distribution of Gamma-Ray Bursts**
Shude Mao, Ramesh Narayan and Tsvi Piran[^1]
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138
*Received .........;*
ABSTRACT
Kouveliotou et al. (1993) recently confirmed that gamma-ray bursts are bimodal in duration. In this paper we compute the statistical properties of the short ($\le 2$ s) and long ($>2$ s) bursts using a method of analysis that makes no assumption regarding the location of the bursts, whether in the Galaxy or at a cosmological distance. We find the 64 ms channel on BATSE to be more sensitive to short bursts and the 1024 ms channel is more sensitive to long bursts. We show that all the currently available data are consistent with the simple hypothesis that both short and long bursts have the same spatial distribution and that within each population the sources are standard candles. The rate of short bursts is $\sim 0.4$ of the rate of long bursts. Although the durations of short and long gamma-ray bursts span several orders of magnitude and the total energy of a typical short burst is smaller than that of a typical long burst by a factor of $\sim 20$, surprisingly the peak luminosities of the two kinds of bursts are equal to within a factor of $\sim 2$.
Introduction
============
The Burst and Transient Source Experiment (BATSE) on board the Compton Gamma-Ray Observatory has demonstrated that the distribution of gamma-ray bursts is isotropic over the sky and bound in the radial direction (Meegan et al. 1992). This strongly indicates that the sources of the bursts are located either at cosmological distances (Paczyński 1991, Dermer 1992, Mao & Paczyński 1992, Piran 1992) or in an extended Galactic halo (Li & Dermer 1992, Duncan, Li, & Thompson 1992). Many models have been proposed for the cosmological scenario (e.g., Eichler, et al., 1989, Piran, Narayan, & Shemi, 1992, Narayan, Paczyński, & Piran 1992, Rees & Mészáros 1992, Usov 1992, Woosley 1993) as well as for the extended halo scenario (Fabian & Podsiadlowski 1993). For a recent review on gamma-ray bursts, see Paczyński (1992).
Recently Kouveliotou et al. (1993) showed that the distribution of durations of gamma-ray bursts is bimodal. In terms of the parameter $\deltat$, which is the time interval during which the integrated counts of a burst go from 5% to 95% of the total integrated counts, the bursts seem to separate cleanly into two distinct groups, with the transition occurring around $\deltat \approx 2$ s. This confirms similar indications from earlier experiments (Cline & Desai 1974, Norris et al. 1984, Dezalay et al. 1992, Hurley 1992), and is the first compelling evidence for distinct sub-classes of gamma-ray bursts. (We refer here to classical gamma-ray bursts, which represent the bulk of the bursts, and do not discuss the class of soft repeaters.)
An obvious and natural question that arises from this result is: what is the relation between the two kinds of bursts? One possibility is that they represent two distinct types of sources. For instance, it is even conceivable that one class of sources is cosmological and that the other is in the Galactic halo. Alternatively, both types of bursts may come from a common source, and the differences may arise merely from variations in the initial conditions of the sources prior to the bursts, or changes in the environment or the viewing angle. We attempt to shed some light on this question by carrying out a statistical comparison of the properties of the two kinds of bursts. We describe the detector channels on BATSE and some selection effects in §2, present the main data analysis in §3, and discuss the implications of the results in §4.
Detector Channels and Selection Effects
=======================================
The burst catalog available in the public domain contains a list of all gamma-ray bursts that triggered the BATSE detectors between April 1991 and March 1992 (Fishman et al. 1993). The catalog provides the angular positions for 260 bursts, the ratio $\cmax/\cmin$ for 241 bursts, where $\cmax$ is the maximum count rate and $\cmin$ is the detection threshold, the fluences and peak count rates for 260 bursts, and durations for 220 bursts.
The trigger mechanism and various selection effects of BATSE have been explained in detail in Fishman (1992). We will repeat the essentials here. The BATSE on-board software tests for bursts by comparing count rates on eight large-area detectors to the threshold levels corresponding to three separate time intervals: 64 ms, 256 ms, and 1024 ms. A burst trigger occurs if the count rate is above the threshold in two or more detectors simultaneously. The thresholds are set by command to a specified number of standard deviations above the background (nominally 5.5 $\sigma$), and the average background rate is computed every 17 s. Due to an unknown technical reason, many bursts have “undetermined” $\cmax/\cmin$ in the 256 ms channel. We therefore ignore this channel in what follows.
For a variety of reasons, BATSE has a variable background as a function of time, so that the detection threshold $\cmin$ does not remain constant. Moreover, there are periods corresponding to “overwrites” when the sensitivity of the detectors is greatly reduced. These effects make an analysis of the $\cmax /\cmin$ data somewhat difficult. In order to have a more uniform sample to work with, we select a constant threshold $\ccut$, and prune the data so as to include only those bursts which satisfy both of the following criteria:
1. $\cmin \le \ccut$,
2. $\cmax \ge \ccut$.
The resulting database corresponds to those bursts which would have been found by a detector that (i) had a constant threshold of $\ccut$, and (ii) was turned on at those times when the real detectors on BATSE had $\cmin \le \ccut$, and was turned off whenever the BATSE sensitivity was poorer than $\ccut$. By cutting the data in this fashion, we are guaranteed to have a sample of bursts with a constant detection threshold and a uniform selection bias. Of course, in the process we lose a few bursts, which causes loss of statistical accuracy. To minimize this, we select $\ccut$ such that the number of bursts retained in the database is maximized. This leads to the choice $\ccut=71$ counts for the 64 ms channel and $\ccut=286$ counts for the 1024 ms channel. Fortunately, only a few bursts are eliminated for these choices of $\ccut$; moreover, the excluded bursts are mostly those that are labeled “overwrites”, and would have been eliminated in any case.
Following Kouveliotou et al. (1993) we define “short bursts” as having $\deltat \le 2$ s, and “long bursts” as having $\deltat >
2$ s. Before going into the main analysis, which we discuss in the next section, we explain first some selection effects associated with the different sensitivities of the 64 ms and 1024 ms channels to the two kinds of bursts. Fig. 1 shows $\ccma$ in the 1024 ms channel vs. $\ccmb$ in the 64 ms channel for the short and long bursts. It is apparent that the 1024 ms channel is more sensitive to long bursts, while the 64 ms channel is more sensitive to short bursts. Both of these effects are quite natural, as we now show.
The noise in the background counts increases, generally, as the square root of the integration time. Therefore the noise is expected to be 4 times greater in the 1024 ms channel than in the 64ms channel. Now, if a burst has a broad luminosity maximum extending over a time interval greater than 1024 ms then there will be 16 times more signal counts in the 1024 ms channel than in the 64 ms channel, and the signal-to-noise ratio will be 4 times greater. Long bursts are likely to display this behavior. On the other hand, if a burst is extremely narrow, with a duration less than 64 ms, then the number of signal counts will be the same in both channels. The most extreme short bursts will correspond to this limit. Based on this argument, we see that the ratio $\ccma$ to $\ccmb$ for the two channels must satisfy $${1 \over 4} <{ \ccma
\over \ccmb } < 4, \eqno (1)$$ with short and long bursts tending towards the lower and upper limit respectively.
Comparing the ratio in eq (1) for the long burst bursts, we find $$R_1=\left\langle { \ccma \over \ccmb } \right\rangle_{\rm long} = 2.5
\pm 0.64 , \eqno (2)$$ where the error estimate reflects the width of the distribution. Since $R_1$ is not very different from the maximum value of 4, we conclude that, to a first approximation, the long bursts tend to have broad nearly constant luminosity profiles. For a more detailed description of the intensity statistics, we note that the maximum luminosity in the 64 ms channel is greater by a factor $\sim 4/2.5=1.6$ than the luminosity in the 1024 channel. This may be interpreted as evidence for a “fractal” behavior in the burst luminosity, $$\bar L (\Delta t)
\propto \Delta t^{-0.2},\eqno (3)$$ where $\bar L(\Delta t)$ represents the maximum luminosity of a burst as measured with a time constant of $\Delta t$.
In the case of the short bursts we find $$R_2=\left\langle {
\ccma \over \ccmb } \right\rangle_{\rm short} = 0.76 \pm 0.48, \eqno
(4)$$ which shows that the 64 ms channel is more sensitive than the 1024 ms channel. This is almost entirely because the longer channel dilutes the signal. As clear evidence of this effect we note that there is a strong correlation between the widths $\deltat$ of the short bursts and the quantity ${\ccma / \ccmb }$ (the correlation coefficient is 0.49).
Data Analysis
=============
In this section we carry out a comparison among various samples of bursts. We work primarily with three samples:
- sample $s_{64}$ consisting of 40 short bursts which were detected in the 64 ms channel,
- sample $l_{1024}$ consisting of 113 long bursts detected in the 1024 ms channel, and
- sample $l_{1024,64}$ consisting of 71 long bursts detected in both the 1024 ms and 64 ms channels.
In comparing different samples, the key observational data we use are the distributions of $\vmax$ of the samples, where $\vmax = (\cmax
/\ccut)^{-3/2}$ (Schmidt et al. 1988). Our analysis is based on a simple hypothesis, namely that all three populations of bursts have the same underlying spatial distribution, and that within each population the sources are standard candles. On this hypothesis, any apparent differences between the samples are just because they are viewed to different distances (or depths) on account of differences in the source luminosity and/or detector sensitivity. Our main result is that all the available data are consistent with this idea.
In our analysis, we make use of the fact that the brighter bursts agree with a homogeneous Euclidean distribution, which is characterized by cumulative counts varying linearly as $\vmax$ and a mean $\vvmax$ equal to 0.5. In contrast, the fainter bursts deviate significantly from such a distribution. Based on this observation, we make the following reasonable assumptions:
\(1) We assume that $\vvmax$ decreases monotonically with increasing depth of a sample. Therefore, if we compare two samples of bursts (with the same spatial distribution by hypothesis), we can say that the sample with the smaller value of $\vvmax$ corresponds to a greater depth or distance. Conversely, if two samples have the same value of $\vvmax$ we say that they correspond to the same distance. As a further check, when $\vvmax$ of two samples agree, we compare their full $\vmax$ distributions by means of the Kolmogorov-Smirnov test (K-S test), and thus investigate whether or not the two populations do indeed have a common spatial distribution.
\(2) If two samples have different $\vmax$ distributions and different mean $\vvmax$, but we suspect that they have intrinsically the same spatial distribution, then we can bring the two samples into agreement by increasing $\ccut$ for the deeper population. In effect, we artificially reduce the sensitivity of the detector corresponding to the deeper sample so that its sensitivity becomes equal to that of the shallower sample. We can think of this operation equivalently as reducing the luminosity of all bursts in the deep sample by a constant factor. Of course the procedure is meaningful only if it leads to agreement in the mean $\vvmax$ and also in the shapes of the $\vmax$ distributions, as discussed in (1) above.
A point that we would emphasize is that our method of analysis is virtually model-free and applies regardless of whether bursts are Galactic or cosmological.
To illustrate the method we consider the two samples of long bursts, $l_{1024}$ and $l_{1024,64}$. The discussion in §2 showed that the 1024 ms channel is more sensitive to long bursts than the 64 ms channel by a factor of 2.5 (eq 2). We therefore expect $l_{1024}$ to correspond to a deeper sample than $l_{1024,64}$. This is confirmed by the mean $\vvmax$ values, which are $0.29 \pm 0.027$ for $l_{1024}$ and $0.38 \pm 0.034$ for $l_{1024,64}$. Both groups deviate significantly from a uniform distribution in Euclidean space, but the deviation is much larger for $l_{1024}$. We can now artificially bring the two populations to the same distance by increasing $\ccut$ for the $l_{1024}$ sample by a factor of 2.5. On doing this we find that the value of $\vvmax$ for $l_{1024}$ becomes $0.37 \pm 0.035$, which is nearly equal to the $\vvmax$ of the $l_{1024,64}$ sample, exactly as expected. Furthermore, the two cumulative $V/V_{max}$ distributions agree very well with each other after this distance correction has been done. The K-S probability (for a worse fit than the one obtained) is 86%, which is excellent. These calculations show that the $l_{1024}$ and $l_{1024,64}$ samples do have the same spatial distribution, with the former being a deeper sample than the latter by a factor of $\sqrt {2.5}$ in luminosity distance. This is no surprise since the two samples have a large number of bursts in common, and moreover we know that there is a good correlation between the signals in the 1024 ms and 64 ms channels for long bursts (cf Fig. 1). The test however demonstrates the validity of the method.
We next proceed to the more interesting test of comparing the short and long bursts. The average $\langle V/V_{max} \rangle $ of the $s_{64}$ sample is $0.31 \pm 0.042$, as compared to $0.29 \pm 0.027$ for $l_{1024}$, which indicates that the two samples correspond to nearly the same distance. We now vary $\ccut$ of the $l_{1024}$ sample and $s_{64}$ sample individually so as to find the range of values over which the $\vvmax$ values of the two samples agree to within $\pm
1\sigma$ (see Fig. 2). We find that $$R_3 = \left({\ccutp \over\ccut}\right )_{l_{1024}}
\left({\ccut \over \ccutp}\right )_{s_{64}} =
1.4^{+1.3}_{-0.6}, \eqno (5)$$ where the value $R_3=1.4$ corresponds to the case when the two $\vvmax$ values are exactly equal.. As defined here, $\sqrt{R_3}$ represents the ratio of the limiting distances of the $l_{1024}$ and $s_{64}$ samples. We should mention in passing that there is a second region of good fit around $R_3\sim 6$, but the number of bursts that survive the cut for this comparison is so small that we do not find the solution convincing. Even the primary solution given in eq (5) suffers to some extent from the limited number of bursts in the two samples.
We now test whether or not the $s_{64}$ and $l_{1024}$ populations are really consistent with the same spatial distribution. For this we do a K-S test to compare the two distributions of $\vmax$ (see Fig. 2). We see that the test indicates fairly convincingly that the two populations do have the same spatial distribution. For instance, the K-S probability is 42% if we keep the two $\ccut$ values unchanged ($R_3=1$) and 50% if we multiply $\ccut$ for the 1024 ms channel by the optimum factor of 1.4 to obtain equality of $\vvmax$. To illustrate the quality of the agreement, we show in Fig. 3 the $\vmax$ distributions corresponding to the case when $R_3=1$. Note how much better the two distributions agree with each other than with the diagonal line which represents the homogeneous Euclidean model. The probability that either of the observed samples is drawn from the Euclidean distribution, is vanishingly small, $< 10^{-4}$ according to the K-S test.
Finally, we compare the $s_{64}$ and $l_{1024,64}$ samples. In analogy with eq (5) we define a corresponding ratio $R_4$ for this comparison, $$R_4\equiv \left({\ccutp \over\ccut}\right )_{s_{64}}
\left({\ccut \over \ccutp}\right )_{l_{1024,64}}. \eqno (6)$$ For consistency with the two previous comparisons, we expect $$R_4={R_1\over R_3}= 1.8^{+2}_{-1}.\eqno (7)$$ To check this, we change $\ccut$ for the $s_{64}$ sample from 71 counts to $\ccutp=1.8\times 71$ counts and compare the modified $s_{64}$ sample with $l_{1024,64}$. The mean $\vvmax$ values of the two samples are $0.41\pm 0.047$ for $s_{64}$ and $0.38\pm 0.034$ for $l_{1024,64}$, showing excellent agreement. Further, the K-S test gives a high probability of 46%, again in good agreement. We thus conclude that the $s_{64}$ and $l_{1024,64}$ samples are consistent with a common spatial distribution, and that the former is deeper than the latter by about $\sqrt{1.8}$ in luminosity distance. (As an aside we mention that, corresponding to the second solution $R_3\sim 6$ mentioned earlier, there is an indication of a solution at $R_4\sim 0.5$, but the reduction in counts in these comparisons is somewhat severe and we are inclined not to take these solutions seriously.)
A very interesting feature of the comparison discussed in the previous paragraph is that $R_4$ directly represents the ratio of the luminosities of the long and the short bursts [*in the same detector*]{}, viz. the 64 ms channel. Since the 64 ms channel corresponds to the shortest time interval used in the triggers, it provides the closest approximation to the instantaneous luminosity of a burst. We thus conclude that [*the maximum instantaneous luminosities of the short and long bursts are nearly equal*]{}, to within a factor $\sim 2$. In fact, the difference in the luminosities may be even less than the value indicated in (7). This is because, in the cosmological scenario, the K-correction (cf. Piran 1992; Mao & Paczyński 1992) depends on the spectral index of the gamma-ray bursts. Although the spectral indices of bursts have large variations, there is some indication that the short bursts are systematically harder than the long ones (Dezalay et al. 1992; Kouveliotou et al. 1993). The effects of this will be to bring the luminosity ratio even closer to unity. Utilizing the fact that we have $\cmin=71$ counts in the 64ms channel, we can [*roughly*]{} estimate the peak luminosity of bursts by assuming a power law spectrum (see eq 10 below): $$L_{\rm
peak}\sim 2 \times 10^{43} (D_{\rm max}/{\rm Mpc})^2 ~ {\rm erg ~ s^{-1}},
\eqno(8)$$ where $D_{\rm max}$ is the maximum distance to which the bursts can be detected in the 64 ms channel.
An important consequence of the comparisons carried out above is that we obtain the relative depths of the short and long burst samples. We can therefore estimate the relative number densities in space of the two kinds of bursts. We find $${n_S\over n_L} \sim 0.4^{+0.4}_{-0.2}. \eqno (9)$$ We should however mention one caveat, namely that BATSE may have missed some very short bursts with durations smaller than 64 ms because of dilution. If this is a substantial effect, then the above ratio may be higher, possibly closer to unity.
We now estimate the ratio of the total energy outputs in the short and long bursts. We first define an effective duration $\Delta t_{\rm eff}$ as the duration of a burst would have if it had a constant count rate $\cmax$ and the same fluence, i.e., $$\Delta
t_{\rm eff} = {S \over \langle E \rangle ~ \cmax}, ~~~ \langle E \rangle
\equiv
{\int_{E_1}^{E_2} E^{-\alpha+1} dE \over \int_{E_1}^{E_2} E^{-\alpha}
dE}, \eqno (10)$$ where $S$ is the fluence in units of $\rm erg ~ cm^{-2}$ in the energy range 50–300 keV (which coincides with the trigger energy range), $\langle E \rangle$ is the mean energy in the energy range of 50-300 keV, $\cmax$ is the maximum count rate in units of $\rm photons ~ cm^{-2} ~ s^{-1}$, and we assume a power law photon number distribution $n(E)~dE \propto E^{-\alpha}~dE$. Adopting $\alpha=2$ (Schaefer et al. 1992), we find that $ \langle \Delta t_{\rm eff}
\rangle \approx 0.4~{\rm s},~12.5~{\rm s}$ for the short and long bursts respectively. Combining this with the luminosity ratio $R_4$, we find the total energy ratio is $E_L/E_S
\approx 20$. It is quite remarkable that the short and long bursts differ by such large factors in their durations and total energy outputs, but yet are so similar in their maximum luminosities. Finally, for completeness, we mention that the short and long bursts are both individually consistent with perfect isotropy.
Discussion
==========
This paper has been motivated by the recent discovery of Kouveliotou et al. (1993) that gamma-ray bursts consist of two distinct subclasses, namely short and long bursts. The main aim of our investigation is to test the simple hypothesis that both populations of bursts have the same underlying space distribution and that within each population the sources are standard candles. Our conclusion is that all the available data are consistent with this hypothesis. Even though the distributions of $\cmax /\ccut$ for the short and long bursts sometimes appear to be different in certain detector channels, we are always able to bring the two populations into agreement by modifying the detector sensitivity in one or the other sample so as to reduce the two samples to the same depth or distance. When we do this, not only do the mean $\vvmax$ values agree, but also the full distributions of $\vmax$ agree when compared by means of the K-S test. Of course, these calculations do not prove our basic hypothesis, particularly since we are hampered by the small number of bursts in the samples but they make the idea quite plausible. We therefore feel that it is unlikely that one type of bursts (say long) is cosmological while the other arises in the halo, or that one is in the halo and the other in the disk (Smith & Lamb 1993). It would be too much of an accident for the two populations to have the same $V/V_{\rm max}$ distributions. Instead, we favor models where the two kinds of bursts arise in the same source. In this case, the large difference in durations between the short and long bursts may be caused by variations in the initial conditions or in the environment of the source or due to differences in the viewing angle.
Once we accept that the two kinds of bursts arise from a common source population, we are able to estimate the relative luminosities and number densities of the two populations. Surprisingly, we find that both the short and long bursts have the same peak luminosity, $L_{\rm
peak}\sim 2 \times 10^{43} (D_{\rm max}/{\rm Mpc})^2 {\rm erg ~ s^{-1}}$, to within a factor of two. The equality of the two peak luminosities is quite remarkable when we consider that the durations of the short and long bursts differ by $\sim 50$ and their total energy outputs differ by $\sim 20$. Incidentally, our estimate of $L_{\rm peak}$ corresponds to the 64 ms channel. On smaller timescales, the peak luminosity will be larger, though not by a significant factor (see eq 3). Further, we estimate the number density of the short bursts to be $\sim 0.4$ times that of the long bursts. If the difference between short and long bursts is due to viewing angle, this ratio gives an estimate of the relative solid angles associated with the two kinds of bursts.
An important point that we would stress is that these results are obtained in an essentially model-independent way. For instance, we do not need to make any assumption on whether the sources are at cosmological distances or in the Galaxy.
The constancy of luminosity in two classes of bursts which differ so much in their other properties might provide a clue to the physical origin of the bursts. Unfortunately, it is virtually impossible to explain the result using the most widely used limiting luminosity in astrophysics, namely the Eddington limit. If we assume that the sources are not dynamically expanding (but see Piran and Shemi, 1993) and take a fixed opacity (e.g. electron scattering), the Eddington limit is proportional to the mass of the source, which in turn is limited by the variability timescale, $\delta t$, i.e., $L_{\rm Edd} < 1.3
\times 10^{40} (\delta t/{\rm ms}) ~ {\rm erg ~ s^{-1}}$. Taking $\delta t
\sim 10$ ms, the characteristic burst luminosity $L_{\rm peak}$ that we have obtained is marginally consistent with the Eddington luminosity limit for sources of mass $\sim 10^3 M_\odot$ located at distances $\sim 100 \rm kpc$ in the Galactic halo. Smaller masses are ruled out by the observed isotropy of the bursts, while larger masses are ruled out by the variability argument. In fact, Bhat et al. (1993) claim to see variability down to $200~\mu$s, which rules out even the $10^3
M_\odot$ scenario.
Another robust limiting luminosity is $L_{\rm max} = c^5/G = 4\times
10^{59} ~{\rm erg ~ s}^{-1}$, which is an absolute upper limit for any source, corresponding to the emission of the entire rest mass within a gravitational light crossing time. If we identify $L_{\rm peak}$ with this limit, then we obtain the luminosity distance to the sources to be $\sim 10^8 ~ {\rm Mpc}$, corresponding to a redshift of $\sim 10^4$, in an $\Omega =1$ Friedmann universe. This is far too large for any known model.
The fact that it is not easy to come up with a physical explanation for the existence of a characteristic peak luminosity for gamma-ray bursts implies that, if the effect is real, it may provide an important and vital clue for understanding the origin of the bursts. Unfortunately, as we have tried to stress, the results have only modest statistical significance at this point. It would be very interesting to repeat the analysis with the complete database of bursts detected by BATSE.
We acknowledge helpful discussions with Bohdan Paczyński. This work was supported in part by NASA grant NAG 5-1904.
References
==========
Figure Captions
===============
[**Fig. 1:**]{} $\cmax/\cmin$ for the 1024 ms channel vs. that for the 64 ms channel. All bursts with $\cmax/\cmin$ available in both the 64 ms and 1024 ms channels are plotted. Open and filled circles correspond to bursts with durations shorter and longer than 2s respectively. Note that $\cmax/\cmin$ in the 1024 ms channel is larger than $\cmax/\cmin$ in the 64 ms channel for almost all the long bursts, while the short bursts tend to have larger $\cmax/\cmin$ in the 64 ms channel.
[**Fig. 2:**]{} The thick line shows the probability values $p_{K-S}$ obtained with the Kolmogorov-Smirnov (K-S) test when the $l_{1024}$ and $s_{64}$ samples are compared versus the quantity $\log R_3$ (see eq 5). Also shown as a thin line is $\Delta/\sigma$ vs. $\log R_3$, where $\Delta$ is defined as the absolute difference of $\vvmax$ between the two samples, and $\sigma$ is the expected standard deviation. The dotted line corresponds to a $1 \sigma$ deviation.
[**Fig. 3:**]{} $V/V_{\rm max}$ for each burst is shown versus the burst’s intensity rank normalized by the total number. The thick and thin lines correspond to the 113 long bursts in the 1024 ms channel (sample $l_{1024}$) and the 40 short bursts in the 64 ms channel (sample $s_{64}$). The K-S probability that these two samples are drawn from the same distribution is 42%. The probabilities that these two samples are drawn from a uniform distribution in a Euclidean space (indicated as a dashed diagonal line) is $< 10^{-4}$.
[^1]: Permanent address: Racah Institute for Physics, The Hebrew University, Jerusalem, Israel
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abstract: |
We present an $ab$-$initio$ density function theory to investigate the electronic and magnetic structures of the bilayer graphene with intercalated atoms C, N, and O. The intercalated atom although initially positioned at the middle site of the bilayer interval will finally be adsorbed to one graphene layer. Both N and O atoms favor the bridge site (i.e. above the carbon-carbon bonding of the lower graphene layer), while the C atom prefers the hollow site (i.e. just above a carbon atom of the lower graphene layer and simultaneously below the center of a carbon hexagon of the upper layer). Concerning the magnetic property, both C and N adatoms can induce itinerant Stoner magnetism by introducing extended or quasilocalized states around the Fermi level. Full spin polarization can be obtained in N-intercalated system and the magnetic moment mainly focuses on the N atom. In C-intercalated system, both the foreign C atom and some carbon atoms of the bilayer graphene are induced to be spin-polarized. N and O atoms can easily get electrons from carbon atoms of bilayer graphene, which leads to Fermi level shifting downward to valence band and thus producing the metallic behavior in bilayer graphene.
****PACS Numbers: 71.15.Mb, 73.43.Cd, 81.05. Uw
author:
- 'S. J. Gong$^{1}$[@G], W. Sheng$^{2}$, Z. Q. Yang$^{2}$, J. H. Chu$^{1,3}$'
title: 'Electronic and magnetic properties of bilayer graphene with intercalated adsorption atoms C, N and O'
---
* ***Introduction:** Since the discovery of monolayer graphene in the year 2004 [@Novo], this two-dimensional material remains in the focus of active research motivated by its novel physical properties and a promising potential for applications[@Tan; @Gusy; @Wang]. Experimental groups have enabled preparation and study of systems with one or a small number of graphene layers [@Geim]. Bilayer graphene, which is made of two stacked graphene layers, is considered to be particularly importance for electronics applications because of its special band structure [@Min]. Coupling of the two monolayer graphene sheets in the usual A-B stacking of bilayer graphene yields pairs of hyperbolically dispersing valence and conduction bands that are split from one another by the interlayer interaction[@Mcca]. The band gap of the bilayer can be easily opened when a difference between the electrostatic potential of the two layer is introduced, either by chemical doping or by applying gate voltage [Zhong,Kuzm,Nils, Nico, Bouk,Ohta,Zhou,Kosh,Yu]{}, which makes graphene channels have a high resistance for the OFF state. Angle-resolved photoemission (ARPES) measurements indicate such a gap in potassium doped bilayer graphene epitaxially grown on SiC [@Ohta], and infrared spectroscopy measurements also detect the similar gap in the electrostatically gated bilayer graphene [@Kuzm; @Nils; @Nico].
In this work, we carried out first-principle calculations and theoretical analysis to explore the electronic and magnetic properties of bilayer graphene with intercalated atoms C, N and O. The calculations were performed using the projector augmented wave (PAW) formalism of density functional theory (DFT) as implemented in the Vienna $ab$ initio simulation package (VASP) [@Kres]. Because generalized gradient approximation (GGA) [Perd]{} gives essentially no bonding between graphene planes and leads to excessively large values of bilayer distance [@Bouk], we performed the calculations within the localized density approximation (LDA). We find that LDA gives rise to a bilayer distance of 3.34 $\mathring{A}$, in good agreement with the experiment value. Some previous investigations used LDA to optimize the structure to get a reasonable bilayer distance, and sequently used GGA to calculate the electronic structure [@Zhong]. To keep consistent, we use LDA in all our calculations for the investigated system. An energy cutoff of 400 eV for plane-wave expansion of the PAWs is used. The model system here consists of a $4\times 4$ supercell with a foreign atom intercalated between the two coupling graphene layers. The supercell parameters are set to be the same as $a=b=9.84$ $\mathring{A}$ in the $xy$ plane ($a$ and $b$ indicate the crystal lattice constants). The Brillouin zone is sampled using a $11\times 11\times 1$ $\Gamma $ centered k-point grid. For geometry optimization, all the internal coordinates are relaxed until the Hellmann-Feynman forces are less than 0.01 eV/$\mathring{A}%
.$ The vacuum thickness along the z axis is 16 $\mathring{A}$ to avoid the interaction between graphene layers of adjacent supercells.
We considered the $\widetilde{\mathbf{A}}$-B Bernal stacking structure for bilayer graphene. Top-view for the bilayer graphene is shown in Fig. 1(a), in which violet color is for the lower layer, and gray color is for the upper layer. Seen from the top-view, ‘$\widetilde{\text{A}}$’ position in the lower layer coincides with ‘B’ position in the upper layer, ‘A’ position in the upper layer is exactly the center of the hexagon of the lower graphene layer from the top-view, and ‘$\widetilde{\text{B}}$’ position in the lower layer is right the center of the hexagon of the upper graphene layer. The foreign atom is intercalated between the upper and lower graphene layers and is initially positioned in the middle position of the bilayer distance. Three kinds of initial positions are considered for each intercalated atom: bridge, top and hollow positions, which are noted in the following by using subscript $^{\text{\textbf{\textquotedblleft }}}$Bri, Top, and Hol when needed, for example, we use N$%
_{\text{Bri}}$ to indicates the bilayer system with N atom positioned at the bridge site. Bridge site is above the carbon-carbon bonding in the lower graphene layer, top site indicates the middle position between the coinciding positions ‘$\widetilde{\text{A}}$’ and ‘B’, and hollow site is just above a carbon atom of the lower graphene layer and simultaneously below the center of a carbon hexagon of the upper layer. The structure relaxation calculations show that, if the intercalated atom is initially positioned at the bridge or hollow site, it will finally be adsorbed to one graphene layer and away from the other layer. While for the top site, the foreign atom will keep being located in** t**he middle position of the bilayer interval.
The binding energy is defined as: $dE=E_{\text{graphene}}+E_{\text{atom}}-E_{%
\text{total}},$ where $E_{\text{graphene}}$ is the energy of the clean bilayer graphene, $E_{\text{atom}}$ stands for the energy of the single foreign atom, and $E_{\text{total}}$ is the total energy of the bilayer graphene with the intercalated atom. The binding energy, the distance between the foreign atom and its nearest C atom, and the interlayer distance of the doped bilayer graphene are illustrated in Table I. Comparing the data in Table I, we find both N and O atoms favor the bridge site, while the C atom prefers the hollow site. No intercalated atom is stable at the top site. At the top position, the relaxation calculations show that nearest C-N bonding is about 1.83 $\mathring{A}$, and C-O bonding is about 1.74 $%
\mathring{A}$, which is much larger than the typical lengths of C-N (1.47 $%
\mathring{A}$)and C-O (1.42 $\mathring{A}$) [@Bond], implying the physisorption rather than the chemisorption. For the hollow site, O atom is adsorbed to the layer which provides the nearest C atom, and away from the hexagonal cental of the other graphene layer. O atom and the nearest C atom form the C=O bonding with the length of 1.39 $\mathring{A}$, smaller than the the length of C-O bonding in the O$_{\text{Bri}}$ structure (1.44 $%
\mathring{A}$). The calculation shows N$_{\text{Hol}}$ structure doesn’t exist because of the negative binding energy. Concerning the bilayer distance, with the foreign atom intercalated in the bilayer space, the distance is enlarged for all the investigated systems.
Figure 1 displays the ground states for C, N and O configurations, where Fig. 1(b), (c) and (d) are the C$_{\text{Hol}}$, N$_{\text{Bri}}$, and O$_{%
\text{Bri}}$ structures, respectively. In the C$_{\text{Hol}}$ system, the optimized interlayer distance is about 3.39 $\mathring{A}$, which is a little larger than the value of 3.34 $\mathring{A}$ for the pure bilayer graphene. Calculations show that C-C bonding in the upper layer nearly keeps unchanged. The carbon atom which is right below the foreign C is pushed down, forming a dumbbell at the saddle point. The foreign C bonds with the adjacent three carbon atoms with bonding length being 1.54 $\mathring{A}$, which is a standard length for $sp^{3}$ hybridization [@Bond]. In N$_{\text{Bri}}$ system, the bilayer distance is enlarged to the value of 3.87 $\mathring{A}$, and the length of C-N is 1.43 $\mathring{A}$, indicating the chemically adsorption not the physically adsorption. The two carbon atoms bonded with N atom are drawn out of the graphene layer, and the C-C bonding is 1.55 $\mathring{A}$, which implies $%
sp^{3}$ hybridization of the two carbon atoms. In O$_{\text{Bri}}$ system, bilayer distance is about 3.76 $\mathring{A}$, length of C-O bonding is 1.44 $\mathring{A}$, and the two carbon atoms bonded with O atom are also drawn out of the graphene layer with the C-C bonding is 1.51 $\mathring{A}$. Obviously, N and O atoms have the similar ground state structure.
Side-view of the charge distributions for C$_{\text{Bri}}$, N$_{\text{Bri}}$, and O$_{\text{Bri}}$ systems are displayed in Fig. 2. Although C$_{\text{%
Bri}}$ structure is not the ground state for C-intercalated system, we show its charge contour together with those of N$_{\text{Bri}}$ and O$_{\text{Bri}%
}$ systems, to provide a clear comparison. We know O atom is lack of two electrons, when it is adsorbed in the bilayer graphene, it strongly interacts with its adjacent carbon atoms and get electrons from them (see the upper panel of Fig. 2). N atom lacks three electrons, it also gets electrons from its adjacent carbon atoms (see the middle panel of Fig.2). C atom lacks four electrons to get saturated, and it tends to share electrons with its adjacent carbon atoms in bilayer graphene (see the lower panel of Fig. 2). All these three atoms show strong interaction with lower graphene layer. Concerning their interactions with the upper graphene layer, judging from Fig. 2, C atom has the strongest interaction with the upper layer, N is the second, and O is the third. We compare the stable positions for C, N and O atoms in monolayer and bilayer graphene, as illustrated in table II. It is clear that N atom has the same stable position in both monolayer and bilayer graphene, so does the O atom. C adatom is stable at the bridge site in monolayer graphene, and Hollow site in bilayer graphene. We believe the different stable positions for C atom results from its interaction with the upper graphene layer.
The C$_{\text{Hol}}$ structure is favored as the ground state for C atom intercalated in the bilayer graphene. The spin-resolved band structures and density of states (DOS) for C$_{\text{Hol}}$ system are shown in Fig. 3. Fig. 3(a) and (b) are the band structures for the majority spin and minority spin, respectively. It is clearly seen that impurity bands for the majority and minority spin components lie, respectively, lower and higher than the Fermi level. In the spin-resolved total DOS (see Fig. 3(c)), two narrow peaks at the opposite side of the Fermi level are observed, indicating the itinerant magnetism triggered by the intercalated C atom. In the following, we will find the itinerant magnetism comes from not only the foreign C atom, but also the carbon atoms of the bilayer graphene. In the orbital-resolved PDOS of the foreign C atom, $s$ state and $p$ state have peaks in the same energy range, which is indicative of the $sp^{3}$ hybridization. The C-C bonding between the foreign C and the nearest carbon atoms with the value of 1.54 $\mathring{A}$, is also a proof of the $sp^{3}$ hybridization, which has been pointed in the previous analysis. When the spin degree of freedom is neglected, our calculation from first principle predicts a twofold degenerate peak at the Fermi level (Fig. 3(e)), but the spin unpolarized state is not the ground state. Including the spin degree of freedom, the balance between the majority and minority spin components will be destroyed. The spin density distribution of the lower graphene layer of the C$_{\text{%
Hol}}$ system are shown in Fig. 3(f). Both the foreign C atom and the pushed down carbon atom are magnetic, yet we cannot see their magnetic moment distribution in Fig. 3(f), because they are not in the lower graphene plane. Seen from the top-view, these two carbon atoms occupy coinciding positions, which are noted in Fig. 3(f) by the letter ‘C’. The three nearest carbon atoms bonded with the foreign C atom are nearly nonmagnetic and the next-near-neighbors are magnetic. On the whole, the total magnetic moment of the C$_{\text{Hol}}$ system is about 1.32 $\mu B$, and the magnetic moment distributions show threefold symmetry, which is similar with the hydrogen adsorption on the graphene plane [@Yazy].
Spin-resolved band structure and density of states for N$_{\text{Bri%
}}$ system are shown in Fig. 4, where Fig. 4(a) and (b) show the band structure for majority and minority spin, respectively, and Fig. 4(c) and (d) display the total DOS of N$_{\text{Bri}}$ system and PDOS of N atom, respectively. In the total DOS, very narrow and sharp peak at the Fermi level is observed. We draw the PDOS of N atom and find the peak arises from from the N adsorption and the N atom is nearly 100% spin polarized. In the band structure, we find the very localized states at the Fermi level in the minority spin band structure. Such quasilocalized states give rise to the strong Stoner magnetism with magnetic moment of 0.65 $\mu $B located at the N atom. In the PDOS of N atom, between the energy -0.5 eV and -1 eV, another narrow peak is obtained. Both in the majority and minority spin band structures, the corresponding flat impurity bands in the energy range from -0.5 eV to -1 eV are observed. In addition, both in the majority and minority band structures, the characteristic conical point at K point still can be clearly identified, implying the bilayer graphene is not strongly purturbed by the N atom. The critical difference is that, in freestanding graphene the Fermi level coincides with the conical point, while the Fermi level obtained in N-intercalated system is shifted downward and becomes below the conic point. A shift downwards (upwards) means the holes (electrons) are donated by the adsorption atom. The manganese doping (electron donor) results in the upward shift of the Fermi level, reported by previous investigations [@Zhong].
Band structure and density of states of O$_{\text{Bri}}$ system are displayed in Fig. 5, where Fig. 5(a), (b) and (c) are the band structure, total density of states, and partial density of states of O atom, respectively. The calculation shows the O$_{\text{Bri}}$ system is nonmagnetic. From the above analysis about N$_{\text{Bri}}$ and C$_{\text{Hol%
}}$ systems, we see the impurity states localized around the Fermi level play an important role in the magnetic properties, and the intercalated atom itself is spin-polarized. In O$_{\text{Bri}}$ system, we do not get such localized states around the Fermi level, and the O atom is nonmagnetic. In the PDOS of O atom, we find the peak nearest to the Fermi level is around the energy of 0.75 eV. From the band structure, we can also see that the impurity state nearest to the Fermi level is at the energy of 0.75 eV. Like N$_{\text{Bri}}$ system, the O$_{\text{Bri}}$ system also undergoes a change from semimetal to metal, because O atom get electrons form graphene, thus the Fermi level shifts downward.
In summary, we calculated the structure, electronic and magnetic properties of the bilayer graphene with intercalated atoms C, N, and O by the $ab$-$%
initio$ density function theory. Impurities even at very low density may bring fruitful physical properties for bilayer graphene system. Structure relaxation shows that the intercalated atom although initially positioned at the middle position of the bilayer distance, will finally be adsorbed to one graphene layer and away from the other layer. The bilayer distance is enlarged for all the investigated adsorption systems. Both N and O atoms are stable at the bridge site, while C atom is stable at the hollow site. Concerning the magnetic property, O-intercalated system is nonmagnetic, while N and C atoms can induce stoner magnetism by introducing extended or quasilocalized states around the Fermi level. Nearly 100% spin polarization is obtained in N-intercalated system and the magnetic moment focuses on the N atom. For C-intercalated system, the magnetic moment distributes on the foreign C atom and certain carbon atoms of the bilayer graphene. Both in N-intercalated and O-intercalated systems, the Fermi level is obviously shift downward, inducing the metallic behavior of the bilayer graphene.
**Acknowledgements:** This work was supported by the the Ministry of Sciences and Technology through the 973-Project (No. 2007CB924901), the National Natural Science Foundation of China (Grant Nos. 60221502 and 1067027), the Grand Foundation of Shanghai Science and Technology (05DJ14003).
[\*]{} E-mail address: [email protected]
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Figure Captions
===============
Fig. 1
: \(a) $\widetilde{\text{A}}$-B Bernal stacking structure for bilayer graphene. (b) C$_{\text{Hol}}$ system, where the atom in violet is the foreign C atom. (c) N$_{\text{Bri}}$ system, where the atom in blue is the foreign N atom. (d) O$_{\text{Bri}}$ system, where the atom in red is the foreign O atom.
Fig. 2
: Side-view of charge contours for C$_{\text{Bri}}$, N$_{\text{Bri}}$ and O$_{\text{Bri}}$ systems, respectively.
Fig. 3
: Electronic and magnetic structures of the C$_{\text{Hol}}$ system. (a) Band structure of majority spin, (b) band structure of minority spin, (c) total density of states, (d)the orbital-resolved DOS for the foreign C atom, (e) density of states for spin unpolarized C$_{\text{Hol}}$ system, (f) distribution of the spin density on the lower graphene layer..
Fig. 4
: Spin-resolved band structure and density of states of N$_{\text{Bri}%
}$ system. (a) Band structure of the majority spin; (b) band structure of the minority spin, (c) total density of states, (d) partial density of states of N atom.
Fig. 5
: Band structure and density of states of O$_{\text{Bri}}$ system. (a) Band structure, (b) total density of states, (c) partial density of states of O atom.
Table I
: The binding energy ($dE$), the length of bonding between the adsorption atom and its nearest C atom ($a_{\text{C-atom}}$), and the bilayer distance of the adsorption system ($dis.$). The stable position for each intercalated atom is noted by box.
Table II
: The stable positions for adsorption atoms C, N, and O in monolayer and bilayer graphene.
|
---
abstract: 'We show how an adversary can emulate a Bell inequality using existing detector control methods if the Bell test is not loophole-free. For a Clauser-Horne-Shimony-Holt inequality, our model fakes a maximum violation predicted by quantum mechanics for a detector efficiency up to the threshold efficiency of about 0.8284. When the inequality is re-calibrated by incorporating non-detection events, our model emulates its exact local bound. Thus existing technologies may allow the adversary to practically subvert quantum protocols all the way up to the local limit, which hints that Bell tests need to be loophole-free for their correct application.'
author:
- Shihan Sajeed
- Nigar Sultana
- Charles Ci Wen Lim
- Vadim Makarov
title: An optimal local model to practically emulate Bell inequalities
---
Introduction
============
More than $50$ years ago, John Stewart Bell showed that any physical theory based on the assumptions of locality (i.e., nothing can communicate faster than light) and realism (i.e., physical properties of an object are fixed and pre-defined) must satisfy a set of statistical criteria called Bell inequalities [@bell1964]. That is, if a Bell-type experiment is performed and the results show a violation of a Bell inequality, then the underlying physical process cannot be explained by a local theory. This kind of tests are called Bell tests and the violation of the inequality is called Bell violation. Since the earlier demonstrations utilizing cascade decays in atoms [@freedman1972; @aspect1981; @aspect1982; @aspect1982a], Bell violations have been observed in tests utilizing nonlinear optical processes [@weihs1998; @giustina2013; @christensen2013; @poh2015], ions [@Rowe2001], neutral atoms [@hofmann2012], Josephson junction [@ansmann2009] and solid state qubits [@pfaff2012]. The implications of the Bell test not only change our understanding of nature, but also find application in device independent (DI) quantum communications [@ekert1991; @mayers1998; @barrett2005], randomness generation and amplification [@colbeck2006; @pironio2010; @colbeck2012], DI-verified quantum computation [@gheorghiu2015; @hajdusek2015], certifying quantum devices [@mayers2004; @mckague2010; @pironio2010] and DI bit commitment [@aharon2016]. Entanglement, a necessary precondition for unconditional security [@lo1999; @curty2004a] in quantum key distribution, can also be certified from the violation of a Bell inequality, independently of the underlying implementation details. This paves the way for the device-independent tests of security [@acin2007; @acin2006]. However, for the observed Bell violation to be conclusive, it is important that the Bell test is loophole-free.
More specifically, a loophole-free Bell test is an entanglement experiment that requires multiple implementation loopholes such as the detection, locality, and measurement-independent loopholes to be closed simultaneously. Here, we focus on the detection loophole, and defer the rest to Ref. [@brunner2014]. In general, the detection loophole is a scenario in which the observed Bell violation (a test statistics) is no longer reliable as the measurement sample and may not be a true representative of the population (i.e., the entire measurement statistics). Crucially, this situation commonly happens in practice as practical detectors have finite detection efficiencies and hence one could end up with samples that are non-representative. While the detection loophole is not an issue for non-adversarial settings, the same is not true for the case of quantum cryptography since an adversary can take advantage of it to come up with a local model to *fake* Bell violations [@gerhardt2011a]. For this reason, much effort has been devoted to closing the detection loophole in practice.
How a local model can theoretically simulate non-local correlations – taking advantage of the detection loophole – has already been reported in the literature [@gisin1999; @larsson1999]. However, methods of experimentally implementing such correlations using practical means have rarely been discussed, despite the importance in practical quantum cryptography. The state-of-the art method is arguably that of Ref. [@gerhardt2011a], where the authors demonstrated how an adversary could implement a local model using existing optical detector control methods to violate a Bell inequality for active basis choice schemes. However, their local model is effective only for a detector efficiency of up to $\eta = 0.5$, while theoretically it is possible fake the inequality for a threshold efficiency of up to $\eta_{\text{T}} = 2 (\sqrt{2}-1) \approx 0.8284$ (here, efficiency $\eta$ refers to the probability that one party observes a conclusive outcome given a measurement is made). In this article, we discuss how to experimentally fake the violation for higher efficiencies. More specifically, we show how existing optical detector control methods [@lydersen2010a; @gerhardt2011; @liu2014] can be exploited to not only *fake* the violation of standard Bell type inequalities all the way up to the threshold efficiency but also simulate the local bound of more general Bell inequalities. Our results point out once again that when Bell tests are performed for certifying randomness, guaranteeing security in quantum communications, or detecting non-locality, they should either be performed with an efficiency at which the test is robust against detection loopholes, or should use the bound given by more general inequalities (for example, \[eqn2\] presented later). Otherwise, existing detection control methods may allow to implement a local model to simulate the results of the test.
The Article is organized as follows. In \[assumptions\], we outline the assumptions and methodology of the Bell test that we consider in this Article. In \[faking\] we present several local models that allow an adversary to implement a practical setup to fake the Bell test or emulate the local bounds given by the inequalities. We conclude in \[conclusion\].
Assumptions for Bell test {#assumptions}
=========================
The experimental setup of a Clauser-Horne-Shimony-Holt (CHSH) Bell test for two parties with binary inputs and outputs [@clauser1969] is shown in \[fig:setup\]. The test assumes that a source of entangled photon pairs sends each member of the pairs to two legitimate parties, Alice and Bob. Alice randomly measures the polarizations along direction $\alpha_0$ or $\alpha_1$ and Bob randomly along $\beta_0$ or $\beta_1$ as shown in \[fig:setup\]. The measurement along a particular direction is performed with the help of a rotatable half wave plate (HWP) followed by a polarization beamsplitter (PBS) and two single photon detectors. This type of analyzer is called an active basis choice analyzer. The possible polarization measurement outcomes expected at Alice and Bob are $P_A \in \{\alpha_0, \alpha_0^\perp, \alpha_1, \alpha_1^\perp\}$ and $P_B \in \{\beta_0, \beta_0^\perp, \beta_1, \beta_1^\perp\}$, and they are mapped into outcomes $\{+,-,+,-\}$.
![Setup for a CHSH test. The measurement angles shown are arbitrary.[]{data-label="fig:setup"}](setup.pdf){width=".8\columnwidth"}
We assume that Alice and Bob are situated far apart, so that the locality loophole does not exist. However, due to the finite efficiency of the detectors and optical losses in the setup, it is not possible to measure the polarization of all the photons. So, the final statistics are calculated from post selected photons, i.e., from the photons that have been detected. For each pair of measurement settings $\{\alpha,\beta\} \in \{00,01,10,11\}$ chosen by Alice and Bob, the correlation function $E(\alpha,\beta)$ is calculated as $$\begin{aligned}
\label{expect}
&& E(\alpha,\beta) =\\ \nonumber
&& \ \ \ \ \ \ \ \frac{N_{\alpha,\beta}(++) + N_{\alpha,\beta}(--) - N_{\alpha,\beta}(+-) - N_{\alpha,\beta}(-+)} {N_{\alpha,\beta}(++) + N_{\alpha,\beta}(--) + N_{\alpha,\beta}(+-) + N_{\alpha,\beta}(-+)},\end{aligned}$$ where $N_{\alpha,\beta}(i,j)$ represents the number of coincidences with outcome $\{i,j\} \in \{++,+-,-+,--\}$ for a particular setting $(\alpha,\beta)$. The associated CHSH Bell inequality is $$\label{eqn1}
\begin{aligned}
S_{\rm{CHSH}}= &\\
E(\alpha_0,&\beta_0) + E(\alpha_1,\beta_0) + E(\alpha_1,\beta_1) - E(\alpha_0,\beta_1)\leq 2.
\end{aligned}$$
Quantum mechanics predicts a maximum violation of $S = 2 \sqrt{2}$ for the setting choice $\alpha_0 = -78.75\degree$, $\alpha_1 = 56.25\degree$, $\beta_0 = 11.25\degree$, $\beta_1 = -33.75\degree$ [@eberhard1993], and even stronger correlations are algebraically possible in theory leading to $S \le 4$ [@popescu1994]. However, as long as the efficiency of a measurement is $\eta =1$, all local models must necessarily satisfy Eq. (\[eqn1\]). Unfortunately, this is not true for $\eta < 1$. In particular, when $\eta$ is less than some threshold $\eta_{\text{T}}$, it is possible to devise local models that violate \[eqn1\]. For the CHSH test described here, $\eta_{\text{T}} = 2(\sqrt{2} -1)\approx 82.84\%$ [@garg87]. In order to avoid this, these tests are performed in the region $\eta > \eta_{\text{T}}$. Note that the CHSH test is not the most robust Bell test as one can further reduce the detection threshold by looking at marginal correlations (or singles statistics). This is given by the Eberhard Bell inequality [@eberhard1993], which has a detection threshold of $\eta_T = 2/3\approx 66.67\%$. Alternatively, one can include the ‘efficiency’ in the inequality and recalibrate it as a function of $\eta$ as [@garg87] $$\begin{aligned}
\label{eqn2}
S'(\eta) &= E(\alpha_0,\beta_0) + E(\alpha_1,\beta_0) + E(\alpha_1,\beta_1) - E(\alpha_0,\beta_1)\\
& \leq \frac{4}{\eta} - 2.
\end{aligned}$$ The recalibrated CHSH Bell inequality gives the local bound of $S'$ as a function of $\eta$, i.e., how much violation is required to certify non-locality for a given efficiency. This is shown by the solid (red) curve in \[fig:result\_ce\]. Note that, when $\eta=1$, \[eqn2\] becomes \[eqn1\] since the post-selected correlation set becomes the entire measurement set. Also, when $\eta < 2/3$, $S'(\eta) >4$, which is not physical. Thus, a local model that can simulate \[eqn2\] for efficiency range $ 2/3 \le \eta \le 1$ would be the optimum model to exploit detection loopholes in a Bell test. We present it in the next section.
Faking Bell inequality with improved efficiency {#faking}
===============================================
![Local bounds for recalibrated inequality $S'$ \[\[eqn2\]\], improved faking model \[\[S\_vs\_n\]\], and perfect faking model. The quantum mechanical bound $2\sqrt{2}$ is also shown. The improved faking model achieves this bound at $\eta \approx 0.6678$ and the perfect model at $\eta = 2 (\sqrt{2}-1)$. The perfect model can fully emulate \[eqn2\] for efficiency range $ 2/3 \le \eta \le 1$.[]{data-label="fig:result_ce"}](S_vs_sp.pdf){width=".8\columnwidth"}
For ease of understanding, we will go step by step. First, we review an existing local model that can fake \[eqn1\] for $\eta \le 1/2$ [@gerhardt2011a] and point out its limitations. Then we propose a modification to this model that enables it to fake \[eqn1\] up to $\eta \le 2/3$. We then present our perfect model that can not only fake \[eqn1\] for $\eta \le 2 (\sqrt{2}-1)$ but also emulate the local bounds given by \[eqn2\]. Since all three models exploit an existing detector control method – bright-light detector control [@lydersen2010a; @gerhardt2011; @liu2014] – we first recap it.
**Bright-light detector control:** Single-photon detectors used in a Bell test may become insensitive to single photons when exposed to bright light [@lydersen2010a; @lydersen2011c]. Even in this mode, they can produce a detection event (‘click’) when additionally exposed to a light pulse of intensity $I$ equal to or higher than a threshold level $I_\text{th}$. This allows an adversary Eve to have control over the detectors by tailoring $I$. For example, if the measurement basis matches that of the incoming light pulse, all of it is incident on a single detector with intensity $I \ge I_\text{th}$ and results in a detection event. However, in case of basis mismatch, the incoming light is split between two detectors with intensity $I/2 < I_\text{th}$ (assuming a conjugate basis) and none of the detectors click. This is how the adversary can have control over detection outcomes. The feasibility of bright-light control has been confirmed numerous times, with both detectors based on avalanche photodiodes [@lydersen2010a; @gerhardt2011; @lydersen2010b; @wiechers2011; @sauge2011; @jogenfors2015; @huang2016] and superconducting nanowires [@lydersen2011c; @tanner2014]. Next, we show how an adversary can exploit it to implement a local-realistic model.
**Conditions for violation:** Let us assume that an arbitrary value of $ |S| \le 4$ needs to be simulated by the local model. Assuming symmetry for each setting combination $(\alpha,\beta)$, this implies $|E| = S/4$. Assuming $N_{\alpha,\beta}(++) = N_{\alpha,\beta}(--) = N_\text{sim}$ and $N_{\alpha,\beta}(+-) = N_{\alpha,\beta}(-+) = N_\text{dif}$, (where $2 N_\text{sim} + 2 N_\text{dif} = 1$), Eq. \[expect\] can be written as $$\frac{N_\text{sim}}{N_\text{dif}} = \frac{1 + E}{1 - E}.
\label{xy_rel}$$ This implies that under the assumptions specified above, an arbitrary correlation value $E$ requires the ratio of similar to different outcomes to follow \[xy\_rel\]. For example, the quantum mechanical prediction of $S = 2 \sqrt{2}$, which corresponds to $E= \pm 1/{\sqrt{2}}$, requires $$N_\text{sim} = (3 \pm 2\sqrt{2}) N_\text{dif}.$$ Below we describe several techniques by which an active attacker can satisfy this condition.
------------ ------------------ ------------------ -------------------- ------------------ ------------------
\[-1.5em\]
$\beta_0$ $\beta_0^\perp$ $\beta_1$ $\beta_1^\perp$
$\alpha_0$ $N_\text{sim}/4$ $ N_\text{dif}/4 $ $N_\text{dif}/4$ $N_\text{sim}/4$
$\alpha_0^\perp$ $N_\text{dif}/4$ $ N_\text{sim}/4 $ $N_\text{sim}/4$ $N_\text{dif}/4$
$\alpha_1$ $N_\text{sim}/4$ $ N_\text{dif}/4 $ $N_\text{sim}/4$ $N_\text{dif}/4$
$\alpha_1^\perp$ $N_\text{dif}/4$ $ N_\text{sim}/4 $ $N_\text{dif}/4$ $N_\text{sim}/4$
------------ ------------------ ------------------ -------------------- ------------------ ------------------
: Probability of each polarization combination generated by the source in the existing faking model [@gerhardt2011a]. They are normalized to maintain $2 N_\text{sim} + 2 N_\text{dif} = 1$.[]{data-label="tab:first-model-polarization-probabilities"}
-------------------- --------- ------- --------- ------- --------- ------- --------- ------- --------- ------- --------- -------
Outcome Prob. Outcome Prob. Outcome Prob. Outcome Prob. Outcome Prob. Outcome Prob.
$\alpha_0 \beta_0$ $++$ $a/2$ $++$ $b/4$ $+-$ $b/4$ $+-$ $a/2$ $+$ $b/4$ $+$ $1/2$
$-+$ $b/4$ $--$ $b/4$ $-$ $b/4$
$\alpha_1 \beta_1$ $++$ $b/4$ $++$ $a/2$ $++$ $a/2$ $++$ $b/4$ $+$ $a/2$ $+$ $1/2$
$-+$ $b/4$ $-+$ $b/4$
-------------------- --------- ------- --------- ------- --------- ------- --------- ------- --------- ------- --------- -------
**Existing model:** A straightforward approach to force the outcomes to follow \[xy\_rel\] is to generate polarization combinations at the source with desired statistics and then force deterministic outcomes during the measurement, as done in Ref. . We assume each polarization combination is generated according to the probabilities given in \[tab:first-model-polarization-probabilities\], where $N_\text{sim}$ and $N_\text{dif}$ obey \[xy\_rel\]. We also assume that the intensity is tailored to bring the bright-light control method into play, i.e., matched bases lead to deterministic outcome with unity probability, while mismatched bases lead to no detection. Let’s consider the case when the source generates polarization combination $\alpha_0 \beta_0$ $(\alpha_0 \beta_1)$ with probability $N_\text{sim}/4$ $(N_\text{dif}/4)$. They result in coincidences only for the setting $\alpha_0 \beta_0$ $(\alpha_0 \beta_1)$ and lead to deterministic similar (similar) outcomes with unity probability. For the remaining three setting choices, no coincidence happens and the outcomes have no effect on the correlation. This is true for all the polarization combinations in \[tab:first-model-polarization-probabilities\]. In this way, it is possible to generate similar and different outcomes with desired probability to match \[xy\_rel\] for any desired value of $E$. A problem with this method, however, is that half of the time the measurement basis does not match the preparation basis and results in no detection. As a result, the efficiency at each side becomes only $\eta= 0.5$. This is a limitation in Ref. . Next, we outline a way to implement an improved local realistic model with a higher detection efficiency.
**Improvement to existing model:** Above we have recapped the existing first method that leads to CHSH parameter $S_1=4$ with an efficiency $\eta_1=0.5$. We now generate a second method that leads to CHSH parameter $S_2 = 2$ with efficiency $\eta_2 = 1$. For this, let’s assume that the source always sends polarization $\alpha~(\beta)$ to Alice (Bob), where $\alpha$ ($\beta$) is polarized at an angle that is midway between $\alpha_0$ and $\alpha_1$ ($\beta_0$ and $\beta_1$). In this case, irrespective of the measurement settings, the input intensity $I$ is split at a ratio of $\cos^{2}(\phi_A): \sin^2(\phi_A) $ between the two detectors in Alice and at $\cos^{2}(\phi_B): \sin^2(\phi_B)$ in Bob. Here, $\phi_A = |\alpha_1 - \alpha_0|/2$ and $\phi_B = |\beta_1 - \beta_0|/2$. Tailoring the intensity to satisfy $I \cos^{2}(\phi) \ge I_\text{th} $ and $I \sin^{2}(\phi) < I_\text{th}$ at the respective sides ensures that only one of the detectors clicks (with outcome $+$), irrespective of the basis choice, and efficiency stays 1. This will result in $E=+1$ for each measurement setting and lead to a CHSH parameter $S_2=2$ with an efficiency $\eta_2 =1$. Note that this method (presented here for its ease of explanation) results in only $++$ outcomes. It can be symmetricized to produce all four outcomes $++$, $+-$, $-+$, $--$, which we omit for brevity.
Thus, we have outlined two independent approaches to control $S$: the first one leads to $S_1 = 4$ with an efficiency $\eta_1 = 0.5$, while the second one leads to $S_2 = 2$ with efficiency $\eta_2 = 1$. An adversary can then use a probabilistic mixture of these two approaches to increase the faking efficiency of the Bell test. With probability $p_1~(p_2 = 1 - p_1)$ she uses the first (second) method. The input intensity needs to be tailored to $2 I_\text{th} > I \ge I_\text{th}/\cos^2(\phi)$ to ensure that the first (second) method leads to detection efficiency of $\eta_1 = 0.5$ ($\eta_2 = 1$) and results in $S_1 = 4$ ($S_2 = 2$). The resultant efficiency as seen by Alice and Bob will be $\eta = \sqrt{p_1 \eta_1^2 + p_2 \eta_2^2}$ and the improved CHSH parameter will be $$S_{\text{imp}} = \frac{p_1 S_1 \eta_1^2 + p_2 S_2 \eta_2^2}{\eta^2}.
\label{S_vs_n}$$ The variation of $S_{\text{imp}}$ with $\eta$ is shown in \[fig:result\_ce\]. The left-most point $(\eta,S_{\text{imp}}) = (0.5,4)$ corresponds to the first method with $p_2 =0$. As $p_2$ is increased, $S_{\text{imp}}$ becomes smaller with increasing efficiency and eventually becomes $(\eta,S_{\text{imp}}) = (1,2)$ at the rightmost point with $p_2 =1$. Quantum mechanical prediction $S =2 \sqrt{2} $ is obtained at $p_2 \approx 0.2612$ and the corresponding efficiency is $\eta \approx 0.6678$. This is still lower than the threshold efficiency limit $\eta_T = 2 (\sqrt{2}-1) \approx 0.8284$ for CHSH inequality. To achieve higher local bounds, one more degree of freedom needs to be introduced, as discussed in our next model.
**Perfect local model:** Now we present a perfect local model that can not only fake a violation of inequality (\[eqn1\]) for $\eta \le 2 (\sqrt{2}-1)$ but also emulate the local bounds given by \[eqn2\] for $2/3 \le \eta \le 1$. For this model, we make three assumptions. First, we assume that the adversary at the source always generates one of the two polarization combinations $\{\alpha_0, \beta_0\}$ and $\{\alpha_1, \beta_1\}$ with equal probability of $1/2$ each. Second, we assume that the adversary chooses the light intensity towards Bob in such a way that they result in a deterministic outcome with unity probability. For the ease of analysis we will assume that the polarizations $\{\beta_0, \beta_1 \}$, at Bob, lead to deterministic outcomes $\{+,+\}$ and $\{-,+\}$ with unity efficiency when measured along $\beta_0$ and $\beta_1$ respectively (however, any other outcomes will also do as long as they are deterministic and unity probability). Third, we assume that at Alice, whenever the measurement basis matches (does not match) that of the incoming light, a deterministic (random) outcome is produced with probability $a~(b)$. Such detector control can be achieved using a method similar to that presented above in the improvement to existing model, as detailed in \[satisfy\_a3\]. For each setting, the possible outcomes at Alice and Bob and the corresponding coincidence probabilities are shown in \[tab:third-model-outcomes\]. For any measurement setting $\{\alpha,\beta\}$, the correlation function $E$ is related to $a$ and $b$ as $$|E| = \frac{\frac{a}{2} + \frac{b}{4} - \frac{b}{4}}{ \frac{a}{2} + 2 \frac{b}{4}} = \frac{a}{a+b},
\label{rel_eab}$$ and the coincidence probability is $$\frac{a}{2} + \frac{b}{2} = \eta^2.
\label{rel_ab2}$$ Solving \[rel\_eab,rel\_ab2\], we get, $$\begin{aligned}
a =& 2 E \eta^2\\
b =& 2 (1-E) \eta^2
\label{ab}
\end{aligned}$$ Thus, to emulate the local bounds in an actual experiment having detector efficiency $\eta$, an adversary can use \[eqn2\] to calculate the maximum correlation value $E$ corresponding to that $\eta$, and then use \[ab\] to set the values of $a,b$. As long as \[ab\] is maintained, the single click probability during the test is equal to $\eta$ and the CHSH value is equal to the bound as shown by the thick black dashed line in \[fig:result\_ce\]. For example, for a Bell test done with detector efficiency $\eta = 2(\sqrt{2}-1)$, the local bound is $S'= 2\sqrt{2}$ according to \[eqn2\]. This can be attained – according to \[ab\] – if $a = 12\sqrt{2} - 16 = 0.97$ and $b= 40-28\sqrt{2} = 0.40$, which leads to $|E| = 1/\sqrt{2}$. Similarly, the local bound of $S'$ \[\[eqn2\]\] can be achieved for any $2/3 \le \eta \le 1$. For $\eta < 2/3$, the local bound becomes unrealistic $S' > 4$ \[\[eqn2\]\], which requires $b$ to be negative, which is impossible in practice. This concludes our local model that can emulate the local bounds given by \[eqn2\] for every value of $ 2/3 \le \eta \le 1$.
Conclusion
==========
Although it is a known fact that a local theory can violate a Bell inequality up to a threshold detection efficiency, it is rarely addressed in the literature how an adversary can actually implement it. In this work, we have shown that the existing detector control method can be exploited to implement a local model that can fake the CHSH Bell inequality \[\[eqn1\]\] up to the threshold efficiency. Our model can also simulate the local bound of the calibrated CHSH Bell inequality \[\[eqn2\]\] for efficiency over 2/3. Our results point out that whenever Bell violations are used for testing less-conventional theories, implementing device-independent quantum secure communication [@acin2007], certifying randomness [@pironio2010] and nonlocality, loophole-free Bell tests [@hensen2015; @giustina2015; @shalm2015] should be performed.
We thank D. A. Graft, G. Adenier, H.-K. Lo, and Y. Zhang for discussions. This work was funded by Industry Canada, CFI, NSERC (programs Discovery and CryptoWorks21), Ontario MRI, US Office of Naval Research, and the Ministry of Education and Science of Russia (program NTI center for quantum communication technologies).
[*Author contributions:*]{} S.S. developed the perfect local model, measured detector characteristics, and wrote the manuscript with input from all authors. N.S. developed the improvement to existing model. C.C.W.L. provided useful inputs and V.M. supervised the study.
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matches mismatches
$a-b$ $I$ $\alpha_0$ $\alpha_0$ x $I \ge I_{\text{th}}$, $I \cos^2(2\phi_0) < I_{\text{th}}$, $I \sin^2(2\phi_0) < I_{\text{th}}$
$b/2$ $I$ $\alpha_0 + \phi_0$ $\alpha_0$ $\alpha_1$ $I \sin^2(\phi_0) < I_{\text{th}} \le I \cos^2(\phi_0)$
$b/2$ $I$ $\alpha_0 - \phi_1$ $\alpha_0$ $\alpha_1^\perp$ $I \sin^2(\phi_1) < I_{\text{th}} \le I \cos^2(\phi_1)$
$1-a$ vacuum x x
------- -------- --------------------- ------------ ------------------ -------------------------------------------------------------------------------------------------
Strategies for controlling and {#satisfy_a3}
===============================
Here we show that regardless of the value of $\alpha_0$ and $\alpha_1$ an adversary can satisfy the assumption that whenever the Alice’s basis matches (does not match) that of the incoming light, a deterministic (random) outcome is produced with probability $a~(b)$. For simplicity, let us assume the case when the adversary sends a light polarized at angle $\alpha_0$ towards Alice (strategies for the other polarizations are similar). Then, with probability $(a-b)$, she sends light polarized at angle $\alpha_0$ which, when measured in the same (different) basis, results in detection (no detection) if intensity is tailored properly (see \[tab:third-model-control-ab\]). With probability $b/2$, she sends the light at an angle midway between $\alpha_0$ and $\alpha_1~(\alpha_1^\perp)$ at angle $\alpha_0 +\phi_1~(\alpha_0 -\phi_1^\perp)$. Here, $\phi_1 = |\alpha_0 -\alpha_1|/2$, $\phi_1^\perp = |\alpha_0 -\alpha_1^\perp|/2$. As a result, when the basis matches, for both the cases, outcome is $\alpha_0$ while for basis mismatch the outcome is $\alpha_1$ and $\alpha_1^\perp$ with probability $b/2$. The condition for this is $I \sin^2\phi < I_{\text{th}} < I \cos^2(\phi)$ for $\phi \in \{\phi_1,\phi_1^\perp \}$ as shown in \[tab:third-model-control-ab\]. For the remaining times (with probability $1-a$), the adversary sends vacuum. Overall, from \[tab:third-model-control-ab\], it can be seen that when the basis matches that of the incoming light, it results in a deterministic outcome with probability $a$; while when the basis mismatches, it results in a random outcome with probability $b$. This supports the practicality of our assumption. Note that this method leads to asymmetric detection efficiency, as Bob’s efficiency is always higher than Alice’s. However, this can be avoided by reversing the roles of Alice and Bob half of the time.
We have so far assumed that the blinded detector is controllable as a step function: for $I < I_\text{th}$ the click probability is 0, and for $I \ge I_\text{th}$ it is 1. This is of course a simplification [@lydersen2010a; @gerhardt2011; @liu2014; @lydersen2010b; @wiechers2011; @lydersen2011c; @sauge2011; @jogenfors2015; @huang2016]. Real detectors have noise, which leads to them having two thresholds $I_\text{never} < I_\text{always}$, with click probability 0 for $I \le I_\text{never}$ and 1 for $I \ge I_\text{always}$. In the range $I_\text{never} < I < I_\text{always}$, the click probability gradually increases from 0 to 1. These thresholds depend on the blinding power and regime. Furthermore, no two detector samples are identical, and require tweaking the faked states to achieve perfect or near-perfect control [@gerhardt2011; @liu2014; @jogenfors2015]. Generally, if the ratio $I_\text{always}/I_\text{never}$ can be made sufficiently small, perfect control can be achieved. These issues are device-specific and should be treated at the implementation stage. However, the ability to obtain an arbitrary click probability by adjusting $I$ may allow an alternative method of controlling $a$ and $b$, as we show below.
Practical detectors, when blinded, gradually increase their click probability from 0 to 1 in a certain range of trigger intensity $I$ [@lydersen2010a; @gerhardt2011; @liu2014; @lydersen2010b; @wiechers2011; @lydersen2011c; @sauge2011; @jogenfors2015; @huang2016]. This can be used to obtain probabilistic detections. To illustrate this, we have measured control characteristics of one avalanche photodiode detector in a commercial QKD system Clavis2 [@idqclavis2specs; @huang2016]. At a particular continuous-wave blinding power, we varied the trigger pulse energy and recorded the corresponding click probability as shown in \[fig:control-trigger-energy-Clavis2\]. The result shows that it is in principle possible for an adversary to select a value of trigger pulse intensity $I$ (without varying the polarization by $\pm \phi$) that in a matching basis leads to click probability $1$ in one detector, and when halved owing to basis mismatch, leads to a random click in either detector with probability $\sim 0.40$. However, some double clicks (i.e., simultaneous clicks in both detectors) will happen in this strategy. Their handling in a Bell test will need to be considered.
![Control characteristics of a detector in commercial quantum key distribution system Clavis2 [@idqclavis2specs; @huang2016], responding to a short trigger pulse atop continuous-wave blinding power of (a) $740~\micro\watt$ and (b) $367~\micro\watt$. Wavelength of light was $\sim 1.55~\micro\meter$.[]{data-label="fig:control-trigger-energy-Clavis2"}](trig_vs_eff.pdf)
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abstract: |
We study the $ABC$ model ($A + B \rightarrow 2B$, $B + C \rightarrow 2C$, $C + A \rightarrow 2A$), and its counterpart: the three–component neutral drift model ($A + B \rightarrow 2A$ or $2B$, $B + C \rightarrow
2B$ or $2C$, $C + A \rightarrow 2C$ or $2A$.) In the former case, the mean field approximation exhibits cyclic behaviour with an amplitude determined by the initial condition. When stochastic phenomena are taken into account the amplitude of oscillations will drift and eventually one and then two of the three species will become extinct. The second model remains stationary for all initial conditions in the mean field approximation, and drifts when stochastic phenomena are considered. We analyzed the distribution of first extinction times of both models by simulations of the Master Equation, and from the point of view of the Fokker-Planck equation. Survival probability vs. time plots suggest an exponential decay. For the neutral model the extinction rate is inversely proportional to the system size, while the cyclic model exhibits anomalous behaviour for small system sizes. In the large system size limit the extinction times for both models will be the same. This result is compatible with the smallest eigenvalue obtained from the numerical solution of the Fokker-Planck equation. We also studied the behaviour of the probability distribution. The exponential decay is found to be robust against certain changes, such as the three reactions having different rates.
author:
- |
Margarita Ifti and Birger Bergersen\
Department of Physics and Astronomy, University of British Columbia\
6224 Agricultural Road, Vancouver, BC, Canada V6T 1Z1
title: 'Survival and Extinction in Cyclic and Neutral Three–Species Systems'
---
Introduction
============
Cyclic phenomena are often ignored when studying epidemiological and evolutionary processes, but nevertheless they can have important consequences, e.g. we contract most infective diseases only once in our lifetime, because our immune system has “memory”. Vaccines are designed based on this knowledge, and they work quite efficiently. However, it is widely accepted that some viruses, such as the flu virus, mutate in order to make themselves unrecognizable by our immune system, and thus be able to reinfect us. Immunity to pertussis (whooping cough) is temporary, and decreases as the time after the most recent pertussis infection increases. The chicken pox (agent VZV—varicella zoster virus) also repeats, because immunity wanes with time. Our model is motivated by this type of situation.
In epidemiological models populations are often categorized in three states: susceptible $(S)$, infected $(I)$ and recovered $(R)$. There exists a vast literature about the so-called $SIR$ models, where the “loss of immunity” step is not considered (e.g. the classic texts by Bailey [@bib1], Anderson and May [@bib2], the review by Hethcote [@bib3]). We consider the case when the “loss of immunity” step is present, otherwise referred to as $SIRS$ models [@bib4; @bib5].
Recently it has been shown that many species of bacteria are able to produce toxic substances that are effective against bacteria of the same species which do not produce a resistance factor against the toxin [@bib6; @bib7]. These bacteria exist in colonies of three possible types: sensitive $(S)$, killer $(K)$ and resistant $(R)$. The killer type produces the toxin, and the resistance factor that protects it from its own toxin, at a metabolism cost. The resistant strain only produces the resistance factor (at a smaller metabolism cost). The sensitive strain produces neither. Clearly, the $S$ colony can be invaded by a $K$ colony, the $K$ can be invaded by $R$, and the $R$ can be invaded by $S$.
In our first model the population is categorized into three “species”: $A$, $B$, $C$, and the rules are such that when an $A$ meets a $B$, it becomes $B$, when $B$ meets $C$, it becomes $C$, when $C$ meets $A$, it becomes $A$. The total population size is conserved. Otherwise this could be seen as a three-party voter model, when the follower of a certain party “converts” a follower of another certain party when they meet.
The reaction is similar to the cyclical “Rock-paper-scissors game”, of which a biological example has been found recently [@bib8; @bib9]. It is played by the males of a lizard species that exist in three versions: the blue throat male defends a territory that contains one female, the orange throat male defends a territory with many females, and the male with a throat with yellow stripes does not defend his own territory, but can sneak into the territory of the orange males and mate with their females. Hence, a “blue” population can be invaded by “orange” males, while an “orange” population is vulnerable to “yellow” males, who on their turn are at an disadvantage against the “blue” males, who defend their territory very well. Many spatial models that relate to such systems have been built [@bib10; @bib11; @bib12; @bib22].
The second model is a three–component version of the famous Kimura model of neutral genetic drift [@bib15; @bib16]. In that case, when individual from two species meet, the offspring may be either of the first or the second species, with equal probability.
Description of the Model
========================
Consider a system in which three species $A$, $B$, $C$ are competing in a way described by the reaction: $A + B \rightarrow 2B$, $B + C \rightarrow
2C$, $C + A \rightarrow 2A$.
The rate equations for this system will be: $$\begin{aligned}
N \frac {d A}{d t} = AC - AB \nonumber \\
N \frac {d B}{d t} = BA - BC \\
N \frac {d C}{d t} = CB - CA \nonumber\end{aligned}$$
with $A + B + C = N =$ const. (we assume the rates are the same, in which case a time–rescale will remove them from the equations.) These equations can be rewritten as:
$$\begin{aligned}
N \frac {d}{d t} \ln A = C - B \nonumber \\
N \frac {d}{d t} \ln B = A - C \\
N \frac {d}{d t} \ln C = B - A \nonumber\end{aligned}$$
which leads to the second conservation rule: $ A B C = H $ = const.
The above model contrasts with the neutral drift model ($A + B
\rightarrow 2A$ or $2B$, $B + C \rightarrow 2B$ or $2C$, $C + A
\rightarrow 2C$ or $2A$) for which:
$$\frac {d A}{d t} = \frac {d B}{d t} = \frac {d C}{d t} = 0$$
Assuming that the system is subject to stochastic noise due to Poisson birth and death processes (intrinsic noise) we get the master equation for the cyclic model:
$$\frac{\partial P(A,B,C,t)}{\partial t} = \frac {1}{N} [(A-1)(C+1)
P(A-1,B,C+1,t) - AC P(A,B,C,t) +$$$$+ (A+1) (B-1) P(A+1,B-1,C,t) -
ABP(A,B,C,t) +$$$$+ (B+1)(C-1) P(A,B+1,C-1,t) - BCP(A,B,C,t)]$$
For the neutral drift case the master equation is:
$$\frac{\partial P(A,B,C,t)}{\partial t} = \frac {1}{N} [\frac {1}{2}
(A-1)(C+1) P(A-1,B,C+1,t) +$$$$+ \frac {1}{2} (A+1)(C-1) P(A+1,B,C-1,t) -
AC P(A,B,C,t) +$$$$+ \frac {1}{2} (A+1) (B-1) P(A+1,B-1,C,t) + \frac
{1}{2} (A-1)(B+1) P(A-1, B+1,C,t) -$$$$- ABP(A,B,C,t) + \frac {1}{2}
(B+1)(C-1) P(A,B+1,C-1,t) +$$$$+ \frac {1}{2} (B-1)(C+1)
P(A,B-1,C+1,t) - BCP(A,B,C,t)]$$
Now we introduce the “shift” operators, defined by
$$\epsilon_{A} f(A,B,C) = f(A+1,B,C)$$
and
$$\epsilon^{-1}_{A} f(A,B,C) = f(A-1,B,C)$$
and likewise for B and C operators. The master equation for the cyclic model now reads:
$$\frac {\partial P(A,B,C,t)}{\partial t} = \frac {1}{N}[(\epsilon_C
\epsilon ^{-1}_A -1) AC + (\epsilon_A \epsilon^{-1}_B -1)AB +(\epsilon_B
\epsilon^{-1}_C -1)BC] P(A,B,C,t)$$
and for the neutral drift one:
$$\frac {\partial P(A,B,C,t)}{\partial t} = \frac {1}{N}[(\frac
{1}{2} (\epsilon_C \epsilon ^{-1}_A + \epsilon_A \epsilon^{-1}_C) -1) AC +
(\frac {1}{2} (\epsilon_A \epsilon^{-1}_B+\epsilon_B \epsilon^{-1}_A)-1)AB
+$$$$+(\frac {1}{2} (\epsilon_B \epsilon^{-1}_C +
\epsilon_C \epsilon^{-1}_B) -1)BC] P(A,B,C,t)$$
Next we transform to the intensive quantities
$$x = \frac {A}{N}, y = \frac {B}{N}, z = \frac {C}{N}$$
We use the system size expansion of Horsthemke and Brenig [@bib17; @bib18]. However, the notation and style is closer to Van Kampen [@bib19; @bib20].
The shift operators become
$$\begin{aligned}
\epsilon_{A(B,C)} = \sum_j \frac {1}{j!} N^{-j} \frac
{\partial^j}{\partial x^j (y^j, z^j)} \\
\epsilon^{-1}_{A(B,C)} = \sum_j \frac {-1^j}{j!} N^{-j} \frac
{\partial^j}{\partial x^j (y^j, z^j)} \nonumber\end{aligned}$$
Further we use the rules of transformation of random variables to define:
$$W(x,y,z,t) = P(A,B,C,t) \cdot N^3$$
Before we go ahead with the expansion, a few comments are necessary. From the rate equations, it is clear that our system does not have one single steady state or limit cycle. Instead, the mean-field solution is completely dependent on the initial conditions, and we do not have a macroscopic solution to expand about, but rather an infinity of neutrally stable cycles (for the competition case) or points (for the neutral model.) The above expansion, otherwise known as Kramers-Moyal expansion, is discussed by van Kampen [@bib19; @bib20]. It is a risky expansion, and it does not work in most cases, mainly because higher derivatives are not small themselves. Also, with time, large fluctuations may occur, and since our system presents itself as a fluctuations–driven one, it becomes even more important to watch out for this sort of complications. In the case of neutral drift, the second order term is the leading one, so we would expect this Ansatz to work for large system sizes. In the cyclic competition model, the first order term does not drive the system out of the (neutrally) stable trajectories, and again it would be the second order term to essentially determine the fate of the system. So, we set out to investigate the applicability of the Kramers-Moyal expansion to the cyclic competition and neutral drift three-species systems.
We transform the master equation into an equation for $W(x,y,z,t)$ by substituting the expressions for the shift operators in it, and further by grouping together the terms of the same order in $N$. The terms of order $N^1$ cancel, and the first terms in the expansion are of order $N^0$; keeping only those gives us the first order equation for the cyclic case:
$$\frac {\partial W}{\partial t} = [(\frac {\partial}{\partial x} - \frac
{\partial} {\partial z}) x z + (\frac {\partial}{\partial y} -
\frac {\partial}{\partial x}) y x + (\frac {\partial}{\partial z}
- \frac {\partial}{\partial y}) z y] W$$
For the neutral drift case the first-order term is identically zero.
The second order term is obtained by considering the terms of order $N^{-1}$ in the expansion of the master equation:
$$\frac {1}{2N} [(\frac {\partial^2}{\partial x^2} -2 \frac {\partial^2}
{\partial x \partial z} + \frac {\partial^2}{\partial z^2}) x z
+ [(\frac {\partial^2}{\partial x^2} -2 \frac {\partial^2}{\partial x
\partial y} + \frac {\partial^2}{\partial y^2}) x y +$$ $$+[(\frac {\partial^2}{\partial y^2} -2 \frac {\partial^2}{\partial y
\partial z} + \frac {\partial^2}{\partial z^2}) y z]W$$
and is identical for both cyclic and neutral drift cases.
The First Order Term
====================
The concentrations $x$, $y$, $z$ of a three–component system are commonly represented by the distances of a point inside an equilateral triangle of unit height from its sides (Fig. 1):
For the neutral drift case the first order equation tells us that in the mean–field approximation the system will remain in the initial state forever.
After some algebra, the first order equation \[13\] for the cyclic system becomes:
$$\frac {\partial W}{\partial t} = \frac {\partial}{\partial x} (xz-xy)W +
\frac {\partial}{\partial y} (xy-yz)W + \frac {\partial}{\partial z}
(yz-xz)W$$
At the centre of the triangle ($x=y=z=1/3$) all three expressions above are zero, so in the first order approximation the system will stay at that state forever. It can be verified that the product $H=x y z$ is a solution of the first-order equation, as is any function of that product. The lines $x y z = $ const. will then represent possible trajectories of the system, which are closed, since there is no term to drive the system out of those trajectories. In this (mean-field) approximation, the cyclic model exhibits global stability with constant system size $N$ [@bib28; @bib29]. The populations of each species oscillate out of phase with one-another, and the amplitude remains constant [@bib21]. These are the same trajectories that are obtained when one solves the rate equations \[1\]. Fig. 2 shows some of those closed trajectories.
The Second Order Term
=====================
The second order term \[14\], which represents the “diffusion term” must be added to the first order equation to give the Fokker-Planck equation for our system. This term is identical for both the cyclic (competing species) and neutral genetic drift cases. As we will see, this makes the long (evolutionary) time fate of both systems essentially the same, at least for reasonably large system sizes.
The trajectories of a given realisation of the system can easily be obtained by simulation: they initially spiral out of the centre, with the populations of each species oscillating out of phase with one–another, so that the total size of the population is conserved. The amplitude drifts until one of the populations becomes zero (species goes extinct). In other words, the slow second order term makes the system cross from a closed (mean-field) trajectory to a neighbouring one. This would suggest an adiabatic approximation. In the neutral drift model, the fluctuations drive the system from one point–solution of the mean-field equations to a neighbouring one.
The computer simulations of both systems start with an equal number of $A$, $B$, $C$, (start at the centre) and generate times for the next possible reaction event with exponential distribution as $- N \cdot \ln
(rn)/AB$, (for $A+B \rightarrow 2B$ reaction, and similarly for the other two reactions,) where $rn$ is a random variable with uniform distribution in $[0,1]$ (this means the events are really independent) [@gibson]. The reaction which occurs first is then picked and the system is updated. The process is repeated until one of the species, and then another one, go extinct. Fig. 3 shows the variation with time of the number of $A$ for a realization of the system in the cyclic competition scenario, and Fig. 4 shows the evolution of the number of $A$ population in the neutral drift case (in both realizations the total population size is $N=600$.) From the time series for the population numbers it looks as if the neutral drift model is some sort of “adiabatic” approximation of the cyclic competition model. If in the neutral drift model the population number is random walking, in the cyclic competition one the amplitude of oscillations is random walking! In other words, if we average the cyclic competition model over the cyclic orbits, we get essentially the same behaviour as the neutral drift model. Then the fluctuations drive both models from a neutrally stable cycle (point) to the neighbouring ones, and these fluctuations remain quite small by virtue of the vicinity of the neighbouring cycle (point.) It is quite funny how we all tend to go around in circles in life, even though the point we end up to is still the same!
We investigated the probability distribution of first extinction times. For that we ran fifty thousands copies of the system for each (different) size of the system, and the number of survivors was plotted vs. time. In the semilog axes we get a straight line, except for the few “rare events”. The slope of this straight line is -3.01 for system size 3000, -2.98 for system size 6000, -2.99 for system size 9000. Fig. 5 shows the plots for these three different system sizes: 3000, 6000, and 9000.
The time dependence of survival probability scales approximately with $1/N$, but there is still a very weak dependence left for small $N$. This can be justified by looking at the Fokker-Planck equation for this system, which contains both a term of order $N^0$ and $N^{-1}$.
To further investigate the behaviour of the system, we looked at the cummulative, conditional on being alive (which is equivalent to a normalization condition,) probability distribution for $H=xyz$. If $P(H<h|$alive$)$ increases linearly with $H$ for small $H$, the extinction rate will be non-zero, and we will get exponential decay. This corresponds to a constant distribution for $P(H|$alive$)$. Fig. 6 shows these plots for $t=1.25$, $t=1.5$, $t=1.75$, and $t=2.0$ in linear axes.
For the neutral drift case the data from the simulations give us a slope of -2.992 for system size 6000, -2.991 for system size 3000. The data collapse when time is rescaled with $1/N$, and that is in agreement with the fact that there is only a term of the order $1/N$ present in the Fokker-Planck equation. In Fig. 7 we show the number of survivors vs. time plots for cyclic (competition) and drift cases together, for system size $N=6000$. They clearly agree.
Probability Distribution at Long Times
======================================
In order to get an expression for $W$ for long times, we looked at our “experimental” data. It is possible to find a complete set of eigenfunctions of the Laplace equation, which vanish at the boundary. Having this complete set of eigenfunctions for the equilateral triangle [@bib23; @bib24], the probability density can be written:
$$W = \sum_{m,n} c_{m,n} {\phi}_{m,n}$$
where ${\phi}_{m,n}$ are the abovementioned symmetric eigenfunctions. The non-normalized eigenfunctions are then obtained from the general expression:
$$\phi_{(m,n)} (x,y) = \sum_{(m,n)} \pm \exp [{\frac { 2 \pi i}{3}} (nx + my)]$$
where we are using the relation $x+y+z=1$ and summation over the index pair $(m,n)$ means summation for $(-n,m-n)$, $(-n,-m)$, $(n-m,-m)$, $(n-m,n)$, $(m,n)$, $(m,m-n)$ [@bib24].
We get a symmetric eigenfunction when $m+n$ is a multiple of 3, $m$ is also a multiple of 3, but $m \neq 2n$, and $n \neq 2m$. [^1]
In order to calculate the expansion coefficients, we took snapshots of the system at given times, and then used the relation:
$$c_{(m,n)} (t) \propto \frac {1}{p} \sum_{l=1}^p { \phi_{(m,n)} (x_l(t),
y_l(t))}$$
where $p$ is the number of experimental points. The time-evolution of the expansion coefficients is given in Fig. 8. The doubly–degenerate functions die out quite soon, while the Lamé symmetric functions persist.
With the coefficients obtained this way we can then construct the probability density at different times. Some snapshots at the $W(t)$ are shown in Fig. 9.
At $t=0$ the $W(0)$ is a $\delta$-function. As time passes, for $t=0.1$ we see the $W$ starts to spread, and it spreads even more for $t=0.2$, becoming a “cake” for very long times ($t=1.5$,) for both the cyclic system and the neutral drift one. The “cake” obtained this way shows some Gibbs oscillations, which are due to the inclusion of only a few terms in the expansion, and the presence of the absorbing boundary. For a uniform probability distribution the expansion coefficients are proportional to $1/m$, where $m$ is the first index in the pair $(m,n)$. It can be seen that our “experimental” expansion coefficients approach those values.
The second–order Fokker–Planck equation for the drift case accepts solutions of the form:
$$W(x,y,z,t) = e^{-\lambda t} W(x,y,z)$$
with $\lambda$ the smallest eigenvalue of the spatial equation. We solved the eigenvalue problem by using the Galerkin method, with the help of Maple, and obtained a value of $\lambda = 3.01$ for the smallest eigenvalue, when we keep the first six eigenfunctions in the expansion. This is in very good agreement with our “experimental” data.
Case When Rates Are Not the Same
================================
Now consider the cyclic (competition) case when the rates of the three equations are not the same, i. e. the rate equations read like:
$$\begin{aligned}
N \frac {d A}{d t} = c_{13} AC - c_{12} AB \nonumber \\
N \frac {d B}{d t} = c_{21} AB - c_{23} BC \\
N \frac {d C}{d t} = c_{32} BC - c_{31} AC \nonumber\end{aligned}$$
We can always rescale the $A, B, C$ so that the matrix $c_{ij}$ be symmetric, and the equations read:
$$\begin{aligned}
N \frac {d A}{d t} = A (\beta C - \gamma B) \nonumber \\
N \frac {d B}{d t} = B (\gamma A - \alpha C) \\
N \frac {d C}{d t} = C (\alpha B - \beta A) \nonumber\end{aligned}$$
Again, $A+B+C=N=$const. By manipulating the rate equations, we can find the second integral of motion to be $A^{\alpha} B^{\beta} C^{\gamma}
=H=$const. The new centre will then be not at the point (1/3, 1/3, 1/3), but at $A = \alpha, B= \beta, C= \gamma$. The lines along which the quantity $A^{\alpha} B^{\beta} C^{\gamma}$ is constant are shown in Fig. 10.
We need to check whether the exponential decay behaviour is universal when the symmetry is broken this way. For this we ran fifty thousand simulations for different combinations of $\alpha$, $\beta$, and $\gamma$, for system size $N=3000$, starting from the new centre ($A=N
\cdot \alpha$, $B=N \cdot \beta$, $C=N \cdot \gamma$.) The number of survivors vs. time plots were again obtained, and those plots show that the exponential decay behaviour is robust. The time scale is now dependent on the initial distance from the boundary, with the equal rates case having the largest distance, and so the largest time scale (and the smallest slope.) Some of those plots are shown in Fig. 11. In that figure, the times for the equal rates case (for which the sum of rates is 3) are multiplied by three, to make them comparable to the times for the unequal rates case, for which the sum of rates is 1.
Conclusions
===========
We have considered an $ABC$ model in both the cyclic competition and neutral genetic drift versions, and studied the long-term behaviour of such a model. The number of the $A, B, C$ species in the cyclic competition case oscillates with a drifting with time amplitude, until one of the species (and then the next one in the cycle) goes extinct. In the neutral drift case the number of the $A, B, C$ species drifts, until one of them becomes zero. In both scenarios the number of survivors vs. time plots show an exponential decay, with the same exponent. The result is verified by writing and solving the Fokker–Planck equation for the second model. Finally, its robustness is checked against variations in the rate of the three different reactions in the system. It is very interesting to note that there is no difference in the time scale for the ensemble of species in cyclic competition and the case of neutral genetic drift.
There is growing concern about the effects of habitat fragmentation in the survival of the species [@bib13]. If the population of a certain species goes extinct in one patch (e.g. a herd, school, swarm) while it still survives in other patches, then what is known as “rescue effect” can prevent global extinction [@bib14; @bib25; @bib26]. Otherwise, the species is doomed to go extinct altogether. The models with individuals of “species” $A$, $B$, and $C$ distributed in a lattice have been studied by Szábo et al. [@bib10; @bib11; @bib22]. If we try to model habitat fragmentation as an ensemble of patches, each of those patches can be considered as one copy of our non-spatial system. However, with continuous migration in and out of the patch, the non-spatial picture described in this paper would be seriously perturbed. One aspect of this immigration and emigration has been recently discussed by Togashi and Kaneko [@bib30]. In a first approximation, the broad range of extinction times could relate to the persistence of species that exist in different “versions” such as certain lizards, or even some kinds of bacteria. Applied to its epidemiological scenario, it would relate to the endemicity of certain diseases, namely those with mutating pathogen, and then the outbreaks of epidemics in certain patches (areas). The many patches will then form a network [@bib27]. We have work in progress which places our $ABC$ ensemble in one of these small worlds with a scale-free topology.
Acknowledgements
================
We thank Nicolaas G. van Kampen for constructive criticism, and Michael Döbeli for helpful and interesting discussions and suggestions.
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[^1]: In \[24\], of all the symmetric eigenfunctions, Pinsky only considers the non-degenerate ones. There are also two-fold degenerate symmetric eigenfunctions, which we verified to be orthogonal to the original eigenfunctions.
|
The vortex phase diagram of the highly-anisotropic high-$T_c$ superconductor Bi$_{2}$Sr$_{2}$CaCu$_2$O$_{8+\delta}$ (BSCCO) has been intensively studied in the past few years. In superconductors, strong supercurrents flow near the surface, which produce nonuniform magnetic-field distribution in the sample. Such nonuniformity broadens the thermodynamic phase transitions and thereby hinders the study of the phase diagram. Also, the surface (or edge) currents produce a geometrical barrier [@geometrical] in flat samples at low fields, which gives rise to a hysteretic behavior [@Majer]. In higher fields, the Bean-Livingston surface barrier is known to be strong in BSCCO and this makes the global properties of the vortex system complicated [@Fuchs-local_ac]. A useful way to get rid of the effects of the surface and the magnetic-field nonuniformity is to measure the electromagnetic properties locally. There have been a number of efforts along this line [@Majer; @Fuchs-local_ac; @Zeldov; @Tamegai; @Ando; @Doyle], and the true nature of the vortex phases of BSCCO is beginning to be fully understood. For example, local magnetization measurements using microscopic Hall probes have found, quite conclusively, the presence of a first-order transition (FOT) of the vortex system [@Zeldov]. With the improvement of instrumentation and crystal quality, it has become clear that the first-order transition can also be determined as a step in the global dc magnetization measured with SQUID magnetometer [@Pastoriza; @Hanaguri; @Watauchi].
The miniature two-coil mutual-inductance technique [@Jeanneret] has been used for the study of the vortex phase diagram of BSCCO [@Ando; @Doyle]. With this technique, a small ac perturbation field is applied near the center of the crystal and therefore the the surface barrier, which hinders vortex entry and exit at the edge, has minimal effect on the measured response. Because of this advantage, a sharp distinct change in the local ac response has been observed [@Ando; @Doyle] and such a feature has been associated with a decoupling transition [@Glazman; @Daemen; @Ikeda] of the vortex lines. It is naturally expected that the decoupling line“ thus determined is identical to the FOT measured by dc magnetization measurements, although there has been no direct comparison between the two phenomenon measured on an identical sample. Since it is known that the first-order transition in the dc magnetization has a critical point and thus disappears above a certain field [@Zeldov], it is intriguing how the decoupling” signal of the miniature two-coil technique transforms at higher fields, above the critical point. In fact, the nature of the vortex matter in the field range above the critical point is still controversial [@Fuchs-Tx; @Forgan; @Horovitz]; since the ac technique can probe the growth of the correlation lengths of the vortex system [@Ando-YBCO], it is expected that the local ac measurement using the two-coil technique gives a new insight into the vortex phase transformations.
In this paper, we present the results of our miniature two-coil measurements and the global dc magnetization measurements on the same crystals. It is found that these two techniques detect the anomaly at the same temperature $T_{FOT}(H)$, directly demonstrating that the two phenomena are of the same origin. We measured crystals with three different dopings and confirmed that the result is reproducible among systems with different anisotropy. In higher fields where the FOT is not observed by the global dc magnetization measurement, a distinct feature is still observable in the local ac response and the position of such feature is weakly frequency dependent. We discuss that the frequency-dependent feature above the critical point is likely to originate from the growth of a short-range order in the vortex system.
The single crystals of BSCCO are grown with a floating-zone method and are carefully annealed and quenched to obtain uniform oxygen content inside the sample. We obtained three different dopings by annealing the crystals at different temperatures in air; annealing at 800$^{\circ}$C for 72 hours gives an optimally-doped sample with $T_c$=91 K (sample A), 650$^{\circ}$C for 100 hours gives a lightly-overdoped sample with $T_c$=88 K (sample B), and 400$^{\circ}$C for 10 days gives an overdoped sample with $T_c$=80 K (sample C). All the samples have the transition width of less than 1.5 K. A tactful quenching at the end of the anneal is essential for obtaining such a narrow transition width. $T_c$ is defined by the onset temperature of the Meissner signal in the dc magnetization measurement. The crystals are cut into platelets with lateral sizes larger than 3 $\times$ 3 mm$^2$ and the thickness of the samples are typically 0.02 mm. We used a very small (0.6 mm diameter) coaxial set of pickup and drive coils for our two-coil mutual-inductance measurements (see the inset to Fig. 1). The details of our technique have been described elsewhere [@Ando; @Ando-YBCO]. The amplitude of the drive current $I_d$ was 7.5, 7.5, and 1.0 mA for the measurements of samples A, B, and C, respectively. The linearity of the measured voltage with respect to $I_d$ was always confirmed. These $I_d$ produce the ac magnetic field of about 0.01 - 0.1 G at the sample. We emphasize that our two-coil geometry mainly induces and detects shielding currents flowing near the [*center*]{} of the sample, while usual ac-susceptibility measurements are most sensitive to shielding currents flowing near the [*edge*]{} of the sample. All the two-coil measurements are done in the field-cooled procedure. The global dc magnetization measurements are done with a Quantum Design SQUID magnetometer equipped with a slow temperature-sweep operation mode.
=7.5cm
Figure 1(a) shows the temperature dependence of the in-phase signals of our two-coil measurement on sample A in 190 G, taken at various frequencies from 3 kHz to 24 kHz. To compare the signals from different frequencies, the data are plotted in the unit of inductance change. It is apparent that there is a frequency-independent step-like change at a temperature $T_d$, which is 68.5 K here. The temperature dependence of the global dc magnetization in the same field is shown in Fig. 1(b), which shows that the FOT is taking place at exactly the same temperature as the step-like change in the two-coil signal.
According to the linear ac-response theory of the vortex system, the ac response is governed by the ac penetration depth $\lambda_{ac}$ [@Brandt; @Coffey]. $\lambda_{ac}$ in our configuration is related to the in-plane resistivity $\rho_{ab}$ in the manner $\rho_{ab}$=Re$(i\omega \mu_0 \lambda_{ac}^2)$ [@Ando-YBCO]. It has been reported that the apparent resistivity measured in the mixed state of BSCCO is largely dominated by the surface current [@Fuchs-local_ac]. Recent measurement of the bulk and surface contributions to the resistivity found [@Fuchs-bulk/strip] that the bulk contribution shows a sharp change at the FOT, while the surface contribution is governed by the surface barrier and shows a broader change. Since our measurement is not sensitive to the edge current, it is expected that $\lambda_{ac}$ of our measurement reflects mainly the bulk resistivity. Therefore, the step-like change in the local ac response is most likely to originate from the reported sharp change in the bulk resistivity [@Fuchs-bulk/strip]. We note that there has been a confusion about the origin of the step-like change in the local ac response measured by the miniature two-coil technique and it was discussed that the source of the sudden change may be related to a change in the $c$-axis resistivity [@Ando; @Doyle].
=6.5cm
Figure 2(a) shows the $T$ dependence of the in-phase signals of our two-coil measurement on sample A in three different magnetic fields. We observed that the sharp step-like change in the two-coil signal becomes broadened when the magnetic field exceeds a certain limit $H_{lim}$; in the case of sample A, the step-like change is observed in up to 400 G, but becomes broadened at 500 G. It was found that this $H_{lim}$ corresponds to the magnetic-field value at the critical point of the FOT; namely, the FOT in the dc magnetization measurement also disappears in fields above $H_{lim}$. Figures 2(b) and 2(c) show that the FOT is observed in the dc magnetization at 400 G but is not detectable at 500 G. This is also a clear evidence that the origin of the step-like change in the two-coil signal is the FOT.
In Fig. 2(a), the 500-G data do not show a step-like change, but clear changes in the slope at two separate temperatures, $T_{k1}$ and $T_{k2}$, are discernible. The signal changes much rapidly between $T_{k1}$ and $T_{k2}$ compared to the temperatures outside of this region, so the data look like that the step-like change at $T_{d}$ is broadened to the temperature region of $T_{k1}<$$T$$<T_{k2}$. Figure 3 shows the in-phase signals of sample A in 600 G, which is above $H_{lim}$, taken at various frequencies. Apparently, $T_{k1}$ and $T_{k2}$ inferred from the 600-G data change with frequency, although the change is small. This indicates that $T_{k1}$ and $T_{k2}$ do not mark a true phase transition but mark a crossover.
=7.0cm
Figures 4(a) and 4(b) show the in- and out-of-phase signals of samples B and C, respectively, in two selected magnetic fields below and above $H_{lim}$. Also in these two samples, the $T$ dependence of the two-coil signals show a step-like change in magnetic fields below $H_{lim}$, while the change is broadened in $H$$>$$H_{lim}$. Figure 5 shows the $T_{d}(H)$ lines for the three samples determined by our two-coil measurements. Clearly, the $T_{d}(H)$ line tends to be steeper for more overdoped samples. The $T_{FOT}$ data obtained from the dc magnetization are also plotted in Fig 5; apparently, $T_{d}(H)$ and $T_{FOT}(H)$ agree very well in all the three samples. The inset to Fig. 5 shows the $T_{d}(H)$ lines together with the $T_{k1}(H)$ and $T_{k2}(H)$ lines at higher fields (determined with 12 kHz), plotted versus normalized temperature $T/T_c$. The $T_{k1}(H)$ and $T_{k2}(H)$ lines are much steeper compared to the $T_{d}(H)$ line.
=6.0cm
After the existence of the first-order transition of the vortex system in BSCCO has been established [@Zeldov], much efforts have been devoted to the clarification of the details of the phase diagram. There have been accumulating evidences that the FOT line is a sublimation line, at which a solid of vortex lines transforms into a gas of pancake vortices [@Matsuda; @Fuchs-dc_flux]. In the $H$-$T$ phase diagram, there are two lines other than the FOT line, called depinning line“ and the second-peak line” [@Khaykovich]. The three lines merge at the critical point; the depinning line separates the low- and high-temperature regions at fields above $H_{lim}$ and the second-peak line separates the high- and low-field regions at low temperatures. Apparently, our $T_{k1}(H)$ and $T_{k2}(H)$ lines are very similar to the depinning line; thus, an examination of the $T_{k1}(H)$ and $T_{k2}(H)$ lines is expected to give an insight into the nature of the depinning line.
Since the step-like change at $T_{d}(H)$ marks an abrupt onset of the long-range correlation in the vortex system, the broadened change between $T_{k1}(H)$ and $T_{k2}(H)$ is expected to indicate an increase of a (short-range) correlation in the vortex system. In general, a probe with higher frequency looks at physics at shorter length scale [@Fisher]; in the case of our local ac response, $\lambda_{ac}(\omega)$ is smaller for larger $\omega$. With decreasing temperature, it is expected that the local ac response shows a qualitative change when the $c$-axis correlation length $L_c$ of the vortex system starts to grow, and another qualitative change at a lower temperature is also expected when $L_c$ becomes comparable to $\lambda_{ac}(\omega)$. This is one possible scenario for what is happening at $T_{k2}$ and $T_{k1}$. The facts that $T_{k1}$ and $T_{k2}$ are dependent on frequency and that a higher frequency gives a higher apparent $T_{k1}$ are consistent with the above scenario.
=7.0cm
Recently, Fuchs [*et al.*]{} used the change in the surface-barrier height for the determination of the vortex phase transformations [@Fuchs-Tx] (note here that the surface barrier is different from the geometrical barrier which is only effective at low fields near $H_{c1}$) and the presence of a new transition line, $T_x$ line, at temperatures higher than the depinning line (and above the FOT line) was suggested. Since it is almost clear that the vortex phase above the FOT line is a gas of pancake vortices at temperatures higher than this new $T_x$ line [@Fuchs-Tx], the existence of the $T_x$ line implies that the depinning line separates a highly disordered entangled vortex solid (low-temperature side) from either (a) disentangled liquid of lines with hexatic order or (b) some kind of solid which consists of an aligned stack of ordered two-dimensional pancake layers [@Fuchs-Tx]. Our data suggests that the latter possibility (b) is more likely, because the growth of the short-range correlation between $T_{k1}$ and $T_{k2}$ has a natural meaning of a growth of the alignment of the pancake layers in the latter picture. Note that we did not observe any feature which can be associated with the $T_x$ line; this is reasonable because the $T_x$ line only manifests itself in a change in the surface barrier, which has little effect on our measurement.
Finally, let us briefly discuss the magnetic-field dependence of $T_{d}$. As has been reported [@Ando; @Doyle], the $T_{d}(H)$ line measured with the two-coil technique can be well fitted with the formula for the decoupling line [@Glazman; @Daemen; @Ikeda]. This is actually a matter of course, because our $T_{d}(H)$ line is identical to the $T_{FOT}$ line and the FOT is most likely to be a sublimation transition, which is essentially a decoupling transition. Fittings of our data to the decoupling formula [@Glazman; @Daemen] $H \simeq H_0(T_c -T_d)/T_d$ give the anisotropy ratio $\gamma$ of $\sim$100, $\sim$85, and $\sim$77 for samples A, B, and C, respectively (the prefactor is given by $H_0\simeq \alpha_D \gamma^2 \phi_0^3/(4\pi \lambda(0))^2 T_c d$, where $\alpha_D \simeq$0.1 is a constant, $d$=15 ${\rm \AA}$ is the spacing between the bilayers, and $\lambda(0) \simeq$2000 ${\rm \AA}$ is the penetration depth).
In summary, we measured the local ac response of three BSCCO crystals (optimally doped, lightly overdoped, and overdoped samples) using a miniature two-coil technique and compared the result with a global dc magnetization measurement. The origin of the step-like change in the two-coil measurement is identified to be the first-order transition (FOT), where the [*bulk*]{} resistivity (which is free from the edge contribution) is reported to show a sharp change [@Fuchs-bulk/strip]. The sudden step-like change in the two-coil signal starts to be broadened at fields above $H_{lim}$, where the FOT is no longer observed. This broadened change takes place between $T_{k1}$ and $T_{k2}$ and these two temperatures are still well defined, although they are frequency dependent. We discuss that the observation of the feature at $T_{k1}$ and $T_{k2}$ is likely to indicate the growth of a short-range correlation of the vortex matter, which gives a clue to identify the nature of the depinning line.
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|
---
abstract: '[Shape-invariant signals under Fourier transform are investigated leading to a class of eigenfunctions for the Fourier operator. The classical uncertainty Gabor-Heisenberg principle is revisited and the concept of isoresolution in joint time-frequency analysis is introduced. It is shown that any Fourier eigenfunction achieve isoresolution. It is shown that an isoresolution wavelet can be derived from each known wavelet family by a suitable scaling. ]{}'
author:
- 'L. R. Soares[^1]'
- 'H. M. de Oliveira[^2]'
- 'R. J. Cintra[^3]'
- 'R. M. Campello de Souza[^4]'
bibliography:
- 'ref-clean.bib'
title: 'Fourier Eigenfunctions, Uncertainty Gabor Principle and Isoresolution Wavelets '
---
@twocolumn
**Keywords**\
[Gabor-Heisenberg inequality, Fourier eigenfunctions, Isoresolution wavelets, time-frequency analysis. ]{}
Preliminaries
=============
The Fourier transform is often interpreted as a linear operator $\mathcal{F}$. An interesting problem in this framework is to find out the eigenfunctions in the language of operators [@HERS64; @SOKORED66; @PEIDIN02]. Let $\mathcal{V}$ be a vector space equipped with a linear transform, $T: \mathcal{V}\to\mathcal{V}$, $\mathbf{v} \mapsto T(\mathbf{v})$. Under the linear transform $T$, eigenfunctions are solutions of $T(\mathbf{v})= \lambda \cdot \mathbf{v}$, which corresponds here to $\operatorname{\mathcal{F}}\{ f(t) \}(\omega) = \lambda \cdot f(\omega)$ where $f \in L^2(\mathbb{R})$ and $\lambda$ is a scalar. They are a quite remarkable class of functions, which preserves the shape under Fourier transform: Both the signal and its spectrum (time and frequency representation) have the same shape. In joint time-frequency representation [@COH95; @QIACHE99] this feature can represent a very good balance between the two domains. It is well known that the Gaussian pulse is a signal whose shape is preserved under the Fourier operator: $$\begin{aligned}
e^{-t^2/2}
\stackrel{\mathcal{F}}{\longleftrightarrow}
\sqrt{2\pi}
\cdot
e^{-\omega^2/2}
.\end{aligned}$$ This can easily be derived by writing $$\begin{aligned}
\frac{1}{\sqrt{2\pi}}
\int_{-\infty}^\infty
e^{-t^2/2}
\cdot
e^{j \omega t}
\operatorname{d}t
=
F(\omega)
.\end{aligned}$$ Deriving this equation and using integral by parts, one notice that: $\frac{\operatorname{d}}{\operatorname{d}\omega}F(\omega)=-\omega F(\omega)$. The solution of the differential equation $\frac{\operatorname{d}}{\operatorname{d}\omega}F(\omega)
+\omega F(\omega)
=
0$ under the initial condition $F(0)=1$ is $F(\omega) = e^{-\omega^2/2}$. It follows promptly that $\lambda = \sqrt{2\pi}$.
The question is: Are there other eigenfunctions? This matter is addressed in the next section. It is worthwhile to bear in mind that some results in this paper are deliberately *non nova, sed nove*.
Shape-invariant Signals: Eigenfunctions of the Fourier Operator
===============================================================
Let $\mathscr{E}\{\cdot\}$ and $\mathscr{O}\{\cdot\}$ denote the functionals that extract the even and odd part of a given signal, respectively.
\[proposition-1\] Let $f(t) \stackrel{\mathcal{F}}{\longleftrightarrow} F(\omega)$ be an arbitrary Fourier transform pair. Then the signal $$\begin{aligned}
h(t)
=
\sqrt{2\pi}
\cdot
\mathscr{E}
\{ f(t) \}
+
\mathscr{E}
\{ F(\omega) \}\end{aligned}$$ is invariant under the Fourier transform. Furthermore, we have that: $
H(\omega)
=
\operatorname{\mathcal{F}}
\{ h(t) \}(\omega)
=
\sqrt{2\pi}
\cdot
h(\omega)$.
It follows from the definition of $h(\cdot)$ that $$\begin{aligned}
2 \cdot h(t)
=
\sqrt{2\pi}
\cdot
\left[
f(t) + f(-t)
\right]
+
\left[
F(t) + F(-t)
\right]
.\end{aligned}$$ Taking the Fourier transform, $$\begin{aligned}
2 \cdot H(\omega)
=
\sqrt{2\pi}
\cdot
\left[
F(\omega) + F(-\omega)
\right]
+
\left[
2\pi f(-\omega) + 2\pi f(\omega)
\right]
.\end{aligned}$$ and the proof follows.
Each even function $f(t) \stackrel{\mathcal{F}}{\longleftrightarrow} F(\omega)$ induces a Fourier invariant $h(t) = \sqrt{2\pi} f(t) + F(t)$.
For instance, the following signals $$\begin{aligned}
h_1(t)
&=
\sqrt{2\pi}
\cdot
\frac{1}{1+t^2}
+
\pi
e^{-|t|}
,
\\
h_2(t)
&=
\sqrt{2\pi} |t|
-
\frac{2}{t^2}\end{aligned}$$ have spectra with similar shape. Another remarkable example is: $$\begin{aligned}
\label{equation-4}
\operatorname{sech}
\left(
\sqrt{\frac{\pi}{2}}
t
\right)
\stackrel{\mathcal{F}}{\longleftrightarrow}
\sqrt{2\pi}
\operatorname{sech}
\left(
\sqrt{\frac{\pi}{2}}
\omega
\right)
,\end{aligned}$$ where $\operatorname{sech}(\cdot)$ is the hyperbolic secant function.
Let $f(t) \stackrel{\mathcal{F}}{\longleftrightarrow} F(\omega)$ be an arbitrary Fourier transform pair. Then the signal $$\begin{aligned}
h(t)
=
\sqrt{2\pi}
\cdot
\mathscr{O}
\left\{
f(t)
\right\}
-
\mathscr{O}
\left\{
F(t)
\right\}\end{aligned}$$ is an invariant under Fourier transform. Furthermore, $\operatorname{\mathcal{F}}\left\{ h(t) \right\} = -\sqrt{2\pi} h(\omega)$.
The proof is similar to the proof of Proposition \[proposition-1\].
Each odd function $f(t) \stackrel{\mathcal{F}}{\longleftrightarrow} F(\omega)$ induces a Fourier invariant $
h(t)
=
\sqrt{2\pi}
f(t)
-
F(t)
$.
Let us now focus on a particular and important class of Fourier invariant, which generates an orthogonal and complete set. To begin with, let us denote by $\mathcal{E}$ the class of eigenfunctions of the Fourier operator defined according to the following proposition.
A signal $f(t)$ is in $\mathcal{E}$ if, and only if, the signal $f$ satisfies the differential equation $
\frac{\operatorname{d}^2}{\operatorname{d}t^2}
f(t)
-
t^2
\cdot
f(t)
=
\kappa
\cdot
f(t)$, for some scalar $\kappa \in \mathbb{C}$.
We begin demonstrating the suffiency. By hypothesis, we have that $$\begin{aligned}
f(t)
\stackrel{\mathcal{F}}{\longleftrightarrow}
\lambda
f(\omega)
.\end{aligned}$$ The properties of time and frequency differentiation for $\mathcal{F}$ give: $$\begin{aligned}
\frac{\operatorname{d}^2}{\operatorname{d}t^2}
f(t)
&
\stackrel{\mathcal{F}}{\longleftrightarrow}
(j\omega)^2
\lambda
f(\omega),
\\
(-jt)^2
f(t)
&
\stackrel{\mathcal{F}}{\longleftrightarrow}
\lambda
\frac{\operatorname{d}^2}{\operatorname{d}\omega^2}
f(\omega)
.\end{aligned}$$ Adding above expressions[^5], we derive $$\begin{aligned}
\frac{\operatorname{d}^2}{\operatorname{d}t^2}
f(t)
- t^2 f(t)
\stackrel{\mathcal{F}}{\longleftrightarrow}
-
\lambda
\left[
\frac{\operatorname{d}^2}{\operatorname{d}\omega^2}
f(\omega)
-
\omega^2
f(\omega)
\right]
.\end{aligned}$$
Thus, the signal $\frac{\operatorname{d}^2}{\operatorname{d}t^2}f(t)- t^2 f(t)$ has also its shape preserved, provided that $f$ itself preserves its shape. Therefore, $\frac{\operatorname{d}^2}{\operatorname{d}t^2}f(t)- t^2 f(t)\in\mathcal{E}$, that is, we are looking for signals such that $\frac{\operatorname{d}^2}{\operatorname{d}t^2}f(t)- t^2 f(t)=\kappa f(t)$, since they have identical eigenvalues.
Now we demonstrate the necessity.
By hypohthesis, the signal $f(t)$ satisfies the differential equation $\frac{\operatorname{d}^2}{\operatorname{d}t^2}f(t)- t^2 f(t)=\kappa f(t)$, $\kappa \in \mathbb{C}$. Applying the operator $\mathcal{F}$, we obtain: $$\begin{aligned}
(j\omega)^2
F(\omega)
+
\frac{\operatorname{d}^2}{\operatorname{d}\omega^2}
F(\omega)
=
\kappa
\lambda
F(\omega)
.\end{aligned}$$ Thus, $
\frac{\operatorname{d}^2}{\operatorname{d}\omega^2}
F(\omega)
-\omega^2
F(\omega)=
\kappa \lambda F(\omega)$, i.e., its spectrum also obeys a similar differential equation. Therefore, $f$ and $F$ have identical shape, since they are solutions of the same differential equation.
The key equation for shape-invariant signal is thus $\frac{\operatorname{d}^2}{\operatorname{d}t^2}f(t)- t^2 f(t)=\kappa f(t)$. Let us try solutions of the form $$\begin{aligned}
f(t)
=
p(t)
e^{-t^2/2}
,\end{aligned}$$ where $p(t)$ is a function to be determined. Therefore, $$\begin{aligned}
\frac{\operatorname{d}^2}{\operatorname{d}t^2}
\left[
p(t)
e^{-t^2/2}
\right]
-
t^2
p(t)
e^{-t^2/2}
=
\kappa
p(t)
e^{-t^2/2}
.\end{aligned}$$ After simple algebraic manipulations, we derive $$\begin{aligned}
\frac{\operatorname{d}^2}{\operatorname{d}t^2}
p(t)
-
2t
\frac{\operatorname{d}}{\operatorname{d}t}
p(t)
+
(\kappa + 1)
p(t)
=
0
,\end{aligned}$$ where $n$ is a integer.
A standard differential equation of the above form [@ABRASTE68] is $$\begin{aligned}
\label{equation-11}
\frac{\operatorname{d}^2}{\operatorname{d}t^2}
p(t)
-
2t
\frac{\operatorname{d}}{\operatorname{d}t}
p(t)
+
2n
p(t)
=
0
,\end{aligned}$$ where $n$ is a integer. Thus, for a suitable choice $\kappa = -(2n+1)$ (eigenvalues), the solutions $p(t)$ are exactly Hermite polynomials [@ABRASTE68], which form a complete orthogonal system. Thus, we have: $$\begin{aligned}
p(t)
=
H_n(t),\end{aligned}$$ where $$\begin{aligned}
H_0(t) &=1,
\\
H_1(t) &=2t,
\\
H_2(t) &=-2+4t^2,
\\
H_3(t) &=-12t+8t^3,
\\
H_4(t) &= 12-48t^2+16t^4,
\\
&\vdots\end{aligned}$$
Possible eigenvalues of the Fourier transform are the four roots of the unit ($\pm1, \pm j$) times $\sqrt{2\pi}$.
Let us denote by $\mathcal{F}^{(n)}$ the operator corresponding to iterate $n$ times the operator $\mathcal{F}$. Let $t
\stackrel{\mathcal{F}}{\longleftrightarrow}
\omega
\stackrel{\mathcal{F}}{\longleftrightarrow}
\omega'
\stackrel{\mathcal{F}}{\longleftrightarrow}
\Omega$ be the Fourier domain variables for the iterate Fourier transform. Observe that, for $f\in \mathcal{E}$, we have: $$\label{equation-13}
\begin{split}
\operatorname{\mathcal{F}}^{(2)}
\left\{
f(t)
\right\}
(\omega')
&=
2\pi
f(-\omega')
,
\\
\operatorname{\mathcal{F}}^{(4)}
\left\{
f(t)
\right\}
(\Omega)
&=
4\pi^2
f(\Omega)
.
\end{split}$$ But, $$\label{equation-14}
\begin{split}
\operatorname{\mathcal{F}}^{(2)}
\left\{
f(t)
\right\}
(\omega')
&=
\lambda^2
f(-\omega')
,
\\
\operatorname{\mathcal{F}}^{(4)}
\left\{
f(t)
\right\}
(\Omega)
&=
\lambda^4
f(\Omega)
.
\end{split}$$ From and , it follows that $\lambda / \sqrt{2\pi} \in \mathbf{C}$ has order 4.
We conclude that $\left\{
\psi_n(t)
=
H_n(t)
e^{-t^2/2}
\right\}_{n=0}^\infty$ are shape-invariant under Fourier operator associated to $\lambda_n = (-j)^n \sqrt{2\pi}$. Therefore, $$\begin{aligned}
\label{equation-15}
H_n(t)
e^{-t^2/2}
\stackrel{\mathcal{F}}{\longleftrightarrow}
(-j)^n
\sqrt{2\pi}
H_n(\omega)
e^{-\omega^2/2}
.\end{aligned}$$ Another interpretation can be derived evoking Rodrigues’ formula [@ABRASTE68]: $$\begin{aligned}
H_n(t)
=
(-1)^n
e^{t^2}
\frac{\operatorname{d}^n}{\operatorname{d}t^n}
e^{-t^2}
.\end{aligned}$$ The 2nd-order differential equation hold by invariant signals is $$\begin{aligned}
\frac{\operatorname{d}^2}{\operatorname{d}x^2}
y
+
(2n+1-x^2)
y
=
0
.\end{aligned}$$ The above differential equation is exactly the celebrated Schrödinger equation for the harmonic oscillator [@BEI67].
Consequences on the Time-Frequency Plane
========================================
Let us now investigate certain consequences of eigenfunctions of the Fourier operator on the time-frequency plane [@COH95; @OLIBAR00].
Let $f(t)$ be a finite energy signal $E$, equipped with Fourier transform, $F(\omega)$. The time and frequency moments of $f$ are defined by: $$\begin{aligned}
\overline{t^n}
&
=
\frac
{
\int_{-\infty}^\infty
f^\ast(t)
t^n
f(t)
\operatorname{d}t
}
{
\int_{-\infty}^\infty
f^\ast(t)
f(t)
\operatorname{d}t
}
,
\\
&
=
\frac{1}{E}
\int_{-\infty}^\infty
t^n
|f(t)|^2
\operatorname{d}t
\\
\overline{\omega^n}
&
=
\frac
{
\int_{-\infty}^\infty
F^\ast(\omega)
\omega^n
F(\omega)
\operatorname{d}\omega
}
{
\int_{-\infty}^\infty
F^\ast(\omega)
F(\omega)
\operatorname{d}\omega
}
\\
&
=
\frac{1}{2\pi E}
\int_{-\infty}^\infty
\omega^n
|F(\omega)|^2
\operatorname{d}\omega
.\end{aligned}$$
By analogy to Probability Theory, the term $|f(t)|^2/E$ denotes a “time-domain” energy density, where $E$ is a normalising factor so as to make the whole integral of the density be equal to the unity. It is customary to deal with the energy spectral density $G(\omega)=|F(\omega)|^2$, whose integral over a frequency band gives the energy content of the signal within such a band. Let us suppose in the sequel, without loss of generality, that $E=1$ (energy normalised signals).
The “effective duration” (respectively the “effective frequency width”) of a signal $f(t)$ (respectively $F(\omega)$) is defined according to: $$\begin{aligned}
\Delta t
&=
\sqrt{2 \pi \overline{(t-\overline{t})^2}}
\quad
\text{r.m.s duration,}
\\
\Delta f
&=
\sqrt{2 \pi \overline{(f-\overline{f})^2}}
\quad
\text{r.m.s bandwidth}
,\end{aligned}$$ where $\Delta t$ and $\Delta f$ correspond to the standard deviation, i.e., spreading measures. However, other common and much handier definitions are $$\begin{aligned}
\Delta_t
&
=
\sqrt{\overline{(t-\overline{t})^2}}
,
\\
\Delta_\omega
&
=
\sqrt{\overline{(f-\overline{f})^2}}
.\end{aligned}$$ Clearly, $\Delta_t = \Delta t / \sqrt{2\pi}$ and $\Delta_\omega = \sqrt{2\pi} \Delta f$.
Revisiting the Gabor Principle
------------------------------
By applying arguments from quantum mechanics [@BEI67], Gabor [@GAB46; @GAB53] derived an uncertainty relation nowadays called Gabor-Heisenberg principle for signals: $\Delta t \cdot \Delta f \geq 1/2$, proving that time and frequency cannot be exactly measured (simultaneously). The Gabor-Heisenberg uncertainty principle states a lower bound on the product $\Delta t \cdot \Delta\omega$, or alternatively: $$\begin{aligned}
\label{equation-20}
\Delta_t\cdot\Delta_\omega \geq 1/2
.\end{aligned}$$
The Gabor lower bound is only achieved by the first invariant signal (eigenfunctions of $\mathcal{F}$ operator).
*Sketch of the proof:* From , the bound is achieved if, and only if, $\frac{\operatorname{d}}{\operatorname{d}t}
f(t)
=
\kappa
t
f(t)$ This condition can be interpreted as: “derivative in time domain” is equivalent to the “’derivative in frequency domain”. Therefore, $$\begin{aligned}
\frac{\operatorname{d}^2}{\operatorname{d}t^2}
f(t)
=
\kappa
\left[
f(t)
+ t
\cdot
\frac{\operatorname{d}}{\operatorname{d}t}
f(t)
\right]
=
\kappa f(t)
+
\kappa^2
t^2 f(t)
.\end{aligned}$$ Simple manipulations yield: $$\begin{aligned}
\frac{\operatorname{d}^2}{\operatorname{d}t^2}
f(t)
-
\kappa
(1+\kappa t^2)
(\kappa t)^2
\cdot
f(t)
=
0
.\end{aligned}$$ The only solutions on $\mathcal{E}$ correspond to $\kappa=\pm1$, i.e., $\frac{\operatorname{d}^2}{\operatorname{d}t^2}
f(t)
+
(1-t^2)f(t)=0$ or $\frac{\operatorname{d}^2}{\operatorname{d}t^2}
f(t)
-
(1+t^2)f(t)=0$.
Any real signal $f(t) \stackrel{\mathcal{F}}{\longleftrightarrow} F(\omega)$ such that $f(t),
\frac{\operatorname{d}}{\operatorname{d}t}f(t),
F(\omega),
\frac{\operatorname{d}}{\operatorname{d}\omega}F(\omega)
\in L^2(\mathbb{R})$ have finite resolutions.
Applying the Parseval-Plancherel Theorem [@ABRASTE68], it follows that $$\begin{aligned}
\int_{-\infty}^\infty
t^2
f^2(t)
\operatorname{d}t
&
=
\int_{-\infty}^\infty
[j t f(t)]
\cdot
[j t f(t)]^\ast
\operatorname{d}t
\\
&
=
\frac{1}{2\pi}
\int_{-\infty}^\infty
\left|
\frac{\operatorname{d}}{\operatorname{d}\omega}
F(\omega)
\right|^2\end{aligned}$$ and $$\begin{aligned}
\int_{-\infty}^\infty
\omega^2
\left|
F^2(\omega)
\right|^2
\operatorname{d}\omega
&
=
\int_{-\infty}^\infty
[j \omega F(\omega)]
\cdot
[j \omega F(\omega)]^\ast
\operatorname{d}\omega
\\
&
=
2\pi
\int_{-\infty}^\infty
\left[
\frac{\operatorname{d}}{\operatorname{d}t}
f(t)
\right]^2
.\end{aligned}$$ Therefore, $$\begin{aligned}
\Delta_t^2
=
\frac{
\int_{-\infty}^\infty
\left|
\frac{\operatorname{d}}{\operatorname{d}\omega}
F(\omega)
\right|^2
\operatorname{d}\omega
}
{
\int_{-\infty}^\infty
\left|
F(\omega)
\right|^2
\operatorname{d}\omega
}
<
\infty
,
\\
\Delta_\omega^2
=
\frac{
\int_{-\infty}^\infty
\left|
\frac{\operatorname{d}}{\operatorname{d}t}
f(t)
\right|^2
\operatorname{d}t
}
{
\int_{-\infty}^\infty
\left|
f(t)
\right|^2
\operatorname{d}\omega
}
<
\infty
.\end{aligned}$$ Thus, the above quantities are given by the square root of the ratio between the energy of the signal derivative and the energy of signal itself. Thus, the resolution for the Fourier invariant signal $\operatorname{sech}(\cdot)$ given by is $$\begin{aligned}
\Delta_t
=
\Delta_\omega
=
\sqrt{\frac{\pi}{6}}
\approx
0.7235987766\ldots\end{aligned}$$ since that $$\begin{gathered}
\int_{-\infty}^\infty
\operatorname{sech}(t)
\operatorname{d}t
=2
,
\\
\int_{-\infty}^\infty
\operatorname{tanh}^2(t)
\operatorname{sech}^2(t)
\operatorname{d}t
=\frac{2}{3}
,
\\
\int_{-\infty}^\infty
\left(
\frac{2 t}{\pi}
\right)^2
\operatorname{sech}^2(t)
\operatorname{d}t
=\frac{2}{3}
,\end{gathered}$$ where $\operatorname{tanh}(\cdot)$ is the hyperbolic tangent function.
\[[[@GAB46]]{}\] Time-frequency uncertainty of Fourier Eigenfunctions $\psi_n^\ast(t) = H_n(t) e^{-t^2/2} e^{j\omega_0 t + \phi_0}$, where $\omega_0$ and $\phi_0$ are constants, attain quantized values of the Gabor-Heisenberg lower bound, i.e. $$\begin{aligned}
\Delta t \cdot \Delta f = \frac{1}{2} \cdot (2n+1)
,
\\
\Delta_t \cdot \Delta_\omega = \frac{1}{2} \cdot (2n+1)
.\end{aligned}$$
That is why Gabor functions are relevant in some problems (e.g. [@OKA98].)
The Concept of Isoresolution Wavelet
====================================
The concept of isoresolution analysis is introduced in this section. According to the Gabor principle, if one increases resolution in one domain, the resolution must decrease in the other domain so as to guarantee the lower bound given by . When analysing signals in joint time-frequency plane, frequently, there is no grounds to assure a better resolution in a domain than in the other domain. As an interesting property, any Fourier eigenfunction achieves isoresolution as it can be seen by the following proposition.
Fourier-invariant signals perform an isoresolution, that is, $\Delta t = \Delta_\omega$
Supposing that $f\in\mathcal{E}$, then $F(\omega) = \lambda f(\omega)$. Therefore: $$\begin{aligned}
\frac{
\int_{-\infty}^\infty
F(\omega)
\omega^2
F(\omega)^\ast
\operatorname{d}\omega
}
{
\int_{-\infty}^\infty
\left|
F(\omega)
\right|^2
\operatorname{d}\omega
}
=
\frac{
\int_{-\infty}^\infty
\omega^2
|\lambda|^2
f^2(\omega)
\operatorname{d}\omega
}
{
\int_{-\infty}^\infty
\left|
\lambda
\right|^2
f^2(\omega)
\operatorname{d}\omega
}\end{aligned}$$ and the proof follows.
Wavelet name Time resolution $\Delta_t$ Frequency resolution $\Delta_\omega$ Isoresolution factor $\sqrt{\Delta_t/\Delta_\omega}$
---------------- ---------------------------- -------------------------------------- ------------------------------------------------------
`Gaus1` 1.500000 1.500000 1.000000
`mexh` 1.166667 2.500000 0.683130
`morl` 0.500002 25.499997 0.140028
`fbsp 2-1-0.5` $\infty$ 14.475133 -
`shan 1-0.5` $\infty$ 13.159733 -
`haar` 0.333333 $\infty$
This is an interesting property for signalling on the joint time-frequency plane.
It is suggested here the changing of the time-frequency resolution by a proper scaling that allows for identical resolution in both domains.
\[proposition-9\] If $\psi(t)$ has effective duration $\Delta t$ and effective bandwidth $\Delta \omega$, then the scaled version $\psi\left( \sqrt{\Delta_t / \Delta_\omega}\,\, t \right)$ achieves isoresolution.
Scaled versions $\psi(at)$, $a\neq0$, have resolutions $\Delta_t/|a|$ and $|a|\cdot \Delta_\omega$, so $|a|$ can be appropriately chosen. The quantity $\sqrt{\Delta_t / \Delta_\omega}$ is refered to as the *isoresolution factor*. The essential idea of isoresolution can be placed in the wavelet structure. Normally, the basic wavelet of a family $\frac{1}{\sqrt{|a|}}\psi\left(\frac{t-b}{a}\right)$ holds the admissibility condition but often does not achieve isoresolution. We propose here to redefine the basic wavelet of a family so as to achieve isoresolution. For instance, the standard Mexican hat wavelet $\psi_\text{Mhat}(t)$ satisfies: $$\begin{aligned}
2(t^2-1)
\cdot
\frac{e^{-t^2/2}}{\sqrt[4]{\pi}\sqrt{3}}
\stackrel{\mathcal{F}}{\longleftrightarrow}
-2
\sqrt{\frac{2}{3}}
\sqrt[4]{\pi}
\omega^2
e^{-\omega^2/2}
.\end{aligned}$$
The isoresolution Mexican hat wavelet can be found applying Proposition \[proposition-9\]: $$\begin{aligned}
\sqrt{\frac{7}{15}}
\cdot
\psi_\text{Mhat}
\left(
\sqrt{\frac{7}{15}}
\,\,
t
\right)
.\end{aligned}$$ For any isoresolution wavelet, the scaling by $a>1$ or $a<1$ corresponds to unbalance resolution in a different way. Table \[table-1\] displays both time and frequency resolution for a few known continuous wavelets: Gaussian derivatives, Mexican hat, Morlet, frequency B-Spline, Shannon, and Haar [@MIS01]. The wavelet `Gaus1` is an invariant wavelet therefore it achieves isoresolution, in accordance to proposition 8. It is valuable to mention that compact support wavelets (in time or frequency) cannot attain isoresolution, since no signal can simultaneously be time and frequency limited [@WOZJAC67].
Perspectives and Closing Remarks
================================
Eigenfunctions of the Fourier operator were investigated and the Gabor principle was revisited defining the concept of isoresolution, i.e, a signal with the same time and frequency resolution. The functions $\left\{ \psi_n(t) \right\}$ (see ) turn up as a very appealing choice for designing representations such as wavelets. It is time to try finding new wavelets starting with . Since they are solutions of a wave equation (2nd order differential equation), our approach (Mathieu [@LIRA03a], Legendre [@LIRA03b], Chebyshev [@CIN03]) can be useful to construct new wavelets: The Quantum Wavelets, or Gabor-Schrödinger wavelets. The construction of new wavelets based on these complete, orthogonal, domain shape-invariant system is currently being investigated. The idea is to adapt the concept of isoresolution in orthogonal multiresolution analysis [@DEO03; @HESSWIK96].
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by CNPq and CAPES. The authors also thank Mr. M. Müller for some motivating discussions.
[^1]: L. R. Soares was with the Grupo de Pesquisa em Comunicações (<span style="font-variant:small-caps;">codec</span>), Departamento de Eletrônica e Sistemas, Universidade Federal de Pernambuco.
[^2]: H. M. de Oliveira was with the Grupo de Pesquisa em Comunicações (<span style="font-variant:small-caps;">codec</span>), Departamento de Eletrônica e Sistemas, Universidade Federal de Pernambuco. Currently he is with the Signal Processing Group, Departamento de Estatística, Universidade Federal de Pernambuco. Email: [[email protected]]([email protected])
[^3]: R. J. Cintra was with the Graduate Program in Electrical Engineering, Universidade Federal de Pernambuco. Currently he is with the Signal Processing Group, Departamento de Estatística, Universidade Federal de Pernambuco. Email: [[email protected]]([email protected])
[^4]: R. M. Campello de Souza is with Departamento de Eletrônica e Sistemas, Universidade Federal de Pernambuco. Email: [[email protected]]([email protected])
[^5]: N.B. Subtracting: $\frac{\operatorname{d}^2}{\operatorname{d}t^2}
f(t)
+ t^2 f(t)
\stackrel{\mathcal{F}}{\longleftrightarrow}
-
\lambda
\left[
\frac{\operatorname{d}^2}{\operatorname{d}\omega^2}
f(\omega)
+
\omega^2
f(\omega)
\right]$.
|
---
author:
- 'A. Monreal-Ibero'
- 'J. R. Walsh'
- 'J. M. Vílchez'
bibliography:
- 'mybib\_aa.bib'
date: 'Received: 4 May 2012 / Accepted: 8 June 2012 '
subtitle: '2D mapping of the physical and chemical properties[^1]'
title: 'The ionized gas in the central region of :'
---
[Blue Compact Dwarf (BCD) galaxies constitute the ideal laboratories to test the interplay between massive star formation and the surrounding gas. As one of the nearest BCD galaxies, NGC 5253 was previously studied with the aim to elucidate in detail the starburst interaction processes. Some open issues regarding the properties of its ionized gas still remain to be addressed.]{} [The 2D structure of the main physical and chemical properties of the ionized gas in the core of NGC 5253 has been studied.]{} [Optical integral field spectroscopy (IFS) data has been obtained with FLAMES Argus and lower resolution gratings of the Giraffe spectrograph.]{} [ We derived 2D maps for different tracers of electron density ($n_e$), electron temperature ($T_e$) and ionization degree. The maps for $n_e$ as traced by [\[O<span style="font-variant:small-caps;">ii</span>\]]{}, [<span style="font-variant:small-caps;">\[Sii\]</span>]{}, [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}, and [\[Ar<span style="font-variant:small-caps;">iv</span>\]]{} line ratios are compatible with a 3D stratified view of the nebula with the highest $n_e$ in the innermost layers and a decrease of $n_e$ outwards. To our knowledge, this is the first time that a $T_e$ map based on [<span style="font-variant:small-caps;">\[Sii\]</span>]{} lines for an extragalactic object is presented. The joint interpretation of the $T_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) and $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}) maps is consistent with a $T_e$ structure in 3D with higher temperatures close to the main ionizing source surrounded by a colder and more diffuse component. The highest ionization degree is found at the peak of emission for the gas with relatively high ionization in the main [GH<span style="font-variant:small-caps;">ii</span>R]{} and lower ionization degree delineating the more extended diffuse component. We derived abundances of oxygen, neon, argon, and nitrogen. Abundances for $O$, $Ne$ and $Ar$ are constant over the mapped area within ${\hbox{\rlap{\lower.55ex\hbox{$\sim$}} \kern-.3em
\raise.4ex \hbox{$<$}}}$0.1 dex. The mean $12+\log(O/H)$ is $8.26\pm0.04$ while the relative abundances of $\log(N/O)$, $\log(Ne/O)$ and $\log(Ar/O)$ were $\sim-1.32\pm0.05$, $-0.65\pm0.03$ and $-2.33\pm0.06$, respectively. There are two locations with enhanced $N/O$. The first ($\log(N/O)\sim-0.95$) occupies an area of about 80 pc$\times$35 pc and is associated to two super star clusters. The second ($\log(N/O)\sim-1.17$), reported here for the first time, is associated to two moderately massive ($2-4\times10^4$ M$_\odot$) and relatively old ($\sim10$ Myr) clusters. A comparison of the $N/O$ map with those produced by strong line methods supports the use of N2O2 over N2S2 in the search for chemical inhomogeneities *within* a galaxy. The results on the localized nitrogen enhancement were used to compile and discuss the factors that affect the complex relationship between Wolf-Rayet stars and $N/O$ excess. ]{}
Introduction
============
Local Blue Compact Dwarf galaxies [BCDs, see @kun00 for a review] constitute ideal laboratories to test how the interaction between massive star formation, gas and dust affects galaxy evolution. Specifically, both winds of massive stars and supernova explosions i) inject mechanical energy to the interstellar medium (ISM), redistributing the gas within the galaxy and therefore, quenching (or igniting) future star formation; ii) eject processed material into the ISM thus causing a chemical enrichment of the galaxy.
NGC 5253 is an example BCD particularly suited for the study of the interaction between gas and massive star formation, since it is relatively close and a wealth of ancillary information is available in all spectral ranges, from X-ray to radio. This galaxy belongs to the Centaurus A / M 83 complex [@kar07] and is suffering a recent burst of star formation. The existence of an <span style="font-variant:small-caps;">Hi</span> plume extending along the optical minor axis [@kob08] supports the idea of the burst being caused by a former encounter with M 83 [@van80]. A wealth of studies at different wavelengths shows that this galaxy is peculiar in several aspects. In particular, high resolution multiband photometry with the *Hubble Space Telescope* (HST) revealed the existence of several relatively young star clusters in the central region of the galaxy with typical masses of $\sim2-120\times10^3$ M$_\odot$ [e.g. @har04]. Among those, two very massive ($\sim1-2\times10^6$ M$_\odot$) Super Star Clusters (SSCs) at the nucleus of the galaxy, separated by $\sim$04 [@alo04] and associated with a very dense compact H<span style="font-variant:small-caps;">ii</span> region, only detected in the radio at 1.3 cm and 2 cm [@tur00], stand out as candidates to be the youngest globular cluster(s) yet observed [@gor01]. Also, @lop12 showed how this galaxy does not satisfy the Schmidt-Kennicutt law of star formation and seems to be slightly metal-deficient in comparison with starbursts of similar baryonic mass. But probably, the most well-known peculiarity of NGC 5253 is the existence of areas with enhanced abundance of nitrogen. This was first reported by @wel70 and confirmed afterwards by several works [e.g. @wal89].
Narrow band imaging showed that as a consequence of its starburst nature, the ionized gas in presents a complex structure that includes filaments and arcs [e.g. @cal04]. Therefore, proper characterization of the properties of the ionized gas in would benefit from high quality two-dimensional spectral mapping over a contiguous area of interest. Nowadays, Integral Field Spectroscopy (IFS) facilities represent the obvious choice to obtain this kind of information and several works devoted to the study of BCDs using this approach have been published in recent years [e.g. @izo06; @gar08; @jam09; @wes10].
In particular, @mon10 [hereafter Paper I] carried out a detailed study of the central region in using IFS data collected with FLAMES [@pas02]. One of the topics studied in more detail there was the kinematics of the ionized gas. We found that the line profiles were complex, needing up to three components to reproduce them. In the main Giant H<span style="font-variant:small-caps;">ii</span> Region ([GH<span style="font-variant:small-caps;">ii</span>R]{}, see Fig. \[apuntado\]), one of them presented supersonic widths while the other two were relatively narrower. Moreover, the broad component presented and excess in nitrogen of $\sim$1.4 times than that of the narrow one [@mon11b]. This was consistent with a scenario where the two SSCs produce an outflow that encounters the previously quiescent gas. Also, we delimited very precisely the area polluted with extra nitrogen, based on the excess of the [<span style="font-variant:small-caps;">\[Nii\]</span>$\lambda$6584/H$\alpha$]{} ratio with respect to [<span style="font-variant:small-caps;">\[Sii\]</span>$\lambda\lambda$6717,6731/H$\alpha$]{} and clearly demonstrated that, at least, some of the Wolf-Rayet (WR) star population cannot be the cause of this enrichment. Moreover, we could resolve a long-standing issue regarding the elusive He<span style="font-variant:small-caps;">ii</span> emission at $\lambda$4686 Å. @cam86 mentioned a possible detection of He<span style="font-variant:small-caps;">ii</span> but this result was not certainly confirmed afterwards. We detected several localized areas with clear He<span style="font-variant:small-caps;">ii</span>$\lambda$4686 detection although, rather puzzlingly, not coincident in general with the area exhibiting extra nitrogen.
With the limited spectral range utilized in that work, we could only determine *relative* nitrogen abundances. Moreover, while interpreting the increase of [<span style="font-variant:small-caps;">\[Nii\]</span>]{}$\lambda$6584/[<span style="font-variant:small-caps;">\[Sii\]</span>]{}$\lambda\lambda$6717,6731 as caused by an enhancement in nitrogen abundance is the most natural explanation and is supported by previous observational evidence [e.g. @kob97], this is not the only alternative. Instead, specific combinations of ionization structure and metallicity gradient could also produce similar observables. Since dwarf galaxies are not expected to have strong metallicity gradients in a similar manner to large spirals, this option seems unlikely. However, it could not be rejected *a priori* in our previous work.
Here, we present the natural continuation of the work in . To overcome the mentioned drawbacks, we obtained new FLAMES observations that allow us to map missing physical properties such as electron temperature ($T_e$) and local degree of excitation. In this manner we are able to: i) map the chemical content in the central part of the galaxy and detect inhomogeneities (if any) in an unbiased manner; ii) evaluate how well relative abundance tracers based on strong emission lines reproduce the values derived from direct measurements; iii) check if the derived physical and chemical structure is consistent with the picture sketched in .
The paper is organized as follows: Sec. \[obsred\] describe the observations and technical details about the data reduction and processing necessary to extract the required information for the analysis. Sec. \[resultados\] contains the results from one Gaussian fitting for the physical (i.e. $T_e$, $n_e$, ionization degree) and chemical (i.e. metallicity and relative abundances) properties both in 2D and in particularly interesting apertures. Sec. \[discusion\] will explore how the derived maps trace the 3D physical structure of the central part of , and the reliability (or limitations) of strong line based tracers to determine the relative abundance in nitrogen. Basic information for can be found in Table 1 in .
Observations and data processing \[obsred\]
===========================================
Observations
------------
Data were obtained with the *Fibre Large Array Multi Element Spectrograph*, FLAMES [@pas02] at Kueyen, Telescope Unit 2 of the 8 m VLT at ESO’s observatory on Paranal. We used the ARGUS Integral Field Unit with the sampling of 0.52$^{\prime\prime}$/lens. This permits coverage of a field of view (fov) of $11\farcs5 \times 7\farcs3$. Utilized gratings were L385.7 (LR1), L427.2 (LR2), and L479.7 (LR3) and L682.2 (LR6).
Data for the two last gratings were obtained in visitor mode on February 10, 2007 (programme 078.B-0043). Details of these observations appear in . In addition to these, data for the L385.7 (LR1) and L427.2 (LR2) were obtained in service mode during June 2009 (programme 383.B-0043). The spectral range, resolving power, exposure time and airmass for each configuration are listed in Table \[log\_observaciones\]. Seeing ranged typically between $0\farcs4$ and $1\farcs2$ with a median($\pm$standard deviation) of $\sim0\farcs8$ $(\pm0\farcs2)$. All the data were taken under clear conditions. In addition, standard sets of calibration files were obtained. These included continuum and ThAr arc lamps exposures as well as frames for the spectrophotometric standard stars HR 7596 and LTT 3218 for the L385.7 (LR1) and L427.2 (LR2) gratings respectively.
Due to a guiding problem, there was a $\sim2\farcs9$ offset between the pointings on runs 078.B-0043 and 383.B-0043. In spite of it, the main emitting regions were covered by all the configurations. The precise area covered in each run is shown in Figure \[apuntado\] which contains the FLAMES fov over-plotted on an HST $B$, [H$\alpha$]{}, $I$ colour image.
[ccccccccc]{} Grating & Spectral range & Resolution & t$_{\mathrm{exp}}$ & Airmass\
& (Å) & & (s) &\
L385.7 & 3610–4081 & 12800 & $21\times895$ & 1.00–1.10\
L427.2 & 3964–4567 & 10200 & $9\times895$ & 1.00–1.09\
\
\
\
Data reduction
--------------
The processing of the data on for run 078.B-0043 is described in . Those for run 383.B-0043 were processed using a combination of the pipeline provided by ESO (version 2.8.9)[^2] via `esorex`, version 3.9.0 and some IRAF[^3] routines.
The corresponding master calibration files were created with the ESO pipeline tasks `gimasterbias`, `gimasterflat` and `giwavecalibration`. In particular, fifteen bias frames were used for the masterbias creation while six and three continuum lamp exposures were needed for the creation of the L385.7 and L427.2 masterflats, respectively. In order to remove the cosmic rays in the target exposures, we combined all the individual frames for a given grating using `imcombine` with IRAF. After that, each combined frame was processed using the ESO pipeline in order to perform bias subtraction, spectral tracing and extraction, wavelength calibration and correction of fibre transmission and data cube creation.
In both settings, four isolated and evenly distributed arc lines were fitted by a Gaussian in each spectrum in order to estimate the uncertainties in the wavelength calibration as well as the instrumental width. Centroids of the lines were determined with an accuracy of $\sim0.005$ Å which, for the covered spectral range, translates into velocities of $\sim$0.3-0.4 km s$^{-1}$. We measured a spectral resolution of FWHM $\sim$0.329 Å and 0.446 Å for the L385.7 and L427.2, respectively. This translates to $\sigma_{instru} \sim$ 10.9 and 13.3 km s$^{-1}$.
In the next step, the sky background was subtracted. For that purpose, we created a good signal-to-noise (S/N) spectrum by averaging the spectra of the sky fibres in each combined frame which was subtracted from every spectrum. Fibers suffering significant contamination due to cross-talk of adjacent object fibres were excluded from this combination. Relative flux calibration was performed within IRAF. The sky in the standard star frames was subtracted following the same methodology as with the science frames and then a spectrum was formed for each calibration star by co-adding all the fibres of each standard star frame. Then, a sensitivity function was determined with the IRAF tasks `standard` and `sensfunc` and science frames were calibrated with `calibrate`.
Finally, since data were taken in clear conditions, it was necessary to place both datacubes on the same (relative) flux scale. We scaled each spectrum in the LR1 mode to the same level as that of LR2 using the H7 and \[Ne<span style="font-variant:small-caps;">iii</span>\]$\lambda$3967 emission lines, which were present in both instrumental set-ups.
Subtracting the emission of the stellar population
--------------------------------------------------
The spectral region analyzed in this work is rich in spectral features caused by the underlying stellar population (e.g. Balmer absorption lines). Their effect on the estimation of the flux in the emission lines for the gas is negligible for the brightest lines and/or those in the area of the bright [GH<span style="font-variant:small-caps;">ii</span>R]{}. However, they may affect the measurements in the area of lower surface brightness and, especially in those spaxels associated with the relatively older clusters ($\sim$10 Myr)[^4] associated to the peak of emission in the continuum $\sharp$3 .
To estimate and correct this effect, we modelled and subtracted out the contribution of the underlying stellar continuum using the STARLIGHT[^5] spectral synthesis code [@cid05; @cid09]. This code reproduces a given observed spectrum by selecting a linear combination of a sub-set of $N_\star$ spectral components from a pre-defined set of base spectra. In our particular case, we utilized as base spectra a set of single star populations from @bru03. These are based on the Padova 2000 evolutionary tracks [@gir00] and assume Salpeter initial mass function between 0.1 and 100 M$_\odot$. Since has a metallicity of Z$\sim$0.3 Z$_\odot$[^6] [e.g. @kob99], we utilized only base spectra with $Z = $0.004 and 0.008. For each metallicity, we selected a set of 18 spectra with ages ranging from 1 Myr to 2.5 Gyr. We allowed for a single extinction for all the base spectra that was modeled as a uniform dust screen with the extinction law by @car89. The stellar spectral energy distribution at each spaxel was independently modelled for each cube. The spectral regions utilized in the fits were 3650-4060 Å and the whole spectral range for the LR1 and LR2 gratings, respectively. All the main outputs for each spaxel were then reformatted in order to create three cubes per grating with the total, gaseous and modelled stellar emission. Additionally, the information associated to auxiliary properties such as stellar extinction in the $V$ band, the stellar velocity and velocity dispersion, the reduced $\chi^2$ and the absolute deviation of the fit (in %) were reformatted and saved as a file suitable for manipulation with standard astronomical software. Hereafter, we will use both terms *map* and *image* to refer to these kind of files.
Representative examples of the achieved subtraction of the stellar component are shown in Fig. \[stellaremissionsubs\] which contains the observed, modeled and subtracted spectra for spaxels in the lowest surface brightness area, at the peak of emission in the continuum $\sharp$3 and in the peak of emission for the ionized gas. The subtraction of the stellar continuum is satisfactory in all three cases. Note that given the spectral resolution of the FLAMES data (larger than the one of the base spectra), some fine structure (e.g. at $\sim$3930 Å) cannot be reproduced. However, this does not affect the recovery of the gaseous emission in which we are primarily interested.
Line fitting and map creation
-----------------------------
As a first step, information from the relevant emission lines for each individual spaxel was obtained by fitting Gaussian functions in a semi-automatic way using the IDL based routine MPFITEXPR [@mar09]. Most of the lines (i.e. [H$\gamma$]{}-H8, [<span style="font-variant:small-caps;">\[Oiii\]</span>]{}$\lambda$4363, [He<span style="font-variant:small-caps;">i</span>]{}$\lambda$4009, [He<span style="font-variant:small-caps;">i</span>]{}$\lambda$4026, [He<span style="font-variant:small-caps;">i</span>]{}$\lambda$4388, [He<span style="font-variant:small-caps;">i</span>]{}$\lambda$4471, [<span style="font-variant:small-caps;">\[Sii\]</span>]{}$\lambda$4074 and [<span style="font-variant:small-caps;">\[Sii\]</span>]{}$\lambda$4076) were strong enough over most of the fov to be fitted independently. Exceptional cases were [\[Ne<span style="font-variant:small-caps;">iii</span>\]]{}$\lambda$3967 and [He<span style="font-variant:small-caps;">i</span>]{}$\lambda$3965 that were fitted simultaneously with H7, to assure a proper deblending of the emission lines and [\[O<span style="font-variant:small-caps;">ii</span>\]]{}$\lambda$3727 and [\[O<span style="font-variant:small-caps;">ii</span>\]]{}$\lambda$3729 that were fitted simultaneously. For the rest of lines, the kinematic results for H7, present in both LR1 and LR2, were utilized as additional constrains. Several (strong) lines showed signs of asymmetries and/or multiple components in their profiles for a large number of spaxels. When possible, multi-component fits to these lines were also performed trying to assure a continuity in the derived physical (and chemical) properties between adjacent spaxels. In the present work, this multi-component fit will only be utilized in the discussion of the electron density structure. Finally, as we did with the fitting of the underlying stellar population, we used the derived quantities together with the position within the data-cube for each spaxel to create a map.
On account of the different pointings between the 078.B-0043 and 382.B-0043 observing runs, an important issue for this analysis was assuring a proper match between maps for quantities derived from the LR3 and LR6 cubes and those of the LR1 and LR2. As a first guess we used the three main peaks in the continuum images as our reference to align the images. However, given the relatively small fov of the IFU, the extended nature of knots $\sharp$1 and $\sharp$3 and that knot $\sharp$2 falls at the edge of the IFU in the LR1 and LR2 data, the derived offset were not accurate enough. To refine this first guess, we created maps for ratios involving lines observed with the two different pointings (e.g. [<span style="font-variant:small-caps;">\[Oiii\]</span>]{}$\lambda$5007/[<span style="font-variant:small-caps;">\[Oiii\]</span>]{}4363) using a grid of offsets centered in our first guess and in steps of 0.5 spaxels (=026$\sim$1/4th seeing). We consider as final offset \[2.5 spa,5.0 spa\] (=\[13,26\]) which was the one providing the best continuity in adjacent spaxels for the variation of the line ratios. If the map of a given quantity was based on information from both runs, when possible, we independently carried out the calculations for 078.B-0043 and 383.B-0043. Only at the end, the offset was applied to the information associated to the 078.B-0043 run before carrying out the final calculation involving data from both runs. In this manner, the number of spatial interpolations was minimized.
[ccccccccc]{} $\lambda_0$ & Ion & $f(\lambda)$ &\
(Å) & & & Knot $\sharp$1 & Knot $\sharp$2 & Knot $\sharp$3\
3669.47 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.268 & 0.17$\pm$0.05 & 0.42$\pm$0.05 & …\
3671.48 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.268 & 0.24$\pm$0.05 & 0.24$\pm$0.06 & …\
3673.76 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.268 & 0.35$\pm$0.05 & 0.46$\pm$0.06 & …\
3676.37 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.265 & 0.53$\pm$0.06 & 0.57$\pm$0.07 & …\
3679.36 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.265 & 0.52$\pm$0.04 & 0.69$\pm$0.11 & …\
3682.81 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.265 & 0.67$\pm$0.05 & 0.77$\pm$0.08 & …\
3686.83 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.263 & 0.75$\pm$0.05 & 0.99$\pm$0.08 & …\
3691.56 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.263 & 0.83$\pm$0.06 & 0.91$\pm$0.06 & …\
3697.15 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.261 & 1.10$\pm$0.08 & 0.91$\pm$0.06 & …\
3703.86 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.261 & 1.30$\pm$0.08 & 1.25$\pm$0.06 & …\
3705.04 & [He<span style="font-variant:small-caps;">i</span>]{}& 0.261 & 0.70$\pm$0.07 & 0.57$\pm$0.06 & …\
3711.97 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.259 & 1.37$\pm$0.08 & 1.71$\pm$0.09 & …\
3721.83 & [\[S<span style="font-variant:small-caps;">iii</span>\]]{}& 0.257 & 2.95$\pm$0.16 & 2.68$\pm$0.09 & …\
3726.03 & [\[O<span style="font-variant:small-caps;">ii</span>\]]{}& 0.255 & 43.82$\pm$2.73 & 75.36$\pm$2.05 & 101.88$\pm$3.97\
3728.82 & [\[O<span style="font-variant:small-caps;">ii</span>\]]{}& 0.255 & 46.35$\pm$2.77 & 93.87$\pm$2.61 & 144.38$\pm$6.84\
3734.17 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.255 & 1.99$\pm$0.14 & 1.99$\pm$0.08 & 1.26$\pm$0.20\
3750.15 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.251 & 2.70$\pm$0.16 & 2.80$\pm$0.10 & 3.24$\pm$0.40\
3770.63 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.247 & 3.34$\pm$0.18 & 3.54$\pm$0.09 & 2.51$\pm$0.32\
3797.90 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.241 & 4.59$\pm$0.25 & 4.69$\pm$0.12 & 4.56$\pm$0.33\
3819.61 & [He<span style="font-variant:small-caps;">i</span>]{}& 0.237 & 0.86$\pm$0.06 & 0.88$\pm$0.04 & 0.65$\pm$0.20\
3835.39 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.235 & 6.61$\pm$0.35 & 6.74$\pm$0.15 & 6.65$\pm$0.48\
3868.75 & [\[Ne<span style="font-variant:small-caps;">iii</span>\]]{}& 0.227 & 48.87$\pm$3.10 & 31.73$\pm$0.93 & 22.34$\pm$0.89\
3889.05 & [<span style="font-variant:small-caps;">Hi</span>]{}+[He<span style="font-variant:small-caps;">i</span>]{}& 0.223 & 16.52$\pm$0.90 & 18.11$\pm$0.43 & 15.91$\pm$0.67\
3964.73 & [He<span style="font-variant:small-caps;">i</span>]{}& 0.209 & 0.57$\pm$0.35 & 0.59$\pm$0.19 & 0.77$\pm$0.45\
3967.46 & [\[Ne<span style="font-variant:small-caps;">iii</span>\]]{}& 0.207 & 15.88$\pm$0.82 & 10.38$\pm$0.31 & 8.77$\pm$0.59\
3970.07 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.207 & 15.80$\pm$0.82 & 15.80$\pm$0.38 & 15.80$\pm$0.72\
4009.22 & [He<span style="font-variant:small-caps;">i</span>]{}& 0.198 & 0.20$\pm$0.03$^{\mathrm{(\ast)}}$ & …& …\
4026.21 & [He<span style="font-variant:small-caps;">i</span>]{}& 0.194 & 1.73$\pm$0.09 & 1.69$\pm$0.05 & 0.81$\pm$0.19\
4068.60 & [<span style="font-variant:small-caps;">\[Sii\]</span>]{}& 0.187 & 1.40$\pm$0.08 & 1.41$\pm$0.05 & 3.28$\pm$0.24\
4076.35 & [<span style="font-variant:small-caps;">\[Sii\]</span>]{}& 0.184 & 0.46$\pm$0.03 & 0.52$\pm$0.03 & 1.25$\pm$0.16\
4097.26 & [O<span style="font-variant:small-caps;">ii</span>]{}& 0.180 & 0.05$\pm$0.01 & …& …\
4101.74 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.180 & 25.79$\pm$1.37 & 25.42$\pm$0.58 & 25.19$\pm$0.89\
4120.82 & [He<span style="font-variant:small-caps;">i</span>]{}& 0.176 & 0.20$\pm$0.02 & 0.09$\pm$0.03 & …\
4143.76 & [He<span style="font-variant:small-caps;">i</span>]{}& 0.172 & 0.23$\pm$0.02 & 0.16$\pm$0.02 & …\
4243.97 & [\[Fe<span style="font-variant:small-caps;">ii</span>\]]{}& 0.153 & 0.03$\pm$0.01 & 0.05$\pm$0.01 & …\
4267.15 & [C<span style="font-variant:small-caps;">ii</span>]{}& 0.147 & 0.06$\pm$0.01 & 0.10$\pm$0.01 & …\
4287.40 & [\[Fe<span style="font-variant:small-caps;">ii</span>\]]{}& 0.144 & 0.12$\pm$0.01 & 0.24$\pm$0.02 & 1.23$\pm$0.15\
4340.47 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.133 & 46.60$\pm$2.87 & 46.60$\pm$1.12 & 46.60$\pm$1.71\
4359.34 & [\[Fe<span style="font-variant:small-caps;">ii</span>\]]{}& 0.129 & 0.06$\pm$0.01 & 0.13$\pm$0.01 & 0.53$\pm$0.10\
4363.21 & [<span style="font-variant:small-caps;">\[Oiii\]</span>]{}& 0.129 & 7.77$\pm$0.39 & 3.70$\pm$0.08 & 2.64$\pm$0.21\
4368.25 & [O<span style="font-variant:small-caps;">i</span>]{}& 0.127 & 0.03$\pm$0.01 & …& …\
4387.93 & [He<span style="font-variant:small-caps;">i</span>]{}& 0.122 & 0.49$\pm$0.03 & 0.45$\pm$0.02 & …\
4413.78 & [\[Fe<span style="font-variant:small-caps;">ii</span>\]]{}& 0.118 & 0.07$\pm$0.01 & 0.01$\pm$0.01 & …\
4416.27 & [\[Fe<span style="font-variant:small-caps;">ii</span>\]]{}& 0.115 & 0.04$\pm$0.01 & 0.03$\pm$0.01 & …\
4437.55 & [He<span style="font-variant:small-caps;">i</span>]{}& 0.110 & 4.48$\pm$0.25 & …& …\
4452.11 & [\[Fe<span style="font-variant:small-caps;">ii</span>\]]{}& 0.108 & 0.13$\pm$0.01 & …& …\
4471.48 & [He<span style="font-variant:small-caps;">i</span>]{}& 0.103 & 0.08$\pm$0.01 & 0.19$\pm$0.01 & 0.36$\pm$0.18\
This line presented a “secondary peak” towards the blue which correspond to a weak [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{} line at $\lambda$4008. At the resolution of the present data a proper deblending of the lines was not feasible.
[ccccccccc]{} $\lambda_0$ & Ion & $f(\lambda)$ &\
(Å) & & & Knot $\sharp$1 & Knot $\sharp$2 & Knot $\sharp$3\
4562.60 & [Mg<span style="font-variant:small-caps;">i</span>\]]{}& 0.080 & 0.11$\pm$0.01 & 0.19$\pm$0.01 &0.86$\pm$ 0.19\
4571.00 & [Mg<span style="font-variant:small-caps;">i</span>\]]{}& 0.078 & 0.12$\pm$0.01 & 0.16$\pm$0.01 &0.74$\pm$ 0.17\
4649.13 & [O<span style="font-variant:small-caps;">ii</span>]{}& 0.057 & 0.08$\pm$0.01 & …& …\
4658.10 & [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}& 0.054 & 0.96$\pm$0.05 & 0.80$\pm$0.03 &2.34$\pm$ 0.19\
4701.53 & [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}& 0.043 & 0.27$\pm$0.01 & 0.23$\pm$0.01 &0.84$\pm$ 0.19\
4711.37 & [\[Ar<span style="font-variant:small-caps;">iv</span>\]]{}& 0.041 & 1.17$\pm$0.06 & 0.19$\pm$0.00 &0.57$\pm$ 0.02\
4713.14 & [He<span style="font-variant:small-caps;">i</span>]{}& 0.041 & 0.65$\pm$0.05 & 0.41$\pm$0.01 &0.19$\pm$ 0.16\
4740.16 & [\[Ar<span style="font-variant:small-caps;">iv</span>\]]{}& 0.032 & 1.18$\pm$0.05 & 0.17$\pm$0.01 & …\
4754.83 & [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}& 0.030 & 0.19$\pm$0.01 & 0.18$\pm$0.01 & …\
4769.60 & [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}& 0.024 & 0.10$\pm$0.01 & 0.10$\pm$0.01 & …\
4814.55 & [\[Fe<span style="font-variant:small-caps;">ii</span>\]]{}& 0.013 & 0.06$\pm$0.01 & …& …\
4861.33 & [<span style="font-variant:small-caps;">Hi</span>]{}& 0.000 & 100.00$\pm$4.36 & 100.00$\pm$1.82 &100.00$\pm$ 2.51\
4881.00 & [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}& $-$0.005 & 0.37$\pm$0.02 & 0.25$\pm$0.02 & …\
4921.93 & [He<span style="font-variant:small-caps;">i</span>]{}& $-$0.016 & 1.14$\pm$0.05 & 1.03$\pm$0.03 & …\
4931.32 & [<span style="font-variant:small-caps;">\[Oiii\]</span>]{}& $-$0.019 & 0.16$\pm$0.01 & 0.16$\pm$0.02 & …\
4958.91 & [<span style="font-variant:small-caps;">\[Oiii\]</span>]{}& $-$0.027 & 232.56$\pm$10.88 & 162.83$\pm$3.25 & 114.29$\pm$ 3.25\
4985.90 & [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}& $-$0.032 & 0.47$\pm$0.03 & 0.64$\pm$0.03 &2.78$\pm$0.26\
5006.84 & [<span style="font-variant:small-caps;">\[Oiii\]</span>]{}& $-$0.040 & 732.56$\pm$34.05 & 508.85$\pm$10.23 & 358.10$\pm$10.25\
5015.68 & [He<span style="font-variant:small-caps;">i</span>]{}& $-$0.043 & 1.87$\pm$0.09 & 2.19$\pm$0.06 &2.11$\pm$0.25\
5041.03 & [<span style="font-variant:small-caps;">Siii</span>]{}& $-$0.049 & 0.24$\pm$0.02 & 0.16$\pm$0.01 & …\
5055.98 & [<span style="font-variant:small-caps;">Siii</span>]{}& $-$0.055 & …& 0.16$\pm$0.01 & …\
6548.03 & [<span style="font-variant:small-caps;">\[Nii\]</span>]{}& $-$0.312 & 10.70$\pm$0.46 & 5.88$\pm$0.16 & 11.33$\pm$ 0.29\
6562.82 & [<span style="font-variant:small-caps;">Hi</span>]{}& $-$0.314 & 400.00$\pm$16.86 & 391.15$\pm$7.37 & 415.24$\pm$ 10.18\
6583.41 & [<span style="font-variant:small-caps;">\[Nii\]</span>]{}& $-$0.316 & 32.67$\pm$1.39 & 17.96$\pm$0.49 & 35.05$\pm$ 0.80\
6678.15 & [He<span style="font-variant:small-caps;">i</span>]{}& $-$0.329 & 4.55$\pm$0.20 & 4.20$\pm$0.08 & …\
6716.47 & [<span style="font-variant:small-caps;">\[Sii\]</span>]{}& $-$0.332 & 13.95$\pm$0.60 & 20.53$\pm$0.41 & 55.24$\pm$1.32\
6730.85 & [<span style="font-variant:small-caps;">\[Sii\]</span>]{}& $-$0.335 & 12.91$\pm$0.55 & 16.55$\pm$0.34 & 40.76$\pm$0.98\
7002.23 & [O<span style="font-variant:small-caps;">i</span>]{}& $-$0.365 & 0.09$\pm$0.01 & 0.10$\pm$0.01 & …\
7065.28 & [He<span style="font-variant:small-caps;">i</span>]{}& $-$0.375 & 7.73$\pm$0.30 & 3.85$\pm$0.08 & 2.83$\pm$0.19\
7135.78 & [\[Ar<span style="font-variant:small-caps;">iii\]</span>]{}& $-$0.384 & 16.34$\pm$0.73 & 12.65$\pm$0.26 & 10.67$\pm$0.30\
[lcccccccc]{} & Knot $\sharp$1 & Knot $\sharp$2 & Knot $\sharp$3\
$E(B-V)$ & 0.31$\pm$0.04 & 0.29$\pm$0.02 & 0.35$\pm$0.04\
$n_e$(cm$^{-3}$) ([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) & 310$^{+150}_{-140}$ & 140$^{+50}_{-50}$ & 30$^{+100}_{-60}$\
$n_e$(cm$^{-3}$) ([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) & 385$^{+150}_{-120}$ & 170$^{+35}_{-30}$ & 70$^{+30}_{-25}$\
$n_e$(cm$^{-3}$) ([\[Ar<span style="font-variant:small-caps;">iv</span>\]]{}) & 4600$^{+1200}_{-1000}$ & 2800$^{+1800}_{-900}$ & …\
$n_e$(cm$^{-3}$) ([\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}) & 270$^{+40}_{-30}$ & 155$^{+20}_{-15}$ & 95$^{+25}_{-20}$\
$T_e$ (K) ([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}) & 11570$^{+420}_{-350}$ & 10280$^{+130}_{-120}$ & 10320$^{+350}_{-300}$\
$T_e$ (K) ([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) & 11360 & 10470 & 10500\
$T_e$ (K) ([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) & 9700$^{+1700}_{-1300}$ & 8700$^{+490}_{-440}$ & 8900$^{+870}_{-680}$\
- Ratios for $n_e$: [\[O<span style="font-variant:small-caps;">ii</span>\]]{}: [\[O<span style="font-variant:small-caps;">ii</span>\]]{}$\lambda$3727/[\[O<span style="font-variant:small-caps;">ii</span>\]]{}$\lambda$3729; [<span style="font-variant:small-caps;">\[Sii\]</span>]{}: [<span style="font-variant:small-caps;">\[Sii\]</span>]{}$\lambda$6717/[<span style="font-variant:small-caps;">\[Sii\]</span>]{}$\lambda$6731; [\[Ar<span style="font-variant:small-caps;">iv</span>\]]{}: [\[Ar<span style="font-variant:small-caps;">iv</span>\]]{}$\lambda$4711/[\[Ar<span style="font-variant:small-caps;">iv</span>\]]{}$\lambda$4740; [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}: [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}$\lambda$4986/[\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}$\lambda$4658\
- Ratios for $T_e$: [<span style="font-variant:small-caps;">\[Oiii\]</span>]{}: [<span style="font-variant:small-caps;">\[Oiii\]</span>]{}$\lambda\lambda$4959,5007/[<span style="font-variant:small-caps;">\[Oiii\]</span>]{}$\lambda$4363; [<span style="font-variant:small-caps;">\[Sii\]</span>]{}: [<span style="font-variant:small-caps;">\[Sii\]</span>]{}$\lambda\lambda$6716,6731/[<span style="font-variant:small-caps;">\[Sii\]</span>]{}$\lambda$4069,4076. Note that $T_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) was not obtained directly from measurements of [\[O<span style="font-variant:small-caps;">ii</span>\]]{} lines. Instead, it was derived from $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}) and $n_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) according to the expressions provided in @gar09.\
[lcccccccc]{} & Knot $\sharp$1 & Knot $\sharp$2 & Knot $\sharp$3\
$12+\log(O^+/H^+)$ & 7.33$\pm$0.10 & 7.73$\pm$0.04 & 7.88$\pm$0.09\
$12+\log(O^{2+}/H^+)$ & 8.18$\pm$0.09 & 8.17$\pm$0.03 & 8.03$\pm$0.07\
$12+\log(O/H)$ & 8.24$\pm$0.09 & 8.32$\pm$0.04 & 8.26$\pm$0.08\
$12+\log(S^+/H^+)$ & 5.71$\pm$0.07 & 5.96$\pm$0.03 & 6.31$\pm$0.05\
$\log(S^+/O^+)$ & $-$1.62$\pm$0.17 & $-$1.77$\pm$0.07 & $-$1.57$\pm$0.14\
$12+\log(N^+/H^+)$ & 6.52$\pm$0.07 & 6.36$\pm$0.03 & 6.62$\pm$0.06\
$12+\log(N/H)$ & 7.43$\pm$0.07 & 6.94$\pm$0.03 & 7.00$\pm$0.06\
ICF ($N^{+}$) & 8.04 & 3.84 & 2.40\
$\log(N/O)$ & $-$0.81$\pm$0.25 & $-$1.37$\pm$0.11 & $-$1.26$\pm$0.22\
$12+\log(Ne^{2+}/H^+)$ & 7.49$\pm$0.10 & 7.50$\pm$0.04 & 7.34$\pm$0.09\
ICF (Ne) & 1.14 & 1.35 & 1.71\
$12+\log(Ne/H)$ & 7.55$\pm$0.10 & 7.63$\pm$0.04 & 7.57$\pm$0.09\
$\log(Ne/O)$ & $-$0.69$\pm$0.28 & $-$0.69$\pm$0.12 & $-$0.69$\pm$0.25\
$12+\log(Ar^{2+}/H^+)$ & 5.89$\pm$0.07 & 5.90$\pm$0.03 & 5.80$\pm$0.05\
$12+\log(Ar^{3+}/H^+)$ & 4.84$\pm$0.08 & 4.15$\pm$0.05 & …\
ICF (Ar)$^{(\ast)}$ & 1.02$\pm$0.04 & 1.07$\pm$0.05 & 1.45$\pm$1.91\
$12+\log(Ar/H)$ & 5.90$\pm$0.22 & 5.93$\pm$0.12 & 5.96$\pm$1.45\
$\log(Ar/O)$ & $-$2.34$\pm$0.30 & $-$2.38$\pm$0.16 & $-$2.30$\pm$1.53\
$12+\log(Fe^{2+}/H^+)$ & 5.58$\pm$0.09 & 5.66$\pm$0.04 & 6.12$\pm$0.09\
$^{(\ast)}$ ICF (Ar$^{2+}$) for knot $\sharp$3.
Results \[resultados\]
======================
Gaseous emission in selected apertures \[selaper\]
--------------------------------------------------
The main aim of the present work is to provide a full 2D characterization of the physical and chemical properties of the ionized gas in the central region of the galaxy. However, there are reasons to perform a more detailed analysis in specific regions. Firstly, we can define very precisely the aperture utilized to extract a given spectrum, avoiding the limitations associated to the positioning of a slit and assuring that the spectrum is spatially associated with a given feature of interest (e.g. the location of a star cluster). As an example, @lop07 characterized the properties of our knot $\sharp$2 (their UV-1). However, neither their positions HII-1 nor HII-2 sample the gas associated to the peak of emission in [H$\alpha$]{}, but some emission associated to the [GH<span style="font-variant:small-caps;">ii</span>R]{} towards the west and south of the peak of emission. Secondly, a comparison of the measurements provided here, with those already existing in the literature serves as a sanity check which will reinforce the results obtained in the 2D mapping. Thirdly, most of the present methodologies used in the analysis of the physical and chemical properties of the ionized gas were developed to study *complete* H<span style="font-variant:small-caps;">ii</span> regions. At present, the astronomical community is still in the process of understanding under which conditions these can be applied to small portions of them [e.g. @per11; @erc12]. The areas analyzed in this section will be closer to this notion of *complete* H<span style="font-variant:small-caps;">ii</span> than individual spaxels. Finally, by combining the signal of a collection of spaxels, a larger signal-to-noise (S/N) ratio is achieved and fainter lines can be detected.
As we saw in Paper I, the stellar emission is dominated by three peaks of emission named as knots $\sharp$1, $\sharp$2 and $\sharp$3 (see Tab. 3 of Paper I to establish the correspondence between this nomenclature and those of previous studies). Note that knot $\sharp$1 is associated to the main [GH<span style="font-variant:small-caps;">ii</span>R]{} marked in Fig. \[apuntado\]. The spectra associated to them have been co-added and extracted in order to determine relative line intensities in their surrounding ionized gas.
Utilized apertures in the LR1 and LR2 data are outlined in Fig. \[estructura\]. For the LR3 and LR6, we applied an offset of 2 and 5 spaxels in the *x* and *y* direction respectively. Since the offset between the two pointings was determined as \[2.5 spa,5.0 spa\], this implies a small difference in the selected apertures from the old and the new data which introduces an extra uncertainty when comparing fluxes of lines belonging to different set of data. In consequence, the line fluxes for the LR3 and the LR6 gratings were directly measured with respect to [H$\beta$]{} and [H$\alpha$]{} while those for the LR1 and LR2 gratings where measured with respect to [H$\gamma$]{} and H7, and then we assumed the theorietical Balmer line intensities obtained from @sto95 for Case B, $T_e=10^4$ K and $n_e=100$ cm$^{-3}$. Most of the lines were measured independently by fitting the emission to a single Gaussian profile using `mpfit`. Exceptionally, lines in the pairs H16 and [He<span style="font-variant:small-caps;">i</span>]{}$\lambda$3705, [\[Fe<span style="font-variant:small-caps;">ii</span>\]]{}$\lambda$4414 and [\[Fe<span style="font-variant:small-caps;">ii</span>\]]{}$\lambda$4416, and [\[Ar<span style="font-variant:small-caps;">iv</span>\]]{}$\lambda$4711 and [He<span style="font-variant:small-caps;">i</span>]{}$\lambda$4713 were simultaneously fitted due to their proximity in wavelength. Similarly, all the three lines [He<span style="font-variant:small-caps;">i</span>]{}$\lambda$3965, [\[Ne<span style="font-variant:small-caps;">iii</span>\]]{}$\lambda$3967 and H7 were fitted at once. Measurements for all detected lines are compiled in Tables \[lineratios\_lr1lr2\] and \[lineratios\_lr3lr6\].
### Physical conditions of the ionized gas \[fiscon\]
Derived physical conditions for the ionized gas are listed in Table \[physiprop\]. The values of reddening were derived from the measured [H$\alpha$]{}/[H$\beta$]{} line ratio following the methodology described in Paper I. Electron temperature ($T_e$) and density ($n_e$) based on [<span style="font-variant:small-caps;">\[Sii\]</span>]{}, [\[O<span style="font-variant:small-caps;">ii</span>\]]{} and [<span style="font-variant:small-caps;">\[Oiii\]</span>]{} emission lines (and thus tracing different layers in the ionization structure) were derived using the expressions provided by @gar09[^7]. For that we proceeded as follows: we assumed an initial $n_e$ of 100 cm$^{-3}$ to obtain a first guess of the different $T_e$’s. The resulting $T_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) and $T_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) were used as input to obtain new estimates of $n_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) and $n_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) and the process was iterated until convergence. Typically one or two iterations were enough. These functions reproduce the predictions of the task `temden`, based on the `fivel` program [@sha95] included in the IRAF package `nebular`. They used the same atomic coefficients as in @per03, except for $O^+$ for which they used the transition probabilities from @zei82 and the collision strengths from @pra76. $T_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) was not independently derived from our data, since the [\[O<span style="font-variant:small-caps;">ii</span>\]]{}$\lambda\lambda$7320,7330 doublet was not covered in these observations. Instead, we derived $T_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) from $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}) and $n_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) using the models presented by @per03. As an improvement to former relations based on modelling [e.g. @sta90], these take into account dependencies on $n_e$. At low densities, for a given $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}), models by @per03 predict only slightly (i.e. ${\hbox{\rlap{\lower.55ex\hbox{$\sim$}} \kern-.3em
\raise.4ex \hbox{$<$}}}2$%) lower temperatures than those by @sta90 while in the densest zones of NGC 5253 (e.g. our knot $\sharp$1), estimated $T_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) can be up to $\sim$17% smaller. For $n_e$([\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}), we used the line ratios tabulated by @kee01, assuming $T_e=12\,000$ K. Note that for the utilized emission lines, the dependence of the derived $n_e$ on the assumed temperature is negligible. Finally, $n_e$([\[Ar<span style="font-variant:small-caps;">iv</span>\]]{}) was derived using directly `temden` and assuming $T_e$([\[Ar<span style="font-variant:small-caps;">iv</span>\]]{})=$T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}).
Derived $T_e$ and $n_e$ for our knots $\sharp$1 and $\sharp$2 agree well within the uncertainties with the values reported by @sid09 for his apertures A and B, respectively. Additional measurements for $T_e$ and $n_e$ in knot $\sharp$2 are provided by @lop07, @kob97 and @gus11. While our derived $T_e$’s are in relatively good agreement, with $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}) and $T_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) being slightly lower and larger respectively than the values reported in these works, there is a discrepancy between the different measurements of $n_e$ in the literature. A comparison of our $n_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) for knot $\sharp$ 1 and the values reported by @gus11 for their apertures C1 and P2 shows a discrepancy between the reported values. Since $n_e$ varies a lot in this area (a factor of $\sim$2-3 on scales of $\sim$05, see Fig. \[mapne\] in Sec. \[densistruc\]), the precise definition of the aperture (i.e. size, position and shape) in each case seems the most plausible explanation for this discrepancy. Our $n_e\sim150$ cm$^{-3}$ in knot $\sharp$2 is consistent within the errors with being below the low density limit as reported by @kob97. However, it is a factor $\sim$2 lower than the values reported by @lop07. While estimated uncertainties could account for the differences in $n_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}), one must resort to other causes, such as differences in the definition of the aperture, to explain the discrepancies in $n_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}).
### Chemical abundances \[abunaper\]
Abundances for the different ionic species are listed in Table \[chemiprop\] and were derived using the relations provided in Appendix B.2 of @gar09. These are appropriate fittings to the results of the IRAF task `ionic` [@sha95], based on the 5-level atom program developed by @der87, and follow the functional form given by @pag92. For the abundances of neon, argon, iron and O$^{++}$, we utilized $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}). For O$^{+}$ and nitrogen, we utilized $T_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}), derived from $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}) and $n_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}). Finally, for sulfur, we took $T_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}). Differences in abundances due to the assumed $T_e$ are discussed below. To derive the total abundances, unseen ionization stages of a given element were taken into account by including the appropriate ionization factor (ICF) for each species when necessary, following the prescriptions provided by @kin94 for all the elements but argon. For this element, we utilized those provided by @izo94.
*Oxygen:* We assumed $O/H = (O^+ + O^{2+})/H^+$. The derived values are in good agreement with those provided in the literature using long-slit data [e.g. @gus11; @sid09; @lop07; @kob97]. The largest differences are for knot $\sharp$1 that shows abundances $\sim$0.1 dex higher than those measured by @gus11. This is however within the uncertainties.
*Sulfur:* Derived values are typically $\sim$0.2 dex larger than those provided by @lop07. Differences can be attributed to the adopted $T_e$. @lop07 use the average of $T_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}), $T_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}), and $T_e$([<span style="font-variant:small-caps;">\[Nii\]</span>]{}) that can be $\sim$2000 K larger than $T_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}). If a $T_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}), derived from the $n_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) and $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}), were utilized, our abundances would become typically $\sim$0.15 dex smaller in agreement with the values reported by these authors. A similar effect is observed when comparing with the results reported by @sid09 who utilized $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}) to derive all the abundances and those reported by @gus11 who used a $T_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) $\sim$1500-2000 K larger than the $T_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) utilized here.
*Nitrogen:* We assumed $N/O=N^+/O^+$, which is an accurate approximation to about $\pm$20% for nebulae with metallicities smaller than that in the LMC [@gar90]. In agreement with previous work, while knots $\sharp$2 and $\sharp$3 present relative abundances in nitrogen within the range expected for galaxies at this metallicity [see e.g. @mol06], the aperture associated to knot $\sharp$1 present a clear excess in nitrogen. Interestingly, although consistent with the $N/O$ expected for galaxies at this metallicity, knot $\sharp$3 has a slightly larger abundance than knot $\sharp$2.
*Neon:* Since the ionization structure is similar to the one of the oxygen, we assumed $Ne^{2+}/Ne = O^{2+}/O$. Reported values agree within the errors with previous measurements in the literature [@gus11; @lop07; @kob97].
*Argon:* Total argon abundance was derived using the ICFs based on the expressions provided by @izo94. Derived ratios are within the range of those previously published [@gus11; @lop07; @sid09] and consistent with a homogeneous $Ar/O$ ratio across the face of the galaxy. *Iron:* Reported values for Fe$^2+$ agree with those given by @lop07.
To recap, abundances of the different ionic species in knot $\sharp$1 and knot $\sharp$2 agree in general with those previously reported [@kob97; @lop07; @sid09]. Abundances in knot $\sharp$3 (not reported so far) are similar to those in knot $\sharp$1 and knot $\sharp$2. With the exception of the relative abundance for $N/O$ in knot $\sharp$1 and maybe knot $\sharp$3, no chemical species seems overabundant in any of the selected apertures. This is in agreement with the relative enrichment in nitrogen previously widely reported.
Mapping electron density and temperature tracers \[mapneyte\]
-------------------------------------------------------------
In , we found an $n_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) gradient declining from the peak of emission in [H$\alpha$]{} outwards and traced the gas at the highest densities by means of the $n_e$([\[Ar<span style="font-variant:small-caps;">iv</span>\]]{}). Also, our multi-component analysis of the [<span style="font-variant:small-caps;">\[Sii\]</span>]{} line profiles tentatively suggested similar densities over the whole face of the main H<span style="font-variant:small-caps;">ii</span> region for the broad component, while the narrow component presented somewhat lower (higher) ratios towards the NW (SE) part of the region. Given the restricted spectral range utilized in that work, the $T_e$ structure could not be derived.
Here, we present maps for two additional tracers of $n_e$: [\[O<span style="font-variant:small-caps;">ii</span>\]]{}$\lambda$3726/[\[O<span style="font-variant:small-caps;">ii</span>\]]{}$\lambda$3729 and [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}$\lambda$4986/[\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}$\lambda$4658. In comparison to [<span style="font-variant:small-caps;">\[Sii\]</span>]{}$\lambda$6717/[<span style="font-variant:small-caps;">\[Sii\]</span>]{}$\lambda$6731, the first ratio is sensitive to a slightly larger range of densities, tracing to lower $n_e$ [@ost06]. Moreover, the [\[O<span style="font-variant:small-caps;">ii</span>\]]{} lines are stronger than those of [<span style="font-variant:small-caps;">\[Sii\]</span>]{}. Therefore, we will be able to more systematically discuss the differences in $n_e$ structure between different kinematic components, as suggested by the [<span style="font-variant:small-caps;">\[Sii\]</span>]{} line ratio. The second ratio is sensitive to an even larger range of densities, from $\sim10^2$ cm$^{-3}$ to ${\hbox{\rlap{\lower.55ex\hbox{$\sim$}} \kern-.3em
\raise.4ex \hbox{$<$}}}10^7$ cm$^{-3}$ [@kee01] and has already been used with success in similar works to this one to reveal very high density locations [@jam09]. Also, we will introduce for the first time maps for two tracers of $T_e$ in this galaxy: [<span style="font-variant:small-caps;">\[Oiii\]</span>]{}$\lambda\lambda$4959,5007/[<span style="font-variant:small-caps;">\[Oiii\]</span>]{}$\lambda$4343 and [<span style="font-variant:small-caps;">\[Sii\]</span>]{}$\lambda\lambda$6717,6731/$\lambda\lambda$4069,4076. By comparing the maps in $n_e$ and $T_e$ according to the different tracers, we will be able to see how the physical properties of the gas are structured in the different ionization layers (see Sec. \[discusion\]).
The maps for our new $n_e$ tracers are presented in Fig. \[mapo2rat\]. For that involving the [\[O<span style="font-variant:small-caps;">ii</span>\]]{} emission lines, large values, associated to relatively large $n_e$, are found in the main [GH<span style="font-variant:small-caps;">ii</span>R]{}. There is a secondary maximum at the location of knot $\sharp$2 and ratios typical of low densities elsewhere. The map for the [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{} line ratios presents the inverse tendencies: the maximun of [\[O<span style="font-variant:small-caps;">ii</span>\]]{} line ratio corresponds to the minimum of the [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{} line ratio. The two overall structures are similar to the one depicted by the [<span style="font-variant:small-caps;">\[Sii\]</span>]{} line ratios .
More interestingly, Fig. \[mapo3ands2rat\] presents the maps for the line ratios tracing the $T_e$. Both of them display similar structure, with the absolute minimum at the location of the two SSCs (i.e. knot $\sharp$1), corresponding to the highest $T_e$, ratios remaining relatively low in the [GH<span style="font-variant:small-caps;">ii</span>R]{}and then tending to higher values in the rest of the fov. As with the selected apertures in Sec. \[fiscon\], on each individual spaxel the predicted $T_e$ from the [<span style="font-variant:small-caps;">\[Oiii\]</span>]{} line ratio differs from the one using the [<span style="font-variant:small-caps;">\[Sii\]</span>]{} lines. We will explore in Sec. \[discusion\] how this is related to the ionization structure of the galaxy.
Mapping tracers of the local ionization degree\[localioni\]
-----------------------------------------------------------
The relation between the [<span style="font-variant:small-caps;">\[Nii\]</span>$\lambda$6584/H$\alpha$]{} and [<span style="font-variant:small-caps;">\[Sii\]</span>$\lambda\lambda$6717,6731/H$\alpha$]{} line ratios in galaxies is not monotonic, even though the $S^+$ and $N^+$ ionization potentials and critical densities are in a similar range. Instead, different ionization mechanisms [shocks, star formation, AGNs, e.g. @mon06; @mon10b] and/or physical and chemical conditions of the gas (metallicity, relative abundances and degree of ionization) trace different loci in the [<span style="font-variant:small-caps;">\[Nii\]</span>$\lambda$6584/H$\alpha$]{} vs. [<span style="font-variant:small-caps;">\[Sii\]</span>$\lambda\lambda$6717,6731/H$\alpha$]{} diagram. In , we attributed the different structure in the [<span style="font-variant:small-caps;">\[Nii\]</span>$\lambda$6584/H$\alpha$]{} and [<span style="font-variant:small-caps;">\[Sii\]</span>$\lambda\lambda$6717,6731/H$\alpha$]{} maps to an excess in nitrogen. This was supported by evidence of star-formation as the main ionization mechanism and previous estimations of the ionization parameter at specific locations based on long-slit measurements [@kob97]. However, certain combinations of specific ionization structures and metallicity gradients could reproduce a similar locus in the [<span style="font-variant:small-caps;">\[Nii\]</span>$\lambda$6584/H$\alpha$]{} vs. [<span style="font-variant:small-caps;">\[Sii\]</span>$\lambda\lambda$6717,6731/H$\alpha$]{} diagram. This possibility can be ruled out with the present data. Here, we present the ionization structure in as traced by different line ratios, while the next section will contain maps for the abundances of different heavy elements (oxygen, nitrogen, neon and argon).
The degree of ionization can be traced by means of ratios of lines of the same element tracing two different ionization states. The upper panel of Fig. \[mapsu\] contains a map for one of these ratios: [<span style="font-variant:small-caps;">\[Oiii\]</span>$\lambda\lambda$4959,5007/<span style="font-variant:small-caps;">\[Oii\]</span>$\lambda\lambda$3726,3729]{}. Note that in order to minimize uncertainties associated to aperture matching, absolute flux calibration and extinction, the [<span style="font-variant:small-caps;">\[Oiii\]</span>]{} and [\[O<span style="font-variant:small-caps;">ii</span>\]]{} lines were measured with respect to [H$\beta$]{} and H7. Then, we assumend the theoretical Balmer line intensities obtained from @sto95 for Case B, $T_e= 10^4$ K and $n_e=$ 100 cm$^{−3}$. The map shows that the ionization structure reproduces the morphology observed for the ionized gas. That is: i) line ratios tracing the highest ionization degree are associated to knot $\sharp$1; ii) relatively high ionization is found in the main [GH<span style="font-variant:small-caps;">ii</span>R]{}; iii) a secondary peak of high ionization is found around knot $\sharp$2; iv) low ionization degree is found in the rest of the fov, where the diffuse component of the ionized gas becomes more relevant.
Also, if the metallicity is known, line ratios like [\[O<span style="font-variant:small-caps;">ii</span>\]]{}$\lambda\lambda$3726,3729/H7 or [<span style="font-variant:small-caps;">\[Sii\]</span>$\lambda\lambda$6717,6731/H$\alpha$]{} can also be used to trace the ionization degree. The map [\[O<span style="font-variant:small-caps;">ii</span>\]]{}$\lambda\lambda$3726,3729/H7 is presented in the lower panel of Fig. \[mapsu\] while that of [<span style="font-variant:small-caps;">\[Sii\]</span>$\lambda\lambda$6717,6731/H$\alpha$]{} was included in . The similar structure in all the three maps suggests a lack of metallicity gradient in the galaxy as will be shown in the next section.
In order to explore whether these three tracers predict consistent ionization degree for a given position, we show in Fig. \[compau\], the relation between the different ratios for each individual spaxel, using [<span style="font-variant:small-caps;">\[Oiii\]</span>$\lambda\lambda$4959,5007/<span style="font-variant:small-caps;">\[Oii\]</span>$\lambda\lambda$3726,3729]{} as reference. Also, we overplotted the locus of equal estimated ionization parameter according to the relations proposed by @dia00 and assuming $Z=0.3\, Z_\odot$ with a black line. For a given location, [<span style="font-variant:small-caps;">\[Oiii\]</span>$\lambda\lambda$4959,5007/<span style="font-variant:small-caps;">\[Oii\]</span>$\lambda\lambda$3726,3729]{} and [\[O<span style="font-variant:small-caps;">ii</span>\]]{}$\lambda\lambda$3726,2729/[H$\beta$]{} predict a similar degree of ionization while the predictions of [<span style="font-variant:small-caps;">\[Sii\]</span>$\lambda\lambda$6717,6731/H$\alpha$]{} would correspond to smaller ionization parameters. A similar result was found in a detailed analysis of NGC 588, a [GH<span style="font-variant:small-caps;">ii</span>R]{} in M 33 [@mon11]. Both, the spatial variations of the different line ratios (Fig. \[mapsu\]) and the observed excess in the [<span style="font-variant:small-caps;">\[Sii\]</span>$\lambda\lambda$6717,6731/H$\alpha$]{} ratio when compared with photoionization models (Fig. \[compau\]) can be jointly explained as a 3D view of the ionization structure of the galaxy. Specifically, for a given spaxel (i.e. a given line of sight), the lower ionization species (e.g. $S^+$), delineate the more extended diffuse component while $O^{++}$ will be confined to different high ionization zones closer to the ionizing sources. Alternatively, the difference between the ratios found here and those predicted by the Díaz et al. models can be attributed to differences between the relative $S/O$ abundances in [$\log(S/O)\sim-1.47$, @kob97; @sid09] and those assumed in the models [$\log(S/O)=-1.71$, @gre89]. A detailed modelling of the ionization structure of the galaxy - out of the scope of this work - could help to disentangle these two possibilities.
\
Mapping abundances of heavy elements \[secabun\]
------------------------------------------------
Following the same methodology as in Sec. \[abunaper\], we derived maps for the abundances of oxygen, nitrogen, neon and argon. These are presented in Fig. \[mapabun\] and show, as expected, a homogeneous distribution within the uncertainties in all the elements but nitrogen. Note that the spaxel-to-spaxel variations in these maps are not dominated by the quality of the data (e.g. S/N ratio), but are inherent to the adopted $T_e$ (and to a much lesser extent $n_e$) as well as the utilized methodology. The most obvious example would be the map showing the distribution of the argon abundance, where a different methodology was adopted to estimate the ICF depending on whether the [\[Ar<span style="font-variant:small-caps;">iv</span>\]]{}$\lambda$4740 emission line was or was not detected . Nevertheless, abundances derived in these two groups differ only by $\sim$0.1 dex, well within the expected uncertainties for this element. Also, $12+\log(O/H)$ marginally anti-correlates with $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}) in the spaxels with lowest surface brightness. The mean ($\pm$standard deviation) metallicity in our fov is $12+\log(O/H)=8.26$ $(\pm0.04)$ while the relative abundances $\log(Ne/O)$ and $\log(Ar/O)$ are $-0.64 (\pm0.03)$ and $-2.32 (\pm0.05)$. Supporting this homogeneity, in all three cases – oxygen, neon and argon – the distribution of the values measured over the fov can be well reproduced by a Gaussian with $\sigma\sim0.03-0.07$ dex and all the values measured in the individual spaxels are consistent with the mean value at the 3$\sigma$ level.
The abundance in nitrogen differs from this pattern. As reported in , there is a roughly elliptical area of $\sim80$ pc$\times35$ pc, associated to the main H<span style="font-variant:small-caps;">ii</span> region and centered on the two massive SSCs, that presents an enhancement of $N/O$. More interestingly, a putative secondary excess in nitrogen not reported so far appears in the vicinity of (but not centered on) knot $\sharp$3, at the area of our fov with the lowest surface brightness.
The existence of a second area with slight $N/O$ enhancement is supported by the histogram of the values measured along the FLAMES fov (not shown). Contrary to the case for $O$, $Ne$ and $Ar$, three Gaussians centered at $\log(N/O)=-1.32$, $-1.17$ and $-0.95$ and with widths $\sigma= 0.05,0.07,0.03$ are needed to reproduce the full distribution. Most of the spaxels are associated to the first Gaussian and trace the typical $N/O$ abundance for . The second largest group, with $\log(N/O)\sim-0.95$, are associated to the area with N-enhancement already reported for the main [GH<span style="font-variant:small-caps;">ii</span>R]{}. Using the criterion that spaxels belonging to two Gaussian trace different abundances if their centers are separated by more than 3$\sigma$, we can conclude that the third group of spaxels, which are associated to the area in the vicinity of knot $\sharp$3, traces a zone with slightly larger, but clearly distinct, $N/O$ value than the typical one for the galaxy. Given that this area is located at the spaxels with the lowest surface brightness, and to reject the possibility of any systematic effect due to a poor S/N ratio in faint lines like [<span style="font-variant:small-caps;">\[Oiii\]</span>]{}$\lambda$4363, we extracted a spectrum by summing up the flux in a rectangular area of 5$\times$6 spaxels at this location. The measured relative N-abundance, $\log(N/O)=-1.15$, is in agreement with the result found on a spaxel-by-spaxel basis.
![Comparison between the predicted $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}) and $T_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}). Spaxels were grouped in the same three bins in $n_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) as in Fig. \[compau\] as it is indicated in the upper right-hand corner of the diagram. \[compate\]](./fg12.ps){width="48.00000%"}
Discussion \[discusion\]
========================
Electron density structure \[densistruc\]
-----------------------------------------
As we discussed in Sec. \[localioni\], for a given spaxel, different ions are associated to different layers along the line of sight. We have presented maps for tracers of electron density based on [<span style="font-variant:small-caps;">\[Sii\]</span>]{}, [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{} and [\[O<span style="font-variant:small-caps;">ii</span>\]]{} emission line ratios . The corresponding density maps were derived using the same methodology as in Sec. \[fiscon\] and appear in Fig. \[mapne\]. Values consistent with being below the low-density limit are found in most of the fov, independently of the utilized tracer, while knot $\sharp$2 presents values somewhat larger ($\sim$190 cm$^{-3}$). The richest density structure is found in the main [GH<span style="font-variant:small-caps;">ii</span>R]{}. Here, all three tracers depict an $n_e$ structure with two peaks: the first one centered at knot $\sharp$1 while the second one at $\sim$26 ($\sim$50 pc) towards the northwest. However the values of the peaks vary depending on the utilized tracer. The largest densities if Fig. \[mapne\] are traced by [<span style="font-variant:small-caps;">\[Sii\]</span>]{}, while [\[O<span style="font-variant:small-caps;">ii</span>\]]{} and [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{} predict densities about 10-25% smaller. Given their ionization potentials, (23.3, 35.1, 30.7 eV for [<span style="font-variant:small-caps;">\[Sii\]</span>]{}, [\[O<span style="font-variant:small-caps;">ii</span>\]]{} and [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}, respectively), in the absence of extinction, $n_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) - for example - traces the density in a layer closer to the ionizing source than $n_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) [see e.g. @gar92]. This can be reversed in conditions of heavy extinction. Even if the intrinsic structure would be the same, lines involved in the determination of $n_e$([\[O<span style="font-variant:small-caps;">ii</span>\]]{}) are bluer (and therefore more sensitive to extinction) than those associated to $n_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) which therefore can probe deeper in the nebula. Since this [GH<span style="font-variant:small-caps;">ii</span>R]{} suffers from relatively high extinction the differences found between the three density maps are consistent with an onion-like structure where inner layers are denser than the outer ones. This is supported by the relative densities found between the two peaks. Also it is consistent with the fact that the [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{} ratio does not trace very high densities as the [\[Ar<span style="font-variant:small-caps;">iv</span>\]]{} ratio did . With an ionization potential of 59.8 eV, [\[Ar<span style="font-variant:small-caps;">iv</span>\]]{} would sample the densest innermost layer of this onion-like structure.
The one Gaussian fitting approach utilized up to now is useful to have a picture of the overall density structure in the galaxy. However, it is known that emission lines in NGC 5253 present complex profiles with asymmetries tracing different kinematics components. For the [GH<span style="font-variant:small-caps;">ii</span>R]{}, we tentatively measured the electron density for what we called the “broad” and “narrow” components on a spaxel-by-spaxel basis in using the [<span style="font-variant:small-caps;">\[Sii\]</span>]{} lines. Results were affected by large uncertainties, mainly due to the deblending procedure but also to the S/N ratio of the spectra. They suggested large and uniform densities for the “broad” component ($n_e\sim470$ cm$^{-3}$), while the “narrow” component presented a decrease of density from the northwest to the southeast. The stronger [\[O<span style="font-variant:small-caps;">ii</span>\]]{} lines offer the possibility of discussing this difference with much reduced uncertainties. Electron density maps from our multi-component analysis are presented in Fig. \[mapnemulti\]. The structure depicted in the map for broad component (bottom panel), with larger densities ($\sim500$ cm$^{-3}$) close to knot $\sharp$1 and decreasing outwards, is consistent with the general picture sketched above (i.e. larger densities closer to the ionizing source). The narrow component (upper panel) present a different structure, with the highest density (${\hbox{\rlap{\lower.55ex\hbox{$\sim$}} \kern-.3em
\raise.4ex \hbox{$<$}}}600$ cm$^{-3}$) at the secondary peak of the [GH<span style="font-variant:small-caps;">ii</span>R]{} at $\sim2\farcs6$ from knot $\sharp$1. This can be understood within the proposed scenario in (see their Fig. 21, where different elements associated with the area of the [GH<span style="font-variant:small-caps;">ii</span>R]{} are shown). There, this narrow component was associated to a shell of previously existing quiescent gas that has been reached by the ionization front. Densities in the shell would be high, if this pre-existing gas had been piled-up by the outflow associated to the broad component.
Electron temperature structure
------------------------------
As it happened with $n_e$, a comparison of $T_e$ maps derived from lines associated to different ions is useful to asses the $T_e$ structure in 3D of . Derived maps for $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}) and $T_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) are presented in Fig. \[mapte\]. At present, maps of $T_e$ based on any tracer are still scarce and most of the times focused on Galactic objects [e.g. @nun12; @tsa08]. Indeed, to our knowledge this is the first time that a map for $T_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) is provided and one of the few existing examples of $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}) maps in extragalactic astronomy. As we pointed out in Sec. \[mapneyte\], the overall structure is the same for $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}) and $T_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) maps. However, $T_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) is smaller than $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}), with typical $T_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{})/$T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}) ranging from $\sim$0.8 at the [GH<span style="font-variant:small-caps;">ii</span>R]{} to $\sim$0.6 in the areas of low surface brightness.
This relation between both temperatures is shown on a spaxel-by-spaxel basis in Fig. \[compate\]. Only those spaxels where the estimated uncertainties for both temperatures are smaller than 40% were considered. Both, the different values for $T_e$([<span style="font-variant:small-caps;">\[Oiii\]</span>]{}) and $T_e$([<span style="font-variant:small-caps;">\[Sii\]</span>]{}) and the larger differences at low surface brightness, can be understood within the frame of a change in the relative contribution along the line of sight of the warmer gas associated to the [GH<span style="font-variant:small-caps;">ii</span>R]{} and a colder and more diffuse gas component that extends further away. The 2D information on the plane of the sky (i.e. the maps in Fig. \[mapte\]) fits also in a satisfactory manner with this interpretation.
Summarizing, both the available information on the plane of the sky (Fig. \[mapte\]) and along the line of sight (Fig. \[compate\]) are consistent with a $T_e$ structure in 3D with higher temperatures close to the ionizing source surrounded by a more diffuse component of ionized gas at lower temperatures.
Validity of strong line methods: $N/O$ relative abundance
---------------------------------------------------------
Ideally, metallicity and relative abundances for the ionized gas in galaxies should be derived in a direct manner. However, this requires the determination of $T_e$, which depends on the detection of faint lines like [<span style="font-variant:small-caps;">\[Oiii\]</span>]{}$\lambda$4363. In extragalactic astronomy, most of the time this is not feasible. Instead, certain combinations of strong emission lines with more or less well established empirical and/or theoretical calibrations should be used [e.g. @per09; @kew08]. A comparison between both methodologies on a spaxel-by-spaxel basis would be useful to test the reliability of the strong line-based methods and identify the cause of possible biases. For example, @mon11 and @rel10 found that metallicity tracers are modulated by the ionization structure. The best-known metallicity tracer would probably be R23 [@pag79]. Unfortunately, with a 12+$\log(O/H)\sim8.28$, falls at the turn-over of the metallicity-R23 relation and therefore, this tracer is not appropriate for metallicity determinations in this galaxy. Alternatively, two widely used tracers are $O3N2=\log(($[<span style="font-variant:small-caps;">\[Oiii\]</span>$\lambda$5007/H$\beta$]{}$)/($[<span style="font-variant:small-caps;">\[Nii\]</span>$\lambda$6584/H$\alpha$]{}$))$ [@all79] and N2=[<span style="font-variant:small-caps;">\[Nii\]</span>$\lambda$6584/H$\alpha$]{} [@den02]. However, both involve [<span style="font-variant:small-caps;">\[Nii\]</span>]{} emission lines, and are therefore, affected by the variation of relative $N/O$ abundance across the galaxy. Thus, is not an adequate example to test the reliability of metallicity determinations based on strong optical lines. Instead, since this galaxy presents a range of $N/O$ abundances, a different test for the corresponding strong line based tracers would be of high interest. This will be discussed in this section. As a baseline, we converted our line ratio maps into relative abundances using the expressions provided by @per09, that relate $N/O$ with N2O2 = $\log$([<span style="font-variant:small-caps;">\[Nii\]</span>]{}$\lambda$6584/[\[O<span style="font-variant:small-caps;">ii</span>\]]{}$\lambda\lambda$3726,3729) and N2S2 =$\log$([<span style="font-variant:small-caps;">\[Nii\]</span>]{}$\lambda$6584/[<span style="font-variant:small-caps;">\[Sii\]</span>]{}$\lambda\lambda$6717,6731) as defined by @kew02b and @sab77.
Maps for the estimated relative $N/O$ abundances are presented in Fig. \[compaabun\]. They show that both tracers, N2O2 and N2S2, detect the main area with N-enhancement. Interestingly enough, only N2O2 is sensitive to the newly discovered area, associated to knot $\sharp$3.
![Comparison between the residuals for the determination of $N/O$ based on the two strong line methods under consideration with respect to the direct one. The locus of equal residuals is indicated with a black line. An orange square marks the position where the three methods agree. Data were divided in three bins of $n_e$ as in Figs. \[compau\] and \[compate\]. \[comparesi\]](./fg14.ps){width="45.00000%"}
However, both tracers fail to predict the correct $N/O$ abundance. This is not unexpected since the utilized relations are valid to interpret global tendencies in a statistically significant sample of galaxies, while $N/O$ abundances for an individual object can depart about $\pm0.3$ and $\pm0.5$ dex from this relation for the N2O2 and N2S2 tracers respectively [see e.g. Figs. 10 and 11 in @per09]. Figure \[comparesi\] shows a comparison of the residuals between the strong line methods and the direct one. Only those spaxels with an estimated uncertainty lower than 0.25 dex in both residuals were included in the comparison. The good correlation between the residuals is consistent with both methods being affected by the same factors. Also, there is an increase of the residual with $n_e$. This also implies an increase with the other physical parameters since a comparison of the different maps presented throughout the paper shows that in general, larger $n_e$ corresponds to brighter areas and with larger $T_e$ and ionization strength. Given the spatial coincidence of these variations, disentangling the relative role of a given physical quantity as the cause of the variations in the residuals is not straightforward for this galaxy and will not be addressed here. Mean ($\pm$standard deviation) for the residuals are $\sim0.11 (\pm0.03)$ and $\sim0.02 (\pm0.12)$ for N2O2 and N2S2, respectively. Therefore, a method based on N2O2 is $\sim$4 times less sensitive to any variation of the physical/chemical properties than the one based on N2S2.
In summary, our comparison of the $N/O$ abundances derived using strong line methods with respect to direct measurements in the search for chemical inhomogeneities within a galaxy, supports the use of that based on N2O2 over that based on N2S2. The first method is sensitive to a wider range of $N/O$ abundances and is more stable against variations of physical conditions within the area of interest.
On the relationship between extra N and WR stars
------------------------------------------------
There are several works in the literature using long-slit observations that suggest a connection between nitrogen overabundance and the presence of WR stars based on the simultaneous detection of WR features and a higher than the expected $N/O$ in a given object [e.g. @wal87; @thu96; @gus00; @pus04; @izo06b; @hag06; @per10; @lop12]. Also, local associations between WR emission and N-enhancements have been found by means of IFS based observations [e.g. @jam09] supporting this connection. However, the general picture is far from this simple one-to-one association. For example, @per11 studied a sample of BCD galaxies that are N-overabundant over large areas supporting the idea of enrichment caused by accretion of less chemically processed gas [see also @keh08]. A similar result was found by @amo12 for a set of BCD galaxies at higher redshift. On top of that, resolution effects also play a important role. Already at distances of $\sim$25-40 Mpc the whole area covered here would have a projected size of the order of the resolution of typical ground based observations ($\sim$10).
We reported in areas with both extra-N and WR emission in , as for example, our knot $\sharp$1 [widely discussed in the literature, e.g. @wal89; @kob97; @sch99; @lop07] together with areas with WR emission but without any chemical anomaly (e.g. our knot $\sharp$2). Moreover, we report in Sec. \[secabun\] a new area presenting an overabundance of nitrogen without any WR emission associated to it. This variety of options (i.e. WR emission and enhanced N, no WR emission and enhanced N, WR emission and no enhanced N, no WR emission and no enhanced N - i.e. the normal case) seems a more natural situation than a one-to-one association since the chemical enrichment of the warm ionized medium due to the material expelled by the massive stars is a complex process. In a simplified manner, this can be divided in three basic steps.
Firstly, stars must bring their processed material to the surface. Recent stellar evolution models show how the inclusion of rotation favors the appearance of processed material at stages as early as the main sequence [i.e. @mae00; @mae05; @mae10]. However they are particularly powerful at the WR, or even Luminous Blue Variable, phases [see @cro07 and references therein]. Also, observational evidence of N-enhancement in WR ring nebulae has been reported [@fer12].
Secondly, this material should be expelled via stellar winds, first, and supernova explosions, afterwards. In the case of star clusters, where stars with a variety of initial masses have a variety of evolutionary paths, the yield of the different elements varies in a non-trivial manner with time [e.g. @mol12]. An extra caveat arises for clusters with stellar masses of ${\hbox{\rlap{\lower.55ex\hbox{$\sim$}} \kern-.3em
\raise.4ex \hbox{$<$}}}10^4$ M$_\odot$ since in this regime the Initial Mass Function is not properly sampled and stochastic effects are important [e.g. @vil10].
Finally, this new material must be disseminated and then diluted in the warm ionized medium which, in due time, would reach a new chemical homogeneity. This is an even less trivial step since on top of the specific characteristics of a given star cluster, factors like the existence and characteristics of neighbouring clusters, presence of cloudlets of gas, degree of the inhomogeneities, etc. are important to properly trace the evolution of the yielded material [e.g. @ten96; @dea02].
Therefore, a unique evolutionary scenario to describe the path from the creation of the new nitrogen to its incorporation to the warm ionized medium seems unlikely. In the following we propose some evolutionary paths for our knots $\sharp$1-3. These scenarios do not intend to be more than reasonable suggestions that compile the constraints derived from our results and which could be tested with detailed modeling.
Regarding our knots $\sharp$2 and $\sharp$3, we hypothesize here about the possibility that they constitute two snapshots of the same evolutionary path. According to @har04 the main clusters associated to them are relatively similar in terms of mass and youth, although those in knot $\sharp$3 are slightly older[^8]. Supporting this youth, candidates to supernova remnants have been found close to both knots [@lab06]. Moreover, only knot $\sharp$2 presented spectral features typical of Wolf-Rayet stars. In this evolutionary path, stars would expel their processed nitrogen at very early stages (${\hbox{\rlap{\lower.55ex\hbox{$\sim$}} \kern-.3em
\raise.4ex \hbox{$<$}}}$5 Myr). The difference between the age of the cluster in knot $\sharp$3 ($\sim$10 Myr) and the moment when the yielding of extra-N took place [as soon as $\sim2.5$ Myr, according to the models of @mol12] sets an upper limit of $\sim$8 Myr for the duration of this process. If rotating O stars were playing a relevant role, this limit could be extended up to the age of clusters. A lower limit can be estimated under the most optimistic (and efficient) assumption: instantaneous incorporation of the yielded material to the warm ionized gas. If, in addition, we assume that the velocity of the ionized gas with respect to knot $\sharp$3 traces the velocity under which the contamination of extra nitrogen propagates through the ISM, the area could be enriched in only $\sim$2 Myr. Therefore, the process of the cooling down and mixing of the yielded material with the ISM over an area of 40-50 pc in diameter should last between $\sim$2 and $\sim$8 Myr. Note that the moment when the newly created material has completely mixed with the previously existing gas, and any chemical inhomogeneity has been erased, should occur much later and cannot be delimited with these observations further than $\sim$10 Myr.
Knot $\sharp$1 should follow a different evolutionary path since at this location both WR features and nitrogen enrichment are found. The knot is associated to two very young and massive super star clusters [e.g. @sch99; @gor01; @alo04], embedded in a very dense and compact nebula [@tur00]. In that sense, they can be seen as nascent clusters that have not managed yet to disperse the cloud of gas where they were born. This implies a very particular set of physical conditions for the warm interstellar medium. Indeed, a giant molecular cloud associated to this region has been reported [@mei02] and we showed through this work that $n_e$ and $T_e$ associated to this region are relatively high, as well as the degree of ionization. Also, the region presents supersonic velocity widths and high extinction . Somehow, the combination of some of these particular conditions has made possible the incorporation of the newly created nitrogen at an earlier stage.
A third example mentioned in previous work provides a different evolutionary path. [@gar12] gives an age of 3.5 Myr and a stellar mass of $\sim5\times10^3$ M$_\odot$ for their knot C in . Moreover, they report a relative abundance of nitrogen $\sim$0.2 dex larger than in their other apertures. No WR feature was detected for this knot (García-Benito, private communication). Clearly, at this mass regime, the yield is dominated by stochastic effects. Given the lack of any WR detection one has to resort to rotating O stars as the cause of the N-enhancement. Contrary to what happened with our knot $\sharp$1 no particularly extreme physical conditions other than a complex inner dust structure were reported for its surrounding ISM. This leaves the open question of how this ISM managed to incorporate this new material in such a short time scale.
All in all, even if there seems to be a connection between WR emission and nitrogen enhancement, as supported by the fact that WR galaxies show an elevated $N/O$ relative to non-WR galaxies [@bri08], local examples like and , where linear spatial resolutions of $\sim$20 pc arcsec$^{-1}$ can be achieved, show how this relationship is complex. Specifically, the case of the knots $\sharp$2 and $\sharp$3 in , where WR emission and N-enhancement are associated to different star clusters separated by only $\sim$90 pc, illustrates the possibility that in galaxies at distances ${\hbox{\rlap{\lower.55ex\hbox{$\sim$}} \kern-.3em
\raise.4ex \hbox{$>$}}}25$ Mpc, the spatial coincidence of WR emission and $N/O$ overabundances does not necessarily imply an intimate association between them based on *cause-effect* relationship. However, this does not reject the possibility that both of them were related due to a common external cause. As an example, in one could identify as this external cause the putative event that triggered the starburst (i.e. the past interaction with M 83).
Conclusions
===========
We have carried out a detailed 2D study of the physical and chemical properties of the ionized gas in the central part of , a very nearby BCD. The area was mapped in a continuous manner with the ARGUS-IFU unit of FLAMES. This work represents the natural continuation of the one presented in . The different maps utilized along the paper as well as the reduced data cubes are available as FITS files from the authors.
The major conclusions can be summarized as follows:
1\. Physical and chemical properties of the ionized gas associated to the main star clusters were derived by extracting spectra in apertures of 9-10 spaxels. Measurements associated to knots $\sharp$1 and $\sharp$2 agree in general with those previously reported. The existing discrepancies are associated to the assumed electron temperature. Abundances for knot $\sharp$3 (not reported so far) are also provided. With the exception of the relative abundance for $N/O$ in knot $\sharp$1 and possible knot $\sharp$3, no chemical species seems overabundant in any of the selected apertures.
2\. Maps of the electron density based on four different tracers - namely [\[O<span style="font-variant:small-caps;">ii</span>\]]{}, [<span style="font-variant:small-caps;">\[Sii\]</span>]{}, [\[Fe<span style="font-variant:small-caps;">iii</span>\]]{}, and [\[Ar<span style="font-variant:small-caps;">iv</span>\]]{} line ratios - were discussed. In all the cases, higher densities are associated to the main [GH<span style="font-variant:small-caps;">ii</span>R]{}. The joint analysis of these maps is consistent with a 3D stratified view of the nebula where the highest densities are located in the innermost layers while density decreases when going outwards.
3\. The 2D $n_e$ structure for the two main kinematic components was also derived using the [\[O<span style="font-variant:small-caps;">ii</span>\]]{} lines as baseline. While the so-called broad component follows the picture described above, there is a change of structure for the narrow one. This fits well with the proposed scenario in where this component was associated to a shell of previously existing material that had been piled up by the outflow associated to the broad component.
4\. Maps for $T_e({\textsc{[S\,ii]}})$ and $T_e({\textsc{[O\,iii]}})$ were derived. To our knowledge, this is the first time that a $T_e({\textsc{[S\,ii]}})$ map for an extragalactic object is presented. Also, we provided with one of the few examples of existing $T_e({\textsc{[O\,iii]}})$ maps up to date. The joint interpretation of the information on the plane of the sky and along the line of sight is consistent with a $T_e$ structure in 3D with higher temperatures close to the main ionizing source surrounded by a colder and more diffuse component. This $T_e$ structure together with the lack of any strong broad component far from the main [GH<span style="font-variant:small-caps;">ii</span>R]{} is in accord with the lack of any clear evidence of shocks playing a dominant role.
5\. Ionization structure was traced by means of the [<span style="font-variant:small-caps;">\[Oiii\]</span>$\lambda\lambda$4959,5007/<span style="font-variant:small-caps;">\[Oii\]</span>$\lambda\lambda$3726,3729]{}, [\[O<span style="font-variant:small-caps;">ii</span>\]]{}$\lambda\lambda$3726,3729/[H$\beta$]{}, and [<span style="font-variant:small-caps;">\[Sii\]</span>]{}$\lambda\lambda$6717,6731/[H$\alpha$]{} ratios. The two first of them predict similar ionization degree while the third one would be typical of lower ionization. A possible 3D interpretation of both the observed structure in the maps for each individual ratio and the discrepancy of [<span style="font-variant:small-caps;">\[Sii\]</span>]{}$\lambda\lambda$6717,6731/[H$\alpha$]{} for individual spaxels is consistent with the lower ionization species (i.e. $S^+$) delineating the more extended diffuse component. 6. Maps for the 2D distribution of abundances for oxygen, neon, argon were derived. All of them are consistent with no chemical inhomogeneities. The derived mean ($\pm$ standard deviation) oxygen abundance is $12+\log(O/H)=8.26\pm0.04$. The mean logarithmic relative abundances for argon and neon were $-0.65\pm0.03$ and $-2.33\pm0.06$, respectively.
7\. In the same manner, a map for the 2D distribution of nitrogen was derived. typically presents a $\log(N/O)$ of $\sim-1.32\pm0.05$. However, there are two locations with enhanced $N/O$. The first one was already reported and characterized. With a $\log(N/O)\sim-0.95$, it occupies an elliptical area of about 80 pc$\times$35 pc and is associated to the two SSCs at the nucleus of the galaxy. The second one is reported here for the first time. It presents a $\log(N/O)\sim-1.17$ and it is associated to two moderately massive ($2-4\times10^4$ M$_\odot$) and relatively old ($\sim10$ Myr) clusters (knot $\sharp$ 3).
8\. The map of $N/O$ relative abundance derived through the direct method was compared with those derived using strong line methods. The comparison supports a method based on N2O2 over a method based on N2S2 in the search of chemical inhomogeneities *within* a galaxy since the first method is sensitive to a wider range of $N/O$ abundances and is more stable against variations of physical conditions within the area of interest.
9\. We utilized the results on the localized detection of WR emission and nitrogen enhancement to compile and discuss the factors that affect the complex relationship between the presence of WR stars and $N/O$ excess. Even if there seems to be such a relationship, a unique scenario describing the path from the production of the new nitrogen to its incorporation into the warm ionized medium seems unlikely. In particular, we use the areas associated to knots $\sharp$2 and $\sharp$3 in as examples of WR emission and N-enhancement that would be perceived as spatially coincident at distances ${\hbox{\rlap{\lower.55ex\hbox{$\sim$}} \kern-.3em
\raise.4ex \hbox{$>$}}}25$ Mpc but that are not intimately associated in a cause-effect fashion. However, this does not reject the possibility that both of them are related due to a common external cause.
This paper has benefited from fruitful conversations during the Workshop “Metals in 3D: New insights from Integral Field Spectroscopy”. We would like to thank in particular to R. García-Benito, B. James, M. Mollá, L. Smith, G. Tenorio-Tagle, and E. Pérez-Montero, M. Relaño as well as to M. Westmoquette and R. Amorín with whom we shared stimulating discussions that helped to improve it. We also thank the referee for the useful comments that have significantly improved the first submitted version of this paper.
Based on observations carried out at the European Southern Observatory, Paranal (Chile), programmes 078.B-0043(A) and 383.B-0043(A). This paper uses the plotting package `jmaplot`, developed by Jesús Maíz-Apellániz, `http://dae45.iaa.csic.es:8080/\simjmaiz/software`. This research made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The STARLIGHT project is supported by the Brazilian agencies CNPq, CAPES, FAPESP and by the France-Brazil CAPES-COFECUB programme.
A. M.-I. is supported by the Spanish Research Council within the programme JAE-Doc, Junta para la Ampliación de Estudios, co-funded by the FSE. This work has been partially funded by the projects AYA2010-21887 from the Spanish PNAYA, CSD2006 - 00070 “1st Science with GTC” from the CONSOLIDER 2010 programme of the Spanish MICINN, and TIC114 Galaxias y Cosmología of the Junta de Andalucía (Spain).
[^1]: Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile (ESO Programme 078.B-0043 and 383.B-0043).
[^2]: http://www.eso.org/projects/dfs/dfs-shared/web/vlt/vlt-instrument-pipelines.html.
[^3]: The Image Reduction and Analysis Facility *IRAF* is distributed by the National Optical Astronomy Observatories which is operated by the association of Universities for Research in Astronomy, Inc. under cooperative agreement with the National Science Foundation.
[^4]: Ages for the clusters associated to this knot were incorrectly quoted in our previous work. The ages estimated by @har04, for their clusters 3 and 5 were 8 and 11 Myr, respectively.
[^5]: http://starlight.ufsc.br/index.php?section=1
[^6]: We assumed 12 + $\log$(O/H)$_\odot$ = 8.66 [@asp04].
[^7]: http://www.iac.es/consolider-ingenio-gtc/index.php?option=com\_content&view=article&id=223:qa-spatially-resolved-study-of-ionized-regions-in-galaxies-at-different-scales&catid=45:tesis&Itemid=65
[^8]: Knot $\sharp$2 corresponds to clusters 4 and 8 in @har04, with stellar masses of $\sim$2.7 and 1.3$\times10^4$ M$_\odot$ and ages of 1 and 5 Myr while knot $\sharp$3 corresponds to their cluster 3 and 5, with masses 4.2 and 2.1$\times10^4$ M$_\odot$ and ages of 8 and 11 Myr, respectively.
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abstract: 'Magneto-transport measurements in a wide GaAs quantum well in which we can tune the Fermi energy ($E_F$) to lie in different Landau levels of the two occupied electric subbands reveal a remarkable pattern for the appearance and disappearance of fractional quantum Hall states at $\nu$ = 10/3, 11/3, 13/3, 14/3, 16/3, and 17/3. The data provide direct evidence that the $q/3$ states are stable and strong even at such high fillings as long as $E_F$ lies in a ground-state ($N=0$) Landau level of either of the two electric subbands, regardless of whether that level belongs to the symmetric or the anti-symmetric subband. Evidently, the node in the out-of-plane direction of the anti-symmetric subband does not de-stabilize the $q/3$ fractional states. On the other hand, when $E_F$ lies in an excited ($N>0$) Landau level of either subband, the wavefunction node(s) in the in-plane direction weaken or completely de-stabilize the $q/3$ fractional quantum Hall states. Our data also show that the $q/3$ states remain stable very near the crossing of two Landau levels belonging to the two subbands, especially if the levels have parallel spins.'
author:
- Yang Liu
- 'J. Shabani'
- 'M. Shayegan'
bibliography:
- 'paper\_v1.bib'
date:
- today
-
title: 'When are the $q/3$ fractional quantum Hall states stable?'
---
Introduction
============
![\[fig:NQWvsWQW\] Longitudinal resistance ($R_{xx}$) vs. perpendicular magnetic field ($B$) traces are shown for electrons confined to: (a) a narrow (well-width $W=$ 30 nm) GaAs quantum well, and (b) a wide ($W=$ 55 nm) quantum well. In (a) FQH states at $\nu=5/2$ and 7/2 can be clearly seen, but the states at $\nu=7/3$ and 8/3 are weak. In contrast, the even-denominator states are absent in (b) but strong FQH states are seen at $\nu=7/3$ and 8/3. Note also the absence of FQH states for $\nu > 4$ in (a). The insets schematically show the positions of the spin-split LLs of the lowest (S) and second (A) electric subbands, as well as the position of $E_F$ at $\nu=3$; the indices $N=0$ and $N=1$ indicate the lowest and the excited LLs, respectively. The subband separation for the trace in (b) is $\Delta=24$ K.](Fig1_edit_1){width="45.00000%"}
The fractional quantum Hall (FQH) effect, [@Tsui.PRL.1982] signaled by the vanishing of the longitudinal resistance and the quantization of the Hall resistance, is the hallmark of an interacting two-dimensional electron system (2DES) in a large perpendicular magnetic field. It is a unique incompressible quantum liquid phase described by the celebrated Laughlin wavefunction. [@Laughlin.PRL.1983] In a standard, single-subband 2DES confined to a low-disorder GaAs quantum well, the FQH effect is most prominently observed at low Landau level (LL) filling factors $\nu<2$, where the Fermi energy ($E_F$) lies in the spin-resolved LLs with the lowest orbital index ($N=0$). [@CF_Jain] The strongest states are seen at the $q/3$ fractional fillings, namely at $\nu=1/3$, 2/3, 4/3, and 5/3. In contrast, as illustrated in Fig. 1(a), when $E_F$ lies in the second ($N=1$) set of LLs ($2<\nu<4$), the equivalent $q/3$ states at $\nu = 7/3, 8/3, 10/3,$ and 11/3 are much weaker. [@Pan.PRL.1999; @Toke.PRB.2005] In yet higher LLs ($\nu>4$), e.g., at $\nu=$13/3, 14/3, 16/3, and 17/3, which correspond to $E_F$ being in the third ($N=2$) set of LLs, the FQH states are essentially absent; [@Lilly.PRL.1999; @Du.SSC.1999; @Gervais.PRL.2004] see Fig. 1(a). This absence is believed to be a result of the larger extent of the electron wavefunction (in the 2D plane) and its extra nodes that modify the (exchange-correlation) interaction effects and favor the stability of various non-uniform charge density states (e.g., stripe phases) over the FQH states. [@TheQHE; @MacDonald.PRB.1986; @Ambrumenil.JPC.1988; @Koulakov.PRL.1996; @Moessner.PRB.1996]
Recently, the FQH effect was examined in a wide GaAs quantum well where two electric subbands are occupied.[@Shabani.PRL.2010] A main finding of Ref. is highlighted in Fig. 1(b): When the Fermi level ($E_F$) lies in the $N=0$ LLs of the anti-symmetric electric subband, the even-denominator FQH states (at $\nu=5/2$ and 7/2) are absent and, instead, strong FQH states are observed at $q/3$ fillings $\nu=$ 7/3, 8/3, 10/3 and 11/3. Here we extend the measurements in this two-subband system and examine the stability of the $q/3$ FQH states at even higher fillings as we tune the position of $E_F$ to lie in different LLs of the two subbands. At a fixed 2DES density, we observe a remarkable pattern of alternating appearance and disappearance of the $q/3$ states as we tune the subband separation and the position of $E_F$. The data demonstrate that the $q/3$ states are stable even at filling factors as high as $\nu=17/3$, as long as $E_F$ lies in a ground state ($N=0$) LL, regardless of whether that LL belongs to the symmetric or anti-symmetric subband.
Sample and experimental details
===============================
{width=".9\textwidth"}
Our sample, grown by molecular beam epitaxy, is a 55 nm-wide GaAs quantum well (QW) bounded on each side by undoped Al$_{0.24}$Ga$_{0.76}$As spacer layers and Si $\delta$-doped layers.[^1] We fitted the sample with an evaporated Ti/Au front-gate and an In back-gate to change the 2D electron density, $n$, and tune the charge distribution symmetry and the occupancy of the two electric subbands, as demonstrated in Fig. \[fig:cartoon\]. This tunability, combined with the very high mobility ($\sim$ 400 m$^2$/Vs) of the sample, is key to our success in probing the strength of the $q/3$ states at high fillings.
![\[fig:waterfall\] Waterfall plot of $R_{xx}$ vs. $B$ taken at a fixed density $n=2.12\times
10^{11}$cm$^{-2}$ as the subband separation ($\Delta$) is increased. The scale for $R_{xx}$ is indicted in the upper left (0 to 1 k$\Omega$). Each trace is shifted vertically so that its zero (of $R_{xx}$) is aligned with its measured value of $\Delta$ which is used as the y-axis of the waterfall plot. Vertical lines mark the field positions of the filling factors, $\nu$. ](waterfall_final){width="45.00000%"}
When the QW in our experiments is ”balanced”, i.e., the charge distribution is symmetric, the occupied subbands are the symmetric (S) and anti-symmetric (A) states (see the lower panels in Figs. 2(a) and (b)). When the QW is ”imbalanced,” the two occupied subbands are no longer symmetric or anti-symmetric; nevertheless, for brevity, we still refer to these as S (ground state) and A (excited state). In our experiments, we carefully control the electron density and charge distribution symmetry in the QW via applying back- and front-gate biases.[@Suen.PRL.1994; @Shabani.PRL.2009] For each pair of gate biases, we measure the occupied subband electron densities from the Fourier transforms of the low-field ($B\le 0.5$ T) Shubnikov-de Haas oscillations. These Fourier transforms, examples of which are shown in Fig. 2(c), exhibit two peaks ($B_S$ and $B_A$) whose frequencies, multiplied by $2e/h$, give the subband densities, $n_S$ and $n_A$. The difference between these densities directly gives the subband separation, $\Delta$, through the expression $\Delta=\frac{\pi\hbar^2}{m^{*}}(n_S-n_A)$, where $m^{*}$ is the electron effective mass. Note that, at a fixed total density, $\Delta$ is smallest when the charge distribution is balanced and it increases as the QW is imbalanced. Figure 2(d) shows the measured $\Delta$ as a function of the charge $\delta n$ transferred between the back and front sides of the QW. Note that we measure $\delta n$ from the change in the sample density induced by the application of either the back-gate or the front-gate bias.
magneto-transport data
======================
{width=".9\textwidth"}
![\[fig:low\_field\_colorful\] An expanded color-scale plot of $R_{xx}$ data at low fields for $n=2.12\times 10^{11} $ cm$^{-2}$. The solid white lines denote $\Delta=iE_C$ for $i=2,3,4,
5$. The dashed white lines represent $\Delta=iE_C\pm E_Z$, using a fixed $g^{\star}=8.8$ (see text).](low_field_colorful){width="40.00000%"}
{width=".9\textwidth"}
![\[fig:hall\_trace\_212\] $R_{xx}$ and $R_{xy}$ traces at high magnetic fields and $T=30$ mK for $n=2.12\times 10^{11} $ cm$^{-2}$ for the “balanced” QW. Fractional quantum Hall states at $\nu=4/3$, 5/3, 7/3, and 8/3 are clearly seen. The upper panels show the LL diagrams and positions of $E_F$ for the indicated fillings.](2_1_balanced){width="45.00000%"}
Figure \[fig:waterfall\] shows a series of longitudinal resistance ($R_{xx}$) vs. magnetic field ($B$) traces taken at a fixed density $n=2.12\times 10^{11} $ cm$^{-2}$ as the subband spacing is increased. The y-axis is $\Delta$, which is measured from the low-field Shubnikov-de Haas oscillations of each trace. The same data are interpolated and presented in a color-scale plot in Fig. \[fig:colorful\](a). In Fig. \[fig:low\_field\_colorful\], we show a color-scale plot of the data in the low field regime.
In Figs. \[fig:waterfall\], \[fig:colorful\](a), and \[fig:low\_field\_colorful\] we observe numerous LL coincidences at various integer filling factors, signaled by a weakening or disappearance of the $R_{xx}$ minimum. For example, the $R_{xx}$ minimum at $\nu=4$ is strong and wide at all values of $\Delta$ except near $\Delta$ = 32 and 58 K, marked by squares in Fig. \[fig:colorful\](a), where it becomes narrow or disappears. Such coincidences can be easily explained in a simple fan diagram of the LL energies in our system as a function of increasing $\Delta$, as schematically shown in Fig. \[fig:colorful\](b). In this figure, we denote an energy level by its subband index (S or A), LL index ($N=0, 1, 2, \cdots$), and spin ($\uparrow$ or $\downarrow$). Also indicated in Fig. \[fig:colorful\](b) are the separations between various levels: the cyclotron energy ($E_C=\hbar
eB/m^{*}$), Zeeman energy ($E_Z=g^{*}\mu_BB$, where $g^{*}$ is the effective Landé g-factor), and $\Delta$. From Fig. \[fig:colorful\](b) it is clear that the condition for observing a LL coincidence at odd fillings is $\Delta=iE_C$, while for coincidences at even fillings, the condition is $\Delta=iE_C\pm E_Z$; in both cases, $i$ is a positive integer.
In Figs. \[fig:colorful\](a) and \[fig:colorful\](b), we have indicated the two coincidences at $\nu=4$ with squares. Note that the coincidences at even fillings correspond to a crossing of two levels with antiparallel spins. In Figs. \[fig:waterfall\] and \[fig:colorful\](a), the coincidences at low, odd fillings (e.g., $\nu=3$ and 5) are not as easy to see at low temperatures since the resistance minima remain strong as the two LLs, which have parallel spins, cross. Such behavior has been reported previously and has been interpreted as a signature of easy-plane ferromagnetism. [@Jungwirth.PRL.1998; @Muraki.PRL.2001; @Vakili.PRL.2006] We note that our data taken at higher temperatures ($T$ = 0.31 K) reveal a weakening of the $\nu=5$ minimum at $\Delta=35$ K, and of the $\nu=3$ minimum at $\Delta=58$ K;[@Shabani.Thesis.2011] these are marked by circles in Fig. \[fig:colorful\](a). The crossings at higher odd fillings are clearly seen in Figs. \[fig:colorful\](a) and \[fig:low\_field\_colorful\]; e.g., the $\nu=7$ minimum disappears at around $\Delta=50$ K, and $\nu=9$ around $\Delta=$ 40 K and 60 K.[^2]
In Figs. \[fig:colorful\](a) and \[fig:low\_field\_colorful\] we include several solid white lines representing $\Delta=iE_C$, assuming GaAs band effective mass of $m^{*}=0.067$ (in units of free electron mass). These lines indeed pass through the positions of the $observed$ LL coincidences for odd fillings, implying that $\Delta$ is not re-normalized at LL coincidences. We note that, with the application of magnetic field, the subband electron occupation might vary because of the finite number of discrete LLs that are occupied. This could lead to a redistribution of charge which in turn could lead to changes in $\Delta$ as a function of magnetic field. At LL coincidences, however, the two crossing LLs which belong to the different subbands are energetically degenerate. If the coincidence occurs at the Fermi energy, electrons can move between the two degenerate LLs so that the subband occupancy and the charge distribution, and therefore $\Delta$, are restored back to their zero-field values. This conjecture is indeed confirmed by self-consistent calculations reported for a two-subband 2D electron system in a perpendicular magnetic field:[@Trott.PRB.1989] While the subband occupancy and $\Delta$ oscillate with field, they equal their zero-field values whenever two LLs belonging to different subbands coincide at $E_F$. We conclude that the field positions of the LL coincidences at $E_F$ are determined by the value of $\Delta$ at $B=0$, and that the lines drawn in Figs. 4(a) and 5 accurately describe the positions of these coincidences.
The dashed lines in Figs. \[fig:colorful\](a) and \[fig:low\_field\_colorful\], represent $\Delta=iE_C\pm E_Z$, $i=1,2,
...$, where $g^{*}$ is chosen as a fitting parameter so that these lines pass through the even-filling coincidences. All the dashed lines in Figs. \[fig:colorful\](a) and \[fig:low\_field\_colorful\] are drawn using $g^{*}=$ 8.8, except for the $\Delta = E_C \pm E_Z$ lines, which are drawn using $g^{*}=$ 8.9 and 7.6, respectively. We conclude that $g^{*}$ is enhanced by a factor of $\sim$ 20 relative to the GaAs band g-factor (0.44). This enhancement is somewhat larger than the values reported for GaAs QWs with two subbands occupied. For example, Muraki $et\ al.$ [@Muraki.PRL.2001] reported a $\sim$ 10-fold enhancement of $g^{*}$ for electrons in a 40 nm-wide QW with $n\sim
3\times 10^{11} $ cm$^{-2}$ while Zhang $et\ al.$ [@Zhang.PRB.2006] measured a $\sim$ 5-fold enhancement in a 24 nm-wide QW with $n\sim 7\times 10^{11} $ cm$^{-2}$. It appears then that the enhancement depends on the QW width and electron density, and a systematic study of the enhancement would be an interesting future project. However, we would like to emphasize that the dashed lines in Figs. \[fig:colorful\](a) and \[fig:low\_field\_colorful\] pass through nearly all of the observed coincidences quite well. Since each of these lines are drown using very similar $g^{*}$, the data imply that the enhancement is nearly independent of the filling factor.[^3]
We now focus on the main finding of our work, namely the correspondence between the stability of the FQH states and the position of $E_F$. Note in Figs. \[fig:waterfall\] and \[fig:colorful\](a) that FQH states are observed only in certain ranges of $\Delta$. For example, the $\nu=10/3$ and 11/3 states are seen in the regions marked by A and C in Fig. \[fig:colorful\](a) but they are essentially absent in the B region. The $\nu=13/3$ and 14/3 states, on the other hand, are absent in regions D and F while they are clearly seen in regions E and G.
To understand this behavior, in the fan diagram of Fig. \[fig:colorful\](b) we have highlighted the position of $E_F$ as a function of $\Delta$ for different filling factors by color-coded lines. Concentrating on the range $3<\nu<4$ (green line in Fig. \[fig:colorful\](b)), at small values of $\Delta$ (region A), $E_F$ lies in the A0$\downarrow$ level. At higher $\Delta$, past the first $\nu=4$ coincidence which occurs when $\Delta=E_C - E_Z$, $E_F$ is in the S1$\uparrow$ level (region B). Once $\Delta$ exceeds $E_C$, $E_F$ lies in the A0$\uparrow$ level (region C) until the second $\nu=4$ coincidence occurs when $\Delta=E_C + E_Z$. Note in Fig. \[fig:colorful\](a) that strong FQH states at $\nu=10/3$ and $11/3$ are seen in regions A and C. From the fan diagram of Fig. \[fig:colorful\](b) it is clear that in these regions $E_F$ is in the $ground$-$state$ ($N=0$) LLs of the asymmetric subband, i.e., A0$\uparrow$ and A0$\downarrow$. In contrast, in region B, where the 10/3 and 11/3 states are essentially absent, $E_F$ lies in an $excited$ ($N=1$) LL, namely, S1$\uparrow$. We conclude that the 10/3 and 11/3 FQH states are stable and strong when $E_F$ lies in a ground-state LL.
The data in the range $4<\nu<5$ corroborate the above conclusion. In Fig. \[fig:colorful\](b) we represent the position of $E_F$ in this filling range by a blue line. In regions E and G, $E_F$ lies in the ground-state LLs of the asymmetric subband (A0$\downarrow$ and A0$\uparrow$), and these regions are indeed where the $\nu=13/3$ and 14/3 FQH states are seen. In regions D and F, on the other hand, $E_F$ is in the excited LLs of the symmetric subband (S1$\uparrow$ and S1$\downarrow$), and the 13/3 and 14/3 FQH states are absent. Data at yet higher fillings ($5<\nu<6$) follow the same trend: FQH states at $\nu=16/3$ and 17/3 are seen in region I when $E_F$ is in the A0$\downarrow$ level,[^4] but they are absent in regions H or J where $E_F$ lies in the S1$\downarrow$ or S2$\uparrow$ levels.
In Fig. \[fig:hall\_trace\_290\] we show additional data for a density of $n=2.90\times 10^{11} $ cm$^{-2}$ in the same QW. Longitudinal and Hall resistance traces are shown in the bottom panels for three different values of $\Delta$, and in each panel the calculated charge distribution (at $B=0$) is also shown. In the top panels, we show the positions of the LLs and $E_F$, corresponding to the filling factors in the bottom panels. In all cases, strong $q/3$ FQH states are observed when $E_F$ lies in the $N=0$ of the A0$\downarrow$ level. Note that the data shown in Fig. \[fig:hall\_trace\_290\] are for asymmetric charge distributions. We would like to emphasize that strong $q/3$ states are also observed for symmetric (“balanced”) charge distributions; e.g., see the bottom trace in Fig. 3, or the traces in Fig. 2(c) of Shabani $et$ $al.$ [@Shabani.PRL.2010]
Next we address the FQH states observed at lower $\nu$ ($< 3$) in our sample. Data are shown for $n=2.12\times 10^{11} $ cm$^{-2}$ for the “balanced” QW ($\Delta=23$ K) in Fig. \[fig:hall\_trace\_212\]; the $R_{xx}$ trace is an extension of the lowest trace shown in Fig. \[fig:waterfall\]. In the range 1 $<\nu<$ 3, strong FQH states are seen at $\nu=$ 4/3, 5/3, 7/3 and 8/3. Data taken at yet higher magnetic fields (not shown) reveal the presence of a very strong FQH state at $\nu$ = 2/3. From the fan diagram of Fig. \[fig:colorful\](b), it is clear that $E_F$ at these fillings lies in an $N=0$ LL, namely, the A0$\uparrow$ ($\nu=$ 7/3 and 8/3), S0$\downarrow$ ($\nu=$ 4/3 and 5/3), or S0$\uparrow$ ($\nu=$ 2/3) levels.[^5]
discussion
==========
Our observations provide direct evidence that the $q/3$ FQH states are strong when $E_F$ resides in a ground-state ($N=0$) LL, regardless of whether that LL belongs to the A or S subband. This finding implies that the node in the wavefunction in the $out$-$of$-$plane$ direction does not significantly de-stabilize the $q/3$ FQH states. On the other hand, when $E_F$ lies in an $N>0$ LL, the wavefunction node(s) in the $in$-$plane$ direction weaken or completely de-stabilize the $q/3$ FQH states. These conclusions are consistent with the data from single-subband samples, [@Pan.PRL.1999; @Lilly.PRL.1999; @Du.SSC.1999; @Gervais.PRL.2004] as well as theoretical calculations. [@TheQHE; @MacDonald.PRB.1986; @Ambrumenil.JPC.1988; @Koulakov.PRL.1996; @Moessner.PRB.1996; @Toke.PRB.2005] In a composite Fermion picture, our data also imply that the lower lying (fully occupied) LLs are essentially inert and the composite Fermions are formed in the partially filled LL where $E_F$ lies. The composite Fermions, however, could have a spin and/or subband degree of freedom, as we briefly discuss in the last paragraph of this section (see also, Ref. ).
Our data also allow us to assess the stability of the FQH states as two LLs approach each other. In Fig. \[fig:colorful\](a) the dashed line denoted $E_C-E_Z$ marks the position of the expected crossing between the A0$\downarrow$ and the S1$\uparrow$ levels, based on the LL coincidence we observe for the $\nu=4$ quantum Hall state. It is clear in Fig. \[fig:colorful\](a) that as we approach this line from the A region, the 10/3 and 11/3 FQH states disappear when $\Delta$ is about 5 K away from $E_C-E_Z$. A similar statement can be made regarding the stability of the 11/3 state as the $E_C+E_Z$ dashed line is approached from the C region, and the stability of the 13/3 and 14/3 states as one approaches the $E_C+E_Z$ line from the G region or the $E_C-E_Z$ line from the E region.[@Note4] Note that what is common to all these observations is that the boundaries marked by the dashed lines correspond to the crossing of two LLs with $antiparallel$ spins.
Data of Fig. \[fig:colorful\](a) suggest that, when the two approaching LLs have $parallel$ spins, the $q/3$ states remain stable even closer to the expected LL crossings. For example, the 10/3 and 11/3 FQH states in region C are stable very close to the boundary (the line marked $E_C$) separating this region from B. Similarly, the 13/3 and 14/3 states are stable in region E close to the $E_C$ line separating E from F. Note that in both cases, i.e., traversing from C to B or from E to F, the two approaching LLs have parallel spins (see Fig. \[fig:colorful\](b)). We conclude that the relative spins of the two approaching LLs also play a role in the stability of the $q/3$ FQH states. It is worth emphasizing that, as is evident from Figs. \[fig:waterfall\] and \[fig:colorful\](a) data, the relative spins of the two approaching LLs also play a crucial role in the stability of the $integer$ quantum Hall (IQH) states. For antiparallel-spin LLs, the IQH state (e.g., at $\nu=4$) becomes very weak or completely disappears, while for the parallel-spin LLs the IQH state (e.g., at $\nu=3$), remains strong. This behavior has been attributed to easy-axis (for an opposite-spin crossing) and easy-plane (for a same-spin crossing) ferromagnetism. [@Muraki.PRL.2001; @Vakili.PRL.2006; @Jungwirth.PRL.1998]
We highlight three further observations. First, strong FQH states at large $q/3$ fillings have been recently observed in very high quality graphene samples.[@Dean.cond.mat.2010] These states qualitatively resemble what we see in our two-subband system. It is tempting to associate the valley degree of freedom in graphene with the subband degree of freedom in our sample. But the LL structure in graphene is of course different from GaAs so it is not obvious if this association is valid. Second, data taken in the $N$ = 1 LL at very low temperatures and in the highest quality, single-subband samples exhibit FQH states at even-denominator fillings $\nu$ = 5/2 and 7/2. [@Willett.PRL.1987; @Pan.PRB.2008] In the traces shown in Fig. \[fig:waterfall\], we do not see any even-denominator states when $N$ = 1, e.g., at $\nu$ = 7/2 in region B where $E_F$ is in the S1$\uparrow$ level. However, in the same sample, at higher densities ($n > 3.4\times 10^{11}$ cm$^{-2}$) and at low temperatures ($T=$ 30 mK), we do indeed observe a FQH state at $\nu$ = 7/2 flanked by very weak 10/3 and 11/3 states when $E_F$ lies in the S1$\uparrow$ level. [@Shabani.PRL.2010]
Third, in the $N=0$ LL, high-quality samples show strong higher-order, odd-denominator FQH states at composite Fermion filling factor sequences such as 2/5, 3/7, 4/9, etc.[@CF_Jain] We do observe a qualitatively similar behavior in our data when $E_F$ is in an $N$ = 0 LL. For example, in region A (Figs. \[fig:waterfall\] and \[fig:colorful\](a)) we see weak but clear minima at $\nu$ = 17/5 next to the 10/3 minimum. Again, at higher densities and low temperatures, such states become more developed. [@Shabani.PRL.2010] In Fig. 1(b), for example, there are strong minima at $\nu=$ 12/5 and 13/5, adjacent to the 7/3 and 8/3 minima, and at 17/5 and 18/5, adjacent to the 10/3 and 11/3 minima. These states, as well as the $q/3$ states, exhibit subtle evolutions even when $E_F$ lies within a fixed $N=0$ LL, consistent with the presence of composite Fermions which have spin and/or subband degrees of freedom.[@Shabani.PRL.2010] A related question concerns the role of charge distribution symmetry in the stability of the $q/3$ states. In other words, in a QW with fixed width, density and filling, and with $E_F$ in a particular $N=0$ LL, how does the strength of given a FQH state at a particular filling vary with charge distribution symmetry. We do not have data to answer this question quantitatively, but the data we present here clearly indicates that a primary factor determining the strength of the $q/3$ FQH states is whether or not $E_F$ lies in an $N=0$ LL.
summary
=======
In conclusion, the position of $E_F$ is what determines the stability of odd-denominator, $q/3$ FQH states at a given filling factor. When $E_F$ lies in a ground-state ($N=0$) LL, the $q/3$ FQH states are stable and strong, regardless of whether that LL belongs to the symmetric or antisymmetric subband. This observation implies that the wavefunction node in the out-of-plane direction is not detrimental to the stability of these FQH states. Also, the $q/3$ FQH states appear to be stable very near the crossing of two LLs, especially if the LLs have parallel spins.
We acknowledge support through the NSF (DMR-0904117 and MRSEC DMR-0819860) for sample fabrication and characterization, and the DOE BES (DE-FG0200-ER45841) for measurements. We thank J. K. Jain and Z. Papic for illuminating discussions.
[^1]: Our sample is the same as the one used in Ref. . Based on our careful measurements of the subband separation ($\Delta$) while imbalancing the QW (see Fig. \[fig:cartoon\]), we conclude that the QW has a width of 55 nm, slightly smaller than 56 nm which was quoted in Ref. . We emphasize that throughout our paper we use the experimentally measured values of $\Delta$, and that the exact width of the QW has no bearing on our conclusions.
[^2]: We note that when the charge distribution is nearly symmetric, LL coincidences at even fillings are also difficult to see at very low temperatures. For example, there is a coincidence at $\nu=8$ at $\Delta\simeq 26$ K but we can only see a weakening of the $R_{xx}$ minimum at $T\gtrsim 0.3$ K.
[^3]: The observation of a significantly enhanced g-factor which is independent of the filling factor has been reproted in the past \[S. J. Papadakis, E. P. De Poortere, and M. Shayegan, Phys. Rev. B **59**, R12743 (1999); Y. P. Shkolnikov, E. P. De Poortere, E. Tutuc, and M. Shayegan, Phys. Rev. Lett. **89**, 226805 (2002)\].
[^4]: \[footnote1\]The $E_C+E_Z$ line going through region I does not correspond to a LL coincidence at the Fermi energy in this region; this should be evident from Fig. \[fig:colorful\](b) diagram. The same is true about the $2E_C-E_Z$ line as it goes through region G.
[^5]: Traces taken at higher values of $\Delta$ reveal that the $\nu=$ 7/3 and 8/3 states remain strong up to the $\nu=$ 3 coincidence. Past this coincidence, the 7/3 and 8/3 states become weaker, consistent with the fact that $E_F$ now lies in an excited LL (the S1 level, see Fig. \[fig:colorful\](b)).
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---
abstract: 'Feasible tomography schemes for large particle numbers must possess, besides an appropriate data acquisition protocol, also an efficient way to reconstruct the density operator from the observed finite data set. Since state reconstruction typically requires the solution of a non-linear large-scale optimization problem, this is a major challenge in the design of scalable tomography schemes. Here we present an efficient state reconstruction scheme for permutationally invariant quantum state tomography. It works for all common state-of-the-art reconstruction principles, including, in particular, maximum likelihood and least squares methods, which are the preferred choices in today’s experiments. This high efficiency is achieved by greatly reducing the dimensionality of the problem employing a particular representation of permutationally invariant states known from spin coupling combined with convex optimization, which has clear advantages regarding speed, control and accuracy in comparison to commonly employed numerical routines. First prototype implementations easily allow reconstruction of a state of 20 qubits in a few minutes on a standard computer.'
address:
- '$^1$ Naturwissenschaftlich-Technische Fakultät, Universit[ä]{}t Siegen, Walter-Flex-Stra[ß]{}e 3, D-57068 Siegen, Germany'
- '$^2$ Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Technikerstra[ß]{}e 21A, A-6020 Innsbruck, Austria'
- '$^3$ Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, E-48080 Bilbao, Spain'
- '$^4$ <span style="font-variant:small-caps;">Ikerbasque</span>, Basque Foundation for Science, E-48011 Bilbao, Spain'
- '$^5$ Wigner Research Centre for Physics, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary'
- '$^6$ Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Stra[ß]{}e 1, D-85748 Garching, Germany'
- '$^7$ Fakultät für Physik, Ludwig-Maximilians-Universität, D-80797 München, Germany'
- '$^8$ Technical University of Denmark, Department of Mathematics, Matematiktorvet Building 303 B, 2800 Kgs. Lyngby, Denmark'
author:
- 'Tobias Moroder$^{1,2}$, Philipp Hyllus$^3$, G[é]{}za T[ó]{}th$^{3,4,5}$, Christian Schwemmer$^{6,7}$, Alexander Niggebaum$^{6,7}$, Stefanie Gaile$^8$, Otfried Gühne$^{1,2}$ and Harald Weinfurter$^{5,6}$'
title: Permutationally invariant state reconstruction
---
Introduction
============
Full information about the experimental state of a quantum system is naturally highly desirable because it enables one to determine the mean value of each observable and thus also of each other property of the quantum state. Abstractly, such a complete description is given for example by the density operator, a positive semidefinite matrix $\rho$ with unit trace. Quantum state tomography [@qse_book] refers to the task to determine the density operator for a previously unknown quantum state by means of appropriate measurements. Via the respective outcomes more and more information about the true state generating the data is collected up to the point where this information uniquely specifies the particular state. Quantum state tomography has successfully been applied in many experiments using different physical systems, including trapped ions [@haeffner05a] or photons [@kiesel07a] as prominent instances.
Unfortunately, tomography comes with a very high price due to the exponential scaling of the number of parameters required to describe composed quantum systems. For an $N$ qubit system the total number of parameters of the associated density operator is $4^N-1$ and any standard tomography protocol is naturally designed to determine all these variables. The most common scheme used in experiments [@james01a] consists of locally measuring in the basis of all different Pauli operators and requires an overall amount of $3^N$ different measurement settings with $2^N$ distinct outcomes each. Other schemes, [*e.g.*]{}, using an informationally complete measurement [@renes04a] locally would require just one setting but, nevertheless, the statistics for $4^N$ different outcomes. Hence the important figure of merit to compare different methods is given by the combination of settings and outcomes.
For such a scaling, the methods rapidly become intractable, already for present state-of-the-art experiments: Recording for example the data of $14$ trapped ions [@monz11a], currently the record for quantum registers, would require about $150$ days, although $100$ measurement outcomes can be collected for a single setting in only about three seconds. In photonic experiments this scaling is even worse because count rates are typically much lower, [*e.g.*]{}, in recent eight-photon experiments [@yao11a; @huang11a] a coincidence of single click events occurs only at the order of minutes, hence it would require about seven years to collect an adequate data set. This directly shows that more sophisticated tomography techniques are mandatory.
New tomography protocols equipped with better scaling behaviour exploit the idea that the measurement scheme is explicitly optimized only for particular kinds of states rather than for all possible ones. If the true unknown state lies within this designed target class then full information about the state can be obtained with much less effort, and if the underlying density operator is not a member then a certificate signals that tomography is impossible in this given case. Recent results along this direction include tomography schemes designed for states with a low rank [@gross10a; @gross11a; @flammia12a], particularly for important low rank states like matrix product [@cramer10a] or multi-scale entanglement renormalization ansatz states [@landon_cardinal12a]. Other schemes include a tomography scheme based on information criteria [@yin11a] or—the topic of this manuscript—states with permutation invariance [@toth10a].
However it needs to be stressed that in any real experiment all these tomography schemes must cope with another, purely statistical challenge: Since only a finite number of measurements can be carried out in any experiment one cannot access the true probabilities predicted by quantum mechanics $p_k=\tr(\rho_{\rm true} M_k)$, operator $M_k$ describing the measurements, but merely relative frequencies $f_k$. Though these deviations might be small, the approximation $f_k\approx p_k$ causes severe problems in the actual state reconstruction process, [*i.e.*]{}, the task to determine the density operator from the observed data. If one naïvely uses the frequencies according to Born’s rule $f_k=\tr(\hat \rho_{\rm lin} M_k)$ and solves for the unknown operator $\hat \rho_{\rm lin}$, then, apart from possible inconsistencies in the set of linear equations, the reconstructed operator $\hat \rho_{\rm lin} \! \not\geq 0$ is often not a valid density operator anymore because some of its eigenvalues are negative. Hence in such cases this reconstruction called linear inversion delivers an unreasonable answer. It should be kept in mind that inconsistencies can also be due to systematic errors, [*e.g.*]{}, if the true measurements are aligned wrongly relative to the respective operator representation [@moroder12a; @rosset12a], but such effects are typically ignored.
Therefore, statistical state reconstruction relies on other principles than linear inversion. In general, these methods require the solution of a non-linear optimization problem, which is much harder to solve than just a system of linear equations. For large system sizes this becomes, besides the exponential scaling of the number of settings and outcomes, a second major problem, again due to the exponential scaling of the number of parameters of the density operator. In fact for the current tomography record of eight ions in a trap [@haeffner05a], this actual reconstruction took even longer than the experiment itself (one week versus a couple of hours). Hence, feasible quantum state tomography schemes for large systems must, in addition to an efficient measurement procedure, possess also an efficient state reconstruction algorithm, otherwise they are not scalable.
In this paper we develop a scalable reconstruction algorithm for the proposed permutationally invariant tomography scheme [@toth10a]. It works for common reconstruction principles, including, among others, maximum likelihood and least squares methods. This scheme becomes possible once more by taking advantage of the particular symmetry of this special kind of states, which provides an efficient and operational way to store, characterize and even process those states. This method enables a large dimension reduction in the underlying optimization problem such that it gets into the feasible regime. The final low dimensional optimization is performed via non-linear convex optimization which offers great advantages in contrast to commonly used numerical routines, in particular regarding numerical stability and accuracy. Already a first prototype implementation of this algorithm allows state reconstruction for $20$ qubits in a few minutes on a standard desktop computer.
The outline of the manuscript is as follows: Section \[sec:background\] summarizes the background on permutationally invariant tomography and on statistical state reconstruction. The key method is explained in Sec. \[sec:method\] and highlighted via examples in Sec. \[sec:examples\], that are generated by our current implementation. Section \[sec:details\] collects all the technical details: the mentioned toolbox, more notes about convex optimization, additional information about the pretest or certificate and the measurement optimization, both addressed for large qubit numbers. Finally, we conclude and provide an outlook on further directions in Sec. \[sec:conclusion\].
Background {#sec:background}
==========
Permutationally invariant tomography {#sec:recap_pitomo}
------------------------------------
Permutationally invariant tomography has been introduced as a scalable reconstruction protocol for multi-qubit systems in Ref. [@toth10a]. It is designed for all states of the system that remain invariant under all possible interchanges of its different particles. Abstractly such a permutationally invariant state $\rho_{\rm PI}$ of $N$ qubits can be expressed in the form, $$\label{eq:PI-state}
\rho_{\rm PI} = \left[ \:\rho \:\right]_{\rm PI} = \frac{1}{N!} \sum_{p\in S_N} V(p)\: \rho \: V(p)^\dag,$$ where $V(p)$ is the unitary operator which permutes the $N$ different subsystems according to the particular permutation $p$ and the summation runs over all possible elements of the permutation group $S_N$. Many important states, like Greenberger-Horne-Zeilinger states or Dicke states fall within this special class.
As shown in Ref. [@toth10a], full information of such states can be obtained by using in total ${N+2 \choose N} =(N^2+3N+2)/2$ different local binary measurement settings, while for each setting only the count rates of $(N+1)$ different outcomes need to be registered. This finally leads to a cubic scaling in contrast to the exponential scaling of standard tomography schemes.
The measurement strategy that attains this number runs as follows: Each setting is described by a unit vector $\hat a \in \mathbbm{R}^3$ which defines associated eigenstates $\ket{i}_a$ of the trace-less operator $\hat a \cdot \vec \sigma$. Each party locally measures in this basis and registers the outcomes “$0$” or “$1$” respectively. The permutationally invariant part can be reconstructed from the collective outcomes, [*i.e.*]{}, only the number of “$0$” or “$1$” results at the different parties matters but not the individual site information. The corresponding coarse-grained measurements are given by $$\begin{aligned}
\label{eq:PI-measurements}
M_k^a&=& \sum_{p^\prime} V(p^\prime) \ket{0}_a\!\bra{0}^{\otimes k} \otimes \ket{1}_a\!\bra{1}^{\otimes N-k} V(p^\prime)^\dag, \\
& = & {N \choose k} \left[ \ket{0}_a\!\bra{0}^{\otimes k} \otimes \ket{1}_a\!\bra{1}^{\otimes N-k}\right]_{\rm PI}\end{aligned}$$ with $\: k=0,\dots,N,$ and where the summation $p^\prime$ is over all permutations that give distinct terms. In total one needs the above stated number of different settings $\hat a$. These settings can be optimized in order to minimize the total variance which provides an advantage in contrast to random selection.
In addition to this measurement strategy there is also a pretest which estimates the “closeness” of the true, unknown state with respect to all permutationally invariant states from just a few measurement results [@toth10a]. This provides a way to test in advance whether permutationally invariant tomography is a good method for the unknown state.
Restricting to the permutationally invariant part of a density operator has already been discussed in the literature; for example for spins in a Stern-Gerlach experiment [@dariano03a] or in terms of the polarization density operator [@karassiov05a; @adamson07a; @adamson08a]. Here, due to the restricted class of possible measurements, only the permutationally invariant part of, in principle distinguishable, particles is accessible [@adamson07a; @adamson08a]. This is a strong conceptual difference to permutationally invariant tomography where one intentionally constrains itself to this invariant part. Nevertheless it should be emphasized that the employed techniques are similar.
Statistical state reconstruction
--------------------------------
Since standard linear inversion of the observed data typically results in unreasonable estimates as explained in the introduction, one employs other principles for actual state reconstruction. In general one uses a certain fit function $F(\rho)$ that penalizes deviations between the observed frequencies $f_k$ and the true probabilities predicted by quantum mechanics $p_k(\rho)=\tr(\rho M_k)$ if the state of the system is $\rho$. The reconstructed density operator $\hat \rho$ is then given by the (often unique) state that minimizes this fit function, $$\label{eq:struct_srecon}
\hat \rho = \arg \: \min_{\rho \geq 0} F(\rho),$$ hence the reconstructed state is precisely the one which best fits the observed data. Since the optimization explicitly restricts to physical density operators this assures validity of the final estimate $\hat \rho$ in contrast to linear inversion. Depending on the functional form of the fit function different reconstruction principles are distinguished. A list of the most common choices is provided in Tab. \[tab:reconF\].
[lc]{} Reconstruction principle
------------------------------------------------------------------------
------------------------------------------------------------------------
& Fit function $F(\rho)$\
------------------------------------------------------------------------
------------------------------------------------------------------------
Maximum Likelihood [@hradil97a] & $-\sum_k f_k \log[ p_k(\rho)]$\
------------------------------------------------------------------------
------------------------------------------------------------------------
Least Squares [@langford07a] & $\sum_k w_k [ f_k - p_k(\rho)]^2$, $\:w_k > 0$\
------------------------------------------------------------------------
------------------------------------------------------------------------
Free Least Squares [@james01a] & $\sum_k 1/p_k(\rho)[ f_k - p_k(\rho)]^2$\
------------------------------------------------------------------------
------------------------------------------------------------------------
Hedged Maximum Likelihood [@blume10a] &$-\sum_k f_k \log[ p_k(\rho)]-\beta \log[\det(\rho)]$, $\:\beta > 0$\
\[tab:reconF\]
The presumably best-known and most often employed method is called maximum likelihood principle [@hradil97a]. Given a set of measured frequencies $f_k$ the maximum likelihood state $\hat \rho_{\rm ml}$ is exactly the one with the highest probability to generate these data. Other common fit functions, usually employed in photonic state reconstruction, are different variants of least squares [@james01a; @langford07a] that originate from the likelihood function using Gaussian approximations for the probabilities. There this is often also called maximum likelihood principle but we distinguish these, indeed different, functions here explicitly. Typically the weights in the least squares function are set to be $w_k=1/f_k$ because $f_k$ represents an estimate of the variance in a multinomial distribution, cf. the free least squares principle. However, this leads to a strong bias if the counts rates are extremal, [*e.g.*]{}, if one of the outcomes is never observed this method naturally leads to difficulties. A method to circumvent this is given by the free least squares function [@james01a] or using improved error analysis for rare events. Let us stress that all these principles have the property that if linear inversion delivers a valid estimate $\hat \rho_{\rm lin} \geq 0$, it is also the estimate given by these reconstruction principles [^1].
Finally, hedged maximum likelihood [@blume10a] represents a method that circumvents low rank state estimates. Via this one obtains more reasonable error bars using parametric bootstrapping methods [@bootstrapping]; for other error estimates we refer to the recently introduced confidence regions for quantum states [@christandl11a; @blume-kohout12a]. In principle many more fit functions are possible, like generic loss functions [@mood], but considering these is out of the scope of this work.
Method {#sec:method}
======
From the previous it is apparent that permutationally invariant state reconstruction requires the solution of $$\label{eq:PI_staterecon}
\hat \rho_{\rm PI} = \arg \: \min_{\rho_{\rm PI} \geq 0} F(\rho_{\rm PI}),$$ for the preferred fit function. This large-scale optimization becomes feasible along the following lines:
First, one reduces the dimensionality of the underlying optimization problem because one cannot work with full density operators anymore. This requires an operational way to characterize permutationally invariant states $\rho_{\rm PI} \geq 0$ over which the optimization is performed, and additionally demands an efficient way to compute probabilities $p_k(\rho_{\rm PI})$ which appear in the fit function. Second, one needs a method to perform the final optimization. We employ convex optimization for this task.
Reduction of the dimensionality
-------------------------------
This reduction relies on an efficient toolbox to handle permutationally invariant states, which exploits the particular symmetry. Here we explain this method and the final structure; for more details see Sec. \[sec:details\_reduction\]. These techniques are well-proven and established; we employ and adapt them here for the permutationally invariant tomography scheme such that we finally reach state reconstruction of larger qubits.
The methods of this toolbox are obtained via the concept of spin coupling that describes how individual, distinguishable spins couple to a combined system if they become indistinguishable. Since we deal with qubits we only need to focus on spin-$1/2$ particles. In the simplest case, two spin-$1/2$ particles can couple to a spin-$1$ system, called the triplet, if both spins are aligned symmetric, or to a spin-$0$ state, the singlet, if the spins are aligned anti-symmetric. Abstractly, this can be denoted as $\mathbbm{C}^2 \otimes \mathbbm{C}^2 = \mathbbm{C}^3 \oplus \mathbbm{C}^1$. If one considers now three spins, then of course all spins can point in the same direction giving a total spin-$3/2$ system. There is also to a spin-$1/2$ system possible if two particles form already a spin-$0$ and the remaining one stays untouched. This can be achieved however by more than one possibility, in fact by two inequivalent choices [^2], and is expressed by $\mathbbm{C}^2 \otimes \mathbbm{C}^2 \otimes \mathbbm{C}^2 = \mathbbm{C}^4 \oplus (\mathbbm{C}^2 \otimes \mathbbm{C}^2)$.
This scheme can be extended to $N$ spin-$1/2$ particles to obtain the following decomposition of the total Hilbert space, $$\mathcal{H}=(\mathbbm{C}^2)^{\otimes N} = \bigoplus_{j=j_{\rm min}}^{N/2} \mathcal{H}_j \otimes \mathcal{K}_j.$$ where the summation runs over different total spin numbers $j=j_{\rm min},j_{\rm min}+1,\dots, N/2$ starting from $j_{\rm min}\in \{0,1/2\}$ depending on whether $N$ is even or odd. Here, $\mathcal{H}_j$ are called the spin Hilbert spaces with dimensions $\dim(\mathcal{H}_j)=2j+1$, while $\mathcal{K}_j$ are referred as multiplicative spaces that account for the different possibilities to obtain a spin-$j$ state. They are generally of a much larger dimension, cf. Eq. (\[eq:dimK\]).
Permutationally invariant states have a simpler form on this Hilbert space decomposition, namely $$\label{eq:PI-state1}
\rho_{\rm PI} = \bigoplus_{j=j_{\rm min}}^{N/2} p_j \rho_j \otimes \frac{\mathbbm{1}}{\dim(\mathcal{K}_j)},$$ with density operators $\rho_j$ called spin states and according probabilities $p_j$. Thus a permutationally invariant density operator only contains non-trivial parts on the spin Hilbert spaces while carrying no information on the multiplicative spaces. Note further that there are no coherences between different spin states. This means that any permutationally invariant state can be parsed into a block structure as schematically depicted in Fig. \[fig:PIstate\]. The main diagonal is built up by unnormalized spin states $\tilde \rho_j=p_j \rho_j / \dim(\mathcal{K}_j)$, which appear several times, the number being equal to the dimension of the corresponding multiplicative space. This block-decomposition represents a natural way to treat permutationally invariant states and has for example been employed already in the aforementioned related works of permutationally invariant tomography [@dariano03a; @karassiov05a; @adamson07a; @adamson08a] but also in other contexts [@cirac99a; @demkowicz05].
![Block decomposition for a generic permutationally invariant state as given by Eq. (\[eq:PI-state1\]) with $\tilde \rho_j=p_j \rho_j / \dim(\mathcal{K}_j)$.[]{data-label="fig:PIstate"}](PI_state1.eps)
This structure shows that if we work with permutationally invariant states we do not need to consider the full density operator but rather that it is sufficient to deal only with this ensemble of spin states. Therefore we identify from now on $$\rho_{\rm PI} \Longleftrightarrow p_j \rho_j,\:j=j_{\rm min},j_{\rm min}+1,\dots, N/2.$$ This provides already an efficient way to store and to visualize such states. More importantly, it also enables an operational way to characterize valid states, since any permutationally invariant operator $\rho_{\rm PI}$ represents a true state if and only if all these spin operators $\rho_j$ are density operators and $p_j$ a probability distribution. This is in contrast to the generalized Bloch vector employed in the original proposal of permutationally invariant tomography [@toth10a] given by Eq. (\[eq:gen\_bloch\_vec\]), which also provides efficient storage and processing of permutationally invariant states, but where Bloch vectors of physical states are not as straightforward to characterize.
Via this identification one can demonstrate once more the origin of the cubic scaling of the permutationally invariant tomography scheme. The largest spin state is of dimension $N+1$ which requires parameters on the order of $N^2$ for characterization. Since one has on the order of $N$ of these states this shows results in a cubic scaling.
Fixing the ensemble of spin states as parametrization it is now required to obtain an efficient procedure to compute the probabilities $p_k^a(\rho_{\rm PI})$ for the optimized measurement scheme. This is achieved as follows: First let us stress that a similar block decomposition as given by Eq. (\[eq:PI-state1\]) holds for all permutationally invariant operators. Hence also the measurements $M_k^a$ given by Eq. (\[eq:PI-measurements\]) can be cast into this form. Using the convention $$\label{eq:PI-meas}
M_k^a = \bigoplus_{j=j_{\rm min}}^{N/2} M_{k,j}^a \otimes \mathbbm{1}$$ leads to $$\label{eq:prob_PI}
\tr(\rho_{\rm PI} M_k^a) = \sum_{j=j_{\rm min}}^{N/2} p_j \tr(\rho_j M_{k,j}^a).$$ Therefore the problem is shifted to the computation of the spin-$j$ operators $M_{k,j}^a$ for each setting $\hat a$. As we show in Prop. \[prop:measurements\] below, using the idea that measurements can be transformed into each other by a local operation $U_a\ket{i} = \ket{i}_a$ this provides the relation $$M_{k,j}^a= W^a_j M_{k,j}^{e_3} W^{a,\dag}_j.$$ Here $M_{k,j}^{e_3}$ corresponds to the measurement in the standard basis (that need to be computed once) and $W_j^a$ is a unitary transformation determined by the rotation $U_a$. This connection is given by $$U_a = \exp(-i \sum_l t_l \sigma_l/2) \Longrightarrow W_j^a = \exp(-i \sum_l t_l S_{l,j})$$ where $S_{l,j}$ stands for the spin operators in dimension $2j+1$. This finally provides the efficient way to compute probabilities.
Optimization
------------
As a second step one still needs to cope with the optimization itself. Although there are different numerical routines for statistical state reconstruction like maximum likelihood [@hardil04a] or least squares [@james01a; @reimpell_thesis], we prefer non-linear convex optimization [@cobook] to obtain the final solution. Quantum state reconstruction problems are known to be convex [@reimpell_thesis; @kosut04a], but convex optimization has hardly been used for this task. However, convex optimization possesses several advantages: First of all it is a systematic approach that works for any convex fit function, including maximum likelihood and least squares. In contrast to other algorithms such as the fixed-point algorithm proposed in Ref. [@hardil04a] it gives a precise stopping condition via an appropriate error control (see, however Ref. [@glancy12a]) and still exploits all the favourable, convex, structure in comparison to re-parametrization ideas as in Ref. [@james01a]. Moreover it is guaranteed to find the global optimum and the obtained accuracy is typically much higher than with other methods.
Quantum state reconstruction as defined via Eq. (\[eq:struct\_srecon\]) can be formulated as a convex optimization problem as follows: All fit functions listed in Tab. \[tab:reconF\] are convex on the set of states. Via a linear parametrization of the density operator $\rho(x)= \mathbbm{1}/\dim(\mathcal{H}) + \sum x_i B_i$, using an appropriate operator basis $B_i$ such that normalization is fulfilled directly, the required optimization problem becomes $$\begin{aligned}
\label{eq:convex_opti}
\min_x && F[\rho(x)] \\
\nonumber
\textrm{s.t.}&& \rho(x) = \frac{\mathbbm{1}}{\dim(\mathcal{H})}+ \sum_i x_i B_i \geq 0,\end{aligned}$$ with a convex objective function $F(x)=F[\rho(x)]$ and a linear matrix inequality as constraint, [*i.e.*]{}, precisely the structure of a non-linear convex optimization problem [@cobook]. For permutationally invariant states one uses $\rho(x)=\oplus_j \bar \rho_j(x)$ with $\bar \rho_j=p_j \rho_j$ by using an appropriate block-diagonal operator basis $B_i$; therefore we continue this discussion with the more general form.
The optimization given by Eq. (\[eq:convex\_opti\]) can be performed for instance with the help of a barrier function [@cobook] [^3]. Rather than considering the constrained problem one solves the unconstrained convex task given by $$\label{eq:uncon_opti}
\min_x F[\rho(x)] - t \log[\det\rho(x)],$$ where the constraint is now directly included in the objective function. This so-called barrier term acts precisely as its name suggests: If one tries to leave the strictly feasible set, [*i.e.*]{}, all parameters $x$ that satisfy $\rho(x) > 0$, one always reaches a point where at least one of the eigenvalues vanishes. Since the barrier term is large within this neighbourhood, in fact singular at the crossing, it penalizes points close to the border and thus ensures that one searches for an optimum well inside the region where the constraint is satisfied. The penalty parameter $t>0$ plays the role of a scaling factor. If it becomes very small the effect of the barrier term becomes negligible within the strictly feasible set and only remains at the border. Therefore a solution of Eq. (\[eq:uncon\_opti\]) with a very small value of $t$ provides an excellent approximation to the real solution. As shown in Sec. \[sec:copti\_details\] this statement can be made more precise by $$\label{eq:slackness}
F[\rho(x_{\rm sol}^t)] - F[\rho(x_{\rm sol})] \leq t \dim(\mathcal{H})$$ which follows from convexity and which relates the true solution $x_{\rm sol}$ of the original problem given by Eq. (\[eq:convex\_opti\]) to the solution $x_{\rm sol}^t$ of the unconstrained problem with penalty parameter $t$. This condition represents the above mentioned error control and serves as a stopping condition, [*i.e.*]{}, as a quantitative error bound for a given $t$. Note that for permutationally invariant tomography $\dim(\mathcal{H})$ is not the dimension of the true $N$-qubit Hilbert space but instead the dimension of $\rho(x)=\oplus_j \bar \rho_j(x)$, [*i.e.*]{}, $\sum_{j=j_{\rm min}} (2j+1)=(N+1)(N+2j_{\rm min}+1)/4$ which increases only quadratically.
Small penalty parameters are approached by an iterative process: For a given starting point $x_{\rm start}^{n}$ and a certain value of the parameter $t_n>0$ one solves Eq. (\[eq:uncon\_opti\]). Its solution will be the starting point for $x_{\rm start}^{n+1}=x_{\rm sol}^{n}$ for the next unconstrained optimization with a lower penalty parameter $t_{n+1}< t_{n}$. As starting point for the first iteration we employ $t_0=1$ and the point $x_{\rm start}^{0}$ corresponding to the totally mixed state. This procedure is repeated until one has reached small enough penalty parameters. The penalty parameter is decreased step-wise. Then each unconstrained problem can be solved very efficiently since one starts already quite close to the true solution.
Let us point out that via the above mentioned barrier method one additionally obtains solutions to the hedged state reconstruction with $\beta=t$ since the unconstrained problem given by Eq. (\[eq:uncon\_opti\]) is precisely the fit function of the hedged version of Tab. \[tab:reconF\].
Finally for comparative purposes we also like to mention the iterative fixed-point algorithm of Ref. [@rehavcek01a]; for a modification see Ref. [@rehacek07a]. It is designed for maximum likelihood estimation and is straightforward to implement since it only requires matrix multiplication, however, it has deficits regarding control and accuracy. For permutationally invariant tomography the algorithm can be adapted as follows: Given a valid iterate $\rho_{\rm PI}^n$ characterized by the ensemble of spin states $\bar \rho^n_j=p_j^n\rho_j^n$ one evaluates the probabilities $p_k^a(\rho_{\rm PI}^n)$ using Eq. (\[eq:prob\_PI\]). Next, one computes the operators $$R^n_j = \sum_{a,k} \frac{f_k^a}{p_k^a(\rho_{\rm PI}^n)} M_{k,j}^a,$$ which determine the next iterate $\bar \rho^{n+1}_j= R^n_j \bar \rho_j^n R^{n\dag}_j/\mathcal{N}$ with $\mathcal{N}=\sum_j \tr(R^n_j \bar\rho_j^n R^{n\dag}_j)$. This iteration is started for example from the totally mixed state and repeated until a sufficiently good solution is obtained.
Examples {#sec:examples}
========
The two methods from the previous section are employed in a prototype implementation under <span style="font-variant:small-caps;">MATLAB</span>, which already enables state reconstruction of about $20$ qubits on a standard desktop computer.
The current algorithm is tested along the following lines: For a randomly generated permutationally invariant state $\rho_{\rm PI}^{\rm true}$ we compute the true probabilities $p_{k,\rm true}^a$ for the chosen measurement settings. Rather than sampling we set the observed frequencies equal to this distribution, [*i.e.*]{}, $f_k^a=p_{k,\rm true}^a$. In this way linear inversion would return the original state, hence also each other reconstruction principle from Tab. \[tab:reconF\] has this state as solution. We now start the algorithm and compare the trace distance between the analytic solution $\rho_{\rm PI}^{\rm true}$ and the state after $n$ iterations $\rho_{\rm PI}^n$. This distance $\frac{1}{2}\tr|\rho_{\rm PI}^{\rm true} - \rho_{\rm PI}^n|$ quantifies the probability with which the two states, the true analytic solution and the iterate after $n$ steps in the algorithm, could be distinguished [@nielsen_chuang].
A typical representative of this process is depicted in Fig. \[fig:example1\] for $12$ qubits using optimized settings. The randomly generated state $\rho_{\rm PI}^{\rm true}$ was chosen to lie at the boundary of the state space since such rank-reduced solutions better resemble the case of state reconstruction of real data. More precisely, each spin state of the true density operator is given by a pure state $\rho^{\rm true}_j=\ket{\psi_j}\bra{\psi_j}$ chosen according to the Haar measure, while the $p_j$ are selected by the symmetric Dirichlet distribution with concentration measure $\alpha=1/2$ [@zyczkowski01]. As apparent the algorithm behaves similar for all three reconstruction principles and rapidly obtains a good solution after about $70$ iterations. The steps in this plot are points where the penalty parameter is reduced by a factor of $10$ starting from $t=1$ and decreased down to $t=10^{-10}$. The slight rise after these points comes from the fact that we plot the trace distance and not the actual function (fit-function plus penalty term) that is minimized.
Figure \[fig:example2\] shows a similar comparison for the maximum likelihood reconstruction of $20$ qubits but now plotted versus algorithm time [^4]. For comparison we include the performance of the iterative fixed-point algorithm, which requires much more iterations in general ($3000$ in this case vs. about $90$ for convex optimization). Let us emphasise that a similar behaviour between these two algorithms appears also for smaller qubit numbers. As one can see, convex optimization delivers a faster and in particular more accurate solution. In contrast, the iterative fixed-point algorithm shows a bad convergence rate although it initially starts off better. This was one of the main reasons for us to switch to convex optimization.
The current algorithm times of this test are listed in Tab. \[tab:current\_performance\] which are averaged over $50$ randomly generated states. Thus already this prototype implementation enables state reconstruction of larger qubit numbers in moderate times. The small time difference between reconstruction principles is because least squares as a quadratic fit function provides some advantages in the implementation. More details about this difference are given in Sec. \[sec:details\].
[ccccc]{}
------------------------------------------------------------------------
------------------------------------------------------------------------
& $N=8$ & $N=12$ & $N=16$ &$N=20$\
------------------------------------------------------------------------
------------------------------------------------------------------------
Maximum Likelihood & & & &\
------------------------------------------------------------------------
------------------------------------------------------------------------
Algorithm test & $8.5 \sec$ & $47 \sec$ & $2.7 \min$ &$9.2 \min$\
------------------------------------------------------------------------
------------------------------------------------------------------------
Simulated experiment & $9.2 \sec$ & $48 \sec$ & $2.9 \min$ &$9.3 \min$\
------------------------------------------------------------------------
------------------------------------------------------------------------
Least Squares & & & &\
------------------------------------------------------------------------
------------------------------------------------------------------------
Algorithm test & $8.4 \sec$ & $39 \sec$ & $2.5 \min$ &$6 \min$\
------------------------------------------------------------------------
------------------------------------------------------------------------
Simulated experiment & $9.2 \sec$ & $43 \sec$ & $2.7 \min$ &$6.7 \min$\
Table \[tab:current\_performance\] also contains the algorithm times for reconstructions using simulated frequencies $f_k^a=n_k^a/N_{\rm r}$. For each setting they are deduced from the count rates $n_k^a$ sampled from a multinomial distribution using the true event distribution $p_{k,\rm true}^a$ and $N_{\rm r}=1000$ repetitions. The true probabilities correspond to the same states as already employed in the algorithm test. From the table one observes that state reconstruction for data with count statistics requires only slightly more time than the algorithm test with the correct probabilities. We attribute this to the fact that a few more iterations are typically required in order to achieve the desired accuracy.
Finally let us perform the reconstruction of a simulated experiment of $N=14$ qubits. Suppose that one intends to create a Dicke state $\ket{D_{k,N}}$ as given by Eq. (\[eq:dicke\_basis\]), but that the preparation suffers from some imperfections such that at best one can prepare states of the form $$\label{eq:aimed_state}
\rho_{\rm dicke} = \sum_{k=0}^{N} {N \choose k} p^k (1-p)^{N-k} \ket{D_{k,N}}\bra{D_{k,N}},$$ where $p=0.6$ characterizes some asymmetry. As the true state prepared in the experiment we now model some further imperfection in the form of an additional small misalignment $U^{\otimes N}$, with $U=\exp(-i \theta \sigma_y/2)$, $\theta=0.2$, and some permutationally invariant noise $\sigma_{\rm PI}$ (chosen via the aforementioned method but using Hilbert-Schmidt instead of the Haar measure), [*i.e.*]{}, $$\label{eq:prepared_state}
\rho_{\rm true}=0.6 \:U^{\otimes N} \rho_{\rm dicke} U^{\dag \otimes N} + 0.4\: \sigma_{\rm PI}.$$ The frequencies are obtained via sampling from the state given by Eq. (\[eq:prepared\_state\]) using intentionally only $N_{\rm r}=200$ repetitions per setting (to see some differences). Finally, we reconstruct the state according to the maximum likelihood principle. Figure \[fig:example\] shows the tomography bar plots of one of these examples for the largest spin state $p_j\rho_{j}$, $j=N/2=7$ for both states. Though this state might be artificial this example should highlight once more that this state reconstruction algorithm works also for realistic data and for qubit sizes where clearly any non-tailored state reconstruction scheme would fail. Moreover, it demonstrates that the spin ensemble $p_j\rho_j$ represents a very convenient graphical representation of such states (compared to a $2^{14} \times 2^{14}$ matrix in this case).
![The real part of the true and reconstructed (according to maximum likelihood) largest spin ensemble $p_j\rho_{j}$, $j=N/2=7$ using the optimal measurement setting. The basis is given by the Dicke basis $\ket{D_{k,14}}$, cf. Eq. (\[eq:dicke\_basis\]).[]{data-label="fig:example"}](beschrfinal2.eps)
Details {#sec:details}
=======
Reduction {#sec:details_reduction}
---------
Let us first give more details regarding the reduction step. This starts by recalling a group theoretic result summarized in the next section, which is then used to show how the stated simplifications with respect to states and measurements are obtained.
### Background {#sec:schur-weyl}
Consider the following two unitary representations defined on the $N$ qubit Hilbert space: The permutation or symmetric group $V(p)$ which is defined by their action onto a standard tensor product basis by $V(p)\ket{i_1, \dots, i_N}=\ket{i_{p^{-1}(1)},\dots, i_{p^{-1}(N)}}$ according to the given permutation $p$, and the tensor product representation $W(U)=U^{\otimes N}$ of the special unitary group. A result known as the Schur-Weyl duality [@simon; @christandl06] states that the action of these two groups is dual, which means that the total Hilbert space can be divided into blocks on which the two representations commute. More precisely one has $$\begin{aligned}
(\mathbbm{C}^2)^{\otimes N} &=& \bigoplus_{j=j_{\rm min}}^{N/2} \mathcal{H}_j \otimes \mathcal{K}_j, \\
\label{eq:rep_symm}
V(p) &=& \bigoplus_{j=j_{\rm min}}^{N/2} \mathbbm{1} \otimes V_j(p), \\
\label{eq:rep_SU(2)}
W(U) &=& \bigoplus_{j=j_{\rm min}}^{N/2} W_j(U) \otimes \mathbbm{1}.\end{aligned}$$ Here $V_j$ and $W_j$ are respective irreducible representations, and $j_{\rm min} \in \{0, 1/2\}$ depending on whether $N$ is even or odd. The dimensions of the appearing Hilbert spaces are $\dim(\mathcal{H}_j)=2j+1$ and $$\label{eq:dimK}
\dim(\mathcal{K}_j) = {N \choose N/2-j} - {N \choose N/2-j-1}$$ for all $j<{N/2}$ and $\dim(\mathcal{K}_{N/2})=1$. Let us note that Eq. (\[eq:rep\_symm\]) already ensures the block-diagonal structure of permutationally invariant operators, while Eq. (\[eq:rep\_SU(2)\]) becomes important for the measurement computation.
A basis of the Hilbert space $\mathcal{H}_j \otimes \mathcal{K}_j$ is formed by the spin states $\ket{j,m,\alpha}=\ket{j,m} \otimes \ket{\alpha_j}$ with $m=-j,\dots,j$ and $\alpha_j=1,\dots,\dim(\mathcal{K}_j)$. These are obtained by starting with the states having the largest spin number $m=j$, which are given by $$\begin{aligned}
\ket{j,j,1}&=&\ket{0}^{\otimes 2j} \otimes \ket{\psi^-}^{\otimes N-2j}, \\
\ket{j,j,\alpha}&=&\sum_p c_{j,p} V(p) \ket{j,j,1}, \end{aligned}$$ for all $\alpha \geq 2$. The coefficients $c_{j,p}$ must ensure that the states $\ket{j,j,\alpha}$ are orthogonal, otherwise their choice is completely free since the detailed structure of different $\alpha$’s is not important. The full basis is obtained by subsequently applying the ladder operator $J_-=\sum_{n=1}^N \sigma_{-;n}$ to decrease the spin number $m$. Here $\sigma_{-;n}$ refers to the operator with $\sigma_-=(\sigma_x-i \sigma_y)/2$ on the $n$-th qubit and identity on the rest. Thus in total the basis becomes $$\ket{j,m,\alpha} = \mathcal{N} J_-^{j-m} \ket{j,j,\alpha},$$ with appropriate normalizations $\mathcal{N}$. Note that the subspace corresponding to the highest spin number $j=N/2$ is also called the symmetric subspace, which contains many important states like Greenberger-Horne-Zeilinger or Dicke states, which using the spin states read as $$\begin{aligned}
\ket{\rm{GHZ}}\!\!&=&\!\frac{1}{\sqrt{2}}\! \left( \ket{0}^{\otimes N} + \ket{1}^{\otimes N}\right) \!=\!\frac{1}{\sqrt{2}}\! \left( \ket{N/2,N/2,1} + \ket{N/2,-N/2,1} \right)\!, \\
\label{eq:dicke_basis}
\ket{D_{k,N}}\!&=& \mathcal{N} \left[ \ket{1}^{\otimes k}\otimes \ket{0}^{\otimes N-k} \right]_{\rm PI}= \ket{N/2, N/2-k,1}.\end{aligned}$$
### Permutationally invariant states and measurement operators
Let us now employ this result in order to derive a generic form for permutationally invariant states; we give the proof for completeness.
\[prop:PIstate\] Any permutationally invariant state of $N$ qubits $\rho_{\rm PI}$ defined via Eq. (\[eq:PI-state\]) can be written as $$\rho_{\rm PI} = \bigoplus_{j=j_{\rm min}}^{N/2} p_j \rho_j \otimes \frac{\mathbbm{1}}{\dim(\mathcal{K}_j)},$$ hence it is fully characterized already by the ensemble $p_j \rho_j$. Moreover $\rho_{\rm PI}$ is a density operator if and only if all $\rho_j$ are density operators and $p_j$ a probability distribution.
The proposition follows using the representation $V(p)$ given by Eq. (\[eq:rep\_symm\]) in the definition of the states Eq. (\[eq:PI-state\]) and then applying Schur’s lemma [@simon; @christandl06]. This lemma states that any linear operator $A$ from $\mathcal{K}_j$ to $\mathcal{K}_i$ which commutes with all elements $p$ of the group $V_i(p) A = A V_j(p)$ must either be zero if $i$ and $j$ label different irreducible representations or $A= c \mathbbm{1}$ if they are unitarily equivalent. Since $A_{\rm PI}=1/N! \sum_p V_i(p) A V_j(p)^\dag$ fulfils this relation one obtains $$\frac{1}{N!} \sum_p V_i(p) A V_j(p)^\dag = \delta_{ij} \: \tr(A) \frac{\mathbbm{1}}{\dim(\mathcal{K}_j)}.$$ The normalization can be checked taking the trace on both sides. Adding appropriate identities provides $$\label{eq:help3}
\frac{1}{N!} \sum_p \mathbbm{1} \otimes V_i(p) B \mathbbm{1} \otimes V_j(p)^\dag = \delta_{ij} \: \tr_{\mathcal{K}_j}( B ) \otimes \frac{\mathbbm{1}}{\dim(\mathcal{K}_j)}$$ where $B$ should now be a linear operator from $\mathcal{H}_j \otimes \mathcal{K}_j$ to $\mathcal{H}_i \otimes \mathcal{K}_i$.
Finally let $P_j$ denote the projector onto $\mathcal{H}_j\otimes \mathcal{K}_j$ and using Eq. (\[eq:help3\]) delivers $$\begin{aligned}
\rho_{\rm PI} &=& \frac{1}{N!}\sum_p V(p) \rho V(p)^\dag = \sum_{i,i^\prime,j,j^\prime} \frac{1}{N!}\sum_p P_i V(p)P_{i^\prime} \rho P_j V(p)^\dag P_{j^\prime} \\
&=& \sum_{i,j} \left\{ \frac{1}{N!} \sum_p P_i[\mathbbm{1} \otimes V_i(p)] P_i \rho_{\rm PI} P_{j}[\mathbbm{1} \otimes V_{j}(p)]^\dag P_{j}\right\}\\
&=& \sum_{i,j} P_i \left[ \delta_{ij} \: \tr_{\mathcal{K}_j}(P_i \rho_{\rm PI} P_{j}) \otimes \frac{\mathbbm{1}}{\dim(\mathcal{K}_j)} \right] P_j \\
&=& \bigoplus_j \tr_{\mathcal{K}_j}(P_j \rho_{\rm PI} P_j) \otimes \frac{\mathbbm{1}}{\dim(\mathcal{K}_j)},\end{aligned}$$ which provides the general structure.
The state characterization part follows because positivity of a block-matrix is equivalent to positivity of each block.
Next let us concentrate on the measurement part. Though the block decomposition follows already from the previous proposition, it is here more important to obtain an efficient computation of each measurement block for the selected setting.
\[prop:measurements\] The POVM elements $M_k^a$ as defined in Eq. (\[eq:PI-measurements\]) for any local setting $\hat a \in \mathbbm{R}^3$ can be decomposed as $M_k^a=\bigoplus_j M_{k,j}^a \otimes \mathbbm{1}$ with $$M_{k,j}^a=W_j(U_a) M^{e_3}_{k,j} W_j(U_a)^\dag.$$ The unitary is given by $W_j(U_a)=\exp(-i \sum_l t_l S_{l,j})$ using the spin operators $S_{l,j}$ in dimension $2j+1$, while the coefficients $t_l$ are determined by $U_a = \exp(-i \sum_l t_l \sigma_l/2)$ which satisfies $\hat a \cdot \vec \sigma= U_a \sigma_z U_{a}^\dag$. For the measurement in the standard basis $\hat a=\hat{e}_3$ one gets $$M^{e_3}_{k,j}= \ket{j,N/2-k}\bra{j,N/2-k}$$ if $-j \leq N/2-k \leq j$ and zero otherwise.
The basic idea is to consider the measurement in an arbitrary local basis $\hat a$ by a rotation followed by the collective measurement in the standard basis. The block decomposition is obtained as follows $$\begin{aligned}
M_k^a &=& U_a^{\otimes N} M_k^{e_3} U_a^{^\dag \otimes N} = W(U_a) \Big[ \bigoplus_j M^{e_3}_{k,j} \otimes \mathbbm{1} \Big] W(U_a)^\dag \\
&=& \bigoplus_j W_j(U_a) M^{e_3}_{k,j} W_j(U_a)^\dag \otimes \mathbbm{1}.\end{aligned}$$ The first step holds because $U_a$ satisfies $\ket{i}_a\!\bra{i}=U_a \ket{i}\bra{i} U_a^\dag$, while the block decomposition of the standard basis measurement $M_k^{e_3}$ is employed afterwards. In the last part one uses the tensor product representation given by Eq. (\[eq:rep\_SU(2)\]).
Since one knows that $W_j$ is irreducible it can be uniquely written in terms of its Lie algebra representation $dW_j$ as $W_j(U_a)=W_j(e^{-i X}) = e^{-i dW_j(X)}$, which is given by the spin operators in this case, [*i.e.*]{}, $dW_j(\sigma_l/2) = S_{l,j}$ [@hall].
Thus it is left to compute the measurement blocks $M^{e_3}_{k,j}$ for the standard basis. Note it is sufficient to evaluate $M_{k,j} \otimes \ket{1_j}\bra{1_j}$ such that one can employ the spin basis states $\ket{j,m,1}$ as introduced in Sec. \[sec:schur-weyl\]. At first note that $M^{e_3}_{k,j}$ exactly contains $k$ projections onto $\ket{0}$, while each basis state $\ket{j,m,1}$ possesses $(N/2+m)$ zeros. Therefore one obtains $M^{e_3}_{k,j} \ket{j,m,1} \propto \delta_{k,N/2+m} \ket{j,m,1}$. Since each POVM has to resolve the identity this is only possible if each $M_{k,j}^{e_3}$ is the stated rank-$1$ projector on the basis states.
Finally one still needs to express $U_a=\exp(-i \sum_l t_l \sigma_l /2)$ for the chosen setting $\hat a \in \mathbbm{R}^3$. Since this can be related to a familiar rotation [@hall] these coefficients can be expressed as $t_l = (\theta \hat n)_l$ via a rotation about an angle $\theta$ around the axis $\hat n$. Since this rotation should turn $\hat e_3$ into $\hat a$ its parameters are given by $$\begin{aligned}
\hat n &=& \frac{\hat e_3 \times \hat a}{\| \hat e_3 \times \hat a \|_2}, \\
\theta &=& \arccos(\hat e_3 \cdot \hat a),\end{aligned}$$ and $\hat n = \hat e_1$ (or any other orthogonal vector) if $\hat a = \pm \hat e_3$.
Convex optimization {#sec:copti_details}
-------------------
In this part we collect some more details regarding the described convex optimization algorithm; for a complete coverage we refer to the book [@cobook].
Each unconstrained optimization given by Eq. (\[eq:uncon\_opti\]) is solved via a damped Newton algorithm. The minimization of $f(x)=F[\rho(x)] - t \log[\det\rho(x)]$ is obtained by an iterative process. In order to determine a search direction at a given iterate $x^n$ one minimizes the quadratic approximation $$f(x^n+\Delta x) \approx f(x^n) + \nabla f(x^n)^T \Delta x + \frac{1}{2} \Delta x^T \nabla^2 f(x^n) \Delta x.$$ This reduces to solving a linear set of equation called the Newton equation $$\nabla^2 f(x^n) \Delta x_{\rm nt} = - \nabla f(x^n),$$ which determines the search direction $\Delta x_{\rm nt}$. The steplength $s$ for the next iterate $x^{n+1}= x^{n} + s \Delta x_{\rm nt}$ is chosen by a backtracking line search. Here one picks the largest $s=\max_{k \in \mathbbm{N}}\beta^k$ with $\beta \in (0,1)$ such that the iterate stays feasible $\rho(x^{n+1})>0$ and that the function value decreases sufficiently, [*i.e.*]{}, $f(x^{n+1}) \leq f(x^n) + \alpha s \nabla f(x^n)^T \Delta x_{\rm nt}$ with $\alpha \in (0,0.5)$. The process is stopped if one has reached an appropriate solution, which can be identified by $\| \nabla f(x^n) \|_2 \leq \epsilon$. If the initial point $x_{\rm start}$ is already sufficiently close to the true solution then the whole algorithm converges quadratically, [*i.e.*]{}, the precision gets doubled at each step.
At this point let us give the gradient and Hessian of the appearing functions. For the barrier term $\psi(x) = - \log [ \det \rho(x)]$ restricted to the positive domain $\rho(x)= \mathbbm{1}/\dim(\mathcal{H})+\sum_i x_i B_i > 0$ one gets [@cobook] $$\begin{aligned}
\label{eq:gradient_barrier}
\frac{\partial \psi(x)}{\partial x_i} &=& - \tr[ \rho(x)^{-1} B_i], \\
\label{eq:hessian_barrier}
\frac{\partial^2 \psi(x)}{\partial x_j\partial x_i} &=& \tr[\rho(x)^{-1} B_j \rho(x)^{-1} B_i].\end{aligned}$$ Equation (\[eq:hessian\_barrier\]) shows that the Hessian of the penalty term $\nabla^2 \psi(x)>0$ is positive definite, such that $\psi(x)$ is indeed convex. The derivatives of the preferred fit function can be computed directly. For instance using the likelihood function $F_{\rm ml}(x)= - \sum_k f_k \log p_k(x)$ with $p_k(x)=\tr[\rho(x) M_k]$ they read $$\begin{aligned}
\frac{\partial F_{\rm ml}(x)}{\partial x_i} &=& - \sum_k \frac{f_k}{p_k(x)} \tr(B_i M_k), \\
\label{eq:hessian_maxlik}
\frac{\partial^2 F_{\rm ml}(x)}{\partial x_j\partial x_i} &=&\sum_k \frac{f_k}{p^2_k(x)} \tr(B_j M_k ) \tr(B_i M_k ).\end{aligned}$$
The bottleneck of such an algorithm is the actual computation of the second derivatives. Although the expansion coefficients of each measurement $\tr(B_j M_k)$ can be computed in advance, it is still necessary to compute Eq. (\[eq:hessian\_maxlik\]) anew at each point $x$ due to the dependence of $p_k(x)$. With respect to that the least squares fit function bears a great advantage since its Hessian is constant, [*i.e.*]{}, $\partial_j\partial_i F_{\rm ls}(x) = 2 \sum_k w_k \tr(B_j M_k ) \tr(B_i M_k )$, such that one saves time on this part.
At last let us comment on the optimality conditions, known as the Karush-Kuhn-Tucker conditions [@cobook]. A given $x^\star$ is the global solution of the convex problem given by Eq. (\[eq:convex\_opti\]) if and only if [^5] there exists an additional Lagrange multiplier $Z^\star$ such that the pair $(x^\star,\Lambda^\star)$ satisfies $$\begin{aligned}
\label{eq:kkt_gradient}
\frac{\partial}{\partial x_i} F(x^\star)-\tr[\Lambda^\star B_i] &=& 0, \;\forall i,\\
\label{eq:kkt_feasibility}
\Lambda^\star \geq 0, \;\;\rho(x^\star) \geq 0, \\
\label{eq:kkt_gap}
\tr[\Lambda^\star \rho(x^\star)] = 0. \end{aligned}$$ Given the solution $x^t_{\rm sol}$ of the corresponding unconstrained problem with penalty parameter $t$ it follows from $\nabla f(x^t_{\rm sol})=0$ using Eq. (\[eq:gradient\_barrier\]) that the gradient conditions are satisfied with $\Lambda_t = t \rho(x^t_{\rm sol})^{-1}$ being the Lagrange multiplier. This pair $(x_{\rm sol}^t,\Lambda_t)$ also satisfies Eq. (\[eq:kkt\_feasibility\]); only the duality gap condition $\tr[\Lambda_t \rho(x^t_{\rm sol})] = t \dim(\mathcal{H}) > 0$ does not hold exactly. However this quantity appears in the following inequality $$\begin{aligned}
F(x^t_{\rm sol}) - \tr[ \Lambda_t \rho(x^t_{\rm sol})] &=& \min_{x: \rho(x) \geq 0} F(x) - \tr[\Lambda_t \rho(x)] \\ &\leq& \min_{x: \rho(x) \geq 0} F(x) = F(x_{\rm sol}).\end{aligned}$$ Here one used that $x_{\rm sol}^t$ is the solution of $F(x)-\tr[\Lambda_t\rho (x)]$ because the gradient vanishes (and the solution is not at the border), and $\tr[\Lambda_t \rho(x)] \geq 0$ for the inequality. This is the stated accuracy given by Eq. (\[eq:slackness\]) which relates the function value of $x^t_{\rm sol}$ to the true solution $x_{\rm sol}$.
Additional tools
----------------
### Optimization of measurement settings {#sec:optimized_settings}
Measurement settings, each described by a unit vector $\hat{a}_i \in \mathbbm{R}^3$ as explained in Sec. \[sec:recap\_pitomo\], are chosen to optimize a figure of merit characterizing how well a given permutationally invariant target state $\rho_{\rm tar}$ can be reconstructed. This is motivated as follows: A permutationally invariant state $\rho_{\rm PI}$ is uniquely described by its generalized Bloch vector [@toth10a] defined as $$\label{eq:gen_bloch_vec}
b_{klmn}=\tr(\left[ \sigma_x^{\otimes k} \otimes \sigma_y^{\otimes l} \otimes \sigma_z^{\otimes m} \otimes \mathbbm{1}^{\otimes n} \right]_{\rm PI} \rho_{\rm PI})$$ with natural numbers satisfying $k+l+m+n=N$. Consequently, one possible figure of merit is given by the total error of all Bloch vector elements, [*i.e.*]{}, more precisely by $$\mathcal{E}^2_{\rm total}(\hat a_i,\rho_{\rm tar})= \sum_{k,l,m,n} {N \choose k,l,m,n} \; \mathcal{E}^2_{b_{klmn}}(\hat a_i,\rho_{\rm tar}).$$ Here the multinomial coefficient weights the error of each Bloch vector by its number of appearance in a generic Pauli product decomposition.
The error of each Bloch vector element must now be related to the performed measurements. For that note that each Bloch vector element can be expressed as $$b_{klmn} = \sum_i c_i^{klmn} \tr( [(\hat{a}_i \cdot \vec \sigma)^{\otimes N-n} \otimes \mathbbm{1}^{\otimes n}]_{\rm PI} \rho_{\rm tar})$$ using appropriate coefficients $c_i^{klmn}$ and the expectation values of $[(\hat{a}_i \cdot \vec \sigma)^{\otimes N-n} \otimes \mathbbm{1}^{\otimes n}]_{\rm PI}$ which can be computed from the coarse-grained measurement outcomes $M_k^{a_i}$ of setting $\hat{a}_i$ as given by Eq. (\[eq:PI-measurements\]) using linear combinations. Assuming independent errors one gets $$\begin{aligned}
\mathcal{E}^2_{b_{klmn}}(\hat a_i,\rho_{\rm tar}) = \sum_i c_i^{klmn} \mathcal{E}^2_{\rho_{\rm tar}} \left( [(\hat{a}_i \cdot \vec \sigma)^{\otimes N-n} \otimes \mathbbm{1}^{\otimes n}]_{\rm PI} \right).\end{aligned}$$ The detailed form of the error expression $\mathcal{E}^2_{\rho_{\rm tar}} \left( [(\hat{a}_i \cdot \vec \sigma)^{\otimes N-n} \otimes \mathbbm{1}^{\otimes n}]_{\rm PI} \right)$ may depend on the actual physical realization, but we assume the following form $$\mathcal{E}^2_{\rho_{\rm tar}} \left( [(\hat{a}_i \cdot \vec \sigma)^{\otimes N-n} \otimes \mathbbm{1}^{\otimes n}]_{\rm PI} \right) =K \Delta_{\rho_{\rm tar}} \left( [(\hat{a}_i \cdot \vec \sigma)^{\otimes N-n} \otimes \mathbbm{1}^{\otimes n}]_{\rm PI} \right),$$ where $\Delta_{\rho}[A]=\tr(\rho A^2)-[\tr(\rho A)]^2$ is the standard variance and $K$ a state-independent factor. This form fits for example well to the common error model in photonic experiments where count rates are assumed to follow a Poissonian distribution. For more details on this derivation we refer to Ref. [@toth10a].
For large qubit numbers $N$ this optimization is carried out iteratively. Starting from randomly chosen measurement directions or from vectors which are uniformly distributed according to some frame potential [@gross07a], one searches for updates according to $$\hat{a}_i^\prime=\frac{p\hat{a}_i^n+(1-p)\hat{r}_i}{\| p\hat{a}_i^n+(1-p)\hat{r}_i \|}.$$ Here $\hat{a}_i^n$ is the current iterate, $p<1$ a probability close to $1$ and $\hat{r}_i$ are randomly chosen unit vectors. If this new set of directions $\hat{a}_i^\prime$ leads to a smaller total error $\mathcal{E}^2_{{\rm total}}(\hat{a}^\prime_i,\rho_{\rm tar})$ than the previous set then this new measurement settings form the next iterate $\hat{a}^{n+1}_i=\hat{a}_i^n$, otherwise this procedure is carried out once more. This process is repeated until the total error does not decrease any more. This way of optimizing the measurements requires a method to compute the total error $\mathcal{E}^2_{{\rm total}}(\hat{a}_i,\rho_{\rm tar})$ for a given set of measurements $\hat{a}_i$. Using the generalized Bloch vector or the spin ensemble this computation can be carried out again efficiently for larger qubit numbers $N$.
Though this algorithm is not proven to attain the true global optimum it is still a straightforward technique to obtain good settings. In the end this is often sufficient, recalling that the true unknown state can deviate from the target state and that one considers “just” an overall single figure of merit. For our simulations we always use the optimized settings for the totally mixed state.
Regarding this point we finally like to add that if one does not employ the minimal number of measurement settings, but rather an over-complete set, [*e.g.*]{}, four times as much settings but four times fewer measurements per setting, then the procedure is quite insensitive to the chosen measurement directions. Hence, in many practical situations the search for optimal directions might not even be necessary and randomly chosen measurement directions suffice equally well.
### Statistical pretest
Via the pretest one estimates the fidelity between the true $\rho_{\rm true}$ and the best permutationally invariant state $F_{\rm PI}(\rho_{\rm true})=\max_{\rho_{\rm PI} \geq 0} \tr(\sqrt{\sqrt{\rho_{\rm true}} \rho_{\rm PI}\sqrt{\rho_{\rm true}} })$ using only measurement results from a few settings $\hat a \in T$, [*e.g.*]{}, employing only $T=\{\hat e_1,\hat e_2,\hat e_3\}$. Depending on this quantity one decides whether permutationally invariant tomography is worth to continue. As explained in detail in Ref. [@toth10a] this fidelity can be bounded by $$\label{eq:pretest}
F_{\rm PI}(\rho_{\rm true}) \geq [\tr(\rho_{\rm true} Z)]^2$$ with an operator $Z=\sum z_k^a M_k^a$ being built up by the performed measurements $M_k^a$ given by Eq. (\[eq:PI-measurements\]) and satisfying $Z \leq P_{\rm sym}$, where $P_{\rm sym}$ denotes the projector onto the symmetric subspace.
The expansion coefficients $z_k^a$ should be optimized to attain the best lower bound. For a given target state $\rho_{\rm tar}$ this problem can be cast into a semidefinite program [@cobook; @toth09a] that can be solved efficiently using standard numerical routines. However for larger qubit numbers one must again employ the block structure of the measurement operators as given by Eq. (\[eq:PI-meas\]) to handle the operator inequality. Note that the projector on the symmetric subspace has a Block structure $P_{{\rm sym},j}=\delta_{j,N/2} \mathbbm{1}$. Then the final problem reads as $$\begin{aligned}
\label{eq:sdp_pretest}
\max_z && \sum_{a \in T,k} z_k^a\tr(\rho_{\rm tar} M_k^a) \\
\nonumber
\textrm{s.t.}&& \sum_{a \in T,k} z_k^a M_{k,j=N/2}^a \leq \mathbbm{1}, \sum_{a \in T,k} z_k^a M_{k,j}^a \leq 0, \forall j < N/2.\end{aligned}$$
If one experimentally implements this pretest one must account for additional statistical fluctuations. For the chosen $Z$ one can employ the sample mean $\bar Z = \sum z_k^a f_k^a$ using the observed frequencies $f_k^a=n_k^a/N_{\rm R}$ in $N_{\rm R}$ repetitions of setting $\hat a$, as an estimate of the true expectation value $\tr(\rho_{\rm true} Z)$. This sample mean $\bar Z$ will fluctuate around the true mean but large deviations will become less likely, such that for an appropriately chosen $\epsilon$ the quantity $\mathrm{sign}(\bar Z -\epsilon) (\bar Z - \epsilon)^2$ is a lower bound to the true fidelity at the desired confidence level. The proof essentially uses the techniques employed in Refs. [@flammia11a; @silva11a].
For any $Z=\sum z_k^a M_k^a \leq P_{\rm sym}$ let $\bar Z = \sum z_k^a n_k^a /N_{\rm R}$ denote the sample mean using $N_{\rm R}$ repetitions for setting $\hat a \in T$. If the data are generated by the state $\rho_{\rm true}$ then $$\mathrm{Prob} \left[ F_{\rm PI}(\rho_{\rm true}) \geq \mathrm{sign}(\bar Z -\epsilon) (\bar Z - \epsilon) ^2 \right] \geq 1-\exp(-2 N_{\rm R} \epsilon^2 / C_z^2)$$ with $C_z^2 = \sum_a \left( z_{\rm max}^s - z_{\rm min}^s \right)^2$ where $z_{\rm max/min}^a$ are the respective optima for setting $\hat a$ over all outcomes $k$.
The given statement follows along $$\begin{aligned}
\mathrm{Prob} &\left[ F_{\rm PI}(\rho_{\rm true}) \geq \mathrm{sign}(\bar Z -\epsilon) (\bar Z - \epsilon) ^2 \right]
\geq \mathrm{Prob} \left[ \tr(\rho_{\rm true} Z) \geq \bar Z -\epsilon \right] \\
&\geq 1 - \mathrm{Prob} \left[ \tr(\rho_{\rm true} Z) \leq \bar Z -\epsilon \right] \geq 1 - \exp(-2 N_{\rm R} \epsilon^2 / C_z^2).\end{aligned}$$ Here the first inequality holds because the set of outcomes satisfying $\{ n_k^a: \tr(\rho_{\rm true} Z) \geq \bar Z -\epsilon\}$ is a subset of $\{ n_k^a: F_{\rm PI}(\rho_{\rm true}) \geq \mathrm{sign}(\bar Z -\epsilon) (\bar Z - \epsilon) ^2\}$ using Eq. (\[eq:pretest\]). In the last inequality we use Hoeffdings tail inequality [@hoeffding] to bound $\mathrm{Prob} \left[ \tr(\rho_{\rm true} Z) \leq \bar Z -\epsilon \right]$.
Note that this pretest can also be applied after the whole tomography scheme in which case the projector $P_{\rm sym}=\sum z_k^a M_k^a$ becomes accessible. Moreover, let us point out that a strong statistical significance, or a low $\epsilon$ respectively, might not be achieved with the best expectation value as given by Eq. (\[eq:sdp\_pretest\]) [@jungnitsch10a]; hence optimizing $Z$ for a rather mixed state is often better.
Finally let us remark that the pretest can be improved by additional projectors, see supplementary material of Ref. [@toth10a]. This leads to the bound $F_{\rm PI}(\rho_{\rm true}) \geq \sum_j p_j^2$ with $p_j$ being the weight of the corresponding spin-$j$ state of the permutationally invariant part of $\rho_{\rm true}$ as given in Eq. (\[eq:PI-state1\]). From this expression one sees that this test only works well for states having a rather large weight on one of these spin states. Others, like the totally mixed state, while clearly being permutationally invariant, are not identified as states close to the permutationally invariant subspace. This is in stark contrast to compressed sensing where the certificate succeeds for the whole class of low rank target states [@gross11a].
### Entanglement and MaxLik-MaxEnt principle
Following the last comment from the previous section, we want to argue that even in the case of an inefficient certificate, permutationally invariant state reconstruction as given by Eq. (\[eq:PI\_staterecon\]) is meaningful. At first we would like to emphasize that the permutationally invariant part of any density operator represents a fair representative to investigate the entanglement properties of the true, unknown state. This is because the transformation given Eq. (\[eq:PI-state\]) can be achieved by local operations and classical communications, whereby the amount of entanglement cannot increase. Thus if the permutationally invariant part of the density operator is entangled this holds true also for the real state.
Second, permutationally invariant state reconstruction also represents the solution of the maximum-likelihood maximum-entropy principle as introduced in Ref. [@teo11a], which goes as follows: If the performed measurements are not tomographically complete then there is, in general, not a single state $\hat \rho_{\rm ml}$ that maximizes the likelihood but rather a complete set of them. In order to single out a “good” representative, Ref. [@teo11a] proposes to choose the state which has the largest entropy, which, according to the Jaynes principle [@jaynes], is the special state for which one has the fewest information.
\[prop:PI&MlikMent\] Using the described permutationally invariant tomography scheme, the reconstructed permutationally invariant state given by Eq. (\[eq:PI\_staterecon\]) (with the likelihood function) is also the solution of the maximum-likelihood maximum-entropy principle.
Since the measurements given by Eq. (\[eq:PI-measurements\]) are invariant and tomographically complete for permutationally invariant states, all density operators with the same spin ensemble as $\hat \rho_{\rm PI}$ have the same maximum likelihood. According to Ref. [@jaynes] the state with maximal entropy and consistent with a given set of expectation values for operators $M^a_k$ has the form $\rho \propto \exp(\sum_{a,k} \lambda^a_k M^a_k)$. The Lagrange multipliers $\lambda^a_k \in \mathbbm{R}$ must be chosen such that the given expectation values match. However, because all $M^a_k$ are permutationally invariant we can employ the block decomposition given by Eq. (\[eq:PI-meas\]) and finally obtain $\exp(\sum_{a,k} \lambda^a_k M^a_k) = \exp(\oplus_j \sum_{a,k} \lambda^a_k M^a_{k,j}\otimes \mathbbm{1}) = \oplus_j \exp(\sum^a_k \lambda^a_k M^a_{k,j}) \otimes \mathbbm{1}$. Hence we obtain the same structure as $\hat \rho_{\rm PI}$, which therefore is also the state with maximum entropy.
Conclusion and outline {#sec:conclusion}
======================
In this manuscript we provided all necessary ingredients to carry out permutationally invariant tomography [@toth10a] in experiments with large qubit numbers. This includes, besides scheme specific tasks like the statistical pretest and the optimization of the measurement settings, in particular the state reconstruction part. Accounting for statistical fluctuations due to a finite amount of data, this reconstruction demands the solution of a non-linear large-scale optimization problem. We achieve this by first using a convenient toolbox to store, characterize and process permutationally invariant states, which largely reduces the dimension of the underlying problem and second by using convex optimization, which is superior compared to commonly used numerical routines in many respects. This makes permutationally invariant tomography a complete tomography method requiring only moderate measurement and data analysis effort.
There are many questions one may pursue in this direction: First, let us stress that the current prototype implementation is still not optimal. As explained, the bottleneck is the computation of the second derivatives, hence we strongly believe that Hessian free-optimization, like quasi-Newton or conjugate gradients [@geiger99a], or the use of other barrier functions more tailored to linear matrix inequalities [@pennon] are likely to push the reconstruction limit further. Second, it is natural to try to exploit other symmetries in the development of “symmetry” tomography protocols, [*i.e.*]{}, tomography should work for all states that remain invariant under the action of a specific group. Clearly any symmetry decreases the number of state dependent parameters, but the challenge is to devise efficient local measurement strategies. Interesting classes here are graph-diagonal [@hein05a] or, more general, locally maximally entangleable states [@kruszynska09a], translation or shift-invariant states [@oconnor01a], or $U^{\otimes N}$ invariant states [@eggeling01a; @cabello03a]. Third, it is worth to investigate to what extent particularly designed state tomography protocols are useful in further tasks, like process tomography for quantum gates. For instance, permutationally invariant tomography might be unable to resolve the Toffoli gate [@monz09a] directly, but since the operation on all $N$ target qubits is symmetric, a permutationally invariant resolution of this subspace (and the additional control qubit) might be sufficient. Finally let us point out that permutationally invariant tomography can be further restricted to the symmetric subspace, which often contains the desired states. This is reasonable since we have seen that the pretest is only good if the unknown state has a large weight in one of the spin states. However since the symmetric subspace grows only linearly with the number of particles, this tomography scheme can analyse even much more qubits efficiently.
We thank M. Kleinmann, B. Kraus, T. Monz, P. Schindler and J. Wehr for stimulating discussions about the topic and technicalities. This work has been supported by the FWF (START prize Y376-N16), the EU (Project Q-ESSENCE and Marie Curie CIG 293993/ENFOQI), the BMBF (Chist-Era Project QUASAR), the ERC (Starting Grant GEDENTQOPT), the QCCC of the Elite Network of Bavaria, the Spanish MICINN (Project No. FIS2009-12773-C02-02), the Basque Government (Project No. IT4720-10), and the support of the National Research Fund of Hungary OTKA (Contract No. K83858).
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[^1]: For the least squares fit functions this follows because $F=0$ in this case and clearly $F\geq 0$ for those functions. In the case of the likelihood it follows from positivity of the classical relative entropy between probability distributions.
[^2]: More precisely, all states of the form $\ket{\psi}=V(p)\ket{\psi^-}\otimes \ket{0}$ with $p$ being any possible permutation, are states of total spin $j=1/2$ and projection $m=1/2$ to the collective spin operators $J_i=\sum_{n=1}^3 \sigma_{i;n}/2$, $\sigma_{i;n}$ being the corresponding Pauli operator on qubit $n$. However as can be checked these states only span a $2$-dimensional subspace.
[^3]: Let us stress that both least squares options can be parsed into a simpler convex problem, called semidefinite program, as for instance shown in Ref. [@reimpell_thesis; @langford07a], but that this does not work with the true maximum likelihood function to our best knowledge.
[^4]: All simulations were performed on an Intel Core i5-650, 3.2 Ghz, 8 GB RAM using MATLAB 7.12.
[^5]: Sufficiency holds under the Slater regularity condition that demands a strictly feasible point $\rho(x)>0$, which naturally holds for state reconstruction problems.
|
---
abstract: 'The blue star reported in the field of the young LMC cluster NGC 1818 by Elson et al. (1998) has the wrong luminosity and radius to be a “luminous white dwarf” member of the cluster. In addition, unless the effective temperature range quoted by the authors is a drastic underestimate, the luminosity is much too low for it to be a cluster member in the post-AGB phase. Other possibilities, including that of binary evolution, are briefly discussed. However, the implication that the massive main sequence turnoff stars in this cluster can produce white dwarfs (instead of neutron stars) from single-star evolution needs to be reconsidered.'
author:
- James Liebert
- and
title: ON THE NATURE OF THE PECULIAR HOT STAR IN THE YOUNG LMC CLUSTER NGC 1818
---
INTRODUCTION
============
Elson et al. (1998) – see also the STScI Press Release of 9 April 1998; http//oposite.stsci.edu/pubinfo/pr/1998/16 – have announced the discovery of “a luminous white dwarf” in the young star cluster NGC 1818 in the Large Magellanic Cloud. The cluster has a main sequence turnoff mass of between 7.6 and 9.0, depending on whether convective core overshooting is assumed in the models. That an even more massive progenitor might have produced a white dwarf would be of potential importance in the determination of the upper limit in initial mass for producing this kind of stellar remnant, and perhaps the lower limits of neutron stars and/or supernovae. Of the previously studied young clusters in the Milky Way Galaxy by Koester & Reimers (1996, and references therein), the highest derived initial stellar mass forming a massive white dwarf is 6.97, and the error bars for this determination are unfortunately large.
Elson et al. found a candidate bluer than the NGC 1818 main sequence in their color magnitude diagrams, whose colors suggested an extremely young (hot) white dwarf. Admittedly it lay outside the cluster core radius, but they pointed out that red giants of the cluster are also found outside the cluster core. Moreover, they argued that the probability of finding a quasar in this small field was 10$^{-3}$. Finally, in a “Note added in proof” a spectrum was mentioned that confirms that the object is a star, and suggests a T$_{eff}$ of 25,000–35,000 K.
A DIFFERENT INTERPRETATION
==========================
Not a White Dwarf Cluster Member
--------------------------------
The designation of this object as a “white dwarf,” even in the title of the paper, is the first issue to be questioned. A white dwarf is defined as an object whose interior is characterized by an electron degenerate equation of state, out to nearly the surface. They are known to have radii of the order 10$^{-2}$ solar. The most luminous white dwarfs are very hot and of relatively low mass (large radius). Of over 300 hot white dwarfs analyzed from the Palomar Green Survey (Liebert et al. 1995), the most luminous in visual magnitude units are still fainter than M$_V\sim$6 (but are also much hotter than 35,000 K). At the distance of the Large Magellanic Cloud this would correspond to an apparent V magnitude of 24.5, compared to V$\sim$18.4 for the NGC 1818 candidate. At the distance of the Large Magellanic Cloud (m-M $\sim$18.5), the candidate has an absolute visual magnitude (M$_V$) near zero. Such an object with a 30,000 K effective temperature has an implied radius of the order of solar.
The real inconsistency in radius becomes worse, if the object is supposed to be a white dwarf formed from the evolution of a massive (intermediate mass) star. It is well known (Weidemann 1990) that clusters with main sequence turnoff masses $\ga$5.0 produce massive white dwarfs $\ga$0.9. Thus, due to their abnormally small radii, the young white dwarfs found in NGC 2516 by Koester & Reimers (1996) and other young clusters have absolute visual magnitudes (M$_V$) no brighter than 10.75. At the LMC distance, this would correspond to an apparent V magnitude of $\ga$29. Note also – since this is relevant to the following subsection – that the estimated masses of the several white dwarfs analyzed in this cluster span 0.85-1.31. Perhaps the best analyzed case (because it is nearest and brightest) is LB 1497 (0349+247), a member of the Pleiades cluster which also has a turnoff mass of $\ga$5.0; its mass is found to be 1.025 based on the gravitational redshift (Wegner, Reid & McMahon 1989), or 1.084 from a model atmospheres analysis by Bergeron, Liebert & Fulbright (1995), similar to those of Koester & Reimers (1996) .
Not a Post-AGB Cluster Member
-----------------------------
One must therefore assume that what Elson et al. (1998) really meant was that their NGC 2818 candidate is an object on its way to [*becoming*]{} a white dwarf – a post-asymptotic giant branch (post-AGB) star. This hypothesis would leave intact their basic conclusion that this cluster of high turnoff mass stars can produce white dwarfs. We note from the previous subsection that, if NGC 2818 were to produce a post-AGB star, its mass should be high ($\ga$0.9) compared to typical (older) stellar remnants. Unfortunately, unless the 25,000–35,000 K temperature range from a spectrum in the “note added” comment is a drastic underestimate, I must argue that the star cannot be a post-AGB member of the cluster.
It has been well known since the work of Paczynski (1971) that the luminosity of a post-AGB star increases rapidly with the core mass – see Iben & Renzini (1983) for a still-timely review. A number of similar calculations (eg. Schönberner 1979, Wood & Faulkner 1986) have verified this correlation. Blöcker & Schönberner (1991) and Blöcker (1995) showed that the post-AGB luminosity for a given mass has some dependence on the structure of the AGB model. The recent calculations nonetheless appear to offer similar predictions for the luminosity of post-AGB cores near 0.9. For example, the 0.836 track for solar composition shown in Blöcker (1995) has log L/ $\sim$4.25 at 30,000 K as it evolves at nearly constant luminosity to very high effective temperatures. A 0.855 track for Z=0.004 by Vassiliadis & Wood (1994) has log L/ $\sim$4.30. Yet the Elson et al. cluster candidate has M$_V$ near zero. At $T_{eff}$ near 30,000 K the bolometric correction (BC$_V$) is approximately -3 (Wesemael et al. 1980, for a log g = 4, pure hydrogen atmosphere). M$_{bol}$ varies slowly with $T_{eff}$ and is therefore not terribly sensitive to the uncertain temperature. The estimated luminosity of the candidate at the cluster distance is therefore only log L/ $\sim$3.0. This value appears to correspond to a core mass of roughly 0.5, requiring extrapolation of published core mass – luminosity relations, and arguably too low for a star to even reach the AGB.
If, on the other hand, the true effective temperature of the NGC 1818 candidate has been drastically underestimated (the “note added” remark) and $T_{eff}$ approached 100,000 K, the BC$_V$ might become large enough for the luminosity to match a massive post-AGB track. For this to be the case, however, any hydrogen lines detected in the spectrum would be extremely weak, since most hydrogen would be ionized.
Possible alternative solutions
------------------------------
One might conclude that the more likely hypothesis is that the object is a foreground hot star of the Galactic halo. An extended horizontal branch (or hot subdwarf) star is a possible explanation: such objects may have T$_{eff}$ near 30,000 K, are in the long-lived phase of core helium-burning, and are characteristic of the metal-poor, halo population. If this interpretation is correct, M$_V$ could be more like +4–5, and the the object might be 5–10 kpc distant.
The authors state, however, that their measured radial velocity is consistent with LMC membership, and renders unlikely the possibility that their candidate could be a foreground halo star. I conclude with a few remarks about the implications of this possibility. In particular, this would mean that a young cluster can produce an unexpected kind of evolved object that is underluminous compared with the post-AGB tracks expected for massive stars.
I speculate that binary evolution might provide a solution. Low mass white dwarfs of $\la$0.5 with interiors apparently composed of helium have been found as companions to stars ranging from low mass main sequence stars to white dwarfs (Marsh, Dhillon & Duck 1995) and millisecond pulsars (Lundgren et al. 1996). What appears to be required is that the progenitor of the white dwarf, during post-main sequence evolution, transfers its envelope to the companion (or loses it) before the mass of the core reaches the amount required for ignition of helium. When no envelope remains, the undermassive progenitor core could leave the red giant branch (RGB), and evolve on a track that is parallel to, but at much lower luminosity than, the post-AGB tracks discussed earlier. The post-RGB tracks evolve much more slowly than the massive stellar tracks (D’Cruz et al. 1996), offering a higher probability of catching a star in this otherwise-rare phase of evolving to a white dwarf.
Were the speculative hypothesis posed in the previous paragraph to be proven correct, it would mean that Elson et al.’s (1998) candidate [*is*]{} becoming a white dwarf in this cluster with a turnoff mass between 7.6 and 9.0. It would [*not*]{}, however, support the stated implications for the upper mass limit of white dwarf formation (and lower limit for neutron star production) from single-star evolution. If a low-mass remnant core can form from interacting binary star evolution, such objects are likely to be rare.
This work was supported by the National Science Foundation through grant AST92-17961.
Blöcker, T. 1995, , 299, 755
Blöcker, T., & Schönberner, D. 1991, , 240, L11
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Elson, R.A., Sigurdsson, S., Hurley, J., Davies, M.B., & Gilmore, G.F. 1998, , 499, L53
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Liebert, J., & Bergeron, P. 1995, in [*White Dwarfs*]{}, ed. D. Koester & K. Werner (Heidelberg: Springer), p. 12
Lundgren, S.C., Cordes, J.M., Foster, R.S., Wolszczan, A. & Camilo, F. 1996, , 458, L33
Marsh, T.R., Dhillon, V.S., & Duck, S.R. 1995, , 275, 828
Paczynski, B. 1971, [*Acta. Astr.*]{}, 21, 417
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Wesemael, F., Auer, L.H., Van Horn, H.M., & Savedoff, M.P. 1980, , 43, 159
Wood, P.R., & Faulkner, D.J. 1986, , 307, 659
|
---
abstract: 'The MEG Experiment searches for a lepton flavour violating decay, [$\mu^+\rightarrow\mathrm{e}^+\gamma\ $]{}, with a branching-ratio sensitivity of $10^{-13}$ in order to explore the parameter region predicted by many theoretical models beyond the Standard Model. Detector construction and the Engineering Run were completed in 2007, and the first Physics Run will be carried out in 2008. In this paper, the prospects of MEG Physics Run in 2008 is described in addition to the experimental overview.'
author:
- 'H. Nishiguchi'
title: Lepton Flavour Violating Muon Decay at MEG
---
Introduction
============
The Standard Model of elementary particle physics is one of the greatest successes of modern science. Based on the principles of gauge symmetries and spontaneous symmetry breaking, everything had been consistently described until experimental evidence for neutrino oscillation was shown by SuperKamiokande for the first time.
Now, [**Lepton Flavour Violation (LFV)**]{} among charged leptons, which has never been observed while the quark mixing and the neutrino oscillations have been experimentally confirmed, is attracting a great deal of attention, since its observation is highly expected by many of well motivated theories beyond the Standard Model[@motivation; @muonreview]. Additionally, it would be a clear evidence of existence of new physics beyond the Standard Model because it is strongly suppressed in the Standard Model. Even if we assume a finite neutrino mass within the Standard Model, [$\mu^+\rightarrow\mathrm{e}^+\gamma\ $]{}could occur with a negligible rate, $\approx 10^{-50}$. In consequence, recent reviews on flavour physics (see Ref.[@motivation] for example) thus indicate high expectations for the next leading [$\mu^+\rightarrow\mathrm{e}^+\gamma\ $]{}search experiment, [**MEG**]{} [@MEG99], which is just starting the physics-data taking in 2008.\
The ambitious goal of the MEG experiment is to search for a [$\mu^+\rightarrow\mathrm{e}^+\gamma\ $]{}decay with an improved sensitivity by at least two orders of magnitude over the current best limit of $\mathcal{B}(\mu^+\rightarrow\mathrm{e}^+\gamma)
<1.2\times10^{-11}$(90%C.L.) [@MEGA]. It is predicted that [$\mu^+\rightarrow\mathrm{e}^+\gamma\ $]{}is naturally causable with a branching ratio just below the current upper limit,$10^{-11}\sim10^{-14}$, by the leading theories for physics beyond the standard model, [*eg.*]{} the Supersymmetric theories of Grand Unification or Supersymmetric Standard Model with the [*seesaw*]{} mechanism (see Ref.[@muonreview] for a review).
The signal of [$\mu^+\rightarrow\mathrm{e}^+\gamma\ $]{}decay is very simple and is characterized by a 2-body final state of a positron and $\gamma$-ray pair emitted in opposite directions with the same energy, 52.8MeV, which corresponds to half the muon mass. There are two major backgrounds in the search for [$\mu^+\rightarrow\mathrm{e}^+\gamma\ $]{}. One is a physics (prompt) background from a radiative muon decay, [$\mu^+\rightarrow\mathrm{e}^+\nu_{\mathrm{e}}\bar{\nu}_{\mu}\gamma\ $]{}, when the positron and the $\gamma$-ray are emitted back-to-back with the two neutrinos carrying off tiny energy. The other background is accidental coincidence of a positron from a normal Michel decay, [$\mu^+\rightarrow\mathrm{e}^+\nu_{\mathrm{e}}\bar{\nu}_{\mu}\ $]{}, with a high energy random photon. The source of high energy $\gamma$ ray is either a radiative decay [$\mu^+\rightarrow\mathrm{e}^+\nu_{\mathrm{e}}\bar{\nu}_{\mu}\gamma\ $]{}, annihilation-in-flight or external bremsstrahlung of a positron. Both are schematically shown in Figure \[signal\_bg\] in addition to the signal kinematics.
![Schematic views of [$\mu^+\rightarrow\mathrm{e}^+\gamma\ $]{}event signature and backgrounds; (Left) [$\mu^+\rightarrow\mathrm{e}^+\gamma\ $]{}Signal, (Centre) Physics Background, (Right) Accidental Background \[signal\_bg\]](signature.eps "fig:"){height="3.4cm"} ![Schematic views of [$\mu^+\rightarrow\mathrm{e}^+\gamma\ $]{}event signature and backgrounds; (Left) [$\mu^+\rightarrow\mathrm{e}^+\gamma\ $]{}Signal, (Centre) Physics Background, (Right) Accidental Background \[signal\_bg\]](radiative_background.eps "fig:"){height="3.4cm"} ![Schematic views of [$\mu^+\rightarrow\mathrm{e}^+\gamma\ $]{}event signature and backgrounds; (Left) [$\mu^+\rightarrow\mathrm{e}^+\gamma\ $]{}Signal, (Centre) Physics Background, (Right) Accidental Background \[signal\_bg\]](accidental_background.eps "fig:"){height="3.4cm"}
The background is primarily dominated by accidental coincidence. Suppressing such an accidental overlap holds the key for leading MEG to a successful conclusion.
The excellent sensitivity of the MEG experiment is enabled by three key elements: (1) the world’s most intense DC muon beam provided at the Paul Scherrer Institute (PSI); (2) an innovative liquid xenon scintillation $\gamma$-ray detector [@Xenon]; (3) a specially designed positron spectrometer with a highly graded magnetic field [@COBRA].
The beam line and the detector construction have been completed in summer 2007, and Beam- and Detector-Engineering run has been carried out right after that. In this engineering run, all the detector-calibration procedures were established. In addition to the calibration, this engineering run provided a lot of information, [*eg.*]{} detector performances, expected number of backgrounds, and what we have to maintain until the first physics data-taking in order to gain an experimental sensitivity as high as possible.
Beam and Detector
=================
In order to fulfill the ambitious goal, the MEG experiment is designed carefully. The MEG detector apparatus consists of the muon beam transport system and the detector system; the Photon Detector and the Positron Spectrometer. A schematic view of the MEG apparatus is shown in Figure \[apparatus\].\
{width="144mm"}
A DC muon beam is the best tool to search for [$\mu^+\rightarrow\mathrm{e}^+\gamma\ $]{} since experimental sensitivity is mainly limited by accidental overlap of background events. The 590 MeV proton cyclotron at the Paul Scherrer Institute (PSI) delivers up to 2.2 mA proton beam, which is the world record for such proton cyclotrons at present (2008). This megawatt accelerator has played a role of [*progenitor*]{} of the most intense DC pion and muon beams and made it possible to measure the rare decays and the search for “classical” forbidden decay modes, [*eg.*]{} [$\mu\rightarrow\mathrm{e}\gamma\ $]{}. The MEG experiment employs this most intense DC muon beam.\
The momentum and direction of positrons are measured precisely by a Positron Spectrometer, which consists of a superconducting solenoidal magnet specially designed to form a highly graded field, an ultimate low-mass drift chamber system, and a precise time measuring counter system [@COBRA]. The Positron Spectrometer has to satisfy several requirements. First, the spectrometer must cope in a stable way with a very high muon rate up to $3\times10^7$ s$^{-1}$. Second, a very-low-mass tracker is required since the momentum resolution is limited primarily by multiple Coulomb scattering. Furthermore, it is also important to minimize the amount of material from the point of view of background suppression in the photon detector. Additionally, excellent bidirectional spacial resolution of the tracker is necessary for both the transverse and longitudinal directions. Finally, excellent timing resolution is also necessary in order to suppress accidental overlap of events.
In order to attain such requirements, we adopted a specially designed solenoidal magnet with a highly graded field. The MEG solenoidal magnet is designed to change its radius between the centre and the outside. This provides a highly graded magnetic field (1.27 T at $z=0$ and decreasing as $|z|$ increases, 0.49 T at $z=1.25$ m, where $z$ is the coordinate along the beam axis) and allows to solve the problems inevitable in a normal uniform solenoidal field. In a uniform solenoidal field, positrons that emitted close to 90[$^{\circ} $ -.4em ]{} undergo many turns in the tracker volume. However, the MEG solenoidal magnetic field can sweep such positrons out of the fiducial tracking volume quickly. In addition, this special magnetic field has yet another advantage. In this specially designed field, positrons with the same absolute momenta follow trajectories with a constant projected bending radius independent of the emission angles while in a uniform solenoidal field the bending radius depends on the emission angle. This allows us to discriminate sharply high momentum signal positrons from the tremendous Michel positron background originating from the muon-stopping target. The Positron Spectrometer therefore does not need to measure the positron trajectory in the small radius region. In other words, the drift chambers can be sensitive only to higher momentum positrons and blind to most of the Michel positrons that can cause accidental coincidences. Thanks to this benefit, this spectrometer can cope within such a highly-irradiated environment.\
While all positrons are confined by the solenoid, the $\gamma$ ray pass through the thin superconducting coil of the spectrometer with $\approx$80% transmission probability, and are detected by an innovative liquid-xenon photon detector [@Xenon]. Scintillation light emitted inside liquid xenon are viewed from all sides by photo-multiplier tubes (PMT) that are immersed in liquid xenon in order to maximize direct light collection. Liquid-xenon scintillator has very high light yield ($\approx$75% of NaI crystal) and fast response, which are the most essential ingredients for precise energy and timing resolutions required for this experiment. A scintillation pulse from xenon is very fast and has a short tail, thereby minimizing the pile-up problem. Distributions of the PMT outputs enable a measurement of the $\gamma$-ray incident position with a few mm accuracy.
Absorption of scintillation light by impurities inside liquid xenon, especially water and oxygen, could significantly degrade the detector performance, although there is no absorption by liquid xenon itself. In order to solve the absorption issue, a purification system that circulates and purifies xenon gas was developed [@gas_purification]. Various studies were carried out using a 100 liter prototype detector with 238 PMTs in order to gain practical experiences in operating such a new device and to prove its excellent performance. The prototype detector was tested by using $\gamma$ rays from laser Compton scattering at National Institute of Advanced Industrial Science and Technology (AIST) in Tsukuba, Japan. Gamma rays with the Compton edge energy of 10, 20 and 40 MeV were generated via backward scattering of laser photons by 800 MeV electron beam in the storage ring of AIST. Another test was carried out at PSI by using the pion charge exchange reaction, $\pi^{-}p\rightarrow\pi^{0}n$, which provides two $\gamma$ rays from the $\pi^{0}$ decay. By tagging back-to-back $\gamma$ rays, monochromatic $\gamma$ rays of 55 MeV and 83 MeV are selected. The energy resolution of 2 %, the timing resolution of 65 ps, and the position resolution of $\approx$4 mm depending on the incident position with respect to the PMT positions were obtained by these beam tests. In addition to the performance estimation, the purification method was established.
Based on the prototype works, the final MEG liquid-xenon photon detector, which is filled with $\approx$900 litres of liquid xenon incorporating 846 PMTs, was built. In order to speed up the purification process for the final detector, a liquid-phase purification system that uses a cryogenic centrifugal fluid pump capable to flow 100 liter of liquid xenon per hour was developed [@liq_purification]. Currently(2008), this is the world’s largest liquid-xenon photon detector.\
All the signals, from PMTs and drift chambers, are individually recorded as digitized waveform by a custom chip called Domino Ring Sampler (DRS) [@DRS]. The PMT signals are digitized at 1.6 GHz sampling speed to obtain a timing resolution of 50 ps by bin interpolation, and the drift chamber signals are digitized at 500 MHz in order to compensate wide drift time distributions. Recording all the waveform may cause difficulties concerning data size, data-acquisition (DAQ) flow speed [*etc.*]{}, however the rewards outweigh the works and difficulties, because waveform digitizing of all channels gives us an excellent handle to identify the pile-up event and to suppress noise that can worsen detector resolutions.
Run 2007 (Engineering Run)
==========================
In summer 2007, construction of all detector components was completed. Then we immediately started the detector operation in a phased manner.
We started several studies for the liquid-xenon photon detector, liquefaction test, liquid xenon transferring test, long term monitoring, stability checks, PMT gain calibrations, and liquid-xenon purification [*etc*]{}. In parallel with this, positron-spectrometer conditioning was performed, drift-chamber gas control test, high-voltage conditioning, wire alignment by using cosmic rays, position-measurement calibration, relative gain calibration [*etc*]{}. After such fundamental studies and conditioning, the muon-beam commissioning was performed with the final detector apparatus. Final focusing of the muon beam, beam profile measurement, and muon rate measurement, were carried out.
After the muon-beam commissioning was completed, we started Engineering Run in October 2007. Figure \[event\] shows a typical example of accidental background events in the Engineering-Run.
![An example of event display of Engineering-Run 2007 \[event\]](event.eps){width="60mm"}
We successfully ran the whole program of the Engineering Run. All the detector components were operated over three months, trigger and DAQ electronics were integrated and data-taking worked at expected event rate, a full set of calibration has been performed. The physics data that has been taken at the Engineering Run was analyzed in the winter shutdown 2007-2008, and its results gave a certain feedback to the detector maintenance and also the offline-analysis development. One of the most important analysis of Engineering Run 2007 are to complete the reconstruction algorithm for all the detector, to evaluate the detector performances, and to estimate the feasible sensitivity of MEG.\
By analyzing the data of engineering-run 2007, we could evaluate all the detector performances and verify the quality of Monte Calro (MC) simulation for the MEG detector apparatus. Unfortunately, the obtained performances were little worse than the design value due to several unsatisfactory conditioning of detectors, [*eg.*]{} approximately 27 % of dead channel of the drift-chamber system degraded the spectrometer resolutions, still remained impurity of liquid xenon deteriorated the photon-detector resolutions, and badly-fabricated internal clock circuit of DRS provided poor timing resolutions for all sub-detectors. Although such bad performances were obtained, we could figure out the sources of deterioration and reproduce phenomenon by the MC incorporating artificially-degraded detector descriptions.
Table \[performances\] summarizes the obtained performances by the Engineering-Run 2007 and the expected performances for the MEG Physics-Run 2008. Expected values are obtained by assuming that all the deterioration clarified in 2007 could be fixed by the maintenance works during winter shutdown of 2007-2008. All resolutions are converted to Full-Width-at-Half-Maximum (FWHM).
[Quantity]{} [[ ]{}Run 2007[ ]{}]{} [[ ]{}Run 2008[ ]{}]{}
------------------------------------ ------------------------ ------------------------
$\gamma$-Energy Resolution (%) 6.5 5.0
$\gamma$-Timing Resolution (ns) 0.27 0.15
$\gamma$-Spatial Resolution (mm) 15 9.0
$\gamma$-Detection Efficiency (%) $>$40 $>$40
e$^+$-Momentum Resolution (%)[ ]{} 2.1 1.1
e$^+$-Timing Resolution (ns) 0.12 0.12
e$^+$-Angular Resolution (mrad) 17 17
e$^+$-Detection Efficiency (%) 39 65
: Detector Performances, Obtained(2007) and Expected(2008) \[performances\]
Run 2008 (Physics Run)
======================
On the basis of the result of the Engineering-Run 2007, we completed various maintenance on each sub-detectors during the winter-shutdown term 2007-2008, and now we are carrying out the final detector conditioning. The MEG Physics Run is being planned to start in summer 2008. We here discuss the feasible sensitivity of the MEG experiment by employing obtained performances and expected improvements.
We are planning to have 20 weeks of beam time in 2008. According to the PSI proton-accelerator operation procedure, 20-weeks beam time is corresponding to 8$\times 10^{6}$ sec. By employing these numbers with expected beam intensity, 3$\times 10^{7}$ sec$^{-1}$, the number of background event for the MEG physics run 2008 is expected to be 0.4.
Finally, let us evaluate the single event sensitivity and the feasible upper limit that will be determined by physics run 2008. By assuming 65 % of positron-detection efficiency, 40 % of $\gamma$-ray detection efficiency, 70 % of selection efficiency, 3$\times 10^{7}$ s$^{-1}$ of muon-beam intensity, $T=8\times 10^{6}$ s of experiment-running time, and $\Omega/4\pi = 0.09$ of detector solid angle that is calculated from the detector geometrical acceptance, the single event sensitivity for the MEG physics run 2008 can be evaluated as $
\mathcal{B}^{2008}_{\mathrm{S.E.S.}}(\mu^{+}\rightarrow\mathrm{e}^{+}\gamma)
= 2.6 \times 10^{-13}.
$ In the case that no candidate is observed, a $2.6\times10^{-13}$ single event sensitivity implies the upper limit on $\mathcal{B}^{2008}
(\mu^{+}\rightarrow\mathrm{e}^+\gamma)$ at the 90 % confidence level as $
< 7.2 \times 10^{-13}
$ for the MEG physics run 2008. This sensitivity will be eventually improved down to $1\times10^{-13}$ by the follow-up data-taking after next year.
[9]{}
|
---
abstract: 'We study arrays of parallel doped semiconductor nanowires in a temperature range where the electrons propagate through the nanowires by phonon assisted hops between localized states. By solving the Random Resistor Network problem, we compute the thermopower $S$, the electrical conductance $G$, and the electronic thermal conductance $K^e$ of the device. We investigate how those quantities depend on the position – which can be tuned with a back gate – of the nanowire impurity band with respect to the equilibrium electrochemical potential. We show that large power factors can be reached near the band edges, when $S$ self-averages to large values while $G$ is small but scales with the number of wires. Calculating the amount of heat exchanged locally between the electrons inside the nanowires and the phonons of the environment, we show that phonons are mainly absorbed near one electrode and emitted near the other when a charge current is driven through the nanowires near their band edges. This phenomenon could be exploited for a field control of the heat exchange between the phonons and the electrons at submicron scales in electronic circuits. It could be also used for cooling hot spots.'
author:
- Riccardo Bosisio
- Cosimo Gorini
- Geneviève Fleury
- 'Jean-Louis Pichard'
bibliography:
- 'PRAppl\_BGFP.bib'
title: |
Using Activated Transport in Parallel Nanowires\
for Energy Harvesting and Hot Spot Cooling
---
Introduction
============
A good thermoelectric machine must be efficient at converting heat into electricity and also must provide a substantial electric output power for practical applications. In the linear response regime, this requires optimizing simultaneously the figure of merit $ZT=S^2GT/(K^e+K^{ph})$ and the power factor $\mathcal{Q}=S^2G$, $T$ being the operating temperature, $S$ the device thermopower, $G$ its electrical conductance, and $K^e$ and $K^{ph}$ its electronic and phononic thermal conductances. In the quest for high performance thermoelectrics, semiconductor nanowires (NWs) are playing a front role,[@Hicks1993; @Curtin2012; @Blanc2013; @Brovman2013; @Stranz2013; @Karg2013] apparently offering the best of three worlds. First, an enhanced $S$ due to strongly broken and gate-tunable particle-hole symmetry.[@Brovman2013; @Roddaro2013; @Moon2013; @Tian2012] Second, a suppressed $K^{ph}$ by virtue of reduced dimensionality.[@Curtin2012; @Blanc2013] Finally, a high power output thanks to scalability, i.e. parallel stacking.[@Hochbaum2008; @Curtin2012; @Stranz2013; @Atashbar2004; @Yerushalmi2007; @Wang2009; @Zhang2010; @Davila2011; @Farrell2012; @Pregl2013]\
The perspective of developing competitive thermoelectric devices with the standard building blocks of the semiconductor industry has raised a great interest in the scientific community over the last decade. On a technological standpoint, much effort has been put into the synthesis of dense NWs arrays with controlled NW diameter, length, doping, and crystal orientation.[@Atashbar2004; @Yerushalmi2007; @Wang2009; @Persson2009; @Zhang2010; @Farrell2012; @Pregl2013] Arrays made out of various semiconductor materials including e.g. Silicon, Silicon Germanium, Indium Arsenide, or Bismuth Telluride have thus been investigated. Versatile measurement platforms have been developed to access the set of thermoelectric coefficients and the feasibility of NW-based thermoelectric modules have been assessed.[@Abramson2004; @Keyani2006; @Hochbaum2008; @Davila2011; @Curtin2012; @Stranz2013] On the theory side, numerous calculations of $S$, $G$, $K^e$ and $K^{ph}$ of various single NWs have been carried out in the ballistic regime of electronic transport [@ODwyer2006; @Markussen2009; @Liang2010; @Gumbs2010; @Neophytou2010; @Wang2014] or in the diffusive regime [@Markussen2009] where a semi-classical Boltzmann approach can be used. [@Lin2000; @Humphrey2005; @Bejenari2008; @Vo2008; @Shi2009; @Bejenari2010; @Neophytou2012; @Ramaya2012; @Neophytou2014; @Bejenari2014; @Curtin2014; @Davoody2014] In two recent works,[@Bosisio20141; @Bosisio20142] we took a different approach by considering the presence of electronic localized states randomly distributed along the NWs and making up an impurity band in the semiconductor band gap. Such states are known to play a leading role in thin nanowires, where localization effects are enhanced by low dimensionality and the system size rapidly exceeds the electron localization length. After a first study devoted to the low temperature coherent regime,[@Bosisio20141] we investigated the phonon-assisted hopping regime [@Bosisio20142] taking place at higher temperatures and usually referred to as Mott activated regime. For a long time, thermoelectric transport in this regime has been somewhat overlooked in the theoretical literature (with the exception of a few older works on bulk semiconductors[@Zvyagin1973; @Zvyagin1991; @Movaghar1981; @Wysokinski1985]). In fact, the problem of thermally-activated thermoelectric transport in NWs has been revisited only recently by Jiang *et al.* in Refs. [@Jiang2012; @Jiang2013]. However the case of gated NWs where band edges are approached has not been considered though band-edge transport, where particle-hole asymmetry is maximal, is acknowledged to be the critical one for thermoelectric conversion [@Mahan1989; @Shakouri2011]. In our previous paper,[@Bosisio20142] we studied the behavior of the thermopower $S$ and of the electrical conductance $G$ of single disordered and gated NWs in the activated regime. We obtained near the band edges a substantial enhancement of the typical thermopower $S_0$ but also, unsurprisingly, a decrease of the typical conductance $G_0$ and large sample-to-sample fluctuations of both $G$ and $S$. This is unsatisfactory if a reliable and efficient thermoelectric device is to be realized.\
In the present paper, we circumvent the latter shortcomings by considering a large set of NWs stacked in parallel in the field effect transistor (FET) device configuration. Besides assessing the opportunities offered by band edge activated transport for energy harvesting, we show that activated transport through such a device can be used for an electrostatic control of the heat exhange between the phonons and the electrons at sub-micron scales: Injecting the carriers through the NWs gives rise to a local cooling \[heating\] effect near the source \[drain\] electrode when the chemical potential of the device probes the lower NWs band edge (and conversely when it probes the upper edge). This opens promising perspectives for a local management of heat and for cooling hot-spots in microelectronics.\
Hereafter, we study arrays of doped semiconductor NWs, arranged in parallel and attached to two electrodes. The NWs can be either suspended or deposited onto an electrically and thermally insulating substrate. A metallic gate beneath the sample is used to vary the carrier density inside the NWs. This corresponds to a setup in the FET configuration, as sketched in Fig. \[fig:sys\]. If the thermopower or the thermal conductances are to be investigated, a heater (not shown in Fig. \[fig:sys\]) is added on one side of the sample to induce a temperature gradient between the electrodes. We focus on a temperature range where the activated regime proposed by Mott \[Variable Range Hopping (VRH) regime\] takes place, assuming *(i)* that phonon-assisted transport occurs between localized states of the NWs impurity band only and *(ii)* that the substrate, or the NWs themselves if they are suspended, act as a phonon bath to which NWs charge carriers are well coupled. We thus consider intermediate temperatures, where the thermal energy $k_BT$ is high enough to allow inelastic hopping between Anderson localized states of different energies (typically a few Kelvin degrees), yet low enough to keep localization effects. Such VRH regime is observed up to room temperatures in three-dimensional amorphous semiconductors [@Shklovskii1984] and very likely up to higher temperatures in the one-dimensional limit. Following Refs. [@Jiang2012; @Jiang2013; @Bosisio20142], we solve numerically the Miller-Abrahams Random Resistor Network problem [@Miller1960] for obtaining $S$, $G$, and $K^e$. This allows us to identify also the regions where heat exchanges between the electrons and the phonons dominantly take place in the activated regime, notably when the chemical potential probes the edges of the NWs impurity band.\
![(Color online) Array of suspended (a) and deposited (b) parallel NWs in the FET configuration. The NWs are drawn in green, the two metallic electrodes in yellow, the substrate in grey and the back gate in dark grey. The blue \[red\] strip in (b) indicates the substrate region that is cooled down \[heated up\] in the phonon-assisted activated regime, when a charge current flows from the left to the right electrode and the gate voltage is tuned so as to probe the lower edge of the NWs impurity band.[]{data-label="fig:sys"}](fig_parallel_sys.pdf){width="\columnwidth"}
The model used throughout the paper is presented in Sec. \[sec\_method\], together with a summary of the method. We find in Sec. \[sec\_scaling\] that once a large set of NWs is stacked in parallel, the strong $G$, $S$ and $K^e$ fluctuations are suppressed. Denoting by $G_0$, $S_0$ and $K^e_0$ the [*typical*]{} values for a single NW, we observe more precisely that the thermopower of a large NW array self-averages ($S\to {S_0}$) while its electrical and electronic thermal conductances $G\to M {G_0}$, $K^e\to M {K^e_0}$ as the number $M$ of wires in parallel increases (see Fig. \[fig:fluctuations\]). Taking full advantage of the gate, we move close to the impurity band edges, where we recently obtained a drastic ${S_0}$ enhancement.[@Bosisio20142] We show in Sec. \[sec\_QZT\] that in this regime a large $S_0$ partly compensates an exponentially small $G_0$, so that substantial values of the power factor ${\mathcal Q}\approx MS_0^2G_0$ can be reached upon stacking plenty of NWs in parallel \[see Fig. \[fig:QZT\](a)\]. Remarkably, the electronic figure of merit $Z_eT=S^2GT/K^e$ is also found to reach promising values $Z_eT\approx 3$ when ${\mathcal Q}$ is maximal \[see Fig. \[fig:QZT\](b)\]. Furthermore, we discuss how the phononic thermal conductance $K^{ph}$ will inevitably reduce the full figure of merit $ZT$ and argue that, even if record high $ZT$ is probably not to be sought in such setups, the latter have the great advantage of offering at once high output power and reasonable efficiency with standard nanotechnology building blocks. The most important result of this paper is given in Sec. \[sec\_hotspots\]. We study how deposited NWs in the FET configuration can be used to manage heat in the substrate, generating hot/cold spots “on demand”. The idea is simple to grasp and relies on the calculation of the local heat exchanges between the NWs electrons and the substrate phonons: When the gate voltage is adjusted such that the equilibrium electrochemical potential $\mu$ (defined in the electronic reservoirs) roughly coincides with one (say the lower) impurity band edge, basically all energy states in the NWs lie above $\mu$. Therefore, if charge carriers injected into the system around $\mu$ are to gain the other end, they need to (on the average) absorb phonons at the entrance so as to jump to available states, and then to release phonons when tunneling out (again at $\mu$). This generates in the nearby substrate regions cold strips near the injecting electrode and hot strips near the drain electrode \[see Figs. \[fig:sys\](b) and \[fig:hotspots\]\]. These strips get scrambled along the nanowires if $\mu$ does not probe the edges of the NWs impurity band. Such reliable and tunable cold spots may be exploited in devising thermal management tools for high-density circuitry, where ever increasing power densities have become a critical issue.[@Vassighi2006] Moreover, the creation/annihilation of the cold/hot strips can be controlled by the back gate voltage. Note that the underlying mechanism governing the physics of VRH transport at the NWs band edges is somewhat reminiscent of the mechanism of “cooling by heating” put forward in Refs. [@Pekola2007; @Rutten2009; @Levy2012; @Marl2012; @Cleuren2012], which also exploits the presence of a third bosonic bath in addition to the two electronic reservoirs. In our case, bosons are phonons provided mainly by the substrate; in other setups, bosons are photons provided by laser illumination (or more simply by the sun for a photovoltaic cell). All those studies fall into the growing category of works dealing with boson-assisted electronic transport that have been shown to open promising perspectives for heat management.
Model and method {#sec_method}
================
Architecture and/or material specific predictions, though very important for practical engineering purposes, are however [*not*]{} our concern at present. On the contrary, our goal is to reach conclusions which are as general as possible, relying on a bare-bone but widely applicable Anderson model devised to capture the essentials of the physics we are interested in. We consider a set of $M$ NWs in parallel. Each NW is modeled as a chain of length $L$ described by a one-dimensional (1D) Anderson tight-binding Hamiltonian with on-site disorder [@Bosisio20142]: \[eq\_modelAnderson1D\] =-t\_[i=1]{}\^[N-1]{}(c\_i\^c\_[i+1]{}+)+\_[i=1]{}\^[N]{}(\_i+V\_g) c\_i\^c\_i. Here $N$ is the number of sites in the chain ($L=Na$ with $a$ lattice spacing), $c^{\dagger}_i$ and $c_i$ are the electron creation and annihilation operators on site $i$ and $t$ is the hopping energy ([*inter*]{}-wire hopping is neglected). We assume that no site can be doubly occupied due to Coulomb repulsion, but otherwise neglect interactions.[@Ambegaokar1971] The site energies $\epsilon_i$ are uncorrelated random numbers uniformly distributed in the interval $[-W/2,W/2]$, while $V_g$ is a constant (tunable) potential due to the back gate. The electronic states are localized at certain positions $x_i$ with localization lengths $\xi_i$ and eigenenergies $E_i$. The $E_i$’s lie within the NW impurity band whose center can be shifted with the gate voltage $V_g$. For simplicity’s sake, we generate randomly the positions $x_i$ along the chain (with a uniform distribution) and assume $\xi_i=\xi(E_i)$, where $\xi(E)$ characterizes the exponential decay of the *typical* conductance $G_0\sim\exp(-2L/\xi)$ of the 1D Anderson model at zero temperature and energy $E$. Analytical expressions giving $\xi(E)$ in the weak disorder limit of the Anderson model are given in Ref. [@Bosisio20141].
The NWs are attached to two electronic reservoirs $L$ and $R$, and to a phonon bath, i.e. the system is in a three-terminal configuration. Particles and heat(energy) can be exchanged with the electrodes, but only heat(energy) with the phonon bath. At equilibrium the whole system is thermalized at a temperature $T$ and both $L$ and $R$ are at electrochemical potential $\mu$ (set to $\mu\equiv 0$, at the band center when $V_g=0$). A voltage and/or temperature bias between the electrodes drives an electron current through the NWs. Hereafter we consider the linear response regime, valid when small biases $\delta\mu\equiv\mu_L-\mu_R$ and $\delta T\equiv T_L-T_R$ are applied.
We study the inelastic activated regime. Following ref.[@Jiang2013], we assume that the charge carriers (say electrons of charge $e$) tunnel elastically from reservoir $\alpha=L,R$ into some localized states $i$ whose energies $E_i$ are located in a window of order $k_BT_\alpha$ around $\mu_\alpha$. They then proceed via phonon-assisted hops to the other end, finally tunneling out. The maximal carriers hop along the NWs is of the order of Mott length $L_M$ in space (or Mott energy $\Delta$ in energy) [@Bosisio20142]. At the lowest temperatures considered in this work, $\xi(\mu)\ll L_M\ll L$ and transport is of Variable Range Hopping (VRH) type. An increasing temperature shortens $L_M$ until $L_M\approx\xi(\mu)$, when the Nearest Neighbors Hopping (NNH) regime is reached. The crossover VRH$\rightarrow$NNH takes place roughly at Mott temperature $T_M$, whose dependence on $V_g$ can be found in Ref. [@Bosisio20142].
The *total* electron and heat currents flowing through the *whole* array are calculated by solving the Random Resistor Network problem.[@Miller1960; @Ambegaokar1971] The method is summarized in Appendix \[app\_rrn\]. It takes as input parameters the rate $\gamma_e$ quantifying the coupling between the NWs (localized) and the reservoirs (extended) states, and the rate $\gamma_{ep}$ measuring the coupling to the NWs and/or substrate phonons. We point out that we go beyond the usual approximation [@Miller1960; @Ambegaokar1971; @Jiang2013] neglecting the $\xi_i$’s variations from state to state \[$\xi_i\approx\xi(\mu)$\], the latter being inappropriate close to the band edges, where $\xi_i$ varies strongly with the energy. Following Ref. [@Bosisio20142] the random resistor network is then solved for $\xi_i\neq\xi_j$. The particle and heat currents thus obtained are related to the small imposed biases $\delta\mu, \delta T$ via the Onsager matrix,[@Callen1985] which gives access to $G$, $K^e$ and $S$.
Scaling of the thermoelectric coefficients with the number of nanowires {#sec_scaling}
=======================================================================
The typical conductance $G_0$ and thermopower $S_0$ of a single NW were studied in Ref. [@Bosisio20142]. They are defined as the [*median*]{} of the distribution of $\ln G$ and $S$, obtained when considering a large statistical ensemble of disorder configurations. In Fig. \[fig:fluctuations\] we show that, if the system is made of a sufficiently large number $M$ of parallel NWs, the *overall* electrical conductance scales as the number of wires times the typical NW value ($G\approx M\, G_0$), while the thermopower averages out to the typical value of a single wire ($S\approx S_0$). For completeness the [*mean*]{} values are also shown and seen to be a less accurate estimate. As expected, convergence is faster at higher temperatures. Identical results have been obtained for the electronic thermal conductance $K^e\approx MK^e_0$ (not shown).
![Convergence of $G/M$ (left, in units of $e^2/\hbar$) and $S$ (right) with the number $M$ of parallel NWs. Symbols correspond to $V_g=1.9\,t$ ([$\circ$]{}), $2.1\,t$ () and $2.3\,t$ () at $k_BT=0.1\,t$, and $V_g=1.9\,t$ at $k_BT=0.5\,t$ (). The horizontal lines indicate the corresponding mean values (dashed lines) and typical values (solid lines) of $\ln G$ and $S$ of a single wire ($M=1$). Parameters: $W=t$, $\gamma_e=\gamma_{ep}=t/\hbar$ and $L=450a$.[]{data-label="fig:fluctuations"}](G_S_fluctuations.pdf){width="\columnwidth"}
Power factor and figure of merit {#sec_QZT}
================================
By stacking a large number $M$ of NWs in parallel, the device power factor can be enhanced $\mathcal{Q}\approx MS_0^2G_0$ [*without*]{} affecting its electronic figure of merit $Z_eT\approx S_0^2G_0T/K_0^e$. Fig. \[fig:QZT\] shows how the asymptotical $\mathcal{Q}/M$ and $Z_eT$ values (reached when $M\gtrsim 100$) depend on the gate voltage $V_g$ and on the temperature $T$. We observe in panel (a) that the power factor is maximum for $\mu$ close to the impurity band edge (black solid line) and for VRH temperatures. This parameter range represents the best compromise between two opposite requirements: maximizing the thermopower (hence favoring low $T$ and large $V_g$) while keeping a reasonable electrical conductance (favoring instead higher $T$ and $V_g\approx 0$). Formulas previously reported,[@Bosisio20142] giving the $T$- and $V_g$-dependence of $G_0$ and $S_0$, let us predict that $\mathcal{Q}$ is maximal when $|S_0|=2k_B/|e|\approx 0.2\,\mathrm{mV}\,\mathrm{K}^{-1}$ (black dashed line). A comparison between panels (a) and (b) of Fig. \[fig:QZT\] reveals that, in the parameter range corresponding to the best power factor ($V_g\sim2.5t, k_BT\sim0.6t$), $Z_eT\simeq 3$, a remarkably large value. Much larger values of $Z_eT$ could be obtained at lower temperatures or far outside the band, but they are not of interest for practical purposes since in those regions $\mathcal{Q}$ is vanishing.
![(Color online) $\mathcal{Q}/M$ in units of $k_B^2/\hbar$ (a) and $Z_eT$ (b) as a function of $T$ and $V_g$. Data are shown in the large $M$ limit ($M=150$) where there is self-averaging. The horizontal lines give $V_g$’s value at which the band edge is probed at $\mu$ (below \[above\] it, one probes the inside \[outside\] of the impurity band). The red dashed lines $T=T_M$ separate the VRH ($T\lesssim T_M$) and the NNH ($T\gtrsim T_M$) regimes. The black dashed line in (a) is the contour along which $S_0=2k_B/e$. Parameters: $W=t$, $\gamma_e=\gamma_{ep}=t/\hbar$ and $L=450a$.[]{data-label="fig:QZT"}](QZT.pdf){width="0.9\columnwidth"}
In Appendix \[app\_size\], $\mathcal{Q}$ and $Z_eT$ are shown to be roughly independent of the NWs length $L$ (for $L\gtrsim L_M$) in the temperature and gate voltage ranges explored in Fig. \[fig:QZT\]. Moreover $\mathcal{Q}/\gamma_e$ and $Z_eT$ are almost independent of the choice of the parameters $\gamma_e$ and $\gamma_{ep}$, provided $\gamma_{ep}\gtrsim\gamma_e$ (see Appendix \[app:depgammaeep\]). When $\gamma_{ep}<\gamma_e$, both quantities are found to be (slightly) reduced.\
Let us now estimate the order-of-magnitude of the device performance. The substrate (or the NWs themselves if they are suspended) is assumed to supply enough phonons to the NWs charge carriers for the condition $\gamma_{ep}\gtrsim\gamma_e$ to hold. Besides, we keep explicit the $\gamma_e$-linear dependence of $\mathcal{Q}$ (and of $K^e_0$ that will soon be needed). $\gamma_e$ depends on the quality of the metal/NW contact. We estimate it to be within the range $0.01-1$ in units of $t/\hbar$, where $t/k_B\approx 150\,\mathrm{K}$ throughout [^1]. This yields $\gamma_e\approx 0.02-2\times 10^{13}\,\mathrm{s}^{-1}$. For the sake of brevity, we introduce the dimensionless number $\tilde{\gamma}_e=\gamma_e\hbar/t$. Focusing on the region of Fig. \[fig:QZT\](a) where the power factor is maximal, we evaluate the typical output power and figure of merit than can be expected. We first notice that power factor $\mathcal{Q}/M\approx4k_B^2/h$ maximum values in Fig. \[fig:QZT\](a), obtained with $\tilde{\gamma}_e=1$, would yield $\mathcal{Q}\approx 7\tilde{\gamma}_e\times 10^{-7}\,\mathrm{W}.\mathrm{K}^{-2}$ for a chip with $M\approx 10^5$ parallel NWs. Since $\mathcal{Q}$ controls the maximal output power $P_{max}$ that can be extracted from the setup as $P_{max}=\mathcal{Q}(\delta T)^2/4$,[@Benenti2013] one expects $P_{max}\approx 20\tilde{\gamma}_e\,\mu\mathrm{W}$ for a small temperature bias $\delta T\approx 10\,\mathrm{K}$. In this region a large value $Z_eT\approx 3$ is obtained, but to estimate the full figure of merit $ZT=Z_eT/(1+K^{ph}/K^{e})$, the phononic part $K^{ph}$ of the thermal conductance must also be taken into account. To limit the reduction of $ZT$ by phonons, the setup configuration with suspended nanowires is preferable \[Fig. \[fig:sys\](a)\]. In this case $K^{ph}\approx MK_0^{nw}$, $K_0^{nw}$ being the typical phononic thermal conductance of a single NW, and has to be compared to $K^{e}\approx MK_0^{e}$. Introducing the corresponding conductivities $\kappa$’s, the ratio $K^{ph}/K^{e}\approx \kappa_0^{nw}/\kappa_0^{e}$ is to be estimated. Our numerical results obtained for 1D NWs show $K_0^{e}\approx 1.5\tilde{\gamma}_e k_Bt/\hbar$ in the range of interest where $\mathcal{Q}$ is maximal and $Z_eT\approx 3$ (at $V_g=2.5t$ and $k_BT=0.6t$, keeping other parameters in Fig. \[fig:QZT\] unchanged). To deduce the corresponding conductivity $\kappa_0^{e}$, the NW aspect ratio must be specified. We consider for instance the case of $1\,\mu\mathrm{m}$-long NWs with a diameter of $20\,\mathrm{nm}$, for which our pure 1D model is expected to hold[^2], at least semi-quantitatively. Thereby we get $\kappa_0^{e}\approx 1\tilde{\gamma}_e\,\mathrm{W}/(\mathrm{K.m})$, while the measured thermal conductivity of Si NWs of similar geometry is $\kappa_0^{nw}\approx 2\,\mathrm{W}/(\mathrm{K.m})$ at $T\approx 100\,\mathrm{K}$.[@Li2003] We thus evaluate for suspended NWs $ZT\approx Z_eT/(1+2/\tilde{\gamma}_e)$, i.e. $ZT\approx 0.01-1$ for $Z_eT\approx 3$ and $\tilde{\gamma}_e=0.01-1$. Those estimations though rough are extremely encouraging as they show us that such a simple and Si-based device shall generate high electrical power from wasted heat (scalable with $M$, for $M$ large enough) with a fair efficiency (independent of $M$, for $M$ large enough).\
Let us note that maximizing $\gamma_e$ is important for achieving high $\mathcal{Q}$ and $ZT$. However at the same time $\gamma_{ep}\gtrsim\gamma_e$ should preferably hold. If the NWs themselves do not ensure a large enough $\gamma_{ep}$, the use of a substrate providing phonons is to be envisaged. Yet, this will add a detrimental contribution $K^{sub}$ to $K^{ph}$. In general the substrate cross-section ($\Sigma^{sub}$) will be substantially larger than the NWs one ($M\Sigma^{nw}$). Thus, even for a good thermal insulator such as SiO$_2$, with thermal conductivity $\kappa^{sub}\approx 0.7\,\mathrm{W}/(\mathrm{K.m})$ at $T\approx 100\,\mathrm{K}$,[@Hung2012] $Z/Z_e=[1+(\kappa^{sub}\Sigma^{sub}+M\kappa_0^{nw}\Sigma^{nw})/M\kappa_0^{e}\Sigma^{nw}]^{-1}\ll1$. Better ratios $Z/Z_e$ could be obtained for substrates with lower $K^{sub}$ (Silica aerogels,[@Hopkins2011] porous silica,[@Scheuerpflug1992] very thin substrate layer) but they will not necessarily guarantee a good value of $\gamma_{ep}$ (and hence of $Z_e$). Clearly, finding a balance between a large $\gamma_{ep}$ and a low $K^{ph}$ is a material engineering optimization problem. Though the presence of a substrate appears detrimental for efficiently harvesting electrical energy from the wasted heat, we shall now see how it could be used for heat management at the nanoscale.
{width="90.00000%"}
Gate-controlled creation/annihilation of cold/hot strips {#sec_hotspots}
========================================================
Hereafter, we consider the deposited setup sketched in Fig. \[fig:sys\](b) and assume a constant temperature $T$ everywhere. An intriguing feature of this three-terminal setup is the possibility to generate/control hot/cold strips close to the substrate boundaries by applying a bias $\delta\mu/e$, if one tunes $V_g$ for probing the NWS band edges. This effect is a direct consequence of the heat exchange mechanism between electrons in the NWs and phonons in the substrate. Indeed, given a pair of localized states $i$ and $j$ inside a NW, with energies $E_i$ and $E_j$ respectively, the heat current absorbed from (or released to) the phonon bath by an electron in the transition $i\to j$ is $I_{ij}^{Q}=\left(E_j-E_i\right)I^N_{ij}$, $I^N_{ij}$ being the hopping particle current between $i$ and $j$.[@Bosisio20142] The overall hopping heat current through each localized state $i$ is then found by summing over all but the $i$-th states: \[eq:IQ\_i\] I\_[i]{}\^[Q]{}=\_j I\_[ij]{}\^Q = \_[j]{} (E\_j-E\_i)I\^N\_[ij]{} with the convention that $I^Q_{i}$ is positive (negative) when it enters (leaves) the NWs at site $i$. Since the energy levels $E_i$ are randomly distributed, the $I^Q_i$’s (and in particular their sign) fluctuate from site to site (see Fig. \[fig:HotSpots\_Vg2.25\_kT0.05\] in Appendix \[app\_lambdaph\] for an illustration). The physically relevant quantities are however not the $I^Q_i$’s, rather their sum within an area $\Lambda_{ph}\times\Lambda_{ph}$, where $\Lambda_{ph}$ is the phonon thermalization length in the substrate (i.e. the length over which a [*local*]{} substrate temperature can be defined, see Appendix \[app\_lambdaph\] for an estimation). Given a point $(x,y)$ and a $\Lambda_{ph}\times\Lambda_{ph}$ area centered around it, such sum is denoted $\mathcal{I}^Q_{x,y}$. If $\mathcal{I}^Q_{x,y}>0$ $[<0]$, a volume $\Lambda_{ph}^3$ of the substrate beneath $(x,y)$ is cooled \[heated\] [^3]. Deeper than $\Lambda_{ph}$ away from the surface, the equilibrium temperature $T$ is reached.\
Fig. \[fig:hotspots\] shows how $\mathcal{I}^Q_{x,y}$ depends on the coordinates $x, y$ in the two-dimensional parallel NW array. Left and right panels show respectively the situation in the absence of a gate voltage, when charge carriers tunnel into/out of NWs at the impurity band *center*, and the opposite situation when a large gate voltage is applied in order to inject/extract carriers at the band *bottom*. In both cases, two values of the temperature are considered (top/bottom panels). All other parameters are fixed. Note that data are plotted for the model introduced in Sec. \[sec\_method\], having estimated $a\approx 3.2\,\mathrm{nm}$, $t/k_B\approx 150\,\mathrm{K}$, and $\Lambda_{ph}\approx 480[240]\,\mathrm{nm}\approx 150[75]a$ for SiO$_2$ substrate at the temperatures considered, $T=0.25[0.5]t/k_B\approx37.5[75]\,\mathrm{K}$. Those estimates are discussed in Appendix \[app\_lambdaph\]. In the left panels of Fig. \[fig:hotspots\], the heat maps show puddles of positive and negative $\mathcal{I}^Q_{x,y}$, corresponding respectively to cooled and heated regions in the substrate below the NW array. They are the signature of random absorption and emission of substrate phonons by the charge carriers, all along their propagation through the NWs around the band center. In the right panels, the regions of positive and negative $\mathcal{I}^Q_{x,y}$ are respectively confined to the NWs entrance and exit. This is due to the fact that charge carriers entering the NWs at $\mu$ around the band bottom find available states to jump to (at a maximal distance $L_M$ in space or $\Delta$ in energy [^4]) only *above* $\mu$. Therefore, they need to absorb phonons to reach higher energies states (blue region). After a few hops, having climbed at higher energies, they continue propagating with equal probabilities of having upward/downward energy hops (white region). On reaching the other end they progressively climb down, i.e. release heat to the substrate (red region), until they reach $\mu$ and tunnel out into the right reservoir. As a consequence, the substrate regions below the NWs extremities are cooled on the source side and heated on the drain side \[see Fig. \[fig:sys\](b)\]. A comparison between top and bottom panels of Fig. \[fig:hotspots\] shows us that the heat maps are not much modified when the temperature is doubled \[from $k_BT=0.25t$ (top) to $k_BT=0.5t$ (bottom)\]. The fact that the surface $\Lambda_{ph}\times\Lambda_{ph}$ inside which the heat currents are summed up is smaller at larger temperature ($\Lambda_{ph}=75a$ at $k_BT=0.5t$ instead of $\Lambda_{ph}=150a$ at $k_BT=0.25t$) is compensated by a smoothing of the $I^Q_i$’s fluctuations. This makes the hot and cold strips still clearly visible and well-defined in the bottom right panel of Fig. \[fig:hotspots\].\
We point out that the maximum values of $\mathcal{I}^Q_{x,y}$ are roughly of the same order of magnitude with or without the gate (see scale bars in Fig. \[fig:hotspots\]). The advantage of using a gate is the ability to split the positive and negative $\mathcal{I}^Q_{x,y}$ regions into two well separated strips in the vicinity of the injection and drain electrodes. One can then imagine to exploit the cold strip in the substrate to cool down a hot part of an electronic circuit put in close proximity. Let us also stress that the assumption of elastic tunneling processes between the electrodes and the NWs is not necessary to observe the gate-induced hot/cold strips. The latter arise from the “climbing” up/down in energy that charge carriers, at $\mu$ far into the electrodes, must undergo in order to hop through the NWs (hopping transport being favored around the impurity band center in the NWs). Though in our model heat exchanges take place only inside the NWs, phonon emission/absorption will actually take place also at the electrodes extremities, roughly within an inelastic relaxation length from the contacts. This has clearly no qualitative impact, as it only amounts to a slight shift/smearing of the hot/cold strips.\
Finally, let us estimate the cooling powers associated to the data shown in Fig. \[fig:hotspots\]. Assuming again $t/k_B\approx 150\,\mathrm{K}$ and $a\approx 3.2\,\mathrm{nm}$, we find that a value of $\mathcal{I}^Q_{x,y}=10^{-3}(t^2/\hbar)$ in Fig. \[fig:hotspots\](bottom) corresponds to a cooling power density of the order of $8.10^{-10}\,\mathrm{W}.\mu\mathrm{m}^{-2}$ at the temperature considered $T=0.5t/k_B\approx 77\,\mathrm{K}$ (the boiling temperature of liquid nitrogen at atmospheric pressure), for which $\Lambda_{ph}\approx 240\,\mathrm{nm}$ in SiO$_2$. We underline that this order of magnitude is obtained for a given set of parameters, in particular for an infinitesimal bias $\delta\mu=10^{-3}t \approx 13\, \mu V$ that guarantees to remain in the linear response regime. It should not be taken in the strict sense but only as a benchmark value to fix ideas. For instance, according to this estimation, one should be able to reach cooling power densities $\approx 6.10^{-8}\,\mathrm{W}.\mu\mathrm{m}^{-2}$ by applying a larger bias $\delta\mu/e \approx 1\,\mathrm{mV}$. To be more specific, we note that the geometry considered in Fig. \[fig:hotspots\] is realized[^5] with a bidimensional array of 150, $5\,\mu$m-long NWs, covered by two $7.2\,\mu$m-long (or longer) metallic electrodes. For this geometry and at $T\approx 77\,\mathrm{K}$, the areas of the cooled and heated regions are approximately $7.2 \times 0.25 \approx 2
\mu\mathrm{m}^{2}$ (see the lower right panel of Fig. \[fig:hotspots\]) but if one considered $1\,\mathrm{cm}$ electrodes covering $2.10^{5}$ NWs, those areas would naturally extend. Thus, for a bias $\delta\mu/e \approx 1\,\mathrm{mV}$ and a temperature $T \approx 77 \,\mathrm{K}$, our setup would allow to take $\approx 0.15\,\mathrm{mW}$ in a strip of $1\,\mathrm{cm}\times 0.25\,\mu\mathrm{m}$ area and $0.25\,\mu\mathrm{m}$ thickness located in the SiO$_2$ substrate below the source electrode and to transfer it in another strip of similar size located at $5\,\mu\mathrm{m}$ away below the drain electrode. Obviously, the longer the NWs, the longer would be the scale of the heat transfer. The larger the bias and the number of used NWs, the larger would be the heat transfer.
Conclusion
==========
The low carrier density of a doped semiconductor can be varied by applying a voltage on a (back, side or front) metallic gate. This led us to study thin and weakly doped semiconductor NWs, where electron transport is activated, instead of thick metallic NWs (with much larger electrical and thermal conductances) where the field effects are negligible. Considering arrays of these NWs in the FET configuration, we have focused our attention on the activated regime which characterizes a very broad temperature domain in amorphous semiconductors.[@Ambegaokar1971] When charge transport between localized states is thermally assisted by phonons, we have shown that the absorption or the emission of phonons in strips located near the source and drain electrodes can be controlled with a back gate. This opens new perspectives for managing heat at submicron scales. By tuning the electrochemical potential $\mu$ near the band edges of the NWs impurity band, we have studied how to take advantage of electron-phonon coupling for energy harvesting and hot spot cooling. Our estimates indicate that large power factors are reachable in these arrays, with good thermoelectric figures of merit. This work was supported by CEA within the DSM-Energy Program (project E112-7-Meso-Therm-DSM). We thank O. Bourgeois, Y. Imry and F. Ladieu for stimulating discussions.
Resolution of the Random Resistor Network problem {#app_rrn}
=================================================
![(Color online) Phonon-assisted hopping transport through the localized states (dots) of a disordered NW connected to two electrodes $L$ and $R$, and to a phonon bath. The electronic reservoirs $L$ and $R$ are thermalized at temperatures $T_{L[R]}$ and held at electrochemical potentials $\mu_{L[R]}$ (their Fermi functions are sketched by the black curves on both sides). A metallic gate (shaded grey plate drawn on top) allows to shift the NW impurity band (blue central region). Here, the gate potential $V_g$ is adjusted such that electrons tunnel in and out of the electronic reservoirs near the lower edge of the impurity band. Therefore, electrons tend to absorb phonons at the entrance in order to reach available states of higher energies, and to emit phonons on the way out. The two wavy arrows indicate the local heat flows between the NW electrons and the phonon bath. They give rise to a pair of cold (blue) and hot (red) spots in the substrate beneath the NW (in the deposited setup configuration).[]{data-label="fig:sketch_sys"}](sketch_sys.pdf){width="\columnwidth"}
Hereafter, we summarize the numerical method used to solve the random-resistor network problem.[@Miller1960; @Ambegaokar1971; @Jiang2013] The three-terminal setup configuration is reminded in Fig. \[fig:sketch\_sys\], with emphasis on the hopping transport mechanism taking place in the NWs. Starting from a set of states $i$ localized at positions $x_i$ inside the NWs, with energies $E_i$ and localization lengths $\xi_i$, we first evaluate the transition rates $\Gamma_{i\alpha}$ from the localized state $i$ to the reservoir $\alpha=L$ or $R$, and $\Gamma_{ij}$ from states $i$ to $j$ within the same wire ([*inter*]{}-wire hopping being neglected). They are given by the Fermi Golden rule as $$\begin{aligned}
\Gamma_{i\alpha}&=\gamma_{i\alpha} f_i [1-f_{\alpha}(E_i)]\label{eq_rates1}\\
\Gamma_{ij}&=\gamma_{ij} f_i (1-f_j) [N_{ij}+\theta(E_i-E_j)]\label{eq_rates2}\end{aligned}$$ where $f_i$ is the occupation probability of state $i$, $f_\alpha(E)=[\exp((E-\mu_\alpha)/k_BT_\alpha)+1]^{-1}$ is the Fermi distribution of reservoir $\alpha$, $N_{ij}=[\exp(|E_j-E_i|/k_BT)-1]^{-1}$ is the probability of having a phonon with energy $|E_j-E_i|$ assisting the hop, and $\theta$ is the Heaviside function. In Eq. , $\gamma_{i\alpha}=\gamma_e\exp(-2x_{i\alpha}/\xi_i)$, $x_{i\alpha}$ denoting the distance of state $i$ from reservoir $\alpha$, and $\gamma_e$ being a constant quantifying the coupling from the localized states in the NW to the extended states in the reservoirs. Usually, $\xi_i\approx\xi(\mu)$ is assumed and the rate $\gamma_{ij}$ in Eq. is simply given by $\gamma_{ij}=\gamma_{ep}\exp(-2x_{ij}/\xi(\mu))$, with $x_{ij}=|x_i-x_j|$ and $\gamma_{ep}$ measuring the electron-phonon coupling. Since this approximation does not hold in the vicinity of the impurity band edges, where the localization lengths vary strongly with the energy, we use a generalized expression for $\gamma_{ij}$ that accounts for the different localization lengths $\xi_i\neq\xi_j$ (see Ref. [@Bosisio20142]).\
By using Eqs. - and imposing charge conservation at each network node $i$, we deduce the $N\,f_i$’s of the $M$ independent NWs. The charge and heat currents flowing from reservoir $\alpha$ to the system can then be calculated as $I^e_{\alpha}=e\,\sum_i I_{\alpha i}$ and $I^Q_{\alpha}=\sum_i I_{\alpha i} (E_i-\mu_{\alpha})$, where $I_{\alpha i}=\Gamma_{\alpha i}-\Gamma_{i\alpha}$ and $e$ is the electron charge. In principle, the heat current $I^Q_P=(1/2)\sum_i I^Q_i$ coming from the phonon bath can be calculated as well but in this work, we only investigated the behavior of the local heat currents $I_{i}^{Q}= \sum_{j} \left(E_j-E_i\right)I^N_{ij}$ with $I^N_{ij}=\Gamma_{ij}-\Gamma_{ji}$. Without loss of generality, we choose the right terminal $R$ as the reference, i.e. we set $\mu_R=\mu$, $T_R=T$ and we impose on the left side $\mu_L=\mu+\delta\mu$, $T_L=T+\delta T$. Using the Onsager formalism, we relate the particle ($I^e_{L}$) and heat ($I^Q_{L}$) currents computed in linear response to the small imposed bias $\delta\mu$ and $\delta T$.[@Callen1985] This allow us to deduce the thermoelectric coefficients $G$, $K^e$ and $S$.
Size Effects {#app_size}
============
We have investigated the effects on the various transport coefficients $G$, $K^e$ and $S$, the power factor $\mathcal{Q}$, and the electronic figure of merit $Z_eT$, of varying the length $L$ of the NWs. The results are shown in Fig. \[fig:sizeeffects\], for three values of the temperatures $k_BT=0.1t,0.5t$ and $1.0t$, and for two configurations corresponding to bulk ($V_g=t$) and edge transport ($V_g=2.5t$). In all cases (except the one for $k_BT=0.1t$ and $V_g=2.5t$, $\text{\small{\color{red}$\bullet$}}$ in Fig. \[fig:sizeeffects\]), electronic transport through the NWs is thermally activated (see Fig.3 in Ref. [@Bosisio20142]) and the results are seen to be essentially size-independent, as expected in the activated regime. In the case identified by $\text{\small{\color{red}$\bullet$}}$ in Fig. \[fig:sizeeffects\] and corresponding to the lowest temperature and the vicinity of the band edge, transport turns out to be achieved by elastic tunneling processes: the electrical conductance becomes size-dependent, which causes the electronic figure of merit $Z_eT$ to decrease roughly as $1/L$. However, being interested in the activated regime and in particular in the regime of temperatures where the power factor is largest ($k_BT\simeq 0.5t$), we can conclude that the size effects on the results shown in this work are completely negligible. Also, we note that the small fluctuations observed especially at the smallest sizes in Fig. \[fig:sizeeffects\] are a consequence of having taken a finite number of parallel NWs (M=150): they would diminish in the limit M$\to\infty$ due to self-averaging.
![(Color online) Behavior of the transport coefficients as a function of the NWs length $L$ (in units of the spacing $a$). Panels show (a) the rescaled electronic contribution to the thermal conductance $K^e$ (in units of $k_Bt/\hbar$), (b) the thermopower $S$ (in units of $k_B/e$), (c) the rescaled power factor $\mathcal{Q}$ (in units of $k_B^2/\hbar$) and (d) the electronic figure of merit $Z_eT$. In all the four panels, data are plotted for $k_BT=0.1t$ (circles), $k_BT=0.5t$ (squares) and $k_BT=t$ (rhombus), in the case of bulk transport ($V_g=t$, empty symbols) and edge transport ($V_g=2.5t$, full symbols). Lines are guides to the eye. Other parameters are fixed to $W=t$ and $\gamma_e=\gamma_{ep}=t/\hbar$.[]{data-label="fig:sizeeffects"}](KSQZT_vs_L.pdf){width="\columnwidth"}
On the dependence on the couplings $\gamma_e$ and $\gamma_{ep}$ {#app:depgammaeep}
===============================================================
In this section, we investigate how the transport coefficients $G$, $K^e$ and $S$, the power factor $\mathcal{Q}=S^2G$ and the electronic figure of merit $Z_eT=S^2GT/K^e$ are modified upon varying the couplings $\gamma_e$ and $\gamma_{ep}$ of the localized states with the electrodes and the phonon bath, respectively. We introduce the notation $\alpha\equiv \gamma_{ep}/\gamma_{e}$. We first notice that if $\alpha$ is kept fixed, the electrical conductance $G$ and the electronic thermal conductance $K^e$ are strictly proportional to $\gamma_e$, while the thermopower $S$ is independent of it. This behavior is a direct consequence of the formulation of the random resistor network problem and can be seen at the stage of writing the equations (see Ref. [@Bosisio20142]), before solving them numerically. Therefore, for any fixed $\alpha$, $\mathcal{Q}/\gamma_e$ and $Z_eT$ are necessarily independent of the choice of $\gamma_e$. We thus find that $G/\gamma_{e}$, $K^e/\gamma_{e}$, $S$, $\mathcal{Q}/\gamma_e$ and $Z_eT$ are functions of the single parameter $\alpha$, and not of the couple of parameters $\gamma_e$ and $\gamma_{ep}$ separately. Those functions are plotted in Fig. \[fig:QZT\_game\_gamep\_full\] for two different temperatures. The conductances, the power factor and the figure of merit increase with $\alpha$ (as long as lack of phonons is a limiting factor to transport through the NWs), while the thermopower decreases. All of them tend to saturate for $\alpha\gtrsim 1$. This shows us, *inter alia*, that $\mathcal{Q}/\gamma_e$ and $Z_eT$ are essentially independent of $\gamma_e$ and $\gamma_{ep}$ if $\gamma_{ep}\gtrsim \gamma_e$ and that they only deviate slowly from this limit if $\gamma_{ep}< \gamma_e$. Such a robustness of $\mathcal{Q}/\gamma_e$ and $Z_eT$ to variations of $\gamma_e$ and $\gamma_{ep}$ reinforces the impact of the results shown in this work.
{width="\textwidth"}
Estimation of the phonon thermalization length {#app_lambdaph}
==============================================
We show in Fig. \[fig:HotSpots\_Vg2.25\_kT0.05\] an example of the map of the *raw* heat currents $I^Q_i$ locally exchanged between the NWs and the substrate \[see Eq. \]. We see that the $I^Q_i$’s fluctuate between positive and negative values at random positions of the substrate, and that no net effect emerges. As discussed in Sec. \[sec\_hotspots\], the formation of the hot and cold strips is a process which becomes visible only upon summing in a single term $\mathcal{I}^Q_{x,y}$ all the contributions $I^Q_i$ coming from states $i$ located within an area $\Lambda_{ph}\times\Lambda_{ph}$ around the point of coordinates $(x,y)$. $\Lambda_{ph}$, which represents the thermalization length of the substrate, is given by the inelastic phonon mean free path: this quantity may be different for different phonon wavelengths, and while it does not change much around room temperatures, it can vary significantly at lower temperatures. It is possible to relate $\Lambda_{ph}$ to the *dominant* phonon wave length [@Pohl2002] as $\Lambda_{ph}=300 \lambda_{ph}^{dom}$, where the coefficient 300 is for SiO$_2$ and may be different for other materials. This allows the calculation of the thermalization length $\Lambda_{ph}$, once $\lambda_{ph}^{dom}$ is known. According to Refs. [@Klitsner1987; @Ziman1996], the latter can be estimated as $$\lambda_{ph}^{dom}\simeq\frac{h v_s }{4.25 k_BT},
\label{eq:lambdaph}$$ where $h$ is the Planck constant. Taking $v_s=5300\,\mathrm{m/s}$ the sound velocity in SiO$_2$,[@Ziman1996] we can easily deduce $\lambda_{ph}^{dom}\simeq 0.2\,\mathrm{nm}$ from which $\Lambda_{ph}\simeq 60\,\mathrm{nm}$ at room temperature $T=300\,\mathrm{K}$. Values of $\Lambda_{ph}$ at other (not vanishing) temperatures follow immediately from the temperature dependence in Eq. . We shall stress that the real values of $\Lambda_{ph}$ may differ from our prediction by a small numerical factor, which however is not important within our qualitative approach. To convert these lengths in the units used in Sec. \[sec\_hotspots\], we assume the average distance between localized states $a\approx 3.2\,\mathrm{nm}$ in highly doped silicon NWs, which together with $t/k_B\approx 150 K$ allows us to estimate for example $\Lambda_{ph}\approx 75 a$ at $T=0.5t/k_B=75\,\mathrm{K}$.
![(Color online) Map of the local heat currents $I^Q_i$ exchanged between the NWs and the substrate at each NWs site $i=(i_x,i_y)$, in units of $10^{-7}t^2/\hbar$, for $k_BT=0.05t$ and $V_g=2.25t$. The horizontal coordinate is the position along the NWs, while the vertical one labels each NW. The presence of hot and cold strips is hidden by the fluctuations. They emerge when the raw $I^Q_i$ data are summed up within areas of size $\Lambda_{ph}\times \Lambda_{ph}$. Parameters: $M=150$, $L=450a$, $W=t$, $\gamma_{e}=\gamma_{ep}=t/\hbar$ and $\delta\mu=10^{-5}t$.[]{data-label="fig:HotSpots_Vg2.25_kT0.05"}](HotSpots_app.pdf){width="0.8\columnwidth"}
[^1]: We estimated $t$ by comparing the band width $4t+W$ in our model to the typical width of the impurity band in highly doped Silicon NWs (see for instance Ref. [@Salleh2011]). Note that the NWs are then depleted by field effect.
[^2]: The use of the 1D model is justified at a semi-quantitative level if the nanowire diameter is smaller than the Mott hopping length $L_M$ (the typical length of an electron hop along the nanowire).
[^3]: Practically, we map the 2D parallel NW array onto a square grid, and for each square of size $\Lambda_{ph}^2$ we calculate the net heat current entering the NWs. For better visibility, data are then smoothed (with a standard Gaussian interpolation) to produce the heat map shown in Fig. \[fig:hotspots\].
[^4]: Here $L_M\approx 10.6a$ and $\Delta\approx 2.4t$ for $k_BT=0.25t$, while $L_M\approx 7.5a$ and $\Delta\approx 3.3t$ for $k_BT=0.5t$.
[^5]: Data shown in Fig. \[fig:hotspots\] result from numerical simulations run for a set of 150 1D NWs (of length $1500a$) separated from each other by a distance $15a$. They are expected to describe the physics of realistic arrays made of 150 NWs covering an area of width $150\times 15a\approx 7.2\mu\mathrm{m}$ and length $1500a\approx 5\mu\mathrm{m}$, taking again $a\approx 3.2\,\mathrm{nm}$. For instance, 150 NWs with $10\,\mathrm{nm}$ diameter and $20\%$ packing density. Other configurations could be considered as well, as long as the NW diameter is small enough for the 1D model to make sense and the packing density does not exceed the typical values reachable experimentally.
|
---
abstract: 'We report detection of nuclear magnetic resonance (NMR) using an anisotropic magnetoresistive (AMR) sensor. A “remote-detection” arrangement was used, in which protons in flowing water were pre-polarized in the field of a superconducting NMR magnet, adiabatically inverted, and subsequently detected with an AMR sensor situated downstream from the magnet and the adiabatic inverter. AMR sensing is well suited for NMR detection in microfluidic “lab-on-a-chip” applications.'
author:
- 'F. Verpillat'
- 'M. P. Ledbetter'
- 'D. Budker'
- 'S. Xu'
- 'D. Michalak'
- 'C. Hilty'
- 'L.-S. Bouchard'
- 'S. Antonijevic'
- 'A. Pines'
title: Detection of nuclear magnetic resonance with an anisotropic magnetoresistive sensor
---
The three essential elements of a nuclear-magnetic-resonance (NMR) or magnetic-resonance-imaging (MRI) experiment, nuclear-spin polarization, encoding, and detection can be spatially separated, which is referred to as remote detection of NMR or MRI [@Mou2003]. One important potential advantage of this approach is that encoding and detection can occur in a near-zero magnetic field; however, conventional inductive detection has poor sensitivity at low frequencies, necessitating the use of alternative techniques for detection. Superconducting quantum-interference devices (SQUIDs) [@Won2002] and alkali-vapor atomic magnetometers [@Xu2006PNAS] have been successfully utilized for this purpose. In this note, we report the use of another novel technology – that of anisotropic magneto-resistive (AMR) sensors [@Tsy2001] – for a remote-NMR experiment. Although less sensitive than SQUIDs or atomic magnetometers (including even the miniature chip-scale atomic magnetometers [@Sch2007]), the all-solid-state AMR sensors do not require cryogenics or vapor-cell heating, and may be particularly fit to microfluidic applications because the sensors are small, inexpensive and can be manufactured as arrays for spatial sensitivity.
![Experimental setup. Water is pre-polarized by flowing it through the magnet; the magnetization is periodically inverted by passing the liquid through the adiabatic fast passage (AFP) module; the magnetization is detected with a gradiometer consisting of two AMR sensors.[]{data-label="setup"}](schema2.eps "fig:"){width="3.5"}\
The experimental setup is shown in Fig. \[setup\]. Tap water, pre-polarized by flowing through a Bruker 17 Tesla magnet, flows through an adiabatic-inversion region, where its polarization is periodically reversed, after which it proceeds to flow past an AMR detector. The adiabatic polarization inverter incorporates a set of coils in anti-Helmoltz configuration to supply a gradient of $\rm
B_z$. A second set of Helmoltz coils is used to apply a $5.5~{\rm
kHz}$ oscillating field in the x direction, resonant with the protons’ Larmor frequency in the center of the inverter. When the oscillating field is on, as the water flows through the device, its magnetization is adiabatically reversed. Switching the oscillating field on and off results in magnetization either parallel or anti-parallel to the bias field. After the adiabatic inverter, the water flows into the detection region consisting of a $0.5-{\rm
cm^3}$ glass ball adjacent to a pair of Honeywell HMC1001 AMR sensors arranged as a gradiometer in order to cancel the common-mode magnetic-field noise. The active part of the sensor is a thin film with an area of about ${\rm 1.5~mm\times 1.5~mm}$ packaged in a chip with dimensions ${\rm 10~mm\times 3.9~mm\times 1.5~mm}$. The manufacturer specifications of the HMC1001 sensor give a single-shot resolution of $40~{\rm \mu G}$ with a read-out rate of 1 kHz, corresponding to a sensitivity of about $1.8~{\rm \mu G/\sqrt{Hz}}$ assuming white noise. In our experimental setup, we realized a sensitivity of about $2.7~{\rm \mu G/\sqrt{Hz}}$ at 20 Hz (per sensor), however the low-frequency performance was considerably worse, on the order of ${\rm 40~\mu G/\sqrt{Hz}}$ at 1 Hz, necessitating long signal averaging. The detection region is housed inside a single layer of magnetic shielding with open ends. The water-carrying tube was 1/16” i.d. and the flow rate was $3.8 ~{\rm
cm^3/s}$, corresponding to an average speed of water of $\approx
2~{\rm m/s}$. The average travel time from the magnet to the inverter is $\approx 1.5~\rm s$, and it is $\approx 0.5~\rm s$ from the inverter to the detector.
![Upper traces – magnetic field detected with the $0.3-{\rm Hz}$ modulation frequency and a fit to a sinusoidal function. Lower trace – the on-off pattern of the adiabatic inverter. The signal was averaged over 20 min.[]{data-label="signal"}](signal_fit.eps){width="3.5"}
Data were recorded on a digital oscilloscope, averaging for about 20 min for modulation frequencies in the range $0.3~{\rm Hz}$ to $1.7~{\rm Hz}$. There was considerable low-frequency drift in the signal (due to either ambient field drifting or intrinsic drift in the magnetometer) and hence we subtracted from the raw data a slow-varying background approximated by a 3rd-order polynomial. The resulting signal for a modulation frequency of $0.3~{\rm Hz}$ is shown in Fig. \[signal\]. Neglecting fluid mixing in the transfer tube and detection ball, one would expect that the signal should be a square wave for low modulation frequencies. However, considerable mixing produced signals well approximated (above the cutoff frequency) by a sinusoid, as indicated by the dashed line in Fig. \[signal\].
All the data were fit to sinusoidal profiles and the resulting amplitudes are shown as a function of frequency in Fig. \[ampl\]. The rapid drop in amplitude is due to mixing of the magnetization as it propagates from the AFP device through the detection ball, effectively integrating the magnetization. A simple model for the spectral response of the system can be obtained in analogy to a low-pass RC filter where the frequency dependence of the signal is $S_0=\alpha M_0/\sqrt{1+(f/f_c)^2}$. Here $\alpha$ is a proportionality constant depending on geometry relating the magnetic field at the sensor to the magnetization of the sample and $f_c$ is a cutoff frequency. Overlaying the data in Fig. \[ampl\] is a fit to this model function with $\alpha M_0=67~{\rm \mu G}$ and $f_c=0.2~{\rm Hz}$.
Significant improvement in sensitivity and bandwidth can be expected in future work. We suspect that the low-frequency performance of our AMR sensor was limited by the open-ended magnetic shields used in the experiment. Optimization of geometry will lead to substantial gains in both sensitivity to nuclear magnetization (by reducing the distance from the sample to the magnetometer), as well as improved bandwidth (by minimizing the volume of the detected water so that less mixing would occur at high frequencies). In principle the detected volume could be a microfluidic channel built into the sensor package, similar to the construction in Ref. [@Pek2004] where magnetic microparticles were detected. The higher bandwidth has the additional benefit of moving the signal above the $1/f$ knee of the sensor. Higher sensitivity and spatial resolution may also be achieved by using an array of sensors as in Ref. [@Wood2005].
![Amplitude of the modulated water signal and a fit to an RC-filter transfer function.[]{data-label="ampl"}](signal_freq.eps){width="3.5"}
We have demonstrated, to our knowledge, the first detection of NMR signals using an anisotropic magneto-resistive sensor. The technique may be useful for spatial localization (MRI), relaxometry, diffusometry or spin labeling in chemical analysis [@Anw2007]. With anticipated future advances in AMR sensors, as well as in related solid-state technologies such as magnetic tunnel junctions, solid-state chip-scale magnetometers may eventually reach the picotesla sensitivity level [@Fer2006]. Incorporation of built-in microfluidic channels at the chip level will allow the construction of dedicated “lab-on-a-chip” devices. With these improvements, room temperature, solid state devices appear to be an inexpensive and robust alternative for detection of both in-situ and remote-detection NMR/MRI without cryogenics.
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abstract: 'The $t$-channel contribution to the difference of electromagnetic polarizabilities of the nucleon, $(\alpha-\beta)^t$, can be quantitatively understood in terms of a $\sigma$-meson pole in the complex $t$-plane of the invariant scattering amplitude $A_1(s,t)$ with properties of the $\sigma$ meson as given by the quark-level Nambu–Jona-Lasinio model (NJL). Equivalently, this quantity may be understood in terms of a cut in the complex $t$-plane where the properties of the $\sigma$ meson are taken from the $\pi\pi\to\sigma\to\pi\pi$, $\gamma\gamma\to\sigma\to\pi\pi$ and $N\bar{N}\to \sigma \to \pi\pi$ reactions. This equivalence may be understood as a sum rule where the properties of the $\sigma$ meson as predicted by the NJL model are related to the $f_0(600)$ particle observed in the three reactions. In the following we describe details of the derivation of $(\alpha-\beta)^t$ making use of predictions of the quark-level NJL model for the $\sigma$-meson mass.'
---
\
Martin Schumacher\
[email protected]\
Zweites Physikalisches Institut der Universität Göttingen, Friedrich-Hund-Platz 1\
D-37077 Göttingen, Germany
[**PACS.**]{} 11.30.Rd Chiral symmetries – 11.55.Fv Dispersion relations – 11.55.Hx Sum rules – 13.60Fz Elastic and Compton scattering – 14.65.Bt Light quarks – 14.70.Bh Photons
Introduction
============
The $\sigma$ meson is an indispensable supplement of the pion [@schwinger57]. In terms of non-strange quarks, the argument is that four $q\bar{q}$ states should correspond to four mesons where the neutral members of the $({\mbox{\boldmath $\pi$}},\sigma)$ isospin quartet have the flavor structures $|\pi^0\rangle=(u\bar{u}-d\bar{d})/{\sqrt{2}}$ and $|\sigma\rangle=(u\bar{u}+d\bar{d})/{\sqrt{2}}$. According to its flavor structure the $\sigma$ meson is a scalar-isoscalar particle with relative angular momentum $L=1$ and spin $S=1$ of the two quarks coupled to $J=0$. Therefore, it has the quantum numbers of the vacuum and, correspondingly, the $\sigma$-field entering into the linear $\sigma$ model (L$\sigma$M) has a nonzero vacuum expectation value $\langle 0|\sigma|0\rangle \neq 0$. This leads to a mass $m_\sigma$ of the $\sigma$ meson which is not quantitatively predicted by the L$\sigma$M, but quite naturally follows from the quark-level Nambu–Jona-Lasinio model (NJL), by adjusting the predictions of this model to the pion decay constant $f_\pi$, the average current-quark mass $m_0=\frac12 (m_u+m_d)$ and to the pion mass $m_\pi$. In this way parameters can be avoided which otherwise would not be precisely determined. On the other hand, the $\sigma$ meson showing up as a broad resonant intermediate state in reactions in which two pions are involved [@eidelman04] may be understood as a $(u\bar{u}+d\bar{d})/{\sqrt{2}}$, $1^3P_0$ core state in a confining potential, coupled to a $(\pi^+\pi^--\pi^0\pi^0+\pi^-\pi^+)/{\sqrt{3}}$ di-pion state in the continuum, where the two pions are in a relative $S$-state with isospin $I=0$. This coupling lowers the average mass as compared to the confined core state and leads to a broad mass distribution. This dual aspect of the $\sigma$ meson leads to two different predictions for the $t$-channel contribution, $(\alpha-\beta)^t$, of the difference of the electric and magnetic polarizability of the nucleon which have been shown [@levchuk05] to agree with each other and with the experimental result. In the present work we give details of the derivation of $(\alpha-\beta)^t$ making use of predictions of the NJL model for the $\sigma$-meson mass.
The quantity $(\alpha-\beta)^t$ predicted from a $\sigma$-meson pole
=====================================================================
Effective field theories are an excellent tool to adapt properties of QCD to the low-energy regime. In applying these effective field theories to phenomena like the polarizability of the nucleon, care has to be taken to find out what aspects of the phenomenon under consideration can be reproduced and where other theoretical tools are more appropriate.
Outline of the problem and arguments in favor of the NJL model
--------------------------------------------------------------
In case of Compton scattering [@lvov93a; @lvov97; @babusci98; @drechsel03; @wissmann04; @schumacher05a; @schumacher05b] two types of degrees of freedom (d.o.f.) of the nucleon have to be taken into account which may be termed $s$-channel d.o.f. and $t$-channel d.o.f. The $s$-channel d.o.f. are those degrees of freedom of the nucleon which also show up in photoabsorption experiments on the nucleon. The main examples are the nonresonant $E_{0^+}$ channel of pion photoproduction corresponding to the “pion cloud”, and the $\Delta$ resonance. The nonresonant $E_{0^+}$ channel is of $E1$ multipolarity and contributes only about $40\%$ of the electric polarizability $\alpha$, in partial contradiction to the frequently stated belief that the “pion cloud” dominates the electric polarizability. The results obtained [@levchuk05] for the $s$-channel contributions are $\alpha^s_p=4.5\pm 0.5$ and $\alpha^s_n=5.1\pm 0.6$ (in units of $10^{-4}{\rm
fm}^3$) for the proton and neutron, respectively, to be compared with the corresponding experimental values $\alpha^{\rm exp}_p=12.0\pm 0.6$ and $\alpha^{\rm exp}_n=12.5\pm 1.7.$ The $\Delta$ resonance is the origin of the by far dominating part of the electromagnetic polarizabilities. It is of $M1$ multipolarity and, therefore, a strong source of paramagnetic polarizability. Here the numbers are $\beta^s_p=9.5\pm 0.5$ and $\beta^s_n=10.1\pm 0.6$ to be compared with $\beta^{\rm exp}_p=1.9\pm 0.6$ and $\beta^{\rm exp}_n=2.7 \pm
1.8$. Apparently, there exists a strong diamagnetic polarizability which cannot have its origin from the $s$-channel d.o.f. For illustration, the $s$-channel d.o.f. are shown in Fig. \[s-channel\].
![ $s$-channel d.o.f.: Photoabsorption cross section separated into multipoles[]{data-label="s-channel"}](figures/absorption-2.eps){width="0.5\linewidth"}
We see the strong $P_{33}(1232)$ ($\Delta$) resonance line and two other prominent lines corresponding to the $D_{13}(1520)$ and the $F_{15}(1680)$ resonances which are sources of $E1$ and $E2$ strength, respectively. The main source of $E1$ strength is due to the nonresonant part of the cross section, which at the lower energies is due to $1\pi$ photoproduction, and due to $2\pi$ $\cdots$ photoproduction at higher energies.
The additional $t$-channel d.o.f. are required by Mandelstam analyticity [@hearn62]. The invariant amplitudes $A_i(s,t)$ are analytical functions of the two variable $s$ and $t$ and, therefore, the singularities of the $t$-channel as well as those of the $s$-channel have to be taken into account. The $t$-channel d.o.f. may be identified with a $(\pi^0,\sigma)$ doublet in the intermediate state where the two mesons are coupled to two photons on the one side and to constituent quarks on the other. In terms of linear polarization the two cases differ from each other by the fact that for the $\pi^0$ meson the directions of linear polarization are perpendicular whereas for the $\sigma$ meson they are parallel. The $\pi^0$ meson corresponds to a pole in the $t$-plane which contributes the dominant part, $\gamma^t_\pi$, of the backward spin polarizability $\gamma_\pi$. The $\sigma$ meson contributes the dominant part, $(\alpha-\beta)^t$, of the difference of electromagnetic polarizabilities $(\alpha-\beta)$. As a broad mass distribution, the $\sigma$ meson corresponds to a cut in the $t$-plane whereas as a particle with a definite mass $m_\sigma$ it corresponds to a pole in analogy to the $\pi^0$ meson case.
As far as effective field theories are concerned only those versions are of interest which contain the $\sigma$ meson explicitly as a particle. This means that the quark-level linear $\sigma$ model (L$\sigma$M) and the quark-level Nambu–Jona-Lasinio model (NJL) are candidates for representing the $t$-channel d.o.f. whereas the nonlinear $\sigma$ model and chiral perturbation theory may be disregarded in connection with the present purpose. The Lagrangian of the L$\sigma$M consistently describes the mechanism of spontaneous symmetry breaking but does not have the capability of predicting the $\sigma$ meson mass $m_\sigma$ on an absolute scale [@gellmann60; @dealfaro73]. This is different in the NJL model [@nambu61; @lurie64; @eguchi76; @vogl91; @klevansky92; @hatsuda94; @bijnens96; @thomas00] which describes dynamical symmetry breaking and, thus, predicts a definite $\sigma$ meson mass, $m_\sigma$, instead of a broad mass distribution. This is the reason why we consider the NJL model as the appropriate candidate for our present study. However, the application of the NJL should be restricted to predictions in connection with the $\sigma$-pole contribution. Applications to other aspects of the electromagnetic polarizabilities are not expected to lead to meaningful results.
The mass $m_\sigma$ of the $\sigma$ meson \[subsection2-2\]
-----------------------------------------------------------
The Lagrangian of the NJL model has been formulated in two equivalent ways [@lurie64; @eguchi76; @vogl91; @klevansky92] $$\begin{aligned}
&&{\cal L}_{\rm NJL}=\bar{\psi}(i{\mbox{$/\!\!\!\partial$}}-m_0) \psi
+ \frac{G}{2}[(\bar{\psi}\psi)^2+(\bar{\psi}i\gamma_5{\mbox{\boldmath$\tau$}}\psi)^2],
\label{NJL1}\\
{\rm and}\hspace{1cm} && \nonumber\\
&&{\cal L'}_ {\rm
NJL}=\bar{\psi}i{\mbox{$/\!\!\!\partial$}}\psi-g\bar{\psi}(\sigma+i\gamma_ 5
{\mbox{\boldmath$\tau$}}\cdot{\mbox{\boldmath $\pi$}})\psi-\frac12\delta\mu^2(\sigma^2+{\mbox{\boldmath $\pi$}}^2)+\frac{gm_0}{G}
\sigma,
\label{NJL2}\\
{\rm where} \hspace{1cm}&&\nonumber\\
&&G=g^2/\delta\mu^2\quad \mbox{and}\quad \delta\mu^2=(m^{\rm cl}_\sigma)^2.
\label{grelations}\end{aligned}$$
![a) Four-fermion theory tadpole diagram, b) bosonized tadpole diagram [@lurie64].[]{data-label="tadpolegraphs"}](figures/tadpole.eps){width="0.5\linewidth"}
Eq. (\[NJL1\]) describes the four-fermion version of the NJL model and eq. (\[NJL2\]) the bosonized version. The quantity $\psi$ denotes the spinor of constituent quarks with two flavors. The quantity $G$ is the coupling constant of the four-fermion version, $g$ the Yukawa coupling constant and $\delta\mu$ a mass parameter entering into the mass counter-term of eq. (\[NJL2\]). The coupling constants $G$, $g$ and the mass parameter $\delta\mu$ are related to each other and to the $\sigma$ meson mass in the chiral limit (cl), $m^{\rm cl}_\sigma$, as given in (\[grelations\]). The relation $\delta\mu^2=(m^{\rm
cl}_\sigma)^2$ can easily be derived by applying spontaneous symmetry breaking to $\delta\mu$ in analogy to spontaneous symmetry breaking predicted by the L$\sigma$M for the mass parameter $\mu$ entering into this model [@dealfaro73]. In the chiral limit ($m_0\to 0$) the version of Eq. (\[NJL2\]) contains the spinor-dependent term in the same way as the L$\sigma$M, whereas the major part of the bosonic $({\mbox{\boldmath $\pi$}},\sigma)$ dependent terms of the L$\sigma$M are absent. This means that with respect to the spinor dependent terms the L$\sigma$M and the NJL model are equivalent whereas only a truncated version of the bosonic part of the L$\sigma$M is produced. The terms in Eqs. (\[NJL1\]) and (\[NJL2\]) containing the average current-quark mass $m_0$ describe explicit symmetry breaking and can be shown to be equivalent. The insight that the L$\sigma$M and the NJL model are basically equivalent dates back to the 1960s and 1970s. Fig. \[tadpolegraphs\] shows the four-fermion theory tadpole diagram and the bosonized tadpole diagram. The underlying idea has been worked out in detail by Eguchi [@eguchi76], by Vogl and Weise [@vogl91] and Delbourgo and Scadron [@delbourgo95]. Both versions can be exploited to make a prediction for $m_\sigma$ on an absolute scale. The NJL model faces the problem that use is made of integrals in momentum space which diverge in the infinite momentum limit. To overcome this problem two different regularization schemes have been developed. The regularization through a cut-off momentum $\Lambda$ restricts the evaluation of the integrals to the low-momentum region. Dimensional regularization treats the integrals without a cut off but makes use of the fact that the difference between two diverging integrals is finite.
### Four-fermion version of the NJL model with regularization through a cut-off parameter $\Lambda$
Using diagrammatic techniques the following equation may be found [@klevansky92; @hatsuda94] $$\begin{aligned}
&&M^*=m_0+ 8\, i\, G N_c \int^{\Lambda}\frac{d^4 p}{(2\pi)^4}
\frac{M^*}{p^2-M^{*2}},
\label{gapdiagram}\\
&&f^2_\pi = -4\,i\,N_cM^{*2} \int^{\Lambda}\frac{d^4p}{
(2\pi)^4}\frac{1}{(p^2-M^{*2})^2},
\label{fpiexpress}\\
&&m^2_\pi=-\frac{m_0}{M^*}\frac{1}{4\,i\,G N_c I(m^2_\pi)},\quad
I(k^2)=\int^{\Lambda}\frac{d^4p}{(2\pi)^4}\frac{1}{[(p+\frac12 k)^2-M^{*2}][
(p-\frac12 k)^2-M^{*2}]}.
\label{pionmass-2}\end{aligned}$$ The expression given in (\[gapdiagram\]) is the gap equation with $M^*$ being the mass of the constituent quark with the contribution $m_0=(m _u+m_d)/2$ of the current quarks included and $N_c=3$ being the number of colors. Eq. (\[fpiexpress\]) represents the pion decay constant having the experimental value $f_\pi=(92.42\pm 0.26)$ MeV [@eidelman04]. The expression given in (\[pionmass-2\]) is equivalent to the Gell-Mann–Oakes–Renner relation [@gellmann68] when formulating (\[pionmass-2\]) in the chiral limit. Using (\[gapdiagram\]) – (\[pionmass-2\]) it is possible to calculate the quantities $M^*$, $f_\pi$ and $m_\pi$ simultaneously and to adjust the parameters $G$ and $\Lambda$ in such a way that the experimental values for $f_\pi$, $m_\pi$ and $m_0$ are reproduced. It is apparent that this procedure leaves no room for an unknown parameter so that the predicted value for $m_\sigma$ is model independent except, of course, for the general use of the theoretical frame provided by the NJL model. Numerical calculations of the type outlined above have been carried out by Hatsuda and Kunihiro [@hatsuda94] applying RPA techniques. The result obtained is $$m_\sigma \simeq 668 \,\,\,{\rm MeV}
\label{numericalresult-2}$$ where $m_\sigma\simeq 2M^*$ has been applied.
### Bosonized NJL model or dynamical L$\sigma$M with dimensional regularization
A second way to predict $m_\sigma$ on an absolute scale introduced by Delbourgo and Scadron [@delbourgo95] is obtained by exploiting Eqs. (\[NJL2\]) and (\[grelations\]). The starting point [@delbourgo95] are representations of the pion decay constant and the gap equation for the constituent-quark mass, M, in the chiral limit $$\begin{aligned}
&&f^{\rm cl}_\pi=-4iN_cgM\int \frac{d^4p}{(2\pi)^4}\frac{1}{(p^2-M^2)^2},
\label{fpicl}\\
&&M=-\frac{8 i N_c g^2}{(m^{\rm cl}_\sigma)^2}
\int\frac{d^4 p }{(2\pi)^4}\frac{M}{p^2-M^2}.
\label{LSMandNJL-2}\end{aligned}$$ The third equation, (\[pionmass-2\]), for the pion mass $m_\pi$ is not needed because this quantity is equal to zero in the chiral limit, $m^{\rm cl}_\pi=0$. Except for a strict application of the chiral limit, these equations differ from those in (\[gapdiagram\]) and (\[fpiexpress\]) by the fact that use has been made of the Goldberger-Treiman relation for the chiral limit $$gf^{\rm cl}_\pi=M
\label{goldtrei}$$ in (\[fpicl\]) and by replacing $G$ by $-g^2/(m^{\rm cl}_\sigma)^2$ in (\[LSMandNJL-2\]). The different signs in front of the integrals in (\[gapdiagram\]) and (\[LSMandNJL-2\]) follow from the fact that different regularization schemes are applied in the two cases [@delbourgo95].
Then, making use of the identity (dimensional regularization [@thomas00; @delbourgo95]) $$-i\frac{(m^{\rm cl}_\sigma)^2}{16N_cg^2}=
\int\frac{d^4 p}{(2\pi)^4}\left[\frac{M^2}{(p^2-M^2)^2}-
\frac{1}{p^2-M^2}\right]
=-\frac{iM^2}{(4\pi)^2}
\label{identity}$$ we arrive at $$(m^{\rm cl}_\sigma)^2=\frac{N_c g^2 M^2}{\pi^2},
\label{sigmamasssquared}$$ and with $m^{\rm cl}_\sigma=2M$ at $$g=g_{\pi qq}=g_{\sigma qq}= 2\pi/\sqrt{N_c}=3.63.
\label{coupling}$$ The $\sigma$-meson mass corresponding to this coupling constant is $$m_\sigma= 666.0\,\,\,{\rm MeV},
\label{sigmamassfinal-2}$$ where use has been made of $m^2_\sigma= (m^{\rm cl}_\sigma)^2 + m^2_\pi$, $f^{\rm cl}_\pi=89.8$ MeV [@nagy04], $M=325.8$ MeV and $m_\pi= 138.0$ MeV. It is satisfying to note that the results given in (\[numericalresult-2\]) and (\[sigmamassfinal-2\]) are in an excellent agreement with each other. Since the procedures to arrive at $m_\sigma$ are quite different in the two cases the good agreement of the two results gives us confidence that value obtained for $m_\sigma$ is on a stable basis.
The two-photon decays of the $f_0(980)$, the $\pi^0$ and the $\sigma$ meson \[Thetwo-photon\]
---------------------------------------------------------------------------------------------
For mesons $P$ having the constituent-quark structure $$|q\bar{q}\rangle=\frac{a|u\bar{u}\rangle + b |d\bar{d}\rangle
+ c |s\bar{s}\rangle}{\sqrt{a^2+b^2+c^2}}
\label{quarkstructure}$$ the two-photon amplitude is given in the generic form $$|M(P\to \gamma\gamma)|= \frac{\alpha_e}{\pi f_P}
N_c\,\sqrt{2}\,\frac{a\, e^2_u+ b\, e^2_d
+ c\,(\hat{m}/m_s)\,e^2_s}{\sqrt{a^2+b^2+c^2}},
\label{twophotonamplitude}$$ where $\alpha_e=1/137.04$, $f_P$ the decay constant of the meson $P$ (see e.g. [@donoghue85; @cooper88]) and $\hat{m}/m_s\approx 1/\sqrt{2}$ the ratio of light and strange constituent quark masses. However, we can use $$f_P \simeq f_\pi
\label{pidecayconst}$$ without a major loss of precision. The reason is that $f_P$ does not depend on the flavor wave function of the meson as can be seen in (\[gapdiagram\]). Small deviations from (\[pidecayconst\]) only occur when there is a strange-quark content in the meson as in case of $\eta$, $\eta'$ and $f_0(980)$, because of the larger current-quark mass of the strange quark. This leads us to the following relations $$\begin{aligned}
&|M(f_0(980)\to\gamma\gamma)|&=\frac{\alpha_e}{\pi f_\pi}N_c \left[
\left(-\frac13\right)^2 \right]=\frac13
\frac{\alpha_e}{\pi f_\pi},\label{width2}\\
&|M(\sigma\to\gamma\gamma)|&=\frac{\alpha_e}{\pi f_\pi}N_c \left[
\left(\frac23\right)^2 + \left(-\frac13\right)^2 \right]=\frac53
\frac{\alpha_e}{\pi f_\pi},\label{width3}\\
&|M(\pi^0\to\gamma\gamma)|&=\frac{\alpha_e}{\pi f_\pi}N_c \left[
\left(\frac23\right)^2 - \left(-\frac13\right)^2 \right]=
\frac{\alpha_e}{\pi f_\pi},\label{width4}\\
&\Gamma_{f_0(980)\to\gamma\gamma}&=\frac{m^3_{f_0(980)}}{64\pi}
|M(f_0(980)\to\gamma\gamma)|^2= 0.33\,\, {\rm keV},\label{width6}\\
&\Gamma_{\sigma\to\gamma\gamma}&=\frac{m^3_\sigma}{64\pi}
|M(\sigma\to\gamma\gamma)|^2=2.6\,\, {\rm keV},\label{width7}\\
&\Gamma_{\pi^0\to\gamma\gamma}&=\frac{m^3_{\pi^0}}{64\pi}
|M(\pi^0\to\gamma\gamma)|^2=7.73\times10^{-3}\,\, {\rm keV},\label{width8}\end{aligned}$$ where $f_P=f_\pi$ is used in all three cases. For the $\sigma$ meson we use the “model independent” $\sigma$ mass, $m_\sigma=666$ MeV, determined in the previous subsection using the method of [@delbourgo95]. The pion decay constant is $f_\pi= (92.43\pm 0.26)$ MeV. For the $\sigma$ meson we then obtain $\Gamma_{\sigma\to\gamma\gamma}=2.6 \,\,\,\mbox{keV}$ which agrees with the experimental result of Boglione and Pennington [@boglione99] $\Gamma_ {\sigma\to\gamma\gamma}=(3.8\pm 1.5)$ keV within the errors. In (\[width3\]) it has been assumed that the coupling of the two photons to the $\sigma$ meson proceeds only through its non-strange quark content and that there are no additional contributions due to meson loops as obtained in the frame of the L$\sigma$M [@beveren02; @scadron04]. From a theoretical point of view this neglect of meson-loop contributions is justified through the use of the NJL model where such additional contributions are absent. For sake of completeness we note that for the $\pi^0$ meson the prediction is $\Gamma_{\pi^0\to\gamma\gamma}=7.73\times 10^{-3}$ keV to be compared with the experimental value $\Gamma_{\pi^0\to\gamma\gamma}=(7.74\pm 0.55)\times 10^{-3}$ keV obtained from the mean lifetime $\tau_{\pi^0}=(8.4\pm 0.6)\times 10^{-17}$ s and the branching ratio $\pi^0\to\gamma\gamma$ of $(98.798\pm 0.032)\%$ [@eidelman04]. We see that for the $\pi^0$ meson as well as for its chiral partner, the $\sigma$ meson, the coupling to two photons may be understood as proceeding through their $q\bar{q}$ internal structures. It is remarkable to note that the prediction for the two-photon width of $f_0(980)$ based on a $|\bar{s}s\rangle$ quark structure is in agreement with the experimental value $\Gamma_{f_0(980)\to\gamma\gamma}=(0.39^{+0.10}_{-0.13} )$ keV given by the particle data group PDG [@eidelman04] and also with $\Gamma_{f_0(980)\to\gamma\gamma}=(0.28^{+0.09}_{-0.13} )$ keV given by Boglione and Pennington [@boglione99].
There was a longstanding discussion whether or not the $q\bar{q}$ structure of scalar mesons with masses below 1 GeV should be replaced by a $(qq)(\bar{q}\bar{q})$ structure [@jaffe76]. This alternative appears to be strongly disfavored by the insight that scalar mesons neither correspond to a $q\bar{q}$ structure nor to a $(qq)(\bar{q}\bar{q})$ structure, but to a $q\bar{q}$ structure component coupled to di-meson states [@beveren86]. Also, the good agreement of the two-photon decay width of the $f_0(980)$ meson predicted on the basis of a $|s\bar{s}\rangle$ structure with the corresponding experimental values may be considered as an argument in favor of this structure.
Polarizabilities and invariant amplitudes
-----------------------------------------
For the discussion of the polarizabilities of the nucleon in terms of Compton scattering the forward direction ($\theta=0$) and the backward direction ($\theta=\pi$) are of special interest. Denoting the spin-independent and spin-dependent amplitudes for the forward and backward direction by $f_0$, $g_0$, $f_\pi$ and $g_\pi$, respectively, we arrive at $$\begin{aligned}
&&f_0(\omega)= -\frac{\omega^2}{2\pi}\left[A_3(\nu,t)+ A_6(\nu,t)
\right],\quad\quad\quad
g_0(\omega)=\frac{\omega^3}{2\pi {m_N}}A_4(\nu,t), \label{T3}\\
&&f_\pi(\omega)=-\frac{\omega\omega'}{2\pi}\left(1+\frac{\omega\omega'}
{{m_N}^2}\right)^{1/2}\left[
A_1(\nu,t) - \frac{t}{4 {m_N}^2}A_5(\nu,t)\right],\label{T4}\\
&&g_\pi(\omega)=-\frac{\omega\omega'}{2\pi {m_N}}\sqrt{\omega\omega'}
\left[
A_2(\nu,t)+ \left(1-\frac{t}{4 {m_N}^2}\right)A_5(\nu,t)\right]
,\label{T5}\\
&&\omega'(\theta=\pi)=\frac{\omega}{1+2\frac{\omega}{{m_N}}},\,\,
\nu=\frac12 (\omega+\omega'),\,\, t(\theta=0)=0,
\,\,t(\theta=\pi)=-4\omega\omega',
\label{T6}\end{aligned}$$ where $A_i$ are the invariant amplitudes in the standard definition [@lvov93a; @lvov97; @babusci98; @drechsel03; @wissmann04; @schumacher05a; @schumacher05b] and $m_N$ the nucleon mass. For the electric, $\alpha$, and magnetic, $\beta$, polarizabilities and the spin polarizabilities $\gamma_0$ and $\gamma_\pi$ for the forward and backward directions, respectively, we obtain the relations $$\begin{aligned}
&&\alpha+\beta = -\frac{1}{2\pi}\left[A^{\rm nB}_3(0,0)+
A^{\rm nB}_6(0,0)\right], \quad
\alpha-\beta = -\frac{1}{2\pi}
\left[A^{\rm nB}_1(0,0)\right], \nonumber\\
&&\gamma_0= \frac{1}{2\pi {m_N}}\left[A^{\rm nB}_4(0,0)
\right], \quad\quad\quad\quad \quad\quad\quad\,\,\,
\gamma_\pi = -\frac{1}{2\pi {m_N}}
\left[A^{\rm nB}_2(0,0)+A^{\rm nB}_5(0,0) \right],
\label{T7}\end{aligned}$$ where $A_i^{\rm nB}$ are the non-Born parts of the invariant amplitudes.
According to Eqs. (\[T3\]) to (\[T7\]) the following linear combinations of invariant amplitudes are of special importance because they contain the physics of the four fundamental sum rules, [ *viz.*]{} the BEFT [@bernabeu74] sum rule for $(\alpha-\beta)$, the LN [@lvov99] sum rule for $\gamma_\pi$, the BL [@baldin60] sum rule for $(\alpha+\beta)$, and the GDH [@gerasimov66] sum rule for the square of the anomalous magnetic moment $\kappa^2$, respectively: $$\begin{aligned}
&& {\tilde A}_1(\nu,t)\equiv A_1(\nu,t)-\frac{t}{4{m_N}^2}A_5(\nu,t),\label{T8}\\
&& {\tilde A}_2(\nu,t)\equiv A_2(\nu,t)+\left(1-\frac{t}{4{m_N}^2}\right)
A_5(\nu,t),\label{T9}\\
&& {\tilde A}_3(\nu,t) \equiv A_{3+6}(\nu,t)\equiv A_3(\nu,t)+
A_6(\nu,t), \label{T10}\\
&& {\tilde A}_4(\nu,t) \equiv A_4(\nu,t).
\label{T11}\end{aligned}$$ Therefore, these linear combinations of invariant amplitudes may be considered as generalized polarizabilities containing the essential physics of the polarizability of the nucleon.
In the following use is made of the relations concerning $\alpha+\beta$, $\alpha-\beta$ and $\gamma_\pi$, whereas the relation concerning $\gamma_0$ is written down only for the sake of completeness.
Fixed-$\theta$ dispersion relations at $\theta=\pi$
---------------------------------------------------
Fixed-$\theta$ dispersion relations have the advantage that a clear-cut separation of the $s$-channel and the $t$-channel is possible, i.e. there definitely is no double counting of empirical input [@hearn62]. In the following we are interested in the backward spin-polarizability $\gamma_\pi$ and the difference of electromagnetic polarizabilities $\alpha-\beta$ which have a firm relation to the invariant amplitudes at $\theta=\pi$ where one can write down [@drechsel03; @hoehler83] a dispersion integral as $$\begin{aligned}
{\rm Re}A_i(s,t)&=& A^{\rm B}_i(s,t)+A^{t-{\rm
pole}}_i(s,t)\nonumber\\
&+& \frac{1}{\pi}\int^\infty_{({m_N}+m_\pi)^2} ds'{\rm
Im}_s A_i(s',\tilde{t})
\left[\frac{1}{s'-s}+\frac{1}{s'-u}-\frac{1}{s'}\right]\nonumber\\
&+&\frac{1}{\pi} \int^\infty_{4m^2_\pi} dt' \frac{{\rm Im}_t
A_i(\tilde{s},t')}{t'-t},
\label{Drechsel(185)}\end{aligned}$$ where ${\rm Im}_s A(s',\tilde{t})$ is evaluated along the hyperbola given by $$s'u'={m_N}^4, \quad s'+\tilde{t}+u'=2{m_N}^2,
\label{Drechsel(186)}$$ and ${\rm Im}_t A_i(\tilde{s},t')$ runs along the path defined by the hyperbola $$\tilde{s}\tilde{u}={m_N}^4, \quad \tilde{s}+t'+\tilde{u}=2{m_N}^2.
\label{Drechsel(187)}$$ The amplitude $A^{t-{\rm pole}}_i(s,t)$ entering into (\[Drechsel(185)\]) describes the contribution of $t$-channel poles to the scattering amplitudes which for pseudoscalar mesons may be written in the form $$A^{{\pi^0}+\eta+\eta'}_2(t)=\frac{g_{{\pi} NN}F_{\pi^0\gamma\gamma}}
{t-m^2_{\pi^0}}\tau_3+\frac{g_{{\eta} NN}F_{\eta\gamma\gamma}}
{t-m^2_{\eta}}+\frac{g_{{\eta'} NN}F_{\eta'\gamma\gamma}}
{t-m^2_{\eta'}},
\label{pseudoscalarPole}$$ where the quantities $g_{\pi NN}$, [*etc.*]{} are the meson-nucleon coupling constants and the quantities $F_{\pi^0 \gamma\gamma}$, [*etc.*]{} the two-photon decay amplitudes. The last term in (\[Drechsel(185)\]) represents the contribution of $t$-channel cuts to the scattering amplitudes which later on will be discussed in connection with the scalar-isoscalar $t$-channel. From (\[pseudoscalarPole\]) the following relation for the $t$-channel part of the backward spin-polarizability may be obtained $$\gamma^t_\pi= \frac{1}{2\pi {m_N}}\left[\frac{g_{{\pi} NN}
F_{\pi^0\gamma\gamma}}
{m^2_{\pi^0}}\tau_3+\frac{g_{{\eta} NN}F_{\eta\gamma\gamma}}
{m^2_{\eta}}+\frac{g_{{\eta'} NN}F_{\eta'\gamma\gamma}}
{m^2_{\eta'}}\right].
\label{tchannelgammapi}$$ The pion-nucleon coupling constant is given by the experimental value $g_{\pi NN}= 13.169\pm 0.057$ [@bugg04]. The corresponding quantities for the $\eta$ and $\eta'$ meson cannot be much different. The reason is that on a quark level and in the chiral limit the coupling constant $g_{\pi qq}$ shows up as a universal proportionality constant between the quantities $f^{\rm cl}_\pi$ and $M$ given in (\[fpicl\]) and (\[LSMandNJL-2\]), respectively. A different problem is the sign of the two-photon decay amplitude where we have $$F_{\pi^0\gamma\gamma}=-|M(\pi^0\to\gamma\gamma)|,\quad
F_{\eta\gamma\gamma}=\pm|M(\eta\to\gamma\gamma)|,\quad
F_{\eta'\gamma\gamma}=\pm|M(\eta'\to\gamma\gamma)|.
\label{signsofamplitudes}$$ The minus sign in case of $F_{\pi^0\gamma\gamma}$ is well established [@terentev73; @lvov97; @lvov99] whereas the other signs are less well known.
The expression (\[tchannelgammapi\]) for the $t$-channel part of the backward spin-polarizability has been tested and found valid. Details may be found in Table \[spinpo\].
1 Spin pol. proton neutron
--- ---------------- ---------------- ---------------- -----------------------------
2 $\gamma_\pi$ $-38.7\pm 1.8$ $+58.6\pm 4.0$ Experiment [@schumacher05b]
3 $\gamma^s_\pi$ $+7.1 \pm 1.8$ $+9.1\pm 1.8$ Sum rule[@lvov99]
4 $\gamma^t_\pi$ $-45.8\pm 2.5$ $+49.5\pm 4.4$ line 2 – line 3
5 $\gamma^t_\pi$ $-46.7$ $+46.7$ $\pi^0$-pole only
6 $\gamma^t_\pi$ $-45.1$ $+48.3$ $\pi^0+\eta+\eta'$-poles
7 $\gamma^t_\pi$ $-48.3$ $+45.1$ $\pi^0+\eta+\eta'$-poles
: Backward spin-polarizability for the proton and the neutron (units $10^{-4} {\rm fm}^4$)
\
a) $\eta$ and $\eta'$ contributions assumed to be positive numbers (line 6),\
b) $\eta$ and $\eta'$ contributions assumed to be negative numbers (line 7)
\[spinpo\]
In [@lvov99] it is proposed to adopt the minus sign for the $\eta$ and $\eta'$ contributions in Eq. (\[signsofamplitudes\]). By comparing line 4 with lines 6 and 7 in Table \[spinpo\] we see that a better agreement with the experimental values is obtained when the plus sign is used.
For $(\alpha-\beta)$ we have the choice to either use the pole representation as appropriate for the $\sigma$ meson as entering into the NJL model or the cut representation as appropriate for the $\sigma$ meson as a broad resonant intermediate state.
![a) $\pi^0$ pole diagram. b) $\sigma$ pole diagram, c) scalar-isoscalar $t$-channel as entering into the BEFT sum rule[]{data-label="poleandcutgraphs"}](figures/SigmaGraphs-1.eps){width="0.5\linewidth"}
Here we first discuss the pole representation (see Fig. 3 b)). Then the $\sigma$ meson may be understood as a $(u\bar{u} + d \bar{d})/\sqrt{2}$ configuration having a definite mass $m_\sigma$ as predicted by the quark-level NJL model. The corresponding amplitude is constructed in analogy to the pseudoscalar pole and given by $$A^{t-{\rm pole}}_1(t)=\frac{g_{\sigma NN}F_{\sigma\gamma\gamma}}{
t-m^2_\sigma},
\label{sigmapole}$$ with $g_{\sigma NN}$ being the $\sigma$-nucleon coupling constant, $F_{\sigma\gamma\gamma}$ the two-photon $\sigma$ decay amplitude and $m_\sigma$ the $\sigma$ mass[^1] . Then some consideration shows that the $t$-channel part of the polarizability difference is given by $$(\alpha-\beta)^t_{p,n}=\frac{g_{\pi NN}F_{\sigma\gamma\gamma}}
{2\pi m^2_\sigma}=\frac{5\alpha_eg_{\pi NN}}{6\pi^2 m^2_\sigma f_\pi}
=15.2\approx \frac{5\alpha_e g^2_{\pi NN}}{6\pi^2m^2_\sigma g_A {m_N}},
\label{alpha-beta-quarks}$$ in units of $10^{-4}{\rm fm}^3$, where $\alpha_e=1/137.04$, $g_{\sigma
NN}\equiv g_{\pi NN}= 13.169\pm 0.057$ [@bugg04], $f_\pi=(92.42\pm0.26)$ MeV, $m_\sigma= 666$ MeV as derived in subsection \[subsection2-2\] using the method of [@delbourgo95]. In (\[alpha-beta-quarks\]) the quark-model prediction $F_{\sigma\gamma\gamma}=+|M(\sigma\to\gamma\gamma)|$ with $N_c=3$ has been used. The identity $g_{\sigma NN}\equiv g_{\pi NN}$ is predicted by the L$\sigma$M and has been experimentally confirmed by Durso et al. [@durso80]. On the r.h.s. of Eq. (\[alpha-beta-quarks\]) use has been made of the approximately valid Goldberger-Treiman relation where $g_A=1.255\pm 0.006$ is the axial vector coupling constant.
An upper limit for the possible correction to $(\alpha-\beta)^t_{p,n}$ as given in (\[alpha-beta-quarks\]) due to the $f_0(980)$ meson can be calculated making the assumption that the coupling constants $g_{f(980) NN}$ and $g_{\pi NN}$ are equal to each other. The result obtained is a possible correction of not more than $9 \%$.
The quantity $(\alpha-\beta)^t$ predicted from the reactions $\pi\pi\to\sigma\to\pi\pi$, $\gamma\gamma\to\sigma\to\pi\pi$ and $N\bar{N}\to\sigma\to\pi\pi$
==========================================================================================================================================================
A different approach to a calculation of $(\alpha-\beta)^t$ which only makes use of experimental data without specific assumptions about the internal structure of the $\sigma$ meson is provided by the BEFT sum rule [@bernabeu74]. Instead of exploiting Eq. (\[alpha-beta-quarks\]) the following relation is considered $$\gamma\gamma\to|\sigma\rangle\to \pi\pi \to |\sigma\rangle
\to N\bar{N},
\label{eq-SUM3}$$ and replaced by $$\begin{aligned}
&&\gamma\gamma\to|\sigma\rangle\to \pi\pi, \label{eq-SUM4a} \\
&&\pi\pi \to |\sigma\rangle\to N\bar{N},
\label{eq-SUM4b}\end{aligned}$$ obtained by means of a $t$-channel cut (see Fig. \[poleandcutgraphs\] c)). The further procedure is to use the unitarity relation $${\rm Im}_t T(\gamma\gamma\to N\bar{N})=\frac12 \sum_n
(2\pi)^4\delta^4(P_n-P_i)\, T(\gamma\gamma\to n)\, T^*(N\bar{N}\to n),
\label{eq-SUM5}$$ where the sum on the right-hand side is taken over all allowed intermediate states $n$ having the same total 4-momentum as the initial state. Furthermore, $\pi\pi$ intermediate states are taken into account where the spin is $J=0$ and the isospin $I=0$. These are the quantum numbers of the intermediate $\sigma$ meson.
If we restrict ourselves in the calculation of the $t$-channel absorptive part to intermediate states with two pions with angular momentum $J\leq 2$, the BEFT sum rule [@bernabeu74] gets a convenient form for calculations: $$\begin{aligned}
(\alpha-\beta)^t_{p,n}&=&
\frac{1}{16 \pi^2}\int^\infty_{4 m^2_\pi}\frac{dt}{t^2}\frac{16}{4{m_N}^2-t}
\left(\frac{t-4m^2_\pi}{t}\right)^{1/2}\Big[f^0_+(t)
F^{0*}_{0}(t)\nonumber\\
&&-\left({m_N}^2-\frac{t}{4}\right)\left(\frac{t}{4}-m^2_\pi\right)
f^2_+(t) F^{2*}_{0}(t)\Big],\label{BackSR}\end{aligned}$$ where $f^{(0,2)}_+(t)$ and $F^{(0,2)}_0(t)$ are the partial-wave helicity amplitudes of the processes $N\bar{N}\to \pi\pi$ and $\pi\pi\to \gamma\gamma$ with angular momentum $J=0$ and $2$, respectively, and isospin $I=0$.
Except for the quantum numbers, properties of the $\sigma$ meson enter through the amplitudes $f^{(0,2)}_+(t)$ and $F^{(0,2)}_0(t)$ corresponding to the reactions (\[eq-SUM4a\]) and (\[eq-SUM4b\]). These amplitudes incorporate the phase $\delta^I_J(s)=\delta^0_0(s)$ which is extracted from $\pi$-$\pi$ scattering data as obtained in $\pi N\to N \pi\pi$ scattering experiments. The information contained in the phase $\delta^0_0(s)$ can be understood as being due to a resonance with the pole parameters [@eidelman04; @colangelo01; @ishida03] $$\sqrt{s^{\rm pole}}\approx (500 -i\, 250)\,{\rm MeV}
\label{eq-SUM6}$$ and the special property of a $90^\circ$ crossing of the phase at $$\sqrt{s}(\delta^0_0=90^\circ)\approx 900 \,{\rm MeV}.
\label{eq-SUM7}$$ The apparent mismatch of the $90^\circ$ phase crossing expected for the resonance part as given by (\[eq-SUM6\]) and the observed $90^\circ$ phase crossing as given by (\[eq-SUM7\]) has led to numerous considerations [@ishida96; @kaloshin04; @bugg03] among which the possible existence of a background [@ishida96] or a second pole at negative mass parameter $m^2$ [@kaloshin04] or one pole with an $s$-dependent width $\Gamma(s)$ [@bugg03] played a role. This discussion shows that a generally accepted model for the $\sigma$ meson as a resonant state is still missing ( see, however, the last entry in [@beveren86] and references therein). Furthermore, the inclusion of the $\pi\pi$ phase relation into the $\gamma\gamma\to\pi\pi$ and $\bar{N}N\to \pi\pi$ amplitudes is not an easy task. This may be the reason that only recently some consistency in the prediction of $(\alpha-\beta)^t$ from the BEFT sum rule [@bernabeu74] has been obtained. This has led to $$(\alpha-\beta)^t_{p,n}= (14.0\pm 2.0)\,\,
\mbox{(Ref.\cite{schumacher05b})},
\,\,\,\,\, 16.46\,\, \mbox{(Ref. \cite{drechsel03})},
\label{BEFT}$$ where the first value has been obtained by Levchuk et al. (see[@schumacher05b]) and the second value by Drechsel et al. [@drechsel03]. The large error given in Ref. [@schumacher05b] takes care of the uncertainties contained in the appropriate choice of experimental data. The arithmetic average of the two results is $(\alpha-\beta)^t_{p,n}= 15.3\pm 1.3$. This number has to be compared with the experimental data $(\alpha-\beta)^t_p=15.1\pm 1.3$, $(\alpha-\beta)^t_n=14.8\pm 2.7$ (see Ref. [@schumacher05b]). We see that the predictions for $(\alpha-\beta)^t$ obtained from the $\sigma$-meson pole based on the NJL model and from the BEFT sum rule lead to agreement with each other and to agreement with experiment.
Discussion
==========
According to our recent analysis [@schumacher05a; @schumacher05b; @levchuk05] the experimental polarizabilities may be summarized in the form given in Table \[tablepolarizabilities\].
--------------------------------------------------------------------------------------------
1 proton neutron
---- -------------------- ---------------------------------- -------------------------------
2 BL sum rule $(\alpha+\beta)_p=13.9\pm 0.3$ $
(\alpha+\beta)_n=15.2\pm 0.5$
3 Compton scattering $(\alpha-\beta)_p=10.1\pm 0.9$ $(\alpha-\beta)_n=9.8\pm 2.5$
4 BEFT sum rule $(\alpha-\beta)^s_p=-5.0\pm 1.0$ $(\alpha-\beta)^s_n=-5.0\pm
1.0$
5 line 3 – line 4 $(\alpha-\beta)^t_p=15.1\pm 1.3$ $(\alpha-\beta)^t_n=14.8\pm
2.7$
6 experiment $\alpha_p=12.0\pm 0.6$ $\alpha_n=12.5 \pm 1.7$
7 $s$-channel only $\alpha^s_p= \,\,\,4.5\pm 0.5$ $
\alpha^s_n=\,\,\,5.1\pm 0.6$
8 $t$-channel only $\alpha^t_p= \,\,\,7.5\pm 0.8$ $
\alpha^t_n=\,\,\,7.4\pm 1.8$
9 experiment $\beta_p=\,\,\,1.9\mp 0.6$ $\beta_n=\,\,2.7\mp 1.8$
10 $s$-channel only $\beta^s_p= \,\,\,9.5\pm 0.5$ $
\beta^s_n=10.1\pm 0.6$
11 $t$-channel only $\beta^t_p= \,\,\,-7.6\pm 0.8$ $
\beta^t_n=-7.4\pm 1.9$
--------------------------------------------------------------------------------------------
: Summary on electromagnetic polarizabilities in units of $10^{-4}{\rm fm}^3$
\[tablepolarizabilities\]
The quantities $\alpha_p,\beta_p,\alpha_n,\beta_n$ are the experimental electric and magnetic polarizabilities for the proton and neutron, respectively. The quantities with an upper label $s$ are the corresponding electric and magnetic polarizabilities where only the $s$-channel degrees of freedom are included. These latter quantities have been obtained by making use of the fact that $(\alpha+\beta)$, when calculated from forward-angle dispersion theory as given by the Baldin or Baldin-Lapidus (BL) sum rule [@baldin60] $$(\alpha+\beta)=\frac{1}{2\pi^2}\int^\infty_{m_\pi+\frac{m^2_\pi}{2{m_N}}}
\frac{\sigma_{\rm tot}(\omega)}{\omega^2}d\omega
\label{BLsumrule}$$ has no $t$-channel contribution, i.e. $(\alpha+\beta)=(\alpha+\beta)^s$, and by using the estimate $(\alpha-\beta)^s_{p,n}=-5.0\pm 1.0$ obtained form the $s$-channel part of the BEFT sum rule [@bernabeu74] $$(\alpha-\beta)^s=\frac{1}{2\pi^2}\int^\infty_{m_\pi+\frac{m^2_\pi}{2{m_N}}}
\sqrt{1+\frac{2\omega}{m_N}}\left[\sigma(\omega,E1,M2,E3,\cdots)
-\sigma(\omega,M1,E2,M3,\cdots)\right]\frac{d\omega}{\omega^2}
\label{BEFTsumrule}$$ both for the proton and the neutron (see [@schumacher05b]). The absence of a $t$-channel contribution to $(\alpha+\beta)$ follows from the observation [@schumacher05b] that the BL sum rule is fulfilled. This observation is explained by the fact that in the forward direction vector-meson dominance converts the $t$-channel contribution into the Regge part of the photoabsorption cross section. The numbers in line 5 of Table \[tablepolarizabilities\] are the $t$-channel contributions to $(\alpha-\beta)$ obtained from the experimental values $(\alpha-\beta)_p=10.1\pm 0.9$ and $(\alpha-\beta)_n=9.8\pm 2.5$ and the estimate for $(\alpha-\beta)^s_{p,n}$. As noted before, we see that the experimental values for $\alpha$ are much larger than the $s$-channel contributions alone, whereas for the magnetic polarizabilities the opposite is true. For the magnetic polarizability it makes sense to identify the large difference between the experimental value and the $s$-channel contribution with the diamagnetic polarizability. This means that we may consider $\beta^t$ as the diamagnetic polarizability.
Certainly, by identifying the expression obtained for $(\alpha-\beta)^t_{p,n}$ in Eq. (\[alpha-beta-quarks\]) with that of Eq. (\[BackSR\]) a sum rule is obtained. This finding is also supported by the two graphs $b)$ and $c)$ in Fig. \[poleandcutgraphs\]. Furthermore, we see in Table \[summary\] that the experimental results obtained for $(\alpha-\beta)^t$ from the experiments, from the $\sigma$-meson pole and from the $\sigma$-meson cut agree with each other and thus give also strong support for the existence and validity of the sum rule.
$(\alpha-\beta)^t_p$ $ (\alpha-\beta)^t_n$
--------------- ---------------------- -----------------------
experiment $15.1 \pm 1.3$ $14.8 \pm 2.7$
$\sigma$-pole $15.2$ $15.2$
BEFT sum rule $15.3\pm 1.3$ $15.3\pm 1.3$
: Difference of electromagnetic polarizabilities $(\alpha-\beta)^t_{p,n}$ in the $t$-channel (in units of $10^{-4}{\rm fm}^3$). The result given for the $\sigma$-meson cut or BEFT sum rule is the arithmetic average of the results of Drechsel et al. [@drechsel03] and Levchuk et al. (see [@levchuk05; @schumacher05b])
\[summary\]
We expect that by studying this sum rule in more detail some more insight into the structure of the $\sigma$ meson is obtained. In such a study the role of the quark-level NJL model would be to describe the $\sigma$ meson as the particle of the $\sigma$ field with a definite mass $m_\sigma =666$ MeV, whereas the BEFT sum rule exploits on-shell properties of the $\sigma$ meson as there is [*e.g.*]{} the phase relation $\delta^0_0(s)$. The sum rule we are proposing has the property of linking on-shell aspects of the $\sigma$ meson with properties of the $\sigma$ meson as the particle of the $\sigma$ field.
As a concluding remark we wish to state that the present paper closes a circle, starting with the work of Hearn and Leader (1962) [@hearn62] where the role of the $t$-channel contributions to the Compton scattering amplitudes has been clarified. In the present work we have shown that the $(\sigma,\pi^0)$ particle doublet is capable of quantitatively reproducing this $t$-channel contribution, with minor additional contributions from the $f_0(980)$, the $\eta$ and the $\eta'$ meson. Since the $t$-channel contribution cannot be interpreted in terms of nucleon resonances or in terms of the meson cloud as viewed in photon-meson production experiments, it is reasonable to interpret this contribution in terms of a short-range property of the constituent quarks, as proposed in [@schumacher05b]. More explicitly this means that each constituent quark $|q\rangle$ of the nucleon is converted into a $|(\sigma,\pi^0)q\rangle$ $t$-channel resonant intermediate state during the Compton scattering process. However, this resonant intermediate state is located in the unphysical region at positive $t$. At $\theta=\pi$ the $\pi^0$ meson is involved in those scattering processes where the incoming and the outgoing photon have perpendicular directions of linear polarization whereas for the $\sigma$ meson the directions of linear polarization are parallel.
Other approaches using model calculations (see [@schumacher05b] for a summary) or diagrammatic techniques [@bernard91; @hildebrandt04; @hemmert98] do not take care of the scalar-isoscalar $t$-channel contribution in a sufficient way. This means that they are important with respect to a test of the underlying theoretical ansatz, but they cannot be considered as a quantitative descriptions of the electromagnetic polarizabilities of the nucleon. It is very interesting to analyze the different strategies contained in these approaches [@bernard91; @hildebrandt04; @hemmert98] to cope with the serious problem caused by the missing scalar-isoscalar $t$-channel in the light of dispersion theory. The most transparent of these strategies are the neglect of the contribution of the $\Delta$ resonance [@bernard91] and the introduction of an empirical counter term of unnatural size [@hildebrandt04].
The strategy of neglecting the $\Delta$ resonance contribution has been analysed by L’vov [@lvov93] in terms of dispersion theory. It has been confirmed that the simultaneous neglect of the scalar-isoscalar $t$-channel and of the $\Delta$ resonance leads to an approximate agreement with the experimental data. Furthermore, it has been confirmed that the use of the heavy baryon approximation ($m_N\to \infty$) leads to an improvement of the agreement with the experimental data, though no good physical reason has been found for such a replacement. Finally, it has been shown [@lvov93] that the evaluation of chiral loops [@bernard91] leads to an approximate agreement with the contributions of the $E_{0^+}$ component to $\alpha$ and $\beta$. However, this is only the case when the empirical CGLN amplitude $E_{0+}$ is replaced by the Born approximation $E^{\rm Born}_{0+}$.
Since there is no definite interpretation available for the empirical counter terms of unnatural sizes [@hildebrandt04], $\delta\alpha$ and $\delta\beta$, it may be allowed to tentatively compare these terms with the present $t$-channel polarizabilities $\alpha^t_{p,n}$ and $\beta^t_{p,n}$ contained in Table \[tablepolarizabilities\]. Qualitatively, these terms $\delta\alpha$ and $\delta\beta$ are introduced to produce diamagnetism. Furthermore, their physical nature is assumed to be related to short-distance phenomena of some kind. These two properties suggest that $\delta\alpha$ and $\delta\beta$ on the one hand and $\alpha^t_{p,n}$ and $\beta^t_{p,n}$ on the other should have some common features or even may be identical. Unfortunately, the numbers obtained empirically, i.e. $\delta\alpha=-5.92\pm 1.36$ and $\delta\beta=-10.68\pm 1.17$, are not in a good agreement with the numbers obtained for $\alpha^t_{p,n}=-\beta^t_{p,n}=7.6$.
Summary
=======
The present paper succeeds in a quantitative derivation of the $\sigma$-meson pole contribution to $(\alpha-\beta)^t$ which formerly was introduced and treated semi-quantitatively by adjusting to experimental data [@lvov97]. Furthermore, there was no strict argument for the pole-structure of this contribution, since the only knowledge about the scalar-isoscalar $t$-channel was obtained from the BEFT sum rule which makes use of a $t$-channel cut and not of a $t$-channel pole. In the present paper we show that all the relevant parameters of the $\sigma$-meson pole, [*viz.*]{} the $\sigma$-meson mass $m_\sigma$ and properties of the two-photon decay width $\Gamma_{\gamma\gamma}$ follow from the NJL model without any adjustable parameter. Making use of previous work [@hatsuda94; @delbourgo95], the mass $m_\sigma$ is obtained in two different versions of the NJL model and two different regularization schemes. In the first case [@hatsuda94] the four-fermion version of the NJL model is used and the regularization scheme makes use of a cut-off parameter $\Lambda$. Both quantities, the cut-off parameter $\Lambda$ and the coupling $G$ are obtained within this regularization scheme by adjusting to empirical data as there are the pion decay constant $f_\pi$, the pion mass $m_\pi$ and the current-quark mass $m_0$. The result is $m_\sigma=668$ MeV. In the second case [@delbourgo95] the bosonized version of the NJL model is applied and the regularization scheme makes use of dimensional scaling, [*i.e.*]{} no cut-off parameter $\Lambda$ is present so that the integrals extends to infinity. Instead, use is made of the fact that the difference of the diverging integrals for the constituent-quark mass $M$ and the pion decay constant $f_\pi$ is finite. The result is $m_\sigma=666$ MeV. It is remarkable and has not been pointed out before that these two completely different treatments of the NJL model perfectly lead to the same result. This gives us confidence that conclusions based on this number for the $\sigma$ meson mass are on a sound basis. In case of the two-photon decay width $\Gamma_{\gamma\gamma}$ the NJL model is used to justify the neglect of couplings of the $\sigma$ meson to two photons via meson loops. Such additional couplings follow from the L$\sigma$M. If taken into account the additional couplings would partly destroy the good agreement between theory and experiment. Therefore, the argument delivered by the NJL model is important.
A further new results obtained is the calculation of the two-photon decay width $\Gamma_{\gamma\gamma}$ of the $f_0(980)$ meson. This calculation is based on a $s\bar{s}$ structure of the $f_0(980)$ meson and leads to agreement with experimental results. We consider this as an argument that indeed the $f_0(980)$ meson and the $\sigma$ meson have a $q\bar{q}$ core and not a $qq\bar{q}\bar{q}$ structure as suggested in other approaches [@jaffe76]. This result is important for the present investigation as well as for the physics of scalar mesons in general. In addition, for the first time the contribution of the $f_0(980)$ meson to $(\alpha-\beta)^t$ has been calculated. A further new result are arguments in favor of positive decay amplitudes $F_{\eta\gamma\gamma}$ and $F_{\eta'\gamma\gamma}$ which formerly were believed to be negative.
Via the two representations of $(\alpha-\beta)^t$ a quantitative link is obtained between the $\sigma$ meson as the particle of the $\sigma$ field and the $\sigma$ meson as showing up as an extremely shortlived intermediate state in reactions where two pions are involved. This observation may be exploited to get insight into the structure of the $\sigma$ meson by constructing a model which makes the relation between the graphs b) and c) in Figure \[poleandcutgraphs\] transparent.
Acknowledgment {#acknowledgment .unnumbered}
==============
The author is indebted to Deutsche Forschungsgemeinschaft for the support of this work through the projects SCHU222 and 436RUS113/510. He thanks M.I. Levchuk, A.I. L’vov and A.I. Milstein for their continuous interest in this work and for many comments and suggestions. He also thanks E. van Beveren, F. Kleefeld, G. Rupp, and M.D. Scadron for many useful comments regarding their work.
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[^1]: Occasionally it has been proposed to modify the pole amplitude given in (\[sigmapole\]) by considering the quantities $g_{\sigma NN}$ and $F_{\sigma\gamma\gamma}$ as $t$-dependent formfactors. Such a procedure, however, is not allowed because it is incompatible with the requirements of dispersion theory.
|
**EVOLUTION OF LOW-MASS HELIUM STARS\
IN SEMIDETACHED BINARIES\
**
L. R. YUNGELSON$ ^*$
*Institute of Astronomy of Russian Academy of Sciences, Moscow*\
Received
We present results of a systematic investigation of the evolution of low-mass (0.35, 0.40 and 0.65) helium donors in semidetached binaries with accretors – white dwarfs. In the initial models of evolutionary sequences abundance of helium in the center $0.1 \aplt Y_c \leq 0.98 $. Results of computations may be applied to the study of the origin and evolutionary state of stars. It is shown that the minimum orbital periods of the systems only weakly depend on the total mass of the system and evolutionary state of the donor at RLOF and are equal to 9-11 min. The scatter in the mass-exchange rates at given in the range $ {\mbox {$P_{\rm orb}$}}_{,min} < {\mbox {$P_{\rm orb}$}}\aplt$ 40 min. does not exceed $\sim2.5$. At $ {\mbox {$P_{\rm orb}$}}\apgt $ 20 min mass-losing stars are weakly degenerate homogeneous cooling objects and abundances of He, C, N, O, Ne in the matter lost by them depends on the extent of He-depletion at RLOF. For the systems which are currently considered as the most probable model candidates for stars with helium donors these abundances are $Y \apgt 0.4$, $ X_{\rm C} \aplt 0.3$, $ X_{\rm O} \aplt 0.25$, $ X_{\rm N} \aplt 0.5\times 10^{-2}$. At ${\mbox {$P_{\rm orb}$}}\apgt $ 40 min. the timescale of mass-loss begins to exceed thermal time-scale of the donors, the latter begin to contract, they become more degenerate and, apparently, “white-dwarf” and “helium-star” populations of stars merge.
*Key words*: stars – variable and peculiar
PACS numbers: 97.10.Cv, 97.10.Me,\
[$ ^*$]{}Email: [email protected]
Introduction {#sec:intro}
============
Non-degenerate helium stars in close binaries (CB) form in so-called “case B” of mass-exchange, when stars with mass $ M \apgt (2.3 - 2.5)$ overflow their Roche lobes in the hydrogen-shell burning stage (Kippenhahn and Weigert, 1967; Paczyński, 1967b). Masses of helium stars are $ \gtrsim 0.32$ (Iben, 1990; Han et al., 2002).
For the current notions on stellar evolution, importance of low-mass helium stars is defined by the possibility of formation of pairs containing white dwarfs or neutron stars accompanied by helium stars in the course of evolution of CB. Orbital periods of such systems may be so short, that angular momentum loss (AML) via gravitational wave radiation (GWR) enables Roche-lobe overflow (RLOF) by helium star before helium exhaustion in the core of the latter and, under proper conditions, stable mass-transfer is possible (Savonije et al., 1986; Iben and Tutukov, 1987; see also Yungelson, 2005a and Postnov and Yungelson, 2006).
Semidetached pairs of neutron stars with helium-star companions were not observed as yet, though, the first computation of the evolution of a CB with a non-degenerate helium donor was performed just for such a pair (Savonije et al. 1986). On the other hand, it is assumed that in the case when in a semidetached binary helium star is accompanied by a white dwarf, such a system may be observed as an ultra-short period cataclysmic variable of the type (Savonije et al., 1986). Other hypothetical scenarios for formation of stars assume stable mass-exchange between white dwarfs (¶, 1967a) or mass-loss by a remnant of low-mass ($\lesssim 1.5$) main-sequence star which overflowed its Roche lobe after exhaustion of a significant fraction of the hydrogen in its core ($ X_{c} \lesssim 0.4$) or even immediately after formation of a low-mass ($ \sim0.01$) helium core (Tutukov et al., 1985; Podsiadlowski et al., 2003)[^1].
AM CVn stars are important for the physics in general, since it is expected that, thanks to their extremely short orbital periods, they, along to the close detached short-period white dwarf pairs, will be among the first objects which will be able to detect space-born gravitational wave antenna *LISA* (Evans et. al., 1987; Nelemans et al., 2004). More, since the distance to some of the stars is known with sufficient accuracy, they could be used as verification binaries for *LISA* (Stroeer and Vecchio, 2006).
stars, despite small number of identified and candidate objects ($\sim20 $, see the list with parameters of the systems and references at
`),`
are of great interest not only because they are potential sources of detectable gravitational waves, but also, for instance, because their formation involves evolution in common envelopes, their accretion disks may consist of helium or helium-carbon-oxygen mixture. stars are potential progenitors of SN Ia (Tutukov and Yungelson, 1979; see also Solheim and Yungelson, 2005), of still hypothetical SN .Ia (Bildsten et al., 2007), and of other explosive phenomena associated with accumulation of helium at the surface of white dwarfs (see, e.g., Tutukov and Yungelson, 1981; Iben and Tutukov, 1991). Note also, that it is expected that $\sim 100$ most tight Galactic stars might be observed both in gravitational waves and in the electromagnetic spectrum (Nelemans et al., 2004).
It is still unclear which of the above-mentioned scenarios for formation of stars acts in the Nature. All scenarios for formation of stars imply that their progenitors pass through one or two common-envelope stages, but since clear understanding of the processes occurring in the latter is absent, the efficiency of common envelopes ejection remains a crucial parameter of scenarios that defines final separation of components and, hence, possibility of formation of a semidetached system. Also, conditions for stable mass transfer after RLOF by a white dwarf are not clear (see, e.g., Marsh et al., 2004; Gokhale et al., 2007; Motl et al., 2007). On the other hand, for instance, if stars descend from strongly evolved hydrogen-rich cataclysmic variables, only a minor fraction of them evolves to $ {\mbox {$P_{\rm orb}$}}<25 $ min. which are typical for a considerable number of the stars; more, hydrogen which has to be present in the spectra of accretion disks of most of such systems is not observed as yet. Nevertheless, it is possible that all formation channels contribute to the population of stars. Resolution of the problem of the origin of stars is also important because observational estimates of their Galactic population are by an order of magnitude below model estimates based on the above-described models (Roelofs et al., 2007a,b). This may point to certain flaws in the understanding of evolution of close binaries. Note, however, that the “deficit” of observed stars may be due to numerous selection effects (see detailed discussion in Roelofs et al., 2007a,c and Anderson et al. 2005, 2008).
Nelemans and Tout (2003) noticed that the chemistry of accretion disk in an type star may serve as an identifier of its origin. If a helium white dwarf serves as the donor, the disk has to contain He and other H-burning products. In the case of helium-star donor, the disk, can, along to He, contain CNO-cycle and He-burning products. If the donor is an evolved main-sequence star, the disk may contain H and H-burning products. In the first case abundances depend on the mass of the white dwarf progenitor, while in other cases it depends also on how far was the star evolved prior to RLOF.
Evolution of CB with low-mass He-donors was computed by Savonije et. al. (1986), Tutukov and Fedorova (1989), Ergma and Fedorova (1990). These papers were focused, mainly, on mass-transfer rates, ranges of orbital periods, since results of computations were applied to X-ray systems and SN Ia progenitors. In the present paper we carry out a systematic investigation of the evolution of helium stars in semidetached binaries as a function of their mass, evolutionary state at RLOF, and the parameters of a binary that contains helium star. Special attention is paid to the chemical composition of the matter lost by helium stars. In the forthcoming study (Nelemans and Yungelson, in prep.) these results will be applied for the analysis of the origin of observed stars.
Method of computations {#sec:model}
======================
For our computations we have used a specially adapted version of P.P. Eggleton evolutionary code (1971; private comm. 2006). Equation of state, opacity tables, and other input data are described by Pols et al. (1995). Information on the latest modifications of the code may be found at `www.ast.cam.ac.uk/stars/`. In the code, abundances of $^1$H, $^4$He, $^{12}$C, $^{14}$N, $^{16}$O, $^{20}$Ne, $^{24}$Mg are computed. The nuclear reactions network is given in Table 1 of Pols et al. (1995). Nuclear reactions rates are taken after Caughlan and Fowler (1988), with exception of $^{12}{\rm C}(\alpha,\gamma)^{16}{\rm O}$ reaction for which the data of Caughlan et al. (1985) are used. Homogeneous helium models have mass fractions of helium Y=0.98, carbon $ X_{\rm C}=$0.00019, nitrogen $ X_{\rm N}=$0.01315, oxygen $ X_{\rm O}=$ 0.00072, neon $X_{\rm Ne}=$0.00185, magnesium $ X_{\rm Mg}=$0.00068.
It was assumed that mass-exchange is conservative. Angular momentum loss via GWR was taken into account using standard Landau and Lifshitz (1971) formula $$\left ( \frac {\dot{J}}{J} \right )_{\rm GWR} = -\frac{32}{5} \, \frac{G^3}{c^5} \frac{M_{\rm He}\, M_{\rm wd} (M_{\rm He} + M_{\rm wd})}{a^{4}}.
\label{eq:gwr}$$ Here and are the masses of components, $a$ is their separation.
Mass loss rate $ \dot{M}_{\rm He}$ is related to $ \dot{J}/J $ as $$\label{eq:mdot}
\frac{\dot{M}_{\rm He}}{M_{\rm He}} =
\left ( \frac{\dot{J}}{J} \right )_{\rm GWR} \times \left [\frac{\zeta (M_{\rm He})}{2} +
\frac{5}{6} - \frac{M_{\rm He}}{M_{\rm wd}} \right]^{-1},$$ where $ \zeta (M_{\rm He}) = d\ln R/d\ln M_{\rm He}$. The term in brackets is $\sim 1$.
Results {#sec:results}
=======
Evolution of helium stars {#sec:evolution}
-------------------------
----- -------- --------- -------- ---------- -------- ------- ---------- -------- ---------- ----------
No. $M_d,$ $ M_a,$ $P_0$, $t_c$, $P_c$, $Y_c$ $t_f$, $P_f$, $M_{df}$ $Y_{sf}$
min. $10^6$yr min. $10^6$yr min.
1 0.35 0.5 20 1.29 15.96 0.977 427.00 42.06 0.027 0.976
2 0.35 0.5 40 15.99 16.24 0.936 400.99 41.52 0.028 0.935
3 0.35 0.5 60 50.80 17.02 0.871 426.05 41.18 0.028 0.870
4 0.35 0.5 80 110.71 18.14 0.774 492.51 40.81 0.027 0.723
5 0.35 0.5 100 202.32 19.71 0.642 615.18 40.35 0.025 0.640
6 0.35 0.5 120 332.36 22.46 0.435 689.67 38.30 0.025 0.428
7 0.35 0.5 140 502.68 34.76 0.178 874.27 37.05 0.022 0.166
8 0.35 0.5 144 557.78 35.00 0.118 840.34 35.31 0\. 024 0.104
9 0.40 0.6 20 0.14 19.51 0.979 348.53 41.90 0.027 0.978
10 0.40 0.6 40 11.68 19.94 0.923 338.05 41.50 0.028 0.920
11 0.40 0.6 60 50.78 17.02 0.871 426.05 41.18 0.028 0.825
12 0.40 0.6 80 84.52 22.42 0.704 476.41 41.63 0.025 0.698
13 0.40 0.6 100 154.47 25.57 0.509 733.73 41.96 0.020 0.495
14 0.40 0.6 120 252.55 29.97 0.228 652.76 38.52 0.021 0.194
15 0.40 0.6 130 315.30 29.71 0.066 528.00 34.08 0.026 0.035
16 0.40 0.8 20 0.07 19.71 0.976 323.79 42.55 0.027 0.977
17 0.40 0.8 40 9.14 20.61 0.933 330.17 42.61 0.027 0.923
18 0.40 0.8 60 30.21 21.67 0.854 358.96 42.51 0.027 0.849
19 0.40 0.8 80 66.95 22.85 0.751 431.17 42.60 0.025 0.744
20 0.40 0.8 100 122.98 25.40 0.601 198.19 30.51 0.043 0.587
21 0.40 0.8 120 201.61 28.65 0.376 568.45 40.12 0.023 0.353
22 0.40 0.8 140 306.39 30.63 0.090 551.02 36.14 0.024 0.057
23 0.65 0.8 35 0.17 34.54 0.976 324.85 43.96 0.027 0.856
24 0.65 0.8 40 2.06 35.29 0.928 340.97 44.04 0.026 0.806
25 0.65 0.8 60 14.76 38.74 0.708 443.33 43.96 0\. 022 0.547
26 0.65 0.8 80 36.29 44.87 0.396 353.64 39.99 0\. 022 0.129
27 0.65 0.8 85 42.83 47.07 0.286 324.29 38.23 0.023 0.0135
28 0.65 0.8 90 50.86 48.56 0.186 69.36 4.31 0.378 0.00
----- -------- --------- -------- ---------- -------- ------- ---------- -------- ---------- ----------
\[tab:tracks\]
Following the model of the Galactic population of stars (Nelemans et al., 2001), we have considered as typical progenitors of helium-donor stars the systems with masses $ M_{\rm He}+M_{\rm wd} $ = (0.35 + 0.5), (0.4 + 0.6), and (0.4 + 0.8). In addition, we have considered a pair (0.65+0.8). If helium-accreting white dwarfs avoid double- (or “edge-lit-”) detonation which may destroy the dwarf, such a system may belong to progenitors of stars, but not to the most “fertile” of them. As the second parameter of computations we considered initial orbital period of the system, i.e., the period of the system immediately after completion of the common envelope stage that resulted in formation of a helium star. The range of ${\mbox {$P_{\rm orb}$}}_{,0}$ for every system was chosen in such a way that the set of computed models included both stars that filled their Roche lobe virtually unevolved and stars that had at the instant of RLOF radii close to the maximum of the radii of low-mass helium stars which are attained at $ Y_c \simeq 0.1 $ (see Table. \[tab:tracks\] and Fig. \[fig:t\_comp\]). The maximum period in the set of initial orbital periods for every computed $ {\mbox {$M_{\rm He}$}}+ {\mbox {$M_{\rm wd}$}}$ pair is, in fact, the limiting initial period which still allows RLOF and formation of an star. Nuclear evolution of He-stars with mass $M \lesssim 0.8 $ terminates after helium burning stage and they evolve directly into white dwarfs (¶, 1971).
The lifetime of helium stars is short. According to our computations for (0.35 - 0.65) mass range $$\label{eq:age}
t_{\mathrm He} \approx 10^{6.95}\,M_{\rm He}^{-4.1}(1+M_{\mathrm He}^{3.74}),$$ where time is in yr, masses in . During He-burning, the radii of helium stars increase by $\sim 30\% $ only. Therefore, the crucial factor which defines the range of the orbital periods of the precursors of stars is AML. In the Nelemans et. al. (2001) model, in $ \simeq 90\% $ of the systems helium stars overflow their Roche lobes during first $ \simeq 50\%$ of their lifetime. Figure \[fig:t\_comp\] shows that this time-span corresponds to the reduction of $Y_c $ to $\simeq 0.5 $. Thus, most of the systems might have initial orbital periods up to 100 – 120 min.
Evolution of mass-losing stars is followed to $M \simeq (0.02 -0.03)\,{\mbox {$M_{\odot}$}}$. Continuation of computations is hampered by the absence of adequate EOS and low-temperature opacity tables in our code. However, for the analysis of the scenarios of formation and evolution of stars which involves information on the elemental abundances in the donor-star this is not important, since the chemical composition of the stars in our case becomes “frozen” even before the minimum of is reached, because of the drop of the temperature in the core and switch-off of nuclear burning (see below).
Figure \[fig:mdot\] shows dependence of the mass-loss rate by Roche-lobes filling helium stars on . The behavior of low-mass components of CB evolving under the influence of AML via GWR, like for “usual” hydrogen-rich cataclysmic variables, is defined by the relation between the timescale of stellar evolution $ \tau_{\rm ev} $, thermal timescale of the star $ \tau_{\rm KH} $ and AML timescale (Faulkner, 1971; ¶, 1981; Savonije et al., 1986). The relation between the timescales itself depends on the mass of the star, its evolutionary state at RLOF and total mass of the binary (see Fig. \[fig:tscales\]). In the system (0.35+0.5), ${\mbox {$P_{\rm orb}$}}_{,0}$=20min. with initially virtually unevolved donor, $ \tau_{\rm KH} > \tau_{\rm GW} $ after the RLOF. Due to AML separation of components and Roche-lobe radius decrease. The star which is not in thermal equilibrium reacts to the mass loss by an increase of mass-loss rate. But since mass-transfer acts in opposite direction (moves components apart), $\dot{M} $ does not change significantly. A considerable fraction of the energy generated by nuclear burning is absorbed in the envelope of the star, resulting in decrease of luminosity and further growth of $ \tau_{\rm KH}$. With decrease of the stellar mass the significance of nuclear burning is rapidly diminishing. convective core disappears, while a surface convective zone appears; during this stage $ \tau_{\rm KH} \gg \tau_{\rm GW} $. As drops, continues to decrease, resulting in increase of . At certain instant the effect of mass-transfer starts to dominate, orbital period reaches the minimum, while – the maximum. With increase of the timescale of AML increases, but since the star becomes more degenerate and has convective envelope, the power of $M-R$ relation is negative and mass loss continues, but rapidly drops.
If the donor is initially more massive and more evolved at the instant of RLOF, it retains thermal equilibrium for a longer time and is initially completely defined by AML. But increases with decrease of , while decreases and, starting from a certain moment, evolution proceeds like in the lower-mass and less-evolved system described above. Evolution of characteristic timescales for the system (0.65+0.8), $P_{\rm orb,0}$ =60min. is similar to that described by Savonije et al. (1986) for a (0.6+1.4) system with $Y_c=0.28$ at RLOF which is usually quoted as “typical”.
The minimum periods of the systems with helium donors are confined to a narrow range of ${\mbox {$P_{\rm orb}$}}= (9.3 -- 10.9)$min, masses of stars at ${\mbox {$P_{\rm orb}$}}_{, min}$ are 0.20– 0.26. The models that overfilled their Roche lobes in the advanced stages of evolution reach lower periods and have at minimum period lower masses than initially less evolved donors. This may be related to the lower abundance of He and enhanced abundance of heavy elements.
Mass-radius relation {#sec:m_r}
--------------------
All existing computations of the evolution of initially non-degenerate helium donors are discontinued at ${\mbox {$M_{\rm He}$}}\simeq 0.02$. However, starting from $ {\mbox {$P_{\rm orb}$}}\simeq 20$min. these stars are, in fact, weakly degenerate homogeneous objects (the parameter of degeneracy $\psi \approx 8.33\times 10^7 \rho_c T_c^{-3/2} \sim 10$). This allows to consider qualitatively their further evolution using results of computations for mass-losing arbitrary degenerate helium dwarfs (Deloye et al., 2007). Deloye and coauthors have shown that, when the mass of the dwarf decreases to $ \simeq 0.01 - 0.03$ , their becomes comparable to the timescale of mass loss and further ${\mbox {$\tau_{\rm KH}$}}< {\mbox {$\tau_{\rm m}$}}$. The stage of adiabatic expansion terminates, the donor gradually cools, becomes more degenerate and $M-R$ relation approaches relation for fully degenerate configurations. Expected picture of evolution is confirmed by Fig. \[fig:mr\] which shows variation of stellar radii with decrease of mass. In addition to our data, we show in Fig. \[fig:mr\] mass-radius relation for degenerate helium white dwarf with initial mass of 0.3 and initial degeneracy parameter $ \log \psi =1.1 $ (the least degenerate initially model, C. Deloye priv. comm.) and $M-R$ relation for zero-temperature He white dwarfs (Deloye et al. 2007). It is seen that the morphology of $M-R$ curves for helium stars and helium dwarfs is similar and, hence, one may expect that the remnants of helium stars will gradually approach the curve for the remnants of helium dwarfs[^2]. Note, Deloye et al. in fact predicted this effect, based on consideration of the physics of cooling objects and they noticed also that some of the evolutionary tracks published by Tutukov and Fedorova (1989) have features described above.
Our results, combined with results of Deloye et al. (2007) suggest that at ${\mbox {$P_{\rm orb}$}}\apgt 40$min. two families of AM CVn stars – the “white dwarf” one and the “helium star” one – merge and the origin of the donors may be then identified by their chemical composition only (see §\[sec:chem\]).
Our conclusions concerning $M-R$ relation may have certain implications for population models of stars. Existing models (Tutukov and Yungelson, 1996; Hils and Bender 2000; Nelemans et al., 2001, 2004) were carried out under assumption that the power of $M-R$ relation is constant, keeps also for $M_{\rm He} \aplt 0.02$ and that the evolution of stars is restricted by the Hubble time only. This assumption allowed to use for modeling Eq. (\[eq:mdot\]) with constant $ \zeta (M_{\rm He})$. Such an approach is, apparently, not justified and one has to use an $M-R$ relation that takes into account growing degeneracy of donors with mass loss.
Figure \[fig:mr\] shows that the $M-R$ relation (in solar units) $$\label{eq:mr_tf}
R \approx 10^{-1.367}M_{\rm He}^{-0.062},$$ suggested by Nelemans et al. (2001) using results of Tutukov and Fedorova (1989) does not agree well with the results of more modern computations and is not valid for $ {\mbox {$P_{\rm orb}$}}\apgt 35$min.
In the period range ${\mbox {$P_{\rm orb}$}}\approx 10-35$ min which hosts 10 out of 17 stars with estimated orbital periods, the radii of the models with initial masses (0.35-0.40) may be approximated as $$\label{eq:mr_lry}
R \approx 10^{-1.478}\left(\frac{P_{\rm orb,0}}{20 {\rm min}}\right)^{-0.05} M_{\rm He}^{-0.16}\left(\frac{0.35}{M_{\rm He,0}}\right)^{0.345}$$ where $P_{\rm orb,0}$ and $M_{\rm He,0}$ are initial orbital period of the system and initial mass of the donor and $M$ and $R$ are in solar units, - in min.
The time spent by a star in the certain range of orbital periods is proportional to $ {\mbox {$P_{\rm orb}$}}/\dot{P}_{\rm orb} $. In Fig. \[fig:pdotp\] we compare dependencies of $ {\mbox {$P_{\rm orb}$}}/\dot{P}_{\rm orb} $ on for the system (0.35+0.50), ${\mbox {$P_{\rm orb}$}}_{,0}=20$ obtained in evolutionary computations and obtained by means of Eqs. (\[eq:mr\_tf\]) and (\[eq:mr\_lry\]). It is evident that in the ${\mbox {$P_{\rm orb}$}}\approx (10-35)$min. range Eq. (\[eq:mr\_tf\]) results in the 20-25% difference in the number of systems. Note, however, that this discrepancy is not very significant if one takes into account all uncertainties involved in the modeling of the population of stars.
Masses of donors in stars {#sec:masses}
--------------------------
Relatively accurate masses of components are derived only for SDSS J0926+3624 with =28.3min, a unique partially eclipsing type system (Marsh et al., 2007): $ M_{\rm wd}=0.84\pm0.05$, $ {\mbox {$M_{\rm He}$}}=0.029\pm0.002$. In the $ P-M $ diagram (Fig. \[fig:pm\]) the donor of SDSS J0926+3624 is located both below the curve corresponding to the initially most evolved nondegenerate helium donors and the curve that describes the least degenerate white dwarfs. Apparently, this system belongs to the latter family of stars (see also Deloye et al., 2007). Note however, that at the moment when =28.3min, the remnants of initially substantially evolved donors become oxygen-neon white dwarfs (see Figs. \[fig:ab035\] – \[fig:ab065\]). Though existing models of the population of stars suggest that the probability of RLOF by a far-evolved helium star is low, only analysis of chemical composition of the donor will be able to identify the scenario of the origin of SDSS J0926+3624.
We show in Fig. \[fig:pm\] also mass estimates for several other stars after Roelofs et al. (2006; 2007b). For the masses of components are based on the kinematical data: ${\mbox {$M_{\rm wd}$}}= (0.68 \pm 0.06)$, ${\mbox {$M_{\rm He}$}}= 0.125\pm 0.012$. For HP Lib, CR Boo, and V803 Cen masses of the donors are derived indirectly, assuming that for these systems one may use the relation between superhump period excess and mass ratio of components derived by Roelofs et al. (2006) for : $ \varepsilon(q) = 0.12q $. For GP Com the range of possible donor mass is based on the limits for its luminosity and assumptions about the nature of the star. The estimated masses of the donors agree with assumption that they descend from nondegenerate helium stars (see also Roelofs et al., 2007a). Note however, that there are no traces of He-burning products in the spectra of these stars.
Spectral lines of nitrogen, neon, and helium were discovered in GP Com (Lambert and Slovak, 1981; Marsh et al., 1991, 1995; Strohmayer, 2004). In particular, Strohmayer found the following abundances: $Y = 0.977\pm 0.002$, $X_{\rm N} = 1.7 \pm 0.1 \times 10^{-2}$, $X_{\rm O} = 2.2 \pm 0.3\times 10^{-3}$, $X_{\rm Ne} = 3.7 \pm 0.2 \times 10^{-3}$, $X_{\rm C} < 2 \times 10^{-3}$, $X_{\rm Mg} < 1.6\times 10^{-4}$. Considerable excess of nitrogen compared to oxygen and carbon points to the enrichment of stellar matter by the products of CNO-cycle. Such abundances may be typical for the remnants of the least massive possible progenitors (${\mbox {$M_{\rm He}$}}_{,0} \sim 0.3$) which overfilled Roche lobe soon after TAMS, before helium started to burn in their interiors. In such stars abundances in the matter lost by them is virtually the same during whole course of evolution and close to the ones found by Strohmayer.
The system (0.65+0.8)$\mathbf{ M_{\odot}, P_{\rm orb,0}=90}$min. {#sec:0650890}
-----------------------------------------------------------------
The system with initial parameters (0.65+0.8), $P_{\rm orb,0}=90$min. is of special interest because of its “nonstandard” evolution which may illustrate transitional behavior between “white dwarf” and “helium star” scenarios of formation of stars. At the instant of RLOF abundances in the center of the donor are $Y \approx 0.19$, $X_{\rm C} \approx 0.45$, $X_{\rm 0} \approx 0.34$, nitrogen is destroyed already. At difference to lower mass stars in which nuclear burning terminates after loss of several hundredth of , in this more massive and hotter star it lasts longer. When the mass decreases to $M_2 \approx 0.52$ and $Y_c$ becomes $\approx 0.008$, the star begins to contract. Mass-exchange interrupts for $\simeq 2.2$ Myrand resumes in the helium-shell burning stage (Fig. \[fig:0650890\]). After the loss of additional $\simeq 0.1$ nuclear burning ceases and the star is now a “hybrid” white dwarf with $\simeq 0.3$ carbon-oxygen-neon core and $\simeq 0.1$ helium envelope with an admixture of C and O. The timescale of AML continues to decrease, while increases. Evolution of the donor is accompanied by its cooling and decrease of radius. The dwarf experiences a stage of adiabatic contraction similar to the one that is typical for the initial stages of mass-exchange in the systems with degenerate donors (Deloye et al., 2007). In the last computed model of the sequence, =4.3 min, =0.375, degeneracy parameter of the center of the star is $\psi \approx 14$. One may expect that further evolution will be qualitatively similar to the evolution of helium white dwarf-donors: contraction phase will be followed by expansion which will continue until mass will decrease to $\simeq 0.01$ and thermal timescale will become comparable to the mas-loss timescale and a stage of contraction to the fully degenerate configuration will ensue. The minimum of the system will be several minutes. Note, that at difference to He white-dwarf donors which retain their original chemical composition during evolution, in the case of He-star descendant under consideration, the donor will become an oxygen-carbon white dwarf, with an admixture of neon $(X_{\rm O} \approx 0.27$, $X_{\rm C} \approx 0.71,$ see Fig. \[fig:t\_comp\]).
Total mass of this system is 1.45. If unstable He-burning at the surface of white dwarf does not decrease its mass below Chandrasekhar limiting mass, such a system may be a progenitor.
Evolution of stellar chemical composition {#sec:chem}
=========================================
Figures \[fig:ab035\] – \[fig:ab065\] show the evolution of chemical composition of matter lost by stars upon RLOF. As we mentioned before, in the stars with initial masses 0.35 and 0.40 nuclear burning terminates soon after RLOF. Simultaneously, disappears the convective core. When is still larger than ${\mbox {$P_{\rm orb}$}}_{, min}$, abundances in the lost matter are virtually constant and correspond to the abundances in the matter that experienced hydrogen burning via CNO-cycle. In the period range from ${\mbox {$P_{\rm orb}$}}_{, min}$ to ${\mbox {$P_{\rm orb}$}}\approx (15 - 20$)min. the convective core of initial model is uncovered. The donors in initially most wide systems are an exception, since they had large cores and chemical composition of the lost matter starts to change already before ${\mbox {$P_{\rm orb}$}}_{,min}$. But this change happens over such a short time that it is highly unlikely to observe a star in this “transition” state. Thus, in the range of that harbors almost all known stars chemical composition of the matter accreted by white dwarf is defined by the extent of helium exhaustion at the instant of RLOF (compare Fig. \[fig:t\_comp\] and Figs. \[fig:ab035\] – \[fig:ab065\]).
As we mentioned above, it is expected that the overwhelming majority of precursors of stars in the “helium star” channel of formation were formed with ${\mbox {$P_{\rm orb}$}}<$ 100 – 120 min. Then, typical abundances in the transferred matter are $Y \apgt 0.4$, $ 2\times 10^{-4} \aplt X_{\rm C} \aplt 0.3$, $ 7\times 10^{-4} \aplt X_{\rm O} \aplt 0.25$, $ 5\times 10^{-4} \aplt X_{\rm N} \aplt 0.5\times 10^{-2}$. However, one cannot exclude that abundance of nitrogen is much lower or it is absent at all; diminished (or zero) $X_{\rm N}$ has to correlate with relatively low $Y$ and enhanced $X_{\rm C}$ and $X_{\rm O}$.
We did not plot in Figs. \[fig:ab035\] – \[fig:ab065\] variations of the Ne abundance. It also changes rapidly by ${\mbox {$P_{\rm orb}$}}_{, min}$ and at ${\mbox {$P_{\rm orb}$}}> $20 min. neon abundance $2 \times 10^{-3} \aplt X_{\rm Ne} \aplt 2 \times 10^{-2}$. Variation of $ X_{\rm Ne} $ is, evidently, too small to serve as identifier of the scenarios of stars formation.
If the donor in an type system was initially a white dwarf, there are no reasons to expect that abundances in the matter lost by it vary in the course of evolution. They must to correspond to the “standard” chemical composition of a stellar core that experienced hydrogen burning, with small variations due to the differences in the masses of precursors of white dwarfs. Similar abundances will have matter lost by He-stars that filled Roche lobes soon after TAMS. However, since initial $ Y \approx 1 $, while initial $ X_{\rm C} \sim 10^{-4} $, $ X_{\rm N} \sim 10^{-2} $, $ X_{\rm O} \sim 10^{-3} $, detection of even slight enrichment in C and O or impoverishment in N may indicate possibility of formation of stars with He-star donors.
Conclusion {#sec:concl}
==========
Above, we presented results of evolutionary computations for semidetached low-mass helium components in the systems with white dwarf accretors. Evolution was considered as conservative in mass, but accompanied by angular momentum loss via gravitational waves radiation. We may summarize our main results as follows.
Evolution of binaries under consideration only weakly depends on the mass of helium star, total mass of the system, and evolutionary state of the donor at the instant of RLOF. The minimum = 9.3 – 10.9min, the masses of the donor at ${\mbox {$P_{\rm orb}$}}_{, min}$ are 0.20– 0.26 (with exception of initially very far-evolved system $(0.65+0.8)$, ${\mbox {$P_{\rm orb}$}}_{,0}=90$min.) which, in fact, represents a transitional case between systems with initially nondegenerate and degenerate donors). In the period range $20 - 40 $ min, which may be adequately described by our models and which hosts a significant fraction of all observed stars, the scatter of for a given does not exceed a factor $ \sim 2.5 $.
Our computations are limited by $ {\mbox {$M_{\rm He}$}}\apgt 0.02$. With further decrease of stellar mass thermal timescale of the donor becomes shorter than the angular momentum loss timescale, the stage of adiabatic donor expansion comes to the end, and the matter of the donor becomes more degenerate as it cools down. Qualitative comparison with the computations for initially arbitrary degenerate helium white dwarfs allows to suggest that at ${\mbox {$P_{\rm orb}$}}\apgt$ (40–45)min “white-dwarf” and “helium-star” families of stars merge and only analysis of the chemical composition of the matter lost by the donor is able to discriminate between different scenarios of formation. Such an analysis is carried out in continuation of the present study. This result also shows that existing theoretical models of the population of stars might need certain revision, since they were carried out under assumption that helium stars evolve till Hubble time with the same $M-R$ relation.
Initial periods of the most typical, according to current understanding, progenitors of stars with initially nondegenerate helium donors are confined to about (20 – 120) min. This implies a very narrow range of initial separations of components (for instance, 2– 7 for $ {\mbox {$M_{\rm He}$}}+ {\mbox {$M_{\rm wd}$}}= 0.35\,{\mbox {$M_{\odot}$}}\ + 0.5\,{\mbox {$M_{\odot}$}}$ pair). This interval is defined by a combination of different parameters. A binary passes through two common envelope stages. The “standard” equation for the variation of orbital separation of components based on energy balance between binding energy of the mass-losing star and orbital energy of the system (Webbink, 1984; de Kool et al., 1987) is $$\frac{a_f}{a_i}= \frac{M_{1,c}}{M_1}\left[ 1+\left( \frac{2}{\alpha\lambda r_{1,L}}\right)
\left( \frac{M_1-M_{1,c}}{M_2}\right) \right]^{-1},$$ where the indexes $ i$ and $f $ label initial and final separation of components, $\alpha$ – is the parameter of common envelope efficiency, $\lambda$ – is the parameter of the binding energy of the stellar envelope, $M_1 $ and $ M_{1,c} $ are initial mass of mass-losing star and the mass of its remnant, $ r_{1,L}$ is dimensionless radius of the star at the beginning of mass transfer, $ M_2 $ is the mass of companion.
For every common-envelope stage the values of $\alpha$ and $\lambda$ are, most probably, different and depend on $a_i $ at the beginning of the stage; the value of $a_f $ after the second common envelope depends on “initial-final” mass relations for progenitors of white dwarf and helium star, which, in their own turn, depend on their evolutionary state at the instant of RLOF. This suggests that formation of stars needs a very fine “tuning” of evolutionary parameters. The treatment of stellar evolution incorporated in the population synthesis codes that predict existence of stars with initially nondegenerate He-donors is, at the moment, inevitably too crude for account of all subtleties of parameters. If stars with confidently established presence of He-burning products signatures in their spectra will not be detected, this may mean that the “necessary” combination of parameters is not realized in the Nature or this combination is such, that all He-donors overflow Roche lobes prior to ignition of He or when He burns very weakly. Such a possibility is suggested by positions of and V803 Cen in the period-mass diagram (Fig. \[fig:pm\]). Since initial abundance of He in stars is close to 1, while abundances of C, O, N are by 2 to 4 orders of magnitude lower, even slight enhancements of $ X_{\rm C} $ and $ X_{\rm O} $ or reduction of $ X_{\rm N} $ may indicate possibility of formation of systems with He-star donors.
Above-discussed scenario of evolution needs some caveats. Evolution of every considered system has two phases – prior and after period minimum. In the systems with $ M_{2,0} \approx (0.35 -0.40)$ and $ {\mbox {$P_{\rm orb}$}}_{,0} \le 120$min. which we consider as typical, in the first stage of mass-exchange $ 2.5 \times 10^{-8} \aplt {\mbox {$\dot{M}$}}\aplt 10^{-7}$. Livne (1990), Livne and Glasner (1991), Livne and Arnett (1995) and other authors have shown that for accretion rate ${\mbox {$\dot{M}$}}\sim 10^{-8}$ accumulation of $ \sim 0.1 $ of He at the surface of CO white dwarf may result in detonation of He which may initiate detonation of carbon in the center of white dwarf and, presumably, SN Ia[^3]. Modeling by means of population synthesis has shown that ELDs may be dominating mechanism of SN Ia in young ($\lesssim 1$Gyr) populations (see, e.g., Branch et al., 1995; Hurley et al., 2002; Yungelson, 2005b). However, models of light-curves (Hoeflich and Khokhlov, 1996; Hoeflich et al., 1996) and spectra (Nugent et al., 1997) of objects experiencing ELDs revealed that they disagree with observations. But this conclusion is by no means final in the absence of more detailed calculations. Double-detonation, if it occurs, may prevent transformation into stars of some ($ \sim 40\%$) of their potential precursors (Nelemans et al., 2001). Note, for instance, that double-detonation may prevent formation of an star with initial parameters $ {\mbox {$M_{\rm He}$}}+ {\mbox {$M_{\rm wd}$}}=$ (0.65+0.80). It is possible that instead of a detonation of a massive He-layer recurrent Nova-scale helium flashes occur (see, e.g., Bildsten et al., 2007 and references therein). The flashes may be accompanied by mass and momentum loss from the system and change the behavior of evolutionary sequences. The problem of unstable He-burning at the surface of accreting white dwarfs and its possible impact on the evolution of stars needs additional investigation.
*Availability of results*: Detailed results of computations may be found at\
`www.inasan.ru/~lry/HELIUM_STARS/`.
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[^1]: For realization of this scenario angular momentum loss by magnetically coupled stellar wind is also necessary.
[^2]: When comparing our results with those of Deloye et. al. one must have in mind the differences in initial masses of models and degeneracy parameters, as well as in the input parameters of different codes.
[^3]: “Double-detonation” or “edge-lit detonation” (ELD).
|
---
abstract: 'We propose a physics-aware machine learning method to time-accurately predict extreme events in a turbulent flow. The method combines two radically different approaches: empirical modelling based on reservoir computing, which learns the chaotic dynamics from data only, and physical modelling based on conservation laws. We show that the combination of the two approaches is able to predict the occurrence and amplitude of extreme events in the self-sustaining process in turbulence—the abrupt transitions from turbulent to quasi-laminar states—which cannot be achieved by using either approach separately. This opens up new possibilities for enhancing synergistically data-driven methods with physical knowledge for the accurate prediction of extreme events in chaotic dynamical systems.'
author:
- 'N.A.K. Doan'
- 'W. Polifke'
- 'L. Magri'
bibliography:
- 'library.bib'
title: 'A physics-aware machine to predict extreme events in turbulence'
---
[*Introduction.–*]{} Extreme events occur in many natural and engineering systems [@Farazmand2019], such as oceanic rogue waves [@Dysthe2008], extreme climate and weather events, e.g., flooding and storm damage [@Easterling2000; @Majda2012], intermittency in turbulence [@Platt1991], and thermoacoustic instabilities in aeroengines and rocket motors [@Lieuwen2006a], to name only a few. In this Letter, we focus on abrupt self-sustaining process events in a turbulent flow [@Moehlis2004]. Turbulent flows are chaotic dynamical systems that are extremely sensitive to small perturbations to the system. This is commonly referred to as the [*butterfly effect*]{} [@Lorenz1963] in chaos theory. Because of the butterfly effect, the time accurate prediction of chaotic flows can only be achieved for a typically short time, which is called the predictability time. This is a roadblock for the [*time-accurate*]{} prediction of extreme events because, after the predictability time, a minuscule difference between the initial conditions, such as floating-point errors, is exponentially amplified. Because of this, the time-accurate prediction of extreme events is still an open problem [@Farazmand2019]. The state-of-the-art in the prediction of extreme events chiefly relies on statistical approaches, e.g., Extreme Value Theory [@Nicodemi2012] and Large Deviation Theory [@Varadhan2008]. These methods characterize the probability of the occurrence of an event and the heavy tail of the probability density function of the observable associated with the event. Notably, Sapsis [@Sapsis2018] combined Large Deviation Theory with data-driven methods to characterize efficiently the heavy tail of the distribution. These statistical methods provide an excellent framework to identify precursors and calculate the probability of extreme events, but they do not provide a robust way to time-accurately predict their occurrence and amplitude. Recently, machine learning and data-driven methods have shown great potential in learning the unpredictable dynamics of chaotic systems. In particular, Echo State Networks [@Jaeger2007; @Lukosevicius2009] (ESNs), which are a class of recurrent neural networks based on reservoir computing, have proved successful in learning the chaotic dynamics beyond the predictability time [@Pathak2018; @Pathak2018a; @Doan2019]. ESNs predict the dynamics of chaotic systems by learning temporal patterns in data only, but the learned solutions may violate physical principles. Turbulent flows, however, must obey physical principles such as momentum and mass conservation. The over-reaching objective of this Letter is to propose a machine learning method that produces physical solutions to predict extreme events in a turbulent flow. We show that constraining the physical principles in the training of the machine is key to the time accurate prediction of an extreme event. [*Turbulent flow model.–*]{} To describe the self-sustaining process in turbulence, we regard the turbulent flow as an autonomous dynamical system $\dot{\bm{y}} = \mathcal{N} (\bm{y})$ with $\bm{y}(0) =\bm{y}_0$, where $\dot{(\;)}$ is the temporal derivative; and $\mathcal{N}$ is a deterministic nonlinear differential operator, which encapsulates the numerical discretization of the spatial derivatives and boundary conditions (if any). The turbulent flow under investigation is incompressible. The domain is a cuboid of size $L_x \times L_y \times L_z$ between two infinite parallel walls at $y=0$ and $y=L_y$, which are periodic in the $x$ and $z$ directions. A sinusoidal volume force is applied in the $y$-direction. The flow is governed by momentum and mass conservation laws, i.e., the Navier-Stokes equations, which were reduced in form by Moehlis, Faisst and Eckhart (MFE) [@Moehlis2004] (see Supplementary Material). The MFE model, which provides the operator $\mathcal{N}$ in the dynamical system formulation, captures the essential features of the transition from turbulence to quasi-laminar states such as the exponential distribution of turbulent lifetimes. The velocity field is decomposed as $\bm{v}({\bm x},t) = \sum_{i=1}^9 a_i (t) \bm{v_i}(\bm{x})$, where $\bm{v_i}(\bm{x})$ are spatial Fourier modes (or combinations of them) [@Moehlis2004]. Hence, the Navier-Stokes equations are projected onto $\bm{v_i}(\bm{x})$ to yield nine ordinary differential equations for the modes’ amplitudes, $a_i$, which are nonlinearly coupled. Consequently, the state vector is ${\bm y}=\{a_i\}_1^9$. All the variables are non-dimensional [@Moehlis2004]. Physically, $\bm{v}_1$ is the laminar profile mode; $\bm{v}_2$ is the streak mode; $\bm{v}_3$ is the downstream vortex mode; $\bm{v}_4$ and $\bm{v}_5$ are the spanwise flow modes; $\bm{v}_6$ and $\bm{v}_7$ are the normal vortex modes; $\bm{v}_8$ is the three-dimensional mode; and $\bm{v}_9$ is the modification of the mean profile caused by turbulence. The flow has a fixed point $a_1=1$, $a_2=...=a_9=0$, which is a laminar state [^1]. The domain size is $L_x=1.75\pi$, $L_y=2$ and $L_z=1.2\pi$. The Reynolds number is $600$. The initial condition is such that the turbulent flow has chaotic bursts between fully turbulent and quasi-laminar states. These are the extreme turbulent events we wish to predict. The governing equations are integrated in time with a 4$^{th}$-order Runge-Kutta scheme [@Press1992] with a time step $\Delta t = 0.25$. This provides the evolution of the nine modes $a_i$ from $t=0$ to $t=30,000$ (Fig. \[fig:MFE\_evol\], top panel). The evolution of the kinetic energy, $k= 0.5 \sum_{i=1}^9 a_i^2$, is shown in Fig. \[fig:MFE\_evol\]. The time is normalized by the largest Lyapunov exponent, $\lambda_{\max}=41$, which was calculated as the average logarithmic error growth rate between two nearby trajectories [@Boffetta2002]. (The Lyapunov time scale is $\lambda_{\max}^{-1}$.) The kinetic energy, $k$, has sudden large peaks, which suddenly burst from smaller chaotic oscillations.
![Kinetic energy, $k$, and velocity field in the mid-$y$ plane. The arrows indicate the in-plane velocity ($x$-$z$ directions), the coloured contour indicates the ouf-of-plane velocity, and the grey box indicates the data used for the training of the PI-ESN. []{data-label="fig:MFE_evol"}](MFE_evolution_v3.eps){width="7cm"}
Each burst is a quasi-relaminarization event, which occurs in three phases (Fig. \[fig:MFE\_evol\]): (i) the originally laminar velocity profile becomes unstable and breaks down into vortices due to the shear imposed by the volume force (panels 5-7); (ii) the vortices align to become streaks (panels 8-9 and 1-2); and (iii) the streaks break down leading to flow relaminarization (panels 3-5). [*Physics-aware reservoir computing.–*]{} To learn the turbulent dynamics, we constrain the physical knowledge of the turbulent flow into a reservoir computing data-driven method based on the Echo State Network [@Jaeger2007; @Lukosevicius2009] (ESN): The Physics-Informed Echo State Network [@Doan2019] (PI-ESN). A schematic is shown in Fig. \[fig:ESN\_schema\]. We have training data with an input time series $\bm{u}(n)\in \mathbb{R}^{N_u}$ and a target time series $\bm{y}(n)\in \mathbb{R}^{N_y}$, where $n=0,1,2,\ldots, N_t$ are the discrete time instants that span from $0$ to $T=N_t\Delta t$. During prediction, the target at time $n$ becomes the input at time $n+1$, i.e., $\bm{u}(n+1)=\bm{y}(n)$. The training of the PI-ESN is achieved by (i) minimizing the error between the prediction, $\widehat{\bm{y}}(n)$, and the target data ${\bm{y}}(n)$ when the PI-ESN is excited with the input, $\bm{u}(n)$ (Fig. \[fig:ESN\_schema\]a), and (ii) enforcing that the prediction does not violate the physical constraints. To enforce (ii), we observe that a solution of the turbulent flow, $\bm{y}=\{a_i\}_1^9$, is such that the [*physical error*]{} (also known as the residual) is zero, i.e., $\mathcal{F}(\bm{y})\equiv \dot{\bm{y}}-\mathcal{N}(\bm{y})=0$. To estimate the physical error beyond the training data, the PI-ESN is looped back to its input (Fig. \[fig:ESN\_schema\]b) to obtain predictions $\lbrace \widehat{\bm{y}} (n_p) \rbrace_{p=1}^{N_p}$ in the time window $(T+\Delta t) \leq t \leq (T+N_p\Delta t)$. The number of collocation points, $N_p$, is user-defined. The physical error $\mathcal{F}({\widehat{\bm{y}}(n_p)})$ is evaluated to train the PI-ESN such that the sum of (i) the physical error between the prediction and the available data from $t=0$ to $t=T$, $E_d$, and (ii) the physical error for $t>T$, $E_p$, is minimized. Mathematically, we wish to find $\widehat{{\bm y}}(n)$ for $n=0,1,\ldots,N_t+N_p$ that minimizes $$\begin{gathered}
E_{tot}^P = \underbrace{\frac{1}{N_t} \sum_{n=1}^{N_t} \lvert\lvert\widehat{{\bm y}} (n) - {\bm y} (n)\lvert\lvert^2 }_{E_{d}} + \underbrace{ \frac{1}{N_p} \sum_{p=1}^{N_p} \lvert\lvert \mathcal{F}(\widehat{{\bm y}}(n_p))\lvert\lvert^2 }_{E_p},
\label{eq:EPhys}\end{gathered}$$ where $\lvert\lvert\cdot\lvert\lvert$ is the Euclidean norm. Note that the PI-ESN is straightforward to implement because it is requires only cheap residual calculations at the collocation points, i.e., it does not require solving for the exact solution. The architecture of the PI-ESN follows that of the ESN, which consists of an input matrix $\bm{W}_{in}\in\mathbb{R}^{N_x \times N_u}$, which is a sparse matrix; a *reservoir* that contains $N_x$ neurons that are connected by the recurrent weight matrix $\bm{W}\in\mathbb{R}^{N_x \times N_x}$, which is another sparse matrix; and the output matrix $\bm{W}_{out}\in\mathbb{R}^{N_y\times N_x}$. The input time series, $\bm{u}(n)$, is connected to the reservoir through $\bm{W}_{in}$ to excite the states of the neurons, $\bm{x}$, as $
\bm{x}(n+1) = \tanh \left( \bm{W} \bm{x}(n) + \bm{W}_{in} \bm{u}(n+1) \right)
$, where $\tanh(\cdot)$ is the activation function. The output of the PI-ESN, $\widehat{\bm{y}}(n)$, is computed by linear combination of the reservoir states as $\widehat{\bm{y}}(n) = \bm{W}_{out} \bm{x}(n)$. The matrices $\bm{W}_{in}$ and $\bm{W}$ are randomly generated and fixed [@Lukosevicius2012]. Only $\bm{W}_{out}$ is trained to minimize . Following [@Pathak2018a], each row of $\bm{W}_{in}$ has only one non-zero element, which is randomly drawn from a uniform distribution over $[-\sigma_{in},\sigma_{in}]$; $\bm{W}$ has an average connectivity $\langle d \rangle$, whose non-zero elements are drawn from a uniform distribution over the interval $[-1,1]$; and $\bm{W}$ is scaled such that its largest eigenvalue is $\Lambda\leq 1$, which ensures the Echo State Property [@Lukosevicius2012].
![PI-ESN during (a) training and (b) prediction.[]{data-label="fig:ESN_schema"}](ESN_schema_v4.eps){width="7cm"}
The training of the PI-ESN is achieved in two steps. First, the network is initialized by an output matrix, $\bm{W}_{out}$, that minimizes a data-only cost functional $E_{tot}^{NP} = E_d + \gamma || \bm{w}_{out,i} ||^2
\label{eq:err_data}$, where $\gamma$ is a Thikonov regularization factor and $\bm{w}_{out,i}$ denotes the $i$-th row of $\bm{W}_{out}$. This is the output matrix of the conventional ESN [@Pathak2018]. Second, the physical error is minimized with the L-BFGS method [@Byrd1995], which is a quasi-Newton optimization algorithm. [*Results.–*]{} A grid search (see Supplementary Material) provides the following set of hyperparameters for tuning, which perform satisfactorily in the range of $N_x=[500, 3000]$ neurons: $\Lambda = 0.9$, $\sigma_{in}=1.0$, $\langle d \rangle = 3$, $\gamma = 10^{-6}$. Only $t=2500$ time units (equivalent to $t^+\approx 61$) in the window $t=[11500, 14000]$ (equivalent to $t^+\approx[280, 341]$ in the grey box of Fig. \[fig:MFE\_evol\]) are used for training. The data beyond this time window is used for validation only. We use $N_p=5000$ collocation points (equivalent to $t=1250$ or $t^+\approx30.5$), which provide a sufficient number of predictions beyond the training data with a relatively low computational time. Fig. \[fig:MFE\_ESN\_comp\] shows the evolution of three representative modes’ amplitudes during the extreme event in the dashed red box in the top panel of Fig. \[fig:MFE\_evol\]. The PI-ESN solution (solid grey line) and the conventional ESN solution (dashed grey line) are computed with a reservoir of $N_x=3000$ units, and they are compared against the exact solution from numerical integration (solid black line). The normalized error between the exact evolution and the machine predictions is computed as $E(n) = \left(||\bm{y}(n) - \widehat{\bm{y}}(n) ||\right)/\sqrt{\frac{1}{N_t}\sum_{n=1}^{N_t} || \bm{y}(n)||}$ [^2]. Although the same training data is used for both the PI-ESN and the conventional ESN, the PI-ESN has a significantly higher extrapolation-in-time capability than the conventional ESN. To compare the performances, we define the predictability horizon as the time required for $E\geq0.2$ from the same initial condition. The predictability horizon of the PI-ESN is $\approx 2$ Lyapunov times longer than the predictability horizon of the conventional ESN. This significant improvement is achieved by enforcing the prior physical knowledge of the turbulent flow, whose evolution has to uncompromisingly fulfil the momentum and mass conservation laws. As shown in Fig. \[fig:MFE\_ESN\_comp\], until $t^+\approx 2.14$, both ESN and PI-ESN accurately predict the flow evolution. The predicted solution from the conventional ESN starts diverging from the exact evolution at $t^+ \approx 3.21$, which leads to a completely different solution during the extreme event. On the other hand, the PI-ESN is able to time-accurately predict the occurrence and the amplitude of the extreme event. After that the event has occurred, the solution diverges because the butterfly effect is significant. The turbulent velocity fields predicted by the conventional ESN and PI-ESN are shown in Fig. \[fig:MFE\_ESN\_comp\_2D\]a,b, respectively, which are evaluated at the same times as the exact solution in panels (3)-(5) of Fig. \[fig:MFE\_evol\]. The bottom rows of Fig. \[fig:MFE\_ESN\_comp\_2D\]a,b show the normalized absolute error between the predicted velocity field and the exact velocity field. The discrepancy in the turbulent velocity field is mainly due to the error on the prediction of the downstream vortex mode, $a_3$ (Fig. \[fig:MFE\_ESN\_comp\]). On one hand, because no physical knowledge is constrained in the conventional ESN, the sign and amplitude of $a_3$ are incorrectly predicted, which means that the out-of-plane velocity evolves in the opposite direction of the exact solution. On the other hand, the PI-ESN is able to predict satisfactorily both the in-plane velocity and the out-of-plane velocity during the extreme event.
![Evolution of $a_1$, $a_2$, $a_3$ during the extreme event of Fig. \[fig:MFE\_evol\]: exact evolution (solid black line), PI-ESN prediction (solid grey line), and conventional ESN prediction (dashed grey line) with reservoirs of $N_x=3000$ neurons. The error of the PI-ESN and ESN predictions is $E$. []{data-label="fig:MFE_ESN_comp"}](evol_comp_gray_v4.eps){width="8.6cm"}
![Evolution of the velocity field (top rows) and the normalized error (bottom rows) in the velocity field in the mid-$y$ plane at the same time instants as panels (3)-(5) of Fig. \[fig:MFE\_ESN\_comp\]. Predictions from (a) the conventional ESN and (b) the PI-ESN. The arrows indicate the in-plane velocity ($x$-$z$ directions) and the coloured contour indicates the out-of-plane velocity. The panels correspond to $t^+ = t\lambda_{\max} \approx 2.14,~3.21,~4.27$ in Fig. \[fig:MFE\_ESN\_comp\]. []{data-label="fig:MFE_ESN_comp_2D"}](evol_comp_2D_v3tick.eps){width="8.6cm"}
[*Robustness.–*]{} To quantitatively assess the robustness of the results, we compute the average predictability horizon of the machines with no further training. We follow the following steps: (i) by inspection of Fig. \[fig:MFE\_evol\], we define events as extreme when their kinetic energy is $k\geq0.1$; (ii) we identify the times when all the extreme events start in the dataset of Fig. \[fig:MFE\_evol\]; (iii) for each time, the exact initial condition at $t^+ \approx 0.61$ just before the time instant in which the extreme events starts is inputted in the PI-ESN and ESN; (iv) the machines are evolved to provide the prediction; and (v) the predictability time is computed by averaging over all the extreme events in the dataset. The results are parameterized with the size of the reservoir, $N_x$ (Fig. \[fig:MFE\_comp\_predRE\]). On one hand, for small reservoirs ($N_x\lesssim2000$), the performances of the ESN and PI-ESN are comparable. This means that the performance is more sensitive to the data cost functional, $E_d$, than the physical error, $E_p$. On the other hand, for larger reservoirs ($N_x\gtrsim2000$), the physical knowledge is fully exploited by the PI-ESN. This means that the performance becomes markedly sensitive to the physical error, $E_p$. This results in a improvement in the average predictability of $\approx 1.5$ Lyapunov times. Because an extreme event takes $\approx1$ Lyapunov time on average, the improved predictability time of the PI-ESN is key to the time-accurate prediction of the occurrence and amplitude of the extreme events.
![Comparison of the average predictability horizons of the PI-ESN and ESN for all the extreme events in the dataset.[]{data-label="fig:MFE_comp_predRE"}](Pred_comp_v2_gray.eps){width="6cm"}
[*Conclusions and perspectives.–*]{} The combination of empirical modelling – based on reservoir computing – with physical modelling – based on conservation laws – enables the time-accurate prediction of extreme events in a turbulent flow. We have compared the performance of a physics-informed echo state network (PI-ESN) and a conventional echo state network (ESN). The difference between the two networks is that the former is a physics-aware machine, whereas the latter is a physics-blind machine because it is trained with data only. In the PI-ESN, the physical error from the conservation laws is minimized beyond the training data. This brings in crucial information, which can be exploited in two ways: (i) with the same amount of available data, the PI-ESN solution is accurate for a longer time than the conventional ESN solution; or (ii) less data is required to obtain the same accuracy as the conventional ESN. In this Letter, we have taken advantage of property (i) for the prediction of extreme events in a turbulent flow. In future applications of physics-aware machines, the approach should be extended to higher dimensional dynamical systems, such as three-dimensional turbulent flows computed by high-fidelity simulations. This is challenging because the reservoir increases the degrees of freedom of the simulation, which can be aided by both massive GPU computations and nonlinear model reduction. Second, the approach should be extended to tackle dynamical systems that contain stochastic processes. This will be useful to filter out the noise from experimental data to use in the training. These aspects are currently investigated in other studies. In conclusion, this study opens up new possibilities for enhancing synergistically data-driven methods with physical knowledge for the accurate prediction of extreme events in chaotic dynamical systems.
The authors acknowledge the support of the TUM Institute for Advanced Study funded by the German Excellence Initiative and the EU 7$^{th}$ Framework Programme n. 291763. L.M. acknowledges support from the Royal Academy of Engineering Research Fellowships Scheme.
[^1]: The Supplementary material reports the initial condition, the expressions for $\bm{v_i}$ and the equations for $a_i$ [@Moehlis2004]. Detailed analysis of the MFE model is available in [@Kim2008; @Joglekar2015].
[^2]: The denominator of the error cannot be zero because the fixed point of the MFE model has $||\bm{y}|| = 1$ and the system has unsteady chaotic oscillations.
|
---
abstract: 'Streamers are a mode of dielectric breakdown of a gas in a strong electric field: A sharp nonlinear ionization wave propagates into a non-ionized gas, leaving a nonequilibrium plasma behind. The ionization avalanche in the tip of the wave is due to free electrons being accelerated in the strong field and ionizing the gas by impact. This chain reaction deeper in the wave is suppressed by the generated free charges screening the field. Simulations of streamers show two widely separated spatial scales: the width of the charged layer where the electron density gradients and the ionization rate are very large (${\cal O}(\mu$m)), and the width of the electrically screened, finger-shaped and ionized region (${\cal O}$(mm)). We thus recently have suggested to analyze first the properties of the charge-ionization-layer on the inner scale on which it is almost planar, and then to understand the streamer shape on the outer scale as the motion of an effective interface as is done in other examples of nonequilibrium pattern formation. The first step thus is the analysis of the inner dynamics of planar streamer fronts. For these, we resolve the long-standing question about what determines the front speed, by applying the modern insights of pattern formation to the streamer equations used in the recent simulations. These include field-driven impact ionization, electron drift and diffusion and the Poisson equation for the electric field. First, in appropriately chosen dimensionless units only one parameter remains to characterize the gas, the dimensionless electron diffusion constant $D$; for typical gases under normal conditions $D\approx0.1$-0.3. Then we determine essentially all relevant properties of planar streamer fronts. Technically, we identify the propagation of streamer fronts as an example of [*front propagation into unstable states*]{}. In terms of the marginal stability scenario we then find: the front approached asymptotically starting from any sufficiently localized initial condition (the “selected front”), is the steepest uniformly translating front solution, which is physical and stable. Negatively charged fronts are selected by linear marginal stability, which allows us to derive their velocity analytically. Positively charged fronts can only propagate due to electron diffusion against the electric field; as a result their behavior is singular in the limit of $D \to 0$. For $D \lesssim 1$, these fronts are selected by nonlinear marginal stability and we have to apply numerical methods for predicting the selected front velocity. For larger $D$, linear marginal stability applies and the velocity can be determined analytically. Numerical integrations of the temporal evolution of planar fronts out of localized initial conditions confirm all our analytical and numerical predictions for the selection. Finally, our general predictions for the selected front velocity and for the degree of ionization of the plasma are in semi-quantitative agreement with recent numerical solutions of three-dimensional streamer propagation. This gives credence to our suggestion, that the front analysis on the inner ($\mu$m) scale yields the moving boundary conditions for a moving “streamer interface”, whose pattern formation is governed by the evolution of the fields on the outer (mm) scale.'
address:
- ' Instituut–Lorentz, Universiteit Leiden, Postbus 9506, 2300 RA Leiden, the Netherlands, '
- ' Université Paris VII, GPS Tour 23, 2 Place Jussieu, 75251 Paris Cedex 05, France '
author:
- Ute Ebert and Wim van Saarloos
- Christiane Caroli
title: Propagation and Structure of Planar Streamer Fronts
---
[2]{}
Introduction
============
Discharges are nonequilibrium ionization processes occurring in initially non-ionized matter exposed to strong electric fields. Depending on the spatio-temporal characteristics of the electric field and on the ionization and charge transport properties of the medium, discharges can assume many different modes of appearance. In particular in gases under approximately normal conditions one distinguishes phenomenologically between stationary modes like arc, glow or dark discharges and transient phenomena like leaders, the initial stages of sparks, and streamers [@Gal; @Gal2; @Raizer; @Lagar; @vasilyak; @footnote1], which occur e.g. in silent discharges [@silent]. The latter nonstationary discharges often form the initial state of a discharge that later on becomes stationary. We will focus here on an essential element of many transient discharge phenomena, the initial field-driven ionization wave.
The conceptually simplest problem of this kind has become known as the streamer problem in a non-attaching gas. It treats the dynamics of the free electrons and positive ions in a homogeneous gas at rest taking the following mechanisms into account: [*(i)*]{} impact ionization, the process in which a free electron accelerated in a strong local field ionizes a neutral molecule, generating a new free electron and a positive ion; [*(ii)*]{} drift and diffusion of charged particles, in particular of the electrons whose mobility is much larger than that of the ions; [*(iii)*]{} the coupling of the electric field to the charges through the Poisson equation of electrostatics.
Recent numerical simulations [@DW; @Vit] of a basic model incorporating these physical ingredients for parameter values appropriate for nitrogen under normal conditions reveal that a streamer consists of a sharp nonlinear ionization front which propagates into a non-ionized gas, leaving a weakly ionized nonequilibrium plasma behind. The underlying mechanism is that in the leading edge of the front the electrons are accelerated by the large imposed electric field; this causes the build-up of an electron avalanche due to impact ionization. The generated free charges eventually screen the field and thus suppress further ionization. It is the nonlinear balance between these two nonequilibrium processes, namely the ionization avalanche and the electric screening, which determines the dynamics of the ionization front and the state of the plasma behind it. In confined geometries, streamers usually have a nontrivial fingerlike shape, as is illustrated by the snapshots in Fig. 1 of streamer dynamics taken from the simulations of Vitello [ *et al.*]{} [@Vit]. As the sharpness of the electron density profiles in Fig. 1 illustrates, the “passive body” of the finger is separated from the external non-ionized gas by a very narrow region — of width of order microns — in which essentially all the action is occurring. This width has to be compared to the size of the filament, which is of order millimeters. It is in this narrow layer that most of the ionization process is taking place. In this same region, there is a nonzero charge density, and consequently, also a very large electric field gradient. These features indicate that there are two different spatial scales in this process, an “inner” scale associated with the thickness of the zone where the ionization takes place, and an “outer” one where the spatial variations are set by the size of the finger and the external experimental geometry. It is precisely for these reasons that accurate simulations are extremely demanding and that they were accomplished only recently by Dhali and Williams [@DW] and by Vitello [*et al.*]{} [@Vit] (See also [@wan]).
Such a separation of scales is strongly reminiscent of what occurs in combustion fronts [@wil; @buc]. A combustion front is a narrow layer of thickness $\ell_{in}$ to which the combustion is essentially confined, while outside of it, the temperature field varies on a much longer scale $\ell_{out}$. Physically, such sharp combustion fronts occur in the limit when the chemical reaction rates involved in the combustion are very fast once a sufficient temperature is reached. It has been shown that, on the basis of an asymptotic expansion to lowest order in the small parameter $\varepsilon= \ell_{in} / \ell_{out}$ using matched asymptotic expansions [@vDyke; @bender], the problem can be analyzed in terms of the propagation of an “effective interface”. More specifically, one first solves the so-called inner problem of a locally [*almost planar*]{} reaction zone. This permits to relate the temperature and chemical composition fields on both sides of the front (at distances $L$ such that $\ell_{in} \ll L \ll
\ell_{out}$) and to determine the local front velocity as a function of local curvature and fields. On the scale of the remaining outer problem, these relations then play the role of boundary conditions and of a kinetic equation for the effective moving interface of zero thickness. Besides in combustion, the technique of asymptotic matching to obtain an effective interface description has also been applied to chemical waves [@Meron], thermal plumes [@ZTB] and to phase field models of solidification [@pf1; @pf2].
In spite of some important differences between combustion and streamer fronts as discussed in the appendix, a similar approach appears possible for streamers. As discussed also in [@short], building on such a reduced description of streamer dynamics appears very desirable, not only because it might make numerical studies much easier, but also because it will allow us to draw upon the knowledge and methods which have been developed in the last decade in the field of interfacial pattern formation and dynamics [@reviews]. The first step towards this goal is to determine the field dependence of the velocity and the ionization and charge profile of a planar front which propagates into the non-ionized region. We thus analyze in this paper the inner problem for a planar streamer front. This allows us to reduce the problem to effectively one dimension. Our analysis clearly identifies the problem of streamer front propagation as an example of front propagation into unstable states. Physically, the instability of the non-ionized gas against charge fluctuations can be traced back to the fact that any small electron density gets amplified by the impact ionization. As is standard for front propagation into unstable states [@benjacob; @vs1; @vs2; @vs3; @Oono], we find that the one-dimensional streamer equations exhibit a one-parameter family of uniformly translating front solutions, parametrized by their velocity. As usual [@benjacob; @vs1; @vs2; @vs3; @Oono], the question is then to decide which of these front solutions is the dynamically selected one, i.e., is the one reached at long times after a localized ionized region has been created by some initial ionization event. The existing knowledge of front propagation into unstable states [@vs1; @vs2] provides us with an educated guess for the selected velocity, which we confirm with the help of numerical studies. Taken together, our results provide an essentially complete solution of the inner problem of planar streamer fronts.
In itself, the idea to analyze the planar fronts of a streamer model is not new — we refer to [@TO; @AT; @Fowler; @Dya] for earlier work. Apart from the fact, that the authors from the seventies [@TO; @AT; @Fowler] investigate different models, which are more inspired by equilibrium concepts (e.g., the ionization behind the front is determined by thermal ionization, where the electron temperature is raised by application of strong electric fields), our work casts new light on this old problem from two different angles:\
First of all, it was empirically noted, that the standard approach to analyze uniformly translating fronts, failed to determine a unique propagation velocity, given the field and the gas parameters. Turcotte and Ong [@TO] clearly state this failure of their theory (this “great defect” of their theory is recalled in Fowler’s reviews [@Fowler]) and suggest, that a unique solution might be determined by a dynamical stability analysis. Albright and Tidman [@AT] then perform such a stability analysis, but not in a systematic way, and they draw incorrect conclusions. D’yakonov and Kachorovskii [@Dya] also find the indeterminacy of the speed of uniformly translating planar fronts, now for an approximated version of our model, and propose to solve this by using the tip radius of the streamer finger as an extra length scale, which they, however, cannot determine. We, in contrast, trace the indeterminacy of the velocity from the analysis of uniformly translating streamer front solutions to the fact, that this is an example of front propagation into unstable states. Applying the concepts explained above, we solve the selection problem for planar fronts without additional assumptions or approximations. We argue that a particular front solution out of a whole family of dynamically stable solutions is selected, because it is the only one compatible with the initial condition of a localized ionization seed.\
Secondly, this result is the first [*ingredient*]{} for studying the formation of patterns, in particular of the tip radius — we do not attempt to model [*global features of the pattern formation*]{} with our planar front analysis. Our approach thus is very different in spirit to the earlier investigations: As also stressed in [@short], in an effective interface description based on a matched asymptotic expansion, the results of weakly curved, almost planar fronts are essentially used [*locally*]{} everywhere in the interface region: They enter the analysis on the outer scale as boundary conditions at the moving interface. It is on this outer scale that pattern formation problems like the size, velocity and shape of the streamer should be analyzed. Once our results on planar fronts will be extended to weakly curved fronts, all the necessary ingredients to tackle these questions appear to be available.
The main results of our present analysis of the streamer equations used in the simulations [@DW; @Vit] can be summarized as follows:
[*(a)*]{} Dimensional analysis shows that in dimensionless units, a single parameter remains to characterize the gas, the dimensionless electron diffusion coefficient $D$ characteristic of the gas \[see Eq. (\[2-8\])\]. For gases under normal conditions, $D$ is small, of order 0.1-0.3.
[*(b)*]{} The length scale set by the electron impact ionization coefficient \[the coefficient $\alpha_0^{-1}$ in Eq. (\[2006\])\] is on the order of microns for nitrogen. For $D \lesssim 1$ the thickness $\ell_{in}$ of the charged layer is on the order of this same ionization length for [*negatively charged streamer fronts*]{} [ *(NSF)*]{} [@nomenclature]. Given that typical streamer diameters found in the simulations are of the order of $1$mm, $\varepsilon =
\ell_{in} / \ell_{out}$ is at most of order $ 10^{-2}$; this justifies an effective interface description of streamer dynamics.
[*(c)*]{} We find that electron diffusion acts as a singular perturbation for [*positively charged streamer fronts (PSF)*]{}: without diffusion, such fronts can not propagate, but with any nonzero $D$, they do. As a result, the behavior is singular in the limit $D
\rightarrow 0$: for $D={\cal O} (1)$, the thickness $\ell_{in}$ is again of order of the ionization length, but for $D\rightarrow 0$ the electron density and its gradients diverge due to the appearance of another smaller length scale (of order $D/ \alpha_0 $).
[*(d)*]{} The electron density generated by the propagating front is again basically set by dimensional analysis for [*NSF*]{}. We calculate for $D \lesssim 1.5$ the dependence of the dimensionless electron density $\sigma^-$ behind the front on the electric field $E^+$ far ahead of our planar front. Our results compare favorably with those extracted from the simulations [@Vit], according to the prescriptions of the theory of matched asymptotic expansions [@vDyke; @bender]. Namely, $E^+$ is [*not*]{} the field value at the electrode position, but the value obtained by extrapolating the slowly varying outer field to the front position. We also calculate the full $D$ and $E^+$ dependence of the electron density $\sigma^-$ behind the front of [*PSF*]{} for $D \lesssim 1.5$.
[*(e)*]{} The dynamically relevant (“selected”) front velocity $v_f$ is a unique function of $E^+$ and $D$. The analysis confirms the strong asymmetry between $NSF$ and $PSF$ also found in the simulations [@DW; @Vit] for fronts propagating into an essentially non-ionized region. The asymmetry is the stronger the smaller $D$ and disappears for $D \gg 1$.
[*(f)*]{} For $NSF$, $v_f$ is given by the so-called linear marginal stability velocity $v^*$ [@vs1] — see Eq. (\[w1\]) below. For parameter values used in the simulations, we find that $v_f$ is typically 30 to 40 % higher than the electron drift velocity just in front of the streamer head, which agrees semi-quantitatively with the findings of Vitello [*et al.*]{} [@Vit].
[*(g)*]{} We find that $PSF$ propagate for any nonzero value of the dimensionless electron diffusion coefficient $D$. Due to the singular behavior as $D \rightarrow 0$, we find that fronts propagate with a unique velocity $v^\dagger$ predicted by the so-called nonlinear marginal stability mechanism [@vs2] for small $D$. For the Townsend expression used in the simulations [@DW; @Vit], this happens below a well-defined field-dependent value of $D$ of order unity \[see Fig. 3\]. Above this threshold value, $PSF$ propagate with the linear marginal stability value $v^*$.
In this paper, our main focus will be on those results that are of greatest interest from the point of view of understanding the generation of low temperature plasmas by the streamer mechanism. We note, however, that the equations for planar streamer fronts \[Eqs. (\[308\]) and (\[309\]) below\] appear to be of interest in their own right. As will be discussed briefly in Section V, our streamers have several features in common with the celebrated nonlinear diffusion equation studied in mathematics [@aw; @eckmann] since the early work of Kolmogorov [*et al.*]{} [@kpp] and Fisher [@fisher]; at the same time, however, they are sufficiently more complicated that they appear to present new challenges from a mathematical point of view.
This paper is organized as follows. In Section II we introduce the basic equations for streamer formation, and perform a dimensional analysis for the inner problem of streamer fronts. In Section III, we discuss the stability of the basic homogeneous states of interest, the homogeneous non-ionized state and the homogeneous weakly ionized state. We also discuss the physical mechanism of streamer formation and the proper initial and boundary conditions to study these in the case of planar fronts, which allow us to simplify the equations describing planar front dynamics. In Section IV we demonstrate that there exists a one-parameter family of uniformly translating fronts characterized by a continuous range of front velocities $v$. We also briefly show how in the case $D=0$, the equations for uniformly translating fronts can be solved analytically. These solutions, which turn out to be useful as a small-$D$ approximation for $NSF$, yield an explicit expression for the electron density $\sigma^-$ behind the $NSF$ in terms of the field $E^+$ just ahead of it. This is followed by an analysis of the general case $D \neq 0$; then the equations can not be solved analytically, but we demonstrate that there still is a one-parameter family of uniformly translating front solutions. For $PSF$, we show that the limit $D \rightarrow 0$ is singular; we discuss this limit in detail and show that it accounts for the strong asymmetry between $PSF$ and $NSF$ for realistic values of $D$. In Section V we then summarize some of the main results [@benjacob; @vs1; @vs2; @vs3; @Oono] concerning the so-called selection problem, the question which particular front solution from the family is reached asymptotically for large times for a large class of initial conditions. Application of these concepts allows us to predict the shape and velocity of the dynamically relevant front solution (the selected front) and the value of the electron density generated behind it. This yields the various selection results for $NSF$ and for $PSF$, summarized in points [*(c)-(g)*]{} above, and leads us to predict that the behavior of $PSF$ in the limit $D\rightarrow 0$ is singular. In Section VI we present numerical simulations of the full partial differential equations for planar streamer dynamics; starting from various initial conditions, we illustrate, that in all cases we have studied the long time dynamics of the system is characterized by a $NSF$ and a $PSF$ whose behavior is in full agreement with our predictions. In the concluding section we finally reflect on our results and on the future steps to be taken to arrive at an effective interface description of streamer dynamics. In an appendix we discuss differences and similarities between combustion and streamer fronts.
Modeling and Dimensional Analysis
=================================
The Minimal Streamer Model
---------------------------
For simulating the dynamical development of streamers out of a macroscopic initial ionization seed in a so-called non-attaching gas like N$_2$ under normal conditions, Dhali and Williams [@DW] and Vitello [*et al.*]{} [@Vit] use the following set of deterministic continuum equations for the electron density $n_e$, the ion density $n_+$ and the electric field ${\cal E}$: balance equations for electrons and ions, $$\begin{aligned}
\label{2011}
\partial_t\;n_e \;+\; \nabla_{\bf R}\cdot{\bf j}_e
&=& source ~,
\\
\label{2012}
\partial_t\;n_+ \;+\; \nabla_{\bf R}\cdot{\bf j}_+
&=& source ~, \end{aligned}$$ where the fact that the two source terms are the same is due to charge conservation in an ionization event; the Poisson equation, $$\label{2013}
\nabla_{\bf R}\cdot {\cal E} = {{e}\over{\varepsilon_0}} \;(n_+ -n_e)
~,$$ and the approximate phenomenological expressions $$\begin{aligned}
\label{2014}
{\bf j}_e &=& - n_e \;\mu_e \;{\cal E} - D_e \;\nabla_{\bf R} \;n_e ~,
\\
\label{2005}
{\bf j}_+ &=& 0~,
\\
\label{2006}
source &=& |n_e \mu_e {\cal E}| \;\alpha_0
\;\mbox{\large{e}}^{\textstyle -E_0/|{\cal E}|}~. \end{aligned}$$ Apart from the fact that we will allow for a slight generalization of Eq. (\[2006\]), these are the equations that we will investigate analytically below.
In these equations, ${\bf j}_e$ and ${\bf j}_+$ are the particle current densities of electrons and positive ions, and $e$ is the absolute value of the electron charge. The (dimensional) spatial coordinates are denoted by $\bf R$, and $\nabla_{\bf R}$ is the gradient with respect to these coordinates. The use of only Poisson’s law of electrostatics, Eq. (\[2013\]), means that all magnetic fields as well as terms in the Maxwell equations associated with time-dependences of the fields, are neglected [@maxwell].
The electron particle current density ${\bf j}_e$ is approximated in (\[2014\]) as the sum of a drift and a diffusion term. Note that this diffusion approximation implies that the electron mean free path must be small with respect to the scale of variation $\ell_{in}$ of the electric field. This condition is just about satisfied for the parameter values taken for $N_2$ in the simulations, except possibly at the highest field values (see also the discussion in the concluding Section VII). The electron drift velocity is taken to be linear in the field ${\cal E}$, with $\mu_e$ the (positive) electron mobility. The electron diffusion coefficient $D_e$ and the mobility $\mu_e$ are treated here as independent coefficients, since they effectively depend on the field strength [@Raizer] (only in the low-field limit are they related by the Einstein relation). More generally, the diffusion coefficient should be replaced by a diffusion tensor, which is diagonal in a reference frame with one axis along the electric field. Its longitudinal component, the only relevant one for planar fronts perpendicular to $E$, is somewhat smaller than the transverse one. Since we will see, that N$_2$ reaches a typical degree of ionization of only $10^{-5}$, density fluctuations of the non-ionized gas can be neglected and the mean free path of the electrons and therefore $\mu_e$ and $D_e$ can be taken as independent of the degree of ionization.
The ionic current is neglected according to Eq. (\[2005\]), since the mobility of ions is at least two orders of magnitude smaller than that of the electrons [@DW]. In particular, for the analysis of the inner scale, that we will perform in the present paper, ${\bf j}_+$ is negligible.
The $source$ (\[2006\]) finally accounts for the creation of free charges by impact ionization. If the product of electric field ${\cal
E}$ and electronic mean free path $\ell_{mfp}$ is large enough, free electrons can gain sufficient kinetic energy to ionize neutral molecules. Accordingly there is a threshold field $|{\cal E}|=E_0
\propto {\ell_{mfp}}^{-1}$. For $|{\cal E}| \gtrsim E_0$ the probability that a scattering event carries at least the ionization energy is large. The effective ionization cross-section $\sigma_{cs}(|{\cal E}|)$ then essentially saturates, while for $|{\cal E}| \ll E_0$ the ionization rate per scattering event is largely suppressed. The $source$-term is given by the ionization rate, which can be calculated as the product of the drift current of free electrons $|n_e\mu_e {\cal E}|$ times the target particle density $n_n$ of the neutral gas times the effective ionization cross-section $\sigma_{cs}(|{\cal E}|)$. Commonly, a phenomenological ionization coefficient $\alpha(|{\cal E}|) = n_n
\sigma_{cs}(|{\cal E}|)$ is used, (which clearly has dimension of inverse length,) whose field threshold behavior in the Townsend approximation $\alpha(|{\cal E}|)=\alpha_0 \exp(-E_0/|{\cal E}|)$ [@Raizer] is expressed by Eq. (\[2006\]). As discussed by Raizer [@Raizer], in the approximation that every collision is ionizing, if the electron carries an energy larger than the ionization energy $I$, we have $$\label{mfp}
\alpha_0 \approx \ell^{-1}_{mfp}~,\quad \mbox{and } \quad E_0 \approx
I/(e\;\ell_{mfp})~.$$ Since in much of our analysis the specific form of $\alpha(|{\cal
E}|)$ is not needed, we will use a slightly more general formulation in Eq. (\[2022\]) below.
In the $source$-term, ionization due to the photons also created in recombination or scattering events, is neglected. This is motivated by the ionization cross-sections due to photons being much smaller than those due to electrons. Note that, if photo-ionization is taken into account, the dynamical equations become nonlocal.
No sink term needs to be included for the analysis of the inner problem, since the recombination length at a degree of ionization of order $10^{-5}$ that we will derive below is very large as compared with the front width $\ell_{in}$. (For this reason, the inner problem is the same for streamers and leaders [@Raizer]: the difference between these discharge modes, which consists in the fact that recombination is non-negligible in the plasma body of leaders, would come into play only when solving, at a later stage, the outer problem.) The fact that the degree of ionization remains small is also the reason that saturation effects are neglected in (\[2006\]).
In contrast to the situation in N$_2$, that is described by our model equations, in attaching gases like O$_2$, a third charged species plays a role, namely negative ions formed by a neutral molecule catching a free electron. For a description of the physics of such attaching gases and simulations thereof, see, e.g., [@nonattach].
The equations above are deterministic. Thermal fluctuations in fact can be neglected, since even an unphysically small ionization energy of 3 eV leads to a Boltzmann factor of $10^{-52}$ at room temperature. Also other stochastic effects are not accounted for in the simulations we compare to. We further discuss possible stochastic effects in the experiments in the conclusion.
Finally, the dynamical system (\[2011\])-(\[2006\]) must be complemented by:
[*(i)*]{} boundary conditions: as will be discussed in detail in Section III, for the problem of front propagation, these are specified by the value $E^+$ of the electric field far ahead of the front, where the total charge density vanishes.
[*(ii)*]{} initial conditions: we ignore the details of the plasma nucleation event (e.g. triggering by radiation from an external source), and assume that at $t=0$ a small well-localized ionization seed is present. The precise meaning, for our problem, of “well-localized” will be made clear in Section V.
Dimensional Analysis
--------------------
In order to identify the physical scales and intrinsic parameters of our problem, we reduce Eqs. (\[2011\])-(\[2006\]) to a dimensionless form. The most natural scale of length and electric field are the ionization length $\ell_{ion}= \alpha_0^{-1}$ and the threshold field $E_0$ of the ionization rate (\[2006\]). The velocity scale is then the electron drift velocity at this field strength, $v_0=\mu_e E_0$, leading to a time unit $t_0=(\alpha_0 \mu_e E_0)^{-1}$, and a charge unit $q_0=\varepsilon_0 \alpha_0 E_0$.
For concreteness, we list here the values of these quantities for N$_2$ at normal pressure, used in the simulations [@DW; @Vit] $$\begin{aligned}
&\alpha_0^{-1} \approx 2.3\;\mu\mbox{m}~ , \quad
& \quad v_0 \approx 7.56 \cdot 10^7\;\mbox{cm/s}~, \nonumber\\
&t_0 \approx 3\cdot10^{-12} \;\mbox{s}~, \quad
& \quad q_0 \approx 4.7 \cdot10^{14}\; e/\mbox{cm}^3~ \label{2026} \\
\nonumber &E_0 \approx 200 \;\mbox{kV/cm} \quad
& \quad \mu_e \approx 380 \;\mbox{cm}^2\mbox{/Vs}~. \end{aligned}$$ We now introduce dimensionless quantities by defining $$\begin{array}{ll}
\nonumber {\bf r}={\bf R}\; \alpha_0 ~, & \tau= t \;/t_0 ~, \\
q=(n_+ - n_e)\; e/ q_0
~,\hspace*{8mm} & \nonumber \sigma= n_e \; e /q_0 ~, \\
\label{2-7} {\bf j} = -{\bf j}_e \;
e / (q_0v_0) ~, & {\bf E} = {\cal E} /E_0 ~.
\end{array}$$ Note that with our definition, ${\bf j}$ now plays the role of a dimensionless [*charge*]{} current. If we furthermore introduce the dimensionless diffusion coefficient $D$ as $$\label{2-8}
D= D_e \alpha_0 / \mu_e E_0~,$$ we obtain what we call the streamer equations $$\begin{aligned}
\label{2022}
\partial_\tau\;\sigma \;-\; \nabla\cdot{\bf j}
&=& \sigma \; f(|{\bf E}|)~,
\\
\label{2023}
\partial_\tau\;q \;+\; \nabla\cdot{\bf j}
&=& 0~,
\\
\label{2024}
q &=& \nabla\cdot{\bf E}~,
\\
\label{2025}
{\bf j} &=& \sigma \;{\bf E} + D \;\nabla \sigma~, \end{aligned}$$ where $\nabla$ denotes the gradient with respect to the dimensionless coordinate [**r**]{}, and where the “ionization function” $$\label{2-13}
f(|{\bf E}|) = |{\bf E}| \; \alpha(|{\bf E}|) / \alpha_0~$$ is assumed to vanish at zero field. Townsend’s expression (\[2006\]) yields: $$\label{2-14}
f_T(|{\bf E}|) = |{\bf E}| \; \exp(-1/ |{\bf E}|)~.$$ In general, we will treat an ionization function with the properties [@footnotef] $$\label{2-17a}
f(0)=0=f'(0)~,\quad \mbox{and} \quad f'(|{\bf E}|) \ge 0
\quad \mbox{for all }~|{\bf E}|~.$$ The dimensionless equations (\[2022\])-(\[2025\]) now depend on only one internal parameter, the dimensionless diffusion coefficient $D$. For the values used in [@DW; @Vit] for N$_2$ under normal conditions, $D\approx 0.1$, while according to the data given by Raizer [@Raizer], for Ne and Ar, $D\approx 0.3$. We believe that typical values are generally in the range 0.1-0.3, since in the approximation (\[mfp\]), $\alpha_0/E_0 \approx I/e$ and since the ratio $D_e/ \mu_e$ appears to be commonly of the order of volts for large fields, while $I$ is typically of the order of several electron volts.
We are now able, solely on the basis of the dimensional analysis above, to make a first semi-quantitative prediction about streamers. We will in practice be interested in external fields $E^+ = {\cal
O}(1)$ (for $E^+ \ll 1$ and $\alpha_0^{-1}$ on the order of micrometers, the electron avalanche process becomes much too ineffective for streamer fronts to develop at reasonably small distances; also our scale separation approach discussed in the introduction might break down). We can therefore expect that, for $D$ values $\lesssim 1$, as is the case for N$_2$, front widths will be of order $\alpha_0^{-1}$, and that in addition the reduced electron density $\sigma^-$ far behind the front on the inner scale will be of order unity as well. This leads one to expect electron densities in the streamer body on the order of $10^{14}$ cm$^{-3}$, in agreement with numerical findings.
Homogeneous Solutions and the Concept of Fronts
===============================================
Homogeneous States and their Stability
--------------------------------------
The first task, when studying in general the propagation of a front, is to identify the nature and stability of the states which the front connects. We expect the invading state, here the ionized one created by the front, to be stable [@footnote5a], while the invaded state can in general either be metastable or unstable. Physically, we of course expect the non-ionized state to be unstable in a non-vanishing field in the present model. (In an attaching gas forming also negative ions, it is conceivable that the non-ionized state is metastable for not too strong fields.)
Equations (\[2022\])-(\[2025\]) immediately yield, that stationary homogeneous states simply are solutions of $$\label{2-15}
\sigma f(|{\bf E}|) =0~.$$ So, these stationary states decompose into two families:
[*(i)*]{} Non-ionized states, with $\sigma=0$, ${\bf E}$ arbitrary: Since the density of free electrons vanishes, no ionization can occur, whatever the value of ${\bf E}$ is. If also the density of ions vanishes, $\nabla \cdot {\bf E}=0$. Since these states correspond to the physical situation far ahead of the front, we label them (+). Moreover, since we will need in particular the case in which the field ahead of the front is constant, we take ${\bf E}^+=$const.
[*(ii)*]{} Completely screened states, labeled ($-$), with ${\rm
E}=0$, $\sigma^-$ arbitrary [@footnote5ab]: Whatever the electron density, for ${\bf E}=0$ impact ionization does not occur and thermal energy is much too small to permit ionization.
Since the steady states we consider as well as the equations of motion are translation invariant in space and time, the eigenstates of the linear perturbations are Fourier modes of the form $$\left( \begin{array}{l} \delta \sigma ({\bf r},t) \\
\delta {\bf E}({\bf r},t)
\end{array} \right)
= \left( \begin{array}{l} \sigma_1 \\ {\bf E}_1 \end{array} \right)
\exp(i{\bf k} \cdot {\bf r} + \omega \tau)~.
\label{2-16}$$ We first investigate the linear stability of the non-ionized state $\sigma^+=0$. Upon linearizing the equations about the zeroth order values $(\sigma^+=0, {\bf E}^+)$, we find two branches of modes:
[*(a)*]{} The first, trivial branch is a zero mode $(\omega=0)$, with $\sigma_1=0$, expressing that the electron density remains zero. This zero mode accounts for the degeneracy of the non-ionized states, i.e., for the fact that there exists a (+) stationary state for each value of ${\bf E}^+$. (For ${\bf E}^+ \neq {\rm const.}$, these zero modes express the degeneracy of all steady states with $q^+=\nabla \cdot
{\bf E}^+$ for any ion density $q^+$ as long as the electron density $\sigma^+$ vanishes.)
[*(b)*]{} The second branch of perturbations is associated with fluctuations carrying a finite electron charge; its dispersion relation is $$\label{2-17}
\omega^+ = i {\bf k}\cdot {\bf E}^+ + f(|{ \bf E}^+|) -Dk^2~,$$ with $i\omega^+ {\bf k} \cdot {\bf E}_1 =(f(|{\bf
E}^+|)-\omega^+)\sigma_1$. The first term on the r.h.s. of (\[2-17\]) simply expresses the fact that the electrons drift, to first order, in the electric field ${\bf E}^+$ with velocity $(- {\bf
E}^+)$. The real part $\Re \omega^+$, the sign of which determines whether fluctuations decay or are amplified, contains a destabilizing term, expressing that any small electron density fluctuation is amplified at rate $f$, and a stabilizing term, due to the diffusive spreading of electron charges. For $k^2<f(|{\bf E}^+|)/D$, $\Re
\omega^+>0$: non-ionized states are unstable against long-wavelength perturbations.
We note, that the single Fourier eigenmodes (\[2-16\]) violate individually the physical constraint that $\sigma$ be positive everywhere. But Eq. (\[2-17\]) also determines the time evolution of physically allowed fluctuations (wavepackets) that are superpositions of these eigenmodes. For example, one easily deduces from it Lozanski’s expression [@Loz] for the time-evolution of a Gaussian-shaped small electron density with arbitrary constants $c_1,c_2>0$, $$\begin{aligned}
\label{2034}
\delta \sigma({\bf r},\tau) &=& c_1\;
\mbox{\large e}^{\textstyle \;f(|{\bf E}^+|)\:\tau }\;\;
(c_2+4D\tau)^{-3/2} \nonumber\\
&& \qquad\mbox{\large e}^{\textstyle -\;
({\bf r}+{\bf E}^+\tau)^2/(c_2+4D\tau) }~, \end{aligned}$$ as long as linearization around the non-ionized state holds. As expected, the center of the spreading packet drifts with velocity $-{\bf E}^+$, while the total number of electrons it contains is amplified at rate $f$ and the wave-packet stays Gaussian, with time-dependent width $c_2+4D\tau$. Such ionization modes derived by linearizing around the non-ionized state are known as electron or ionization avalanches in the gas discharge literature.
We now perform the same linear stability analysis for the completely screened states $(\sigma^-={\rm const.},{\bf E}^-=0)$. The fact that $f'(0)=0$ from Eq. (\[2-17a\]) assures that the linear perturbations are not affected by ionization; the dynamics thus evolves with conserved particle densities.
Again, due to the existence of a continuous family of screened stationary states, parametrized by $\sigma^-$, the spectrum contains a branch of $\omega=0$ modes. For the nontrivial branch, the dispersion relation is given by $$\omega^- = -\sigma^- -Dk^2~,
\label{2-19}$$ while the eigendirection of such a perturbation is given by $$\label{2-20}
\sigma_1 + i {\bf k}\cdot {\bf E}_1 =\sigma_1 + q_1 =0~.$$ Since $(\sigma_1+q_1)$ is the dimensionless ion density of the linear mode, Eq. (\[2-20\]) simply expresses the fact that ions are completely immobile in our model.
Equation (\[2-19\]) expresses the fact that the completely screened ($-$) states are stable, the decay of perturbations being due to the added stabilizing effects of overdamped plasmons $(-\sigma^-)$ and electron diffusion. The ${\bf k}\to 0$ limit of the plasmon mode leads to dielectric screening [@panof].
The Mechanism of Front Creation
-------------------------------
Let us now investigate the dynamical evolution of an initial state in which the electron and ion densities vanish everywhere except in a small localized region. An example of such localized initial conditions is an initially Gaussian electron density, as in the simulations [@DW; @Vit] — under what circumstances initial conditions are sufficiently localized will become clear later. As long as the electron and ion densities are small enough, we can neglect in linear approximation the changes in the field as we did above when linearizing about the non-ionized state. As a result, both densities will grow due to impact ionization. If this were the only mechanism, the space charge would remain unchanged and the ionization would continue indefinitely. However, the electrons are mobile, and at the same time they start to drift in the direction opposite to the electric field $ {\bf E}$. If we neglect for the moment the diffusion, this drift has two effects: First of all, the electrons start to drift in the direction of the anode. Impact ionization then starts in previously non-ionized regions as well, so the ionized region expands towards the anode. Secondly, as the electrons drift while the ions stay put (on the fast time scale), a charge separation occurs which tends to suppress the field strength in the ionized region. When the size of the initial perturbation and/or the time during which the avalanche has built up are large enough, the screening of the field becomes almost complete in the ionized region so that ionization stops there. The behavior in this region can be described by linearizing around the screened state as done above. After an electrically screened body of the ionized region has developed, the initial ionization avalanche is said to have developed into a streamer. Thus streamer fronts are strongly nonlinear and determined by two competing mechanisms, which dynamically balance each other: the ionization process which is strongest at the leading edge and the screening of the field due to the free charges which increases towards the rear end of the front. This balance also explains our finding that the ionization length and the screening length in the plasma behind the $NSF$ are of the same order of magnitude for field values that are not too small. Technically speaking, the challenge in constructing the full front is to connect the two regimes linearized about the homogeneous states in an appropriate way through the nonlinear regime of the front.
In the above discussion, we have neglected electron diffusion. In this case the $NSF$ propagates towards the anode with a velocity that is at least the drift velocity of the electrons in the local electric field. The $PSF$, in contrast, is moving in the direction [*opposite*]{} to the drift of the only mobile species, the electrons. Its space charge is formed by the ions staying put, while the electrons are drawn into the ionized body. Propagation of a $PSF$ is therefore only possible if the electron diffusion current overcompensates the drift current. This in turn implies that if the diffusion coefficient $D$ is small, electron density gradients must be extremely steep. From this discussion it already becomes evident — and we will derive this below — that for an $NSF$, diffusion is a small correction for $D \ll 1$, since drift and diffusion currents are acting in parallel directions. In $PSF$, however, diffusion has to overcome the drift, and as a result in this case the limit of vanishing diffusion is very singular. We will see in Section IV that this manifests itself through the emergence of a new inner length scale $D/ \alpha_0 = D_e/(\mu_e E_0)$, the diffusion length associated with the electron drift velocity.
Of course, a charged front only screens the normal component of the electric field. This is why electric screening is efficient in the head of the streamer, while the field penetrates in the body of a single streamer in the simulations [@DW; @Vit]. Our planar front analysis thus serves as a first approximation for the mechanisms in the moving tip of the streamer finger.
The one-dimensional Streamer Equations
--------------------------------------
Let us now restrict our analysis to the case of plane fronts perpendicular to a constant electric field. Of course, in practice planar streamer fronts will be unstable to deformations along the front (very much like in the Mullins-Sekerka instability in crystal growth [@reviews]), but as explained in the introduction, the planar front analysis is a first step towards understanding the dynamics on the inner length scale $\alpha_0^{-1}$ and time scale $t_0$. As such, it is the first basic ingredient for deriving an effective interface model on scales $\gg \alpha_0^{-1}$.
We choose the $x$ axis as parallel to the field and perpendicular to the planar front so that $ {\bf E} = E \hat{\bf x}$ and $ \nabla =
\hat{\bf x}\;\partial_x$. From the point of view of matched asymptotic expansions, the electric field in the non-ionized region before the front will vary adiabatically slowly on the “inner” time scale $\tau$ of the front, the timescale on which the front propagates over a distance comparable to its width, because the length scales of the outer problem determining the changes of $E$ are assumed to be much larger than the inner scale $\alpha_0^{-1}$. For our study of the inner problem, we thus take the asymptotic field value ${\bf E}^+$ in the unionized region constant in time. Furthermore, we will use the convention that the unionized initial state into which the front propagates is at the right towards large positive values of $x$, so that there $$\begin{aligned}
\label{304}
\label{2041}
\sigma \to \sigma^+ = 0~,\;\; q \to q^+ = 0~,\;\; &&E \to E^+~~,
\quad \partial_{\tau} E^+=0~,
\nonumber\\
&&\mbox{for } x \to +\infty~, \end{aligned}$$ which motivates now the use of the superscript +. We emphasize again that “$x \to +\infty$” should be interpreted on the length scale $\alpha_0^{-1}$ of the inner problem in the sense of matched asymptotic expansions [@vDyke; @bender]. Far behind the front, i.e., for $x \to -\infty$, the discussion of Section II leads us to expect a homogeneous stable state $$\begin{aligned}
\label{305}
\label{2042}
\sigma \to \sigma^- \ne 0~,\quad q \to q^- = 0~,\quad &&E \to 0~,
\nonumber\\
&&\mbox{for } x \to - \infty~. \end{aligned}$$ Which value $\sigma^-$ will be dynamically selected and what the corresponding front velocity and profile are, for a given fixed value of the electric field $E^+$ before the front, is the selection problem, we aim to solve.
The boundary condition (\[304\]) allows an important simplification of the equations in one dimension: If we insert (\[2024\]) into (\[2023\]), we obtain $$\label{306}
\partial_\tau\;E + j \;=\; h(\tau)~,$$ where $h(\tau)$ is an arbitrary function of time which is constant in space. In view of the boundary condition (\[304\]), $h(\tau)$ vanishes at $x \to \infty$ and thus everywhere. For planar fronts, the model Eqs. (\[2022\])-(\[2025\]) then reduce to $$\begin{aligned}
\label{307}
\partial_\tau\;\sigma
&=& \partial_x\;(\sigma \;E) \;+\; D \;{\partial_x}^2 \sigma \;+\;
\sigma \; f(|E|)~,
\\
\label{308}
\partial_\tau\;E
&=& \;-\;\sigma\;E\;-\;D\;\partial_x\; \sigma~, \end{aligned}$$ with space charge and electric current given by $$\label{309}
q\;=\;\partial_x\; E \qquad\mbox{and}\qquad
j\;=\;\sigma\;E\;+\;D\;\partial_x \;\sigma~.$$ We will refer to this set of equations as to [*the one-dimensional streamer equations*]{}. They are the basic equations of this paper, on which the rest of our analysis will be based.
Eq. (\[308\]) implies that the field decays behind the front, if no strong density gradients act against it. As we shall see later when we will discuss our simulation results in Section VI, such strong density gradients often occur during the transient regime before a $PSF$ emerges. Once, however, a front has approached an approximately uniformly translating state, the electron density $\sigma^-$ behind the front is almost homogeneous and the field behind the front then decays to zero on a time scale $1/ \sigma^-$ according to (\[308\]). Note, that the local decay of the field for any nonzero electron density is due to electrodynamics of conserved quantities that continues also after the impact ionization has been suppressed.
We finally note that in the limit where the diffusion is small ($D \ll
1$), it is easy to identify the crossover time from the linear avalanche regime to that of streamer propagation in the case that the initial electron density is small and nonzero only in a very narrow localized region. As explained in the beginning of this section, in the avalanche regime we can neglect the changes in the background field $E^+$ due to the build-up of the charges. The evolution of the electron density is then described by the linearized version of (\[307\]), a linear equation with drift, diffusion and growth. Hence, if the initial electron density is, e.g., Gaussian, the electron density will according to (\[2034\]) remain a Gaussian profile, whose maximum drifts with a velocity $|{\bf E}| $ in the direction opposite to the field and whose amplitude grows exponentially as $\exp(f(E^+)\tau)$ . In other words, if the total initial electron charge is $N_e(0) = \int dx
\; \sigma(x,0)$, then the total number of electrons in this avalanche regime grows as $N_e(\tau)=N_e(0) \exp(f(E^+)\tau)$. Likewise, the total ion charge grows exponentially, but if both the diffusion constant and the width and amplitude of the initial perturbation are small, the electron drift will separate the negative electron charge and the positive ion charge almost completely. The crossover to the nonlinear streamer regime will therefore occur when the total charge in the positively and negatively charged regions is big enough that screening of the field becomes appreciable, i.e., at a time $\tau_c$ when $$\label{309a}
N_e(\tau_c) \approx |E^+|~,~~~\Rightarrow~~~ \tau_c \approx
{{1}\over{f(E^+)}} \ln[|E^+|/N_e(0)]~.$$
Uniformly Translating Front Solutions
=====================================
Above we already have introduced the idea, that fronts asymptotically approach some shape, which is independent of the initial conditions. This is based on our experience [@benjacob; @vs1; @vs2; @vs3; @Oono] with other examples of front propagation into unstable states that the front will acquire some asymptotic shape and velocity in the long time limit, which will be the same (“universal”) for a large class of “sufficiently localized” initial conditions that comprise most physically relevant initial states. This property is often referred to as the front selection problem. Our subsequent analysis will therefore follow the usual strategy in examples of this type: We will first show in this section that there generally is a one-parameter family of front solutions. In Section V we then summarize our present understanding of the front selection problem, and on the basis of this predict the properties of the selected streamer front. The numerical simulations that confirm our predictions are presented in Section VI.
Uniformly translating fronts with velocity $v$ are stationary in a coordinate system moving with velocity $v$. If we denote this comoving coordinate by $\xi=x-v\tau$, the partial differential equations ([ *pde’s*]{}) (\[308\]) and (\[309\]) in this coordinate system become $$\begin{array}{rcl}
\label{3011}
\left.\partial_\tau\;\sigma\right|_\xi &=& v \;\partial_\xi\;\sigma
\;+\; \partial_\xi\; (\sigma\;E) \;+\; D\;{\partial_\xi}^2\sigma
\;+\; \sigma\;f(|E|)~ ,
\\
\left.\partial_\tau\;E\right|_\xi &=& v\;\partial_\xi\;E\;-\;\sigma\;E
\;-\; D\;\partial_\xi\;\sigma ~.
\end{array}$$ A front translating uniformly with velocity $v$ in the [*fixed*]{} frame $x$ is stationary in this comoving frame, $\left.\partial_\tau\;\sigma\right|_\xi \;=\; 0 \;=\;
\left.\partial_\tau\;E\right|_\xi$. As a result, the corresponding front profiles are solutions of the ordinary differential equations ([*ode’s*]{}). (We continue to use partial differential signs $\partial_\xi$ even though the uniformly translating solutions are functions of the variable $\xi$ only.) $$\begin{array}{rcrcrcccc}
\label{3013}
D\;{\partial_\xi}^2\sigma &+& (v+E)\;\partial_\xi \sigma &+&
\sigma\;\partial_\xi E &+& \sigma\;f(|E|) &=& 0~,
\nonumber \\
& & D\;\partial_\xi \sigma &-& v\;\partial_\xi E &+& \sigma \;E &=&
0~,
\nonumber \\
\end{array}$$ These equations are analyzed below. Both for $D=0$ and for $D\neq
0$, they admit solutions for a range of values of the velocity, so we are indeed faced with the question of front selection.
It is important to realize that not all the exact uniformly translating front solutions of these [*ode’s*]{} correspond to physically relevant solutions. In particular, any physical electron density $\sigma$ needs to be non-negative ($\sigma \geq 0$), but as we shall see the set (\[3013\]) admits $PSF$ solutions where $\sigma$ goes negative. We expect these solutions to be unstable (in accord with the “nonlinear marginal stability” scenario [@vs2]), and also not te be approachable from an initial condition with $\sigma \ge
0$. Hence they are neither dynamically nor physically relevant. Furthermore, note that in our model the ion density $q_i$ ($=\rho+\sigma$) can only increase due to impact ionization \[Eqs. (\[2022\]) and (\[2023\]) imply $\partial_{\tau} q_i =
\sigma f(E) \geq 0$\]. With our convention that the non-ionized state is on the right, this implies that uniformly receding front solutions with $v \leq 0$ are unphysical. We will therefore call a uniformly translating front solution physical if $$\label{3015}
v >0 \quad \mbox{and} \quad \sigma(\xi) \ge 0 \quad \mbox{for all}~\xi~.$$
$D=0$ Front Solutions
---------------------
In contrast to the case $D \neq 0$, where we can derive properties of uniformly translating fronts only either qualitatively by discussing flows in phase space or quantitatively by numerical integration, Eqs. (\[3013\]) for $D=0$ can be integrated explicitly. Doing so, we derive a simple explicit expression for the electron density $\sigma^-$ behind the front in terms of the field $E^+$ before the front; this analysis generalizes an earlier result of D’yakonov and Kachorovskii [@Dya], and explicitly illustrates the existence of a family of uniformly translating solutions. For $NSF$, these results extend smoothly to the case $D\neq 0$: The electron density $\sigma^-(E^+)$ derived for $D=0$ will turn out to be a good approximation for $D\lesssim 1$, and the small overshoot of $\sigma$ at the rear end of the front visible in the three-dimensional simulations in Fig. 1(c), is also recovered for $D=0$. For $PSF$, on the other hand, we will see that $D$ acts as a singular perturbation, so that the class of $D=0$ $PSF$-solutions that we derive here is not relevant for the $PSF$ selection problem for $D\lesssim 1$.
The [*ode’s*]{} describing uniformly translating fronts for vanishing diffusion are found by putting $D=0$ in (\[3013\]). These equations then become $$\begin{aligned}
\label{401}
\partial_\xi\;\Big[ \;(v+E)\;\sigma\;\Big] &=& -\; \sigma\;f(|E|)~,
\\
\label{402}
v\;\partial_\xi\;\ln |E| &=& \sigma ~ . \end{aligned}$$ Upon insertion of the l.h.s. of Eq. (\[402\]) for $\sigma$ in the r.h.s. of Eq. (\[401\]), this equation can then be expressed as a complete derivative by writing $$\label{403}
\partial_\xi\;\left[
\;(v+E)\;\sigma \;+\; v\;\int_c^{|E|} dx\;\frac{f(|x|)}{x} \;\right]
\;=\; 0~.$$ For physical fronts with $v > 0$ and $\sigma \ge 0$ \[see (\[3015\])\], we see from (\[402\]), that $E$ is a monotonic function of $\xi$, $$\begin{aligned}
\label{404}
\mbox{sgn}\;\Big(\partial_\xi E(\xi)\Big) \;=\; &&\mbox{sgn}\;q(\xi)
\;=\; \mbox{sgn}\;E(\xi) \;=\; \mbox{sgn}\;E^+
\nonumber\\
&&\mbox{for all }\xi~ . \end{aligned}$$ This allows us to use $E$ as a coordinate instead of $\xi$. According to (\[2041\]), before the front at $\xi \to \infty$ the electron density vanishes, so $\sigma^+= \sigma[E^+] = 0$. Eqs. (\[403\]) and (\[404\]) together then determine $\sigma$ as a function of $E$ as $$\label{406}
\sigma[E] \;=\; \frac{v}{v+E}\;\;\rho_{E^+}[E] ~,$$ with the function $$\label{407}
\rho_{E^+}[E]\;=\; \int_{|E|}^{|E^+|} dx\;\frac{f(|x|)}{x} \;=\;
\rho_{|E^+|}[|E|] ~~(\geq 0)~.$$ The function $\rho_{E^+}[E]$ is nothing but the ion density, as can be deduced by inserting $q = \partial_\xi E$ into (\[402\]) and equating the charge density $q$ with $\rho - \sigma$. The ion density $\rho$ for $D=0$ turns out to be a function of $E$ and $E^+$ only, and to be independent of the particular front shape parametrized by $v$.
The fields $\sigma$, $\rho$ and $E$ as a function of $\xi$ can be found by solving the implicit equation for $E=E(\xi)$ $$\label{408}
\partial_\xi\;\ln\;|E| \;=\; \frac{\rho_{|E^+|}[|E|]}{v+E}~,$$ which can be derived from Eqs. (\[402\]) and (\[407\]).
Eq. (\[406\]) immediately shows that physically allowable solutions with $\sigma \geq 0$ and $v>0$ must have $v+E \ge 0 $ for all field values. Because of the monotonicity of $E$ as a function of $\xi$, this is automatically satisfied for $PSF$ with $E^+>0$, but for $NSF$ this implies in particular that $v+E^+ \ge 0$; together with $v> 0$ we thus have for physical fronts $$\label{409}
v \;\ge\; \mbox{max} \;\Big[\; 0\; , \;-E^+\; \Big]~.$$ In physical terms, the condition $v \ge -E^+$ expresses, that the velocity of uniformly translating fronts must be at least the drift velocity $-E^+$ of free electrons in the leading edge of the front, where the field is strongest. (Remember, that (\[404\]) implies, that the field is monotonic in space.)
For all values of $v$ obeying the inequality (\[409\]) the solutions of (\[406\]) and (\[408\]) are proper, physically allowable solutions for fixed $E^+$; within the context of the present model, this illustrates a general feature of front propagation into unstable states, namely that there exists a family of front solutions parametrized by the velocity [@footnote6].
In Fig. 2(a), we plot the solution (\[406\]) for $\sigma$ as a function of $E$ for the fixed value of the velocity $v=2$ in the case that the impact ionization function $f(E)$ is given by the Townsend expression $f_T(E)$ of (\[2-14\]) as in the numerical simulations [@DW; @Vit]. Note that in this representation, the state behind the front at $\xi=- \infty$ corresponds to a point on the $\sigma $ axis, and that the front solution $\sigma(\xi)$, $E(\xi)$ is represented in this diagram by the flow along one of the trajectories towards either the positive $E$-axis for $PSF$ or the negative $E$-axis for $NSF$ for $\xi \rightarrow \infty$. Note furthermore that $\sigma$ overshoots the value $\sigma^-$ ($=\sigma(\xi \rightarrow -\infty$)) in the case of $NSF$. This property as well as the monotonicity of $\sigma[E]$ and accordingly of $\sigma(\xi)$ for positive fronts, follows immediately from Eq. (\[406\]). For $NSF$, it can also be observed in the three-dimensional simulations of Vitello [*et al.*]{} [@Vit], shown in Fig. 1(c).
The smallest $E^+$ for which a front solution with $v=2$ is shown in Fig. 2(a), is $E^+ = -1.999$. For this value of $E^+$, $\sigma[E]$ continues to increase till $E \approx E^+$ and then suddenly decays to zero. A short analytical investigation of (\[406\]) shows, that this behavior develops into a discontinuity of $\sigma[E]$ at the point $E=E^+$ for $v = -E^+$. $\sigma[E]$ then increases monotonically up to $f(E^+)$ for $E \downarrow E^+$ and then jumps to zero discontinuously at $E^+$. This shock-like behavior stays unchanged under a parameter change to $\sigma(\xi)$. It is further discussed and motivated in Section V.
An immediate consequence of Eqs. (\[406\]) and (\[407\]) for the electron and ion density is that the value $\sigma^-$ behind the front (where $E \rightarrow 0$) is a simple function of the value $E^+$ of the field ahead of the streamer profile: $$\label{4010}
\sigma^-(E^+) \;=\; \rho_{|E^+|}[0]\;=\; \int_0^{|E^+|}
dx\;\frac{f(|x|)}{x}$$ The virtue of this expression for the electron density $\sigma^-$ far behind the front as well as of the expression (\[407\]) for the ion density $\rho$ throughout the whole front, is that it is independent of the velocity $v$, hence independent of whichever front profile is selected, provided that the limit $D\rightarrow 0$ is smooth. We shall see later that this $D=0$ result remains relatively accurate for $NSF$ fronts with $D\lesssim 1$, and compare it to the results of the simulations [@DW; @Vit] in Section VI. For $PSF$, on the other hand, the above result will turn out to be less relevant due to the non-perturbative nature of the limit $D\rightarrow 0$ in this case.
For the Townsend function $f_T(|x|)$ \[Eq. (\[2-14\])\] the function $\sigma^-(E^+)$ can be expressed as $$\begin{aligned}
\label{4011}
\sigma^-(E^+)_T &\;=\;& |E^+| \;E_2\Big(|E^+|^{-1}\Big)
\nonumber\\
&\;=\;& f_T(|E^+|)-E_1\Big(|E^+|^{-1}\Big)~, \end{aligned}$$ where $E_n(z)$ is the exponential integral [@Abramo].
We finally note that the second form of (\[4011\]) shows that $\sigma^-$ approaches $f_T$ for large fields, since $f_T \gg E_1 $ for $E^+\gg 1$. For $E^+$ of order unity, $\sigma$ and $f_T$ are still of the same order, and this shows (for small $D$) that the growth rate (\[2-17\]) of long wavelength unstable modes in the unionized state is comparable to the damping rate (\[2-19\]) of stable modes in the plasma behind a $NSF$. For small fields, the strict bounds on $E_2$ [@Abramo] show that $\sigma^- \approx E^+ f_T(E^+)$, so that the approximate equivalence of these two time scales does not hold for $E^+ \ll 1$, but in the small field range our starting model is not very realistic anyway, because of the neglect of stabilizing recombination terms.
$D\neq0$ Front Solutions
------------------------
For $D\neq 0$, we can not obtain the uniformly translating solutions analytically. Moreover, perturbation theory around the $D=0$ case is not simply possible, as $D$ appears in front of the highest derivative in Eq. (\[3013\]), so the diffusion term acts as a singular perturbation. As a consequence, Eqs. (\[3013\]) reduce to a set of two coupled first order [*ode’s*]{} for $D=0$, while three are required for $D\neq 0$. However, we can still easily demonstrate the existence of a one-parameter family of uniformly translating front solutions for $D\neq 0$ through standard counting arguments for [ *ode’s*]{}. Building on the results of such an analysis, the solutions can then be constructed by integrating numerically in a stable direction, using so-called “shooting methods” [@press].
To perform the analysis, it is convenient to write the equations as a set of three coupled first order [*ode’s*]{}. There is some freedom for the choice of the third variable: The standard choice would be $\sigma' = \partial_{\xi} \sigma $, but for the discussion of the singular limit as well as for numerical stability, the charge density $q$ has turned out to be the most convenient choice. The [ *ode’s*]{} (\[3013\]) then become $$\begin{aligned}
\nonumber
\partial_\xi\; \sigma
&=& - \; {{\sigma E -v q}\over{D}}~,
\\
\label{503}
\partial_\xi \;E
&=& \;\;\; q ~,
\\
\nonumber
\partial_\xi\;q
&=& - \;\frac{\sigma f(|E|) }{v}\;+\; \frac {\sigma E -v q}{D}~.\end{aligned}$$ Just as we thought of the profiles for $D=0$ as describing flow in a two-dimensional $(\sigma,E)$ phase space, we can now think of Eqs. (\[503\]) as describing a flow in a three-dimensional ($\sigma,
E,q$) phase space. The velocity $v$ just plays the role of a parameter in the flow equations, while $\xi$ again plays the role of a time-like variable — see the sketch in Fig. 2(b).
The steady states of the full [*pde’s*]{} discussed in Section III correspond to fixed points of the flow: the points $(\sigma,0,0)$ on the $\sigma$-axis are fixed points of the flow (\[503\]) and correspond to homogeneously ionized plasma states, while the $E$-axis is a line of fixed points ($0,E,0$) each of which corresponds to a non-ionized state with $\sigma=\sigma' =0$ and $E\neq0$.
A uniformly translating front solution now corresponds to the existence of a trajectory in this phase space that starts at “time” $\xi=-\infty$ on the $\sigma$-axis and flows to the $E$-axis for $\xi
\rightarrow \infty$. The multiplicity of such solutions (i.e., whether they exist as discrete sets, or, e.g., as a one- or two-parameter family) can be determined as follows. If we linearize the flow near an arbitrary point $(\sigma^-,0,0)$ on the $\sigma$-axis by writing $(\sigma,0,0)$ $= (\sigma^-,0,0)$ $+ A \exp({-\Lambda^- \xi})$, we find the eigenvalue equation $$\label{5010}
\Lambda^- \; \left( {\Lambda^-}^2 - \Lambda^-\;\frac{v}{D} -
\frac{\sigma^-}{D} \right) \;=\; 0 ~.$$ The fact that there is a zero eigenmode is a consequence of the fact that the $\sigma$-axis is a line of fixed points. For the two nontrivial eigenvalues (which correspond to the linearized modes (\[2-19\]) about the ionized and screened region by equating $i {\bf k}\cdot \hat{\bf x}=-\Lambda^-$ and $\omega^- = \Lambda^-v$) we have $$\label{5012}
\Lambda^-_\pm \;=\; \frac{v \pm
\sqrt{v^2+4D\sigma^-}}{2D}~.$$ The eigenvalue $\Lambda^-_+$ is positive, and hence gives a decaying exponential; thus points along the corresponding eigendirection flow into the $\sigma$-axis as $\xi$ increases. The eigenvalue $\Lambda^-_-$, on the other hand, is negative and hence corresponds to an unstable eigendirection, with flow away from the axis. This implies that at each point $(\sigma^-,0,0)$ on the $\sigma$-axis, there is, for fixed $v$, a unique eigendirection $(-\Lambda_-^-,1,\Lambda^-_-)E_1$ along which the flow is away from the axis. This flow can be followed in two anti-parallel directions, determined by the sign of $E_1$. The one flowing towards positive values of $E$ is the beginning of a $PSF$ front profile, the one flowing towards negative $E$ is the beginning of an $NSF$ profile. From these eigendirections, one derives that for $PSF$ with field perturbations $E_1>0$, the electron density decreases close to the $\sigma$ axis, while for $NSF$ it increases. Accordingly, before reaching $\sigma=0$ for $\xi
\rightarrow \infty$, a $NSF$ profile has at least one maximum of $\sigma$, while a negative one can be (and is) monotonic. This generalizes our result for $D=0$, and is consistent with the findings of Vitello [*et al.*]{} [@Vit] shown in Fig. 1(c). The physical origin of the maximum of $\sigma$ in the rear end of the $NSF$ profile is the screening of the field: Due to the low ionization rate in an already fairly suppressed field the ion density has already almost acquired its final value, so the electron density has to overshoot its asymptotic value $\sigma^-$ so as to make $\partial_\xi E <0$. The screening behind a $PSF$ happens by suppressing the electron density faster than the ion density for increasing $\xi$, and so there $\sigma $ is monotonic.
Let us now investigate the stability of the flow near a point $(0,E^+,0)$ on the $E$-axis. Upon linearizing the flow equations (\[503\]) and writing the $\xi$ dependence of the perturbations in the form $\exp(-\Lambda^+ \xi)$, we find the eigenvalue equation $$\label{506}
\Lambda^+ \; \left( {\Lambda^+}^2 - \Lambda^+\;\frac{v+E^+}{D} +
\frac{f(|E^+|)}{D} \right) \;=\; 0 ~.$$ Again, there is a zero eigenvalue due to the fact that the whole $E$-axis is a line of fixed points. The two nontrivial eigenvalues are $$\label{509}
\Lambda^+_\pm \;=\; \frac{v+E^+ \pm
\sqrt{(v+E^+)^2-4Df(|E^+|)}}{2D}~.$$ These eigenvalues can be related to (\[2-17\]) in the same way as (\[5010\]) could be related to (\[2-19\]). For $v+E^+ >0$, the real parts of these eigenvalues are always positive, so that both eigendirections are stable. In other words, for $E^+ > -v$, all points near the $E$-axis flow towards this axis — in slightly more technical terms: there is a two-dimensional stable manifold flowing into each of these points on the $E$-axis. For $E^+ < -v$, the flow is away from the $E$-axis, and fronts with $v+E^+<0$ can not be constructed. This generalizes (\[409\]) to $D\neq 0$.
The existence of a one-parameter family of fronts with velocity $v>
-E^+$ can now simply be understood as follows. As we saw before, there is one unique $PSF$ and one unique $NSF$ trajectory flowing out of each point on the $\sigma$-axis for fixed $v$ and $D$. Since the flow defined by Eqs (\[503\]) is continuous, we can expect each trajectory to extend smoothly [@footnote6b]. Once the flow gets near the $E$-axis, we know from the above analysis that the trajectory will be attracted completely to the axis, provided $v$ is large enough. Thus, for each $\sigma^-$ and $v$, there will exist two unique trajectories, i.e., a unique $PSF$ solution and a unique $NSF$ solution. Since each of these trajectories flows into a unique point on the $E$-axis, the flow equations implicitly define a unique relation of the form $\sigma^-=\sigma^-(v,E^+)$ for each of the two types of fronts. For a given value of $E^+$, we thus have a one-parameter family of front solutions, parametrized by $v$.
There are two important properties of the front solutions associated with their asymptotic large $\xi$ behavior. First of all, we note that according to (\[509\]) the eigenvalues $\Lambda^+_\pm$ are only real for $$\label{508}
v \;\ge\; v^* \equiv - E^+ + 2 \;\sqrt{D\;f(|E^+|)}~.$$ This implies that the corresponding front profiles can certainly not approach the asymptotic state $\sigma=0$ ahead of the front in a monotonic way for $v<v^*$: When the eigenvalues are complex, the front profiles have an oscillatory tail of the form $\exp[-(\Re \Lambda^+)\xi] \cos[(\Im
\Lambda^+) \xi) $. Clearly, this violates the physical condition that the electron density $\sigma$ should remain positive, so solutions with $-E^+ < v < v^*$ are physically excluded: $v^*$ denotes, in the present case, the smallest velocity of physically allowable uniformly translating front solutions.
The identification of $v^*$ as a bound on the velocity of physically allowed front profiles depends only on the structure of the eigenvalues $\Lambda^+$ associated with the [*linear*]{} flow near unstable states. There is a second, [*nonlinear*]{}, way in which the range of physically allowed values of $v$ can be bounded. To understand this, note that for any $v \geq v^*$, the asymptotic decay of $\sigma(\xi)$ for $\xi \rightarrow \infty$ for a uniformly translating profile will be $$\label{508a}
\sigma(\xi) = A_- \;\mbox{\large e}^{\textstyle -\Lambda^+_- \xi}
\; + \; A_+ \;\mbox{\large e}^{\textstyle -\Lambda^+_+ \xi}
+ \mbox{h.o.t.}$$ with real coefficients $A_-$ and $A_+$. Here, h.o.t. stands for higher powers of the two exponentials generated when expanding the equations to higher than linear order in the variables. Clearly, the smallest eigenvalue $\Lambda^+_-$ governs the asymptotic decay of the profile provided $A_- \neq 0$. That $A_-$ will generically be nonzero for an arbitrary velocity $v$ follows again from the counting argument above for the flow in phase space: Each $PSF$ and $NSF$ trajectory flowing out of a point on the $\sigma$-axis is unique, and hence there is no freedom to impose an additional condition $A_-=0$ close to the $E$-axis. Furthermore, the coefficients $A_-$ and $A_+$ depend on the full global nonlinear behavior of the flow, and hence they depend implicitly on $v$.
There might exist, however, particular velocities $v^{part} > v^*$, for which $$\label{420a}
A_-(v^{part})=0~.$$ For discussing these we invoke again a continuity argument for the front properties as a function of $v$. We expect a very slowly decaying, nearly homogeneous uniformly translating front solution to have a non-negative density everywhere, and to have a very large velocity, since the velocity of a profile is essentially inversely proportional to its slope in the limit of small slopes. (So indeed the roots $\Lambda_-^-$ given by (\[5012\]) and $\Lambda^+_-$ given by (\[509\]) vanish in the limit that $v$ becomes large.) So for large $v$ we expect to find physical solutions. These are characterized by $A_->0$ in the leading edge of the front. Decreasing $v$ continuously, we either reach $v=v^*$ smoothly with still $A_->0$, or we reach the first particular velocity, $v_1^{part}$, where $A_-$ vanishes. In the latter case, we expect by continuity $A_-(v)<0$ for $v<v^{part}_1$. This implies that then $\sigma$ approaches zero from below, i.e., that the front solution is unphysical. Below the next $v_2^{part}$, we expect the electron density to develop two zero’s and so forth. The largest $v^{part}$, if it exists, thus plays the role of the [*nonlinear front velocity*]{} $v^\dagger$ [@vs3], $$\label{4052}
v^\dagger = \mbox{max}\; \{\; v^{part}\; |\; A_-(v^{part})=0 \; \}$$ for a given $E^+$. (Note that if $\Lambda^+_- < 0.5 \Lambda^+_+$, the higher order terms in (\[508a\]) of order $\exp( -2 \Lambda^+_- \xi)$ are actually larger than the second term $\exp(-\Lambda^+_+\xi)$. This does not change our argument, though, as the prefactor of this second order term will vanish if $A_-$ vanishes.)
At the velocity $v=v^\dagger$ or at any $v=v^{part}$, the flow in phase space approaches the $E$-axis along the eigenvector where the flow is most rapidly contracting. The trajectory corresponding to the nonlinear front solution is therefore more appropriately referred to as a strongly heteroclinic orbit, where heteroclinic indicates that it is a trajectory from one fixed point to another one. The existence and properties of strongly heteroclinic orbits have recently been under active investigation [@footnote8].
Such a velocity $v^\dagger$, if it exists, bounds the continuum of velocities of physical uniformly translating solutions from below, and thus replaces the earlier bound $v^*$ derived from linearizing the equations in the leading edge of the front.
Nonlinear Front Solutions for PSF
---------------------------------
For $NSF$, the bounding velocity $v^*$ given by (\[508\]) is always positive. Moreover, by integrating the flow equations (\[503\]) numerically and searching for particular solutions for which, according to (\[420a\]), $A_-(v^{part})=0$, we have convinced ourselves that there are no such solutions for any $D\neq 0$ and $E^+<0$. Hence, the smallest velocity of physical $NSF$ solutions is always $v^*$, for any value of the parameters.
For $PSF$, on the other hand, the situation is very different, since $v^*<0$ for $(E^+)^2 >4Df(|E^+|)$ — for the Townsend function (\[2-14\]), this happens for $D\le 0.25E^+e^{1/E^+}$, hence for any $E^+$ for $D\le 0.68$. In particular for $PSF$ at small $D$ the question therefore arises whether there are nonlinear front solutions defined by (\[420a\]) and (\[4052\]) with $v^\dagger >0$. The results of a numerical search for such solutions are shown in Fig. 3, as a function of $D$ and $E^+$. Below the full line in this diagram, there exists indeed a nonlinear front $v^\dagger >v^*$, whereas above this line $v^*$ denotes the smallest velocity of physical front solutions. While these results have been obtained numerically, the existence of a single (unique) particular solution with $A_-(v^{part})=0$ in the limit $D\rightarrow 0$ can be demonstrated analytically. Since a full discussion of these results will be given elsewhere [@ebert], we confine ourselves here to a brief outline of the arguments that also demonstrate the singular nature of these solutions for $D\rightarrow 0$.
If we take the limit $D\rightarrow 0$ with $v$ fixed, assuming no nontrivial scaling of the variables $\sigma$, $E$ and $q$ and of the spatial coordinate $\xi$, Eqs. (\[503\]) can easily be shown to reduce to those studied in Section IV.A for $D=0$. Hence, we can recover in this way the family of front solutions obtained there. [*Any*]{} particular solution, on the other hand, for which $A_-(v^{part})=0$, decays according to (\[508a\]) as $\exp(-\Lambda^+_+ \xi)$ as $\xi
\rightarrow \infty$. Since $\Lambda^+_+ \propto D^{-1}$ for $D\rightarrow 0$, such a particular front solution becomes extremely steep as $D\rightarrow 0$: its gradients diverge as $1/D$ and so that the diffusion term can still overcome the drift term as $D\rightarrow
0$. That the velocity of such a solution must also have a nontrivial scaling in this limit can be seen from the third equation of (\[503\]), written in the form $$\label{4053}
\partial_\xi\;q
= \sigma \; \left( -\; \frac{ f(|E|) }{v} + \frac{E}{D} \right) \;-\;
\frac {v}{D}\; q~.$$ Any [*nontrivial*]{} scaling of this equation in the limit $D\rightarrow 0$ can only occur if the first term between brackets remains of the same order as the other two, which diverge as $1/D$. This is only possible if $v$ scales as $D$. In this limit, the third term can then be neglected, and since $\partial_\xi q$ has to change sign in the front region (as the charge density $q$ vanishes as $\xi \rightarrow \pm
\infty$), there must be an intermediate value $\hat{E} <E^+$ of the field for which $v=Df(|\hat{E}|)/\hat{E}$.
Now that we know the scaling of the spatial gradient of the velocity of such particular front profiles for $D\rightarrow 0$, one easily convinces oneself that the electron and charge density of these solutions must [*diverge*]{} as $1/D$ in this limit. To study the existence of such possible solutions, it is therefore convenient to introduce new variables and coordinates according to $$\label{4054}
x=D\tilde{x}~,\;\; v=D\tilde{v}~,\;\; \xi=D\tilde{\xi}~,\;\;
\sigma=\tilde{\sigma}/D~,\;\; q=\tilde{q}/D~,$$ with $E$ and $\tau$ unchanged. In these new variables, the flow equations (\[503\]) become $$\begin{aligned}
\nonumber
\partial_{\tilde{\xi}}\; \tilde{\sigma}
&=& - \; \tilde{\sigma}\; E + D \; \tilde{v}\; \tilde{q},
\\
\label{4055}
\partial_{\tilde{\xi}} \; E
&=&\tilde{ q} ~,
\\
\nonumber
\partial_{\tilde{\xi}}\;\tilde{q}
&=& \;\tilde{\sigma} \left( E- \frac{f(|E|)}{\tilde{v}}\right) \; - \;
D \; \tilde{v}\; \tilde{ q}~. \end{aligned}$$ The limit $D\rightarrow 0$ can now be taken simply by leaving out the term $D\tilde{v}\tilde{q}$ in the first and last equation. We will show elsewhere [@ebert] that the resulting equations have [*one unique*]{} physical front solution thus fixing one particular value of the scaled velocity $\tilde{v}_1$ and in view of the scaling (\[4054\]) and the scaling of the eigenvalues $\Lambda^+_{\pm}$, this solution must have $A_-(\tilde{v}_1)=0$. This solution is therefore precisely the $D \rightarrow0$ limit of the nonlinear front solution with velocity $v^\dagger =\tilde{v}_1 D$. Furthermore, since the limit $D\rightarrow 0$ is smooth for Eqs. (\[4055\]), this shows that there exists a nonlinear front solution with $v^\dagger>0$ for [*any*]{} $E^+$ and nonzero but small $D$. Due to the singular scaling (\[4054\]), the corresponding front solutions are determined by [*ode’s*]{} that have a different structure from those studied for $D=0$ in Section IV.A, and therefore these nonlinear front solutions can not be obtained perturbatively from the latter class of solutions — of course, the latter class of solutions still exists for $D\neq 0$, in agreement with the counting arguments given earlier, but these now correspond to a singular limit of Eqs. (\[4055\])! The significance of these nonlinear front solutions lies in the fact that they will turn out to be the selected fronts that dominate the dynamics of $PSF$ in the physically important range $0.1 \lesssim D \lesssim 0.3$.
The nonlinear front solution can be constructed numerically very easily by integrating Eqs. (\[4055\]) using standard numerical “double shooting” routines [@press]. Fig. 4 shows our numerical results for the smallest physical velocity, max($v^\dagger,v^*$) in the case that the ionization function is given by the Townsend expression. The scaled velocities $v^\dagger/D$ and $v^*/D$ are plotted; in agreement with our arguments above the scaled velocity $v^\dagger/D$ of the nonlinear front solution approaches a finite limit as $D\rightarrow 0$. Furthermore, the ratio $v^\dagger/D$ hardly varies with $D$ in the physical range $0.1 \lesssim D \lesssim 0.3$, and for small fields $E^+$, the scaled velocity $v^\dagger/D$ becomes exponentially small, in agreement with the bound $v^\dagger/D < E^+
\exp(-1/E^+)$ that follows from the observations discussed after Eq. (\[4053\]) above.
We finally note that our numerical routines have not only allowed us to obtain the results show in Figs. 3 and 4, but have also enabled us to verify numerically all the statements made above about the multiplicity of solutions, the parameter ranges for physical fronts, the monotony properties, the singular behavior of the small $D$ $PSF$ limit, and the persistence of the family of front solutions for $D\rightarrow 0$.
Selection of the Asymptotic Front
=================================
Front Propagation into Unstable States
--------------------------------------
We have seen that the non-ionized state into which the streamer fronts propagate, is an unstable state, that the homogeneous weakly ionized plasma is a stable state, and that there is a family of uniformly translating front solutions connecting the two. The existence of a family of front solutions is a generic feature of front propagation into unstable states. We, therefore, briefly recall what is known in the literature for analogous problems and then translate this experience to the streamer problem. The prototype equation for studies of this type of front propagation is $$\label{aw}
\label{3016}
\partial_t u = \partial_x^2 u + g(u)~,$$ where $g(u)$ is some nonlinear function which satisfies $$\label{gcond}
\label{3017}
g(0)=0~, ~~~g(1)=0~,~~~ g'(0) >0 ~,~~~
g'(1)<0~.$$ Note that these relations imply that the “state” $u=0$ is unstable, and that the “state” $u=1$ is stable. The study of the propagation of fronts into the unstable state $u=0$ in this equation dates back to the early work of Kolmogorov [*et al.*]{} [@kpp] and Fisher [@fisher] in the context of population dynamics. Later Gel’fand [@comb1] studied a particular example of a function $g(u)$ motivated by combustion. The mathematical research on this equation culminated in the work by Aronson and Weinberger [@aw], who rigorously solved the front propagation problem for (\[aw\]). In particular, they proved that any initial perturbation, that is nonzero only in a finite part of space, approaches a unique uniformly translating front solution with velocity $v_f$ in the long time limit. If $g''(u)<0$ for all $u$, $v_f$ equals $v^*=2\sqrt{g'(0)}$ (derived from linearizing in the tip of the front), while for general $g(u)$, $v_f$ approaches either $v^*$ or some $v^\dagger>v^*$. We refer to the literature for a detailed discussion of this work [@aw; @eckmann].
The velocities $v^*$ and $v^\dagger$ of the above problem directly correspond to our $v^*$ (\[508\]) and $v^\dagger$ (\[4052\]), since they are also the smallest velocities, which still allow for uniformly translating fronts with $u\ge 0$ everywhere. So if $u$ is interpreted as a population density or a chemical concentration, the selected front for every sufficiently localized initial state is the slowest physical uniformly translating front. In other interpretations no physical constraints bind $u$ to positive values. Nevertheless the selected velocity stays the same. In this case, one can prove that every front with smaller velocity is dynamically unstable [@benjacob], i.e., that the selected front is marginally stable. The slowest physical or stable solution, which is selected, coincides with the steepest physical or stable one.
In the last decade, it has been recognized that several aspects of the front selection problem encountered for the nonlinear diffusion equation (\[aw\]), seem to have more general validity. Certain scenarios, justified by heuristic arguments but lacking a detailed mathematical proof, were formulated and numerically tested on more complicated [*pde*]{}’s that were often of higher order in the spatial derivatives [@deelanger; @benjacob; @vs1; @vs2; @vs3; @Oono]. Some of the equations studied lead to non-uniformly translating fronts that leave a nontrivial spatially periodic state behind [@deelanger; @benjacob; @vs2; @vs3; @deevs]. A particular scenario is the one distinguishing between the so-called [*linear marginal stability*]{} regime where $v_f=v^*$ and the [*nonlinear marginal stability*]{} regime where $v_f=v^\dagger$ [@benjacob; @vs1; @vs2; @vs3]. These names stem from the fact that in this formulation, the two regimes of front selection are related to the stability properties of the front solutions — in both cases, the selected front separates stable front solutions from unstable ones. Applied to (\[3016\]), this scenario just provides an intuitive explanation of all the well-known mathematical results. For plasma physicists, it is worth mentioning that dynamics in the linear marginal stability regime is related to that determined by the “pinch point analysis” which was developed in plasma physics in the late fifties [@bers; @LLX; @vs2].
Predictions for Streamer Fronts
-------------------------------
By extending the arguments in the appendix of [@vs2], one may show that in the streamer case just like in the case of the above problem (\[3016\]), all physical solutions, i.e., all solutions with $u\ge0$ resp. $\sigma\ge0$ everywhere, are stable. For a detailed discussion, we refer to [@ebert]. It can be argued [@vs1; @vs2], and proven for (\[aw\]) [@benjacob], that a sufficiently localized initial condition will approach the physical uniformly translating front, which is closest in “phase space”, i.e., the steepest one. Both for (\[3016\]) and for the streamer equations, the steepest uniformly translating physical front is uniquely defined. It is also the slowest one.
We can immediately prove this when initially $\sigma(x,\tau=0)=0$ for $x>x_c$ for streamer fronts with $D=0$: In general, there is a front solution for every $v\ge\max[0,-E^+]$, but now the only way in which the electrons can enter the range $x>x_c$ is through electron drift with velocity $-E^+$. Clearly, therefore, the asymptotic front speed of a $NSF$ can only be $-E^+$, while a $PSF$ can not propagate at all. If the initial electron density, however, decays exponentially, the local electron density grows by drift and ionization, and the front can move quicker than $-E^+>0$ for a $NSF$.
For $D\ne0$, we will here only conjecture the analogous statements, and we will test them numerically in Section VI:
1. [*Selected front velocity.*]{} If the initial conditions are sufficiently localized, [*the selected front is the slowest physically acceptable front solution*]{}, i.e., the slowest front profile for which $\sigma(\xi)\ge0$ for all $\xi$. In view of the discussion of Section IV, this means that the selected front velocity $v_f$ is predicted to be $$\label{w1}
v_f= v^* = -E^+ +2 \sqrt{D f(|E^+|)} ~,$$ except when there exists a nonlinear front solution satisfying (\[4052\]): In that case $$\label{w2}
v_f=v^\dagger ~.$$ Note that the result (\[w1\]) ($v^*$ is the [*Linear Marginal Stability*]{} value in the terminology of [@vs1; @vs2]) is an explicit expression for $v_f$ in terms of parameters associated with the linear instability of the unstable state only. On the other hand, the existence of a nonlinear front and the value of $v^\dagger$ (the [*Nonlinear Marginal Stability*]{} value) depends on the whole nonlinear behavior of the flow equations (\[503\]).
2. [*Localized initial conditions.*]{} Initial conditions are sufficiently localized if their spatial decay is faster than the asymptotic decay associated with the smallest eigenvalue of the selected profile, i.e., if $$\begin{aligned}
\label{w3}
\sigma(x,\tau=0) &<& C \;\mbox{\large e}^{\textstyle
-\Lambda^+_-(v^*) x} ~, \quad\mbox{or} \\
\label{w4}
\sigma(x,\tau=0) &<& C \;\mbox{\large e}^{\textstyle
-\Lambda^+_-(v^\dagger) x} ~, \\
&&\mbox{for } x\rightarrow \infty ~, \nonumber\end{aligned}$$ depending on whether the selected front is $v^*$ or $v^\dagger$. Here $C$ is an arbitrary constant, and $\Lambda^+_-(v^*)$ ($= \Lambda^+_+(v^*) $) and $\Lambda^+_-(v^\dagger)$ are given by (\[509\]).
3. [*Non-localized initial conditions.*]{} If an initial condition does not obey (\[w3\]) or (\[w4\]), faster front speeds are possible. In particular, if initially $\sigma(x,\tau=0) \sim
\exp(-\Lambda x)$, with $\Lambda < \Lambda^+_-(v^*)$ or $\Lambda <
\Lambda^+_-(v^\dagger)$, whichever regime applies, then the front speed is given by $$\label{w5}
v= -E^+ +D \Lambda + \frac{f(|E^+|)}{\Lambda} ~,$$ which is obtained by solving (\[506\]) for $v$ in terms of $\Lambda$.
We now combine the analytic and numeric findings from Section IV with the selection rules above to quantitative predictions for asymptotic fronts, which evolve from sufficiently localized initial conditions, in the case that the impact ionization is given by the Townsend expression (\[2-14\]):
- For $NSF$, we numerically have not found any nonlinear fronts for any $D$ and $E^+$, so our simple yet powerful prediction is that for $NSF$ $v_f=v^*$ with $v^*$ given by (\[w1\]). In principle it is possible that for other ionization functions $f(E)$ than the Townsend function (\[2-14\]), there can be nonlinear front solutions also in the $NSF$ regime. In practice, we expect, however, that this will not be the case for physically reasonable functions $f(E)$, i.e., for functions consistent with (\[2-17a\]).
Once the predicted velocities are known, the value $\sigma^-$ of the electron density behind the streamer head is obtained from the numerical integration of the flow equations. The results of these calculations are shown in Fig. 5$(a)$. Since for $NSF$, the limit $D\to0$ is smooth, also $\sigma^-$ depends only weakly on $D$ for $D \lesssim 1$, so that the $D=0$ prediction (\[4011\]) is quite accurate for realistic values of the diffusion coefficient.
At the predicted values of the selected front velocity, the width of the front region can be obtained directly from our numerical solutions of the flow equations. We have somewhat arbitrarily defined the width $w$ as the distance between the points where $\sigma$ is 90% and 10% of the value $\sigma^-$. As Fig. 6 shows for $NSF$ fronts with $D=0.1$, this front width is typically of order 3 for field values of order unity. This confirms again that in the small $D$ limit the impact ionization length $\alpha_0^{-1}$ sets the inner scale of streamer fronts. Furthermore, we find that our numerical data are well fitted by the expression $w \approx 6/
\Lambda^+_\pm(v^*)$, which shows that the front width simply scales with the spatial decay rate $\Lambda_+^+(v^*)=\Lambda^+_-(v^*)$ of the streamer profile in the leading edge. $NSF$ fronts always have a maximum of the electron density within the front.
- As we saw in Section IV, for $PSF$ with $D\lesssim 0.9$, there always is a nonlinear front solution with velocity $v^\dagger
>v^*$. The prediction is that in this range the selected front solution is the nonlinear front solution, i.e., $v_f=v^\dagger$. Values of $v^\dagger$ as as function of $D$ and for several values of $E^+$ were already given in Fig. 4. We also saw before that these nonlinear front solutions are singular in the limit $D\ll 1$, where $v^\dagger \approx D \tilde{v}^\dagger(D=0)$ and $\sigma^{-} \approx \tilde{\sigma}^-(D=0)/D$. The resulting predictions for $\sigma^-$ are shown in Fig. 5$(b)$.
The fact that the dimensionless inner decay length of these nonlinear fronts scales as $D$, implies that the physical decay length of such solutions is $D/ \alpha_0= D_e/(\mu_e E_0)$, i.e., is given by the electron diffusion length. However, since simultaneously the electron density $\sigma^-$ diverges as $1/D$, the total front width $w$ defined above still approaches a finite limit as $D\rightarrow 0$ in units of the ionization length $\alpha_0^{-1}$.
We finally note that the front propagation problem posed by the one-dimensional streamer equations has a number of interesting differences and similarities with the Aronson Weinberger front propagation problem (\[aw\]). In particular, it can be hoped, that techniques of strict bounds developed for the time development of these fronts [@aw] as well as for the nonlinear front velocity $v^\dagger$ [@footnote8] might be also applicable to planar streamer fronts.
Numerical tests of the predictions
==================================
We have tested the predictions listed in Section V by numerically integrating the [*pde’s*]{} (\[307\]), (\[308\]) forward in time. Our computer program is a finite difference code with a time integration which is based on a semi-implicit method.
We have performed an extensive search through parameter space, varying $D$ between 0.02 and 3, and $|E^+|$ between 0.3 and 10. All our numerical studies of the dynamics fully confirm our predictions for fronts, and therefore we only present a sample of our results that illustrate the important features.
All the simulations of the initial value problem we present in the remaining figures, have initially a field $E=-1$ constant in space. We keep the field constant in time in the non-ionized region. The simulations of Fig. 7 – 10 start with the same localized initial ionization seed, a Gaussian profile for the electron density, $\sigma(x,t=0) = 0.01 \exp -(x-x_0)^2$. Fig. 7 shows a run for $D=1$ and times $t=0$ – 130 in time steps $\Delta t = 2$. As can be seen, the small ionization seed near $x_0=50$ initially grows while drifting to the right in accord with Eq. (\[2034\]). At time $t = {\cal O}(20)$, the ionization is strong enough that field saturation sets in and two asymmetrically propagating fronts emerge. The one propagating to the right develops into a uniformly propagating $NSF$ with velocity $v^*=2.21$ [@asymptotics] and degree of ionization behind the front $\sigma^-=0.130$. The maximum value of $\sigma$ in the rear part of the front is $\sigma_{max} =
0.150$. At the same time, a structure develops on the left, which at time $t=130$ not yet has reached a stationary form, and which eventually will develop into a $PSF$. (Note that propagation to the left into a negative field $-E^+$ corresponds to a $PSF$ front moving to the right towards $x\rightarrow \infty$ in a field $+E^+$). How the $PSF$ actually reaches a uniformly translating profile, is shown in Fig. 8, where the development for $x_0=150$ and otherwise identical initial and boundary conditions is followed in time steps of $\Delta
t = 10$ during the time $t=0$ – 500. An asymptotic velocity of $v^\dagger = 0.22$ and a degree of ionization $\sigma^-=0.43$ is reached. Note the huge difference in the degree of ionization and in the front velocity already for the unrealistically large value of the diffusion constant $D=1$.
The predictions from Section V for the selected uniformly translating fronts for $D=1$ and $E^+=\pm 1$ yield for the $NSF$ $v^* = 2.213$ and $\sigma^- = 0.129$, and for the $PSF$ $v^\dagger = 0.2199$ and $\sigma^- = 0.432$. They thus correctly predict the simulations of the initial value problem shown in Figs. 7 and 8 within the accuracy given. Note that for the velocity $v^\dagger$ of the $PSF$ and for the degrees of ionization $\sigma^-$ both behind the $PSF$ and the $NSF$ this fact also shows the relative accuracy of the two very different numerical methods used, while for the velocity $v^*$ of the $NSF$ the numerical integration of the initial value problem exactly reproduces the analytic result.
As $D$ decreases, both the structures within the fronts and the asymmetry between $NSF$ and $PSF$ become more pronounced. We illustrate this in Figs. 9 and 10 with the temporal development starting from the same initial perturbation as before, but now for $D=0.1$, the value corresponding to the parameter values of the earlier three-dimensional simulations [@DW; @Vit]. The time ranges in each plot are chosen appropriately for seeing the $NSF$ and the $PSF$ evolve into a uniformly translating state. Fig. 9 shows a perturbation (initially localized at $x_0=50$) evolving during time $t=0$ – 130 in time steps $\Delta t = 2$. Except for the smaller diffusion constant and the stretched $x$ axis, the situation is thus identical with that of Fig. 7. The $NSF$ on the right propagates with a somewhat smaller velocity $v^*=1.39$, leaving a slightly higher ionization $\sigma^-=0.147$ behind. The maximum $\sigma_{max}=0.199$ is relatively higher, since diffusional smoothening of structures is less pronounced. On the time scales of Fig. 9, the left front does not propagate, but retracts into an apparently immobile structure. The electrons drift with the field into the ionized region, leaving a layer of screening ions behind. Thus the electrons and the field are almost separated such that ionization on this side almost cannot occur. Eventually few electrons will reach the nonzero field region by diffusion and slowly build up a higher ionization and ultimately a propagating $PSF$. That a $PSF$ actually emerges, is shown in Fig. 10. Only times $t=4000$ - 8000 in time steps of $\Delta t =100$ after the initial perturbation at $t=0$ and $x_0=60$ are shown. The front propagates with velocity $v^\dagger=0.0149$ leaving behind an ionization $\sigma^-=6.32$. The numerical values predicted in Section V are $v^*=1.384$ and $\sigma^-=0.144$ for $NSF$, and $v^\dagger=0.0146$ and $\sigma^-=6.234$ for $PSF$. The remaining numerical discrepancy of maximally 2% could be resolved by choosing a still smaller gridsize in Figs. 9 and 10. Comparison of the $PSF$ for $D=1$ and $D=0.1$ indicates that the time it takes such a front to build up, rapidly increases with decreasing $D$, but we have not pursued the scaling of the transient time with $D$.
We finally show in Fig. 11 the evolution of streamer fronts starting from non-localized initial conditions, i.e., not obeying the bounds (\[w3\]) or (\[w4\]) for $D=0.1$. We used an initial electron density profile $\sigma(x,t=0)= 0.01/ (2\:\cosh\Lambda(x-200))$ with $\Lambda=0.25$ and an initial field $E=-1$. At these values, for the $NSF$, $\Lambda^{+}_{-}(v^*)=1.918$ and for the $PSF$, $\Lambda^{+}_{-}(v^\dagger)=0.3766$. In this case, the bounds (\[w3\]) or (\[w4\]) are indeed violated for both fronts, and Eq. (\[w5\]) predicts a $PSF$ with velocity $v=0.497>v^\dagger=0.0146$ and an $NSF$ with velocity $v=2.497>v^*=1.384$. The simulations find the fronts propagating with velocities 0.50 and 2.50, respectively. The ionization behind the $NSF$ is $\sigma^-=0.149$ and behind the $PSF$ $\sigma^-=0.158$, so that now both are comparable to each other and to $\sigma^-(D=0)=0.1485$ found from Fig. 5(a). Note that the diffusion constant is identical with that of Figs. 9 and 10, the only difference being the extended initial perturbation.
The simulations confirm that streamer front propagation is indeed correctly described by the marginal stability scenario, which in the present case amounts to the statement that the slowest physical velocity is selected, whenever one starts from sufficiently localized initial conditions, just as for the simpler case (\[aw\]).
Conclusions and Outlook
=======================
The analysis in this paper fully supports the validity of an effective interface approach suggested by the results of the full three-dimensional simulations of Dhali and Williams and of Vitello [*et al.*]{} [@DW; @Vit]. This emerges from our detailed study of the associated one-dimensional problem, which yields the following results:
- After a very brief stage of transient exponential amplification of the initial ionized seed, the growing streamer evolves into an electrically screened plasma body separating two narrow fronts which propagate into the non-ionized outer region. We show that these two fronts correspond, for all practical purposes, to translating profiles which propagate independently. This entails that the separation of spatial scales between an inner front and an outer one, set by the global geometry, is indeed justified.
- This enables us to draw upon the existing knowledge about front propagation into unstable states and thus to provide definite predictions about:
$\bullet$ the relationship $v_f(E^+)$ between the velocity of a planar streamer front and the value of the electric field ahead of it, and
$\bullet$ the value of the degree of ionization of the plasma created by the front, $\sigma^-(E^+)$.
These predictions, although only valid as such in the absence of front curvature, still compare very favorably with the numerical results of Ref. [@Vit]. The two values of $\sigma^- (E^+)$ on the axis of Fig.1$(a)$ and 1$(b)$ behind the curved fronts of the 3D simulations [@Vit] (with the convention that $E^+$ should be understood as the electric field value extrapolated from the external non-ionized region to the front position) are plotted in Fig. 5$(a)$. Without adjustable parameters our one-dimensional predictions for $\sigma^-(E^+)$ are well within a factor of 2 from the 3D simulations. Likewise, the velocity values for $v_f(E^+)$ even agree to about 20%.
Moreover, our analysis shows that $NSF$ and $PSF$ propagate in this model and for realistic values of the reduced diffusion coefficient $D$, in a very asymmetric manner:
$\bullet$ $NSF$ rapidly reach a regime of uniform propagation — typically, on the scale of several tens of time units, i.e., in less than $10^{-10}$s. Their velocity is slightly larger than the electron drift velocity in the field $E^+$.
$\bullet$ This is to be contrasted with the dynamics of $PSF$: For realistic $D$-values, of order 0.1-0.3, they approach uniform translation considerably more slowly than $NSF$ — typically on the time scale of $10^{-8}$s. Moreover, their asymptotic velocity is also much smaller than $v_f^{NSF}$. It obeys the inequality $v_f^{PSF}<DE^+
\exp(-1/E^+)$ [@ebert]. Finally, while the widths of $PSF$ and $NSF$ are comparable, the degree of ionization behind $PSF$ is much larger (up to a hundred times for $D=0.1$) than that behind $NSF$.
These results answer the question of whether $PSF$ do or do not propagate, while explaining why the simulations of Vitello [*et al.*]{} [@Vit] could not yield a definite answer — most probably, because, although their total width is of order $\alpha_0^{-1}$, their true inner length scale, as defined by the steepness of the profile, was too small to be resolved by their grid size. (Note that the apparent symmetry between $PSF$ and $NSF$ found in earlier simulations [@DW] is to be related to the fact that there propagation into a pre-ionized medium (with initial electron density of $10^8/ \mbox{cm}^3$) is studied, and possibly also due to the use of a poorly resolving grid.)
It was observed empirically by Dhali and Williams [@DW] that in the three-dimensional simulations, the dielectric relaxation time in the plasma behind the front was of the same order as the intrinsic time scale set by the front motion. Our analysis shows, that this was no accident: It is a manifestation of the fact that the balance of the growth mechanism (impact ionization) and the stabilization mechanism (screening) leads to a single time scale $t_0=(\alpha_0 \mu_e E_0)^{-1}$ for a $NSF$ and for the relaxation behind it for fields of order $E_0$. Since our dimensionless value of $\sigma^-$ is the inverse dielectric relaxation time, it is of order unity (or slightly smaller) for fields $E^+ \approx -1$.
Of course, the above results should only be considered as a first step towards a realistic treatment of streamer propagation. They will have to be developed and extended along two different directions:
[*(i)*]{} [*Predictions of patterns within the present model and comparison with the simulations:*]{} Within the frame of the present continuous and fully deterministic model, we here have only considered the restricted case of a one-dimensional geometry. This enabled us to demonstrate that the concept of effective interfaces does apply to streamers. This approach will now have to be extended to the description of curved fronts. As also discussed in [@short], one will then be equiped with a reduced formulation, valid on the outer scale, which will permit us to study real three-dimensional streamers as pattern-forming systems, as was done, e.g., for viscous fingers and dendritic solidification fronts [@reviews]. This should provide a direct approach to the question of dielectric patterns, alternative to the phenomenological $DLA$-inspired dielectric breakdown models [@pietronero].
[*(ii)*]{} [*Possibly, extensions of the model will be necessary to predict real experiments:*]{} We have based our analysis on the minimal model as defined in Section II. It contains several restricting simplifications. A first step in the improvement of the model would be to include the field dependence of the transport coefficients $D_e$ and $\mu_e$. It is clear that this will not modify our qualitative analysis, as, e.g., the counting argument for the existence of front solutions in Section IV depends only on the linearization about the stable and unstable states. Moreover, the qualitative asymmetry between the $NSF$ and $PSF$ will persist as these result from the asymmetry of the electron drift. Quantitatively, the value of $v^*$, the selected value of $NSF$, will simply be given by (\[w1\]) with the transport coefficient and ionization rate evaluated at the field value $E^+$. The slow transient build-up and small speed of $PSF$ could be affected quantitatively by ionic motion, but from this effect, we expect no major qualitative differences.\
Finally, it should be kept in mind that our continuum equations are only valid on length scales larger than the mean free path $\ell_{mfp}$. On the other hand, we find for the strongest field values appearing in the simulations (which are much larger than the values of the field across the gap, due to the enhancement near the streamer tip), that the front width decreases down to about $3
\alpha_0^{-1}\approx 3 \ell_{mfp}$ in the approximation (\[mfp\]). In such limits, nonlocality of the transport and ionization effects begin to play a role. In addition, under these conditions, a typical volume of size $\ell_{mfp}^3$ contains only of the order of 1000 electrons for the parameter values (\[2026\]) used in the simulations. Fluctuations are then likely to become non-negligible. In principle, treating these effects calls for a full kinetic description. This is probably out of reach for the moment, but one might want to mimic the main features of these effects by introducing stochastic terms in the equations. These also could mimic photo-ionization somewhat before the front due to photons released in the impact ionization events, or the natural homogeneous background ionization due to radioactivity and cosmic radiation. Investigation of their relevance for branching of dielectric breakdown patterns might help to understand the asymmetry between the macroscopic patterns of discharges propagating into a positive or a negative field [@maan].
In conclusion, our analysis opens the way to a microscopically based interface approach to discharges that seems promising for building a coherent framework for the analysis of breakdown patterns of various degrees of complexity.
WvS gratefully acknowledges hospitality of the Université Paris VII, where this work was initiated. UE thanks F. Döbele and A. Stampa for valuable discussions about streamer experiments and plasma physics. Her work was made possible by the Priority Program Non-Linear Systems of the Dutch Science Foundation NWO. We also gratefully acknowledge financial support by NWO and the Lorentz Fund for visits of UE and WvS to the Université Paris VII. Finally, we like to thank P.A. Vitello for making copies of figures from [@Vit] available, which appear here as Fig. 1.
Differences and similarities between combustion and streamer fronts {#differences-and-similarities-between-combustion-and-streamer-fronts .unnumbered}
===================================================================
In the introduction, we draw on the similarity between the streamer problem and other problems like combustion, chemical waves, thermal plumes, phase field models, [*etc.*]{}, to motivate the development of an effective interface approach. Of these problems, streamer propagation is most closely analogous to combustion, in that the strong nonlinearity of the reaction rates (the combustion rate and the ionization rate) is an important factor in giving rise to front development in flames and streamers, respectively. There are important differences as well, however, and since several interface techniques were originally developed in the context of combustion[@wil; @buc; @pf1], we highlight some of the differences and similarities here:
[*(a)*]{} In combustion the reaction rate depends strongly on the temperature, whose outer dynamics is governed by a diffusion equation of the form $\partial_t T\!=\!\nabla^2 T$, while for streamers the ionization rate depends strongly on the field $|{\bf E}|$, with ${\bf
E}$ the [*gradient*]{} of the potential $\Phi$ that obeys the Laplace equation $\nabla^2\Phi \approx 0$ in the outer region where the total charge density vanishes. This field strength ${\bf E}$ varies strongly in the streamer front, since the increased screening resulting from the rising electron density suppresses ${\bf E}$ — and hence the ionization rate — to zero. In combustion, on the other hand, the temperature hardly varies throughout the combustion zone.
[*(b)*]{} Combustion fronts are essentially fronts progating into a metastable state, because the front has to supply the heat that increases the temperature and hence the reaction rate, while streamer front propagation is an example of front propagation into unstable states, where the reaction starts for any nonvanishing electron density.
[*(c)*]{} In a flame front typically the temperature remains high enough that all the reactions proceed to saturation: all the combustable material burns. The temperature difference between the flame front and the background is then essentially determined by conservation (conversion) of energy. In typical streamer fronts, on the other hand, the field ${\bf E}$ is suppressed long before saturation effects start to play a role, and hence the ionization level behind the front is set by the internal dynamics of the front rather than by conservation laws (i.e., the gas density).
[*(d)*]{} The electron drift $-\mu_e{\bf E}$ has no clear analogue in combustion.
[*(e)*]{} Finally, the relevant asymptotic expansion for streamers is not quite like the “activation energy asymptotics” of combustion [@wil; @buc], since we consider here fields strengths that are comparable to the characteristic field scale $E_0$ of the ionization rate given in Eq. (\[2006\]) before the front, whereas in combustion activation energy asymptotics is often appropriate since the flame temperature remains much smaller than the chemical activation energy. For streamers, an analysis like activation energy asymptotics is appropriate in the limit of small fields $|{\bf E}|\ll E_0$. Of course, in streamers the rapid variation of the field ${\bf E}$ in the front region, and hence the rapid suppression of the ionization rate, looks, at first sight, similar to the suppression of the chemical reaction rate with decreasing temperature in flames. However, in flames this is due to the strongly nonlinear dependence of the reaction rate on temperature before the front, (so that a slight suppression of the temperature reduces the reaction rate dramatically), while in streamers in large external fields of order $E_0$ this is due to the fact that the field itself is reduced significantly behind the streamer front, as a result of screening.
I. Gallimberti, J. Phys. (France) [**40**]{}, Colloque C7, 193 (1979). A. Bondiou and I. Gallimberti, J. Phys. D.: Appl. Phys. [**27**]{} 1252 (1994). Y.P. Raizer, [*Gas Discharge Physics*]{} (Springer, Berlin, 1991). A.N. Lagarkov and I.M. Rutkevich, [*Ionization Waves in Electrical Breakdown of Gases*]{} (Springer, New York, 1994) L.M. Vasilyak, S.V. Kostyuchenko, N.N. Kudryavtsev and I.V. Filyugin, Physics–Uspekhi [**37**]{}, 247 (1994). The distinction between the various discharge phenomena seems to vary between authors. See, e.g., D. Braun, U. Küchler and G. Pietsch, J. Phys. D: Appl. Phys. [**24**]{}, 564 (1991); W.R. Rutgers, in: [*Proc. XXI Int. Conf. on Phenomena in Ionized Gases*]{}, ed.: AG Plasmaphysik, Bochum, Düsseldorf, Essen, Jülich (RUB, Bochum, 1993); U. Kogelschatz, in: [*Proc. X Int. Conf. on Gas Discharges and their Applications*]{}, ed. W.T. Williams (Swansea, UK: Univ. College Swansea, 1992), p. 972, vol. 2. S.K. Dhali and P.F. Williams, Phys. Rev. A [**31**]{}, 1219 (1985) and J. Appl. Phys. [**62**]{}, 4696 (1987). P.A. Vitello, B.M. Penetrante, and J.N. Bardsley, Phys. Rev. E [**49**]{}, 5574 (1994). See also P.A. Vitello, B.M. Penetrante, and J.N. Bardsley, in [*Non-Thermal Plasma Techniques for Pollution Control*]{}, ed, B.M. Penetrante and S.E. Schultheis (Springer, Heidelberg, 1993). M C. Wang and E.E. Kunhardt, Phys. Rev. A [**42**]{}, 2366 (1990). See references in this article to earlier numerical studies by Kunhardt and co-workers. F.A. Williams, [*Combustion Theory*]{}, (Benjamin/Cummings, Menlo Park, 1985). J.D. Buckmaster and G.S.S. Ludford, [*Theory of Laminar Flames*]{}, (Cambridge University Press, Cambridge, 1982), or: J.D. Buckmaster and G.S.S. Ludford, [*Lectures on Mathematical Combustion*]{}, (SIAM, Philadelphia, 1983); a nice elementary account of several of the essential ingredients of flame fronts and their instabilities can be found in J.D. Buckmaster, Physica [**12D**]{}, 173 (1984). M. van Dyke, [*Perturbation Methods in Fluid Dynamics*]{} (Parabolic, Stanford, 1975). C.M. Bender and S.A. Orszag, [*Advanced Mathematical Methods for Scientists and Engineers*]{} (McGraw-Hill, New York, 1978). E. Meron, Phys. Rep. [**218**]{}, 1 (1992). G. Zocchi, P. Tabeling and M. Ben-Amar, Phys. Rev. Lett. [**69**]{}, 601 (1992), and M. Ben-Amar, Phys. Fluids [**A4**]{}, 2641 (1992). P. Fife, [*Dynamics of Internal Layers and Diffusive Interfaces*]{} (SIAM, Philadelphia, 1988). A. Karma and W.J. Rappel, Phys. Rev. E [**53**]{}, R3107 (1996) and references therein. U. Ebert, W. van Saarloos, and C. Caroli \[submitted\]. Some important reviews of recent advances are: J.S. Langer in [*Chance and Matter*]{}, ed. J. Souletie (North-Holland, Amsterdam, 1987); D.A. Kessler, J. Koplik and H. Levine, Adv. Physics [**37**]{}, 255 (1988); P. Pelcé, [*Dynamics of Curved Fronts*]{} (Academic, Boston, 1988); and Y. Pomeau and M. Ben Amar, in: [*Solids far from Equilibrium*]{}, C. Godreche, Ed. (Cambridge University Press, Cambridge, 1992). E. Ben-Jacob, H.R. Brandt, G. Dee, L. Kramer, and J.S. Langer, Physica [**14D**]{}, 348 (1985). W. van Saarloos, Phys. Rev. A [**37**]{}, 211 (1988). W. van Saarloos, Phys. Rev. A [**39**]{}, 6367 (1989). W. van Saarloos and P.C. Hohenberg, Physica D [**56**]{}, 303 (1992); errata: Physica D [**69**]{}, 209 (1993). G.C. Paquette and Y. Oono, Phys. Rev. E [**49**]{}, 2368 (1994); G.C. Paquette, L.-Y. Chen, N. Goldenfeld and Y. Oono, Phys. Rev. Lett. [**72**]{}, 76 (1994). D.L. Turcotte, and R.S.B. Ong, J. Plasma Phys. [**2**]{}, 145 (1968). N.W. Albright, and D.A. Tidman, Phys. Fluids [**15**]{}, 86 (1972). R.G. Fowler, Advances in Electronics and Electron Physics [**35**]{}, 1 (1974); and [**41**]{}, 1 (1976). M.I. D’yakonov and V.Y. Kachorovskii, Sov. Phys. JETP [**68**]{}, 1070 (1989). We have been informed (V. Shumeiko, private communication with WvS), that D’yakonov has in a talk also pointed out the strong similarity between streamers and other problems of pattern formation such as dendrites. Instead of distinguishing the two different types of streamers with the acronyms [*ADS (anode directed streamer)*]{} and [*CDS (cathode directed streamer)*]{} as is sometimes done, we prefer to use [*NSF (negative streamer front)*]{} and [*PSF (positive streamer front)*]{}, as this wording is more appropriate for an effective interface approach. D.G. Aronson and H.F. Weinberger, in [*Partial Differential Equations and Related Topics*]{}, ed.: J.A. Goldstein (Springer, Heidelberg, 1975); Adv. Math. [**30**]{}, 33 (1978). P. Collet and J.-P. Eckmann, [*Instabilities and Fronts in Extended Systems*]{} (Princeton University Press, Princeton, 1990). A. Kolmogorov, I. Petrovsky, and N. Piskunov, Bull. Univ. Moskou, Ser. Internat., Sec. A [**1**]{}, 1 (1937). R.A. Fisher, Ann. Eugenics [**7**]{}, 355 (1937). A short calculation shows, that a homogeneous current $|{\bf j}|=1$ generates magnetic fields, whose Lorentz force on the free charges moving with the drift velocity $\mu_e E_0$ becomes comparable to the force of an electric field of order $E_0$, when the current extends over a region of width $\ell_B = \alpha_0^{-1} c^2 / (\mu_e E_0)^2
= 1.6\cdot10^5\;\alpha_0^{-1} \approx 0.4$ m for the parameters of [@DW; @Vit]. In practice, streamer dimensions are much smaller than $\ell_B$, and hence the magnetic fields can generally be neglected. S.K. Dhali and A.P. Pal, J. Appl. Phys., 1355 (1988). It is useful to realize that our model is inaccurate for small fields, since any nonzero electron density is unstable in our model for any field strength, due in particular to the neglect of recombination terms. A similar situation occurs in combustion; there, the exponential reaction rate is sometimes modified by introducing a so-called switch-on-temperature, below which the combustion rate is zero. As discussed in [@buc], the results do no strongly depend on the switch-on-temperature provided it is low enough, and an asymptotic expansion in fact nicely circumvents the introduction of this temperature. Similar observations hold for the streamer problem: A crude way to circumvent the introduction of recombination terms would be to introduce a threshold-field $E_{thr}$ below which $f(|{\bf E}|)$ vanishes. Essentially all of our analysis applies to such cases as well, and the results do not depend sensitively on the value of this field. One should be aware that it is possible that the state generated by the front is actually unstable or metastable. In such cases the front is usually followed by a second front invading the unstable or metastable state. See, e.g., K. Nozaki and N. Bekki, Phys. Rev. Lett. [**51**]{} 2171 (1983), and F.J. Elmer, J. Burns and H. Suhl, Europhys. Lett. [**22**]{} 399 (1993). The discussion is easily generalized to the case in which the ionization function $f(|{\bf E}|)=0$ in an interval ${
E} \in [0,E_{thr}]$ — compare Ref. [@footnotef]. In this case, ionization is absent for any field $E < E_{thr}$, so that there is a larger multiplicity of stationary states of the model, and so that for fields below this threshold the dynamics is due to conventional electrodynamics with conserved particle densities. This generalization, however, does not affect our conclusions about front propagation qualitatively: It is easy to convince oneself that the state generated after long times by a propagating front is still the one with ${\bf E}=0$, since as long as the field behind the front is nonzero but of the same sign as $E^+$, charge is pumped into the streamer head from behind in such a way that the field jump across the streamer head increases in time. Since $E^+$ is fixed, this implies that the field behind the streamer head decays. E.D. Lozanskii, Zh. Tekh. Fiz. [**38**]{}, 1563 (1968) \[Sov. Phys. Tech. Phys. [**13**]{}, 1269 (1969)\]. W.K.H. Panofsky and M. Phillips, [*Classical Electricity and Magnetism*]{}, (Addison–Wesley, Reading, Massachusetts, 1962), p. 122. This is argued in appendix A of [@vs1]: For equations like ours that admit uniformly translating front solutions, one generally expects a [*one-parameter*]{} family of front solutions, a feature well known for the case of the nonlinear diffusion equation discussed briefly in Section V. For pattern forming fronts propagating into an unstable state, one typically has a [*two parameter*]{} family of fronts — see, e.g., [@vs3] or P. Collet and J.-P. Eckmann, Commun. Math. Phys. [**107**]{}, 39 (1986), although a general proof is lacking. M. Abramowitz and I.A. Stegun, [*Handbook of Mathematical Functions*]{} (Dover, New York, 1972). W.H. Press, B.R. Flannery, S.A. Teukolsky and W.T. Vetterling, [*Numerical Recipes*]{} (Cambridge University Press, Cambridge, 1986). Of course, in principle in a three-dimensional dynamical system, one can have limit cycles and other (strange) attractors. Limit cycles would in our case correspond to spatially periodic ionization waves, but the occurrence of such solutions is clearly excluded physically by the fact the ion density must be a monotonically increasing function of $\xi$. Other types of nontrivial behavior of the flow are excluded for the same reason. See, e.g., R.D. Benguria and M.C. Depassier, Phys. Rev. Lett. [**73**]{}, 2272 (1994); and A. Goriely, Phys. Rev. Lett. [**75**]{}, 2047 (1995). U. Ebert \[in preparation\]. I.M. Gel’fand, Usp. Mat. Nauk [**14**]{}, No. 2 (86) 87 (1959) \[Am. Math. Soc. Transl. Ser. 2 [**29**]{} 295 (1963)\]. G.T. Dee and J.S. Langer, Phys. Rev. Lett. [ **50**]{}, 383 (1983). G.T. Dee and W. van Saarloos, Phys. Rev. Lett. [**60**]{}, 264 (1988). For a review, see A. Bers, in: [*Handbook of Plasma Physics*]{}, ed.: M.N. Rosenbluth and R.Z. Sagdeev (North-Holland, Amsterdam, 1983), Vol. 1. E.M. Lifshitz and L.P. Pitaevskii, [ *Physical Kinetics*]{}, Course of Theoretical Physics Vol. 10 (Pergamon, New York, 1981), Chap. VI. By tracking the position $x_f(\tau)$ of the point where $\sigma(x_f(\tau),\tau)=0.1 \sigma^-$, we followed the convergence of the local velocity $v_f(\tau)=dx(\tau)/d\tau$ towards $v^*$, and found it to be consistent with the formula $v_f(\tau)=v^*
-1.5D/(\sqrt{f|E^+|}\tau) + \cdots $ derived in [@vs2]. L. Niemeyer, L. Pietronero and H.J. Wiessmann, Phys. Rev. Lett. [**52**]{}, 1033 (1984). J.C. Maan, B. Willing, and P. Uhlig, Physica B [**204**]{}, 311 (1995).
(8,12.7) (-.5,5) (3.5,5) (-17,0) (1.2,6.6)[$(a)$]{} (5.2,6.6)[$(b)$]{} (2.0,1.3)[$(c)$]{}
FIG. 1. Results of the numerical simulations of the full three-dimensional streamer equations (\[2011\])-(\[2006\]) of Vitello [*et al.*]{}, reprinted from Fig. 1 and 10 in [@Vit]. $(a)$ Negative streamer propagating downwards towards the anode. Electrodes are planar and located at z$=0$ and 0.5 cm; the voltage between the electrodes is 25 kV, which in the absence of the streamer amounts to a constant electric field $|{\cal E}|=E_0/4$. The system continues sidewards sufficiently far to make the lateral boundaries irrelevant. The streamer is assumed to be cylindersymmetric. The dimensionless diffusion constant is $D=0.1$. Each line indicates an increase of $n_e$ by a factor 10; densities of $10^3 -
10^{14}$ cm$^{-3}$ can be seen (initial background ionization: $1$ cm$^{-3}$). Shape at time 4.75 ns after an initial ionization seed was placed near the upper electrode. $(b)$ Shape at time 5.5 ns. $(c)$ Logarithmic electron $n_e$ and total charge $n_s$ density along the symmetry axis of $(b)$. Solid line: $n_e$; dot-dashed: $|n_s|$ for $n_s>0$; dotted: $|n_s|$ for $n_s<0$. Note the exponential increase of the densities on $\mu$m scale within the front as well as maximum of both densities in the rear part of the front. Courtesy of P.A. Vitello.
(8,5.5) (-.2,-2) (3.0,-.5) (.1,4.2)[$(a)$]{} (4.1,5)[$(b)$]{} (2.9,3.8)[$\sigma$]{} (5.65,0.5)[$E$]{} (6.1,4.8)[$\sigma$]{} (7.7,3.3)[$E$]{} (6.3,2.2)[$q$]{}
FIG. 2. $(a)$ Uniformly translating fronts for $D=0$ and $v=2$ shown as flows in the two-dimensional $(E,\sigma)$ phase space. Out of each point $\sigma^-$ on the $\sigma$ axis, there is a $PSF$ flowing to the right and a $NSF$ to the left. Both reach the same value $|E^+|$ on the horizontal axis, which also is independent of $v$. Note, that $NSF$ have a maximum of $\sigma$ within the front, while $PSF$ have monotonic $\sigma$. Note also, that no physical fronts (i.e., with $\sigma \ge 0$ everywhere) reach a value $E^+<-v=-2$, in agreement with Eq. (\[409\]). $(b)$ Sketch of a uniformly translating $PSF$ and $NSF$ for $D \ne 0$ as a flow in three-dimensional $(E,\sigma,q)$ phase space. The thick curves indicate the trajectories, while the thin ones show their projection into the $\sigma=0$ and $q=0$ planes. For fixed $v$, there is at each point of the $\sigma$ axis still only one outgoing vector, which can be followed in two antiparallel directions. The $E$ axis is fully attractive and will always be reached.
(8,5) =6.5cm (.8,.3)
FIG. 3. Phase diagram for $PSF$ as a function of $D$ and $E^+$. Above the solid line the lowest speed of physical front solutions is given by $v^*$, below the line $v^\dagger$ corresponds to the smallest speed of physical front solutions. Accordingly, the selected front speed is $v^*$ above the solid line ([*linear marginal stability regime*]{}), and $v^\dagger$ below the solid line ([*nonlinear marginal stability regime*]{}). The dotted curve indicates $v^*=0$ and is a lower bound for the cross-over to $v^\dagger$ behaviour of the selected fronts.
(8,5.5) (.5,.5)
FIG. 4. $\tilde{v}^\dagger = v^\dagger/D$ (solid) and $\tilde{v}^* = v^*/D$ (dashed lines) as a function of $D$ for $E^+ =$ 0.3 – 1.0 in steps of 0.1, and for $E^+=$ 1.0 – 2.0 in steps of 0.2. $\tilde{v}^\dagger$ depends only weakly on $D$, i.e., the physical front velocity $v^\dagger$ is approximately proportional to $D$. At $\tilde{v}^\dagger(E^+,D_{cr}(E^+))=\tilde{v}^*(E^+,D_{cr}(E^+))$, the selected front crosses over from $v^\dagger$ to $v^*$; the $v^\dagger$ solutions disappear. Plotting $D_{cr}(E^+)$ in the $(E^+,D)$ plane yields the solid curve in the phase diagram of Fig. 3, while the zeros of $v^*$ determine the dotted curve in Fig. 3.
(8,10.5) (.5,.5)
FIG. 5. Electron density $\sigma^-$ behind the planar selected front as a function of the field $E^+$ before the front for several $D$; dotted: $v^*$ fronts; solid: $v^\dagger$ fronts. $(a)$ $NSF$: For $v^*$ fronts, $\sigma^-$ depends but weakly on $D$. Results for $D=$ 0, 1, 3 are shown. Crosses: Extrapolation of $\sigma^-(E^+)$ for $D=0.1$ for the curved fronts of the 3D simulations [@Vit]. $(b)$ $PSF$ results for $D=$ 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 1, 2, 3. For $v^\dagger$ fronts, $\tilde{\sigma}^- = const. + {\cal O}(D)$, i.e., $\sigma^-(E^+)$ is approximately proportional to $1/D$.
(8,5) (.5,.5)
FIG. 6. Width $w$ of the front profile (measured between points with 0.1 $\sigma^-$ and 0.9 $\sigma^-$) as a function of $E^+$ for the selected $NSF$ fronts with $D=0.1$. The dashed line is given by $w=6/ \Lambda^+_-(v^*)$.
(8,9.5) =6.5cm (.7,-.5)
FIG. 7. Numerical integration of the time evolution given by Eqs. (\[307\]) and (\[308\]) for $D=1.0$ in a constant background field $E=-1$ (numerical gridsize $\Delta x=0.1$ and timestep $\Delta \tau=0.05$, initial perturbation at $x_0=50$). Initial condition at $t=0$: small charge-neutral, ionized region of Gaussian shape depicted by the lowest line. Each new line corresponds to a time step $\Delta t =2$ and the upper line to $t=130$. $(a)$ The electron density $\sigma(x,t)$ initially grows and then, after field screening in the middle sets in, develops into a $NSF$ propagating to the right and a $PSF$ propagating to the left. $(b)$ The electric field $E(x,t)$ stays $E=-1$ in the non-ionized region and becomes dynamically screened to zero in the ionized region.
(8,5) =6.5cm (.7,.5)
FIG. 8. Emergence of the uniformly translating $PSF$ on the left in the system of Fig. 7. Conditions identical to Fig. 7 except for $x_0=150$ and different numerical gridsize ($\Delta x=0.05$ and $\Delta \tau= 0.01$). $\sigma(x,t)$ is shown in time range $t=0$ – 500 in time steps $\Delta t = 10$. Note the difference in the duration of the transient regimes, in the propagation velocities of $PSF$ and $NSF$, and in the degrees of ionization behind these.
(8,4.5) (-.8,-1.5)
FIG. 9. Identical with Fig. 7$(a)$, except that here $D=0.1$. Time range also $t=0$ – 130 in steps of $\Delta t = 2$. The $NSF$ has sharper contours and propagates slower than for $D=1$, the $PSF$ appears not to develop.
(8,5) =6.5cm (.7,.5)
FIG. 10. Emergence of the uniformly translating $PSF$ on the left for $D=0.1$. Initial conditions identical with Fig. 9. The time range $t=4000$ – 8000 after an initial perturbation at $t=0$ and $x_0=60$ is shown in time steps of $\Delta t = 100$. (Numerical gridsize $\Delta x = 0.01$ and $\Delta \tau = 0.5$.)
(8,5) =6.5cm (.7,.5)
FIG. 11. A non-localized initial condition with $\Lambda = 0.25$ as described in the text; otherwise, the situation is like in Fig. 9, and $D=0.1$.
|
---
author:
- 'Daniel Egli[^1] and Zhou Gang[^2]'
title: Some Hamiltonian Models of Friction II
---
.1in
$^{\ast,\dagger}$Institute for Theoretical Physics, ETH Zurich, CH-8093, Zürich, Switzerland
Introduction
============
In [@Froehlich10] a model of quantum friction is introduced. A tracer particle is coupled to a bath of identical bosons. It is heuristically motivated that the regime of a very dense but weakly interacting bose gas, and heavy tracer particle, corresponds to a classical limit, and the model reduces to a classical Hamiltonian system for $(X,P)$, the position and momentum of the tracer particle, and $\beta(x)$, the fluctuation from the mean density $\rho_0$ of the bosons.
The resulting equations are $$\begin{aligned}
\dot{X_t}=&\frac{P_t}{M},\quad\quad\\
\dot{P_t}=&-\nabla_{X}V(X_t)-g\int\nabla_{X}W(x-X_t)(|\beta_t(x)|^2
+2\sqrt{\frac{\rho_0}{g^2}}Re\beta_t(x)) dx, \label{XPeqns'}\\
i\dot{\beta}_t(x)=&(-\frac{1}{2m}\Delta+gW(x-X_t))
\beta_t(x)+\sqrt{\rho_0}W(x-X_t) \nonumber\\
+&\kappa[\phi *(|\beta_t|^2+2\sqrt{\frac{\rho_0}{g^2}}
{\mathrm{Re}\,}\beta_t)](x)(\beta_t(x)+\sqrt{\frac{\rho_0}{g^2}})\,, \label{betaeqn}\end{aligned}$$ where $V$ is an external potential affecting only the tracer particle, $W$ is a two-body potential modelling the interaction between tracer particle and the medium, $\phi$ is a two-body potential modelling the interaction between medium particles, and $g,\kappa$ are coupling constants. See [@Froehlich10; @Froehlich102] for a detailed description of the model.
These equations are Hamiltonian with Hamilton functional $$\begin{aligned}
H(X,P,\beta,\bar{\beta}) =& \frac{P^2}{2M}+V(X)+\int \left[\frac{1}{2m}|\nabla\beta(x)|^2+gW(x-X)(|\beta(x)|^2+2\sqrt{\frac{\rho_0}{g^2}}{\mathrm{Re}\,}\beta(x))\right]\d x\\
+&\frac{\kappa}{2} \iint(|\beta(x)|^2+2\sqrt{\frac{\rho_0}{g^2}}{\mathrm{Re}\,}\beta(x))\phi(x-y)(|\beta(y)|^2+2\sqrt{\frac{\rho_0}{g^2}}{\mathrm{Re}\,}\beta(y))\d x\d y\,,\end{aligned}$$ and the standard symplectic form $P'\cdot X-P\cdot X'+2\i\,{\mathrm{Im}\,}\int\bar{\beta}\beta'$.\
In [@Froehlich102], the model of a non-interacting medium without external forcing ($\kappa=0, V=0$) and a tracer particle that is weakly coupled to the medium ($g\to 0$) is studied. In the present paper, we go one step further and restore the full coupling of the tracer particle to the medium, that is, we consider the following parameter regime: $$\kappa=0\ \text{and}\ g\neq 0.$$ The case of an interacting medium ($\kappa\neq 0$) will hopefully be treated in a forthcoming paper.
The equations take the form $$\begin{aligned}
\label{equations}
\dot{X}_t&=\frac{P_t}{M}\nonumber\\
\dot{P}_t&=-\partial_xV(X_t)-g\int_{\RR^3}\partial_xW^{X_t}\(|\beta_t(x)|^2+2\sqrt{\frac{\rho_0}{g^2}}{\mathrm{Re}\,}\beta_t(x)\)\d x\\
\i\dot{\beta}_t(x)&=h^{X_t}\beta_t+\sqrt{\rho_0}W^{X_t}\,,\nonumber\end{aligned}$$ where $h^{X_t}:=-\frac{1}{2m}\Delta+gW^{X_t}$ and $W^{X_t}(x):=W(x-X_t)$.
In the main part of the paper we consider the cases the external potential $V$ vanishes, and prove that the tracer particle experiences friction and is decelerated to a full stop. We prove a lower bound for the strength of this friction mechanism, namely $|P_t|\leq c t^{-1-\eps}$, $t\to\infty$, for some explicit $\eps>0$ depending on the initial conditions. At large times, the medium is shown to exhibit the expected behavior: It forms a “splash” that follows the motion of the tracer particle. Remarkably, even though initial conditions $\beta_0$ can be chosen to be very small (in $L^2$-sense), the splash that the medium forms eventually is not square integrable. This is a consequence of the fact that we chose the medium to be non-interacting. This fact is also responsible for making it difficult to “guess” the right asymptotic behaviour of $|P_t|$ on a heuristic level. See [@Froehlich10; @Froehlich102] for a more thorough discussion.
The second author, together with his collaborators, considered in [@Froehlich102] the problem with $\kappa=g=0$. They found completely analogous results. Nevertheless, our findings are interesting in their own right as we treat a particle coupled fully to the medium (as opposed to a weak coupling limit), which is usually a much harder problem. The main technical difference is that the generator of time evolution of the reservoir, $h^{X_t}=-\Delta+gW^{X_t}$, depends on time, for $g\neq 0$, through the position, $X_t$, of the particle.
The remainder of the paper is organized as follows. In section 2, we present the main mathematical result. In section 3, we analyze the local well-posedness. In section 4, we recast the equations in a more convenient form; in particular, we expand the time-dependent propagator around its value at some fixed large time, and we split the equation for $P_t$ into linear and non-linear parts. In section 5, we apply a contraction principle to prove the existence of the solution $P_t$ with the desired decay, and in section 6 we prove the main theorem. Technical proofs of various propositions used along the way have been relegated to the appendix.
Main theorem
============
In order to be able to state a precise theorem, introduce the continuous, monotonically increasing function $\Omega:(-\infty,1)\to \RR^+$, $$\begin{aligned}
\label{eq:defOmega}
\Omega(\delta):=\frac{1}{\pi}\int_0^1\frac{1}{1+(1-r)^\pez}(1-r)^\mez\(\frac{1}{1-2\delta}(r^\mez-r^{-\delta})+r^{\pez-\delta}\)\d r\,.\end{aligned}$$ By numerical simulation we find that there exists a constant $\delta^*\simeq 0.66$ such that $$\begin{aligned}
\Omega(\delta^*)=1.\end{aligned}$$ Moreover, for any constant $\delta<\delta^*$, we have $$\Omega(\delta)<1.$$ For the system of equations (\[equations\]) we prove the following main theorem,
\[thm:main\]Suppose that in the external potential vanishes, $V\equiv 0$, and the potential $W$ is smooth, spherically symmetric, decays rapidly at $|x|=\infty$, and satisfies $$\begin{aligned}
|\widehat{W}(0)|\neq 0\,.
\end{aligned}$$ Then, for any $\delta\in I:=(\frac{1}{2},\delta^*)$ there exists a $g_0>0$ and an $\eps_0>0$ such that if $0\leq g\leq g_0$ and ${\lVert \x^5\beta_0 \rVert}_2,|P_0|\leq\eps_0$ and ${\lVert \x^3\partial_x\beta_0 \rVert}_2<\infty$ then $$\begin{aligned}
\label{momentumdecay}
|P_t|\leq ct^{-\frac{1}{2}-\delta} \;\textrm{as $t\to\infty$,}
\end{aligned}$$ and $$\begin{aligned}
\label{deltadecay}
\lim_{t\to \infty}{\lVert \x^{-3}(\beta_t+\sqrt{\rho_0}(h^{X_t})^{-1}W^{X_t}) \rVert}_2=0\,.\end{aligned}$$ In particular, the particle comes to rest after a finite distance: There is an $X_\infty\in\RR^3$ such that $X_t\to X_\infty$, and $$\begin{aligned}
\beta_t \to -\sqrt{\rho_0}(h^{X_\infty})^{-1}W^{X_\infty}\notin L^2(\RR^3)\,.\end{aligned}$$
The theorem will be proved in section \[mainproof\].
Now we present the main difficulties in the proof and the strategies of overcoming them. Similar to what was proved in [@Froehlich102], we start with decomposing the equation for $\dot{P}_t$ into a linear and a non-linear part. Part of the linear equations can be solved explicitly, and we use the solution to rewrite the equation for $P_t$ in terms of this solution and the non-linear part. Since we expect that the momentum $P_t$ decays for large times $t$ it is reasonable to assume that eventually the dynamics is dominated by the linear part. The detailed knowledge of the decay properties of the solution to the linear part and standard dispersive estimates enable us to use a contraction principle to establish the claim. It is recommended that the reader consult [@Froehlich102] for a more precise outline of the general strategy which is almost identical to the present case.
There is one major technical difference to the model studied in [@Froehlich102], namely that the generator of time evolution, $h^{X_t}=-\Delta+gW^{X_t}$, depends on time through the position, $X_t$, of the particle. Mathematically, this makes it more involved to cancel various terms by symmetry considerations, and, as an additional complication, the generator of translations, $\partial_x$, no longer commutes with $h^{X_t}$. We treat this as follows. Since we expect that the particle will come to rest at some $X_\infty\in\RR^3$, we expand the propagator $U(t,s)$ gererated by $ h^{X_t}=-\Delta+gW^{X_t}$ around the “instantaneous” propagator, $\e^{-\i h^{X_t} t}$, at some large time $t$ where $$\begin{aligned}
\e^{-\i h^{X_t} t}=\e^{-\i h^{X_T} t}\big|_{t=T}\end{aligned}$$ is to be understood. By Duhamel’s principle we obtain $$\begin{aligned}
U(t,0)=\e^{-\i h^{X_t} t}-\i\int_0^t\e^{-\i h^{X_t}(t-s)}(X_s-X_t)\cdot\partial_xW^{X_t}\e^{-\i h^{X_t} s}\d s+\dots\,.\end{aligned}$$
To facilitate later discussions we rescale the equation such that $$2m=1, \ |\widehat{W}(0)|=1.$$
The local well-posedness
========================
In this section we discuss the local well-posedness of solutions to equation system .
Apply Duhamel’s principle on the last equation of to obtain $$\begin{aligned}
\label{eq:localEx}
\beta_{t}=U(t,0)\beta_0-\i\sqrt{\rho_0}\int_{0}^{t}U(t,s)W^{X_{s}}\ ds\,,\end{aligned}$$ where $U(t,s)$ is the propagator generated by the operators $h^{X_{\tau}}=-\Delta+gW^{X_{\tau}},\ \tau\in [s,t]$. Since the right hand side does not depend on $\beta_{\cdot},$ one can see that for any given trajectory $X_{\cdot}$ there exists a solution $\beta_{\cdot}$, with $\beta_{t}\in L^{2}(\mathbb{R}^3)$ for any time $t\in [0,\infty).$
To simplify the problem we plug into . Based on the discussion above, the local existence of the solution is transformed into the local existence of the trajectory and momentum. The latter can be achieved by a standard iteration technique. The proof is simple, but tedious. Hence we omit the details here.
Furthermore, we observe that if $\beta_0\equiv 0$ and $P_0=0$ then $X_t=X_0$ and $P_t=0$ is a solution. This, together with the local well-posedness, implies that a small solution exists in a large time interval. This is the content of the next theorem.
\[thm:localwp\]The equation (\[equations\]) is locally well-posed: If $P_0\in\RR^3$ and $\x^3\beta_0\in L^2(\RR^3)$ then there exists a time $T_{\rm loc}=T_{\rm loc}(|P_0|,{\lVert \x^3\beta_0 \rVert}_2)$ such that $|P_t|<\infty$ for any time $t\leq T_{\rm loc}$. Moreover, for any $T_{\rm loc}>0$ there exists an $\eps_0(T_{\rm loc})$ such that if $|P_0|,{\lVert \x^3\beta_0 \rVert}_2\leq \eps_0(T_{\rm loc})$ then $P$ satisfies the estimate $$\begin{aligned}
\label{eq:localdecay}
|P_t|\leq T_{\rm loc}^{-2} \quad t\in [0,T_{\rm loc}]\,.
\end{aligned}$$
Reformulation of the problem
============================
We begin with presenting the main difficulties and the strategies in overcoming them. Equation , obtained from the last equation in , does not help directly in our analysis. To illustrate the difficulty we plug into the right hand side of the second equation of . One of the terms we obtain is $$\begin{aligned}
\Psi_t:=\rho_0 {\mathrm{Re}\,}\i\langle \partial_{x}W^{X_{t}}, \int_{0}^{t} U(t,s)W^{X_{s}}\rangle\ \d s.\end{aligned}$$ In order to show $P_t\rightarrow 0$ as $t\rightarrow \infty$, we have to prove that this term is small. To this end, we will prove that $|X_t-X_s|$ is small, which yields $$\begin{aligned}
U(t,s)W^{X_{s}}=\e^{\i(t-s)h^{X_{t}}}W^{X_{t}}+O(|X_t-X_s|).\end{aligned}$$ Put this into the expression for $\Psi$ to find that the contribution of the first term is zero because $W$ is a spherically symmetric function. To make this rigorous and to further control the terms in $O(|X_t-X_s|)$ we Duhamel-expand the term $U(t,s)W^{X_{s}}$ around $\e^{\i(t-s)h^{X_{t}}}W^{X_{t}}$ to a certain order.
However, as it turns out, it is not convenient to work on the term $U(t,s)W^{X_{s}}$ as some of the information becomes hard to see. In what follows we present a different approach with the Duhamel expansion still being the underlying idea.
Since the generator $h^{X_t}$ depends on the position $X_t$ of the particle, we expand it around its value at a position $X_T$ for any fixed time $T>0$. Define $\bar{\beta}^X:=-\sqrt{\rho_0}(h^X)^{-1}W^X$ and introduce a new function $\delta_t=\delta_{t,T}$ by $$\begin{aligned}
\label{eq:decomp}
\beta_t-\bar{\beta}^{X_T}-\sqrt{\rho_0}\sum_{|\alpha|=1}^{N_0}\frac{1}{\alpha!}(X_t-X_T)^\alpha\partial_x^\alpha(\h)^{-1}\W
=:\delta_t\,.\end{aligned}$$ Here the positive integer $N_0$ is defined as $$\begin{aligned}
\label{eq:N0}
N_0:=\min\{n\in\NN:(n+1)(\delta-\frac{1}{2})\geq \frac{3}{2}+\delta\}.\end{aligned}$$ Recall the constant $\delta$ in Theorem \[thm:main\]. The motivation for choosing $N_0$ as above is that if we can prove $|P_t|=O(t^{-\frac{1}{2}-\delta})$, then $|X_t-X_T|^{N_0+1}=O(t^{-\frac{3}{2}})$.
Now, $\delta_t$ satisfies the equation $$\begin{aligned}
\label{eq:delta}
\i\dot{\delta}_t&=\h\delta_t+g(W^{X_t}-\W)\delta_t-\i\frac{\sqrt{\rho_0}}{M}P_t\cdot\sum_{|\alpha|=1}^{N_0}\frac{1}{\alpha!}\alpha(X_t-X_T)^{\alpha-1}\partial_x^\alpha(\h)^{-1}\W -G_1\nonumber\\
\delta_0&=\beta_0-\bar{\beta}^{X_T}-\sqrt{\rho_0}\sum_{|\alpha|=1}^{N_0}\frac{1}{\alpha!}(X_0-X_T)^\alpha\partial_x^\alpha(\h)^{-1}\W\,,\end{aligned}$$ where $\alpha X^{\alpha-1}$ means the vector $X=(\alpha_1X^{(\alpha_1-1,\alpha_2,\alpha_3)},\alpha_2X^{(\alpha_1,\alpha_2-1,\alpha_3)},\alpha_3X^{(\alpha_1,\alpha_2,\alpha_3-1)})$, and the term $G_1$ is defined as $$\begin{aligned}
\label{eq:G1}
G_1:=h^{X_t}r_{N_0}\,,\end{aligned}$$ with $r_{N_0}$ defined by $$\begin{aligned}
\label{eq:barbeta}
\bar{\beta}^{X_t}=:\bar{\beta}^{X_T}+\sqrt{\rho_0}\sum_{|\alpha|\leq N} \frac{1}{\alpha!}(X_t-X_T)^\alpha\partial_x^\alpha(\h)^{-1}\W+r_{N_0}\,,\end{aligned}$$ and estimated in the following lemma. Define an estimating function $\mu:\ \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ by $$\begin{aligned}
\label{majorant}
\mu(t):=\max_{0\leq s\leq t}(1+s)^{\frac{1}{2}+\delta}|P_s|\,.\end{aligned}$$ and recall the definition of $N_0$ in .
\[lem:taylor\] If $\mu(t)\leq 1$ in the interval $[0,T],$ then in the same interval the function $r_{N_0}$ in satisfies the estimate $$\begin{aligned}
\label{eq:rN}
{\lVert \x^3h^{X_t}r_{N_0} \rVert}_2\leq C_{N_0}|X_t-X_T|^{N_0+1}.
\end{aligned}$$ $$\begin{aligned}
\label{eq:rN2}
\|\langle x\rangle^{-3}r_{N_0}\|_2 \leq C_{N_0}|X_t-X_T|^{N_0+1}.
\end{aligned}$$
Use the fact $r_{N_0}(s)$ is the remainder term in the Taylor expansion of $(h^{X_s})^{-1}W^{X_s}$ to write the expression as $$\begin{aligned}
r_{N_0}(s)=(-1)^{N_0+1}\int_t^s\int_t^{s_1}\dots\int_t^{s_{N_0}}\partial_x^{j_1}\dots\partial_x^{j_{N_0+1}}(h^{X_{s_{N_0+1}}})^{-1}W^{X_{s_{N_0+1}}}\dot{X}_{s_{N_0+1}}^{j_{N_0+1}}\dots \dot{X}_{s_1}^{j_{1}}\d \underline{s}\,.\end{aligned}$$ The claim follows immediately by Taylor-expanding the function $\bar{\beta}_{t}$ around $\bar{\beta}_{T}$ in the vector variable $X_{t}-X_{T}$. To control the remainder we have used the fact that $(h^X)^{-1}$ is a bounded operator from $L^{2,3}$ to $L^{2,-3}$, and the exponential decay of $W$.
Using Duhamel’s principle we can rewrite $\delta_t$ in the form $$\begin{aligned}
\label{eq:deltaduha1}
&\delta_t&&=&&\e^{-\i\h t}\delta_0-\i g\int_0^t\e^{-\i\h(t-s)}[W^{X_s}-\W]\delta_s\d s\nonumber\\
&&&&&-\frac{\sqrt{\rho_0}}{M}\sum_{|\alpha|=1}^{N_0}\frac{1}{\alpha!}\int_0^t\e^{-\i\h (t-s)}\partial_x^\alpha(\h)^{-1}\W P_s\alpha(X_s-X_T)^{\alpha-1}\d s
+\i\int_0^t\e^{-\i\h (t-s)}G_1(s)\d s\,.\end{aligned}$$ The function $\delta_t$ admits the following estimate:
\[prop:delta\] If $\mu(T)\leq 1$ then for any $\tau\leq T$ we have $$\begin{aligned}
\label{eq:bounddelta}
{\lVert \x^{-3}\delta_\tau \rVert}_2\lesssim (1+\tau)^\mez\,.
\end{aligned}$$
The proposition will be proved in Appendix \[sec:ControlRem\].
In what follows we derive an equation for $\dot{P}_t$. To this end, we rewrite equation for $\delta_t$ as $$\begin{aligned}
\label{eq:deltaduha2}
&\delta_t&&=&&\e^{-\i\h t}\sqrt{\rho_0}(X_0-X_T)\cdot\partial_x(\h)^{-1}\W-\frac{\sqrt{\rho_0}}{M}\int_0^t\e^{-\i\h (t-s)}P_s\cdot\partial_x(\h)^{-1}\W \d s\nonumber\\
&&&&&+\e^{-\i\h t}(\beta_0-\bar{\beta}^{X_T}+\sqrt{\rho_0}\sum_{|\alpha|=2}^{N_0}\frac{1}{\alpha!}(X_0-X_T)^\alpha\partial_x^\alpha(\h)^{-1}\W)\nonumber\\
&&&&&-\i g\int_0^t\e^{-\i\h(t-s)}[W^{X_s}-\W]\delta_s\d s\nonumber\\
&&&&&-\frac{\sqrt{\rho_0}}{M}\sum_{|\alpha|=2}^{N_0}\frac{1}{\alpha!}\int_0^t\e^{-\i\h (t-s)}\partial_x^\alpha(\h)^{-1}\W P_s\alpha(X_s-X_T)^{\alpha-1}\d s
+\i\int_0^t\e^{-\i\h (t-s)}G_1(s)\d s\nonumber\\
&&&=:&&\sum_{n=1}^6D_n(t)\,,\end{aligned}$$ where $D_1$ and $D_2$ will be the main terms (being linear in $P_t$) in the equation for $\dot{P}_t$, whereas $D_3$ through $D_6$ will constitute remainder terms.
Recalling (\[equations\]) and using $\beta_T=\bar{\beta}^{X_T}+\delta_T$ we thus arrive at the following equation for $\dot{P}_t$, where we evaluate at $t=T$ to effect the cancelations due to spherical symmetry, which is only perfect when all centers agree: $$\begin{aligned}
\dot{P}_t\big|_{t=T}=&-2\rho_0{\mathrm{Re}\,}{\langle{\partial_x W^{X_T}} \mspace{2mu}, {\e^{-\i\h T}(X_0-X_T)\cdot\partial_x(\h)^{-1}\W}\rangle}\nonumber\\
&-2g\sqrt{\rho_0}{\mathrm{Re}\,}{\langle{\bar{\beta}^{X_T}\partial_x W^{X_T}} \mspace{2mu}, {\e^{-\i\h T}(X_0-X_T)\cdot\partial_x(\h)^{-1}\W}\rangle}\nonumber\\
&+2\frac{\rho_0}{M}{\mathrm{Re}\,}{\langle{\partial_x W^{X_T}} \mspace{2mu}, {\int_0^T\e^{-\i\h (T-s)}P_s\cdot\partial_x(\h)^{-1}\W \d s}\rangle}\nonumber\\
&+2g\frac{\sqrt{\rho_0}}{M}{\mathrm{Re}\,}{\langle{\bar{\beta}^{X_T}\partial_x W^{X_T}} \mspace{2mu}, {\int_0^T\e^{-\i\h (T-s)}P_s\cdot\partial_x(\h)^{-1}\W \d s}\rangle}\nonumber\\
&+ R(P,T)\,,\end{aligned}$$ with $R(P,T)$ defined as $$\begin{aligned}
\label{eq:remainder}
R(P,T)=&-2\sqrt{\rho_0}{\langle{(1+\frac{g}{\sqrt{\rho_0}}\bar{\beta}^{X_T})\partial_x\W} \mspace{2mu}, {\sum_{n=3}^6D_n}\rangle}
-g{\langle{\partial_x\W} \mspace{2mu}, {|\delta_T|^2}\rangle}\\
=&\sum_{k=3}^7\tilde{D}_{k}\,,\nonumber\end{aligned}$$ where the $\tilde{D}_{k}$ are naturally defined. By shifting the center of integration and using the spherical symmetry of $W$ the above equation is equivalent to $(k=1,2,3)$ $$\begin{aligned}
\dot{P}^{(k)}_T=&-2\rho_0{\mathrm{Re}\,}{\langle{(1+\frac{g}{\sqrt{\rho_0}}\bar{\beta})\partial_{x_k} W} \mspace{2mu}, {\e^{-\i h T}(X_0-X_T)_k\partial_{x_k}( h)^{-1} W}\rangle}\nonumber\\
&+2\frac{\rho_0}{M}{\mathrm{Re}\,}{\langle{(1+\frac{g}{\sqrt{\rho_0}}\bar{\beta})\partial_{x_k} W} \mspace{2mu}, {\int_0^T\e^{-\i h (T-s)}P^{(k)}_s\partial_{x_k}(h)^{-1} W \d s}\rangle}\nonumber\\
&+ R(P,T)_k\,.\end{aligned}$$ Since $T>0$ is arbitrary we have $$\begin{aligned}
\label{eq:mainequation}
\dot{P}_t=L(P)(t)+R(P,t)\,,\end{aligned}$$ where $L(P)$ is defined as $$\begin{aligned}
L(P):=\begin{pmatrix}L(P^{(1)})\\L(P^{(2)})\\L(P^{(3)})\end{pmatrix}\,.\end{aligned}$$ **Remark:** From now on, we will write $t$ for $T$ for esthetic reasons.\
The existence of the solution in the infinite time interval
===========================================================
It is hard to derive a decay estimate for $P_t$ directly from (\[eq:mainequation\]). In what follows we will rearrange terms until a fixed point theorem becomes applicable.
We will express the solution of the full equation (\[eq:mainequation\]) in terms of the solution $K(t)$ of one part of the linear equation, $$\begin{aligned}
\label{eq:k}
\dot{K}(t)&=Z{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\int_0^t\e^{-\i h (t-s)}K(s)\partial_{x_1}(h)^{-1} W\d s}\rangle}\,,\\
K(0)&=1.\end{aligned}$$ Here the constant $Z\in \mathbb{R}^{+}$ is defined as $$Z:=2\frac{\rho_0}{M}.$$ In Appendix \[appendix:k\] we prove the following lemma,
\[lem:k\] Let $K(t)$ be a solution of equation (\[eq:k\]) with $K(0)=1$. Then there exist real constants $C_1,\ C_2$ such that as $t\to\infty$ $$\begin{aligned}
\label{eq:k-estimate}
ZK(t)=\frac{3}{\sqrt{2}}\pi^{-\frac{5}{2}}(1+C_1g)t^\mez+C_2 t^{-1}+O(t^{-\frac{3}{2}})\,.
\end{aligned}$$
With $K(t)$ at hand, we can write the Duhamel-like formula $$\begin{aligned}
\label{eq:temPt}
P_t=K(t)P_0+Z\int_0^tK(t-s){\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}\int_0^sP_{s_1}\d s+\int_0^tK(t-s)R(P,s)\d s\,.\end{aligned}$$ We now manipulate and to obtain an effective equation for $P_{t}.$ Since the procedure is very similar to [@Froehlich102] we will go through the steps quickly.
We integrate both sides of from $0$ to $t$, then multiply by $K(t)$ to obtain $$\begin{aligned}
K(t)P_{t}=K(t)P_0+K(t)\int_{0}^{t}\Phi(s) \d s\end{aligned}$$ where $\Phi(s)$ stands for various terms on the right hand side of . Now we use this equation to subtract , then manipulate the linear terms of $P_{t}$ in and $\Phi$ and use the observation ${\mathrm{Re}\,}\langle[1-gh^{-1}W]\partial_{x_1}W,\ (ih)^{-1}\partial_{x_1}h^{-1}W \rangle=0$ to find $$\begin{aligned}
\label{eq:contraction1}
P_t=\frac{1}{1-K(t)}A(P)(t)+\frac{1}{1-K(t)}\int_0^t[K(t-s)-K(t)]R(P,s)\d s\,,\end{aligned}$$ where the linear operator $A$ is defined by $$\begin{aligned}
\label{eq:defina}
A(P)(t)=&-Z\int_0^t[K(t-s)-K(t)]{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}\int_s^tP_{s_1}\d s_1\d s\nonumber\\
&+ZK(t){\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {(-\i h)^{-1}\int_0^t[\e^{-\i h(t-s)}-\e^{-\i h t}]P_s\d s \partial_{x_1}(h)^{-1}W}\rangle}\\
&+Z\int_0^tK(t-s){\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}\d s\int_0^tP_{s_1}\d s_1\,.\nonumber\end{aligned}$$
In the rest of the paper we focus on studying . We start with casting the equation in a Banach space setting, so that a fixed point theorem applies. In order to rewrite the equation for $P_t$ as the integral equation (\[eq:contraction1\]) we had to divide by $1-K(t)$, which needs some care for small values of $t$ since $K(t)\to 1$ as $t\to 0$. But because we know from Lemma \[lem:k\] that $K(t)\to 0$ for $t\to\infty$, it suffices to wait long enough before dividing by $1-K(t)$. Therefore, we divide the time interval $[0,\infty)$ into two parts $[0,T_{\rm loc})$ and $[T_{\rm loc},\infty)$. Introduce a family of Banach spaces that reflects the self-consistent assumption $P_t=O(t^{\mez-\delta})$, $$\begin{aligned}
B_{\delta,T_{\rm loc}}:=\{f:t^{\pez+\delta} f\in L^\infty[T_{\rm loc},\infty)\}\end{aligned}$$ with norm $$\begin{aligned}
{\lVert f \rVert}_{\delta,T_{\rm loc}}:={\lVert t^{\pez+\delta}f \rVert}_\infty\,.\end{aligned}$$ On the finite interval $[0,T_{\rm loc})$ we can use standard existence and uniqueness results to solve (\[eq:temPt\]), and for the infinite interval $[T_{\rm loc},\infty)$ we use a fixed point theorem. Introduce the Heaviside function $\chi_{T_{\rm loc}}:=\11_{[0,T_{\rm loc})}$ and rewrite (\[eq:contraction1\]) as $$\begin{aligned}
\label{eq:contraction2}
P_t=\Upsilon((1-\chi_{T_{\rm loc}})P)(t)+G_t,\end{aligned}$$ where $$\begin{aligned}
\Upsilon((1-\chi_{T_{\rm loc}})P)(t)&:=\frac{1}{1-K(t)}A((1-\chi_{T_{\rm loc}})P)(t)+\frac{1}{1-K(t)}\int_0^t[K(t-s)-K(t)][R(P,s)-R(\chi_{T_{\rm loc}}P,s)]\d s\\
G_t&:=\frac{1}{1-K(t)}A(\chi_{T_{\rm loc}}P)(t)+\frac{1}{1-K(t)}\int_0^t[K(t-s)-K(t)]R(\chi_{T_{\rm loc}}P,s)\d s\,.\end{aligned}$$
Now we present the strategy of applying the fixed point theorem. To this end two criteria have to be verified: the nonlinear operator $\Upsilon$ maps a small neighborhood of $0$, in the space $B_{\delta,T},$ into itself and is contractive; the function $G_{t}$ is sufficiently small in the space $B_{\delta,T}$.
The following two propositions, to be proven in the appendix, show that for $T_{\rm loc}$ large enough, $\Upsilon((1-\chi_{T_{\rm loc}})P)(t):B_{\delta,T_{\rm loc}}\to B_{\delta,T_{\rm loc}}$ is indeed a contraction, and $G_t$ is small in $B_{\delta,T_{\rm loc}}$ if the initial conditions for $P$ and $\beta$ are small enough, which will allow us to prove the main theorem. Recall the definition of $\Omega$ from .
\[prop:contraction-lemma\]There is an $M>0$ such that for $T_{\rm loc}\geq M$ the mapping $\Upsilon((1-\chi_{T_{\rm loc}})P)(t):B_{\delta,T_{\rm loc}}\to B_{\delta,T_{\rm loc}}$ is a contraction, or more precisely:
- For any function $q\in B_{\delta,T_{\rm loc}}$ $$\begin{aligned}
t^{\pez+\delta} \big|\frac{1}{1-K(t)}A((1-\chi_{T_{\rm loc}})q_t)\big|\leq [\frac{1}{\pi}\Omega(\delta)+\eps(T_{\rm loc})+O(g)]{\lVert q_t \rVert}_{\delta,T_{\rm loc}}\,,
\end{aligned}$$ where $\eps(T_{\rm loc})\to 0$ as $T_{\rm loc}\to\infty$.
- Recall that the solution $P$ exists in the time interval $[0,T_{\rm loc}]$ according to Theorem \[thm:localwp\]. Suppose that $Q_1,Q_2:[0,\infty)\to\RR^3$ are two functions satisfying $$\begin{aligned}
Q_1(t)=Q_2(t)=P_t \quad \textrm{for $t\in[0,T_{\rm loc}]$}\,,
\end{aligned}$$ and in the interval $[T_{\rm loc},\infty)$ $$\begin{aligned}
{\lVert Q_1 \rVert}_{\delta,T_{\rm loc}},{\lVert Q_2 \rVert}_{\delta,T_{\rm loc}}\ll 1\,.\end{aligned}$$ Then,
$$\begin{aligned}
t^{\pez+\delta}\big|\frac{1}{1-K(t)}\int_0^t[K(s-t)-K(t)][R(Q_1,s)-R(Q_2,s)]\d s \big|\lesssim \({\lVert Q_1 \rVert}_{\delta,T_{\rm loc}}+{\lVert Q_2 \rVert}_{\delta,T_{\rm loc}}\){\lVert Q_1-Q_2 \rVert}_{\delta,T_{\rm loc}}\,.
\end{aligned}$$
\[prop:contraction-lemma2\]Suppose that $T_{\rm loc}\geq M$ (see Proposition \[prop:contraction-lemma\]) and $|P_0|,{\lVert \x^3\beta_0 \rVert}_2\leq\eps_0(T_{\rm loc})$ (see Theorem \[thm:localwp\]). Then $G_t$ is in the Banach space $B_{\delta,T_{\rm loc}}$, and its norm is small. Specifically, for any $t\geq T_{\rm loc}$ $$\begin{array}{ccc}
t^{\pez+\delta}\big|\frac{1}{1-K(t)}A(\chi_{T_{\rm loc}}P)(t)\big|&\leq&\eps(T_{\rm loc})\\
t^{\pez+\delta}\big|\frac{1}{1-K(t)}\int_0^t[K(t-s)-K(t)]R(\chi_{T_{\rm loc}}P,s)\d s\big|&\leq&\eps(T_{\rm loc})\,,
\end{array}$$ with $\eps(T_{\rm loc})\to 0$ as $T_{\rm loc}\to\infty$.
#### Key ideas
As we have stated above, the proof of these propositions can be found in the appendix, but we want to give here the key ideas.
To prove that $G_t=G(\chi_{T_{\rm loc}}P_{\cdot})(t)$ is small we need to choose the initial conditions suitably small. For notice that $G_t$ is defined in terms of $\chi_{T_{\rm loc}}P_{\cdot}$ and the initial condition $\beta_0$; moreover, Theorem \[thm:localwp\] states that for $|P_0|$ and ${\lVert \x^3\beta_0 \rVert}_2$ small enough we have $|P_t|\leq T_{\rm loc}^{-2}$ for any $0\leq t\leq T_{\rm loc}$. This makes it plausible that we can prove ${\lVert G_t \rVert}_{\delta, T_{\rm loc}}\to 0$ as $T_{\rm loc}\to \infty$.
The proof of Proposition \[prop:contraction-lemma\] is more involved because $\Upsilon_t=\Upsilon((1-\chi_{T_{\rm loc}})P_{\cdot})(t)$ is defined in terms of the infinite time trajectory, $\chi_{[T_{\rm loc},\infty)}P_{\cdot}$. For brevity, write $$\begin{aligned}
f(s):={\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}\,,\end{aligned}$$ so that the first term in the definition of $A(P)(t)$ takes the form $$\begin{aligned}
\Gamma_1((1-\chi_{T_{\rm loc}})P_\cdot):=-Z\int_0^t[K(t-s)-K(t)]f(s)\int_s^t(1-\chi_{T_{\rm loc}})P_{s_1}\d s_1\d s\,.\end{aligned}$$ We already know the decay of $K(t)=O(t^\mez)$ from Lemma \[lem:k\], and standard dispersive estimates give $$\begin{aligned}
f(t)=C t^\mdz(1+\widetilde{C}g)+o(t^{-\frac{3}{2}})\,,\end{aligned}$$ for some explicit constant $C$. This, combined with the self-consistent assumption $P_t=O(t^{\mez-\delta})$, is enough to prove $$\begin{aligned}
{\lVert \Gamma_1((1-\chi_{T_{\rm loc}})P_\cdot) \rVert}_{\delta,T_{\rm loc}}\leq [\Omega_1(\delta)+\epsilon(T_{loc})]{\lVert P \rVert}_{\delta,T_{\rm loc}} \,,\end{aligned}$$ where the constant $\Omega_1(\delta)$ is defined as $\Omega_1(\delta):=\frac{1}{(1-2\delta)\pi}\int_0^1\frac{1}{1+(1-r)^\pez}(1-r)^\mez[r^\mez-r^{-\delta}]\d r$, and $\epsilon(T_{loc})\rightarrow 0$ as $T_{loc}\rightarrow \infty.$ A similar computation can be made for the second term in the definition of $A(P)(t)$ with $\Omega_1(\delta)$ replaced by $\Omega_2(\delta):=\frac{1}{\pi}\int_0^1\frac{1}{1+(1-r)^\pez}(1-r)^\mez r^{\pez-\delta}\d r\,.$ In the end, $\delta$ has to be chosen such that $\Omega_1(\delta)+\Omega_2(\delta)<1$ in order to effect a contraction.
The third term in the definition of $A(P)(t)$ can be rewritten as $$\begin{aligned}
\Gamma_3((1-\chi_{T_{\rm loc}})P_{\cdot}):&=Z\int_0^tK(t-s)f(s)\d s\int_0^t(1-\chi_{T_{\rm loc}})P_{s_1}\d s_1\\&=Z\int_0^tK(s)f(t-s)\d s\int_0^t(1-\chi_{T_{\rm loc}})P_{s_1}\d s_1\\
&=\dot{K}(t)\int_0^t(1-\chi_{T_{\rm loc}})P_{s_1}\d s_1\,,\end{aligned}$$ and by a modification of the proof of Lemma \[lem:k\] we prove that $\dot{K}(t)=O(t^\mdz)$, so that it is again straight forward to establish $$\begin{aligned}
{\lVert \Gamma_3((1-\chi_{T_{\rm loc}})P_\cdot) \rVert}_{\delta,T_{\rm loc}}\leq \eps(T_{\rm loc}){\lVert P \rVert}_{\delta,T_{\rm loc}} \,,\end{aligned}$$ where $\eps(T_{\rm loc})\to 0$ as $T_{\rm loc}\to\infty$.
The proof of point (2) of Proposition \[prop:contraction-lemma\] involves lengthy computations the core of which are the propagator estimates proved in Appendix \[sec:propagator-estimates\].
Proof of Main Theorem \[thm:main\] {#mainproof}
===================================
As discussed before, we divide the time interval $[0,\infty)$ into two parts, $[0,T_{\rm loc})$ and $[T_{\rm loc},\infty)$ with $T_{\rm loc}$ being a large constant. The existence of the solution in the finite domain was proven in Theorem \[thm:localwp\]. For the infinite domain, Propositions \[prop:contraction-lemma\] and \[prop:contraction-lemma2\] enable us to apply the contraction lemma on (\[eq:contraction2\]) to see that there exists a small solution $P$ in the space $B_{\delta,T_{\rm loc}}$. By the definition of $B_{\delta,T_{\rm loc}}$ we have proven (\[momentumdecay\]).
To prove it is sufficient to prove that $$\|\langle x\rangle^{-3}\delta_{T,T}\|_{2}\lesssim T^{-\frac{1}{2}}\ \text{for any}\ T\geq 0$$ where the function $\delta_{t}=\delta_{t,T}$ is defined in . This has been proved in Proposition \[prop:delta\].
The existence of $X_{\infty}$ is resulted by the fact $P_{\cdot}=O(t^{-\frac{1}{2}-\delta})$ is integrable in the region $[T_{\rm loc},\infty).$
The proof is complete. $\square$
Proof of Proposition \[prop:delta\] {#sec:ControlRem}
====================================
For any time $\tau\leq T$ we apply Duhamel’s principle to rewrite (\[eq:delta\]) as $$\begin{aligned}
\label{eq:deltaduha}
&\delta_\tau&&=&&\e^{-\i\h\tau}\delta_0-\i g\int_0^\tau\e^{-\i\h(\tau-s)}[W^{X_s}-\W]\delta_s\d s\nonumber\\
&&&&&-\frac{\sqrt{\rho_0}}{M}\sum_{|\alpha|=1}^{N_0}\frac{1}{\alpha!}\int_0^\tau\e^{-\i\h (\tau-s)}\partial_x^\alpha(\h)^{-1}\W P_s\alpha(X_s-X_T)^{\alpha-1}\d s
+\i\int_0^\tau\e^{-\i\h (\tau-s)}G_1(s)\d s\nonumber\\
&&&=:&&\sum_{n=1}^4B_n\,.
\end{aligned}$$ Now we estimate each term on the right hand side of . Recall the definition of $\mu(T)$ in (\[majorant\]) and the assumption $\mu(T)\leq 1$. By the definition of $\delta_0$ and the propagator estimates of Proposition \[prop:propagator\] we have $$\begin{aligned}
&{\lVert \x^{-3}B_1 \rVert}_2&&=&&{\lVert \x^{-3}\e^{-\i\h \tau}[\beta_0-\bar{\beta}^{X_T}-\sqrt{\rho_0}\sum_{|\alpha|=1}^{N_0}\frac{1}{\alpha!}(X_0-X_T)^\alpha\partial_x^\alpha(\h)^{-1}\W] \rVert}_2\label{eq:b111}\\
&&&\leq &&{\lVert \x^{-3}\e^{-\i\h \tau}\beta_0 \rVert}_2+{\lVert \x^{-3}\e^{-\i\h \tau}\bar{\beta}^{X_T} \rVert}_2\nonumber\\
&&&&&+\sqrt{\rho_0}\sum_{|\alpha|=1}^{N_0}\frac{1}{\alpha!}|X_0-X_T|^\alpha{\lVert \x^{-3}\e^{-\i\h \tau}\partial_x^\alpha(\h)^{-1}\W \rVert}_2\nonumber\\
&&&\lesssim&&(1+\tau)^\mdz{\lVert \x^3\beta_0 \rVert}_2+(1+\tau)^\mez+(1+\tau)^\mdz\mu(T)\nonumber\end{aligned}$$ where in the third line we used the fact $$\begin{aligned}
|X_0-X_T|\leq \int_0^T|P_s|\d s\lesssim \mu(T)\,.\end{aligned}$$ For the last line we recall the overarching hypothesis of Theorem \[thm:main\] ${\lVert \x^3\beta_0 \rVert}_2\leq \eps_0$.\
For $B_3$ we have $$\begin{aligned}
{\lVert \x^{-3}B_3 \rVert}_2\lesssim&\mu(T)\int_0^\tau(1+\tau-s)^\mdz(1+s)^\mfv\d s\\
\lesssim &\mu(T)(1+\tau)^\mfv\,;\end{aligned}$$ recall that we only consider $\delta\in(\frac{1}{2},\delta^*)$ and $\delta^*<1$. Similarly for $B_4$, $$\begin{aligned}
{\lVert \x^{-3}B_4 \rVert}_2\lesssim&\mu^{N_0+1}(T)\int_0^\tau(1+\tau-s)^\mdz(1+s)^\mdz\d s\\
\lesssim &\mu^{N_0+1}(T)(1+\tau)^\mdz\,.\end{aligned}$$ Since $B_2$ depends on $\delta_\tau$, we have to proceed in a different way. Define the function $Q$ by $$\begin{aligned}
Q(\tau):=\max_{0\leq s\leq \tau\leq T}(1+s)^\pez{\lVert \x^{-3}\delta_s \rVert}_2\,.\end{aligned}$$ Then $B_2$ admits the estimate $$\begin{aligned}
{\lVert \x^{-3}B_2 \rVert}_2\lesssim&g\int_0^\tau(1+\tau-s)^\mdz{\lVert \x^3(\W-W^{X_s})\delta_s \rVert}_2\d s\\
\lesssim &gQ(\tau)\int_0^\tau(1+\tau-s)^\mdz |X_t-X_s|\ (1+s)^\mez\d s\\
\lesssim & gQ(\tau)\mu(\tau)\int_{0}^{\tau}(1+\tau-s)^\mdz\ [(1+s)^{\frac{1}{2}-\delta}-(1+\tau)^{\frac{1}{2}-\delta}]\ (1+s)^\mez\d s\\
\lesssim & gQ(\tau)\mu(\tau) (1+\tau)^{-\frac{1}{2}-\delta}\end{aligned}$$ In the first line, we used the fact $$\begin{aligned}
|\x^3W^{X_\cdot}|\lesssim\x^{-3}\,,\end{aligned}$$ which holds since $|X_\cdot|$ is bounded, and in the last step Lemma \[LM:convo\] has been used.
Collecting the estimates above we find $$\begin{aligned}
(1+\tau)^\pez{\lVert \x^{-3}\delta_\tau \rVert}\lesssim gQ(\tau)+1+\eps_0+\mu(T)\,,\end{aligned}$$ which by definition of $Q(\tau)$ yields for any $0\leq \tau\leq T$ $$\begin{aligned}
Q(\tau)\lesssim gQ(\tau)+1+\eps_0+\mu(T)\,.\end{aligned}$$ As $g$ is small we obtain $$\begin{aligned}
Q(\tau)\lesssim 1+\eps_0+\mu(T)\lesssim 1\,,\end{aligned}$$ which is the desired estimate.
The proof is complete.
$\square$
In the proof of the following result has been used.
\[LM:convo\] $$\begin{aligned}
\int_{0}^{t}(1+t-s)^{-\frac{3}{2}} (s^{\frac{1}{2}-\delta}-t^{\frac{1}{2}-\delta})(1+s)^\mez\d s\lesssim & (1+t)^{-\frac{1}{2}-\delta}.\end{aligned}$$
We start with deriving a convenient form $$\begin{aligned}
(t^{\pez-\delta}-s^{\pez-\delta})=-t^{\pez-\delta}s^{\pez-\delta}(t^{\delta-\pez}-s^{\delta-\pez})\,.\end{aligned}$$ To estimate the term $t^{\delta-\pez}-s^{\delta-\pez}$ we consider two different regimes, $0\leq s\leq \frac{t}{2}$ and $\frac{t}{2}\leq s\leq t$. In the first regime we use direct estimate, for the second we use Taylor expansion to find that for $s\leq t$ and any $\eps>0$ there exists a constant $c(\epsilon)>0$ $$\begin{aligned}
t^\eps-s^\eps\leq c(\epsilon)\frac{t-s}{t^{1-\eps}}\end{aligned}$$ This implies $$\begin{aligned}
\int_{0}^{t}(1+t-s)^{-\frac{3}{2}} (s^{\frac{1}{2}-\delta}-t^{\frac{1}{2}-\delta})(1+s)^\mez\d s\lesssim &
\int_0^t(1+t-s)^{-\frac{1}{2}} t^{-1}s^{\pez-\delta}(1+s)^\mez\d s\\
\leq & t^{-1}\int_0^t(t-s)^{-\frac{1}{2}} s^{-\delta}\d s\\
\leq &t^{-\frac{1}{2}-\delta} \int_{0}^{1}(1-s)^{-\frac{1}{2}} s^{-\delta}\ ds\\
\lesssim & t^{-\frac{1}{2}-\delta}\end{aligned}$$ where in the second step we rescaled variable $s\rightarrow t s$ and in the last step we used $\delta<1.$
To remove the singularity at $t=0$ we use a direct estimate on the expression to prove it is bound for $t\leq 1.$
The proof is complete.
The following result will be used later. Define a new function $\phi$ by $$\begin{aligned}
\label{eq:difPhi}
\phi_t:=\delta_t+\e^{\i h^{X_{T}}t}\bar{\beta}^{X_{T}}.\end{aligned}$$
\[cor:oneterm\] $$\begin{aligned}
\|\langle x\rangle^{-3}\phi_t\|_2\lesssim (1+t)^{-\frac{1}{2}-\delta}\end{aligned}$$
The proof is based on an improvement of the proof of Proposition . Observe that the only term not of order $t^{-\frac{1}{2}-\delta}$ (or smaller) is $-\e^{\i h^{X_{T}}t}\bar{\beta}^{X_{T}}$, see . Recall that $\delta<1.$ Hence by removing this term we obtain the desired estimate.
The proof is complete.
Proof of Lemma \[lem:k\] {#appendix:k}
========================
We follow the strategy of [@Froehlich102]. Define $Z:=\frac{2\rho_0}{M}$ and a function $G:\RR\to\CC$ by $$\begin{aligned}
\label{eq:A1}
G(k+\i0):=&\frac{\i}{2}{\langle{(h+k+\i0)^{-1}\partial_{x_1}(h)^{-1}W} \mspace{2mu}, {[1-g(h)^{-1}W]\partial_{x_1} W}\rangle}\nonumber\\
-&\frac{\i}{2}{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {(h-k-\i0)^{-1}\partial_{x_1}(h)^{-1} W}\rangle}\end{aligned}$$ Now, we relate $G$ to the solution $K$:
\[lem111\]The solution $K$ of (\[eq:k\]) takes the form $$\begin{aligned}
K(t)&=-\frac{1}{\pi}\int_{-\infty}^\infty{\mathrm{Re}\,}\frac{1}{\i k+ZG(k+\i0)}\cos kt\d k.
\end{aligned}$$ In particular, $$\begin{aligned}
K(t)=0 \qquad \textrm{for $t<0$}.\end{aligned}$$
The proof of Lemma \[lem111\] is done as in [@Froehlich102] and is not repeated here. With this explicit expression for $K$, we can prove Lemma \[lem:k\] with the help of the following lemma,
\[lem:G\]The function $G(k+\i0)$ satisfies $$\begin{aligned}
G(k+\i0)=\begin{cases}(\i-1)\frac{\pi^2}{3}(1+O(g))k^\pez+C_1k+O(k^\pdz)& \textrm{if $k>0$}\\
(-\i-1)\frac{\pi^2}{3}(1+O(g))|k|^\pez+C_2 k+O(|k|^\pdz)& \textrm{if $k<0$}\end{cases},
\end{aligned}$$ with $C_1,\ C_2$ being some constants.
Lemma \[lem:G\] is proven at the end of this section.
Decompose $K(t)$ into two parts, $$\begin{aligned}
K(t)=K_+(t)+K_-(t)\,,
\end{aligned}$$ with $$\begin{aligned}
K_+(t):=-\frac{1}{\pi}\int_0^\infty{\mathrm{Re}\,}\frac{1}{\i k+ZG(k+\i0)}\cos kt\d k\end{aligned}$$ and $$\begin{aligned}
K_-(t):=-\frac{1}{\pi}\int_{-\infty}^0{\mathrm{Re}\,}\frac{1}{\i k+ZG(k+\i0)}\cos kt\d k\end{aligned}$$ Define a new function $g:\RR^+\to\RR$ by $$\begin{aligned}
|k|^\mez g(|k|^\pez):&=-\frac{1}{\pi}{\mathrm{Re}\,}\frac{1}{\i k+ZG(k+\i0)}\\
&=-\frac{1}{Z\pi} \frac{{\mathrm{Re}\,}G}{(\frac{k}{Z}+{\mathrm{Im}\,}G)^2+({\mathrm{Re}\,}G)^2}\\
&=\frac{3(1+O(g))}{2\pi^3Z} |k|^\mez(1+O(k^\pez))\,,\end{aligned}$$ where we used the explicit form of $G(k+\i0)$ of Lemma \[lem:G\]. By construction, the function $g$ is smooth on $[0,\infty)$ and satisfies (because $G(k)$ is bounded as $k\to\infty$) $$\begin{aligned}
|g(\rho)|\leq C(1+\rho)^{-3}\,.\end{aligned}$$ Compute directly to obtain $$\begin{aligned}
K_+(t)&=\int_0^\infty k^\mez g(k^\pez)\cos kt\d k\\
&=2\int_0^\infty g(\rho)\cos(\rho^2 t)\d\rho\\
&=2g(0)\int_0^\infty \cos(\rho^2 t)+D\end{aligned}$$ with $D$ defined as $$\begin{aligned}
D:=2\int_0^\infty[g(\rho)-g(0)]\cos(\rho^2 t)\d\rho\,.\end{aligned}$$ The first term on the right hand side is the dominant one: $$\begin{aligned}
2g(0)\int_0^\infty\cos(\rho^2 t)\d\rho=2g(0)t^\mez\int_0^\infty \cos x^2\d x = \frac{3(1+O(g))}{2\sqrt{2}Z}\pi^{-\frac{5}{2}} t^\mez\,,\end{aligned}$$ where we used the Fresnel integral $\int_0^\infty \cos x^2\d x=(\pi/8)^\pez$.
We prove now that $D$ is a correction of order $t^\mdz$. This implies $$\begin{aligned}
K_+=\frac{3(1+O(g))}{2\sqrt{2}Z}\pi^{-\frac{5}{2}} t^\mez+O(t^{-1})\,.\end{aligned}$$ Since we find by completely analogous computation $$\begin{aligned}
K_-=\frac{3(1+O(g))}{2\sqrt{2}Z}\pi^{-\frac{5}{2}} t^\mez+O(t^{-1})\end{aligned}$$ the claim follows.
To estimate $D$ we first integrate by parts: $$\begin{aligned}
|D|&=t^{-1}|\int_0^\infty \rho^{-1}[g(\rho)-g(0)]\partial_\rho\sin(\rho^2t)\d\rho|\\
&=t^{-1}|\int_0^\infty H(\rho)\sin(\rho^2t)\d\rho|\end{aligned}$$ with $H(\rho):=\partial_\rho(\rho^{-1}[g(\rho)-g(0)])$ a smooth function satisfying $|H(\rho)|\lesssim (1+\rho)^{-2}$. Write $H(\rho)=H(0)+\rho[\rho^{-1}(H(\rho)-H(0))]$ and perform again integration by parts to obtain $$\begin{aligned}
|D|=t^{-1}|H(0)||\int_0^\infty\sin(\rho^2t)\d\rho|+\frac{1}{2}t^{-2}\lim_{\rho\to0}\frac{|H(\rho)-H(0)|}{\rho}+\frac{1}{2}t^{-2}|\int_0^\infty\partial_\rho[\rho^{-1}(H(\rho)-H(0))]|\d\rho\,.\end{aligned}$$ The first term on the right hand side can be computed explicitely, $$\begin{aligned}
t^{-1}|H(0)||\int_0^\infty\sin(\rho^2t)\d\rho|=t^\mdz|H(0)|\sqrt{\frac{\pi}{8}}\,,\end{aligned}$$ and the second term is obviously of order $t^{-2}$. The last term is controlled by $$\begin{aligned}
t^{-2}\int_0^\infty(1+\rho)^{-2}\d\rho\lesssim t^{-2}\end{aligned}$$ by the fact that $|\partial_\rho[\rho^{-1}(H(\rho)-H(0))]|\lesssim (1+\rho)^{-2}$.
The basic idea is to expand $(h+k)^{-1}$ around $h^{-1}$. By classical results, see e.g. [@jensen79], if the constant $|g|$ in $h=-\Delta+gW$ is sufficiently small and $W$ decays sufficiently fast at $\infty,$ then $h$ has no zero-resonance or eigenvectors. This together with the discussions above and results in [@jensen79] implies that $$\begin{aligned}
\label{eq:expansion}
(h+k)^{-1}=B_0+\zeta B_1+\zeta^2 B_2+O(\zeta^3)\,,
\end{aligned}$$ in the topology of $\mathcal{B}(L^{2,3},L^{2,-3})$, $B_i$ being operators in $\mathcal{B}(L^{2,3},L^{2,-3})$, namely $$\begin{aligned}
B_0&=(1+(-\Delta)^{-1}gW)^{-1}(-\Delta)^{-1}\\
B_1&=\frac{1}{4\pi}{\langle{\cdot} \mspace{2mu}, {(1+(-\Delta)^{-1}gW)1}\rangle}(1+(-\Delta)^{-1}gW)1\,\end{aligned}$$ and the variable $\zeta$ is defined by $\zeta:=k^{\frac{1}{2}},$ where $k$ is in the domain $\mathbb{C}\backslash\mathbb{R}^+,$ and $k^{\frac{1}{2}}=k^{\frac{1}{2}}>0$ for $k>0.$
A minor difficulty in the present situation is that we cannot apply the expansion directly because $\partial_{x_1}h^{-1}W\notin L^{2,3}$. To make still applicable we observe that for any $k\not=0$, $g\in \mathbb{R}$ $$\begin{aligned}
\label{eq:simpleMan}
(-\Delta+gW)^{-1}(-\Delta+gW+k\pm 0i)^{-1}=\frac{1}{k}[(-\Delta+gW)^{-1}-(-\Delta+gW+k\pm 0i)^{-1}].\end{aligned}$$ There is one more minor difficulty: In applying this equation to study $G(k+i0)$ it might happen that $G(k+i0)$ is singular at $k=0.$ For this we use the presence $\partial_{x}$ and the fact that $W$ is spherically symmetric, or simply using symmetries, to prove that the singular terms in $G(k+i0)$ are identically zero.
In the next we carry out the ideas presented above.
Observe that $G(k+i0)$ contains two term, denote the two terms by $G_1(k+i0)$ and $G_2(k+i0)$, i.e., $$G(k+i0)=G_{1}(k+i0)-G_2(k+i0).$$ We start with studying $G_{1}(k+i0).$ Define a function $W_1$ by $$\begin{aligned}
\partial_{x_1}W_1:=[1-g(h)^{-1}W]\partial_{x_1} W\,.\end{aligned}$$ Clearly, $W_1$ is spherically symmetric and rapidly decaying. Use the second resolvent identity to rewrite the first term of $G(k)$, $$\begin{aligned}
G_1(k+i0)=&\frac{\i}{2}{\langle{[(-\Delta + k+\i0)^{-1}-(h + k+\i0)^{-1}gW(-\Delta + k+\i0)^{-1}]\partial_{x_1}[(-\Delta)^{-1}-(-\Delta)^{-1}gWh^{-1}]W} \mspace{2mu}, {\partial_{x_1}W_1}\rangle}\nonumber\\
=&\frac{\i}{2}{\langle{(-\Delta + k+\i0)^{-1}\partial_{x_1}(-\Delta)^{-1}W} \mspace{2mu}, {\partial_{x_1}W_1}\rangle}\nonumber\\
&-\frac{\i}{2}{\langle{(-\Delta + k+\i0)^{-1}\partial_{x_1}(-\Delta)^{-1}gWh^{-1}W} \mspace{2mu}, {\partial_{x_1}W_1}\rangle}\nonumber\\
&+\frac{\i}{2}{\langle{(h + k+\i0)^{-1}gW(-\Delta + k+\i0)^{-1}\partial_{x_1}(-\Delta)^{-1}W} \mspace{2mu}, {\partial_{x_1}W_1}\rangle}\nonumber\\
&-\frac{\i}{2}{\langle{(h + k+\i0)^{-1}gW(-\Delta + k+\i0)^{-1}\partial_{x_1}(-\Delta)^{-1}gWh^{-1}W} \mspace{2mu}, {\partial_{x_1}W_1}\rangle}\nonumber\\
=&A_1+A_2+A_3+A_4\end{aligned}$$ where the terms $A_l,\ l=1,2,3,4,$ are naturally defined as the four terms on the right hand side.
We claim that there exist constants $C_{l,n},\ l=1,2,3,4\ n=1,2,3$ such that $$\begin{aligned}
\label{eq:claimTaylor}
A_{l}=C_{l,1}+C_{l,2}k^{\frac{1}{2}}+C_{l,3}k+O(k^{\frac{3}{2}})\end{aligned}$$ and moreover the constants $C_{l,n},\ l\not=1$ are of order $O(g)$.
Suppose the claim holds. Then by the same techniques we prove the second term $G_2(k+i0)$ in $G(k+i0)$ can also be expanded in a form similar to . It is not difficult to see that the constant terms cancel each other if $G(k+\i0)$ is defined for the point $k=0$ (which we will prove), which makes $G(0+i0)=0.$ For the term of order $k^{\frac{1}{2}}$ we will compute the coefficient in the proof of the claim.
We start with proving for $l=1.$ $B_1$ is the main term in the sense that only $B_1$ does not contain the small factor $g$. We rewrite it as $$\begin{aligned}
\frac{\i}{2}{\langle{(-\Delta + k+\i0)^{-1}\partial_{x_1}(-\Delta)^{-1}W} \mspace{2mu}, {\partial_{x_1}W_1}\rangle}=\frac{\i}{6}{\langle{(-\Delta + k+\i0)^{-1}W} \mspace{2mu}, {W_1}\rangle}\,,\end{aligned}$$ for which (\[eq:expansion\]) becomes applicable. The constant term in the expansion vanishes in the difference (\[eq:A1\]). For the $k^{1/2}$-term we get (consider first $k>0$) $$\begin{aligned}
&\frac{1}{24\pi}k^{1/2}(\i-1)\;{\langle{W} \mspace{2mu}, {(1+(-\Delta)^{-1}gW}\rangle}{\langle{(1+(-\Delta)^{-1}gW} \mspace{2mu}, {W_1}\rangle}\\
=&\frac{1}{24\pi}k^{1/2}(\i-1)\big[{\langle{W} \mspace{2mu}, {1}\rangle}{\langle{1} \mspace{2mu}, {W}\rangle}+O(g)\big]\,,\end{aligned}$$ where we used $W_1=W+O(g)$ in ${\lVert \cdot \rVert}_\infty$. Using ${\langle{W} \mspace{2mu}, {1}\rangle}={\langle{1} \mspace{2mu}, {W}\rangle}=(2\pi)^\pdz\widehat{W}(0)$ the last line equals $$\begin{aligned}
\frac{\pi^2}{3}k^{1/2}(\i-1)(1+O(g)) \,.\end{aligned}$$
The case $l=2$ is treated analogously and gives a contribution of order $k^\pez O(g)$.
For the case $l=3$ we rewrite the expression as $$\begin{aligned}
\label{eq:a31}
A_3=&\frac{\i}{2}{\langle{(h + k+\i0)^{-1}gW(-\Delta + k+\i0)^{-1}\partial_{x_1}(-\Delta)^{-1}W} \mspace{2mu}, {\partial_{x_1}W_1}\rangle}\nonumber\\
=&g\frac{\i}{2}{\langle{(-\Delta + k+\i0)^{-1}\partial_{x_1}(-\Delta)^{-1}W} \mspace{2mu}, {W(h + k-\i0)^{-1}\partial_{x_1}W_1}\rangle}\,.\end{aligned}$$ Now we apply (with $g=0$) and above to expand the expression in $k^{\frac{1}{2}}.$ Observe that the singular terms $k^{-1}$ and $k^{-\frac{1}{2}}$ vanish by symmetry. Hence we have proved
The term $A_4$ can be treated in the exact same way and yields a contribution of order $k^\pez O(g^2)$.
Proof of Point (1) of Proposition \[prop:contraction-lemma\]
============================================================
Similar to [@Froehlich102] we decompose the linear operator $A$, defined in (\[eq:defina\]), as follows, $$\begin{aligned}
A((1-\chi_{T_{\rm loc}})q):=\sum_{k=1}^3\Gamma_k((1-\chi_{T_{\rm loc}})q)\,,\end{aligned}$$ where the terms $\Gamma_k$ are defined as $$\begin{aligned}
\Gamma_1&:=-Z\int_0^t[K(t-s)-K(t)]{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}\int_s^tq_{s_1}(1-\chi_{T_{\rm loc}}(s_1))\d s_1\d s \\
\Gamma_2&:=Z\int_0^tK(t-s){\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}\d s\int_0^tq_{s_1}(1-\chi_{T_{\rm loc}}(s_1))\d s_1 \\
\Gamma_3&:=ZK(t){\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {(-\i h)^{-1}\int_0^t[\e^{-\i h(t-s)}-\e^{-\i h t}]q_s(1-\chi_{T_{\rm loc}}(s))\d s \partial_{x_1}(h)^{-1}W}\rangle} \,.\end{aligned}$$
Before the actual estimate we define two continuous functions $\Omega_{1},\ \Omega_{2}:\ (-\infty,1)\rightarrow \mathbb{R}^{+}$ by $$\begin{aligned}
\Omega_1(\delta)&:=\frac{1}{(1-2\delta)\pi}\int_0^1\frac{1}{1+(1-r)^\pez}(1-r)^\mez[r^\mez-r^{-\delta}]\d r\label{omega1}\\
\Omega_2(\delta)&:=\frac{1}{\pi}\int_0^1\frac{1}{1+(1-r)^\pez}(1-r)^\mez r^{\pez-\delta}\d r\,.\label{omega2}\end{aligned}$$ Recall the function $\Omega(\delta)$ introduced before Theorem \[thm:main\]. It is given by the sum $\Omega(\delta)=\Omega_1(\delta)+\Omega_2(\delta)$, and we compute $$\begin{aligned}
\Omega(\delta)=\frac{1}{\pi d(2d-1)}+\frac{1}{2\sqrt{\pi}}\(\frac{2\Gamma(\pez-\delta)}{\Gamma(1-\delta)}-\frac{\Gamma(-\delta)}{\Gamma(\pdz-\delta)}\)\,.\end{aligned}$$ Note that $\Omega(\delta)$ has only apparent singularities at $\delta=0,\pez$. It is a continuous, monotonically increasing function $$\begin{aligned}
\Omega:(-\infty,1)\to\RR^+\end{aligned}$$ with the following properties $$\begin{aligned}
\lim_{\delta\to -\infty} \Omega(\delta)&=0\\
\Omega(0)&=1-\frac{\log 4}{\pi}\simeq 0.56\\
\Omega(\pez)&=1+\frac{\log 4-2}{\pi}\simeq 0.8\,.\\
\lim_{\delta\to 1}\Omega(\delta)&=\infty\end{aligned}$$ Numerical analysis suggests that $\Omega(\delta)<1$ for all $\delta<0.66$.
Point (1) of Proposition \[prop:contraction-lemma\] is covered by the following lemma,
If $q_t\in B_{\delta,T_{\rm loc}}$ then there is a small constant $\eps(T_{\rm loc})$ satisfying $\eps(\infty)=0$ such that $$\begin{aligned}
|\Gamma_1|&\leq t^{\mez-\delta}[\Omega_1(\delta)+\eps(T_{\rm loc})](1+O(g)){\lVert q \rVert}_{\delta,T_{\rm loc}}\\
|\Gamma_3|&\leq t^{\mez-\delta}[\Omega_2(\delta)+\eps(T_{\rm loc})](1+O(g)){\lVert q \rVert}_{\delta,T_{\rm loc}}\\
|\Gamma_2|&\leq t^{\mez-\delta}\eps(T_{\rm loc}){\lVert q \rVert}_{\delta,T_{\rm loc}}\,,
\end{aligned}$$
We start with $\Gamma_2$. The second term in the product is easy to estimate, $$\begin{aligned}
\label{directEst}
|\int_0^tq_{s_1}(1-\chi_{T_{\rm loc}}(s_1))\d s_1|\leq \int_{0}^{t}(1-\chi_{T_{\rm loc}}(s_1)) s_1^{-\frac{1}{2}-\delta}\ ds_1{\lVert q \rVert}_{\delta,T_{\rm loc}} \lesssim {\lVert q \rVert}_{\delta,T_{\rm loc}}\,,\end{aligned}$$ where we used the fact that $-\frac{1}{2}-\delta>1$, hence $s^{-\frac{1}{2}-\delta}$ is integrable in $[T_{\rm loc},\infty )$. The first term is estimated as follows. Apply the Fourier transform to the convolution function, then inverse Fourier transform to find $$\begin{aligned}
Z\int_0^tK(t-s){\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}\d s=\frac{1}{2\pi}\int_{-\infty}^\infty\frac{F(k)}{\i k+ZG(k+\i0)}\e^{-\i kt}\d k \,,\end{aligned}$$ where $F(k)$ is defined as $$\begin{aligned}
F(k):&=Z\int_0^\infty\e^{\i ks}{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}\d s\\
&=\frac{Z}{2}\int_0^\infty\e^{\i ks}[{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}+{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {\e^{\i h s}\partial_{x_1}(h)^{-1}W}\rangle}]\d s\\
&=\frac{Z}{2}[{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {(-ih+ik-0)^{-1}\partial_{x_1}(h)^{-1}W}\rangle}+{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {(ih+ik-0)^{-1}\partial_{x_1}(h)^{-1}W}\rangle}]\\
&=-ZG(k+\i0)\,.\end{aligned}$$ Around $k=0$, the term $\frac{F(k)}{\i k+ZG(k+\i0)}$ has the expansion $$\begin{aligned}
\frac{F(k)}{\i k+ZG(k+\i0)}=-1+Ck^\pez+O(k)\,.\end{aligned}$$ The constant term does not contribute, as is seen by integration by parts, $$\begin{aligned}
\int_{-\infty}^\infty\frac{F(k)}{\i k+ZG(k+\i0)}\e^{-\i kt}\d k =\int_{-\infty}^\infty\frac{1}{\i t}\partial_k\(\frac{F(k)}{\i k+ZG(k+\i0)}\)\e^{-\i kt}\d k \,,\end{aligned}$$ and the Fourier transform of $k^\pez$ is of order $t^\mdz$. The detailed computations are identical to [@Froehlich102] and thus omitted. We obtain $$\begin{aligned}
\label{eq:threehalf}
|Z\int_0^tK(t-s){\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}\d s|\lesssim (1+t)^\mdz\,.\end{aligned}$$ Combining and , we have the desired estimate $$\begin{aligned}
|\Gamma_2|\lesssim (1+t)^\mdz {\lVert q \rVert}_{\delta,T_{\rm loc}}\leq T_{\rm loc}^\mez(1+t)^{\mez-\delta}{\lVert q \rVert}_{\delta,T_{\rm loc}}\,.\end{aligned}$$ Here the fact $\delta<1$ has been used.
Now, we turn to $\Gamma_1$. Recall the asymptotic expression for $K$ in Lemma (\[eq:k-estimate\]), $$\begin{aligned}
\label{eq:zKt}
ZK(t)=\frac{3}{\sqrt{2}}\pi^{-\frac{5}{2}}(1+Cg)t^\mez+O(t^{-1})\,,\end{aligned}$$ and that for ${\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h t}\partial_{x_1}(h)^{-1}W}\rangle}$ in below.
To estimate $\Gamma_1$ we take the leading order terms $\widetilde{K},\widetilde{M},\widetilde{\Gamma}_1$ to approximate the functions in and , and $\Gamma_1$, $$\begin{aligned}
Z\widetilde{K}(t)&:=\frac{3}{\sqrt{2}}\pi^{-\frac{5}{2}}t^\mez(1+Cg)\\
\widetilde{M}&:=-\frac{1}{3\sqrt{2}}\pi^{\pdz}t^\mdz(1+\widetilde{C}g)\\
\widetilde{\Gamma}_1&:=-Z\int_0^t[\widetilde{K}(t-s)-\widetilde{K}(s)]\widetilde{M}(s)\int_s^tq_{s_1}[1-\chi_T(s_1)]\d s_1\d s\,.\end{aligned}$$ Compute directly to obtain $$\begin{aligned}
|\widetilde{\Gamma}_1|&\leq \frac{1+O(g)}{2\pi}\int_0^t[(t-s)^\mez-t^\mez]s^\mdz\int_s^t|q_{s_1}|\d s_1\d s\\
&\leq \frac{1+O(g)}{(1-2\delta)\pi}\int_0^t[(t-s)^\mez-t^\mez]s^\mdz(t^{\pez-\delta}-s^{\pez-\delta})\d s {\lVert q_t \rVert}_{\delta,T}\\
&=\frac{1+O(g)}{(1-2\delta)\pi}\int_0^t(t-s)^\mez t^\mez\frac{1}{(t-s)^\pez+t^\pez}s^\mez(t^{\pez-\delta}-s^{\pez-\delta})\d s {\lVert q_t \rVert}_{\delta,T}\,,\end{aligned}$$ Change variables $s=rt$ to obtain $$\begin{aligned}
\label{eq:hatG1}
|\widetilde{\Gamma}_1|\leq t^{\mez-\delta}(1+O(g))\Omega_1(\delta){\lVert q_t \rVert}_{\delta,T}\,,\end{aligned}$$ where the constant $\Omega_1$ is defined in (\[omega1\]).
In what follows we estimate $\Gamma_1-\widetilde{\Gamma}_1$. Divide the integration region $[0,t]$ into three parts, $[0,T_{\rm loc}^{\frac{1}{3}}]$,$[T_{\rm loc}^{\frac{1}{3}},t-T_{\rm loc}^{\frac{1}{3}}]$, and $[t-T_{\rm loc}^{\frac{1}{3}},t]$ and denote these parts by $I_{k},\ k=1,2,3$, i.e. $$\begin{aligned}
\label{eq:I123}
\Gamma_1-\widetilde{\Gamma}_1=I_1+I_2+I_3.\end{aligned}$$ We start with estimating $I_1:$ $$\begin{aligned}
I_1:=&Z\int_0^{T_{\rm loc}^{\frac{1}{3}}}[K(t-s)-K(s)]{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}\int_s^tq_{s_1}[1-\chi_{T_{\rm loc}}(s_1)]\d s_1\d s\\
-&Z\int_0^{T_{\rm loc}^{\frac{1}{3}}}[\widetilde{K}(t-s)-\widetilde{K}(s)]\widetilde{M}(s)\int_s^tq_{s_1}[1-\chi_{T_{\rm loc}}(s_1)]\d s_1\d s\,.\end{aligned}$$ For the term inside the integral we have $$\begin{aligned}
\label{eq:KKK}
&|K(t-s)-K(t)|,|\widetilde{K}(t-s)-\widetilde{K}(t)|\nonumber\\
\lesssim &t^\mez(t-s)^\mez\frac{s}{t^\pez+(t-s^\pez)}+(t-s)^\mdz-t^\mdz\\
\lesssim &t^\mdz(1+s)\nonumber\end{aligned}$$ because $s\leq T_{\rm loc}^{\frac{1}{3}}$, and $t\geq T_{\rm loc}$. And consequently $$\begin{aligned}
|K(t-s)-K(t)|{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h t}\partial_{x_1}(h)^{-1}W}\rangle}+|\widetilde{K}(t-s)-\widetilde{K}(t)||\widetilde{M}(s)|\lesssim t^\mdz s^\mez(1+O(g))\,.\end{aligned}$$ Plug this into $I_1$ to obtain $$\begin{aligned}
\label{eq:estI1}
|I_1|\lesssim t^{-\frac{3}{2}}(1+O(g))\int_0^{T_{\rm loc}^{\frac{1}{3}}}s^\mez [s^{\frac{1}{2}-\delta}- t^{\frac{1}{2}-\delta}]\d s{\lVert q_t \rVert}_{\delta,T_{\rm loc}}=t^{-\frac{3}{2}}2T_{\rm loc}^{\frac{1}{6}}{\lVert q_t \rVert}_{\delta,T_{\rm loc}}\leq t^{\mez-\delta}T_{\rm loc}^{-\frac{1}{3}}{\lVert q_t \rVert}_{\delta,T_{\rm loc}}\,,\end{aligned}$$ where, recall that we only consider the regime $t\geq T_{\rm loc}\gg 1$
Now we turn to $I_2,$ which is defined by $$\begin{aligned}
I_2:=&Z\int_{T_{\rm loc}^{\frac{1}{3}}}^{t-T_{\rm loc}^{\frac{1}{3}}}[K(t-s)-K(t)]{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle} \int_s^t(1-\chi_{T_{\rm loc}}(s_1))q_{s_1}\d s_1\d s\\
-&Z\int_{T_{\rm loc}^{\frac{1}{3}}}^{t-T_{\rm loc}^{\frac{1}{3}}}[\widetilde{K}(t-s)-\widetilde{K}(t)]\widetilde{M}(s)\int_s^t(1-\chi_{T_{\rm loc}}(s_1))q_{s_1}\d s_1\d s\\
=&Z\int_{T_{\rm loc}^{\frac{1}{3}}}^{t-T_{\rm loc}^{\frac{1}{3}}}[(K(t-s)-K(t))-(\widetilde{K}(t-s)-\widetilde{K}(t))]{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}\times\\
& \int_s^t(1-\chi_{T_{\rm loc}}(s_1))q_{s_1}\d s_1\d s\\
+&Z\int_{T_{\rm loc}^{\frac{1}{3}}}^{t-T_{\rm loc}^{\frac{1}{3}}}[\widetilde{K}(t-s)-\widetilde{K}(t)][{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}-\widetilde{M}(s)]
\int_s^t(1-\chi_{T_{\rm loc}}(s_1))q_{s_1}\d s_1\d s\,.\end{aligned}$$ For the terms inside the integral we use the following estimates $$\begin{aligned}
|K(t-s)-\widetilde{K}(t-s)|\lesssim& (1+t-s)^{-1},\\
|K(t)-\widetilde{K}(t)|\lesssim& (1+t)^{-1},\\
|\widetilde{K}(t-s)-\widetilde{K}(t)|\lesssim &t^{-1}(t-s)^\mez s,\\\end{aligned}$$ implied by and $$\begin{aligned}
|{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}-\widetilde{M}(s)|\lesssim &s^{-\frac{5}{2}}\,,\end{aligned}$$ by and $$\begin{aligned}
|\int_{s}^{t} (1-\chi_{T_{\rm loc}}(s_1))q_{s_1}\d s_1|\lesssim {\lVert q_t \rVert}_{\delta,T_{\rm loc}}[s^{\frac{1}{2}-\delta}-t^{\frac{1}{2}-\delta}].\end{aligned}$$ to obtain $$\begin{aligned}
\label{eq:estI2}
|I_2|\lesssim T_{\rm loc}^{-\frac{1}{3}} t^{\mez-\delta}{\lVert q_t \rVert}_{\delta,T_{\rm loc}}\,.\end{aligned}$$ For $I_3$, we have $$\begin{aligned}
{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}\lesssim t^\mdz\,,\end{aligned}$$ and hence $$\begin{aligned}
|I_3|=&|Z\int_{t-T_{\rm loc}^{\frac{1}{3}}}^t[K(t-s)-K(t)]{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle} \int_s^t(1-\chi_{T_{\rm loc}}(s_1))q_{s_1}\d s_1\d s\nonumber\\
-&Z\int_{t-T_{\rm loc}^{\frac{1}{3}}}^t[\widetilde{K}(t-s)-\widetilde{K}(t)]\widetilde{M}(s)\int_s^t(1-\chi_{T_{\rm loc}}(s_1))q_{s_1}\d s_1\d s|\nonumber\\
\lesssim &\int_{t-T_{\rm loc}^{\frac{1}{3}}}^t(|t-s|^\mez+t^\mez)\d s \;t^{-1-\delta}{\lVert q_t \rVert}_{\delta,T_{\rm loc}}\nonumber\\
\lesssim & T_{\rm loc}^{1/6}t^{-1-\delta}{\lVert q_t \rVert}_{\delta,T_{\rm loc}}\nonumber\\
\leq &T_{\rm loc}^{-1/3}t^{\mez-\delta}{\lVert q_t \rVert}_{\delta,T_{\rm loc}}\,\label{eq:estI3}.\end{aligned}$$ Putting , , , and together, we have shown that $$\begin{aligned}
|\Gamma_1|\lesssim t^{\mez-\delta}[\Omega_1(\delta)+\eps(T_{\rm loc})](1+O(g)){\lVert q_t \rVert}_{\delta,T_{\rm loc}}\,,\end{aligned}$$ where $\eps(T_{\rm loc})\to 0$, as $T_{\rm loc}\to\infty$.\
Finally, we turn to $\Gamma_3$. Similar to the strategy in estimating $\Gamma_1$, we start with retrieving the main part. Define a new function $\widetilde{V}$ to approximate the function $V(t):={\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {(-\i h)^{-1}\e^{-\i h t}\partial_{x_1}(h)^{-1}W}\rangle}$ when $t$ is large, $$\begin{aligned}
\widetilde{V}:=\frac{\sqrt{2}}{3}\pi^\pdz(1+\widetilde{C}g) t^\mez \,.\end{aligned}$$ To see this we simply integrate .
Now, define an approximation $\widetilde{\Gamma}_3$ of $\Gamma_3$, $$\begin{aligned}
\widetilde{\Gamma}_3:=Z\widetilde{K}(t)\int_0^t[\widetilde{V}(t-s)-\widetilde{V}(t)](1-\chi_{T_{\rm loc}}(s))q_s\d s\end{aligned}$$ Compute $$\begin{aligned}
|\widetilde{\Gamma}_3|\leq &t^\mez\frac{1}{\pi}(1+O(g))\int_0^t[(t-s)^\mez-t^\mez]s^{\mez-\delta}\d s{\lVert q_t \rVert}_{\delta,T_{\rm loc}}\\
=&t^{-1}\frac{1}{\pi}(1+O(g))\int_0^t\frac{s}{(t-s)^\pez+t^\pez}(t-s)^\mez s^{\mez-\delta}\d s {\lVert q_t \rVert}_{\delta,T_{\rm loc}}\,.\end{aligned}$$ The change variables $s=tr$ yields $$\begin{aligned}
\label{eq:estG3}
|\widetilde{\Gamma}_3|\leq t^{\mez-\delta}\frac{1}{\pi}(1+O(g))\int_0^1(1-r)^\mez\frac{1}{1+(1-r)^\pez}r^{\pez-\delta}\d r {\lVert q_t \rVert}_{\delta,T_{\rm loc}}=t^{\mez-\delta}\Omega_2(\delta)(1+O(g)){\lVert q_t \rVert}_{\delta,T_{\rm loc}}\,.\end{aligned}$$ For the difference $\Gamma_3-\widetilde{\Gamma}_3$ we use almost the same techniques in to obtain $$\begin{aligned}
\label{eq:diffG3}
|\Gamma_3-\widetilde{\Gamma}_3|
\lesssim&t^{-1-\delta}(1+O(g))){\lVert q_t \rVert}_{\delta,T_{\rm loc}}\leq (1+O(g))T_{\rm loc}^\mez t^{\mez-\delta}{\lVert q_t \rVert}_{\delta,T_{\rm loc}}\,.\end{aligned}$$ Putting and together, we have shown $$\begin{aligned}
|\Gamma_3|\lesssim t^{\mez-\delta}[\Omega_2(\delta)+\eps(T_{\rm loc})](1+O(g)){\lVert q_t \rVert}_{\delta,T_{\rm loc}}\,,\end{aligned}$$ which finishes the proof.
Proof of Point (2) of Proposition \[prop:contraction-lemma\] {#sec:Point2}
============================================================
We start with a different results, then show it implies Point (2).
Let $Q_1$ and $Q_2$ be as in Proposition \[prop:contraction-lemma\], and recall the definition of $R(P,t)$ in (\[eq:remainder\]) and the definitions of $\tilde{D}_{k},\ k=1,2,\cdots,7$ in . If $|P_0|\leq T_{\rm loc}^{-2}$ then, for $k=3,4,5,6,7$, $$\begin{aligned}
\label{eq:nonlinear}
|\widetilde{D}_k(Q_1)-\widetilde{D}_k(Q_2)|(t)\leq t^{-1-\delta}[\eps(T_{\rm loc})+g]{\lVert Q_1-Q_2 \rVert}_{\delta,T_{\rm loc}}({\lVert Q_1 \rVert}_{\delta,T_{\rm loc}}+{\lVert Q_2 \rVert}_{\delta,T_{\rm loc}})\,.
\end{aligned}$$
The lengthy proof is divided into the next few subsections.
In the following we use to prove Point (2): We start with analyzing the function $K$ of . Lemma \[lem:k\] implies that $$\begin{aligned}
|K(t-s)-K(t)|\lesssim (1+t-s)^\mez(1+t)^{-1}s+(1+t-s)^{-\frac{3}{2}}+(1+t)^{-\frac{3}{2}}\,.\end{aligned}$$ Compute directly and use the fact $|\widetilde{D}_k(Q_1)-\widetilde{D}_k(Q_2)|(t)=0$ for $t\leq T_{\rm loc}$ to obtain the desired estimate $$\begin{aligned}
\int_{0}^{t} |K(t-s)-K(t)| |\widetilde{D}_k(Q_1)-\widetilde{D}_k(Q_2)|(s)\ ds\leq t^{-\frac{1}{2}-\delta}\eps(T_{\rm loc}){\lVert Q_1-Q_2 \rVert}_{\delta,T_{\rm loc}}({\lVert Q_1 \rVert}_{\delta,T_{\rm loc}}+{\lVert Q_2 \rVert}_{\delta,T_{\rm loc}})\,.\end{aligned}$$
The term $\widetilde{D}_3$
--------------------------
Because of the spherical symmetry of $W$ certain terms in the definition vanish. For notational convenience, we define $$\begin{aligned}
(1+\frac{g}{\sqrt{\rho_0}}\bar{\beta}^{X_t})\partial_x W^{X_t}:=V^{X_t}\,.\end{aligned}$$ Note that $V$ is rapidly decaying. This makes $\widetilde{D}_3$ take the form $$\begin{aligned}
\widetilde{D}_3(t)&=-2\sqrt{\rho_0}{\mathrm{Re}\,}{\langle{V^{X_t}} \mspace{2mu}, {\e^{-\i h^{X_t} t}\beta_0}\rangle}-2\rho_0{\mathrm{Re}\,}{\langle{V^{X_t}} \mspace{2mu}, {\e^{-\i h^{X_t} t}\sum_{|\alpha|=2}^{N_0}\frac{1}{\alpha!}(X_0-X_t)^\alpha\partial_x^\alpha(h^{X_t})^{-1}W^{X_t}}\rangle}\\
&=-2\sqrt{\rho_0}{\mathrm{Re}\,}{\langle{\e^{\i h t}V} \mspace{2mu}, {\beta_0^{-X_t}}\rangle}-2\rho_0{\mathrm{Re}\,}{\langle{\e^{-\i h^{X_t} t}V^{X_t}} \mspace{2mu}, {\sum_{|\alpha|=2}^{N_0}\frac{1}{\alpha!}(X_0-X_t)^\alpha\partial_x^\alpha(h^{X_t})^{-1}W^{X_t}}\rangle}\\
&=:\widetilde{D}_{31}(X_t)+\widetilde{D}_{32}(X_t)\,.\end{aligned}$$ Compute directly to obtain $$\begin{aligned}
|\widetilde{D}_{3,1}(X_t)-\widetilde{D}_{3,1}(\tilde{X}_t)|\leq &|{\langle{\e^{\i h t}V} \mspace{2mu}, {(\beta_0^{-X_t}-\beta_0^{-\widetilde{X}_t}}\rangle}|\\
\leq & t^{-\frac{5}{2}}{\lVert \x^{5}(\beta_0^{-X_t}-\beta_0^{-\widetilde{X}_t}) \rVert}_2\\
\lesssim & t^{-\frac{5}{2}}{\lVert \x^{5}(\beta_0-\beta_0^{\int_{T_{\rm loc}}^t (Q_1-Q_2)(s)\d s}) \rVert}_2\\
=&t^{-\frac{5}{2}}{\lVert \x^{5}\int_{T_{\rm loc}}^t\partial_s\beta_0^{\int_{T_{\rm loc}}^s (Q_1-Q_2)(s_1)\d s_1}\d s \rVert}_2\\
=&t^{-\frac{5}{2}}{\lVert \x^{5}\partial_x\beta_0\cdot\int_{T_{\rm loc}}^t(Q_1-Q_2)(s)\d s \rVert}_2\\
\leq& t^{-\frac{5}{2}}{\lVert \x^{5}\partial_x\beta_0 \rVert}_2\int_{T_{\rm loc}}^t|Q_1-Q_2|(s)\d s\\
\lesssim &t^{-1-\delta} \epsilon(T_{loc}){\lVert Q_1-Q_2 \rVert}_{\delta,T_{\rm loc}}\end{aligned}$$ where, in the second step we used the propagator estimate , and we used the facts $X_t=X_{T_{\rm loc}}+\int_{T_{\rm loc}}^tQ_1(s)\d s$ and $\widetilde{X}_t=X_{T_{\rm loc}}+\int_{T_{\rm loc}}^tQ_2(s)\d s$, the fact $t\geq T\gg 1$, and ${\lVert \x^{3}\partial_x\beta_0 \rVert}_2\leq 1$ in the Main Theorem \[thm:main\].
The term $\widetilde{D}_{32}$ is treated similarly: $$\begin{aligned}
|\widetilde{D}_{32}(Q_1)-\widetilde{D}_{32}(Q_2)|(t)&\lesssim |{\langle{V} \mspace{2mu}, {\e^{-\i h t}\sum_{|\alpha|=2}^{N_0}(\widetilde{X}_t-X_t)^\alpha\partial_x^\alpha(h)^{-1}W}\rangle}|\\
&=|{\langle{V} \mspace{2mu}, {\e^{-\i h t}\sum_{|\alpha|=3,\ |\alpha|\ \text{is odd}}^{N_0}(\widetilde{X}_t-X_t)^\alpha\partial_x^\alpha(h)^{-1}W}\rangle}|\\
&\lesssim t^{-\frac{5}{2}} \sum_{|\alpha|=2}^{N_0} \int_{t_{\rm loc}}^t|[Q_1(s)-Q_2(s)]^\alpha|\\
&\lesssim t^{-1-\delta}\epsilon(T_{loc})({\lVert Q_1^2 \rVert}_{\delta,T_{\rm loc}}+{\lVert Q_2^2 \rVert}_{\delta,T_{\rm loc}}){\lVert Q_1-Q_2 \rVert}_{\delta,T_{\rm loc}},\end{aligned}$$ where in the second step the fact that $W$ is spherically symmetric was used to remove the terms with $|\alpha|$ even. In the third step was used.
So the claim follows for $\widetilde{D}_3$ in .
The term $\widetilde{D}_4$ and $\widetilde{D}_5$
------------------------------------------------
In what follows we only estimate $\widetilde{D}_4$. The analysis of $\widetilde{D}_5$ is almost identical, hence omitted.
Consider $$\begin{aligned}
\widetilde{D}_4(t)=&-\i g{\langle{V^{X_t}} \mspace{2mu}, {\int_0^t\e^{-\i h^{X_t}(t-s)}[W^{X_s}-W^{X_t}]\delta_s\d s}\rangle}\\
=&-\i g{\langle{V} \mspace{2mu}, {\int_0^t\e^{-\i h (t-s)}[W^{X_s-X_t}-W]\delta_s^{-X_t}\d s}\rangle}\end{aligned}$$ Observe that the term depends on $Q$, namely $(X_s-X_t)$ and $\delta_s^{-X_t}$. Consequently $$\begin{aligned}
|\widetilde{D}_{4}(Q_1)-\widetilde{D}_{4}(Q_2)|(t)
&\lesssim g|{\langle{V} \mspace{2mu}, {\int_0^t\e^{-\i h(t-s)} [W^{X_s-X_t}-W^{\widetilde{X}_s-\widetilde{X}_t}]\delta^{-X_t}_s\d s}\rangle}|\\
&+g|{\langle{V} \mspace{2mu}, {\int_0^t\e^{-\i h(t-s)} [W^{\widetilde{X}_s-\widetilde{X}_t}-W](\delta^{-X_t}_s-\widetilde{\delta}^{-\widetilde{X}_t}_s)\d s}\rangle}|\\
&=:B_1+B_2\end{aligned}$$ where $B_1$ and $B_2$ are defined naturally. For $B_1$, by the decay estimate in Proposition \[eq:classical\], we obtain $$\begin{aligned}
B_1&\lesssim g\int_0^t|\widetilde{X}_t-\widetilde{X}_s+X_s-X_t|(1+t-s)^{-\frac{3}{2}}{\lVert \x^3\partial_xW\delta^{-X_t}_s \rVert}_2\d s\\
&\lesssim g\int_0^t|\widetilde{X}_t-\widetilde{X}_s+X_s-X_t|(1+t-s)^{-\frac{3}{2}}(1+s)^\mez\d s\,.\end{aligned}$$ Both $Q_1$ and $Q_2$ are in $B_{\delta,T_{\rm loc}}$, and thus $$\begin{aligned}
|\widetilde{X}_t-\widetilde{X}_s+X_s-X_t|&=|\int_s^t[Q_1-Q_2](s_1)\d s_1|\leq {\lVert Q_1-Q_2 \rVert}_{\delta,T_{\rm loc}}\int_s^ts_1^{\mez-\delta}\d s_1\\
&\lesssim{\lVert Q_1-Q_2 \rVert}_{\delta,T_{\rm loc}}(t^{\pez-\delta}-s^{\pez-\delta}).\end{aligned}$$
Apply Lemma \[LM:convo\] to obtain $$\begin{aligned}
\label{eq:mira}
B_{1}(t) \lesssim g t^{-1-\delta}{\lVert Q_1-Q_2 \rVert}_{\delta,T_{\rm loc}}\,.\end{aligned}$$ For $B_2$, since $\delta_s$ depends on $Q$, we have $$\begin{aligned}
B_2
\lesssim& g \int_0^t(1+t-s)^{-\frac{3}{2}}|{\widetilde{X}}_t-{\widetilde{X}}_s|{\lVert \x^3\partial_xW(\delta^{-X_t}_s-\widetilde{\delta}^{-\widetilde{X}_t}_s) \rVert}_2\d s\nonumber\\
\lesssim & g \|Q_2\|\int_0^t(1+t-s)^{-\frac{3}{2}}((1+s)^{\frac{1}{2}-\delta}-(1+t)^{\frac{1}{2}-\delta}){\lVert \x^3\partial_xW(\delta^{-X_t}_s-\widetilde{\delta}^{-\widetilde{X}_t}_s) \rVert}_2\d s\nonumber\end{aligned}$$ In Lemma \[LM:diff\] below we prove $$\begin{aligned}
\label{eq:claim1}
{\lVert \x^3\partial_xW(\delta^{-X_t}_s-\widetilde{\delta}^{-\widetilde{X}_t}_s) \rVert}_2\lesssim (1+s)^{-\frac{1}{2}-\delta}{\lVert Q_1-Q_2 \rVert}_{\delta,T_{\rm loc}}\,\end{aligned}$$ which together with a weaker version of Lemma \[LM:convo\] implies the desired estimate.
The proof is complete.
$\square$
The following result has been used in .
\[LM:diff\] $$\begin{aligned}
{\lVert \x^6\partial_xW(\delta^{-X_t}_s-\widetilde{\delta}^{-\widetilde{X}_t}_s) \rVert}_2\lesssim (1+s)^{-\frac{1}{2}-\delta}{\lVert Q_1-Q_2 \rVert}_{\delta,T_{\rm loc}}.\end{aligned}$$
Recall the definition of $\delta_{t,T}, \ t\leq T,$ from . We start with deriving an equation for $\delta^{-X_{t}}_s=\delta^{-X_{t}}_{s,t}, \ s\leq t,$ from : $$\begin{aligned}
&\delta_{s,t}^{-X_t}&&=&&\e^{-\i h s}\sqrt{\rho_0}(X_0-X_t)\cdot\partial_x h^{-1} W-\frac{\sqrt{\rho_0}}{M}\int_0^s\e^{-\i h (s-s_1)}P_{s_1}\cdot\partial_x h^{-1} W \d s_1\nonumber\\
&&&&&+\e^{-\i h s}[\beta_0^{-X_t}-\bar{\beta}+\sqrt{\rho_0}\sum_{|\alpha|=2}^{N_0}\frac{1}{\alpha!}(X_0-X_t)^\alpha\partial_x^\alpha h^{-1} W]\label{eq:diff2}\\
&&&&&-\i g\int_0^s \e^{-\i h(s-s_1)}[W^{X_s-X_t}-W]\delta_{s_1,t}^{-X_{t}}\d s_1\nonumber\\
&&&&&-\frac{\sqrt{\rho_0}}{M}\sum_{|\alpha|=2}^{N_0}\frac{1}{\alpha!}\int_0^s \e^{-\i h (s-s_1)}\partial_x^\alpha h^{-1} W P_{s_1}\alpha(X_{s_1}-X_T)^{\alpha-1}\d s_1
+\i\int_0^s \e^{-\i h (s-s_1)}[G_1(s_1)]^{-X_t}\d s_1\nonumber\end{aligned}$$ Introduce a new function $\eta$, $$\begin{aligned}
\eta_s:=\delta^{-X_t}_s-\widetilde{\delta}^{-\widetilde{X}_t}_s\,,\end{aligned}$$ which satisfies a new equation $$\begin{aligned}
\eta_s=&\e^{-\i h s}\sqrt{\rho_0}[(X_0-X_t)-(\tilde{X}_0-\tilde{X}_t)]\cdot\partial_x h^{-1} W-\frac{\sqrt{\rho_0}}{M}\int_0^s\e^{-\i h (s-s_1)}[Q_1(s_1)-Q_2(s_1)]\cdot\partial_x h^{-1} W \d s_1\\ \label{eq:etas}
&+\e^{-\i h s}[\beta_0^{-X_t}-\beta_0^{-\tilde{X}_t}]+\sqrt{\rho_0}\sum_{|\alpha|=2}^{N_0}\frac{1}{\alpha!}[(X_0-X_t)^\alpha-(\tilde{X}_0-\tilde{X}_t)^{\alpha}]\partial_x^\alpha h^{-1} W]\nonumber\\
&-\i g\int_0^s \e^{-\i h(s-s_1)}[W^{X_s-X_t}-W]\eta_{s_1}\d s_1-\i g\int_0^s \e^{-\i h(s-s_1)}[W^{X_s-X_t}-W^{\tilde{X_s}-\tilde{X_t}}]\widetilde{\delta}_{s_1,t}^{-X_{t}}\d s_1\nonumber\\
&-\frac{\sqrt{\rho_0}}{M}\sum_{|\alpha|=2}^{N_0}\frac{1}{\alpha!}\int_0^s \e^{-\i h (s-s_1)}\partial_x^\alpha h^{-1} W [Q_1(s_1)-Q_2(s_1)]\alpha(X_{s_1}-X_T)^{\alpha-1}\d s_1\nonumber\\
&-\frac{\sqrt{\rho_0}}{M}\sum_{|\alpha|=2}^{N_0}\frac{1}{\alpha!}\int_0^s \e^{-\i h (s-s_1)}\partial_x^\alpha h^{-1} W Q_{2}(s_1)\alpha [(X_{s_1}-X_T)^{\alpha-1}-(\tilde{X}_{s_1}-\tilde{X}_T)^{\alpha-1}]\d s_1\nonumber\\
&+\i\int_0^s \e^{-\i h (s-s_1)}{[G_1^{-X_t}(s_1)-\tilde{G}_1^{-\tilde{X}_t}(s_1)]}\d s_1.\end{aligned}$$
Here the function $\tilde{G}_1$ is defined in the same way as $G_1$ (see ), the only difference is that it depends on the new trajectory $\tilde{X}.$
What is left is essentially to improve the proof of Proposition \[prop:delta\]. The difference between and is that the term $-e^{ih^{X_{T}} t}\bar{\beta}^{X_{T}}$ is not present. This makes the proof almost identical to that of Corollary \[cor:oneterm\], hence we omit the details here.
The proof is complete.
The term $\widetilde{D}_6$
--------------------------
Consider finally $$\begin{aligned}
\i{\langle{V^{X_t}} \mspace{2mu}, {\int_0^t\e^{-\i h^{X_t} (t-s)}G_1(s)\d s}\rangle}\,,\end{aligned}$$ so that $$\begin{aligned}
|\widetilde{D}_{6}(Q_1)-\widetilde{D}_{6}(Q_2)|(t)=&|{\langle{\partial_xW} \mspace{2mu}, {\int_0^t\e^{-\i h (t-s)}(h^{X_s-X_t}r^{-X_t}_{N_0}-h^{{\widetilde{X}}_s-{\widetilde{X}}_t}\widetilde{r}^{-{\widetilde{X}}_t}_{N_0})\d s}\rangle}|\\
\leq &|{\langle{\partial_xW} \mspace{2mu}, {\int_0^t\e^{-\i h (t-s)}(h^{X_s-X_t}-h^{{\widetilde{X}}_s-{\widetilde{X}}_t})r^{-X_t}_{N_0}(s)\d s}\rangle}|\\
&+|{\langle{\partial_xW} \mspace{2mu}, {\int_0^t\e^{-\i h (t-s)}h^{{\widetilde{X}}_s-{\widetilde{X}}_t}(r^{-X_t}_{N_0}-\widetilde{r}^{-{\widetilde{X}}_t}_{N_0})(s)\d s}\rangle}|\\
=&|{\langle{\e^{-\i h (t-s)}\partial_xW} \mspace{2mu}, {\int_0^t(h^{X_s-X_t}-h^{{\widetilde{X}}_s-{\widetilde{X}}_t})r^{-X_t}_{N_0}(s)\d s}\rangle}|\\
&+|{\langle{\e^{-\i h (t-s)}\partial_xW} \mspace{2mu}, {\int_0^t h^{{\widetilde{X}}_s-{\widetilde{X}}_t}(r^{-X_t}_{N_0}-\widetilde{r}^{-{\widetilde{X}}_t}_{N_0})(s)\d s}\rangle}|\\
=&D_1+D_2.\end{aligned}$$
For $D_1$, use $$\begin{aligned}
h^{X_s-X_t}-h^{{\widetilde{X}}_s-{\widetilde{X}}_t}=g(W^{X_s-X_t}-W^{{\widetilde{X}}_s-{\widetilde{X}}_t})\end{aligned}$$ and $$\begin{aligned}
\|\langle x\rangle^{-5}\e^{-\i h t}\partial_xW\|_2\lesssim (1+|t|)^{-\frac{5}{2}}\end{aligned}$$ of to obtain $$\begin{aligned}
D_1\lesssim &g\int_0^t(1+t-s)^{-\frac{5}{2}}{\lVert \x^5(W^{X_s-X_t}-W^{{\widetilde{X}}_s-{\widetilde{X}}_t}) r^{-X_t}_{N_0} \rVert}_2\d s\\
\lesssim &g\int_0^t(1+t-s)^{-\frac{5}{2}}(|X_s-X_t|){\lVert \x^{-5}r^{-X_t}_{N_0} \rVert}_2\d s\\
\lesssim &g{\lVert Q_1-Q_2 \rVert}_{T_{\rm loc},\delta}\int_0^t(1+t-s)^{-\frac{5}{2}}(1+s)^{\pez-\delta}(1+s)^\mdz\d s\\
\lesssim &g{\lVert Q_1-Q_2 \rVert}_{T_{\rm loc},\delta}t^{-1-\delta}\,,\end{aligned}$$ where in the second step we used $$\begin{aligned}
{\lVert \x^{-3}r_{N_0}(s) \rVert}_2\lesssim (1+s)^\mdz\,\end{aligned}$$ of .
Now we turn to $D_2.$ The key part is to estimate $M(t,s):=h^{{\widetilde{X}}_s-{\widetilde{X}}_t}(r^{-X_t}_{N_0}-\widetilde{r}^{-{\widetilde{X}}_t}_{N_0}).$ By methods similar to the proof of we obtain $$\begin{aligned}
\label{eq:claim}
\| \langle x\rangle^{5}M(t,s)\|_{2}\lesssim (1+s)^{-1-\delta}({\lVert Q_1 \rVert}_{\delta,T_{\rm loc}}+{\lVert Q_2 \rVert}_{\delta,T_{\rm loc}}){\lVert Q_1-Q_2 \rVert}_{\delta,T_{\rm loc}}.\end{aligned}$$ This together with implies that $$\begin{aligned}
|D_2|\lesssim \int_{0}^{t}(1+t-s)^{-\frac{5}{2}}(1+s)^{-1-\delta}({\lVert Q_1 \rVert}_{\delta,T_{\rm loc}}+{\lVert Q_2 \rVert}_{\delta,T_{\rm loc}}){\lVert Q_1-Q_2 \rVert}_{\delta,T_{\rm loc}}.\end{aligned}$$
The Term $\tilde{D}_{7}$
------------------------
The last step is to incorporate the term $g{\langle{\partial_xW^{X_t}} \mspace{2mu}, {|\delta_t|^2}\rangle}=
g{\langle{\partial_xW} \mspace{2mu}, {|\delta_t^{-X_t}|^2}\rangle}$. Compute directly to obtain $$\begin{aligned}
|\delta_t^{-X_t}(Q_1)|^2-|\delta_t^{-\tilde{X}_t}(Q_2)|^2=(\delta_t^{-X_t}(Q_1)-\delta_t(Q_2))[\delta^{-X_t}_t]^*(Q_1)
+\delta^{-\tilde{X}_t}_t(Q_2)([\delta^{-X_t}_t]^*(Q_1)-[\delta^{-\tilde{X}_t}_t]^*(Q_2))\,.\end{aligned}$$ Apply Lemma \[LM:diff\] and use the estimate $\|\langle x\rangle^{-3}\delta_t^{-X_t}\|_2, \ \|\langle x\rangle^{-3}\delta_t^{-\tilde{X}_t}\|_2\lesssim (1+t)^{-\frac{1}{2}}$ of Proposition \[prop:delta\] to obtain the desired estimate $$\begin{aligned}
g|{\langle{\partial_xW^{X_t}} \mspace{2mu}, {|\delta_t(Q_1)|^2-|\delta_t(Q_2)|^2}\rangle}|&\lesssim g{\lVert \x^3\partial_xW^{X_t}\delta_t(Q_1)-\delta_t(Q_2) \rVert}_2|{\lVert \x^{-3}\delta_t^*(Q_1) \rVert}_2\\
&\lesssim g({\lVert Q_1 \rVert}_{T_{\rm loc},\delta}+{\lVert Q_2 \rVert}_{T_{\rm loc},\delta}){\lVert Q_1-Q_2 \rVert}_{T_{\rm loc},\delta}t^{-\delta-1}\,.\end{aligned}$$
Proof of Proposition \[prop:contraction-lemma2\]
================================================
As discussed at the beginning of Section \[sec:Point2\] it is sufficient for us to prove the following result.
$$\begin{aligned}
|A(\chi_{T_{\rm loc}}(P))|\leq \eps(T_{\rm loc})t^{\mez-\delta}\,.\end{aligned}$$
Recall the definition of $A$ in (\[eq:defina\]) and the local existence estimate $|P_t|\leq T_{\rm loc}^{-2}$ for $t\in [0,T_{\rm loc}]$. Then compute $$\begin{aligned}
|A(\chi_{T_{\rm loc}} P)|\leq &Z\eps(T_{\rm loc})[\int_0^{T_{\rm loc}}|K(t-s)-K(t)||{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}|\d s\\
+&|\int_0^tK(t-s){\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {\e^{-\i h s}\partial_{x_1}(h)^{-1}W}\rangle}\d s|\\
+&|K(t)|\int_0^{T_{\rm loc}}|{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {(-\i h)^{-1}[\e^{-\i h(t-s)}-\e^{-\i h t}]\partial_{x_1}(h)^{-1}W}\rangle}\d s]\,.
\end{aligned}$$ As proved in (\[eq:threehalf\]) the second term on the right hand side is of order $t^\mdz$. For the third term, we have by a computation similar to (\[eq:KKK\]) $$\begin{aligned}
|{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1}W} \mspace{2mu}, {(-\i h)^{-1}[\e^{-\i h(t-s)}-\e^{-\i h t}]\partial_{x_1}(h)^{-1}W}\rangle}|\lesssim (1+t-s)^\mez(1+t)^{-1}s+(1+t-s)^\mdz\,.\end{aligned}$$ So we obtain $$\begin{aligned}
|A(\chi_{T_{\rm loc}} P)|\lesssim &\eps(T_{\rm loc})[(1+t)^{-1}\int_0^{T_{\rm loc}}(1+t-s)^\mez\frac{s}{(1+s)^\pdz}\d s+\int_0^{T_{\rm loc}}(1+t-s)^\mdz(1+s)^\mdz\d s\\
&+(1+t)^\mdz+(1+t)^\mdz\int_0^{T_{\rm loc}}(1+t-s)^\mez s\d s+(1+t)^\mez\int_0^{T_{\rm loc}}(1+t-s)^\mdz\d s]\\
&\leq \eps(T_{\rm loc})(1+t)^{\mez-\delta}\,,\end{aligned}$$ where $\eps(T_{\rm loc})\to 0$ as $T_{\rm loc}\to\infty$.
We are left with proving $$\begin{aligned}
\int_0^t[K(t-s)-K(t)]\widetilde{D}_k(\chi_{T_{\rm loc}}P,s)\d s\big|\leq t^{\mez-\delta}\eps(T_{\rm loc})\,,\end{aligned}$$ for $k=3,4,5,6,7$, recall that we only consider $t\geq T_{loc}\gg 1$. As in the proof of Proposition \[prop:contraction-lemma\], all we have to show is $$\begin{aligned}
|\widetilde{D}_k(\chi_{T_{\rm loc}}P,s)|\leq (1+t)^{-1-\delta}\eps(T_{\rm loc})\,.\end{aligned}$$ The estimates are very similar to the ones in the proof of Proposition \[prop:contraction-lemma\], so we will do only three of them, namely $\widetilde{D}_3,$ $\widetilde{D}_4$ and $\widetilde{D}_7.$
Apply propagator estimates in Proposition \[prop:propagator\] to estimate $\widetilde{D}_3$ $$\begin{aligned}
|\widetilde{D}_3(\chi_{T_{\rm loc}}P,t)|&\lesssim |{\langle{\partial_xW^{X_t}} \mspace{2mu}, {\e^{-\i h^{X_t} t}\beta_0}\rangle}|+|{\langle{\partial_xW^{X_t}} \mspace{2mu}, {\e^{-\i h^{X_t} t}\sum_{|\alpha|=2}^{N_0}\frac{1}{\alpha!}(X_0-X_t)^\alpha\partial_x^\alpha( h^{X_t})^{-1}W^{X_t}}\rangle}|\\
\lesssim& (1+\|\langle x\rangle^{5}\beta_0\|_2) t^{-\frac{5}{2}}\leq (1+t)^{-1-\delta}\eps(T_{\rm loc}) (1+\|\langle x\rangle^{5}\beta_0\|_2)\,.\end{aligned}$$
For $\widetilde{D}_4$, observe $$\begin{aligned}
|\widetilde{D}_4(\chi_{T_{\rm loc}}P,t)|&\lesssim g |{\langle{\partial_xW^{X_t}} \mspace{2mu}, {\int_0^{T_{\rm loc}}\e^{-\i h^{X_t}(t-s)}\partial_xW^{X_t}\cdot (X_s-X_{T_{\rm loc}})\delta_s\d s}\rangle}|\\
&\lesssim g \int_0^{T_{\rm loc}}(1+t-s)^{-\frac{5}{2}}\int_s^{T_{\rm loc}}|P_{s_1}|\d s_1{\lVert \x^{-3}\delta_s^{-X_t} \rVert}_2\d s\,.\end{aligned}$$ Because of the local existence estimate , we have $$\begin{aligned}
\int_s^{T_{\rm loc}}|P_{s_1}|\d s_1\leq T_{\rm loc}^{-2}(T_{\rm loc}-s)\,.\end{aligned}$$ So we can estimate $$\begin{aligned}
|\widetilde{D}_4(\chi_{T_{\rm loc}}P,t)|&\lesssim g T_{\rm loc}^{-2}\int_0^{T_{\rm loc}}(1+t-s)^{-\frac{5}{2}}(T_{\rm loc}-s)(1+s)^\mez\d s\lesssim gT_{\rm loc}^\mez (1+t)^{-\frac{1}{2}-\delta}\,.\end{aligned}$$ Here in the last inequality two regimes have been considered: $t\in [T_{\rm loc},2T_{\rm loc}]$ and $t>2T_{\rm loc}.$ In the first one we bound $(1+t-s)^{-\frac{5}{2}}(T_{\rm loc}-s)$ by $(1+t-s)^{-\frac{3}{2}};$ and in the second bound $(1+t-s)^{-\frac{5}{2}}(T_{\rm loc}-s)$ by $t^{-\frac{5}{2}}T_{\rm loc}$ for $s\in [0,T_{\rm loc}].$
Now we turn to $\widetilde{D}_7.$ We estimate the term $g{\langle{\partial_xW^{X_t}} \mspace{2mu}, {|\delta_t|^2(\chi_{T_{\rm loc}}P)}\rangle}$ by applying Corollary \[cor:oneterm\]. Observe that $\partial_xW^{X_t}\perp |e^{-h^{X_{T}}}\bar{\beta}^{X_{T}}|^2$ by the fact the latter is spherically symmetric. This together with the estimate on $\phi_t$ and the fact $\|\langle x\rangle^{-3}e^{-h^{X_{T}}}\bar{\beta}^{X_{T}}\|_{2}\lesssim (1+t)^{-\frac{1}{2}}$, implies the desired estimate $$\begin{aligned}
|g{\langle{\partial_xW^{X_t}} \mspace{2mu}, {\delta_t(\chi_{T_{\rm loc}}P)\delta^*_t(\chi_{T_{\rm loc}}P)}\rangle}|\lesssim g (1+t)^{-1-\delta}\,.\end{aligned}$$ $\square$
Propagator Estimates {#sec:propagator-estimates}
====================
In this section we prove the propagator estimates used throughout the article. Define $h:=-\Delta+gW$, with $g\in\RR$ small and $W(x)=W(|x|)$.
\[prop:propagator\]If $W:\RR^3\to\RR$ is a smooth function and decays exponentially fast at $\infty$ we have $$\begin{aligned}
\label{eq:classical}
{\lVert \x^{-3}\e^{\i th}(h)^{-1+\eps}\phi \rVert}_2&\leq C(1+t)^{\mez(1+2\eps)}{\lVert \x^3\phi \rVert}_2\,,\quad \eps\in[0,1]\\
{\lVert \x^{-5}\e^{\i th}\partial_x(h)^{-1}W \rVert}_2&\leq C(1+t)^\mdz\label{eq:nonclass1}\\
{\lVert \x^{-5}\e^{\i th}\partial_xW \rVert}_2&\leq C(1+t)^{-\frac{5}{2}}\label{eq:nonclass11}\\
{\lVert \x^{-5}\e^{\i th}\partial_x^\alpha(h)^{-1}W \rVert}_2&\leq C(1+t)^{-\frac{5}{2}}\label{eq:nonclass2}\,,\quad |\alpha|\geq 3,\ \alpha\ \text{is odd}\,.\end{aligned}$$
Estimate is a classic result, see e.g. [@jensen79]. In the proof of the remaining assertions we will use the following
\[lem:prop\]For any smooth, spherically symmetric and fast decaying function $\varphi$ we have $$\begin{aligned}
{\lVert \x^{-5}\e^{-\i t\Delta}\partial_x(-\Delta)^{-1}\varphi \rVert}_2\leq C(1+t)^\mdz{\lVert \x^4\varphi \rVert}_2\,.
\end{aligned}$$
By Fourier transform we obtain $$\begin{aligned}
\e^{-\i t\Delta}\partial_x(-\Delta)^{-1}\varphi=C\int_{\RR^3}\e^{\i k\cdot x}\e^{\i t|k|^2}\frac{k}{|k|^2}\hat{\varphi}(k)\d k\,,\end{aligned}$$ for some constant $C\in \RR$. Since $\varphi$ is spherically symmetric, so is $\hat{\varphi}$. Using polar coordinates ($\RR^3\ni k=\rho g(\alpha,\vartheta)$) we find $$\begin{aligned}
\e^{-\i t\Delta}\partial_x(-\Delta)^{-1}\varphi&=C\int_{-1}^1\int_0^{2\pi}\int_0^\infty\e^{\i\rho |x|\cos\vartheta} \e^{\i t\rho^2}\rho g(\alpha,\vartheta)\hat{\varphi}(\rho)\d\rho\d\alpha\d\cos\vartheta\\
&=C\int_{-1}^1\int_0^{2\pi}\int_0^\infty\e^{\i t\rho^2}\rho g(\alpha,\vartheta)\hat{\varphi}(\rho)\d\rho\d\alpha\d\cos\vartheta\\
&+C\int_{-1}^1\int_0^{2\pi}\int_0^\infty\frac{\e^{\i\rho |x|\cos\vartheta}-1}{\rho}\rho^2\e^{\i t\rho^2}g(\alpha,\vartheta)\hat{\varphi}(\rho)\d\rho\d\alpha\d\cos\vartheta\\
&=C\int_{-1}^1\int_0^{2\pi}\int_0^\infty\frac{\e^{\i\rho |x|\cos\vartheta}-1}{\rho}\rho^2\e^{\i t\rho^2}g(\alpha,\vartheta)\hat{\varphi}(\rho)\d\rho\d\alpha\d\cos\vartheta\,,\end{aligned}$$ as the unit vector $g(\alpha,\vartheta)$ averages to zero over the unit sphere. Denote by $f_x(\rho)$ the smooth function $\frac{\e^{\i\rho |x|\cos\vartheta}-1}{\rho}$ and evaluate the $\rho$-integral by scaling $\rho\to t^\pez\rho$ as follows: $$\begin{aligned}
\label{eq:rho1}
\int_0^\infty f_x(\rho)\rho^2\e^{\i t\rho^2}\hat{\varphi}(\rho)\d\rho=t^\mdz\int_0^\infty f_x(\rho t^\mez)\rho^2\e^{\i \rho^2}\hat{\varphi}(\rho t^\mez)\d\rho \,.\end{aligned}$$ Since $\rho^2\e^{\i \rho^2}$ is not integrable we integrate by parts which yields $$\begin{aligned}
&t^\mdz\int_0^\infty f_x(\rho t^\mez)\rho^2\e^{\i \rho^2}\hat{\varphi}(\rho t^\mez)\d\rho=-t^\mdz\frac{1}{2\i}\int_0^\infty \e^{\i \rho^2}\partial_\rho(\rho f_x(\rho t^\mez)\hat{\varphi}(\rho t^\mez))\d\rho\\
=&-t^\mdz\frac{1}{2\i}\int_0^\infty \e^{\i \rho^2} f_x(\rho t^\mez)\hat{\varphi}(\rho t^\mez)\d\rho-t^\mdz \frac{1}{2\i}\int_0^\infty \e^{\i \rho^2}\rho \partial_\rho (f_x(\rho t^\mez)\hat{\varphi}(\rho t^\mez))\d\rho\,.\end{aligned}$$ The first term on the second line is easily seen to be given by $$\begin{aligned}
t^\mdz\frac{1}{2\i}\int_0^\infty \e^{\i \rho^2} f_x(\rho t^\mez)\hat{\varphi}(\rho t^\mez)\d\rho=Ct^\mdz (f_x(0)\hat{\varphi}(0)+o(1))=Ct^\mdz (\cos\vartheta|x|\hat{\varphi}(0)+o(1))\,,\end{aligned}$$ where $o(1)$ is short for $o(1)\,,\; t\to\infty$. In the second term we integrate by parts again to get $$\begin{aligned}
&t^\mdz \frac{1}{2\i}\int_0^\infty \e^{\i \rho^2}\rho \partial_\rho (f_x(\rho t^\mez)\hat{\varphi}(\rho t^\mez))\d\rho\\
=&-Ct^\mdz (f'_x(0)\hat{\varphi}(0)+f_x(0)\hat{\varphi}'(0))
-t^\mdz\frac{1}{2\i} \int_0^\infty \e^{\i \rho^2}\partial^2_\rho (f_x(\rho t^\mez)\hat{\varphi}(\rho t^\mez))\d\rho \,.\end{aligned}$$ The last term is given by $$\begin{aligned}
t^\mdz\frac{1}{2\i} \int_0^\infty \e^{\i \rho^2}\partial^2_\rho (f_x(\rho t^\mez)\hat{\varphi}(\rho t^\mez))\d\rho=Ct^\mdz (\partial^2_r\bigg|_{r=0} f_x(r)\hat{\varphi}(r)+o(1)) \,.\end{aligned}$$ Summarizing, we have shown $$\begin{aligned}
|\e^{-\i t\Delta}\partial_x(-\Delta)^{-1}\varphi (x)|\leq Ct^\mdz (|x|^3 {\lVert y^2\varphi \rVert}_1+o(1))\,,\end{aligned}$$ because $f''_x(0)=-\i\cos^3\vartheta|x|^3$, and $\hat{\varphi}''(0)=\int y^2\varphi \d^3 y$. Using Hölder’s inequality we arrive at $$\begin{aligned}
{\lVert \x^{-5}\e^{-\i t\Delta}\partial_x(-\Delta)^{-1}\varphi \rVert}_2\leq C t^\mdz{\lVert \x^4\varphi \rVert}_2\,,\end{aligned}$$ which is the claim.
The proof of (\[eq:nonclass11\]) uses the spherical symmetry of $W$ and the Kato-Jensen expansion of the propagator [@jensen79], $$\begin{aligned}
\e^{-\i ht}=t^\mdz B_1+t^{-\frac{5}{2}}B_2+\dots\end{aligned}$$ in $\mathcal{B}(L^{2,-5},L^{2,5})$ (this expansion holds if $h=-\Delta+gW$ has no negative eigenvalues and no zero resonance, which is the case for our choice of $gW$). Here, $$\begin{aligned}
B_1(\cdot) = C{\langle{\cdot} \mspace{2mu}, {(1+(-\Delta)^{-1}gW)^{-1}1}\rangle}(1+(-\Delta)^{-1}gW)^{-1}1\,,\end{aligned}$$ so $B_1(\partial_xW)=0$ because of the spherical symmetry of $W$. We obtain $$\begin{aligned}
{\lVert \x^{-5}\e^{\i ht}\partial_xW \rVert}_2\leq C(1+t)^{-\frac{5}{2}}\,,\end{aligned}$$ which is (\[eq:nonclass11\]).
To prove (\[eq:nonclass1\]) define the function $$\begin{aligned}
\xi:=(1+gW(-\Delta)^{-1})^{-1}W\,.\end{aligned}$$ The function $\xi$ is spherically symmetric, and from the equation $(-\Delta)^{-1}\xi=h^{-1}W$ we get $$\begin{aligned}
\xi=(-\Delta)h^{-1}W=W-gWh^{-1}W\,,\end{aligned}$$ from wich it is easy to see that $\xi$ decays exponentially fast at $\infty$, since $h^{-1}$ is a bounded operator $\mathscr{H}^{2,s}\to \mathscr{H}^{2,-s}$ for $s>\frac{1}{2}$.\
By Duhamel’s principle, we rewrite $\e^{\i th}\partial_xh^{-1}W$ as $$\begin{aligned}
\e^{\i th}\partial_xh^{-1}W &= \e^{\i th}\partial_x(-\Delta)^{-1}\xi\\
&=\e^{-\i t\Delta}\partial_x(-\Delta)^{-1}\xi+\int_0^t\e^{\i h(t-s)}gW\e^{-\i s\Delta}\partial_x(-\Delta)^{-1}\xi\d s\,.\end{aligned}$$ The desired estimate follows from (\[eq:classical\]) and Lemma \[lem:prop\].
The proof for (\[eq:nonclass2\]) proceeds along the very same lines as the one for (\[eq:nonclass1\]). The integral over the angle function $g(\phi,\vartheta)$ ($\RR^3\ni k^\alpha=\rho^{|\alpha|}g(\phi,\vartheta)$) still vanishes because $|\alpha|$ is odd, and the additional (at least) two powers of $\rho$ in (\[eq:rho1\]) give the desired $t^{-\frac{5}{2}}$-decay. The increased number of integrations by parts poses no problem since ${\lVert \x^nW \rVert}_2<\infty$ for any $n$.
The proof is complete.
$\square$
The following result has been used several times.
$$\begin{aligned}
\label{eq:help1}
{\mathrm{Re}\,}{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h t}\partial_{x_1}(h)^{-1}W}\rangle}&=\frac{1}{3}{\mathrm{Re}\,}{\langle{W} \mspace{2mu}, {\e^{\i\Delta t}W}\rangle}+O(gt^\mdz)\nonumber\\
&=-\frac{1}{3\sqrt{2}}\pi^{\pdz}t^\mdz(1+\widetilde{C}g)+O(t^{-\frac{5}{2}})\,.\end{aligned}$$
The results in [@jensen79] implies that the expression is of the form $C_1 t^{-\frac{1}{2}}+C_2 t^{-\frac{3}{2}}+O( t^{-\frac{5}{2}})$ as $t\rightarrow \infty.$ It is very involved to compute each constant. Instead in what follows we show that $C_1=0$ and $C_2=-\frac{1}{3\sqrt{2}}\pi^{\pdz}+O(g).$
To prove this we expand $h^{-1}$ to be $(-\Delta)^{-1}+g(-\Delta)^{-1}W h^{-1}$ and obtain $$\begin{aligned}
{\langle{[1-g(h)^{-1}W]\partial_{x_1} W} \mspace{2mu}, {\e^{-\i h t}\partial_{x_1}(h)^{-1}W}\rangle}&={\langle{\partial_{x_1}W} \mspace{2mu}, {\e^{\i\Delta t}\partial_{x_1}(-\Delta)^{-1}W}\rangle}-g{\langle{(h^{-1}W)\partial_{x_1} W} \mspace{2mu}, {\e^{\i\Delta t}\partial_{x_1}(-\Delta)^{-1}W}\rangle}\\
-&g{\langle{V} \mspace{2mu}, {\e^{\i\Delta t}\partial_{x_1}(-\Delta)^{-1}Wh^{-1}W}\rangle}+g{\langle{V} \mspace{2mu}, {\int_0^t\e^{\i\Delta(t-s)}W\e^{-\i hs}\partial_{x_1}h^{-1}W}\rangle}\,,\end{aligned}$$ where we used the abbreviation $$\begin{aligned}
V=[1-g(h)^{-1}W]\partial_{x_1} W\,.\end{aligned}$$ The first term is the main term $$\begin{aligned}
{\langle{\partial_{x_1}W} \mspace{2mu}, {\e^{\i\Delta t}\partial_{x_1}(-\Delta)^{-1}W}\rangle}=\frac{1}{3}{\langle{W} \mspace{2mu}, {\e^{\i\Delta t}W}\rangle}=-\frac{1}{3\sqrt{2}}\pi^{\pdz}t^\mdz+O(t^{-\frac{5}{2}})\,\end{aligned}$$ which is obtain by Fourier transformation and integration by parts, see [@Froehlich102]. For the various other terms on the right hand side we use the propagator estimates in Proposition \[prop:propagator\] to estimate: $$\begin{aligned}
g|{\langle{(h^{-1}W)\partial_{x_1} W} \mspace{2mu}, {\e^{\i\Delta t}\partial_{x_1}(-\Delta)^{-1}W}\rangle}|\leq g t^\mdz\end{aligned}$$ for the third $$\begin{aligned}
&g|{\langle{V} \mspace{2mu}, {\e^{\i\Delta t}\partial_{x_1}(-\Delta)^{-1}Wh^{-1}W}\rangle}|\leq g t^\mdz\\\end{aligned}$$ and for the last $$\begin{aligned}
g|{\langle{V} \mspace{2mu}, {\int_0^t\e^{\i\Delta(t-s)}W\e^{-\i hs}\partial_{x_1}h^{-1}W}\rangle}|&\leq g {\lVert \x^3V \rVert}_2\int_0^t(1+t-s)^\mdz{\lVert \x^3W\e^{-\i hs}\partial_{x_1}h^{-1}W \rVert}_2\d s\\
\lesssim&g\int_0^t(1+t-s)^\mdz(1+s)^\mdz{\lVert \x^4W \rVert}_2\d s\lesssim gt^\mdz\,.\end{aligned}$$ This concludes the proof of (\[eq:help1\]).
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Professors Jürg Fröhlich and Israel Michael Sigal for suggesting this project and very useful discussions.
[FSSG10]{}
Jürg Fröhlich, Zhou Gang, and Avy Soffer, *Some Hamiltonian Models of Friction*, to appear in Journal of Mathematical Physics.
Jürg Fröhlich, Israel Michael Sigal, Avi Soffer, and Zhou Gang, *Hamiltonian Evolution of Particles Coupled to a Dispersive Medium*, to be published, 2010.
Arne Jensen and Tosio Kato, *Spectral Properties of Schrödinger Operators and Time-Decay of the Wave Functions*, Duke Mathematical Journal **46** (1979), no. 3, 583–611.
Kenji Yajima, *The $W^{k,p}$-continuity of Wave Operators for Schrödinger Operators*, Proc. Japan Acad. **69** (1993), 94–98.
[^1]: [email protected]
[^2]: [email protected]
|
---
abstract: 'We report the discovery of two massive eccentric systems with BRITE data, $\tau$ Ori and $\tau$ Lib, showing heartbeat effects close to the periastron passage. $\tau$ Lib exhibits shallow eclipses that will soon vanish due to the apsidal motion in the system. In neither system, tidally excited oscillations were detected.'
author:
- Andrzej Pigulski
- 'Monika K. Kamińska'
- Krzysztof Kamiński
- Ernst Paunzen
- Jan Budaj
- Theodor Pribulla
- 'Pascal J. Torres'
- Ivanka Stateva
- Ewa Niemczura
- Marek Skarka
- Filiz Kahraman Aliçavuş
- Matej Sekeráš
- Mathieu van der Swaelmen
- Martin Vaňko
- Leonardo Vanzi
- Ana Borisova
- 'Krzysztof He[ł]{}miniak'
- Fahri Aliçavuş
- Wojciech Dimitrov
- Jakub Tokarek
- Aliz Derekas
- Daniela Fernández
- Zoltan Garai
- Mirela Napetova
- Richard Komžík
- Thibault Merle
- Milena Ratajczak
- 'Noel D. Richardson & Ritter Observing Team'
- Eiji Kambe
- Nobuharu Ukita
- the BRITE Team
title: |
Ori and Lib:\
Two new massive heartbeat binaries
---
Following the discovery by [@2012ApJ...753...86T] of 17 stars in highly eccentric systems that show heartbeat effect during periastron passage and sometimes tidally excited oscillations (TEOs), there is a growing interest in the study of such systems. The first massive heartbeat system, $\iota$ Ori, was recently discovered by [@2017MNRAS.467.2494P]. We announce the discovery of two more relatively massive eccentric binaries showing heartbeat signals, $\tau$ Ori and $\tau$ Lib.
$\tau$ Orionis (HD 34503, $V =$ 3.6 mag) is a visual quadruple system, in which the brightest component A is an evolved mid B-type star classified as B5 III. Variability of the radial velocity of $\tau$ Ori A was found over a century ago [@1907ApJ....25R..59F], but up to now no orbital period was derived. In numerous studies, the star was used as an MK standard for spectral type B5III. It was not known to be variable in brightness. $\tau$ Librae (HD139365, $V =$ 3.6 mag), a member of the Sco-Cen association, is a double-lined spectroscopic binary classified as B3V+B5. The variability of the radial velocity of the star was found by [@1926PASP...38..132M]. Similarly to $\tau$ Ori, the star was not known as photometrically variable prior to the BRITE observations.
The two stars were observed by BRITE-Constellation [@2014PASP..126..573W; @2016PASP..128l5001P] in 2013–2016 ($\tau$ Ori) and in 2015 ($\tau$ Lib). The photometry, obtained by means of the pipeline presented by , was subsequently corrected for instrumental effects and analyzed. For $\tau$ Ori, it revealed four short brightenings which we interpreted as a possible heartbeat signal in the vicinity of the periastron passage. The blue- and red-filter BRITE light curves of $\tau$ Ori phased with the derived orbital period of 90.3 d are shown in Fig. \[fig:tau2\], top left panel. For $\tau$ Lib, the analysis revealed periodic variability with a period of 3.45 d, again showing a heartbeat shape, with a possibility of a very shallow ($\sim$3 mmag) eclipse (Fig. \[fig:tau2\], bottom left panel). The eclipse was confirmed in the Solar Mass Ejection Imager (SMEI, @2004SoPh..225..177J) observations made between 2003 and 2010, in which it is much deeper (10–14 mmag) than in 2015. This changing shape of the eclipse can be interpreted in terms of apsidal motion.
![Left panels: BRITE observations of $\tau$ Ori (top) and $\tau$ Lib (bottom) in the blue (filled circles) and red (open circles) bands phased with their orbital periods: 90.3 d and 3.4501 d, respectively. Note the shallow eclipse at phase 0.0 for $\tau$ Lib. Right panels: Radial-velocity curves for the primary (filled circles) and the secondary (open circles) components of $\tau$ Ori (top) and $\tau$ Lib (bottom). The solid- and dashed-line curves stand for the preliminary orbital solution for the primary and secondary components, respectively.[]{data-label="fig:tau2"}](Tau2-HB-RV){width="84.50000%"}
The discovery of a heartbeat in $\tau$ Ori and $\tau$ Lib prompted us to organize a spectroscopic campaign. It resulted in over 260 spectra of $\tau$ Ori and 42 spectra of $\tau$ Lib. The radial-velocity curves (Fig. \[fig:tau2\], right panels) show changes typical of eccentric systems. Both stars are double-lined spectroscopic binaries. $\tau$ Ori is a system consisting of two stars with practically the same mass (mass ratio $q =$ 0.997) in a highly eccentric orbit. The preliminary solution of the radial-velocity curve resulted in $P_{\rm orb} =$ 90.29 d, $e =$ 0.834, and $\omega =$ 156$^{\rm o}$. According to [@2012ApJ...753...86T], the observed shape of the heartbeat corresponds to an inclination of about 30$^{\rm o}$, which implies component masses of about 6 M$_\odot$, consistent with the spectral classification. The preliminary spectroscopic parameters derived from the fit shown in Fig. \[fig:tau2\] are the following: $P_{\rm orb} =$ 3.4501 d, $e =$ 0.276, and $\omega =$ 155.4$^{\rm o}$. The inclination estimated at about 66$^{\rm o}$ allows to derive masses of 6.6 and 5.3 M$_\odot$. The difference between the BRITE and SMEI light curves can be explained by the change of the longitude of periastron. The estimated period of apsidal motion amounts to about 350 years. The eclipses will vanish in a few years. The star will become again eclipsing around 2050, when the secondary eclipse will start to be visible.
No tidally excited oscillations with amplitudes exceeding 0.35 mmag were found in the BRITE data of the two stars under consideration. A detailed analysis of the presented data will be published elsewhere.
[8]{} \[1\][\#1]{} \[1\][`#1`]{} \[2\]\[\][[\#2](#2)]{}
, E. B., ** **25**, 59 (1907)
, B. V., et al., ** **225**, 177 (2004)
, J. H., ** **38**, 132 (1926)
, H., et al., ** **128**, 12, 125001 (2016)
, H., et al., ** **467**, 2494 (2017)
, A., et al., ** **605**, A26 (2017)
, S. E., et al., ** **753**, 86 (2012)
, W. W., et al., ** **126**, 573 (2014)
|
---
abstract: 'The leading locally observable effect of a long-wavelength metric perturbation corresponds to a tidal field. We derive the tidal field induced by scalar, vector, and tensor perturbations, and use second order perturbation theory to calculate the effect on the locally measured small-scale density fluctuations. For sub-horizon scalar perturbations, we recover the standard perturbation theory result ($F_2$ kernel). For tensor modes of wavenumber $k_L$, we find that effects persist for $k_L\tau \gg 1$, i.e. even long after the gravitational wave has entered the horizon and redshifted away, i.e. it is a “fossil” effect. We then use these results, combined with the “ruler perturbations” of [@stdruler], to predict the observed distortion of the small-scale matter correlation function induced by a long-wavelength tensor mode. We also estimate the observed signal in the B mode of the cosmic shear from a gravitational wave background, including both tidal (intrinsic alignment) and projection (lensing) effects. The non-vanishing tidal effect in the $k_L\tau \gg 1$ limit significantly increases the intrinsic alignment contribution to shear B modes, especially at low redshifts $z \lesssim 2$.'
author:
- Fabian Schmidt
- Enrico Pajer
- Matias Zaldarriaga
bibliography:
- 'GW.bib'
title: 'Large-Scale Structure and Gravitational Waves III: Tidal Effects'
---
Introduction {#sec:intro}
============
Cosmological perturbation theory is a robust pillar on which our interpretation of cosmological observations rests. Although the linear results are by now part of textbook material, second and higher order effects have not yet been comprehensively computed. Given the ever increasing amount and precision of observations, many of these effects have already been or soon will be detected, providing strong motivation for a growing body of work (see [@2013JCAP...11..015C; @PietroniPeloso; @Creminelli:2013poa; @Kehagias1; @Valageas; @Creminelli:2013mca; @Bartolo:2010rw; @Giddings:2011zd; @Kehagias2; @Kehagias3] for recent developments).
While independent in linear perturbation theory, scalar, vector, and tensor modes are coupled at second order. This leads to interesting effects which have only recently been begun to be explored. While vector modes decay at linear order also on super horizon scales, tensors are conserved and might therefore have survived since the very early universe, thereby providing us with a unique opportunity to peek almost directly into those early stages of cosmological evolution. Specifically, a measurement of a scale-invariant background of gravitational waves would provide strong support for the inflationary paradigm, and tell us about the energy scale of inflation.
In this paper, we show that the leading locally observable effect of a long-wavelength perturbation $k_L$ (be it scalar, vector, or tensor) on small-scale density perturbations with $k_{S}\gg k_{L}$ is given by an effective tidal field, and we derive the resulting contribution to the density field at second order. Our formalism thus captures any purely gravitational coupling at leading order in $k_L/k_S$. The most well-known case is a long-wavelength density (scalar) perturbation, whose effect on small-scale fluctuations is given by standard second-order perturbation theory (specifically the $F_2$ kernel). As a check of our formalism and computations, we re-derive this standard result. Our results are new for tidal fields of vector origin, which, to the best of our knowledge, have not been previously considered in the literature. For the case in which the tidal field is generated by gravitational waves (tensor modes), first estimates of the tidal effects were given in [@MasuiPen; @GWshear]. More recently, a detailed computation has been presented in [@dai/etal:13] assuming matter domination. Comparison with these previous works is in order. In [@MasuiPen], the anisotropy in the short-scale power spectrum was estimated to be of order $h^{(0)}_{ij}k_{S}^{i}k_{S}^{j}/k_{S}^{2}$, where $h^{(0)}_{ij}$ is the spatial metric perturbation at early times. Although this agrees with our final result and that of [@dai/etal:13] up to factors of order one, it does not capture the time dependence of the effect which sheds light on the physical origin of the effect as we will see momentarily. In [@GWshear] it was roughly estimated that the “intrinsic” shape correlations of galaxies induced by tensor modes are proportional to the instantaneous amplitude of the tensor mode tidal field, while we will argue that it should be more accurately given by a certain time integral over the tidal field. Let us defer galaxy alignments for the moment and consider the anisotropy of small-scale density statistics. We reproduce the results of [@dai/etal:13] in the matter dominated regime ($k_L \lesssim 0.01\iMpch$). However, we argue that a signal of comparable size will come from smaller scale tensor modes that entered the horizon during radiation domination. Our treatment of radiation domination neglects perturbations in radiation, which we will study in a separate publication. This additional effect will however not change the above conclusion. Let us stress that by dividing the effects into separately observable pieces as we will describe now, we believe our approach makes the physics of the tensor mode effects intuitive and transparent.
Before presenting our results, we briefly summarize our methodology. The purely gravitational effect of long wavelength metric perturbations on short scales can be captured in a convenient and physically transparent way adopting a series of different coordinates [@conformalfermi]. Given some set of long-wavelength primordial metric perturbations $h(k_{L})$, one can define *conformal Fermi Normal Coordinates* ([$\overline{\mathrm{FNC}}$]{}) at all times along any chosen timelike geodesic. In these local coordinates, the metric is FLRW along the central geodesic with all physical effects due to $h(k_{L})$ being encoded in corrections to the metric at order $(\nabla\nabla h)\,x^2$. Results obtained in these coordinates have a clear physical interpretation as corresponding to what a local freely falling observer moving along the central geodesic would measure.
Symbol Relation Meaning
------------------------------------- -------------------------------------------------------------------------------------------------- ----------------------------------------------------------
${\mathcal{H}}$ = $a' / a = a H$ comoving Hubble scale
$\delta$ $ =\rho_{m}({{\mathbf{x}}})/\bar \rho_{m}-1 $ matter density perturbation
$\Phi_{s}$ [Eq. (\[eq:FNCmetric\])]{} Newtonian gauge potential in absence of tidal field
$t_{ij}, T$ $t_{ij}({\mathbf{0}};\tau) = T(k_L,\tau)\, t_{ij}^{(0)}({\mathbf{0}})$ tidal field $t_{ij}$ and its transfer function $T$
$\alpha, \beta, \gamma$ [Eq. (\[eq:d2tLPT2\])]{} functions of $k$ and $\tau$
$V,F,D_{\sigma_{1}},D_{\sigma_{2}}$ [Eq. (\[eq:Fdef\])]{}, [Eq. (\[eq:Vdef\])]{}, [Eq. (\[eq:Dsigma1\])]{}, [Eq. (\[eq:Dsigma2\])]{} functions of $k$ and $\tau$
${\mathbf{s}}$ ${{\mathbf{x}}}({\mathbf{q}},\tau) = {\mathbf{q}} + {\mathbf{s}}({\mathbf{q}},\tau)$ Lagrangian displacement vector
${\mathbf{q}}$ ${{\mathbf{x}}}({\mathbf{q}},\tau) = {\mathbf{q}} + {\mathbf{s}}({\mathbf{q}},\tau)$ Lagrangian coordinate
${\mathbf{x}}$ ${{\mathbf{x}}}({\mathbf{q}},\tau) = {\mathbf{q}} + {\mathbf{s}}({\mathbf{q}},\tau)$ Eulerian coordinate
$u^i$ [Eq. (\[eq:udef\])]{} peculiar velocity
$\theta$ $= \partial_{i} u^{i}$ peculiar velocity divergence
subscripts $1,2,\dots$ $\delta= \delta_{1}+\delta_{2}+\dots$ (matter) perturbations at linear, second, ... order
subscripts $t$ or $s$ $\delta_{1}=\delta_{1,s}+\delta_{1,t}$ contributions at 0’th and first order in the tidal field
subscripts $L$ or $S$ $k_{L}\ll k_{S}$ Long $L$ and Short $S$ wavelengths
superscript $(0)$ $h_{ij}^{(0)}=h_{ij}(\tau_{0})$ quantity evaluated at “initial” time $\tau_{0}$
In the end, one is typically interested in how the perturbations $h(k_L)$ affect measurements by an observer far away, e.g. on Earth. This requires computing *projection effects*, which we define as the mapping from local [$\overline{\mathrm{FNC}}$]{} coordinates to the coordinates chosen by the observer. We will perform this mapping using the results of [@stdruler]. An advantage of this methodology is that all results at all points of the computation represent physical observables. This is to be contrasted with using global coordinates, in which case unphysical contributions from different parts of the computation need to cancel each other before the final observable result is obtained.
Our main result for tensor (and vector) perturbations can be summarized as follows. Given a primordial spatial metric perturbation[^1] $h_{ij}^{(0)}$ at position ${{\mathbf{x}}}$ (traceless by assumption) with comoving wavenumber $k_L$, the density field ${\delta}_{2,t}$ at conformal time $\tau$ is given in terms of the linear density field ${\delta}_{1,s}$ at the same time by & \_[2,t]{}(,) \[eq:d2tintro\]\
& = h\_[ij]{}\^[(0)]{}() \_[1,s]{}(,). The notation ${\delta}_{2,t}$ indicates that this is the correction to the linear density field induced at second order by vector and tensor modes (see [Tab. \[tab:not\]]{} for a summary of our notation). Here the coefficients $\alpha,\,\beta$ are functions of the wavenumber $k_L$ and conformal time $\tau$. In particular, for $k_L\tau \ll 1$, i.e. when the long-wavelength modes are superhorizon, $\alpha$ and $\beta$ go to zero as demanded by the equivalence principle. More interesting is the opposite limit $k_L\tau \gg 1$, when the tensor or vector mode has long entered the horizon and decayed. We will see that, despite what one might have expected, the coefficients $\alpha$ and $\beta$ do not vanish in this limit but rather asymptote to constant values $\alpha_\infty,\,\beta_\infty$. Thus, the small-scale density field preserves the knowledge of the primordial vector or tensor modes which have long decayed away. Such a feature was called *fossil effect* in [@MasuiPen], and we will adopt this name for the $k_L\tau \gg 1$ limit of the tidal effects. Notice that the fossil effect is generated at the time of horizon re-entry of the tensor or vector mode. For modes of observational interest, this happens at $z<10^{5}$, much after inflation and reheating. In this sense, although the size of the effect is proportional to the primordial tensor modes, the physical coupling between tensors and scalars that we discuss in this work is generated in the late universe.
The two terms in [Eq. (\[eq:d2tintro\])]{} correspond to different physical effects. The first term ($\propto\alpha$) indicates the effect of the tidal field on the evolution of small-scale fluctuations. The second term ($\propto\beta$) on the other hand encodes the effect of the displacement of matter by the long-wavelength tidal field. In other words, it is the effect of the tidal field on the mapping from Lagrangian to Eulerian positions.
The most obvious observational consequence of the effect described in [Eq. (\[eq:d2tintro\])]{} is an *anisotropic distortion of the local small-scale power spectrum*, whose fractional amplitude is given by & - 1 \[eq:Pkintro\]\
&= 2 k\_S\^i k\_S\^j h\_[ij]{}\^[(0)]{} . Given that we consider the regime $k_L/k_S \ll 1$, this expression can then be used to derive the contribution to the tensor-scalar-scalar bispectrum ${\langle}h_{ij}^{(0)}({{\mathbf{k}}}_L) {\delta}({{\mathbf{k}}}_S) {\delta}({{\mathbf{k}}}_S) {\rangle}$ in the squeezed limit. Of course, this bispectrum is only accessible observationally if one has an independent measurement of the vector or tensor modes. Alternatively, if no indepedent estimate of $h_{ij}^{(0)}$ is available, [Eq. (\[eq:Pkintro\])]{} also describes a specific, anisotropic contribution to the collapsed limit of the four-point function of the density field [@jeong/kamionkowski:2012], which can be measured without any external data sets. The specific case studied by [@jeong/kamionkowski:2012] was the 21cm emission at high redshifts. However, the same idea, applied to long-wavelength scalar tidal fields, also underlies the tidal field reconstruction of [@pen/etal:12].
Another probe of anisotropic small-scale fluctuations is the *alignment of dark matter halos*, that are known to orient along long-wavelength tidal fields, and that in turn influence the orientations of galaxies within them [@Peebles:69; @CatelanKamionkowskiBlandford; @HirataSeljak:04]. This can be observed through the shapes of galaxy images, as measured in large area weak lensing shear surveys. The preferential alignment of galaxy images with large-scale tidal fields is known as *intrinsic alignments*. @GWshear were the first to point out that tensor modes are also expected to contribute to intrinsic alignments. As shown there, the observed galaxy shape correlations induced by tensor modes are in fact expected to be dominated by the alignment contribution. The reason for this is that the tensor-mode lensing effect (part of the projection effects discussed above) is strongly suppressed as the propagating gravitational waves fail to produce a coherent deflection along the light cone as scalar perturbations do.
However, the estimates of the alignment effect in [@GWshear] did not take into account the qualitatively different evolution with time of tensor with respect to scalar tidal fields. Specifically, they assumed that the alignment is proportional to the instantaneous tidal field at the time of observation of the galaxy. On the other hand, here we found, in agreement with [@dai/etal:13], that the tidal effect on small-scale fluctuations comes instead from a time integral over the past history of the tensor mode that peaks at horizon crossing $k_{L}\simeq {\mathcal{H}}$ but remains constant afterwards. In [Sec. \[sec:IA\]]{} of this paper, we use these results to provide a more accurate estimate of the intrinsic alignment by tensor modes. We find a shear B-mode power spectrum of $l(l+1)C^{BB}_{\gamma}(l)/2\pi \sim \mathrm{few} \times 10^{-12}$ over a wide range of scales and redshifts (see [Fig. \[fig:Cl\]]{}).
The remainder of the paper derives the coefficient functions $\alpha$ and $\beta$, and presents these applications in more detail. The outline of the paper is as follows. In [Sec. \[sec:FNC\]]{} we review Fermi Normal Coordinates ([$\mathrm{FNC}$]{}) and their conformal analog ([$\overline{\mathrm{FNC}}$]{}) introduced in [@conformalfermi] (leaving details to [App. \[app:FNC\]]{}) and derive a general expression for the local tidal field. In [Sec. \[sec:LPT\]]{} we solve for the effect of the tidal field on short scale density fluctuations using Lagrangian Perturbation Theory (the equivalent Eulerian derivation can be found in [App. \[app:eulerian\]]{}). We give explicit applications of our general results to long-wavelength scalar, vector and tensor perturbations in [Sec. \[sec:scalar\]]{} and [Sec. \[sec:tensor\]]{}. We discuss projection effects and the distortion of small-scale correlations observed on Earth in [Sec. \[sec:xilocal\]]{}. Finally, in [Sec. \[sec:IA\]]{} we derive the implications of our results for lensing shear surveys.
Our notation is summarized in [Tab. \[tab:not\]]{}. For our numerical results, we adopt a flat $\Lambda$CDM cosmology with ${\Omega_{m0}}= 1-\Omega_{\Lambda 0} = 0.3$ and $h=0.72$.
Fermi Normal Coordinates {#sec:FNC}
========================
Our framework for the computation of gravitational tidal effects is the conformal Fermi normal coordinate ([$\overline{\mathrm{FNC}}$]{}) frame, which was first introduced in @conformalfermi and which we review in this section.
Consider a perturbed FRW metric given by g\_(,)=a()\^[2]{}, \[eq:metrichij\] with $\eta_{\mu\nu}$ the mostly positive Minkowski-space metric and $h_{\mu\nu}\ll 1$ some set of small perturbations. We want now to study the effect of the interactions between long ($k_{L}$) and short ($k_{S}$) wavelength perturbations in $h$, assuming $k_{L}\ll k_{S}$. For this purpose we consider a region around the timelike geodesic of a comoving observer governed by the metric [Eq. (\[eq:metrichij\])]{}, with approximate size $k_{S}^{-1}$ on a certain spatial slice around the geodesic. We can then construct a coordinate frame $\{\bar x_F^\mu\}$ with spatial origin corresponding to this central geodesic in which the metric is Friedmann-Robertson-Walker along the central geodesic, with corrections going as the spatial distance from the geodesic squared (the explicit form is given in [@conformalfermi], see also [App. \[app:FNC\]]{}) *at all times* $\bar x_F^0$ g\_\^F = a\^2(x\^0\[|x\_F\^0\]) . \[eq:gmnF\] Note that these coordinates are not globally valid, but apply in a “spaghetti-shaped” region of spacetime around the central geodesic. We call the frame described by the coordinates $\{\bar x_F^\mu\}$ the *conformal* Fermi Normal Coordinate frame ([$\overline{\mathrm{FNC}}$]{}), as it is a generalization of the Fermi Normal Coordinates ([$\mathrm{FNC}$]{}) first introduced by [@ManasseMisner] (and recently applied in cosmology in [@BaldaufEtal; @stdruler; @GWshear]). We give here a brief overview of its properties and refer the reader to [@conformalfermi] for further details. When the size $\sim k_S^{-1}$ of the region considered is much smaller than the horizon, then we can do a further simple coordinate transformation to recover the standard FNC. However, unlike the standard FNC, [$\overline{\mathrm{FNC}}$]{} are also applicable if the region considered is superhorizon.
In case of the standard FNC, the $\O(x_F^2)$ corrections are given by the Riemann tensor evaluated along the central geodesic. For the metric [Eq. (\[eq:metrichij\])]{} (at linear order in $h$), this includes terms of order $H^2$, $H \nabla h$, and $\nabla\nabla h$ (here $\nabla$ stands for either a space or time derivative). As we show in [App. \[app:FNC\]]{}, the $\O(\bar x_F^2)$ corrections in [$\overline{\mathrm{FNC}}$]{} on the other hand come from two sources: first, there are contributions of order $\nabla\nabla h$ from the Riemann tensor of the *conformal* metric $\eta_{\mu\nu}+h_{\mu\nu}$, which agree with the corresponding terms in [$\mathrm{FNC}$]{} (up to factors of $a$ from the leading order relation $x_F^i = a\,\bar x_F^i$ between spatial [$\mathrm{FNC}$]{} and [$\overline{\mathrm{FNC}}$]{}). Second, there are terms of order $H \nabla h$ which enter through the $a^2$ prefactor due to the transformation of the time coordinate. Again, these agree with the terms of the same type in [$\mathrm{FNC}$]{}. Finally, the $\O(H^2 x_F^2)$ terms disappear in [$\overline{\mathrm{FNC}}$]{}, since we have explicitly kept the $a^2$ prefactor.
Thus, the [$\overline{\mathrm{FNC}}$]{} correspond to the natural *comoving* coordinates an observer moving along the central geodesic would choose. In fact, the coordinates chosen to interpret cosmological observations from Earth are essentially [$\overline{\mathrm{FNC}}$]{} constructed for the geodesic of the Solar System and with the size of the patch given by our current Hubble horizon, thereby removing effects of all super-horizon modes $k_L < H_0$, where again $k_L$ is the *comoving* wavenumber of the long-wavelength perturbation. As mentioned above, the advantage of the [$\overline{\mathrm{FNC}}$]{} frame over [$\mathrm{FNC}$]{} is that the corrections to the FRW metric are always of order $(\bar x_F k_L)^2$ \[so in case of the [$\overline{\mathrm{FNC}}$]{} around Earth, the parameter is $(k_L H_0)^2$\]. This allows one to follow the given region around the central geodesic back to early times where it was larger than the horizon $H^{-1}$ (at which point the regular FNC become invalid). The [$\overline{\mathrm{FNC}}$]{} frame is useful for studying the gravitational interaction of perturbations in cosmology whenever there is a hierarchy between long and short modes, so that the effect of the long modes can be studied neglecting corrections of higher order in $k_L/k_S$.
Non-relativistic limit {#sec:FNCA}
----------------------
Note that since we consider scalar[^2] perturbations that are much smaller in scale than tensor perturbations $k_{S}\gg k_{L}$, and since the effect of the latter only becomes relevant as $k_L\gtrsim {\mathcal{H}}$, for all practical purposes we can restrict our analysis to when the short scalar fluctuations are well within the horizon, $k_{S}\gg {\mathcal{H}}$. Additionally, in this work we will restrict ourselves to the dynamics of non-relativistic matter. These two assumptions allow us to use the standard pseudo-Newtonian limit usually adopted in the theory of large-scale structure to describe the gravitational dynamics on short scales.
Throughout, we neglect the effect of perturbations in the radiation component, which is not correct in general: radiation interacts gravitationally with matter at all times. Also, before recombination, radiation couples tightly to baryons. There are two regimes in which we can neglect the effect of radiation on matter. The first is during matter domination, i.e. for $a \gg a_{\rm eq} = \Omega_{r0}/{\Omega_{m0}}$ (and hence after recombination) when the energy density of radiation has redshifted away and its gravitational coupling with matter is very small. The second is during radiation domination on scales smaller than the dissipation scale, where the perturbations in the electron-baryon-photon fluid are completely erased. Hence our results will be applicable in these two regimes. On the other hand, the scales that enter during radiation domination but are larger than the dissipation scale are quite interesting observationally, and deserve a separate study with a full relativistic treatment. We defer this to future work.
Consider objects in the vicinity of the central geodesic around which the [$\overline{\mathrm{FNC}}$]{} are constructed. If these objects are slow moving, i.e. if their velocities relative to that of the central geodesic are much smaller than the speed of light, then their dynamics are governed up to order $v/c$ by $g_{00}^F$. In the following, we will assume that in global coordinates $h_{0i}=0$, which applies to vector and scalar perturbations in popular gauges as well as tensor perturbations. In this case, $g_{00}^F$ is given by (see [App. \[app:FNC\]]{}) g\_[00]{}\^F =& -a\_F\^2([|\_[F]{}]{}) \[eq:FNCmetric\]\
t\_[ij]{}([|\_[F]{}]{}) =& 12 . \[eq:tij\] Here, and throughout, primed denote derivatives with respect to $\tau$. In [Eq. (\[eq:tij\])]{}, all occurences of $h$ are evaluated at a given proper time ${\bar \tau_{F}}$ along the central geodesic (i.e. at $\bar{{\mathbf{x}}}_F=0$ in [$\overline{\mathrm{FNC}}$]{}). $a_F$ is the locally measured scale factor given by a\_F([|\_[F]{}]{}) a(([|\_[F]{}]{},0))= a([|\_[F]{}]{}+ 12 \^[[|\_[F]{}]{}]{} h\_[00]{}(0,)d). The apparent unphysical dependence on a metric perturbation $h_{00}$ (without any derivative) is simply because we are referring to an unobservable “background” scale factor $a(\tau)$ here. As discussed in more detail in [App. \[app:FNC\]]{}, this corresponds to an unobservable shift in the time coordinate. What *is* observable is the different proper time ${\bar \tau_{F}}$ of different regions on a constant-observed-redshift surface. This is part of the “projection effects” we will discuss in [Sec. \[sec:proj\]]{}.
There are two contributions to $g_{00}^F$: the first, $\Phi_s$ is the potential sourced by the small-scale scalar perturbations. The second, $t_{ij}$, is the tidal tensor induced by the long-wavelength metric perturbations. Note that $t_{ij}$ has dimension 1/length$^2$. We will work to linear order in $t_{ij}$ throughout. Note that we allow for a non-zero trace component $t_i^{\, i}$, which will permit us to consider a long-wavelength density perturbation in addition to vectors and tensors. There are corrections to [Eq. (\[eq:FNCmetric\])]{} of order $(\bar x_F^i)^3$ which we neglect as they are suppressed by $k_L/k_S$.
The effect of tensor and, for common gauges such as Newtonian and synchronous gauges, that of vector modes is encoded in the $h_{ij}$ contribution to $t_{ij}$. This contribution agrees with that derived in [@GWshear], taking into account that the tidal field in the latter paper is given in terms of physical coordinates ${{\mathbf{r}}}_F = a \bar{{\mathbf{x}}}_F$. We see that in order to have an effect, these modes have to evolve in time; specifically, superhorizon modes which are conserved will have no effect. On the other hand, scalar long-wavelength perturbations are encoded by the $h_{00} = -2 \Phi_L$ contribution, where $\Phi_L$ is the long-wavelength potential perturbation in Newtonian gauge. Note that the tidal effect is proportional to the second *spatial* derivative of $h_{00}$: this reflects the fact that spatially constant and pure-gradient potential perturbations cannot have an observable impact by way of the equivalence principle.
In the following, we will work exclusively in the [$\overline{\mathrm{FNC}}$]{} frame. Therefore we simplify the notation in what follows: |x\_F\^0 & [\
]{}|x\_F\^i & x\^i [\
]{}a\_F & a. By separating the linearized Einstein equations into long- and short-wavelength parts and transforming the short-wavelength part to [$\overline{\mathrm{FNC}}$]{}, once can show that up to corrections of order $k_L/k_S$, the small-scale potential $\Phi_s$ satisfies the standard Poisson equation in comoving coordinates, \^2 \_s = 4G a\^2 = 32 [\_[m0]{}]{}H\_0\^2 a\^[-1]{} = 32 [\_m]{}() \^2 . \[eq:poisson\] where $G$ is Newton’s constant, ${\mathcal{H}}= a'/a$, $\rho$ is the matter density with homogenous average $\bar \rho$ and contrast $\delta\equiv \delta \rho/\bar \rho-1$.
We define a transfer function $T(k_L,\tau)$ of the tidal field as follows, t\_[ij]{}(0;) = T(k\_L,) t\_[ij]{}\^[(0)]{}(0), \[eq:transfer\] which in general depends on the wavenumber $k_L$ of the long-wavelength perturbation. We will give the transfer function explicitly later on when dealing with the scalar and tensor tidal fields separately. In the following, we will often suppress the argument $k_L$ as it does not enter in the derivation otherwise. Throughout, we will assume that the source term for the second order density goes to zero at early times, i.e. $a(\tau) t_{ij}(\tau) \stackrel{\tau\to 0}{=} 0$; more specifically, we assume that the small-scale fluctuations have settled into the growing mode by the time $a(\tau) t_{ij}(\tau)$ becomes non-negligible. This again is only valid if the wavelength of the small-scale fluctuations is sufficiently smaller than that of the tidal field.
Lagrangian derivation of tidal effects {#sec:LPT}
======================================
In this section, we derive the effects of the external field on small-scale density perturbations using the Lagrangian approach. A Eulerian derivation which arrives at the same result is given in [App. \[app:eulerian\]]{}. For simplicity, we will assume an Einstein-de Sitter Universe in this section. This is applicable to external tidal fields which become relevant during matter domination, e.g. those induced by long-wavelength modes that enter the horizon during matter domination, i.e. with wave numbers $H_0^{-1} \ll k_L < k_{\rm eq}$. We extend the derivation to include tidal fields that become relevant during radiation domination and $\Lambda$ domination in [Sec. \[sec:tensorRD\]]{}.
The comoving Eulerian coordinate ${{\mathbf{x}}}$ at conformal time $\tau$ is related to the Lagrangian coordinate ${\mathbf{q}}$ by the displacement ${\mathbf{s}}$, (q,) = q + š(q,). In Lagrangian perturbation theory (LPT), we adopt the single-stream approximation, in which case the Eulerian fractional matter overdensity ${\delta}({{\mathbf{x}}},\tau)$ is given by \[(,),\] = |1 + M(q,)|\^[-1]{} - 1, \[eq:dLPT\] where ${\mathbf{M}}_{ij}$ is the deformation tensor, M\_[ij]{}(q,) = s\^j(q,). Note that $\partial_q^i = \partial_x^i + {\mathbf{M}}^i_{\ j} \partial_q^j$. The evolution equation for ${\mathbf{s}}$ is simply the equation of motion of a particle in comoving units, s”\^i(,) + s’\^i(,) =& - \_x\^i \_[(),]{} , \[eq:seom\] where primes denote derivatives with respect to conformal time $\tau$. Using [Eq. (\[eq:dLPT\])]{}, the Poisson equation [Eq. (\[eq:poisson\])]{} becomes at second order & \_x\^2\_s(()) = 32 [\_m]{}\^2 (()) [\
]{}& = 32 [\_m]{}\^2 \_. \[eq:Phis\] We are interested in the leading effect of $t_{ij}$ on the density in Eulerian space [Eq. (\[eq:dLPT\])]{}. For this, we decompose the displacement as š = š\_s + š\_t, where ${\mathbf{s}}_s$ is the scalar contribution which remains when setting $t_{ij}$ to zero. Correspondingly, we will use ${\mathbf{M}}_s,\,{\mathbf{M}}_t$. We will further perform a perturbative expansion in ${\mathbf{s}}$. Specifically, we consider the linear displacement, which uniquely separates into scalar and tensor pieces ${\mathbf{s}}_{1,s},\,{\mathbf{s}}_{1,t}$, and the quadratic mixed contribution from the coupling of ${\mathbf{s}}_s$ and ${\mathbf{s}}_t$, denoted as ${\mathbf{s}}_{2,t}$ (the contributions of order $(s_{1,s})^2$ lead to the standard second order LPT result). Without loss of generality, we set ${\mathbf{s}}_{1,s}({{\mathbf{q}}}=0,\tau) = {\mathbf{0}} = {\mathbf{s}}_{1,t}({{\mathbf{q}}}=0,\tau)$ at some time of interest $\tau$, so that *at linear order* the origin coincides in both Eulerian and Lagrangian coordinates.
Linear solutions
----------------
The linearized version of [Eq. (\[eq:seom\])]{} becomes, separated into scalar and tensor parts, s\_[1,s]{}”\^i(,) + s\_[1,s]{}’\^i(,) =& 32 \^2 \_[qj]{} s\_[1,s]{}\^j(,) [\
]{}s\_[1,t]{}”\^i(,) + s\_[1,t]{}’\^i(,) =& - 12 \_q\^i . \[eq:seomlin\] Since this is at linear order, we have set ${{\mathbf{x}}}= {{\mathbf{q}}}$. Assuming only the growing mode is present in the initial conditions, the first equation can be integrated to give s\_[1,s]{}\^i(,) = - \_[1,s]{}(,) = - a() \_[1,s]{}(,\_0), \[eq:sLs\] where $a(\tau_0)=1$. The equation for $s_{1,t}^i$, rewritten as (\^2 s\_[1,t]{}’\^i)’ = - \^2 T() t\^[(0)i]{}\_[ k]{} q\^k , can be integrated to give s\_[1,t]{}\^i(,) =& - F() t\^[(0)i]{}\_[ k]{} q\^k \[eq:sLt\]\
F() & \_0\^ \_0\^[’]{} d” a(”) T(”). \[eq:Fdef\] Note that T() = 1[a()]{} ’, and that [Eqs. (\[eq:sLt\])–(\[eq:Fdef\])]{} are valid for a general expansion history. In parallel with the linear scalar density ${\delta}_{1,s}$, we define \_[1,t]{}(,) = - \_[qi]{} s\^i\_[1,t]{}() = F() t\^[(0)i]{}\_[ i]{}. \[eq:dLt\] If $t_k^{\ k}=0$, ${\rm Tr}\,{\mathbf{M}}_{1,t} = 0$ and there is no first-order contribution to the density (${\delta}_{1,t}=0$) as expected.
Second-order solution
---------------------
The equation for the second-order displacement $s_{2,t}$ is obtained by collecting all second-order pieces from the right-hand side of [Eq. (\[eq:seom\])]{}. As described above, we will only consider the coupling of $s_{1,s}$ with $s_{1,t}$.
Taking the divergence with respect to ${{\mathbf{q}}}$ of the equation for ${\mathbf{s}}$ \[[Eq. (\[eq:seom\])]{}\] yields ” + ’ =& -\_x\^2 \_s - M\_[ij]{} \_x\^i \_x\^j \_s [\
]{}& - M\_[ij]{} \_x\^i \_x\^j , \[eq:sigeom\] where ${\sigma}= \partial_{q\,i} s^i$ and the r.h.s. is evaluated at ${{\mathbf{x}}}({{\mathbf{q}}})$. Inserting the Poisson equation at second order \[[Eq. (\[eq:Phis\])]{}\], writing ${\mathbf{M}} = {\mathbf{M}}_s + {\mathbf{M}}_t$, and subtracting the equation for ${\mathbf{s}}_{1,t}$ \[[Eq. (\[eq:seomlin\])]{}\] leads to the following equation for ${\sigma}_{2,t}$: \_[2,t]{}” + \_[2,t]{}’ =& 32 [\_m]{}\^2 [[Tr]{}]{}M\_[2,t]{} \[eq:s2Heom\]\
& -32 [\_m]{}\^2 [[Tr]{}]{}M\_[1,s]{} [[Tr]{}]{}M\_[1,t]{} - M\_[1,s]{}\^[ij]{} t\_[ij]{}, where on the r.h.s. all contributions are evaluated at ${{\mathbf{q}}}$. Note that the second term in [Eq. (\[eq:sigeom\])]{} has canceled with the term $\propto {{\rm Tr}\,}{\mathbf{M}}_s \cdot {\mathbf{M}}_t$ from the second-order density. We obtain & \_[2,t]{}” + \_[2,t]{}’ - 32 [\_m]{}\^2 \_[2,t]{} [\
]{}& = -32 [\_m]{}\^2 \_[1,s]{} \_[1,t]{}|\_[,]{} + ( \_[1,s]{})\_[,]{} t\_[ij]{}(). \[eq:s2t1\] Specializing to Einstein-de Sitter, the previous equation becomes & \_[2,t]{}”(,) + 2 \_[2,t]{}’(,) - 6[\^2]{}\_[2,t]{}(,) = (,) [\
]{}& (,) = - 32 H\_0\^2 a() \_[1,s]{}(,\_0) F() t\^[(0)i]{}\_[ i]{} [\
]{}& + a() T() ( \_[1,s]{}(,\_0)) t\_[ij]{}\^[(0)]{} . The growing and decaying modes of this equation correspond to ${\sigma}_{2,t} \propto \tau^2$ and ${\sigma}_{2,t} \propto \tau^{-3}$, respectively, and the solution is \_[2,t]{}(,) = \_0\^d’ 15(,’). By assumption (see the last paragraph of [Sec. \[sec:FNCA\]]{}), $\Sigma({{\mathbf{q}}},\tau) \to 0$ as $\tau\to 0$, and we have fixed the boundary conditions so that both ${\sigma}_{2,t}$ and ${\sigma}_{2,t}'$ vanish in this limit as well. We then obtain \_[2,t]{}(,) =& D\_[1]{}() ( \_[1,s]{}(,)) t\_[ij]{}\^[(0)]{} [\
]{}& - 32 D\_[2]{}() \_[1,s]{}(,) t\_[ i]{}\^[(0)i]{}, where the coefficient functions are given by V() & \_0\^ d’ F’(’) \[eq:Vdef\]\
D\_[1]{}() & 1[a()]{}\_0\^d’ 15a(’) T(’) [\
]{}=& 15 { F() + 4 V() } \[eq:Dsigma1\]\
D\_[2]{}() & \_0\^d’ 15F(’) . \[eq:Dsigma2\] Let us now derive the *Eulerian* density in the presence of $t_{ij}$. This is defined as \_[t]{}() = () - ()|\_[t\_[ij]{}=0]{}. \[eq:dHdef\] Using [Eq. (\[eq:dLPT\])]{}, we have & \_t(()) = \_[1,t]{}() + \_[2,t]{}(()) [\
]{}& = \_[1,t]{}() - \_[2,t]{}() + \_[1,t]{}() \_[1,s]{}() + [[Tr]{}]{}(M\_[1,s]{}M\_[1,t]{})\_.However, there is a further subtlety in [Eq. (\[eq:dHdef\])]{}: we want to compare the overdensities at the same *Eulerian* position ${{\mathbf{x}}}$. The *Lagrangian* coordinate that corresponds to this position is different for $t_{ij} \neq 0$ and $t_{ij}=0$. More precisely, we have (at linear order which suffices for this purpose) = + š\_[1,s]{} + š\_[1,t]{}, whereas ${\delta}_{1,s}$ as defined here gives the density at \_s = + š\_[1,s]{} = - š\_[1,t]{}. Thus, the contribution to the Eulerian density by tensor modes is \_[t]{}(,) =& \_[1,t]{}() - \_[2,t]{}() + \_[1,t]{}() \_[1,s]{}() [\
]{}& + [[Tr]{}]{}(M\_[1,s]{}M\_[1,t]{})\_ - (-\_s)\_\_[1,s]{}()|\_ [\
]{}=& \_[1,t]{}() - \_[2,t]{}() + \_[1,t]{}() \_[1,s]{}() [\
]{}& + [[Tr]{}]{}(M\_[1,s]{}M\_[1,t]{})\_ - s\_[1,t]{}\^i() \_[qi]{} \_[1,s]{}(). \[eq:dtLPT1\] Note that we can replace ${{\mathbf{q}}}$ with ${{\mathbf{x}}}$ at this order. Inserting the expressions derived above, we obtain for the second order contribution \_[2,t]{}(,) =& t\_[ij]{}\^[(0)]{}\_[1,s]{}(, ). \[eq:d2tLPT1\] To recap, the first two terms here come from the modified second-order evolution of the scalar fluctuations, due to the presence of the external tidal field proper (first term) and the additional contribution to the matter density (second term). The next two terms are due to the non-linear relation between displacement and density, which entails a coupling of the linear displacements due to small-scale scalar and external tidal displacements. Finally, the last term is directly proportional to the displacement by the external tidal field, and encodes the fact that we evaluate the small-scale perturbations at different relative (Eulerian) positions than we would have in the absence of $t_{ij}$.
We can now bring [Eq. (\[eq:d2tLPT1\])]{} into the form of [Eq. (\[eq:d2tintro\])]{} in [Sec. \[sec:intro\]]{}: & \_[2,t]{}(,) [\
]{}& = t\_[ij]{}\^[(0)]{}() \_[1,s]{}(,), \[eq:d2tLPT\] where () =& 45 { F() - V() } [\
]{}() =& F() [\
]{}[\
]{}() =& 32 D\_[2]{}() + F() .\[eq:d2tLPT2\] Again, there is a dependence on $k_L$ which we have not written here that enters through the tranfer function in [Eq. (\[eq:Fdef\])]{} and [Eqs. (\[eq:Vdef\])–(\[eq:Dsigma2\])]{}.
Scalar tidal field {#sec:scalar}
==================
In this section, we consider a tidal field $t_{ij}$ induced by a long-wavelength density perturbation ${\delta}_{1,L}$. In principle we could perform this computation for any $k_{L}\ll k_{S}$. On the other hand, our goal is to make contact with standard results, thus providing a non-trivial check of our formulae. Therefore we will assume to be in matter domination and that the long mode is well inside the horizon, namely $k_L \gg {\mathcal{H}}$. In this case, [Eq. (\[eq:d2tLPT\])]{} should reduce to the standard expression ($F_{2}$ kernel of [@bernardeau/etal:02]) for the second order density perturbation in the limit that one mode is much longer than the other.
Before discussing the detailed calculation it is useful to present a heuristic but intuitive derivation of the main result of this section. We want to predict the structure of the second order density field $\delta_{2}$. Up to a potentially time-dependent factor, this is the same as the quadratic source terms in the Eulerian equation of motion for $\delta$, [Eq. (\[eq:d2eomcomp3\])]{}. To avoid proliferation of $\nabla^{-2}$ it is convenient to use the Newtonian potential and its derivatives as fundamental building block $\Phi\sim \left( {\mathcal{H}}^{2}/k^{2} \right) \delta$, instead of $\delta$ itself. Then, we should write all possible second order terms using just $\Phi$ and spatial derivatives, since time derivatives are very small during matter domination (zero in exact EdS). First we notice that $\Phi$ must appear with at least one derivative (a constant $\Phi$ cannot lead to any physical effect); the term with one derivative contains a locally-unobservable uniform acceleration or bulk flow as we will discuss below. Second, since ${\delta}$ is a scalar we need to have an even total number of derivatives (lest we are left with uncontracted indices). The lowest number of derivatives is then two but the term $\partial_{i}\Phi \partial_{i}\Phi $ cannot appear because the equation for $\delta$ has an additional spatial derivative with respect to the Euler equation (conservation of momentum in [Eq. (\[eq:euler\])]{}) where each $\Phi$ should appear with at least one spatial derivative. Hence the allowed second order terms have at least four derivatives: \_[i]{}\_[j]{}\_[i]{}\_[j]{},\^[2]{}\^[2]{}, \_[i]{}\_[i]{}\^[2]{}.\[eq:vici\] Terms with a higher number of spatial derivatives are suppressed by $k/k_{\rm NL}$ where $k_{\rm NL}$ is the cutoff of the hydrodynamic theory or fluid approximation (see e.g. [@Baumann:2010tm]) and we can safely neglect here as long as we are interested in the mildly (as opposed to fully) non-linear regime. The terms in [Eq. (\[eq:vici\])]{} are indeed those appearing in the well-known $F_2$ kernel. We can further massage these terms by applying to the case at hand. We take one perturbation to be long and one short, dropping terms suppressed by $k_{L}/k_{S}$. We are then left with only three terms: \[eq:guess1\] \_[i]{}\_[j]{}\_[L]{}\_[i]{}\_[j]{}\_[S]{},\^[2]{}\_[L]{}\^[2]{}\_[S]{}, \_[i]{}\_[L]{} \_[i]{}\^[2]{}\_[S]{}. The presence of $\partial_{i}\Phi_{L}$ in an observable quantity like $\delta_{2}$ tells us immediately that this expression is valid in some set of global coordinates that are *not* free falling. In order to derive the second order effects that a local free falling observer would measure, we expand $\partial_{i}\Phi_{L}$ around the observer’s geodesic \[eq:guess2\] \_[i]{}\_[L]{}(x)=\_[i]{}\_[L]{}(0)+\_[i]{}\_[j]{}\_[L]{}(0)x\^[j]{}+…The first term in this expression, once contracted with $\partial_{i}\partial^{2}\Phi_{S}$, represents the effect of the bulk flow since it can be thought of as arising from expanding $\partial^{2}\Phi_{S}(x+U_{L} t)$ at linear order in $u_{L}$ and using $u_{L}^{i}\sim \partial^{i}\Phi_{L}/{\mathcal{H}}$. This effect is present in global coordinates \[as used in standard perturbation theory, see [Eq. (\[eq:d2tscalar\])]{} below\] but can be removed by a boost into the free falling local frame where $\partial_{i}\Phi$ vanishes.
Now that we have built some intuition, let us move to the detailed derivation. Using [Eq. (\[eq:tij\])]{} and the fact that in global Newtonian coordinates h\_[00]{}=-2\_[L]{},h\_[ij]{}=-2\_[L]{}\_[ij]{}, we find t\_[ij]{}=- -\_[L,ij]{}, since $\Phi$ is constant during matter domination. The transfer function \[[Eq. (\[eq:transfer\])]{}\] is then $T(k_L,\tau) = $ const and we can choose for simplicity $T(\tau,k_{L})=1$. Then one finds F() =& 23 H\_0\^[-2]{} a() [\
]{}V() =& D\_[2]{}() = 4[21]{} H\_0\^[-2]{} a() . Further, evaluating the Poisson equation at $\tau_0$ where $a(\tau_0)=1$, we have t\_[ij]{}\^[(0)]{} = 32 H\_0\^2 \_[1,L]{}(\_0). [Eq. (\[eq:d2tLPT\])]{} then yields & \_[2,s]{}(,) = 47 ( \_[1,L]{}()) ( \_[1,s]{}(,)) [\
]{}&+ 7 \_[1,L]{}() \_[1,s]{}(, ) + ( \_[1,L]{}()) x\^i \^j \_[1,s]{}(, ) \[eq:d2tscalar1\] . This expression contains precisely the terms we predicted in [Eq. (\[eq:guess1\])]{} and [Eq. (\[eq:guess2\])]{}, without the bulk flow \[first term in [Eq. (\[eq:guess2\])]{}\]. In order to compare with the standard perturbation theory result, we have to transform from the local [$\overline{\mathrm{FNC}}$]{} (free falling) frame to global coordinates, which corresponds to adding back in the bulk flow. We can hence replace ( \_[1,L]{}) x\^j \_[1,L]{}. \[eq:d1L\] Further, note that the other permutation corresponding to this term, i.e. $(\partial_i/\nabla^2 {\delta}_{1,s})\partial_i {\delta}_{1,L}$ is not included in our derivation since it involves the third derivative of the long-wavelength potential, i.e. it is suppressed by $k_L/k_S$. Finally, in our treatment we have split ${\delta}_1 = {\delta}_{1,L} + {\delta}_{1,s}$ so that the second order solution contains two permutations $L\leftrightarrow s$ in the quadratic source terms. Thus, if we express the result in terms of ${\delta}_1$, we need to divide [Eq. (\[eq:d2tscalar1\])]{} by two. We finally obtain & \_[2]{}(,) = 27 ( \_[1]{}(,)) ( \_[1]{}(,)) [\
]{}& + 7 \_[1]{}(,) \_[1]{}(, ) + ( \_[1]{}(,)) \^i \_[1]{}(, ) \[eq:d2tscalar\] This is easily seen to be identical to the standard second order scalar density perturbation, which is usally expressed in Fourier space through the $F_2$ kernel (e.g., [@bernardeau/etal:02]).
Tensor and vector modes {#sec:tensor}
=======================
In this section, we describe the effect of vector and tensor metric perturbations on the growth of small-scale fluctuations. We will focus mostly on tensor modes. All models of inflation at sufficiently high energy scale predict an approximately scale-invariant background of gravitational waves, providing strong motivation to study their effects. On the other hand, it is very difficult to devise a mechanism which produces long-wavelength vector modes while satisfying all cosmological constraints e.g. from the CMB. Our formalism is not suited to treat effects of very small scale vector modes which could have been produced in the early Universe, since we always consider scalar perturbations with wavelengths much smaller than those of the vector modes (in any case, the effect of small scale vector modes is hard to observe unless they are coupled to long-wavelength fluctuations). However, all of the following results immediately apply to vector modes should the need arise; the only modification is in the transfer function of the long-wavelength tidal field \[[Eqs. (\[eq:Dhtensor\])–(\[eq:Dhvector\])]{}\].
Tidal field
-----------
Since we work to linear order in the long-wavelength perturbations, any scalar long-wavelength perturbations in $h_{\mu\nu}$ decouple and need not be considered. We thus set $h_{00} = 0$ in [Eq. (\[eq:tij\])]{}, and the long-wavelength tensor and vector modes are contained in $h_{ij}$.
We define a general transfer function $D_h(\tau)$ so that h\_[ij]{}(0,) = D\_h() h\_[ij]{}\^[(0)]{} , where $h_{ij}^{(0)} = h_{ij}({\mathbf{0}},\tau=0)$ is the primordial amplitude of the metric perturbation and $D_h(\tau\to 0) = 1$. For a single Fourier mode $k_L$ tensor perturbation during matter domination, this transfer function is given by D\_[h,T]{}() =& 3 , \[eq:Dhtensor\] while the decay of vector modes produced instantaneously at a time $\tau_*$ is described by D\_[h,V]{}() = , \[eq:Dhvector\] and $D_{h,V}(\tau) = 0 $ for $\tau < \tau_*$.
![*Top panel:* the function $F(\tau)$ \[[Eq. (\[eq:FVtensor\])]{}\], corresponding to $\beta(\tau)$ in [Eq. (\[eq:d2ttensor\])]{}, for tensor modes of various wavenumbers $k_L$ as function of scale factor. *Bottom panel:* coefficient $\alpha(\tau)$ in [Eq. (\[eq:d2ttensor\])]{}, as a function of $a$ for tensor modes with the same wavenumbers $k_L$ as in the upper panel. For wavenumbers entering the horizon during matter domination, this reduces to [Eq. (\[eq:alphaMD\])]{}. All results were obtained by numerical integration of the linear and second order equations (see [App. \[app:numerics\]]{}) for a flat $\Lambda$CDM cosmology. []{data-label="fig:FDsigma"}](F_vs_a "fig:"){width="48.00000%"} ![*Top panel:* the function $F(\tau)$ \[[Eq. (\[eq:FVtensor\])]{}\], corresponding to $\beta(\tau)$ in [Eq. (\[eq:d2ttensor\])]{}, for tensor modes of various wavenumbers $k_L$ as function of scale factor. *Bottom panel:* coefficient $\alpha(\tau)$ in [Eq. (\[eq:d2ttensor\])]{}, as a function of $a$ for tensor modes with the same wavenumbers $k_L$ as in the upper panel. For wavenumbers entering the horizon during matter domination, this reduces to [Eq. (\[eq:alphaMD\])]{}. All results were obtained by numerical integration of the linear and second order equations (see [App. \[app:numerics\]]{}) for a flat $\Lambda$CDM cosmology. []{data-label="fig:FDsigma"}](d2t_vs_a "fig:"){width="48.00000%"}
Second order density {#sec:tensord}
--------------------
[Eq. (\[eq:tij\])]{} yields a tidal transfer function of T() = -12 a\^[-1]{} (a D\_h’)’ , so that F() =& - 12 [\
]{}V() =& - 12 \_0\^d’ D\_h’(’), \[eq:FVtensor\] where we have used that $D_h(0) = 1$. Inserting this into [Eq. (\[eq:d2tLPT\])]{}, and making use of $h_i^{\ i} = 0$ yields the second order density induced by tensor or vector mode tidal fields at time $\tau$, in terms of the linear small-scale density field ${\delta}_{1,s}$ and the *primordial* tensor mode amplitude, or vector mode at production, respectively, $h_{ij}^{(0)}$: \_[2,t]{}(,) =& h\_[ij]{}\^[(0)]{}() \_[1,s]{}(,) \[eq:d2ttensor\] () =& 45 { F() - V() } \[eq:alphaMD\]\
=& +18 +6,\
() =& F()\[eq:beta\]\
=&+3 +3 .This is [Eq. (\[eq:d2tintro\])]{} and the main analytical result of the paper. In order to elucidate its properties, we will consider tensor modes of fixed wavenumber $k_L$ (we will briefly discuss vector modes below). In the limit of $k_L\tau \to 0$, that is before the tensor mode enters the horizon, we have F() 1[20]{} (k\_L)\^2; V() 1[70]{} (k\_L)\^2, so that [Eq. (\[eq:d2ttensor\])]{} becomes \_[2,t]{}(,) & (k\_L)\^2 h\_[ij]{}\^[(0)]{} 15\_[1,s]{}(, ). Thus, the lowest order effect of a long-wavelength tensor mode with wavenumber $k_L$ comes in at order $(k_L\tau)^2$, as expected.
A further interesting limit to consider is $k_L\tau \gg 1$, that is long after horizon crossing of the tensor mode. In this limit, $D_h \to 0$, signifying that the tensor mode has decayed away. Thus, $V(\tau) \to 0$ while $F(\tau) \to 1/2$. [Eq. (\[eq:d2ttensor\])]{} becomes in this “fossil” limit \_[2,t]{}(,) & h\_[ij]{}\^[(0)]{}\_[1,s]{}(, ). \[eq:d2tfossil\] The first term in [Eq. (\[eq:d2tfossil\])]{} is the tidal interaction in the strict sense of the word. The second term corresponds to the effect of an anisotropic expansion rescaling the physical coordinates. This term has the same form as that derived by [@MasuiPen], the difference being that here this is just one specific limit of a more general $k_{L}$-dependent expression. One way of visualizing this term is to imagine a set of freely falling test particles in a Universe that is unperturbed apart from the tensor mode, which are initially (at $k_L\tau \ll 1$) arranged as a spherical shell ${{\mathbf{x}}}^2 = r^2$. Then, their distribution at a later time is given by \^2 - 2 F() h\_[ij]{}\^[(0)]{}x\^i x\^j = r\^2, that is the particles have been rearranged into an ellipsoid. Note that the coefficient $F(\tau)$ of this term is valid for a general expansion history, provided the corresponding transfer function $D_h(\tau)$ is used in [Eq. (\[eq:Fdef\])]{}. On the other hand, the tidal interaction term in [Eq. (\[eq:d2ttensor\])]{} does depend on the background cosmology; for example, $V(\tau)$ depends on the behavior of the decaying scalar mode.
Finally, we point out that [Eq. (\[eq:d2ttensor\])]{} agrees with the results of App. D in [@dai/etal:13], which were derived for matter domination as well. Specifically, from Eq. (D19) in that paper we see that their function $2 \mathcal{S}_N(K)$ is equal to $\alpha$ as defined in [Eq. (\[eq:alphaMD\])]{}.
The functions $\beta(\tau) = F(\tau)$ and $\alpha(\tau)$ are shown in the upper and lower panel, respectively, of [Fig. \[fig:FDsigma\]]{} as function of scale factor for various tensor mode wavenumbers $k_L$. As expected, $F(\tau)$ asymptotes to $1/2$ long after horizon entry of the tensor mode, while $\alpha(\tau)$ asymptotes to $2/5$ for modes entering the horizon during matter domination as assumed in this derivation (see curve for $k_L = 0.01 \iMpch$ in the lower panel of [Fig. \[fig:FDsigma\]]{}). We will consider the case of tensor modes entering during radiation ($k_L \gg 0.01 \iMpch$) and $\Lambda$ domination ($k_L \ll 0.01 \iMpch$) in the next section.
[Fig. \[fig:d2t\]]{} shows $\alpha(k_L,\tau),\,\beta(k_L,\tau)$ vs $k_L$ for two different values of the scale factor. Modes of wavenumber approaching ${\mathcal{H}}$ have not decayed yet resulting in the oscillatory features around $k_L \sim 10^{-3} h\,{\rm Mpc}^{-1}$. The effect of modes of even longer wavelength ($k_L \ll 10^{-3} h\,{\rm Mpc}^{-1}$) is strongly suppressed, since they have not entered the current horizon yet. On the other hand, modes with $k_L \gg {\mathcal{H}}$ asymptote to a redshift-independent value, which for $\beta$ is $1/2$ at all wavenumbers, while for $\alpha$ this value is $2/5$ for a narrow range of scales for which the approximation of matter domination applies.
![Coefficient functions $\alpha(k_L,\tau),\,\beta(k_L,\tau)$ in [Eq. (\[eq:d2ttensor\])]{} as function of $k_L$ for fixed scale factors $a(\tau) = 1$ and $a(\tau) = 1/3$. The same $\Lambda$CDM cosmology as in [Fig. \[fig:FDsigma\]]{} was assumed.[]{data-label="fig:d2t"}](d2t_vs_kT){width="48.00000%"}
Including radiation and $\Lambda$ {#sec:tensorRD}
---------------------------------
The analytical results derived in the previous sections apply only for tensor modes which enter the horizon during matter domination. In order to extend the applicable range in $k_L$, we perform a numerical integration of the linear and second order equations, including radiation and a cosmological constant. The details are given in [App. \[app:numerics\]]{}. [Figs. \[fig:FDsigma\]–\[fig:d2t\]]{} in fact show these numerical results. Note that we have completely neglected the effects of the baryon-photon fluid before recombination. Thus, the results are only strictly valid for scalar perturbations with wavelength shorter than the dissipation scale for which the perturbations in the baryon-photon fluid are erased by viscosity.
![Coefficient function $\alpha(k_L,\tau_0)$ as in [Fig. \[fig:d2t\]]{} (green dashed), and for a flat matter dominated (Einstein-de Sitter, EdS) Universe (blue solid). The black dotted line shows the EdS result plotted at the same $k_L\tau$ as $\Lambda$CDM (see text). All curves are for $a(\tau_0)=1$.[]{data-label="fig:d2tLambda"}](d2t_vs_kT_LCDM){width="48.00000%"}
Let us first consider the effects of radiation, important for tensor modes with $k_L \gtrsim 0.01 h {\rm Mpc}^{-1}$. On intermediate scales, there is a suppression of the second order density field, while on very small scales, we see a logarithmic increase of the effect. It is possible to qualitatively understand these trends through an analytical derivation in pure radiation domination, analogous to [Sec. \[sec:tensord\]]{}, which is described in [App. \[app:RD\]]{}. During radiation domination, both scalar and tensor modes evolve differently than during matter domination, with the scalar growth being proportional to $\ln \tau \propto \ln a$ rather than $a(\tau)$. The derivation of [Sec. \[sec:LPT\]]{} can easily be adapted to this case (again assuming that the scalar perturbation is in the growing mode), leading to the result \_[RD]{}() =& V\^[RD]{}() [\
]{}\_[RD]{}() =& F(), where the coefficient function $V^{\rm RD}(\tau)$ is defined in [Eq. (\[eq:VRDdef\])]{} while $F(\tau)$ is still given by [Eq. (\[eq:FVtensor\])]{} (with the appropriate $D_h$ for radiation domination). Compared to the result for matter domination, note that the contribution to $\alpha(\tau)$, $4/5\, F(\tau)$ from the second order density (fourth term in [Eq. (\[eq:dtLPT1\])]{}) has canceled with a contribution of opposite sign in the contribution from second order evolution. This leads to the suppression on intermediate scales relative to the matter domination result. The second difference to the case of matter domination is that $V^{\rm RD}(\tau)$ continues to evolve logarithmically even as $k_L\tau \gg 1$, explaining the trend seen for $k_L \geq 10 h {\rm Mpc}^{-1}$ in [Fig. \[fig:d2t\]]{}.
We now turn to the effects of $\Lambda$. The main effect of $\Lambda$ is to modify the linear scalar growth factor and the scale factor-conformal time relation. This is illustrated in [Fig. \[fig:d2tLambda\]]{}, where we also show the result for a Universe with $\Lambda=0$, with rescaled $k_L$ (dotted line) so as to show results at the same value of $k_L\tau$ in both cases. We see agreement to better than 5%. Note that in the $\Lambda=0$ case we have rescaled radiation to keep $\Omega_{r 0}/{\Omega_{m0}}$ and thus $a_{\rm eq}$ fixed.
In summary, while the presence of radiation and cosmological constant change the results in detail, the key result is that $\alpha$ remains in the range of $0.2-0.5$ over the entire range of accessible tensor wavenumbers, and the range of redshifts relevant for large-scale structure. Thus, we do not confirm the expectation of [@dai/etal:13] who argued that the effect should be strongly suppressed for modes $k_L \gtrsim 0.01 \iMpch$.
Impact on small-scale density statistics {#sec:xilocal}
========================================
Consider a region of size $R \ll 1/k_L$ over which a tensor mode $h_{ij}$ with wavenumber $k_L$ can be considered spatially constant (as described in [Sec. \[sec:FNC\]]{}, this region is entirely contained within the conformal Fermi patch corresponding to the tensor mode). Corrections will be suppressed by powers of $k_L/k_S$, where $k_S$ is the wavenumber of the scalar perturbation considered. We now show how the results for the second-order density ${\delta}_{2,t}$ from the previous section predict a modification of the small-scale correlation function $\xi_F({{\mathbf{r}}})$ of the matter density field measured locally, that is in the [$\overline{\mathrm{FNC}}$]{} frame.
We define the local two-point function in [$\overline{\mathrm{FNC}}$]{} $\xi_F({{\mathbf{r}}},\tau)$ through \_F(,) = \_F(0,) \_F(,) , \[eq:xihdef\] where \_F(,) = \_[1,s]{}(,) + \_[2,t]{}(,). Analogously, we define the linear matter correlation function $\xi_{1,s}$, i.e. that of ${\delta}_{1,s}$. By assumption $r \ll 1/k_L$. Inserting [Eq. (\[eq:d2ttensor\])]{} yields \_F(,) =& \_[1,s]{}(r,) [\
]{}& + h\_[ij]{}\^[(0)]{}\_[1,s]{}(r,) [\
]{}= \_[1,s]{}(r,) + & h\_[ij]{}\^[(0)]{} r\^i r\^j, \[eq:xitensor\] where primes denote derivatives with respect to $r$ and \_[\^[-2]{} \_[1,s]{}]{}(r,) = - k\^[-2]{} P\_[1,s]{}(k,) e\^[i ]{}. Note that the same result is obtained when defining [Eq. (\[eq:xihdef\])]{} as ${\langle}{\delta}_F(-{{\mathbf{r}}}/2){\delta}_F({{\mathbf{r}}}/2){\rangle}$. We can Fourier transform [Eq. (\[eq:xitensor\])]{} to obtain the local power spectrum. This parallels the calculation in App. B of [@conformalfermi] and yields P\_(,|h) - P\_(k,) =& k\^i k\^j h\_[ij]{}\^[(0)]{} P\_(k,)[\
]{}&. This is [Eq. (\[eq:Pkintro\])]{}.
[Eq. (\[eq:xitensor\])]{} applies to the matter density field. In reality, one observes the statistics of some tracer of matter which is generally biased, i.e. whose clustering properties are not identical to those of matter. The simplest case is a linear local bias. Then, [Eq. (\[eq:xitensor\])]{} remains valid for the tracers if $\xi_{1,s}$ is replaced with \_[1,s,g]{}(r) = b\_1\^2\_[1,s]{}(r), where $b_1$ is the linear bias. Two conditions need to be fulfilled for this to be valid: first, the matter correlation $\xi_{1,s}(r)$ on the scale $r$ has to be small, so that corrections of order $[b_2 \xi_{1,s}(r)]^2$ are negligible, where $b_2$ is the quadratic bias of the tracer. Second, any non-locality in the relation between tracer number density and matter density has to be restricted to scales much smaller than $r$. These two conditions are likely to be satisfied by the 21cm emission from the dark ages which was considered in [@MasuiPen; @jeong/kamionkowski:2012]. On the other hand, for galaxy surveys at lower redshifts, these conditions are not met in general, necessitating a higher order perturbative treatment, which we will not attempt here.
Projection effects {#sec:proj}
------------------
[Eq. (\[eq:xitensor\])]{} gives the effect of tensor modes on small-scale correlations that would be observed locally by a comoving observer, i.e. in the observer’s [$\overline{\mathrm{FNC}}$]{} frame. In the end however, we want to know the effect on the small-scale correlations as observed from Earth, specifically through the arrival directions and redshifts of photons. We thus need to add the effects of the mapping from [$\overline{\mathrm{FNC}}$]{}(source location) to [$\overline{\mathrm{FNC}}$]{}(Earth), which we refer to as “projection effects”.[^3] Note that since the two frames are uniquely defined (at the relevant order), these projection effects are gauge-invariant. As described in Sec. IV of [@conformalfermi], there are three ingredients to this mapping:
- The transformation of the correlation scale ${{\mathbf{r}}}$ through the cosmic ruler perturbations derived in [@stdruler].
- A shift in time from fixed proper time at [$\overline{\mathrm{FNC}}$]{}(source) to fixed observed redshift on Earth, as derived in [@Tpaper].
- A modulation of the mean observed density within the patch over which the small-scale correlation is measured. The modulation of the observed density of a general tracer by tensor modes was derived in [@GWpaper].
As shown in [App. \[app:proj\]]{}, it is straightforward to collect all these results to obtain (, |h) &= \_F(;), where again $\xi_F({{\mathbf{r}}},\tau)$ is the small-scale correlation function in the [$\overline{\mathrm{FNC}}$]{} frame. Here, metric perturbations outside integrals are evaluated at the source, while metric perturbations inside integrals are evaluated on the past lightcone in the background (see [App. \[app:proj\]]{} for details). Further, $h_\parallel \equiv h_{ij}{\hat{n}}^i {\hat{n}}^j$, ${{\mathbf{\hat{n}}}}$ is the unit vector along the line of sight, and ${\tilde{\chi}}\equiv {\bar{\chi}}({\tilde{z}})$ where ${\bar{\chi}}(z)$ is the comoving distance-redshift relation in the background and ${\tilde{z}}$ is the observed redshift. ${\Delta}x^i$, defined in [Eqs. (\[eq:Dxpar\])–(\[eq:Dxperp\])]{} denote the displacement of the true source position from the apparent position. $\M_T$ is the gauge-invariant magnification produced by the tensor mode and is given in [Eq. (\[eq:MT\])]{}. Further, there are two tracer-dependent parameters: the magnification bias parameter $\Q$, given in the simplest case of a sharp flux-limited survey by $\Q = - d\ln \bar{n}_g/d\ln f_{\rm cut}$; and the paramater $b_e$, which quantifies the redshift evolution of the comoving number density of tracers through [Eq. (\[eq:btdef\])]{}.
We now insert [Eq. (\[eq:xitensor\])]{} for $\xi_F$ in [Eq. (\[eq:xiproj\])]{}, which effectively results in adding up all tensor mode effects. We then obtain (, |h) =& \_[1,s]{}(r;). For biased tracers, we again just need to replace $\xi_{1,s}$ with $b_1^2 \xi_{1,s}$ as long as linear local bias is sufficient. This result agrees with [@dai/etal:13], with the exception of the magnification bias contribution $\Q \M_T$ which they did not include.
Shear correlation and intrinsic alignments {#sec:IA}
==========================================
We now turn to the signature of tensor modes in galaxy shear surveys. As discussed in [@DodelsonEtal; @GWshear], the effect of tensor modes on photon geodesics leads to a contribution to the weak lensing shear which can be measured through the correlations of galaxy shapes (second moments). Specifically, a gravitational wave background contributes to the parity-odd B-mode component, which does not receive any scalar contribution at linear order and thus provides a window to search for gravitational waves.
The tensor-mode contribution to the shear, which is a tracefree symmetric tensor on the sky (with two independent components) is given by [@GWshear] \^[proj]{}\_[ij]{} = -12 ¶\_i\^[ k]{}¶\_j\^[ l]{} h\_[kl]{} - \_[(i]{} x\_[j)]{}, \[eq:shear\] where $\P_{kl} = {\delta}_{kl} - {\hat{n}}_k {\hat{n}}_l$ is the projection operator onto the sky plane, $\partial_{\perp i} = \P_i^{\ j}\partial_j$, and the displacement ${\Delta}x_\perp$ is given in [Eq. (\[eq:Dxperp\])]{}.
[Eq. (\[eq:shear\])]{} gives the contribution to galaxy shape correlations induced by the mapping from the source’s [$\overline{\mathrm{FNC}}$]{} frame to observed coordinates, in analogy with the projection effects discussed in [Sec. \[sec:proj\]]{} \[in fact, ${\gamma}_{ij}$ is part of the terms appearing in the first line of [Eq. (\[eq:xiproj\])]{}\]. However, we expect that the effect of tensor modes on the local density field described by ${\delta}_{2,t}$ \[[Eq. (\[eq:d2ttensor\])]{}\] also affects galaxy shapes; that is, there is also a correlation of galaxy shapes with the local tidal field in the [$\overline{\mathrm{FNC}}$]{} frame, leading to a contribution to shape correlations which we will denote as ${\gamma}^{\rm IA}_{ij}$. In the terminology of weak lensing shear, this effect is referred to as *intrinsic alignment*. Just as in [Sec. \[sec:proj\]]{}, the *observed* correlation of galaxy shapes is then given by the sum of the two contributions, \^[obs]{}\_[ij]{} = \^[proj]{}\_[ij]{} + \^[IA]{}\_[ij]{}. We now consider the intrinsic alignment contribution in more detail. It is difficult to predict the amplitude of the alignment of galaxy shapes with the large-scale tidal field from first principles. For tensor modes, this effect was first considered by @GWshear, who used observations of alignments with scalar tidal fields to estimate the effect for gravitational waves. Using our results from [Sec. \[sec:tensor\]]{}, we can elaborate a bit more on this effect. We have found above that the second order density field induced by the (trace-free component of) a *scalar* tidal field is given by \_[2,s]{} = 27 ( \_[1,L]{}(,) ) \_[1,s]{}(, ), while that of a tensor mode is given by \_[2,t]{} = (k\_[L]{}, ) h\_[ij]{}\^[(0)]{} \_[1,s]{}(, ). One possible way to estimate the intrinsic alignment by tensor modes is to assume that the alignment scales as the second order density perturbation induced by the external tidal field. The alignment by scalar tidal fields has observationally been measured at low redshift for elliptical galaxies [@HirataEtal:07; @OkumuraJing; @BlazekEtal]. The scalar intrinsic alignment contribution to the shear can be parametrized as \^[IA,s]{}\_[ij]{}(,) = -[\_[m0]{}]{}C a\^[-1]{}(\_P) ¶\_i\^[ k]{} ¶\_j\^[ l]{} ( (,\_P) ), where $\tau$ is the observation epoch while $\tau_P$ is the epoch at which the tidal field is evaluated. Note that since scalar tidal fields do not evolve strongly, the precise value of $\tau_P$ does not change results by more than 30%. In the following, we will choose $\tau_P = \tau$. For the $z=0.2-0.4$ luminous red galaxies (LRG) studied in [@OkumuraJing; @BlazekEtal], $\tilde C = C_1\rho_{\rm cr,0} \approx 0.12$. Thus, by matching the second order density induced by scalar and tensor tidal fields at the observation epoch, we arrive at the following estimate for the tensor contribution to intrinsic alignments: \^[IA,t]{}\_[ij]{}(,) = - [\_[m0]{}]{}C 72 ¶\_i\^[ k]{} ¶\_j\^[ l]{} h\_[kl]{}\^[(0)]{}() . \[eq:gammaIAt\]
![Tensor mode contribution to the $B$ mode angular shear power spectrum from a gravitational wave background with tensor-to-scalar ratio $r=0.1$. The blue solid lines show the result using the matching of the second order density \[[Eq. (\[eq:gammaIAt\])]{}\], green dashed lines show the result when using the instantaneous tensor tidal field as adopted in [@GWshear], and black dotted lines show the result in the absence of intrinsic alignment, i.e. only including the lensing contribution. In all cases, thick lines are for a source redshift of $z=2$, while thin lines are for $z=0.8$.[]{data-label="fig:Cl"}](Cl_BB){width="49.00000%"}
[Fig. \[fig:Cl\]]{} shows the resulting predicted angular power spectrum of the B-mode of the shear, assuming an almost scale-invariant gravitational wave background with tensor-to-scalar ratio $r=0.1$ (that is, exactly one half of the value adopted in [@GWshear], with otherwise identical parameters). The solid lines lines show the result obtained using the matching relation [Eq. (\[eq:gammaIAt\])]{} at observed redshift $z = 0.8$ (thin) and $z=2$ (thick), respectively. Here, we have used the numerical results for $\Lambda$CDM ([App. \[app:numerics\]]{}) for the coefficient $\alpha(k_L, \tau)$, as shown in [Fig. \[fig:d2t\]]{}. The dotted lines in [Fig. \[fig:Cl\]]{} show the result for $\tilde C=0$, i.e. when only the lensing (projection) effect contributes and any alignment effect of the tensor mode tidal field is absent. Clearly, the lensing effect is at least one order of magnitude smaller than the estimated alignment effect, and moreover drops much more rapidly towards higher $\ell$ (smaller scales).
The long-dashed lines in [Fig. \[fig:Cl\]]{} show the previous prescription adopted in [@GWshear], which relates the shear to the instantaneous tidal field induced by tensor modes, \^[IA,t]{}\_[ij]{}(,) = C ¶\_i\^[ k]{} ¶\_j\^[ l]{} t\_[kl]{}\^[tensor]{}(, ), \[eq:gammaIAprev\] for the same redshifts. Unlike [Eq. (\[eq:gammaIAt\])]{}, this relation only yields a contribution to the shear for tensor modes that have recently entered the horizon. For a relatively high source redshift of $z=2$, the prediction from the second order density-matching is a factor of $2-4$ larger than the prediction using [Eq. (\[eq:gammaIAprev\])]{} on large scales, well within the uncertainty of these rough estimates, while at lower source redshifts the difference becomes much larger (factor of $\sim 30$ at $z=0.8$). This is because the effect has more time to build up in [Eq. (\[eq:gammaIAt\])]{}, i.e. a wider range in wavenumber contributes to the alignment signal. Consequently, in this ansatz the redshift-dependence of the alignment contribution is significantly weaker (it is almost entirely due to the $a^{-1}(\tau)$ prefactor in [Eq. (\[eq:gammaIAt\])]{}, given the redshift independence of the tidal imprint on the second order density field for $k_L \gg {\mathcal{H}}$). The redshift dependence is weaker at the lowest $\ell$, because very large scale tensor modes that have recently entered the horizon have had less time to build up an effect at earlier times. Finally, we point out that the redshift dependence of the alignment strength of galaxies is very uncertain at this point and likely depends strongly on the particular galaxy sample considered.
Nevertheless, the fact that the signal is much larger at low source redshifts than what was estimated in [@GWshear] greatly improves the observational prospects for detecting this effect. Note also the slow suppression of the signal for $l \gtrsim 100$ of the alignment signal predicted by [Eq. (\[eq:gammaIAt\])]{}, which is due to the suppression of the second order density in the range $0.01 < k_L [h/{\rm Mpc}] < 1$ \[[Fig. \[fig:d2t\]]{}\]; however, given the order-of-magnitude uncertainty of this estimate such details of the predicted signal should be taken with a grain of salt.
Finally, in our prediction of ${\gamma}^{\rm IA}_{ij}$, we have only considered the first term in [Eq. (\[eq:d2tscalar\])]{} and [Eq. (\[eq:d2ttensor\])]{}, respectively. There might also be a contribution from the differential displacement \[second term in [Eq. (\[eq:d2ttensor\])]{} and last two terms in [Eq. (\[eq:d2tscalar\])]{}\] to the orientation of galaxies in both scalar and tensor cases, but investigating the relative contribution of this term goes beyond the scope of this paper.
Conclusions {#sec:concl}
===========
In this paper we have computed the effect of long-wavelength perturbations on the dynamics of short-wavelength matter inhomogeneities. We made crucial use of conformal Fermi Normal Coordinates ([$\overline{\mathrm{FNC}}$]{}) in order to isolate the physical effects and remove unobservable coordinate artifacts. This ensures that the results of the various steps of our computations are individually observable. Our formalism can be applied to scalar, vector and tensor long-wavelength perturbations. The case of scalars is well-known and provides a nice check of our results. For vector and tensor perturbations we find that the effect on the short-wavelength matter inhomogeneity is given by [Eq. (\[eq:d2tintro\])]{}, with the consequent anisotropy in the power spectrum given by [Eq. (\[eq:Pkintro\])]{}. Interestingly, these effects remain of order $h_{ij}^{(0)}$ in the $k_L\tau \gg 1$ (fossil) limit, even for wavenumbers $k_L$ which have entered the horizon during radiation domination.
All these results are given in the reference frame of a comoving observer. We have also computed projection effects necessary to make contact with what is observed from Earth. The projected 2-point function of matter (or linearly biased tracers) is given in [Eq. (\[eq:xitot\])]{} as an example, but other projected quantities can be straightforwardly computed as well. Although we have not done so, this can also be easily coupled to the signal-to-noise forecasts for 21cm correlations performed in [@BookMKFS; @jeong/kamionkowski:2012]. Further, it would be interesting to study other large-scale structure tracers as probes of this effect, and cross-correlations between them and the cosmic microwave background.
A further important pertinent observable is cosmic shear, i.e. correlations of galaxy shapes. Here, the locally observable (“tidal”) effects and projection effects are commonly known as “intrinsic alignments” and “lensing”, respectively. We have made the rough approximation that the tidal alignment of galaxies scales as the anisotropic contribution to the second order density field. The resulting odd parity B-mode power spectrum, to which there are no scalar contributions at linear order, is shown in [Fig. \[fig:Cl\]]{}. Interestingly, the tidal effects are much stronger than the lensing effects for tensor modes, while the converse is true for standard scalar density perturbations. Moreover, we have found that the residual effect in the density field which remains when $k_T\tau \gg 1$ \[[Eq. (\[eq:d2tfossil\])]{}\] greatly enhances the expected signal for source galaxies at low redshifts ($z < 2$).
The assumption that galaxy alignments scale with the anisotropic part of the second order density field can only be seen as a rough approximation. It would thus be interesting to perform N-body simulations with an external tidal field imposed, which mimics the effect of a long-wavelength gravitational wave. One could then study the alignment of dark matter halos and substructure with this external tidal field.
Another approximation we have made throughout is to neglect the effects of the baryon-photon fluid before recombination. This issue can be studied using the same tools as presented here. However, since the fluid is relativistic, one also has to take into account the metric components $g_{0i}^F$ and $g_{ij}^F$. We leave this for future work as well.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are happy to thank L. Dai, D. Jeong and M. Kamionkowski for helpful discussions. We further thank K. Masui for discussions and pointing out a factor of 2 mistake in [Sec. \[sec:xilocal\]]{}. E. P. was supported in part by the Department of Energy grant DE-FG02-91ER-40671. F. S. acknowledges support at Princeton by NASA through Einstein Postdoctoral Fellowship grant number PF2-130100 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060. M. Z. is supported in part by the National Science Foundation grants PHY-0855425, AST-0907969, PHY-1213563 and by the David $\&$ Lucile Packard Foundation.
Derivation of [Eqs. (\[eq:FNCmetric\])–(\[eq:tij\])]{} {#app:FNC}
======================================================
This section deals with the transformation of the metric from a set of global coordinates [Eq. (\[eq:metrichij\])]{} to [$\overline{\mathrm{FNC}}$]{}, specifically the time-time component. We will set $h_{0i}=0$ and only keep $h_{00}$ and $h_{ij}$, since this is sufficient to treat scalar perturbations in all widespread gauge choices as well as tensor perturbations. The relation between global coordinates and [$\overline{\mathrm{FNC}}$]{} is then [@conformalfermi] x\^0(|x\^\_F) =& |x\^0\_F + 12 \_0\^[|x\^0\_F]{} h\_[00]{}() d + u\_i |x\_F\^i - 14 h’\_[ij]{} |x\_F\^i |x\_F\^j + Ø\[(|x\_F\^i)\^3\] \[eq:FNCbtrans0\]\
x\^k(|x\^\_F) =& u\^k (|x\_F\^0 - \_F) + |x\_F\^k - 12 h\^k\_[ i]{} |x\_F\^i - 14 |x\_F\^i |x\_F\^j + Ø\[(|x\_F\^i)\^3\], \[eq:FNCbtransi\] where $u^i$ is the coordinate velocity of the central geodesic, and all perturbations are evaluated at the central geodesic, i.e. at $\bar{{\mathbf{x}}}_F = 0$. First, let us verify that using this coordinate transform, the conformal metric $\eta_{\mu\nu} + h_{\mu\nu}$ becomes $\eta_{\mu\nu} + \O([\bar x_F^i]^2)$, i.e. Eq. (14) in [@conformalfermi]. We have =& 1 + 12 h\_[00]{}(0,|x\_F\^0) + u\_i’ |x\_F\^i [\
]{} =& u\^i. \[eq:FNCderiv\] Neglecting terms of order $(u/c)^2$, the central point has a four-velocity \^= = (1 + 12 h\_[00]{},u\^i ) which follows the geodesic equation for the metric $\eta_{\mu\nu}+h_{\mu\nu}$ whose spatial components are u’\^i = - \^i\_[00]{} (1 + Ø(h))\^2 = 12 \^i h\_[00]{}. We are interested in the time-time component of the conformal metric, which we denote $\bar g_{00}$. In global coordinates, it is given by |g\_[00]{}(x) = \_[00]{} + h\_[00]{}(x) = -1 + h\_[00]{}(0,) + \_i h\_[00]{}(0,) x\^i + 12 \_i \_j h\_[00]{}(0,) x\^i x\^j + Ø\[(x\^i)\^3\], where we have chosen the global coordinate origin to coincide with that of the [$\overline{\mathrm{FNC}}$]{} frame at the specific time considered. In going to [$\overline{\mathrm{FNC}}$]{}, the second term is immediately canceled by the second term in [Eq. (\[eq:FNCbtrans0\])]{}, while the third, gradient term is canceled by the $u^i$ term in [Eq. (\[eq:FNCbtrans0\])]{} together with the geodesic equation. Thus, we obtain for the 00 component of the conformal metric in [$\overline{\mathrm{FNC}}$]{} coordinates |g\_[00]{}\^F = \_[00]{} + 12 \_i \_j h\_[00]{}|x\_F\^i |x\_F\^j + 12 h\_[ij]{}” |x\_F\^i |x\_F\^j, where again $h$ is evaluated at ${\mathbf{0}}$. Here, we have replaced $x^i \to \bar x_F^i$ at this order. This is Eq. (14) of [@conformalfermi].
We now consider physical metric, that is $g_{\mu\nu} = a^2(\tau)[\eta_{\mu\nu} + h_{\mu\nu}]$. First, the origin of the [$\overline{\mathrm{FNC}}$]{} frame is now constrained to follow a geodesic in the physical metric. The four-velocity is given by (e.g., [@stdruler]) \^= = a\^[-1]{}(x\^0) (1 + 12 h\_[00]{}(x), u\^i(x) ). The relevant Christoffel components are \^i\_[00]{} =& 12 a\^[-2]{} = -12 \^i h\_[00]{} [\
]{}\^i\_[0j]{} =& 12 a\^[-2]{} = \^i\_[ j]{} + Ø(h) = \^i\_[ j]{} + Ø(h). Since $\Gamma^i_{0j}$ multiplies $u$, we do not have to include the $h$ terms there. We then obtain for the $i$-component of $\uu^\mu$ () = (\^0) ()’ =& - \^i\_[00]{} (\^0)\^2 - 2\^i\_[0j]{} \^0 \^j = a\^[-2]{} [\
]{} u’\^i =& 12 \^i h\_[00]{} - u\^i. \[eq:geod\] Note that this agrees with the Euler equation in [Eq. (\[eq:euler\])]{} once linearized. [Eq. (\[eq:FNCderiv\])]{} is still valid, since $u^i$ in [Eq. (\[eq:FNCderiv\])]{} is defined as $dP^i/dx^0$ and = = = u\^i, recalling that $u^i$ is first order. We now transform $g_{00}$ to $g_{00}^F$ in [$\overline{\mathrm{FNC}}$]{}. Using [Eq. (\[eq:geod\])]{}, we obtain g\_[00]{}\^F = a\^2(x\^0\[|x\_F\]) . We expand the prefactor $a^2\left(x^0[\bar x_F]\right)$ as follows: a\^2(x\^0\[|x\_F\]) = a\^2(|x\_F\^0 + 12h\_[00]{} d) . Putting everything together, the $00$ component of the physical metric in [$\overline{\mathrm{FNC}}$]{} becomes g\_[00]{}\^F =& a\_F\^2(|x\_F\^0) \[eq:g00F\]\
a\_F(|x\_F\^0) & a(|x\_F\^0 + 12h\_[00]{}(0,) d). \[eq:aF\] This corresponds to [Eqs. (\[eq:FNCmetric\])–(\[eq:tij\])]{}. $g_{00}^F$ is, apart from the scale factor, clearly in the FNC form, with corrections going as spatial distance from the central geodesic squared. The fact that the scale factor is evaluated at $\bar x_F^0 + \frac12\int h_{00}({\mathbf{0}},\tau) d\tau$ might surprise at first. Note however that by construction, at $\bar{{\mathbf{x}}}_F=0$ the coordinate $\bar x_0^F$ is the proper time along the central geodesic with respect to the metric $\eta_{\mu\nu} + h_{\mu\nu}$, so that t\_F \^[|x\_F\^0]{} ad= \^[|x\_F\^0]{} a\_F() d is the proper time (with respect to the physical metric $g_{\mu\nu}$) along the central geodesic. Thus, evaluating [Eq. (\[eq:g00F\])]{} at $\bar{{\mathbf{x}}}_F=0$ yields dt\_F = a\_F(|x\_F\^0) d|x\_F\^0. In other words, $\bar x_F^0$ has a clear interpretation as “conformal proper time”. In particular for $h_{00}=0$, $t_F = t$, where $t$ is the time coordinate, and $\bar x_F^0 = \tau$, where $\tau$ is the corresponding conformal time. For $h_{00} \neq 0$, the non-trivial argument of the scale factor in [Eq. (\[eq:aF\])]{} expresses the fact that constant-proper-time surfaces are *not* constant-scale-factor surfaces. If we transform to the standard FNC frame, i.e. to physical rather than comoving coordinates, we obtain metric corrections of the form $H^2 {{\mathbf{x}}}_F^2$, $\dot H {{\mathbf{x}}}_F^2$, where $H$ is the Hubble rate evaluated at the time coordinate corresponding to the given proper time along the central geodesic, i.e. at $\bar x_F^0 + \frac12 \int h_{00}({\mathbf{0}},\tau)d\tau$ just as in the scale factor above. The apparent unphysical dependence on a metric perturbation $h_{00}$ (without any derivative) is simply because we are referring to an unobservable “background” scale factor here. A local observer moving along the central geodesic will simply measure the Hubble rate as a function of his/her proper time (this is in fact is how we define our background scale factor in practice). Thus, the scale factor multiplying the metric [Eq. (\[eq:g00F\])]{} is the scale factor that would *locally be reconstructed* from the measured Hubble rate, hence our notation of $a_F(\bar x_F^0)$ in [Eq. (\[eq:FNCmetric\])]{}.
Eulerian derivation {#app:eulerian}
===================
In this section we present an independent derivation of the main results of [Sec. \[sec:LPT\]]{} using Eulerian perturbation theory. We define the peculiar velocity through ǔ = = a(t) = a v - ,v = . \[eq:udef\] The continuity and Euler equations for an ideal fluid are then given by ’ + =& 0 \[eq:cont\]\
ǔ’ + (ǔ) ǔ + ǔ =& -- 1( v), \[eq:euler\] where ${\sigma}_{ij}$ is the stress tensor of the fluid including pressure (here, $[{{\mathbf{\nabla}}}(\rho{\mathbf{{\sigma}}})]^i = \partial_j(\rho \sigma^{ij})$). In the following, we will set ${\sigma}_{ij}=0$. Separating density and velocity into parts zeroth and first order in $t_{ij}$, we write = \_s+\_t; ǔ =& ǔ\_s + ǔ\_t, where ${\delta}_s,\,{\mathbf{u}}_s; {\delta}_t,\,{\mathbf{u}}_t$ satisfy \_s’ + =& 0 [\
]{}ǔ’\_s + (ǔ\_s) ǔ\_s + ǔ\_s =& -\_s \[eq:eulers\]\
\_t’ + ǔ\_t + =& 0 [\
]{}\^i =& - T() t\^[(0)i]{}\_[ j]{} x\^j . \[eq:eulerH\] The linearized equations are easily seen to be equivalent to the corresponding Lagrangian equations. Hence, we can make use of [Eqs. (\[eq:sLs\])–(\[eq:dLt\])]{} for the linear solutions. In particular, u\_[1,s]{}\^i(,) =& - a’() \_[1,s]{}(,\_0) [\
]{}u\_[1,t]{}\^i(,) =& -F’() t\_[ij]{}\^[(0)]{} x\^j [\
]{}\_[1,t]{}(,) =& F() t\_[i]{}\^[(0)i]{}.
Second order solution
---------------------
As in the Lagrangian derivation ([Sec. \[sec:LPT\]]{}), we work in a perturbative expansion in all of ${\delta}_s,\,{\mathbf{u}}_s; {\delta}_t,\,{\mathbf{u}}_t$, i.e. ǔ\_t =& ǔ\_[1,t]{} + ǔ\_[2,t]{} + , and derive the leading corrections ${\delta}_{2,t},\,{\mathbf{u}}_{2,t}$. These corrections obey the equations \_[2,t]{}’ + ǔ\_[2,t]{} =& - \[eq:contH1\]\
ǔ\_[2,t]{}’ + (ǔ\_[1,s]{}) ǔ\_[1,t]{} + (ǔ\_[1,t]{}) ǔ\_[1,s]{} + ǔ\_[2,t]{} =& -\_s\^[(2)]{}. \[eq:eulerH1\] Taking the divergence of the second equation, assuming Einstein-de Sitter, and introducing $\theta_{2,t} = {{\mathbf{\nabla}}}\cdot {\mathbf{u}}_{2,t}$ allows us to write these equations as \_[2,t]{}’ + \_[2,t]{} = - S\_1 [\
]{}\_[2,t]{}’ + \_[2,t]{} + 32 \^2 \_[2,t]{} =& - S\_2 [\
]{}S\_1 =& -\_[1,s]{}\^[(0)]{}[\
]{}S\_2 =& [\
]{}=& a’ F’ t\_[ij]{}\^[(0)]{} . Here, ${\delta}_{1,s}^{(0)}$ stands for the linear scalar density at a reference time $\tau_0$, i.e. with the growth taken out. We now take the derivative with respect to $\tau$ of the continuity equation and insert the Euler equation for $\theta_{2,t}'$. This yields \_[2,t]{}” + \_[2,t]{}’ - 32 \^2 \_[2,t]{} = - \[S\_1’ + S\_1\] + S\_[2,t]{} = -1a \[a S\_1\]’ + S\_2, \[eq:d2eomcomp3\] which can also be written as \_[2,t]{}” + 2\_[2,t]{}’ - 6[\^2]{} \_[2,t]{} =& -1a \[a S\_1\]’ + S\_2 [\
]{}=& t\_[ij]{}\^[(0)]{} { x\^i \^j + 2 a’ F’ + 1a ’ \^[ij]{} } \_[1,s]{}\^[(0)]{} . The Green’s function for this ODE, with the boundary conditions ${\delta}_{2,t}(0) = {\delta}_{2,t}'(0) = 0$, is G(,’) = 15 ( - ) (-’). We then have \_0\^d’ G(,’) a’ F’ =& 25 a() \[F()-V()\] [\
]{}\_0\^d’ G(,’) a F” =& 15 a() \[-F() + 6 V()\] [\
]{}\_0\^d’ G(,’) 1a ’ =& a() . Putting everything together, we obtain \_[2,t]{}(,) =& t\_[ij]{}\^[(0)]{} { F() x\^i \^j + 45 + \^[ij]{} } \_[1,s]{}(,). This agrees exactly with the result of the Lagrangian derivation, [Eq. (\[eq:d2tLPT\])]{}.
Numerical evaluation for $\Lambda$CDM {#app:numerics}
=====================================
This section presents the equations used to numerically evaluate the tensor mode contribution for $\Lambda$CDM, i.e. including a cosmological constant in addition to matter and radiation. The numerical results are shown in [Figs. \[fig:FDsigma\]–\[fig:d2tLambda\]]{}. We have (assuming flatness) H\^2 = H\_0\^2 =: H\_0\^2 E\^2(a), \[eq:HLCDM\] where $a=1$ today and \_[0]{} + [\_[m0]{}]{}+ \_[r 0]{} = 1. The $\Omega_{X 0}$ refer to fractions of the critical density today. The relation between $a$ and the dimensionless conformal time $y = \tilde H_0 \tau$ needs to be solved numerically through y(a) = \_0\^a . However, deep in radiation and matter domination, we use the analytical result obtained when neglecting the $\Lambda$ term in [Eq. (\[eq:HLCDM\])]{}: y(a) =& 2[\_[m0]{}]{}\^[-1/2]{} ( - ) .
### Linear solutions
Transforming the tensor mode equation to $y$, we obtain D\_h”(y) + 2 f\_H(y) D\_h’(y) + D\_h(y) = 0 \[eq:Dheom\]\
f\_H(y) = a(y) E(a(y))[\
]{}D\_h(0) = 1;D\_h’(0) = 0, where for the remainder of this section primes stand for derivatives with respect to $y$. The Poisson equation is now \^2 = 32 [\_[m0]{}]{}H\_0\^2 a\^[-1]{}(y) (y) . We define a linear growth factor $D_{1,s}$ for scalar perturbations through \_[1,s]{}(,) = D\_[1,s]{}() \_[1,s]{}(,\_0), which satisfies D\_[1,s]{}”(y) + f\_H(y) D\_[1,s]{}’(y) - 32 [\_[m0]{}]{}a\^[-1]{}(y) D\_[1,s]{}(y) = 0 with boundary conditions D\_[1,s]{}(0) = 0; D\_[1,s]{}(y\_0) = 1. In order to enforce these boundary conditions, we integrate the growth equation from some $y_{\rm min}$ deep in radiation domination, with initial conditions D\_[1,s]{}(y\_[min]{}) = C; D\_[1,s]{}’(y\_[min]{}) = , and adjust $C$ so that $D_{1,s}(y_0) = 1$ where $a(y_0)=1$.
### Second order solution
In order to solve for ${\sigma}_{2,t}$, we transform [Eq. (\[eq:s2t1\])]{} from $\tau$ to $y$, yielding \_[2,t]{}”(,y) + f\_H(y) \_[2,t]{}’(,y) - 32 [\_[m0]{}]{}a\^[-1]{}(y) \_[2,t]{}(,y) =& (,y) [\
]{}(,y) =& -12 a\^[-1]{} D\_[1,s]{}(y) \_0() [\
]{}\_0() =& ( \_[1,s]{}(,y\_0)) h\_[ij]{}\^[(0)]{} . \[eq:s2eom\] As before, we start integrating at $y_{\rm min}$ where $k_L \tau_{\rm min} = k_L/\tilde H_0 \:y_{\rm min}$ is sufficiently small so that the right-hand side can be set to zero. The initial conditions for ${\sigma}_{2,t}$ are then \_[2,t]{}(y\_[min]{}) = 0;\_[2,t]{}’(y\_[min]{}) = 0. We then solve [Eq. (\[eq:s2eom\])]{} numerically using a fourth-order Runge-Kutta scheme with adaptive step size.
Radiation domination {#app:RD}
====================
We now consider the case of dark matter during pure radiation domination (RD) in the Lagrangian treatment of [Sec. \[sec:LPT\]]{}. In addition to clarifying the reason for the behavior of the tensor-scalar coupling shown in [Fig. \[fig:d2t\]]{}, these results are also used for the initial conditions of the numerical integration described in [App. \[app:numerics\]]{}. We have [\_m]{}= 0; (a) = H\_0 a\^[-1]{}; = H\_0\^[-1]{} a; = \^[-1]{}, \[eq:RD\] where $\tilde H_0$ is the Hubble constant at some reference time during RD where $a(\tau_0)=1$.
Linear solutions
----------------
Since ${\Omega_m}=0$, the $q$-divergence of the linearized scalar EOM becomes \_[1,s]{}”(,) + 1 \_[1,s]{}’(,) =& 0. The growing mode corresponds to ${\sigma}_{1,s} \propto \ln \tau$, while the decaying mode is ${\sigma}_{1,s} = $ const. In the following, we will again assume that the scalar perturbations have settled in the growing mode by the time when the tidal field $t_{ij}$ becomes relevant. Since the growth is only logarithmic in $\tau$ rather than polynomial as in matter domination, this is a much stronger restriction.
We normalize the density perturbation ${\delta}_{1,s}({{\mathbf{q}}},\tau)$ to its value at horizon crossing ${\mathcal{H}}_* = 1/\tau_* = k_S$: \_[1,s]{}(,) = (k\_S ) \_[1,s]{}\^[H]{}()(k\_S1), which again is only valid if $k_S\tau \gg 1$. We then have s\_[1,s]{}\^i(,) = - (k\_S ) \_[1,s]{}\^[H]{}(). For the tidal field, [Eqs. (\[eq:sLt\])–(\[eq:dLt\])]{} are valid for a general expansion history.
Second-order solution
---------------------
Again, since ${\Omega_m}=0$, the source terms of the Poisson equation vanish, and [Eq. (\[eq:s2t1\])]{} simplifies to \_[2,t]{}” + \_[2,t]{}’ = - M\_[1,s]{}\^[ij]{} t\_[ij]{}, \[eq:s2t1RD\] where on the r.h.s. all contributions are evaluated at ${{\mathbf{q}}}$ and $\tau$. In reality, ${\Omega_m}$ is of course never exactly zero; thus, our results assume that $t^i_{\ i}$ is not dramatically enhanced so that the prefactor of ${\Omega_m}$ sufficiently suppresses the first source term in [Eq. (\[eq:s2t1\])]{} over the second one. We obtain \_[2,t]{}”(,) + 1 \_[2,t]{}’(,) =& (,) [\
]{}(,) = (k\_S ) T() \_0() =& (k\_S ) T() ( \_[1,s]{}\^[H]{}()) t\_[ij]{}\^[(0)]{} . The growing and decaying modes are again ${\sigma}_{2,t} \propto \ln \tau$ and ${\sigma}_{2,t} \propto $ const, respectively. The solution for this equation with the appropriate boundary condtions is given by \_[2,t]{}(,) = \_0\^d’ ’ () (,’). We then obtain using integration by parts (and $F(\tau\to0) = 0$) to obtain \_[2,t]{}(,) =& D\_[1]{}() \_0()\
D\_[1]{}() =& \_[1/k\_S]{}\^d’ ’ () (k\_S ’) T(’) [\
]{}=& F() (k\_S) - 2 \_[1/k\_S]{}\^ F(’).Thus, \_[2,t]{}(,) = t\_[ij]{}\^[(0)]{} ( \_[1,s]{}(,)) , \[eq:s2tRD\] where $F$ is defined as before \[[Eq. (\[eq:Fdef\])]{}\] and we have introduced V\^[RD]{}() =& 2 \[k\_S\]\^[-1]{}\_[1/k\_S]{}\^ F(’) . \[eq:VRDdef\] As in [Sec. \[sec:LPT\]]{}, the total contribution to the Eulerian density induced by the external tidal field is then given by [Eq. (\[eq:dtLPT1\])]{}, which yields for the second order part \_[2,t]{}(, ) =& t\_[ij]{}\^[(0)]{} \_[1,s]{}(, ) [\
]{}=& t\_[ij]{}\^[(0)]{}\_[1,s]{}(, ). \[eq:d2tRD\] This has very similar structure to the result in matter domination \[[Eq. (\[eq:d2tLPT\])]{}\], the key difference being that the coefficient of the tidal term $\partial^i\partial^j/\nabla^2 {\delta}_{1,s}$ only involves the function $V^{\rm RD}$ rather than $F$.
Tensor modes {#app:tensorRD}
------------
As in [Sec. \[sec:tensor\]]{}, we have $F(\tau) = \frac12 \left[1 - D_h(\tau)\right]$, where in RD D\_h() = . The function $V^{\rm RD}(\tau)$ becomes V\^[RD]{}() =& \[k\_S\]\^[-1]{}\_[k\_L/k\_S]{}\^[k\_L]{} x (1-x) [\
]{}=& \[k\_S\]\^[-1]{} \_[k\_L/k\_S]{}\^[k\_L ]{}. In the $k_L\tau\to\infty$ limit, $F(\tau) \to 1/2$ just as in matter domination. On the other hand, $V^{\rm RD}(\tau)$ becomes V\^[RD]{}() & \[k\_S\]\^[-1]{} = . For $k_L \tau \gg k_S/k_L$, $V^{\rm RD}$ logarithmically approaches 1 from below. [Eq. (\[eq:d2tRD\])]{} becomes in this limit \_[2,t]{}(,) =& h\_[ij]{}\^[(0)]{}\_[1,s]{}(, ). \[eq:d2tfossilRD\] Thus, there is a non-zero second-order density for $k_L\tau\gg 1$ during radiation domination as well. Moreover, the only difference to the corresponding result for matter domination \[[Eq. (\[eq:d2tfossil\])]{}\] is the numerical coefficient of the first term ($2/5$ in MD, order 1 in RD depending on $k_L\tau$), and the fact that it evolves logarithmically with $\tau$.
Projection effects {#app:proj}
==================
We now derive the projection effect contribution to the observed local small-scale correlation function $\xi_{\delta}({{\mathbf{r}}},\tau | h)$. Here, ${{\mathbf{r}}}$ and $\tau$ are the observationally inferred comoving separation and conformal time, respectively. We make no particular assumptions about the nature of the tracers which are used to measure the small-scale correlation function. We will only consider the tensor (or vector) case here, so that $h_{00} = 0 = h_{0i}$.
We begin with Eq. (45) in [@conformalfermi], which gives (, |h) =& \_F(;) \[eq:xiproj1\] where $x$ is the inferred spacetime position of the center of the region in which the correlation function is measured, and $\xi_F({{\mathbf{r}}},\tau)$ is the correlation function in the local [$\overline{\mathrm{FNC}}$]{} frame. Further, $a_{ij}$ is the distortion of the *standard ruler* defined by the correlation function, $\T$ is the shift, in terms of the logarithm of the scale factor, between constant-proper-time and constant-observed-redshift surfaces, and $c$ is the perturbation to the observed number density of the tracer induced by the tensor mode. We now consider each of these ingredients in turn. Note that each term in [Eq. (\[eq:xiproj1\])]{} is gauge-invariant and in principle independently observable.
First, the ruler distortion is most naturally decomposed as a\_[ij]{} =& \_i \_j + \_[(i]{} ¶\_[j)k]{} \^k + ¶\_[ik]{} ¶\_[jl]{} \^[kl]{},\[eq:rulerpert\] where $\P^{ij} = {\delta}^{ij} - {\hat{n}}^i{\hat{n}}^j$ is the projection operator perpendicular to the line of sight ${\hat{n}}^i$. $\C,\,\B_i,$ and $\A_{ij}$ are the gauge-invariant ruler perturbations defined in [@stdruler], which, when specialized to a metric with $h_{00} = 0 = h_{0i}$ are given by =& - a - 12 h\_- \_ x\_[\
]{}\_i =& -¶\_i\^[ j]{} h\_[jk]{} \^k - \^k \_[i]{} x\_k - \_ x\_[i]{} [\
]{}\_[ij]{} =& - a ¶\_[ij]{} - 12 ¶\_i\^[ k]{}¶\_j\^[ l]{} h\_[kl]{} - 12 (¶\_[jk]{} \_[i]{} + ¶\_[ik]{} \_[j]{}) x\^k. \[eq:coeff\] The displacements ${\Delta}x^i,\,{\Delta}\ln a$ are also given in [@stdruler] and again specializing to purely spatial metric perturbations x\_=& - 12\_0\^ d h\_- a \[eq:Dxpar\]\
x\_\^i =& 12 ¶\^[ij]{} (h\_[jk]{})\_o \^k - \_0\^ d. \[eq:Dxperp\] The perturbation to the scale factor at emission is given by a =& 12 \_0\^ d h\_’ . \[eq:Dlna\] In [Eqs. (\[eq:coeff\])–(\[eq:Dlna\])]{}, metric perturbations outside integrals are evaluated at the source, unless they are marked by a subscript $o$, in which case they are evaluated at the observer. Metric perturbations inside integrals are evaluated on the past lightcone in the background, i.e. at x\^i = \^i ;x\^0 = \_o - , where $\tau_o$ is the conformal time at observation. Primes denote derivatives with respect to $\tau$, and ${\tilde{\chi}}\equiv {\bar{\chi}}({\tilde{z}})$ where ${\bar{\chi}}(z)$ is the comoving distance-redshift relation in the background and ${\tilde{z}}$ is the observed redshift. Further, \_\^i = ¶\^[ij]{} \_j ; h\_ = h\_[ij]{} \^i \^j. The decomposition given by [Eq. (\[eq:rulerpert\])]{} allows us to easily derive the distortions along and perpendicular to the line of sight in 3D space. However, writing the expressions in Cartesian form leads to more compact expressions. Using that, for an arbitrary symmetric tensor $a_{ij}$, a\_[ij]{} = a\_\_i \_j + 2 \_[(i]{} ¶\_[j)]{}\^[ k]{} \^l a\_[kl]{} + ¶\_i\^[ k]{} ¶\_j\^[ l]{} a\_[kl]{}, [Eq. (\[eq:rulerpert\])]{} becomes a\_[ij]{} =& -a \_[ij]{} - 12 h\_[ij]{} - \_[(i]{} x\_[j)]{} = -12 \_0\^ d h\_’ \_[ij]{} - 12 h\_[ij]{} - \_[(i]{} x\_[j)]{}. \[eq:aij3D\] The first equality can also be read off directly from Eq. (30) in [@stdruler], when setting $v^i=0$. Note however, that as discussed in [@gaugePk; @stdruler], the derivative along the line of sight is really a derivative with respect to observed redshift, i.e. along the past light cone: \^i \_i x\^k x\^k = ()\^[-1]{} x\^k. This subtlety is somewhat glossed over in the notation [Eq. (\[eq:aij3D\])]{}, which does not make explicit the fundamental difference between line-of-sight and transverse directions.
We further need $\T$, which was derived in [@Tpaper] and is given in our case by =& a = 12\_0\^ d h\_’.
Finally, the observed fractional number density perturbation of tracers induced by tensor modes was derived in [@GWpaper] (note that ${\delta}z$ in that paper is equal to ${\Delta}\ln a$ defined above). It is given by c =& b\_e a + \_i x\^i + \_T , \[eq:ct\] where the magnification induced by a tensor mode is \_T = - 2a +h\_- + 2 , \[eq:MT\] and the convergence is & -\_[i]{}x\_\^i = 54 h\_[o]{} -12 h\_ - 12 \_0\^ d - 14 \^2\_\_0\^ d h\_. \[eq:kappa\] Here $\nabla^2_\Omega = {\tilde{\chi}}^2 \nabla_\perp^2$ denotes the Laplacian on the unit 2-sphere. The Jacobian in [Eq. (\[eq:ct\])]{} is then given by \_i x\^i = \_ x\_+ - 2 . \[eq:jacobian\] Note again the subtlety in the Cartesian notation. The number density modulation is governed by two tracer-dependent parameters: the magnification bias parameter $\Q$, given in the simplest case of a sharp flux-limited survey by $\Q = - d\ln \bar{n}_g/d\ln f_{\rm cut}$; and the paramater $b_e$, which quantifies the redshift evolution of the comoving number density of tracers through b\_e |\_ = - (1+)|\_ . \[eq:btdef\]
Putting everything together, the observed local two-point correlation function becomes (, |h) = \_F(;). \[eq:xiproj2\]
[^1]: For vector perturbations, we here assume instantaneous generation at some initial time.
[^2]: The framework of the previous section is valid to study any type of small scale perturbations but in this paper we will focus on short *scalar* perturbations.
[^3]: As long as the region over which we measure small-scale correlations is much smaller than the horizon, which is always the case in practice, the distinction between [$\mathrm{FNC}$]{} and [$\overline{\mathrm{FNC}}$]{} is irrelevant ([Sec. \[sec:FNC\]]{}).
|
---
abstract: 'We report [*Chandra*]{} observations of a double X-ray source in the $z=0.1569$ galaxy [SDSS J171544.05+600835.7]{}. The galaxy was initially identified as a dual AGN candidate based on the double-peaked [ $\,$]{}emission lines, with a line-of-sight velocity separation of 350 km s$^{-1}$, in its Sloan Digital Sky Survey spectrum. We used the Kast Spectrograph at Lick Observatory to obtain two longslit spectra of the galaxy at two different position angles, which reveal that the two AGN emission components have not only a velocity offset, but also a projected spatial offset of 1.9 $h^{-1}_{70}$ kpc on the sky. [*Chandra*]{}/ACIS observations of two X-ray sources with the same spatial offset and orientation as the optical emission suggest the galaxy most likely contains Compton-thick dual AGN, although the observations could also be explained by AGN jets. Deeper X-ray observations that reveal Fe K lines, if present, would distinguish between the two scenarios. The observations of a double X-ray source in [SDSS J171544.05+600835.7 $\,$]{}are a proof of concept for a new, systematic detection method that selects promising dual AGN candidates from ground-based spectroscopy that exhibits both velocity and spatial offsets in the AGN emission features.'
author:
- 'Julia M. Comerford, David Pooley, Brian F. Gerke, and Greg M. Madejski'
title: |
Chandra Observations of a 1.9 kpc Separation Double X-ray Source\
in a Candidate Dual AGN Galaxy at $\MakeLowercase{z}=0.16$
---
[^1]
Introduction {#intro}
============
A wealth of observations have shown that galaxy mergers are common and that nearly all galaxies host a central supermassive black hole (SMBH). Consequently, some galaxies must host two SMBHs as the result of recent mergers. These are known as dual SMBHs for the first $\sim 100$ Myr after the merger when they are at separations $\simgt 1$ kpc [@BE80.1; @MI01.1]. Dual-SMBH systems are an important testing ground for theories of galaxy formation and evolution. For example, simulations predict that quasar feedback in mergers can have extreme effects on star formation [@SP05.1] and that the core-cusp division in nuclear stellar distributions may be caused by the scouring effects of dual SMBHs [@MI02.2; @LA07.1]. A statistical study of dual SMBHs and their host galaxies would thus have important implications for theories of galaxy formation and SMBH growth.
Dual SMBHs are observable when sufficient gas accretes onto them to power dual active galactic nuclei (AGN). While there have been identifications of hundreds of binary quasar pairs at separations $>10$ kpc (e.g., @HE06.1 [@MY07.1; @MY08.1; @GR10.2]), as well as a handful of galaxy pairs where each galaxy hosts an AGN [@BA04.2; @GU05.1; @PI10.1], very few AGN pairs have been observed in the next evolutionary stage where they coexist at kpc-scale separations in the same merger-remnant galaxy. To date, the confirmed dual AGN were identified by radio or X-ray resolution of two AGN with separations of 7 kpc in the $z=0.02$ double radio source 3C 75 at the center of the galaxy cluster Abell 400 [@HU06.1], 4 kpc in the $z=0.05$ ultraluminous infrared galaxy Mrk 463 [@BI08.1], and 0.7 kpc in the $z=0.02$ ultraluminous infrared galaxy NGC 6240 [@KO03.1].
In recent years dual AGN candidates have been selected as galaxies with double-peaked AGN emission lines, first in the DEEP2 Galaxy Redshift Survey [@GE07.2; @CO09.1] and later in the Sloan Digital Sky Survey (SDSS; @WA09.1 [@LI10.1; @SM10.1]). Although a double-peaked line profile is expected for dual AGN, it could also be produced by gas kinematics in a single AGN; follow-up observations are necessary to distinguish between the two scenarios. Follow-up optical longslit spectroscopy, near-infrared imaging, and adaptive optics imaging have added circumstantial evidence that many double-peaked systems may indeed be dual AGN [@FU10.1; @LI10.2; @SH10.1; @GR11.1; @RO11.1], but direct resolution of two separate AGN is required for direct evidence of dual AGN.
![Segments of the two-dimensional Lick/Kast spectra at position angle $32^\circ.9$ east of north (top) and position angle $120^\circ.7$ east of north (middle) and the SDSS spectrum (bottom) for [SDSS J171544.05+600835.7]{}. Each spectrum is shifted to the rest frame of the host galaxy, and centered on the rest wavelength of [ $\,$]{}(dotted vertical line). In the Lick/Kast spectra, night-sky emission features have been subtracted and both vertical axes span $11\farcs7$ (31.8 $h^{-1}_{70}$ kpc at the $z=0.1569$ redshift of the galaxy). The double peaks in [ $\,$]{}in the SDSS spectrum correspond to spatially-offset double emission features in the Lick/Kast spectra, suggesting the presence of dual AGN with a line-of-sight velocity separation of 350 km s$^{-1}$ and a projected spatial separation of 1.9 $h^{-1}_{70}$ kpc (or 0$\farcs$68) on the sky. [*Chandra*]{} observations support the presence of dual AGN.[]{data-label="fig:spectra"}](fig1.eps){width="3.5in"}
Here we present such evidence for dual AGN in the form of [*Chandra*]{}/ACIS observations of two AGN sources separated by 1.9 $h^{-1}_{70}$ kpc in the $z=0.1569$ galaxy [SDSS J171544.05+600835.7]{}. This galaxy was first identified as a dual AGN candidate by its double-peaked Type 2 AGN spectrum in SDSS, where the [ $\,$]{}peaks are separated by 350 km s$^{-1}$ [@LI10.1; @SM10.1]. Our follow-up Lick/Kast longslit spectra of the galaxy show two [ $\,$]{}emission components separated on the sky by 1.9 $h^{-1}_{70}$ kpc, or 0$\farcs$68, an angular separation resolvable by [*Chandra*]{}. The follow-up [*Chandra*]{} observations reveal double X-ray sources with the same spatial separation and orientation on the sky as the two [ $\,$]{}emission components, indicating that [SDSS J171544.05+600835.7 $\,$]{}likely hosts Compton-thick dual AGN. We assume a Hubble constant $H_0 =70 \,$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_m=0.3$, and $\Omega_\Lambda=0.7$ throughout, and all distances are given in physical (not comoving) units.
Observations and Analysis
=========================
SDSS Spectrum
-------------
The SDSS spectrum of [SDSS J171544.05+600835.7 $\,$]{}exhibits several double-peaked AGN emission lines, where one peak is blueshifted and one is redshifted relative to the systemic redshift of the host galaxy and the [ $\,$]{}peaks have a line-of-sight velocity separation of 350 km s$^{-1}$ (@LI10.1 [@SM10.1]; Figure \[fig:spectra\]). We fit two Gaussians each to the continuum-subtracted , , , and [ $\,$]{}line profiles and used the areas under the best-fit Gaussians as estimates of the line fluxes of the redshifted and blueshifted components of each line. This yields line flux ratios of ${\mbox{[\ion{O}{3}] $\lambda$5007}}/{\mbox{H$\beta$}}=4.4 \pm 1.4$ and ${\mbox{[\ion{N}{2}] $\lambda$6584}}/{\mbox{H$\alpha$}}=1.1 \pm 0.2$ for the blueshifted components, and ${\mbox{[\ion{O}{3}] $\lambda$5007}}/{\mbox{H$\beta$}}=8.6 \pm 3.1$ and ${\mbox{[\ion{N}{2}] $\lambda$6584}}/{\mbox{H$\alpha$}}=0.5 \pm 0.1$ for the redshifted components, where the uncertainties are derived from propagating the errors in the parameters of the best-fit Gaussians. These line flux ratios clearly indicate that both the redshifted and the blueshifted emission components are produced by AGN [@BA81.1; @KE06.1].
Lick/Kast Longslit Spectra {#lick}
--------------------------
The SDSS fiber spectrum carries no spatial information about the source or sources of the emission producing the double-peaked lines. Because this spatial information can help distinguish whether the emission is produced by two spatially-offset AGN or gas kinematics from a single AGN, we obtained follow-up slit spectroscopy of the galaxy.
We used the Kast Spectrograph on the Lick 3-m telescope to obtain spectra of the galaxy with a 1200 lines mm$^{-1}$ grating on UT 2009 August 17. To determine the orientation of the AGN emission components on the plane of the sky, we observed the galaxy at two different position angles, $32^\circ.9$ east of north and $120^\circ.7$ east of north. At each position angle we took three 1200 s exposures, and each spectrum spans the wavelength range 4790 – 6200 Å. The data were reduced following standard procedures in IRAF and IDL.
The spectra at both position angles reveal two distinct emission components in , , and [ $\,$]{}separated in both velocity and spatial position (Figure \[fig:spectra\]). The [ $\,$]{}emission has the highest signal-to-noise ratio, which enables the most precise separation measurements. For each spectrum we determine the projected spatial separation between the two [ $\,$]{}emission features by measuring the spatial centroid of each emission component individually, then we combine the projected separation measurements at both position angles to determine the spatial separation and position angle on the sky (for details see Comerford et al., in prep.). We find the two [ $\,$]{}emission components have a projected separation on the sky of $1.86 \pm 0.41$ $h^{-1}_{70}$ kpc, or 0$\farcs$684 $\pm$ 0$\farcs$151, and the position angle on the sky is $145^\circ.6$ east of north.
{width="32.00000%"}0.02{width="32.00000%"}0.02{width="32.00000%"}
Chandra Observations {#chandra}
--------------------
[SDSS J171544.05+600835.7 $\,$]{}was observed with the [*Chandra X-ray Observatory*]{} on 2011 March 17 beginning at 16:22 UT with an exposure time of 29669 s. The observation was taken with the telescope aimpoint on the Advanced CCD Imaging Spectrometer (ACIS) S3 chip in “timed exposure” mode and telemetered to the ground in “faint” mode. The observation was reduced using the latest [*Chandra*]{} software (CIAO4.3) and the most recent set of calibration files (CALDB4.4.1). The data were reprocessed with the “chandra\_repro” script using the subpixel event repositioning algorithm of [@LI04.4]. Intervals of strong background flaring were searched for, but none were found.
We made a sky image of the field of [SDSS J171544.05+600835.7 $\,$]{}at a resolution of 00492 per pixel with events in the 0.3–8.0 keV energy range, and we fit two-dimensional models to that image. All fits were performed in Sherpa [@FR01.2] using modified @CA79.1 statistics (“cstat” in Sherpa) and the @NE65.1 optimization method (“simplex” in Sherpa). A family of fits was performed, using a grid of parameter starting points to ensure a proper sampling of the multidimensional fit space.
We fit a two-component source model with a fixed background (based on a source-free region near SDSSJ171544.05+600835.7). The source model was a $\beta$ profile, which is a two-dimensional Lorentzian with a varying power law of the form $I(r) = A(1+(r/r_0)^2)^{-\alpha}$ and is a good match to the [*Chandra*]{} PSF. Based on other work [@PO09.1], the power law index $\alpha$ was tied to the $r_0$ parameter, and both components were required to have the same $r_0$. Their positions and amplitudes were unconstrained. The best fit amplitudes are 0.49[$^{_{+0.23}}_{^{-0.16}}$]{} and 0.21[$^{_{+0.12}}_{^{-0.08}}$]{} counts pixel$^{-1}$ for the northern and southern sources, respectively, and where the southern source is detected at 3.7$\sigma$.
The two X-ray components are separated by $1.85 \pm 0.22$ $h^{-1}_{70}$ kpc, or $0\farcs68\pm0\farcs08$, at a position angle of $147^\circ \pm 9^\circ$ east of north, where both the separation and the position angle are consistent with those measured for the two [ $\,$]{}emission components in the Lick/Kast longslit spectra (§ \[lick\]). The best fit model is shown in Figure \[fig:xrayimage\].
To estimate the fluxes of the two components, we cannot reliably extract spectra of and make response files for the two separately. We therefore extracted a spectrum of both sources together and use the results of our two-dimensional image fits to assign appropriate fractions of the total flux to each component. We fit the unbinned spectrum in Sherpa using cstat statistics and the simplex method. The spectral model was a simple absorbed power law with the column density constrained to be at least the Galactic value of $n_H={\ensuremath{{2.6}\!\times\!10^{20}}}~{\mbox{cm$^{-2}$}}$ [@DI90.1]. No additional absorption was preferred in the fit, and the best fit power law index was $1.9\pm0.2$. The total unabsorbed 0.5–8 keV flux is [${1.5\pm0.4}\!\times\!10^{-14}$]{} [$\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}$]{}. The uncertainty was calculated using the “sample\_energy\_flux” tool in Sherpa, which takes into account uncertainties in all model parameters. Using the results of our two-dimensional image fit, the northern component has a best-fit flux of $F_{0.5-8}={\ensuremath{{1.1}\!\times\!10^{-14}}}~{\ensuremath{\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}}}$, and the southern component has $F_{0.5-8}={\ensuremath{{4.4}\!\times\!10^{-15}}}~{\ensuremath{\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}}}$.
The 2–10 keV fluxes are $F_{2-10}=6.4 \pm 3.1 \times 10^{-15}~{\ensuremath{\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}}}$ for the northern component and $F_{2-10}=2.7 \pm 1.5 \times 10^{-15}~{\ensuremath{\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}}}$ for the southern component, which we compare to the 2–10 keV fluxes predicted from the [ $\,$]{}fluxes using the scaling relation for Type 2 AGN in [@HE05.1]. The predicted 2–10 keV fluxes are $3.3 \pm 7.9 \times 10^{-14}~{\ensuremath{\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}}}$ and $2.1 \pm 5.1 \times 10^{-14}~{\ensuremath{\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}}}$ for the redshifted and blueshifted components of , respectively. The measured 2–10 keV fluxes are hence a factor of several lower than but within the broad uncertainties of the predictions from [@HE05.1].
Although we cannot fit separate spectra for the two components, we can extract the counts in small regions centered on them and form hardness ratios, defined as $\mathrm{HR}=(H-S)/(H+S)$ where $H$ is the number of counts in the 2–8 keV range and $S$ is the number of counts in the 0.5–2 keV range. Counts were extracted from 025 radius regions centered on each source, yielding 20 counts from the northern source with $\mathrm{HR}=-0.57{\ensuremath{^{_{+0.15}}_{^{-0.19}}}}$ and 10 counts from the southern source with $\mathrm{HR}=-0.37{\ensuremath{^{_{+0.24}}_{^{-0.30}}}}$. Uncertainties on the hardness ratios were calculated using the Bayesian Estimation of Hardness Ratios package [@PA06.1]. Judging from the measured hardness ratios, neither source is as hard as would be expected for moderately absorbed (but not Compton-thick) AGN, although the signal-to-noise ratio is very modest.
![SDSS image of [SDSS J171544.05+600835.7]{}. Unlike the dual AGN host galaxies known to date, this galaxy has no indication of extreme star formation or an unusual morphology. This image reveals no tidal features or companions, and no structure on scales $\simgt0\farcs1$ is reported from $K'$-band adaptive optics imaging [@FU10.1].[]{data-label="fig:sdss"}](fig3.eps){width="2.5in"}
Keck/LGSAO and SDSS Imaging
---------------------------
If [SDSS J171544.05+600835.7 $\,$]{}has dual AGN separated by 1.9 $h^{-1}_{70}$ kpc that are the result of a galaxy merger, we might expect the two AGN to be coincident with two stellar components that are the remnants of the progenitor galaxies in the merger. In fact, [SDSS J171544.05+600835.7 $\,$]{}was one of 50 SDSS galaxies with double-peaked [ $\,$]{}lines that was targeted for Keck II laser guide-star adaptive-optics (LGSAO) observations in [@FU10.1], which found that 16 of these galaxies exhibited double stellar components with separations 0.6 – 12 $h^{-1}_{70}$ kpc (or 0.2 – 3$^{\prime\prime}$), suggestive of merging systems. While [@FU10.1] do not show the image of [SDSS J171544.05+600835.7]{}, they report that the Keck/LGSAO $K'$-band image reveals only a single component with $K'=14.8$ and classify the galaxy as “isolated". The PSF FWHMs for their observations were $0\farcs065$ to $0\farcs130$, suggesting that [SDSS J171544.05+600835.7 $\,$]{}may have no substructure at $\simgt0\farcs1$ visible in these $K'$-band observations.
Based on its SDSS photometry, [SDSS J171544.05+600835.7 $\,$]{}has a rest-frame $u-r$ color of 2.62 and an absolute $r$-band magnitude of -22.0, placing it on the red sequence of SDSS galaxies [@BA04.3]. The SDSS image of the galaxy shows no signatures of galaxy interaction (Figure \[fig:sdss\]).
Interpretations
===============
Here we explore the physical mechanisms that could explain the observations of double AGN emission components and X-ray sources in [SDSS J171544.05+600835.7]{}.
The sources’ high X-ray luminosities lead us to consider the possibility that they are ultra-luminous X-ray sources (ULXs), which are variable off-nuclear X-ray sources. Some ULXs are also candidate intermediate mass black holes (e.g., @KA03.4 [@FA09.1]). However, estimates of the black hole mass from the optical spectra place [SDSS J171544.05+600835.7 $\,$]{}in the supermassive black hole regime (Fu et al. 2010). Further, the measured [ $\,$]{}fluxes and luminosities are well above those typically measured in ULXs (e.g., @PO10.1 [@CS11.1]).
Another possibility is that the galaxy hosts dual AGN that are the result of a triple SMBH interaction, where a gravitational slingshot effect [@SA74.1] ejected the least massive SMBH (corresponding to the southern AGN) while the remaining two SMBHs merged (producing the northern AGN). The remaining stellar component associated with the southern AGN could be too faint to appear in the Keck/LGSAO observations. However, the optical observations suggest that both AGN have associated narrow-line regions and it is unclear how the ejected SMBH would maintain its narrow-line region.
We note also that if there is only a single source visible in the Keck/LGSAO image, a second stellar component could be either obscured or too faint to be within the detection threshold of the Keck/LGSAO observation. If there are two AGN with associated stellar components, the stellar component accompanying the southern AGN is more likely to be obscured since it is harder and consequently more absorbed than the northern AGN. If the southern AGN is obscured and the northern AGN is not, the overall extinction for the system could be skewed towards the low value measured in § \[chandra\].
Another explanation for our observations is that the galaxy hosts AGN jets that produce both [ $\,$]{}and X-rays from a combination of photoionization from the AGN and collisional ionization from the jets (e.g., @KR09.1 [@BI10.1]). For this scenario to explain our observations, either the AGN is completely obscured and only the jets are visible or one of the sources in our observations is the AGN while the other is the foreground jet and the background jet is obscured. Both scenarios rely on significant obscuration in part of the galaxy, but this could be consistent with the low column density we measured if the rest of the galaxy is relatively unobscured. However, the jet scenario typically produces a bright core and a much fainter jet, whereas our observations show two sources that differ by only a factor of 2 in 2–10 keV luminosities.
The observations may also be explained by dual AGN that are Compton thick. The X-ray spectra alone – if confirmed – exclude Compton-intermediate AGN, where the absorption would correspond to a column of $\sim 10^{22} - 10^{24}$ cm$^{-2}$, as there the observed X-ray flux would be dominated by partially (photoelectrically) absorbed continuum that is detectable as a hard X-ray spectrum. Instead the data suggest that both sources are either unabsorbed or the absorption is severe, essentially resulting in a Compton-thick spectrum. The optical spectra seem to exclude classical, unabsorbed Type 1 AGN, so we are left with a Compton-thick scenario, with $\tau_{\rm Thomson}$ of at least a few (column $> 10^{25}$ cm$^{-2}$). There, not only the primary nuclear soft X-ray flux is photoelectrically absorbed, but even the hard X-rays are suppressed by Compton opacity, and we detect only primary soft X-rays scattered back to our line of sight, as is the case in, e.g., NGC 1068 and other Compton-thick AGN. The measured 2–10 keV luminosities for the northern and southern components are within the range of values measured for Compton-thick AGN (e.g., @LE06.1). The Compton-thick scenario is also consistent with the low column density of the galaxy, since the column density measurement presupposed that the source was not Compton thick.
We conclude that SDSS J171544.05+600835.7 most likely hosts Compton-thick dual AGN, because it is the scenario that is most consistent with the existing data. AGN jets might also explain the observations, and deeper X-ray observations could distinguish between these two possibilities. These more sensitive X-ray measurements would enable a test of the Compton-thick scenario, since better spectral measurements in the soft X-ray band could provide a possible measurement of the Fe K line, which is seen in heavily absorbed AGN (for a discussion, see, e.g., @LE06.1). Sensitive hard X-ray measurements would provide much better constraints on the absorbing column. [*Hubble Space Telescope*]{} narrow-band imaging of the [ $\,$]{}emission could also show whether it has the biconical morphology expected for AGN jets.
Conclusions
===========
We report observations of a double X-ray source with 1.9 $h^{-1}_{70}$ kpc, or 0$\farcs$68, projected spatial separation in the $z=0.1569$ candidate dual AGN galaxy [SDSS J171544.05+600835.7]{}. This Seyfert 2 galaxy exhibits double-peaked [ $\,$]{}emission lines with 350 km s$^{-1}$ line-of-sight velocity separation in its SDSS spectrum, and our follow-up Lick/Kast longslit spectra show two 1.9 $h^{-1}_{70}$ kpc separation [ $\,$]{}emission components. While the velocity and spatial offsets provide circumstantial evidence for dual AGN, these features could also be produced by gas kinematics from a single AGN. The [*Chandra*]{} observations bolster the evidence for dual AGN, by revealing two X-ray components suggestive of Compton-thick AGN with the same spatial separation and orientation as the two sources of optical [ $\,$]{}emission.
To date, dual AGN have typically been identified serendipitously because of the interesting characteristics of their host galaxies. These host galaxies include ultraluminous infrared galaxies [@KO03.1; @BI08.1], a double radio source at the center of a galaxy cluster [@HU06.1], and a host galaxy with double bright nuclei and a tidal tail [@CO09.3]. [SDSS J171544.05+600835.7 $\,$]{}is unlike these systems because there is nothing particularly noteworthy about the galaxy, which is a seemingly ordinary red sequence galaxy without tidal features visible in SDSS imaging or substructure reported in adaptive optics imaging. Our observations suggest that dual AGN may be more ubiquitous and not limited to only galaxies with extreme star formation or unusual morphologies.
We have introduced a systematic, observational method for selecting promising dual AGN candidates, which have until now have only been identified through serendipitous discoveries of individual systems. The method consists of three steps: 1) select dual AGN candidates as objects whose spectra exhibit double-peaked AGN emission lines in SDSS or other spectroscopic surveys of galaxies; 2) conduct follow-up longslit spectroscopy of the dual AGN candidates; 3) if the follow-up longslit spectra reveal an object has two spatially-distinct AGN emission components, use follow-up X-ray or radio observations to identify whether the object is a dual AGN. [SDSS J171544.05+600835.7 $\,$]{}is the first object for which this technique has been demonstrated, and our observations show it most likely hosts dual AGN; deeper X-ray observations would provide the definitive evidence. Future observations will determine the general applicability of this systematic method for selecting dual AGN.
J.M.C. acknowledges insightful discussions with Jenny Greene, as well as support from a W.J. McDonald Postdoctoral Fellowship. The Texas Cosmology Center is supported by the College of Natural Sciences and the Department of Astronomy at the University of Texas at Austin and the McDonald Observatory. B.F.G. and G.M.M. were supported by the U.S. Department of Energy under contract number DE-AC02-76SF00515.
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[^1]: $^3$W.J. McDonald Postdoctoral Fellow
|
---
abstract: 'We construct a classical action for a system of $N$ point-like sources which carry SU(2) non-Abelian charges coupled to non-Abelian Chern-Simons gauge fields, and develop a quantum mechanics for them. Adopting the coherent state quantization and solving the Gauss’ constraint in an appropriately chosen gauge, we obtain a quantum mechanical Hamiltonian given in terms of the Knizhnik-Zamolodchikov connection. Then we study the non-Abelian Aharonov-Bohm effect, employing the obtained Hamiltonian for two-particle sector. An explicit evaluation of the differential cross section for the non-Abelian Aharonov-Bohm scattering is given.'
author:
- 'Taejin Lee[@tlee]'
- 'Phillial Oh[@poh]'
---
Non-Abelian Chern-Simons Quantum Mechanics
and
Non-Abelian Aharonov-Bohm Effect
Department of Physics, Kangwon National University, Chuncheon 200-701, KOREA
Department of Physics, Sung Kyun Kwan University, Suwon 440-746, KOREA
Introduction {#sec:intro}
============
Since Aharonov and Bohm discussed in their celebrated paper [@ahbo] the significance of the phase that arises from the charge-flux interaction, now known as the Aharonov-Bohm effect, it has been one of the most important subjects in both experimental and theoretical physics [@peshkin]. The subject spreaded diverse branches and recently has developed into the anyon physics [@anyon] which has wide range of applications, including the fractional quantum Hall effect [@frac] and the high-$T_c$ super-conductivity [@high].
In this paper, we discuss in some detail the latest development in this direction, that is, the non-Abelian Chern-Simons quantum mechanics [@lo1; @lo2] and the non-Abelian Aharonov-Bohm effect [@wilwu; @ver]. Recently we proposed a classical action for a system of the non-Abelian Chern-Simons (NACS) particles and developed a quantum mechanical description for them. The NACS particles are point-like sources which carry non-Abelian charges and interact with each other through the non-Abelian Chern-Simons term [@des]. Their interaction is essentially the non-Abelian Aharonov-Bohm effect, just as the interaction between anyons is the Aharonov-Bohm effect.
The non-Abelian Aharonov-Bohm effect, however, is not a newly raised issue: It was considered long ago by Wu and Yang [@wuyang] and its details were studied in a test-particle framework by Horváthy [@hor]. Wu and Yang proposed a gedanken experiment to test the existence of the $SU(2)$ isotopic spin gauge field: They consider a scattering of the beam of protons, neutrons or their mixture around a cylinder where an isotopic magnetic flux is confined. A rotating cylinder made of heavy elements with a neutron excess was suggested as a source of the non-Abelian magnetic flux. The existence of the isotopic $SU(2)$ gauge field would be manifested through the non-integrable phase factor, upon which the outcome of experiment depends.
Recently it has been in the limelight again as it becomes clear that the NACS particle and the non-Abelian Aharonov-Bohm effect might be realized in various physical phenomena such as the fractional quantum Hall effect, cosmic strings [@cosmic], and the gravitational scattering in (2+1) dimensions [@3dgr].
The NACS particles are in a sense non-Abelian generalization of anyons. Their physical properties are similar to those of anyons, but can be certainly distinguished from. While anyons acquire fractional spins and satisfy exotic statistics, the NACS particles acquire fractional but rational spins and exhibit yet more generalized braid (or non-Abelian) statistics [@froh]. In this respect, it is quite interesting to apply the NACS particles to those physical phenomena, in some generalized context, where anyons play important roles, and to scrutinize the outcome. Such attempts have been made in refs.[@wen; @moo; @balfra] where the possibility for the NACS particles to be realized in the fractional quantum Hall effect, the paradigm of anyons, is discussed.
The NACS particles may manifest their existence also in somewhat different circumstances. Such examples include the vortices in (2+1) dimensions and the cosmic strings in (3+1) dimensions which are formed when some gauge group $G$ is broken down via Higgs mechanism to some discrete non-Abelian subgroup $H$ [@discrete]. Since interactions between these objects are characterized by the holonomies, equivalents of Wu and Yang’s non-integrable phase factors, associated with the windings around themselves, their interactions are effectively non-Abelian Aharonov-Bohm effect. The point-like sources which gravitationally interact with each other in (2+1) dimensions also belong to the examples. Since the (2+1) dimensional gravity is equivalent to the non-Abelian Chern-Simons theory of group $ISO(2,1)$ [@wittgr], they are NACS particles with the internal symmetry group of $ISO(2,1)$. Accordingly the (2+1) dimensional gravity can be also discribed in the framework of the non-Abelian Aharonov-Bohm effect [@3dgr].
The rest of the paper is organized as follows. In Sec. II we discuss how to construct a classical action describing non-Abelian Chern-Simons particles and analyze its constraint algebra. In Sec. III we apply the coherent state quantization to the Chern-Simons gauge fields and introduce a gauge fixing condition, called holomorphic gauge, where the Gauss’ constraints are explicitly solved. Then in Sec. IV we obtain a quantum mechanical description of the NACS particles and show that they obey braid statistics. Sec. V is devoted to the two body problem in non-Abelian Chern-Simons quantum mechanics. In Sec. VI we evaluate the differential cross section of the non-Abelian Aharonov-Bohm scattering. In Sec VII we conclude the paper with a brief summary and a discussion on applications of the non-Abelian Chern-Simons quantum mechanics.
Non-Abelian Chern-Simons Particles {#sec:nacsp}
==================================
A classical description of point-like sources with non-Abelian (isospin) charges can be given in terms of their spatial coordinates and the isospin vectors which transform under the adjoint representation of the internal symmetry group. Endowing them with the corresponding non-Abelian magnetic fluxes to make them the NACS particles is done by introducing the non-Abelian Chern-Simons term and minimally coupling their isospin charges with the Chern-Simons gauge fields. To be specific and to avoid unnecessary burden, we take the internal symmetry group to be $SU(2)$ throughout the paper. (For discussions on the NACS particles with $SU(N)$, $N\ge 2$ isospin charges, consult ref. [@lo2].)
Although there are several equivalent ways [@bal78; @bal90; @alek] to define the isospin degrees of freedom, we find it convenient to define the isospin vectors directly on the reduced phase space $S^2$ for the $SU(2)$ internal symmetry group as follows [@oh] $$Q^1_\alpha= J_\alpha \sin \theta_\alpha \cos\phi_\alpha,\quad Q^2_\alpha=
J_\alpha\sin \theta_\alpha \sin \phi_\alpha,\quad Q^3_\alpha =
J_\alpha\cos\theta_\alpha \label{iso}$$ where $\theta_\alpha,\, \phi_\alpha$ are the coordinates of the internal $S^2$ and $J_\alpha$ is a constant. Denoting the spatial coordinates of the particles by ${\bf q}_\alpha$, $\alpha=1,2,\cdots,N$, we can construct the classical Lagrangean as $$L = \sum_\alpha\left(-{1 \over 2} m_\alpha \dot{\bf
q}_\alpha^2 +J_\alpha \cos \theta_\alpha \dot{\phi}_\alpha\right)
-\kappa\int
d^2 {\bf x} \,\epsilon^{\mu\nu\lambda} {\rm tr}\left(A_\mu \partial_\nu
A_\lambda +{2\over 3} A_\mu A_\nu A_\lambda\right)$$ $$+\int d^2{\bf x}\sum_\alpha \left(A^a_i(t,{\bf x}) \dot
q^i_\alpha +A^a_0(t, {\bf x})\right) Q^a_\alpha \delta ({\bf
x}-{\bf q}_\alpha)\label{lag}$$ Here $4\pi\kappa = {\rm integer}$, $A_\mu=A_\mu^a T^a$, $[T^a, T^b] = \epsilon^{abc} T^c$, ${\rm tr} (T^a T^b) = -1/2 \delta^{ab}$, and the space-time signature is $(+,-,-)$.
With the defining equations for the isovectors Eq.(\[iso\]), we obtain the following Euler-Lagrangean equations $$m_\alpha \ddot{q}_{\alpha i} = - (F^a_{ij}({\bf q}_\alpha)
\dot{q}_\alpha^j + F^a_{i0}({\bf q}_\alpha)) Q^a_\alpha, \label{eul1}$$ $$\dot{Q}^a_\alpha = -\epsilon^{abc} (A^b_i({\bf q}_\alpha) \dot{q}^i_\alpha+
A^b_0({\bf q}_\alpha)) Q^c_\alpha, \label{eul2}$$ $${\kappa \over 2} \epsilon^{ij} F^a_{ij} ({\bf x}) = -\sum_\alpha Q^a_\alpha
\delta({\bf x}-{\bf q}_\alpha), \label{gau}$$ $$\kappa \epsilon^{ij} F^a_{j0}({\bf x})
= -\sum_\alpha Q^a_\alpha \dot{q}^i_\alpha \delta({\bf x}-{\bf
q}_\alpha) \label{eul4}$$ where $F^a_{ij}= \partial_i A^a_j -\partial_j A^a_i
+\epsilon^{abc} A^b_i A^c_j$. The first two equations are Wong’s equations [@wong] and the third one corresponds to the Gauss’ law constraint which tells us that the NACS particle of isospin charge $Q^a_\alpha$ carries the magnetic flux $-Q^a_\alpha/\kappa$; $$\Phi_{\rm m} = \frac{1}{2}\int_{B_\alpha}
\epsilon^{ij}F^a_{ij}({\bf x}) d^2x = -\frac{1}{\kappa} Q^a_\alpha$$ where $B_\alpha$ denotes a small patch covering the position of the $\alpha$-th NACS particle.
Since the part of Lagrangean for the isospin degrees of freedom is usually of first order, second class constraints arise in the procedure of canonical quantization. These second class constraints are entangled with the first class constraints which generate gauge symmetries and it makes the constraint analysis difficult [@bal78; @bal90]. One of the advantages we have, to define the isovectors as in Eq.(\[iso\]) is that it is easy to avoid the second class constraints from the outset by judiciously defining the Poisson bracket [@ber; @fadd]. Introducing the canonical momenta $p^i_\alpha$, $$p^i_\alpha ={\partial L \over \partial
\dot q_{i\alpha}}= m_\alpha \dot{q}^i_\alpha+ A^{a i} ({\bf q}_\alpha)
Q^a_\alpha$$ we can convert the given Lagrangean Eq.(\[lag\]) to a first order Lagrangean $$L = \sum_\alpha\left(p^i_\alpha \dot q_{i\alpha} + J_\alpha
\cos \theta_\alpha \dot \phi_\alpha\right) +\int d^2{\bf x}\left(
{\kappa\over 2}\epsilon^{ij}\dot A^a_i A^a_j\right) - H,$$ $$H = H_0 -\int d^2 {\bf x} \Biggl[ A^a_0\left( {\kappa \over 2}
\epsilon^{ij} F^a_{ij} +\sum_\alpha Q^a_\alpha \delta({\bf x}- {\bf
q}_\alpha)\right)\Biggr]$$ $$H_0 = \sum_\alpha {1\over 2 m_\alpha}\left(p^i_\alpha-A^{ai}({\bf q}_\alpha)
Q^a_\alpha\right)^2. \label{first}$$
This first order Lagrangean can be rewritten as [@fadd] $$L= a_I({\xi})\dot\xi^I -H(\xi)\label{forder}$$ and the Euler-Lagrangean equations from Eq.(\[forder\]) as $$f_{IJ}(\xi)\dot\xi^J = \frac{\partial}{\partial \xi^I} H(\xi),$$ $$f_{IJ}(\xi)=\frac{\partial}{\partial \xi^I}
a_J(\xi)-\frac{\partial}{\partial \xi^J}
a_I(\xi)$$ where $\xi^I$ denote collectively the canonical variables, $p^i_\alpha$, $q^i_\alpha$, $\theta_\alpha$, $\phi_\alpha$, $A^a_i({\bf x})$. $f_{IJ}(\xi)$ defines the pre-symplectic two form $f$ by $$f=\frac{1}{2}f_{IJ}(\xi)d\xi^I d\xi^J = da(\xi)$$ where $a(\xi)$ is the canonical one form, $a(\xi) = a_I(\xi)d\xi^I$. For the given first order Lagrangean Eq.(\[first\]), we find $$f = dp^i dq^i - J\sin \theta d\theta d\phi -\int d^2{\bf x}\frac{\kappa}{2}
\epsilon^{ij} \delta A^a_i({\bf x}) \delta A^a_j ({\bf x}) \label{pre}$$ where we suppress the indices labeling the particles. When the matrix $(f_{IJ})$ is non-singular, i.e., has its inverse $(f^{IJ})$ as in Eq.(\[pre\]), the Poisson bracket is taken as $$\{F(\xi), G(\xi)\} = f^{IJ}(\xi)\frac{\partial F(\xi)}{\partial \xi^I}
\frac{\partial G(\xi)}{\partial \xi^J}\label{poiss}$$ and the Euler-Lagrangean equations can be expressed as $$\dot \xi^I = \{ \xi^I, H\} = f^{IJ} \frac{\partial}{\partial \xi^J} H.$$
From Eqs.(\[pre\]) and (\[poiss\]), we can define the Poisson bracket as $$\{F, G\}= \sum_\alpha\left[\left({\partial F
\over \partial q^i_\alpha}{\partial G \over \partial p_{\alpha i}}-
{\partial F \over \partial p_{\alpha i}}{\partial G \over \partial
q^i_\alpha}\right) -{1\over J_\alpha\sin\theta_\alpha}\left({\partial F
\over \partial \phi_\alpha}{\partial G \over \partial \theta_\alpha}-
{\partial F \over \partial \theta_\alpha}{\partial G \over \partial
\phi_\alpha}\right)\right]$$ $$+\int d^2 {\bf x} \left({1\over\kappa}\epsilon_{ij}
{\delta F \over \delta A^a_i
({\bf x})}{\delta G \over \delta A^a_j ({\bf x})}\right)\label{poi}$$ and the fundamental commutators as $$\{q^i_\alpha, p_{\beta j}\} = \delta^i_j\delta_{\alpha\beta},\qquad
\{Q^a_\alpha,Q^b_\beta\} =\epsilon^{abc}
Q^c_\alpha\delta_{\alpha\beta}$$ $$\{A^a_i({\bf x}), A^b_j({\bf y})\} = {1\over \kappa}\epsilon_{ij}
\delta({\bf x}- {\bf y})\delta^{ab}.$$
As expected, the Gauss’ law constraints $$\Phi^a = {\kappa \over 2}\epsilon^{ij} F^a_{ij} ({\bf x}) +
\sum_\alpha Q^a_\alpha \delta({\bf x}- {\bf q}_\alpha) =
0\label{gaus}$$ form $SU(2)$ algebra and no further constraints arise $$\{\Phi^a ({\bf x}),\Phi^b ({\bf y})\} =
\epsilon^{abc} \Phi^c\delta({\bf x}-{\bf y}), \qquad
\{H_0, \Phi^a ({\bf x})\} = 0.$$ In contrast to the approach of Balachandran [*et al.*]{} [@bal78; @bal90], the second class constraints never appear in ours.
If one can solve the Gauss’ constraints explicitly, one can describe the dynamics of the NACS particles solely by the quantum mechanical Hamiltonian $H_0$ in Eq.(\[first\]) with $A^a_i({\bf q}_\alpha)$ which are determined through the Gauss’ constraints. Thus it is desirable to solve the Gauss’ constraints explicitly, if possible. We may attempt to solve the constraints in Coulomb or covariant gauges, but find it difficult. However, as we shall see, there are two gauge conditions where the constraints have explicit solutions. The first one is the axial gauge condition. Considering the nature of (2+1) dimensions, one can easily understand how the axial gauge works. Choosing an axial gauge condition, say, $A^a_1=0$, we get a solution for the Gauss’ constraint [@gua] $$A^a_1({\bf x})=0,\quad A^a_2({\bf x})=-\frac{1}{\kappa}\sum_\alpha
Q^a_\alpha\theta(x-x_\alpha)\delta(y-y_\alpha)+f(y) \label{axial}$$ where $f(y)$ is an arbitrary function of $y$. Unfortunately, it is rather awkward to describe the dynamics of the NACS particles with the axial gauge solution because of the strings attached on them. (Analyzing the the well-known anyon quantum mechanics in the axial gauge [@kapu] may give us an idea. But we will not pursue it here.) The alternative is the holomorphic gauge condition which we shall discuss in the following section.
Coherent State Quantization and Holomorphic Gauge {#sec: coherent}
=================================================
If one adopts the coherent state quantization [@klauder] for the gauge fields, one can choose a better gauge condition. We recall that $A^a_1$ and $A^a_2$ are canonical conjugates to each other, i.e., $\{A^a_1({\bf x}), A^b_2({\bf y})\}
=\kappa\delta^{ab}\delta({\bf x}-{\bf y})$, or upon quantization $[\hat A^a_1({\bf x}), \hat A^b_2({\bf y})]
=i\kappa\delta^{ab}\delta({\bf x}-{\bf y})$. This suggests for us to define ‘creation’ and ‘annihilation’ operators $${\cal A}^{a \dagger} =
\sqrt{\frac{1}{2\kappa}}\left(\hat A^a_1 -i \hat
A^a_2\right),\quad {\cal A}^{a} =
\sqrt{\frac{1}{2\kappa}}\left(\hat A^a_1 +i \hat
A^a_2\right),$$ $$\left[{\cal A}^{a}({\bf
x}), {\cal A}^{b \dagger} ({\bf y})\right]
=\delta^{ab}\delta({\bf x}-{\bf y})$$ and to construct coherent states $$\vert A_{\bar z}> \equiv \exp\left(\sqrt{\frac{\kappa}{2}}\int d^2 {\bf x}
A^a_{\bar z} {\cal A}^{a\dagger}\right)\vert 0>$$ and their adjoints $$<A_z\vert \equiv <0\vert \exp\left(\sqrt{\frac{\kappa}{2}}\int d^2 {\bf x}
A^a_{z} {\cal A}^{a}\right).$$ Then we have $$<A_z\vert A_{\bar z}> = \exp\left(\frac{\kappa}{2}\int d^2{\bf x}
A^a_z A^a_{\bar z}\right)$$ and the resolution of identity $$\int D A_z D A_{\bar z} \exp\left(-\frac{\kappa}{2}\int d^2{\bf x} A^a_z
A^a_{\bar z}\right)\vert A_{\bar z}><A_z\vert = I.\label{resol}$$ Partitioning the time interval $[t_1, t_2]$ into many pieces and repeatedly using the resolution of identity Eq.(\[resol\]), we obtain a functional integral representation for the transition amplitude (in the sector of gauge fields) $$<A_z^f, t_f\vert A_{\bar z}^i, t_i>=
\int D A_z D A_{\bar z} \exp\Biggl\{\frac{\kappa}{2}
\int d^2{\bf x} \left(A^f_z
A^f_{\bar z} +A^i_z A^i_{\bar z} \right)+\label{trgauge}$$ $$i\int d^2{\bf x} \int^{t_2}_{t_1} dt
\left[\frac{i\kappa}{2}\left(A^a_{z} \dot A^a_{\bar z}
-\dot A^a_{z} A^a_{\bar
z}\right) -H(A_z, A_{\bar z})\right]\Biggr\}$$ where $H(A_z, A_{\bar z})$ is obtained by substituting ${\cal A}^{a\dagger}\rightarrow A^a_z$, ${\cal A}^{a}\rightarrow
A^a_{\bar z}$ in the given Hamiltonian $H({\cal A}^{a\dagger}, {\cal
A}^{a})$. It is worthwhile to note that $A^a_z$ and $A^a_{\bar z}$ must be treated as independent variables [@brown].
In order to get the path integral representation of the physical transition amplitude $Z$, now we include the particle sector $$Z=\int D p^z D q^{\bar z} D p^{\bar z} D
q^z D\cos\theta D\phi D A_z D A_{\bar z} D A_0\label{ampli}$$ $$\exp\left\{
-\kappa i \int d^2 z\left(A^f_{\bar z} A^f_z+A^i_{\bar z} A^i_z\right)\right\}
\exp\{i\int^{t_f}_{t_i} dt L\},$$ $$L= \sum_\alpha\left(p^{\bar z}_\alpha \dot z_{\alpha}+
p^{z}_\alpha \dot{{\bar z}}_{\alpha}+ J_\alpha \cos\theta_\alpha
\dot\phi_\alpha\right) +\int d^2 z \left(
{\kappa\over 2}\left(\dot A^a_z A^a_{\bar z} -\dot A^a_{\bar z} A^a_{
z}\right) + A^a_0 \Phi^a \right) - H,$$ $$H = \sum_\alpha {2\over
m_\alpha}\left(p^{\bar z}_\alpha-A^{a}_z(z_\alpha, \bar z_\alpha)
Q^a_\alpha\right) \left(p^{z}_\alpha-A^{a}_{\bar z}(z_\alpha, \bar z_\alpha)
Q^a_\alpha\right)$$ where $\Phi^a(z)=\kappa F^a_{z\bar z}+\sum_\alpha
Q^a_\alpha\delta (z-z_\alpha)=0.$ For the sake of convenience, we employed in Eq.(\[ampli\]) complex coordinates where $$z = x+ iy,\quad \bar z = x- iy,$$ $$z_\alpha = q^1_\alpha + i q^2_\alpha,\quad \bar z_\alpha = q^1_\alpha - i
q^2_\alpha.$$ Adopting the coherent state quantization for the Chern-Simons gauge fields [@ems; @bos] effectively extends the gauge orbit space in that $A^a_z$ and $A^a_{\bar z}$ are treated as independent variables [@brown]. It implies concurrently that there are wider class of gauge fixing conditions available in the frame-work of the coherent state quantization: We can choose, $A^a_{\bar z}=0$, as a gauge fixing condition. This gauge fixing condition may be called holomorphic gauge condition. In this gauge, the Gauss’ constraint reduces to $$\Phi^a(z)=-\kappa \partial_{\bar z} A^a_z +\sum_\alpha
Q^a_\alpha\delta(z-z_\alpha)=0 \label{gauss}$$ and has an explicit solution $$A^a_{\bar z} (z, \bar z)= 0,\quad
A^a_z (z, \bar z) = {i\over 2\pi \kappa}\sum_\alpha Q^a_\alpha
{1\over z -z_\alpha}\label{sol}$$ which is less singular that that in the axial gauge Eq.(\[axial\]). As we shall see, it is amusing to observe that the dynamics of the NACS particles lies entirely in the holomorphic sector of $z$ in the holomorphic gauge.
Reaching this point, one may realize that the holomorphic gauge may not be connected to the more conventional gauges such as Coulomb and axial gauges by canonical gauge transformations and may worry about the equivalence of the path integral, representing physical amplitude in the holomorphic gauge to those in the conventional ones. This worry can be taken care of by the BRST formulation [@brst] as discussed in ref.[@lo1]: The equivalence of the path integral in the holomorphic gauge to those in conventional gauges follows from the Fradkin and Vilkovisky theorem [@brst; @fradkin]. In the frame-work of the coherent state quantization, the holomorphic gauge condition is a legitimate gauge fixing condition.
Employing the coherent state quantization for the gauge fields, we write the path integral representing the physical transition amplitude [@fadsla] as $$Z=\int D p^z D q^{\bar z} D p^{\bar z} D
q^z D\cos\theta D\phi D A_z D A_{\bar z} \delta(\Phi) \delta(\chi) \det
\{\chi,\Phi\}$$ $$\exp\left\{
-\kappa i \int d^2z\left(A^f_{\bar z} A^f_z+A^i_{\bar z} A^i_z\right)\right\}
\exp\{i\int^{t_f}_{t_i} dt (K - H)\},\label{action}$$ $$K = \sum_\alpha\left(p^{\bar z}_\alpha \dot z_{\alpha}+
p^{z}_\alpha \dot{{\bar z}}_{\alpha}+
J_\alpha \cos \theta_\alpha \dot \phi_\alpha\right)
+\int d^2 z {\kappa\over 2}\left(\dot A^a_z A^a_{\bar z} -\dot A^a_{\bar z}
A^a_{ z}\right)$$ where $\chi = 0$ is the gauge condition. In the holomorphic gauge $\chi = A^a_{\bar z}$ and $\det \{\chi,\Phi\}= \det
\partial_{\bar z}$. Unless we consider a nontrivial topology for the two dimensional space, $\det\partial_{\bar z}$ is not singular. Since it is independent of dynamical variables, it can be dropped from the path integral. Therefore we find that the path integral is now reduced to $$Z =\int D p^z D p^{\bar z} D
q^z D q^{\bar z} D\cos \theta D \phi
D A_z D A_{\bar z} \delta(A^a_{\bar z}) \delta
(\Phi^a) \exp\{i\int dt(K - H)\}.$$
Chern-Simons Quantum Mechanics in Holomorphic Gauge {#csqm}
===================================================
The quantum mechanical description of the NACS particles is obtained if we integrate out the field variables. By use of the given solution in the holomorphic gauge Eq.(\[sol\]), we get the path integral expressed only in terms of the quantum mechanical variables $$Z =\int D p^z D p^{\bar z} D
q^z D q^{\bar z} D\cos \theta D \phi
\exp\{i\int dt(K - H)\}, \label{qmpath}$$ $$K = \sum_\alpha\left(p^{\bar z}_\alpha \dot z_{\alpha}+
p^{z}_\alpha \dot{{\bar z}}_{\alpha}+
J_\alpha \cos \theta_\alpha \dot \phi_\alpha\right),$$ $$H = \sum_\alpha {2\over m_\alpha}
p^z_\alpha \left( p^{\bar z}_\alpha - A^a_z (
z_\alpha,\bar z_\alpha) Q^a_\alpha\right)$$ where $A^a_z(z,\bar z)$ denotes the holomorphic gauge solution Eq.(\[sol\]).
The operator formulation of the Chern-Simons quantum mechanics follows from the observation that the path integral Eq.(\[qmpath\]) is equivalently expressed as $$Z = <\eta_f\vert \exp\{ -i \hat{H}(t_f-t_i)\} \vert \eta_i>,\label{qmop}$$ $$\hat{H} = \sum_\alpha {2\over m_\alpha}
\hat p^z_\alpha \left(\hat p^{\bar z}_\alpha - \hat A^a_z (
z_\alpha) \hat Q^a_\alpha\right).$$ The variables with ‘ $\hat{}$ ’ denote quantum operators of which commutators are given by $$[ {\bar z}_\alpha, \hat
p^{z}_\alpha] = i,\quad [{z}_\alpha, \hat p^{\bar z}_\alpha]= i,\quad
[\hat Q^a_\alpha,\hat Q^b_\beta] =i\epsilon^{abc} \hat
Q^c_\alpha \delta_{\alpha\beta}.$$ The gauge field $ \hat A^a_z$ here stands for the operator version of the solution Eq.(\[sol\]). Since the iso-vector operators $\hat Q^a$’s satisfy the $SU(2)$ algebra, they can be represented by $SU(2)$ some generators, say, $T^a_j$ in a representation of isospin $j$. For example, when $j=1/2$, $\hat Q^a$’s are represented by the Pauli matrices $\tau^a/2$ and the state vector $|\eta>$ by an iso-spin doublet. We suppressed the isospin indices in Eq.(\[qmop\]). In passing, note that $\hat Q^2_\alpha |\eta>=
J^2_\alpha|\eta> = j_\alpha(j_\alpha+1)|\eta>$, $j_\alpha\in {\bf Z}_{n+1/2}$.
We now conclude that the dynamics of the NACS particles are determined by the Hamiltonian $\hat H$ $$\hat {H} = -\sum_\alpha {1\over m_\alpha}\left(\nabla_{\bar
z_\alpha}\nabla_{z_\alpha} +\nabla_{z_\alpha}\nabla_{\bar
z_\alpha}\right)$$ $$\nabla_{z_\alpha} ={\partial\over \partial z_\alpha} +{1\over 2\pi
\kappa}\left( \sum_{\beta\not=\alpha}
\hat Q^a_\alpha \hat Q^a_\beta {1\over
z_\alpha -z_\beta}+\hat Q^2_\alpha a_z (z_\alpha)\right)\label{ham}$$ $$\nabla_{\bar z_\alpha} ={\partial\over \partial \bar z_\alpha}$$ where $a_z (z_\alpha)=\lim_{z\rightarrow z_\alpha} 1/(z-z_\alpha)$ and it needs an appropriate regularization. The second term and the third term in $\nabla_{z_\alpha}$ describe mutual and self interactions of NACS particles respectively. The Hamiltonian Eq.(\[ham\]) without the self interaction terms has been conjectured and applied to the non-Abelian Aharonov-Bohm effect by Verlinde [@ver].
The mutual interaction terms in $\hat H$ are responsible for the non-Abelian statistics. This becomes clear as we remove the interaction terms by some singular non-unitary transformation, [*i.e.*]{} casting the wave function $\Psi_h$ for the NACS particles in the holomorphic gauge into the following form $$\Psi_h(z_1,\dots,z_N) = U^{-1}(z_1,\dots,z_N)
\Psi_a (z_1,\dots,z_N),\label{wave}$$ where $U^{-1}(z_1,\dots,z_N)$ satisfies the Knizhnik-Zamolodchikov (KZ) equation [@kz] $$\left({\partial\over \partial z_\alpha} + {1\over 2\pi
\kappa} \sum_{\beta\not=\alpha} \hat Q^a_\alpha \hat Q^a_\beta {1\over
z_\alpha -z_\beta}\right) U^{-1}(z_1,\dots,z_N) =0.\label{kzeq}$$ The Hamiltonian for $\Psi_a$ is a free Hamiltonian $\hat H=
-\sum_\alpha \frac{ 2}{m_\alpha}
(\partial_{\bar z_\alpha}\partial_{z_\alpha})$.
In Eq.(\[wave\]) we suppress the effects of the self interaction for the following reason. The self interaction terms can be also removed in a similar way as the mutual interaction terms are removed. As a result, we may have a factor $$\exp\left(-{1\over 2\pi \kappa}\sum_\alpha \lim_{z\rightarrow z_\alpha}\int^z
{\hat Q^2_\alpha \over z^\prime-z_\alpha} dz^\prime\right)\label{factor}$$ in the r.h.s. of Eq.(\[wave\]). This factor is not well defined until we specify how to take the limit $z\rightarrow z_\alpha$. Here we take the limit near $z_\alpha$ along $z=z_\alpha+t\epsilon_\alpha$ where $\epsilon_\alpha$ is a complex number with $|\epsilon_\alpha|=1$ and $t$ is a real parameter to be taken $0$ in the limit. Then the factor Eq.(\[factor\]) is factorized into a divergent piece $$\lim_{t\rightarrow 0}\prod_\alpha \exp\left(
-\frac{1}{2\pi\kappa}\hat Q^2_\alpha
\ln t\right)\label{facd}$$ and a regular one $$\prod_\alpha \exp\left(-\frac{i}{2\pi\kappa}\hat Q^2_\alpha
\arg\epsilon_\alpha\right).\label{facr}$$ Since the divergent piece Eq.(\[facd\]) can be absorbed into a normalization constant of the wave function, the quantum mechanical Hamiltonian for the NACS particles does not contain any singularity. We may further absorb the constant regular factor Eq.(\[facr\]) into the phase of the wave function. Therefore we may safely remove the self interaction terms in the nonrelativistic Hamiltonian Eq.(\[ham\]) for the NACS particles as in the case for the anyons [@jackiw]. However, in the context of the relativistic quantum mechanics for the NACS particles, the regular factor Eq.(\[facr\]) cannot be no longer ignored: Noting that the regularization procedure described above corresponds to the framing of the knots [@witt] which represent the world lines of the relativistic NACS particles, we may identify the anomalous spins of the NACS particles as $2j_\alpha (j_\alpha+1)/4\pi\kappa$ from the additional phase which the wave function acquires due to the regular factor under $2\pi$ rotation.
The KZ equation has a formal solution which is expressed as a path ordered line integral in the $N$-dimensional complex space $$U^{-1}(z_1,\dots,z_N) = P \exp\left[-{1\over 2\pi\kappa} \int_\Gamma
\sum_\alpha d\zeta^\alpha \sum_{\beta\not=\alpha}
\hat Q^a_\alpha \hat Q^a_\beta
{1\over \zeta_\alpha -\zeta_\beta}\right],\label{kzsol}$$ where $\Gamma$ is a path in the $N$-dimensional complex space with one end point fixed (reference point) and the other being $\zeta_f = (z_1,\dots,z_N)$. Explicit evaluation of the above formal expression will give the monodromy matrices. We see that $\Psi_a$ obeys the braid statistics, due to the transformation function $U(z_1,\dots,z_N)$ while $\Psi_h$ satisfies ordinary statistics. In analogy with the Abelian Chern-Simons particle theory we may call $\Psi_a$ the NACS particle wave function in the anyon gauge. The two descriptions for the NACS particle - one in the holomorphic gauge with $\Psi_h$ and $H$ in Eq.(\[ham\]) and the other in the anyon gauge with $\Psi_a$ and the free Hamiltonian - are equivalent and the transformation function between two gauges is given by $U(z_1,\dots,z_N)$ Eq.(\[kzsol\]). It also defines the inner product in the holomorphic gauge $$<\Psi_1 |\Psi_2> = \int d^{2N}\zeta \Psi_1(\zeta)^\dagger U^\dagger
(\zeta) U(\zeta) \Psi_2 (\zeta)$$ where $\zeta = (z_1,\dots,z_N)$. The Hamiltonian in the holomorphic gauge Eq.(\[ham\]) is certainly Hermitian with this inner product $$\nabla^\dagger_{z_\alpha}=-\nabla_{\bar z_\alpha},\quad
\nabla^\dagger_{\bar z_\alpha}=-\nabla_{z_\alpha},\quad
\hat H^\dagger = \hat H.$$
When $N=2$, the KZ equation is particularly simple and the solution for the KZ equation can be easily obtained $$U(z_1,z_2)= \exp\left[\hat Q^a_1
\hat Q^a_2\frac{1}{2\pi\kappa}\ln(z_1-z_2)\right].$$ Therefore if we exchange the positions of two NACS particles along an oriented path as depicted by Fig. 1, the wave function in the anyon gauge transforms as $$\Psi_a(z_1,z_2) \rightarrow \Psi_a(z_2,z_1) = \exp\left(\frac{\hat Q^a_1
\hat Q^a_2}{2\kappa}i\right) \Psi_a(z_1,z_2).$$ The operator ${\cal R}_{\alpha\beta}=\exp\left(\frac{\hat Q^a_\alpha\hat
Q^a_\beta}{2\kappa}i\right)$ is called the braid operator or half-monodromy which satisfies the Yang-Baxter equation and exhibits some character of the non-Abelian statistics. If we wind one NACS particle around the other, [*i.e.*]{} doubly exchange, the transformation of the wave function of the system is given by the monodromy operator ${\cal M}_{\alpha\beta}=
({\cal R}_{\alpha\beta})^2$, $$\Psi_a(z_1,z_2) \rightarrow
\exp\left(\frac{\hat
Q^a_1 \hat Q^a_2}{\kappa}i\right) \Psi_a(z_1,z_2).$$
Two Body Problem in Chern-Simons Quantum Mechanics
==================================================
Being equipped with the non-Abelian Chern-Simons quantum mechanics discussed in the previous section, we now specifically consider a system of two NACS particles. Although two body system does not reveal all the novel features of the NACS particles, it certainly enables us to capture some essence of them. Since two body problem is presently the only one known to have explicit solutions, it is worth while to explore the two body system in some detail. This section also serves as a preparatory stage for the discussions on the non-Abelian Aharonov-Bohm effect in the following section.
The system of two NACS particles is described by the Hamiltonian $$\hat {H} = -\sum_{\alpha=1}^{2}{1\over m_\alpha}\left(\nabla_{\bar
z_\alpha}\nabla_{z_\alpha} +\nabla_{z_\alpha}\nabla_{\bar
z_\alpha}\right)$$ $$\nabla_{z_1} ={\partial\over \partial z_1} +\Omega{1\over
z_1 -z_2}, \quad
\nabla_{\bar z_1} ={\partial\over \partial \bar z_1},\label{two}$$ $$\nabla_{z_2} ={\partial\over \partial z_2} +\Omega{1\over
z_2 -z_1}, \quad
\nabla_{\bar z_2} ={\partial\over \partial \bar z_2}.$$ Note that $\Omega$ is a block-diagonal matrix $$\Omega= \hat Q^a_1\hat Q^a_2 / (2\pi\kappa)=\frac{1}{4\pi\kappa}
\left((\hat Q_1+\hat Q_2)^2-(\hat Q_1)^2-(\hat Q_2)^2\right)\label{omega}$$ $$=\sum_j\frac{1}{4\pi\kappa}\left(j(j+1)-j_1(j_1+1)-j_2(j_2+1)\right)\otimes
{I}_j$$ and the half-monodromy is written as $${\cal R}=\exp(i\Omega\pi)\label{half}$$ where $|j_1-j_2|\le j\le j_1+j_2$ and ${ I}_j$ is an identity matrix in the subspace of total isospin $j$ which is spanned by $\{|m|\le j;
|j,m>\}$. In Eq. (\[omega\]), $j(j+1)$, $j_1(j_1+1)$, and $j_2(j_2+1)$ are eigenvalues of $(\hat Q_1+\hat Q_2)^2$, $(\hat Q_1)^2$, and $(\hat Q_2)^2$ respectively.
Upon introducing the center of mass and relative coordinates $Z = (z_1+z_2)/2,\quad z = z_1 -z_2,$ the two body Hamiltonian Eq.(\[two\]) becomes $$\hat H = -{1\over 2\mu} \partial_Z \partial_{\bar Z}
-\frac{1}{\mu}(\nabla_z\nabla_{\bar z} +\nabla_{\bar z}\nabla_z),$$ $$\nabla_z = \partial_z +\frac{\Omega}{z},\quad \nabla_{\bar z} =
\partial_{\bar z}$$ where we take the mass of NACS particles, $m_1 = m_2 =2\mu$ for simplicity. As usual, the motion of center of mass coordinates is not dynamical and is decoupled from the motion of relative coordinate. If we write the wave function for the two body system as $\Psi(z,\bar z, Z,\bar Z, t) = \Psi_{\rm
CM} (Z,\bar Z,t) \psi(z,\bar z,t)$, the Schrödinger equation for the system becomes $$-{1\over 2\mu} \partial_Z \partial_{\bar Z} \Psi_{\rm CM} (Z,\bar Z,t) =
i\frac{\partial \Psi_{\rm CM} (Z,\bar Z, t)}{\partial t}$$ $$-\frac{1}{\mu}(\nabla_z\nabla_{\bar z} +\nabla_{\bar z}\nabla_z)
\psi(z,\bar z,t) = i\frac{\partial \psi(z,\bar z,t)}{\partial
t}.\label{hamrel}$$
Since the center-of-mass Hamiltonian $\hat H_{\rm CM} =
-{1\over 2\mu} \partial_Z \partial_{\bar Z}$ has a set of eigenstates $\{e^{i\left({\bf K}\cdot {\bf R}-(K^2/2\mu)t\right)}\}$, the two body problem now reduces to finding energy eigenfunctions $ \psi_n(z,\bar z)$ of the Hamiltonian for the relative motion $\hat H_{\rm rel}$ $$\hat H_{\rm rel} = -\frac{1}{\mu}(\nabla_z\nabla_{\bar z} +
\nabla_{\bar z}\nabla_z),\label{eigen}$$ $$\hat H_{\rm rel} \psi_n(z,\bar z) =
E_n \psi_n(z,\bar z).$$ If the energy eigenfunctions $\psi_n(z,\bar z)$ are found, the energy eigenstates of the system of two NACS particles are written as $$\{\Psi_{K,n}(z,\bar z, Z,\bar Z, t) =
e^{-i\left(E_n+K^2/2\mu\right)t} e^{i{\bf K}\cdot
{\bf R}}\psi_n(z,\bar z)\}.$$
To obtain the energy eigenfunctions for $\hat H_{\rm rel}$ we find it useful to take a similarity transformation given by $$\hat H_{\rm rel} \longrightarrow \hat H_{\rm rel}^\prime =
G^{-1} \hat H_{\rm rel} G,$$ $$\psi_n(z,\bar z) \longrightarrow
\psi_n^\prime(z,\bar z) =G^{-1} \psi_n(z,\bar z).$$ The following observations help us to solve the Schrödinger equation for the relative motion. Firstly, the transformed Hamiltonian $\hat H_{\rm
rel}^\prime$ has the same set of eigenvalues $\{E_n\}$ as $\hat H_{\rm rel}$ $$\hat H_{\rm rel}^\prime \psi_n^\prime(z,\bar z) = E_n
\psi_n^\prime(z,\bar z).\label{treigen}$$ Secondly, if we choose the transformation function $G$ to be $$G (z,\bar z) = \exp\left(-\frac{\Omega}{2}\ln(z\bar z)\right),$$ the inner product between the states becomes usual and transformed Hamiltonian $\hat H_{\rm rel}^\prime$ becomes manifestly Hermitian $$\hat H_{\rm rel}^\prime = -\frac{1}{\mu}
(\nabla_z^\prime\nabla_{\bar z}^\prime +
\nabla_{\bar z}^\prime\nabla_z^\prime),\label{trham}$$ $$\nabla_z^\prime=\partial_z + \frac{\Omega}{2}\frac{1}{z},\quad
\nabla_{\bar z}^\prime=\partial_{\bar z}
-\frac{\Omega}{2}\frac{1}{\bar z}.$$
The Hamiltonian $\hat H_{\rm rel}^\prime$ resembles the Hamiltonian for the relative motion of two anyon system in Coulomb gauge. Indeed it may be regarded as the Hamiltonian for the system of two NACS particles in Coulomb gauge in a sense. As we mentioned earlier, in general, the Gauss’ constraint cannot be solved explicitly in Coulomb gauge in that the constraint is nonlinear in this gauge. However, this is not the case for the one particle sector. Suppose that one NACS particle is located at the origin. One can easily find that the Gauss’ constraint Eq.(\[gaus\]) which is now simplified to be $$\Phi^a = {\kappa \over 2}\epsilon^{ij} F^a_{ij} ({\bf x}) +
Q^a \delta({\bf x}) =0\label{gausone}$$ has an explicit solution in the Coulomb gauge $$A^a_i ({\bf x}) = \epsilon_{ij} \frac{Q^a}{2\pi\kappa}\frac{x^j}{r^2},\quad
\partial_i A^a_i ({\bf x}) =0.$$ Therefore, if we neglect the self interaction, we can construct a Hamiltonian for the system of two NACS particles as follows $$\hat H_C=\sum_{\alpha=1, 2} {1\over 2 m_\alpha}\left(p^i_\alpha-A^{ai}({\bf
q}_\alpha) Q^a_\alpha\right)^2,$$ $$A^a_i ({\bf q}_1) =\epsilon_{ij}\frac{Q^a_2}{2\pi\kappa}\frac{q^j_1-q^j_2}
{\left({\bf q}_1-{\bf q}_2\right)^2},$$ $$A^a_i ({\bf q}_2) =-\epsilon_{ij}\frac{Q^a_1}{2\pi\kappa}\frac{q^j_1-q^j_2}
{\left({\bf q}_1-{\bf q}_2\right)^2}.$$ Certainly, the Hamiltonian for the relativistic motion which follows from $\hat
H_C$ coincides with $\hat H^\prime_{\rm rel}$. This shows that the description of the two-body system in the test particle frame-work is valid.
Rewriting the Hamiltonian $\hat H_{\rm rel}^\prime$ in the polar coordinates and projecting it onto the subspace of total isospin $j$, we have $$\hat H_{\rm rel}^\prime =
-\frac{1}{2\mu}\left[\frac{\partial^2}{\partial r^2}+
\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\left(\frac{\partial}
{\partial \theta}+i\omega_j\right)^2\right].\label{hampr}$$ where $\Omega|j,m>=\omega_j|j,m>$ and $$\omega_j=\frac{1}{4\pi\kappa}
\left(j(j+1)-j_1(j_1+1)-j_2(j_2+1)\right).\label{eigenv}$$ As is well-known in the case of Aharonov-Bohm scattering [@ahbo], the Hamiltonian $\hat H_{\rm rel}^\prime$ has eigenfunctions $$\{ n\in {\bf
Z}; e^{in\theta} J_{n+\omega_j}(kr), e^{in\theta} J_{-n-\omega_j}(kr)\}$$ with an eigenvalue $E_k = \frac{\hbar^2 k^2}{2\mu}$. Requiring the solution to be regular at the origin, we get a set of eigenfunctions $$\{ n\in {\bf Z}; e^{in\theta} J_{|n+\omega_j|}(kr)\}.\label{col}$$ Subsequently, we have eigenfunctions in the holomorphic gauge $$\{ n\in {\bf Z}; \exp\left(-\frac{\omega_j}{2}\ln z\bar
z\right) e^{in\theta} J_{|n+\omega_j|}(kr)\}\label{hol}$$ and eigenfunctions in the anyon gauge $$\{ n\in {\bf Z}; \exp\left(i\omega_j\theta\right) e^{in\theta}
J_{|n+\omega_j|}(kr)\}.\label{any}$$ Note that the eigenfunctions in the anyon gauge are multivalued.
Non-Abelian Aharonov-Bohm Effect {#sec:naab}
================================
Having obtained the Hamiltonian for the relative motion Eq.(\[col\]), we can proceed to our discussion on the non-Abelian Aharonov-Bohm scattering in parallel to the discussion on the Aharonov-Bohm scattering in ref. [@ahbo]. From the discussion on the two body system in the previous section, it is clear that the state of the system can be characterized by $|{\bf k}>\otimes |j_1,j_2;m_1,m_2>$ where $|{\bf k}>$ and $|j_1,j_2;m_1,m_2>$ describe the momentum and isospin state of the system respectively.
Now let us examine the structure of the Hamiltonian $\hat H_{\rm rel}^\prime$, Eq.(\[hampr\]). The Hamiltonian commutes with the operators $(\hat Q_1+\hat Q_2)^2$, $(\hat Q_1)^2$, and $(\hat Q_2)^2$ and it implies that the scattering amplitude $f({\bf k}^\prime, {\bf k})$ has the following structure $$f({\bf k}^\prime, {\bf
k})_{\{m_1^\prime,m_2^\prime;m_1,m_2\}}=\sum_{j=|j_1-j_2|}^{j_1+j_2}
\sum_{m=-j}^{j}\label{scatt}$$ $$<j_1,j_2;m_1^\prime,m_2^\prime|j_1,j_2;j,m>f_j
({\bf k}^\prime, {\bf k}) <j_1,j_2;j,m|j_1,j_2;m_1,m_2>$$ where $<j_1,j_2;j,m|j_1,j_2;m_1,m_2>$ is the well-known Clebsch-Gordan coefficient. We may call $f_j ({\bf k}^\prime, {\bf k})$ the scattering amplitude of isospin $j$ channel. Here we assume that two NACS particles are distinguishable and will consider the collision between indistinguishable NACS particles after a little.
In order to evaluate $f_j ({\bf k}^\prime, {\bf k})$, we only need to consider the channel of isospin $j$ where the dynamics of the system is governed by $\hat H_{\rm rel}^\prime$, Eq.(\[hampr\]). The most general solution of the Schrödinger equation $$\hat H_{\rm rel}^\prime \psi = \frac{\hbar^2 k^2}{2\mu} \psi$$ is given as a superposition of the eigenfunctions Eq.(\[col\]) $$\psi = \sum_n a_n e^{in\theta} J_{|n+\omega_j|}(kr) \label{gen}$$ and the scattering solution corresponds to the solution which has a spatial asymptotics of $$e^{ikr\cos\theta-i\omega_j \theta} +
f_j(k,\theta)\frac{e^{ikr}}{\sqrt{r}}.\label{scat}$$ Fig. 2 depicts scattering of two NACS particles in the center of mass frame. Due to the long range nature of the Aharonov-Bohm effect, the portion of the incident wave is not a plane wave as in the case of the Coulomb scattering [@gott].
Comparing the asymptotics of the solution Eq.(\[gen\]) $$\psi \longrightarrow \sum_n a_n e^{in\theta} \sqrt{\frac{2}{\pi kr}}
\cos\left(kr-\frac{|n+\omega_j|\pi}{2}-\frac{\pi}{4}\right)$$ with the asymptotics of the expansion of the plane wave in angular momentum eigenstates $$e^{ikr\cos\theta}=\sum_n i^n e^{in\theta} J_n(kr)$$ $$\longrightarrow\sum_n i^n e^{in\theta} \sqrt{\frac{2}{\pi kr}}
\cos\left(kr-\frac{n\pi}{2}-\frac{\pi}{4}\right)$$ suggests that $a_n = \exp\left[\pi i(n- |n+\omega_j|/2)\right]$. Thus the scattering solution is $$\psi=\sum^\infty_{n=-[\omega_j]} e^{i\pi(n-\omega_j)/2} e^{in\theta}
J_{n+\omega_j}(kr)+$$ $$\sum^{-[\omega_j]-1}_{-\infty }e^{-i\pi(n-\omega_j)/2}
e^{in\theta}J_{-n-\omega_j}(kr).$$
We can obtain the spatial asymptotics of $\psi$ either by use of a differential equation [@ahbo; @hagen] or by use of the Schläfli contour representation of Bessel function [@jackiw] $$\psi \longrightarrow e^{ikr\cos\theta-i\omega_j\theta}
+\frac{e^{-i\omega_j\pi}}{\sqrt{2\pi k i}} e^{-i\left([\omega_j]+\frac{1}{2}
\right)\theta}\frac{\sin\omega_j\pi}{\sin\theta
/2}\frac{e^{ikr}}{\sqrt{r}}.\label{asym}$$ Then the scattering amplitude of isospin $j$ channel [@comm] is read as follows $$f_j(k,\theta)= \frac{e^{-i\omega_j\pi}}{\sqrt{2\pi k i}}
e^{-i\left([\omega_j]+\frac{1}{2}\right)\theta}\frac{\sin\omega_j\pi}
{\sin\theta/2}.\label{scatj}$$
The scattering amplitude $f_j ({\bf k}^\prime, {\bf k})$ follows from Eq.(\[scatt\]) and Eq.(\[scatj\]). This result may be expressed more succinctly if we define a matrix ${\cal F}({\bf k}^\prime, {\bf k})$ in the isospin space $$f({\bf k}^\prime, {\bf
k})_{\{m_1^\prime,m_2^\prime;m_1,m_2\}}=
<j_1,j_2;m_1^\prime,m_2^\prime|{\cal F}({\bf k}^\prime, {\bf k})
|j_1,j_2;m_1,m_2>.$$ (Since $j_1$ and $j_2$ are fixed, they will be omitted in the formulae hereafter.) We find that ${\cal F}({\bf k}^\prime, {\bf k})$ can be written as $${\cal F}({\bf k}^\prime, {\bf k})= \frac{e^{-i\Omega\pi}}{\sqrt{2\pi k i}}
e^{-i\left([\Omega]+\frac{1}{2}\right)\theta}
\frac{\sin\Omega\pi}{\sin\theta/2}\label{scatm}$$ where $[\alpha]$ denotes the integer such that $0\le \alpha -[\alpha]< 1$ and $[\Omega]$ should be understood as $[\Omega]|j,m>=[\omega_j]|j,m>$. In terms of the half-monodromy matrix Eq.(\[half\]) or monodromy ${\cal M}={\cal R}^2$, it may be expressed as $${\cal F}({\bf k}^\prime, {\bf k})=\frac{1}{\sqrt{2\pi k i}}{\cal R}^{-1}
({\cal R}^{-1}-{\cal R})
\frac{e^{-i[\Omega]\theta}}{1-e^{i\theta}}\label{scatmm}$$ $$=\frac{1}{\sqrt{2\pi k i}}({\cal M}^{-1}-1)
\frac{e^{-i[\Omega]\theta}}{1-e^{i\theta}}.$$ This result is similar to that obtained by Lo and Preskill [@lopre], but does not completely agree with. Note in particular that the obtained scattering amplitude Eq.(\[scatm\]) is single valued in contrast to theirs. However, as we shall see, both scattering amplitudes yield the same ‘inclusive’ cross section.
The differential cross section for the scattering process $|m_1,m_2>\rightarrow |m^\prime_1,m^\prime_2>$ is then given by $$\frac{d\sigma}{d\theta}\left(|m_1,m_2>\rightarrow |m^\prime_1,m^\prime_2>
\right) = \Bigm|<m^\prime_1,m^\prime_2|{\cal F}(k,\theta)|m_1,m_2>\Bigm|^2.$$ Writing down its explicit expression, we have $$\frac{d\sigma}{d\theta}\left(|m_1,m_2>\rightarrow |m^\prime_1,m^\prime_2>
\right) =-\frac{1}{8\pi k} \sum_{j,m}\sum_{j^\prime, m^\prime}
<m^\prime_1,m^\prime_2|j,m>$$ $$<j,m|m_1,m_2>
<m_1,m_2|j^\prime,m^\prime><j^\prime,m^\prime|m^\prime_1,m^\prime_2>$$ $$\left(e^{-2i\omega_j\pi}-1\right)\left(e^{-2i\omega_{j^\prime}\pi}-1\right)
\frac{e^{-i([\omega_j]-[\omega_{j^\prime}])\theta}}{\sin^2\theta/2}.$$ If we do not measure the isospin orientations of particles after scattering, the cross section is given by $$\frac{d\sigma}{d\theta}\left(|m_1,m_2>\rightarrow {\rm all}\right)
=<m_1,m_2|{\cal F}^\dagger(k,\theta){\cal F}(k,\theta)|m_1,m_2>\label{incl}$$ $$=\frac{1}{2\pi k} \frac{<m_1,m_2|\sin^2 \Omega\pi|m_1,m_2>}{\sin^2
\theta/2}$$ $$=\frac{1}{4\pi k} \frac{1}{\sin^2 \theta/2}\left(1-{\rm
Re}<m_1,m_2|{\cal M} |m_1,m_2>\right).$$ This is called inclusive cross section in ref.[@lopre]. The formula for the inclusive cross Eq.(\[incl\]) was obtained first by Verlinde [@ver] and was confirmed recently by Lo and Preskill [@lopre].
If two NACS particles share the same physical properties and belong to the same isospin multiplet, [*i.e.*]{} $j_1=j_2$ (but $m_1$ may differ from $m_2$), it is appropriate to regard them indistinguishable [@lopre]. In such a case, two body system is described by the symmetrized wave function $$\Psi(z,\bar z, Z,\bar Z, t) = \frac{1}{\sqrt{2}}\Psi_{\rm
CM} (Z,\bar Z,t)\left(\psi(z,\bar z,t) +\psi(-z,-\bar
z,t)\right).\label{symm}$$ (We describe the NACS particles in the holomorphic gauge as bosons interacting with each other through the non-Abelian Chern-Simons gauge field. Alternatively we may describe the NACS particles in terms of the fermions, then we must choose the antisymmetrized wave function.) Going through the same analysis given in sections V and VI with the symmetrized wave function Eq.(\[symm\]), we find that the scattering amplitude $f_j^I ({\bf k}^\prime, {\bf k})$ for the scattering of indistinguishable NACS particles in the isopsin $j$ channel is given in terms of $f_j ({\bf k}^\prime, {\bf k})$ Eq.(\[scatj\]) by $$f_j^I (k,\theta)= f_j(k,\theta)+ f_j(k,\pi-\theta)$$ $$=\frac{e^{-i\omega_j\pi}}{\sqrt{2\pi k i}}
e^{-i\left([\omega_j]+\frac{1}{2}\right)\theta}\frac{\sin\omega_j\pi}
{\sin\theta/2}\left(1+e^{i\left([\omega_j]+1/2\right)(2\theta-\pi)}\tan
\theta/2\right).$$\[scatb\] Then we obtain the scattering amplitude matrix $${\cal F}^I({\bf k}^\prime, {\bf k})=
{\cal F}({\bf k}^\prime, {\bf k})+{\cal F}(-{\bf k}^\prime, {\bf k})$$ $$=\frac{1}{\sqrt{2\pi k i}}({\cal M}^{-1}-1)
\frac{e^{-i[\Omega]\theta}}{1-e^{i\theta}}\left(1+
e^{i\left([\Omega]+1/2\right)(2\theta-\pi)}\tan
\theta/2\right)$$ $$= {\cal F}({\bf k}^\prime, {\bf k}) \left(1+
e^{i\left([\Omega]+1/2\right)(2\theta-\pi)}\tan
\theta/2\right).$$ This yields the differential cross section for the indistinguishable NACS particles $$\frac{d\sigma^I}{d\theta}\left(|m_1,m_2>\rightarrow {\rm all}\right)=
<m_1,m_2|{\cal F}^{I\dagger}(k,\theta)
{\cal F}^I(k,\theta)|m_1,m_2>\label{incli}$$ $$=<m_1,m_2|{\cal F}^{\dagger}(k,\theta){\cal
F}(k,\theta)\left\{\sec^2\theta/2-2\cos\left((2[\Omega]+1)\theta\right)
\tan\theta/2\right\}|m_1,m_2>.$$
Conclusions and Discussion {#concl}
==========================
Description of NACS particles often involves field degrees of freedom, since they are usually described as isospin particles of which isospin charges are coupled to non-Abelian Chern-Simons gauge fields. However, it is not necessary to rely upon the non-Abelian Chern-Simons gauge fields to discuss the quantum mechanics of NACS particles. The non-Abelian Chern-Simons gauge fields are to be completely determined in principle in terms of the quantum mechanical degrees of freedom of the isopsin particles through the Gauss’s law constraints, i.e., they themselves do not have dynamical degrees of freedom. Their only role is to introduce a topological interaction between the isospin particles, endowing them with non-Abelian magnetic fluxes. Thus, if we solve the Gauss’ law constraints explicitly, we are able to describe the dynamics of NACS particles solely by a quantum mechanical Hamiltonian.
In this paper we have shown that the Gauss’ law constraints can be solved explicitly in the framework of coherent state quantizaion in an appropriately chosen gauge and discussed the dynamics of the NACS particles, based upon the obtained quantum mechanical Hamiltonian, especially the non-Abelian Aharonov-Bohm scattering: In the framework of the coherent state quantization, we are able to choose the holomorphic gauge condition to fix the gauge degrees of freedom thanks to the enlarged gauge orbit space and obtain the quantum mechanical Hamiltonian which governs the dynamics of the NACS particles through the KZ equation entirely in the holomorphic sector. The obtained quantum mechanical Hamiltonian enabled us to discuss the scattering of NACS particles in parallel to the discussion of the Aharonov-Bohm scattering. Evaluating the differential scattering cross section, we confirmed, yet in a more concrete way, the results which were previously obtained by Verlinde [@ver] and by Lo and Preskill [@lopre]. However, scattering amplitude which we obtained differs from theirs in details, in particular, it is single valued in contrast to that of Lo and Preskill. We conceive that the gauge condition chosen implicitly in ref.[@lopre] be different from ours and it may make such a difference. A close comparison between the two approaches would be worthwhile.
Incorporating an additional $U(1)$ electromagnetic field into the NACS particle theory, we can construct the model which has been proposed to describe some fractional quantum Hall states called singlet quantum Hall effect states [@wen; @moo; @balfra]. The non-Abelian quantum mechanics developed in this paper will be certainly useful to explore their physical properties. The proposed non-Abelian Chern-Simons quantum mechanics also gives us immediate helps in discussing [@lo; @tlee93] some subjects, in more generalized context, which have been previously discussed in the anyon theory such as construction of exact many-body wavefunctions [@exact] and evaluation of the second virial coefficient [@virial]. Although only the NACS particles with $SU(2)$ symmetry group is discussed in this paper, our discussion on the NACS particles is not limited to the case of $SU(2)$. Many of formal properties of NACS particles with $SU(2)$ are expected to be shared among those with other symmetry groups. For an example, the formal expression of the scattering matrix in terms of the monodromy Eq.(\[scatmm\]) may hold in general. A construction of the quantum mechanical Hamiltonian for NACS particles with $SU(N)$, $N\ge 2$ symmetry group is presented in ref.[@lo2] and analyses on the NACS particles with more general symmetry groups are currently in progress [@lo3]. As we mentioned in the introduction, the NACS particles make their appearances in various circumstances. For instance, they may appear as gravitationally interacting point-like sources in (2+1) dimensions, as non-Abelian vortices in (2+1) dimensions, or cosmic string with some discrete non-Abelian charges in (3+1) dimensions. Since the coherent state quantization and the holomorphic gauge fixing apply also to the models describing those NACS particles of various kinds, it will be certainly interesting to develop the non-Abelian Chern-Simons quantum mechanics for them and to discuss the related subjects in the framework given in this paper.
TL was supported in part by the KOSEF and PO was supported by the KOSEF through C.T.P. at S.N.U. TL would like to thank professor R. Jackiw for valuable comments. We also thank professor C. Lee and professor Y. M. Cho for useful discussions.
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abstract: 'The recently introduced hierarchical regular networks HN3 and HN4 are analyzed in detail. We use renormalization group arguments to show that HN3, a 3-regular planar graph, has a diameter growing as $\sqrt{N}$ with the system size, and random walks on HN3 exhibit super-diffusion with an anomalous exponent $d_{w}=2-\log_{2}(\phi)\approx1.306$, where $\phi=\left(\sqrt{5}+1\right)/2=1.618\ldots$ is the “golden ratio.” In contrast, HN4, a non-planar 4-regular graph, has a diameter that grows slower than any power of $N$, yet, fast than any power of $\ln N$. In an annealed approximation we can show that diffusive transport on HN4 occurs ballistically ($d_{w}=1$). Walkers on both graphs possess a first-return probability with a power law tail characterized by an exponent $\mu=2-1/d_{w}$. It is shown explicitly that recurrence properties on HN3 depend on the starting site.'
author:
- Stefan Boettcher
- Bruno Goncalves
- Julian Azaret
bibliography:
- '/Users/stb/Boettcher.bib'
title: Geometry and Dynamics for Hierarchical Regular Networks
---
Introduction
============
Networks with a sufficiently intricate structure to exhibit not-trivial properties for statistical models but sufficiently simple to reveal analytical insights are few and far between. Familiar examples are random graphs [@ER; @Bollobas], the hierarchical lattices [@Berker79] originating in the Migdal-Kadanoff bond-moving scheme [@Migdal76; @Kadanoff76], fractal lattices [@Mandelbrot82], or scale-free networks [@Barabasi01; @Andrade05; @Zhang07]. Scale-free networks and random graphs can elucidate mean-field properties only, whereas hierarchical lattices often provide excellent results for complex statistical models in low dimensions [@Southern77] but do not possess a mean-field limit. Fractal lattices provide tantalizing access to dynamical systems in non-integer dimensions but can not be tuned across dimensions. From the perspective of statistical physics, it could be desirable to have a network that combines solvability for low-dimensional systems with mean-field properties in such a way that one could interpolate between either extreme.
We have recently introduced a set of networks that overlay a lattice backbone with regular long-range links [@SWPRL; @SWN], similar to small-world networks [@Watts98] but hierarchical and without randomness. These networks have a recursive construction but retain a fixed, regular degree. The hierarchical sequence of long-distance links occur in a pattern reminiscent of the tower-of-hanoi sequence [@SWN]. Therefore, we have dubbed them “Hanoi-Networks” and abbreviated them as HN3 and HN4, since one is 3-regular and the other 4-regular. While almost identical, both types of networks lead to very distinct behaviors, as revealed by our studies here. Future work will focus on detailed studies of Ising models on these networks. In Ref. [@SWPRL] we have shown already that spin models on HN4 have the desired properties mentioned above. Here, we analyze in great detail diffusive transport on these networks, for which especially HN3 proves to possess a rich behavior.
The paper is structured as follows. In the next section, we introduce the networks and discuss various key design aspects. In Sec. \[sec:Graph-Structure\], we explore geometric aspects of the networks such as their diameter. Sec. \[sec:Random-Walks\] contains our analysis of random walks on both networks, starting with a review of the dynamic renormalization group in the context of a simple one-dimensional random walk which facilitates an efficient discussion of the equivalent but far more involved calculation for HN3. Unfortunately, the same approach does not seem to apply to HN4, so we conclude the section with a derivation of a moment-generating equation in Fourier space for walks on HN4 directly from the master equation and an annealed approximation. We conclude this paper in Sec. \[sec:Conclusions\], indicating various future projects that can be developed from the work presented here.
Network Design\[sec:Network-Design\]
====================================
Both networks we are discussing in this paper consist of a simple geometric backbone, a one-dimensional line of $N=2^{k}$ sites, either infinite $\left(-2^{k}\leq n\leq2^{k},\, k\to\infty\right)$, semi-infinite $\left(0\leq n\leq2^{k},\, k\to\infty\right)$, or closed into a ring. Each site on the one-dimensional lattice backbone is connected to its nearest neighbor. (In general, the following procedure also works with a higher-dimensional lattice.)
To generate the small-world hierarchy in these graphs, consider parameterizing any integer $n$ (except for zero) *uniquely* in terms of two other integers $(i,j)$, $i\geq0$, via $$\begin{aligned}
n & = & 2^{i}\left(2j+1\right).
\label{eq:numbering}\end{aligned}$$ Here, $i$ denotes the level in the hierarchy whereas $j$ labels consecutive sites within each hierarchy. For instance, $i=0$ refers to all odd integers, $i=1$ to all integers once divisible by 2 (i. e., 2, 6, 10,...), and so on. In these networks, aside from the backbone, each site is also connected with (one or both) of its nearest neighbors *within* the hierarchy. For example, we obtain the 3-regular network HN3 (best done on a semi-infinite line) by connecting first all nearest neighbors along the backbone, but in addition also 1 to 3, 5 to 7, 9 to 11, etc, for $i=0$, next 2 to 6, 10 to 14, etc, for $i=1$, and 4 to 12, 20 to 28, etc, for $i=2$, and so on, as depicted in Fig. \[fig:3hanoi\]. Correspondingly, HN4 is obtained in the same manner, but connecting to *both* neighbors in the hierarchy, i. e., 1 to 3, 3 to 5, 5 to 7, etc, for $i=0$, 2 to 6, 6 to 10, etc, for $i=1$, and so forth. For this network it is clearly preferable to extend the line to $-\infty<n<\infty$ and also connect -1 to 1 , -2 to 2, etc, as well as all negative integers in the above pattern, see Fig. \[fig:4hanoi\]. The site with index zero, not being covered by Eq. (\[eq:numbering\]), is clearly a special place, either on the boundary of the HN3 or in the center of the HN4. It is easy to generalize these graphs, for instance, by putting the structure on a ring with periodic boundary conditions, which may require special treatment of the highest level in the hierarchy on finite rings.
![Depiction of HN3 on a semi-infinite line. The leftmost site here is $n=0$, which requires special treatment. The entire network can be made 3-regular with a self-loop at $n=0$. Note that HN3 is planar.[]{data-label="fig:3hanoi"}](hanoi3)
![Depiction of HN4 on an infinite line. The center site is $n=0$, which requires special treatment. The entire network becomes 4-regular with a self-loop at $n=0$. Note that HN4 is non-planar.[]{data-label="fig:4hanoi"}](hanoi4)
Random walks on these networks have fascinating properties due to their fractal nature and their long-range links as shown, for instance, in Fig. \[fig:HN3RW-3D\].
Network Geometry\[sec:Graph-Structure\]
=======================================
Distance Measure on HN3\[sub:Distance-measure-on\]
--------------------------------------------------
For HN3, it is simple to determine geometric properties, for instance, its diameter $d$, which is the longest of the shortest paths between any two sites, to wit, on a finite graph of size $N=N_{k}=2^{k}$ for $k\to\infty$ . Clearly, $d$ in this case would be the end-to-end distance between sites $n=0$ and $n=N$ with the smallest number of hops. Using a sequence of networks for $k=2,4,6,\ldots$, the diameter-path looks like a Koch curve, see Fig. \[fig:Koch3hanoi\]. We can define the path $\Pi_{k}$ as a sequence of jumps reaching from one end of the $N=2^{k}$-long graph to the other via $$\begin{aligned}
\Pi_{0} & = & 1,\nonumber \\
\Pi_{2} & = & 1-2-1=\Pi_{0}-2-\Pi_{0},\nonumber \\
\Pi_{4} & = & 1-2-1-8-1-2-1=\Pi_{2}-8-\Pi_{2},\nonumber \\
& \dots & , \\
\Pi_{k+2} & = & \Pi_{k}-2^{k+1}-\Pi_{k},\nonumber
\label{eq:recurPath3}\end{aligned}$$ with an obvious notation of using “-” to string together a sequence of ever more complex moves. (While somewhat redundant here, introducing this notation will prove useful for HN4 below.) Hence, the length $d_{k}$ of each path $\Pi_{k}$ is given by $$\begin{aligned}
d_{k+2} & = & 2d_{k}+1\qquad {\rm for}\qquad N_{k+2}=4N_{k},
\label{eq:recurDia3}\end{aligned}$$ thus, $$d\sim\sqrt{N}.
\label{eq:3dia}$$ This property is demonstrated also in Fig. \[fig:3dia\]. In some ways, this property is reminiscent of a square-lattice consisting of $N$ lattice sites. The diameter (=diagonal) of this square is also $\sim\sqrt{N}$. In this sense, presuming that each link corresponds to a unit distance, HN3 is a fractal lattice of dimension 2, i. e. filling the plane. As shown in Fig. \[fig:HN3neighbors\], we also find that the number of sites that can be reached from a given site grows quadratically with the number of jumps allowed. It should be noted, though, that we are employing the one-dimensional lattice backbone as our metric to measure distances in the random walks in Sec. \[sec:Random-Walks\], which enforces a trivial fractal dimension of $d_{f}=1$. We conclude that, while interesting in its own right, HN3 is far from any mean-field behavior for which we would expect typical distances to depend only on (some power of) $\ln N$.
![ Sequence of shortest end-to-end paths (=diameter, thick black lines) for HN3 of size $N=2^{k}$, $k=2,4,8$. (Note that every second step in the hierarchical development has been omitted. At level $k=3,5,\ldots$ the shortest path is the same as at $k-1$, except each linear segment counts for two steps.) Whenever the system size $N$ increases by a factor of 4, the diameter $d$ increases by a factor of $\sim2$, leading to Eq. (\[eq:3dia\]). []{data-label="fig:Koch3hanoi"}](hanoiKoch3_2 "fig:") ![ Sequence of shortest end-to-end paths (=diameter, thick black lines) for HN3 of size $N=2^{k}$, $k=2,4,8$. (Note that every second step in the hierarchical development has been omitted. At level $k=3,5,\ldots$ the shortest path is the same as at $k-1$, except each linear segment counts for two steps.) Whenever the system size $N$ increases by a factor of 4, the diameter $d$ increases by a factor of $\sim2$, leading to Eq. (\[eq:3dia\]). []{data-label="fig:Koch3hanoi"}](hanoiKoch3_4 "fig:") ![ Sequence of shortest end-to-end paths (=diameter, thick black lines) for HN3 of size $N=2^{k}$, $k=2,4,8$. (Note that every second step in the hierarchical development has been omitted. At level $k=3,5,\ldots$ the shortest path is the same as at $k-1$, except each linear segment counts for two steps.) Whenever the system size $N$ increases by a factor of 4, the diameter $d$ increases by a factor of $\sim2$, leading to Eq. (\[eq:3dia\]). []{data-label="fig:Koch3hanoi"}](hanoiKoch3_6 "fig:")
![Plot of the shortest path length between the origin of HN3 and the $l$th site on two networks of extend $N=3\:2^{11}$ and $N=3\:2^{12}$. In both sets of data, we plot the path-distance relative to the root of the separation between site $l$ and the origin $(l=0)$ along the linear backbone. Then, all rescaled distances fluctuate around a constant mean. Those fluctuations are very fractal, their self-similarity becoming apparent when super-imposing the data for both sizes $N$ on a relative distance scale with $l/N$. []{data-label="fig:3dia"}](dist_scal)
![Plot of the number of sites (or “neighborhood”) $S_{d}$ that can be reached from a given site within less than $d$ jumps on HN3. Averaged over many starting sites, $S_{d}/d^{2}$ slowly converges to a constant, demonstrating that $S_{d}$ grows quadratically with $d$. Note that some features due to the hierarchical structure remain even after averaging over sites, such as the peaks at $d=192,$ 384, 768, etc.[]{data-label="fig:HN3neighbors"}](neighbors)
Distance Measure on HN4\[sub:Distance-Measure-on\]
--------------------------------------------------
The situation is more interesting for the HN4. Using the notation from Eq. (\[eq:recurPath3\]), we have $$\begin{aligned}
\Pi_{0} & = & 1,\nonumber \\
\Pi_{1} & = & 1-1,\nonumber\\
\Pi_{2} & = & 1-2-1=\Pi_{0}-2-\Pi_{0},\nonumber \\
\Pi_{3} & = & 1-2-2-2-1=\Pi_{0}-3\times2-\Pi_{0},\\
\Pi_{4} & = & 1-1-4-4-4-1-1=\Pi_{1}-3\times4-\Pi_{1},\nonumber \\
\Pi_{5} & = & 1-2-1-8-8-8-1-2-1\nonumber\\
&=&\Pi_{2}-3\times8-\Pi_{2},\nonumber \\
\Pi_{6} & = & 1-2-1-7\times8-1-2-1\nonumber\\
&=&\Pi_{2}-7\times8-\Pi_{2},\nonumber
\label{eq:Path4}\end{aligned}$$ and so on. Due to degeneracies at each level (which we have not listed), one has to proceed to many more levels in the hierarchy to discern the relevant pattern. In fact, any pattern evolves for an increasing number of levels before it gets taken over by a new one, with two patterns creating degeneracies at the crossover. Finally, we get (setting the degeneracies aside) $$\begin{aligned}
\Pi_{k} & = & \begin{cases}
\Pi_{k-2}-1\times2^{k-1}-\Pi_{k-2}, & (k=2),\\
\Pi_{k-3}-3\times2^{k-2}-\Pi_{k-3}, & (2<k\leq5),\\
\Pi_{k-4}-7\times2^{k-3}-\Pi_{k-4}, & (5<k\leq9),\\
\Pi_{k-5}-15\times2^{k-4}-\Pi_{k-5}, & (9<k\leq14),\\
\ldots,\end{cases}
\label{eq:recurPath4}\end{aligned}$$ and so on.
Note that the paths here do *not* search out the longest possible jump, as in Eq. (\[eq:recurPath3\]). Instead, the paths reach quickly to some intermediate level and follow *consecutive* jumps at that level before trailing off in the end. As we will see repeatedly, this is the main distinguishing feature discriminating between HN3 and HN4: Once a level is reached in the HN4, the entire graph can be traversed at *that* level, while in the HN3 any transport *must* climb down to lower levels (or merely jump back on that level), see Figs. \[fig:3hanoi\]-\[fig:4hanoi\].
![Plot of the shortest end-to-end path length for HN4 networks of increasing backbone sizes $N=2^{k}$, obtained by simulation. Note the piecewise-linear shape of the graph, which is reflected in Eq. (\[eq:dkgrecur\]), for example. In turn, Eq. (\[eq:crossover\]) concerns only the “bends” between each consecutive linear segment $g$.[]{data-label="fig:dgk"}](RingWidth2)
Corresponding to Eq. (\[eq:recurPath4\]), we obtain for the end-to-end shortest paths (=diameters $d_{k}$) $$\begin{aligned}
d_{k} & = & \begin{cases}
2d_{0}+1, & (k=2),\\
2d_{k-3}+3, & (2<k\leq5),\\
2d_{k-4}+7, & (5<k\leq9),\\
\ldots,\end{cases}
\label{eq:dkrecur}\end{aligned}$$ which we can generalize into a single statement introducing a “generation” index $g\geq2$, $$\begin{aligned}
d_{k}^{g} & = &
2d_{k-g}^{g-1}+\left(2^{g-1}-1\right),\qquad\left(l_{g-1}<k\leq
l_{g}\right),
\label{eq:dkgrecur}\end{aligned}$$ defining $l_{1}=1$, where in general $$\begin{aligned}
l_{g} & = & l_{g-1}+g,\qquad l_{2}=2,
\label{eq:crossover}\end{aligned}$$ demarcates the crossover point between the generations, see Fig. \[fig:dgk\]. Eq. (\[eq:crossover\]) easily yields $$\begin{aligned}
l_{g} & = & \frac{1}{2}g\left(g+1\right)-1\qquad\left(g\geq2\right).
\label{eq:crossoversol}\end{aligned}$$ To obtain the asymptotic behavior for $d_{k}$, instead of solving Eq. (\[eq:dkgrecur\]) for all $k$, we note that exactly on the crossover points $k=l_{g}$ (i. e., $k-g=l_{g-1}$) we have $$\begin{aligned}
d_{l_{g}}^{g} & = & 2d_{l_{g-1}}^{g-1}+\left(2^{g-1}-1\right).
\label{eq:crossoverrecur}\end{aligned}$$ Defining $e_{g}=2^{g}d_{l_{g}}^{g}$, we get $$\begin{aligned}
e_{g} & = & e_{g-1}+\frac{1}{2}-2^{-g},\end{aligned}$$ which is easily summed up to give $$\begin{aligned}
d_{l_{g}}^{g} & = & \left(g-1\right)2^{g-1}+1.
\label{eq:dkgsol}\end{aligned}$$ Remembering that $k=l_{g}$ and, from Eq. (\[eq:crossoversol\]), that $g\sim\sqrt{2l_{g}}\sim\sqrt{2k}\sim\sqrt{\log_{2}N^{2}}$, we finally get $$\begin{aligned}
d_{k} & \sim &\frac{1}{2}
\sqrt{\log_{2}N^{2}}\,\,2^{\sqrt{\log_{2}N^{2}}}\qquad\left(N\to\infty\right)
\label{eq:dkasymp}\end{aligned}$$ for the diameter of the HN4. As we would expect that the diameter (or rather, the average of shortest paths) in a small-world graph should behave as $d\sim\log N$, it is instructive to rewrite Eq. (\[eq:dkasymp\]) as $$\begin{aligned}
d_{k} & \sim & \left(\log_{2}N\right)^{\alpha}\quad{\rm with}
\quad\alpha\sim\frac{\sqrt{2\log_{2}N^{}}}{\log_{2}\log_{2}N}+\frac{1}{2}.
\label{eq:alpha}\end{aligned}$$ Technically, of course, $\alpha$ diverges with $N$ and the diameter grows faster than any power of $\log_{2}N$ but less than any however-small power of $N$, unlike Eq. (\[eq:3dia\]). In reality, though, $\alpha$ varies only very slowly with $N$, ranging merely from $\alpha\approx1.5$ to 3 over *fifty* decades, see Fig. \[fig:4dia\].
![Plot of the system-size dependence of the exponent $\alpha=\alpha(N)$ defined in Eq. (\[eq:alpha\]). The solid curve is the exact value based on Eq. (\[eq:dkgsol\]), and the dashed curve is the asymptotic approximation also given in Eq. (\[eq:alpha\]). This approximation is always an upper bound. Note that $\alpha(N)$ barely varies within a factor of 2 over 50 orders of magnitude.[]{data-label="fig:4dia"}](alphaHN4)
Random Walks\[sec:Random-Walks\]
================================
In the following we will study random walks on HN3 and HN4. For simplicity, we have focused in our simulations only on two elementary observables, the mean displacement with time, $\left\langle
\left|l\right|\right\rangle \sim t^{1/d_{w}}$, and the first-return time distribution $Q(\Delta t)\sim\Delta t^{-\mu}$. All walks are controlled by one parameter, $p$, which determines the probability of a walker to step off the lattice into the direction of a long-range jump. In particular, on HN3 a walker steps off the lattice with probability $p$ and jumps either to the left or right neighbor with probability $\left(1-p\right)/2$, whereas on HN4 long-range jumps to either the left or right occur with probability $p/2$ instead. In both cases, we should return to a simple one-dimensional nearest-neighbor walk for $p\to0$, where $d_{w}=2$ and $\mu=3/2$, although we would expect that limit to be singular. Here, we only consider uniform values of $p$, independent of the sites (or level of hierarchy). With a bit more algebra, the following considerations could be extended to values that for each site depend of the level of hierarchy, $p=p(i)$, for instance.
Renormalization Group for Random Walks
--------------------------------------
Here, we analyze the spatio-temporal rescaling of simple random walks with nearest-neighbor jumps along the available links using the renormalization group. First, we review the method using as a simple example the well-known one-dimensional walk. Walks on such a graph obey simple diffusion, $d_{w}=2$, which implies that each rescaling of space entails a rescaling of time according to $$\begin{aligned}
N\to N'=2N & \qquad & T\to T'=2^{d_{w}}T=4T.
\label{eq:1dWalkscale}\end{aligned}$$ This is synonymous with the asymptotic form with the mean-square displacement $$\left\langle \left|l\right|\right\rangle \sim t^{1/d_{w}},
\label{MSDeq}$$ which defines the anomalous dimension of the walk in terms of the exponent $d_{w}$. While the result for the one-dimensional walk can be easily obtained with much simpler means, it serves as a pedagogical example of calculating first-passage and first-return times using RG on more complex structures. Later on, reference to this presentation will allow us to avoid excessive algebra.
### RG for the 1d random walk\[sub:RG-for-2RW\]
As a pedagogical example, we present here the theory as it will be applied in Sec. \[sub:RG-for-3RW\]. We consider a biased random walk on a finite one-dimensional line. The master equations for such a random walk on a lattice of length $N=2^{K+1}$ with reflecting boundaries are given by
$$\begin{aligned}
P_{0,t+1} & = & p\, P_{1,t},\nonumber \\
P_{1,t+1} & = & P_{0,t}+p\, P_{2,t},\nonumber \\
P_{l,t+1} & = & (1-p)\, P_{l-1,t}+p\, P_{l+1,t}\qquad(2\leq l\leq
N-2),\nonumber \\
P_{N-1,t+1} & = & (1-p)\, P_{N-2,t}+P_{N,t},\nonumber \\
P_{N,t+1} & = & (1-p)\, P_{N-1,t},
\label{eq:1dRW}\end{aligned}$$
where $P_{l,t}$ denotes the probability of a walker to be at site $l$ at time $t$, $p$ is the probability expressing the biasing for left or right hops. Since we want to start the walks at time $t=0$ at the origin $l=0$, these equations have the initial condition $$\begin{aligned}
P_{l,0} & = & \delta_{l,0}.
\label{eq:1dRWinit}\end{aligned}$$
To facilitate renormalization this non-equilibrium process, we introduce a generating function $$\begin{aligned}
\tilde{P}_{l}(z) & = & \sum_{t=0}^{\infty}P_{l,t}z^{t}
\label{eq:generator}\end{aligned}$$ for all $0\leq l\leq N$. Incorporating the initial condition in Eq. (\[eq:1dRWinit\]), Eqs. (\[eq:1dRW\]) transform into $$\begin{aligned}
\tilde{P}_{0} & = & a\,\tilde{P}_{1}+1,\nonumber \\
\tilde{P}_{1} & = & c\,\tilde{P}_{0}+a\,\tilde{P}_{2},\nonumber \\
\tilde{P}_{l} & = & b\,\tilde{P}_{l-1}+a\,\tilde{P}_{l+1}\qquad(2\leq l\leq N-2),\nonumber \\
\tilde{P}_{N-1} & = & b\,\tilde{P}_{N-2}+d\,\tilde{P}_{N},\nonumber \\
\tilde{P}_{N} & = & b\,\tilde{P}_{N-1},\label{eq:1dRWgen}\end{aligned}$$ where we have inserted generalized “hopping rates” in preparation for the RG. Initially, at the $k=0$th RG step, it is $$\begin{aligned}
a^{(0)} & = & p\, z,\nonumber \\
b^{(0)} & = & (1-p)\, z,\label{eq:RGinitpara}\\
c^{(0)} & = & z,\nonumber \\
d^{(0)} & = & z,\nonumber \end{aligned}$$ which provides a sufficient number of renormalizable parameters that are potentially required to consider special sites at both boundaries.
A single step of applying the RG consists of solving Eqs. (\[eq:1dRWgen\]) for $\tilde{P_{l}}$ with odd values of $l$ (which is trivial here, as they are already expressed explicitly in terms of even ones) and eliminating them from the equations for the even $l$. After that elimination, we can rewrite the equations for even $l$ as $$\begin{aligned}
\tilde{P}_{0} & = & \frac{a^{2}}{1-ac}\,\tilde{P}_{2}+\frac{1}{1-ac},\nonumber \\
\tilde{P}_{2} & = & \frac{bc}{1-2ab}\,\tilde{P}_{0}+\frac{a^{2}}{1-2ab}\,\tilde{P}_{4},\nonumber \\
\tilde{P}_{2l} & = & \frac{b^{2}}{1-2ab}\,\tilde{P}_{2l-2}+\frac{a^{2}}{1-2ab}\,\tilde{P}_{2l+2}\quad\left(2\leq l\leq\frac{N}{2}-2\right),\nonumber \\
\tilde{P}_{N-2} & = & \frac{b^{2}}{1-2ab}\,\tilde{P}_{N-4}+\frac{ad}{1-2ab}\,\tilde{P}_{N},\nonumber \\
\tilde{P}_{N} & = & \frac{b^{2}}{1-bd}\,\tilde{P}_{N-2}.\label{eq:1dRWgen-reno}\end{aligned}$$ Comparing these equations with Eqs. (\[eq:1dRWgen\]) allows to extract the RG recursion equations. [\[]{}Note that superscripts referring the $k$th RG step have been suppressed thus far in Eqs. (\[eq:1dRWgen\]) and (\[eq:1dRWgen-reno\]).\]
Before we analyze the first return time at the boundary specifically, we can use the equation for bulk sites $l$ in (\[eq:1dRWgen-reno\]) to extract already the diffusion exponent $d_{w}$. A comparison of the respective coefficients in Eqs. (\[eq:1dRWgen\]) and (\[eq:1dRWgen-reno\]) yields $$\begin{aligned}
a^{(k+1)}&=&\frac{\left(a^{(k)}\right)^{2}}{1-2a^{(k)}b^{(k)}},\nonumber\\
b^{(k+1)}&=&\frac{\left(b^{(k)}\right)^{2}}{1-2a^{(k)}b^{(k)}}.
\label{eq:RGbulk}\end{aligned}$$ These recursions converge for $k\to\infty$ towards fixed points $\left(a^{*},b^{*}\right)$ that characterize the dynamics in the infinite-time limit (which corresponds to the limit of $z\to1^{-}$). The trivial fixed point $a^{*}=b^{*}=0$ is unphysical, as it can not be reached from the initial conditions in (\[eq:RGinitpara\]) for any choice of $p$ (and $z=1$). The physical fixed points are $\left(a^{*},b^{*}\right)=\left(1,0\right)$, which is reached for any bias $p>\frac{1}{2}$, or $\left(a^{*},b^{*}\right)=\left(0,1\right)$, reached for $p<\frac{1}{2}$; finally, $\left(a^{*},b^{*}\right)=\left(\frac{1}{2},\frac{1}{2}\right)$ can only be reached by entirely unbiased walks for $p=\frac{1}{2}$. To explore the behavior for large but finite times, we expand the RG recursions in (\[eq:RGbulk\]) to first order in $\epsilon=1-z$ by writing for $y\in\left\{ a,b\right\} $: $$\begin{aligned}
y^{(k)} & \sim & y^{*}+y_{1}^{(k)}\epsilon+\ldots.
\label{eq:epsilon1}\end{aligned}$$ Inserting the Ansatz in Eq. (\[eq:epsilon1\]) into the recursions in Eqs. (\[eq:RGbulk\]), we obtain near the fixed point with $a^{*}=b^{*}=\frac{1}{2}$: $$\begin{aligned}
a_{1}^{(k+1)} & = & 3a_{1}^{(k)}+b_{1}^{(k)},\nonumber \\
b_{1}^{(k+1)} & = & a_{1}^{(k)}+3b_{1}^{(k)},
\label{eq:1dRWunbias}\end{aligned}$$ with the result that $$\begin{aligned}
a_{1}^{(k)} & = & b_{1}^{(k)}\propto4^{k}.\end{aligned}$$ This implies that as space rescales by a factor of 2 (i. e., eliminating all odd-index sites), time rescales by a factor of 4, as indicated in Eq. (\[eq:1dWalkscale\]) for an unbiased random walk, leading to $d_{w}=2$. The same analysis for either of the biased fixed points yields that, for example, $a^{(k)}\equiv0$ beyond any power of $\epsilon$ and, with the Ansatz $b^{(k)}\sim1+b_{1}^{(k)}\epsilon$ in Eqs. (\[eq:RGbulk\]), $b_{1}^{(k)}\propto2^{k}$. Following the interpretation in Eq. (\[eq:1dWalkscale\]), this would imply $d_{w}=1$ and we find the familiar result that with the slightest bias, i. e., $p<\frac{1}{2}$ or $p>\frac{1}{2}$, the motion at large length and time scales is dominated by the constant-velocity drift upon reaching the bulk.
In this scenario of a bias, average first-return times are clearly system-size independent constants: A walker with a bias *towards* the origin ($p>\frac{1}{2}$) will drift back recurrently after only small excursions; a walker with a bias *away* from the origin ($p<\frac{1}{2}$) returns at most a finite number of times in short order until the drift eventually carries it away without further recurrence. In the following, we therefore focus exclusively on the unbiased case $p=\frac{1}{2}$. Then, we can equate $a=b$ at every step, to get from Eq. (\[eq:RGbulk\]): $$\begin{aligned}
a^{(k+1)}=\frac{\left(a^{(k)}\right)^{2}}{1-2\left(a^{(k)}\right)^{2}}.
\label{eq:RGbulkunbias}\end{aligned}$$
To derive the return-time behavior, we have to examine Eqs. (\[eq:1dRWgen-reno\]) more closely. Comparing coefficients also in the boundary terms leads to $$\begin{aligned}
c^{(k+1)}&=&\frac{a^{(k)}c^{(k)}}{1-2\left(a^{(k)}\right)^{2}},\nonumber\\
d^{(k+1)}&=&\frac{a^{(k)}d^{(k)}}{1-2\left(a^{(k)}\right)^{2}}.
\label{eq:RGupper}\end{aligned}$$ For large $k$, both $c^{(k)}$ and $d^{(k)}$ are entrained with $a^{(k)}$, and we obtain a consistent and closed set of relations for all coefficients in Eqs. (\[eq:1dRWgen\]) and (\[eq:1dRWgen-reno\]) by identifying $c=d=2a$. Further renormalizing $$\begin{aligned}
\tilde{P}_{l}^{(k+1)} & = &
\left[1-2\left(a^{(k)}\right)^{2}\right]\,\tilde{P}_{2l}^{(k)}
\label{eq:amplitude-reno}\end{aligned}$$ ensures invariance of the constant term at the lower boundary that originated from the unit initial condition in Eq. (\[eq:1dRWinit\]).
After $k=K$ RG steps, the system has reduced to $$\begin{aligned}
\tilde{P}_{0}^{(K)} & = & a^{(K)}\tilde{P}_{1}^{(K)}+1,\nonumber \\
\tilde{P}_{1}^{(K)} & = & 2a^{(K)}\tilde{P}_{0}^{(K)}+2a^{(K)}\tilde{P}_{2}^{(K)},\label{eq:1d-elementary}\\
\tilde{P}_{2}^{(K)} & = & a^{(K)}\tilde{P}_{1}^{(K)},\nonumber \end{aligned}$$ which yields $$\begin{aligned}
\tilde{P}_{0}^{(K)} & = &
\frac{1-2\left(a^{(K)}\right)^{2}}{1-4\left(a^{(K)}\right)^{2}}.
\label{eq:P0K1d}\end{aligned}$$ Using Eq. (\[eq:amplitude-reno\]) in turn obtains $$\begin{aligned}
\tilde{P}_{0}^{(0)} & = &
\frac{\tilde{P}_{0}^{(K)}}{\prod_{k=0}^{K-1}\left[1-2\left(a^{(k)}\right)^{2}\right]}\nonumber\\
&=&\frac{1-2\left(a^{(K)}\right)^{2}}{1-4\left(a^{(K)}\right)^{2}}\:\prod_{k=0}^{K-1}\frac{1}{\left[1-2\left(a^{(k)}\right)^{2}\right]}.
\label{eq:P001d}\end{aligned}$$ It is a well-known fact that the generating functions for being at the origin, $\tilde{P}_{0}^{(0)}$, and for the first-return probability to the same site, $\tilde{Q}_{0}$, satisfy the following simple relation [@Redner01]: $$\begin{aligned}
\tilde{Q}_{l} & = & 1-\frac{1}{\tilde{P}_{l}^{(0)}}.
\label{eq:Q}\end{aligned}$$ Note that a recurrent walk (with $\tilde{Q}_{0}=1$) requires that $ $ $\tilde{P}_{0}^{(0)}$ diverges at long times (i. e. $z\to1^{-}$). In our one-dimensional walk here, it is $a^{(K)}\to a^{*}=\frac{1}{2}$, on behalf of which the denominator of $\tilde{P}_{0}^{(0)}$ in Eq. (\[eq:P001d\]) has a zero, making the walk recurrent. In more detail, it is $$\begin{aligned}
\tilde{Q}_{0} & = &
1-\frac{1-4\left(a^{(K)}\right)^{2}}{1-2\left(a^{(K)}\right)^{2}}\:\prod_{k=0}^{K-1}\left[1-2\left(a^{(k)}\right)^{2}\right].
\label{eq:Q1d}\end{aligned}$$
We intend to extract the exponent $\mu$ for the first-return probability distribution, which on a finite but large system of size $N\to\infty$ behaves as $$\begin{aligned}
Q_{0}(t) & \sim & t^{-\mu}e^{-t/\tau_{N}}\qquad(t\to\infty),
\label{eq:Qt}\end{aligned}$$ where $\tau_{N}$ is a cut-off time scale that diverges in some form with $N$. As the physics of the return probabilities at any finite time should not change if the system size becomes infinite independently, it must be $\mu>1$ for $Q_{0}(t)$ to remain normalizable. Based on that observation [@Redner01], we need to calculate the first two moments of $Q_{0}(t)$, corresponding to an expansion of $\tilde{Q}_{0}(z)$ to 2nd order in $\epsilon=1-z$, to extract $\mu$. Since $\mu>1$, the normalization integral $$\begin{aligned}
{\cal N} & \sim & \int^{\infty}dt\, t^{-\mu}e^{-t/\tau_{N}}\sim O(1)
\label{eq:taunorm}\end{aligned}$$ is dominated by its behavior for small $t$, which is irrelevant in detail except for the fact that it makes the integral become a constant independent of $\tau_{N}$. If we further assume that $\mu<2$, then for all $m\geq1$, the integrals for those $m$th moments *do* diverge with $\tau_{N}$, $$\begin{aligned}
\left\langle t^{m}\right\rangle _{N} & \sim\frac{1}{{\cal N}} &
\int_{0}^{\infty}dt\, t^{m-\mu}e^{-t/\tau_{N}}\sim\tau_{N}^{m+1-\mu}.
\label{eq:taumoment}\end{aligned}$$ From the ratio of $\left\langle t\right\rangle _{N}$ and $\left\langle t^{2}\right\rangle _{N}$, we obtain then $$\begin{aligned}
\mu & =\lim_{N\to\infty} & 2+\frac{1}{1-\frac{\left\langle
t^{2}\right\rangle _{N}}{\left\langle t\right\rangle _{N}}}.
\label{eq:mu}\end{aligned}$$
Luckily, due to the leading zero in $1-4\left(a^{(K)}\right)^{2}$, any other factor in Eq. (\[eq:Q1d\]) only needs to be expanded to first order in $\epsilon$. Extending the Ansatz in Eq. (\[eq:epsilon1\]) for $a^{(k)}$ to 2nd order, we obtain here: $$\begin{aligned}
a^{(K)} & \sim & \frac{1}{2}-\frac{1}{2}\times4^{K}\epsilon+5\times16^{K}\epsilon^{2}+\ldots,\\
1-4\left(a^{(K)}\right)^{2} & \sim & 8\times4^{K}\epsilon+116\times16^{K}\epsilon^{2}+\ldots,\\
1-2\left(a^{(K)}\right)^{2} & \sim & \frac{1}{2}+4\times4^{K}\epsilon+\ldots,\\
\prod_{k=0}^{K-1}\left[1-2\left(a^{(k)}\right)^{2}\right] & \sim & \prod_{k=0}^{K-1}\left[\frac{1}{2}+4\times4^{k}\epsilon+\ldots\right],\\
& \sim & 2^{-K}\left[1+8\epsilon\sum_{k=0}^{K-1}4^{k}+\ldots\right],\\
& \sim & 2^{-K}+\frac{8}{3}\,2^{K}\epsilon+\ldots.\end{aligned}$$ Inserting these expressions into Eq. (\[eq:Q1d\]), we get $$\begin{aligned}
\tilde{Q}_{0} & \sim &
1-16\times2^{K}\epsilon+\frac{952}{3}\times8^{K}\epsilon^{2}+\ldots.
\label{eq:Q1dasymp}\end{aligned}$$ The moments of $Q_{0}(t)$ are obtained via derivatives of $\tilde{Q}_{0}(z)$, i. e. $$\begin{aligned}
\left\langle t^{m}\right\rangle & = &
\left[z\,\frac{d}{dz}\right]^{m}\tilde{Q}_{0}(z)\vert_{z=1}.
\label{eq:def-moments}\end{aligned}$$ Applied to Eq. (\[eq:Q1dasymp\]), we calculate for the dominant asymptotics of the moments: $$\begin{aligned}
\left\langle t\right\rangle _{K} & \sim & 2^{K}\sim N,\\
\left\langle t^{2}\right\rangle _{K} & \sim & 8^{K}\sim N^{3},\end{aligned}$$ which from Eq. (\[eq:mu\]) leads to the familiar first-return exponent of a one-dimensional walk, $$\begin{aligned}
\mu & = & \frac{3}{2}.
\label{eq:1dmu}\end{aligned}$$
### RG for the random walk on HN3\[sub:RG-for-3RW\]
We follow the discussion for one-dimensional walkers above to model diffusion on HN3. The lesson of the previous section is that we only need to consider the bulk equations to extract the diffusion exponent $d_{w}$, and supplement with the calculation on the final system at the end of the $k=K$th step of the RG to obtain the first-return exponent for a system of size $N=2^{K+1}$. We can section the line into segments centered around sites $l$ with $i=1$ in Eq. (\[eq:numbering\]), i. e. $n=2(2j+1)$. Such a site $l=n$ is surrounded by two sites of odd index, which are *mutually* linked. Furthermore, $n$ is linked by a long-distance jump to a site also of type $i=1$ at $l=n\pm4$ in the neighboring segment, where the direction does not matter here. The sites $l=n\pm2$, which are shared at the boundary between adjacent segments also have even index, but their value of $i\geq2$ is undetermined and irrelevant for the immediate RG step, as they have a long-distance jump to some sites $l=m_{\pm}$ at least eight sites away.
In each segment in the bulk, the master-equation reads $$\begin{aligned}
P_{n+2,t+1} & = & \frac{1-p}{2}\left[P_{n+3,t}+P_{n+1,t}\right]+p\, P_{m_{+},t},\nonumber \\
P_{n+1,t+1} & = & \frac{1-p}{2}\left[P_{n+2,t}+P_{n,t}\right]+p\, P_{n-1,t},\nonumber \\
P_{n,t+1} & = & \frac{1-p}{2}\left[P_{n+1,t}+P_{n-1,t}\right]+p\, P_{n\pm4,t},\label{eq:3RW}\\
P_{n-1,t+1} & = & \frac{1-p}{2}\left[P_{n,t}+P_{n-2,t}\right]+p\, P_{n+1,t},\nonumber\\
P_{n-2,t+1} & = & \frac{1-p}{2}\left[P_{n-1,t}+P_{n-3,t}\right]+p\,
P_{m_{-},t}.\nonumber\end{aligned}$$ Here, $p$ is the (uniform) probability for a walker to take a long-range jump, whereas $(1-p)/2$ is the probability to jump either left or right towards a nearest-neighbor site along the backbone. Without restriction of generality, let us assume that we happened to let all walks start from a site within this segment. Note that, unlike for the $1d$-walk above, there are *three* distinct types of sites even in the bulk of this problem: $P_{n\pm2,t}$, $P_{n\pm1,t}$, and $P_{n,t}$. In principle, one could expect three distinct return-time behaviors as a result. We will demonstrate below that, indeed, different sites possess differences in their return-time behavior, depending on their level in the hierarchy, but all scale with the same exponent $\mu$. Here, we choose to have the walk start on site $n-2$ within this segment: $$\begin{aligned}
P_{l,0} & = & \delta_{l,n-2}.
\label{eq:3RWinit}\end{aligned}$$
Using the generating function in Eq. (\[eq:generator\]) on Eqs. (\[eq:3RW\]-\[eq:3RWinit\]) yields: $$\begin{aligned}
\tilde{P}_{n+2} & = & a\left[\tilde{P}_{n+3}+\tilde{P}_{n+1}\right]+c\left[\tilde{P}_{n+4}+\tilde{P}_{n}\right]+p_{2}\,\tilde{P}_{m_{+}},\nonumber \\
\tilde{P}_{n+1} & = & b\left[\tilde{P}_{n+2}+\tilde{P}_{n}\right]+p_{1}\,\tilde{P}_{n-1},\nonumber \\
\tilde{P}_{n} & = & a\left[\tilde{P}_{n+1}+\tilde{P}_{n-1}\right]+c\left[\tilde{P}_{n+2}+\tilde{P}_{n-2}\right]+p_{2}\,\tilde{P}_{n\pm4},\nonumber\\
\tilde{P}_{n-1} & = & b\left[\tilde{P}_{n}+\tilde{P}_{n-2}\right]+p_{1}\,\tilde{P}_{n+1},\label{eq:3RWgen} \\
\tilde{P}_{n-2} & = &
a\left[\tilde{P}_{n-1}+\tilde{P}_{n-3}\right]+c\left[\tilde{P}_{n}+\tilde{P}_{n-4}\right]+p_{2}\,\tilde{P}_{m_{-}}+1,\nonumber \end{aligned}$$ where we have again absorbed the parameters $p$ and $z$ into general “hoping rates”, which are initially $$\begin{aligned}
a^{(0)} & = & b^{(0)}=\frac{z}{2}\left(1-p\right),\nonumber \\
c^{(0)} & = & 0,\label{eq:3parainit}\\
p_{1}^{(0)} & = & p_{2}^{(0)}=z\, p.\nonumber \end{aligned}$$ Note that we have added new terms with a parameter $c$, which is zero initially. As Fig. \[fig:RG3RW\] depicts, such links are not present in the original network HN3, but have to be taken into account during the RG process. In all, a surprising number of parameters, five in all, is required even for the most symmetric set of initial conditions to obtain a closed set of RG recursion equations. Unlike for the one-dimensional walk in Eq. (\[eq:1dRWgen\]), the parameters $a$ and $b$ here do not express a directional bias or drift along the backbone. Instead, they are necessary merely to distinguish between hops out of (currently) odd and even sites, respectively, as shown in Fig. \[fig:RG3RW\].
![Depiction of the (exact) RG step for random walks on HN3. Hopping rates from one site to another along a link are labeled at the originating site. The RG step consists of tracing out odd-labeled variables $\tilde{P}_{n\pm1}$ in the top graph and expressing the renormalized rates $(a',b',c',p_{1}',p_{2}')$ on the bottom in terms of the ones $(a,b,c,p_{1},p_{2})$ from the top. The node $\tilde{P}_{n}$, bridged by a (dotted) link between $\tilde{P}_{n-1}$ and $\tilde{P}_{n+1}$, is special as it *must* have $n=2(2j+1)$ and is to be decimated at the following RG step, justifying the designation of $p_{1}'$. Note that the original graph in Fig. \[fig:3hanoi\] does not have the green, dashed links with hopping rates $(c,c')$, which *emerge* during the RG recursion. []{data-label="fig:RG3RW"}](RG3RWnew "fig:") ![Depiction of the (exact) RG step for random walks on HN3. Hopping rates from one site to another along a link are labeled at the originating site. The RG step consists of tracing out odd-labeled variables $\tilde{P}_{n\pm1}$ in the top graph and expressing the renormalized rates $(a',b',c',p_{1}',p_{2}')$ on the bottom in terms of the ones $(a,b,c,p_{1},p_{2})$ from the top. The node $\tilde{P}_{n}$, bridged by a (dotted) link between $\tilde{P}_{n-1}$ and $\tilde{P}_{n+1}$, is special as it *must* have $n=2(2j+1)$ and is to be decimated at the following RG step, justifying the designation of $p_{1}'$. Note that the original graph in Fig. \[fig:3hanoi\] does not have the green, dashed links with hopping rates $(c,c')$, which *emerge* during the RG recursion. []{data-label="fig:RG3RW"}](RG3RWnew_after "fig:")
The RG update step consist of eliminating from these five equations those two that refer to an odd index, $n\pm1$. As a first step, adding the two relations referring to the indices $n\pm1$, we obtain $$\begin{aligned}
\tilde{P}_{n+1}+\tilde{P}_{n-1} & = &
\frac{b}{1-p_{1}}\left[\tilde{P}_{n+2}+2\tilde{P}_{n}+\tilde{P}_{n-2}\right],\end{aligned}$$ which allows us to eliminate any reference to odd sites $n\pm1$ from the middle relation in (\[eq:3RWgen\]). Furthermore, solving for $\tilde{P}_{n\pm1}$ explicitly, $$\begin{aligned}
\tilde{P}_{n\pm1} & = &
\frac{b}{1-p_{1}^{2}}\,\tilde{P}_{n\pm2}+\frac{bp_{1}}{1-p_{1}^{2}}\,\tilde{P}_{n\mp2}+\frac{b}{1-p_{1}}\,\tilde{P}_{n},\end{aligned}$$ and inserting into the relations for $\tilde{P}_{n\pm2}$ in Eq. (\[eq:3RWgen\]) results in $$\begin{aligned}
\tilde{P}_{n+2} & = & \frac{\left[ab+c\left(1-p_{1}\right)\right]\left(1+p_{1}\right)}{1-p_{1}^{2}-2ab}\left[\tilde{P}_{n+4}+\tilde{P}_{n}\right]\nonumber \\
&&\quad+\frac{abp_{1}}{1-p_{1}^{2}-2ab}\left[\tilde{P}_{n+6}+\tilde{P}_{n-2}\right]\nonumber \\
&&\quad+\frac{p_{2}\left(1-p_{1}^{2}\right)}{1-p_{1}^{2}-2ab}\,\tilde{P}_{m_{+}},\nonumber\\
\tilde{P}_{n} & = & \frac{ab+c\left(1-p_{1}\right)}{1-p_{1}-2ab}\left[\tilde{P}_{n+2}+\tilde{P}_{n-2}\right]\nonumber \\
&&\quad+\frac{p_{2}\left(1-p_{1}\right)}{1-p_{1}-2ab}\,\tilde{P}_{n\pm4},\label{eq:3RWgen_reno}\\
\tilde{P}_{n-2} & = & \frac{\left[ab+c\left(1-p_{1}\right)\right]\left(1+p_{1}\right)}{1-p_{1}^{2}-2ab}\left[\tilde{P}_{n}+\tilde{P}_{n-4}\right]\nonumber \\
&&\quad+\frac{abp_{1}}{1-p_{1}^{2}-2ab}\left[\tilde{P}_{n+2}+\tilde{P}_{n-6}\right]\nonumber \\
& &
\quad+\frac{p_{2}\left(1-p_{1}^{2}\right)}{1-p_{1}^{2}-2ab}\,\tilde{P}_{m_{-}}+\frac{1-p_{1}^{2}}{1-p_{1}^{2}-2ab}.\nonumber\end{aligned}$$ Given that $n$ is only once divisible by 2, either $n-2$ or $n+2$ must be divisible by 2 at most twice, and we assume without restriction of generality that $n+2$ satisfies this description, whereas $n-2$ is divisible by a higher power of 2. (This also selects the upper sign in the index “$n\pm4$”). As in Eq. (\[eq:amplitude-reno\]), we now add the proper superscript for the $k$th RG step and obtain the renormalized generating functions at step $k+1$ as $$\begin{aligned}
\tilde{P}_{l}^{(k+1)} &
=\frac{1-\left(p_{1}^{(k)}\right)^{2}-2a^{(k)}b^{(k)}}{1-\left(p_{1}^{(k)}\right)^{2}}
& \tilde{P}_{2l}^{(k)}.
\label{eq:3RWampli-reno}\end{aligned}$$ The proportionality factor in this relation arises from the necessity to preserve the unity of the initial condition, as in Eq. (\[eq:amplitude-reno\]) above. Note that we would have obtained the *identical* factor, if we had chosen to start the walker from the central site $n$ instead of $n+2$. Starting at a site like $n\pm1$, in turn, would have provided a different factor, $$\begin{aligned}
\tilde{P}_{l}^{(k+1)} & = &
\frac{1-p_{1}^{(k)}-2a^{(k)}b^{(k)}}{1-p_{1}^{(k)}}\,\tilde{P}_{2l}^{(k)},
\label{eq:3RWampli-reno2}\end{aligned}$$ potentially leading to a distinct return-time behavior. Yet, asymptotically for large times and distances both alternatives prove identical to sufficiently high order as to not affect the scaling discussed below.
Proceeding with the RG step, the role of the central site in the new segments is then specified via $\frac{n+2}{2}\to n$ such that $\tilde{P}_{n-2}^{(k)}\to\tilde{P}_{n-2}^{(k+1)}$, $\tilde{P}_{n}^{(k)}\to\tilde{P}_{n-1}^{(k+1)}$, $\tilde{P}_{n+2}^{(k)}\to\tilde{P}_{n}^{(k+1)}$, and correspondingly for the functions on the left-hand side of Eqs. (\[eq:3RWgen\_reno\]), which now reads $$\begin{aligned}
\tilde{P}_{n}^{(k+1)} & = & a^{(k+1)}\left[\tilde{P}_{n+1}^{(k+1)}+\tilde{P}_{n-1}^{(k+1)}\right]\nonumber \\
&&\quad+c^{(k+1)}\left[\tilde{P}_{n+2}^{(k+1)}+\tilde{P}_{n-2}^{(k+1)}\right]+p_{2}^{(k+1)}\,\tilde{P}_{n\pm4}^{(k+1)},\nonumber \\
\tilde{P}_{n-1}^{(k+1)} & = & b^{(k+1)}\left[\tilde{P}_{n}^{(k+1)}+\tilde{P}_{n-2}^{(k+1)}\right]+p_{1}^{(k+1)}\,\tilde{P}_{n+1}^{(k+1)},\nonumber\\
\tilde{P}_{n-2}^{(k+1)} & = & a^{(k+1)}\left[\tilde{P}_{n-1}^{(k+1)}+\tilde{P}_{n-3}^{(k+1)}\right]\label{eq:3RWgen_reno_new} \\
&&\quad+c^{(k+1)}\left[\tilde{P}_{n}^{(k+1)}+\tilde{P}_{n-4}^{(k+1)}\right]\nonumber \\
&&\quad+p_{2}^{(k+1)}\,\tilde{P}_{m_{-}}^{(k+1)}+1,\nonumber\end{aligned}$$ These equations have *exactly* the desired form of the corresponding unrenormalized ones in Eqs. (\[eq:3RWgen\]), necessitating renomalization recursions for the parameters of the form $$\begin{aligned}
a^{(k+1)} & = & \frac{\left[a^{(k)}b^{(k)}+c^{(k)}\left(1-p_{1}^{(k)}\right)\right]\left(1+p_{1}^{(k)}\right)}{1-\left(p_{1}^{(k)}\right)^{2}-2a^{(k)}b^{(k)}},\nonumber \\
\nonumber \\b^{(k+1)} & = & \frac{a^{(k)}b^{(k)}+c^{(k)}\left(1-p_{1}^{(k)}\right)}{1-p_{1}^{(k)}-2a^{(k)}b^{(k)}},\nonumber \\
\nonumber \\c^{(k+1)} & = & \frac{a^{(k)}b^{(k)}p_{1}^{(k)}}{1-\left(p_{1}^{(k)}\right)^{2}-2a^{(k)}b^{(k)}},\label{eq:RG3RWfp}\\
\nonumber \\p_{1}^{(k+1)} & = & \frac{p_{2}^{(k)}\left(1-p_{1}^{(k)}\right)}{1-p_{1}^{(k)}-2a^{(k)}b^{(k)}},\nonumber \\
\nonumber \\p_{2}^{(k+1)} & = &
\frac{p_{2}^{(k)}\left[1-\left(p_{1}^{(k)}\right)^{2}\right]}{1-\left(p_{1}^{(k)}\right)^{2}-2a^{(k)}b^{(k)}}.\nonumber \end{aligned}$$ The analysis of the fixed points for $k\to\infty$ of Eqs. (\[eq:RG3RWfp\]) is surprisingly subtle. Of course, we obtain a rather simple fixed point for the choice of $p=0$, which eliminates all long-range jumps. Then, $c^{(k)}$, $p_{1}^{(k)}$, and $p_{2}^{(k)}$ vanish for $k=0$ in the initial conditions in (\[eq:3parainit\]) and remain zero for all $k>0$, according to Eqs. (\[eq:RG3RWfp\]). As a consequence, the distinction between $a^{(k)}$ and $b^{(k)}$ disappears and both recursions reduce *exactly* to that for the unbiased one-dimensional walk in Eq. (\[eq:RGbulkunbias\]) with $a^{*}=b^{*}=\frac{1}{2}$, leading to ordinary diffusion with $d_{w}=2$ and $\mu=\frac{3}{2}$, as discussed in Sec. \[sub:RG-for-2RW\]. Clearly, this fixed point is unstable with respect to variations in $p$.
For *any* probability $p>0$ inserted in Eqs. (\[eq:3parainit\]), the recursions in (\[eq:RG3RWfp\]) evolve towards an apparent fixed point at $a^{*}=b^{*}=c^{*}=0$ and $p_{1}^{*}=p_{2}^{*}=1$. But these recursions are *singular* at such a fixed point, requiring a more detailed investigation. If we choose an arbitrarily small $\delta>0$ and set $p=1-\delta$, we find that for all $k\geq0$ it is $a^{(k)}\sim b^{(k)}\sim c^{(k)}=O(\delta)$ and $p_{1}^{(k)}\sim p_{2}^{(k)}=1-O(\delta)$. Hence, setting $\delta=0$ in the end indeed validates this fixed point. Yet, the physics of this fixed point, corresponding to choosing $p=1$, is trivial and does not reflect the numerical observations: Strictly for $p=1$ there is *no* transport at all along the backbone, and any walker is confined forever to jump back-and-forth on the first long-range link it accesses, which would imply $d_{w}=\infty$ in Eq. (\[MSDeq\]). For any choice of $0<p<1$, no matter how close to unity $p$ gets, at long-enough times the walker “escapes” along the backbone sufficiently often to explore ever-longer jumps both, to prevent confinement and to exceed ordinary diffusion. Thus, we must conclude that even this confinement fixed point is unstable and there has to be a third fixed point, at least.
To find this fixed point, we have to move beyond looking at the stationary behavior ($k=\infty$) of Eqs. (\[eq:RG3RWfp\]). Although Eqs. (\[eq:RG3RWfp\]) represent a five-dimensional parameter space, there do not appear to be any further stationary points reachable from the initial conditions in Eqs. (\[eq:3parainit\]) aside from those two already discussed. As all flow appears to converge towards the singular confinement fixed point, we make an Ansatz inserted into Eqs. (\[eq:RG3RWfp\]) for $k\gg1$ that explores asymptotically the boundary layer [@BO] in its vicinity: $$\begin{aligned}
y^{(k)} & \sim & A_{y}\alpha^{-k}\quad\left(y\in\left\{
a,b,c,1-p_{1},1-p_{2}\right\} \right),
\label{eq:Ansatz0}\end{aligned}$$ with the assumption that $\alpha>1$. Expanding Eqs. (\[eq:RG3RWfp\]) to leading order in $\alpha^{-k}$, we find an over-determined system of equations
$$\begin{aligned}
\frac{1}{\alpha}A_{a} & = & \frac{1}{\alpha}A_{b}=\frac{A_{a}^{2}}{A_{1-p_{1}}}+A_{c},\nonumber \\
\frac{1}{\alpha}A_{c} & = & \frac{A_{a}^{2}}{2A_{1-p_{1}}},\label{eq:A}\\
\frac{1}{\alpha}A_{1-p_{1}} & = & A_{1-p_{2}}-\frac{2A_{a}^{2}}{A_{1-p_{1}}},\nonumber \\
\frac{1}{\alpha}A_{1-p_{2}} & = & A_{1-p_{1}}-\frac{A_{a}^{2}}{A_{1-p_{1}}}.\nonumber \end{aligned}$$
Exercising the freedom to choose $A_{a}=1$, we find$$\begin{aligned}
A_{b} & = & A_{a}=1,\nonumber \\
A_{c} & = & \frac{1}{2\phi},\nonumber \\
A_{1-p_{1}} & = & 2,\label{eq:As}\\
A_{1-p_{2}} & = & \phi^{2},\nonumber \end{aligned}$$ where $$\phi=\frac{\sqrt{5}+1}{2}=1.6180\ldots
\label{eq:goldensection}$$ is the “golden ratio” [@Livio03], and obtain the eigenvalue equation $$\alpha^{3}\left(\alpha+3\right)=8.
\label{eq:eigenvalue}$$ It has a unique solution that satisfies the condition on the boundary layer, $\alpha>1$, namely $$\begin{aligned}
\alpha & = & \frac{2}{\phi}=1.2361\ldots.
\label{eq:alpha1}\end{aligned}$$
Thus, we found another fixed point, lurking in the boundary layer of the confinement fixed point but with very distinct physical properties from it. Each reduction of the system size by a factor of 2 is accompanied by a rescaling of the hopping parameters by a factor of $\alpha^{-1}$, bringing them closer to confinement, yet, leaving just enough room to escape and find still longer jumps. In fact, at this point of the analysis it is not even obvious whether these ever less frequent escapes from total confinement ultimately would result in sub-diffusive, normal, or super-diffusive behavior.
The leading-order Ansatz in Eq. (\[eq:Ansatz0\]) merely provided the existence of a third fixed point at infinite times. Extending the analysis to include finite-time corrections (i. e., $\epsilon=1-z\ll1$), we include first-order corrections, $$\begin{aligned}
y^{(k)} & \sim & A_{y}\alpha^{-k}\left\{ 1+\epsilon
B_{y}\beta^{k}+\ldots\right\} ,
\label{eq:Ansatz}\end{aligned}$$ expand the recursions in (\[eq:RG3RWfp\]) in $\epsilon$, and then also use the fact that $\alpha^{-k}$ is small. Using the leading-order constants $A_{y}$ in Eqs. (\[eq:As\]), also the next-leading constants $B_{y}$ are determined self-consistently. We extract, again uniquely, $$\beta=2\alpha=\frac{4}{\phi}
\label{eq:beta}$$ and find, choosing $B_{a}=1$, $$\begin{aligned}
B_{b} & = & B_{a}=1,\nonumber \\
B_{c} & = & \frac{4}{5}\,\phi,\nonumber \\
B_{1-p_{1}} & = & -\frac{6}{5},\label{eq:Bs}\\
B_{1-p_{2}} & = & -\frac{8}{5\phi^{2}}.\nonumber \end{aligned}$$ Accordingly, time rescales as $$\begin{aligned}
T & \to & T'=\frac{4}{\phi}\, T,
\label{eq:Tscal}\end{aligned}$$ and we obtain from Eq. (\[MSDeq\]) with $T\sim L^{d_{w}}$ for the diffusion exponent for HN3: $$\begin{aligned}
d_{w} & = & 2-\log_{2}\phi=1.30576\ldots.
\label{eq:D-expo}\end{aligned}$$ Our simulations in Fig. \[fig:MSDextra\] are in excellent agreement with this result for $d_{w}$.
![Plot of the results from simulations of the mean-square displacement of random walks on HN3 displayed in Fig. \[fig:3hanoi\]. More than $10^{7}$ walks were evolved up to $t_{{\rm max}}=10^{6}$ steps to measure $\langle r^{2}\rangle_{t}$. The data is extrapolated according to Eq. (\[MSDeq\]), such that the intercept on the vertical axis determines $d_{w}$ asymptotically. The exact result from Eq. (\[eq:D-expo\]) is indicated by the arrow. []{data-label="fig:MSDextra"}](3H_MSDextra)
To extract the scaling behavior of the first-return distribution $Q_{0}(t)$ defined in Eq. (\[eq:Qt\]), we proceed similar to the discussion in Sec. \[sub:RG-for-2RW\]. Here, we will find that different sites behave differently with respect to their return-time behavior, which is not surprising as the hierarchy of long-range jumps restricts translational invariance along the backbone. As in the discussion of first returns in Sec. \[sub:RG-for-2RW\], we anticipate that we need an expansion of the parameters to order $\epsilon^{2}$, i. e., we extend the Ansatz in Eq. (\[eq:Ansatz\]) even further. We find that it is sufficient to use $$\begin{aligned}
y^{(k)} & \sim & A_{y}\alpha^{-k}\left\{ 1+\epsilon B_{y}\beta^{k}+\epsilon^{2}C_{y}\beta^{2k}+\ldots\right\}\nonumber \\
& \sim & A_{y}\left\{ \alpha^{-k}+\epsilon
B_{y}2^{k}+\epsilon^{2}C_{y}\left(4\alpha\right)^{k}+\ldots\right\}
\label{eq:Ansatz2}\end{aligned}$$ with $$\begin{aligned}
C_{a} & = & C_{b}=\frac{121}{25}\,\frac{\phi}{16},\nonumber \\
C_{c} & = & \frac{121}{25}\,\frac{\phi^{2}}{16},\nonumber \\
C_{1-p_{1}} & = & -\frac{121}{25}\,\frac{\phi}{16},\label{eq:Cs}\\
C_{1-p_{2}} & = & -\frac{121}{25}\,\frac{1}{16\phi},\nonumber \end{aligned}$$ where we have also dropped at each order in $\epsilon$ terms that grow less than the leading exponential in $k$. For the coefficients $C_{y}$ there is no freedom to choose, as they are a result of quadratic terms of the previous order, $\epsilon B_{y}\beta^{k}$; other terms with that freedom are subdominant.
As for Eq. (\[eq:1d-elementary\]), after $k=K-1$ RG steps, the system has reduced to an elementary graph consisting of a single segment like that one shown in Fig. \[fig:RG3RW\] (left) with $n=2$ but with $p_{2}=0$. Now, all even sites ($l=0,2,4$) are no longer connected to a long-range jump. (Jumps of rate $c$ do not count as long-range, since $c\to0$ for $k\to\infty$, whereas $p_{1}\sim p_{2}\to1$.) We therefore consider two (extreme) possibilities: (1) Returns to a starting point at the boundary ($l=0,N$) or central site ($l=N/2$) on the network, and (2) returns to a site ($l=N/4$ or $l=3N/4$) with the longest-possible long-range jump on the original network.
First, we consider case (1) of starting at a boundary site, say, $l=0$. It is easy to show that the central site $l=N/2$ behaves identical to those on the boundary, and furthermore, even making the boundary sites more accessible be using periodic boundary conditions does not change the conclusion. We solve the system of equations $$\begin{aligned}
\tilde{P}_{0} & = & a\,\tilde{P}_{1}+c\,\tilde{P}_{2}+1,\nonumber \\
\tilde{P}_{1} & = & 2b\,\tilde{P}_{0}+b\,\tilde{P}_{2}+p_{1}\,\tilde{P}_{3},\nonumber \\
\tilde{P}_{2} & = & a\left[\tilde{P}_{1}+\tilde{P}_{3}\right]+2c\left[\tilde{P}_{0}+\tilde{P}_{4}\right],\label{eq:3RW-elemetary}\\
\tilde{P}_{3} & = & b\,\tilde{P}_{2}+2b\,\tilde{P}_{4}+p_{1}\,\tilde{P}_{1},\nonumber \\
\tilde{P}_{4} & = & a\,\tilde{P}_{3}+c\,\tilde{P}_{2},\nonumber \end{aligned}$$ where we have suppressed the superscript $^{(K-1)}$ on the generating function and the parameters alike. [\[]{}As we have learned from the $1d$-walk in Eq. (\[eq:1d-elementary\]), the hopping parameters originating from the boundary sites $l=0$ and $l=4$ have to be doubled due to the reflecting boundaries.\] The solution for $\tilde{P}_{0}$ of this system of equations yields $$\begin{aligned}
&&\tilde{P}_{0} =\label{eq:3RW-P0} \\
&&\frac{\left(1-2c^{2}\right)\left(1-p_{1}^{2}\right)-2ab\left[1+\left(1+2c\right)\left(1+p_{1}\right)\right]+2a^{2}b^{2}}{\left[\left(1-p_{1}\right)\left(1-2c\right)-4ab\right](1+2c)\left(1+p_{1}-2ab\right)}.\nonumber\end{aligned}$$ Notice that *both,* numerator and denominator, decay asymptotically like $\sim1-p_{1}^{2}$ near the stable fixed point. Dividing out that behavior, both behave as $1+y^{(K)}$ with $y^{(K)}$ as in Eq. (\[eq:Ansatz2\]), where $y$ here stands for a mix of coefficient that results from multiplying out the terms in numerator and denominator, respectively. Since we do not expect any spurious cancellations, the ratio of these two expressions *also* results in the form $1+y^{(K)}$. Hence, making superscripts reappear, we find from Eq. (\[eq:Ansatz2\]) $$\begin{aligned}
\frac{1}{\tilde{P}_{0}^{(K-1)}} & \sim & 1+{\cal
B}\,\epsilon\,2^{K}+{\cal
C}\,\epsilon^{2}\left(4\alpha\right)^{K}+\ldots,
\label{eq:3RW-Porigin}\end{aligned}$$ where we have also dropped the $\alpha^{-K}$-term in order $\epsilon^{0}$ and marked constants that are unimportant for the scaling with $K$ by calligraphy script. Using Eq. (\[eq:3RWampli-reno\]) inserted into the relation for $\tilde{Q}_{0}$ in Eq. (\[eq:Q\]), we find $$\begin{aligned}
\tilde{Q}_{0} & = & 1-\frac{1}{\tilde{P}_{0}^{(0)}}\nonumber \\
& = &
1-\frac{1}{\tilde{P}_{0}^{(K-1)}}\,\prod_{k=0}^{K-2}\left[1-\frac{2a^{(k)}b^{(k)}}{1-\left(p_{1}^{(k)}\right)^{2}}\right].
\label{eq:3RW-Qorigin}\end{aligned}$$ Ignoring at most a finite number of factors in the product (hence, missing an overall constant ${\cal A}$), we can expand the remaining factors in the product using the asymptotic expansion in Eq. (\[eq:Ansatz2\]) for $1\ll k\leq K-2\to\infty$: $$\begin{aligned}
&&\prod_{k=0}^{K-2}\left[1-\frac{2a^{(k)}b^{(k)}}{1-\left(p_{1}^{(k)}\right)^{2}}\right] \nonumber \\
&& \sim {\cal A}\prod_{k\gg1}^{K-2}\left[1-\alpha^{-k}+{\cal D}\,\epsilon\,2^{k}+{\cal E}\epsilon^{2}\left(4\alpha\right)^{k}+\ldots\right],\nonumber \\
& &\sim {\cal A}\left[1-{\cal D}\,\epsilon\sum_{k\gg1}^{K-2}2^{k}+{\cal E}\,\epsilon^{2}\sum_{k\gg1}^{K-2}\left(4\alpha\right)^{k}+\ldots\right],\nonumber \\
& &\sim {\cal A}+{\cal F}\,\epsilon\,2^{K}+{\cal
G}\,\epsilon^{2}\left(4\alpha\right)^{K}+\ldots.
\label{eq:Product}\end{aligned}$$ Note that $0<{\cal A}<1$, as each factor in the product must be between $\frac{1}{2}$ and 1 for any choice the probability $p$. Inserting Eqs. (\[eq:3RW-Porigin\]) and (\[eq:Product\]) into Eq. (\[eq:3RW-Qorigin\]), we obtain $$\begin{aligned}
\tilde{Q}_{0} & \sim & \left(1-{\cal A}\right)+{\cal
H}\,\epsilon\,2^{K}+{\cal
I}\,\epsilon^{2}\left(4\alpha\right)^{K}+\ldots.
\label{eq:non-recurQ}\end{aligned}$$ This result implies first that returns to sites like $l=0,N/2,N$ are *not* recurrent because walks may escape forever with a finite probability ${\cal A}$. Finally, an application of Eq. (\[eq:def-moments\]) produces $$\begin{aligned}
\left\langle t\right\rangle _{K} & \sim & 2^{K}\sim N,\\
\left\langle t^{2}\right\rangle _{K} & \sim &
\left(4\alpha\right)^{K}\sim N^{\log_{2}\left(4\alpha\right)},\end{aligned}$$ hence, with Eq. (\[eq:mu\]) and $\alpha=2/\phi$: $$\begin{aligned}
\mu & = & 2-\frac{1}{2-\log_{2}\left(\phi\right)}=1.23416\ldots.
\label{eq:3RWmu}\end{aligned}$$ A relation of this form, $\mu=2-1/d_{w}$, is commonly found for Lévy flights [@Metzler04], and the result is again borne out by our simulations, see Fig. \[fig:FR\].
![Plot of the probability $Q(\Delta t)$ of first returns to the origin after $\Delta t$ update steps on a system of unlimited size. Data was collected for three different walks on HN3 with $p=0.1$ (circles), $p=0.3$ (squares), and $p=0.8$ (diamonds). The data with the smallest and largest $p$ exhibit strong transient effects. The exact result in Eq. (\[eq:3RWmu\]), $\mu=1.234\ldots$, is indicated by the dashed line. []{data-label="fig:FR"}](P_F)
Case (2), referring to walkers starting near the longest jump in the network, is represented by the system $$\begin{aligned}
\tilde{P}_{0} & = & a\,\tilde{P}_{1}+c\,\tilde{P}_{2},\nonumber \\
\tilde{P}_{1} & = & 2b\,\tilde{P}_{0}+b\,\tilde{P}_{2}+p_{1}\,\tilde{P}_{3}+1,\nonumber \\
\tilde{P}_{2} & = & a\left[\tilde{P}_{1}+\tilde{P}_{3}\right]+2c\left[\tilde{P}_{0}+\tilde{P}_{4}\right],\label{eq:3RW-elemetary-long}\\
\tilde{P}_{3} & = & b\,\tilde{P}_{2}+2b\,\tilde{P}_{4}+p_{1}\,\tilde{P}_{1},\nonumber \\
\tilde{P}_{4} & = & a\,\tilde{P}_{3}+c\,\tilde{P}_{2},\nonumber \end{aligned}$$ with the constant term from the initial conditions now near an odd site, say, $l=1$. Solving for the launch site of the walk, we find $$\begin{aligned}
\tilde{P}_{1} & = &
\frac{1-2c-ab\left(3-2c\right)}{\left[\left(1-p_{1}\right)\left(1-2c\right)-4ab\right]\left(1+p_{1}-2ab\right)}.
\label{eq:3RW-P1}\end{aligned}$$ Note that in this case *only* the denominator of $\tilde{P}_{1}=\tilde{P}_{1}^{(K-1)}$ decays with $1-p_{1}$ on approaching the stable fixed point which will ensure that $\tilde{P}_{N/4}^{(0)}$ diverges, making this process recurrent on behalf of Eq. (\[eq:Q\]). In contrast to Eq. (\[eq:3RW-Porigin\]), we find here $$\begin{aligned}
&&\frac{1}{\tilde{P}_{1}^{(K-1)}}
\label{eq:3RW-P-long}\\
&& \sim \left(1-p_{1}\right)\left[1+{\cal J}\,\epsilon\,2^{K}+{\cal K}\epsilon^{2}\left(4\alpha\right)^{K}+\ldots\right],\nonumber\\
&& \sim A_{1-p_{1}}\alpha^{-K+1}\left[1+{\cal J}\,\epsilon\,2^{K}+{\cal K}\epsilon^{2}\left(4\alpha\right)^{K}+\ldots\right]\nonumber\\
&& \quad\left[1+\epsilon B_{1-p_{1}}\left(2\alpha\right)^{K-1}+\epsilon^{2}C_{1-p_{1}}\left(2\alpha\right)^{2K-2}+\ldots\right],\nonumber\\
&& \sim \left(2\alpha\right)\alpha^{-K}\left[1+{\cal
L}\,\epsilon\,\left(2\alpha\right)^{K}+{\cal
M}\,\epsilon^{2}\left(2\alpha\right)^{2K}+\ldots\right],\nonumber\end{aligned}$$ where the last step is justified by the observation that the leading terms order-by-order in $\epsilon$ in the first square bracket dominate over those in the second.
We have to apply Eq. (\[eq:3RWampli-reno2\]) for this case to obtain $$\begin{aligned}
\tilde{P}_{\frac{N}{4}}^{(0)} & = & \tilde{P}_{1}^{(K-1)}\,\prod_{k=0}^{K-2}\frac{1-p_{1}^{(k)}}{1-p_{1}^{(k)}-2a^{(k)}b^{(k)}}\end{aligned}$$ and from Eq. (\[eq:Q\]):$$\begin{aligned}
\tilde{Q}_{\frac{N}{4}} & = & 1-\frac{1}{\tilde{P}_{\frac{N}{4}}^{(0)}},\nonumber \\
& = & 1-\frac{1}{\tilde{P}_{1}^{(K-1)}}\,\prod_{k=0}^{K-2}\left[1-\frac{2a^{(k)}b^{(k)}}{1-p_{1}^{(k)}}\right].\label{eq:QN4}\end{aligned}$$ The product in Eq. (\[eq:QN4\]) behaves identically to that discussed above in Eq. (\[eq:Product\]) and hence will not alter the overall form of the expansion when multiplying $1/\tilde{P}_{1}^{(K-1)}$ in Eq. (\[eq:3RW-P-long\]). Then we get$$\begin{aligned}
\tilde{Q}_{\frac{N}{4}} & \sim & 1-\left(2\alpha\right)\alpha^{-K}\left[1+{\cal N}\,\epsilon\,\left(2\alpha\right)^{K}+{\cal O}\epsilon^{2}\left(2\alpha\right)^{2K}+\ldots\right],\nonumber \\
& \sim & 1-\left(2\alpha\right)\alpha^{-K}+{\cal P}\,\epsilon\,2^{K}+{\cal Q}\,\epsilon^{2}\left(4\alpha\right)^{K}+\ldots.\label{eq:Q-recur}\end{aligned}$$ This result is almost identical to that for returns to the origin above in Eq. (\[eq:non-recurQ\]), except that sites at the longest jump in the system are recurrent, i. e., $\tilde{Q}_{N/4}\equiv1$ for large systems ($K\to\infty$) and times ($\epsilon=1-z\to0$). (Presumably recurrence will gradually degrade from the strictly recurrent sites at the highest level in the hierarchy to those at the lowest.) Yet, the scaling of return times in form of the exponent $\mu$ is described by Eq. (\[eq:3RWmu\]) for *all* sites.
Generating Function for Random Walks on HN4\[sub:RG-for-4RW\]
-------------------------------------------------------------
Next, we consider a random walk on HN4. The “master-equation” [@Redner01] for the probability of the walker to be at site $n$, as defined in Eq. (\[eq:numbering\]), at time $t$ is given by $$\begin{aligned}
P_{n,t} & = & \frac{1-p}{2}\left[P_{n-1,t-1}+P_{n+1,t-1}\right]\nonumber \\ & &
\quad+\frac{p}{2}\left[P_{n-2^{i+1},t-1}+P_{n+2^{i+1},t-1}\right],
\label{eq:4RW}\end{aligned}$$ where $p$ is the probability to make a long-range jump. (Throughout, we considered $p$ uniform, independent of $n$ or $t$). To make the connection between $n$ and $i$ explicit, we rewrite Eq. (\[eq:4RW\]) as $$\begin{aligned}
P_{2^{i}\left(2j+1\right),t} & = &
\frac{1-p}{2}\left[P_{2^{i}\left(2j+1\right)-1,t-1}+P_{2^{i}\left(2j+1\right)+1,t-1}\right]\nonumber\\
& &+\frac{p}{2}\left[P_{2^{i}\left(2j-1\right),t-1}+P_{2^{i}\left(2j+3\right),t-1}\right],\nonumber \\
P_{0,t} & = & \frac{1-p}{2}\left[P_{-1,t-1}+P_{1,t-1}\right]+p\,
P_{0,t-1}.
\label{eq:new4RW}\end{aligned}$$ Note that the case $n=0$ is not covered by Eq. (\[eq:numbering\]) and, hence, must be treated separately. Here, we choose the site $n=0$ to be the only one connected to itself such that HN4 is 4-regular throughout, as depicted in Fig. \[fig:4hanoi\].
It is straightforward to apply the generating function in Eq. (\[eq:generator\]) again, assuming, for simplicity, the initial condition $$\begin{aligned}
P_{n,0} & = & \delta_{n,0}.
\label{eq:RW4init}\end{aligned}$$ We obtain $$\begin{aligned}
\tilde{P}{}_{2^{i}\left(2j+1\right)} & = & \frac{1-p}{2}\, z\,\left[\tilde{P}_{2^{i}\left(2j+1\right)-1}+\tilde{P}_{2^{i}\left(2j+1\right)+1}\right]\nonumber\\
& &+\frac{p}{2}\, z\,\left[\tilde{P}_{2^{i}\left(2j-1\right)}+\tilde{P}_{2^{i}\left(2j+3\right)}\right],\nonumber \\
\tilde{P}_{0}-1 & = & \frac{1-p}{2}\,
z\,\left[\tilde{P}_{-1}+\tilde{P}_{1,}\right]+p\, z\,\tilde{P}_{0}.
\label{eq:new4RWgen}\end{aligned}$$ While the overall structure of this problem is even more symmetric than for HN3 in Sec. \[sub:RG-for-3RW\], a RG treatment does not seem possible in this case. Tracing out all odd sites would immediately interconnect *all* other remaining sites. (The resulting infinite set of coupled equations may have certain symmetry properties that would lend themselves for a recursive treatment. We have not yet explored such a possibility.)
In contrast to an ordinary lattice, say, it is also not straightforward to solve this equation by a Fourier transform such as $$\begin{aligned}
F(z,\phi) & = & \sum_{n=-\infty}^{\infty}\tilde{P}_{n}(x)\, e^{n\phi
I},
\label{eq:Fourier}\end{aligned}$$ defining $I=\sqrt{-1}$. Considering the $1-p$ terms, originating from nearest-neighbor jumps, and the $p$ terms, originating from long-range jumps, in Eqs. (\[eq:new4RWgen\]) separately provides for regularly-space patterns, level-by-level in the hierarchy. But the mixing of nearest-neighbor and long-range jumps destroys this regularity. Hence, we resort to transforming the Eqs. (\[eq:new4RWgen\]) in each level $i$ with a partial transform, $$\Pi_{i}(z,\phi) =
\sum_{j=-\infty}^{\infty}\tilde{P}_{2^{i}(2j+1)}(z)\,\exp\left\{
2^{i}\left(2j+1\right)\phi I\right\} .
\label{eq:level-transform}$$ Inserting Eq. (\[eq:level-transform\]) and application of the general theorem $$\begin{aligned}
\sum_{i=1}^{\infty}\sum_{j=-\infty}^{\infty}f_{2^{i}\left(2j+1\right)\pm1}
& = & \sum_{j=-\infty}^{\infty}f_{2j+1}-f_{\pm1},
\label{eq:theorem}\end{aligned}$$ which results because $2^{i}(2j+1)$ for $i\geq1$ exactly runs over all even numbers $\not=0$ and over all odd numbers for $i=0$, yields for Eqs. (\[eq:new4RWgen\]): $$\begin{aligned}
&&\sum_{i=1}^{\infty}\Pi_{i}\left[1-pz\cos\left(2^{i+1}\phi\right)\right]\nonumber\\
&& \quad = (1-p)z\cos\phi\Pi_{0}+1-(1-pz)\tilde{P}_{0},\nonumber \\
\nonumber \\
&&\Pi_{0}\left[1-pz\cos\left(2\phi\right)\right]
\label{eq:4RWtransformed}\\
&& \quad =
(1-p)z\cos\phi\sum_{i=1}^{\infty}\Pi_{i}+(1-p)z\cos\phi\tilde{P}_{0}.\nonumber\end{aligned}$$ We can combine both relations to get $$\begin{aligned}
1&=&\left[1-zp-(1-p)z\cos\phi\right]F\nonumber\\
&& \quad+zp\sum_{i=0}^{\infty}\left[1-\cos\left(2^{i+1}\phi\right)\right]\Pi_{i},
\label{eq:F-Pi}\end{aligned}$$ using Eqs. (\[eq:Fourier\]) and (\[eq:level-transform\]) to eliminate $\tilde{P}_{0}$ via $$\begin{aligned}
\sum_{i=0}^{\infty}\Pi_{i} & = & \sum_{i=0}^{\infty}\sum_{j=-\infty}^{\infty}\tilde{P}_{2^{i}(2j+1)}\,\exp\left\{ 2^{i}\left(2j+1\right)\phi I\right\} ,\nonumber\\
& = & \sum_{n=-\infty}^{\infty}\tilde{P}_{n}\, e^{n\phi I}-\tilde{P}_{0},\\
& = & F-\tilde{P}_{0}.\nonumber\end{aligned}$$ At this point, there does not seem to be any further progress possible on Eq. (\[eq:F-Pi\]), due to the term $\sum_{i}\Pi_{i}\cos\left(2^{i+1}\phi\right)$, which resembles a Weierstrass function [@Hughes81]. At best, on could try to extract information about the moments of the walk, $$\begin{aligned}
\left\langle n^{k}\right\rangle _{t} & = & \sum_{n=-\infty}^{\infty}n^{k}P_{n,t},\end{aligned}$$ via the moment-generating function$$\begin{aligned}
M_{k}(z) & = & \sum_{t=0}^{\infty}\left\langle n^{k}\right\rangle _{t}\, z^{t},\nonumber \\
& = & \left[-I\partial_{\phi}\right]^{k}F(z,\phi)|_{\phi=0}.\label{eq:Moment-gen}\end{aligned}$$ Note that the 2nd moment $M_{2}(z)$ already would provide the exponent $d_{w}$ on behalf of the definition in Eq. (\[MSDeq\]). [\[]{}All odd moments vanish, of course, as Eq. (\[eq:F-Pi\]) is even in $\phi$.\] The 0th moment, setting $\phi=0$ in Eq. (\[eq:F-Pi\]), simply results in $$\begin{aligned}
M_{0}(z) & = & \frac{1}{1-z},\end{aligned}$$ which just demonstrates that everything is properly normalized, $ $${\cal N}_{t}=\left\langle n^{0}\right\rangle _{t}=\sum_{n=-\infty}^{\infty}P_{n,t}=1$, at all times $t$. But already the 2nd moment would lead to terms containing $\sum_{i}\Pi_{i}4^{i}$, which we can not account for, even at $\phi=0$ and in the limit $z\to1^{-}$.
Instead, we note that the long-time behavior is dominated by the long-range jumps, as discussed for HN3 in Sec. \[sub:RG-for-3RW\]. To simplify matters, we set $p=1/2$ here, although any finite probability would lead to the same conclusions. We make an “annealed” approximation, i. e., we assume that we happen to be at some site $n$ in Eq. (\[eq:numbering\]) with probability $1/2^{i}$, corresponding to the relative frequency of such a site, yet independent of time or history. This ignores the fact that in the network geometry a long jump of length $2^{i}$ can be followed *only* by another jump of that length or a jump of unit length, and that many intervening steps are necessary to make a jump of length $2^{i+1}$, for instance. Here, at each instant the walker jumps a distance $2^{i}$ left or right irrespectively with probability $1/2^{i+1}$, and we can write $$\begin{aligned}
{\cal P}_{n,t} & = & \sum_{n'}T_{n,n'}{\cal P}_{n',t-1}\label{eq:Transfer}\end{aligned}$$ with $$\begin{aligned}
T_{n,n'} & = & \frac{a-1}{2a}\sum_{i=0}^{\infty}a^{-i}\left(\delta_{n-n',b^{i}}+\delta_{n-n',-b^{i}}\right),\label{eq:T}\end{aligned}$$ where $a=b=2$. Eqs. (\[eq:Transfer\]-\[eq:T\]) are identical to the Weierstrass random walk discussed in Refs. [@Hughes81; @Shlesinger93] for arbitrary $1<a<b^{2}$. There, it was shown that $d_{w}=\ln(a)/\ln(b)$, which leads to the conclusion that $d_{w}=1$ in Eq. (\[MSDeq\]) for HN4, as has been predicted (with logarithmic corrections) on the basis of numerical simulations in Ref. [@SWPRL].
Conclusions\[sec:Conclusions\]
==============================
We have show how the powerful tools of the dynamic renormalization group [@Redner01] allow to dissect this intricate random walk problem on the planar network HN3 with a “hidden” fixed point. Indeed, using a boundary-layer analysis, we unravel the irregular singularity of the dominant fixed point in a five-dimensional parameter space, resulting in a set of exact, non-trivial exponents describing super-diffusive transport. Adding just one more link to each site, we obtain a non-planar network HN4 which possess an even higher degree of symmetry, yet, for which we can only develop an equation for the generator and an alternative “annealed” treatment which provides results that are consistent with simulations. (We believe that a proper exploitation of the symmetry in HN4, which eludes us here, will ultimately make exact results possible.)
Aside from the singular fixed point, HN3 serves further as an instructive example for a network in which nodes have heterogeneous recurrence properties. The diffusion exponent $d_{w}$ is larger than the fractal dimension $d_{f}=1$ of the lattice backbone that the walk is embedded in, which usually implies recurrence [@Bollt05; @Condamin07]. Here, the near-confined state of the walk favors recurrences to sites in higher levels of the hierarchy, although the associated first-return exponent is the same for all sites for the time distribution of any given return.
We should also mention that our results for HN3 can have an alternative interpretation. If we ignore the one-dimensional lattice backbone and instead consider the network as graph without particular embedding, then Eq. (\[eq:3dia\]) for the diameter, or more specifically the average growth in neighborhood $S_{d}\sim d^{2}$ with jump-distance $d$ found in Fig. \[fig:HN3neighbors\], implies that the fractal dimension for that graph is $d_{f}=2$. The RG would discover the then-obscured asymmetry between the backbone and long-range jumps (even when starting with $p=1/3$) and lead to the same analysis. Yet, with all distances now being (on average) measured as the square-root of their separation along the backbone, also the mean-square displacement in Eq. (\[MSDeq\]) needs to be reevaluated, yielding a diffusion exponent twice its previous value, $d_{w}=2(2-\log_{2}\phi)=2.61\ldots$. In this interpretation, $d_{w}>d_{f}$ still applies, but walks are now sub-diffusive in this measure. Of course, an exponent that is independent of such a metric, like the purely event-base first return probability, does *not* change. In turn, the relation between $d_{w}$ and $\mu$ fails, consistent with the fact that the walk can no longer be considered a Lévy flight.
Finally, our results suggest that many other interesting transport phenomena, such as voter models, exclusion processes, or self-organized critical phenomena can be fruitfully studied on these networks, which are sufficiently complex for interesting results but sufficiently simple to be tractable. Especially in light of the tremendous interest in complex dynamics on designed structures, we hope that these networks can make a useful contribution [@Barabasi01; @Andrade05; @Hinczewski06; @Zhang07].
|
---
abstract: |
The majority of Internet traffic is caused by a relatively small number of flows (so-called elephant flows). This phenomenon can be exploited to facilitate traffic engineering: resource-costly individual flow forwarding entries can be created only for elephants, while serving mice over shortest paths.
Although this idea already appeared as a part of proposed TE systems, it was not examined by itself. It remains unknown what extent of flow table occupancy and operations number reduction can be achieved, how to select thresholds or sampling rates to cover the desired fraction of traffic or how to detect elephants with low computational and memory overhead.
In this paper, we use reproducible traffic models obtained from 30-day-long campus/residential trace covering 4 billion flows to answer these questions. The most important finding is that simple packet sampling performs surprisingly well on realistic traffic, reducing the number of flow entries by a factor up to 400 with the aim to cover 80% of the traffic. Its superb performance and negligible overhead questions the need for more sophisticated algorithms. We also provide an open-source software package allowing the replication of our experiments or the performing of similar evaluations for other algorithms or flow distributions.
author:
-
bibliography:
- './bib.bib'
title: 'Evaluation of Elephant-based Algorithms for Flow Table Reduction under Realistic Traffic Distributions '
---
flows, elephant, mice, heavy hitter, SDN, traffic engineering, sampling
Introduction
============
It is widely believed that the distribution of flow length and size in the Internet follows the Pareto principle: the majority of traffic is comprised of a relatively small number of flows. Such flows are called *elephant flows*. The remaining flows, which are large in number but carry very little traffic, are called *mice flows*. In practice, flow length and size distributions are even more long-tailed than the original Pareto rule (80/20) assumed. According to recent analysis, 80% of traffic is caused by only 0.2-0.4% of flows [@megyesi2013analysis] [@flows-agh].
The above-described phenomenon can be exploited to facilitate traffic engineering, QoS provisioning or security monitoring. For example, network operators often limit the rate of elephant flows or put them in low-priority queues in order to protect mice flows, which are usually associated with delay-sensitive services. However, the application which can benefit from the long-tailed nature of Internet traffic the most is flow-based traffic engineering. Per-packet routing imposes significant limitations. Due to routing loop prevention constrains, only a subset of disjoint paths existing between selected nodes in the network can be used [@dual]. Adaptive (load-sensitive) routing is also impossible in a per-packet approach, as the dynamic alteration of link costs leads to instability (route flapping) which ultimately deteriorates network performance; this has been shown by early ARPANET pitfalls [@Bertsekas82] and definitively proven in [@wang1992analysis].
Flow-based routing can overcome these problems by maintaining separate per-flow forwarding entries. It allows flows between the same endpoints to follow any number of alternative paths. Furthermore, paths for subsequent flows can be chosen with a current or predicted network load in mind, effectively resulting not only in multipath routing, but also in adaptive routing. Adaptive routing of flows is also more stable than selecting paths at the packet level, since the load on each link fluctuate more slowly, as has been shown in [@FAMTAR_IMPLEMENTATION]. All of this improves network utilisation and reduces the need for link oversubscription, as more traffic can be served using existing infrastructure.
Despite continuous technological advancements, the number of simultaneous flows in networks still overwhelms the capacities of switch flow tables [@shen2019powerful]. Moreover, in the case of centralised control plane usage, controller throughput can impose additional limits on the rate of incoming flows. This is confirmed in practice. Despite the reactive approach being taught as a primary mode during SDN courses, the real deployments like Google’s B4 [@jain2013b4] are limited to proactive systems. Such systems forward packets according to predefined, per-subnet shortest-path entries. Specific entries are created for heavy-hitter aggregates, which are detected basing on the out-of-band traffic analysis or external information (for example, notification concerning expected migration between datacentres). This means that existing SDN deployments are limited to proprietary inter-DC WANs. In case of networks which operator does not control endpoints, like public WAN, such an information is not available. A possible solution is to focus on dynamically identified elephant flows. This should significantly reduce the number of flow entries while simultaneously keeping most traffic covered by TE mechanisms.
The problem is *early* identification of whether a particular flow is, or more precisely will become, an elephant. Issues related to elephant flow detection have been the subject of many papers. However, most works focus on post factum identification and have been mainly directed in the context of flow accounting and network performance or security monitoring. In the case of traffic engineering, detection accuracy is not the most important issue. Instead, the focus should be put on the moment of identification, and particularly on the amount of traffic transmitted by the flows after their classification as elephants (i.e. when they have individual entries) and the resulting reduction in flow table occupancy. A vast number of proposed algorithms have not been analysed from that point of view.
Moreover, performance of any elephant-related mechanisms strictly depends upon flow length and size distribution and the definition of elephant flow. Different papers make different assumptions on this matter. In particular, traffic distributions used for evaluation differ vastly. Some papers make over-simplistic and arbitrary assumptions by considering constant elephant-to-mice ratio and sizes, which often do not correspond with reality. Other papers use distributions obtained from real traffic traces; however, they are either unreproducible or simplified to a single distribution function.
The aim of this paper is to fill this gap in research and provide a thorough analysis of the performance of elephant detection algorithms from the point of view of SDN traffic engineering. The key point of our research is the use of realistic, accurate and reproducible flow length and size distributions. We analyse the performance of three algorithms:
- *first*, which assumes a priori knowledge about flow length/size and classifies it accordingly since its first packet
- *threshold*, which classifies a flow as an elephant after transmitting a predefined amount of packets or bytes
- *sampling*, which performs packet sampling and classifies flows in a probabilistic manner
It should be noted that the *first* algorithm is impossible to achieve in reality: it is not possible to know in advance how long a flow will be. Nevertheless, it is interesting from an analytical point of view because it can provide upper boundaries for the performance of other solutions.
These algorithms are not particularly novel. Instead the contribution of this paper lies in:
- analysis of parameters relevant to traffic engineering in the context of SDN: fraction of traffic covered, reduction of number of flow entry operations (and thus a controller traffic) and reduction of flow table occupancy
- use of realistic and accurate distribution mixtures obtained from 30-day-long 4-billion flow trace of campus/residential traffic, which is many orders of magnitude more than in previous research
- reproducibility of the research, as both the distribution mixtures and code used for analysis are provided as an open-source package
The most important and surprising finding is, however, the performance of the *sampling* algorithm. It introduces a negligible overhead to the packet processing pipeline and does not require any memory (unlike counters or bloom filters), yet when applied to realistic traffic, it can reduce the number of flow entries by a factor of 400, while still maintaining 80% of the traffic covered by individual flow entries. Thus, it can provide a lower bound for the performance of other solutions and in many cases, it can eliminate the need for more sophisticated algorithms altogether.
We believe that this work will set a framework for the analysis and comparison of flow table occupancy reduction algorithms based on the distinction between mice and elephant flows and provide upper and lower bounds for the performance of such algorithms. We acknowledge that various networks can have different distributions; therefore, we an provide open-source software package allowing both the replication of our experiments and the performing of similar evaluations for other algorithms or flow distributions [@github-flow-models].
Related works
=============
The idea of performing adaptive routing only for elephant flows while keeping mice on the shortest paths is not new. According to our knowledge, it was first proposed in 1999 in [@rexford-long-flows]. The authors of that paper, however, did not solve the problem of detecting elephant flows. They propose the usage of per-flow counters or timers, which is pointless considering that our goal is the reduction in the number of tracked flows and flow table operations. Moreover, their analysis is based on one-week trace collected in 1997, which is both outdated and too short. A similar approach was proposed in [@sarrar2012leveraging], but concerned the top destination IP prefixes (so-called *heavy hitters*) instead of 5-tuple flows.
This approach has been recently reiterated, specifically in the SDN context. The general idea is to initially install shortest path wildcard entries and monitor the traffic in order to identify elephant flows. After identification, the controller can compute alternative non-congested paths for them based on the global network view and install individual entries for these most significant flows in order to load-balance traffic.
Hedera [@al2010hedera] was proposed as a dynamic flow scheduling system for datacentres, aimed at going beyond ECMP limitations. By default, all flows are load-balanced onto ECMP paths. Such a path is used until the flow grows and meets a predefined threshold rate. After reaching the threshold, elephants are rerouted in mid-connection onto flow-specific paths, computed dynamically by the controller. Hedera assumes that the edge switches collect flow statistics for all flows using OpenFlow counters. This means that it actually only reduces non-edge switches overhead, while the edge switches still has to maintain individual entries for all flows.
In 2011, Curtis et al. presented a system called Mahout [@curtis2011mahout]. Unlike Hedera, it performs the elephant detection at the end hosts by monitoring socket buffers (via a shim layer in the OS). After reaching a predefined threshold, it marks subsequent packets of flow using an in-band signalling mechanism. The switches in the network are configured to forward these marked packets to the controller, which as with Hedera computes the best path and installs flow-specific entries in switches. With that approach, monitoring overhead can be completely eliminated from the switches and controller, as elephant detection is moved to end hosts, which must be modifed.
Benson et al. in their MicroTE [@benson2011microte] paper propose a similar approach, although not based precisely on elephant detection. Each server in the network is provided with a kernel module to monitor and predict the traffic it sends. The MicroTE controller aggregates these statistics and computes optimal paths for such a traffic (called *predictable*). Weighted ECMP load-balancing is used for the remainder of traffic. Both Mahout and MicroTE require end host modifications, which means that they are usable only in the context of private datacentres.
DevoFlow [@curtis2011devoflow] is another example of a complete TE system based on modified Openflow switches, which key feature is the reduction of OpenFlow overhead by focusing on significant flows. In order to detect these flows, DevoFlow explores both *threshold* and *sampling* approaches, which is similar to this paper. However, the traffic model used in DevoFlow was a datacentre workload, which considerably differs from the residential/ISP load. Moreover, the authors “reverse engineered” the flow distributions they used for evaluation from plots presented in another paper and did not make them available, which makes their results both inaccurate and unreproducible.
In addition to the above, the DevoFlow paper authors do not analyse the amount of traffic covered. Instead, the only performance indicator they provide is the aggregate throughput of the whole network, which also depends on topology, demands matrix and routing decisions. Only absolute values of the number of flow entries are provided, so the reduction of table occupancy also cannot be determined. Only three thresholds/sampling probabilities are analysed, whereas our paper provides analysis for the continuous spectrum of values.
A similar system to DevoFlow is proposed in [@xu2017scalable]. It detects elephant flows only on edge switches with the use of a bloom filters variant, called *randomised counter sharing*. However, the traffic model used for the evaluation of this mechanism is oversimplified: the authors assume the power law for the flow-size distribution, where 20% of all flows account for 80% of traffic volume. This is, as shown in [@megyesi2013analysis] and [@flows-agh], far from reality.
OpenSample [@suh2014opensample] is a TE system based on the rerouting of elephant flows, which are detected by sampling packet headers on switches with sFlow. The authors claim that by using sFlow with TCP sequence numbers, it can achieve a low latency of measurements with a high degree of accuracy. Unlike Mahout and MicroTE, OpenSample can be implemented without end host modifications and unlike Hedera, it does not require the use of expensive OpenFlow counters. The authors provide analysis of the percentage of traffic covered after detection and after rerouting. However, the used traffic model is extremely simplified. They consider only two classes of flows, short flows with an exponential distribution of mean 1 MB size, and long flows with the same exponential distribution but a 1 GB mean flow size. The paper does not analyse flow table occupancy reduction.
Planck [@rasley2014planck] is another system, similar to OpenSample. It deserves special attention because it does not use sFlow for packet sampling. Instead, the authors propose the use of a port mirroring feature to redirect all packets to a single monitoring port. Because the total traffic forwarded through the switch usually exceeds the capacity of the monitoring port, some packets are dropped, which effectively provides a packet sample. Such an approach has several advantages over sFlow, specifically, the reduced load of switch CPU and significantly lower latencies. As a result of this, the Planck-based traffic engineering system can reroute congested flows within milliseconds, which can improve its overall performance.
The following surveys provide a good overview of SDN traffic engineering systems: [@akyildiz2014roadmap] [@abbasi2016traffic]. The use of packet sampling for traffic engineering purposes is indicated in [@cern-sampling-report], which also provides a good overview of other packet sampling techniques.
The use of packet sampling for elephant flow identification (but not for traffic engineering purposes) was the subject of [@Estan:2002:NDT:964725.633056], [@estan2003new], [@mori2004identifying], [@mori2010characterizing], [@guang2008online], [@zhang2010identifying] and [@ros2018high]. A two-stage sampling scheme is proposed in [@afek2018detecting]. An even more complicated, multi-stage algorithm based on packet sampling and correlation-based flow classification is presented in [@tang2019elephant].
It has to be noted that several other, non-sampling based techniques for elephant/heavy-hitters detection and flow table occupancy reduction have been proposed. This includes flow table compression and entry aggregation [@liu2010tcam] [@7810727] [@dulinski2020mpls], entry caching [@katta2016cacheflow], label-based switching [@huang2018software], use of multiple hash tables [@sivaraman2017heavy] [@wang2019distributed] [@da2018ideafix] and a variety of bloom filters or sketching-based approaches [@CAFARO2019770] [@huang2017sketchvisor] [@basat2017optimal] [@10.1145/3359989.3365408].
Moreover, in the wake of growing machine learning popularity, it recently started being employed as well. This encompass unsupervised [@kiran2019understanding] [@da2019predicting] [@dtorres] [@8990807] [@9045216] and supervised approaches [@10.4108/icst.iniscom.2015.258274] [@7549048] [@7785324] [@8868201], including deep reinforcement learning [@mu2018sdn]. With this paper, we argue that this is an overkill as satisfying results can be achieved with simple sampling techniques.
Most of the works mentioned in previous paragraphs focus, however, on the issue from the network monitoring or accounting point of view. The main evaluated parameters are related to detection accuracy; it includes FPR (false positive ratio) and FNR (false negative ratio) and the precision of flow length/size estimation from the sample. We, on the other hand, focus on parameters essential from the traffic engineering point of view: the average flow table occupancy reduction and the amount of traffic covered since detection.
Used traffic model
==================
Flow size and length distributions have a crucial impact on the performance of any elephant-related algorithms. Unfortunately, most papers make oversimplified assumptions on this matter. Those which use real empirical distributions do not publish them, making their research irreproducible.
In this paper we use flow length and size distributions from the `agh_2015` dataset presented in [@flows-agh]. These are based on traffic traces collected on the outgoing interface of campus/residential network over a period of 30 days and consisting of more than 4 billion flows. Both the timespan of the collection and the number of flows are many orders of magnitude higher than in previously published flow statistics. These distributions are in line with selected values of CAIDA and BME traces presented in [@megyesi2013analysis], which confirms their credibility.
To provide an overview of the properties of the traffic model used, we present its parameters in Tables \[tab-meta\] and \[tab-protocol\] and cumulative distribution functions (CDF). Figures \[cdf-length\] and \[cdf-size\] show CDFs of the number of flows and the total amount of packets and bytes as functions of flow length (number of packets) and flow size (amount of bytes), respectively.
------------------------- -------------- ----------------
**Based on dataset** agh\_2015
**Flow definition** 5-tuple unidirectional
**L2 technology** Ethernet
**Inactive timeout** 15 seconds
**Collection duration** 30 days
**Average flow length** 78.578370 packets
**Average flow size** 68410.894128 bytes
**Average packet size** 870.607188 bytes
------------------------- -------------- ----------------
: Metadata of used flow model [@flows-agh]
\[tab-meta\]
**TCP** **UDP** **Other**
------------- --------- --------- -----------
**Flows** 53.85 43.09 3.06
**Packets** 83.51 16.01 0.48
**Octets** 88.57 11.27 0.15
: Traffic shares by transport layer protocol (%) [@flows-agh]
\[tab-protocol\]
![Distributions in function of flow length (number of packets) [@flows-agh].[]{data-label="cdf-length"}](figures/cdf-length)
![Distributions in function of flow size (number of bytes) [@flows-agh].[]{data-label="cdf-size"}](figures/cdf-size)
--------------------------- ---------- ------------ ----------
**Flows of length up to**
(lr)[2-4]{} of flows of packets of bytes
1 47.8326 0.6087 0.1047
2 65.3421 1.0544 0.1728
4 74.8933 1.4696 0.2537
8 84.1319 2.1958 0.4412
16 90.5756 3.1633 0.7830
100 97.3478 6.3895 2.4322
1000 99.4544 14.3271 8.0737
10000 99.8922 30.3569 22.8925
100000 99.9896 67.8990 61.0966
1000000 99.9998 93.5945 92.1873
10000000 100.0000 98.4079 98.0464
--------------------------- ---------- ------------ ----------
: Selected values of empirical distributions (length) [@flows-agh]
\[tab-cdf-length\]
------------------------- ---------- ------------ ----------
**Flows of size up to**
(lr)[2-4]{} of flows of packets of bytes
64 4.3082 0.0548 0.0040
128 32.3376 0.4196 0.0424
256 56.8711 0.9477 0.1030
512 71.1101 1.4143 0.1780
1024 79.0397 1.9054 0.2622
6400 92.2723 3.9583 0.7748
64000 97.9108 8.6001 2.3759
640000 99.4866 18.3175 6.8464
6400000 99.8868 38.9130 18.3467
64000000 99.9802 64.1497 44.5602
640000000 99.9994 91.2676 86.7492
------------------------- ---------- ------------ ----------
: Selected values of empirical distributions (size) [@flows-agh]
\[tab-cdf-size\]
Additionally, the authors of [@flows-agh] provide histogram CSV files containing full data of all distributions; however, more important from our point of view are accurately fitted distribution mixture models. These allow an analytical calculation of all the performance parameters of the evaluated algorithms.
Algorithms
==========
First
-----
In the *first* algorithm, a flow entry is created on the arrival of the flow’s first packet when the length/size of flow will exceed the selected threshold. Thus, it assumes that based on the first packet, the length/size of the whole flow can be determined. This is impossible in practice. However, it is still worth evaluating such an approach because it can provide an upper bound for performance of all elephant detection algorithms. In particular it applies to algorithms which attempt to predict the flow length/size on the basis of header values, like machine learning based approaches. The flowchart of the *first* algorithm is show in Figure \[schema-first\].
![The *first* algorithm.[]{data-label="schema-first"}](figures/schema-first)
Threshold
---------
As already mentioned, in practice switches cannot know in advance whether a newly appeared flow will eventually become an elephant or a mice flow. Thus, they cannot create an entry for this flow when its first packet appears. Instead, flow entry can be created when the amount of traffic or the number of transmitted packets exceeds a certain elephant detection threshold. The most trivial approach is to use per-flow counters on each switch. A counter reaching a threshold would cause a flow entry to be created. This is the outline of *threshold* algorithm, which is also shown in Figure \[schema-threshold\].
![The *threshold* algorithm.[]{data-label="schema-threshold"}](figures/schema-threshold)
Such an algorithm is possible to be implemented in practice; however, it has obvious drawbacks. Per-flow counters must be stored and updated with each packet. In the case of OpenFlow, flow entries aimed at packet counting use the same memory as any other flow entry, so this would not yield any improvements in terms of flow table usage. In the case of other data plane technologies, the low-overhead implementation of accurate per-flow counters may be possible, as shown by TurboFlow for P4 [@turboflow].
The results of the *threshold* algorithm can also provide an upper bound for all algorithms, which are based on some kind of inexact counting, including bloom filters and sketching-based approaches.
Sampling
--------
An alternative approach is to use sampling. Packets without entries in flow tables can be randomly sampled with some probability $ p $. If a packet is sampled, a new flow entry is created and subsequent packets of that flow are forwarded in accordance with it, without being sampled. Otherwise, the packet is forwarded basing on aggregated (usually ECMP) entry without the creation of an flow entry and sampling is performed for the rest of the packets until an flow entry is created. We call this *sampling* algorithm, which is shown in Figure \[schema-sampling\].
![The *sampling* algorithm.[]{data-label="schema-sampling"}](figures/schema-sampling)
The probability that a flow has an entry in the flow entry after reaching $ n $ packets is given by:
$$p_{total} = 1 - (1 - p)^n$$
where *n* is the number of packets.\
Figure \[graph-probability\] presents $ p_{total} $ for two selected values of $ p $.
![Probability that a flow has an entry in flow table after reaching $ n $ packets[]{data-label="graph-probability"}](figures/probability)
This approach is conceptually similar to the “sample and hold” technique presented in [@Estan:2002:NDT:964725.633056], although it was proposed for monitoring rather then for traffic engineering purposes. The advantage of this method is that it is stateless, i.e. there is no need to store and update any kind of counters. Additionally, it has a negligible performance impact as computation of a random number can be performed using a hardware random generator or a simple software pseudorandom generator with a few CPU cycles.
Alternatively, systematic sampling can be performed: 1 in N packets can be sampled, where $ N = \frac{1}{p} $. Systematic sampling has an even lower computational overhead, but it can be suboptimal when sampling periodic populations. However, the performance of systematic and random sampling is not distinguishable in case of links where the degree of multiplexing of flows is high [@choi2005accuracy].
The *sampling* algorithm can also be used to sample flows according to their size. It is enough to scale the sampling probability for each packet proportionally to its size. In our simulation, we scaled provided sampling probabilities by relative packet size:
$$p_{scaled} = p \cdot \frac{packet\_size}{min\_packet\_size}$$
In this way, larger packets have greater probability of being sampled. Such an approach is called Non-Uniform Probabilistic Sampling in RFC 5475 [@rfc5475].
We acknowledge that sampling can also introduce overheads. For example sFlow routes sampled packets through the switch CPU, which limits sampling rate and introduce latencies. To solve that problem, low-overhead port mirroring based approach can be used, as proposed in Planck system [@rasley2014planck].
In our calculations and simulations, we assumed that sampling is performed only on an edge switches (i.e. any given packet is sampled only once, when it enters the network). An alternative setup would be a network in which all switches sample packets independently (i.e. a packet is sampled on all switches on its path). In such case, the effective sampling probability value used to read algorithm performance from the results tables has to be calculated using the following equation:
$$sampling\_probability = 1 - (1 - p)^{avg\_path\_len}$$
where:
- *p* is the sampling probability set on a single switch and
- *avg\_path\_len* is the average length of path in the network.\
Simulations
===========
In this section, we present the results of the evaluation of the *first*, *threshold* and *sampling* algorithms. The following parameters are analysed:
- *flow coverage* – the percentage of flows which were detected by the algorithm (i.e. an individual flow entry was created for them)
- *traffic coverage* – the percentage of traffic (bytes) in the network which were transmitted by flows after their detection (i.e. when they had an individual entry)
- *flow table occupancy* – the average number of entries in the flow table relative to keeping entries for all flows
In addition to the above, we present reciprocals of *flow coverage* and *occupancy* values, which may provide a better indicator of gain given by the algorithms:
- *operations reduction* – tells us by what factor the number of flow entry additions/removals (and thus the controller traffic) can be reduced
- *occupancy reduction* – tells us by what factor the average number of flow entries in tables can be reduced
For selected values of thresholds (in cases of *first* and *threshold* algorithms) and sampling probability (in the case of *sampling* algorithm) we performed packet-level simulations. Packets were randomly generated, basing on the distribution mixtures from the used traffic model. For each value of threshold/sampling probability, we performed the experiment five times with different random seeds, each time generating 1 billion flows and calculated mean values from these five runs. All algorithms are evaluated using both flow length (number of packets) and flow size (amount of bytes) as elephant classification criterion. Results of these simulations are shown in Tables \[tab-length\] and \[tab-size\].
In the case of *first* algorithm, flows which have a length/size that is higher than the threshold are classified as elephants and are added to the flow table on their first packet. Shorter/smaller flows are classified as mice and are never added to flow table. It can be seen that in the case of the *first* algorithm, operations number and occupancy reduction factors are the equal.
In the case of *threshold* algorithm, a flow is classified as elephant and added to the flow table when the number of packets/bytes it transmits exceeds the threshold. The average reduction of flow table occupancy is always higher than the reduction of the number of flow entry operations. This is expected as flows are added not with their first packet but with some subsequent packet, so they occupy the flow table for only a fraction of their lifetime. Unfortunately, the same applies to traffic (packets or bytes) coverage. In order to achieve similar traffic coverage, a lower threshold value has to be used than in the *first* approach, which also results in lower table occupancy reduction.
In the case of *sampling* algorithm, similarly as in *threshold* algorithm, the average reduction of flow table size is higher than the reduction of the number of flow entry operations, for the same reason. It can be also seen that, in general, decreasing sampling probability results in an exponential increase in the flow table occupancy reduction, but only in a linear decrease in traffic coverage.
------------------------------------- ---------- ------------ ------------ ---------- ------------ ------------ ---------- ---------- ------------ -----------
(lr)[2-4]{} (lr)[5-7]{} (l)[9-11]{} Traffic Operations Occupancy Traffic Operations Occupancy Traffic Operations Occupancy
coverage reduction reduction coverage reduction reduction coverage reduction reduction
(%) (x) (x) (%) (x) (x) (%) (x) (x)
1 99.89 1.92 1.92 99.71 1.92 2.60 1.00 100.00 1.00 1.00
2 99.82 2.88 2.88 99.52 2.88 4.06 5.00e-01 99.77 1.41 1.54
4 99.74 3.89 3.89 99.23 3.89 6.16 2.50e-01 99.47 2.04 2.41
8 99.56 5.99 5.99 98.77 5.99 10.28 1.25e-01 99.04 3.00 3.81
16 99.22 10.40 10.40 98.10 10.40 17.71 6.25e-02 98.43 4.53 6.09
32 98.75 17.32 17.32 97.16 17.32 29.15 3.12e-02 97.61 6.93 9.74
64 97.99 28.33 28.33 95.87 28.33 46.66 1.56e-02 96.46 10.83 15.78
128 96.99 44.05 44.05 94.16 44.05 73.62 7.81e-03 94.97 16.95 25.42
256 95.65 69.57 69.57 91.88 69.57 119.93 3.90e-03 92.96 26.88 41.30
512 93.79 115.98 115.98 88.88 115.98 198.05 1.95e-03 90.37 42.21 66.07
1024 91.44 191.38 191.38 84.96 191.38 318.15 9.76e-04 86.93 67.57 107.39
2048 88.45 300.49 300.49 79.73 300.49 503.95 4.88e-04 82.52 105.88 170.58
4096 84.16 469.59 469.59 72.77 469.59 827.40 2.44e-04 76.41 169.31 276.96
8192 77.78 775.64 775.64 64.01 775.64 1462.54 1.22e-04 69.26 271.58 453.25
16384 69.37 1399.51 1399.51 53.83 1399.51 2834.49 6.10e-05 61.21 431.17 735.66
32768 59.27 2794.15 2794.15 42.60 2794.15 6069.15 3.05e-05 50.30 727.99 1271.26
65536 47.29 6201.41 6201.41 31.09 6201.41 14399.51 1.52e-05 40.64 1229.39 2197.34
131072 34.27 15345.62 15345.62 20.65 15345.62 37977.86 7.62e-06 30.39 2283.27 4198.14
262144 22.41 42262.61 42262.61 12.47 42262.61 111279.85 3.81e-06 19.61 4994.85 9425.16
524288 13.26 130950.45 130950.45 6.84 130950.45 367074.49 1.90e-06 14.21 8402.36 16061.45
1048576 7.09 456577.43 456577.43 3.37 456577.43 1365306.53 9.53e-07 9.95 14669.12 28322.65
2097152 3.37 1799949.44 1799949.44 1.49 1799949.44 5604593.17 4.76e-07 6.21 27264.93 53215.32
------------------------------------- ---------- ------------ ------------ ---------- ------------ ------------ ---------- ---------- ------------ -----------
\[tab-length\]
------------------------------------- ---------- ------------ ----------- ---------- ------------ ------------ ---------- ---------- ------------ -----------
(lr)[2-4]{} (lr)[5-7]{} (l)[9-11]{} Traffic Operations Occupancy Traffic Operations Occupancy Traffic Operations Occupancy
coverage reduction reduction coverage reduction reduction coverage reduction reduction
(%) (x) (x) (%) (x) (x) (%) (x) (x)
64 100.00 1.04 1.04 99.90 1.04 1.58 1.00 100.00 1.00 1.00
128 99.95 1.53 1.53 99.83 1.53 2.45 5.00e-01 99.98 1.12 1.14
256 99.89 2.34 2.34 99.73 2.34 3.58 2.50e-01 99.90 1.49 1.61
512 99.82 3.43 3.43 99.59 3.43 5.12 1.25e-01 99.76 2.06 2.38
1024 99.73 4.76 4.76 99.41 4.76 7.23 6.25e-02 99.57 2.86 3.51
2048 99.60 6.74 6.74 99.14 6.74 10.60 3.12e-02 99.32 4.01 5.18
4096 99.38 10.03 10.03 98.77 10.03 15.83 1.56e-02 98.98 5.67 7.66
8192 99.10 15.02 15.02 98.28 15.02 23.52 7.81e-03 98.54 8.08 11.32
16384 98.73 22.18 22.18 97.62 22.18 34.87 3.90e-03 97.94 11.69 16.85
32768 98.22 32.86 32.86 96.73 32.86 52.09 1.95e-03 97.14 16.99 25.07
65536 97.52 49.23 49.23 95.53 49.23 78.37 9.76e-04 96.08 24.87 37.35
131072 96.58 74.00 74.00 93.93 74.00 118.38 4.88e-04 94.70 36.29 55.38
262144 95.32 111.45 111.45 91.80 111.45 180.52 2.44e-04 92.84 53.78 83.23
524288 93.60 170.39 170.39 89.00 170.39 279.56 1.22e-04 90.43 79.91 125.35
1048576 91.29 265.01 265.01 85.37 265.01 437.69 6.10e-05 87.36 119.62 189.44
2097152 88.29 414.00 414.00 80.71 414.00 690.84 3.05e-05 83.22 182.75 292.31
4194304 84.41 650.85 650.85 74.77 650.85 1112.19 1.52e-05 77.99 275.65 448.04
8388608 79.21 1053.94 1053.94 67.30 1053.94 1847.74 7.62e-06 71.80 415.82 685.60
16777216 72.51 1759.10 1759.10 58.24 1759.10 3152.64 3.81e-06 64.19 659.01 1096.09
33554432 64.46 2945.28 2945.28 47.41 2945.28 5653.28 1.90e-06 53.81 1052.41 1793.71
67108864 53.75 5289.09 5289.09 34.77 5289.09 11787.26 9.53e-07 43.64 1804.65 3155.34
134217728 38.69 12142.54 12142.54 22.15 12142.54 31499.08 4.76e-07 32.17 3176.87 5777.40
268435456 23.50 36943.89 36943.89 12.63 36943.89 102107.04 2.38e-07 22.36 5640.73 10524.92
536870912 13.20 124827.05 124827.05 6.77 124827.05 350449.76 1.19e-07 14.24 9935.61 18946.22
1073741824 7.02 434921.71 434921.71 3.33 434921.71 1273242.43 5.96e-08 10.50 17532.00 33966.28
------------------------------------- ---------- ------------ ----------- ---------- ------------ ------------ ---------- ---------- ------------ -----------
\[tab-size\]
Comparison
==========
We use distribution mixture equations provided in [@flows-agh] to calculate the performance of analysed algorithms for continuous spectrum of threshold/sampling probability values, which is impossible with simulations. This allows straightforward plotting and comparison of algorithms performance against each other.
Calculated performance indicators of all algorithms are shown in Figure \[fig-absolute\]. Reduction of flow table occupancy and number of operations is presented on y-axis (logarithmic). The x-axis (linear) is desired traffic coverage. In occupancy calculation, we assumed that average packet interarrival time is the same for all flows (it does not depend on flow length/size) and its mean value does not change during flow lifetime. The calculated results are in line with selected values obtained in simulations, which confirms their correctness.
It can be seen that the *first* algorithm achieves the best performance. For any target traffic coverage, it gives the largest reduction, both in flow table occupancy and operations number. While its implementation is impossible in practice, the results are still valuable as they provide an upper boundary for all flow table usage reduction algorithms based on elephant/mice differentiation. Because of that, in Figure \[fig-to-first\], we show performance of all algorithms relative to it.
---------------------------------------------------------------------------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- -------- ---------
(lr)[2-7]{} (l)[8-13]{}
(lr)[2-3]{} (lr)[4-5]{} (lr)[6-7]{} (lr)[8-9]{} (lr)[10-11]{} (l)[12-13]{} length size length size length size length size length size length size
99 14.45 17.78 8.70 13.21 4.13 7.73 14.45 17.78 5.23 8.35 3.22 5.58
95 93.53 126.08 63.96 93.45 26.94 53.39 93.53 126.08 38.65 58.63 17.91 34.69
90 250.20 343.15 178.11 253.00 73.74 139.11 250.20 343.15 103.74 154.56 46.94 88.06
80 692.14 1005.94 514.81 758.40 224.90 401.11 692.14 1005.94 306.48 452.87 138.53 247.26
75 1084.61 1535.06 760.02 1130.81 333.03 585.87 1084.61 1535.06 435.56 661.15 202.26 357.31
50 5794.48 6673.91 4080.75 5208.51 1410.50 2379.13 5794.48 6673.91 1946.72 2753.58 801.93 1369.27
---------------------------------------------------------------------------- --------- --------- --------- --------- --------- --------- --------- --------- --------- --------- -------- ---------
\[tab-calculated\]
Occupancy in the case of *threshold* algorithm is approximately 1.5 times higher than with the *first* and the number of flow entries operations is 2-3 times higher. Its usability is limited in OpenFlow switches, but low-overhead implementation may be possible in other dataplane technologies (such as P4). Apart from that, it provides an upper boundary for all algorithms based on some kind of inexact counting, for example, bloom filters or sketches.
In the case of of *sampling* algorithm, occupancy is 2-4 times higher and operations number is 3-7 times higher than in the *first* case. However, the numbers are still high. For example, with a target of 80% traffic coverage, which we believe is a fair target in TE case, it can reduce the number of flow table entries by a factor of 225 (packet sampling) or 401 (size-based non-uniform sampling). The performance of *sampling* algorithm is surprisingly good and stays within the same order of magnitude as performance of the *first* and *threshold* algorithms. However, unlike them, it is trivial to implement, has negligible computational overhead and does not require any memory. We argue that in many use cases it can be sufficient and eliminate the need for more sophisticated (especially machine learning based) solutions.
Finally, in Table \[tab-calculated\], we present results of all algorithms for selected traffic coverage values. All algorithms achieve better performance when *size* is use as threshold/sampling base. This can be attributed to the fact that longer flows have larger average packet size.
Further research
================
We performed our research using the most accurate publicly available distribution mixtures. However, we acknowledge that results depend strictly on flow length and size distributions and they may vary between different networks. Therefore, it would be valuable to repeat similar evaluations using different distributions. We have provided an open source implementation of our evaluation tools, which can be used for that purpose:
<https://github.com/piotrjurkiewicz/flow-models>
In this paper, we use flow sizes to calculate the traffic coverage. However, from the traffic engineering point of view, it is more important to maximize the coverage of flow rates. It has been shown that there is a correlation between flow sizes and rates [@zhang2002characteristics]. Therefore, the idea that high-rate flows could be identified using flow sizes is theoretically correct [@mori2004identifying]. Nevertheless, it may be interesting to directly use flow rates for traffic coverage calculations.
Similarly, we assumed that packet interarrival time is the same for all flows and does not depend on flow length/size, in other words that the flow duration is proportional to flow length/size. Using real flow durations to calculate occupancy would give more accurate results. However, the calculation of flow rate and duration distributions requires accurate timestamps. Hardware NetFlow agents usually cannot assign accurate timestamps to generated flow records [@hofstede2014flow]. Therefore, packet traces instead flow records should be analysed in order to obtain accurate flow rate and duration distributions.
In our research we followed the NetFlow flow definition, with 15 seconds inactive timeout. It will be interesting to perform similar evaluations for flows with other timeouts, especially subsecond timeouts (so-called *flowlets*). Flowlet-based traffic engineering is an interesting concept, since the number of simultaneous flowlets is several orders of magnitude lower than the number of simultaneous flows. However, we also expect that gains from elephant-mice flow differentiation will be lower for flowlets.
Another interesting research direction is the usage of elephant-based traffic engineering in distributed systems. In centralised approaches, all switches send detected elephants to the central controller, which installs flow-specific paths on all switches at once. In distributed systems, the installation of elephant flow-specific entry would have to be coordinated between all switches without usage of the central controller.
Implementation aspects are also very important. The exact counting of packets of all candidate flows is resource intensive in OpenFlow switches. Novel dataplane technologies, like P4 or eBPF can allow the implementation of low-overhead per-flow counters [@turboflow]. An interesting alternative is inexact counting of significant queue contributors based on count-min sketches, which was proven to run on P4 switch at line rate and identify elephants with high accuracy and low latency [@10.1145/3359989.3365408].
Moving to the *sampling* algorithm, it has to be said that it cannot be achieved using solely OpenFlow as the standard does not permit the creation of rules with probabilistic actions (however, the Open vSwitch kernel module allows such rules with its Netlink API (`OVS_ACTION_ATTR_SAMPLE`)). In order to sample packets on hardware OpenFlow switches, out-of-band mechanisms like sFlow have to be used. High performance and low latency out-of-band sampling mechanisms alternative to sFlow, like [@rasley2014planck], are also an interesting research topic.
Conclusions
===========
The contribution of this paper is fourfold. This is the first paper that examines flow table occupancy reduction approaches from the SDN traffic engineering point of view using accurate and reproducible flow length and size distribution mixtures. The accuracy of used distributions have a crucial impact on results.
Secondly, the results of the *first* and *threshold* algorithms provide upper bounds for whole classes of flow table usage reduction algorithms. Such a bounds were not previously available.
Thirdly, we discovered the surprisingly good performance of the *sampling* algorithm when applied to realistic traffic. Having it mind its simplicity and negligible overhead, this questions the need for usage of more sophisticated algorithms.
Finally, we have made all evaluation tools open source, allowing to replication of our experiments with minimal effort. What is more important is that these tools can be also used to perform similar evaluations for other algorithms or with other flow distributions.
We believe that our paper will set the baseline for the performance of flow table occupancy reduction algorithms. We also hope that methodology and tools presented in this paper will be used in evaluation of more sophisticated algorithms, which are frequently proposed nowadays, finally making it possible to compare evaluation results from different papers.
Acknowledgment {#acknowledgment .unnumbered}
==============
The research was carried out with the support of the project “Intelligent management of traffic in multi-layer Software-Defined Networks” founded by the Polish National Science Centre under project no. 2017/25/B/ST6/02186.
|
---
abstract: 'We proved that any Lorentz transformation of 2-torus $T^2$ is Anosov automorphism. One completely describes admissible parameters of Lorentz transformations and their arithmetical properties. One proved that an admissible speed light parameter has a countable spectra accumulating to this parameter.'
author:
- 'S. Aranson'
- 'E. Zhuzhoma'
title: On arithmetical and dynamical properties of Lorentz maps of the torus
---
Introduction
============
Lorentz manifold $M$ is a pseudo-Riemannian manifold with the signature $(1, n-1)$, where $n = \dim M$. This means that Lorentzian metric is defined in each fiber $T_pM$, $p\in M$, of the tangent bundle by the nondegenerate bilinear form $T_pM\times T_pM\to \mathbb R$ with the diagonal type $(-, +\ldots +)$. In contrast with a Riemannian metric which can be defined on any smooth compact manifold, a Lorentzian metric is defined with certain topological conditions. For example, a compact manifold endowed with a Lorentzian metric must have a zero Euler characteristic [@BeemEhrlich-book81]. Isometry groups of compact Lorentzian manifolds are considered in [@AdamsStuch97], [@DAmbra88] ,where there are many references on other articles. 2-dimensional torus is a unique closed orientable surface admitting the structure of Lorentzian manifold. Moreover, the torus admits the Lorentzian metric of constant zero curvature. It is well known that any Lorentzian manifold with a metric of constant zero curvature is locally isomorphic to the Minkowski space-time [@Wolf-book72]. Therefore, it is natural to consider the isometry group of the 2-torus that is endowed by the Minkowski metric. The group isometry of such Lorentzian torus is richer, in sense, than the isometry group of Riemannian torus. The later consists of the rigid Euclidean shifts and the isometry exchanging “meridian" and “parallel". In addition with rigid Euclidean shifts, the Lorentzian torus contains the 2-parametric family of Lorentz transformations. Lorentz maps play an important role in electrodynamics, Special Theory of Relativity, and other physic branches, and they have the interest itself.
In this paper, we study dynamical properties of Lorentz transformations on the 2-torus. We completely describe the admissible parameters of Lorentz transformations and their arithmetical properties. At the end of the paper we present the algorithm for calculating a spectra of fixed admissible speed light parameter.
[*Acknowledgment*]{}. The research was partially supported by RFFI grant 02-01-00098. We thanks Ludmila Litinskaya and Igor Aranson for valuable discusions. This work was done while the second author was visiting Rennes 1 University (IRMAR) in March-Mai 2004. He thanks the support CNRS which made this visit possible. He would like to thank Anton Zorich and Vadim Kaimanovich for their hospitality.
Statement of main results
=========================
Let us represent the 2-torus $T^2$ as the factor-space $\mathbb{R}^2/\mathbb{Z}^2$ where $\mathbb{R}^2$ is Euclidean plane endowed with coordinates $(x,t)$, and $\mathbb{Z}^2$ is the group of covering transformations that consists of integer shifts. The natural projection $\pi : \mathbb{R}^2\to \mathbb{R}^2/\mathbb{Z}^2$ is a universal cover. This covering map and usual Euclidean metric on $\mathbb{R}^2$ induces on $T^2$ the structure of a flat Riemannian manifold. Denote such torus by $T^2_E$. The Minkowski metric generated by the form $ds^2 = -dx^2 + c^2dt^2$ induces on $T^2$ the metric of Lorentzian manifold of constant zero curvature. Denote such torus by $T^2_L$.
A homeomorphism $T^2_L\to T^2_L$ is called [*Lorentz transformation*]{} if its covering map is of the following type $$\label{lorentz-map}
\overline x = \frac{x-Vt}{\sqrt{1-\frac{V^2}{c^2}}},\quad
\overline t = \frac{t-\frac{V}{c^2}x}{\sqrt{1-\frac{V^2}{c^2}}},\quad {\mbox where}\quad 0<\vert V\vert < c.$$ Such a homeomorphism is denoted by $\phi_{V,c}$. The parameters $(V,c)$ for which there exist a Lorentz transformation $\phi_{V,c}$ are called [*admissible*]{}. Sometimes we’ll say that the pair $(V,c)$ is admissible. It is easy to see that the set of admissible pairs $(V,c)$ is not empty. One can check that a Lorentzian transformation is an isometry of $T^2_L$.
The following theorem describes dynamical properties of Lorentz transformations on the both $T^2_E$ and $T^2_L$.
\[dynamic-properties\] Every Lorentz transformation $\phi_{V,c}$ is an Anosov hyperbolic automorphism of $T^2_E$. Any point with rational coordinates is periodic. Other points are dense under $\phi_{V,c}$ on the both $T^2_E$ and $T^2_L$.
Note that if the pair $(V,c)$ is admissible, then the pair $(-V,c)$ is also admissible. For every fixed admissible $c$, the set of values $V$ that form the admissible pairs $(V,c)$ is called a [*spectra*]{} of $c$. The following theorem describes arithmetical properties of admissible parameters and specters.
\[arithmetical-properties\] A positive value is an admissible (light speed) $c$ if and only if this value is an irrational algebraic number which is the square root from a rational number (thus, on the positive halfline, there are the dense subset of admissible $c$). Moreover,
1. A positive value is an admissible (relative speed) $V$ if and only if this value is a rational of the type $\frac{n}{m}$ such that $p = \frac{m^2-1}{n}$ is an integer, where $m$, $n\in \mathbb{N}$ are coprime.
2. For every admissible $V$, there is a unique admissible $c\stackrel{\rm def}{=}c(V)$ such that they form the admissible pair $(V, c(V))$.
3. Given any admissible $c$, there is a countable spectra $$\ldots , -V_i, \ldots , -V_1, V_1, \ldots , V_i, \ldots$$ with $c = c(V_i)$ for every $i\in \mathbb{N}$.
4. Each spectra $\ldots , c(-V_i), \ldots , c(-V_1), c(V_1), \ldots , c(V_i), \ldots $ has exactly two points $-c(V_i)$, $c(V_i)$ of accumulation.
5. Specters that correspond to different light speeds are mutually disjoint.
As a consequence of items (3) and (4) of theorem \[arithmetical-properties\], we get
The set of admissible $V$ is dense on the real line.
Proof of main theorems
======================
First of all, let us consider when a homeomorphism ${\overline }\phi_{V,c}: \mathbb{R}^2\to \mathbb{R}^2$ is a covering map for some homeomorphism $\phi_{V,c}$ of the torus. Due to (\[lorentz-map\]), the matrix $$\mathbf{A} =
\left(
\begin{array}{cc}
a & -aV \\
-\frac{aV}{c^2} & a
\end{array}
\right)$$ must be unimodal and integer, where $a = \frac{1}{\sqrt{1-\frac{V^2}{c^2}}}$. The right calculation gives that $\det A = 1$. Then the unique condition is that the elements of $\mathbf A$ are integers. Denote $$\label{redenote}
a\stackrel{\rm def}{=}m,\quad -aV^2\stackrel{\rm def}{=}-n,\quad -\frac{aV^2}{c^2}\stackrel{\rm def}{=}-p.$$ It follows from $0 < \vert V\vert < c$ that $m\ge 2$. Since a pair $(V,c)$ is admissible if and only if the pair $(-V,c)$ is also admissible, later on, we assume $V > 0$. Hence, $m$, $n$, and $p$ are natural number that satisfies the following equation $$\label{Pell-equation}
m^2 - np = 1\quad m\ge 2, \quad n, p\in \mathbb{N}.$$ If we know the solution $(m,n,p)$ of (\[Pell-equation\]), then one can find the parameters $c$ and $V$: $$\label{cV-from-mnp}
c = \sqrt{\frac{n}{p}},\quad V = \frac{n}{m},$$ and vise versa, given parameters $c$ and $V$, one can define the solution $(m,n,p)$ of (\[Pell-equation\]): $$\label{mnp-from-cV}
m = \frac{1}{\sqrt{1-\frac{V^2}{c^2}}},\quad n = \frac{V}{\sqrt{1-\frac{V^2}{c^2}}},\quad
p = \frac{V}{c^2\sqrt{1-\frac{V^2}{c^2}}}.$$ Thus, the existence of admissible parameters reduses to the existence of integer positive solutions of equation (\[Pell-equation\]).
\[solutions-exist\] There are infinitely many admissible pairs $(c_m,V_m)$, $(c_{-m},V_{-m})$ such that $$c_m, V_m \to +\infty ,\quad c_{-m}, V_{-m} \to 0\quad \mbox{as}\quad m\to +\infty .$$
[*Proof*]{}. Equation (\[Pell-equation\]) has infinitely many integer solutions $(m,n,1)$, where $m\ge 2$, $n = m^2 - 1$. It follows from (\[cV-from-mnp\]) and (\[Pell-equation\]) that the corresponding admissible parameters $c_m = \sqrt{m^2-1} $ and $V_m = \frac{m^2-1}{m}$ tend to $+\infty$ as $ m\to +\infty $. Similarly, (\[Pell-equation\]) has infinitely many integer solutions $(m,1,p)$, where $m\ge 2$, $p = m^2 - 1$. The corresponding admissible parameters $ c_{-m} = \frac{1}{m^2-1} $ and $V_{-m} = \frac{1}{m}$ tend to $0$ as $ m\to +\infty $. This proves the lemma. $\Box$ As a consequence, there are infinitely many admissible parameters $c$ and $V$.
\[c-is-irrational\] Any admissible $c$ is algebraic irrational, and any admissible $V$ is rational.
[*Proof*]{}. It follows from (\[Pell-equation\]) and (\[cV-from-mnp\]) that $c = \frac{\sqrt{m^2-1}}{p}$. This implies that $c$ is algebraic irrational. The rationality of $V$ follows from (\[cV-from-mnp\]). $\Box$
[*Proof of theorem \[dynamic-properties\].*]{} The straight calculation gives that the eigenvalues of the matrix $\mathbf A$ are $$\lambda _1 = \frac{\sqrt{1-\beta ^2}}{1+\beta}\in (0;1),\quad
\lambda _2 = \frac{\sqrt{1-\beta ^2}}{1-\beta}\in (1;+\infty ),$$ where $ \beta = \frac{V}{c} $. The corresponding eigenvectors are $$\vec e_1 = (c\mu , \mu ),\quad \vec e_2 = (-c\mu , \mu ),$$ where $\mu = const$. By lemma \[c-is-irrational\], $c$ is irrational. This implies that the corresponding transformation $\phi_{V,c}$ is an Anosov hyperbolic automorphism of $T^2_E$. Since $\phi_{V,c}$ is linear, any point with rational coordinates is periodic, and other points are dense under $\phi_{V,c}$ on the both $T^2_E$ and $T^2_L$ [@Robinson-book99]. The theorem is proved. $\Box$. In order to proof theorem \[arithmetical-properties\], we consider properties of solutions $(m,n,p)$ of equation (\[Pell-equation\]) in details.
\[coprime\] Let $(m,n,p)$ is a solution of equation (\[Pell-equation\]). Then every pair $(m,n)$, $(m,p)$, and $(m,np)$ are coprime numbers.
[*Proof*]{}. First, let us prove that the numbers $m$, $n$ are coprime. Suppose the contradiction. Then there is the integer $k\neq 0$ such that $m = km_0$, $ n = kn_0 $. Deviding the quality $m^2 - np = 1$ on $k$, we get $$m\frac{m}{k} - p\frac{n}{k} = \frac{1}{k}$$ or $ m_0m - n_0p = \frac{1}{k}$. Since the left side is an integer and the right side is fractional, we get the contradiction.
The proof is similar for co-primeness of $m$ and $p$. It follows from the quality $m^2 - np = 1$ that $m$ and $np$ are also coprime. $\Box$ Admissible parameters $c$ and $V$ are said to be corresponding to the solution $(m,n,p)$ of equation (\[Pell-equation\]) if the relations (\[cV-from-mnp\]), (\[mnp-from-cV\]) hold.
\[absolutly-different-for-c\] Let $(m,n,p)$, $(m^{\prime},n^{\prime},p^{\prime})$ are different solutions of equation (\[Pell-equation\]) corresponding to the same admissible $c$ (that is, the solutions don’t coinside in at least one position). Then $m\neq m^{\prime}$, $n\neq n^{\prime}$,$p\neq p^{\prime}$.
[*Proof*]{}. By (\[cV-from-mnp\]), we have $$n = c^2p,\quad n^{\prime} = c^2p^{\prime}.$$ If $p = p^{\prime}$, then $n = n^{\prime}$. Due to (\[Pell-equation\]), $m^2 = (m^{\prime})^2$ and hence, $m = m^{\prime}$. If $p \neq p^{\prime}$, then $n \neq n^{\prime}$. According to (\[Pell-equation\]) and (\[cV-from-mnp\]), $$m^2 - np = m^2 - c^2p^2 = 1 = (m^{\prime})^2 - n^{\prime}p^{\prime} = (m^{\prime})^2 - c^2(p^{\prime})^2.$$ Hence, $p \neq p^{\prime}$ implies $m^2 \neq (m^{\prime})^2$ and $m \neq m^{\prime}$. $\Box$
\[unique-for-V\] Given an admissible $V$, there is a unique solution $(m,n,p)$ of equation (\[Pell-equation\]).
[*Proof*]{}. Due to (\[cV-from-mnp\]), we have $$V = \frac{n}{m} = \frac{n^{\prime}}{m^{\prime}}.$$ By lemma \[coprime\], the pairs $(m,n)$, $(m^{\prime},n^{\prime})$ are coprime. Hence, $m = m^{\prime}$, $n = n^{\prime}$. It follows from (\[Pell-equation\]) that $p = p^{\prime}$. $\Box$
\[fixed-V-correspond-one-c\] Given an admissible $V$, there is a unique admissible $c$ such that the pair $(c,V)$ is admissible.
Corollary \[fixed-V-correspond-one-c\] means that the parameter $c$ is a function of $V$, $c = c(V)$, that is, the parameter $V$ defines uniquely Lorentzian transformation of the torus. In contrast with lemma \[unique-for-V\], the following lemma means that a fixed admissible $c$ correspond infinitely many triples $(m,n,p)$.
\[fixed-c-correspond-infinitely-many-triples\] Given an admissible $c$, there are infinitely many different solutions $(m,n,p)$ of equation (\[Pell-equation\]).
[*Proof*]{}. Let us fix $c\stackrel{\rm def}{=}C_0$ corresponding to some solution $(M_0,N_0,P_0)$ of (\[Pell-equation\]), $ C_0 = \sqrt{\frac{N_0}{P_0}} = \frac{\sqrt{M^2-1}}{P_0} $. We have to find infinitely many different solutions $(m,n,p)$ of (\[Pell-equation\]) that satisfy the last equality with replacing $(M_0,N_0,P_0)$ by $(m,n,p)$.
Consider the equation $$\label{real-Pell-equation}
m^2 - ap^2 = 1,\quad \mbox{where}\quad a = \frac{N_0}{P_0}.$$ Note that (\[real-Pell-equation\]) in general is not a Pell’s equation because $a$ is not necessary a natural number (see case 2) below). Since we try to solve equation (\[Pell-equation\]) with the condition $ \frac{n}{p} = \frac{N_0}{P_0} $, it is enough to prove that (\[real-Pell-equation\]) has infinitely many different integer solutions $(m,p)$, $m\ge 2$, $p\in \mathbb{N}$. There are two cases: 1) $a$ is a natural number; 2) $a$ is not a natural number. In the case 1), $ \sqrt{a} = C_0 $ is irrational, according to lemma \[c-is-irrational\]. Therefore, (\[real-Pell-equation\]) is actually a Pell’s equation (see, for example [@Sierpinski-book64]). Due to theorems 10.9.1 and 10.9.2 [@Sierpinski-book64], a Pell’s equation has infinitely many (nontrivial) positive solutions $(m_i,p_i)$. This gives the infinitely many different solutions $(m_i,n_i,p_i)$ of equation (\[Pell-equation\]), where $ n_i = N_0\frac{p_i}{P_0} $.
In the case 2), replacing $a$ in (\[real-Pell-equation\]) by $ a = \frac{N_0}{P_0}$, we get $ m^2 - \frac{N_0}{P_0}p^2 = 1 $ or $$\label{another-real-Pell-equation}
m^2 - (N_0P_0)\left(\frac{p}{P_0}\right)^2 = 1.$$ This is a Pell’s equation with irrational $$\sqrt{N_0P_0} = P_0\frac{N_0}{P_0} = C_0P_0$$ since $C_0$ is irrational (lemma \[c-is-irrational\]). Hence, there are infinitely many (nontrivial) positive solutions $ (m_i,P_i) $, where $ P_i = \frac{p}{P_0} $. Again, this gives the infinitely many different solutions $(m_i,n_i,p_i)$ of equation (\[Pell-equation\]), where $ p_i = P_iP_0 $, $ n_i = N_0P_i = N_0\frac{p_i}{P_0} $. This completes the proof. $\Box$
\[spectra-is-infinite\] Every admissible $c$ has a countable spectra.
\[points-of-accumulation\] Let $ V_1, V_2, \ldots , V_i, \ldots $ is the positive part of the spectra of $c$, $c = c(V_i)$. Then the set $ \{V_1, V_2, \ldots , V_i, \ldots \} $ has a unique point of accumulation which is $c$.
[*Proof*]{}. Since the set $ \{V_1, V_2, \ldots , V_i, \ldots \} $ is countable, it has at least one point of accumulation, say $ w\in [0;c] $. We have to show that $ w = c $. By lemma \[fixed-c-correspond-infinitely-many-triples\], there are infinitely many different solutions $(m,n,p)$ of equation (\[Pell-equation\]) corresponding to $c$. Due to (\[Pell-equation\]) and (\[cV-from-mnp\]), $$\frac{V_i}{c} = \frac{\sqrt{m^2_i-1}}{m_i}\quad \mbox{for every}\quad i\ge 1.$$ If the sequence $ \{m_i\} $ is bounded, then there only finitely many $ V_i $. This contradicts to corollary \[spectra-is-infinite\]. Then, for any unbounded subsequence $ m_{i_k}\to +\infty $, $ \frac{V_{i_k}}{c}\to 1 $ as $ k\to \infty $. Hence, $ V_{i_k}\to c $ as $ k\to \infty $. This proves the lemma. $\Box$
\[c-many\] Let $ C $ be a positive irrational number, which is the square root from some rational number, say $ \frac{N}{P} $. Then $ C $ is an admissible (light speed) parameter.
[*Proof*]{}. By condition, $ C^2 = \frac{N}{P} $. Without loss of generality, we can assume that $ N $, $ P\in \mathbb{N} $. Consider the equation $$\label{again-Pell-equation}
m^2 - dr^2 = 1,\quad \mbox{where}\quad d = NP.$$ Since $C$ is irrational, $CP = \sqrt{NP} = \sqrt{d}$ is also irrational. Hence, (\[again-Pell-equation\]) is a Pell equation that has infinitely many positive (nontrivial) solutions $(m,r)$. Given the solution $(m_0,r_0)$, put $n = r_0N$, $p = r_0P$. Then $$m_0^2 - np = m_0^2 - r_0^2NP = m_0^2 - dr_0^2 = 1.$$ Another words, $(m_0,n,p)$ is a positive (nontrivial) solution of equation (\[Pell-equation\]). Since $ C = \sqrt{\frac{N}{P}} = \sqrt{\frac{n}{p}} $, $ C $ is admissible. The lemma is proved. $\Box$
The set of admissible (light speed) parameters $c$ is dense on the real positeve halfline.
[*Proof of theorem \[arithmetical-properties\]*]{} follows from the lemmas \[c-is-irrational\], \[coprime\] - \[c-many\] above.
Calculation of spectra
======================
Here, given admissible parameters (light speed) $c$, we present the algorithm calculating the corresponding spectra $ \ldots , -V_i, \ldots , -V_1, V_1, V_2, \ldots , V_i, \ldots $, where $c = c(V_i)$. Later on, $c$ if fixed. It is enough to present such algorithm for the positive part $ \{V_1, V_2, \ldots , V_i, \ldots \} $ of the spectra. Due to lemma \[points-of-accumulation\], $c$ is a unique point of accumulation for this positive part. Therefore, without loss of generality, we can assume that $$V_1 < V_2 < \ldots < V_i < V_{i+1} < \ldots .$$ According to (\[cV-from-mnp\]) and (\[mnp-from-cV\]), any admissible pair $(c, V_i)$ corespond uniquely the triples $(m_i, n_i, p_i)$ of positive integer solutions of (\[Pell-equation\]), such that $$\label{another-cV-from-mnp}
c = \sqrt{\frac{n_i}{p_i}},\quad V_i = \frac{n_i}{m_i},$$ $$\label{another-mnp-from-cV}
m_i = \frac{1}{\sqrt{1-\frac{V_i^2}{c^2}}},\quad n_i = \frac{V_i}{\sqrt{1-\frac{V_i^2}{c^2}}},\quad
p_i = \frac{V_i}{c^2\sqrt{1-\frac{V_i^2}{c^2}}},$$ $$\label{another-Pell-equation}
m_i^2 - n_ip_i = 1\quad m_i\ge 2, \quad n_i, p_i\in \mathbb{N}.$$ In the set of arbitrary triples $(m, n, p)$, we introduce a partial order as follows. Say $$(m, n, p) < (m^{\prime}, n^{\prime}, p^{\prime})\quad \mbox{iff}
\quad m < m^{\prime}, n < n^{\prime}, p < p^{\prime}.$$ We also shall use a partial order for doubles: $$(m, n) < (m^{\prime}, n^{\prime})\quad \mbox{iff}
\quad m < m^{\prime}, n < n^{\prime}.$$
\[ordering\] Let $(m_i, n_i, p_i)$ corresponds to $(c, V_i)$. Then $$(m_1, n_1, p_1) < \ldots < (m_i, n_i, p_i) < (m_{i+1}, n_{i+1}, p_{i+1}) < \ldots .$$
[*Proof*]{}. It follows from (\[another-mnp-from-cV\]) and (\[another-Pell-equation\]) that $$V_i^2 = \frac{m_i^2-1}{m_i^2}c^2 < V_{i+1}^2 = \frac{m_{i+1}^2-1}{m_{i+1}^2}c^2.$$ Hence, $ \frac{m_i^2-1}{m_i^2} < \frac{m_{i+1}^2-1}{m_{i+1}^2}$, and $m_i < m_{i+1}$. Since $ V_i = \frac{n_i}{m_i} < V_{i+1} = \frac{n_{i+1}}{m_{i+1}} $, $n_i < n _{i+1}$. Similarly, $ c^2 = \frac{n_i}{p_i} = \frac{n_{i+1}}{p_{i+1}} $ implies $p_i < p_{i+1}$. $\Box$ Thus, to define the spectra for the given $c$ it is sufficient to find the first and minimal triple $ (m_1, n_1, p_1) $ and represent a recurrent formula in calculating $ (m_i, n_i, p_i) $, $i\ge 2$. Due to (\[another-cV-from-mnp\]), each $ (m_i, n_i, p_i) $ will correspond to admissible $V_i = \frac{n_i}{m_i}$ with $c = c(V_i) = \sqrt{\frac{n_i}{p_i}}$.
According to theorem \[arithmetical-properties\], $ c^2 = \frac{n_*}{p_*} $ for some $n_*$, $p_*\in \mathbb{N}$. Without loss of generality, we can assume that $n_*$, $p_*$ are coprime (otherwise, we divide $n_*$, $p_*$ on a common multiplier). Consider the equation $$\label{one-more-Pell-equation}
m^2 - dr^2 = 1,\quad \mbox{where}\quad d = c^2p^2_*.$$ Since $c$ is irrational, $cp_* = \sqrt{c^2p^2_*} = \sqrt{d}$ is also irrational. Hence, (\[one-more-Pell-equation\]) is a Pell equation. Due to [@Sierpinski-book64], (\[one-more-Pell-equation\]) has a minimal (in the set of natural numbers) solution, say $(m_0,r_0)$. Below we recall how one can get all solutions of Pell’s equation and they have an order structure..
Put $ m_1 = m_0 $, $n_1 = r_0n_*$, $p_1 = r_0p_*$. Then $$m_1^2 - n_1p_1 = m_0^2 - r_0^2(n_*p_*) = m_0^2 - (c^2p_*^2)r_0^2 = m_0^2 - dr_0^2 = 1.$$ Moreover, $ c^2 = \frac{n_*}{p_*} = \frac{n_1}{p_1} $. Hence, the triple $ (m_1, n_1, p_1) $ correspond to some term in the spectra of $c$. Let us show that $ (m_1, n_1, p_1) $ is a minimal triple.
Suppose the contrary. Then there is the triple $ (m^{\prime},n^{\prime},p^{\prime}) < (m_1, n_1, p_1)$ with $ c^2 = \frac{n^{\prime}}{p^{\prime}} $, $ (m^{\prime})^2 - n^{\prime}p^{\prime} = 1 $. We have $ \frac{n^{\prime}}{p^{\prime}} = \frac{n_*}{p_*} $, and so $ n^{\prime} = \frac{p^{\prime}}{p_*}n_* $. Since $n_*$ and $p_*$ are coprime, $ \frac{p^{\prime}}{p_*} $ is a natural number, say $ r^{\prime} $. The straight calculation shows that $ (m^{\prime}, r^{\prime}) $ is a solution of (\[one-more-Pell-equation\]): $$(m^{\prime})^2 - (c^2p^2_*)(r^{\prime})^2 = (m^{\prime})^2 - (c^2p^2_*)(\frac{p^{\prime}}{p_*})^2 =
(m^{\prime})^2 - c^2(p^{\prime})^2 = (m^{\prime})^2 - n^{\prime}p^{\prime} = 1.$$ Since $ (m_1, p_1) $ is the minimal solution of (\[one-more-Pell-equation\]), $ m_1 < m^{\prime} $. This contradicts to inequality $ (m^{\prime},n^{\prime},p^{\prime}) < (m_1, n_1, p_1)$ that means, in particular, $ m_1 > m^{\prime} $.
[**Resume**]{}. To get the minimal triple $ (m_1, n_1, p_1) $, one need to take the minimal (in the set of natural numbers) solution $(m_0,r_0)$ of (\[one-more-Pell-equation\]). Then $$m_1 = m_0 ,\quad n_1 = r_0n_*,\quad p_1 = r_0p_*$$ gives the minimal triple $ (m_1, n_1, p_1) $. This triple corresponds to the minimal term $V_1$ of the spectra of $c$ by (\[another-cV-from-mnp\]).
Remark, that if $ c^2 = \frac{n_*}{p_*} $ with coprime $n_*$, $p_*$ then, in general, $n_1\neq n_*$ and $p_1\neq p_*$. For example, if $ c^2 = \frac{1}{2} $, then $ (m_1, n_1, p_1) = (3,2,4)$.
Let us recall the solution of Pell’s equation (\[one-more-Pell-equation\]). Due to theorem 3.5.7 [@Sierpinski-book64], if the period of the continued fraction of $d$ consists of an even numbers $s$ of terms, then the numerator and the denominator of the $(ns-1)$-th convergent, $n\in \mathbb{N}$, form a solution of (\[one-more-Pell-equation\]). Moreover, all the solutions are obtained in this way. From this we see that the solution in the least natural numbers is given by the $(s-1)$-th convergent. Since numerators and denominators form increasing sequences, the solutions of (\[one-more-Pell-equation\]) are endowed with the natural order: $$(P_{s-1},Q_{s-1}) < (P_{2s-1},Q_{2s-1}) < \ldots (P_{ns-1},Q_{ns-1}) < \ldots .$$ Due to theorem 3.5.8 [@Sierpinski-book64], if the period of the continued fraction of $d$ consists of an odd numbers $s$ of terms, then the numerator and the denominator of the $(2ns-1)$-th convergent, $n\in \mathbb{N}$, form a solution of (\[one-more-Pell-equation\]). In this case, all the solutions are also obtained in this way. From this we see that the solution in the least natural numbers is given by the $(2s-1)$-th convergent. Again, the solutions of (\[one-more-Pell-equation\]) are endowed with the natural order: $$(P_{2s-1},Q_{2s-1}) < (P_{4s-1},Q_{4s-1}) < \ldots (P_{2ns-1},Q_{2ns-1}) < \ldots .$$
It is easy to see that the system of equations $ m^2 - np = 1 $, $ c^2 = \frac{n}{p} $ is equivalent to the system $ m^2 - c^2p^2 = 1 $, $ n = c^2p $. Therefore, we consider the equation $$\label{quasi-Pell}
X^2 - c^2Y^2 = 1.$$ Let us introduce an operation $\bigotimes$ on pairs of numbers as follows: $$(u,v)\bigotimes (x,y) = (ux + c^2vy, vx + uy).$$
\[structure-of-solutions\] Let $ (u,v)$ and $(x,y) $ are solutions (not necessary, integer) of (\[quasi-Pell\]). Then $ (u,v)\bigotimes (x,y) = (ux + c^2vy, vx + uy) $ is the solution of (\[quasi-Pell\]) as well.
[*Proof*]{}. The straight calculation gives $ (ux + c^2vy)^2 - c^2(vx + uy)^2 = (x^2 - c^2y^2)(u^2 - c^2v^2) = 1$. $\Box$
\[1-from-structure-of-solutions\] Let $ (m,p)$ and $(m^{\prime},p^{\prime}) $ are integer solutions (not necessary, positive) of (\[quasi-Pell\]). Then $(m,p)\bigotimes (m^{\prime},p^{\prime})$ is the solution of (\[quasi-Pell\]) as well.
Taking in mind that the minimal solution $(m_1,p_1)$ of equation (\[quasi-Pell\]) consists of positive $m_1$ and $p_1$, we get
\[2-from-structure-of-solutions\] Let $ (m_1,p_1)$ be the minimal solution of (\[quasi-Pell\]). Then $$\underbrace{(m_1,p_1)\bigotimes \cdots \bigotimes (m_1,p_1)}_{k \mbox {times }}
\stackrel{\rm def}{=}(m_1,p_1)^k$$ is the solution of (\[quasi-Pell\]) as well. Moreover $$(m_1,p_1) < (m_1,p_1)^2 < \ldots (m_1,p_1)^k < \ldots .$$
Consider the transformation $G: \mathbb{R}^2 \to \mathbb{R}^2$ of the type $$x^{\prime} = m_1x - c^2p_1y, \quad y^{\prime} = -p_1x + m_1y,$$ where $ (m_1,p_1)$ is the minimal solution of (\[quasi-Pell\]). The determinant of $G$ equals 1, so $G$ is an orientation preserving diffeomorphism of $\mathbb{R}^2$. Moreover, due to lemma \[structure-of-solutions\], the hyperbolas $x^2 - c^2y^2 = 1$ denoted by $H$ is invariant under $G$.
Let $M_k$ be the point with the coordinates $ (m_1,p_1)^k $. According to corollary \[2-from-structure-of-solutions\], $M_k\in H$ for every $k\in \mathbb{N}$. The point $M_0$ correspond to the trivial solution $(1,0)$ of $(m^{\prime},p^{\prime})$, and is the vertex of right side branch of the hyperbolas $H$.
\[all-solutions\] Let $(m^{\prime},p^{\prime})$ be a nontrivial positive integer solution of (\[quasi-Pell\]). Then $(m^{\prime},p^{\prime})$ is $ (m_1,p_1)^k $ for some $k\in \mathbb{N}$.
[*Proof*]{}. Suppose the contrary. Then $ (m_1,p_1)^k < (m^{\prime},p^{\prime}) < (m_1,p_1)^{k+1}$ for some $k\in \mathbb{N}$. This means that the point $M^{\prime}$ with the coordinates $(m^{\prime},p^{\prime})$ is between points $M_k$, $M_{k+1}$ on $H$. Recall that $H$ is invariant under $G$ and, hence, under $G^{-k}$. Under the diffeomorphism $G^{-k}$, the points $M_k$, $M_{k+1}$ are mapped into the points $M_0$ and $M_1$ respectively. Since $G$ preserves orientation, $M^{\prime}$ is mapped to the point $G^{-k}(M^{\prime})$ between $M_0$ and $M_1$. Due to corollary \[1-from-structure-of-solutions\], $G^{-k}(M^{\prime})$ corresponds to an integer solution of (\[quasi-Pell\]). Since $G^{-k}(M^{\prime})$ is between $M_0$ and $M_1$, this solution is nontrivial, positive, and less that $(m_1,p_1)$. This contradiction proves the lemma. $\Box$ As a consequence of lemma \[all-solutions\], we get that the triples $ (m_k, n_k, p_k) $ are obtained from the following relations: $$(m_k, p_k) = (m_1,p_1)^k, \quad n_k = \frac{m_k^2-1}{p_k}, \quad k\in \mathbb{N}.$$ Let us get direct recurrent formulas.
\[recurrent-formulas\] $ (m_k, p_k) = (m_1,p_1)^k $ iff $$\label{RecurrentFormulas}
m_k = m_1m_{k-1} + c^2p_1p_{k-1}, \quad p_k = m_1p_{k-1} + p_1m_{k-1}, \quad k\ge 1,$$ where $m_0 = 1$, $p_0 = 0$.
[*Proof*]{}. Let $ (m_k, p_k) = (m_1,p_1)^k $. Obviously, (\[RecurrentFormulas\]) takes place for $k = 1$. Suppose (\[RecurrentFormulas\]) are true for $1$, $\ldots$, $k-1 \ge 1$, and show that (\[RecurrentFormulas\]) is true for $k$. By induction suggestion, we have $$m_{k-1} = m_1m_{k-2} + c^2p_1p_{k-2}, \quad p_{k-1} = m_1p_{k-2} + p_1m_{k-2}.$$ Since $ (m_k, p_k) = (m_1,p_1)^k = (m_1,p_1)\bigotimes (m_1,p_1)^{k-1} = (m_1,p_1)\bigotimes (m_{k-1},p_{k-1}) $, (\[RecurrentFormulas\]) follows by definition of $\bigotimes$. The converse statement is proved similarly. $\Box$ [**Resume.**]{} After finding the minimal triple $ (m_1,n_1p_1) $, the recurrent formulas $$m_k = m_1m_{k-1} + c^2p_1p_{k-1}, \quad p_k = m_1p_{k-1} + p_1m_{k-1}, \quad
n_k = \frac{m_k^2-1}{p_k}$$ give the $k$-th triple $ (m_k, n_k, p_k) $. This triple corresponds to the $k$-th term $(V_{-k}, V_k) = (-V_k, V_k)$ of the spectra of $c$ by (\[another-cV-from-mnp\]): $ V_k = \frac{n_k}{m_k} $.
Remark that the relation $ (m_k, p_k) = (m_1,p_1)^k $ is equivalent to $$m_k + cp_k = (m_1 + cp_1)^k .$$ This can be proved similarly to the proof of lemma \[recurrent-formulas\].
[99]{}
The isometry groups of Lorentz manifolds I, II. [*Invent. Math.*]{}, [**129**]{}(1997), 239-261; 263-287.
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Isometry groups of Lorentz manifolds I, II. [*Invent. Math.*]{}, [**95**]{}(1988), 555-565.
, second edition. CRC Press, [**1999**]{}.
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Bull. Amer. Math. Soc., 1967, 73, 1, 741-817.
. Univ. of Carolina, Berkley, [**1972**]{}.
2875 COWLEY WAY (1015), SAN DIEGO, CA 92110, USA
[*E-mail address*]{}: [email protected]
DEPARTMENT OF APPL. MATHEMATICS, NIZHNY NOVGOROD STATE TECHNICAL UNIVERSITY, NIZHNY NOVGOROD, RUSSIA
[*E-mail address*]{}: [email protected]
[*Current e-mail address*]{}: [email protected]
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---
abstract: 'An increase in the dimension of Hilbert space for quantum key distribution (QKD) can decrease its fidelity requirements while also increasing its bandwidth. A significant obstacle for QKD with qudits ($d\geq 3$) has been an efficient and practical quantum state sorter for photons with complex wavefronts. We propose such a sorter based on a multiplexed thick hologram constructed from photo-thermal refractive glass. We validate this approach using coupled-mode theory to simulate a holographic sorter for states spanned by three planewaves. The utility of such a sorter for broader quantum information processing applications can be substantial.'
author:
- 'Warner A. Miller$^{1,2,*}$ and Mark T. Gruneisen$^{2,}$'
date: 'September 30 ,2008'
title: 'Efficient photon sorter in a high-dimensional Hilbert space'
---
We concern ourselves here with the secure distribution of a one time only key from a sender (Alice) to a receiver (Bob). The three elements of any quantum key distribution (QKD) system are: (1) Alice must be able to prepare at will a single photon state chosen from a set of mutually-unbiased bases (MUB) [@Schwinger:60; @Wootters:89], (2) each of these quantum amplitudes must be propagated from Alice to Bob with reasonable fidelity, and finally (3) Bob must have the ability to choose between one or another of the MUBs and, if he is lucky, be able to efficiently sort and detect each of these photon states. This QKD scenario has been exhaustively studied in the literature and is replete with security proofs for numerous protocols [@Bennett:84; @Ekert:91; @Ekert:00]. These security proofs have been extended in many cases to higher-dimensional Hilbert spaces [@Cerf:02], and all of the protocols have been or are currently being implemented successfully [@Groblacher:06].
Conventional realizations of QKD today involve transmitting heavily-attenuated laser pulses from Alice to Bob and encoding qubit information in each packet by utilizing the spin of the photon. This allows Alice and Bob, who are suitably authenticated, the possibility to establish and share an arbitrarily-secure one-time only key between them. Here they have access to a two-dimensional Hilbert space and can therefore form three MUBs each with two orthogonal polarization states. Such a six-state QKD scheme [@Bruss:98] has limited bandwidth and optical fidelity constraints. These constraints can be ameliorated by extending the QKD to higher-dimensional Hilbert space [@Cerf:02].
The potential of extending photon-based QKD to higher dimensions was made possible in 1992 when Allen et. al.[@Allen:92] showed that Laguerre-Gaussian light beams possessed a quantized orbital angular momentum (OAM) of $l\hbar$ per photon. This opened up an arbitrarily high dimensional quantum space to a single photon[@Allen:03]. Following this discovery Mair et. al. [@Mair:01; @Oemrawsingh:04] unequivocally demonstrated the quantum nature of photon OAM by showing that pairs of OAM photons can be entangled using parametric down conversion. Shortly thereafter, Molina-Terriza et. al. [@Molina-Terriza:02] introduced a scheme to prepare photons in multidimensional vector states of OAM commencing OAM QKD. Recently a practical method has been demonstrated to produce arbitrary OAM MUB states using computer-generated holography with a single spatial light modulator (SLM) [@Gruneisen:08].
While the advantage of OAM QKD lies in its ability to increase bandwidth while simultaneously tolerating a higher bit error rate (BER) [@Cerf:02], two potential problems confront this approach. First, such transverse photon wave functions are more fragile in propagation than the photon’s spin [@Paterson:05; @Aksenov:08], and the divergence of the states ($\propto\! \sqrt{l}$) may require larger apertures. Despite this, multi-conjugate adaptive optical communication channels may be able to ameliorate these problems [@Paterson:05b; @Tyler:08]. A second obstacle involves the efficient sorting of OAM MUB-state photons with small Fock-state quantum numbers, and the paper will address this particular problem. Currently, the only solution to this problem is the use if a cascaded Mach-Zehnder interferometric system [@Xue:01; @Leach:04; @Zou:05]. Proposed systems of this kind have been demonstrated only for 4-dimensions and are simply not practical. Other approaches that use crossed thin diffraction gratings are not efficient enough to establish a secure key.
We focus here on the efficient sorting of single photons with arbitrary complex wavefronts. Ideally, what is needed is an OAM version of a polarizer, i.e. a single optical element, one per MUB basis, that can efficiently sort each of the qu$d$it states in that basis while equally distributing every other qu$d$it state. Thick holographic gratings fortunately produce high diffraction efficiency in the first order [@Goodman:05; @Kogelnik:69; @Case:75]. If several predominant diffracted orders are required, as is the case for sorting, several independent fringe structures can exist in the emulsion. Such multiplexed holograms have been used for multiple-beam splitters and recombiners [@Case:75] and more recently for wide-angle beam steering[@Ciapurin:06]. In this paper we propose such a MUB-state sorter based on a multiplexed thick holographic element constructed from commercially available photo thermal refractive (PTR) glass [@Efimov:99]. Due to the unique properties of PTR glass the grating’s thickness can approach several [*mm*]{} and be highly Bragg selective. There is evidence that such sorters can be highly efficient, $> 95\%$ [@Ciapurin:06]. Our simulations presented here and empirical data on thick Bragg gratings indicate that they may provide an adequate solution to this critical and long standing problem.
Before we describe our proposed thick holographic MUB sorter we will briefly focus our attention on twelve-state QKD and work in a $3$-dimensional Hilbert space. This is one more dimension than that available to photon polarization states and serves to illustrate our approach. Nevertheless, our work is equally applicable to higher dimensional Hilbert spaces. Its limitations will require further investigation.
In 3-dimensions there are a maximum of four MUBs which we refer to here as $MUB_1$, $MUB_2$, $MUB_3$ and $MUB_4$. Each of these orthonormal bases contain three state vectors. If we identify $|a\rangle$, $|b\rangle$ and $|c\rangle$ as the orthonormal ket vectors of $MUB_1$, then the other nine qu$d$it states from the other three MUB bases are specific linear combinations of these (Table \[table:1\]).
[| c |c | c | c | ]{} $MUB_1$ & $MUB_2$ & $MUB_3$ & $MUB_4$\
$ \begin{array}{l}
|a \rangle \\
|b \rangle \\
|c \rangle
\end{array} $ & $ \begin{array}{lllll}
|a_2 \rangle & \propto & | a \rangle + & | b \rangle + & | c \rangle \\
|b_2 \rangle & \propto & | a \rangle + z & | b \rangle + z^2 & | c \rangle \\
|c_2 \rangle & \propto & | a \rangle + z^2 & | b \rangle + z & | c \rangle
\end{array} $ & $ \begin{array}{lllll}
|a_3 \rangle & \propto & | a \rangle + & | b \rangle + z & | c \rangle \\
|b_3 \rangle & \propto & | a \rangle + z & | b \rangle + & | c \rangle \\
|c_3 \rangle & \propto & | a \rangle + z^2 & | b \rangle + z^2 & | c \rangle
\end{array} $ & $ \begin{array}{lllll}
|a_4 \rangle & \propto & | a \rangle + & | b \rangle + z^2 & | c \rangle \\
|b_4 \rangle & \propto & | a \rangle + z & | b \rangle + z & | c \rangle \\
|c_4 \rangle & \propto & | a \rangle + z^2 & | b \rangle + & | c \rangle \\ \hline
\end{array} $\
For our application we can freely choose as our $MUB_1$ any three pure OAM states ($|a\rangle$, $|b\rangle$ and $|c\rangle$) corresponding to an angular momentum, $l_a=a\hbar$, $l_b=b\hbar$, and $l_c=c\hbar$ with integers $a$, $b$ and $c$ being the azimuthal quantum numbers. For the purpose of our calculations we can simplify our analysis and retain the physical content by quantizing in the space of linear momentum (k-QKD) rather than in angular momentum (OAM-QKD). As a result we can freely choose as $MUB_1$ any three non co-linear planewaves. In this case, our three integer quantum numbers will be the number of waves of tilt of these planewaves with respect to the normal of the holographic emulsion of aperture $D$. These waves correspond to a transverse linear momentum $p^x_a = a \hbar k^x$, $p^x_b = b \hbar k^x$ and $p^x_c = c
\hbar k^x$; respectively. Here, $k^x = k \lambda/D$ is the x-component of a plane wave of frequency with one wave of tilt ($\tau\! \sim \lambda/D$). In the frame of the hologram and in units where the speed of light is unity, the components of the 4-momentum ($p = \{ p^t,p^x, p^y, p^z
\}$) of each of our three photons can be expressed in terms of their transverse momentum $k^x$ and wavenumber ($k$). $$\begin{aligned}
p_{a} &= \hbar k_a &= a \hbar k^x \{1,1,0,\sqrt{(k/a k^x)^2-1}\},\\
p_{b} & = \hbar k_b &= b \hbar k^x \{1,1,0,\sqrt{(k/b k^x)^2-1}\}, \\
p_{c} &= \hbar k_c &= c \hbar k^x \{1,1,0,\sqrt{(k/c k^x)^2-1}\}. \end{aligned}$$ In the remainder of this paper the transverse linear momentum wavenumbers represent our three quantum numbers for k-QKD. These three planewaves define our first MUB. $$MUB_1 = \left\{ | a \rangle, | b \rangle, | c \rangle \right\}.$$ Each of these states represents a transverse Fourier mode of a photon; they are orthogonal ($\langle i | j \rangle = \delta_{i,j}$) and define our 3-dimensional Hilbert space. The other nine MUB states (Table \[table:1\]) can be obtained from these by linear superposition and will represent wavefronts with both amplitude and phase variations.
For each MUB we consider a multiplexed thick holographic sorter, i. e. a triple-exposed grating structure formed by the incoherent superposition of three gratings within a single emulsion. Here, each grating is formed by the superposition of the respective MUB state and its own unique plane reference wave. In this paper we concentrate on the construction of the $MUB_4$ sorter (Fig. \[fig:vh\]), as the other three MUB sorters will be of similar design.
![An illustration of our proposed thick holographic $MUB_4$ sorter. The three signal waves are the appropriate linear superpositions of the three qu$d$it states of $MUB_4$ shown in Table \[table:1\].[]{data-label="fig:vh"}](figure1.eps){width="0.8\linewidth"}
We construct the $MUB_4$ sorting in three steps. First, we record the interference pattern of our first signal wave $| a_4 \rangle$ with a corresponding reference planewave, $| r_1\rangle$ having $r_1\! \gg\! 1$ waves of tilt. After this initial recording is complete, we then record the interference pattern of our second signal state from $MUB_4$, namely $| b_4 \rangle$ with a second reference planewave $\langle | r_2\rangle$. Finally, we record a third independent set of fringe patterns by interfering the qu$d$it signal state $| c_4
\rangle$ with a third reference planewave $| r_3 \rangle$. This produces a triple-multiplexed hologram.
We show that the hologram described above faithfully represents a quantum projection operator for $MUB_4$. $$\label{po}
{\cal P}_4 = | r_1 \rangle\langle a_4 | + | r_2 \rangle\langle b_4 | +
| r_3 \rangle\langle c_4 |$$ Its operation on any one of the 12 MUB states should produce the desired result. In other words, if the hologram is illuminated by MUB state $| a_4 \rangle$, $| b_4 \rangle$ or $| c_4 \rangle$ it should produce a planewave in state $| r_1 \rangle$, $| r_2 \rangle$ or $| r_3 \rangle$, respectively. If it is illuminated by any of the other nine MUB states it should then produce an equally weighted response into all three reference states, e.g. $$\langle r_i | {\cal P}_4 | b_3 \rangle = \frac{1}{\sqrt{3}},\ \forall i\in\{1,2,3\}.$$
The unique property of PTR glass with its bulk index of refraction, $n_0=1.4865$, and depth of modulation, $\Delta n/n_0= 336\, ppm$, place it squarely in the realms of scalar diffraction theory and coupled-mode (CM) theory. Furthermore, a PTR hologram can be thick $L\! \sim\! D\sim\! 1\, cm$ with Bragg-plane periods, $\Lambda \!
\sim\! \lambda$. Consequently, their Bragg selectivity $\sim \!
\Lambda/L$ can approach the diffraction limit of one wave of tilt across its aperture [@Ciapurin:06]. For $\lambda\!
=\! 1085\, nm$ and $D\! \sim\! L\sim 1\, cm$ this yields a minimal divergence of our three signal waves of a few $arcsec$.
To examine the $MUB_4$ sorter we will follow closely the CM approach of Kogelnik [@Kogelnik:69] and the notation used by Case [@Case:75] to solve the scalar wave equation for polarization perpendicular to the plane of incidence. $$\label{swe}
\nabla^2 E_y + k^2 E_y = 0$$ Here, the linearly-polarized electric field $E_y(x,z)$ of frequency $\nu$ is assumed to be independent of $y$. Following CM theory we keep only primary modes for the electric field. These are the transverse harmonic modes given by the k-vectors $\vec k_1$, $\vec
k_2$, $\vec k_3$, $\vec k_a$, $\vec k_b$ and $\vec k_c$ associated to planewave reference states $|r_1\rangle$, $|r_2\rangle$, $|r_3\rangle$ and $MUB_1$ signal states $|a\rangle$, $|b\rangle$, $|c\rangle$, respectively. $$\begin{aligned}
E_y (x,z) & = R_1(z) \exp^{\vec k_1 \cdot \vec r} +
R_2(z) \exp^{\vec k_2 \cdot \vec r} +
R_3(z) \exp^{\vec k_3 \cdot \vec r} \\
& + S_a(z) \exp^{\vec k_a \cdot \vec r} +
S_b(z) \exp^{\vec k_b \cdot \vec r} +
S_c(z) \exp^{\vec k_c \cdot \vec r}. \end{aligned}$$ Here, the six mode amplitudes, $\{R_i\}$ and $\{S_i\}$ are only functions of $z$. They are set initially to $\{R_i(z=0)\} =
\{1,1,1\}$ and to the corresponding amplitude and phase factors of one of the twelve corresponding signal states shown in Table \[table:1\]. For example, signal state $|c_4>$ one would set $\{S_i\} = 1/\sqrt{3} \{ 1,z^2,1\}$. The wavenumber $k(x,z)$ of Eq. \[swe\] represents the three incoherently recorded gratings mentioned above, $$k = n(x,z) k_0 = \underbrace{n_0 k_0}_{\beta} \left( 1 +
\frac{\Delta n}{n_0}
\frac{\left(I_{R1} + I_{R2} + I_{R3}\right)}{6(1+\sqrt{3})} \right),$$ where $I_{Ri}$ is the intensity modulation of the $i^{th}$ grating, e.g. $$I_{R3} = 2 - |e^{ik_3\cdot r} + \frac{1}{\sqrt{3}}\left( e^{ik_a\cdot r} + z e^{ik_b\cdot r}+ e^{ik_c\cdot r} \right) |^2.$$ Assuming that the interaction between the diffracted orders is slow, we can neglect second order terms and arrive at the CM equations for the mode amplitudes, $$\left( \begin{array}{c}
R'_1 \\
R'_2 \\
R'_3 \\
S'_a \\
S'_b \\
S'_c
\end{array} \right) = i \kappa^2
\left( \begin{array}{c c c c c c }
0 & 0 & 0 & \frac{1}{\rho_1} & \frac{1}{\rho_1} & \frac{z}{\rho_1} \\
0 & 0 & 0 & \frac{1}{\rho_2} & \frac{z^*}{\rho_2} & \frac{z^*}{\rho_2} \\
0 & 0 & 0 & \frac{1}{\rho_1} & \frac{z}{\rho_1} & \frac{1}{\rho_1} \\
\frac{1}{\sigma_a} & \frac{1}{\sigma_a} & \frac{1}{\sigma_a} & 0 & 0 & 0 \\
\frac{1}{\sigma_b} & \frac{z}{\sigma_b} & \frac{z^*}{\sigma_b} & 0 & 0 & 0 \\
\frac{z^*}{\sigma_c} & \frac{z}{\sigma_c} & \frac{1}{\sigma_c} & 0 & 0 & 0
\end{array} \right)
\left( \begin{array}{c}
R_1 \\
R_2 \\
R_3 \\
S_a \\
S_b \\
S_c
\end{array} \right),$$ where $\kappa^2 \equiv \left(\frac{\beta^2}{6(3+\sqrt{3})}\right)
\left( \frac{\Delta n}{n}\right)$, and $\rho_i = k^z_i$ and $\sigma_j=k^z_j$ are the $z$-components of the wave vectors for the reference and signal states, respectively. The solution of this equation for each of the twelve initial signal states (MUB states) are shown in Fig. \[fig:MUB4\] and faithfully reproduce the desired projection operator of Eq. \[po\]. We independently examined the far-field pattern for such gratings using a finite difference time domain solution of Maxwell’s equations and observed that the CW assumptions were valid, i.e. only the primary modes were dominant, and there were no relevant polarization changes in the field.
{width="0.71\linewidth"}
While the analysis presented here suggests that a high efficiency single optical element sorter is feasible with commercially available materials and holographic recording techniques, further work is needed. First, we need to understand the quantum nature of such an element. Second, we need to examine its sensitivity to alignment (linear and rotational) and examine the wavelength scaling issues involved in recording OAM photons. Our first step will be to produce a 3-state k-QKD $MUB_1$ sorter. Our goal is to test its performance using states generated by a single phase SLM [@Gruneisen:08]. If the MUB-state sorters described here can be produced, they should have far more utility in quantum information processing than just QKD, e.g. as an essential element in linear quantum computing [@Knill:01].
We wish to acknowledge important discussions with Glenn Tyler, Robert Boyd, Leonid Glebov, Geoff Anderson, David Reicher and Raymond Dymale. We are also grateful for the advice provided by Angela Guzman, Grigoriy Kreymerman, William Rhodes, Chris Beetle, and Ayman Sweiti of the FAU Quantum Optics group. We thank Anna Miller for reviewing this letter. This work was supported by the Air Force Office of Scientific Research, and by AFRL/RDSE under the IPA program.
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|
---
author:
- 'Wen-Loong Ma$^{1}$, Yizhar Or$^{2}$ and Aaron D. Ames$^{3}$ [^1][^2][^3] [^4]'
bibliography:
- 'citation.bib'
title: |
**Dynamic Walking on Slippery Surfaces:\
[Demonstrating Stable Bipedal Gaits with Planned Ground Slippage$^*$]{}**
---
=1
[^1]: $^{1}$W. Ma is with Mechanical Engineering, California Institute of Technology, Pasadena, CA, USA. [[email protected]]{}
[^2]: $^{2}$Y. Or is with the faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa, Israel. [[email protected]]{}
[^3]: $^{3}$A. Ames is with the faculty of Mechanical Engineering and Control + Dynamical Systems, California Institute of Technology, Pasadena, CA, USA. [[email protected]]{}
[^4]: $*$This work is supported by NSF grant 1724464, 1544332, 1724457, and Disney Research LA. The work has been conducted while Y. Or was hosted by A. D. Ames and AMBER lab at Caltech during his sabbatical leave from the Technion.
|
---
abstract: 'High resolution soft X-ray spectroscopy of the prototype accretion disk wind quasar, PDS 456, is presented. Here, the [[*XMM–Newton*]{}]{} RGS spectra are analyzed from the large 2013–2014 [[*XMM–Newton*]{}]{} campaign, consisting of 5 observations of approximately 100ks in length. During the last observation (hereafter OBS.E), the quasar is at a minimum flux level and broad absorption line profiles are revealed in the soft X-ray band, with typical velocity widths of $\sigma_{\rm v}\sim 10,000$kms$^{-1}$. During a period of higher flux in the 3rd and 4th observations (OBS.C and D, respectively), a very broad absorption trough is also present above 1keV. From fitting the absorption lines with models of photoionized absorption spectra, the inferred outflow velocities lie in the range $\sim 0.1-0.2c$. The absorption lines likely originate from He and H-like neon and L-shell iron at these energies. Comparison with earlier archival data of PDS 456 also reveals similar absorption structure near 1 keV in a 40ks observation in 2001, and generally the absorption lines appear most apparent when the spectrum is more absorbed overall. The presence of the soft X-ray broad absorption lines is also independently confirmed from an analysis of the [[*XMM–Newton*]{}]{} EPIC spectra below 2keV. We suggest that the soft X-ray absorption profiles could be associated with a lower ionization and possibly clumpy phase of the accretion disk wind, where the latter is known to be present in this quasar from its well studied iron K absorption profile and where the wind velocity reaches a typical value of 0.3$c$.'
author:
- 'J. N. Reeves, V. Braito, E. Nardini, E. Behar, P. T. O’Brien, F. Tombesi, T. J. Turner, M. T. Costa'
title: 'Discovery of Broad Soft X-ray Absorption Lines from the Quasar Wind in PDS 456'
---
Introduction
============
The masses of supermassive black holes (SMBHs, with $M_{\rm BH}=10^{6}-10^{9}M_{\odot}$) are known to correlate with galaxy bulge mass (Magorrian et al. 1998) and even more tightly with the stellar velocity dispersion on kpc scales; the so-called $M-\sigma$ relation (Ferrarese & Merritt 2000, Gebhardt et al. 2000). This implies a co-evolution between SMBHs and their host galaxies, although the exact mechanism linking the two remained unclear. A potential mechanism responsible for this co-evolution came with the discovery of extremely energetic outflows from the black holes powering the most luminous Active Galactic Nuclei (AGN) and quasars (Pounds et al. 2003; Reeves et al. 2003; Chartas et al. 2002, 2003; Tombesi et al. 2010; Gofford et al. 2013). At high redshifts, such winds would have provided the necessary mechanical feedback that both controlled the formation of stellar bulges and simultaneously regulated SMBH growth, leaving the observed $M-\sigma$ relation as a record of the process (Silk & Rees 1998; King 2003; Di Matteo et al. 2005).
In the local Universe ($z < 0.3$), the most powerful and best characterized X-ray wind observed so far in an AGN is hosted by PDS456, a nearby ($z = 0.184$) radio-quiet quasar identified only in 1997 (Torres et al. 1997). PDS456 is a remarkable object in many respects. The optical and near-infrared spectra show Balmer and Paschen lines with broad wings (full-width at zero intensity of $>$30,000 kms$^{-1}$; Simpson et al. 1999), while in the $HST$/STIS UV spectrum the C<span style="font-variant:small-caps;">iv</span>$\lambda1549$Å emission line is blueshifted by $\sim$5,000 kms$^{-1}$, and a tentative absorption trough extends from $\sim$14,000 to 24,000 kms$^{-1}$ bluewards of the Ly$\alpha$ rest-frame energy (O’Brien et al. 2005). The bolometric luminosity of PDS456, of the order of $L_{\rm bol}=10^{47}$ergs$^{-1}$ (Reeves et al. 2000; Yun et al. 2004), is more typical of a source at the peak of the quasar epoch ($z \sim 2$–3), when AGN feedback is thought to have played a major role in the evolution of galaxies. No direct measurement is available for the mass of the central black hole, but this can be estimated from the SMBH/host galaxy scaling relations to be $\sim 1$–$2 \times 10^9 {\hbox{$\rm\thinspace M_{\odot}$}}$ (Nardini et al. 2015), thus implying that the black hole in PDS456 is accreting at a substantial fraction of the Eddington rate. Under these physical conditions, the photon momentum flux can contribute to the driving of massive accretion-disk winds (several ${\rm M}_{\odot}$ yr$^{-1}$), provided that the nuclear environment is sufficiently opaque to the continuum radiation (e.g. King 2010; see also Hagino et al. 2015).
The clear presence of strong absorption above 7 keV in PDS456 was revealed by a short (40-ks) [[*XMM–Newton*]{}]{} observation in 2001. If attributed to iron K-shell absorption, such a feature would have arisen in a high velocity outflow, requiring a large column density of highly ionized matter (Reeves et al. 2003). An unusual absorption trough was also found in the soft X-ray band near 1 keV (see also Behar et al. 2010). The detection of fast Fe K absorption has been confirmed in subsequent [*Suzaku*]{} campaigns in 2007, 2011 and 2013, which implied an outflow velocity of $\sim0.3\,c$ (Reeves et al. 2009; Reeves et al. 2014; Gofford et al. 2014). The broad, blueshifted emission and absorption profiles in the UV could be then potentially associated with the decelerating phase of the wind out to large scales.
PDS456 was observed again with [[*XMM–Newton*]{}]{} in a series of five observations between August 2013 and February 2014, with the first four sequences carried out over about a month and the last one six months later, in order to sample the spectral variations over different time scales. All the observations were complemented by the simultaneous high-energy spectra collected by [*NuSTAR*]{}, which provide a valuable broadband view that extends from the optical/UV to the hard X-rays. In this campaign, Nardini et al. (2015) were able to detect a persistent P-Cygni like profile from highly ionized iron, thus establishing the wide-angle character of the disk wind in PDS456. This proved to be important in accurately determining the overall energy budget of the outflow in PDS456, made possible through establishing that the overall wind solid angle was a significant fraction of $4\pi$ steradian. Thus the inner disk wind expels matter at rates close to Eddington, with a kinetic power a significant fraction of the bolometric output of the quasar, comfortably exceeding the typical values thought to be significant for quasar mode feedback (Hopkins & Elvis 2010). In many respects, PDS456 strongly resembles the two fast outflows recently measured in two other luminous obscured quasars, IRASF11119+3257 (Tombesi et al. 2015) and Mrk231 (Feruglio et al. 2015). Here the initial accretion disk wind is likely critical for evacuating matter from the central regions of these systems during their post merger phases, with the eventual fate of the gas seen at large ($\sim$kpc) scales through massive ($\sim1000$yr$^{-1}$) energy conserving molecular outflows.
However to date there have been very few detections of fast AGN outflows in the soft X-ray band (see Pounds 2014; Longinotti et al. 2015; Gupta et al. 2013, 2015 for some possible recent examples). Here we present the soft X-ray spectroscopy of PDS456 obtained from the [[*XMM–Newton*]{}]{} Reflection Grating Spectrometer (RGS; den Herder et al. 2001) as part of the extended 2013–2014 campaign. In the subsequent analysis we will show the detection of broad soft X-ray absorption profiles in PDS456, which may be potentially associated to the fast wind measured in the iron K band. In Section2, we outline the [[*XMM–Newton*]{}]{} observations and data reduction, in Section3 the overall form of the mean RGS spectrum is discussed, while Section 4 presents the detection of the soft X-ray absorption profiles in the individual RGS observations. In Section 5 photoionization modeling of the putative wind in the soft X-ray band is presented, while Section 6 discusses the origins of the absorbing gas from a clumpy, multi phase accretion disk wind. Throughout this work we assume the concordance cosmological values of H$_{\rm 0}$=70kms$^{-1}$Mpc$^{-1}$ and $\Omega_{\Lambda_{\rm 0}}=0.73$, and errors are quoted at 90% confidence ($\Delta\chi^{2}=2.7$) for one parameter of interest. In the spectral analysis, we adopt a conversion between energy and wavelength of $E = (12.3984$ Å/$\lambda)$keV.
XMM-Newton Observations of PDS 456
==================================
Five [[*XMM–Newton*]{}]{} observations of PDS 456, of at least 100ks in duration, were performed over a six month time period in 2013–2014. The EPIC-pn and MOS instruments were operating in Large Window mode and with the Thin filter applied. The observations are listed in Table1 and are described in more detail in Nardini et al. (2015). The first four observations (hereafter OBS.A–D) were performed within a four week period (with separations of $4-10$days between observations), while the fifth observation (OBS.E) was performed about 5 months later and caught the AGN in the lowest flux state of all the campaign (see Table1). The [[*XMM–Newton*]{}]{} data have been processed and cleaned using the Science Analysis Software (<span style="font-variant:small-caps;">sas</span> v14.0.0), and analyzed using standard software packages (<span style="font-variant:small-caps;">ftools</span> v6.17, <span style="font-variant:small-caps;">xspec</span> v12.9).
For the scientific analysis of this paper we concentrated on the RGS data, which have the highest spectral resolution in the soft X-ray band, and we compared the RGS results with the EPIC MOS and pn data for consistency. The RGS data have been reduced using the standard <span style="font-variant:small-caps;">sas</span> task [*rgsproc*]{} and the most recent calibration files. High background time intervals have been filtered out applying a threshold of 0.2 counts s$^{-1}$ on the background event files. OBS.A was affected by severe telemetry issues and about 28 ks into the observation, even if the data were still recorded, the RGS telemetry became corrupted. The main effect is not a simple loss of the data but an incorrect count rate for each of the RGS spectra. As it was not possible to recover the ODF files and as the subsequent RGS spectra were not suitable for analysis, hereafter we concentrate only on OBS.B–E from this campaign. Nonetheless during OBS.A, from an analysis of the unaffected EPIC spectra, PDS456 was caught in a bright and less obscured state, which was mainly featureless in the soft X-ray band (see Nardini et al. 2015).
For each of the remaining four observations we first checked that the RGS1 and RGS2 spectra were in good agreement, typically to within the 3% level, and we subsequently combined them with the <span style="font-variant:small-caps;">sas</span> task [*rgscombine*]{}. The spectra were analyzed over the 0.45–2.0keV energy range; below 0.45keV the spectra are noisy due to the Galactic absorption column towards PDS456. During these observations the two RGS collected typically $\sim
8600 - 13600$ net counts per observation (see Table 1 for details). Two of the observations (OBS.C and OBS.D), which were separated by only 4 days and show a very similar spectral shape and count rate, were further combined using [*rgscombine*]{} into a single spectrum (OBS.CD). Finally, in order to investigate the mean spectral properties with a higher signal to noise spectrum, we combined all the four available observations obtained during this campaign.
In the initial spectral fitting, we first adopted a spectral binning of $\Delta\lambda=0.1$Å for the RGS spectra, which just slightly under-samples the FWHM spectral resolution of $\Delta\lambda=0.06-0.08$Å. At this binning, the spectra have $>20$ counts per bin and thus $\chi^{2}$ minimization was used for all the spectral fitting. Note that in the subsequent analysis we also considered a finer binning (of $\Delta\lambda=0.05$Å) in order to search for any narrow features in the combined RGS1+2 spectra, as well as a coarser binning (of $\Delta\lambda=0.2$Å) for modeling the broad spectral features present in some of the individual sequences.
The [[*XMM–Newton*]{}]{} EPIC data were filtered for high background time intervals, which yields net exposure times as listed in Table 1. For the analysis we concentrated on the MOS spectra, as they offer a better spectral resolution in the soft band compared to the pn data; however, we did check the pn spectra for consistency in each observation. The MOS1+2 source and background spectra were extracted using a circular region with a radius of $35''$ and two circular regions with the same radius, respectively. Response matrices and ancillary response files at the source position were created using the <span style="font-variant:small-caps;">sas</span> tasks *arfgen* and *rmfgen*. We then combined for each exposure the MOS1 and MOS2 spectra, after verifying that the individual spectra were consistent. As for the RGS data, after checking that the MOS spectra for OBS.C and OBS.D were similar, we combined them into a single spectrum.
The MOS source spectra were then binned with constant energy intervals of 15 eV. This oversamples the spectral resolution of the EPIC MOS detectors below 2 keV, where the FWHM is $\sim 50$eV (60 eV) at 0.5 keV (1 keV), by only factor of 3–4. We note that adopting this binning allows us to simultaneously fit both the MOS and the RGS data with a similar sampling. The source is bright enough to collect between $0.54-0.79$countss$^{-1}$ for the combined MOS1+2 spectra over the $0.5-2$ keV band.
In order to study the long term spectral variability of the PDS456 soft X-ray spectra, the RGS and EPIC spectra from the 2001 and 2007 archival observations were also extracted, (see Table1). The data were reduced in an identical way as to the 2013–2014 observations. The 2001 observation consisted of a single exposure of $\sim 40$ks in duration, but at a higher overall count rate compared to the 2013–2014 observations, while the 2007 observations consisted of two sequences over two consecutive [[*XMM–Newton*]{}]{} orbits. These latter two 2007 spectra were consistent in spectral shape, with just a simple offset in flux between them, so they were combined to give a single RGS spectrum from 2007.
Initial Mean RGS Spectral Analysis
==================================
Initially we combined all of the 2013–2014 RGS observations of PDS456 (i.e., OBS.B through to OBS.E), in order to construct a single, time-averaged spectrum. Although the spectrum of PDS456 is time variable, the idea is to have an initial parameterization of both the continuum and local Galactic absorption before proceeding with the analysis of the individual sequences. The mean RGS spectrum was fitted with a simple power-law plus blackbody continuum form, where the blackbody is required to account for the soft excess towards lower energies seen in this AGN. The mean spectrum was initially binned in constant wavelength bins of $\Delta\lambda=0.1$Å over the 0.45–2.0keV (or 6.2–28.0Å) range. A Galactic absorption column from 21-cm measurements of $N_{\rm H}=2.4\times10^{21}$cm$^{-2}$ (Dickey & Lockman 1990; Kalberla et al. 2005) was adopted at first, modeled with the `tbnew` ISM absorption model in [xspec]{} using the cross–sections and abundances of Wilms et al. (2000). However, in subsequent fits, the Galactic $N_{\rm H}$ value was allowed to vary, along with the relative abundance of Oxygen (compared to H) in order to ensure a good fit in the region around the neutral O edge.
This simple continuum form returned a photon index of $\Gamma=2.1\pm0.2$, with a blackbody temperature of $kT = 102^{+15}_{-10}$eV, while the Galactic column was found to be in excess of the 21-cm value with $N_{\rm H}=(3.9\pm0.6)\times10^{21}$cm$^{-2}$. The resulting fit is shown in Figure1, where the Galactic absorption model reproduces well the structure observed around the neutral O edge at $\sim$0.5keV. The relative O abundance was found to be higher by a factor of $A_{\rm O}=1.27\pm0.12$ compared to the O/H abundance of $4.9\times10^{-4}$ reported in Wilms et al. (2000). The absorbed 0.5–2.0keV soft X-ray flux resulting from this model is $2.14\times10^{-12}$ergcm$^{-2}$s$^{-1}$. Note that if instead the Solar abundances of Grevesse & Sauval (1998) are adopted, which present a higher abundance of O, then the relative O abundance is consistent with Solar ($A_{\rm O}=0.95\pm0.09$). While this model reproduces the overall shape of the soft X-ray continuum, the fit statistic is relatively poor with a resulting reduced chi-squared of $\chi^{2}/\nu = 287.7/205$, suggesting the presence of additional spectral complexity. An almost identical fit is obtained if a somewhat different form for the soft excess is used, e.g. a power-law plus Comptonized-disk spectrum.
The mean spectrum was also binned at a finer resolution of $\Delta\lambda=0.05$Å per bin, in order to better sample the resolution of the RGS over the spectral range of interest. The motivation for this is to check whether there is any signature of a warm absorber from low velocity, low to moderately ionized outflowing gas, which is commonly detected in the high resolution soft X-ray spectra of nearby Seyfert 1 galaxies (Kaastra et al. 2000, Sako et al. 2001, Kaspi et al. 2002). Furthermore any such low velocity (and distant) component of a warm absorber would not be expected to strongly vary between the observations. Thus with the high signal to noise of the mean 464ks net exposure, any narrow absorption (or emission) lines from distant and less variable photoionized gas should be readily apparent in the spectrum close to the expected rest-frame energies of the strongest atomic lines in the soft X-ray band. The residuals to the above absorbed power-law plus blackbody fit are shown in Figure 2, which is plotted in the rest frame of PDS456. It is clear that upon inspection of the spectra, no strong narrow absorption or emission lines are observed close to the expected positions of the strongest lines, such as the He-$\alpha$ and Ly-$\alpha$ resonance lines of O, Ne, Mg or Si, or at the energy of the Fe M-shell UTA.
In order to place a formal limit on the column of any warm absorber towards PDS456, the spectrum was modeled with an <span style="font-variant:small-caps;">xstar</span> absorption grid with a turbulence velocity of $\sigma=300$kms$^{-1}$, adopting Solar abundances (Grevesse & Sauval 1998) and allowing for a modest outflow velocity of up to 5,000kms$^{-1}$. The above baseline continuum was used. The addition of this warm absorber component resulted in little improvement in the overall fit. Formally we can place an upper-limit on the column density of between $N_{\rm H} < 0.6\times10^{20}$cm$^{-2}$ and $N_{\rm H} < 1.7\times10^{20}$cm$^{-2}$ for an ionization parameter[^1] in the range from $\log \xi=0-2$, which covers the typical range of ionization seen in the soft X-ray warm absorbers in Seyfert 1s. Thus the presence of a low velocity, low ionization warm absorber appears ruled out in PDS456.
Although no conventional signature of a warm absorber, in the form of narrow absorption lines, appears in these RGS spectra, the finely binned mean spectrum does reveal broad residual structure over the energy range from 0.8–1.6keV. Indeed, fitting with the above baseline continuum results in a very poor fit, with $\chi^{2}/\nu = 573/406$ and a corresponding null hypothesis probability of $8.6\times10^{-8}$. In the following sections we investigate whether these broad features may represent the soft X-ray signature of the known fast wind towards PDS456 detected at iron K (e.g. Reeves et al. 2009; Nardini et al. 2015). In the subsequent spectral fitting we adopt the coarse spectral binning of $\Delta\lambda=0.2$Å to model these broad profiles, as no narrow features are required by the data.
Analysis of Individual RGS Sequences
====================================
We then proceeded to investigate each of the RGS spectra collected during the 2013–2014 campaign. A first inspection of the individual RGS spectra shows that there are remarkable differences between OBS.B (upper and black data points in the upper panel of Figure \[fluxed\]) and OBS.E (blue data points). The 0.5–2 keV spectral shapes appear broadly similar, with some variability of the intensity of the emerging continuum. In addition, the appearance of a deep trough at $\sim 1$ keV in OBS.E (observed frame) suggests that the variations could at least in part be caused by a variable soft X-ray absorber. On the other hand, the spectra obtained during OBS.C and OBS.D are almost identical (see Figure \[fluxed\] lower panel) and are intermediate in flux between the OBS.B and OBS.E. Thus in order to investigate the main driver of the soft X-ray spectral variability, we proceeded to fit each of the main spectral states (i.e. OBS.B, OBS.CD and OBS.E) independently. For the continuum we adopted the blackbody plus power-law continuum model found with the analysis of the mean RGS spectrum, allowing the ${N_{\mbox{\scriptsize H}}}$ of the Galactic absorption to vary, as well as the continuum parameters.
We found that, while for OBS.CD and OBS.E the photon index of the primary power law component is consistent with $\Gamma=2.1$, during OBS.B the spectrum requires a steeper power-law component ($\Gamma \sim 2.5\pm 0.2$). Statistically the fit is poor for most of the sequences when compared to the above continuum model, i.e. $\chi^2/\nu= 149.9/100 $ and $\chi^2/\nu= 170.2/100 $, for OBS.CD and OBS.E, respectively (see Table 2). In contrast, the continuum model can reproduce the overall shape of the RGS spectrum OBS.B ($\chi^2/\nu= 131.2/100$); indeed, as seen in Figure \[sequences\] (top panel) OBS.B appears almost featureless in the RGS.
However, the continuum model leaves clear residuals in the 0.9–1.3 keV energy range for OBS.CD and OBS.E. Broad and deep residuals are present in both of these spectra (Figure \[sequences\], second and third panels), where three main absorption troughs are detected. During OBS.CD a very broad absorption feature is present near 1.2keV, while during OBS.E two absorption troughs emerge in the residuals at $\sim 1.0$keV and $\sim 1.17$keV, respectively. In particular, a closer inspection of the residuals of the OBS.E spectrum to the best fit continuum model (see Figure \[pcyg\]) unveils complex absorption structure around 1keV.
Modeling and Identification of the Individual Profiles
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To parameterize these profiles we added several Gaussian absorption and emission lines to the baseline continuum model. We imposed an initial selection criteria, whereby any line in the RGS spectra was added to the baseline model only if its addition yielded an improvement in the fit statistic of at least $\Delta \chi^2= 9.2$ (for 2 interesting parameters) or $\Delta \chi^2= 11.3$ (for 3 interesting parameters), equivalent to the 99% significance level in $\chi^{2}$ statistics. Furthermore we impose an additional constraint, whereby any individual line in the RGS should be independently confirmed corresponding to at least the same improvement in $\Delta\chi^{2}$ in the MOS (see Section4.3). Thus requiring a possible line detection in the RGS to be independently confirmed in the MOS, at a self consistent energy and flux, reduces the likelihood that the line in question is spurious due to photon noise. Subsequently the parameters of these Gaussian lines selected in the RGS are listed in Table \[Gauss\], as well as the F-test probabilities associated with the addition of each line to the baseline model. Their statistical significances, which are typically confirmed at the $5\sigma$ level or higher once the MOS data are accounted for, are evaluated more throughly in the Appendix.
As noted above, in OBS.CD we detected a strong ($EW=42 \pm 13$ eV) and very broad ($\sigma \sim 100$ eV) absorption line at $E=1174\pm 38$ eV, with an apparent velocity width of $\sigma_{\rm v}=28000{\ensuremath{^{+13000}_{-9000}}}$km s$^{-1}$. We attempt to place a tentative identification on this absorption feature. The $\sim 1.2$keV absorption line could be identified with blue-shifted ($v_{\rm out} \sim 0.2\,c$) Ne<span style="font-variant:small-caps;">x</span> Ly$\alpha$ ($E_\mathrm{Lab}=1.022$keV), perhaps blended with iron L absorption in the ionization range from Fe[<span style="font-variant:small-caps;">xx–xxiv</span>]{}. Alternatively, associating this broad absorption line with a single feature, but with a smaller blueshift, would require an identification with the Fe<span style="font-variant:small-caps;">xxiv</span> $2s\rightarrow 3p$ at $E_\mathrm{Lab}=1.167$keV. However, this would imply a single very broad line seen in isolation, without a strong contribution from other Fe L lines, e.g. Fe<span style="font-variant:small-caps;">xxiii</span> at $E_\mathrm{Lab}=1.125$keV and lower ionization ions. Furthermore, as we will discuss later in Section 5, modeling this absorption profile with a grid of photoionized absorption spectra prefers the solution with a fast and highly ionized wind, noting that part of the large velocity width of the broad trough could be explained with a blend of these lines. Finally a narrower absorption line is also apparent at 846eV in OBS.CD, and although it is much weaker than the broad 1.2keV profile, it is also independently confirmed in the MOS (see Section 4.3). While the identification of this isolated line is uncertain, we note that if it is tentatively associated to absorption from O<span style="font-variant:small-caps;">viii</span> Ly$\alpha$ (at $E_\mathrm{Lab}=0.654$keV), its outflow velocity would be consistent with that found at iron K, at $\sim 0.3\,c$.
The appearance of complex soft X-ray absorption structure near 1keV in OBSE is perhaps not unexpected, especially as this RGS sequence appears the most absorbed and at the lowest flux of all the 2013–2014 [[*XMM–Newton*]{}]{} data sets. As shown in Figure\[pcyg\], two deep absorption lines (at $E\sim 1.01$keV and $E\sim 1.17 $keV) and an emission line at $E\sim 0.9$keV are formally required by the data (see Table2). Indeed, the addition of the two Gaussian absorption lines results in the fit statistic improving by $\Delta \chi^2=44.3$ for $\Delta\nu=5$. Assuming that the absorption line profiles have the same width, then the two lines are found to be resolved with a common width of $\sigma=41^{+14}_{-12}$eV (or $\sigma_{\rm v}=12000^{+4000}_{-3500}$kms$^{-1}$).
One likely identification of the emission/absorption line pair in OBS.E at 913eV and 1016eV may be with Ne<span style="font-variant:small-caps;">ix</span> (at $E_\mathrm{Lab}=0.905$ keV), which would then require the absorption line to be blueshifted with respect to the emission line component. Alternatively, if the absorption at 1016eV is separately associated with Ne<span style="font-variant:small-caps;">x</span> Ly$\alpha$ (at $E_\mathrm{Lab}=1.022$keV), then this would require little or no velocity shift. However, the second absorption line at 1166eV only requires a blueshift if it is associated with Ne<span style="font-variant:small-caps;">x</span> Lyman-$\alpha$, alternatively it could be associated to Fe<span style="font-variant:small-caps;">xxiv</span> $2s\rightarrow3p$ without requiring a blueshift. Thus the precise identification of these 1keV absorption lines are difficult to determine on an ad-hoc basis from fitting simple Gaussian profiles. Their most likely origin will be discussed further when we present the self consistent photoionization modeling of the OBS.E spectrum in Section5.3. Nonetheless, regardless of their possible identification, the detection of the broad absorption profiles in the soft X-ray band in both OBS.E and OBS.CD may suggest the presence of a new absorption zone with somewhat lower velocity and/or ionization compared to the well-known ionized wind established at iron K.\
Comparison to the 2001 and 2007 *XMM–Newton* Observations
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Following the results of the 2013–2014 observations, we re-analyzed the RGS spectra collected during the past [[*XMM–Newton*]{}]{} observations of PDS456, when the quasar was observed in the two extreme states: a highly obscured one (2001; Reeves et al. 2003) and an unobscured state (2007; Behar et al. 2010). For the continuum model we again adopted the best fit found with the analysis of the mean RGS spectrum, allowing the ${N_{\mbox{\scriptsize H}}}$ of the Galactic absorption to vary as well as allowing the continuum parameters to adjust. For the 2001 observation we included an additional neutral partial covering absorber in the continuum model; indeed, during this observation, even though the intrinsic flux of PDS456 is relatively high (see Table1), the AGN appeared heavily obscured, with the presence of strong spectral curvature over the 1–10keV band (Reeves et al. 2003). This curvature is especially evident in the EPIC MOS data (see Section 4.3). Without this absorber the derived photon index, although poorly constrained, is extremely hard ($\Gamma\sim 1.5$), and its extrapolation above 2 keV lies well above the EPIC spectra. On the other hand, the 2007 spectrum required a steeper photon index ($\Gamma=2.4-2.5$), indicating a lack of intrinsic absorption.
The residuals for the 2001 and 2007 RGS data to the baseline continuum model are shown in the two lower panels of Figure \[sequences\]. Similarly to the OBS.E spectrum during the 2013–2014 campaign, the 2001 observation tracks the presence of the highly ionized wind. The presence of absorption near 1keV in the 2001 observation was first noted in this data set by Reeves et al. (2003). A highly significant broad absorption profile is apparent in the residuals at $E=1061\pm 11$ eV, with an equivalent width of $EW\sim 37 $eV, while the improvement in the fit statistic upon adding this absorption line is $\Delta \chi^2/\Delta\nu=42.0/3$ (see Table 2). The profile is resolved with a width of $\sigma=42^{+13}_{-11}$eV (or $\sigma_{\rm v}=11900{\ensuremath{^{+3700}_{-3100}}}$km s$^{-1}$), as per the similar broad profiles in OBS.E. As discussed above, one plausible identification is with mildly blueshifted Ne<span style="font-variant:small-caps;">x</span> Ly$\alpha$ (again notwithstanding any possible contribution from iron L absorption). At face value this suggests the presence of a lower velocity zone of the wind, which is investigated further in Section5. In contrast to the 2001 spectrum, the 2007 observations show little evidence for intrinsic absorption and appear to be featureless (see also Behar et al. 2010).
Consistency check with EPIC-MOS soft X-ray spectra
--------------------------------------------------
In this section we perform a consistency check on the lines detected in the RGS with the EPIC MOS spectra. Indeed, given the strength and breadth of the absorption features detected in the RGS spectra, we may expect them to be detectable at the MOS CCD spectral resolution. We thus considered the combined MOS1+2 spectra for each observation and, while we fitted only the 0.5–2keV energy range where the soft X-ray features occur, we also checked that the best fit models provide a good representation of the overall X-ray continuum to higher energies. The continuum model found from the analysis of the RGS spectra was adopted, but again allowing the parameters to adjust. As noted above, the 2001 MOS observation also shows pronounced curvature between 1.2–2.0keV, which is apparent in the residuals of this spectrum in Figure \[MOS\], and was accounted for by including a neutral partial covering absorber.
The residuals of the five MOS spectra (OBS.B, CD, E, 2001 and 2007) to the best fit continuum models are shown in Figure \[MOS\]. Several absorption profiles are clearly visible (which are labelled on each of the corresponding spectral panels), and they are generally in good agreement with the features present in the RGS in Figure 4. In particular, the broad absorption profiles present during OBS.CD and OBS.E are confirmed at high significance, whereby both the broad profile at $\sim1.2$ keV (OBS.CD) and the complex absorption structure between 0.9–1.2keV (OBS.E) emerge at high signal to noise in the MOS data. Likewise the broad absorption trough detected at 1.06keV in the 2001 dataset, is also confirmed in the MOS spectra, with self consistent parameters. On the other hand, the featureless nature of the 2007 spectrum is also confirmed in the MOS, with no obvious residuals present in the spectrum. The results of the Gaussian line fitting to the MOS spectra are subsequently summarized in Table 3.
As a further check of the consistency between the RGS and MOS detections of the absorption lines, we generated and overlaid confidence contours for each of the lines that are independently detected in each of the RGS and MOS spectra. As an illustration we show two of the examples from the OBS.E and 2001 sequences in Figure7. The upper panel shows the 68%, 90% and 99% confidence contours (for 2 parameters of interest) of line energy against flux for the $\sim1.16$keV absorption line detected in the OBS.E spectra. The contours show the close agreement between the two detectors, with the absorption line confirmed in both the RGS and MOS spectra. Generally the line fluxes are more tightly constrained in the MOS, which is likely due to the higher signal to noise and better continuum determination afforded by the MOS spectra. The lower panel of Figure7 shows the comparison between line energy and width (in terms of $\sigma$) for the $\sim 1$keV broad absorption trough in the 2001 observation. The contours show that the broadening of the line is independently measured by both RGS and MOS, with a width of $\sigma\sim40$eV (or $\sigma=12000$kms$^{-1}$). The only marginal disagreement at the 90% level is in the line energy, which is offset by $\sim20$eV, although this is likely to lie within the uncertainty of the absolute energy scale of the MOS detectors.
Modeling with Photoionized Absorption Models
============================================
Having established the consistent detection of the absorption lines between the RGS and MOS spectra, in this next section we model the absorption (and any emission) from these spectra with a self consistent [xstar]{} photoionization model (Kallman & Bautista 2001). The goals of the modeling are to determine the likely identifications of the absorption profiles and to reveal the properties and possible outflow velocities of the soft X-ray absorption zones.
Generation of Tabulated Photoionized Models
-------------------------------------------
In order to generate the <span style="font-variant:small-caps;">xstar</span> grids for the modeling, we used the input SED defined from a simultaneous fit to the [[*XMM–Newton*]{}]{} OM, EPIC and [*NuSTAR*]{} data, taken from the 2013–2014 campaign. The details of the SED modeling can be found in Nardini et al. (2015); in summary the SED can be parameterized by a double broken power law from between the optical–UV, UV to soft X-ray and X-ray portions of the spectrum. In the optical-UV band covered by the OM (2–10eV), the spectrum has $\Gamma \simeq 0.7$ (so rising in $\nu F_{\nu}$ space), between the UV and soft X-rays (10–500eV) the SED declines steeply with $\Gamma \sim
3.3$, and above 500eV, the intrinsic photon index is $\Gamma \simeq 2.4$. As per Nardini et al. (2015), the total 1–1000Rydberg ionizing luminosity is $5\times10^{46} -1\times10^{47}$ergs$^{-1}$, depending on the exact SED model adopted. Note that the main impact upon using this SED form as the input photoionizing continuum is in the ionization balance of the gas; when compared to a model computed with a standard $\Gamma=2$ X-ray power-law continuum, the ionization parameter will be comparatively higher, as there are a larger number of input UV and soft X-ray ionizing photons in the SED form, resulting in a higher ionizing luminosity over the 1–1000Rydberg band. This is particularly critical in the case of PDS456, which has a steep UV to X-ray continuum; as a result, the ionization parameter can be an order of magnitude higher when compared to a model produced with a flat $\Gamma=2$ illuminating continuum.
Subsequently, a tabulated grid of <span style="font-variant:small-caps;">xstar</span> models was calculated adopting this double broken power-law SED, with an input ionizing luminosity of $1\times10^{47}$ergs$^{-1}$ and an electron density of $n_{\rm e}=10^{8}$cm$^{-3}$, although the output spectra are largely insensitive to the density that is chosen. The grid of models also covers a wide range of parameter space, from $5\times10^{20} < N_{\rm H} <2.5\times10^{23}$cm$^{-2}$ in column density (in steps of $(\log N_{\rm H})=0.14$) and from $2 < \log\xi < 7$ (in steps of $(\log\xi)=0.2$) in terms of the ionization parameter. The subsequent output spectra were calculated over the 0.1–20keV range, over 10,000 energy steps. A turbulence velocity of $\sigma_{\rm turb}=5,000$kms$^{-1}$ was chosen for the final grid of models, as it was found that much lower turbulences could not match the breadth or equivalent width of the soft X-ray absorption features, while for higher turbulences, the absorption profiles became too broad to model the observed features seen near to 1keV in the RGS spectra.
Given the good agreement between the RGS and MOS for the [[*XMM–Newton*]{}]{} sequences, for each sequence they were fitted simultaneously, allowing for a cross normalization factor (typically within $\pm5$% of 1.0) between the two instruments. This has the benefit of utilizing both the high spectral resolution of the RGS and the higher signal to noise of the MOS data. The spectra were fitted between 0.6–2.0keV, avoiding the energy band below 0.6keV in order to minimize any calibration uncertainties associated with the O edge in both detectors. We did not directly fit the MOS spectra above 2keV, as our motivation was to study the properties of the soft X-ray absorption. However, as a consistency check we did extrapolate the spectra above 2keV to see whether, to first order, the model reproduced the continuum form above 2keV. The continuum model adopted was the same as in the Gaussian fitting, i.e. a powerlaw plus blackbody component absorbed by Galactic absorption. Generally, the photon index of the power law was fixed in all of the sequences to $\Gamma=2.1$, however in OBS.B and 2007 this was found to be steeper with $\Gamma=2.4-2.5$. In the 2001 sequence, which has a strongly absorbed spectrum, an additional neutral partial covering absorption component (with $N_{\rm H}=4.8^{+1.1}_{-0.9}
\times 10^{22}$cm$^{-2}$ and covering fraction $f_{\rm cov}=0.34\pm0.02$) was applied to the soft X-ray power law, as otherwise the photon index was found to be unusually hard with $\Gamma\sim1$. We note that a similar partial covering component has been found in some of the previous [*Suzaku*]{} observations of PDS456, when the spectrum becomes unusually hard (Reeves et al. 2014, Gofford et al. 2014, Matzeu et al. 2016).
Fitting Methodology
-------------------
The <span style="font-variant:small-caps;">xstar</span> absorption grids were then applied to each of the sequences allowing the column density, ionization parameter and outflow velocity to vary independently, as well as the continuum parameters. In order to find the best fit solution for each absorption zone as fitted to each sequence in velocity space, we first varied the outflow velocity from $-0.3\,c$ (where the negative sign here denotes blueshift) up to $+0.3c$, in steps of $0.005c$, while allowing the ionization and column to adjust at each step in the fitting. Thus no a priori assumption is made about the likely velocity of any of the absorbers fitted to each sequence and the wide range in velocity space searched ensures that the global minimum is found, allowing for any possible zero velocity solution as well as outflow or inflow. This procedure was then repeated for each absorption zone which was subsequently added to each sequence, applying the criteria that the addition of each absorption grid to the model required an improvement in fit statistic corresponding to at least the 99.9% confidence level. Once the global minimum for each absorption grid was found, we then conducted a finer search in velocity, ionization and column density in order to find the exact minimum and uncertainties associated with each parameter.
As an example, we describe in detail how this was applied to the OBS.E spectra, while the same fitting procedure was applied to all of the sequences. The initial starting point was with the baseline continuum model, with only the Galactic absorption component present (and no intrinsic absorption), which yielded a poor fit statistic of $\chi^{2}/\nu=289.0/161$. A single ionized zone of absorption was then added to the model and the $\chi^{2}$ space searched by stepping through the fit at each point in the outflow velocity, as described above. Figure 8 (upper panel) shows the resulting fit statistic against outflow velocity for this absorber. A clear minimum is found at $v=-0.17\pm0.01c$, while a zero velocity solution is ruled out at $>99.99$% confidence. As a result of the addition of this single absorber with an outflow velocity of $-0.17c$, the fit statistic improved to $\chi^{2}/\nu=206/158$. The corresponding F-test null hypothesis upon adding the first zone of absorption to the model (compared to the model with no absorption) is very low, with $P_{\rm f}=1.3\times10^{-11}$. A second absorption grid was then added to the model, to test whether the fit improved further upon its addition and the velocity space of the new absorber was searched for any minimum in $\chi^{2}$. The result of adding this second absorption grid to the model is shown in Figure8 (lower panel), which also shows a well defined minimum, but at a lower outflow velocity of $v=-0.06\pm0.01c$. The fit statistic also significantly improved upon the addition of the second absorption grid, with a resulting reduced chi-squared of $\chi^{2}/\nu=181/155$. The corresponding F-test null hypothesis upon adding the second zone of absorption to the model (compared to the model with only one absorber) is then $P_{\rm f}=1.6\times10^{-4}$. The addition of further absorption grids to the OBS.E spectra did not significantly improve the fit.
Thus the OBS.E spectrum requires two absorption zones, one with a faster outflow velocity ($-0.17c$), while both zones formally require an outflow velocity at $>99.9$% confidence. The resulting best-fit absorption parameters for both OBS.E as well as the other spectral sequences are shown in Table4, after applying the same fitting procedure as described above. In general, the soft X-ray absorption can primarily be modeled with two main <span style="font-variant:small-caps;">xstar</span> zones (labeled as zones 1 and 2 respectively in Table 4), both of which are required to be outflowing with respect to the systemic velocity of PDS456, while we refer to zone 2 as having the higher velocity of the two zones.
Results of Photoionization Modeling to OBS.E
--------------------------------------------
### A Self-consistent Emission and Absorption Model
In addition to the two absorption zones described above, the fit to OBS.E improves further to $\chi^{2}/\nu = 143.2/150$, by adding two broad Gaussian emission profiles. The emission line centroids are at $922\pm9$eV and $1066\pm16$eV and are apparent in the MOS and RGS spectra for OBS.E (see Figure 9), while the lines have fluxes of $7.5^{+4.0}_{-2.9}\times10^{-5}$ and $3.2^{+2.2}_{-1.8}\times10^{-5}$photonscm$^{-2}$s$^{-1}$, respectively (or equivalent widths of $17\pm8$eV and $11\pm7$eV). Note that strong 0.92keV line emission has also been detected previously in some of the [[*Suzaku*]{}]{} XIS spectra of PDS456, see for instance Figure2 in Reeves et al. (2009). The line width, assuming a common width between the lines, is $\sigma=20\pm8$eV (or equivalently, $\sigma_{\rm v}=6,500\pm2,600$kms$^{-1}$).
In order to model the emission, we removed the adhoc Gaussian emission lines from the model and instead fitted the emission using a self consistent grid of photoionized emission model spectra produced by <span style="font-variant:small-caps;">xstar</span>, with the same input parameters and turbulence velocity ($\sigma_{\rm turb}=5,000$kms$^{-1}$) as per the absorption grids. Like for the absorption modeling, two emission grids were added to the model, with the column densities of the two emission zones tied to the corresponding values in the two absorption zones. The ionization, outflow velocity and normalization (which is proportional to luminosity) of the two emission grids were allowed to vary. The parameters of the two emission zones are reported in Table5 as applied to OBS.E. The most significant emission zone is zone1, which is also the lower ionization of the two zones ($\log\xi = 2.8\pm0.3$) and yields an improvement in fit statistic of $\Delta \chi^{2}/\Delta\nu = 23.0/3$ upon its addition to the baseline absorption model, while in comparison the second and more highly ionized zone ($\log\xi\sim4.6$) is only marginally significant ($\Delta \chi^{2}/\Delta\nu = 12.0/3$).
This final <span style="font-variant:small-caps;">xstar</span> model provides a good fit to the simultaneous RGS and MOS spectra in OBS.E, with an overall fit statistic of $\chi^{2}/\nu = 146.5/149$. The resulting fit is shown in Figure 9, panel (a), with the <span style="font-variant:small-caps;">xstar</span> model able to reproduce the absorption features, i.e., the two broad absorption troughs at rest frame energies of $\sim1.0$keV and $\sim1.2$keV. While the emitters do not necessarily have to be physically associated to the corresponding absorption zones, the emission is nonetheless able to reproduce the excess soft X-ray emission. In particular, the lower ionization emission zone 1 accounts for the stronger emission line observed near 920eV. Emission zone 1 also predicts an emission feature near to 0.8keV, which is consistent with the spectrum shown in Figure9. Emission zone 2 has a more subtle effect, mainly accounting for the weaker emission line near 1.06keV. If this best-fit model to the soft X-ray spectrum of OBS.E is extrapolated up to higher energies using the MOS data, the agreement is good, with the model reproducing well the shape and level of the continuum above 2keV, as is seen in Figure 9, panel (b). The remaining residuals at high energies are due to the iron K absorption profile, as reported in Nardini et al. (2015).
### Properties of the Soft X-ray Emission and Absorption
In terms of the absorber properties, zone 1 has a lower column and lower ionization compared to zone 2 (see Table 4 for parameter details), as well as a lower outflow velocity of $v_{\rm out}/c=-0.064\pm0.012$ (compared to $v_{\rm out}/c=-0.17\pm0.01$ in zone 2), and primarily accounts for the lowest energy of the two absorption troughs. Note that a non-zero outflow velocity is required for both zones in the fit, as is also described in the fitting procedure described in Section5.2 (see Figure 8). With the emitter included, if the velocity of both zones is forced to zero and the model refitted allowing the other absorber parameters to vary ($N_{\rm H}$, $\log\xi$), then the fit is substantially worse compared to the case where the outflow velocities are allowed to vary ($\chi^{2}/\nu = 181.2/152$ versus $\chi^{2}/\nu = 143.2/150$). On the other hand, if we do allow the velocity to vary, but assume that it is the same for both zones, then the fit statistic is also worse than before ($\chi^{2}/\nu = 176.4/151$), as adjusting the ionization alone is not enough to explain the different energy centroids of the absorption profiles. Thus the two absorber zones are required to be distinct in velocity space, as well as in column/ionization.
In Figure 10 the relative contribution of each absorption zone is shown, in terms of the fraction of the continuum that is absorbed against energy. The main contribution towards zone 1 arises from Ne<span style="font-variant:small-caps;">ix–x</span>, as well as absorption from L-shell Fe from ions in the range Fe<span style="font-variant:small-caps;">xix–xx</span>; the blend of these lines together with a modest net blueshift produces the overall broad absorption trough which is centered near to 1keV in the rest frame. The contribution towards zone 2 is similar, giving a blend of lines from Ne<span style="font-variant:small-caps;">ix–x</span> and Fe<span style="font-variant:small-caps;">xx–xiv</span>; then the higher energy of this absorption blend (centered near to 1.2keV), arises primarily from the higher outflow velocity and somewhat from the higher ionization of this zone.
Note that although either the 1 keV or 1.2 keV absorption troughs in OBS.E could at first sight be associated with absorption from Ne<span style="font-variant:small-caps;">x</span> (at 1.02keV) or Fe<span style="font-variant:small-caps;">xxiv</span> (at 1.16keV) with a smaller blue-shift, the self consistent [xstar]{} modeling presented above appears to rule this out. This is because it is difficult to produce absorption from these species in isolation, without a contribution from other, lower energy, lines. For instance for the higher velocity zone, although the Fe<span style="font-variant:small-caps;">xxiv</span> $2s\rightarrow3p$ absorption can contribute towards the broad absorption profile at 1.2keV, the other lower energy L-shell Fe lines near to 1keV also strongly contribute, reducing the overall centroid energy of the absorption trough produced in the [xstar]{} model (see Figure 10 for their relative contributions). The net result is that an overall blue-shift of the absorption profile is required to match its observed energy in the actual spectrum.
In terms of the emission, the strongest contribution is from zone 1, which has the lowest ionization parameter. The outflow velocities of the two emission zones are lower than their respective absorption zones, i.e., the outflow velocity of zone 1 is $v_{\rm out}=-0.04\pm0.02c$. This could be consistent with the emission expected from a bi-conical outflow, as emission can be observed across different sightlines which will appear to have a lower net outflow velocity than along the direct line of sight to the observer. The emission from the near side of the outflow may be preferentially observed in this case, if the emission from the far (receding) side of the wind is obscured through a greater column of matter. This may be the case if, for instance, the receding wind is on the far side of the accretion disk.
The relative contribution of the photoionized emission zones to the soft X-ray spectrum of PDS456 are shown in Figure 11. The dominant contribution towards the emission is from zone 1 (Figure 11, red curve) and the majority of the emission arises from O<span style="font-variant:small-caps;">vii</span> and O<span style="font-variant:small-caps;">viii</span>. In particular, the emission near to 0.9keV is mainly accounted for by blueshifted O<span style="font-variant:small-caps;">viii</span> Radiative Recombination Continuum (RRC) at 0.87keV, along with some contribution from the forbidden line of Ne<span style="font-variant:small-caps;">ix</span>. At first sight, the O<span style="font-variant:small-caps;">viii</span> RRC appears stronger than the O<span style="font-variant:small-caps;">viii</span> Ly$\alpha$ line at lower energies. However, this is just due to the latter emission being more suppressed by the Galactic absorption towards PDS456, as this lower energy line falls just above the deep neutral O<span style="font-variant:small-caps;">i</span> edge as seen in the observed frame. In contrast, the higher ionization ($\log\xi=4.6^{+0.6}_{-0.4}$) zone 2 is much weaker (Figure 11, blue curve), and its main contribution is to reinforce the emission near 1keV, mainly due to blueshifted emission from Ne<span style="font-variant:small-caps;">ix</span> and Ne<span style="font-variant:small-caps;">x</span>.
Extension to the other *XMM–Newton* observations
------------------------------------------------
The absorption modeling was then extended to the other 2013–2014 sequences, as well as the 2001 and 2007 [[*XMM–Newton*]{}]{} observations. In these observations, the soft X-ray emission is less prominent, as the continuum fluxes are generally higher than in OBS.E, thus we subsequently concentrated on modeling only the absorption zones. The zone 2 absorber appears common to all of the data sets, except for the bright and featureless 2007 spectrum, where only an upper limit can be placed on the column. Indeed, the ionization parameter of this zone is consistent within errors for all the data sets, varying only in the narrow range from $\log\xi=4.04-4.19$ (see Table4). The main differences arise in the column density, which is higher in the more absorbed OBS.E ($N_{\rm H}=1.5\pm0.4 \times 10^{22}$cm$^{-2}$) and 2001 ($N_{\rm H}=2.2^{+0.7}_{-0.6} \times 10^{22}$cm$^{-2}$) sequences and weakest in the least absorbed OBS.B and 2007 observations. Indeed in the bright, continuum dominated 2007 observations, a low upper-limit can be placed on the column density of $N_{\rm H}< 0.23
\times 10^{22}$cm$^{-2}$. Likewise, although the RGS data is not available for OBS.A, on the basis of the EPIC/MOS data we can still place an upper-limit on the column density of $N_{\rm H}<0.35\times10^{22}$cm$^{-2}$ for zone 2, consistent with the absorption being weaker while the continuum flux was brighter during this observation.
The outflow velocity is strongly variable between some of sequences, e.g. $v_{\rm out}/c=-0.27\pm0.01$ in OBS.CD versus $v_{\rm out}/c=-0.17\pm0.01$ in OBS.E. This is illustrated in Figure 12, which shows the confidence contours in the $N_{\rm H}$ versus outflow velocity plane for zone 2 in each of the B, CD, E and 2001 sequences, the best fit parameters clearly differing along the velocity axis. Thus, while the column determines the depths of the broad absorption troughs, the outflow velocity determines its blueshift. Indeed, in the CD observation it can be seen (e.g. from Figure 6) that the broad absorption trough is shallower and shifted to higher energies compared to observation E.
On the other hand, while the lower velocity zone 1 is required to reproduce the lower energy absorption troughs in OBS.E and 2001 near to 1keV, it is not required in any of the other observations and thus appears rather sporadic. Zone 3 is included for completeness and represents a possible high ionization phase of the wind, similar to what is seen at iron K, with a similar ionization level of $\log \xi = 5.5$ and a velocity of $-0.3\,c$ (i.e., Nardini et al. 2015). At this level of ionization, in the soft X-ray band, there is only a small imprint from zone 3 in the spectrum, which is mainly in the form of a O<span style="font-variant:small-caps;">viii</span> Ly$\alpha$ absorption line blueshifted to above 0.8keV (see the lower panel of Figure 10), while only trace amounts of absorption due to H-like Ne and Mg are predicted. Due to its transparency at soft X-rays, this highest ionization zone is therefore only formally required in the OBS.CD spectrum, mainly to reproduce the weak absorption feature detected near to 0.84keV in the RGS and MOS data (see Tables2 and 3).
Discussion
==========
The Detection of Fast Soft X-ray Absorbers
------------------------------------------
Although the numbers of confirmed fast outflows have significantly increased through systematic studies of AGN in the iron K band (Tombesi et al. 2010, Gofford et al. 2013), to date there have been relatively few detections of fast outflows in the soft X-ray band. Here, the detection of broad soft X-ray absorption line profiles associated with the fast wind in PDS456 represents one of the few cases where the presence of an ultra fast outflow has been established in the form of highly blueshifted absorption in both the Fe K and soft X-rays bands. Other notable recent examples include the nearby QSO, PG1211+143 (Pounds 2014) and in the NLS1, IRAS17020+4544 (Longinotti et al. 2015). In both of these cases, the observations suggest that the outflow is more complex than a single-zone medium, with multiple ionization and/or velocity components. In PG1211+143, at least two different velocity components are present, both in the iron K band as well as soft X-rays (Pounds et al. 2016), while in the NLS1 IRAS17020+4544, a complex fast ($v_{\rm out}=23,000-33,000$kms$^{-1}$) soft X-ray absorber is present, covering at least 3 zones in terms of ionization parameter.
In PDS456, while the velocity of the soft X-ray absorber (with $v_{\rm out}=0.17-0.27\,c$ for zone 2) is similar to the highly ionized iron K absorption in this AGN ($v_{\rm out}=0.25-0.3\,c$), the column density and ionization is up to two orders of magnitude lower than the iron K absorber, i.e., $\log\xi \sim4$ and $N_{\rm H}=10^{22}$cm$^{-2}$ at soft X-rays versus $\log \xi \sim 6$ and $N_{\rm H}=10^{24}$cm$^{-2}$ at iron K. This also suggests that the outflow is more complex than what would be expected for a simple homogeneous radial outflow, where the density varies with radius as $n\propto r^{-2}$. Thus, as $\xi=L/nr^{2}$, in the radial case the ionization should remain approximately constant along the outflow. A density profile with a power-law distribution flatter than $n(r) \propto r^{-2}$ could account for a decrease in ionization with absorber radial distance, and indeed it has been suggested from the distribution of warm absorbing gas amongst several Seyfert 1s (Behar 2009, Tombesi et al. 2013). Another likely possibility, especially given the rapid variability of the soft X-ray absorption seen here, is a clumpy, multi phase wind. Thus denser (and more compact) clumps or filaments could coexist within a smoother highly ionized outflow to explain the ionization gradients. Indeed a recent, albeit slower ($v\sim0.01c$), multi-phase disk wind was recently revealed in the changing look Seyfert galaxy, NGC1365 (Braito et al. 2014), with very rapid changes in soft X-ray absorption (with $\Delta N_{\rm H}\sim10^{23}$cm$^{-2}$ in $\Delta t <100$ks), requiring a clumpy outflowing medium.
The Soft X-ray Outflow
----------------------
We now explore the properties of the soft X-ray absorber within the context of a clumpy wind model. In order to place a radial constraint on the absorbing gas, its variability timescale is considered. We concentrate on zone 2, which is the best determined of the absorption zones and is relatively constant in ionization (see Table4). During the first four observations in the 2013–2014 campaign, there is a subtle variation in the column density from $<0.35\times10^{22}$cm$^{-2}$ (OBS.A) to $0.70^{+0.22}_{-0.21}\times10^{22}$cm$^{-2}$ (OBS.CD), i.e. a slight increase over a 3 week timescale. A stronger variation occurs over the subsequent 5-month period between OBS.A–D and OBS.E, with an increase in column density to $1.5\pm0.4\times10^{22}$cm$^{-2}$, while the velocity of the absorber also decreases from $0.27\pm0.01\,c$ to $0.17\pm0.01\,c$. The more subtle absorption variability between OBS.A and OBS.D in 2013 likely corresponds to the passage of the same absorbing system (with the same outflow velocity) across the line of sight. However, the stronger variations in $N_{\rm H}$ and outflow velocity between this and OBS.E (as well as in 2001) likely correspond to different absorption systems (with 2007 being unabsorbed). Therefore we set a plausible timescale for the absorption variability, in terms of the passage of clumps or filaments of gas across our line of sight, of between $\Delta t = 10^{6} - 10^{7}$s, i.e., weeks to months. It is also likely that some variability can occur on shorter timescales, as found during the low flux 2013 [*Suzaku*]{} observations, where changes in the soft X-ray spectrum on timescales of ${\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}}100$ks could be accounted for by variability of a partially covering absorber (Matzeu et al. 2016).
The transverse motion of the absorbing systems moving across the line of sight is assumed to be due to the overall Keplerian rotation of the gas within the wind, with the observed outflow velocity being due to the bulk motion of the wind towards us. The Keplerian velocity at a radius $R$ is simply $v_{\rm K}^{2}=c^{2}/r_{\rm g}$, where $r_{\rm g} = R / R_{\rm g}$ is the radial distance from the black hole in gravitational units. The column through an (approximately spherical) absorbing cloud can be expressed as $N_{\rm H} \sim n_{\rm H} \Delta R$, where its size is $\Delta R = v_{\rm K} \Delta t$. The number density of an absorbing cloud is then given as: $$n_{\rm H} \sim \frac{N_{\rm H}}{\Delta R} = \frac{N_{\rm H} r_{\rm g}^{1/2}}{c\Delta t}.$$ The definition of the ionization parameter gives $n_{\rm H} \sim n_{\rm e} =
L_{\rm ion}/\xi R^{2}$, thus equating these densities yields the radial distance of the clouds: $$r_{\rm g}^{5/2} = \frac{L_{\rm ion}}{\xi} \frac{c^{5}\Delta t}{N_{\rm H}} (GM_{\rm BH})^{-2}.$$ For the zone 2 absorber $\log\xi \sim 4$ and $N_{\rm H} \sim 10^{22}$cm$^{-2}$, while for PDS456 $L_{\rm ion} \sim 10^{47}$ergs$^{-1}$ and $M_{\rm BH}\sim10^{9}{\hbox{$\rm\thinspace M_{\odot}$}}$. Then for $\Delta t = 10^{6}-10^{7}$s, $R \sim 4000-11000R_{\rm g} = 0.7-1.7 \times 10^{18}
\sim 10^{18}$cm. At this distance, and for the ionization of the gas, then the number density is $n_{\rm H}\sim10^{7}$cm$^{-3}$, while the radial extent of the clouds is $\Delta R\sim10^{15}$cm. So the outflow clouds can be fairly compact (approximately $10 R_{\rm g}$), and also capable of partially covering the X-ray source.
At a distance of $R\sim10^{18}$cm, the Keplerian velocity is $v_{\rm K}\sim4000$kms$^{-1}$. This agrees well with the measured velocity widths of the soft X-ray emission lines, with $\sigma_{\rm v}=6,500\pm2,600$kms$^{-1}$ (e.g. from OBS.E), or the required velocity broadening ($\sigma=5,000$kms$^{-1}$) in the <span style="font-variant:small-caps;">xstar</span> modeling. Furthermore, this matches well the velocity widths of the broad UV emission lines in PDS456, where the FWHM of the Ly$\alpha$ and C<span style="font-variant:small-caps;">iv</span> lines are 12,000kms$^{-1}$ and 15,000kms$^{-1}$, respectively, and where the C<span style="font-variant:small-caps;">iv</span> profile shows one of the largest blueshifts amongst quasars of $\sim5,000$kms$^{-1}$ (O’Brien et al. 2005; compare to Richards et al. 2002). Thus the location of the soft X-ray outflow appears coincident with the BLR in this AGN, and indeed some of the more ionized (and less dense) BLR clouds could be responsible for the soft X-ray absorption.
Alternatively, we can assume instead that the soft X-ray absorber arises from a wind with a smooth radial profile out to large distances. In that case, from integrating through the flow, then the radial extent of the wind is given by $R_{\rm smooth} =
L_{\rm ion} / N_{\rm H} \xi \sim 10^{21}$cm for the above outflow parameters. Such a large-scale outflow up to a kpc in extent, with a rather low density (where $n_{\rm H}\sim10$cm$^{-3}$), would not be able to explain the variability of the absorber on weeks to months timescales. Furthermore, the velocity widths resulting from gas on these large scales is likely to be rather small, of $\sigma \sim 100$kms$^{-1}$, whereas (as seen in Figure2) there is no narrow component of emission/absorption in PDS456 that could be associated to a distant warm absorber. We conclude that the fast, soft X-ray absorber in PDS456 arises from clumpy gas on smaller (BLR) scales, and may exist co-spatially as a higher density (but lower ionization) phase of the fast, accretion disk wind.
The Soft X-ray Emission
-----------------------
The luminosity of the soft X-ray line emission can also be used to calculate the global covering factor of the outflowing gas. From the photoionization modeling, the normalization (or flux), $\kappa$, of each of the emission components is defined by [xstar]{} in terms of: $$\kappa = f\frac{L_{38}}{D_{\rm kpc}^2}$$ where $L_{38}$ is the ionizing luminosity in units of $10^{38}$ergs$^{-1}$, $D_{\rm kpc}$ is the distance to the quasar in kpc. Here $f$ is the covering fraction of the gas with respect to the total solid angle, where $f = \Omega / 4\pi$. For a spherical shell of gas, $f=1$, while $L$ is the quasar luminosity that illuminates the photoionized shell. Thus by comparing the predicted normalisation ($\kappa$) for a fully covering shell of gas illuminated by a luminosity $L$ versus the observed normalization ($\kappa_{\rm xstar}$) determined from the photoionization modeling, the covering fraction of the gas can be estimated. For PDS456, with $L=10^{47}$ergs$^{-1}$ at a luminosity distance of $D=860$Mpc, for a spherical shell the expected [xstar]{} normalization is $\kappa=1.35\times10^{-3}$. Compared to the observed normalization factors reported in Table5, the covering fraction of strongest lower ionization (zone1) of emitting gas is $f=0.44\pm0.22$.
The soft X-ray emitting (and absorbing) gas thus covers a substantial fraction of $4\pi$ steradian, consistent with the result obtained by Nardini et al. (2015) for the high ionization wind measured at Fe K, which was found to cover at least $2\pi$steradian solid angle. This is consistent with the picture of the soft X-ray outflowing gas being embedded within the wide angle high ionization disk wind. Note that, despite the large covering factor of the soft X-ray gas, its [*volume*]{} filling factor is likely smaller. This is consistent with a geometry where the absorbing clouds intercept a large enough fraction of the sightlines from the X-ray source, but individually are compact enough to occupy a small enough region in volume.
Indeed, the volume filling factor can be estimated by considering the emission measure of the gas. To estimate the emission measure, we consider the strong O<span style="font-variant:small-caps;">viii</span> RRC emission, with a flux of $\sim7.5\times10^{-5}$photonscm$^{-2}$s$^{-1}$, as determined from the fit to the [[*XMM–Newton*]{}]{} OBS.E spectrum (see Section 5.2). This corresponds to a luminosity of $L_{\rm O\,VIII} \sim 6\times10^{51}$photonss$^{-1}$ at the distance of PDS456. The recombination coefficient for O<span style="font-variant:small-caps;">viii</span> is $\alpha_{r}=1.2\times10^{-11}$cm$^{3}$s$^{-1}$, for a temperature of $kT\sim10$eV (Verner & Ferland 1996). The overall emission measure can then be calculated from: $${\rm EM} = \frac{L_{\rm O\,VIII}}{\alpha_r A_{\rm O}f_if_r}$$ where $A_{\rm O}$ is the abundance of Oxygen, $f_i$ is the ionic fraction of the parent ion (fully ionized Oxygen) and $f_r$ is the fraction of recombinations that occur direct to the ground state. Here we take $f_i\sim0.5$ given the ionization state of the gas, $f_r\sim0.3$ for the fraction of recombinations direct to ground, while an O abundance of $4.9\times10^{-4}$ is assumed (Wilms et al. 2000, Asplund et al. 2009). This gives an estimated emission measure for the soft X-ray emitting gas of ${\rm EM}\sim6\times10^{66}$cm$^{-3}$.
In comparison we can also calculate the emission measure by assuming the gas clouds occupy a spherical region of radius $R$, with a mean density $n$ and a volume filling factor of $V_{\rm f}$. For the emission measure:- $${\rm EM} = \int n^2 {\rm d}V \sim \frac{4}{3}\pi R^{3} V_{\rm f} n^{2}.$$ Taking a radius of $R=10^{18}$cm (consistent with the line velocity widths) and an emitter ionization of $\log\xi\sim3$ (for the lower ionization zone 1) gives a density of $n=10^8$cm$^{-3}$ for the emitting clumps. By comparison of the above expression with the emission measure estimated from the line emission, this implies a volume filling factor of $V_{f}\sim10^{-4}$, consistent with our expectations of a clumpy medium. A higher fraction of $V_{\rm f}\sim1$ only occurs at much larger kpc scales (and at correspondingly lower densities, as $nR^{2}=L_{\rm ion}/\xi$), resulting in a smooth medium, but inconsistent with the observed absorption variability. Note that the clumps probably exist as quasi-spherical clouds, rather than long strands or filaments of material, in order to satisfy the criteria of a low volume filling factor but a high covering factor.
Outflow Energetics
------------------
Following Nardini et al. (2015), we also estimate the mass outflow rate from: $$\dot{M}_{\rm out}=1.2m_{\rm p}\times 4\pi f v_{\rm out} N_{\rm H} R$$ where $f$ is the covering fraction and for the outflow, $v_{\rm out}=0.2c$, $N_{\rm H}=10^{22}$cm$^{-2}$, $R=10^{18}$cm and $f\sim0.4$ from above. As the above equation is expressed in terms of the column density, it is independent of any volume filling factor, as the observed $N_{\rm H}$ through the line of sight is the same regardless of whether the outflowing matter is smooth or clumpy. This yields a mass outflow rate of $\dot{M}_{\rm out}=6\times10^{26}$gs$^{-1}$ (or $\sim 10 M_{\odot}$yr$^{-1}$), with a corresponding kinetic power of $\dot{E}_{K}\sim10^{46}$ergs$^{-1}$ (or approximately 10% of the bolometric luminosity). This is consistent with the estimates from Nardini et al. (2015) for the highly ionized zone of the wind. This could suggest either that the lower ionization clumps form further out along the wind from the high ionization matter, or that the less ionized matter carries an energetically similar component as per the highly ionized part of the disk wind.
The overall view of the wind in PDS 456
---------------------------------------
PDS456 is the first known AGN where it has been possible to resolve broad absorption profiles (with typical velocity widths of $\sigma\sim5,000$kms$^{-1}$) in the soft X-ray band. In contrast, the absorption lines detected from the few fast outflows that appear to be present at soft X-rays (e.g. Gupta et al. 2013, 2015, Longinotti et al. 2015, Pounds et al. 2016), appear to result from narrower systems. Indeed, the detection of both the broad soft X-ray absorption lines and the fast P-Cygni-like profile at iron K may suggest that PDS456 is a higher ionization, X-ray analogue of the Broad Absorption Line (BAL) quasars commonly known to occur in the UV (Turnshek et al. 1988, Weymann et al. 1991). Indeed fast X-ray counterparts of the UV outflows have also been detected in some of the BAL quasars (Chartas et al. 2002, 2003). However unlike the UV outflows, which are likely to be line driven, the X-ray outflows are generally faster and higher ionization. Indeed they may instead be driven by other mechanisms, such as by continuum radiation, i.e. Compton scattering driven winds (King & Pounds 2003), or by magneto hydrodynamical processes (Fukumura et al. 2010).
One possibility is that the broad soft X-ray absorption lines in PDS456 may become more apparent when the overall continuum is more absorbed. This appears to be the case in the current RGS observations, where the quasar spectrum is generally more featureless at higher flux levels, while the absorption structure becomes apparent when the continuum flux is attenuated. A similar situation may have occurred during the recent extended campaign on the nearby Seyfert 1, NGC 5548 (Kaastra et al. 2014), when broad and blueshifted UV BAL profiles unexpectedly emerged when the soft X-ray flux was heavily suppressed by a partial covering X-ray absorber. In PDS456, the X-ray spectrum is also highly variable and similar to NGC5548, as well as to other AGN such as NGC3516 (Markowitz et al. 2008, Turner et al. 2008), NGC1365 (Risaliti et al. 2009) or ESO323-G077 (Miniutti et al. 2014), which exhibit pronounced absorption variability at soft X-rays.
Indeed a prolonged absorption event may have occurred in PDS456 months prior to the 2013 [[*XMM–Newton*]{}]{} observations, during an extended low flux observation with [*Suzaku*]{} in February–March 2013 (see Gofford et al. 2014; Matzeu et al. 2016 for details of these observations). During the 2013 [*Suzaku*]{} observations, the soft X-ray flux was lower by about a factor of $\times 10$, when compared to the relatively unobscured level observed in [[*XMM–Newton*]{}]{} OBS.B just six months later (Figure13). This suggests the presence of a variable, high column density obscuring medium in PDS456, with $N_{\rm H}>10^{23}$cm$^{-2}$. Several months later, during the last of the current [[*XMM–Newton*]{}]{} observations in February 2014 (OBS.E), the soft X-ray flux below 2keV started to decline again when compared to OBS.B, coincident with when the broad soft X-ray absorption features emerged in the RGS spectra. The more moderate level of obscuration during OBS.E (with $N_{\rm H}\sim10^{22}$cm$^{-2}$), likely accounts for the decrease in soft X-ray flux between OBS.B and OBS.E, while the overall level of the hard X-ray continuum above 2keV remained unchanged.
The overall view of the wind in PDS456 thus appears more complex than a simple, homogeneous radial outflow. From the above considerations it appears that the soft X-ray outflow represents a denser, variable and clumpy absorber, likely embedded within the fast, less dense, higher ionization phase of the wide-angle wind responsible for the persistent iron K absorption profile. The fast, high ionization phase of the wind is likely launched from the inner accretion disk (at distances within $\sim100\,R_{\rm g}$ or $\sim10^{16}$cm from the black hole, see Nardini et al. 2015), while the clumps within the outflow appear to manifest over larger scales, at around $R\sim10^{18}$cm, and are perhaps coincident with density perturbations within the wind. Indeed, hydrodynamical simulations naturally predict the existence of time variable density variations and streamlines within the wind (Proga, Stone & Kallmann 2000), which could then lead to the observed X-ray absorption variability. The denser clumps responsible for the soft X-ray absorption appear radially coincident with the expected location of the AGN broad emission line region, while they may also be responsible for the soft X-ray re-emission from the wind. The denser clouds may also be sufficiently compact ($\Delta R\sim10\,R_{\rm g}$) to act as the putative partial covering X-ray absorber, and may thus be responsible for the strong (order of magnitude) absorption variability seen at soft X-rays towards this quasar (Gofford et al. 2014, Matzeu et al. 2016). Indeed, in AGN in general, a complex topology of clouds surrounding the X-ray continuum source may be able to account for the general distribution of broadband X-ray spectral shapes observed from both the type I and type II AGN population (Tatum et al. 2013, 2016).
Future, high resolution observations may prove crucial in revealing the structure of the accretion disk wind seen in PDS456, as well as in other quasars, and the different wind phases that are present over different scales. Furthermore, any coincidence between the UV and the soft X-ray absorber in PDS456 can be established by simultaneous soft X-ray observations (i.e., with [[*XMM–Newton*]{}]{} RGS or [*Chandra*]{} LETG) and the UV (via [*HST*]{}/COS), which, as for the case of NGC5548, can determine how the UV line profiles respond to changes in the soft X-ray absorber.
Acknowledgements
================
J.N. Reeves, E. Nardini, P.T. O’Brien and M.T. Costa acknowledge the financial support of STFC. J.N. Reeves also acknowledges NASA grant number NNX15AF12G, while T.J. Turner acknowledges NASA grant number NNX13AM27G. E. Behar received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 655324, and from the I-CORE program of the Planning and Budgeting Committee (grant number 1937/12). This research is based on observations obtained with [[*XMM–Newton*]{}]{}, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.
Appendix: Statistical Significance of the Absorption Line Detections
====================================================================
Here we quantify and discuss the statistical significance of the absorption line detections in the RGS and MOS spectra, which are reported in Section4 of the main article. As an initial guide to the significance of the line detections, Tables 2 and 3 give the F-test null probabilities ($P_{\rm F}$) for the addition of each of the lines to the RGS and MOS spectra, noting that the F-test assumes that the prior distributions of the uncertainties are Gaussian. Nonetheless this gives an indication of the significance level. Considering the RGS alone, the most significant lines are the broad absorption profiles in 2001 at 1.06keV and in the OBS.E spectrum at 1.16keV, where $P_{\rm F}= 1.3\times10^{-5}$ and $P_{\rm F}= 9.0\times10^{-6}$ respectively, which exceeds the $4\sigma$ level. The weakest two features (the emission line associated to OBS.E, and the weak 850eV absorption line in OBS.CD) are confirmed at about the 99% level in the RGS alone.
However when considering the MOS spectra alone, where the statistics are highest, the null probabilities are even smaller, e.g. at the level of $P_{\rm F}\sim10^{-11}$ for some of the lines, which is at the $7\sigma$ confidence level. Indeed the likelihood of any false positives in both RGS and MOS is very small, as they would also have to occur at a consistent energy between both detectors. For instance the above 1.16keV absorption line in OBS.E has null probabilities of $P_{\rm F}= 9.0\times10^{-6}$ (RGS) and $P_{\rm F}= 7.4\times10^{-6}$ (MOS) for each detector, and thus the resultant probability of a false detection occurring in both detectors at the same energy is extremely small. This is also the case for the other RGS lines; indeed when considering their multiplicative probabilities of being independently detected in both the RGS and MOS, then the line detections are all subsequently confirmed at the $5\sigma$ or higher level.
Indeed as a further guide to the whether the spectra can be adequately accounted for by a simple baseline continuum without lines (which is the null hypothesis case), we also quote the reduced chi-squared for the baseline model and subsequent false probability resulting from this in Tables 2 or 3. For each spectrum, the fit statistic is extremely poor without the addition of any absorption lines to the spectra and the baseline continuum alone is not an adequate description of the data.
Monte Carlo simulations have then been used to assess the significance of the line detections (Protassov et al. 2002). This is essentially a means to calibrate a test statistic, whereby the prior uncertainty distribution is not known. As a test case, we ran the simulations on the OBS.E spectra, as this has the lowest continuum flux and thus may be more likely to yield a higher rate of false positives. The Monte Carlo simulations were performed independently for both the RGS and MOS exposures. For the null hypothesis model, we adopted the best fit baseline model for the continuum, with no absorption features, and we simulated 1000 spectra with the photon statistics expected for the same exposure times of the actual observation. In order to account for the uncertainties in the continuum model, each of the simulated spectra was then fitted with the null hypothesis model and a new simulation was performed with the new best fit as the continuum model; see Porquet et al. (2004) and Markowitz, Reeves & Braito (2006) for similar examples of applying this method.
Each of these simulated spectra was then fitted with the null hypothesis model to obtain a baseline $\chi^2$ value, fitted over the full 0.4-2.0keV band. The spectra were systematically searched for any line-like deviations over the 0.8–1.5keV (rest-frame) energy range, which is the most likely band for the lines to be found in the real data. To achieve this, we added a Gaussian line component and stepped through each energy interval in the spectra, with a step size of 7 eV and re-fitted the spectrum at each step. Thus by searching the spectra for any false positives over a wide energy range, no a priori assumption is made about the initial line energy and the Monte Carlo simulations represent a blind trial. Note that the step size over which the simulations were searched was chosen in order to match the binning used in the actual RGS and MOS spectra. We also assumed a line width of $\sim40$eV, similar to the typical value in the actual data.
We then obtained for each RGS simulated spectrum a minimum $\chi ^{2}$ and from these created a distribution of 1000 simulated values of the maximum $|\Delta \chi^{2}|$ values obtained from each spectrum (compared to the null hypothesis model). We then constructed two independent cumulative frequency distributions of the $|\Delta \chi^{2}|$ values expected for a blind line search in both the MOS and RGS spectra.
Only 1 RGS fake spectrum had a $|\Delta \chi^{2}| \ge 17.3$, which corresponds to the lowest significance case of the reported $|\Delta\chi^2|$ values for the observed absorption lines in the actual OBS.E spectrum (at 1.02keV, see Table2). Thus the inferred rate of false detections is $P_{\rm F}=1\times10^{-3}$, or alternatively the statistical significance of the detection is 99.9%. We then inspected the $\Delta \chi^{2}$ distribution of the MOS simulated spectra and we found that none of the spectra has a $|\Delta \chi^{2}| \ge 33.4$, which is the level of the fit improvement in the actual MOS spectrum for the same 1keV absorption line. Indeed the highest $|\Delta\chi^2|$ deviation found from the MOS simulations is 18.8; thus the false detection rate in the MOS OBS.E spectra is $P_{\rm F}<10^{-3}$. Hence even without requiring the false detections to occur at the same energy in the MOS and RGS (i.e. within $\pm 20 $eV as above), the Monte Carlo simulations demonstrate that the absorption features in OBSE have a combined false probability of $P_{\rm F}<10^{-6}$, given that the false detections would have to occur in both the RGS and MOS. Furthermore, given the lines occur at the same energy in the actual RGS and MOS spectra, this confirms that the features are likely intrinsic to the AGN and are not a statistical artifact.
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. \[tbnew\]
[lccccccc]{}
\
2013 OBS.A & 0721010201 & RGS1+2 & 2013-08-27 04:23:26 & – & – & –\
& & MOS1+2 & & 105.8 & $1.176\pm0.003$ & 3.80\
2013 OBS.B & 0721010301 & RGS1+2 & 2013-09-06 03:06:51 & 109.3 & $0.124\pm0.001$ & 2.52\
& & MOS1+2 & & 105.6 & $0.786\pm0.003$ & 2.58\
2013 OBS.C & 0721010401 & RGS1+2 & 2013-09-15 18:30:00 & 115.5 & $0.110\pm0.001$ & 2.28\
& & MOS1+2 & & 111.2 & $0.733\pm0.002$ & 2.42\
2013 OBS.D & 0721010501 & RGS1+2 & 2013-09-20 02:29:39 & 108.8 & $0.107\pm0.001$ & 2.25\
& & MOS1+2 & & 107.7 & $0.719\pm0.002$ & 2.38\
2014 OBS.E & 0721010601 & RGS1+2 & 2014-02-26 07:45:26 & 131.0 & $0.066\pm0.002$ & 1.58\
& & MOS1+2 & & 114.9 & $0.538\pm0.002$ & 1.67\
Mean B–E & – & RGS1+2 & – & 464.6 & $0.102\pm0.001$ & 2.14\
\
2001 & 0041160101 & RGS1+2 & 2001-02-26 09:44:02 & 43.4 & $0.209\pm0.003$ & 4.01\
& & MOS1+2 & & 43.7 & $1.315\pm0.005$ & 4.11\
2007 & 0501580101 & RGS1+2 & 2007-09-12 01:07:18 & 90.5 & $0.199\pm0.002$ & 4.25\
& & MOS1+2 & &87.6& $1.462\pm0.004$ & 4.42\
2007 & 0501580102 & RGS1+2 & 2007-09-14 01:46:07 & 89.1 & $0.126\pm0.002$ & 2.67\
& & MOS1+2 & & 86.1 & $0.950\pm0.003$ & 2.86\
\[observations\]
[lccccccc]{}\
ABS1 &$846{\ensuremath{^{+6}_{-6}}}$ & $6.5^{+6.0}_{-2.8}$ & $2,300^{+2,100}_{-1,000}$ &$-5.0{\ensuremath{^{+2.4}_{-2.4}}}$ & $-4.6{\ensuremath{^{+2.2}_{-2.2}}}$ &17.0/3 & $8.2\times10^{-3}$\
ABS2 &$1174{\ensuremath{^{+47}_{-58}}}$ & $109{\ensuremath{^{+49}_{-35}}}$ & $28,000^{+13,000}_{-9,000}$ & $-13.0{\ensuremath{^{+4.0}_{-4.0}}} $& $-41{\ensuremath{^{+13}_{-13}}}$ & 25.2/3 & $7.1\times10^{-4}$\
&&&&&&\
\
ABS1 &$1016{\ensuremath{^{+19}_{-18}}}$ & $41{\ensuremath{^{+14}_{-12}}}$ & $12,000^{+4,000}_{-3,500}$ &$-11.2{\ensuremath{^{+8.0}_{-5.4}}}$ & $-27.3{\ensuremath{^{+19.3}_{-13.2}}}$ & 17.3/3 & $1.4\times10^{-3}$\
ABS2 &$1166{\ensuremath{^{+19}_{-20}}}$ & $41^{\rm t}$ & &$-6.7{\ensuremath{^{+4.1}_{-2.8}}} $& $-31.3{\ensuremath{^{+19.1}_{-13.0}}}$ & 27.0/2 & $9.0\times10^{-6}$\
EMIS1 &$913{\ensuremath{^{+16}_{-15}}}$ & $18{\ensuremath{^{+13}_{-8}}}$ & $6,000^{+4,000}_{-3,000}$ &$5.2{\ensuremath{^{+4.3}_{-3.2}}} $& $9.8{\ensuremath{^{+8.1}_{-6.0}}}$ & 11.6/3 & 0.013\
&&&&& &\
\
ABS1 &$1061{\ensuremath{^{+11}_{-11}}}$ & $42{\ensuremath{^{+13}_{-11}}}$ & $11,900^{+3,700}_{-3,100}$ & $-40.6{\ensuremath{^{+12.4}_{-16.8}}} $& $-37{\ensuremath{^{+11}_{-15}}}$ & 42.0/3 & $1.32\times10^{-5}$\
\
\[Gauss\]
[lcccccc]{}
\
ABS1 & $ 839 {\ensuremath{^{+ 16}_{-4 }}}$ & $-8.2{\ensuremath{^{+ 4.2}_{-3.5 }}}$ & $-7.6 {\ensuremath{^{+4.0 }_{-3.2 }}}$ & $10^{\rm f}$ & 18/2 & $1.2\times10^{-3}$\
ABS2 & $ 1253{\ensuremath{^{+18 }_{-18 }}}$ & $-2.1 {\ensuremath{^{+ 0.7}_{- 0.7}}}$ &$-7.7 {\ensuremath{^{+2.5 }_{-2.6 }}}$ & $100^{\rm f}$ & 34.1/2 & $5.5\times10^{-6}$\
&&& &\
\
ABS1 &$ 845{\ensuremath{^{+ 16}_{-15 }}}$ &$ -4.7{\ensuremath{^{+ 2.5}_{-2.3 }}}$ &$-4.9 {\ensuremath{^{+2.6}_{-2.4 }}}$ & $10^{\rm f}$ & 13.6/2 & $3.9\times10^{-3}$\
ABS2 &$1249 {\ensuremath{^{+ 21}_{- 21}}}$ &$ -6.8{\ensuremath{^{+1.3 }_{-1.3 }}}$ &$-24.2{\ensuremath{^{+4.6 }_{-4.6 }}}$ & $110^{\rm f}$ & 71.0/2 & $4.6\times10^{-11}$\
&&& &\
\
ABS1 &$ 998{\ensuremath{^{+ 17}_{-16 }}}$&$-6.2 {\ensuremath{^{+2.1 }_{- 1.8}}}$&$-16.8{\ensuremath{^{+ 5.7}_{-5.2 }}}$ & $45^{+30}_{-15}$ & 38.7/3 & $5.8\times10^{-6}$\
ABS2 &$ 1172{\ensuremath{^{+13 }_{- 14}}}$&$-4.0 {\ensuremath{^{+0.9 }_{-0.9 }}}$&$-19.2 {\ensuremath{^{+4.4 }_{-4.2 }}}$ & $45^{\rm t}$ & 33.4/2 & $7.4\times10^{-6}$\
EMIS1 & $933\pm23$ & $3.2^{+1.7}_{-1.6}$ & $7.1^{+3.1}_{-3.0}$ & $10^{\rm f}$ & 44.7/2 & $2.4\times10^{-7}$\
&&& &\
\
ABS1 &$1083 {\ensuremath{^{+8 }_{-8 }}}$&$-21.1 {\ensuremath{^{+3.7 }_{-3.2 }}}$&$-32.2 {\ensuremath{^{+5.7 }_{-4.9 }}}$ & $47^{+11}_{-10}$ & 79.1/3 & $1.8\times10^{-11}$\
ABS2 &$ 1254{\ensuremath{^{+ 12}_{-13 }}}$&$-8.6 {\ensuremath{^{+ 1.7}_{-1.6 }}}$&$-22.9 {\ensuremath{^{+ 4.6}_{- 4.2}}}$ & $47^{\rm t}$ & 74.7/2 & $1.1\times10^{-11}$\
&&& &\
\
\[MOSgauss\]
[ccccccc]{}
Zone 1 & ${N_{\mbox{\scriptsize H}}}^{\rm a}$ &$-$&$-$&$0.61{\ensuremath{^{+0.25}_{-0.20}}}$&$1.9{\ensuremath{^{+0.5 }_{-0.5 }}}$&$-$\
& $\log \xi ^{\rm b}$ & $-$&$-$ &$3.65 {\ensuremath{^{+0.13 }_{-0.13 }}}$&$4.09{\ensuremath{^{+0.07 }_{- 0.10}}}$ &$-$\
& $v_{\mathrm{out}}/c$ &$-$ & $-$&$-0.064 {\ensuremath{^{+0.012 }_{-0.012 }}}$ &$-0.082 {\ensuremath{^{+0.01 }_{-0.01 }}}$&$-$\
& $\Delta \chi^2$ & $-$&$-$ &24.9&31.8 &$-$\
Zone 2 & ${N_{\mbox{\scriptsize H}}}^{\rm a}$ &$0.52{\ensuremath{^{+0.29 }_{- 0.20}}}$ & $0.70 {\ensuremath{^{+ 0.22}_{-0.21}}}$&$1.50{\ensuremath{^{+0.40}_{-0.45}}}$&$2.2 {\ensuremath{^{+ 0.7}_{-0.6 }}}$ &$<0.23$\
& $\log \xi^{\rm b}$ & $4.04{\ensuremath{^{+0.13 }_{- 0.29}}}$& $ 4.18{\ensuremath{^{+ 0.05}_{- 0.08}}}$&$4.18 {\ensuremath{^{+0.08 }_{-0.11 }}}$&$ 4.19{\ensuremath{^{+0.07 }_{-0.08 }}}$ &$ 4.10^f$\
& $v_{\mathrm{out}}/c$ &$-0.254 {\ensuremath{^{+ 0.011}_{-0.013 }}}$ &$ -0.267{\ensuremath{^{+0.006}_{- 0.007}}}$ &$ -0.170{\ensuremath{^{+0.012 }_{-0.011 }}}$& $-0.237{\ensuremath{^{+0.013 }_{-0.013 }}}$ & $-0.177{\ensuremath{^{+ 0.015}_{-0.011 }}}$\
& $\Delta \chi^2$ & 50.4& 80.5&83.0&20.2 &6.7\
Zone 3 & ${N_{\mbox{\scriptsize H}}}^{\rm a}$ & $<31$ & $23 {\ensuremath{^{+ 8}_{-8}}}$ & $<37$ & $<46$ &$<22$\
& $\log \xi^{\rm b}$ & $5.5^{\rm f}$ & $5.5^{\rm f}$& $5.5^{\rm f}$ & $5.5^{\rm f}$ & $5.5^{\rm f}$\
& $v_{\mathrm{out}}/c$ & $-0.3^{\rm f}$ & $-0.30{\ensuremath{^{+0.01}_{- 0.01}}}$ & $-0.3^{\rm f}$ & $-0.3^{\rm f}$ & $-0.3^{\rm f}$\
& $\Delta \chi^2$ & – & 20.0 & – & – & 10.9\
\
\[xstar\_sequences\]
[lcc]{}
$N_{\rm H}$$^{\rm a}$ & 0.6$^{\rm f}$ & 1.5$^{\rm f}$\
$\log\xi$$^{\rm b}$ & $2.8\pm0.3$ & $4.6^{+0.6}_{-0.4}$\
$v_{\rm out}/c$ & $-0.040\pm0.018$ & $-0.08\pm0.02$\
$\kappa_{\rm xstar}$$^{\rm c}$ & $6.0\pm3.0 \times10^{-4}$ & $<1.2\times10^{-3}$\
$f=\Omega/4\pi$$^{\rm d}$ & $0.44\pm0.22$ & $<0.88$\
\[photoionized\]
[^1]: The ionization parameter is defined as $\xi=L_{\rm ion}/n_{\rm e} r^{2}$, where $n_{\rm e}$ is the electron density, $r$ is the radial distance between the X-ray emitter and the ionized gas and $L_{\rm ion}$ is the ionizing luminosity over the 1–1000Rydberg range. The units of $\xi$ are subsequently ergcms$^{-1}$.
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Subsets and Splits